Change to GIF version
|
|
Metamath Proof Explorer |
This is the Unicode version. Change to GIF version |
|
| Ref | Description |
| idi 1 | (_Note_: This inference r... |
| a1ii 2 | (_Note_: This inference r... |
| mp2 9 | A double modus ponens infe... |
| mp2b 10 | A double modus ponens infe... |
| a1i 11 | Inference introducing an a... |
| 2a1i 12 | Inference introducing two ... |
| ax1w 13 | Weakening of ~ ax-1 . As ... |
| mp1i 14 | Inference detaching an ant... |
| a2i 15 | Inference distributing an ... |
| mpd 16 | A modus ponens deduction. ... |
| imim2i 17 | Inference adding common an... |
| syl 18 | An inference version of th... |
| 3syl 19 | Inference chaining two syl... |
| 4syl 20 | Inference chaining three s... |
| mpi 21 | A nested modus ponens infe... |
| mpisyl 22 | A syllogism combined with ... |
| id 23 | Principle of identity. Th... |
| idALT 24 | Alternate proof of ~ id . ... |
| idd 25 | Principle of identity ~ id... |
| a1d 26 | Deduction introducing an e... |
| 2a1d 27 | Deduction introducing two ... |
| a1i13 28 | Add two antecedents to a w... |
| 2a1 29 | A double form of ~ ax-1 . ... |
| a2d 30 | Deduction distributing an ... |
| sylcom 31 | Syllogism inference with c... |
| syl5com 32 | Syllogism inference with c... |
| com12 33 | Inference that swaps (comm... |
| syl11 34 | A syllogism inference. Co... |
| syl5 35 | A syllogism rule of infere... |
| syl6 36 | A syllogism rule of infere... |
| syl56 37 | Combine ~ syl5 and ~ syl6 ... |
| syl6com 38 | Syllogism inference with c... |
| mpcom 39 | Modus ponens inference wit... |
| syli 40 | Syllogism inference with c... |
| syl2im 41 | Replace two antecedents. ... |
| syl2imc 42 | A commuted version of ~ sy... |
| pm2.27 43 | This theorem, sometimes ca... |
| mpdd 44 | A nested modus ponens dedu... |
| mpid 45 | A nested modus ponens dedu... |
| mpdi 46 | A nested modus ponens dedu... |
| mpii 47 | A doubly nested modus pone... |
| syld 48 | Syllogism deduction. Dedu... |
| syldc 49 | Syllogism deduction. Comm... |
| mp2d 50 | A double modus ponens dedu... |
| a1dd 51 | Double deduction introduci... |
| 2a1dd 52 | Double deduction introduci... |
| pm2.43i 53 | Inference absorbing redund... |
| pm2.43d 54 | Deduction absorbing redund... |
| pm2.43a 55 | Inference absorbing redund... |
| pm2.43b 56 | Inference absorbing redund... |
| pm2.43 57 | Absorption of redundant an... |
| imim2d 58 | Deduction adding nested an... |
| imim2 59 | A closed form of syllogism... |
| embantd 60 | Deduction embedding an ant... |
| 3syld 61 | Triple syllogism deduction... |
| sylsyld 62 | A double syllogism inferen... |
| imim12i 63 | Inference joining two impl... |
| imim1i 64 | Inference adding common co... |
| imim3i 65 | Inference adding three nes... |
| sylc 66 | A syllogism inference comb... |
| syl3c 67 | A syllogism inference comb... |
| syl6mpi 68 | A syllogism inference. (C... |
| mpsyl 69 | Modus ponens combined with... |
| mpsylsyld 70 | Modus ponens combined with... |
| syl6c 71 | Inference combining ~ syl6... |
| syl6ci 72 | A syllogism inference comb... |
| syldd 73 | Nested syllogism deduction... |
| syl5d 74 | A nested syllogism deducti... |
| syl7 75 | A syllogism rule of infere... |
| syl6d 76 | A nested syllogism deducti... |
| syl8 77 | A syllogism rule of infere... |
| syl9 78 | A nested syllogism inferen... |
| syl9r 79 | A nested syllogism inferen... |
| syl10 80 | A nested syllogism inferen... |
| a1ddd 81 | Triple deduction introduci... |
| imim12d 82 | Deduction combining antece... |
| imim1d 83 | Deduction adding nested co... |
| imim1 84 | A closed form of syllogism... |
| pm2.83 85 | Theorem *2.83 of [Whitehea... |
| peirceroll 86 | Over minimal implicational... |
| com23 87 | Commutation of antecedents... |
| com3r 88 | Commutation of antecedents... |
| com13 89 | Commutation of antecedents... |
| com3l 90 | Commutation of antecedents... |
| pm2.04 91 | Swap antecedents. Theorem... |
| com34 92 | Commutation of antecedents... |
| com4l 93 | Commutation of antecedents... |
| com4t 94 | Commutation of antecedents... |
| com4r 95 | Commutation of antecedents... |
| com24 96 | Commutation of antecedents... |
| com14 97 | Commutation of antecedents... |
| com45 98 | Commutation of antecedents... |
| com35 99 | Commutation of antecedents... |
| com25 100 | Commutation of antecedents... |
| com5l 101 | Commutation of antecedents... |
| com15 102 | Commutation of antecedents... |
| com52l 103 | Commutation of antecedents... |
| com52r 104 | Commutation of antecedents... |
| com5r 105 | Commutation of antecedents... |
| imim12 106 | Closed form of ~ imim12i a... |
| jarr 107 | Elimination of a nested an... |
| jarri 108 | Inference associated with ... |
| pm2.86d 109 | Deduction associated with ... |
| pm2.86 110 | Converse of Axiom ~ ax-2 .... |
| pm2.86i 111 | Inference associated with ... |
| loolin 112 | The Linearity Axiom of the... |
| loowoz 113 | An alternate for the Linea... |
| con4 114 | Alias for ~ ax-3 to be use... |
| con4i 115 | Inference associated with ... |
| con4d 116 | Deduction associated with ... |
| mt4 117 | The rule of modus tollens.... |
| mt4d 118 | Modus tollens deduction. ... |
| mt4i 119 | Modus tollens inference. ... |
| pm2.21i 120 | A contradiction implies an... |
| pm2.24ii 121 | A contradiction implies an... |
| pm2.21d 122 | A contradiction implies an... |
| pm2.21ddALT 123 | Alternate proof of ~ pm2.2... |
| pm2.21 124 | From a wff and its negatio... |
| pm2.24 125 | Theorem *2.24 of [Whitehea... |
| jarl 126 | Elimination of a nested an... |
| jarli 127 | Inference associated with ... |
| pm2.18d 128 | Deduction form of the Clav... |
| pm2.18 129 | Clavius law, or "consequen... |
| pm2.18i 130 | Inference associated with ... |
| notnotr 131 | Double negation eliminatio... |
| notnotri 132 | Inference associated with ... |
| notnotriALT 133 | Alternate proof of ~ notno... |
| notnotrd 134 | Deduction associated with ... |
| con2d 135 | A contraposition deduction... |
| con2 136 | Contraposition. Theorem *... |
| mt2d 137 | Modus tollens deduction. ... |
| mt2i 138 | Modus tollens inference. ... |
| nsyl3 139 | A negated syllogism infere... |
| con2i 140 | A contraposition inference... |
| nsyl 141 | A negated syllogism infere... |
| nsyl2 142 | A negated syllogism infere... |
| notnot 143 | Double negation introducti... |
| notnoti 144 | Inference associated with ... |
| notnotd 145 | Deduction associated with ... |
| con1d 146 | A contraposition deduction... |
| con1 147 | Contraposition. Theorem *... |
| con1i 148 | A contraposition inference... |
| mt3d 149 | Modus tollens deduction. ... |
| mt3i 150 | Modus tollens inference. ... |
| pm2.24i 151 | Inference associated with ... |
| pm2.24d 152 | Deduction form of ~ pm2.24... |
| con3d 153 | A contraposition deduction... |
| con3 154 | Contraposition. Theorem *... |
| con3i 155 | A contraposition inference... |
| con3rr3 156 | Rotate through consequent ... |
| nsyld 157 | A negated syllogism deduct... |
| nsyli 158 | A negated syllogism infere... |
| nsyl4 159 | A negated syllogism infere... |
| nsyl5 160 | A negated syllogism infere... |
| pm3.2im 161 | Theorem *3.2 of [Whitehead... |
| jc 162 | Deduction joining the cons... |
| jcn 163 | Theorem joining the conseq... |
| jcnd 164 | Deduction joining the cons... |
| impi 165 | An importation inference. ... |
| expi 166 | An exportation inference. ... |
| simprim 167 | Simplification. Similar t... |
| simplim 168 | Simplification. Similar t... |
| pm2.5g 169 | General instance of Theore... |
| pm2.5 170 | Theorem *2.5 of [Whitehead... |
| conax1 171 | Contrapositive of ~ ax-1 .... |
| conax1k 172 | Weakening of ~ conax1 . G... |
| pm2.51 173 | Theorem *2.51 of [Whitehea... |
| pm2.52 174 | Theorem *2.52 of [Whitehea... |
| pm2.521g 175 | A general instance of Theo... |
| pm2.521g2 176 | A general instance of Theo... |
| pm2.521 177 | Theorem *2.521 of [Whitehe... |
| expt 178 | Exportation theorem ~ pm3.... |
| exptOLD 179 | Obsolete version of ~ expt... |
| impt 180 | Importation theorem ~ pm3.... |
| pm2.61d 181 | Deduction eliminating an a... |
| pm2.61d1 182 | Inference eliminating an a... |
| pm2.61d2 183 | Inference eliminating an a... |
| pm2.61i 184 | Inference eliminating an a... |
| pm2.61ii 185 | Inference eliminating two ... |
| pm2.61nii 186 | Inference eliminating two ... |
| pm2.61iii 187 | Inference eliminating thre... |
| ja 188 | Inference joining the ante... |
| jad 189 | Deduction form of ~ ja . ... |
| pm2.01 190 | Weak Clavius law. If a fo... |
| pm2.01i 191 | Inference associated with ... |
| pm2.01d 192 | Deduction based on reducti... |
| pm2.6 193 | Theorem *2.6 of [Whitehead... |
| pm2.61 194 | Theorem *2.61 of [Whitehea... |
| pm2.65 195 | Theorem *2.65 of [Whitehea... |
| pm2.65i 196 | Inference for proof by con... |
| pm2.65iOLD 197 | Obsolete version of ~ pm2.... |
| pm2.21dd 198 | A contradiction implies an... |
| pm2.65d 199 | Deduction for proof by con... |
| mto 200 | The rule of modus tollens.... |
| mtod 201 | Modus tollens deduction. ... |
| mtoi 202 | Modus tollens inference. ... |
| mt2 203 | A rule similar to modus to... |
| mt3 204 | A rule similar to modus to... |
| peirce 205 | Peirce's axiom. A non-int... |
| looinv 206 | The Inversion Axiom of the... |
| bijust0 207 | A self-implication (see ~ ... |
| bijust 208 | Theorem used to justify th... |
| impbi 211 | Property of the biconditio... |
| impbii 212 | Infer an equivalence from ... |
| impbidd 213 | Deduce an equivalence from... |
| impbid21d 214 | Deduce an equivalence from... |
| impbid 215 | Deduce an equivalence from... |
| dfbi1 216 | Relate the biconditional c... |
| dfbi1ALT 217 | Alternate proof of ~ dfbi1... |
| biimp 218 | Property of the biconditio... |
| biimpi 219 | Infer an implication from ... |
| sylbi 220 | A mixed syllogism inferenc... |
| sylib 221 | A mixed syllogism inferenc... |
| sylbb 222 | A mixed syllogism inferenc... |
| biimpr 223 | Property of the biconditio... |
| bicom1 224 | Commutative law for the bi... |
| bicom 225 | Commutative law for the bi... |
| bicomd 226 | Commute two sides of a bic... |
| bicomi 227 | Inference from commutative... |
| impbid1 228 | Infer an equivalence from ... |
| impbid2 229 | Infer an equivalence from ... |
| impcon4bid 230 | A variation on ~ impbid wi... |
| biimpri 231 | Infer a converse implicati... |
| biimpd 232 | Deduce an implication from... |
| mpbi 233 | An inference from a bicond... |
| mpbir 234 | An inference from a bicond... |
| mpbid 235 | A deduction from a bicondi... |
| mpbii 236 | An inference from a nested... |
| sylibr 237 | A mixed syllogism inferenc... |
| sylbir 238 | A mixed syllogism inferenc... |
| sylbbr 239 | A mixed syllogism inferenc... |
| sylbb1 240 | A mixed syllogism inferenc... |
| sylbb2 241 | A mixed syllogism inferenc... |
| sylibd 242 | A syllogism deduction. (C... |
| sylbid 243 | A syllogism deduction. (C... |
| mpbidi 244 | A deduction from a bicondi... |
| biimtrid 245 | A mixed syllogism inferenc... |
| biimtrrid 246 | A mixed syllogism inferenc... |
| imbitrid 247 | A mixed syllogism inferenc... |
| syl5ibcom 248 | A mixed syllogism inferenc... |
| imbitrrid 249 | A mixed syllogism inferenc... |
| syl5ibrcom 250 | A mixed syllogism inferenc... |
| biimprd 251 | Deduce a converse implicat... |
| biimpcd 252 | Deduce a commuted implicat... |
| biimprcd 253 | Deduce a converse commuted... |
| imbitrdi 254 | A mixed syllogism inferenc... |
| imbitrrdi 255 | A mixed syllogism inferenc... |
| biimtrdi 256 | A mixed syllogism inferenc... |
| biimtrrdi 257 | A mixed syllogism inferenc... |
| syl7bi 258 | A mixed syllogism inferenc... |
| syl8ib 259 | A syllogism rule of infere... |
| mpbird 260 | A deduction from a bicondi... |
| mpbiri 261 | An inference from a nested... |
| sylibrd 262 | A syllogism deduction. (C... |
| sylbird 263 | A syllogism deduction. (C... |
| biid 264 | Principle of identity for ... |
| biidd 265 | Principle of identity with... |
| pm5.1im 266 | Two propositions are equiv... |
| 2th 267 | Two truths are equivalent.... |
| 2thd 268 | Two truths are equivalent.... |
| monothetic 269 | Two self-implications (see... |
| ibi 270 | Inference that converts a ... |
| ibir 271 | Inference that converts a ... |
| ibd 272 | Deduction that converts a ... |
| pm5.74 273 | Distribution of implicatio... |
| pm5.74i 274 | Distribution of implicatio... |
| pm5.74ri 275 | Distribution of implicatio... |
| pm5.74d 276 | Distribution of implicatio... |
| pm5.74rd 277 | Distribution of implicatio... |
| bitri 278 | An inference from transiti... |
| bitr2i 279 | An inference from transiti... |
| bitr3i 280 | An inference from transiti... |
| bitr4i 281 | An inference from transiti... |
| bitrd 282 | Deduction form of ~ bitri ... |
| bitr2d 283 | Deduction form of ~ bitr2i... |
| bitr3d 284 | Deduction form of ~ bitr3i... |
| bitr4d 285 | Deduction form of ~ bitr4i... |
| bitrid 286 | A syllogism inference from... |
| bitr2id 287 | A syllogism inference from... |
| bitr3id 288 | A syllogism inference from... |
| bitr3di 289 | A syllogism inference from... |
| bitrdi 290 | A syllogism inference from... |
| bitr2di 291 | A syllogism inference from... |
| bitr4di 292 | A syllogism inference from... |
| bitr4id 293 | A syllogism inference from... |
| 3imtr3i 294 | A mixed syllogism inferenc... |
| 3imtr4i 295 | A mixed syllogism inferenc... |
| 3imtr3d 296 | More general version of ~ ... |
| 3imtr4d 297 | More general version of ~ ... |
| 3imtr3g 298 | More general version of ~ ... |
| 3imtr4g 299 | More general version of ~ ... |
| 3bitri 300 | A chained inference from t... |
| 3bitrri 301 | A chained inference from t... |
| 3bitr2i 302 | A chained inference from t... |
| 3bitr2ri 303 | A chained inference from t... |
| 3bitr3i 304 | A chained inference from t... |
| 3bitr3ri 305 | A chained inference from t... |
| 3bitr4i 306 | A chained inference from t... |
| 3bitr4ri 307 | A chained inference from t... |
| 3bitrd 308 | Deduction from transitivit... |
| 3bitrrd 309 | Deduction from transitivit... |
| 3bitr2d 310 | Deduction from transitivit... |
| 3bitr2rd 311 | Deduction from transitivit... |
| 3bitr3d 312 | Deduction from transitivit... |
| 3bitr3rd 313 | Deduction from transitivit... |
| 3bitr4d 314 | Deduction from transitivit... |
| 3bitr4rd 315 | Deduction from transitivit... |
| 3bitr3g 316 | More general version of ~ ... |
| 3bitr4g 317 | More general version of ~ ... |
| notnotb 318 | Double negation. Theorem ... |
| con34b 319 | A biconditional form of co... |
| con4bid 320 | A contraposition deduction... |
| notbid 321 | Deduction negating both si... |
| notbi 322 | Contraposition. Theorem *... |
| notbii 323 | Negate both sides of a log... |
| con4bii 324 | A contraposition inference... |
| mtbi 325 | An inference from a bicond... |
| mtbir 326 | An inference from a bicond... |
| mtbid 327 | A deduction from a bicondi... |
| mtbird 328 | A deduction from a bicondi... |
| mtbii 329 | An inference from a bicond... |
| mtbiri 330 | An inference from a bicond... |
| sylnib 331 | A mixed syllogism inferenc... |
| sylnibr 332 | A mixed syllogism inferenc... |
| sylnbi 333 | A mixed syllogism inferenc... |
| sylnbir 334 | A mixed syllogism inferenc... |
| xchnxbi 335 | Replacement of a subexpres... |
| xchnxbir 336 | Replacement of a subexpres... |
| xchbinx 337 | Replacement of a subexpres... |
| xchbinxr 338 | Replacement of a subexpres... |
| imbi2i 339 | Introduce an antecedent to... |
| bibi2i 340 | Inference adding a bicondi... |
| bibi1i 341 | Inference adding a bicondi... |
| bibi12i 342 | The equivalence of two equ... |
| imbi2d 343 | Deduction adding an antece... |
| imbi1d 344 | Deduction adding a consequ... |
| bibi2d 345 | Deduction adding a bicondi... |
| bibi1d 346 | Deduction adding a bicondi... |
| imbi12d 347 | Deduction joining two equi... |
| bibi12d 348 | Deduction joining two equi... |
| imbi12 349 | Closed form of ~ imbi12i .... |
| imbi1 350 | Theorem *4.84 of [Whitehea... |
| imbi2 351 | Theorem *4.85 of [Whitehea... |
| imbi1i 352 | Introduce a consequent to ... |
| imbi12i 353 | Join two logical equivalen... |
| bibi1 354 | Theorem *4.86 of [Whitehea... |
| bitr3 355 | Closed nested implication ... |
| con2bi 356 | Contraposition. Theorem *... |
| con2bid 357 | A contraposition deduction... |
| con1bid 358 | A contraposition deduction... |
| con1bii 359 | A contraposition inference... |
| con2bii 360 | A contraposition inference... |
| con1b 361 | Contraposition. Bidirecti... |
| con2b 362 | Contraposition. Bidirecti... |
| biimt 363 | A wff is equivalent to its... |
| pm5.5 364 | Theorem *5.5 of [Whitehead... |
| a1bi 365 | Inference introducing a th... |
| mt2bi 366 | A false consequent falsifi... |
| mtt 367 | Modus-tollens-like theorem... |
| imnot 368 | If a proposition is false,... |
| pm5.501 369 | Theorem *5.501 of [Whitehe... |
| ibib 370 | Implication in terms of im... |
| ibibr 371 | Implication in terms of im... |
| tbt 372 | A wff is equivalent to its... |
| nbn2 373 | The negation of a wff is e... |
| bibif 374 | Transfer negation via an e... |
| nbn 375 | The negation of a wff is e... |
| nbn3 376 | Transfer falsehood via equ... |
| pm5.21im 377 | Two propositions are equiv... |
| 2false 378 | Two falsehoods are equival... |
| 2falsed 379 | Two falsehoods are equival... |
| pm5.21ni 380 | Two propositions implying ... |
| pm5.21nii 381 | Eliminate an antecedent im... |
| pm5.21ndd 382 | Eliminate an antecedent im... |
| bija 383 | Combine antecedents into a... |
| pm5.18 384 | Theorem *5.18 of [Whitehea... |
| xor3 385 | Two ways to express "exclu... |
| nbbn 386 | Move negation outside of b... |
| nbbnOLD 387 | Obsolete version of ~ nbbn... |
| biass 388 | Associative law for the bi... |
| birot 389 | Rotation of the arguments ... |
| biluk 390 | Lukasiewicz's shortest axi... |
| pm5.19 391 | Theorem *5.19 of [Whitehea... |
| bi2.04 392 | Logical equivalence of com... |
| pm5.4 393 | Antecedent absorption impl... |
| imdi 394 | Distributive law for impli... |
| pm5.41 395 | Theorem *5.41 of [Whitehea... |
| imbibi 396 | The antecedent of one side... |
| imbibiOLD 397 | Obsolete version of ~ imbi... |
| pm4.8 398 | Theorem *4.8 of [Whitehead... |
| pm4.81 399 | A formula is equivalent to... |
| imim21b 400 | Simplify an implication be... |
| pm4.63 403 | Theorem *4.63 of [Whitehea... |
| pm4.67 404 | Theorem *4.67 of [Whitehea... |
| imnan 405 | Express an implication in ... |
| imnani 406 | Infer an implication from ... |
| iman 407 | Implication in terms of co... |
| pm3.24 408 | Law of noncontradiction. ... |
| annim 409 | Express a conjunction in t... |
| pm4.61 410 | Theorem *4.61 of [Whitehea... |
| pm4.65 411 | Theorem *4.65 of [Whitehea... |
| imp 412 | Importation inference. (C... |
| impcom 413 | Importation inference with... |
| con3dimp 414 | Variant of ~ con3d with im... |
| mpnanrd 415 | Eliminate the right side o... |
| impd 416 | Importation deduction. (C... |
| impcomd 417 | Importation deduction with... |
| ex 418 | Exportation inference. (T... |
| expcom 419 | Exportation inference with... |
| expdcom 420 | Commuted form of ~ expd . ... |
| expd 421 | Exportation deduction. (C... |
| expcomd 422 | Deduction form of ~ expcom... |
| imp31 423 | An importation inference. ... |
| imp32 424 | An importation inference. ... |
| exp31 425 | An exportation inference. ... |
| exp32 426 | An exportation inference. ... |
| imp4b 427 | An importation inference. ... |
| imp4a 428 | An importation inference. ... |
| imp4c 429 | An importation inference. ... |
| imp4d 430 | An importation inference. ... |
| imp41 431 | An importation inference. ... |
| imp42 432 | An importation inference. ... |
| imp43 433 | An importation inference. ... |
| imp44 434 | An importation inference. ... |
| imp45 435 | An importation inference. ... |
| exp4b 436 | An exportation inference. ... |
| exp4a 437 | An exportation inference. ... |
| exp4c 438 | An exportation inference. ... |
| exp4d 439 | An exportation inference. ... |
| exp41 440 | An exportation inference. ... |
| exp42 441 | An exportation inference. ... |
| exp43 442 | An exportation inference. ... |
| exp44 443 | An exportation inference. ... |
| exp45 444 | An exportation inference. ... |
| imp5d 445 | An importation inference. ... |
| imp5a 446 | An importation inference. ... |
| imp5g 447 | An importation inference. ... |
| imp55 448 | An importation inference. ... |
| imp511 449 | An importation inference. ... |
| exp5c 450 | An exportation inference. ... |
| exp5j 451 | An exportation inference. ... |
| exp5l 452 | An exportation inference. ... |
| exp53 453 | An exportation inference. ... |
| pm3.3 454 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 455 | Theorem *3.31 (Imp) of [Wh... |
| impexp 456 | Import-export theorem. Pa... |
| impancom 457 | Mixed importation/commutat... |
| expdimp 458 | A deduction version of exp... |
| expimpd 459 | Exportation followed by a ... |
| impr 460 | Import a wff into a right ... |
| impl 461 | Export a wff from a left c... |
| expr 462 | Export a wff from a right ... |
| expl 463 | Export a wff from a left c... |
| ancoms 464 | Inference commuting conjun... |
| pm3.22 465 | Theorem *3.22 of [Whitehea... |
| ancom 466 | Commutative law for conjun... |
| ancomd 467 | Commutation of conjuncts i... |
| biancomi 468 | Commuting conjunction in a... |
| biancomd 469 | Commuting conjunction in a... |
| ancomst 470 | Closed form of ~ ancoms . ... |
| ancomsd 471 | Deduction commuting conjun... |
| anasss 472 | Associative law for conjun... |
| anassrs 473 | Associative law for conjun... |
| anass 474 | Associative law for conjun... |
| pm3.2 475 | Join antecedents with conj... |
| pm3.2i 476 | Infer conjunction of premi... |
| pm3.21 477 | Join antecedents with conj... |
| pm3.43i 478 | Nested conjunction of ante... |
| pm3.43 479 | Theorem *3.43 (Comp) of [W... |
| dfbi2 480 | A theorem similar to the s... |
| dfbi 481 | Definition ~ df-bi rewritt... |
| biimpa 482 | Importation inference from... |
| biimpar 483 | Importation inference from... |
| biimpac 484 | Importation inference from... |
| biimparc 485 | Importation inference from... |
| adantr 486 | Inference adding a conjunc... |
| adantl 487 | Inference adding a conjunc... |
| simpl 488 | Elimination of a conjunct.... |
| simpli 489 | Inference eliminating a co... |
| simpr 490 | Elimination of a conjunct.... |
| simpri 491 | Inference eliminating a co... |
| intnan 492 | Introduction of conjunct i... |
| intnanr 493 | Introduction of conjunct i... |
| intnand 494 | Introduction of conjunct i... |
| intnanrd 495 | Introduction of conjunct i... |
| adantld 496 | Deduction adding a conjunc... |
| adantrd 497 | Deduction adding a conjunc... |
| pm3.41 498 | Theorem *3.41 of [Whitehea... |
| pm3.42 499 | Theorem *3.42 of [Whitehea... |
| simpld 500 | Deduction eliminating a co... |
| simprd 501 | Deduction eliminating a co... |
| simplbi 502 | Deduction eliminating a co... |
| simprbi 503 | Deduction eliminating a co... |
| simprbda 504 | Deduction eliminating a co... |
| simplbda 505 | Deduction eliminating a co... |
| simplbi2 506 | Deduction eliminating a co... |
| simplbi2comt 507 | Closed form of ~ simplbi2c... |
| simplbi2com 508 | A deduction eliminating a ... |
| birani 509 | Inference adding a conjunc... |
| bilani 510 | Inference adding a conjunc... |
| biranri 511 | Inference adding a conjunc... |
| bilanri 512 | Inference adding a conjunc... |
| simpl2im 513 | Implication from an elimin... |
| simplbiim 514 | Implication from an elimin... |
| impel 515 | An inference for implicati... |
| mpan9 516 | Modus ponens conjoining di... |
| sylan9 517 | Nested syllogism inference... |
| sylan9r 518 | Nested syllogism inference... |
| sylan9bb 519 | Nested syllogism inference... |
| sylan9bbr 520 | Nested syllogism inference... |
| jca 521 | Deduce conjunction of the ... |
| jcad 522 | Deduction conjoining the c... |
| jca2 523 | Inference conjoining the c... |
| jca31 524 | Join three consequents. (... |
| jca32 525 | Join three consequents. (... |
| jcai 526 | Deduction replacing implic... |
| jcab 527 | Distributive law for impli... |
| pm4.76 528 | Theorem *4.76 of [Whitehea... |
| jctil 529 | Inference conjoining a the... |
| jctir 530 | Inference conjoining a the... |
| jccir 531 | Inference conjoining a con... |
| jccil 532 | Inference conjoining a con... |
| jctl 533 | Inference conjoining a the... |
| jctr 534 | Inference conjoining a the... |
| jctild 535 | Deduction conjoining a the... |
| jctird 536 | Deduction conjoining a the... |
| iba 537 | Introduction of antecedent... |
| ibar 538 | Introduction of antecedent... |
| biantru 539 | A wff is equivalent to its... |
| biantrur 540 | A wff is equivalent to its... |
| biantrud 541 | A wff is equivalent to its... |
| biantrurd 542 | A wff is equivalent to its... |
| bianfi 543 | A wff conjoined with false... |
| bianfd 544 | A wff conjoined with false... |
| baib 545 | Move conjunction outside o... |
| baibr 546 | Move conjunction outside o... |
| rbaibr 547 | Move conjunction outside o... |
| rbaib 548 | Move conjunction outside o... |
| baibd 549 | Move conjunction outside o... |
| rbaibd 550 | Move conjunction outside o... |
| bianabs 551 | Absorb a hypothesis into t... |
| pm5.44 552 | Theorem *5.44 of [Whitehea... |
| pm5.42 553 | Theorem *5.42 of [Whitehea... |
| ancl 554 | Conjoin antecedent to left... |
| anclb 555 | Conjoin antecedent to left... |
| ancr 556 | Conjoin antecedent to righ... |
| ancrb 557 | Conjoin antecedent to righ... |
| ancli 558 | Deduction conjoining antec... |
| ancri 559 | Deduction conjoining antec... |
| ancld 560 | Deduction conjoining antec... |
| ancrd 561 | Deduction conjoining antec... |
| impac 562 | Importation with conjuncti... |
| anc2l 563 | Conjoin antecedent to left... |
| anc2r 564 | Conjoin antecedent to righ... |
| anc2li 565 | Deduction conjoining antec... |
| anc2ri 566 | Deduction conjoining antec... |
| pm4.71 567 | Implication in terms of bi... |
| pm4.71r 568 | Implication in terms of bi... |
| pm4.71i 569 | Inference converting an im... |
| pm4.71ri 570 | Inference converting an im... |
| pm4.71d 571 | Deduction converting an im... |
| pm4.71rd 572 | Deduction converting an im... |
| pm4.71da 573 | Deduction converting a bic... |
| pm4.24 574 | Theorem *4.24 of [Whitehea... |
| anidm 575 | Idempotent law for conjunc... |
| anidmdbi 576 | Conjunction idempotence wi... |
| anidms 577 | Inference from idempotent ... |
| imdistan 578 | Distribution of implicatio... |
| imdistani 579 | Distribution of implicatio... |
| imdistanri 580 | Distribution of implicatio... |
| imdistand 581 | Distribution of implicatio... |
| imdistanda 582 | Distribution of implicatio... |
| pm5.3 583 | Theorem *5.3 of [Whitehead... |
| pm5.32 584 | Distribution of implicatio... |
| pm5.32i 585 | Distribution of implicatio... |
| pm5.32ri 586 | Distribution of implicatio... |
| bianim 587 | Exchanging conjunction in ... |
| pm5.32d 588 | Distribution of implicatio... |
| pm5.32rd 589 | Distribution of implicatio... |
| pm5.32da 590 | Distribution of implicatio... |
| bian1d 591 | Adding a superfluous conju... |
| sylan 592 | A syllogism inference. (C... |
| sylanb 593 | A syllogism inference. (C... |
| sylanbr 594 | A syllogism inference. (C... |
| sylanbrc 595 | Syllogism inference. (Con... |
| syl2anc 596 | Syllogism inference combin... |
| syl2anc2 597 | Double syllogism inference... |
| sylancl 598 | Syllogism inference combin... |
| sylancr 599 | Syllogism inference combin... |
| sylancom 600 | Syllogism inference with c... |
| sylanblc 601 | Syllogism inference combin... |
| sylanblrc 602 | Syllogism inference combin... |
| syldan 603 | A syllogism deduction with... |
| sylbida 604 | A syllogism deduction. (C... |
| sylan2 605 | A syllogism inference. (C... |
| sylan2b 606 | A syllogism inference. (C... |
| sylan2br 607 | A syllogism inference. (C... |
| syl2an 608 | A double syllogism inferen... |
| syl2anr 609 | A double syllogism inferen... |
| syl2anb 610 | A double syllogism inferen... |
| syl2anbr 611 | A double syllogism inferen... |
| sylancb 612 | A syllogism inference comb... |
| sylancbr 613 | A syllogism inference comb... |
| syldanl 614 | A syllogism deduction with... |
| syland 615 | A syllogism deduction. (C... |
| sylani 616 | A syllogism inference. (C... |
| sylan2d 617 | A syllogism deduction. (C... |
| sylan2i 618 | A syllogism inference. (C... |
| syl2ani 619 | A syllogism inference. (C... |
| syl2and 620 | A syllogism deduction. (C... |
| anim12d 621 | Conjoin antecedents and co... |
| anim12d1 622 | Variant of ~ anim12d where... |
| anim1d 623 | Add a conjunct to right of... |
| anim2d 624 | Add a conjunct to left of ... |
| anim12i 625 | Conjoin antecedents and co... |
| anim12ci 626 | Variant of ~ anim12i with ... |
| anim1i 627 | Introduce conjunct to both... |
| anim1ci 628 | Introduce conjunct to both... |
| anim2i 629 | Introduce conjunct to both... |
| anim12ii 630 | Conjoin antecedents and co... |
| anim12dan 631 | Conjoin antecedents and co... |
| im2anan9 632 | Deduction joining nested i... |
| im2anan9r 633 | Deduction joining nested i... |
| pm3.45 634 | Theorem *3.45 (Fact) of [W... |
| anbi2i 635 | Introduce a left conjunct ... |
| anbi1i 636 | Introduce a right conjunct... |
| anbi2ci 637 | Variant of ~ anbi2i with c... |
| anbi1ci 638 | Variant of ~ anbi1i with c... |
| bianbi 639 | Exchanging conjunction in ... |
| anbi12i 640 | Conjoin both sides of two ... |
| anbi12ci 641 | Variant of ~ anbi12i with ... |
| anbi2d 642 | Deduction adding a left co... |
| anbi1d 643 | Deduction adding a right c... |
| anbi12d 644 | Deduction joining two equi... |
| anbi1 645 | Introduce a right conjunct... |
| anbi2 646 | Introduce a left conjunct ... |
| anbi1cd 647 | Introduce a proposition as... |
| an2anr 648 | Double commutation in conj... |
| pm4.38 649 | Theorem *4.38 of [Whitehea... |
| bi2anan9 650 | Deduction joining two equi... |
| bi2anan9r 651 | Deduction joining two equi... |
| bi2bian9 652 | Deduction joining two bico... |
| anbiim 653 | Adding biconditional when ... |
| anbiimOLD 654 | Obsolete version of ~ anbi... |
| bianass 655 | An inference to merge two ... |
| bianassc 656 | An inference to merge two ... |
| an21 657 | Swap two conjuncts. (Cont... |
| an12 658 | Swap two conjuncts. Note ... |
| an32 659 | A rearrangement of conjunc... |
| an13 660 | A rearrangement of conjunc... |
| an31 661 | A rearrangement of conjunc... |
| an12s 662 | Swap two conjuncts in ante... |
| ancom2s 663 | Inference commuting a nest... |
| an13s 664 | Swap two conjuncts in ante... |
| an32s 665 | Swap two conjuncts in ante... |
| ancom1s 666 | Inference commuting a nest... |
| an31s 667 | Swap two conjuncts in ante... |
| anass1rs 668 | Commutative-associative la... |
| an4 669 | Rearrangement of 4 conjunc... |
| an42 670 | Rearrangement of 4 conjunc... |
| an43 671 | Rearrangement of 4 conjunc... |
| an3 672 | A rearrangement of conjunc... |
| an4s 673 | Inference rearranging 4 co... |
| an42s 674 | Inference rearranging 4 co... |
| anabs1 675 | Absorption into embedded c... |
| anabs5 676 | Absorption into embedded c... |
| anabs7 677 | Absorption into embedded c... |
| anabsan 678 | Absorption of antecedent w... |
| anabss1 679 | Absorption of antecedent i... |
| anabss4 680 | Absorption of antecedent i... |
| anabss5 681 | Absorption of antecedent i... |
| anabsi5 682 | Absorption of antecedent i... |
| anabsi6 683 | Absorption of antecedent i... |
| anabsi7 684 | Absorption of antecedent i... |
| anabsi8 685 | Absorption of antecedent i... |
| anabss7 686 | Absorption of antecedent i... |
| anabsan2 687 | Absorption of antecedent w... |
| anabss3 688 | Absorption of antecedent i... |
| anandi 689 | Distribution of conjunctio... |
| anandir 690 | Distribution of conjunctio... |
| anandis 691 | Inference that undistribut... |
| anandirs 692 | Inference that undistribut... |
| sylanl1 693 | A syllogism inference. (C... |
| sylanl2 694 | A syllogism inference. (C... |
| sylanr1 695 | A syllogism inference. (C... |
| sylanr2 696 | A syllogism inference. (C... |
| syl6an 697 | A syllogism deduction comb... |
| syl2an2r 698 | ~ syl2anr with antecedents... |
| syl2an2 699 | ~ syl2an with antecedents ... |
| mpdan 700 | An inference based on modu... |
| mpancom 701 | An inference based on modu... |
| mpidan 702 | A deduction which "stacks"... |
| mpan 703 | An inference based on modu... |
| mpan2 704 | An inference based on modu... |
| mp2an 705 | An inference based on modu... |
| mp4an 706 | An inference based on modu... |
| mpan2d 707 | A deduction based on modus... |
| mpand 708 | A deduction based on modus... |
| mpani 709 | An inference based on modu... |
| mpan2i 710 | An inference based on modu... |
| mp2ani 711 | An inference based on modu... |
| mp2and 712 | A deduction based on modus... |
| mpanl1 713 | An inference based on modu... |
| mpanl2 714 | An inference based on modu... |
| mpanl12 715 | An inference based on modu... |
| mpanr1 716 | An inference based on modu... |
| mpanr2 717 | An inference based on modu... |
| mpanr12 718 | An inference based on modu... |
| mpanlr1 719 | An inference based on modu... |
| mpbirand 720 | Detach truth from conjunct... |
| mpbiran2d 721 | Detach truth from conjunct... |
| mpbiran 722 | Detach truth from conjunct... |
| mpbiran2 723 | Detach truth from conjunct... |
| mpbir2an 724 | Detach a conjunction of tr... |
| mpbi2and 725 | Detach a conjunction of tr... |
| mpbir2and 726 | Detach a conjunction of tr... |
| adantll 727 | Deduction adding a conjunc... |
| adantlr 728 | Deduction adding a conjunc... |
| adantrl 729 | Deduction adding a conjunc... |
| adantrr 730 | Deduction adding a conjunc... |
| adantlll 731 | Deduction adding a conjunc... |
| adantllr 732 | Deduction adding a conjunc... |
| adantlrl 733 | Deduction adding a conjunc... |
| adantlrr 734 | Deduction adding a conjunc... |
| adantrll 735 | Deduction adding a conjunc... |
| adantrlr 736 | Deduction adding a conjunc... |
| adantrrl 737 | Deduction adding a conjunc... |
| adantrrr 738 | Deduction adding a conjunc... |
| ad2antrr 739 | Deduction adding two conju... |
| ad2antlr 740 | Deduction adding two conju... |
| ad2antrl 741 | Deduction adding two conju... |
| ad2antll 742 | Deduction adding conjuncts... |
| ad3antrrr 743 | Deduction adding three con... |
| ad3antlr 744 | Deduction adding three con... |
| ad4antr 745 | Deduction adding 4 conjunc... |
| ad4antlr 746 | Deduction adding 4 conjunc... |
| ad5antr 747 | Deduction adding 5 conjunc... |
| ad5antlr 748 | Deduction adding 5 conjunc... |
| ad6antr 749 | Deduction adding 6 conjunc... |
| ad6antlr 750 | Deduction adding 6 conjunc... |
| ad7antr 751 | Deduction adding 7 conjunc... |
| ad7antlr 752 | Deduction adding 7 conjunc... |
| ad8antr 753 | Deduction adding 8 conjunc... |
| ad8antlr 754 | Deduction adding 8 conjunc... |
| ad9antr 755 | Deduction adding 9 conjunc... |
| ad9antlr 756 | Deduction adding 9 conjunc... |
| ad10antr 757 | Deduction adding 10 conjun... |
| ad10antlr 758 | Deduction adding 10 conjun... |
| ad2ant2l 759 | Deduction adding two conju... |
| ad2ant2r 760 | Deduction adding two conju... |
| ad2ant2lr 761 | Deduction adding two conju... |
| ad2ant2rl 762 | Deduction adding two conju... |
| adantl3r 763 | Deduction adding 1 conjunc... |
| ad4ant13 764 | Deduction adding conjuncts... |
| ad4ant14 765 | Deduction adding conjuncts... |
| ad4ant23 766 | Deduction adding conjuncts... |
| ad4ant24 767 | Deduction adding conjuncts... |
| adantl4r 768 | Deduction adding 1 conjunc... |
| ad5ant13 769 | Deduction adding conjuncts... |
| ad5ant14 770 | Deduction adding conjuncts... |
| ad5ant15 771 | Deduction adding conjuncts... |
| ad5ant23 772 | Deduction adding conjuncts... |
| ad5ant24 773 | Deduction adding conjuncts... |
| ad5ant25 774 | Deduction adding conjuncts... |
| adantl5r 775 | Deduction adding 1 conjunc... |
| adantl6r 776 | Deduction adding 1 conjunc... |
| pm3.33 777 | Theorem *3.33 (Syll) of [W... |
| pm3.34 778 | Theorem *3.34 (Syll) of [W... |
| simpll 779 | Simplification of a conjun... |
| simplld 780 | Deduction form of ~ simpll... |
| simplr 781 | Simplification of a conjun... |
| simplrd 782 | Deduction eliminating a do... |
| simprl 783 | Simplification of a conjun... |
| simprld 784 | Deduction eliminating a do... |
| simprr 785 | Simplification of a conjun... |
| simprrd 786 | Deduction form of ~ simprr... |
| simplll 787 | Simplification of a conjun... |
| simpllr 788 | Simplification of a conjun... |
| simplrl 789 | Simplification of a conjun... |
| simplrr 790 | Simplification of a conjun... |
| simprll 791 | Simplification of a conjun... |
| simprlr 792 | Simplification of a conjun... |
| simprrl 793 | Simplification of a conjun... |
| simprrr 794 | Simplification of a conjun... |
| simp-4l 795 | Simplification of a conjun... |
| simp-4r 796 | Simplification of a conjun... |
| simp-5l 797 | Simplification of a conjun... |
| simp-5r 798 | Simplification of a conjun... |
| simp-6l 799 | Simplification of a conjun... |
| simp-6r 800 | Simplification of a conjun... |
| simp-7l 801 | Simplification of a conjun... |
| simp-7r 802 | Simplification of a conjun... |
| simp-8l 803 | Simplification of a conjun... |
| simp-8r 804 | Simplification of a conjun... |
| simp-9l 805 | Simplification of a conjun... |
| simp-9r 806 | Simplification of a conjun... |
| simp-10l 807 | Simplification of a conjun... |
| simp-10r 808 | Simplification of a conjun... |
| simp-11l 809 | Simplification of a conjun... |
| simp-11r 810 | Simplification of a conjun... |
| pm2.01da 811 | Deduction based on reducti... |
| pm2.18da 812 | Deduction based on reducti... |
| impbida 813 | Deduce an equivalence from... |
| pm5.21nd 814 | Eliminate an antecedent im... |
| pm3.35 815 | Conjunctive detachment. T... |
| pm5.74da 816 | Distribution of implicatio... |
| bitr 817 | Theorem *4.22 of [Whitehea... |
| biantr 818 | A transitive law of equiva... |
| pm4.14 819 | Theorem *4.14 of [Whitehea... |
| pm3.37 820 | Theorem *3.37 (Transp) of ... |
| anim12 821 | Conjoin antecedents and co... |
| pm3.4 822 | Conjunction implies implic... |
| exbiri 823 | Inference form of ~ exbir ... |
| pm2.61ian 824 | Elimination of an antecede... |
| pm2.61dan 825 | Elimination of an antecede... |
| pm2.61ddan 826 | Elimination of two anteced... |
| pm2.61dda 827 | Elimination of two anteced... |
| mtand 828 | A modus tollens deduction.... |
| pm2.65da 829 | Deduction for proof by con... |
| condan 830 | Proof by contradiction. (... |
| biadan 831 | An implication is equivale... |
| biadani 832 | Inference associated with ... |
| biadaniALT 833 | Alternate proof of ~ biada... |
| biadanii 834 | Inference associated with ... |
| biadanid 835 | Deduction associated with ... |
| pm5.1 836 | Two propositions are equiv... |
| pm5.21 837 | Two propositions are equiv... |
| pm5.35 838 | Theorem *5.35 of [Whitehea... |
| abai 839 | Introduce one conjunct as ... |
| abab 840 | Introduce one conjunct as ... |
| pm4.45im 841 | Conjunction with implicati... |
| impimprbi 842 | An implication and its rev... |
| nan 843 | Theorem to move a conjunct... |
| pm5.31 844 | Theorem *5.31 of [Whitehea... |
| pm5.31r 845 | Variant of ~ pm5.31 . (Co... |
| pm4.15 846 | Theorem *4.15 of [Whitehea... |
| pm5.36 847 | Theorem *5.36 of [Whitehea... |
| annotanannot 848 | A conjunction with a negat... |
| pm5.33 849 | Theorem *5.33 of [Whitehea... |
| syl12anc 850 | Syllogism combined with co... |
| syl21anc 851 | Syllogism combined with co... |
| syl22anc 852 | Syllogism combined with co... |
| bibiad 853 | Eliminate an hypothesis ` ... |
| syl1111anc 854 | Four-hypothesis eliminatio... |
| syldbl2 855 | Stacked hypotheseis implie... |
| mpsyl4anc 856 | An elimination deduction. ... |
| pm4.87 857 | Theorem *4.87 of [Whitehea... |
| bimsc1 858 | Removal of conjunct from o... |
| a2and 859 | Deduction distributing a c... |
| animpimp2impd 860 | Deduction deriving nested ... |
| pm4.64 863 | Theorem *4.64 of [Whitehea... |
| pm4.66 864 | Theorem *4.66 of [Whitehea... |
| pm2.53 865 | Theorem *2.53 of [Whitehea... |
| pm2.54 866 | Theorem *2.54 of [Whitehea... |
| imor 867 | Implication in terms of di... |
| imori 868 | Infer disjunction from imp... |
| imorri 869 | Infer implication from dis... |
| pm4.62 870 | Theorem *4.62 of [Whitehea... |
| jaoi 871 | Inference disjoining the a... |
| jao1i 872 | Add a disjunct in the ante... |
| jaod 873 | Deduction disjoining the a... |
| mpjaod 874 | Eliminate a disjunction in... |
| ori 875 | Infer implication from dis... |
| orri 876 | Infer disjunction from imp... |
| orrd 877 | Deduce disjunction from im... |
| ord 878 | Deduce implication from di... |
| orci 879 | Deduction introducing a di... |
| olci 880 | Deduction introducing a di... |
| orc 881 | Introduction of a disjunct... |
| olc 882 | Introduction of a disjunct... |
| pm1.4 883 | Axiom *1.4 of [WhiteheadRu... |
| orcom 884 | Commutative law for disjun... |
| orcomd 885 | Commutation of disjuncts i... |
| orcoms 886 | Commutation of disjuncts i... |
| orcd 887 | Deduction introducing a di... |
| olcd 888 | Deduction introducing a di... |
| orcs 889 | Deduction eliminating disj... |
| olcs 890 | Deduction eliminating disj... |
| olcnd 891 | A lemma for Conjunctive No... |
| orcnd 892 | A lemma for Conjunctive No... |
| mtord 893 | A modus tollens deduction ... |
| pm3.2ni 894 | Infer negated disjunction ... |
| pm2.45 895 | Theorem *2.45 of [Whitehea... |
| pm2.46 896 | Theorem *2.46 of [Whitehea... |
| pm2.47 897 | Theorem *2.47 of [Whitehea... |
| pm2.48 898 | Theorem *2.48 of [Whitehea... |
| pm2.49 899 | Theorem *2.49 of [Whitehea... |
| norbi 900 | If neither of two proposit... |
| nbior 901 | If two propositions are no... |
| orel1 902 | Elimination of disjunction... |
| pm2.25 903 | Theorem *2.25 of [Whitehea... |
| orel2 904 | Elimination of disjunction... |
| pm2.67-2 905 | Slight generalization of T... |
| pm2.67 906 | Theorem *2.67 of [Whitehea... |
| curryax 907 | A non-intuitionistic posit... |
| exmid 908 | Law of excluded middle, al... |
| exmidd 909 | Law of excluded middle in ... |
| pm2.1 910 | Theorem *2.1 of [Whitehead... |
| pm2.13 911 | Theorem *2.13 of [Whitehea... |
| pm2.621 912 | Theorem *2.621 of [Whitehe... |
| pm2.62 913 | Theorem *2.62 of [Whitehea... |
| pm2.68 914 | Theorem *2.68 of [Whitehea... |
| dfor2 915 | Logical 'or' expressed in ... |
| pm2.07 916 | Theorem *2.07 of [Whitehea... |
| pm1.2 917 | Axiom *1.2 of [WhiteheadRu... |
| oridm 918 | Idempotent law for disjunc... |
| pm4.25 919 | Theorem *4.25 of [Whitehea... |
| pm2.4 920 | Theorem *2.4 of [Whitehead... |
| pm2.41 921 | Theorem *2.41 of [Whitehea... |
| orim12i 922 | Disjoin antecedents and co... |
| orim1i 923 | Introduce disjunct to both... |
| orim2i 924 | Introduce disjunct to both... |
| orim12dALT 925 | Alternate proof of ~ orim1... |
| orbi2i 926 | Inference adding a left di... |
| orbi1i 927 | Inference adding a right d... |
| orbi12i 928 | Infer the disjunction of t... |
| orbi2d 929 | Deduction adding a left di... |
| orbi1d 930 | Deduction adding a right d... |
| orbi1 931 | Theorem *4.37 of [Whitehea... |
| orbi12d 932 | Deduction joining two equi... |
| pm1.5 933 | Axiom *1.5 (Assoc) of [Whi... |
| or12 934 | Swap two disjuncts. (Cont... |
| orass 935 | Associative law for disjun... |
| pm2.31 936 | Theorem *2.31 of [Whitehea... |
| pm2.32 937 | Theorem *2.32 of [Whitehea... |
| pm2.3 938 | Theorem *2.3 of [Whitehead... |
| or32 939 | A rearrangement of disjunc... |
| or4 940 | Rearrangement of 4 disjunc... |
| or42 941 | Rearrangement of 4 disjunc... |
| orordi 942 | Distribution of disjunctio... |
| orordir 943 | Distribution of disjunctio... |
| orimdi 944 | Disjunction distributes ov... |
| pm2.76 945 | Theorem *2.76 of [Whitehea... |
| pm2.85 946 | Theorem *2.85 of [Whitehea... |
| pm2.75 947 | Theorem *2.75 of [Whitehea... |
| pm4.78 948 | Implication distributes ov... |
| biort 949 | A disjunction with a true ... |
| biorf 950 | A wff is equivalent to its... |
| biortn 951 | A wff is equivalent to its... |
| biorfi 952 | The dual of ~ biorf is not... |
| biorfri 953 | A wff is equivalent to its... |
| pm2.26 954 | Theorem *2.26 of [Whitehea... |
| pm2.63 955 | Theorem *2.63 of [Whitehea... |
| pm2.64 956 | Theorem *2.64 of [Whitehea... |
| pm2.42 957 | Theorem *2.42 of [Whitehea... |
| pm5.11g 958 | A general instance of Theo... |
| pm5.11 959 | Theorem *5.11 of [Whitehea... |
| pm5.12 960 | Theorem *5.12 of [Whitehea... |
| pm5.14 961 | Theorem *5.14 of [Whitehea... |
| pm5.13 962 | Theorem *5.13 of [Whitehea... |
| pm5.55 963 | Theorem *5.55 of [Whitehea... |
| pm4.72 964 | Implication in terms of bi... |
| imimorb 965 | Simplify an implication be... |
| oibabs 966 | Absorption of disjunction ... |
| orbidi 967 | Disjunction distributes ov... |
| pm5.7 968 | Disjunction distributes ov... |
| jaao 969 | Inference conjoining and d... |
| jaoa 970 | Inference disjoining and c... |
| jaoian 971 | Inference disjoining the a... |
| jaodan 972 | Deduction disjoining the a... |
| mpjaodan 973 | Eliminate a disjunction in... |
| pm3.44 974 | Theorem *3.44 of [Whitehea... |
| jao 975 | Disjunction of antecedents... |
| jaob 976 | Disjunction of antecedents... |
| pm4.77 977 | Theorem *4.77 of [Whitehea... |
| pm3.48 978 | Theorem *3.48 of [Whitehea... |
| orim12d 979 | Disjoin antecedents and co... |
| orim12da 980 | Deduce a disjunction from ... |
| orim1d 981 | Disjoin antecedents and co... |
| orim2d 982 | Disjoin antecedents and co... |
| orim2 983 | Axiom *1.6 (Sum) of [White... |
| pm2.38 984 | Theorem *2.38 of [Whitehea... |
| pm2.36 985 | Theorem *2.36 of [Whitehea... |
| pm2.37 986 | Theorem *2.37 of [Whitehea... |
| pm2.81 987 | Theorem *2.81 of [Whitehea... |
| pm2.8 988 | Theorem *2.8 of [Whitehead... |
| pm2.73 989 | Theorem *2.73 of [Whitehea... |
| pm2.74 990 | Theorem *2.74 of [Whitehea... |
| pm2.82 991 | Theorem *2.82 of [Whitehea... |
| pm4.39 992 | Theorem *4.39 of [Whitehea... |
| animorl 993 | Conjunction implies disjun... |
| animorr 994 | Conjunction implies disjun... |
| animorlr 995 | Conjunction implies disjun... |
| animorrl 996 | Conjunction implies disjun... |
| ianor 997 | Negated conjunction in ter... |
| anor 998 | Conjunction in terms of di... |
| ioran 999 | Negated disjunction in ter... |
| pm4.52 1000 | Theorem *4.52 of [Whitehea... |
| pm4.53 1001 | Theorem *4.53 of [Whitehea... |
| pm4.54 1002 | Theorem *4.54 of [Whitehea... |
| pm4.55 1003 | Theorem *4.55 of [Whitehea... |
| pm4.56 1004 | Theorem *4.56 of [Whitehea... |
| oran 1005 | Disjunction in terms of co... |
| pm4.57 1006 | Theorem *4.57 of [Whitehea... |
| pm3.1 1007 | Theorem *3.1 of [Whitehead... |
| pm3.11 1008 | Theorem *3.11 of [Whitehea... |
| pm3.12 1009 | Theorem *3.12 of [Whitehea... |
| pm3.13 1010 | Theorem *3.13 of [Whitehea... |
| pm3.14 1011 | Theorem *3.14 of [Whitehea... |
| pm4.44 1012 | Theorem *4.44 of [Whitehea... |
| pm4.45 1013 | Theorem *4.45 of [Whitehea... |
| orabs 1014 | Absorption of redundant in... |
| oranabs 1015 | Absorb a disjunct into a c... |
| pm5.61 1016 | Theorem *5.61 of [Whitehea... |
| pm5.6 1017 | Conjunction in antecedent ... |
| orcanai 1018 | Change disjunction in cons... |
| orsild 1019 | A lemma for not-or-not eli... |
| orsird 1020 | A lemma for not-or-not eli... |
| pm4.79 1021 | Theorem *4.79 of [Whitehea... |
| pm5.53 1022 | Theorem *5.53 of [Whitehea... |
| ordi 1023 | Distributive law for disju... |
| ordir 1024 | Distributive law for disju... |
| andi 1025 | Distributive law for conju... |
| andir 1026 | Distributive law for conju... |
| orddi 1027 | Double distributive law fo... |
| anddi 1028 | Double distributive law fo... |
| pm5.17 1029 | Theorem *5.17 of [Whitehea... |
| pm5.15 1030 | Theorem *5.15 of [Whitehea... |
| pm5.16 1031 | Theorem *5.16 of [Whitehea... |
| xor 1032 | Two ways to express exclus... |
| nbi2 1033 | Two ways to express "exclu... |
| xordi 1034 | Conjunction distributes ov... |
| pm5.54 1035 | Theorem *5.54 of [Whitehea... |
| pm5.62 1036 | Theorem *5.62 of [Whitehea... |
| pm5.63 1037 | Theorem *5.63 of [Whitehea... |
| niabn 1038 | Miscellaneous inference re... |
| ninba 1039 | Miscellaneous inference re... |
| pm4.43 1040 | Theorem *4.43 of [Whitehea... |
| pm4.82 1041 | Theorem *4.82 of [Whitehea... |
| pm4.83 1042 | Theorem *4.83 of [Whitehea... |
| pclem6 1043 | Negation inferred from emb... |
| bigolden 1044 | Dijkstra-Scholten's Golden... |
| pm5.71 1045 | Theorem *5.71 of [Whitehea... |
| pm5.75 1046 | Theorem *5.75 of [Whitehea... |
| ecase2d 1047 | Deduction for elimination ... |
| ecase3 1048 | Inference for elimination ... |
| ecase 1049 | Inference for elimination ... |
| ecase3d 1050 | Deduction for elimination ... |
| ecased 1051 | Deduction for elimination ... |
| ecase3ad 1052 | Deduction for elimination ... |
| ccase 1053 | Inference for combining ca... |
| ccased 1054 | Deduction for combining ca... |
| ccase2 1055 | Inference for combining ca... |
| 4cases 1056 | Inference eliminating two ... |
| 4casesdan 1057 | Deduction eliminating two ... |
| cases 1058 | Case disjunction according... |
| dedlem0a 1059 | Lemma for an alternate ver... |
| dedlem0b 1060 | Lemma for an alternate ver... |
| dedlema 1061 | Lemma for weak deduction t... |
| dedlemb 1062 | Lemma for weak deduction t... |
| cases2 1063 | Case disjunction according... |
| cases2ALT 1064 | Alternate proof of ~ cases... |
| dfbi3 1065 | An alternate definition of... |
| pm5.24 1066 | Theorem *5.24 of [Whitehea... |
| 4exmid 1067 | The disjunction of the fou... |
| consensus 1068 | The consensus theorem. Th... |
| pm4.42 1069 | Theorem *4.42 of [Whitehea... |
| prlem1 1070 | A specialized lemma for se... |
| prlem2 1071 | A specialized lemma for se... |
| oplem1 1072 | A specialized lemma for se... |
| dn1 1073 | A single axiom for Boolean... |
| bianir 1074 | A closed form of ~ mpbir ,... |
| jaoi2 1075 | Inference removing a negat... |
| jaoi3 1076 | Inference separating a dis... |
| ornld 1077 | Selecting one statement fr... |
| dfifp2 1080 | Alternate definition of th... |
| dfifp3 1081 | Alternate definition of th... |
| dfifp4 1082 | Alternate definition of th... |
| dfifp5 1083 | Alternate definition of th... |
| dfifp6 1084 | Alternate definition of th... |
| dfifp7 1085 | Alternate definition of th... |
| ifpdfbi 1086 | Define the biconditional a... |
| ifpdfbiOLD 1087 | Obsolete version of ~ ifpd... |
| anifp 1088 | The conditional operator i... |
| ifpor 1089 | The conditional operator i... |
| ifpn 1090 | Conditional operator for t... |
| ifptru 1091 | Value of the conditional o... |
| ifpfal 1092 | Value of the conditional o... |
| ifpid 1093 | Value of the conditional o... |
| casesifp 1094 | Version of ~ cases express... |
| ifpbi123d 1095 | Equivalence deduction for ... |
| ifpbi23d 1096 | Equivalence deduction for ... |
| ifpimpda 1097 | Separation of the values o... |
| 1fpid3 1098 | The value of the condition... |
| elimh 1099 | Hypothesis builder for the... |
| dedt 1100 | The weak deduction theorem... |
| con3ALT 1101 | Proof of ~ con3 from its a... |
| 3orass 1106 | Associative law for triple... |
| 3orel1 1107 | Partial elimination of a t... |
| 3orrot 1108 | Rotation law for triple di... |
| 3orcoma 1109 | Commutation law for triple... |
| 3orcomb 1110 | Commutation law for triple... |
| 3anass 1111 | Associative law for triple... |
| 3anan12 1112 | Convert triple conjunction... |
| 3anan32 1113 | Convert triple conjunction... |
| 3anan32OLD 1114 | Obsolete version of ~ 3ana... |
| 3ancoma 1115 | Commutation law for triple... |
| 3ancomb 1116 | Commutation law for triple... |
| 3anrot 1117 | Rotation law for triple co... |
| 3anrev 1118 | Reversal law for triple co... |
| anandi3 1119 | Distribution of triple con... |
| anandi3r 1120 | Distribution of triple con... |
| 3anidm 1121 | Idempotent law for conjunc... |
| 3an4anass 1122 | Associative law for four c... |
| 3ioran 1123 | Negated triple disjunction... |
| 3ianor 1124 | Negated triple conjunction... |
| 3anor 1125 | Triple conjunction express... |
| 3oran 1126 | Triple disjunction in term... |
| 3impa 1127 | Importation from double to... |
| 3imp 1128 | Importation inference. (C... |
| 3imp31 1129 | The importation inference ... |
| 3imp231 1130 | Importation inference. (C... |
| 3imp21 1131 | The importation inference ... |
| 3impb 1132 | Importation from double to... |
| bi23imp13 1133 | ~ 3imp with middle implica... |
| 3impib 1134 | Importation to triple conj... |
| 3impia 1135 | Importation to triple conj... |
| 3expa 1136 | Exportation from triple to... |
| 3exp 1137 | Exportation inference. (C... |
| 3expb 1138 | Exportation from triple to... |
| 3expia 1139 | Exportation from triple co... |
| 3expib 1140 | Exportation from triple co... |
| 3com12 1141 | Commutation in antecedent.... |
| 3com13 1142 | Commutation in antecedent.... |
| 3comr 1143 | Commutation in antecedent.... |
| 3com23 1144 | Commutation in antecedent.... |
| 3coml 1145 | Commutation in antecedent.... |
| 3jca 1146 | Join consequents with conj... |
| 3jcad 1147 | Deduction conjoining the c... |
| 3adant1 1148 | Deduction adding a conjunc... |
| 3adant2 1149 | Deduction adding a conjunc... |
| 3adant3 1150 | Deduction adding a conjunc... |
| 3ad2ant1 1151 | Deduction adding conjuncts... |
| 3ad2ant2 1152 | Deduction adding conjuncts... |
| 3ad2ant3 1153 | Deduction adding conjuncts... |
| simp1 1154 | Simplification of triple c... |
| simp2 1155 | Simplification of triple c... |
| simp3 1156 | Simplification of triple c... |
| simp1i 1157 | Infer a conjunct from a tr... |
| simp2i 1158 | Infer a conjunct from a tr... |
| simp3i 1159 | Infer a conjunct from a tr... |
| simp1d 1160 | Deduce a conjunct from a t... |
| simp2d 1161 | Deduce a conjunct from a t... |
| simp3d 1162 | Deduce a conjunct from a t... |
| simp1bi 1163 | Deduce a conjunct from a t... |
| simp2bi 1164 | Deduce a conjunct from a t... |
| simp3bi 1165 | Deduce a conjunct from a t... |
| 3simpa 1166 | Simplification of triple c... |
| 3simpb 1167 | Simplification of triple c... |
| 3simpc 1168 | Simplification of triple c... |
| 3anim123i 1169 | Join antecedents and conse... |
| 3anim1i 1170 | Add two conjuncts to antec... |
| 3anim2i 1171 | Add two conjuncts to antec... |
| 3anim3i 1172 | Add two conjuncts to antec... |
| 3anbi123i 1173 | Join 3 biconditionals with... |
| 3orbi123i 1174 | Join 3 biconditionals with... |
| 3anbi1i 1175 | Inference adding two conju... |
| 3anbi2i 1176 | Inference adding two conju... |
| 3anbi3i 1177 | Inference adding two conju... |
| syl3an 1178 | A triple syllogism inferen... |
| syl3anb 1179 | A triple syllogism inferen... |
| syl3anbr 1180 | A triple syllogism inferen... |
| syl3an1 1181 | A syllogism inference. (C... |
| syl3an2 1182 | A syllogism inference. (C... |
| syl3an3 1183 | A syllogism inference. (C... |
| syl3an132 1184 | ~ syl2an with antecedents ... |
| 3adantl1 1185 | Deduction adding a conjunc... |
| 3adantl2 1186 | Deduction adding a conjunc... |
| 3adantl3 1187 | Deduction adding a conjunc... |
| 3adantr1 1188 | Deduction adding a conjunc... |
| 3adantr2 1189 | Deduction adding a conjunc... |
| 3adantr3 1190 | Deduction adding a conjunc... |
| ad4ant123 1191 | Deduction adding conjuncts... |
| ad4ant124 1192 | Deduction adding conjuncts... |
| ad4ant134 1193 | Deduction adding conjuncts... |
| ad4ant234 1194 | Deduction adding conjuncts... |
| 3adant1l 1195 | Deduction adding a conjunc... |
| 3adant1r 1196 | Deduction adding a conjunc... |
| 3adant2l 1197 | Deduction adding a conjunc... |
| 3adant2r 1198 | Deduction adding a conjunc... |
| 3adant3l 1199 | Deduction adding a conjunc... |
| 3adant3r 1200 | Deduction adding a conjunc... |
| 3adant3r1 1201 | Deduction adding a conjunc... |
| 3adant3r2 1202 | Deduction adding a conjunc... |
| 3adant3r3 1203 | Deduction adding a conjunc... |
| 3ad2antl1 1204 | Deduction adding conjuncts... |
| 3ad2antl2 1205 | Deduction adding conjuncts... |
| 3ad2antl3 1206 | Deduction adding conjuncts... |
| 3ad2antr1 1207 | Deduction adding conjuncts... |
| 3ad2antr2 1208 | Deduction adding conjuncts... |
| 3ad2antr3 1209 | Deduction adding conjuncts... |
| simpl1 1210 | Simplification of conjunct... |
| simpl2 1211 | Simplification of conjunct... |
| simpl3 1212 | Simplification of conjunct... |
| simpr1 1213 | Simplification of conjunct... |
| simpr2 1214 | Simplification of conjunct... |
| simpr3 1215 | Simplification of conjunct... |
| simp1l 1216 | Simplification of triple c... |
| simp1r 1217 | Simplification of triple c... |
| simp2l 1218 | Simplification of triple c... |
| simp2r 1219 | Simplification of triple c... |
| simp3l 1220 | Simplification of triple c... |
| simp3r 1221 | Simplification of triple c... |
| simp11 1222 | Simplification of doubly t... |
| simp12 1223 | Simplification of doubly t... |
| simp13 1224 | Simplification of doubly t... |
| simp21 1225 | Simplification of doubly t... |
| simp22 1226 | Simplification of doubly t... |
| simp23 1227 | Simplification of doubly t... |
| simp31 1228 | Simplification of doubly t... |
| simp32 1229 | Simplification of doubly t... |
| simp33 1230 | Simplification of doubly t... |
| simpll1 1231 | Simplification of conjunct... |
| simpll2 1232 | Simplification of conjunct... |
| simpll3 1233 | Simplification of conjunct... |
| simplr1 1234 | Simplification of conjunct... |
| simplr2 1235 | Simplification of conjunct... |
| simplr3 1236 | Simplification of conjunct... |
| simprl1 1237 | Simplification of conjunct... |
| simprl2 1238 | Simplification of conjunct... |
| simprl3 1239 | Simplification of conjunct... |
| simprr1 1240 | Simplification of conjunct... |
| simprr2 1241 | Simplification of conjunct... |
| simprr3 1242 | Simplification of conjunct... |
| simpl1l 1243 | Simplification of conjunct... |
| simpl1r 1244 | Simplification of conjunct... |
| simpl2l 1245 | Simplification of conjunct... |
| simpl2r 1246 | Simplification of conjunct... |
| simpl3l 1247 | Simplification of conjunct... |
| simpl3r 1248 | Simplification of conjunct... |
| simpr1l 1249 | Simplification of conjunct... |
| simpr1r 1250 | Simplification of conjunct... |
| simpr2l 1251 | Simplification of conjunct... |
| simpr2r 1252 | Simplification of conjunct... |
| simpr3l 1253 | Simplification of conjunct... |
| simpr3r 1254 | Simplification of conjunct... |
| simp1ll 1255 | Simplification of conjunct... |
| simp1lr 1256 | Simplification of conjunct... |
| simp1rl 1257 | Simplification of conjunct... |
| simp1rr 1258 | Simplification of conjunct... |
| simp2ll 1259 | Simplification of conjunct... |
| simp2lr 1260 | Simplification of conjunct... |
| simp2rl 1261 | Simplification of conjunct... |
| simp2rr 1262 | Simplification of conjunct... |
| simp3ll 1263 | Simplification of conjunct... |
| simp3lr 1264 | Simplification of conjunct... |
| simp3rl 1265 | Simplification of conjunct... |
| simp3rr 1266 | Simplification of conjunct... |
| simpl11 1267 | Simplification of conjunct... |
| simpl12 1268 | Simplification of conjunct... |
| simpl13 1269 | Simplification of conjunct... |
| simpl21 1270 | Simplification of conjunct... |
| simpl22 1271 | Simplification of conjunct... |
| simpl23 1272 | Simplification of conjunct... |
| simpl31 1273 | Simplification of conjunct... |
| simpl32 1274 | Simplification of conjunct... |
| simpl33 1275 | Simplification of conjunct... |
| simpr11 1276 | Simplification of conjunct... |
| simpr12 1277 | Simplification of conjunct... |
| simpr13 1278 | Simplification of conjunct... |
| simpr21 1279 | Simplification of conjunct... |
| simpr22 1280 | Simplification of conjunct... |
| simpr23 1281 | Simplification of conjunct... |
| simpr31 1282 | Simplification of conjunct... |
| simpr32 1283 | Simplification of conjunct... |
| simpr33 1284 | Simplification of conjunct... |
| simp1l1 1285 | Simplification of conjunct... |
| simp1l2 1286 | Simplification of conjunct... |
| simp1l3 1287 | Simplification of conjunct... |
| simp1r1 1288 | Simplification of conjunct... |
| simp1r2 1289 | Simplification of conjunct... |
| simp1r3 1290 | Simplification of conjunct... |
| simp2l1 1291 | Simplification of conjunct... |
| simp2l2 1292 | Simplification of conjunct... |
| simp2l3 1293 | Simplification of conjunct... |
| simp2r1 1294 | Simplification of conjunct... |
| simp2r2 1295 | Simplification of conjunct... |
| simp2r3 1296 | Simplification of conjunct... |
| simp3l1 1297 | Simplification of conjunct... |
| simp3l2 1298 | Simplification of conjunct... |
| simp3l3 1299 | Simplification of conjunct... |
| simp3r1 1300 | Simplification of conjunct... |
| simp3r2 1301 | Simplification of conjunct... |
| simp3r3 1302 | Simplification of conjunct... |
| simp11l 1303 | Simplification of conjunct... |
| simp11r 1304 | Simplification of conjunct... |
| simp12l 1305 | Simplification of conjunct... |
| simp12r 1306 | Simplification of conjunct... |
| simp13l 1307 | Simplification of conjunct... |
| simp13r 1308 | Simplification of conjunct... |
| simp21l 1309 | Simplification of conjunct... |
| simp21r 1310 | Simplification of conjunct... |
| simp22l 1311 | Simplification of conjunct... |
| simp22r 1312 | Simplification of conjunct... |
| simp23l 1313 | Simplification of conjunct... |
| simp23r 1314 | Simplification of conjunct... |
| simp31l 1315 | Simplification of conjunct... |
| simp31r 1316 | Simplification of conjunct... |
| simp32l 1317 | Simplification of conjunct... |
| simp32r 1318 | Simplification of conjunct... |
| simp33l 1319 | Simplification of conjunct... |
| simp33r 1320 | Simplification of conjunct... |
| simp111 1321 | Simplification of conjunct... |
| simp112 1322 | Simplification of conjunct... |
| simp113 1323 | Simplification of conjunct... |
| simp121 1324 | Simplification of conjunct... |
| simp122 1325 | Simplification of conjunct... |
| simp123 1326 | Simplification of conjunct... |
| simp131 1327 | Simplification of conjunct... |
| simp132 1328 | Simplification of conjunct... |
| simp133 1329 | Simplification of conjunct... |
| simp211 1330 | Simplification of conjunct... |
| simp212 1331 | Simplification of conjunct... |
| simp213 1332 | Simplification of conjunct... |
| simp221 1333 | Simplification of conjunct... |
| simp222 1334 | Simplification of conjunct... |
| simp223 1335 | Simplification of conjunct... |
| simp231 1336 | Simplification of conjunct... |
| simp232 1337 | Simplification of conjunct... |
| simp233 1338 | Simplification of conjunct... |
| simp311 1339 | Simplification of conjunct... |
| simp312 1340 | Simplification of conjunct... |
| simp313 1341 | Simplification of conjunct... |
| simp321 1342 | Simplification of conjunct... |
| simp322 1343 | Simplification of conjunct... |
| simp323 1344 | Simplification of conjunct... |
| simp331 1345 | Simplification of conjunct... |
| simp332 1346 | Simplification of conjunct... |
| simp333 1347 | Simplification of conjunct... |
| 3anibar 1348 | Remove a hypothesis from t... |
| 3mix1 1349 | Introduction in triple dis... |
| 3mix2 1350 | Introduction in triple dis... |
| 3mix3 1351 | Introduction in triple dis... |
| 3mix1i 1352 | Introduction in triple dis... |
| 3mix2i 1353 | Introduction in triple dis... |
| 3mix3i 1354 | Introduction in triple dis... |
| 3mix1d 1355 | Deduction introducing trip... |
| 3mix2d 1356 | Deduction introducing trip... |
| 3mix3d 1357 | Deduction introducing trip... |
| 3pm3.2i 1358 | Infer conjunction of premi... |
| pm3.2an3 1359 | Version of ~ pm3.2 for a t... |
| mpbir3an 1360 | Detach a conjunction of tr... |
| mpbir3and 1361 | Detach a conjunction of tr... |
| syl3anbrc 1362 | Syllogism inference. (Con... |
| syl21anbrc 1363 | Syllogism inference. (Con... |
| 3imp3i2an 1364 | An elimination deduction. ... |
| ex3 1365 | Apply ~ ex to a hypothesis... |
| 3imp1 1366 | Importation to left triple... |
| 3impd 1367 | Importation deduction for ... |
| 3imp2 1368 | Importation to right tripl... |
| 3impdi 1369 | Importation inference (und... |
| 3impdir 1370 | Importation inference (und... |
| 3exp1 1371 | Exportation from left trip... |
| 3expd 1372 | Exportation deduction for ... |
| 3exp2 1373 | Exportation from right tri... |
| exp5o 1374 | A triple exportation infer... |
| exp516 1375 | A triple exportation infer... |
| exp520 1376 | A triple exportation infer... |
| 3impexp 1377 | Version of ~ impexp for a ... |
| 3an1rs 1378 | Swap conjuncts. (Contribu... |
| 13an22anass 1379 | Associative law for four c... |
| 13an22anassOLD 1380 | Obsolete version of ~ 13an... |
| 3anasss 1381 | Associative law for conjun... |
| 3anassrs 1382 | Associative law for conjun... |
| 4anpull2 1383 | An equivalence of two four... |
| 4anpull2OLD 1384 | Obsolete version of ~ 4anp... |
| ad5ant245 1385 | Deduction adding conjuncts... |
| ad5ant234 1386 | Deduction adding conjuncts... |
| ad5ant235 1387 | Deduction adding conjuncts... |
| ad5ant123 1388 | Deduction adding conjuncts... |
| ad5ant124 1389 | Deduction adding conjuncts... |
| ad5ant124OLD 1390 | Obsolete version of ~ ad5a... |
| ad5ant125 1391 | Deduction adding conjuncts... |
| ad5ant125OLD 1392 | Obsolete version of ~ ad5a... |
| ad5ant134 1393 | Deduction adding conjuncts... |
| ad5ant134OLD 1394 | Obsolete version of ~ ad5a... |
| ad5ant135 1395 | Deduction adding conjuncts... |
| ad5ant135OLD 1396 | Obsolete version of ~ ad5a... |
| ad5ant145 1397 | Deduction adding conjuncts... |
| ad5ant2345 1398 | Deduction adding conjuncts... |
| syl3anc 1399 | Syllogism combined with co... |
| syl13anc 1400 | Syllogism combined with co... |
| syl31anc 1401 | Syllogism combined with co... |
| syl112anc 1402 | Syllogism combined with co... |
| syl121anc 1403 | Syllogism combined with co... |
| syl211anc 1404 | Syllogism combined with co... |
| syl23anc 1405 | Syllogism combined with co... |
| syl32anc 1406 | Syllogism combined with co... |
| syl122anc 1407 | Syllogism combined with co... |
| syl212anc 1408 | Syllogism combined with co... |
| syl221anc 1409 | Syllogism combined with co... |
| syl113anc 1410 | Syllogism combined with co... |
| syl131anc 1411 | Syllogism combined with co... |
| syl311anc 1412 | Syllogism combined with co... |
| syl33anc 1413 | Syllogism combined with co... |
| syl222anc 1414 | Syllogism combined with co... |
| syl123anc 1415 | Syllogism combined with co... |
| syl132anc 1416 | Syllogism combined with co... |
| syl213anc 1417 | Syllogism combined with co... |
| syl231anc 1418 | Syllogism combined with co... |
| syl312anc 1419 | Syllogism combined with co... |
| syl321anc 1420 | Syllogism combined with co... |
| syl133anc 1421 | Syllogism combined with co... |
| syl313anc 1422 | Syllogism combined with co... |
| syl331anc 1423 | Syllogism combined with co... |
| syl223anc 1424 | Syllogism combined with co... |
| syl232anc 1425 | Syllogism combined with co... |
| syl322anc 1426 | Syllogism combined with co... |
| syl233anc 1427 | Syllogism combined with co... |
| syl323anc 1428 | Syllogism combined with co... |
| syl332anc 1429 | Syllogism combined with co... |
| syl333anc 1430 | A syllogism inference comb... |
| syl3an1b 1431 | A syllogism inference. (C... |
| syl3an2b 1432 | A syllogism inference. (C... |
| syl3an3b 1433 | A syllogism inference. (C... |
| syl3an1br 1434 | A syllogism inference. (C... |
| syl3an2br 1435 | A syllogism inference. (C... |
| syl3an3br 1436 | A syllogism inference. (C... |
| syld3an3 1437 | A syllogism inference. (C... |
| syld3an1 1438 | A syllogism inference. (C... |
| syld3an2 1439 | A syllogism inference. (C... |
| syl3anl1 1440 | A syllogism inference. (C... |
| syl3anl2 1441 | A syllogism inference. (C... |
| syl3anl3 1442 | A syllogism inference. (C... |
| syl3anl 1443 | A triple syllogism inferen... |
| syl3anr1 1444 | A syllogism inference. (C... |
| syl3anr2 1445 | A syllogism inference. (C... |
| syl3anr3 1446 | A syllogism inference. (C... |
| 3anidm12 1447 | Inference from idempotent ... |
| 3anidm13 1448 | Inference from idempotent ... |
| 3anidm23 1449 | Inference from idempotent ... |
| syl2an3an 1450 | ~ syl3an with antecedents ... |
| syl2an23an 1451 | Deduction related to ~ syl... |
| 3ori 1452 | Infer implication from tri... |
| 3jao 1453 | Disjunction of three antec... |
| 3jaob 1454 | Disjunction of three antec... |
| 3jaoi 1455 | Disjunction of three antec... |
| 3jaoiOLD 1456 | Obsolete version of ~ 3jao... |
| 3jaod 1457 | Disjunction of three antec... |
| 3jaoian 1458 | Disjunction of three antec... |
| 3jaodan 1459 | Disjunction of three antec... |
| mpjao3dan 1460 | Eliminate a three-way disj... |
| 3jaao 1461 | Inference conjoining and d... |
| 3jaaoOLD 1462 | Obsolete version of ~ 3jaa... |
| syl3an9b 1463 | Nested syllogism inference... |
| 3orbi123d 1464 | Deduction joining 3 equiva... |
| 3anbi123d 1465 | Deduction joining 3 equiva... |
| 3anbi12d 1466 | Deduction conjoining and a... |
| 3anbi13d 1467 | Deduction conjoining and a... |
| 3anbi23d 1468 | Deduction conjoining and a... |
| 3anbi1d 1469 | Deduction adding conjuncts... |
| 3anbi2d 1470 | Deduction adding conjuncts... |
| 3anbi3d 1471 | Deduction adding conjuncts... |
| 3anim123d 1472 | Deduction joining 3 implic... |
| 3orim123d 1473 | Deduction joining 3 implic... |
| 3orim123da 1474 | Disjoin antecedents and co... |
| an6 1475 | Rearrangement of 6 conjunc... |
| 3an6 1476 | Analogue of ~ an4 for trip... |
| 3or6 1477 | Analogue of ~ or4 for trip... |
| mp3an1 1478 | An inference based on modu... |
| mp3an2 1479 | An inference based on modu... |
| mp3an3 1480 | An inference based on modu... |
| mp3an12 1481 | An inference based on modu... |
| mp3an13 1482 | An inference based on modu... |
| mp3an23 1483 | An inference based on modu... |
| mp3an1i 1484 | An inference based on modu... |
| mp3anl1 1485 | An inference based on modu... |
| mp3anl2 1486 | An inference based on modu... |
| mp3anl3 1487 | An inference based on modu... |
| mp3anr1 1488 | An inference based on modu... |
| mp3anr2 1489 | An inference based on modu... |
| mp3anr3 1490 | An inference based on modu... |
| mp3an 1491 | An inference based on modu... |
| mpd3an3 1492 | An inference based on modu... |
| mpd3an23 1493 | An inference based on modu... |
| mp3and 1494 | A deduction based on modus... |
| mp3an12i 1495 | ~ mp3an with antecedents i... |
| mp3an2i 1496 | ~ mp3an with antecedents i... |
| mp3an3an 1497 | ~ mp3an with antecedents i... |
| mp3an2ani 1498 | An elimination deduction. ... |
| biimp3a 1499 | Infer implication from a l... |
| biimp3ar 1500 | Infer implication from a l... |
| 3anandis 1501 | Inference that undistribut... |
| 3anandirs 1502 | Inference that undistribut... |
| ecase13d 1503 | Deduction for elimination ... |
| ecase23d 1504 | Deduction for elimination ... |
| ecase33d 1505 | Deduction for elimination ... |
| 3ecase 1506 | Inference for elimination ... |
| 3bior1fd 1507 | A disjunction is equivalen... |
| 3bior1fand 1508 | A disjunction is equivalen... |
| 3bior2fd 1509 | A wff is equivalent to its... |
| 3biant1d 1510 | A conjunction is equivalen... |
| intn3an1d 1511 | Introduction of a triple c... |
| intn3an2d 1512 | Introduction of a triple c... |
| intn3an3d 1513 | Introduction of a triple c... |
| an3andi 1514 | Distribution of conjunctio... |
| an33rean 1515 | Rearrange a 9-fold conjunc... |
| 3orel2 1516 | Partial elimination of a t... |
| 3orel3 1517 | Partial elimination of a t... |
| 3orel13 1518 | Elimination of two disjunc... |
| 3pm3.2ni 1519 | Triple negated disjunction... |
| an42ds 1520 | Inference exchanging the l... |
| nanan 1523 | Conjunction in terms of al... |
| dfnan2 1524 | Alternative denial in term... |
| nanor 1525 | Alternative denial in term... |
| nancom 1526 | Alternative denial is comm... |
| nannan 1527 | Nested alternative denials... |
| nanim 1528 | Implication in terms of al... |
| nannot 1529 | Negation in terms of alter... |
| nanbi 1530 | Biconditional in terms of ... |
| nanbi1 1531 | Introduce a right anti-con... |
| nanbi2 1532 | Introduce a left anti-conj... |
| nanbi12 1533 | Join two logical equivalen... |
| nanbi1i 1534 | Introduce a right anti-con... |
| nanbi2i 1535 | Introduce a left anti-conj... |
| nanbi12i 1536 | Join two logical equivalen... |
| nanbi1d 1537 | Introduce a right anti-con... |
| nanbi2d 1538 | Introduce a left anti-conj... |
| nanbi12d 1539 | Join two logical equivalen... |
| nanass 1540 | A characterization of when... |
| xnor 1543 | Two ways to write XNOR (ex... |
| xorcom 1544 | The connector ` \/_ ` is c... |
| xorass 1545 | The connector ` \/_ ` is a... |
| excxor 1546 | This tautology shows that ... |
| xor2 1547 | Two ways to express "exclu... |
| xoror 1548 | Exclusive disjunction impl... |
| xornan 1549 | Exclusive disjunction impl... |
| xornan2 1550 | XOR implies NAND (written ... |
| xorneg2 1551 | The connector ` \/_ ` is n... |
| xorneg1 1552 | The connector ` \/_ ` is n... |
| xorneg 1553 | The connector ` \/_ ` is u... |
| xorbi12i 1554 | Equality property for excl... |
| xorbi12d 1555 | Equality property for excl... |
| anxordi 1556 | Conjunction distributes ov... |
| xorexmid 1557 | Exclusive-or variant of th... |
| norcom 1560 | The connector ` -\/ ` is c... |
| nornot 1561 | ` -. ` is expressible via ... |
| noran 1562 | ` /\ ` is expressible via ... |
| noror 1563 | ` \/ ` is expressible via ... |
| norasslem1 1564 | This lemma shows the equiv... |
| norasslem2 1565 | This lemma specializes ~ b... |
| norasslem3 1566 | This lemma specializes ~ b... |
| norass 1567 | A characterization of when... |
| trujust 1572 | Soundness justification th... |
| tru 1574 | The truth value ` T. ` is ... |
| dftru2 1575 | An alternate definition of... |
| trut 1576 | A proposition is equivalen... |
| mptru 1577 | Eliminate ` T. ` as an ant... |
| tbtru 1578 | A proposition is equivalen... |
| bitru 1579 | A theorem is equivalent to... |
| trud 1580 | Anything implies ` T. ` . ... |
| truan 1581 | True can be removed from a... |
| fal 1584 | The truth value ` F. ` is ... |
| nbfal 1585 | The negation of a proposit... |
| bifal 1586 | A contradiction is equival... |
| falim 1587 | The truth value ` F. ` imp... |
| falimd 1588 | The truth value ` F. ` imp... |
| dfnot 1589 | Given falsum ` F. ` , we c... |
| inegd 1590 | Negation introduction rule... |
| efald 1591 | Deduction based on reducti... |
| pm2.21fal 1592 | If a wff and its negation ... |
| truimtru 1593 | A ` -> ` identity. (Contr... |
| truimfal 1594 | A ` -> ` identity. (Contr... |
| falimtru 1595 | A ` -> ` identity. (Contr... |
| falimfal 1596 | A ` -> ` identity. (Contr... |
| nottru 1597 | A ` -. ` identity. (Contr... |
| notfal 1598 | A ` -. ` identity. (Contr... |
| trubitru 1599 | A ` <-> ` identity. (Cont... |
| falbitru 1600 | A ` <-> ` identity. (Cont... |
| trubifal 1601 | A ` <-> ` identity. (Cont... |
| falbifal 1602 | A ` <-> ` identity. (Cont... |
| truantru 1603 | A ` /\ ` identity. (Contr... |
| truanfal 1604 | A ` /\ ` identity. (Contr... |
| falantru 1605 | A ` /\ ` identity. (Contr... |
| falanfal 1606 | A ` /\ ` identity. (Contr... |
| truortru 1607 | A ` \/ ` identity. (Contr... |
| truorfal 1608 | A ` \/ ` identity. (Contr... |
| falortru 1609 | A ` \/ ` identity. (Contr... |
| falorfal 1610 | A ` \/ ` identity. (Contr... |
| trunantru 1611 | A ` -/\ ` identity. (Cont... |
| trunanfal 1612 | A ` -/\ ` identity. (Cont... |
| falnantru 1613 | A ` -/\ ` identity. (Cont... |
| falnanfal 1614 | A ` -/\ ` identity. (Cont... |
| truxortru 1615 | A ` \/_ ` identity. (Cont... |
| truxorfal 1616 | A ` \/_ ` identity. (Cont... |
| falxortru 1617 | A ` \/_ ` identity. (Cont... |
| falxorfal 1618 | A ` \/_ ` identity. (Cont... |
| trunortru 1619 | A ` -\/ ` identity. (Cont... |
| trunorfal 1620 | A ` -\/ ` identity. (Cont... |
| falnortru 1621 | A ` -\/ ` identity. (Cont... |
| falnorfal 1622 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1625 | Equality theorem for the a... |
| hadbi123i 1626 | Equality theorem for the a... |
| hadass 1627 | Associative law for the ad... |
| hadbi 1628 | The adder sum is the same ... |
| hadcoma 1629 | Commutative law for the ad... |
| hadcomb 1630 | Commutative law for the ad... |
| hadrot 1631 | Rotation law for the adder... |
| hadnot 1632 | The adder sum distributes ... |
| had1 1633 | If the first input is true... |
| had0 1634 | If the first input is fals... |
| had1OLD 1635 | Obsolete version of ~ had1... |
| had0OLD 1636 | Obsolete version of ~ had0... |
| hadifp 1637 | The value of the adder sum... |
| hadifpOLD 1638 | Obsolete version of ~ hadi... |
| cador 1641 | The adder carry in disjunc... |
| cadan 1642 | The adder carry in conjunc... |
| cadbi123d 1643 | Equality theorem for the a... |
| cadbi123i 1644 | Equality theorem for the a... |
| cadcoma 1645 | Commutative law for the ad... |
| cadcomb 1646 | Commutative law for the ad... |
| cadrot 1647 | Rotation law for the adder... |
| cadnot 1648 | The adder carry distribute... |
| cad11 1649 | If (at least) two inputs a... |
| cad1 1650 | If one input is true, then... |
| cad0 1651 | If one input is false, the... |
| cadifp 1652 | The value of the carry is,... |
| cadtru 1653 | The adder carry is true as... |
| minimp 1654 | A single axiom for minimal... |
| minimp-syllsimp 1655 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1656 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1657 | Derivation of a commuted f... |
| minimp-ax2 1658 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1659 | Derivation of ~ pm2.43 (al... |
| impsingle 1660 | The shortest single axiom ... |
| impsingle-step4 1661 | Derivation of impsingle-st... |
| impsingle-step8 1662 | Derivation of impsingle-st... |
| impsingle-ax1 1663 | Derivation of impsingle-ax... |
| impsingle-step15 1664 | Derivation of impsingle-st... |
| impsingle-step18 1665 | Derivation of impsingle-st... |
| impsingle-step19 1666 | Derivation of impsingle-st... |
| impsingle-step20 1667 | Derivation of impsingle-st... |
| impsingle-step21 1668 | Derivation of impsingle-st... |
| impsingle-step22 1669 | Derivation of impsingle-st... |
| impsingle-step25 1670 | Derivation of impsingle-st... |
| impsingle-imim1 1671 | Derivation of impsingle-im... |
| impsingle-peirce 1672 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1673 | Derivation of ~ ax-2 from ... |
| meredith 1674 | Carew Meredith's sole axio... |
| merlem1 1675 | Step 3 of Meredith's proof... |
| merlem2 1676 | Step 4 of Meredith's proof... |
| merlem3 1677 | Step 7 of Meredith's proof... |
| merlem4 1678 | Step 8 of Meredith's proof... |
| merlem5 1679 | Step 11 of Meredith's proo... |
| merlem6 1680 | Step 12 of Meredith's proo... |
| merlem7 1681 | Between steps 14 and 15 of... |
| merlem8 1682 | Step 15 of Meredith's proo... |
| merlem9 1683 | Step 18 of Meredith's proo... |
| merlem10 1684 | Step 19 of Meredith's proo... |
| merlem11 1685 | Step 20 of Meredith's proo... |
| merlem12 1686 | Step 28 of Meredith's proo... |
| merlem13 1687 | Step 35 of Meredith's proo... |
| luk-1 1688 | 1 of 3 axioms for proposit... |
| luk-2 1689 | 2 of 3 axioms for proposit... |
| luk-3 1690 | 3 of 3 axioms for proposit... |
| luklem1 1691 | Used to rederive standard ... |
| luklem2 1692 | Used to rederive standard ... |
| luklem3 1693 | Used to rederive standard ... |
| luklem4 1694 | Used to rederive standard ... |
| luklem5 1695 | Used to rederive standard ... |
| luklem6 1696 | Used to rederive standard ... |
| luklem7 1697 | Used to rederive standard ... |
| luklem8 1698 | Used to rederive standard ... |
| ax1 1699 | Standard propositional axi... |
| ax2 1700 | Standard propositional axi... |
| ax3 1701 | Standard propositional axi... |
| nic-dfim 1702 | This theorem "defines" imp... |
| nic-dfneg 1703 | This theorem "defines" neg... |
| nic-mp 1704 | Derive Nicod's rule of mod... |
| nic-mpALT 1705 | A direct proof of ~ nic-mp... |
| nic-ax 1706 | Nicod's axiom derived from... |
| nic-axALT 1707 | A direct proof of ~ nic-ax... |
| nic-imp 1708 | Inference for ~ nic-mp usi... |
| nic-idlem1 1709 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1710 | Lemma for ~ nic-id . Infe... |
| nic-id 1711 | Theorem ~ id expressed wit... |
| nic-swap 1712 | The connector ` -/\ ` is s... |
| nic-isw1 1713 | Inference version of ~ nic... |
| nic-isw2 1714 | Inference for swapping nes... |
| nic-iimp1 1715 | Inference version of ~ nic... |
| nic-iimp2 1716 | Inference version of ~ nic... |
| nic-idel 1717 | Inference to remove the tr... |
| nic-ich 1718 | Chained inference. (Contr... |
| nic-idbl 1719 | Double the terms. Since d... |
| nic-bijust 1720 | Biconditional justificatio... |
| nic-bi1 1721 | Inference to extract one s... |
| nic-bi2 1722 | Inference to extract the o... |
| nic-stdmp 1723 | Derive the standard modus ... |
| nic-luk1 1724 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1725 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1726 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1727 | This alternative axiom for... |
| lukshefth1 1728 | Lemma for ~ renicax . (Co... |
| lukshefth2 1729 | Lemma for ~ renicax . (Co... |
| renicax 1730 | A rederivation of ~ nic-ax... |
| tbw-bijust 1731 | Justification for ~ tbw-ne... |
| tbw-negdf 1732 | The definition of negation... |
| tbw-ax1 1733 | The first of four axioms i... |
| tbw-ax2 1734 | The second of four axioms ... |
| tbw-ax3 1735 | The third of four axioms i... |
| tbw-ax4 1736 | The fourth of four axioms ... |
| tbwsyl 1737 | Used to rederive the Lukas... |
| tbwlem1 1738 | Used to rederive the Lukas... |
| tbwlem2 1739 | Used to rederive the Lukas... |
| tbwlem3 1740 | Used to rederive the Lukas... |
| tbwlem4 1741 | Used to rederive the Lukas... |
| tbwlem5 1742 | Used to rederive the Lukas... |
| re1luk1 1743 | ~ luk-1 derived from the T... |
| re1luk2 1744 | ~ luk-2 derived from the T... |
| re1luk3 1745 | ~ luk-3 derived from the T... |
| merco1 1746 | A single axiom for proposi... |
| merco1lem1 1747 | Used to rederive the Tarsk... |
| retbwax4 1748 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1749 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1750 | Used to rederive the Tarsk... |
| merco1lem3 1751 | Used to rederive the Tarsk... |
| merco1lem4 1752 | Used to rederive the Tarsk... |
| merco1lem5 1753 | Used to rederive the Tarsk... |
| merco1lem6 1754 | Used to rederive the Tarsk... |
| merco1lem7 1755 | Used to rederive the Tarsk... |
| retbwax3 1756 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1757 | Used to rederive the Tarsk... |
| merco1lem9 1758 | Used to rederive the Tarsk... |
| merco1lem10 1759 | Used to rederive the Tarsk... |
| merco1lem11 1760 | Used to rederive the Tarsk... |
| merco1lem12 1761 | Used to rederive the Tarsk... |
| merco1lem13 1762 | Used to rederive the Tarsk... |
| merco1lem14 1763 | Used to rederive the Tarsk... |
| merco1lem15 1764 | Used to rederive the Tarsk... |
| merco1lem16 1765 | Used to rederive the Tarsk... |
| merco1lem17 1766 | Used to rederive the Tarsk... |
| merco1lem18 1767 | Used to rederive the Tarsk... |
| retbwax1 1768 | ~ tbw-ax1 rederived from ~... |
| merco2 1769 | A single axiom for proposi... |
| mercolem1 1770 | Used to rederive the Tarsk... |
| mercolem2 1771 | Used to rederive the Tarsk... |
| mercolem3 1772 | Used to rederive the Tarsk... |
| mercolem4 1773 | Used to rederive the Tarsk... |
| mercolem5 1774 | Used to rederive the Tarsk... |
| mercolem6 1775 | Used to rederive the Tarsk... |
| mercolem7 1776 | Used to rederive the Tarsk... |
| mercolem8 1777 | Used to rederive the Tarsk... |
| re1tbw1 1778 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1779 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1780 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1781 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1782 | Justification for ~ rb-imd... |
| rb-imdf 1783 | The definition of implicat... |
| anmp 1784 | Modus ponens for ` { \/ , ... |
| rb-ax1 1785 | The first of four axioms i... |
| rb-ax2 1786 | The second of four axioms ... |
| rb-ax3 1787 | The third of four axioms i... |
| rb-ax4 1788 | The fourth of four axioms ... |
| rbsyl 1789 | Used to rederive the Lukas... |
| rblem1 1790 | Used to rederive the Lukas... |
| rblem2 1791 | Used to rederive the Lukas... |
| rblem3 1792 | Used to rederive the Lukas... |
| rblem4 1793 | Used to rederive the Lukas... |
| rblem5 1794 | Used to rederive the Lukas... |
| rblem6 1795 | Used to rederive the Lukas... |
| rblem7 1796 | Used to rederive the Lukas... |
| re1axmp 1797 | ~ ax-mp derived from Russe... |
| re2luk1 1798 | ~ luk-1 derived from Russe... |
| re2luk2 1799 | ~ luk-2 derived from Russe... |
| re2luk3 1800 | ~ luk-3 derived from Russe... |
| mptnan 1801 | Modus ponendo tollens 1, o... |
| mptxor 1802 | Modus ponendo tollens 2, o... |
| mtpor 1803 | Modus tollendo ponens (inc... |
| mtpxor 1804 | Modus tollendo ponens (ori... |
| stoic1a 1805 | Stoic logic Thema 1 (part ... |
| stoic1b 1806 | Stoic logic Thema 1 (part ... |
| stoic2a 1807 | Stoic logic Thema 2 versio... |
| stoic2b 1808 | Stoic logic Thema 2 versio... |
| stoic3 1809 | Stoic logic Thema 3. Stat... |
| stoic4a 1810 | Stoic logic Thema 4 versio... |
| stoic4b 1811 | Stoic logic Thema 4 versio... |
| alnex 1814 | Universal quantification o... |
| eximal 1815 | An equivalence between an ... |
| nf2 1818 | Alternate definition of no... |
| nf3 1819 | Alternate definition of no... |
| nf4 1820 | Alternate definition of no... |
| nfi 1821 | Deduce that ` x ` is not f... |
| nfri 1822 | Consequence of the definit... |
| nfd 1823 | Deduce that ` x ` is not f... |
| nfrd 1824 | Consequence of the definit... |
| nftht 1825 | Closed form of ~ nfth . (... |
| nfntht 1826 | Closed form of ~ nfnth . ... |
| nfntht2 1827 | Closed form of ~ nfnth . ... |
| gen2 1829 | Generalization applied twi... |
| mpg 1830 | Modus ponens combined with... |
| mpgbi 1831 | Modus ponens on biconditio... |
| mpgbir 1832 | Modus ponens on biconditio... |
| nex 1833 | Generalization rule for ne... |
| nfth 1834 | No variable is (effectivel... |
| nfnth 1835 | No variable is (effectivel... |
| hbth 1836 | No variable is (effectivel... |
| nftru 1837 | The true constant has no f... |
| nffal 1838 | The false constant has no ... |
| sptruw 1839 | Version of ~ sp when ` ph ... |
| altru 1840 | For all sets, ` T. ` is tr... |
| alfal 1841 | For all sets, ` -. F. ` is... |
| alim 1843 | Restatement of Axiom ~ ax-... |
| alimi 1844 | Inference quantifying both... |
| 2alimi 1845 | Inference doubly quantifyi... |
| ala1 1846 | Add an antecedent in a uni... |
| al2im 1847 | Closed form of ~ al2imi . ... |
| al2imi 1848 | Inference quantifying ante... |
| alanimi 1849 | Variant of ~ al2imi with c... |
| alimdh 1850 | Deduction form of Theorem ... |
| albi 1851 | Theorem 19.15 of [Margaris... |
| albii 1852 | Inference adding universal... |
| 2albii 1853 | Inference adding two unive... |
| 3albii 1854 | Inference adding three uni... |
| sylgt 1855 | Closed form of ~ sylg . (... |
| sylg 1856 | A syllogism combined with ... |
| alrimih 1857 | Inference form of Theorem ... |
| hbxfrbi 1858 | A utility lemma to transfe... |
| alex 1859 | Universal quantifier in te... |
| exnal 1860 | Existential quantification... |
| 2nalexn 1861 | Part of theorem *11.5 in [... |
| 2exnaln 1862 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1863 | Theorem *11.25 in [Whitehe... |
| alimex 1864 | An equivalence between an ... |
| aleximi 1865 | A variant of ~ al2imi : in... |
| alexbii 1866 | Biconditional form of ~ al... |
| exim 1867 | Theorem 19.22 of [Margaris... |
| eximi 1868 | Inference adding existenti... |
| 2eximi 1869 | Inference adding two exist... |
| eximii 1870 | Inference associated with ... |
| exa1 1871 | Add an antecedent in an ex... |
| 19.38 1872 | Theorem 19.38 of [Margaris... |
| 19.38a 1873 | Under a nonfreeness hypoth... |
| 19.38b 1874 | Under a nonfreeness hypoth... |
| imnang 1875 | Quantified implication in ... |
| alinexa 1876 | A transformation of quanti... |
| exnalimn 1877 | Existential quantification... |
| alexn 1878 | A relationship between two... |
| 2exnexn 1879 | Theorem *11.51 in [Whitehe... |
| exbi 1880 | Theorem 19.18 of [Margaris... |
| exbii 1881 | Inference adding existenti... |
| 2exbii 1882 | Inference adding two exist... |
| 3exbii 1883 | Inference adding three exi... |
| nfbiit 1884 | Equivalence theorem for th... |
| nfbii 1885 | Equality theorem for the n... |
| nfxfr 1886 | A utility lemma to transfe... |
| nfxfrd 1887 | A utility lemma to transfe... |
| nfnbi 1888 | A variable is nonfree in a... |
| nfnt 1889 | If a variable is nonfree i... |
| nfn 1890 | Inference associated with ... |
| nfnd 1891 | Deduction associated with ... |
| exanali 1892 | A transformation of quanti... |
| 2exanali 1893 | Theorem *11.521 in [Whiteh... |
| exancom 1894 | Commutation of conjunction... |
| exan 1895 | Place a conjunct in the sc... |
| alrimdh 1896 | Deduction form of Theorem ... |
| eximdh 1897 | Deduction from Theorem 19.... |
| nexdh 1898 | Deduction for generalizati... |
| albidh 1899 | Formula-building rule for ... |
| exbidh 1900 | Formula-building rule for ... |
| exsimpl 1901 | Simplification of an exist... |
| exsimpr 1902 | Simplification of an exist... |
| 19.26 1903 | Theorem 19.26 of [Margaris... |
| 19.26-2 1904 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1905 | Theorem ~ 19.26 with tripl... |
| hban 1906 | If ` x ` is not free in ` ... |
| hb3an 1907 | If ` x ` is not free in ` ... |
| 19.29 1908 | Theorem 19.29 of [Margaris... |
| 19.29r 1909 | Variation of ~ 19.29 . (C... |
| 19.29r2 1910 | Variation of ~ 19.29r with... |
| 19.29x 1911 | Variation of ~ 19.29 with ... |
| 19.35 1912 | Theorem 19.35 of [Margaris... |
| 19.35i 1913 | Inference associated with ... |
| 19.35ri 1914 | Inference associated with ... |
| 19.25 1915 | Theorem 19.25 of [Margaris... |
| 19.30 1916 | Theorem 19.30 of [Margaris... |
| 19.43 1917 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1918 | Obsolete proof of ~ 19.43 ... |
| 19.33 1919 | Theorem 19.33 of [Margaris... |
| 19.33b 1920 | The antecedent provides a ... |
| 19.40 1921 | Theorem 19.40 of [Margaris... |
| 19.40-2 1922 | Theorem *11.42 in [Whitehe... |
| 19.40b 1923 | The antecedent provides a ... |
| albiim 1924 | Split a biconditional and ... |
| 2albiim 1925 | Split a biconditional and ... |
| exintrbi 1926 | Add/remove a conjunct in t... |
| exintr 1927 | Introduce a conjunct in th... |
| alsyl 1928 | Universally quantified and... |
| nfimd 1929 | If in a context ` x ` is n... |
| nfimt 1930 | Closed form of ~ nfim and ... |
| nfim 1931 | If ` x ` is not free in ` ... |
| nfand 1932 | If in a context ` x ` is n... |
| nf3and 1933 | Deduction form of bound-va... |
| nfan 1934 | If ` x ` is not free in ` ... |
| nfnan 1935 | If ` x ` is not free in ` ... |
| nf3an 1936 | If ` x ` is not free in ` ... |
| nfbid 1937 | If in a context ` x ` is n... |
| nfbi 1938 | If ` x ` is not free in ` ... |
| nfor 1939 | If ` x ` is not free in ` ... |
| nf3or 1940 | If ` x ` is not free in ` ... |
| empty 1941 | Two characterizations of t... |
| emptyex 1942 | On the empty domain, any e... |
| emptyal 1943 | On the empty domain, any u... |
| emptynf 1944 | On the empty domain, any v... |
| ax5d 1946 | Version of ~ ax-5 with ant... |
| ax5e 1947 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1948 | If a formula holds for som... |
| nfv 1949 | If ` x ` is not present in... |
| nfvd 1950 | ~ nfv with antecedent. Us... |
| alimdv 1951 | Deduction form of Theorem ... |
| eximdv 1952 | Deduction form of Theorem ... |
| 2alimdv 1953 | Deduction form of Theorem ... |
| 2eximdv 1954 | Deduction form of Theorem ... |
| albidv 1955 | Formula-building rule for ... |
| exbidv 1956 | Formula-building rule for ... |
| nfbidv 1957 | An equality theorem for no... |
| 2albidv 1958 | Formula-building rule for ... |
| 2exbidv 1959 | Formula-building rule for ... |
| 3exbidv 1960 | Formula-building rule for ... |
| 4exbidv 1961 | Formula-building rule for ... |
| alrimiv 1962 | Inference form of Theorem ... |
| alrimivv 1963 | Inference form of Theorem ... |
| alrimdv 1964 | Deduction form of Theorem ... |
| exlimiv 1965 | Inference form of Theorem ... |
| exlimiiv 1966 | Inference (Rule C) associa... |
| exlimivv 1967 | Inference form of Theorem ... |
| exlimdv 1968 | Deduction form of Theorem ... |
| exlimdvv 1969 | Deduction form of Theorem ... |
| exlimddv 1970 | Existential elimination ru... |
| nexdv 1971 | Deduction for generalizati... |
| 2ax5 1972 | Quantification of two vari... |
| stdpc5v 1973 | Version of ~ stdpc5 with a... |
| 19.21v 1974 | Version of ~ 19.21 with a ... |
| 19.32v 1975 | Version of ~ 19.32 with a ... |
| 19.31v 1976 | Version of ~ 19.31 with a ... |
| 19.23v 1977 | Version of ~ 19.23 with a ... |
| 19.23vv 1978 | Theorem ~ 19.23v extended ... |
| pm11.53v 1979 | Version of ~ pm11.53 with ... |
| 19.36imv 1980 | One direction of ~ 19.36v ... |
| 19.36iv 1981 | Inference associated with ... |
| 19.37imv 1982 | One direction of ~ 19.37v ... |
| 19.37iv 1983 | Inference associated with ... |
| 19.41v 1984 | Version of ~ 19.41 with a ... |
| 19.41vv 1985 | Version of ~ 19.41 with tw... |
| 19.41vvv 1986 | Version of ~ 19.41 with th... |
| 19.41vvvv 1987 | Version of ~ 19.41 with fo... |
| 19.42v 1988 | Version of ~ 19.42 with a ... |
| exdistr 1989 | Distribution of existentia... |
| exdistrv 1990 | Distribute a pair of exist... |
| 4exdistrv 1991 | Distribute two pairs of ex... |
| 19.42vv 1992 | Version of ~ 19.42 with tw... |
| exdistr2 1993 | Distribution of existentia... |
| 19.42vvv 1994 | Version of ~ 19.42 with th... |
| 3exdistr 1995 | Distribution of existentia... |
| 4exdistr 1996 | Distribution of existentia... |
| weq 1997 | Extend wff definition to i... |
| speimfw 1998 | Specialization, with addit... |
| speimfwALT 1999 | Alternate proof of ~ speim... |
| spimfw 2000 | Specialization, with addit... |
| ax12i 2001 | Inference that has ~ ax-12... |
| ax6v 2003 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 2004 | At least one individual ex... |
| spimw 2005 | Specialization. Lemma 8 o... |
| spimew 2006 | Existential introduction, ... |
| speiv 2007 | Inference from existential... |
| speivw 2008 | Version of ~ spei with a d... |
| exgen 2009 | Rule of existential genera... |
| extru 2010 | There exists a variable su... |
| 19.2 2011 | Theorem 19.2 of [Margaris]... |
| 19.2d 2012 | Deduction associated with ... |
| 19.8w 2013 | Weak version of ~ 19.8a an... |
| spnfw 2014 | Weak version of ~ sp . Us... |
| spfalw 2015 | Version of ~ sp when ` ph ... |
| spvw 2016 | Version of ~ sp when ` x `... |
| 19.3v 2017 | Version of ~ 19.3 with a d... |
| 19.8v 2018 | Version of ~ 19.8a with a ... |
| 19.9v 2019 | Version of ~ 19.9 with a d... |
| spimevw 2020 | Existential introduction, ... |
| spimvw 2021 | A weak form of specializat... |
| spsv 2022 | Generalization of antecede... |
| spvv 2023 | Specialization, using impl... |
| chvarvv 2024 | Implicit substitution of `... |
| 19.39 2025 | Theorem 19.39 of [Margaris... |
| 19.24 2026 | Theorem 19.24 of [Margaris... |
| 19.34 2027 | Theorem 19.34 of [Margaris... |
| 19.36v 2028 | Version of ~ 19.36 with a ... |
| 19.12vvv 2029 | Version of ~ 19.12vv with ... |
| 19.27v 2030 | Version of ~ 19.27 with a ... |
| 19.28v 2031 | Version of ~ 19.28 with a ... |
| 19.37v 2032 | Version of ~ 19.37 with a ... |
| 19.44v 2033 | Version of ~ 19.44 with a ... |
| 19.45v 2034 | Version of ~ 19.45 with a ... |
| equs4v 2035 | Version of ~ equs4 with a ... |
| alequexv 2036 | Version of ~ equs4v with i... |
| exsbim 2037 | One direction of the equiv... |
| equsv 2038 | If a formula does not cont... |
| equsalvw 2039 | Version of ~ equsalv with ... |
| equsexvw 2040 | Version of ~ equsexv with ... |
| cbvaliw 2041 | Change bound variable. Us... |
| cbvalivw 2042 | Change bound variable. Us... |
| ax7v 2044 | Weakened version of ~ ax-7... |
| ax7v1 2045 | First of two weakened vers... |
| ax7v2 2046 | Second of two weakened ver... |
| equid 2047 | Identity law for equality.... |
| nfequid 2048 | Bound-variable hypothesis ... |
| equcomiv 2049 | Weaker form of ~ equcomi w... |
| ax6evr 2050 | A commuted form of ~ ax6ev... |
| ax7 2051 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2052 | Commutative law for equali... |
| equcom 2053 | Commutative law for equali... |
| equcomd 2054 | Deduction form of ~ equcom... |
| equcoms 2055 | An inference commuting equ... |
| equtr 2056 | A transitive law for equal... |
| equtrr 2057 | A transitive law for equal... |
| equeuclr 2058 | Commuted version of ~ eque... |
| equeucl 2059 | Equality is a left-Euclide... |
| equequ1 2060 | An equivalence law for equ... |
| equequ2 2061 | An equivalence law for equ... |
| equtr2 2062 | Equality is a left-Euclide... |
| stdpc6 2063 | One of the two equality ax... |
| equvinv 2064 | A variable introduction la... |
| equvinva 2065 | A modified version of the ... |
| equvelv 2066 | A biconditional form of ~ ... |
| ax13b 2067 | An equivalence between two... |
| spfw 2068 | Weak version of ~ sp . Us... |
| spw 2069 | Weak version of the specia... |
| cbvalw 2070 | Change bound variable. Us... |
| cbvalvw 2071 | Change bound variable. Us... |
| cbvexvw 2072 | Change bound variable. Us... |
| cbvaldvaw 2073 | Rule used to change the bo... |
| cbvexdvaw 2074 | Rule used to change the bo... |
| cbval2vw 2075 | Rule used to change bound ... |
| cbvex2vw 2076 | Rule used to change bound ... |
| cbvex4vw 2077 | Rule used to change bound ... |
| alcomimw 2078 | Weak version of ~ ax-11 . ... |
| excomimw 2079 | Weak version of ~ excomim ... |
| alcomw 2080 | Weak version of ~ alcom an... |
| excomw 2081 | Weak version of ~ excom an... |
| hbn1fw 2082 | Weak version of ~ ax-10 fr... |
| hbn1w 2083 | Weak version of ~ hbn1 . ... |
| hba1w 2084 | Weak version of ~ hba1 . ... |
| hbe1w 2085 | Weak version of ~ hbe1 . ... |
| hbalw 2086 | Weak version of ~ hbal . ... |
| 19.8aw 2087 | If a formula is true, then... |
| exexw 2088 | Existential quantification... |
| spaev 2089 | A special instance of ~ sp... |
| cbvaev 2090 | Change bound variable in a... |
| aevlem0 2091 | Lemma for ~ aevlem . Inst... |
| aevlem 2092 | Lemma for ~ aev and ~ axc1... |
| aeveq 2093 | The antecedent ` A. x x = ... |
| aev 2094 | A "distinctor elimination"... |
| aev2 2095 | A version of ~ aev with tw... |
| hbaev 2096 | All variables are effectiv... |
| naev 2097 | If some set variables can ... |
| naev2 2098 | Generalization of ~ hbnaev... |
| hbnaev 2099 | Any variable is free in ` ... |
| sbjust 2100 | Justification theorem for ... |
| dfsb 2103 | Simplify definition ~ df-s... |
| sbtlem 2104 | In the case of ~ sbt , the... |
| sbt 2105 | A substitution into a theo... |
| sbtru 2106 | The result of substituting... |
| stdpc4 2107 | The specialization axiom o... |
| sbtALT 2108 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2109 | A double specialization us... |
| sbi1 2110 | Distribute substitution ov... |
| spsbim 2111 | Distribute substitution ov... |
| spsbbi 2112 | Biconditional property for... |
| sbimi 2113 | Distribute substitution ov... |
| sb2imi 2114 | Distribute substitution ov... |
| sbbii 2115 | Infer substitution into bo... |
| 2sbbii 2116 | Infer double substitution ... |
| sbimdv 2117 | Deduction substituting bot... |
| sbbidv 2118 | Deduction substituting bot... |
| sban 2119 | Conjunction inside and out... |
| sb3an 2120 | Threefold conjunction insi... |
| spsbe 2121 | Existential generalization... |
| sbequ 2122 | Equality property for subs... |
| sbequiOLD 2123 | Obsolete version of ~ sbeq... |
| sb6 2124 | Alternate definition of su... |
| 2sb6 2125 | Equivalence for double sub... |
| sb1v 2126 | One direction of ~ sb5 , p... |
| sbv 2127 | Substitution for a variabl... |
| sbcom4 2128 | Commutativity law for subs... |
| pm11.07 2129 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2130 | Substitution in an implica... |
| sbrimvwOLD 2131 | Obsolete version of ~ sbri... |
| sbbiiev 2132 | An equivalence of substitu... |
| sbievw 2133 | Conversion of implicit sub... |
| sbiedvw 2134 | Conversion of implicit sub... |
| 2sbievw 2135 | Conversion of double impli... |
| sbcom3vv 2136 | Substituting ` y ` for ` x... |
| sbievw2 2137 | ~ sbievw applied twice, av... |
| sbco2vv 2138 | A composition law for subs... |
| cbvsbv 2139 | Change the bound variable ... |
| sbco4lem 2140 | Lemma for ~ sbco4 . It re... |
| sbco4 2141 | Two ways of exchanging two... |
| equsb3 2142 | Substitution in an equalit... |
| equsb3r 2143 | Substitution applied to th... |
| equsb1v 2144 | Substitution applied to an... |
| nsb 2145 | Any substitution in an alw... |
| sbn1 2146 | One direction of ~ sbn , u... |
| wel 2148 | Extend wff definition to i... |
| ax8v 2150 | Weakened version of ~ ax-8... |
| ax8v1 2151 | First of two weakened vers... |
| ax8v2 2152 | Second of two weakened ver... |
| ax8 2153 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2154 | An identity law for the no... |
| elsb1 2155 | Substitution for the first... |
| cleljust 2156 | When the class variables i... |
| ax9v 2158 | Weakened version of ~ ax-9... |
| ax9v1 2159 | First of two weakened vers... |
| ax9v2 2160 | Second of two weakened ver... |
| ax9 2161 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2162 | An identity law for the no... |
| elequ2g 2163 | A form of ~ elequ2 with a ... |
| elsb2 2164 | Substitution for the secon... |
| elequ12 2165 | An identity law for the no... |
| ru0 2166 | The FOL statement used in ... |
| ax6dgen 2167 | Tarski's system uses the w... |
| ax10w 2168 | Weak version of ~ ax-10 fr... |
| ax11w 2169 | Weak version of ~ ax-11 fr... |
| ax11dgen 2170 | Degenerate instance of ~ a... |
| ax12wlem 2171 | Lemma for weak version of ... |
| ax12w 2172 | Weak version of ~ ax-12 fr... |
| ax12dgen 2173 | Degenerate instance of ~ a... |
| ax12wdemo 2174 | Example of an application ... |
| ax13w 2175 | Weak version (principal in... |
| ax13dgen1 2176 | Degenerate instance of ~ a... |
| ax13dgen2 2177 | Degenerate instance of ~ a... |
| ax13dgen3 2178 | Degenerate instance of ~ a... |
| ax13dgen4 2179 | Degenerate instance of ~ a... |
| hbn1 2181 | Alias for ~ ax-10 to be us... |
| hbe1 2182 | The setvar ` x ` is not fr... |
| hbe1a 2183 | Dual statement of ~ hbe1 .... |
| nf5di 2184 | Since the converse holds b... |
| nf5-1 2185 | One direction of ~ nf5 can... |
| nf5i 2186 | Deduce that ` x ` is not f... |
| nf5dh 2187 | Deduce that ` x ` is not f... |
| nf5dv 2188 | Apply the definition of no... |
| nfnaew 2189 | All variables are effectiv... |
| nfe1 2190 | The setvar ` x ` is not fr... |
| nfa1 2191 | The setvar ` x ` is not fr... |
| nfna1 2192 | A convenience theorem part... |
| nfia1 2193 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2194 | The setvar ` x ` is not fr... |
| modal5 2195 | The analogue in our predic... |
| nfs1v 2196 | The setvar ` x ` is not fr... |
| alcoms 2198 | Swap quantifiers in an ant... |
| alcom 2199 | Theorem 19.5 of [Margaris]... |
| alrot3 2200 | Theorem *11.21 in [Whitehe... |
| alrot4 2201 | Rotate four universal quan... |
| excom 2202 | Theorem 19.11 of [Margaris... |
| excomim 2203 | One direction of Theorem 1... |
| excom13 2204 | Swap 1st and 3rd existenti... |
| exrot3 2205 | Rotate existential quantif... |
| exrot4 2206 | Rotate existential quantif... |
| hbal 2207 | If ` x ` is not free in ` ... |
| hbald 2208 | Deduction form of bound-va... |
| sbal 2209 | Move universal quantifier ... |
| sbalv 2210 | Quantify with new variable... |
| hbsbw 2211 | If ` z ` is not free in ` ... |
| sbcom2 2212 | Commutativity law for subs... |
| nfa2 2213 | Lemma 24 of [Monk2] p. 114... |
| nfexhe 2214 | Version of ~ nfex with the... |
| nfexa2 2215 | An inner universal quantif... |
| ax12v 2217 | This is essentially Axiom ... |
| ax12v2 2218 | It is possible to remove a... |
| ax12ev2 2219 | Version of ~ ax12v2 rewrit... |
| ax12ev2c 2220 | A commuted form of ~ ax12e... |
| 19.8a 2221 | If a wff is true, it is tr... |
| 19.8ad 2222 | If a wff is true, it is tr... |
| sp 2223 | Specialization. A univers... |
| spi 2224 | Inference rule of universa... |
| sps 2225 | Generalization of antecede... |
| 2sp 2226 | A double specialization (s... |
| spsd 2227 | Deduction generalizing ant... |
| 19.2g 2228 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2229 | Inference form of ~ 19.21 ... |
| 19.21bbi 2230 | Inference removing two uni... |
| 19.23bi 2231 | Inference form of Theorem ... |
| nexr 2232 | Inference associated with ... |
| qexmid 2233 | Quantified excluded middle... |
| nf5r 2234 | Consequence of the definit... |
| nf5ri 2235 | Consequence of the definit... |
| nf5rd 2236 | Consequence of the definit... |
| spimedv 2237 | Deduction version of ~ spi... |
| spimefv 2238 | Version of ~ spime with a ... |
| nfim1 2239 | A closed form of ~ nfim . ... |
| nfan1 2240 | A closed form of ~ nfan . ... |
| 19.3t 2241 | Closed form of ~ 19.3 and ... |
| 19.3 2242 | A wff may be quantified wi... |
| 19.9d 2243 | A deduction version of one... |
| 19.9t 2244 | Closed form of ~ 19.9 and ... |
| 19.9 2245 | A wff may be existentially... |
| 19.21t 2246 | Closed form of Theorem 19.... |
| 19.21 2247 | Theorem 19.21 of [Margaris... |
| stdpc5 2248 | An axiom scheme of standar... |
| 19.21-2 2249 | Version of ~ 19.21 with tw... |
| 19.23t 2250 | Closed form of Theorem 19.... |
| 19.23 2251 | Theorem 19.23 of [Margaris... |
| alimd 2252 | Deduction form of Theorem ... |
| alrimi 2253 | Inference form of Theorem ... |
| alrimdd 2254 | Deduction form of Theorem ... |
| alrimd 2255 | Deduction form of Theorem ... |
| eximd 2256 | Deduction form of Theorem ... |
| exlimi 2257 | Inference associated with ... |
| exlimd 2258 | Deduction form of Theorem ... |
| exlimimdd 2259 | Existential elimination ru... |
| exlimdd 2260 | Existential elimination ru... |
| nexd 2261 | Deduction for generalizati... |
| albid 2262 | Formula-building rule for ... |
| exbid 2263 | Formula-building rule for ... |
| nfbidf 2264 | An equality theorem for ef... |
| 19.16 2265 | Theorem 19.16 of [Margaris... |
| 19.17 2266 | Theorem 19.17 of [Margaris... |
| 19.27 2267 | Theorem 19.27 of [Margaris... |
| 19.28 2268 | Theorem 19.28 of [Margaris... |
| 19.19 2269 | Theorem 19.19 of [Margaris... |
| 19.36 2270 | Theorem 19.36 of [Margaris... |
| 19.36i 2271 | Inference associated with ... |
| 19.37 2272 | Theorem 19.37 of [Margaris... |
| 19.32 2273 | Theorem 19.32 of [Margaris... |
| 19.31 2274 | Theorem 19.31 of [Margaris... |
| 19.41 2275 | Theorem 19.41 of [Margaris... |
| 19.42 2276 | Theorem 19.42 of [Margaris... |
| 19.44 2277 | Theorem 19.44 of [Margaris... |
| 19.45 2278 | Theorem 19.45 of [Margaris... |
| spimfv 2279 | Specialization, using impl... |
| chvarfv 2280 | Implicit substitution of `... |
| cbv3v2 2281 | Version of ~ cbv3 with two... |
| sbalex 2282 | Equivalence of two ways to... |
| sb4av 2283 | Version of ~ sb4a with a d... |
| sbimd 2284 | Deduction substituting bot... |
| sbbid 2285 | Deduction substituting bot... |
| 2sbbid 2286 | Deduction doubly substitut... |
| sbequ1 2287 | An equality theorem for su... |
| sbequ2 2288 | An equality theorem for su... |
| stdpc7 2289 | One of the two equality ax... |
| sbequ12 2290 | An equality theorem for su... |
| sbequ12r 2291 | An equality theorem for su... |
| sbelx 2292 | Elimination of substitutio... |
| sbequ12a 2293 | An equality theorem for su... |
| sbid 2294 | An identity theorem for su... |
| sbcov 2295 | A composition law for subs... |
| sb6a 2296 | Equivalence for substituti... |
| sbid2vw 2297 | Reverting substitution yie... |
| axc16g 2298 | Generalization of ~ axc16 ... |
| axc16 2299 | Proof of older axiom ~ ax-... |
| axc16gb 2300 | Biconditional strengthenin... |
| axc16nf 2301 | If ~ dtru is false, then t... |
| axc11v 2302 | Version of ~ axc11 with a ... |
| axc11rv 2303 | Version of ~ axc11r with a... |
| drsb2 2304 | Formula-building lemma for... |
| equsalv 2305 | An equivalence related to ... |
| equsexv 2306 | An equivalence related to ... |
| sbft 2307 | Substitution has no effect... |
| sbf 2308 | Substitution for a variabl... |
| sbf2 2309 | Substitution has no effect... |
| sbh 2310 | Substitution for a variabl... |
| hbs1 2311 | The setvar ` x ` is not fr... |
| nfs1f 2312 | If ` x ` is not free in ` ... |
| sb5 2313 | Alternate definition of su... |
| equs5av 2314 | A property related to subs... |
| 2sb5 2315 | Equivalence for double sub... |
| dfsb7 2316 | An alternate definition of... |
| sbn 2317 | Negation inside and outsid... |
| sbex 2318 | Move existential quantifie... |
| nf5 2319 | Alternate definition of ~ ... |
| nf6 2320 | An alternate definition of... |
| nf5d 2321 | Deduce that ` x ` is not f... |
| nf5diOLD 2322 | Obsolete version of ~ nf5d... |
| 19.9h 2323 | A wff may be existentially... |
| 19.21h 2324 | Theorem 19.21 of [Margaris... |
| 19.23h 2325 | Theorem 19.23 of [Margaris... |
| exlimih 2326 | Inference associated with ... |
| exlimdh 2327 | Deduction form of Theorem ... |
| equsalhw 2328 | Version of ~ equsalh with ... |
| equsexhv 2329 | An equivalence related to ... |
| hba1 2330 | The setvar ` x ` is not fr... |
| hbnt 2331 | Closed theorem version of ... |
| hbn 2332 | If ` x ` is not free in ` ... |
| hbnd 2333 | Deduction form of bound-va... |
| hbim1 2334 | A closed form of ~ hbim . ... |
| hbimd 2335 | Deduction form of bound-va... |
| hbim 2336 | If ` x ` is not free in ` ... |
| hbanOLD 2337 | Obsolete version of ~ hban... |
| hb3anOLD 2338 | Obsolete version of ~ hb3a... |
| sbi2 2339 | Introduction of implicatio... |
| sbim 2340 | Implication inside and out... |
| sbrim 2341 | Substitution in an implica... |
| sblim 2342 | Substitution in an implica... |
| sbor 2343 | Disjunction inside and out... |
| sbbi 2344 | Equivalence inside and out... |
| sblbis 2345 | Introduce left bicondition... |
| sbrbis 2346 | Introduce right biconditio... |
| sbrbif 2347 | Introduce right biconditio... |
| sbnf 2348 | Move nonfree predicate in ... |
| sbiev 2349 | Conversion of implicit sub... |
| sbiedw 2350 | Conversion of implicit sub... |
| axc7 2351 | Show that the original axi... |
| axc7e 2352 | Abbreviated version of ~ a... |
| modal-b 2353 | The analogue in our predic... |
| 19.9ht 2354 | A closed version of ~ 19.9... |
| axc4 2355 | Show that the original axi... |
| axc4i 2356 | Inference version of ~ axc... |
| nfal 2357 | If ` x ` is not free in ` ... |
| nfex 2358 | If ` x ` is not free in ` ... |
| hbex 2359 | If ` x ` is not free in ` ... |
| nfnf 2360 | If ` x ` is not free in ` ... |
| 19.12 2361 | Theorem 19.12 of [Margaris... |
| nfald 2362 | Deduction form of ~ nfal .... |
| nfexd 2363 | If ` x ` is not free in ` ... |
| nfsbv 2364 | If ` z ` is not free in ` ... |
| sbco2v 2365 | A composition law for subs... |
| aaan 2366 | Distribute universal quant... |
| eeor 2367 | Distribute existential qua... |
| cbv3v 2368 | Rule used to change bound ... |
| cbv1v 2369 | Rule used to change bound ... |
| cbv2w 2370 | Rule used to change bound ... |
| cbvaldw 2371 | Deduction used to change b... |
| cbvexdw 2372 | Deduction used to change b... |
| cbv3hv 2373 | Rule used to change bound ... |
| cbvalv1 2374 | Rule used to change bound ... |
| cbvexv1 2375 | Rule used to change bound ... |
| cbval2v 2376 | Rule used to change bound ... |
| cbvex2v 2377 | Rule used to change bound ... |
| dvelimhw 2378 | Proof of ~ dvelimh without... |
| pm11.53 2379 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2380 | Special case of ~ 19.12 wh... |
| eean 2381 | Distribute existential qua... |
| eeanv 2382 | Distribute a pair of exist... |
| eeeanv 2383 | Distribute three existenti... |
| ee4anv 2384 | Distribute two pairs of ex... |
| sb8v 2385 | Substitution of variable i... |
| sb8f 2386 | Substitution of variable i... |
| sb8ef 2387 | Substitution of variable i... |
| 2sb8ef 2388 | An equivalent expression f... |
| sb6rfv 2389 | Reversed substitution. Ve... |
| sbnf2 2390 | Two ways of expressing " `... |
| exsb 2391 | An equivalent expression f... |
| 2exsb 2392 | An equivalent expression f... |
| sbbib 2393 | Reversal of substitution. ... |
| sbbibvv 2394 | Reversal of substitution. ... |
| cbvsbvf 2395 | Change the bound variable ... |
| cleljustALT 2396 | Alternate proof of ~ clelj... |
| cleljustALT2 2397 | Alternate proof of ~ clelj... |
| equs5aALT 2398 | Alternate proof of ~ equs5... |
| equs5eALT 2399 | Alternate proof of ~ equs5... |
| axc11r 2400 | Same as ~ axc11 but with r... |
| dral1v 2401 | Formula-building lemma for... |
| drex1v 2402 | Formula-building lemma for... |
| drnf1v 2403 | Formula-building lemma for... |
| ax13v 2405 | A weaker version of ~ ax-1... |
| ax13lem1 2406 | A version of ~ ax13v with ... |
| ax13 2407 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2408 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2409 | An equation between setvar... |
| dveeq2 2410 | Quantifier introduction wh... |
| nfeqf1 2411 | An equation between setvar... |
| dveeq1 2412 | Quantifier introduction wh... |
| nfeqf 2413 | A variable is effectively ... |
| axc9 2414 | Derive set.mm's original ~... |
| ax6e 2415 | At least one individual ex... |
| ax6 2416 | Theorem showing that ~ ax-... |
| axc10 2417 | Show that the original axi... |
| spimt 2418 | Closed theorem form of ~ s... |
| spim 2419 | Specialization, using impl... |
| spimed 2420 | Deduction version of ~ spi... |
| spime 2421 | Existential introduction, ... |
| spimv 2422 | A version of ~ spim with a... |
| spimvALT 2423 | Alternate proof of ~ spimv... |
| spimev 2424 | Distinct-variable version ... |
| spv 2425 | Specialization, using impl... |
| spei 2426 | Inference from existential... |
| chvar 2427 | Implicit substitution of `... |
| chvarv 2428 | Implicit substitution of `... |
| cbv3 2429 | Rule used to change bound ... |
| cbval 2430 | Rule used to change bound ... |
| cbvex 2431 | Rule used to change bound ... |
| cbvalv 2432 | Rule used to change bound ... |
| cbvexv 2433 | Rule used to change bound ... |
| cbv1 2434 | Rule used to change bound ... |
| cbv2 2435 | Rule used to change bound ... |
| cbv3h 2436 | Rule used to change bound ... |
| cbv1h 2437 | Rule used to change bound ... |
| cbv2h 2438 | Rule used to change bound ... |
| cbvald 2439 | Deduction used to change b... |
| cbvexd 2440 | Deduction used to change b... |
| cbvaldva 2441 | Rule used to change the bo... |
| cbvexdva 2442 | Rule used to change the bo... |
| cbval2 2443 | Rule used to change bound ... |
| cbvex2 2444 | Rule used to change bound ... |
| cbval2vv 2445 | Rule used to change bound ... |
| cbvex2vv 2446 | Rule used to change bound ... |
| cbvex4v 2447 | Rule used to change bound ... |
| equs4 2448 | Lemma used in proofs of im... |
| equsal 2449 | An equivalence related to ... |
| equsex 2450 | An equivalence related to ... |
| equsexALT 2451 | Alternate proof of ~ equse... |
| equsalh 2452 | An equivalence related to ... |
| equsexh 2453 | An equivalence related to ... |
| axc15 2454 | Derivation of set.mm's ori... |
| ax12 2455 | Rederivation of Axiom ~ ax... |
| ax12b 2456 | A bidirectional version of... |
| ax13ALT 2457 | Alternate proof of ~ ax13 ... |
| axc11n 2458 | Derive set.mm's original ~... |
| aecom 2459 | Commutation law for identi... |
| aecoms 2460 | A commutation rule for ide... |
| naecoms 2461 | A commutation rule for dis... |
| axc11 2462 | Show that ~ ax-c11 can be ... |
| hbae 2463 | All variables are effectiv... |
| hbnae 2464 | All variables are effectiv... |
| nfae 2465 | All variables are effectiv... |
| nfnae 2466 | All variables are effectiv... |
| hbnaes 2467 | Rule that applies ~ hbnae ... |
| axc16i 2468 | Inference with ~ axc16 as ... |
| axc16nfALT 2469 | Alternate proof of ~ axc16... |
| dral2 2470 | Formula-building lemma for... |
| dral1 2471 | Formula-building lemma for... |
| dral1ALT 2472 | Alternate proof of ~ dral1... |
| drex1 2473 | Formula-building lemma for... |
| drex2 2474 | Formula-building lemma for... |
| drnf1 2475 | Formula-building lemma for... |
| drnf2 2476 | Formula-building lemma for... |
| nfald2 2477 | Variation on ~ nfald which... |
| nfexd2 2478 | Variation on ~ nfexd which... |
| exdistrf 2479 | Distribution of existentia... |
| dvelimf 2480 | Version of ~ dvelimv witho... |
| dvelimdf 2481 | Deduction form of ~ dvelim... |
| dvelimh 2482 | Version of ~ dvelim withou... |
| dvelim 2483 | This theorem can be used t... |
| dvelimv 2484 | Similar to ~ dvelim with f... |
| dvelimnf 2485 | Version of ~ dvelim using ... |
| dveeq2ALT 2486 | Alternate proof of ~ dveeq... |
| equvini 2487 | A variable introduction la... |
| equvel 2488 | A variable elimination law... |
| equs5a 2489 | A property related to subs... |
| equs5e 2490 | A property related to subs... |
| equs45f 2491 | Two ways of expressing sub... |
| equs5 2492 | Lemma used in proofs of su... |
| dveel1 2493 | Quantifier introduction wh... |
| dveel2 2494 | Quantifier introduction wh... |
| axc14 2495 | Axiom ~ ax-c14 is redundan... |
| sb6x 2496 | Equivalence involving subs... |
| sbequ5 2497 | Substitution does not chan... |
| sbequ6 2498 | Substitution does not chan... |
| sb5rf 2499 | Reversed substitution. Us... |
| sb6rf 2500 | Reversed substitution. Fo... |
| ax12vALT 2501 | Alternate proof of ~ ax12v... |
| 2ax6elem 2502 | We can always find values ... |
| 2ax6e 2503 | We can always find values ... |
| 2sb5rf 2504 | Reversed double substituti... |
| 2sb6rf 2505 | Reversed double substituti... |
| sbel2x 2506 | Elimination of double subs... |
| sb4b 2507 | Simplified definition of s... |
| sb3b 2508 | Simplified definition of s... |
| sb3 2509 | One direction of a simplif... |
| sb1 2510 | One direction of a simplif... |
| sb2 2511 | One direction of a simplif... |
| sb4a 2512 | A version of one implicati... |
| dfsb1 2513 | Alternate definition of su... |
| hbsb2 2514 | Bound-variable hypothesis ... |
| nfsb2 2515 | Bound-variable hypothesis ... |
| hbsb2a 2516 | Special case of a bound-va... |
| sb4e 2517 | One direction of a simplif... |
| hbsb2e 2518 | Special case of a bound-va... |
| hbsb3 2519 | If ` y ` is not free in ` ... |
| nfs1 2520 | If ` y ` is not free in ` ... |
| axc16ALT 2521 | Alternate proof of ~ axc16... |
| axc16gALT 2522 | Alternate proof of ~ axc16... |
| equsb1 2523 | Substitution applied to an... |
| equsb2 2524 | Substitution applied to an... |
| dfsb2 2525 | An alternate definition of... |
| dfsb3 2526 | An alternate definition of... |
| drsb1 2527 | Formula-building lemma for... |
| sb2ae 2528 | In the case of two success... |
| sb6f 2529 | Equivalence for substituti... |
| sb5f 2530 | Equivalence for substituti... |
| nfsb4t 2531 | A variable not free in a p... |
| nfsb4 2532 | A variable not free in a p... |
| sbequ8 2533 | Elimination of equality fr... |
| sbie 2534 | Conversion of implicit sub... |
| sbied 2535 | Conversion of implicit sub... |
| sbiedv 2536 | Conversion of implicit sub... |
| 2sbiev 2537 | Conversion of double impli... |
| sbcom3 2538 | Substituting ` y ` for ` x... |
| sbco 2539 | A composition law for subs... |
| sbid2 2540 | An identity law for substi... |
| sbid2v 2541 | An identity law for substi... |
| sbidm 2542 | An idempotent law for subs... |
| sbco2 2543 | A composition law for subs... |
| sbco2d 2544 | A composition law for subs... |
| sbco3 2545 | A composition law for subs... |
| sbcom 2546 | A commutativity law for su... |
| sbtrt 2547 | Partially closed form of ~... |
| sbtr 2548 | A partial converse to ~ sb... |
| sb8 2549 | Substitution of variable i... |
| sb8e 2550 | Substitution of variable i... |
| sb9 2551 | Commutation of quantificat... |
| sb9i 2552 | Commutation of quantificat... |
| sbhb 2553 | Two ways of expressing " `... |
| nfsbd 2554 | Deduction version of ~ nfs... |
| nfsb 2555 | If ` z ` is not free in ` ... |
| hbsb 2556 | If ` z ` is not free in ` ... |
| sb7f 2557 | This version of ~ dfsb7 do... |
| sb7h 2558 | This version of ~ dfsb7 do... |
| sb10f 2559 | Hao Wang's identity axiom ... |
| sbal1 2560 | Check out ~ sbal for a ver... |
| sbal2 2561 | Move quantifier in and out... |
| 2sb8e 2562 | An equivalent expression f... |
| dfmoeu 2563 | An elementary proof of ~ m... |
| dfeumo 2564 | An elementary proof showin... |
| mojust 2566 | Soundness justification th... |
| dfmo 2568 | Simplify definition ~ df-m... |
| nexmo 2569 | Nonexistence implies uniqu... |
| exmo 2570 | Any proposition holds for ... |
| moabs 2571 | Absorption of existence co... |
| moim 2572 | The at-most-one quantifier... |
| moimi 2573 | The at-most-one quantifier... |
| moimdv 2574 | The at-most-one quantifier... |
| mobi 2575 | Equivalence theorem for th... |
| mobii 2576 | Formula-building rule for ... |
| mobidv 2577 | Formula-building rule for ... |
| mobid 2578 | Formula-building rule for ... |
| moa1 2579 | If an implication holds fo... |
| moan 2580 | "At most one" is still the... |
| moani 2581 | "At most one" is still tru... |
| moor 2582 | "At most one" is still the... |
| mooran1 2583 | "At most one" imports disj... |
| mooran2 2584 | "At most one" exports disj... |
| nfmo1 2585 | Bound-variable hypothesis ... |
| nfmod2 2586 | Bound-variable hypothesis ... |
| nfmodv 2587 | Bound-variable hypothesis ... |
| nfmov 2588 | Bound-variable hypothesis ... |
| nfmod 2589 | Bound-variable hypothesis ... |
| nfmo 2590 | Bound-variable hypothesis ... |
| mof 2591 | Version of ~ df-mo with di... |
| mo3 2592 | Alternate definition of th... |
| mo 2593 | Equivalent definitions of ... |
| mo4 2594 | At-most-one quantifier exp... |
| mo4f 2595 | At-most-one quantifier exp... |
| eu3v 2598 | An alternate way to expres... |
| eujust 2599 | Soundness justification th... |
| eujustALT 2600 | Alternate proof of ~ eujus... |
| eu6lem 2601 | Lemma of ~ eu6im . A diss... |
| eu6 2602 | Alternate definition of th... |
| eu6im 2603 | One direction of ~ eu6 nee... |
| euf 2604 | Version of ~ eu6 with disj... |
| euex 2605 | Existential uniqueness imp... |
| eumo 2606 | Existential uniqueness imp... |
| eumoi 2607 | Uniqueness inferred from e... |
| exmoeub 2608 | Existence implies that uni... |
| exmoeu 2609 | Existence is equivalent to... |
| moeuex 2610 | Uniqueness implies that ex... |
| moeu 2611 | Uniqueness is equivalent t... |
| eubi 2612 | Equivalence theorem for th... |
| eubii 2613 | Introduce unique existenti... |
| eubidv 2614 | Formula-building rule for ... |
| eubid 2615 | Formula-building rule for ... |
| nfeu1ALT 2616 | Alternate version of ~ nfe... |
| nfeu1 2617 | Bound-variable hypothesis ... |
| nfeud2 2618 | Bound-variable hypothesis ... |
| nfeudw 2619 | Bound-variable hypothesis ... |
| nfeud 2620 | Bound-variable hypothesis ... |
| nfeuw 2621 | Bound-variable hypothesis ... |
| nfeu 2622 | Bound-variable hypothesis ... |
| dfeu 2623 | Rederive ~ df-eu from the ... |
| dfmo2 2624 | Rederive ~ df-mo from the ... |
| euequ 2625 | There exists a unique set ... |
| sb8eulem 2626 | Lemma. Factor out the com... |
| sb8euv 2627 | Variable substitution in u... |
| sb8eu 2628 | Variable substitution in u... |
| sb8mo 2629 | Variable substitution for ... |
| cbvmovw 2630 | Change bound variable. Us... |
| cbvmow 2631 | Rule used to change bound ... |
| cbvmo 2632 | Rule used to change bound ... |
| cbveuvw 2633 | Change bound variable. Us... |
| cbveuw 2634 | Version of ~ cbveu with a ... |
| cbveu 2635 | Rule used to change bound ... |
| cbveuALT 2636 | Alternative proof of ~ cbv... |
| eu2 2637 | An alternate way of defini... |
| eu1 2638 | An alternate way to expres... |
| euor 2639 | Introduce a disjunct into ... |
| euorv 2640 | Introduce a disjunct into ... |
| euor2 2641 | Introduce or eliminate a d... |
| sbmo 2642 | Substitution into an at-mo... |
| eu4 2643 | Uniqueness using implicit ... |
| euimmo 2644 | Existential uniqueness imp... |
| euim 2645 | Add unique existential qua... |
| moanimlem 2646 | Factor out the common proo... |
| moanimv 2647 | Introduction of a conjunct... |
| moanim 2648 | Introduction of a conjunct... |
| euan 2649 | Introduction of a conjunct... |
| moanmo 2650 | Nested at-most-one quantif... |
| moaneu 2651 | Nested at-most-one and uni... |
| euanv 2652 | Introduction of a conjunct... |
| mopick 2653 | "At most one" picks a vari... |
| moexexlem 2654 | Factor out the proof skele... |
| 2moexv 2655 | Double quantification with... |
| moexexvw 2656 | "At most one" double quant... |
| 2moswapv 2657 | A condition allowing to sw... |
| 2euswapv 2658 | A condition allowing to sw... |
| 2euexv 2659 | Double quantification with... |
| 2exeuv 2660 | Double existential uniquen... |
| eupick 2661 | Existential uniqueness "pi... |
| eupicka 2662 | Version of ~ eupick with c... |
| eupickb 2663 | Existential uniqueness "pi... |
| eupickbi 2664 | Theorem *14.26 in [Whitehe... |
| mopick2 2665 | "At most one" can show the... |
| moexex 2666 | "At most one" double quant... |
| moexexv 2667 | "At most one" double quant... |
| 2moex 2668 | Double quantification with... |
| 2euex 2669 | Double quantification with... |
| 2eumo 2670 | Nested unique existential ... |
| 2eu2ex 2671 | Double existential uniquen... |
| 2moswap 2672 | A condition allowing to sw... |
| 2euswap 2673 | A condition allowing to sw... |
| 2exeu 2674 | Double existential uniquen... |
| 2mo2 2675 | Two ways of expressing "th... |
| 2mo 2676 | Two ways of expressing "th... |
| 2mos 2677 | Double "there exists at mo... |
| 2eu1 2678 | Double existential uniquen... |
| 2eu1v 2679 | Double existential uniquen... |
| 2eu2 2680 | Double existential uniquen... |
| 2eu3 2681 | Double existential uniquen... |
| 2eu4 2682 | This theorem provides us w... |
| 2eu5 2683 | An alternate definition of... |
| 2eu6 2684 | Two equivalent expressions... |
| 2eu7 2685 | Two equivalent expressions... |
| 2eu8 2686 | Two equivalent expressions... |
| euae 2687 | Two ways to express "exact... |
| exists1 2688 | Two ways to express "exact... |
| exists2 2689 | A condition implying that ... |
| barbara 2690 | "Barbara", one of the fund... |
| celarent 2691 | "Celarent", one of the syl... |
| darii 2692 | "Darii", one of the syllog... |
| dariiALT 2693 | Alternate proof of ~ darii... |
| ferio 2694 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2695 | Lemma for ~ barbari and th... |
| barbari 2696 | "Barbari", one of the syll... |
| barbariALT 2697 | Alternate proof of ~ barba... |
| celaront 2698 | "Celaront", one of the syl... |
| cesare 2699 | "Cesare", one of the syllo... |
| camestres 2700 | "Camestres", one of the sy... |
| festino 2701 | "Festino", one of the syll... |
| festinoALT 2702 | Alternate proof of ~ festi... |
| baroco 2703 | "Baroco", one of the syllo... |
| barocoALT 2704 | Alternate proof of ~ festi... |
| cesaro 2705 | "Cesaro", one of the syllo... |
| camestros 2706 | "Camestros", one of the sy... |
| datisi 2707 | "Datisi", one of the syllo... |
| disamis 2708 | "Disamis", one of the syll... |
| ferison 2709 | "Ferison", one of the syll... |
| bocardo 2710 | "Bocardo", one of the syll... |
| darapti 2711 | "Darapti", one of the syll... |
| daraptiALT 2712 | Alternate proof of ~ darap... |
| felapton 2713 | "Felapton", one of the syl... |
| calemes 2714 | "Calemes", one of the syll... |
| dimatis 2715 | "Dimatis", one of the syll... |
| fresison 2716 | "Fresison", one of the syl... |
| calemos 2717 | "Calemos", one of the syll... |
| fesapo 2718 | "Fesapo", one of the syllo... |
| bamalip 2719 | "Bamalip", one of the syll... |
| axia1 2720 | Left 'and' elimination (in... |
| axia2 2721 | Right 'and' elimination (i... |
| axia3 2722 | 'And' introduction (intuit... |
| axin1 2723 | 'Not' introduction (intuit... |
| axin2 2724 | 'Not' elimination (intuiti... |
| axio 2725 | Definition of 'or' (intuit... |
| axi4 2726 | Specialization (intuitioni... |
| axi5r 2727 | Converse of ~ axc4 (intuit... |
| axial 2728 | The setvar ` x ` is not fr... |
| axie1 2729 | The setvar ` x ` is not fr... |
| axie2 2730 | A key property of existent... |
| axi9 2731 | Axiom of existence (intuit... |
| axi10 2732 | Axiom of Quantifier Substi... |
| axi12 2733 | Axiom of Quantifier Introd... |
| axbnd 2734 | Axiom of Bundling (intuiti... |
| axexte 2736 | The axiom of extensionalit... |
| axextg 2737 | A generalization of the ax... |
| axextb 2738 | A bidirectional version of... |
| axextmo 2739 | There exists at most one s... |
| nulmo 2740 | There exists at most one e... |
| eleq1ab 2743 | Extension (in the sense of... |
| cleljustab 2744 | Extension of ~ cleljust fr... |
| abid 2745 | Simplification of class ab... |
| vexwt 2746 | A standard theorem of pred... |
| vexw 2747 | If ` ph ` is a theorem, th... |
| vextru 2748 | Every setvar is a member o... |
| nfsab1 2749 | Bound-variable hypothesis ... |
| hbab1 2750 | Bound-variable hypothesis ... |
| hbab 2751 | Bound-variable hypothesis ... |
| hbabg 2752 | Bound-variable hypothesis ... |
| nfsab 2753 | Bound-variable hypothesis ... |
| nfsabg 2754 | Bound-variable hypothesis ... |
| dfcleq 2756 | The defining characterizat... |
| cvjust 2757 | Every set is a class. Pro... |
| ax9ALT 2758 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2759 | A weaker version of ~ eleq... |
| eqriv 2760 | Infer equality of classes ... |
| eqrdv 2761 | Deduce equality of classes... |
| eqrdav 2762 | Deduce equality of classes... |
| eqid 2763 | Law of identity (reflexivi... |
| eqidd 2764 | Class identity law with an... |
| eqeq1d 2765 | Deduction from equality to... |
| eqeq1dALT 2766 | Alternate proof of ~ eqeq1... |
| eqeq1 2767 | Equality implies equivalen... |
| eqeq1i 2768 | Inference from equality to... |
| eqcomd 2769 | Deduction from commutative... |
| eqcom 2770 | Commutative law for class ... |
| eqcoms 2771 | Inference applying commuta... |
| eqcomi 2772 | Inference from commutative... |
| neqcomd 2773 | Commute an inequality. (C... |
| eqeq2d 2774 | Deduction from equality to... |
| eqeq2 2775 | Equality implies equivalen... |
| eqeq2i 2776 | Inference from equality to... |
| eqeqan12d 2777 | A useful inference for sub... |
| eqeqan12rd 2778 | A useful inference for sub... |
| eqeq12d 2779 | A useful inference for sub... |
| eqeq12 2780 | Equality relationship amon... |
| eqeq12i 2781 | A useful inference for sub... |
| eqeqan12dALT 2782 | Alternate proof of ~ eqeqa... |
| eqtr 2783 | Transitive law for class e... |
| eqtr2 2784 | A transitive law for class... |
| eqtr3 2785 | A transitive law for class... |
| eqtri 2786 | An equality transitivity i... |
| eqtr2i 2787 | An equality transitivity i... |
| eqtr3i 2788 | An equality transitivity i... |
| eqtr4i 2789 | An equality transitivity i... |
| 3eqtri 2790 | An inference from three ch... |
| 3eqtrri 2791 | An inference from three ch... |
| 3eqtr2i 2792 | An inference from three ch... |
| 3eqtr2ri 2793 | An inference from three ch... |
| 3eqtr3i 2794 | An inference from three ch... |
| 3eqtr3ri 2795 | An inference from three ch... |
| 3eqtr4i 2796 | An inference from three ch... |
| 3eqtr4ri 2797 | An inference from three ch... |
| eqtrd 2798 | An equality transitivity d... |
| eqtr2d 2799 | An equality transitivity d... |
| eqtr3d 2800 | An equality transitivity e... |
| eqtr4d 2801 | An equality transitivity e... |
| 3eqtrd 2802 | A deduction from three cha... |
| 3eqtrrd 2803 | A deduction from three cha... |
| 3eqtr2d 2804 | A deduction from three cha... |
| 3eqtr2rd 2805 | A deduction from three cha... |
| 3eqtr3d 2806 | A deduction from three cha... |
| 3eqtr3rd 2807 | A deduction from three cha... |
| 3eqtr4d 2808 | A deduction from three cha... |
| 3eqtr4rd 2809 | A deduction from three cha... |
| eqtrid 2810 | An equality transitivity d... |
| eqtr2id 2811 | An equality transitivity d... |
| eqtr3id 2812 | An equality transitivity d... |
| eqtr3di 2813 | An equality transitivity d... |
| eqtrdi 2814 | An equality transitivity d... |
| eqtr2di 2815 | An equality transitivity d... |
| eqtr4di 2816 | An equality transitivity d... |
| eqtr4id 2817 | An equality transitivity d... |
| sylan9eq 2818 | An equality transitivity d... |
| sylan9req 2819 | An equality transitivity d... |
| sylan9eqr 2820 | An equality transitivity d... |
| 3eqtr3g 2821 | A chained equality inferen... |
| 3eqtr3a 2822 | A chained equality inferen... |
| 3eqtr4g 2823 | A chained equality inferen... |
| 3eqtr4a 2824 | A chained equality inferen... |
| eq2tri 2825 | A compound transitive infe... |
| iseqsetvlem 2826 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2827 | Alternate proof of ~ iseqs... |
| abbi 2828 | Equivalent formulas yield ... |
| abbidv 2829 | Equivalent wff's yield equ... |
| abbii 2830 | Equivalent wff's yield equ... |
| abbid 2831 | Equivalent wff's yield equ... |
| abbib 2832 | Equal class abstractions r... |
| abbibOLD 2833 | Obsolete version of ~ abbi... |
| cbvabv 2834 | Rule used to change bound ... |
| cbvabw 2835 | Rule used to change bound ... |
| cbvab 2836 | Rule used to change bound ... |
| eqabbw 2837 | Version of ~ eqabb using i... |
| eqabcbw 2838 | Version of ~ eqabcb using ... |
| dfclel 2840 | Characterization of the el... |
| elex2 2841 | If a class contains anothe... |
| issettru 2842 | Weak version of ~ isset . ... |
| iseqsetv-clel 2843 | Alternate proof of ~ iseqs... |
| issetlem 2844 | Lemma for ~ elisset and ~ ... |
| elissetv 2845 | An element of a class exis... |
| elisset 2846 | An element of a class exis... |
| eleq1w 2847 | Weaker version of ~ eleq1 ... |
| eleq2w 2848 | Weaker version of ~ eleq2 ... |
| eleq1d 2849 | Deduction from equality to... |
| eleq2d 2850 | Deduction from equality to... |
| eleq2dALT 2851 | Alternate proof of ~ eleq2... |
| eleq1 2852 | Equality implies equivalen... |
| eleq2 2853 | Equality implies equivalen... |
| eleq12 2854 | Equality implies equivalen... |
| eleq1i 2855 | Inference from equality to... |
| eleq2i 2856 | Inference from equality to... |
| eleq12i 2857 | Inference from equality to... |
| eleq12d 2858 | Deduction from equality to... |
| eleq1a 2859 | A transitive-type law rela... |
| eqeltri 2860 | Substitution of equal clas... |
| eqeltrri 2861 | Substitution of equal clas... |
| eleqtri 2862 | Substitution of equal clas... |
| eleqtrri 2863 | Substitution of equal clas... |
| eqeltrd 2864 | Substitution of equal clas... |
| eqeltrrd 2865 | Deduction that substitutes... |
| eleqtrd 2866 | Deduction that substitutes... |
| eleqtrrd 2867 | Deduction that substitutes... |
| eqeltrid 2868 | A membership and equality ... |
| eqeltrrid 2869 | A membership and equality ... |
| eleqtrid 2870 | A membership and equality ... |
| eleqtrrid 2871 | A membership and equality ... |
| eqeltrdi 2872 | A membership and equality ... |
| eqeltrrdi 2873 | A membership and equality ... |
| eleqtrdi 2874 | A membership and equality ... |
| eleqtrrdi 2875 | A membership and equality ... |
| 3eltr3i 2876 | Substitution of equal clas... |
| 3eltr4i 2877 | Substitution of equal clas... |
| 3eltr3d 2878 | Substitution of equal clas... |
| 3eltr4d 2879 | Substitution of equal clas... |
| 3eltr3g 2880 | Substitution of equal clas... |
| 3eltr4g 2881 | Substitution of equal clas... |
| eleq2s 2882 | Substitution of equal clas... |
| eqneltri 2883 | If a class is not an eleme... |
| eqneltrd 2884 | If a class is not an eleme... |
| eqneltrrd 2885 | If a class is not an eleme... |
| neleqtrd 2886 | If a class is not an eleme... |
| neleqtrrd 2887 | If a class is not an eleme... |
| nelneq 2888 | A way of showing two class... |
| nelneq2 2889 | A way of showing two class... |
| eqsb1 2890 | Substitution for the left-... |
| clelsb1 2891 | Substitution for the first... |
| clelsb2 2892 | Substitution for the secon... |
| cleqh 2893 | Establish equality between... |
| hbxfreq 2894 | A utility lemma to transfe... |
| hblem 2895 | Change the free variable o... |
| hblemg 2896 | Change the free variable o... |
| eqabdv 2897 | Deduction from a wff to a ... |
| eqabcdv 2898 | Deduction from a wff to a ... |
| eqabi 2899 | Equality of a class variab... |
| abid1 2900 | Every class is equal to a ... |
| abid2 2901 | A simplification of class ... |
| eqab 2902 | One direction of ~ eqabb i... |
| eqabb 2903 | Equality of a class variab... |
| eqabcb 2904 | Equality of a class variab... |
| eqabrd 2905 | Equality of a class variab... |
| eqabri 2906 | Equality of a class variab... |
| eqabcri 2907 | Equality of a class variab... |
| clelab 2908 | Membership of a class vari... |
| clabel 2909 | Membership of a class abst... |
| sbab 2910 | The right-hand side of the... |
| nfcjust 2912 | Justification theorem for ... |
| nfci 2914 | Deduce that a class ` A ` ... |
| nfcii 2915 | Deduce that a class ` A ` ... |
| nfcr 2916 | Consequence of the not-fre... |
| nfcrALT 2917 | Alternate version of ~ nfc... |
| nfcri 2918 | Consequence of the not-fre... |
| nfcd 2919 | Deduce that a class ` A ` ... |
| nfcrd 2920 | Consequence of the not-fre... |
| nfcrii 2921 | Consequence of the not-fre... |
| nfceqdf 2922 | An equality theorem for ef... |
| nfceqi 2923 | Equality theorem for class... |
| nfcxfr 2924 | A utility lemma to transfe... |
| nfcxfrd 2925 | A utility lemma to transfe... |
| nfcv 2926 | If ` x ` is disjoint from ... |
| nfcvd 2927 | If ` x ` is disjoint from ... |
| nfab1 2928 | Bound-variable hypothesis ... |
| nfnfc1 2929 | The setvar ` x ` is bound ... |
| clelsb1fw 2930 | Substitution for the first... |
| clelsb1f 2931 | Substitution for the first... |
| nfab 2932 | Bound-variable hypothesis ... |
| nfabg 2933 | Bound-variable hypothesis ... |
| nfaba1 2934 | Bound-variable hypothesis ... |
| nfaba1g 2935 | Bound-variable hypothesis ... |
| nfeqd 2936 | Hypothesis builder for equ... |
| nfeld 2937 | Hypothesis builder for ele... |
| nfnfc 2938 | Hypothesis builder for ` F... |
| nfeq 2939 | Hypothesis builder for equ... |
| nfel 2940 | Hypothesis builder for ele... |
| nfeq1 2941 | Hypothesis builder for equ... |
| nfel1 2942 | Hypothesis builder for ele... |
| nfeq2 2943 | Hypothesis builder for equ... |
| nfel2 2944 | Hypothesis builder for ele... |
| drnfc1 2945 | Formula-building lemma for... |
| drnfc2 2946 | Formula-building lemma for... |
| nfabdw 2947 | Bound-variable hypothesis ... |
| nfabd 2948 | Bound-variable hypothesis ... |
| nfabd2 2949 | Bound-variable hypothesis ... |
| dvelimdc 2950 | Deduction form of ~ dvelim... |
| dvelimc 2951 | Version of ~ dvelim for cl... |
| nfcvf 2952 | If ` x ` and ` y ` are dis... |
| nfcvf2 2953 | If ` x ` and ` y ` are dis... |
| cleqf 2954 | Establish equality between... |
| eqabf 2955 | Equality of a class variab... |
| abid2f 2956 | A simplification of class ... |
| abid2fOLD 2957 | Obsolete version of ~ abid... |
| sbabel 2958 | Theorem to move a substitu... |
| neii 2961 | Inference associated with ... |
| neir 2962 | Inference associated with ... |
| nne 2963 | Negation of inequality. (... |
| neneqd 2964 | Deduction eliminating ineq... |
| neneq 2965 | From inequality to non-equ... |
| neqned 2966 | If it is not the case that... |
| neqne 2967 | From non-equality to inequ... |
| neirr 2968 | No class is unequal to its... |
| exmidne 2969 | Excluded middle with equal... |
| eqneqall 2970 | A contradiction concerning... |
| nonconne 2971 | Law of noncontradiction wi... |
| necon3ad 2972 | Contrapositive law deducti... |
| necon3bd 2973 | Contrapositive law deducti... |
| necon2ad 2974 | Contrapositive inference f... |
| necon2bd 2975 | Contrapositive inference f... |
| necon1ad 2976 | Contrapositive deduction f... |
| necon1bd 2977 | Contrapositive deduction f... |
| necon4ad 2978 | Contrapositive inference f... |
| necon4bd 2979 | Contrapositive inference f... |
| necon3d 2980 | Contrapositive law deducti... |
| necon1d 2981 | Contrapositive law deducti... |
| necon2d 2982 | Contrapositive inference f... |
| necon4d 2983 | Contrapositive inference f... |
| necon3ai 2984 | Contrapositive inference f... |
| necon3bi 2985 | Contrapositive inference f... |
| necon1ai 2986 | Contrapositive inference f... |
| necon1bi 2987 | Contrapositive inference f... |
| necon2ai 2988 | Contrapositive inference f... |
| necon2bi 2989 | Contrapositive inference f... |
| necon4ai 2990 | Contrapositive inference f... |
| necon3i 2991 | Contrapositive inference f... |
| necon1i 2992 | Contrapositive inference f... |
| necon2i 2993 | Contrapositive inference f... |
| necon4i 2994 | Contrapositive inference f... |
| necon3abid 2995 | Deduction from equality to... |
| necon3bbid 2996 | Deduction from equality to... |
| necon1abid 2997 | Contrapositive deduction f... |
| necon1bbid 2998 | Contrapositive inference f... |
| necon4abid 2999 | Contrapositive law deducti... |
| necon4bbid 3000 | Contrapositive law deducti... |
| necon2abid 3001 | Contrapositive deduction f... |
| necon2bbid 3002 | Contrapositive deduction f... |
| necon3bid 3003 | Deduction from equality to... |
| necon4bid 3004 | Contrapositive law deducti... |
| necon3abii 3005 | Deduction from equality to... |
| necon3bbii 3006 | Deduction from equality to... |
| necon1abii 3007 | Contrapositive inference f... |
| necon1bbii 3008 | Contrapositive inference f... |
| necon2abii 3009 | Contrapositive inference f... |
| necon2bbii 3010 | Contrapositive inference f... |
| necon3bii 3011 | Inference from equality to... |
| necom 3012 | Commutation of inequality.... |
| necomi 3013 | Inference from commutative... |
| necomd 3014 | Deduction from commutative... |
| nesym 3015 | Characterization of inequa... |
| nesymi 3016 | Inference associated with ... |
| nesymir 3017 | Inference associated with ... |
| neeq1d 3018 | Deduction for inequality. ... |
| neeq2d 3019 | Deduction for inequality. ... |
| neeq12d 3020 | Deduction for inequality. ... |
| neeq1 3021 | Equality theorem for inequ... |
| neeq2 3022 | Equality theorem for inequ... |
| neeq1i 3023 | Inference for inequality. ... |
| neeq2i 3024 | Inference for inequality. ... |
| neeq12i 3025 | Inference for inequality. ... |
| eqnetrd 3026 | Substitution of equal clas... |
| eqnetrrd 3027 | Substitution of equal clas... |
| neeqtrd 3028 | Substitution of equal clas... |
| eqnetri 3029 | Substitution of equal clas... |
| eqnetrri 3030 | Substitution of equal clas... |
| neeqtri 3031 | Substitution of equal clas... |
| neeqtrri 3032 | Substitution of equal clas... |
| neeqtrrd 3033 | Substitution of equal clas... |
| eqnetrrid 3034 | A chained equality inferen... |
| 3netr3d 3035 | Substitution of equality i... |
| 3netr4d 3036 | Substitution of equality i... |
| 3netr3g 3037 | Substitution of equality i... |
| 3netr4g 3038 | Substitution of equality i... |
| nebi 3039 | Contraposition law for ine... |
| pm13.18 3040 | Theorem *13.18 in [Whitehe... |
| pm13.181 3041 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3042 | Inference eliminating an i... |
| pm2.21ddne 3043 | A contradiction implies an... |
| pm2.61ne 3044 | Deduction eliminating an i... |
| pm2.61dne 3045 | Deduction eliminating an i... |
| pm2.61dane 3046 | Deduction eliminating an i... |
| pm2.61da2ne 3047 | Deduction eliminating two ... |
| pm2.61da3ne 3048 | Deduction eliminating thre... |
| pm2.61iine 3049 | Equality version of ~ pm2.... |
| mteqand 3050 | A modus tollens deduction ... |
| neor 3051 | Logical OR with an equalit... |
| neanior 3052 | A De Morgan's law for ineq... |
| ne3anior 3053 | A De Morgan's law for ineq... |
| neorian 3054 | A De Morgan's law for ineq... |
| nemtbir 3055 | An inference from an inequ... |
| nelne1 3056 | Two classes are different ... |
| nelne2 3057 | Two classes are different ... |
| elnelneqd 3058 | Two classes are not equal ... |
| elnelneq2d 3059 | Two classes are not equal ... |
| nelelne 3060 | Two classes are different ... |
| neneor 3061 | If two classes are differe... |
| nfne 3062 | Bound-variable hypothesis ... |
| nfned 3063 | Bound-variable hypothesis ... |
| nabbib 3064 | Not equivalent wff's corre... |
| neli 3067 | Inference associated with ... |
| nelir 3068 | Inference associated with ... |
| nelcon3d 3069 | Contrapositive law deducti... |
| neleq12d 3070 | Equality theorem for negat... |
| neleq1 3071 | Equality theorem for negat... |
| neleq2 3072 | Equality theorem for negat... |
| nfnel 3073 | Bound-variable hypothesis ... |
| nfneld 3074 | Bound-variable hypothesis ... |
| nnel 3075 | Negation of negated member... |
| elnelne1 3076 | Two classes are different ... |
| elnelne2 3077 | Two classes are different ... |
| pm2.24nel 3078 | A contradiction concerning... |
| pm2.61danel 3079 | Deduction eliminating an e... |
| rgen 3082 | Generalization rule for re... |
| ralel 3083 | All elements of a class ar... |
| rgenw 3084 | Generalization rule for re... |
| rgen2w 3085 | Generalization rule for re... |
| mprg 3086 | Modus ponens combined with... |
| mprgbir 3087 | Modus ponens on biconditio... |
| ralrid 3088 | Sufficient condition for t... |
| raln 3089 | Restricted universally qua... |
| ralnex 3092 | Relationship between restr... |
| dfrex2 3093 | Relationship between restr... |
| nrex 3094 | Inference adding restricte... |
| alral 3095 | Universal quantification i... |
| rexex 3096 | Restricted existence impli... |
| rextru 3097 | Two ways of expressing tha... |
| ralimi2 3098 | Inference quantifying both... |
| reximi2 3099 | Inference quantifying both... |
| ralimia 3100 | Inference quantifying both... |
| reximia 3101 | Inference quantifying both... |
| ralimiaa 3102 | Inference quantifying both... |
| ralimi 3103 | Inference quantifying both... |
| reximi 3104 | Inference quantifying both... |
| ral2imi 3105 | Inference quantifying ante... |
| ralim 3106 | Distribution of restricted... |
| rexim 3107 | Theorem 19.22 of [Margaris... |
| ralbii2 3108 | Inference adding different... |
| rexbii2 3109 | Inference adding different... |
| ralbiia 3110 | Inference adding restricte... |
| rexbiia 3111 | Inference adding restricte... |
| ralbii 3112 | Inference adding restricte... |
| rexbii 3113 | Inference adding restricte... |
| ralanid 3114 | Cancellation law for restr... |
| rexanid 3115 | Cancellation law for restr... |
| ralcom3 3116 | A commutation law for rest... |
| dfral2 3117 | Relationship between restr... |
| rexnal 3118 | Relationship between restr... |
| ralinexa 3119 | A transformation of restri... |
| rexanali 3120 | A transformation of restri... |
| ralbi 3121 | Distribute a restricted un... |
| rexbi 3122 | Distribute restricted quan... |
| ralrexbid 3123 | Formula-building rule for ... |
| r19.35 3124 | Restricted quantifier vers... |
| r19.26m 3125 | Version of ~ 19.26 and ~ r... |
| r19.26 3126 | Restricted quantifier vers... |
| r19.26-3 3127 | Version of ~ r19.26 with t... |
| ralbiim 3128 | Split a biconditional and ... |
| r19.29 3129 | Restricted quantifier vers... |
| r19.29r 3130 | Restricted quantifier vers... |
| r19.29imd 3131 | Theorem 19.29 of [Margaris... |
| r19.40 3132 | Restricted quantifier vers... |
| r19.30 3133 | Restricted quantifier vers... |
| r19.43 3134 | Restricted quantifier vers... |
| 3r19.43 3135 | Restricted quantifier vers... |
| 2ralimi 3136 | Inference quantifying both... |
| 3ralimi 3137 | Inference quantifying both... |
| 4ralimi 3138 | Inference quantifying both... |
| 5ralimi 3139 | Inference quantifying both... |
| 6ralimi 3140 | Inference quantifying both... |
| 2ralbii 3141 | Inference adding two restr... |
| 2rexbii 3142 | Inference adding two restr... |
| 3ralbii 3143 | Inference adding three res... |
| 4ralbii 3144 | Inference adding four rest... |
| 2ralbiim 3145 | Split a biconditional and ... |
| ralnex2 3146 | Relationship between two r... |
| ralnex3 3147 | Relationship between three... |
| rexnal2 3148 | Relationship between two r... |
| rexnal3 3149 | Relationship between three... |
| nrexralim 3150 | Negation of a complex pred... |
| r19.26-2 3151 | Restricted quantifier vers... |
| 2r19.29 3152 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3153 | Theorem 19.29 of [Margaris... |
| r2allem 3154 | Lemma factoring out common... |
| r2exlem 3155 | Lemma factoring out common... |
| hbralrimi 3156 | Inference from Theorem 19.... |
| ralrimiv 3157 | Inference from Theorem 19.... |
| ralrimiva 3158 | Inference from Theorem 19.... |
| rexlimiva 3159 | Inference from Theorem 19.... |
| rexlimiv 3160 | Inference from Theorem 19.... |
| nrexdv 3161 | Deduction adding restricte... |
| ralrimivw 3162 | Inference from Theorem 19.... |
| rexlimivw 3163 | Weaker version of ~ rexlim... |
| ralrimdv 3164 | Inference from Theorem 19.... |
| rexlimdv 3165 | Inference from Theorem 19.... |
| ralrimdva 3166 | Inference from Theorem 19.... |
| rexlimdva 3167 | Inference from Theorem 19.... |
| rexlimdvaa 3168 | Inference from Theorem 19.... |
| rexlimdva2 3169 | Inference from Theorem 19.... |
| r19.29an 3170 | A commonly used pattern in... |
| rexlimdv3a 3171 | Inference from Theorem 19.... |
| rexlimdvw 3172 | Inference from Theorem 19.... |
| rexlimddv 3173 | Restricted existential eli... |
| r19.29a 3174 | A commonly used pattern in... |
| ralimdv2 3175 | Inference quantifying both... |
| reximdv2 3176 | Deduction quantifying both... |
| reximdvai 3177 | Deduction quantifying both... |
| ralimdva 3178 | Deduction quantifying both... |
| reximdva 3179 | Deduction quantifying both... |
| ralimdv 3180 | Deduction quantifying both... |
| reximdv 3181 | Deduction from Theorem 19.... |
| reximddv 3182 | Deduction from Theorem 19.... |
| reximddv3 3183 | Deduction from Theorem 19.... |
| reximssdv 3184 | Derivation of a restricted... |
| ralbidv2 3185 | Formula-building rule for ... |
| rexbidv2 3186 | Formula-building rule for ... |
| ralbidva 3187 | Formula-building rule for ... |
| rexbidva 3188 | Formula-building rule for ... |
| ralbidv 3189 | Formula-building rule for ... |
| rexbidv 3190 | Formula-building rule for ... |
| r19.21v 3191 | Restricted quantifier vers... |
| r19.37v 3192 | Restricted quantifier vers... |
| r19.23v 3193 | Restricted quantifier vers... |
| r19.36v 3194 | Restricted quantifier vers... |
| r19.27v 3195 | Restricted quantitifer ver... |
| r19.41v 3196 | Restricted quantifier vers... |
| r19.28v 3197 | Restricted quantifier vers... |
| r19.42v 3198 | Restricted quantifier vers... |
| r19.32v 3199 | Restricted quantifier vers... |
| r19.45v 3200 | Restricted quantifier vers... |
| r19.44v 3201 | One direction of a restric... |
| r2al 3202 | Double restricted universa... |
| r2ex 3203 | Double restricted existent... |
| r3al 3204 | Triple restricted universa... |
| r3ex 3205 | Triple existential quantif... |
| rgen2 3206 | Generalization rule for re... |
| ralrimivv 3207 | Inference from Theorem 19.... |
| rexlimivv 3208 | Inference from Theorem 19.... |
| ralrimivva 3209 | Inference from Theorem 19.... |
| ralrimdvv 3210 | Inference from Theorem 19.... |
| rgen3 3211 | Generalization rule for re... |
| ralrimivvva 3212 | Inference from Theorem 19.... |
| ralimdvva 3213 | Deduction doubly quantifyi... |
| reximdvva 3214 | Deduction doubly quantifyi... |
| ralimdvv 3215 | Deduction doubly quantifyi... |
| ralimdvvOLD 3216 | Obsolete version of ~ rali... |
| ralimd4v 3217 | Deduction quadrupally quan... |
| ralimd4vOLD 3218 | Obsolete version of ~ rali... |
| ralimd6v 3219 | Deduction sextupally quant... |
| ralimd6vOLD 3220 | Obsolete version of ~ rali... |
| ralrimdvva 3221 | Inference from Theorem 19.... |
| rexlimdvv 3222 | Inference from Theorem 19.... |
| rexlimdvva 3223 | Inference from Theorem 19.... |
| rexlimdvvva 3224 | Inference from Theorem 19.... |
| reximddv2 3225 | Double deduction from Theo... |
| r19.29vva 3226 | A commonly used pattern ba... |
| 2rexbiia 3227 | Inference adding two restr... |
| 2ralbidva 3228 | Formula-building rule for ... |
| 2rexbidva 3229 | Formula-building rule for ... |
| 2ralbidv 3230 | Formula-building rule for ... |
| 2rexbidv 3231 | Formula-building rule for ... |
| rexralbidv 3232 | Formula-building rule for ... |
| 3ralbidv 3233 | Formula-building rule for ... |
| 4ralbidv 3234 | Formula-building rule for ... |
| 6ralbidv 3235 | Formula-building rule for ... |
| r19.41vv 3236 | Version of ~ r19.41v with ... |
| reeanlem 3237 | Lemma factoring out common... |
| reeanv 3238 | Rearrange restricted exist... |
| 3reeanv 3239 | Rearrange three restricted... |
| 2ralor 3240 | Distribute restricted univ... |
| risset 3241 | Two ways to say " ` A ` be... |
| nelb 3242 | A definition of ` -. A e. ... |
| rspw 3243 | Restricted specialization.... |
| cbvralvw 3244 | Change the bound variable ... |
| cbvrexvw 3245 | Change the bound variable ... |
| cbvraldva 3246 | Rule used to change the bo... |
| cbvrexdva 3247 | Rule used to change the bo... |
| cbvral2vw 3248 | Change bound variables of ... |
| cbvrex2vw 3249 | Change bound variables of ... |
| cbvral3vw 3250 | Change bound variables of ... |
| cbvral4vw 3251 | Change bound variables of ... |
| cbvral6vw 3252 | Change bound variables of ... |
| cbvral8vw 3253 | Change bound variables of ... |
| rsp 3254 | Restricted specialization.... |
| rspa 3255 | Restricted specialization.... |
| rspe 3256 | Restricted specialization.... |
| rspec 3257 | Specialization rule for re... |
| r19.21bi 3258 | Inference from Theorem 19.... |
| r19.21be 3259 | Inference from Theorem 19.... |
| r19.21t 3260 | Restricted quantifier vers... |
| r19.21 3261 | Restricted quantifier vers... |
| r19.23t 3262 | Closed theorem form of ~ r... |
| r19.23 3263 | Restricted quantifier vers... |
| ralrimi 3264 | Inference from Theorem 19.... |
| ralrimia 3265 | Inference from Theorem 19.... |
| rexlimi 3266 | Restricted quantifier vers... |
| ralimdaa 3267 | Deduction quantifying both... |
| reximdai 3268 | Deduction from Theorem 19.... |
| r19.37 3269 | Restricted quantifier vers... |
| r19.41 3270 | Restricted quantifier vers... |
| ralrimd 3271 | Inference from Theorem 19.... |
| rexlimd2 3272 | Version of ~ rexlimd with ... |
| rexlimd 3273 | Deduction form of ~ rexlim... |
| r19.29af2 3274 | A commonly used pattern ba... |
| r19.29af 3275 | A commonly used pattern ba... |
| reximd2a 3276 | Deduction quantifying both... |
| ralbida 3277 | Formula-building rule for ... |
| rexbida 3278 | Formula-building rule for ... |
| ralbid 3279 | Formula-building rule for ... |
| rexbid 3280 | Formula-building rule for ... |
| rexbidvALT 3281 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3282 | Alternate proof of ~ rexbi... |
| rsp2 3283 | Restricted specialization,... |
| rsp2e 3284 | Restricted specialization.... |
| rspec2 3285 | Specialization rule for re... |
| rspec3 3286 | Specialization rule for re... |
| r2alf 3287 | Double restricted universa... |
| r2exf 3288 | Double restricted existent... |
| 2ralbida 3289 | Formula-building rule for ... |
| nfra1 3290 | The setvar ` x ` is not fr... |
| nfre1 3291 | The setvar ` x ` is not fr... |
| ralcom4 3292 | Commutation of restricted ... |
| rexcom4 3293 | Commutation of restricted ... |
| ralcom 3294 | Commutation of restricted ... |
| rexcom 3295 | Commutation of restricted ... |
| rexcom4a 3296 | Specialized existential co... |
| ralrot3 3297 | Rotate three restricted un... |
| ralcom13 3298 | Swap first and third restr... |
| rexcom13 3299 | Swap first and third restr... |
| rexrot4 3300 | Rotate four restricted exi... |
| 2ex2rexrot 3301 | Rotate two existential qua... |
| nfra2w 3302 | Similar to Lemma 24 of [Mo... |
| hbra1 3303 | The setvar ` x ` is not fr... |
| ralcomf 3304 | Commutation of restricted ... |
| rexcomf 3305 | Commutation of restricted ... |
| cbvralfw 3306 | Rule used to change bound ... |
| cbvrexfw 3307 | Rule used to change bound ... |
| cbvralw 3308 | Rule used to change bound ... |
| cbvrexw 3309 | Rule used to change bound ... |
| hbral 3310 | Bound-variable hypothesis ... |
| nfraldw 3311 | Deduction version of ~ nfr... |
| nfrexdw 3312 | Deduction version of ~ nfr... |
| nfralw 3313 | Bound-variable hypothesis ... |
| nfrexw 3314 | Bound-variable hypothesis ... |
| r19.12 3315 | Restricted quantifier vers... |
| reean 3316 | Rearrange restricted exist... |
| cbvralsvw 3317 | Change bound variable by u... |
| cbvrexsvw 3318 | Change bound variable by u... |
| rexeq 3319 | Equality theorem for restr... |
| raleq 3320 | Equality theorem for restr... |
| raleqi 3321 | Equality inference for res... |
| rexeqi 3322 | Equality inference for res... |
| raleqdv 3323 | Equality deduction for res... |
| rexeqdv 3324 | Equality deduction for res... |
| raleqtrdv 3325 | Substitution of equal clas... |
| rexeqtrdv 3326 | Substitution of equal clas... |
| raleqtrrdv 3327 | Substitution of equal clas... |
| rexeqtrrdv 3328 | Substitution of equal clas... |
| raleqbidva 3329 | Equality deduction for res... |
| rexeqbidva 3330 | Equality deduction for res... |
| raleqbidvv 3331 | Version of ~ raleqbidv wit... |
| rexeqbidvv 3332 | Version of ~ rexeqbidv wit... |
| raleqbi1dv 3333 | Equality deduction for res... |
| rexeqbi1dv 3334 | Equality deduction for res... |
| raleleq 3335 | All elements of a class ar... |
| raleqbii 3336 | Equality deduction for res... |
| rexeqbii 3337 | Equality deduction for res... |
| raleqbidv 3338 | Equality deduction for res... |
| rexeqbidv 3339 | Equality deduction for res... |
| cbvraldva2 3340 | Rule used to change the bo... |
| cbvrexdva2 3341 | Rule used to change the bo... |
| sbralie 3342 | Implicit to explicit subst... |
| sbralieALT 3343 | Alternative shorter proof ... |
| sbralieOLD 3344 | Obsolete version of ~ sbra... |
| raleqf 3345 | Equality theorem for restr... |
| rexeqf 3346 | Equality theorem for restr... |
| raleqbid 3347 | Equality deduction for res... |
| rexeqbid 3348 | Equality deduction for res... |
| cbvralf 3349 | Rule used to change bound ... |
| cbvrexf 3350 | Rule used to change bound ... |
| cbvral 3351 | Rule used to change bound ... |
| cbvrex 3352 | Rule used to change bound ... |
| cbvralv 3353 | Change the bound variable ... |
| cbvrexv 3354 | Change the bound variable ... |
| cbvralsv 3355 | Change bound variable by u... |
| cbvrexsv 3356 | Change bound variable by u... |
| cbvral2v 3357 | Change bound variables of ... |
| cbvrex2v 3358 | Change bound variables of ... |
| cbvral3v 3359 | Change bound variables of ... |
| rgen2a 3360 | Generalization rule for re... |
| nfrald 3361 | Deduction version of ~ nfr... |
| nfrexd 3362 | Deduction version of ~ nfr... |
| nfral 3363 | Bound-variable hypothesis ... |
| nfrex 3364 | Bound-variable hypothesis ... |
| nfra2 3365 | Similar to Lemma 24 of [Mo... |
| ralcom2 3366 | Commutation of restricted ... |
| reu5 3371 | Restricted uniqueness in t... |
| reurmo 3372 | Restricted existential uni... |
| reurex 3373 | Restricted unique existenc... |
| mormo 3374 | Unrestricted "at most one"... |
| rmobiia 3375 | Formula-building rule for ... |
| reubiia 3376 | Formula-building rule for ... |
| rmobii 3377 | Formula-building rule for ... |
| reubii 3378 | Formula-building rule for ... |
| rmoanid 3379 | Cancellation law for restr... |
| reuanid 3380 | Cancellation law for restr... |
| 2reu2rex 3381 | Double restricted existent... |
| rmobidva 3382 | Formula-building rule for ... |
| reubidva 3383 | Formula-building rule for ... |
| rmobidv 3384 | Formula-building rule for ... |
| reubidv 3385 | Formula-building rule for ... |
| reueubd 3386 | Restricted existential uni... |
| rmo5 3387 | Restricted "at most one" i... |
| nrexrmo 3388 | Nonexistence implies restr... |
| moel 3389 | "At most one" element in a... |
| cbvrmovw 3390 | Change the bound variable ... |
| cbvreuvw 3391 | Change the bound variable ... |
| rmobida 3392 | Formula-building rule for ... |
| reubida 3393 | Formula-building rule for ... |
| cbvrmow 3394 | Change the bound variable ... |
| cbvreuw 3395 | Change the bound variable ... |
| nfrmo1 3396 | The setvar ` x ` is not fr... |
| nfreu1 3397 | The setvar ` x ` is not fr... |
| nfrmow 3398 | Bound-variable hypothesis ... |
| nfreuw 3399 | Bound-variable hypothesis ... |
| rmoeq1 3400 | Equality theorem for restr... |
| reueq1 3401 | Equality theorem for restr... |
| rmoeqd 3402 | Equality deduction for res... |
| reueqd 3403 | Equality deduction for res... |
| reueqdv 3404 | Formula-building rule for ... |
| reueqbidv 3405 | Formula-building rule for ... |
| rmoeq1f 3406 | Equality theorem for restr... |
| reueq1f 3407 | Equality theorem for restr... |
| cbvreu 3408 | Change the bound variable ... |
| cbvrmo 3409 | Change the bound variable ... |
| cbvrmov 3410 | Change the bound variable ... |
| cbvreuv 3411 | Change the bound variable ... |
| nfrmod 3412 | Deduction version of ~ nfr... |
| nfreud 3413 | Deduction version of ~ nfr... |
| nfrmo 3414 | Bound-variable hypothesis ... |
| nfreu 3415 | Bound-variable hypothesis ... |
| rabbidva2 3418 | Equivalent wff's yield equ... |
| rabbia2 3419 | Equivalent wff's yield equ... |
| rabbiia 3420 | Equivalent formulas yield ... |
| rabbii 3421 | Equivalent wff's correspon... |
| rabbidva 3422 | Equivalent wff's yield equ... |
| rabbidv 3423 | Equivalent wff's yield equ... |
| rabbieq 3424 | Equivalent wff's correspon... |
| rabswap 3425 | Swap with a membership rel... |
| cbvrabv 3426 | Rule to change the bound v... |
| rabeqcda 3427 | When ` ps ` is always true... |
| rabeqc 3428 | A restricted class abstrac... |
| rabeqi 3429 | Equality theorem for restr... |
| rabeq 3430 | Equality theorem for restr... |
| rabeqdv 3431 | Equality of restricted cla... |
| rabeqbidva 3432 | Equality of restricted cla... |
| rabeqbidv 3433 | Equality of restricted cla... |
| rabrabi 3434 | Abstract builder restricte... |
| nfrab1 3435 | The abstraction variable i... |
| rabid 3436 | An "identity" law of concr... |
| rabidim1 3437 | Membership in a restricted... |
| reqabi 3438 | Inference from equality of... |
| rabrab 3439 | Abstract builder restricte... |
| rabbida4 3440 | Version of ~ rabbidva2 wit... |
| rabbida 3441 | Equivalent wff's yield equ... |
| rabbid 3442 | Version of ~ rabbidv with ... |
| rabeqd 3443 | Deduction form of ~ rabeq ... |
| rabeqbida 3444 | Version of ~ rabeqbidva wi... |
| rabbi 3445 | Equivalent wff's correspon... |
| rabid2f 3446 | An "identity" law for rest... |
| rabid2im 3447 | One direction of ~ rabid2 ... |
| rabid2 3448 | An "identity" law for rest... |
| rabeqf 3449 | Equality theorem for restr... |
| cbvrabw 3450 | Rule to change the bound v... |
| nfrabw 3451 | A variable not free in a w... |
| nfrab 3452 | A variable not free in a w... |
| cbvrab 3453 | Rule to change the bound v... |
| vjust 3455 | Justification theorem for ... |
| dfv2 3457 | Alternate definition of th... |
| vex 3458 | All setvar variables are s... |
| elv 3459 | If a proposition is implie... |
| elvd 3460 | If a proposition is implie... |
| el2v 3461 | If a proposition is implie... |
| el3v 3462 | If a proposition is implie... |
| el3v3 3463 | If a proposition is implie... |
| eqv 3464 | The universe contains ever... |
| eqvf 3465 | The universe contains ever... |
| abv 3466 | The class of sets verifyin... |
| abvALT 3467 | Alternate proof of ~ abv ,... |
| isset 3468 | Two ways to express that "... |
| cbvexeqsetf 3469 | The expression ` E. x x = ... |
| issetft 3470 | Closed theorem form of ~ i... |
| issetf 3471 | A version of ~ isset that ... |
| isseti 3472 | A way to say " ` A ` is a ... |
| issetri 3473 | A way to say " ` A ` is a ... |
| eqvisset 3474 | A class equal to a variabl... |
| elex 3475 | If a class is a member of ... |
| elexi 3476 | If a class is a member of ... |
| elexd 3477 | If a class is a member of ... |
| elex22 3478 | If two classes each contai... |
| prcnel 3479 | A proper class doesn't bel... |
| ralv 3480 | A universal quantifier res... |
| rexv 3481 | An existential quantifier ... |
| reuv 3482 | A unique existential quant... |
| rmov 3483 | An at-most-one quantifier ... |
| rabab 3484 | A class abstraction restri... |
| rexcom4b 3485 | Specialized existential co... |
| ceqsal1t 3486 | One direction of ~ ceqsalt... |
| ceqsalt 3487 | Closed theorem version of ... |
| ceqsralt 3488 | Restricted quantifier vers... |
| ceqsalg 3489 | A representation of explic... |
| ceqsalgALT 3490 | Alternate proof of ~ ceqsa... |
| ceqsal 3491 | A representation of explic... |
| ceqsalALT 3492 | A representation of explic... |
| ceqsalv 3493 | A representation of explic... |
| ceqsralv 3494 | Restricted quantifier vers... |
| gencl 3495 | Implicit substitution for ... |
| 2gencl 3496 | Implicit substitution for ... |
| 3gencl 3497 | Implicit substitution for ... |
| cgsexg 3498 | Implicit substitution infe... |
| cgsex2g 3499 | Implicit substitution infe... |
| cgsex4g 3500 | An implicit substitution i... |
| ceqsex 3501 | Elimination of an existent... |
| ceqsexv 3502 | Elimination of an existent... |
| ceqsexv2d 3503 | Elimination of an existent... |
| ceqsex2 3504 | Elimination of two existen... |
| ceqsex2v 3505 | Elimination of two existen... |
| ceqsex3v 3506 | Elimination of three exist... |
| ceqsex4v 3507 | Elimination of four existe... |
| ceqsex6v 3508 | Elimination of six existen... |
| ceqsex8v 3509 | Elimination of eight exist... |
| gencbvex 3510 | Change of bound variable u... |
| gencbvex2 3511 | Restatement of ~ gencbvex ... |
| gencbval 3512 | Change of bound variable u... |
| sbhypf 3513 | Introduce an explicit subs... |
| spcimgft 3514 | Closed theorem form of ~ s... |
| spcimgfi1 3515 | A closed version of ~ spci... |
| spcgft 3516 | A closed version of ~ spcg... |
| spcimgf 3517 | Rule of specialization, us... |
| spcimegf 3518 | Existential specialization... |
| vtoclgft 3519 | Closed theorem form of ~ v... |
| vtocleg 3520 | Implicit substitution of a... |
| vtoclg 3521 | Implicit substitution of a... |
| vtocle 3522 | Implicit substitution of a... |
| vtoclbg 3523 | Implicit substitution of a... |
| vtocl 3524 | Implicit substitution of a... |
| vtocldf 3525 | Implicit substitution of a... |
| vtocld 3526 | Implicit substitution of a... |
| vtocl2d 3527 | Implicit substitution of t... |
| vtoclef 3528 | Implicit substitution of a... |
| vtoclf 3529 | Implicit substitution of a... |
| vtocl2 3530 | Implicit substitution of c... |
| vtocl3 3531 | Implicit substitution of c... |
| vtoclb 3532 | Implicit substitution of a... |
| vtoclgf 3533 | Implicit substitution of a... |
| vtoclg1f 3534 | Version of ~ vtoclgf with ... |
| vtocl2gf 3535 | Implicit substitution of a... |
| vtocl3gf 3536 | Implicit substitution of a... |
| vtocl2g 3537 | Implicit substitution of 2... |
| vtocl3g 3538 | Implicit substitution of a... |
| vtoclgaf 3539 | Implicit substitution of a... |
| vtoclga 3540 | Implicit substitution of a... |
| vtocl2ga 3541 | Implicit substitution of 2... |
| vtocl2gaf 3542 | Implicit substitution of 2... |
| vtocl3gaf 3543 | Implicit substitution of 3... |
| vtocl3ga 3544 | Implicit substitution of 3... |
| vtocl4g 3545 | Implicit substitution of 4... |
| vtocl4ga 3546 | Implicit substitution of 4... |
| vtoclegft 3547 | Implicit substitution of a... |
| vtoclri 3548 | Implicit substitution of a... |
| spcgf 3549 | Rule of specialization, us... |
| spcegf 3550 | Existential specialization... |
| spcimdv 3551 | Restricted specialization,... |
| spcdv 3552 | Rule of specialization, us... |
| spcimedv 3553 | Restricted existential spe... |
| spcgv 3554 | Rule of specialization, us... |
| spcegv 3555 | Existential specialization... |
| spcedv 3556 | Existential specialization... |
| spc2egv 3557 | Existential specialization... |
| spc2gv 3558 | Specialization with two qu... |
| spc2ed 3559 | Existential specialization... |
| spc2d 3560 | Specialization with 2 quan... |
| spc3egv 3561 | Existential specialization... |
| spc3gv 3562 | Specialization with three ... |
| spcv 3563 | Rule of specialization, us... |
| spcev 3564 | Existential specialization... |
| spc2ev 3565 | Existential specialization... |
| rspct 3566 | A closed version of ~ rspc... |
| rspcdf 3567 | Restricted specialization,... |
| rspc 3568 | Restricted specialization,... |
| rspce 3569 | Restricted existential spe... |
| rspcimdv 3570 | Restricted specialization,... |
| rspcimedv 3571 | Restricted existential spe... |
| rspcdv 3572 | Restricted specialization,... |
| rspcedv 3573 | Restricted existential spe... |
| rspcebdv 3574 | Restricted existential spe... |
| rspcdv2 3575 | Restricted specialization,... |
| rspcv 3576 | Restricted specialization,... |
| rspccv 3577 | Restricted specialization,... |
| rspcva 3578 | Restricted specialization,... |
| rspccva 3579 | Restricted specialization,... |
| rspcev 3580 | Restricted existential spe... |
| rspcdva 3581 | Restricted specialization,... |
| rspcedvd 3582 | Restricted existential spe... |
| rspcedvdw 3583 | Version of ~ rspcedvd wher... |
| rspceb2dv 3584 | Restricted existential spe... |
| rspcime 3585 | Prove a restricted existen... |
| rspceaimv 3586 | Restricted existential spe... |
| rspcedeqvd 3587 | Restricted existential spe... |
| rspc2 3588 | Restricted specialization ... |
| rspc2gv 3589 | Restricted specialization ... |
| rspc2v 3590 | 2-variable restricted spec... |
| rspc2va 3591 | 2-variable restricted spec... |
| rspc2ev 3592 | 2-variable restricted exis... |
| 2rspcedvdw 3593 | Double application of ~ rs... |
| rspc2dv 3594 | 2-variable restricted spec... |
| rspc3v 3595 | 3-variable restricted spec... |
| rspc3ev 3596 | 3-variable restricted exis... |
| 3rspcedvdw 3597 | Triple application of ~ rs... |
| rspc3dv 3598 | 3-variable restricted spec... |
| rspc4v 3599 | 4-variable restricted spec... |
| rspc6v 3600 | 6-variable restricted spec... |
| rspc8v 3601 | 8-variable restricted spec... |
| rspceeqv 3602 | Restricted existential spe... |
| ralxpxfr2d 3603 | Transfer a universal quant... |
| rexraleqim 3604 | Statement following from e... |
| eqvincg 3605 | A variable introduction la... |
| eqvinc 3606 | A variable introduction la... |
| eqvincf 3607 | A variable introduction la... |
| alexeqg 3608 | Two ways to express substi... |
| ceqex 3609 | Equality implies equivalen... |
| ceqsexg 3610 | A representation of explic... |
| ceqsexgv 3611 | Elimination of an existent... |
| ceqsrexv 3612 | Elimination of a restricte... |
| ceqsrexbv 3613 | Elimination of a restricte... |
| ceqsralbv 3614 | Elimination of a restricte... |
| ceqsrex2v 3615 | Elimination of a restricte... |
| clel2g 3616 | Alternate definition of me... |
| clel2 3617 | Alternate definition of me... |
| clel3g 3618 | Alternate definition of me... |
| clel3 3619 | Alternate definition of me... |
| clel4g 3620 | Alternate definition of me... |
| clel4 3621 | Alternate definition of me... |
| clel5 3622 | Alternate definition of cl... |
| pm13.183 3623 | Compare theorem *13.183 in... |
| rr19.3v 3624 | Restricted quantifier vers... |
| rr19.28v 3625 | Restricted quantifier vers... |
| elab6g 3626 | Membership in a class abst... |
| elabd2 3627 | Membership in a class abst... |
| elabd3 3628 | Membership in a class abst... |
| elabgt 3629 | Membership in a class abst... |
| elabgtOLD 3630 | Obsolete version of ~ elab... |
| elabgf 3631 | Membership in a class abst... |
| elabf 3632 | Membership in a class abst... |
| elabg 3633 | Membership in a class abst... |
| elabgw 3634 | Membership in a class abst... |
| elab2gw 3635 | Membership in a class abst... |
| elab 3636 | Membership in a class abst... |
| elab2g 3637 | Membership in a class abst... |
| elabd 3638 | Explicit demonstration the... |
| elab2 3639 | Membership in a class abst... |
| elab4g 3640 | Membership in a class abst... |
| elab3gf 3641 | Membership in a class abst... |
| elab3g 3642 | Membership in a class abst... |
| elab3 3643 | Membership in a class abst... |
| elrabi 3644 | Implication for the member... |
| elrabf 3645 | Membership in a restricted... |
| rabtru 3646 | Abstract builder using the... |
| elrab3t 3647 | Membership in a restricted... |
| elrab 3648 | Membership in a restricted... |
| elrab3 3649 | Membership in a restricted... |
| elrabd 3650 | Membership in a restricted... |
| elrabrd 3651 | Deduction version of ~ elr... |
| elrab2 3652 | Membership in a restricted... |
| elrab2w 3653 | Membership in a restricted... |
| ralab 3654 | Universal quantification o... |
| ralrab 3655 | Universal quantification o... |
| rexab 3656 | Existential quantification... |
| rexrab 3657 | Existential quantification... |
| ralab2 3658 | Universal quantification o... |
| ralrab2 3659 | Universal quantification o... |
| rexab2 3660 | Existential quantification... |
| rexrab2 3661 | Existential quantification... |
| reurab 3662 | Restricted existential uni... |
| abidnf 3663 | Identity used to create cl... |
| dedhb 3664 | A deduction theorem for co... |
| class2seteq 3665 | Writing a set as a class a... |
| nelrdva 3666 | Deduce negative membership... |
| eqeu 3667 | A condition which implies ... |
| moeq 3668 | There exists at most one s... |
| eueq 3669 | A class is a set if and on... |
| eueqi 3670 | There exists a unique set ... |
| eueq2 3671 | Equality has existential u... |
| eueq3 3672 | Equality has existential u... |
| moeq3 3673 | "At most one" property of ... |
| mosub 3674 | "At most one" remains true... |
| mo2icl 3675 | Theorem for inferring "at ... |
| mob2 3676 | Consequence of "at most on... |
| moi2 3677 | Consequence of "at most on... |
| mob 3678 | Equality implied by "at mo... |
| moi 3679 | Equality implied by "at mo... |
| morex 3680 | Derive membership from uni... |
| euxfr2w 3681 | Transfer existential uniqu... |
| euxfrw 3682 | Transfer existential uniqu... |
| euxfr2 3683 | Transfer existential uniqu... |
| euxfr 3684 | Transfer existential uniqu... |
| euind 3685 | Existential uniqueness via... |
| reu2 3686 | A way to express restricte... |
| reu6 3687 | A way to express restricte... |
| reu3 3688 | A way to express restricte... |
| reu6i 3689 | A condition which implies ... |
| eqreu 3690 | A condition which implies ... |
| rmo4 3691 | Restricted "at most one" u... |
| reu4 3692 | Restricted uniqueness usin... |
| reu7 3693 | Restricted uniqueness usin... |
| reu8 3694 | Restricted uniqueness usin... |
| rmo3f 3695 | Restricted "at most one" u... |
| rmo4f 3696 | Restricted "at most one" u... |
| reu2eqd 3697 | Deduce equality from restr... |
| reueq 3698 | Equality has existential u... |
| rmoeq 3699 | Equality's restricted exis... |
| rmoan 3700 | Restricted "at most one" s... |
| rmoim 3701 | Restricted "at most one" i... |
| rmoimia 3702 | Restricted "at most one" i... |
| rmoimi 3703 | Restricted "at most one" i... |
| rmoimi2 3704 | Restricted "at most one" i... |
| 2reu5a 3705 | Double restricted existent... |
| reuimrmo 3706 | Restricted uniqueness impl... |
| 2reuswap 3707 | A condition allowing swap ... |
| 2reuswap2 3708 | A condition allowing swap ... |
| reuxfrd 3709 | Transfer existential uniqu... |
| reuxfr 3710 | Transfer existential uniqu... |
| reuxfr1d 3711 | Transfer existential uniqu... |
| reuxfr1ds 3712 | Transfer existential uniqu... |
| reuxfr1 3713 | Transfer existential uniqu... |
| reuind 3714 | Existential uniqueness via... |
| 2rmorex 3715 | Double restricted quantifi... |
| 2reu5lem1 3716 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3717 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3718 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3719 | Double restricted existent... |
| 2reurmo 3720 | Double restricted quantifi... |
| 2reurex 3721 | Double restricted quantifi... |
| 2rmoswap 3722 | A condition allowing to sw... |
| 2rexreu 3723 | Double restricted existent... |
| cdeqi 3726 | Deduce conditional equalit... |
| cdeqri 3727 | Property of conditional eq... |
| cdeqth 3728 | Deduce conditional equalit... |
| cdeqnot 3729 | Distribute conditional equ... |
| cdeqal 3730 | Distribute conditional equ... |
| cdeqab 3731 | Distribute conditional equ... |
| cdeqal1 3732 | Distribute conditional equ... |
| cdeqab1 3733 | Distribute conditional equ... |
| cdeqim 3734 | Distribute conditional equ... |
| cdeqcv 3735 | Conditional equality for s... |
| cdeqeq 3736 | Distribute conditional equ... |
| cdeqel 3737 | Distribute conditional equ... |
| nfcdeq 3738 | If we have a conditional e... |
| nfccdeq 3739 | Variation of ~ nfcdeq for ... |
| rru 3740 | Relative version of Russel... |
| ru 3741 | Russell's Paradox. Propos... |
| dfsbcq 3744 | Proper substitution of a c... |
| dfsbcq2 3745 | This theorem, which is sim... |
| sbsbc 3746 | Show that ~ df-sb and ~ df... |
| sbceq1d 3747 | Equality theorem for class... |
| sbceq1dd 3748 | Equality theorem for class... |
| sbceqbid 3749 | Equality theorem for class... |
| sbc8g 3750 | This is the closest we can... |
| sbc2or 3751 | The disjunction of two equ... |
| sbcex 3752 | By our definition of prope... |
| sbceq1a 3753 | Equality theorem for class... |
| sbceq2a 3754 | Equality theorem for class... |
| spsbc 3755 | Specialization: if a formu... |
| spsbcd 3756 | Specialization: if a formu... |
| sbcth 3757 | A substitution into a theo... |
| sbcthdv 3758 | Deduction version of ~ sbc... |
| sbcid 3759 | An identity theorem for su... |
| nfsbc1d 3760 | Deduction version of ~ nfs... |
| nfsbc1 3761 | Bound-variable hypothesis ... |
| nfsbc1v 3762 | Bound-variable hypothesis ... |
| nfsbcdw 3763 | Deduction version of ~ nfs... |
| nfsbcw 3764 | Bound-variable hypothesis ... |
| sbccow 3765 | A composition law for clas... |
| nfsbcd 3766 | Deduction version of ~ nfs... |
| nfsbc 3767 | Bound-variable hypothesis ... |
| sbcco 3768 | A composition law for clas... |
| sbcco2 3769 | A composition law for clas... |
| sbc5 3770 | An equivalence for class s... |
| sbc5ALT 3771 | Alternate proof of ~ sbc5 ... |
| sbc6g 3772 | An equivalence for class s... |
| sbc6 3773 | An equivalence for class s... |
| sbc7 3774 | An equivalence for class s... |
| cbvsbcw 3775 | Change bound variables in ... |
| cbvsbcvw 3776 | Change the bound variable ... |
| cbvsbc 3777 | Change bound variables in ... |
| cbvsbcv 3778 | Change the bound variable ... |
| sbciegft 3779 | Conversion of implicit sub... |
| sbciegf 3780 | Conversion of implicit sub... |
| sbcieg 3781 | Conversion of implicit sub... |
| sbcie2g 3782 | Conversion of implicit sub... |
| sbcie 3783 | Conversion of implicit sub... |
| sbciedf 3784 | Conversion of implicit sub... |
| sbcied 3785 | Conversion of implicit sub... |
| sbcied2 3786 | Conversion of implicit sub... |
| elrabsf 3787 | Membership in a restricted... |
| eqsbc1 3788 | Substitution for the left-... |
| sbcng 3789 | Move negation in and out o... |
| sbcimg 3790 | Distribution of class subs... |
| sbcan 3791 | Distribution of class subs... |
| sbcor 3792 | Distribution of class subs... |
| sbcbig 3793 | Distribution of class subs... |
| sbcn1 3794 | Move negation in and out o... |
| sbcim1 3795 | Distribution of class subs... |
| sbcbid 3796 | Formula-building deduction... |
| sbcbidv 3797 | Formula-building deduction... |
| sbcbii 3798 | Formula-building inference... |
| sbcbi1 3799 | Distribution of class subs... |
| sbcbi2 3800 | Substituting into equivale... |
| sbcal 3801 | Move universal quantifier ... |
| sbcex2 3802 | Move existential quantifie... |
| sbceqal 3803 | Class version of one impli... |
| sbeqalb 3804 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3805 | Substitution for the right... |
| sbc3an 3806 | Distribution of class subs... |
| sbcel1v 3807 | Class substitution into a ... |
| sbcel2gv 3808 | Class substitution into a ... |
| sbcel21v 3809 | Class substitution into a ... |
| sbcimdv 3810 | Substitution analogue of T... |
| sbctt 3811 | Substitution for a variabl... |
| sbcgf 3812 | Substitution for a variabl... |
| sbc19.21g 3813 | Substitution for a variabl... |
| sbcg 3814 | Substitution for a variabl... |
| sbcgfi 3815 | Substitution for a variabl... |
| sbcfal 3816 | Substituting a class in th... |
| sbc2iegf 3817 | Conversion of implicit sub... |
| sbc2ie 3818 | Conversion of implicit sub... |
| sbc2iedv 3819 | Conversion of implicit sub... |
| sbc3ie 3820 | Conversion of implicit sub... |
| sbccomlem 3821 | Lemma for ~ sbccom . (Con... |
| sbccom 3822 | Commutative law for double... |
| sbcralt 3823 | Interchange class substitu... |
| sbcrext 3824 | Interchange class substitu... |
| sbcralg 3825 | Interchange class substitu... |
| sbcrex 3826 | Interchange class substitu... |
| sbcreu 3827 | Interchange class substitu... |
| reu8nf 3828 | Restricted uniqueness usin... |
| sbcabel 3829 | Interchange class substitu... |
| rspsbc 3830 | Restricted quantifier vers... |
| rspsbca 3831 | Restricted quantifier vers... |
| rspesbca 3832 | Existence form of ~ rspsbc... |
| spesbc 3833 | Existence form of ~ spsbc ... |
| spesbcd 3834 | form of ~ spsbc . (Contri... |
| sbcth2 3835 | A substitution into a theo... |
| ra4v 3836 | Version of ~ ra4 with a di... |
| ra4 3837 | Restricted quantifier vers... |
| rmo2 3838 | Alternate definition of re... |
| rmo2i 3839 | Condition implying restric... |
| rmo3 3840 | Restricted "at most one" u... |
| rmob 3841 | Consequence of "at most on... |
| rmoi 3842 | Consequence of "at most on... |
| nrmod 3843 | Deduce the negation of a r... |
| rmob2 3844 | Consequence of "restricted... |
| rmoi2 3845 | Consequence of "restricted... |
| rmoanim 3846 | Introduction of a conjunct... |
| rmoanimALT 3847 | Alternate proof of ~ rmoan... |
| reuan 3848 | Introduction of a conjunct... |
| 2reu1 3849 | Double restricted existent... |
| 2reu2 3850 | Double restricted existent... |
| csb2 3853 | Alternate expression for t... |
| csbeq1 3854 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3855 | Equality deduction for pro... |
| csbeq2 3856 | Substituting into equivale... |
| csbeq2d 3857 | Formula-building deduction... |
| csbeq2dv 3858 | Formula-building deduction... |
| csbeq2i 3859 | Formula-building inference... |
| csbeq12dv 3860 | Formula-building inference... |
| cbvcsbw 3861 | Change bound variables in ... |
| cbvcsb 3862 | Change bound variables in ... |
| cbvcsbv 3863 | Change the bound variable ... |
| csbid 3864 | Analogue of ~ sbid for pro... |
| csbeq1a 3865 | Equality theorem for prope... |
| csbcow 3866 | Composition law for chaine... |
| csbco 3867 | Composition law for chaine... |
| csbtt 3868 | Substitution doesn't affec... |
| csbconstgf 3869 | Substitution doesn't affec... |
| csbconstg 3870 | Substitution doesn't affec... |
| csbgfi 3871 | Substitution for a variabl... |
| csbconstgi 3872 | The proper substitution of... |
| nfcsb1d 3873 | Bound-variable hypothesis ... |
| nfcsb1 3874 | Bound-variable hypothesis ... |
| nfcsb1v 3875 | Bound-variable hypothesis ... |
| nfcsbd 3876 | Deduction version of ~ nfc... |
| nfcsbw 3877 | Bound-variable hypothesis ... |
| nfcsb 3878 | Bound-variable hypothesis ... |
| csbhypf 3879 | Introduce an explicit subs... |
| csbiebt 3880 | Conversion of implicit sub... |
| csbiedf 3881 | Conversion of implicit sub... |
| csbieb 3882 | Bidirectional conversion b... |
| csbiebg 3883 | Bidirectional conversion b... |
| csbiegf 3884 | Conversion of implicit sub... |
| csbief 3885 | Conversion of implicit sub... |
| csbie 3886 | Conversion of implicit sub... |
| csbied 3887 | Conversion of implicit sub... |
| csbied2 3888 | Conversion of implicit sub... |
| csbie2t 3889 | Conversion of implicit sub... |
| csbie2 3890 | Conversion of implicit sub... |
| csbie2g 3891 | Conversion of implicit sub... |
| cbvrabcsfw 3892 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3893 | A more general version of ... |
| cbvrexcsf 3894 | A more general version of ... |
| cbvreucsf 3895 | A more general version of ... |
| cbvrabcsf 3896 | A more general version of ... |
| cbvralv2 3897 | Rule used to change the bo... |
| cbvrexv2 3898 | Rule used to change the bo... |
| rspc2vd 3899 | Deduction version of 2-var... |
| dfss3 3902 | Alternate definition of su... |
| dfss6 3903 | Alternate definition of su... |
| dfssf 3904 | Equivalence for subclass r... |
| dfss3f 3905 | Equivalence for subclass r... |
| nfss 3906 | If ` x ` is not free in ` ... |
| ssel 3907 | Membership relationships f... |
| ssel2 3908 | Membership relationships f... |
| sseli 3909 | Membership implication fro... |
| sselii 3910 | Membership inference from ... |
| sselid 3911 | Membership inference from ... |
| sseld 3912 | Membership deduction from ... |
| sselda 3913 | Membership deduction from ... |
| sseldd 3914 | Membership inference from ... |
| ssneld 3915 | If a class is not in anoth... |
| ssneldd 3916 | If an element is not in a ... |
| ssriv 3917 | Inference based on subclas... |
| ssrd 3918 | Deduction based on subclas... |
| ssrdv 3919 | Deduction based on subclas... |
| sstr2 3920 | Transitivity of subclass r... |
| sstr 3921 | Transitivity of subclass r... |
| sstri 3922 | Subclass transitivity infe... |
| sstrd 3923 | Subclass transitivity dedu... |
| sstrid 3924 | Subclass transitivity dedu... |
| sstrdi 3925 | Subclass transitivity dedu... |
| sylan9ss 3926 | A subclass transitivity de... |
| sylan9ssr 3927 | A subclass transitivity de... |
| eqss 3928 | The subclass relationship ... |
| eqssi 3929 | Infer equality from two su... |
| eqssd 3930 | Equality deduction from tw... |
| sssseq 3931 | If a class is a subclass o... |
| eqrd 3932 | Deduce equality of classes... |
| eqri 3933 | Infer equality of classes ... |
| eqelssd 3934 | Equality deduction from su... |
| ssid 3935 | Any class is a subclass of... |
| ssidd 3936 | Weakening of ~ ssid . (Co... |
| ssv 3937 | Any class is a subclass of... |
| sseq1 3938 | Equality theorem for subcl... |
| sseq2 3939 | Equality theorem for the s... |
| sseq12 3940 | Equality theorem for the s... |
| sseq1i 3941 | An equality inference for ... |
| sseq2i 3942 | An equality inference for ... |
| sseq12i 3943 | An equality inference for ... |
| sseq1d 3944 | An equality deduction for ... |
| sseq2d 3945 | An equality deduction for ... |
| sseq12d 3946 | An equality deduction for ... |
| eqsstrd 3947 | Substitution of equality i... |
| eqsstrrd 3948 | Substitution of equality i... |
| sseqtrd 3949 | Substitution of equality i... |
| sseqtrrd 3950 | Substitution of equality i... |
| eqsstrid 3951 | A chained subclass and equ... |
| eqsstrrid 3952 | A chained subclass and equ... |
| sseqtrdi 3953 | A chained subclass and equ... |
| sseqtrrdi 3954 | A chained subclass and equ... |
| sseqtrid 3955 | Subclass transitivity dedu... |
| sseqtrrid 3956 | Subclass transitivity dedu... |
| eqsstrdi 3957 | A chained subclass and equ... |
| eqsstrrdi 3958 | A chained subclass and equ... |
| eqsstri 3959 | Substitution of equality i... |
| eqsstrri 3960 | Substitution of equality i... |
| sseqtri 3961 | Substitution of equality i... |
| sseqtrri 3962 | Substitution of equality i... |
| 3sstr3i 3963 | Substitution of equality i... |
| 3sstr4i 3964 | Substitution of equality i... |
| 3sstr3g 3965 | Substitution of equality i... |
| 3sstr4g 3966 | Substitution of equality i... |
| 3sstr3d 3967 | Substitution of equality i... |
| 3sstr4d 3968 | Substitution of equality i... |
| eqimssd 3969 | Equality implies inclusion... |
| eqimsscd 3970 | Equality implies inclusion... |
| eqimss 3971 | Equality implies inclusion... |
| eqimss2 3972 | Equality implies inclusion... |
| eqimssi 3973 | Infer subclass relationshi... |
| eqimss2i 3974 | Infer subclass relationshi... |
| nssne1 3975 | Two classes are different ... |
| nssne2 3976 | Two classes are different ... |
| nss 3977 | Negation of subclass relat... |
| nssrex 3978 | Negation of subclass relat... |
| nelss 3979 | Demonstrate by witnesses t... |
| ssrexf 3980 | Restricted existential qua... |
| ssrmof 3981 | "At most one" existential ... |
| ssralv 3982 | Quantification restricted ... |
| ssrexv 3983 | Existential quantification... |
| ss2ralv 3984 | Two quantifications restri... |
| ss2rexv 3985 | Two existential quantifica... |
| ralss 3986 | Restricted universal quant... |
| rexss 3987 | Restricted existential qua... |
| ss2abim 3988 | Class abstractions in a su... |
| ss2ab 3989 | Class abstractions in a su... |
| abss 3990 | Class abstraction in a sub... |
| ssab 3991 | Subclass of a class abstra... |
| ssabral 3992 | The relation for a subclas... |
| ss2abdv 3993 | Deduction of abstraction s... |
| ss2abi 3994 | Inference of abstraction s... |
| abssdv 3995 | Deduction of abstraction s... |
| abssi 3996 | Inference of abstraction s... |
| ss2rab 3997 | Restricted abstraction cla... |
| rabss 3998 | Restricted class abstracti... |
| ssrab 3999 | Subclass of a restricted c... |
| ss2rabd 4000 | Subclass of a restricted c... |
| ssrabdv 4001 | Subclass of a restricted c... |
| rabssdv 4002 | Subclass of a restricted c... |
| ss2rabdv 4003 | Deduction of restricted ab... |
| ss2rabi 4004 | Inference of restricted ab... |
| rabss2 4005 | Subclass law for restricte... |
| rabss2OLD 4006 | Obsolete version of ~ rabs... |
| ssab2 4007 | Subclass relation for the ... |
| ssrab2 4008 | Subclass relation for a re... |
| rabss3d 4009 | Subclass law for restricte... |
| ssrab3 4010 | Subclass relation for a re... |
| rabssrabd 4011 | Subclass of a restricted c... |
| ssrabeq 4012 | If the restricting class o... |
| rabssab 4013 | A restricted class is a su... |
| eqrrabd 4014 | Deduce equality with a res... |
| uniiunlem 4015 | A subset relationship usef... |
| dfpss2 4018 | Alternate definition of pr... |
| dfpss3 4019 | Alternate definition of pr... |
| psseq1 4020 | Equality theorem for prope... |
| psseq2 4021 | Equality theorem for prope... |
| psseq1i 4022 | An equality inference for ... |
| psseq2i 4023 | An equality inference for ... |
| psseq12i 4024 | An equality inference for ... |
| psseq1d 4025 | An equality deduction for ... |
| psseq2d 4026 | An equality deduction for ... |
| psseq12d 4027 | An equality deduction for ... |
| pssss 4028 | A proper subclass is a sub... |
| pssne 4029 | Two classes in a proper su... |
| pssssd 4030 | Deduce subclass from prope... |
| pssned 4031 | Proper subclasses are uneq... |
| sspss 4032 | Subclass in terms of prope... |
| pssirr 4033 | Proper subclass is irrefle... |
| pssirrOLD 4034 | Obsolete version of ~ pssi... |
| pssn2lp 4035 | Proper subclass has no 2-c... |
| sspsstri 4036 | Two ways of stating tricho... |
| ssnpss 4037 | Partial trichotomy law for... |
| psstr 4038 | Transitive law for proper ... |
| sspsstr 4039 | Transitive law for subclas... |
| psssstr 4040 | Transitive law for subclas... |
| psstrd 4041 | Proper subclass inclusion ... |
| sspsstrd 4042 | Transitivity involving sub... |
| psssstrd 4043 | Transitivity involving sub... |
| npss 4044 | A class is not a proper su... |
| ssnelpss 4045 | A subclass missing a membe... |
| ssnelpssd 4046 | Subclass inclusion with on... |
| ssexnelpss 4047 | If there is an element of ... |
| difjust 4049 | Soundness justification th... |
| dfdif2 4051 | Alternate definition of cl... |
| eldif 4052 | Expansion of membership in... |
| eldifd 4053 | If a class is in one class... |
| eldifad 4054 | If a class is in the diffe... |
| eldifbd 4055 | If a class is in the diffe... |
| elneeldif 4056 | The elements of a set diff... |
| velcomp 4057 | Characterization of setvar... |
| dfdif3 4058 | Alternate definition of cl... |
| difeq1 4059 | Equality theorem for class... |
| difeq2 4060 | Equality theorem for class... |
| difeq12 4061 | Equality theorem for class... |
| difeq1i 4062 | Inference adding differenc... |
| difeq2i 4063 | Inference adding differenc... |
| difeq12i 4064 | Equality inference for cla... |
| difeq1d 4065 | Deduction adding differenc... |
| difeq2d 4066 | Deduction adding differenc... |
| difeq12d 4067 | Equality deduction for cla... |
| difeqri 4068 | Inference from membership ... |
| nfdif 4069 | Bound-variable hypothesis ... |
| eldifi 4070 | Implication of membership ... |
| eldifn 4071 | Implication of membership ... |
| elndif 4072 | A set does not belong to a... |
| neldif 4073 | Implication of membership ... |
| difdif 4074 | Double class difference. ... |
| difss 4075 | Subclass relationship for ... |
| difssd 4076 | A difference of two classe... |
| difss2 4077 | If a class is contained in... |
| difss2d 4078 | If a class is contained in... |
| ssdifss 4079 | Preservation of a subclass... |
| ddif 4080 | Double complement under un... |
| ssconb 4081 | Contraposition law for sub... |
| sscon 4082 | Contraposition law for sub... |
| ssdif 4083 | Difference law for subsets... |
| ssdifd 4084 | If ` A ` is contained in `... |
| sscond 4085 | If ` A ` is contained in `... |
| ssdifssd 4086 | If ` A ` is contained in `... |
| ssdif2d 4087 | If ` A ` is contained in `... |
| raldifb 4088 | Restricted universal quant... |
| rexdifi 4089 | Restricted existential qua... |
| complss 4090 | Complementation reverses i... |
| compleq 4091 | Two classes are equal if a... |
| injust 4093 | Soundness justification th... |
| dfin5 4095 | Alternate definition for t... |
| elin 4096 | Expansion of membership in... |
| dfss2 4097 | Alternate definition of th... |
| dfss 4098 | Variant of subclass defini... |
| elini 4099 | Membership in an intersect... |
| elind 4100 | Deduce membership in an in... |
| elinel1 4101 | Membership in an intersect... |
| elinel2 4102 | Membership in an intersect... |
| elin2 4103 | Membership in a class defi... |
| elin1d 4104 | Elementhood in the first s... |
| elin2d 4105 | Elementhood in the first s... |
| elin3 4106 | Membership in a class defi... |
| nel1nelin 4107 | Membership in an intersect... |
| nel2nelin 4108 | Membership in an intersect... |
| incom 4109 | Commutative law for inters... |
| ineqcom 4110 | Two ways of expressing tha... |
| ineqcomi 4111 | Two ways of expressing tha... |
| ineqri 4112 | Inference from membership ... |
| ineq1 4113 | Equality theorem for inter... |
| ineq2 4114 | Equality theorem for inter... |
| ineq12 4115 | Equality theorem for inter... |
| ineq1i 4116 | Equality inference for int... |
| ineq2i 4117 | Equality inference for int... |
| ineq12i 4118 | Equality inference for int... |
| ineq1d 4119 | Equality deduction for int... |
| ineq2d 4120 | Equality deduction for int... |
| ineq12d 4121 | Equality deduction for int... |
| ineqan12d 4122 | Equality deduction for int... |
| sseqin2 4123 | A relationship between sub... |
| nfin 4124 | Bound-variable hypothesis ... |
| rabbi2dva 4125 | Deduction from a wff to a ... |
| inidm 4126 | Idempotent law for interse... |
| inass 4127 | Associative law for inters... |
| in12 4128 | A rearrangement of interse... |
| in32 4129 | A rearrangement of interse... |
| in13 4130 | A rearrangement of interse... |
| in31 4131 | A rearrangement of interse... |
| inrot 4132 | Rotate the intersection of... |
| in4 4133 | Rearrangement of intersect... |
| inindi 4134 | Intersection distributes o... |
| inindir 4135 | Intersection distributes o... |
| inss1 4136 | The intersection of two cl... |
| inss2 4137 | The intersection of two cl... |
| ssin 4138 | Subclass of intersection. ... |
| ssini 4139 | An inference showing that ... |
| ssind 4140 | A deduction showing that a... |
| ssrin 4141 | Add right intersection to ... |
| sslin 4142 | Add left intersection to s... |
| ssrind 4143 | Add right intersection to ... |
| ss2in 4144 | Intersection of subclasses... |
| ssinss1 4145 | Intersection preserves sub... |
| ssinss1OLD 4146 | Obsolete version of ~ ssin... |
| ssinss1d 4147 | Intersection preserves sub... |
| inss 4148 | Inclusion of an intersecti... |
| ralin 4149 | Restricted universal quant... |
| rexin 4150 | Restricted existential qua... |
| dfss7 4151 | Alternate definition of su... |
| unjust 4153 | Soundness justification th... |
| elun 4155 | Expansion of membership in... |
| elunnel1 4156 | A member of a union that i... |
| elunnel2 4157 | A member of a union that i... |
| uneqri 4158 | Inference from membership ... |
| unidm 4159 | Idempotent law for union o... |
| uncom 4160 | Commutative law for union ... |
| equncom 4161 | If a class equals the unio... |
| equncomi 4162 | Inference form of ~ equnco... |
| uneq1 4163 | Equality theorem for the u... |
| uneq2 4164 | Equality theorem for the u... |
| uneq12 4165 | Equality theorem for the u... |
| uneq1i 4166 | Inference adding union to ... |
| uneq2i 4167 | Inference adding union to ... |
| uneq12i 4168 | Equality inference for the... |
| uneq1d 4169 | Deduction adding union to ... |
| uneq2d 4170 | Deduction adding union to ... |
| uneq12d 4171 | Equality deduction for the... |
| nfun 4172 | Bound-variable hypothesis ... |
| unass 4173 | Associative law for union ... |
| un12 4174 | A rearrangement of union. ... |
| un23 4175 | A rearrangement of union. ... |
| un4 4176 | A rearrangement of the uni... |
| unundi 4177 | Union distributes over its... |
| unundir 4178 | Union distributes over its... |
| ssun1 4179 | Subclass relationship for ... |
| ssun2 4180 | Subclass relationship for ... |
| ssun3 4181 | Subclass law for union of ... |
| ssun4 4182 | Subclass law for union of ... |
| elun1 4183 | Membership law for union o... |
| elun2 4184 | Membership law for union o... |
| elunant 4185 | A statement is true for ev... |
| unss1 4186 | Subclass law for union of ... |
| ssequn1 4187 | A relationship between sub... |
| unss2 4188 | Subclass law for union of ... |
| unss12 4189 | Subclass law for union of ... |
| ssequn2 4190 | A relationship between sub... |
| unss 4191 | The union of two subclasse... |
| unssi 4192 | An inference showing the u... |
| unssd 4193 | A deduction showing the un... |
| unssad 4194 | If ` ( A u. B ) ` is conta... |
| unssbd 4195 | If ` ( A u. B ) ` is conta... |
| ssun 4196 | A condition that implies i... |
| rexun 4197 | Restricted existential qua... |
| ralunb 4198 | Restricted quantification ... |
| ralun 4199 | Restricted quantification ... |
| symdifcom 4202 | Symmetric difference is co... |
| symdifeq1 4203 | Equality theorem for symme... |
| symdifeq2 4204 | Equality theorem for symme... |
| nfsymdif 4205 | Hypothesis builder for sym... |
| elsymdif 4206 | Membership in a symmetric ... |
| dfsymdif4 4207 | Alternate definition of th... |
| elsymdifxor 4208 | Membership in a symmetric ... |
| dfsymdif2 4209 | Alternate definition of th... |
| symdifass 4210 | Symmetric difference is as... |
| difsssymdif 4211 | The symmetric difference c... |
| difsymssdifssd 4212 | If the symmetric differenc... |
| unabs 4213 | Absorption law for union. ... |
| inabs 4214 | Absorption law for interse... |
| nssinpss 4215 | Negation of subclass expre... |
| nsspssun 4216 | Negation of subclass expre... |
| dfss4 4217 | Subclass defined in terms ... |
| dfun2 4218 | An alternate definition of... |
| dfin2 4219 | An alternate definition of... |
| difin 4220 | Difference with intersecti... |
| ssdifim 4221 | Implication of a class dif... |
| ssdifsym 4222 | Symmetric class difference... |
| dfss5 4223 | Alternate definition of su... |
| dfun3 4224 | Union defined in terms of ... |
| dfin3 4225 | Intersection defined in te... |
| dfin4 4226 | Alternate definition of th... |
| invdif 4227 | Intersection with universa... |
| indif 4228 | Intersection with class di... |
| indif2 4229 | Bring an intersection in a... |
| indif1 4230 | Bring an intersection in a... |
| indifcom 4231 | Commutation law for inters... |
| indi 4232 | Distributive law for inter... |
| undi 4233 | Distributive law for union... |
| indir 4234 | Distributive law for inter... |
| undir 4235 | Distributive law for union... |
| unineq 4236 | Infer equality from equali... |
| uneqin 4237 | Equality of union and inte... |
| difundi 4238 | Distributive law for class... |
| difundir 4239 | Distributive law for class... |
| difindi 4240 | Distributive law for class... |
| difindir 4241 | Distributive law for class... |
| indifdi 4242 | Distribute intersection ov... |
| indifdir 4243 | Distribute intersection ov... |
| difdif2 4244 | Class difference by a clas... |
| undm 4245 | De Morgan's law for union.... |
| indm 4246 | De Morgan's law for inters... |
| difun1 4247 | A relationship involving d... |
| undif3 4248 | An equality involving clas... |
| difin2 4249 | Represent a class differen... |
| dif32 4250 | Swap second and third argu... |
| difabs 4251 | Absorption-like law for cl... |
| sscon34b 4252 | Relative complementation r... |
| rcompleq 4253 | Two subclasses are equal i... |
| dfsymdif3 4254 | Alternate definition of th... |
| unabw 4255 | Union of two class abstrac... |
| unab 4256 | Union of two class abstrac... |
| inab 4257 | Intersection of two class ... |
| difab 4258 | Difference of two class ab... |
| abanssl 4259 | A class abstraction with a... |
| abanssr 4260 | A class abstraction with a... |
| notabw 4261 | A class abstraction define... |
| notab 4262 | A class abstraction define... |
| unrab 4263 | Union of two restricted cl... |
| inrab 4264 | Intersection of two restri... |
| inrab2 4265 | Intersection with a restri... |
| difrab 4266 | Difference of two restrict... |
| dfrab3 4267 | Alternate definition of re... |
| dfrab2 4268 | Alternate definition of re... |
| rabdif 4269 | Move difference in and out... |
| notrab 4270 | Complementation of restric... |
| dfrab3ss 4271 | Restricted class abstracti... |
| rabun2 4272 | Abstraction restricted to ... |
| reuun2 4273 | Transfer uniqueness to a s... |
| reuss2 4274 | Transfer uniqueness to a s... |
| reuss 4275 | Transfer uniqueness to a s... |
| reuun1 4276 | Transfer uniqueness to a s... |
| reupick 4277 | Restricted uniqueness "pic... |
| reupick3 4278 | Restricted uniqueness "pic... |
| reupick2 4279 | Restricted uniqueness "pic... |
| euelss 4280 | Transfer uniqueness of an ... |
| dfnul4 4283 | Alternate definition of th... |
| dfnul2 4284 | Alternate definition of th... |
| dfnul3 4285 | Alternate definition of th... |
| noel 4286 | The empty set has no eleme... |
| noelOLD 4287 | Obsolete version of ~ noel... |
| nel02 4288 | The empty set has no eleme... |
| n0i 4289 | If a class has elements, t... |
| ne0i 4290 | If a class has elements, t... |
| ne0d 4291 | Deduction form of ~ ne0i .... |
| n0ii 4292 | If a class has elements, t... |
| ne0ii 4293 | If a class has elements, t... |
| vn0 4294 | The universal class is not... |
| vn0OLD 4295 | Obsolete version of ~ vn0 ... |
| vn0ALT 4296 | Alternate proof of ~ vn0 .... |
| eq0f 4297 | A class is equal to the em... |
| neq0f 4298 | A class is not empty if an... |
| n0f 4299 | A class is nonempty if and... |
| eq0 4300 | A class is equal to the em... |
| eq0ALT 4301 | Alternate proof of ~ eq0 .... |
| neq0 4302 | A class is not empty if an... |
| n0 4303 | A class is nonempty if and... |
| n0limd 4304 | Deduction rule for nonempt... |
| nel0 4305 | From the general negation ... |
| reximdva0 4306 | Restricted existence deduc... |
| rspn0 4307 | Specialization for restric... |
| n0rex 4308 | There is an element in a n... |
| ssn0rex 4309 | There is an element in a c... |
| n0moeu 4310 | A case of equivalence of "... |
| rex0 4311 | Vacuous restricted existen... |
| reu0 4312 | Vacuous restricted uniquen... |
| rmo0 4313 | Vacuous restricted at-most... |
| 0el 4314 | Membership of the empty se... |
| n0el 4315 | Negated membership of the ... |
| eqeuel 4316 | A condition which implies ... |
| ssdif0 4317 | Subclass expressed in term... |
| difn0 4318 | If the difference of two s... |
| pssdifn0 4319 | A proper subclass has a no... |
| pssdif 4320 | A proper subclass has a no... |
| ndisj 4321 | Express that an intersecti... |
| inn0f 4322 | A nonempty intersection. ... |
| inn0 4323 | A nonempty intersection. ... |
| difin0ss 4324 | Difference, intersection, ... |
| inssdif0 4325 | Intersection, subclass, an... |
| inssdif0OLD 4326 | Obsolete version of ~ inss... |
| inindif 4327 | The intersection and class... |
| difid 4328 | The difference between a c... |
| difidALT 4329 | Alternate proof of ~ difid... |
| dif0 4330 | The difference between a c... |
| ab0w 4331 | The class of sets verifyin... |
| ab0 4332 | The class of sets verifyin... |
| ab0ALT 4333 | Alternate proof of ~ ab0 ,... |
| dfnf5 4334 | Characterization of nonfre... |
| ab0orv 4335 | The class abstraction defi... |
| ab0orvALT 4336 | Alternate proof of ~ ab0or... |
| abn0 4337 | Nonempty class abstraction... |
| rab0 4338 | Any restricted class abstr... |
| rab0OLD 4339 | Obsolete version of ~ rab0... |
| rabeq0w 4340 | Condition for a restricted... |
| rabeq0 4341 | Condition for a restricted... |
| rabn0 4342 | Nonempty restricted class ... |
| rabxm 4343 | Law of excluded middle, in... |
| rabnc 4344 | Law of noncontradiction, i... |
| elneldisj 4345 | The set of elements ` s ` ... |
| elnelun 4346 | The union of the set of el... |
| un0 4347 | The union of a class with ... |
| in0 4348 | The intersection of a clas... |
| 0un 4349 | The union of the empty set... |
| 0in 4350 | The intersection of the em... |
| inv1 4351 | The intersection of a clas... |
| unv 4352 | The union of a class with ... |
| 0ss 4353 | The empty set is a subset ... |
| ss0b 4354 | Any subset of the empty se... |
| ss0 4355 | Any subset of the empty se... |
| sseq0b 4356 | The only subclass of the e... |
| sseq0 4357 | A subclass of an empty cla... |
| ssn0 4358 | A class with a nonempty su... |
| 0dif 4359 | The difference between the... |
| un00 4360 | Two classes are both empty... |
| vss 4361 | Only the universal class h... |
| vvin 4362 | Two classes are both the u... |
| 0pss 4363 | The empty set is a proper ... |
| npss0 4364 | No set is a proper subset ... |
| pssv 4365 | Any non-universal class is... |
| nvpss 4366 | No class strictly includes... |
| abf 4367 | A class abstraction determ... |
| eq0rdv 4368 | Deduction for equality to ... |
| eq0rdvALT 4369 | Alternate proof of ~ eq0rd... |
| csbprc 4370 | The proper substitution of... |
| csb0 4371 | The proper substitution of... |
| sbcel12 4372 | Distribute proper substitu... |
| sbceqg 4373 | Distribute proper substitu... |
| sbceqi 4374 | Distribution of class subs... |
| sbcnel12g 4375 | Distribute proper substitu... |
| sbcne12 4376 | Distribute proper substitu... |
| sbcel1g 4377 | Move proper substitution i... |
| sbceq1g 4378 | Move proper substitution t... |
| sbcel2 4379 | Move proper substitution i... |
| sbceq2g 4380 | Move proper substitution t... |
| csbcom 4381 | Commutative law for double... |
| sbcnestgfw 4382 | Nest the composition of tw... |
| csbnestgfw 4383 | Nest the composition of tw... |
| sbcnestgw 4384 | Nest the composition of tw... |
| csbnestgw 4385 | Nest the composition of tw... |
| sbcco3gw 4386 | Composition of two substit... |
| sbcnestgf 4387 | Nest the composition of tw... |
| csbnestgf 4388 | Nest the composition of tw... |
| sbcnestg 4389 | Nest the composition of tw... |
| csbnestg 4390 | Nest the composition of tw... |
| sbcco3g 4391 | Composition of two substit... |
| csbco3g 4392 | Composition of two class s... |
| csbnest1g 4393 | Nest the composition of tw... |
| csbidm 4394 | Idempotent law for class s... |
| csbvarg 4395 | The proper substitution of... |
| csbvargi 4396 | The proper substitution of... |
| sbccsb 4397 | Substitution into a wff ex... |
| sbccsb2 4398 | Substitution into a wff ex... |
| rspcsbela 4399 | Special case related to ~ ... |
| sbnfc2 4400 | Two ways of expressing " `... |
| csbab 4401 | Move substitution into a c... |
| csbun 4402 | Distribution of class subs... |
| csbin 4403 | Distribute proper substitu... |
| csbie2df 4404 | Conversion of implicit sub... |
| 2nreu 4405 | If there are two different... |
| disj 4406 | Two ways of saying that tw... |
| disjr 4407 | Two ways of saying that tw... |
| disj1 4408 | Two ways of saying that tw... |
| reldisj 4409 | Two ways of saying that tw... |
| disj3 4410 | Two ways of saying that tw... |
| disjne 4411 | Members of disjoint sets a... |
| disjeq0 4412 | Two disjoint sets are equa... |
| disjel 4413 | A set can't belong to both... |
| disj2 4414 | Two ways of saying that tw... |
| disj4 4415 | Two ways of saying that tw... |
| ssdisj 4416 | Intersection with a subcla... |
| disjpss 4417 | A class is a proper subset... |
| undisj1 4418 | The union of disjoint clas... |
| undisj2 4419 | The union of disjoint clas... |
| ssindif0 4420 | Subclass expressed in term... |
| inelcm 4421 | The intersection of classe... |
| minel 4422 | A minimum element of a cla... |
| undif4 4423 | Distribute union over diff... |
| disjssun 4424 | Subset relation for disjoi... |
| vdif0 4425 | Universal class equality i... |
| difrab0eq 4426 | If the difference between ... |
| pssnel 4427 | A proper subclass has a me... |
| disjdifg 4428 | A class does not intersect... |
| disjdif 4429 | A class and its relative c... |
| disjdifr 4430 | A class and its relative c... |
| difin0 4431 | The difference of a class ... |
| unvdif 4432 | The union of a class and i... |
| undif1 4433 | Absorption of difference b... |
| undif2 4434 | Absorption of difference b... |
| srcmpltd 4435 | If a statement is true for... |
| prsrcmpltd 4436 | If a statement is true for... |
| undifabs 4437 | Absorption of difference b... |
| inundif 4438 | The intersection and class... |
| disjdif2 4439 | The difference of a class ... |
| difun2 4440 | Absorption of union by dif... |
| undif 4441 | Union of complementary par... |
| undifr 4442 | Union of complementary par... |
| undif5 4443 | An equality involving clas... |
| ssdifin0 4444 | A subset of a difference d... |
| ssdifeq0 4445 | A class is a subclass of i... |
| ssundif 4446 | A condition equivalent to ... |
| difcom 4447 | Swap the arguments of a cl... |
| pssdifcom1 4448 | Two ways to express overla... |
| pssdifcom2 4449 | Two ways to express non-co... |
| difdifdir 4450 | Distributive law for class... |
| uneqdifeq 4451 | Two ways to say that ` A `... |
| raldifeq 4452 | Equality theorem for restr... |
| rzal 4453 | Vacuous quantification is ... |
| rzalALT 4454 | Alternate proof of ~ rzal ... |
| rexn0 4455 | Restricted existential qua... |
| ralf0 4456 | The quantification of a fa... |
| ral0 4457 | Vacuous universal quantifi... |
| r19.2z 4458 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4459 | A response to the notion t... |
| r19.3rz 4460 | Restricted quantification ... |
| r19.28z 4461 | Restricted quantifier vers... |
| r19.3rzv 4462 | Restricted quantification ... |
| r19.3rzvOLD 4463 | Obsolete version of ~ r19.... |
| r19.9rzv 4464 | Restricted quantification ... |
| r19.28zv 4465 | Restricted quantifier vers... |
| r19.37zv 4466 | Restricted quantifier vers... |
| r19.45zv 4467 | Restricted version of Theo... |
| r19.44zv 4468 | Restricted version of Theo... |
| r19.27z 4469 | Restricted quantifier vers... |
| r19.27zv 4470 | Restricted quantifier vers... |
| r19.36zv 4471 | Restricted quantifier vers... |
| ralnralall 4472 | A contradiction concerning... |
| falseral0 4473 | A false statement can only... |
| falseral0OLD 4474 | Obsolete version of ~ fals... |
| ralidmw 4475 | Idempotent law for restric... |
| ralidm 4476 | Idempotent law for restric... |
| raaan 4477 | Rearrange restricted quant... |
| raaanv 4478 | Rearrange restricted quant... |
| sbss 4479 | Set substitution into the ... |
| sbcssg 4480 | Distribute proper substitu... |
| raaan2 4481 | Rearrange restricted quant... |
| 2reu4lem 4482 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4483 | Definition of double restr... |
| csbdif 4484 | Distribution of class subs... |
| dfif2 4487 | An alternate definition of... |
| dfif6 4488 | An alternate definition of... |
| ifeq1 4489 | Equality theorem for condi... |
| ifeq2 4490 | Equality theorem for condi... |
| iftrue 4491 | Value of the conditional o... |
| iftruei 4492 | Inference associated with ... |
| iftrued 4493 | Value of the conditional o... |
| iffalse 4494 | Value of the conditional o... |
| iffalsei 4495 | Inference associated with ... |
| iffalsed 4496 | Value of the conditional o... |
| ifnefalse 4497 | When values are unequal, b... |
| iftrueb 4498 | When the branches are not ... |
| ifsb 4499 | Distribute a function over... |
| dfif3 4500 | Alternate definition of th... |
| dfif4 4501 | Alternate definition of th... |
| dfif5 4502 | Alternate definition of th... |
| ifssun 4503 | A conditional class is inc... |
| ifeq12 4504 | Equality theorem for condi... |
| ifeq1d 4505 | Equality deduction for con... |
| ifeq2d 4506 | Equality deduction for con... |
| ifeq12d 4507 | Equality deduction for con... |
| ifbi 4508 | Equivalence theorem for co... |
| ifbid 4509 | Equivalence deduction for ... |
| ifbieq1d 4510 | Equivalence/equality deduc... |
| ifbieq2i 4511 | Equivalence/equality infer... |
| ifbieq2d 4512 | Equivalence/equality deduc... |
| ifbieq12i 4513 | Equivalence deduction for ... |
| ifbieq12d 4514 | Equivalence deduction for ... |
| nfifd 4515 | Deduction form of ~ nfif .... |
| nfif 4516 | Bound-variable hypothesis ... |
| ifeq1da 4517 | Conditional equality. (Co... |
| ifeq2da 4518 | Conditional equality. (Co... |
| ifeq12da 4519 | Equivalence deduction for ... |
| ifbieq12d2 4520 | Equivalence deduction for ... |
| ifclda 4521 | Conditional closure. (Con... |
| ifeqda 4522 | Separation of the values o... |
| elimif 4523 | Elimination of a condition... |
| ifbothda 4524 | A wff ` th ` containing a ... |
| ifboth 4525 | A wff ` th ` containing a ... |
| ifid 4526 | Identical true and false a... |
| eqif 4527 | Expansion of an equality w... |
| ifval 4528 | Another expression of the ... |
| elif 4529 | Membership in a conditiona... |
| ifel 4530 | Membership of a conditiona... |
| ifcl 4531 | Membership (closure) of a ... |
| ifcld 4532 | Membership (closure) of a ... |
| ifcli 4533 | Inference associated with ... |
| ifexd 4534 | Existence of the condition... |
| ifexg 4535 | Existence of the condition... |
| ifex 4536 | Existence of the condition... |
| ifeqor 4537 | The possible values of a c... |
| ifnot 4538 | Negating the first argumen... |
| ifan 4539 | Rewrite a conjunction in a... |
| ifor 4540 | Rewrite a disjunction in a... |
| 2if2 4541 | Resolve two nested conditi... |
| ifcomnan 4542 | Commute the conditions in ... |
| csbif 4543 | Distribute proper substitu... |
| dedth 4544 | Weak deduction theorem tha... |
| dedth2h 4545 | Weak deduction theorem eli... |
| dedth3h 4546 | Weak deduction theorem eli... |
| dedth4h 4547 | Weak deduction theorem eli... |
| dedth2v 4548 | Weak deduction theorem for... |
| dedth3v 4549 | Weak deduction theorem for... |
| dedth4v 4550 | Weak deduction theorem for... |
| elimhyp 4551 | Eliminate a hypothesis con... |
| elimhyp2v 4552 | Eliminate a hypothesis con... |
| elimhyp3v 4553 | Eliminate a hypothesis con... |
| elimhyp4v 4554 | Eliminate a hypothesis con... |
| elimel 4555 | Eliminate a membership hyp... |
| elimdhyp 4556 | Version of ~ elimhyp where... |
| keephyp 4557 | Transform a hypothesis ` p... |
| keephyp2v 4558 | Keep a hypothesis containi... |
| keephyp3v 4559 | Keep a hypothesis containi... |
| pwjust 4561 | Soundness justification th... |
| elpwg 4563 | Membership in a power clas... |
| elpw 4564 | Membership in a power clas... |
| velpw 4565 | Setvar variable membership... |
| elpwd 4566 | Membership in a power clas... |
| elpwi 4567 | Subset relation implied by... |
| elpwb 4568 | Characterization of the el... |
| elpwid 4569 | An element of a power clas... |
| elelpwi 4570 | If ` A ` belongs to a part... |
| sspw 4571 | The powerclass preserves i... |
| sspwi 4572 | The powerclass preserves i... |
| sspwd 4573 | The powerclass preserves i... |
| pweq 4574 | Equality theorem for power... |
| pweqALT 4575 | Alternate proof of ~ pweq ... |
| pweqi 4576 | Equality inference for pow... |
| pweqd 4577 | Equality deduction for pow... |
| pwv 4578 | The power class of the uni... |
| pwunss 4579 | The power class of the uni... |
| nfpw 4580 | Bound-variable hypothesis ... |
| pwidg 4581 | A set is an element of its... |
| pwidgOLD 4582 | Obsolete version of ~ pwid... |
| pwidb 4583 | A class is an element of i... |
| pwid 4584 | A set is a member of its p... |
| pwss 4585 | Subclass relationship for ... |
| pwundif 4586 | Break up the power class o... |
| snjust 4587 | Soundness justification th... |
| sneq 4594 | Equality theorem for singl... |
| sneqi 4595 | Equality inference for sin... |
| sneqd 4596 | Equality deduction for sin... |
| dfsn2 4597 | Alternate definition of si... |
| elsng 4598 | There is exactly one eleme... |
| elsn 4599 | There is exactly one eleme... |
| velsn 4600 | There is only one element ... |
| elsni 4601 | There is at most one eleme... |
| elsnd 4602 | There is at most one eleme... |
| rabsneq 4603 | Equality of class abstract... |
| absn 4604 | Condition for a class abst... |
| dfpr2 4605 | Alternate definition of a ... |
| dfsn2ALT 4606 | Alternate definition of si... |
| elprg 4607 | A member of a pair of clas... |
| elpri 4608 | If a class is an element o... |
| elpr 4609 | A member of a pair of clas... |
| elpr2g 4610 | A member of a pair of sets... |
| elpr2 4611 | A member of a pair of sets... |
| elprn1 4612 | A member of an unordered p... |
| elprn2 4613 | A member of an unordered p... |
| nelpr2 4614 | If a class is not an eleme... |
| nelpr1 4615 | If a class is not an eleme... |
| nelpri 4616 | If an element doesn't matc... |
| prneli 4617 | If an element doesn't matc... |
| nelprd 4618 | If an element doesn't matc... |
| eldifpr 4619 | Membership in a set with t... |
| rexdifpr 4620 | Restricted existential qua... |
| snidg 4621 | A set is a member of its s... |
| snidb 4622 | A class is a set iff it is... |
| snid 4623 | A set is a member of its s... |
| vsnid 4624 | A setvar variable is a mem... |
| elsn2g 4625 | There is exactly one eleme... |
| elsn2 4626 | There is exactly one eleme... |
| nelsn 4627 | If a class is not equal to... |
| rabeqsn 4628 | Conditions for a restricte... |
| rabsssn 4629 | Conditions for a restricte... |
| rabeqsnd 4630 | Conditions for a restricte... |
| ralsnsg 4631 | Substitution expressed in ... |
| rexsns 4632 | Restricted existential qua... |
| rexsngf 4633 | Restricted existential qua... |
| ralsngf 4634 | Restricted universal quant... |
| reusngf 4635 | Restricted existential uni... |
| ralsng 4636 | Substitution expressed in ... |
| rexsng 4637 | Restricted existential qua... |
| reusng 4638 | Restricted existential uni... |
| 2ralsng 4639 | Substitution expressed in ... |
| rexreusng 4640 | Restricted existential uni... |
| exsnrex 4641 | There is a set being the e... |
| ralsn 4642 | Convert a universal quanti... |
| rexsn 4643 | Convert an existential qua... |
| elunsn 4644 | Elementhood in a union wit... |
| elpwunsn 4645 | Membership in an extension... |
| eqoreldif 4646 | An element of a set is eit... |
| eltpg 4647 | Members of an unordered tr... |
| eldiftp 4648 | Membership in a set with t... |
| eltpi 4649 | A member of an unordered t... |
| eltp 4650 | A member of an unordered t... |
| el7g 4651 | Members of a set with seve... |
| dftp2 4652 | Alternate definition of un... |
| nfpr 4653 | Bound-variable hypothesis ... |
| ifpr 4654 | Membership of a conditiona... |
| ralprgf 4655 | Convert a restricted unive... |
| rexprgf 4656 | Convert a restricted exist... |
| ralprg 4657 | Convert a restricted unive... |
| rexprg 4658 | Convert a restricted exist... |
| raltpg 4659 | Convert a restricted unive... |
| rextpg 4660 | Convert a restricted exist... |
| ralpr 4661 | Convert a restricted unive... |
| rexpr 4662 | Convert a restricted exist... |
| reuprg0 4663 | Convert a restricted exist... |
| reuprg 4664 | Convert a restricted exist... |
| reurexprg 4665 | Convert a restricted exist... |
| raltp 4666 | Convert a universal quanti... |
| rextp 4667 | Convert an existential qua... |
| nfsn 4668 | Bound-variable hypothesis ... |
| csbsng 4669 | Distribute proper substitu... |
| csbprg 4670 | Distribute proper substitu... |
| elinsn 4671 | If the intersection of two... |
| disjsn 4672 | Intersection with the sing... |
| disjsn2 4673 | Two distinct singletons ar... |
| disjpr2 4674 | Two completely distinct un... |
| disjprsn 4675 | The disjoint intersection ... |
| disjtpsn 4676 | The disjoint intersection ... |
| disjtp2 4677 | Two completely distinct un... |
| snprc 4678 | The singleton of a proper ... |
| snnzb 4679 | A singleton is nonempty if... |
| rmosn 4680 | A restricted at-most-one q... |
| r19.12sn 4681 | Special case of ~ r19.12 w... |
| rabsn 4682 | Condition where a restrict... |
| rabsnifsb 4683 | A restricted class abstrac... |
| rabsnif 4684 | A restricted class abstrac... |
| rabrsn 4685 | A restricted class abstrac... |
| euabsn2 4686 | Another way to express exi... |
| euabsn 4687 | Another way to express exi... |
| reusn 4688 | A way to express restricte... |
| absneu 4689 | Restricted existential uni... |
| rabsneu 4690 | Restricted existential uni... |
| eusn 4691 | Two ways to express " ` A ... |
| rabsnt 4692 | Truth implied by equality ... |
| prcom 4693 | Commutative law for unorde... |
| preq1 4694 | Equality theorem for unord... |
| preq2 4695 | Equality theorem for unord... |
| preq12 4696 | Equality theorem for unord... |
| preq1i 4697 | Equality inference for uno... |
| preq2i 4698 | Equality inference for uno... |
| preq12i 4699 | Equality inference for uno... |
| preq1d 4700 | Equality deduction for uno... |
| preq2d 4701 | Equality deduction for uno... |
| preq12d 4702 | Equality deduction for uno... |
| tpeq1 4703 | Equality theorem for unord... |
| tpeq2 4704 | Equality theorem for unord... |
| tpeq3 4705 | Equality theorem for unord... |
| tpeq1d 4706 | Equality theorem for unord... |
| tpeq2d 4707 | Equality theorem for unord... |
| tpeq3d 4708 | Equality theorem for unord... |
| tpeq123d 4709 | Equality theorem for unord... |
| tprot 4710 | Rotation of the elements o... |
| tpcoma 4711 | Swap 1st and 2nd members o... |
| tpcomb 4712 | Swap 2nd and 3rd members o... |
| tpass 4713 | Split off the first elemen... |
| qdass 4714 | Two ways to write an unord... |
| qdassr 4715 | Two ways to write an unord... |
| tpidm12 4716 | Unordered triple ` { A , A... |
| tpidm13 4717 | Unordered triple ` { A , B... |
| tpidm23 4718 | Unordered triple ` { A , B... |
| tpidm 4719 | Unordered triple ` { A , A... |
| tppreq3 4720 | An unordered triple is an ... |
| prid1g 4721 | An unordered pair contains... |
| prid2g 4722 | An unordered pair contains... |
| prid1 4723 | An unordered pair contains... |
| prid2 4724 | An unordered pair contains... |
| ifpprsnss 4725 | An unordered pair is a sin... |
| prprc1 4726 | A proper class vanishes in... |
| prprc2 4727 | A proper class vanishes in... |
| prprc 4728 | An unordered pair containi... |
| tpid1 4729 | One of the three elements ... |
| tpid1g 4730 | Closed theorem form of ~ t... |
| tpid2 4731 | One of the three elements ... |
| tpid2g 4732 | Closed theorem form of ~ t... |
| tpid3g 4733 | Closed theorem form of ~ t... |
| tpid3 4734 | One of the three elements ... |
| snnzg 4735 | The singleton of a set is ... |
| snn0d 4736 | The singleton of a set is ... |
| snnz 4737 | The singleton of a set is ... |
| prnz 4738 | A pair containing a set is... |
| prnzg 4739 | A pair containing a set is... |
| tpnz 4740 | An unordered triple contai... |
| tpnzd 4741 | An unordered triple contai... |
| raltpd 4742 | Convert a universal quanti... |
| snssb 4743 | Characterization of the in... |
| snssg 4744 | The singleton formed on a ... |
| snss 4745 | The singleton of an elemen... |
| snssi 4746 | The singleton of an elemen... |
| snssd 4747 | The singleton of an elemen... |
| eldifsn 4748 | Membership in a set with a... |
| eldifsnbd 4749 | Membership in a set with a... |
| eldifsnd 4750 | Membership in a set with a... |
| ssdifsn 4751 | Subset of a set with an el... |
| elpwdifsn 4752 | A subset of a set is an el... |
| eldifsni 4753 | Membership in a set with a... |
| eldifsnneq 4754 | An element of a difference... |
| neldifsn 4755 | The class ` A ` is not in ... |
| neldifsnd 4756 | The class ` A ` is not in ... |
| rexdifsn 4757 | Restricted existential qua... |
| raldifsni 4758 | Rearrangement of a propert... |
| raldifsnb 4759 | Restricted universal quant... |
| eldifvsn 4760 | A set is an element of the... |
| difsn 4761 | An element not in a set ca... |
| difprsnss 4762 | Removal of a singleton fro... |
| difprsn1 4763 | Removal of a singleton fro... |
| difprsn2 4764 | Removal of a singleton fro... |
| diftpsn3 4765 | Removal of a singleton fro... |
| difpr 4766 | Removing two elements as p... |
| tpprceq3 4767 | An unordered triple is an ... |
| tppreqb 4768 | An unordered triple is an ... |
| difsnb 4769 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4770 | ` ( B \ { A } ) ` is a pro... |
| difsnid 4771 | If we remove a single elem... |
| eldifeldifsn 4772 | An element of a difference... |
| snsspr1 4773 | A singleton is a subset of... |
| snsspr2 4774 | A singleton is a subset of... |
| snsstp1 4775 | A singleton is a subset of... |
| snsstp2 4776 | A singleton is a subset of... |
| snsstp3 4777 | A singleton is a subset of... |
| prssg 4778 | A pair of elements of a cl... |
| prss 4779 | A pair of elements of a cl... |
| prssi 4780 | A pair of elements of a cl... |
| prssd 4781 | Deduction version of ~ prs... |
| prsspwg 4782 | An unordered pair belongs ... |
| ssprss 4783 | A pair as subset of a pair... |
| ssprsseq 4784 | A proper pair is a subset ... |
| sssn 4785 | The subsets of a singleton... |
| ssunsn2 4786 | The property of being sand... |
| ssunsn 4787 | Possible values for a set ... |
| eqsn 4788 | Two ways to express that a... |
| eqsnd 4789 | Deduce that a set is a sin... |
| issn 4790 | A sufficient condition for... |
| n0snor2el 4791 | A nonempty set is either a... |
| ssunpr 4792 | Possible values for a set ... |
| sspr 4793 | The subsets of a pair. (C... |
| sstp 4794 | The subsets of an unordere... |
| tpss 4795 | An unordered triple of ele... |
| tpssi 4796 | An unordered triple of ele... |
| sneqrg 4797 | Closed form of ~ sneqr . ... |
| sneqr 4798 | If the singletons of two s... |
| snsssn 4799 | If a singleton is a subset... |
| mosneq 4800 | There exists at most one s... |
| sneqbg 4801 | Two singletons of sets are... |
| snsspw 4802 | The singleton of a class i... |
| prsspw 4803 | An unordered pair belongs ... |
| preq1b 4804 | Biconditional equality lem... |
| preq2b 4805 | Biconditional equality lem... |
| preqr1 4806 | Reverse equality lemma for... |
| preqr2 4807 | Reverse equality lemma for... |
| preq12b 4808 | Equality relationship for ... |
| opthpr 4809 | An unordered pair has the ... |
| preqr1g 4810 | Reverse equality lemma for... |
| preq12bg 4811 | Closed form of ~ preq12b .... |
| prneimg 4812 | Two pairs are not equal if... |
| prneimg2 4813 | Two pairs are not equal if... |
| prnebg 4814 | A (proper) pair is not equ... |
| pr1eqbg 4815 | A (proper) pair is equal t... |
| pr1nebg 4816 | A (proper) pair is not equ... |
| preqsnd 4817 | Equivalence for a pair equ... |
| prnesn 4818 | A proper unordered pair is... |
| prneprprc 4819 | A proper unordered pair is... |
| preqsn 4820 | Equivalence for a pair equ... |
| preq12nebg 4821 | Equality relationship for ... |
| prel12g 4822 | Equality of two unordered ... |
| opthprneg 4823 | An unordered pair has the ... |
| elpreqprlem 4824 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4825 | Equality and membership ru... |
| elpreqprb 4826 | A set is an element of an ... |
| elpr2elpr 4827 | For an element ` A ` of an... |
| ralunsn 4828 | Restricted quantification ... |
| 2ralunsn 4829 | Double restricted quantifi... |
| pwsn 4830 | The power set of a singlet... |
| pwpr 4831 | The power set of an unorde... |
| pwtp 4832 | The power set of an unorde... |
| pw0 4833 | Compute the power set of t... |
| pwpw0 4834 | Compute the power set of t... |
| pwpwpw0 4835 | Compute the power set of t... |
| prproe 4836 | For an element of a proper... |
| 3elpr2eq 4837 | If there are three element... |
| dfopif 4840 | Rewrite ~ df-op using ` if... |
| dfopg 4841 | Value of the ordered pair ... |
| dfop 4842 | Value of an ordered pair w... |
| opeq1 4843 | Equality theorem for order... |
| opeq2 4844 | Equality theorem for order... |
| opeq12 4845 | Equality theorem for order... |
| opeq1i 4846 | Equality inference for ord... |
| opeq2i 4847 | Equality inference for ord... |
| opeq12i 4848 | Equality inference for ord... |
| opeq1d 4849 | Equality deduction for ord... |
| opeq2d 4850 | Equality deduction for ord... |
| opeq12d 4851 | Equality deduction for ord... |
| nfop 4852 | Bound-variable hypothesis ... |
| nfopd 4853 | Deduction version of bound... |
| csbopg 4854 | Distribution of class subs... |
| opidg 4855 | The ordered pair ` <. A , ... |
| opid 4856 | The ordered pair ` <. A , ... |
| opprc 4857 | Expansion of an ordered pa... |
| opprc1 4858 | Expansion of an ordered pa... |
| opprc2 4859 | Expansion of an ordered pa... |
| oprcl 4860 | If an ordered pair has an ... |
| oteq1 4863 | Equality theorem for order... |
| oteq2 4864 | Equality theorem for order... |
| oteq3 4865 | Equality theorem for order... |
| oteq1d 4866 | Equality deduction for ord... |
| oteq2d 4867 | Equality deduction for ord... |
| oteq3d 4868 | Equality deduction for ord... |
| oteq123d 4869 | Equality deduction for ord... |
| dfuni2 4872 | Alternate definition of cl... |
| eluni 4873 | Membership in class union.... |
| eluni2 4874 | Membership in class union.... |
| elunii 4875 | Membership in class union.... |
| nfunid 4876 | Deduction version of ~ nfu... |
| nfuni 4877 | Bound-variable hypothesis ... |
| uniss 4878 | Subclass relationship for ... |
| unissi 4879 | Subclass relationship for ... |
| unissd 4880 | Subclass relationship for ... |
| unieq 4881 | Equality theorem for class... |
| unieqi 4882 | Inference of equality of t... |
| unieqd 4883 | Deduction of equality of t... |
| eluniab 4884 | Membership in union of a c... |
| elunirab 4885 | Membership in union of a c... |
| uniprg 4886 | The union of a pair is the... |
| unipr 4887 | The union of a pair is the... |
| unisng 4888 | A set equals the union of ... |
| unisn 4889 | A set equals the union of ... |
| unisnv 4890 | A set equals the union of ... |
| unisn3 4891 | Union of a singleton in th... |
| dfnfc2 4892 | An alternative statement o... |
| uniun 4893 | The class union of the uni... |
| uniin 4894 | The class union of the int... |
| uniinOLD 4895 | Obsolete version of ~ unii... |
| ssuni 4896 | Subclass relationship for ... |
| uni0b 4897 | The union of a set is empt... |
| uni0c 4898 | The union of a set is empt... |
| uni0 4899 | The union of the empty set... |
| uni0OLD 4900 | Obsolete version of ~ uni0... |
| csbuni 4901 | Distribute proper substitu... |
| elssuni 4902 | An element of a class is a... |
| unissel 4903 | Condition turning a subcla... |
| unissb 4904 | Relationship involving mem... |
| uniss2 4905 | A subclass condition on th... |
| unidif 4906 | If the difference ` A \ B ... |
| ssunieq 4907 | Relationship implying unio... |
| unimax 4908 | Any member of a class is t... |
| pwuni 4909 | A class is a subclass of t... |
| dfint2 4912 | Alternate definition of cl... |
| inteq 4913 | Equality law for intersect... |
| inteqi 4914 | Equality inference for cla... |
| inteqd 4915 | Equality deduction for cla... |
| elint 4916 | Membership in class inters... |
| elint2 4917 | Membership in class inters... |
| elintg 4918 | Membership in class inters... |
| elinti 4919 | Membership in class inters... |
| nfint 4920 | Bound-variable hypothesis ... |
| elintabg 4921 | Two ways of saying a set i... |
| elintab 4922 | Membership in the intersec... |
| elintrab 4923 | Membership in the intersec... |
| elintrabg 4924 | Membership in the intersec... |
| int0 4925 | The intersection of the em... |
| intss1 4926 | An element of a class incl... |
| ssint 4927 | Subclass of a class inters... |
| ssintab 4928 | Subclass of the intersecti... |
| ssintub 4929 | Subclass of the least uppe... |
| ssmin 4930 | Subclass of the minimum va... |
| intmin 4931 | Any member of a class is t... |
| intss 4932 | Intersection of subclasses... |
| intssuni 4933 | The intersection of a none... |
| ssintrab 4934 | Subclass of the intersecti... |
| unissint 4935 | If the union of a class is... |
| intssuni2 4936 | Subclass relationship for ... |
| intminss 4937 | Under subset ordering, the... |
| intmin2 4938 | Any set is the smallest of... |
| intmin3 4939 | Under subset ordering, the... |
| intmin4 4940 | Elimination of a conjunct ... |
| intab 4941 | The intersection of a spec... |
| int0el 4942 | The intersection of a clas... |
| intun 4943 | The class intersection of ... |
| intprg 4944 | The intersection of a pair... |
| intpr 4945 | The intersection of a pair... |
| intsng 4946 | Intersection of a singleto... |
| intsn 4947 | The intersection of a sing... |
| uniintsn 4948 | Two ways to express " ` A ... |
| uniintab 4949 | The union and the intersec... |
| intunsn 4950 | Theorem joining a singleto... |
| rint0 4951 | Relative intersection of a... |
| elrint 4952 | Membership in a restricted... |
| elrint2 4953 | Membership in a restricted... |
| eliun 4958 | Membership in indexed unio... |
| eliin 4959 | Membership in indexed inte... |
| eliuni 4960 | Membership in an indexed u... |
| eliund 4961 | Membership in indexed unio... |
| iuncom 4962 | Commutation of indexed uni... |
| iuncom4 4963 | Commutation of union with ... |
| iunconst 4964 | Indexed union of a constan... |
| iinconst 4965 | Indexed intersection of a ... |
| iuneqconst 4966 | Indexed union of identical... |
| iuniin 4967 | Law combining indexed unio... |
| iinssiun 4968 | An indexed intersection is... |
| iunss1 4969 | Subclass theorem for index... |
| iinss1 4970 | Subclass theorem for index... |
| iuneq1 4971 | Equality theorem for index... |
| iineq1 4972 | Equality theorem for index... |
| ss2iun 4973 | Subclass theorem for index... |
| iuneq2 4974 | Equality theorem for index... |
| iineq2 4975 | Equality theorem for index... |
| iuneq2i 4976 | Equality inference for ind... |
| iineq2i 4977 | Equality inference for ind... |
| iineq2d 4978 | Equality deduction for ind... |
| iuneq2dv 4979 | Equality deduction for ind... |
| iineq2dv 4980 | Equality deduction for ind... |
| iuneq12df 4981 | Equality deduction for ind... |
| iuneq1d 4982 | Equality theorem for index... |
| iuneq12d 4983 | Equality deduction for ind... |
| iuneq2d 4984 | Equality deduction for ind... |
| nfiun 4985 | Bound-variable hypothesis ... |
| nfiin 4986 | Bound-variable hypothesis ... |
| nfiung 4987 | Bound-variable hypothesis ... |
| nfiing 4988 | Bound-variable hypothesis ... |
| nfiu1 4989 | Bound-variable hypothesis ... |
| nfii1 4990 | Bound-variable hypothesis ... |
| dfiun2g 4991 | Alternate definition of in... |
| dfiin2g 4992 | Alternate definition of in... |
| dfiun2 4993 | Alternate definition of in... |
| dfiin2 4994 | Alternate definition of in... |
| dfiunv2 4995 | Define double indexed unio... |
| cbviun 4996 | Rule used to change the bo... |
| cbviin 4997 | Change bound variables in ... |
| cbviung 4998 | Rule used to change the bo... |
| cbviing 4999 | Change bound variables in ... |
| cbviunv 5000 | Rule used to change the bo... |
| cbviinv 5001 | Change bound variables in ... |
| cbviunvg 5002 | Rule used to change the bo... |
| cbviinvg 5003 | Change bound variables in ... |
| iunssf 5004 | Subset theorem for an inde... |
| iunssfOLD 5005 | Obsolete version of ~ iuns... |
| iunss 5006 | Subset theorem for an inde... |
| iunssOLD 5007 | Obsolete version of ~ iuns... |
| ssiun 5008 | Subset implication for an ... |
| ssiun2 5009 | Identity law for subset of... |
| ssiun2s 5010 | Subset relationship for an... |
| iunss2 5011 | A subclass condition on th... |
| iunssd 5012 | Subset theorem for an inde... |
| iunab 5013 | The indexed union of a cla... |
| iunrab 5014 | The indexed union of a res... |
| iunxdif2 5015 | Indexed union with a class... |
| ssiinf 5016 | Subset theorem for an inde... |
| ssiin 5017 | Subset theorem for an inde... |
| iinss 5018 | Subset implication for an ... |
| iinss2 5019 | An indexed intersection is... |
| uniiun 5020 | Class union in terms of in... |
| intiin 5021 | Class intersection in term... |
| iunid 5022 | An indexed union of single... |
| iun0 5023 | An indexed union of the em... |
| 0iun 5024 | An empty indexed union is ... |
| 0iin 5025 | An empty indexed intersect... |
| viin 5026 | Indexed intersection with ... |
| iunsn 5027 | Indexed union of a singlet... |
| iunn0 5028 | There is a nonempty class ... |
| iinab 5029 | Indexed intersection of a ... |
| iinrab 5030 | Indexed intersection of a ... |
| iinrab2 5031 | Indexed intersection of a ... |
| iunin2 5032 | Indexed union of intersect... |
| iunin1 5033 | Indexed union of intersect... |
| iinun2 5034 | Indexed intersection of un... |
| iundif2 5035 | Indexed union of class dif... |
| uniin1 5036 | Union of intersection. Ge... |
| uniin2 5037 | Union of intersection. Ge... |
| iindif1 5038 | Indexed intersection of cl... |
| 2iunin 5039 | Rearrange indexed unions o... |
| iindif2 5040 | Indexed intersection of cl... |
| iinin2 5041 | Indexed intersection of in... |
| iinin1 5042 | Indexed intersection of in... |
| iinvdif 5043 | The indexed intersection o... |
| elriin 5044 | Elementhood in a relative ... |
| riin0 5045 | Relative intersection of a... |
| riinn0 5046 | Relative intersection of a... |
| riinrab 5047 | Relative intersection of a... |
| symdif0 5048 | Symmetric difference with ... |
| symdifv 5049 | The symmetric difference w... |
| symdifid 5050 | The symmetric difference o... |
| iinxsng 5051 | A singleton index picks ou... |
| iinxprg 5052 | Indexed intersection with ... |
| iunxsng 5053 | A singleton index picks ou... |
| iunxsn 5054 | A singleton index picks ou... |
| iunxsngf 5055 | A singleton index picks ou... |
| iunun 5056 | Separate a union in an ind... |
| iunxun 5057 | Separate a union in the in... |
| iunxdif3 5058 | An indexed union where som... |
| iunxprg 5059 | A pair index picks out two... |
| iunxiun 5060 | Separate an indexed union ... |
| iinuni 5061 | A relationship involving u... |
| iununi 5062 | A relationship involving u... |
| sspwuni 5063 | Subclass relationship for ... |
| pwssb 5064 | Two ways to express a coll... |
| elpwpw 5065 | Characterization of the el... |
| pwpwab 5066 | The double power class wri... |
| pwpwssunieq 5067 | The class of sets whose un... |
| elpwuni 5068 | Relationship for power cla... |
| iinpw 5069 | The power class of an inte... |
| iunpwss 5070 | Inclusion of an indexed un... |
| intss2 5071 | A nonempty intersection of... |
| rintn0 5072 | Relative intersection of a... |
| dfdisj2 5075 | Alternate definition for d... |
| disjss2 5076 | If each element of a colle... |
| disjeq2 5077 | Equality theorem for disjo... |
| disjeq2dv 5078 | Equality deduction for dis... |
| disjss1 5079 | A subset of a disjoint col... |
| disjeq1 5080 | Equality theorem for disjo... |
| disjeq1d 5081 | Equality theorem for disjo... |
| disjeq12d 5082 | Equality theorem for disjo... |
| cbvdisj 5083 | Change bound variables in ... |
| cbvdisjv 5084 | Change bound variables in ... |
| nfdisjw 5085 | Bound-variable hypothesis ... |
| nfdisj 5086 | Bound-variable hypothesis ... |
| nfdisj1 5087 | Bound-variable hypothesis ... |
| disjor 5088 | Two ways to say that a col... |
| disjors 5089 | Two ways to say that a col... |
| disji2 5090 | Property of a disjoint col... |
| disji 5091 | Property of a disjoint col... |
| invdisj 5092 | If there is a function ` C... |
| invdisjrab 5093 | The restricted class abstr... |
| disjiun 5094 | A disjoint collection yiel... |
| disjord 5095 | Conditions for a collectio... |
| disjiunb 5096 | Two ways to say that a col... |
| disjiund 5097 | Conditions for a collectio... |
| sndisj 5098 | Any collection of singleto... |
| 0disj 5099 | Any collection of empty se... |
| disjxsn 5100 | A singleton collection is ... |
| disjx0 5101 | An empty collection is dis... |
| disjprg 5102 | A pair collection is disjo... |
| disjxiun 5103 | An indexed union of a disj... |
| disjxun 5104 | The union of two disjoint ... |
| disjss3 5105 | Expand a disjoint collecti... |
| breq 5108 | Equality theorem for binar... |
| breq1 5109 | Equality theorem for a bin... |
| breq2 5110 | Equality theorem for a bin... |
| breq12 5111 | Equality theorem for a bin... |
| breqi 5112 | Equality inference for bin... |
| breq1i 5113 | Equality inference for a b... |
| breq2i 5114 | Equality inference for a b... |
| breq12i 5115 | Equality inference for a b... |
| breq1d 5116 | Equality deduction for a b... |
| breqd 5117 | Equality deduction for a b... |
| breq2d 5118 | Equality deduction for a b... |
| breq12d 5119 | Equality deduction for a b... |
| breq123d 5120 | Equality deduction for a b... |
| breqdi 5121 | Equality deduction for a b... |
| breqan12d 5122 | Equality deduction for a b... |
| breqan12rd 5123 | Equality deduction for a b... |
| breq1dd 5124 | Equality deduction for a b... |
| breq2dd 5125 | Equality deduction for a b... |
| eqnbrtrd 5126 | Substitution of equal clas... |
| nbrne1 5127 | Two classes are different ... |
| nbrne2 5128 | Two classes are different ... |
| eqbrtri 5129 | Substitution of equal clas... |
| eqbrtrd 5130 | Substitution of equal clas... |
| eqbrtrri 5131 | Substitution of equal clas... |
| eqbrtrrd 5132 | Substitution of equal clas... |
| breqtri 5133 | Substitution of equal clas... |
| breqtrd 5134 | Substitution of equal clas... |
| breqtrri 5135 | Substitution of equal clas... |
| breqtrrd 5136 | Substitution of equal clas... |
| 3brtr3i 5137 | Substitution of equality i... |
| 3brtr4i 5138 | Substitution of equality i... |
| 3brtr3d 5139 | Substitution of equality i... |
| 3brtr4d 5140 | Substitution of equality i... |
| 3brtr3g 5141 | Substitution of equality i... |
| 3brtr4g 5142 | Substitution of equality i... |
| eqbrtrid 5143 | A chained equality inferen... |
| eqbrtrrid 5144 | A chained equality inferen... |
| breqtrid 5145 | A chained equality inferen... |
| breqtrrid 5146 | A chained equality inferen... |
| eqbrtrdi 5147 | A chained equality inferen... |
| eqbrtrrdi 5148 | A chained equality inferen... |
| breqtrdi 5149 | A chained equality inferen... |
| breqtrrdi 5150 | A chained equality inferen... |
| ssbrd 5151 | Deduction from a subclass ... |
| ssbr 5152 | Implication from a subclas... |
| ssbri 5153 | Inference from a subclass ... |
| nfbrd 5154 | Deduction version of bound... |
| nfbr 5155 | Bound-variable hypothesis ... |
| brab1 5156 | Relationship between a bin... |
| br0 5157 | The empty binary relation ... |
| brne0 5158 | If two sets are in a binar... |
| brun 5159 | The union of two binary re... |
| brin 5160 | The intersection of two re... |
| brdif 5161 | The difference of two bina... |
| brsymdif 5162 | Characterization of the sy... |
| sbcbr123 5163 | Move substitution in and o... |
| sbcbr 5164 | Move substitution in and o... |
| sbcbr12g 5165 | Move substitution in and o... |
| sbcbr1g 5166 | Move substitution in and o... |
| sbcbr2g 5167 | Move substitution in and o... |
| brralrspcev 5168 | Restricted existential spe... |
| brimralrspcev 5169 | Restricted existential spe... |
| opabss 5172 | The collection of ordered ... |
| opabbid 5173 | Equivalent wff's yield equ... |
| opabbidv 5174 | Equivalent wff's yield equ... |
| opabbii 5175 | Equivalent wff's yield equ... |
| nfopabd 5176 | Bound-variable hypothesis ... |
| nfopab 5177 | Bound-variable hypothesis ... |
| nfopab1 5178 | The first abstraction vari... |
| nfopab2 5179 | The second abstraction var... |
| cbvopab 5180 | Rule used to change bound ... |
| cbvopabv 5181 | Rule used to change bound ... |
| cbvopab1 5182 | Change first bound variabl... |
| cbvopab1g 5183 | Change first bound variabl... |
| cbvopab2 5184 | Change second bound variab... |
| cbvopab1s 5185 | Change first bound variabl... |
| cbvopab1v 5186 | Rule used to change the fi... |
| cbvopab2v 5187 | Rule used to change the se... |
| unopab 5188 | Union of two ordered pair ... |
| mpteq12da 5191 | An equality inference for ... |
| mpteq12df 5192 | An equality inference for ... |
| mpteq12f 5193 | An equality theorem for th... |
| mpteq12dva 5194 | An equality inference for ... |
| mpteq12dv 5195 | An equality inference for ... |
| mpteq12 5196 | An equality theorem for th... |
| mpteq1 5197 | An equality theorem for th... |
| mpteq1d 5198 | An equality theorem for th... |
| mpteq1i 5199 | An equality theorem for th... |
| mpteq2da 5200 | Slightly more general equa... |
| mpteq2dva 5201 | Slightly more general equa... |
| mpteq2dv 5202 | An equality inference for ... |
| mpteq2ia 5203 | An equality inference for ... |
| mpteq2i 5204 | An equality inference for ... |
| mpteq12i 5205 | An equality inference for ... |
| nfmpt 5206 | Bound-variable hypothesis ... |
| nfmpt1 5207 | Bound-variable hypothesis ... |
| cbvmptf 5208 | Rule to change the bound v... |
| cbvmptfg 5209 | Rule to change the bound v... |
| cbvmpt 5210 | Rule to change the bound v... |
| cbvmptg 5211 | Rule to change the bound v... |
| cbvmptv 5212 | Rule to change the bound v... |
| cbvmptvg 5213 | Rule to change the bound v... |
| mptv 5214 | Function with universal do... |
| dftr2 5217 | An alternate way of defini... |
| dftr2c 5218 | Variant of ~ dftr2 with co... |
| dftr5 5219 | An alternate way of defini... |
| dftr3 5220 | An alternate way of defini... |
| dftr4 5221 | An alternate way of defini... |
| treq 5222 | Equality theorem for the t... |
| trel 5223 | In a transitive class, the... |
| trel3 5224 | In a transitive class, the... |
| trss 5225 | An element of a transitive... |
| trun 5226 | The union of transitive cl... |
| trin 5227 | The intersection of transi... |
| tr0 5228 | The empty set is transitiv... |
| trv 5229 | The universe is transitive... |
| triun 5230 | An indexed union of a clas... |
| truni 5231 | The union of a class of tr... |
| triin 5232 | An indexed intersection of... |
| trint 5233 | The intersection of a clas... |
| trintss 5234 | Any nonempty transitive cl... |
| axrep1 5236 | The version of the Axiom o... |
| axreplem 5237 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5238 | Axiom of Replacement expre... |
| axrep3 5239 | Axiom of Replacement sligh... |
| axrep4v 5240 | Version of ~ axrep4 with a... |
| axrep4 5241 | A more traditional version... |
| axrep5 5242 | Axiom of Replacement (simi... |
| axrep6 5243 | A condensed form of ~ ax-r... |
| replem 5244 | A lemma for variants of th... |
| zfrep6 5245 | A version of the Axiom of ... |
| axrep6g 5246 | ~ axrep6 in class notation... |
| zfrepclf 5247 | An inference based on the ... |
| zfrep3cl 5248 | An inference based on the ... |
| zfrep4 5249 | A version of Replacement u... |
| axsepgfromrep 5250 | A more general version ~ a... |
| axsep 5251 | Axiom scheme of separation... |
| axsepg 5253 | A more general version of ... |
| sepg 5254 | Version of the axiom of se... |
| sepgi 5255 | Inference associated with ... |
| zfausclOLD 5256 | Obsolete version of ~ sepg... |
| sepexlem 5257 | Lemma for ~ sepex . Use ~... |
| sepex 5258 | Convert implication to equ... |
| sepexi 5259 | Convert implication to equ... |
| ax6vsep 5260 | Derive ~ ax6v (a weakened ... |
| axnulALT 5261 | Alternate proof of ~ axnul... |
| axnul 5262 | The Null Set Axiom of ZF s... |
| 0ex 5264 | The Null Set Axiom of ZF s... |
| al0ssb 5265 | The empty set is the uniqu... |
| sseliALT 5266 | Alternate proof of ~ sseli... |
| csbexg 5267 | The existence of proper su... |
| csbex 5268 | The existence of proper su... |
| unisn2 5269 | A version of ~ unisn witho... |
| exnelv 5270 | For any set ` x ` , there ... |
| nalset 5271 | No set contains all sets. ... |
| nalsetOLD 5272 | Obsolete version of ~ nals... |
| vneqv 5273 | The universal class is not... |
| vnex 5274 | The universal class does n... |
| vnexOLD 5275 | Obsolete proof of ~ vnex a... |
| nvel 5276 | The universal class does n... |
| nvelim 5277 | If a class is the universa... |
| vprc 5278 | The universal class is not... |
| vprcOLD 5279 | Obsolete proof of ~ vprc ,... |
| nvelOLD 5280 | Obsolete proof of ~ nvel ,... |
| inex1 5281 | Separation Scheme (Aussond... |
| inex2 5282 | Separation Scheme (Aussond... |
| inex1g 5283 | Closed-form, generalized S... |
| inex2g 5284 | Sufficient condition for a... |
| ssexg 5285 | A subclass of a set is a s... |
| ssex 5286 | A subclass of a set is a s... |
| ssexOLD 5287 | Obsolete version of ~ ssex... |
| ssexi 5288 | A subclass of a set is a s... |
| ssexgOLD 5289 | Obsolete version of ~ ssex... |
| ssexd 5290 | A subclass of a set is a s... |
| abexd 5291 | Conditions for a class abs... |
| abex 5292 | Conditions for a class abs... |
| prcssprc 5293 | The superclass of a proper... |
| sselpwd 5294 | Membership in a power set.... |
| difexg 5295 | Existence of a difference.... |
| difexi 5296 | Existence of a difference,... |
| difexd 5297 | Existence of a difference.... |
| sepab 5298 | Separation Scheme (Aussond... |
| elpw2g 5299 | Membership in a power clas... |
| elpw2 5300 | Membership in a power clas... |
| elpwi2 5301 | Membership in a power clas... |
| rabelpw 5302 | A restricted class abstrac... |
| rabexg 5303 | Separation Scheme in terms... |
| rabex 5304 | Separation Scheme in terms... |
| rabexd 5305 | Separation Scheme in terms... |
| rabex2 5306 | Separation Scheme in terms... |
| rab2ex 5307 | A class abstraction based ... |
| elssabg 5308 | Membership in a class abst... |
| intex 5309 | The intersection of a none... |
| intnex 5310 | If a class intersection is... |
| intexab 5311 | The intersection of a none... |
| intexrab 5312 | The intersection of a none... |
| iinexg 5313 | The existence of a class i... |
| intabs 5314 | Absorption of a redundant ... |
| inuni 5315 | The intersection of a unio... |
| axpweq 5316 | Two equivalent ways to exp... |
| pwnss 5317 | The power set of a set is ... |
| pwne 5318 | No set equals its power se... |
| difelpw 5319 | A difference is an element... |
| class2set 5320 | The class of elements of `... |
| 0elpw 5321 | Every power class contains... |
| pwne0 5322 | A power class is never emp... |
| 0nep0 5323 | The empty set and its powe... |
| 0inp0 5324 | Something cannot be equal ... |
| unidif0 5325 | The removal of the empty s... |
| unidif0OLD 5326 | Obsolete version of ~ unid... |
| eqsnuniex 5327 | If a class is equal to the... |
| iin0 5328 | An indexed intersection of... |
| notsep 5329 | In the Separation Scheme ~... |
| intv 5330 | The intersection of the un... |
| zfpow 5332 | Axiom of Power Sets expres... |
| axpow2 5333 | A variant of the Axiom of ... |
| axpow3 5334 | A variant of the Axiom of ... |
| elALT2 5335 | Alternate proof of ~ el us... |
| dtruALT2 5336 | Alternate proof of ~ dtru ... |
| dtrucor 5337 | Corollary of ~ dtru . Thi... |
| dtrucor2 5338 | The theorem form of the de... |
| dvdemo1 5339 | Demonstration of a theorem... |
| dvdemo2 5340 | Demonstration of a theorem... |
| nfnid 5341 | A setvar variable is not f... |
| nfcvb 5342 | The "distinctor" expressio... |
| vpwex 5343 | Power set axiom: the power... |
| pwexg 5344 | Power set axiom expressed ... |
| pwexd 5345 | Deduction version of the p... |
| pwex 5346 | Power set axiom expressed ... |
| pwel 5347 | Quantitative version of ~ ... |
| abssexg 5348 | Existence of a class of su... |
| snexALT 5349 | Alternate proof of ~ snex ... |
| p0ex 5350 | The power set of the empty... |
| p0exALT 5351 | Alternate proof of ~ p0ex ... |
| pp0ex 5352 | The power set of the power... |
| ord3ex 5353 | The ordinal number 3 is a ... |
| dtruALT 5354 | Alternate proof of ~ dtru ... |
| axc16b 5355 | This theorem shows that Ax... |
| eunex 5356 | Existential uniqueness imp... |
| eusv1 5357 | Two ways to express single... |
| eusvnf 5358 | Even if ` x ` is free in `... |
| eusvnfb 5359 | Two ways to say that ` A (... |
| eusv2i 5360 | Two ways to express single... |
| eusv2nf 5361 | Two ways to express single... |
| eusv2 5362 | Two ways to express single... |
| reusv1 5363 | Two ways to express single... |
| reusv2lem1 5364 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5365 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5366 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5367 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5368 | Lemma for ~ reusv2 . (Con... |
| reusv2 5369 | Two ways to express single... |
| reusv3i 5370 | Two ways of expressing exi... |
| reusv3 5371 | Two ways to express single... |
| eusv4 5372 | Two ways to express single... |
| alxfr 5373 | Transfer universal quantif... |
| ralxfrd 5374 | Transfer universal quantif... |
| rexxfrd 5375 | Transfer existential quant... |
| ralxfr2d 5376 | Transfer universal quantif... |
| rexxfr2d 5377 | Transfer existential quant... |
| ralxfrd2 5378 | Transfer universal quantif... |
| rexxfrd2 5379 | Transfer existence from a ... |
| ralxfr 5380 | Transfer universal quantif... |
| ralxfrALT 5381 | Alternate proof of ~ ralxf... |
| rexxfr 5382 | Transfer existence from a ... |
| rabxfrd 5383 | Membership in a restricted... |
| rabxfr 5384 | Membership in a restricted... |
| reuhypd 5385 | A theorem useful for elimi... |
| reuhyp 5386 | A theorem useful for elimi... |
| zfpair 5387 | The Axiom of Pairing of Ze... |
| axprALT 5388 | Alternate proof of ~ axpr ... |
| axprlem1 5389 | Lemma for ~ axpr . There ... |
| axprlem2 5390 | Lemma for ~ axpr . There ... |
| axprlem3 5391 | Lemma for ~ axpr . Elimin... |
| axprlem4 5392 | Lemma for ~ axpr . If an ... |
| axpr 5393 | Unabbreviated version of t... |
| axprlem1OLD 5394 | Obsolete version of ~ axpr... |
| zfpair2 5396 | Derive the abbreviated ver... |
| vsnex 5397 | A singleton built on a set... |
| axprglem 5398 | Lemma for ~ axprg . (Cont... |
| axprg 5399 | Derive The Axiom of Pairin... |
| prex 5400 | The Axiom of Pairing using... |
| snex 5401 | A singleton is a set. The... |
| snexg 5402 | A singleton built on a set... |
| snexgALT 5403 | Alternate proof of ~ snexg... |
| snexOLD 5404 | Obsolete version of ~ snex... |
| prexOLD 5405 | Obsolete version of ~ prex... |
| exel 5406 | There exist two sets, one ... |
| exexneq 5407 | There exist two different ... |
| exneq 5408 | Given any set (the " ` y `... |
| dtru 5409 | Given any set (the " ` y `... |
| el 5410 | Any set is an element of s... |
| el.OLD 5411 | Obsolete version of ~ el a... |
| sels 5412 | If a class is a set, then ... |
| selsALT 5413 | Alternate proof of ~ sels ... |
| elALT 5414 | Alternate proof of ~ el , ... |
| snelpwg 5415 | A singleton of a set is a ... |
| snelpwi 5416 | If a set is a member of a ... |
| snelpw 5417 | A singleton of a set is a ... |
| prelpw 5418 | An unordered pair of two s... |
| prelpwi 5419 | If two sets are members of... |
| rext 5420 | A theorem similar to exten... |
| sspwb 5421 | The powerclass constructio... |
| unipw 5422 | A class equals the union o... |
| univ 5423 | The union of the universe ... |
| pwtr 5424 | A class is transitive iff ... |
| ssextss 5425 | An extensionality-like pri... |
| ssext 5426 | An extensionality-like pri... |
| nssss 5427 | Negation of subclass relat... |
| pweqb 5428 | Classes are equal if and o... |
| intidg 5429 | The intersection of all se... |
| moabex 5430 | "At most one" existence im... |
| moabexOLD 5431 | Obsolete version of ~ moab... |
| rmorabex 5432 | Restricted "at most one" e... |
| euabex 5433 | The abstraction of a wff w... |
| nnullss 5434 | A nonempty class (even if ... |
| exss 5435 | Restricted existence in a ... |
| opex 5436 | An ordered pair of classes... |
| opexOLD 5437 | Obsolete version of ~ opex... |
| otex 5438 | An ordered triple of class... |
| elopg 5439 | Characterization of the el... |
| elop 5440 | Characterization of the el... |
| opi1 5441 | One of the two elements in... |
| opi2 5442 | One of the two elements of... |
| opeluu 5443 | Each member of an ordered ... |
| op1stb 5444 | Extract the first member o... |
| brv 5445 | Two classes are always in ... |
| opnz 5446 | An ordered pair is nonempt... |
| opnzi 5447 | An ordered pair is nonempt... |
| opth1 5448 | Equality of the first memb... |
| opth 5449 | The ordered pair theorem. ... |
| opthg 5450 | Ordered pair theorem. ` C ... |
| opth1g 5451 | Equality of the first memb... |
| opthg2 5452 | Ordered pair theorem. (Co... |
| opth2 5453 | Ordered pair theorem. (Co... |
| opthneg 5454 | Two ordered pairs are not ... |
| opthne 5455 | Two ordered pairs are not ... |
| otth2 5456 | Ordered triple theorem, wi... |
| otth 5457 | Ordered triple theorem. (... |
| otthg 5458 | Ordered triple theorem, cl... |
| otthne 5459 | Contrapositive of the orde... |
| eqvinop 5460 | A variable introduction la... |
| eqvinot 5461 | A variable introduction la... |
| sbcop1 5462 | The proper substitution of... |
| sbcop 5463 | The proper substitution of... |
| copsexgw 5464 | Version of ~ copsexg with ... |
| copsexgwOLD 5465 | Obsolete version of ~ cops... |
| copsexg 5466 | Substitution of class ` A ... |
| cotsexgw 5467 | Substitution of class ` A ... |
| copsex2t 5468 | Closed theorem form of ~ c... |
| copsex2g 5469 | Implicit substitution infe... |
| copsex2dv 5470 | Implicit substitution dedu... |
| copsex4g 5471 | An implicit substitution i... |
| 0nelop 5472 | A property of ordered pair... |
| opwo0id 5473 | An ordered pair is equal t... |
| opeqex 5474 | Equivalence of existence i... |
| oteqex2 5475 | Equivalence of existence i... |
| oteqex 5476 | Equivalence of existence i... |
| opcom 5477 | An ordered pair commutes i... |
| moop2 5478 | "At most one" property of ... |
| opeqsng 5479 | Equivalence for an ordered... |
| opeqsn 5480 | Equivalence for an ordered... |
| opeqpr 5481 | Equivalence for an ordered... |
| snopeqop 5482 | Equivalence for an ordered... |
| propeqop 5483 | Equivalence for an ordered... |
| propssopi 5484 | If a pair of ordered pairs... |
| snopeqopsnid 5485 | Equivalence for an ordered... |
| mosubopt 5486 | "At most one" remains true... |
| mosubop 5487 | "At most one" remains true... |
| mosubott 5488 | "At most one" remains true... |
| euop2 5489 | Transfer existential uniqu... |
| euotd 5490 | Prove existential uniquene... |
| opthwiener 5491 | Justification theorem for ... |
| uniop 5492 | The union of an ordered pa... |
| uniopel 5493 | Ordered pair membership is... |
| opthhausdorff 5494 | Justification theorem for ... |
| opthhausdorff0 5495 | Justification theorem for ... |
| otsndisj 5496 | The singletons consisting ... |
| otiunsndisj 5497 | The union of singletons co... |
| iunopeqop 5498 | Implication of an ordered ... |
| iunopeqopOLD 5499 | Obsolete version of ~ iuno... |
| brsnop 5500 | Binary relation for an ord... |
| brtp 5501 | A necessary and sufficient... |
| opabidw 5502 | The law of concretion. Sp... |
| opabid 5503 | The law of concretion. Sp... |
| elopabw 5504 | Membership in a class abst... |
| elopab 5505 | Membership in a class abst... |
| rexopabb 5506 | Restricted existential qua... |
| vopelopabsb 5507 | The law of concretion in t... |
| opelopabsb 5508 | The law of concretion in t... |
| brabsb 5509 | The law of concretion in t... |
| opelopabt 5510 | Closed theorem form of ~ o... |
| opelopabga 5511 | The law of concretion. Th... |
| brabga 5512 | The law of concretion for ... |
| opelopab2a 5513 | Ordered pair membership in... |
| opelopaba 5514 | The law of concretion. Th... |
| braba 5515 | The law of concretion for ... |
| brab2d 5516 | Expressing that two sets a... |
| opelopabg 5517 | The law of concretion. Th... |
| brabg 5518 | The law of concretion for ... |
| opelopabgf 5519 | The law of concretion. Th... |
| opelopab2 5520 | Ordered pair membership in... |
| opelopab 5521 | The law of concretion. Th... |
| brab 5522 | The law of concretion for ... |
| opelopabaf 5523 | The law of concretion. Th... |
| opelopabf 5524 | The law of concretion. Th... |
| ssopab2 5525 | Equivalence of ordered pai... |
| ssopab2bw 5526 | Equivalence of ordered pai... |
| eqopab2bw 5527 | Equivalence of ordered pai... |
| ssopab2b 5528 | Equivalence of ordered pai... |
| ssopab2i 5529 | Inference of ordered pair ... |
| ssopab2dv 5530 | Inference of ordered pair ... |
| eqopab2b 5531 | Equivalence of ordered pai... |
| opabn0 5532 | Nonempty ordered pair clas... |
| opab0 5533 | Empty ordered pair class a... |
| csbopab 5534 | Move substitution into a c... |
| csbopabw 5535 | Move substitution into a c... |
| csbmpt12 5536 | Move substitution into a m... |
| csbmpt2 5537 | Move substitution into the... |
| iunopab 5538 | Move indexed union inside ... |
| elopabr 5539 | Membership in an ordered-p... |
| elopabran 5540 | Membership in an ordered-p... |
| rbropapd 5541 | Properties of a pair in an... |
| rbropap 5542 | Properties of a pair in a ... |
| 2rbropap 5543 | Properties of a pair in a ... |
| 0nelopab 5544 | The empty set is not an el... |
| brabv 5545 | If two classes are in a re... |
| pwin 5546 | The power class of the int... |
| pwssun 5547 | The power class of the uni... |
| pwun 5548 | The power class of the uni... |
| dfid4 5551 | The identity function expr... |
| dfid2 5552 | Alternate definition of th... |
| dfid3 5553 | A stronger version of ~ df... |
| epelg 5556 | The membership relation an... |
| epeli 5557 | The membership relation an... |
| epel 5558 | The membership relation an... |
| 0sn0ep 5559 | An example for the members... |
| epn0 5560 | The membership relation is... |
| poss 5565 | Subset theorem for the par... |
| poeq1 5566 | Equality theorem for parti... |
| poeq2 5567 | Equality theorem for parti... |
| poeq12d 5568 | Equality deduction for par... |
| nfpo 5569 | Bound-variable hypothesis ... |
| nfso 5570 | Bound-variable hypothesis ... |
| pocl 5571 | Characteristic properties ... |
| ispod 5572 | Sufficient conditions for ... |
| swopolem 5573 | Perform the substitutions ... |
| swopo 5574 | A strict weak order is a p... |
| poirr 5575 | A partial order is irrefle... |
| potr 5576 | A partial order is a trans... |
| po2nr 5577 | A partial order has no 2-c... |
| po3nr 5578 | A partial order has no 3-c... |
| po2ne 5579 | Two sets related by a part... |
| po0 5580 | Any relation is a partial ... |
| pofun 5581 | The inverse image of a par... |
| sopo 5582 | A strict linear order is a... |
| soss 5583 | Subset theorem for the str... |
| soeq1 5584 | Equality theorem for the s... |
| soeq2 5585 | Equality theorem for the s... |
| soeq12d 5586 | Equality deduction for tot... |
| sonr 5587 | A strict order relation is... |
| sotr 5588 | A strict order relation is... |
| sotrd 5589 | Transitivity law for stric... |
| solin 5590 | A strict order relation is... |
| so2nr 5591 | A strict order relation ha... |
| so3nr 5592 | A strict order relation ha... |
| sotric 5593 | A strict order relation sa... |
| sotrieq 5594 | Trichotomy law for strict ... |
| sotrieq2 5595 | Trichotomy law for strict ... |
| soasym 5596 | Asymmetry law for strict o... |
| sotr2 5597 | A transitivity relation. ... |
| issod 5598 | An irreflexive, transitive... |
| issoi 5599 | An irreflexive, transitive... |
| isso2i 5600 | Deduce strict ordering fro... |
| so0 5601 | Any relation is a strict o... |
| somo 5602 | A totally ordered set has ... |
| sotrine 5603 | Trichotomy law for strict ... |
| sotr3 5604 | Transitivity law for stric... |
| dffr6 5611 | Alternate definition of ~ ... |
| frd 5612 | A nonempty subset of an ` ... |
| fri 5613 | A nonempty subset of an ` ... |
| seex 5614 | The ` R ` -preimage of an ... |
| exse 5615 | Any relation on a set is s... |
| dffr2 5616 | Alternate definition of we... |
| dffr2ALT 5617 | Alternate proof of ~ dffr2... |
| frc 5618 | Property of well-founded r... |
| frss 5619 | Subset theorem for the wel... |
| sess1 5620 | Subset theorem for the set... |
| sess2 5621 | Subset theorem for the set... |
| freq1 5622 | Equality theorem for the w... |
| freq2 5623 | Equality theorem for the w... |
| freq12d 5624 | Equality deduction for wel... |
| seeq1 5625 | Equality theorem for the s... |
| seeq2 5626 | Equality theorem for the s... |
| seeq12d 5627 | Equality deduction for the... |
| nffr 5628 | Bound-variable hypothesis ... |
| nfse 5629 | Bound-variable hypothesis ... |
| nfwe 5630 | Bound-variable hypothesis ... |
| frirr 5631 | A well-founded relation is... |
| fr2nr 5632 | A well-founded relation ha... |
| fr0 5633 | Any relation is well-found... |
| frminex 5634 | If an element of a well-fo... |
| efrirr 5635 | A well-founded class does ... |
| efrn2lp 5636 | A well-founded class conta... |
| epse 5637 | The membership relation is... |
| tz7.2 5638 | Similar to Theorem 7.2 of ... |
| dfepfr 5639 | An alternate way of saying... |
| epfrc 5640 | A subset of a well-founded... |
| wess 5641 | Subset theorem for the wel... |
| weeq1 5642 | Equality theorem for the w... |
| weeq2 5643 | Equality theorem for the w... |
| weeq12d 5644 | Equality deduction for wel... |
| wefr 5645 | A well-ordering is well-fo... |
| weso 5646 | A well-ordering is a stric... |
| wecmpep 5647 | The elements of a class we... |
| wetrep 5648 | On a class well-ordered by... |
| wefrc 5649 | A nonempty subclass of a c... |
| we0 5650 | Any relation is a well-ord... |
| wereu 5651 | A nonempty subset of an ` ... |
| wereu2 5652 | A nonempty subclass of an ... |
| xpeq1 5669 | Equality theorem for Carte... |
| xpss12 5670 | Subset theorem for Cartesi... |
| xpss 5671 | A Cartesian product is inc... |
| inxpssres 5672 | Intersection with a Cartes... |
| relxp 5673 | A Cartesian product is a r... |
| xpss1 5674 | Subset relation for Cartes... |
| xpss2 5675 | Subset relation for Cartes... |
| xpeq2 5676 | Equality theorem for Carte... |
| elxpi 5677 | Membership in a Cartesian ... |
| elxp 5678 | Membership in a Cartesian ... |
| elxp2 5679 | Membership in a Cartesian ... |
| xpeq12 5680 | Equality theorem for Carte... |
| xpeq1i 5681 | Equality inference for Car... |
| xpeq2i 5682 | Equality inference for Car... |
| xpeq12i 5683 | Equality inference for Car... |
| xpeq1d 5684 | Equality deduction for Car... |
| xpeq2d 5685 | Equality deduction for Car... |
| xpeq12d 5686 | Equality deduction for Car... |
| sqxpeqd 5687 | Equality deduction for a C... |
| nfxp 5688 | Bound-variable hypothesis ... |
| 0nelxp 5689 | The empty set is not a mem... |
| 0nelelxp 5690 | A member of a Cartesian pr... |
| opelxp 5691 | Ordered pair membership in... |
| opelxpi 5692 | Ordered pair membership in... |
| opelxpii 5693 | Ordered pair membership in... |
| opelxpd 5694 | Ordered pair membership in... |
| opelvv 5695 | Ordered pair membership in... |
| opelvvg 5696 | Ordered pair membership in... |
| opelxp1 5697 | The first member of an ord... |
| opelxp2 5698 | The second member of an or... |
| otelxp 5699 | Ordered triple membership ... |
| otelxp1 5700 | The first member of an ord... |
| otel3xp 5701 | An ordered triple is an el... |
| opabssxpd 5702 | An ordered-pair class abst... |
| rabxp 5703 | Class abstraction restrict... |
| brxp 5704 | Binary relation on a Carte... |
| pwvrel 5705 | A set is a binary relation... |
| pwvabrel 5706 | The powerclass of the cart... |
| brrelex12 5707 | Two classes related by a b... |
| brrelex1 5708 | If two classes are related... |
| brrelex2 5709 | If two classes are related... |
| brrelex12i 5710 | Two classes that are relat... |
| brrelex1i 5711 | The first argument of a bi... |
| brrelex2i 5712 | The second argument of a b... |
| nprrel12 5713 | Proper classes are not rel... |
| nprrel 5714 | No proper class is related... |
| 0nelrel0 5715 | A binary relation does not... |
| 0nelrel 5716 | A binary relation does not... |
| fconstmpt 5717 | Representation of a consta... |
| vtoclr 5718 | Variable to class conversi... |
| opthprc 5719 | Justification theorem for ... |
| brel 5720 | Two things in a binary rel... |
| elxp3 5721 | Membership in a Cartesian ... |
| opeliunxp 5722 | Membership in a union of C... |
| opeliun2xp 5723 | Membership of an ordered p... |
| xpundi 5724 | Distributive law for Carte... |
| xpundir 5725 | Distributive law for Carte... |
| xpiundi 5726 | Distributive law for Carte... |
| xpiundir 5727 | Distributive law for Carte... |
| iunxpconst 5728 | Membership in a union of C... |
| xpun 5729 | The Cartesian product of t... |
| elvv 5730 | Membership in universal cl... |
| elvvOLD 5731 | Obsolete version of ~ elvv... |
| elvvv 5732 | Membership in universal cl... |
| elvvuni 5733 | An ordered pair contains i... |
| brinxp2 5734 | Intersection of binary rel... |
| brinxp 5735 | Intersection of binary rel... |
| opelinxp 5736 | Ordered pair element in an... |
| poinxp 5737 | Intersection of partial or... |
| soinxp 5738 | Intersection of total orde... |
| frinxp 5739 | Intersection of well-found... |
| seinxp 5740 | Intersection of set-like r... |
| weinxp 5741 | Intersection of well-order... |
| posn 5742 | Partial ordering of a sing... |
| sosn 5743 | Strict ordering on a singl... |
| frsn 5744 | Founded relation on a sing... |
| wesn 5745 | Well-ordering of a singlet... |
| elopaelxp 5746 | Membership in an ordered-p... |
| bropaex12 5747 | Two classes related by an ... |
| opabssxp 5748 | An abstraction relation is... |
| brab2a 5749 | The law of concretion for ... |
| optocl 5750 | Implicit substitution of c... |
| optoclOLD 5751 | Obsolete version of ~ opto... |
| 2optocl 5752 | Implicit substitution of c... |
| 3optocl 5753 | Implicit substitution of c... |
| opbrop 5754 | Ordered pair membership in... |
| 0xp 5755 | The Cartesian product with... |
| xp0 5756 | The Cartesian product with... |
| csbxp 5757 | Distribute proper substitu... |
| releq 5758 | Equality theorem for the r... |
| releqi 5759 | Equality inference for the... |
| releqd 5760 | Equality deduction for the... |
| nfrel 5761 | Bound-variable hypothesis ... |
| sbcrel 5762 | Distribute proper substitu... |
| relss 5763 | Subclass theorem for relat... |
| ssrel 5764 | A subclass relationship de... |
| eqrel 5765 | Extensionality principle f... |
| ssrel2 5766 | A subclass relationship de... |
| ssrel3 5767 | Subclass relation in anoth... |
| relssi 5768 | Inference from subclass pr... |
| relssdv 5769 | Deduction from subclass pr... |
| eqrelriv 5770 | Inference from extensional... |
| eqrelriiv 5771 | Inference from extensional... |
| eqbrriv 5772 | Inference from extensional... |
| eqrelrdv 5773 | Deduce equality of relatio... |
| eqbrrdv 5774 | Deduction from extensional... |
| eqbrrdiv 5775 | Deduction from extensional... |
| eqrelrdv2 5776 | A version of ~ eqrelrdv . ... |
| ssrelrel 5777 | A subclass relationship de... |
| eqrelrel 5778 | Extensionality principle f... |
| elrel 5779 | A member of a relation is ... |
| elrelb 5780 | A member of a relation exp... |
| rel0 5781 | The empty set is a relatio... |
| nrelv 5782 | The universal class is not... |
| nrelvOLD 5783 | Obsolete version of ~ nrel... |
| relsng 5784 | A singleton is a relation ... |
| relsnb 5785 | An at-most-singleton is a ... |
| relsnopg 5786 | A singleton of an ordered ... |
| relsn 5787 | A singleton is a relation ... |
| relsnop 5788 | A singleton of an ordered ... |
| copsex2gb 5789 | Implicit substitution infe... |
| copsex2ga 5790 | Implicit substitution infe... |
| elopaba 5791 | Membership in an ordered-p... |
| xpsspw 5792 | A Cartesian product is inc... |
| unixpss 5793 | The double class union of ... |
| relun 5794 | The union of two relations... |
| relin1 5795 | The intersection with a re... |
| relin2 5796 | The intersection with a re... |
| relinxp 5797 | Intersection with a Cartes... |
| reldif 5798 | A difference cutting down ... |
| reliun 5799 | An indexed union is a rela... |
| reliin 5800 | An indexed intersection is... |
| reluni 5801 | The union of a class is a ... |
| relint 5802 | The intersection of a clas... |
| relopabiv 5803 | Alternate proof of ~ relop... |
| relopabvOLD 5804 | Obsolete version of ~ relo... |
| relopabi 5805 | A class of ordered pairs i... |
| relopabiOLD 5806 | Obsolete version of ~ relo... |
| relopabiALT 5807 | Alternate proof of ~ relop... |
| relopab 5808 | A class of ordered pairs i... |
| mptrel 5809 | The maps-to notation alway... |
| reli 5810 | The identity relation is a... |
| rele 5811 | The membership relation is... |
| opabid2 5812 | A relation expressed as an... |
| inopab 5813 | Intersection of two ordere... |
| difopab 5814 | Difference of two ordered-... |
| inxp 5815 | Intersection of two Cartes... |
| xpindi 5816 | Distributive law for Carte... |
| xpindir 5817 | Distributive law for Carte... |
| xpiindi 5818 | Distributive law for Carte... |
| xpriindi 5819 | Distributive law for Carte... |
| eliunxp 5820 | Membership in a union of C... |
| opeliunxp2 5821 | Membership in a union of C... |
| raliunxp 5822 | Write a double restricted ... |
| rexiunxp 5823 | Write a double restricted ... |
| ralxp 5824 | Universal quantification r... |
| rexxp 5825 | Existential quantification... |
| el2xptp 5826 | A member of a nested Carte... |
| exopxfr 5827 | Transfer ordered-pair exis... |
| exopxfr2 5828 | Transfer ordered-pair exis... |
| djussxp 5829 | Disjoint union is a subset... |
| ralxpf 5830 | Version of ~ ralxp with bo... |
| rexxpf 5831 | Version of ~ rexxp with bo... |
| iunxpf 5832 | Indexed union on a Cartesi... |
| opabbi2dv 5833 | Deduce equality of a relat... |
| relop 5834 | A necessary and sufficient... |
| ideqg 5835 | For sets, the identity rel... |
| ideq 5836 | For sets, the identity rel... |
| ididg 5837 | A set is identical to itse... |
| issetid 5838 | Two ways of expressing set... |
| coss1 5839 | Subclass theorem for compo... |
| coss2 5840 | Subclass theorem for compo... |
| coeq1 5841 | Equality theorem for compo... |
| coeq2 5842 | Equality theorem for compo... |
| coeq1i 5843 | Equality inference for com... |
| coeq2i 5844 | Equality inference for com... |
| coeq1d 5845 | Equality deduction for com... |
| coeq2d 5846 | Equality deduction for com... |
| coeq12i 5847 | Equality inference for com... |
| coeq12d 5848 | Equality deduction for com... |
| nfco 5849 | Bound-variable hypothesis ... |
| brcog 5850 | Ordered pair membership in... |
| opelco2g 5851 | Ordered pair membership in... |
| brcogw 5852 | Ordered pair membership in... |
| brcoi 5853 | Ordered pair membership in... |
| eqbrrdva 5854 | Deduction from extensional... |
| brco 5855 | Binary relation on a compo... |
| opelco 5856 | Ordered pair membership in... |
| cnvss 5857 | Subset theorem for convers... |
| cnveq 5858 | Equality theorem for conve... |
| cnveqi 5859 | Equality inference for con... |
| cnveqd 5860 | Equality deduction for con... |
| elcnv 5861 | Membership in a converse r... |
| elcnv2 5862 | Membership in a converse r... |
| nfcnv 5863 | Bound-variable hypothesis ... |
| brcnvg 5864 | The converse of a binary r... |
| opelcnvg 5865 | Ordered-pair membership in... |
| opelcnv 5866 | Ordered-pair membership in... |
| brcnv 5867 | The converse of a binary r... |
| cnv0 5868 | The converse of the empty ... |
| cnv0OLD 5869 | Obsolete version of ~ cnv0... |
| cnvi 5870 | The converse of the identi... |
| csbcnv 5871 | Move class substitution in... |
| csbcnvOLD 5872 | Obsolete version of ~ csbc... |
| csbcnvgALTOLD 5873 | Obsolete version of ~ csbc... |
| cnvco 5874 | Distributive law of conver... |
| cnvuni 5875 | The converse of a class un... |
| dfdm3 5876 | Alternate definition of do... |
| dfrn2 5877 | Alternate definition of ra... |
| dfrn3 5878 | Alternate definition of ra... |
| elrn2g 5879 | Membership in a range. (C... |
| elrng 5880 | Membership in a range. (C... |
| elrn2 5881 | Membership in a range. (C... |
| elrn 5882 | Membership in a range. (C... |
| ssrelrn 5883 | If a relation is a subset ... |
| dfdm4 5884 | Alternate definition of do... |
| dfdmf 5885 | Definition of domain, usin... |
| csbdm 5886 | Distribute proper substitu... |
| eldmg 5887 | Domain membership. Theore... |
| eldm2g 5888 | Domain membership. Theore... |
| eldm 5889 | Membership in a domain. T... |
| eldm2 5890 | Membership in a domain. T... |
| dmss 5891 | Subset theorem for domain.... |
| dmeq 5892 | Equality theorem for domai... |
| dmeqi 5893 | Equality inference for dom... |
| dmeqd 5894 | Equality deduction for dom... |
| opeldmd 5895 | Membership of first of an ... |
| opeldm 5896 | Membership of first of an ... |
| breldm 5897 | Membership of first of a b... |
| breldmg 5898 | Membership of first of a b... |
| dmun 5899 | The domain of a union is t... |
| dmin 5900 | The domain of an intersect... |
| breldmd 5901 | Membership of first of a b... |
| dmiun 5902 | The domain of an indexed u... |
| dmuni 5903 | The domain of a union. Pa... |
| dmopab 5904 | The domain of a class of o... |
| dmopabelb 5905 | A set is an element of the... |
| dmopab2rex 5906 | The domain of an ordered p... |
| dmopabss 5907 | Upper bound for the domain... |
| dmopab3 5908 | The domain of a restricted... |
| dm0 5909 | The domain of the empty se... |
| dmi 5910 | The domain of the identity... |
| dmv 5911 | The domain of the universe... |
| dmep 5912 | The domain of the membersh... |
| dm0rn0 5913 | An empty domain is equival... |
| dm0rn0OLD 5914 | Obsolete version of ~ dm0r... |
| rn0 5915 | The range of the empty set... |
| rnep 5916 | The range of the membershi... |
| reldm0 5917 | A relation is empty iff it... |
| dmxp 5918 | The domain of a Cartesian ... |
| dmxpid 5919 | The domain of a Cartesian ... |
| dmxpin 5920 | The domain of the intersec... |
| xpid11 5921 | The Cartesian square is a ... |
| dmcnvcnv 5922 | The domain of the double c... |
| rncnvcnv 5923 | The range of the double co... |
| elreldm 5924 | The first member of an ord... |
| rneq 5925 | Equality theorem for range... |
| rneqi 5926 | Equality inference for ran... |
| rneqd 5927 | Equality deduction for ran... |
| rnss 5928 | Subset theorem for range. ... |
| rnssi 5929 | Subclass inference for ran... |
| brelrng 5930 | The second argument of a b... |
| brelrn 5931 | The second argument of a b... |
| opelrn 5932 | Membership of second membe... |
| releldm 5933 | The first argument of a bi... |
| relelrn 5934 | The second argument of a b... |
| releldmb 5935 | Membership in a domain. (... |
| relelrnb 5936 | Membership in a range. (C... |
| releldmi 5937 | The first argument of a bi... |
| relelrni 5938 | The second argument of a b... |
| dfrnf 5939 | Definition of range, using... |
| nfdm 5940 | Bound-variable hypothesis ... |
| nfrn 5941 | Bound-variable hypothesis ... |
| dmiin 5942 | Domain of an intersection.... |
| rnopab 5943 | The range of a class of or... |
| rnopabss 5944 | Upper bound for the range ... |
| rnopab3 5945 | The range of a restricted ... |
| rnmpt 5946 | The range of a function in... |
| elrnmpt 5947 | The range of a function in... |
| elrnmpt1s 5948 | Elementhood in an image se... |
| elrnmpt1 5949 | Elementhood in an image se... |
| elrnmptg 5950 | Membership in the range of... |
| elrnmpti 5951 | Membership in the range of... |
| elrnmptd 5952 | The range of a function in... |
| elrnmpt1d 5953 | Elementhood in an image se... |
| elrnmptdv 5954 | Elementhood in the range o... |
| elrnmpt2d 5955 | Elementhood in the range o... |
| nelrnmpt 5956 | Non-membership in the rang... |
| dfiun3g 5957 | Alternate definition of in... |
| dfiin3g 5958 | Alternate definition of in... |
| dfiun3 5959 | Alternate definition of in... |
| dfiin3 5960 | Alternate definition of in... |
| riinint 5961 | Express a relative indexed... |
| relrn0 5962 | A relation is empty iff it... |
| dmrnssfld 5963 | The domain and range of a ... |
| dmcoss 5964 | Domain of a composition. ... |
| dmcossOLD 5965 | Obsolete version of ~ dmco... |
| rncoss 5966 | Range of a composition. (... |
| dmcosseq 5967 | Domain of a composition. ... |
| dmcosseqOLD 5968 | Obsolete version of ~ dmco... |
| dmcoeq 5969 | Domain of a composition. ... |
| dmcocnv1 5970 | The domain of a class post... |
| dmcocnv2 5971 | The domain of a class prec... |
| rncoeq 5972 | Range of a composition. (... |
| rncocnv2 5973 | The range of a class preco... |
| rncocnv1 5974 | The range of a class postc... |
| reseq1 5975 | Equality theorem for restr... |
| reseq2 5976 | Equality theorem for restr... |
| reseq1i 5977 | Equality inference for res... |
| reseq2i 5978 | Equality inference for res... |
| reseq12i 5979 | Equality inference for res... |
| reseq1d 5980 | Equality deduction for res... |
| reseq2d 5981 | Equality deduction for res... |
| reseq12d 5982 | Equality deduction for res... |
| nfres 5983 | Bound-variable hypothesis ... |
| csbres 5984 | Distribute proper substitu... |
| res0 5985 | A restriction to the empty... |
| dfres3 5986 | Alternate definition of re... |
| opelres 5987 | Ordered pair elementhood i... |
| brres 5988 | Binary relation on a restr... |
| opelresi 5989 | Ordered pair membership in... |
| brresi 5990 | Binary relation on a restr... |
| opres 5991 | Ordered pair membership in... |
| resieq 5992 | A restricted identity rela... |
| opelidres 5993 | ` <. A , A >. ` belongs to... |
| resres 5994 | The restriction of a restr... |
| resundi 5995 | Distributive law for restr... |
| resundir 5996 | Distributive law for restr... |
| resindi 5997 | Class restriction distribu... |
| resindir 5998 | Class restriction distribu... |
| inres 5999 | Move intersection into cla... |
| resdifcom 6000 | Commutative law for restri... |
| resiun1 6001 | Distribution of restrictio... |
| resiun2 6002 | Distribution of restrictio... |
| resss 6003 | A class includes its restr... |
| rescom 6004 | Commutative law for restri... |
| ssres 6005 | Subclass theorem for restr... |
| ssres2 6006 | Subclass theorem for restr... |
| relres 6007 | A restriction is a relatio... |
| relresOLD 6008 | Obsolete version of ~ relr... |
| resabs1 6009 | Absorption law for restric... |
| resabs1i 6010 | Absorption law for restric... |
| resabs1d 6011 | Absorption law for restric... |
| resabs2 6012 | Absorption law for restric... |
| residm 6013 | Idempotent law for restric... |
| dmresss 6014 | The domain of a restrictio... |
| dmres 6015 | The domain of a restrictio... |
| ssdmres 6016 | A domain restricted to a s... |
| dmresexg 6017 | The domain of a restrictio... |
| rnresss 6018 | The range of a restriction... |
| xpssres 6019 | Restriction of a constant ... |
| elinxp 6020 | Membership in an intersect... |
| elres 6021 | Membership in a restrictio... |
| elsnres 6022 | Membership in restriction ... |
| relssres 6023 | Simplification law for res... |
| dmressnsn 6024 | The domain of a restrictio... |
| eldmressnsn 6025 | The element of the domain ... |
| eldmeldmressn 6026 | An element of the domain (... |
| resdm 6027 | A relation restricted to i... |
| resexg 6028 | The restriction of a set i... |
| resexd 6029 | The restriction of a set i... |
| resex 6030 | The restriction of a set i... |
| resindm 6031 | When restricting a class, ... |
| resindmOLD 6032 | Obsolete version of ~ resi... |
| resdmdfsn 6033 | Restricting a class to its... |
| resdmdfsnOLD 6034 | Obsolete version of ~ resd... |
| reldmun 6035 | Split a relation into two ... |
| reldisjunOLD 6036 | Obsolete version of ~ reld... |
| relresdm1 6037 | Restriction of a disjoint ... |
| resopab 6038 | Restriction of a class abs... |
| iss 6039 | A subclass of the identity... |
| resopab2 6040 | Restriction of a class abs... |
| resmpt 6041 | Restriction of the mapping... |
| resmpt3 6042 | Unconditional restriction ... |
| resmptf 6043 | Restriction of the mapping... |
| resmptd 6044 | Restriction of the mapping... |
| dfres2 6045 | Alternate definition of th... |
| mptss 6046 | Sufficient condition for i... |
| elimampt 6047 | Membership in the image of... |
| elidinxp 6048 | Characterization of the el... |
| elidinxpid 6049 | Characterization of the el... |
| elrid 6050 | Characterization of the el... |
| idinxpres 6051 | The intersection of the id... |
| idinxpresid 6052 | The intersection of the id... |
| idssxp 6053 | A diagonal set as a subset... |
| opabresid 6054 | The restricted identity re... |
| mptresid 6055 | The restricted identity re... |
| dmresi 6056 | The domain of a restricted... |
| restidsing 6057 | Restriction of the identit... |
| iresn0n0 6058 | The identity function rest... |
| imaeq1 6059 | Equality theorem for image... |
| imaeq2 6060 | Equality theorem for image... |
| imaeq1i 6061 | Equality theorem for image... |
| imaeq2i 6062 | Equality theorem for image... |
| imaeq1d 6063 | Equality theorem for image... |
| imaeq2d 6064 | Equality theorem for image... |
| imaeq12d 6065 | Equality theorem for image... |
| imass1 6066 | Subset theorem for image. ... |
| imass2 6067 | Subset theorem for image. ... |
| resima 6068 | A restriction to an image.... |
| resima2 6069 | Image under a restricted c... |
| dfima2 6070 | Alternate definition of im... |
| dfima3 6071 | Alternate definition of im... |
| elimag 6072 | Membership in an image. T... |
| elima 6073 | Membership in an image. T... |
| elima2 6074 | Membership in an image. T... |
| elima3 6075 | Membership in an image. T... |
| nfima 6076 | Bound-variable hypothesis ... |
| nfimad 6077 | Deduction version of bound... |
| dfrn7 6078 | The range is equal to the ... |
| imadmrn 6079 | The image of the domain of... |
| imadmrnOLD 6080 | Obsolete version of ~ imad... |
| imassrnOLD 6081 | Obsolete version of ~ imas... |
| mptima 6082 | Image by a function in map... |
| mptimass 6083 | Image by a function in map... |
| imai 6084 | Image by the identity rela... |
| rnresi 6085 | The range of the restricte... |
| resiima 6086 | The image of a restriction... |
| ima0 6087 | Image of the empty set. T... |
| csbima12 6088 | Move class substitution in... |
| imadisj 6089 | A class whose image under ... |
| imadisjlnd 6090 | Deduction form of one nega... |
| cnvimassrndm 6091 | The preimage of a superset... |
| cnvimarndm 6092 | The preimage of the range ... |
| cnvimarndmOLD 6093 | Obsolete version of ~ cnvi... |
| imasng 6094 | The image of a singleton. ... |
| relimasn 6095 | The image of a singleton. ... |
| elrelimasn 6096 | Elementhood in the image o... |
| elimasng1 6097 | Membership in an image of ... |
| elimasn1 6098 | Membership in an image of ... |
| elimasng 6099 | Membership in an image of ... |
| elimasn 6100 | Membership in an image of ... |
| elimasni 6101 | Membership in an image of ... |
| args 6102 | Two ways to express the cl... |
| elinisegg 6103 | Membership in the inverse ... |
| eliniseg 6104 | Membership in the inverse ... |
| epin 6105 | Any set is equal to its pr... |
| epini 6106 | Any set is equal to its pr... |
| iniseg 6107 | An idiom that signifies an... |
| inisegn0 6108 | Nonemptiness of an initial... |
| dffr3 6109 | Alternate definition of we... |
| dfse2 6110 | Alternate definition of se... |
| ndmima 6111 | The image of a singleton o... |
| relcnv 6112 | A converse is a relation. ... |
| relbrcnvg 6113 | When ` R ` is a relation, ... |
| eliniseg2 6114 | Eliminate the class existe... |
| relbrcnv 6115 | When ` R ` is a relation, ... |
| relco 6116 | A composition is a relatio... |
| cotrg 6117 | Two ways of saying that th... |
| cotr 6118 | Two ways of saying a relat... |
| idrefALT 6119 | Alternate proof of ~ idref... |
| cnvsym 6120 | Two ways of saying a relat... |
| intasym 6121 | Two ways of saying a relat... |
| asymref 6122 | Two ways of saying a relat... |
| asymref2 6123 | Two ways of saying a relat... |
| intirr 6124 | Two ways of saying a relat... |
| brcodir 6125 | Two ways of saying that tw... |
| codir 6126 | Two ways of saying a relat... |
| qfto 6127 | A quantifier-free way of e... |
| xpidtr 6128 | A Cartesian square is a tr... |
| trin2 6129 | The intersection of two tr... |
| poirr2 6130 | A partial order is irrefle... |
| trinxp 6131 | The relation induced by a ... |
| soirri 6132 | A strict order relation is... |
| sotri 6133 | A strict order relation is... |
| son2lpi 6134 | A strict order relation ha... |
| sotri2 6135 | A transitivity relation. ... |
| sotri3 6136 | A transitivity relation. ... |
| poleloe 6137 | Express "less than or equa... |
| poltletr 6138 | Transitive law for general... |
| somin1 6139 | Property of a minimum in a... |
| somincom 6140 | Commutativity of minimum i... |
| somin2 6141 | Property of a minimum in a... |
| soltmin 6142 | Being less than a minimum,... |
| cnvopab 6143 | The converse of a class ab... |
| mptcnv 6144 | The converse of a mapping ... |
| cnvun 6145 | The converse of a union is... |
| cnvdif 6146 | Distributive law for conve... |
| cnvin 6147 | Distributive law for conve... |
| rnun 6148 | Distributive law for range... |
| rnin 6149 | The range of an intersecti... |
| rninOLD 6150 | Obsolete version of ~ rnin... |
| rniun 6151 | The range of an indexed un... |
| rnuni 6152 | The range of a union. Par... |
| imaundi 6153 | Distributive law for image... |
| imaundir 6154 | The image of a union. (Co... |
| dminss 6155 | An upper bound for interse... |
| imainss 6156 | An upper bound for interse... |
| inimass 6157 | The image of an intersecti... |
| inimasn 6158 | The intersection of the im... |
| cnvxp 6159 | The converse of a Cartesia... |
| cnvxpOLD 6160 | Obsolete version of ~ cnvx... |
| xp0OLD 6161 | Obsolete version of ~ xp0 ... |
| xpnz 6162 | The Cartesian product of n... |
| xpeq0 6163 | At least one member of an ... |
| xpdisj1 6164 | Cartesian products with di... |
| xpdisj2 6165 | Cartesian products with di... |
| xpsndisj 6166 | Cartesian products with tw... |
| difxp 6167 | Difference of Cartesian pr... |
| difxp1 6168 | Difference law for Cartesi... |
| difxp2 6169 | Difference law for Cartesi... |
| djudisj 6170 | Disjoint unions with disjo... |
| xpdifid 6171 | The set of distinct couple... |
| xpdifcnvepel 6172 | The set of couples in a Ca... |
| resdisj 6173 | A double restriction to di... |
| rnxp 6174 | The range of a Cartesian p... |
| dmxpss 6175 | The domain of a Cartesian ... |
| rnxpss 6176 | The range of a Cartesian p... |
| rnxpid 6177 | The range of a Cartesian s... |
| ssxpb 6178 | A Cartesian product subcla... |
| xp11 6179 | The Cartesian product of n... |
| xpcan 6180 | Cancellation law for Carte... |
| xpcan2 6181 | Cancellation law for Carte... |
| ssrnres 6182 | Two ways to express surjec... |
| rninxp 6183 | Two ways to express surjec... |
| dminxp 6184 | Two ways to express totali... |
| imainrect 6185 | Image by a restricted and ... |
| xpima 6186 | Direct image by a Cartesia... |
| xpima1 6187 | Direct image by a Cartesia... |
| xpima2 6188 | Direct image by a Cartesia... |
| xpimasn 6189 | Direct image of a singleto... |
| sossfld 6190 | The base set of a strict o... |
| sofld 6191 | The base set of a nonempty... |
| cnvcnv3 6192 | The set of all ordered pai... |
| dfrel2 6193 | Alternate definition of re... |
| dfrel4v 6194 | A relation can be expresse... |
| dfrel4 6195 | A relation can be expresse... |
| cnvcnv 6196 | The double converse of a c... |
| cnvcnv2 6197 | The double converse of a c... |
| cnvcnvss 6198 | The double converse of a c... |
| cnvcnvssOLD 6199 | Obsolete version of ~ cnvc... |
| cnvrescnv 6200 | Two ways to express the co... |
| cnvssb 6201 | Subclass theorem for conve... |
| cnveqb 6202 | Equality theorem for conve... |
| cnveq0 6203 | A relation is empty iff it... |
| cnveq0b 6204 | A class is empty iff it is... |
| dfrel3 6205 | Alternate definition of re... |
| elid 6206 | Characterization of the el... |
| dmresv 6207 | The domain of a universal ... |
| rnresv 6208 | The range of a universal r... |
| dfrn4 6209 | Range defined in terms of ... |
| imassrn 6210 | Any image by a class is in... |
| cnvimass 6211 | Any preimage by a class is... |
| cnvimassrndmOLD 6212 | Obsolete version of ~ cnvi... |
| 0ima 6213 | Image under the empty rela... |
| imadifssrn 6214 | Condition for the range of... |
| imadifssranOLD 6215 | Obsolete version of ~ imad... |
| imadifssranOLDOLD 6216 | Obsolete version of ~ imad... |
| csbrn 6217 | Distribute proper substitu... |
| rescnvcnv 6218 | The restriction of the dou... |
| cnvcnvres 6219 | The double converse of the... |
| imacnvcnv 6220 | The image of the double co... |
| dmsnn0 6221 | The domain of a singleton ... |
| rnsnn0 6222 | The range of a singleton i... |
| dmsn0 6223 | The domain of the singleto... |
| cnvsn0 6224 | The converse of the single... |
| dmsn0el 6225 | The domain of a singleton ... |
| relsn2 6226 | A singleton is a relation ... |
| dmsnopg 6227 | The domain of a singleton ... |
| dmsnopss 6228 | The domain of a singleton ... |
| dmpropg 6229 | The domain of an unordered... |
| dmsnop 6230 | The domain of a singleton ... |
| dmprop 6231 | The domain of an unordered... |
| dmtpop 6232 | The domain of an unordered... |
| cnvcnvsn 6233 | Double converse of a singl... |
| dmsnsnsn 6234 | The domain of the singleto... |
| rnsnopg 6235 | The range of a singleton o... |
| rnpropg 6236 | The range of a pair of ord... |
| cnvsng 6237 | Converse of a singleton of... |
| rnsnop 6238 | The range of a singleton o... |
| op1sta 6239 | Extract the first member o... |
| cnvsn 6240 | Converse of a singleton of... |
| op2ndb 6241 | Extract the second member ... |
| op2nda 6242 | Extract the second member ... |
| opswap 6243 | Swap the members of an ord... |
| cnvresima 6244 | An image under the convers... |
| resdm2 6245 | A class restricted to its ... |
| resdmres 6246 | Restriction to the domain ... |
| resresdm 6247 | A restriction by an arbitr... |
| imadmres 6248 | The image of the domain of... |
| resdmss 6249 | Subset relationship for th... |
| resdifdi 6250 | Distributive law for restr... |
| resdifdir 6251 | Distributive law for restr... |
| mptpreima 6252 | The preimage of a function... |
| mptiniseg 6253 | Converse singleton image o... |
| dmmpt 6254 | The domain of the mapping ... |
| dmmptss 6255 | The domain of a mapping is... |
| dmmptg 6256 | The domain of the mapping ... |
| rnmpt0f 6257 | The range of a function in... |
| rnmptn0 6258 | The range of a function in... |
| dfco2 6259 | Alternate definition of a ... |
| dfco2a 6260 | Generalization of ~ dfco2 ... |
| coundi 6261 | Class composition distribu... |
| coundir 6262 | Class composition distribu... |
| cores 6263 | Restricted first member of... |
| resco 6264 | Associative law for the re... |
| imaco 6265 | Image of the composition o... |
| rnco 6266 | The range of the compositi... |
| rncoOLD 6267 | Obsolete version of ~ rnco... |
| rnco2 6268 | The range of the compositi... |
| dmco 6269 | The domain of a compositio... |
| coeq0 6270 | A composition of two relat... |
| coiun 6271 | Composition with an indexe... |
| cocnvcnv1 6272 | A composition is not affec... |
| cocnvcnv2 6273 | A composition is not affec... |
| cores2 6274 | Absorption of a reverse (p... |
| idresdmsscocnv1 6275 | A class postcomposed with ... |
| idresrnsscocnv2 6276 | A class precomposed with i... |
| co02 6277 | Composition with the empty... |
| co01 6278 | Composition with the empty... |
| coi1 6279 | Composition with the ident... |
| coi2 6280 | Composition with the ident... |
| cnvresid 6281 | A restricted identity rela... |
| coires1 6282 | Precomposing with a restri... |
| coires2 6283 | Postcomposing with a restr... |
| coass 6284 | Associative law for class ... |
| coassOLD 6285 | Obsolete version of ~ coas... |
| relcnvtrg 6286 | Subclass law for converse ... |
| relcnvtrgOLD 6287 | Obsolete form of ~ relcnvt... |
| relcnvtrOLD 6288 | Obsolete form of ~ relcnvt... |
| relssdmrn 6289 | A relation is included in ... |
| resssxp 6290 | If the ` R ` -image of a c... |
| cnvssrndm 6291 | The converse is a subset o... |
| cossxp 6292 | Composition as a subset of... |
| relrelss 6293 | Two ways to describe the s... |
| unielrel 6294 | The membership relation fo... |
| relfld 6295 | The double union of a rela... |
| relresfld 6296 | Restriction of a relation ... |
| relresfldOLD 6297 | Obsolete version of ~ relr... |
| relcoi2 6298 | Composition with the ident... |
| relcoi1 6299 | Composition with the ident... |
| unidmrn 6300 | The double union of the co... |
| relcnvfld 6301 | if ` R ` is a relation, it... |
| dfdm2 6302 | Alternate definition of do... |
| unixp 6303 | The double class union of ... |
| unixp0 6304 | A Cartesian product is emp... |
| unixpid 6305 | Field of a Cartesian squar... |
| ressn 6306 | Restriction of a class to ... |
| cnviin 6307 | The converse of an interse... |
| cnvpo 6308 | The converse of a partial ... |
| cnvso 6309 | The converse of a strict o... |
| xpco 6310 | Composition of two Cartesi... |
| xpcoid 6311 | Composition of two Cartesi... |
| elsnxp 6312 | Membership in a Cartesian ... |
| reu3op 6313 | There is a unique ordered ... |
| reuop 6314 | There is a unique ordered ... |
| opreu2reurex 6315 | There is a unique ordered ... |
| opreu2reu 6316 | If there is a unique order... |
| dfpo2 6317 | Quantifier-free definition... |
| csbcog 6318 | Distribute proper substitu... |
| snres0 6319 | Condition for restriction ... |
| imaindm 6320 | The image is unaffected by... |
| predeq123 6323 | Equality theorem for the p... |
| predeq1 6324 | Equality theorem for the p... |
| predeq2 6325 | Equality theorem for the p... |
| predeq3 6326 | Equality theorem for the p... |
| nfpred 6327 | Bound-variable hypothesis ... |
| csbpredg 6328 | Move class substitution in... |
| predpredss 6329 | If ` A ` is a subset of ` ... |
| predss 6330 | The predecessor class of `... |
| sspred 6331 | Another subset/predecessor... |
| dfpred2 6332 | An alternate definition of... |
| dfpred3 6333 | An alternate definition of... |
| dfpred3g 6334 | An alternate definition of... |
| elpredgg 6335 | Membership in a predecesso... |
| elpredg 6336 | Membership in a predecesso... |
| elpredimg 6337 | Membership in a predecesso... |
| elpredim 6338 | Membership in a predecesso... |
| elpred 6339 | Membership in a predecesso... |
| predexg 6340 | The predecessor class exis... |
| dffr4 6341 | Alternate definition of we... |
| predel 6342 | Membership in the predeces... |
| predtrss 6343 | If ` R ` is transitive ove... |
| predpo 6344 | Property of the predecesso... |
| predso 6345 | Property of the predecesso... |
| setlikespec 6346 | If ` R ` is set-like in ` ... |
| predidm 6347 | Idempotent law for the pre... |
| predin 6348 | Intersection law for prede... |
| predun 6349 | Union law for predecessor ... |
| preddif 6350 | Difference law for predece... |
| predep 6351 | The predecessor under the ... |
| trpred 6352 | The class of predecessors ... |
| preddowncl 6353 | A property of classes that... |
| predpoirr 6354 | Given a partial ordering, ... |
| predfrirr 6355 | Given a well-founded relat... |
| pred0 6356 | The predecessor class over... |
| dfse3 6357 | Alternate definition of se... |
| predrelss 6358 | Subset carries from relati... |
| predprc 6359 | The predecessor of a prope... |
| predres 6360 | Predecessor class is unaff... |
| frpomin 6361 | Every nonempty (possibly p... |
| frpomin2 6362 | Every nonempty (possibly p... |
| frpoind 6363 | The principle of well-foun... |
| frpoinsg 6364 | Well-Founded Induction Sch... |
| frpoins2fg 6365 | Well-Founded Induction sch... |
| frpoins2g 6366 | Well-Founded Induction sch... |
| frpoins3g 6367 | Well-Founded Induction sch... |
| tz6.26 6368 | All nonempty subclasses of... |
| tz6.26i 6369 | All nonempty subclasses of... |
| wfi 6370 | The Principle of Well-Orde... |
| wfii 6371 | The Principle of Well-Orde... |
| wfisg 6372 | Well-Ordered Induction Sch... |
| wfis 6373 | Well-Ordered Induction Sch... |
| wfis2fg 6374 | Well-Ordered Induction Sch... |
| wfis2f 6375 | Well-Ordered Induction sch... |
| wfis2g 6376 | Well-Ordered Induction Sch... |
| wfis2 6377 | Well-Ordered Induction sch... |
| wfis3 6378 | Well-Ordered Induction sch... |
| ordeq 6387 | Equality theorem for the o... |
| elong 6388 | An ordinal number is an or... |
| elon 6389 | An ordinal number is an or... |
| eloni 6390 | An ordinal number has the ... |
| elon2 6391 | An ordinal number is an or... |
| limeq 6392 | Equality theorem for the l... |
| ordwe 6393 | Membership well-orders eve... |
| ordtr 6394 | An ordinal class is transi... |
| ordfr 6395 | Membership is well-founded... |
| ordelss 6396 | An element of an ordinal c... |
| trssord 6397 | A transitive subclass of a... |
| ordirr 6398 | No ordinal class is a memb... |
| nordeq 6399 | A member of an ordinal cla... |
| ordn2lp 6400 | An ordinal class cannot be... |
| tz7.5 6401 | A nonempty subclass of an ... |
| ordelord 6402 | An element of an ordinal c... |
| tron 6403 | The class of all ordinal n... |
| ordelon 6404 | An element of an ordinal c... |
| onelon 6405 | An element of an ordinal n... |
| tz7.7 6406 | A transitive class belongs... |
| ordelssne 6407 | For ordinal classes, membe... |
| ordelpss 6408 | For ordinal classes, membe... |
| ordpss 6409 | ~ ordelpss with an anteced... |
| ordsseleq 6410 | For ordinal classes, inclu... |
| ordin 6411 | The intersection of two or... |
| onin 6412 | The intersection of two or... |
| ordtri3or 6413 | A trichotomy law for ordin... |
| ordtri1 6414 | A trichotomy law for ordin... |
| ontri1 6415 | A trichotomy law for ordin... |
| ordtri2 6416 | A trichotomy law for ordin... |
| ordtri3 6417 | A trichotomy law for ordin... |
| ordtri4 6418 | A trichotomy law for ordin... |
| orddisj 6419 | An ordinal class and its s... |
| onfr 6420 | The ordinal class is well-... |
| onelpss 6421 | Relationship between membe... |
| onsseleq 6422 | Relationship between subse... |
| onelss 6423 | An element of an ordinal n... |
| oneltri 6424 | The elementhood relation o... |
| ordtr1 6425 | Transitive law for ordinal... |
| ordtr2 6426 | Transitive law for ordinal... |
| ordtr3 6427 | Transitive law for ordinal... |
| ontr1 6428 | Transitive law for ordinal... |
| ontr2 6429 | Transitive law for ordinal... |
| onelssex 6430 | Ordinal less than is equiv... |
| ordunidif 6431 | The union of an ordinal st... |
| ordintdif 6432 | If ` B ` is smaller than `... |
| onintss 6433 | If a property is true for ... |
| oneqmini 6434 | A way to show that an ordi... |
| ord0 6435 | The empty set is an ordina... |
| 0elon 6436 | The empty set is an ordina... |
| ord0eln0 6437 | A nonempty ordinal contain... |
| on0eln0 6438 | An ordinal number contains... |
| dflim2 6439 | An alternate definition of... |
| inton 6440 | The intersection of the cl... |
| nlim0 6441 | The empty set is not a lim... |
| limord 6442 | A limit ordinal is ordinal... |
| limuni 6443 | A limit ordinal is its own... |
| limuni2 6444 | The union of a limit ordin... |
| 0ellim 6445 | A limit ordinal contains t... |
| limelon 6446 | A limit ordinal class that... |
| onn0 6447 | The class of all ordinal n... |
| suceqd 6448 | Deduction associated with ... |
| suceq 6449 | Equality of successors. (... |
| elsuci 6450 | Membership in a successor.... |
| elsucg 6451 | Membership in a successor.... |
| elsuc2g 6452 | Variant of membership in a... |
| elsuc 6453 | Membership in a successor.... |
| elsuc2 6454 | Membership in a successor.... |
| nfsuc 6455 | Bound-variable hypothesis ... |
| elelsuc 6456 | Membership in a successor.... |
| sucel 6457 | Membership of a successor ... |
| suc0 6458 | The successor of the empty... |
| sucprc 6459 | A proper class is its own ... |
| unisucs 6460 | The union of the successor... |
| unisucg 6461 | A transitive class is equa... |
| unisuc 6462 | A transitive class is equa... |
| sssucid 6463 | A class is included in its... |
| sucidg 6464 | Part of Proposition 7.23 o... |
| sucid 6465 | A set belongs to its succe... |
| nsuceq0 6466 | No successor is empty. (C... |
| eqelsuc 6467 | A set belongs to the succe... |
| iunsuc 6468 | Inductive definition for t... |
| suctr 6469 | The successor of a transit... |
| trsuc 6470 | A set whose successor belo... |
| trsucss 6471 | A member of the successor ... |
| ordsssuc 6472 | An ordinal is a subset of ... |
| onsssuc 6473 | A subset of an ordinal num... |
| ordsssuc2 6474 | An ordinal subset of an or... |
| onmindif 6475 | When its successor is subt... |
| ordnbtwn 6476 | There is no set between an... |
| onnbtwn 6477 | There is no set between an... |
| sucssel 6478 | A set whose successor is a... |
| orddif 6479 | Ordinal derived from its s... |
| orduniss 6480 | An ordinal class includes ... |
| ordtri2or 6481 | A trichotomy law for ordin... |
| ordtri2or2 6482 | A trichotomy law for ordin... |
| ordtri2or3 6483 | A consequence of total ord... |
| ordelinel 6484 | The intersection of two or... |
| ordssun 6485 | Property of a subclass of ... |
| ordequn 6486 | The maximum (i.e. union) o... |
| ordun 6487 | The maximum (i.e., union) ... |
| onunel 6488 | The union of two ordinals ... |
| ordunisssuc 6489 | A subclass relationship fo... |
| suc11 6490 | The successor operation be... |
| onun2 6491 | The union of two ordinals ... |
| ontr 6492 | An ordinal number is a tra... |
| onunisuc 6493 | An ordinal number is equal... |
| onordi 6494 | An ordinal number is an or... |
| onirri 6495 | An ordinal number is not a... |
| oneli 6496 | A member of an ordinal num... |
| onelssi 6497 | A member of an ordinal num... |
| onssneli 6498 | An ordering law for ordina... |
| onssnel2i 6499 | An ordering law for ordina... |
| onelini 6500 | An element of an ordinal n... |
| oneluni 6501 | An ordinal number equals i... |
| onunisuci 6502 | An ordinal number is equal... |
| onsseli 6503 | Subset is equivalent to me... |
| onun2i 6504 | The union of two ordinal n... |
| unizlim 6505 | An ordinal equal to its ow... |
| on0eqel 6506 | An ordinal number either e... |
| snsn0non 6507 | The singleton of the singl... |
| onxpdisj 6508 | Ordinal numbers and ordere... |
| onnev 6509 | The class of ordinal numbe... |
| iotajust 6511 | Soundness justification th... |
| dfiota2 6513 | Alternate definition for d... |
| nfiota1 6514 | Bound-variable hypothesis ... |
| nfiotadw 6515 | Deduction version of ~ nfi... |
| nfiotaw 6516 | Bound-variable hypothesis ... |
| nfiotad 6517 | Deduction version of ~ nfi... |
| nfiota 6518 | Bound-variable hypothesis ... |
| cbviotaw 6519 | Change bound variables in ... |
| cbviotavw 6520 | Change bound variables in ... |
| cbviota 6521 | Change bound variables in ... |
| cbviotav 6522 | Change bound variables in ... |
| sb8iota 6523 | Variable substitution in d... |
| iotaeq 6524 | Equality theorem for descr... |
| iotabi 6525 | Equivalence theorem for de... |
| uniabio 6526 | Part of Theorem 8.17 in [Q... |
| iotaval2 6527 | Version of ~ iotaval using... |
| iotauni2 6528 | Version of ~ iotauni using... |
| iotanul2 6529 | Version of ~ iotanul using... |
| iotaval 6530 | Theorem 8.19 in [Quine] p.... |
| iotain 6531 | Equivalence between two di... |
| iotassuni 6532 | The ` iota ` class is a su... |
| iotaex 6533 | Theorem 8.23 in [Quine] p.... |
| iotauni 6534 | Equivalence between two di... |
| iotaint 6535 | Equivalence between two di... |
| iota1 6536 | Property of iota. (Contri... |
| iotanul 6537 | Theorem 8.22 in [Quine] p.... |
| iota4 6538 | Theorem *14.22 in [Whitehe... |
| iota4an 6539 | Theorem *14.23 in [Whitehe... |
| iota5 6540 | A method for computing iot... |
| iotabidv 6541 | Formula-building deduction... |
| iotabii 6542 | Formula-building deduction... |
| iotacl 6543 | Membership law for descrip... |
| iota2df 6544 | A condition that allows to... |
| iota2d 6545 | A condition that allows to... |
| iota2 6546 | The unique element such th... |
| iotan0 6547 | Representation of "the uni... |
| iotan0OLD 6548 | Obsolete version of ~ iota... |
| sniota 6549 | A class abstraction with a... |
| dfiota4 6550 | The ` iota ` operation usi... |
| csbiota 6551 | Class substitution within ... |
| dffun2 6568 | Alternate definition of a ... |
| dffun6 6569 | Alternate definition of a ... |
| dffun3 6570 | Alternate definition of fu... |
| dffun4 6571 | Alternate definition of a ... |
| dffun5 6572 | Alternate definition of fu... |
| dffun6f 6573 | Definition of function, us... |
| funmo 6574 | A function has at most one... |
| fununiq 6575 | Functions are single-value... |
| funrel 6576 | A function is a relation. ... |
| 0nelfun 6577 | A function does not contai... |
| funss 6578 | Subclass theorem for funct... |
| funeq 6579 | Equality theorem for funct... |
| funeqi 6580 | Equality inference for the... |
| funeqd 6581 | Equality deduction for the... |
| nffun 6582 | Bound-variable hypothesis ... |
| sbcfung 6583 | Distribute proper substitu... |
| sbcfungOLD 6584 | Obsolete version of ~ sbcf... |
| funeu 6585 | There is exactly one value... |
| funeu2 6586 | There is exactly one value... |
| dffun7 6587 | Alternate definition of a ... |
| dffun8 6588 | Alternate definition of a ... |
| dffun9 6589 | Alternate definition of a ... |
| funfn 6590 | A class is a function if a... |
| funfnd 6591 | A function is a function o... |
| funi 6592 | The identity relation is a... |
| nfunv 6593 | The universal class is not... |
| funopg 6594 | A Kuratowski ordered pair ... |
| funopab 6595 | A class of ordered pairs i... |
| funopabeq 6596 | A class of ordered pairs o... |
| funopab4 6597 | A class of ordered pairs o... |
| funmpt 6598 | A function in maps-to nota... |
| funmpt2 6599 | Functionality of a class g... |
| funco 6600 | The composition of two fun... |
| funresfunco 6601 | Composition of two functio... |
| funres 6602 | A restriction of a functio... |
| funresd 6603 | A restriction of a functio... |
| funssres 6604 | The restriction of a funct... |
| fun2ssres 6605 | Equality of restrictions o... |
| funun 6606 | The union of functions wit... |
| fununmo 6607 | If the union of classes is... |
| fununfun 6608 | If the union of classes is... |
| fununfunOLD 6609 | Obsolete version of ~ funu... |
| fundif 6610 | A function with removed el... |
| fundifOLD 6611 | Obsolete version of ~ fund... |
| funcnvsn 6612 | The converse singleton of ... |
| funsng 6613 | A singleton of an ordered ... |
| fnsng 6614 | Functionality and domain o... |
| funsn 6615 | A singleton of an ordered ... |
| funprg 6616 | A set of two pairs is a fu... |
| funtpg 6617 | A set of three pairs is a ... |
| funpr 6618 | A function with a domain o... |
| funtp 6619 | A function with a domain o... |
| fnsn 6620 | Functionality and domain o... |
| fnprg 6621 | Function with a domain of ... |
| fntpg 6622 | Function with a domain of ... |
| fntp 6623 | A function with a domain o... |
| funcnvpr 6624 | The converse pair of order... |
| funcnvtp 6625 | The converse triple of ord... |
| funcnvqp 6626 | The converse quadruple of ... |
| fun0 6627 | The empty set is a functio... |
| funcnv0 6628 | The converse of the empty ... |
| funcnvcnv 6629 | The double converse of a f... |
| funcnv2 6630 | A simpler equivalence for ... |
| funcnv 6631 | The converse of a class is... |
| funcnv3 6632 | A condition showing a clas... |
| fun2cnv 6633 | The double converse of a c... |
| svrelfun 6634 | A single-valued relation i... |
| fncnv 6635 | Single-rootedness (see ~ f... |
| fun11 6636 | Two ways of stating that `... |
| fununi 6637 | The union of a chain (with... |
| funin 6638 | The intersection with a fu... |
| funres11 6639 | The restriction of a one-t... |
| funcnvres 6640 | The converse of a restrict... |
| cnvresidOLD 6641 | Obsolete version of ~ cnvr... |
| funcnvres2 6642 | The converse of a restrict... |
| funimacnv 6643 | The image of the preimage ... |
| funimass1 6644 | A kind of contraposition l... |
| funimass2 6645 | A kind of contraposition l... |
| imadif 6646 | The image of a difference ... |
| imain 6647 | The image of an intersecti... |
| funimaexg 6648 | Axiom of Replacement using... |
| funimaex 6649 | The image of a set under a... |
| isarep1 6650 | Part of a study of the Axi... |
| isarep2 6651 | Part of a study of the Axi... |
| fneq1 6652 | Equality theorem for funct... |
| fneq2 6653 | Equality theorem for funct... |
| fneq1d 6654 | Equality deduction for fun... |
| fneq2d 6655 | Equality deduction for fun... |
| fneq12d 6656 | Equality deduction for fun... |
| fneq12 6657 | Equality theorem for funct... |
| fneq1i 6658 | Equality inference for fun... |
| fneq2i 6659 | Equality inference for fun... |
| nffn 6660 | Bound-variable hypothesis ... |
| fnfun 6661 | A function with domain is ... |
| fnfund 6662 | A function with domain is ... |
| fnrel 6663 | A function with domain is ... |
| fndm 6664 | The domain of a function. ... |
| fndmi 6665 | The domain of a function. ... |
| fndmd 6666 | The domain of a function. ... |
| funfni 6667 | Inference to convert a fun... |
| fndmu 6668 | A function has a unique do... |
| fnbr 6669 | The first argument of bina... |
| fnop 6670 | The first argument of an o... |
| fneu 6671 | There is exactly one value... |
| fneu2 6672 | There is exactly one value... |
| fnunres1 6673 | Restriction of a disjoint ... |
| fnunres2 6674 | Restriction of a disjoint ... |
| fnun 6675 | The union of two functions... |
| fnund 6676 | The union of two functions... |
| fnunop 6677 | Extension of a function wi... |
| fncofn 6678 | Composition of a function ... |
| fnco 6679 | Composition of two functio... |
| fnresdm 6680 | A function does not change... |
| fnresdisj 6681 | A function restricted to a... |
| 2elresin 6682 | Membership in two function... |
| fnssresb 6683 | Restriction of a function ... |
| fnssres 6684 | Restriction of a function ... |
| fnssresd 6685 | Restriction of a function ... |
| fnresin1 6686 | Restriction of a function'... |
| fnresin2 6687 | Restriction of a function'... |
| fnres 6688 | An equivalence for functio... |
| idfn 6689 | The identity relation is a... |
| fnresi 6690 | The restricted identity re... |
| fnima 6691 | The image of a function's ... |
| fn0 6692 | A function with empty doma... |
| fnimadisj 6693 | A class that is disjoint w... |
| fnimaeq0 6694 | Images under a function ne... |
| dfmpt3 6695 | Alternate definition for t... |
| mptfnf 6696 | The maps-to notation defin... |
| fnmptf 6697 | The maps-to notation defin... |
| fnopabg 6698 | Functionality and domain o... |
| fnopab 6699 | Functionality and domain o... |
| mptfng 6700 | The maps-to notation defin... |
| fnmpt 6701 | The maps-to notation defin... |
| fnmptd 6702 | The maps-to notation defin... |
| mpt0 6703 | A mapping operation with e... |
| fnmpti 6704 | The maps-to notation defin... |
| dmmpti 6705 | Domain of the mapping oper... |
| dmmptd 6706 | The domain of the mapping ... |
| mptun 6707 | Union of mappings which ar... |
| partfun 6708 | Rewrite a function defined... |
| feq1 6709 | Equality theorem for funct... |
| feq2 6710 | Equality theorem for funct... |
| feq3 6711 | Equality theorem for funct... |
| feq23 6712 | Equality theorem for funct... |
| feq1d 6713 | Equality deduction for fun... |
| feq1dd 6714 | Equality deduction for fun... |
| feq2d 6715 | Equality deduction for fun... |
| feq3d 6716 | Equality deduction for fun... |
| feq2dd 6717 | Equality deduction for fun... |
| feq3dd 6718 | Equality deduction for fun... |
| feq12d 6719 | Equality deduction for fun... |
| feq123d 6720 | Equality deduction for fun... |
| feq123 6721 | Equality theorem for funct... |
| feq1i 6722 | Equality inference for fun... |
| feq2i 6723 | Equality inference for fun... |
| feq12i 6724 | Equality inference for fun... |
| feq23i 6725 | Equality inference for fun... |
| feq23d 6726 | Equality deduction for fun... |
| nff 6727 | Bound-variable hypothesis ... |
| sbcfng 6728 | Distribute proper substitu... |
| sbcfg 6729 | Distribute proper substitu... |
| elimf 6730 | Eliminate a mapping hypoth... |
| ffn 6731 | A mapping is a function wi... |
| ffnd 6732 | A mapping is a function wi... |
| dffn2 6733 | Any function is a mapping ... |
| ffun 6734 | A mapping is a function. ... |
| ffunOLD 6735 | Obsolete version of ~ ffun... |
| ffund 6736 | A mapping is a function, d... |
| frel 6737 | A mapping is a relation. ... |
| freld 6738 | A mapping is a relation. ... |
| frn 6739 | The range of a mapping. (... |
| frnd 6740 | Deduction form of ~ frn . ... |
| fdm 6741 | The domain of a mapping. ... |
| fdmd 6742 | Deduction form of ~ fdm . ... |
| fdmi 6743 | Inference associated with ... |
| dffn3 6744 | A function maps to its ran... |
| ffrn 6745 | A function maps to its ran... |
| ffrnb 6746 | Characterization of a func... |
| ffrnbd 6747 | A function maps to its ran... |
| fss 6748 | Expanding the codomain of ... |
| fssd 6749 | Expanding the codomain of ... |
| fssdmd 6750 | Expressing that a class is... |
| fssdm 6751 | Expressing that a class is... |
| fimass 6752 | The image of a class under... |
| fimassd 6753 | The image of a class is a ... |
| fimacnv 6754 | The preimage of the codoma... |
| fcof 6755 | Composition of a function ... |
| fco 6756 | Composition of two functio... |
| fcod 6757 | Composition of two mapping... |
| fco2 6758 | Functionality of a composi... |
| fssxp 6759 | A mapping is a class of or... |
| funssxp 6760 | Two ways of specifying a p... |
| ffdm 6761 | A mapping is a partial fun... |
| ffdmd 6762 | The domain of a function. ... |
| fdmrn 6763 | A different way to write `... |
| funcofd 6764 | Composition of two functio... |
| opelf 6765 | The members of an ordered ... |
| fun 6766 | The union of two functions... |
| fun2 6767 | The union of two functions... |
| fun2d 6768 | The union of functions wit... |
| fnfco 6769 | Composition of two functio... |
| fssres 6770 | Restriction of a function ... |
| fssresd 6771 | Restriction of a function ... |
| fssres2 6772 | Restriction of a restricte... |
| fresin 6773 | An identity for the mappin... |
| resasplit 6774 | If two functions agree on ... |
| fresaun 6775 | The union of two functions... |
| fresaunres2 6776 | From the union of two func... |
| fresaunres1 6777 | From the union of two func... |
| fcoi1 6778 | Composition of a mapping a... |
| fcoi2 6779 | Composition of restricted ... |
| feu 6780 | There is exactly one value... |
| fcnvres 6781 | The converse of a restrict... |
| fimacnvdisj 6782 | The preimage of a class di... |
| fint 6783 | Function into an intersect... |
| fin 6784 | Mapping into an intersecti... |
| f0 6785 | The empty function. (Cont... |
| f00 6786 | A class is a function with... |
| f0bi 6787 | A function with empty doma... |
| f0dom0 6788 | A function is empty iff it... |
| f0rn0 6789 | If there is no element in ... |
| fconst 6790 | A Cartesian product with a... |
| fconstg 6791 | A Cartesian product with a... |
| fnconstg 6792 | A Cartesian product with a... |
| fconst6g 6793 | Constant function with loo... |
| fconst6 6794 | A constant function as a m... |
| f1eq1 6795 | Equality theorem for one-t... |
| f1eq2 6796 | Equality theorem for one-t... |
| f1eq3 6797 | Equality theorem for one-t... |
| nff1 6798 | Bound-variable hypothesis ... |
| dff12 6799 | Alternate definition of a ... |
| f1f 6800 | A one-to-one mapping is a ... |
| f1fn 6801 | A one-to-one mapping is a ... |
| f1fun 6802 | A one-to-one mapping is a ... |
| f1funOLD 6803 | Obsolete version of ~ f1fu... |
| f1rel 6804 | A one-to-one onto mapping ... |
| f1relOLD 6805 | Obsolete version of ~ f1re... |
| f1dm 6806 | The domain of a one-to-one... |
| f1ss 6807 | A function that is one-to-... |
| f1ssr 6808 | A function that is one-to-... |
| f1ssres 6809 | A function that is one-to-... |
| f1resf1 6810 | The restriction of an inje... |
| f1cnvcnv 6811 | Two ways to express that a... |
| f1cof1 6812 | Composition of two one-to-... |
| f1co 6813 | Composition of one-to-one ... |
| foeq1 6814 | Equality theorem for onto ... |
| foeq2 6815 | Equality theorem for onto ... |
| foeq3 6816 | Equality theorem for onto ... |
| nffo 6817 | Bound-variable hypothesis ... |
| fof 6818 | An onto mapping is a mappi... |
| fofun 6819 | An onto mapping is a funct... |
| fofn 6820 | An onto mapping is a funct... |
| forn 6821 | The codomain of an onto fu... |
| dffo2 6822 | Alternate definition of an... |
| foima 6823 | The image of the domain of... |
| dffn4 6824 | A function maps onto its r... |
| funforn 6825 | A function maps its domain... |
| fodmrnu 6826 | An onto function has uniqu... |
| fimadmfo 6827 | A function is a function o... |
| fores 6828 | Restriction of an onto fun... |
| fimadmfoALT 6829 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6830 | The preimage of the codoma... |
| focofo 6831 | Composition of onto functi... |
| foco 6832 | Composition of onto functi... |
| foconst 6833 | A nonzero constant functio... |
| f1oeq1 6834 | Equality theorem for one-t... |
| f1oeq2 6835 | Equality theorem for one-t... |
| f1oeq3 6836 | Equality theorem for one-t... |
| f1oeq23 6837 | Equality theorem for one-t... |
| f1eq123d 6838 | Equality deduction for one... |
| foeq123d 6839 | Equality deduction for ont... |
| f1oeq123d 6840 | Equality deduction for one... |
| f1oeq1d 6841 | Equality deduction for one... |
| f1oeq2d 6842 | Equality deduction for one... |
| f1oeq3d 6843 | Equality deduction for one... |
| nff1o 6844 | Bound-variable hypothesis ... |
| f1of1 6845 | A one-to-one onto mapping ... |
| f1of 6846 | A one-to-one onto mapping ... |
| f1ofn 6847 | A one-to-one onto mapping ... |
| f1ofun 6848 | A one-to-one onto mapping ... |
| f1orel 6849 | A one-to-one onto mapping ... |
| f1odm 6850 | The domain of a one-to-one... |
| f1odmOLD 6851 | Obsolete version of ~ f1od... |
| dff1o2 6852 | Alternate definition of on... |
| dff1o3 6853 | Alternate definition of on... |
| f1ofo 6854 | A one-to-one onto function... |
| dff1o4 6855 | Alternate definition of on... |
| dff1o5 6856 | Alternate definition of on... |
| f1orn 6857 | A one-to-one function maps... |
| f1f1orn 6858 | A one-to-one function maps... |
| f1ocnv 6859 | The converse of a one-to-o... |
| f1ocnvb 6860 | A relation is a one-to-one... |
| f1ores 6861 | The restriction of a one-t... |
| f1orescnv 6862 | The converse of a one-to-o... |
| f1imacnv 6863 | Preimage of an image. (Co... |
| foimacnv 6864 | A reverse version of ~ f1i... |
| foun 6865 | The union of two onto func... |
| f1oun 6866 | The union of two one-to-on... |
| f1un 6867 | The union of two one-to-on... |
| resdif 6868 | The restriction of a one-t... |
| resin 6869 | The restriction of a one-t... |
| f1oco 6870 | Composition of one-to-one ... |
| f1cnv 6871 | The converse of an injecti... |
| funcocnv2 6872 | Precomposition with the co... |
| funcocnv2OLD 6873 | Obsolete version of ~ func... |
| fococnv2 6874 | The composition of an onto... |
| f1ococnv2 6875 | The composition of a one-t... |
| f1cocnv2 6876 | Composition of an injectiv... |
| f1ococnv1 6877 | The composition of a one-t... |
| f1cocnv1 6878 | Composition of an injectiv... |
| funcoeqres 6879 | Express a constraint on a ... |
| f1ssf1 6880 | A subset of an injective f... |
| f10 6881 | The empty set maps one-to-... |
| f10d 6882 | The empty set maps one-to-... |
| f1o00 6883 | One-to-one onto mapping of... |
| fo00 6884 | Onto mapping of the empty ... |
| f1o0 6885 | One-to-one onto mapping of... |
| f1oi 6886 | A restriction of the ident... |
| f1oiOLD 6887 | Obsolete version of ~ f1oi... |
| f1ovi 6888 | The identity relation is a... |
| f1osn 6889 | A singleton of an ordered ... |
| f1osng 6890 | A singleton of an ordered ... |
| f1sng 6891 | A singleton of an ordered ... |
| fsnd 6892 | A singleton of an ordered ... |
| f1oprswap 6893 | A two-element swap is a bi... |
| f1oprg 6894 | An unordered pair of order... |
| tz6.12-2 6895 | Function value when ` F ` ... |
| tz6.12-2OLD 6896 | Obsolete version of ~ tz6.... |
| fveu 6897 | The value of a function at... |
| brprcneu 6898 | If ` A ` is a proper class... |
| brprcneuALT 6899 | Alternate proof of ~ brprc... |
| fvprc 6900 | A function's value at a pr... |
| fvprcALT 6901 | Alternate proof of ~ fvprc... |
| rnfvprc 6902 | The range of a function va... |
| fv2 6903 | Alternate definition of fu... |
| dffv3 6904 | A definition of function v... |
| dffv4 6905 | The previous definition of... |
| elfv 6906 | Membership in a function v... |
| fveq1 6907 | Equality theorem for funct... |
| fveq2 6908 | Equality theorem for funct... |
| fveq1i 6909 | Equality inference for fun... |
| fveq1d 6910 | Equality deduction for fun... |
| fveq2i 6911 | Equality inference for fun... |
| fveq2d 6912 | Equality deduction for fun... |
| 2fveq3 6913 | Equality theorem for neste... |
| fveq12i 6914 | Equality deduction for fun... |
| fveq12d 6915 | Equality deduction for fun... |
| fveqeq2d 6916 | Equality deduction for fun... |
| fveqeq2 6917 | Equality deduction for fun... |
| nffv 6918 | Bound-variable hypothesis ... |
| nffvmpt1 6919 | Bound-variable hypothesis ... |
| nffvd 6920 | Deduction version of bound... |
| fvex 6921 | The value of a class exist... |
| fvexi 6922 | The value of a class exist... |
| fvexd 6923 | The value of a class exist... |
| fvif 6924 | Move a conditional outside... |
| iffv 6925 | Move a conditional outside... |
| fv3 6926 | Alternate definition of th... |
| fvres 6927 | The value of a restricted ... |
| fvresd 6928 | The value of a restricted ... |
| funssfv 6929 | The value of a member of t... |
| tz6.12c 6930 | Corollary of Theorem 6.12(... |
| tz6.12-1 6931 | Function value. Theorem 6... |
| tz6.12 6932 | Function value. Theorem 6... |
| tz6.12f 6933 | Function value, using boun... |
| tz6.12i 6934 | Corollary of Theorem 6.12(... |
| fvbr0 6935 | Two possibilities for the ... |
| fvrn0 6936 | A function value is a memb... |
| fvn0fvelrn 6937 | If the value of a function... |
| elfvunirn 6938 | A function value is a subs... |
| fvssunirn 6939 | The result of a function v... |
| ndmfv 6940 | The value of a class outsi... |
| ndmfvrcl 6941 | Reverse closure law for fu... |
| elfvdm 6942 | If a function value has a ... |
| elfvex 6943 | If a function value has a ... |
| elfvexd 6944 | If a function value has a ... |
| eliman0 6945 | A nonempty function value ... |
| nfvres 6946 | The value of a non-member ... |
| nfunsn 6947 | If the restriction of a cl... |
| fvfundmfvn0 6948 | If the "value of a class" ... |
| 0fv 6949 | Function value of the empt... |
| fv2prc 6950 | A function value of a func... |
| elfv2ex 6951 | If a function value of a f... |
| fveqres 6952 | Equal values imply equal v... |
| csbfv12 6953 | Move class substitution in... |
| csbfv2g 6954 | Move class substitution in... |
| csbfv 6955 | Substitution for a functio... |
| funbrfv 6956 | The second argument of a b... |
| funopfv 6957 | The second element in an o... |
| fnbrfvb 6958 | Equivalence of function va... |
| fnopfvb 6959 | Equivalence of function va... |
| fvelima2 6960 | Function value in an image... |
| funbrfvb 6961 | Equivalence of function va... |
| funopfvb 6962 | Equivalence of function va... |
| fnbrfvb2 6963 | Version of ~ fnbrfvb for f... |
| fdmeu 6964 | There is exactly one codom... |
| funbrfv2b 6965 | Function value in terms of... |
| dffn5 6966 | Representation of a functi... |
| fnrnfv 6967 | The range of a function ex... |
| fvelrnb 6968 | A member of a function's r... |
| foelcdmi 6969 | A member of a surjective f... |
| dfimafn 6970 | Alternate definition of th... |
| dfimafn2 6971 | Alternate definition of th... |
| funimass4 6972 | Membership relation for th... |
| fvelima 6973 | Function value in an image... |
| funimassd 6974 | Sufficient condition for t... |
| fvelimad 6975 | Function value in an image... |
| feqmptd 6976 | Deduction form of ~ dffn5 ... |
| feqresmpt 6977 | Express a restricted funct... |
| feqmptdf 6978 | Deduction form of ~ dffn5f... |
| dffn5f 6979 | Representation of a functi... |
| fvelimab 6980 | Function value in an image... |
| fvelimabd 6981 | Deduction form of ~ fvelim... |
| fimarab 6982 | Expressing the image of a ... |
| unima 6983 | Image of a union. (Contri... |
| fvi 6984 | The value of the identity ... |
| fviss 6985 | The value of the identity ... |
| fniinfv 6986 | The indexed intersection o... |
| fnsnfv 6987 | Singleton of function valu... |
| opabiotafun 6988 | Define a function whose va... |
| opabiotadm 6989 | Define a function whose va... |
| opabiota 6990 | Define a function whose va... |
| fnimapr 6991 | The image of a pair under ... |
| fnimatpd 6992 | The image of an unordered ... |
| ssimaex 6993 | The existence of a subimag... |
| ssimaexg 6994 | The existence of a subimag... |
| funfv 6995 | A simplified expression fo... |
| funfv2 6996 | The value of a function. ... |
| funfv2f 6997 | The value of a function. ... |
| fvun 6998 | Value of the union of two ... |
| fvun1 6999 | The value of a union when ... |
| fvun2 7000 | The value of a union when ... |
| fvun1d 7001 | The value of a union when ... |
| fvun2d 7002 | The value of a union when ... |
| dffv2 7003 | Alternate definition of fu... |
| dmfco 7004 | Domains of a function comp... |
| fvco2 7005 | Value of a function compos... |
| fvco 7006 | Value of a function compos... |
| fvcod 7007 | Value of a function compos... |
| fvco3 7008 | Value of a function compos... |
| fvco3d 7009 | Value of a function compos... |
| fvco4i 7010 | Conditions for a compositi... |
| fvopab3g 7011 | Value of a function given ... |
| fvopab3ig 7012 | Value of a function given ... |
| brfvopabrbr 7013 | The binary relation of a f... |
| fvmptg 7014 | Value of a function given ... |
| fvmpti 7015 | Value of a function given ... |
| fvmpt 7016 | Value of a function given ... |
| fvmpt2f 7017 | Value of a function given ... |
| funcnvmpt 7018 | Condition for a function i... |
| fvtresfn 7019 | Functionality of a tuple-r... |
| fvmpts 7020 | Value of a function given ... |
| fvmpt3 7021 | Value of a function given ... |
| fvmpt3i 7022 | Value of a function given ... |
| fvmptdf 7023 | Deduction version of ~ fvm... |
| fvmptd 7024 | Deduction version of ~ fvm... |
| fvmptd2 7025 | Deduction version of ~ fvm... |
| mptrcl 7026 | Reverse closure for a mapp... |
| fvmpt2i 7027 | Value of a function given ... |
| fvmpt2 7028 | Value of a function given ... |
| fvmptss 7029 | If all the values of the m... |
| fvmpt2d 7030 | Deduction version of ~ fvm... |
| fvmptex 7031 | Express a function ` F ` w... |
| fvmptd3f 7032 | Alternate deduction versio... |
| fvmptd2f 7033 | Alternate deduction versio... |
| fvmptdv 7034 | Alternate deduction versio... |
| fvmptdv2 7035 | Alternate deduction versio... |
| mpteqb 7036 | Bidirectional equality the... |
| fvmptt 7037 | Closed theorem form of ~ f... |
| fvmptf 7038 | Value of a function given ... |
| fvmptnf 7039 | The value of a function gi... |
| fvmptd3 7040 | Deduction version of ~ fvm... |
| fvmptd4 7041 | Deduction version of ~ fvm... |
| fvmptn 7042 | This somewhat non-intuitiv... |
| fvmptss2 7043 | A mapping always evaluates... |
| elfvmptrab1w 7044 | Implications for the value... |
| elfvmptrab1 7045 | Implications for the value... |
| elfvmptrab 7046 | Implications for the value... |
| fvopab4ndm 7047 | Value of a function given ... |
| fvmptndm 7048 | Value of a function given ... |
| fvmptrabfv 7049 | Value of a function mappin... |
| fvopab5 7050 | The value of a function th... |
| fvopab6 7051 | Value of a function given ... |
| eqfnfv 7052 | Equality of functions is d... |
| eqfnfv2 7053 | Equality of functions is d... |
| eqfnfv3 7054 | Derive equality of functio... |
| eqfnfvd 7055 | Deduction for equality of ... |
| eqfnfv2f 7056 | Equality of functions is d... |
| fsneq 7057 | Equality condition for two... |
| eqfunfv 7058 | Equality of functions is d... |
| eqfnun 7059 | Two functions on ` A u. B ... |
| fvreseq0 7060 | Equality of restricted fun... |
| fvreseq1 7061 | Equality of a function res... |
| fvreseq 7062 | Equality of restricted fun... |
| fnmptfvd 7063 | A function with a given do... |
| fndmdif 7064 | Two ways to express the lo... |
| fndmdifcom 7065 | The difference set between... |
| fndmdifeq0 7066 | The difference set of two ... |
| fndmin 7067 | Two ways to express the lo... |
| fneqeql 7068 | Two functions are equal if... |
| fneqeql2 7069 | Two functions are equal if... |
| fnreseql 7070 | Two functions are equal on... |
| chfnrn 7071 | The range of a choice func... |
| funfvop 7072 | Ordered pair with function... |
| funfvbrb 7073 | Two ways to say that ` A `... |
| fvimacnvi 7074 | A member of a preimage is ... |
| fvimacnv 7075 | The argument of a function... |
| funimass3 7076 | A kind of contraposition l... |
| funimass5 7077 | A subclass of a preimage i... |
| funconstss 7078 | Two ways of specifying tha... |
| fvimacnvALT 7079 | Alternate proof of ~ fvima... |
| elpreima 7080 | Membership in the preimage... |
| elpreimaOLD 7081 | Obsolete version of ~ elpr... |
| elpreimad 7082 | Membership in the preimage... |
| fniniseg 7083 | Membership in the preimage... |
| fncnvima2 7084 | Inverse images under funct... |
| fniniseg2 7085 | Inverse point images under... |
| unpreima 7086 | Preimage of a union. (Con... |
| inpreima 7087 | Preimage of an intersectio... |
| difpreima 7088 | Preimage of a difference. ... |
| respreima 7089 | The preimage of a restrict... |
| cnvimainrn 7090 | The preimage of the inters... |
| sspreima 7091 | The preimage of a subset i... |
| iunpreima 7092 | Preimage of an indexed uni... |
| iinpreima 7093 | Preimage of an intersectio... |
| intpreima 7094 | Preimage of an intersectio... |
| fimacnvinrn 7095 | Taking the converse image ... |
| fimacnvinrn2 7096 | Taking the converse image ... |
| rescnvimafod 7097 | The restriction of a funct... |
| fvn0ssdmfun 7098 | If a class' function value... |
| fnopfv 7099 | Ordered pair with function... |
| fvelrn 7100 | A function's value belongs... |
| nelrnfvne 7101 | A function value cannot be... |
| fveqdmss 7102 | If the empty set is not co... |
| fveqressseq 7103 | If the empty set is not co... |
| fnfvelrn 7104 | A function's value belongs... |
| ffvelcdm 7105 | A function's value belongs... |
| fnfvelrnd 7106 | A function's value belongs... |
| ffvelcdmi 7107 | A function's value belongs... |
| ffvelcdmda 7108 | A function's value belongs... |
| ffvelcdmd 7109 | A function's value belongs... |
| feldmfvelcdm 7110 | A class is an element of t... |
| rexrn 7111 | Restricted existential qua... |
| ralrn 7112 | Restricted universal quant... |
| elrnrexdm 7113 | For any element in the ran... |
| elrnrexdmb 7114 | For any element in the ran... |
| eldmrexrn 7115 | For any element in the dom... |
| eldmrexrnb 7116 | For any element in the dom... |
| fvcofneq 7117 | The values of two function... |
| ralrnmptw 7118 | A restricted quantifier ov... |
| rexrnmptw 7119 | A restricted quantifier ov... |
| ralrnmpt 7120 | A restricted quantifier ov... |
| rexrnmpt 7121 | A restricted quantifier ov... |
| f0cli 7122 | Unconditional closure of a... |
| dff2 7123 | Alternate definition of a ... |
| dff3 7124 | Alternate definition of a ... |
| dff4 7125 | Alternate definition of a ... |
| dffo3 7126 | An onto mapping expressed ... |
| dffo4 7127 | Alternate definition of an... |
| dffo5 7128 | Alternate definition of an... |
| exfo 7129 | A relation equivalent to t... |
| dffo3f 7130 | An onto mapping expressed ... |
| foelrn 7131 | Property of a surjective f... |
| foelrnf 7132 | Property of a surjective f... |
| foco2 7133 | If a composition of two fu... |
| fmpt 7134 | Functionality of the mappi... |
| f1ompt 7135 | Express bijection for a ma... |
| fmpti 7136 | Functionality of the mappi... |
| fvmptelcdm 7137 | The value of a function at... |
| fmptd 7138 | Domain and codomain of the... |
| fmpttd 7139 | Version of ~ fmptd with in... |
| fmpt3d 7140 | Domain and codomain of the... |
| fmptdf 7141 | A version of ~ fmptd using... |
| fompt 7142 | Express being onto for a m... |
| ffnfv 7143 | A function maps to a class... |
| ffnfvf 7144 | A function maps to a class... |
| fnfvrnss 7145 | An upper bound for range d... |
| fcdmssb 7146 | A function is a function i... |
| rnmptss 7147 | The range of an operation ... |
| rnmptssd 7148 | The range of a function gi... |
| fmpt2d 7149 | Domain and codomain of the... |
| ffvresb 7150 | A necessary and sufficient... |
| fssrescdmd 7151 | Restriction of a function ... |
| f1oresrab 7152 | Build a bijection between ... |
| f1ossf1o 7153 | Restricting a bijection, w... |
| fmptco 7154 | Composition of two functio... |
| fmptcof 7155 | Version of ~ fmptco where ... |
| fmptcos 7156 | Composition of two functio... |
| cofmpt 7157 | Express composition of a m... |
| fcompt 7158 | Express composition of two... |
| fcoconst 7159 | Composition with a constan... |
| fsn 7160 | A function maps a singleto... |
| fsn2 7161 | A function that maps a sin... |
| fsng 7162 | A function maps a singleto... |
| fsn2g 7163 | A function that maps a sin... |
| xpsng 7164 | The Cartesian product of t... |
| xpsn 7165 | The Cartesian product of t... |
| xpprsng 7166 | The Cartesian product of a... |
| xpsnprg 7167 | The Cartesian product of a... |
| xpsntpg 7168 | The Cartesian product of a... |
| f1o2sn 7169 | A singleton consisting in ... |
| residpr 7170 | Restriction of the identit... |
| dfmpt 7171 | Alternate definition for t... |
| fnasrn 7172 | A function expressed as th... |
| idref 7173 | Two ways to state that a r... |
| funiun 7174 | A function is a union of s... |
| funopsn 7175 | If a function is an ordere... |
| funopsnOLD 7176 | Obsolete version of ~ funo... |
| funop 7177 | An ordered pair is a funct... |
| funopdmsn 7178 | The domain of a function w... |
| funsndifnop 7179 | A singleton of an ordered ... |
| funsneqopb 7180 | A singleton of an ordered ... |
| ressnop0 7181 | If ` A ` is not in ` C ` ,... |
| fpr 7182 | A function with a domain o... |
| fprg 7183 | A function with a domain o... |
| ftpg 7184 | A function with a domain o... |
| ftp 7185 | A function with a domain o... |
| fnressn 7186 | A function restricted to a... |
| funressn 7187 | A function restricted to a... |
| fressnfv 7188 | The value of a function re... |
| fvrnressn 7189 | If the value of a function... |
| fvressn 7190 | The value of a function re... |
| fvconst 7191 | The value of a constant fu... |
| fnsnr 7192 | If a class belongs to a fu... |
| fnsnbg 7193 | A function's domain is a s... |
| fnsnb 7194 | A function whose domain is... |
| fmptsn 7195 | Express a singleton functi... |
| fmptsng 7196 | Express a singleton functi... |
| fmptsnd 7197 | Express a singleton functi... |
| fmptap 7198 | Append an additional value... |
| fmptapd 7199 | Append an additional value... |
| fmptpr 7200 | Express a pair function in... |
| fvresi 7201 | The value of a restricted ... |
| fninfp 7202 | Express the class of fixed... |
| fnelfp 7203 | Property of a fixed point ... |
| fndifnfp 7204 | Express the class of non-f... |
| fnelnfp 7205 | Property of a non-fixed po... |
| fnnfpeq0 7206 | A function is the identity... |
| fvunsn 7207 | Remove an ordered pair not... |
| fvsng 7208 | The value of a singleton o... |
| fvsn 7209 | The value of a singleton o... |
| fvsnun1 7210 | The value of a function wi... |
| fvsnun2 7211 | The value of a function wi... |
| fnsnsplit 7212 | Split a function into a si... |
| fsnunf 7213 | Adjoining a point to a fun... |
| fsnunf2 7214 | Adjoining a point to a pun... |
| fsnunfv 7215 | Recover the added point fr... |
| fsnunres 7216 | Recover the original funct... |
| funresdfunsn 7217 | Restricting a function to ... |
| fvpr1g 7218 | The value of a function wi... |
| fvpr2g 7219 | The value of a function wi... |
| fvpr1 7220 | The value of a function wi... |
| fvpr2 7221 | The value of a function wi... |
| fprb 7222 | A condition for functionho... |
| fvtp1 7223 | The first value of a funct... |
| fvtp2 7224 | The second value of a func... |
| fvtp3 7225 | The third value of a funct... |
| fvtp1g 7226 | The value of a function wi... |
| fvtp2g 7227 | The value of a function wi... |
| fvtp3g 7228 | The value of a function wi... |
| fvtp0 7229 | The undefined value of a f... |
| tpres 7230 | An unordered triple of ord... |
| fvconst2g 7231 | The value of a constant fu... |
| fconst2g 7232 | A constant function expres... |
| fvconst2 7233 | The value of a constant fu... |
| fconst2 7234 | A constant function expres... |
| fconst5 7235 | Two ways to express that a... |
| rnmptc 7236 | Range of a constant functi... |
| fnprb 7237 | A function whose domain ha... |
| fntpb 7238 | A function whose domain ha... |
| fnpr2g 7239 | A function whose domain ha... |
| fpr2g 7240 | A function that maps a pai... |
| fconstfv 7241 | A constant function expres... |
| fconst3 7242 | Two ways to express a cons... |
| fconst4 7243 | Two ways to express a cons... |
| resfunexg 7244 | The restriction of a funct... |
| resiexd 7245 | The restriction of the ide... |
| fnex 7246 | If the domain of a functio... |
| fnexd 7247 | If the domain of a functio... |
| funex 7248 | If the domain of a functio... |
| opabex 7249 | Existence of a function ex... |
| mptexg 7250 | If the domain of a functio... |
| mptexgf 7251 | If the domain of a functio... |
| mptex 7252 | If the domain of a functio... |
| mptexd 7253 | If the domain of a functio... |
| mptrabex 7254 | If the domain of a functio... |
| fex 7255 | If the domain of a mapping... |
| fexd 7256 | If the domain of a mapping... |
| mptfvmpt 7257 | A function in maps-to nota... |
| eufnfv 7258 | A function is uniquely det... |
| funfvima 7259 | A function's value in a pr... |
| funfvima2 7260 | A function's value in an i... |
| funfvima2d 7261 | A function's value in a pr... |
| fnfvima 7262 | The function value of an o... |
| fnfvimad 7263 | A function's value belongs... |
| resfvresima 7264 | The value of the function ... |
| funfvima3 7265 | A class including a functi... |
| ralima 7266 | Universal quantification u... |
| rexima 7267 | Existential quantification... |
| fvclss 7268 | Upper bound for the class ... |
| elabrex 7269 | Elementhood in an image se... |
| elabrexg 7270 | Elementhood in an image se... |
| abrexco 7271 | Composition of two image m... |
| imaiun 7272 | The image of an indexed un... |
| imauni 7273 | The image of a union is th... |
| fniunfv 7274 | The indexed union of a fun... |
| funiunfv 7275 | The indexed union of a fun... |
| funiunfvf 7276 | The indexed union of a fun... |
| eluniima 7277 | Membership in the union of... |
| elunirn 7278 | Membership in the union of... |
| elunirnALT 7279 | Alternate proof of ~ eluni... |
| fnunirn 7280 | Membership in a union of s... |
| dff13 7281 | A one-to-one function in t... |
| dff13f 7282 | A one-to-one function in t... |
| f1veqaeq 7283 | If the values of a one-to-... |
| f1cofveqaeq 7284 | If the values of a composi... |
| f1cofveqaeqALT 7285 | Alternate proof of ~ f1cof... |
| dff14i 7286 | A one-to-one function maps... |
| 2f1fvneq 7287 | If two one-to-one function... |
| f1mpt 7288 | Express injection for a ma... |
| f1fveq 7289 | Equality of function value... |
| f1elima 7290 | Membership in the image of... |
| f1imass 7291 | Taking images under a one-... |
| f1imaeq 7292 | Taking images under a one-... |
| f1imapss 7293 | Taking images under a one-... |
| fpropnf1 7294 | A function, given by an un... |
| f1dom3fv3dif 7295 | The function values for a ... |
| f1dom3el3dif 7296 | The codomain of a 1-1 func... |
| dff14a 7297 | A one-to-one function in t... |
| dff14b 7298 | A one-to-one function in t... |
| dff15 7299 | A one-to-one function in t... |
| f1resveqaeq 7300 | If a function restricted t... |
| f1resrcmplf1dlem 7301 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 7302 | If a function's restrictio... |
| f1ounsn 7303 | Extension of a bijection b... |
| f12dfv 7304 | A one-to-one function with... |
| f13dfv 7305 | A one-to-one function with... |
| dff1o6 7306 | A one-to-one onto function... |
| f1ocnvfv1 7307 | The converse value of the ... |
| f1ocnvfv2 7308 | The value of the converse ... |
| f1ocnvfv 7309 | Relationship between the v... |
| f1ocnvfvb 7310 | Relationship between the v... |
| nvof1o 7311 | An involution is a bijecti... |
| nvocnv 7312 | The converse of an involut... |
| f1cdmsn 7313 | If a one-to-one function w... |
| fsnex 7314 | Relate a function with a s... |
| f1prex 7315 | Relate a one-to-one functi... |
| f1ocnvdm 7316 | The value of the converse ... |
| f1ocnvfvrneq 7317 | If the values of a one-to-... |
| fcof1 7318 | An application is injectiv... |
| fcofo 7319 | An application is surjecti... |
| cbvfo 7320 | Change bound variable betw... |
| cbvexfo 7321 | Change bound variable betw... |
| cocan1 7322 | An injection is left-cance... |
| cocan2 7323 | A surjection is right-canc... |
| fcof1oinvd 7324 | Show that a function is th... |
| fcof1od 7325 | A function is bijective if... |
| 2fcoidinvd 7326 | Show that a function is th... |
| fcof1o 7327 | Show that two functions ar... |
| 2fvcoidd 7328 | Show that the composition ... |
| 2fvidf1od 7329 | A function is bijective if... |
| 2fvidinvd 7330 | Show that two functions ar... |
| foeqcnvco 7331 | Condition for function equ... |
| f1eqcocnv 7332 | Condition for function equ... |
| fveqf1o 7333 | Given a bijection ` F ` , ... |
| f1ocoima 7334 | The composition of two bij... |
| nf1const 7335 | A constant function from a... |
| nf1oconst 7336 | A constant function from a... |
| f1ofvswap 7337 | Swapping two values in a b... |
| fvf1pr 7338 | Values of a one-to-one fun... |
| fliftrel 7339 | ` F ` , a function lift, i... |
| fliftel 7340 | Elementhood in the relatio... |
| fliftel1 7341 | Elementhood in the relatio... |
| fliftcnv 7342 | Converse of the relation `... |
| fliftfun 7343 | The function ` F ` is the ... |
| fliftfund 7344 | The function ` F ` is the ... |
| fliftfuns 7345 | The function ` F ` is the ... |
| fliftf 7346 | The domain and range of th... |
| fliftval 7347 | The value of the function ... |
| isoeq1 7348 | Equality theorem for isomo... |
| isoeq2 7349 | Equality theorem for isomo... |
| isoeq3 7350 | Equality theorem for isomo... |
| isoeq4 7351 | Equality theorem for isomo... |
| isoeq5 7352 | Equality theorem for isomo... |
| nfiso 7353 | Bound-variable hypothesis ... |
| isof1o 7354 | An isomorphism is a one-to... |
| isof1oidb 7355 | A function is a bijection ... |
| isof1oopb 7356 | A function is a bijection ... |
| isorel 7357 | An isomorphism connects bi... |
| soisores 7358 | Express the condition of i... |
| soisoi 7359 | Infer isomorphism from one... |
| isoid 7360 | Identity law for isomorphi... |
| isocnv 7361 | Converse law for isomorphi... |
| isocnv2 7362 | Converse law for isomorphi... |
| isocnv3 7363 | Complementation law for is... |
| isores2 7364 | An isomorphism from one we... |
| isores1 7365 | An isomorphism from one we... |
| isores3 7366 | Induced isomorphism on a s... |
| isotr 7367 | Composition (transitive) l... |
| isomin 7368 | Isomorphisms preserve mini... |
| isoini 7369 | Isomorphisms preserve init... |
| isoini2 7370 | Isomorphisms are isomorphi... |
| isofrlem 7371 | Lemma for ~ isofr . (Cont... |
| isoselem 7372 | Lemma for ~ isose . (Cont... |
| isofr 7373 | An isomorphism preserves w... |
| isose 7374 | An isomorphism preserves s... |
| isofr2 7375 | A weak form of ~ isofr tha... |
| isopolem 7376 | Lemma for ~ isopo . (Cont... |
| isopo 7377 | An isomorphism preserves t... |
| isosolem 7378 | Lemma for ~ isoso . (Cont... |
| isoso 7379 | An isomorphism preserves t... |
| isowe 7380 | An isomorphism preserves t... |
| isowe2 7381 | A weak form of ~ isowe tha... |
| f1oiso 7382 | Any one-to-one onto functi... |
| f1oiso2 7383 | Any one-to-one onto functi... |
| f1owe 7384 | Well-ordering of isomorphi... |
| f1oweOLD 7385 | Obsolete version of f1owe ... |
| f1we 7386 | Pull back a well ordering ... |
| weniso 7387 | A set-like well-ordering h... |
| weisoeq 7388 | Thus, there is at most one... |
| weisoeq2 7389 | Thus, there is at most one... |
| knatar 7390 | The Knaster-Tarski theorem... |
| fvresval 7391 | The value of a restricted ... |
| funeldmb 7392 | If ` (/) ` is not part of ... |
| eqfunresadj 7393 | Law for adjoining an eleme... |
| eqfunressuc 7394 | Law for equality of restri... |
| fnssintima 7395 | Condition for subset of an... |
| fnimasnd 7396 | The image of a function by... |
| canth 7397 | No set ` A ` is equinumero... |
| ncanth 7398 | Cantor's theorem fails for... |
| riotaeqdv 7401 | Formula-building deduction... |
| riotabidv 7402 | Formula-building deduction... |
| riotaeqbidv 7403 | Equality deduction for res... |
| riotaex 7404 | Restricted iota is a set. ... |
| riotav 7405 | An iota restricted to the ... |
| riotauni 7406 | Restricted iota in terms o... |
| nfriota1 7407 | The abstraction variable i... |
| nfriotadw 7408 | Deduction version of ~ nfr... |
| cbvriotaw 7409 | Change bound variable in a... |
| cbvriotavw 7410 | Change bound variable in a... |
| nfriotad 7411 | Deduction version of ~ nfr... |
| nfriota 7412 | A variable not free in a w... |
| cbvriota 7413 | Change bound variable in a... |
| cbvriotav 7414 | Change bound variable in a... |
| csbriota 7415 | Interchange class substitu... |
| riotacl2 7416 | Membership law for "the un... |
| riotacl 7417 | Closure of restricted iota... |
| riotasbc 7418 | Substitution law for descr... |
| riotabidva 7419 | Equivalent wff's yield equ... |
| riotabiia 7420 | Equivalent wff's yield equ... |
| riota1 7421 | Property of restricted iot... |
| riota1a 7422 | Property of iota. (Contri... |
| riota2df 7423 | A deduction version of ~ r... |
| riota2f 7424 | This theorem shows a condi... |
| riota2 7425 | This theorem shows a condi... |
| riotaeqimp 7426 | If two restricted iota des... |
| riotaprop 7427 | Properties of a restricted... |
| riota5f 7428 | A method for computing res... |
| riota5 7429 | A method for computing res... |
| riotass2 7430 | Restriction of a unique el... |
| riotass 7431 | Restriction of a unique el... |
| moriotass 7432 | Restriction of a unique el... |
| snriota 7433 | A restricted class abstrac... |
| riotaxfrd 7434 | Change the variable ` x ` ... |
| eusvobj2 7435 | Specify the same property ... |
| eusvobj1 7436 | Specify the same object in... |
| f1ofveu 7437 | There is one domain elemen... |
| f1ocnvfv3 7438 | Value of the converse of a... |
| riotaund 7439 | Restricted iota equals the... |
| riotassuni 7440 | The restricted iota class ... |
| riotaclb 7441 | Bidirectional closure of r... |
| riotarab 7442 | Restricted iota of a restr... |
| oveq 7449 | Equality theorem for opera... |
| oveq1 7450 | Equality theorem for opera... |
| oveq2 7451 | Equality theorem for opera... |
| oveq12 7452 | Equality theorem for opera... |
| oveq1i 7453 | Equality inference for ope... |
| oveq2i 7454 | Equality inference for ope... |
| oveq12i 7455 | Equality inference for ope... |
| oveqi 7456 | Equality inference for ope... |
| oveq123i 7457 | Equality inference for ope... |
| oveq1d 7458 | Equality deduction for ope... |
| oveq2d 7459 | Equality deduction for ope... |
| oveqd 7460 | Equality deduction for ope... |
| oveq12d 7461 | Equality deduction for ope... |
| oveqan12d 7462 | Equality deduction for ope... |
| oveqan12rd 7463 | Equality deduction for ope... |
| oveq123d 7464 | Equality deduction for ope... |
| fvoveq1d 7465 | Equality deduction for nes... |
| fvoveq1 7466 | Equality theorem for neste... |
| ovanraleqv 7467 | Equality theorem for a con... |
| imbrov2fvoveq 7468 | Equality theorem for neste... |
| ovrspc2v 7469 | If an operation value is a... |
| oveqrspc2v 7470 | Restricted specialization ... |
| oveqdr 7471 | Equality of two operations... |
| nfovd 7472 | Deduction version of bound... |
| nfov 7473 | Bound-variable hypothesis ... |
| oprabidw 7474 | The law of concretion. Sp... |
| oprabid 7475 | The law of concretion. Sp... |
| ovex 7476 | The result of an operation... |
| ovexi 7477 | The result of an operation... |
| ovexd 7478 | The result of an operation... |
| ovssunirn 7479 | The result of an operation... |
| 0ov 7480 | Operation value of the emp... |
| ovprc 7481 | The value of an operation ... |
| ovprc1 7482 | The value of an operation ... |
| ovprc2 7483 | The value of an operation ... |
| ovrcl 7484 | Reverse closure for an ope... |
| elfvov1 7485 | Utility theorem: reverse c... |
| elfvov2 7486 | Utility theorem: reverse c... |
| csbov123 7487 | Move class substitution in... |
| csbov 7488 | Move class substitution in... |
| csbov12g 7489 | Move class substitution in... |
| csbov1g 7490 | Move class substitution in... |
| csbov2g 7491 | Move class substitution in... |
| rspceov 7492 | A frequently used special ... |
| elovimad 7493 | Elementhood of the image s... |
| fnbrovb 7494 | Value of a binary operatio... |
| fnotovb 7495 | Equivalence of operation v... |
| opabbrex 7496 | A collection of ordered pa... |
| opabresex2 7497 | Restrictions of a collecti... |
| fvmptopab 7498 | The function value of a ma... |
| f1opr 7499 | Condition for an operation... |
| brfvopab 7500 | The classes involved in a ... |
| dfoprab2 7501 | Class abstraction for oper... |
| reloprab 7502 | An operation class abstrac... |
| oprabv 7503 | If a pair and a class are ... |
| nfoprab1 7504 | The abstraction variables ... |
| nfoprab2 7505 | The abstraction variables ... |
| nfoprab3 7506 | The abstraction variables ... |
| nfoprab 7507 | Bound-variable hypothesis ... |
| oprabbid 7508 | Equivalent wff's yield equ... |
| oprabbidv 7509 | Equivalent wff's yield equ... |
| oprabbii 7510 | Equivalent wff's yield equ... |
| ssoprab2 7511 | Equivalence of ordered pai... |
| ssoprab2b 7512 | Equivalence of ordered pai... |
| eqoprab2bw 7513 | Equivalence of ordered pai... |
| eqoprab2b 7514 | Equivalence of ordered pai... |
| mpoeq123 7515 | An equality theorem for th... |
| mpoeq12 7516 | An equality theorem for th... |
| mpoeq123dva 7517 | An equality deduction for ... |
| mpoeq123dv 7518 | An equality deduction for ... |
| mpoeq123i 7519 | An equality inference for ... |
| mpoeq3dva 7520 | Slightly more general equa... |
| mpoeq3ia 7521 | An equality inference for ... |
| mpoeq3dv 7522 | An equality deduction for ... |
| nfmpo1 7523 | Bound-variable hypothesis ... |
| nfmpo2 7524 | Bound-variable hypothesis ... |
| nfmpo 7525 | Bound-variable hypothesis ... |
| 0mpo0 7526 | A mapping operation with e... |
| mpo0v 7527 | A mapping operation with e... |
| mpo0 7528 | A mapping operation with e... |
| oprab4 7529 | Two ways to state the doma... |
| cbvoprab1 7530 | Rule used to change first ... |
| cbvoprab2 7531 | Change the second bound va... |
| cbvoprab12 7532 | Rule used to change first ... |
| cbvoprab12v 7533 | Rule used to change first ... |
| cbvoprab3 7534 | Rule used to change the th... |
| cbvoprab3v 7535 | Rule used to change the th... |
| cbvmpox 7536 | Rule to change the bound v... |
| cbvmpo 7537 | Rule to change the bound v... |
| cbvmpov 7538 | Rule to change the bound v... |
| elimdelov 7539 | Eliminate a hypothesis whi... |
| brif1 7540 | Move a relation inside and... |
| ovif 7541 | Move a conditional outside... |
| ovif2 7542 | Move a conditional outside... |
| ovif12 7543 | Move a conditional outside... |
| ifov 7544 | Move a conditional outside... |
| ifmpt2v 7545 | Move a conditional inside ... |
| dmoprab 7546 | The domain of an operation... |
| dmoprabss 7547 | The domain of an operation... |
| rnoprab 7548 | The range of an operation ... |
| rnoprab2 7549 | The range of a restricted ... |
| reldmoprab 7550 | The domain of an operation... |
| oprabss 7551 | Structure of an operation ... |
| eloprabga 7552 | The law of concretion for ... |
| eloprabg 7553 | The law of concretion for ... |
| ssoprab2i 7554 | Inference of operation cla... |
| mpov 7555 | Operation with universal d... |
| mpomptx 7556 | Express a two-argument fun... |
| mpompt 7557 | Express a two-argument fun... |
| mpodifsnif 7558 | A mapping with two argumen... |
| mposnif 7559 | A mapping with two argumen... |
| fconstmpo 7560 | Representation of a consta... |
| resoprab 7561 | Restriction of an operatio... |
| resoprab2 7562 | Restriction of an operator... |
| resmpo 7563 | Restriction of the mapping... |
| funoprabg 7564 | "At most one" is a suffici... |
| funoprab 7565 | "At most one" is a suffici... |
| fnoprabg 7566 | Functionality and domain o... |
| mpofun 7567 | The maps-to notation for a... |
| fnoprab 7568 | Functionality and domain o... |
| ffnov 7569 | An operation maps to a cla... |
| fovcld 7570 | Closure law for an operati... |
| fovcl 7571 | Closure law for an operati... |
| eqfnov 7572 | Equality of two operations... |
| eqfnov2 7573 | Two operators with the sam... |
| fnov 7574 | Representation of a functi... |
| mpo2eqb 7575 | Bidirectional equality the... |
| reldmmpo 7576 | The domain of an operation... |
| rnmpo 7577 | The range of an operation ... |
| rnmposs 7578 | The range of an operation ... |
| elrnmpog 7579 | Membership in the range of... |
| elrnmpo 7580 | Membership in the range of... |
| elimampo 7581 | Membership in the image of... |
| elrnmpores 7582 | Membership in the range of... |
| ralrnmpo 7583 | A restricted quantifier ov... |
| rexrnmpo 7584 | A restricted quantifier ov... |
| ovid 7585 | The value of an operation ... |
| ovidig 7586 | The value of an operation ... |
| ovidi 7587 | The value of an operation ... |
| ov 7588 | The value of an operation ... |
| ovigg 7589 | The value of an operation ... |
| ovig 7590 | The value of an operation ... |
| ovmpt4g 7591 | Value of a function given ... |
| ovmpos 7592 | Value of a function given ... |
| ov2gf 7593 | The value of an operation ... |
| ovmpodxf 7594 | Value of an operation give... |
| ovmpodx 7595 | Value of an operation give... |
| ovmpod 7596 | Value of an operation give... |
| ovmpox 7597 | The value of an operation ... |
| ovmpoga 7598 | Value of an operation give... |
| ovmpoa 7599 | Value of an operation give... |
| ovmpodf 7600 | Alternate deduction versio... |
| ovmpodv 7601 | Alternate deduction versio... |
| ovmpodv2 7602 | Alternate deduction versio... |
| ovmpog 7603 | Value of an operation give... |
| ovmpo 7604 | Value of an operation give... |
| ovmpot 7605 | The value of an operation ... |
| fvmpopr2d 7606 | Value of an operation give... |
| ov3 7607 | The value of an operation ... |
| ov6g 7608 | The value of an operation ... |
| ovg 7609 | The value of an operation ... |
| ovres 7610 | The value of a restricted ... |
| ovresd 7611 | Lemma for converting metri... |
| oprres 7612 | The restriction of an oper... |
| ovn0ssdmfun 7613 | If a class' operation valu... |
| oprssov 7614 | The value of a member of t... |
| fovcdm 7615 | An operation's value belon... |
| fovcdmda 7616 | An operation's value belon... |
| fovcdmd 7617 | An operation's value belon... |
| fnrnov 7618 | The range of an operation ... |
| foov 7619 | An onto mapping of an oper... |
| fnovrn 7620 | An operation's value belon... |
| ovelrn 7621 | A member of an operation's... |
| funimassov 7622 | Membership relation for th... |
| ovelimab 7623 | Operation value in an imag... |
| ovima0 7624 | An operation value is a me... |
| ovconst2 7625 | The value of a constant op... |
| oprssdm 7626 | Domain of closure of an op... |
| nssdmovg 7627 | The value of an operation ... |
| ndmovg 7628 | The value of an operation ... |
| ndmov 7629 | The value of an operation ... |
| ndmovcl 7630 | The closure of an operatio... |
| ndmovrcl 7631 | Reverse closure law, when ... |
| ndmovcom 7632 | Any operation is commutati... |
| ndmovass 7633 | Any operation is associati... |
| ndmovdistr 7634 | Any operation is distribut... |
| ndmovord 7635 | Elimination of redundant a... |
| ndmovordi 7636 | Elimination of redundant a... |
| caovclg 7637 | Convert an operation closu... |
| caovcld 7638 | Convert an operation closu... |
| caovcl 7639 | Convert an operation closu... |
| caovcomg 7640 | Convert an operation commu... |
| caovcomd 7641 | Convert an operation commu... |
| caovcom 7642 | Convert an operation commu... |
| caovassg 7643 | Convert an operation assoc... |
| caovassd 7644 | Convert an operation assoc... |
| caovass 7645 | Convert an operation assoc... |
| caovcang 7646 | Convert an operation cance... |
| caovcand 7647 | Convert an operation cance... |
| caovcanrd 7648 | Commute the arguments of a... |
| caovcan 7649 | Convert an operation cance... |
| caovordig 7650 | Convert an operation order... |
| caovordid 7651 | Convert an operation order... |
| caovordg 7652 | Convert an operation order... |
| caovordd 7653 | Convert an operation order... |
| caovord2d 7654 | Operation ordering law wit... |
| caovord3d 7655 | Ordering law. (Contribute... |
| caovord 7656 | Convert an operation order... |
| caovord2 7657 | Operation ordering law wit... |
| caovord3 7658 | Ordering law. (Contribute... |
| caovdig 7659 | Convert an operation distr... |
| caovdid 7660 | Convert an operation distr... |
| caovdir2d 7661 | Convert an operation distr... |
| caovdirg 7662 | Convert an operation rever... |
| caovdird 7663 | Convert an operation distr... |
| caovdi 7664 | Convert an operation distr... |
| caov32d 7665 | Rearrange arguments in a c... |
| caov12d 7666 | Rearrange arguments in a c... |
| caov31d 7667 | Rearrange arguments in a c... |
| caov13d 7668 | Rearrange arguments in a c... |
| caov4d 7669 | Rearrange arguments in a c... |
| caov411d 7670 | Rearrange arguments in a c... |
| caov42d 7671 | Rearrange arguments in a c... |
| caov32 7672 | Rearrange arguments in a c... |
| caov12 7673 | Rearrange arguments in a c... |
| caov31 7674 | Rearrange arguments in a c... |
| caov13 7675 | Rearrange arguments in a c... |
| caov4 7676 | Rearrange arguments in a c... |
| caov411 7677 | Rearrange arguments in a c... |
| caov42 7678 | Rearrange arguments in a c... |
| caovdir 7679 | Reverse distributive law. ... |
| caovdilem 7680 | Lemma used by real number ... |
| caovlem2 7681 | Lemma used in real number ... |
| caovmo 7682 | Uniqueness of inverse elem... |
| imaeqexov 7683 | Substitute an operation va... |
| imaeqalov 7684 | Substitute an operation va... |
| mpondm0 7685 | The value of an operation ... |
| elmpocl 7686 | If a two-parameter class i... |
| elmpocl1 7687 | If a two-parameter class i... |
| elmpocl2 7688 | If a two-parameter class i... |
| elovmpod 7689 | Utility lemma for two-para... |
| elovmpo 7690 | Utility lemma for two-para... |
| elovmporab 7691 | Implications for the value... |
| elovmporab1w 7692 | Implications for the value... |
| elovmporab1 7693 | Implications for the value... |
| 2mpo0 7694 | If the operation value of ... |
| relmptopab 7695 | Any function to sets of or... |
| f1ocnvd 7696 | Describe an implicit one-t... |
| f1od 7697 | Describe an implicit one-t... |
| f1ocnv2d 7698 | Describe an implicit one-t... |
| f1o2d 7699 | Describe an implicit one-t... |
| f1opw2 7700 | A one-to-one mapping induc... |
| f1opw 7701 | A one-to-one mapping induc... |
| elovmpt3imp 7702 | If the value of a function... |
| ovmpt3rab1 7703 | The value of an operation ... |
| ovmpt3rabdm 7704 | If the value of a function... |
| elovmpt3rab1 7705 | Implications for the value... |
| elovmpt3rab 7706 | Implications for the value... |
| mpt3mpt 7709 | Express a three-argument f... |
| mpt3eqdv 7710 | An equality deduction for ... |
| funmpt3 7711 | A function in maps-to nota... |
| mpt3fvd 7712 | Value of a three-argument ... |
| mpt3fvotd 7713 | Value of a three-argument ... |
| mpt3fvot2d 7714 | Value of a three-argument ... |
| ofeqd 7719 | Equality theorem for funct... |
| ofeq 7720 | Equality theorem for funct... |
| ofreq 7721 | Equality theorem for funct... |
| ofexg 7722 | A function operation restr... |
| nfof 7723 | Hypothesis builder for fun... |
| nfofr 7724 | Hypothesis builder for fun... |
| ofrfvalg 7725 | Value of a relation applie... |
| offval 7726 | Value of an operation appl... |
| ofrfval 7727 | Value of a relation applie... |
| ofval 7728 | Evaluate a function operat... |
| ofrval 7729 | Exhibit a function relatio... |
| offn 7730 | The function operation pro... |
| offun 7731 | The function operation pro... |
| offval2f 7732 | The function operation exp... |
| ofmresval 7733 | Value of a restriction of ... |
| fnfvof 7734 | Function value of a pointw... |
| off 7735 | The function operation pro... |
| ofres 7736 | Restrict the operands of a... |
| offval2 7737 | The function operation exp... |
| ofrfval2 7738 | The function relation acti... |
| offvalfv 7739 | The function operation exp... |
| ofmpteq 7740 | Value of a pointwise opera... |
| coof 7741 | The composition of a _homo... |
| ofco 7742 | The composition of a funct... |
| offveq 7743 | Convert an identity of the... |
| offveqb 7744 | Equivalent expressions for... |
| ofc1 7745 | Left operation by a consta... |
| ofc2 7746 | Right operation by a const... |
| ofc12 7747 | Function operation on two ... |
| caofref 7748 | Transfer a reflexive law t... |
| caofinvl 7749 | Transfer a left inverse la... |
| caofid0l 7750 | Transfer a left identity l... |
| caofid0r 7751 | Transfer a right identity ... |
| caofid1 7752 | Transfer a right absorptio... |
| caofid2 7753 | Transfer a right absorptio... |
| caofcom 7754 | Transfer a commutative law... |
| caofidlcan 7755 | Transfer a cancellation/id... |
| caofrss 7756 | Transfer a relation subset... |
| caofass 7757 | Transfer an associative la... |
| caoftrn 7758 | Transfer a transitivity la... |
| caofdi 7759 | Transfer a distributive la... |
| caofdir 7760 | Transfer a reverse distrib... |
| caonncan 7761 | Transfer ~ nncan -shaped l... |
| relrpss 7764 | The proper subset relation... |
| brrpssg 7765 | The proper subset relation... |
| brrpss 7766 | The proper subset relation... |
| porpss 7767 | Every class is partially o... |
| sorpss 7768 | Express strict ordering un... |
| sorpssi 7769 | Property of a chain of set... |
| sorpssun 7770 | A chain of sets is closed ... |
| sorpssin 7771 | A chain of sets is closed ... |
| sorpssuni 7772 | In a chain of sets, a maxi... |
| sorpssint 7773 | In a chain of sets, a mini... |
| sorpsscmpl 7774 | The componentwise compleme... |
| zfun 7776 | Axiom of Union expressed w... |
| axun2 7777 | A variant of the Axiom of ... |
| uniex2 7778 | The Axiom of Union using t... |
| uniex2OLD 7779 | Obsolete version of ~ unie... |
| vuniex 7780 | The union of a setvar is a... |
| uniexg 7781 | The ZF Axiom of Union in c... |
| uniex 7782 | The Axiom of Union in clas... |
| uniexd 7783 | Deduction version of the Z... |
| unexg 7784 | The union of two sets is a... |
| unex 7785 | The union of two sets is a... |
| tpex 7786 | An unordered triple of cla... |
| unexb 7787 | Existence of union is equi... |
| xpexg 7788 | The Cartesian product of t... |
| xpexd 7789 | The Cartesian product of t... |
| 3xpexg 7790 | The Cartesian product of t... |
| xpex 7791 | The Cartesian product of t... |
| unexd 7792 | The union of two sets is a... |
| sqxpexg 7793 | The Cartesian square of a ... |
| abnexg 7794 | Sufficient condition for a... |
| abnex 7795 | Sufficient condition for a... |
| snnex 7796 | The class of all singleton... |
| pwnex 7797 | The class of all power set... |
| difex2 7798 | If the subtrahend of a cla... |
| difsnexi 7799 | If the difference of a cla... |
| uniuni 7800 | Expression for double unio... |
| uniexr 7801 | Converse of the Axiom of U... |
| uniexb 7802 | The Axiom of Union and its... |
| pwexr 7803 | Converse of the Axiom of P... |
| pwexb 7804 | The Axiom of Power Sets an... |
| elpwpwel 7805 | A class belongs to a doubl... |
| eldifpw 7806 | Membership in a power clas... |
| elpwun 7807 | Membership in the power cl... |
| pwuncl 7808 | Power classes are closed u... |
| iunpw 7809 | An indexed union of a powe... |
| fr3nr 7810 | A well-founded relation ha... |
| epne3 7811 | A well-founded class conta... |
| dfwe2 7812 | Alternate definition of we... |
| epweon 7813 | The membership relation we... |
| epweonALT 7814 | Alternate proof of ~ epweo... |
| ordon 7815 | The class of all ordinal n... |
| onprc 7816 | No set contains all ordina... |
| ssorduni 7817 | The union of a class of or... |
| ssonuni 7818 | The union of a set of ordi... |
| ssonunii 7819 | The union of a set of ordi... |
| ordeleqon 7820 | A way to express the ordin... |
| ordsson 7821 | Any ordinal class is a sub... |
| dford5 7822 | A class is ordinal iff it ... |
| onss 7823 | An ordinal number is a sub... |
| predon 7824 | The predecessor of an ordi... |
| ssonprc 7825 | Two ways of saying a class... |
| onuni 7826 | The union of an ordinal nu... |
| orduni 7827 | The union of an ordinal cl... |
| onint 7828 | The intersection (infimum)... |
| onint0 7829 | The intersection of a clas... |
| onssmin 7830 | A nonempty class of ordina... |
| onminesb 7831 | If a property is true for ... |
| onminsb 7832 | If a property is true for ... |
| oninton 7833 | The intersection of a none... |
| onintrab 7834 | The intersection of a clas... |
| onintrab2 7835 | An existence condition equ... |
| onnmin 7836 | No member of a set of ordi... |
| onnminsb 7837 | An ordinal number smaller ... |
| oneqmin 7838 | A way to show that an ordi... |
| uniordint 7839 | The union of a set of ordi... |
| onminex 7840 | If a wff is true for an or... |
| sucon 7841 | The class of all ordinal n... |
| sucexb 7842 | A successor exists iff its... |
| sucexg 7843 | The successor of a set is ... |
| sucex 7844 | The successor of a set is ... |
| onmindif2 7845 | The minimum of a class of ... |
| ordsuci 7846 | The successor of an ordina... |
| sucexeloni 7847 | If the successor of an ord... |
| onsuc 7848 | The successor of an ordina... |
| ordsuc 7849 | A class is ordinal if and ... |
| ordpwsuc 7850 | The collection of ordinals... |
| onpwsuc 7851 | The collection of ordinal ... |
| onsucb 7852 | A class is an ordinal numb... |
| ordsucss 7853 | The successor of an elemen... |
| onpsssuc 7854 | An ordinal number is a pro... |
| ordelsuc 7855 | A set belongs to an ordina... |
| onsucmin 7856 | The successor of an ordina... |
| ordsucelsuc 7857 | Membership is inherited by... |
| ordsucsssuc 7858 | The subclass relationship ... |
| ordsucuniel 7859 | Given an element ` A ` of ... |
| ordsucun 7860 | The successor of the maxim... |
| ordunpr 7861 | The maximum of two ordinal... |
| ordunel 7862 | The maximum of two ordinal... |
| onsucuni 7863 | A class of ordinal numbers... |
| ordsucuni 7864 | An ordinal class is a subc... |
| orduniorsuc 7865 | An ordinal class is either... |
| unon 7866 | The class of all ordinal n... |
| ordunisuc 7867 | An ordinal class is equal ... |
| orduniss2 7868 | The union of the ordinal s... |
| onsucuni2 7869 | A successor ordinal is the... |
| 0elsuc 7870 | The successor of an ordina... |
| limon 7871 | The class of ordinal numbe... |
| onuniorsuc 7872 | An ordinal number is eithe... |
| onssi 7873 | An ordinal number is a sub... |
| onsuci 7874 | The successor of an ordina... |
| onuninsuci 7875 | An ordinal is equal to its... |
| onsucssi 7876 | A set belongs to an ordina... |
| nlimsucg 7877 | A successor is not a limit... |
| orduninsuc 7878 | An ordinal class is equal ... |
| ordunisuc2 7879 | An ordinal equal to its un... |
| ordzsl 7880 | An ordinal is zero, a succ... |
| onzsl 7881 | An ordinal number is zero,... |
| dflim3 7882 | An alternate definition of... |
| dflim4 7883 | An alternate definition of... |
| limsuc 7884 | The successor of a member ... |
| limsssuc 7885 | A class includes a limit o... |
| nlimon 7886 | Two ways to express the cl... |
| limuni3 7887 | The union of a nonempty cl... |
| tfi 7888 | The Principle of Transfini... |
| tfisg 7889 | A closed form of ~ tfis . ... |
| tfis 7890 | Transfinite Induction Sche... |
| tfis2f 7891 | Transfinite Induction Sche... |
| tfis2 7892 | Transfinite Induction Sche... |
| tfis3 7893 | Transfinite Induction Sche... |
| tfisi 7894 | A transfinite induction sc... |
| tfinds 7895 | Principle of Transfinite I... |
| tfindsg 7896 | Transfinite Induction (inf... |
| tfindsg2 7897 | Transfinite Induction (inf... |
| tfindes 7898 | Transfinite Induction with... |
| tfinds2 7899 | Transfinite Induction (inf... |
| tfinds3 7900 | Principle of Transfinite I... |
| dfom2 7903 | An alternate definition of... |
| elom 7904 | Membership in omega. The ... |
| omsson 7905 | Omega is a subset of ` On ... |
| limomss 7906 | The class of natural numbe... |
| nnon 7907 | A natural number is an ord... |
| nnoni 7908 | A natural number is an ord... |
| nnord 7909 | A natural number is ordina... |
| trom 7910 | The class of finite ordina... |
| ordom 7911 | The class of finite ordina... |
| elnn 7912 | A member of a natural numb... |
| omon 7913 | The class of natural numbe... |
| omelon2 7914 | Omega is an ordinal number... |
| nnlim 7915 | A natural number is not a ... |
| omssnlim 7916 | The class of natural numbe... |
| limom 7917 | Omega is a limit ordinal. ... |
| peano2b 7918 | A class belongs to omega i... |
| nnsuc 7919 | A nonzero natural number i... |
| omsucne 7920 | A natural number is not th... |
| ssnlim 7921 | An ordinal subclass of non... |
| omsinds 7922 | Strong (or "total") induct... |
| omun 7923 | The union of two finite or... |
| peano1 7924 | Zero is a natural number. ... |
| peano2 7925 | The successor of any natur... |
| peano3 7926 | The successor of any natur... |
| peano3OLD 7927 | Obsolete version of ~ pean... |
| peano4 7928 | Two natural numbers are eq... |
| peano5 7929 | The induction postulate: a... |
| nn0suc 7930 | A natural number is either... |
| find 7931 | The Principle of Finite In... |
| finds 7932 | Principle of Finite Induct... |
| findsg 7933 | Principle of Finite Induct... |
| finds2 7934 | Principle of Finite Induct... |
| finds1 7935 | Principle of Finite Induct... |
| findes 7936 | Finite induction with expl... |
| dmexg 7937 | The domain of a set is a s... |
| rnexg 7938 | The range of a set is a se... |
| dmexd 7939 | The domain of a set is a s... |
| fndmexd 7940 | If a function is a set, it... |
| dmfex 7941 | If a mapping is a set, its... |
| fndmexb 7942 | The domain of a function i... |
| fdmexb 7943 | The domain of a function i... |
| dmfexALT 7944 | Alternate proof of ~ dmfex... |
| dmex 7945 | The domain of a set is a s... |
| rnex 7946 | The range of a set is a se... |
| iprc 7947 | The identity function is a... |
| resiexg 7948 | The existence of a restric... |
| imaexg 7949 | The image of a set is a se... |
| imaex 7950 | The image of a set is a se... |
| rnexd 7951 | The range of a set is a se... |
| imaexd 7952 | The image of a set is a se... |
| exse2 7953 | Any set relation is set-li... |
| xpexr 7954 | If a Cartesian product is ... |
| xpexr2 7955 | If a nonempty Cartesian pr... |
| xpexcnv 7956 | A condition where the conv... |
| soex 7957 | If the relation in a stric... |
| elxp4 7958 | Membership in a Cartesian ... |
| elxp5 7959 | Membership in a Cartesian ... |
| cnvexg 7960 | The converse of a set is a... |
| cnvex 7961 | The converse of a set is a... |
| relcnvexb 7962 | A relation is a set iff it... |
| f1oexrnex 7963 | If the range of a 1-1 onto... |
| f1oexbi 7964 | There is a one-to-one onto... |
| coexg 7965 | The composition of two set... |
| coex 7966 | The composition of two set... |
| coexd 7967 | The composition of two set... |
| funcnvuni 7968 | The union of a chain (with... |
| fun11uni 7969 | The union of a chain (with... |
| resf1extb 7970 | Extension of an injection ... |
| resf1ext2b 7971 | Extension of an injection ... |
| fex2 7972 | A function with bounded do... |
| fabexd 7973 | Existence of a set of func... |
| fabexg 7974 | Existence of a set of func... |
| fabex 7975 | Existence of a set of func... |
| mapex 7976 | The class of all functions... |
| f1oabexg 7977 | The class of all 1-1-onto ... |
| fiunlem 7978 | Lemma for ~ fiun and ~ f1i... |
| fiun 7979 | The union of a chain (with... |
| f1iun 7980 | The union of a chain (with... |
| fviunfun 7981 | The function value of an i... |
| ffoss 7982 | Relationship between a map... |
| f11o 7983 | Relationship between one-t... |
| resfunexgALT 7984 | Alternate proof of ~ resfu... |
| cofunexg 7985 | Existence of a composition... |
| cofunex2g 7986 | Existence of a composition... |
| fnexALT 7987 | Alternate proof of ~ fnex ... |
| funexw 7988 | Weak version of ~ funex th... |
| mptexw 7989 | Weak version of ~ mptex th... |
| funrnex 7990 | If the domain of a functio... |
| zfrep6OLD 7991 | Obsolete proof of ~ zfrep6... |
| focdmex 7992 | If the domain of an onto f... |
| f1dmex 7993 | If the codomain of a one-t... |
| f1ovv 7994 | The codomain/range of a 1-... |
| fvclex 7995 | Existence of the class of ... |
| fvresex 7996 | Existence of the class of ... |
| abrexexg 7997 | Existence of a class abstr... |
| abrexex 7998 | Existence of a class abstr... |
| iunexg 7999 | The existence of an indexe... |
| abrexex2g 8000 | Existence of an existentia... |
| opabex3d 8001 | Existence of an ordered pa... |
| opabex3rd 8002 | Existence of an ordered pa... |
| opabex3 8003 | Existence of an ordered pa... |
| iunex 8004 | The existence of an indexe... |
| abrexex2 8005 | Existence of an existentia... |
| abexssex 8006 | Existence of a class abstr... |
| abexex 8007 | A condition where a class ... |
| f1oweALT 8008 | Alternate proof of one dir... |
| wemoiso 8009 | Thus, there is at most one... |
| wemoiso2 8010 | Thus, there is at most one... |
| oprabexd 8011 | Existence of an operator a... |
| oprabex 8012 | Existence of an operation ... |
| oprabex3 8013 | Existence of an operation ... |
| oprabrexex2 8014 | Existence of an existentia... |
| ab2rexex 8015 | Existence of a class abstr... |
| ab2rexex2 8016 | Existence of an existentia... |
| xpexgALT 8017 | Alternate proof of ~ xpexg... |
| offval3 8018 | General value of ` ( F oF ... |
| offres 8019 | Pointwise combination comm... |
| ofmres 8020 | Equivalent expressions for... |
| ofmresex 8021 | Existence of a restriction... |
| mptcnfimad 8022 | The converse of a mapping ... |
| 1stval 8027 | The value of the function ... |
| 2ndval 8028 | The value of the function ... |
| 1stnpr 8029 | Value of the first-member ... |
| 2ndnpr 8030 | Value of the second-member... |
| 1st0 8031 | The value of the first-mem... |
| 2nd0 8032 | The value of the second-me... |
| op1st 8033 | Extract the first member o... |
| op2nd 8034 | Extract the second member ... |
| op1std 8035 | Extract the first member o... |
| op2ndd 8036 | Extract the second member ... |
| op1stg 8037 | Extract the first member o... |
| op2ndg 8038 | Extract the second member ... |
| ot1stg 8039 | Extract the first member o... |
| ot2ndg 8040 | Extract the second member ... |
| ot3rdg 8041 | Extract the third member o... |
| 1stval2 8042 | Alternate value of the fun... |
| 2ndval2 8043 | Alternate value of the fun... |
| oteqimp 8044 | The components of an order... |
| fo1st 8045 | The ` 1st ` function maps ... |
| fo2nd 8046 | The ` 2nd ` function maps ... |
| br1steqg 8047 | Uniqueness condition for t... |
| br2ndeqg 8048 | Uniqueness condition for t... |
| f1stres 8049 | Mapping of a restriction o... |
| f2ndres 8050 | Mapping of a restriction o... |
| fo1stres 8051 | Onto mapping of a restrict... |
| fo2ndres 8052 | Onto mapping of a restrict... |
| 1st2val 8053 | Value of an alternate defi... |
| 2nd2val 8054 | Value of an alternate defi... |
| 1stcof 8055 | Composition of the first m... |
| 2ndcof 8056 | Composition of the second ... |
| xp1st 8057 | Location of the first elem... |
| xp2nd 8058 | Location of the second ele... |
| elxp6 8059 | Membership in a Cartesian ... |
| elxp7 8060 | Membership in a Cartesian ... |
| eqopi 8061 | Equality with an ordered p... |
| xp2 8062 | Representation of Cartesia... |
| unielxp 8063 | The membership relation fo... |
| 1st2nd2 8064 | Reconstruction of a member... |
| 1st2ndb 8065 | Reconstruction of an order... |
| xpopth 8066 | An ordered pair theorem fo... |
| eqop 8067 | Two ways to express equali... |
| eqop2 8068 | Two ways to express equali... |
| op1steq 8069 | Two ways of expressing tha... |
| opreuopreu 8070 | There is a unique ordered ... |
| el2xptp0 8071 | A member of a nested Carte... |
| el2xpss 8072 | Version of ~ elrel for tri... |
| 2nd1st 8073 | Swap the members of an ord... |
| 1st2nd 8074 | Reconstruction of a member... |
| 1stdm 8075 | The first ordered pair com... |
| 2ndrn 8076 | The second ordered pair co... |
| 1st2ndbr 8077 | Express an element of a re... |
| releldm2 8078 | Two ways of expressing mem... |
| reldm 8079 | An expression for the doma... |
| reldmOLD 8080 | Obsolete version of ~ reld... |
| releldmdifi 8081 | One way of expressing memb... |
| funfv1st2nd 8082 | The function value for the... |
| funelss 8083 | If the first component of ... |
| funeldmdif 8084 | Two ways of expressing mem... |
| sbcopeq1a 8085 | Equality theorem for subst... |
| csbopeq1a 8086 | Equality theorem for subst... |
| sbcoteq1a 8087 | Equality theorem for subst... |
| dfopab2 8088 | A way to define an ordered... |
| dfoprab3s 8089 | A way to define an operati... |
| dfoprab3 8090 | Operation class abstractio... |
| dfoprab4 8091 | Operation class abstractio... |
| dfoprab4f 8092 | Operation class abstractio... |
| opabex2 8093 | Condition for an operation... |
| opabn1stprc 8094 | An ordered-pair class abst... |
| opiota 8095 | The property of a uniquely... |
| cnvoprab 8096 | The converse of a class ab... |
| dfxp3 8097 | Define the Cartesian produ... |
| elopabi 8098 | A consequence of membershi... |
| eloprabi 8099 | A consequence of membershi... |
| mpomptsx 8100 | Express a two-argument fun... |
| mpompts 8101 | Express a two-argument fun... |
| dmmpossx 8102 | The domain of a mapping is... |
| fmpox 8103 | Functionality, domain and ... |
| fmpo 8104 | Functionality, domain and ... |
| fnmpo 8105 | Functionality and domain o... |
| fnmpoi 8106 | Functionality and domain o... |
| dmmpo 8107 | Domain of a class given by... |
| fmpodg 8108 | Domain and codomain of the... |
| fmpod 8109 | Domain and codomain of the... |
| ovmpoelrn 8110 | An operation's value belon... |
| dmmpoga 8111 | Domain of an operation giv... |
| dmmpog 8112 | Domain of an operation giv... |
| mpoexxg 8113 | Existence of an operation ... |
| mpoexg 8114 | Existence of an operation ... |
| mpoexga 8115 | If the domain of an operat... |
| mpoexw 8116 | Weak version of ~ mpoex th... |
| mpoex 8117 | If the domain of an operat... |
| mpoexd 8118 | Existence of an operation ... |
| mptmpoopabbrd 8119 | The operation value of a f... |
| mptmpoopabovd 8120 | The operation value of a f... |
| el2mpocsbcl 8121 | If the operation value of ... |
| el2mpocl 8122 | If the operation value of ... |
| fnmpoovd 8123 | A function with a Cartesia... |
| offval22 8124 | The function operation exp... |
| brovpreldm 8125 | If a binary relation holds... |
| bropopvvv 8126 | If a binary relation holds... |
| bropfvvvvlem 8127 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8128 | If a binary relation holds... |
| ovmptss 8129 | If all the values of the m... |
| ovmptssOLD 8130 | Obsolete version of ~ ovmp... |
| relmpoopab 8131 | Any function to sets of or... |
| fmpoco 8132 | Composition of two functio... |
| oprabco 8133 | Composition of a function ... |
| oprab2co 8134 | Composition of operator ab... |
| f1od2 8135 | Sufficient condition for a... |
| df1st2 8136 | An alternate possible defi... |
| df2nd2 8137 | An alternate possible defi... |
| 1stconst 8138 | The mapping of a restricti... |
| 2ndconst 8139 | The mapping of a restricti... |
| dfmpo 8140 | Alternate definition for t... |
| mposn 8141 | An operation (in maps-to n... |
| curry1 8142 | Composition with ` ``' ( 2... |
| curry1val 8143 | The value of a curried fun... |
| curry1f 8144 | Functionality of a curried... |
| curry2 8145 | Composition with ` ``' ( 1... |
| curry2f 8146 | Functionality of a curried... |
| curry2val 8147 | The value of a curried fun... |
| cnvf1olem 8148 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8149 | Describe a function that m... |
| fparlem1 8150 | Lemma for ~ fpar . (Contr... |
| fparlem2 8151 | Lemma for ~ fpar . (Contr... |
| fparlem3 8152 | Lemma for ~ fpar . (Contr... |
| fparlem4 8153 | Lemma for ~ fpar . (Contr... |
| fpar 8154 | Merge two functions in par... |
| fsplit 8155 | A function that can be use... |
| fsplitfpar 8156 | Merge two functions with a... |
| offsplitfpar 8157 | Express the function opera... |
| f2ndf 8158 | The ` 2nd ` (second compon... |
| fo2ndf 8159 | The ` 2nd ` (second compon... |
| f1o2ndf1 8160 | The ` 2nd ` (second compon... |
| opco1 8161 | Value of an operation prec... |
| opco2 8162 | Value of an operation prec... |
| opco1i 8163 | Inference form of ~ opco1 ... |
| mpof1o2d 8164 | Sufficient condition for a... |
| frxp 8165 | A lexicographical ordering... |
| xporderlem 8166 | Lemma for lexicographical ... |
| poxp 8167 | A lexicographical ordering... |
| soxp 8168 | A lexicographical ordering... |
| wexp 8169 | A lexicographical ordering... |
| fnwelem 8170 | Lemma for ~ fnwe . (Contr... |
| fnwe 8171 | A variant on lexicographic... |
| fnwe2lem1 8172 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 8173 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem3 8174 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem4 8175 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 8176 | A well-ordering can be con... |
| fnse 8177 | Condition for the well-ord... |
| fvproj 8178 | Value of a function on ord... |
| fimaproj 8179 | Image of a cartesian produ... |
| ralxpes 8180 | A version of ~ ralxp with ... |
| ralxp3f 8181 | Restricted for all over a ... |
| ralxp3 8182 | Restricted for all over a ... |
| ralxp3es 8183 | Restricted for-all over a ... |
| frpoins3xpg 8184 | Special case of founded pa... |
| frpoins3xp3g 8185 | Special case of founded pa... |
| xpord2lem 8186 | Lemma for Cartesian produc... |
| poxp2 8187 | Another way of partially o... |
| frxp2 8188 | Another way of giving a we... |
| xpord2pred 8189 | Calculate the predecessor ... |
| sexp2 8190 | Condition for the relation... |
| xpord2indlem 8191 | Induction over the Cartesi... |
| xpord2ind 8192 | Induction over the Cartesi... |
| xpord3lem 8193 | Lemma for triple ordering.... |
| poxp3 8194 | Triple Cartesian product p... |
| frxp3 8195 | Give well-foundedness over... |
| xpord3pred 8196 | Calculate the predecsessor... |
| sexp3 8197 | Show that the triple order... |
| xpord3inddlem 8198 | Induction over the triple ... |
| xpord3indd 8199 | Induction over the triple ... |
| xpord3ind 8200 | Induction over the triple ... |
| orderseqlem 8201 | Lemma for ~ poseq and ~ so... |
| poseq 8202 | A partial ordering of ordi... |
| soseq 8203 | A linear ordering of ordin... |
| suppval 8206 | The value of the operation... |
| supp0prc 8207 | The support of a class is ... |
| suppvalbr 8208 | The value of the operation... |
| suppvalbrOLD 8209 | Obsolete version of ~ supp... |
| supp0 8210 | The support of the empty s... |
| suppval1 8211 | The value of the operation... |
| suppvalfng 8212 | The value of the operation... |
| suppvalfn 8213 | The value of the operation... |
| elsuppfng 8214 | An element of the support ... |
| elsuppfn 8215 | An element of the support ... |
| fvdifsupp 8216 | Function value is zero out... |
| cnvimadfsn 8217 | The support of functions "... |
| suppimacnvss 8218 | The support of functions "... |
| suppimacnv 8219 | Support sets of functions ... |
| fsuppeq 8220 | Two ways of writing the su... |
| fsuppeqg 8221 | Version of ~ fsuppeq avoid... |
| suppssdm 8222 | The support of a function ... |
| suppsnop 8223 | The support of a singleton... |
| snopsuppss 8224 | The support of a singleton... |
| fvn0elsupp 8225 | If the function value for ... |
| fvn0elsuppb 8226 | The function value for a g... |
| rexsupp 8227 | Existential quantification... |
| ressuppss 8228 | The support of the restric... |
| suppun 8229 | The support of a class/fun... |
| ressuppssdif 8230 | The support of the restric... |
| mptsuppdifd 8231 | The support of a function ... |
| mptsuppd 8232 | The support of a function ... |
| extmptsuppeq 8233 | The support of an extended... |
| suppfnss 8234 | The support of a function ... |
| funsssuppss 8235 | The support of a function ... |
| fnsuppres 8236 | Two ways to express restri... |
| fnsuppeq0 8237 | The support of a function ... |
| fczsupp0 8238 | The support of a constant ... |
| suppss 8239 | Show that the support of a... |
| suppssr 8240 | A function is zero outside... |
| suppssrg 8241 | A function is zero outside... |
| suppssov1 8242 | Formula building theorem f... |
| suppssov2 8243 | Formula building theorem f... |
| suppssof1 8244 | Formula building theorem f... |
| suppss2 8245 | Show that the support of a... |
| suppsssn 8246 | Show that the support of a... |
| suppssfv 8247 | Formula building theorem f... |
| suppofssd 8248 | Condition for the support ... |
| suppofss1d 8249 | Condition for the support ... |
| suppofss2d 8250 | Condition for the support ... |
| suppco 8251 | The support of the composi... |
| suppcoss 8252 | The support of the composi... |
| supp0cosupp0 8253 | The support of the composi... |
| imacosupp 8254 | The image of the support o... |
| opeliunxp2f 8255 | Membership in a union of C... |
| mpoxeldm 8256 | If there is an element of ... |
| mpoxneldm 8257 | If the first argument of a... |
| mpoxopn0yelv 8258 | If there is an element of ... |
| mpoxopynvov0g 8259 | If the second argument of ... |
| mpoxopxnop0 8260 | If the first argument of a... |
| mpoxopx0ov0 8261 | If the first argument of a... |
| mpoxopxprcov0 8262 | If the components of the f... |
| mpoxopynvov0 8263 | If the second argument of ... |
| mpoxopoveq 8264 | Value of an operation give... |
| mpoxopovel 8265 | Element of the value of an... |
| mpoxopoveqd 8266 | Value of an operation give... |
| brovex 8267 | A binary relation of the v... |
| brovmpoex 8268 | A binary relation of the v... |
| sprmpod 8269 | The extension of a binary ... |
| tposss 8272 | Subset theorem for transpo... |
| tposeq 8273 | Equality theorem for trans... |
| tposeqd 8274 | Equality theorem for trans... |
| tposssxp 8275 | The transposition is a sub... |
| reltpos 8276 | The transposition is a rel... |
| brtpos2 8277 | Value of the transposition... |
| brtpos0 8278 | The behavior of ` tpos ` w... |
| reldmtpos 8279 | Necessary and sufficient c... |
| brtpos 8280 | The transposition swaps ar... |
| ottpos 8281 | The transposition swaps th... |
| relbrtpos 8282 | The transposition swaps ar... |
| dmtpos 8283 | The domain of ` tpos F ` w... |
| rntpos 8284 | The range of ` tpos F ` wh... |
| tposexg 8285 | The transposition of a set... |
| ovtpos 8286 | The transposition swaps th... |
| tposfun 8287 | The transposition of a fun... |
| dftpos2 8288 | Alternate definition of ` ... |
| dftpos3 8289 | Alternate definition of ` ... |
| dftpos4 8290 | Alternate definition of ` ... |
| tpostpos 8291 | Value of the double transp... |
| tpostpos2 8292 | Value of the double transp... |
| tposfn2 8293 | The domain of a transposit... |
| tposfo2 8294 | Condition for a surjective... |
| tposf2 8295 | The domain and codomain of... |
| tposf12 8296 | Condition for an injective... |
| tposf1o2 8297 | Condition of a bijective t... |
| tposfo 8298 | The domain and codomain/ra... |
| tposf 8299 | The domain and codomain of... |
| tposfn 8300 | Functionality of a transpo... |
| tpos0 8301 | Transposition of the empty... |
| tposco 8302 | Transposition of a composi... |
| tpossym 8303 | Two ways to say a function... |
| tposeqi 8304 | Equality theorem for trans... |
| tposex 8305 | A transposition is a set. ... |
| nftpos 8306 | Hypothesis builder for tra... |
| tposoprab 8307 | Transposition of a class o... |
| tposmpo 8308 | Transposition of a two-arg... |
| tposconst 8309 | The transposition of a con... |
| mpocurryd 8314 | The currying of an operati... |
| mpocurryvald 8315 | The value of a curried ope... |
| fvmpocurryd 8316 | The value of the value of ... |
| pwuninel2 8319 | Proof of ~ pwuninel under ... |
| pwuninel 8320 | The powerclass of the unio... |
| pwuninelOLD 8321 | Obsolete version of ~ pwun... |
| undefval 8322 | Value of the undefined val... |
| undefnel2 8323 | The undefined value genera... |
| undefnel 8324 | The undefined value genera... |
| undefne0 8325 | The undefined value genera... |
| frecseq123 8328 | Equality theorem for the w... |
| nffrecs 8329 | Bound-variable hypothesis ... |
| csbfrecsg 8330 | Move class substitution in... |
| fpr3g 8331 | Functions defined by well-... |
| frrlem1 8332 | Lemma for well-founded rec... |
| frrlem2 8333 | Lemma for well-founded rec... |
| frrlem3 8334 | Lemma for well-founded rec... |
| frrlem4 8335 | Lemma for well-founded rec... |
| frrlem5 8336 | Lemma for well-founded rec... |
| frrlem6 8337 | Lemma for well-founded rec... |
| frrlem7 8338 | Lemma for well-founded rec... |
| frrlem8 8339 | Lemma for well-founded rec... |
| frrlem9 8340 | Lemma for well-founded rec... |
| frrlem10 8341 | Lemma for well-founded rec... |
| frrlem11 8342 | Lemma for well-founded rec... |
| frrlem12 8343 | Lemma for well-founded rec... |
| frrlem13 8344 | Lemma for well-founded rec... |
| frrlem14 8345 | Lemma for well-founded rec... |
| fprlem1 8346 | Lemma for well-founded rec... |
| fprlem2 8347 | Lemma for well-founded rec... |
| fpr2a 8348 | Weak version of ~ fpr2 whi... |
| fpr1 8349 | Law of well-founded recurs... |
| fpr2 8350 | Law of well-founded recurs... |
| fpr3 8351 | Law of well-founded recurs... |
| frrrel 8352 | Show without using the axi... |
| frrdmss 8353 | Show without using the axi... |
| frrdmcl 8354 | Show without using the axi... |
| fprfung 8355 | A "function" defined by we... |
| fprresex 8356 | The restriction of a funct... |
| wrecseq123 8359 | General equality theorem f... |
| nfwrecs 8360 | Bound-variable hypothesis ... |
| wrecseq1 8361 | Equality theorem for the w... |
| wrecseq2 8362 | Equality theorem for the w... |
| wrecseq3 8363 | Equality theorem for the w... |
| csbwrecsg 8364 | Move class substitution in... |
| wfr3g 8365 | Functions defined by well-... |
| wfrrel 8366 | The well-ordered recursion... |
| wfrdmss 8367 | The domain of the well-ord... |
| wfrdmcl 8368 | The predecessor class of a... |
| wfrfun 8369 | The "function" generated b... |
| wfrresex 8370 | Show without using the axi... |
| wfr2a 8371 | A weak version of ~ wfr2 w... |
| wfr1 8372 | The Principle of Well-Orde... |
| wfr2 8373 | The Principle of Well-Orde... |
| wfr3 8374 | The principle of Well-Orde... |
| iunon 8375 | The indexed union of a set... |
| iinon 8376 | The nonempty indexed inter... |
| onfununi 8377 | A property of functions on... |
| onovuni 8378 | A variant of ~ onfununi fo... |
| onoviun 8379 | A variant of ~ onovuni wit... |
| onnseq 8380 | There are no length ` _om ... |
| dfsmo2 8383 | Alternate definition of a ... |
| issmo 8384 | Conditions for which ` A `... |
| issmo2 8385 | Alternate definition of a ... |
| smoeq 8386 | Equality theorem for stric... |
| smodm 8387 | The domain of a strictly m... |
| smores 8388 | A strictly monotone functi... |
| smores3 8389 | A strictly monotone functi... |
| smores2 8390 | A strictly monotone ordina... |
| smodm2 8391 | The domain of a strictly m... |
| smofvon2 8392 | The function values of a s... |
| iordsmo 8393 | The identity relation rest... |
| smo0 8394 | The empty set is a strictl... |
| smofvon 8395 | If ` B ` is a strictly mon... |
| smoel 8396 | If ` x ` is less than ` y ... |
| smoiun 8397 | The value of a strictly mo... |
| smoiso 8398 | If ` F ` is an isomorphism... |
| smoel2 8399 | A strictly monotone ordina... |
| smo11 8400 | A strictly monotone ordina... |
| smoord 8401 | A strictly monotone ordina... |
| smoword 8402 | A strictly monotone ordina... |
| smogt 8403 | A strictly monotone ordina... |
| smocdmdom 8404 | The codomain of a strictly... |
| smoiso2 8405 | The strictly monotone ordi... |
| dfrecs3 8408 | The old definition of tran... |
| recseq 8409 | Equality theorem for ` rec... |
| nfrecs 8410 | Bound-variable hypothesis ... |
| tfrlem1 8411 | A technical lemma for tran... |
| tfrlem3a 8412 | Lemma for transfinite recu... |
| tfrlem3 8413 | Lemma for transfinite recu... |
| tfrlem4 8414 | Lemma for transfinite recu... |
| tfrlem5 8415 | Lemma for transfinite recu... |
| recsfval 8416 | Lemma for transfinite recu... |
| tfrlem6 8417 | Lemma for transfinite recu... |
| tfrlem6OLD 8418 | Obsolete version of ~ tfrl... |
| tfrlem7 8419 | Lemma for transfinite recu... |
| tfrlem8 8420 | Lemma for transfinite recu... |
| tfrlem9 8421 | Lemma for transfinite recu... |
| tfrlem9a 8422 | Lemma for transfinite recu... |
| tfrlem10 8423 | Lemma for transfinite recu... |
| tfrlem11 8424 | Lemma for transfinite recu... |
| tfrlem12 8425 | Lemma for transfinite recu... |
| tfrlem13 8426 | Lemma for transfinite recu... |
| tfrlem14 8427 | Lemma for transfinite recu... |
| tfrlem15 8428 | Lemma for transfinite recu... |
| tfrlem16 8429 | Lemma for finite recursion... |
| tfr1a 8430 | A weak version of ~ tfr1 w... |
| tfr2a 8431 | A weak version of ~ tfr2 w... |
| tfr2b 8432 | Without assuming ~ ax-rep ... |
| tfr1 8433 | Principle of Transfinite R... |
| tfr2 8434 | Principle of Transfinite R... |
| tfr3 8435 | Principle of Transfinite R... |
| tfr1ALT 8436 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8437 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8438 | Alternate proof of ~ tfr3 ... |
| recsfnon 8439 | Strong transfinite recursi... |
| recsval 8440 | Strong transfinite recursi... |
| tz7.44lem1 8441 | The ordered pair abstracti... |
| tz7.44-1 8442 | The value of ` F ` at ` (/... |
| tz7.44-2 8443 | The value of ` F ` at a su... |
| tz7.44-3 8444 | The value of ` F ` at a li... |
| rdgeq1 8447 | Equality theorem for the r... |
| rdgeq2 8448 | Equality theorem for the r... |
| rdgeq12 8449 | Equality theorem for the r... |
| nfrdg 8450 | Bound-variable hypothesis ... |
| rdglem1 8451 | Lemma used with the recurs... |
| rdgfun 8452 | The recursive definition g... |
| rdgdmlim 8453 | The domain of the recursiv... |
| rdgfnon 8454 | The recursive definition g... |
| rdgvalg 8455 | Value of the recursive def... |
| rdgval 8456 | Value of the recursive def... |
| rdg0 8457 | The initial value of the r... |
| rdgseg 8458 | The initial segments of th... |
| rdgsucg 8459 | The value of the recursive... |
| rdgsuc 8460 | The value of the recursive... |
| rdglimg 8461 | The value of the recursive... |
| rdglim 8462 | The value of the recursive... |
| rdg0g 8463 | The initial value of the r... |
| rdgsucmptf 8464 | The value of the recursive... |
| rdgsucmptnf 8465 | The value of the recursive... |
| rdgsucmpt2 8466 | This version of ~ rdgsucmp... |
| rdgsucmpt 8467 | The value of the recursive... |
| rdglim2 8468 | The value of the recursive... |
| rdglim2a 8469 | The value of the recursive... |
| rdg0n 8470 | If ` A ` is a proper class... |
| frfnom 8471 | The function generated by ... |
| fr0g 8472 | The initial value resultin... |
| frsuc 8473 | The successor value result... |
| frsucmpt 8474 | The successor value result... |
| frsucmptn 8475 | The value of the finite re... |
| frsucmpt2 8476 | The successor value result... |
| onelfvnef1 8477 | A sufficient condition for... |
| tz7.48lem 8478 | A way of showing an ordina... |
| tz7.48lemOLD 8479 | Obsolete version of ~ tz7.... |
| tz7.48-2 8480 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8481 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8482 | Proposition 7.48(3) of [Ta... |
| tz7.49 8483 | Proposition 7.49 of [Takeu... |
| tz7.49c 8484 | Corollary of Proposition 7... |
| seqomlem0 8487 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8488 | Lemma for ` seqom ` . The... |
| seqomlem2 8489 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8490 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8491 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8492 | Equality theorem for ` seq... |
| fnseqom 8493 | An index-aware recursive d... |
| seqom0g 8494 | Value of an index-aware re... |
| seqomsuc 8495 | Value of an index-aware re... |
| omsucelsucb 8496 | Membership is inherited by... |
| df1o2 8511 | Expanded value of the ordi... |
| df2o3 8512 | Expanded value of the ordi... |
| df2o2 8513 | Expanded value of the ordi... |
| 1oex 8514 | Ordinal 1 is a set. (Cont... |
| 1oelpr 8515 | ` 1o ` is an element of ` ... |
| 2oex 8516 | ` 2o ` is a set. (Contrib... |
| 1on 8517 | Ordinal 1 is an ordinal nu... |
| 2on 8518 | Ordinal 2 is an ordinal nu... |
| 2on0 8519 | Ordinal two is not zero. ... |
| ord3 8520 | Ordinal 3 is an ordinal cl... |
| 3on 8521 | Ordinal 3 is an ordinal nu... |
| 4on 8522 | Ordinal 4 is an ordinal nu... |
| 1n0 8523 | Ordinal one is not equal t... |
| 1n0OLD 8524 | Obsolete version of ~ 1n0 ... |
| nlim1 8525 | 1 is not a limit ordinal. ... |
| nlim2 8526 | 2 is not a limit ordinal. ... |
| xp01disj 8527 | Cartesian products with th... |
| xp01disjl 8528 | Cartesian products with th... |
| ordgt0ge1 8529 | Two ways to express that a... |
| ordge1n0 8530 | An ordinal greater than or... |
| el1o 8531 | Membership in ordinal one.... |
| ord1eln01 8532 | An ordinal that is not 0 o... |
| ord2eln012 8533 | An ordinal that is not 0, ... |
| 1ellim 8534 | A limit ordinal contains 1... |
| 2ellim 8535 | A limit ordinal contains 2... |
| dif1o 8536 | Two ways to say that ` A `... |
| ondif1 8537 | Two ways to say that ` A `... |
| ondif2 8538 | Two ways to say that ` A `... |
| 2oconcl 8539 | Closure of the pair swappi... |
| 0lt1o 8540 | Ordinal zero is less than ... |
| dif20el 8541 | An ordinal greater than on... |
| 0we1 8542 | The empty set is a well-or... |
| brwitnlem 8543 | Lemma for relations which ... |
| fnoa 8544 | Functionality and domain o... |
| fnom 8545 | Functionality and domain o... |
| fnoe 8546 | Functionality and domain o... |
| oav 8547 | Value of ordinal addition.... |
| omv 8548 | Value of ordinal multiplic... |
| oe0lem 8549 | A helper lemma for ~ oe0 a... |
| oev 8550 | Value of ordinal exponenti... |
| oevn0 8551 | Value of ordinal exponenti... |
| oa0 8552 | Addition with zero. Propo... |
| om0 8553 | Ordinal multiplication wit... |
| oe0m 8554 | Value of zero raised to an... |
| om0x 8555 | Ordinal multiplication wit... |
| oe0m0 8556 | Ordinal exponentiation wit... |
| oe0m1 8557 | Ordinal exponentiation wit... |
| oe0 8558 | Ordinal exponentiation wit... |
| oev2 8559 | Alternate value of ordinal... |
| oasuc 8560 | Addition with successor. ... |
| oesuclem 8561 | Lemma for ~ oesuc . (Cont... |
| omsuc 8562 | Multiplication with succes... |
| oesuc 8563 | Ordinal exponentiation wit... |
| onasuc 8564 | Addition with successor. ... |
| onmsuc 8565 | Multiplication with succes... |
| onesuc 8566 | Exponentiation with a succ... |
| oa1suc 8567 | Addition with 1 is same as... |
| oalim 8568 | Ordinal addition with a li... |
| omlim 8569 | Ordinal multiplication wit... |
| oelim 8570 | Ordinal exponentiation wit... |
| oacl 8571 | Closure law for ordinal ad... |
| omcl 8572 | Closure law for ordinal mu... |
| oecl 8573 | Closure law for ordinal ex... |
| oa0r 8574 | Ordinal addition with zero... |
| om0r 8575 | Ordinal multiplication wit... |
| o1p1e2 8576 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8577 | 2 + 2 = 4 for ordinal numb... |
| om1 8578 | Ordinal multiplication wit... |
| om1r 8579 | Ordinal multiplication wit... |
| oe1 8580 | Ordinal exponentiation wit... |
| oe1m 8581 | Ordinal exponentiation wit... |
| oaordi 8582 | Ordering property of ordin... |
| oaord 8583 | Ordering property of ordin... |
| oacan 8584 | Left cancellation law for ... |
| oaword 8585 | Weak ordering property of ... |
| oawordri 8586 | Weak ordering property of ... |
| oaord1 8587 | An ordinal is less than it... |
| oaword1 8588 | An ordinal is less than or... |
| oaword2 8589 | An ordinal is less than or... |
| oawordeulem 8590 | Lemma for ~ oawordex . (C... |
| oawordeu 8591 | Existence theorem for weak... |
| oawordexr 8592 | Existence theorem for weak... |
| oawordex 8593 | Existence theorem for weak... |
| oaordex 8594 | Existence theorem for orde... |
| oa00 8595 | An ordinal sum is zero iff... |
| oalimcl 8596 | The ordinal sum with a lim... |
| oaass 8597 | Ordinal addition is associ... |
| oarec 8598 | Recursive definition of or... |
| oaf1o 8599 | Left addition by a constan... |
| oacomf1olem 8600 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8601 | Define a bijection from ` ... |
| omordi 8602 | Ordering property of ordin... |
| omord2 8603 | Ordering property of ordin... |
| omord 8604 | Ordering property of ordin... |
| omcan 8605 | Left cancellation law for ... |
| omword 8606 | Weak ordering property of ... |
| omwordi 8607 | Weak ordering property of ... |
| omwordri 8608 | Weak ordering property of ... |
| omword1 8609 | An ordinal is less than or... |
| omword2 8610 | An ordinal is less than or... |
| om00 8611 | The product of two ordinal... |
| om00el 8612 | The product of two nonzero... |
| omordlim 8613 | Ordering involving the pro... |
| omlimcl 8614 | The product of any nonzero... |
| odi 8615 | Distributive law for ordin... |
| omass 8616 | Multiplication of ordinal ... |
| oneo 8617 | If an ordinal number is ev... |
| omeulem1 8618 | Lemma for ~ omeu : existen... |
| omeulem2 8619 | Lemma for ~ omeu : uniquen... |
| omopth2 8620 | An ordered pair-like theor... |
| omeu 8621 | The division algorithm for... |
| om2 8622 | Two ways to double an ordi... |
| oen0 8623 | Ordinal exponentiation wit... |
| oeordi 8624 | Ordering law for ordinal e... |
| oeord 8625 | Ordering property of ordin... |
| oecan 8626 | Left cancellation law for ... |
| oeword 8627 | Weak ordering property of ... |
| oewordi 8628 | Weak ordering property of ... |
| oewordri 8629 | Weak ordering property of ... |
| oeworde 8630 | Ordinal exponentiation com... |
| oeordsuc 8631 | Ordering property of ordin... |
| oelim2 8632 | Ordinal exponentiation wit... |
| oeoalem 8633 | Lemma for ~ oeoa . (Contr... |
| oeoa 8634 | Sum of exponents law for o... |
| oeoelem 8635 | Lemma for ~ oeoe . (Contr... |
| oeoe 8636 | Product of exponents law f... |
| oelimcl 8637 | The ordinal exponential wi... |
| oeeulem 8638 | Lemma for ~ oeeu . (Contr... |
| oeeui 8639 | The division algorithm for... |
| oeeu 8640 | The division algorithm for... |
| nna0 8641 | Addition with zero. Theor... |
| nnm0 8642 | Multiplication with zero. ... |
| nnasuc 8643 | Addition with successor. ... |
| nnmsuc 8644 | Multiplication with succes... |
| nnesuc 8645 | Exponentiation with a succ... |
| nna0r 8646 | Addition to zero. Remark ... |
| nnm0r 8647 | Multiplication with zero. ... |
| nnacl 8648 | Closure of addition of nat... |
| nnmcl 8649 | Closure of multiplication ... |
| nnecl 8650 | Closure of exponentiation ... |
| nnacli 8651 | ` _om ` is closed under ad... |
| nnmcli 8652 | ` _om ` is closed under mu... |
| nnarcl 8653 | Reverse closure law for ad... |
| nnacom 8654 | Addition of natural number... |
| nnaordi 8655 | Ordering property of addit... |
| nnaord 8656 | Ordering property of addit... |
| nnaordr 8657 | Ordering property of addit... |
| nnawordi 8658 | Adding to both sides of an... |
| nnaass 8659 | Addition of natural number... |
| nndi 8660 | Distributive law for natur... |
| nnmass 8661 | Multiplication of natural ... |
| nnmsucr 8662 | Multiplication with succes... |
| nnmcom 8663 | Multiplication of natural ... |
| nnaword 8664 | Weak ordering property of ... |
| nnacan 8665 | Cancellation law for addit... |
| nnaword1 8666 | Weak ordering property of ... |
| nnaword2 8667 | Weak ordering property of ... |
| nnmordi 8668 | Ordering property of multi... |
| nnmord 8669 | Ordering property of multi... |
| nnmword 8670 | Weak ordering property of ... |
| nnmcan 8671 | Cancellation law for multi... |
| nnmwordi 8672 | Weak ordering property of ... |
| nnmwordri 8673 | Weak ordering property of ... |
| nnawordex 8674 | Equivalence for weak order... |
| nnaordex 8675 | Equivalence for ordering. ... |
| nnaordex2 8676 | Equivalence for ordering. ... |
| 1onn 8677 | The ordinal 1 is a natural... |
| 1onnALT 8678 | Shorter proof of ~ 1onn us... |
| 2onn 8679 | The ordinal 2 is a natural... |
| 2onnALT 8680 | Shorter proof of ~ 2onn us... |
| 3onn 8681 | The ordinal 3 is a natural... |
| 4onn 8682 | The ordinal 4 is a natural... |
| 1one2o 8683 | Ordinal one is not ordinal... |
| oaabslem 8684 | Lemma for ~ oaabs . (Cont... |
| oaabs 8685 | Ordinal addition absorbs a... |
| oaabs2 8686 | The absorption law ~ oaabs... |
| omabslem 8687 | Lemma for ~ omabs . (Cont... |
| omabs 8688 | Ordinal multiplication is ... |
| nnm1 8689 | Multiply an element of ` _... |
| nnm2 8690 | Multiply an element of ` _... |
| nn2m 8691 | Multiply an element of ` _... |
| nnneo 8692 | If a natural number is eve... |
| nneob 8693 | A natural number is even i... |
| omsmolem 8694 | Lemma for ~ omsmo . (Cont... |
| omsmo 8695 | A strictly monotonic ordin... |
| omopthlem1 8696 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8697 | Lemma for ~ omopthi . (Co... |
| omopthi 8698 | An ordered pair theorem fo... |
| omopth 8699 | An ordered pair theorem fo... |
| nnasmo 8700 | There is at most one left ... |
| eldifsucnn 8701 | Condition for membership i... |
| on2recsfn 8704 | Show that double recursion... |
| on2recsov 8705 | Calculate the value of the... |
| on2ind 8706 | Double induction over ordi... |
| on3ind 8707 | Triple induction over ordi... |
| coflton 8708 | Cofinality theorem for ord... |
| cofon1 8709 | Cofinality theorem for ord... |
| cofon2 8710 | Cofinality theorem for ord... |
| cofonr 8711 | Inverse cofinality law for... |
| naddfn 8712 | Natural addition is a func... |
| naddcllem 8713 | Lemma for ordinal addition... |
| naddcl 8714 | Closure law for natural ad... |
| naddov 8715 | The value of natural addit... |
| naddov2 8716 | Alternate expression for n... |
| naddcld 8717 | Closure law for natural ad... |
| naddov3 8718 | Alternate expression for n... |
| naddf 8719 | Function statement for nat... |
| naddcom 8720 | Natural addition is commut... |
| naddrid 8721 | Ordinal zero is the additi... |
| naddlid 8722 | Ordinal zero is the additi... |
| naddssim 8723 | Ordinal less-than-or-equal... |
| naddelim 8724 | Ordinal less-than is prese... |
| naddel1 8725 | Ordinal less-than is not a... |
| naddel2 8726 | Ordinal less-than is not a... |
| naddss1 8727 | Ordinal less-than-or-equal... |
| naddss2 8728 | Ordinal less-than-or-equal... |
| naddword1 8729 | Weak-ordering principle fo... |
| naddword2 8730 | Weak-ordering principle fo... |
| naddunif 8731 | Uniformity theorem for nat... |
| naddasslem1 8732 | Lemma for ~ naddass . Exp... |
| naddasslem2 8733 | Lemma for ~ naddass . Exp... |
| naddass 8734 | Natural ordinal addition i... |
| nadd32 8735 | Commutative/associative la... |
| nadd4 8736 | Rearragement of terms in a... |
| nadd42 8737 | Rearragement of terms in a... |
| naddel12 8738 | Natural addition to both s... |
| naddsuc2 8739 | Natural addition with succ... |
| naddoa 8740 | Natural addition of a natu... |
| omnaddcl 8741 | The naturals are closed un... |
| dfer2 8746 | Alternate definition of eq... |
| dfec2 8748 | Alternate definition of ` ... |
| ecexg 8749 | An equivalence class modul... |
| ecexr 8750 | A nonempty equivalence cla... |
| dfqs2 8752 | Alternate definition of qu... |
| ereq1 8753 | Equality theorem for equiv... |
| ereq2 8754 | Equality theorem for equiv... |
| errel 8755 | An equivalence relation is... |
| erdm 8756 | The domain of an equivalen... |
| ercl 8757 | Elementhood in the field o... |
| ersym 8758 | An equivalence relation is... |
| ercl2 8759 | Elementhood in the field o... |
| ersymb 8760 | An equivalence relation is... |
| ertr 8761 | An equivalence relation is... |
| ertrd 8762 | A transitivity relation fo... |
| ertr2d 8763 | A transitivity relation fo... |
| ertr3d 8764 | A transitivity relation fo... |
| ertr4d 8765 | A transitivity relation fo... |
| erref 8766 | An equivalence relation is... |
| ercnv 8767 | The converse of an equival... |
| errn 8768 | The range and domain of an... |
| erssxp 8769 | An equivalence relation is... |
| erex 8770 | An equivalence relation is... |
| erexb 8771 | An equivalence relation is... |
| iserd 8772 | A reflexive, symmetric, tr... |
| iseri 8773 | A reflexive, symmetric, tr... |
| iseriALT 8774 | Alternate proof of ~ iseri... |
| brinxper 8775 | Conditions for a reflexive... |
| brdifun 8776 | Evaluate the incomparabili... |
| swoer 8777 | Incomparability under a st... |
| swoord1 8778 | The incomparability equiva... |
| swoord2 8779 | The incomparability equiva... |
| swoso 8780 | If the incomparability rel... |
| eqerlem 8781 | Lemma for ~ eqer . (Contr... |
| eqer 8782 | Equivalence relation invol... |
| ider 8783 | The identity relation is a... |
| 0er 8784 | The empty set is an equiva... |
| eceq1 8785 | Equality theorem for equiv... |
| eceq1d 8786 | Equality theorem for equiv... |
| eceq2 8787 | Equality theorem for equiv... |
| eceq2i 8788 | Equality theorem for the `... |
| eceq2d 8789 | Equality theorem for the `... |
| elecg 8790 | Membership in an equivalen... |
| ecref 8791 | All elements are in their ... |
| elec 8792 | Membership in an equivalen... |
| relelec 8793 | Membership in an equivalen... |
| elecres 8794 | Elementhood in the restric... |
| elecreseq 8795 | The restricted coset of ` ... |
| elecex 8796 | Condition for a coset to b... |
| ecss 8797 | An equivalence class is a ... |
| ecdmn0 8798 | A representative of a none... |
| ereldm 8799 | Equality of equivalence cl... |
| erth 8800 | Basic property of equivale... |
| erth2 8801 | Basic property of equivale... |
| erthi 8802 | Basic property of equivale... |
| erdisj 8803 | Equivalence classes do not... |
| ecidsn 8804 | An equivalence class modul... |
| qseq1 8805 | Equality theorem for quoti... |
| qseq2 8806 | Equality theorem for quoti... |
| qseq2i 8807 | Equality theorem for quoti... |
| qseq1d 8808 | Equality theorem for quoti... |
| qseq2d 8809 | Equality theorem for quoti... |
| qseq12 8810 | Equality theorem for quoti... |
| 0qs 8811 | Quotient set with the empt... |
| elqsg 8812 | Closed form of ~ elqs . (... |
| elqs 8813 | Membership in a quotient s... |
| elqsi 8814 | Membership in a quotient s... |
| elqsecl 8815 | Membership in a quotient s... |
| ecelqs 8816 | Membership of an equivalen... |
| ecelqsw 8817 | Membership of an equivalen... |
| ecelqsi 8818 | Membership of an equivalen... |
| ecopqsi 8819 | "Closure" law for equivale... |
| qsexg 8820 | A quotient set exists. (C... |
| qsex 8821 | A quotient set exists. (C... |
| uniqs 8822 | The union of a quotient se... |
| uniqsw 8823 | The union of a quotient se... |
| qsss 8824 | A quotient set is a set of... |
| uniqs2 8825 | The union of a quotient se... |
| snecg 8826 | The singleton of a coset i... |
| snec 8827 | The singleton of an equiva... |
| ecqs 8828 | Equivalence class in terms... |
| ecid 8829 | A set is equal to its cose... |
| qsid 8830 | A set is equal to its quot... |
| ectocld 8831 | Implicit substitution of c... |
| ectocl 8832 | Implicit substitution of c... |
| elqsn0 8833 | A quotient set does not co... |
| ecelqsdm 8834 | Membership of an equivalen... |
| ecelqsdmb 8835 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8836 | ` R ` -coset in its domain... |
| xpider 8837 | A Cartesian square is an e... |
| iiner 8838 | The intersection of a none... |
| riiner 8839 | The relative intersection ... |
| erinxp 8840 | A restricted equivalence r... |
| ecinxp 8841 | Restrict the relation in a... |
| qsinxp 8842 | Restrict the equivalence r... |
| qsdisj 8843 | Members of a quotient set ... |
| qsdisj2 8844 | A quotient set is a disjoi... |
| qsel 8845 | If an element of a quotien... |
| uniinqs 8846 | Class union distributes ov... |
| qliftlem 8847 | Lemma for theorems about a... |
| qliftrel 8848 | ` F ` , a function lift, i... |
| qliftel 8849 | Elementhood in the relatio... |
| qliftel1 8850 | Elementhood in the relatio... |
| qliftfun 8851 | The function ` F ` is the ... |
| qliftfund 8852 | The function ` F ` is the ... |
| qliftfuns 8853 | The function ` F ` is the ... |
| qliftf 8854 | The domain and codomain of... |
| qliftval 8855 | The value of the function ... |
| ecoptocl 8856 | Implicit substitution of c... |
| 2ecoptocl 8857 | Implicit substitution of c... |
| 3ecoptocl 8858 | Implicit substitution of c... |
| brecop 8859 | Binary relation on a quoti... |
| brecop2 8860 | Binary relation on a quoti... |
| eroveu 8861 | Lemma for ~ erov and ~ ero... |
| erovlem 8862 | Lemma for ~ erov and ~ ero... |
| erov 8863 | The value of an operation ... |
| eroprf 8864 | Functionality of an operat... |
| erov2 8865 | The value of an operation ... |
| eroprf2 8866 | Functionality of an operat... |
| ecopoveq 8867 | This is the first of sever... |
| ecopovsym 8868 | Assuming the operation ` F... |
| ecopovtrn 8869 | Assuming that operation ` ... |
| ecopover 8870 | Assuming that operation ` ... |
| eceqoveq 8871 | Equality of equivalence re... |
| ecovcom 8872 | Lemma used to transfer a c... |
| ecovass 8873 | Lemma used to transfer an ... |
| ecovdi 8874 | Lemma used to transfer a d... |
| mapprc 8879 | When ` A ` is a proper cla... |
| pmex 8880 | The class of all partial f... |
| fnmap 8881 | Set exponentiation has a u... |
| fnpm 8882 | Partial function exponenti... |
| reldmmap 8883 | Set exponentiation is a we... |
| mapvalg 8884 | The value of set exponenti... |
| pmvalg 8885 | The value of the partial m... |
| mapval 8886 | The value of set exponenti... |
| elmapg 8887 | Membership relation for se... |
| elmapd 8888 | Deduction form of ~ elmapg... |
| elmapdd 8889 | Deduction associated with ... |
| mapdm0 8890 | The empty set is the only ... |
| elpmg 8891 | The predicate "is a partia... |
| elpm2g 8892 | The predicate "is a partia... |
| elpm2r 8893 | Sufficient condition for b... |
| elpmi 8894 | A partial function is a fu... |
| pmfun 8895 | A partial function is a fu... |
| elmapex 8896 | Eliminate antecedent for m... |
| elmapi 8897 | A mapping is a function, f... |
| elmaprd 8898 | Deduction associated with ... |
| elmaprdOLD 8899 | Obsolete version of ~ elma... |
| mapfset 8900 | If ` B ` is a set, the val... |
| mapssfset 8901 | The value of the set expon... |
| mapfoss 8902 | The value of the set expon... |
| fsetsspwxp 8903 | The class of all functions... |
| fset0 8904 | The set of functions from ... |
| fsetdmprc0 8905 | The set of functions with ... |
| fsetex 8906 | The set of functions betwe... |
| f1setex 8907 | The set of injections betw... |
| fosetex 8908 | The set of surjections bet... |
| f1osetex 8909 | The set of bijections betw... |
| fsetfcdm 8910 | The class of functions wit... |
| fsetfocdm 8911 | The class of functions wit... |
| fsetprcnex 8912 | The class of all functions... |
| fsetcdmex 8913 | The class of all functions... |
| fsetexb 8914 | The class of all functions... |
| elmapfn 8915 | A mapping is a function wi... |
| elmapfun 8916 | A mapping is always a func... |
| cureq 8917 | Equality theorem for curry... |
| curf 8918 | Functional property of cur... |
| uncf 8919 | Functional property of unc... |
| curfv 8920 | Value of currying. (Contr... |
| uncov 8921 | Value of uncurrying. (Con... |
| elmapssres 8922 | A restricted mapping is a ... |
| elmapssresd 8923 | A restricted mapping is a ... |
| fpmg 8924 | A total function is a part... |
| pmss12g 8925 | Subset relation for the se... |
| pmresg 8926 | Elementhood of a restricte... |
| elmap 8927 | Membership relation for se... |
| mapval2 8928 | Alternate expression for t... |
| elpm 8929 | The predicate "is a partia... |
| elpm2 8930 | The predicate "is a partia... |
| fpm 8931 | A total function is a part... |
| mapsspm 8932 | Set exponentiation is a su... |
| pmsspw 8933 | Partial maps are a subset ... |
| mapsspw 8934 | Set exponentiation is a su... |
| mapfvd 8935 | The value of a function th... |
| elmapresaun 8936 | ~ fresaun transposed to ma... |
| fvmptmap 8937 | Special case of ~ fvmpt fo... |
| map0e 8938 | Set exponentiation with an... |
| map0b 8939 | Set exponentiation with an... |
| map0g 8940 | Set exponentiation is empt... |
| 0map0sn0 8941 | The set of mappings of the... |
| mapsnd 8942 | The value of set exponenti... |
| map0 8943 | Set exponentiation is empt... |
| mapsn 8944 | The value of set exponenti... |
| mapss 8945 | Subset inheritance for set... |
| fdiagfn 8946 | Functionality of the diago... |
| fvdiagfn 8947 | Functionality of the diago... |
| mapsnconst 8948 | Every singleton map is a c... |
| mapsncnv 8949 | Expression for the inverse... |
| mapsnf1o2 8950 | Explicit bijection between... |
| mapsnf1o3 8951 | Explicit bijection in the ... |
| ralxpmap 8952 | Quantification over functi... |
| dfixp 8955 | Eliminate the expression `... |
| ixpsnval 8956 | The value of an infinite C... |
| elixp2 8957 | Membership in an infinite ... |
| fvixp 8958 | Projection of a factor of ... |
| ixpfn 8959 | A nuple is a function. (C... |
| elixp 8960 | Membership in an infinite ... |
| elixpconst 8961 | Membership in an infinite ... |
| ixpconstg 8962 | Infinite Cartesian product... |
| ixpconst 8963 | Infinite Cartesian product... |
| ixpeq1 8964 | Equality theorem for infin... |
| ixpeq1d 8965 | Equality theorem for infin... |
| ss2ixp 8966 | Subclass theorem for infin... |
| ixpeq2 8967 | Equality theorem for infin... |
| ixpeq2dva 8968 | Equality theorem for infin... |
| ixpeq2dv 8969 | Equality theorem for infin... |
| cbvixp 8970 | Change bound variable in a... |
| cbvixpv 8971 | Change bound variable in a... |
| nfixpw 8972 | Bound-variable hypothesis ... |
| nfixp 8973 | Bound-variable hypothesis ... |
| nfixp1 8974 | The index variable in an i... |
| ixpprc 8975 | A cartesian product of pro... |
| ixpf 8976 | A member of an infinite Ca... |
| uniixp 8977 | The union of an infinite C... |
| ixpexg 8978 | The existence of an infini... |
| ixpin 8979 | The intersection of two in... |
| ixpiin 8980 | The indexed intersection o... |
| ixpint 8981 | The intersection of a coll... |
| ixp0x 8982 | An infinite Cartesian prod... |
| ixpssmap2g 8983 | An infinite Cartesian prod... |
| ixpssmapg 8984 | An infinite Cartesian prod... |
| 0elixp 8985 | Membership of the empty se... |
| ixpn0 8986 | The infinite Cartesian pro... |
| ixp0 8987 | The infinite Cartesian pro... |
| ixpssmap 8988 | An infinite Cartesian prod... |
| resixp 8989 | Restriction of an element ... |
| undifixp 8990 | Union of two projections o... |
| mptelixpg 8991 | Condition for an explicit ... |
| resixpfo 8992 | Restriction of elements of... |
| elixpsn 8993 | Membership in a class of s... |
| ixpsnf1o 8994 | A bijection between a clas... |
| mapsnf1o 8995 | A bijection between a set ... |
| boxriin 8996 | A rectangular subset of a ... |
| boxcutc 8997 | The relative complement of... |
| relen 9006 | Equinumerosity is a relati... |
| reldom 9007 | Dominance is a relation. ... |
| relsdom 9008 | Strict dominance is a rela... |
| encv 9009 | If two classes are equinum... |
| breng 9010 | Equinumerosity relation. ... |
| bren 9011 | Equinumerosity relation. ... |
| brdom2g 9012 | Dominance relation. This ... |
| brdomg 9013 | Dominance relation. (Cont... |
| brdomi 9014 | Dominance relation. (Cont... |
| brdom 9015 | Dominance relation. (Cont... |
| domen 9016 | Dominance in terms of equi... |
| domeng 9017 | Dominance in terms of equi... |
| ctex 9018 | A countable set is a set. ... |
| f1oen4g 9019 | The domain and range of a ... |
| f1dom4g 9020 | The domain of a one-to-one... |
| f1oen3g 9021 | The domain and range of a ... |
| f1dom3g 9022 | The domain of a one-to-one... |
| f1oen2g 9023 | The domain and range of a ... |
| f1dom2g 9024 | The domain of a one-to-one... |
| f1oeng 9025 | The domain and range of a ... |
| f1domg 9026 | The domain of a one-to-one... |
| f1oen 9027 | The domain and range of a ... |
| f1dom 9028 | The domain of a one-to-one... |
| brsdom 9029 | Strict dominance relation,... |
| isfi 9030 | Express " ` A ` is finite"... |
| enssdom 9031 | Equinumerosity implies dom... |
| enssdomOLD 9032 | Obsolete version of ~ enss... |
| dfdom2 9033 | Alternate definition of do... |
| endom 9034 | Equinumerosity implies dom... |
| sdomdom 9035 | Strict dominance implies d... |
| sdomnen 9036 | Strict dominance implies n... |
| brdom2 9037 | Dominance in terms of stri... |
| bren2 9038 | Equinumerosity expressed i... |
| enrefg 9039 | Equinumerosity is reflexiv... |
| enref 9040 | Equinumerosity is reflexiv... |
| eqeng 9041 | Equality implies equinumer... |
| domrefg 9042 | Dominance is reflexive. (... |
| en2d 9043 | Equinumerosity inference f... |
| en3d 9044 | Equinumerosity inference f... |
| en2i 9045 | Equinumerosity inference f... |
| en3i 9046 | Equinumerosity inference f... |
| dom2lem 9047 | A mapping (first hypothesi... |
| dom2d 9048 | A mapping (first hypothesi... |
| dom3d 9049 | A mapping (first hypothesi... |
| dom2 9050 | A mapping (first hypothesi... |
| dom3 9051 | A mapping (first hypothesi... |
| idssen 9052 | Equality implies equinumer... |
| domssl 9053 | If ` A ` is a subset of ` ... |
| domssr 9054 | If ` C ` is a superset of ... |
| ssdomg 9055 | A set dominates its subset... |
| ener 9056 | Equinumerosity is an equiv... |
| ensymb 9057 | Symmetry of equinumerosity... |
| ensym 9058 | Symmetry of equinumerosity... |
| ensymi 9059 | Symmetry of equinumerosity... |
| ensymd 9060 | Symmetry of equinumerosity... |
| entr 9061 | Transitivity of equinumero... |
| domtr 9062 | Transitivity of dominance ... |
| entri 9063 | A chained equinumerosity i... |
| entr2i 9064 | A chained equinumerosity i... |
| entr3i 9065 | A chained equinumerosity i... |
| entr4i 9066 | A chained equinumerosity i... |
| endomtr 9067 | Transitivity of equinumero... |
| domentr 9068 | Transitivity of dominance ... |
| f1imaeng 9069 | If a function is one-to-on... |
| f1imaen2g 9070 | If a function is one-to-on... |
| f1imaen3g 9071 | If a set function is one-t... |
| f1imaen 9072 | If a function is one-to-on... |
| en0 9073 | The empty set is equinumer... |
| en0ALT 9074 | Shorter proof of ~ en0 , d... |
| en0r 9075 | The empty set is equinumer... |
| ensn1 9076 | A singleton is equinumerou... |
| ensn1g 9077 | A singleton is equinumerou... |
| enpr1g 9078 | ` { A , A } ` has only one... |
| en1 9079 | A set is equinumerous to o... |
| en1b 9080 | A set is equinumerous to o... |
| reuen1 9081 | Two ways to express "exact... |
| euen1 9082 | Two ways to express "exact... |
| euen1b 9083 | Two ways to express " ` A ... |
| funen1cnv 9084 | If a function is equinumer... |
| en1uniel 9085 | A singleton contains its s... |
| 2dom 9086 | A set that dominates ordin... |
| fundmen 9087 | A function is equinumerous... |
| fundmeng 9088 | A function is equinumerous... |
| cnven 9089 | A relational set is equinu... |
| cnvct 9090 | If a set is countable, so ... |
| fndmeng 9091 | A function is equinumerate... |
| mapsnend 9092 | Set exponentiation to a si... |
| mapsnen 9093 | Set exponentiation to a si... |
| snmapen 9094 | Set exponentiation: a sing... |
| snmapen1 9095 | Set exponentiation: a sing... |
| map1 9096 | Set exponentiation: ordina... |
| en2sn 9097 | Two singletons are equinum... |
| 0fi 9098 | The empty set is finite. ... |
| snfi 9099 | A singleton is finite. (C... |
| fiprc 9100 | The class of finite sets i... |
| unen 9101 | Equinumerosity of union of... |
| enrefnn 9102 | Equinumerosity is reflexiv... |
| en2prd 9103 | Two proper unordered pairs... |
| enpr2d 9104 | A pair with distinct eleme... |
| ssct 9105 | Any subset of a countable ... |
| difsnen 9106 | All decrements of a set ar... |
| domdifsn 9107 | Dominance over a set with ... |
| xpsnen 9108 | A set is equinumerous to i... |
| xpsneng 9109 | A set is equinumerous to i... |
| xp1en 9110 | One times a cardinal numbe... |
| endisj 9111 | Any two sets are equinumer... |
| undom 9112 | Dominance law for union. ... |
| xpcomf1o 9113 | The canonical bijection fr... |
| xpcomco 9114 | Composition with the bijec... |
| xpcomen 9115 | Commutative law for equinu... |
| xpcomeng 9116 | Commutative law for equinu... |
| xpsnen2g 9117 | A set is equinumerous to i... |
| xpassen 9118 | Associative law for equinu... |
| xpdom2 9119 | Dominance law for Cartesia... |
| xpdom2g 9120 | Dominance law for Cartesia... |
| xpdom1g 9121 | Dominance law for Cartesia... |
| xpdom3 9122 | A set is dominated by its ... |
| xpdom1 9123 | Dominance law for Cartesia... |
| domunsncan 9124 | A singleton cancellation l... |
| omxpenlem 9125 | Lemma for ~ omxpen . (Con... |
| omxpen 9126 | The cardinal and ordinal p... |
| omf1o 9127 | Construct an explicit bije... |
| pw2f1olem 9128 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9129 | The power set of a set is ... |
| pw2eng 9130 | The power set of a set is ... |
| pw2en 9131 | The power set of a set is ... |
| fopwdom 9132 | Covering implies injection... |
| enfixsn 9133 | Given two equipollent sets... |
| sbthlem1 9134 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9135 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9136 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9137 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9138 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9139 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9140 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9141 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9142 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9143 | Lemma for ~ sbth . (Contr... |
| sbth 9144 | Schroeder-Bernstein Theore... |
| sbthb 9145 | Schroeder-Bernstein Theore... |
| sbthcl 9146 | Schroeder-Bernstein Theore... |
| dfsdom2 9147 | Alternate definition of st... |
| brsdom2 9148 | Alternate definition of st... |
| sdomnsym 9149 | Strict dominance is asymme... |
| domnsym 9150 | Theorem 22(i) of [Suppes] ... |
| 0domg 9151 | Any set dominates the empt... |
| dom0 9152 | A set dominated by the emp... |
| 0sdomg 9153 | A set strictly dominates t... |
| 0dom 9154 | Any set dominates the empt... |
| 0sdom 9155 | A set strictly dominates t... |
| sdom0 9156 | The empty set does not str... |
| sdomdomtr 9157 | Transitivity of strict dom... |
| sdomentr 9158 | Transitivity of strict dom... |
| domsdomtr 9159 | Transitivity of dominance ... |
| ensdomtr 9160 | Transitivity of equinumero... |
| sdomirr 9161 | Strict dominance is irrefl... |
| sdomtr 9162 | Strict dominance is transi... |
| sdomn2lp 9163 | Strict dominance has no 2-... |
| enen1 9164 | Equality-like theorem for ... |
| enen2 9165 | Equality-like theorem for ... |
| domen1 9166 | Equality-like theorem for ... |
| domen2 9167 | Equality-like theorem for ... |
| sdomen1 9168 | Equality-like theorem for ... |
| sdomen2 9169 | Equality-like theorem for ... |
| domtriord 9170 | Dominance is trichotomous ... |
| sdomel 9171 | For ordinals, strict domin... |
| sdomdif 9172 | The difference of a set fr... |
| onsdominel 9173 | An ordinal with more eleme... |
| domunsn 9174 | Dominance over a set with ... |
| fodomr 9175 | There exists a mapping fro... |
| pwdom 9176 | Injection of sets implies ... |
| canth2 9177 | Cantor's Theorem. No set ... |
| canth2g 9178 | Cantor's theorem with the ... |
| 2pwuninel 9179 | The power set of the power... |
| 2pwne 9180 | No set equals the power se... |
| disjen 9181 | A stronger form of ~ pwuni... |
| disjenex 9182 | Existence version of ~ dis... |
| domss2 9183 | A corollary of ~ disjenex ... |
| domssex2 9184 | A corollary of ~ disjenex ... |
| domssex 9185 | Weakening of ~ domssex2 to... |
| xpf1o 9186 | Construct a bijection on a... |
| xpen 9187 | Equinumerosity law for Car... |
| mapen 9188 | Two set exponentiations ar... |
| mapdom1 9189 | Order-preserving property ... |
| mapxpen 9190 | Equinumerosity law for dou... |
| xpmapenlem 9191 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9192 | Equinumerosity law for set... |
| mapunen 9193 | Equinumerosity law for set... |
| map2xp 9194 | A cardinal power with expo... |
| mapdom2 9195 | Order-preserving property ... |
| mapdom3 9196 | Set exponentiation dominat... |
| pwen 9197 | If two sets are equinumero... |
| ssenen 9198 | Equinumerosity of equinume... |
| limenpsi 9199 | A limit ordinal is equinum... |
| limensuci 9200 | A limit ordinal is equinum... |
| limensuc 9201 | A limit ordinal is equinum... |
| infensuc 9202 | Any infinite ordinal is eq... |
| dif1enlem 9203 | Lemma for ~ rexdif1en and ... |
| rexdif1en 9204 | If a set is equinumerous t... |
| dif1en 9205 | If a set ` A ` is equinume... |
| dif1ennn 9206 | If a set ` A ` is equinume... |
| findcard 9207 | Schema for induction on th... |
| findcard2 9208 | Schema for induction on th... |
| findcard2s 9209 | Variation of ~ findcard2 r... |
| findcard2d 9210 | Deduction version of ~ fin... |
| nnfi 9211 | Natural numbers are finite... |
| pssnn 9212 | A proper subset of a natur... |
| ssnnfi 9213 | A subset of a natural numb... |
| unfi 9214 | The union of two finite se... |
| unfid 9215 | The union of two finite se... |
| ssfi 9216 | A subset of a finite set i... |
| ssfiALT 9217 | Shorter proof of ~ ssfi us... |
| diffi 9218 | If ` A ` is finite, ` ( A ... |
| cnvfi 9219 | If a set is finite, its co... |
| pwssfi 9220 | Every element of the power... |
| fnfi 9221 | A version of ~ fnex for fi... |
| f1oenfi 9222 | If the domain of a one-to-... |
| f1oenfirn 9223 | If the range of a one-to-o... |
| f1domfi 9224 | If the codomain of a one-t... |
| f1domfi2 9225 | If the domain of a one-to-... |
| enreffi 9226 | Equinumerosity is reflexiv... |
| ensymfib 9227 | Symmetry of equinumerosity... |
| entrfil 9228 | Transitivity of equinumero... |
| enfii 9229 | A set equinumerous to a fi... |
| enfi 9230 | Equinumerous sets have the... |
| enfiALT 9231 | Shorter proof of ~ enfi us... |
| domfi 9232 | A set dominated by a finit... |
| entrfi 9233 | Transitivity of equinumero... |
| entrfir 9234 | Transitivity of equinumero... |
| domtrfil 9235 | Transitivity of dominance ... |
| domtrfi 9236 | Transitivity of dominance ... |
| domtrfir 9237 | Transitivity of dominance ... |
| f1imaenfi 9238 | If a function is one-to-on... |
| ssdomfi 9239 | A finite set dominates its... |
| ssdomfi2 9240 | A set dominates its finite... |
| sbthfilem 9241 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9242 | Schroeder-Bernstein Theore... |
| domnsymfi 9243 | If a set dominates a finit... |
| sdomdomtrfi 9244 | Transitivity of strict dom... |
| domsdomtrfi 9245 | Transitivity of dominance ... |
| sucdom2 9246 | Strict dominance of a set ... |
| phplem1 9247 | Lemma for Pigeonhole Princ... |
| phplem2 9248 | Lemma for Pigeonhole Princ... |
| nneneq 9249 | Two equinumerous natural n... |
| php 9250 | Pigeonhole Principle. A n... |
| php2 9251 | Corollary of Pigeonhole Pr... |
| php3 9252 | Corollary of Pigeonhole Pr... |
| php4 9253 | Corollary of the Pigeonhol... |
| php5 9254 | Corollary of the Pigeonhol... |
| phpeqd 9255 | Corollary of the Pigeonhol... |
| nndomog 9256 | Cardinal ordering agrees w... |
| onomeneq 9257 | An ordinal number equinume... |
| onfin 9258 | An ordinal number is finit... |
| ordfin 9259 | A generalization of ~ onfi... |
| onfin2 9260 | A set is a natural number ... |
| nndomo 9261 | Cardinal ordering agrees w... |
| nnsdomo 9262 | Cardinal ordering agrees w... |
| sucdom 9263 | Strict dominance of a set ... |
| snnen2o 9264 | A singleton ` { A } ` is n... |
| 0sdom1dom 9265 | Strict dominance over 0 is... |
| 0sdom1domALT 9266 | Alternate proof of ~ 0sdom... |
| 1sdom2 9267 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9268 | Alternate proof of ~ 1sdom... |
| sdom1 9269 | A set has less than one me... |
| modom 9270 | Two ways to express "at mo... |
| modom2 9271 | Two ways to express "at mo... |
| rex2dom 9272 | A set that has at least 2 ... |
| 1sdom2dom 9273 | Strict dominance over 1 is... |
| 1sdom 9274 | A set that strictly domina... |
| unxpdomlem1 9275 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9276 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9277 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9278 | Cartesian product dominate... |
| unxpdom2 9279 | Corollary of ~ unxpdom . ... |
| sucxpdom 9280 | Cartesian product dominate... |
| pssinf 9281 | A set equinumerous to a pr... |
| fisseneq 9282 | A finite set is equal to i... |
| ominf 9283 | The set of natural numbers... |
| isinf 9284 | Any set that is not finite... |
| fineqvlem 9285 | Lemma for ~ fineqv . (Con... |
| fineqv 9286 | If the Axiom of Infinity i... |
| xpfir 9287 | The components of a nonemp... |
| ssfid 9288 | A subset of a finite set i... |
| infi 9289 | The intersection of two se... |
| rabfi 9290 | A restricted class built f... |
| finresfin 9291 | The restriction of a finit... |
| f1finf1o 9292 | Any injection from one fin... |
| nfielex 9293 | If a class is not finite, ... |
| en1eqsn 9294 | A set with one element is ... |
| en1eqsnbi 9295 | A set containing an elemen... |
| dif1ennnALT 9296 | Alternate proof of ~ dif1e... |
| enp1ilem 9297 | Lemma for uses of ~ enp1i ... |
| enp1i 9298 | Proof induction for ~ en2 ... |
| en2 9299 | A set equinumerous to ordi... |
| en3 9300 | A set equinumerous to ordi... |
| en4 9301 | A set equinumerous to ordi... |
| findcard3 9302 | Schema for strong inductio... |
| ac6sfi 9303 | A version of ~ ac6s for fi... |
| frfi 9304 | A partial order is well-fo... |
| fimax2g 9305 | A finite set has a maximum... |
| fimaxg 9306 | A finite set has a maximum... |
| fisupg 9307 | Lemma showing existence an... |
| wofi 9308 | A total order on a finite ... |
| ordunifi 9309 | The maximum of a finite co... |
| fissorduni 9310 | The union (supremum) of a ... |
| nnunifi 9311 | The union (supremum) of a ... |
| unblem1 9312 | Lemma for ~ unbnn . After... |
| unblem2 9313 | Lemma for ~ unbnn . The v... |
| unblem3 9314 | Lemma for ~ unbnn . The v... |
| unblem4 9315 | Lemma for ~ unbnn . The f... |
| unbnn 9316 | Any unbounded subset of na... |
| unbnn2 9317 | Version of ~ unbnn that do... |
| isfinite2 9318 | Any set strictly dominated... |
| nnsdomg 9319 | Omega strictly dominates a... |
| isfiniteg 9320 | A set is finite iff it is ... |
| infsdomnn 9321 | An infinite set strictly d... |
| infn0 9322 | An infinite set is not emp... |
| infn0ALT 9323 | Shorter proof of ~ infn0 u... |
| fin2inf 9324 | This (useless) theorem, wh... |
| unfilem1 9325 | Lemma for proving that the... |
| unfilem2 9326 | Lemma for proving that the... |
| unfilem3 9327 | Lemma for proving that the... |
| unfir 9328 | If a union is finite, the ... |
| unfib 9329 | A union is finite if and o... |
| unfi2 9330 | The union of two finite se... |
| difinf 9331 | An infinite set ` A ` minu... |
| fodomfi 9332 | An onto function implies d... |
| fofi 9333 | If an onto function has a ... |
| f1fi 9334 | If a 1-to-1 function has a... |
| imafi 9335 | Images of finite sets are ... |
| pwfir 9336 | If the power set of a set ... |
| pwfilem 9337 | Lemma for ~ pwfi . (Contr... |
| pwfi 9338 | The power set of a finite ... |
| xpfi 9339 | The Cartesian product of t... |
| 3xpfi 9340 | The Cartesian product of t... |
| domunfican 9341 | A finite set union cancell... |
| infcntss 9342 | Every infinite set has a d... |
| prfi 9343 | An unordered pair is finit... |
| prfiALT 9344 | Shorter proof of ~ prfi us... |
| tpfi 9345 | An unordered triple is fin... |
| fiint 9346 | Equivalent ways of stating... |
| fodomfir 9347 | There exists a mapping fro... |
| fodomfib 9348 | Equivalence of an onto map... |
| fofinf1o 9349 | Any surjection from one fi... |
| rneqdmfinf1o 9350 | Any function from a finite... |
| fidomdm 9351 | Any finite set dominates i... |
| dmfi 9352 | The domain of a finite set... |
| fundmfibi 9353 | A function is finite if an... |
| resfnfinfin 9354 | The restriction of a funct... |
| residfi 9355 | A restricted identity func... |
| cnvfiALT 9356 | Shorter proof of ~ cnvfi u... |
| rnfi 9357 | The range of a finite set ... |
| f1dmvrnfibi 9358 | A one-to-one function whos... |
| f1vrnfibi 9359 | A one-to-one function whic... |
| iunfi 9360 | The finite union of finite... |
| unifi 9361 | The finite union of finite... |
| unifi2 9362 | The finite union of finite... |
| infssuni 9363 | If an infinite set ` A ` i... |
| unirnffid 9364 | The union of the range of ... |
| mapfi 9365 | Set exponentiation of fini... |
| ixpfi 9366 | A Cartesian product of fin... |
| ixpfi2 9367 | A Cartesian product of fin... |
| mptfi 9368 | A finite mapping set is fi... |
| abrexfi 9369 | An image set from a finite... |
| cnvimamptfin 9370 | Any preimage by a mapping ... |
| elfpw 9371 | Membership in a class of f... |
| unifpw 9372 | A set is the union of its ... |
| f1opwfi 9373 | A one-to-one mapping induc... |
| fissuni 9374 | A finite subset of a union... |
| fipreima 9375 | Given a finite subset ` A ... |
| finsschain 9376 | A finite subset of the uni... |
| indexfi 9377 | If for every element of a ... |
| imafi2 9378 | The image by a finite set ... |
| unifi3 9379 | If a union is finite, then... |
| tfsnfin2 9380 | A transfinite sequence is ... |
| relfsupp 9383 | The property of a function... |
| relprcnfsupp 9384 | A proper class is never fi... |
| isfsupp 9385 | The property of a class to... |
| isfsuppd 9386 | Deduction form of ~ isfsup... |
| funisfsupp 9387 | The property of a function... |
| fsuppimp 9388 | Implications of a class be... |
| fsuppimpd 9389 | A finitely supported funct... |
| fsuppfund 9390 | A finitely supported funct... |
| fisuppfi 9391 | A function on a finite set... |
| finnzfsuppd 9392 | If a function is zero outs... |
| fdmfisuppfi 9393 | The support of a function ... |
| fdmfifsupp 9394 | A function with a finite d... |
| fsuppmptdm 9395 | A mapping with a finite do... |
| fndmfisuppfi 9396 | The support of a function ... |
| fndmfifsupp 9397 | A function with a finite d... |
| suppeqfsuppbi 9398 | If two functions have the ... |
| suppssfifsupp 9399 | If the support of a functi... |
| fsuppsssupp 9400 | If the support of a functi... |
| fsuppsssuppgd 9401 | If the support of a functi... |
| fsuppss 9402 | A subset of a finitely sup... |
| fsuppssov1 9403 | Formula building theorem f... |
| fsuppxpfi 9404 | The cartesian product of t... |
| fczfsuppd 9405 | A constant function with v... |
| fsuppun 9406 | The union of two finitely ... |
| fsuppunfi 9407 | The union of the support o... |
| fsuppunbi 9408 | If the union of two classe... |
| 0fsupp 9409 | The empty set is a finitel... |
| snopfsupp 9410 | A singleton containing an ... |
| funsnfsupp 9411 | Finite support for a funct... |
| fsuppres 9412 | The restriction of a finit... |
| fmptssfisupp 9413 | The restriction of a mappi... |
| ressuppfi 9414 | If the support of the rest... |
| resfsupp 9415 | If the restriction of a fu... |
| resfifsupp 9416 | The restriction of a funct... |
| ffsuppbi 9417 | Two ways of saying that a ... |
| fsuppmptif 9418 | A function mapping an argu... |
| sniffsupp 9419 | A function mapping all but... |
| fsuppcolem 9420 | Lemma for ~ fsuppco . For... |
| fsuppco 9421 | The composition of a 1-1 f... |
| fsuppco2 9422 | The composition of a funct... |
| fsuppcor 9423 | The composition of a funct... |
| mapfienlem1 9424 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9425 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9426 | Lemma 3 for ~ mapfien . (... |
| mapfien 9427 | A bijection of the base se... |
| mapfien2 9428 | Equinumerousity relation f... |
| fival 9431 | The set of all the finite ... |
| elfi 9432 | Specific properties of an ... |
| elfi2 9433 | The empty intersection nee... |
| elfir 9434 | Sufficient condition for a... |
| intrnfi 9435 | Sufficient condition for t... |
| iinfi 9436 | An indexed intersection of... |
| inelfi 9437 | The intersection of two se... |
| ssfii 9438 | Any element of a set ` A `... |
| fi0 9439 | The set of finite intersec... |
| fieq0 9440 | A set is empty iff the cla... |
| fiin 9441 | The elements of ` ( fi `` ... |
| dffi2 9442 | The set of finite intersec... |
| fiss 9443 | Subset relationship for fu... |
| inficl 9444 | A set which is closed unde... |
| fipwuni 9445 | The set of finite intersec... |
| fisn 9446 | A singleton is closed unde... |
| fiuni 9447 | The union of the finite in... |
| fipwss 9448 | If a set is a family of su... |
| elfiun 9449 | A finite intersection of e... |
| dffi3 9450 | The set of finite intersec... |
| fifo 9451 | Describe a surjection from... |
| marypha1lem 9452 | Core induction for Philip ... |
| marypha1 9453 | (Philip) Hall's marriage t... |
| marypha2lem1 9454 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9455 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9456 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9457 | Lemma for ~ marypha2 . Pr... |
| marypha2 9458 | Version of ~ marypha1 usin... |
| dfsup2 9463 | Quantifier-free definition... |
| supeq1 9464 | Equality theorem for supre... |
| supeq1d 9465 | Equality deduction for sup... |
| supeq1i 9466 | Equality inference for sup... |
| supeq2 9467 | Equality theorem for supre... |
| supeq3 9468 | Equality theorem for supre... |
| supeq123d 9469 | Equality deduction for sup... |
| nfsup 9470 | Hypothesis builder for sup... |
| supmo 9471 | Any class ` B ` has at mos... |
| supexd 9472 | A supremum is a set. (Con... |
| supeu 9473 | A supremum is unique. Sim... |
| supval2 9474 | Alternate expression for t... |
| eqsup 9475 | Sufficient condition for a... |
| eqsupd 9476 | Sufficient condition for a... |
| supcl 9477 | A supremum belongs to its ... |
| supub 9478 | A supremum is an upper bou... |
| suplub 9479 | A supremum is the least up... |
| suplub2 9480 | Bidirectional form of ~ su... |
| supnub 9481 | An upper bound is not less... |
| supssd 9482 | Inequality deduction for s... |
| supex 9483 | A supremum is a set. (Con... |
| sup00 9484 | The supremum under an empt... |
| sup0riota 9485 | The supremum of an empty s... |
| sup0 9486 | The supremum of an empty s... |
| supmax 9487 | The greatest element of a ... |
| fisup2g 9488 | A finite set satisfies the... |
| fisupcl 9489 | A nonempty finite set cont... |
| supgtoreq 9490 | The supremum of a finite s... |
| suppr 9491 | The supremum of a pair. (... |
| supsn 9492 | The supremum of a singleto... |
| supisolem 9493 | Lemma for ~ supiso . (Con... |
| supisoex 9494 | Lemma for ~ supiso . (Con... |
| supiso 9495 | Image of a supremum under ... |
| infeq1 9496 | Equality theorem for infim... |
| infeq1d 9497 | Equality deduction for inf... |
| infeq1i 9498 | Equality inference for inf... |
| infeq2 9499 | Equality theorem for infim... |
| infeq3 9500 | Equality theorem for infim... |
| infeq123d 9501 | Equality deduction for inf... |
| nfinf 9502 | Hypothesis builder for inf... |
| infexd 9503 | An infimum is a set. (Con... |
| eqinf 9504 | Sufficient condition for a... |
| eqinfd 9505 | Sufficient condition for a... |
| infval 9506 | Alternate expression for t... |
| infcllem 9507 | Lemma for ~ infcl , ~ infl... |
| infcl 9508 | An infimum belongs to its ... |
| inflb 9509 | An infimum is a lower boun... |
| infglb 9510 | An infimum is the greatest... |
| infglbb 9511 | Bidirectional form of ~ in... |
| infnlb 9512 | A lower bound is not great... |
| infssd 9513 | Inequality deduction for i... |
| infex 9514 | An infimum is a set. (Con... |
| infmin 9515 | The smallest element of a ... |
| infmo 9516 | Any class ` B ` has at mos... |
| infeu 9517 | An infimum is unique. (Co... |
| fimin2g 9518 | A finite set has a minimum... |
| fiming 9519 | A finite set has a minimum... |
| fiinfg 9520 | Lemma showing existence an... |
| fiinf2g 9521 | A finite set satisfies the... |
| fiinfcl 9522 | A nonempty finite set cont... |
| infltoreq 9523 | The infimum of a finite se... |
| infpr 9524 | The infimum of a pair. (C... |
| infsupprpr 9525 | The infimum of a proper pa... |
| infsn 9526 | The infimum of a singleton... |
| inf00 9527 | The infimum regarding an e... |
| infempty 9528 | The infimum of an empty se... |
| infiso 9529 | Image of an infimum under ... |
| dfoi 9532 | Rewrite ~ df-oi with abbre... |
| oieq1 9533 | Equality theorem for ordin... |
| oieq2 9534 | Equality theorem for ordin... |
| nfoi 9535 | Hypothesis builder for ord... |
| ordiso2 9536 | Generalize ~ ordiso to pro... |
| ordiso 9537 | Order-isomorphic ordinal n... |
| ordtypecbv 9538 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9539 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9540 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9541 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9542 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9543 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9544 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9545 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9546 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9547 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9548 | Lemma for ~ ordtype . Usi... |
| oi0 9549 | Definition of the ordinal ... |
| oicl 9550 | The order type of the well... |
| oif 9551 | The order isomorphism of t... |
| oiiso2 9552 | The order isomorphism of t... |
| ordtype 9553 | For any set-like well-orde... |
| oiiniseg 9554 | ` ran F ` is an initial se... |
| ordtype2 9555 | For any set-like well-orde... |
| oiexg 9556 | The order isomorphism on a... |
| oion 9557 | The order type of the well... |
| oiiso 9558 | The order isomorphism of t... |
| oien 9559 | The order type of a well-o... |
| oieu 9560 | Uniqueness of the unique o... |
| oismo 9561 | When ` A ` is a subclass o... |
| oiid 9562 | The order type of an ordin... |
| hartogslem1 9563 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9564 | Lemma for ~ hartogs . (Co... |
| hartogs 9565 | The class of ordinals domi... |
| wofib 9566 | The only sets which are we... |
| wemaplem1 9567 | Value of the lexicographic... |
| wemaplem2 9568 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9569 | Lemma for ~ wemapso . Tra... |
| wemappo 9570 | Construct lexicographic or... |
| wemapsolem 9571 | Lemma for ~ wemapso . (Co... |
| wemapso 9572 | Construct lexicographic or... |
| wemapso2lem 9573 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9574 | An alternative to having a... |
| card2on 9575 | The alternate definition o... |
| card2inf 9576 | The alternate definition o... |
| harf 9579 | Functionality of the Harto... |
| harcl 9580 | Values of the Hartogs func... |
| harval 9581 | Function value of the Hart... |
| elharval 9582 | The Hartogs number of a se... |
| harndom 9583 | The Hartogs number of a se... |
| harword 9584 | Weak ordering property of ... |
| relwdom 9587 | Weak dominance is a relati... |
| brwdom 9588 | Property of weak dominance... |
| brwdomi 9589 | Property of weak dominance... |
| brwdomn0 9590 | Weak dominance over nonemp... |
| 0wdom 9591 | Any set weakly dominates t... |
| fowdom 9592 | An onto function implies w... |
| wdomref 9593 | Reflexivity of weak domina... |
| brwdom2 9594 | Alternate characterization... |
| domwdom 9595 | Weak dominance is implied ... |
| wdomtr 9596 | Transitivity of weak domin... |
| wdomen1 9597 | Equality-like theorem for ... |
| wdomen2 9598 | Equality-like theorem for ... |
| wdompwdom 9599 | Weak dominance strengthens... |
| canthwdom 9600 | Cantor's Theorem, stated u... |
| wdom2d 9601 | Deduce weak dominance from... |
| wdomd 9602 | Deduce weak dominance from... |
| brwdom3 9603 | Condition for weak dominan... |
| brwdom3i 9604 | Weak dominance implies exi... |
| unwdomg 9605 | Weak dominance of a (disjo... |
| xpwdomg 9606 | Weak dominance of a Cartes... |
| wdomima2g 9607 | A set is weakly dominant o... |
| wdomimag 9608 | A set is weakly dominant o... |
| unxpwdom2 9609 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9610 | If a Cartesian product is ... |
| ixpiunwdom 9611 | Describe an onto function ... |
| harwdom 9612 | The value of the Hartogs f... |
| axreg2 9614 | Axiom of Regularity expres... |
| zfregcl 9615 | The Axiom of Regularity wi... |
| zfregclOLD 9616 | Obsolete version of ~ zfre... |
| zfreg 9617 | The Axiom of Regularity us... |
| elirrv 9618 | The membership relation is... |
| elirrvOLD 9619 | Obsolete version of ~ elir... |
| elirrvOLDOLD 9620 | Obsolete version of ~ elir... |
| elirr 9621 | No class is a member of it... |
| elneq 9622 | A class is not equal to an... |
| nelaneq 9623 | A class is not an element ... |
| nelaneqOLD 9624 | Obsolete version of ~ nela... |
| nelaneqOLDOLD 9625 | Obsolete version of ~ nela... |
| epinid0 9626 | The membership relation an... |
| sucprcreg 9627 | A class is equal to its su... |
| sucprcregOLD 9628 | Obsolete version of ~ sucp... |
| ruv 9629 | The Russell class is equal... |
| ruALT 9630 | Alternate proof of ~ ru , ... |
| disjcsn 9631 | A class is disjoint from i... |
| zfregfr 9632 | The membership relation is... |
| elirrvALT 9633 | Alternate proof of ~ elirr... |
| en2lp 9634 | No class has 2-cycle membe... |
| elnanel 9635 | Two classes are not elemen... |
| cnvepnep 9636 | The membership (epsilon) r... |
| epnsym 9637 | The membership (epsilon) r... |
| elnotel 9638 | A class cannot be an eleme... |
| elnel 9639 | A class cannot be an eleme... |
| en3lplem1 9640 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9641 | Lemma for ~ en3lp . (Cont... |
| en3lp 9642 | No class has 3-cycle membe... |
| preleqg 9643 | Equality of two unordered ... |
| preleq 9644 | Equality of two unordered ... |
| preleqALT 9645 | Alternate proof of ~ prele... |
| opthreg 9646 | Theorem for alternate repr... |
| suc11reg 9647 | The successor operation be... |
| dford2 9648 | Assuming ~ ax-reg , an ord... |
| inf0 9649 | Existence of ` _om ` impli... |
| inf1 9650 | Variation of Axiom of Infi... |
| inf2 9651 | Variation of Axiom of Infi... |
| inf3lema 9652 | Lemma for our Axiom of Inf... |
| inf3lemb 9653 | Lemma for our Axiom of Inf... |
| inf3lemc 9654 | Lemma for our Axiom of Inf... |
| inf3lemd 9655 | Lemma for our Axiom of Inf... |
| inf3lem1 9656 | Lemma for our Axiom of Inf... |
| inf3lem2 9657 | Lemma for our Axiom of Inf... |
| inf3lem3 9658 | Lemma for our Axiom of Inf... |
| inf3lem4 9659 | Lemma for our Axiom of Inf... |
| inf3lem5 9660 | Lemma for our Axiom of Inf... |
| inf3lem6 9661 | Lemma for our Axiom of Inf... |
| inf3lem7 9662 | Lemma for our Axiom of Inf... |
| inf3 9663 | Our Axiom of Infinity ~ ax... |
| infeq5i 9664 | Half of ~ infeq5 . (Contr... |
| infeq5 9665 | The statement "there exist... |
| zfinf 9667 | Axiom of Infinity expresse... |
| axinf2 9668 | A standard version of Axio... |
| zfinf2 9670 | A standard version of the ... |
| omex 9671 | The existence of omega (th... |
| axinf 9672 | The first version of the A... |
| inf5 9673 | The statement "there exist... |
| omelon 9674 | Omega is an ordinal number... |
| dfom3 9675 | The class of natural numbe... |
| elom3 9676 | A simplification of ~ elom... |
| dfom4 9677 | A simplification of ~ df-o... |
| dfom5 9678 | ` _om ` is the smallest li... |
| oancom 9679 | Ordinal addition is not co... |
| isfinite 9680 | A set is finite iff it is ... |
| fict 9681 | A finite set is countable ... |
| nnsdom 9682 | A natural number is strict... |
| omenps 9683 | Omega is equinumerous to a... |
| omensuc 9684 | The set of natural numbers... |
| infdifsn 9685 | Removing a singleton from ... |
| infdiffi 9686 | Removing a finite set from... |
| unbnn3 9687 | Any unbounded subset of na... |
| noinfep 9688 | Using the Axiom of Regular... |
| cantnffval 9691 | The value of the Cantor no... |
| cantnfdm 9692 | The domain of the Cantor n... |
| cantnfvalf 9693 | Lemma for ~ cantnf . The ... |
| cantnfs 9694 | Elementhood in the set of ... |
| cantnfcl 9695 | Basic properties of the or... |
| cantnfval 9696 | The value of the Cantor no... |
| cantnfval2 9697 | Alternate expression for t... |
| cantnfsuc 9698 | The value of the recursive... |
| cantnfle 9699 | A lower bound on the ` CNF... |
| cantnflt 9700 | An upper bound on the part... |
| cantnflt2 9701 | An upper bound on the ` CN... |
| cantnff 9702 | The ` CNF ` function is a ... |
| cantnf0 9703 | The value of the zero func... |
| cantnfrescl 9704 | A function is finitely sup... |
| cantnfres 9705 | The ` CNF ` function respe... |
| cantnfp1lem1 9706 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9707 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9708 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9709 | If ` F ` is created by add... |
| oemapso 9710 | The relation ` T ` is a st... |
| oemapval 9711 | Value of the relation ` T ... |
| oemapvali 9712 | If ` F < G ` , then there ... |
| cantnflem1a 9713 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9714 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9715 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9716 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9717 | Lemma for ~ cantnf . This... |
| cantnflem2 9718 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9719 | Lemma for ~ cantnf . Here... |
| cantnflem4 9720 | Lemma for ~ cantnf . Comp... |
| cantnf 9721 | The Cantor Normal Form the... |
| oemapwe 9722 | The lexicographic order on... |
| cantnffval2 9723 | An alternate definition of... |
| cantnff1o 9724 | Simplify the isomorphism o... |
| wemapwe 9725 | Construct lexicographic or... |
| oef1o 9726 | A bijection of the base se... |
| cnfcomlem 9727 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9728 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9729 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9730 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9731 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9732 | Any infinite ordinal ` B `... |
| cnfcom3clem 9733 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9734 | Wrap the construction of ~... |
| ttrcleq 9737 | Equality theorem for trans... |
| nfttrcld 9738 | Bound variable hypothesis ... |
| nfttrcl 9739 | Bound variable hypothesis ... |
| relttrcl 9740 | The transitive closure of ... |
| brttrcl 9741 | Characterization of elemen... |
| brttrcl2 9742 | Characterization of elemen... |
| ssttrcl 9743 | If ` R ` is a relation, th... |
| ttrcltr 9744 | The transitive closure of ... |
| ttrclresv 9745 | The transitive closure of ... |
| ttrclco 9746 | Composition law for the tr... |
| cottrcl 9747 | Composition law for the tr... |
| ttrclss 9748 | If ` R ` is a subclass of ... |
| dmttrcl 9749 | The domain of a transitive... |
| rnttrcl 9750 | The range of a transitive ... |
| ttrclexg 9751 | If ` R ` is a set, then so... |
| dfttrcl2 9752 | When ` R ` is a set and a ... |
| ttrclselem1 9753 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9754 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9755 | If ` R ` is set-like over ... |
| trcl 9756 | For any set ` A ` , show t... |
| tz9.1 9757 | Every set has a transitive... |
| tz9.1c 9758 | Alternate expression for t... |
| epfrs 9759 | The strong form of the Axi... |
| zfregs 9760 | The strong form of the Axi... |
| zfregs2 9761 | Alternate strong form of t... |
| tcvalg 9764 | Value of the transitive cl... |
| tcid 9765 | Defining property of the t... |
| tctr 9766 | Defining property of the t... |
| tcmin 9767 | Defining property of the t... |
| tc2 9768 | A variant of the definitio... |
| tcsni 9769 | The transitive closure of ... |
| tcss 9770 | The transitive closure fun... |
| tcel 9771 | The transitive closure fun... |
| tcidm 9772 | The transitive closure fun... |
| tc0 9773 | The transitive closure of ... |
| tc00 9774 | The transitive closure is ... |
| setind 9775 | Set (epsilon) induction. ... |
| setind2 9776 | Set (epsilon) induction, s... |
| setinds 9777 | Principle of set induction... |
| setinds2f 9778 | ` _E ` induction schema, u... |
| setinds2 9779 | ` _E ` induction schema, u... |
| frmin 9780 | Every (possibly proper) su... |
| frind 9781 | A subclass of a well-found... |
| frinsg 9782 | Well-Founded Induction Sch... |
| frins 9783 | Well-Founded Induction Sch... |
| frins2f 9784 | Well-Founded Induction sch... |
| frins2 9785 | Well-Founded Induction sch... |
| frins3 9786 | Well-Founded Induction sch... |
| frr3g 9787 | Functions defined by well-... |
| frrlem15 9788 | Lemma for general well-fou... |
| frrlem16 9789 | Lemma for general well-fou... |
| frr1 9790 | Law of general well-founde... |
| frr2 9791 | Law of general well-founde... |
| frr3 9792 | Law of general well-founde... |
| r1funlimOLD 9797 | The cumulative hierarchy o... |
| r1fun 9798 | The cumulative hierarchy o... |
| r1dmlim 9799 | The domain of the cumulati... |
| r1fnon 9800 | The cumulative hierarchy o... |
| rnr1 9801 | The range of the cumulativ... |
| r10 9802 | Value of the cumulative hi... |
| r1sucg 9803 | Value of the cumulative hi... |
| r1suc 9804 | Value of the cumulative hi... |
| r1limg 9805 | Value of the cumulative hi... |
| r1lim 9806 | Value of the cumulative hi... |
| r1fin 9807 | The first ` _om ` stages o... |
| r1sdom 9808 | Each stage of the cumulati... |
| r111 9809 | The cumulative hierarchy o... |
| r1tr 9810 | Each stage of the cumulati... |
| r1tr2 9811 | Each stage of the cumulati... |
| r1ordg 9812 | Ordering relation for the ... |
| r1ord3g 9813 | Ordering relation for the ... |
| r1ord 9814 | Ordering relation for the ... |
| r1ord2 9815 | Ordering relation for the ... |
| r1ord3 9816 | Ordering relation for the ... |
| r1sssuc 9817 | A stage of the cumulative ... |
| r1pwss 9818 | Each stage of the cumulati... |
| r1sscl 9819 | Each stage of the cumulati... |
| r1val1 9820 | The value of the cumulativ... |
| tz9.12lem1 9821 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9822 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9823 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9824 | A set is well-founded if a... |
| tz9.13 9825 | Every set is well-founded,... |
| tz9.13g 9826 | Every set is well-founded,... |
| rankwflemb 9827 | Two ways of expressing tha... |
| rankwflembOLD 9828 | Obsolete version of ~ rank... |
| rankf 9829 | The domain and codomain of... |
| rankon 9830 | The rank of a set is an or... |
| r1elwf 9831 | Any element of (any stage ... |
| rankvalb 9832 | Value of the rank function... |
| rankr1ai 9833 | One direction of ~ rankr1a... |
| rankvaln 9834 | Value of the rank function... |
| rankidb 9835 | Identity law for the rank ... |
| rankdmr1 9836 | A rank is a member of the ... |
| rankr1ag 9837 | A version of ~ rankr1a tha... |
| rankr1bg 9838 | A relationship between ran... |
| r1rankidb 9839 | Any set is a subset of the... |
| r1elssi 9840 | The range of the ` R1 ` fu... |
| r1elss 9841 | The range of the ` R1 ` fu... |
| pwwf 9842 | A power set is well-founde... |
| sswf 9843 | A subset of a well-founded... |
| snwf 9844 | A singleton is well-founde... |
| unwf 9845 | A binary union is well-fou... |
| prwf 9846 | An unordered pair is well-... |
| opwf 9847 | An ordered pair is well-fo... |
| unir1 9848 | The cumulative hierarchy o... |
| jech9.3 9849 | Every set belongs to some ... |
| jech9.3OLD 9850 | Obsolete version of ~ jech... |
| rankwflem 9851 | Every set is well-founded,... |
| rankval 9852 | Value of the rank function... |
| rankvalg 9853 | Value of the rank function... |
| rankval2 9854 | Value of an alternate defi... |
| uniwf 9855 | A union is well-founded if... |
| rankr1clem 9856 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9857 | A relationship between the... |
| rankidn 9858 | A relationship between the... |
| rankpwi 9859 | The rank of a power set. ... |
| rankelb 9860 | The membership relation is... |
| wfelirr 9861 | A well-founded set is not ... |
| rankval2b 9862 | Value of an alternate defi... |
| rankval3b 9863 | The value of the rank func... |
| ranksnb 9864 | The rank of a singleton. ... |
| rankonidlem 9865 | Lemma for ~ rankonid . (C... |
| rankonid 9866 | The rank of an ordinal num... |
| onwf 9867 | The ordinals are all well-... |
| r1wf 9868 | Each stage in the cumulati... |
| elwf 9869 | An element of a well-found... |
| onssr1 9870 | Initial segments of the or... |
| rankr1g 9871 | A relationship between the... |
| rankid 9872 | Identity law for the rank ... |
| rankr1 9873 | A relationship between the... |
| ssrankr1 9874 | A relationship between an ... |
| rankr1a 9875 | A relationship between ran... |
| r1val2 9876 | The value of the cumulativ... |
| r1val3 9877 | The value of the cumulativ... |
| rankel 9878 | The membership relation is... |
| rankelg 9879 | The membership relation is... |
| rankval3 9880 | The value of the rank func... |
| bndrank 9881 | Any class whose elements h... |
| unbndrank 9882 | The elements of a proper c... |
| rankpw 9883 | The rank of the powerset i... |
| rankpwg 9884 | The rank of the powerset i... |
| ranklim 9885 | The rank of a set belongs ... |
| r1pw 9886 | A set is in a given stage ... |
| r1pwALT 9887 | Alternate shorter proof of... |
| r1pwcl 9888 | The cumulative hierarchy o... |
| rankssb 9889 | The subset relation is inh... |
| rankss 9890 | The subset relation is inh... |
| rankunb 9891 | The rank of the union of t... |
| rankprb 9892 | The rank of an unordered p... |
| rankopb 9893 | The rank of an ordered pai... |
| rankuni2b 9894 | The value of the rank func... |
| ranksn 9895 | The rank of a singleton. ... |
| rankuni2 9896 | The rank of a union. Part... |
| rankun 9897 | The rank of the union of t... |
| rankpr 9898 | The rank of an unordered p... |
| rankop 9899 | The rank of an ordered pai... |
| rankung 9900 | The rank of the union of t... |
| ranksng 9901 | The rank of a singleton. ... |
| r1rankid 9902 | Any set is a subset of the... |
| rankeq0b 9903 | A set is empty iff its ran... |
| rankeq0 9904 | A set is empty iff its ran... |
| rankr1id 9905 | The rank of the hierarchy ... |
| rankuni 9906 | The rank of a union. Part... |
| rankval4b 9907 | The rank of a set is the s... |
| rankr1b 9908 | A relationship between ran... |
| ranksuc 9909 | The rank of a successor. ... |
| rankuniss 9910 | Upper bound of the rank of... |
| rankval4 9911 | The rank of a set is the s... |
| rankbnd 9912 | The rank of a set is bound... |
| rankbnd2 9913 | The rank of a set is bound... |
| rankc1 9914 | A relationship that can be... |
| rankc2 9915 | A relationship that can be... |
| rankelun 9916 | Rank membership is inherit... |
| rankelpr 9917 | Rank membership is inherit... |
| rankelop 9918 | Rank membership is inherit... |
| rankxpl 9919 | A lower bound on the rank ... |
| rankxpu 9920 | An upper bound on the rank... |
| rankfu 9921 | An upper bound on the rank... |
| rankmapu 9922 | An upper bound on the rank... |
| rankxplim 9923 | The rank of a Cartesian pr... |
| rankxplim2 9924 | If the rank of a Cartesian... |
| rankxplim3 9925 | The rank of a Cartesian pr... |
| rankxpsuc 9926 | The rank of a Cartesian pr... |
| tcwf 9927 | The transitive closure fun... |
| tcrank 9928 | This theorem expresses two... |
| rankfilimbi 9929 | If all elements of a finit... |
| r1filimi 9930 | If all elements of a finit... |
| dfhf2 9933 | Alternate definition of th... |
| elhf 9934 | Membership in the heredita... |
| elhfOLD 9935 | Obsolete version of ~ elhf... |
| hffi 9936 | Hereditarily finite sets a... |
| elhf2 9937 | Alternate form of membersh... |
| elhf2g 9938 | Hereditarily finiteness vi... |
| elhf4 9939 | A set is hereditarily fini... |
| elhf3 9940 | A set is hereditarily fini... |
| hfelhf 9941 | Any member of a hereditari... |
| hfsshf 9942 | Any subset of a hereditari... |
| hfelhfOLD 9943 | Obsolete version of ~ elhf... |
| 0hf 9944 | The empty set is a heredit... |
| hfun 9945 | The union of two hereditar... |
| hfunOLD 9946 | Obsolete version of ~ hfun... |
| hfsn 9947 | The singleton of a heredit... |
| hfsnOLD 9948 | Obsolete version of ~ hfsn... |
| hfadj 9949 | Adjoining one hereditarily... |
| elhf3OLD 9950 | Obsolete version of ~ elhf... |
| hfuni 9951 | The union of a hereditaril... |
| hfuniOLD 9952 | Obsolete version of ~ hfun... |
| hfpw 9953 | The power class of a hered... |
| hfpwOLD 9954 | Obsolete version of ~ hfpw... |
| scotteqd 9957 | Equality theorem for the S... |
| scotteq 9958 | Closed form of ~ scotteqd ... |
| nfscott 9959 | Bound-variable hypothesis ... |
| scottex 9960 | Scott's trick produces a s... |
| scottexOLD 9961 | Obsolete version of ~ scot... |
| scottss 9962 | Scott's trick produces a s... |
| scott0 9963 | Applying Scott's trick to ... |
| scott0b 9964 | Applying Scott's trick yie... |
| scott0OLD 9965 | Obsolete version of ~ scot... |
| scottabf 9966 | Value of the Scott operati... |
| scottab 9967 | Value of the Scott operati... |
| scottabes 9968 | Value of the Scott operati... |
| elscottab 9969 | An element of the output o... |
| scottexsOLD 9970 | Obsolete theorem as of 27-... |
| scott0bs 9971 | Theorem scheme version of ... |
| scott0bsOLD 9972 | Obsolete version of ~ scot... |
| scottex2OLD 9973 | Obsolete version of ~ scot... |
| scotteld 9974 | The Scott operation sends ... |
| scottelrankd 9975 | Property of a Scott's tric... |
| scottrankd 9976 | Rank of a nonempty Scott's... |
| cplem1 9977 | Lemma for the Collection P... |
| cplem1OLD 9978 | Obsolete version of ~ cple... |
| cplem2 9979 | Lemma for the Collection P... |
| cplem2OLD 9980 | Obsolete version of ~ cple... |
| cp 9981 | Collection Principle. Thi... |
| bnd 9982 | A very strong generalizati... |
| bnd2 9983 | A variant of the Boundedne... |
| kardex 9984 | The collection of all sets... |
| kardexOLD 9985 | Obsolete version of ~ kard... |
| karden 9986 | If we allow the Axiom of R... |
| kardenOLD 9987 | Obsolete version of ~ kard... |
| htalem 9988 | Lemma for defining an emul... |
| hta 9989 | A ZFC emulation of Hilbert... |
| htaOLD 9990 | Obsolete version of ~ hta ... |
| setrec1lem1 9993 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 9994 | Lemma for ~ setrec1 . If ... |
| bnd2d 9995 | Deduction form of ~ bnd2 .... |
| setrec1lem3 9996 | Lemma for ~ setrec1 . If ... |
| spcdvw 9997 | A version of ~ spcdv where... |
| setrec1lem4 9998 | Lemma for ~ setrec1 . If ... |
| setrec1 9999 | This is the first of two f... |
| setrec2fun 10000 | This is the second of two ... |
| setrec2lem1 10001 | Lemma for ~ setrec2 . The... |
| dffun3f 10002 | Alternate definition of fu... |
| setrec2lem2 10003 | Lemma for ~ setrec2 . The... |
| setrec2 10004 | This is the second of two ... |
| setrec2v 10005 | Version of ~ setrec2 with ... |
| djueq12 10012 | Equality theorem for disjo... |
| djueq1 10013 | Equality theorem for disjo... |
| djueq2 10014 | Equality theorem for disjo... |
| nfdju 10015 | Bound-variable hypothesis ... |
| djuex 10016 | The disjoint union of sets... |
| djuexb 10017 | The disjoint union of two ... |
| djulcl 10018 | Left closure of disjoint u... |
| djurcl 10019 | Right closure of disjoint ... |
| djulf1o 10020 | The left injection functio... |
| djurf1o 10021 | The right injection functi... |
| inlresf 10022 | The left injection restric... |
| inlresf1 10023 | The left injection restric... |
| inrresf 10024 | The right injection restri... |
| inrresf1 10025 | The right injection restri... |
| djuin 10026 | The images of any classes ... |
| djur 10027 | A member of a disjoint uni... |
| djuss 10028 | A disjoint union is a subc... |
| djuunxp 10029 | The union of a disjoint un... |
| djuexALT 10030 | Alternate proof of ~ djuex... |
| eldju1st 10031 | The first component of an ... |
| eldju2ndl 10032 | The second component of an... |
| eldju2ndr 10033 | The second component of an... |
| djuun 10034 | The disjoint union of two ... |
| 1stinl 10035 | The first component of the... |
| 2ndinl 10036 | The second component of th... |
| 1stinr 10037 | The first component of the... |
| 2ndinr 10038 | The second component of th... |
| updjudhf 10039 | The mapping of an element ... |
| updjudhcoinlf 10040 | The composition of the map... |
| updjudhcoinrg 10041 | The composition of the map... |
| updjud 10042 | Universal property of the ... |
| cardf2 10051 | The cardinality function i... |
| cardon 10052 | The cardinal number of a s... |
| isnum2 10053 | A way to express well-orde... |
| isnumi 10054 | A set equinumerous to an o... |
| ennum 10055 | Equinumerous sets are equi... |
| finnum 10056 | Every finite set is numera... |
| onenon 10057 | Every ordinal number is nu... |
| tskwe 10058 | A Tarski set is well-order... |
| xpnum 10059 | The cartesian product of n... |
| cardval3 10060 | An alternate definition of... |
| cardid2 10061 | Any numerable set is equin... |
| isnum3 10062 | A set is numerable iff it ... |
| oncardval 10063 | The value of the cardinal ... |
| oncardid 10064 | Any ordinal number is equi... |
| cardonle 10065 | The cardinal of an ordinal... |
| card0 10066 | The cardinality of the emp... |
| cardidm 10067 | The cardinality function i... |
| oncard 10068 | A set is a cardinal number... |
| ficardom 10069 | The cardinal number of a f... |
| ficardid 10070 | A finite set is equinumero... |
| cardnn 10071 | The cardinality of a natur... |
| cardnueq0 10072 | The empty set is the only ... |
| cardne 10073 | No member of a cardinal nu... |
| carden2a 10074 | If two sets have equal non... |
| carden2b 10075 | If two sets are equinumero... |
| card1 10076 | A set has cardinality one ... |
| cardsn 10077 | A singleton has cardinalit... |
| carddomi2 10078 | Two sets have the dominanc... |
| sdomsdomcardi 10079 | A set strictly dominates i... |
| cardlim 10080 | An infinite cardinal is a ... |
| cardsdomelir 10081 | A cardinal strictly domina... |
| cardsdomel 10082 | A cardinal strictly domina... |
| iscard 10083 | Two ways to express the pr... |
| iscard2 10084 | Two ways to express the pr... |
| carddom2 10085 | Two numerable sets have th... |
| harcard 10086 | The class of ordinal numbe... |
| cardprclem 10087 | Lemma for ~ cardprc . (Co... |
| cardprc 10088 | The class of all cardinal ... |
| carduni 10089 | The union of a set of card... |
| cardiun 10090 | The indexed union of a set... |
| cardennn 10091 | If ` A ` is equinumerous t... |
| cardsucinf 10092 | The cardinality of the suc... |
| cardsucnn 10093 | The cardinality of the suc... |
| cardom 10094 | The set of natural numbers... |
| carden2 10095 | Two numerable sets are equ... |
| cardsdom2 10096 | A numerable set is strictl... |
| domtri2 10097 | Trichotomy of dominance fo... |
| nnsdomel 10098 | Strict dominance and eleme... |
| cardval2 10099 | An alternate version of th... |
| isinffi 10100 | An infinite set contains s... |
| fidomtri 10101 | Trichotomy of dominance wi... |
| fidomtri2 10102 | Trichotomy of dominance wi... |
| harsdom 10103 | The Hartogs number of a we... |
| onsdom 10104 | Any well-orderable set is ... |
| harval2 10105 | An alternate expression fo... |
| harsucnn 10106 | The next cardinal after a ... |
| cardmin2 10107 | The smallest ordinal that ... |
| pm54.43lem 10108 | In Theorem *54.43 of [Whit... |
| pm54.43 10109 | Theorem *54.43 of [Whitehe... |
| enpr2 10110 | An unordered pair with dis... |
| pr2ne 10111 | If an unordered pair has t... |
| prdom2 10112 | An unordered pair has at m... |
| en2eqpr 10113 | Building a set with two el... |
| en2eleq 10114 | Express a set of pair card... |
| en2other2 10115 | Taking the other element t... |
| dif1card 10116 | The cardinality of a nonem... |
| leweon 10117 | Lexicographical order is a... |
| r0weon 10118 | A set-like well-ordering o... |
| infxpenlem 10119 | Lemma for ~ infxpen . (Co... |
| infxpen 10120 | Every infinite ordinal is ... |
| xpomen 10121 | The Cartesian product of o... |
| xpct 10122 | The cartesian product of t... |
| infxpidm2 10123 | Every infinite well-ordera... |
| infxpenc 10124 | A canonical version of ~ i... |
| infxpenc2lem1 10125 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10126 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10127 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10128 | Existence form of ~ infxpe... |
| iunmapdisj 10129 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10130 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10131 | Lemma for ~ fseqen . (Con... |
| fseqdom 10132 | One half of ~ fseqen . (C... |
| fseqen 10133 | A set that is equinumerous... |
| infpwfidom 10134 | The collection of finite s... |
| dfac8alem 10135 | Lemma for ~ dfac8a . If t... |
| dfac8a 10136 | Numeration theorem: every ... |
| dfac8b 10137 | The well-ordering theorem:... |
| dfac8clem 10138 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10139 | If the union of a set is w... |
| ac10ct 10140 | A proof of the well-orderi... |
| ween 10141 | A set is numerable iff it ... |
| ac5num 10142 | A version of ~ ac5b with t... |
| ondomen 10143 | If a set is dominated by a... |
| numdom 10144 | A set dominated by a numer... |
| ssnum 10145 | A subset of a numerable se... |
| onssnum 10146 | All subsets of the ordinal... |
| indcardi 10147 | Indirect strong induction ... |
| acnrcl 10148 | Reverse closure for the ch... |
| acneq 10149 | Equality theorem for the c... |
| isacn 10150 | The property of being a ch... |
| acni 10151 | The property of being a ch... |
| acni2 10152 | The property of being a ch... |
| acni3 10153 | The property of being a ch... |
| acnlem 10154 | Construct a mapping satisf... |
| numacn 10155 | A well-orderable set has c... |
| finacn 10156 | Every set has finite choic... |
| acndom 10157 | A set with long choice seq... |
| acnnum 10158 | A set ` X ` which has choi... |
| acnen 10159 | The class of choice sets o... |
| acndom2 10160 | A set smaller than one wit... |
| acnen2 10161 | The class of sets with cho... |
| fodomacn 10162 | A version of ~ fodom that ... |
| fodomnum 10163 | A version of ~ fodom that ... |
| fonum 10164 | A surjection maps numerabl... |
| numwdom 10165 | A surjection maps numerabl... |
| fodomfi2 10166 | Onto functions define domi... |
| wdomfil 10167 | Weak dominance agrees with... |
| infpwfien 10168 | Any infinite well-orderabl... |
| inffien 10169 | The set of finite intersec... |
| wdomnumr 10170 | Weak dominance agrees with... |
| alephfnon 10171 | The aleph function is a fu... |
| aleph0 10172 | The first infinite cardina... |
| alephlim 10173 | Value of the aleph functio... |
| alephsuc 10174 | Value of the aleph functio... |
| alephon 10175 | An aleph is an ordinal num... |
| alephcard 10176 | Every aleph is a cardinal ... |
| alephnbtwn 10177 | No cardinal can be sandwic... |
| alephnbtwn2 10178 | No set has equinumerosity ... |
| alephordilem1 10179 | Lemma for ~ alephordi . (... |
| alephordi 10180 | Strict ordering property o... |
| alephord 10181 | Ordering property of the a... |
| alephord2 10182 | Ordering property of the a... |
| alephord2i 10183 | Ordering property of the a... |
| alephord3 10184 | Ordering property of the a... |
| alephsucdom 10185 | A set dominated by an alep... |
| alephsuc2 10186 | An alternate representatio... |
| alephdom 10187 | Relationship between inclu... |
| alephgeom 10188 | Every aleph is greater tha... |
| alephislim 10189 | Every aleph is a limit ord... |
| aleph11 10190 | The aleph function is one-... |
| alephf1 10191 | The aleph function is a on... |
| alephsdom 10192 | If an ordinal is smaller t... |
| alephdom2 10193 | A dominated initial ordina... |
| alephle 10194 | The argument of the aleph ... |
| cardaleph 10195 | Given any transfinite card... |
| cardalephex 10196 | Every transfinite cardinal... |
| infenaleph 10197 | An infinite numerable set ... |
| isinfcard 10198 | Two ways to express the pr... |
| iscard3 10199 | Two ways to express the pr... |
| cardnum 10200 | Two ways to express the cl... |
| alephinit 10201 | An infinite initial ordina... |
| carduniima 10202 | The union of the image of ... |
| cardinfima 10203 | If a mapping to cardinals ... |
| alephiso 10204 | Aleph is an order isomorph... |
| alephprc 10205 | The class of all transfini... |
| alephsson 10206 | The class of transfinite c... |
| unialeph 10207 | The union of the class of ... |
| alephsmo 10208 | The aleph function is stri... |
| alephf1ALT 10209 | Alternate proof of ~ aleph... |
| alephfplem1 10210 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10211 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10212 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10213 | Lemma for ~ alephfp . (Co... |
| alephfp 10214 | The aleph function has a f... |
| alephfp2 10215 | The aleph function has at ... |
| alephval3 10216 | An alternate way to expres... |
| alephsucpw2 10217 | The power set of an aleph ... |
| mappwen 10218 | Power rule for cardinal ar... |
| finnisoeu 10219 | A finite totally ordered s... |
| iunfictbso 10220 | Countability of a countabl... |
| aceq1 10223 | Equivalence of two version... |
| aceq0 10224 | Equivalence of two version... |
| aceq2 10225 | Equivalence of two version... |
| aceq3lem 10226 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10227 | Equivalence of two version... |
| dfac4 10228 | Equivalence of two version... |
| dfac5lem1 10229 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10230 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10231 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10232 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10233 | Lemma for ~ dfac5 . (Cont... |
| dfac5 10234 | Equivalence of two version... |
| dfac2a 10235 | Our Axiom of Choice (in th... |
| dfac2b 10236 | Axiom of Choice (first for... |
| dfac2 10237 | Axiom of Choice (first for... |
| dfac7 10238 | Equivalence of the Axiom o... |
| dfac0 10239 | Equivalence of two version... |
| dfac1 10240 | Equivalence of two version... |
| dfac8 10241 | A proof of the equivalency... |
| dfac9 10242 | Equivalence of the axiom o... |
| dfac10 10243 | Axiom of Choice equivalent... |
| dfac10c 10244 | Axiom of Choice equivalent... |
| dfac10b 10245 | Axiom of Choice equivalent... |
| acacni 10246 | A choice equivalent: every... |
| dfacacn 10247 | A choice equivalent: every... |
| dfac13 10248 | The axiom of choice holds ... |
| dfac12lem1 10249 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10250 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10251 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10252 | The axiom of choice holds ... |
| dfac12k 10253 | Equivalence of ~ dfac12 an... |
| dfac12a 10254 | The axiom of choice holds ... |
| dfac12 10255 | The axiom of choice holds ... |
| kmlem1 10256 | Lemma for 5-quantifier AC ... |
| kmlem2 10257 | Lemma for 5-quantifier AC ... |
| kmlem3 10258 | Lemma for 5-quantifier AC ... |
| kmlem4 10259 | Lemma for 5-quantifier AC ... |
| kmlem5 10260 | Lemma for 5-quantifier AC ... |
| kmlem6 10261 | Lemma for 5-quantifier AC ... |
| kmlem7 10262 | Lemma for 5-quantifier AC ... |
| kmlem8 10263 | Lemma for 5-quantifier AC ... |
| kmlem9 10264 | Lemma for 5-quantifier AC ... |
| kmlem10 10265 | Lemma for 5-quantifier AC ... |
| kmlem11 10266 | Lemma for 5-quantifier AC ... |
| kmlem12 10267 | Lemma for 5-quantifier AC ... |
| kmlem13 10268 | Lemma for 5-quantifier AC ... |
| kmlem14 10269 | Lemma for 5-quantifier AC ... |
| kmlem15 10270 | Lemma for 5-quantifier AC ... |
| kmlem16 10271 | Lemma for 5-quantifier AC ... |
| dfackm 10272 | Equivalence of the Axiom o... |
| undjudom 10273 | Cardinal addition dominate... |
| endjudisj 10274 | Equinumerosity of a disjoi... |
| djuen 10275 | Disjoint unions of equinum... |
| djuenun 10276 | Disjoint union is equinume... |
| dju1en 10277 | Cardinal addition with car... |
| dju1dif 10278 | Adding and subtracting one... |
| dju1p1e2 10279 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10280 | Alternate proof of ~ dju1p... |
| dju0en 10281 | Cardinal addition with car... |
| xp2dju 10282 | Two times a cardinal numbe... |
| djucomen 10283 | Commutative law for cardin... |
| djuassen 10284 | Associative law for cardin... |
| xpdjuen 10285 | Cardinal multiplication di... |
| mapdjuen 10286 | Sum of exponents law for c... |
| pwdjuen 10287 | Sum of exponents law for c... |
| djudom1 10288 | Ordering law for cardinal ... |
| djudom2 10289 | Ordering law for cardinal ... |
| djudoml 10290 | A set is dominated by its ... |
| djuxpdom 10291 | Cartesian product dominate... |
| djufi 10292 | The disjoint union of two ... |
| cdainflem 10293 | Any partition of omega int... |
| djuinf 10294 | A set is infinite iff the ... |
| infdju1 10295 | An infinite set is equinum... |
| pwdju1 10296 | The sum of a powerset with... |
| pwdjuidm 10297 | If the natural numbers inj... |
| djulepw 10298 | If ` A ` is idempotent und... |
| onadju 10299 | The cardinal and ordinal s... |
| cardadju 10300 | The cardinal sum is equinu... |
| djunum 10301 | The disjoint union of two ... |
| unnum 10302 | The union of two numerable... |
| nnadju 10303 | The cardinal and ordinal s... |
| nnadjuALT 10304 | Shorter proof of ~ nnadju ... |
| ficardadju 10305 | The disjoint union of fini... |
| ficardun 10306 | The cardinality of the uni... |
| ficardun2 10307 | The cardinality of the uni... |
| pwsdompw 10308 | Lemma for ~ domtriom . Th... |
| unctb 10309 | The union of two countable... |
| infdjuabs 10310 | Absorption law for additio... |
| infunabs 10311 | An infinite set is equinum... |
| infdju 10312 | The sum of two cardinal nu... |
| infdif 10313 | The cardinality of an infi... |
| infdif2 10314 | Cardinality ordering for a... |
| infxpdom 10315 | Dominance law for multipli... |
| infxpabs 10316 | Absorption law for multipl... |
| infunsdom1 10317 | The union of two sets that... |
| infunsdom 10318 | The union of two sets that... |
| infxp 10319 | Absorption law for multipl... |
| pwdjudom 10320 | A property of dominance ov... |
| infpss 10321 | Every infinite set has an ... |
| infmap2 10322 | An exponentiation law for ... |
| ackbij2lem1 10323 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10324 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10325 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10326 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10327 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10328 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10329 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10330 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10331 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10332 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10333 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10334 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10335 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10336 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10337 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10338 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10339 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10340 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10341 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10342 | The Ackermann bijection, p... |
| ackbij1b 10343 | The Ackermann bijection, p... |
| ackbij2lem2 10344 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10345 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10346 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10347 | The Ackermann bijection, p... |
| hfom 10348 | The set of hereditarily fi... |
| fictb 10349 | A set is countable iff its... |
| cflem 10350 | A lemma used to simplify c... |
| cfval 10351 | Value of the cofinality fu... |
| cff 10352 | Cofinality is a function o... |
| cfub 10353 | An upper bound on cofinali... |
| cflm 10354 | Value of the cofinality fu... |
| cf0 10355 | Value of the cofinality fu... |
| cardcf 10356 | Cofinality is a cardinal n... |
| cflecard 10357 | Cofinality is bounded by t... |
| cfle 10358 | Cofinality is bounded by i... |
| cfon 10359 | The cofinality of any set ... |
| cfonOLD 10360 | Obsolete version of ~ cfon... |
| cfeq0 10361 | Only the ordinal zero has ... |
| cfsuc 10362 | Value of the cofinality fu... |
| cff1 10363 | There is always a map from... |
| cfflb 10364 | If there is a cofinal map ... |
| cfval2 10365 | Another expression for the... |
| coflim 10366 | A simpler expression for t... |
| cflim3 10367 | Another expression for the... |
| cflim2 10368 | The cofinality function is... |
| cfom 10369 | Value of the cofinality fu... |
| cfss 10370 | There is a cofinal subset ... |
| cfslb 10371 | Any cofinal subset of ` A ... |
| cfslbn 10372 | Any subset of ` A ` smalle... |
| cfslb2n 10373 | Any small collection of sm... |
| cofsmo 10374 | Any cofinal map implies th... |
| cfsmolem 10375 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10376 | The map in ~ cff1 can be a... |
| cfcoflem 10377 | Lemma for ~ cfcof , showin... |
| coftr 10378 | If there is a cofinal map ... |
| cfcof 10379 | If there is a cofinal map ... |
| cfidm 10380 | The cofinality function is... |
| alephsing 10381 | The cofinality of a limit ... |
| sornom 10382 | The range of a single-step... |
| isfin1a 10397 | Definition of a Ia-finite ... |
| fin1ai 10398 | Property of a Ia-finite se... |
| isfin2 10399 | Definition of a II-finite ... |
| fin2i 10400 | Property of a II-finite se... |
| isfin3 10401 | Definition of a III-finite... |
| isfin4 10402 | Definition of a IV-finite ... |
| fin4i 10403 | Infer that a set is IV-inf... |
| isfin5 10404 | Definition of a V-finite s... |
| isfin6 10405 | Definition of a VI-finite ... |
| isfin7 10406 | Definition of a VII-finite... |
| sdom2en01 10407 | A set with less than two e... |
| infpssrlem1 10408 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10409 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10410 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10411 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10412 | Lemma for ~ infpssr . (Co... |
| infpssr 10413 | Dedekind infinity implies ... |
| fin4en1 10414 | Dedekind finite is a cardi... |
| ssfin4 10415 | Dedekind finite sets have ... |
| domfin4 10416 | A set dominated by a Dedek... |
| ominf4 10417 | ` _om ` is Dedekind infini... |
| infpssALT 10418 | Alternate proof of ~ infps... |
| isfin4-2 10419 | Alternate definition of IV... |
| isfin4p1 10420 | Alternate definition of IV... |
| fin23lem7 10421 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10422 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10423 | A II-finite set contains m... |
| isfin2-2 10424 | ` Fin2 ` expressed in term... |
| ssfin2 10425 | A subset of a II-finite se... |
| enfin2i 10426 | II-finiteness is a cardina... |
| fin23lem24 10427 | Lemma for ~ fin23 . In a ... |
| fincssdom 10428 | In a chain of finite sets,... |
| fin23lem25 10429 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10430 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10431 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10432 | Lemma for ~ fin23 but coul... |
| fin23lem27 10433 | The mapping constructed in... |
| isfin3ds 10434 | Property of a III-finite s... |
| ssfin3ds 10435 | A subset of a III-finite s... |
| fin23lem12 10436 | The beginning of the proof... |
| fin23lem13 10437 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10438 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10439 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10440 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10441 | Lemma for ~ fin23 . The f... |
| fin23lem20 10442 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10443 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10444 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10445 | Lemma for ~ fin23 . The r... |
| fin23lem29 10446 | Lemma for ~ fin23 . The r... |
| fin23lem30 10447 | Lemma for ~ fin23 . The r... |
| fin23lem31 10448 | Lemma for ~ fin23 . The r... |
| fin23lem32 10449 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10450 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10451 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10452 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10453 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10454 | Lemma for ~ fin23 . The c... |
| fin23lem39 10455 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10456 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10457 | Lemma for ~ fin23 . A set... |
| isf32lem1 10458 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10459 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10460 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10461 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10462 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10463 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10464 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10465 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10466 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10467 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10468 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10469 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10470 | One half of ~ isfin3-2 . ... |
| isf33lem 10471 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10472 | Weakly Dedekind-infinite s... |
| isfin3-3 10473 | Weakly Dedekind-infinite s... |
| fin33i 10474 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10475 | Lemma for ~ compsscnv . (... |
| compsscnv 10476 | Complementation on a power... |
| isf34lem1 10477 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10478 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10479 | Complementation is an anti... |
| isf34lem3 10480 | Lemma for ~ isfin3-4 . (C... |
| compss 10481 | Express image under of the... |
| isf34lem4 10482 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10483 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10484 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10485 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10486 | Inference from ~ isfin3-4 ... |
| isfin3-4 10487 | Weakly Dedekind-infinite s... |
| fin11a 10488 | Every I-finite set is Ia-f... |
| enfin1ai 10489 | Ia-finiteness is a cardina... |
| isfin1-2 10490 | A set is finite in the usu... |
| isfin1-3 10491 | A set is I-finite iff ever... |
| isfin1-4 10492 | A set is I-finite iff ever... |
| dffin1-5 10493 | Compact quantifier-free ve... |
| fin23 10494 | Every II-finite set (every... |
| fin34 10495 | Every III-finite set is IV... |
| isfin5-2 10496 | Alternate definition of V-... |
| fin45 10497 | Every IV-finite set is V-f... |
| fin56 10498 | Every V-finite set is VI-f... |
| fin17 10499 | Every I-finite set is VII-... |
| fin67 10500 | Every VI-finite set is VII... |
| isfin7-2 10501 | A set is VII-finite iff it... |
| fin71num 10502 | A well-orderable set is VI... |
| dffin7-2 10503 | Class form of ~ isfin7-2 .... |
| dfacfin7 10504 | Axiom of Choice equivalent... |
| fin1a2lem1 10505 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10506 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10507 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10508 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10509 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10510 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10511 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10512 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10513 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10514 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10515 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10516 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10517 | Lemma for ~ fin1a2 . (Con... |
| fin12 10518 | Weak theorem which skips I... |
| fin1a2s 10519 | An II-infinite set can hav... |
| fin1a2 10520 | Every Ia-finite set is II-... |
| itunifval 10521 | Function value of iterated... |
| itunifn 10522 | Functionality of the itera... |
| ituni0 10523 | A zero-fold iterated union... |
| itunisuc 10524 | Successor iterated union. ... |
| itunitc1 10525 | Each union iterate is a me... |
| itunitc 10526 | The union of all union ite... |
| ituniiun 10527 | Unwrap an iterated union f... |
| hsmexlem7 10528 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10529 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10530 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10531 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10532 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10533 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10534 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10535 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10536 | Lemma for ~ hsmex . (Cont... |
| hsmex 10537 | The collection of heredita... |
| hsmex2 10538 | The set of hereditary size... |
| hsmex3 10539 | The set of hereditary size... |
| axcc2lem 10541 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10542 | A possibly more useful ver... |
| axcc3 10543 | A possibly more useful ver... |
| axcc4 10544 | A version of ~ axcc3 that ... |
| acncc 10545 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10546 | Relax the constraint on ~ ... |
| domtriomlem 10547 | Lemma for ~ domtriom . (C... |
| domtriom 10548 | Trichotomy of equinumerosi... |
| fin41 10549 | Under countable choice, th... |
| dominf 10550 | A nonempty set that is a s... |
| dcomex 10552 | The Axiom of Dependent Cho... |
| axdc2lem 10553 | Lemma for ~ axdc2 . We co... |
| axdc2 10554 | An apparent strengthening ... |
| axdc3lem 10555 | The class ` S ` of finite ... |
| axdc3lem2 10556 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10557 | Simple substitution lemma ... |
| axdc3lem4 10558 | Lemma for ~ axdc3 . We ha... |
| axdc3 10559 | Dependent Choice. Axiom D... |
| axdc4lem 10560 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10561 | A more general version of ... |
| axcclem 10562 | Lemma for ~ axcc . (Contr... |
| axcc 10563 | Although CC can be proven ... |
| zfac 10565 | Axiom of Choice expressed ... |
| ac2 10566 | Axiom of Choice equivalent... |
| ac3 10567 | Axiom of Choice using abbr... |
| axac3 10569 | This theorem asserts that ... |
| ackm 10570 | A remarkable equivalent to... |
| axac2 10571 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10572 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10573 | Apply a choice equivalent.... |
| cardeqv 10574 | All sets are well-orderabl... |
| numth3 10575 | All sets are well-orderabl... |
| numth2 10576 | Numeration theorem: any se... |
| numth 10577 | Numeration theorem: every ... |
| ac7 10578 | An Axiom of Choice equival... |
| ac7g 10579 | An Axiom of Choice equival... |
| ac4 10580 | Equivalent of Axiom of Cho... |
| ac4c 10581 | Equivalent of Axiom of Cho... |
| ac5 10582 | An Axiom of Choice equival... |
| ac5b 10583 | Equivalent of Axiom of Cho... |
| ac6num 10584 | A version of ~ ac6 which t... |
| ac6 10585 | Equivalent of Axiom of Cho... |
| ac6c4 10586 | Equivalent of Axiom of Cho... |
| ac6c5 10587 | Equivalent of Axiom of Cho... |
| ac9 10588 | An Axiom of Choice equival... |
| ac6s 10589 | Equivalent of Axiom of Cho... |
| ac6n 10590 | Equivalent of Axiom of Cho... |
| ac6s2 10591 | Generalization of the Axio... |
| ac6s3 10592 | Generalization of the Axio... |
| ac6sg 10593 | ~ ac6s with sethood as ant... |
| ac6sf 10594 | Version of ~ ac6 with boun... |
| ac6s4 10595 | Generalization of the Axio... |
| ac6s5 10596 | Generalization of the Axio... |
| ac8 10597 | An Axiom of Choice equival... |
| ac9s 10598 | An Axiom of Choice equival... |
| numthcor 10599 | Any set is strictly domina... |
| weth 10600 | Well-ordering theorem: any... |
| zorn2lem1 10601 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10602 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10603 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10604 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10605 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10606 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10607 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10608 | Zorn's Lemma of [Monk1] p.... |
| zorng 10609 | Zorn's Lemma. If the unio... |
| zornn0g 10610 | Variant of Zorn's lemma ~ ... |
| zorn2 10611 | Zorn's Lemma of [Monk1] p.... |
| zorn 10612 | Zorn's Lemma. If the unio... |
| zornn0 10613 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10614 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10615 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10616 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10617 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10618 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10619 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10620 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10621 | The Teichmüller-Tukey... |
| ttukeyg 10622 | The Teichmüller-Tukey... |
| ttukey 10623 | The Teichmüller-Tukey... |
| axdclem 10624 | Lemma for ~ axdc . (Contr... |
| axdclem2 10625 | Lemma for ~ axdc . Using ... |
| axdc 10626 | This theorem derives ~ ax-... |
| fodomg 10627 | An onto function implies d... |
| fodom 10628 | An onto function implies d... |
| dmct 10629 | The domain of a countable ... |
| dmctOLD 10630 | Obsolete version of ~ dmct... |
| rnct 10631 | The range of a countable s... |
| fodomb 10632 | Equivalence of an onto map... |
| wdomac 10633 | When assuming AC, weak and... |
| brdom3 10634 | Equivalence to a dominance... |
| brdom5 10635 | An equivalence to a domina... |
| brdom4 10636 | An equivalence to a domina... |
| brdom7disj 10637 | An equivalence to a domina... |
| brdom6disj 10638 | An equivalence to a domina... |
| fin71ac 10639 | Once we allow AC, the "str... |
| imadomg 10640 | An image of a function und... |
| imadomnum 10641 | A version of ~ imadomg tha... |
| fimact 10642 | The image by a function of... |
| fimactOLD 10643 | Obsolete version of ~ fima... |
| fnrndomnum 10644 | A version of ~ fnrndomg th... |
| fnrndomg 10645 | The range of a function is... |
| fnrndomgOLD 10646 | Obsolete version of ~ fnrn... |
| fnct 10647 | If the domain of a functio... |
| fnctOLD 10648 | Obsolete version of ~ fnct... |
| mptct 10649 | A countable mapping set is... |
| iunfo 10650 | Existence of an onto funct... |
| iundom2g 10651 | An upper bound for the car... |
| iundomg 10652 | An upper bound for the car... |
| iundom 10653 | An upper bound for the car... |
| unidom 10654 | An upper bound for the car... |
| uniimadom 10655 | An upper bound for the car... |
| uniimadomf 10656 | An upper bound for the car... |
| cardval 10657 | The value of the cardinal ... |
| cardid 10658 | Any set is equinumerous to... |
| cardidg 10659 | Any set is equinumerous to... |
| cardidd 10660 | Any set is equinumerous to... |
| cardf 10661 | The cardinality function i... |
| carden 10662 | Two sets are equinumerous ... |
| cardeq0 10663 | Only the empty set has car... |
| unsnen 10664 | Equinumerosity of a set wi... |
| carddom 10665 | Two sets have the dominanc... |
| cardsdom 10666 | Two sets have the strict d... |
| domtri 10667 | Trichotomy law for dominan... |
| entric 10668 | Trichotomy of equinumerosi... |
| entri2 10669 | Trichotomy of dominance an... |
| entri3 10670 | Trichotomy of dominance. ... |
| sdomsdomcard 10671 | A set strictly dominates i... |
| canth3 10672 | Cantor's theorem in terms ... |
| infxpidm 10673 | Every infinite class is eq... |
| ondomon 10674 | The class of ordinals domi... |
| cardmin 10675 | The smallest ordinal that ... |
| ficard 10676 | A set is finite iff its ca... |
| infinfg 10677 | Equivalence between two in... |
| infinf 10678 | Equivalence between two in... |
| unirnfdomd 10679 | The union of the range of ... |
| konigthlem 10680 | Lemma for ~ konigth . (Co... |
| konigth 10681 | Konig's Theorem. If ` m (... |
| alephsucpw 10682 | The power set of an aleph ... |
| aleph1 10683 | The set exponentiation of ... |
| alephval2 10684 | An alternate way to expres... |
| dominfac 10685 | A nonempty set that is a s... |
| iunctb 10686 | The countable union of cou... |
| unictb 10687 | The countable union of cou... |
| infmap 10688 | An exponentiation law for ... |
| alephadd 10689 | The sum of two alephs is t... |
| alephmul 10690 | The product of two alephs ... |
| alephexp1 10691 | An exponentiation law for ... |
| alephsuc3 10692 | An alternate representatio... |
| alephexp2 10693 | An expression equinumerous... |
| alephreg 10694 | A successor aleph is regul... |
| pwcfsdom 10695 | A corollary of Konig's The... |
| cfpwsdom 10696 | A corollary of Konig's The... |
| alephom 10697 | From ~ canth2 , we know th... |
| smobeth 10698 | The beth function is stric... |
| nd1 10699 | A lemma for proving condit... |
| nd2 10700 | A lemma for proving condit... |
| nd3 10701 | A lemma for proving condit... |
| nd4 10702 | A lemma for proving condit... |
| axextnd 10703 | A version of the Axiom of ... |
| axrepndlem1 10704 | Lemma for the Axiom of Rep... |
| axrepndlem2 10705 | Lemma for the Axiom of Rep... |
| axrepnd 10706 | A version of the Axiom of ... |
| axunndlem1 10707 | Lemma for the Axiom of Uni... |
| axunnd 10708 | A version of the Axiom of ... |
| axpowndlem1 10709 | Lemma for the Axiom of Pow... |
| axpowndlem2 10710 | Lemma for the Axiom of Pow... |
| axpowndlem3 10711 | Lemma for the Axiom of Pow... |
| axpowndlem4 10712 | Lemma for the Axiom of Pow... |
| axpownd 10713 | A version of the Axiom of ... |
| axregndlem1 10714 | Lemma for the Axiom of Reg... |
| axregndlem2 10715 | Lemma for the Axiom of Reg... |
| axregnd 10716 | A version of the Axiom of ... |
| axinfndlem1 10717 | Lemma for the Axiom of Inf... |
| axinfnd 10718 | A version of the Axiom of ... |
| axacndlem1 10719 | Lemma for the Axiom of Cho... |
| axacndlem2 10720 | Lemma for the Axiom of Cho... |
| axacndlem3 10721 | Lemma for the Axiom of Cho... |
| axacndlem4 10722 | Lemma for the Axiom of Cho... |
| axacndlem5 10723 | Lemma for the Axiom of Cho... |
| axacnd 10724 | A version of the Axiom of ... |
| zfcndext 10725 | Axiom of Extensionality ~ ... |
| zfcndrep 10726 | Axiom of Replacement ~ ax-... |
| zfcndun 10727 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10728 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10729 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10730 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10731 | Axiom of Choice ~ ax-ac , ... |
| elgch 10734 | Elementhood in the collect... |
| fingch 10735 | A finite set is a GCH-set.... |
| gchi 10736 | The only GCH-sets which ha... |
| gchen1 10737 | If ` A <_ B < ~P A ` , and... |
| gchen2 10738 | If ` A < B <_ ~P A ` , and... |
| gchor 10739 | If ` A <_ B <_ ~P A ` , an... |
| engch 10740 | The property of being a GC... |
| gchdomtri 10741 | Under certain conditions, ... |
| fpwwe2cbv 10742 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10743 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10744 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10745 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10746 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10747 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10748 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10749 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10750 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10751 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10752 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10753 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10754 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10755 | Given any function ` F ` f... |
| fpwwecbv 10756 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10757 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10758 | Given any function ` F ` f... |
| canth4 10759 | An "effective" form of Can... |
| canthnumlem 10760 | Lemma for ~ canthnum . (C... |
| canthnum 10761 | The set of well-orderable ... |
| canthwelem 10762 | Lemma for ~ canthwe . (Co... |
| canthwe 10763 | The set of well-orders of ... |
| canthp1lem1 10764 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10765 | Lemma for ~ canthp1 . (Co... |
| canthp1 10766 | A slightly stronger form o... |
| finngch 10767 | The exclusion of finite se... |
| gchdju1 10768 | An infinite GCH-set is ide... |
| gchinf 10769 | An infinite GCH-set is Ded... |
| pwfseqlem1 10770 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10771 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10772 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10773 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10774 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10775 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10776 | The powerset of a Dedekind... |
| pwxpndom2 10777 | The powerset of a Dedekind... |
| pwxpndom 10778 | The powerset of a Dedekind... |
| pwdjundom 10779 | The powerset of a Dedekind... |
| gchdjuidm 10780 | An infinite GCH-set is ide... |
| gchxpidm 10781 | An infinite GCH-set is ide... |
| gchpwdom 10782 | A relationship between dom... |
| gchaleph 10783 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10784 | If ` ( aleph `` A ) ` and ... |
| hargch 10785 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10786 | If ` ( aleph `` suc A ) ` ... |
| gch2 10787 | It is sufficient to requir... |
| gch3 10788 | An equivalent formulation ... |
| gch-kn 10789 | The equivalence of two ver... |
| gchaclem 10790 | Lemma for ~ gchac (obsolet... |
| gchhar 10791 | A "local" form of ~ gchac ... |
| gchacg 10792 | A "local" form of ~ gchac ... |
| gchac 10793 | The Generalized Continuum ... |
| elwina 10798 | Conditions of weak inacces... |
| elina 10799 | Conditions of strong inacc... |
| winaon 10800 | A weakly inaccessible card... |
| inawinalem 10801 | Lemma for ~ inawina . (Co... |
| inawina 10802 | Every strongly inaccessibl... |
| omina 10803 | ` _om ` is a strongly inac... |
| winacard 10804 | A weakly inaccessible card... |
| winainflem 10805 | A weakly inaccessible card... |
| winainf 10806 | A weakly inaccessible card... |
| winalim 10807 | A weakly inaccessible card... |
| winalim2 10808 | A nontrivial weakly inacce... |
| winafp 10809 | A nontrivial weakly inacce... |
| winafpi 10810 | This theorem, which states... |
| gchina 10811 | Assuming the GCH, weakly a... |
| iswun 10816 | Properties of a weak unive... |
| wuntr 10817 | A weak universe is transit... |
| wununi 10818 | A weak universe is closed ... |
| wunpw 10819 | A weak universe is closed ... |
| wunelss 10820 | The elements of a weak uni... |
| wunpr 10821 | A weak universe is closed ... |
| wunun 10822 | A weak universe is closed ... |
| wuntp 10823 | A weak universe is closed ... |
| wunss 10824 | A weak universe is closed ... |
| wunin 10825 | A weak universe is closed ... |
| wundif 10826 | A weak universe is closed ... |
| wunint 10827 | A weak universe is closed ... |
| wunsn 10828 | A weak universe is closed ... |
| wunsuc 10829 | A weak universe is closed ... |
| wun0 10830 | A weak universe contains t... |
| wunr1om 10831 | A weak universe is infinit... |
| wunom 10832 | A weak universe contains a... |
| wunfi 10833 | A weak universe contains a... |
| wunop 10834 | A weak universe is closed ... |
| wunot 10835 | A weak universe is closed ... |
| wunxp 10836 | A weak universe is closed ... |
| wunpm 10837 | A weak universe is closed ... |
| wunmap 10838 | A weak universe is closed ... |
| wunf 10839 | A weak universe is closed ... |
| wundm 10840 | A weak universe is closed ... |
| wunrn 10841 | A weak universe is closed ... |
| wuncnv 10842 | A weak universe is closed ... |
| wunres 10843 | A weak universe is closed ... |
| wunfv 10844 | A weak universe is closed ... |
| wunco 10845 | A weak universe is closed ... |
| wuntpos 10846 | A weak universe is closed ... |
| intwun 10847 | The intersection of a coll... |
| r1limwun 10848 | Each limit stage in the cu... |
| r1wunlim 10849 | The stages of the cumulati... |
| wunex2 10850 | Construct a weak universe ... |
| wunex 10851 | Construct a weak universe ... |
| uniwun 10852 | Every set is contained in ... |
| wunex3 10853 | Construct a weak universe ... |
| wuncval 10854 | Value of the weak universe... |
| wuncid 10855 | The weak universe closure ... |
| wunccl 10856 | The weak universe closure ... |
| wuncss 10857 | The weak universe closure ... |
| wuncidm 10858 | The weak universe closure ... |
| wuncval2 10859 | Our earlier expression for... |
| eltskg 10862 | Properties of a Tarski cla... |
| eltsk2g 10863 | Properties of a Tarski cla... |
| tskpwss 10864 | First axiom of a Tarski cl... |
| tskpw 10865 | Second axiom of a Tarski c... |
| tsken 10866 | Third axiom of a Tarski cl... |
| 0tsk 10867 | The empty set is a (transi... |
| tsksdom 10868 | An element of a Tarski cla... |
| tskssel 10869 | A part of a Tarski class s... |
| tskss 10870 | The subsets of an element ... |
| tskin 10871 | The intersection of two el... |
| tsksn 10872 | A singleton of an element ... |
| tsktrss 10873 | A transitive element of a ... |
| tsksuc 10874 | If an element of a Tarski ... |
| tsk0 10875 | A nonempty Tarski class co... |
| tsk1 10876 | One is an element of a non... |
| tsk2 10877 | Two is an element of a non... |
| 2domtsk 10878 | If a Tarski class is not e... |
| tskr1om 10879 | A nonempty Tarski class is... |
| tskhf 10880 | A nonempty Tarski class co... |
| tskinf 10881 | A nonempty Tarski class is... |
| tskpr 10882 | If ` A ` and ` B ` are mem... |
| tskop 10883 | If ` A ` and ` B ` are mem... |
| tskxpss 10884 | A Cartesian product of two... |
| tskwe2 10885 | A Tarski class is well-ord... |
| inttsk 10886 | The intersection of a coll... |
| inar1 10887 | ` ( R1 `` A ) ` for ` A ` ... |
| hfomALT 10888 | Alternate proof of ~ hfom ... |
| rankcf 10889 | Any set must be at least a... |
| inatsk 10890 | ` ( R1 `` A ) ` for ` A ` ... |
| hftsk 10891 | The set of hereditarily fi... |
| tskord 10892 | A Tarski class contains al... |
| tskcard 10893 | An even more direct relati... |
| r1tskina 10894 | There is a direct relation... |
| tskuni 10895 | The union of an element of... |
| tskwun 10896 | A nonempty transitive Tars... |
| tskint 10897 | The intersection of an ele... |
| tskun 10898 | The union of two elements ... |
| tskxp 10899 | The Cartesian product of t... |
| tskmap 10900 | Set exponentiation is an e... |
| tskurn 10901 | A transitive Tarski class ... |
| elgrug 10904 | Properties of a Grothendie... |
| grutr 10905 | A Grothendieck universe is... |
| gruelss 10906 | A Grothendieck universe is... |
| grupw 10907 | A Grothendieck universe co... |
| gruss 10908 | Any subset of an element o... |
| grupr 10909 | A Grothendieck universe co... |
| gruurn 10910 | A Grothendieck universe co... |
| gruiun 10911 | If ` B ( x ) ` is a family... |
| gruuni 10912 | A Grothendieck universe co... |
| grurn 10913 | A Grothendieck universe co... |
| gruima 10914 | A Grothendieck universe co... |
| gruel 10915 | Any element of an element ... |
| grusn 10916 | A Grothendieck universe co... |
| gruop 10917 | A Grothendieck universe co... |
| gruun 10918 | A Grothendieck universe co... |
| gruxp 10919 | A Grothendieck universe co... |
| grumap 10920 | A Grothendieck universe co... |
| gruixp 10921 | A Grothendieck universe co... |
| gruiin 10922 | A Grothendieck universe co... |
| gruf 10923 | A Grothendieck universe co... |
| gruen 10924 | A Grothendieck universe co... |
| gruwun 10925 | A nonempty Grothendieck un... |
| intgru 10926 | The intersection of a fami... |
| ingru 10927 | The intersection of a univ... |
| wfgru 10928 | The wellfounded part of a ... |
| grudomon 10929 | Each ordinal that is compa... |
| gruina 10930 | If a Grothendieck universe... |
| grur1a 10931 | A characterization of Grot... |
| grur1 10932 | A characterization of Grot... |
| grutsk1 10933 | Grothendieck universes are... |
| grutsk 10934 | Grothendieck universes are... |
| axgroth5 10936 | The Tarski-Grothendieck ax... |
| axgroth2 10937 | Alternate version of the T... |
| grothpw 10938 | Derive the Axiom of Power ... |
| grothpwex 10939 | Derive the Axiom of Power ... |
| axgroth6 10940 | The Tarski-Grothendieck ax... |
| grothomex 10941 | The Tarski-Grothendieck Ax... |
| grothac 10942 | The Tarski-Grothendieck Ax... |
| axgroth3 10943 | Alternate version of the T... |
| axgroth4 10944 | Alternate version of the T... |
| grothprimlem 10945 | Lemma for ~ grothprim . E... |
| grothprim 10946 | The Tarski-Grothendieck Ax... |
| grothtsk 10947 | The Tarski-Grothendieck Ax... |
| inaprc 10948 | An equivalent to the Tarsk... |
| tskmval 10951 | Value of our tarski map. ... |
| tskmid 10952 | The set ` A ` is an elemen... |
| tskmcl 10953 | A Tarski class that contai... |
| sstskm 10954 | Being a part of ` ( tarski... |
| eltskm 10955 | Belonging to ` ( tarskiMap... |
| elni 10988 | Membership in the class of... |
| elni2 10989 | Membership in the class of... |
| pinn 10990 | A positive integer is a na... |
| pion 10991 | A positive integer is an o... |
| piord 10992 | A positive integer is ordi... |
| niex 10993 | The class of positive inte... |
| 0npi 10994 | The empty set is not a pos... |
| 1pi 10995 | Ordinal 'one' is a positiv... |
| addpiord 10996 | Positive integer addition ... |
| mulpiord 10997 | Positive integer multiplic... |
| mulidpi 10998 | 1 is an identity element f... |
| ltpiord 10999 | Positive integer 'less tha... |
| ltsopi 11000 | Positive integer 'less tha... |
| ltrelpi 11001 | Positive integer 'less tha... |
| dmaddpi 11002 | Domain of addition on posi... |
| dmmulpi 11003 | Domain of multiplication o... |
| addclpi 11004 | Closure of addition of pos... |
| mulclpi 11005 | Closure of multiplication ... |
| addcompi 11006 | Addition of positive integ... |
| addasspi 11007 | Addition of positive integ... |
| mulcompi 11008 | Multiplication of positive... |
| mulasspi 11009 | Multiplication of positive... |
| distrpi 11010 | Multiplication of positive... |
| addcanpi 11011 | Addition cancellation law ... |
| mulcanpi 11012 | Multiplication cancellatio... |
| addnidpi 11013 | There is no identity eleme... |
| ltexpi 11014 | Ordering on positive integ... |
| ltapi 11015 | Ordering property of addit... |
| ltmpi 11016 | Ordering property of multi... |
| 1lt2pi 11017 | One is less than two (one ... |
| nlt1pi 11018 | No positive integer is les... |
| indpi 11019 | Principle of Finite Induct... |
| enqbreq 11031 | Equivalence relation for p... |
| enqbreq2 11032 | Equivalence relation for p... |
| enqer 11033 | The equivalence relation f... |
| enqex 11034 | The equivalence relation f... |
| nqex 11035 | The class of positive frac... |
| 0nnq 11036 | The empty set is not a pos... |
| elpqn 11037 | Each positive fraction is ... |
| ltrelnq 11038 | Positive fraction 'less th... |
| pinq 11039 | The representatives of pos... |
| 1nq 11040 | The positive fraction 'one... |
| nqereu 11041 | There is a unique element ... |
| nqerf 11042 | Corollary of ~ nqereu : th... |
| nqercl 11043 | Corollary of ~ nqereu : cl... |
| nqerrel 11044 | Any member of ` ( N. X. N.... |
| nqerid 11045 | Corollary of ~ nqereu : th... |
| enqeq 11046 | Corollary of ~ nqereu : if... |
| nqereq 11047 | The function ` /Q ` acts a... |
| addpipq2 11048 | Addition of positive fract... |
| addpipq 11049 | Addition of positive fract... |
| addpqnq 11050 | Addition of positive fract... |
| mulpipq2 11051 | Multiplication of positive... |
| mulpipq 11052 | Multiplication of positive... |
| mulpqnq 11053 | Multiplication of positive... |
| ordpipq 11054 | Ordering of positive fract... |
| ordpinq 11055 | Ordering of positive fract... |
| addpqf 11056 | Closure of addition on pos... |
| addclnq 11057 | Closure of addition on pos... |
| mulpqf 11058 | Closure of multiplication ... |
| mulclnq 11059 | Closure of multiplication ... |
| addnqf 11060 | Domain of addition on posi... |
| mulnqf 11061 | Domain of multiplication o... |
| addcompq 11062 | Addition of positive fract... |
| addcomnq 11063 | Addition of positive fract... |
| mulcompq 11064 | Multiplication of positive... |
| mulcomnq 11065 | Multiplication of positive... |
| adderpqlem 11066 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 11067 | Lemma for ~ mulerpq . (Co... |
| adderpq 11068 | Addition is compatible wit... |
| mulerpq 11069 | Multiplication is compatib... |
| addassnq 11070 | Addition of positive fract... |
| mulassnq 11071 | Multiplication of positive... |
| mulcanenq 11072 | Lemma for distributive law... |
| distrnq 11073 | Multiplication of positive... |
| 1nqenq 11074 | The equivalence class of r... |
| mulidnq 11075 | Multiplication identity el... |
| recmulnq 11076 | Relationship between recip... |
| recidnq 11077 | A positive fraction times ... |
| recclnq 11078 | Closure law for positive f... |
| recrecnq 11079 | Reciprocal of reciprocal o... |
| dmrecnq 11080 | Domain of reciprocal on po... |
| ltsonq 11081 | 'Less than' is a strict or... |
| lterpq 11082 | Compatibility of ordering ... |
| ltanq 11083 | Ordering property of addit... |
| ltmnq 11084 | Ordering property of multi... |
| 1lt2nq 11085 | One is less than two (one ... |
| ltaddnq 11086 | The sum of two fractions i... |
| ltexnq 11087 | Ordering on positive fract... |
| halfnq 11088 | One-half of any positive f... |
| nsmallnq 11089 | The is no smallest positiv... |
| ltbtwnnq 11090 | There exists a number betw... |
| ltrnq 11091 | Ordering property of recip... |
| archnq 11092 | For any fraction, there is... |
| npex 11098 | The class of positive real... |
| elnp 11099 | Membership in positive rea... |
| elnpi 11100 | Membership in positive rea... |
| prn0 11101 | A positive real is not emp... |
| prpssnq 11102 | A positive real is a subse... |
| elprnq 11103 | A positive real is a set o... |
| 0npr 11104 | The empty set is not a pos... |
| prcdnq 11105 | A positive real is closed ... |
| prub 11106 | A positive fraction not in... |
| prnmax 11107 | A positive real has no lar... |
| npomex 11108 | A simplifying observation,... |
| prnmadd 11109 | A positive real has no lar... |
| ltrelpr 11110 | Positive real 'less than' ... |
| genpv 11111 | Value of general operation... |
| genpelv 11112 | Membership in value of gen... |
| genpprecl 11113 | Pre-closure law for genera... |
| genpdm 11114 | Domain of general operatio... |
| genpn0 11115 | The result of an operation... |
| genpss 11116 | The result of an operation... |
| genpnnp 11117 | The result of an operation... |
| genpcd 11118 | Downward closure of an ope... |
| genpnmax 11119 | An operation on positive r... |
| genpcl 11120 | Closure of an operation on... |
| genpass 11121 | Associativity of an operat... |
| plpv 11122 | Value of addition on posit... |
| mpv 11123 | Value of multiplication on... |
| dmplp 11124 | Domain of addition on posi... |
| dmmp 11125 | Domain of multiplication o... |
| nqpr 11126 | The canonical embedding of... |
| 1pr 11127 | The positive real number '... |
| addclprlem1 11128 | Lemma to prove downward cl... |
| addclprlem2 11129 | Lemma to prove downward cl... |
| addclpr 11130 | Closure of addition on pos... |
| mulclprlem 11131 | Lemma to prove downward cl... |
| mulclpr 11132 | Closure of multiplication ... |
| addcompr 11133 | Addition of positive reals... |
| addasspr 11134 | Addition of positive reals... |
| mulcompr 11135 | Multiplication of positive... |
| mulasspr 11136 | Multiplication of positive... |
| distrlem1pr 11137 | Lemma for distributive law... |
| distrlem4pr 11138 | Lemma for distributive law... |
| distrlem5pr 11139 | Lemma for distributive law... |
| distrpr 11140 | Multiplication of positive... |
| 1idpr 11141 | 1 is an identity element f... |
| ltprord 11142 | Positive real 'less than' ... |
| psslinpr 11143 | Proper subset is a linear ... |
| ltsopr 11144 | Positive real 'less than' ... |
| prlem934 11145 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11146 | The sum of two positive re... |
| ltaddpr2 11147 | The sum of two positive re... |
| ltexprlem1 11148 | Lemma for Proposition 9-3.... |
| ltexprlem2 11149 | Lemma for Proposition 9-3.... |
| ltexprlem3 11150 | Lemma for Proposition 9-3.... |
| ltexprlem4 11151 | Lemma for Proposition 9-3.... |
| ltexprlem5 11152 | Lemma for Proposition 9-3.... |
| ltexprlem6 11153 | Lemma for Proposition 9-3.... |
| ltexprlem7 11154 | Lemma for Proposition 9-3.... |
| ltexpri 11155 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11156 | Lemma for Proposition 9-3.... |
| ltapr 11157 | Ordering property of addit... |
| addcanpr 11158 | Addition cancellation law ... |
| prlem936 11159 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11160 | Lemma for Proposition 9-3.... |
| reclem3pr 11161 | Lemma for Proposition 9-3.... |
| reclem4pr 11162 | Lemma for Proposition 9-3.... |
| recexpr 11163 | The reciprocal of a positi... |
| suplem1pr 11164 | The union of a nonempty, b... |
| suplem2pr 11165 | The union of a set of posi... |
| supexpr 11166 | The union of a nonempty, b... |
| enrer 11175 | The equivalence relation f... |
| nrex1 11176 | The class of signed reals ... |
| enrbreq 11177 | Equivalence relation for s... |
| enreceq 11178 | Equivalence class equality... |
| enrex 11179 | The equivalence relation f... |
| ltrelsr 11180 | Signed real 'less than' is... |
| addcmpblnr 11181 | Lemma showing compatibilit... |
| mulcmpblnrlem 11182 | Lemma used in lemma showin... |
| mulcmpblnr 11183 | Lemma showing compatibilit... |
| prsrlem1 11184 | Decomposing signed reals i... |
| addsrmo 11185 | There is at most one resul... |
| mulsrmo 11186 | There is at most one resul... |
| addsrpr 11187 | Addition of signed reals i... |
| mulsrpr 11188 | Multiplication of signed r... |
| ltsrpr 11189 | Ordering of signed reals i... |
| gt0srpr 11190 | Greater than zero in terms... |
| 0nsr 11191 | The empty set is not a sig... |
| 0r 11192 | The constant ` 0R ` is a s... |
| 1sr 11193 | The constant ` 1R ` is a s... |
| m1r 11194 | The constant ` -1R ` is a ... |
| addclsr 11195 | Closure of addition on sig... |
| mulclsr 11196 | Closure of multiplication ... |
| dmaddsr 11197 | Domain of addition on sign... |
| dmmulsr 11198 | Domain of multiplication o... |
| addcomsr 11199 | Addition of signed reals i... |
| addasssr 11200 | Addition of signed reals i... |
| mulcomsr 11201 | Multiplication of signed r... |
| mulasssr 11202 | Multiplication of signed r... |
| distrsr 11203 | Multiplication of signed r... |
| m1p1sr 11204 | Minus one plus one is zero... |
| m1m1sr 11205 | Minus one times minus one ... |
| ltsosr 11206 | Signed real 'less than' is... |
| 0lt1sr 11207 | 0 is less than 1 for signe... |
| 1ne0sr 11208 | 1 and 0 are distinct for s... |
| 0idsr 11209 | The signed real number 0 i... |
| 1idsr 11210 | 1 is an identity element f... |
| 00sr 11211 | A signed real times 0 is 0... |
| ltasr 11212 | Ordering property of addit... |
| pn0sr 11213 | A signed real plus its neg... |
| negexsr 11214 | Existence of negative sign... |
| recexsrlem 11215 | The reciprocal of a positi... |
| addgt0sr 11216 | The sum of two positive si... |
| mulgt0sr 11217 | The product of two positiv... |
| sqgt0sr 11218 | The square of a nonzero si... |
| recexsr 11219 | The reciprocal of a nonzer... |
| mappsrpr 11220 | Mapping from positive sign... |
| ltpsrpr 11221 | Mapping of order from posi... |
| map2psrpr 11222 | Equivalence for positive s... |
| supsrlem 11223 | Lemma for supremum theorem... |
| supsr 11224 | A nonempty, bounded set of... |
| opelcn 11241 | Ordered pair membership in... |
| opelreal 11242 | Ordered pair membership in... |
| elreal 11243 | Membership in class of rea... |
| elreal2 11244 | Ordered pair membership in... |
| 0ncn 11245 | The empty set is not a com... |
| ltrelre 11246 | 'Less than' is a relation ... |
| addcnsr 11247 | Addition of complex number... |
| mulcnsr 11248 | Multiplication of complex ... |
| eqresr 11249 | Equality of real numbers i... |
| addresr 11250 | Addition of real numbers i... |
| mulresr 11251 | Multiplication of real num... |
| ltresr 11252 | Ordering of real subset of... |
| ltresr2 11253 | Ordering of real subset of... |
| dfcnqs 11254 | Technical trick to permit ... |
| addcnsrec 11255 | Technical trick to permit ... |
| mulcnsrec 11256 | Technical trick to permit ... |
| axaddf 11257 | Addition is an operation o... |
| axmulf 11258 | Multiplication is an opera... |
| axcnex 11259 | The complex numbers form a... |
| axresscn 11260 | The real numbers are a sub... |
| ax1cn 11261 | 1 is a complex number. Ax... |
| axicn 11262 | ` _i ` is a complex number... |
| axaddcl 11263 | Closure law for addition o... |
| axaddrcl 11264 | Closure law for addition i... |
| axmulcl 11265 | Closure law for multiplica... |
| axmulrcl 11266 | Closure law for multiplica... |
| axmulcom 11267 | Multiplication of complex ... |
| axaddass 11268 | Addition of complex number... |
| axmulass 11269 | Multiplication of complex ... |
| axdistr 11270 | Distributive law for compl... |
| axi2m1 11271 | i-squared equals -1 (expre... |
| ax1ne0 11272 | 1 and 0 are distinct. Axi... |
| ax1rid 11273 | ` 1 ` is an identity eleme... |
| axrnegex 11274 | Existence of negative of r... |
| axrrecex 11275 | Existence of reciprocal of... |
| axcnre 11276 | A complex number can be ex... |
| axpre-lttri 11277 | Ordering on reals satisfie... |
| axpre-lttrn 11278 | Ordering on reals is trans... |
| axpre-ltadd 11279 | Ordering property of addit... |
| axpre-mulgt0 11280 | The product of two positiv... |
| axpre-sup 11281 | A nonempty, bounded-above ... |
| wuncn 11282 | A weak universe containing... |
| cnex 11308 | Alias for ~ ax-cnex . See... |
| addcl 11309 | Alias for ~ ax-addcl , for... |
| readdcl 11310 | Alias for ~ ax-addrcl , fo... |
| mulcl 11311 | Alias for ~ ax-mulcl , for... |
| remulcl 11312 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11313 | Alias for ~ ax-mulcom , fo... |
| addass 11314 | Alias for ~ ax-addass , fo... |
| mulass 11315 | Alias for ~ ax-mulass , fo... |
| adddi 11316 | Alias for ~ ax-distr , for... |
| recn 11317 | A real number is a complex... |
| reex 11318 | The real numbers form a se... |
| reelprrecn 11319 | Reals are a subset of the ... |
| cnelprrecn 11320 | Complex numbers are a subs... |
| mpoaddf 11321 | Addition is an operation o... |
| mpomulf 11322 | Multiplication is an opera... |
| elimne0 11323 | Hypothesis for weak deduct... |
| adddir 11324 | Distributive law for compl... |
| 0cn 11325 | Zero is a complex number. ... |
| 0cnd 11326 | Zero is a complex number, ... |
| c0ex 11327 | Zero is a set. (Contribut... |
| 0elpr01 11328 | 0 is an element of ` { 0 ,... |
| 1cnd 11329 | One is a complex number, d... |
| 1ex 11330 | One is a set. (Contribute... |
| 1elpr01 11331 | 1 is an element of ` { 0 ,... |
| cnre 11332 | Alias for ~ ax-cnre , for ... |
| mulrid 11333 | The number 1 is an identit... |
| mullid 11334 | Identity law for multiplic... |
| 1re 11335 | The number 1 is real. Thi... |
| 1red 11336 | The number 1 is real, dedu... |
| 0re 11337 | The number 0 is real. Rem... |
| 0red 11338 | The number 0 is real, dedu... |
| pr01ssre 11339 | The pair ` { 0 , 1 } ` is ... |
| mulridi 11340 | Identity law for multiplic... |
| mullidi 11341 | Identity law for multiplic... |
| addcli 11342 | Closure law for addition. ... |
| mulcli 11343 | Closure law for multiplica... |
| mulcomi 11344 | Commutative law for multip... |
| mulcomli 11345 | Commutative law for multip... |
| addassi 11346 | Associative law for additi... |
| mulassi 11347 | Associative law for multip... |
| adddii 11348 | Distributive law (left-dis... |
| adddiri 11349 | Distributive law (right-di... |
| recni 11350 | A real number is a complex... |
| readdcli 11351 | Closure law for addition o... |
| remulcli 11352 | Closure law for multiplica... |
| mulridd 11353 | Identity law for multiplic... |
| mullidd 11354 | Identity law for multiplic... |
| addcld 11355 | Closure law for addition. ... |
| mulcld 11356 | Closure law for multiplica... |
| mulcomd 11357 | Commutative law for multip... |
| addassd 11358 | Associative law for additi... |
| mulassd 11359 | Associative law for multip... |
| adddid 11360 | Distributive law (left-dis... |
| adddird 11361 | Distributive law (right-di... |
| adddirp1d 11362 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11363 | Join AB+CB into (A+C) on L... |
| recnd 11364 | Deduction from real number... |
| readdcld 11365 | Closure law for addition o... |
| remulcld 11366 | Closure law for multiplica... |
| pnfnre 11377 | Plus infinity is not a rea... |
| pnfnre2 11378 | Plus infinity is not a rea... |
| mnfnre 11379 | Minus infinity is not a re... |
| ressxr 11380 | The standard reals are a s... |
| rexpssxrxp 11381 | The Cartesian product of s... |
| rexr 11382 | A standard real is an exte... |
| 0xr 11383 | Zero is an extended real. ... |
| renepnf 11384 | No (finite) real equals pl... |
| renemnf 11385 | No real equals minus infin... |
| rexrd 11386 | A standard real is an exte... |
| renepnfd 11387 | No (finite) real equals pl... |
| renemnfd 11388 | No real equals minus infin... |
| pnfex 11389 | Plus infinity exists. (Co... |
| pnfxr 11390 | Plus infinity belongs to t... |
| pnfnemnf 11391 | Plus and minus infinity ar... |
| mnfnepnf 11392 | Minus and plus infinity ar... |
| mnfxr 11393 | Minus infinity belongs to ... |
| rexri 11394 | A standard real is an exte... |
| 1xr 11395 | ` 1 ` is an extended real ... |
| renfdisj 11396 | The reals and the infiniti... |
| ltrelxr 11397 | "Less than" is a relation ... |
| ltrel 11398 | "Less than" is a relation.... |
| lerelxr 11399 | "Less than or equal to" is... |
| lerel 11400 | "Less than or equal to" is... |
| xrlenlt 11401 | "Less than or equal to" ex... |
| xrlenltd 11402 | "Less than or equal to" ex... |
| xrltnle 11403 | "Less than" expressed in t... |
| xrltnled 11404 | 'Less than' in terms of 'l... |
| xrnltled 11405 | "Not less than" implies "l... |
| ssxr 11406 | The three (non-exclusive) ... |
| ltxrlt 11407 | The standard less-than ` <... |
| axlttri 11408 | Ordering on reals satisfie... |
| axlttrn 11409 | Ordering on reals is trans... |
| axltadd 11410 | Ordering property of addit... |
| axmulgt0 11411 | The product of two positiv... |
| axsup 11412 | A nonempty, bounded-above ... |
| lttr 11413 | Alias for ~ axlttrn , for ... |
| mulgt0 11414 | The product of two positiv... |
| lenlt 11415 | 'Less than or equal to' ex... |
| ltnle 11416 | 'Less than' expressed in t... |
| ltso 11417 | 'Less than' is a strict or... |
| gtso 11418 | 'Greater than' is a strict... |
| lttri2 11419 | Consequence of trichotomy.... |
| lttri3 11420 | Trichotomy law for 'less t... |
| lttri4 11421 | Trichotomy law for 'less t... |
| letri3 11422 | Trichotomy law. (Contribu... |
| leloe 11423 | 'Less than or equal to' ex... |
| eqlelt 11424 | Equality in terms of 'less... |
| ltle 11425 | 'Less than' implies 'less ... |
| leltne 11426 | 'Less than or equal to' im... |
| lelttr 11427 | Transitive law. (Contribu... |
| leltletr 11428 | Transitive law, weaker for... |
| ltletr 11429 | Transitive law. (Contribu... |
| ltleletr 11430 | Transitive law, weaker for... |
| letr 11431 | Transitive law. (Contribu... |
| ltnr 11432 | 'Less than' is irreflexive... |
| leid 11433 | 'Less than or equal to' is... |
| ltne 11434 | 'Less than' implies not eq... |
| ltnsym 11435 | 'Less than' is not symmetr... |
| ltnsym2 11436 | 'Less than' is antisymmetr... |
| letric 11437 | Trichotomy law. (Contribu... |
| ltlen 11438 | 'Less than' expressed in t... |
| eqle 11439 | Equality implies 'less tha... |
| eqled 11440 | Equality implies 'less tha... |
| ltadd2 11441 | Addition to both sides of ... |
| ne0gt0 11442 | A nonzero nonnegative numb... |
| lecasei 11443 | Ordering elimination by ca... |
| lelttric 11444 | Trichotomy law. (Contribu... |
| ltlecasei 11445 | Ordering elimination by ca... |
| ltnri 11446 | 'Less than' is irreflexive... |
| eqlei 11447 | Equality implies 'less tha... |
| eqlei2 11448 | Equality implies 'less tha... |
| gtneii 11449 | 'Less than' implies not eq... |
| ltneii 11450 | 'Greater than' implies not... |
| lttri2i 11451 | Consequence of trichotomy.... |
| lttri3i 11452 | Consequence of trichotomy.... |
| letri3i 11453 | Consequence of trichotomy.... |
| leloei 11454 | 'Less than or equal to' in... |
| ltleni 11455 | 'Less than' expressed in t... |
| ltnsymi 11456 | 'Less than' is not symmetr... |
| lenlti 11457 | 'Less than or equal to' in... |
| ltnlei 11458 | 'Less than' in terms of 'l... |
| ltlei 11459 | 'Less than' implies 'less ... |
| ltleii 11460 | 'Less than' implies 'less ... |
| ltnei 11461 | 'Less than' implies not eq... |
| letrii 11462 | Trichotomy law for 'less t... |
| lttri 11463 | 'Less than' is transitive.... |
| lelttri 11464 | 'Less than or equal to', '... |
| ltletri 11465 | 'Less than', 'less than or... |
| letri 11466 | 'Less than or equal to' is... |
| le2tri3i 11467 | Extended trichotomy law fo... |
| ltadd2i 11468 | Addition to both sides of ... |
| mulgt0i 11469 | The product of two positiv... |
| mulgt0ii 11470 | The product of two positiv... |
| ltnrd 11471 | 'Less than' is irreflexive... |
| gtned 11472 | 'Less than' implies not eq... |
| ltned 11473 | 'Greater than' implies not... |
| ne0gt0d 11474 | A nonzero nonnegative numb... |
| lttrid 11475 | Ordering on reals satisfie... |
| lttri2d 11476 | Consequence of trichotomy.... |
| lttri3d 11477 | Consequence of trichotomy.... |
| lttri4d 11478 | Trichotomy law for 'less t... |
| letri3d 11479 | Consequence of trichotomy.... |
| leloed 11480 | 'Less than or equal to' in... |
| eqleltd 11481 | Equality in terms of 'less... |
| ltlend 11482 | 'Less than' expressed in t... |
| lenltd 11483 | 'Less than or equal to' in... |
| ltnled 11484 | 'Less than' in terms of 'l... |
| ltled 11485 | 'Less than' implies 'less ... |
| ltnsymd 11486 | 'Less than' implies 'less ... |
| nltled 11487 | 'Not less than ' implies '... |
| lensymd 11488 | 'Less than or equal to' im... |
| letrid 11489 | Trichotomy law for 'less t... |
| leltned 11490 | 'Less than or equal to' im... |
| leneltd 11491 | 'Less than or equal to' an... |
| mulgt0d 11492 | The product of two positiv... |
| ltadd2d 11493 | Addition to both sides of ... |
| letrd 11494 | Transitive law deduction f... |
| lelttrd 11495 | Transitive law deduction f... |
| ltadd2dd 11496 | Addition to both sides of ... |
| ltletrd 11497 | Transitive law deduction f... |
| lttrd 11498 | Transitive law deduction f... |
| lelttrdi 11499 | If a number is less than a... |
| dedekind 11500 | The Dedekind cut theorem. ... |
| dedekindle 11501 | The Dedekind cut theorem, ... |
| mul12 11502 | Commutative/associative la... |
| mul32 11503 | Commutative/associative la... |
| mul31 11504 | Commutative/associative la... |
| mul4 11505 | Rearrangement of 4 factors... |
| mul4r 11506 | Rearrangement of 4 factors... |
| muladd11 11507 | A simple product of sums e... |
| 1p1times 11508 | Two times a number. (Cont... |
| peano2cn 11509 | A theorem for complex numb... |
| peano2re 11510 | A theorem for reals analog... |
| readdcan 11511 | Cancellation law for addit... |
| 00id 11512 | ` 0 ` is its own additive ... |
| mul02lem1 11513 | Lemma for ~ mul02 . If an... |
| mul02lem2 11514 | Lemma for ~ mul02 . Zero ... |
| mul02 11515 | Multiplication by ` 0 ` . ... |
| mul01 11516 | Multiplication by ` 0 ` . ... |
| addrid 11517 | ` 0 ` is an additive ident... |
| cnegex 11518 | Existence of the negative ... |
| cnegex2 11519 | Existence of a left invers... |
| addlid 11520 | ` 0 ` is a left identity f... |
| addcan 11521 | Cancellation law for addit... |
| addcan2 11522 | Cancellation law for addit... |
| addcom 11523 | Addition is commutative. ... |
| addridi 11524 | ` 0 ` is an additive ident... |
| addlidi 11525 | ` 0 ` is a left identity f... |
| mul02i 11526 | Multiplication by 0. Theo... |
| mul01i 11527 | Multiplication by ` 0 ` . ... |
| addcomi 11528 | Addition is commutative. ... |
| addcomli 11529 | Addition is commutative. ... |
| addcani 11530 | Cancellation law for addit... |
| addcan2i 11531 | Cancellation law for addit... |
| mul12i 11532 | Commutative/associative la... |
| mul32i 11533 | Commutative/associative la... |
| mul4i 11534 | Rearrangement of 4 factors... |
| mul02d 11535 | Multiplication by 0. Theo... |
| mul01d 11536 | Multiplication by ` 0 ` . ... |
| addridd 11537 | ` 0 ` is an additive ident... |
| addlidd 11538 | ` 0 ` is a left identity f... |
| addcomd 11539 | Addition is commutative. ... |
| addcand 11540 | Cancellation law for addit... |
| addcan2d 11541 | Cancellation law for addit... |
| addcanad 11542 | Cancelling a term on the l... |
| addcan2ad 11543 | Cancelling a term on the r... |
| addneintrd 11544 | Introducing a term on the ... |
| addneintr2d 11545 | Introducing a term on the ... |
| mul12d 11546 | Commutative/associative la... |
| mul32d 11547 | Commutative/associative la... |
| mul31d 11548 | Commutative/associative la... |
| mul4d 11549 | Rearrangement of 4 factors... |
| muladd11r 11550 | A simple product of sums e... |
| comraddd 11551 | Commute RHS addition, in d... |
| comraddi 11552 | Commute RHS addition. See... |
| ltaddneg 11553 | Adding a negative number t... |
| ltaddnegr 11554 | Adding a negative number t... |
| add12 11555 | Commutative/associative la... |
| add32 11556 | Commutative/associative la... |
| add32r 11557 | Commutative/associative la... |
| add4 11558 | Rearrangement of 4 terms i... |
| add42 11559 | Rearrangement of 4 terms i... |
| add12i 11560 | Commutative/associative la... |
| add32i 11561 | Commutative/associative la... |
| add4i 11562 | Rearrangement of 4 terms i... |
| add42i 11563 | Rearrangement of 4 terms i... |
| add12d 11564 | Commutative/associative la... |
| add32d 11565 | Commutative/associative la... |
| add4d 11566 | Rearrangement of 4 terms i... |
| add42d 11567 | Rearrangement of 4 terms i... |
| 0cnALT 11572 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11573 | Alternate proof of ~ 0cnAL... |
| negeu 11574 | Existential uniqueness of ... |
| subval 11575 | Value of subtraction, whic... |
| negeq 11576 | Equality theorem for negat... |
| negeqi 11577 | Equality inference for neg... |
| negeqd 11578 | Equality deduction for neg... |
| nfnegd 11579 | Deduction version of ~ nfn... |
| nfneg 11580 | Bound-variable hypothesis ... |
| csbnegg 11581 | Move class substitution in... |
| negex 11582 | A negative is a set. (Con... |
| subcl 11583 | Closure law for subtractio... |
| negcl 11584 | Closure law for negative. ... |
| negicn 11585 | ` -u _i ` is a complex num... |
| subf 11586 | Subtraction is an operatio... |
| subadd 11587 | Relationship between subtr... |
| subadd2 11588 | Relationship between subtr... |
| subsub23 11589 | Swap subtrahend and result... |
| pncan 11590 | Cancellation law for subtr... |
| pncan2 11591 | Cancellation law for subtr... |
| pncan3 11592 | Subtraction and addition o... |
| npcan 11593 | Cancellation law for subtr... |
| addsubass 11594 | Associative-type law for a... |
| addsub 11595 | Law for addition and subtr... |
| subadd23 11596 | Commutative/associative la... |
| addsub12 11597 | Commutative/associative la... |
| 2addsub 11598 | Law for subtraction and ad... |
| addsubeq4 11599 | Relation between sums and ... |
| pncan3oi 11600 | Subtraction and addition o... |
| mvrraddi 11601 | Move the right term in a s... |
| mvrladdi 11602 | Move the left term in a su... |
| mvlladdi 11603 | Move the left term in a su... |
| subid 11604 | Subtraction of a number fr... |
| subid1 11605 | Identity law for subtracti... |
| npncan 11606 | Cancellation law for subtr... |
| nppcan 11607 | Cancellation law for subtr... |
| nnpcan 11608 | Cancellation law for subtr... |
| nppcan3 11609 | Cancellation law for subtr... |
| subcan2 11610 | Cancellation law for subtr... |
| subeq0 11611 | If the difference between ... |
| npncan2 11612 | Cancellation law for subtr... |
| subsub2 11613 | Law for double subtraction... |
| nncan 11614 | Cancellation law for subtr... |
| subsub 11615 | Law for double subtraction... |
| nppcan2 11616 | Cancellation law for subtr... |
| subsub3 11617 | Law for double subtraction... |
| subsub4 11618 | Law for double subtraction... |
| sub32 11619 | Swap the second and third ... |
| nnncan 11620 | Cancellation law for subtr... |
| nnncan1 11621 | Cancellation law for subtr... |
| nnncan2 11622 | Cancellation law for subtr... |
| npncan3 11623 | Cancellation law for subtr... |
| pnpcan 11624 | Cancellation law for mixed... |
| pnpcan2 11625 | Cancellation law for mixed... |
| pnncan 11626 | Cancellation law for mixed... |
| ppncan 11627 | Cancellation law for mixed... |
| addsub4 11628 | Rearrangement of 4 terms i... |
| subadd4 11629 | Rearrangement of 4 terms i... |
| sub4 11630 | Rearrangement of 4 terms i... |
| neg0 11631 | Minus 0 equals 0. (Contri... |
| negid 11632 | Addition of a number and i... |
| negsub 11633 | Relationship between subtr... |
| subneg 11634 | Relationship between subtr... |
| negneg 11635 | A number is equal to the n... |
| neg11 11636 | Negative is one-to-one. (... |
| negcon1 11637 | Negative contraposition la... |
| negcon2 11638 | Negative contraposition la... |
| negeq0 11639 | A number is zero iff its n... |
| subcan 11640 | Cancellation law for subtr... |
| negsubdi 11641 | Distribution of negative o... |
| negdi 11642 | Distribution of negative o... |
| negdi2 11643 | Distribution of negative o... |
| negsubdi2 11644 | Distribution of negative o... |
| neg2sub 11645 | Relationship between subtr... |
| renegcli 11646 | Closure law for negative o... |
| resubcli 11647 | Closure law for subtractio... |
| renegcl 11648 | Closure law for negative o... |
| resubcl 11649 | Closure law for subtractio... |
| negreb 11650 | The negative of a real is ... |
| peano2cnm 11651 | "Reverse" second Peano pos... |
| peano2rem 11652 | "Reverse" second Peano pos... |
| negcli 11653 | Closure law for negative. ... |
| negidi 11654 | Addition of a number and i... |
| negnegi 11655 | A number is equal to the n... |
| subidi 11656 | Subtraction of a number fr... |
| subid1i 11657 | Identity law for subtracti... |
| negne0bi 11658 | A number is nonzero iff it... |
| negrebi 11659 | The negative of a real is ... |
| negne0i 11660 | The negative of a nonzero ... |
| subcli 11661 | Closure law for subtractio... |
| pncan3i 11662 | Subtraction and addition o... |
| negsubi 11663 | Relationship between subtr... |
| subnegi 11664 | Relationship between subtr... |
| subeq0i 11665 | If the difference between ... |
| neg11i 11666 | Negative is one-to-one. (... |
| negcon1i 11667 | Negative contraposition la... |
| negcon2i 11668 | Negative contraposition la... |
| negdii 11669 | Distribution of negative o... |
| negsubdii 11670 | Distribution of negative o... |
| negsubdi2i 11671 | Distribution of negative o... |
| subaddi 11672 | Relationship between subtr... |
| subadd2i 11673 | Relationship between subtr... |
| subaddrii 11674 | Relationship between subtr... |
| subsub23i 11675 | Swap subtrahend and result... |
| addsubassi 11676 | Associative-type law for s... |
| addsubi 11677 | Law for subtraction and ad... |
| subcani 11678 | Cancellation law for subtr... |
| subcan2i 11679 | Cancellation law for subtr... |
| pnncani 11680 | Cancellation law for mixed... |
| addsub4i 11681 | Rearrangement of 4 terms i... |
| 0reALT 11682 | Alternate proof of ~ 0re .... |
| negcld 11683 | Closure law for negative. ... |
| subidd 11684 | Subtraction of a number fr... |
| subid1d 11685 | Identity law for subtracti... |
| negidd 11686 | Addition of a number and i... |
| negnegd 11687 | A number is equal to the n... |
| negeq0d 11688 | A number is zero iff its n... |
| negne0bd 11689 | A number is nonzero iff it... |
| negcon1d 11690 | Contraposition law for una... |
| negcon1ad 11691 | Contraposition law for una... |
| neg11ad 11692 | The negatives of two compl... |
| negned 11693 | If two complex numbers are... |
| negne0d 11694 | The negative of a nonzero ... |
| negrebd 11695 | The negative of a real is ... |
| subcld 11696 | Closure law for subtractio... |
| pncand 11697 | Cancellation law for subtr... |
| pncan2d 11698 | Cancellation law for subtr... |
| pncan3d 11699 | Subtraction and addition o... |
| npcand 11700 | Cancellation law for subtr... |
| nncand 11701 | Cancellation law for subtr... |
| negsubd 11702 | Relationship between subtr... |
| subnegd 11703 | Relationship between subtr... |
| subeq0ad 11704 | The difference of two comp... |
| subeq0d 11705 | If the difference between ... |
| subne0d 11706 | Two unequal numbers have n... |
| subne0ad 11707 | If the difference of two c... |
| neg11d 11708 | If the difference between ... |
| negdid 11709 | Distribution of negative o... |
| negdi2d 11710 | Distribution of negative o... |
| negsubdid 11711 | Distribution of negative o... |
| negsubdi2d 11712 | Distribution of negative o... |
| neg2subd 11713 | Relationship between subtr... |
| subaddd 11714 | Relationship between subtr... |
| subadd2d 11715 | Relationship between subtr... |
| addsubassd 11716 | Associative-type law for s... |
| addsubd 11717 | Law for subtraction and ad... |
| subadd23d 11718 | Commutative/associative la... |
| addsub12d 11719 | Commutative/associative la... |
| npncand 11720 | Cancellation law for subtr... |
| nppcand 11721 | Cancellation law for subtr... |
| nppcan2d 11722 | Cancellation law for subtr... |
| nppcan3d 11723 | Cancellation law for subtr... |
| subsubd 11724 | Law for double subtraction... |
| subsub2d 11725 | Law for double subtraction... |
| subsub3d 11726 | Law for double subtraction... |
| subsub4d 11727 | Law for double subtraction... |
| sub32d 11728 | Swap the second and third ... |
| nnncand 11729 | Cancellation law for subtr... |
| nnncan1d 11730 | Cancellation law for subtr... |
| nnncan2d 11731 | Cancellation law for subtr... |
| npncan3d 11732 | Cancellation law for subtr... |
| pnpcand 11733 | Cancellation law for mixed... |
| pnpcan2d 11734 | Cancellation law for mixed... |
| pnncand 11735 | Cancellation law for mixed... |
| ppncand 11736 | Cancellation law for mixed... |
| subcand 11737 | Cancellation law for subtr... |
| subcan2d 11738 | Cancellation law for subtr... |
| subcanad 11739 | Cancellation law for subtr... |
| subneintrd 11740 | Introducing subtraction on... |
| subcan2ad 11741 | Cancellation law for subtr... |
| subneintr2d 11742 | Introducing subtraction on... |
| addsub4d 11743 | Rearrangement of 4 terms i... |
| subadd4d 11744 | Rearrangement of 4 terms i... |
| sub4d 11745 | Rearrangement of 4 terms i... |
| 2addsubd 11746 | Law for subtraction and ad... |
| addsubeq4d 11747 | Relation between sums and ... |
| subsubadd23 11748 | Swap the second and the th... |
| addsubsub23 11749 | Swap the second and the th... |
| subeqxfrd 11750 | Transfer two terms of a su... |
| mvlraddd 11751 | Move the right term in a s... |
| mvlladdd 11752 | Move the left term in a su... |
| mvlladdcd 11753 | Rotate the variables right... |
| mvrraddd 11754 | Move the right term in a s... |
| mvrladdd 11755 | Move the left term in a su... |
| mvrrsubd 11756 | Move a subtraction in the ... |
| assraddsubd 11757 | Associate RHS addition-sub... |
| subaddeqd 11758 | Transfer two terms of a su... |
| addlsub 11759 | Left-subtraction: Subtrac... |
| addrsub 11760 | Right-subtraction: Subtra... |
| subexsub 11761 | A subtraction law: Exchan... |
| addid0 11762 | If adding a number to a an... |
| addn0nid 11763 | Adding a nonzero number to... |
| pnpncand 11764 | Addition/subtraction cance... |
| subeqrev 11765 | Reverse the order of subtr... |
| addeq0 11766 | Two complex numbers add up... |
| pncan1 11767 | Cancellation law for addit... |
| npcan1 11768 | Cancellation law for subtr... |
| subeq0bd 11769 | If two complex numbers are... |
| renegcld 11770 | Closure law for negative o... |
| resubcld 11771 | Closure law for subtractio... |
| negn0 11772 | The image under negation o... |
| negf1o 11773 | Negation is an isomorphism... |
| kcnktkm1cn 11774 | k times k minus 1 is a com... |
| muladd 11775 | Product of two sums. (Con... |
| subdi 11776 | Distribution of multiplica... |
| subdir 11777 | Distribution of multiplica... |
| ine0 11778 | The imaginary unit ` _i ` ... |
| mulneg1 11779 | Product with negative is n... |
| mulneg2 11780 | The product with a negativ... |
| mulneg12 11781 | Swap the negative sign in ... |
| mul2neg 11782 | Product of two negatives. ... |
| submul2 11783 | Convert a subtraction to a... |
| mulm1 11784 | Product with minus one is ... |
| addneg1mul 11785 | Addition with product with... |
| mulsub 11786 | Product of two differences... |
| mulsub2 11787 | Swap the order of subtract... |
| mulm1i 11788 | Product with minus one is ... |
| mulneg1i 11789 | Product with negative is n... |
| mulneg2i 11790 | Product with negative is n... |
| mul2negi 11791 | Product of two negatives. ... |
| subdii 11792 | Distribution of multiplica... |
| subdiri 11793 | Distribution of multiplica... |
| muladdi 11794 | Product of two sums. (Con... |
| mulm1d 11795 | Product with minus one is ... |
| mulneg1d 11796 | Product with negative is n... |
| mulneg2d 11797 | Product with negative is n... |
| mul2negd 11798 | Product of two negatives. ... |
| subdid 11799 | Distribution of multiplica... |
| subdird 11800 | Distribution of multiplica... |
| muladdd 11801 | Product of two sums. (Con... |
| mulsubd 11802 | Product of two differences... |
| muls1d 11803 | Multiplication by one minu... |
| mulsubfacd 11804 | Multiplication followed by... |
| addmulsub 11805 | The product of a sum and a... |
| subaddmulsub 11806 | The difference with a prod... |
| mulsubaddmulsub 11807 | A special difference of a ... |
| gt0ne0 11808 | Positive implies nonzero. ... |
| lt0ne0 11809 | A number which is less tha... |
| ltadd1 11810 | Addition to both sides of ... |
| leadd1 11811 | Addition to both sides of ... |
| leadd2 11812 | Addition to both sides of ... |
| ltsubadd 11813 | 'Less than' relationship b... |
| ltsubadd2 11814 | 'Less than' relationship b... |
| lesubadd 11815 | 'Less than or equal to' re... |
| lesubadd2 11816 | 'Less than or equal to' re... |
| ltaddsub 11817 | 'Less than' relationship b... |
| ltaddsub2 11818 | 'Less than' relationship b... |
| leaddsub 11819 | 'Less than or equal to' re... |
| leaddsub2 11820 | 'Less than or equal to' re... |
| suble 11821 | Swap subtrahends in an ine... |
| lesub 11822 | Swap subtrahends in an ine... |
| ltsub23 11823 | 'Less than' relationship b... |
| ltsub13 11824 | 'Less than' relationship b... |
| le2add 11825 | Adding both sides of two '... |
| ltleadd 11826 | Adding both sides of two o... |
| leltadd 11827 | Adding both sides of two o... |
| lt2add 11828 | Adding both sides of two '... |
| addgt0 11829 | The sum of 2 positive numb... |
| addgegt0 11830 | The sum of nonnegative and... |
| addgtge0 11831 | The sum of nonnegative and... |
| addge0 11832 | The sum of 2 nonnegative n... |
| ltaddpos 11833 | Adding a positive number t... |
| ltaddpos2 11834 | Adding a positive number t... |
| ltsubpos 11835 | Subtracting a positive num... |
| posdif 11836 | Comparison of two numbers ... |
| lesub1 11837 | Subtraction from both side... |
| lesub2 11838 | Subtraction of both sides ... |
| ltsub1 11839 | Subtraction from both side... |
| ltsub2 11840 | Subtraction of both sides ... |
| lt2sub 11841 | Subtracting both sides of ... |
| le2sub 11842 | Subtracting both sides of ... |
| ltneg 11843 | Negative of both sides of ... |
| ltnegcon1 11844 | Contraposition of negative... |
| ltnegcon2 11845 | Contraposition of negative... |
| leneg 11846 | Negative of both sides of ... |
| lenegcon1 11847 | Contraposition of negative... |
| lenegcon2 11848 | Contraposition of negative... |
| lt0neg1 11849 | Comparison of a number and... |
| lt0neg2 11850 | Comparison of a number and... |
| le0neg1 11851 | Comparison of a number and... |
| le0neg2 11852 | Comparison of a number and... |
| addge01 11853 | A number is less than or e... |
| addge02 11854 | A number is less than or e... |
| add20 11855 | Two nonnegative numbers ar... |
| subge0 11856 | Nonnegative subtraction. ... |
| suble0 11857 | Nonpositive subtraction. ... |
| leaddle0 11858 | The sum of a real number a... |
| subge02 11859 | Nonnegative subtraction. ... |
| lesub0 11860 | Lemma to show a nonnegativ... |
| mulge0 11861 | The product of two nonnega... |
| mullt0 11862 | The product of two negativ... |
| msqgt0 11863 | A nonzero square is positi... |
| msqge0 11864 | A square is nonnegative. ... |
| 0lt1 11865 | 0 is less than 1. Theorem... |
| 0le1 11866 | 0 is less than or equal to... |
| relin01 11867 | An interval law for less t... |
| ltordlem 11868 | Lemma for ~ ltord1 . (Con... |
| ltord1 11869 | Infer an ordering relation... |
| leord1 11870 | Infer an ordering relation... |
| eqord1 11871 | A strictly increasing real... |
| ltord2 11872 | Infer an ordering relation... |
| leord2 11873 | Infer an ordering relation... |
| eqord2 11874 | A strictly decreasing real... |
| wloglei 11875 | Form of ~ wlogle where bot... |
| wlogle 11876 | If the predicate ` ch ( x ... |
| leidi 11877 | 'Less than or equal to' is... |
| gt0ne0i 11878 | Positive means nonzero (us... |
| gt0ne0ii 11879 | Positive implies nonzero. ... |
| msqgt0i 11880 | A nonzero square is positi... |
| msqge0i 11881 | A square is nonnegative. ... |
| addgt0i 11882 | Addition of 2 positive num... |
| addge0i 11883 | Addition of 2 nonnegative ... |
| addgegt0i 11884 | Addition of nonnegative an... |
| addgt0ii 11885 | Addition of 2 positive num... |
| add20i 11886 | Two nonnegative numbers ar... |
| ltnegi 11887 | Negative of both sides of ... |
| lenegi 11888 | Negative of both sides of ... |
| ltnegcon2i 11889 | Contraposition of negative... |
| mulge0i 11890 | The product of two nonnega... |
| lesub0i 11891 | Lemma to show a nonnegativ... |
| ltaddposi 11892 | Adding a positive number t... |
| posdifi 11893 | Comparison of two numbers ... |
| ltnegcon1i 11894 | Contraposition of negative... |
| lenegcon1i 11895 | Contraposition of negative... |
| subge0i 11896 | Nonnegative subtraction. ... |
| ltadd1i 11897 | Addition to both sides of ... |
| leadd1i 11898 | Addition to both sides of ... |
| leadd2i 11899 | Addition to both sides of ... |
| ltsubaddi 11900 | 'Less than' relationship b... |
| lesubaddi 11901 | 'Less than or equal to' re... |
| ltsubadd2i 11902 | 'Less than' relationship b... |
| lesubadd2i 11903 | 'Less than or equal to' re... |
| ltaddsubi 11904 | 'Less than' relationship b... |
| lt2addi 11905 | Adding both side of two in... |
| le2addi 11906 | Adding both side of two in... |
| gt0ne0d 11907 | Positive implies nonzero. ... |
| lt0ne0d 11908 | Something less than zero i... |
| leidd 11909 | 'Less than or equal to' is... |
| msqgt0d 11910 | A nonzero square is positi... |
| msqge0d 11911 | A square is nonnegative. ... |
| lt0neg1d 11912 | Comparison of a number and... |
| lt0neg2d 11913 | Comparison of a number and... |
| le0neg1d 11914 | Comparison of a number and... |
| le0neg2d 11915 | Comparison of a number and... |
| addgegt0d 11916 | Addition of nonnegative an... |
| addgtge0d 11917 | Addition of positive and n... |
| addgt0d 11918 | Addition of 2 positive num... |
| addge0d 11919 | Addition of 2 nonnegative ... |
| mulge0d 11920 | The product of two nonnega... |
| ltnegd 11921 | Negative of both sides of ... |
| lenegd 11922 | Negative of both sides of ... |
| ltnegcon1d 11923 | Contraposition of negative... |
| ltnegcon2d 11924 | Contraposition of negative... |
| lenegcon1d 11925 | Contraposition of negative... |
| lenegcon2d 11926 | Contraposition of negative... |
| ltaddposd 11927 | Adding a positive number t... |
| ltaddpos2d 11928 | Adding a positive number t... |
| ltsubposd 11929 | Subtracting a positive num... |
| posdifd 11930 | Comparison of two numbers ... |
| addge01d 11931 | A number is less than or e... |
| addge02d 11932 | A number is less than or e... |
| subge0d 11933 | Nonnegative subtraction. ... |
| suble0d 11934 | Nonpositive subtraction. ... |
| subge02d 11935 | Nonnegative subtraction. ... |
| ltadd1d 11936 | Addition to both sides of ... |
| leadd1d 11937 | Addition to both sides of ... |
| leadd2d 11938 | Addition to both sides of ... |
| ltsubaddd 11939 | 'Less than' relationship b... |
| lesubaddd 11940 | 'Less than or equal to' re... |
| ltsubadd2d 11941 | 'Less than' relationship b... |
| lesubadd2d 11942 | 'Less than or equal to' re... |
| ltaddsubd 11943 | 'Less than' relationship b... |
| ltaddsub2d 11944 | 'Less than' relationship b... |
| leaddsub2d 11945 | 'Less than or equal to' re... |
| subled 11946 | Swap subtrahends in an ine... |
| lesubd 11947 | Swap subtrahends in an ine... |
| ltsub23d 11948 | 'Less than' relationship b... |
| ltsub13d 11949 | 'Less than' relationship b... |
| lesub1d 11950 | Subtraction from both side... |
| lesub2d 11951 | Subtraction of both sides ... |
| ltsub1d 11952 | Subtraction from both side... |
| ltsub2d 11953 | Subtraction of both sides ... |
| ltadd1dd 11954 | Addition to both sides of ... |
| ltsub1dd 11955 | Subtraction from both side... |
| ltsub2dd 11956 | Subtraction of both sides ... |
| leadd1dd 11957 | Addition to both sides of ... |
| leadd2dd 11958 | Addition to both sides of ... |
| lesub1dd 11959 | Subtraction from both side... |
| lesub2dd 11960 | Subtraction of both sides ... |
| lesub3d 11961 | The result of subtracting ... |
| le2addd 11962 | Adding both side of two in... |
| le2subd 11963 | Subtracting both sides of ... |
| ltleaddd 11964 | Adding both sides of two o... |
| leltaddd 11965 | Adding both sides of two o... |
| lt2addd 11966 | Adding both side of two in... |
| lt2subd 11967 | Subtracting both sides of ... |
| possumd 11968 | Condition for a positive s... |
| sublt0d 11969 | When a subtraction gives a... |
| ltaddsublt 11970 | Addition and subtraction o... |
| 1le1 11971 | One is less than or equal ... |
| ixi 11972 | ` _i ` times itself is min... |
| recextlem1 11973 | Lemma for ~ recex . (Cont... |
| recextlem2 11974 | Lemma for ~ recex . (Cont... |
| recex 11975 | Existence of reciprocal of... |
| mulcand 11976 | Cancellation law for multi... |
| mulcan2d 11977 | Cancellation law for multi... |
| mulcanad 11978 | Cancellation of a nonzero ... |
| mulcan2ad 11979 | Cancellation of a nonzero ... |
| mulcan 11980 | Cancellation law for multi... |
| mulcan2 11981 | Cancellation law for multi... |
| mulcani 11982 | Cancellation law for multi... |
| mul0or 11983 | If a product is zero, one ... |
| mulne0b 11984 | The product of two nonzero... |
| mulne0 11985 | The product of two nonzero... |
| mulne0i 11986 | The product of two nonzero... |
| muleqadd 11987 | Property of numbers whose ... |
| receu 11988 | Existential uniqueness of ... |
| mulnzcnf 11989 | Multiplication maps nonzer... |
| mul0ori 11990 | If a product is zero, one ... |
| mul0ord 11991 | If a product is zero, one ... |
| msq0i 11992 | A number is zero iff its s... |
| msq0d 11993 | A number is zero iff its s... |
| mulne0bd 11994 | The product of two nonzero... |
| mulne0d 11995 | The product of two nonzero... |
| mulcan1g 11996 | A generalized form of the ... |
| mulcan2g 11997 | A generalized form of the ... |
| mulne0bad 11998 | A factor of a nonzero comp... |
| mulne0bbd 11999 | A factor of a nonzero comp... |
| 1div0 12002 | You can't divide by zero, ... |
| divval 12003 | Value of division: if ` A ... |
| divmul 12004 | Relationship between divis... |
| divmul2 12005 | Relationship between divis... |
| divmul3 12006 | Relationship between divis... |
| divcl 12007 | Closure law for division. ... |
| reccl 12008 | Closure law for reciprocal... |
| divcan2 12009 | A cancellation law for div... |
| divcan1 12010 | A cancellation law for div... |
| diveq0 12011 | A ratio is zero iff the nu... |
| divne0b 12012 | The ratio of nonzero numbe... |
| divne0 12013 | The ratio of nonzero numbe... |
| recne0 12014 | The reciprocal of a nonzer... |
| recid 12015 | Multiplication of a number... |
| recid2 12016 | Multiplication of a number... |
| divrec 12017 | Relationship between divis... |
| divrec2 12018 | Relationship between divis... |
| divass 12019 | An associative law for div... |
| div23 12020 | A commutative/associative ... |
| div32 12021 | A commutative/associative ... |
| div13 12022 | A commutative/associative ... |
| div12 12023 | A commutative/associative ... |
| divmulass 12024 | An associative law for div... |
| divmulasscom 12025 | An associative/commutative... |
| divdir 12026 | Distribution of division o... |
| divcan3 12027 | A cancellation law for div... |
| divcan4 12028 | A cancellation law for div... |
| div11 12029 | One-to-one relationship fo... |
| diveq1 12030 | Equality in terms of unit ... |
| divid 12031 | A number divided by itself... |
| div0 12032 | Division into zero is zero... |
| div1 12033 | A number divided by 1 is i... |
| 1div1e1 12034 | 1 divided by 1 is 1. (Con... |
| divneg 12035 | Move negative sign inside ... |
| muldivdir 12036 | Distribution of division o... |
| divsubdir 12037 | Distribution of division o... |
| muldivdid 12038 | Distribution of division o... |
| subdivcomb1 12039 | Bring a term in a subtract... |
| subdivcomb2 12040 | Bring a term in a subtract... |
| recrec 12041 | A number is equal to the r... |
| rec11 12042 | Reciprocal is one-to-one. ... |
| rec11r 12043 | Mutual reciprocals. (Cont... |
| divmuldiv 12044 | Multiplication of two rati... |
| divdivdiv 12045 | Division of two ratios. T... |
| divcan5 12046 | Cancellation of common fac... |
| divmul13 12047 | Swap the denominators in t... |
| divmul24 12048 | Swap the numerators in the... |
| divmuleq 12049 | Cross-multiply in an equal... |
| recdiv 12050 | The reciprocal of a ratio.... |
| divcan6 12051 | Cancellation of inverted f... |
| divdiv32 12052 | Swap denominators in a div... |
| divcan7 12053 | Cancel equal divisors in a... |
| dmdcan 12054 | Cancellation law for divis... |
| divdiv1 12055 | Division into a fraction. ... |
| divdiv2 12056 | Division by a fraction. (... |
| recdiv2 12057 | Division into a reciprocal... |
| ddcan 12058 | Cancellation in a double d... |
| divadddiv 12059 | Addition of two ratios. T... |
| divsubdiv 12060 | Subtraction of two ratios.... |
| conjmul 12061 | Two numbers whose reciproc... |
| rereccl 12062 | Closure law for reciprocal... |
| redivcl 12063 | Closure law for division o... |
| eqneg 12064 | A number equal to its nega... |
| eqnegd 12065 | A complex number equals it... |
| eqnegad 12066 | If a complex number equals... |
| div2neg 12067 | Quotient of two negatives.... |
| divneg2 12068 | Move negative sign inside ... |
| recclzi 12069 | Closure law for reciprocal... |
| recne0zi 12070 | The reciprocal of a nonzer... |
| recidzi 12071 | Multiplication of a number... |
| div1i 12072 | A number divided by 1 is i... |
| eqnegi 12073 | A number equal to its nega... |
| reccli 12074 | Closure law for reciprocal... |
| recidi 12075 | Multiplication of a number... |
| recreci 12076 | A number is equal to the r... |
| dividi 12077 | A number divided by itself... |
| div0i 12078 | Division into zero is zero... |
| divclzi 12079 | Closure law for division. ... |
| divcan1zi 12080 | A cancellation law for div... |
| divcan2zi 12081 | A cancellation law for div... |
| divreczi 12082 | Relationship between divis... |
| divcan3zi 12083 | A cancellation law for div... |
| divcan4zi 12084 | A cancellation law for div... |
| rec11i 12085 | Reciprocal is one-to-one. ... |
| divcli 12086 | Closure law for division. ... |
| divcan2i 12087 | A cancellation law for div... |
| divcan1i 12088 | A cancellation law for div... |
| divreci 12089 | Relationship between divis... |
| divcan3i 12090 | A cancellation law for div... |
| divcan4i 12091 | A cancellation law for div... |
| divne0i 12092 | The ratio of nonzero numbe... |
| rec11ii 12093 | Reciprocal is one-to-one. ... |
| divasszi 12094 | An associative law for div... |
| divmulzi 12095 | Relationship between divis... |
| divdirzi 12096 | Distribution of division o... |
| divdiv23zi 12097 | Swap denominators in a div... |
| divmuli 12098 | Relationship between divis... |
| divdiv32i 12099 | Swap denominators in a div... |
| divassi 12100 | An associative law for div... |
| divdiri 12101 | Distribution of division o... |
| div23i 12102 | A commutative/associative ... |
| div11i 12103 | One-to-one relationship fo... |
| divmuldivi 12104 | Multiplication of two rati... |
| divmul13i 12105 | Swap denominators of two r... |
| divadddivi 12106 | Addition of two ratios. T... |
| divdivdivi 12107 | Division of two ratios. T... |
| rerecclzi 12108 | Closure law for reciprocal... |
| rereccli 12109 | Closure law for reciprocal... |
| redivclzi 12110 | Closure law for division o... |
| redivcli 12111 | Closure law for division o... |
| div1d 12112 | A number divided by 1 is i... |
| reccld 12113 | Closure law for reciprocal... |
| recne0d 12114 | The reciprocal of a nonzer... |
| recidd 12115 | Multiplication of a number... |
| recid2d 12116 | Multiplication of a number... |
| recrecd 12117 | A number is equal to the r... |
| dividd 12118 | A number divided by itself... |
| div0d 12119 | Division into zero is zero... |
| divcld 12120 | Closure law for division. ... |
| divcan1d 12121 | A cancellation law for div... |
| divcan2d 12122 | A cancellation law for div... |
| divrecd 12123 | Relationship between divis... |
| divrec2d 12124 | Relationship between divis... |
| divcan3d 12125 | A cancellation law for div... |
| divcan4d 12126 | A cancellation law for div... |
| diveq0d 12127 | A ratio is zero iff the nu... |
| diveq1d 12128 | Equality in terms of unit ... |
| diveq1ad 12129 | The quotient of two comple... |
| diveq0ad 12130 | A fraction of complex numb... |
| divne1d 12131 | If two complex numbers are... |
| divne0bd 12132 | A ratio is zero iff the nu... |
| divnegd 12133 | Move negative sign inside ... |
| divneg2d 12134 | Move negative sign inside ... |
| div2negd 12135 | Quotient of two negatives.... |
| divne0d 12136 | The ratio of nonzero numbe... |
| recdivd 12137 | The reciprocal of a ratio.... |
| recdiv2d 12138 | Division into a reciprocal... |
| divcan6d 12139 | Cancellation of inverted f... |
| ddcand 12140 | Cancellation in a double d... |
| rec11d 12141 | Reciprocal is one-to-one. ... |
| divmuld 12142 | Relationship between divis... |
| div32d 12143 | A commutative/associative ... |
| div13d 12144 | A commutative/associative ... |
| divdiv32d 12145 | Swap denominators in a div... |
| divcan5d 12146 | Cancellation of common fac... |
| divcan5rd 12147 | Cancellation of common fac... |
| divcan7d 12148 | Cancel equal divisors in a... |
| dmdcand 12149 | Cancellation law for divis... |
| dmdcan2d 12150 | Cancellation law for divis... |
| divdiv1d 12151 | Division into a fraction. ... |
| divdiv2d 12152 | Division by a fraction. (... |
| divmul2d 12153 | Relationship between divis... |
| divmul3d 12154 | Relationship between divis... |
| divassd 12155 | An associative law for div... |
| div12d 12156 | A commutative/associative ... |
| div23d 12157 | A commutative/associative ... |
| divdird 12158 | Distribution of division o... |
| divsubdird 12159 | Distribution of division o... |
| div11d 12160 | One-to-one relationship fo... |
| divmuldivd 12161 | Multiplication of two rati... |
| divmul13d 12162 | Swap denominators of two r... |
| divmul24d 12163 | Swap the numerators in the... |
| divadddivd 12164 | Addition of two ratios. T... |
| divsubdivd 12165 | Subtraction of two ratios.... |
| divmuleqd 12166 | Cross-multiply in an equal... |
| divdivdivd 12167 | Division of two ratios. T... |
| diveq1bd 12168 | If two complex numbers are... |
| div2sub 12169 | Swap the order of subtract... |
| div2subd 12170 | Swap subtrahend and minuen... |
| rereccld 12171 | Closure law for reciprocal... |
| redivcld 12172 | Closure law for division o... |
| subrecd 12173 | Subtraction of reciprocals... |
| subrec 12174 | Subtraction of reciprocals... |
| subreci 12175 | Subtraction of reciprocals... |
| mvllmuld 12176 | Move the left term in a pr... |
| mvllmuli 12177 | Move the left term in a pr... |
| ldiv 12178 | Left-division. (Contribut... |
| rdiv 12179 | Right-division. (Contribu... |
| mdiv 12180 | A division law. (Contribu... |
| lineq 12181 | Solution of a (scalar) lin... |
| elimgt0 12182 | Hypothesis for weak deduct... |
| elimge0 12183 | Hypothesis for weak deduct... |
| ltp1 12184 | A number is less than itse... |
| lep1 12185 | A number is less than or e... |
| ltm1 12186 | A number minus 1 is less t... |
| lem1 12187 | A number minus 1 is less t... |
| letrp1 12188 | A transitive property of '... |
| p1le 12189 | A transitive property of p... |
| recgt0 12190 | The reciprocal of a positi... |
| prodgt0 12191 | Infer that a multiplicand ... |
| prodgt02 12192 | Infer that a multiplier is... |
| ltmul1a 12193 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12194 | Multiplication of both sid... |
| ltmul2 12195 | Multiplication of both sid... |
| lemul1 12196 | Multiplication of both sid... |
| lemul2 12197 | Multiplication of both sid... |
| lemul1a 12198 | Multiplication of both sid... |
| lemul2a 12199 | Multiplication of both sid... |
| ltmul12a 12200 | Comparison of product of t... |
| lemul12b 12201 | Comparison of product of t... |
| lemul12a 12202 | Comparison of product of t... |
| ltmulgt11 12203 | Multiplication by a number... |
| ltmulgt12 12204 | Multiplication by a number... |
| mulgt1 12205 | The product of two numbers... |
| lemulge11 12206 | Multiplication by a number... |
| lemulge12 12207 | Multiplication by a number... |
| ltdiv1 12208 | Division of both sides of ... |
| lediv1 12209 | Division of both sides of ... |
| gt0div 12210 | Division of a positive num... |
| ge0div 12211 | Division of a nonnegative ... |
| divgt0 12212 | The ratio of two positive ... |
| divge0 12213 | The ratio of nonnegative a... |
| mulge0b 12214 | A condition for multiplica... |
| mulle0b 12215 | A condition for multiplica... |
| mulsuble0b 12216 | A condition for multiplica... |
| ltmuldiv 12217 | 'Less than' relationship b... |
| ltmuldiv2 12218 | 'Less than' relationship b... |
| ltdivmul 12219 | 'Less than' relationship b... |
| ledivmul 12220 | 'Less than or equal to' re... |
| ltdivmul2 12221 | 'Less than' relationship b... |
| lt2mul2div 12222 | 'Less than' relationship b... |
| ledivmul2 12223 | 'Less than or equal to' re... |
| lemuldiv 12224 | 'Less than or equal' relat... |
| lemuldiv2 12225 | 'Less than or equal' relat... |
| ltrec 12226 | The reciprocal of both sid... |
| lerec 12227 | The reciprocal of both sid... |
| lt2msq1 12228 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12229 | Two nonnegative numbers co... |
| ltdiv2 12230 | Division of a positive num... |
| ltrec1 12231 | Reciprocal swap in a 'less... |
| lerec2 12232 | Reciprocal swap in a 'less... |
| ledivdiv 12233 | Invert ratios of positive ... |
| lediv2 12234 | Division of a positive num... |
| ltdiv23 12235 | Swap denominator with othe... |
| lediv23 12236 | Swap denominator with othe... |
| lediv12a 12237 | Comparison of ratio of two... |
| lediv2a 12238 | Division of both sides of ... |
| reclt1 12239 | The reciprocal of a positi... |
| recgt1 12240 | The reciprocal of a positi... |
| recgt1i 12241 | The reciprocal of a number... |
| recp1lt1 12242 | Construct a number less th... |
| recreclt 12243 | Given a positive number ` ... |
| le2msq 12244 | The square function on non... |
| msq11 12245 | The square of a nonnegativ... |
| ledivp1 12246 | "Less than or equal to" an... |
| squeeze0 12247 | If a nonnegative number is... |
| ltp1i 12248 | A number is less than itse... |
| recgt0i 12249 | The reciprocal of a positi... |
| recgt0ii 12250 | The reciprocal of a positi... |
| prodgt0i 12251 | Infer that a multiplicand ... |
| divgt0i 12252 | The ratio of two positive ... |
| divge0i 12253 | The ratio of nonnegative a... |
| ltreci 12254 | The reciprocal of both sid... |
| lereci 12255 | The reciprocal of both sid... |
| lt2msqi 12256 | The square function on non... |
| le2msqi 12257 | The square function on non... |
| msq11i 12258 | The square of a nonnegativ... |
| divgt0i2i 12259 | The ratio of two positive ... |
| ltrecii 12260 | The reciprocal of both sid... |
| divgt0ii 12261 | The ratio of two positive ... |
| ltmul1i 12262 | Multiplication of both sid... |
| ltdiv1i 12263 | Division of both sides of ... |
| ltmuldivi 12264 | 'Less than' relationship b... |
| ltmul2i 12265 | Multiplication of both sid... |
| lemul1i 12266 | Multiplication of both sid... |
| lemul2i 12267 | Multiplication of both sid... |
| ltdiv23i 12268 | Swap denominator with othe... |
| ledivp1i 12269 | "Less than or equal to" an... |
| ltdivp1i 12270 | Less-than and division rel... |
| ltdiv23ii 12271 | Swap denominator with othe... |
| ltmul1ii 12272 | Multiplication of both sid... |
| ltdiv1ii 12273 | Division of both sides of ... |
| ltp1d 12274 | A number is less than itse... |
| lep1d 12275 | A number is less than or e... |
| ltm1d 12276 | A number minus 1 is less t... |
| lem1d 12277 | A number minus 1 is less t... |
| recgt0d 12278 | The reciprocal of a positi... |
| divgt0d 12279 | The ratio of two positive ... |
| mulgt1d 12280 | The product of two numbers... |
| lemulge11d 12281 | Multiplication by a number... |
| lemulge12d 12282 | Multiplication by a number... |
| lemul1ad 12283 | Multiplication of both sid... |
| lemul2ad 12284 | Multiplication of both sid... |
| ltmul12ad 12285 | Comparison of product of t... |
| lemul12ad 12286 | Comparison of product of t... |
| lemul12bd 12287 | Comparison of product of t... |
| fimaxre 12288 | A finite set of real numbe... |
| fimaxre2 12289 | A nonempty finite set of r... |
| fimaxre3 12290 | A nonempty finite set of r... |
| fiminre 12291 | A nonempty finite set of r... |
| fiminre2 12292 | A nonempty finite set of r... |
| negfi 12293 | The negation of a finite s... |
| lbreu 12294 | If a set of reals contains... |
| lbcl 12295 | If a set of reals contains... |
| lble 12296 | If a set of reals contains... |
| lbinf 12297 | If a set of reals contains... |
| lbinfcl 12298 | If a set of reals contains... |
| lbinfle 12299 | If a set of reals contains... |
| sup2 12300 | A nonempty, bounded-above ... |
| sup3 12301 | A version of the completen... |
| infm3lem 12302 | Lemma for ~ infm3 . (Cont... |
| infm3 12303 | The completeness axiom for... |
| suprcl 12304 | Closure of supremum of a n... |
| suprub 12305 | A member of a nonempty bou... |
| suprubd 12306 | Natural deduction form of ... |
| suprcld 12307 | Natural deduction form of ... |
| suprlub 12308 | The supremum of a nonempty... |
| suprnub 12309 | An upper bound is not less... |
| suprleub 12310 | The supremum of a nonempty... |
| supaddc 12311 | The supremum function dist... |
| supadd 12312 | The supremum function dist... |
| supmul1 12313 | The supremum function dist... |
| supmullem1 12314 | Lemma for ~ supmul . (Con... |
| supmullem2 12315 | Lemma for ~ supmul . (Con... |
| supmul 12316 | The supremum function dist... |
| sup3ii 12317 | A version of the completen... |
| suprclii 12318 | Closure of supremum of a n... |
| suprubii 12319 | A member of a nonempty bou... |
| suprlubii 12320 | The supremum of a nonempty... |
| suprnubii 12321 | An upper bound is not less... |
| suprleubii 12322 | The supremum of a nonempty... |
| riotaneg 12323 | The negative of the unique... |
| negiso 12324 | Negation is an order anti-... |
| dfinfre 12325 | The infimum of a set of re... |
| infrecl 12326 | Closure of infimum of a no... |
| infrenegsup 12327 | The infimum of a set of re... |
| infregelb 12328 | Any lower bound of a nonem... |
| infrelb 12329 | If a nonempty set of real ... |
| infrefilb 12330 | The infimum of a finite se... |
| supfirege 12331 | The supremum of a finite s... |
| neg1cn 12332 | -1 is a complex number. (... |
| neg1rr 12333 | -1 is a real number. (Con... |
| neg1ne0 12334 | -1 is nonzero. (Contribut... |
| neg1lt0 12335 | -1 is less than 0. (Contr... |
| negneg1e1 12336 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12337 | The imaginary unit ` _i ` ... |
| rimul 12338 | A real number times the im... |
| cru 12339 | The representation of comp... |
| crne0 12340 | The real representation of... |
| creur 12341 | The real part of a complex... |
| creui 12342 | The imaginary part of a co... |
| cju 12343 | The complex conjugate of a... |
| ofsubeq0 12344 | Function analogue of ~ sub... |
| ofnegsub 12345 | Function analogue of ~ neg... |
| ofsubge0 12346 | Function analogue of ~ sub... |
| indv 12349 | Value of the indicator fun... |
| indval 12350 | Value of the indicator fun... |
| indval0 12351 | The indicator function gen... |
| indval2 12352 | Alternate value of the ind... |
| indf 12353 | An indicator function as a... |
| indfval 12354 | Value of the indicator fun... |
| fvindre 12355 | The range of the indicator... |
| ind1 12356 | Value of the indicator fun... |
| ind0 12357 | Value of the indicator fun... |
| ind1a 12358 | Value of the indicator fun... |
| indconst0 12359 | Indicator of the empty set... |
| indconst1 12360 | Indicator of the whole set... |
| indpi1 12361 | Preimage of the singleton ... |
| nnexALT 12364 | Alternate proof of ~ nnex ... |
| peano5nni 12365 | Peano's inductive postulat... |
| nnssre 12366 | The positive integers are ... |
| nnsscn 12367 | The positive integers are ... |
| nnex 12368 | The set of positive intege... |
| nnre 12369 | A positive integer is a re... |
| nncn 12370 | A positive integer is a co... |
| nnrei 12371 | A positive integer is a re... |
| nncni 12372 | A positive integer is a co... |
| 1nn 12373 | Peano postulate: 1 is a po... |
| peano2nn 12374 | Peano postulate: a success... |
| dfnn2 12375 | Alternate definition of th... |
| dfnn3 12376 | Alternate definition of th... |
| nnred 12377 | A positive integer is a re... |
| nncnd 12378 | A positive integer is a co... |
| peano2nnd 12379 | Peano postulate: a success... |
| nnind 12380 | Principle of Mathematical ... |
| nnindALT 12381 | Principle of Mathematical ... |
| nnindd 12382 | Principle of Mathematical ... |
| nn1m1nn 12383 | Every positive integer is ... |
| nn1suc 12384 | If a statement holds for 1... |
| nnaddcl 12385 | Closure of addition of pos... |
| nnmulcl 12386 | Closure of multiplication ... |
| nnmulcli 12387 | Closure of multiplication ... |
| nnadd1com 12388 | Addition with 1 is commuta... |
| nnaddcom 12389 | Addition is commutative fo... |
| nnaddcomli 12390 | Version of ~ addcomli for ... |
| nnmtmip 12391 | "Minus times minus is plus... |
| nn2ge 12392 | There exists a positive in... |
| nnge1 12393 | A positive integer is one ... |
| nngt1ne1 12394 | A positive integer is grea... |
| nnle1eq1 12395 | A positive integer is less... |
| nngt0 12396 | A positive integer is posi... |
| nnnlt1 12397 | A positive integer is not ... |
| nnnle0 12398 | A positive integer is not ... |
| nnne0 12399 | A positive integer is nonz... |
| nnneneg 12400 | No positive integer is equ... |
| 0nnn 12401 | Zero is not a positive int... |
| 0nnnALT 12402 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12403 | Alternate version of ~ nnn... |
| nngt0i 12404 | A positive integer is posi... |
| nnne0i 12405 | A positive integer is nonz... |
| nndivre 12406 | The quotient of a real and... |
| nnrecre 12407 | The reciprocal of a positi... |
| nnrecgt0 12408 | The reciprocal of a positi... |
| nnsub 12409 | Subtraction of positive in... |
| nnsubi 12410 | Subtraction of positive in... |
| nndiv 12411 | Two ways to express " ` A ... |
| nndivtr 12412 | Transitive property of div... |
| nnge1d 12413 | A positive integer is one ... |
| nngt0d 12414 | A positive integer is posi... |
| nnne0d 12415 | A positive integer is nonz... |
| nnrecred 12416 | The reciprocal of a positi... |
| nnaddcld 12417 | Closure of addition of pos... |
| nnmulcld 12418 | Closure of multiplication ... |
| nndivred 12419 | A positive integer is one ... |
| 1t1e1ALT 12420 | Alternate proof of ~ 1t1e1... |
| nnadddir 12421 | Right-distributivity for n... |
| nnmul1com 12422 | Multiplication with 1 is c... |
| nnmulcom 12423 | Multiplication is commutat... |
| 1eltp012 12440 | 1 is an element of ` { 0 ,... |
| 0ne1 12441 | Zero is different from one... |
| 1m1e0 12442 | One minus one equals zero.... |
| 2nn 12443 | 2 is a positive integer. ... |
| 2re 12444 | The number 2 is real. (Co... |
| 2cn 12445 | The number 2 is a complex ... |
| 2cnALT 12446 | Alternate proof of ~ 2cn .... |
| 2ex 12447 | The number 2 is a set. (C... |
| 2cnd 12448 | The number 2 is a complex ... |
| 3nn 12449 | 3 is a positive integer. ... |
| 3re 12450 | The number 3 is real. (Co... |
| 3cn 12451 | The number 3 is a complex ... |
| 3ex 12452 | The number 3 is a set. (C... |
| 4nn 12453 | 4 is a positive integer. ... |
| 4re 12454 | The number 4 is real. (Co... |
| 4cn 12455 | The number 4 is a complex ... |
| 5nn 12456 | 5 is a positive integer. ... |
| 5re 12457 | The number 5 is real. (Co... |
| 5cn 12458 | The number 5 is a complex ... |
| 6nn 12459 | 6 is a positive integer. ... |
| 6re 12460 | The number 6 is real. (Co... |
| 6cn 12461 | The number 6 is a complex ... |
| 7nn 12462 | 7 is a positive integer. ... |
| 7re 12463 | The number 7 is real. (Co... |
| 7cn 12464 | The number 7 is a complex ... |
| 8nn 12465 | 8 is a positive integer. ... |
| 8re 12466 | The number 8 is real. (Co... |
| 8cn 12467 | The number 8 is a complex ... |
| 9nn 12468 | 9 is a positive integer. ... |
| 9re 12469 | The number 9 is real. (Co... |
| 9cn 12470 | The number 9 is a complex ... |
| 0le0 12471 | Zero is nonnegative. (Con... |
| 0le2 12472 | The number 0 is less than ... |
| 0le2OLD 12473 | Obsolete version of ~ 0le2... |
| 2pos 12474 | The number 2 is positive. ... |
| 2posOLD 12475 | Obsolete version of ~ 2pos... |
| 2ne0 12476 | The number 2 is nonzero. ... |
| 2thalfe1 12477 | 2 times one half equals 1.... |
| 3pos 12478 | The number 3 is positive. ... |
| 3ne0 12479 | The number 3 is nonzero. ... |
| 4pos 12480 | The number 4 is positive. ... |
| 4ne0 12481 | The number 4 is nonzero. ... |
| 5pos 12482 | The number 5 is positive. ... |
| 6pos 12483 | The number 6 is positive. ... |
| 7pos 12484 | The number 7 is positive. ... |
| 8pos 12485 | The number 8 is positive. ... |
| 9pos 12486 | The number 9 is positive. ... |
| 1pneg1e0 12487 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12488 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12489 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12490 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12491 | Function value at ` N + 1 ... |
| 1p0e1 12492 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12493 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12494 | 2 - 1 = 1. The result is ... |
| 2m1e1OLD 12495 | Obsolete version of ~ 2m1e... |
| 1e2m1 12496 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12497 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12498 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12499 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12500 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12501 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12502 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12503 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12504 | Two plus two equals four. ... |
| 2times 12505 | Two times a number. (Cont... |
| times2 12506 | A number times 2. (Contri... |
| 2timesi 12507 | Two times a number. (Cont... |
| times2i 12508 | A number times 2. (Contri... |
| 2txmxeqx 12509 | Two times a complex number... |
| 2div2e1 12510 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12511 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12512 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12513 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12514 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12515 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12516 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12517 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12518 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12519 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12520 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12521 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12522 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12523 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12524 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12525 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12526 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12527 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12528 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12529 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12530 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12531 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12532 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12533 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12534 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12535 | 3 times 2 equals 6. (Cont... |
| 2t3e6 12536 | 2 times 3 equals 6. (Cont... |
| 3t3e9 12537 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12538 | 4 times 2 equals 8. (Cont... |
| 2t4e8 12539 | 2 times 4 equals 8. (Cont... |
| 2t0e0 12540 | 2 times 0 equals 0. (Cont... |
| 4div2e2 12541 | One half of four is two. ... |
| 1lt2 12542 | 1 is less than 2. (Contri... |
| 2lt3 12543 | 2 is less than 3. (Contri... |
| 2le3 12544 | 2 is less than or equal to... |
| 1lt3 12545 | 1 is less than 3. (Contri... |
| 3lt4 12546 | 3 is less than 4. (Contri... |
| 2lt4 12547 | 2 is less than 4. (Contri... |
| 1lt4 12548 | 1 is less than 4. (Contri... |
| 4lt5 12549 | 4 is less than 5. (Contri... |
| 3lt5 12550 | 3 is less than 5. (Contri... |
| 2lt5 12551 | 2 is less than 5. (Contri... |
| 1lt5 12552 | 1 is less than 5. (Contri... |
| 5lt6 12553 | 5 is less than 6. (Contri... |
| 4lt6 12554 | 4 is less than 6. (Contri... |
| 3lt6 12555 | 3 is less than 6. (Contri... |
| 2lt6 12556 | 2 is less than 6. (Contri... |
| 1lt6 12557 | 1 is less than 6. (Contri... |
| 6lt7 12558 | 6 is less than 7. (Contri... |
| 5lt7 12559 | 5 is less than 7. (Contri... |
| 4lt7 12560 | 4 is less than 7. (Contri... |
| 3lt7 12561 | 3 is less than 7. (Contri... |
| 2lt7 12562 | 2 is less than 7. (Contri... |
| 1lt7 12563 | 1 is less than 7. (Contri... |
| 7lt8 12564 | 7 is less than 8. (Contri... |
| 6lt8 12565 | 6 is less than 8. (Contri... |
| 5lt8 12566 | 5 is less than 8. (Contri... |
| 4lt8 12567 | 4 is less than 8. (Contri... |
| 3lt8 12568 | 3 is less than 8. (Contri... |
| 2lt8 12569 | 2 is less than 8. (Contri... |
| 1lt8 12570 | 1 is less than 8. (Contri... |
| 8lt9 12571 | 8 is less than 9. (Contri... |
| 7lt9 12572 | 7 is less than 9. (Contri... |
| 6lt9 12573 | 6 is less than 9. (Contri... |
| 5lt9 12574 | 5 is less than 9. (Contri... |
| 4lt9 12575 | 4 is less than 9. (Contri... |
| 3lt9 12576 | 3 is less than 9. (Contri... |
| 2lt9 12577 | 2 is less than 9. (Contri... |
| 1lt9 12578 | 1 is less than 9. (Contri... |
| 0ne2 12579 | 0 is not equal to 2. (Con... |
| 1ne2 12580 | 1 is not equal to 2. (Con... |
| 1le2 12581 | 1 is less than or equal to... |
| 2cnne0 12582 | 2 is a nonzero complex num... |
| 2rene0 12583 | 2 is a nonzero real number... |
| 1le3 12584 | 1 is less than or equal to... |
| neg1mulneg1e1 12585 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12586 | One-half is real. (Contri... |
| halfcn 12587 | One-half is a complex numb... |
| halfgt0 12588 | One-half is greater than z... |
| halfge0 12589 | One-half is not negative. ... |
| halflt1 12590 | One-half is less than one.... |
| 2halves 12591 | Two halves make a whole. ... |
| 1mhlfehlf 12592 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12593 | An eighth of four thirds i... |
| halfthird 12594 | Half minus a third. (Cont... |
| halfpm6th 12595 | One half plus or minus one... |
| it0e0 12596 | i times 0 equals 0. (Cont... |
| 2mulicn 12597 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12598 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12599 | Closure of half of a numbe... |
| rehalfcl 12600 | Real closure of half. (Co... |
| half0 12601 | Half of a number is zero i... |
| halfpos2 12602 | A number is positive iff i... |
| halfpos 12603 | A positive number is great... |
| halfnneg2 12604 | A number is nonnegative if... |
| halfaddsubcl 12605 | Closure of half-sum and ha... |
| halfaddsub 12606 | Sum and difference of half... |
| subhalfhalf 12607 | Subtracting the half of a ... |
| lt2halves 12608 | A sum is less than the who... |
| addltmul 12609 | Sum is less than product f... |
| nominpos 12610 | There is no smallest posit... |
| avglt1 12611 | Ordering property for aver... |
| avglt2 12612 | Ordering property for aver... |
| avgle1 12613 | Ordering property for aver... |
| avgle2 12614 | Ordering property for aver... |
| avgle 12615 | The average of two numbers... |
| 2timesd 12616 | Two times a number. (Cont... |
| times2d 12617 | A number times 2. (Contri... |
| halfcld 12618 | Closure of half of a numbe... |
| 2halvesd 12619 | Two halves make a whole. ... |
| rehalfcld 12620 | Real closure of half. (Co... |
| lt2halvesd 12621 | A sum is less than the who... |
| rehalfcli 12622 | Half a real number is real... |
| lt2addmuld 12623 | If two real numbers are le... |
| add1p1 12624 | Adding two times 1 to a nu... |
| sub1m1 12625 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12626 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12627 | A complex number increased... |
| div4p1lem1div2 12628 | An integer greater than 5,... |
| nnunb 12629 | The set of positive intege... |
| arch 12630 | Archimedean property of re... |
| nnrecl 12631 | There exists a positive in... |
| bndndx 12632 | A bounded real sequence ` ... |
| elnn0 12635 | Nonnegative integers expre... |
| nnssnn0 12636 | Positive naturals are a su... |
| nn0ssre 12637 | Nonnegative integers are a... |
| nn0sscn 12638 | Nonnegative integers are a... |
| nn0ex 12639 | The set of nonnegative int... |
| nnnn0 12640 | A positive integer is a no... |
| nnnn0i 12641 | A positive integer is a no... |
| nn0re 12642 | A nonnegative integer is a... |
| nn0cn 12643 | A nonnegative integer is a... |
| nn0rei 12644 | A nonnegative integer is a... |
| nn0cni 12645 | A nonnegative integer is a... |
| dfn2 12646 | The set of positive intege... |
| elnnne0 12647 | The positive integer prope... |
| 0nn0 12648 | 0 is a nonnegative integer... |
| 1nn0 12649 | 1 is a nonnegative integer... |
| 2nn0 12650 | 2 is a nonnegative integer... |
| 3nn0 12651 | 3 is a nonnegative integer... |
| 4nn0 12652 | 4 is a nonnegative integer... |
| 5nn0 12653 | 5 is a nonnegative integer... |
| 6nn0 12654 | 6 is a nonnegative integer... |
| 7nn0 12655 | 7 is a nonnegative integer... |
| 8nn0 12656 | 8 is a nonnegative integer... |
| 9nn0 12657 | 9 is a nonnegative integer... |
| nn0ge0 12658 | A nonnegative integer is g... |
| nn0nlt0 12659 | A nonnegative integer is n... |
| nn0ge0i 12660 | Nonnegative integers are n... |
| nn0le0eq0 12661 | A nonnegative integer is l... |
| nn0p1gt0 12662 | A nonnegative integer incr... |
| nnnn0addcl 12663 | A positive integer plus a ... |
| nn0nnaddcl 12664 | A nonnegative integer plus... |
| 0mnnnnn0 12665 | The result of subtracting ... |
| un0addcl 12666 | If ` S ` is closed under a... |
| un0mulcl 12667 | If ` S ` is closed under m... |
| nn0addcl 12668 | Closure of addition of non... |
| nn0mulcl 12669 | Closure of multiplication ... |
| nn0addcli 12670 | Closure of addition of non... |
| nn0mulcli 12671 | Closure of multiplication ... |
| nn0p1nn 12672 | A nonnegative integer plus... |
| peano2nn0 12673 | Second Peano postulate for... |
| nnm1nn0 12674 | A positive integer minus 1... |
| elnn0nn 12675 | The nonnegative integer pr... |
| elnnnn0 12676 | The positive integer prope... |
| elnnnn0b 12677 | The positive integer prope... |
| elnnnn0c 12678 | The positive integer prope... |
| nn0addge1 12679 | A number is less than or e... |
| nn0addge2 12680 | A number is less than or e... |
| nn0addge1i 12681 | A number is less than or e... |
| nn0addge2i 12682 | A number is less than or e... |
| nn0sub 12683 | Subtraction of nonnegative... |
| ltsubnn0 12684 | Subtracting a nonnegative ... |
| nn0negleid 12685 | A nonnegative integer is g... |
| difgtsumgt 12686 | If the difference of a rea... |
| nn0le2x 12687 | A nonnegative integer is l... |
| nn0le2xi 12688 | A nonnegative integer is l... |
| nn0lele2xi 12689 | 'Less than or equal to' im... |
| fcdmnn0supp 12690 | Two ways to write the supp... |
| fcdmnn0fsupp 12691 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12692 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12693 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12694 | A positive integer is a no... |
| nn0red 12695 | A nonnegative integer is a... |
| nn0cnd 12696 | A nonnegative integer is a... |
| nn0ge0d 12697 | A nonnegative integer is g... |
| nn0addcld 12698 | Closure of addition of non... |
| nn0mulcld 12699 | Closure of multiplication ... |
| nn0readdcl 12700 | Closure law for addition o... |
| nn0n0n1ge2 12701 | A nonnegative integer whic... |
| nn0n0n1ge2b 12702 | A nonnegative integer is n... |
| nn0ge2m1nn 12703 | If a nonnegative integer i... |
| nn0ge2m1nn0 12704 | If a nonnegative integer i... |
| nn0nndivcl 12705 | Closure law for dividing o... |
| elxnn0 12708 | An extended nonnegative in... |
| nn0ssxnn0 12709 | The standard nonnegative i... |
| nn0xnn0 12710 | A standard nonnegative int... |
| xnn0xr 12711 | An extended nonnegative in... |
| 0xnn0 12712 | Zero is an extended nonneg... |
| pnf0xnn0 12713 | Positive infinity is an ex... |
| nn0nepnf 12714 | No standard nonnegative in... |
| nn0xnn0d 12715 | A standard nonnegative int... |
| nn0nepnfd 12716 | No standard nonnegative in... |
| xnn0nemnf 12717 | No extended nonnegative in... |
| xnn0xrnemnf 12718 | The extended nonnegative i... |
| xnn0nnn0pnf 12719 | An extended nonnegative in... |
| elz 12722 | Membership in the set of i... |
| nnnegz 12723 | The negative of a positive... |
| zre 12724 | An integer is a real. (Co... |
| zcn 12725 | An integer is a complex nu... |
| zrei 12726 | An integer is a real numbe... |
| zssre 12727 | The integers are a subset ... |
| zsscn 12728 | The integers are a subset ... |
| zex 12729 | The set of integers exists... |
| elnnz 12730 | Positive integer property ... |
| 0z 12731 | Zero is an integer. (Cont... |
| 0zd 12732 | Zero is an integer, deduct... |
| elnn0z 12733 | Nonnegative integer proper... |
| elznn0nn 12734 | Integer property expressed... |
| elznn0 12735 | Integer property expressed... |
| elznn 12736 | Integer property expressed... |
| zle0orge1 12737 | There is no integer in the... |
| elz2 12738 | Membership in the set of i... |
| dfz2 12739 | Alternative definition of ... |
| zexALT 12740 | Alternate proof of ~ zex .... |
| nnz 12741 | A positive integer is an i... |
| nnssz 12742 | Positive integers are a su... |
| nn0ssz 12743 | Nonnegative integers are a... |
| nn0z 12744 | A nonnegative integer is a... |
| nn0zd 12745 | A nonnegative integer is a... |
| nnzd 12746 | A positive integer is an i... |
| nnzi 12747 | A positive integer is an i... |
| nn0zi 12748 | A nonnegative integer is a... |
| elnnz1 12749 | Positive integer property ... |
| znnnlt1 12750 | An integer is not a positi... |
| nnzrab 12751 | Positive integers expresse... |
| nn0zrab 12752 | Nonnegative integers expre... |
| 1z 12753 | One is an integer. (Contr... |
| 1zzd 12754 | One is an integer, deducti... |
| 2z 12755 | 2 is an integer. (Contrib... |
| 3z 12756 | 3 is an integer. (Contrib... |
| 4z 12757 | 4 is an integer. (Contrib... |
| znegcl 12758 | Closure law for negative i... |
| neg1z 12759 | -1 is an integer. (Contri... |
| znegclb 12760 | A complex number is an int... |
| nn0negz 12761 | The negative of a nonnegat... |
| nn0negzi 12762 | The negative of a nonnegat... |
| zaddcl 12763 | Closure of addition of int... |
| peano2z 12764 | Second Peano postulate gen... |
| zsubcl 12765 | Closure of subtraction of ... |
| peano2zm 12766 | "Reverse" second Peano pos... |
| zletr 12767 | Transitive law of ordering... |
| zrevaddcl 12768 | Reverse closure law for ad... |
| znnsub 12769 | The positive difference of... |
| znn0sub 12770 | The nonnegative difference... |
| nzadd 12771 | The sum of a real number n... |
| zmulcl 12772 | Closure of multiplication ... |
| zltp1le 12773 | Integer ordering relation.... |
| zleltp1 12774 | Integer ordering relation.... |
| zlem1lt 12775 | Integer ordering relation.... |
| zltlem1 12776 | Integer ordering relation.... |
| zltlem1d 12777 | Integer ordering relation,... |
| zltp1led 12778 | Integer ordering relation,... |
| zgt0ge1 12779 | An integer greater than ` ... |
| 0nn0m1nnn0 12780 | A number is zero if and on... |
| nnleltp1 12781 | Positive integer ordering ... |
| nnltp1le 12782 | Positive integer ordering ... |
| nnaddm1cl 12783 | Closure of addition of pos... |
| nn0ltp1le 12784 | Nonnegative integer orderi... |
| nn0leltp1 12785 | Nonnegative integer orderi... |
| nn0ltlem1 12786 | Nonnegative integer orderi... |
| nn0sub2 12787 | Subtraction of nonnegative... |
| nn0lt10b 12788 | A nonnegative integer less... |
| nn0lt2 12789 | A nonnegative integer less... |
| nn0le2is012 12790 | A nonnegative integer whic... |
| nn0lem1lt 12791 | Nonnegative integer orderi... |
| nnlem1lt 12792 | Positive integer ordering ... |
| nnltlem1 12793 | Positive integer ordering ... |
| nnm1ge0 12794 | A positive integer decreas... |
| nn0ge0div 12795 | Division of a nonnegative ... |
| zdiv 12796 | Two ways to express " ` M ... |
| zdivadd 12797 | Property of divisibility: ... |
| zdivmul 12798 | Property of divisibility: ... |
| zextle 12799 | An extensionality-like pro... |
| zextlt 12800 | An extensionality-like pro... |
| recnz 12801 | The reciprocal of a number... |
| btwnnz 12802 | A number between an intege... |
| gtndiv 12803 | A larger number does not d... |
| halfnz 12804 | One-half is not an integer... |
| 3halfnz 12805 | Three halves is not an int... |
| suprzcl 12806 | The supremum of a bounded-... |
| prime 12807 | Two ways to express " ` A ... |
| msqznn 12808 | The square of a nonzero in... |
| zneo 12809 | No even integer equals an ... |
| nneo 12810 | A positive integer is even... |
| nneoi 12811 | A positive integer is even... |
| zeo 12812 | An integer is even or odd.... |
| zeo2 12813 | An integer is even or odd ... |
| peano2uz2 12814 | Second Peano postulate for... |
| peano5uzi 12815 | Peano's inductive postulat... |
| peano5uzti 12816 | Peano's inductive postulat... |
| dfuzi 12817 | An expression for the uppe... |
| uzind 12818 | Induction on the upper int... |
| uzind2 12819 | Induction on the upper int... |
| uzind3 12820 | Induction on the upper int... |
| nn0ind 12821 | Principle of Mathematical ... |
| nn0indALT 12822 | Principle of Mathematical ... |
| nn0indd 12823 | Principle of Mathematical ... |
| fzind 12824 | Induction on the integers ... |
| fnn0ind 12825 | Induction on the integers ... |
| nn0ind-raph 12826 | Principle of Mathematical ... |
| zindd 12827 | Principle of Mathematical ... |
| fzindd 12828 | Induction on the integers ... |
| btwnz 12829 | Any real number can be san... |
| zred 12830 | An integer is a real numbe... |
| zcnd 12831 | An integer is a complex nu... |
| znegcld 12832 | Closure law for negative i... |
| peano2zd 12833 | Deduction from second Pean... |
| zaddcld 12834 | Closure of addition of int... |
| zsubcld 12835 | Closure of subtraction of ... |
| zmulcld 12836 | Closure of multiplication ... |
| znnn0nn 12837 | The negative of a negative... |
| zadd2cl 12838 | Increasing an integer by 2... |
| zriotaneg 12839 | The negative of the unique... |
| suprfinzcl 12840 | The supremum of a nonempty... |
| 9p1e10 12843 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12844 | Version of the definition ... |
| decex 12845 | A decimal number is a set.... |
| deceq1 12846 | Equality theorem for the d... |
| deceq2 12847 | Equality theorem for the d... |
| deceq1i 12848 | Equality theorem for the d... |
| deceq2i 12849 | Equality theorem for the d... |
| deceq12i 12850 | Equality theorem for the d... |
| numnncl 12851 | Closure for a numeral (wit... |
| num0u 12852 | Add a zero in the units pl... |
| num0h 12853 | Add a zero in the higher p... |
| numcl 12854 | Closure for a decimal inte... |
| numsuc 12855 | The successor of a decimal... |
| deccl 12856 | Closure for a numeral. (C... |
| 11nn0 12857 | 11 is a nonnegative intege... |
| 12nn0 12858 | 12 is a nonnegative intege... |
| 16nn0 12859 | 16 is a nonnegative intege... |
| 25nn0 12860 | 25 is a nonnegative intege... |
| 10nn 12861 | 10 is a positive integer. ... |
| 10pos 12862 | The number 10 is positive.... |
| 10nn0 12863 | 10 is a nonnegative intege... |
| 10re 12864 | The number 10 is real. (C... |
| decnncl 12865 | Closure for a numeral. (C... |
| 11nn 12866 | 11 is a positive integer. ... |
| dec0u 12867 | Add a zero in the units pl... |
| dec0h 12868 | Add a zero in the higher p... |
| numnncl2 12869 | Closure for a decimal inte... |
| decnncl2 12870 | Closure for a decimal inte... |
| numlt 12871 | Comparing two decimal inte... |
| numltc 12872 | Comparing two decimal inte... |
| le9lt10 12873 | A "decimal digit" (i.e. a ... |
| declt 12874 | Comparing two decimal inte... |
| decltc 12875 | Comparing two decimal inte... |
| declth 12876 | Comparing two decimal inte... |
| decsuc 12877 | The successor of a decimal... |
| 3declth 12878 | Comparing two decimal inte... |
| 3decltc 12879 | Comparing two decimal inte... |
| decle 12880 | Comparing two decimal inte... |
| decleh 12881 | Comparing two decimal inte... |
| declei 12882 | Comparing a digit to a dec... |
| numlti 12883 | Comparing a digit to a dec... |
| declti 12884 | Comparing a digit to a dec... |
| decltdi 12885 | Comparing a digit to a dec... |
| numsucc 12886 | The successor of a decimal... |
| decsucc 12887 | The successor of a decimal... |
| 1e0p1 12888 | The successor of zero. (C... |
| dec10p 12889 | Ten plus an integer. (Con... |
| numma 12890 | Perform a multiply-add of ... |
| nummac 12891 | Perform a multiply-add of ... |
| numma2c 12892 | Perform a multiply-add of ... |
| numadd 12893 | Add two decimal integers `... |
| numaddc 12894 | Add two decimal integers `... |
| nummul1c 12895 | The product of a decimal i... |
| nummul2c 12896 | The product of a decimal i... |
| decma 12897 | Perform a multiply-add of ... |
| decmac 12898 | Perform a multiply-add of ... |
| decma2c 12899 | Perform a multiply-add of ... |
| decadd 12900 | Add two numerals ` M ` and... |
| decaddc 12901 | Add two numerals ` M ` and... |
| decaddc2 12902 | Add two numerals ` M ` and... |
| decrmanc 12903 | Perform a multiply-add of ... |
| decrmac 12904 | Perform a multiply-add of ... |
| decaddm10 12905 | The sum of two multiples o... |
| decaddi 12906 | Add two numerals ` M ` and... |
| decaddci 12907 | Add two numerals ` M ` and... |
| decaddci2 12908 | Add two numerals ` M ` and... |
| decsubi 12909 | Difference between a numer... |
| decmul1 12910 | The product of a numeral w... |
| decmul1c 12911 | The product of a numeral w... |
| decmul2c 12912 | The product of a numeral w... |
| decmulnc 12913 | The product of a numeral w... |
| 11multnc 12914 | The product of 11 (as nume... |
| decmul10add 12915 | A multiplication of a numb... |
| 6p5lem 12916 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12917 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12918 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12919 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12920 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12921 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12922 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12923 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12924 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12925 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12926 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12927 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12928 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12929 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12930 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12931 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12932 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12933 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12934 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12935 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12936 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12937 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12938 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12939 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12940 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12941 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12942 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12943 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12944 | 4 times 3 equals 12. (Con... |
| 4t4e16 12945 | 4 times 4 equals 16. (Con... |
| 5t2e10 12946 | 5 times 2 equals 10. (Con... |
| 5t3e15 12947 | 5 times 3 equals 15. (Con... |
| 5t4e20 12948 | 5 times 4 equals 20. (Con... |
| 5t5e25 12949 | 5 times 5 equals 25. (Con... |
| 6t2e12 12950 | 6 times 2 equals 12. (Con... |
| 6t3e18 12951 | 6 times 3 equals 18. (Con... |
| 6t4e24 12952 | 6 times 4 equals 24. (Con... |
| 6t5e30 12953 | 6 times 5 equals 30. (Con... |
| 6t6e36 12954 | 6 times 6 equals 36. (Con... |
| 7t2e14 12955 | 7 times 2 equals 14. (Con... |
| 7t3e21 12956 | 7 times 3 equals 21. (Con... |
| 7t4e28 12957 | 7 times 4 equals 28. (Con... |
| 7t5e35 12958 | 7 times 5 equals 35. (Con... |
| 7t6e42 12959 | 7 times 6 equals 42. (Con... |
| 7t7e49 12960 | 7 times 7 equals 49. (Con... |
| 8t2e16 12961 | 8 times 2 equals 16. (Con... |
| 8t3e24 12962 | 8 times 3 equals 24. (Con... |
| 8t4e32 12963 | 8 times 4 equals 32. (Con... |
| 8t5e40 12964 | 8 times 5 equals 40. (Con... |
| 8t6e48 12965 | 8 times 6 equals 48. (Con... |
| 8t7e56 12966 | 8 times 7 equals 56. (Con... |
| 8t8e64 12967 | 8 times 8 equals 64. (Con... |
| 9t2e18 12968 | 9 times 2 equals 18. (Con... |
| 9t3e27 12969 | 9 times 3 equals 27. (Con... |
| 9t4e36 12970 | 9 times 4 equals 36. (Con... |
| 9t5e45 12971 | 9 times 5 equals 45. (Con... |
| 9t6e54 12972 | 9 times 6 equals 54. (Con... |
| 9t7e63 12973 | 9 times 7 equals 63. (Con... |
| 9t8e72 12974 | 9 times 8 equals 72. (Con... |
| 9t9e81 12975 | 9 times 9 equals 81. (Con... |
| 9t11e99 12976 | 9 times 11 equals 99. (Co... |
| 9t11e99OLD 12977 | Obsolete version of ~ 9t11... |
| 9lt10 12978 | 9 is less than 10. (Contr... |
| 8lt10 12979 | 8 is less than 10. (Contr... |
| 7lt10 12980 | 7 is less than 10. (Contr... |
| 6lt10 12981 | 6 is less than 10. (Contr... |
| 5lt10 12982 | 5 is less than 10. (Contr... |
| 4lt10 12983 | 4 is less than 10. (Contr... |
| 3lt10 12984 | 3 is less than 10. (Contr... |
| 2lt10 12985 | 2 is less than 10. (Contr... |
| 1lt10 12986 | 1 is less than 10. (Contr... |
| 1lt10OLD 12987 | Obsolete version of ~ 1lt1... |
| decbin0 12988 | Decompose base 4 into base... |
| decbin2 12989 | Decompose base 4 into base... |
| decbin3 12990 | Decompose base 4 into base... |
| 5recm6rec 12991 | One fifth minus one sixth.... |
| uzval 12994 | The value of the upper int... |
| uzf 12995 | The domain and codomain of... |
| eluz1 12996 | Membership in the upper se... |
| eluzel2 12997 | Implication of membership ... |
| eluz2 12998 | Membership in an upper set... |
| eluzmn 12999 | Membership in an earlier u... |
| eluz1i 13000 | Membership in an upper set... |
| eluzuzle 13001 | An integer in an upper set... |
| eluzelz 13002 | A member of an upper set o... |
| eluzelre 13003 | A member of an upper set o... |
| eluzelcn 13004 | A member of an upper set o... |
| eluzle 13005 | Implication of membership ... |
| eluz 13006 | Membership in an upper set... |
| uzid 13007 | Membership of the least me... |
| uzidd 13008 | Membership of the least me... |
| uzn0 13009 | The upper integers are all... |
| uztrn 13010 | Transitive law for sets of... |
| uztrn2 13011 | Transitive law for sets of... |
| uzneg 13012 | Contraposition law for upp... |
| uzssz 13013 | An upper set of integers i... |
| uzssre 13014 | An upper set of integers i... |
| uzss 13015 | Subset relationship for tw... |
| uztric 13016 | Totality of the ordering r... |
| uz11 13017 | The upper integers functio... |
| eluzp1m1 13018 | Membership in the next upp... |
| eluzp1l 13019 | Strict ordering implied by... |
| eluzp1p1 13020 | Membership in the next upp... |
| eluzadd 13021 | Membership in a later uppe... |
| eluzsub 13022 | Membership in an earlier u... |
| eluzaddi 13023 | Membership in a later uppe... |
| eluzsubi 13024 | Membership in an earlier u... |
| subeluzsub 13025 | Membership of a difference... |
| uzm1 13026 | Choices for an element of ... |
| uznn0sub 13027 | The nonnegative difference... |
| uzin 13028 | Intersection of two upper ... |
| uzp1 13029 | Choices for an element of ... |
| nn0uz 13030 | Nonnegative integers expre... |
| nnuz 13031 | Positive integers expresse... |
| elnnuz 13032 | A positive integer express... |
| elnn0uz 13033 | A nonnegative integer expr... |
| 1eluzge0 13034 | 1 is an integer greater th... |
| 2eluzge0 13035 | 2 is an integer greater th... |
| 2eluzge1 13036 | 2 is an integer greater th... |
| 5eluz3 13037 | 5 is an integer greater th... |
| uzuzle23 13038 | An integer greater than or... |
| uzuzle24 13039 | An integer greater than or... |
| uzuzle34 13040 | An integer greater than or... |
| uzuzle35 13041 | An integer greater than or... |
| eluz2nn 13042 | An integer greater than or... |
| eluz3nn 13043 | An integer greater than or... |
| eluz4nn 13044 | An integer greater than or... |
| eluz5nn 13045 | An integer greater than or... |
| eluzge2nn0 13046 | If an integer is greater t... |
| eluz2n0 13047 | An integer greater than or... |
| uz3m2nn 13048 | An integer greater than or... |
| uznnssnn 13049 | The upper integers startin... |
| raluz 13050 | Restricted universal quant... |
| raluz2 13051 | Restricted universal quant... |
| rexuz 13052 | Restricted existential qua... |
| rexuz2 13053 | Restricted existential qua... |
| 2rexuz 13054 | Double existential quantif... |
| peano2uz 13055 | Second Peano postulate for... |
| peano2uzs 13056 | Second Peano postulate for... |
| peano2uzr 13057 | Reversed second Peano axio... |
| uzaddcl 13058 | Addition closure law for a... |
| nn0pzuz 13059 | The sum of a nonnegative i... |
| uzind4 13060 | Induction on the upper set... |
| uzind4ALT 13061 | Induction on the upper set... |
| uzind4s 13062 | Induction on the upper set... |
| uzind4s2 13063 | Induction on the upper set... |
| uzind4i 13064 | Induction on the upper int... |
| uzwo 13065 | Well-ordering principle: a... |
| uzwo2 13066 | Well-ordering principle: a... |
| nnwo 13067 | Well-ordering principle: a... |
| nnwof 13068 | Well-ordering principle: a... |
| nnwos 13069 | Well-ordering principle: a... |
| indstr 13070 | Strong Mathematical Induct... |
| eluznn0 13071 | Membership in a nonnegativ... |
| eluznn 13072 | Membership in a positive u... |
| eluz2b1 13073 | Two ways to say "an intege... |
| eluz2gt1 13074 | An integer greater than or... |
| eluz2b2 13075 | Two ways to say "an intege... |
| eluz2b3 13076 | Two ways to say "an intege... |
| uz2m1nn 13077 | One less than an integer g... |
| 1nuz2 13078 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 13079 | A positive integer is eith... |
| uz2mulcl 13080 | Closure of multiplication ... |
| indstr2 13081 | Strong Mathematical Induct... |
| uzinfi 13082 | Extract the lower bound of... |
| nninf 13083 | The infimum of the set of ... |
| nn0inf 13084 | The infimum of the set of ... |
| infssuzle 13085 | The infimum of a subset of... |
| infssuzcl 13086 | The infimum of a subset of... |
| ublbneg 13087 | The image under negation o... |
| eqreznegel 13088 | Two ways to express the im... |
| supminf 13089 | The supremum of a bounded-... |
| lbzbi 13090 | If a set of reals is bound... |
| zsupss 13091 | Any nonempty bounded subse... |
| suprzcl2 13092 | The supremum of a bounded-... |
| suprzub 13093 | The supremum of a bounded-... |
| uzsupss 13094 | Any bounded subset of an u... |
| nn01to3 13095 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 13096 | Alternate proof of ~ nn0ge... |
| uzwo3 13097 | Well-ordering principle: a... |
| zmin 13098 | There is a unique smallest... |
| zmax 13099 | There is a unique largest ... |
| zbtwnre 13100 | There is a unique integer ... |
| rebtwnz 13101 | There is a unique greatest... |
| elq 13104 | Membership in the set of r... |
| qmulz 13105 | If ` A ` is rational, then... |
| znq 13106 | The ratio of an integer an... |
| qre 13107 | A rational number is a rea... |
| zq 13108 | An integer is a rational n... |
| qred 13109 | A rational number is a rea... |
| zssq 13110 | The integers are a subset ... |
| nn0ssq 13111 | The nonnegative integers a... |
| nnssq 13112 | The positive integers are ... |
| qssre 13113 | The rationals are a subset... |
| qsscn 13114 | The rationals are a subset... |
| qex 13115 | The set of rational number... |
| nnq 13116 | A positive integer is rati... |
| qcn 13117 | A rational number is a com... |
| qexALT 13118 | Alternate proof of ~ qex .... |
| 1q 13119 | The number 1 is rational. ... |
| qaddcl 13120 | Closure of addition of rat... |
| qnegcl 13121 | Closure law for the negati... |
| qmulcl 13122 | Closure of multiplication ... |
| qsubcl 13123 | Closure of subtraction of ... |
| qreccl 13124 | Closure of reciprocal of r... |
| qdivcl 13125 | Closure of division of rat... |
| qrevaddcl 13126 | Reverse closure law for ad... |
| nnrecq 13127 | The reciprocal of a positi... |
| irradd 13128 | The sum of an irrational n... |
| irrmul 13129 | The product of an irration... |
| elpq 13130 | A positive rational is the... |
| elpqb 13131 | A class is a positive rati... |
| rpnnen1lem2 13132 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 13133 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 13134 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 13135 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 13136 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 13137 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 13138 | One half of ~ rpnnen , whe... |
| reexALT 13139 | Alternate proof of ~ reex ... |
| cnref1o 13140 | There is a natural one-to-... |
| cnexALT 13141 | The set of complex numbers... |
| xrex 13142 | The set of extended reals ... |
| mpoaddex 13143 | The addition operation is ... |
| addex 13144 | The addition operation is ... |
| mpomulex 13145 | The multiplication operati... |
| mulex 13146 | The multiplication operati... |
| elrp 13149 | Membership in the set of p... |
| elrpii 13150 | Membership in the set of p... |
| 1rp 13151 | 1 is a positive real. (Co... |
| 2rp 13152 | 2 is a positive real. (Co... |
| 3rp 13153 | 3 is a positive real. (Co... |
| 5rp 13154 | 5 is a positive real. (Co... |
| rpssre 13155 | The positive reals are a s... |
| rpre 13156 | A positive real is a real.... |
| rpxr 13157 | A positive real is an exte... |
| rpcn 13158 | A positive real is a compl... |
| nnrp 13159 | A positive integer is a po... |
| rpgt0 13160 | A positive real is greater... |
| rpge0 13161 | A positive real is greater... |
| rpregt0 13162 | A positive real is a posit... |
| rprege0 13163 | A positive real is a nonne... |
| rpne0 13164 | A positive real is nonzero... |
| rprene0 13165 | A positive real is a nonze... |
| rpcnne0 13166 | A positive real is a nonze... |
| neglt 13167 | The negative of a positive... |
| rpcndif0 13168 | A positive real number is ... |
| ralrp 13169 | Quantification over positi... |
| rexrp 13170 | Quantification over positi... |
| rpaddcl 13171 | Closure law for addition o... |
| rpmulcl 13172 | Closure law for multiplica... |
| rpmtmip 13173 | "Minus times minus is plus... |
| rpdivcl 13174 | Closure law for division o... |
| rpreccl 13175 | Closure law for reciprocat... |
| rphalfcl 13176 | Closure law for half of a ... |
| rpgecl 13177 | A number greater than or e... |
| rphalflt 13178 | Half of a positive real is... |
| rerpdivcl 13179 | Closure law for division o... |
| ge0p1rp 13180 | A nonnegative number plus ... |
| rpneg 13181 | Either a nonzero real or i... |
| negelrp 13182 | Elementhood of a negation ... |
| negelrpd 13183 | The negation of a negative... |
| 0nrp 13184 | Zero is not a positive rea... |
| ltsubrp 13185 | Subtracting a positive rea... |
| ltaddrp 13186 | Adding a positive number t... |
| difrp 13187 | Two ways to say one number... |
| elrpd 13188 | Membership in the set of p... |
| nnrpd 13189 | A positive integer is a po... |
| zgt1rpn0n1 13190 | An integer greater than 1 ... |
| rpred 13191 | A positive real is a real.... |
| rpxrd 13192 | A positive real is an exte... |
| rpcnd 13193 | A positive real is a compl... |
| rpgt0d 13194 | A positive real is greater... |
| rpge0d 13195 | A positive real is greater... |
| rpne0d 13196 | A positive real is nonzero... |
| rpregt0d 13197 | A positive real is real an... |
| rprege0d 13198 | A positive real is real an... |
| rprene0d 13199 | A positive real is a nonze... |
| rpcnne0d 13200 | A positive real is a nonze... |
| rpreccld 13201 | Closure law for reciprocat... |
| rprecred 13202 | Closure law for reciprocat... |
| rphalfcld 13203 | Closure law for half of a ... |
| reclt1d 13204 | The reciprocal of a positi... |
| recgt1d 13205 | The reciprocal of a positi... |
| rpaddcld 13206 | Closure law for addition o... |
| rpmulcld 13207 | Closure law for multiplica... |
| rpdivcld 13208 | Closure law for division o... |
| ltrecd 13209 | The reciprocal of both sid... |
| lerecd 13210 | The reciprocal of both sid... |
| ltrec1d 13211 | Reciprocal swap in a 'less... |
| lerec2d 13212 | Reciprocal swap in a 'less... |
| lediv2ad 13213 | Division of both sides of ... |
| ltdiv2d 13214 | Division of a positive num... |
| lediv2d 13215 | Division of a positive num... |
| ledivdivd 13216 | Invert ratios of positive ... |
| divge1 13217 | The ratio of a number over... |
| divlt1lt 13218 | A real number divided by a... |
| divle1le 13219 | A real number divided by a... |
| ledivge1le 13220 | If a number is less than o... |
| ge0p1rpd 13221 | A nonnegative number plus ... |
| rerpdivcld 13222 | Closure law for division o... |
| ltsubrpd 13223 | Subtracting a positive rea... |
| ltaddrpd 13224 | Adding a positive number t... |
| ltaddrp2d 13225 | Adding a positive number t... |
| ltmulgt11d 13226 | Multiplication by a number... |
| ltmulgt12d 13227 | Multiplication by a number... |
| gt0divd 13228 | Division of a positive num... |
| ge0divd 13229 | Division of a nonnegative ... |
| rpgecld 13230 | A number greater than or e... |
| divge0d 13231 | The ratio of nonnegative a... |
| ltmul1d 13232 | The ratio of nonnegative a... |
| ltmul2d 13233 | Multiplication of both sid... |
| lemul1d 13234 | Multiplication of both sid... |
| lemul2d 13235 | Multiplication of both sid... |
| ltdiv1d 13236 | Division of both sides of ... |
| lediv1d 13237 | Division of both sides of ... |
| ltmuldivd 13238 | 'Less than' relationship b... |
| ltmuldiv2d 13239 | 'Less than' relationship b... |
| lemuldivd 13240 | 'Less than or equal to' re... |
| lemuldiv2d 13241 | 'Less than or equal to' re... |
| ltdivmuld 13242 | 'Less than' relationship b... |
| ltdivmul2d 13243 | 'Less than' relationship b... |
| ledivmuld 13244 | 'Less than or equal to' re... |
| ledivmul2d 13245 | 'Less than or equal to' re... |
| ltmul1dd 13246 | The ratio of nonnegative a... |
| ltmul2dd 13247 | Multiplication of both sid... |
| ltdiv1dd 13248 | Division of both sides of ... |
| lediv1dd 13249 | Division of both sides of ... |
| lediv12ad 13250 | Comparison of ratio of two... |
| mul2lt0rlt0 13251 | If the result of a multipl... |
| mul2lt0rgt0 13252 | If the result of a multipl... |
| mul2lt0llt0 13253 | If the result of a multipl... |
| mul2lt0lgt0 13254 | If the result of a multipl... |
| mul2lt0bi 13255 | If the result of a multipl... |
| prodge0rd 13256 | Infer that a multiplicand ... |
| prodge0ld 13257 | Infer that a multiplier is... |
| ltdiv23d 13258 | Swap denominator with othe... |
| lediv23d 13259 | Swap denominator with othe... |
| lt2mul2divd 13260 | The ratio of nonnegative a... |
| nnledivrp 13261 | Division of a positive int... |
| nn0ledivnn 13262 | Division of a nonnegative ... |
| addlelt 13263 | If the sum of a real numbe... |
| ge2halflem1 13264 | Half of an integer greater... |
| ltxr 13271 | The 'less than' binary rel... |
| elxr 13272 | Membership in the set of e... |
| xrnemnf 13273 | An extended real other tha... |
| xrnepnf 13274 | An extended real other tha... |
| xrltnr 13275 | The extended real 'less th... |
| ltpnf 13276 | Any (finite) real is less ... |
| ltpnfd 13277 | Any (finite) real is less ... |
| 0ltpnf 13278 | Zero is less than plus inf... |
| mnflt 13279 | Minus infinity is less tha... |
| mnfltd 13280 | Minus infinity is less tha... |
| mnflt0 13281 | Minus infinity is less tha... |
| mnfltpnf 13282 | Minus infinity is less tha... |
| mnfltxr 13283 | Minus infinity is less tha... |
| pnfnlt 13284 | No extended real is greate... |
| nltmnf 13285 | No extended real is less t... |
| pnfge 13286 | Plus infinity is an upper ... |
| pnfged 13287 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13288 | An extended nonnegative in... |
| 0lepnf 13289 | 0 less than or equal to po... |
| xnn0ge0 13290 | An extended nonnegative in... |
| mnfle 13291 | Minus infinity is less tha... |
| mnfled 13292 | Minus infinity is less tha... |
| xrltnsym 13293 | Ordering on the extended r... |
| xrltnsym2 13294 | 'Less than' is antisymmetr... |
| xrlttri 13295 | Ordering on the extended r... |
| xrlttr 13296 | Ordering on the extended r... |
| xrltso 13297 | 'Less than' is a strict or... |
| xrlttri2 13298 | Trichotomy law for 'less t... |
| xrlttri3 13299 | Trichotomy law for 'less t... |
| xrleloe 13300 | 'Less than or equal' expre... |
| xrleltne 13301 | 'Less than or equal to' im... |
| xrltlen 13302 | 'Less than' expressed in t... |
| dfle2 13303 | Alternative definition of ... |
| dflt2 13304 | Alternative definition of ... |
| xrltle 13305 | 'Less than' implies 'less ... |
| xrltled 13306 | 'Less than' implies 'less ... |
| xrleid 13307 | 'Less than or equal to' is... |
| xrleidd 13308 | 'Less than or equal to' is... |
| xrletri 13309 | Trichotomy law for extende... |
| xrletri3 13310 | Trichotomy law for extende... |
| xrletrid 13311 | Trichotomy law for extende... |
| xrlelttr 13312 | Transitive law for orderin... |
| xrltletr 13313 | Transitive law for orderin... |
| xrletr 13314 | Transitive law for orderin... |
| xrlttrd 13315 | Transitive law for orderin... |
| xrlelttrd 13316 | Transitive law for orderin... |
| xrltletrd 13317 | Transitive law for orderin... |
| xrletrd 13318 | Transitive law for orderin... |
| xrltne 13319 | 'Less than' implies not eq... |
| xrgtned 13320 | 'Greater than' implies not... |
| nltpnft 13321 | An extended real is not le... |
| xgepnf 13322 | An extended real which is ... |
| ngtmnft 13323 | An extended real is not gr... |
| xlemnf 13324 | An extended real which is ... |
| xrrebnd 13325 | An extended real is real i... |
| xrre 13326 | A way of proving that an e... |
| xrre2 13327 | An extended real between t... |
| xrre3 13328 | A way of proving that an e... |
| ge0gtmnf 13329 | A nonnegative extended rea... |
| ge0nemnf 13330 | A nonnegative extended rea... |
| xrrege0 13331 | A nonnegative extended rea... |
| xrmax1 13332 | An extended real is less t... |
| xrmax2 13333 | An extended real is less t... |
| xrmin1 13334 | The minimum of two extende... |
| xrmin2 13335 | The minimum of two extende... |
| xrmaxeq 13336 | The maximum of two extende... |
| xrmineq 13337 | The minimum of two extende... |
| xrmaxlt 13338 | Two ways of saying the max... |
| xrltmin 13339 | Two ways of saying an exte... |
| xrmaxle 13340 | Two ways of saying the max... |
| xrlemin 13341 | Two ways of saying a numbe... |
| max1 13342 | A number is less than or e... |
| max1ALT 13343 | A number is less than or e... |
| max2 13344 | A number is less than or e... |
| 2resupmax 13345 | The supremum of two real n... |
| min1 13346 | The minimum of two numbers... |
| min2 13347 | The minimum of two numbers... |
| maxle 13348 | Two ways of saying the max... |
| lemin 13349 | Two ways of saying a numbe... |
| maxlt 13350 | Two ways of saying the max... |
| ltmin 13351 | Two ways of saying a numbe... |
| lemaxle 13352 | A real number which is les... |
| max0sub 13353 | Decompose a real number in... |
| ifle 13354 | An if statement transforms... |
| z2ge 13355 | There exists an integer gr... |
| qbtwnre 13356 | The rational numbers are d... |
| qbtwnxr 13357 | The rational numbers are d... |
| qsqueeze 13358 | If a nonnegative real is l... |
| qextltlem 13359 | Lemma for ~ qextlt and qex... |
| qextlt 13360 | An extensionality-like pro... |
| qextle 13361 | An extensionality-like pro... |
| xralrple 13362 | Show that ` A ` is less th... |
| alrple 13363 | Show that ` A ` is less th... |
| xnegeq 13364 | Equality of two extended n... |
| xnegex 13365 | A negative extended real e... |
| xnegpnf 13366 | Minus ` +oo ` . Remark of... |
| xnegmnf 13367 | Minus ` -oo ` . Remark of... |
| rexneg 13368 | Minus a real number. Rema... |
| xneg0 13369 | The negative of zero. (Co... |
| xnegcl 13370 | Closure of extended real n... |
| xnegneg 13371 | Extended real version of ~... |
| xneg11 13372 | Extended real version of ~... |
| xltnegi 13373 | Forward direction of ~ xlt... |
| xltneg 13374 | Extended real version of ~... |
| xleneg 13375 | Extended real version of ~... |
| xlt0neg1 13376 | Extended real version of ~... |
| xlt0neg2 13377 | Extended real version of ~... |
| xle0neg1 13378 | Extended real version of ~... |
| xle0neg2 13379 | Extended real version of ~... |
| xaddval 13380 | Value of the extended real... |
| xaddf 13381 | The extended real addition... |
| xmulval 13382 | Value of the extended real... |
| xaddpnf1 13383 | Addition of positive infin... |
| xaddpnf2 13384 | Addition of positive infin... |
| xaddmnf1 13385 | Addition of negative infin... |
| xaddmnf2 13386 | Addition of negative infin... |
| pnfaddmnf 13387 | Addition of positive and n... |
| mnfaddpnf 13388 | Addition of negative and p... |
| rexadd 13389 | The extended real addition... |
| rexsub 13390 | Extended real subtraction ... |
| rexaddd 13391 | The extended real addition... |
| xnn0xaddcl 13392 | The extended nonnegative i... |
| xaddnemnf 13393 | Closure of extended real a... |
| xaddnepnf 13394 | Closure of extended real a... |
| xnegid 13395 | Extended real version of ~... |
| xaddcl 13396 | The extended real addition... |
| xaddcom 13397 | The extended real addition... |
| xaddrid 13398 | Extended real version of ~... |
| xaddlid 13399 | Extended real version of ~... |
| xaddridd 13400 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13401 | Extended nonnegative integ... |
| xnn0lenn0nn0 13402 | An extended nonnegative in... |
| xnn0le2is012 13403 | An extended nonnegative in... |
| xnn0xadd0 13404 | The sum of two extended no... |
| xnegdi 13405 | Extended real version of ~... |
| xaddass 13406 | Associativity of extended ... |
| xaddass2 13407 | Associativity of extended ... |
| xpncan 13408 | Extended real version of ~... |
| xnpcan 13409 | Extended real version of ~... |
| xleadd1a 13410 | Extended real version of ~... |
| xleadd2a 13411 | Commuted form of ~ xleadd1... |
| xleadd1 13412 | Weakened version of ~ xlea... |
| xltadd1 13413 | Extended real version of ~... |
| xltadd2 13414 | Extended real version of ~... |
| xaddge0 13415 | The sum of nonnegative ext... |
| xle2add 13416 | Extended real version of ~... |
| xlt2add 13417 | Extended real version of ~... |
| xsubge0 13418 | Extended real version of ~... |
| xposdif 13419 | Extended real version of ~... |
| xlesubadd 13420 | Under certain conditions, ... |
| xmullem 13421 | Lemma for ~ rexmul . (Con... |
| xmullem2 13422 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13423 | Extended real multiplicati... |
| xmul01 13424 | Extended real version of ~... |
| xmul02 13425 | Extended real version of ~... |
| xmulneg1 13426 | Extended real version of ~... |
| xmulneg2 13427 | Extended real version of ~... |
| rexmul 13428 | The extended real multipli... |
| xmulf 13429 | The extended real multipli... |
| xmulcl 13430 | Closure of extended real m... |
| xmulpnf1 13431 | Multiplication by plus inf... |
| xmulpnf2 13432 | Multiplication by plus inf... |
| xmulmnf1 13433 | Multiplication by minus in... |
| xmulmnf2 13434 | Multiplication by minus in... |
| xmulpnf1n 13435 | Multiplication by plus inf... |
| xmulrid 13436 | Extended real version of ~... |
| xmullid 13437 | Extended real version of ~... |
| xmulm1 13438 | Extended real version of ~... |
| xmulasslem2 13439 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13440 | Extended real version of ~... |
| xmulge0 13441 | Extended real version of ~... |
| xmulasslem 13442 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13443 | Lemma for ~ xmulass . (Co... |
| xmulass 13444 | Associativity of the exten... |
| xlemul1a 13445 | Extended real version of ~... |
| xlemul2a 13446 | Extended real version of ~... |
| xlemul1 13447 | Extended real version of ~... |
| xlemul2 13448 | Extended real version of ~... |
| xltmul1 13449 | Extended real version of ~... |
| xltmul2 13450 | Extended real version of ~... |
| xadddilem 13451 | Lemma for ~ xadddi . (Con... |
| xadddi 13452 | Distributive property for ... |
| xadddir 13453 | Commuted version of ~ xadd... |
| xadddi2 13454 | The assumption that the mu... |
| xadddi2r 13455 | Commuted version of ~ xadd... |
| x2times 13456 | Extended real version of ~... |
| xnegcld 13457 | Closure of extended real n... |
| xaddcld 13458 | The extended real addition... |
| xmulcld 13459 | Closure of extended real m... |
| xadd4d 13460 | Rearrangement of 4 terms i... |
| xnn0add4d 13461 | Rearrangement of 4 terms i... |
| xrsupexmnf 13462 | Adding minus infinity to a... |
| xrinfmexpnf 13463 | Adding plus infinity to a ... |
| xrsupsslem 13464 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13465 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13466 | Any subset of extended rea... |
| xrinfmss 13467 | Any subset of extended rea... |
| xrinfmss2 13468 | Any subset of extended rea... |
| xrub 13469 | By quantifying only over r... |
| supxr 13470 | The supremum of a set of e... |
| supxr2 13471 | The supremum of a set of e... |
| supxrcl 13472 | The supremum of an arbitra... |
| supxrun 13473 | The supremum of the union ... |
| supxrmnf 13474 | Adding minus infinity to a... |
| supxrpnf 13475 | The supremum of a set of e... |
| supxrunb1 13476 | The supremum of an unbound... |
| supxrunb2 13477 | The supremum of an unbound... |
| supxrbnd1 13478 | The supremum of a bounded-... |
| supxrbnd2 13479 | The supremum of a bounded-... |
| xrsup0 13480 | The supremum of an empty s... |
| supxrub 13481 | A member of a set of exten... |
| supxrlub 13482 | The supremum of a set of e... |
| supxrleub 13483 | The supremum of a set of e... |
| supxrre 13484 | The real and extended real... |
| supxrbnd 13485 | The supremum of a bounded-... |
| supxrgtmnf 13486 | The supremum of a nonempty... |
| supxrre1 13487 | The supremum of a nonempty... |
| supxrre2 13488 | The supremum of a nonempty... |
| supxrss 13489 | Smaller sets of extended r... |
| xrsupssd 13490 | Inequality deduction for s... |
| infxrcl 13491 | The infimum of an arbitrar... |
| infxrlb 13492 | A member of a set of exten... |
| infxrgelb 13493 | The infimum of a set of ex... |
| infxrre 13494 | The real and extended real... |
| infxrmnf 13495 | The infinimum of a set of ... |
| xrinf0 13496 | The infimum of the empty s... |
| infxrss 13497 | Larger sets of extended re... |
| reltre 13498 | For all real numbers there... |
| rpltrp 13499 | For all positive real numb... |
| reltxrnmnf 13500 | For all extended real numb... |
| infmremnf 13501 | The infimum of the reals i... |
| infmrp1 13502 | The infimum of the positiv... |
| ixxval 13511 | Value of the interval func... |
| elixx1 13512 | Membership in an interval ... |
| ixxf 13513 | The set of intervals of ex... |
| ixxex 13514 | The set of intervals of ex... |
| ixxssxr 13515 | The set of intervals of ex... |
| elixx3g 13516 | Membership in a set of ope... |
| ixxssixx 13517 | An interval is a subset of... |
| ixxdisj 13518 | Split an interval into dis... |
| ixxun 13519 | Split an interval into two... |
| ixxin 13520 | Intersection of two interv... |
| ixxss1 13521 | Subset relationship for in... |
| ixxss2 13522 | Subset relationship for in... |
| ixxss12 13523 | Subset relationship for in... |
| ixxub 13524 | Extract the upper bound of... |
| ixxlb 13525 | Extract the lower bound of... |
| iooex 13526 | The set of open intervals ... |
| iooval 13527 | Value of the open interval... |
| ioo0 13528 | An empty open interval of ... |
| ioon0 13529 | An open interval of extend... |
| ndmioo 13530 | The open interval function... |
| iooid 13531 | An open interval with iden... |
| elioo3g 13532 | Membership in a set of ope... |
| elioore 13533 | A member of an open interv... |
| lbioo 13534 | An open interval does not ... |
| ubioo 13535 | An open interval does not ... |
| iooval2 13536 | Value of the open interval... |
| iooin 13537 | Intersection of two open i... |
| iooss1 13538 | Subset relationship for op... |
| iooss2 13539 | Subset relationship for op... |
| iocval 13540 | Value of the open-below, c... |
| icoval 13541 | Value of the closed-below,... |
| iccval 13542 | Value of the closed interv... |
| elioo1 13543 | Membership in an open inte... |
| elioo2 13544 | Membership in an open inte... |
| elioc1 13545 | Membership in an open-belo... |
| elico1 13546 | Membership in a closed-bel... |
| elicc1 13547 | Membership in a closed int... |
| iccid 13548 | A closed interval with ide... |
| ico0 13549 | An empty open interval of ... |
| ioc0 13550 | An empty open interval of ... |
| icc0 13551 | An empty closed interval o... |
| dfrp2 13552 | Alternate definition of th... |
| elicod 13553 | Membership in a left-close... |
| icogelb 13554 | An element of a left-close... |
| icogelbd 13555 | An element of a left-close... |
| elicore 13556 | A member of a left-closed ... |
| ubioc1 13557 | The upper bound belongs to... |
| lbico1 13558 | The lower bound belongs to... |
| iccleub 13559 | An element of a closed int... |
| iccgelb 13560 | An element of a closed int... |
| elioo5 13561 | Membership in an open inte... |
| eliooxr 13562 | A nonempty open interval s... |
| eliooord 13563 | Ordering implied by a memb... |
| elioo4g 13564 | Membership in an open inte... |
| ioossre 13565 | An open interval is a set ... |
| ioosscn 13566 | An open interval is a set ... |
| elioc2 13567 | Membership in an open-belo... |
| elico2 13568 | Membership in a closed-bel... |
| elicc2 13569 | Membership in a closed rea... |
| elicc2i 13570 | Inference for membership i... |
| elicc4 13571 | Membership in a closed rea... |
| iccss 13572 | Condition for a closed int... |
| iccssioo 13573 | Condition for a closed int... |
| icossico 13574 | Condition for a closed-bel... |
| iccss2 13575 | Condition for a closed int... |
| iccssico 13576 | Condition for a closed int... |
| iccssioo2 13577 | Condition for a closed int... |
| iccssico2 13578 | Condition for a closed int... |
| icossico2d 13579 | Condition for a closed-bel... |
| ioomax 13580 | The open interval from min... |
| iccmax 13581 | The closed interval from m... |
| ioopos 13582 | The set of positive reals ... |
| ioorp 13583 | The set of positive reals ... |
| iooshf 13584 | Shift the arguments of the... |
| iocssre 13585 | A closed-above interval wi... |
| icossre 13586 | A closed-below interval wi... |
| iccssre 13587 | A closed real interval is ... |
| iccssxr 13588 | A closed interval is a set... |
| iocssxr 13589 | An open-below, closed-abov... |
| icossxr 13590 | A closed-below, open-above... |
| ioossicc 13591 | An open interval is a subs... |
| iccssred 13592 | A closed real interval is ... |
| eliccxr 13593 | A member of a closed inter... |
| icossicc 13594 | A closed-below, open-above... |
| iocssicc 13595 | A closed-above, open-below... |
| ioossico 13596 | An open interval is a subs... |
| iocssioo 13597 | Condition for a closed int... |
| icossioo 13598 | Condition for a closed int... |
| ioossioo 13599 | Condition for an open inte... |
| iccsupr 13600 | A nonempty subset of a clo... |
| elioopnf 13601 | Membership in an unbounded... |
| elioomnf 13602 | Membership in an unbounded... |
| elicopnf 13603 | Membership in a closed unb... |
| repos 13604 | Two ways of saying that a ... |
| ioof 13605 | The set of open intervals ... |
| iccf 13606 | The set of closed interval... |
| unirnioo 13607 | The union of the range of ... |
| dfioo2 13608 | Alternate definition of th... |
| ioorebas 13609 | Open intervals are element... |
| xrge0neqmnf 13610 | A nonnegative extended rea... |
| xrge0nre 13611 | An extended real which is ... |
| elrege0 13612 | The predicate "is a nonneg... |
| nn0rp0 13613 | A nonnegative integer is a... |
| rge0ssre 13614 | Nonnegative real numbers a... |
| elxrge0 13615 | Elementhood in the set of ... |
| 0e0icopnf 13616 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13617 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13618 | The nonnegative reals are ... |
| ge0mulcl 13619 | The nonnegative reals are ... |
| ge0xaddcl 13620 | The nonnegative reals are ... |
| ge0xmulcl 13621 | The nonnegative extended r... |
| lbicc2 13622 | The lower bound of a close... |
| ubicc2 13623 | The upper bound of a close... |
| elicc01 13624 | Membership in the closed r... |
| elunitrn 13625 | The closed unit interval i... |
| elunitcn 13626 | The closed unit interval i... |
| 0elunit 13627 | Zero is an element of the ... |
| 1elunit 13628 | One is an element of the c... |
| iooneg 13629 | Membership in a negated op... |
| iccneg 13630 | Membership in a negated cl... |
| icoshft 13631 | A shifted real is a member... |
| icoshftf1o 13632 | Shifting a closed-below, o... |
| icoun 13633 | The union of two adjacent ... |
| icodisj 13634 | Adjacent left-closed right... |
| ioounsn 13635 | The union of an open inter... |
| snunioo 13636 | The closure of one end of ... |
| snunico 13637 | The closure of the open en... |
| snunioc 13638 | The closure of the open en... |
| prunioo 13639 | The closure of an open rea... |
| ioodisj 13640 | If the upper bound of one ... |
| ioojoin 13641 | Join two open intervals to... |
| difreicc 13642 | The class difference of ` ... |
| iccsplit 13643 | Split a closed interval in... |
| iccshftr 13644 | Membership in a shifted in... |
| iccshftri 13645 | Membership in a shifted in... |
| iccshftl 13646 | Membership in a shifted in... |
| iccshftli 13647 | Membership in a shifted in... |
| iccdil 13648 | Membership in a dilated in... |
| iccdili 13649 | Membership in a dilated in... |
| icccntr 13650 | Membership in a contracted... |
| icccntri 13651 | Membership in a contracted... |
| divelunit 13652 | A condition for a ratio to... |
| lincmb01cmp 13653 | A linear combination of tw... |
| iccf1o 13654 | Describe a bijection from ... |
| iccen 13655 | Any nontrivial closed inte... |
| xov1plusxeqvd 13656 | A complex number ` X ` is ... |
| unitssre 13657 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13658 | The closed unit interval i... |
| supicc 13659 | Supremum of a bounded set ... |
| supiccub 13660 | The supremum of a bounded ... |
| supicclub 13661 | The supremum of a bounded ... |
| supicclub2 13662 | The supremum of a bounded ... |
| zltaddlt1le 13663 | The sum of an integer and ... |
| xnn0xrge0 13664 | An extended nonnegative in... |
| nnge2recico01 13665 | The reciprocal of an integ... |
| fzval 13668 | The value of a finite set ... |
| fzval2 13669 | An alternative way of expr... |
| fzf 13670 | Establish the domain and c... |
| elfz1 13671 | Membership in a finite set... |
| elfz 13672 | Membership in a finite set... |
| elfz2 13673 | Membership in a finite set... |
| elfzd 13674 | Membership in a finite set... |
| elfz5 13675 | Membership in a finite set... |
| elfz4 13676 | Membership in a finite set... |
| elfzuzb 13677 | Membership in a finite set... |
| eluzfz 13678 | Membership in a finite set... |
| elfzuz 13679 | A member of a finite set o... |
| elfzuz3 13680 | Membership in a finite set... |
| elfzel2 13681 | Membership in a finite set... |
| elfzel1 13682 | Membership in a finite set... |
| elfzelz 13683 | A member of a finite set o... |
| elfzelzd 13684 | A member of a finite set o... |
| fzssz 13685 | A finite sequence of integ... |
| fzssre 13686 | A finite sequence of integ... |
| elfzle1 13687 | A member of a finite set o... |
| elfzle2 13688 | A member of a finite set o... |
| elfzuz2 13689 | Implication of membership ... |
| elfzle3 13690 | Membership in a finite set... |
| eluzfz1 13691 | Membership in a finite set... |
| eluzfz2 13692 | Membership in a finite set... |
| eluzfz2b 13693 | Membership in a finite set... |
| elfz3 13694 | Membership in a finite set... |
| elfz1eq 13695 | Membership in a finite set... |
| elfzubelfz 13696 | If there is a member in a ... |
| peano2fzr 13697 | A Peano-postulate-like the... |
| fzn0 13698 | Properties of a finite int... |
| fz0 13699 | A finite set of sequential... |
| fzn 13700 | A finite set of sequential... |
| fzen 13701 | A shifted finite set of se... |
| fz1n 13702 | A 1-based finite set of se... |
| 0nelfz1 13703 | 0 is not an element of a f... |
| 0fz1 13704 | Two ways to say a finite 1... |
| fz10 13705 | There are no integers betw... |
| fz00m1 13706 | There are no integers betw... |
| uzsubsubfz 13707 | Membership of an integer g... |
| uzsubsubfz1 13708 | Membership of an integer g... |
| ige3m2fz 13709 | Membership of an integer g... |
| fzsplit2 13710 | Split a finite interval of... |
| fzsplit 13711 | Split a finite interval of... |
| fzdisj 13712 | Condition for two finite i... |
| fz01en 13713 | 0-based and 1-based finite... |
| elfznn 13714 | A member of a finite set o... |
| elfz1end 13715 | A nonempty finite range of... |
| fz1ssnn 13716 | A finite set of positive i... |
| fznn0sub 13717 | Subtraction closure for a ... |
| fzmmmeqm 13718 | Subtracting the difference... |
| fzaddel 13719 | Membership of a sum in a f... |
| fzadd2 13720 | Membership of a sum in a f... |
| fzsubel 13721 | Membership of a difference... |
| fzopth 13722 | A finite set of sequential... |
| fzass4 13723 | Two ways to express a nond... |
| fzss1 13724 | Subset relationship for fi... |
| fzss2 13725 | Subset relationship for fi... |
| fzssuz 13726 | A finite set of sequential... |
| fzsn 13727 | A finite interval of integ... |
| fzssp1 13728 | Subset relationship for fi... |
| fzssnn 13729 | Finite sets of sequential ... |
| ssfzunsnext 13730 | A subset of a finite seque... |
| ssfzunsn 13731 | A subset of a finite seque... |
| fzsuc 13732 | Join a successor to the en... |
| fzpred 13733 | Join a predecessor to the ... |
| fzpreddisj 13734 | A finite set of sequential... |
| elfzp1 13735 | Append an element to a fin... |
| fzp1ss 13736 | Subset relationship for fi... |
| fzelp1 13737 | Membership in a set of seq... |
| fzp1elp1 13738 | Add one to an element of a... |
| fznatpl1 13739 | Shift membership in a fini... |
| fzpr 13740 | A finite interval of integ... |
| fztp 13741 | A finite interval of integ... |
| fz12pr 13742 | An integer range between 1... |
| fzsuc2 13743 | Join a successor to the en... |
| fzp1disj 13744 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13745 | Remove a successor from th... |
| fzprval 13746 | Two ways of defining the f... |
| fztpval 13747 | Two ways of defining the f... |
| fzrev 13748 | Reversal of start and end ... |
| fzrev2 13749 | Reversal of start and end ... |
| fzrev2i 13750 | Reversal of start and end ... |
| fzrev3 13751 | The "complement" of a memb... |
| fzrev3i 13752 | The "complement" of a memb... |
| fznn 13753 | Finite set of sequential i... |
| elfz1b 13754 | Membership in a 1-based fi... |
| elfz1uz 13755 | Membership in a 1-based fi... |
| elfzm11 13756 | Membership in a finite set... |
| uzsplit 13757 | Express an upper integer s... |
| uzdisj 13758 | The first ` N ` elements o... |
| fseq1p1m1 13759 | Add/remove an item to/from... |
| fseq1m1p1 13760 | Add/remove an item to/from... |
| fz1sbc 13761 | Quantification over a one-... |
| elfzp1b 13762 | An integer is a member of ... |
| elfzm1b 13763 | An integer is a member of ... |
| elfzp12 13764 | Options for membership in ... |
| fzne1 13765 | Elementhood in a finite se... |
| fzdif1 13766 | Split the first element of... |
| fz0dif1 13767 | Split the first element of... |
| fzm1 13768 | Choices for an element of ... |
| fzneuz 13769 | No finite set of sequentia... |
| fznuz 13770 | Disjointness of the upper ... |
| uznfz 13771 | Disjointness of the upper ... |
| fzp1nel 13772 | One plus the upper bound o... |
| fzrevral 13773 | Reversal of scanning order... |
| fzrevral2 13774 | Reversal of scanning order... |
| fzrevral3 13775 | Reversal of scanning order... |
| fzshftral 13776 | Shift the scanning order i... |
| ige2m1fz1 13777 | Membership of an integer g... |
| ige2m1fz 13778 | Membership in a 0-based fi... |
| elfz2nn0 13779 | Membership in a finite set... |
| fznn0 13780 | Characterization of a fini... |
| elfznn0 13781 | A member of a finite set o... |
| elfz3nn0 13782 | The upper bound of a nonem... |
| fz0ssnn0 13783 | Finite sets of sequential ... |
| fz1ssfz0 13784 | Subset relationship for fi... |
| 0elfz 13785 | 0 is an element of a finit... |
| nn0fz0 13786 | A nonnegative integer is a... |
| elfz0add 13787 | An element of a finite set... |
| fz0sn 13788 | An integer range from 0 to... |
| fz0tp 13789 | An integer range from 0 to... |
| fz0to3un2pr 13790 | An integer range from 0 to... |
| fz0to4untppr 13791 | An integer range from 0 to... |
| fz0to5un2tp 13792 | An integer range from 0 to... |
| elfz0ubfz0 13793 | An element of a finite set... |
| elfz0fzfz0 13794 | A member of a finite set o... |
| fz0fzelfz0 13795 | If a member of a finite se... |
| fznn0sub2 13796 | Subtraction closure for a ... |
| uzsubfz0 13797 | Membership of an integer g... |
| fz0fzdiffz0 13798 | The difference of an integ... |
| elfzmlbm 13799 | Subtracting the lower boun... |
| elfzmlbp 13800 | Subtracting the lower boun... |
| fzctr 13801 | Lemma for theorems about t... |
| difelfzle 13802 | The difference of two inte... |
| difelfznle 13803 | The difference of two inte... |
| nn0split 13804 | Express the set of nonnega... |
| nn0disj 13805 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13806 | A finite set of sequential... |
| fvffz0 13807 | The function value of a fu... |
| 1fv 13808 | A function on a singleton.... |
| 4fvwrd4 13809 | The first four function va... |
| 2ffzeq 13810 | Two functions over 0-based... |
| preduz 13811 | The value of the predecess... |
| prednn 13812 | The value of the predecess... |
| prednn0 13813 | The value of the predecess... |
| predfz 13814 | Calculate the predecessor ... |
| fzof 13817 | Functionality of the half-... |
| elfzoel1 13818 | Reverse closure for half-o... |
| elfzoel2 13819 | Reverse closure for half-o... |
| elfzoelz 13820 | Reverse closure for half-o... |
| fzoval 13821 | Value of the half-open int... |
| elfzo 13822 | Membership in a half-open ... |
| elfzo2 13823 | Membership in a half-open ... |
| elfzod 13824 | Membership in a half-open ... |
| elfzouz 13825 | Membership in a half-open ... |
| nelfzo 13826 | An integer not being a mem... |
| fzolb 13827 | The left endpoint of a hal... |
| fzolb2 13828 | The left endpoint of a hal... |
| elfzole1 13829 | A member in a half-open in... |
| elfzolt2 13830 | A member in a half-open in... |
| elfzolt3 13831 | Membership in a half-open ... |
| elfzolt2b 13832 | A member in a half-open in... |
| elfzolt3b 13833 | Membership in a half-open ... |
| elfzop1le2 13834 | A member in a half-open in... |
| fzonel 13835 | A half-open range does not... |
| elfzouz2 13836 | The upper bound of a half-... |
| elfzofz 13837 | A half-open range is conta... |
| elfzo3 13838 | Express membership in a ha... |
| fzon0 13839 | A half-open integer interv... |
| fzossfz 13840 | A half-open range is conta... |
| fzossz 13841 | A half-open integer interv... |
| fzon 13842 | A half-open set of sequent... |
| fzo0n 13843 | A half-open range of nonne... |
| fzonlt0 13844 | A half-open integer range ... |
| fzo0 13845 | Half-open sets with equal ... |
| fzonnsub 13846 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13847 | If ` M < N ` then ` N - M ... |
| fzoss1 13848 | Subset relationship for ha... |
| fzoss2 13849 | Subset relationship for ha... |
| fzossrbm1 13850 | Subset of a half-open rang... |
| fzo0ss1 13851 | Subset relationship for ha... |
| fzossnn0 13852 | A half-open integer range ... |
| fzospliti 13853 | One direction of splitting... |
| fzosplit 13854 | Split a half-open integer ... |
| fzodisj 13855 | Abutting half-open integer... |
| fzouzsplit 13856 | Split an upper integer set... |
| fzouzdisj 13857 | A half-open integer range ... |
| fzoun 13858 | A half-open integer range ... |
| fzodisjsn 13859 | A half-open integer range ... |
| prinfzo0 13860 | The intersection of a half... |
| lbfzo0 13861 | An integer is strictly gre... |
| elfzo0 13862 | Membership in a half-open ... |
| elfzo0z 13863 | Membership in a half-open ... |
| nn0p1elfzo 13864 | A nonnegative integer incr... |
| elfzo0le 13865 | A member in a half-open ra... |
| elfzolem1 13866 | A member in a half-open in... |
| elfzo0subge1 13867 | The difference of the uppe... |
| elfzo0suble 13868 | The difference of the uppe... |
| elfzonn0 13869 | A member of a half-open ra... |
| fzonmapblen 13870 | The result of subtracting ... |
| fzofzim 13871 | If a nonnegative integer i... |
| fz1fzo0m1 13872 | Translation of one between... |
| fzossnn 13873 | Half-open integer ranges s... |
| elfzo1 13874 | Membership in a half-open ... |
| fzo1lb 13875 | 1 is the left endpoint of ... |
| 1elfzo1 13876 | 1 is in a half-open range ... |
| fzo1fzo0n0 13877 | An integer between 1 and a... |
| fzo0n0 13878 | A half-open integer range ... |
| fzoaddel 13879 | Translate membership in a ... |
| fzo0addel 13880 | Translate membership in a ... |
| fzo0addelr 13881 | Translate membership in a ... |
| fzoaddel2 13882 | Translate membership in a ... |
| elfzoextl 13883 | Membership of an integer i... |
| elfzoext 13884 | Membership of an integer i... |
| elincfzoext 13885 | Membership of an increased... |
| fzosubel 13886 | Translate membership in a ... |
| fzosubel2 13887 | Membership in a translated... |
| fzosubel3 13888 | Membership in a translated... |
| eluzgtdifelfzo 13889 | Membership of the differen... |
| ige2m2fzo 13890 | Membership of an integer g... |
| fzocatel 13891 | Translate membership in a ... |
| ubmelfzo 13892 | If an integer in a 1-based... |
| elfzodifsumelfzo 13893 | If an integer is in a half... |
| elfzom1elp1fzo 13894 | Membership of an integer i... |
| elfzom1elfzo 13895 | Membership in a half-open ... |
| fzval3 13896 | Expressing a closed intege... |
| fz0add1fz1 13897 | Translate membership in a ... |
| fzosn 13898 | Expressing a singleton as ... |
| elfzomin 13899 | Membership of an integer i... |
| zpnn0elfzo 13900 | Membership of an integer i... |
| zpnn0elfzo1 13901 | Membership of an integer i... |
| fzosplitsnm1 13902 | Removing a singleton from ... |
| elfzonlteqm1 13903 | If an element of a half-op... |
| fzonn0p1 13904 | A nonnegative integer is a... |
| fzossfzop1 13905 | A half-open range of nonne... |
| fzonn0p1p1 13906 | If a nonnegative integer i... |
| elfzom1p1elfzo 13907 | Increasing an element of a... |
| fzo0ssnn0 13908 | Half-open integer ranges s... |
| fzo01 13909 | Expressing the singleton o... |
| fzo12sn 13910 | A 1-based half-open intege... |
| fzo13pr 13911 | A 1-based half-open intege... |
| fzo0to2pr 13912 | A half-open integer range ... |
| fz01pr 13913 | An integer range between 0... |
| fzo0to3tp 13914 | A half-open integer range ... |
| fzo0to42pr 13915 | A half-open integer range ... |
| fzo1to4tp 13916 | A half-open integer range ... |
| fzo0sn0fzo1 13917 | A half-open range of nonne... |
| elfzo0l 13918 | A member of a half-open ra... |
| fzoend 13919 | The endpoint of a half-ope... |
| fzo0end 13920 | The endpoint of a zero-bas... |
| ssfzo12 13921 | Subset relationship for ha... |
| ssfzoulel 13922 | If a half-open integer ran... |
| ssfzo12bi 13923 | Subset relationship for ha... |
| fzoopth 13924 | A half-open integer range ... |
| ubmelm1fzo 13925 | The result of subtracting ... |
| fzofzp1 13926 | If a point is in a half-op... |
| fzofzp1b 13927 | If a point is in a half-op... |
| elfzom1b 13928 | An integer is a member of ... |
| elfzom1elp1fzo1 13929 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13930 | Membership of a positive i... |
| elfzonelfzo 13931 | If an element of a half-op... |
| elfzodif0 13932 | If an integer ` M ` is in ... |
| fzonfzoufzol 13933 | If an element of a half-op... |
| elfzomelpfzo 13934 | An integer increased by an... |
| elfznelfzo 13935 | A value in a finite set of... |
| elfznelfzob 13936 | A value in a finite set of... |
| peano2fzor 13937 | A Peano-postulate-like the... |
| fzosplitsn 13938 | Extending a half-open rang... |
| fzosplitpr 13939 | Extending a half-open inte... |
| fzosplitprm1 13940 | Extending a half-open inte... |
| fzosplitsni 13941 | Membership in a half-open ... |
| fzisfzounsn 13942 | A finite interval of integ... |
| elfzr 13943 | A member of a finite inter... |
| elfzlmr 13944 | A member of a finite inter... |
| elfz0lmr 13945 | A member of a finite inter... |
| fzone1 13946 | Elementhood in a half-open... |
| fzom1ne1 13947 | Elementhood in a half-open... |
| fzostep1 13948 | Two possibilities for a nu... |
| fzoshftral 13949 | Shift the scanning order i... |
| fzind2 13950 | Induction on the integers ... |
| fvinim0ffz 13951 | The function values for th... |
| injresinjlem 13952 | Lemma for ~ injresinj . (... |
| injresinj 13953 | A function whose restricti... |
| f1resfz0f1d 13954 | If a function with a seque... |
| subfzo0 13955 | The difference between two... |
| fvf1tp 13956 | Values of a one-to-one fun... |
| flval 13961 | Value of the floor (greate... |
| flcl 13962 | The floor (greatest intege... |
| reflcl 13963 | The floor (greatest intege... |
| fllelt 13964 | A basic property of the fl... |
| flcld 13965 | The floor (greatest intege... |
| flle 13966 | A basic property of the fl... |
| flltp1 13967 | A basic property of the fl... |
| fllep1 13968 | A basic property of the fl... |
| fraclt1 13969 | The fractional part of a r... |
| fracle1 13970 | The fractional part of a r... |
| fracge0 13971 | The fractional part of a r... |
| flge 13972 | The floor function value i... |
| fllt 13973 | The floor function value i... |
| flflp1 13974 | Move floor function betwee... |
| flid 13975 | An integer is its own floo... |
| flidm 13976 | The floor function is idem... |
| flidz 13977 | A real number equals its f... |
| flltnz 13978 | The floor of a non-integer... |
| flwordi 13979 | Ordering relation for the ... |
| flword2 13980 | Ordering relation for the ... |
| flval2 13981 | An alternate way to define... |
| flval3 13982 | An alternate way to define... |
| flbi 13983 | A condition equivalent to ... |
| flbi2 13984 | A condition equivalent to ... |
| adddivflid 13985 | The floor of a sum of an i... |
| ico01fl0 13986 | The floor of a real number... |
| flge0nn0 13987 | The floor of a number grea... |
| flge1nn 13988 | The floor of a number grea... |
| fldivnn0 13989 | The floor function of a di... |
| refldivcl 13990 | The floor function of a di... |
| divfl0 13991 | The floor of a fraction is... |
| fladdz 13992 | An integer can be moved in... |
| flzadd 13993 | An integer can be moved in... |
| flmulnn0 13994 | Move a nonnegative integer... |
| btwnzge0 13995 | A real bounded between an ... |
| 2tnp1ge0ge0 13996 | Two times an integer plus ... |
| flhalf 13997 | Ordering relation for the ... |
| fldivle 13998 | The floor function of a di... |
| fldivnn0le 13999 | The floor function of a di... |
| flltdivnn0lt 14000 | The floor function of a di... |
| ltdifltdiv 14001 | If the dividend of a divis... |
| fldiv4p1lem1div2 14002 | The floor of an integer eq... |
| fldiv4lem1div2uz2 14003 | The floor of an integer gr... |
| fldiv4lem1div2 14004 | The floor of a positive in... |
| ceilval 14005 | The value of the ceiling f... |
| dfceil2 14006 | Alternative definition of ... |
| ceilval2 14007 | The value of the ceiling f... |
| ceicl 14008 | The ceiling function retur... |
| ceilcl 14009 | Closure of the ceiling fun... |
| ceilcld 14010 | Closure of the ceiling fun... |
| ceige 14011 | The ceiling of a real numb... |
| ceilge 14012 | The ceiling of a real numb... |
| ceilged 14013 | The ceiling of a real numb... |
| ceim1l 14014 | One less than the ceiling ... |
| ceilm1lt 14015 | One less than the ceiling ... |
| ceile 14016 | The ceiling of a real numb... |
| ceille 14017 | The ceiling of a real numb... |
| ceilid 14018 | An integer is its own ceil... |
| ceilidz 14019 | A real number equals its c... |
| flleceil 14020 | The floor of a real number... |
| fleqceilz 14021 | A real number is an intege... |
| quoremz 14022 | Quotient and remainder of ... |
| quoremnn0 14023 | Quotient and remainder of ... |
| quoremnn0ALT 14024 | Alternate proof of ~ quore... |
| intfrac2 14025 | Decompose a real into inte... |
| intfracq 14026 | Decompose a rational numbe... |
| fldiv 14027 | Cancellation of the embedd... |
| fldiv2 14028 | Cancellation of an embedde... |
| fznnfl 14029 | Finite set of sequential i... |
| uzsup 14030 | An upper set of integers i... |
| ioopnfsup 14031 | An upper set of reals is u... |
| icopnfsup 14032 | An upper set of reals is u... |
| rpsup 14033 | The positive reals are unb... |
| resup 14034 | The real numbers are unbou... |
| xrsup 14035 | The extended real numbers ... |
| modval 14038 | The value of the modulo op... |
| modvalr 14039 | The value of the modulo op... |
| modcl 14040 | Closure law for the modulo... |
| flpmodeq 14041 | Partition of a division in... |
| modcld 14042 | Closure law for the modulo... |
| mod0 14043 | ` A mod B ` is zero iff ` ... |
| mulmod0 14044 | The product of an integer ... |
| negmod0 14045 | ` A ` is divisible by ` B ... |
| modge0 14046 | The modulo operation is no... |
| modlt 14047 | The modulo operation is le... |
| modelico 14048 | Modular reduction produces... |
| moddiffl 14049 | Value of the modulo operat... |
| moddifz 14050 | The modulo operation diffe... |
| modfrac 14051 | The fractional part of a n... |
| flmod 14052 | The floor function express... |
| intfrac 14053 | Break a number into its in... |
| zmod10 14054 | An integer modulo 1 is 0. ... |
| zmod1congr 14055 | Two arbitrary integers are... |
| modmulnn 14056 | Move a positive integer in... |
| modvalp1 14057 | The value of the modulo op... |
| zmodcl 14058 | Closure law for the modulo... |
| zmodcld 14059 | Closure law for the modulo... |
| zmodfz 14060 | An integer mod ` B ` lies ... |
| zmodfzo 14061 | An integer mod ` B ` lies ... |
| zmodfzp1 14062 | An integer mod ` B ` lies ... |
| modid 14063 | Identity law for modulo. ... |
| modid0 14064 | A positive real number mod... |
| modid2 14065 | Identity law for modulo. ... |
| zmodid2 14066 | Identity law for modulo re... |
| zmodidfzo 14067 | Identity law for modulo re... |
| zmodidfzoimp 14068 | Identity law for modulo re... |
| 0mod 14069 | Special case: 0 modulo a p... |
| 1mod 14070 | Special case: 1 modulo a r... |
| modabs 14071 | Absorption law for modulo.... |
| modabs2 14072 | Absorption law for modulo.... |
| modcyc 14073 | The modulo operation is pe... |
| modcyc2 14074 | The modulo operation is pe... |
| modadd1 14075 | Addition property of the m... |
| modaddb 14076 | Addition property of the m... |
| modaddid 14077 | The sums of two nonnegativ... |
| modaddabs 14078 | Absorption law for modulo.... |
| modaddmod 14079 | The sum of a real number m... |
| muladdmodid 14080 | The sum of a positive real... |
| mulp1mod1 14081 | The product of an integer ... |
| muladdmod 14082 | A real number is the sum o... |
| modmuladd 14083 | Decomposition of an intege... |
| modmuladdim 14084 | Implication of a decomposi... |
| modmuladdnn0 14085 | Implication of a decomposi... |
| negmod 14086 | The negation of a number m... |
| m1modnnsub1 14087 | Minus one modulo a positiv... |
| m1modge3gt1 14088 | Minus one modulo an intege... |
| addmodid 14089 | The sum of a positive inte... |
| addmodidr 14090 | The sum of a positive inte... |
| modadd2mod 14091 | The sum of a real number m... |
| modm1p1mod0 14092 | If a real number modulo a ... |
| modltm1p1mod 14093 | If a real number modulo a ... |
| modmul1 14094 | Multiplication property of... |
| modmul12d 14095 | Multiplication property of... |
| modnegd 14096 | Negation property of the m... |
| modadd12d 14097 | Additive property of the m... |
| modsub12d 14098 | Subtraction property of th... |
| modsubmod 14099 | The difference of a real n... |
| modsubmodmod 14100 | The difference of a real n... |
| 2txmodxeq0 14101 | Two times a positive real ... |
| 2submod 14102 | If a real number is betwee... |
| modifeq2int 14103 | If a nonnegative integer i... |
| modaddmodup 14104 | The sum of an integer modu... |
| modaddmodlo 14105 | The sum of an integer modu... |
| modmulmod 14106 | The product of a real numb... |
| modmulmodr 14107 | The product of an integer ... |
| modaddmulmod 14108 | The sum of a real number a... |
| moddi 14109 | Distribute multiplication ... |
| modsubdir 14110 | Distribute the modulo oper... |
| modeqmodmin 14111 | A real number equals the d... |
| modirr 14112 | A number modulo an irratio... |
| modfzo0difsn 14113 | For a number within a half... |
| modsumfzodifsn 14114 | The sum of a number within... |
| modlteq 14115 | Two nonnegative integers l... |
| addmodlteq 14116 | Two nonnegative integers l... |
| om2uz0i 14117 | The mapping ` G ` is a one... |
| om2uzsuci 14118 | The value of ` G ` (see ~ ... |
| om2uzuzi 14119 | The value ` G ` (see ~ om2... |
| om2uzlti 14120 | Less-than relation for ` G... |
| om2uzlt2i 14121 | The mapping ` G ` (see ~ o... |
| om2uzrani 14122 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 14123 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 14124 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 14125 | An alternative definition ... |
| om2uzrdg 14126 | A helper lemma for the val... |
| uzrdglem 14127 | A helper lemma for the val... |
| uzrdgfni 14128 | The recursive definition g... |
| uzrdg0i 14129 | Initial value of a recursi... |
| uzrdgsuci 14130 | Successor value of a recur... |
| ltweuz 14131 | ` < ` is a well-founded re... |
| ltwenn 14132 | Less than well-orders the ... |
| ltwefz 14133 | Less than well-orders a se... |
| uzenom 14134 | An upper integer set is de... |
| uzinf 14135 | An upper integer set is in... |
| nnnfi 14136 | The set of positive intege... |
| uzrdgxfr 14137 | Transfer the value of the ... |
| fzennn 14138 | The cardinality of a finit... |
| fzen2 14139 | The cardinality of a finit... |
| cardfz 14140 | The cardinality of a finit... |
| hashgf1o 14141 | ` G ` maps ` _om ` one-to-... |
| fzfi 14142 | A finite interval of integ... |
| fzfid 14143 | Commonly used special case... |
| fzofi 14144 | Half-open integer sets are... |
| fsequb 14145 | The values of a finite rea... |
| fsequb2 14146 | The values of a finite rea... |
| fseqsupcl 14147 | The values of a finite rea... |
| fseqsupubi 14148 | The values of a finite rea... |
| nn0ennn 14149 | The nonnegative integers a... |
| nnenom 14150 | The set of positive intege... |
| nnct 14151 | ` NN ` is countable. (Con... |
| uzindi 14152 | Indirect strong induction ... |
| axdc4uzlem 14153 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 14154 | A version of ~ axdc4 that ... |
| ssnn0fi 14155 | A subset of the nonnegativ... |
| rabssnn0fi 14156 | A subset of the nonnegativ... |
| uzsinds 14157 | Strong (or "total") induct... |
| nnsinds 14158 | Strong (or "total") induct... |
| nn0sinds 14159 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 14160 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 14161 | If all functions of a fini... |
| fsuppmapnn0fiubex 14162 | If all functions of a fini... |
| fsuppmapnn0fiub0 14163 | If all functions of a fini... |
| suppssfz 14164 | Condition for a function o... |
| fsuppmapnn0ub 14165 | If a function over the non... |
| fsuppmapnn0fz 14166 | If a function over the non... |
| mptnn0fsupp 14167 | A mapping from the nonnega... |
| mptnn0fsuppd 14168 | A mapping from the nonnega... |
| mptnn0fsuppr 14169 | A finitely supported mappi... |
| f13idfv 14170 | A one-to-one function with... |
| seqex 14173 | Existence of the sequence ... |
| seqeq1 14174 | Equality theorem for the s... |
| seqeq2 14175 | Equality theorem for the s... |
| seqeq3 14176 | Equality theorem for the s... |
| seqeq1d 14177 | Equality deduction for the... |
| seqeq2d 14178 | Equality deduction for the... |
| seqeq3d 14179 | Equality deduction for the... |
| seqeq123d 14180 | Equality deduction for the... |
| nfseq 14181 | Hypothesis builder for the... |
| seqval 14182 | Value of the sequence buil... |
| seqfn 14183 | The sequence builder funct... |
| seq1 14184 | Value of the sequence buil... |
| seq1i 14185 | Value of the sequence buil... |
| seqp1 14186 | Value of the sequence buil... |
| seqexw 14187 | Weak version of ~ seqex th... |
| seqp1d 14188 | Value of the sequence buil... |
| seqm1 14189 | Value of the sequence buil... |
| seqcl2 14190 | Closure properties of the ... |
| seqf2 14191 | Range of the recursive seq... |
| seqcl 14192 | Closure properties of the ... |
| seqf 14193 | Range of the recursive seq... |
| seqfveq2 14194 | Equality of sequences. (C... |
| seqfeq2 14195 | Equality of sequences. (C... |
| seqfveq 14196 | Equality of sequences. (C... |
| seqfeq 14197 | Equality of sequences. (C... |
| seqshft2 14198 | Shifting the index set of ... |
| seqres 14199 | Restricting its characteri... |
| serf 14200 | An infinite series of comp... |
| serfre 14201 | An infinite series of real... |
| monoord 14202 | Ordering relation for a mo... |
| monoord2 14203 | Ordering relation for a mo... |
| sermono 14204 | The partial sums in an inf... |
| seqsplit 14205 | Split a sequence into two ... |
| seq1p 14206 | Removing the first term fr... |
| seqcaopr3 14207 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14208 | The sum of two infinite se... |
| seqcaopr 14209 | The sum of two infinite se... |
| seqf1olem2a 14210 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14211 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14212 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14213 | Rearrange a sum via an arb... |
| seradd 14214 | The sum of two infinite se... |
| sersub 14215 | The difference of two infi... |
| seqid3 14216 | A sequence that consists e... |
| seqid 14217 | Discarding the first few t... |
| seqid2 14218 | The last few partial sums ... |
| seqhomo 14219 | Apply a homomorphism to a ... |
| seqz 14220 | If the operation ` .+ ` ha... |
| seqfeq4 14221 | Equality of series under d... |
| seqfeq3 14222 | Equality of series under d... |
| seqdistr 14223 | The distributive property ... |
| ser0 14224 | The value of the partial s... |
| ser0f 14225 | A zero-valued infinite ser... |
| serge0 14226 | A finite sum of nonnegativ... |
| serle 14227 | Comparison of partial sums... |
| ser1const 14228 | Value of the partial serie... |
| seqof 14229 | Distribute function operat... |
| seqof2 14230 | Distribute function operat... |
| expval 14233 | Value of exponentiation to... |
| expnnval 14234 | Value of exponentiation to... |
| exp0 14235 | Value of a complex number ... |
| 0exp0e1 14236 | The zeroth power of zero e... |
| exp1 14237 | Value of a complex number ... |
| expp1 14238 | Value of a complex number ... |
| expneg 14239 | Value of a complex number ... |
| expneg2 14240 | Value of a complex number ... |
| expn1 14241 | A complex number raised to... |
| expcllem 14242 | Lemma for proving nonnegat... |
| expcl2lem 14243 | Lemma for proving integer ... |
| nnexpcl 14244 | Closure of exponentiation ... |
| nn0expcl 14245 | Closure of exponentiation ... |
| zexpcl 14246 | Closure of exponentiation ... |
| qexpcl 14247 | Closure of exponentiation ... |
| reexpcl 14248 | Closure of exponentiation ... |
| expcl 14249 | Closure law for nonnegativ... |
| rpexpcl 14250 | Closure law for integer ex... |
| qexpclz 14251 | Closure of integer exponen... |
| reexpclz 14252 | Closure of integer exponen... |
| expclzlem 14253 | Lemma for ~ expclz . (Con... |
| expclz 14254 | Closure law for integer ex... |
| m1expcl2 14255 | Closure of integer exponen... |
| m1expcl 14256 | Closure of exponentiation ... |
| zexpcld 14257 | Closure of exponentiation ... |
| nn0expcli 14258 | Closure of exponentiation ... |
| nn0sqcl 14259 | The square of a nonnegativ... |
| expm1t 14260 | Exponentiation in terms of... |
| 1exp 14261 | Value of 1 raised to an in... |
| expeq0 14262 | A positive integer power i... |
| expne0 14263 | A positive integer power i... |
| expne0i 14264 | An integer power is nonzer... |
| expgt0 14265 | A positive real raised to ... |
| expnegz 14266 | Value of a nonzero complex... |
| 0exp 14267 | Value of zero raised to a ... |
| expge0 14268 | A nonnegative real raised ... |
| expge1 14269 | A real greater than or equ... |
| expgt1 14270 | A real greater than 1 rais... |
| mulexp 14271 | Nonnegative integer expone... |
| mulexpz 14272 | Integer exponentiation of ... |
| exprec 14273 | Integer exponentiation of ... |
| expadd 14274 | Sum of exponents law for n... |
| expaddzlem 14275 | Lemma for ~ expaddz . (Co... |
| expaddz 14276 | Sum of exponents law for i... |
| expmul 14277 | Product of exponents law f... |
| expmulz 14278 | Product of exponents law f... |
| m1expeven 14279 | Exponentiation of negative... |
| expsub 14280 | Exponent subtraction law f... |
| expp1z 14281 | Value of a nonzero complex... |
| expm1 14282 | Value of a nonzero complex... |
| expdiv 14283 | Nonnegative integer expone... |
| sqval 14284 | Value of the square of a c... |
| sqneg 14285 | The square of the negative... |
| sqnegd 14286 | The square of the negative... |
| sqsubswap 14287 | Swap the order of subtract... |
| sqcl 14288 | Closure of square. (Contr... |
| sqmul 14289 | Distribution of squaring o... |
| sqeq0 14290 | A complex number is zero i... |
| sqdiv 14291 | Distribution of squaring o... |
| sqdivid 14292 | The square of a nonzero co... |
| sqne0 14293 | A complex number is nonzer... |
| resqcl 14294 | Closure of squaring in rea... |
| resqcld 14295 | Closure of squaring in rea... |
| sqgt0 14296 | The square of a nonzero re... |
| sqn0rp 14297 | The square of a nonzero re... |
| nnsqcl 14298 | The positive naturals are ... |
| zsqcl 14299 | Integers are closed under ... |
| qsqcl 14300 | The square of a rational i... |
| sq11 14301 | The square function is one... |
| nn0sq11 14302 | The square function is one... |
| lt2sq 14303 | The square function is inc... |
| le2sq 14304 | The square function is non... |
| le2sq2 14305 | The square function is non... |
| sqge0 14306 | The square of a real is no... |
| sqge0d 14307 | The square of a real is no... |
| zsqcl2 14308 | The square of an integer i... |
| 0expd 14309 | Value of zero raised to a ... |
| exp0d 14310 | Value of a complex number ... |
| exp1d 14311 | Value of a complex number ... |
| expeq0d 14312 | If a positive integer powe... |
| sqvald 14313 | Value of square. Inferenc... |
| sqcld 14314 | Closure of square. (Contr... |
| sqeq0d 14315 | A number is zero iff its s... |
| expcld 14316 | Closure law for nonnegativ... |
| expp1d 14317 | Value of a complex number ... |
| expaddd 14318 | Sum of exponents law for n... |
| expmuld 14319 | Product of exponents law f... |
| sqrecd 14320 | Square of reciprocal is re... |
| expclzd 14321 | Closure law for integer ex... |
| expne0d 14322 | A nonnegative integer powe... |
| expnegd 14323 | Value of a nonzero complex... |
| exprecd 14324 | An integer power of a reci... |
| expp1zd 14325 | Value of a nonzero complex... |
| expm1d 14326 | Value of a nonzero complex... |
| expsubd 14327 | Exponent subtraction law f... |
| sqmuld 14328 | Distribution of squaring o... |
| sqdivd 14329 | Distribution of squaring o... |
| expdivd 14330 | Nonnegative integer expone... |
| mulexpd 14331 | Nonnegative integer expone... |
| znsqcld 14332 | The square of a nonzero in... |
| reexpcld 14333 | Closure of exponentiation ... |
| expge0d 14334 | A nonnegative real raised ... |
| expge1d 14335 | A real greater than or equ... |
| ltexp2a 14336 | Exponent ordering relation... |
| expmordi 14337 | Base ordering relationship... |
| rpexpmord 14338 | Base ordering relationship... |
| expcan 14339 | Cancellation law for integ... |
| ltexp2 14340 | Strict ordering law for ex... |
| leexp2 14341 | Ordering law for exponenti... |
| leexp2a 14342 | Weak ordering relationship... |
| ltexp2r 14343 | The integer powers of a fi... |
| leexp2r 14344 | Weak ordering relationship... |
| leexp1a 14345 | Weak base ordering relatio... |
| leexp1ad 14346 | Weak base ordering relatio... |
| exple1 14347 | A real between 0 and 1 inc... |
| expubnd 14348 | An upper bound on ` A ^ N ... |
| sumsqeq0 14349 | The sum of two squres of r... |
| sqvali 14350 | Value of square. Inferenc... |
| sqcli 14351 | Closure of square. (Contr... |
| sqeq0i 14352 | A complex number is zero i... |
| sqrecii 14353 | The square of a reciprocal... |
| sqmuli 14354 | Distribution of squaring o... |
| sqdivi 14355 | Distribution of squaring o... |
| resqcli 14356 | Closure of square in reals... |
| sqgt0i 14357 | The square of a nonzero re... |
| sqge0i 14358 | The square of a real is no... |
| lt2sqi 14359 | The square function on non... |
| le2sqi 14360 | The square function on non... |
| sq11i 14361 | The square function is one... |
| sq0 14362 | The square of 0 is 0. (Co... |
| sq0i 14363 | If a number is zero, then ... |
| sq0id 14364 | If a number is zero, then ... |
| sq1 14365 | The square of 1 is 1. (Co... |
| neg1sqe1 14366 | The square of ` -u 1 ` is ... |
| sq2 14367 | The square of 2 is 4. (Co... |
| sq3 14368 | The square of 3 is 9. (Co... |
| sq4e2t8 14369 | The square of 4 is 2 times... |
| cu2 14370 | The cube of 2 is 8. (Cont... |
| exp4sqsq 14371 | Raising a number to the fo... |
| irec 14372 | The reciprocal of ` _i ` .... |
| i2 14373 | ` _i ` squared. (Contribu... |
| i3 14374 | ` _i ` cubed. (Contribute... |
| i4 14375 | ` _i ` to the fourth power... |
| nnlesq 14376 | A positive integer is less... |
| zzlesq 14377 | An integer is less than or... |
| iexpcyc 14378 | Taking ` _i ` to the ` K `... |
| expnass 14379 | A counterexample showing t... |
| sqlecan 14380 | Cancel one factor of a squ... |
| subsq 14381 | Factor the difference of t... |
| subsq2 14382 | Express the difference of ... |
| binom2i 14383 | The square of a binomial. ... |
| subsqi 14384 | Factor the difference of t... |
| sqeqori 14385 | The squares of two complex... |
| subsq0i 14386 | The two solutions to the d... |
| sqeqor 14387 | The squares of two complex... |
| binom2 14388 | The square of a binomial. ... |
| binom2d 14389 | Deduction form of ~ binom2... |
| binom21 14390 | Special case of ~ binom2 w... |
| binom2sub 14391 | Expand the square of a sub... |
| binom2sub1 14392 | Special case of ~ binom2su... |
| binom2subi 14393 | Expand the square of a sub... |
| mulbinom2 14394 | The square of a binomial w... |
| binom3 14395 | The cube of a binomial. (... |
| sq01 14396 | If a complex number equals... |
| zesq 14397 | An integer is even iff its... |
| nnesq 14398 | A positive integer is even... |
| crreczi 14399 | Reciprocal of a complex nu... |
| bernneq 14400 | Bernoulli's inequality, du... |
| bernneq2 14401 | Variation of Bernoulli's i... |
| bernneq3 14402 | A corollary of ~ bernneq .... |
| expnbnd 14403 | Exponentiation with a base... |
| expnlbnd 14404 | The reciprocal of exponent... |
| expnlbnd2 14405 | The reciprocal of exponent... |
| expmulnbnd 14406 | Exponentiation with a base... |
| digit2 14407 | Two ways to express the ` ... |
| digit1 14408 | Two ways to express the ` ... |
| modexp 14409 | Exponentiation property of... |
| discr1 14410 | A nonnegative quadratic fo... |
| discr 14411 | If a quadratic polynomial ... |
| expnngt1 14412 | If an integer power with a... |
| expnngt1b 14413 | An integer power with an i... |
| sqoddm1div8 14414 | A squared odd number minus... |
| nnsqcld 14415 | The naturals are closed un... |
| nnexpcld 14416 | Closure of exponentiation ... |
| nn0expcld 14417 | Closure of exponentiation ... |
| rpexpcld 14418 | Closure law for exponentia... |
| ltexp2rd 14419 | The power of a positive nu... |
| reexpclzd 14420 | Closure of exponentiation ... |
| sqgt0d 14421 | The square of a nonzero re... |
| ltexp2d 14422 | Ordering relationship for ... |
| leexp2d 14423 | Ordering law for exponenti... |
| expcand 14424 | Ordering relationship for ... |
| leexp2ad 14425 | Ordering relationship for ... |
| leexp2rd 14426 | Ordering relationship for ... |
| lt2sqd 14427 | The square function on non... |
| le2sqd 14428 | The square function on non... |
| sq11d 14429 | The square function is one... |
| ltexp1d 14430 | Elevating to a positive po... |
| ltexp1dd 14431 | Raising both sides of 'les... |
| exp11nnd 14432 | The function elevating non... |
| mulsubdivbinom2 14433 | The square of a binomial w... |
| muldivbinom2 14434 | The square of a binomial w... |
| sq10 14435 | The square of 10 is 100. ... |
| sq10e99m1 14436 | The square of 10 is 99 plu... |
| 3dec 14437 | A "decimal constructor" wh... |
| nn0le2msqi 14438 | The square function on non... |
| nn0opthlem1 14439 | A rather pretty lemma for ... |
| nn0opthlem2 14440 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14441 | An ordered pair theorem fo... |
| nn0opth2i 14442 | An ordered pair theorem fo... |
| nn0opth2 14443 | An ordered pair theorem fo... |
| facnn 14446 | Value of the factorial fun... |
| fac0 14447 | The factorial of 0. (Cont... |
| fac1 14448 | The factorial of 1. (Cont... |
| facp1 14449 | The factorial of a success... |
| fac2 14450 | The factorial of 2. (Cont... |
| fac3 14451 | The factorial of 3. (Cont... |
| fac4 14452 | The factorial of 4. (Cont... |
| facnn2 14453 | Value of the factorial fun... |
| faccl 14454 | Closure of the factorial f... |
| faccld 14455 | Closure of the factorial f... |
| facmapnn 14456 | The factorial function res... |
| facne0 14457 | The factorial function is ... |
| facdiv 14458 | A positive integer divides... |
| facndiv 14459 | No positive integer (great... |
| facwordi 14460 | Ordering property of facto... |
| faclbnd 14461 | A lower bound for the fact... |
| faclbnd2 14462 | A lower bound for the fact... |
| faclbnd3 14463 | A lower bound for the fact... |
| faclbnd4lem1 14464 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14465 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14466 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14467 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14468 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14469 | The factorial function gro... |
| faclbnd6 14470 | Geometric lower bound for ... |
| facubnd 14471 | An upper bound for the fac... |
| facavg 14472 | The product of two factori... |
| bcval 14475 | Value of the binomial coef... |
| bcval2 14476 | Value of the binomial coef... |
| bcval3 14477 | Value of the binomial coef... |
| bcval4 14478 | Value of the binomial coef... |
| bcrpcl 14479 | Closure of the binomial co... |
| bccmpl 14480 | "Complementing" its second... |
| bcn0 14481 | ` N ` choose 0 is 1. Rema... |
| bc0k 14482 | The binomial coefficient "... |
| bcnn 14483 | ` N ` choose ` N ` is 1. ... |
| bcn1 14484 | Binomial coefficient: ` N ... |
| bcnp1n 14485 | Binomial coefficient: ` N ... |
| bcm1k 14486 | The proportion of one bino... |
| bcp1n 14487 | The proportion of one bino... |
| bcp1nk 14488 | The proportion of one bino... |
| bcval5 14489 | Write out the top and bott... |
| bcn2 14490 | Binomial coefficient: ` N ... |
| bcp1m1 14491 | Compute the binomial coeff... |
| bcpasc 14492 | Pascal's rule for the bino... |
| bccl 14493 | A binomial coefficient, in... |
| bccl2 14494 | A binomial coefficient, in... |
| bcn2m1 14495 | Compute the binomial coeff... |
| bcn2p1 14496 | Compute the binomial coeff... |
| permnn 14497 | The number of permutations... |
| bcnm1 14498 | The binomial coefficient o... |
| 4bc3eq4 14499 | The value of four choose t... |
| 4bc2eq6 14500 | The value of four choose t... |
| hashkf 14503 | The finite part of the siz... |
| hashgval 14504 | The value of the ` # ` fun... |
| hashginv 14505 | The converse of ` G ` maps... |
| hashinf 14506 | The value of the ` # ` fun... |
| hashbnd 14507 | If ` A ` has size bounded ... |
| hashfxnn0 14508 | The size function is a fun... |
| hashf 14509 | The size function maps all... |
| hashxnn0 14510 | The value of the hash func... |
| hashresfn 14511 | Restriction of the domain ... |
| dmhashres 14512 | Restriction of the domain ... |
| hashnn0pnf 14513 | The value of the hash func... |
| hashnnn0genn0 14514 | If the size of a set is no... |
| hashnemnf 14515 | The size of a set is never... |
| hashv01gt1 14516 | The size of a set is eithe... |
| hashfz1 14517 | The set ` ( 1 ... N ) ` ha... |
| hashen 14518 | Two finite sets have the s... |
| hasheni 14519 | Equinumerous sets have the... |
| hasheqf1o 14520 | The size of two finite set... |
| fiinfnf1o 14521 | There is no bijection betw... |
| hasheqf1oi 14522 | The size of two sets is eq... |
| hashf1rn 14523 | The size of a finite set w... |
| hasheqf1od 14524 | The size of two sets is eq... |
| fz1eqb 14525 | Two possibly-empty 1-based... |
| hashcard 14526 | The size function of the c... |
| hashcl 14527 | Closure of the ` # ` funct... |
| hashxrcl 14528 | Extended real closure of t... |
| hashclb 14529 | Reverse closure of the ` #... |
| nfile 14530 | The size of any infinite s... |
| hashvnfin 14531 | A set of finite size is a ... |
| hashnfinnn0 14532 | The size of an infinite se... |
| isfinite4 14533 | A finite set is equinumero... |
| hasheq0 14534 | Two ways of saying a set i... |
| hashneq0 14535 | Two ways of saying a set i... |
| hashgt0n0 14536 | If the size of a set is gr... |
| hashnncl 14537 | Positive natural closure o... |
| hash0 14538 | The empty set has size zer... |
| hashelne0d 14539 | A set with an element has ... |
| hashsng 14540 | The size of a singleton. ... |
| hashen1 14541 | A set has size 1 if and on... |
| hash1elsn 14542 | A set of size 1 with a kno... |
| hashrabrsn 14543 | The size of a restricted c... |
| hashrabsn01 14544 | The size of a restricted c... |
| hashrabsn1 14545 | If the size of a restricte... |
| hashfn 14546 | A function is equinumerous... |
| fseq1hash 14547 | The value of the size func... |
| hashgadd 14548 | ` G ` maps ordinal additio... |
| hashgval2 14549 | A short expression for the... |
| hashdom 14550 | Dominance relation for the... |
| hashdomi 14551 | Non-strict order relation ... |
| hashsdom 14552 | Strict dominance relation ... |
| hashun 14553 | The size of the union of d... |
| hashun2 14554 | The size of the union of f... |
| hashun3 14555 | The size of the union of f... |
| hashinfxadd 14556 | The extended real addition... |
| hashunx 14557 | The size of the union of d... |
| hashge0 14558 | The cardinality of a set i... |
| hashgt0 14559 | The cardinality of a nonem... |
| hashge1 14560 | The cardinality of a nonem... |
| 1elfz0hash 14561 | 1 is an element of the fin... |
| hashnn0n0nn 14562 | If a nonnegative integer i... |
| hashunsng 14563 | The size of the union of a... |
| hashunsngx 14564 | The size of the union of a... |
| hashunsnggt 14565 | The size of a set is great... |
| hashprg 14566 | The size of an unordered p... |
| elprchashprn2 14567 | If one element of an unord... |
| hashprb 14568 | The size of an unordered p... |
| hashprdifel 14569 | The elements of an unorder... |
| prhash2ex 14570 | There is (at least) one se... |
| hashle00 14571 | If the size of a set is le... |
| hashgt0elex 14572 | If the size of a set is gr... |
| hashgt0elexb 14573 | The size of a set is great... |
| hashp1i 14574 | Size of a finite ordinal. ... |
| hash1 14575 | Size of a finite ordinal. ... |
| hash2 14576 | Size of a finite ordinal. ... |
| hash3 14577 | Size of a finite ordinal. ... |
| hash4 14578 | Size of a finite ordinal. ... |
| pr0hash2ex 14579 | There is (at least) one se... |
| hashss 14580 | The size of a subset is le... |
| hashpss 14581 | The size of a proper subse... |
| prsshashgt1 14582 | The size of a superset of ... |
| hashin 14583 | The size of the intersecti... |
| hashssdif 14584 | The size of the difference... |
| hashdif 14585 | The size of the difference... |
| hashdifsn 14586 | The size of the difference... |
| hashdifpr 14587 | The size of the difference... |
| hashsn01 14588 | The size of a singleton is... |
| hashsnle1 14589 | The size of a singleton is... |
| hashsnlei 14590 | Get an upper bound on a co... |
| hash1snb 14591 | The size of a set is 1 if ... |
| euhash1 14592 | The size of a set is 1 in ... |
| hash1n0 14593 | If the size of a set is 1 ... |
| hashgt12el 14594 | In a set with more than on... |
| hashgt12el2 14595 | In a set with more than on... |
| hashgt23el 14596 | A set with more than two e... |
| hashunlei 14597 | Get an upper bound on a co... |
| hashsslei 14598 | Get an upper bound on a co... |
| hashfz 14599 | Value of the numeric cardi... |
| fzsdom2 14600 | Condition for finite range... |
| hashfzo 14601 | Cardinality of a half-open... |
| hashfzo0 14602 | Cardinality of a half-open... |
| hashfzp1 14603 | Value of the numeric cardi... |
| hashfz0 14604 | Value of the numeric cardi... |
| hashxplem 14605 | Lemma for ~ hashxp . (Con... |
| hashxp 14606 | The size of the Cartesian ... |
| hashmap 14607 | The size of the set expone... |
| hashpw 14608 | The size of the power set ... |
| hashfun 14609 | A finite set is a function... |
| hashres 14610 | The number of elements of ... |
| hashreshashfun 14611 | The number of elements of ... |
| hashimarn 14612 | The size of the image of a... |
| hashimarni 14613 | If the size of the image o... |
| hashfundm 14614 | The size of a set function... |
| hashf1dmrn 14615 | The size of the domain of ... |
| hashf1dmcdm 14616 | The size of the domain of ... |
| resunimafz0 14617 | TODO-AV: Revise using ` F... |
| fnfz0hash 14618 | The size of a function on ... |
| ffz0hash 14619 | The size of a function on ... |
| fnfz0hashnn0 14620 | The size of a function on ... |
| ffzo0hash 14621 | The size of a function on ... |
| fnfzo0hash 14622 | The size of a function on ... |
| fnfzo0hashnn0 14623 | The value of the size func... |
| hashbclem 14624 | Lemma for ~ hashbc : induc... |
| hashbc 14625 | The binomial coefficient c... |
| hashfacen 14626 | The number of bijections b... |
| hashf1lem1 14627 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14628 | Lemma for ~ hashf1 . (Con... |
| hashf1 14629 | The permutation number ` |... |
| hashfac 14630 | A factorial counts the num... |
| leiso 14631 | Two ways to write a strict... |
| leisorel 14632 | Version of ~ isorel for st... |
| fz1isolem 14633 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14634 | Any finite ordered set has... |
| ishashinf 14635 | Any set that is not finite... |
| seqcoll 14636 | The function ` F ` contain... |
| seqcoll2 14637 | The function ` F ` contain... |
| phphashd 14638 | Corollary of the Pigeonhol... |
| phphashrd 14639 | Corollary of the Pigeonhol... |
| hashprlei 14640 | An unordered pair has at m... |
| hash2pr 14641 | A set of size two is an un... |
| hash2prde 14642 | A set of size two is an un... |
| hash2exprb 14643 | A set of size two is an un... |
| hash2prb 14644 | A set of size two is a pro... |
| prprrab 14645 | The set of proper pairs of... |
| nehash2 14646 | The cardinality of a set w... |
| hash2prd 14647 | A set of size two is an un... |
| hash2pwpr 14648 | If the size of a subset of... |
| hashle2pr 14649 | A nonempty set of size les... |
| hashle2prv 14650 | A nonempty subset of a pow... |
| pr2pwpr 14651 | The set of subsets of a pa... |
| hashge2el2dif 14652 | A set with size at least 2... |
| hashge2el2difr 14653 | A set with at least 2 diff... |
| hashge2el2difb 14654 | A set has size at least 2 ... |
| hashdmpropge2 14655 | The size of the domain of ... |
| hashtplei 14656 | An unordered triple has at... |
| hashtpg 14657 | The size of an unordered t... |
| hash7g 14658 | The size of an unordered s... |
| hashge3el3dif 14659 | A set with size at least 3... |
| elss2prb 14660 | An element of the set of s... |
| hash2sspr 14661 | A subset of size two is an... |
| exprelprel 14662 | If there is an element of ... |
| hash3tr 14663 | A set of size three is an ... |
| hash1to3 14664 | If the size of a set is be... |
| hash3tpde 14665 | A set of size three is an ... |
| hash3tpexb 14666 | A set of size three is an ... |
| hash3tpb 14667 | A set of size three is a p... |
| tpf1ofv0 14668 | The value of a one-to-one ... |
| tpf1ofv1 14669 | The value of a one-to-one ... |
| tpf1ofv2 14670 | The value of a one-to-one ... |
| tpf 14671 | A function into a (proper)... |
| tpfo 14672 | A function onto a (proper)... |
| tpf1o 14673 | A bijection onto a (proper... |
| fundmge2nop0 14674 | A function with a domain c... |
| fundmge2nop 14675 | A function with a domain c... |
| fun2dmnop0 14676 | A function with a domain c... |
| fun2dmnop 14677 | A function with a domain c... |
| hashdifsnp1 14678 | If the size of a set is a ... |
| fi1uzind 14679 | Properties of an ordered p... |
| brfi1uzind 14680 | Properties of a binary rel... |
| brfi1ind 14681 | Properties of a binary rel... |
| brfi1indALT 14682 | Alternate proof of ~ brfi1... |
| opfi1uzind 14683 | Properties of an ordered p... |
| opfi1ind 14684 | Properties of an ordered p... |
| iswrd 14687 | Property of being a word o... |
| wrdval 14688 | Value of the set of words ... |
| iswrdi 14689 | A zero-based sequence is a... |
| wrdf 14690 | A word is a zero-based seq... |
| wrdfd 14691 | A word is a zero-based seq... |
| iswrdb 14692 | A word over an alphabet is... |
| wrddm 14693 | The indices of a word (i.e... |
| sswrd 14694 | The set of words respects ... |
| snopiswrd 14695 | A singleton of an ordered ... |
| wrdexg 14696 | The set of words over a se... |
| wrdexb 14697 | The set of words over a se... |
| wrdexi 14698 | The set of words over a se... |
| wrdsymbcl 14699 | A symbol within a word ove... |
| wrdfn 14700 | A word is a function with ... |
| wrdv 14701 | A word over an alphabet is... |
| wrdlndm 14702 | The length of a word is no... |
| iswrdsymb 14703 | An arbitrary word is a wor... |
| wrdfin 14704 | A word is a finite set. (... |
| lencl 14705 | The length of a word is a ... |
| lennncl 14706 | The length of a nonempty w... |
| wrdffz 14707 | A word is a function from ... |
| wrdeq 14708 | Equality theorem for the s... |
| wrdeqi 14709 | Equality theorem for the s... |
| iswrddm0 14710 | A function with empty doma... |
| wrd0 14711 | The empty set is a word (t... |
| 0wrd0 14712 | The empty word is the only... |
| ffz0iswrd 14713 | A sequence with zero-based... |
| wrdsymb 14714 | A word is a word over the ... |
| nfwrd 14715 | Hypothesis builder for ` W... |
| csbwrdg 14716 | Class substitution for the... |
| wrdnval 14717 | Words of a fixed length ar... |
| wrdmap 14718 | Words as a mapping. (Cont... |
| hashwrdn 14719 | If there is only a finite ... |
| wrdnfi 14720 | If there is only a finite ... |
| wrdsymb0 14721 | A symbol at a position "ou... |
| wrdlenge1n0 14722 | A word with length at leas... |
| len0nnbi 14723 | The length of a word is a ... |
| wrdlenge2n0 14724 | A word with length at leas... |
| wrdsymb1 14725 | The first symbol of a none... |
| wrdlen1 14726 | A word of length 1 starts ... |
| fstwrdne 14727 | The first symbol of a none... |
| fstwrdne0 14728 | The first symbol of a none... |
| eqwrd 14729 | Two words are equal iff th... |
| elovmpowrd 14730 | Implications for the value... |
| elovmptnn0wrd 14731 | Implications for the value... |
| wrdred1 14732 | A word truncated by a symb... |
| wrdred1hash 14733 | The length of a word trunc... |
| lsw 14736 | Extract the last symbol of... |
| lsw0 14737 | The last symbol of an empt... |
| lsw0g 14738 | The last symbol of an empt... |
| lsw1 14739 | The last symbol of a word ... |
| lswcl 14740 | Closure of the last symbol... |
| lswlgt0cl 14741 | The last symbol of a nonem... |
| ccatfn 14744 | The concatenation operator... |
| ccatfval 14745 | Value of the concatenation... |
| ccatcl 14746 | The concatenation of two w... |
| ccatlen 14747 | The length of a concatenat... |
| ccat0 14748 | The concatenation of two w... |
| ccatval1 14749 | Value of a symbol in the l... |
| ccatval2 14750 | Value of a symbol in the r... |
| ccatval3 14751 | Value of a symbol in the r... |
| elfzelfzccat 14752 | An element of a finite set... |
| ccatvalfn 14753 | The concatenation of two w... |
| ccatdmss 14754 | The domain of a concatenat... |
| ccatsymb 14755 | The symbol at a given posi... |
| ccatfv0 14756 | The first symbol of a conc... |
| ccatval1lsw 14757 | The last symbol of the lef... |
| ccatval21sw 14758 | The first symbol of the ri... |
| ccatlid 14759 | Concatenation of a word by... |
| ccatrid 14760 | Concatenation of a word by... |
| ccatass 14761 | Associative law for concat... |
| ccatrn 14762 | The range of a concatenate... |
| ccatf1 14763 | Conditions for a concatena... |
| ccatidid 14764 | Concatenation of the empty... |
| lswccatn0lsw 14765 | The last symbol of a word ... |
| lswccat0lsw 14766 | The last symbol of a word ... |
| ccatalpha 14767 | A concatenation of two arb... |
| ccatrcl1 14768 | Reverse closure of a conca... |
| ids1 14771 | Identity function protecti... |
| s1val 14772 | Value of a singleton word.... |
| s1rn 14773 | The range of a singleton w... |
| s1eq 14774 | Equality theorem for a sin... |
| s1eqd 14775 | Equality theorem for a sin... |
| s1cl 14776 | A singleton word is a word... |
| s1cld 14777 | A singleton word is a word... |
| s1prc 14778 | Value of a singleton word ... |
| s1cli 14779 | A singleton word is a word... |
| s1len 14780 | Length of a singleton word... |
| s1nz 14781 | A singleton word is not th... |
| s1dm 14782 | The domain of a singleton ... |
| s1mapd 14783 | A singleton word is a func... |
| s1f1 14784 | Conditions for a length 1 ... |
| s1dmALT 14785 | Alternate version of ~ s1d... |
| s1fv 14786 | Sole symbol of a singleton... |
| lsws1 14787 | The last symbol of a singl... |
| eqs1 14788 | A word of length 1 is a si... |
| wrdl1exs1 14789 | A word of length 1 is a si... |
| wrdl1s1 14790 | A word of length 1 is a si... |
| s111 14791 | The singleton word functio... |
| ccatws1cl 14792 | The concatenation of a wor... |
| ccatws1clv 14793 | The concatenation of a wor... |
| ccat2s1cl 14794 | The concatenation of two s... |
| ccats1alpha 14795 | A concatenation of a word ... |
| ccatws1len 14796 | The length of the concaten... |
| ccatws1lenp1b 14797 | The length of a word is ` ... |
| wrdlenccats1lenm1 14798 | The length of a word is th... |
| ccat2s1len 14799 | The length of the concaten... |
| ccatw2s1cl 14800 | The concatenation of a wor... |
| ccatw2s1len 14801 | The length of the concaten... |
| ccats1val1 14802 | Value of a symbol in the l... |
| ccats1val2 14803 | Value of the symbol concat... |
| ccat1st1st 14804 | The first symbol of a word... |
| ccat2s1p1 14805 | Extract the first of two c... |
| ccat2s1p2 14806 | Extract the second of two ... |
| ccatw2s1ass 14807 | Associative law for a conc... |
| ccatws1n0 14808 | The concatenation of a wor... |
| ccatws1ls 14809 | The last symbol of the con... |
| lswccats1 14810 | The last symbol of a word ... |
| lswccats1fst 14811 | The last symbol of a nonem... |
| ccatw2s1p1 14812 | Extract the symbol of the ... |
| ccatw2s1p2 14813 | Extract the second of two ... |
| ccat2s1fvw 14814 | Extract a symbol of a word... |
| ccat2s1fst 14815 | The first symbol of the co... |
| swrdnznd 14818 | The value of a subword ope... |
| swrdval 14819 | Value of a subword. (Cont... |
| swrd00 14820 | A zero length substring. ... |
| swrdcl 14821 | Closure of the subword ext... |
| swrdval2 14822 | Value of the subword extra... |
| swrdlen 14823 | Length of an extracted sub... |
| swrdfv 14824 | A symbol in an extracted s... |
| swrdfv0 14825 | The first symbol in an ext... |
| swrdf 14826 | A subword of a word is a f... |
| swrdf1 14827 | Condition for a subword to... |
| swrdvalfn 14828 | Value of the subword extra... |
| swrdrn 14829 | The range of a subword of ... |
| swrdrn3 14830 | Express the range of a sub... |
| swrdlend 14831 | The value of the subword e... |
| swrdnd 14832 | The value of the subword e... |
| swrdnd2 14833 | Value of the subword extra... |
| swrdnnn0nd 14834 | The value of a subword ope... |
| swrdnd0 14835 | The value of a subword ope... |
| swrd0 14836 | A subword of an empty set ... |
| swrdrlen 14837 | Length of a right-anchored... |
| swrdlen2 14838 | Length of an extracted sub... |
| swrdfv2 14839 | A symbol in an extracted s... |
| swrdwrdsymb 14840 | A subword is a word over t... |
| swrdsb0eq 14841 | Two subwords with the same... |
| swrdsbslen 14842 | Two subwords with the same... |
| swrdspsleq 14843 | Two words have a common su... |
| swrds1 14844 | Extract a single symbol fr... |
| swrdlsw 14845 | Extract the last single sy... |
| ccatswrd 14846 | Joining two adjacent subwo... |
| swrdccat2 14847 | Recover the right half of ... |
| pfxnndmnd 14850 | The value of a prefix oper... |
| pfxval 14851 | Value of a prefix operatio... |
| pfx00 14852 | The zero length prefix is ... |
| pfx0 14853 | A prefix of an empty set i... |
| pfxval0 14854 | Value of a prefix operatio... |
| pfxcl 14855 | Closure of the prefix extr... |
| pfxmpt 14856 | Value of the prefix extrac... |
| pfxres 14857 | Value of the prefix extrac... |
| pfxf 14858 | A prefix of a word is a fu... |
| pfxfn 14859 | Value of the prefix extrac... |
| pfxfv 14860 | A symbol in a prefix of a ... |
| pfxlen 14861 | Length of a prefix. (Cont... |
| pfxid 14862 | A word is a prefix of itse... |
| pfxrn 14863 | The range of a prefix of a... |
| pfxn0 14864 | A prefix consisting of at ... |
| pfxnd 14865 | The value of a prefix oper... |
| pfxnd0 14866 | The value of a prefix oper... |
| pfxwrdsymb 14867 | A prefix of a word is a wo... |
| addlenpfx 14868 | The sum of the lengths of ... |
| pfxfv0 14869 | The first symbol of a pref... |
| pfxtrcfv 14870 | A symbol in a word truncat... |
| pfxtrcfv0 14871 | The first symbol in a word... |
| pfxfvlsw 14872 | The last symbol in a nonem... |
| pfxeq 14873 | The prefixes of two words ... |
| pfxtrcfvl 14874 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14875 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14876 | Two (nonempty) words are e... |
| disjwrdpfx 14877 | Sets of words are disjoint... |
| ccatpfx 14878 | Concatenating a prefix wit... |
| pfxccat1 14879 | Recover the left half of a... |
| pfx1 14880 | The prefix of length one o... |
| swrdswrdlem 14881 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14882 | A subword of a subword is ... |
| pfxswrd 14883 | A prefix of a subword is a... |
| swrdpfx 14884 | A subword of a prefix is a... |
| pfxpfx 14885 | A prefix of a prefix is a ... |
| pfxpfxid 14886 | A prefix of a prefix with ... |
| pfxcctswrd 14887 | The concatenation of the p... |
| lenpfxcctswrd 14888 | The length of the concaten... |
| lenrevpfxcctswrd 14889 | The length of the concaten... |
| pfxlswccat 14890 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14891 | The last symbol of a word ... |
| ccats1pfxeqrex 14892 | There exists a symbol such... |
| ccatopth 14893 | An ~ opth -like theorem fo... |
| ccatopth2 14894 | An ~ opth -like theorem fo... |
| ccatlcan 14895 | Concatenation of words is ... |
| ccatrcan 14896 | Concatenation of words is ... |
| wrdeqs1cat 14897 | Decompose a nonempty word ... |
| cats1un 14898 | Express a word with an ext... |
| wrdind 14899 | Perform induction over the... |
| wrd2ind 14900 | Perform induction over the... |
| swrdccatfn 14901 | The subword of a concatena... |
| swrdccatin1 14902 | The subword of a concatena... |
| pfxccatin12lem4 14903 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14904 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14905 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14906 | The subword of a concatena... |
| pfxccatin12lem2c 14907 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14908 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14909 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14910 | The subword of a concatena... |
| pfxccat3 14911 | The subword of a concatena... |
| swrdccat 14912 | The subword of a concatena... |
| pfxccatpfx1 14913 | A prefix of a concatenatio... |
| pfxccatpfx2 14914 | A prefix of a concatenatio... |
| pfxccat3a 14915 | A prefix of a concatenatio... |
| swrdccat3blem 14916 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14917 | A suffix of a concatenatio... |
| pfxccatid 14918 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14919 | A word is a prefix of a wo... |
| swrdccatin1d 14920 | The subword of a concatena... |
| swrdccatin2d 14921 | The subword of a concatena... |
| pfxccatin12d 14922 | The subword of a concatena... |
| reuccatpfxs1lem 14923 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14924 | There is a unique word hav... |
| reuccatpfxs1v 14925 | There is a unique word hav... |
| splval 14928 | Value of the substring rep... |
| splcl 14929 | Closure of the substring r... |
| splid 14930 | Splicing a subword for the... |
| spllen 14931 | The length of a splice. (... |
| splfv1 14932 | Symbols to the left of a s... |
| splfv2a 14933 | Symbols within the replace... |
| splval2 14934 | Value of a splice, assumin... |
| revval 14937 | Value of the word reversin... |
| revcl 14938 | The reverse of a word is a... |
| revlen 14939 | The reverse of a word has ... |
| revfv 14940 | Reverse of a word at a poi... |
| rev0 14941 | The empty word is its own ... |
| revs1 14942 | Singleton words are their ... |
| revccat 14943 | Antiautomorphic property o... |
| revrev 14944 | Reversal is an involution ... |
| revpfxsfxrev 14945 | The reverse of a prefix of... |
| swrdrevpfx 14946 | A subword expressed in ter... |
| reps 14949 | Construct a function mappi... |
| repsundef 14950 | A function mapping a half-... |
| repsconst 14951 | Construct a function mappi... |
| repsf 14952 | The constructed function m... |
| repswsymb 14953 | The symbols of a "repeated... |
| repsw 14954 | A function mapping a half-... |
| repswlen 14955 | The length of a "repeated ... |
| repsw0 14956 | The "repeated symbol word"... |
| repsdf2 14957 | Alternative definition of ... |
| repswsymball 14958 | All the symbols of a "repe... |
| repswsymballbi 14959 | A word is a "repeated symb... |
| repswfsts 14960 | The first symbol of a none... |
| repswlsw 14961 | The last symbol of a nonem... |
| repsw1 14962 | The "repeated symbol word"... |
| repswswrd 14963 | A subword of a "repeated s... |
| repswpfx 14964 | A prefix of a repeated sym... |
| repswccat 14965 | The concatenation of two "... |
| repswrevw 14966 | The reverse of a "repeated... |
| cshfn 14969 | Perform a cyclical shift f... |
| cshword 14970 | Perform a cyclical shift f... |
| cshnz 14971 | A cyclical shift is the em... |
| 0csh0 14972 | Cyclically shifting an emp... |
| cshw0 14973 | A word cyclically shifted ... |
| cshwmodn 14974 | Cyclically shifting a word... |
| cshwsublen 14975 | Cyclically shifting a word... |
| cshwn 14976 | A word cyclically shifted ... |
| cshwcl 14977 | A cyclically shifted word ... |
| cshwlen 14978 | The length of a cyclically... |
| cshwf 14979 | A cyclically shifted word ... |
| cshwfn 14980 | A cyclically shifted word ... |
| cshwrn 14981 | The range of a cyclically ... |
| cshwidxmod 14982 | The symbol at a given inde... |
| cshwidxmodr 14983 | The symbol at a given inde... |
| cshwidx0mod 14984 | The symbol at index 0 of a... |
| cshwidx0 14985 | The symbol at index 0 of a... |
| cshwidxm1 14986 | The symbol at index ((n-N)... |
| cshwidxm 14987 | The symbol at index (n-N) ... |
| cshwidxn 14988 | The symbol at index (n-1) ... |
| cshf1 14989 | Cyclically shifting a word... |
| cshinj 14990 | If a word is injectiv (reg... |
| repswcshw 14991 | A cyclically shifted "repe... |
| 2cshw 14992 | Cyclically shifting a word... |
| 2cshwid 14993 | Cyclically shifting a word... |
| lswcshw 14994 | The last symbol of a word ... |
| 2cshwcom 14995 | Cyclically shifting a word... |
| cshwleneq 14996 | If the results of cyclical... |
| 3cshw 14997 | Cyclically shifting a word... |
| cshweqdif2 14998 | If cyclically shifting two... |
| cshweqdifid 14999 | If cyclically shifting a w... |
| cshweqrep 15000 | If cyclically shifting a w... |
| cshw1 15001 | If cyclically shifting a w... |
| cshw1repsw 15002 | If cyclically shifting a w... |
| cshwsexa 15003 | The class of (different!) ... |
| 2cshwcshw 15004 | If a word is a cyclically ... |
| scshwfzeqfzo 15005 | For a nonempty word the se... |
| cshwcshid 15006 | A cyclically shifted word ... |
| cshwcsh2id 15007 | A cyclically shifted word ... |
| cshimadifsn 15008 | The image of a cyclically ... |
| cshimadifsn0 15009 | The image of a cyclically ... |
| wrdco 15010 | Mapping a word by a functi... |
| lenco 15011 | Length of a mapped word is... |
| s1co 15012 | Mapping of a singleton wor... |
| revco 15013 | Mapping of words (i.e., a ... |
| ccatco 15014 | Mapping of words commutes ... |
| cshco 15015 | Mapping of words commutes ... |
| swrdco 15016 | Mapping of words commutes ... |
| pfxco 15017 | Mapping of words commutes ... |
| lswco 15018 | Mapping of (nonempty) word... |
| repsco 15019 | Mapping of words commutes ... |
| cats1cld 15034 | Closure of concatenation w... |
| cats1co 15035 | Closure of concatenation w... |
| cats1cli 15036 | Closure of concatenation w... |
| cats1fvn 15037 | The last symbol of a conca... |
| cats1fv 15038 | A symbol other than the la... |
| cats1len 15039 | The length of concatenatio... |
| cats1cat 15040 | Closure of concatenation w... |
| cats2cat 15041 | Closure of concatenation o... |
| s2eqd 15042 | Equality theorem for a dou... |
| s3eqd 15043 | Equality theorem for a len... |
| s4eqd 15044 | Equality theorem for a len... |
| s5eqd 15045 | Equality theorem for a len... |
| s6eqd 15046 | Equality theorem for a len... |
| s7eqd 15047 | Equality theorem for a len... |
| s8eqd 15048 | Equality theorem for a len... |
| s3eq2 15049 | Equality theorem for a len... |
| s2cld 15050 | A doubleton word is a word... |
| s3cld 15051 | A length 3 string is a wor... |
| s4cld 15052 | A length 4 string is a wor... |
| s5cld 15053 | A length 5 string is a wor... |
| s6cld 15054 | A length 6 string is a wor... |
| s7cld 15055 | A length 7 string is a wor... |
| s8cld 15056 | A length 8 string is a wor... |
| s2cl 15057 | A doubleton word is a word... |
| s3cl 15058 | A length 3 string is a wor... |
| s2cli 15059 | A doubleton word is a word... |
| s3cli 15060 | A length 3 string is a wor... |
| s4cli 15061 | A length 4 string is a wor... |
| s5cli 15062 | A length 5 string is a wor... |
| s6cli 15063 | A length 6 string is a wor... |
| s7cli 15064 | A length 7 string is a wor... |
| s8cli 15065 | A length 8 string is a wor... |
| s2fv0 15066 | Extract the first symbol f... |
| s2fv1 15067 | Extract the second symbol ... |
| s2len 15068 | The length of a doubleton ... |
| s2dm 15069 | The domain of a doubleton ... |
| s2mapd 15070 | A doubleton word is a func... |
| s3fv0 15071 | Extract the first symbol f... |
| s3fv1 15072 | Extract the second symbol ... |
| s3fv2 15073 | Extract the third symbol f... |
| s3len 15074 | The length of a length 3 s... |
| s4fv0 15075 | Extract the first symbol f... |
| s4fv1 15076 | Extract the second symbol ... |
| s4fv2 15077 | Extract the third symbol f... |
| s4fv3 15078 | Extract the fourth symbol ... |
| s4len 15079 | The length of a length 4 s... |
| s5len 15080 | The length of a length 5 s... |
| s6len 15081 | The length of a length 6 s... |
| s7len 15082 | The length of a length 7 s... |
| s8len 15083 | The length of a length 8 s... |
| lsws2 15084 | The last symbol of a doubl... |
| lsws3 15085 | The last symbol of a 3 let... |
| lsws4 15086 | The last symbol of a 4 let... |
| s2prop 15087 | A length 2 word is an unor... |
| s2dmALT 15088 | Alternate version of ~ s2d... |
| s3tpop 15089 | A length 3 word is an unor... |
| s4prop 15090 | A length 4 word is a union... |
| s3fn 15091 | A length 3 word is a funct... |
| funcnvs1 15092 | The converse of a singleto... |
| funcnvs2 15093 | The converse of a length 2... |
| funcnvs3 15094 | The converse of a length 3... |
| funcnvs4 15095 | The converse of a length 4... |
| s2f1o 15096 | A length 2 word with mutua... |
| f1oun2prg 15097 | A union of unordered pairs... |
| s4f1o 15098 | A length 4 word with mutua... |
| s4dom 15099 | The domain of a length 4 w... |
| s2co 15100 | Mapping a doubleton word b... |
| s3co 15101 | Mapping a length 3 string ... |
| s0s1 15102 | Concatenation of fixed len... |
| s1s2 15103 | Concatenation of fixed len... |
| s1s3 15104 | Concatenation of fixed len... |
| s1s4 15105 | Concatenation of fixed len... |
| s1s5 15106 | Concatenation of fixed len... |
| s1s6 15107 | Concatenation of fixed len... |
| s1s7 15108 | Concatenation of fixed len... |
| s2s2 15109 | Concatenation of fixed len... |
| s4s2 15110 | Concatenation of fixed len... |
| s4s3 15111 | Concatenation of fixed len... |
| s4s4 15112 | Concatenation of fixed len... |
| s3s4 15113 | Concatenation of fixed len... |
| s2s5 15114 | Concatenation of fixed len... |
| s5s2 15115 | Concatenation of fixed len... |
| s2eq2s1eq 15116 | Two length 2 words are equ... |
| s2eq2seq 15117 | Two length 2 words are equ... |
| s3eqs2s1eq 15118 | Two length 3 words are equ... |
| s3eq3seq 15119 | Two length 3 words are equ... |
| swrds2 15120 | Extract two adjacent symbo... |
| swrds2m 15121 | Extract two adjacent symbo... |
| wrdlen2i 15122 | Implications of a word of ... |
| wrd2pr2op 15123 | A word of length two repre... |
| wrdlen2 15124 | A word of length two. (Co... |
| wrdlen2s2 15125 | A word of length two as do... |
| wrdl2exs2 15126 | A word of length two is a ... |
| pfx2 15127 | A prefix of length two. (... |
| wrd3tpop 15128 | A word of length three rep... |
| wrdlen3s3 15129 | A word of length three as ... |
| s3rex 15130 | Membership in a family of ... |
| s3rexrd 15131 | Converse of ~ s3rex , dedu... |
| repsw2 15132 | The "repeated symbol word"... |
| repsw3 15133 | The "repeated symbol word"... |
| swrd2lsw 15134 | Extract the last two symbo... |
| 2swrd2eqwrdeq 15135 | Two words of length at lea... |
| ccatw2s1ccatws2 15136 | The concatenation of a wor... |
| ccat2s1fvwALT 15137 | Alternate proof of ~ ccat2... |
| wwlktovf 15138 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 15139 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 15140 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 15141 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 15142 | There is a bijection betwe... |
| eqwrds3 15143 | A word is equal with a len... |
| wrdl3s3 15144 | A word of length 3 is a le... |
| s2rn 15145 | Range of a length 2 string... |
| s3rn 15146 | Range of a length 3 string... |
| s7rn 15147 | Range of a length 7 string... |
| s7f1o 15148 | A length 7 word with mutua... |
| s3sndisj 15149 | The singletons consisting ... |
| s3iunsndisj 15150 | The union of singletons co... |
| ofccat 15151 | Letterwise operations on w... |
| ofs1 15152 | Letterwise operations on a... |
| ofs2 15153 | Letterwise operations on a... |
| coss12d 15154 | Subset deduction for compo... |
| trrelssd 15155 | The composition of subclas... |
| xpcogend 15156 | The most interesting case ... |
| xpcoidgend 15157 | If two classes are not dis... |
| cotr2g 15158 | Two ways of saying that th... |
| cotr2 15159 | Two ways of saying a relat... |
| cotr3 15160 | Two ways of saying a relat... |
| coemptyd 15161 | Deduction about compositio... |
| xptrrel 15162 | The cross product is alway... |
| 0trrel 15163 | The empty class is a trans... |
| cleq1lem 15164 | Equality implies bijection... |
| cleq1 15165 | Equality of relations impl... |
| clsslem 15166 | The closure of a subclass ... |
| trcleq1 15171 | Equality of relations impl... |
| trclsslem 15172 | The transitive closure (as... |
| trcleq2lem 15173 | Equality implies bijection... |
| cvbtrcl 15174 | Change of bound variable i... |
| trcleq12lem 15175 | Equality implies bijection... |
| trclexlem 15176 | Existence of relation impl... |
| trclublem 15177 | If a relation exists then ... |
| trclubi 15178 | The Cartesian product of t... |
| trclubgi 15179 | The union with the Cartesi... |
| trclub 15180 | The Cartesian product of t... |
| trclubg 15181 | The union with the Cartesi... |
| trclfv 15182 | The transitive closure of ... |
| brintclab 15183 | Two ways to express a bina... |
| brtrclfv 15184 | Two ways of expressing the... |
| brcnvtrclfv 15185 | Two ways of expressing the... |
| brtrclfvcnv 15186 | Two ways of expressing the... |
| brcnvtrclfvcnv 15187 | Two ways of expressing the... |
| trclfvss 15188 | The transitive closure (as... |
| trclfvub 15189 | The transitive closure of ... |
| trclfvlb 15190 | The transitive closure of ... |
| trclfvcotr 15191 | The transitive closure of ... |
| trclfvlb2 15192 | The transitive closure of ... |
| trclfvlb3 15193 | The transitive closure of ... |
| cotrtrclfv 15194 | The transitive closure of ... |
| trclidm 15195 | The transitive closure of ... |
| trclun 15196 | Transitive closure of a un... |
| trclfvg 15197 | The value of the transitiv... |
| trclfvcotrg 15198 | The value of the transitiv... |
| reltrclfv 15199 | The transitive closure of ... |
| dmtrclfv 15200 | The domain of the transiti... |
| reldmrelexp 15203 | The domain of the repeated... |
| relexp0g 15204 | A relation composed zero t... |
| relexp0 15205 | A relation composed zero t... |
| relexp0d 15206 | A relation composed zero t... |
| relexpsucnnr 15207 | A reduction for relation e... |
| relexp1g 15208 | A relation composed once i... |
| dfid5 15209 | Identity relation is equal... |
| dfid6 15210 | Identity relation expresse... |
| relexp1d 15211 | A relation composed once i... |
| relexpsucnnl 15212 | A reduction for relation e... |
| relexpsucl 15213 | A reduction for relation e... |
| relexpsucr 15214 | A reduction for relation e... |
| relexpsucrd 15215 | A reduction for relation e... |
| relexpsucld 15216 | A reduction for relation e... |
| relexpcnv 15217 | Commutation of converse an... |
| relexpcnvd 15218 | Commutation of converse an... |
| relexp0rel 15219 | The exponentiation of a cl... |
| relexprelg 15220 | The exponentiation of a cl... |
| relexprel 15221 | The exponentiation of a re... |
| relexpreld 15222 | The exponentiation of a re... |
| relexpnndm 15223 | The domain of an exponenti... |
| relexpdmg 15224 | The domain of an exponenti... |
| relexpdm 15225 | The domain of an exponenti... |
| relexpdmd 15226 | The domain of an exponenti... |
| relexpnnrn 15227 | The range of an exponentia... |
| relexprng 15228 | The range of an exponentia... |
| relexprn 15229 | The range of an exponentia... |
| relexprnd 15230 | The range of an exponentia... |
| relexpfld 15231 | The field of an exponentia... |
| relexpfldd 15232 | The field of an exponentia... |
| relexpaddnn 15233 | Relation composition becom... |
| relexpuzrel 15234 | The exponentiation of a cl... |
| relexpaddg 15235 | Relation composition becom... |
| relexpaddd 15236 | Relation composition becom... |
| rtrclreclem1 15239 | The reflexive, transitive ... |
| dfrtrclrec2 15240 | If two elements are connec... |
| rtrclreclem2 15241 | The reflexive, transitive ... |
| rtrclreclem3 15242 | The reflexive, transitive ... |
| rtrclreclem4 15243 | The reflexive, transitive ... |
| dfrtrcl2 15244 | The two definitions ` t* `... |
| relexpindlem 15245 | Principle of transitive in... |
| relexpind 15246 | Principle of transitive in... |
| rtrclind 15247 | Principle of transitive in... |
| shftlem 15250 | Two ways to write a shifte... |
| shftuz 15251 | A shift of the upper integ... |
| shftfval 15252 | The value of the sequence ... |
| shftdm 15253 | Domain of a relation shift... |
| shftfib 15254 | Value of a fiber of the re... |
| shftfn 15255 | Functionality and domain o... |
| shftval 15256 | Value of a sequence shifte... |
| shftval2 15257 | Value of a sequence shifte... |
| shftval3 15258 | Value of a sequence shifte... |
| shftval4 15259 | Value of a sequence shifte... |
| shftval5 15260 | Value of a shifted sequenc... |
| shftf 15261 | Functionality of a shifted... |
| 2shfti 15262 | Composite shift operations... |
| shftidt2 15263 | Identity law for the shift... |
| shftidt 15264 | Identity law for the shift... |
| shftcan1 15265 | Cancellation law for the s... |
| shftcan2 15266 | Cancellation law for the s... |
| seqshft 15267 | Shifting the index set of ... |
| sgnval 15270 | Value of the signum functi... |
| sgn0 15271 | The signum of 0 is 0. (Co... |
| sgnp 15272 | The signum of a positive e... |
| sgnrrp 15273 | The signum of a positive r... |
| sgn1 15274 | The signum of 1 is 1. (Co... |
| sgnpnf 15275 | The signum of ` +oo ` is 1... |
| sgnn 15276 | The signum of a negative e... |
| sgnmnf 15277 | The signum of ` -oo ` is -... |
| sgndm 15278 | The domain of the signum f... |
| sgncl 15279 | Closure of the signum: Th... |
| sgnrn 15280 | The range of the signum fu... |
| sgnfo 15281 | The signum function as ont... |
| sgnneg 15282 | Negation of the signum. (... |
| sgn3da 15283 | A conditional containing a... |
| sgnclre 15284 | Closure of the signum for ... |
| sgn0bi 15285 | Zero signum. (Contributed... |
| sgnnbi 15286 | Negative signum. (Contrib... |
| sgnpbi 15287 | Positive signum. (Contrib... |
| sgnsub 15288 | Signum of a difference wit... |
| sgnmul 15289 | Signum of a product. (Con... |
| sgnmulrp2 15290 | Multiplication by a positi... |
| sgnmulsgn 15291 | If two real numbers are of... |
| cjval 15298 | The value of the conjugate... |
| cjth 15299 | The defining property of t... |
| cjf 15300 | Domain and codomain of the... |
| cjcl 15301 | The conjugate of a complex... |
| reval 15302 | The value of the real part... |
| imval 15303 | The value of the imaginary... |
| imre 15304 | The imaginary part of a co... |
| reim 15305 | The real part of a complex... |
| recl 15306 | The real part of a complex... |
| imcl 15307 | The imaginary part of a co... |
| ref 15308 | Domain and codomain of the... |
| imf 15309 | Domain and codomain of the... |
| crre 15310 | The real part of a complex... |
| crim 15311 | The real part of a complex... |
| replim 15312 | Reconstruct a complex numb... |
| remim 15313 | Value of the conjugate of ... |
| reim0 15314 | The imaginary part of a re... |
| reim0b 15315 | A number is real iff its i... |
| rereb 15316 | A number is real iff it eq... |
| mulre 15317 | A product with a nonzero r... |
| rere 15318 | A real number equals its r... |
| cjreb 15319 | A number is real iff it eq... |
| recj 15320 | Real part of a complex con... |
| reneg 15321 | Real part of negative. (C... |
| readd 15322 | Real part distributes over... |
| resub 15323 | Real part distributes over... |
| remullem 15324 | Lemma for ~ remul , ~ immu... |
| remul 15325 | Real part of a product. (... |
| remul2 15326 | Real part of a product. (... |
| rediv 15327 | Real part of a division. ... |
| imcj 15328 | Imaginary part of a comple... |
| imneg 15329 | The imaginary part of a ne... |
| imadd 15330 | Imaginary part distributes... |
| imsub 15331 | Imaginary part distributes... |
| immul 15332 | Imaginary part of a produc... |
| immul2 15333 | Imaginary part of a produc... |
| imdiv 15334 | Imaginary part of a divisi... |
| cjre 15335 | A real number equals its c... |
| cjcj 15336 | The conjugate of the conju... |
| cjadd 15337 | Complex conjugate distribu... |
| cjmul 15338 | Complex conjugate distribu... |
| ipcnval 15339 | Standard inner product on ... |
| cjmulrcl 15340 | A complex number times its... |
| cjmulval 15341 | A complex number times its... |
| cjmulge0 15342 | A complex number times its... |
| cjneg 15343 | Complex conjugate of negat... |
| addcj 15344 | A number plus its conjugat... |
| cjsub 15345 | Complex conjugate distribu... |
| cjexp 15346 | Complex conjugate of posit... |
| imval2 15347 | The imaginary part of a nu... |
| re0 15348 | The real part of zero. (C... |
| im0 15349 | The imaginary part of zero... |
| re1 15350 | The real part of one. (Co... |
| im1 15351 | The imaginary part of one.... |
| rei 15352 | The real part of ` _i ` . ... |
| imi 15353 | The imaginary part of ` _i... |
| cj0 15354 | The conjugate of zero. (C... |
| cji 15355 | The complex conjugate of t... |
| cjreim 15356 | The conjugate of a represe... |
| cjreim2 15357 | The conjugate of the repre... |
| cj11 15358 | Complex conjugate is a one... |
| cjne0 15359 | A number is nonzero iff it... |
| cjdiv 15360 | Complex conjugate distribu... |
| cnrecnv 15361 | The inverse to the canonic... |
| sqeqd 15362 | A deduction for showing tw... |
| recli 15363 | The real part of a complex... |
| imcli 15364 | The imaginary part of a co... |
| cjcli 15365 | Closure law for complex co... |
| replimi 15366 | Construct a complex number... |
| cjcji 15367 | The conjugate of the conju... |
| reim0bi 15368 | A number is real iff its i... |
| rerebi 15369 | A real number equals its r... |
| cjrebi 15370 | A number is real iff it eq... |
| recji 15371 | Real part of a complex con... |
| imcji 15372 | Imaginary part of a comple... |
| cjmulrcli 15373 | A complex number times its... |
| cjmulvali 15374 | A complex number times its... |
| cjmulge0i 15375 | A complex number times its... |
| renegi 15376 | Real part of negative. (C... |
| imnegi 15377 | Imaginary part of negative... |
| cjnegi 15378 | Complex conjugate of negat... |
| addcji 15379 | A number plus its conjugat... |
| readdi 15380 | Real part distributes over... |
| imaddi 15381 | Imaginary part distributes... |
| remuli 15382 | Real part of a product. (... |
| immuli 15383 | Imaginary part of a produc... |
| cjaddi 15384 | Complex conjugate distribu... |
| cjmuli 15385 | Complex conjugate distribu... |
| ipcni 15386 | Standard inner product on ... |
| cjdivi 15387 | Complex conjugate distribu... |
| crrei 15388 | The real part of a complex... |
| crimi 15389 | The imaginary part of a co... |
| recld 15390 | The real part of a complex... |
| imcld 15391 | The imaginary part of a co... |
| cjcld 15392 | Closure law for complex co... |
| replimd 15393 | Construct a complex number... |
| remimd 15394 | Value of the conjugate of ... |
| cjcjd 15395 | The conjugate of the conju... |
| reim0bd 15396 | A number is real iff its i... |
| rerebd 15397 | A real number equals its r... |
| cjrebd 15398 | A number is real iff it eq... |
| cjne0d 15399 | A number is nonzero iff it... |
| recjd 15400 | Real part of a complex con... |
| imcjd 15401 | Imaginary part of a comple... |
| cjmulrcld 15402 | A complex number times its... |
| cjmulvald 15403 | A complex number times its... |
| cjmulge0d 15404 | A complex number times its... |
| renegd 15405 | Real part of negative. (C... |
| imnegd 15406 | Imaginary part of negative... |
| cjnegd 15407 | Complex conjugate of negat... |
| addcjd 15408 | A number plus its conjugat... |
| cjexpd 15409 | Complex conjugate of posit... |
| readdd 15410 | Real part distributes over... |
| imaddd 15411 | Imaginary part distributes... |
| resubd 15412 | Real part distributes over... |
| imsubd 15413 | Imaginary part distributes... |
| remuld 15414 | Real part of a product. (... |
| immuld 15415 | Imaginary part of a produc... |
| cjaddd 15416 | Complex conjugate distribu... |
| cjmuld 15417 | Complex conjugate distribu... |
| ipcnd 15418 | Standard inner product on ... |
| cjdivd 15419 | Complex conjugate distribu... |
| rered 15420 | A real number equals its r... |
| reim0d 15421 | The imaginary part of a re... |
| cjred 15422 | A real number equals its c... |
| remul2d 15423 | Real part of a product. (... |
| immul2d 15424 | Imaginary part of a produc... |
| redivd 15425 | Real part of a division. ... |
| imdivd 15426 | Imaginary part of a divisi... |
| crred 15427 | The real part of a complex... |
| crimd 15428 | The imaginary part of a co... |
| sqrtval 15433 | Value of square root funct... |
| absval 15434 | The absolute value (modulu... |
| rennim 15435 | A real number does not lie... |
| cnpart 15436 | The specification of restr... |
| sqrt0 15437 | The square root of zero is... |
| 01sqrexlem1 15438 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15439 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15440 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15441 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15442 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15443 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15444 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15445 | Existence of a square root... |
| resqrex 15446 | Existence of a square root... |
| sqrmo 15447 | Uniqueness for the square ... |
| resqreu 15448 | Existence and uniqueness f... |
| resqrtcl 15449 | Closure of the square root... |
| resqrtthlem 15450 | Lemma for ~ resqrtth . (C... |
| resqrtth 15451 | Square root theorem over t... |
| remsqsqrt 15452 | Square of square root. (C... |
| sqrtge0 15453 | The square root function i... |
| sqrtgt0 15454 | The square root function i... |
| sqrtmul 15455 | Square root distributes ov... |
| sqrtle 15456 | Square root is monotonic. ... |
| sqrtlt 15457 | Square root is strictly mo... |
| sqrt11 15458 | The square root function i... |
| sqrt00 15459 | A square root is zero iff ... |
| rpsqrtcl 15460 | The square root of a posit... |
| sqrtdiv 15461 | Square root distributes ov... |
| sqrtneglem 15462 | The square root of a negat... |
| sqrtneg 15463 | The square root of a negat... |
| sqrtsq2 15464 | Relationship between squar... |
| sqrtsq 15465 | Square root of square. (C... |
| sqrtmsq 15466 | Square root of square. (C... |
| sqrt1 15467 | The square root of 1 is 1.... |
| sqrt4 15468 | The square root of 4 is 2.... |
| sqrt9 15469 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15470 | The square root of 2 is bo... |
| sqrtm1 15471 | The imaginary unit is the ... |
| nn0sqeq1 15472 | A natural number with squa... |
| absneg 15473 | Absolute value of the nega... |
| abscl 15474 | Real closure of absolute v... |
| abscj 15475 | The absolute value of a nu... |
| absvalsq 15476 | Square of value of absolut... |
| absvalsq2 15477 | Square of value of absolut... |
| sqabsadd 15478 | Square of absolute value o... |
| sqabssub 15479 | Square of absolute value o... |
| absval2 15480 | Value of absolute value fu... |
| abs0 15481 | The absolute value of 0. ... |
| absi 15482 | The absolute value of the ... |
| absge0 15483 | Absolute value is nonnegat... |
| absrpcl 15484 | The absolute value of a no... |
| abs00 15485 | The absolute value of a nu... |
| abs00ad 15486 | A complex number is zero i... |
| abs00bd 15487 | If a complex number is zer... |
| absreimsq 15488 | Square of the absolute val... |
| absreim 15489 | Absolute value of a number... |
| absmul 15490 | Absolute value distributes... |
| absdiv 15491 | Absolute value distributes... |
| absid 15492 | A nonnegative number is it... |
| abs1 15493 | The absolute value of one ... |
| absnid 15494 | For a negative number, its... |
| leabs 15495 | A real number is less than... |
| absor 15496 | The absolute value of a re... |
| absre 15497 | Absolute value of a real n... |
| absresq 15498 | Square of the absolute val... |
| absmod0 15499 | ` A ` is divisible by ` B ... |
| absexp 15500 | Absolute value of positive... |
| absexpz 15501 | Absolute value of integer ... |
| abssq 15502 | Square can be moved in and... |
| sqabs 15503 | The squares of two reals a... |
| absrele 15504 | The absolute value of a co... |
| absimle 15505 | The absolute value of a co... |
| max0add 15506 | The sum of the positive an... |
| absz 15507 | A real number is an intege... |
| nn0abscl 15508 | The absolute value of an i... |
| zabscl 15509 | The absolute value of an i... |
| zabs0b 15510 | An integer has an absolute... |
| abslt 15511 | Absolute value and 'less t... |
| absle 15512 | Absolute value and 'less t... |
| abssubne0 15513 | If the absolute value of a... |
| absdiflt 15514 | The absolute value of a di... |
| absdifle 15515 | The absolute value of a di... |
| elicc4abs 15516 | Membership in a symmetric ... |
| lenegsq 15517 | Comparison to a nonnegativ... |
| releabs 15518 | The real part of a number ... |
| recval 15519 | Reciprocal expressed with ... |
| absidm 15520 | The absolute value functio... |
| absgt0 15521 | The absolute value of a no... |
| nnabscl 15522 | The absolute value of a no... |
| abssub 15523 | Swapping order of subtract... |
| abssubge0 15524 | Absolute value of a nonneg... |
| abssuble0 15525 | Absolute value of a nonpos... |
| absmax 15526 | The maximum of two numbers... |
| abstri 15527 | Triangle inequality for ab... |
| abs3dif 15528 | Absolute value of differen... |
| abs2dif 15529 | Difference of absolute val... |
| abs2dif2 15530 | Difference of absolute val... |
| abs2difabs 15531 | Absolute value of differen... |
| abs1m 15532 | For any complex number, th... |
| recan 15533 | Cancellation law involving... |
| absf 15534 | Mapping domain and codomai... |
| abs3lem 15535 | Lemma involving absolute v... |
| abslem2 15536 | Lemma involving absolute v... |
| rddif 15537 | The difference between a r... |
| absrdbnd 15538 | Bound on the absolute valu... |
| fzomaxdiflem 15539 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15540 | A bound on the separation ... |
| uzin2 15541 | The upper integers are clo... |
| rexanuz 15542 | Combine two different uppe... |
| rexanre 15543 | Combine two different uppe... |
| rexfiuz 15544 | Combine finitely many diff... |
| rexuz3 15545 | Restrict the base of the u... |
| rexanuz2 15546 | Combine two different uppe... |
| r19.29uz 15547 | A version of ~ 19.29 for u... |
| r19.2uz 15548 | A version of ~ r19.2z for ... |
| rexuzre 15549 | Convert an upper real quan... |
| rexico 15550 | Restrict the base of an up... |
| cau3lem 15551 | Lemma for ~ cau3 . (Contr... |
| cau3 15552 | Convert between three-quan... |
| cau4 15553 | Change the base of a Cauch... |
| caubnd2 15554 | A Cauchy sequence of compl... |
| caubnd 15555 | A Cauchy sequence of compl... |
| sqreulem 15556 | Lemma for ~ sqreu : write ... |
| sqreu 15557 | Existence and uniqueness f... |
| sqrtcl 15558 | Closure of the square root... |
| sqrtthlem 15559 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15560 | Mapping domain and codomai... |
| sqrtth 15561 | Square root theorem over t... |
| sqrtrege0 15562 | The square root function m... |
| eqsqrtor 15563 | Solve an equation containi... |
| eqsqrtd 15564 | A deduction for showing th... |
| eqsqrt2d 15565 | A deduction for showing th... |
| amgm2 15566 | Arithmetic-geometric mean ... |
| sqrtthi 15567 | Square root theorem. Theo... |
| sqrtcli 15568 | The square root of a nonne... |
| sqrtgt0i 15569 | The square root of a posit... |
| sqrtmsqi 15570 | Square root of square. (C... |
| sqrtsqi 15571 | Square root of square. (C... |
| sqsqrti 15572 | Square of square root. (C... |
| sqrtge0i 15573 | The square root of a nonne... |
| absidi 15574 | A nonnegative number is it... |
| absnidi 15575 | A negative number is the n... |
| leabsi 15576 | A real number is less than... |
| absori 15577 | The absolute value of a re... |
| absrei 15578 | Absolute value of a real n... |
| sqrtpclii 15579 | The square root of a posit... |
| sqrtgt0ii 15580 | The square root of a posit... |
| sqrt11i 15581 | The square root function i... |
| sqrtmuli 15582 | Square root distributes ov... |
| sqrtmulii 15583 | Square root distributes ov... |
| sqrtmsq2i 15584 | Relationship between squar... |
| sqrtlei 15585 | Square root is monotonic. ... |
| sqrtlti 15586 | Square root is strictly mo... |
| abslti 15587 | Absolute value and 'less t... |
| abslei 15588 | Absolute value and 'less t... |
| cnsqrt00 15589 | A square root of a complex... |
| absvalsqi 15590 | Square of value of absolut... |
| absvalsq2i 15591 | Square of value of absolut... |
| abscli 15592 | Real closure of absolute v... |
| absge0i 15593 | Absolute value is nonnegat... |
| absval2i 15594 | Value of absolute value fu... |
| abs00i 15595 | The absolute value of a nu... |
| absgt0i 15596 | The absolute value of a no... |
| absnegi 15597 | Absolute value of negative... |
| abscji 15598 | The absolute value of a nu... |
| releabsi 15599 | The real part of a number ... |
| abssubi 15600 | Swapping order of subtract... |
| absmuli 15601 | Absolute value distributes... |
| sqabsaddi 15602 | Square of absolute value o... |
| sqabssubi 15603 | Square of absolute value o... |
| absdivzi 15604 | Absolute value distributes... |
| abstrii 15605 | Triangle inequality for ab... |
| abs3difi 15606 | Absolute value of differen... |
| abs3lemi 15607 | Lemma involving absolute v... |
| rpsqrtcld 15608 | The square root of a posit... |
| sqrtgt0d 15609 | The square root of a posit... |
| absnidd 15610 | A negative number is the n... |
| leabsd 15611 | A real number is less than... |
| absord 15612 | The absolute value of a re... |
| absred 15613 | Absolute value of a real n... |
| resqrtcld 15614 | The square root of a nonne... |
| sqrtmsqd 15615 | Square root of square. (C... |
| sqrtsqd 15616 | Square root of square. (C... |
| sqrtge0d 15617 | The square root of a nonne... |
| sqrtnegd 15618 | The square root of a negat... |
| absidd 15619 | A nonnegative number is it... |
| sqrtdivd 15620 | Square root distributes ov... |
| sqrtmuld 15621 | Square root distributes ov... |
| sqrtsq2d 15622 | Relationship between squar... |
| sqrtled 15623 | Square root is monotonic. ... |
| sqrtltd 15624 | Square root is strictly mo... |
| sqr11d 15625 | The square root function i... |
| nn0absid 15626 | A nonnegative integer is i... |
| nn0absidi 15627 | A nonnegative integer is i... |
| absltd 15628 | Absolute value and 'less t... |
| absled 15629 | Absolute value and 'less t... |
| abssubge0d 15630 | Absolute value of a nonneg... |
| abssuble0d 15631 | Absolute value of a nonpos... |
| absdifltd 15632 | The absolute value of a di... |
| absdifled 15633 | The absolute value of a di... |
| icodiamlt 15634 | Two elements in a half-ope... |
| abscld 15635 | Real closure of absolute v... |
| sqrtcld 15636 | Closure of the square root... |
| sqrtrege0d 15637 | The real part of the squar... |
| sqsqrtd 15638 | Square root theorem. Theo... |
| msqsqrtd 15639 | Square root theorem. Theo... |
| sqr00d 15640 | A square root is zero iff ... |
| absvalsqd 15641 | Square of value of absolut... |
| absvalsq2d 15642 | Square of value of absolut... |
| absge0d 15643 | Absolute value is nonnegat... |
| absval2d 15644 | Value of absolute value fu... |
| abs00d 15645 | The absolute value of a nu... |
| absne0d 15646 | The absolute value of a nu... |
| absrpcld 15647 | The absolute value of a no... |
| absnegd 15648 | Absolute value of negative... |
| abscjd 15649 | The absolute value of a nu... |
| releabsd 15650 | The real part of a number ... |
| absexpd 15651 | Absolute value of positive... |
| abssubd 15652 | Swapping order of subtract... |
| absmuld 15653 | Absolute value distributes... |
| absdivd 15654 | Absolute value distributes... |
| abstrid 15655 | Triangle inequality for ab... |
| abs2difd 15656 | Difference of absolute val... |
| abs2dif2d 15657 | Difference of absolute val... |
| abs2difabsd 15658 | Absolute value of differen... |
| abs3difd 15659 | Absolute value of differen... |
| abs3lemd 15660 | Lemma involving absolute v... |
| reusq0 15661 | A complex number is the sq... |
| bhmafibid1cn 15662 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15663 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15664 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15665 | The Brahmagupta-Fibonacci ... |
| limsupgord 15668 | Ordering property of the s... |
| limsupcl 15669 | Closure of the superior li... |
| limsupval 15670 | The superior limit of an i... |
| limsupgf 15671 | Closure of the superior li... |
| limsupgval 15672 | Value of the superior limi... |
| limsupgle 15673 | The defining property of t... |
| limsuple 15674 | The defining property of t... |
| limsuplt 15675 | The defining property of t... |
| limsupval2 15676 | The superior limit, relati... |
| limsupgre 15677 | If a sequence of real numb... |
| limsupbnd1 15678 | If a sequence is eventuall... |
| limsupbnd2 15679 | If a sequence is eventuall... |
| climrel 15688 | The limit relation is a re... |
| rlimrel 15689 | The limit relation is a re... |
| clim 15690 | Express the predicate: Th... |
| rlim 15691 | Express the predicate: Th... |
| rlim2 15692 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15693 | Use strictly less-than in ... |
| rlim3 15694 | Restrict the range of the ... |
| climcl 15695 | Closure of the limit of a ... |
| rlimpm 15696 | Closure of a function with... |
| rlimf 15697 | Closure of a function with... |
| rlimss 15698 | Domain closure of a functi... |
| rlimcl 15699 | Closure of the limit of a ... |
| clim2 15700 | Express the predicate: Th... |
| clim2c 15701 | Express the predicate ` F ... |
| clim0 15702 | Express the predicate ` F ... |
| clim0c 15703 | Express the predicate ` F ... |
| rlim0 15704 | Express the predicate ` B ... |
| rlim0lt 15705 | Use strictly less-than in ... |
| climi 15706 | Convergence of a sequence ... |
| climi2 15707 | Convergence of a sequence ... |
| climi0 15708 | Convergence of a sequence ... |
| rlimi 15709 | Convergence at infinity of... |
| rlimi2 15710 | Convergence at infinity of... |
| ello1 15711 | Elementhood in the set of ... |
| ello12 15712 | Elementhood in the set of ... |
| ello12r 15713 | Sufficient condition for e... |
| lo1f 15714 | An eventually upper bounde... |
| lo1dm 15715 | An eventually upper bounde... |
| lo1bdd 15716 | The defining property of a... |
| ello1mpt 15717 | Elementhood in the set of ... |
| ello1mpt2 15718 | Elementhood in the set of ... |
| ello1d 15719 | Sufficient condition for e... |
| lo1bdd2 15720 | If an eventually bounded f... |
| lo1bddrp 15721 | Refine ~ o1bdd2 to give a ... |
| elo1 15722 | Elementhood in the set of ... |
| elo12 15723 | Elementhood in the set of ... |
| elo12r 15724 | Sufficient condition for e... |
| o1f 15725 | An eventually bounded func... |
| o1dm 15726 | An eventually bounded func... |
| o1bdd 15727 | The defining property of a... |
| lo1o1 15728 | A function is eventually b... |
| lo1o12 15729 | A function is eventually b... |
| elo1mpt 15730 | Elementhood in the set of ... |
| elo1mpt2 15731 | Elementhood in the set of ... |
| elo1d 15732 | Sufficient condition for e... |
| o1lo1 15733 | A real function is eventua... |
| o1lo12 15734 | A lower bounded real funct... |
| o1lo1d 15735 | A real eventually bounded ... |
| icco1 15736 | Derive eventual boundednes... |
| o1bdd2 15737 | If an eventually bounded f... |
| o1bddrp 15738 | Refine ~ o1bdd2 to give a ... |
| climconst 15739 | An (eventually) constant s... |
| rlimconst 15740 | A constant sequence conver... |
| rlimclim1 15741 | Forward direction of ~ rli... |
| rlimclim 15742 | A sequence on an upper int... |
| climrlim2 15743 | Produce a real limit from ... |
| climconst2 15744 | A constant sequence conver... |
| climz 15745 | The zero sequence converge... |
| rlimuni 15746 | A real function whose doma... |
| rlimdm 15747 | Two ways to express that a... |
| climuni 15748 | An infinite sequence of co... |
| fclim 15749 | The limit relation is func... |
| climdm 15750 | Two ways to express that a... |
| climeu 15751 | An infinite sequence of co... |
| climreu 15752 | An infinite sequence of co... |
| climmo 15753 | An infinite sequence of co... |
| rlimres 15754 | The restriction of a funct... |
| lo1res 15755 | The restriction of an even... |
| o1res 15756 | The restriction of an even... |
| rlimres2 15757 | The restriction of a funct... |
| lo1res2 15758 | The restriction of a funct... |
| o1res2 15759 | The restriction of a funct... |
| lo1resb 15760 | The restriction of a funct... |
| rlimresb 15761 | The restriction of a funct... |
| o1resb 15762 | The restriction of a funct... |
| climeq 15763 | Two functions that are eve... |
| lo1eq 15764 | Two functions that are eve... |
| rlimeq 15765 | Two functions that are eve... |
| o1eq 15766 | Two functions that are eve... |
| climmpt 15767 | Exhibit a function ` G ` w... |
| 2clim 15768 | If two sequences converge ... |
| climmpt2 15769 | Relate an integer limit on... |
| climshftlem 15770 | A shifted function converg... |
| climres 15771 | A function restricted to u... |
| climshft 15772 | A shifted function converg... |
| serclim0 15773 | The zero series converges ... |
| rlimcld2 15774 | If ` D ` is a closed set i... |
| rlimrege0 15775 | The limit of a sequence of... |
| rlimrecl 15776 | The limit of a real sequen... |
| rlimge0 15777 | The limit of a sequence of... |
| climshft2 15778 | A shifted function converg... |
| climrecl 15779 | The limit of a convergent ... |
| climge0 15780 | A nonnegative sequence con... |
| climabs0 15781 | Convergence to zero of the... |
| o1co 15782 | Sufficient condition for t... |
| o1compt 15783 | Sufficient condition for t... |
| rlimcn1 15784 | Image of a limit under a c... |
| rlimcn1b 15785 | Image of a limit under a c... |
| rlimcn3 15786 | Image of a limit under a c... |
| rlimcn2 15787 | Image of a limit under a c... |
| climcn1 15788 | Image of a limit under a c... |
| climcn2 15789 | Image of a limit under a c... |
| addcn2 15790 | Complex number addition is... |
| subcn2 15791 | Complex number subtraction... |
| mulcn2 15792 | Complex number multiplicat... |
| reccn2 15793 | The reciprocal function is... |
| cn1lem 15794 | A sufficient condition for... |
| abscn2 15795 | The absolute value functio... |
| cjcn2 15796 | The complex conjugate func... |
| recn2 15797 | The real part function is ... |
| imcn2 15798 | The imaginary part functio... |
| climcn1lem 15799 | The limit of a continuous ... |
| climabs 15800 | Limit of the absolute valu... |
| climcj 15801 | Limit of the complex conju... |
| climre 15802 | Limit of the real part of ... |
| climim 15803 | Limit of the imaginary par... |
| rlimmptrcl 15804 | Reverse closure for a real... |
| rlimabs 15805 | Limit of the absolute valu... |
| rlimcj 15806 | Limit of the complex conju... |
| rlimre 15807 | Limit of the real part of ... |
| rlimim 15808 | Limit of the imaginary par... |
| o1of2 15809 | Show that a binary operati... |
| o1add 15810 | The sum of two eventually ... |
| o1mul 15811 | The product of two eventua... |
| o1sub 15812 | The difference of two even... |
| rlimo1 15813 | Any function with a finite... |
| rlimdmo1 15814 | A convergent function is e... |
| o1rlimmul 15815 | The product of an eventual... |
| o1const 15816 | A constant function is eve... |
| lo1const 15817 | A constant function is eve... |
| lo1mptrcl 15818 | Reverse closure for an eve... |
| o1mptrcl 15819 | Reverse closure for an eve... |
| o1add2 15820 | The sum of two eventually ... |
| o1mul2 15821 | The product of two eventua... |
| o1sub2 15822 | The product of two eventua... |
| lo1add 15823 | The sum of two eventually ... |
| lo1mul 15824 | The product of an eventual... |
| lo1mul2 15825 | The product of an eventual... |
| o1dif 15826 | If the difference of two f... |
| lo1sub 15827 | The difference of an event... |
| climadd 15828 | Limit of the sum of two co... |
| climmul 15829 | Limit of the product of tw... |
| climsub 15830 | Limit of the difference of... |
| climaddc1 15831 | Limit of a constant ` C ` ... |
| climaddc2 15832 | Limit of a constant ` C ` ... |
| climmulc2 15833 | Limit of a sequence multip... |
| climsubc1 15834 | Limit of a constant ` C ` ... |
| climsubc2 15835 | Limit of a constant ` C ` ... |
| climle 15836 | Comparison of the limits o... |
| climsqz 15837 | Convergence of a sequence ... |
| climsqz2 15838 | Convergence of a sequence ... |
| rlimadd 15839 | Limit of the sum of two co... |
| rlimsub 15840 | Limit of the difference of... |
| rlimmul 15841 | Limit of the product of tw... |
| rlimdiv 15842 | Limit of the quotient of t... |
| rlimneg 15843 | Limit of the negative of a... |
| rlimle 15844 | Comparison of the limits o... |
| rlimsqzlem 15845 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15846 | Convergence of a sequence ... |
| rlimsqz2 15847 | Convergence of a sequence ... |
| lo1le 15848 | Transfer eventual upper bo... |
| o1le 15849 | Transfer eventual boundedn... |
| rlimno1 15850 | A function whose inverse c... |
| clim2ser 15851 | The limit of an infinite s... |
| clim2ser2 15852 | The limit of an infinite s... |
| iserex 15853 | An infinite series converg... |
| isermulc2 15854 | Multiplication of an infin... |
| climlec2 15855 | Comparison of a constant t... |
| iserle 15856 | Comparison of the limits o... |
| iserge0 15857 | The limit of an infinite s... |
| climub 15858 | The limit of a monotonic s... |
| climserle 15859 | The partial sums of a conv... |
| isershft 15860 | Index shift of the limit o... |
| isercolllem1 15861 | Lemma for ~ isercoll . (C... |
| isercolllem2 15862 | Lemma for ~ isercoll . (C... |
| isercolllem3 15863 | Lemma for ~ isercoll . (C... |
| isercoll 15864 | Rearrange an infinite seri... |
| isercoll2 15865 | Generalize ~ isercoll so t... |
| climsup 15866 | A bounded monotonic sequen... |
| climcau 15867 | A converging sequence of c... |
| climbdd 15868 | A converging sequence of c... |
| caucvgrlem 15869 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15870 | A Cauchy sequence of real ... |
| caucvgrlem2 15871 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15872 | A Cauchy sequence of compl... |
| caurcvg 15873 | A Cauchy sequence of real ... |
| caurcvg2 15874 | A Cauchy sequence of real ... |
| caucvg 15875 | A Cauchy sequence of compl... |
| caucvgb 15876 | A function is convergent i... |
| serf0 15877 | If an infinite series conv... |
| iseraltlem1 15878 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15879 | Lemma for ~ iseralt . The... |
| iseraltlem3 15880 | Lemma for ~ iseralt . Fro... |
| iseralt 15881 | The alternating series tes... |
| sumex 15884 | A sum is a set. (Contribu... |
| sumeq1 15885 | Equality theorem for a sum... |
| nfsum1 15886 | Bound-variable hypothesis ... |
| nfsum 15887 | Bound-variable hypothesis ... |
| sumeq2w 15888 | Equality theorem for sum, ... |
| sumeq2ii 15889 | Equality theorem for sum, ... |
| sumeq2 15890 | Equality theorem for sum. ... |
| cbvsum 15891 | Change bound variable in a... |
| cbvsumv 15892 | Change bound variable in a... |
| sumeq1i 15893 | Equality inference for sum... |
| sumeq2i 15894 | Equality inference for sum... |
| sumeq12i 15895 | Equality inference for sum... |
| sumeq1d 15896 | Equality deduction for sum... |
| sumeq2d 15897 | Equality deduction for sum... |
| sumeq2dv 15898 | Equality deduction for sum... |
| sumeq2sdv 15899 | Equality deduction for sum... |
| 2sumeq2dv 15900 | Equality deduction for dou... |
| sumeq12dv 15901 | Equality deduction for sum... |
| sumeq12rdv 15902 | Equality deduction for sum... |
| sum2id 15903 | The second class argument ... |
| sumfc 15904 | A lemma to facilitate conv... |
| fz1f1o 15905 | A lemma for working with f... |
| sumrblem 15906 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15907 | The sequence of partial su... |
| sumrb 15908 | Rebase the starting point ... |
| summolem3 15909 | Lemma for ~ summo . (Cont... |
| summolem2a 15910 | Lemma for ~ summo . (Cont... |
| summolem2 15911 | Lemma for ~ summo . (Cont... |
| summo 15912 | A sum has at most one limi... |
| zsum 15913 | Series sum with index set ... |
| isum 15914 | Series sum with an upper i... |
| fsum 15915 | The value of a sum over a ... |
| sum0 15916 | Any sum over the empty set... |
| sumz 15917 | Any sum of zero over a sum... |
| fsumf1o 15918 | Re-index a finite sum usin... |
| sumss 15919 | Change the index set to a ... |
| fsumss 15920 | Change the index set to a ... |
| sumss2 15921 | Change the index set of a ... |
| fsumcvg2 15922 | The sequence of partial su... |
| fsumsers 15923 | Special case of series sum... |
| fsumcvg3 15924 | A finite sum is convergent... |
| fsumser 15925 | A finite sum expressed in ... |
| fsumcl2lem 15926 | - Lemma for finite sum clo... |
| fsumcllem 15927 | - Lemma for finite sum clo... |
| fsumcl 15928 | Closure of a finite sum of... |
| fsumrecl 15929 | Closure of a finite sum of... |
| fsumzcl 15930 | Closure of a finite sum of... |
| fsumnn0cl 15931 | Closure of a finite sum of... |
| fsumrpcl 15932 | Closure of a finite sum of... |
| fsumclf 15933 | Closure of a finite sum of... |
| fsumzcl2 15934 | A finite sum with integer ... |
| fsumadd 15935 | The sum of two finite sums... |
| fsumsplit 15936 | Split a sum into two parts... |
| fsumsplitf 15937 | Split a sum into two parts... |
| sumsnf 15938 | A sum of a singleton is th... |
| fsumsplitsn 15939 | Separate out a term in a f... |
| fsumsplit1 15940 | Separate out a term in a f... |
| sumsn 15941 | A sum of a singleton is th... |
| fsum1 15942 | The finite sum of ` A ( k ... |
| sumpr 15943 | A sum over a pair is the s... |
| sumtp 15944 | A sum over a triple is the... |
| sumsns 15945 | A sum of a singleton is th... |
| fsumm1 15946 | Separate out the last term... |
| fzosump1 15947 | Separate out the last term... |
| fsum1p 15948 | Separate out the first ter... |
| fsummsnunz 15949 | A finite sum all of whose ... |
| fsumsplitsnun 15950 | Separate out a term in a f... |
| fsump1 15951 | The addition of the next t... |
| isumclim 15952 | An infinite sum equals the... |
| isumclim2 15953 | A converging series conver... |
| isumclim3 15954 | The sequence of partial fi... |
| sumnul 15955 | The sum of a non-convergen... |
| isumcl 15956 | The sum of a converging in... |
| isummulc2 15957 | An infinite sum multiplied... |
| isummulc1 15958 | An infinite sum multiplied... |
| isumdivc 15959 | An infinite sum divided by... |
| isumrecl 15960 | The sum of a converging in... |
| isumge0 15961 | An infinite sum of nonnega... |
| isumadd 15962 | Addition of infinite sums.... |
| sumsplit 15963 | Split a sum into two parts... |
| fsump1i 15964 | Optimized version of ~ fsu... |
| fsum2dlem 15965 | Lemma for ~ fsum2d - induc... |
| fsum2d 15966 | Write a double sum as a su... |
| fsumxp 15967 | Combine two sums into a si... |
| fsumcnv 15968 | Transform a region of summ... |
| fsumcom2 15969 | Interchange order of summa... |
| fsumcom 15970 | Interchange order of summa... |
| fsum0diaglem 15971 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15972 | Two ways to express "the s... |
| mptfzshft 15973 | 1-1 onto function in maps-... |
| fsumrev 15974 | Reversal of a finite sum. ... |
| fsumshft 15975 | Index shift of a finite su... |
| fsumshftm 15976 | Negative index shift of a ... |
| fsumrev2 15977 | Reversal of a finite sum. ... |
| fsum0diag2 15978 | Two ways to express "the s... |
| fsummulc2 15979 | A finite sum multiplied by... |
| fsummulc1 15980 | A finite sum multiplied by... |
| fsumdivc 15981 | A finite sum divided by a ... |
| fsumneg 15982 | Negation of a finite sum. ... |
| fsumsub 15983 | Split a finite sum over a ... |
| fsum2mul 15984 | Separate the nested sum of... |
| fsumconst 15985 | The sum of constant terms ... |
| fsumconst1 15986 | The sum of 1 over a finite... |
| fsumdifsnconst 15987 | The sum of constant terms ... |
| modfsummodslem1 15988 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15989 | Induction step for ~ modfs... |
| modfsummod 15990 | A finite sum modulo a posi... |
| fsumge0 15991 | If all of the terms of a f... |
| fsumless 15992 | A shorter sum of nonnegati... |
| fsumge1 15993 | A sum of nonnegative numbe... |
| fsum00 15994 | A sum of nonnegative numbe... |
| fsumle 15995 | If all of the terms of fin... |
| fsumlt 15996 | If every term in one finit... |
| fsumabs 15997 | Generalized triangle inequ... |
| telfsumo 15998 | Sum of a telescoping serie... |
| telfsumo2 15999 | Sum of a telescoping serie... |
| telfsum 16000 | Sum of a telescoping serie... |
| telfsum2 16001 | Sum of a telescoping serie... |
| fsumparts 16002 | Summation by parts. (Cont... |
| fsumrelem 16003 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 16004 | The real part of a sum. (... |
| fsumim 16005 | The imaginary part of a su... |
| fsumcj 16006 | The complex conjugate of a... |
| fsumrlim 16007 | Limit of a finite sum of c... |
| fsumo1 16008 | The finite sum of eventual... |
| o1fsum 16009 | If ` A ( k ) ` is O(1), th... |
| seqabs 16010 | Generalized triangle inequ... |
| iserabs 16011 | Generalized triangle inequ... |
| cvgcmp 16012 | A comparison test for conv... |
| cvgcmpub 16013 | An upper bound for the lim... |
| cvgcmpce 16014 | A comparison test for conv... |
| abscvgcvg 16015 | An absolutely convergent s... |
| climfsum 16016 | Limit of a finite sum of c... |
| fsumiun 16017 | Sum over a disjoint indexe... |
| hashiun 16018 | The cardinality of a disjo... |
| hash2iun 16019 | The cardinality of a neste... |
| hash2iun1dif1 16020 | The cardinality of a neste... |
| hashrabrex 16021 | The number of elements in ... |
| hashuni 16022 | The cardinality of a disjo... |
| qshash 16023 | The cardinality of a set w... |
| indsum 16024 | Finite sum of a product wi... |
| indsumhash 16025 | The finite sum of the indi... |
| ackbijnn 16026 | Translate the Ackermann bi... |
| binomlem 16027 | Lemma for ~ binom (binomia... |
| binom 16028 | The binomial theorem: ` ( ... |
| binom1p 16029 | Special case of the binomi... |
| binom11 16030 | Special case of the binomi... |
| binom1dif 16031 | A summation for the differ... |
| bcxmaslem1 16032 | Lemma for ~ bcxmas . (Con... |
| bcxmas 16033 | Parallel summation (Christ... |
| incexclem 16034 | Lemma for ~ incexc . (Con... |
| incexc 16035 | The inclusion/exclusion pr... |
| incexc2 16036 | The inclusion/exclusion pr... |
| isumshft 16037 | Index shift of an infinite... |
| isumsplit 16038 | Split off the first ` N ` ... |
| isum1p 16039 | The infinite sum of a conv... |
| isumnn0nn 16040 | Sum from 0 to infinity in ... |
| isumrpcl 16041 | The infinite sum of positi... |
| isumle 16042 | Comparison of two infinite... |
| isumless 16043 | A finite sum of nonnegativ... |
| isumsup2 16044 | An infinite sum of nonnega... |
| isumsup 16045 | An infinite sum of nonnega... |
| isumltss 16046 | A partial sum of a series ... |
| climcndslem1 16047 | Lemma for ~ climcnds : bou... |
| climcndslem2 16048 | Lemma for ~ climcnds : bou... |
| climcnds 16049 | The Cauchy condensation te... |
| divrcnv 16050 | The sequence of reciprocal... |
| divcnv 16051 | The sequence of reciprocal... |
| flo1 16052 | The floor function satisfi... |
| divcnvshft 16053 | Limit of a ratio function.... |
| supcvg 16054 | Extract a sequence ` f ` i... |
| infcvgaux1i 16055 | Auxiliary theorem for appl... |
| infcvgaux2i 16056 | Auxiliary theorem for appl... |
| harmonic 16057 | The harmonic series ` H ` ... |
| arisum 16058 | Arithmetic series sum of t... |
| arisum2 16059 | Arithmetic series sum of t... |
| trireciplem 16060 | Lemma for ~ trirecip . Sh... |
| trirecip 16061 | The sum of the reciprocals... |
| expcnv 16062 | A sequence of powers of a ... |
| explecnv 16063 | A sequence of terms conver... |
| geoserg 16064 | The value of the finite ge... |
| geoser 16065 | The value of the finite ge... |
| pwdif 16066 | The difference of two numb... |
| pwm1geoser 16067 | The n-th power of a number... |
| geolim 16068 | The partial sums in the in... |
| geolim2 16069 | The partial sums in the ge... |
| georeclim 16070 | The limit of a geometric s... |
| geo2sum 16071 | The value of the finite ge... |
| geo2sum2 16072 | The value of the finite ge... |
| geo2lim 16073 | The value of the infinite ... |
| geomulcvg 16074 | The geometric series conve... |
| geoisum 16075 | The infinite sum of ` 1 + ... |
| geoisumr 16076 | The infinite sum of recipr... |
| geoisum1 16077 | The infinite sum of ` A ^ ... |
| geoisum1c 16078 | The infinite sum of ` A x.... |
| 0.999... 16079 | The recurring decimal 0.99... |
| geoihalfsum 16080 | Prove that the infinite ge... |
| cvgrat 16081 | Ratio test for convergence... |
| mertenslem1 16082 | Lemma for ~ mertens . (Co... |
| mertenslem2 16083 | Lemma for ~ mertens . (Co... |
| mertens 16084 | Mertens' theorem. If ` A ... |
| prodf 16085 | An infinite product of com... |
| clim2prod 16086 | The limit of an infinite p... |
| clim2div 16087 | The limit of an infinite p... |
| prodfmul 16088 | The product of two infinit... |
| prodf1 16089 | The value of the partial p... |
| prodf1f 16090 | A one-valued infinite prod... |
| prodfclim1 16091 | The constant one product c... |
| prodfn0 16092 | No term of a nonzero infin... |
| prodfrec 16093 | The reciprocal of an infin... |
| prodfdiv 16094 | The quotient of two infini... |
| ntrivcvg 16095 | A non-trivially converging... |
| ntrivcvgn0 16096 | A product that converges t... |
| ntrivcvgfvn0 16097 | Any value of a product seq... |
| ntrivcvgtail 16098 | A tail of a non-trivially ... |
| ntrivcvgmullem 16099 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 16100 | The product of two non-tri... |
| prodex 16103 | A product is a set. (Cont... |
| prodeq1f 16104 | Equality theorem for a pro... |
| prodeq1 16105 | Equality theorem for a pro... |
| nfcprod1 16106 | Bound-variable hypothesis ... |
| nfcprod 16107 | Bound-variable hypothesis ... |
| prodeq2w 16108 | Equality theorem for produ... |
| prodeq2ii 16109 | Equality theorem for produ... |
| prodeq2 16110 | Equality theorem for produ... |
| cbvprod 16111 | Change bound variable in a... |
| cbvprodv 16112 | Change bound variable in a... |
| cbvprodi 16113 | Change bound variable in a... |
| prodeq1i 16114 | Equality inference for pro... |
| prodeq2i 16115 | Equality inference for pro... |
| prodeq12i 16116 | Equality inference for pro... |
| prodeq1d 16117 | Equality deduction for pro... |
| prodeq2d 16118 | Equality deduction for pro... |
| prodeq2dv 16119 | Equality deduction for pro... |
| prodeq2sdv 16120 | Equality deduction for pro... |
| 2cprodeq2dv 16121 | Equality deduction for dou... |
| prodeq12dv 16122 | Equality deduction for pro... |
| prodeq12rdv 16123 | Equality deduction for pro... |
| prod2id 16124 | The second class argument ... |
| prodrblem 16125 | Lemma for ~ prodrb . (Con... |
| fprodcvg 16126 | The sequence of partial pr... |
| prodrblem2 16127 | Lemma for ~ prodrb . (Con... |
| prodrb 16128 | Rebase the starting point ... |
| prodmolem3 16129 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 16130 | Lemma for ~ prodmo . (Con... |
| prodmolem2 16131 | Lemma for ~ prodmo . (Con... |
| prodmo 16132 | A product has at most one ... |
| zprod 16133 | Series product with index ... |
| iprod 16134 | Series product with an upp... |
| zprodn0 16135 | Nonzero series product wit... |
| iprodn0 16136 | Nonzero series product wit... |
| fprod 16137 | The value of a product ove... |
| fprodntriv 16138 | A non-triviality lemma for... |
| prod0 16139 | A product over the empty s... |
| prod1 16140 | Any product of one over a ... |
| prodfc 16141 | A lemma to facilitate conv... |
| fprodf1o 16142 | Re-index a finite product ... |
| prodss 16143 | Change the index set to a ... |
| fprodss 16144 | Change the index set to a ... |
| fprodser 16145 | A finite product expressed... |
| fprodcl2lem 16146 | Finite product closure lem... |
| fprodcllem 16147 | Finite product closure lem... |
| fprodcl 16148 | Closure of a finite produc... |
| fprodrecl 16149 | Closure of a finite produc... |
| fprodzcl 16150 | Closure of a finite produc... |
| fprodnncl 16151 | Closure of a finite produc... |
| fprodrpcl 16152 | Closure of a finite produc... |
| fprodnn0cl 16153 | Closure of a finite produc... |
| fprodcllemf 16154 | Finite product closure lem... |
| fprodreclf 16155 | Closure of a finite produc... |
| fprodmul 16156 | The product of two finite ... |
| fproddiv 16157 | The quotient of two finite... |
| prodsn 16158 | A product of a singleton i... |
| fprod1 16159 | A finite product of only o... |
| prodsnf 16160 | A product of a singleton i... |
| climprod1 16161 | The limit of a product ove... |
| fprodsplit 16162 | Split a finite product int... |
| fprodm1 16163 | Separate out the last term... |
| fprod1p 16164 | Separate out the first ter... |
| fprodp1 16165 | Multiply in the last term ... |
| fprodm1s 16166 | Separate out the last term... |
| fprodp1s 16167 | Multiply in the last term ... |
| prodsns 16168 | A product of the singleton... |
| fprodfac 16169 | Factorial using product no... |
| fprodabs 16170 | The absolute value of a fi... |
| fprodeq0 16171 | Any finite product contain... |
| fprodshft 16172 | Shift the index of a finit... |
| fprodrev 16173 | Reversal of a finite produ... |
| fprodconst 16174 | The product of constant te... |
| fprodn0 16175 | A finite product of nonzer... |
| fprod2dlem 16176 | Lemma for ~ fprod2d - indu... |
| fprod2d 16177 | Write a double product as ... |
| fprodxp 16178 | Combine two products into ... |
| fprodcnv 16179 | Transform a product region... |
| fprodcom2 16180 | Interchange order of multi... |
| fprodcom 16181 | Interchange product order.... |
| fprod0diag 16182 | Two ways to express "the p... |
| fproddivf 16183 | The quotient of two finite... |
| fprodsplitf 16184 | Split a finite product int... |
| fprodsplitsn 16185 | Separate out a term in a f... |
| fprodsplit1f 16186 | Separate out a term in a f... |
| fprodn0f 16187 | A finite product of nonzer... |
| fprodclf 16188 | Closure of a finite produc... |
| fprodge0 16189 | If all the terms of a fini... |
| fprodeq0g 16190 | Any finite product contain... |
| fprodge1 16191 | If all of the terms of a f... |
| fprodle 16192 | If all the terms of two fi... |
| fprodmodd 16193 | If all factors of two fini... |
| iprodclim 16194 | An infinite product equals... |
| iprodclim2 16195 | A converging product conve... |
| iprodclim3 16196 | The sequence of partial fi... |
| iprodcl 16197 | The product of a non-trivi... |
| iprodrecl 16198 | The product of a non-trivi... |
| iprodmul 16199 | Multiplication of infinite... |
| risefacval 16204 | The value of the rising fa... |
| fallfacval 16205 | The value of the falling f... |
| risefacval2 16206 | One-based value of rising ... |
| fallfacval2 16207 | One-based value of falling... |
| fallfacval3 16208 | A product representation o... |
| risefaccllem 16209 | Lemma for rising factorial... |
| fallfaccllem 16210 | Lemma for falling factoria... |
| risefaccl 16211 | Closure law for rising fac... |
| fallfaccl 16212 | Closure law for falling fa... |
| rerisefaccl 16213 | Closure law for rising fac... |
| refallfaccl 16214 | Closure law for falling fa... |
| nnrisefaccl 16215 | Closure law for rising fac... |
| zrisefaccl 16216 | Closure law for rising fac... |
| zfallfaccl 16217 | Closure law for falling fa... |
| nn0risefaccl 16218 | Closure law for rising fac... |
| rprisefaccl 16219 | Closure law for rising fac... |
| risefallfac 16220 | A relationship between ris... |
| fallrisefac 16221 | A relationship between fal... |
| risefac0 16222 | The value of the rising fa... |
| fallfac0 16223 | The value of the falling f... |
| risefacp1 16224 | The value of the rising fa... |
| fallfacp1 16225 | The value of the falling f... |
| risefacp1d 16226 | The value of the rising fa... |
| fallfacp1d 16227 | The value of the falling f... |
| risefac1 16228 | The value of rising factor... |
| fallfac1 16229 | The value of falling facto... |
| risefacfac 16230 | Relate rising factorial to... |
| fallfacfwd 16231 | The forward difference of ... |
| 0fallfac 16232 | The value of the zero fall... |
| 0risefac 16233 | The value of the zero risi... |
| binomfallfaclem1 16234 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16235 | Lemma for ~ binomfallfac .... |
| binomfallfac 16236 | A version of the binomial ... |
| binomrisefac 16237 | A version of the binomial ... |
| fallfacval4 16238 | Represent the falling fact... |
| bcfallfac 16239 | Binomial coefficient in te... |
| fallfacfac 16240 | Relate falling factorial t... |
| bpolylem 16243 | Lemma for ~ bpolyval . (C... |
| bpolyval 16244 | The value of the Bernoulli... |
| bpoly0 16245 | The value of the Bernoulli... |
| bpoly1 16246 | The value of the Bernoulli... |
| bpolycl 16247 | Closure law for Bernoulli ... |
| bpolysum 16248 | A sum for Bernoulli polyno... |
| bpolydiflem 16249 | Lemma for ~ bpolydif . (C... |
| bpolydif 16250 | Calculate the difference b... |
| fsumkthpow 16251 | A closed-form expression f... |
| bpoly2 16252 | The Bernoulli polynomials ... |
| bpoly3 16253 | The Bernoulli polynomials ... |
| bpoly4 16254 | The Bernoulli polynomials ... |
| fsumcube 16255 | Express the sum of cubes i... |
| eftcl 16268 | Closure of a term in the s... |
| reeftcl 16269 | The terms of the series ex... |
| eftabs 16270 | The absolute value of a te... |
| eftval 16271 | The value of a term in the... |
| efcllem 16272 | Lemma for ~ efcl . The se... |
| ef0lem 16273 | The series defining the ex... |
| efval 16274 | Value of the exponential f... |
| esum 16275 | Value of Euler's constant ... |
| eff 16276 | Domain and codomain of the... |
| efcl 16277 | Closure law for the expone... |
| efcld 16278 | Closure law for the expone... |
| efval2 16279 | Value of the exponential f... |
| efcvg 16280 | The series that defines th... |
| efcvgfsum 16281 | Exponential function conve... |
| reefcl 16282 | The exponential function i... |
| reefcld 16283 | The exponential function i... |
| ere 16284 | Euler's constant ` _e ` = ... |
| ege2le3 16285 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16286 | Value of the exponential f... |
| efcj 16287 | The exponential of a compl... |
| efaddlem 16288 | Lemma for ~ efadd (exponen... |
| efadd 16289 | Sum of exponents law for e... |
| fprodefsum 16290 | Move the exponential funct... |
| efcan 16291 | Cancellation law for expon... |
| efne0d 16292 | The exponential of a compl... |
| efne0 16293 | The exponential of a compl... |
| efne0OLD 16294 | Obsolete version of ~ efne... |
| efneg 16295 | The exponential of the opp... |
| eff2 16296 | The exponential function m... |
| efsub 16297 | Difference of exponents la... |
| efexp 16298 | The exponential of an inte... |
| efzval 16299 | Value of the exponential f... |
| efgt0 16300 | The exponential of a real ... |
| rpefcl 16301 | The exponential of a real ... |
| rpefcld 16302 | The exponential of a real ... |
| eftlcvg 16303 | The tail series of the exp... |
| eftlcl 16304 | Closure of the sum of an i... |
| reeftlcl 16305 | Closure of the sum of an i... |
| eftlub 16306 | An upper bound on the abso... |
| efsep 16307 | Separate out the next term... |
| effsumlt 16308 | The partial sums of the se... |
| eft0val 16309 | The value of the first ter... |
| ef4p 16310 | Separate out the first fou... |
| efgt1p2 16311 | The exponential of a posit... |
| efgt1p 16312 | The exponential of a posit... |
| efgt1 16313 | The exponential of a posit... |
| eflt 16314 | The exponential function o... |
| efle 16315 | The exponential function o... |
| reef11 16316 | The exponential function o... |
| reeff1 16317 | The exponential function m... |
| eflegeo 16318 | The exponential function o... |
| sinval 16319 | Value of the sine function... |
| cosval 16320 | Value of the cosine functi... |
| sinf 16321 | Domain and codomain of the... |
| cosf 16322 | Domain and codomain of the... |
| sincl 16323 | Closure of the sine functi... |
| coscl 16324 | Closure of the cosine func... |
| tanval 16325 | Value of the tangent funct... |
| tancl 16326 | The closure of the tangent... |
| sincld 16327 | Closure of the sine functi... |
| coscld 16328 | Closure of the cosine func... |
| tancld 16329 | Closure of the tangent fun... |
| tanval2 16330 | Express the tangent functi... |
| tanval3 16331 | Express the tangent functi... |
| resinval 16332 | The sine of a real number ... |
| recosval 16333 | The cosine of a real numbe... |
| efi4p 16334 | Separate out the first fou... |
| resin4p 16335 | Separate out the first fou... |
| recos4p 16336 | Separate out the first fou... |
| resincl 16337 | The sine of a real number ... |
| recoscl 16338 | The cosine of a real numbe... |
| retancl 16339 | The closure of the tangent... |
| resincld 16340 | Closure of the sine functi... |
| recoscld 16341 | Closure of the cosine func... |
| retancld 16342 | Closure of the tangent fun... |
| sinneg 16343 | The sine of a negative is ... |
| cosneg 16344 | The cosines of a number an... |
| tanneg 16345 | The tangent of a negative ... |
| sin0 16346 | Value of the sine function... |
| cos0 16347 | Value of the cosine functi... |
| tan0 16348 | The value of the tangent f... |
| efival 16349 | The exponential function i... |
| efmival 16350 | The exponential function i... |
| sinhval 16351 | Value of the hyperbolic si... |
| coshval 16352 | Value of the hyperbolic co... |
| resinhcl 16353 | The hyperbolic sine of a r... |
| rpcoshcl 16354 | The hyperbolic cosine of a... |
| recoshcl 16355 | The hyperbolic cosine of a... |
| retanhcl 16356 | The hyperbolic tangent of ... |
| tanhlt1 16357 | The hyperbolic tangent of ... |
| tanhbnd 16358 | The hyperbolic tangent of ... |
| efeul 16359 | Eulerian representation of... |
| efieq 16360 | The exponentials of two im... |
| sinadd 16361 | Addition formula for sine.... |
| cosadd 16362 | Addition formula for cosin... |
| tanaddlem 16363 | A useful intermediate step... |
| tanadd 16364 | Addition formula for tange... |
| sinsub 16365 | Sine of difference. (Cont... |
| cossub 16366 | Cosine of difference. (Co... |
| addsin 16367 | Sum of sines. (Contribute... |
| subsin 16368 | Difference of sines. (Con... |
| sinmul 16369 | Product of sines can be re... |
| cosmul 16370 | Product of cosines can be ... |
| addcos 16371 | Sum of cosines. (Contribu... |
| subcos 16372 | Difference of cosines. (C... |
| sincossq 16373 | Sine squared plus cosine s... |
| sin2t 16374 | Double-angle formula for s... |
| cos2t 16375 | Double-angle formula for c... |
| cos2tsin 16376 | Double-angle formula for c... |
| sinbnd 16377 | The sine of a real number ... |
| cosbnd 16378 | The cosine of a real numbe... |
| sinbnd2 16379 | The sine of a real number ... |
| cosbnd2 16380 | The cosine of a real numbe... |
| ef01bndlem 16381 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16382 | Bounds on the sine of a po... |
| cos01bnd 16383 | Bounds on the cosine of a ... |
| cos1bnd 16384 | Bounds on the cosine of 1.... |
| cos2bnd 16385 | Bounds on the cosine of 2.... |
| sinltx 16386 | The sine of a positive rea... |
| sin01gt0 16387 | The sine of a positive rea... |
| cos01gt0 16388 | The cosine of a positive r... |
| sin02gt0 16389 | The sine of a positive rea... |
| sincos1sgn 16390 | The signs of the sine and ... |
| sincos2sgn 16391 | The signs of the sine and ... |
| sin4lt0 16392 | The sine of 4 is negative.... |
| absefi 16393 | The absolute value of the ... |
| absef 16394 | The absolute value of the ... |
| absefib 16395 | A complex number is real i... |
| efieq1re 16396 | A number whose imaginary e... |
| demoivre 16397 | De Moivre's Formula. Proo... |
| demoivreALT 16398 | Alternate proof of ~ demoi... |
| eirrlem 16401 | Lemma for ~ eirr . (Contr... |
| eirr 16402 | ` _e ` is irrational. (Co... |
| egt2lt3 16403 | Euler's constant ` _e ` = ... |
| epos 16404 | Euler's constant ` _e ` is... |
| epr 16405 | Euler's constant ` _e ` is... |
| ene0 16406 | ` _e ` is not 0. (Contrib... |
| ene1 16407 | ` _e ` is not 1. (Contrib... |
| xpnnen 16408 | The Cartesian product of t... |
| znnen 16409 | The set of integers and th... |
| qnnen 16410 | The rational numbers are c... |
| rpnnen2lem1 16411 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16412 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16413 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16414 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16415 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16416 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16417 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16418 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16419 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16420 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16421 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16422 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16423 | The other half of ~ rpnnen... |
| rpnnen 16424 | The cardinality of the con... |
| rexpen 16425 | The real numbers are equin... |
| cpnnen 16426 | The complex numbers are eq... |
| rucALT 16427 | Alternate proof of ~ ruc .... |
| ruclem1 16428 | Lemma for ~ ruc (the reals... |
| ruclem2 16429 | Lemma for ~ ruc . Orderin... |
| ruclem3 16430 | Lemma for ~ ruc . The con... |
| ruclem4 16431 | Lemma for ~ ruc . Initial... |
| ruclem6 16432 | Lemma for ~ ruc . Domain ... |
| ruclem7 16433 | Lemma for ~ ruc . Success... |
| ruclem8 16434 | Lemma for ~ ruc . The int... |
| ruclem9 16435 | Lemma for ~ ruc . The fir... |
| ruclem10 16436 | Lemma for ~ ruc . Every f... |
| ruclem11 16437 | Lemma for ~ ruc . Closure... |
| ruclem12 16438 | Lemma for ~ ruc . The sup... |
| ruclem13 16439 | Lemma for ~ ruc . There i... |
| ruc 16440 | The set of positive intege... |
| resdomq 16441 | The set of rationals is st... |
| aleph1re 16442 | There are at least aleph-o... |
| aleph1irr 16443 | There are at least aleph-o... |
| cnso 16444 | The complex numbers can be... |
| sqrt2irrlem 16445 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16446 | The square root of 2 is ir... |
| sqrt2re 16447 | The square root of 2 exist... |
| sqrt2irr0 16448 | The square root of 2 is an... |
| nthruc 16449 | The sequence ` NN ` , ` ZZ... |
| nthruz 16450 | The sequence ` NN ` , ` NN... |
| divides 16453 | Define the divides relatio... |
| dvdsval2 16454 | One nonzero integer divide... |
| dvdsval3 16455 | One nonzero integer divide... |
| dvdszrcl 16456 | Reverse closure for the di... |
| dvdsmod0 16457 | If a positive integer divi... |
| p1modz1 16458 | If a number greater than 1... |
| dvdsmodexp 16459 | If a positive integer divi... |
| nndivdvds 16460 | Strong form of ~ dvdsval2 ... |
| nndivides 16461 | Definition of the divides ... |
| moddvds 16462 | Two ways to say ` A == B `... |
| modm1div 16463 | An integer greater than on... |
| addmulmodb 16464 | An integer plus a product ... |
| dvds0lem 16465 | A lemma to assist theorems... |
| dvds1lem 16466 | A lemma to assist theorems... |
| dvds2lem 16467 | A lemma to assist theorems... |
| iddvds 16468 | An integer divides itself.... |
| 1dvds 16469 | 1 divides any integer. Th... |
| dvds0 16470 | Any integer divides 0. Th... |
| negdvdsb 16471 | An integer divides another... |
| dvdsnegb 16472 | An integer divides another... |
| absdvdsb 16473 | An integer divides another... |
| dvdsabsb 16474 | An integer divides another... |
| 0dvds 16475 | Only 0 is divisible by 0. ... |
| dvdsmul1 16476 | An integer divides a multi... |
| dvdsmul2 16477 | An integer divides a multi... |
| iddvdsexp 16478 | An integer divides a posit... |
| muldvds1 16479 | If a product divides an in... |
| muldvds2 16480 | If a product divides an in... |
| dvdscmul 16481 | Multiplication by a consta... |
| dvdsmulc 16482 | Multiplication by a consta... |
| dvdscmulr 16483 | Cancellation law for the d... |
| dvdsmulcr 16484 | Cancellation law for the d... |
| summodnegmod 16485 | The sum of two integers mo... |
| difmod0 16486 | The difference of two inte... |
| modmulconst 16487 | Constant multiplication in... |
| dvds2ln 16488 | If an integer divides each... |
| dvds2add 16489 | If an integer divides each... |
| dvds2sub 16490 | If an integer divides each... |
| dvds2addd 16491 | Deduction form of ~ dvds2a... |
| dvds2subd 16492 | Deduction form of ~ dvds2s... |
| dvdstr 16493 | The divides relation is tr... |
| dvdstrd 16494 | The divides relation is tr... |
| dvdsmultr1 16495 | If an integer divides anot... |
| dvdsmultr1d 16496 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16497 | If an integer divides anot... |
| dvdsmultr2d 16498 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16499 | If an integer divides eith... |
| dvdssub2 16500 | If an integer divides a di... |
| dvdsadd 16501 | An integer divides another... |
| dvdsaddr 16502 | An integer divides another... |
| dvdssub 16503 | An integer divides another... |
| dvdssubr 16504 | An integer divides another... |
| dvdsadd2b 16505 | Adding a multiple of the b... |
| dvdsaddre2b 16506 | Adding a multiple of the b... |
| fsumdvds 16507 | If every term in a sum is ... |
| dvdslelem 16508 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16509 | The divisors of a positive... |
| dvdsleabs 16510 | The divisors of a nonzero ... |
| dvdsleabs2 16511 | Transfer divisibility to a... |
| dvdsabseq 16512 | If two integers divide eac... |
| dvdseq 16513 | If two nonnegative integer... |
| divconjdvds 16514 | If a nonzero integer ` M `... |
| dvdsdivcl 16515 | The complement of a diviso... |
| dvdsflip 16516 | An involution of the divis... |
| dvdsssfz1 16517 | The set of divisors of a n... |
| dvds1 16518 | The only nonnegative integ... |
| alzdvds 16519 | Only 0 is divisible by all... |
| dvdsext 16520 | Poset extensionality for d... |
| fzm1ndvds 16521 | No number from ` 1 ` to ` ... |
| fzo0dvdseq 16522 | Zero is the only one of th... |
| fzocongeq 16523 | Two different elements of ... |
| addmodlteqALT 16524 | Two nonnegative integers l... |
| dvdsfac 16525 | A positive integer divides... |
| dvdsexp2im 16526 | If an integer divides anot... |
| dvdsexp 16527 | A power divides a power wi... |
| dvdsmod 16528 | Any number ` K ` whose mod... |
| mulmoddvds 16529 | If an integer is divisible... |
| 3dvds 16530 | A rule for divisibility by... |
| 3dvdsdec 16531 | A decimal number is divisi... |
| 3dvds2dec 16532 | A decimal number is divisi... |
| fprodfvdvdsd 16533 | A finite product of intege... |
| fproddvdsd 16534 | A finite product of intege... |
| evenelz 16535 | An even number is an integ... |
| zeo3 16536 | An integer is even or odd.... |
| zeo4 16537 | An integer is even or odd ... |
| zeneo 16538 | No even integer equals an ... |
| odd2np1lem 16539 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16540 | An integer is odd iff it i... |
| even2n 16541 | An integer is even iff it ... |
| oddm1even 16542 | An integer is odd iff its ... |
| oddp1even 16543 | An integer is odd iff its ... |
| oexpneg 16544 | The exponential of the neg... |
| mod2eq0even 16545 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16546 | An integer is 1 modulo 2 i... |
| oddnn02np1 16547 | A nonnegative integer is o... |
| oddge22np1 16548 | An integer greater than on... |
| evennn02n 16549 | A nonnegative integer is e... |
| evennn2n 16550 | A positive integer is even... |
| 2tp1odd 16551 | A number which is twice an... |
| mulsucdiv2z 16552 | An integer multiplied with... |
| sqoddm1div8z 16553 | A squared odd number minus... |
| 2teven 16554 | A number which is twice an... |
| zeo5 16555 | An integer is either even ... |
| evend2 16556 | An integer is even iff its... |
| oddp1d2 16557 | An integer is odd iff its ... |
| zob 16558 | Alternate characterization... |
| oddm1d2 16559 | An integer is odd iff its ... |
| ltoddhalfle 16560 | An integer is less than ha... |
| halfleoddlt 16561 | An integer is greater than... |
| opoe 16562 | The sum of two odds is eve... |
| omoe 16563 | The difference of two odds... |
| opeo 16564 | The sum of an odd and an e... |
| omeo 16565 | The difference of an odd a... |
| z0even 16566 | 2 divides 0. That means 0... |
| n2dvds1 16567 | 2 does not divide 1. That... |
| n2dvdsm1 16568 | 2 does not divide -1. Tha... |
| z2even 16569 | 2 divides 2. That means 2... |
| n2dvds3 16570 | 2 does not divide 3. That... |
| z4even 16571 | 2 divides 4. That means 4... |
| 4dvdseven 16572 | An integer which is divisi... |
| m1expe 16573 | Exponentiation of -1 by an... |
| m1expo 16574 | Exponentiation of -1 by an... |
| m1exp1 16575 | Exponentiation of negative... |
| nn0enne 16576 | A positive integer is an e... |
| nn0ehalf 16577 | The half of an even nonneg... |
| nnehalf 16578 | The half of an even positi... |
| nn0onn 16579 | An odd nonnegative integer... |
| nn0o1gt2 16580 | An odd nonnegative integer... |
| nno 16581 | An alternate characterizat... |
| nn0o 16582 | An alternate characterizat... |
| nn0ob 16583 | Alternate characterization... |
| nn0oddm1d2 16584 | A positive integer is odd ... |
| nnoddm1d2 16585 | A positive integer is odd ... |
| sumeven 16586 | If every term in a sum is ... |
| sumodd 16587 | If every term in a sum is ... |
| evensumodd 16588 | If every term in a sum wit... |
| oddsumodd 16589 | If every term in a sum wit... |
| pwp1fsum 16590 | The n-th power of a number... |
| oddpwp1fsum 16591 | An odd power of a number i... |
| divalglem0 16592 | Lemma for ~ divalg . (Con... |
| divalglem1 16593 | Lemma for ~ divalg . (Con... |
| divalglem2 16594 | Lemma for ~ divalg . (Con... |
| divalglem4 16595 | Lemma for ~ divalg . (Con... |
| divalglem5 16596 | Lemma for ~ divalg . (Con... |
| divalglem6 16597 | Lemma for ~ divalg . (Con... |
| divalglem7 16598 | Lemma for ~ divalg . (Con... |
| divalglem8 16599 | Lemma for ~ divalg . (Con... |
| divalglem9 16600 | Lemma for ~ divalg . (Con... |
| divalglem10 16601 | Lemma for ~ divalg . (Con... |
| divalg 16602 | The division algorithm (th... |
| divalgb 16603 | Express the division algor... |
| divalg2 16604 | The division algorithm (th... |
| divalgmod 16605 | The result of the ` mod ` ... |
| divalgmodcl 16606 | The result of the ` mod ` ... |
| modremain 16607 | The result of the modulo o... |
| ndvdssub 16608 | Corollary of the division ... |
| ndvdsadd 16609 | Corollary of the division ... |
| ndvdsp1 16610 | Special case of ~ ndvdsadd... |
| ndvdsi 16611 | A quick test for non-divis... |
| 5ndvds3 16612 | 5 does not divide 3. (Con... |
| 5ndvds6 16613 | 5 does not divide 6. (Con... |
| flodddiv4 16614 | The floor of an odd intege... |
| fldivndvdslt 16615 | The floor of an integer di... |
| flodddiv4lt 16616 | The floor of an odd number... |
| flodddiv4t2lthalf 16617 | The floor of an odd number... |
| bitsfval 16622 | Expand the definition of t... |
| bitsval 16623 | Expand the definition of t... |
| bitsval2 16624 | Expand the definition of t... |
| bitsss 16625 | The set of bits of an inte... |
| bitsf 16626 | The ` bits ` function is a... |
| bits0 16627 | Value of the zeroth bit. ... |
| bits0e 16628 | The zeroth bit of an even ... |
| bits0o 16629 | The zeroth bit of an odd n... |
| bitsp1 16630 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16631 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16632 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16633 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16634 | The bits of a number are a... |
| bitsmod 16635 | Truncating the bit sequenc... |
| bitsfi 16636 | Every number is associated... |
| bitscmp 16637 | The bit complement of ` N ... |
| 0bits 16638 | The bits of zero. (Contri... |
| m1bits 16639 | The bits of negative one. ... |
| bitsinv1lem 16640 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16641 | There is an explicit inver... |
| bitsinv2 16642 | There is an explicit inver... |
| bitsf1ocnv 16643 | The ` bits ` function rest... |
| bitsf1o 16644 | The ` bits ` function rest... |
| bitsf1 16645 | The ` bits ` function is a... |
| 2ebits 16646 | The bits of a power of two... |
| bitsinv 16647 | The inverse of the ` bits ... |
| bitsinvp1 16648 | Recursive definition of th... |
| sadadd2lem2 16649 | The core of the proof of ~... |
| sadfval 16651 | Define the addition of two... |
| sadcf 16652 | The carry sequence is a se... |
| sadc0 16653 | The initial element of the... |
| sadcp1 16654 | The carry sequence (which ... |
| sadval 16655 | The full adder sequence is... |
| sadcaddlem 16656 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16657 | Non-recursive definition o... |
| sadadd2lem 16658 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16659 | Sum of initial segments of... |
| sadadd3 16660 | Sum of initial segments of... |
| sadcl 16661 | The sum of two sequences i... |
| sadcom 16662 | The adder sequence functio... |
| saddisjlem 16663 | Lemma for ~ sadadd . (Con... |
| saddisj 16664 | The sum of disjoint sequen... |
| sadaddlem 16665 | Lemma for ~ sadadd . (Con... |
| sadadd 16666 | For sequences that corresp... |
| sadid1 16667 | The adder sequence functio... |
| sadid2 16668 | The adder sequence functio... |
| sadasslem 16669 | Lemma for ~ sadass . (Con... |
| sadass 16670 | Sequence addition is assoc... |
| sadeq 16671 | Any element of a sequence ... |
| bitsres 16672 | Restrict the bits of a num... |
| bitsuz 16673 | The bits of a number are a... |
| bitsshft 16674 | Shifting a bit sequence to... |
| smufval 16676 | The multiplication of two ... |
| smupf 16677 | The sequence of partial su... |
| smup0 16678 | The initial element of the... |
| smupp1 16679 | The initial element of the... |
| smuval 16680 | Define the addition of two... |
| smuval2 16681 | The partial sum sequence s... |
| smupvallem 16682 | If ` A ` only has elements... |
| smucl 16683 | The product of two sequenc... |
| smu01lem 16684 | Lemma for ~ smu01 and ~ sm... |
| smu01 16685 | Multiplication of a sequen... |
| smu02 16686 | Multiplication of a sequen... |
| smupval 16687 | Rewrite the elements of th... |
| smup1 16688 | Rewrite ~ smupp1 using onl... |
| smueqlem 16689 | Any element of a sequence ... |
| smueq 16690 | Any element of a sequence ... |
| smumullem 16691 | Lemma for ~ smumul . (Con... |
| smumul 16692 | For sequences that corresp... |
| gcdval 16695 | The value of the ` gcd ` o... |
| gcd0val 16696 | The value, by convention, ... |
| gcdn0val 16697 | The value of the ` gcd ` o... |
| gcdcllem1 16698 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16699 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16700 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16701 | Closure of the ` gcd ` ope... |
| gcddvds 16702 | The gcd of two integers di... |
| dvdslegcd 16703 | An integer which divides b... |
| nndvdslegcd 16704 | A positive integer which d... |
| gcdcl 16705 | Closure of the ` gcd ` ope... |
| gcdnncl 16706 | Closure of the ` gcd ` ope... |
| gcdcld 16707 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16708 | Closure of the ` gcd ` ope... |
| gcdle1d 16709 | The greatest common diviso... |
| gcdle2d 16710 | The greatest common diviso... |
| zeqzmulgcd 16711 | An integer is the product ... |
| divgcdz 16712 | An integer divided by the ... |
| gcdf 16713 | Domain and codomain of the... |
| gcdcom 16714 | The ` gcd ` operator is co... |
| gcdcomd 16715 | The ` gcd ` operator is co... |
| divgcdnn 16716 | A positive integer divided... |
| divgcdnnr 16717 | A positive integer divided... |
| gcdeq0 16718 | The gcd of two integers is... |
| gcdn0gt0 16719 | The gcd of two integers is... |
| gcd0id 16720 | The gcd of 0 and an intege... |
| gcdid0 16721 | The gcd of an integer and ... |
| nn0gcdid0 16722 | The gcd of a nonnegative i... |
| gcdneg 16723 | Negating one operand of th... |
| neggcd 16724 | Negating one operand of th... |
| gcdaddmlem 16725 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16726 | Adding a multiple of one o... |
| gcdadd 16727 | The gcd of two numbers is ... |
| gcdid 16728 | The gcd of a number and it... |
| gcdnn0id 16729 | The gcd of a nonnegative i... |
| gcd1 16730 | The gcd of a number with 1... |
| gcdabs1 16731 | The gcd of the absolute va... |
| gcdabs2 16732 | The gcd of the absolute va... |
| gcdabs 16733 | The gcd of two integers is... |
| modgcd 16734 | The gcd remains unchanged ... |
| 1gcd 16735 | The gcd of one and an inte... |
| gcdmultipled 16736 | The greatest common diviso... |
| gcdmultiplez 16737 | The gcd of a multiple of a... |
| gcdmultiple 16738 | The gcd of a multiple of a... |
| dvdsgcdidd 16739 | The greatest common diviso... |
| 6gcd4e2 16740 | The greatest common diviso... |
| bezoutlem1 16741 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16742 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16743 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16744 | Lemma for ~ bezout . (Con... |
| bezout 16745 | Bézout's identity: ... |
| dvdsgcd 16746 | An integer which divides e... |
| dvdsgcdb 16747 | Biconditional form of ~ dv... |
| dfgcd2 16748 | Alternate definition of th... |
| gcdass 16749 | Associative law for ` gcd ... |
| mulgcd 16750 | Distribute multiplication ... |
| absmulgcd 16751 | Distribute absolute value ... |
| mulgcdr 16752 | Reverse distribution law f... |
| gcddiv 16753 | Division law for the ` gcd... |
| gcdzeq 16754 | A positive integer ` A ` i... |
| gcdeq 16755 | ` A ` is equal to its gcd ... |
| dvdssqim 16756 | Unidirectional form of ~ d... |
| dvdsexpim 16757 | If two numbers are divisib... |
| dvdsexpad 16758 | Deduction associated with ... |
| dvdsmulgcd 16759 | A divisibility equivalent ... |
| rpmulgcd 16760 | If ` K ` and ` M ` are rel... |
| rplpwr 16761 | If ` A ` and ` B ` are rel... |
| rprpwr 16762 | If ` A ` and ` B ` are rel... |
| rppwr 16763 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16764 | If ` A ` and ` B ` are rel... |
| sqgcd 16765 | Square distributes over gc... |
| expgcd 16766 | Exponentiation distributes... |
| nn0expgcd 16767 | Exponentiation distributes... |
| zexpgcd 16768 | Exponentiation distributes... |
| dvdsexpnn 16769 | Two positive integers are ... |
| dvdssqnn 16770 | Two positive integers are ... |
| dvdssq 16771 | Two integers are divisible... |
| bezoutr 16772 | Partial converse to ~ bezo... |
| bezoutr1 16773 | Converse of ~ bezout for w... |
| nn0seqcvgd 16774 | A strictly-decreasing nonn... |
| seq1st 16775 | A sequence whose iteration... |
| algr0 16776 | The value of the algorithm... |
| algrf 16777 | An algorithm is a step fun... |
| algrp1 16778 | The value of the algorithm... |
| alginv 16779 | If ` I ` is an invariant o... |
| algcvg 16780 | One way to prove that an a... |
| algcvgblem 16781 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16782 | Two ways of expressing tha... |
| algcvga 16783 | The countdown function ` C... |
| algfx 16784 | If ` F ` reaches a fixed p... |
| eucalgval2 16785 | The value of the step func... |
| eucalgval 16786 | Euclid's Algorithm ~ eucal... |
| eucalgf 16787 | Domain and codomain of the... |
| eucalginv 16788 | The invariant of the step ... |
| eucalglt 16789 | The second member of the s... |
| eucalgcvga 16790 | Once Euclid's Algorithm ha... |
| eucalg 16791 | Euclid's Algorithm compute... |
| lcmval 16796 | Value of the ` lcm ` opera... |
| lcmcom 16797 | The ` lcm ` operator is co... |
| lcm0val 16798 | The value, by convention, ... |
| lcmn0val 16799 | The value of the ` lcm ` o... |
| lcmcllem 16800 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16801 | Closure of the ` lcm ` ope... |
| dvdslcm 16802 | The lcm of two integers is... |
| lcmledvds 16803 | A positive integer which b... |
| lcmeq0 16804 | The lcm of two integers is... |
| lcmcl 16805 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16806 | The greatest common diviso... |
| lcmneg 16807 | Negating one operand of th... |
| neglcm 16808 | Negating one operand of th... |
| lcmabs 16809 | The lcm of two integers is... |
| lcmgcdlem 16810 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16811 | The product of two numbers... |
| lcmdvds 16812 | The lcm of two integers di... |
| lcmid 16813 | The lcm of an integer and ... |
| lcm1 16814 | The lcm of an integer and ... |
| lcmgcdnn 16815 | The product of two positiv... |
| lcmgcdeq 16816 | Two integers' absolute val... |
| lcmdvdsb 16817 | Biconditional form of ~ lc... |
| lcmass 16818 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16819 | The least common multiple ... |
| 6lcm4e12 16820 | The least common multiple ... |
| absproddvds 16821 | The absolute value of the ... |
| absprodnn 16822 | The absolute value of the ... |
| fissn0dvds 16823 | For each finite subset of ... |
| fissn0dvdsn0 16824 | For each finite subset of ... |
| lcmfval 16825 | Value of the ` _lcm ` func... |
| lcmf0val 16826 | The value, by convention, ... |
| lcmfn0val 16827 | The value of the ` _lcm ` ... |
| lcmfnnval 16828 | The value of the ` _lcm ` ... |
| lcmfcllem 16829 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16830 | Closure of the ` _lcm ` fu... |
| lcmfpr 16831 | The value of the ` _lcm ` ... |
| lcmfcl 16832 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16833 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16834 | The least common multiple ... |
| dvdslcmf 16835 | The least common multiple ... |
| lcmfledvds 16836 | A positive integer which i... |
| lcmf 16837 | Characterization of the le... |
| lcmf0 16838 | The least common multiple ... |
| lcmfsn 16839 | The least common multiple ... |
| lcmftp 16840 | The least common multiple ... |
| lcmfunsnlem1 16841 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16842 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16843 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16844 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16845 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16846 | The least common multiple ... |
| lcmfdvdsb 16847 | Biconditional form of ~ lc... |
| lcmfunsn 16848 | The ` _lcm ` function for ... |
| lcmfun 16849 | The ` _lcm ` function for ... |
| lcmfass 16850 | Associative law for the ` ... |
| lcmf2a3a4e12 16851 | The least common multiple ... |
| lcmflefac 16852 | The least common multiple ... |
| coprmgcdb 16853 | Two positive integers are ... |
| ncoprmgcdne1b 16854 | Two positive integers are ... |
| ncoprmgcdgt1b 16855 | Two positive integers are ... |
| coprmdvds1 16856 | If two positive integers a... |
| coprmdvds 16857 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16858 | If an integer is divisible... |
| mulgcddvds 16859 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16860 | If ` M ` is relatively pri... |
| qredeq 16861 | Two equal reduced fraction... |
| qredeu 16862 | Every rational number has ... |
| rpmul 16863 | If ` K ` is relatively pri... |
| rpdvds 16864 | If ` K ` is relatively pri... |
| coprmprod 16865 | The product of the element... |
| coprmproddvdslem 16866 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16867 | If a positive integer is d... |
| congr 16868 | Definition of congruence b... |
| divgcdcoprm0 16869 | Integers divided by gcd ar... |
| divgcdcoprmex 16870 | Integers divided by gcd ar... |
| cncongr1 16871 | One direction of the bicon... |
| cncongr2 16872 | The other direction of the... |
| cncongr 16873 | Cancellability of Congruen... |
| cncongrcoprm 16874 | Corollary 1 of Cancellabil... |
| isprm 16877 | The predicate "is a prime ... |
| prmnn 16878 | A prime number is a positi... |
| prmz 16879 | A prime number is an integ... |
| prmssnn 16880 | The prime numbers are a su... |
| prmex 16881 | The set of prime numbers e... |
| 0nprm 16882 | 0 is not a prime number. ... |
| 1nprm 16883 | 1 is not a prime number. ... |
| 1idssfct 16884 | The positive divisors of a... |
| isprm2lem 16885 | Lemma for ~ isprm2 . (Con... |
| isprm2 16886 | The predicate "is a prime ... |
| isprm3 16887 | The predicate "is a prime ... |
| isprm4 16888 | The predicate "is a prime ... |
| prmind2 16889 | A variation on ~ prmind as... |
| prmind 16890 | Perform induction over the... |
| dvdsprime 16891 | If ` M ` divides a prime, ... |
| nprm 16892 | A product of two integers ... |
| nprmi 16893 | An inference for composite... |
| dvdsnprmd 16894 | If a number is divisible b... |
| prm2orodd 16895 | A prime number is either 2... |
| 2prm 16896 | 2 is a prime number. (Con... |
| 2mulprm 16897 | A multiple of two is prime... |
| 3prm 16898 | 3 is a prime number. (Con... |
| 4nprm 16899 | 4 is not a prime number. ... |
| prmuz2 16900 | A prime number is an integ... |
| prmssuz2 16901 | The primes are integers gr... |
| prmgt1 16902 | A prime number is an integ... |
| prmm2nn0 16903 | Subtracting 2 from a prime... |
| oddprmgt2 16904 | An odd prime is greater th... |
| oddprmge3 16905 | An odd prime is greater th... |
| ge2nprmge4 16906 | A composite integer greate... |
| sqnprm 16907 | A square is never prime. ... |
| dvdsprm 16908 | An integer greater than or... |
| exprmfct 16909 | Every integer greater than... |
| prmdvdsfz 16910 | Each integer greater than ... |
| nprmdvds1 16911 | No prime number divides 1.... |
| isprm5 16912 | One need only check prime ... |
| isprm7 16913 | One need only check prime ... |
| maxprmfct 16914 | The set of prime factors o... |
| divgcdodd 16915 | Either ` A / ( A gcd B ) `... |
| coprm 16916 | A prime number either divi... |
| prmrp 16917 | Unequal prime numbers are ... |
| euclemma 16918 | Euclid's lemma. A prime n... |
| isprm6 16919 | A number is prime iff it s... |
| prmdvdsexp 16920 | A prime divides a positive... |
| prmdvdsexpb 16921 | A prime divides a positive... |
| prmdvdsexpr 16922 | If a prime divides a nonne... |
| prmdvdssq 16923 | Condition for a prime divi... |
| prmexpb 16924 | Two positive prime powers ... |
| prmfac1 16925 | The factorial of a number ... |
| dvdszzq 16926 | Divisibility for an intege... |
| rpexp 16927 | If two numbers ` A ` and `... |
| rpexp1i 16928 | Relative primality passes ... |
| rpexp12i 16929 | Relative primality passes ... |
| prmndvdsfaclt 16930 | A prime number does not di... |
| prmdvdsbc 16931 | Condition for a prime numb... |
| prmdvdsncoprmbd 16932 | Two positive integers are ... |
| ncoprmlnprm 16933 | If two positive integers a... |
| cncongrprm 16934 | Corollary 2 of Cancellabil... |
| isevengcd2 16935 | The predicate "is an even ... |
| isoddgcd1 16936 | The predicate "is an odd n... |
| 3lcm2e6 16937 | The least common multiple ... |
| qnumval 16942 | Value of the canonical num... |
| qdenval 16943 | Value of the canonical den... |
| qnumdencl 16944 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16945 | The canonical numerator of... |
| qdencl 16946 | The canonical denominator ... |
| fnum 16947 | Canonical numerator define... |
| fden 16948 | Canonical denominator defi... |
| qnumdenbi 16949 | Two numbers are the canoni... |
| qnumdencoprm 16950 | The canonical representati... |
| qeqnumdivden 16951 | Recover a rational number ... |
| qmuldeneqnum 16952 | Multiplying a rational by ... |
| divnumden 16953 | Calculate the reduced form... |
| divdenle 16954 | Reducing a quotient never ... |
| qnumgt0 16955 | A rational is positive iff... |
| qgt0numnn 16956 | A rational is positive iff... |
| nn0gcdsq 16957 | Squaring commutes with GCD... |
| zgcdsq 16958 | ~ nn0gcdsq extended to int... |
| numdensq 16959 | Squaring a rational square... |
| numsq 16960 | Square commutes with canon... |
| densq 16961 | Square commutes with canon... |
| qden1elz 16962 | A rational is an integer i... |
| zsqrtelqelz 16963 | If an integer has a ration... |
| nonsq 16964 | Any integer strictly betwe... |
| posqsqznn 16965 | When a positive rational s... |
| numdenexp 16966 | Elevating a rational numbe... |
| numexp 16967 | Elevating to a nonnegative... |
| denexp 16968 | Elevating to a nonnegative... |
| phival 16973 | Value of the Euler ` phi `... |
| phicl2 16974 | Bounds and closure for the... |
| phicl 16975 | Closure for the value of t... |
| phibndlem 16976 | Lemma for ~ phibnd . (Con... |
| phibnd 16977 | A slightly tighter bound o... |
| phicld 16978 | Closure for the value of t... |
| phi1 16979 | Value of the Euler ` phi `... |
| dfphi2 16980 | Alternate definition of th... |
| hashdvds 16981 | The number of numbers in a... |
| phiprmpw 16982 | Value of the Euler ` phi `... |
| phiprm 16983 | Value of the Euler ` phi `... |
| crth 16984 | The Chinese Remainder Theo... |
| phimullem 16985 | Lemma for ~ phimul . (Con... |
| phimul 16986 | The Euler ` phi ` function... |
| eulerthlem1 16987 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16988 | Lemma for ~ eulerth . (Co... |
| eulerth 16989 | Euler's theorem, a general... |
| fermltl 16990 | Fermat's little theorem. ... |
| prmdiv 16991 | Show an explicit expressio... |
| prmdiveq 16992 | The modular inverse of ` A... |
| prmdivdiv 16993 | The (modular) inverse of t... |
| hashgcdlem 16994 | A correspondence between e... |
| dvdsfi 16995 | A natural number has finit... |
| hashgcdeq 16996 | Number of initial positive... |
| phisum 16997 | The divisor sum identity o... |
| odzval 16998 | Value of the order functio... |
| odzcllem 16999 | - Lemma for ~ odzcl , show... |
| odzcl 17000 | The order of a group eleme... |
| odzid 17001 | Any element raised to the ... |
| odzdvds 17002 | The only powers of ` A ` t... |
| odzphi 17003 | The order of any group ele... |
| modprm1div 17004 | A prime number divides an ... |
| m1dvdsndvds 17005 | If an integer minus 1 is d... |
| modprminv 17006 | Show an explicit expressio... |
| modprminveq 17007 | The modular inverse of ` A... |
| vfermltl 17008 | Variant of Fermat's little... |
| vfermltlALT 17009 | Alternate proof of ~ vferm... |
| powm2modprm 17010 | If an integer minus 1 is d... |
| reumodprminv 17011 | For any prime number and f... |
| modprm0 17012 | For two positive integers ... |
| nnnn0modprm0 17013 | For a positive integer and... |
| modprmn0modprm0 17014 | For an integer not being 0... |
| coprimeprodsq 17015 | If three numbers are copri... |
| coprimeprodsq2 17016 | If three numbers are copri... |
| oddprm 17017 | A prime not equal to ` 2 `... |
| nnoddn2prm 17018 | A prime not equal to ` 2 `... |
| oddn2prm 17019 | A prime not equal to ` 2 `... |
| nnoddn2prmb 17020 | A number is a prime number... |
| prm23lt5 17021 | A prime less than 5 is eit... |
| prm23ge5 17022 | A prime is either 2 or 3 o... |
| pythagtriplem1 17023 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 17024 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 17025 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 17026 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 17027 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 17028 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 17029 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 17030 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 17031 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 17032 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 17033 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 17034 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 17035 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 17036 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 17037 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 17038 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 17039 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 17040 | Lemma for ~ pythagtrip . ... |
| pythagtrip 17041 | Parameterize the Pythagore... |
| iserodd 17042 | Collect the odd terms in a... |
| pclem 17045 | - Lemma for the prime powe... |
| pcprecl 17046 | Closure of the prime power... |
| pcprendvds 17047 | Non-divisibility property ... |
| pcprendvds2 17048 | Non-divisibility property ... |
| pcpre1 17049 | Value of the prime power p... |
| pcpremul 17050 | Multiplicative property of... |
| pcval 17051 | The value of the prime pow... |
| pceulem 17052 | Lemma for ~ pceu . (Contr... |
| pceu 17053 | Uniqueness for the prime p... |
| pczpre 17054 | Connect the prime count pr... |
| pczcl 17055 | Closure of the prime power... |
| pccl 17056 | Closure of the prime power... |
| pccld 17057 | Closure of the prime power... |
| pcmul 17058 | Multiplication property of... |
| pcdiv 17059 | Division property of the p... |
| pcqmul 17060 | Multiplication property of... |
| pc0 17061 | The value of the prime pow... |
| pc1 17062 | Value of the prime count f... |
| pcqcl 17063 | Closure of the general pri... |
| pcqdiv 17064 | Division property of the p... |
| pcrec 17065 | Prime power of a reciproca... |
| pcexp 17066 | Prime power of an exponent... |
| pcxnn0cl 17067 | Extended nonnegative integ... |
| pcxcl 17068 | Extended real closure of t... |
| pcge0 17069 | The prime count of an inte... |
| pczdvds 17070 | Defining property of the p... |
| pcdvds 17071 | Defining property of the p... |
| pczndvds 17072 | Defining property of the p... |
| pcndvds 17073 | Defining property of the p... |
| pczndvds2 17074 | The remainder after dividi... |
| pcndvds2 17075 | The remainder after dividi... |
| pcdvdsb 17076 | ` P ^ A ` divides ` N ` if... |
| pcelnn 17077 | There are a positive numbe... |
| pceq0 17078 | There are zero powers of a... |
| pcidlem 17079 | The prime count of a prime... |
| pcid 17080 | The prime count of a prime... |
| pcneg 17081 | The prime count of a negat... |
| pcabs 17082 | The prime count of an abso... |
| pcdvdstr 17083 | The prime count increases ... |
| pcgcd1 17084 | The prime count of a GCD i... |
| pcgcd 17085 | The prime count of a GCD i... |
| pc2dvds 17086 | A characterization of divi... |
| pc11 17087 | The prime count function, ... |
| pcz 17088 | The prime count function c... |
| pcprmpw2 17089 | Self-referential expressio... |
| pcprmpw 17090 | Self-referential expressio... |
| dvdsprmpweq 17091 | If a positive integer divi... |
| dvdsprmpweqnn 17092 | If an integer greater than... |
| dvdsprmpweqle 17093 | If a positive integer divi... |
| difsqpwdvds 17094 | If the difference of two s... |
| pcaddlem 17095 | Lemma for ~ pcadd . The o... |
| pcadd 17096 | An inequality for the prim... |
| pcadd2 17097 | The inequality of ~ pcadd ... |
| pcmptcl 17098 | Closure for the prime powe... |
| pcmpt 17099 | Construct a function with ... |
| pcmpt2 17100 | Dividing two prime count m... |
| pcmptdvds 17101 | The partial products of th... |
| pcprod 17102 | The product of the primes ... |
| sumhash 17103 | The sum of 1 over a set is... |
| fldivp1 17104 | The difference between the... |
| pcfaclem 17105 | Lemma for ~ pcfac . (Cont... |
| pcfac 17106 | Calculate the prime count ... |
| pcbc 17107 | Calculate the prime count ... |
| qexpz 17108 | If a power of a rational n... |
| expnprm 17109 | A second or higher power o... |
| oddprmdvds 17110 | Every positive integer whi... |
| prmpwdvds 17111 | A relation involving divis... |
| pockthlem 17112 | Lemma for ~ pockthg . (Co... |
| pockthg 17113 | The generalized Pocklingto... |
| pockthi 17114 | Pocklington's theorem, whi... |
| unbenlem 17115 | Lemma for ~ unben . (Cont... |
| unben 17116 | An unbounded set of positi... |
| infpnlem1 17117 | Lemma for ~ infpn . The s... |
| infpnlem2 17118 | Lemma for ~ infpn . For a... |
| infpn 17119 | There exist infinitely man... |
| infpn2 17120 | There exist infinitely man... |
| prmunb 17121 | The primes are unbounded. ... |
| prminf 17122 | There are an infinite numb... |
| prmreclem1 17123 | Lemma for ~ prmrec . Prop... |
| prmreclem2 17124 | Lemma for ~ prmrec . Ther... |
| prmreclem3 17125 | Lemma for ~ prmrec . The ... |
| prmreclem4 17126 | Lemma for ~ prmrec . Show... |
| prmreclem5 17127 | Lemma for ~ prmrec . Here... |
| prmreclem6 17128 | Lemma for ~ prmrec . If t... |
| prmrec 17129 | The sum of the reciprocals... |
| 1arithlem1 17130 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 17131 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 17132 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 17133 | Lemma for ~ 1arith . (Con... |
| 1arith 17134 | Fundamental theorem of ari... |
| 1arith2 17135 | Fundamental theorem of ari... |
| elgz 17138 | Elementhood in the gaussia... |
| gzcn 17139 | A gaussian integer is a co... |
| zgz 17140 | An integer is a gaussian i... |
| igz 17141 | ` _i ` is a gaussian integ... |
| gznegcl 17142 | The gaussian integers are ... |
| gzcjcl 17143 | The gaussian integers are ... |
| gzaddcl 17144 | The gaussian integers are ... |
| gzmulcl 17145 | The gaussian integers are ... |
| gzreim 17146 | Construct a gaussian integ... |
| gzsubcl 17147 | The gaussian integers are ... |
| gzabssqcl 17148 | The squared norm of a gaus... |
| 4sqlem5 17149 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 17150 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 17151 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 17152 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 17153 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 17154 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 17155 | Lemma for ~ 4sq . The set... |
| 4sqlem2 17156 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 17157 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 17158 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 17159 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 17160 | Lemma for ~ mul4sq : algeb... |
| mul4sq 17161 | Euler's four-square identi... |
| 4sqlem11 17162 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 17163 | Lemma for ~ 4sq . For any... |
| 4sqlem13 17164 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 17165 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 17166 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 17167 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 17168 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 17169 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 17170 | Lemma for ~ 4sq . The pro... |
| 4sq 17171 | Lagrange's four-square the... |
| vdwapfval 17178 | Define the arithmetic prog... |
| vdwapf 17179 | The arithmetic progression... |
| vdwapval 17180 | Value of the arithmetic pr... |
| vdwapun 17181 | Remove the first element o... |
| vdwapid1 17182 | The first element of an ar... |
| vdwap0 17183 | Value of a length-1 arithm... |
| vdwap1 17184 | Value of a length-1 arithm... |
| vdwmc 17185 | The predicate " The ` <. R... |
| vdwmc2 17186 | Expand out the definition ... |
| vdwpc 17187 | The predicate " The colori... |
| vdwlem1 17188 | Lemma for ~ vdw . (Contri... |
| vdwlem2 17189 | Lemma for ~ vdw . (Contri... |
| vdwlem3 17190 | Lemma for ~ vdw . (Contri... |
| vdwlem4 17191 | Lemma for ~ vdw . (Contri... |
| vdwlem5 17192 | Lemma for ~ vdw . (Contri... |
| vdwlem6 17193 | Lemma for ~ vdw . (Contri... |
| vdwlem7 17194 | Lemma for ~ vdw . (Contri... |
| vdwlem8 17195 | Lemma for ~ vdw . (Contri... |
| vdwlem9 17196 | Lemma for ~ vdw . (Contri... |
| vdwlem10 17197 | Lemma for ~ vdw . Set up ... |
| vdwlem11 17198 | Lemma for ~ vdw . (Contri... |
| vdwlem12 17199 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 17200 | Lemma for ~ vdw . Main in... |
| vdw 17201 | Van der Waerden's theorem.... |
| vdwnnlem1 17202 | Corollary of ~ vdw , and l... |
| vdwnnlem2 17203 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 17204 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 17205 | Van der Waerden's theorem,... |
| ramtlecl 17207 | The set ` T ` of numbers w... |
| hashbcval 17209 | Value of the "binomial set... |
| hashbccl 17210 | The binomial set is a fini... |
| hashbcss 17211 | Subset relation for the bi... |
| hashbc0 17212 | The set of subsets of size... |
| hashbc2 17213 | The size of the binomial s... |
| 0hashbc 17214 | There are no subsets of th... |
| ramval 17215 | The value of the Ramsey nu... |
| ramcl2lem 17216 | Lemma for extended real cl... |
| ramtcl 17217 | The Ramsey number has the ... |
| ramtcl2 17218 | The Ramsey number is an in... |
| ramtub 17219 | The Ramsey number is a low... |
| ramub 17220 | The Ramsey number is a low... |
| ramub2 17221 | It is sufficient to check ... |
| rami 17222 | The defining property of a... |
| ramcl2 17223 | The Ramsey number is eithe... |
| ramxrcl 17224 | The Ramsey number is an ex... |
| ramubcl 17225 | If the Ramsey number is up... |
| ramlb 17226 | Establish a lower bound on... |
| 0ram 17227 | The Ramsey number when ` M... |
| 0ram2 17228 | The Ramsey number when ` M... |
| ram0 17229 | The Ramsey number when ` R... |
| 0ramcl 17230 | Lemma for ~ ramcl : Exist... |
| ramz2 17231 | The Ramsey number when ` F... |
| ramz 17232 | The Ramsey number when ` F... |
| ramub1lem1 17233 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 17234 | Lemma for ~ ramub1 . (Con... |
| ramub1 17235 | Inductive step for Ramsey'... |
| ramcl 17236 | Ramsey's theorem: the Rams... |
| ramsey 17237 | Ramsey's theorem with the ... |
| prmoval 17240 | Value of the primorial fun... |
| prmocl 17241 | Closure of the primorial f... |
| prmone0 17242 | The primorial function is ... |
| prmo0 17243 | The primorial of 0. (Cont... |
| prmo1 17244 | The primorial of 1. (Cont... |
| prmop1 17245 | The primorial of a success... |
| prmonn2 17246 | Value of the primorial fun... |
| prmo2 17247 | The primorial of 2. (Cont... |
| prmo3 17248 | The primorial of 3. (Cont... |
| prmdvdsprmo 17249 | The primorial of a number ... |
| prmdvdsprmop 17250 | The primorial of a number ... |
| fvprmselelfz 17251 | The value of the prime sel... |
| fvprmselgcd1 17252 | The greatest common diviso... |
| prmolefac 17253 | The primorial of a positiv... |
| prmodvdslcmf 17254 | The primorial of a nonnega... |
| prmolelcmf 17255 | The primorial of a positiv... |
| prmgaplem1 17256 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17257 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17258 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17259 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17260 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17261 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17262 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17263 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17264 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17265 | Lemma for ~ prmgap . (Con... |
| prmgap 17266 | The prime gap theorem: for... |
| prmgaplcm 17267 | Alternate proof of ~ prmga... |
| prmgapprmolem 17268 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17269 | Alternate proof of ~ prmga... |
| dec2dvds 17270 | Divisibility by two is obv... |
| dec5dvds 17271 | Divisibility by five is ob... |
| dec5dvds2 17272 | Divisibility by five is ob... |
| dec5nprm 17273 | A decimal number greater t... |
| dec2nprm 17274 | A decimal number greater t... |
| modxai 17275 | Add exponents in a power m... |
| mod2xi 17276 | Double exponents in a powe... |
| modxp1i 17277 | Add one to an exponent in ... |
| mod2xnegi 17278 | Version of ~ mod2xi where ... |
| modsubi 17279 | Subtract from within a mod... |
| gcdi 17280 | Calculate a GCD via Euclid... |
| gcdmodi 17281 | Calculate a GCD via Euclid... |
| numexp0 17282 | Calculate an integer power... |
| numexp1 17283 | Calculate an integer power... |
| numexpp1 17284 | Calculate an integer power... |
| numexp2x 17285 | Double an integer power. ... |
| decsplit0b 17286 | Split a decimal number int... |
| decsplit0 17287 | Split a decimal number int... |
| decsplit1 17288 | Split a decimal number int... |
| decsplit 17289 | Split a decimal number int... |
| karatsuba 17290 | The Karatsuba multiplicati... |
| 2exp4 17291 | Two to the fourth power is... |
| 2exp5 17292 | Two to the fifth power is ... |
| 2exp6 17293 | Two to the sixth power is ... |
| 2exp7 17294 | Two to the seventh power i... |
| 2exp8 17295 | Two to the eighth power is... |
| 2exp11 17296 | Two to the eleventh power ... |
| 2exp16 17297 | Two to the sixteenth power... |
| 3exp3 17298 | Three to the third power i... |
| 2expltfac 17299 | The factorial grows faster... |
| cshwsidrepsw 17300 | If cyclically shifting a w... |
| cshwsidrepswmod0 17301 | If cyclically shifting a w... |
| cshwshashlem1 17302 | If cyclically shifting a w... |
| cshwshashlem2 17303 | If cyclically shifting a w... |
| cshwshashlem3 17304 | If cyclically shifting a w... |
| cshwsdisj 17305 | The singletons resulting b... |
| cshwsiun 17306 | The set of (different!) wo... |
| cshwsex 17307 | The class of (different!) ... |
| cshws0 17308 | The size of the set of (di... |
| cshwrepswhash1 17309 | The size of the set of (di... |
| cshwshashnsame 17310 | If a word (not consisting ... |
| cshwshash 17311 | If a word has a length bei... |
| prmlem0 17312 | Lemma for ~ prmlem1a and ~... |
| prmlem1a 17313 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1 17314 | A quick proof skeleton to ... |
| 5prm 17315 | 5 is a prime number. (Con... |
| 6nprm 17316 | 6 is not a prime number. ... |
| 7prm 17317 | 7 is a prime number. (Con... |
| 8nprm 17318 | 8 is not a prime number. ... |
| 9nprm 17319 | 9 is not a prime number. ... |
| 10nprm 17320 | 10 is not a prime number. ... |
| 10nprmOLD 17321 | Obsolete version of ~ 10np... |
| 11prm 17322 | 11 is a prime number. (Co... |
| 13prm 17323 | 13 is a prime number. (Co... |
| 17prm 17324 | 17 is a prime number. (Co... |
| 19prm 17325 | 19 is a prime number. (Co... |
| 23prm 17326 | 23 is a prime number. (Co... |
| prmlem2 17327 | Our last proving session g... |
| 37prm 17328 | 37 is a prime number. (Co... |
| 43prm 17329 | 43 is a prime number. (Co... |
| 83prm 17330 | 83 is a prime number. (Co... |
| 139prm 17331 | 139 is a prime number. (C... |
| 163prm 17332 | 163 is a prime number. (C... |
| 317prm 17333 | 317 is a prime number. (C... |
| 631prm 17334 | 631 is a prime number. (C... |
| prmo4 17335 | The primorial of 4. (Cont... |
| prmo5 17336 | The primorial of 5. (Cont... |
| prmo6 17337 | The primorial of 6. (Cont... |
| 1259lem1 17338 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17339 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17340 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17341 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17342 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17343 | 1259 is a prime number. (... |
| 2503lem1 17344 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17345 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17346 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17347 | 2503 is a prime number. (... |
| 4001lem1 17348 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17349 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17350 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17351 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17352 | 4001 is a prime number. (... |
| brstruct 17355 | The structure relation is ... |
| isstruct2 17356 | The property of being a st... |
| structex 17357 | A structure is a set. (Co... |
| structn0fun 17358 | A structure without the em... |
| isstruct 17359 | The property of being a st... |
| structcnvcnv 17360 | Two ways to express the re... |
| structfung 17361 | The converse of the conver... |
| structfun 17362 | Convert between two kinds ... |
| structfn 17363 | Convert between two kinds ... |
| strleun 17364 | Combine two structures int... |
| strle1 17365 | Make a structure from a si... |
| strle2 17366 | Make a structure from a pa... |
| strle3 17367 | Make a structure from a tr... |
| sbcie2s 17368 | A special version of class... |
| sbcie3s 17369 | A special version of class... |
| reldmsets 17372 | The structure override ope... |
| setsvalg 17373 | Value of the structure rep... |
| setsval 17374 | Value of the structure rep... |
| fvsetsid 17375 | The value of the structure... |
| fsets 17376 | The structure replacement ... |
| setsdm 17377 | The domain of a structure ... |
| setsfun 17378 | A structure with replaceme... |
| setsfun0 17379 | A structure with replaceme... |
| setsn0fun 17380 | The value of the structure... |
| setsstruct2 17381 | An extensible structure wi... |
| setsexstruct2 17382 | An extensible structure wi... |
| setsstruct 17383 | An extensible structure wi... |
| wunsets 17384 | Closure of structure repla... |
| setsres 17385 | The structure replacement ... |
| setsabs 17386 | Replacing the same compone... |
| setscom 17387 | Different components can b... |
| sloteq 17390 | Equality theorem for the `... |
| slotfn 17391 | A slot is a function on se... |
| strfvnd 17392 | Deduction version of ~ str... |
| strfvn 17393 | Value of a structure compo... |
| strfvss 17394 | A structure component extr... |
| wunstr 17395 | Closure of a structure ind... |
| str0 17396 | All components of the empt... |
| strfvi 17397 | Structure slot extractors ... |
| fveqprc 17398 | Lemma for showing the equa... |
| oveqprc 17399 | Lemma for showing the equa... |
| wunndx 17402 | Closure of the index extra... |
| ndxarg 17403 | Get the numeric argument f... |
| ndxid 17404 | A structure component extr... |
| strndxid 17405 | The value of a structure c... |
| setsidvald 17406 | Value of the structure rep... |
| strfvd 17407 | Deduction version of ~ str... |
| strfv2d 17408 | Deduction version of ~ str... |
| strfv2 17409 | A variation on ~ strfv to ... |
| strfv 17410 | Extract a structure compon... |
| strfv3 17411 | Variant on ~ strfv for lar... |
| strssd 17412 | Deduction version of ~ str... |
| strss 17413 | Propagate component extrac... |
| setsid 17414 | Value of the structure rep... |
| setsnid 17415 | Value of the structure rep... |
| baseval 17418 | Value of the base set extr... |
| baseid 17419 | Utility theorem: index-ind... |
| basfn 17420 | The base set extractor is ... |
| base0 17421 | The base set of the empty ... |
| elbasfv 17422 | Utility theorem: reverse c... |
| elbasov 17423 | Utility theorem: reverse c... |
| strov2rcl 17424 | Partial reverse closure fo... |
| basendx 17425 | Index value of the base se... |
| basendxnn 17426 | The index value of the bas... |
| basndxelwund 17427 | The index of the base set ... |
| basprssdmsets 17428 | The pair of the base index... |
| opelstrbas 17429 | The base set of a structur... |
| 1strstr 17430 | A constructed one-slot str... |
| 1strbas 17431 | The base set of a construc... |
| 1strwunbndx 17432 | A constructed one-slot str... |
| 1strwun 17433 | A constructed one-slot str... |
| 2strstr 17434 | A constructed two-slot str... |
| 2strbas 17435 | The base set of a construc... |
| 2strop 17436 | The other slot of a constr... |
| reldmress 17439 | The structure restriction ... |
| ressval 17440 | Value of structure restric... |
| ressid2 17441 | General behavior of trivia... |
| ressval2 17442 | Value of nontrivial struct... |
| ressbas 17443 | Base set of a structure re... |
| ressbasssg 17444 | The base set of a restrict... |
| ressbas2 17445 | Base set of a structure re... |
| ressbasss 17446 | The base set of a restrict... |
| ressbasssOLD 17447 | Obsolete version of ~ ress... |
| ressbasss2 17448 | The base set of a restrict... |
| resseqnbas 17449 | The components of an exten... |
| ress0 17450 | All restrictions of the em... |
| ressid 17451 | Behavior of trivial restri... |
| ressinbas 17452 | Restriction only cares abo... |
| ressval3d 17453 | Value of structure restric... |
| ressress 17454 | Restriction composition la... |
| ressabs 17455 | Restriction absorption law... |
| wunress 17456 | Closure of structure restr... |
| plusgndx 17483 | Index value of the ~ df-pl... |
| plusgid 17484 | Utility theorem: index-ind... |
| plusgndxnn 17485 | The index of the slot for ... |
| basendxltplusgndx 17486 | The index of the slot for ... |
| basendxnplusgndx 17487 | The slot for the base set ... |
| grpstr 17488 | A constructed group is a s... |
| grpbase 17489 | The base set of a construc... |
| grpplusg 17490 | The operation of a constru... |
| ressplusg 17491 | ` +g ` is unaffected by re... |
| grpbasex 17492 | The base of an explicitly ... |
| grpplusgx 17493 | The operation of an explic... |
| mulrndx 17494 | Index value of the ~ df-mu... |
| mulridx 17495 | Utility theorem: index-ind... |
| basendxnmulrndx 17496 | The slot for the base set ... |
| plusgndxnmulrndx 17497 | The slot for the group (ad... |
| rngstr 17498 | A constructed ring is a st... |
| rngbase 17499 | The base set of a construc... |
| rngplusg 17500 | The additive operation of ... |
| rngmulr 17501 | The multiplicative operati... |
| starvndx 17502 | Index value of the ~ df-st... |
| starvid 17503 | Utility theorem: index-ind... |
| starvndxnbasendx 17504 | The slot for the involutio... |
| starvndxnplusgndx 17505 | The slot for the involutio... |
| starvndxnmulrndx 17506 | The slot for the involutio... |
| ressmulr 17507 | ` .r ` is unaffected by re... |
| ressstarv 17508 | ` *r ` is unaffected by re... |
| srngstr 17509 | A constructed star ring is... |
| srngbase 17510 | The base set of a construc... |
| srngplusg 17511 | The addition operation of ... |
| srngmulr 17512 | The multiplication operati... |
| srnginvl 17513 | The involution function of... |
| scandx 17514 | Index value of the ~ df-sc... |
| scaid 17515 | Utility theorem: index-ind... |
| scandxnbasendx 17516 | The slot for the scalar is... |
| scandxnplusgndx 17517 | The slot for the scalar fi... |
| scandxnmulrndx 17518 | The slot for the scalar fi... |
| vscandx 17519 | Index value of the ~ df-vs... |
| vscaid 17520 | Utility theorem: index-ind... |
| vscandxnbasendx 17521 | The slot for the scalar pr... |
| vscandxnplusgndx 17522 | The slot for the scalar pr... |
| vscandxnmulrndx 17523 | The slot for the scalar pr... |
| vscandxnscandx 17524 | The slot for the scalar pr... |
| lmodstr 17525 | A constructed left module ... |
| lmodbase 17526 | The base set of a construc... |
| lmodplusg 17527 | The additive operation of ... |
| lmodsca 17528 | The set of scalars of a co... |
| lmodvsca 17529 | The scalar product operati... |
| ipndx 17530 | Index value of the ~ df-ip... |
| ipid 17531 | Utility theorem: index-ind... |
| ipndxnbasendx 17532 | The slot for the inner pro... |
| ipndxnplusgndx 17533 | The slot for the inner pro... |
| ipndxnmulrndx 17534 | The slot for the inner pro... |
| slotsdifipndx 17535 | The slot for the scalar is... |
| ipsstr 17536 | Lemma to shorten proofs of... |
| ipsbase 17537 | The base set of a construc... |
| ipsaddg 17538 | The additive operation of ... |
| ipsmulr 17539 | The multiplicative operati... |
| ipssca 17540 | The set of scalars of a co... |
| ipsvsca 17541 | The scalar product operati... |
| ipsip 17542 | The multiplicative operati... |
| resssca 17543 | ` Scalar ` is unaffected b... |
| ressvsca 17544 | ` .s ` is unaffected by re... |
| ressip 17545 | The inner product is unaff... |
| phlstr 17546 | A constructed pre-Hilbert ... |
| phlbase 17547 | The base set of a construc... |
| phlplusg 17548 | The additive operation of ... |
| phlsca 17549 | The ring of scalars of a c... |
| phlvsca 17550 | The scalar product operati... |
| phlip 17551 | The inner product (Hermiti... |
| tsetndx 17552 | Index value of the ~ df-ts... |
| tsetid 17553 | Utility theorem: index-ind... |
| tsetndxnn 17554 | The index of the slot for ... |
| basendxlttsetndx 17555 | The index of the slot for ... |
| tsetndxnbasendx 17556 | The slot for the topology ... |
| tsetndxnplusgndx 17557 | The slot for the topology ... |
| tsetndxnmulrndx 17558 | The slot for the topology ... |
| tsetndxnstarvndx 17559 | The slot for the topology ... |
| slotstnscsi 17560 | The slots ` Scalar ` , ` .... |
| topgrpstr 17561 | A constructed topological ... |
| topgrpbas 17562 | The base set of a construc... |
| topgrpplusg 17563 | The additive operation of ... |
| topgrptset 17564 | The topology of a construc... |
| resstset 17565 | ` TopSet ` is unaffected b... |
| plendx 17566 | Index value of the ~ df-pl... |
| pleid 17567 | Utility theorem: self-refe... |
| plendxnn 17568 | The index value of the ord... |
| basendxltplendx 17569 | The index value of the ` B... |
| plendxnbasendx 17570 | The slot for the order is ... |
| plendxnplusgndx 17571 | The slot for the "less tha... |
| plendxnmulrndx 17572 | The slot for the "less tha... |
| plendxnscandx 17573 | The slot for the "less tha... |
| plendxnvscandx 17574 | The slot for the "less tha... |
| slotsdifplendx 17575 | The index of the slot for ... |
| otpsstr 17576 | Functionality of a topolog... |
| otpsbas 17577 | The base set of a topologi... |
| otpstset 17578 | The open sets of a topolog... |
| otpsle 17579 | The order of a topological... |
| ressle 17580 | ` le ` is unaffected by re... |
| ocndx 17581 | Index value of the ~ df-oc... |
| ocid 17582 | Utility theorem: index-ind... |
| basendxnocndx 17583 | The slot for the orthocomp... |
| plendxnocndx 17584 | The slot for the orthocomp... |
| dsndx 17585 | Index value of the ~ df-ds... |
| dsid 17586 | Utility theorem: index-ind... |
| dsndxnn 17587 | The index of the slot for ... |
| basendxltdsndx 17588 | The index of the slot for ... |
| dsndxnbasendx 17589 | The slot for the distance ... |
| dsndxnplusgndx 17590 | The slot for the distance ... |
| dsndxnmulrndx 17591 | The slot for the distance ... |
| slotsdnscsi 17592 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17593 | The slot for the distance ... |
| slotsdifdsndx 17594 | The index of the slot for ... |
| unifndx 17595 | Index value of the ~ df-un... |
| unifid 17596 | Utility theorem: index-ind... |
| unifndxnn 17597 | The index of the slot for ... |
| basendxltunifndx 17598 | The index of the slot for ... |
| unifndxnbasendx 17599 | The slot for the uniform s... |
| unifndxntsetndx 17600 | The slot for the uniform s... |
| slotsdifunifndx 17601 | The index of the slot for ... |
| ressunif 17602 | ` UnifSet ` is unaffected ... |
| odrngstr 17603 | Functionality of an ordere... |
| odrngbas 17604 | The base set of an ordered... |
| odrngplusg 17605 | The addition operation of ... |
| odrngmulr 17606 | The multiplication operati... |
| odrngtset 17607 | The open sets of an ordere... |
| odrngle 17608 | The order of an ordered me... |
| odrngds 17609 | The metric of an ordered m... |
| ressds 17610 | ` dist ` is unaffected by ... |
| homndx 17611 | Index value of the ~ df-ho... |
| homid 17612 | Utility theorem: index-ind... |
| ccondx 17613 | Index value of the ~ df-cc... |
| ccoid 17614 | Utility theorem: index-ind... |
| slotsbhcdif 17615 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17616 | The index of the slot for ... |
| slotsdifocndx 17617 | The index of the slot for ... |
| resshom 17618 | ` Hom ` is unaffected by r... |
| ressco 17619 | ` comp ` is unaffected by ... |
| restfn 17624 | The subspace topology oper... |
| topnfn 17625 | The topology extractor fun... |
| restval 17626 | The subspace topology indu... |
| elrest 17627 | The predicate "is an open ... |
| elrestr 17628 | Sufficient condition for b... |
| 0rest 17629 | Value of the structure res... |
| restid2 17630 | The subspace topology over... |
| restsspw 17631 | The subspace topology is a... |
| firest 17632 | The finite intersections o... |
| restid 17633 | The subspace topology of t... |
| topnval 17634 | Value of the topology extr... |
| topnid 17635 | Value of the topology extr... |
| topnpropd 17636 | The topology extractor fun... |
| reldmprds 17648 | The structure product is a... |
| prdsbasex 17650 | Lemma for structure produc... |
| imasvalstr 17651 | An image structure value i... |
| prdsvalstr 17652 | Structure product value is... |
| prdsbaslem 17653 | Lemma for ~ prdsbas and si... |
| prdsvallem 17654 | Lemma for ~ prdsval . (Co... |
| prdsval 17655 | Value of the structure pro... |
| prdssca 17656 | Scalar ring of a structure... |
| prdsbas 17657 | Base set of a structure pr... |
| prdsplusg 17658 | Addition in a structure pr... |
| prdsmulr 17659 | Multiplication in a struct... |
| prdsvsca 17660 | Scalar multiplication in a... |
| prdsip 17661 | Inner product in a structu... |
| prdsle 17662 | Structure product weak ord... |
| prdsless 17663 | Closure of the order relat... |
| prdsds 17664 | Structure product distance... |
| prdsdsfn 17665 | Structure product distance... |
| prdstset 17666 | Structure product topology... |
| prdshom 17667 | Structure product hom-sets... |
| prdsco 17668 | Structure product composit... |
| prdsbas2 17669 | The base set of a structur... |
| prdsbasmpt 17670 | A constructed tuple is a p... |
| prdsbasfn 17671 | Points in the structure pr... |
| prdsbasprj 17672 | Each point in a structure ... |
| prdsplusgval 17673 | Value of a componentwise s... |
| prdsplusgfval 17674 | Value of a structure produ... |
| prdsmulrval 17675 | Value of a componentwise r... |
| prdsmulrfval 17676 | Value of a structure produ... |
| prdsleval 17677 | Value of the product order... |
| prdsdsval 17678 | Value of the metric in a s... |
| prdsvscaval 17679 | Scalar multiplication in a... |
| prdsvscafval 17680 | Scalar multiplication of a... |
| prdsbas3 17681 | The base set of an indexed... |
| prdsbasmpt2 17682 | A constructed tuple is a p... |
| prdsbascl 17683 | An element of the base has... |
| prdsdsval2 17684 | Value of the metric in a s... |
| prdsdsval3 17685 | Value of the metric in a s... |
| pwsval 17686 | Value of a structure power... |
| pwsbas 17687 | Base set of a structure po... |
| pwselbasb 17688 | Membership in the base set... |
| pwselbas 17689 | An element of a structure ... |
| pwselbasr 17690 | The reverse direction of ~... |
| pwsplusgval 17691 | Value of addition in a str... |
| pwsmulrval 17692 | Value of multiplication in... |
| pwsle 17693 | Ordering in a structure po... |
| pwsleval 17694 | Ordering in a structure po... |
| pwsvscafval 17695 | Scalar multiplication in a... |
| pwsvscaval 17696 | Scalar multiplication of a... |
| pwssca 17697 | The ring of scalars of a s... |
| pwsdiagel 17698 | Membership of diagonal ele... |
| pwssnf1o 17699 | Triviality of singleton po... |
| imasval 17712 | Value of an image structur... |
| imasbas 17713 | The base set of an image s... |
| imasds 17714 | The distance function of a... |
| imasdsfn 17715 | The distance function is a... |
| imasdsval 17716 | The distance function of a... |
| imasdsval2 17717 | The distance function of a... |
| imasplusg 17718 | The group operation in an ... |
| imasmulr 17719 | The ring multiplication in... |
| imassca 17720 | The scalar field of an ima... |
| imasvsca 17721 | The scalar multiplication ... |
| imasip 17722 | The inner product of an im... |
| imastset 17723 | The topology of an image s... |
| imasle 17724 | The ordering of an image s... |
| f1ocpbllem 17725 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17726 | An injection is compatible... |
| f1ovscpbl 17727 | An injection is compatible... |
| f1olecpbl 17728 | An injection is compatible... |
| imasaddfnlem 17729 | The image structure operat... |
| imasaddvallem 17730 | The operation of an image ... |
| imasaddflem 17731 | The image set operations a... |
| imasaddfn 17732 | The image structure's grou... |
| imasaddval 17733 | The value of an image stru... |
| imasaddf 17734 | The image structure's grou... |
| imasmulfn 17735 | The image structure's ring... |
| imasmulval 17736 | The value of an image stru... |
| imasmulf 17737 | The image structure's ring... |
| imasvscafn 17738 | The image structure's scal... |
| imasvscaval 17739 | The value of an image stru... |
| imasvscaf 17740 | The image structure's scal... |
| imasless 17741 | The order relation defined... |
| imasleval 17742 | The value of the image str... |
| qusval 17743 | Value of a quotient struct... |
| quslem 17744 | The function in ~ qusval i... |
| qusin 17745 | Restrict the equivalence r... |
| qusbas 17746 | Base set of a quotient str... |
| quss 17747 | The scalar field of a quot... |
| divsfval 17748 | Value of the function in ~... |
| ercpbllem 17749 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17750 | Translate the function com... |
| erlecpbl 17751 | Translate the relation com... |
| qusaddvallem 17752 | Value of an operation defi... |
| qusaddflem 17753 | The operation of a quotien... |
| qusaddval 17754 | The addition in a quotient... |
| qusaddf 17755 | The addition in a quotient... |
| qusmulval 17756 | The multiplication in a qu... |
| qusmulf 17757 | The multiplication in a qu... |
| fnpr2o 17758 | Function with a domain of ... |
| fnpr2ob 17759 | Biconditional version of ~... |
| fvpr0o 17760 | The value of a function wi... |
| fvpr1o 17761 | The value of a function wi... |
| fvprif 17762 | The value of the pair func... |
| xpsfrnel 17763 | Elementhood in the target ... |
| xpsfeq 17764 | A function on ` 2o ` is de... |
| xpsfrnel2 17765 | Elementhood in the target ... |
| xpscf 17766 | Equivalent condition for t... |
| xpsfval 17767 | The value of the function ... |
| xpsff1o 17768 | The function appearing in ... |
| xpsfrn 17769 | A short expression for the... |
| xpsff1o2 17770 | The function appearing in ... |
| xpsval 17771 | Value of the binary struct... |
| xpsrnbas 17772 | The indexed structure prod... |
| xpsbas 17773 | The base set of the binary... |
| xpsaddlem 17774 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17775 | Value of the addition oper... |
| xpsmul 17776 | Value of the multiplicatio... |
| xpssca 17777 | Value of the scalar field ... |
| xpsvsca 17778 | Value of the scalar multip... |
| xpsless 17779 | Closure of the ordering in... |
| xpsle 17780 | Value of the ordering in a... |
| ismre 17789 | Property of being a Moore ... |
| fnmre 17790 | The Moore collection gener... |
| mresspw 17791 | A Moore collection is a su... |
| mress 17792 | A Moore-closed subset is a... |
| mre1cl 17793 | In any Moore collection th... |
| mreintcl 17794 | A nonempty collection of c... |
| mreiincl 17795 | A nonempty indexed interse... |
| mrerintcl 17796 | The relative intersection ... |
| mreriincl 17797 | The relative intersection ... |
| mreincl 17798 | Two closed sets have a clo... |
| mreuni 17799 | Since the entire base set ... |
| mreunirn 17800 | Two ways to express the no... |
| ismred 17801 | Properties that determine ... |
| ismred2 17802 | Properties that determine ... |
| mremre 17803 | The Moore collections of s... |
| submre 17804 | The subcollection of a clo... |
| xrsle 17805 | The ordering of the extend... |
| xrge0le 17806 | The "less than or equal to... |
| xrsbas 17807 | The base set of the extend... |
| xrge0base 17808 | The base of the extended n... |
| mrcflem 17809 | The domain and codomain of... |
| fnmrc 17810 | Moore-closure is a well-be... |
| mrcfval 17811 | Value of the function expr... |
| mrcf 17812 | The Moore closure is a fun... |
| mrcval 17813 | Evaluation of the Moore cl... |
| mrccl 17814 | The Moore closure of a set... |
| mrcsncl 17815 | The Moore closure of a sin... |
| mrcid 17816 | The closure of a closed se... |
| mrcssv 17817 | The closure of a set is a ... |
| mrcidb 17818 | A set is closed iff it is ... |
| mrcss 17819 | Closure preserves subset o... |
| mrcssid 17820 | The closure of a set is a ... |
| mrcidb2 17821 | A set is closed iff it con... |
| mrcidm 17822 | The closure operation is i... |
| mrcsscl 17823 | The closure is the minimal... |
| mrcuni 17824 | Idempotence of closure und... |
| mrcun 17825 | Idempotence of closure und... |
| mrcssvd 17826 | The Moore closure of a set... |
| mrcssd 17827 | Moore closure preserves su... |
| mrcssidd 17828 | A set is contained in its ... |
| mrcidmd 17829 | Moore closure is idempoten... |
| mressmrcd 17830 | In a Moore system, if a se... |
| submrc 17831 | In a closure system which ... |
| mrieqvlemd 17832 | In a Moore system, if ` Y ... |
| mrisval 17833 | Value of the set of indepe... |
| ismri 17834 | Criterion for a set to be ... |
| ismri2 17835 | Criterion for a subset of ... |
| ismri2d 17836 | Criterion for a subset of ... |
| ismri2dd 17837 | Definition of independence... |
| mriss 17838 | An independent set of a Mo... |
| mrissd 17839 | An independent set of a Mo... |
| ismri2dad 17840 | Consequence of a set in a ... |
| mrieqvd 17841 | In a Moore system, a set i... |
| mrieqv2d 17842 | In a Moore system, a set i... |
| mrissmrcd 17843 | In a Moore system, if an i... |
| mrissmrid 17844 | In a Moore system, subsets... |
| mreexd 17845 | In a Moore system, the clo... |
| mreexmrid 17846 | In a Moore system whose cl... |
| mreexexlemd 17847 | This lemma is used to gene... |
| mreexexlem2d 17848 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17849 | Base case of the induction... |
| mreexexlem4d 17850 | Induction step of the indu... |
| mreexexd 17851 | Exchange-type theorem. In... |
| mreexdomd 17852 | In a Moore system whose cl... |
| mreexfidimd 17853 | In a Moore system whose cl... |
| isacs 17854 | A set is an algebraic clos... |
| acsmre 17855 | Algebraic closure systems ... |
| isacs2 17856 | In the definition of an al... |
| acsfiel 17857 | A set is closed in an alge... |
| acsfiel2 17858 | A set is closed in an alge... |
| acsmred 17859 | An algebraic closure syste... |
| isacs1i 17860 | A closure system determine... |
| mreacs 17861 | Algebraicity is a composab... |
| acsfn 17862 | Algebraicity of a conditio... |
| acsfn0 17863 | Algebraicity of a point cl... |
| acsfn1 17864 | Algebraicity of a one-argu... |
| acsfn1c 17865 | Algebraicity of a one-argu... |
| acsfn2 17866 | Algebraicity of a two-argu... |
| iscat 17875 | The predicate "is a catego... |
| iscatd 17876 | Properties that determine ... |
| catidex 17877 | Each object in a category ... |
| catideu 17878 | Each object in a category ... |
| cidfval 17879 | Each object in a category ... |
| cidval 17880 | Each object in a category ... |
| cidffn 17881 | The identity arrow constru... |
| cidfn 17882 | The identity arrow operato... |
| catidd 17883 | Deduce the identity arrow ... |
| iscatd2 17884 | Version of ~ iscatd with a... |
| catidcl 17885 | Each object in a category ... |
| catlid 17886 | Left identity property of ... |
| catrid 17887 | Right identity property of... |
| catcocl 17888 | Closure of a composition a... |
| catass 17889 | Associativity of compositi... |
| catcone0 17890 | Composition of non-empty h... |
| 0catg 17891 | Any structure with an empt... |
| 0cat 17892 | The empty set is a categor... |
| homffval 17893 | Value of the functionalize... |
| fnhomeqhomf 17894 | If the Hom-set operation i... |
| homfval 17895 | Value of the functionalize... |
| homffn 17896 | The functionalized Hom-set... |
| homfeq 17897 | Condition for two categori... |
| homfeqd 17898 | If two structures have the... |
| homfeqbas 17899 | Deduce equality of base se... |
| homfeqval 17900 | Value of the functionalize... |
| comfffval 17901 | Value of the functionalize... |
| comffval 17902 | Value of the functionalize... |
| comfval 17903 | Value of the functionalize... |
| comfffval2 17904 | Value of the functionalize... |
| comffval2 17905 | Value of the functionalize... |
| comfval2 17906 | Value of the functionalize... |
| comfffn 17907 | The functionalized composi... |
| comffn 17908 | The functionalized composi... |
| comfeq 17909 | Condition for two categori... |
| comfeqd 17910 | Condition for two categori... |
| comfeqval 17911 | Equality of two compositio... |
| catpropd 17912 | Two structures with the sa... |
| cidpropd 17913 | Two structures with the sa... |
| oppcval 17916 | Value of the opposite cate... |
| oppchomfval 17917 | Hom-sets of the opposite c... |
| oppchom 17918 | Hom-sets of the opposite c... |
| oppccofval 17919 | Composition in the opposit... |
| oppcco 17920 | Composition in the opposit... |
| oppcbas 17921 | Base set of an opposite ca... |
| oppccatid 17922 | Lemma for ~ oppccat . (Co... |
| oppchomf 17923 | Hom-sets of the opposite c... |
| oppcid 17924 | Identity function of an op... |
| oppccat 17925 | An opposite category is a ... |
| 2oppcbas 17926 | The double opposite catego... |
| 2oppchomf 17927 | The double opposite catego... |
| 2oppccomf 17928 | The double opposite catego... |
| oppchomfpropd 17929 | If two categories have the... |
| oppccomfpropd 17930 | If two categories have the... |
| oppccatf 17931 | ` oppCat ` restricted to `... |
| monfval 17936 | Definition of a monomorphi... |
| ismon 17937 | Definition of a monomorphi... |
| ismon2 17938 | Write out the monomorphism... |
| monhom 17939 | A monomorphism is a morphi... |
| moni 17940 | Property of a monomorphism... |
| monpropd 17941 | If two categories have the... |
| oppcmon 17942 | A monomorphism in the oppo... |
| oppcepi 17943 | An epimorphism in the oppo... |
| isepi 17944 | Definition of an epimorphi... |
| isepi2 17945 | Write out the epimorphism ... |
| epihom 17946 | An epimorphism is a morphi... |
| epii 17947 | Property of an epimorphism... |
| sectffval 17954 | Value of the section opera... |
| sectfval 17955 | Value of the section relat... |
| sectss 17956 | The section relation is a ... |
| issect 17957 | The property " ` F ` is a ... |
| issect2 17958 | Property of being a sectio... |
| sectcan 17959 | If ` G ` is a section of `... |
| sectco 17960 | Composition of two section... |
| isofval 17961 | Function value of the func... |
| invffval 17962 | Value of the inverse relat... |
| invfval 17963 | Value of the inverse relat... |
| isinv 17964 | Value of the inverse relat... |
| invss 17965 | The inverse relation is a ... |
| invsym 17966 | The inverse relation is sy... |
| invsym2 17967 | The inverse relation is sy... |
| invfun 17968 | The inverse relation is a ... |
| isoval 17969 | The isomorphisms are the d... |
| inviso1 17970 | If ` G ` is an inverse to ... |
| inviso2 17971 | If ` G ` is an inverse to ... |
| invf 17972 | The inverse relation is a ... |
| invf1o 17973 | The inverse relation is a ... |
| invinv 17974 | The inverse of the inverse... |
| invco 17975 | The composition of two iso... |
| dfiso2 17976 | Alternate definition of an... |
| dfiso3 17977 | Alternate definition of an... |
| inveq 17978 | If there are two inverses ... |
| isofn 17979 | The function value of the ... |
| isohom 17980 | An isomorphism is a homomo... |
| isoco 17981 | The composition of two iso... |
| oppcsect 17982 | A section in the opposite ... |
| oppcsect2 17983 | A section in the opposite ... |
| oppcinv 17984 | An inverse in the opposite... |
| oppciso 17985 | An isomorphism in the oppo... |
| sectmon 17986 | If ` F ` is a section of `... |
| monsect 17987 | If ` F ` is a monomorphism... |
| sectepi 17988 | If ` F ` is a section of `... |
| episect 17989 | If ` F ` is an epimorphism... |
| sectid 17990 | The identity is a section ... |
| invid 17991 | The inverse of the identit... |
| idiso 17992 | The identity is an isomorp... |
| idinv 17993 | The inverse of the identit... |
| invisoinvl 17994 | The inverse of an isomorph... |
| invisoinvr 17995 | The inverse of an isomorph... |
| invcoisoid 17996 | The inverse of an isomorph... |
| isocoinvid 17997 | The inverse of an isomorph... |
| rcaninv 17998 | Right cancellation of an i... |
| cicfval 18001 | The set of isomorphic obje... |
| brcic 18002 | The relation "is isomorphi... |
| cic 18003 | Objects ` X ` and ` Y ` in... |
| brcici 18004 | Prove that two objects are... |
| cicref 18005 | Isomorphism is reflexive. ... |
| ciclcl 18006 | Isomorphism implies the le... |
| cicrcl 18007 | Isomorphism implies the ri... |
| cicsym 18008 | Isomorphism is symmetric. ... |
| cictr 18009 | Isomorphism is transitive.... |
| cicer 18010 | Isomorphism is an equivale... |
| sscrel 18017 | The subcategory subset rel... |
| brssc 18018 | The subcategory subset rel... |
| sscpwex 18019 | An analogue of ~ pwex for ... |
| subcrcl 18020 | Reverse closure for the su... |
| sscfn1 18021 | The subcategory subset rel... |
| sscfn2 18022 | The subcategory subset rel... |
| ssclem 18023 | Lemma for ~ ssc1 and simil... |
| isssc 18024 | Value of the subcategory s... |
| ssc1 18025 | Infer subset relation on o... |
| ssc2 18026 | Infer subset relation on m... |
| sscres 18027 | Any function restricted to... |
| sscid 18028 | The subcategory subset rel... |
| ssctr 18029 | The subcategory subset rel... |
| ssceq 18030 | The subcategory subset rel... |
| rescval 18031 | Value of the category rest... |
| rescval2 18032 | Value of the category rest... |
| rescbas 18033 | Base set of the category r... |
| reschom 18034 | Hom-sets of the category r... |
| reschomf 18035 | Hom-sets of the category r... |
| rescco 18036 | Composition in the categor... |
| rescabs 18037 | Restriction absorption law... |
| rescabs2 18038 | Restriction absorption law... |
| issubc 18039 | Elementhood in the set of ... |
| issubc2 18040 | Elementhood in the set of ... |
| 0ssc 18041 | For any category ` C ` , t... |
| 0subcat 18042 | For any category ` C ` , t... |
| catsubcat 18043 | For any category ` C ` , `... |
| subcssc 18044 | An element in the set of s... |
| subcfn 18045 | An element in the set of s... |
| subcss1 18046 | The objects of a subcatego... |
| subcss2 18047 | The morphisms of a subcate... |
| subcidcl 18048 | The identity of the origin... |
| subccocl 18049 | A subcategory is closed un... |
| subccatid 18050 | A subcategory is a categor... |
| subcid 18051 | The identity in a subcateg... |
| subccat 18052 | A subcategory is a categor... |
| issubc3 18053 | Alternate definition of a ... |
| fullsubc 18054 | The full subcategory gener... |
| fullresc 18055 | The category formed by str... |
| resscat 18056 | A category restricted to a... |
| subsubc 18057 | A subcategory of a subcate... |
| relfunc 18066 | The set of functors is a r... |
| funcrcl 18067 | Reverse closure for a func... |
| isfunc 18068 | Value of the set of functo... |
| isfuncd 18069 | Deduce that an operation i... |
| funcf1 18070 | The object part of a funct... |
| funcixp 18071 | The morphism part of a fun... |
| funcf2 18072 | The morphism part of a fun... |
| funcfn2 18073 | The morphism part of a fun... |
| funcid 18074 | A functor maps each identi... |
| funcco 18075 | A functor maps composition... |
| funcsect 18076 | The image of a section und... |
| funcinv 18077 | The image of an inverse un... |
| funciso 18078 | The image of an isomorphis... |
| funcoppc 18079 | A functor on categories yi... |
| idfuval 18080 | Value of the identity func... |
| idfu2nd 18081 | Value of the morphism part... |
| idfu2 18082 | Value of the morphism part... |
| idfu1st 18083 | Value of the object part o... |
| idfu1 18084 | Value of the object part o... |
| idfucl 18085 | The identity functor is a ... |
| cofuval 18086 | Value of the composition o... |
| cofu1st 18087 | Value of the object part o... |
| cofu1 18088 | Value of the object part o... |
| cofu2nd 18089 | Value of the morphism part... |
| cofu2 18090 | Value of the morphism part... |
| cofuval2 18091 | Value of the composition o... |
| cofucl 18092 | The composition of two fun... |
| cofuass 18093 | Functor composition is ass... |
| cofulid 18094 | The identity functor is a ... |
| cofurid 18095 | The identity functor is a ... |
| resfval 18096 | Value of the functor restr... |
| resfval2 18097 | Value of the functor restr... |
| resf1st 18098 | Value of the functor restr... |
| resf2nd 18099 | Value of the functor restr... |
| funcres 18100 | A functor restricted to a ... |
| funcres2b 18101 | Condition for a functor to... |
| funcres2 18102 | A functor into a restricte... |
| idfusubc0 18103 | The identity functor for a... |
| idfusubc 18104 | The identity functor for a... |
| wunfunc 18105 | A weak universe is closed ... |
| funcpropd 18106 | If two categories have the... |
| funcres2c 18107 | Condition for a functor to... |
| fullfunc 18112 | A full functor is a functo... |
| fthfunc 18113 | A faithful functor is a fu... |
| relfull 18114 | The set of full functors i... |
| relfth 18115 | The set of faithful functo... |
| isfull 18116 | Value of the set of full f... |
| isfull2 18117 | Equivalent condition for a... |
| fullfo 18118 | The morphism map of a full... |
| fulli 18119 | The morphism map of a full... |
| isfth 18120 | Value of the set of faithf... |
| isfth2 18121 | Equivalent condition for a... |
| isffth2 18122 | A fully faithful functor i... |
| fthf1 18123 | The morphism map of a fait... |
| fthi 18124 | The morphism map of a fait... |
| ffthf1o 18125 | The morphism map of a full... |
| fullpropd 18126 | If two categories have the... |
| fthpropd 18127 | If two categories have the... |
| fulloppc 18128 | The opposite functor of a ... |
| fthoppc 18129 | The opposite functor of a ... |
| ffthoppc 18130 | The opposite functor of a ... |
| fthsect 18131 | A faithful functor reflect... |
| fthinv 18132 | A faithful functor reflect... |
| fthmon 18133 | A faithful functor reflect... |
| fthepi 18134 | A faithful functor reflect... |
| ffthiso 18135 | A fully faithful functor r... |
| fthres2b 18136 | Condition for a faithful f... |
| fthres2c 18137 | Condition for a faithful f... |
| fthres2 18138 | A faithful functor into a ... |
| idffth 18139 | The identity functor is a ... |
| cofull 18140 | The composition of two ful... |
| cofth 18141 | The composition of two fai... |
| coffth 18142 | The composition of two ful... |
| rescfth 18143 | The inclusion functor from... |
| ressffth 18144 | The inclusion functor from... |
| fullres2c 18145 | Condition for a full funct... |
| ffthres2c 18146 | Condition for a fully fait... |
| inclfusubc 18147 | The "inclusion functor" fr... |
| fnfuc 18152 | The ` FuncCat ` operation ... |
| natfval 18153 | Value of the function givi... |
| isnat 18154 | Property of being a natura... |
| isnat2 18155 | Property of being a natura... |
| natffn 18156 | The natural transformation... |
| natrcl 18157 | Reverse closure for a natu... |
| nat1st2nd 18158 | Rewrite the natural transf... |
| natixp 18159 | A natural transformation i... |
| natcl 18160 | A component of a natural t... |
| natfn 18161 | A natural transformation i... |
| nati 18162 | Naturality property of a n... |
| wunnat 18163 | A weak universe is closed ... |
| catstr 18164 | A category structure is a ... |
| fucval 18165 | Value of the functor categ... |
| fuccofval 18166 | Value of the functor categ... |
| fucbas 18167 | The objects of the functor... |
| fuchom 18168 | The morphisms in the funct... |
| fucco 18169 | Value of the composition o... |
| fuccoval 18170 | Value of the functor categ... |
| fuccocl 18171 | The composition of two nat... |
| fucidcl 18172 | The identity natural trans... |
| fuclid 18173 | Left identity of natural t... |
| fucrid 18174 | Right identity of natural ... |
| fucass 18175 | Associativity of natural t... |
| fuccatid 18176 | The functor category is a ... |
| fuccat 18177 | The functor category is a ... |
| fucid 18178 | The identity morphism in t... |
| fucsect 18179 | Two natural transformation... |
| fucinv 18180 | Two natural transformation... |
| invfuc 18181 | If ` V ( x ) ` is an inver... |
| fuciso 18182 | A natural transformation i... |
| natpropd 18183 | If two categories have the... |
| fucpropd 18184 | If two categories have the... |
| initofn 18191 | ` InitO ` is a function on... |
| termofn 18192 | ` TermO ` is a function on... |
| zeroofn 18193 | ` ZeroO ` is a function on... |
| initorcl 18194 | Reverse closure for an ini... |
| termorcl 18195 | Reverse closure for a term... |
| zeroorcl 18196 | Reverse closure for a zero... |
| initoval 18197 | The value of the initial o... |
| termoval 18198 | The value of the terminal ... |
| zerooval 18199 | The value of the zero obje... |
| isinito 18200 | The predicate "is an initi... |
| istermo 18201 | The predicate "is a termin... |
| iszeroo 18202 | The predicate "is a zero o... |
| isinitoi 18203 | Implication of a class bei... |
| istermoi 18204 | Implication of a class bei... |
| initoid 18205 | For an initial object, the... |
| termoid 18206 | For a terminal object, the... |
| dfinito2 18207 | An initial object is a ter... |
| dftermo2 18208 | A terminal object is an in... |
| dfinito3 18209 | An alternate definition of... |
| dftermo3 18210 | An alternate definition of... |
| initoo 18211 | An initial object is an ob... |
| termoo 18212 | A terminal object is an ob... |
| iszeroi 18213 | Implication of a class bei... |
| 2initoinv 18214 | Morphisms between two init... |
| initoeu1 18215 | Initial objects are essent... |
| initoeu1w 18216 | Initial objects are essent... |
| initoeu2lem0 18217 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 18218 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 18219 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 18220 | Initial objects are essent... |
| 2termoinv 18221 | Morphisms between two term... |
| termoeu1 18222 | Terminal objects are essen... |
| termoeu1w 18223 | Terminal objects are essen... |
| homarcl 18232 | Reverse closure for an arr... |
| homafval 18233 | Value of the disjointified... |
| homaf 18234 | Functionality of the disjo... |
| homaval 18235 | Value of the disjointified... |
| elhoma 18236 | Value of the disjointified... |
| elhomai 18237 | Produce an arrow from a mo... |
| elhomai2 18238 | Produce an arrow from a mo... |
| homarcl2 18239 | Reverse closure for the do... |
| homarel 18240 | An arrow is an ordered pai... |
| homa1 18241 | The first component of an ... |
| homahom2 18242 | The second component of an... |
| homahom 18243 | The second component of an... |
| homadm 18244 | The domain of an arrow wit... |
| homacd 18245 | The codomain of an arrow w... |
| homadmcd 18246 | Decompose an arrow into do... |
| arwval 18247 | The set of arrows is the u... |
| arwrcl 18248 | The first component of an ... |
| arwhoma 18249 | An arrow is contained in t... |
| homarw 18250 | A hom-set is a subset of t... |
| arwdm 18251 | The domain of an arrow is ... |
| arwcd 18252 | The codomain of an arrow i... |
| dmaf 18253 | The domain function is a f... |
| cdaf 18254 | The codomain function is a... |
| arwhom 18255 | The second component of an... |
| arwdmcd 18256 | Decompose an arrow into do... |
| idafval 18261 | Value of the identity arro... |
| idaval 18262 | Value of the identity arro... |
| ida2 18263 | Morphism part of the ident... |
| idahom 18264 | Domain and codomain of the... |
| idadm 18265 | Domain of the identity arr... |
| idacd 18266 | Codomain of the identity a... |
| idaf 18267 | The identity arrow functio... |
| coafval 18268 | The value of the compositi... |
| eldmcoa 18269 | A pair ` <. G , F >. ` is ... |
| dmcoass 18270 | The domain of composition ... |
| homdmcoa 18271 | If ` F : X --> Y ` and ` G... |
| coaval 18272 | Value of composition for c... |
| coa2 18273 | The morphism part of arrow... |
| coahom 18274 | The composition of two com... |
| coapm 18275 | Composition of arrows is a... |
| arwlid 18276 | Left identity of a categor... |
| arwrid 18277 | Right identity of a catego... |
| arwass 18278 | Associativity of compositi... |
| setcval 18281 | Value of the category of s... |
| setcbas 18282 | Set of objects of the cate... |
| setchomfval 18283 | Set of arrows of the categ... |
| setchom 18284 | Set of arrows of the categ... |
| elsetchom 18285 | A morphism of sets is a fu... |
| setccofval 18286 | Composition in the categor... |
| setcco 18287 | Composition in the categor... |
| setccatid 18288 | Lemma for ~ setccat . (Co... |
| setccat 18289 | The category of sets is a ... |
| setcid 18290 | The identity arrow in the ... |
| setcmon 18291 | A monomorphism of sets is ... |
| setcepi 18292 | An epimorphism of sets is ... |
| setcsect 18293 | A section in the category ... |
| setcinv 18294 | An inverse in the category... |
| setciso 18295 | An isomorphism in the cate... |
| resssetc 18296 | The restriction of the cat... |
| funcsetcres2 18297 | A functor into a smaller c... |
| setc2obas 18298 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18299 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18300 | The category of sets in a ... |
| cat1 18301 | The definition of category... |
| catcval 18304 | Value of the category of c... |
| catcbas 18305 | Set of objects of the cate... |
| catchomfval 18306 | Set of arrows of the categ... |
| catchom 18307 | Set of arrows of the categ... |
| catccofval 18308 | Composition in the categor... |
| catcco 18309 | Composition in the categor... |
| catccatid 18310 | Lemma for ~ catccat . (Co... |
| catcid 18311 | The identity arrow in the ... |
| catccat 18312 | The category of categories... |
| resscatc 18313 | The restriction of the cat... |
| catcisolem 18314 | Lemma for ~ catciso . (Co... |
| catciso 18315 | A functor is an isomorphis... |
| catcbascl 18316 | An element of the base set... |
| catcslotelcl 18317 | A slot entry of an element... |
| catcbaselcl 18318 | The base set of an element... |
| catchomcl 18319 | The Hom-set of an element ... |
| catcccocl 18320 | The composition operation ... |
| catcoppccl 18321 | The category of categories... |
| catcfuccl 18322 | The category of categories... |
| fncnvimaeqv 18323 | The inverse images of the ... |
| bascnvimaeqv 18324 | The inverse image of the u... |
| estrcval 18327 | Value of the category of e... |
| estrcbas 18328 | Set of objects of the cate... |
| estrchomfval 18329 | Set of morphisms ("arrows"... |
| estrchom 18330 | The morphisms between exte... |
| elestrchom 18331 | A morphism between extensi... |
| estrccofval 18332 | Composition in the categor... |
| estrcco 18333 | Composition in the categor... |
| estrcbasbas 18334 | An element of the base set... |
| estrccatid 18335 | Lemma for ~ estrccat . (C... |
| estrccat 18336 | The category of extensible... |
| estrcid 18337 | The identity arrow in the ... |
| estrchomfn 18338 | The Hom-set operation in t... |
| estrchomfeqhom 18339 | The functionalized Hom-set... |
| estrreslem1 18340 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18341 | Lemma 2 for ~ estrres . (... |
| estrres 18342 | Any restriction of a categ... |
| funcestrcsetclem1 18343 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18344 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18345 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18346 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18347 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18348 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18349 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18350 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18351 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18352 | The "natural forgetful fun... |
| fthestrcsetc 18353 | The "natural forgetful fun... |
| fullestrcsetc 18354 | The "natural forgetful fun... |
| equivestrcsetc 18355 | The "natural forgetful fun... |
| setc1strwun 18356 | A constructed one-slot str... |
| funcsetcestrclem1 18357 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18358 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18359 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18360 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18361 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18362 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18363 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18364 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18365 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18366 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18367 | The "embedding functor" fr... |
| fthsetcestrc 18368 | The "embedding functor" fr... |
| fullsetcestrc 18369 | The "embedding functor" fr... |
| embedsetcestrc 18370 | The "embedding functor" fr... |
| fnxpc 18379 | The binary product of cate... |
| xpcval 18380 | Value of the binary produc... |
| xpcbas 18381 | Set of objects of the bina... |
| xpchomfval 18382 | Set of morphisms of the bi... |
| xpchom 18383 | Set of morphisms of the bi... |
| relxpchom 18384 | A hom-set in the binary pr... |
| xpccofval 18385 | Value of composition in th... |
| xpcco 18386 | Value of composition in th... |
| xpcco1st 18387 | Value of composition in th... |
| xpcco2nd 18388 | Value of composition in th... |
| xpchom2 18389 | Value of the set of morphi... |
| xpcco2 18390 | Value of composition in th... |
| xpccatid 18391 | The product of two categor... |
| xpcid 18392 | The identity morphism in t... |
| xpccat 18393 | The product of two categor... |
| 1stfval 18394 | Value of the first project... |
| 1stf1 18395 | Value of the first project... |
| 1stf2 18396 | Value of the first project... |
| 2ndfval 18397 | Value of the first project... |
| 2ndf1 18398 | Value of the first project... |
| 2ndf2 18399 | Value of the first project... |
| 1stfcl 18400 | The first projection funct... |
| 2ndfcl 18401 | The second projection func... |
| prfval 18402 | Value of the pairing funct... |
| prf1 18403 | Value of the pairing funct... |
| prf2fval 18404 | Value of the pairing funct... |
| prf2 18405 | Value of the pairing funct... |
| prfcl 18406 | The pairing of functors ` ... |
| prf1st 18407 | Cancellation of pairing wi... |
| prf2nd 18408 | Cancellation of pairing wi... |
| 1st2ndprf 18409 | Break a functor into a pro... |
| catcxpccl 18410 | The category of categories... |
| xpcpropd 18411 | If two categories have the... |
| evlfval 18420 | Value of the evaluation fu... |
| evlf2 18421 | Value of the evaluation fu... |
| evlf2val 18422 | Value of the evaluation na... |
| evlf1 18423 | Value of the evaluation fu... |
| evlfcllem 18424 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18425 | The evaluation functor is ... |
| curfval 18426 | Value of the curry functor... |
| curf1fval 18427 | Value of the object part o... |
| curf1 18428 | Value of the object part o... |
| curf11 18429 | Value of the double evalua... |
| curf12 18430 | The partially evaluated cu... |
| curf1cl 18431 | The partially evaluated cu... |
| curf2 18432 | Value of the curry functor... |
| curf2val 18433 | Value of a component of th... |
| curf2cl 18434 | The curry functor at a mor... |
| curfcl 18435 | The curry functor of a fun... |
| curfpropd 18436 | If two categories have the... |
| uncfval 18437 | Value of the uncurry funct... |
| uncfcl 18438 | The uncurry operation take... |
| uncf1 18439 | Value of the uncurry funct... |
| uncf2 18440 | Value of the uncurry funct... |
| curfuncf 18441 | Cancellation of curry with... |
| uncfcurf 18442 | Cancellation of uncurry wi... |
| diagval 18443 | Define the diagonal functo... |
| diagcl 18444 | The diagonal functor is a ... |
| diag1cl 18445 | The constant functor of ` ... |
| diag11 18446 | Value of the constant func... |
| diag12 18447 | Value of the constant func... |
| diag2 18448 | Value of the diagonal func... |
| diag2cl 18449 | The diagonal functor at a ... |
| curf2ndf 18450 | As shown in ~ diagval , th... |
| hofval 18455 | Value of the Hom functor, ... |
| hof1fval 18456 | The object part of the Hom... |
| hof1 18457 | The object part of the Hom... |
| hof2fval 18458 | The morphism part of the H... |
| hof2val 18459 | The morphism part of the H... |
| hof2 18460 | The morphism part of the H... |
| hofcllem 18461 | Lemma for ~ hofcl . (Cont... |
| hofcl 18462 | Closure of the Hom functor... |
| oppchofcl 18463 | Closure of the opposite Ho... |
| yonval 18464 | Value of the Yoneda embedd... |
| yoncl 18465 | The Yoneda embedding is a ... |
| yon1cl 18466 | The Yoneda embedding at an... |
| yon11 18467 | Value of the Yoneda embedd... |
| yon12 18468 | Value of the Yoneda embedd... |
| yon2 18469 | Value of the Yoneda embedd... |
| hofpropd 18470 | If two categories have the... |
| yonpropd 18471 | If two categories have the... |
| oppcyon 18472 | Value of the opposite Yone... |
| oyoncl 18473 | The opposite Yoneda embedd... |
| oyon1cl 18474 | The opposite Yoneda embedd... |
| yonedalem1 18475 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18476 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18477 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18478 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18479 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18480 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18481 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18482 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18483 | Lemma for ~ yoneda . (Con... |
| yonedainv 18484 | The Yoneda Lemma with expl... |
| yonffthlem 18485 | Lemma for ~ yonffth . (Co... |
| yoneda 18486 | The Yoneda Lemma. There i... |
| yonffth 18487 | The Yoneda Lemma. The Yon... |
| yoniso 18488 | If the codomain is recover... |
| oduval 18491 | Value of an order dual str... |
| oduleval 18492 | Value of the less-equal re... |
| oduleg 18493 | Truth of the less-equal re... |
| odubas 18494 | Base set of an order dual ... |
| isprs 18499 | Property of being a preord... |
| prslem 18500 | Lemma for ~ prsref and ~ p... |
| prsref 18501 | "Less than or equal to" is... |
| prstr 18502 | "Less than or equal to" is... |
| oduprs 18503 | Being a proset is a self-d... |
| isdrs 18504 | Property of being a direct... |
| drsdir 18505 | Direction of a directed se... |
| drsprs 18506 | A directed set is a proset... |
| drsbn0 18507 | The base of a directed set... |
| drsdirfi 18508 | Any _finite_ number of ele... |
| isdrs2 18509 | Directed sets may be defin... |
| ispos 18517 | The predicate "is a poset"... |
| ispos2 18518 | A poset is an antisymmetri... |
| posprs 18519 | A poset is a proset. (Con... |
| posi 18520 | Lemma for poset properties... |
| posref 18521 | A poset ordering is reflex... |
| posasymb 18522 | A poset ordering is asymme... |
| postr 18523 | A poset ordering is transi... |
| 0pos 18524 | Technical lemma to simplif... |
| isposd 18525 | Properties that determine ... |
| isposi 18526 | Properties that determine ... |
| isposix 18527 | Properties that determine ... |
| pospropd 18528 | Posethood is determined on... |
| odupos 18529 | Being a poset is a self-du... |
| oduposb 18530 | Being a poset is a self-du... |
| pltfval 18532 | Value of the less-than rel... |
| pltval 18533 | Less-than relation. ( ~ d... |
| pltle 18534 | "Less than" implies "less ... |
| pltne 18535 | The "less than" relation i... |
| pltirr 18536 | The "less than" relation i... |
| pleval2i 18537 | One direction of ~ pleval2... |
| pleval2 18538 | "Less than or equal to" in... |
| pltnle 18539 | "Less than" implies not co... |
| pltval3 18540 | Alternate expression for t... |
| pltnlt 18541 | The less-than relation imp... |
| pltn2lp 18542 | The less-than relation has... |
| plttr 18543 | The less-than relation is ... |
| pltletr 18544 | Transitive law for chained... |
| plelttr 18545 | Transitive law for chained... |
| pospo 18546 | Write a poset structure in... |
| lubfval 18551 | Value of the least upper b... |
| lubdm 18552 | Domain of the least upper ... |
| lubfun 18553 | The LUB is a function. (C... |
| lubeldm 18554 | Member of the domain of th... |
| lubelss 18555 | A member of the domain of ... |
| lubeu 18556 | Unique existence proper of... |
| lubval 18557 | Value of the least upper b... |
| lubcl 18558 | The least upper bound func... |
| lubprop 18559 | Properties of greatest low... |
| luble 18560 | The greatest lower bound i... |
| lublecllem 18561 | Lemma for ~ lublecl and ~ ... |
| lublecl 18562 | The set of all elements le... |
| lubid 18563 | The LUB of elements less t... |
| glbfval 18564 | Value of the greatest lowe... |
| glbdm 18565 | Domain of the greatest low... |
| glbfun 18566 | The GLB is a function. (C... |
| glbeldm 18567 | Member of the domain of th... |
| glbelss 18568 | A member of the domain of ... |
| glbeu 18569 | Unique existence proper of... |
| glbval 18570 | Value of the greatest lowe... |
| glbcl 18571 | The least upper bound func... |
| glbprop 18572 | Properties of greatest low... |
| glble 18573 | The greatest lower bound i... |
| joinfval 18574 | Value of join function for... |
| joinfval2 18575 | Value of join function for... |
| joindm 18576 | Domain of join function fo... |
| joindef 18577 | Two ways to say that a joi... |
| joinval 18578 | Join value. Since both si... |
| joincl 18579 | Closure of join of element... |
| joindmss 18580 | Subset property of domain ... |
| joinval2lem 18581 | Lemma for ~ joinval2 and ~... |
| joinval2 18582 | Value of join for a poset ... |
| joineu 18583 | Uniqueness of join of elem... |
| joinlem 18584 | Lemma for join properties.... |
| lejoin1 18585 | A join's first argument is... |
| lejoin2 18586 | A join's second argument i... |
| joinle 18587 | A join is less than or equ... |
| meetfval 18588 | Value of meet function for... |
| meetfval2 18589 | Value of meet function for... |
| meetdm 18590 | Domain of meet function fo... |
| meetdef 18591 | Two ways to say that a mee... |
| meetval 18592 | Meet value. Since both si... |
| meetcl 18593 | Closure of meet of element... |
| meetdmss 18594 | Subset property of domain ... |
| meetval2lem 18595 | Lemma for ~ meetval2 and ~... |
| meetval2 18596 | Value of meet for a poset ... |
| meeteu 18597 | Uniqueness of meet of elem... |
| meetlem 18598 | Lemma for meet properties.... |
| lemeet1 18599 | A meet's first argument is... |
| lemeet2 18600 | A meet's second argument i... |
| meetle 18601 | A meet is less than or equ... |
| joincomALT 18602 | The join of a poset is com... |
| joincom 18603 | The join of a poset is com... |
| meetcomALT 18604 | The meet of a poset is com... |
| meetcom 18605 | The meet of a poset is com... |
| join0 18606 | Lemma for ~ odumeet . (Co... |
| meet0 18607 | Lemma for ~ odujoin . (Co... |
| odulub 18608 | Least upper bounds in a du... |
| odujoin 18609 | Joins in a dual order are ... |
| oduglb 18610 | Greatest lower bounds in a... |
| odumeet 18611 | Meets in a dual order are ... |
| poslubmo 18612 | Least upper bounds in a po... |
| posglbmo 18613 | Greatest lower bounds in a... |
| poslubd 18614 | Properties which determine... |
| poslubdg 18615 | Properties which determine... |
| posglbdg 18616 | Properties which determine... |
| istos 18619 | The predicate "is a toset"... |
| tosso 18620 | Write the totally ordered ... |
| tospos 18621 | A Toset is a Poset. (Cont... |
| tleile 18622 | In a Toset, any two elemen... |
| tltnle 18623 | In a Toset, "less than" is... |
| p0val 18628 | Value of poset zero. (Con... |
| p1val 18629 | Value of poset zero. (Con... |
| p0le 18630 | Any element is less than o... |
| ple1 18631 | Any element is less than o... |
| resspos 18632 | The restriction of a Poset... |
| resstos 18633 | The restriction of a Toset... |
| islat 18636 | The predicate "is a lattic... |
| odulatb 18637 | Being a lattice is self-du... |
| odulat 18638 | Being a lattice is self-du... |
| latcl2 18639 | The join and meet of any t... |
| latlem 18640 | Lemma for lattice properti... |
| latpos 18641 | A lattice is a poset. (Co... |
| latjcl 18642 | Closure of join operation ... |
| latmcl 18643 | Closure of meet operation ... |
| latref 18644 | A lattice ordering is refl... |
| latasymb 18645 | A lattice ordering is asym... |
| latasym 18646 | A lattice ordering is asym... |
| lattr 18647 | A lattice ordering is tran... |
| latasymd 18648 | Deduce equality from latti... |
| lattrd 18649 | A lattice ordering is tran... |
| latjcom 18650 | The join of a lattice comm... |
| latlej1 18651 | A join's first argument is... |
| latlej2 18652 | A join's second argument i... |
| latjle12 18653 | A join is less than or equ... |
| latleeqj1 18654 | "Less than or equal to" in... |
| latleeqj2 18655 | "Less than or equal to" in... |
| latjlej1 18656 | Add join to both sides of ... |
| latjlej2 18657 | Add join to both sides of ... |
| latjlej12 18658 | Add join to both sides of ... |
| latnlej 18659 | An idiom to express that a... |
| latnlej1l 18660 | An idiom to express that a... |
| latnlej1r 18661 | An idiom to express that a... |
| latnlej2 18662 | An idiom to express that a... |
| latnlej2l 18663 | An idiom to express that a... |
| latnlej2r 18664 | An idiom to express that a... |
| latjidm 18665 | Lattice join is idempotent... |
| latmcom 18666 | The join of a lattice comm... |
| latmle1 18667 | A meet is less than or equ... |
| latmle2 18668 | A meet is less than or equ... |
| latlem12 18669 | An element is less than or... |
| latleeqm1 18670 | "Less than or equal to" in... |
| latleeqm2 18671 | "Less than or equal to" in... |
| latmlem1 18672 | Add meet to both sides of ... |
| latmlem2 18673 | Add meet to both sides of ... |
| latmlem12 18674 | Add join to both sides of ... |
| latnlemlt 18675 | Negation of "less than or ... |
| latnle 18676 | Equivalent expressions for... |
| latmidm 18677 | Lattice meet is idempotent... |
| latabs1 18678 | Lattice absorption law. F... |
| latabs2 18679 | Lattice absorption law. F... |
| latledi 18680 | An ortholattice is distrib... |
| latmlej11 18681 | Ordering of a meet and joi... |
| latmlej12 18682 | Ordering of a meet and joi... |
| latmlej21 18683 | Ordering of a meet and joi... |
| latmlej22 18684 | Ordering of a meet and joi... |
| lubsn 18685 | The least upper bound of a... |
| latjass 18686 | Lattice join is associativ... |
| latj12 18687 | Swap 1st and 2nd members o... |
| latj32 18688 | Swap 2nd and 3rd members o... |
| latj13 18689 | Swap 1st and 3rd members o... |
| latj31 18690 | Swap 2nd and 3rd members o... |
| latjrot 18691 | Rotate lattice join of 3 c... |
| latj4 18692 | Rearrangement of lattice j... |
| latj4rot 18693 | Rotate lattice join of 4 c... |
| latjjdi 18694 | Lattice join distributes o... |
| latjjdir 18695 | Lattice join distributes o... |
| mod1ile 18696 | The weak direction of the ... |
| mod2ile 18697 | The weak direction of the ... |
| latmass 18698 | Lattice meet is associativ... |
| latdisdlem 18699 | Lemma for ~ latdisd . (Co... |
| latdisd 18700 | In a lattice, joins distri... |
| isclat 18703 | The predicate "is a comple... |
| clatpos 18704 | A complete lattice is a po... |
| clatlem 18705 | Lemma for properties of a ... |
| clatlubcl 18706 | Any subset of the base set... |
| clatlubcl2 18707 | Any subset of the base set... |
| clatglbcl 18708 | Any subset of the base set... |
| clatglbcl2 18709 | Any subset of the base set... |
| oduclatb 18710 | Being a complete lattice i... |
| clatl 18711 | A complete lattice is a la... |
| isglbd 18712 | Properties that determine ... |
| lublem 18713 | Lemma for the least upper ... |
| lubub 18714 | The LUB of a complete latt... |
| lubl 18715 | The LUB of a complete latt... |
| lubss 18716 | Subset law for least upper... |
| lubel 18717 | An element of a set is les... |
| lubun 18718 | The LUB of a union. (Cont... |
| clatglb 18719 | Properties of greatest low... |
| clatglble 18720 | The greatest lower bound i... |
| clatleglb 18721 | Two ways of expressing "le... |
| clatglbss 18722 | Subset law for greatest lo... |
| isdlat 18725 | Property of being a distri... |
| dlatmjdi 18726 | In a distributive lattice,... |
| dlatl 18727 | A distributive lattice is ... |
| odudlatb 18728 | The dual of a distributive... |
| dlatjmdi 18729 | In a distributive lattice,... |
| ipostr 18732 | The structure of ~ df-ipo ... |
| ipoval 18733 | Value of the inclusion pos... |
| ipobas 18734 | Base set of the inclusion ... |
| ipolerval 18735 | Relation of the inclusion ... |
| ipotset 18736 | Topology of the inclusion ... |
| ipole 18737 | Weak order condition of th... |
| ipolt 18738 | Strict order condition of ... |
| ipopos 18739 | The inclusion poset on a f... |
| isipodrs 18740 | Condition for a family of ... |
| ipodrscl 18741 | Direction by inclusion as ... |
| ipodrsfi 18742 | Finite upper bound propert... |
| fpwipodrs 18743 | The finite subsets of any ... |
| ipodrsima 18744 | The monotone image of a di... |
| isacs3lem 18745 | An algebraic closure syste... |
| acsdrsel 18746 | An algebraic closure syste... |
| isacs4lem 18747 | In a closure system in whi... |
| isacs5lem 18748 | If closure commutes with d... |
| acsdrscl 18749 | In an algebraic closure sy... |
| acsficl 18750 | A closure in an algebraic ... |
| isacs5 18751 | A closure system is algebr... |
| isacs4 18752 | A closure system is algebr... |
| isacs3 18753 | A closure system is algebr... |
| acsficld 18754 | In an algebraic closure sy... |
| acsficl2d 18755 | In an algebraic closure sy... |
| acsfiindd 18756 | In an algebraic closure sy... |
| acsmapd 18757 | In an algebraic closure sy... |
| acsmap2d 18758 | In an algebraic closure sy... |
| acsinfd 18759 | In an algebraic closure sy... |
| acsdomd 18760 | In an algebraic closure sy... |
| acsinfdimd 18761 | In an algebraic closure sy... |
| acsexdimd 18762 | In an algebraic closure sy... |
| mrelatglb 18763 | Greatest lower bounds in a... |
| mrelatglb0 18764 | The empty intersection in ... |
| mrelatlub 18765 | Least upper bounds in a Mo... |
| mreclatBAD 18766 | A Moore space is a complet... |
| isps 18771 | The predicate "is a poset"... |
| psrel 18772 | A poset is a relation. (C... |
| psref2 18773 | A poset is antisymmetric a... |
| pstr2 18774 | A poset is transitive. (C... |
| pslem 18775 | Lemma for ~ psref and othe... |
| psdmrn 18776 | The domain and range of a ... |
| psref 18777 | A poset is reflexive. (Co... |
| psrn 18778 | The range of a poset equal... |
| psasym 18779 | A poset is antisymmetric. ... |
| pstr 18780 | A poset is transitive. (C... |
| cnvps 18781 | The converse of a poset is... |
| cnvpsb 18782 | The converse of a poset is... |
| psss 18783 | Any subset of a partially ... |
| psssdm2 18784 | Field of a subposet. (Con... |
| psssdm 18785 | Field of a subposet. (Con... |
| istsr 18786 | The predicate is a toset. ... |
| istsr2 18787 | The predicate is a toset. ... |
| tsrlin 18788 | A toset is a linear order.... |
| tsrlemax 18789 | Two ways of saying a numbe... |
| tsrps 18790 | A toset is a poset. (Cont... |
| cnvtsr 18791 | The converse of a toset is... |
| tsrss 18792 | Any subset of a totally or... |
| ledm 18793 | The domain of ` <_ ` is ` ... |
| lern 18794 | The range of ` <_ ` is ` R... |
| lefld 18795 | The field of the 'less or ... |
| letsr 18796 | The "less than or equal to... |
| isdir 18801 | A condition for a relation... |
| reldir 18802 | A direction is a relation.... |
| dirdm 18803 | A direction's domain is eq... |
| dirref 18804 | A direction is reflexive. ... |
| dirtr 18805 | A direction is transitive.... |
| dirge 18806 | For any two elements of a ... |
| tsrdir 18807 | A totally ordered set is a... |
| ischn 18810 | Property of being a chain.... |
| chnwrd 18811 | A chain is an ordered sequ... |
| chnltm1 18812 | Basic property of a chain.... |
| pfxchn 18813 | A prefix of a chain is sti... |
| nfchnd 18814 | Bound-variable hypothesis ... |
| chneq1 18815 | Equality theorem for chain... |
| chneq2 18816 | Equality theorem for chain... |
| chneq12 18817 | Equality theorem for chain... |
| chnrss 18818 | Chains under a relation ar... |
| chndss 18819 | Chains with an alphabet ar... |
| chnrdss 18820 | Subset theorem for chains.... |
| chnexg 18821 | Chains with a set given fo... |
| nulchn 18822 | Empty set is an increasing... |
| s1chn 18823 | A singleton word is always... |
| chnind 18824 | Induction over a chain. S... |
| chnub 18825 | In a chain, the last eleme... |
| chnlt 18826 | Compare any two elements i... |
| chnso 18827 | A chain induces a total or... |
| chnccats1 18828 | Extend a chain with a sing... |
| chnccat 18829 | Concatenate two chains. (... |
| chnrev 18830 | Reverse of a chain is chai... |
| chnflenfi 18831 | There is a finite number o... |
| chnf 18832 | A chain is a zero-based fi... |
| chnpof1 18833 | A chain under relation whi... |
| chnpoadomd 18834 | A chain under relation whi... |
| chnpolleha 18835 | A chain under relation whi... |
| chnpolfz 18836 | Provided that chain's rela... |
| chnfi 18837 | There is a finite number o... |
| chninf 18838 | There is an infinite numbe... |
| chnfibg 18839 | Given a partial order, the... |
| ex-chn1 18840 | Example: a doubleton of tw... |
| ex-chn2 18841 | Example: sequence <" ZZ NN... |
| ismgm 18846 | The predicate "is a magma"... |
| ismgmn0 18847 | The predicate "is a magma"... |
| mgmcl 18848 | Closure of the operation o... |
| isnmgm 18849 | A condition for a structur... |
| mgmsscl 18850 | If the base set of a magma... |
| plusffval 18851 | The group addition operati... |
| plusfval 18852 | The group addition operati... |
| plusfeq 18853 | If the addition operation ... |
| plusffn 18854 | The group addition operati... |
| mgmplusf 18855 | The group addition functio... |
| mgmn0plusgf 18856 | The restriction of the gro... |
| mgmn0plusgplusf 18857 | The group addition functio... |
| mgmpropd 18858 | If two structures have the... |
| ismgmd 18859 | Deduce a magma from its pr... |
| issstrmgm 18860 | Characterize a substructur... |
| intopsn 18861 | The internal operation for... |
| mgmb1mgm1 18862 | The only magma with a base... |
| mgm0 18863 | Any set with an empty base... |
| mgm0b 18864 | The structure with an empt... |
| mgm1 18865 | The structure with one ele... |
| opifismgm 18866 | A structure with a group a... |
| mgmidmo 18867 | A two-sided identity eleme... |
| mgmideud 18868 | Uniqueness of the left and... |
| grpidval 18869 | The value of the group ide... |
| idvalriota 18870 | The unique value of the gr... |
| grpidpropd 18871 | If two structures have the... |
| fn0g 18872 | The group identity extract... |
| 0g0 18873 | The identity element funct... |
| ismgmid 18874 | Conditions for a class to ... |
| mgmidcl 18875 | The identity element of a ... |
| mgmlrid 18876 | The identity element of a ... |
| 0gisid 18877 | In a structure with an ide... |
| ismgmid2 18878 | Show that a given element ... |
| lidrideqd 18879 | If there is a left and rig... |
| lidrididd 18880 | If there is a left and rig... |
| grpidd 18881 | Deduce the identity elemen... |
| mgmidsssn0 18882 | Property of the set of ide... |
| grpinvalem 18883 | Lemma for ~ grpinva . (Co... |
| grpinva 18884 | Deduce right inverse from ... |
| grprida 18885 | Deduce right identity from... |
| mgmidpfod 18886 | The operation of a magma w... |
| mgmidprnd 18887 | Range of an operation with... |
| mgmfod 18888 | The operation of a magma w... |
| idressidex0 18889 | The restriction of a struc... |
| idressidex 18890 | The restriction of a struc... |
| idressid 18891 | The restriction of a struc... |
| imasmgm2 18892 | The image structure of a u... |
| qusmgm 18893 | Prove that a quotient stru... |
| gsumvalx 18894 | Expand out the substitutio... |
| gsumval 18895 | Expand out the substitutio... |
| gsumpropd 18896 | The group sum depends only... |
| gsumpropd2lem 18897 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18898 | A stronger version of ~ gs... |
| gsummgmpropd 18899 | A stronger version of ~ gs... |
| gsumress 18900 | The group sum in a substru... |
| gsumval1 18901 | Value of the group sum ope... |
| gsum0 18902 | Value of the empty group s... |
| gsumval2a 18903 | Value of the group sum ope... |
| gsumval2 18904 | Value of the group sum ope... |
| gsumsplit1r 18905 | Splitting off the rightmos... |
| gsumprval 18906 | Value of the group sum ope... |
| gsumpr12val 18907 | Value of the group sum ope... |
| mgmhmrcl 18912 | Reverse closure of a magma... |
| submgmrcl 18913 | Reverse closure for submag... |
| ismgmhm 18914 | Property of a magma homomo... |
| mgmhmf 18915 | A magma homomorphism is a ... |
| mgmhmpropd 18916 | Magma homomorphism depends... |
| mgmhmlin 18917 | A magma homomorphism prese... |
| mgmhmf1o 18918 | A magma homomorphism is bi... |
| idmgmhm 18919 | The identity homomorphism ... |
| issubmgm 18920 | Expand definition of a sub... |
| issubmgm2 18921 | Submagmas are subsets that... |
| rabsubmgmd 18922 | Deduction for proving that... |
| submgmss 18923 | Submagmas are subsets of t... |
| submgmid 18924 | Every magma is trivially a... |
| submgmcl 18925 | Submagmas are closed under... |
| submgmmgm 18926 | Submagmas are themselves m... |
| submgmbas 18927 | The base set of a submagma... |
| subsubmgm 18928 | A submagma of a submagma i... |
| resmgmhm 18929 | Restriction of a magma hom... |
| resmgmhm2 18930 | One direction of ~ resmgmh... |
| resmgmhm2b 18931 | Restriction of the codomai... |
| mgmhmco 18932 | The composition of magma h... |
| mgmhmima 18933 | The homomorphic image of a... |
| mgmhmeql 18934 | The equalizer of two magma... |
| submgmacs 18935 | Submagmas are an algebraic... |
| issgrp 18938 | The predicate "is a semigr... |
| issgrpv 18939 | The predicate "is a semigr... |
| issgrpn0 18940 | The predicate "is a semigr... |
| isnsgrp 18941 | A condition for a structur... |
| sgrpmgm 18942 | A semigroup is a magma. (... |
| sgrpass 18943 | A semigroup operation is a... |
| sgrpcl 18944 | Closure of the operation o... |
| sgrp0 18945 | Any set with an empty base... |
| sgrp0b 18946 | The structure with an empt... |
| sgrp1 18947 | The structure with one ele... |
| issgrpd 18948 | Deduce a semigroup from it... |
| sgrppropd 18949 | If two structures are sets... |
| prdsplusgsgrpcl 18950 | Structure product pointwis... |
| prdssgrpd 18951 | The product of a family of... |
| ismnddef 18954 | The predicate "is a monoid... |
| ismnd 18955 | The predicate "is a monoid... |
| isnmnd 18956 | A condition for a structur... |
| sgrpidmnd 18957 | A semigroup with an identi... |
| mndsgrp 18958 | A monoid is a semigroup. ... |
| mndmgm 18959 | A monoid is a magma. (Con... |
| mndcl 18960 | Closure of the operation o... |
| mndass 18961 | A monoid operation is asso... |
| mndid 18962 | A monoid has a two-sided i... |
| mndideu 18963 | The two-sided identity ele... |
| mndideuOLD 18964 | Obsolete version of ~ mndi... |
| mnd32g 18965 | Commutative/associative la... |
| mnd12g 18966 | Commutative/associative la... |
| mnd4g 18967 | Commutative/associative la... |
| mndidcl 18968 | The identity element of a ... |
| mndbn0 18969 | The base set of a monoid i... |
| hashfinmndnn 18970 | A finite monoid has positi... |
| mndplusf 18971 | The group addition operati... |
| mndlrid 18972 | A monoid's identity elemen... |
| mndlid 18973 | The identity element of a ... |
| mndrid 18974 | The identity element of a ... |
| ismndd 18975 | Deduce a monoid from its p... |
| mndpfo 18976 | The addition operation of ... |
| mndfo 18977 | The addition operation of ... |
| mndpfoOLD 18978 | Obsolete version of ~ mndp... |
| mndfoOLD 18979 | Obsolete version of ~ mndf... |
| mndpropd 18980 | If two structures have the... |
| mndprop 18981 | If two structures have the... |
| issubmnd 18982 | Characterize a submonoid b... |
| ress0g 18983 | ` 0g ` is unaffected by re... |
| ress0gOLD 18984 | Obsolete version of ~ ress... |
| submnd0 18985 | The zero of a submonoid is... |
| submnd0OLD 18986 | Obsolete version of ~ subm... |
| mndinvmod 18987 | Uniqueness of an inverse e... |
| mndpsuppss 18988 | The support of a mapping o... |
| mndpsuppfi 18989 | The support of a mapping o... |
| mndpfsupp 18990 | A mapping of a scalar mult... |
| prdsplusgcl 18991 | Structure product pointwis... |
| prdsidlem 18992 | Characterization of identi... |
| prdsmndd 18993 | The product of a family of... |
| prds0g 18994 | The identity in a product ... |
| pwsmnd 18995 | The structure power of a m... |
| pws0g 18996 | The identity in a structur... |
| imasmnd2 18997 | The image structure of a m... |
| imasmnd 18998 | The image structure of a m... |
| imasmndf1 18999 | The image of a monoid unde... |
| xpsmnd 19000 | The binary product of mono... |
| xpsmnd0 19001 | The identity element of a ... |
| mnd1 19002 | The (smallest) structure r... |
| mnd1id 19003 | The singleton element of a... |
| qusmnd 19004 | Prove that a quotient stru... |
| ismhm 19009 | Property of a monoid homom... |
| ismhmd 19010 | Deduction version of ~ ism... |
| mhmrcl1 19011 | Reverse closure of a monoi... |
| mhmrcl2 19012 | Reverse closure of a monoi... |
| mhmf 19013 | A monoid homomorphism is a... |
| ismhm0 19014 | Property of a monoid homom... |
| mhmismgmhm 19015 | Each monoid homomorphism i... |
| mhmpropd 19016 | Monoid homomorphism depend... |
| mhmlin 19017 | A monoid homomorphism comm... |
| mhm0 19018 | A monoid homomorphism pres... |
| idmhm 19019 | The identity homomorphism ... |
| mhmf1o 19020 | A monoid homomorphism is b... |
| mndvcl 19021 | Tuple-wise additive closur... |
| mndvass 19022 | Tuple-wise associativity i... |
| mndvlid 19023 | Tuple-wise left identity i... |
| mndvrid 19024 | Tuple-wise right identity ... |
| mhmvlin 19025 | Tuple extension of monoid ... |
| submrcl 19026 | Reverse closure for submon... |
| issubm 19027 | Expand definition of a sub... |
| issubm2 19028 | Submonoids are subsets tha... |
| issubmndb 19029 | The submonoid predicate. ... |
| issubmd 19030 | Deduction for proving a su... |
| mndissubm 19031 | If the base set of a monoi... |
| resmndismnd 19032 | If the base set of a monoi... |
| submss 19033 | Submonoids are subsets of ... |
| submid 19034 | Every monoid is trivially ... |
| subm0cl 19035 | Submonoids contain zero. ... |
| submcl 19036 | Submonoids are closed unde... |
| submcld 19037 | Submonoids are closed unde... |
| submmnd 19038 | Submonoids are themselves ... |
| submbas 19039 | The base set of a submonoi... |
| subm0 19040 | Submonoids have the same i... |
| subsubm 19041 | A submonoid of a submonoid... |
| 0subm 19042 | The zero submonoid of an a... |
| insubm 19043 | The intersection of two su... |
| 0mhm 19044 | The constant zero linear f... |
| resmhm 19045 | Restriction of a monoid ho... |
| resmhm2 19046 | One direction of ~ resmhm2... |
| resmhm2b 19047 | Restriction of the codomai... |
| mhmco 19048 | The composition of monoid ... |
| mhmimalem 19049 | Lemma for ~ mhmima and sim... |
| mhmima 19050 | The homomorphic image of a... |
| mhmeql 19051 | The equalizer of two monoi... |
| submacs 19052 | Submonoids are an algebrai... |
| mndind 19053 | Induction in a monoid. In... |
| prdspjmhm 19054 | A projection from a produc... |
| pwspjmhm 19055 | A projection from a struct... |
| pwsdiagmhm 19056 | Diagonal monoid homomorphi... |
| pwsco1mhm 19057 | Right composition with a f... |
| pwsco2mhm 19058 | Left composition with a mo... |
| gsumvallem2 19059 | Lemma for properties of th... |
| gsumsubm 19060 | Evaluate a group sum in a ... |
| gsumz 19061 | Value of a group sum over ... |
| gsumwsubmcl 19062 | Closure of the composite i... |
| gsumws1 19063 | A singleton composite reco... |
| gsumwcl 19064 | Closure of the composite o... |
| gsumsgrpccat 19065 | Homomorphic property of no... |
| gsumccat 19066 | Homomorphic property of co... |
| gsumws2 19067 | Valuation of a pair in a m... |
| gsumccatsn 19068 | Homomorphic property of co... |
| gsumspl 19069 | The primary purpose of the... |
| gsumwmhm 19070 | Behavior of homomorphisms ... |
| gsumwspan 19071 | The submonoid generated by... |
| frmdval 19076 | Value of the free monoid c... |
| frmdbas 19077 | The base set of a free mon... |
| frmdelbas 19078 | An element of the base set... |
| frmdplusg 19079 | The monoid operation of a ... |
| frmdadd 19080 | Value of the monoid operat... |
| vrmdfval 19081 | The canonical injection fr... |
| vrmdval 19082 | The value of the generatin... |
| vrmdf 19083 | The mapping from the index... |
| frmdmnd 19084 | A free monoid is a monoid.... |
| frmd0 19085 | The identity of the free m... |
| frmdsssubm 19086 | The set of words taking va... |
| frmdgsum 19087 | Any word in a free monoid ... |
| frmdss2 19088 | A subset of generators is ... |
| frmdup1 19089 | Any assignment of the gene... |
| frmdup2 19090 | The evaluation map has the... |
| frmdup3lem 19091 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 19092 | Universal property of the ... |
| efmnd 19095 | The monoid of endofunction... |
| efmndbas 19096 | The base set of the monoid... |
| efmndbasabf 19097 | The base set of the monoid... |
| elefmndbas 19098 | Two ways of saying a funct... |
| elefmndbas2 19099 | Two ways of saying a funct... |
| efmndbasf 19100 | Elements in the monoid of ... |
| efmndhash 19101 | The monoid of endofunction... |
| efmndbasfi 19102 | The monoid of endofunction... |
| efmndfv 19103 | The function value of an e... |
| efmndtset 19104 | The topology of the monoid... |
| efmndplusg 19105 | The group operation of a m... |
| efmndov 19106 | The value of the group ope... |
| efmndcl 19107 | The group operation of the... |
| efmndtopn 19108 | The topology of the monoid... |
| symggrplem 19109 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 19110 | The monoid of endofunction... |
| efmndsgrp 19111 | The monoid of endofunction... |
| ielefmnd 19112 | The identity function rest... |
| efmndid 19113 | The identity function rest... |
| efmndmnd 19114 | The monoid of endofunction... |
| efmnd0nmnd 19115 | Even the monoid of endofun... |
| efmndbas0 19116 | The base set of the monoid... |
| efmnd1hash 19117 | The monoid of endofunction... |
| efmnd1bas 19118 | The monoid of endofunction... |
| efmnd2hash 19119 | The monoid of endofunction... |
| submefmnd 19120 | If the base set of a monoi... |
| sursubmefmnd 19121 | The set of surjective endo... |
| injsubmefmnd 19122 | The set of injective endof... |
| idressubmefmnd 19123 | The singleton containing o... |
| idresefmnd 19124 | The structure with the sin... |
| smndex1ibas 19125 | The modulo function ` I ` ... |
| smndex1iidm 19126 | The modulo function ` I ` ... |
| smndex1gbas 19127 | The constant functions ` (... |
| smndex1gbasOLD 19128 | Obsolete version of ~ smnd... |
| smndex1gid 19129 | The composition of a const... |
| smndex1gidOLD 19130 | Obsolete version of ~ smnd... |
| smndex1igid 19131 | The composition of the mod... |
| smndex1igidOLD 19132 | Obsolete version of ~ smnd... |
| smndex1basss 19133 | The modulo function ` I ` ... |
| smndex1bas 19134 | The base set of the monoid... |
| smndex1mgm 19135 | The monoid of endofunction... |
| smndex1sgrp 19136 | The monoid of endofunction... |
| smndex1mndlem 19137 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 19138 | The monoid of endofunction... |
| smndex1id 19139 | The modulo function ` I ` ... |
| smndex1n0mnd 19140 | The identity of the monoid... |
| nsmndex1 19141 | The base set ` B ` of the ... |
| smndex2dbas 19142 | The doubling function ` D ... |
| smndex2dnrinv 19143 | The doubling function ` D ... |
| smndex2hbas 19144 | The halving functions ` H ... |
| smndex2dlinvh 19145 | The halving functions ` H ... |
| mgm2nsgrplem1 19146 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 19147 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 19148 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 19149 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 19150 | A small magma (with two el... |
| sgrp2nmndlem1 19151 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 19152 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 19153 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 19154 | A small semigroup (with tw... |
| sgrp2rid2ex 19155 | A small semigroup (with tw... |
| sgrp2nmndlem4 19156 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 19157 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 19158 | A small semigroup (with tw... |
| mgmnsgrpex 19159 | There is a magma which is ... |
| sgrpnmndex 19160 | There is a semigroup which... |
| sgrpssmgm 19161 | The class of all semigroup... |
| mndsssgrp 19162 | The class of all monoids i... |
| degenmgmopdm 19163 | The domain of the operatio... |
| degenmgmbas 19164 | The base set of a degenera... |
| degenmgmnfn 19165 | The operation of a degener... |
| degenmgm 19166 | A degenerate magma: althou... |
| degenmgm2opdm 19167 | The domain of the operatio... |
| degenmgm2nfun 19168 | The operation of a second ... |
| degenmgm2 19169 | A degenerate magma: althou... |
| pwmndgplus 19170 | The operation of the monoi... |
| pwmndid 19171 | The identity of the monoid... |
| pwmnd 19172 | The power set of a class `... |
| isgrp 19179 | The predicate "is a group"... |
| grpmnd 19180 | A group is a monoid. (Con... |
| grpcl 19181 | Closure of the operation o... |
| grpass 19182 | A group operation is assoc... |
| grpinvex 19183 | Every member of a group ha... |
| grpideu 19184 | The two-sided identity ele... |
| grpassd 19185 | A group operation is assoc... |
| grpmndd 19186 | A group is a monoid. (Con... |
| grpcld 19187 | Closure of the operation o... |
| grpplusf 19188 | The group addition operati... |
| grpplusfo 19189 | The group addition operati... |
| resgrpplusfrn 19190 | The underlying set of a gr... |
| grppropd 19191 | If two structures have the... |
| grpprop 19192 | If two structures have the... |
| grppropstr 19193 | Generalize a specific 2-el... |
| grpss 19194 | Show that a structure exte... |
| isgrpd2e 19195 | Deduce a group from its pr... |
| isgrpd2 19196 | Deduce a group from its pr... |
| isgrpde 19197 | Deduce a group from its pr... |
| isgrpd 19198 | Deduce a group from its pr... |
| isgrpi 19199 | Properties that determine ... |
| grpsgrp 19200 | A group is a semigroup. (... |
| grpmgmd 19201 | A group is a magma, deduct... |
| dfgrp2 19202 | Alternate definition of a ... |
| dfgrp2e 19203 | Alternate definition of a ... |
| isgrpix 19204 | Properties that determine ... |
| grpidcl 19205 | The identity element of a ... |
| grpbn0 19206 | The base set of a group is... |
| grplid 19207 | The identity element of a ... |
| grprid 19208 | The identity element of a ... |
| grplidd 19209 | The identity element of a ... |
| grpridd 19210 | The identity element of a ... |
| grpn0 19211 | A group is not empty. (Co... |
| hashfingrpnn 19212 | A finite group has positiv... |
| grprcan 19213 | Right cancellation law for... |
| grpinveu 19214 | The left inverse element o... |
| grpid 19215 | Two ways of saying that an... |
| isgrpid2 19216 | Properties showing that an... |
| grpidd2 19217 | Deduce the identity elemen... |
| grpinvfval 19218 | The inverse function of a ... |
| grpinvfvalALT 19219 | Shorter proof of ~ grpinvf... |
| grpinvval 19220 | The inverse of a group ele... |
| grpinvfn 19221 | Functionality of the group... |
| grpinvfvi 19222 | The group inverse function... |
| grpsubfval 19223 | Group subtraction (divisio... |
| grpsubfvalALT 19224 | Shorter proof of ~ grpsubf... |
| grpsubval 19225 | Group subtraction (divisio... |
| grpinvf 19226 | The group inversion operat... |
| grpinvcl 19227 | A group element's inverse ... |
| grpinvcld 19228 | A group element's inverse ... |
| grplinv 19229 | The left inverse of a grou... |
| grprinv 19230 | The right inverse of a gro... |
| grpinvid1 19231 | The inverse of a group ele... |
| grpinvid2 19232 | The inverse of a group ele... |
| isgrpinv 19233 | Properties showing that a ... |
| grplinvd 19234 | The left inverse of a grou... |
| grprinvd 19235 | The right inverse of a gro... |
| grplrinv 19236 | In a group, every member h... |
| grpidinv2 19237 | A group's properties using... |
| grpidinv 19238 | A group has a left and rig... |
| grpinvid 19239 | The inverse of the identit... |
| grplcan 19240 | Left cancellation law for ... |
| grpasscan1 19241 | An associative cancellatio... |
| grpasscan2 19242 | An associative cancellatio... |
| grpidrcan 19243 | If right adding an element... |
| grpidlcan 19244 | If left adding an element ... |
| grpinvinv 19245 | Double inverse law for gro... |
| grpinvcnv 19246 | The group inverse is its o... |
| grpinv11 19247 | The group inverse is one-t... |
| grpinvf1o 19248 | The group inverse is a one... |
| grpinvnz 19249 | The inverse of a nonzero g... |
| grpinvnzcl 19250 | The inverse of a nonzero g... |
| grpsubinv 19251 | Subtraction of an inverse.... |
| grplmulf1o 19252 | Left multiplication by a g... |
| grpraddf1o 19253 | Right addition by a group ... |
| grpinvpropd 19254 | If two structures have the... |
| grpidssd 19255 | If the base set of a group... |
| grpinvssd 19256 | If the base set of a group... |
| grpinvadd 19257 | The inverse of the group o... |
| grpsubf 19258 | Functionality of group sub... |
| grpsubcl 19259 | Closure of group subtracti... |
| grpsubrcan 19260 | Right cancellation law for... |
| grpinvsub 19261 | Inverse of a group subtrac... |
| grpinvval2 19262 | A ~ df-neg -like equation ... |
| grpsubid 19263 | Subtraction of a group ele... |
| grpsubid1 19264 | Subtraction of the identit... |
| grpsubeq0 19265 | If the difference between ... |
| grpsubadd0sub 19266 | Subtraction expressed as a... |
| grpsubadd 19267 | Relationship between group... |
| grpsubsub 19268 | Double group subtraction. ... |
| grpaddsubass 19269 | Associative-type law for g... |
| grppncan 19270 | Cancellation law for subtr... |
| grpnpcan 19271 | Cancellation law for subtr... |
| grpsubsub4 19272 | Double group subtraction (... |
| grppnpcan2 19273 | Cancellation law for mixed... |
| grpnpncan 19274 | Cancellation law for group... |
| grpnpncan0 19275 | Cancellation law for group... |
| grpnnncan2 19276 | Cancellation law for group... |
| dfgrp3lem 19277 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 19278 | Alternate definition of a ... |
| dfgrp3e 19279 | Alternate definition of a ... |
| grplactfval 19280 | The left group action of e... |
| grplactval 19281 | The value of the left grou... |
| grplactcnv 19282 | The left group action of e... |
| grplactf1o 19283 | The left group action of e... |
| grpsubpropd 19284 | Weak property deduction fo... |
| grpsubpropd2 19285 | Strong property deduction ... |
| grp1 19286 | The (smallest) structure r... |
| grp1inv 19287 | The inverse function of th... |
| prdsinvlem 19288 | Characterization of invers... |
| prdsgrpd 19289 | The product of a family of... |
| prdsinvgd 19290 | Negation in a product of g... |
| pwsgrp 19291 | A structure power of a gro... |
| pwsinvg 19292 | Negation in a structure po... |
| pwssub 19293 | Subtraction in a structure... |
| imasgrp2 19294 | The image structure of a g... |
| imasgrp 19295 | The image structure of a g... |
| imasgrpf1 19296 | The image of a group under... |
| qusgrp2 19297 | Prove that a quotient stru... |
| xpsgrp 19298 | The binary product of grou... |
| xpsinv 19299 | Value of the negation oper... |
| xpsgrpsub 19300 | Value of the subtraction o... |
| mhmlem 19301 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19302 | A surjective monoid morphi... |
| mhmmnd 19303 | The image of a monoid ` G ... |
| mhmfmhm 19304 | The function fulfilling th... |
| ghmgrp 19305 | The image of a group ` G `... |
| mulgfval 19308 | Group multiple (exponentia... |
| mulgfvalALT 19309 | Shorter proof of ~ mulgfva... |
| mulgval 19310 | Value of the group multipl... |
| mulgfn 19311 | Functionality of the group... |
| mulgfvi 19312 | The group multiple operati... |
| mulg0 19313 | Group multiple (exponentia... |
| mulgnn 19314 | Group multiple (exponentia... |
| ressmulgnn 19315 | Values for the group multi... |
| ressmulgnn0 19316 | Values for the group multi... |
| ressmulgnnd 19317 | Values for the group multi... |
| mulgnngsum 19318 | Group multiple (exponentia... |
| mulgnn0gsum 19319 | Group multiple (exponentia... |
| mulg1 19320 | Group multiple (exponentia... |
| mulgnnp1 19321 | Group multiple (exponentia... |
| mulg2 19322 | Group multiple (exponentia... |
| mulgnegnn 19323 | Group multiple (exponentia... |
| mulgnn0p1 19324 | Group multiple (exponentia... |
| mulgnnsubcl 19325 | Closure of the group multi... |
| mulgnn0subcl 19326 | Closure of the group multi... |
| mulgsubcl 19327 | Closure of the group multi... |
| mulgnncl 19328 | Closure of the group multi... |
| mulgnn0cl 19329 | Closure of the group multi... |
| mulgcl 19330 | Closure of the group multi... |
| mulgneg 19331 | Group multiple (exponentia... |
| mulgnegneg 19332 | The inverse of a negative ... |
| mulgm1 19333 | Group multiple (exponentia... |
| mulgnn0cld 19334 | Closure of the group multi... |
| mulgcld 19335 | Deduction associated with ... |
| mulgaddcomlem 19336 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19337 | The group multiple operato... |
| mulginvcom 19338 | The group multiple operato... |
| mulginvinv 19339 | The group multiple operato... |
| mulgnn0z 19340 | A group multiple of the id... |
| mulgz 19341 | A group multiple of the id... |
| mulgnndir 19342 | Sum of group multiples, fo... |
| mulgnn0dir 19343 | Sum of group multiples, ge... |
| mulgdirlem 19344 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19345 | Sum of group multiples, ge... |
| mulgp1 19346 | Group multiple (exponentia... |
| mulgneg2 19347 | Group multiple (exponentia... |
| mulgnnass 19348 | Product of group multiples... |
| mulgnn0ass 19349 | Product of group multiples... |
| mulgass 19350 | Product of group multiples... |
| mulgassr 19351 | Reversed product of group ... |
| mulgmodid 19352 | Casting out multiples of t... |
| mulgsubdir 19353 | Distribution of group mult... |
| mhmmulg 19354 | A homomorphism of monoids ... |
| mulgpropd 19355 | Two structures with the sa... |
| submmulgcl 19356 | Closure of the group multi... |
| submmulg 19357 | A group multiple is the sa... |
| pwsmulg 19358 | Value of a group multiple ... |
| issubg 19365 | The subgroup predicate. (... |
| subgss 19366 | A subgroup is a subset. (... |
| subgid 19367 | A group is a subgroup of i... |
| subggrp 19368 | A subgroup is a group. (C... |
| subgbas 19369 | The base of the restricted... |
| subgrcl 19370 | Reverse closure for the su... |
| subg0 19371 | A subgroup of a group must... |
| subginv 19372 | The inverse of an element ... |
| subg0cl 19373 | The group identity is an e... |
| subginvcl 19374 | The inverse of an element ... |
| subgcl 19375 | A subgroup is closed under... |
| subgcld 19376 | A subgroup is closed under... |
| subgsubcl 19377 | A subgroup is closed under... |
| subgsub 19378 | The subtraction of element... |
| subgmulgcl 19379 | Closure of the group multi... |
| subgmulg 19380 | A group multiple is the sa... |
| issubg2 19381 | Characterize the subgroups... |
| issubgrpd2 19382 | Prove a subgroup by closur... |
| issubgrpd 19383 | Prove a subgroup by closur... |
| issubg3 19384 | A subgroup is a symmetric ... |
| issubg4 19385 | A subgroup is a nonempty s... |
| grpissubg 19386 | If the base set of a group... |
| resgrpisgrp 19387 | If the base set of a group... |
| subgsubm 19388 | A subgroup is a submonoid.... |
| subsubg 19389 | A subgroup of a subgroup i... |
| subgint 19390 | The intersection of a none... |
| 0subg 19391 | The zero subgroup of an ar... |
| trivsubgd 19392 | The only subgroup of a tri... |
| trivsubgsnd 19393 | The only subgroup of a tri... |
| isnsg 19394 | Property of being a normal... |
| isnsg2 19395 | Weaken the condition of ~ ... |
| nsgbi 19396 | Defining property of a nor... |
| nsgsubg 19397 | A normal subgroup is a sub... |
| nsgconj 19398 | The conjugation of an elem... |
| isnsg3 19399 | A subgroup is normal iff t... |
| subgacs 19400 | Subgroups are an algebraic... |
| nsgacs 19401 | Normal subgroups form an a... |
| elnmz 19402 | Elementhood in the normali... |
| nmzbi 19403 | Defining property of the n... |
| nmzsubg 19404 | The normalizer N_G(S) of a... |
| ssnmz 19405 | A subgroup is a subset of ... |
| isnsg4 19406 | A subgroup is normal iff i... |
| nmznsg 19407 | Any subgroup is a normal s... |
| 0nsg 19408 | The zero subgroup is norma... |
| nsgid 19409 | The whole group is a norma... |
| 0idnsgd 19410 | The whole group and the ze... |
| trivnsgd 19411 | The only normal subgroup o... |
| triv1nsgd 19412 | A trivial group has exactl... |
| 1nsgtrivd 19413 | A group with exactly one n... |
| releqg 19414 | The left coset equivalence... |
| ecxpid 19415 | The equivalence class of a... |
| qsxpid 19416 | The quotient set of a cart... |
| eqgfval 19417 | Value of the subgroup left... |
| eqgval 19418 | Value of the subgroup left... |
| eqger 19419 | The subgroup coset equival... |
| eqglact 19420 | A left coset can be expres... |
| eqgid 19421 | The left coset containing ... |
| eqgen 19422 | Each coset is equipotent t... |
| eqgcpbl 19423 | The subgroup coset equival... |
| qusxpid 19424 | The Group quotient equival... |
| qustriv 19425 | The quotient of a group ` ... |
| qustrivr 19426 | Converse of ~ qustriv . (... |
| eqg0el 19427 | Equivalence class of a quo... |
| quselbas 19428 | Membership in the base set... |
| quseccl0 19429 | Closure of the quotient ma... |
| qusgrp 19430 | If ` Y ` is a normal subgr... |
| quseccl 19431 | Closure of the quotient ma... |
| qusadd 19432 | Value of the group operati... |
| qus0 19433 | Value of the group identit... |
| qusinv 19434 | Value of the group inverse... |
| qussub 19435 | Value of the group subtrac... |
| ecqusaddd 19436 | Addition of equivalence cl... |
| ecqusaddcl 19437 | Closure of the addition in... |
| lagsubg2 19438 | Lagrange's theorem for fin... |
| lagsubg 19439 | Lagrange's theorem for Gro... |
| eqg0subg 19440 | The coset equivalence rela... |
| eqg0subgecsn 19441 | The equivalence classes mo... |
| qus0subgbas 19442 | The base set of a quotient... |
| qus0subgadd 19443 | The addition in a quotient... |
| cycsubmel 19444 | Characterization of an ele... |
| cycsubmcl 19445 | The set of nonnegative int... |
| cycsubm 19446 | The set of nonnegative int... |
| cyccom 19447 | Condition for an operation... |
| cycsubmcom 19448 | The operation of a monoid ... |
| cycsubggend 19449 | The cyclic subgroup genera... |
| cycsubgcl 19450 | The set of integer powers ... |
| cycsubgss 19451 | The cyclic subgroup genera... |
| cycsubg 19452 | The cyclic group generated... |
| cycsubgcld 19453 | The cyclic subgroup genera... |
| cycsubg2 19454 | The subgroup generated by ... |
| cycsubg2cl 19455 | Any multiple of an element... |
| reldmghm 19458 | Lemma for group homomorphi... |
| isghm 19459 | Property of being a homomo... |
| isghm3 19460 | Property of a group homomo... |
| ghmgrp1 19461 | A group homomorphism is on... |
| ghmgrp2 19462 | A group homomorphism is on... |
| ghmf 19463 | A group homomorphism is a ... |
| ghmlin 19464 | A homomorphism of groups i... |
| ghmid 19465 | A homomorphism of groups p... |
| ghminv 19466 | A homomorphism of groups p... |
| ghmsub 19467 | Linearity of subtraction t... |
| isghmd 19468 | Deduction for a group homo... |
| ghmmhm 19469 | A group homomorphism is a ... |
| ghmmhmb 19470 | Group homomorphisms and mo... |
| ghmmulg 19471 | A group homomorphism prese... |
| ghmrn 19472 | The range of a homomorphis... |
| 0ghm 19473 | The constant zero linear f... |
| idghm 19474 | The identity homomorphism ... |
| resghm 19475 | Restriction of a homomorph... |
| resghm2 19476 | One direction of ~ resghm2... |
| resghm2b 19477 | Restriction of the codomai... |
| ghmghmrn 19478 | A group homomorphism from ... |
| ghmco 19479 | The composition of group h... |
| ghmima 19480 | The image of a subgroup un... |
| ghmpreima 19481 | The inverse image of a sub... |
| ghmeql 19482 | The equalizer of two group... |
| ghmnsgima 19483 | The image of a normal subg... |
| ghmnsgpreima 19484 | The inverse image of a nor... |
| ghmker 19485 | The kernel of a homomorphi... |
| ghmeqker 19486 | Two source points map to t... |
| pwsdiagghm 19487 | Diagonal homomorphism into... |
| f1ghm0to0 19488 | If a group homomorphism ` ... |
| ghmf1 19489 | Two ways of saying a group... |
| kerf1ghm 19490 | A group homomorphism ` F `... |
| ghmf1o 19491 | A bijective group homomorp... |
| conjghm 19492 | Conjugation is an automorp... |
| conjsubg 19493 | A conjugated subgroup is a... |
| conjsubgen 19494 | A conjugated subgroup is e... |
| conjnmz 19495 | A subgroup is unchanged un... |
| conjnmzb 19496 | Alternative condition for ... |
| conjnsg 19497 | A normal subgroup is uncha... |
| qusghm 19498 | If ` Y ` is a normal subgr... |
| ghmpropd 19499 | Group homomorphism depends... |
| gimfn 19504 | The group isomorphism func... |
| isgim 19505 | An isomorphism of groups i... |
| gimf1o 19506 | An isomorphism of groups i... |
| gimghm 19507 | An isomorphism of groups i... |
| isgim2 19508 | A group isomorphism is a h... |
| subggim 19509 | Behavior of subgroups unde... |
| gimcnv 19510 | The converse of a group is... |
| gimco 19511 | The composition of group i... |
| gim0to0 19512 | A group isomorphism maps t... |
| imasgim 19513 | A relabeling of the elemen... |
| brgic 19514 | The relation "is isomorphi... |
| brgici 19515 | Prove isomorphic by an exp... |
| gicref 19516 | Isomorphism is reflexive. ... |
| giclcl 19517 | Isomorphism implies the le... |
| gicrcl 19518 | Isomorphism implies the ri... |
| gicsym 19519 | Isomorphism is symmetric. ... |
| gictr 19520 | Isomorphism is transitive.... |
| gicer 19521 | Isomorphism is an equivale... |
| gicen 19522 | Isomorphic groups have equ... |
| gicsubgen 19523 | A less trivial example of ... |
| ghmqusnsglem1 19524 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19525 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19526 | The mapping ` H ` induced ... |
| ghmquskerlem1 19527 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19528 | In the case of theorem ~ g... |
| ghmquskerlem2 19529 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19530 | The mapping ` H ` induced ... |
| ghmqusker 19531 | A surjective group homomor... |
| gicqusker 19532 | The image ` H ` of a group... |
| isga 19535 | The predicate "is a (left)... |
| gagrp 19536 | The left argument of a gro... |
| gaset 19537 | The right argument of a gr... |
| gagrpid 19538 | The identity of the group ... |
| gaf 19539 | The mapping of the group a... |
| gafo 19540 | A group action is onto its... |
| gaass 19541 | An "associative" property ... |
| ga0 19542 | The action of a group on t... |
| gaid 19543 | The trivial action of a gr... |
| subgga 19544 | A subgroup acts on its par... |
| gass 19545 | A subset of a group action... |
| gasubg 19546 | The restriction of a group... |
| gaid2 19547 | A group operation is a lef... |
| galcan 19548 | The action of a particular... |
| gacan 19549 | Group inverses cancel in a... |
| gapm 19550 | The action of a particular... |
| gaorb 19551 | The orbit equivalence rela... |
| gaorber 19552 | The orbit equivalence rela... |
| gastacl 19553 | The stabilizer subgroup in... |
| gastacos 19554 | Write the coset relation f... |
| orbstafun 19555 | Existence and uniqueness f... |
| orbstaval 19556 | Value of the function at a... |
| orbsta 19557 | The Orbit-Stabilizer theor... |
| orbsta2 19558 | Relation between the size ... |
| cntrval 19563 | Substitute definition of t... |
| cntzfval 19564 | First level substitution f... |
| cntzval 19565 | Definition substitution fo... |
| elcntz 19566 | Elementhood in the central... |
| cntzel 19567 | Membership in a centralize... |
| cntzsnval 19568 | Special substitution for t... |
| elcntzsn 19569 | Value of the centralizer o... |
| sscntz 19570 | A centralizer expression f... |
| cntzrcl 19571 | Reverse closure for elemen... |
| cntzssv 19572 | The centralizer is uncondi... |
| cntzi 19573 | Membership in a centralize... |
| elcntr 19574 | Elementhood in the center ... |
| cntrss 19575 | The center is a subset of ... |
| cntri 19576 | Defining property of the c... |
| resscntz 19577 | Centralizer in a substruct... |
| cntzsgrpcl 19578 | Centralizers are closed un... |
| cntz2ss 19579 | Centralizers reverse the s... |
| cntzrec 19580 | Reciprocity relationship f... |
| cntziinsn 19581 | Express any centralizer as... |
| cntzsubm 19582 | Centralizers in a monoid a... |
| cntzsubg 19583 | Centralizers in a group ar... |
| cntzidss 19584 | If the elements of ` S ` c... |
| cntzmhm 19585 | Centralizers in a monoid a... |
| cntzmhm2 19586 | Centralizers in a monoid a... |
| cntrsubgnsg 19587 | A central subgroup is norm... |
| cntrnsg 19588 | The center of a group is a... |
| oppgval 19591 | Value of the opposite grou... |
| oppgplusfval 19592 | Value of the addition oper... |
| oppgplus 19593 | Value of the addition oper... |
| setsplusg 19594 | The other components of an... |
| oppgbas 19595 | Base set of an opposite gr... |
| oppgtset 19596 | Topology of an opposite gr... |
| oppgtopn 19597 | Topology of an opposite gr... |
| oppgmnd 19598 | The opposite of a monoid i... |
| oppgmndb 19599 | Bidirectional form of ~ op... |
| oppgid 19600 | Zero in a monoid is a symm... |
| oppggrp 19601 | The opposite of a group is... |
| oppggrpb 19602 | Bidirectional form of ~ op... |
| oppginv 19603 | Inverses in a group are a ... |
| invoppggim 19604 | The inverse is an antiauto... |
| oppggic 19605 | Every group is (naturally)... |
| oppgsubm 19606 | Being a submonoid is a sym... |
| oppgsubg 19607 | Being a subgroup is a symm... |
| oppgcntz 19608 | A centralizer in a group i... |
| oppgcntr 19609 | The center of a group is t... |
| gsumwrev 19610 | A sum in an opposite monoi... |
| oppgle 19611 | less-than relation of an o... |
| oppglt 19612 | less-than relation of an o... |
| symgval 19615 | The value of the symmetric... |
| symgbas 19616 | The base set of the symmet... |
| elsymgbas2 19617 | Two ways of saying a funct... |
| elsymgbas 19618 | Two ways of saying a funct... |
| symgbasf1o 19619 | Elements in the symmetric ... |
| symgbasf 19620 | A permutation (element of ... |
| symgbasmap 19621 | A permutation (element of ... |
| symghash 19622 | The symmetric group on ` n... |
| symgbasfi 19623 | The symmetric group on a f... |
| symgfv 19624 | The function value of a pe... |
| symgfvne 19625 | The function values of a p... |
| symgressbas 19626 | The symmetric group on ` A... |
| symgplusg 19627 | The group operation of a s... |
| symgov 19628 | The value of the group ope... |
| symgcl 19629 | The group operation of the... |
| idresperm 19630 | The identity function rest... |
| symgmov1 19631 | For a permutation of a set... |
| symgmov2 19632 | For a permutation of a set... |
| symgbas0 19633 | The base set of the symmet... |
| symg1hash 19634 | The symmetric group on a s... |
| symg1bas 19635 | The symmetric group on a s... |
| symg2hash 19636 | The symmetric group on a (... |
| symg2bas 19637 | The symmetric group on a p... |
| 0symgefmndeq 19638 | The symmetric group on the... |
| snsymgefmndeq 19639 | The symmetric group on a s... |
| symgpssefmnd 19640 | For a set ` A ` with more ... |
| symgvalstruct 19641 | The value of the symmetric... |
| symgsubmefmnd 19642 | The symmetric group on a s... |
| symgtset 19643 | The topology of the symmet... |
| symggrp 19644 | The symmetric group on a s... |
| symgid 19645 | The group identity element... |
| symginv 19646 | The group inverse in the s... |
| symgsubmefmndALT 19647 | The symmetric group on a s... |
| galactghm 19648 | The currying of a group ac... |
| lactghmga 19649 | The converse of ~ galactgh... |
| symgtopn 19650 | The topology of the symmet... |
| symgga 19651 | The symmetric group induce... |
| pgrpsubgsymgbi 19652 | Every permutation group is... |
| pgrpsubgsymg 19653 | Every permutation group is... |
| idressubgsymg 19654 | The singleton containing o... |
| idrespermg 19655 | The structure with the sin... |
| cayleylem1 19656 | Lemma for ~ cayley . (Con... |
| cayleylem2 19657 | Lemma for ~ cayley . (Con... |
| cayley 19658 | Cayley's Theorem (construc... |
| cayleyth 19659 | Cayley's Theorem (existenc... |
| symgfix2 19660 | If a permutation does not ... |
| symgextf 19661 | The extension of a permuta... |
| symgextfv 19662 | The function value of the ... |
| symgextfve 19663 | The function value of the ... |
| symgextf1lem 19664 | Lemma for ~ symgextf1 . (... |
| symgextf1 19665 | The extension of a permuta... |
| symgextfo 19666 | The extension of a permuta... |
| symgextf1o 19667 | The extension of a permuta... |
| symgextsymg 19668 | The extension of a permuta... |
| symgextres 19669 | The restriction of the ext... |
| gsumccatsymgsn 19670 | Homomorphic property of co... |
| gsmsymgrfixlem1 19671 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19672 | The composition of permuta... |
| fvcosymgeq 19673 | The values of two composit... |
| gsmsymgreqlem1 19674 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19675 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19676 | Two combination of permuta... |
| symgfixelq 19677 | A permutation of a set fix... |
| symgfixels 19678 | The restriction of a permu... |
| symgfixelsi 19679 | The restriction of a permu... |
| symgfixf 19680 | The mapping of a permutati... |
| symgfixf1 19681 | The mapping of a permutati... |
| symgfixfolem1 19682 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19683 | The mapping of a permutati... |
| symgfixf1o 19684 | The mapping of a permutati... |
| f1omvdmvd 19687 | A permutation of any class... |
| f1omvdcnv 19688 | A permutation and its inve... |
| mvdco 19689 | Composing two permutations... |
| f1omvdconj 19690 | Conjugation of a permutati... |
| f1otrspeq 19691 | A transposition is charact... |
| f1omvdco2 19692 | If exactly one of two perm... |
| f1omvdco3 19693 | If a point is moved by exa... |
| pmtrfval 19694 | The function generating tr... |
| pmtrval 19695 | A generated transposition,... |
| pmtrfv 19696 | General value of mapping a... |
| pmtrprfv 19697 | In a transposition of two ... |
| pmtrprfv3 19698 | In a transposition of two ... |
| pmtrf 19699 | Functionality of a transpo... |
| pmtrmvd 19700 | A transposition moves prec... |
| pmtrrn 19701 | Transposing two points giv... |
| pmtrfrn 19702 | A transposition (as a kind... |
| pmtrffv 19703 | Mapping of a point under a... |
| pmtrrn2 19704 | For any transposition ther... |
| pmtrfinv 19705 | A transposition function i... |
| pmtrfmvdn0 19706 | A transposition moves at l... |
| pmtrff1o 19707 | A transposition function i... |
| pmtrfcnv 19708 | A transposition function i... |
| pmtrfb 19709 | An intrinsic characterizat... |
| pmtrfconj 19710 | Any conjugate of a transpo... |
| symgsssg 19711 | The symmetric group has su... |
| symgfisg 19712 | The symmetric group has a ... |
| symgtrf 19713 | Transpositions are element... |
| symggen 19714 | The span of the transposit... |
| symggen2 19715 | A finite permutation group... |
| symgtrinv 19716 | To invert a permutation re... |
| pmtr3ncomlem1 19717 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19718 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19719 | Transpositions over sets w... |
| pmtrdifellem1 19720 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19721 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19722 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19723 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19724 | A transposition of element... |
| pmtrdifwrdellem1 19725 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19726 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19727 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19728 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19729 | A sequence of transpositio... |
| pmtrdifwrdel2 19730 | A sequence of transpositio... |
| pmtrprfval 19731 | The transpositions on a pa... |
| pmtrprfvalrn 19732 | The range of the transposi... |
| psgnunilem1 19737 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19738 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19739 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19740 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19741 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19742 | Addition and subtraction o... |
| psgnuni 19743 | If the same permutation ca... |
| psgnfval 19744 | Function definition of the... |
| psgnfn 19745 | Functionality and domain o... |
| psgndmsubg 19746 | The finitary permutations ... |
| psgneldm 19747 | Property of being a finita... |
| psgneldm2 19748 | The finitary permutations ... |
| psgneldm2i 19749 | A sequence of transpositio... |
| psgneu 19750 | A finitary permutation has... |
| psgnval 19751 | Value of the permutation s... |
| psgnvali 19752 | A finitary permutation has... |
| psgnvalii 19753 | Any representation of a pe... |
| psgnpmtr 19754 | All transpositions are odd... |
| psgn0fv0 19755 | The permutation sign funct... |
| sygbasnfpfi 19756 | The class of non-fixed poi... |
| psgnfvalfi 19757 | Function definition of the... |
| psgnvalfi 19758 | Value of the permutation s... |
| psgnran 19759 | The range of the permutati... |
| gsmtrcl 19760 | The group sum of transposi... |
| psgnfitr 19761 | A permutation of a finite ... |
| psgnfieu 19762 | A permutation of a finite ... |
| pmtrsn 19763 | The value of the transposi... |
| psgnsn 19764 | The permutation sign funct... |
| psgnprfval 19765 | The permutation sign funct... |
| psgnprfval1 19766 | The permutation sign of th... |
| psgnprfval2 19767 | The permutation sign of th... |
| odfval 19776 | Value of the order functio... |
| odfvalALT 19777 | Shorter proof of ~ odfval ... |
| odval 19778 | Second substitution for th... |
| odlem1 19779 | The group element order is... |
| odcl 19780 | The order of a group eleme... |
| odf 19781 | Functionality of the group... |
| odid 19782 | Any element to the power o... |
| odlem2 19783 | Any positive annihilator o... |
| odmodnn0 19784 | Reduce the argument of a g... |
| mndodconglem 19785 | Lemma for ~ mndodcong . (... |
| mndodcong 19786 | If two multipliers are con... |
| mndodcongi 19787 | If two multipliers are con... |
| oddvdsnn0 19788 | The only multiples of ` A ... |
| odnncl 19789 | If a nonzero multiple of a... |
| odmod 19790 | Reduce the argument of a g... |
| oddvds 19791 | The only multiples of ` A ... |
| oddvdsi 19792 | Any group element is annih... |
| odcong 19793 | If two multipliers are con... |
| odeq 19794 | The ~ oddvds property uniq... |
| odval2 19795 | A non-conditional definiti... |
| odcld 19796 | The order of a group eleme... |
| odm1inv 19797 | The (order-1)th multiple o... |
| odmulgid 19798 | A relationship between the... |
| odmulg2 19799 | The order of a multiple di... |
| odmulg 19800 | Relationship between the o... |
| odmulgeq 19801 | A multiple of a point of f... |
| odbezout 19802 | If ` N ` is coprime to the... |
| od1 19803 | The order of the group ide... |
| odeq1 19804 | The group identity is the ... |
| odinv 19805 | The order of the inverse o... |
| odf1 19806 | The multiples of an elemen... |
| odinf 19807 | The multiples of an elemen... |
| dfod2 19808 | An alternative definition ... |
| odcl2 19809 | The order of an element of... |
| oddvds2 19810 | The order of an element of... |
| finodsubmsubg 19811 | A submonoid whose elements... |
| 0subgALT 19812 | A shorter proof of ~ 0subg... |
| submod 19813 | The order of an element is... |
| subgod 19814 | The order of an element is... |
| odsubdvds 19815 | The order of an element of... |
| odf1o1 19816 | An element with zero order... |
| odf1o2 19817 | An element with nonzero or... |
| odhash 19818 | An element of zero order g... |
| odhash2 19819 | If an element has nonzero ... |
| odhash3 19820 | An element which generates... |
| odngen 19821 | A cyclic subgroup of size ... |
| gexval 19822 | Value of the exponent of a... |
| gexlem1 19823 | The group element order is... |
| gexcl 19824 | The exponent of a group is... |
| gexid 19825 | Any element to the power o... |
| gexlem2 19826 | Any positive annihilator o... |
| gexdvdsi 19827 | Any group element is annih... |
| gexdvds 19828 | The only ` N ` that annihi... |
| gexdvds2 19829 | An integer divides the gro... |
| gexod 19830 | Any group element is annih... |
| gexcl3 19831 | If the order of every grou... |
| gexnnod 19832 | Every group element has fi... |
| gexcl2 19833 | The exponent of a finite g... |
| gexdvds3 19834 | The exponent of a finite g... |
| gex1 19835 | A group or monoid has expo... |
| ispgp 19836 | A group is a ` P ` -group ... |
| pgpprm 19837 | Reverse closure for the fi... |
| pgpgrp 19838 | Reverse closure for the se... |
| pgpfi1 19839 | A finite group with order ... |
| pgp0 19840 | The identity subgroup is a... |
| subgpgp 19841 | A subgroup of a p-group is... |
| sylow1lem1 19842 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19843 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19844 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19845 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19846 | Lemma for ~ sylow1 . Usin... |
| sylow1 19847 | Sylow's first theorem. If... |
| odcau 19848 | Cauchy's theorem for the o... |
| pgpfi 19849 | The converse to ~ pgpfi1 .... |
| pgpfi2 19850 | Alternate version of ~ pgp... |
| pgphash 19851 | The order of a p-group. (... |
| isslw 19852 | The property of being a Sy... |
| slwprm 19853 | Reverse closure for the fi... |
| slwsubg 19854 | A Sylow ` P ` -subgroup is... |
| slwispgp 19855 | Defining property of a Syl... |
| slwpss 19856 | A proper superset of a Syl... |
| slwpgp 19857 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19858 | Every ` P ` -subgroup is c... |
| slwn0 19859 | Every finite group contain... |
| subgslw 19860 | A Sylow subgroup that is c... |
| sylow2alem1 19861 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19862 | Lemma for ~ sylow2a . All... |
| sylow2a 19863 | A named lemma of Sylow's s... |
| sylow2blem1 19864 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19865 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19866 | Sylow's second theorem. P... |
| sylow2b 19867 | Sylow's second theorem. A... |
| slwhash 19868 | A sylow subgroup has cardi... |
| fislw 19869 | The sylow subgroups of a f... |
| sylow2 19870 | Sylow's second theorem. S... |
| sylow3lem1 19871 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19872 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19873 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19874 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19875 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19876 | Lemma for ~ sylow3 , secon... |
| sylow3 19877 | Sylow's third theorem. Th... |
| lsmfval 19882 | The subgroup sum function ... |
| lsmvalx 19883 | Subspace sum value (for a ... |
| lsmelvalx 19884 | Subspace sum membership (f... |
| lsmelvalix 19885 | Subspace sum membership (f... |
| oppglsm 19886 | The subspace sum operation... |
| lsmssv 19887 | Subgroup sum is a subset o... |
| lsmless1x 19888 | Subset implies subgroup su... |
| lsmless2x 19889 | Subset implies subgroup su... |
| lsmub1x 19890 | Subgroup sum is an upper b... |
| lsmub2x 19891 | Subgroup sum is an upper b... |
| lsmval 19892 | Subgroup sum value (for a ... |
| lsmelval 19893 | Subgroup sum membership (f... |
| lsmelvali 19894 | Subgroup sum membership (f... |
| lsmelvalm 19895 | Subgroup sum membership an... |
| lsmelvalmi 19896 | Membership of vector subtr... |
| lsmsubm 19897 | The sum of two commuting s... |
| lsmsubg 19898 | The sum of two commuting s... |
| lsmcom2 19899 | Subgroup sum commutes. (C... |
| smndlsmidm 19900 | The direct product is idem... |
| lsmub1 19901 | Subgroup sum is an upper b... |
| lsmub2 19902 | Subgroup sum is an upper b... |
| lsmunss 19903 | Union of subgroups is a su... |
| lsmless1 19904 | Subset implies subgroup su... |
| lsmless2 19905 | Subset implies subgroup su... |
| lsmless12 19906 | Subset implies subgroup su... |
| lsmidm 19907 | Subgroup sum is idempotent... |
| lsmlub 19908 | The least upper bound prop... |
| lsmss1 19909 | Subgroup sum with a subset... |
| lsmss1b 19910 | Subgroup sum with a subset... |
| lsmss2 19911 | Subgroup sum with a subset... |
| lsmss2b 19912 | Subgroup sum with a subset... |
| lsmass 19913 | Subgroup sum is associativ... |
| mndlsmidm 19914 | Subgroup sum is idempotent... |
| lsm01 19915 | Subgroup sum with the zero... |
| lsm02 19916 | Subgroup sum with the zero... |
| subglsm 19917 | The subgroup sum evaluated... |
| lssnle 19918 | Equivalent expressions for... |
| lsmmod 19919 | The modular law holds for ... |
| lsmmod2 19920 | Modular law dual for subgr... |
| lsmpropd 19921 | If two structures have the... |
| cntzrecd 19922 | Commute the "subgroups com... |
| lsmcntz 19923 | The "subgroups commute" pr... |
| lsmcntzr 19924 | The "subgroups commute" pr... |
| lsmdisj 19925 | Disjointness from a subgro... |
| lsmdisj2 19926 | Association of the disjoin... |
| lsmdisj3 19927 | Association of the disjoin... |
| lsmdisjr 19928 | Disjointness from a subgro... |
| lsmdisj2r 19929 | Association of the disjoin... |
| lsmdisj3r 19930 | Association of the disjoin... |
| lsmdisj2a 19931 | Association of the disjoin... |
| lsmdisj2b 19932 | Association of the disjoin... |
| lsmdisj3a 19933 | Association of the disjoin... |
| lsmdisj3b 19934 | Association of the disjoin... |
| subgdisj1 19935 | Vectors belonging to disjo... |
| subgdisj2 19936 | Vectors belonging to disjo... |
| subgdisjb 19937 | Vectors belonging to disjo... |
| pj1fval 19938 | The left projection functi... |
| pj1val 19939 | The left projection functi... |
| pj1eu 19940 | Uniqueness of a left proje... |
| pj1f 19941 | The left projection functi... |
| pj2f 19942 | The right projection funct... |
| pj1id 19943 | Any element of a direct su... |
| pj1eq 19944 | Any element of a direct su... |
| pj1lid 19945 | The left projection functi... |
| pj1rid 19946 | The left projection functi... |
| pj1ghm 19947 | The left projection functi... |
| pj1ghm2 19948 | The left projection functi... |
| lsmhash 19949 | The order of the direct pr... |
| efgmval 19956 | Value of the formal invers... |
| efgmf 19957 | The formal inverse operati... |
| efgmnvl 19958 | The inversion function on ... |
| efgrcl 19959 | Lemma for ~ efgval . (Con... |
| efglem 19960 | Lemma for ~ efgval . (Con... |
| efgval 19961 | Value of the free group co... |
| efger 19962 | Value of the free group co... |
| efgi 19963 | Value of the free group co... |
| efgi0 19964 | Value of the free group co... |
| efgi1 19965 | Value of the free group co... |
| efgtf 19966 | Value of the free group co... |
| efgtval 19967 | Value of the extension fun... |
| efgval2 19968 | Value of the free group co... |
| efgi2 19969 | Value of the free group co... |
| efgtlen 19970 | Value of the free group co... |
| efginvrel2 19971 | The inverse of the reverse... |
| efginvrel1 19972 | The inverse of the reverse... |
| efgsf 19973 | Value of the auxiliary fun... |
| efgsdm 19974 | Elementhood in the domain ... |
| efgsval 19975 | Value of the auxiliary fun... |
| efgsdmi 19976 | Property of the last link ... |
| efgsval2 19977 | Value of the auxiliary fun... |
| efgsrel 19978 | The start and end of any e... |
| efgs1 19979 | A singleton of an irreduci... |
| efgs1b 19980 | Every extension sequence e... |
| efgsp1 19981 | If ` F ` is an extension s... |
| efgsres 19982 | An initial segment of an e... |
| efgsfo 19983 | For any word, there is a s... |
| efgredlema 19984 | The reduced word that form... |
| efgredlemf 19985 | Lemma for ~ efgredleme . ... |
| efgredlemg 19986 | Lemma for ~ efgred . (Con... |
| efgredleme 19987 | Lemma for ~ efgred . (Con... |
| efgredlemd 19988 | The reduced word that form... |
| efgredlemc 19989 | The reduced word that form... |
| efgredlemb 19990 | The reduced word that form... |
| efgredlem 19991 | The reduced word that form... |
| efgred 19992 | The reduced word that form... |
| efgrelexlema 19993 | If two words ` A , B ` are... |
| efgrelexlemb 19994 | If two words ` A , B ` are... |
| efgrelex 19995 | If two words ` A , B ` are... |
| efgredeu 19996 | There is a unique reduced ... |
| efgred2 19997 | Two extension sequences ha... |
| efgcpbllema 19998 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19999 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 20000 | Two extension sequences ha... |
| efgcpbl2 20001 | Two extension sequences ha... |
| frgpval 20002 | Value of the free group co... |
| frgpcpbl 20003 | Compatibility of the group... |
| frgp0 20004 | The free group is a group.... |
| frgpeccl 20005 | Closure of the quotient ma... |
| frgpgrp 20006 | The free group is a group.... |
| frgpadd 20007 | Addition in the free group... |
| frgpinv 20008 | The inverse of an element ... |
| frgpmhm 20009 | The "natural map" from wor... |
| vrgpfval 20010 | The canonical injection fr... |
| vrgpval 20011 | The value of the generatin... |
| vrgpf 20012 | The mapping from the index... |
| vrgpinv 20013 | The inverse of a generatin... |
| frgpuptf 20014 | Any assignment of the gene... |
| frgpuptinv 20015 | Any assignment of the gene... |
| frgpuplem 20016 | Any assignment of the gene... |
| frgpupf 20017 | Any assignment of the gene... |
| frgpupval 20018 | Any assignment of the gene... |
| frgpup1 20019 | Any assignment of the gene... |
| frgpup2 20020 | The evaluation map has the... |
| frgpup3lem 20021 | The evaluation map has the... |
| frgpup3 20022 | Universal property of the ... |
| 0frgp 20023 | The free group on zero gen... |
| isabl 20028 | The predicate "is an Abeli... |
| ablgrp 20029 | An Abelian group is a grou... |
| ablgrpd 20030 | An Abelian group is a grou... |
| ablcmn 20031 | An Abelian group is a comm... |
| ablcmnd 20032 | An Abelian group is a comm... |
| iscmn 20033 | The predicate "is a commut... |
| isabl2 20034 | The predicate "is an Abeli... |
| cmnpropd 20035 | If two structures have the... |
| ablpropd 20036 | If two structures have the... |
| ablprop 20037 | If two structures have the... |
| iscmnd 20038 | Properties that determine ... |
| isabld 20039 | Properties that determine ... |
| isabli 20040 | Properties that determine ... |
| cmnmnd 20041 | A commutative monoid is a ... |
| cmncom 20042 | A commutative monoid is co... |
| ablcom 20043 | An Abelian group operation... |
| cmn32 20044 | Commutative/associative la... |
| cmn4 20045 | Commutative/associative la... |
| cmn12 20046 | Commutative/associative la... |
| abl32 20047 | Commutative/associative la... |
| cmnmndd 20048 | A commutative monoid is a ... |
| cmnbascntr 20049 | The base set of a commutat... |
| rinvmod 20050 | Uniqueness of a right inve... |
| ablinvadd 20051 | The inverse of an Abelian ... |
| ablsub2inv 20052 | Abelian group subtraction ... |
| ablsubadd 20053 | Relationship between Abeli... |
| ablsub4 20054 | Commutative/associative su... |
| abladdsub4 20055 | Abelian group addition/sub... |
| abladdsub 20056 | Associative-type law for g... |
| ablsubadd23 20057 | Commutative/associative la... |
| ablsubaddsub 20058 | Double subtraction and add... |
| ablpncan2 20059 | Cancellation law for subtr... |
| ablpncan3 20060 | A cancellation law for Abe... |
| ablsubsub 20061 | Law for double subtraction... |
| ablsubsub4 20062 | Law for double subtraction... |
| ablpnpcan 20063 | Cancellation law for mixed... |
| ablnncan 20064 | Cancellation law for group... |
| ablsub32 20065 | Swap the second and third ... |
| ablnnncan 20066 | Cancellation law for group... |
| ablnnncan1 20067 | Cancellation law for group... |
| ablsubsub23 20068 | Swap subtrahend and result... |
| mulgnn0di 20069 | Group multiple of a sum, f... |
| mulgdi 20070 | Group multiple of a sum. ... |
| mulgmhm 20071 | The map from ` x ` to ` n ... |
| mulgghm 20072 | The map from ` x ` to ` n ... |
| mulgsubdi 20073 | Group multiple of a differ... |
| ghmfghm 20074 | The function fulfilling th... |
| ghmcmn 20075 | The image of a commutative... |
| ghmabl 20076 | The image of an abelian gr... |
| invghm 20077 | The inversion map is a gro... |
| eqgabl 20078 | Value of the subgroup cose... |
| qusecsub 20079 | Two subgroup cosets are eq... |
| subgabl 20080 | A subgroup of an abelian g... |
| subcmn 20081 | A submonoid of a commutati... |
| submcmn 20082 | A submonoid of a commutati... |
| submcmn2 20083 | A submonoid is commutative... |
| cntzcmn 20084 | The centralizer of any sub... |
| cntzcmnss 20085 | Any subset in a commutativ... |
| cntrcmnd 20086 | The center of a monoid is ... |
| cntrabl 20087 | The center of a group is a... |
| cntzspan 20088 | If the generators commute,... |
| cntzcmnf 20089 | Discharge the centralizer ... |
| ghmplusg 20090 | The pointwise sum of two l... |
| ablnsg 20091 | Every subgroup of an abeli... |
| odadd1 20092 | The order of a product in ... |
| odadd2 20093 | The order of a product in ... |
| odadd 20094 | The order of a product is ... |
| gex2abl 20095 | A group with exponent 2 (o... |
| gexexlem 20096 | Lemma for ~ gexex . (Cont... |
| gexex 20097 | In an abelian group with f... |
| torsubg 20098 | The set of all elements of... |
| oddvdssubg 20099 | The set of all elements wh... |
| lsmcomx 20100 | Subgroup sum commutes (ext... |
| ablcntzd 20101 | All subgroups in an abelia... |
| lsmcom 20102 | Subgroup sum commutes. (C... |
| lsmsubg2 20103 | The sum of two subgroups i... |
| lsm4 20104 | Commutative/associative la... |
| prdscmnd 20105 | The product of a family of... |
| prdsabld 20106 | The product of a family of... |
| pwscmn 20107 | The structure power on a c... |
| pwsabl 20108 | The structure power on an ... |
| qusabl 20109 | If ` Y ` is a subgroup of ... |
| abl1 20110 | The (smallest) structure r... |
| abln0 20111 | Abelian groups (and theref... |
| cnaddablx 20112 | The complex numbers are an... |
| cnaddabl 20113 | The complex numbers are an... |
| cnaddid 20114 | The group identity element... |
| cnaddinv 20115 | Value of the group inverse... |
| zaddablx 20116 | The integers are an Abelia... |
| frgpnabllem1 20117 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 20118 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 20119 | The free group on two or m... |
| imasabl 20120 | The image structure of an ... |
| iscyg 20123 | Definition of a cyclic gro... |
| iscyggen 20124 | The property of being a cy... |
| iscyggen2 20125 | The property of being a cy... |
| iscyg2 20126 | A cyclic group is a group ... |
| cyggeninv 20127 | The inverse of a cyclic ge... |
| cyggenod 20128 | An element is the generato... |
| cyggenod2 20129 | In an infinite cyclic grou... |
| iscyg3 20130 | Definition of a cyclic gro... |
| iscygd 20131 | Definition of a cyclic gro... |
| iscygodd 20132 | Show that a group with an ... |
| cycsubmcmn 20133 | The set of nonnegative int... |
| cyggrp 20134 | A cyclic group is a group.... |
| cygabl 20135 | A cyclic group is abelian.... |
| cygctb 20136 | A cyclic group is countabl... |
| 0cyg 20137 | The trivial group is cycli... |
| prmcyg 20138 | A group with prime order i... |
| lt6abl 20139 | A group with fewer than ` ... |
| ghmcyg 20140 | The image of a cyclic grou... |
| cyggex2 20141 | The exponent of a cyclic g... |
| cyggex 20142 | The exponent of a finite c... |
| cyggexb 20143 | A finite abelian group is ... |
| giccyg 20144 | Cyclicity is a group prope... |
| cycsubgcyg 20145 | The cyclic subgroup genera... |
| cycsubgcyg2 20146 | The cyclic subgroup genera... |
| gsumval3a 20147 | Value of the group sum ope... |
| gsumval3eu 20148 | The group sum as defined i... |
| gsumval3lem1 20149 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 20150 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 20151 | Value of the group sum ope... |
| gsumcllem 20152 | Lemma for ~ gsumcl and rel... |
| gsumzres 20153 | Extend a finite group sum ... |
| gsumzcl2 20154 | Closure of a finite group ... |
| gsumzcl 20155 | Closure of a finite group ... |
| gsumzf1o 20156 | Re-index a finite group su... |
| gsumres 20157 | Extend a finite group sum ... |
| gsumcl2 20158 | Closure of a finite group ... |
| gsumcl 20159 | Closure of a finite group ... |
| gsumf1o 20160 | Re-index a finite group su... |
| gsumreidx 20161 | Re-index a finite group su... |
| gsumzsubmcl 20162 | Closure of a group sum in ... |
| gsumsubmcl 20163 | Closure of a group sum in ... |
| gsumsubgcl 20164 | Closure of a group sum in ... |
| gsumzaddlem 20165 | The sum of two group sums.... |
| gsumzadd 20166 | The sum of two group sums.... |
| gsumadd 20167 | The sum of two group sums.... |
| gsummptfsadd 20168 | The sum of two group sums ... |
| gsummptfidmadd 20169 | The sum of two group sums ... |
| gsummptfidmadd2 20170 | The sum of two group sums ... |
| gsumzsplit 20171 | Split a group sum into two... |
| gsumsplit 20172 | Split a group sum into two... |
| gsumsplit2 20173 | Split a group sum into two... |
| gsummptfidmsplit 20174 | Split a group sum expresse... |
| gsummptfidmsplitres 20175 | Split a group sum expresse... |
| gsummptfzsplit 20176 | Split a group sum expresse... |
| gsummptfzsplitl 20177 | Split a group sum expresse... |
| gsumconst 20178 | Sum of a constant series. ... |
| gsumconstf 20179 | Sum of a constant series. ... |
| gsummptshft 20180 | Index shift of a finite gr... |
| gsumzmhm 20181 | Apply a group homomorphism... |
| gsummhm 20182 | Apply a group homomorphism... |
| gsummhm2 20183 | Apply a group homomorphism... |
| gsummptmhm 20184 | Apply a group homomorphism... |
| gsummulglem 20185 | Lemma for ~ gsummulg and ~... |
| gsummulg 20186 | Nonnegative multiple of a ... |
| gsummulgz 20187 | Integer multiple of a grou... |
| gsumzoppg 20188 | The opposite of a group su... |
| gsumzinv 20189 | Inverse of a group sum. (... |
| gsuminv 20190 | Inverse of a group sum. (... |
| gsummptfidminv 20191 | Inverse of a group sum exp... |
| gsumsub 20192 | The difference of two grou... |
| gsummptfssub 20193 | The difference of two grou... |
| gsummptfidmsub 20194 | The difference of two grou... |
| gsumsnfd 20195 | Group sum of a singleton, ... |
| gsumsnd 20196 | Group sum of a singleton, ... |
| gsumsnf 20197 | Group sum of a singleton, ... |
| gsumsn 20198 | Group sum of a singleton. ... |
| gsumpr 20199 | Group sum of a pair. (Con... |
| gsumzunsnd 20200 | Append an element to a fin... |
| gsumunsnfd 20201 | Append an element to a fin... |
| gsumunsnd 20202 | Append an element to a fin... |
| gsumunsnf 20203 | Append an element to a fin... |
| gsumunsn 20204 | Append an element to a fin... |
| gsumdifsnd 20205 | Extract a summand from a f... |
| gsumpt 20206 | Sum of a family that is no... |
| gsummptf1o 20207 | Re-index a finite group su... |
| gsummptun 20208 | Group sum of a disjoint un... |
| gsummpt1n0 20209 | If only one summand in a f... |
| gsummptif1n0 20210 | If only one summand in a f... |
| gsummptcl 20211 | Closure of a finite group ... |
| gsummptfif1o 20212 | Re-index a finite group su... |
| gsummptfzcl 20213 | Closure of a finite group ... |
| gsum2dlem1 20214 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 20215 | Lemma for ~ gsum2d . (Con... |
| gsum2d 20216 | Write a sum over a two-dim... |
| gsum2d2lem 20217 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 20218 | Write a group sum over a t... |
| gsumcom2 20219 | Two-dimensional commutatio... |
| gsumxp 20220 | Write a group sum over a c... |
| gsumcom 20221 | Commute the arguments of a... |
| gsumcom3 20222 | A commutative law for fini... |
| gsumcom3fi 20223 | A commutative law for fini... |
| gsumxp2 20224 | Write a group sum over a c... |
| prdsgsum 20225 | Finite commutative sums in... |
| pwsgsum 20226 | Finite commutative sums in... |
| fsfnn0gsumfsffz 20227 | Replacing a finitely suppo... |
| nn0gsumfz 20228 | Replacing a finitely suppo... |
| nn0gsumfz0 20229 | Replacing a finitely suppo... |
| gsummptnn0fz 20230 | A final group sum over a f... |
| gsummptnn0fzfv 20231 | A final group sum over a f... |
| telgsumfzslem 20232 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 20233 | Telescoping group sum rang... |
| telgsumfz 20234 | Telescoping group sum rang... |
| telgsumfz0s 20235 | Telescoping finite group s... |
| telgsumfz0 20236 | Telescoping finite group s... |
| telgsums 20237 | Telescoping finitely suppo... |
| telgsum 20238 | Telescoping finitely suppo... |
| reldmdprd 20243 | The domain of the internal... |
| dmdprd 20244 | The domain of definition o... |
| dmdprdd 20245 | Show that a given family i... |
| dprddomprc 20246 | A family of subgroups inde... |
| dprddomcld 20247 | If a family of subgroups i... |
| dprdval0prc 20248 | The internal direct produc... |
| dprdval 20249 | The value of the internal ... |
| eldprd 20250 | A class ` A ` is an intern... |
| dprdgrp 20251 | Reverse closure for the in... |
| dprdf 20252 | The function ` S ` is a fa... |
| dprdf2 20253 | The function ` S ` is a fa... |
| dprdcntz 20254 | The function ` S ` is a fa... |
| dprddisj 20255 | The function ` S ` is a fa... |
| dprdw 20256 | The property of being a fi... |
| dprdwd 20257 | A mapping being a finitely... |
| dprdff 20258 | A finitely supported funct... |
| dprdfcl 20259 | A finitely supported funct... |
| dprdffsupp 20260 | A finitely supported funct... |
| dprdfcntz 20261 | A function on the elements... |
| dprdssv 20262 | The internal direct produc... |
| dprdfid 20263 | A function mapping all but... |
| eldprdi 20264 | The domain of definition o... |
| dprdfinv 20265 | Take the inverse of a grou... |
| dprdfadd 20266 | Take the sum of group sums... |
| dprdfsub 20267 | Take the difference of gro... |
| dprdfeq0 20268 | The zero function is the o... |
| dprdf11 20269 | Two group sums over a dire... |
| dprdsubg 20270 | The internal direct produc... |
| dprdub 20271 | Each factor is a subset of... |
| dprdlub 20272 | The direct product is smal... |
| dprdspan 20273 | The direct product is the ... |
| dprdres 20274 | Restriction of a direct pr... |
| dprdss 20275 | Create a direct product by... |
| dprdz 20276 | A family consisting entire... |
| dprd0 20277 | The empty family is an int... |
| dprdf1o 20278 | Rearrange the index set of... |
| dprdf1 20279 | Rearrange the index set of... |
| subgdmdprd 20280 | A direct product in a subg... |
| subgdprd 20281 | A direct product in a subg... |
| dprdsn 20282 | A singleton family is an i... |
| dmdprdsplitlem 20283 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 20284 | The function ` S ` is a fa... |
| dprddisj2 20285 | The function ` S ` is a fa... |
| dprd2dlem2 20286 | The direct product of a co... |
| dprd2dlem1 20287 | The direct product of a co... |
| dprd2da 20288 | The direct product of a co... |
| dprd2db 20289 | The direct product of a co... |
| dprd2d2 20290 | The direct product of a co... |
| dmdprdsplit2lem 20291 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 20292 | The direct product splits ... |
| dmdprdsplit 20293 | The direct product splits ... |
| dprdsplit 20294 | The direct product is the ... |
| dmdprdpr 20295 | A singleton family is an i... |
| dprdpr 20296 | A singleton family is an i... |
| dpjlem 20297 | Lemma for theorems about d... |
| dpjcntz 20298 | The two subgroups that app... |
| dpjdisj 20299 | The two subgroups that app... |
| dpjlsm 20300 | The two subgroups that app... |
| dpjfval 20301 | Value of the direct produc... |
| dpjval 20302 | Value of the direct produc... |
| dpjf 20303 | The ` X ` -th index projec... |
| dpjidcl 20304 | The key property of projec... |
| dpjeq 20305 | Decompose a group sum into... |
| dpjid 20306 | The key property of projec... |
| dpjlid 20307 | The ` X ` -th index projec... |
| dpjrid 20308 | The ` Y ` -th index projec... |
| dpjghm 20309 | The direct product is the ... |
| dpjghm2 20310 | The direct product is the ... |
| ablfacrplem 20311 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20312 | A finite abelian group who... |
| ablfacrp2 20313 | The factors ` K , L ` of ~... |
| ablfac1lem 20314 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20315 | The factors of ~ ablfac1b ... |
| ablfac1b 20316 | Any abelian group is the d... |
| ablfac1c 20317 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20318 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20319 | The factorization of ~ abl... |
| pgpfac1lem1 20320 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20321 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20322 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20323 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20324 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20325 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20326 | Factorization of a finite ... |
| pgpfaclem1 20327 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20328 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20329 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20330 | Full factorization of a fi... |
| ablfaclem1 20331 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20332 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20333 | Lemma for ~ ablfac . (Con... |
| ablfac 20334 | The Fundamental Theorem of... |
| ablfac2 20335 | Choose generators for each... |
| issimpg 20338 | The predicate "is a simple... |
| issimpgd 20339 | Deduce a simple group from... |
| simpggrp 20340 | A simple group is a group.... |
| simpggrpd 20341 | A simple group is a group.... |
| simpg2nsg 20342 | A simple group has two nor... |
| trivnsimpgd 20343 | Trivial groups are not sim... |
| simpgntrivd 20344 | Simple groups are nontrivi... |
| simpgnideld 20345 | A simple group contains a ... |
| simpgnsgd 20346 | The only normal subgroups ... |
| simpgnsgeqd 20347 | A normal subgroup of a sim... |
| 2nsgsimpgd 20348 | If any normal subgroup of ... |
| simpgnsgbid 20349 | A nontrivial group is simp... |
| ablsimpnosubgd 20350 | A subgroup of an abelian s... |
| ablsimpg1gend 20351 | An abelian simple group is... |
| ablsimpgcygd 20352 | An abelian simple group is... |
| ablsimpgfindlem1 20353 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20354 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20355 | Relationship between the o... |
| ablsimpgfind 20356 | An abelian simple group is... |
| fincygsubgd 20357 | The subgroup referenced in... |
| fincygsubgodd 20358 | Calculate the order of a s... |
| fincygsubgodexd 20359 | A finite cyclic group has ... |
| prmgrpsimpgd 20360 | A group of prime order is ... |
| ablsimpgprmd 20361 | An abelian simple group ha... |
| ablsimpgd 20362 | An abelian group is simple... |
| isomnd 20367 | A (left) ordered monoid is... |
| isogrp 20368 | A (left-)ordered group is ... |
| ogrpgrp 20369 | A left-ordered group is a ... |
| omndmnd 20370 | A left-ordered monoid is a... |
| omndtos 20371 | A left-ordered monoid is a... |
| omndadd 20372 | In an ordered monoid, the ... |
| omndaddr 20373 | In a right ordered monoid,... |
| omndadd2d 20374 | In a commutative left orde... |
| omndadd2rd 20375 | In a left- and right- orde... |
| submomnd 20376 | A submonoid of an ordered ... |
| omndmul2 20377 | In an ordered monoid, the ... |
| omndmul3 20378 | In an ordered monoid, the ... |
| omndmul 20379 | In a commutative ordered m... |
| ogrpinv0le 20380 | In an ordered group, the o... |
| ogrpsub 20381 | In an ordered group, the o... |
| ogrpaddlt 20382 | In an ordered group, stric... |
| ogrpaddltbi 20383 | In a right ordered group, ... |
| ogrpaddltrd 20384 | In a right ordered group, ... |
| ogrpaddltrbid 20385 | In a right ordered group, ... |
| ogrpsublt 20386 | In an ordered group, stric... |
| ogrpinv0lt 20387 | In an ordered group, the o... |
| ogrpinvlt 20388 | In an ordered group, the o... |
| gsumle 20389 | A finite sum in an ordered... |
| fnmgp 20392 | The multiplicative group o... |
| mgpval 20393 | Value of the multiplicatio... |
| mgpplusg 20394 | Value of the group operati... |
| mgpbas 20395 | Base set of the multiplica... |
| mgpsca 20396 | The multiplication monoid ... |
| mgptset 20397 | Topology component of the ... |
| mgptopn 20398 | Topology of the multiplica... |
| mgpds 20399 | Distance function of the m... |
| mgpress 20400 | Subgroup commutes with the... |
| prdsmgp 20401 | The multiplicative monoid ... |
| elmgplsm 20402 | Membership in a product of... |
| elmgplsmd 20403 | Membership in a product of... |
| isrng 20406 | The predicate "is a non-un... |
| rngabl 20407 | A non-unital ring is an (a... |
| rngmgp 20408 | A non-unital ring is a sem... |
| rngmgpf 20409 | Restricted functionality o... |
| rnggrp 20410 | A non-unital ring is a (ad... |
| rngass 20411 | Associative law for the mu... |
| rngdi 20412 | Distributive law for the m... |
| rngdir 20413 | Distributive law for the m... |
| rngacl 20414 | Closure of the addition op... |
| rng0cl 20415 | The zero element of a non-... |
| rngcl 20416 | Closure of the multiplicat... |
| rnglz 20417 | The zero of a non-unital r... |
| rngrz 20418 | The zero of a non-unital r... |
| rngmneg1 20419 | Negation of a product in a... |
| rngmneg2 20420 | Negation of a product in a... |
| rngm2neg 20421 | Double negation of a produ... |
| rngansg 20422 | Every additive subgroup of... |
| rngsubdi 20423 | Ring multiplication distri... |
| rngsubdir 20424 | Ring multiplication distri... |
| isrngd 20425 | Properties that determine ... |
| rngpropd 20426 | If two structures have the... |
| prdsmulrngcl 20427 | Closure of the multiplicat... |
| prdsrngd 20428 | A product of non-unital ri... |
| imasrng 20429 | The image structure of a n... |
| imasrngf1 20430 | The image of a non-unital ... |
| xpsrngd 20431 | A product of two non-unita... |
| qusrng 20432 | The quotient structure of ... |
| rng1zrlem 20433 | Lemma for ~ rng1zr and ~ s... |
| rng1zr 20434 | The only ring with a base ... |
| rngen1zr 20435 | The only ring with one ele... |
| rngen1zr0 20436 | The only ring with one ele... |
| ringidval 20439 | The value of the unity ele... |
| dfur2 20440 | The multiplicative identit... |
| ringurd 20441 | Deduce the unity element o... |
| issrg 20444 | The predicate "is a semiri... |
| srgcmn 20445 | A semiring is a commutativ... |
| srgmnd 20446 | A semiring is a monoid. (... |
| srgmgp 20447 | A semiring is a monoid und... |
| srgdilem 20448 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20449 | Closure of the multiplicat... |
| srgass 20450 | Associative law for the mu... |
| srgideu 20451 | The unity element of a sem... |
| srgfcl 20452 | Functionality of the multi... |
| srgdi 20453 | Distributive law for the m... |
| srgdir 20454 | Distributive law for the m... |
| srgidcl 20455 | The unity element of a sem... |
| srg0cl 20456 | The zero element of a semi... |
| srgidmlem 20457 | Lemma for ~ srglidm and ~ ... |
| srglidm 20458 | The unity element of a sem... |
| srgridm 20459 | The unity element of a sem... |
| issrgid 20460 | Properties showing that an... |
| srgacl 20461 | Closure of the addition op... |
| srgcom 20462 | Commutativity of the addit... |
| srgrz 20463 | The zero of a semiring is ... |
| srglz 20464 | The zero of a semiring is ... |
| srgisid 20465 | In a semiring, the only le... |
| o2timesd 20466 | An element of a ring-like ... |
| rglcom4d 20467 | Restricted commutativity o... |
| srgo2times 20468 | A semiring element plus it... |
| srgcom4lem 20469 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20470 | Restricted commutativity o... |
| srg1zr 20471 | The only semiring with a b... |
| srgen1zr0 20472 | The only semiring with one... |
| srgmulgass 20473 | An associative property be... |
| srgpcomp 20474 | If two elements of a semir... |
| srgpcompp 20475 | If two elements of a semir... |
| srgpcomppsc 20476 | If two elements of a semir... |
| srglmhm 20477 | Left-multiplication in a s... |
| srgrmhm 20478 | Right-multiplication in a ... |
| srgsummulcr 20479 | A finite semiring sum mult... |
| sgsummulcl 20480 | A finite semiring sum mult... |
| srg1expzeq1 20481 | The exponentiation (by a n... |
| srgbinomlem1 20482 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20483 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20484 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20485 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20486 | Lemma for ~ srgbinom . In... |
| srgbinom 20487 | The binomial theorem for c... |
| csrgbinom 20488 | The binomial theorem for c... |
| isring 20493 | The predicate "is a (unita... |
| ringgrp 20494 | A ring is a group. (Contr... |
| ringmgp 20495 | A ring is a monoid under m... |
| iscrng 20496 | A commutative ring is a ri... |
| crngmgp 20497 | A commutative ring's multi... |
| ringbn0 20498 | The base set of a ring is ... |
| ringgrpd 20499 | A ring is a group. (Contr... |
| ringmnd 20500 | A ring is a monoid under a... |
| ringmgm 20501 | A ring is a magma. (Contr... |
| crngring 20502 | A commutative ring is a ri... |
| crngringd 20503 | A commutative ring is a ri... |
| crnggrpd 20504 | A commutative ring is a gr... |
| mgpf 20505 | Restricted functionality o... |
| ringdilem 20506 | Properties of a unital rin... |
| ringcl 20507 | Closure of the multiplicat... |
| crngcom 20508 | A commutative ring's multi... |
| iscrng2 20509 | A commutative ring is a ri... |
| ringass 20510 | Associative law for multip... |
| ringideu 20511 | The unity element of a rin... |
| crngcomd 20512 | Multiplication is commutat... |
| crngbascntr 20513 | The base set of a commutat... |
| ringcld 20514 | Closure of the multiplicat... |
| ringassd 20515 | Associative law for multip... |
| crng12d 20516 | Commutative/associative la... |
| crng32d 20517 | Commutative/associative la... |
| crng4 20518 | Commutative/associative la... |
| ringdi 20519 | Distributive law for the m... |
| ringdir 20520 | Distributive law for the m... |
| ringdid 20521 | Distributive law for the m... |
| ringdird 20522 | Distributive law for the m... |
| ringdi22 20523 | Expand the product of two ... |
| ringidcl 20524 | The unity element of a rin... |
| ringidcld 20525 | The unity element of a rin... |
| ring0cl 20526 | The zero element of a ring... |
| ringidmlem 20527 | Lemma for ~ ringlidm and ~... |
| ringlidm 20528 | The unity element of a rin... |
| ringridm 20529 | The unity element of a rin... |
| isringid 20530 | Properties showing that an... |
| ringlidmd 20531 | The unity element of a rin... |
| ringridmd 20532 | The unity element of a rin... |
| ringid 20533 | The multiplication operati... |
| ringo2times 20534 | A ring element plus itself... |
| ringadd2 20535 | A ring element plus itself... |
| ringidss 20536 | A subset of the multiplica... |
| ringacl 20537 | Closure of the addition op... |
| ringcomlem 20538 | Lemma for ~ ringcom . Thi... |
| ringcom 20539 | Commutativity of the addit... |
| ringabl 20540 | A ring is an Abelian group... |
| ringcmn 20541 | A ring is a commutative mo... |
| ringabld 20542 | A ring is an Abelian group... |
| ringcmnd 20543 | A ring is a commutative mo... |
| ringrng 20544 | A unital ring is a non-uni... |
| ringssrng 20545 | The unital rings are non-u... |
| isringrng 20546 | The predicate "is a unital... |
| dfring2 20547 | The predicate "is a unital... |
| dfring3 20548 | The predicate "is a (unita... |
| ringpropd 20549 | If two structures have the... |
| crngpropd 20550 | If two structures have the... |
| ringprop 20551 | If two structures have the... |
| isringd 20552 | Properties that determine ... |
| iscrngd 20553 | Properties that determine ... |
| ringlz 20554 | The zero of a unital ring ... |
| ringrz 20555 | The zero of a unital ring ... |
| ringlzd 20556 | The zero of a unital ring ... |
| ringrzd 20557 | The zero of a unital ring ... |
| ringsrg 20558 | Any ring is also a semirin... |
| ring1eq0 20559 | If one and zero are equal,... |
| ring1ne0 20560 | If a ring has at least two... |
| ringinvnz1ne0 20561 | In a unital ring, a left i... |
| ringinvnzdiv 20562 | In a unital ring, a left i... |
| ringnegl 20563 | Negation in a ring is the ... |
| ringnegr 20564 | Negation in a ring is the ... |
| ringmneg1 20565 | Negation of a product in a... |
| ringmneg2 20566 | Negation of a product in a... |
| ringm2neg 20567 | Double negation of a produ... |
| ringsubdi 20568 | Ring multiplication distri... |
| ringsubdir 20569 | Ring multiplication distri... |
| mulgass2 20570 | An associative property be... |
| ring1 20571 | The (smallest) structure r... |
| ringn0 20572 | Rings exist. (Contributed... |
| ringlghm 20573 | Left-multiplication in a r... |
| ringrghm 20574 | Right-multiplication in a ... |
| gsummulc1 20575 | A finite ring sum multipli... |
| gsummulc2 20576 | A finite ring sum multipli... |
| gsummgp0 20577 | If one factor in a finite ... |
| gsumdixp 20578 | Distribute a binary produc... |
| prdsmulrcl 20579 | A structure product of rin... |
| prdsringd 20580 | A product of rings is a ri... |
| prdscrngd 20581 | A product of commutative r... |
| prds1 20582 | Value of the ring unity in... |
| pwsring 20583 | A structure power of a rin... |
| pws1 20584 | Value of the ring unity in... |
| pwscrng 20585 | A structure power of a com... |
| pwsmgp 20586 | The multiplicative group o... |
| pwspjmhmmgpd 20587 | The projection given by ~ ... |
| pwsexpg 20588 | Value of a group exponenti... |
| pwsgprod 20589 | Finite products in a power... |
| imasring 20590 | The image structure of a r... |
| imasringf1 20591 | The image of a ring under ... |
| xpsringd 20592 | A product of two rings is ... |
| xpsring1d 20593 | The multiplicative identit... |
| qusring2 20594 | The quotient structure of ... |
| crngbinom 20595 | The binomial theorem for c... |
| opprval 20598 | Value of the opposite ring... |
| opprmulfval 20599 | Value of the multiplicatio... |
| opprmul 20600 | Value of the multiplicatio... |
| crngoppr 20601 | In a commutative ring, the... |
| opprlem 20602 | Lemma for ~ opprbas and ~ ... |
| opprbas 20603 | Base set of an opposite ri... |
| oppradd 20604 | Addition operation of an o... |
| opprrng 20605 | An opposite non-unital rin... |
| opprrngb 20606 | A class is a non-unital ri... |
| opprring 20607 | An opposite ring is a ring... |
| opprringb 20608 | Bidirectional form of ~ op... |
| oppr0 20609 | Additive identity of an op... |
| oppr1 20610 | Multiplicative identity of... |
| opprneg 20611 | The negative function in a... |
| opprsubg 20612 | Being a subgroup is a symm... |
| mulgass3 20613 | An associative property be... |
| reldvdsr 20620 | The divides relation is a ... |
| dvdsrval 20621 | Value of the divides relat... |
| dvdsr 20622 | Value of the divides relat... |
| dvdsr2 20623 | Value of the divides relat... |
| dvdsrmul 20624 | A left-multiple of ` X ` i... |
| dvdsrcl 20625 | Closure of a dividing elem... |
| dvdsrcl2 20626 | Closure of a dividing elem... |
| dvdsrid 20627 | An element in a (unital) r... |
| dvdsrtr 20628 | Divisibility is transitive... |
| dvdsrmul1 20629 | The divisibility relation ... |
| dvdsrneg 20630 | An element divides its neg... |
| dvdsr01 20631 | In a ring, zero is divisib... |
| dvdsr02 20632 | Only zero is divisible by ... |
| isunit 20633 | Property of being a unit o... |
| 1unit 20634 | The multiplicative identit... |
| unitcl 20635 | A unit is an element of th... |
| unitss 20636 | The set of units is contai... |
| opprunit 20637 | Being a unit is a symmetri... |
| crngunit 20638 | Property of being a unit i... |
| dvdsunit 20639 | A divisor of a unit is a u... |
| unitmulcl 20640 | The product of units is a ... |
| unitmulclb 20641 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20642 | The base set of the group ... |
| unitgrp 20643 | The group of units is a gr... |
| unitabl 20644 | The group of units of a co... |
| unitgrpid 20645 | The identity of the group ... |
| unitsubm 20646 | The group of units is a su... |
| invrfval 20649 | Multiplicative inverse fun... |
| unitinvcl 20650 | The inverse of a unit exis... |
| unitinvinv 20651 | The inverse of the inverse... |
| ringinvcl 20652 | The inverse of a unit is a... |
| unitlinv 20653 | A unit times its inverse i... |
| unitrinv 20654 | A unit times its inverse i... |
| 1rinv 20655 | The inverse of the ring un... |
| 0unit 20656 | The additive identity is a... |
| unitnegcl 20657 | The negative of a unit is ... |
| ringunitnzdiv 20658 | In a unitary ring, a unit ... |
| ring1nzdiv 20659 | In a unitary ring, the rin... |
| dvrfval 20662 | Division operation in a ri... |
| dvrval 20663 | Division operation in a ri... |
| dvrcl 20664 | Closure of division operat... |
| unitdvcl 20665 | The units are closed under... |
| dvrid 20666 | A ring element divided by ... |
| dvr1 20667 | A ring element divided by ... |
| dvrass 20668 | An associative law for div... |
| dvrcan1 20669 | A cancellation law for div... |
| dvrcan3 20670 | A cancellation law for div... |
| dvreq1 20671 | Equality in terms of ratio... |
| dvrdir 20672 | Distributive law for the d... |
| rdivmuldivd 20673 | Multiplication of two rati... |
| ringinvdv 20674 | Write the inverse function... |
| rngidpropd 20675 | The ring unity depends onl... |
| dvdsrpropd 20676 | The divisibility relation ... |
| unitpropd 20677 | The set of units depends o... |
| invrpropd 20678 | The ring inverse function ... |
| isirred 20679 | An irreducible element of ... |
| isnirred 20680 | The property of being a no... |
| isirred2 20681 | Expand out the class diffe... |
| opprirred 20682 | Irreducibility is symmetri... |
| irredn0 20683 | The additive identity is n... |
| irredcl 20684 | An irreducible element is ... |
| irrednu 20685 | An irreducible element is ... |
| irredn1 20686 | The multiplicative identit... |
| irredrmul 20687 | The product of an irreduci... |
| irredlmul 20688 | The product of a unit and ... |
| irredmul 20689 | If product of two elements... |
| irredneg 20690 | The negative of an irreduc... |
| irrednegb 20691 | An element is irreducible ... |
| rnghmrcl 20698 | Reverse closure of a non-u... |
| rnghmfn 20699 | The mapping of two non-uni... |
| rnghmval 20700 | The set of the non-unital ... |
| isrnghm 20701 | A function is a non-unital... |
| isrnghmmul 20702 | A function is a non-unital... |
| rnghmmgmhm 20703 | A non-unital ring homomorp... |
| rnghmval2 20704 | The non-unital ring homomo... |
| isrngim 20705 | An isomorphism of non-unit... |
| rngimrcl 20706 | Reverse closure for an iso... |
| rnghmghm 20707 | A non-unital ring homomorp... |
| rnghmf 20708 | A ring homomorphism is a f... |
| rnghmmul 20709 | A homomorphism of non-unit... |
| isrnghm2d 20710 | Demonstration of non-unita... |
| isrnghmd 20711 | Demonstration of non-unita... |
| rnghmf1o 20712 | A non-unital ring homomorp... |
| isrngim2 20713 | An isomorphism of non-unit... |
| rngimf1o 20714 | An isomorphism of non-unit... |
| rngimrnghm 20715 | An isomorphism of non-unit... |
| rngimcnv 20716 | The converse of an isomorp... |
| rnghmco 20717 | The composition of non-uni... |
| idrnghm 20718 | The identity homomorphism ... |
| c0mgm 20719 | The constant mapping to ze... |
| c0mhm 20720 | The constant mapping to ze... |
| c0ghm 20721 | The constant mapping to ze... |
| c0snmgmhm 20722 | The constant mapping to ze... |
| c0snmhm 20723 | The constant mapping to ze... |
| c0snghm 20724 | The constant mapping to ze... |
| rngisomfv1 20725 | If there is a non-unital r... |
| rngisom1 20726 | If there is a non-unital r... |
| rngisomring 20727 | If there is a non-unital r... |
| rngisomring1 20728 | If there is a non-unital r... |
| dfrhm2 20734 | The property of a ring hom... |
| rhmval0 20735 | The set of ring homomorphi... |
| isrhm0 20736 | The predicate "is a ring h... |
| rhmrcl1 20737 | Reverse closure of a ring ... |
| rhmrcl2 20738 | Reverse closure of a ring ... |
| isrhm 20739 | A function is a ring homom... |
| rhmmhm 20740 | A ring homomorphism is a h... |
| rhmisrnghm 20741 | Each unital ring homomorph... |
| rimrcl 20742 | Reverse closure for an iso... |
| isrim0 20743 | A ring isomorphism is a ho... |
| rhmghm 20744 | A ring homomorphism is an ... |
| rhmf 20745 | A ring homomorphism is a f... |
| rhmcl 20746 | Closure law for a ring hom... |
| rimcnv 20747 | The converse of a ring iso... |
| rhmadd 20748 | Ring homomorphisms preserv... |
| rhmsub 20749 | Ring homomorphisms preserv... |
| rhmmul 20750 | A homomorphism of rings pr... |
| isrhm2d 20751 | Demonstration of ring homo... |
| isrhmd 20752 | Demonstration of ring homo... |
| rhm0 20753 | A ring homomorphism preser... |
| rhm1 20754 | Ring homomorphisms are req... |
| idrhm 20755 | The identity homomorphism ... |
| crngrhmfo 20756 | The image of a surjective ... |
| rhmf1o 20757 | A ring homomorphism is bij... |
| isrim 20758 | An isomorphism of rings is... |
| rimf1o 20759 | An isomorphism of rings is... |
| rimval 20760 | The set of ring isomorphis... |
| rimrhm 20761 | A ring isomorphism is a ho... |
| rimrcl1 20762 | Reverse closure of a ring ... |
| rimrcl2 20763 | Reverse closure of a ring ... |
| rimgim 20764 | An isomorphism of rings is... |
| rimisrngim 20765 | Each unital ring isomorphi... |
| rhmfn 20766 | The mapping of two rings t... |
| rimfn 20767 | The mapping of two rings t... |
| rhmval 20768 | The ring homomorphisms bet... |
| rhmco 20769 | The composition of ring ho... |
| rhmkerinj 20770 | A ring homomorphism is inj... |
| pwsco1rhm 20771 | Right composition with a f... |
| pwsco2rhm 20772 | Left composition with a ri... |
| ricrel 20774 | The domain of the ring iso... |
| brric 20775 | The relation "is isomorphi... |
| brrici 20776 | Prove isomorphic by an exp... |
| rimco 20777 | The composition of ring is... |
| ricref 20778 | Ring isomorphism is reflex... |
| riclcl 20779 | Ring isomorphism implies t... |
| ricrcl 20780 | Ring isomorphism implies t... |
| ricsym 20781 | Ring isomorphism is symmet... |
| rictr 20782 | Ring isomorphism is transi... |
| isbrric2 20783 | The relation "is isomorphi... |
| brric2 20784 | The ring isomorphism relat... |
| ricgic 20785 | If two rings are (ring) is... |
| ricer 20786 | Ring isomorphism is an equ... |
| dfric2 20787 | Alternate definition of th... |
| rhmdvdsr 20788 | A ring homomorphism preser... |
| rhmopp 20789 | A ring homomorphism is als... |
| elrhmunit 20790 | Ring homomorphisms preserv... |
| rhmunitinv 20791 | Ring homomorphisms preserv... |
| isnzr 20794 | Property of a nonzero ring... |
| nzrnz 20795 | One and zero are different... |
| nzrring 20796 | A nonzero ring is a ring. ... |
| nzrringOLD 20797 | Obsolete version of ~ nzrr... |
| isnzr2 20798 | Equivalent characterizatio... |
| drnglidl1ne0 20799 | In a nonzero ring, the zer... |
| isnzr2hash 20800 | Equivalent characterizatio... |
| nzrpropd 20801 | If two structures have the... |
| opprnzrb 20802 | The opposite of a nonzero ... |
| opprnzr 20803 | The opposite of a nonzero ... |
| ringelnzr 20804 | A ring is nonzero if it ha... |
| nzrunit 20805 | A unit is nonzero in any n... |
| 0ringnnzr 20806 | A ring is a zero ring iff ... |
| 0ring 20807 | If a ring has only one ele... |
| 0ringdif 20808 | A zero ring is a ring whic... |
| 0ringbas 20809 | The base set of a zero rin... |
| 0ring01eq 20810 | In a ring with only one el... |
| 01eq0ring 20811 | If the zero and the identi... |
| 01eq0ringOLD 20812 | Obsolete version of ~ 01eq... |
| 0ring01eqbi2 20813 | In a ring, ` 0 = 1 ` iff t... |
| 0ring01eqbi 20814 | In a unital ring the zero ... |
| 0ring1eq0 20815 | In a zero ring, a ring whi... |
| c0rhm 20816 | The constant mapping to ze... |
| c0rnghm 20817 | The constant mapping to ze... |
| zrrnghm 20818 | The constant mapping to ze... |
| nrhmzr 20819 | There is no ring homomorph... |
| islring 20822 | The predicate "is a local ... |
| lringnzr 20823 | A local ring is a nonzero ... |
| lringring 20824 | A local ring is a ring. (... |
| lringnz 20825 | A local ring is a nonzero ... |
| lringuplu 20826 | If the sum of two elements... |
| issubrng 20829 | The subring of non-unital ... |
| subrngss 20830 | A subring is a subset. (C... |
| subrngid 20831 | Every non-unital ring is a... |
| subrngrng 20832 | A subring is a non-unital ... |
| subrngrcl 20833 | Reverse closure for a subr... |
| subrngsubg 20834 | A subring is a subgroup. ... |
| subrngringnsg 20835 | A subring is a normal subg... |
| subrngbas 20836 | Base set of a subring stru... |
| subrng0 20837 | A subring always has the s... |
| subrngacl 20838 | A subring is closed under ... |
| subrngmcl 20839 | A subring is closed under ... |
| issubrng2 20840 | Characterize the subrings ... |
| opprsubrng 20841 | Being a subring is a symme... |
| subrngint 20842 | The intersection of a none... |
| subrngin 20843 | The intersection of two su... |
| subrngmre 20844 | The subrings of a non-unit... |
| subsubrng 20845 | A subring of a subring is ... |
| subsubrng2 20846 | The set of subrings of a s... |
| rhmimasubrnglem 20847 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20848 | The homomorphic image of a... |
| cntzsubrng 20849 | Centralizers in a non-unit... |
| subrngpropd 20850 | If two structures have the... |
| issubrg 20853 | The subring predicate. (C... |
| subrgss 20854 | A subring is a subset. (C... |
| subrgid 20855 | Every ring is a subring of... |
| subrgring 20856 | A subring is a ring. (Con... |
| subrgcrng 20857 | A subring of a commutative... |
| subrgrcl 20858 | Reverse closure for a subr... |
| subrgsubg 20859 | A subring is a subgroup. ... |
| subrgsubrng 20860 | A subring of a unital ring... |
| subrg0 20861 | A subring always has the s... |
| subrg1cl 20862 | A subring contains the mul... |
| subrgbas 20863 | Base set of a subring stru... |
| subrg1 20864 | A subring always has the s... |
| subrgacl 20865 | A subring is closed under ... |
| subrgmcl 20866 | A subring is closed under ... |
| subrgsubm 20867 | A subring is a submonoid o... |
| subrgdvds 20868 | If an element divides anot... |
| subrguss 20869 | A unit of a subring is a u... |
| subrginv 20870 | A subring always has the s... |
| subrgdv 20871 | A subring always has the s... |
| subrgunit 20872 | An element of a ring is a ... |
| subrgugrp 20873 | The units of a subring for... |
| issubrg2 20874 | Characterize the subrings ... |
| opprsubrg 20875 | Being a subring is a symme... |
| subrgnzr 20876 | A subring of a nonzero rin... |
| subrgint 20877 | The intersection of a none... |
| subrgin 20878 | The intersection of two su... |
| subrgmre 20879 | The subrings of a ring are... |
| subsubrg 20880 | A subring of a subring is ... |
| subsubrg2 20881 | The set of subrings of a s... |
| issubrg3 20882 | A subring is an additive s... |
| resrhm 20883 | Restriction of a ring homo... |
| resrhm2b 20884 | Restriction of the codomai... |
| rhmeql 20885 | The equalizer of two ring ... |
| rhmima 20886 | The homomorphic image of a... |
| rnrhmsubrg 20887 | The range of a ring homomo... |
| cntzsubr 20888 | Centralizers in a ring are... |
| pwsdiagrhm 20889 | Diagonal homomorphism into... |
| subrgpropd 20890 | If two structures have the... |
| rhmpropd 20891 | Ring homomorphism depends ... |
| rgspnval 20894 | Value of the ring-span of ... |
| rgspncl 20895 | The ring-span of a set is ... |
| rgspnssid 20896 | The ring-span of a set con... |
| rgspnmin 20897 | The ring-span is contained... |
| rngcval 20900 | Value of the category of n... |
| rnghmresfn 20901 | The class of non-unital ri... |
| rnghmresel 20902 | An element of the non-unit... |
| rngcbas 20903 | Set of objects of the cate... |
| rngchomfval 20904 | Set of arrows of the categ... |
| rngchom 20905 | Set of arrows of the categ... |
| elrngchom 20906 | A morphism of non-unital r... |
| rngchomfeqhom 20907 | The functionalized Hom-set... |
| rngccofval 20908 | Composition in the categor... |
| rngcco 20909 | Composition in the categor... |
| dfrngc2 20910 | Alternate definition of th... |
| rnghmsscmap2 20911 | The non-unital ring homomo... |
| rnghmsscmap 20912 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20913 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20914 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20915 | The non-unital ring homomo... |
| rngccat 20916 | The category of non-unital... |
| rngcid 20917 | The identity arrow in the ... |
| rngcsect 20918 | A section in the category ... |
| rngcinv 20919 | An inverse in the category... |
| rngciso 20920 | An isomorphism in the cate... |
| rngcifuestrc 20921 | The "inclusion functor" fr... |
| funcrngcsetc 20922 | The "natural forgetful fun... |
| funcrngcsetcALT 20923 | Alternate proof of ~ funcr... |
| zrinitorngc 20924 | The zero ring is an initia... |
| zrtermorngc 20925 | The zero ring is a termina... |
| zrzeroorngc 20926 | The zero ring is a zero ob... |
| ringcval 20929 | Value of the category of u... |
| rhmresfn 20930 | The class of unital ring h... |
| rhmresel 20931 | An element of the unital r... |
| ringcbas 20932 | Set of objects of the cate... |
| ringchomfval 20933 | Set of arrows of the categ... |
| ringchom 20934 | Set of arrows of the categ... |
| elringchom 20935 | A morphism of unital rings... |
| ringchomfeqhom 20936 | The functionalized Hom-set... |
| ringccofval 20937 | Composition in the categor... |
| ringcco 20938 | Composition in the categor... |
| dfringc2 20939 | Alternate definition of th... |
| rhmsscmap2 20940 | The unital ring homomorphi... |
| rhmsscmap 20941 | The unital ring homomorphi... |
| rhmsubcsetclem1 20942 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20943 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20944 | The unital ring homomorphi... |
| ringccat 20945 | The category of unital rin... |
| ringcid 20946 | The identity arrow in the ... |
| rhmsscrnghm 20947 | The unital ring homomorphi... |
| rhmsubcrngclem1 20948 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20949 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20950 | The unital ring homomorphi... |
| rngcresringcat 20951 | The restriction of the cat... |
| ringcsect 20952 | A section in the category ... |
| ringcinv 20953 | An inverse in the category... |
| ringciso 20954 | An isomorphism in the cate... |
| ringcbasbas 20955 | An element of the base set... |
| funcringcsetc 20956 | The "natural forgetful fun... |
| zrtermoringc 20957 | The zero ring is a termina... |
| zrninitoringc 20958 | The zero ring is not an in... |
| srhmsubclem1 20959 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20960 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20961 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20962 | According to ~ df-subc , t... |
| sringcat 20963 | The restriction of the cat... |
| crhmsubc 20964 | According to ~ df-subc , t... |
| cringcat 20965 | The restriction of the cat... |
| rngcrescrhm 20966 | The category of non-unital... |
| rhmsubclem1 20967 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20968 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20969 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20970 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20971 | According to ~ df-subc , t... |
| rhmsubccat 20972 | The restriction of the cat... |
| rrgval 20979 | Value of the set or left-r... |
| isrrg 20980 | Membership in the set of l... |
| rrgeq0i 20981 | Property of a left-regular... |
| rrgeq0 20982 | Left-multiplication by a l... |
| rrgsupp 20983 | Left multiplication by a l... |
| rrgss 20984 | Left-regular elements are ... |
| unitrrg 20985 | Units are regular elements... |
| rrgnz 20986 | In a nonzero ring, the zer... |
| isdomn 20987 | Expand definition of a dom... |
| domnnzr 20988 | A domain is a nonzero ring... |
| domnring 20989 | A domain is a ring. (Cont... |
| domneq0 20990 | In a domain, a product is ... |
| domnmuln0 20991 | In a domain, a product of ... |
| isdomn5 20992 | The equivalence between th... |
| isdomn2 20993 | A ring is a domain iff all... |
| domnrrg 20994 | In a domain, a nonzero ele... |
| isdomn6 20995 | A ring is a domain iff the... |
| isdomn3 20996 | Nonzero elements form a mu... |
| isdomn4 20997 | A ring is a domain iff it ... |
| opprdomnb 20998 | A class is a domain if and... |
| opprdomn 20999 | The opposite of a domain i... |
| isdomn4r 21000 | A ring is a domain iff it ... |
| domnlcanb 21001 | Left-cancellation law for ... |
| domnlcan 21002 | Left-cancellation law for ... |
| domnrcanb 21003 | Right-cancellation law for... |
| domnrcan 21004 | Right-cancellation law for... |
| domneq0r 21005 | Right multiplication by a ... |
| isidom 21006 | An integral domain is a co... |
| idomdomd 21007 | An integral domain is a do... |
| idomcringd 21008 | An integral domain is a co... |
| idomringd 21009 | An integral domain is a ri... |
| isdrng 21014 | The predicate "is a divisi... |
| drngunit 21015 | Elementhood in the set of ... |
| drngui 21016 | The set of units of a divi... |
| drngring 21017 | A division ring is a ring.... |
| drngringd 21018 | A division ring is a ring.... |
| drnggrpd 21019 | A division ring is a group... |
| drnggrp 21020 | A division ring is a group... |
| ringinveu 21021 | If a ring unit element ` X... |
| isdrng4 21022 | A division ring is a ring ... |
| isfld 21023 | A field is a commutative d... |
| flddrngd 21024 | A field is a division ring... |
| fldcrngd 21025 | A field is a commutative r... |
| drngprops 21026 | Properties of a division r... |
| isdrng2 21027 | A division ring can equiva... |
| drngprop 21028 | If two structures have the... |
| drngmgp 21029 | A division ring contains a... |
| drngid 21030 | A division ring's unity is... |
| drngunz 21031 | A division ring's unity is... |
| drngnzr 21032 | A division ring is a nonze... |
| drngdomn 21033 | A division ring is a domai... |
| isdrng3lem0 21034 | Lemma for ~ isdrng3 : The... |
| isdrng3lem1 21035 | Lemma for ~ isdrng3 (for t... |
| isdrng3lem2 21036 | Lemma for ~ isdrng3 (for t... |
| isdrng3 21037 | A division ring is a ring ... |
| isdrng5 21038 | A division ring is a ring ... |
| drngmcl 21039 | The product of two nonzero... |
| drngid2 21040 | Properties showing that an... |
| drnginvrcl 21041 | Closure of the multiplicat... |
| drnginvrn0 21042 | The multiplicative inverse... |
| drnginvrcld 21043 | Closure of the multiplicat... |
| drnginvrl 21044 | Property of the multiplica... |
| drnginvrr 21045 | Property of the multiplica... |
| drnginvrld 21046 | Property of the multiplica... |
| drnginvrrd 21047 | Property of the multiplica... |
| drngmul0or 21048 | A product is zero iff one ... |
| drngmulne0 21049 | A product is nonzero iff b... |
| drngmuleq0 21050 | An element is zero iff its... |
| opprdrng 21051 | The opposite of a division... |
| isdrngd 21052 | Properties that characteri... |
| isdrngrd 21053 | Properties that characteri... |
| isdrngdOLD 21054 | Obsolete version of ~ isdr... |
| isdrngrdOLD 21055 | Obsolete version of ~ isdr... |
| zrdrng 21056 | A zero ring is not a divis... |
| drngpropd 21057 | If two structures have the... |
| fldpropd 21058 | If two structures have the... |
| fldidom 21059 | A field is an integral dom... |
| fidomndrnglem 21060 | Lemma for ~ fidomndrng . ... |
| fidomndrng 21061 | A finite domain is a divis... |
| fiidomfld 21062 | A finite integral domain i... |
| rng1nnzr 21063 | The (smallest) structure r... |
| ring1zr 21064 | The only unital ring with ... |
| ringen1zr0 21065 | The only unital ring with ... |
| rng1nfld 21066 | The zero ring is not a fie... |
| issubdrg 21067 | Characterize the subfields... |
| drhmsubc 21068 | According to ~ df-subc , t... |
| drngcat 21069 | The restriction of the cat... |
| fldcat 21070 | The restriction of the cat... |
| fldc 21071 | The restriction of the cat... |
| fldhmsubc 21072 | According to ~ df-subc , t... |
| issdrg 21075 | Property of a division sub... |
| sdrgrcl 21076 | Reverse closure for a sub-... |
| sdrgdrng 21077 | A sub-division-ring is a d... |
| sdrgsubrg 21078 | A sub-division-ring is a s... |
| sdrgid 21079 | Every division ring is a d... |
| sdrgss 21080 | A division subring is a su... |
| sdrgbas 21081 | Base set of a sub-division... |
| issdrg2 21082 | Property of a division sub... |
| sdrgunit 21083 | A unit of a sub-division-r... |
| imadrhmcl 21084 | The image of a (nontrivial... |
| fldsdrgfld 21085 | A sub-division-ring of a f... |
| acsfn1p 21086 | Construction of a closure ... |
| subrgacs 21087 | Closure property of subrin... |
| sdrgacs 21088 | Closure property of divisi... |
| cntzsdrg 21089 | Centralizers in division r... |
| subdrgint 21090 | The intersection of a none... |
| sdrgint 21091 | The intersection of a none... |
| primefld 21092 | The smallest sub division ... |
| primefld0cl 21093 | The prime field contains t... |
| primefld1cl 21094 | The prime field contains t... |
| abvfval 21097 | Value of the set of absolu... |
| isabv 21098 | Elementhood in the set of ... |
| isabvd 21099 | Properties that determine ... |
| abvrcl 21100 | Reverse closure for the ab... |
| abvfge0 21101 | An absolute value is a fun... |
| abvf 21102 | An absolute value is a fun... |
| abvcl 21103 | An absolute value is a fun... |
| abvge0 21104 | The absolute value of a nu... |
| abveq0 21105 | The value of an absolute v... |
| abvne0 21106 | The absolute value of a no... |
| abvgt0 21107 | The absolute value of a no... |
| abvmul 21108 | An absolute value distribu... |
| abvtri 21109 | An absolute value satisfie... |
| abv0 21110 | The absolute value of zero... |
| abv1z 21111 | The absolute value of one ... |
| abv1 21112 | The absolute value of one ... |
| abvneg 21113 | The absolute value of a ne... |
| abvsubtri 21114 | An absolute value satisfie... |
| abvrec 21115 | The absolute value distrib... |
| abvdiv 21116 | The absolute value distrib... |
| abvdom 21117 | Any ring with an absolute ... |
| abvres 21118 | The restriction of an abso... |
| abvtrivd 21119 | The trivial absolute value... |
| abvtrivg 21120 | The trivial absolute value... |
| abvtriv 21121 | The trivial absolute value... |
| abvpropd 21122 | If two structures have the... |
| abvn0b 21123 | Another characterization o... |
| staffval 21128 | The functionalization of t... |
| stafval 21129 | The functionalization of t... |
| staffn 21130 | The functionalization is e... |
| issrng 21131 | The predicate "is a star r... |
| srngrhm 21132 | The involution function in... |
| srngring 21133 | A star ring is a ring. (C... |
| srngcnv 21134 | The involution function in... |
| srngf1o 21135 | The involution function in... |
| srngcl 21136 | The involution function in... |
| srngnvl 21137 | The involution function in... |
| srngadd 21138 | The involution function in... |
| srngmul 21139 | The involution function in... |
| srng1 21140 | The conjugate of the ring ... |
| srng0 21141 | The conjugate of the ring ... |
| issrngd 21142 | Properties that determine ... |
| idsrngd 21143 | A commutative ring is a st... |
| isorng 21148 | An ordered ring is a ring ... |
| orngring 21149 | An ordered ring is a ring.... |
| orngogrp 21150 | An ordered ring is an orde... |
| isofld 21151 | An ordered field is a fiel... |
| orngmul 21152 | In an ordered ring, the or... |
| orngsqr 21153 | In an ordered ring, all sq... |
| ornglmulle 21154 | In an ordered ring, multip... |
| orngrmulle 21155 | In an ordered ring, multip... |
| ornglmullt 21156 | In an ordered ring, multip... |
| orngrmullt 21157 | In an ordered ring, multip... |
| orngmullt 21158 | In an ordered ring, the st... |
| ofldfld 21159 | An ordered field is a fiel... |
| ofldtos 21160 | An ordered field is a tota... |
| orng0le1 21161 | In an ordered ring, the ri... |
| ofldlt1 21162 | In an ordered field, the r... |
| suborng 21163 | Every subring of an ordere... |
| subofld 21164 | Every subfield of an order... |
| islmod 21169 | The predicate "is a left m... |
| lmodlema 21170 | Lemma for properties of a ... |
| islmodd 21171 | Properties that determine ... |
| lmodgrp 21172 | A left module is a group. ... |
| lmodring 21173 | The scalar component of a ... |
| lmodfgrp 21174 | The scalar component of a ... |
| lmodgrpd 21175 | A left module is a group. ... |
| lmodbn0 21176 | The base set of a left mod... |
| lmodacl 21177 | Closure of ring addition f... |
| lmodmcl 21178 | Closure of ring multiplica... |
| lmodsn0 21179 | The set of scalars in a le... |
| lmodvacl 21180 | Closure of vector addition... |
| lmodass 21181 | Left module vector sum is ... |
| lmodlcan 21182 | Left cancellation law for ... |
| lmodvscl 21183 | Closure of scalar product ... |
| lmodvscld 21184 | Closure of scalar product ... |
| scaffval 21185 | The scalar multiplication ... |
| scafval 21186 | The scalar multiplication ... |
| scafeq 21187 | If the scalar multiplicati... |
| scaffn 21188 | The scalar multiplication ... |
| lmodscaf 21189 | The scalar multiplication ... |
| lmodvsdi 21190 | Distributive law for scala... |
| lmodvsdir 21191 | Distributive law for scala... |
| lmodvsass 21192 | In a module, scalar multip... |
| lmod0cl 21193 | The ring zero in a left mo... |
| lmod1cl 21194 | The ring unity in a left m... |
| lmodvs1 21195 | Scalar product with the ri... |
| lmod0vcl 21196 | The zero vector is a vecto... |
| lmod0vlid 21197 | Left identity law for the ... |
| lmod0vrid 21198 | Right identity law for the... |
| lmod0vid 21199 | Identity equivalent to the... |
| lmod0vs 21200 | Zero times a vector is the... |
| lmodvs0 21201 | Anything times the zero ve... |
| lmodvsmmulgdi 21202 | Distributive law for a gro... |
| lmodfopnelem1 21203 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 21204 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 21205 | The (functionalized) opera... |
| lcomf 21206 | A linear-combination sum i... |
| lcomfsupp 21207 | A linear-combination sum i... |
| lmodvnegcl 21208 | Closure of vector negative... |
| lmodvnegid 21209 | Addition of a vector with ... |
| lmodvneg1 21210 | Minus 1 times a vector is ... |
| lmodvsneg 21211 | Multiplication of a vector... |
| lmodvsubcl 21212 | Closure of vector subtract... |
| lmodcom 21213 | Left module vector sum is ... |
| lmodabl 21214 | A left module is an abelia... |
| lmodcmn 21215 | A left module is a commuta... |
| lmodnegadd 21216 | Distribute negation throug... |
| lmod4 21217 | Commutative/associative la... |
| lmodvsubadd 21218 | Relationship between vecto... |
| lmodvaddsub4 21219 | Vector addition/subtractio... |
| lmodvpncan 21220 | Addition/subtraction cance... |
| lmodvnpcan 21221 | Cancellation law for vecto... |
| lmodvsubval2 21222 | Value of vector subtractio... |
| lmodsubvs 21223 | Subtraction of a scalar pr... |
| lmodsubdi 21224 | Scalar multiplication dist... |
| lmodsubdir 21225 | Scalar multiplication dist... |
| lmodsubeq0 21226 | If the difference between ... |
| lmodsubid 21227 | Subtraction of a vector fr... |
| lmodvsghm 21228 | Scalar multiplication of t... |
| lmodprop2d 21229 | If two structures have the... |
| lmodpropd 21230 | If two structures have the... |
| gsumvsmul 21231 | Pull a scalar multiplicati... |
| mptscmfsupp0 21232 | A mapping to a scalar prod... |
| mptscmfsuppd 21233 | A function mapping to a sc... |
| rmodislmodlem 21234 | Lemma for ~ rmodislmod . ... |
| rmodislmod 21235 | The right module ` R ` ind... |
| lssset 21238 | The set of all (not necess... |
| islss 21239 | The predicate "is a subspa... |
| islssd 21240 | Properties that determine ... |
| lssss 21241 | A subspace is a set of vec... |
| lssel 21242 | A subspace member is a vec... |
| lss1 21243 | The set of vectors in a le... |
| lssuni 21244 | The union of all subspaces... |
| lssn0 21245 | A subspace is not empty. ... |
| 00lss 21246 | The empty structure has no... |
| lsscl 21247 | Closure property of a subs... |
| lssvacl 21248 | Closure of vector addition... |
| lssvsubcl 21249 | Closure of vector subtract... |
| lssvancl1 21250 | Non-closure: if one vector... |
| lssvancl2 21251 | Non-closure: if one vector... |
| lss0cl 21252 | The zero vector belongs to... |
| lsssn0 21253 | The singleton of the zero ... |
| lss0ss 21254 | The zero subspace is inclu... |
| lssle0 21255 | No subspace is smaller tha... |
| lssne0 21256 | A nonzero subspace has a n... |
| lssvneln0 21257 | A vector ` X ` which doesn... |
| lssneln0 21258 | A vector ` X ` which doesn... |
| lssssr 21259 | Conclude subspace ordering... |
| lssvscl 21260 | Closure of scalar product ... |
| lssvnegcl 21261 | Closure of negative vector... |
| lsssubg 21262 | All subspaces are subgroup... |
| lsssssubg 21263 | All subspaces are subgroup... |
| islss3 21264 | A linear subspace of a mod... |
| lsslmod 21265 | A submodule is a module. ... |
| lsslss 21266 | The subspaces of a subspac... |
| islss4 21267 | A linear subspace is a sub... |
| lss1d 21268 | One-dimensional subspace (... |
| lssintcl 21269 | The intersection of a none... |
| lssincl 21270 | The intersection of two su... |
| lssmre 21271 | The subspaces of a module ... |
| lssacs 21272 | Submodules are an algebrai... |
| prdsvscacl 21273 | Pointwise scalar multiplic... |
| prdslmodd 21274 | The product of a family of... |
| pwslmod 21275 | A structure power of a lef... |
| lspfval 21278 | The span function for a le... |
| lspf 21279 | The span function on a lef... |
| lspval 21280 | The span of a set of vecto... |
| lspcl 21281 | The span of a set of vecto... |
| lspsncl 21282 | The span of a singleton is... |
| lspprcl 21283 | The span of a pair is a su... |
| lsptpcl 21284 | The span of an unordered t... |
| lspsnsubg 21285 | The span of a singleton is... |
| 00lsp 21286 | ~ fvco4i lemma for linear ... |
| lspid 21287 | The span of a subspace is ... |
| lspssv 21288 | A span is a set of vectors... |
| lspss 21289 | Span preserves subset orde... |
| lspssid 21290 | A set of vectors is a subs... |
| lspidm 21291 | The span of a set of vecto... |
| lspun 21292 | The span of union is the s... |
| lspssp 21293 | If a set of vectors is a s... |
| mrclsp 21294 | Moore closure generalizes ... |
| lspsnss 21295 | The span of the singleton ... |
| ellspsn3 21296 | A member of the span of th... |
| lspprss 21297 | The span of a pair of vect... |
| lspsnid 21298 | A vector belongs to the sp... |
| ellspsn6 21299 | Relationship between a vec... |
| ellspsn5b 21300 | Relationship between a vec... |
| ellspsn5 21301 | Relationship between a vec... |
| lspprid1 21302 | A member of a pair of vect... |
| lspprid2 21303 | A member of a pair of vect... |
| lspprvacl 21304 | The sum of two vectors bel... |
| lssats2 21305 | A way to express atomistic... |
| ellspsni 21306 | A scalar product with a ve... |
| lspsn 21307 | Span of the singleton of a... |
| ellspsn 21308 | Member of span of the sing... |
| lspsnvsi 21309 | Span of a scalar product o... |
| lspsnss2 21310 | Comparable spans of single... |
| lspsnneg 21311 | Negation does not change t... |
| lspsnsub 21312 | Swapping subtraction order... |
| lspsn0 21313 | Span of the singleton of t... |
| lsp0 21314 | Span of the empty set. (C... |
| lspuni0 21315 | Union of the span of the e... |
| lspun0 21316 | The span of a union with t... |
| lspsneq0 21317 | Span of the singleton is t... |
| lspsneq0b 21318 | Equal singleton spans impl... |
| lmodindp1 21319 | Two independent (non-colin... |
| lsslsp 21320 | Spans in submodules corres... |
| lss0v 21321 | The zero vector in a submo... |
| lsspropd 21322 | If two structures have the... |
| lsppropd 21323 | If two structures have the... |
| reldmlmhm 21330 | Lemma for module homomorph... |
| lmimfn 21331 | Lemma for module isomorphi... |
| islmhm 21332 | Property of being a homomo... |
| islmhm3 21333 | Property of a module homom... |
| lmhmlem 21334 | Non-quantified consequence... |
| lmhmsca 21335 | A homomorphism of left mod... |
| lmghm 21336 | A homomorphism of left mod... |
| lmhmlmod2 21337 | A homomorphism of left mod... |
| lmhmlmod1 21338 | A homomorphism of left mod... |
| lmhmf 21339 | A homomorphism of left mod... |
| lmhmlin 21340 | A homomorphism of left mod... |
| lmodvsinv 21341 | Multiplication of a vector... |
| lmodvsinv2 21342 | Multiplying a negated vect... |
| islmhm2 21343 | A one-equation proof of li... |
| islmhmd 21344 | Deduction for a module hom... |
| 0lmhm 21345 | The constant zero linear f... |
| idlmhm 21346 | The identity function on a... |
| invlmhm 21347 | The negative function on a... |
| lmhmco 21348 | The composition of two mod... |
| lmhmplusg 21349 | The pointwise sum of two l... |
| lmhmvsca 21350 | The pointwise scalar produ... |
| lmhmf1o 21351 | A bijective module homomor... |
| lmhmima 21352 | The image of a subspace un... |
| lmhmpreima 21353 | The inverse image of a sub... |
| lmhmlsp 21354 | Homomorphisms preserve spa... |
| lmhmrnlss 21355 | The range of a homomorphis... |
| lmhmkerlss 21356 | The kernel of a homomorphi... |
| reslmhm 21357 | Restriction of a homomorph... |
| reslmhm2 21358 | Expansion of the codomain ... |
| reslmhm2b 21359 | Expansion of the codomain ... |
| lmhmeql 21360 | The equalizer of two modul... |
| lspextmo 21361 | A linear function is compl... |
| pwsdiaglmhm 21362 | Diagonal homomorphism into... |
| pwssplit0 21363 | Splitting for structure po... |
| pwssplit1 21364 | Splitting for structure po... |
| pwssplit2 21365 | Splitting for structure po... |
| pwssplit3 21366 | Splitting for structure po... |
| islmim 21367 | An isomorphism of left mod... |
| lmimf1o 21368 | An isomorphism of left mod... |
| lmimlmhm 21369 | An isomorphism of modules ... |
| lmimgim 21370 | An isomorphism of modules ... |
| islmim2 21371 | An isomorphism of left mod... |
| lmimcnv 21372 | The converse of a bijectiv... |
| brlmic 21373 | The relation "is isomorphi... |
| brlmici 21374 | Prove isomorphic by an exp... |
| lmiclcl 21375 | Isomorphism implies the le... |
| lmicrcl 21376 | Isomorphism implies the ri... |
| lmicsym 21377 | Module isomorphism is symm... |
| lmhmpropd 21378 | Module homomorphism depend... |
| islbs 21381 | The predicate " ` B ` is a... |
| lbsss 21382 | A basis is a set of vector... |
| lbsel 21383 | An element of a basis is a... |
| lbssp 21384 | The span of a basis is the... |
| lbsind 21385 | A basis is linearly indepe... |
| lbsind2 21386 | A basis is linearly indepe... |
| lbspss 21387 | No proper subset of a basi... |
| lsmcl 21388 | The sum of two subspaces i... |
| lsmspsn 21389 | Member of subspace sum of ... |
| lsmelval2 21390 | Subspace sum membership in... |
| lsmsp 21391 | Subspace sum in terms of s... |
| lsmsp2 21392 | Subspace sum of spans of s... |
| lsmssspx 21393 | Subspace sum (in its exten... |
| lsmpr 21394 | The span of a pair of vect... |
| lsppreli 21395 | A vector expressed as a su... |
| lsmelpr 21396 | Two ways to say that a vec... |
| lsppr0 21397 | The span of a vector paire... |
| lsppr 21398 | Span of a pair of vectors.... |
| lspprel 21399 | Member of the span of a pa... |
| lspprabs 21400 | Absorption of vector sum i... |
| lspvadd 21401 | The span of a vector sum i... |
| lspsntri 21402 | Triangle-type inequality f... |
| lspsntrim 21403 | Triangle-type inequality f... |
| lbspropd 21404 | If two structures have the... |
| pj1lmhm 21405 | The left projection functi... |
| pj1lmhm2 21406 | The left projection functi... |
| islvec 21409 | The predicate "is a left v... |
| lvecdrng 21410 | The set of scalars of a le... |
| lveclmod 21411 | A left vector space is a l... |
| lveclmodd 21412 | A vector space is a left m... |
| lvecgrpd 21413 | A vector space is a group.... |
| lsslvec 21414 | A vector subspace is a vec... |
| lmhmlvec 21415 | The property for modules t... |
| lvecvs0or 21416 | If a scalar product is zer... |
| lvecvsn0 21417 | A scalar product is nonzer... |
| lssvs0or 21418 | If a scalar product belong... |
| lvecvscan 21419 | Cancellation law for scala... |
| lvecvscan2 21420 | Cancellation law for scala... |
| lvecinv 21421 | Invert coefficient of scal... |
| lspsnvs 21422 | A nonzero scalar product d... |
| lspsneleq 21423 | Membership relation that i... |
| lspsncmp 21424 | Comparable spans of nonzer... |
| lspsnne1 21425 | Two ways to express that v... |
| lspsnne2 21426 | Two ways to express that v... |
| lspsnnecom 21427 | Swap two vectors with diff... |
| lspabs2 21428 | Absorption law for span of... |
| lspabs3 21429 | Absorption law for span of... |
| lspsneq 21430 | Equal spans of singletons ... |
| lspsneu 21431 | Nonzero vectors with equal... |
| ellspsn4 21432 | A member of the span of th... |
| lspdisj 21433 | The span of a vector not i... |
| lspdisjb 21434 | A nonzero vector is not in... |
| lspdisj2 21435 | Unequal spans are disjoint... |
| lspfixed 21436 | Show membership in the spa... |
| lspexch 21437 | Exchange property for span... |
| lspexchn1 21438 | Exchange property for span... |
| lspexchn2 21439 | Exchange property for span... |
| lspindpi 21440 | Partial independence prope... |
| lspindp1 21441 | Alternate way to say 3 vec... |
| lspindp2l 21442 | Alternate way to say 3 vec... |
| lspindp2 21443 | Alternate way to say 3 vec... |
| lspindp3 21444 | Independence of 2 vectors ... |
| lspindp4 21445 | (Partial) independence of ... |
| lvecindp 21446 | Compute the ` X ` coeffici... |
| lvecindp2 21447 | Sums of independent vector... |
| lspsnsubn0 21448 | Unequal singleton spans im... |
| lsmcv 21449 | Subspace sum has the cover... |
| lspsolvlem 21450 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21451 | If ` X ` is in the span of... |
| lssacsex 21452 | In a vector space, subspac... |
| lspsnat 21453 | There is no subspace stric... |
| lspsncv0 21454 | The span of a singleton co... |
| lsppratlem1 21455 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21456 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21457 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21458 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21459 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21460 | Lemma for ~ lspprat . Neg... |
| lspprat 21461 | A proper subspace of the s... |
| islbs2 21462 | An equivalent formulation ... |
| islbs3 21463 | An equivalent formulation ... |
| lbsacsbs 21464 | Being a basis in a vector ... |
| lvecdim 21465 | The dimension theorem for ... |
| lbsextlem1 21466 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21467 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21468 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21469 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21470 | For any linearly independe... |
| lbsext 21471 | For any linearly independe... |
| lbsexg 21472 | Every vector space has a b... |
| lbsex 21473 | Every vector space has a b... |
| lvecprop2d 21474 | If two structures have the... |
| lvecpropd 21475 | If two structures have the... |
| sraval 21480 | Lemma for ~ srabase throug... |
| sralem 21481 | Lemma for ~ srabase and si... |
| srabase 21482 | Base set of a subring alge... |
| sraaddg 21483 | Additive operation of a su... |
| sramulr 21484 | Multiplicative operation o... |
| srasca 21485 | The set of scalars of a su... |
| sravsca 21486 | The scalar product operati... |
| sraip 21487 | The inner product operatio... |
| sratset 21488 | Topology component of a su... |
| sratopn 21489 | Topology component of a su... |
| srads 21490 | Distance function of a sub... |
| sraring 21491 | Condition for a subring al... |
| sralmod 21492 | The subring algebra is a l... |
| sralmod0 21493 | The subring module inherit... |
| issubrgd 21494 | Prove a subring by closure... |
| rlmfn 21495 | ` ringLMod ` is a function... |
| rlmval 21496 | Value of the ring module. ... |
| rlmval2 21497 | Value of the ring module e... |
| rlmbas 21498 | Base set of the ring modul... |
| rlmplusg 21499 | Vector addition in the rin... |
| rlm0 21500 | Zero vector in the ring mo... |
| rlmsub 21501 | Subtraction in the ring mo... |
| rlmmulr 21502 | Ring multiplication in the... |
| rlmsca 21503 | Scalars in the ring module... |
| rlmsca2 21504 | Scalars in the ring module... |
| rlmvsca 21505 | Scalar multiplication in t... |
| rlmtopn 21506 | Topology component of the ... |
| rlmds 21507 | Metric component of the ri... |
| rlmlmod 21508 | The ring module is a modul... |
| rlmlvec 21509 | The ring module over a div... |
| rlmlsm 21510 | Subgroup sum of the ring m... |
| rlmvneg 21511 | Vector negation in the rin... |
| rlmscaf 21512 | Functionalized scalar mult... |
| ixpsnbasval 21513 | The value of an infinite C... |
| lidlval 21518 | Value of the set of ring i... |
| rspval 21519 | Value of the ring span fun... |
| lidlss 21520 | An ideal is a subset of th... |
| lidlbasel 21521 | An element of an ideal is ... |
| lidlssbas 21522 | The base set of the restri... |
| lidlbas 21523 | A (left) ideal of a ring i... |
| islidl 21524 | Predicate of being a (left... |
| rnglidlmcl 21525 | A (left) ideal containing ... |
| rngridlmcl 21526 | A right ideal (which is a ... |
| dflidl2rng 21527 | Alternate (the usual textb... |
| isridlrng 21528 | A right ideal is a left id... |
| lidl0cl 21529 | An ideal contains 0. (Con... |
| lidlacl 21530 | An ideal is closed under a... |
| lidlnegcl 21531 | An ideal contains negative... |
| lidlsubg 21532 | An ideal is a subgroup of ... |
| lidlsubcl 21533 | An ideal is closed under s... |
| lidlmcl 21534 | An ideal is closed under l... |
| lidl1el 21535 | An ideal contains 1 iff it... |
| lidlmcld 21536 | An ideal is closed under l... |
| dflidl2 21537 | Alternate (the usual textb... |
| lidl0ALT 21538 | Alternate proof for ~ lidl... |
| rnglidl0 21539 | Every non-unital ring cont... |
| lidl0 21540 | Every ring contains a zero... |
| lidl1ALT 21541 | Alternate proof for ~ lidl... |
| rnglidl1 21542 | The base set of every non-... |
| lidl1 21543 | Every ring contains a unit... |
| 0ringidl 21544 | The zero ideal is the only... |
| lidlunin0 21545 | The union of a nonempty su... |
| unichnlidl 21546 | The union of a nonempty ch... |
| lidlacs 21547 | The ideal system is an alg... |
| rspcl 21548 | The span of a set of ring ... |
| rspssid 21549 | The span of a set of ring ... |
| rsp1 21550 | The span of the identity e... |
| rsp0 21551 | The span of the zero eleme... |
| rspssp 21552 | The ideal span of a set of... |
| rspvalint 21553 | The ideal generated by a s... |
| rspprop 21554 | Properties of a class to b... |
| elrspsn 21555 | Membership in a principal ... |
| rspsn0 21556 | A principal ideal (an idea... |
| rspsnid 21557 | A principal ideal contains... |
| pidlnz 21558 | A principal ideal generate... |
| mrcrsp 21559 | Moore closure generalizes ... |
| lidlnz 21560 | A nonzero ideal contains a... |
| drngnidl 21561 | A division ring has only t... |
| lidlrsppropd 21562 | The left ideals and ring s... |
| rnglidlmmgm 21563 | The multiplicative group o... |
| rnglidlmsgrp 21564 | The multiplicative group o... |
| rnglidlrng 21565 | A (left) ideal of a non-un... |
| lidlnsg 21566 | An ideal is a normal subgr... |
| lsmidllsp 21567 | The sum of two ideals is t... |
| lsmidl 21568 | The sum of two ideals is a... |
| drngidl 21569 | A nonzero ring is a divisi... |
| isfieldidl 21570 | Determine if a ring is a f... |
| isfieldidl2 21571 | Determine if a ring is a f... |
| 2idlval 21574 | Definition of a two-sided ... |
| isridl 21575 | A right ideal is a left id... |
| 2idlelb 21576 | Membership in a two-sided ... |
| 2idllidld 21577 | A two-sided ideal is a lef... |
| 2idlridld 21578 | A two-sided ideal is a rig... |
| 2idl1el 21579 | A two-sided ideal contains... |
| df2idl2rng 21580 | Alternate (the usual textb... |
| df2idl2 21581 | Alternate (the usual textb... |
| ridl0 21582 | Every ring contains a zero... |
| ridl1 21583 | Every ring contains a unit... |
| 2idl0 21584 | Every ring contains a zero... |
| 2idl1 21585 | Every ring contains a unit... |
| 2idlss 21586 | A two-sided ideal is a sub... |
| 2idlbas 21587 | The base set of a two-side... |
| 2idlelbas 21588 | The base set of a two-side... |
| rng2idlsubrng 21589 | A two-sided ideal of a non... |
| rng2idlnsg 21590 | A two-sided ideal of a non... |
| rng2idl0 21591 | The zero (additive identit... |
| rng2idlsubgsubrng 21592 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21593 | A two-sided ideal of a non... |
| rng2idlsubg0 21594 | The zero (additive identit... |
| 2idlcpblrng 21595 | The coset equivalence rela... |
| 2idlcpbl 21596 | The coset equivalence rela... |
| qus2idrng 21597 | The quotient of a non-unit... |
| qus1 21598 | The multiplicative identit... |
| qusring 21599 | If ` S ` is a two-sided id... |
| qusrhm 21600 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21601 | The preimage of an ideal b... |
| kerlidl 21602 | The kernel of a ring homom... |
| ker2idl 21603 | The kernel of a ring homom... |
| qusmul2idl 21604 | Value of the ring operatio... |
| crngridl 21605 | In a commutative ring, the... |
| crng2idl 21606 | In a commutative ring, a t... |
| df2idl2crng 21607 | The predicate "is an ideal... |
| qusmulrng 21608 | Value of the multiplicatio... |
| quscrng 21609 | The quotient of a commutat... |
| qusmulcrng 21610 | Value of the ring operatio... |
| rhmqusnsg 21611 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21612 | The ring unity of a two-si... |
| rngqiprngghmlem1 21613 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21614 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21615 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21616 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21617 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21618 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21619 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21620 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21621 | The base set of the produc... |
| rngqiprng 21622 | The product of the quotien... |
| rngqiprngimf 21623 | ` F ` is a function from (... |
| rngqiprngimfv 21624 | The value of the function ... |
| rngqiprngghm 21625 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21626 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21627 | ` F ` is a function from (... |
| rngqiprnglin 21628 | ` F ` is linear with respe... |
| rngqiprngho 21629 | ` F ` is a homomorphism of... |
| rngqiprngim 21630 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21631 | The unity of a two-sided i... |
| rngringbdlem1 21632 | In a unital ring, the quot... |
| rngringbdlem2 21633 | A non-unital ring is unita... |
| rngringbd 21634 | A non-unital ring is unita... |
| ring2idlqus 21635 | For every unital ring ther... |
| ring2idlqusb 21636 | A non-unital ring is unita... |
| rngqiprngfulem1 21637 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21638 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21639 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21640 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21641 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21642 | The ring unity of the prod... |
| rngqiprngfu 21643 | The function value of ` F ... |
| rngqiprngu 21644 | If a non-unital ring has a... |
| ring2idlqus1 21645 | If a non-unital ring has a... |
| prmidlval 21648 | The class of prime ideals ... |
| isprmidl 21649 | The predicate "is a prime ... |
| prmidlnr 21650 | A prime ideal is a proper ... |
| prmidl 21651 | The main property of a pri... |
| prmidl2 21652 | A condition that shows an ... |
| idlmulssprm 21653 | Let ` P ` be a prime ideal... |
| pridln1 21654 | A proper ideal cannot cont... |
| prmidlidl 21655 | A prime ideal is an ideal.... |
| prmidlssidl 21656 | Prime ideals as a subset o... |
| cringm4 21657 | Commutative/associative la... |
| isprmidlc 21658 | The predicate "is prime id... |
| prmidlc 21659 | Property of a prime ideal ... |
| prmidlc2 21660 | Property of a prime ideal ... |
| cmprmidlmcl 21661 | The complement of a prime ... |
| prmidlprop 21662 | Property of prime ideals. ... |
| 0ringprmidl 21663 | The trivial ring does not ... |
| prmidl0 21664 | The zero ideal of a commut... |
| rhmpreimaprmidl 21665 | The preimage of a prime id... |
| qsidomlem1 21666 | If the quotient ring of a ... |
| qsidomlem2 21667 | A quotient by a prime idea... |
| qsidom 21668 | An ideal ` I ` in the comm... |
| qsnzr 21669 | A quotient of a nonzero ri... |
| ssdifidllem 21670 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 21671 | Let ` R ` be a ring, and l... |
| ssdifidlprm 21672 | If the set ` S ` of ~ ssdi... |
| prmidlsubm 21673 | The complement of a prime ... |
| lpival 21678 | Value of the set of princi... |
| islpidl 21679 | Property of being a princi... |
| lpi0 21680 | The zero ideal is always p... |
| lpi1 21681 | The unit ideal is always p... |
| islpir 21682 | Principal ideal rings are ... |
| lpiss 21683 | Principal ideals are a sub... |
| islpir2 21684 | Principal ideal rings are ... |
| lpirring 21685 | Principal ideal rings are ... |
| drnglpir 21686 | Division rings are princip... |
| rspsn 21687 | Membership in principal id... |
| lidldvgen 21688 | An element generates an id... |
| lpigen 21689 | An ideal is principal iff ... |
| cnfldstr 21710 | The field of complex numbe... |
| cnfldex 21711 | The field of complex numbe... |
| cnfldbas 21712 | The base set of the field ... |
| mpocnfldadd 21713 | The addition operation of ... |
| cnfldadd 21714 | The addition operation of ... |
| mpocnfldmul 21715 | The multiplication operati... |
| cnfldmul 21716 | The multiplication operati... |
| cnfldcj 21717 | The conjugation operation ... |
| cnfldtset 21718 | The topology component of ... |
| cnfldle 21719 | The ordering of the field ... |
| cnfldds 21720 | The metric of the field of... |
| cnfldunif 21721 | The uniform structure comp... |
| cnfldfun 21722 | The field of complex numbe... |
| cnfldfunALT 21723 | The field of complex numbe... |
| xrsstr 21724 | The extended real structur... |
| xrsex 21725 | The extended real structur... |
| xrsadd 21726 | The addition operation of ... |
| xrsmul 21727 | The multiplication operati... |
| xrstset 21728 | The topology component of ... |
| cncrng 21729 | The complex numbers form a... |
| cnring 21730 | The complex numbers form a... |
| xrsmcmn 21731 | The "multiplicative group"... |
| cnfld0 21732 | Zero is the zero element o... |
| cnfld1 21733 | One is the unity element o... |
| cnfldneg 21734 | The additive inverse in th... |
| cnfldplusf 21735 | The functionalized additio... |
| cnfldsub 21736 | The subtraction operator i... |
| cndrng 21737 | The complex numbers form a... |
| cnflddiv 21738 | The division operation in ... |
| cnfldinv 21739 | The multiplicative inverse... |
| cnfldmulg 21740 | The group multiple functio... |
| cnfldexp 21741 | The exponentiation operato... |
| cnsrng 21742 | The complex numbers form a... |
| xrsmgm 21743 | The "additive group" of th... |
| xrsnsgrp 21744 | The "additive group" of th... |
| xrsmgmdifsgrp 21745 | The "additive group" of th... |
| xrsds 21746 | The metric of the extended... |
| xrsdsval 21747 | The metric of the extended... |
| xrsdsreval 21748 | The metric of the extended... |
| xrsdsreclblem 21749 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21750 | The metric of the extended... |
| cnsubmlem 21751 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21752 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21753 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 21754 | Lemma for ~ resubdrg and f... |
| qsubdrg 21755 | The rational numbers form ... |
| zsubrg 21756 | The integers form a subrin... |
| gzsubrg 21757 | The gaussian integers form... |
| nn0subm 21758 | The nonnegative integers f... |
| rege0subm 21759 | The nonnegative reals form... |
| absabv 21760 | The regular absolute value... |
| zsssubrg 21761 | The integers are a subset ... |
| qsssubdrg 21762 | The rational numbers are a... |
| cnsubrg 21763 | There are no subrings of t... |
| cnmgpabl 21764 | The unit group of the comp... |
| cnmgpid 21765 | The group identity element... |
| cnmsubglem 21766 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21767 | The positive reals form a ... |
| gzrngunitlem 21768 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21769 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21770 | Relate a group sum on ` CC... |
| regsumfsum 21771 | Relate a group sum on ` ( ... |
| expmhm 21772 | Exponentiation is a monoid... |
| nn0srg 21773 | The nonnegative integers f... |
| rge0srg 21774 | The nonnegative real numbe... |
| xrge0plusg 21775 | The additive law of the ex... |
| xrs1mnd 21776 | The extended real numbers,... |
| xrs10 21777 | The zero of the extended r... |
| xrs1cmn 21778 | The extended real numbers ... |
| xrge0subm 21779 | The nonnegative extended r... |
| xrge0cmn 21780 | The nonnegative extended r... |
| xrge0omnd 21781 | The nonnegative extended r... |
| zringcrng 21784 | The ring of integers is a ... |
| zringring 21785 | The ring of integers is a ... |
| zringrng 21786 | The ring of integers is a ... |
| zringabl 21787 | The ring of integers is an... |
| zringgrp 21788 | The ring of integers is an... |
| zringbas 21789 | The integers are the base ... |
| zringplusg 21790 | The addition operation of ... |
| zringsub 21791 | The subtraction of element... |
| zringmulg 21792 | The multiplication (group ... |
| zringmulr 21793 | The multiplication operati... |
| zring0 21794 | The zero element of the ri... |
| zring1 21795 | The unity element of the r... |
| zringnzr 21796 | The ring of integers is a ... |
| dvdsrzring 21797 | Ring divisibility in the r... |
| zringlpirlem1 21798 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21799 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21800 | Lemma for ~ zringlpir . A... |
| zringinvg 21801 | The additive inverse of an... |
| zringunit 21802 | The units of ` ZZ ` are th... |
| zringlpir 21803 | The integers are a princip... |
| zringndrg 21804 | The integers are not a div... |
| zringcyg 21805 | The integers are a cyclic ... |
| zringsubgval 21806 | Subtraction in the ring of... |
| zringmpg 21807 | The multiplicative group o... |
| prmirredlem 21808 | A positive integer is irre... |
| dfprm2 21809 | The positive irreducible e... |
| prmirred 21810 | The irreducible elements o... |
| expghm 21811 | Exponentiation is a group ... |
| mulgghm2 21812 | The powers of a group elem... |
| mulgrhm 21813 | The powers of the element ... |
| mulgrhm2 21814 | The powers of the element ... |
| irinitoringc 21815 | The ring of integers is an... |
| nzerooringczr 21816 | There is no zero object in... |
| pzriprnglem1 21817 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21818 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21819 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21820 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21821 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21822 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21823 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21824 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21825 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21826 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21827 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21828 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21829 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21830 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21831 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21832 | The ring unity of the ring... |
| pzriprng 21833 | The non-unital ring ` ( ZZ... |
| pzriprng1 21834 | The ring unity of the ring... |
| zrhval 21843 | Define the unique homomorp... |
| zrhval2 21844 | Alternate value of the ` Z... |
| zrhmulg 21845 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21846 | The ` ZRHom ` homomorphism... |
| zrhrhm 21847 | The ` ZRHom ` homomorphism... |
| zrh1 21848 | Interpretation of 1 in a r... |
| zrh0 21849 | Interpretation of 0 in a r... |
| zrhpropd 21850 | The ` ZZ ` ring homomorphi... |
| zlmval 21851 | Augment an abelian group w... |
| zlmlem 21852 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21853 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21854 | Group operation of a ` ZZ ... |
| zlmmulr 21855 | Ring operation of a ` ZZ `... |
| zlmsca 21856 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21857 | Scalar multiplication oper... |
| zlmlmod 21858 | The ` ZZ ` -module operati... |
| chrval 21859 | Definition substitution of... |
| chrcl 21860 | Closure of the characteris... |
| chrid 21861 | The canonical ` ZZ ` ring ... |
| chrdvds 21862 | The ` ZZ ` ring homomorphi... |
| chrcong 21863 | If two integers are congru... |
| dvdschrmulg 21864 | In a ring, any multiple of... |
| fermltlchr 21865 | A generalization of Fermat... |
| chrnzr 21866 | Nonzero rings are precisel... |
| chrrhm 21867 | The characteristic restric... |
| domnchr 21868 | The characteristic of a do... |
| znlidl 21869 | The set ` n ZZ ` is an ide... |
| zncrng2 21870 | Making a commutative ring ... |
| znval 21871 | The value of the ` Z/nZ ` ... |
| znle 21872 | The value of the ` Z/nZ ` ... |
| znval2 21873 | Self-referential expressio... |
| znbaslem 21874 | Lemma for ~ znbas . (Cont... |
| znbas2 21875 | The base set of ` Z/nZ ` i... |
| znadd 21876 | The additive structure of ... |
| znmul 21877 | The multiplicative structu... |
| znzrh 21878 | The ` ZZ ` ring homomorphi... |
| znbas 21879 | The base set of ` Z/nZ ` s... |
| zncrng 21880 | ` Z/nZ ` is a commutative ... |
| znzrh2 21881 | The ` ZZ ` ring homomorphi... |
| znzrhval 21882 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21883 | The ` ZZ ` ring homomorphi... |
| zncyg 21884 | The group ` ZZ / n ZZ ` is... |
| zndvds 21885 | Express equality of equiva... |
| zndvds0 21886 | Special case of ~ zndvds w... |
| znf1o 21887 | The function ` F ` enumera... |
| zzngim 21888 | The ` ZZ ` ring homomorphi... |
| znle2 21889 | The ordering of the ` Z/nZ... |
| znleval 21890 | The ordering of the ` Z/nZ... |
| znleval2 21891 | The ordering of the ` Z/nZ... |
| zntoslem 21892 | Lemma for ~ zntos . (Cont... |
| zntos 21893 | The ` Z/nZ ` structure is ... |
| znhash 21894 | The ` Z/nZ ` structure has... |
| znfi 21895 | The ` Z/nZ ` structure is ... |
| znfld 21896 | The ` Z/nZ ` structure is ... |
| znidomb 21897 | The ` Z/nZ ` structure is ... |
| znchr 21898 | Cyclic rings are defined b... |
| znunit 21899 | The units of ` Z/nZ ` are ... |
| znunithash 21900 | The size of the unit group... |
| znrrg 21901 | The regular elements of ` ... |
| cygznlem1 21902 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21903 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21904 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21905 | A cyclic group with ` n ` ... |
| cygzn 21906 | A cyclic group with ` n ` ... |
| cygth 21907 | The "fundamental theorem o... |
| cyggic 21908 | Cyclic groups are isomorph... |
| frgpcyg 21909 | A free group is cyclic iff... |
| freshmansdream 21910 | For a prime number ` P ` ,... |
| frobrhm 21911 | In a commutative ring with... |
| ofldchr 21912 | The characteristic of an o... |
| cnmsgnsubg 21913 | The signs form a multiplic... |
| cnmsgnbas 21914 | The base set of the sign s... |
| cnmsgngrp 21915 | The group of signs under m... |
| psgnghm 21916 | The sign is a homomorphism... |
| psgnghm2 21917 | The sign is a homomorphism... |
| psgninv 21918 | The sign of a permutation ... |
| psgnco 21919 | Multiplicativity of the pe... |
| zrhpsgnmhm 21920 | Embedding of permutation s... |
| zrhpsgninv 21921 | The embedded sign of a per... |
| evpmss 21922 | Even permutations are perm... |
| psgnevpmb 21923 | A class is an even permuta... |
| psgnodpm 21924 | A permutation which is odd... |
| psgnevpm 21925 | A permutation which is eve... |
| psgnodpmr 21926 | If a permutation has sign ... |
| zrhpsgnevpm 21927 | The sign of an even permut... |
| zrhpsgnodpm 21928 | The sign of an odd permuta... |
| cofipsgn 21929 | Composition of any class `... |
| zrhpsgnelbas 21930 | Embedding of permutation s... |
| zrhcopsgnelbas 21931 | Embedding of permutation s... |
| evpmodpmf1o 21932 | The function for performin... |
| pmtrodpm 21933 | A transposition is an odd ... |
| psgnfix1 21934 | A permutation of a finite ... |
| psgnfix2 21935 | A permutation of a finite ... |
| psgndiflemB 21936 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21937 | Lemma 2 for ~ psgndif . (... |
| psgndif 21938 | Embedding of permutation s... |
| copsgndif 21939 | Embedding of permutation s... |
| rebase 21942 | The base of the field of r... |
| remulg 21943 | The multiplication (group ... |
| resubdrg 21944 | The real numbers form a di... |
| resubgval 21945 | Subtraction in the field o... |
| replusg 21946 | The addition operation of ... |
| remulr 21947 | The multiplication operati... |
| re0g 21948 | The zero element of the fi... |
| re1r 21949 | The unity element of the f... |
| rele2 21950 | The ordering relation of t... |
| relt 21951 | The ordering relation of t... |
| reds 21952 | The distance of the field ... |
| redvr 21953 | The division operation of ... |
| retos 21954 | The real numbers are a tot... |
| refld 21955 | The real numbers form a fi... |
| refldcj 21956 | The conjugation operation ... |
| resrng 21957 | The real numbers form a st... |
| regsumsupp 21958 | The group sum over the rea... |
| rzgrp 21959 | The quotient group ` RR / ... |
| isphl 21964 | The predicate "is a genera... |
| phllvec 21965 | A pre-Hilbert space is a l... |
| phllmod 21966 | A pre-Hilbert space is a l... |
| phlsrng 21967 | The scalar ring of a pre-H... |
| phllmhm 21968 | The inner product of a pre... |
| ipcl 21969 | Closure of the inner produ... |
| ipcj 21970 | Conjugate of an inner prod... |
| iporthcom 21971 | Orthogonality (meaning inn... |
| ip0l 21972 | Inner product with a zero ... |
| ip0r 21973 | Inner product with a zero ... |
| ipeq0 21974 | The inner product of a vec... |
| ipdir 21975 | Distributive law for inner... |
| ipdi 21976 | Distributive law for inner... |
| ip2di 21977 | Distributive law for inner... |
| ipsubdir 21978 | Distributive law for inner... |
| ipsubdi 21979 | Distributive law for inner... |
| ip2subdi 21980 | Distributive law for inner... |
| ipass 21981 | Associative law for inner ... |
| ipassr 21982 | "Associative" law for seco... |
| ipassr2 21983 | "Associative" law for inne... |
| ipffval 21984 | The inner product operatio... |
| ipfval 21985 | The inner product operatio... |
| ipfeq 21986 | If the inner product opera... |
| ipffn 21987 | The inner product operatio... |
| phlipf 21988 | The inner product operatio... |
| ip2eq 21989 | Two vectors are equal iff ... |
| isphld 21990 | Properties that determine ... |
| phlpropd 21991 | If two structures have the... |
| ssipeq 21992 | The inner product on a sub... |
| phssipval 21993 | The inner product on a sub... |
| phssip 21994 | The inner product (as a fu... |
| phlssphl 21995 | A subspace of an inner pro... |
| ocvfval 22002 | The orthocomplement operat... |
| ocvval 22003 | Value of the orthocompleme... |
| elocv 22004 | Elementhood in the orthoco... |
| ocvi 22005 | Property of a member of th... |
| ocvss 22006 | The orthocomplement of a s... |
| ocvocv 22007 | A set is contained in its ... |
| ocvlss 22008 | The orthocomplement of a s... |
| ocv2ss 22009 | Orthocomplements reverse s... |
| ocvin 22010 | An orthocomplement has tri... |
| ocvsscon 22011 | Two ways to say that ` S `... |
| ocvlsp 22012 | The orthocomplement of a l... |
| ocv0 22013 | The orthocomplement of the... |
| ocvz 22014 | The orthocomplement of the... |
| ocv1 22015 | The orthocomplement of the... |
| unocv 22016 | The orthocomplement of a u... |
| iunocv 22017 | The orthocomplement of an ... |
| cssval 22018 | The set of closed subspace... |
| iscss 22019 | The predicate "is a closed... |
| cssi 22020 | Property of a closed subsp... |
| cssss 22021 | A closed subspace is a sub... |
| iscss2 22022 | It is sufficient to prove ... |
| ocvcss 22023 | The orthocomplement of any... |
| cssincl 22024 | The zero subspace is a clo... |
| css0 22025 | The zero subspace is a clo... |
| css1 22026 | The whole space is a close... |
| csslss 22027 | A closed subspace of a pre... |
| lsmcss 22028 | A subset of a pre-Hilbert ... |
| cssmre 22029 | The closed subspaces of a ... |
| mrccss 22030 | The Moore closure correspo... |
| thlval 22031 | Value of the Hilbert latti... |
| thlbas 22032 | Base set of the Hilbert la... |
| thlle 22033 | Ordering on the Hilbert la... |
| thlleval 22034 | Ordering on the Hilbert la... |
| thloc 22035 | Orthocomplement on the Hil... |
| pjfval 22042 | The value of the projectio... |
| pjdm 22043 | A subspace is in the domai... |
| pjpm 22044 | The projection map is a pa... |
| pjfval2 22045 | Value of the projection ma... |
| pjval 22046 | Value of the projection ma... |
| pjdm2 22047 | A subspace is in the domai... |
| pjff 22048 | A projection is a linear o... |
| pjf 22049 | A projection is a function... |
| pjf2 22050 | A projection is a function... |
| pjfo 22051 | A projection is a surjecti... |
| pjcss 22052 | A projection subspace is a... |
| ocvpj 22053 | The orthocomplement of a p... |
| ishil 22054 | The predicate "is a Hilber... |
| ishil2 22055 | The predicate "is a Hilber... |
| isobs 22056 | The predicate "is an ortho... |
| obsip 22057 | The inner product of two e... |
| obsipid 22058 | A basis element has length... |
| obsrcl 22059 | Reverse closure for an ort... |
| obsss 22060 | An orthonormal basis is a ... |
| obsne0 22061 | A basis element is nonzero... |
| obsocv 22062 | An orthonormal basis has t... |
| obs2ocv 22063 | The double orthocomplement... |
| obselocv 22064 | A basis element is in the ... |
| obs2ss 22065 | A basis has no proper subs... |
| obslbs 22066 | An orthogonal basis is a l... |
| reldmdsmm 22069 | The direct sum is a well-b... |
| dsmmval 22070 | Value of the module direct... |
| dsmmbase 22071 | Base set of the module dir... |
| dsmmval2 22072 | Self-referential definitio... |
| dsmmbas2 22073 | Base set of the direct sum... |
| dsmmfi 22074 | For finite products, the d... |
| dsmmelbas 22075 | Membership in the finitely... |
| dsmm0cl 22076 | The all-zero vector is con... |
| dsmmacl 22077 | The finite hull is closed ... |
| prdsinvgd2 22078 | Negation of a single coord... |
| dsmmsubg 22079 | The finite hull of a produ... |
| dsmmlss 22080 | The finite hull of a produ... |
| dsmmlmod 22081 | The direct sum of a family... |
| frlmval 22084 | Value of the "free module"... |
| frlmlmod 22085 | The free module is a modul... |
| frlmpws 22086 | The free module as a restr... |
| frlmlss 22087 | The base set of the free m... |
| frlmpwsfi 22088 | The finite free module is ... |
| frlmsca 22089 | The ring of scalars of a f... |
| frlm0 22090 | Zero in a free module (rin... |
| frlmbas 22091 | Base set of the free modul... |
| frlmelbas 22092 | Membership in the base set... |
| frlmrcl 22093 | If a free module is inhabi... |
| frlmbasfsupp 22094 | Elements of the free modul... |
| frlmbasmap 22095 | Elements of the free modul... |
| frlmbasf 22096 | Elements of the free modul... |
| frlmlvec 22097 | The free module over a div... |
| frlmfibas 22098 | The base set of the finite... |
| elfrlmbasn0 22099 | If the dimension of a free... |
| frlmplusgval 22100 | Addition in a free module.... |
| frlmsubgval 22101 | Subtraction in a free modu... |
| frlmvscafval 22102 | Scalar multiplication in a... |
| frlmvplusgvalc 22103 | Coordinates of a sum with ... |
| frlmvscaval 22104 | Coordinates of a scalar mu... |
| frlmplusgvalb 22105 | Addition in a free module ... |
| frlmvscavalb 22106 | Scalar multiplication in a... |
| frlmvplusgscavalb 22107 | Addition combined with sca... |
| frlmgsum 22108 | Finite commutative sums in... |
| frlmsplit2 22109 | Restriction is homomorphic... |
| frlmsslss 22110 | A subset of a free module ... |
| frlmsslss2 22111 | A subset of a free module ... |
| frlmbas3 22112 | An element of the base set... |
| mpofrlmd 22113 | Elements of the free modul... |
| frlmip 22114 | The inner product of a fre... |
| frlmipval 22115 | The inner product of a fre... |
| frlmphllem 22116 | Lemma for ~ frlmphl . (Co... |
| frlmphl 22117 | Conditions for a free modu... |
| uvcfval 22120 | Value of the unit-vector g... |
| uvcval 22121 | Value of a single unit vec... |
| uvcvval 22122 | Value of a unit vector coo... |
| uvcvvcl 22123 | A coordinate of a unit vec... |
| uvcvvcl2 22124 | A unit vector coordinate i... |
| uvcvv1 22125 | The unit vector is one at ... |
| uvcvv0 22126 | The unit vector is zero at... |
| uvcff 22127 | Domain and codomain of the... |
| uvcf1 22128 | In a nonzero ring, each un... |
| uvcresum 22129 | Any element of a free modu... |
| frlmssuvc1 22130 | A scalar multiple of a uni... |
| frlmssuvc2 22131 | A nonzero scalar multiple ... |
| frlmsslsp 22132 | A subset of a free module ... |
| frlmlbs 22133 | The unit vectors comprise ... |
| frlmup1 22134 | Any assignment of unit vec... |
| frlmup2 22135 | The evaluation map has the... |
| frlmup3 22136 | The range of such an evalu... |
| frlmup4 22137 | Universal property of the ... |
| ellspd 22138 | The elements of the span o... |
| elfilspd 22139 | Simplified version of ~ el... |
| rellindf 22144 | The independent-family pre... |
| islinds 22145 | Property of an independent... |
| linds1 22146 | An independent set of vect... |
| linds2 22147 | An independent set of vect... |
| islindf 22148 | Property of an independent... |
| islinds2 22149 | Expanded property of an in... |
| islindf2 22150 | Property of an independent... |
| lindff 22151 | Functional property of a l... |
| lindfind 22152 | A linearly independent fam... |
| lindsind 22153 | A linearly independent set... |
| lindfind2 22154 | In a linearly independent ... |
| lindsind2 22155 | In a linearly independent ... |
| lindff1 22156 | A linearly independent fam... |
| lindfrn 22157 | The range of an independen... |
| f1lindf 22158 | Rearranging and deleting e... |
| lindfres 22159 | Any restriction of an inde... |
| lindsss 22160 | Any subset of an independe... |
| f1linds 22161 | A family constructed from ... |
| islindf3 22162 | In a nonzero ring, indepen... |
| lindfmm 22163 | Linear independence of a f... |
| lindsmm 22164 | Linear independence of a s... |
| lindsmm2 22165 | The monomorphic image of a... |
| lsslindf 22166 | Linear independence is unc... |
| lsslinds 22167 | Linear independence is unc... |
| islbs4 22168 | A basis is an independent ... |
| lbslinds 22169 | A basis is independent. (... |
| islinds3 22170 | A subset is linearly indep... |
| islinds4 22171 | A set is independent in a ... |
| lmimlbs 22172 | The isomorphic image of a ... |
| lmiclbs 22173 | Having a basis is an isomo... |
| islindf4 22174 | A family is independent if... |
| islindf5 22175 | A family is independent if... |
| indlcim 22176 | An independent, spanning f... |
| lbslcic 22177 | A module with a basis is i... |
| lmisfree 22178 | A module has a basis iff i... |
| lvecisfrlm 22179 | Every vector space is isom... |
| lmimco 22180 | The composition of two iso... |
| lmictra 22181 | Module isomorphism is tran... |
| uvcf1o 22182 | In a nonzero ring, the map... |
| uvcendim 22183 | In a nonzero ring, the num... |
| frlmisfrlm 22184 | A free module is isomorphi... |
| frlmiscvec 22185 | Every free module is isomo... |
| lindsdom 22186 | A linearly independent set... |
| lindsenlbs 22187 | A maximal linearly indepen... |
| isassa 22194 | The properties of an assoc... |
| assalem 22195 | The properties of an assoc... |
| assaass 22196 | Left-associative property ... |
| assaassr 22197 | Right-associative property... |
| assalmod 22198 | An associative algebra is ... |
| assaring 22199 | An associative algebra is ... |
| assasca 22200 | The scalars of an associat... |
| assa2ass 22201 | Left- and right-associativ... |
| assa2ass2 22202 | Left- and right-associativ... |
| isassad 22203 | Sufficient condition for b... |
| issubassa3 22204 | A subring that is also a s... |
| issubassa 22205 | The subalgebras of an asso... |
| sraassab 22206 | A subring algebra is an as... |
| sraassa 22207 | The subring algebra over a... |
| rlmassa 22208 | The ring module over a com... |
| assapropd 22209 | If two structures have the... |
| aspval 22210 | Value of the algebraic clo... |
| asplss 22211 | The algebraic span of a se... |
| aspid 22212 | The algebraic span of a su... |
| aspsubrg 22213 | The algebraic span of a se... |
| aspss 22214 | Span preserves subset orde... |
| aspssid 22215 | A set of vectors is a subs... |
| asclfval 22216 | Function value of the alge... |
| asclval 22217 | Value of a mapped algebra ... |
| asclfn 22218 | Unconditional functionalit... |
| asclf 22219 | The algebra scalar lifting... |
| asclghm 22220 | The algebra scalar lifting... |
| asclelbas 22221 | Lifted scalars are in the ... |
| ascl0 22222 | The scalar 0 embedded into... |
| ascl1 22223 | The scalar 1 embedded into... |
| asclmul1 22224 | Left multiplication by a l... |
| asclmul2 22225 | Right multiplication by a ... |
| ascldimul 22226 | The algebra scalar lifting... |
| asclinvg 22227 | The group inverse (negatio... |
| asclrhm 22228 | The algebra scalar lifting... |
| rnascl 22229 | The set of lifted scalars ... |
| issubassa2 22230 | A subring of a unital alge... |
| rnasclsubrg 22231 | The scalar multiples of th... |
| rnasclmulcl 22232 | (Vector) multiplication is... |
| rnasclassa 22233 | The scalar multiples of th... |
| ressascl 22234 | The lifting of scalars is ... |
| asclpropd 22235 | If two structures have the... |
| aspval2 22236 | The algebraic closure is t... |
| assamulgscmlem1 22237 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 22238 | Lemma for ~ assamulgscm (i... |
| assamulgscm 22239 | Exponentiation of a scalar... |
| asclmulg 22240 | Apply group multiplication... |
| zlmassa 22241 | The ` ZZ ` -module operati... |
| reldmpsr 22252 | The multivariate power ser... |
| psrval 22253 | Value of the multivariate ... |
| psrvalstr 22254 | The multivariate power ser... |
| psrbag 22255 | Elementhood in the set of ... |
| psrbagf 22256 | A finite bag is a function... |
| psrbagfsupp 22257 | Finite bags have finite su... |
| snifpsrbag 22258 | A bag containing one eleme... |
| fczpsrbag 22259 | The constant function equa... |
| psrbaglesupp 22260 | The support of a dominated... |
| psrbaglecl 22261 | The set of finite bags is ... |
| psrbagaddcl 22262 | The sum of two finite bags... |
| psrbagcon 22263 | The analogue of the statem... |
| psrbaglefi 22264 | There are finitely many ba... |
| psrbagconcl 22265 | The complement of a bag is... |
| psrbagleadd1 22266 | The analogue of " ` X <_ F... |
| psrbagconf1o 22267 | Bag complementation is a b... |
| psrbagres 22268 | Restrict a bag of variable... |
| gsumbagdiaglem 22269 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 22270 | Two-dimensional commutatio... |
| psrass1lem 22271 | A group sum commutation us... |
| psrbas 22272 | The base set of the multiv... |
| psrelbas 22273 | An element of the set of p... |
| psrelbasfun 22274 | An element of the set of p... |
| psrplusg 22275 | The addition operation of ... |
| psradd 22276 | The addition operation of ... |
| psraddcl 22277 | Closure of the power serie... |
| rhmpsrlem1 22278 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 22279 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 22280 | The multiplication operati... |
| psrmulfval 22281 | The multiplication operati... |
| psrmulval 22282 | The multiplication operati... |
| psrmulcllem 22283 | Closure of the power serie... |
| psrmulcl 22284 | Closure of the power serie... |
| psrsca 22285 | The scalar field of the mu... |
| psrvscafval 22286 | The scalar multiplication ... |
| psrvsca 22287 | The scalar multiplication ... |
| psrvscaval 22288 | The scalar multiplication ... |
| psrvscacl 22289 | Closure of the power serie... |
| psr0cl 22290 | The zero element of the ri... |
| psr0lid 22291 | The zero element of the ri... |
| psrnegcl 22292 | The negative function in t... |
| psrlinv 22293 | The negative function in t... |
| psrgrp 22294 | The ring of power series i... |
| psr0 22295 | The zero element of the ri... |
| psrneg 22296 | The negative function of t... |
| psrlmod 22297 | The ring of power series i... |
| psr1cl 22298 | The identity element of th... |
| psrlidm 22299 | The identity element of th... |
| psrridm 22300 | The identity element of th... |
| psrass1 22301 | Associative identity for t... |
| psrdi 22302 | Distributive law for the r... |
| psrdir 22303 | Distributive law for the r... |
| psrass23l 22304 | Associative identity for t... |
| psrcom 22305 | Commutative law for the ri... |
| psrass23 22306 | Associative identities for... |
| psrring 22307 | The ring of power series i... |
| psr1 22308 | The identity element of th... |
| psrcrng 22309 | The ring of power series i... |
| psrassa 22310 | The ring of power series i... |
| resspsrbas 22311 | A restricted power series ... |
| resspsradd 22312 | A restricted power series ... |
| resspsrmul 22313 | A restricted power series ... |
| resspsrvsca 22314 | A restricted power series ... |
| subrgpsr 22315 | A subring of the base ring... |
| psrascl 22316 | Value of the scalar inject... |
| psrasclcl 22317 | A scalar is lifted into a ... |
| mvrfval 22318 | Value of the generating el... |
| mvrval 22319 | Value of the generating el... |
| mvrval2 22320 | Value of the generating el... |
| mvrid 22321 | The ` X i ` -th coefficien... |
| mvrf 22322 | The power series variable ... |
| mvrf1 22323 | The power series variable ... |
| mvrcl2 22324 | A power series variable is... |
| reldmmpl 22325 | The multivariate polynomia... |
| mplval 22326 | Value of the set of multiv... |
| mplbas 22327 | Base set of the set of mul... |
| mplelbas 22328 | Property of being a polyno... |
| mvrcl 22329 | A power series variable is... |
| mvrf2 22330 | The power series/polynomia... |
| mplrcl 22331 | Reverse closure for the po... |
| mplelsfi 22332 | A polynomial treated as a ... |
| mplval2 22333 | Self-referential expressio... |
| mplbasss 22334 | The set of polynomials is ... |
| mplelf 22335 | A polynomial is defined as... |
| mplsubglem 22336 | If ` A ` is an ideal of se... |
| mpllsslem 22337 | If ` A ` is an ideal of su... |
| mplsubglem2 22338 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 22339 | The set of polynomials is ... |
| mpllss 22340 | The set of polynomials is ... |
| mplsubrglem 22341 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 22342 | The set of polynomials is ... |
| mpl0 22343 | The zero polynomial. (Con... |
| mplplusg 22344 | Value of addition in a pol... |
| mplmulr 22345 | Value of multiplication in... |
| mpladd 22346 | The addition operation on ... |
| mplneg 22347 | The negative function on m... |
| mplmul 22348 | The multiplication operati... |
| mpl1 22349 | The identity element of th... |
| mplsca 22350 | The scalar field of a mult... |
| mplvsca2 22351 | The scalar multiplication ... |
| mplvsca 22352 | The scalar multiplication ... |
| mplvscaval 22353 | The scalar multiplication ... |
| mplgrp 22354 | The polynomial ring is a g... |
| mpllmod 22355 | The polynomial ring is a l... |
| mplring 22356 | The polynomial ring is a r... |
| mpllvec 22357 | The polynomial ring is a v... |
| mplcrng 22358 | The polynomial ring is a c... |
| mplassa 22359 | The polynomial ring is an ... |
| mplringd 22360 | The polynomial ring is a r... |
| mplcrngd 22361 | The polynomial ring is a c... |
| mpllmodd 22362 | The polynomial ring is a l... |
| mplascl0 22363 | The zero scalar as a polyn... |
| mplascl1 22364 | The one scalar as a polyno... |
| ressmplbas2 22365 | The base set of a restrict... |
| ressmplbas 22366 | A restricted polynomial al... |
| ressmpladd 22367 | A restricted polynomial al... |
| ressmplmul 22368 | A restricted polynomial al... |
| ressmplvsca 22369 | A restricted power series ... |
| subrgmpl 22370 | A subring of the base ring... |
| mplsubrgcl 22371 | An element of a polynomial... |
| subrgmvr 22372 | The variables in a subring... |
| subrgmvrf 22373 | The variables in a polynom... |
| mplmon 22374 | A monomial is a polynomial... |
| mplmonmul 22375 | The product of two monomia... |
| mplcoe1 22376 | Decompose a polynomial int... |
| mplcoe3 22377 | Decompose a monomial in on... |
| mplcoe5lem 22378 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 22379 | Decompose a monomial into ... |
| mplcoe2 22380 | Decompose a monomial into ... |
| mplbas2 22381 | An alternative expression ... |
| ltbval 22382 | Value of the well-order on... |
| ltbwe 22383 | The finite bag order is a ... |
| reldmopsr 22384 | Lemma for ordered power se... |
| opsrval 22385 | The value of the "ordered ... |
| opsrle 22386 | An alternative expression ... |
| opsrval2 22387 | Self-referential expressio... |
| opsrbaslem 22388 | Get a component of the ord... |
| opsrbas 22389 | The base set of the ordere... |
| opsrplusg 22390 | The addition operation of ... |
| opsrmulr 22391 | The multiplication operati... |
| opsrvsca 22392 | The scalar product operati... |
| opsrsca 22393 | The scalar ring of the ord... |
| opsrtoslem1 22394 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22395 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22396 | The ordered power series s... |
| opsrso 22397 | The ordered power series s... |
| opsrcrng 22398 | The ring of ordered power ... |
| opsrassa 22399 | The ring of ordered power ... |
| mplmon2 22400 | Express a scaled monomial.... |
| psrbag0 22401 | The empty bag is a bag. (... |
| psrbagsn 22402 | A singleton bag is a bag. ... |
| mplascl 22403 | Value of the scalar inject... |
| mplasclf 22404 | The scalar injection is a ... |
| subrgascl 22405 | The scalar injection funct... |
| subrgasclcl 22406 | The scalars in a polynomia... |
| mplmon2cl 22407 | A scaled monomial is a pol... |
| mplmon2mul 22408 | Product of scaled monomial... |
| mplind 22409 | Prove a property of polyno... |
| mplcoe4 22410 | Decompose a polynomial int... |
| evlslem4 22415 | The support of a tensor pr... |
| psrbagev1 22416 | A bag of multipliers provi... |
| psrbagev2 22417 | Closure of a sum using a b... |
| evlslem2 22418 | A linear function on the p... |
| evlslem3 22419 | Lemma for ~ evlseu . Poly... |
| evlslem6 22420 | Lemma for ~ evlseu . Fini... |
| evlslem1 22421 | Lemma for ~ evlseu , give ... |
| evlseu 22422 | For a given interpretation... |
| reldmevls 22423 | Well-behaved binary operat... |
| mpfrcl 22424 | Reverse closure for the se... |
| evlsval 22425 | Value of the polynomial ev... |
| evlsval2 22426 | Characterizing properties ... |
| evlsrhm 22427 | Polynomial evaluation is a... |
| evlsval3 22428 | Give a formula for the pol... |
| evlsvval 22429 | Give a formula for the eva... |
| evlsvvvallem 22430 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 22431 | Lemma for theorems using ~... |
| evlsvvval 22432 | Give a formula for the eva... |
| evlssca 22433 | Polynomial evaluation maps... |
| evlsvar 22434 | Polynomial evaluation maps... |
| evlsgsumadd 22435 | Polynomial evaluation maps... |
| evlsgsummul 22436 | Polynomial evaluation maps... |
| evlspw 22437 | Polynomial evaluation for ... |
| evlsvarpw 22438 | Polynomial evaluation for ... |
| evlval 22439 | Value of the simple/same r... |
| evlrhm 22440 | The simple evaluation map ... |
| evlcl 22441 | A polynomial over the ring... |
| evladdval 22442 | Polynomial evaluation buil... |
| evlmulval 22443 | Polynomial evaluation buil... |
| evlsscasrng 22444 | The evaluation of a scalar... |
| evlsca 22445 | Simple polynomial evaluati... |
| evlsvarsrng 22446 | The evaluation of the vari... |
| evlvar 22447 | Simple polynomial evaluati... |
| mpfconst 22448 | Constants are multivariate... |
| mpfproj 22449 | Projections are multivaria... |
| mpfsubrg 22450 | Polynomial functions are a... |
| mpff 22451 | Polynomial functions are f... |
| mpfaddcl 22452 | The sum of multivariate po... |
| mpfmulcl 22453 | The product of multivariat... |
| mpfind 22454 | Prove a property of polyno... |
| selvffval 22457 | Value of the "variable sel... |
| selvfval 22458 | Value of the "variable sel... |
| selvval 22459 | Value of the "variable sel... |
| mhmcompl 22460 | The composition of a monoi... |
| mplmapghm 22461 | The function ` H ` mapping... |
| mhmcoaddmpl 22462 | Show that the ring homomor... |
| rhmcomulmpl 22463 | Show that the ring homomor... |
| evlscl 22464 | A polynomial over the ring... |
| evlsscaval 22465 | Polynomial evaluation buil... |
| evlsvarval 22466 | Polynomial evaluation buil... |
| evlsexpval 22467 | Polynomial evaluation buil... |
| evlsaddval 22468 | Polynomial evaluation buil... |
| evlsmulval 22469 | Polynomial evaluation buil... |
| evlsmaprhm 22470 | The function ` F ` mapping... |
| evlsevl 22471 | Evaluation in a subring is... |
| evlvvval 22472 | Give a formula for the eva... |
| selvcllem1 22473 | ` T ` is an associative al... |
| selvcllem2 22474 | ` D ` is a ring homomorphi... |
| selvcllem3 22475 | The third argument passed ... |
| selvcllemh 22476 | Apply the third argument (... |
| selvcllem4 22477 | The fourth argument passed... |
| selvcllem5 22478 | The fifth argument passed ... |
| selvcl 22479 | Closure of the "variable s... |
| selvval2 22480 | Value of the "variable sel... |
| selvvvval 22481 | Recover the original polyn... |
| selvadd 22482 | The "variable selection" f... |
| selvmul 22483 | The "variable selection" f... |
| reldmmhp 22488 | The domain of the homogene... |
| mhpfval 22489 | Value of the "homogeneous ... |
| mhpval 22490 | Value of the "homogeneous ... |
| ismhp 22491 | Property of being a homoge... |
| ismhp2 22492 | Deduce a homogeneous polyn... |
| ismhp3 22493 | A polynomial is homogeneou... |
| mhprcl 22494 | Reverse closure for homoge... |
| mhpmpl 22495 | A homogeneous polynomial i... |
| mhpdeg 22496 | All nonzero terms of a hom... |
| mhp0cl 22497 | The zero polynomial is hom... |
| mhpsclcl 22498 | A scalar (or constant) pol... |
| mhpvarcl 22499 | A power series variable is... |
| mhpmulcl 22500 | A product of homogeneous p... |
| mhppwdeg 22501 | Degree of a homogeneous po... |
| mhpaddcl 22502 | Homogeneous polynomials ar... |
| mhpinvcl 22503 | Homogeneous polynomials ar... |
| mhpsubg 22504 | Homogeneous polynomials fo... |
| mhpvscacl 22505 | Homogeneous polynomials ar... |
| mhplss 22506 | Homogeneous polynomials fo... |
| psdffval 22508 | Value of the power series ... |
| psdfval 22509 | Give a map between power s... |
| psdval 22510 | Evaluate the partial deriv... |
| psdcoef 22511 | Coefficient of a term of t... |
| psdcl 22512 | The derivative of a power ... |
| psdmplcl 22513 | The derivative of a polyno... |
| psdadd 22514 | The derivative of a sum is... |
| psdvsca 22515 | The derivative of a scaled... |
| psdmullem 22516 | Lemma for ~ psdmul . Tran... |
| psdmul 22517 | Product rule for power ser... |
| psd1 22518 | The derivative of one is z... |
| psdascl 22519 | The derivative of a consta... |
| psdmvr 22520 | The partial derivative of ... |
| psdpw 22521 | Power rule for partial der... |
| psr1baslem 22533 | The set of finite bags on ... |
| psr1val 22534 | Value of the ring of univa... |
| psr1crng 22535 | The ring of univariate pow... |
| psr1assa 22536 | The ring of univariate pow... |
| psr1tos 22537 | The ordered power series s... |
| psr1bas2 22538 | The base set of the ring o... |
| psr1bas 22539 | The base set of the ring o... |
| vr1val 22540 | The value of the generator... |
| vr1cl2 22541 | The variable ` X ` is a me... |
| ply1val 22542 | The value of the set of un... |
| ply1bas 22543 | The value of the base set ... |
| ply1lss 22544 | Univariate polynomials for... |
| ply1subrg 22545 | Univariate polynomials for... |
| ply1crng 22546 | The ring of univariate pol... |
| ply1assa 22547 | The ring of univariate pol... |
| psr1bascl 22548 | A univariate power series ... |
| psr1basf 22549 | Univariate power series ba... |
| ply1basf 22550 | Univariate polynomial base... |
| ply1bascl 22551 | A univariate polynomial is... |
| ply1bascl2 22552 | A univariate polynomial is... |
| coe1fval 22553 | Value of the univariate po... |
| coe1fv 22554 | Value of an evaluated coef... |
| fvcoe1 22555 | Value of a multivariate co... |
| coe1fval3 22556 | Univariate power series co... |
| coe1f2 22557 | Functionality of univariat... |
| coe1fval2 22558 | Univariate polynomial coef... |
| coe1f 22559 | Functionality of univariat... |
| coe1fvalcl 22560 | A coefficient of a univari... |
| coe1sfi 22561 | Finite support of univaria... |
| coe1fsupp 22562 | The coefficient vector of ... |
| mptcoe1fsupp 22563 | A mapping involving coeffi... |
| coe1ae0 22564 | The coefficient vector of ... |
| vr1cl 22565 | The generator of a univari... |
| opsr0 22566 | Zero in the ordered power ... |
| opsr1 22567 | One in the ordered power s... |
| psr1plusg 22568 | Value of addition in a uni... |
| psr1vsca 22569 | Value of scalar multiplica... |
| psr1mulr 22570 | Value of multiplication in... |
| ply1plusg 22571 | Value of addition in a uni... |
| ply1vsca 22572 | Value of scalar multiplica... |
| ply1mulr 22573 | Value of multiplication in... |
| ply1ass23l 22574 | Associative identity with ... |
| ressply1bas2 22575 | The base set of a restrict... |
| ressply1bas 22576 | A restricted polynomial al... |
| ressply1add 22577 | A restricted polynomial al... |
| ressply1mul 22578 | A restricted polynomial al... |
| ressply1vsca 22579 | A restricted power series ... |
| subrgply1 22580 | A subring of the base ring... |
| gsumply1subr 22581 | Evaluate a group sum in a ... |
| psrbaspropd 22582 | Property deduction for pow... |
| psrplusgpropd 22583 | Property deduction for pow... |
| mplbaspropd 22584 | Property deduction for pol... |
| psropprmul 22585 | Reversing multiplication i... |
| ply1opprmul 22586 | Reversing multiplication i... |
| 00ply1bas 22587 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22588 | Protection compatibility o... |
| ply1plusgfvi 22589 | Protection compatibility o... |
| ply1baspropd 22590 | Property deduction for uni... |
| ply1plusgpropd 22591 | Property deduction for uni... |
| opsrring 22592 | Ordered power series form ... |
| opsrlmod 22593 | Ordered power series form ... |
| psr1ring 22594 | Univariate power series fo... |
| ply1ring 22595 | Univariate polynomials for... |
| psr1lmod 22596 | Univariate power series fo... |
| psr1sca 22597 | Scalars of a univariate po... |
| psr1sca2 22598 | Scalars of a univariate po... |
| ply1lmod 22599 | Univariate polynomials for... |
| ply1sca 22600 | Scalars of a univariate po... |
| ply1sca2 22601 | Scalars of a univariate po... |
| ply1ascl0 22602 | The zero scalar as a polyn... |
| ply1ascl1 22603 | The multiplicative identit... |
| ply1mpl0 22604 | The univariate polynomial ... |
| ply10s0 22605 | Zero times a univariate po... |
| ply1mpl1 22606 | The univariate polynomial ... |
| ply1ascl 22607 | The univariate polynomial ... |
| subrg1ascl 22608 | The scalar injection funct... |
| subrg1asclcl 22609 | The scalars in a polynomia... |
| subrgvr1 22610 | The variables in a subring... |
| subrgvr1cl 22611 | The variables in a polynom... |
| coe1z 22612 | The coefficient vector of ... |
| coe1add 22613 | The coefficient vector of ... |
| coe1addfv 22614 | A particular coefficient o... |
| coe1subfv 22615 | A particular coefficient o... |
| coe1mul2lem1 22616 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22617 | An equivalence for ~ coe1m... |
| coe1mul2 22618 | The coefficient vector of ... |
| coe1mul 22619 | The coefficient vector of ... |
| ply1moncl 22620 | Closure of the expression ... |
| ply1tmcl 22621 | Closure of the expression ... |
| coe1tm 22622 | Coefficient vector of a po... |
| coe1tmfv1 22623 | Nonzero coefficient of a p... |
| coe1tmfv2 22624 | Zero coefficient of a poly... |
| coe1tmmul2 22625 | Coefficient vector of a po... |
| coe1tmmul 22626 | Coefficient vector of a po... |
| coe1tmmul2fv 22627 | Function value of a right-... |
| coe1pwmul 22628 | Coefficient vector of a po... |
| coe1pwmulfv 22629 | Function value of a right-... |
| ply1scltm 22630 | A scalar is a term with ze... |
| coe1sclmul 22631 | Coefficient vector of a po... |
| coe1sclmulfv 22632 | A single coefficient of a ... |
| coe1sclmul2 22633 | Coefficient vector of a po... |
| ply1sclf 22634 | A scalar polynomial is a p... |
| ply1sclcl 22635 | The value of the algebra s... |
| coe1scl 22636 | Coefficient vector of a sc... |
| ply1sclid 22637 | Recover the base scalar fr... |
| ply1sclf1 22638 | The polynomial scalar func... |
| ply1scl0 22639 | The zero scalar is zero. ... |
| ply1scln0 22640 | Nonzero scalars create non... |
| ply1scl1 22641 | The one scalar is the unit... |
| coe1id 22642 | Coefficient vector of the ... |
| ply1idvr1 22643 | The identity of a polynomi... |
| cply1mul 22644 | The product of two constan... |
| ply1coefsupp 22645 | The decomposition of a uni... |
| ply1coe 22646 | Decompose a univariate pol... |
| eqcoe1ply1eq 22647 | Two polynomials over the s... |
| ply1coe1eq 22648 | Two polynomials over the s... |
| cply1coe0 22649 | All but the first coeffici... |
| cply1coe0bi 22650 | A polynomial is constant (... |
| coe1fzgsumdlem 22651 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22652 | Value of an evaluated coef... |
| ply1scleq 22653 | Equality of a constant pol... |
| ply1chr 22654 | The characteristic of a po... |
| gsumsmonply1 22655 | A finite group sum of scal... |
| gsummoncoe1 22656 | A coefficient of the polyn... |
| gsumply1eq 22657 | Two univariate polynomials... |
| lply1binom 22658 | The binomial theorem for l... |
| lply1binomsc 22659 | The binomial theorem for l... |
| ply1fermltlchr 22660 | Fermat's little theorem fo... |
| reldmevls1 22665 | Well-behaved binary operat... |
| ply1frcl 22666 | Reverse closure for the se... |
| evls1fval 22667 | Value of the univariate po... |
| evls1val 22668 | Value of the univariate po... |
| evls1rhmlem 22669 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22670 | Polynomial evaluation is a... |
| evls1sca 22671 | Univariate polynomial eval... |
| evls1gsumadd 22672 | Univariate polynomial eval... |
| evls1gsummul 22673 | Univariate polynomial eval... |
| evls1pw 22674 | Univariate polynomial eval... |
| evls1varpw 22675 | Univariate polynomial eval... |
| evl1fval 22676 | Value of the simple/same r... |
| evl1val 22677 | Value of the simple/same r... |
| evl1fval1lem 22678 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22679 | Value of the simple/same r... |
| evl1rhm 22680 | Polynomial evaluation is a... |
| fveval1fvcl 22681 | The function value of the ... |
| evl1sca 22682 | Polynomial evaluation maps... |
| evl1scad 22683 | Polynomial evaluation buil... |
| evl1var 22684 | Polynomial evaluation maps... |
| evl1vard 22685 | Polynomial evaluation buil... |
| evls1var 22686 | Univariate polynomial eval... |
| evls1scasrng 22687 | The evaluation of a scalar... |
| evls1varsrng 22688 | The evaluation of the vari... |
| evl1addd 22689 | Polynomial evaluation buil... |
| evl1subd 22690 | Polynomial evaluation buil... |
| evl1muld 22691 | Polynomial evaluation buil... |
| evl1vsd 22692 | Polynomial evaluation buil... |
| evl1expd 22693 | Polynomial evaluation buil... |
| pf1const 22694 | Constants are polynomial f... |
| pf1id 22695 | The identity is a polynomi... |
| pf1subrg 22696 | Polynomial functions are a... |
| pf1rcl 22697 | Reverse closure for the se... |
| pf1f 22698 | Polynomial functions are f... |
| mpfpf1 22699 | Convert a multivariate pol... |
| pf1mpf 22700 | Convert a univariate polyn... |
| pf1addcl 22701 | The sum of multivariate po... |
| pf1mulcl 22702 | The product of multivariat... |
| pf1ind 22703 | Prove a property of polyno... |
| evl1gsumdlem 22704 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22705 | Polynomial evaluation buil... |
| evl1gsumadd 22706 | Univariate polynomial eval... |
| evl1gsumaddval 22707 | Value of a univariate poly... |
| evl1gsummul 22708 | Univariate polynomial eval... |
| evl1varpw 22709 | Univariate polynomial eval... |
| evl1varpwval 22710 | Value of a univariate poly... |
| evl1scvarpw 22711 | Univariate polynomial eval... |
| evl1scvarpwval 22712 | Value of a univariate poly... |
| evl1gsummon 22713 | Value of a univariate poly... |
| evls1scafv 22714 | Value of the univariate po... |
| evls1expd 22715 | Univariate polynomial eval... |
| evls1varpwval 22716 | Univariate polynomial eval... |
| evls1fpws 22717 | Evaluation of a univariate... |
| ressply1evl 22718 | Evaluation of a univariate... |
| evls1addd 22719 | Univariate polynomial eval... |
| evls1muld 22720 | Univariate polynomial eval... |
| evls1vsca 22721 | Univariate polynomial eval... |
| asclply1subcl 22722 | Closure of the algebra sca... |
| evls1fvcl 22723 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22724 | The function ` F ` mapping... |
| evls1maplmhm 22725 | The function ` F ` mapping... |
| evls1maprnss 22726 | The function ` F ` mapping... |
| evl1maprhm 22727 | The function ` F ` mapping... |
| rhmmpl 22728 | Provide a ring homomorphis... |
| ply1vscl 22729 | Closure of scalar multipli... |
| mhmcoply1 22730 | The composition of a monoi... |
| rhmply1 22731 | Provide a ring homomorphis... |
| rhmply1vr1 22732 | A ring homomorphism betwee... |
| rhmply1vsca 22733 | Apply a ring homomorphism ... |
| rhmply1mon 22734 | Apply a ring homomorphism ... |
| mamufval 22737 | Functional value of the ma... |
| mamuval 22738 | Multiplication of two matr... |
| mamufv 22739 | A cell in the multiplicati... |
| mamudm 22740 | The domain of the matrix m... |
| mamufacex 22741 | Every solution of the equa... |
| mamures 22742 | Rows in a matrix product a... |
| grpvlinv 22743 | Tuple-wise left inverse in... |
| grpvrinv 22744 | Tuple-wise right inverse i... |
| ringvcl 22745 | Tuple-wise multiplication ... |
| mamucl 22746 | Operation closure of matri... |
| mamuass 22747 | Matrix multiplication is a... |
| mamudi 22748 | Matrix multiplication dist... |
| mamudir 22749 | Matrix multiplication dist... |
| mamuvs1 22750 | Matrix multiplication dist... |
| mamuvs2 22751 | Matrix multiplication dist... |
| matbas0pc 22754 | There is no matrix with a ... |
| matbas0 22755 | There is no matrix for a n... |
| matval 22756 | Value of the matrix algebr... |
| matrcl 22757 | Reverse closure for the ma... |
| matbas 22758 | The matrix ring has the sa... |
| matplusg 22759 | The matrix ring has the sa... |
| matsca 22760 | The matrix ring has the sa... |
| matvsca 22761 | The matrix ring has the sa... |
| mat0 22762 | The matrix ring has the sa... |
| matinvg 22763 | The matrix ring has the sa... |
| mat0op 22764 | Value of a zero matrix as ... |
| matsca2 22765 | The scalars of the matrix ... |
| matbas2 22766 | The base set of the matrix... |
| matbas2i 22767 | A matrix is a function. (... |
| matbas2d 22768 | The base set of the matrix... |
| eqmat 22769 | Two square matrices of the... |
| matecl 22770 | Each entry (according to W... |
| matecld 22771 | Each entry (according to W... |
| matplusg2 22772 | Addition in the matrix rin... |
| matvsca2 22773 | Scalar multiplication in t... |
| matlmod 22774 | The matrix ring is a linea... |
| matgrp 22775 | The matrix ring is a group... |
| matvscl 22776 | Closure of the scalar mult... |
| matsubg 22777 | The matrix ring has the sa... |
| matplusgcell 22778 | Addition in the matrix rin... |
| matsubgcell 22779 | Subtraction in the matrix ... |
| matinvgcell 22780 | Additive inversion in the ... |
| matvscacell 22781 | Scalar multiplication in t... |
| matgsum 22782 | Finite commutative sums in... |
| matmulr 22783 | Multiplication in the matr... |
| mamumat1cl 22784 | The identity matrix (as op... |
| mat1comp 22785 | The components of the iden... |
| mamulid 22786 | The identity matrix (as op... |
| mamurid 22787 | The identity matrix (as op... |
| matring 22788 | Existence of the matrix ri... |
| matassa 22789 | Existence of the matrix al... |
| matmulcell 22790 | Multiplication in the matr... |
| mpomatmul 22791 | Multiplication of two N x ... |
| mat1 22792 | Value of an identity matri... |
| mat1ov 22793 | Entries of an identity mat... |
| mat1bas 22794 | The identity matrix is a m... |
| matsc 22795 | The identity matrix multip... |
| ofco2 22796 | Distribution law for the f... |
| oftpos 22797 | The transposition of the v... |
| mattposcl 22798 | The transpose of a square ... |
| mattpostpos 22799 | The transpose of the trans... |
| mattposvs 22800 | The transposition of a mat... |
| mattpos1 22801 | The transposition of the i... |
| tposmap 22802 | The transposition of an I ... |
| mamutpos 22803 | Behavior of transposes in ... |
| mattposm 22804 | Multiplying two transposed... |
| matgsumcl 22805 | Closure of a group sum ove... |
| madetsumid 22806 | The identity summand in th... |
| matepmcl 22807 | Each entry of a matrix wit... |
| matepm2cl 22808 | Each entry of a matrix wit... |
| madetsmelbas 22809 | A summand of the determina... |
| madetsmelbas2 22810 | A summand of the determina... |
| mat0dimbas0 22811 | The empty set is the one a... |
| mat0dim0 22812 | The zero of the algebra of... |
| mat0dimid 22813 | The identity of the algebr... |
| mat0dimscm 22814 | The scalar multiplication ... |
| mat0dimcrng 22815 | The algebra of matrices wi... |
| mat1dimelbas 22816 | A matrix with dimension 1 ... |
| mat1dimbas 22817 | A matrix with dimension 1 ... |
| mat1dim0 22818 | The zero of the algebra of... |
| mat1dimid 22819 | The identity of the algebr... |
| mat1dimscm 22820 | The scalar multiplication ... |
| mat1dimmul 22821 | The ring multiplication in... |
| mat1dimcrng 22822 | The algebra of matrices wi... |
| mat1f1o 22823 | There is a 1-1 function fr... |
| mat1rhmval 22824 | The value of the ring homo... |
| mat1rhmelval 22825 | The value of the ring homo... |
| mat1rhmcl 22826 | The value of the ring homo... |
| mat1f 22827 | There is a function from a... |
| mat1ghm 22828 | There is a group homomorph... |
| mat1mhm 22829 | There is a monoid homomorp... |
| mat1rhm 22830 | There is a ring homomorphi... |
| mat1rngiso 22831 | There is a ring isomorphis... |
| mat1ric 22832 | A ring is isomorphic to th... |
| dmatval 22837 | The set of ` N ` x ` N ` d... |
| dmatel 22838 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22839 | An ` N ` x ` N ` diagonal ... |
| dmatid 22840 | The identity matrix is a d... |
| dmatelnd 22841 | An extradiagonal entry of ... |
| dmatmul 22842 | The product of two diagona... |
| dmatsubcl 22843 | The difference of two diag... |
| dmatsgrp 22844 | The set of diagonal matric... |
| dmatmulcl 22845 | The product of two diagona... |
| dmatsrng 22846 | The set of diagonal matric... |
| dmatcrng 22847 | The subring of diagonal ma... |
| dmatscmcl 22848 | The multiplication of a di... |
| scmatval 22849 | The set of ` N ` x ` N ` s... |
| scmatel 22850 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22851 | A scalar matrix can be exp... |
| scmatscmide 22852 | An entry of a scalar matri... |
| scmatscmiddistr 22853 | Distributive law for scala... |
| scmatmat 22854 | An ` N ` x ` N ` scalar ma... |
| scmate 22855 | An entry of an ` N ` x ` N... |
| scmatmats 22856 | The set of an ` N ` x ` N ... |
| scmateALT 22857 | Alternate proof of ~ scmat... |
| scmatscm 22858 | The multiplication of a ma... |
| scmatid 22859 | The identity matrix is a s... |
| scmatdmat 22860 | A scalar matrix is a diago... |
| scmataddcl 22861 | The sum of two scalar matr... |
| scmatsubcl 22862 | The difference of two scal... |
| scmatmulcl 22863 | The product of two scalar ... |
| scmatsgrp 22864 | The set of scalar matrices... |
| scmatsrng 22865 | The set of scalar matrices... |
| scmatcrng 22866 | The subring of scalar matr... |
| scmatsgrp1 22867 | The set of scalar matrices... |
| scmatsrng1 22868 | The set of scalar matrices... |
| smatvscl 22869 | Closure of the scalar mult... |
| scmatlss 22870 | The set of scalar matrices... |
| scmatstrbas 22871 | The set of scalar matrices... |
| scmatrhmval 22872 | The value of the ring homo... |
| scmatrhmcl 22873 | The value of the ring homo... |
| scmatf 22874 | There is a function from a... |
| scmatfo 22875 | There is a function from a... |
| scmatf1 22876 | There is a 1-1 function fr... |
| scmatf1o 22877 | There is a bijection betwe... |
| scmatghm 22878 | There is a group homomorph... |
| scmatmhm 22879 | There is a monoid homomorp... |
| scmatrhm 22880 | There is a ring homomorphi... |
| scmatrngiso 22881 | There is a ring isomorphis... |
| scmatric 22882 | A ring is isomorphic to ev... |
| mat0scmat 22883 | The empty matrix over a ri... |
| mat1scmat 22884 | A 1-dimensional matrix ove... |
| mvmulfval 22887 | Functional value of the ma... |
| mvmulval 22888 | Multiplication of a vector... |
| mvmulfv 22889 | A cell/element in the vect... |
| mavmulval 22890 | Multiplication of a vector... |
| mavmulfv 22891 | A cell/element in the vect... |
| mavmulcl 22892 | Multiplication of an NxN m... |
| 1mavmul 22893 | Multiplication of the iden... |
| mavmulass 22894 | Associativity of the multi... |
| mavmuldm 22895 | The domain of the matrix v... |
| mavmulsolcl 22896 | Every solution of the equa... |
| mavmul0 22897 | Multiplication of a 0-dime... |
| mavmul0g 22898 | The result of the 0-dimens... |
| mvmumamul1 22899 | The multiplication of an M... |
| mavmumamul1 22900 | The multiplication of an N... |
| marrepfval 22905 | First substitution for the... |
| marrepval0 22906 | Second substitution for th... |
| marrepval 22907 | Third substitution for the... |
| marrepeval 22908 | An entry of a matrix with ... |
| marrepcl 22909 | Closure of the row replace... |
| marepvfval 22910 | First substitution for the... |
| marepvval0 22911 | Second substitution for th... |
| marepvval 22912 | Third substitution for the... |
| marepveval 22913 | An entry of a matrix with ... |
| marepvcl 22914 | Closure of the column repl... |
| ma1repvcl 22915 | Closure of the column repl... |
| ma1repveval 22916 | An entry of an identity ma... |
| mulmarep1el 22917 | Element by element multipl... |
| mulmarep1gsum1 22918 | The sum of element by elem... |
| mulmarep1gsum2 22919 | The sum of element by elem... |
| 1marepvmarrepid 22920 | Replacing the ith row by 0... |
| submabas 22923 | Any subset of the index se... |
| submafval 22924 | First substitution for a s... |
| submaval0 22925 | Second substitution for a ... |
| submaval 22926 | Third substitution for a s... |
| submaeval 22927 | An entry of a submatrix of... |
| 1marepvsma1 22928 | The submatrix of the ident... |
| mdetfval 22931 | First substitution for the... |
| mdetleib 22932 | Full substitution of our d... |
| mdetleib2 22933 | Leibniz' formula can also ... |
| nfimdetndef 22934 | The determinant is not def... |
| mdetfval1 22935 | First substitution of an a... |
| mdetleib1 22936 | Full substitution of an al... |
| mdet0pr 22937 | The determinant function f... |
| mdet0f1o 22938 | The determinant function f... |
| mdet0fv0 22939 | The determinant of the emp... |
| mdetf 22940 | Functionality of the deter... |
| mdetcl 22941 | The determinant evaluates ... |
| m1detdiag 22942 | The determinant of a 1-dim... |
| mdetdiaglem 22943 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22944 | The determinant of a diago... |
| mdetdiagid 22945 | The determinant of a diago... |
| mdet1 22946 | The determinant of the ide... |
| mdetrlin 22947 | The determinant function i... |
| mdetrsca 22948 | The determinant function i... |
| mdetrsca2 22949 | The determinant function i... |
| mdetr0 22950 | The determinant of a matri... |
| mdet0 22951 | The determinant of the zer... |
| mdetrlin2 22952 | The determinant function i... |
| mdetralt 22953 | The determinant function i... |
| mdetralt2 22954 | The determinant function i... |
| mdetero 22955 | The determinant function i... |
| mdettpos 22956 | Determinant is invariant u... |
| mdetunilem1 22957 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22958 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22959 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22960 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22961 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22962 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22963 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22964 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22965 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22966 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22967 | According to the definitio... |
| mdetmul 22968 | Multiplicativity of the de... |
| m2detleiblem1 22969 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22970 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22971 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22972 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22973 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22974 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22975 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22976 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22981 | Lemma for ~ maducoeval2 . ... |
| madufval 22982 | First substitution for the... |
| maduval 22983 | Second substitution for th... |
| maducoeval 22984 | An entry of the adjunct (c... |
| maducoeval2 22985 | An entry of the adjunct (c... |
| maduf 22986 | Creating the adjunct of ma... |
| madutpos 22987 | The adjuct of a transposed... |
| madugsum 22988 | The determinant of a matri... |
| madurid 22989 | Multiplying a matrix with ... |
| madulid 22990 | Multiplying the adjunct of... |
| minmar1fval 22991 | First substitution for the... |
| minmar1val0 22992 | Second substitution for th... |
| minmar1val 22993 | Third substitution for the... |
| minmar1eval 22994 | An entry of a matrix for a... |
| minmar1marrep 22995 | The minor matrix is a spec... |
| minmar1cl 22996 | Closure of the row replace... |
| maducoevalmin1 22997 | The coefficients of an adj... |
| symgmatr01lem 22998 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22999 | Applying a permutation tha... |
| gsummatr01lem1 23000 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 23001 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 23002 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 23003 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 23004 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 23005 | Replacing a row of a squar... |
| smadiadetlem0 23006 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 23007 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 23008 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 23009 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 23010 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 23011 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 23012 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 23013 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 23014 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 23015 | The determinant of a subma... |
| smadiadetglem1 23016 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 23017 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 23018 | The determinant of a squar... |
| smadiadetg0 23019 | Lemma for ~ smadiadetr : v... |
| smadiadetr 23020 | The determinant of a squar... |
| invrvald 23021 | If a matrix multiplied wit... |
| matinv 23022 | The inverse of a matrix is... |
| matunit 23023 | A matrix is a unit in the ... |
| matunitlindflem1 23024 | One direction of ~ matunit... |
| matunitlindflem2 23025 | One direction of ~ matunit... |
| matunitlindf 23026 | A matrix over a field is i... |
| slesolvec 23027 | Every solution of a system... |
| slesolinv 23028 | The solution of a system o... |
| slesolinvbi 23029 | The solution of a system o... |
| slesolex 23030 | Every system of linear equ... |
| cramerimplem1 23031 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 23032 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 23033 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 23034 | One direction of Cramer's ... |
| cramerlem1 23035 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 23036 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 23037 | Lemma 3 for ~ cramer . (C... |
| cramer0 23038 | Special case of Cramer's r... |
| cramer 23039 | Cramer's rule. According ... |
| pmatring 23040 | The set of polynomial matr... |
| pmatlmod 23041 | The set of polynomial matr... |
| pmatassa 23042 | The set of polynomial matr... |
| pmat0op 23043 | The zero polynomial matrix... |
| pmat1op 23044 | The identity polynomial ma... |
| pmat1ovd 23045 | Entries of the identity po... |
| pmat0opsc 23046 | The zero polynomial matrix... |
| pmat1opsc 23047 | The identity polynomial ma... |
| pmat1ovscd 23048 | Entries of the identity po... |
| pmatcoe1fsupp 23049 | For a polynomial matrix th... |
| 1pmatscmul 23050 | The scalar product of the ... |
| cpmat 23057 | Value of the constructor o... |
| cpmatpmat 23058 | A constant polynomial matr... |
| cpmatel 23059 | Property of a constant pol... |
| cpmatelimp 23060 | Implication of a set being... |
| cpmatel2 23061 | Another property of a cons... |
| cpmatelimp2 23062 | Another implication of a s... |
| 1elcpmat 23063 | The identity of the ring o... |
| cpmatacl 23064 | The set of all constant po... |
| cpmatinvcl 23065 | The set of all constant po... |
| cpmatmcllem 23066 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 23067 | The set of all constant po... |
| cpmatsubgpmat 23068 | The set of all constant po... |
| cpmatsrgpmat 23069 | The set of all constant po... |
| 0elcpmat 23070 | The zero of the ring of al... |
| mat2pmatfval 23071 | Value of the matrix transf... |
| mat2pmatval 23072 | The result of a matrix tra... |
| mat2pmatvalel 23073 | A (matrix) element of the ... |
| mat2pmatbas 23074 | The result of a matrix tra... |
| mat2pmatbas0 23075 | The result of a matrix tra... |
| mat2pmatf 23076 | The matrix transformation ... |
| mat2pmatf1 23077 | The matrix transformation ... |
| mat2pmatghm 23078 | The transformation of matr... |
| mat2pmatmul 23079 | The transformation of matr... |
| mat2pmat1 23080 | The transformation of the ... |
| mat2pmatmhm 23081 | The transformation of matr... |
| mat2pmatrhm 23082 | The transformation of matr... |
| mat2pmatlin 23083 | The transformation of matr... |
| 0mat2pmat 23084 | The transformed zero matri... |
| idmatidpmat 23085 | The transformed identity m... |
| d0mat2pmat 23086 | The transformed empty set ... |
| d1mat2pmat 23087 | The transformation of a ma... |
| mat2pmatscmxcl 23088 | A transformed matrix multi... |
| m2cpm 23089 | The result of a matrix tra... |
| m2cpmf 23090 | The matrix transformation ... |
| m2cpmf1 23091 | The matrix transformation ... |
| m2cpmghm 23092 | The transformation of matr... |
| m2cpmmhm 23093 | The transformation of matr... |
| m2cpmrhm 23094 | The transformation of matr... |
| m2pmfzmap 23095 | The transformed values of ... |
| m2pmfzgsumcl 23096 | Closure of the sum of scal... |
| cpm2mfval 23097 | Value of the inverse matri... |
| cpm2mval 23098 | The result of an inverse m... |
| cpm2mvalel 23099 | A (matrix) element of the ... |
| cpm2mf 23100 | The inverse matrix transfo... |
| m2cpminvid 23101 | The inverse transformation... |
| m2cpminvid2lem 23102 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 23103 | The transformation applied... |
| m2cpmfo 23104 | The matrix transformation ... |
| m2cpmf1o 23105 | The matrix transformation ... |
| m2cpmrngiso 23106 | The transformation of matr... |
| matcpmric 23107 | The ring of matrices over ... |
| m2cpminv 23108 | The inverse matrix transfo... |
| m2cpminv0 23109 | The inverse matrix transfo... |
| decpmatval0 23112 | The matrix consisting of t... |
| decpmatval 23113 | The matrix consisting of t... |
| decpmate 23114 | An entry of the matrix con... |
| decpmatcl 23115 | Closure of the decompositi... |
| decpmataa0 23116 | The matrix consisting of t... |
| decpmatfsupp 23117 | The mapping to the matrice... |
| decpmatid 23118 | The matrix consisting of t... |
| decpmatmullem 23119 | Lemma for ~ decpmatmul . ... |
| decpmatmul 23120 | The matrix consisting of t... |
| decpmatmulsumfsupp 23121 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 23122 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 23123 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 23124 | Write a polynomial matrix ... |
| pmatcollpw2lem 23125 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 23126 | Write a polynomial matrix ... |
| monmatcollpw 23127 | The matrix consisting of t... |
| pmatcollpwlem 23128 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 23129 | Write a polynomial matrix ... |
| pmatcollpwfi 23130 | Write a polynomial matrix ... |
| pmatcollpw3lem 23131 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 23132 | Write a polynomial matrix ... |
| pmatcollpw3fi 23133 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 23134 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 23135 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 23136 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 23137 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 23138 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 23139 | Write a scalar matrix over... |
| pm2mpf1lem 23142 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 23143 | Value of the transformatio... |
| pm2mpfval 23144 | A polynomial matrix transf... |
| pm2mpcl 23145 | The transformation of poly... |
| pm2mpf 23146 | The transformation of poly... |
| pm2mpf1 23147 | The transformation of poly... |
| pm2mpcoe1 23148 | A coefficient of the polyn... |
| idpm2idmp 23149 | The transformation of the ... |
| mptcoe1matfsupp 23150 | The mapping extracting the... |
| mply1topmatcllem 23151 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 23152 | A polynomial over matrices... |
| mply1topmatcl 23153 | A polynomial over matrices... |
| mp2pm2mplem1 23154 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 23155 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 23156 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 23157 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 23158 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 23159 | A polynomial over matrices... |
| pm2mpghmlem2 23160 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 23161 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 23162 | The transformation of poly... |
| pm2mpf1o 23163 | The transformation of poly... |
| pm2mpghm 23164 | The transformation of poly... |
| pm2mpgrpiso 23165 | The transformation of poly... |
| pm2mpmhmlem1 23166 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 23167 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 23168 | The transformation of poly... |
| pm2mprhm 23169 | The transformation of poly... |
| pm2mprngiso 23170 | The transformation of poly... |
| pmmpric 23171 | The ring of polynomial mat... |
| monmat2matmon 23172 | The transformation of a po... |
| pm2mp 23173 | The transformation of a su... |
| chmatcl 23176 | Closure of the characteris... |
| chmatval 23177 | The entries of the charact... |
| chpmatfval 23178 | Value of the characteristi... |
| chpmatval 23179 | The characteristic polynom... |
| chpmatply1 23180 | The characteristic polynom... |
| chpmatval2 23181 | The characteristic polynom... |
| chpmat0d 23182 | The characteristic polynom... |
| chpmat1dlem 23183 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 23184 | The characteristic polynom... |
| chpdmatlem0 23185 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 23186 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 23187 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 23188 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 23189 | The characteristic polynom... |
| chpscmat 23190 | The characteristic polynom... |
| chpscmat0 23191 | The characteristic polynom... |
| chpscmatgsumbin 23192 | The characteristic polynom... |
| chpscmatgsummon 23193 | The characteristic polynom... |
| chp0mat 23194 | The characteristic polynom... |
| chpidmat 23195 | The characteristic polynom... |
| chmaidscmat 23196 | The characteristic polynom... |
| fvmptnn04if 23197 | The function values of a m... |
| fvmptnn04ifa 23198 | The function value of a ma... |
| fvmptnn04ifb 23199 | The function value of a ma... |
| fvmptnn04ifc 23200 | The function value of a ma... |
| fvmptnn04ifd 23201 | The function value of a ma... |
| chfacfisf 23202 | The "characteristic factor... |
| chfacfisfcpmat 23203 | The "characteristic factor... |
| chfacffsupp 23204 | The "characteristic factor... |
| chfacfscmulcl 23205 | Closure of a scaled value ... |
| chfacfscmul0 23206 | A scaled value of the "cha... |
| chfacfscmulfsupp 23207 | A mapping of scaled values... |
| chfacfscmulgsum 23208 | Breaking up a sum of value... |
| chfacfpmmulcl 23209 | Closure of the value of th... |
| chfacfpmmul0 23210 | The value of the "characte... |
| chfacfpmmulfsupp 23211 | A mapping of values of the... |
| chfacfpmmulgsum 23212 | Breaking up a sum of value... |
| chfacfpmmulgsum2 23213 | Breaking up a sum of value... |
| cayhamlem1 23214 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 23215 | The right-hand fundamental... |
| cpmidgsum 23216 | Representation of the iden... |
| cpmidgsumm2pm 23217 | Representation of the iden... |
| cpmidpmatlem1 23218 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 23219 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 23220 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 23221 | Representation of the iden... |
| cpmadugsumlemB 23222 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 23223 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 23224 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 23225 | The product of the charact... |
| cpmadugsum 23226 | The product of the charact... |
| cpmidgsum2 23227 | Representation of the iden... |
| cpmidg2sum 23228 | Equality of two sums repre... |
| cpmadumatpolylem1 23229 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 23230 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 23231 | The product of the charact... |
| cayhamlem2 23232 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 23233 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 23234 | The coefficients of the ch... |
| cayhamlem3 23235 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 23236 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 23237 | The Cayley-Hamilton theore... |
| cayleyhamilton 23238 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 23239 | Alternate proof of ~ cayle... |
| cayleyhamilton1 23240 | The Cayley-Hamilton theore... |
| istopg 23243 | Express the predicate " ` ... |
| istop2g 23244 | Express the predicate " ` ... |
| uniopn 23245 | The union of a subset of a... |
| iunopn 23246 | The indexed union of a sub... |
| inopn 23247 | The intersection of two op... |
| fitop 23248 | A topology is closed under... |
| fiinopn 23249 | The intersection of a none... |
| iinopn 23250 | The intersection of a none... |
| unopn 23251 | The union of two open sets... |
| 0opn 23252 | The empty set is an open s... |
| 0ntop 23253 | The empty set is not a top... |
| topopn 23254 | The underlying set of a to... |
| eltopss 23255 | A member of a topology is ... |
| riinopn 23256 | A finite indexed relative ... |
| rintopn 23257 | A finite relative intersec... |
| istopon 23260 | Property of being a topolo... |
| topontop 23261 | A topology on a given base... |
| toponuni 23262 | The base set of a topology... |
| topontopi 23263 | A topology on a given base... |
| toponunii 23264 | The base set of a topology... |
| toptopon 23265 | Alternative definition of ... |
| toptopon2 23266 | A topology is the same thi... |
| topontopon 23267 | A topology on a set is a t... |
| funtopon 23268 | The class ` TopOn ` is a f... |
| toponrestid 23269 | Given a topology on a set,... |
| toponsspwpw 23270 | The set of topologies on a... |
| dmtopon 23271 | The domain of ` TopOn ` is... |
| fntopon 23272 | The class ` TopOn ` is a f... |
| toprntopon 23273 | A topology is the same thi... |
| toponmax 23274 | The base set of a topology... |
| toponss 23275 | A member of a topology is ... |
| toponcom 23276 | If ` K ` is a topology on ... |
| toponcomb 23277 | Biconditional form of ~ to... |
| topgele 23278 | The topologies over the sa... |
| topsn 23279 | The only topology on a sin... |
| istps 23282 | Express the predicate "is ... |
| istps2 23283 | Express the predicate "is ... |
| tpsuni 23284 | The base set of a topologi... |
| tpstop 23285 | The topology extractor on ... |
| tpspropd 23286 | A topological space depend... |
| tpsprop2d 23287 | A topological space depend... |
| topontopn 23288 | Express the predicate "is ... |
| tsettps 23289 | If the topology component ... |
| istpsi 23290 | Properties that determine ... |
| eltpsg 23291 | Properties that determine ... |
| eltpsi 23292 | Properties that determine ... |
| isbasisg 23295 | Express the predicate "the... |
| isbasis2g 23296 | Express the predicate "the... |
| isbasis3g 23297 | Express the predicate "the... |
| basis1 23298 | Property of a basis. (Con... |
| basis2 23299 | Property of a basis. (Con... |
| fiinbas 23300 | If a set is closed under f... |
| basdif0 23301 | A basis is not affected by... |
| baspartn 23302 | A disjoint system of sets ... |
| tgval 23303 | The topology generated by ... |
| tgval2 23304 | Definition of a topology g... |
| eltg 23305 | Membership in a topology g... |
| eltg2 23306 | Membership in a topology g... |
| eltg2b 23307 | Membership in a topology g... |
| eltg4i 23308 | An open set in a topology ... |
| eltg3i 23309 | The union of a set of basi... |
| eltg3 23310 | Membership in a topology g... |
| tgval3 23311 | Alternate expression for t... |
| tg1 23312 | Property of a member of a ... |
| tg2 23313 | Property of a member of a ... |
| bastg 23314 | A member of a basis is a s... |
| unitg 23315 | The topology generated by ... |
| tgss 23316 | Subset relation for genera... |
| tgcl 23317 | Show that a basis generate... |
| tgclb 23318 | The property ~ tgcl can be... |
| tgtopon 23319 | A basis generates a topolo... |
| topbas 23320 | A topology is its own basi... |
| tgtop 23321 | A topology is its own basi... |
| eltop 23322 | Membership in a topology, ... |
| eltop2 23323 | Membership in a topology. ... |
| eltop3 23324 | Membership in a topology. ... |
| fibas 23325 | A collection of finite int... |
| tgdom 23326 | A space has no more open s... |
| tgiun 23327 | The indexed union of a set... |
| tgidm 23328 | The topology generator fun... |
| bastop 23329 | Two ways to express that a... |
| tgtop11 23330 | The topology generation fu... |
| 0top 23331 | The singleton of the empty... |
| en1top 23332 | ` { (/) } ` is the only to... |
| en2top 23333 | If a topology has two elem... |
| tgss3 23334 | A criterion for determinin... |
| tgss2 23335 | A criterion for determinin... |
| basgen 23336 | Given a topology ` J ` , s... |
| basgen2 23337 | Given a topology ` J ` , s... |
| 2basgen 23338 | Conditions that determine ... |
| tgfiss 23339 | If a subbase is included i... |
| tgdif0 23340 | A generated topology is no... |
| bastop1 23341 | A subset of a topology is ... |
| bastop2 23342 | A version of ~ bastop1 tha... |
| distop 23343 | The discrete topology on a... |
| topnex 23344 | The class of all topologie... |
| distopon 23345 | The discrete topology on a... |
| sn0topon 23346 | The singleton of the empty... |
| sn0top 23347 | The singleton of the empty... |
| indislem 23348 | A lemma to eliminate some ... |
| indistopon 23349 | The indiscrete topology on... |
| indistop 23350 | The indiscrete topology on... |
| indisuni 23351 | The base set of the indisc... |
| fctop 23352 | The finite complement topo... |
| fctop2 23353 | The finite complement topo... |
| cctop 23354 | The countable complement t... |
| ppttop 23355 | The particular point topol... |
| pptbas 23356 | The particular point topol... |
| epttop 23357 | The excluded point topolog... |
| indistpsx 23358 | The indiscrete topology on... |
| indistps 23359 | The indiscrete topology on... |
| indistps2 23360 | The indiscrete topology on... |
| indistpsALT 23361 | The indiscrete topology on... |
| indistps2ALT 23362 | The indiscrete topology on... |
| distps 23363 | The discrete topology on a... |
| fncld 23370 | The closed-set generator i... |
| cldval 23371 | The set of closed sets of ... |
| ntrfval 23372 | The interior function on t... |
| clsfval 23373 | The closure function on th... |
| cldrcl 23374 | Reverse closure of the clo... |
| iscld 23375 | The predicate "the class `... |
| iscld2 23376 | A subset of the underlying... |
| cldss 23377 | A closed set is a subset o... |
| cldss2 23378 | The set of closed sets is ... |
| cldopn 23379 | The complement of a closed... |
| isopn2 23380 | A subset of the underlying... |
| opncld 23381 | The complement of an open ... |
| difopn 23382 | The difference of a closed... |
| topcld 23383 | The underlying set of a to... |
| ntrval 23384 | The interior of a subset o... |
| clsval 23385 | The closure of a subset of... |
| 0cld 23386 | The empty set is closed. ... |
| iincld 23387 | The indexed intersection o... |
| intcld 23388 | The intersection of a set ... |
| uncld 23389 | The union of two closed se... |
| cldcls 23390 | A closed subset equals its... |
| incld 23391 | The intersection of two cl... |
| riincld 23392 | An indexed relative inters... |
| iuncld 23393 | A finite indexed union of ... |
| unicld 23394 | A finite union of closed s... |
| clscld 23395 | The closure of a subset of... |
| clsf 23396 | The closure function is a ... |
| ntropn 23397 | The interior of a subset o... |
| clsval2 23398 | Express closure in terms o... |
| ntrval2 23399 | Interior expressed in term... |
| ntrdif 23400 | An interior of a complemen... |
| clsdif 23401 | A closure of a complement ... |
| clsss 23402 | Subset relationship for cl... |
| ntrss 23403 | Subset relationship for in... |
| sscls 23404 | A subset of a topology's u... |
| ntrss2 23405 | A subset includes its inte... |
| ssntr 23406 | An open subset of a set is... |
| clsss3 23407 | The closure of a subset of... |
| ntrss3 23408 | The interior of a subset o... |
| ntrin 23409 | A pairwise intersection of... |
| cmclsopn 23410 | The complement of a closur... |
| cmntrcld 23411 | The complement of an inter... |
| iscld3 23412 | A subset is closed iff it ... |
| iscld4 23413 | A subset is closed iff it ... |
| isopn3 23414 | A subset is open iff it eq... |
| clsidm 23415 | The closure operation is i... |
| ntridm 23416 | The interior operation is ... |
| clstop 23417 | The closure of a topology'... |
| ntrtop 23418 | The interior of a topology... |
| 0ntr 23419 | A subset with an empty int... |
| clsss2 23420 | If a subset is included in... |
| elcls 23421 | Membership in a closure. ... |
| elcls2 23422 | Membership in a closure. ... |
| clsndisj 23423 | Any open set containing a ... |
| ntrcls0 23424 | A subset whose closure has... |
| ntreq0 23425 | Two ways to say that a sub... |
| cldmre 23426 | The closed sets of a topol... |
| mrccls 23427 | Moore closure generalizes ... |
| cls0 23428 | The closure of the empty s... |
| ntr0 23429 | The interior of the empty ... |
| isopn3i 23430 | An open subset equals its ... |
| elcls3 23431 | Membership in a closure in... |
| opncldf1 23432 | A bijection useful for con... |
| opncldf2 23433 | The values of the open-clo... |
| opncldf3 23434 | The values of the converse... |
| isclo 23435 | A set ` A ` is clopen iff ... |
| isclo2 23436 | A set ` A ` is clopen iff ... |
| discld 23437 | The open sets of a discret... |
| sn0cld 23438 | The closed sets of the top... |
| indiscld 23439 | The closed sets of an indi... |
| mretopd 23440 | A Moore collection which i... |
| toponmre 23441 | The topologies over a give... |
| cldmreon 23442 | The closed sets of a topol... |
| iscldtop 23443 | A family is the closed set... |
| mreclatdemoBAD 23444 | The closed subspaces of a ... |
| neifval 23447 | Value of the neighborhood ... |
| neif 23448 | The neighborhood function ... |
| neiss2 23449 | A set with a neighborhood ... |
| neival 23450 | Value of the set of neighb... |
| isnei 23451 | The predicate "the class `... |
| neiint 23452 | An intuitive definition of... |
| isneip 23453 | The predicate "the class `... |
| neii1 23454 | A neighborhood is included... |
| neisspw 23455 | The neighborhoods of any s... |
| neii2 23456 | Property of a neighborhood... |
| neiss 23457 | Any neighborhood of a set ... |
| ssnei 23458 | A set is included in any o... |
| elnei 23459 | A point belongs to any of ... |
| 0nnei 23460 | The empty set is not a nei... |
| neips 23461 | A neighborhood of a set is... |
| opnneissb 23462 | An open set is a neighborh... |
| opnssneib 23463 | Any superset of an open se... |
| ssnei2 23464 | Any subset ` M ` of ` X ` ... |
| neindisj 23465 | Any neighborhood of an ele... |
| opnneiss 23466 | An open set is a neighborh... |
| opnneip 23467 | An open set is a neighborh... |
| opnnei 23468 | A set is open iff it is a ... |
| tpnei 23469 | The underlying set of a to... |
| neiuni 23470 | The union of the neighborh... |
| neindisj2 23471 | A point ` P ` belongs to t... |
| topssnei 23472 | A finer topology has more ... |
| innei 23473 | The intersection of two ne... |
| opnneiid 23474 | Only an open set is a neig... |
| neissex 23475 | For any neighborhood ` N `... |
| 0nei 23476 | The empty set is a neighbo... |
| neipeltop 23477 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23478 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23479 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23480 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23481 | If, to each element ` P ` ... |
| lpfval 23486 | The limit point function o... |
| lpval 23487 | The set of limit points of... |
| islp 23488 | The predicate "the class `... |
| lpsscls 23489 | The limit points of a subs... |
| lpss 23490 | The limit points of a subs... |
| lpdifsn 23491 | ` P ` is a limit point of ... |
| lpss3 23492 | Subset relationship for li... |
| islp2 23493 | The predicate " ` P ` is a... |
| islp3 23494 | The predicate " ` P ` is a... |
| maxlp 23495 | A point is a limit point o... |
| clslp 23496 | The closure of a subset of... |
| islpi 23497 | A point belonging to a set... |
| cldlp 23498 | A subset of a topological ... |
| isperf 23499 | Definition of a perfect sp... |
| isperf2 23500 | Definition of a perfect sp... |
| isperf3 23501 | A perfect space is a topol... |
| perflp 23502 | The limit points of a perf... |
| perfi 23503 | Property of a perfect spac... |
| perftop 23504 | A perfect space is a topol... |
| restrcl 23505 | Reverse closure for the su... |
| restbas 23506 | A subspace topology basis ... |
| tgrest 23507 | A subspace can be generate... |
| resttop 23508 | A subspace topology is a t... |
| resttopon 23509 | A subspace topology is a t... |
| restuni 23510 | The underlying set of a su... |
| stoig 23511 | The topological space buil... |
| restco 23512 | Composition of subspaces. ... |
| restabs 23513 | Equivalence of being a sub... |
| restin 23514 | When the subspace region i... |
| restuni2 23515 | The underlying set of a su... |
| resttopon2 23516 | The underlying set of a su... |
| rest0 23517 | The subspace topology indu... |
| restsn 23518 | The only subspace topology... |
| restsn2 23519 | The subspace topology indu... |
| restcld 23520 | A closed set of a subspace... |
| restcldi 23521 | A closed set is closed in ... |
| restcldr 23522 | A set which is closed in t... |
| restopnb 23523 | If ` B ` is an open subset... |
| ssrest 23524 | If ` K ` is a finer topolo... |
| restopn2 23525 | If ` A ` is open, then ` B... |
| restdis 23526 | A subspace of a discrete t... |
| restfpw 23527 | The restriction of the set... |
| neitr 23528 | The neighborhood of a trac... |
| restcls 23529 | A closure in a subspace to... |
| restntr 23530 | An interior in a subspace ... |
| restlp 23531 | The limit points of a subs... |
| restperf 23532 | Perfection of a subspace. ... |
| perfopn 23533 | An open subset of a perfec... |
| resstopn 23534 | The topology of a restrict... |
| resstps 23535 | A restricted topological s... |
| ordtbaslem 23536 | Lemma for ~ ordtbas . In ... |
| ordtval 23537 | Value of the order topolog... |
| ordtuni 23538 | Value of the order topolog... |
| ordtbas2 23539 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23540 | In a total order, the fini... |
| ordttopon 23541 | Value of the order topolog... |
| ordtopn1 23542 | An upward ray ` ( P , +oo ... |
| ordtopn2 23543 | A downward ray ` ( -oo , P... |
| ordtopn3 23544 | An open interval ` ( A , B... |
| ordtcld1 23545 | A downward ray ` ( -oo , P... |
| ordtcld2 23546 | An upward ray ` [ P , +oo ... |
| ordtcld3 23547 | A closed interval ` [ A , ... |
| ordttop 23548 | The order topology is a to... |
| ordtcnv 23549 | The order dual generates t... |
| ordtrest 23550 | The subspace topology of a... |
| ordtrest2lem 23551 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23552 | An interval-closed set ` A... |
| letopon 23553 | The topology of the extend... |
| letop 23554 | The topology of the extend... |
| letopuni 23555 | The topology of the extend... |
| xrstopn 23556 | The topology component of ... |
| xrstps 23557 | The extended real number s... |
| leordtvallem1 23558 | Lemma for ~ leordtval . (... |
| leordtvallem2 23559 | Lemma for ~ leordtval . (... |
| leordtval2 23560 | The topology of the extend... |
| leordtval 23561 | The topology of the extend... |
| iccordt 23562 | A closed interval is close... |
| iocpnfordt 23563 | An unbounded above open in... |
| icomnfordt 23564 | An unbounded above open in... |
| iooordt 23565 | An open interval is open i... |
| reordt 23566 | The real numbers are an op... |
| lecldbas 23567 | The set of closed interval... |
| pnfnei 23568 | A neighborhood of ` +oo ` ... |
| mnfnei 23569 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23570 | The restriction of the les... |
| ordtresticc 23571 | The restriction of the les... |
| lmrel 23578 | The topological space conv... |
| lmrcl 23579 | Reverse closure for the co... |
| lmfval 23580 | The relation "sequence ` f... |
| cnfval 23581 | The set of all continuous ... |
| cnpfval 23582 | The function mapping the p... |
| iscn 23583 | The predicate "the class `... |
| cnpval 23584 | The set of all functions f... |
| iscnp 23585 | The predicate "the class `... |
| iscn2 23586 | The predicate "the class `... |
| iscnp2 23587 | The predicate "the class `... |
| cntop1 23588 | Reverse closure for a cont... |
| cntop2 23589 | Reverse closure for a cont... |
| cnptop1 23590 | Reverse closure for a func... |
| cnptop2 23591 | Reverse closure for a func... |
| iscnp3 23592 | The predicate "the class `... |
| cnprcl 23593 | Reverse closure for a func... |
| cnf 23594 | A continuous function is a... |
| cnpf 23595 | A continuous function at p... |
| cnpcl 23596 | The value of a continuous ... |
| cnf2 23597 | A continuous function is a... |
| cnpf2 23598 | A continuous function at p... |
| cnprcl2 23599 | Reverse closure for a func... |
| tgcn 23600 | The continuity predicate w... |
| tgcnp 23601 | The "continuous at a point... |
| subbascn 23602 | The continuity predicate w... |
| ssidcn 23603 | The identity function is a... |
| cnpimaex 23604 | Property of a function con... |
| idcn 23605 | A restricted identity func... |
| lmbr 23606 | Express the binary relatio... |
| lmbr2 23607 | Express the binary relatio... |
| lmbrf 23608 | Express the binary relatio... |
| lmconst 23609 | A constant sequence conver... |
| lmcvg 23610 | Convergence property of a ... |
| iscnp4 23611 | The predicate "the class `... |
| cnpnei 23612 | A condition for continuity... |
| cnima 23613 | An open subset of the codo... |
| cnco 23614 | The composition of two con... |
| cnpco 23615 | The composition of a funct... |
| cnclima 23616 | A closed subset of the cod... |
| iscncl 23617 | A characterization of a co... |
| cncls2i 23618 | Property of the preimage o... |
| cnntri 23619 | Property of the preimage o... |
| cnclsi 23620 | Property of the image of a... |
| cncls2 23621 | Continuity in terms of clo... |
| cncls 23622 | Continuity in terms of clo... |
| cnntr 23623 | Continuity in terms of int... |
| cnss1 23624 | If the topology ` K ` is f... |
| cnss2 23625 | If the topology ` K ` is f... |
| cncnpi 23626 | A continuous function is c... |
| cnsscnp 23627 | The set of continuous func... |
| cncnp 23628 | A continuous function is c... |
| cncnp2 23629 | A continuous function is c... |
| cnnei 23630 | Continuity in terms of nei... |
| cnconst2 23631 | A constant function is con... |
| cnconst 23632 | A constant function is con... |
| cnrest 23633 | Continuity of a restrictio... |
| cnrest2 23634 | Equivalence of continuity ... |
| cnrest2r 23635 | Equivalence of continuity ... |
| cnpresti 23636 | One direction of ~ cnprest... |
| cnprest 23637 | Equivalence of continuity ... |
| cnprest2 23638 | Equivalence of point-conti... |
| cndis 23639 | Every function is continuo... |
| cnindis 23640 | Every function is continuo... |
| cnpdis 23641 | If ` A ` is an isolated po... |
| paste 23642 | Pasting lemma. If ` A ` a... |
| lmfpm 23643 | If ` F ` converges, then `... |
| lmfss 23644 | Inclusion of a function ha... |
| lmcl 23645 | Closure of a limit. (Cont... |
| lmss 23646 | Limit on a subspace. (Con... |
| sslm 23647 | A finer topology has fewer... |
| lmres 23648 | A function converges iff i... |
| lmff 23649 | If ` F ` converges, there ... |
| lmcls 23650 | Any convergent sequence of... |
| lmcld 23651 | Any convergent sequence of... |
| lmcnp 23652 | The image of a convergent ... |
| lmcn 23653 | The image of a convergent ... |
| ist0 23668 | The predicate "is a T_0 sp... |
| ist1 23669 | The predicate "is a T_1 sp... |
| ishaus 23670 | The predicate "is a Hausdo... |
| iscnrm 23671 | The property of being comp... |
| t0sep 23672 | Any two topologically indi... |
| t0dist 23673 | Any two distinct points in... |
| t1sncld 23674 | In a T_1 space, singletons... |
| t1ficld 23675 | In a T_1 space, finite set... |
| hausnei 23676 | Neighborhood property of a... |
| t0top 23677 | A T_0 space is a topologic... |
| t1top 23678 | A T_1 space is a topologic... |
| haustop 23679 | A Hausdorff space is a top... |
| isreg 23680 | The predicate "is a regula... |
| regtop 23681 | A regular space is a topol... |
| regsep 23682 | In a regular space, every ... |
| isnrm 23683 | The predicate "is a normal... |
| nrmtop 23684 | A normal space is a topolo... |
| cnrmtop 23685 | A completely normal space ... |
| iscnrm2 23686 | The property of being comp... |
| ispnrm 23687 | The property of being perf... |
| pnrmnrm 23688 | A perfectly normal space i... |
| pnrmtop 23689 | A perfectly normal space i... |
| pnrmcld 23690 | A closed set in a perfectl... |
| pnrmopn 23691 | An open set in a perfectly... |
| ist0-2 23692 | The predicate "is a T_0 sp... |
| ist0-3 23693 | The predicate "is a T_0 sp... |
| cnt0 23694 | The preimage of a T_0 topo... |
| ist1-2 23695 | An alternate characterizat... |
| t1t0 23696 | A T_1 space is a T_0 space... |
| ist1-3 23697 | A space is T_1 iff every p... |
| cnt1 23698 | The preimage of a T_1 topo... |
| ishaus2 23699 | Express the predicate " ` ... |
| haust1 23700 | A Hausdorff space is a T_1... |
| hausnei2 23701 | The Hausdorff condition st... |
| cnhaus 23702 | The preimage of a Hausdorf... |
| nrmsep3 23703 | In a normal space, given a... |
| nrmsep2 23704 | In a normal space, any two... |
| nrmsep 23705 | In a normal space, disjoin... |
| isnrm2 23706 | An alternate characterizat... |
| isnrm3 23707 | A topological space is nor... |
| cnrmi 23708 | A subspace of a completely... |
| cnrmnrm 23709 | A completely normal space ... |
| restcnrm 23710 | A subspace of a completely... |
| resthauslem 23711 | Lemma for ~ resthaus and s... |
| lpcls 23712 | The limit points of the cl... |
| perfcls 23713 | A subset of a perfect spac... |
| restt0 23714 | A subspace of a T_0 topolo... |
| restt1 23715 | A subspace of a T_1 topolo... |
| resthaus 23716 | A subspace of a Hausdorff ... |
| t1sep2 23717 | Any two points in a T_1 sp... |
| t1sep 23718 | Any two distinct points in... |
| sncld 23719 | A singleton is closed in a... |
| sshauslem 23720 | Lemma for ~ sshaus and sim... |
| sst0 23721 | A topology finer than a T_... |
| sst1 23722 | A topology finer than a T_... |
| sshaus 23723 | A topology finer than a Ha... |
| regsep2 23724 | In a regular space, a clos... |
| isreg2 23725 | A topological space is reg... |
| dnsconst 23726 | If a continuous mapping to... |
| ordtt1 23727 | The order topology is T_1 ... |
| lmmo 23728 | A sequence in a Hausdorff ... |
| lmfun 23729 | The convergence relation i... |
| dishaus 23730 | A discrete topology is Hau... |
| ordthauslem 23731 | Lemma for ~ ordthaus . (C... |
| ordthaus 23732 | The order topology of a to... |
| xrhaus 23733 | The topology of the extend... |
| iscmp 23736 | The predicate "is a compac... |
| cmpcov 23737 | An open cover of a compact... |
| cmpcov2 23738 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23739 | Combine ~ cmpcov with ~ ac... |
| cncmp 23740 | Compactness is respected b... |
| fincmp 23741 | A finite topology is compa... |
| 0cmp 23742 | The singleton of the empty... |
| cmptop 23743 | A compact topology is a to... |
| rncmp 23744 | The image of a compact set... |
| imacmp 23745 | The image of a compact set... |
| discmp 23746 | A discrete topology is com... |
| cmpsublem 23747 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23748 | Two equivalent ways of des... |
| tgcmp 23749 | A topology generated by a ... |
| cmpcld 23750 | A closed subset of a compa... |
| uncmp 23751 | The union of two compact s... |
| fiuncmp 23752 | A finite union of compact ... |
| sscmp 23753 | A subset of a compact topo... |
| hauscmplem 23754 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23755 | A compact subspace of a T2... |
| cmpfi 23756 | If a topology is compact a... |
| cmpfii 23757 | In a compact topology, a s... |
| bwth 23758 | The glorious Bolzano-Weier... |
| isconn 23761 | The predicate ` J ` is a c... |
| isconn2 23762 | The predicate ` J ` is a c... |
| connclo 23763 | The only nonempty clopen s... |
| conndisj 23764 | If a topology is connected... |
| conntop 23765 | A connected topology is a ... |
| indisconn 23766 | The indiscrete topology (o... |
| dfconn2 23767 | An alternate definition of... |
| connsuba 23768 | Connectedness for a subspa... |
| connsub 23769 | Two equivalent ways of say... |
| cnconn 23770 | Connectedness is respected... |
| nconnsubb 23771 | Disconnectedness for a sub... |
| connsubclo 23772 | If a clopen set meets a co... |
| connima 23773 | The image of a connected s... |
| conncn 23774 | A continuous function from... |
| iunconnlem 23775 | Lemma for ~ iunconn . (Co... |
| iunconn 23776 | The indexed union of conne... |
| unconn 23777 | The union of two connected... |
| clsconn 23778 | The closure of a connected... |
| conncompid 23779 | The connected component co... |
| conncompconn 23780 | The connected component co... |
| conncompss 23781 | The connected component co... |
| conncompcld 23782 | The connected component co... |
| conncompclo 23783 | The connected component co... |
| t1connperf 23784 | A connected T_1 space is p... |
| is1stc 23789 | The predicate "is a first-... |
| is1stc2 23790 | An equivalent way of sayin... |
| 1stctop 23791 | A first-countable topology... |
| 1stcclb 23792 | A property of points in a ... |
| 1stcfb 23793 | For any point ` A ` in a f... |
| is2ndc 23794 | The property of being seco... |
| 2ndctop 23795 | A second-countable topolog... |
| 2ndci 23796 | A countable basis generate... |
| 2ndcsb 23797 | Having a countable subbase... |
| 2ndcredom 23798 | A second-countable space h... |
| 2ndc1stc 23799 | A second-countable space i... |
| 1stcrestlem 23800 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23801 | A subspace of a first-coun... |
| 2ndcrest 23802 | A subspace of a second-cou... |
| abrexct 23803 | An image set of a countabl... |
| 2ndcctbss 23804 | If a topology is second-co... |
| 2ndcdisj 23805 | Any disjoint family of ope... |
| 2ndcdisj2 23806 | Any disjoint collection of... |
| 2ndcomap 23807 | A surjective continuous op... |
| 2ndcsep 23808 | A second-countable topolog... |
| dis2ndc 23809 | A discrete space is second... |
| 1stcelcls 23810 | A point belongs to the clo... |
| 1stccnp 23811 | A mapping is continuous at... |
| 1stccn 23812 | A mapping ` X --> Y ` , wh... |
| islly 23817 | The property of being a lo... |
| isnlly 23818 | The property of being an n... |
| llyeq 23819 | Equality theorem for the `... |
| nllyeq 23820 | Equality theorem for the `... |
| llytop 23821 | A locally ` A ` space is a... |
| nllytop 23822 | A locally ` A ` space is a... |
| llyi 23823 | The property of a locally ... |
| nllyi 23824 | The property of an n-local... |
| nlly2i 23825 | Eliminate the neighborhood... |
| llynlly 23826 | A locally ` A ` space is n... |
| llyssnlly 23827 | A locally ` A ` space is n... |
| llyss 23828 | The "locally" predicate re... |
| nllyss 23829 | The "n-locally" predicate ... |
| subislly 23830 | The property of a subspace... |
| restnlly 23831 | If the property ` A ` pass... |
| restlly 23832 | If the property ` A ` pass... |
| islly2 23833 | An alternative expression ... |
| llyrest 23834 | An open subspace of a loca... |
| nllyrest 23835 | An open subspace of an n-l... |
| loclly 23836 | If ` A ` is a local proper... |
| llyidm 23837 | Idempotence of the "locall... |
| nllyidm 23838 | Idempotence of the "n-loca... |
| toplly 23839 | A topology is locally a to... |
| topnlly 23840 | A topology is n-locally a ... |
| hauslly 23841 | A Hausdorff space is local... |
| hausnlly 23842 | A Hausdorff space is n-loc... |
| hausllycmp 23843 | A compact Hausdorff space ... |
| cldllycmp 23844 | A closed subspace of a loc... |
| lly1stc 23845 | First-countability is a lo... |
| dislly 23846 | The discrete space ` ~P X ... |
| disllycmp 23847 | A discrete space is locall... |
| dis1stc 23848 | A discrete space is first-... |
| hausmapdom 23849 | If ` X ` is a first-counta... |
| hauspwdom 23850 | Simplify the cardinal ` A ... |
| refrel 23857 | Refinement is a relation. ... |
| isref 23858 | The property of being a re... |
| refbas 23859 | A refinement covers the sa... |
| refssex 23860 | Every set in a refinement ... |
| ssref 23861 | A subcover is a refinement... |
| refref 23862 | Reflexivity of refinement.... |
| reftr 23863 | Refinement is transitive. ... |
| refun0 23864 | Adding the empty set prese... |
| isptfin 23865 | The statement "is a point-... |
| islocfin 23866 | The statement "is a locall... |
| finptfin 23867 | A finite cover is a point-... |
| ptfinfin 23868 | A point covered by a point... |
| finlocfin 23869 | A finite cover of a topolo... |
| locfintop 23870 | A locally finite cover cov... |
| locfinbas 23871 | A locally finite cover mus... |
| locfinnei 23872 | A point covered by a local... |
| lfinpfin 23873 | A locally finite cover is ... |
| lfinun 23874 | Adding a finite set preser... |
| locfincmp 23875 | For a compact space, the l... |
| unisngl 23876 | Taking the union of the se... |
| dissnref 23877 | The set of singletons is a... |
| dissnlocfin 23878 | The set of singletons is l... |
| locfindis 23879 | The locally finite covers ... |
| locfincf 23880 | A locally finite cover in ... |
| comppfsc 23881 | A space where every open c... |
| kgenval 23884 | Value of the compact gener... |
| elkgen 23885 | Value of the compact gener... |
| kgeni 23886 | Property of the open sets ... |
| kgentopon 23887 | The compact generator gene... |
| kgenuni 23888 | The base set of the compac... |
| kgenftop 23889 | The compact generator gene... |
| kgenf 23890 | The compact generator is a... |
| kgentop 23891 | A compactly generated spac... |
| kgenss 23892 | The compact generator gene... |
| kgenhaus 23893 | The compact generator gene... |
| kgencmp 23894 | The compact generator topo... |
| kgencmp2 23895 | The compact generator topo... |
| kgenidm 23896 | The compact generator is i... |
| iskgen2 23897 | A space is compactly gener... |
| iskgen3 23898 | Derive the usual definitio... |
| llycmpkgen2 23899 | A locally compact space is... |
| cmpkgen 23900 | A compact space is compact... |
| llycmpkgen 23901 | A locally compact space is... |
| 1stckgenlem 23902 | The one-point compactifica... |
| 1stckgen 23903 | A first-countable space is... |
| kgen2ss 23904 | The compact generator pres... |
| kgencn 23905 | A function from a compactl... |
| kgencn2 23906 | A function ` F : J --> K `... |
| kgencn3 23907 | The set of continuous func... |
| kgen2cn 23908 | A continuous function is a... |
| txval 23913 | Value of the binary topolo... |
| txuni2 23914 | The underlying set of the ... |
| txbasex 23915 | The basis for the product ... |
| txbas 23916 | The set of Cartesian produ... |
| eltx 23917 | A set in a product is open... |
| txtop 23918 | The product of two topolog... |
| ptval 23919 | The value of the product t... |
| ptpjpre1 23920 | The preimage of a projecti... |
| elpt 23921 | Elementhood in the bases o... |
| elptr 23922 | A basic open set in the pr... |
| elptr2 23923 | A basic open set in the pr... |
| ptbasid 23924 | The base set of the produc... |
| ptuni2 23925 | The base set for the produ... |
| ptbasin 23926 | The basis for a product to... |
| ptbasin2 23927 | The basis for a product to... |
| ptbas 23928 | The basis for a product to... |
| ptpjpre2 23929 | The basis for a product to... |
| ptbasfi 23930 | The basis for the product ... |
| pttop 23931 | The product topology is a ... |
| ptopn 23932 | A basic open set in the pr... |
| ptopn2 23933 | A sub-basic open set in th... |
| xkotf 23934 | Functionality of function ... |
| xkobval 23935 | Alternative expression for... |
| xkoval 23936 | Value of the compact-open ... |
| xkotop 23937 | The compact-open topology ... |
| xkoopn 23938 | A basic open set of the co... |
| txtopi 23939 | The product of two topolog... |
| txtopon 23940 | The underlying set of the ... |
| txuni 23941 | The underlying set of the ... |
| txunii 23942 | The underlying set of the ... |
| ptuni 23943 | The base set for the produ... |
| ptunimpt 23944 | Base set of a product topo... |
| pttopon 23945 | The base set for the produ... |
| pttoponconst 23946 | The base set for a product... |
| ptuniconst 23947 | The base set for a product... |
| xkouni 23948 | The base set of the compac... |
| xkotopon 23949 | The base set of the compac... |
| ptval2 23950 | The value of the product t... |
| txopn 23951 | The product of two open se... |
| txcld 23952 | The product of two closed ... |
| txcls 23953 | Closure of a rectangle in ... |
| txss12 23954 | Subset property of the top... |
| txbasval 23955 | It is sufficient to consid... |
| neitx 23956 | The Cartesian product of t... |
| txcnpi 23957 | Continuity of a two-argume... |
| tx1cn 23958 | Continuity of the first pr... |
| tx2cn 23959 | Continuity of the second p... |
| ptpjcn 23960 | Continuity of a projection... |
| ptpjopn 23961 | The projection map is an o... |
| ptcld 23962 | A closed box in the produc... |
| ptcldmpt 23963 | A closed box in the produc... |
| ptclsg 23964 | The closure of a box in th... |
| ptcls 23965 | The closure of a box in th... |
| dfac14lem 23966 | Lemma for ~ dfac14 . By e... |
| dfac14 23967 | Theorem ~ ptcls is an equi... |
| xkoccn 23968 | The "constant function" fu... |
| txcnp 23969 | If two functions are conti... |
| ptcnplem 23970 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23971 | If every projection of a f... |
| upxp 23972 | Universal property of the ... |
| txcnmpt 23973 | A map into the product of ... |
| uptx 23974 | Universal property of the ... |
| txcn 23975 | A map into the product of ... |
| ptcn 23976 | If every projection of a f... |
| prdstopn 23977 | Topology of a structure pr... |
| prdstps 23978 | A structure product of top... |
| pwstps 23979 | A structure power of a top... |
| txrest 23980 | The subspace of a topologi... |
| txdis 23981 | The topological product of... |
| txindislem 23982 | Lemma for ~ txindis . (Co... |
| txindis 23983 | The topological product of... |
| txdis1cn 23984 | A function is jointly cont... |
| txlly 23985 | If the property ` A ` is p... |
| txnlly 23986 | If the property ` A ` is p... |
| pthaus 23987 | The product of a collectio... |
| ptrescn 23988 | Restriction is a continuou... |
| txtube 23989 | The "tube lemma". If ` X ... |
| txcmplem1 23990 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23991 | Lemma for ~ txcmp . (Cont... |
| txcmp 23992 | The topological product of... |
| txcmpb 23993 | The topological product of... |
| hausdiag 23994 | A topology is Hausdorff if... |
| hauseqlcld 23995 | In a Hausdorff topology, t... |
| txhaus 23996 | The topological product of... |
| txlm 23997 | Two sequences converge iff... |
| lmcn2 23998 | The image of a convergent ... |
| tx1stc 23999 | The topological product of... |
| tx2ndc 24000 | The topological product of... |
| txkgen 24001 | The topological product of... |
| xkohaus 24002 | If the codomain space is H... |
| xkoptsub 24003 | The compact-open topology ... |
| xkopt 24004 | The compact-open topology ... |
| xkopjcn 24005 | Continuity of a projection... |
| xkoco1cn 24006 | If ` F ` is a continuous f... |
| xkoco2cn 24007 | If ` F ` is a continuous f... |
| xkococnlem 24008 | Continuity of the composit... |
| xkococn 24009 | Continuity of the composit... |
| cnmptid 24010 | The identity function is c... |
| cnmptc 24011 | A constant function is con... |
| cnmpt11 24012 | The composition of continu... |
| cnmpt11f 24013 | The composition of continu... |
| cnmpt1t 24014 | The composition of continu... |
| cnmpt12f 24015 | The composition of continu... |
| cnmpt12 24016 | The composition of continu... |
| cnmpt1st 24017 | The projection onto the fi... |
| cnmpt2nd 24018 | The projection onto the se... |
| cnmpt2c 24019 | A constant function is con... |
| cnmpt21 24020 | The composition of continu... |
| cnmpt21f 24021 | The composition of continu... |
| cnmpt2t 24022 | The composition of continu... |
| cnmpt22 24023 | The composition of continu... |
| cnmpt22f 24024 | The composition of continu... |
| cnmpt1res 24025 | The restriction of a conti... |
| cnmpt2res 24026 | The restriction of a conti... |
| cnmptcom 24027 | The argument converse of a... |
| cnmptkc 24028 | The curried first projecti... |
| cnmptkp 24029 | The evaluation of the inne... |
| cnmptk1 24030 | The composition of a curri... |
| cnmpt1k 24031 | The composition of a one-a... |
| cnmptkk 24032 | The composition of two cur... |
| xkofvcn 24033 | Joint continuity of the fu... |
| cnmptk1p 24034 | The evaluation of a currie... |
| cnmptk2 24035 | The uncurrying of a currie... |
| xkoinjcn 24036 | Continuity of "injection",... |
| cnmpt2k 24037 | The currying of a two-argu... |
| txconn 24038 | The topological product of... |
| imasnopn 24039 | If a relation graph is ope... |
| imasncld 24040 | If a relation graph is clo... |
| imasncls 24041 | If a relation graph is clo... |
| qtopval 24044 | Value of the quotient topo... |
| qtopval2 24045 | Value of the quotient topo... |
| elqtop 24046 | Value of the quotient topo... |
| qtopres 24047 | The quotient topology is u... |
| qtoptop2 24048 | The quotient topology is a... |
| qtoptop 24049 | The quotient topology is a... |
| elqtop2 24050 | Value of the quotient topo... |
| qtopuni 24051 | The base set of the quotie... |
| elqtop3 24052 | Value of the quotient topo... |
| qtoptopon 24053 | The base set of the quotie... |
| qtopid 24054 | A quotient map is a contin... |
| idqtop 24055 | The quotient topology indu... |
| qtopcmplem 24056 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 24057 | A quotient of a compact sp... |
| qtopconn 24058 | A quotient of a connected ... |
| qtopkgen 24059 | A quotient of a compactly ... |
| basqtop 24060 | An injection maps bases to... |
| tgqtop 24061 | An injection maps generate... |
| qtopcld 24062 | The property of being a cl... |
| qtopcn 24063 | Universal property of a qu... |
| qtopss 24064 | A surjective continuous fu... |
| qtopeu 24065 | Universal property of the ... |
| qtoprest 24066 | If ` A ` is a saturated op... |
| qtopomap 24067 | If ` F ` is a surjective c... |
| qtopcmap 24068 | If ` F ` is a surjective c... |
| imastopn 24069 | The topology of an image s... |
| imastps 24070 | The image of a topological... |
| qustps 24071 | A quotient structure is a ... |
| kqfval 24072 | Value of the function appe... |
| kqfeq 24073 | Two points in the Kolmogor... |
| kqffn 24074 | The topological indistingu... |
| kqval 24075 | Value of the quotient topo... |
| kqtopon 24076 | The Kolmogorov quotient is... |
| kqid 24077 | The topological indistingu... |
| ist0-4 24078 | The topological indistingu... |
| kqfvima 24079 | When the image set is open... |
| kqsat 24080 | Any open set is saturated ... |
| kqdisj 24081 | A version of ~ imain for t... |
| kqcldsat 24082 | Any closed set is saturate... |
| kqopn 24083 | The topological indistingu... |
| kqcld 24084 | The topological indistingu... |
| kqt0lem 24085 | Lemma for ~ kqt0 . (Contr... |
| isr0 24086 | The property " ` J ` is an... |
| r0cld 24087 | The analogue of the T_1 ax... |
| regr1lem 24088 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 24089 | A Kolmogorov quotient of a... |
| kqreglem1 24090 | A Kolmogorov quotient of a... |
| kqreglem2 24091 | If the Kolmogorov quotient... |
| kqnrmlem1 24092 | A Kolmogorov quotient of a... |
| kqnrmlem2 24093 | If the Kolmogorov quotient... |
| kqtop 24094 | The Kolmogorov quotient is... |
| kqt0 24095 | The Kolmogorov quotient is... |
| kqf 24096 | The Kolmogorov quotient is... |
| r0sep 24097 | The separation property of... |
| nrmr0reg 24098 | A normal R_0 space is also... |
| regr1 24099 | A regular space is R_1, wh... |
| kqreg 24100 | The Kolmogorov quotient of... |
| kqnrm 24101 | The Kolmogorov quotient of... |
| hmeofn 24106 | The set of homeomorphisms ... |
| hmeofval 24107 | The set of all the homeomo... |
| ishmeo 24108 | The predicate F is a homeo... |
| hmeocn 24109 | A homeomorphism is continu... |
| hmeocnvcn 24110 | The converse of a homeomor... |
| hmeocnv 24111 | The converse of a homeomor... |
| hmeof1o2 24112 | A homeomorphism is a 1-1-o... |
| hmeof1o 24113 | A homeomorphism is a 1-1-o... |
| hmeoima 24114 | The image of an open set b... |
| hmeoopn 24115 | Homeomorphisms preserve op... |
| hmeocld 24116 | Homeomorphisms preserve cl... |
| hmeocls 24117 | Homeomorphisms preserve cl... |
| hmeontr 24118 | Homeomorphisms preserve in... |
| hmeoimaf1o 24119 | The function mapping open ... |
| hmeores 24120 | The restriction of a homeo... |
| hmeoco 24121 | The composite of two homeo... |
| idhmeo 24122 | The identity function is a... |
| hmeocnvb 24123 | The converse of a homeomor... |
| hmeoqtop 24124 | A homeomorphism is a quoti... |
| hmph 24125 | Express the predicate ` J ... |
| hmphi 24126 | If there is a homeomorphis... |
| hmphtop 24127 | Reverse closure for the ho... |
| hmphtop1 24128 | The relation "being homeom... |
| hmphtop2 24129 | The relation "being homeom... |
| hmphref 24130 | "Is homeomorphic to" is re... |
| hmphsym 24131 | "Is homeomorphic to" is sy... |
| hmphtr 24132 | "Is homeomorphic to" is tr... |
| hmpher 24133 | "Is homeomorphic to" is an... |
| hmphen 24134 | Homeomorphisms preserve th... |
| hmphsymb 24135 | "Is homeomorphic to" is sy... |
| haushmphlem 24136 | Lemma for ~ haushmph and s... |
| cmphmph 24137 | Compactness is a topologic... |
| connhmph 24138 | Connectedness is a topolog... |
| t0hmph 24139 | T_0 is a topological prope... |
| t1hmph 24140 | T_1 is a topological prope... |
| haushmph 24141 | Hausdorff-ness is a topolo... |
| reghmph 24142 | Regularity is a topologica... |
| nrmhmph 24143 | Normality is a topological... |
| hmph0 24144 | A topology homeomorphic to... |
| hmphdis 24145 | Homeomorphisms preserve to... |
| hmphindis 24146 | Homeomorphisms preserve to... |
| indishmph 24147 | Equinumerous sets equipped... |
| hmphen2 24148 | Homeomorphisms preserve th... |
| cmphaushmeo 24149 | A continuous bijection fro... |
| ordthmeolem 24150 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 24151 | An order isomorphism is a ... |
| txhmeo 24152 | Lift a pair of homeomorphi... |
| txswaphmeolem 24153 | Show inverse for the "swap... |
| txswaphmeo 24154 | There is a homeomorphism f... |
| pt1hmeo 24155 | The canonical homeomorphis... |
| ptuncnv 24156 | Exhibit the converse funct... |
| ptunhmeo 24157 | Define a homeomorphism fro... |
| xpstopnlem1 24158 | The function ` F ` used in... |
| xpstps 24159 | A binary product of topolo... |
| xpstopnlem2 24160 | Lemma for ~ xpstopn . (Co... |
| xpstopn 24161 | The topology on a binary p... |
| ptcmpfi 24162 | A topological product of f... |
| xkocnv 24163 | The inverse of the "curryi... |
| xkohmeo 24164 | The Exponential Law for to... |
| qtopf1 24165 | If a quotient map is injec... |
| qtophmeo 24166 | If two functions on a base... |
| t0kq 24167 | A topological space is T_0... |
| kqhmph 24168 | A topological space is T_0... |
| ist1-5lem 24169 | Lemma for ~ ist1-5 and sim... |
| t1r0 24170 | A T_1 space is R_0. That ... |
| ist1-5 24171 | A topological space is T_1... |
| ishaus3 24172 | A topological space is Hau... |
| nrmreg 24173 | A normal T_1 space is regu... |
| reghaus 24174 | A regular T_0 space is Hau... |
| nrmhaus 24175 | A T_1 normal space is Haus... |
| elmptrab 24176 | Membership in a one-parame... |
| elmptrab2 24177 | Membership in a one-parame... |
| isfbas 24178 | The predicate " ` F ` is a... |
| fbasne0 24179 | There are no empty filter ... |
| 0nelfb 24180 | No filter base contains th... |
| fbsspw 24181 | A filter base on a set is ... |
| fbelss 24182 | An element of the filter b... |
| fbdmn0 24183 | The domain of a filter bas... |
| isfbas2 24184 | The predicate " ` F ` is a... |
| fbasssin 24185 | A filter base contains sub... |
| fbssfi 24186 | A filter base contains sub... |
| fbssint 24187 | A filter base contains sub... |
| fbncp 24188 | A filter base does not con... |
| fbun 24189 | A necessary and sufficient... |
| fbfinnfr 24190 | No filter base containing ... |
| opnfbas 24191 | The collection of open sup... |
| trfbas2 24192 | Conditions for the trace o... |
| trfbas 24193 | Conditions for the trace o... |
| isfil 24196 | The predicate "is a filter... |
| filfbas 24197 | A filter is a filter base.... |
| 0nelfil 24198 | The empty set doesn't belo... |
| fileln0 24199 | An element of a filter is ... |
| filsspw 24200 | A filter is a subset of th... |
| filelss 24201 | An element of a filter is ... |
| filss 24202 | A filter is closed under t... |
| filin 24203 | A filter is closed under t... |
| filtop 24204 | The underlying set belongs... |
| isfil2 24205 | Derive the standard axioms... |
| isfildlem 24206 | Lemma for ~ isfild . (Con... |
| isfild 24207 | Sufficient condition for a... |
| filfi 24208 | A filter is closed under t... |
| filinn0 24209 | The intersection of two el... |
| filintn0 24210 | A filter has the finite in... |
| filn0 24211 | The empty set is not a fil... |
| infil 24212 | The intersection of two fi... |
| snfil 24213 | A singleton is a filter. ... |
| fbasweak 24214 | A filter base on any set i... |
| snfbas 24215 | Condition for a singleton ... |
| fsubbas 24216 | A condition for a set to g... |
| fbasfip 24217 | A filter base has the fini... |
| fbunfip 24218 | A helpful lemma for showin... |
| fgval 24219 | The filter generating clas... |
| elfg 24220 | A condition for elements o... |
| ssfg 24221 | A filter base is a subset ... |
| fgss 24222 | A bigger base generates a ... |
| fgss2 24223 | A condition for a filter t... |
| fgfil 24224 | A filter generates itself.... |
| elfilss 24225 | An element belongs to a fi... |
| filfinnfr 24226 | No filter containing a fin... |
| fgcl 24227 | A generated filter is a fi... |
| fgabs 24228 | Absorption law for filter ... |
| neifil 24229 | The neighborhoods of a non... |
| filunibas 24230 | Recover the base set from ... |
| filunirn 24231 | Two ways to express a filt... |
| filconn 24232 | A filter gives rise to a c... |
| fbasrn 24233 | Given a filter on a domain... |
| filuni 24234 | The union of a nonempty se... |
| trfil1 24235 | Conditions for the trace o... |
| trfil2 24236 | Conditions for the trace o... |
| trfil3 24237 | Conditions for the trace o... |
| trfilss 24238 | If ` A ` is a member of th... |
| fgtr 24239 | If ` A ` is a member of th... |
| trfg 24240 | The trace operation and th... |
| trnei 24241 | The trace, over a set ` A ... |
| cfinfil 24242 | Relative complements of th... |
| csdfil 24243 | The set of all elements wh... |
| supfil 24244 | The supersets of a nonempt... |
| zfbas 24245 | The set of upper sets of i... |
| uzrest 24246 | The restriction of the set... |
| uzfbas 24247 | The set of upper sets of i... |
| isufil 24252 | The property of being an u... |
| ufilfil 24253 | An ultrafilter is a filter... |
| ufilss 24254 | For any subset of the base... |
| ufilb 24255 | The complement is in an ul... |
| ufilmax 24256 | Any filter finer than an u... |
| isufil2 24257 | The maximal property of an... |
| ufprim 24258 | An ultrafilter is a prime ... |
| trufil 24259 | Conditions for the trace o... |
| filssufilg 24260 | A filter is contained in s... |
| filssufil 24261 | A filter is contained in s... |
| isufl 24262 | Define the (strong) ultraf... |
| ufli 24263 | Property of a set that sat... |
| numufl 24264 | Consequence of ~ filssufil... |
| fiufl 24265 | A finite set satisfies the... |
| acufl 24266 | The axiom of choice implie... |
| ssufl 24267 | If ` Y ` is a subset of ` ... |
| ufileu 24268 | If the ultrafilter contain... |
| filufint 24269 | A filter is equal to the i... |
| uffix 24270 | Lemma for ~ fixufil and ~ ... |
| fixufil 24271 | The condition describing a... |
| uffixfr 24272 | An ultrafilter is either f... |
| uffix2 24273 | A classification of fixed ... |
| uffixsn 24274 | The singleton of the gener... |
| ufildom1 24275 | An ultrafilter is generate... |
| uffinfix 24276 | An ultrafilter containing ... |
| cfinufil 24277 | An ultrafilter is free iff... |
| ufinffr 24278 | An infinite subset is cont... |
| ufilen 24279 | Any infinite set has an ul... |
| ufildr 24280 | An ultrafilter gives rise ... |
| fin1aufil 24281 | There are no definable fre... |
| fmval 24292 | Introduce a function that ... |
| fmfil 24293 | A mapping filter is a filt... |
| fmf 24294 | Pushing-forward via a func... |
| fmss 24295 | A finer filter produces a ... |
| elfm 24296 | An element of a mapping fi... |
| elfm2 24297 | An element of a mapping fi... |
| fmfg 24298 | The image filter of a filt... |
| elfm3 24299 | An alternate formulation o... |
| imaelfm 24300 | An image of a filter eleme... |
| rnelfmlem 24301 | Lemma for ~ rnelfm . (Con... |
| rnelfm 24302 | A condition for a filter t... |
| fmfnfmlem1 24303 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 24304 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 24305 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 24306 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 24307 | A filter finer than an ima... |
| fmufil 24308 | An image filter of an ultr... |
| fmid 24309 | The filter map applied to ... |
| fmco 24310 | Composition of image filte... |
| ufldom 24311 | The ultrafilter lemma prop... |
| flimval 24312 | The set of limit points of... |
| elflim2 24313 | The predicate "is a limit ... |
| flimtop 24314 | Reverse closure for the li... |
| flimneiss 24315 | A filter contains the neig... |
| flimnei 24316 | A filter contains all of t... |
| flimelbas 24317 | A limit point of a filter ... |
| flimfil 24318 | Reverse closure for the li... |
| flimtopon 24319 | Reverse closure for the li... |
| elflim 24320 | The predicate "is a limit ... |
| flimss2 24321 | A limit point of a filter ... |
| flimss1 24322 | A limit point of a filter ... |
| neiflim 24323 | A point is a limit point o... |
| flimopn 24324 | The condition for being a ... |
| fbflim 24325 | A condition for a filter t... |
| fbflim2 24326 | A condition for a filter b... |
| flimclsi 24327 | The convergent points of a... |
| hausflimlem 24328 | If ` A ` and ` B ` are bot... |
| hausflimi 24329 | One direction of ~ hausfli... |
| hausflim 24330 | A condition for a topology... |
| flimcf 24331 | Fineness is properly chara... |
| flimrest 24332 | The set of limit points in... |
| flimclslem 24333 | Lemma for ~ flimcls . (Co... |
| flimcls 24334 | Closure in terms of filter... |
| flimsncls 24335 | If ` A ` is a limit point ... |
| hauspwpwf1 24336 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 24337 | If ` X ` is a Hausdorff sp... |
| flffval 24338 | Given a topology and a fil... |
| flfval 24339 | Given a function from a fi... |
| flfnei 24340 | The property of being a li... |
| flfneii 24341 | A neighborhood of a limit ... |
| isflf 24342 | The property of being a li... |
| flfelbas 24343 | A limit point of a functio... |
| flffbas 24344 | Limit points of a function... |
| flftg 24345 | Limit points of a function... |
| hausflf 24346 | If a function has its valu... |
| hausflf2 24347 | If a convergent function h... |
| cnpflfi 24348 | Forward direction of ~ cnp... |
| cnpflf2 24349 | ` F ` is continuous at poi... |
| cnpflf 24350 | Continuity of a function a... |
| cnflf 24351 | A function is continuous i... |
| cnflf2 24352 | A function is continuous i... |
| flfcnp 24353 | A continuous function pres... |
| lmflf 24354 | The topological limit rela... |
| txflf 24355 | Two sequences converge in ... |
| flfcnp2 24356 | The image of a convergent ... |
| fclsval 24357 | The set of all cluster poi... |
| isfcls 24358 | A cluster point of a filte... |
| fclsfil 24359 | Reverse closure for the cl... |
| fclstop 24360 | Reverse closure for the cl... |
| fclstopon 24361 | Reverse closure for the cl... |
| isfcls2 24362 | A cluster point of a filte... |
| fclsopn 24363 | Write the cluster point co... |
| fclsopni 24364 | An open neighborhood of a ... |
| fclselbas 24365 | A cluster point is in the ... |
| fclsneii 24366 | A neighborhood of a cluste... |
| fclssscls 24367 | The set of cluster points ... |
| fclsnei 24368 | Cluster points in terms of... |
| supnfcls 24369 | The filter of supersets of... |
| fclsbas 24370 | Cluster points in terms of... |
| fclsss1 24371 | A finer topology has fewer... |
| fclsss2 24372 | A finer filter has fewer c... |
| fclsrest 24373 | The set of cluster points ... |
| fclscf 24374 | Characterization of finene... |
| flimfcls 24375 | A limit point is a cluster... |
| fclsfnflim 24376 | A filter clusters at a poi... |
| flimfnfcls 24377 | A filter converges to a po... |
| fclscmpi 24378 | Forward direction of ~ fcl... |
| fclscmp 24379 | A space is compact iff eve... |
| uffclsflim 24380 | The cluster points of an u... |
| ufilcmp 24381 | A space is compact iff eve... |
| fcfval 24382 | The set of cluster points ... |
| isfcf 24383 | The property of being a cl... |
| fcfnei 24384 | The property of being a cl... |
| fcfelbas 24385 | A cluster point of a funct... |
| fcfneii 24386 | A neighborhood of a cluste... |
| flfssfcf 24387 | A limit point of a functio... |
| uffcfflf 24388 | If the domain filter is an... |
| cnpfcfi 24389 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 24390 | A function ` F ` is contin... |
| cnfcf 24391 | Continuity of a function i... |
| flfcntr 24392 | A continuous function's va... |
| alexsublem 24393 | Lemma for ~ alexsub . (Co... |
| alexsub 24394 | The Alexander Subbase Theo... |
| alexsubb 24395 | Biconditional form of the ... |
| alexsubALTlem1 24396 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 24397 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 24398 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 24399 | Lemma for ~ alexsubALT . ... |
| alexsubALT 24400 | The Alexander Subbase Theo... |
| ptcmplem1 24401 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 24402 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 24403 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 24404 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 24405 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24406 | Tychonoff's theorem: The ... |
| ptcmp 24407 | Tychonoff's theorem: The ... |
| cnextval 24410 | The function applying cont... |
| cnextfval 24411 | The continuous extension o... |
| cnextrel 24412 | In the general case, a con... |
| cnextfun 24413 | If the target space is Hau... |
| cnextfvval 24414 | The value of the continuou... |
| cnextf 24415 | Extension by continuity. ... |
| cnextcn 24416 | Extension by continuity. ... |
| cnextfres1 24417 | ` F ` and its extension by... |
| cnextfres 24418 | ` F ` and its extension by... |
| istmd 24423 | The predicate "is a topolo... |
| tmdmnd 24424 | A topological monoid is a ... |
| tmdtps 24425 | A topological monoid is a ... |
| istgp 24426 | The predicate "is a topolo... |
| tgpgrp 24427 | A topological group is a g... |
| tgptmd 24428 | A topological group is a t... |
| tgptps 24429 | A topological group is a t... |
| tmdtopon 24430 | The topology of a topologi... |
| tgptopon 24431 | The topology of a topologi... |
| tmdcn 24432 | In a topological monoid, t... |
| tgpcn 24433 | In a topological group, th... |
| tgpinv 24434 | In a topological group, th... |
| grpinvhmeo 24435 | The inverse function in a ... |
| cnmpt1plusg 24436 | Continuity of the group su... |
| cnmpt2plusg 24437 | Continuity of the group su... |
| tmdcn2 24438 | Write out the definition o... |
| tgpsubcn 24439 | In a topological group, th... |
| istgp2 24440 | A group with a topology is... |
| tmdmulg 24441 | In a topological monoid, t... |
| tgpmulg 24442 | In a topological group, th... |
| tgpmulg2 24443 | In a topological monoid, t... |
| tmdgsum 24444 | In a topological monoid, t... |
| tmdgsum2 24445 | For any neighborhood ` U `... |
| oppgtmd 24446 | The opposite of a topologi... |
| oppgtgp 24447 | The opposite of a topologi... |
| distgp 24448 | Any group equipped with th... |
| indistgp 24449 | Any group equipped with th... |
| efmndtmd 24450 | The monoid of endofunction... |
| tmdlactcn 24451 | The left group action of e... |
| tgplacthmeo 24452 | The left group action of e... |
| submtmd 24453 | A submonoid of a topologic... |
| subgtgp 24454 | A subgroup of a topologica... |
| symgtgp 24455 | The symmetric group is a t... |
| subgntr 24456 | A subgroup of a topologica... |
| opnsubg 24457 | An open subgroup of a topo... |
| clssubg 24458 | The closure of a subgroup ... |
| clsnsg 24459 | The closure of a normal su... |
| cldsubg 24460 | A subgroup of finite index... |
| tgpconncompeqg 24461 | The connected component co... |
| tgpconncomp 24462 | The identity component, th... |
| tgpconncompss 24463 | The identity component is ... |
| ghmcnp 24464 | A group homomorphism on to... |
| snclseqg 24465 | The coset of the closure o... |
| tgphaus 24466 | A topological group is Hau... |
| tgpt1 24467 | Hausdorff and T1 are equiv... |
| tgpt0 24468 | Hausdorff and T0 are equiv... |
| qustgpopn 24469 | A quotient map in a topolo... |
| qustgplem 24470 | Lemma for ~ qustgp . (Con... |
| qustgp 24471 | The quotient of a topologi... |
| qustgphaus 24472 | The quotient of a topologi... |
| prdstmdd 24473 | The product of a family of... |
| prdstgpd 24474 | The product of a family of... |
| tsmsfbas 24477 | The collection of all sets... |
| tsmslem1 24478 | The finite partial sums of... |
| tsmsval2 24479 | Definition of the topologi... |
| tsmsval 24480 | Definition of the topologi... |
| tsmspropd 24481 | The group sum depends only... |
| eltsms 24482 | The property of being a su... |
| tsmsi 24483 | The property of being a su... |
| tsmscl 24484 | A sum in a topological gro... |
| haustsms 24485 | In a Hausdorff topological... |
| haustsms2 24486 | In a Hausdorff topological... |
| tsmscls 24487 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24488 | The convergent points of a... |
| tsmsid 24489 | If a sum is finite, the us... |
| haustsmsid 24490 | In a Hausdorff topological... |
| tsms0 24491 | The sum of zero is zero. ... |
| tsmssubm 24492 | Evaluate an infinite group... |
| tsmsres 24493 | Extend an infinite group s... |
| tsmsf1o 24494 | Re-index an infinite group... |
| tsmsmhm 24495 | Apply a continuous group h... |
| tsmsadd 24496 | The sum of two infinite gr... |
| tsmsinv 24497 | Inverse of an infinite gro... |
| tsmssub 24498 | The difference of two infi... |
| tgptsmscls 24499 | A sum in a topological gro... |
| tgptsmscld 24500 | The set of limit points to... |
| tsmssplit 24501 | Split a topological group ... |
| tsmsxplem1 24502 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24503 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24504 | Write a sum over a two-dim... |
| istrg 24513 | Express the predicate " ` ... |
| trgtmd 24514 | The multiplicative monoid ... |
| istdrg 24515 | Express the predicate " ` ... |
| tdrgunit 24516 | The unit group of a topolo... |
| trgtgp 24517 | A topological ring is a to... |
| trgtmd2 24518 | A topological ring is a to... |
| trgtps 24519 | A topological ring is a to... |
| trgring 24520 | A topological ring is a ri... |
| trggrp 24521 | A topological ring is a gr... |
| tdrgtrg 24522 | A topological division rin... |
| tdrgdrng 24523 | A topological division rin... |
| tdrgring 24524 | A topological division rin... |
| tdrgtmd 24525 | A topological division rin... |
| tdrgtps 24526 | A topological division rin... |
| istdrg2 24527 | A topological-ring divisio... |
| mulrcn 24528 | The functionalization of t... |
| invrcn2 24529 | The multiplicative inverse... |
| invrcn 24530 | The multiplicative inverse... |
| cnmpt1mulr 24531 | Continuity of ring multipl... |
| cnmpt2mulr 24532 | Continuity of ring multipl... |
| dvrcn 24533 | The division function is c... |
| istlm 24534 | The predicate " ` W ` is a... |
| vscacn 24535 | The scalar multiplication ... |
| tlmtmd 24536 | A topological module is a ... |
| tlmtps 24537 | A topological module is a ... |
| tlmlmod 24538 | A topological module is a ... |
| tlmtrg 24539 | The scalar ring of a topol... |
| tlmscatps 24540 | The scalar ring of a topol... |
| istvc 24541 | A topological vector space... |
| tvctdrg 24542 | The scalar field of a topo... |
| cnmpt1vsca 24543 | Continuity of scalar multi... |
| cnmpt2vsca 24544 | Continuity of scalar multi... |
| tlmtgp 24545 | A topological vector space... |
| tvctlm 24546 | A topological vector space... |
| tvclmod 24547 | A topological vector space... |
| tvclvec 24548 | A topological vector space... |
| ustfn 24551 | The defined uniform struct... |
| ustval 24552 | The class of all uniform s... |
| isust 24553 | The predicate " ` U ` is a... |
| ustssxp 24554 | Entourages are subsets of ... |
| ustssel 24555 | A uniform structure is upw... |
| ustbasel 24556 | The full set is always an ... |
| ustincl 24557 | A uniform structure is clo... |
| ustdiag 24558 | The diagonal set is includ... |
| ustinvel 24559 | If ` V ` is an entourage, ... |
| ustexhalf 24560 | For each entourage ` V ` t... |
| ustrel 24561 | The elements of uniform st... |
| ustfilxp 24562 | A uniform structure on a n... |
| ustne0 24563 | A uniform structure cannot... |
| ustssco 24564 | In an uniform structure, a... |
| ustexsym 24565 | In an uniform structure, f... |
| ustex2sym 24566 | In an uniform structure, f... |
| ustex3sym 24567 | In an uniform structure, f... |
| ustref 24568 | Any element of the base se... |
| ust0 24569 | The unique uniform structu... |
| ustn0 24570 | The empty set is not an un... |
| ustund 24571 | If two intersecting sets `... |
| ustelimasn 24572 | Any point ` A ` is near en... |
| ustneism 24573 | For a point ` A ` in ` X `... |
| ustbas2 24574 | Second direction for ~ ust... |
| ustuni 24575 | The set union of a uniform... |
| ustbas 24576 | Recover the base of an uni... |
| ustimasn 24577 | Lemma for ~ ustuqtop . (C... |
| trust 24578 | The trace of a uniform str... |
| utopval 24581 | The topology induced by a ... |
| elutop 24582 | Open sets in the topology ... |
| utoptop 24583 | The topology induced by a ... |
| utopbas 24584 | The base of the topology i... |
| utoptopon 24585 | Topology induced by a unif... |
| restutop 24586 | Restriction of a topology ... |
| restutopopn 24587 | The restriction of the top... |
| ustuqtoplem 24588 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24589 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24590 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24591 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24592 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24593 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24594 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24595 | For a given uniform struct... |
| utopsnneiplem 24596 | The neighborhoods of a poi... |
| utopsnneip 24597 | The neighborhoods of a poi... |
| utopsnnei 24598 | Images of singletons by en... |
| utop2nei 24599 | For any symmetrical entour... |
| utop3cls 24600 | Relation between a topolog... |
| utopreg 24601 | All Hausdorff uniform spac... |
| ussval 24608 | The uniform structure on u... |
| ussid 24609 | In case the base of the ` ... |
| isusp 24610 | The predicate ` W ` is a u... |
| ressuss 24611 | Value of the uniform struc... |
| ressust 24612 | The uniform structure of a... |
| ressusp 24613 | The restriction of a unifo... |
| tusval 24614 | The value of the uniform s... |
| tuslem 24615 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24616 | The base set of a construc... |
| tusunif 24617 | The uniform structure of a... |
| tususs 24618 | The uniform structure of a... |
| tustopn 24619 | The topology induced by a ... |
| tususp 24620 | A constructed uniform spac... |
| tustps 24621 | A constructed uniform spac... |
| uspreg 24622 | If a uniform space is Haus... |
| ucnval 24625 | The set of all uniformly c... |
| isucn 24626 | The predicate " ` F ` is a... |
| isucn2 24627 | The predicate " ` F ` is a... |
| ucnimalem 24628 | Reformulate the ` G ` func... |
| ucnima 24629 | An equivalent statement of... |
| ucnprima 24630 | The preimage by a uniforml... |
| iducn 24631 | The identity is uniformly ... |
| cstucnd 24632 | A constant function is uni... |
| ucncn 24633 | Uniform continuity implies... |
| iscfilu 24636 | The predicate " ` F ` is a... |
| cfilufbas 24637 | A Cauchy filter base is a ... |
| cfiluexsm 24638 | For a Cauchy filter base a... |
| fmucndlem 24639 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24640 | The image of a Cauchy filt... |
| cfilufg 24641 | The filter generated by a ... |
| trcfilu 24642 | Condition for the trace of... |
| cfiluweak 24643 | A Cauchy filter base is al... |
| neipcfilu 24644 | In an uniform space, a nei... |
| iscusp 24647 | The predicate " ` W ` is a... |
| cuspusp 24648 | A complete uniform space i... |
| cuspcvg 24649 | In a complete uniform spac... |
| iscusp2 24650 | The predicate " ` W ` is a... |
| cnextucn 24651 | Extension by continuity. ... |
| ucnextcn 24652 | Extension by continuity. ... |
| ispsmet 24653 | Express the predicate " ` ... |
| psmetdmdm 24654 | Recover the base set from ... |
| psmetf 24655 | The distance function of a... |
| psmetcl 24656 | Closure of the distance fu... |
| psmet0 24657 | The distance function of a... |
| psmettri2 24658 | Triangle inequality for th... |
| psmetsym 24659 | The distance function of a... |
| psmettri 24660 | Triangle inequality for th... |
| psmetge0 24661 | The distance function of a... |
| psmetxrge0 24662 | The distance function of a... |
| psmetres2 24663 | Restriction of a pseudomet... |
| psmetlecl 24664 | Real closure of an extende... |
| distspace 24665 | A set ` X ` together with ... |
| ismet 24672 | Express the predicate " ` ... |
| isxmet 24673 | Express the predicate " ` ... |
| ismeti 24674 | Properties that determine ... |
| isxmetd 24675 | Properties that determine ... |
| isxmet2d 24676 | It is safe to only require... |
| metflem 24677 | Lemma for ~ metf and other... |
| xmetf 24678 | Mapping of the distance fu... |
| metf 24679 | Mapping of the distance fu... |
| xmetcl 24680 | Closure of the distance fu... |
| metcl 24681 | Closure of the distance fu... |
| ismet2 24682 | An extended metric is a me... |
| metxmet 24683 | A metric is an extended me... |
| xmetdmdm 24684 | Recover the base set from ... |
| metdmdm 24685 | Recover the base set from ... |
| xmetunirn 24686 | Two ways to express an ext... |
| xmeteq0 24687 | The value of an extended m... |
| meteq0 24688 | The value of a metric is z... |
| xmettri2 24689 | Triangle inequality for th... |
| mettri2 24690 | Triangle inequality for th... |
| xmet0 24691 | The distance function of a... |
| met0 24692 | The distance function of a... |
| xmetge0 24693 | The distance function of a... |
| metge0 24694 | The distance function of a... |
| xmetlecl 24695 | Real closure of an extende... |
| xmetsym 24696 | The distance function of a... |
| xmetpsmet 24697 | An extended metric is a ps... |
| xmettpos 24698 | The distance function of a... |
| metsym 24699 | The distance function of a... |
| xmettri 24700 | Triangle inequality for th... |
| mettri 24701 | Triangle inequality for th... |
| xmettri3 24702 | Triangle inequality for th... |
| mettri3 24703 | Triangle inequality for th... |
| xmetrtri 24704 | One half of the reverse tr... |
| xmetrtri2 24705 | The reverse triangle inequ... |
| metrtri 24706 | Reverse triangle inequalit... |
| xmetgt0 24707 | The distance function of a... |
| metgt0 24708 | The distance function of a... |
| metn0 24709 | A metric space is nonempty... |
| xmetres2 24710 | Restriction of an extended... |
| metreslem 24711 | Lemma for ~ metres . (Con... |
| metres2 24712 | Lemma for ~ metres . (Con... |
| xmetres 24713 | A restriction of an extend... |
| metres 24714 | A restriction of a metric ... |
| 0met 24715 | The empty metric. (Contri... |
| prdsdsf 24716 | The product metric is a fu... |
| prdsxmetlem 24717 | The product metric is an e... |
| prdsxmet 24718 | The product metric is an e... |
| prdsmet 24719 | The product metric is a me... |
| ressprdsds 24720 | Restriction of a product m... |
| resspwsds 24721 | Restriction of a power met... |
| imasdsf1olem 24722 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24723 | The distance function is t... |
| imasf1oxmet 24724 | The image of an extended m... |
| imasf1omet 24725 | The image of a metric is a... |
| xpsdsfn 24726 | Closure of the metric in a... |
| xpsdsfn2 24727 | Closure of the metric in a... |
| xpsxmetlem 24728 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24729 | A product metric of extend... |
| xpsdsval 24730 | Value of the metric in a b... |
| xpsmet 24731 | The direct product of two ... |
| blfvalps 24732 | The value of the ball func... |
| blfval 24733 | The value of the ball func... |
| blvalps 24734 | The ball around a point ` ... |
| blval 24735 | The ball around a point ` ... |
| elblps 24736 | Membership in a ball. (Co... |
| elbl 24737 | Membership in a ball. (Co... |
| elbl2ps 24738 | Membership in a ball. (Co... |
| elbl2 24739 | Membership in a ball. (Co... |
| elbl3ps 24740 | Membership in a ball, with... |
| elbl3 24741 | Membership in a ball, with... |
| blcomps 24742 | Commute the arguments to t... |
| blcom 24743 | Commute the arguments to t... |
| xblpnfps 24744 | The infinity ball in an ex... |
| xblpnf 24745 | The infinity ball in an ex... |
| blpnf 24746 | The infinity ball in a sta... |
| bldisj 24747 | Two balls are disjoint if ... |
| blgt0 24748 | A nonempty ball implies th... |
| bl2in 24749 | Two balls are disjoint if ... |
| xblss2ps 24750 | One ball is contained in a... |
| xblss2 24751 | One ball is contained in a... |
| blss2ps 24752 | One ball is contained in a... |
| blss2 24753 | One ball is contained in a... |
| blhalf 24754 | A ball of radius ` R / 2 `... |
| blfps 24755 | Mapping of a ball. (Contr... |
| blf 24756 | Mapping of a ball. (Contr... |
| blrnps 24757 | Membership in the range of... |
| blrn 24758 | Membership in the range of... |
| xblcntrps 24759 | A ball contains its center... |
| xblcntr 24760 | A ball contains its center... |
| blcntrps 24761 | A ball contains its center... |
| blcntr 24762 | A ball contains its center... |
| xbln0 24763 | A ball is nonempty iff the... |
| bln0 24764 | A ball is not empty. (Con... |
| blelrnps 24765 | A ball belongs to the set ... |
| blelrn 24766 | A ball belongs to the set ... |
| blssm 24767 | A ball is a subset of the ... |
| unirnblps 24768 | The union of the set of ba... |
| unirnbl 24769 | The union of the set of ba... |
| blin 24770 | The intersection of two ba... |
| ssblps 24771 | The size of a ball increas... |
| ssbl 24772 | The size of a ball increas... |
| blssps 24773 | Any point ` P ` in a ball ... |
| blss 24774 | Any point ` P ` in a ball ... |
| blssexps 24775 | Two ways to express the ex... |
| blssex 24776 | Two ways to express the ex... |
| ssblex 24777 | A nested ball exists whose... |
| blin2 24778 | Given any two balls and a ... |
| blbas 24779 | The balls of a metric spac... |
| blres 24780 | A ball in a restricted met... |
| xmeterval 24781 | Value of the "finitely sep... |
| xmeter 24782 | The "finitely separated" r... |
| xmetec 24783 | The equivalence classes un... |
| blssec 24784 | A ball centered at ` P ` i... |
| blpnfctr 24785 | The infinity ball in an ex... |
| xmetresbl 24786 | An extended metric restric... |
| mopnval 24787 | An open set is a subset of... |
| mopntopon 24788 | The set of open sets of a ... |
| mopntop 24789 | The set of open sets of a ... |
| mopnuni 24790 | The union of all open sets... |
| elmopn 24791 | The defining property of a... |
| mopnfss 24792 | The family of open sets of... |
| mopnm 24793 | The base set of a metric s... |
| elmopn2 24794 | A defining property of an ... |
| mopnss 24795 | An open set of a metric sp... |
| isxms 24796 | Express the predicate " ` ... |
| isxms2 24797 | Express the predicate " ` ... |
| isms 24798 | Express the predicate " ` ... |
| isms2 24799 | Express the predicate " ` ... |
| xmstopn 24800 | The topology component of ... |
| mstopn 24801 | The topology component of ... |
| xmstps 24802 | An extended metric space i... |
| msxms 24803 | A metric space is an exten... |
| mstps 24804 | A metric space is a topolo... |
| xmsxmet 24805 | The distance function, sui... |
| msmet 24806 | The distance function, sui... |
| msf 24807 | The distance function of a... |
| xmsxmet2 24808 | The distance function, sui... |
| msmet2 24809 | The distance function, sui... |
| mscl 24810 | Closure of the distance fu... |
| xmscl 24811 | Closure of the distance fu... |
| xmsge0 24812 | The distance function in a... |
| xmseq0 24813 | The distance between two p... |
| xmssym 24814 | The distance function in a... |
| xmstri2 24815 | Triangle inequality for th... |
| mstri2 24816 | Triangle inequality for th... |
| xmstri 24817 | Triangle inequality for th... |
| mstri 24818 | Triangle inequality for th... |
| xmstri3 24819 | Triangle inequality for th... |
| mstri3 24820 | Triangle inequality for th... |
| msrtri 24821 | Reverse triangle inequalit... |
| xmspropd 24822 | Property deduction for an ... |
| mspropd 24823 | Property deduction for a m... |
| setsmsbas 24824 | The base set of a construc... |
| setsmsds 24825 | The distance function of a... |
| setsmstset 24826 | The topology of a construc... |
| setsmstopn 24827 | The topology of a construc... |
| setsxms 24828 | The constructed metric spa... |
| setsms 24829 | The constructed metric spa... |
| tmsval 24830 | For any metric there is an... |
| tmslem 24831 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24832 | The base set of a construc... |
| tmsds 24833 | The metric of a constructe... |
| tmstopn 24834 | The topology of a construc... |
| tmsxms 24835 | The constructed metric spa... |
| tmsms 24836 | The constructed metric spa... |
| imasf1obl 24837 | The image of a metric spac... |
| imasf1oxms 24838 | The image of a metric spac... |
| imasf1oms 24839 | The image of a metric spac... |
| prdsbl 24840 | A ball in the product metr... |
| mopni 24841 | An open set of a metric sp... |
| mopni2 24842 | An open set of a metric sp... |
| mopni3 24843 | An open set of a metric sp... |
| blssopn 24844 | The balls of a metric spac... |
| unimopn 24845 | The union of a collection ... |
| mopnin 24846 | The intersection of two op... |
| mopn0 24847 | The empty set is an open s... |
| rnblopn 24848 | A ball of a metric space i... |
| blopn 24849 | A ball of a metric space i... |
| neibl 24850 | The neighborhoods around a... |
| blnei 24851 | A ball around a point is a... |
| lpbl 24852 | Every ball around a limit ... |
| blsscls2 24853 | A smaller closed ball is c... |
| blcld 24854 | A "closed ball" in a metri... |
| blcls 24855 | The closure of an open bal... |
| blsscls 24856 | If two concentric balls ha... |
| metss 24857 | Two ways of saying that me... |
| metequiv 24858 | Two ways of saying that tw... |
| metequiv2 24859 | If there is a sequence of ... |
| metss2lem 24860 | Lemma for ~ metss2 . (Con... |
| metss2 24861 | If the metric ` D ` is "st... |
| comet 24862 | The composition of an exte... |
| stdbdmetval 24863 | Value of the standard boun... |
| stdbdxmet 24864 | The standard bounded metri... |
| stdbdmet 24865 | The standard bounded metri... |
| stdbdbl 24866 | The standard bounded metri... |
| stdbdmopn 24867 | The standard bounded metri... |
| mopnex 24868 | The topology generated by ... |
| methaus 24869 | The topology generated by ... |
| met1stc 24870 | The topology generated by ... |
| met2ndci 24871 | A separable metric space (... |
| met2ndc 24872 | A metric space is second-c... |
| metrest 24873 | Two alternate formulations... |
| ressxms 24874 | The restriction of a metri... |
| ressms 24875 | The restriction of a metri... |
| prdsmslem1 24876 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24877 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24878 | Lemma for ~ prdsxms . The... |
| prdsxms 24879 | The indexed product struct... |
| prdsms 24880 | The indexed product struct... |
| pwsxms 24881 | A power of an extended met... |
| pwsms 24882 | A power of a metric space ... |
| xpsxms 24883 | A binary product of metric... |
| xpsms 24884 | A binary product of metric... |
| tmsxps 24885 | Express the product of two... |
| tmsxpsmopn 24886 | Express the product of two... |
| tmsxpsval 24887 | Value of the product of tw... |
| tmsxpsval2 24888 | Value of the product of tw... |
| metcnp3 24889 | Two ways to express that `... |
| metcnp 24890 | Two ways to say a mapping ... |
| metcnp2 24891 | Two ways to say a mapping ... |
| metcn 24892 | Two ways to say a mapping ... |
| metcnpi 24893 | Epsilon-delta property of ... |
| metcnpi2 24894 | Epsilon-delta property of ... |
| metcnpi3 24895 | Epsilon-delta property of ... |
| txmetcnp 24896 | Continuity of a binary ope... |
| txmetcn 24897 | Continuity of a binary ope... |
| metuval 24898 | Value of the uniform struc... |
| metustel 24899 | Define a filter base ` F `... |
| metustss 24900 | Range of the elements of t... |
| metustrel 24901 | Elements of the filter bas... |
| metustto 24902 | Any two elements of the fi... |
| metustid 24903 | The identity diagonal is i... |
| metustsym 24904 | Elements of the filter bas... |
| metustexhalf 24905 | For any element ` A ` of t... |
| metustfbas 24906 | The filter base generated ... |
| metust 24907 | The uniform structure gene... |
| cfilucfil 24908 | Given a metric ` D ` and a... |
| metuust 24909 | The uniform structure gene... |
| cfilucfil2 24910 | Given a metric ` D ` and a... |
| blval2 24911 | The ball around a point ` ... |
| elbl4 24912 | Membership in a ball, alte... |
| metuel 24913 | Elementhood in the uniform... |
| metuel2 24914 | Elementhood in the uniform... |
| metustbl 24915 | The "section" image of an ... |
| psmetutop 24916 | The topology induced by a ... |
| xmetutop 24917 | The topology induced by a ... |
| xmsusp 24918 | If the uniform set of a me... |
| restmetu 24919 | The uniform structure gene... |
| metucn 24920 | Uniform continuity in metr... |
| dscmet 24921 | The discrete metric on any... |
| dscopn 24922 | The discrete metric genera... |
| nrmmetd 24923 | Show that a group norm gen... |
| abvmet 24924 | An absolute value ` F ` ge... |
| nmfval 24937 | The value of the norm func... |
| nmval 24938 | The value of the norm as t... |
| nmfval0 24939 | The value of the norm func... |
| nmfval2 24940 | The value of the norm func... |
| nmval2 24941 | The value of the norm on a... |
| nmf2 24942 | The norm on a metric group... |
| nmpropd 24943 | Weak property deduction fo... |
| nmpropd2 24944 | Strong property deduction ... |
| isngp 24945 | The property of being a no... |
| isngp2 24946 | The property of being a no... |
| isngp3 24947 | The property of being a no... |
| ngpgrp 24948 | A normed group is a group.... |
| ngpms 24949 | A normed group is a metric... |
| ngpxms 24950 | A normed group is an exten... |
| ngptps 24951 | A normed group is a topolo... |
| ngpmet 24952 | The (induced) metric of a ... |
| ngpds 24953 | Value of the distance func... |
| ngpdsr 24954 | Value of the distance func... |
| ngpds2 24955 | Write the distance between... |
| ngpds2r 24956 | Write the distance between... |
| ngpds3 24957 | Write the distance between... |
| ngpds3r 24958 | Write the distance between... |
| ngprcan 24959 | Cancel right addition insi... |
| ngplcan 24960 | Cancel left addition insid... |
| isngp4 24961 | Express the property of be... |
| ngpinvds 24962 | Two elements are the same ... |
| ngpsubcan 24963 | Cancel right subtraction i... |
| nmf 24964 | The norm on a normed group... |
| nmcl 24965 | The norm of a normed group... |
| nmge0 24966 | The norm of a normed group... |
| nmeq0 24967 | The identity is the only e... |
| nmne0 24968 | The norm of a nonzero elem... |
| nmrpcl 24969 | The norm of a nonzero elem... |
| nminv 24970 | The norm of a negated elem... |
| nmmtri 24971 | The triangle inequality fo... |
| nmsub 24972 | The norm of the difference... |
| nmrtri 24973 | Reverse triangle inequalit... |
| nm2dif 24974 | Inequality for the differe... |
| nmtri 24975 | The triangle inequality fo... |
| nmtri2 24976 | Triangle inequality for th... |
| ngpi 24977 | The properties of a normed... |
| nm0 24978 | Norm of the identity eleme... |
| nmgt0 24979 | The norm of a nonzero elem... |
| sgrim 24980 | The induced metric on a su... |
| sgrimval 24981 | The induced metric on a su... |
| subgnm 24982 | The norm in a subgroup. (... |
| subgnm2 24983 | A substructure assigns the... |
| subgngp 24984 | A normed group restricted ... |
| ngptgp 24985 | A normed abelian group is ... |
| ngppropd 24986 | Property deduction for a n... |
| reldmtng 24987 | The function ` toNrmGrp ` ... |
| tngval 24988 | Value of the function whic... |
| tnglem 24989 | Lemma for ~ tngbas and sim... |
| tngbas 24990 | The base set of a structur... |
| tngplusg 24991 | The group addition of a st... |
| tng0 24992 | The group identity of a st... |
| tngmulr 24993 | The ring multiplication of... |
| tngsca 24994 | The scalar ring of a struc... |
| tngvsca 24995 | The scalar multiplication ... |
| tngip 24996 | The inner product operatio... |
| tngds 24997 | The metric function of a s... |
| tngtset 24998 | The topology generated by ... |
| tngtopn 24999 | The topology generated by ... |
| tngnm 25000 | The topology generated by ... |
| tngngp2 25001 | A norm turns a group into ... |
| tngngpd 25002 | Derive the axioms for a no... |
| tngngp 25003 | Derive the axioms for a no... |
| tnggrpr 25004 | If a structure equipped wi... |
| tngngp3 25005 | Alternate definition of a ... |
| nrmtngdist 25006 | The augmentation of a norm... |
| nrmtngnrm 25007 | The augmentation of a norm... |
| tngngpim 25008 | The induced metric of a no... |
| isnrg 25009 | A normed ring is a ring wi... |
| nrgabv 25010 | The norm of a normed ring ... |
| nrgngp 25011 | A normed ring is a normed ... |
| nrgring 25012 | A normed ring is a ring. ... |
| nmmul 25013 | The norm of a product in a... |
| nrgdsdi 25014 | Distribute a distance calc... |
| nrgdsdir 25015 | Distribute a distance calc... |
| nm1 25016 | The norm of one in a nonze... |
| unitnmn0 25017 | The norm of a unit is nonz... |
| nminvr 25018 | The norm of an inverse in ... |
| nmdvr 25019 | The norm of a division in ... |
| nrgdomn 25020 | A nonzero normed ring is a... |
| nrgtgp 25021 | A normed ring is a topolog... |
| subrgnrg 25022 | A normed ring restricted t... |
| tngnrg 25023 | Given any absolute value o... |
| isnlm 25024 | A normed (left) module is ... |
| nmvs 25025 | Defining property of a nor... |
| nlmngp 25026 | A normed module is a norme... |
| nlmlmod 25027 | A normed module is a left ... |
| nlmnrg 25028 | The scalar component of a ... |
| nlmngp2 25029 | The scalar component of a ... |
| nlmdsdi 25030 | Distribute a distance calc... |
| nlmdsdir 25031 | Distribute a distance calc... |
| nlmmul0or 25032 | If a scalar product is zer... |
| sranlm 25033 | The subring algebra over a... |
| nlmvscnlem2 25034 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 25035 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 25036 | The scalar multiplication ... |
| rlmnlm 25037 | The ring module over a nor... |
| rlmnm 25038 | The norm function in the r... |
| nrgtrg 25039 | A normed ring is a topolog... |
| nrginvrcnlem 25040 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 25041 | The ring inverse function ... |
| nrgtdrg 25042 | A normed division ring is ... |
| nlmtlm 25043 | A normed module is a topol... |
| isnvc 25044 | A normed vector space is j... |
| nvcnlm 25045 | A normed vector space is a... |
| nvclvec 25046 | A normed vector space is a... |
| nvclmod 25047 | A normed vector space is a... |
| isnvc2 25048 | A normed vector space is j... |
| nvctvc 25049 | A normed vector space is a... |
| lssnlm 25050 | A subspace of a normed mod... |
| lssnvc 25051 | A subspace of a normed vec... |
| rlmnvc 25052 | The ring module over a nor... |
| ngpocelbl 25053 | Membership of an off-cente... |
| nmoffn 25060 | The function producing ope... |
| reldmnghm 25061 | Lemma for normed group hom... |
| reldmnmhm 25062 | Lemma for module homomorph... |
| nmofval 25063 | Value of the operator norm... |
| nmoval 25064 | Value of the operator norm... |
| nmogelb 25065 | Property of the operator n... |
| nmolb 25066 | Any upper bound on the val... |
| nmolb2d 25067 | Any upper bound on the val... |
| nmof 25068 | The operator norm is a fun... |
| nmocl 25069 | The operator norm of an op... |
| nmoge0 25070 | The operator norm of an op... |
| nghmfval 25071 | A normed group homomorphis... |
| isnghm 25072 | A normed group homomorphis... |
| isnghm2 25073 | A normed group homomorphis... |
| isnghm3 25074 | A normed group homomorphis... |
| bddnghm 25075 | A bounded group homomorphi... |
| nghmcl 25076 | A normed group homomorphis... |
| nmoi 25077 | The operator norm achieves... |
| nmoix 25078 | The operator norm is a bou... |
| nmoi2 25079 | The operator norm is a bou... |
| nmoleub 25080 | The operator norm, defined... |
| nghmrcl1 25081 | Reverse closure for a norm... |
| nghmrcl2 25082 | Reverse closure for a norm... |
| nghmghm 25083 | A normed group homomorphis... |
| nmo0 25084 | The operator norm of the z... |
| nmoeq0 25085 | The operator norm is zero ... |
| nmoco 25086 | An upper bound on the oper... |
| nghmco 25087 | The composition of normed ... |
| nmotri 25088 | Triangle inequality for th... |
| nghmplusg 25089 | The sum of two bounded lin... |
| 0nghm 25090 | The zero operator is a nor... |
| nmoid 25091 | The operator norm of the i... |
| idnghm 25092 | The identity operator is a... |
| nmods 25093 | Upper bound for the distan... |
| nghmcn 25094 | A normed group homomorphis... |
| isnmhm 25095 | A normed module homomorphi... |
| nmhmrcl1 25096 | Reverse closure for a norm... |
| nmhmrcl2 25097 | Reverse closure for a norm... |
| nmhmlmhm 25098 | A normed module homomorphi... |
| nmhmnghm 25099 | A normed module homomorphi... |
| nmhmghm 25100 | A normed module homomorphi... |
| isnmhm2 25101 | A normed module homomorphi... |
| nmhmcl 25102 | A normed module homomorphi... |
| idnmhm 25103 | The identity operator is a... |
| 0nmhm 25104 | The zero operator is a bou... |
| nmhmco 25105 | The composition of bounded... |
| nmhmplusg 25106 | The sum of two bounded lin... |
| qtopbaslem 25107 | The set of open intervals ... |
| qtopbas 25108 | The set of open intervals ... |
| retopbas 25109 | A basis for the standard t... |
| retop 25110 | The standard topology on t... |
| uniretop 25111 | The underlying set of the ... |
| retopon 25112 | The standard topology on t... |
| retps 25113 | The standard topological s... |
| iooretop 25114 | Open intervals are open se... |
| icccld 25115 | Closed intervals are close... |
| icopnfcld 25116 | Right-unbounded closed int... |
| iocmnfcld 25117 | Left-unbounded closed inte... |
| qdensere 25118 | ` QQ ` is dense in the sta... |
| cnmetdval 25119 | Value of the distance func... |
| cnmet 25120 | The absolute value metric ... |
| cnxmet 25121 | The absolute value metric ... |
| cnbl0 25122 | Two ways to write the open... |
| cnblcld 25123 | Two ways to write the clos... |
| cnfldms 25124 | The complex number field i... |
| cnfldxms 25125 | The complex number field i... |
| cnfldtps 25126 | The complex number field i... |
| cnfldnm 25127 | The norm of the field of c... |
| cnngp 25128 | The complex numbers form a... |
| cnnrg 25129 | The complex numbers form a... |
| cnfldtopn 25130 | The topology of the comple... |
| cnfldtopon 25131 | The topology of the comple... |
| cnfldtop 25132 | The topology of the comple... |
| cnfldhaus 25133 | The topology of the comple... |
| unicntop 25134 | The underlying set of the ... |
| cnopn 25135 | The set of complex numbers... |
| cnn0opn 25136 | The set of nonzero complex... |
| zringnrg 25137 | The ring of integers is a ... |
| remetdval 25138 | Value of the distance func... |
| remet 25139 | The absolute value metric ... |
| rexmet 25140 | The absolute value metric ... |
| bl2ioo 25141 | A ball in terms of an open... |
| ioo2bl 25142 | An open interval of reals ... |
| ioo2blex 25143 | An open interval of reals ... |
| blssioo 25144 | The balls of the standard ... |
| tgioo 25145 | The topology generated by ... |
| qdensere2 25146 | ` QQ ` is dense in ` RR ` ... |
| blcvx 25147 | An open ball in the comple... |
| rehaus 25148 | The standard topology on t... |
| tgqioo 25149 | The topology generated by ... |
| re2ndc 25150 | The standard topology on t... |
| resubmet 25151 | The subspace topology indu... |
| tgioo2 25152 | The standard topology on t... |
| rerest 25153 | The subspace topology indu... |
| tgioo4 25154 | The standard topology on t... |
| tgioo3 25155 | The standard topology on t... |
| xrtgioo 25156 | The topology on the extend... |
| xrrest 25157 | The subspace topology indu... |
| xrrest2 25158 | The subspace topology indu... |
| xrsxmet 25159 | The metric on the extended... |
| xrsdsre 25160 | The metric on the extended... |
| xrsblre 25161 | Any ball of the metric of ... |
| xrsmopn 25162 | The metric on the extended... |
| zcld 25163 | The integers are a closed ... |
| recld2 25164 | The real numbers are a clo... |
| zcld2 25165 | The integers are a closed ... |
| zdis 25166 | The integers are a discret... |
| sszcld 25167 | Every subset of the intege... |
| reperflem 25168 | A subset of the real numbe... |
| reperf 25169 | The real numbers are a per... |
| cnperf 25170 | The complex numbers are a ... |
| iccntr 25171 | The interior of a closed i... |
| icccmplem1 25172 | Lemma for ~ icccmp . (Con... |
| icccmplem2 25173 | Lemma for ~ icccmp . (Con... |
| icccmplem3 25174 | Lemma for ~ icccmp . (Con... |
| icccmp 25175 | A closed interval in ` RR ... |
| reconnlem1 25176 | Lemma for ~ reconn . Conn... |
| reconnlem2 25177 | Lemma for ~ reconn . (Con... |
| reconn 25178 | A subset of the reals is c... |
| retopconn 25179 | Corollary of ~ reconn . T... |
| iccconn 25180 | A closed interval is conne... |
| opnreen 25181 | Every nonempty open set is... |
| rectbntr0 25182 | A countable subset of the ... |
| xrge0gsumle 25183 | A finite sum in the nonneg... |
| xrge0tsms 25184 | Any finite or infinite sum... |
| xrge0tsms2 25185 | Any finite or infinite sum... |
| metdcnlem 25186 | The metric function of a m... |
| xmetdcn2 25187 | The metric function of an ... |
| xmetdcn 25188 | The metric function of an ... |
| metdcn2 25189 | The metric function of a m... |
| metdcn 25190 | The metric function of a m... |
| msdcn 25191 | The metric function of a m... |
| cnmpt1ds 25192 | Continuity of the metric f... |
| cnmpt2ds 25193 | Continuity of the metric f... |
| nmcn 25194 | The norm of a normed group... |
| ngnmcncn 25195 | The norm of a normed group... |
| abscn 25196 | The absolute value functio... |
| metdsval 25197 | Value of the "distance to ... |
| metdsf 25198 | The distance from a point ... |
| metdsge 25199 | The distance from the poin... |
| metds0 25200 | If a point is in a set, it... |
| metdstri 25201 | A generalization of the tr... |
| metdsle 25202 | The distance from a point ... |
| metdsre 25203 | The distance from a point ... |
| metdseq0 25204 | The distance from a point ... |
| metdscnlem 25205 | Lemma for ~ metdscn . (Co... |
| metdscn 25206 | The function ` F ` which g... |
| metdscn2 25207 | The function ` F ` which g... |
| metnrmlem1a 25208 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 25209 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 25210 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 25211 | Lemma for ~ metnrm . (Con... |
| metnrm 25212 | A metric space is normal. ... |
| metreg 25213 | A metric space is regular.... |
| addcnlem 25214 | Lemma for ~ addcn , ~ subc... |
| addcn 25215 | Complex number addition is... |
| subcn 25216 | Complex number subtraction... |
| mulcn 25217 | Complex number multiplicat... |
| mpomulcn 25218 | Complex number multiplicat... |
| divcn 25219 | Complex number division is... |
| cnfldtgp 25220 | The complex numbers form a... |
| fsumcn 25221 | A finite sum of functions ... |
| fsum2cn 25222 | Version of ~ fsumcn for tw... |
| expcn 25223 | The power function on comp... |
| divccn 25224 | Division by a nonzero cons... |
| sqcn 25225 | The square function on com... |
| iitopon 25230 | The unit interval is a top... |
| iitop 25231 | The unit interval is a top... |
| iiuni 25232 | The base set of the unit i... |
| dfii2 25233 | Alternate definition of th... |
| dfii3 25234 | Alternate definition of th... |
| dfii4 25235 | Alternate definition of th... |
| dfii5 25236 | The unit interval expresse... |
| iicmp 25237 | The unit interval is compa... |
| iiconn 25238 | The unit interval is conne... |
| cncfval 25239 | The value of the continuou... |
| elcncf 25240 | Membership in the set of c... |
| elcncf2 25241 | Version of ~ elcncf with a... |
| cncfrss 25242 | Reverse closure of the con... |
| cncfrss2 25243 | Reverse closure of the con... |
| cncff 25244 | A continuous complex funct... |
| cncfi 25245 | Defining property of a con... |
| elcncf1di 25246 | Membership in the set of c... |
| elcncf1ii 25247 | Membership in the set of c... |
| rescncf 25248 | A continuous complex funct... |
| cncfcdm 25249 | Change the codomain of a c... |
| cncfss 25250 | The set of continuous func... |
| climcncf 25251 | Image of a limit under a c... |
| abscncf 25252 | Absolute value is continuo... |
| recncf 25253 | Real part is continuous. ... |
| imcncf 25254 | Imaginary part is continuo... |
| cjcncf 25255 | Complex conjugate is conti... |
| mulc1cncf 25256 | Multiplication by a consta... |
| divccncf 25257 | Division by a constant is ... |
| cncfco 25258 | The composition of two con... |
| cncfcompt2 25259 | Composition of continuous ... |
| cncfmet 25260 | Relate complex function co... |
| cncfcn 25261 | Relate complex function co... |
| cncfcn1 25262 | Relate complex function co... |
| cncfmptc 25263 | A constant function is a c... |
| cncfmptid 25264 | The identity function is a... |
| cncfmpt1f 25265 | Composition of continuous ... |
| cncfmpt2f 25266 | Composition of continuous ... |
| cncfmpt2ss 25267 | Composition of continuous ... |
| addccncf 25268 | Adding a constant is a con... |
| idcncf 25269 | The identity function is a... |
| sub1cncf 25270 | Subtracting a constant is ... |
| sub2cncf 25271 | Subtraction from a constan... |
| cdivcncf 25272 | Division with a constant n... |
| negcncf 25273 | The negative function is c... |
| negfcncf 25274 | The negative of a continuo... |
| abscncfALT 25275 | Absolute value is continuo... |
| cncfcnvcn 25276 | Rewrite ~ cmphaushmeo for ... |
| expcncf 25277 | The power function on comp... |
| cnmptre 25278 | Lemma for ~ iirevcn and re... |
| cnmpopc 25279 | Piecewise definition of a ... |
| iirev 25280 | Reverse the unit interval.... |
| iirevcn 25281 | The reversion function is ... |
| iihalf1 25282 | Map the first half of ` II... |
| iihalf1cn 25283 | The first half function is... |
| iihalf2 25284 | Map the second half of ` I... |
| iihalf2cn 25285 | The second half function i... |
| elii1 25286 | Divide the unit interval i... |
| elii2 25287 | Divide the unit interval i... |
| iimulcl 25288 | The unit interval is close... |
| iimulcn 25289 | Multiplication is a contin... |
| icoopnst 25290 | A half-open interval start... |
| iocopnst 25291 | A half-open interval endin... |
| icchmeo 25292 | The natural bijection from... |
| icopnfcnv 25293 | Define a bijection from ` ... |
| icopnfhmeo 25294 | The defined bijection from... |
| iccpnfcnv 25295 | Define a bijection from ` ... |
| iccpnfhmeo 25296 | The defined bijection from... |
| xrhmeo 25297 | The bijection from ` [ -u ... |
| xrhmph 25298 | The extended reals are hom... |
| xrcmp 25299 | The topology of the extend... |
| xrconn 25300 | The topology of the extend... |
| icccvx 25301 | A linear combination of tw... |
| oprpiece1res1 25302 | Restriction to the first p... |
| oprpiece1res2 25303 | Restriction to the second ... |
| cnrehmeo 25304 | The canonical bijection fr... |
| cnheiborlem 25305 | Lemma for ~ cnheibor . (C... |
| cnheibor 25306 | Heine-Borel theorem for co... |
| cnllycmp 25307 | The topology on the comple... |
| rellycmp 25308 | The topology on the reals ... |
| bndth 25309 | The Boundedness Theorem. ... |
| evth 25310 | The Extreme Value Theorem.... |
| evth2 25311 | The Extreme Value Theorem,... |
| lebnumlem1 25312 | Lemma for ~ lebnum . The ... |
| lebnumlem2 25313 | Lemma for ~ lebnum . As a... |
| lebnumlem3 25314 | Lemma for ~ lebnum . By t... |
| lebnum 25315 | The Lebesgue number lemma,... |
| xlebnum 25316 | Generalize ~ lebnum to ext... |
| lebnumii 25317 | Specialize the Lebesgue nu... |
| ishtpy 25323 | Membership in the class of... |
| htpycn 25324 | A homotopy is a continuous... |
| htpyi 25325 | A homotopy evaluated at it... |
| ishtpyd 25326 | Deduction for membership i... |
| htpycom 25327 | Given a homotopy from ` F ... |
| htpyid 25328 | A homotopy from a function... |
| htpyco1 25329 | Compose a homotopy with a ... |
| htpyco2 25330 | Compose a homotopy with a ... |
| htpycc 25331 | Concatenate two homotopies... |
| isphtpy 25332 | Membership in the class of... |
| phtpyhtpy 25333 | A path homotopy is a homot... |
| phtpycn 25334 | A path homotopy is a conti... |
| phtpyi 25335 | Membership in the class of... |
| phtpy01 25336 | Two path-homotopic paths h... |
| isphtpyd 25337 | Deduction for membership i... |
| isphtpy2d 25338 | Deduction for membership i... |
| phtpycom 25339 | Given a homotopy from ` F ... |
| phtpyid 25340 | A homotopy from a path to ... |
| phtpyco2 25341 | Compose a path homotopy wi... |
| phtpycc 25342 | Concatenate two path homot... |
| phtpcrel 25344 | The path homotopy relation... |
| isphtpc 25345 | The relation "is path homo... |
| phtpcer 25346 | Path homotopy is an equiva... |
| phtpc01 25347 | Path homotopic paths have ... |
| reparphti 25348 | Lemma for ~ reparpht . (C... |
| reparpht 25349 | Reparametrization lemma. ... |
| phtpcco2 25350 | Compose a path homotopy wi... |
| pcofval 25361 | The value of the path conc... |
| pcoval 25362 | The concatenation of two p... |
| pcovalg 25363 | Evaluate the concatenation... |
| pcoval1 25364 | Evaluate the concatenation... |
| pco0 25365 | The starting point of a pa... |
| pco1 25366 | The ending point of a path... |
| pcoval2 25367 | Evaluate the concatenation... |
| pcocn 25368 | The concatenation of two p... |
| copco 25369 | The composition of a conca... |
| pcohtpylem 25370 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 25371 | Homotopy invariance of pat... |
| pcoptcl 25372 | A constant function is a p... |
| pcopt 25373 | Concatenation with a point... |
| pcopt2 25374 | Concatenation with a point... |
| pcoass 25375 | Order of concatenation doe... |
| pcorevcl 25376 | Closure for a reversed pat... |
| pcorevlem 25377 | Lemma for ~ pcorev . Prov... |
| pcorev 25378 | Concatenation with the rev... |
| pcorev2 25379 | Concatenation with the rev... |
| pcophtb 25380 | The path homotopy equivale... |
| om1val 25381 | The definition of the loop... |
| om1bas 25382 | The base set of the loop s... |
| om1elbas 25383 | Elementhood in the base se... |
| om1addcl 25384 | Closure of the group opera... |
| om1plusg 25385 | The group operation (which... |
| om1tset 25386 | The topology of the loop s... |
| om1opn 25387 | The topology of the loop s... |
| pi1val 25388 | The definition of the fund... |
| pi1bas 25389 | The base set of the fundam... |
| pi1blem 25390 | Lemma for ~ pi1buni . (Co... |
| pi1buni 25391 | Another way to write the l... |
| pi1bas2 25392 | The base set of the fundam... |
| pi1eluni 25393 | Elementhood in the base se... |
| pi1bas3 25394 | The base set of the fundam... |
| pi1cpbl 25395 | The group operation, loop ... |
| elpi1 25396 | The elements of the fundam... |
| elpi1i 25397 | The elements of the fundam... |
| pi1addf 25398 | The group operation of ` p... |
| pi1addval 25399 | The concatenation of two p... |
| pi1grplem 25400 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25401 | The fundamental group is a... |
| pi1id 25402 | The identity element of th... |
| pi1inv 25403 | An inverse in the fundamen... |
| pi1xfrf 25404 | Functionality of the loop ... |
| pi1xfrval 25405 | The value of the loop tran... |
| pi1xfr 25406 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25407 | Given a path ` F ` between... |
| pi1xfrcnv 25408 | Given a path ` F ` between... |
| pi1xfrgim 25409 | The mapping ` G ` between ... |
| pi1cof 25410 | Functionality of the loop ... |
| pi1coval 25411 | The value of the loop tran... |
| pi1coghm 25412 | The mapping ` G ` between ... |
| isclm 25415 | A subcomplex module is a l... |
| clmsca 25416 | The ring of scalars ` F ` ... |
| clmsubrg 25417 | The base set of the ring o... |
| clmlmod 25418 | A subcomplex module is a l... |
| clmgrp 25419 | A subcomplex module is an ... |
| clmabl 25420 | A subcomplex module is an ... |
| clmring 25421 | The scalar ring of a subco... |
| clmfgrp 25422 | The scalar ring of a subco... |
| clm0 25423 | The zero of the scalar rin... |
| clm1 25424 | The identity of the scalar... |
| clmadd 25425 | The addition of the scalar... |
| clmmul 25426 | The multiplication of the ... |
| clmcj 25427 | The conjugation of the sca... |
| isclmi 25428 | Reverse direction of ~ isc... |
| clmzss 25429 | The scalar ring of a subco... |
| clmsscn 25430 | The scalar ring of a subco... |
| clmsub 25431 | Subtraction in the scalar ... |
| clmneg 25432 | Negation in the scalar rin... |
| clmneg1 25433 | Minus one is in the scalar... |
| clmabs 25434 | Norm in the scalar ring of... |
| clmacl 25435 | Closure of ring addition f... |
| clmmcl 25436 | Closure of ring multiplica... |
| clmsubcl 25437 | Closure of ring subtractio... |
| lmhmclm 25438 | The domain of a linear ope... |
| clmvscl 25439 | Closure of scalar product ... |
| clmvsass 25440 | Scalar multiplication is a... |
| clmvscom 25441 | Commutative law for the sc... |
| clmvsdir 25442 | Distributive law for scala... |
| clmvsdi 25443 | Distributive law for scala... |
| clmvs1 25444 | Scalar product with ring u... |
| clmvs2 25445 | A vector plus itself is tw... |
| clm0vs 25446 | Zero times a vector is the... |
| clmopfne 25447 | The (functionalized) opera... |
| isclmp 25448 | The predicate "is a subcom... |
| isclmi0 25449 | Properties that determine ... |
| clmvneg1 25450 | Minus 1 times a vector is ... |
| clmvsneg 25451 | Multiplication of a vector... |
| clmmulg 25452 | The group multiple functio... |
| clmsubdir 25453 | Scalar multiplication dist... |
| clmpm1dir 25454 | Subtractive distributive l... |
| clmnegneg 25455 | Double negative of a vecto... |
| clmnegsubdi2 25456 | Distribution of negative o... |
| clmsub4 25457 | Rearrangement of 4 terms i... |
| clmvsrinv 25458 | A vector minus itself. (C... |
| clmvslinv 25459 | Minus a vector plus itself... |
| clmvsubval 25460 | Value of vector subtractio... |
| clmvsubval2 25461 | Value of vector subtractio... |
| clmvz 25462 | Two ways to express the ne... |
| zlmclm 25463 | The ` ZZ ` -module operati... |
| clmzlmvsca 25464 | The scalar product of a su... |
| nmoleub2lem 25465 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25466 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25467 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25468 | The operator norm is the s... |
| nmoleub2b 25469 | The operator norm is the s... |
| nmoleub3 25470 | The operator norm is the s... |
| nmhmcn 25471 | A linear operator over a n... |
| cmodscexp 25472 | The powers of ` _i ` belon... |
| cmodscmulexp 25473 | The scalar product of a ve... |
| cvslvec 25476 | A subcomplex vector space ... |
| cvsclm 25477 | A subcomplex vector space ... |
| iscvs 25478 | A subcomplex vector space ... |
| iscvsp 25479 | The predicate "is a subcom... |
| iscvsi 25480 | Properties that determine ... |
| cvsi 25481 | The properties of a subcom... |
| cvsunit 25482 | Unit group of the scalar r... |
| cvsdiv 25483 | Division of the scalar rin... |
| cvsdivcl 25484 | The scalar field of a subc... |
| cvsmuleqdivd 25485 | An equality involving rati... |
| cvsdiveqd 25486 | An equality involving rati... |
| cnlmodlem1 25487 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25488 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25489 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25490 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25491 | The set of complex numbers... |
| cnstrcvs 25492 | The set of complex numbers... |
| cnrbas 25493 | The set of complex numbers... |
| cnrlmod 25494 | The complex left module of... |
| cnrlvec 25495 | The complex left module of... |
| cncvs 25496 | The complex left module of... |
| recvs 25497 | The field of the real numb... |
| qcvs 25498 | The field of rational numb... |
| zclmncvs 25499 | The ring of integers as le... |
| isncvsngp 25500 | A normed subcomplex vector... |
| isncvsngpd 25501 | Properties that determine ... |
| ncvsi 25502 | The properties of a normed... |
| ncvsprp 25503 | Proportionality property o... |
| ncvsge0 25504 | The norm of a scalar produ... |
| ncvsm1 25505 | The norm of the opposite o... |
| ncvsdif 25506 | The norm of the difference... |
| ncvspi 25507 | The norm of a vector plus ... |
| ncvs1 25508 | From any nonzero vector of... |
| cnrnvc 25509 | The module of complex numb... |
| cnncvs 25510 | The module of complex numb... |
| cnnm 25511 | The norm of the normed sub... |
| ncvspds 25512 | Value of the distance func... |
| cnindmet 25513 | The metric induced on the ... |
| cnncvsaddassdemo 25514 | Derive the associative law... |
| cnncvsmulassdemo 25515 | Derive the associative law... |
| cnncvsabsnegdemo 25516 | Derive the absolute value ... |
| iscph 25521 | A subcomplex pre-Hilbert s... |
| cphphl 25522 | A subcomplex pre-Hilbert s... |
| cphnlm 25523 | A subcomplex pre-Hilbert s... |
| cphngp 25524 | A subcomplex pre-Hilbert s... |
| cphlmod 25525 | A subcomplex pre-Hilbert s... |
| cphlvec 25526 | A subcomplex pre-Hilbert s... |
| cphnvc 25527 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25528 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25529 | Lemma for ~ cphreccl . (C... |
| cphsca 25530 | A subcomplex pre-Hilbert s... |
| cphsubrg 25531 | The scalar field of a subc... |
| cphreccl 25532 | The scalar field of a subc... |
| cphdivcl 25533 | The scalar field of a subc... |
| cphcjcl 25534 | The scalar field of a subc... |
| cphsqrtcl 25535 | The scalar field of a subc... |
| cphabscl 25536 | The scalar field of a subc... |
| cphsqrtcl2 25537 | The scalar field of a subc... |
| cphsqrtcl3 25538 | If the scalar field of a s... |
| cphqss 25539 | The scalar field of a subc... |
| cphclm 25540 | A subcomplex pre-Hilbert s... |
| cphnmvs 25541 | Norm of a scalar product. ... |
| cphipcl 25542 | An inner product is a memb... |
| cphnmfval 25543 | The value of the norm in a... |
| cphnm 25544 | The square of the norm is ... |
| nmsq 25545 | The square of the norm is ... |
| cphnmf 25546 | The norm of a vector is a ... |
| cphnmcl 25547 | The norm of a vector is a ... |
| reipcl 25548 | An inner product of an ele... |
| ipge0 25549 | The inner product in a sub... |
| cphipcj 25550 | Conjugate of an inner prod... |
| cphipipcj 25551 | An inner product times its... |
| cphorthcom 25552 | Orthogonality (meaning inn... |
| cphip0l 25553 | Inner product with a zero ... |
| cphip0r 25554 | Inner product with a zero ... |
| cphipeq0 25555 | The inner product of a vec... |
| cphdir 25556 | Distributive law for inner... |
| cphdi 25557 | Distributive law for inner... |
| cph2di 25558 | Distributive law for inner... |
| cphsubdir 25559 | Distributive law for inner... |
| cphsubdi 25560 | Distributive law for inner... |
| cph2subdi 25561 | Distributive law for inner... |
| cphass 25562 | Associative law for inner ... |
| cphassr 25563 | "Associative" law for seco... |
| cph2ass 25564 | Move scalar multiplication... |
| cphassi 25565 | Associative law for the fi... |
| cphassir 25566 | "Associative" law for the ... |
| cphpyth 25567 | The pythagorean theorem fo... |
| tcphex 25568 | Lemma for ~ tcphbas and si... |
| tcphval 25569 | Define a function to augme... |
| tcphbas 25570 | The base set of a subcompl... |
| tchplusg 25571 | The addition operation of ... |
| tcphsub 25572 | The subtraction operation ... |
| tcphmulr 25573 | The ring operation of a su... |
| tcphsca 25574 | The scalar field of a subc... |
| tcphvsca 25575 | The scalar multiplication ... |
| tcphip 25576 | The inner product of a sub... |
| tcphtopn 25577 | The topology of a subcompl... |
| tcphphl 25578 | Augmentation of a subcompl... |
| tchnmfval 25579 | The norm of a subcomplex p... |
| tcphnmval 25580 | The norm of a subcomplex p... |
| cphtcphnm 25581 | The norm of a norm-augment... |
| tcphds 25582 | The distance of a pre-Hilb... |
| phclm 25583 | A pre-Hilbert space whose ... |
| tcphcphlem3 25584 | Lemma for ~ tcphcph : real... |
| ipcau2 25585 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25586 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25587 | Lemma for ~ tcphcph : homo... |
| tcphcph 25588 | The standard definition of... |
| ipcau 25589 | The Cauchy-Schwarz inequal... |
| nmparlem 25590 | Lemma for ~ nmpar . (Cont... |
| nmpar 25591 | A subcomplex pre-Hilbert s... |
| cphipval2 25592 | Value of the inner product... |
| 4cphipval2 25593 | Four times the inner produ... |
| cphipval 25594 | Value of the inner product... |
| ipcnlem2 25595 | The inner product operatio... |
| ipcnlem1 25596 | The inner product operatio... |
| ipcn 25597 | The inner product operatio... |
| cnmpt1ip 25598 | Continuity of inner produc... |
| cnmpt2ip 25599 | Continuity of inner produc... |
| csscld 25600 | A "closed subspace" in a s... |
| clsocv 25601 | The orthogonal complement ... |
| cphsscph 25602 | A subspace of a subcomplex... |
| lmmbr 25609 | Express the binary relatio... |
| lmmbr2 25610 | Express the binary relatio... |
| lmmbr3 25611 | Express the binary relatio... |
| lmmcvg 25612 | Convergence property of a ... |
| lmmbrf 25613 | Express the binary relatio... |
| lmnn 25614 | A condition that implies c... |
| cfilfval 25615 | The set of Cauchy filters ... |
| iscfil 25616 | The property of being a Ca... |
| iscfil2 25617 | The property of being a Ca... |
| cfilfil 25618 | A Cauchy filter is a filte... |
| cfili 25619 | Property of a Cauchy filte... |
| cfil3i 25620 | A Cauchy filter contains b... |
| cfilss 25621 | A filter finer than a Cauc... |
| fgcfil 25622 | The Cauchy filter conditio... |
| fmcfil 25623 | The Cauchy filter conditio... |
| iscfil3 25624 | A filter is Cauchy iff it ... |
| cfilfcls 25625 | Similar to ultrafilters ( ... |
| caufval 25626 | The set of Cauchy sequence... |
| iscau 25627 | Express the property " ` F... |
| iscau2 25628 | Express the property " ` F... |
| iscau3 25629 | Express the Cauchy sequenc... |
| iscau4 25630 | Express the property " ` F... |
| iscauf 25631 | Express the property " ` F... |
| caun0 25632 | A metric with a Cauchy seq... |
| caufpm 25633 | Inclusion of a Cauchy sequ... |
| caucfil 25634 | A Cauchy sequence predicat... |
| iscmet 25635 | The property " ` D ` is a ... |
| cmetcvg 25636 | The convergence of a Cauch... |
| cmetmet 25637 | A complete metric space is... |
| cmetmeti 25638 | A complete metric space is... |
| cmetcaulem 25639 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25640 | The convergence of a Cauch... |
| iscmet3lem3 25641 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25642 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25643 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25644 | The property " ` D ` is a ... |
| iscmet2 25645 | A metric ` D ` is complete... |
| cfilresi 25646 | A Cauchy filter on a metri... |
| cfilres 25647 | Cauchy filter on a metric ... |
| caussi 25648 | Cauchy sequence on a metri... |
| causs 25649 | Cauchy sequence on a metri... |
| equivcfil 25650 | If the metric ` D ` is "st... |
| equivcau 25651 | If the metric ` D ` is "st... |
| lmle 25652 | If the distance from each ... |
| nglmle 25653 | If the norm of each member... |
| lmclim 25654 | Relate a limit on the metr... |
| lmclimf 25655 | Relate a limit on the metr... |
| metelcls 25656 | A point belongs to the clo... |
| metcld 25657 | A subset of a metric space... |
| metcld2 25658 | A subset of a metric space... |
| caubl 25659 | Sufficient condition to en... |
| caublcls 25660 | The convergent point of a ... |
| metcnp4 25661 | Two ways to say a mapping ... |
| metcn4 25662 | Two ways to say a mapping ... |
| iscmet3i 25663 | Properties that determine ... |
| lmcau 25664 | Every convergent sequence ... |
| flimcfil 25665 | Every convergent filter in... |
| metsscmetcld 25666 | A complete subspace of a m... |
| cmetss 25667 | A subspace of a complete m... |
| equivcmet 25668 | If two metrics are strongl... |
| relcmpcmet 25669 | If ` D ` is a metric space... |
| cmpcmet 25670 | A compact metric space is ... |
| cfilucfil3 25671 | Given a metric ` D ` and a... |
| cfilucfil4 25672 | Given a metric ` D ` and a... |
| cncmet 25673 | The set of complex numbers... |
| recmet 25674 | The real numbers are a com... |
| bcthlem1 25675 | Lemma for ~ bcth . Substi... |
| bcthlem2 25676 | Lemma for ~ bcth . The ba... |
| bcthlem3 25677 | Lemma for ~ bcth . The li... |
| bcthlem4 25678 | Lemma for ~ bcth . Given ... |
| bcthlem5 25679 | Lemma for ~ bcth . The pr... |
| bcth 25680 | Baire's Category Theorem. ... |
| bcth2 25681 | Baire's Category Theorem, ... |
| bcth3 25682 | Baire's Category Theorem, ... |
| isbn 25689 | A Banach space is a normed... |
| bnsca 25690 | The scalar field of a Bana... |
| bnnvc 25691 | A Banach space is a normed... |
| bnnlm 25692 | A Banach space is a normed... |
| bnngp 25693 | A Banach space is a normed... |
| bnlmod 25694 | A Banach space is a left m... |
| bncms 25695 | A Banach space is a comple... |
| iscms 25696 | A complete metric space is... |
| cmscmet 25697 | The induced metric on a co... |
| bncmet 25698 | The induced metric on Bana... |
| cmsms 25699 | A complete metric space is... |
| cmspropd 25700 | Property deduction for a c... |
| cmssmscld 25701 | The restriction of a metri... |
| cmsss 25702 | The restriction of a compl... |
| lssbn 25703 | A subspace of a Banach spa... |
| cmetcusp1 25704 | If the uniform set of a co... |
| cmetcusp 25705 | The uniform space generate... |
| cncms 25706 | The field of complex numbe... |
| cnflduss 25707 | The uniform structure of t... |
| cnfldcusp 25708 | The field of complex numbe... |
| resscdrg 25709 | The real numbers are a sub... |
| cncdrg 25710 | The only complete subfield... |
| srabn 25711 | The subring algebra over a... |
| rlmbn 25712 | The ring module over a com... |
| ishl 25713 | The predicate "is a subcom... |
| hlbn 25714 | Every subcomplex Hilbert s... |
| hlcph 25715 | Every subcomplex Hilbert s... |
| hlphl 25716 | Every subcomplex Hilbert s... |
| hlcms 25717 | Every subcomplex Hilbert s... |
| hlprlem 25718 | Lemma for ~ hlpr . (Contr... |
| hlress 25719 | The scalar field of a subc... |
| hlpr 25720 | The scalar field of a subc... |
| ishl2 25721 | A Hilbert space is a compl... |
| cphssphl 25722 | A Banach subspace of a sub... |
| cmslssbn 25723 | A complete linear subspace... |
| cmscsscms 25724 | A closed subspace of a com... |
| bncssbn 25725 | A closed subspace of a Ban... |
| cssbn 25726 | A complete subspace of a n... |
| csschl 25727 | A complete subspace of a c... |
| cmslsschl 25728 | A complete linear subspace... |
| chlcsschl 25729 | A closed subspace of a sub... |
| retopn 25730 | The topology of the real n... |
| recms 25731 | The real numbers form a co... |
| reust 25732 | The Uniform structure of t... |
| recusp 25733 | The real numbers form a co... |
| rrxval 25738 | Value of the generalized E... |
| rrxbase 25739 | The base of the generalize... |
| rrxprds 25740 | Expand the definition of t... |
| rrxip 25741 | The inner product of the g... |
| rrxnm 25742 | The norm of the generalize... |
| rrxcph 25743 | Generalized Euclidean real... |
| rrxds 25744 | The distance over generali... |
| rrxvsca 25745 | The scalar product over ge... |
| rrxplusgvscavalb 25746 | The result of the addition... |
| rrxsca 25747 | The field of real numbers ... |
| rrx0 25748 | The zero ("origin") in a g... |
| rrx0el 25749 | The zero ("origin") in a g... |
| csbren 25750 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25751 | Triangle inequality in R^n... |
| rrxf 25752 | Euclidean vectors as funct... |
| rrxfsupp 25753 | Euclidean vectors are of f... |
| rrxsuppss 25754 | Support of Euclidean vecto... |
| rrxmvallem 25755 | Support of the function us... |
| rrxmval 25756 | The value of the Euclidean... |
| rrxmfval 25757 | The value of the Euclidean... |
| rrxmetlem 25758 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25759 | Euclidean space is a metri... |
| rrxdstprj1 25760 | The distance between two p... |
| rrxbasefi 25761 | The base of the generalize... |
| rrxdsfi 25762 | The distance over generali... |
| rrxmetfi 25763 | Euclidean space is a metri... |
| rrxdsfival 25764 | The value of the Euclidean... |
| ehlval 25765 | Value of the Euclidean spa... |
| ehlbase 25766 | The base of the Euclidean ... |
| ehl0base 25767 | The base of the Euclidean ... |
| ehl0 25768 | The Euclidean space of dim... |
| ehleudis 25769 | The Euclidean distance fun... |
| ehleudisval 25770 | The value of the Euclidean... |
| ehl1eudis 25771 | The Euclidean distance fun... |
| ehl1eudisval 25772 | The value of the Euclidean... |
| ehl2eudis 25773 | The Euclidean distance fun... |
| ehl2eudisval 25774 | The value of the Euclidean... |
| minveclem1 25775 | Lemma for ~ minvec . The ... |
| minveclem4c 25776 | Lemma for ~ minvec . The ... |
| minveclem2 25777 | Lemma for ~ minvec . Any ... |
| minveclem3a 25778 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25779 | Lemma for ~ minvec . The ... |
| minveclem3 25780 | Lemma for ~ minvec . The ... |
| minveclem4a 25781 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25782 | Lemma for ~ minvec . The ... |
| minveclem4 25783 | Lemma for ~ minvec . The ... |
| minveclem5 25784 | Lemma for ~ minvec . Disc... |
| minveclem6 25785 | Lemma for ~ minvec . Any ... |
| minveclem7 25786 | Lemma for ~ minvec . Sinc... |
| minvec 25787 | Minimizing vector theorem,... |
| pjthlem1 25788 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25789 | Lemma for ~ pjth . (Contr... |
| pjth 25790 | Projection Theorem: Any H... |
| pjth2 25791 | Projection Theorem with ab... |
| cldcss 25792 | Corollary of the Projectio... |
| cldcss2 25793 | Corollary of the Projectio... |
| hlhil 25794 | Corollary of the Projectio... |
| addcncf 25795 | The addition of two contin... |
| subcncf 25796 | The subtraction of two con... |
| mulcncf 25797 | The multiplication of two ... |
| divcncf 25798 | The quotient of two contin... |
| pmltpclem1 25799 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25800 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25801 | Any function on the reals ... |
| ivthlem1 25802 | Lemma for ~ ivth . The se... |
| ivthlem2 25803 | Lemma for ~ ivth . Show t... |
| ivthlem3 25804 | Lemma for ~ ivth , the int... |
| ivth 25805 | The intermediate value the... |
| ivth2 25806 | The intermediate value the... |
| ivthle 25807 | The intermediate value the... |
| ivthle2 25808 | The intermediate value the... |
| ivthicc 25809 | The interval between any t... |
| evthicc 25810 | Specialization of the Extr... |
| evthicc2 25811 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25812 | A continuous function on a... |
| ovolfcl 25817 | Closure for the interval e... |
| ovolfioo 25818 | Unpack the interval coveri... |
| ovolficc 25819 | Unpack the interval coveri... |
| ovolficcss 25820 | Any (closed) interval cove... |
| ovolfsval 25821 | The value of the interval ... |
| ovolfsf 25822 | Closure for the interval l... |
| ovolsf 25823 | Closure for the partial su... |
| ovolval 25824 | The value of the outer mea... |
| elovolmlem 25825 | Lemma for ~ elovolm and re... |
| elovolm 25826 | Elementhood in the set ` M... |
| elovolmr 25827 | Sufficient condition for e... |
| ovolmge0 25828 | The set ` M ` is composed ... |
| ovolcl 25829 | The volume of a set is an ... |
| ovollb 25830 | The outer volume is a lowe... |
| ovolgelb 25831 | The outer volume is the gr... |
| ovolge0 25832 | The volume of a set is alw... |
| ovolf 25833 | The domain and codomain of... |
| ovollecl 25834 | If an outer volume is boun... |
| ovolsslem 25835 | Lemma for ~ ovolss . (Con... |
| ovolss 25836 | The volume of a set is mon... |
| ovolsscl 25837 | If a set is contained in a... |
| ovolssnul 25838 | A subset of a nullset is n... |
| ovollb2lem 25839 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25840 | It is often more convenien... |
| ovolctb 25841 | The volume of a denumerabl... |
| ovolq 25842 | The rational numbers have ... |
| ovolctb2 25843 | The volume of a countable ... |
| ovol0 25844 | The empty set has 0 outer ... |
| ovolfi 25845 | A finite set has 0 outer L... |
| ovolsn 25846 | A singleton has 0 outer Le... |
| ovolunlem1a 25847 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25848 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25849 | Lemma for ~ ovolun . (Con... |
| ovolun 25850 | The Lebesgue outer measure... |
| ovolunnul 25851 | Adding a nullset does not ... |
| ovolfiniun 25852 | The Lebesgue outer measure... |
| ovoliunlem1 25853 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25854 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25855 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25856 | The Lebesgue outer measure... |
| ovoliun2 25857 | The Lebesgue outer measure... |
| ovoliunnul 25858 | A countable union of nulls... |
| shft2rab 25859 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25860 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25861 | Lemma for ~ ovolshft . (C... |
| ovolshft 25862 | The Lebesgue outer measure... |
| sca2rab 25863 | If ` B ` is a scale of ` A... |
| ovolscalem1 25864 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25865 | Lemma for ~ ovolshft . (C... |
| ovolsca 25866 | The Lebesgue outer measure... |
| ovolicc1 25867 | The measure of a closed in... |
| ovolicc2lem1 25868 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25869 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25870 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25871 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25872 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25873 | The measure of a closed in... |
| ovolicc 25874 | The measure of a closed in... |
| ovolicopnf 25875 | The measure of a right-unb... |
| ovolre 25876 | The measure of the real nu... |
| ismbl 25877 | The predicate " ` A ` is L... |
| ismbl2 25878 | From ~ ovolun , it suffice... |
| volres 25879 | A self-referencing abbrevi... |
| volf 25880 | The domain and codomain of... |
| mblvol 25881 | The volume of a measurable... |
| mblss 25882 | A measurable set is a subs... |
| mblsplit 25883 | The defining property of m... |
| volss 25884 | The Lebesgue measure is mo... |
| cmmbl 25885 | The complement of a measur... |
| nulmbl 25886 | A nullset is measurable. ... |
| nulmbl2 25887 | A set of outer measure zer... |
| unmbl 25888 | A union of measurable sets... |
| shftmbl 25889 | A shift of a measurable se... |
| 0mbl 25890 | The empty set is measurabl... |
| rembl 25891 | The set of all real number... |
| unidmvol 25892 | The union of the Lebesgue ... |
| inmbl 25893 | An intersection of measura... |
| difmbl 25894 | A difference of measurable... |
| finiunmbl 25895 | A finite union of measurab... |
| volun 25896 | The Lebesgue measure funct... |
| volinun 25897 | Addition of non-disjoint s... |
| volfiniun 25898 | The volume of a disjoint f... |
| iundisj 25899 | Rewrite a countable union ... |
| iundisj2 25900 | A disjoint union is disjoi... |
| voliunlem1 25901 | Lemma for ~ voliun . (Con... |
| voliunlem2 25902 | Lemma for ~ voliun . (Con... |
| voliunlem3 25903 | Lemma for ~ voliun . (Con... |
| iunmbl 25904 | The measurable sets are cl... |
| voliun 25905 | The Lebesgue measure funct... |
| volsuplem 25906 | Lemma for ~ volsup . (Con... |
| volsup 25907 | The volume of the limit of... |
| iunmbl2 25908 | The measurable sets are cl... |
| ioombl1lem1 25909 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25910 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25911 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25912 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25913 | An open right-unbounded in... |
| icombl1 25914 | A closed unbounded-above i... |
| icombl 25915 | A closed-below, open-above... |
| ioombl 25916 | An open real interval is m... |
| iccmbl 25917 | A closed real interval is ... |
| iccvolcl 25918 | A closed real interval has... |
| ovolioo 25919 | The measure of an open int... |
| volioo 25920 | The measure of an open int... |
| ioovolcl 25921 | An open real interval has ... |
| ovolfs2 25922 | Alternative expression for... |
| ioorcl2 25923 | An open interval with fini... |
| ioorf 25924 | Define a function from ope... |
| ioorval 25925 | Define a function from ope... |
| ioorinv2 25926 | The function ` F ` is an "... |
| ioorinv 25927 | The function ` F ` is an "... |
| ioorcl 25928 | The function ` F ` does no... |
| uniiccdif 25929 | A union of closed interval... |
| uniioovol 25930 | A disjoint union of open i... |
| uniiccvol 25931 | An almost-disjoint union o... |
| uniioombllem1 25932 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25933 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25934 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25935 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25936 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25937 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25938 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25939 | Lemma for ~ uniioombl . (... |
| uniioombl 25940 | A disjoint union of open i... |
| uniiccmbl 25941 | An almost-disjoint union o... |
| dyadf 25942 | The function ` F ` returns... |
| dyadval 25943 | Value of the dyadic ration... |
| dyadovol 25944 | Volume of a dyadic rationa... |
| dyadss 25945 | Two closed dyadic rational... |
| dyaddisjlem 25946 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25947 | Two closed dyadic rational... |
| dyadmaxlem 25948 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25949 | Any nonempty set of dyadic... |
| dyadmbllem 25950 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25951 | Any union of dyadic ration... |
| opnmbllem 25952 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25953 | All open sets are measurab... |
| opnmblALT 25954 | All open sets are measurab... |
| subopnmbl 25955 | Sets which are open in a m... |
| volsup2 25956 | The volume of ` A ` is the... |
| volcn 25957 | The function formed by res... |
| volivth 25958 | The Intermediate Value The... |
| vitalilem1 25959 | Lemma for ~ vitali . (Con... |
| vitalilem2 25960 | Lemma for ~ vitali . (Con... |
| vitalilem3 25961 | Lemma for ~ vitali . (Con... |
| vitalilem4 25962 | Lemma for ~ vitali . (Con... |
| vitalilem5 25963 | Lemma for ~ vitali . (Con... |
| vitali 25964 | If the reals can be well-o... |
| ismbf1 25975 | The predicate " ` F ` is a... |
| mbff 25976 | A measurable function is a... |
| mbfdm 25977 | The domain of a measurable... |
| mbfconstlem 25978 | Lemma for ~ mbfconst and r... |
| ismbf 25979 | The predicate " ` F ` is a... |
| ismbfcn 25980 | A complex function is meas... |
| mbfima 25981 | Definitional property of a... |
| mbfimaicc 25982 | The preimage of any closed... |
| mbfimasn 25983 | The preimage of a point un... |
| mbfconst 25984 | A constant function is mea... |
| mbf0 25985 | The empty function is meas... |
| mbfid 25986 | The identity function is m... |
| mbfmptcl 25987 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25988 | The domain of a measurable... |
| ismbfcn2 25989 | A complex function is meas... |
| ismbfd 25990 | Deduction to prove measura... |
| ismbf2d 25991 | Deduction to prove measura... |
| mbfeqalem1 25992 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25993 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25994 | If two functions are equal... |
| mbfres 25995 | The restriction of a measu... |
| mbfres2 25996 | Measurability of a piecewi... |
| mbfss 25997 | Change the domain of a mea... |
| mbfmulc2lem 25998 | Multiplication by a consta... |
| mbfmulc2re 25999 | Multiplication by a consta... |
| mbfmax 26000 | The maximum of two functio... |
| mbfneg 26001 | The negative of a measurab... |
| mbfpos 26002 | The positive part of a mea... |
| mbfposr 26003 | Converse to ~ mbfpos . (C... |
| mbfposb 26004 | A function is measurable i... |
| ismbf3d 26005 | Simplified form of ~ ismbf... |
| mbfimaopnlem 26006 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 26007 | The preimage of any open s... |
| mbfimaopn2 26008 | The preimage of any set op... |
| cncombf 26009 | The composition of a conti... |
| cnmbf 26010 | A continuous function is m... |
| mbfaddlem 26011 | The sum of two measurable ... |
| mbfadd 26012 | The sum of two measurable ... |
| mbfsub 26013 | The difference of two meas... |
| mbfmulc2 26014 | A complex constant times a... |
| mbfsup 26015 | The supremum of a sequence... |
| mbfinf 26016 | The infimum of a sequence ... |
| mbflimsup 26017 | The limit supremum of a se... |
| mbflimlem 26018 | The pointwise limit of a s... |
| mbflim 26019 | The pointwise limit of a s... |
| 0pval 26022 | The zero function evaluate... |
| 0plef 26023 | Two ways to say that the f... |
| 0pledm 26024 | Adjust the domain of the l... |
| isi1f 26025 | The predicate " ` F ` is a... |
| i1fmbf 26026 | Simple functions are measu... |
| i1ff 26027 | A simple function is a fun... |
| i1frn 26028 | A simple function has fini... |
| i1fima 26029 | Any preimage of a simple f... |
| i1fima2 26030 | Any preimage of a simple f... |
| i1fima2sn 26031 | Preimage of a singleton. ... |
| i1fd 26032 | A simplified set of assump... |
| i1f0rn 26033 | Any simple function takes ... |
| itg1val 26034 | The value of the integral ... |
| itg1val2 26035 | The value of the integral ... |
| itg1cl 26036 | Closure of the integral on... |
| itg1ge0 26037 | Closure of the integral on... |
| i1f0 26038 | The zero function is simpl... |
| itg10 26039 | The zero function has zero... |
| i1f1lem 26040 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 26041 | Base case simple functions... |
| itg11 26042 | The integral of an indicat... |
| itg1addlem1 26043 | Decompose a preimage, whic... |
| i1faddlem 26044 | Decompose the preimage of ... |
| i1fmullem 26045 | Decompose the preimage of ... |
| i1fadd 26046 | The sum of two simple func... |
| i1fmul 26047 | The pointwise product of t... |
| itg1addlem2 26048 | Lemma for ~ itg1add . The... |
| itg1addlem3 26049 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 26050 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 26051 | Lemma for ~ itg1add . (Co... |
| itg1add 26052 | The integral of a sum of s... |
| i1fmulclem 26053 | Decompose the preimage of ... |
| i1fmulc 26054 | A nonnegative constant tim... |
| itg1mulc 26055 | The integral of a constant... |
| i1fres 26056 | The "restriction" of a sim... |
| i1fpos 26057 | The positive part of a sim... |
| i1fposd 26058 | Deduction form of ~ i1fpos... |
| i1fsub 26059 | The difference of two simp... |
| itg1sub 26060 | The integral of a differen... |
| itg10a 26061 | The integral of a simple f... |
| itg1ge0a 26062 | The integral of an almost ... |
| itg1lea 26063 | Approximate version of ~ i... |
| itg1le 26064 | If one simple function dom... |
| itg1climres 26065 | Restricting the simple fun... |
| mbfi1fseqlem1 26066 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 26067 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 26068 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 26069 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 26070 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 26071 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 26072 | A characterization of meas... |
| mbfi1flimlem 26073 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 26074 | Any real measurable functi... |
| mbfmullem2 26075 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 26076 | Lemma for ~ mbfmul . (Con... |
| mbfmul 26077 | The product of two measura... |
| itg2lcl 26078 | The set of lower sums is a... |
| itg2val 26079 | Value of the integral on n... |
| itg2l 26080 | Elementhood in the set ` L... |
| itg2lr 26081 | Sufficient condition for e... |
| xrge0f 26082 | A real function is a nonne... |
| itg2cl 26083 | The integral of a nonnegat... |
| itg2ub 26084 | The integral of a nonnegat... |
| itg2leub 26085 | Any upper bound on the int... |
| itg2ge0 26086 | The integral of a nonnegat... |
| itg2itg1 26087 | The integral of a nonnegat... |
| itg20 26088 | The integral of the zero f... |
| itg2lecl 26089 | If an ` S.2 ` integral is ... |
| itg2le 26090 | If one function dominates ... |
| itg2const 26091 | Integral of a constant fun... |
| itg2const2 26092 | When the base set of a con... |
| itg2seq 26093 | Definitional property of t... |
| itg2uba 26094 | Approximate version of ~ i... |
| itg2lea 26095 | Approximate version of ~ i... |
| itg2eqa 26096 | Approximate equality of in... |
| itg2mulclem 26097 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 26098 | The integral of a nonnegat... |
| itg2splitlem 26099 | Lemma for ~ itg2split . (... |
| itg2split 26100 | The ` S.2 ` integral split... |
| itg2monolem1 26101 | Lemma for ~ itg2mono . We... |
| itg2monolem2 26102 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 26103 | Lemma for ~ itg2mono . (C... |
| itg2mono 26104 | The Monotone Convergence T... |
| itg2i1fseqle 26105 | Subject to the conditions ... |
| itg2i1fseq 26106 | Subject to the conditions ... |
| itg2i1fseq2 26107 | In an extension to the res... |
| itg2i1fseq3 26108 | Special case of ~ itg2i1fs... |
| itg2addlem 26109 | Lemma for ~ itg2add . (Co... |
| itg2add 26110 | The ` S.2 ` integral is li... |
| itg2gt0 26111 | If the function ` F ` is s... |
| itg2cnlem1 26112 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 26113 | Lemma for ~ itgcn . (Cont... |
| itg2cn 26114 | A sort of absolute continu... |
| ibllem 26115 | Conditioned equality theor... |
| isibl 26116 | The predicate " ` F ` is i... |
| isibl2 26117 | The predicate " ` F ` is i... |
| iblmbf 26118 | An integrable function is ... |
| iblitg 26119 | If a function is integrabl... |
| dfitg 26120 | Evaluate the class substit... |
| itgex 26121 | An integral is a set. (Co... |
| itgeq1f 26122 | Equality theorem for an in... |
| itgeq1 26123 | Equality theorem for an in... |
| nfitg1 26124 | Bound-variable hypothesis ... |
| nfitg 26125 | Bound-variable hypothesis ... |
| cbvitg 26126 | Change bound variable in a... |
| cbvitgv 26127 | Change bound variable in a... |
| itgeq2 26128 | Equality theorem for an in... |
| itgresr 26129 | The domain of an integral ... |
| itg0 26130 | The integral of anything o... |
| itgz 26131 | The integral of zero on an... |
| itgeq2dv 26132 | Equality theorem for an in... |
| itgmpt 26133 | Change bound variable in a... |
| itgcl 26134 | The integral of an integra... |
| itgvallem 26135 | Substitution lemma. (Cont... |
| itgvallem3 26136 | Lemma for ~ itgposval and ... |
| ibl0 26137 | The zero function is integ... |
| iblcnlem1 26138 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 26139 | Expand out the universal q... |
| itgcnlem 26140 | Expand out the sum in ~ df... |
| iblrelem 26141 | Integrability of a real fu... |
| iblposlem 26142 | Lemma for ~ iblpos . (Con... |
| iblpos 26143 | Integrability of a nonnega... |
| iblre 26144 | Integrability of a real fu... |
| itgrevallem1 26145 | Lemma for ~ itgposval and ... |
| itgposval 26146 | The integral of a nonnegat... |
| itgreval 26147 | Decompose the integral of ... |
| itgrecl 26148 | Real closure of an integra... |
| iblcn 26149 | Integrability of a complex... |
| itgcnval 26150 | Decompose the integral of ... |
| itgre 26151 | Real part of an integral. ... |
| itgim 26152 | Imaginary part of an integ... |
| iblneg 26153 | The negative of an integra... |
| itgneg 26154 | Negation of an integral. ... |
| iblss 26155 | A subset of an integrable ... |
| iblss2 26156 | Change the domain of an in... |
| itgitg2 26157 | Transfer an integral using... |
| i1fibl 26158 | A simple function is integ... |
| itgitg1 26159 | Transfer an integral using... |
| itgle 26160 | Monotonicity of an integra... |
| itgge0 26161 | The integral of a positive... |
| itgss 26162 | Expand the set of an integ... |
| itgss2 26163 | Expand the set of an integ... |
| itgeqa 26164 | Approximate equality of in... |
| itgss3 26165 | Expand the set of an integ... |
| itgioo 26166 | Equality of integrals on o... |
| itgless 26167 | Expand the integral of a n... |
| iblconst 26168 | A constant function is int... |
| itgconst 26169 | Integral of a constant fun... |
| ibladdlem 26170 | Lemma for ~ ibladd . (Con... |
| ibladd 26171 | Add two integrals over the... |
| iblsub 26172 | Subtract two integrals ove... |
| itgaddlem1 26173 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 26174 | Lemma for ~ itgadd . (Con... |
| itgadd 26175 | Add two integrals over the... |
| itgsub 26176 | Subtract two integrals ove... |
| itgfsum 26177 | Take a finite sum of integ... |
| iblabslem 26178 | Lemma for ~ iblabs . (Con... |
| iblabs 26179 | The absolute value of an i... |
| iblabsr 26180 | A measurable function is i... |
| iblmulc2 26181 | Multiply an integral by a ... |
| itgmulc2lem1 26182 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 26183 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 26184 | Multiply an integral by a ... |
| itgabs 26185 | The triangle inequality fo... |
| itgsplit 26186 | The ` S. ` integral splits... |
| itgspliticc 26187 | The ` S. ` integral splits... |
| itgsplitioo 26188 | The ` S. ` integral splits... |
| bddmulibl 26189 | A bounded function times a... |
| bddibl 26190 | A bounded function is inte... |
| cniccibl 26191 | A continuous function on a... |
| bddiblnc 26192 | Choice-free proof of ~ bdd... |
| cnicciblnc 26193 | Choice-free proof of ~ cni... |
| itggt0 26194 | The integral of a strictly... |
| itgcn 26195 | Transfer ~ itg2cn to the f... |
| ditgeq1 26198 | Equality theorem for the d... |
| ditgeq2 26199 | Equality theorem for the d... |
| ditgeq3 26200 | Equality theorem for the d... |
| ditgeq3dv 26201 | Equality theorem for the d... |
| ditgex 26202 | A directed integral is a s... |
| ditg0 26203 | Value of the directed inte... |
| cbvditg 26204 | Change bound variable in a... |
| cbvditgv 26205 | Change bound variable in a... |
| ditgpos 26206 | Value of the directed inte... |
| ditgneg 26207 | Value of the directed inte... |
| ditgcl 26208 | Closure of a directed inte... |
| ditgswap 26209 | Reverse a directed integra... |
| ditgsplitlem 26210 | Lemma for ~ ditgsplit . (... |
| ditgsplit 26211 | This theorem is the raison... |
| reldv 26220 | The derivative function is... |
| limcvallem 26221 | Lemma for ~ ellimc . (Con... |
| limcfval 26222 | Value and set bounds on th... |
| ellimc 26223 | Value of the limit predica... |
| limcrcl 26224 | Reverse closure for the li... |
| limccl 26225 | Closure of the limit opera... |
| limcdif 26226 | It suffices to consider fu... |
| ellimc2 26227 | Write the definition of a ... |
| limcnlp 26228 | If ` B ` is not a limit po... |
| ellimc3 26229 | Write the epsilon-delta de... |
| limcflflem 26230 | Lemma for ~ limcflf . (Co... |
| limcflf 26231 | The limit operator can be ... |
| limcmo 26232 | If ` B ` is a limit point ... |
| limcmpt 26233 | Express the limit operator... |
| limcmpt2 26234 | Express the limit operator... |
| limcresi 26235 | Any limit of ` F ` is also... |
| limcres 26236 | If ` B ` is an interior po... |
| cnplimc 26237 | A function is continuous a... |
| cnlimc 26238 | ` F ` is a continuous func... |
| cnlimci 26239 | If ` F ` is a continuous f... |
| cnmptlimc 26240 | If ` F ` is a continuous f... |
| limccnp 26241 | If the limit of ` F ` at `... |
| limccnp2 26242 | The image of a convergent ... |
| limcco 26243 | Composition of two limits.... |
| limciun 26244 | A point is a limit of ` F ... |
| limcun 26245 | A point is a limit of ` F ... |
| dvlem 26246 | Closure for a difference q... |
| dvfval 26247 | Value and set bounds on th... |
| eldv 26248 | The differentiable predica... |
| dvcl 26249 | The derivative function ta... |
| dvbssntr 26250 | The set of differentiable ... |
| dvbss 26251 | The set of differentiable ... |
| dvbsss 26252 | The set of differentiable ... |
| perfdvf 26253 | The derivative is a functi... |
| recnprss 26254 | Both ` RR ` and ` CC ` are... |
| recnperf 26255 | Both ` RR ` and ` CC ` are... |
| dvfg 26256 | Explicitly write out the f... |
| dvf 26257 | The derivative is a functi... |
| dvfcn 26258 | The derivative is a functi... |
| dvreslem 26259 | Lemma for ~ dvres . (Cont... |
| dvres2lem 26260 | Lemma for ~ dvres2 . (Con... |
| dvres 26261 | Restriction of a derivativ... |
| dvres2 26262 | Restriction of the base se... |
| dvres3 26263 | Restriction of a complex d... |
| dvres3a 26264 | Restriction of a complex d... |
| dvidlem 26265 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 26266 | Derivative of a function r... |
| dvconst 26267 | Derivative of a constant f... |
| dvid 26268 | Derivative of the identity... |
| dvcnp 26269 | The difference quotient is... |
| dvcnp2 26270 | A function is continuous a... |
| dvcn 26271 | A differentiable function ... |
| dvnfval 26272 | Value of the iterated deri... |
| dvnff 26273 | The iterated derivative is... |
| dvn0 26274 | Zero times iterated deriva... |
| dvnp1 26275 | Successor iterated derivat... |
| dvn1 26276 | One times iterated derivat... |
| dvnf 26277 | The N-times derivative is ... |
| dvnbss 26278 | The set of N-times differe... |
| dvnadd 26279 | The ` N ` -th derivative o... |
| dvn2bss 26280 | An N-times differentiable ... |
| dvnres 26281 | Multiple derivative versio... |
| cpnfval 26282 | Condition for n-times cont... |
| fncpn 26283 | The ` C^n ` object is a fu... |
| elcpn 26284 | Condition for n-times cont... |
| cpnord 26285 | ` C^n ` conditions are ord... |
| cpncn 26286 | A ` C^n ` function is cont... |
| cpnres 26287 | The restriction of a ` C^n... |
| dvaddbr 26288 | The sum rule for derivativ... |
| dvmulbr 26289 | The product rule for deriv... |
| dvadd 26290 | The sum rule for derivativ... |
| dvmul 26291 | The product rule for deriv... |
| dvaddf 26292 | The sum rule for everywher... |
| dvmulf 26293 | The product rule for every... |
| dvcmul 26294 | The product rule when one ... |
| dvcmulf 26295 | The product rule when one ... |
| dvcobr 26296 | The chain rule for derivat... |
| dvco 26297 | The chain rule for derivat... |
| dvcof 26298 | The chain rule for everywh... |
| dvcjbr 26299 | The derivative of the conj... |
| dvcj 26300 | The derivative of the conj... |
| dvfre 26301 | The derivative of a real f... |
| dvnfre 26302 | The ` N ` -th derivative o... |
| dvexp 26303 | Derivative of a power func... |
| dvexp2 26304 | Derivative of an exponenti... |
| dvrec 26305 | Derivative of the reciproc... |
| dvmptres3 26306 | Function-builder for deriv... |
| dvmptid 26307 | Function-builder for deriv... |
| dvmptc 26308 | Function-builder for deriv... |
| dvmptcl 26309 | Closure lemma for ~ dvmptc... |
| dvmptadd 26310 | Function-builder for deriv... |
| dvmptmul 26311 | Function-builder for deriv... |
| dvmptres2 26312 | Function-builder for deriv... |
| dvmptres 26313 | Function-builder for deriv... |
| dvmptcmul 26314 | Function-builder for deriv... |
| dvmptdivc 26315 | Function-builder for deriv... |
| dvmptneg 26316 | Function-builder for deriv... |
| dvmptsub 26317 | Function-builder for deriv... |
| dvmptcj 26318 | Function-builder for deriv... |
| dvmptre 26319 | Function-builder for deriv... |
| dvmptim 26320 | Function-builder for deriv... |
| dvmptntr 26321 | Function-builder for deriv... |
| dvmptco 26322 | Function-builder for deriv... |
| dvrecg 26323 | Derivative of the reciproc... |
| dvmptdiv 26324 | Function-builder for deriv... |
| dvmptfsum 26325 | Function-builder for deriv... |
| dvcnvlem 26326 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 26327 | A weak version of ~ dvcnvr... |
| dvexp3 26328 | Derivative of an exponenti... |
| dveflem 26329 | Derivative of the exponent... |
| dvef 26330 | Derivative of the exponent... |
| dvsincos 26331 | Derivative of the sine and... |
| dvsin 26332 | Derivative of the sine fun... |
| dvcos 26333 | Derivative of the cosine f... |
| dvferm1lem 26334 | Lemma for ~ dvferm . (Con... |
| dvferm1 26335 | One-sided version of ~ dvf... |
| dvferm2lem 26336 | Lemma for ~ dvferm . (Con... |
| dvferm2 26337 | One-sided version of ~ dvf... |
| dvferm 26338 | Fermat's theorem on statio... |
| rollelem 26339 | Lemma for ~ rolle . (Cont... |
| rolle 26340 | Rolle's theorem. If ` F `... |
| cmvth 26341 | Cauchy's Mean Value Theore... |
| mvth 26342 | The Mean Value Theorem. I... |
| dvlip 26343 | A function with derivative... |
| dvlipcn 26344 | A complex function with de... |
| dvlip2 26345 | Combine the results of ~ d... |
| c1liplem1 26346 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 26347 | C^1 functions are Lipschit... |
| c1lip2 26348 | C^1 functions are Lipschit... |
| c1lip3 26349 | C^1 functions are Lipschit... |
| dveq0 26350 | If a continuous function h... |
| dv11cn 26351 | Two functions defined on a... |
| dvgt0lem1 26352 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 26353 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 26354 | A function on a closed int... |
| dvlt0 26355 | A function on a closed int... |
| dvge0 26356 | A function on a closed int... |
| dvle 26357 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 26358 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 26359 | Lemma for ~ dvivth . (Con... |
| dvivth 26360 | Darboux' theorem, or the i... |
| dvne0 26361 | A function on a closed int... |
| dvne0f1 26362 | A function on a closed int... |
| lhop1lem 26363 | Lemma for ~ lhop1 . (Cont... |
| lhop1 26364 | L'Hôpital's Rule for... |
| lhop2 26365 | L'Hôpital's Rule for... |
| lhop 26366 | L'Hôpital's Rule. I... |
| dvcnvrelem1 26367 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 26368 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 26369 | The derivative rule for in... |
| dvcvx 26370 | A real function with stric... |
| dvfsumle 26371 | Compare a finite sum to an... |
| dvfsumge 26372 | Compare a finite sum to an... |
| dvfsumabs 26373 | Compare a finite sum to an... |
| dvmptrecl 26374 | Real closure of a derivati... |
| dvfsumrlimf 26375 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 26376 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 26377 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 26378 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 26379 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 26380 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 26381 | Compare a finite sum to an... |
| dvfsumrlim2 26382 | Compare a finite sum to an... |
| dvfsumrlim3 26383 | Conjoin the statements of ... |
| dvfsum2 26384 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26385 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26386 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26387 | The Fundamental Theorem of... |
| ftc1lem3 26388 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26389 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26390 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26391 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26392 | The Fundamental Theorem of... |
| ftc1cn 26393 | Strengthen the assumptions... |
| ftc2 26394 | The Fundamental Theorem of... |
| ftc2ditglem 26395 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26396 | Directed integral analogue... |
| itgparts 26397 | Integration by parts. If ... |
| itgsubstlem 26398 | Lemma for ~ itgsubst . (C... |
| itgsubst 26399 | Integration by ` u ` -subs... |
| itgpowd 26400 | The integral of a monomial... |
| reldmmdeg 26405 | Multivariate degree is a b... |
| tdeglem1 26406 | Functionality of the total... |
| tdeglem3 26407 | Additivity of the total de... |
| tdeglem4 26408 | There is only one multi-in... |
| tdeglem2 26409 | Simplification of total de... |
| mdegfval 26410 | Value of the multivariate ... |
| mdegval 26411 | Value of the multivariate ... |
| mdegleb 26412 | Property of being of limit... |
| mdeglt 26413 | If there is an upper limit... |
| mdegldg 26414 | A nonzero polynomial has s... |
| mdegxrcl 26415 | Closure of polynomial degr... |
| mdegxrf 26416 | Functionality of polynomia... |
| mdegcl 26417 | Sharp closure for multivar... |
| mdeg0 26418 | Degree of the zero polynom... |
| mdegnn0cl 26419 | Degree of a nonzero polyno... |
| degltlem1 26420 | Theorem on arithmetic of e... |
| degltp1le 26421 | Theorem on arithmetic of e... |
| mdegaddle 26422 | The degree of a sum is at ... |
| mdegvscale 26423 | The degree of a scalar mul... |
| mdegvsca 26424 | The degree of a scalar mul... |
| mdegle0 26425 | A polynomial has nonpositi... |
| mdegmullem 26426 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26427 | The multivariate degree of... |
| deg1fval 26428 | Relate univariate polynomi... |
| deg1xrf 26429 | Functionality of univariat... |
| deg1xrcl 26430 | Closure of univariate poly... |
| deg1cl 26431 | Sharp closure of univariat... |
| mdegpropd 26432 | Property deduction for pol... |
| deg1fvi 26433 | Univariate polynomial degr... |
| deg1propd 26434 | Property deduction for pol... |
| deg1z 26435 | Degree of the zero univari... |
| deg1nn0cl 26436 | Degree of a nonzero univar... |
| deg1n0ima 26437 | Degree image of a set of p... |
| deg1nn0clb 26438 | A polynomial is nonzero if... |
| deg1lt0 26439 | A polynomial is zero iff i... |
| deg1ldg 26440 | A nonzero univariate polyn... |
| deg1ldgn 26441 | An index at which a polyno... |
| deg1ldgdomn 26442 | A nonzero univariate polyn... |
| deg1leb 26443 | Property of being of limit... |
| deg1val 26444 | Value of the univariate de... |
| deg1lt 26445 | If the degree of a univari... |
| deg1ge 26446 | Conversely, a nonzero coef... |
| coe1mul3 26447 | The coefficient vector of ... |
| coe1mul4 26448 | Value of the "leading" coe... |
| deg1addle 26449 | The degree of a sum is at ... |
| deg1addle2 26450 | If both factors have degre... |
| deg1add 26451 | Exact degree of a sum of t... |
| deg1vscale 26452 | The degree of a scalar tim... |
| deg1vsca 26453 | The degree of a scalar tim... |
| deg1invg 26454 | The degree of the negated ... |
| deg1suble 26455 | The degree of a difference... |
| deg1sub 26456 | Exact degree of a differen... |
| deg1mulle2 26457 | Produce a bound on the pro... |
| deg1sublt 26458 | Subtraction of two polynom... |
| deg1le0 26459 | A polynomial has nonpositi... |
| deg1sclle 26460 | A scalar polynomial has no... |
| deg1scl 26461 | A nonzero scalar polynomia... |
| deg1mul2 26462 | Degree of multiplication o... |
| deg1mul 26463 | Degree of multiplication o... |
| deg1mul3 26464 | Degree of multiplication o... |
| deg1mul3le 26465 | Degree of multiplication o... |
| deg1tmle 26466 | Limiting degree of a polyn... |
| deg1tm 26467 | Exact degree of a polynomi... |
| deg1pwle 26468 | Limiting degree of a varia... |
| deg1pw 26469 | Exact degree of a variable... |
| ply1nz 26470 | Univariate polynomials ove... |
| ply1nzb 26471 | Univariate polynomials are... |
| ply1domn 26472 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26473 | The ring of univariate pol... |
| ply1divmo 26484 | Uniqueness of a quotient i... |
| ply1divex 26485 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26486 | The division algorithm for... |
| ply1divalg2 26487 | Reverse the order of multi... |
| uc1pval 26488 | Value of the set of unitic... |
| isuc1p 26489 | Being a unitic polynomial.... |
| mon1pval 26490 | Value of the set of monic ... |
| ismon1p 26491 | Being a monic polynomial. ... |
| uc1pcl 26492 | Unitic polynomials are pol... |
| mon1pcl 26493 | Monic polynomials are poly... |
| uc1pn0 26494 | Unitic polynomials are not... |
| mon1pn0 26495 | Monic polynomials are not ... |
| uc1pdeg 26496 | Unitic polynomials have no... |
| uc1pldg 26497 | Unitic polynomials have un... |
| mon1pldg 26498 | Unitic polynomials have on... |
| mon1puc1p 26499 | Monic polynomials are unit... |
| uc1pmon1p 26500 | Make a unitic polynomial m... |
| deg1submon1p 26501 | The difference of two moni... |
| mon1pid 26502 | Monicity and degree of the... |
| q1pval 26503 | Value of the univariate po... |
| q1peqb 26504 | Characterizing property of... |
| q1pcl 26505 | Closure of the quotient by... |
| r1pval 26506 | Value of the polynomial re... |
| r1pcl 26507 | Closure of remainder follo... |
| r1pdeglt 26508 | The remainder has a degree... |
| r1pid 26509 | Express the original polyn... |
| r1pid2 26510 | Identity law for polynomia... |
| dvdsq1p 26511 | Divisibility in a polynomi... |
| dvdsr1p 26512 | Divisibility in a polynomi... |
| ply1remlem 26513 | A term of the form ` x - N... |
| ply1rem 26514 | The polynomial remainder t... |
| facth1 26515 | The factor theorem and its... |
| fta1glem1 26516 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26517 | Lemma for ~ fta1g . (Cont... |
| fta1g 26518 | The one-sided fundamental ... |
| fta1blem 26519 | Lemma for ~ fta1b . (Cont... |
| fta1b 26520 | The assumption that ` R ` ... |
| idomrootle 26521 | No element of an integral ... |
| drnguc1p 26522 | Over a division ring, all ... |
| ig1peu 26523 | There is a unique monic po... |
| ig1pval 26524 | Substitutions for the poly... |
| ig1pval2 26525 | Generator of the zero idea... |
| ig1pval3 26526 | Characterizing properties ... |
| ig1pcl 26527 | The monic generator of an ... |
| ig1pdvds 26528 | The monic generator of an ... |
| ig1prsp 26529 | Any ideal of polynomials o... |
| ply1lpir 26530 | The ring of polynomials ov... |
| ply1pid 26531 | The polynomials over a fie... |
| plyco0 26540 | Two ways to say that a fun... |
| plyval 26541 | Value of the polynomial se... |
| plybss 26542 | Reverse closure of the par... |
| elply 26543 | Definition of a polynomial... |
| elply2 26544 | The coefficient function c... |
| plyun0 26545 | The set of polynomials is ... |
| plyf 26546 | A polynomial is a function... |
| plyss 26547 | The polynomial set functio... |
| plyssc 26548 | Every polynomial ring is c... |
| elplyr 26549 | Sufficient condition for e... |
| elplyd 26550 | Sufficient condition for e... |
| ply1termlem 26551 | Lemma for ~ ply1term . (C... |
| ply1term 26552 | A one-term polynomial. (C... |
| plypow 26553 | A power is a polynomial. ... |
| plyconst 26554 | A constant function is a p... |
| ne0p 26555 | A test to show that a poly... |
| ply0 26556 | The zero function is a pol... |
| plyid 26557 | The identity function is a... |
| idpfv 26558 | Value of the identity poly... |
| plyeq0lem 26559 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26560 | If a polynomial is zero at... |
| plypf1 26561 | Write the set of complex p... |
| plyaddlem1 26562 | Derive the coefficient fun... |
| plymullem1 26563 | Derive the coefficient fun... |
| plyaddlem 26564 | Lemma for ~ plyadd . (Con... |
| plymullem 26565 | Lemma for ~ plymul . (Con... |
| plyadd 26566 | The sum of two polynomials... |
| plymul 26567 | The product of two polynom... |
| plysub 26568 | The difference of two poly... |
| plyaddcl 26569 | The sum of two polynomials... |
| plymulcl 26570 | The product of two polynom... |
| plysubcl 26571 | The difference of two poly... |
| coeval 26572 | Value of the coefficient f... |
| coeeulem 26573 | Lemma for ~ coeeu . (Cont... |
| coeeu 26574 | Uniqueness of the coeffici... |
| coelem 26575 | Lemma for properties of th... |
| coeeq 26576 | If ` A ` satisfies the pro... |
| dgrval 26577 | Value of the degree functi... |
| dgrlem 26578 | Lemma for ~ dgrcl and simi... |
| coef 26579 | The domain and codomain of... |
| coef2 26580 | The domain and codomain of... |
| coef3 26581 | The domain and codomain of... |
| dgrcl 26582 | The degree of any polynomi... |
| dgrub 26583 | If the ` M ` -th coefficie... |
| dgrub2 26584 | All the coefficients above... |
| dgrlb 26585 | If all the coefficients ab... |
| coeidlem 26586 | Lemma for ~ coeid . (Cont... |
| coeid 26587 | Reconstruct a polynomial a... |
| coeid2 26588 | Reconstruct a polynomial a... |
| coeid3 26589 | Reconstruct a polynomial a... |
| plyco 26590 | The composition of two pol... |
| coeeq2 26591 | Compute the coefficient fu... |
| dgrle 26592 | Given an explicit expressi... |
| dgreq 26593 | If the highest term in a p... |
| 0dgr 26594 | A constant function has de... |
| 0dgrb 26595 | A function has degree zero... |
| dgrnznn 26596 | A nonzero polynomial with ... |
| coefv0 26597 | The result of evaluating a... |
| coeaddlem 26598 | Lemma for ~ coeadd and ~ d... |
| coemullem 26599 | Lemma for ~ coemul and ~ d... |
| coeadd 26600 | The coefficient function o... |
| coemul 26601 | A coefficient of a product... |
| coe11 26602 | The coefficient function i... |
| coemulhi 26603 | The leading coefficient of... |
| coemulc 26604 | The coefficient function i... |
| coe0 26605 | The coefficients of the ze... |
| coesub 26606 | The coefficient function o... |
| coe1termlem 26607 | The coefficient function o... |
| coe1term 26608 | The coefficient function o... |
| dgr1term 26609 | The degree of a monomial. ... |
| plycn 26610 | A polynomial is a continuo... |
| dgr0 26611 | The degree of the zero pol... |
| coeidp 26612 | The coefficients of the id... |
| dgrid 26613 | The degree of the identity... |
| dgreq0 26614 | The leading coefficient of... |
| dgrlt 26615 | Two ways to say that the d... |
| dgradd 26616 | The degree of a sum of pol... |
| dgradd2 26617 | The degree of a sum of pol... |
| dgrmul2 26618 | The degree of a product of... |
| dgrmul 26619 | The degree of a product of... |
| dgrmulc 26620 | Scalar multiplication by a... |
| dgrsub 26621 | The degree of a difference... |
| dgrcolem1 26622 | The degree of a compositio... |
| dgrcolem2 26623 | Lemma for ~ dgrco . (Cont... |
| dgrco 26624 | The degree of a compositio... |
| plycjlem 26625 | Lemma for ~ plycj and ~ co... |
| plycj 26626 | The double conjugation of ... |
| coecj 26627 | Double conjugation of a po... |
| plyrecj 26628 | A polynomial with real coe... |
| plymul0or 26629 | Polynomial multiplication ... |
| ofmulrt 26630 | The set of roots of a prod... |
| plymul02 26631 | Product of a polynomial wi... |
| plyn0mulidp 26632 | Coefficients of a non-zero... |
| plymulidp 26633 | Coefficients of a polynomi... |
| plyreres 26634 | Real-coefficient polynomia... |
| dvply1 26635 | Derivative of a polynomial... |
| dvply2g 26636 | The derivative of a polyno... |
| dvply2 26637 | The derivative of a polyno... |
| dvnply2 26638 | Polynomials have polynomia... |
| dvnply 26639 | Polynomials have polynomia... |
| plycpn 26640 | Polynomials are smooth. (... |
| quotval 26643 | Value of the quotient func... |
| plydivlem1 26644 | Lemma for ~ plydivalg . (... |
| plydivlem2 26645 | Lemma for ~ plydivalg . (... |
| plydivlem3 26646 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26647 | Lemma for ~ plydivex . In... |
| plydivex 26648 | Lemma for ~ plydivalg . (... |
| plydiveu 26649 | Lemma for ~ plydivalg . (... |
| plydivalg 26650 | The division algorithm on ... |
| quotlem 26651 | Lemma for properties of th... |
| quotcl 26652 | The quotient of two polyno... |
| quotcl2 26653 | Closure of the quotient fu... |
| quotdgr 26654 | Remainder property of the ... |
| plyremlem 26655 | Closure of a linear factor... |
| plyrem 26656 | The polynomial remainder t... |
| facth 26657 | The factor theorem. If a ... |
| fta1lem 26658 | Lemma for ~ fta1 . (Contr... |
| fta1 26659 | The easy direction of the ... |
| rnplynfin 26660 | The range of a nonconstant... |
| plyconz 26661 | The composition of a nonze... |
| quotcan 26662 | Exact division with a mult... |
| vieta1lem1 26663 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26664 | Lemma for ~ vieta1 : induc... |
| vieta1 26665 | The first-order Vieta's fo... |
| plyexmo 26666 | An infinite set of values ... |
| elaa 26669 | Elementhood in the set of ... |
| aacn 26670 | An algebraic number is a c... |
| aasscn 26671 | The algebraic numbers are ... |
| elqaalem1 26672 | Lemma for ~ elqaa . The f... |
| elqaalem2 26673 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26674 | Lemma for ~ elqaa . (Cont... |
| elqaa 26675 | The set of numbers generat... |
| preimaaa 26676 | An element of the preimage... |
| qaa 26677 | Every rational number is a... |
| qssaa 26678 | The rational numbers are c... |
| 0aa 26679 | Zero is algebraic. (Contr... |
| 1aa 26680 | One is algebraic. (Contri... |
| iaa 26681 | The imaginary unit is alge... |
| iaaOLD 26682 | Obsolete version of ~ iaa ... |
| aareccl 26683 | The reciprocal of an algeb... |
| aacjcl 26684 | The conjugate of an algebr... |
| aannenlem1 26685 | Lemma for ~ aannen . (Con... |
| aannenlem2 26686 | Lemma for ~ aannen . (Con... |
| aannenlem3 26687 | The algebraic numbers are ... |
| aannen 26688 | The algebraic numbers are ... |
| aalioulem1 26689 | Lemma for ~ aaliou . An i... |
| aalioulem2 26690 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26691 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26692 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26693 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26694 | Lemma for ~ aaliou . (Con... |
| aaliou 26695 | Liouville's theorem on dio... |
| geolim3 26696 | Geometric series convergen... |
| aaliou2 26697 | Liouville's approximation ... |
| aaliou2b 26698 | Liouville's approximation ... |
| aaliou3lem1 26699 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26700 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26701 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26702 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26703 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26704 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26705 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26706 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26707 | Example of a "Liouville nu... |
| aaliou3 26708 | Example of a "Liouville nu... |
| aaliou3r 26709 | The sum presented above is... |
| taylfvallem1 26714 | Lemma for ~ taylfval . (C... |
| taylfvallem 26715 | Lemma for ~ taylfval . (C... |
| taylfval 26716 | Define the Taylor polynomi... |
| eltayl 26717 | Value of the Taylor series... |
| taylf 26718 | The Taylor series defines ... |
| tayl0 26719 | The Taylor series is alway... |
| taylplem1 26720 | Lemma for ~ taylpfval and ... |
| taylplem2 26721 | Lemma for ~ taylpfval and ... |
| taylpfval 26722 | Define the Taylor polynomi... |
| taylpf 26723 | The Taylor polynomial is a... |
| taylpval 26724 | Value of the Taylor polyno... |
| taylply2 26725 | The Taylor polynomial is a... |
| taylply 26726 | The Taylor polynomial is a... |
| dvtaylp 26727 | The derivative of the Tayl... |
| dvntaylp 26728 | The ` M ` -th derivative o... |
| dvntaylp0 26729 | The first ` N ` derivative... |
| taylthlem1 26730 | Lemma for ~ taylth . This... |
| taylthlem2 26731 | Lemma for ~ taylth . (Con... |
| taylth 26732 | Taylor's theorem. The Tay... |
| ulmrel 26735 | The uniform limit relation... |
| ulmscl 26736 | Closure of the base set in... |
| ulmval 26737 | Express the predicate: Th... |
| ulmcl 26738 | Closure of a uniform limit... |
| ulmf 26739 | Closure of a uniform limit... |
| ulmpm 26740 | Closure of a uniform limit... |
| ulmf2 26741 | Closure of a uniform limit... |
| ulm2 26742 | Simplify ~ ulmval when ` F... |
| ulmi 26743 | The uniform limit property... |
| ulmclm 26744 | A uniform limit of functio... |
| ulmres 26745 | A sequence of functions co... |
| ulmshftlem 26746 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26747 | A sequence of functions co... |
| ulm0 26748 | Every function converges u... |
| ulmuni 26749 | A sequence of functions un... |
| ulmdm 26750 | Two ways to express that a... |
| ulmcaulem 26751 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26752 | A sequence of functions co... |
| ulmcau2 26753 | A sequence of functions co... |
| ulmss 26754 | A uniform limit of functio... |
| ulmbdd 26755 | A uniform limit of bounded... |
| ulmcn 26756 | A uniform limit of continu... |
| ulmdvlem1 26757 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26758 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26759 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26760 | If ` F ` is a sequence of ... |
| mtest 26761 | The Weierstrass M-test. I... |
| mtestbdd 26762 | Given the hypotheses of th... |
| mbfulm 26763 | A uniform limit of measura... |
| iblulm 26764 | A uniform limit of integra... |
| itgulm 26765 | A uniform limit of integra... |
| itgulm2 26766 | A uniform limit of integra... |
| pserval 26767 | Value of the function ` G ... |
| pserval2 26768 | Value of the function ` G ... |
| psergf 26769 | The sequence of terms in t... |
| radcnvlem1 26770 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26771 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26772 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26773 | Zero is always a convergen... |
| radcnvcl 26774 | The radius of convergence ... |
| radcnvlt1 26775 | If ` X ` is within the ope... |
| radcnvlt2 26776 | If ` X ` is within the ope... |
| radcnvle 26777 | If ` X ` is a convergent p... |
| dvradcnv 26778 | The radius of convergence ... |
| pserulm 26779 | If ` S ` is a region conta... |
| psercn2 26780 | Since by ~ pserulm the ser... |
| psercnlem2 26781 | Lemma for ~ psercn . (Con... |
| psercnlem1 26782 | Lemma for ~ psercn . (Con... |
| psercn 26783 | An infinite series converg... |
| pserdvlem1 26784 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26785 | Lemma for ~ pserdv . (Con... |
| pserdv 26786 | The derivative of a power ... |
| pserdv2 26787 | The derivative of a power ... |
| abelthlem1 26788 | Lemma for ~ abelth . (Con... |
| abelthlem2 26789 | Lemma for ~ abelth . The ... |
| abelthlem3 26790 | Lemma for ~ abelth . (Con... |
| abelthlem4 26791 | Lemma for ~ abelth . (Con... |
| abelthlem5 26792 | Lemma for ~ abelth . (Con... |
| abelthlem6 26793 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26794 | Lemma for ~ abelth . (Con... |
| abelthlem7 26795 | Lemma for ~ abelth . (Con... |
| abelthlem8 26796 | Lemma for ~ abelth . (Con... |
| abelthlem9 26797 | Lemma for ~ abelth . By a... |
| abelth 26798 | Abel's theorem. If the po... |
| abelth2 26799 | Abel's theorem, restricted... |
| efcn 26800 | The exponential function i... |
| sincn 26801 | Sine is continuous. (Cont... |
| coscn 26802 | Cosine is continuous. (Co... |
| reeff1olem 26803 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26804 | The real exponential funct... |
| reefiso 26805 | The exponential function o... |
| efcvx 26806 | The exponential function o... |
| reefgim 26807 | The exponential function i... |
| pilem1 26808 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26809 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26810 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26811 | ` _pi ` is between 2 and 4... |
| sinpi 26812 | The sine of ` _pi ` is 0. ... |
| pire 26813 | ` _pi ` is a real number. ... |
| 2pire 26814 | ` ( 2 x. _pi ) ` is a real... |
| picn 26815 | ` _pi ` is a complex numbe... |
| 2picn 26816 | ` ( 2 x. _pi ) ` is a comp... |
| pipos 26817 | ` _pi ` is positive. (Con... |
| pige0 26818 | ` _pi ` is nonnegative. (... |
| pine0 26819 | ` _pi ` is nonzero. (Cont... |
| pirp 26820 | ` _pi ` is a positive real... |
| negpicn 26821 | ` -u _pi ` is a complex nu... |
| sinhalfpilem 26822 | Lemma for ~ sinhalfpi and ... |
| halfpire 26823 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26824 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26825 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26826 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26827 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26828 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26829 | The cosine of ` -u _pi / 2... |
| efhalfpi 26830 | The exponential of ` _i _p... |
| cospi 26831 | The cosine of ` _pi ` is `... |
| efipi 26832 | The exponential of ` _i x.... |
| eulerid 26833 | Euler's identity. (Contri... |
| sin2pi 26834 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26835 | The cosine of ` 2 _pi ` is... |
| ef2pi 26836 | The exponential of ` 2 _pi... |
| ef2kpi 26837 | If ` K ` is an integer, th... |
| efper 26838 | The exponential function i... |
| sinperlem 26839 | Lemma for ~ sinper and ~ c... |
| sinper 26840 | The sine function is perio... |
| cosper 26841 | The cosine function is per... |
| sin2kpi 26842 | If ` K ` is an integer, th... |
| cos2kpi 26843 | If ` K ` is an integer, th... |
| sin2pim 26844 | Sine of a number subtracte... |
| cos2pim 26845 | Cosine of a number subtrac... |
| sinmpi 26846 | Sine of a number less ` _p... |
| cosmpi 26847 | Cosine of a number less ` ... |
| sinppi 26848 | Sine of a number plus ` _p... |
| cosppi 26849 | Cosine of a number plus ` ... |
| efimpi 26850 | The exponential function a... |
| sinhalfpip 26851 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26852 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26853 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26854 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26855 | Ptolemy's Theorem. This t... |
| sincosq1lem 26856 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26857 | The signs of the sine and ... |
| sincosq2sgn 26858 | The signs of the sine and ... |
| sincosq3sgn 26859 | The signs of the sine and ... |
| sincosq4sgn 26860 | The signs of the sine and ... |
| coseq00topi 26861 | Location of the zeroes of ... |
| coseq0negpitopi 26862 | Location of the zeroes of ... |
| tanrpcl 26863 | Positive real closure of t... |
| tangtx 26864 | The tangent function is gr... |
| tanabsge 26865 | The tangent function is gr... |
| sinq12gt0 26866 | The sine of a number stric... |
| sinq12ge0 26867 | The sine of a number betwe... |
| sinq34lt0t 26868 | The sine of a number stric... |
| cosq14gt0 26869 | The cosine of a number str... |
| cosq14ge0 26870 | The cosine of a number bet... |
| sincosq1eq 26871 | Complementarity of the sin... |
| sincos4thpi 26872 | The sine and cosine of ` _... |
| tan4thpi 26873 | The tangent of ` _pi / 4 `... |
| sincos6thpi 26874 | The sine and cosine of ` _... |
| sincos3rdpi 26875 | The sine and cosine of ` _... |
| pigt3 26876 | ` _pi ` is greater than 3.... |
| pige3 26877 | ` _pi ` is greater than or... |
| pige3ALT 26878 | Alternate proof of ~ pige3... |
| abssinper 26879 | The absolute value of sine... |
| sinkpi 26880 | The sine of an integer mul... |
| coskpi 26881 | The absolute value of the ... |
| sineq0 26882 | A complex number whose sin... |
| coseq1 26883 | A complex number whose cos... |
| cos02pilt1 26884 | Cosine is less than one be... |
| cosq34lt1 26885 | Cosine is less than one in... |
| efeq1 26886 | A complex number whose exp... |
| cosne0 26887 | The cosine function has no... |
| cosordlem 26888 | Lemma for ~ cosord . (Con... |
| cosord 26889 | Cosine is decreasing over ... |
| cos0pilt1 26890 | Cosine is between minus on... |
| cos11 26891 | Cosine is one-to-one over ... |
| sinord 26892 | Sine is increasing over th... |
| recosf1o 26893 | The cosine function is a b... |
| resinf1o 26894 | The sine function is a bij... |
| tanord1 26895 | The tangent function is st... |
| tanord 26896 | The tangent function is st... |
| tanregt0 26897 | The real part of the tange... |
| negpitopissre 26898 | The interval ` ( -u _pi (,... |
| efgh 26899 | The exponential function o... |
| efif1olem1 26900 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26901 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26902 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26903 | The exponential function o... |
| efif1o 26904 | The exponential function o... |
| efifo 26905 | The exponential function o... |
| eff1olem 26906 | The exponential function m... |
| eff1o 26907 | The exponential function m... |
| efabl 26908 | The image of a subgroup of... |
| efsubm 26909 | The image of a subgroup of... |
| circgrp 26910 | The circle group ` T ` is ... |
| circsubm 26911 | The circle group ` T ` is ... |
| logrn 26916 | The range of the natural l... |
| ellogrn 26917 | Write out the property ` A... |
| dflog2 26918 | The natural logarithm func... |
| relogrn 26919 | The range of the natural l... |
| logrncn 26920 | The range of the natural l... |
| eff1o2 26921 | The exponential function r... |
| logf1o 26922 | The natural logarithm func... |
| dfrelog 26923 | The natural logarithm func... |
| relogf1o 26924 | The natural logarithm func... |
| logrncl 26925 | Closure of the natural log... |
| logcl 26926 | Closure of the natural log... |
| logimcl 26927 | Closure of the imaginary p... |
| logcld 26928 | The logarithm of a nonzero... |
| logimcld 26929 | The imaginary part of the ... |
| logimclad 26930 | The imaginary part of the ... |
| abslogimle 26931 | The imaginary part of the ... |
| logrnaddcl 26932 | The range of the natural l... |
| relogcl 26933 | Closure of the natural log... |
| eflog 26934 | Relationship between the n... |
| logeq0im1 26935 | If the logarithm of a numb... |
| logccne0 26936 | The logarithm isn't 0 if i... |
| logne0 26937 | Logarithm of a non-1 posit... |
| reeflog 26938 | Relationship between the n... |
| logef 26939 | Relationship between the n... |
| relogef 26940 | Relationship between the n... |
| logeftb 26941 | Relationship between the n... |
| relogeftb 26942 | Relationship between the n... |
| log1 26943 | The natural logarithm of `... |
| loge 26944 | The natural logarithm of `... |
| logi 26945 | The natural logarithm of `... |
| logneg 26946 | The natural logarithm of a... |
| logm1 26947 | The natural logarithm of n... |
| lognegb 26948 | If a number has imaginary ... |
| relogoprlem 26949 | Lemma for ~ relogmul and ~... |
| relogmul 26950 | The natural logarithm of t... |
| relogdiv 26951 | The natural logarithm of t... |
| explog 26952 | Exponentiation of a nonzer... |
| reexplog 26953 | Exponentiation of a positi... |
| relogexp 26954 | The natural logarithm of p... |
| relog 26955 | Real part of a logarithm. ... |
| relogiso 26956 | The natural logarithm func... |
| reloggim 26957 | The natural logarithm is a... |
| logltb 26958 | The natural logarithm func... |
| logfac 26959 | The logarithm of a factori... |
| eflogeq 26960 | Solve an equation involvin... |
| logleb 26961 | Natural logarithm preserve... |
| rplogcl 26962 | Closure of the logarithm f... |
| logge0 26963 | The logarithm of a number ... |
| logcj 26964 | The natural logarithm dist... |
| efiarg 26965 | The exponential of the "ar... |
| cosargd 26966 | The cosine of the argument... |
| cosarg0d 26967 | The cosine of the argument... |
| argregt0 26968 | Closure of the argument of... |
| argrege0 26969 | Closure of the argument of... |
| argimgt0 26970 | Closure of the argument of... |
| argimlt0 26971 | Closure of the argument of... |
| logimul 26972 | Multiplying a number by ` ... |
| logneg2 26973 | The logarithm of the negat... |
| logmul2 26974 | Generalization of ~ relogm... |
| logdiv2 26975 | Generalization of ~ relogd... |
| abslogle 26976 | Bound on the magnitude of ... |
| tanarg 26977 | The basic relation between... |
| logdivlti 26978 | The ` log x / x ` function... |
| logdivlt 26979 | The ` log x / x ` function... |
| logdivle 26980 | The ` log x / x ` function... |
| relogcld 26981 | Closure of the natural log... |
| reeflogd 26982 | Relationship between the n... |
| relogmuld 26983 | The natural logarithm of t... |
| relogdivd 26984 | The natural logarithm of t... |
| logled 26985 | Natural logarithm preserve... |
| relogefd 26986 | Relationship between the n... |
| rplogcld 26987 | Closure of the logarithm f... |
| logge0d 26988 | The logarithm of a number ... |
| logge0b 26989 | The logarithm of a number ... |
| loggt0b 26990 | The logarithm of a number ... |
| logle1b 26991 | The logarithm of a number ... |
| loglt1b 26992 | The logarithm of a number ... |
| divlogrlim 26993 | The inverse logarithm func... |
| logno1 26994 | The logarithm function is ... |
| dvrelog 26995 | The derivative of the real... |
| relogcn 26996 | The real logarithm functio... |
| ellogdm 26997 | Elementhood in the "contin... |
| logdmn0 26998 | A number in the continuous... |
| logdmnrp 26999 | A number in the continuous... |
| logdmss 27000 | The continuity domain of `... |
| logcnlem2 27001 | Lemma for ~ logcn . (Cont... |
| logcnlem3 27002 | Lemma for ~ logcn . (Cont... |
| logcnlem4 27003 | Lemma for ~ logcn . (Cont... |
| logcnlem5 27004 | Lemma for ~ logcn . (Cont... |
| logcn 27005 | The logarithm function is ... |
| dvloglem 27006 | Lemma for ~ dvlog . (Cont... |
| logdmopn 27007 | The "continuous domain" of... |
| logf1o2 27008 | The logarithm maps its con... |
| dvlog 27009 | The derivative of the comp... |
| dvlog2lem 27010 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 27011 | The derivative of the comp... |
| advlog 27012 | The antiderivative of the ... |
| advlogexp 27013 | The antiderivative of a po... |
| efopnlem1 27014 | Lemma for ~ efopn . (Cont... |
| efopnlem2 27015 | Lemma for ~ efopn . (Cont... |
| efopn 27016 | The exponential map is an ... |
| logtayllem 27017 | Lemma for ~ logtayl . (Co... |
| logtayl 27018 | The Taylor series for ` -u... |
| logtaylsum 27019 | The Taylor series for ` -u... |
| logtayl2 27020 | Power series expression fo... |
| logccv 27021 | The natural logarithm func... |
| cxpval 27022 | Value of the complex power... |
| cxpef 27023 | Value of the complex power... |
| 0cxp 27024 | Value of the complex power... |
| cxpexpz 27025 | Relate the complex power f... |
| cxpexp 27026 | Relate the complex power f... |
| logcxp 27027 | Logarithm of a complex pow... |
| cxp0 27028 | Value of the complex power... |
| cxp1 27029 | Value of the complex power... |
| 1cxp 27030 | Value of the complex power... |
| ecxp 27031 | Write the exponential func... |
| cxpcl 27032 | Closure of the complex pow... |
| recxpcl 27033 | Real closure of the comple... |
| rpcxpcl 27034 | Positive real closure of t... |
| cxpne0 27035 | Complex exponentiation is ... |
| cxpeq0 27036 | Complex exponentiation is ... |
| cxpadd 27037 | Sum of exponents law for c... |
| cxpp1 27038 | Value of a nonzero complex... |
| cxpneg 27039 | Value of a complex number ... |
| cxpsub 27040 | Exponent subtraction law f... |
| cxpge0 27041 | Nonnegative exponentiation... |
| mulcxplem 27042 | Lemma for ~ mulcxp . (Con... |
| mulcxp 27043 | Complex exponentiation of ... |
| cxprec 27044 | Complex exponentiation of ... |
| divcxp 27045 | Complex exponentiation of ... |
| cxpmul 27046 | Product of exponents law f... |
| cxpmul2 27047 | Product of exponents law f... |
| cxproot 27048 | The complex power function... |
| cxpmul2z 27049 | Generalize ~ cxpmul2 to ne... |
| abscxp 27050 | Absolute value of a power,... |
| abscxp2 27051 | Absolute value of a power,... |
| cxplt 27052 | Ordering property for comp... |
| cxple 27053 | Ordering property for comp... |
| cxplea 27054 | Ordering property for comp... |
| cxple2 27055 | Ordering property for comp... |
| cxplt2 27056 | Ordering property for comp... |
| cxple2a 27057 | Ordering property for comp... |
| cxplt3 27058 | Ordering property for comp... |
| cxple3 27059 | Ordering property for comp... |
| cxpsqrtlem 27060 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 27061 | The complex exponential fu... |
| logsqrt 27062 | Logarithm of a square root... |
| cxp0d 27063 | Value of the complex power... |
| cxp1d 27064 | Value of the complex power... |
| 1cxpd 27065 | Value of the complex power... |
| cxpcld 27066 | Closure of the complex pow... |
| cxpmul2d 27067 | Product of exponents law f... |
| 0cxpd 27068 | Value of the complex power... |
| cxpexpzd 27069 | Relate the complex power f... |
| cxpefd 27070 | Value of the complex power... |
| cxpne0d 27071 | Complex exponentiation is ... |
| cxpp1d 27072 | Value of a nonzero complex... |
| cxpnegd 27073 | Value of a complex number ... |
| cxpmul2zd 27074 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 27075 | Sum of exponents law for c... |
| cxpsubd 27076 | Exponent subtraction law f... |
| cxpltd 27077 | Ordering property for comp... |
| cxpled 27078 | Ordering property for comp... |
| cxplead 27079 | Ordering property for comp... |
| divcxpd 27080 | Complex exponentiation of ... |
| recxpcld 27081 | Positive real closure of t... |
| cxpge0d 27082 | Nonnegative exponentiation... |
| cxple2ad 27083 | Ordering property for comp... |
| cxplt2d 27084 | Ordering property for comp... |
| cxple2d 27085 | Ordering property for comp... |
| mulcxpd 27086 | Complex exponentiation of ... |
| recxpf1lem 27087 | Complex exponentiation on ... |
| cxpsqrtth 27088 | Square root theorem over t... |
| 2irrexpq 27089 | There exist irrational num... |
| cxprecd 27090 | Complex exponentiation of ... |
| rpcxpcld 27091 | Positive real closure of t... |
| logcxpd 27092 | Logarithm of a complex pow... |
| cxplt3d 27093 | Ordering property for comp... |
| cxple3d 27094 | Ordering property for comp... |
| cxpmuld 27095 | Product of exponents law f... |
| cxpgt0d 27096 | A positive real raised to ... |
| cxpcom 27097 | Commutative law for real e... |
| dvcxp1 27098 | The derivative of a comple... |
| dvcxp2 27099 | The derivative of a comple... |
| dvsqrt 27100 | The derivative of the real... |
| dvcncxp1 27101 | Derivative of complex powe... |
| dvcnsqrt 27102 | Derivative of square root ... |
| cxpcn 27103 | Domain of continuity of th... |
| cxpcn2 27104 | Continuity of the complex ... |
| cxpcn3lem 27105 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 27106 | Extend continuity of the c... |
| resqrtcn 27107 | Continuity of the real squ... |
| sqrtcn 27108 | Continuity of the square r... |
| cxpaddlelem 27109 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 27110 | Ordering property for comp... |
| abscxpbnd 27111 | Bound on the absolute valu... |
| root1id 27112 | Property of an ` N ` -th r... |
| root1eq1 27113 | The only powers of an ` N ... |
| root1cj 27114 | Within the ` N ` -th roots... |
| cxpeq 27115 | Solve an equation involvin... |
| zrtelqelz 27116 | If the ` N ` -th root of a... |
| zrtdvds 27117 | A positive integer root di... |
| rtprmirr 27118 | The root of a prime number... |
| loglesqrt 27119 | An upper bound on the loga... |
| logreclem 27120 | Symmetry of the natural lo... |
| logrec 27121 | Logarithm of a reciprocal ... |
| logbval 27124 | Define the value of the ` ... |
| logbcl 27125 | General logarithm closure.... |
| logbid1 27126 | General logarithm is 1 whe... |
| logb1 27127 | The logarithm of ` 1 ` to ... |
| elogb 27128 | The general logarithm of a... |
| logbchbase 27129 | Change of base for logarit... |
| relogbval 27130 | Value of the general logar... |
| relogbcl 27131 | Closure of the general log... |
| relogbzcl 27132 | Closure of the general log... |
| relogbreexp 27133 | Power law for the general ... |
| relogbzexp 27134 | Power law for the general ... |
| relogbmul 27135 | The logarithm of the produ... |
| relogbmulexp 27136 | The logarithm of the produ... |
| relogbdiv 27137 | The logarithm of the quoti... |
| relogbexp 27138 | Identity law for general l... |
| nnlogbexp 27139 | Identity law for general l... |
| logbrec 27140 | Logarithm of a reciprocal ... |
| logbleb 27141 | The general logarithm func... |
| logblt 27142 | The general logarithm func... |
| relogbcxp 27143 | Identity law for the gener... |
| cxplogb 27144 | Identity law for the gener... |
| relogbcxpb 27145 | The logarithm is the inver... |
| logbmpt 27146 | The general logarithm to a... |
| logbf 27147 | The general logarithm to a... |
| logbfval 27148 | The general logarithm of a... |
| relogbf 27149 | The general logarithm to a... |
| logblog 27150 | The general logarithm to t... |
| logbgt0b 27151 | The logarithm of a positiv... |
| logbgcd1irr 27152 | The logarithm of an intege... |
| 2logb9irr 27153 | Example for ~ logbgcd1irr ... |
| logbprmirr 27154 | The logarithm of a prime t... |
| 2logb3irr 27155 | Example for ~ logbprmirr .... |
| 2logb9irrALT 27156 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 27157 | The square root of two to ... |
| 2irrexpqALT 27158 | Alternate proof of ~ 2irre... |
| angval 27159 | Define the angle function,... |
| angcan 27160 | Cancel a constant multipli... |
| angneg 27161 | Cancel a negative sign in ... |
| angvald 27162 | The (signed) angle between... |
| angcld 27163 | The (signed) angle between... |
| angrteqvd 27164 | Two vectors are at a right... |
| cosangneg2d 27165 | The cosine of the angle be... |
| angrtmuld 27166 | Perpendicularity of two ve... |
| ang180lem1 27167 | Lemma for ~ ang180 . Show... |
| ang180lem2 27168 | Lemma for ~ ang180 . Show... |
| ang180lem3 27169 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 27170 | Lemma for ~ ang180 . Redu... |
| ang180lem5 27171 | Lemma for ~ ang180 : Redu... |
| ang180 27172 | The sum of angles ` m A B ... |
| lawcoslem1 27173 | Lemma for ~ lawcos . Here... |
| lawcos 27174 | Law of cosines (also known... |
| pythag 27175 | Pythagorean theorem. Give... |
| isosctrlem1 27176 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 27177 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 27178 | Lemma for ~ isosctr . Cor... |
| isosctr 27179 | Isosceles triangle theorem... |
| ssscongptld 27180 | If two triangles have equa... |
| affineequiv 27181 | Equivalence between two wa... |
| affineequiv2 27182 | Equivalence between two wa... |
| affineequiv3 27183 | Equivalence between two wa... |
| affineequiv4 27184 | Equivalence between two wa... |
| affineequivne 27185 | Equivalence between two wa... |
| angpieqvdlem 27186 | Equivalence used in the pr... |
| angpieqvdlem2 27187 | Equivalence used in ~ angp... |
| angpined 27188 | If the angle at ABC is ` _... |
| angpieqvd 27189 | The angle ABC is ` _pi ` i... |
| chordthmlem 27190 | If ` M ` is the midpoint o... |
| chordthmlem2 27191 | If M is the midpoint of AB... |
| chordthmlem3 27192 | If M is the midpoint of AB... |
| chordthmlem4 27193 | If P is on the segment AB ... |
| chordthmlem5 27194 | If P is on the segment AB ... |
| chordthm 27195 | The intersecting chords th... |
| heron 27196 | Heron's formula gives the ... |
| quad2 27197 | The quadratic equation, wi... |
| quad 27198 | The quadratic equation. (... |
| 1cubrlem 27199 | The cube roots of unity. ... |
| 1cubr 27200 | The cube roots of unity. ... |
| dcubic1lem 27201 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 27202 | Reverse direction of ~ dcu... |
| dcubic1 27203 | Forward direction of ~ dcu... |
| dcubic 27204 | Solutions to the depressed... |
| mcubic 27205 | Solutions to a monic cubic... |
| cubic2 27206 | The solution to the genera... |
| cubic 27207 | The cubic equation, which ... |
| binom4 27208 | Work out a quartic binomia... |
| dquartlem1 27209 | Lemma for ~ dquart . (Con... |
| dquartlem2 27210 | Lemma for ~ dquart . (Con... |
| dquart 27211 | Solve a depressed quartic ... |
| quart1cl 27212 | Closure lemmas for ~ quart... |
| quart1lem 27213 | Lemma for ~ quart1 . (Con... |
| quart1 27214 | Depress a quartic equation... |
| quartlem1 27215 | Lemma for ~ quart . (Cont... |
| quartlem2 27216 | Closure lemmas for ~ quart... |
| quartlem3 27217 | Closure lemmas for ~ quart... |
| quartlem4 27218 | Closure lemmas for ~ quart... |
| quart 27219 | The quartic equation, writ... |
| asinlem 27226 | The argument to the logari... |
| asinlem2 27227 | The argument to the logari... |
| asinlem3a 27228 | Lemma for ~ asinlem3 . (C... |
| asinlem3 27229 | The argument to the logari... |
| asinf 27230 | Domain and codomain of the... |
| asincl 27231 | Closure for the arcsin fun... |
| acosf 27232 | Domain and codoamin of the... |
| acoscl 27233 | Closure for the arccos fun... |
| atandm 27234 | Since the property is a li... |
| atandm2 27235 | This form of ~ atandm is a... |
| atandm3 27236 | A compact form of ~ atandm... |
| atandm4 27237 | A compact form of ~ atandm... |
| atanf 27238 | Domain and codoamin of the... |
| atancl 27239 | Closure for the arctan fun... |
| asinval 27240 | Value of the arcsin functi... |
| acosval 27241 | Value of the arccos functi... |
| atanval 27242 | Value of the arctan functi... |
| atanre 27243 | A real number is in the do... |
| asinneg 27244 | The arcsine function is od... |
| acosneg 27245 | The negative symmetry rela... |
| efiasin 27246 | The exponential of the arc... |
| sinasin 27247 | The arcsine function is an... |
| cosacos 27248 | The arccosine function is ... |
| asinsinlem 27249 | Lemma for ~ asinsin . (Co... |
| asinsin 27250 | The arcsine function compo... |
| acoscos 27251 | The arccosine function is ... |
| asin1 27252 | The arcsine of ` 1 ` is ` ... |
| acos1 27253 | The arccosine of ` 1 ` is ... |
| reasinsin 27254 | The arcsine function compo... |
| asinsinb 27255 | Relationship between sine ... |
| acoscosb 27256 | Relationship between cosin... |
| asinbnd 27257 | The arcsine function has r... |
| acosbnd 27258 | The arccosine function has... |
| asinrebnd 27259 | Bounds on the arcsine func... |
| asinrecl 27260 | The arcsine function is re... |
| acosrecl 27261 | The arccosine function is ... |
| cosasin 27262 | The cosine of the arcsine ... |
| sinacos 27263 | The sine of the arccosine ... |
| atandmneg 27264 | The domain of the arctange... |
| atanneg 27265 | The arctangent function is... |
| atan0 27266 | The arctangent of zero is ... |
| atandmcj 27267 | The arctangent function di... |
| atancj 27268 | The arctangent function di... |
| atanrecl 27269 | The arctangent function is... |
| efiatan 27270 | Value of the exponential o... |
| atanlogaddlem 27271 | Lemma for ~ atanlogadd . ... |
| atanlogadd 27272 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 27273 | Lemma for ~ atanlogsub . ... |
| atanlogsub 27274 | A variation on ~ atanlogad... |
| efiatan2 27275 | Value of the exponential o... |
| 2efiatan 27276 | Value of the exponential o... |
| tanatan 27277 | The arctangent function is... |
| atandmtan 27278 | The tangent function has r... |
| cosatan 27279 | The cosine of an arctangen... |
| cosatanne0 27280 | The arctangent function ha... |
| atantan 27281 | The arctangent function is... |
| atantanb 27282 | Relationship between tange... |
| atanbndlem 27283 | Lemma for ~ atanbnd . (Co... |
| atanbnd 27284 | The arctangent function is... |
| atanord 27285 | The arctangent function is... |
| atan1 27286 | The arctangent of ` 1 ` is... |
| bndatandm 27287 | A point in the open unit d... |
| atans 27288 | The "domain of continuity"... |
| atans2 27289 | It suffices to show that `... |
| atansopn 27290 | The domain of continuity o... |
| atansssdm 27291 | The domain of continuity o... |
| ressatans 27292 | The real number line is a ... |
| dvatan 27293 | The derivative of the arct... |
| atancn 27294 | The arctangent is a contin... |
| atantayl 27295 | The Taylor series for ` ar... |
| atantayl2 27296 | The Taylor series for ` ar... |
| atantayl3 27297 | The Taylor series for ` ar... |
| leibpilem1 27298 | Lemma for ~ leibpi . (Con... |
| leibpilem2 27299 | The Leibniz formula for ` ... |
| leibpi 27300 | The Leibniz formula for ` ... |
| leibpisum 27301 | The Leibniz formula for ` ... |
| log2cnv 27302 | Using the Taylor series fo... |
| log2tlbnd 27303 | Bound the error term in th... |
| log2ublem1 27304 | Lemma for ~ log2ub . The ... |
| log2ublem2 27305 | Lemma for ~ log2ub . (Con... |
| log2ublem3 27306 | Lemma for ~ log2ub . In d... |
| log2ub 27307 | ` log 2 ` is less than ` 2... |
| log2le1 27308 | ` log 2 ` is less than ` 1... |
| birthdaylem1 27309 | Lemma for ~ birthday . (C... |
| birthdaylem2 27310 | For general ` N ` and ` K ... |
| birthdaylem3 27311 | For general ` N ` and ` K ... |
| birthday 27312 | The Birthday Problem. The... |
| dmarea 27315 | The domain of the area fun... |
| areambl 27316 | The fibers of a measurable... |
| areass 27317 | A measurable region is a s... |
| dfarea 27318 | Rewrite ~ df-area self-ref... |
| areaf 27319 | Area measurement is a func... |
| areacl 27320 | The area of a measurable r... |
| areage0 27321 | The area of a measurable r... |
| areaval 27322 | The area of a measurable r... |
| rlimcnp 27323 | Relate a limit of a real-v... |
| rlimcnp2 27324 | Relate a limit of a real-v... |
| rlimcnp3 27325 | Relate a limit of a real-v... |
| xrlimcnp 27326 | Relate a limit of a real-v... |
| efrlim 27327 | The limit of the sequence ... |
| dfef2 27328 | The limit of the sequence ... |
| cxplim 27329 | A power to a negative expo... |
| sqrtlim 27330 | The inverse square root fu... |
| rlimcxp 27331 | Any power to a positive ex... |
| o1cxp 27332 | An eventually bounded func... |
| cxp2limlem 27333 | A linear factor grows slow... |
| cxp2lim 27334 | Any power grows slower tha... |
| cxploglim 27335 | The logarithm grows slower... |
| cxploglim2 27336 | Every power of the logarit... |
| divsqrtsumlem 27337 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 27338 | The function ` F ` used in... |
| divsqrsum 27339 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 27340 | A bound on the distance of... |
| divsqrtsumo1 27341 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 27342 | Closure of a 0-1 linear co... |
| scvxcvx 27343 | A strictly convex function... |
| jensenlem1 27344 | Lemma for ~ jensen . (Con... |
| jensenlem2 27345 | Lemma for ~ jensen . (Con... |
| jensen 27346 | Jensen's inequality, a fin... |
| amgmlem 27347 | Lemma for ~ amgm . (Contr... |
| amgm 27348 | Inequality of arithmetic a... |
| logdifbnd 27351 | Bound on the difference of... |
| logdiflbnd 27352 | Lower bound on the differe... |
| emcllem1 27353 | Lemma for ~ emcl . The se... |
| emcllem2 27354 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 27355 | Lemma for ~ emcl . The fu... |
| emcllem4 27356 | Lemma for ~ emcl . The di... |
| emcllem5 27357 | Lemma for ~ emcl . The pa... |
| emcllem6 27358 | Lemma for ~ emcl . By the... |
| emcllem7 27359 | Lemma for ~ emcl and ~ har... |
| emcl 27360 | Closure and bounds for the... |
| harmonicbnd 27361 | A bound on the harmonic se... |
| harmonicbnd2 27362 | A bound on the harmonic se... |
| emre 27363 | The Euler-Mascheroni const... |
| emgt0 27364 | The Euler-Mascheroni const... |
| harmonicbnd3 27365 | A bound on the harmonic se... |
| harmoniclbnd 27366 | A bound on the harmonic se... |
| harmonicubnd 27367 | A bound on the harmonic se... |
| harmonicbnd4 27368 | The asymptotic behavior of... |
| fsumharmonic 27369 | Bound a finite sum based o... |
| zetacvg 27372 | The zeta series is converg... |
| eldmgm 27379 | Elementhood in the set of ... |
| dmgmaddn0 27380 | If ` A ` is not a nonposit... |
| dmlogdmgm 27381 | If ` A ` is in the continu... |
| rpdmgm 27382 | A positive real number is ... |
| dmgmn0 27383 | If ` A ` is not a nonposit... |
| dmgmaddnn0 27384 | If ` A ` is not a nonposit... |
| dmgmdivn0 27385 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 27386 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 27387 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 27388 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 27389 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 27390 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 27391 | The series ` G ` is unifor... |
| lgamgulm 27392 | The series ` G ` is unifor... |
| lgamgulm2 27393 | Rewrite the limit of the s... |
| lgambdd 27394 | The log-Gamma function is ... |
| lgamucov 27395 | The ` U ` regions used in ... |
| lgamucov2 27396 | The ` U ` regions used in ... |
| lgamcvglem 27397 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27398 | The log-Gamma function is ... |
| lgamf 27399 | The log-Gamma function is ... |
| gamf 27400 | The Gamma function is a co... |
| gamcl 27401 | The exponential of the log... |
| eflgam 27402 | The exponential of the log... |
| gamne0 27403 | The Gamma function is neve... |
| igamval 27404 | Value of the inverse Gamma... |
| igamz 27405 | Value of the inverse Gamma... |
| igamgam 27406 | Value of the inverse Gamma... |
| igamlgam 27407 | Value of the inverse Gamma... |
| igamf 27408 | Closure of the inverse Gam... |
| igamcl 27409 | Closure of the inverse Gam... |
| gamigam 27410 | The Gamma function is the ... |
| lgamcvg 27411 | The series ` G ` converges... |
| lgamcvg2 27412 | The series ` G ` converges... |
| gamcvg 27413 | The pointwise exponential ... |
| lgamp1 27414 | The functional equation of... |
| gamp1 27415 | The functional equation of... |
| gamcvg2lem 27416 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27417 | An infinite product expres... |
| regamcl 27418 | The Gamma function is real... |
| relgamcl 27419 | The log-Gamma function is ... |
| rpgamcl 27420 | The log-Gamma function is ... |
| lgam1 27421 | The log-Gamma function at ... |
| gam1 27422 | The log-Gamma function at ... |
| facgam 27423 | The Gamma function general... |
| gamfac 27424 | The Gamma function general... |
| wilthlem1 27425 | The only elements that are... |
| wilthlem2 27426 | Lemma for ~ wilth : induct... |
| wilthlem3 27427 | Lemma for ~ wilth . Here ... |
| wilth 27428 | Wilson's theorem. A numbe... |
| wilthimp 27429 | The forward implication of... |
| ftalem1 27430 | Lemma for ~ fta : "growth... |
| ftalem2 27431 | Lemma for ~ fta . There e... |
| ftalem3 27432 | Lemma for ~ fta . There e... |
| ftalem4 27433 | Lemma for ~ fta : Closure... |
| ftalem5 27434 | Lemma for ~ fta : Main pr... |
| ftalem6 27435 | Lemma for ~ fta : Dischar... |
| ftalem7 27436 | Lemma for ~ fta . Shift t... |
| fta 27437 | The Fundamental Theorem of... |
| basellem1 27438 | Lemma for ~ basel . Closu... |
| basellem2 27439 | Lemma for ~ basel . Show ... |
| basellem3 27440 | Lemma for ~ basel . Using... |
| basellem4 27441 | Lemma for ~ basel . By ~ ... |
| basellem5 27442 | Lemma for ~ basel . Using... |
| basellem6 27443 | Lemma for ~ basel . The f... |
| basellem7 27444 | Lemma for ~ basel . The f... |
| basellem8 27445 | Lemma for ~ basel . The f... |
| basellem9 27446 | Lemma for ~ basel . Since... |
| basel 27447 | The sum of the inverse squ... |
| efnnfsumcl 27460 | Finite sum closure in the ... |
| ppisval 27461 | The set of primes less tha... |
| ppisval2 27462 | The set of primes less tha... |
| ppifi 27463 | The set of primes less tha... |
| prmdvdsfi 27464 | The set of prime divisors ... |
| chtf 27465 | Domain and codoamin of the... |
| chtcl 27466 | Real closure of the Chebys... |
| chtval 27467 | Value of the Chebyshev fun... |
| efchtcl 27468 | The Chebyshev function is ... |
| chtge0 27469 | The Chebyshev function is ... |
| vmaval 27470 | Value of the von Mangoldt ... |
| isppw 27471 | Two ways to say that ` A `... |
| isppw2 27472 | Two ways to say that ` A `... |
| vmappw 27473 | Value of the von Mangoldt ... |
| vmaprm 27474 | Value of the von Mangoldt ... |
| vmacl 27475 | Closure for the von Mangol... |
| vmaf 27476 | Functionality of the von M... |
| efvmacl 27477 | The von Mangoldt is closed... |
| vmage0 27478 | The von Mangoldt function ... |
| chpval 27479 | Value of the second Chebys... |
| chpf 27480 | Functionality of the secon... |
| chpcl 27481 | Closure for the second Che... |
| efchpcl 27482 | The second Chebyshev funct... |
| chpge0 27483 | The second Chebyshev funct... |
| ppival 27484 | Value of the prime-countin... |
| ppival2 27485 | Value of the prime-countin... |
| ppival2g 27486 | Value of the prime-countin... |
| ppif 27487 | Domain and codomain of the... |
| ppicl 27488 | Real closure of the prime-... |
| muval 27489 | The value of the Möbi... |
| muval1 27490 | The value of the Möbi... |
| muval2 27491 | The value of the Möbi... |
| isnsqf 27492 | Two ways to say that a num... |
| issqf 27493 | Two ways to say that a num... |
| sqfpc 27494 | The prime count of a squar... |
| dvdssqf 27495 | A divisor of a squarefree ... |
| sqf11 27496 | A squarefree number is com... |
| muf 27497 | The Möbius function i... |
| mucl 27498 | Closure of the Möbius... |
| sgmval 27499 | The value of the divisor f... |
| sgmval2 27500 | The value of the divisor f... |
| 0sgm 27501 | The value of the sum-of-di... |
| sgmf 27502 | The divisor function is a ... |
| sgmcl 27503 | Closure of the divisor fun... |
| sgmnncl 27504 | Closure of the divisor fun... |
| mule1 27505 | The Möbius function t... |
| chtfl 27506 | The Chebyshev function doe... |
| chpfl 27507 | The second Chebyshev funct... |
| ppiprm 27508 | The prime-counting functio... |
| ppinprm 27509 | The prime-counting functio... |
| chtprm 27510 | The Chebyshev function at ... |
| chtnprm 27511 | The Chebyshev function at ... |
| chpp1 27512 | The second Chebyshev funct... |
| chtwordi 27513 | The Chebyshev function is ... |
| chpwordi 27514 | The second Chebyshev funct... |
| chtdif 27515 | The difference of the Cheb... |
| efchtdvds 27516 | The exponentiated Chebyshe... |
| ppifl 27517 | The prime-counting functio... |
| ppip1le 27518 | The prime-counting functio... |
| ppiwordi 27519 | The prime-counting functio... |
| ppidif 27520 | The difference of the prim... |
| ppi1 27521 | The prime-counting functio... |
| cht1 27522 | The Chebyshev function at ... |
| vma1 27523 | The von Mangoldt function ... |
| chp1 27524 | The second Chebyshev funct... |
| ppi1i 27525 | Inference form of ~ ppiprm... |
| ppi2i 27526 | Inference form of ~ ppinpr... |
| ppi2 27527 | The prime-counting functio... |
| ppi3 27528 | The prime-counting functio... |
| cht2 27529 | The Chebyshev function at ... |
| cht3 27530 | The Chebyshev function at ... |
| ppinncl 27531 | Closure of the prime-count... |
| chtrpcl 27532 | Closure of the Chebyshev f... |
| ppieq0 27533 | The prime-counting functio... |
| ppiltx 27534 | The prime-counting functio... |
| prmorcht 27535 | Relate the primorial (prod... |
| mumullem1 27536 | Lemma for ~ mumul . A mul... |
| mumullem2 27537 | Lemma for ~ mumul . The p... |
| mumul 27538 | The Möbius function i... |
| sqff1o 27539 | There is a bijection from ... |
| fsumdvdsdiaglem 27540 | A "diagonal commutation" o... |
| fsumdvdsdiag 27541 | A "diagonal commutation" o... |
| fsumdvdscom 27542 | A double commutation of di... |
| dvdsppwf1o 27543 | A bijection between the di... |
| dvdsflf1o 27544 | A bijection from the numbe... |
| dvdsflsumcom 27545 | A sum commutation from ` s... |
| fsumfldivdiaglem 27546 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27547 | The right-hand side of ~ d... |
| musum 27548 | The sum of the Möbius... |
| musumsum 27549 | Evaluate a collapsing sum ... |
| muinv 27550 | The Möbius inversion ... |
| mpodvdsmulf1o 27551 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27552 | Product of two divisor sum... |
| dvdsmulf1o 27553 | If ` M ` and ` N ` are two... |
| sgmppw 27554 | The value of the divisor f... |
| 0sgmppw 27555 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27556 | The sum of divisors for a ... |
| 1sgm2ppw 27557 | The sum of the divisors of... |
| sgmmul 27558 | The divisor function for f... |
| ppiublem1 27559 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27560 | A prime greater than ` 3 `... |
| ppiub 27561 | An upper bound on the prim... |
| vmalelog 27562 | The von Mangoldt function ... |
| chtlepsi 27563 | The first Chebyshev functi... |
| chprpcl 27564 | Closure of the second Cheb... |
| chpeq0 27565 | The second Chebyshev funct... |
| chteq0 27566 | The first Chebyshev functi... |
| chtleppi 27567 | Upper bound on the ` theta... |
| chtublem 27568 | Lemma for ~ chtub . (Cont... |
| chtub 27569 | An upper bound on the Cheb... |
| fsumvma 27570 | Rewrite a sum over the von... |
| fsumvma2 27571 | Apply ~ fsumvma for the co... |
| pclogsum 27572 | The logarithmic analogue o... |
| vmasum 27573 | The sum of the von Mangold... |
| logfac2 27574 | Another expression for the... |
| chpval2 27575 | Express the second Chebysh... |
| chpchtsum 27576 | The second Chebyshev funct... |
| chpub 27577 | An upper bound on the seco... |
| logfacubnd 27578 | A simple upper bound on th... |
| logfaclbnd 27579 | A lower bound on the logar... |
| logfacbnd3 27580 | Show the stronger statemen... |
| logfacrlim 27581 | Combine the estimates ~ lo... |
| logexprlim 27582 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27583 | Write out ~ logfacrlim as ... |
| mersenne 27584 | A Mersenne prime is a prim... |
| perfect1 27585 | Euclid's contribution to t... |
| perfectlem1 27586 | Lemma for ~ perfect . (Co... |
| perfectlem2 27587 | Lemma for ~ perfect . (Co... |
| perfect 27588 | The Euclid-Euler theorem, ... |
| dchrval 27591 | Value of the group of Diri... |
| dchrbas 27592 | Base set of the group of D... |
| dchrelbas 27593 | A Dirichlet character is a... |
| dchrelbas2 27594 | A Dirichlet character is a... |
| dchrelbas3 27595 | A Dirichlet character is a... |
| dchrelbasd 27596 | A Dirichlet character is a... |
| dchrrcl 27597 | Reverse closure for a Diri... |
| dchrmhm 27598 | A Dirichlet character is a... |
| dchrf 27599 | A Dirichlet character is a... |
| dchrelbas4 27600 | A Dirichlet character is a... |
| dchrzrh1 27601 | Value of a Dirichlet chara... |
| dchrzrhcl 27602 | A Dirichlet character take... |
| dchrzrhmul 27603 | A Dirichlet character is c... |
| dchrplusg 27604 | Group operation on the gro... |
| dchrmul 27605 | Group operation on the gro... |
| dchrmulcl 27606 | Closure of the group opera... |
| dchrn0 27607 | A Dirichlet character is n... |
| dchr1cl 27608 | Closure of the principal D... |
| dchrmullid 27609 | Left identity for the prin... |
| dchrinvcl 27610 | Closure of the group inver... |
| dchrabl 27611 | The set of Dirichlet chara... |
| dchrfi 27612 | The group of Dirichlet cha... |
| dchrghm 27613 | A Dirichlet character rest... |
| dchr1 27614 | Value of the principal Dir... |
| dchreq 27615 | A Dirichlet character is d... |
| dchrresb 27616 | A Dirichlet character is d... |
| dchrabs 27617 | A Dirichlet character take... |
| dchrinv 27618 | The inverse of a Dirichlet... |
| dchrabs2 27619 | A Dirichlet character take... |
| dchr1re 27620 | The principal Dirichlet ch... |
| dchrptlem1 27621 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27622 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27623 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27624 | For any element other than... |
| dchrsum2 27625 | An orthogonality relation ... |
| dchrsum 27626 | An orthogonality relation ... |
| sumdchr2 27627 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27628 | There are exactly ` phi ( ... |
| sumdchr 27629 | An orthogonality relation ... |
| dchr2sum 27630 | An orthogonality relation ... |
| sum2dchr 27631 | An orthogonality relation ... |
| bcctr 27632 | Value of the central binom... |
| pcbcctr 27633 | Prime count of a central b... |
| bcmono 27634 | The binomial coefficient i... |
| bcmax 27635 | The binomial coefficient t... |
| bcp1ctr 27636 | Ratio of two central binom... |
| bclbnd 27637 | A bound on the binomial co... |
| efexple 27638 | Convert a bound on a power... |
| bpos1lem 27639 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27640 | Bertrand's postulate, chec... |
| bposlem1 27641 | An upper bound on the prim... |
| bposlem2 27642 | There are no odd primes in... |
| bposlem3 27643 | Lemma for ~ bpos . Since ... |
| bposlem4 27644 | Lemma for ~ bpos . (Contr... |
| bposlem5 27645 | Lemma for ~ bpos . Bound ... |
| bposlem6 27646 | Lemma for ~ bpos . By usi... |
| bposlem7 27647 | Lemma for ~ bpos . The fu... |
| bposlem8 27648 | Lemma for ~ bpos . Show t... |
| bposlem9 27649 | Lemma for ~ bpos . Derive... |
| bpos 27650 | Bertrand's postulate: ther... |
| zabsle1 27653 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27654 | When ` a ` is coprime to t... |
| lgslem2 27655 | The set ` Z ` of all integ... |
| lgslem3 27656 | The set ` Z ` of all integ... |
| lgslem4 27657 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27658 | Value of the Legendre symb... |
| lgsfval 27659 | Value of the function ` F ... |
| lgsfcl2 27660 | The function ` F ` is clos... |
| lgscllem 27661 | The Legendre symbol is an ... |
| lgsfcl 27662 | Closure of the function ` ... |
| lgsfle1 27663 | The function ` F ` has mag... |
| lgsval2lem 27664 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27665 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27666 | The Legendre symbol is an ... |
| lgs0 27667 | The Legendre symbol when t... |
| lgscl 27668 | The Legendre symbol is an ... |
| lgsle1 27669 | The Legendre symbol has ab... |
| lgsval2 27670 | The Legendre symbol at a p... |
| lgs2 27671 | The Legendre symbol at ` 2... |
| lgsval3 27672 | The Legendre symbol at an ... |
| lgsvalmod 27673 | The Legendre symbol is equ... |
| lgsval4 27674 | Restate ~ lgsval for nonze... |
| lgsfcl3 27675 | Closure of the function ` ... |
| lgsval4a 27676 | Same as ~ lgsval4 for posi... |
| lgscl1 27677 | The value of the Legendre ... |
| lgsneg 27678 | The Legendre symbol is eit... |
| lgsneg1 27679 | The Legendre symbol for no... |
| lgsmod 27680 | The Legendre (Jacobi) symb... |
| lgsdilem 27681 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27682 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27683 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27684 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27685 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27686 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27687 | The Legendre symbol is com... |
| lgsdirprm 27688 | The Legendre symbol is com... |
| lgsdir 27689 | The Legendre symbol is com... |
| lgsdilem2 27690 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27691 | The Legendre symbol is com... |
| lgsne0 27692 | The Legendre symbol is non... |
| lgsabs1 27693 | The Legendre symbol is non... |
| lgssq 27694 | The Legendre symbol at a s... |
| lgssq2 27695 | The Legendre symbol at a s... |
| lgsprme0 27696 | The Legendre symbol at any... |
| 1lgs 27697 | The Legendre symbol at ` 1... |
| lgs1 27698 | The Legendre symbol at ` 1... |
| lgsmodeq 27699 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27700 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27701 | Variation on ~ lgsdir vali... |
| lgsdinn0 27702 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27703 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27704 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27705 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27706 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27707 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27708 | The Legendre symbol for od... |
| lgsqrmod 27709 | If the Legendre symbol of ... |
| lgsqrmodndvds 27710 | If the Legendre symbol of ... |
| lgsdchrval 27711 | The Legendre symbol functi... |
| lgsdchr 27712 | The Legendre symbol functi... |
| gausslemma2dlem0a 27713 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27714 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27715 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27716 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27717 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27718 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27719 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27720 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27721 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27722 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27723 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27724 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27725 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27726 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27727 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27728 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27729 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27730 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27731 | Gauss' Lemma (see also the... |
| lgseisenlem1 27732 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27733 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27734 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27735 | Lemma for ~ lgseisen . (C... |
| lgseisen 27736 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27737 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27738 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27739 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27740 | The Law of Quadratic Recip... |
| lgsquad2lem1 27741 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27742 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27743 | Extend ~ lgsquad to coprim... |
| lgsquad3 27744 | Extend ~ lgsquad2 to integ... |
| m1lgs 27745 | The first supplement to th... |
| 2lgslem1a1 27746 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27747 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27748 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27749 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27750 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27751 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27752 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27753 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27754 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27755 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27756 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27757 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27758 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27759 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27760 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27761 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27762 | The Legendre symbol for ` ... |
| 2lgslem4 27763 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27764 | The second supplement to t... |
| 2lgsoddprmlem1 27765 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27766 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27767 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27768 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27769 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27770 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27771 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27772 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27773 | The second supplement to t... |
| 2sqlem1 27774 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27775 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27776 | Fibonacci's identity (actu... |
| 2sqlem3 27777 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27778 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27779 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27780 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27781 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27782 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27783 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27784 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27785 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27786 | Lemma for ~ 2sq . (Contri... |
| 2sq 27787 | All primes of the form ` 4... |
| 2sqblem 27788 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27789 | The converse to ~ 2sq . (... |
| 2sq2 27790 | ` 2 ` is the sum of square... |
| 2sqn0 27791 | If the sum of two squares ... |
| 2sqcoprm 27792 | If the sum of two squares ... |
| 2sqmod 27793 | Given two decompositions o... |
| 2sqmo 27794 | There exists at most one d... |
| 2sqnn0 27795 | All primes of the form ` 4... |
| 2sqnn 27796 | All primes of the form ` 4... |
| addsq2reu 27797 | For each complex number ` ... |
| addsqn2reu 27798 | For each complex number ` ... |
| addsqrexnreu 27799 | For each complex number, t... |
| addsqnreup 27800 | There is no unique decompo... |
| addsq2nreurex 27801 | For each complex number ` ... |
| addsqn2reurex2 27802 | For each complex number ` ... |
| 2sqreulem1 27803 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27804 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27805 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27806 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27807 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27808 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27809 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27810 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27811 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27812 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27813 | There exists a unique deco... |
| 2sqreunn 27814 | There exists a unique deco... |
| 2sqreult 27815 | There exists a unique deco... |
| 2sqreultb 27816 | There exists a unique deco... |
| 2sqreunnlt 27817 | There exists a unique deco... |
| 2sqreunnltb 27818 | There exists a unique deco... |
| 2sqreuop 27819 | There exists a unique deco... |
| 2sqreuopnn 27820 | There exists a unique deco... |
| 2sqreuoplt 27821 | There exists a unique deco... |
| 2sqreuopltb 27822 | There exists a unique deco... |
| 2sqreuopnnlt 27823 | There exists a unique deco... |
| 2sqreuopnnltb 27824 | There exists a unique deco... |
| 2sqreuopb 27825 | There exists a unique deco... |
| chebbnd1lem1 27826 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27827 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27828 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27829 | The Chebyshev bound: The ... |
| chtppilimlem1 27830 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27831 | Lemma for ~ chtppilim . (... |
| chtppilim 27832 | The ` theta ` function is ... |
| chto1ub 27833 | The ` theta ` function is ... |
| chebbnd2 27834 | The Chebyshev bound, part ... |
| chto1lb 27835 | The ` theta ` function is ... |
| chpchtlim 27836 | The ` psi ` and ` theta ` ... |
| chpo1ub 27837 | The ` psi ` function is up... |
| chpo1ubb 27838 | The ` psi ` function is up... |
| vmadivsum 27839 | The sum of the von Mangold... |
| vmadivsumb 27840 | Give a total bound on the ... |
| rplogsumlem1 27841 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27842 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27843 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27844 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27845 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27846 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27847 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27848 | Lemma for ~ dchrisum . Le... |
| dchrisum 27849 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27850 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27851 | The sum of the Möbius... |
| dchrvmasumlem1 27852 | An alternative expression ... |
| dchrvmasum2lem 27853 | Give an expression for ` l... |
| dchrvmasum2if 27854 | Combine the results of ~ d... |
| dchrvmasumlem2 27855 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27856 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27857 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27858 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27859 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27860 | An asymptotic approximatio... |
| dchrvmaeq0 27861 | The set ` W ` is the colle... |
| dchrisum0fval 27862 | Value of the function ` F ... |
| dchrisum0fmul 27863 | The function ` F ` , the d... |
| dchrisum0ff 27864 | The function ` F ` is a re... |
| dchrisum0flblem1 27865 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27866 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27867 | The divisor sum of a real ... |
| dchrisum0fno1 27868 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27869 | A partial result along the... |
| dchrisum0re 27870 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27871 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27872 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27873 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27874 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27875 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27876 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27877 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27878 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27879 | The sum of the Möbius... |
| dchrvmasumlem 27880 | The sum of the Möbius... |
| dchrmusum 27881 | The sum of the Möbius... |
| dchrvmasum 27882 | The sum of the von Mangold... |
| rpvmasum 27883 | The sum of the von Mangold... |
| rplogsum 27884 | The sum of ` log p / p ` o... |
| dirith2 27885 | Dirichlet's theorem: there... |
| dirith 27886 | Dirichlet's theorem: there... |
| mudivsum 27887 | Asymptotic formula for ` s... |
| mulogsumlem 27888 | Lemma for ~ mulogsum . (C... |
| mulogsum 27889 | Asymptotic formula for ... |
| logdivsum 27890 | Asymptotic analysis of ... |
| mulog2sumlem1 27891 | Asymptotic formula for ... |
| mulog2sumlem2 27892 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27893 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27894 | Asymptotic formula for ... |
| vmalogdivsum2 27895 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27896 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27897 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27898 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27899 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27900 | The Möbius inverse of... |
| log2sumbnd 27901 | Bound on the difference be... |
| selberglem1 27902 | Lemma for ~ selberg . Est... |
| selberglem2 27903 | Lemma for ~ selberg . (Co... |
| selberglem3 27904 | Lemma for ~ selberg . Est... |
| selberg 27905 | Selberg's symmetry formula... |
| selbergb 27906 | Convert eventual boundedne... |
| selberg2lem 27907 | Lemma for ~ selberg2 . Eq... |
| selberg2 27908 | Selberg's symmetry formula... |
| selberg2b 27909 | Convert eventual boundedne... |
| chpdifbndlem1 27910 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27911 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27912 | A bound on the difference ... |
| logdivbnd 27913 | A bound on a sum of logs, ... |
| selberg3lem1 27914 | Introduce a log weighting ... |
| selberg3lem2 27915 | Lemma for ~ selberg3 . Eq... |
| selberg3 27916 | Introduce a log weighting ... |
| selberg4lem1 27917 | Lemma for ~ selberg4 . Eq... |
| selberg4 27918 | The Selberg symmetry formu... |
| pntrval 27919 | Define the residual of the... |
| pntrf 27920 | Functionality of the resid... |
| pntrmax 27921 | There is a bound on the re... |
| pntrsumo1 27922 | A bound on a sum over ` R ... |
| pntrsumbnd 27923 | A bound on a sum over ` R ... |
| pntrsumbnd2 27924 | A bound on a sum over ` R ... |
| selbergr 27925 | Selberg's symmetry formula... |
| selberg3r 27926 | Selberg's symmetry formula... |
| selberg4r 27927 | Selberg's symmetry formula... |
| selberg34r 27928 | The sum of ~ selberg3r and... |
| pntsval 27929 | Define the "Selberg functi... |
| pntsf 27930 | Functionality of the Selbe... |
| selbergs 27931 | Selberg's symmetry formula... |
| selbergsb 27932 | Selberg's symmetry formula... |
| pntsval2 27933 | The Selberg function can b... |
| pntrlog2bndlem1 27934 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27935 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27936 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27937 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27938 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27939 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27940 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27941 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27942 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27943 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27944 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27945 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27946 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27947 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27948 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27949 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27950 | Lemma for ~ pnt . Establi... |
| pntlemd 27951 | Lemma for ~ pnt . Closure... |
| pntlemc 27952 | Lemma for ~ pnt . Closure... |
| pntlema 27953 | Lemma for ~ pnt . Closure... |
| pntlemb 27954 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27955 | Lemma for ~ pnt . Closure... |
| pntlemh 27956 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27957 | Lemma for ~ pnt . The "na... |
| pntlemq 27958 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27959 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27960 | Lemma for ~ pnt . The ind... |
| pntlemi 27961 | Lemma for ~ pnt . Elimina... |
| pntlemf 27962 | Lemma for ~ pnt . Add up ... |
| pntlemk 27963 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27964 | Lemma for ~ pnt . Combine... |
| pntleme 27965 | Lemma for ~ pnt . Package... |
| pntlem3 27966 | Lemma for ~ pnt . Equatio... |
| pntlemp 27967 | Lemma for ~ pnt . Wrappin... |
| pntleml 27968 | Lemma for ~ pnt . Equatio... |
| pnt3 27969 | The Prime Number Theorem, ... |
| pnt2 27970 | The Prime Number Theorem, ... |
| pnt 27971 | The Prime Number Theorem: ... |
| abvcxp 27972 | Raising an absolute value ... |
| padicfval 27973 | Value of the p-adic absolu... |
| padicval 27974 | Value of the p-adic absolu... |
| ostth2lem1 27975 | Lemma for ~ ostth2 , altho... |
| qrngbas 27976 | The base set of the field ... |
| qdrng 27977 | The rationals form a divis... |
| qrng0 27978 | The zero element of the fi... |
| qrng1 27979 | The unity element of the f... |
| qrngneg 27980 | The additive inverse in th... |
| qrngdiv 27981 | The division operation in ... |
| qabvle 27982 | By using induction on ` N ... |
| qabvexp 27983 | Induct the product rule ~ ... |
| ostthlem1 27984 | Lemma for ~ ostth . If tw... |
| ostthlem2 27985 | Lemma for ~ ostth . Refin... |
| qabsabv 27986 | The regular absolute value... |
| padicabv 27987 | The p-adic absolute value ... |
| padicabvf 27988 | The p-adic absolute value ... |
| padicabvcxp 27989 | All positive powers of the... |
| ostth1 27990 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27991 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27992 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27993 | Lemma for ~ ostth2 . (Con... |
| ostth2 27994 | - Lemma for ~ ostth : regu... |
| ostth3 27995 | - Lemma for ~ ostth : p-ad... |
| ostth 27996 | Ostrowski's theorem, which... |
| infdesc 27997 | Infinite descent. The hyp... |
| fltdiv 27998 | A counterexample to FLT st... |
| flt0 27999 | A counterexample for FLT d... |
| fltdvdsabdvdsc 28000 | Any factor of both ` A ` a... |
| fltabcoprmex 28001 | A counterexample to FLT im... |
| fltaccoprm 28002 | A counterexample to FLT wi... |
| fltbccoprm 28003 | A counterexample to FLT wi... |
| fltabcoprm 28004 | A counterexample to FLT wi... |
| fltne 28005 | If a counterexample to FLT... |
| flt4lem1 28006 | Satisfy the antecedent use... |
| flt4lem2 28007 | If ` A ` is even, ` B ` is... |
| flt4lem3 28008 | Equivalent to ~ pythagtrip... |
| flt4lem4 28009 | If the product of two copr... |
| flt4lem5 28010 | In the context of the lemm... |
| flt4lem5elem 28011 | Version of ~ fltaccoprm an... |
| flt4lem5a 28012 | Part 1 of Equation 1 of ... |
| flt4lem5b 28013 | Part 2 of Equation 1 of ... |
| flt4lem5c 28014 | Part 2 of Equation 2 of ... |
| flt4lem5d 28015 | Part 3 of Equation 2 of ... |
| flt4lem5e 28016 | Satisfy the hypotheses of ... |
| flt4lem5f 28017 | Final equation of ~... |
| flt4lem6 28018 | Remove shared factors in a... |
| flt4lem7 28019 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 28020 | Strengthening of Fermat's ... |
| flt4 28021 | Fermat's last theorem for ... |
| flt4ALT 28022 | Fermat's last theorem for ... |
| fltoprmlem1 28023 | Lemma 1 for ~ fltoprm : E... |
| fltoprmlem2 28024 | Lemma 2 for ~ fltoprm : I... |
| fltoprm 28025 | Fermat's last theorem hold... |
| fltoprmgt3 28026 | Fermat's last theorem hold... |
| elno 28033 | Membership in the surreals... |
| ltsval 28034 | The value of the surreal l... |
| bdayval 28035 | The value of the birthday ... |
| nofun 28036 | A surreal is a function. ... |
| nodmon 28037 | The domain of a surreal is... |
| norn 28038 | The range of a surreal is ... |
| nofnbday 28039 | A surreal is a function ov... |
| nodmord 28040 | The domain of a surreal ha... |
| elno2 28041 | An alternative condition f... |
| elno3 28042 | Another condition for memb... |
| ltsval2 28043 | Alternate expression for s... |
| nofv 28044 | The function value of a su... |
| nosgnn0 28045 | ` (/) ` is not a surreal s... |
| nosgnn0i 28046 | If ` X ` is a surreal sign... |
| noreson 28047 | The restriction of a surre... |
| ltsintdifex 28048 |
If ` A |
| ltsres 28049 | If the restrictions of two... |
| noxp1o 28050 | The Cartesian product of a... |
| noseponlem 28051 | Lemma for ~ nosepon . Con... |
| nosepon 28052 | Given two unequal surreals... |
| noextend 28053 | Extending a surreal by one... |
| noextendseq 28054 | Extend a surreal by a sequ... |
| noextenddif 28055 | Calculate the place where ... |
| noextendlt 28056 | Extending a surreal with a... |
| noextendgt 28057 | Extending a surreal with a... |
| nolesgn2o 28058 | Given ` A ` less-than or e... |
| nolesgn2ores 28059 | Given ` A ` less-than or e... |
| nogesgn1o 28060 | Given ` A ` greater than o... |
| nogesgn1ores 28061 | Given ` A ` greater than o... |
| ltssolem1 28062 | Lemma for ~ ltsso . The "... |
| ltsso 28063 | Less-than totally orders t... |
| bdayfo 28064 | The birthday function maps... |
| fvnobday 28065 | The value of a surreal at ... |
| nosepnelem 28066 | Lemma for ~ nosepne . (Co... |
| nosepne 28067 | The value of two non-equal... |
| nosep1o 28068 | If the value of a surreal ... |
| nosep2o 28069 | If the value of a surreal ... |
| nosepdmlem 28070 | Lemma for ~ nosepdm . (Co... |
| nosepdm 28071 | The first place two surrea... |
| nosepeq 28072 | The values of two surreals... |
| nosepssdm 28073 | Given two non-equal surrea... |
| nodenselem4 28074 | Lemma for ~ nodense . Sho... |
| nodenselem5 28075 | Lemma for ~ nodense . If ... |
| nodenselem6 28076 | The restriction of a surre... |
| nodenselem7 28077 | Lemma for ~ nodense . ` A ... |
| nodenselem8 28078 | Lemma for ~ nodense . Giv... |
| nodense 28079 | Given two distinct surreal... |
| bdayimaon 28080 | Lemma for full-eta propert... |
| nolt02olem 28081 | Lemma for ~ nolt02o . If ... |
| nolt02o 28082 | Given ` A ` less-than ` B ... |
| nogt01o 28083 | Given ` A ` greater than `... |
| noresle 28084 | Restriction law for surrea... |
| nomaxmo 28085 | A class of surreals has at... |
| nominmo 28086 | A class of surreals has at... |
| nosupprefixmo 28087 | In any class of surreals, ... |
| noinfprefixmo 28088 | In any class of surreals, ... |
| nosupcbv 28089 | Lemma to change bound vari... |
| nosupno 28090 | The next several theorems ... |
| nosupdm 28091 | The domain of the surreal ... |
| nosupbday 28092 | Birthday bounding law for ... |
| nosupfv 28093 | The value of surreal supre... |
| nosupres 28094 | A restriction law for surr... |
| nosupbnd1lem1 28095 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 28096 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 28097 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 28098 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 28099 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 28100 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 28101 | Bounding law from below fo... |
| nosupbnd2lem1 28102 | Bounding law from above wh... |
| nosupbnd2 28103 | Bounding law from above fo... |
| noinfcbv 28104 | Change bound variables for... |
| noinfno 28105 | The next several theorems ... |
| noinfdm 28106 | Next, we calculate the dom... |
| noinfbday 28107 | Birthday bounding law for ... |
| noinffv 28108 | The value of surreal infim... |
| noinfres 28109 | The restriction of surreal... |
| noinfbnd1lem1 28110 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 28111 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 28112 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 28113 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 28114 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 28115 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 28116 | Bounding law from above fo... |
| noinfbnd2lem1 28117 | Bounding law from below wh... |
| noinfbnd2 28118 | Bounding law from below fo... |
| nosupinfsep 28119 | Given two sets of surreals... |
| noetasuplem1 28120 | Lemma for ~ noeta . Estab... |
| noetasuplem2 28121 | Lemma for ~ noeta . The r... |
| noetasuplem3 28122 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 28123 | Lemma for ~ noeta . When ... |
| noetainflem1 28124 | Lemma for ~ noeta . Estab... |
| noetainflem2 28125 | Lemma for ~ noeta . The r... |
| noetainflem3 28126 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 28127 | Lemma for ~ noeta . If ` ... |
| noetalem1 28128 | Lemma for ~ noeta . Eithe... |
| noetalem2 28129 | Lemma for ~ noeta . The f... |
| noeta 28130 | The full-eta axiom for the... |
| ltsirr 28133 | Surreal less-than is irref... |
| ltstr 28134 | Surreal less-than is trans... |
| ltsasym 28135 | Surreal less-than is asymm... |
| ltslin 28136 | Surreal less-than obeys tr... |
| ltstrieq2 28137 | Trichotomy law for surreal... |
| ltstrine 28138 | Trichotomy law for surreal... |
| lenlts 28139 | Surreal less-than or equal... |
| ltnles 28140 | Surreal less-than in terms... |
| lesloe 28141 | Surreal less-than or equal... |
| lestri3 28142 | Trichotomy law for surreal... |
| lesnltd 28143 | Surreal less-than or equal... |
| ltsnled 28144 | Surreal less-than in terms... |
| lesloed 28145 | Surreal less-than or equal... |
| lestri3d 28146 | Trichotomy law for surreal... |
| ltlestr 28147 | Surreal transitive law. (... |
| leltstr 28148 | Surreal transitive law. (... |
| lestr 28149 | Surreal transitive law. (... |
| ltstrd 28150 | Surreal less-than is trans... |
| ltlestrd 28151 | Surreal less-than is trans... |
| leltstrd 28152 | Surreal less-than is trans... |
| lestrd 28153 | Surreal less-than or equal... |
| lesid 28154 | Surreal less-than or equal... |
| lestric 28155 | Surreal trichotomy law. (... |
| maxs1 28156 | A surreal is less than or ... |
| maxs2 28157 | A surreal is less than or ... |
| mins1 28158 | The minimum of two surreal... |
| mins2 28159 | The minimum of two surreal... |
| ltlesd 28160 | Surreal less-than implies ... |
| ltsne 28161 | Surreal less-than implies ... |
| ltlesnd 28162 | Surreal less-than in terms... |
| bdayfun 28163 | The birthday function is a... |
| bdayfn 28164 | The birthday function is a... |
| bdaydm 28165 | The birthday function's do... |
| bdaydmOLD 28166 | Obsolete version of ~ bday... |
| bdayrn 28167 | The birthday function's ra... |
| bdayon 28168 | The value of the birthday ... |
| nobdaymin 28169 | Any non-empty class of sur... |
| nocvxminlem 28170 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 28171 | Given a nonempty convex cl... |
| noprc 28172 | The surreal numbers are a ... |
| noeta2 28177 | A version of ~ noeta with ... |
| brslts 28178 | Binary relation form of th... |
| sltsex1 28179 | The first argument of surr... |
| sltsex2 28180 | The second argument of sur... |
| sltsss1 28181 | The first argument of surr... |
| sltsss2 28182 | The second argument of sur... |
| sltssep 28183 | The separation property of... |
| sltsd 28184 | Deduce surreal set less-th... |
| sltssnb 28185 | Surreal set less-than of t... |
| sltssn 28186 | Surreal set less-than of t... |
| sltssepc 28187 | Two elements of separated ... |
| sltssepcd 28188 | Two elements of separated ... |
| ssslts1 28189 | Relation between surreal s... |
| ssslts2 28190 | Relation between surreal s... |
| nulslts 28191 | The empty set is less-than... |
| nulsgts 28192 | The empty set is greater t... |
| nulsltsd 28193 | The empty set is less-than... |
| nulsgtsd 28194 | The empty set is greater t... |
| conway 28195 | Conway's Simplicity Theore... |
| cutsval 28196 | The value of the surreal c... |
| cutcuts 28197 | Cut properties of the surr... |
| cutscl 28198 | Closure law for surreal cu... |
| cutscld 28199 | Closure law for surreal cu... |
| cutbday 28200 | The birthday of the surrea... |
| eqcuts 28201 | Condition for equality to ... |
| eqcuts2 28202 | Condition for equality to ... |
| sltstr 28203 | Transitive law for surreal... |
| sltsun1 28204 | Union law for surreal set ... |
| sltsun2 28205 | Union law for surreal set ... |
| cutsun12 28206 | Union law for surreal cuts... |
| dmcuts 28207 | The domain of the surreal ... |
| cutsf 28208 | Functionality statement fo... |
| etaslts 28209 | A restatement of ~ noeta u... |
| etaslts2 28210 | A version of ~ etaslts wit... |
| cutbdaybnd 28211 | An upper bound on the birt... |
| cutbdaybnd2 28212 | An upper bound on the birt... |
| cutbdaybnd2lim 28213 | An upper bound on the birt... |
| cutbdaylt 28214 | If a surreal lies in a gap... |
| lesrec 28215 | A comparison law for surre... |
| lesrecd 28216 | A comparison law for surre... |
| ltsrec 28217 | A comparison law for surre... |
| ltsrecd 28218 | A comparison law for surre... |
| sltsdisj 28219 | If ` A ` preceeds ` B ` , ... |
| eqcuts3 28220 | A variant of the simplicit... |
| 0no 28225 | Surreal zero is a surreal.... |
| 1no 28226 | Surreal one is a surreal. ... |
| bday0 28227 | Calculate the birthday of ... |
| 0lt1s 28228 | Surreal zero is less than ... |
| bday0b 28229 | The only surreal with birt... |
| bday1 28230 | The birthday of surreal on... |
| cuteq0 28231 | Condition for a surreal cu... |
| cutneg 28232 | The simplest number greate... |
| cuteq1 28233 | Condition for a surreal cu... |
| gt0ne0s 28234 | A positive surreal is not ... |
| gt0ne0sd 28235 | A positive surreal is not ... |
| 1ne0s 28236 | Surreal zero does not equa... |
| rightge0 28237 | A surreal is non-negative ... |
| madeval 28248 | The value of the made by f... |
| madeval2 28249 | Alternative characterizati... |
| oldval 28250 | The value of the old optio... |
| newval 28251 | The value of the new optio... |
| madef 28252 | The made function is a fun... |
| oldf 28253 | The older function is a fu... |
| newf 28254 | The new function is a func... |
| old0 28255 | No surreal is older than `... |
| madessno 28256 | Made sets are surreals. (... |
| oldssno 28257 | Old sets are surreals. (C... |
| newssno 28258 | New sets are surreals. (C... |
| madeno 28259 | An element of a made set i... |
| oldno 28260 | An element of an old set i... |
| newno 28261 | An element of a new set is... |
| madenod 28262 | An element of a made set i... |
| oldnod 28263 | An element of an old set i... |
| newnod 28264 | An element of a new set is... |
| leftval 28265 | The value of the left opti... |
| rightval 28266 | The value of the right opt... |
| elleft 28267 | Membership in the left set... |
| elright 28268 | Membership in the right se... |
| leftlt 28269 | A member of a surreal's le... |
| rightgt 28270 | A member of a surreal's ri... |
| leftf 28271 | The functionality of the l... |
| rightf 28272 | The functionality of the r... |
| elmade 28273 | Membership in the made fun... |
| elmade2 28274 | Membership in the made fun... |
| elold 28275 | Membership in an old set. ... |
| sltsleft 28276 | A surreal is greater than ... |
| sltsright 28277 | A surreal is less than its... |
| lltr 28278 | The left options of a surr... |
| made0 28279 | The only surreal made on d... |
| new0 28280 | The only surreal new on da... |
| old1 28281 | The only surreal older tha... |
| madess 28282 | If ` A ` is less than or e... |
| oldssmade 28283 | The older-than set is a su... |
| oldmade 28284 | An element of an old set i... |
| oldmaded 28285 | An element of an old set i... |
| oldss 28286 | If ` A ` is less than or e... |
| leftssold 28287 | The left options are a sub... |
| rightssold 28288 | The right options are a su... |
| leftssno 28289 | The left set of a surreal ... |
| rightssno 28290 | The right set of a surreal... |
| leftold 28291 | An element of a left set i... |
| rightold 28292 | An element of a right set ... |
| leftno 28293 | An element of a left set i... |
| rightno 28294 | An element of a right set ... |
| leftoldd 28295 | An element of a left set i... |
| leftnod 28296 | An element of a left set i... |
| rightoldd 28297 | An element of a right set ... |
| rightnod 28298 | An element of a right set ... |
| madecut 28299 | Given a section that is a ... |
| madeun 28300 | The made set is the union ... |
| madeoldsuc 28301 | The made set is the old se... |
| oldsuc 28302 | The value of the old set a... |
| oldlim 28303 | The value of the old set a... |
| madebdayim 28304 | If a surreal is a member o... |
| oldbdayim 28305 | If ` X ` is in the old set... |
| oldirr 28306 | No surreal is a member of ... |
| leftirr 28307 | No surreal is a member of ... |
| rightirr 28308 | No surreal is a member of ... |
| left0s 28309 | The left set of ` 0s ` is ... |
| right0s 28310 | The right set of ` 0s ` is... |
| left1s 28311 | The left set of ` 1s ` is ... |
| right1s 28312 | The right set of ` 1s ` is... |
| lrold 28313 | The union of the left and ... |
| madebdaylemold 28314 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 28315 | Lemma for ~ madebday . If... |
| madebday 28316 | A surreal is part of the s... |
| oldbday 28317 | A surreal is part of the s... |
| newbday 28318 | A surreal is an element of... |
| newbdayim 28319 | One direction of the bicon... |
| lrcut 28320 | A surreal is equal to the ... |
| cutsfo 28321 | The surreal cut function i... |
| ltsn0 28322 | If ` X ` is less than ` Y ... |
| lruneq 28323 | If two surreals share a bi... |
| ltslpss 28324 | If two surreals share a bi... |
| leslss 28325 | If two surreals ` A ` and ... |
| 0elold 28326 | Zero is in the old set of ... |
| 0elleft 28327 | Zero is in the left set of... |
| 0elright 28328 | Zero is in the right set o... |
| madefi 28329 | The made set of an ordinal... |
| oldfi 28330 | The old set of an ordinal ... |
| bdayiun 28331 | The birthday of a surreal ... |
| bdayle 28332 | A condition for bounding a... |
| sltsbday 28333 | Birthday comparison rule f... |
| cofslts 28334 | If every element of ` A ` ... |
| coinitslts 28335 | If ` B ` is coinitial with... |
| cofcut1 28336 | If ` C ` is cofinal with `... |
| cofcut1d 28337 | If ` C ` is cofinal with `... |
| cofcut2 28338 | If ` A ` and ` C ` are mut... |
| cofcut2d 28339 | If ` A ` and ` C ` are mut... |
| cofcutr 28340 | If ` X ` is the cut of ` A... |
| cofcutr1d 28341 | If ` X ` is the cut of ` A... |
| cofcutr2d 28342 | If ` X ` is the cut of ` A... |
| cofcutrtime 28343 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 28344 | If ` X ` is a timely cut o... |
| cofcutrtime2d 28345 | If ` X ` is a timely cut o... |
| cofss 28346 | Cofinality for a subset. ... |
| coiniss 28347 | Coinitiality for a subset.... |
| cutlt 28348 | Eliminating all elements b... |
| cutpos 28349 | Reduce the elements of a c... |
| cutmax 28350 | If ` A ` has a maximum, th... |
| cutmin 28351 | If ` B ` has a minimum, th... |
| cutminmax 28352 | If the left set of ` X ` h... |
| lrrecval 28355 | The next step in the devel... |
| lrrecval2 28356 | Next, we establish an alte... |
| lrrecpo 28357 | Now, we establish that ` R... |
| lrrecse 28358 | Next, we show that ` R ` i... |
| lrrecfr 28359 | Now we show that ` R ` is ... |
| lrrecpred 28360 | Finally, we calculate the ... |
| noinds 28361 | Induction principle for a ... |
| norecfn 28362 | Surreal recursion over one... |
| norecov 28363 | Calculate the value of the... |
| noxpordpo 28366 | To get through most of the... |
| noxpordfr 28367 | Next we establish the foun... |
| noxpordse 28368 | Next we establish the set-... |
| noxpordpred 28369 | Next we calculate the pred... |
| no2indlesm 28370 | Double induction on surrea... |
| no2inds 28371 | Double induction on surrea... |
| norec2fn 28372 | The double-recursion opera... |
| norec2ov 28373 | The value of the double-re... |
| no3inds 28374 | Triple induction over surr... |
| addsfn 28377 | Surreal addition is a func... |
| addsval 28378 | The value of surreal addit... |
| addsval2 28379 | The value of surreal addit... |
| addsrid 28380 | Surreal addition to zero i... |
| addsridd 28381 | Surreal addition to zero i... |
| addscom 28382 | Surreal addition is commut... |
| addscomd 28383 | Surreal addition is commut... |
| addslid 28384 | Surreal addition to zero i... |
| addsproplem1 28385 | Lemma for surreal addition... |
| addsproplem2 28386 | Lemma for surreal addition... |
| addsproplem3 28387 | Lemma for surreal addition... |
| addsproplem4 28388 | Lemma for surreal addition... |
| addsproplem5 28389 | Lemma for surreal addition... |
| addsproplem6 28390 | Lemma for surreal addition... |
| addsproplem7 28391 | Lemma for surreal addition... |
| addsprop 28392 | Inductively show that surr... |
| addcutslem 28393 | Lemma for ~ addcuts . Sho... |
| addcuts 28394 | Demonstrate the cut proper... |
| addcuts2 28395 | Show that the cut involved... |
| addscld 28396 | Surreal numbers are closed... |
| addscl 28397 | Surreal numbers are closed... |
| addsf 28398 | Function statement for sur... |
| addsfo 28399 | Surreal addition is onto. ... |
| peano2no 28400 | A theorem for surreals tha... |
| ltadds1im 28401 | Surreal less-than is prese... |
| ltadds2im 28402 | Surreal less-than is prese... |
| leadds1im 28403 | Surreal less-than or equal... |
| leadds2im 28404 | Surreal less-than or equal... |
| leadds1 28405 | Addition to both sides of ... |
| leadds2 28406 | Addition to both sides of ... |
| ltadds2 28407 | Addition to both sides of ... |
| ltadds1 28408 | Addition to both sides of ... |
| addscan2 28409 | Cancellation law for surre... |
| addscan1 28410 | Cancellation law for surre... |
| leadds1d 28411 | Addition to both sides of ... |
| leadds2d 28412 | Addition to both sides of ... |
| ltadds2d 28413 | Addition to both sides of ... |
| ltadds1d 28414 | Addition to both sides of ... |
| addscan2d 28415 | Cancellation law for surre... |
| addscan1d 28416 | Cancellation law for surre... |
| addsuniflem 28417 | Lemma for ~ addsunif . St... |
| addsunif 28418 | Uniformity theorem for sur... |
| addsasslem1 28419 | Lemma for addition associa... |
| addsasslem2 28420 | Lemma for addition associa... |
| addsass 28421 | Surreal addition is associ... |
| addsassd 28422 | Surreal addition is associ... |
| adds32d 28423 | Commutative/associative la... |
| adds12d 28424 | Commutative/associative la... |
| adds4d 28425 | Rearrangement of four term... |
| adds42d 28426 | Rearrangement of four term... |
| ltaddspos1d 28427 | Addition of a positive num... |
| ltaddspos2d 28428 | Addition of a positive num... |
| lt2addsd 28429 | Adding both sides of two s... |
| addsgt0d 28430 | The sum of two positive su... |
| ltsp1d 28431 | A surreal is less than its... |
| addsge01d 28432 | A surreal is less-than or ... |
| addbdaylem 28433 | Lemma for ~ addbday . (Co... |
| addbday 28434 | The birthday of the sum of... |
| negsfn 28439 | Surreal negation is a func... |
| subsfn 28440 | Surreal subtraction is a f... |
| negsval 28441 | The value of the surreal n... |
| neg0s 28442 | Negative surreal zero is s... |
| neg1s 28443 | An expression for negative... |
| negsproplem1 28444 | Lemma for surreal negation... |
| negsproplem2 28445 | Lemma for surreal negation... |
| negsproplem3 28446 | Lemma for surreal negation... |
| negsproplem4 28447 | Lemma for surreal negation... |
| negsproplem5 28448 | Lemma for surreal negation... |
| negsproplem6 28449 | Lemma for surreal negation... |
| negsproplem7 28450 | Lemma for surreal negation... |
| negsprop 28451 | Show closure and ordering ... |
| negscl 28452 | The surreals are closed un... |
| negscld 28453 | The surreals are closed un... |
| ltnegsim 28454 | The forward direction of t... |
| negcut 28455 | The cut properties of surr... |
| negcut2 28456 | The cut that defines surre... |
| negsid 28457 | Surreal addition of a numb... |
| negsidd 28458 | Surreal addition of a numb... |
| negsex 28459 | Every surreal has a negati... |
| negnegs 28460 | A surreal is equal to the ... |
| ltnegs 28461 | Negative of both sides of ... |
| lenegs 28462 | Negative of both sides of ... |
| ltnegsd 28463 | Negative of both sides of ... |
| lenegsd 28464 | Negative of both sides of ... |
| negs11 28465 | Surreal negation is one-to... |
| negsdi 28466 | Distribution of surreal ne... |
| lt0negs2d 28467 | Comparison of a surreal an... |
| negsf 28468 | Function statement for sur... |
| negsfo 28469 | Function statement for sur... |
| negsf1o 28470 | Surreal negation is a bije... |
| negsunif 28471 | Uniformity property for su... |
| negbdaylem 28472 | Lemma for ~ negbday . Bou... |
| negbday 28473 | Negation of a surreal numb... |
| negleft 28474 | The left set of the negati... |
| negright 28475 | The right set of the negat... |
| subsval 28476 | The value of surreal subtr... |
| subsvald 28477 | The value of surreal subtr... |
| subscl 28478 | Closure law for surreal su... |
| subscld 28479 | Closure law for surreal su... |
| subsf 28480 | Function statement for sur... |
| subsfo 28481 | Surreal subtraction is an ... |
| negsval2 28482 | Surreal negation in terms ... |
| negsval2d 28483 | Surreal negation in terms ... |
| subsid1 28484 | Identity law for subtracti... |
| subsid 28485 | Subtraction of a surreal f... |
| subadds 28486 | Relationship between addit... |
| subaddsd 28487 | Relationship between addit... |
| pncans 28488 | Cancellation law for surre... |
| pncan3s 28489 | Subtraction and addition o... |
| pncan2s 28490 | Cancellation law for surre... |
| npcans 28491 | Cancellation law for surre... |
| ltsubs1 28492 | Subtraction from both side... |
| ltsubs2 28493 | Subtraction from both side... |
| ltsubs1d 28494 | Subtraction from both side... |
| ltsubs2d 28495 | Subtraction from both side... |
| negsubsdi2d 28496 | Distribution of negative o... |
| addsubsassd 28497 | Associative-type law for s... |
| addsubsd 28498 | Law for surreal addition a... |
| ltsubsubsbd 28499 | Equivalence for the surrea... |
| ltsubsubs2bd 28500 | Equivalence for the surrea... |
| ltsubsubs3bd 28501 | Equivalence for the surrea... |
| lesubsubsbd 28502 | Equivalence for the surrea... |
| lesubsubs2bd 28503 | Equivalence for the surrea... |
| lesubsubs3bd 28504 | Equivalence for the surrea... |
| ltsubaddsd 28505 | Surreal less-than relation... |
| ltsubadds2d 28506 | Surreal less-than relation... |
| ltaddsubsd 28507 | Surreal less-than relation... |
| ltaddsubs2d 28508 | Surreal less-than relation... |
| lesubaddsd 28509 | Surreal less-than or equal... |
| subsubs4d 28510 | Law for double surreal sub... |
| subsubs2d 28511 | Law for double surreal sub... |
| lesubsd 28512 | Swap subtrahends in a surr... |
| nncansd 28513 | Cancellation law for surre... |
| posdifsd 28514 | Comparison of two surreals... |
| ltsubsposd 28515 | Subtraction of a positive ... |
| subsge0d 28516 | Non-negative subtraction. ... |
| addsubs4d 28517 | Rearrangement of four term... |
| ltsm1d 28518 | A surreal is greater than ... |
| subscan1d 28519 | Cancellation law for surre... |
| subscan2d 28520 | Cancellation law for surre... |
| subseq0d 28521 | The difference between two... |
| mulsfn 28524 | Surreal multiplication is ... |
| mulsval 28525 | The value of surreal multi... |
| mulsval2lem 28526 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28527 | The value of surreal multi... |
| muls01 28528 | Surreal multiplication by ... |
| mulsrid 28529 | Surreal one is a right ide... |
| mulsridd 28530 | Surreal one is a right ide... |
| mulsproplemcbv 28531 | Lemma for surreal multipli... |
| mulsproplem1 28532 | Lemma for surreal multipli... |
| mulsproplem2 28533 | Lemma for surreal multipli... |
| mulsproplem3 28534 | Lemma for surreal multipli... |
| mulsproplem4 28535 | Lemma for surreal multipli... |
| mulsproplem5 28536 | Lemma for surreal multipli... |
| mulsproplem6 28537 | Lemma for surreal multipli... |
| mulsproplem7 28538 | Lemma for surreal multipli... |
| mulsproplem8 28539 | Lemma for surreal multipli... |
| mulsproplem9 28540 | Lemma for surreal multipli... |
| mulsproplem10 28541 | Lemma for surreal multipli... |
| mulsproplem11 28542 | Lemma for surreal multipli... |
| mulsproplem12 28543 | Lemma for surreal multipli... |
| mulsproplem13 28544 | Lemma for surreal multipli... |
| mulsproplem14 28545 | Lemma for surreal multipli... |
| mulsprop 28546 | Surreals are closed under ... |
| mulcutlem 28547 | Lemma for ~ mulcut . Stat... |
| mulcut 28548 | Show the cut properties of... |
| mulcut2 28549 | Show that the cut involved... |
| mulscl 28550 | The surreals are closed un... |
| mulscld 28551 | The surreals are closed un... |
| ltmuls 28552 | An ordering relationship f... |
| ltmulsd 28553 | An ordering relationship f... |
| lemulsd 28554 | An ordering relationship f... |
| mulscom 28555 | Surreal multiplication is ... |
| mulscomd 28556 | Surreal multiplication is ... |
| muls02 28557 | Surreal multiplication by ... |
| mulslid 28558 | Surreal one is a left iden... |
| mulslidd 28559 | Surreal one is a left iden... |
| mulsgt0 28560 | The product of two positiv... |
| mulsgt0d 28561 | The product of two positiv... |
| mulsge0d 28562 | The product of two non-neg... |
| sltmuls1 28563 | One surreal set less-than ... |
| sltmuls2 28564 | One surreal set less-than ... |
| mulsuniflem 28565 | Lemma for ~ mulsunif . St... |
| mulsunif 28566 | Surreal multiplication has... |
| addsdilem1 28567 | Lemma for surreal distribu... |
| addsdilem2 28568 | Lemma for surreal distribu... |
| addsdilem3 28569 | Lemma for ~ addsdi . Show... |
| addsdilem4 28570 | Lemma for ~ addsdi . Show... |
| addsdi 28571 | Distributive law for surre... |
| addsdid 28572 | Distributive law for surre... |
| addsdird 28573 | Distributive law for surre... |
| subsdid 28574 | Distribution of surreal mu... |
| subsdird 28575 | Distribution of surreal mu... |
| mulnegs1d 28576 | Product with negative is n... |
| mulnegs2d 28577 | Product with negative is n... |
| mul2negsd 28578 | Surreal product of two neg... |
| mulsasslem1 28579 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28580 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28581 | Lemma for ~ mulsass . Dem... |
| mulsass 28582 | Associative law for surrea... |
| mulsassd 28583 | Associative law for surrea... |
| muls4d 28584 | Rearrangement of four surr... |
| mulsunif2lem 28585 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28586 | Alternate expression for s... |
| ltmuls2 28587 | Multiplication of both sid... |
| ltmuls2d 28588 | Multiplication of both sid... |
| ltmuls1d 28589 | Multiplication of both sid... |
| lemuls2d 28590 | Multiplication of both sid... |
| lemuls1d 28591 | Multiplication of both sid... |
| ltmulnegs1d 28592 | Multiplication of both sid... |
| ltmulnegs2d 28593 | Multiplication of both sid... |
| mulscan2dlem 28594 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28595 | Cancellation of surreal mu... |
| mulscan1d 28596 | Cancellation of surreal mu... |
| muls12d 28597 | Commutative/associative la... |
| lemuls1ad 28598 | Multiplication of both sid... |
| ltmuls12ad 28599 | Comparison of the product ... |
| divsmo 28600 | Uniqueness of surreal inve... |
| muls0ord 28601 | If a surreal product is ze... |
| mulsne0bd 28602 | The product of two nonzero... |
| divsval 28605 | The value of surreal divis... |
| norecdiv 28606 | If a surreal has a recipro... |
| noreceuw 28607 | If a surreal has a recipro... |
| recsne0 28608 | If a surreal has a recipro... |
| divmulsw 28609 | Relationship between surre... |
| divmulswd 28610 | Relationship between surre... |
| divsclw 28611 | Weak division closure law.... |
| divsclwd 28612 | Weak division closure law.... |
| divscan2wd 28613 | A weak cancellation law fo... |
| divscan1wd 28614 | A weak cancellation law fo... |
| ltdivmulswd 28615 | Surreal less-than relation... |
| ltdivmuls2wd 28616 | Surreal less-than relation... |
| ltmuldivswd 28617 | Surreal less-than relation... |
| ltmuldivs2wd 28618 | Surreal less-than relation... |
| divsasswd 28619 | An associative law for sur... |
| divs1 28620 | A surreal divided by one i... |
| divs1d 28621 | A surreal divided by one i... |
| precsexlemcbv 28622 | Lemma for surreal reciproc... |
| precsexlem1 28623 | Lemma for surreal reciproc... |
| precsexlem2 28624 | Lemma for surreal reciproc... |
| precsexlem3 28625 | Lemma for surreal reciproc... |
| precsexlem4 28626 | Lemma for surreal reciproc... |
| precsexlem5 28627 | Lemma for surreal reciproc... |
| precsexlem6 28628 | Lemma for surreal reciproc... |
| precsexlem7 28629 | Lemma for surreal reciproc... |
| precsexlem8 28630 | Lemma for surreal reciproc... |
| precsexlem9 28631 | Lemma for surreal reciproc... |
| precsexlem10 28632 | Lemma for surreal reciproc... |
| precsexlem11 28633 | Lemma for surreal reciproc... |
| precsex 28634 | Every positive surreal has... |
| recsex 28635 | A nonzero surreal has a re... |
| recsexd 28636 | A nonzero surreal has a re... |
| divmuls 28637 | Relationship between surre... |
| divmulsd 28638 | Relationship between surre... |
| divscl 28639 | Surreal division closure l... |
| divscld 28640 | Surreal division closure l... |
| divscan2d 28641 | A cancellation law for sur... |
| divscan1d 28642 | A cancellation law for sur... |
| ltdivmulsd 28643 | Surreal less-than relation... |
| ltdivmuls2d 28644 | Surreal less-than relation... |
| ltmuldivsd 28645 | Surreal less-than relation... |
| ltmuldivs2d 28646 | Surreal less-than relation... |
| divsassd 28647 | An associative law for sur... |
| divmuldivsd 28648 | Multiplication of two surr... |
| divdivs1d 28649 | Surreal division into a fr... |
| divsrecd 28650 | Relationship between surre... |
| divsdird 28651 | Distribution of surreal di... |
| divscan3d 28652 | A cancellation law for sur... |
| abssval 28655 | The value of surreal absol... |
| absscl 28656 | Closure law for surreal ab... |
| abssid 28657 | The absolute value of a no... |
| abs0s 28658 | The absolute value of surr... |
| abssnid 28659 | For a negative surreal, it... |
| absmuls 28660 | Surreal absolute value dis... |
| abssge0 28661 | The absolute value of a su... |
| abssor 28662 | The absolute value of a su... |
| absnegs 28663 | Surreal absolute value of ... |
| leabss 28664 | A surreal is less than or ... |
| abslts 28665 | Surreal absolute value and... |
| abssubs 28666 | Swapping order of surreal ... |
| elons 28669 | Membership in the class of... |
| onssno 28670 | The surreal ordinals are a... |
| onno 28671 | A surreal ordinal is a sur... |
| 0ons 28672 | Surreal zero is a surreal ... |
| 1ons 28673 | Surreal one is a surreal o... |
| elons2 28674 | A surreal is ordinal iff i... |
| elons2d 28675 | The cut of any set of surr... |
| onleft 28676 | The left set of a surreal ... |
| ltonold 28677 | The class of ordinals less... |
| ltonsex 28678 | The class of ordinals less... |
| oncutleft 28679 | A surreal ordinal is equal... |
| oncutlt 28680 | A surreal ordinal is the s... |
| bday11on 28681 | The birthday function is o... |
| onnolt 28682 | If a surreal ordinal is le... |
| onlts 28683 | Less-than is the same as b... |
| onles 28684 | Less-than or equal is the ... |
| onltsd 28685 | Less-than is the same as b... |
| onlesd 28686 | Less-than or equal is the ... |
| oniso 28687 | The birthday function rest... |
| onswe 28688 | Surreal less-than well-ord... |
| onsse 28689 | Surreal less-than is set-l... |
| onsis 28690 | Transfinite induction sche... |
| ons2ind 28691 | Double induction schema fo... |
| bdayons 28692 | The birthday of a surreal ... |
| onaddscl 28693 | The surreal ordinals are c... |
| onmulscl 28694 | The surreal ordinals are c... |
| addonbday 28695 | The birthday of the sum of... |
| peano2ons 28696 | The successor of a surreal... |
| onsbnd 28697 | The surreals of a given bi... |
| onsbnd2 28698 | The surreals of a given bi... |
| seqsex 28701 | Existence of the surreal s... |
| seqseq123d 28702 | Equality deduction for the... |
| nfseqs 28703 | Hypothesis builder for the... |
| seqsval 28704 | The value of the surreal s... |
| noseqex 28705 | The next several theorems ... |
| noseq0 28706 | The surreal ` A ` is a mem... |
| noseqp1 28707 | One plus an element of ` Z... |
| noseqind 28708 | Peano's inductive postulat... |
| noseqinds 28709 | Induction schema for surre... |
| noseqssno 28710 | A surreal sequence is a su... |
| noseqno 28711 | An element of a surreal se... |
| om2noseq0 28712 | The mapping ` G ` is a one... |
| om2noseqsuc 28713 | The value of ` G ` at a su... |
| om2noseqfo 28714 | Function statement for ` G... |
| om2noseqlt 28715 | Surreal less-than relation... |
| om2noseqlt2 28716 | The mapping ` G ` preserve... |
| om2noseqf1o 28717 | ` G ` is a bijection. (Co... |
| om2noseqiso 28718 | ` G ` is an isomorphism fr... |
| om2noseqoi 28719 | An alternative definition ... |
| om2noseqrdg 28720 | A helper lemma for the val... |
| noseqrdglem 28721 | A helper lemma for the val... |
| noseqrdgfn 28722 | The recursive definition g... |
| noseqrdg0 28723 | Initial value of a recursi... |
| noseqrdgsuc 28724 | Successor value of a recur... |
| seqsfn 28725 | The surreal sequence build... |
| seqs1 28726 | The value of the surreal s... |
| seqsp1 28727 | The value of the surreal s... |
| n0sexg 28732 | The set of all non-negativ... |
| n0sex 28733 | The set of all non-negativ... |
| nnsex 28734 | The set of all positive su... |
| peano5n0s 28735 | Peano's inductive postulat... |
| n0ssno 28736 | The non-negative surreal i... |
| nnssn0s 28737 | The positive surreal integ... |
| nnssno 28738 | The positive surreal integ... |
| n0no 28739 | A non-negative surreal int... |
| nnno 28740 | A positive surreal integer... |
| n0nod 28741 | A non-negative surreal int... |
| nnnod 28742 | A positive surreal integer... |
| nnn0s 28743 | A positive surreal integer... |
| nnn0sd 28744 | A positive surreal integer... |
| 0n0s 28745 | Peano postulate: ` 0s ` is... |
| peano2n0s 28746 | Peano postulate: the succe... |
| peano2n0sd 28747 | Peano postulate: the succe... |
| dfn0s2 28748 | Alternate definition of th... |
| n0sind 28749 | Principle of Mathematical ... |
| n0cut 28750 | A cut form for non-negativ... |
| n0cut2 28751 | A cut form for the success... |
| n0on 28752 | A surreal natural is a sur... |
| nnne0s 28753 | A surreal positive integer... |
| n0sge0 28754 | A non-negative integer is ... |
| nnsgt0 28755 | A positive integer is grea... |
| elnns 28756 | Membership in the positive... |
| elnns2 28757 | A positive surreal integer... |
| n0s0suc 28758 | A non-negative surreal int... |
| nnsge1 28759 | A positive surreal integer... |
| n0addscl 28760 | The non-negative surreal i... |
| n0mulscl 28761 | The non-negative surreal i... |
| nnaddscl 28762 | The positive surreal integ... |
| nnmulscl 28763 | The positive surreal integ... |
| 1n0s 28764 | Surreal one is a non-negat... |
| 1nns 28765 | Surreal one is a positive ... |
| peano2nns 28766 | Peano postulate for positi... |
| nnsrecgt0d 28767 | The reciprocal of a positi... |
| n0bday 28768 | A non-negative surreal int... |
| n0ssoldg 28769 | The non-negative surreal i... |
| n0ssold 28770 | The non-negative surreal i... |
| n0fincut 28771 | The simplest number greate... |
| onsfi 28772 | A surreal ordinal with a f... |
| eln0s2 28773 | A non-negative surreal int... |
| onltn0s 28774 | A surreal ordinal that is ... |
| n0cutlt 28775 | A non-negative surreal int... |
| seqn0sfn 28776 | The surreal sequence build... |
| eln0s 28777 | A non-negative surreal int... |
| n0s0m1 28778 | Every non-negative surreal... |
| n0subs 28779 | Subtraction of non-negativ... |
| n0subs2 28780 | Subtraction of non-negativ... |
| n0ltsp1le 28781 | Non-negative surreal order... |
| n0lesltp1 28782 | Non-negative surreal order... |
| n0lesm1lt 28783 | Non-negative surreal order... |
| n0lts1e0 28784 | A non-negative surreal int... |
| bdayn0p1 28785 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28786 | The birthday function rest... |
| n0p1nns 28787 | One plus a non-negative su... |
| dfnns2 28788 | Alternate definition of th... |
| nnsind 28789 | Principle of Mathematical ... |
| nn1m1nns 28790 | Every positive surreal int... |
| nnm1n0s 28791 | A positive surreal integer... |
| eucliddivs 28792 | Euclid's division lemma fo... |
| oldfib 28793 | The old set of an ordinal ... |
| zsex 28796 | The surreal integers form ... |
| zssno 28797 | The surreal integers are a... |
| zno 28798 | A surreal integer is a sur... |
| znod 28799 | A surreal integer is a sur... |
| elzs 28800 | Membership in the set of s... |
| nnzsubs 28801 | The difference of two surr... |
| nnzs 28802 | A positive surreal integer... |
| nnzsd 28803 | A positive surreal integer... |
| 0zs 28804 | Zero is a surreal integer.... |
| n0zs 28805 | A non-negative surreal int... |
| n0zsd 28806 | A non-negative surreal int... |
| 1zs 28807 | One is a surreal integer. ... |
| znegscl 28808 | The surreal integers are c... |
| znegscld 28809 | The surreal integers are c... |
| zaddscl 28810 | The surreal integers are c... |
| zaddscld 28811 | The surreal integers are c... |
| zsubscld 28812 | The surreal integers are c... |
| zmulscld 28813 | The surreal integers are c... |
| elzn0s 28814 | A surreal integer is a sur... |
| elzs2 28815 | A surreal integer is eithe... |
| eln0zs 28816 | Non-negative surreal integ... |
| elnnzs 28817 | Positive surreal integer p... |
| elznns 28818 | Surreal integer property e... |
| zn0subs 28819 | The non-negative differenc... |
| peano5uzs 28820 | Peano's inductive postulat... |
| uzsind 28821 | Induction on the upper sur... |
| zsbday 28822 | A surreal integer has a fi... |
| zcuts 28823 | A cut expression for surre... |
| zcuts0 28824 | Either the left or right s... |
| zsoring 28825 | The surreal integers form ... |
| 1p1e2s 28832 | One plus one is two. Surr... |
| no2times 28833 | Version of ~ 2times for su... |
| 2nns 28834 | Surreal two is a surreal n... |
| 2no 28835 | Surreal two is a surreal n... |
| 2ne0s 28836 | Surreal two is nonzero. (... |
| n0seo 28837 | A non-negative surreal int... |
| zseo 28838 | A surreal integer is eithe... |
| twocut 28839 | Two times the cut of zero ... |
| nohalf 28840 | An explicit expression for... |
| expsval 28841 | The value of surreal expon... |
| expnnsval 28842 | Value of surreal exponenti... |
| exps0 28843 | Surreal exponentiation to ... |
| exps1 28844 | Surreal exponentiation to ... |
| expsp1 28845 | Value of a surreal number ... |
| expscllem 28846 | Lemma for proving non-nega... |
| expscl 28847 | Closure law for surreal ex... |
| n0expscl 28848 | Closure law for non-negati... |
| nnexpscl 28849 | Closure law for positive s... |
| zexpscl 28850 | Closure law for surreal in... |
| expadds 28851 | Sum of exponents law for s... |
| expsne0 28852 | A non-negative surreal int... |
| expsgt0 28853 | A non-negative surreal int... |
| pw2recs 28854 | Any power of two has a mul... |
| pw2divscld 28855 | Division closure for power... |
| pw2divmulsd 28856 | Relationship between surre... |
| pw2divscan3d 28857 | Cancellation law for surre... |
| pw2divscan2d 28858 | A cancellation law for sur... |
| pw2divsassd 28859 | An associative law for div... |
| pw2divscan4d 28860 | Cancellation law for divis... |
| pw2gt0divsd 28861 | Division of a positive sur... |
| pw2ge0divsd 28862 | Divison of a non-negative ... |
| pw2divsrecd 28863 | Relationship between surre... |
| pw2divsdird 28864 | Distribution of surreal di... |
| pw2divsnegd 28865 | Move negative sign inside ... |
| pw2ltdivmulsd 28866 | Surreal less-than relation... |
| pw2ltmuldivs2d 28867 | Surreal less-than relation... |
| pw2ltsdiv1d 28868 | Surreal less-than relation... |
| avglts1d 28869 | Ordering property for aver... |
| avglts2d 28870 | Ordering property for aver... |
| pw2divs0d 28871 | Division into zero is zero... |
| pw2divsidd 28872 | Identity law for division ... |
| pw2ltdivmuls2d 28873 | Surreal less-than relation... |
| halfcut 28874 | Relate the cut of twice of... |
| addhalfcut 28875 | The cut of a surreal non-n... |
| pw2cut 28876 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28877 | Simplify ~ pw2cut in the c... |
| pw2cut2 28878 | Cut expression for powers ... |
| bdaypw2n0bndlem 28879 | Lemma for ~ bdaypw2n0bnd .... |
| bdaypw2n0bnd 28880 | Upper bound for the birthd... |
| bdaypw2bnd 28881 | Birthday bounding rule for... |
| bdayfinbndcbv 28882 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem1 28883 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem2 28884 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbnd 28885 | Given a non-negative integ... |
| z12bdaylem1 28886 | Lemma for ~ z12bday . Pro... |
| z12bdaylem2 28887 | Lemma for ~ z12bday . Sho... |
| elz12s 28888 | Membership in the dyadic f... |
| elz12si 28889 | Inference form of membersh... |
| z12sex 28890 | The class of dyadic fracti... |
| zz12s 28891 | A surreal integer is a dya... |
| z12no 28892 | A dyadic is a surreal. (C... |
| z12addscl 28893 | The dyadics are closed und... |
| z12negscl 28894 | The dyadics are closed und... |
| z12subscl 28895 | The dyadics are closed und... |
| z12shalf 28896 | Half of a dyadic is a dyad... |
| z12negsclb 28897 | A surreal is a dyadic frac... |
| z12zsodd 28898 | A dyadic fraction is eithe... |
| z12sge0 28899 | An expression for non-nega... |
| z12bdaylem 28900 | Lemma for ~ z12bday . Han... |
| z12bday 28901 | A dyadic fraction has a fi... |
| bdayfinlem 28902 | Lemma for ~ bdayfin . Han... |
| bdayfin 28903 | A surreal has a finite bir... |
| dfz12s2 28904 | The set of dyadic fraction... |
| elreno 28907 | Membership in the set of s... |
| reno 28908 | A surreal real is a surrea... |
| renod 28909 | A surreal real is a surrea... |
| recut 28910 | The cut involved in defini... |
| elreno2 28911 | Alternate characterization... |
| 0reno 28912 | Surreal zero is a surreal ... |
| 1reno 28913 | Surreal one is a surreal r... |
| renegscl 28914 | The surreal reals are clos... |
| readdscl 28915 | The surreal reals are clos... |
| remulscllem1 28916 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28917 | Lemma for ~ remulscl . Bo... |
| remulscl 28918 | The surreal reals are clos... |
| itvndx 28929 | Index value of the Interva... |
| lngndx 28930 | Index value of the "line" ... |
| itvid 28931 | Utility theorem: index-ind... |
| lngid 28932 | Utility theorem: index-ind... |
| slotsinbpsd 28933 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28934 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28935 | The slot for the line is n... |
| trkgstr 28936 | Functionality of a Tarski ... |
| trkgbas 28937 | The base set of a Tarski g... |
| trkgdist 28938 | The measure of a distance ... |
| trkgitv 28939 | The congruence relation in... |
| istrkgc 28946 | Property of being a Tarski... |
| istrkgb 28947 | Property of being a Tarski... |
| istrkgcb 28948 | Property of being a Tarski... |
| istrkge 28949 | Property of fulfilling Euc... |
| istrkgl 28950 | Building lines from the se... |
| istrkgld 28951 | Property of fulfilling the... |
| istrkg2ld 28952 | Property of fulfilling the... |
| istrkg3ld 28953 | Property of fulfilling the... |
| axtgcgrrflx 28954 | Axiom of reflexivity of co... |
| axtgcgrid 28955 | Axiom of identity of congr... |
| axtgsegcon 28956 | Axiom of segment construct... |
| axtg5seg 28957 | Five segments axiom, Axiom... |
| axtgbtwnid 28958 | Identity of Betweenness. ... |
| axtgpasch 28959 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28960 | Axiom of Continuity. Axio... |
| axtgcont 28961 | Axiom of Continuity. Axio... |
| axtglowdim2 28962 | Lower dimension axiom for ... |
| axtgupdim2 28963 | Upper dimension axiom for ... |
| axtgeucl 28964 | Euclid's Axiom. Axiom A10... |
| tgjustf 28965 | Given any function ` F ` ,... |
| tgjustr 28966 | Given any equivalence rela... |
| tgjustc1 28967 | A justification for using ... |
| tgjustc2 28968 | A justification for using ... |
| tgcgrcomimp 28969 | Congruence commutes on the... |
| tgcgrcomr 28970 | Congruence commutes on the... |
| tgcgrcoml 28971 | Congruence commutes on the... |
| tgcgrcomlr 28972 | Congruence commutes on bot... |
| tgcgreqb 28973 | Congruence and equality. ... |
| tgcgreq 28974 | Congruence and equality. ... |
| tgcgrneq 28975 | Congruence and equality. ... |
| tgcgrtriv 28976 | Degenerate segments are co... |
| tgcgrextend 28977 | Link congruence over a pai... |
| tgsegconeq 28978 | Two points that satisfy th... |
| tgsegconeu 28979 | The point constructed in ~... |
| tgbtwntriv2 28980 | Betweenness always holds f... |
| tgbtwncom 28981 | Betweenness commutes. The... |
| tgbtwncomb 28982 | Betweenness commutes, bico... |
| tgbtwnne 28983 | Betweenness and inequality... |
| tgbtwntriv1 28984 | Betweenness always holds f... |
| tgbtwnswapid 28985 | If you can swap the first ... |
| tgbtwnintr 28986 | Inner transitivity law for... |
| tgbtwnexch3 28987 | Exchange the first endpoin... |
| tgbtwnouttr2 28988 | Outer transitivity law for... |
| tgbtwnexch2 28989 | Exchange the outer point o... |
| tgbtwnouttr 28990 | Outer transitivity law for... |
| tgbtwnexch 28991 | Outer transitivity law for... |
| tgtrisegint 28992 | A line segment between two... |
| tglowdim1 28993 | Lower dimension axiom for ... |
| tglowdim1i 28994 | Lower dimension axiom for ... |
| tgldimor 28995 | Excluded-middle like state... |
| tgldim0eq 28996 | In dimension zero, any two... |
| tgldim0itv 28997 | In dimension zero, any two... |
| tgldim0cgr 28998 | In dimension zero, any two... |
| tgbtwndiff 28999 | There is always a ` c ` di... |
| tgdim01 29000 | In geometries of dimension... |
| tgifscgr 29001 | Inner five segment congrue... |
| tgcgrsub 29002 | Removing identical parts f... |
| iscgrg 29005 | The congruence property fo... |
| iscgrgd 29006 | The property for two seque... |
| iscgrglt 29007 | The property for two seque... |
| trgcgrg 29008 | The property for two trian... |
| trgcgr 29009 | Triangle congruence. (Con... |
| ercgrg 29010 | The shape congruence relat... |
| tgcgrxfr 29011 | A line segment can be divi... |
| cgr3id 29012 | Reflexivity law for three-... |
| cgr3simp1 29013 | Deduce segment congruence ... |
| cgr3simp2 29014 | Deduce segment congruence ... |
| cgr3simp3 29015 | Deduce segment congruence ... |
| cgr3swap12 29016 | Permutation law for three-... |
| cgr3swap23 29017 | Permutation law for three-... |
| cgr3swap13 29018 | Permutation law for three-... |
| cgr3rotr 29019 | Permutation law for three-... |
| cgr3rotl 29020 | Permutation law for three-... |
| trgcgrcom 29021 | Commutative law for three-... |
| cgr3tr 29022 | Transitivity law for three... |
| tgbtwnxfr 29023 | A condition for extending ... |
| tgcgr4 29024 | Two quadrilaterals to be c... |
| isismt 29027 | Property of being an isome... |
| ismot 29028 | Property of being an isome... |
| motcgr 29029 | Property of a motion: dist... |
| idmot 29030 | The identity is a motion. ... |
| motf1o 29031 | Motions are bijections. (... |
| motcl 29032 | Closure of motions. (Cont... |
| motco 29033 | The composition of two mot... |
| cnvmot 29034 | The converse of a motion i... |
| motplusg 29035 | The operation for motions ... |
| motgrp 29036 | The motions of a geometry ... |
| motcgrg 29037 | Property of a motion: dist... |
| motcgr3 29038 | Property of a motion: dist... |
| tglng 29039 | Lines of a Tarski Geometry... |
| tglnfn 29040 | Lines as functions. (Cont... |
| tglnunirn 29041 | Lines are sets of points. ... |
| tglnpt 29042 | Lines are sets of points. ... |
| tglngne 29043 | It takes two different poi... |
| tglngval 29044 | The line going through poi... |
| tglnssp 29045 | Lines are subset of the ge... |
| tgellng 29046 | Property of lying on the l... |
| tgcolg 29047 | We choose the notation ` (... |
| btwncolg1 29048 | Betweenness implies coline... |
| btwncolg2 29049 | Betweenness implies coline... |
| btwncolg3 29050 | Betweenness implies coline... |
| colcom 29051 | Swapping the points defini... |
| colrot1 29052 | Rotating the points defini... |
| colrot2 29053 | Rotating the points defini... |
| ncolcom 29054 | Swapping non-colinear poin... |
| ncolrot1 29055 | Rotating non-colinear poin... |
| ncolrot2 29056 | Rotating non-colinear poin... |
| tgdim01ln 29057 | In geometries of dimension... |
| ncoltgdim2 29058 | If there are three non-col... |
| lnxfr 29059 | Transfer law for colineari... |
| lnext 29060 | Extend a line with a missi... |
| tgfscgr 29061 | Congruence law for the gen... |
| lncgr 29062 | Congruence rule for lines.... |
| lnid 29063 | Identity law for points on... |
| tgidinside 29064 | Law for finding a point in... |
| tgbtwnconn1lem1 29065 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 29066 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 29067 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 29068 | Connectivity law for betwe... |
| tgbtwnconn2 29069 | Another connectivity law f... |
| tgbtwnconn3 29070 | Inner connectivity law for... |
| tgbtwnconnln3 29071 | Derive colinearity from be... |
| tgbtwnconn22 29072 | Double connectivity law fo... |
| tgbtwnconnln1 29073 | Derive colinearity from be... |
| tgbtwnconnln2 29074 | Derive colinearity from be... |
| legval 29077 | Value of the less-than rel... |
| legov 29078 | Value of the less-than rel... |
| legov2 29079 | An equivalent definition o... |
| legid 29080 | Reflexivity of the less-th... |
| btwnleg 29081 | Betweenness implies less-t... |
| legtrd 29082 | Transitivity of the less-t... |
| legtri3 29083 | Equality from the less-tha... |
| legtrid 29084 | Trichotomy law for the les... |
| leg0 29085 | Degenerated (zero-length) ... |
| legeq 29086 | Deduce equality from "less... |
| legbtwn 29087 | Deduce betweenness from "l... |
| tgcgrsub2 29088 | Removing identical parts f... |
| ltgseg 29089 | The set ` E ` denotes the ... |
| ltgov 29090 | Strict "shorter than" geom... |
| legov3 29091 | An equivalent definition o... |
| legso 29092 | The "shorter than" relatio... |
| ishlg2 29095 | Alternate version of ~ ish... |
| hlgrcl1 29096 | Reverse closure for rays. ... |
| hlgrcl2 29097 | Reverse closure for rays. ... |
| ishlg 29098 | Rays : Definition 6.1 of ... |
| hlcomb 29099 | The half-line relation is ... |
| hlcomd 29100 | The half-line relation is ... |
| hlne1 29101 | The half-line relation imp... |
| hlne2 29102 | The half-line relation imp... |
| hlln 29103 | The half-line relation imp... |
| hleqnid 29104 | The endpoint does not belo... |
| hlid 29105 | The half-line relation is ... |
| hltr 29106 | The half-line relation is ... |
| hlbtwn 29107 | Betweenness is a sufficien... |
| btwnhl1 29108 | Deduce half-line from betw... |
| btwnhl2 29109 | Deduce half-line from betw... |
| btwnhl 29110 | Swap betweenness for a hal... |
| lnhl 29111 | Either a point ` C ` on th... |
| hlcgrex 29112 | Construct a point on a hal... |
| hlcgreulem 29113 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 29114 | The point constructed in ~... |
| hlcgreq 29115 | A constructed point on a h... |
| tghlsub 29116 | Removing identical parts f... |
| btwnlng1 29117 | Betweenness implies coline... |
| btwnlng2 29118 | Betweenness implies coline... |
| btwnlng3 29119 | Betweenness implies coline... |
| lncom 29120 | Swapping the points defini... |
| lnrot1 29121 | Rotating the points defini... |
| lnrot2 29122 | Rotating the points defini... |
| ncolne1 29123 | Non-colinear points are di... |
| ncolne2 29124 | Non-colinear points are di... |
| tgisline 29125 | The property of being a pr... |
| tglnne 29126 | It takes two different poi... |
| tglndim0 29127 | There are no lines in dime... |
| tgelrnln 29128 | The property of being a pr... |
| tglineeltr 29129 | Transitivity law for lines... |
| tglineelsb2 29130 | If ` S ` lies on PQ , then... |
| tglinerflx1 29131 | Reflexivity law for line m... |
| tglinerflx2 29132 | Reflexivity law for line m... |
| tglinecom 29133 | Commutativity law for line... |
| tglinethru 29134 | If ` A ` is a line contain... |
| tghilberti1 29135 | There is a line through an... |
| tghilberti2 29136 | There is at most one line ... |
| tglinethrueu 29137 | There is a unique line goi... |
| tglinesseq 29138 | If a line is a subset of a... |
| tglnne0 29139 | A line ` A ` has at least ... |
| tglineintmo 29140 | Two distinct lines interse... |
| tglineineq 29141 | Two distinct lines interse... |
| tglineinsn 29142 | If two distinct lines inte... |
| tglineneq 29143 | Given three non-colinear p... |
| tglineinteq 29144 | Two distinct lines interse... |
| ncolncol 29145 | Deduce non-colinearity fro... |
| coltr 29146 | A transitivity law for col... |
| coltr3 29147 | A transitivity law for col... |
| colline 29148 | Three points are colinear ... |
| tglowdim2l 29149 | Reformulation of the lower... |
| tglowdim2ln 29150 | There is always one point ... |
| tglnpt2 29151 | Find a second point on a l... |
| tglnpt3 29152 | Find a third point on a li... |
| tglnpt4 29153 | Find a second point on a l... |
| mirreu3 29156 | Existential uniqueness of ... |
| mirval 29157 | Value of the point inversi... |
| mirfv 29158 | Value of the point inversi... |
| mircgr 29159 | Property of the image by t... |
| mirbtwn 29160 | Property of the image by t... |
| ismir 29161 | Property of the image by t... |
| mirf 29162 | Point inversion as functio... |
| mircl 29163 | Closure of the point inver... |
| mirmir 29164 | The point inversion functi... |
| mircom 29165 | Variation on ~ mirmir . (... |
| mirreu 29166 | Any point has a unique ant... |
| mireq 29167 | Equality deduction for poi... |
| mirinv 29168 | The only invariant point o... |
| mirne 29169 | Mirror of non-center point... |
| mircinv 29170 | The center point is invari... |
| mirf1o 29171 | The point inversion functi... |
| miriso 29172 | The point inversion functi... |
| mirbtwni 29173 | Point inversion preserves ... |
| mirbtwnb 29174 | Point inversion preserves ... |
| mircgrs 29175 | Point inversion preserves ... |
| mirmir2 29176 | Point inversion of a point... |
| mirmot 29177 | Point investion is a motio... |
| mirln 29178 | If two points are on the s... |
| mirln2 29179 | If a point and its mirror ... |
| mirconn 29180 | Point inversion of connect... |
| mirhl 29181 | If two points ` X ` and ` ... |
| mirbtwnhl 29182 | If the center of the point... |
| mirhl2 29183 | Deduce half-line relation ... |
| mircgrextend 29184 | Link congruence over a pai... |
| mirtrcgr 29185 | Point inversion of one poi... |
| mirauto 29186 | Point inversion preserves ... |
| miduniq 29187 | Uniqueness of the middle p... |
| miduniq1 29188 | Uniqueness of the middle p... |
| miduniq2 29189 | If two point inversions co... |
| colmid 29190 | Colinearity and equidistan... |
| symquadlem 29191 | Lemma of the symmetrical q... |
| krippenlem 29192 | Lemma for ~ krippen . We ... |
| krippen 29193 | Krippenlemma (German for c... |
| midexlem 29194 | Lemma for the existence of... |
| symquadprlnglem 29195 | Lemma for ~ symquadprlngle... |
| mirleqb 29196 | Equality theorem for point... |
| mirlni 29197 | The mirror of a point ` X ... |
| israg 29202 | Property for 3 points A, B... |
| ragcom 29203 | Commutative rule for right... |
| ragcol 29204 | The right angle property i... |
| ragmir 29205 | Right angle property is pr... |
| mirrag 29206 | Right angle is conserved b... |
| ragtrivb 29207 | Trivial right angle. Theo... |
| ragflat2 29208 | Deduce equality from two r... |
| ragflat 29209 | Deduce equality from two r... |
| ragtriva 29210 | Trivial right angle. Theo... |
| ragflat3 29211 | Right angle and colinearit... |
| ragcgr 29212 | Right angle and colinearit... |
| motrag 29213 | Right angles are preserved... |
| ragncol 29214 | Right angle implies non-co... |
| perpln1 29215 | Derive a line from perpend... |
| perpln2 29216 | Derive a line from perpend... |
| isperp 29217 | Property for 2 lines A, B ... |
| perpcom 29218 | The "perpendicular" relati... |
| perpneq 29219 | Two perpendicular lines ar... |
| isperp2 29220 | Property for 2 lines A, B,... |
| isperp2d 29221 | One direction of ~ isperp2... |
| ragperp 29222 | Deduce that two lines are ... |
| footexALT 29223 | Alternative version of ~ f... |
| footexlem1 29224 | Lemma for ~ footex . (Con... |
| footexlem2 29225 | Lemma for ~ footex . (Con... |
| footex 29226 | From a point ` C ` outside... |
| foot 29227 | From a point ` C ` outside... |
| footne 29228 | Uniqueness of the foot poi... |
| footeq 29229 | Uniqueness of the foot poi... |
| perpin 29230 | If two lines ` A ` and ` B... |
| hlperpnel 29231 | A point on a half-line whi... |
| perprag 29232 | Deduce a right angle from ... |
| perpdragALT 29233 | Deduce a right angle from ... |
| perpdrag 29234 | Deduce a right angle from ... |
| colperp 29235 | Deduce a perpendicularity ... |
| colperpexlem1 29236 | Lemma for ~ colperp . Fir... |
| colperpexlem2 29237 | Lemma for ~ colperpex . S... |
| colperpexlem3 29238 | Lemma for ~ colperpex . C... |
| colperpex 29239 | In dimension 2 and above, ... |
| mideulem2 29240 | Lemma for ~ opphllem , whi... |
| opphllem 29241 | Lemma 8.24 of [Schwabhause... |
| mideulem 29242 | Lemma for ~ mideu . We ca... |
| midex 29243 | Existence of the midpoint,... |
| mideu 29244 | Existence and uniqueness o... |
| islnopp 29245 | The property for two point... |
| islnoppd 29246 | Deduce that ` A ` and ` B ... |
| oppne1 29247 | Points lying on opposite s... |
| oppne2 29248 | Points lying on opposite s... |
| oppne3 29249 | Points lying on opposite s... |
| oppcom 29250 | Commutativity rule for "op... |
| opptgdim2 29251 | If two points opposite to ... |
| oppnid 29252 | The "opposite to a line" r... |
| opphllem1 29253 | Lemma for ~ opphl . (Cont... |
| opphllem2 29254 | Lemma for ~ opphl . Lemma... |
| opphllem3 29255 | Lemma for ~ opphl : We as... |
| opphllem4 29256 | Lemma for ~ opphl . (Cont... |
| opphllem5 29257 | Second part of Lemma 9.4 o... |
| opphllem6 29258 | First part of Lemma 9.4 of... |
| oppperpex 29259 | Restating ~ colperpex usin... |
| opphl 29260 | If two points ` A ` and ` ... |
| lnoppinn0 29261 | The segment between two po... |
| oppmir 29262 | The mirror point with rega... |
| outpasch 29263 | Axiom of Pasch, outer form... |
| hlpasch 29264 | An application of the axio... |
| ishpg 29267 | Value of the half-plane re... |
| hpgbr 29268 | Half-planes : property for... |
| hpgne1 29269 | Points on the open half pl... |
| hpgne2 29270 | Points on the open half pl... |
| lnopp2hpgb 29271 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 29272 | If two points lie on the o... |
| hpgerlem 29273 | Lemma for the proof that t... |
| hpgid 29274 | The half-plane relation is... |
| hpgcom 29275 | The half-plane relation is... |
| hpgtr 29276 | The half-plane relation is... |
| colopp 29277 | Opposite sides of a line f... |
| colhp 29278 | Half-plane relation for co... |
| hphl 29279 | If two points are on the s... |
| hlopp 29280 | If two points ` X ` and ` ... |
| tgplnfn 29283 | The plane generating funct... |
| tgelrnpln 29284 | The property of being a pl... |
| plngval 29285 | The plane defined by a lin... |
| isplng 29286 | The property of being a pl... |
| plngrnssp 29287 | Planes are sets of points.... |
| elplng 29288 | Elementhood in the plane d... |
| plngssp 29289 | Planes are sets of points.... |
| elplngid 29290 | The point ` R ` is itself ... |
| elplnglnid 29291 | The line ` A ` itself is a... |
| lnincplng 29292 | If two lines ` A ` and ` B... |
| plngcplem 29293 | Lemma for ~ plngcp . (Con... |
| plngcp 29294 | The plane defined by a lin... |
| plngrotlem1 29295 | Lemma for ~ plngrot . (Co... |
| plngrotlem2 29296 | Lemma for ~ plngrot . (Co... |
| plngrotlem3 29297 | Lemma for ~ plngrot . (Co... |
| plngrot 29298 | The plane defined by a lin... |
| lnssplnglem 29299 | Lemma for ~ lnssplng . (C... |
| lnssplng 29300 | A line defined by two poin... |
| lnssplng1 29301 | A line defined by two poin... |
| plngmiropp 29302 | Given a line ` A ` and a p... |
| mirplncl 29303 | The mirror of a point with... |
| hpgssplng 29304 | Any point ` X ` on a half ... |
| plng3p 29305 | If ` H ` is a plane contai... |
| nhpmirhp 29306 | If a point ` Z ` is on the... |
| midf 29311 | Midpoint as a function. (... |
| midcl 29312 | Closure of the midpoint. ... |
| ismidb 29313 | Property of the midpoint. ... |
| midbtwn 29314 | Betweenness of midpoint. ... |
| midcgr 29315 | Congruence of midpoint. (... |
| midid 29316 | Midpoint of a null segment... |
| midcom 29317 | Commutativity rule for the... |
| mirmid 29318 | Point inversion preserves ... |
| lmieu 29319 | Uniqueness of the line mir... |
| lmif 29320 | Line mirror as a function.... |
| lmicl 29321 | Closure of the line mirror... |
| islmib 29322 | Property of the line mirro... |
| lmicom 29323 | The line mirroring functio... |
| lmilmi 29324 | Line mirroring is an invol... |
| lmireu 29325 | Any point has a unique ant... |
| lmieq 29326 | Equality deduction for lin... |
| lmiinv 29327 | The invariants of the line... |
| lmicinv 29328 | The mirroring line is an i... |
| lmimid 29329 | If we have a right angle, ... |
| lmif1o 29330 | The line mirroring functio... |
| lmiisolem 29331 | Lemma for ~ lmiiso . (Con... |
| lmiiso 29332 | The line mirroring functio... |
| lmimot 29333 | Line mirroring is a motion... |
| symquadmid 29334 | In a symmetrical quadrilat... |
| hypcgrlem1 29335 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 29336 | Lemma for ~ hypcgr , case ... |
| hypcgr 29337 | If the catheti of two righ... |
| lmiopp 29338 | Line mirroring produces po... |
| lnperpex 29339 | Existence of a perpendicul... |
| lnperpexs 29340 | Existence of a perpendicul... |
| trgcopy 29341 | Triangle construction: a c... |
| trgcopyeulem 29342 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 29343 | Triangle construction: a c... |
| iscgra 29346 | Property for two angles AB... |
| iscgra1 29347 | A special version of ~ isc... |
| iscgrad 29348 | Sufficient conditions for ... |
| cgrane1 29349 | Angles imply inequality. ... |
| cgrane2 29350 | Angles imply inequality. ... |
| cgrane3 29351 | Angles imply inequality. ... |
| cgrane4 29352 | Angles imply inequality. ... |
| cgrahl1 29353 | Angle congruence is indepe... |
| cgrahl2 29354 | Angle congruence is indepe... |
| cgracgr 29355 | First direction of proposi... |
| cgraid 29356 | Angle congruence is reflex... |
| cgraswap 29357 | Swap rays in a congruence ... |
| cgrcgra 29358 | Triangle congruence implie... |
| cgracom 29359 | Angle congruence commutes.... |
| cgratr 29360 | Angle congruence is transi... |
| zerocgra 29361 | Zero angles are congruent.... |
| flatcgra 29362 | Flat angles are congruent.... |
| cgraswaplr 29363 | Swap both side of angle co... |
| cgrabtwn 29364 | Angle congruence preserves... |
| cgrahl 29365 | Angle congruence preserves... |
| cgracol 29366 | Angle congruence preserves... |
| cgrancol 29367 | Angle congruence preserves... |
| dfcgra2 29368 | This is the full statement... |
| sacgr 29369 | Supplementary angles of co... |
| oacgr 29370 | Vertical angle theorem. V... |
| acopy 29371 | Angle construction. Theor... |
| acopyeu 29372 | Angle construction. Theor... |
| ragcgra 29373 | Right angles are congruent... |
| cgrarag 29374 | Any angle ` <" A B C "> ` ... |
| ragsupplcgra 29375 | An angle ` <" X Y Z "> ` i... |
| ragraghl 29376 | Drawing two right angles a... |
| perpeqlem 29377 | Lemma for ~ perpeq . (Con... |
| perpeq 29378 | Uniqueness of the perpendi... |
| tgaaddcpbllem1 29379 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbllem2 29380 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbllem3 29381 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbl 29382 | The angular addition is co... |
| tgaaddcpbl2 29383 | The angular addition is co... |
| isinag 29387 | Property for point ` X ` t... |
| isinagd 29388 | Sufficient conditions for ... |
| inagflat 29389 | Any point lies in a flat a... |
| inagswap 29390 | Swap the order of the half... |
| inagne1 29391 | Deduce inequality from the... |
| inagne2 29392 | Deduce inequality from the... |
| inagne3 29393 | Deduce inequality from the... |
| inaghl 29394 | The "point lie in angle" r... |
| isleag 29396 | Geometrical "less than" pr... |
| isleagd 29397 | Sufficient condition for "... |
| leagne1 29398 | Deduce inequality from the... |
| leagne2 29399 | Deduce inequality from the... |
| leagne3 29400 | Deduce inequality from the... |
| leagne4 29401 | Deduce inequality from the... |
| cgrg3col4 29402 | Lemma 11.28 of [Schwabhaus... |
| elcgrabasi 29405 | Helper theorem for the mem... |
| elcgrabasrd 29406 | Helper theorem for the mem... |
| cgraer 29407 | The angle congruence relat... |
| cgrabasimass 29408 | The angle congruence relat... |
| angmgmaddeu1 29409 | There exists a unique poin... |
| angmgmaddeu2 29410 | Existence of a unique poin... |
| angmgmaddeu3 29411 | Existence of a unique poin... |
| angmgmaddeu4 29412 | There exists a unique poin... |
| angmgmaddeu5 29413 | There exists a unique poin... |
| angmgmaddeu6 29414 | There exists a unique poin... |
| angmgmaddeu7 29415 | There exists a unique poin... |
| angmgmaddov1lem 29416 | Lemma for ~ angmgmaddov1 .... |
| angmgmaddov2lem 29417 | Lemma for ~ angmgmaddov2 .... |
| angmgmaddov1 29418 | Value of the addition oper... |
| angmgmaddov2 29419 | Value of the addition oper... |
| angmgmaddcpbl 29420 | Addition of angles is comp... |
| angmgmaddcl 29421 | Closure of the addition of... |
| angmgmaddlid 29422 | The left identity element ... |
| angmgmaddrid 29423 | The right identity element... |
| angmgmval 29424 | Explicit the value of the ... |
| angmgmlem 29425 | Lemma for ~ angmgm . (Con... |
| angmgm0g 29426 | The identity element of th... |
| angmgm 29427 | The angle addition magma i... |
| angmgmbas 29428 | The base set of the angle ... |
| tgsas1 29429 | First congruence theorem: ... |
| tgsas 29430 | First congruence theorem: ... |
| tgsas2 29431 | First congruence theorem: ... |
| tgsas3 29432 | First congruence theorem: ... |
| tgasa1 29433 | Second congruence theorem:... |
| tgasa 29434 | Second congruence theorem:... |
| tgsss1 29435 | Third congruence theorem: ... |
| tgsss2 29436 | Third congruence theorem: ... |
| tgsss3 29437 | Third congruence theorem: ... |
| dfcgrg2 29438 | Congruence for two triangl... |
| isoas 29439 | Congruence theorem for iso... |
| iseqlg 29442 | Property of a triangle bei... |
| iseqlgd 29443 | Condition for a triangle t... |
| brprlng 29446 | Property of two lines ` A ... |
| prlngd 29447 | Deduce parallelism between... |
| prlngref 29448 | Parallelism is reflexive. ... |
| prlngsym 29449 | Parallelism is symmetric. ... |
| prlngrcl1 29450 | Reverse closure for parall... |
| prlngrcl2 29451 | Reverse closure for parall... |
| prlngin0 29452 | Two parallel lines do not ... |
| prlngpln 29453 | Two parallel lines are on ... |
| prlnghpg 29454 | If two lines ` A ` and ` B... |
| dfprlng2 29455 | Alternate definition of (s... |
| dfprlng3 29456 | Alternate definition of (s... |
| prlngpln3 29457 | Two parallel lines are on ... |
| perpprlng 29458 | If two lines ` A ` and ` B... |
| prlngex 29459 | There exists at least one ... |
| prlngmolem1 29460 | Lemma for ~ prlngmo : Con... |
| prlngmolem2 29461 | Lemma for ~ prlngmo . (Co... |
| prlngmo 29462 | Playfair's axiom. Given a... |
| prlngeu 29463 | Given a line ` A ` and a p... |
| prlngmo2 29464 | Playfair's axiom, without ... |
| prlngeq 29465 | Playfair's axiom, written ... |
| prlngpln4 29466 | Building a parallel line c... |
| prlngplngtr 29467 | Transitivity of parallelis... |
| prlnginn0 29468 | A line ` C ` intersecting ... |
| prlngmid2 29469 | If the midpoints of two se... |
| symquadprlng 29470 | Symmetrical quadrilaterals... |
| prlngsymquadlem 29471 | Lemma for ~ prlngsymquad .... |
| prlngsymquad 29472 | All parallelograms are sym... |
| prlngsymquadopp 29473 | In parallelograms, opposin... |
| quadcgrprlng 29474 | Nontrivial quadrilaterals ... |
| tgaltai 29475 | Parallelism implies altern... |
| f1otrgds 29476 | Convenient lemma for ~ f1o... |
| f1otrgitv 29477 | Convenient lemma for ~ f1o... |
| f1otrg 29478 | A bijection between bases ... |
| f1otrge 29479 | A bijection between bases ... |
| ttgval 29482 | Define a function to augme... |
| ttglem 29483 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 29484 | The base set of a subcompl... |
| ttgplusg 29485 | The addition operation of ... |
| ttgsub 29486 | The subtraction operation ... |
| ttgvsca 29487 | The scalar product of a su... |
| ttgds 29488 | The metric of a subcomplex... |
| ttgitvval 29489 | Betweenness for a subcompl... |
| ttgelitv 29490 | Betweenness for a subcompl... |
| ttgbtwnid 29491 | Any subcomplex module equi... |
| ttgcontlem1 29492 | Lemma for % ttgcont . (Co... |
| xmstrkgc 29493 | Any metric space fulfills ... |
| cchhllem 29494 | Lemma for chlbas and chlvs... |
| elee 29501 | Membership in a Euclidean ... |
| mptelee 29502 | A condition for a mapping ... |
| mpteleeOLD 29503 | Obsolete version of ~ mpte... |
| eleenn 29504 | If ` A ` is in ` ( EE `` N... |
| eleei 29505 | The forward direction of ~... |
| eedimeq 29506 | A point belongs to at most... |
| brbtwn 29507 | The binary relation form o... |
| brcgr 29508 | The binary relation form o... |
| fveere 29509 | The function value of a po... |
| fveecn 29510 | The function value of a po... |
| eqeefv 29511 | Two points are equal iff t... |
| eqeelen 29512 | Two points are equal iff t... |
| brbtwn2 29513 | Alternate characterization... |
| colinearalglem1 29514 | Lemma for ~ colinearalg . ... |
| colinearalglem2 29515 | Lemma for ~ colinearalg . ... |
| colinearalglem3 29516 | Lemma for ~ colinearalg . ... |
| colinearalglem4 29517 | Lemma for ~ colinearalg . ... |
| colinearalg 29518 | An algebraic characterizat... |
| eleesub 29519 | Membership of a subtractio... |
| eleesubd 29520 | Membership of a subtractio... |
| axdimuniq 29521 | The unique dimension axiom... |
| axcgrrflx 29522 | ` A ` is as far from ` B `... |
| axcgrtr 29523 | Congruence is transitive. ... |
| axcgrid 29524 | If there is no distance be... |
| axsegconlem1 29525 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 29526 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 29527 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 29528 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 29529 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 29530 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 29531 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 29532 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 29533 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 29534 | Lemma for ~ axsegcon . Sh... |
| axsegcon 29535 | Any segment ` A B ` can be... |
| ax5seglem1 29536 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 29537 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 29538 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 29539 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 29540 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 29541 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 29542 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 29543 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 29544 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 29545 | Lemma for ~ ax5seg . Take... |
| ax5seg 29546 | The five segment axiom. T... |
| axbtwnid 29547 | Points are indivisible. T... |
| axpaschlem 29548 | Lemma for ~ axpasch . Set... |
| axpasch 29549 | The inner Pasch axiom. Ta... |
| axlowdimlem1 29550 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 29551 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 29552 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 29553 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 29554 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 29555 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 29556 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 29557 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 29558 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 29559 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 29560 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 29561 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 29562 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 29563 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 29564 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 29565 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 29566 | Lemma for ~ axlowdim . Es... |
| axlowdim1 29567 | The lower dimension axiom ... |
| axlowdim2 29568 | The lower two-dimensional ... |
| axlowdim 29569 | The general lower dimensio... |
| axeuclidlem 29570 | Lemma for ~ axeuclid . Ha... |
| axeuclid 29571 | Euclid's axiom. Take an a... |
| axcontlem1 29572 | Lemma for ~ axcont . Chan... |
| axcontlem2 29573 | Lemma for ~ axcont . The ... |
| axcontlem3 29574 | Lemma for ~ axcont . Give... |
| axcontlem4 29575 | Lemma for ~ axcont . Give... |
| axcontlem5 29576 | Lemma for ~ axcont . Comp... |
| axcontlem6 29577 | Lemma for ~ axcont . Stat... |
| axcontlem7 29578 | Lemma for ~ axcont . Give... |
| axcontlem8 29579 | Lemma for ~ axcont . A po... |
| axcontlem9 29580 | Lemma for ~ axcont . Give... |
| axcontlem10 29581 | Lemma for ~ axcont . Give... |
| axcontlem11 29582 | Lemma for ~ axcont . Elim... |
| axcontlem12 29583 | Lemma for ~ axcont . Elim... |
| axcont 29584 | The axiom of continuity. ... |
| eengv 29587 | The value of the Euclidean... |
| eengstr 29588 | The Euclidean geometry as ... |
| eengbas 29589 | The Base of the Euclidean ... |
| ebtwntg 29590 | The betweenness relation u... |
| ecgrtg 29591 | The congruence relation us... |
| elntg 29592 | The line definition in the... |
| elntg2 29593 | The line definition in the... |
| eengtrkg 29594 | The geometry structure for... |
| eengtrkge 29595 | The geometry structure for... |
| edgfid 29598 | Utility theorem: index-ind... |
| edgfndx 29599 | Index value of the ~ df-ed... |
| edgfndxnn 29600 | The index value of the edg... |
| edgfndxid 29601 | The value of the edge func... |
| basendxltedgfndx 29602 | The index value of the ` B... |
| basendxnedgfndx 29603 | The slots ` Base ` and ` .... |
| vtxval 29608 | The set of vertices of a g... |
| iedgval 29609 | The set of indexed edges o... |
| 1vgrex 29610 | A graph with at least one ... |
| opvtxval 29611 | The set of vertices of a g... |
| opvtxfv 29612 | The set of vertices of a g... |
| opvtxov 29613 | The set of vertices of a g... |
| opiedgval 29614 | The set of indexed edges o... |
| opiedgfv 29615 | The set of indexed edges o... |
| opiedgov 29616 | The set of indexed edges o... |
| opvtxfvi 29617 | The set of vertices of a g... |
| opiedgfvi 29618 | The set of indexed edges o... |
| funvtxdmge2val 29619 | The set of vertices of an ... |
| funiedgdmge2val 29620 | The set of indexed edges o... |
| funvtxdm2val 29621 | The set of vertices of an ... |
| funiedgdm2val 29622 | The set of indexed edges o... |
| funvtxval0 29623 | The set of vertices of an ... |
| basvtxval 29624 | The set of vertices of a g... |
| edgfiedgval 29625 | The set of indexed edges o... |
| funvtxval 29626 | The set of vertices of a g... |
| funiedgval 29627 | The set of indexed edges o... |
| structvtxvallem 29628 | Lemma for ~ structvtxval a... |
| structvtxval 29629 | The set of vertices of an ... |
| structiedg0val 29630 | The set of indexed edges o... |
| structgrssvtxlem 29631 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29632 | The set of vertices of a g... |
| structgrssiedg 29633 | The set of indexed edges o... |
| struct2grstr 29634 | A graph represented as an ... |
| struct2grvtx 29635 | The set of vertices of a g... |
| struct2griedg 29636 | The set of indexed edges o... |
| graop 29637 | Any representation of a gr... |
| grastruct 29638 | Any representation of a gr... |
| gropd 29639 | If any representation of a... |
| grstructd 29640 | If any representation of a... |
| gropeld 29641 | If any representation of a... |
| grstructeld 29642 | If any representation of a... |
| setsvtx 29643 | The vertices of a structur... |
| setsiedg 29644 | The (indexed) edges of a s... |
| snstrvtxval 29645 | The set of vertices of a g... |
| snstriedgval 29646 | The set of indexed edges o... |
| vtxval0 29647 | Degenerated case 1 for ver... |
| iedgval0 29648 | Degenerated case 1 for edg... |
| vtxvalsnop 29649 | Degenerated case 2 for ver... |
| iedgvalsnop 29650 | Degenerated case 2 for edg... |
| vtxval3sn 29651 | Degenerated case 3 for ver... |
| iedgval3sn 29652 | Degenerated case 3 for edg... |
| vtxvalprc 29653 | Degenerated case 4 for ver... |
| iedgvalprc 29654 | Degenerated case 4 for edg... |
| edgval 29657 | The edges of a graph. (Co... |
| iedgedg 29658 | An indexed edge is an edge... |
| edgopval 29659 | The edges of a graph repre... |
| edgov 29660 | The edges of a graph repre... |
| edgstruct 29661 | The edges of a graph repre... |
| edgiedgb 29662 | A set is an edge iff it is... |
| edg0iedg0 29663 | There is no edge in a grap... |
| isuhgr 29668 | The predicate "is an undir... |
| isushgr 29669 | The predicate "is an undir... |
| uhgrf 29670 | The edge function of an un... |
| ushgrf 29671 | The edge function of an un... |
| uhgrss 29672 | An edge is a subset of ver... |
| uhgreq12g 29673 | If two sets have the same ... |
| uhgrfun 29674 | The edge function of an un... |
| uhgrn0 29675 | An edge is a nonempty subs... |
| lpvtx 29676 | The endpoints of a loop (w... |
| ushgruhgr 29677 | An undirected simple hyper... |
| isuhgrop 29678 | The property of being an u... |
| uhgr0e 29679 | The empty graph, with vert... |
| uhgr0vb 29680 | The null graph, with no ve... |
| uhgr0 29681 | The null graph represented... |
| uhgrun 29682 | The union ` U ` of two (un... |
| uhgrunop 29683 | The union of two (undirect... |
| ushgrun 29684 | The union ` U ` of two (un... |
| ushgrunop 29685 | The union of two (undirect... |
| uhgrstrrepe 29686 | Replacing (or adding) the ... |
| incistruhgr 29687 | An _incidence structure_ `... |
| isupgr 29692 | The property of being an u... |
| wrdupgr 29693 | The property of being an u... |
| upgrf 29694 | The edge function of an un... |
| upgrfn 29695 | The edge function of an un... |
| upgrss 29696 | An edge is a subset of ver... |
| upgrn0 29697 | An edge is a nonempty subs... |
| upgrle 29698 | An edge of an undirected p... |
| upgrfi 29699 | An edge is a finite subset... |
| upgrex 29700 | An edge is an unordered pa... |
| upgrbi 29701 | Show that an unordered pai... |
| upgrop 29702 | A pseudograph represented ... |
| isumgr 29703 | The property of being an u... |
| isumgrs 29704 | The simplified property of... |
| wrdumgr 29705 | The property of being an u... |
| umgrf 29706 | The edge function of an un... |
| umgrfn 29707 | The edge function of an un... |
| umgredg2 29708 | An edge of a multigraph ha... |
| umgrbi 29709 | Show that an unordered pai... |
| upgruhgr 29710 | An undirected pseudograph ... |
| umgrupgr 29711 | An undirected multigraph i... |
| umgruhgr 29712 | An undirected multigraph i... |
| upgrle2 29713 | An edge of an undirected p... |
| umgrnloopv 29714 | In a multigraph, there is ... |
| umgredgprv 29715 | In a multigraph, an edge i... |
| umgrnloop 29716 | In a multigraph, there is ... |
| umgrnloop0 29717 | A multigraph has no loops.... |
| umgr0e 29718 | The empty graph, with vert... |
| upgr0e 29719 | The empty graph, with vert... |
| upgr1elem 29720 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29721 | A pseudograph with one edg... |
| upgr0eop 29722 | The empty graph, with vert... |
| upgr1eop 29723 | A pseudograph with one edg... |
| upgr0eopALT 29724 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29725 | Alternate proof of ~ upgr1... |
| upgrun 29726 | The union ` U ` of two pse... |
| upgrunop 29727 | The union of two pseudogra... |
| umgrun 29728 | The union ` U ` of two mul... |
| umgrunop 29729 | The union of two multigrap... |
| umgrislfupgrlem 29730 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29731 | A multigraph is a loop-fre... |
| lfgredgge2 29732 | An edge of a loop-free gra... |
| lfgrnloop 29733 | A loop-free graph has no l... |
| uhgredgiedgb 29734 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29735 | A hypergraph has no edges ... |
| uhgredgn0 29736 | An edge of a hypergraph is... |
| edguhgr 29737 | An edge of a hypergraph is... |
| uhgredgrnv 29738 | An edge of a hypergraph co... |
| uhgredgss 29739 | The set of edges of a hype... |
| upgredgss 29740 | The set of edges of a pseu... |
| umgredgss 29741 | The set of edges of a mult... |
| edgupgr 29742 | Properties of an edge of a... |
| edgumgr 29743 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29744 | In a hypergraph, a vertex ... |
| upgredg 29745 | For each edge in a pseudog... |
| umgredg 29746 | For each edge in a multigr... |
| upgrpredgv 29747 | An edge of a pseudograph a... |
| umgrpredgv 29748 | An edge of a multigraph al... |
| upgredg2vtx 29749 | For a vertex incident to a... |
| upgredgpr 29750 | If a proper pair (of verti... |
| edglnl 29751 | The edges incident with a ... |
| numedglnl 29752 | The number of edges incide... |
| umgredgne 29753 | An edge of a multigraph al... |
| umgrnloop2 29754 | A multigraph has no loops.... |
| umgredgnlp 29755 | An edge of a multigraph is... |
| lfuhgr 29756 | A hypergraph is loop-free ... |
| lfuhgr2 29757 | A hypergraph is loop-free ... |
| lfuhgr3 29758 | A hypergraph is loop-free ... |
| isuspgr 29763 | The property of being a si... |
| isusgr 29764 | The property of being a si... |
| uspgrf 29765 | The edge function of a sim... |
| usgrf 29766 | The edge function of a sim... |
| isusgrs 29767 | The property of being a si... |
| usgrfs 29768 | The edge function of a sim... |
| usgrfun 29769 | The edge function of a sim... |
| usgredgss 29770 | The set of edges of a simp... |
| edgusgr 29771 | An edge of a simple graph ... |
| isuspgrop 29772 | The property of being an u... |
| isusgrop 29773 | The property of being an u... |
| usgrop 29774 | A simple graph represented... |
| isausgr 29775 | The property of an ordered... |
| ausgrusgrb 29776 | The equivalence of the def... |
| usgrausgri 29777 | A simple graph represented... |
| ausgrumgri 29778 | If an alternatively define... |
| ausgrusgri 29779 | The equivalence of the def... |
| usgrausgrb 29780 | The equivalence of the def... |
| usgredgop 29781 | An edge of a simple graph ... |
| usgrf1o 29782 | The edge function of a sim... |
| usgrf1 29783 | The edge function of a sim... |
| uspgrf1oedg 29784 | The edge function of a sim... |
| usgrss 29785 | An edge is a subset of ver... |
| uspgredgiedg 29786 | In a simple pseudograph, f... |
| uspgriedgedg 29787 | In a simple pseudograph, f... |
| uspgrushgr 29788 | A simple pseudograph is an... |
| uspgrupgr 29789 | A simple pseudograph is an... |
| uspgrupgrushgr 29790 | A graph is a simple pseudo... |
| usgruspgr 29791 | A simple graph is a simple... |
| usgrumgr 29792 | A simple graph is an undir... |
| usgrumgruspgr 29793 | A graph is a simple graph ... |
| usgruspgrb 29794 | A class is a simple graph ... |
| uspgruhgr 29795 | An undirected simple pseud... |
| usgrupgr 29796 | A simple graph is an undir... |
| usgruhgr 29797 | A simple graph is an undir... |
| usgrislfuspgr 29798 | A simple graph is a loop-f... |
| uspgrun 29799 | The union ` U ` of two sim... |
| uspgrunop 29800 | The union of two simple ps... |
| usgrun 29801 | The union ` U ` of two sim... |
| usgrunop 29802 | The union of two simple gr... |
| usgredg2 29803 | The value of the "edge fun... |
| usgredg2ALT 29804 | Alternate proof of ~ usgre... |
| usgredgprv 29805 | In a simple graph, an edge... |
| usgredgprvALT 29806 | Alternate proof of ~ usgre... |
| usgredgppr 29807 | An edge of a simple graph ... |
| usgrpredgv 29808 | An edge of a simple graph ... |
| edgssv2 29809 | An edge of a simple graph ... |
| usgredg 29810 | For each edge in a simple ... |
| usgrnloopv 29811 | In a simple graph, there i... |
| usgrnloopvALT 29812 | Alternate proof of ~ usgrn... |
| usgrnloop 29813 | In a simple graph, there i... |
| usgrnloopALT 29814 | Alternate proof of ~ usgrn... |
| usgrnloop0 29815 | A simple graph has no loop... |
| usgrnloop0ALT 29816 | Alternate proof of ~ usgrn... |
| usgredgne 29817 | An edge of a simple graph ... |
| usgrf1oedg 29818 | The edge function of a sim... |
| uhgr2edg 29819 | If a vertex is adjacent to... |
| umgr2edg 29820 | If a vertex is adjacent to... |
| usgr2edg 29821 | If a vertex is adjacent to... |
| umgr2edg1 29822 | If a vertex is adjacent to... |
| usgr2edg1 29823 | If a vertex is adjacent to... |
| umgrvad2edg 29824 | If a vertex is adjacent to... |
| umgr2edgneu 29825 | If a vertex is adjacent to... |
| usgrsizedg 29826 | In a simple graph, the siz... |
| usgredg3 29827 | The value of the "edge fun... |
| usgredg4 29828 | For a vertex incident to a... |
| usgredgreu 29829 | For a vertex incident to a... |
| usgredg2vtx 29830 | For a vertex incident to a... |
| uspgredg2vtxeu 29831 | For a vertex incident to a... |
| usgredg2vtxeu 29832 | For a vertex incident to a... |
| usgredg2vtxeuALT 29833 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29834 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29835 | In a simple pseudograph, t... |
| usgredg2vlem1 29836 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29837 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29838 | In a simple graph, the map... |
| usgriedgleord 29839 | Alternate version of ~ usg... |
| ushgredgedg 29840 | In a simple hypergraph the... |
| usgredgedg 29841 | In a simple graph there is... |
| ushgredgedgloop 29842 | In a simple hypergraph the... |
| uspgredgleord 29843 | In a simple pseudograph th... |
| usgredgleord 29844 | In a simple graph the numb... |
| usgredgleordALT 29845 | Alternate proof for ~ usgr... |
| usgrstrrepe 29846 | Replacing (or adding) the ... |
| usgr0e 29847 | The empty graph, with vert... |
| usgr0vb 29848 | The null graph, with no ve... |
| uhgr0v0e 29849 | The null graph, with no ve... |
| uhgr0vsize0 29850 | The size of a hypergraph w... |
| uhgr0edgfi 29851 | A graph of order 0 (i.e. w... |
| usgr0v 29852 | The null graph, with no ve... |
| uhgr0vusgr 29853 | The null graph, with no ve... |
| usgr0 29854 | The null graph represented... |
| uspgr1e 29855 | A simple pseudograph with ... |
| usgr1e 29856 | A simple graph with one ed... |
| usgr0eop 29857 | The empty graph, with vert... |
| uspgr1eop 29858 | A simple pseudograph with ... |
| uspgr1ewop 29859 | A simple pseudograph with ... |
| uspgr1v1eop 29860 | A simple pseudograph with ... |
| usgr1eop 29861 | A simple graph with (at le... |
| uspgr2v1e2w 29862 | A simple pseudograph with ... |
| usgr2v1e2w 29863 | A simple graph with two ve... |
| edg0usgr 29864 | A class without edges is a... |
| lfuhgr1v0e 29865 | A loop-free hypergraph wit... |
| usgr1vr 29866 | A simple graph with one ve... |
| usgr1v 29867 | A class with one (or no) v... |
| usgr1v0edg 29868 | A class with one (or no) v... |
| usgrexmpldifpr 29869 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29870 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29871 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29872 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29873 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29874 | ` G ` is a simple graph of... |
| griedg0prc 29875 | The class of empty graphs ... |
| griedg0ssusgr 29876 | The class of all simple gr... |
| usgrprc 29877 | The class of simple graphs... |
| relsubgr 29880 | The class of the subgraph ... |
| subgrv 29881 | If a class is a subgraph o... |
| issubgr 29882 | The property of a set to b... |
| issubgr2 29883 | The property of a set to b... |
| subgrprop 29884 | The properties of a subgra... |
| subgrprop2 29885 | The properties of a subgra... |
| uhgrissubgr 29886 | The property of a hypergra... |
| subgrprop3 29887 | The properties of a subgra... |
| egrsubgr 29888 | An empty graph consisting ... |
| 0grsubgr 29889 | The null graph (represente... |
| 0uhgrsubgr 29890 | The null graph (as hypergr... |
| uhgrsubgrself 29891 | A hypergraph is a subgraph... |
| subgrfun 29892 | The edge function of a sub... |
| subgruhgrfun 29893 | The edge function of a sub... |
| subgreldmiedg 29894 | An element of the domain o... |
| subgruhgredgd 29895 | An edge of a subgraph of a... |
| subumgredg2 29896 | An edge of a subgraph of a... |
| subuhgr 29897 | A subgraph of a hypergraph... |
| subupgr 29898 | A subgraph of a pseudograp... |
| subumgr 29899 | A subgraph of a multigraph... |
| subusgr 29900 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29901 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29902 | A spanning subgraph ` S ` ... |
| uhgrspan 29903 | A spanning subgraph ` S ` ... |
| upgrspan 29904 | A spanning subgraph ` S ` ... |
| umgrspan 29905 | A spanning subgraph ` S ` ... |
| usgrspan 29906 | A spanning subgraph ` S ` ... |
| uhgrspanop 29907 | A spanning subgraph of a h... |
| upgrspanop 29908 | A spanning subgraph of a p... |
| umgrspanop 29909 | A spanning subgraph of a m... |
| usgrspanop 29910 | A spanning subgraph of a s... |
| uhgrspan1lem1 29911 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29912 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29913 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29914 | The induced subgraph ` S `... |
| upgrreslem 29915 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29916 | Lemma for ~ umgrres and ~ ... |
| upgrres 29917 | A subgraph obtained by rem... |
| umgrres 29918 | A subgraph obtained by rem... |
| usgrres 29919 | A subgraph obtained by rem... |
| upgrres1lem1 29920 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29921 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29922 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29923 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29924 | A pseudograph obtained by ... |
| umgrres1 29925 | A multigraph obtained by r... |
| usgrres1 29926 | Restricting a simple graph... |
| isfusgr 29929 | The property of being a fi... |
| fusgrvtxfi 29930 | A finite simple graph has ... |
| isfusgrf1 29931 | The property of being a fi... |
| isfusgrcl 29932 | The property of being a fi... |
| fusgrusgr 29933 | A finite simple graph is a... |
| opfusgr 29934 | A finite simple graph repr... |
| usgredgffibi 29935 | The number of edges in a s... |
| fusgredgfi 29936 | In a finite simple graph t... |
| usgr1v0e 29937 | The size of a (finite) sim... |
| usgrfilem 29938 | In a finite simple graph, ... |
| fusgrfisbase 29939 | Induction base for ~ fusgr... |
| fusgrfisstep 29940 | Induction step in ~ fusgrf... |
| fusgrfis 29941 | A finite simple graph is o... |
| fusgrfupgrfs 29942 | A finite simple graph is a... |
| nbgrprc0 29945 | The set of neighbors is em... |
| nbgrcl 29946 | If a class ` X ` has at le... |
| nbgrval 29947 | The set of neighbors of a ... |
| dfnbgr2 29948 | Alternate definition of th... |
| dfnbgr3 29949 | Alternate definition of th... |
| nbgrnvtx0 29950 | If a class ` X ` is not a ... |
| nbgrel 29951 | Characterization of a neig... |
| nbgrisvtx 29952 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29953 | The neighbors of a vertex ... |
| nbuhgr 29954 | The set of neighbors of a ... |
| nbupgr 29955 | The set of neighbors of a ... |
| nbupgrel 29956 | A neighbor of a vertex in ... |
| nbumgrvtx 29957 | The set of neighbors of a ... |
| nbumgr 29958 | The set of neighbors of an... |
| nbusgrvtx 29959 | The set of neighbors of a ... |
| nbusgr 29960 | The set of neighbors of an... |
| nbgr2vtx1edg 29961 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29962 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29963 | If a hypergraph has two ve... |
| nbusgreledg 29964 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29965 | A vertex which is not endp... |
| nbgr0vtx 29966 | In a null graph (with no v... |
| nbgr0edglem 29967 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29968 | In an empty graph (with no... |
| nbgr1vtx 29969 | In a graph with one vertex... |
| nbgrnself 29970 | A vertex in a graph is not... |
| nbgrnself2 29971 | A class ` X ` is not a nei... |
| nbgrssovtx 29972 | The neighbors of a vertex ... |
| nbgrssvwo2 29973 | The neighbors of a vertex ... |
| nbgrsym 29974 | In a graph, the neighborho... |
| nbupgrres 29975 | The neighborhood of a vert... |
| usgrnbcnvfv 29976 | Applying the edge function... |
| nbusgredgeu 29977 | For each neighbor of a ver... |
| edgnbusgreu 29978 | For each edge incident to ... |
| nbusgredgeu0 29979 | For each neighbor of a ver... |
| nbusgrf1o0 29980 | The mapping of neighbors o... |
| nbusgrf1o1 29981 | The set of neighbors of a ... |
| nbusgrf1o 29982 | The set of neighbors of a ... |
| nbedgusgr 29983 | The number of neighbors of... |
| edgusgrnbfin 29984 | The number of neighbors of... |
| nbusgrfi 29985 | The class of neighbors of ... |
| nbfiusgrfi 29986 | The class of neighbors of ... |
| hashnbusgrnn0 29987 | The number of neighbors of... |
| nbfusgrlevtxm1 29988 | The number of neighbors of... |
| nbfusgrlevtxm2 29989 | If there is a vertex which... |
| nbusgrvtxm1 29990 | If the number of neighbors... |
| nb3grprlem1 29991 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29992 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29993 | The neighbors of a vertex ... |
| nb3grpr2 29994 | The neighbors of a vertex ... |
| nb3gr2nb 29995 | If the neighbors of two ve... |
| uvtxval 29998 | The set of all universal v... |
| uvtxel 29999 | A universal vertex, i.e. a... |
| uvtxisvtx 30000 | A universal vertex is a ve... |
| uvtxssvtx 30001 | The set of the universal v... |
| vtxnbuvtx 30002 | A universal vertex has all... |
| uvtxnbgrss 30003 | A universal vertex has all... |
| uvtxnbgrvtx 30004 | A universal vertex is neig... |
| uvtx0 30005 | There is no universal vert... |
| isuvtx 30006 | The set of all universal v... |
| uvtxel1 30007 | Characterization of a univ... |
| uvtx01vtx 30008 | If a graph/class has no ed... |
| uvtx2vtx1edg 30009 | If a graph has two vertice... |
| uvtx2vtx1edgb 30010 | If a hypergraph has two ve... |
| uvtxnbgr 30011 | A universal vertex has all... |
| uvtxnbgrb 30012 | A vertex is universal iff ... |
| uvtxusgr 30013 | The set of all universal v... |
| uvtxusgrel 30014 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 30015 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 30016 | If the number of neighbors... |
| uvtxnbvtxm1 30017 | A universal vertex has ` n... |
| nbupgruvtxres 30018 | The neighborhood of a univ... |
| uvtxupgrres 30019 | A universal vertex is univ... |
| cplgruvtxb 30024 | A graph ` G ` is complete ... |
| prcliscplgr 30025 | A proper class (representi... |
| iscplgr 30026 | The property of being a co... |
| iscplgrnb 30027 | A graph is complete iff al... |
| iscplgredg 30028 | A graph ` G ` is complete ... |
| iscusgr 30029 | The property of being a co... |
| cusgrusgr 30030 | A complete simple graph is... |
| cusgrcplgr 30031 | A complete simple graph is... |
| iscusgrvtx 30032 | A simple graph is complete... |
| cusgruvtxb 30033 | A simple graph is complete... |
| iscusgredg 30034 | A simple graph is complete... |
| cusgredg 30035 | In a complete simple graph... |
| cplgr0 30036 | The null graph (with no ve... |
| cusgr0 30037 | The null graph (with no ve... |
| cplgr0v 30038 | A null graph (with no vert... |
| cusgr0v 30039 | A graph with no vertices a... |
| cplgr1vlem 30040 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 30041 | A graph with one vertex is... |
| cusgr1v 30042 | A graph with one vertex an... |
| cplgr2v 30043 | An undirected hypergraph w... |
| cplgr2vpr 30044 | An undirected hypergraph w... |
| nbcplgr 30045 | In a complete graph, each ... |
| cplgr3v 30046 | A pseudograph with three (... |
| cusgr3vnbpr 30047 | The neighbors of a vertex ... |
| cplgrop 30048 | A complete graph represent... |
| cusgrop 30049 | A complete simple graph re... |
| cusgrexilem1 30050 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 30051 | Lemma for ~ usgrexi . (Co... |
| usgrexi 30052 | An arbitrary set regarded ... |
| cusgrexilem2 30053 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 30054 | An arbitrary set ` V ` reg... |
| cusgrexg 30055 | For each set there is a se... |
| structtousgr 30056 | Any (extensible) structure... |
| structtocusgr 30057 | Any (extensible) structure... |
| cffldtocusgr 30058 | The field of complex numbe... |
| cusgrres 30059 | Restricting a complete sim... |
| cusgrsizeindb0 30060 | Base case of the induction... |
| cusgrsizeindb1 30061 | Base case of the induction... |
| cusgrsizeindslem 30062 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 30063 | Part 1 of induction step i... |
| cusgrsize2inds 30064 | Induction step in ~ cusgrs... |
| cusgrsize 30065 | The size of a finite compl... |
| cusgrfilem1 30066 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 30067 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 30068 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 30069 | If the size of a complete ... |
| usgredgsscusgredg 30070 | A simple graph is a subgra... |
| usgrsscusgr 30071 | A simple graph is a subgra... |
| sizusglecusglem1 30072 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 30073 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 30074 | The size of a simple graph... |
| fusgrmaxsize 30075 | The maximum size of a fini... |
| vtxdgfval 30078 | The value of the vertex de... |
| vtxdgval 30079 | The degree of a vertex. (... |
| vtxdgfival 30080 | The degree of a vertex for... |
| vtxdgop 30081 | The vertex degree expresse... |
| vtxdgf 30082 | The vertex degree function... |
| vtxdgelxnn0 30083 | The degree of a vertex is ... |
| vtxdg0v 30084 | The degree of a vertex in ... |
| vtxdg0e 30085 | The degree of a vertex in ... |
| vtxdgfisnn0 30086 | The degree of a vertex in ... |
| vtxdgfisf 30087 | The vertex degree function... |
| vtxdeqd 30088 | Equality theorem for the v... |
| vtxduhgr0e 30089 | The degree of a vertex in ... |
| vtxdlfuhgr1v 30090 | The degree of the vertex i... |
| vdumgr0 30091 | A vertex in a multigraph h... |
| vtxdun 30092 | The degree of a vertex in ... |
| vtxdfiun 30093 | The degree of a vertex in ... |
| vtxduhgrun 30094 | The degree of a vertex in ... |
| vtxduhgrfiun 30095 | The degree of a vertex in ... |
| vtxdlfgrval 30096 | The value of the vertex de... |
| vtxdumgrval 30097 | The value of the vertex de... |
| vtxdusgrval 30098 | The value of the vertex de... |
| vtxd0nedgb 30099 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 30100 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 30101 | The value of the vertex de... |
| vtxdusgrfvedg 30102 | The value of the vertex de... |
| vtxduhgr0nedg 30103 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 30104 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 30105 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 30106 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 30107 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 30108 | The vertex degree function... |
| vtxdgfusgr 30109 | In a finite simple graph, ... |
| fusgrn0degnn0 30110 | In a nonempty, finite grap... |
| 1loopgruspgr 30111 | A graph with one edge whic... |
| 1loopgredg 30112 | The set of edges in a grap... |
| 1loopgrnb0 30113 | In a graph (simple pseudog... |
| 1loopgrvd2 30114 | The vertex degree of a one... |
| 1loopgrvd0 30115 | The vertex degree of a one... |
| 1hevtxdg0 30116 | The vertex degree of verte... |
| 1hevtxdg1 30117 | The vertex degree of verte... |
| 1hegrvtxdg1 30118 | The vertex degree of a gra... |
| 1hegrvtxdg1r 30119 | The vertex degree of a gra... |
| 1egrvtxdg1 30120 | The vertex degree of a one... |
| 1egrvtxdg1r 30121 | The vertex degree of a one... |
| 1egrvtxdg0 30122 | The vertex degree of a one... |
| p1evtxdeqlem 30123 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 30124 | If an edge ` E ` which doe... |
| p1evtxdp1 30125 | If an edge ` E ` (not bein... |
| uspgrloopvtx 30126 | The set of vertices in a g... |
| uspgrloopvtxel 30127 | A vertex in a graph (simpl... |
| uspgrloopiedg 30128 | The set of edges in a grap... |
| uspgrloopedg 30129 | The set of edges in a grap... |
| uspgrloopnb0 30130 | In a graph (simple pseudog... |
| uspgrloopvd2 30131 | The vertex degree of a one... |
| umgr2v2evtx 30132 | The set of vertices in a m... |
| umgr2v2evtxel 30133 | A vertex in a multigraph w... |
| umgr2v2eiedg 30134 | The edge function in a mul... |
| umgr2v2eedg 30135 | The set of edges in a mult... |
| umgr2v2e 30136 | A multigraph with two edge... |
| umgr2v2enb1 30137 | In a multigraph with two e... |
| umgr2v2evd2 30138 | In a multigraph with two e... |
| hashnbusgrvd 30139 | In a simple graph, the num... |
| usgruvtxvdb 30140 | In a finite simple graph w... |
| vdiscusgrb 30141 | A finite simple graph with... |
| vdiscusgr 30142 | In a finite complete simpl... |
| vtxdusgradjvtx 30143 | The degree of a vertex in ... |
| usgrvd0nedg 30144 | If a vertex in a simple gr... |
| uhgrvd00 30145 | If every vertex in a hyper... |
| usgrvd00 30146 | If every vertex in a simpl... |
| vdegp1ai 30147 | The induction step for a v... |
| vdegp1bi 30148 | The induction step for a v... |
| vdegp1ci 30149 | The induction step for a v... |
| vtxdginducedm1lem1 30150 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 30151 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 30152 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 30153 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 30154 | The degree of a vertex ` v... |
| vtxdginducedm1fi 30155 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 30156 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 30157 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 30158 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 30159 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 30160 | Induction step of ~ finsum... |
| finsumvtxdg2size 30161 | The sum of the degrees of ... |
| fusgr1th 30162 | The sum of the degrees of ... |
| finsumvtxdgeven 30163 | The sum of the degrees of ... |
| vtxdgoddnumeven 30164 | The number of vertices of ... |
| fusgrvtxdgonume 30165 | The number of vertices of ... |
| isrgr 30170 | The property of a class be... |
| rgrprop 30171 | The properties of a k-regu... |
| isrusgr 30172 | The property of being a k-... |
| rusgrprop 30173 | The properties of a k-regu... |
| rusgrrgr 30174 | A k-regular simple graph i... |
| rusgrusgr 30175 | A k-regular simple graph i... |
| finrusgrfusgr 30176 | A finite regular simple gr... |
| isrusgr0 30177 | The property of being a k-... |
| rusgrprop0 30178 | The properties of a k-regu... |
| usgreqdrusgr 30179 | If all vertices in a simpl... |
| fusgrregdegfi 30180 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 30181 | If all vertices in a nonem... |
| frusgrnn0 30182 | In a nonempty finite k-reg... |
| 0edg0rgr 30183 | A graph is 0-regular if it... |
| uhgr0edg0rgr 30184 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 30185 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 30186 | A simple graph is 0-regula... |
| 0vtxrgr 30187 | A null graph (with no vert... |
| 0vtxrusgr 30188 | A graph with no vertices a... |
| 0uhgrrusgr 30189 | The null graph as hypergra... |
| 0grrusgr 30190 | The null graph represented... |
| 0grrgr 30191 | The null graph represented... |
| cusgrrusgr 30192 | A complete simple graph wi... |
| cusgrm1rusgr 30193 | A finite simple graph with... |
| rusgrpropnb 30194 | The properties of a k-regu... |
| rusgrpropedg 30195 | The properties of a k-regu... |
| rusgrpropadjvtx 30196 | The properties of a k-regu... |
| rusgrnumwrdl2 30197 | In a k-regular simple grap... |
| rusgr1vtxlem 30198 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 30199 | If a k-regular simple grap... |
| rgrusgrprc 30200 | The class of 0-regular sim... |
| rusgrprc 30201 | The class of 0-regular sim... |
| rgrprc 30202 | The class of 0-regular gra... |
| rgrprcx 30203 | The class of 0-regular gra... |
| rgrx0ndm 30204 | 0 is not in the domain of ... |
| rgrx0nd 30205 | The potentially alternativ... |
| ewlksfval 30212 | The set of s-walks of edge... |
| isewlk 30213 | Conditions for a function ... |
| ewlkprop 30214 | Properties of an s-walk of... |
| ewlkinedg 30215 | The intersection (common v... |
| ewlkle 30216 | An s-walk of edges is also... |
| upgrewlkle2 30217 | In a pseudograph, there is... |
| wkslem1 30218 | Lemma 1 for walks to subst... |
| wkslem2 30219 | Lemma 2 for walks to subst... |
| wksfval 30220 | The set of walks (in an un... |
| iswlk 30221 | Properties of a pair of fu... |
| wlkprop 30222 | Properties of a walk. (Co... |
| wlkv 30223 | The classes involved in a ... |
| iswlkg 30224 | Generalization of ~ iswlk ... |
| wlkf 30225 | The mapping enumerating th... |
| wlkcl 30226 | A walk has length ` # ( F ... |
| wlkp 30227 | The mapping enumerating th... |
| wlkpwrd 30228 | The sequence of vertices o... |
| wlklenvp1 30229 | The number of vertices of ... |
| wksv 30230 | The class of walks is a se... |
| wlkn0 30231 | The sequence of vertices o... |
| wlklenvm1 30232 | The number of edges of a w... |
| ifpsnprss 30233 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 30234 | Each pair of adjacent vert... |
| wlkvtxiedg 30235 | The vertices of a walk are... |
| relwlk 30236 | The set ` ( Walks `` G ) `... |
| wlkvv 30237 | If there is at least one w... |
| wlkop 30238 | A walk is an ordered pair.... |
| wlkcpr 30239 | A walk as class with two c... |
| wlk2f 30240 | If there is a walk ` W ` t... |
| wlkcomp 30241 | A walk expressed by proper... |
| wlkcompim 30242 | Implications for the prope... |
| wlkelwrd 30243 | The components of a walk a... |
| wlkeq 30244 | Conditions for two walks (... |
| edginwlk 30245 | The value of the edge func... |
| upgredginwlk 30246 | The value of the edge func... |
| iedginwlk 30247 | The value of the edge func... |
| wlkl1loop 30248 | A walk of length 1 from a ... |
| wlk1walk 30249 | A walk is a 1-walk "on the... |
| wlk1ewlk 30250 | A walk is an s-walk "on th... |
| upgriswlk 30251 | Properties of a pair of fu... |
| upgrwlkedg 30252 | The edges of a walk in a p... |
| upgrwlkcompim 30253 | Implications for the prope... |
| wlkvtxedg 30254 | The vertices of a walk are... |
| upgrwlkvtxedg 30255 | The pairs of connected ver... |
| uspgr2wlkeq 30256 | Conditions for two walks w... |
| uspgr2wlkeq2 30257 | Conditions for two walks w... |
| uspgr2wlkeqi 30258 | Conditions for two walks w... |
| umgrwlknloop 30259 | In a multigraph, each walk... |
| wlkv0 30260 | If there is a walk in the ... |
| g0wlk0 30261 | There is no walk in a null... |
| 0wlk0 30262 | There is no walk for the e... |
| wlk0prc 30263 | There is no walk in a null... |
| wlklenvclwlk 30264 | The number of vertices in ... |
| wlkson 30265 | The set of walks between t... |
| iswlkon 30266 | Properties of a pair of fu... |
| wlkonprop 30267 | Properties of a walk betwe... |
| wlkpvtx 30268 | A walk connects vertices. ... |
| wlkepvtx 30269 | The endpoints of a walk ar... |
| wlkoniswlk 30270 | A walk between two vertice... |
| wlkonwlk 30271 | A walk is a walk between i... |
| wlkonwlk1l 30272 | A walk is a walk from its ... |
| wlksoneq1eq2 30273 | Two walks with identical s... |
| wlkonl1iedg 30274 | If there is a walk between... |
| wlkon2n0 30275 | The length of a walk betwe... |
| 2wlklem 30276 | Lemma for theorems for wal... |
| upgr2wlk 30277 | Properties of a pair of fu... |
| wlkreslem 30278 | Lemma for ~ wlkres . (Con... |
| wlkres 30279 | The restriction ` <. H , Q... |
| redwlklem 30280 | Lemma for ~ redwlk . (Con... |
| redwlk 30281 | A walk ending at the last ... |
| wlkp1lem1 30282 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 30283 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 30284 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 30285 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 30286 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 30287 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 30288 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 30289 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 30290 | Append one path segment (e... |
| wlkdlem1 30291 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 30292 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 30293 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 30294 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 30295 | Two words representing a w... |
| pfxwlk 30296 | A prefix of a walk is a wa... |
| revwlk 30297 | The reverse of a walk is a... |
| swrdwlk 30298 | Two matching subwords of a... |
| subgrwlk 30299 | If a walk exists in a subg... |
| lfgrwlkprop 30300 | Two adjacent vertices in a... |
| lfgriswlk 30301 | Conditions for a pair of f... |
| lfgrwlknloop 30302 | In a loop-free graph, each... |
| reltrls 30307 | The set ` ( Trails `` G ) ... |
| trlsfval 30308 | The set of trails (in an u... |
| istrl 30309 | Conditions for a pair of c... |
| trliswlk 30310 | A trail is a walk. (Contr... |
| trlf1 30311 | The enumeration ` F ` of a... |
| trlreslem 30312 | Lemma for ~ trlres . Form... |
| trlres 30313 | The restriction ` <. H , Q... |
| upgrtrls 30314 | The set of trails in a pse... |
| upgristrl 30315 | Properties of a pair of fu... |
| upgrf1istrl 30316 | Properties of a pair of a ... |
| wksonproplem 30317 | Lemma for theorems for pro... |
| trlsonfval 30318 | The set of trails between ... |
| istrlson 30319 | Properties of a pair of fu... |
| trlsonprop 30320 | Properties of a trail betw... |
| trlsonistrl 30321 | A trail between two vertic... |
| trlsonwlkon 30322 | A trail between two vertic... |
| trlontrl 30323 | A trail is a trail between... |
| subgrtrl 30324 | If a trail exists in a sub... |
| relpths 30333 | The set ` ( Paths `` G ) `... |
| pthsfval 30334 | The set of paths (in an un... |
| spthsfval 30335 | The set of simple paths (i... |
| ispth 30336 | Conditions for a pair of c... |
| isspth 30337 | Conditions for a pair of c... |
| pthistrl 30338 | A path is a trail (in an u... |
| spthispth 30339 | A simple path is a path (i... |
| pthiswlk 30340 | A path is a walk (in an un... |
| spthiswlk 30341 | A simple path is a walk (i... |
| pthdivtx 30342 | The inner vertices of a pa... |
| pthdadjvtx 30343 | The adjacent vertices of a... |
| dfpth2 30344 | Alternate definition for a... |
| pthhashvtx 30345 | A graph containing a path ... |
| pthdifv 30346 | The vertices of a path are... |
| 2pthnloop 30347 | A path of length at least ... |
| upgr2pthnlp 30348 | A path of length at least ... |
| spthdifv 30349 | The vertices of a simple p... |
| spthdep 30350 | A simple path (at least of... |
| pthdepisspth 30351 | A path with different star... |
| upgrwlkdvdelem 30352 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 30353 | In a pseudograph, all edge... |
| upgrspthswlk 30354 | The set of simple paths in... |
| upgrwlkdvspth 30355 | A walk consisting of diffe... |
| pthsonfval 30356 | The set of paths between t... |
| spthson 30357 | The set of simple paths be... |
| ispthson 30358 | Properties of a pair of fu... |
| isspthson 30359 | Properties of a pair of fu... |
| pthsonprop 30360 | Properties of a path betwe... |
| spthonprop 30361 | Properties of a simple pat... |
| pthonispth 30362 | A path between two vertice... |
| pthontrlon 30363 | A path between two vertice... |
| pthonpth 30364 | A path is a path between i... |
| isspthonpth 30365 | A pair of functions is a s... |
| spthonisspth 30366 | A simple path between to v... |
| spthonpthon 30367 | A simple path between two ... |
| spthonepeq 30368 | The endpoints of a simple ... |
| uhgrwkspthlem1 30369 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 30370 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 30371 | Any walk of length 1 betwe... |
| usgr2wlkneq 30372 | The vertices and edges are... |
| usgr2wlkspthlem1 30373 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 30374 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 30375 | In a simple graph, any wal... |
| usgr2trlncl 30376 | In a simple graph, any tra... |
| usgr2trlspth 30377 | In a simple graph, any tra... |
| usgr2pthspth 30378 | In a simple graph, any pat... |
| usgr2pthlem 30379 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 30380 | In a simple graph, there i... |
| usgr2pth0 30381 | In a simply graph, there i... |
| pthdlem1 30382 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 30383 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 30384 | Lemma 2 for ~ pthd . (Con... |
| pthd 30385 | Two words representing a t... |
| subgrpth 30386 | If a path exists in a subg... |
| clwlks 30389 | The set of closed walks (i... |
| isclwlk 30390 | A pair of functions repres... |
| clwlkiswlk 30391 | A closed walk is a walk (i... |
| clwlkwlk 30392 | Closed walks are walks (in... |
| clwlkswks 30393 | Closed walks are walks (in... |
| isclwlke 30394 | Properties of a pair of fu... |
| isclwlkupgr 30395 | Properties of a pair of fu... |
| clwlkcomp 30396 | A closed walk expressed by... |
| clwlkcompim 30397 | Implications for the prope... |
| upgrclwlkcompim 30398 | Implications for the prope... |
| clwlkcompbp 30399 | Basic properties of the co... |
| clwlkl1loop 30400 | A closed walk of length 1 ... |
| crcts 30405 | The set of circuits (in an... |
| cycls 30406 | The set of cycles (in an u... |
| iscrct 30407 | Sufficient and necessary c... |
| iscycl 30408 | Sufficient and necessary c... |
| crctprop 30409 | The properties of a circui... |
| cyclprop 30410 | The properties of a cycle:... |
| crctisclwlk 30411 | A circuit is a closed walk... |
| crctistrl 30412 | A circuit is a trail. (Co... |
| crctiswlk 30413 | A circuit is a walk. (Con... |
| cyclispth 30414 | A cycle is a path. (Contr... |
| subgrcycl 30415 | If a cycle exists in a sub... |
| cycliswlk 30416 | A cycle is a walk. (Contr... |
| cycliscrct 30417 | A cycle is a circuit. (Co... |
| cyclnumvtx 30418 | The number of vertices of ... |
| cyclnspth 30419 | A (non-trivial) cycle is n... |
| pthisspthorcycl 30420 | A path is either a simple ... |
| pthspthcyc 30421 | A pair ` <. F , P >. ` rep... |
| spthcycl 30422 | A walk is a trivial path i... |
| cyclispthon 30423 | A cycle is a path starting... |
| lfgrn1cycl 30424 | In a loop-free graph there... |
| usgr2trlncrct 30425 | In a simple graph, any tra... |
| umgrn1cycl 30426 | In a multigraph graph (wit... |
| uspgrn2crct 30427 | In a simple pseudograph th... |
| usgrn2cycl 30428 | In a simple graph there ar... |
| crctcshwlkn0lem1 30429 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 30430 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 30431 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 30432 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 30433 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 30434 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 30435 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 30436 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 30437 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 30438 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 30439 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 30440 | Cyclically shifting the in... |
| crctcshwlk 30441 | Cyclically shifting the in... |
| crctcshtrl 30442 | Cyclically shifting the in... |
| crctcsh 30443 | Cyclically shifting the in... |
| wwlks 30454 | The set of walks (in an un... |
| iswwlks 30455 | A word over the set of ver... |
| wwlksn 30456 | The set of walks (in an un... |
| iswwlksn 30457 | A word over the set of ver... |
| wwlksnprcl 30458 | Derivation of the length o... |
| iswwlksnx 30459 | Properties of a word to re... |
| wwlkbp 30460 | Basic properties of a walk... |
| wwlknbp 30461 | Basic properties of a walk... |
| wwlknp 30462 | Properties of a set being ... |
| wwlknbp1 30463 | Other basic properties of ... |
| wwlknvtx 30464 | The symbols of a word ` W ... |
| wwlknllvtx 30465 | If a word ` W ` represents... |
| wwlknlsw 30466 | If a word represents a wal... |
| wspthsn 30467 | The set of simple paths of... |
| iswspthn 30468 | An element of the set of s... |
| wspthnp 30469 | Properties of a set being ... |
| wwlksnon 30470 | The set of walks of a fixe... |
| wspthsnon 30471 | The set of simple paths of... |
| iswwlksnon 30472 | The set of walks of a fixe... |
| wwlksnon0 30473 | Sufficient conditions for ... |
| wwlksonvtx 30474 | If a word ` W ` represents... |
| iswspthsnon 30475 | The set of simple paths of... |
| wwlknon 30476 | An element of the set of w... |
| wspthnon 30477 | An element of the set of s... |
| wspthnonp 30478 | Properties of a set being ... |
| wspthneq1eq2 30479 | Two simple paths with iden... |
| wwlksn0s 30480 | The set of all walks as wo... |
| wwlkssswrd 30481 | Walks (represented by word... |
| wwlksn0 30482 | A walk of length 0 is repr... |
| 0enwwlksnge1 30483 | In graphs without edges, t... |
| wwlkswwlksn 30484 | A walk of a fixed length a... |
| wwlkssswwlksn 30485 | The walks of a fixed lengt... |
| wlkiswwlks1 30486 | The sequence of vertices i... |
| wlklnwwlkln1 30487 | The sequence of vertices i... |
| wlkiswwlks2lem1 30488 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 30489 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 30490 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 30491 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 30492 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 30493 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 30494 | A walk as word corresponds... |
| wlkiswwlks 30495 | A walk as word corresponds... |
| wlkiswwlksupgr2 30496 | A walk as word corresponds... |
| wlkiswwlkupgr 30497 | A walk as word corresponds... |
| wlkswwlksf1o 30498 | The mapping of (ordinary) ... |
| wlkswwlksen 30499 | The set of walks as words ... |
| wwlksm1edg 30500 | Removing the trailing edge... |
| wlklnwwlkln2lem 30501 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 30502 | A walk of length ` N ` as ... |
| wlklnwwlkn 30503 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 30504 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 30505 | A walk of length ` N ` as ... |
| wlknewwlksn 30506 | If a walk in a pseudograph... |
| wlknwwlksnbij 30507 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 30508 | In a simple pseudograph, t... |
| wlknwwlksneqs 30509 | The set of walks of a fixe... |
| wwlkseq 30510 | Equality of two walks (as ... |
| wwlksnred 30511 | Reduction of a walk (as wo... |
| wwlksnext 30512 | Extension of a walk (as wo... |
| wwlksnextbi 30513 | Extension of a walk (as wo... |
| wwlksnredwwlkn 30514 | For each walk (as word) of... |
| wwlksnredwwlkn0 30515 | For each walk (as word) of... |
| wwlksnextwrd 30516 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 30517 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 30518 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 30519 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 30520 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 30521 | There is a bijection betwe... |
| wwlksnexthasheq 30522 | The number of the extensio... |
| disjxwwlksn 30523 | Sets of walks (as words) e... |
| wwlksnndef 30524 | Conditions for ` WWalksN `... |
| wwlksnfi 30525 | The number of walks repres... |
| wlksnfi 30526 | The number of walks of fix... |
| wlksnwwlknvbij 30527 | There is a bijection betwe... |
| wwlksnextproplem1 30528 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 30529 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 30530 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 30531 | Adding additional properti... |
| disjxwwlkn 30532 | Sets of walks (as words) e... |
| hashwwlksnext 30533 | Number of walks (as words)... |
| wwlksnwwlksnon 30534 | A walk of fixed length is ... |
| wspthsnwspthsnon 30535 | A simple path of fixed len... |
| wspthsnonn0vne 30536 | If the set of simple paths... |
| wspthsswwlkn 30537 | The set of simple paths of... |
| wspthnfi 30538 | In a finite graph, the set... |
| wwlksnonfi 30539 | In a finite graph, the set... |
| wspthsswwlknon 30540 | The set of simple paths of... |
| wspthnonfi 30541 | In a finite graph, the set... |
| wspniunwspnon 30542 | The set of nonempty simple... |
| wspn0 30543 | If there are no vertices, ... |
| 2wlkdlem1 30544 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 30545 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 30546 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 30547 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 30548 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 30549 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 30550 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 30551 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 30552 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 30553 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 30554 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 30555 | Construction of a walk fro... |
| 2wlkond 30556 | A walk of length 2 from on... |
| 2trld 30557 | Construction of a trail fr... |
| 2trlond 30558 | A trail of length 2 from o... |
| 2pthd 30559 | A path of length 2 from on... |
| 2spthd 30560 | A simple path of length 2 ... |
| 2pthond 30561 | A simple path of length 2 ... |
| 2pthon3v 30562 | For a vertex adjacent to t... |
| umgr2adedgwlklem 30563 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 30564 | In a multigraph, two adjac... |
| umgr2adedgwlkon 30565 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 30566 | Alternate proof for ~ umgr... |
| umgr2adedgspth 30567 | In a multigraph, two adjac... |
| umgr2wlk 30568 | In a multigraph, there is ... |
| umgr2wlkon 30569 | For each pair of adjacent ... |
| elwwlks2s3 30570 | A walk of length 2 as word... |
| midwwlks2s3 30571 | There is a vertex between ... |
| wwlks2onv 30572 | If a length 3 string repre... |
| elwwlks2ons3im 30573 | A walk as word of length 2... |
| elwwlks2ons3 30574 | For each walk of length 2 ... |
| s3wwlks2on 30575 | A length 3 string which re... |
| sps3wwlks2on 30576 | A length 3 string which re... |
| usgrwwlks2on 30577 | A walk of length 2 between... |
| umgrwwlks2on 30578 | A walk of length 2 between... |
| wwlks2onsym 30579 | There is a walk of length ... |
| elwwlks2on 30580 | A walk of length 2 between... |
| elwspths2on 30581 | A simple path of length 2 ... |
| elwspths2onw 30582 | A simple path of length 2 ... |
| wpthswwlks2on 30583 | For two different vertices... |
| 2wspdisj 30584 | All simple paths of length... |
| 2wspiundisj 30585 | All simple paths of length... |
| usgr2wspthons3 30586 | A simple path of length 2 ... |
| usgr2wspthon 30587 | A simple path of length 2 ... |
| elwwlks2 30588 | A walk of length 2 between... |
| elwspths2spth 30589 | A simple path of length 2 ... |
| rusgrnumwwlkl1 30590 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 30591 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 30592 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 30593 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 30594 | Induction base 1 for ~ rus... |
| rusgr0edg 30595 | Special case for graphs wi... |
| rusgrnumwwlks 30596 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30597 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30598 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30599 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30600 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30601 | In a ` K `-regular graph, ... |
| clwwlk 30604 | The set of closed walks (i... |
| isclwwlk 30605 | Properties of a word to re... |
| clwwlkbp 30606 | Basic properties of a clos... |
| clwwlkgt0 30607 | There is no empty closed w... |
| clwwlksswrd 30608 | Closed walks (represented ... |
| clwwlk1loop 30609 | A closed walk of length 1 ... |
| clwwlkccatlem 30610 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30611 | The concatenation of two w... |
| umgrclwwlkge2 30612 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30613 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30614 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30615 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30616 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30617 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30618 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30619 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30620 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30621 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30622 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30623 | A closed walk as word of l... |
| clwlkclwwlk2 30624 | A closed walk corresponds ... |
| clwlkclwwlkflem 30625 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30626 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30627 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30628 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30629 | ` F ` is a function from t... |
| clwlkclwwlkfo 30630 | ` F ` is a function from t... |
| clwlkclwwlkf1 30631 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30632 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30633 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30634 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30635 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30636 | Cyclically shifting a clos... |
| clwwisshclwwsn 30637 | Cyclically shifting a clos... |
| erclwwlkrel 30638 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30639 | Two classes are equivalent... |
| erclwwlkeqlen 30640 | If two classes are equival... |
| erclwwlkref 30641 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30642 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30643 | ` .~ ` is a transitive rel... |
| erclwwlk 30644 | ` .~ ` is an equivalence r... |
| clwwlkn 30647 | The set of closed walks of... |
| isclwwlkn 30648 | A word over the set of ver... |
| clwwlkn0 30649 | There is no closed walk of... |
| clwwlkneq0 30650 | Sufficient conditions for ... |
| clwwlkclwwlkn 30651 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30652 | The closed walks of a fixe... |
| clwwlknlen 30653 | The length of a word repre... |
| clwwlknnn 30654 | The length of a closed wal... |
| clwwlknwrd 30655 | A closed walk of a fixed l... |
| clwwlknbp 30656 | Basic properties of a clos... |
| isclwwlknx 30657 | Characterization of a word... |
| clwwlknp 30658 | Properties of a set being ... |
| clwwlknwwlksn 30659 | A word representing a clos... |
| clwwlknlbonbgr1 30660 | The last but one vertex in... |
| clwwlkinwwlk 30661 | If the initial vertex of a... |
| clwwlkn1 30662 | A closed walk of length 1 ... |
| loopclwwlkn1b 30663 | The singleton word consist... |
| clwwlkn1loopb 30664 | A word represents a closed... |
| clwwlkn2 30665 | A closed walk of length 2 ... |
| clwwlknfi 30666 | If there is only a finite ... |
| clwwlkel 30667 | Obtaining a closed walk (a... |
| clwwlkf 30668 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30669 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30670 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30671 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30672 | F is a 1-1 onto function, ... |
| clwwlken 30673 | The set of closed walks of... |
| clwwlknwwlkncl 30674 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30675 | A nonempty word over verti... |
| clwwlknwwlksnb 30676 | A word over vertices repre... |
| clwwlkext2edg 30677 | If a word concatenated wit... |
| wwlksext2clwwlk 30678 | If a word represents a wal... |
| wwlksubclwwlk 30679 | Any prefix of a word repre... |
| clwwnisshclwwsn 30680 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30681 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30682 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30683 | The set of cyclical shifts... |
| clwwlknccat 30684 | The concatenation of two w... |
| umgr2cwwk2dif 30685 | If a word represents a clo... |
| umgr2cwwkdifex 30686 | If a word represents a clo... |
| erclwwlknrel 30687 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30688 | Two classes are equivalent... |
| erclwwlkneqlen 30689 | If two classes are equival... |
| erclwwlknref 30690 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30691 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30692 | ` .~ ` is a transitive rel... |
| erclwwlkn 30693 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30694 | The quotient set of the se... |
| hashclwwlkn0 30695 | The number of closed walks... |
| eclclwwlkn1 30696 | An equivalence class accor... |
| eleclclwwlkn 30697 | A member of an equivalence... |
| hashecclwwlkn1 30698 | The size of every equivale... |
| umgrhashecclwwlk 30699 | The size of every equivale... |
| fusgrhashclwwlkn 30700 | The size of the set of clo... |
| clwwlkndivn 30701 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30702 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30703 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30704 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30705 | There is a one-to-one onto... |
| clwlkssizeeq 30706 | The size of the set of clo... |
| clwlksndivn 30707 | The size of the set of clo... |
| clwwlknonmpo 30710 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30711 | The set of closed walks on... |
| isclwwlknon 30712 | A word over the set of ver... |
| clwwlk0on0 30713 | There is no word over the ... |
| clwwlknon0 30714 | Sufficient conditions for ... |
| clwwlknonfin 30715 | In a finite graph ` G ` , ... |
| clwwlknonel 30716 | Characterization of a word... |
| clwwlknonccat 30717 | The concatenation of two w... |
| clwwlknon1 30718 | The set of closed walks on... |
| clwwlknon1loop 30719 | If there is a loop at vert... |
| clwwlknon1nloop 30720 | If there is no loop at ver... |
| clwwlknon1sn 30721 | The set of (closed) walks ... |
| clwwlknon1le1 30722 | There is at most one (clos... |
| clwwlknon2 30723 | The set of closed walks on... |
| clwwlknon2x 30724 | The set of closed walks on... |
| s2elclwwlknon2 30725 | Sufficient conditions of a... |
| clwwlknon2num 30726 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30727 | A word over vertices repre... |
| clwwlknonex2lem1 30728 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30729 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30730 | Extending a closed walk ` ... |
| clwwlknonex2e 30731 | Extending a closed walk ` ... |
| clwwlknondisj 30732 | The sets of closed walks o... |
| clwwlknun 30733 | The set of closed walks of... |
| clwwlkvbij 30734 | There is a bijection betwe... |
| 0ewlk 30735 | The empty set (empty seque... |
| 1ewlk 30736 | A sequence of 1 edge is an... |
| 0wlk 30737 | A pair of an empty set (of... |
| is0wlk 30738 | A pair of an empty set (of... |
| 0wlkonlem1 30739 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30740 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30741 | A walk of length 0 from a ... |
| 0wlkons1 30742 | A walk of length 0 from a ... |
| 0trl 30743 | A pair of an empty set (of... |
| is0trl 30744 | A pair of an empty set (of... |
| 0trlon 30745 | A trail of length 0 from a... |
| 0pth 30746 | A pair of an empty set (of... |
| 0spth 30747 | A pair of an empty set (of... |
| 0pthon 30748 | A path of length 0 from a ... |
| 0pthon1 30749 | A path of length 0 from a ... |
| 0pthonv 30750 | For each vertex there is a... |
| 0clwlk 30751 | A pair of an empty set (of... |
| 0clwlkv 30752 | Any vertex (more precisely... |
| 0clwlk0 30753 | There is no closed walk in... |
| 0crct 30754 | A pair of an empty set (of... |
| 0cycl 30755 | A pair of an empty set (of... |
| 1pthdlem1 30756 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30757 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30758 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30759 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30760 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30761 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30762 | In a graph with two vertic... |
| 1trld 30763 | In a graph with two vertic... |
| 1pthd 30764 | In a graph with two vertic... |
| 1pthond 30765 | In a graph with two vertic... |
| upgr1wlkdlem1 30766 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30767 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30768 | In a pseudograph with two ... |
| upgr1trld 30769 | In a pseudograph with two ... |
| upgr1pthd 30770 | In a pseudograph with two ... |
| upgr1pthond 30771 | In a pseudograph with two ... |
| lppthon 30772 | A loop (which is an edge a... |
| lp1cycl 30773 | A loop (which is an edge a... |
| loop1cycl 30774 | A hypergraph has a cycle o... |
| 2cycld 30775 | Construction of a 2-cycle ... |
| umgr2cycllem 30776 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 30777 | A multigraph with two dist... |
| dfacycgr1 30780 | An alternate definition of... |
| isacycgr 30781 | The property of being an a... |
| isacycgr1 30782 | The property of being an a... |
| acycgrcycl 30783 | Any cycle in an acyclic gr... |
| 1pthon2v 30784 | For each pair of adjacent ... |
| 1pthon2ve 30785 | For each pair of adjacent ... |
| wlk2v2elem1 30786 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30787 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30788 | In a graph with two vertic... |
| ntrl2v2e 30789 | A walk which is not a trai... |
| 3wlkdlem1 30790 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30791 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30792 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30793 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30794 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30795 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30796 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30797 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30798 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30799 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30800 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30801 | Construction of a walk fro... |
| 3wlkond 30802 | A walk of length 3 from on... |
| 3trld 30803 | Construction of a trail fr... |
| 3trlond 30804 | A trail of length 3 from o... |
| 3pthd 30805 | A path of length 3 from on... |
| 3pthond 30806 | A path of length 3 from on... |
| 3spthd 30807 | A simple path of length 3 ... |
| 3spthond 30808 | A simple path of length 3 ... |
| 3cycld 30809 | Construction of a 3-cycle ... |
| 3cyclpd 30810 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30811 | If there is a cycle of len... |
| uhgr3cyclexlem 30812 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30813 | If there are three differe... |
| umgr3cyclex 30814 | If there are three (differ... |
| umgr3v3e3cycl 30815 | If and only if there is a ... |
| upgr4cycl4dv4e 30816 | If there is a cycle of len... |
| dfconngr1 30819 | Alternative definition of ... |
| isconngr 30820 | The property of being a co... |
| isconngr1 30821 | The property of being a co... |
| cusconngr 30822 | A complete hypergraph is c... |
| 0conngr 30823 | A graph without vertices i... |
| 0vconngr 30824 | A graph without vertices i... |
| 1conngr 30825 | A graph with (at most) one... |
| conngrv2edg 30826 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30827 | A vertex in a connected mu... |
| releupth 30830 | The set ` ( EulerPaths `` ... |
| eupths 30831 | The Eulerian paths on the ... |
| iseupth 30832 | The property " ` <. F , P ... |
| iseupthf1o 30833 | The property " ` <. F , P ... |
| eupthi 30834 | Properties of an Eulerian ... |
| eupthf1o 30835 | The ` F ` function in an E... |
| eupthfi 30836 | Any graph with an Eulerian... |
| eupthseg 30837 | The ` N ` -th edge in an e... |
| upgriseupth 30838 | The property " ` <. F , P ... |
| upgreupthi 30839 | Properties of an Eulerian ... |
| upgreupthseg 30840 | The ` N ` -th edge in an e... |
| eupthcl 30841 | An Eulerian path has lengt... |
| eupthistrl 30842 | An Eulerian path is a trai... |
| eupthiswlk 30843 | An Eulerian path is a walk... |
| eupthpf 30844 | The ` P ` function in an E... |
| eupth0 30845 | There is an Eulerian path ... |
| eupthres 30846 | The restriction ` <. H , Q... |
| eupthp1 30847 | Append one path segment to... |
| eupth2eucrct 30848 | Append one path segment to... |
| eupth2lem1 30849 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30850 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30851 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30852 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30853 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30854 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30855 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30856 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30857 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30858 | The effect on vertex degre... |
| eupth2lem3lem1 30859 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30860 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30861 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30862 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30863 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30864 | Formerly part of proof of ... |
| eupth2lem3lem7 30865 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30866 | Formerly part of proof of ... |
| eupth2lem3 30867 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30868 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30869 | Lemma for ~ eupth2 (induct... |
| eupth2 30870 | The only vertices of odd d... |
| eulerpathpr 30871 | A graph with an Eulerian p... |
| eulerpath 30872 | A pseudograph with an Eule... |
| eulercrct 30873 | A pseudograph with an Eule... |
| eucrctshift 30874 | Cyclically shifting the in... |
| eucrct2eupth1 30875 | Removing one edge ` ( I ``... |
| eucrct2eupth 30876 | Removing one edge ` ( I ``... |
| konigsbergvtx 30877 | The set of vertices of the... |
| konigsbergiedg 30878 | The indexed edges of the K... |
| konigsbergiedgw 30879 | The indexed edges of the K... |
| konigsbergssiedgwpr 30880 | Each subset of the indexed... |
| konigsbergssiedgw 30881 | Each subset of the indexed... |
| konigsbergumgr 30882 | The Königsberg graph ... |
| konigsberglem1 30883 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30884 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30885 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30886 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30887 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30888 | The Königsberg Bridge... |
| isfrgr 30891 | The property of being a fr... |
| frgrusgr 30892 | A friendship graph is a si... |
| frgr0v 30893 | Any null graph (set with n... |
| frgr0vb 30894 | Any null graph (without ve... |
| frgruhgr0v 30895 | Any null graph (without ve... |
| frgr0 30896 | The null graph (graph with... |
| frcond1 30897 | The friendship condition: ... |
| frcond2 30898 | The friendship condition: ... |
| frgreu 30899 | Variant of ~ frcond2 : An... |
| frcond3 30900 | The friendship condition, ... |
| frcond4 30901 | The friendship condition, ... |
| frgr1v 30902 | Any graph with (at most) o... |
| nfrgr2v 30903 | Any graph with two (differ... |
| frgr3vlem1 30904 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30905 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30906 | Any graph with three verti... |
| 1vwmgr 30907 | Every graph with one verte... |
| 3vfriswmgrlem 30908 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30909 | Every friendship graph wit... |
| 1to2vfriswmgr 30910 | Every friendship graph wit... |
| 1to3vfriswmgr 30911 | Every friendship graph wit... |
| 1to3vfriendship 30912 | The friendship theorem for... |
| 2pthfrgrrn 30913 | Between any two (different... |
| 2pthfrgrrn2 30914 | Between any two (different... |
| 2pthfrgr 30915 | Between any two (different... |
| 3cyclfrgrrn1 30916 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30917 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30918 | Every vertex in a friendsh... |
| 3cyclfrgr 30919 | Every vertex in a friendsh... |
| 4cycl2v2nb 30920 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30921 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30922 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30923 | A graph with a 4-cycle is ... |
| frgrnbnb 30924 | If two neighbors ` U ` and... |
| frgrconngr 30925 | A friendship graph is conn... |
| vdgn0frgrv2 30926 | A vertex in a friendship g... |
| vdgn1frgrv2 30927 | Any vertex in a friendship... |
| vdgn1frgrv3 30928 | Any vertex in a friendship... |
| vdgfrgrgt2 30929 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30930 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30931 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30932 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30933 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30934 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30935 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30936 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30937 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30938 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30939 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30940 | In a friendship graph, two... |
| frgrwopreglem4a 30941 | In a friendship graph any ... |
| frgrwopreglem5a 30942 | If a friendship graph has ... |
| frgrwopreglem1 30943 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30944 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30945 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30946 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30947 | According to statement 5 i... |
| frgrwopregbsn 30948 | According to statement 5 i... |
| frgrwopreg1 30949 | According to statement 5 i... |
| frgrwopreg2 30950 | According to statement 5 i... |
| frgrwopreglem5lem 30951 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30952 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30953 | Alternate direct proof of ... |
| frgrwopreg 30954 | In a friendship graph ther... |
| frgrregorufr0 30955 | In a friendship graph ther... |
| frgrregorufr 30956 | If there is a vertex havin... |
| frgrregorufrg 30957 | If there is a vertex havin... |
| frgr2wwlkeu 30958 | For two different vertices... |
| frgr2wwlkn0 30959 | In a friendship graph, the... |
| frgr2wwlk1 30960 | In a friendship graph, the... |
| frgr2wsp1 30961 | In a friendship graph, the... |
| frgr2wwlkeqm 30962 | If there is a (simple) pat... |
| frgrhash2wsp 30963 | The number of simple paths... |
| fusgreg2wsplem 30964 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30965 | The set of paths of length... |
| fusgreghash2wspv 30966 | According to statement 7 i... |
| fusgreg2wsp 30967 | In a finite simple graph, ... |
| 2wspmdisj 30968 | The sets of paths of lengt... |
| fusgreghash2wsp 30969 | In a finite k-regular grap... |
| frrusgrord0lem 30970 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30971 | If a nonempty finite frien... |
| frrusgrord 30972 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30973 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30974 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30975 | If the initial vertex of a... |
| clwwnonrepclwwnon 30976 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30977 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30978 | Value of operation ` C ` ,... |
| 2clwwlk2 30979 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30980 | Characterization of an ele... |
| 2clwwlk2clwwlk 30981 | An element of the value of... |
| numclwwlk1lem2foalem 30982 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30983 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30984 | Characterization of an ele... |
| numclwwlk1lem2foa 30985 | Going forth and back from ... |
| numclwwlk1lem2f 30986 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30987 | Value of the function ` T ... |
| numclwwlk1lem2f1 30988 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30989 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30990 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30991 | The set of double loops of... |
| numclwwlk1 30992 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30993 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30994 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30995 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30996 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30997 | The sets of the two repres... |
| wlkl0 30998 | There is exactly one walk ... |
| clwlknon2num 30999 | There are k walks of lengt... |
| numclwlk1lem1 31000 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 31001 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 31002 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 31003 | Value of operation ` H ` ,... |
| numclwwlkovh 31004 | Value of operation ` H ` ,... |
| numclwwlkovq 31005 | Value of operation ` Q ` ,... |
| numclwwlkqhash 31006 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 31007 | In a friendship graph, for... |
| numclwlk2lem2f 31008 | ` R ` is a function mappin... |
| numclwlk2lem2fv 31009 | Value of the function ` R ... |
| numclwlk2lem2f1o 31010 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 31011 | In a friendship graph, the... |
| numclwwlk2 31012 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 31013 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 31014 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 31015 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 31016 | Statement 12 in [Huneke] p... |
| numclwwlk4 31017 | The total number of closed... |
| numclwwlk5lem 31018 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 31019 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 31020 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 31021 | For a prime divisor ` P ` ... |
| numclwwlk7 31022 | Statement 14 in [Huneke] p... |
| numclwwlk8 31023 | The size of the set of clo... |
| frgrreggt1 31024 | If a finite nonempty frien... |
| frgrreg 31025 | If a finite nonempty frien... |
| frgrregord013 31026 | If a finite friendship gra... |
| frgrregord13 31027 | If a nonempty finite frien... |
| frgrogt3nreg 31028 | If a finite friendship gra... |
| friendshipgt3 31029 | The friendship theorem for... |
| friendship 31030 | The friendship theorem: I... |
| conventions 31031 |
H... |
| conventions-labels 31032 |
... |
| conventions-comments 31033 |
... |
| natded 31034 | Here are typical n... |
| ex-natded5.2 31035 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 31036 | A more efficient proof of ... |
| ex-natded5.2i 31037 | The same as ~ ex-natded5.2... |
| ex-natded5.3 31038 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 31039 | A more efficient proof of ... |
| ex-natded5.3i 31040 | The same as ~ ex-natded5.3... |
| ex-natded5.5 31041 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 31042 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 31043 | A more efficient proof of ... |
| ex-natded5.8 31044 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 31045 | A more efficient proof of ... |
| ex-natded5.13 31046 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 31047 | A more efficient proof of ... |
| ex-natded9.20 31048 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 31049 | A more efficient proof of ... |
| ex-natded9.26 31050 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 31051 | A more efficient proof of ... |
| ex-or 31052 | Example for ~ df-or . Exa... |
| ex-an 31053 | Example for ~ df-an . Exa... |
| ex-dif 31054 | Example for ~ df-dif . Ex... |
| ex-un 31055 | Example for ~ df-un . Exa... |
| ex-in 31056 | Example for ~ df-in . Exa... |
| ex-uni 31057 | Example for ~ df-uni . Ex... |
| ex-ss 31058 | Example for ~ df-ss . Exa... |
| ex-pss 31059 | Example for ~ df-pss . Ex... |
| ex-pw 31060 | Example for ~ df-pw . Exa... |
| ex-pr 31061 | Example for ~ df-pr . (Co... |
| ex-br 31062 | Example for ~ df-br . Exa... |
| ex-opab 31063 | Example for ~ df-opab . E... |
| ex-eprel 31064 | Example for ~ df-eprel . ... |
| ex-id 31065 | Example for ~ df-id . Exa... |
| ex-po 31066 | Example for ~ df-po . Exa... |
| ex-xp 31067 | Example for ~ df-xp . Exa... |
| ex-cnv 31068 | Example for ~ df-cnv . Ex... |
| ex-co 31069 | Example for ~ df-co . Exa... |
| ex-dm 31070 | Example for ~ df-dm . Exa... |
| ex-rn 31071 | Example for ~ df-rn . Exa... |
| ex-res 31072 | Example for ~ df-res . Ex... |
| ex-ima 31073 | Example for ~ df-ima . Ex... |
| ex-fv 31074 | Example for ~ df-fv . Exa... |
| ex-1st 31075 | Example for ~ df-1st . Ex... |
| ex-2nd 31076 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 31077 | Example for ~ df-dec , 100... |
| ex-fl 31078 | Example for ~ df-fl . Exa... |
| ex-ceil 31079 | Example for ~ df-ceil . (... |
| ex-mod 31080 | Example for ~ df-mod . (C... |
| ex-exp 31081 | Example for ~ df-exp . (C... |
| ex-fac 31082 | Example for ~ df-fac . (C... |
| ex-bc 31083 | Example for ~ df-bc . (Co... |
| ex-hash 31084 | Example for ~ df-hash . (... |
| ex-sqrt 31085 | Example for ~ df-sqrt . (... |
| ex-abs 31086 | Example for ~ df-abs . (C... |
| ex-dvds 31087 | Example for ~ df-dvds : 3 ... |
| ex-gcd 31088 | Example for ~ df-gcd . (C... |
| ex-lcm 31089 | Example for ~ df-lcm . (C... |
| ex-prmo 31090 | Example for ~ df-prmo : ` ... |
| aevdemo 31091 | Proof illustrating the com... |
| ex-ind-dvds 31092 | Example of a proof by indu... |
| ex-fpar 31093 | Formalized example provide... |
| avril1 31094 | Poisson d'Avril's Theorem.... |
| 2bornot2b 31095 | The law of excluded middle... |
| helloworld 31096 | The classic "Hello world" ... |
| 1p1e2apr1 31097 | One plus one equals two. ... |
| eqid1 31098 | Law of identity (reflexivi... |
| 1div0apr 31099 | Division by zero is forbid... |
| topnfbey 31100 | Nothing seems to be imposs... |
| 9p10ne21 31101 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 31102 | 9 + 10 equals 21. This as... |
| nrt2irr 31104 | The ` N ` -th root of 2 is... |
| nowisdomv 31105 | One's wisdom on matters of... |
| isplig 31108 | The predicate "is a planar... |
| ispligb 31109 | The predicate "is a planar... |
| tncp 31110 | In any planar incidence ge... |
| l2p 31111 | For any line in a planar i... |
| lpni 31112 | For any line in a planar i... |
| nsnlplig 31113 | There is no "one-point lin... |
| nsnlpligALT 31114 | Alternate version of ~ nsn... |
| n0lplig 31115 | There is no "empty line" i... |
| n0lpligALT 31116 | Alternate version of ~ n0l... |
| eulplig 31117 | Through two distinct point... |
| pliguhgr 31118 | Any planar incidence geome... |
| dummylink 31119 | Alias for ~ a1ii that may ... |
| id1 31120 | Alias for ~ idALT that may... |
| isgrpo 31129 | The predicate "is a group ... |
| isgrpoi 31130 | Properties that determine ... |
| grpofo 31131 | A group operation maps ont... |
| grpocl 31132 | Closure law for a group op... |
| grpolidinv 31133 | A group has a left identit... |
| grpon0 31134 | The base set of a group is... |
| grpoass 31135 | A group operation is assoc... |
| grpoidinvlem1 31136 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 31137 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 31138 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 31139 | Lemma for ~ grpoidinv . (... |
| grpoidinv 31140 | A group has a left and rig... |
| grpoideu 31141 | The left identity element ... |
| grporndm 31142 | A group's range in terms o... |
| 0ngrp 31143 | The empty set is not a gro... |
| gidval 31144 | The value of the identity ... |
| grpoidval 31145 | Lemma for ~ grpoidcl and o... |
| grpoidcl 31146 | The identity element of a ... |
| grpoidinv2 31147 | A group's properties using... |
| grpolid 31148 | The identity element of a ... |
| grporid 31149 | The identity element of a ... |
| grporcan 31150 | Right cancellation law for... |
| grpoinveu 31151 | The left inverse element o... |
| grpoid 31152 | Two ways of saying that an... |
| grporn 31153 | The range of a group opera... |
| grpoinvfval 31154 | The inverse function of a ... |
| grpoinvval 31155 | The inverse of a group ele... |
| grpoinvcl 31156 | A group element's inverse ... |
| grpoinv 31157 | The properties of a group ... |
| grpolinv 31158 | The left inverse of a grou... |
| grporinv 31159 | The right inverse of a gro... |
| grpoinvid1 31160 | The inverse of a group ele... |
| grpoinvid2 31161 | The inverse of a group ele... |
| grpolcan 31162 | Left cancellation law for ... |
| grpo2inv 31163 | Double inverse law for gro... |
| grpoinvf 31164 | Mapping of the inverse fun... |
| grpoinvop 31165 | The inverse of the group o... |
| grpodivfval 31166 | Group division (or subtrac... |
| grpodivval 31167 | Group division (or subtrac... |
| grpodivinv 31168 | Group division by an inver... |
| grpoinvdiv 31169 | Inverse of a group divisio... |
| grpodivf 31170 | Mapping for group division... |
| grpodivcl 31171 | Closure of group division ... |
| grpodivdiv 31172 | Double group division. (C... |
| grpomuldivass 31173 | Associative-type law for m... |
| grpodivid 31174 | Division of a group member... |
| grponpcan 31175 | Cancellation law for group... |
| isablo 31178 | The predicate "is an Abeli... |
| ablogrpo 31179 | An Abelian group operation... |
| ablocom 31180 | An Abelian group operation... |
| ablo32 31181 | Commutative/associative la... |
| ablo4 31182 | Commutative/associative la... |
| isabloi 31183 | Properties that determine ... |
| ablomuldiv 31184 | Law for group multiplicati... |
| ablodivdiv 31185 | Law for double group divis... |
| ablodivdiv4 31186 | Law for double group divis... |
| ablodiv32 31187 | Swap the second and third ... |
| ablonncan 31188 | Cancellation law for group... |
| ablonnncan1 31189 | Cancellation law for group... |
| vcrel 31192 | The class of all complex v... |
| vciOLD 31193 | Obsolete version of ~ cvsi... |
| vcsm 31194 | Functionality of the scala... |
| vccl 31195 | Closure of the scalar prod... |
| vcidOLD 31196 | Identity element for the s... |
| vcdi 31197 | Distributive law for the s... |
| vcdir 31198 | Distributive law for the s... |
| vcass 31199 | Associative law for the sc... |
| vc2OLD 31200 | A vector plus itself is tw... |
| vcablo 31201 | Vector addition is an Abel... |
| vcgrp 31202 | Vector addition is a group... |
| vclcan 31203 | Left cancellation law for ... |
| vczcl 31204 | The zero vector is a vecto... |
| vc0rid 31205 | The zero vector is a right... |
| vc0 31206 | Zero times a vector is the... |
| vcz 31207 | Anything times the zero ve... |
| vcm 31208 | Minus 1 times a vector is ... |
| isvclem 31209 | Lemma for ~ isvcOLD . (Co... |
| vcex 31210 | The components of a comple... |
| isvcOLD 31211 | The predicate "is a comple... |
| isvciOLD 31212 | Properties that determine ... |
| cnaddabloOLD 31213 | Obsolete version of ~ cnad... |
| cnidOLD 31214 | Obsolete version of ~ cnad... |
| cncvcOLD 31215 | Obsolete version of ~ cncv... |
| nvss 31225 | Structure of the class of ... |
| nvvcop 31226 | A normed complex vector sp... |
| nvrel 31234 | The class of all normed co... |
| vafval 31235 | Value of the function for ... |
| bafval 31236 | Value of the function for ... |
| smfval 31237 | Value of the function for ... |
| 0vfval 31238 | Value of the function for ... |
| nmcvfval 31239 | Value of the norm function... |
| nvop2 31240 | A normed complex vector sp... |
| nvvop 31241 | The vector space component... |
| isnvlem 31242 | Lemma for ~ isnv . (Contr... |
| nvex 31243 | The components of a normed... |
| isnv 31244 | The predicate "is a normed... |
| isnvi 31245 | Properties that determine ... |
| nvi 31246 | The properties of a normed... |
| nvvc 31247 | The vector space component... |
| nvablo 31248 | The vector addition operat... |
| nvgrp 31249 | The vector addition operat... |
| nvgf 31250 | Mapping for the vector add... |
| nvsf 31251 | Mapping for the scalar mul... |
| nvgcl 31252 | Closure law for the vector... |
| nvcom 31253 | The vector addition (group... |
| nvass 31254 | The vector addition (group... |
| nvadd32 31255 | Commutative/associative la... |
| nvrcan 31256 | Right cancellation law for... |
| nvadd4 31257 | Rearrangement of 4 terms i... |
| nvscl 31258 | Closure law for the scalar... |
| nvsid 31259 | Identity element for the s... |
| nvsass 31260 | Associative law for the sc... |
| nvscom 31261 | Commutative law for the sc... |
| nvdi 31262 | Distributive law for the s... |
| nvdir 31263 | Distributive law for the s... |
| nv2 31264 | A vector plus itself is tw... |
| vsfval 31265 | Value of the function for ... |
| nvzcl 31266 | Closure law for the zero v... |
| nv0rid 31267 | The zero vector is a right... |
| nv0lid 31268 | The zero vector is a left ... |
| nv0 31269 | Zero times a vector is the... |
| nvsz 31270 | Anything times the zero ve... |
| nvinv 31271 | Minus 1 times a vector is ... |
| nvinvfval 31272 | Function for the negative ... |
| nvm 31273 | Vector subtraction in term... |
| nvmval 31274 | Value of vector subtractio... |
| nvmval2 31275 | Value of vector subtractio... |
| nvmfval 31276 | Value of the function for ... |
| nvmf 31277 | Mapping for the vector sub... |
| nvmcl 31278 | Closure law for the vector... |
| nvnnncan1 31279 | Cancellation law for vecto... |
| nvmdi 31280 | Distributive law for scala... |
| nvnegneg 31281 | Double negative of a vecto... |
| nvmul0or 31282 | If a scalar product is zer... |
| nvrinv 31283 | A vector minus itself. (C... |
| nvlinv 31284 | Minus a vector plus itself... |
| nvpncan2 31285 | Cancellation law for vecto... |
| nvpncan 31286 | Cancellation law for vecto... |
| nvaddsub 31287 | Commutative/associative la... |
| nvnpcan 31288 | Cancellation law for a nor... |
| nvaddsub4 31289 | Rearrangement of 4 terms i... |
| nvmeq0 31290 | The difference between two... |
| nvmid 31291 | A vector minus itself is t... |
| nvf 31292 | Mapping for the norm funct... |
| nvcl 31293 | The norm of a normed compl... |
| nvcli 31294 | The norm of a normed compl... |
| nvs 31295 | Proportionality property o... |
| nvsge0 31296 | The norm of a scalar produ... |
| nvm1 31297 | The norm of the negative o... |
| nvdif 31298 | The norm of the difference... |
| nvpi 31299 | The norm of a vector plus ... |
| nvz0 31300 | The norm of a zero vector ... |
| nvz 31301 | The norm of a vector is ze... |
| nvtri 31302 | Triangle inequality for th... |
| nvmtri 31303 | Triangle inequality for th... |
| nvabs 31304 | Norm difference property o... |
| nvge0 31305 | The norm of a normed compl... |
| nvgt0 31306 | A nonzero norm is positive... |
| nv1 31307 | From any nonzero vector, c... |
| nvop 31308 | A complex inner product sp... |
| cnnv 31309 | The set of complex numbers... |
| cnnvg 31310 | The vector addition (group... |
| cnnvba 31311 | The base set of the normed... |
| cnnvs 31312 | The scalar product operati... |
| cnnvnm 31313 | The norm operation of the ... |
| cnnvm 31314 | The vector subtraction ope... |
| elimnv 31315 | Hypothesis elimination lem... |
| elimnvu 31316 | Hypothesis elimination lem... |
| imsval 31317 | Value of the induced metri... |
| imsdval 31318 | Value of the induced metri... |
| imsdval2 31319 | Value of the distance func... |
| nvnd 31320 | The norm of a normed compl... |
| imsdf 31321 | Mapping for the induced me... |
| imsmetlem 31322 | Lemma for ~ imsmet . (Con... |
| imsmet 31323 | The induced metric of a no... |
| imsxmet 31324 | The induced metric of a no... |
| cnims 31325 | The metric induced on the ... |
| vacn 31326 | Vector addition is jointly... |
| nmcvcn 31327 | The norm of a normed compl... |
| nmcnc 31328 | The norm of a normed compl... |
| smcnlem 31329 | Lemma for ~ smcn . (Contr... |
| smcn 31330 | Scalar multiplication is j... |
| vmcn 31331 | Vector subtraction is join... |
| dipfval 31334 | The inner product function... |
| ipval 31335 | Value of the inner product... |
| ipval2lem2 31336 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 31337 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 31338 | Lemma for ~ ipval3 . (Con... |
| ipval2 31339 | Expansion of the inner pro... |
| 4ipval2 31340 | Four times the inner produ... |
| ipval3 31341 | Expansion of the inner pro... |
| ipidsq 31342 | The inner product of a vec... |
| ipnm 31343 | Norm expressed in terms of... |
| dipcl 31344 | An inner product is a comp... |
| ipf 31345 | Mapping for the inner prod... |
| dipcj 31346 | The complex conjugate of a... |
| ipipcj 31347 | An inner product times its... |
| diporthcom 31348 | Orthogonality (meaning inn... |
| dip0r 31349 | Inner product with a zero ... |
| dip0l 31350 | Inner product with a zero ... |
| ipz 31351 | The inner product of a vec... |
| dipcn 31352 | Inner product is jointly c... |
| sspval 31355 | The set of all subspaces o... |
| isssp 31356 | The predicate "is a subspa... |
| sspid 31357 | A normed complex vector sp... |
| sspnv 31358 | A subspace is a normed com... |
| sspba 31359 | The base set of a subspace... |
| sspg 31360 | Vector addition on a subsp... |
| sspgval 31361 | Vector addition on a subsp... |
| ssps 31362 | Scalar multiplication on a... |
| sspsval 31363 | Scalar multiplication on a... |
| sspmlem 31364 | Lemma for ~ sspm and other... |
| sspmval 31365 | Vector addition on a subsp... |
| sspm 31366 | Vector subtraction on a su... |
| sspz 31367 | The zero vector of a subsp... |
| sspn 31368 | The norm on a subspace is ... |
| sspnval 31369 | The norm on a subspace in ... |
| sspimsval 31370 | The induced metric on a su... |
| sspims 31371 | The induced metric on a su... |
| lnoval 31384 | The set of linear operator... |
| islno 31385 | The predicate "is a linear... |
| lnolin 31386 | Basic linearity property o... |
| lnof 31387 | A linear operator is a map... |
| lno0 31388 | The value of a linear oper... |
| lnocoi 31389 | The composition of two lin... |
| lnoadd 31390 | Addition property of a lin... |
| lnosub 31391 | Subtraction property of a ... |
| lnomul 31392 | Scalar multiplication prop... |
| nvo00 31393 | Two ways to express a zero... |
| nmoofval 31394 | The operator norm function... |
| nmooval 31395 | The operator norm function... |
| nmosetre 31396 | The set in the supremum of... |
| nmosetn0 31397 | The set in the supremum of... |
| nmoxr 31398 | The norm of an operator is... |
| nmooge0 31399 | The norm of an operator is... |
| nmorepnf 31400 | The norm of an operator is... |
| nmoreltpnf 31401 | The norm of any operator i... |
| nmogtmnf 31402 | The norm of an operator is... |
| nmoolb 31403 | A lower bound for an opera... |
| nmoubi 31404 | An upper bound for an oper... |
| nmoub3i 31405 | An upper bound for an oper... |
| nmoub2i 31406 | An upper bound for an oper... |
| nmobndi 31407 | Two ways to express that a... |
| nmounbi 31408 | Two ways two express that ... |
| nmounbseqi 31409 | An unbounded operator dete... |
| nmounbseqiALT 31410 | Alternate shorter proof of... |
| nmobndseqi 31411 | A bounded sequence determi... |
| nmobndseqiALT 31412 | Alternate shorter proof of... |
| bloval 31413 | The class of bounded linea... |
| isblo 31414 | The predicate "is a bounde... |
| isblo2 31415 | The predicate "is a bounde... |
| bloln 31416 | A bounded operator is a li... |
| blof 31417 | A bounded operator is an o... |
| nmblore 31418 | The norm of a bounded oper... |
| 0ofval 31419 | The zero operator between ... |
| 0oval 31420 | Value of the zero operator... |
| 0oo 31421 | The zero operator is an op... |
| 0lno 31422 | The zero operator is linea... |
| nmoo0 31423 | The operator norm of the z... |
| 0blo 31424 | The zero operator is a bou... |
| nmlno0lem 31425 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 31426 | The norm of a linear opera... |
| nmlno0 31427 | The norm of a linear opera... |
| nmlnoubi 31428 | An upper bound for the ope... |
| nmlnogt0 31429 | The norm of a nonzero line... |
| lnon0 31430 | The domain of a nonzero li... |
| nmblolbii 31431 | A lower bound for the norm... |
| nmblolbi 31432 | A lower bound for the norm... |
| isblo3i 31433 | The predicate "is a bounde... |
| blo3i 31434 | Properties that determine ... |
| blometi 31435 | Upper bound for the distan... |
| blocnilem 31436 | Lemma for ~ blocni and ~ l... |
| blocni 31437 | A linear operator is conti... |
| lnocni 31438 | If a linear operator is co... |
| blocn 31439 | A linear operator is conti... |
| blocn2 31440 | A bounded linear operator ... |
| ajfval 31441 | The adjoint function. (Co... |
| hmoval 31442 | The set of Hermitian (self... |
| ishmo 31443 | The predicate "is a hermit... |
| phnv 31446 | Every complex inner produc... |
| phrel 31447 | The class of all complex i... |
| phnvi 31448 | Every complex inner produc... |
| isphg 31449 | The predicate "is a comple... |
| phop 31450 | A complex inner product sp... |
| cncph 31451 | The set of complex numbers... |
| elimph 31452 | Hypothesis elimination lem... |
| elimphu 31453 | Hypothesis elimination lem... |
| isph 31454 | The predicate "is an inner... |
| phpar2 31455 | The parallelogram law for ... |
| phpar 31456 | The parallelogram law for ... |
| ip0i 31457 | A slight variant of Equati... |
| ip1ilem 31458 | Lemma for ~ ip1i . (Contr... |
| ip1i 31459 | Equation 6.47 of [Ponnusam... |
| ip2i 31460 | Equation 6.48 of [Ponnusam... |
| ipdirilem 31461 | Lemma for ~ ipdiri . (Con... |
| ipdiri 31462 | Distributive law for inner... |
| ipasslem1 31463 | Lemma for ~ ipassi . Show... |
| ipasslem2 31464 | Lemma for ~ ipassi . Show... |
| ipasslem3 31465 | Lemma for ~ ipassi . Show... |
| ipasslem4 31466 | Lemma for ~ ipassi . Show... |
| ipasslem5 31467 | Lemma for ~ ipassi . Show... |
| ipasslem7 31468 | Lemma for ~ ipassi . Show... |
| ipasslem8 31469 | Lemma for ~ ipassi . By ~... |
| ipasslem9 31470 | Lemma for ~ ipassi . Conc... |
| ipasslem10 31471 | Lemma for ~ ipassi . Show... |
| ipasslem11 31472 | Lemma for ~ ipassi . Show... |
| ipassi 31473 | Associative law for inner ... |
| dipdir 31474 | Distributive law for inner... |
| dipdi 31475 | Distributive law for inner... |
| ip2dii 31476 | Inner product of two sums.... |
| dipass 31477 | Associative law for inner ... |
| dipassr 31478 | "Associative" law for seco... |
| dipassr2 31479 | "Associative" law for inne... |
| dipsubdir 31480 | Distributive law for inner... |
| dipsubdi 31481 | Distributive law for inner... |
| pythi 31482 | The Pythagorean theorem fo... |
| siilem1 31483 | Lemma for ~ sii . (Contri... |
| siilem2 31484 | Lemma for ~ sii . (Contri... |
| siii 31485 | Inference from ~ sii . (C... |
| sii 31486 | Obsolete version of ~ ipca... |
| ipblnfi 31487 | A function ` F ` generated... |
| ip2eqi 31488 | Two vectors are equal iff ... |
| phoeqi 31489 | A condition implying that ... |
| ajmoi 31490 | Every operator has at most... |
| ajfuni 31491 | The adjoint function is a ... |
| ajfun 31492 | The adjoint function is a ... |
| ajval 31493 | Value of the adjoint funct... |
| iscbn 31496 | A complex Banach space is ... |
| cbncms 31497 | The induced metric on comp... |
| bnnv 31498 | Every complex Banach space... |
| bnrel 31499 | The class of all complex B... |
| bnsscmcl 31500 | A subspace of a Banach spa... |
| cnbn 31501 | The set of complex numbers... |
| ubthlem1 31502 | Lemma for ~ ubth . The fu... |
| ubthlem2 31503 | Lemma for ~ ubth . Given ... |
| ubthlem3 31504 | Lemma for ~ ubth . Prove ... |
| ubth 31505 | Uniform Boundedness Theore... |
| minvecolem1 31506 | Lemma for ~ minveco . The... |
| minvecolem2 31507 | Lemma for ~ minveco . Any... |
| minvecolem3 31508 | Lemma for ~ minveco . The... |
| minvecolem4a 31509 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 31510 | Lemma for ~ minveco . The... |
| minvecolem4c 31511 | Lemma for ~ minveco . The... |
| minvecolem4 31512 | Lemma for ~ minveco . The... |
| minvecolem5 31513 | Lemma for ~ minveco . Dis... |
| minvecolem6 31514 | Lemma for ~ minveco . Any... |
| minvecolem7 31515 | Lemma for ~ minveco . Sin... |
| minveco 31516 | Minimizing vector theorem,... |
| ishlo 31519 | The predicate "is a comple... |
| hlobn 31520 | Every complex Hilbert spac... |
| hlph 31521 | Every complex Hilbert spac... |
| hlrel 31522 | The class of all complex H... |
| hlnv 31523 | Every complex Hilbert spac... |
| hlnvi 31524 | Every complex Hilbert spac... |
| hlvc 31525 | Every complex Hilbert spac... |
| hlcmet 31526 | The induced metric on a co... |
| hlmet 31527 | The induced metric on a co... |
| hlpar2 31528 | The parallelogram law sati... |
| hlpar 31529 | The parallelogram law sati... |
| hlex 31530 | The base set of a Hilbert ... |
| hladdf 31531 | Mapping for Hilbert space ... |
| hlcom 31532 | Hilbert space vector addit... |
| hlass 31533 | Hilbert space vector addit... |
| hl0cl 31534 | The Hilbert space zero vec... |
| hladdid 31535 | Hilbert space addition wit... |
| hlmulf 31536 | Mapping for Hilbert space ... |
| hlmulid 31537 | Hilbert space scalar multi... |
| hlmulass 31538 | Hilbert space scalar multi... |
| hldi 31539 | Hilbert space scalar multi... |
| hldir 31540 | Hilbert space scalar multi... |
| hlmul0 31541 | Hilbert space scalar multi... |
| hlipf 31542 | Mapping for Hilbert space ... |
| hlipcj 31543 | Conjugate law for Hilbert ... |
| hlipdir 31544 | Distributive law for Hilbe... |
| hlipass 31545 | Associative law for Hilber... |
| hlipgt0 31546 | The inner product of a Hil... |
| hlcompl 31547 | Completeness of a Hilbert ... |
| cnchl 31548 | The set of complex numbers... |
| htthlem 31549 | Lemma for ~ htth . The co... |
| htth 31550 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 31606 | The group (addition) opera... |
| h2hsm 31607 | The scalar product operati... |
| h2hnm 31608 | The norm function of Hilbe... |
| h2hvs 31609 | The vector subtraction ope... |
| h2hmetdval 31610 | Value of the distance func... |
| h2hcau 31611 | The Cauchy sequences of Hi... |
| h2hlm 31612 | The limit sequences of Hil... |
| axhilex-zf 31613 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31614 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31615 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31616 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31617 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31618 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31619 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31620 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31621 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31622 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31623 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31624 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31625 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31626 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31627 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31628 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31629 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31630 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31643 | The Hilbert space scalar p... |
| hvaddcl 31644 | Closure of vector addition... |
| hvmulcl 31645 | Closure of scalar multipli... |
| hvmulcli 31646 | Closure inference for scal... |
| hvsubf 31647 | Mapping domain and codomai... |
| hvsubval 31648 | Value of vector subtractio... |
| hvsubcl 31649 | Closure of vector subtract... |
| hvaddcli 31650 | Closure of vector addition... |
| hvcomi 31651 | Commutation of vector addi... |
| hvsubvali 31652 | Value of vector subtractio... |
| hvsubcli 31653 | Closure of vector subtract... |
| ifhvhv0 31654 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31655 | Addition with the zero vec... |
| hvmul0 31656 | Scalar multiplication with... |
| hvmul0or 31657 | If a scalar product is zer... |
| hvsubid 31658 | Subtraction of a vector fr... |
| hvnegid 31659 | Addition of negative of a ... |
| hv2neg 31660 | Two ways to express the ne... |
| hvaddlidi 31661 | Addition with the zero vec... |
| hvnegidi 31662 | Addition of negative of a ... |
| hv2negi 31663 | Two ways to express the ne... |
| hvm1neg 31664 | Convert minus one times a ... |
| hvaddsubval 31665 | Value of vector addition i... |
| hvadd32 31666 | Commutative/associative la... |
| hvadd12 31667 | Commutative/associative la... |
| hvadd4 31668 | Hilbert vector space addit... |
| hvsub4 31669 | Hilbert vector space addit... |
| hvaddsub12 31670 | Commutative/associative la... |
| hvpncan 31671 | Addition/subtraction cance... |
| hvpncan2 31672 | Addition/subtraction cance... |
| hvaddsubass 31673 | Associativity of sum and d... |
| hvpncan3 31674 | Subtraction and addition o... |
| hvmulcom 31675 | Scalar multiplication comm... |
| hvsubass 31676 | Hilbert vector space assoc... |
| hvsub32 31677 | Hilbert vector space commu... |
| hvmulassi 31678 | Scalar multiplication asso... |
| hvmulcomi 31679 | Scalar multiplication comm... |
| hvmul2negi 31680 | Double negative in scalar ... |
| hvsubdistr1 31681 | Scalar multiplication dist... |
| hvsubdistr2 31682 | Scalar multiplication dist... |
| hvdistr1i 31683 | Scalar multiplication dist... |
| hvsubdistr1i 31684 | Scalar multiplication dist... |
| hvassi 31685 | Hilbert vector space assoc... |
| hvadd32i 31686 | Hilbert vector space commu... |
| hvsubassi 31687 | Hilbert vector space assoc... |
| hvsub32i 31688 | Hilbert vector space commu... |
| hvadd12i 31689 | Hilbert vector space commu... |
| hvadd4i 31690 | Hilbert vector space addit... |
| hvsubsub4i 31691 | Hilbert vector space addit... |
| hvsubsub4 31692 | Hilbert vector space addit... |
| hv2times 31693 | Two times a vector. (Cont... |
| hvnegdii 31694 | Distribution of negative o... |
| hvsubeq0i 31695 | If the difference between ... |
| hvsubcan2i 31696 | Vector cancellation law. ... |
| hvaddcani 31697 | Cancellation law for vecto... |
| hvsubaddi 31698 | Relationship between vecto... |
| hvnegdi 31699 | Distribution of negative o... |
| hvsubeq0 31700 | If the difference between ... |
| hvaddeq0 31701 | If the sum of two vectors ... |
| hvaddcan 31702 | Cancellation law for vecto... |
| hvaddcan2 31703 | Cancellation law for vecto... |
| hvmulcan 31704 | Cancellation law for scala... |
| hvmulcan2 31705 | Cancellation law for scala... |
| hvsubcan 31706 | Cancellation law for vecto... |
| hvsubcan2 31707 | Cancellation law for vecto... |
| hvsub0 31708 | Subtraction of a zero vect... |
| hvsubadd 31709 | Relationship between vecto... |
| hvaddsub4 31710 | Hilbert vector space addit... |
| hicl 31712 | Closure of inner product. ... |
| hicli 31713 | Closure inference for inne... |
| his5 31718 | Associative law for inner ... |
| his52 31719 | Associative law for inner ... |
| his35 31720 | Move scalar multiplication... |
| his35i 31721 | Move scalar multiplication... |
| his7 31722 | Distributive law for inner... |
| hiassdi 31723 | Distributive/associative l... |
| his2sub 31724 | Distributive law for inner... |
| his2sub2 31725 | Distributive law for inner... |
| hire 31726 | A necessary and sufficient... |
| hiidrcl 31727 | Real closure of inner prod... |
| hi01 31728 | Inner product with the 0 v... |
| hi02 31729 | Inner product with the 0 v... |
| hiidge0 31730 | Inner product with self is... |
| his6 31731 | Zero inner product with se... |
| his1i 31732 | Conjugate law for inner pr... |
| abshicom 31733 | Commuted inner products ha... |
| hial0 31734 | A vector whose inner produ... |
| hial02 31735 | A vector whose inner produ... |
| hisubcomi 31736 | Two vector subtractions si... |
| hi2eq 31737 | Lemma used to prove equali... |
| hial2eq 31738 | Two vectors whose inner pr... |
| hial2eq2 31739 | Two vectors whose inner pr... |
| orthcom 31740 | Orthogonality commutes. (... |
| normlem0 31741 | Lemma used to derive prope... |
| normlem1 31742 | Lemma used to derive prope... |
| normlem2 31743 | Lemma used to derive prope... |
| normlem3 31744 | Lemma used to derive prope... |
| normlem4 31745 | Lemma used to derive prope... |
| normlem5 31746 | Lemma used to derive prope... |
| normlem6 31747 | Lemma used to derive prope... |
| normlem7 31748 | Lemma used to derive prope... |
| normlem8 31749 | Lemma used to derive prope... |
| normlem9 31750 | Lemma used to derive prope... |
| normlem7tALT 31751 | Lemma used to derive prope... |
| bcseqi 31752 | Equality case of Bunjakova... |
| normlem9at 31753 | Lemma used to derive prope... |
| dfhnorm2 31754 | Alternate definition of th... |
| normf 31755 | The norm function maps fro... |
| normval 31756 | The value of the norm of a... |
| normcl 31757 | Real closure of the norm o... |
| normge0 31758 | The norm of a vector is no... |
| normgt0 31759 | The norm of nonzero vector... |
| norm0 31760 | The norm of a zero vector.... |
| norm-i 31761 | Theorem 3.3(i) of [Beran] ... |
| normne0 31762 | A norm is nonzero iff its ... |
| normcli 31763 | Real closure of the norm o... |
| normsqi 31764 | The square of a norm. (Co... |
| norm-i-i 31765 | Theorem 3.3(i) of [Beran] ... |
| normsq 31766 | The square of a norm. (Co... |
| normsub0i 31767 | Two vectors are equal iff ... |
| normsub0 31768 | Two vectors are equal iff ... |
| norm-ii-i 31769 | Triangle inequality for no... |
| norm-ii 31770 | Triangle inequality for no... |
| norm-iii-i 31771 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31772 | Theorem 3.3(iii) of [Beran... |
| normsubi 31773 | Negative doesn't change th... |
| normpythi 31774 | Analogy to Pythagorean the... |
| normsub 31775 | Swapping order of subtract... |
| normneg 31776 | The norm of a vector equal... |
| normpyth 31777 | Analogy to Pythagorean the... |
| normpyc 31778 | Corollary to Pythagorean t... |
| norm3difi 31779 | Norm of differences around... |
| norm3adifii 31780 | Norm of differences around... |
| norm3lem 31781 | Lemma involving norm of di... |
| norm3dif 31782 | Norm of differences around... |
| norm3dif2 31783 | Norm of differences around... |
| norm3lemt 31784 | Lemma involving norm of di... |
| norm3adifi 31785 | Norm of differences around... |
| normpari 31786 | Parallelogram law for norm... |
| normpar 31787 | Parallelogram law for norm... |
| normpar2i 31788 | Corollary of parallelogram... |
| polid2i 31789 | Generalized polarization i... |
| polidi 31790 | Polarization identity. Re... |
| polid 31791 | Polarization identity. Re... |
| hilablo 31792 | Hilbert space vector addit... |
| hilid 31793 | The group identity element... |
| hilvc 31794 | Hilbert space is a complex... |
| hilnormi 31795 | Hilbert space norm in term... |
| hilhhi 31796 | Deduce the structure of Hi... |
| hhnv 31797 | Hilbert space is a normed ... |
| hhva 31798 | The group (addition) opera... |
| hhba 31799 | The base set of Hilbert sp... |
| hh0v 31800 | The zero vector of Hilbert... |
| hhsm 31801 | The scalar product operati... |
| hhvs 31802 | The vector subtraction ope... |
| hhnm 31803 | The norm function of Hilbe... |
| hhims 31804 | The induced metric of Hilb... |
| hhims2 31805 | Hilbert space distance met... |
| hhmet 31806 | The induced metric of Hilb... |
| hhxmet 31807 | The induced metric of Hilb... |
| hhmetdval 31808 | Value of the distance func... |
| hhip 31809 | The inner product operatio... |
| hhph 31810 | The Hilbert space of the H... |
| bcsiALT 31811 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31812 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31813 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31814 | Corollary of the Bunjakova... |
| bcs3 31815 | Corollary of the Bunjakova... |
| hcau 31816 | Member of the set of Cauch... |
| hcauseq 31817 | A Cauchy sequences on a Hi... |
| hcaucvg 31818 | A Cauchy sequence on a Hil... |
| seq1hcau 31819 | A sequence on a Hilbert sp... |
| hlimi 31820 | Express the predicate: Th... |
| hlimseqi 31821 | A sequence with a limit on... |
| hlimveci 31822 | Closure of the limit of a ... |
| hlimconvi 31823 | Convergence of a sequence ... |
| hlim2 31824 | The limit of a sequence on... |
| hlimadd 31825 | Limit of the sum of two se... |
| hilmet 31826 | The Hilbert space norm det... |
| hilxmet 31827 | The Hilbert space norm det... |
| hilmetdval 31828 | Value of the distance func... |
| hilims 31829 | Hilbert space distance met... |
| hhcau 31830 | The Cauchy sequences of Hi... |
| hhlm 31831 | The limit sequences of Hil... |
| hhcmpl 31832 | Lemma used for derivation ... |
| hilcompl 31833 | Lemma used for derivation ... |
| hhcms 31835 | The Hilbert space induced ... |
| hhhl 31836 | The Hilbert space structur... |
| hilcms 31837 | The Hilbert space norm det... |
| hilhl 31838 | The Hilbert space of the H... |
| issh 31840 | Subspace ` H ` of a Hilber... |
| issh2 31841 | Subspace ` H ` of a Hilber... |
| shss 31842 | A subspace is a subset of ... |
| shel 31843 | A member of a subspace of ... |
| shex 31844 | The set of subspaces of a ... |
| shssii 31845 | A closed subspace of a Hil... |
| sheli 31846 | A member of a subspace of ... |
| shelii 31847 | A member of a subspace of ... |
| sh0 31848 | The zero vector belongs to... |
| shaddcl 31849 | Closure of vector addition... |
| shmulcl 31850 | Closure of vector scalar m... |
| issh3 31851 | Subspace ` H ` of a Hilber... |
| shsubcl 31852 | Closure of vector subtract... |
| isch 31854 | Closed subspace ` H ` of a... |
| isch2 31855 | Closed subspace ` H ` of a... |
| chsh 31856 | A closed subspace is a sub... |
| chsssh 31857 | Closed subspaces are subsp... |
| chex 31858 | The set of closed subspace... |
| chshii 31859 | A closed subspace is a sub... |
| ch0 31860 | The zero vector belongs to... |
| chss 31861 | A closed subspace of a Hil... |
| chel 31862 | A member of a closed subsp... |
| chssii 31863 | A closed subspace of a Hil... |
| cheli 31864 | A member of a closed subsp... |
| chelii 31865 | A member of a closed subsp... |
| chlimi 31866 | The limit property of a cl... |
| hlim0 31867 | The zero sequence in Hilbe... |
| hlimcaui 31868 | If a sequence in Hilbert s... |
| hlimf 31869 | Function-like behavior of ... |
| hlimuni 31870 | A Hilbert space sequence c... |
| hlimreui 31871 | The limit of a Hilbert spa... |
| hlimeui 31872 | The limit of a Hilbert spa... |
| isch3 31873 | A Hilbert subspace is clos... |
| chcompl 31874 | Completeness of a closed s... |
| helch 31875 | The Hilbert lattice one (w... |
| ifchhv 31876 | Prove ` if ( A e. CH , A ,... |
| helsh 31877 | Hilbert space is a subspac... |
| shsspwh 31878 | Subspaces are subsets of H... |
| chsspwh 31879 | Closed subspaces are subse... |
| hsn0elch 31880 | The zero subspace belongs ... |
| norm1 31881 | From any nonzero Hilbert s... |
| norm1exi 31882 | A normalized vector exists... |
| norm1hex 31883 | A normalized vector can ex... |
| elch0 31886 | Membership in zero for clo... |
| h0elch 31887 | The zero subspace is a clo... |
| h0elsh 31888 | The zero subspace is a sub... |
| hhssva 31889 | The vector addition operat... |
| hhsssm 31890 | The scalar multiplication ... |
| hhssnm 31891 | The norm operation on a su... |
| issubgoilem 31892 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31893 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31894 | Abelian group property of ... |
| hhssablo 31895 | Abelian group property of ... |
| hhssnv 31896 | Normed complex vector spac... |
| hhssnvt 31897 | Normed complex vector spac... |
| hhsst 31898 | A member of ` SH ` is a su... |
| hhshsslem1 31899 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31900 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31901 | The predicate " ` H ` is a... |
| hhsssh2 31902 | The predicate " ` H ` is a... |
| hhssba 31903 | The base set of a subspace... |
| hhssvs 31904 | The vector subtraction ope... |
| hhssvsf 31905 | Mapping of the vector subt... |
| hhssims 31906 | Induced metric of a subspa... |
| hhssims2 31907 | Induced metric of a subspa... |
| hhssmet 31908 | Induced metric of a subspa... |
| hhssmetdval 31909 | Value of the distance func... |
| hhsscms 31910 | The induced metric of a cl... |
| hhssbnOLD 31911 | Obsolete version of ~ cssb... |
| ocval 31912 | Value of orthogonal comple... |
| ocel 31913 | Membership in orthogonal c... |
| shocel 31914 | Membership in orthogonal c... |
| ocsh 31915 | The orthogonal complement ... |
| shocsh 31916 | The orthogonal complement ... |
| ocss 31917 | An orthogonal complement i... |
| shocss 31918 | An orthogonal complement i... |
| occon 31919 | Contraposition law for ort... |
| occon2 31920 | Double contraposition for ... |
| occon2i 31921 | Double contraposition for ... |
| oc0 31922 | The zero vector belongs to... |
| ocorth 31923 | Members of a subset and it... |
| shocorth 31924 | Members of a subspace and ... |
| ococss 31925 | Inclusion in complement of... |
| shococss 31926 | Inclusion in complement of... |
| shorth 31927 | Members of orthogonal subs... |
| ocin 31928 | Intersection of a Hilbert ... |
| occon3 31929 | Hilbert lattice contraposi... |
| ocnel 31930 | A nonzero vector in the co... |
| chocvali 31931 | Value of the orthogonal co... |
| shuni 31932 | Two subspaces with trivial... |
| chocunii 31933 | Lemma for uniqueness part ... |
| pjhthmo 31934 | Projection Theorem, unique... |
| occllem 31935 | Lemma for ~ occl . (Contr... |
| occl 31936 | Closure of complement of H... |
| shoccl 31937 | Closure of complement of H... |
| choccl 31938 | Closure of complement of H... |
| choccli 31939 | Closure of ` CH ` orthocom... |
| shsval 31944 | Value of subspace sum of t... |
| shsss 31945 | The subspace sum is a subs... |
| shsel 31946 | Membership in the subspace... |
| shsel3 31947 | Membership in the subspace... |
| shseli 31948 | Membership in subspace sum... |
| shscli 31949 | Closure of subspace sum. ... |
| shscl 31950 | Closure of subspace sum. ... |
| shscom 31951 | Commutative law for subspa... |
| shsva 31952 | Vector sum belongs to subs... |
| shsel1 31953 | A subspace sum contains a ... |
| shsel2 31954 | A subspace sum contains a ... |
| shsvs 31955 | Vector subtraction belongs... |
| shsub1 31956 | Subspace sum is an upper b... |
| shsub2 31957 | Subspace sum is an upper b... |
| choc0 31958 | The orthocomplement of the... |
| choc1 31959 | The orthocomplement of the... |
| chocnul 31960 | Orthogonal complement of t... |
| shintcli 31961 | Closure of intersection of... |
| shintcl 31962 | The intersection of a none... |
| chintcli 31963 | The intersection of a none... |
| chintcl 31964 | The intersection (infimum)... |
| spanval 31965 | Value of the linear span o... |
| hsupval 31966 | Value of supremum of set o... |
| chsupval 31967 | The value of the supremum ... |
| spancl 31968 | The span of a subset of Hi... |
| elspancl 31969 | A member of a span is a ve... |
| shsupcl 31970 | Closure of the subspace su... |
| hsupcl 31971 | Closure of supremum of set... |
| chsupcl 31972 | Closure of supremum of sub... |
| hsupss 31973 | Subset relation for suprem... |
| chsupss 31974 | Subset relation for suprem... |
| hsupunss 31975 | The union of a set of Hilb... |
| chsupunss 31976 | The union of a set of clos... |
| spanss2 31977 | A subset of Hilbert space ... |
| shsupunss 31978 | The union of a set of subs... |
| spanid 31979 | A subspace of Hilbert spac... |
| spanss 31980 | Ordering relationship for ... |
| spanssoc 31981 | The span of a subset of Hi... |
| sshjval 31982 | Value of join for subsets ... |
| shjval 31983 | Value of join in ` SH ` . ... |
| chjval 31984 | Value of join in ` CH ` . ... |
| chjvali 31985 | Value of join in ` CH ` . ... |
| sshjval3 31986 | Value of join for subsets ... |
| sshjcl 31987 | Closure of join for subset... |
| shjcl 31988 | Closure of join in ` SH ` ... |
| chjcl 31989 | Closure of join in ` CH ` ... |
| shjcom 31990 | Commutative law for Hilber... |
| shless 31991 | Subset implies subset of s... |
| shlej1 31992 | Add disjunct to both sides... |
| shlej2 31993 | Add disjunct to both sides... |
| shincli 31994 | Closure of intersection of... |
| shscomi 31995 | Commutative law for subspa... |
| shsvai 31996 | Vector sum belongs to subs... |
| shsel1i 31997 | A subspace sum contains a ... |
| shsel2i 31998 | A subspace sum contains a ... |
| shsvsi 31999 | Vector subtraction belongs... |
| shunssi 32000 | Union is smaller than subs... |
| shunssji 32001 | Union is smaller than Hilb... |
| shsleji 32002 | Subspace sum is smaller th... |
| shjcomi 32003 | Commutative law for join i... |
| shsub1i 32004 | Subspace sum is an upper b... |
| shsub2i 32005 | Subspace sum is an upper b... |
| shub1i 32006 | Hilbert lattice join is an... |
| shjcli 32007 | Closure of ` CH ` join. (... |
| shjshcli 32008 | ` SH ` closure of join. (... |
| shlessi 32009 | Subset implies subset of s... |
| shlej1i 32010 | Add disjunct to both sides... |
| shlej2i 32011 | Add disjunct to both sides... |
| shslej 32012 | Subspace sum is smaller th... |
| shincl 32013 | Closure of intersection of... |
| shub1 32014 | Hilbert lattice join is an... |
| shub2 32015 | A subspace is a subset of ... |
| shsidmi 32016 | Idempotent law for Hilbert... |
| shslubi 32017 | The least upper bound law ... |
| shlesb1i 32018 | Hilbert lattice ordering i... |
| shsval2i 32019 | An alternate way to expres... |
| shsval3i 32020 | An alternate way to expres... |
| shmodsi 32021 | The modular law holds for ... |
| shmodi 32022 | The modular law is implied... |
| pjhthlem1 32023 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 32024 | Lemma for ~ pjhth . (Cont... |
| pjhth 32025 | Projection Theorem: Any H... |
| pjhtheu 32026 | Projection Theorem: Any H... |
| pjhfval 32028 | The value of the projectio... |
| pjhval 32029 | Value of a projection. (C... |
| pjpreeq 32030 | Equality with a projection... |
| pjeq 32031 | Equality with a projection... |
| axpjcl 32032 | Closure of a projection in... |
| pjhcl 32033 | Closure of a projection in... |
| omlsilem 32034 | Lemma for orthomodular law... |
| omlsii 32035 | Subspace inference form of... |
| omlsi 32036 | Subspace form of orthomodu... |
| ococi 32037 | Complement of complement o... |
| ococ 32038 | Complement of complement o... |
| dfch2 32039 | Alternate definition of th... |
| ococin 32040 | The double complement is t... |
| hsupval2 32041 | Alternate definition of su... |
| chsupval2 32042 | The value of the supremum ... |
| sshjval2 32043 | Value of join in the set o... |
| chsupid 32044 | A subspace is the supremum... |
| chsupsn 32045 | Value of supremum of subse... |
| shlub 32046 | Hilbert lattice join is th... |
| shlubi 32047 | Hilbert lattice join is th... |
| pjhtheu2 32048 | Uniqueness of ` y ` for th... |
| pjcli 32049 | Closure of a projection in... |
| pjhcli 32050 | Closure of a projection in... |
| pjpjpre 32051 | Decomposition of a vector ... |
| axpjpj 32052 | Decomposition of a vector ... |
| pjclii 32053 | Closure of a projection in... |
| pjhclii 32054 | Closure of a projection in... |
| pjpj0i 32055 | Decomposition of a vector ... |
| pjpji 32056 | Decomposition of a vector ... |
| pjpjhth 32057 | Projection Theorem: Any H... |
| pjpjhthi 32058 | Projection Theorem: Any H... |
| pjop 32059 | Orthocomplement projection... |
| pjpo 32060 | Projection in terms of ort... |
| pjopi 32061 | Orthocomplement projection... |
| pjpoi 32062 | Projection in terms of ort... |
| pjoc1i 32063 | Projection of a vector in ... |
| pjchi 32064 | Projection of a vector in ... |
| pjoccl 32065 | The part of a vector that ... |
| pjoc1 32066 | Projection of a vector in ... |
| pjomli 32067 | Subspace form of orthomodu... |
| pjoml 32068 | Subspace form of orthomodu... |
| pjococi 32069 | Proof of orthocomplement t... |
| pjoc2i 32070 | Projection of a vector in ... |
| pjoc2 32071 | Projection of a vector in ... |
| sh0le 32072 | The zero subspace is the s... |
| ch0le 32073 | The zero subspace is the s... |
| shle0 32074 | No subspace is smaller tha... |
| chle0 32075 | No Hilbert lattice element... |
| chnlen0 32076 | A Hilbert lattice element ... |
| ch0pss 32077 | The zero subspace is a pro... |
| orthin 32078 | The intersection of orthog... |
| ssjo 32079 | The lattice join of a subs... |
| shne0i 32080 | A nonzero subspace has a n... |
| shs0i 32081 | Hilbert subspace sum with ... |
| shs00i 32082 | Two subspaces are zero iff... |
| ch0lei 32083 | The closed subspace zero i... |
| chle0i 32084 | No Hilbert closed subspace... |
| chne0i 32085 | A nonzero closed subspace ... |
| chocini 32086 | Intersection of a closed s... |
| chj0i 32087 | Join with lattice zero in ... |
| chm1i 32088 | Meet with lattice one in `... |
| chjcli 32089 | Closure of ` CH ` join. (... |
| chsleji 32090 | Subspace sum is smaller th... |
| chseli 32091 | Membership in subspace sum... |
| chincli 32092 | Closure of Hilbert lattice... |
| chsscon3i 32093 | Hilbert lattice contraposi... |
| chsscon1i 32094 | Hilbert lattice contraposi... |
| chsscon2i 32095 | Hilbert lattice contraposi... |
| chcon2i 32096 | Hilbert lattice contraposi... |
| chcon1i 32097 | Hilbert lattice contraposi... |
| chcon3i 32098 | Hilbert lattice contraposi... |
| chunssji 32099 | Union is smaller than ` CH... |
| chjcomi 32100 | Commutative law for join i... |
| chub1i 32101 | ` CH ` join is an upper bo... |
| chub2i 32102 | ` CH ` join is an upper bo... |
| chlubi 32103 | Hilbert lattice join is th... |
| chlubii 32104 | Hilbert lattice join is th... |
| chlej1i 32105 | Add join to both sides of ... |
| chlej2i 32106 | Add join to both sides of ... |
| chlej12i 32107 | Add join to both sides of ... |
| chlejb1i 32108 | Hilbert lattice ordering i... |
| chdmm1i 32109 | De Morgan's law for meet i... |
| chdmm2i 32110 | De Morgan's law for meet i... |
| chdmm3i 32111 | De Morgan's law for meet i... |
| chdmm4i 32112 | De Morgan's law for meet i... |
| chdmj1i 32113 | De Morgan's law for join i... |
| chdmj2i 32114 | De Morgan's law for join i... |
| chdmj3i 32115 | De Morgan's law for join i... |
| chdmj4i 32116 | De Morgan's law for join i... |
| chnlei 32117 | Equivalent expressions for... |
| chjassi 32118 | Associative law for Hilber... |
| chj00i 32119 | Two Hilbert lattice elemen... |
| chjoi 32120 | The join of a closed subsp... |
| chj1i 32121 | Join with Hilbert lattice ... |
| chm0i 32122 | Meet with Hilbert lattice ... |
| chm0 32123 | Meet with Hilbert lattice ... |
| shjshsi 32124 | Hilbert lattice join equal... |
| shjshseli 32125 | A closed subspace sum equa... |
| chne0 32126 | A nonzero closed subspace ... |
| chocin 32127 | Intersection of a closed s... |
| chssoc 32128 | A closed subspace less tha... |
| chj0 32129 | Join with Hilbert lattice ... |
| chslej 32130 | Subspace sum is smaller th... |
| chincl 32131 | Closure of Hilbert lattice... |
| chsscon3 32132 | Hilbert lattice contraposi... |
| chsscon1 32133 | Hilbert lattice contraposi... |
| chsscon2 32134 | Hilbert lattice contraposi... |
| chpsscon3 32135 | Hilbert lattice contraposi... |
| chpsscon1 32136 | Hilbert lattice contraposi... |
| chpsscon2 32137 | Hilbert lattice contraposi... |
| chjcom 32138 | Commutative law for Hilber... |
| chub1 32139 | Hilbert lattice join is gr... |
| chub2 32140 | Hilbert lattice join is gr... |
| chlub 32141 | Hilbert lattice join is th... |
| chlej1 32142 | Add join to both sides of ... |
| chlej2 32143 | Add join to both sides of ... |
| chlejb1 32144 | Hilbert lattice ordering i... |
| chlejb2 32145 | Hilbert lattice ordering i... |
| chnle 32146 | Equivalent expressions for... |
| chjo 32147 | The join of a closed subsp... |
| chabs1 32148 | Hilbert lattice absorption... |
| chabs2 32149 | Hilbert lattice absorption... |
| chabs1i 32150 | Hilbert lattice absorption... |
| chabs2i 32151 | Hilbert lattice absorption... |
| chjidm 32152 | Idempotent law for Hilbert... |
| chjidmi 32153 | Idempotent law for Hilbert... |
| chj12i 32154 | A rearrangement of Hilbert... |
| chj4i 32155 | Rearrangement of the join ... |
| chjjdiri 32156 | Hilbert lattice join distr... |
| chdmm1 32157 | De Morgan's law for meet i... |
| chdmm2 32158 | De Morgan's law for meet i... |
| chdmm3 32159 | De Morgan's law for meet i... |
| chdmm4 32160 | De Morgan's law for meet i... |
| chdmj1 32161 | De Morgan's law for join i... |
| chdmj2 32162 | De Morgan's law for join i... |
| chdmj3 32163 | De Morgan's law for join i... |
| chdmj4 32164 | De Morgan's law for join i... |
| chjass 32165 | Associative law for Hilber... |
| chj12 32166 | A rearrangement of Hilbert... |
| chj4 32167 | Rearrangement of the join ... |
| ledii 32168 | An ortholattice is distrib... |
| lediri 32169 | An ortholattice is distrib... |
| lejdii 32170 | An ortholattice is distrib... |
| lejdiri 32171 | An ortholattice is distrib... |
| ledi 32172 | An ortholattice is distrib... |
| spansn0 32173 | The span of the singleton ... |
| span0 32174 | The span of the empty set ... |
| elspani 32175 | Membership in the span of ... |
| spanuni 32176 | The span of a union is the... |
| spanun 32177 | The span of a union is the... |
| sshhococi 32178 | The join of two Hilbert sp... |
| hne0 32179 | Hilbert space has a nonzer... |
| chsup0 32180 | The supremum of the empty ... |
| h1deoi 32181 | Membership in orthocomplem... |
| h1dei 32182 | Membership in 1-dimensiona... |
| h1did 32183 | A generating vector belong... |
| h1dn0 32184 | A nonzero vector generates... |
| h1de2i 32185 | Membership in 1-dimensiona... |
| h1de2bi 32186 | Membership in 1-dimensiona... |
| h1de2ctlem 32187 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 32188 | Membership in 1-dimensiona... |
| spansni 32189 | The span of a singleton in... |
| elspansni 32190 | Membership in the span of ... |
| spansn 32191 | The span of a singleton in... |
| spansnch 32192 | The span of a Hilbert spac... |
| spansnsh 32193 | The span of a Hilbert spac... |
| spansnchi 32194 | The span of a singleton in... |
| spansnid 32195 | A vector belongs to the sp... |
| spansnmul 32196 | A scalar product with a ve... |
| elspansncl 32197 | A member of a span of a si... |
| elspansn 32198 | Membership in the span of ... |
| elspansn2 32199 | Membership in the span of ... |
| spansncol 32200 | The singletons of collinea... |
| spansneleqi 32201 | Membership relation implie... |
| spansneleq 32202 | Membership relation that i... |
| spansnss 32203 | The span of the singleton ... |
| elspansn3 32204 | A member of the span of th... |
| elspansn4 32205 | A span membership conditio... |
| elspansn5 32206 | A vector belonging to both... |
| spansnss2 32207 | The span of the singleton ... |
| normcan 32208 | Cancellation-type law that... |
| pjspansn 32209 | A projection on the span o... |
| spansnpji 32210 | A subset of Hilbert space ... |
| spanunsni 32211 | The span of the union of a... |
| spanpr 32212 | The span of a pair of vect... |
| h1datomi 32213 | A 1-dimensional subspace i... |
| h1datom 32214 | A 1-dimensional subspace i... |
| cmbr 32216 | Binary relation expressing... |
| pjoml2i 32217 | Variation of orthomodular ... |
| pjoml3i 32218 | Variation of orthomodular ... |
| pjoml4i 32219 | Variation of orthomodular ... |
| pjoml5i 32220 | The orthomodular law. Rem... |
| pjoml6i 32221 | An equivalent of the ortho... |
| cmbri 32222 | Binary relation expressing... |
| cmcmlem 32223 | Commutation is symmetric. ... |
| cmcmi 32224 | Commutation is symmetric. ... |
| cmcm2i 32225 | Commutation with orthocomp... |
| cmcm3i 32226 | Commutation with orthocomp... |
| cmcm4i 32227 | Commutation with orthocomp... |
| cmbr2i 32228 | Alternate definition of th... |
| cmcmii 32229 | Commutation is symmetric. ... |
| cmcm2ii 32230 | Commutation with orthocomp... |
| cmcm3ii 32231 | Commutation with orthocomp... |
| cmbr3i 32232 | Alternate definition for t... |
| cmbr4i 32233 | Alternate definition for t... |
| lecmi 32234 | Comparable Hilbert lattice... |
| lecmii 32235 | Comparable Hilbert lattice... |
| cmj1i 32236 | A Hilbert lattice element ... |
| cmj2i 32237 | A Hilbert lattice element ... |
| cmm1i 32238 | A Hilbert lattice element ... |
| cmm2i 32239 | A Hilbert lattice element ... |
| cmbr3 32240 | Alternate definition for t... |
| cm0 32241 | The zero Hilbert lattice e... |
| cmidi 32242 | The commutes relation is r... |
| pjoml2 32243 | Variation of orthomodular ... |
| pjoml3 32244 | Variation of orthomodular ... |
| pjoml5 32245 | The orthomodular law. Rem... |
| cmcm 32246 | Commutation is symmetric. ... |
| cmcm3 32247 | Commutation with orthocomp... |
| cmcm2 32248 | Commutation with orthocomp... |
| lecm 32249 | Comparable Hilbert lattice... |
| fh1 32250 | Foulis-Holland Theorem. I... |
| fh2 32251 | Foulis-Holland Theorem. I... |
| cm2j 32252 | A lattice element that com... |
| fh1i 32253 | Foulis-Holland Theorem. I... |
| fh2i 32254 | Foulis-Holland Theorem. I... |
| fh3i 32255 | Variation of the Foulis-Ho... |
| fh4i 32256 | Variation of the Foulis-Ho... |
| cm2ji 32257 | A lattice element that com... |
| cm2mi 32258 | A lattice element that com... |
| qlax1i 32259 | One of the equations showi... |
| qlax2i 32260 | One of the equations showi... |
| qlax3i 32261 | One of the equations showi... |
| qlax4i 32262 | One of the equations showi... |
| qlax5i 32263 | One of the equations showi... |
| qlaxr1i 32264 | One of the conditions show... |
| qlaxr2i 32265 | One of the conditions show... |
| qlaxr4i 32266 | One of the conditions show... |
| qlaxr5i 32267 | One of the conditions show... |
| qlaxr3i 32268 | A variation of the orthomo... |
| chscllem1 32269 | Lemma for ~ chscl . (Cont... |
| chscllem2 32270 | Lemma for ~ chscl . (Cont... |
| chscllem3 32271 | Lemma for ~ chscl . (Cont... |
| chscllem4 32272 | Lemma for ~ chscl . (Cont... |
| chscl 32273 | The subspace sum of two cl... |
| osumi 32274 | If two closed subspaces of... |
| osumcori 32275 | Corollary of ~ osumi . (C... |
| osumcor2i 32276 | Corollary of ~ osumi , sho... |
| osum 32277 | If two closed subspaces of... |
| spansnji 32278 | The subspace sum of a clos... |
| spansnj 32279 | The subspace sum of a clos... |
| spansnscl 32280 | The subspace sum of a clos... |
| sumspansn 32281 | The sum of two vectors bel... |
| spansnm0i 32282 | The meet of different one-... |
| nonbooli 32283 | A Hilbert lattice with two... |
| spansncvi 32284 | Hilbert space has the cove... |
| spansncv 32285 | Hilbert space has the cove... |
| 5oalem1 32286 | Lemma for orthoarguesian l... |
| 5oalem2 32287 | Lemma for orthoarguesian l... |
| 5oalem3 32288 | Lemma for orthoarguesian l... |
| 5oalem4 32289 | Lemma for orthoarguesian l... |
| 5oalem5 32290 | Lemma for orthoarguesian l... |
| 5oalem6 32291 | Lemma for orthoarguesian l... |
| 5oalem7 32292 | Lemma for orthoarguesian l... |
| 5oai 32293 | Orthoarguesian law 5OA. Th... |
| 3oalem1 32294 | Lemma for 3OA (weak) ortho... |
| 3oalem2 32295 | Lemma for 3OA (weak) ortho... |
| 3oalem3 32296 | Lemma for 3OA (weak) ortho... |
| 3oalem4 32297 | Lemma for 3OA (weak) ortho... |
| 3oalem5 32298 | Lemma for 3OA (weak) ortho... |
| 3oalem6 32299 | Lemma for 3OA (weak) ortho... |
| 3oai 32300 | 3OA (weak) orthoarguesian ... |
| pjorthi 32301 | Projection components on o... |
| pjch1 32302 | Property of identity proje... |
| pjo 32303 | The orthogonal projection.... |
| pjcompi 32304 | Component of a projection.... |
| pjidmi 32305 | A projection is idempotent... |
| pjadjii 32306 | A projection is self-adjoi... |
| pjaddii 32307 | Projection of vector sum i... |
| pjinormii 32308 | The inner product of a pro... |
| pjmulii 32309 | Projection of (scalar) pro... |
| pjsubii 32310 | Projection of vector diffe... |
| pjsslem 32311 | Lemma for subset relations... |
| pjss2i 32312 | Subset relationship for pr... |
| pjssmii 32313 | Projection meet property. ... |
| pjssge0ii 32314 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 32315 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 32316 | The projection on a subspa... |
| pjadji 32317 | A projection is self-adjoi... |
| pjaddi 32318 | Projection of vector sum i... |
| pjinormi 32319 | The inner product of a pro... |
| pjsubi 32320 | Projection of vector diffe... |
| pjmuli 32321 | Projection of scalar produ... |
| pjige0i 32322 | The inner product of a pro... |
| pjige0 32323 | The inner product of a pro... |
| pjcjt2 32324 | The projection on a subspa... |
| pj0i 32325 | The projection of the zero... |
| pjch 32326 | Projection of a vector in ... |
| pjid 32327 | The projection of a vector... |
| pjvec 32328 | The set of vectors belongi... |
| pjocvec 32329 | The set of vectors belongi... |
| pjocini 32330 | Membership of projection i... |
| pjini 32331 | Membership of projection i... |
| pjjsi 32332 | A sufficient condition for... |
| pjfni 32333 | Functionality of a project... |
| pjrni 32334 | The range of a projection.... |
| pjfoi 32335 | A projection maps onto its... |
| pjfi 32336 | The mapping of a projectio... |
| pjvi 32337 | The value of a projection ... |
| pjhfo 32338 | A projection maps onto its... |
| pjrn 32339 | The range of a projection.... |
| pjhf 32340 | The mapping of a projectio... |
| pjfn 32341 | Functionality of a project... |
| pjsumi 32342 | The projection on a subspa... |
| pj11i 32343 | One-to-one correspondence ... |
| pjdsi 32344 | Vector decomposition into ... |
| pjds3i 32345 | Vector decomposition into ... |
| pj11 32346 | One-to-one correspondence ... |
| pjmfn 32347 | Functionality of the proje... |
| pjmf1 32348 | The projector function map... |
| pjoi0 32349 | The inner product of proje... |
| pjoi0i 32350 | The inner product of proje... |
| pjopythi 32351 | Pythagorean theorem for pr... |
| pjopyth 32352 | Pythagorean theorem for pr... |
| pjnormi 32353 | The norm of the projection... |
| pjpythi 32354 | Pythagorean theorem for pr... |
| pjneli 32355 | If a vector does not belon... |
| pjnorm 32356 | The norm of the projection... |
| pjpyth 32357 | Pythagorean theorem for pr... |
| pjnel 32358 | If a vector does not belon... |
| pjnorm2 32359 | A vector belongs to the su... |
| mayete3i 32360 | Mayet's equation E_3. Par... |
| mayetes3i 32361 | Mayet's equation E^*_3, de... |
| hosmval 32367 | Value of the sum of two Hi... |
| hommval 32368 | Value of the scalar produc... |
| hodmval 32369 | Value of the difference of... |
| hfsmval 32370 | Value of the sum of two Hi... |
| hfmmval 32371 | Value of the scalar produc... |
| hosval 32372 | Value of the sum of two Hi... |
| homval 32373 | Value of the scalar produc... |
| hodval 32374 | Value of the difference of... |
| hfsval 32375 | Value of the sum of two Hi... |
| hfmval 32376 | Value of the scalar produc... |
| hoscl 32377 | Closure of the sum of two ... |
| homcl 32378 | Closure of the scalar prod... |
| hodcl 32379 | Closure of the difference ... |
| ho0val 32382 | Value of the zero Hilbert ... |
| ho0f 32383 | Functionality of the zero ... |
| df0op2 32384 | Alternate definition of Hi... |
| dfiop2 32385 | Alternate definition of Hi... |
| hoif 32386 | Functionality of the Hilbe... |
| hoival 32387 | The value of the Hilbert s... |
| hoico1 32388 | Composition with the Hilbe... |
| hoico2 32389 | Composition with the Hilbe... |
| hoaddcl 32390 | The sum of Hilbert space o... |
| homulcl 32391 | The scalar product of a Hi... |
| hoeq 32392 | Equality of Hilbert space ... |
| hoeqi 32393 | Equality of Hilbert space ... |
| hoscli 32394 | Closure of Hilbert space o... |
| hodcli 32395 | Closure of Hilbert space o... |
| hocoi 32396 | Composition of Hilbert spa... |
| hococli 32397 | Closure of composition of ... |
| hocofi 32398 | Mapping of composition of ... |
| hocofni 32399 | Functionality of compositi... |
| hoaddcli 32400 | Mapping of sum of Hilbert ... |
| hosubcli 32401 | Mapping of difference of H... |
| hoaddfni 32402 | Functionality of sum of Hi... |
| hosubfni 32403 | Functionality of differenc... |
| hoaddcomi 32404 | Commutativity of sum of Hi... |
| hosubcl 32405 | Mapping of difference of H... |
| hoaddcom 32406 | Commutativity of sum of Hi... |
| hodsi 32407 | Relationship between Hilbe... |
| hoaddassi 32408 | Associativity of sum of Hi... |
| hoadd12i 32409 | Commutative/associative la... |
| hoadd32i 32410 | Commutative/associative la... |
| hocadddiri 32411 | Distributive law for Hilbe... |
| hocsubdiri 32412 | Distributive law for Hilbe... |
| ho2coi 32413 | Double composition of Hilb... |
| hoaddass 32414 | Associativity of sum of Hi... |
| hoadd32 32415 | Commutative/associative la... |
| hoadd4 32416 | Rearrangement of 4 terms i... |
| hocsubdir 32417 | Distributive law for Hilbe... |
| hoaddridi 32418 | Sum of a Hilbert space ope... |
| hodidi 32419 | Difference of a Hilbert sp... |
| ho0coi 32420 | Composition of the zero op... |
| hoid1i 32421 | Composition of Hilbert spa... |
| hoid1ri 32422 | Composition of Hilbert spa... |
| hoaddrid 32423 | Sum of a Hilbert space ope... |
| hodid 32424 | Difference of a Hilbert sp... |
| hon0 32425 | A Hilbert space operator i... |
| hodseqi 32426 | Subtraction and addition o... |
| ho0subi 32427 | Subtraction of Hilbert spa... |
| honegsubi 32428 | Relationship between Hilbe... |
| ho0sub 32429 | Subtraction of Hilbert spa... |
| hosubid1 32430 | The zero operator subtract... |
| honegsub 32431 | Relationship between Hilbe... |
| homullid 32432 | An operator equals its sca... |
| homco1 32433 | Associative law for scalar... |
| homulass 32434 | Scalar product associative... |
| hoadddi 32435 | Scalar product distributiv... |
| hoadddir 32436 | Scalar product reverse dis... |
| homul12 32437 | Swap first and second fact... |
| honegneg 32438 | Double negative of a Hilbe... |
| hosubneg 32439 | Relationship between opera... |
| hosubdi 32440 | Scalar product distributiv... |
| honegdi 32441 | Distribution of negative o... |
| honegsubdi 32442 | Distribution of negative o... |
| honegsubdi2 32443 | Distribution of negative o... |
| hosubsub2 32444 | Law for double subtraction... |
| hosub4 32445 | Rearrangement of 4 terms i... |
| hosubadd4 32446 | Rearrangement of 4 terms i... |
| hoaddsubass 32447 | Associative-type law for a... |
| hoaddsub 32448 | Law for operator addition ... |
| hosubsub 32449 | Law for double subtraction... |
| hosubsub4 32450 | Law for double subtraction... |
| ho2times 32451 | Two times a Hilbert space ... |
| hoaddsubassi 32452 | Associativity of sum and d... |
| hoaddsubi 32453 | Law for sum and difference... |
| hosd1i 32454 | Hilbert space operator sum... |
| hosd2i 32455 | Hilbert space operator sum... |
| hopncani 32456 | Hilbert space operator can... |
| honpcani 32457 | Hilbert space operator can... |
| hosubeq0i 32458 | If the difference between ... |
| honpncani 32459 | Hilbert space operator can... |
| ho01i 32460 | A condition implying that ... |
| ho02i 32461 | A condition implying that ... |
| hoeq1 32462 | A condition implying that ... |
| hoeq2 32463 | A condition implying that ... |
| adjmo 32464 | Every Hilbert space operat... |
| adjsym 32465 | Symmetry property of an ad... |
| eigrei 32466 | A necessary and sufficient... |
| eigre 32467 | A necessary and sufficient... |
| eigposi 32468 | A sufficient condition (fi... |
| eigorthi 32469 | A necessary and sufficient... |
| eigorth 32470 | A necessary and sufficient... |
| nmopval 32488 | Value of the norm of a Hil... |
| elcnop 32489 | Property defining a contin... |
| ellnop 32490 | Property defining a linear... |
| lnopf 32491 | A linear Hilbert space ope... |
| elbdop 32492 | Property defining a bounde... |
| bdopln 32493 | A bounded linear Hilbert s... |
| bdopf 32494 | A bounded linear Hilbert s... |
| nmopsetretALT 32495 | The set in the supremum of... |
| nmopsetretHIL 32496 | The set in the supremum of... |
| nmopsetn0 32497 | The set in the supremum of... |
| nmopxr 32498 | The norm of a Hilbert spac... |
| nmoprepnf 32499 | The norm of a Hilbert spac... |
| nmopgtmnf 32500 | The norm of a Hilbert spac... |
| nmopreltpnf 32501 | The norm of a Hilbert spac... |
| nmopre 32502 | The norm of a bounded oper... |
| elbdop2 32503 | Property defining a bounde... |
| elunop 32504 | Property defining a unitar... |
| elhmop 32505 | Property defining a Hermit... |
| hmopf 32506 | A Hermitian operator is a ... |
| hmopex 32507 | The class of Hermitian ope... |
| nmfnval 32508 | Value of the norm of a Hil... |
| nmfnsetre 32509 | The set in the supremum of... |
| nmfnsetn0 32510 | The set in the supremum of... |
| nmfnxr 32511 | The norm of any Hilbert sp... |
| nmfnrepnf 32512 | The norm of a Hilbert spac... |
| nlfnval 32513 | Value of the null space of... |
| elcnfn 32514 | Property defining a contin... |
| ellnfn 32515 | Property defining a linear... |
| lnfnf 32516 | A linear Hilbert space fun... |
| dfadj2 32517 | Alternate definition of th... |
| funadj 32518 | Functionality of the adjoi... |
| dmadjss 32519 | The domain of the adjoint ... |
| dmadjop 32520 | A member of the domain of ... |
| adjeu 32521 | Elementhood in the domain ... |
| adjval 32522 | Value of the adjoint funct... |
| adjval2 32523 | Value of the adjoint funct... |
| cnvadj 32524 | The adjoint function equal... |
| funcnvadj 32525 | The converse of the adjoin... |
| adj1o 32526 | The adjoint function maps ... |
| dmadjrn 32527 | The adjoint of an operator... |
| eigvecval 32528 | The set of eigenvectors of... |
| eigvalfval 32529 | The eigenvalues of eigenve... |
| specval 32530 | The value of the spectrum ... |
| speccl 32531 | The spectrum of an operato... |
| hhlnoi 32532 | The linear operators of Hi... |
| hhnmoi 32533 | The norm of an operator in... |
| hhbloi 32534 | A bounded linear operator ... |
| hh0oi 32535 | The zero operator in Hilbe... |
| hhcno 32536 | The continuous operators o... |
| hhcnf 32537 | The continuous functionals... |
| dmadjrnb 32538 | The adjoint of an operator... |
| nmoplb 32539 | A lower bound for an opera... |
| nmopub 32540 | An upper bound for an oper... |
| nmopub2tALT 32541 | An upper bound for an oper... |
| nmopub2tHIL 32542 | An upper bound for an oper... |
| nmopge0 32543 | The norm of any Hilbert sp... |
| nmopgt0 32544 | A linear Hilbert space ope... |
| cnopc 32545 | Basic continuity property ... |
| lnopl 32546 | Basic linearity property o... |
| unop 32547 | Basic inner product proper... |
| unopf1o 32548 | A unitary operator in Hilb... |
| unopnorm 32549 | A unitary operator is idem... |
| cnvunop 32550 | The inverse (converse) of ... |
| unopadj 32551 | The inverse (converse) of ... |
| unoplin 32552 | A unitary operator is line... |
| counop 32553 | The composition of two uni... |
| hmop 32554 | Basic inner product proper... |
| hmopre 32555 | The inner product of the v... |
| nmfnlb 32556 | A lower bound for a functi... |
| nmfnleub 32557 | An upper bound for the nor... |
| nmfnleub2 32558 | An upper bound for the nor... |
| nmfnge0 32559 | The norm of any Hilbert sp... |
| elnlfn 32560 | Membership in the null spa... |
| elnlfn2 32561 | Membership in the null spa... |
| cnfnc 32562 | Basic continuity property ... |
| lnfnl 32563 | Basic linearity property o... |
| adjcl 32564 | Closure of the adjoint of ... |
| adj1 32565 | Property of an adjoint Hil... |
| adj2 32566 | Property of an adjoint Hil... |
| adjeq 32567 | A property that determines... |
| adjadj 32568 | Double adjoint. Theorem 3... |
| adjvalval 32569 | Value of the value of the ... |
| unopadj2 32570 | The adjoint of a unitary o... |
| hmopadj 32571 | A Hermitian operator is se... |
| hmdmadj 32572 | Every Hermitian operator h... |
| hmopadj2 32573 | An operator is Hermitian i... |
| hmoplin 32574 | A Hermitian operator is li... |
| brafval 32575 | The bra of a vector, expre... |
| braval 32576 | A bra-ket juxtaposition, e... |
| braadd 32577 | Linearity property of bra ... |
| bramul 32578 | Linearity property of bra ... |
| brafn 32579 | The bra function is a func... |
| bralnfn 32580 | The Dirac bra function is ... |
| bracl 32581 | Closure of the bra functio... |
| bra0 32582 | The Dirac bra of the zero ... |
| brafnmul 32583 | Anti-linearity property of... |
| kbfval 32584 | The outer product of two v... |
| kbop 32585 | The outer product of two v... |
| kbval 32586 | The value of the operator ... |
| kbmul 32587 | Multiplication property of... |
| kbpj 32588 | If a vector ` A ` has norm... |
| eleigvec 32589 | Membership in the set of e... |
| eleigvec2 32590 | Membership in the set of e... |
| eleigveccl 32591 | Closure of an eigenvector ... |
| eigvalval 32592 | The eigenvalue of an eigen... |
| eigvalcl 32593 | An eigenvalue is a complex... |
| eigvec1 32594 | Property of an eigenvector... |
| eighmre 32595 | The eigenvalues of a Hermi... |
| eighmorth 32596 | Eigenvectors of a Hermitia... |
| nmopnegi 32597 | Value of the norm of the n... |
| lnop0 32598 | The value of a linear Hilb... |
| lnopmul 32599 | Multiplicative property of... |
| lnopli 32600 | Basic scalar product prope... |
| lnopfi 32601 | A linear Hilbert space ope... |
| lnop0i 32602 | The value of a linear Hilb... |
| lnopaddi 32603 | Additive property of a lin... |
| lnopmuli 32604 | Multiplicative property of... |
| lnopaddmuli 32605 | Sum/product property of a ... |
| lnopsubi 32606 | Subtraction property for a... |
| lnopsubmuli 32607 | Subtraction/product proper... |
| lnopmulsubi 32608 | Product/subtraction proper... |
| homco2 32609 | Move a scalar product out ... |
| idunop 32610 | The identity function (res... |
| 0cnop 32611 | The identically zero funct... |
| 0cnfn 32612 | The identically zero funct... |
| idcnop 32613 | The identity function (res... |
| idhmop 32614 | The Hilbert space identity... |
| 0hmop 32615 | The identically zero funct... |
| 0lnop 32616 | The identically zero funct... |
| 0lnfn 32617 | The identically zero funct... |
| nmop0 32618 | The norm of the zero opera... |
| nmfn0 32619 | The norm of the identicall... |
| hmopbdoptHIL 32620 | A Hermitian operator is a ... |
| hoddii 32621 | Distributive law for Hilbe... |
| hoddi 32622 | Distributive law for Hilbe... |
| nmop0h 32623 | The norm of any operator o... |
| idlnop 32624 | The identity function (res... |
| 0bdop 32625 | The identically zero opera... |
| adj0 32626 | Adjoint of the zero operat... |
| nmlnop0iALT 32627 | A linear operator with a z... |
| nmlnop0iHIL 32628 | A linear operator with a z... |
| nmlnopgt0i 32629 | A linear Hilbert space ope... |
| nmlnop0 32630 | A linear operator with a z... |
| nmlnopne0 32631 | A linear operator with a n... |
| lnopmi 32632 | The scalar product of a li... |
| lnophsi 32633 | The sum of two linear oper... |
| lnophdi 32634 | The difference of two line... |
| lnopcoi 32635 | The composition of two lin... |
| lnopco0i 32636 | The composition of a linea... |
| lnopeq0lem1 32637 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 32638 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 32639 | A condition implying that ... |
| lnopeqi 32640 | Two linear Hilbert space o... |
| lnopeq 32641 | Two linear Hilbert space o... |
| lnopunilem1 32642 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 32643 | Lemma for ~ lnopunii . (C... |
| lnopunii 32644 | If a linear operator (whos... |
| elunop2 32645 | An operator is unitary iff... |
| nmopun 32646 | Norm of a unitary Hilbert ... |
| unopbd 32647 | A unitary operator is a bo... |
| lnophmlem1 32648 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 32649 | Lemma for ~ lnophmi . (Co... |
| lnophmi 32650 | A linear operator is Hermi... |
| lnophm 32651 | A linear operator is Hermi... |
| hmops 32652 | The sum of two Hermitian o... |
| hmopm 32653 | The scalar product of a He... |
| hmopd 32654 | The difference of two Herm... |
| hmopco 32655 | The composition of two com... |
| nmbdoplbi 32656 | A lower bound for the norm... |
| nmbdoplb 32657 | A lower bound for the norm... |
| nmcexi 32658 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 32659 | The norm of a continuous l... |
| nmcoplbi 32660 | A lower bound for the norm... |
| nmcopex 32661 | The norm of a continuous l... |
| nmcoplb 32662 | A lower bound for the norm... |
| nmophmi 32663 | The norm of the scalar pro... |
| bdophmi 32664 | The scalar product of a bo... |
| lnconi 32665 | Lemma for ~ lnopconi and ~... |
| lnopconi 32666 | A condition equivalent to ... |
| lnopcon 32667 | A condition equivalent to ... |
| lnopcnbd 32668 | A linear operator is conti... |
| lncnopbd 32669 | A continuous linear operat... |
| lncnbd 32670 | A continuous linear operat... |
| lnopcnre 32671 | A linear operator is conti... |
| lnfnli 32672 | Basic property of a linear... |
| lnfnfi 32673 | A linear Hilbert space fun... |
| lnfn0i 32674 | The value of a linear Hilb... |
| lnfnaddi 32675 | Additive property of a lin... |
| lnfnmuli 32676 | Multiplicative property of... |
| lnfnaddmuli 32677 | Sum/product property of a ... |
| lnfnsubi 32678 | Subtraction property for a... |
| lnfn0 32679 | The value of a linear Hilb... |
| lnfnmul 32680 | Multiplicative property of... |
| nmbdfnlbi 32681 | A lower bound for the norm... |
| nmbdfnlb 32682 | A lower bound for the norm... |
| nmcfnexi 32683 | The norm of a continuous l... |
| nmcfnlbi 32684 | A lower bound for the norm... |
| nmcfnex 32685 | The norm of a continuous l... |
| nmcfnlb 32686 | A lower bound of the norm ... |
| lnfnconi 32687 | A condition equivalent to ... |
| lnfncon 32688 | A condition equivalent to ... |
| lnfncnbd 32689 | A linear functional is con... |
| imaelshi 32690 | The image of a subspace un... |
| rnelshi 32691 | The range of a linear oper... |
| nlelshi 32692 | The null space of a linear... |
| nlelchi 32693 | The null space of a contin... |
| riesz3i 32694 | A continuous linear functi... |
| riesz4i 32695 | A continuous linear functi... |
| riesz4 32696 | A continuous linear functi... |
| riesz1 32697 | Part 1 of the Riesz repres... |
| riesz2 32698 | Part 2 of the Riesz repres... |
| cnlnadjlem1 32699 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 32700 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 32701 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 32702 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32703 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32704 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32705 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32706 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32707 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32708 | Every continuous linear op... |
| cnlnadjeui 32709 | Every continuous linear op... |
| cnlnadjeu 32710 | Every continuous linear op... |
| cnlnadj 32711 | Every continuous linear op... |
| cnlnssadj 32712 | Every continuous linear Hi... |
| bdopssadj 32713 | Every bounded linear Hilbe... |
| bdopadj 32714 | Every bounded linear Hilbe... |
| adjbdln 32715 | The adjoint of a bounded l... |
| adjbdlnb 32716 | An operator is bounded and... |
| adjbd1o 32717 | The mapping of adjoints of... |
| adjlnop 32718 | The adjoint of an operator... |
| adjsslnop 32719 | Every operator with an adj... |
| nmopadjlei 32720 | Property of the norm of an... |
| nmopadjlem 32721 | Lemma for ~ nmopadji . (C... |
| nmopadji 32722 | Property of the norm of an... |
| adjeq0 32723 | An operator is zero iff it... |
| adjmul 32724 | The adjoint of the scalar ... |
| adjadd 32725 | The adjoint of the sum of ... |
| nmoptrii 32726 | Triangle inequality for th... |
| nmopcoi 32727 | Upper bound for the norm o... |
| bdophsi 32728 | The sum of two bounded lin... |
| bdophdi 32729 | The difference between two... |
| bdopcoi 32730 | The composition of two bou... |
| nmoptri2i 32731 | Triangle-type inequality f... |
| adjcoi 32732 | The adjoint of a compositi... |
| nmopcoadji 32733 | The norm of an operator co... |
| nmopcoadj2i 32734 | The norm of an operator co... |
| nmopcoadj0i 32735 | An operator composed with ... |
| unierri 32736 | If we approximate a chain ... |
| branmfn 32737 | The norm of the bra functi... |
| brabn 32738 | The bra of a vector is a b... |
| rnbra 32739 | The set of bras equals the... |
| bra11 32740 | The bra function maps vect... |
| bracnln 32741 | A bra is a continuous line... |
| cnvbraval 32742 | Value of the converse of t... |
| cnvbracl 32743 | Closure of the converse of... |
| cnvbrabra 32744 | The converse bra of the br... |
| bracnvbra 32745 | The bra of the converse br... |
| bracnlnval 32746 | The vector that a continuo... |
| cnvbramul 32747 | Multiplication property of... |
| kbass1 32748 | Dirac bra-ket associative ... |
| kbass2 32749 | Dirac bra-ket associative ... |
| kbass3 32750 | Dirac bra-ket associative ... |
| kbass4 32751 | Dirac bra-ket associative ... |
| kbass5 32752 | Dirac bra-ket associative ... |
| kbass6 32753 | Dirac bra-ket associative ... |
| leopg 32754 | Ordering relation for posi... |
| leop 32755 | Ordering relation for oper... |
| leop2 32756 | Ordering relation for oper... |
| leop3 32757 | Operator ordering in terms... |
| leoppos 32758 | Binary relation defining a... |
| leoprf2 32759 | The ordering relation for ... |
| leoprf 32760 | The ordering relation for ... |
| leopsq 32761 | The square of a Hermitian ... |
| 0leop 32762 | The zero operator is a pos... |
| idleop 32763 | The identity operator is a... |
| leopadd 32764 | The sum of two positive op... |
| leopmuli 32765 | The scalar product of a no... |
| leopmul 32766 | The scalar product of a po... |
| leopmul2i 32767 | Scalar product applied to ... |
| leoptri 32768 | The positive operator orde... |
| leoptr 32769 | The positive operator orde... |
| leopnmid 32770 | A bounded Hermitian operat... |
| nmopleid 32771 | A nonzero, bounded Hermiti... |
| opsqrlem1 32772 | Lemma for opsqri . (Contr... |
| opsqrlem2 32773 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32774 | Lemma for opsqri . (Contr... |
| opsqrlem4 32775 | Lemma for opsqri . (Contr... |
| opsqrlem5 32776 | Lemma for opsqri . (Contr... |
| opsqrlem6 32777 | Lemma for opsqri . (Contr... |
| pjhmopi 32778 | A projector is a Hermitian... |
| pjlnopi 32779 | A projector is a linear op... |
| pjnmopi 32780 | The operator norm of a pro... |
| pjbdlni 32781 | A projector is a bounded l... |
| pjhmop 32782 | A projection is a Hermitia... |
| hmopidmchi 32783 | An idempotent Hermitian op... |
| hmopidmpji 32784 | An idempotent Hermitian op... |
| hmopidmch 32785 | An idempotent Hermitian op... |
| hmopidmpj 32786 | An idempotent Hermitian op... |
| pjsdii 32787 | Distributive law for Hilbe... |
| pjddii 32788 | Distributive law for Hilbe... |
| pjsdi2i 32789 | Chained distributive law f... |
| pjcoi 32790 | Composition of projections... |
| pjcocli 32791 | Closure of composition of ... |
| pjcohcli 32792 | Closure of composition of ... |
| pjadjcoi 32793 | Adjoint of composition of ... |
| pjcofni 32794 | Functionality of compositi... |
| pjss1coi 32795 | Subset relationship for pr... |
| pjss2coi 32796 | Subset relationship for pr... |
| pjssmi 32797 | Projection meet property. ... |
| pjssge0i 32798 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32799 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32800 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32801 | Composition of projections... |
| pjscji 32802 | The projection of orthogon... |
| pjssumi 32803 | The projection on a subspa... |
| pjssposi 32804 | Projector ordering can be ... |
| pjordi 32805 | The definition of projecto... |
| pjssdif2i 32806 | The projection subspace of... |
| pjssdif1i 32807 | A necessary and sufficient... |
| pjimai 32808 | The image of a projection.... |
| pjidmcoi 32809 | A projection is idempotent... |
| pjoccoi 32810 | Composition of projections... |
| pjtoi 32811 | Subspace sum of projection... |
| pjoci 32812 | Projection of orthocomplem... |
| pjidmco 32813 | A projection operator is i... |
| dfpjop 32814 | Definition of projection o... |
| pjhmopidm 32815 | Two ways to express the se... |
| elpjidm 32816 | A projection operator is i... |
| elpjhmop 32817 | A projection operator is H... |
| 0leopj 32818 | A projector is a positive ... |
| pjadj2 32819 | A projector is self-adjoin... |
| pjadj3 32820 | A projector is self-adjoin... |
| elpjch 32821 | Reconstruction of the subs... |
| elpjrn 32822 | Reconstruction of the subs... |
| pjinvari 32823 | A closed subspace ` H ` wi... |
| pjin1i 32824 | Lemma for Theorem 1.22 of ... |
| pjin2i 32825 | Lemma for Theorem 1.22 of ... |
| pjin3i 32826 | Lemma for Theorem 1.22 of ... |
| pjclem1 32827 | Lemma for projection commu... |
| pjclem2 32828 | Lemma for projection commu... |
| pjclem3 32829 | Lemma for projection commu... |
| pjclem4a 32830 | Lemma for projection commu... |
| pjclem4 32831 | Lemma for projection commu... |
| pjci 32832 | Two subspaces commute iff ... |
| pjcmul1i 32833 | A necessary and sufficient... |
| pjcmul2i 32834 | The projection subspace of... |
| pjcohocli 32835 | Closure of composition of ... |
| pjadj2coi 32836 | Adjoint of double composit... |
| pj2cocli 32837 | Closure of double composit... |
| pj3lem1 32838 | Lemma for projection tripl... |
| pj3si 32839 | Stronger projection triple... |
| pj3i 32840 | Projection triplet theorem... |
| pj3cor1i 32841 | Projection triplet corolla... |
| pjs14i 32842 | Theorem S-14 of Watanabe, ... |
| isst 32845 | Property of a state. (Con... |
| ishst 32846 | Property of a complex Hilb... |
| sticl 32847 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32848 | Real closure of the value ... |
| hstcl 32849 | Closure of the value of a ... |
| hst1a 32850 | Unit value of a Hilbert-sp... |
| hstel2 32851 | Properties of a Hilbert-sp... |
| hstorth 32852 | Orthogonality property of ... |
| hstosum 32853 | Orthogonal sum property of... |
| hstoc 32854 | Sum of a Hilbert-space-val... |
| hstnmoc 32855 | Sum of norms of a Hilbert-... |
| stge0 32856 | The value of a state is no... |
| stle1 32857 | The value of a state is le... |
| hstle1 32858 | The norm of the value of a... |
| hst1h 32859 | The norm of a Hilbert-spac... |
| hst0h 32860 | The norm of a Hilbert-spac... |
| hstpyth 32861 | Pythagorean property of a ... |
| hstle 32862 | Ordering property of a Hil... |
| hstles 32863 | Ordering property of a Hil... |
| hstoh 32864 | A Hilbert-space-valued sta... |
| hst0 32865 | A Hilbert-space-valued sta... |
| sthil 32866 | The value of a state at th... |
| stj 32867 | The value of a state on a ... |
| sto1i 32868 | The state of a subspace pl... |
| sto2i 32869 | The state of the orthocomp... |
| stge1i 32870 | If a state is greater than... |
| stle0i 32871 | If a state is less than or... |
| stlei 32872 | Ordering law for states. ... |
| stlesi 32873 | Ordering law for states. ... |
| stji1i 32874 | Join of components of Sasa... |
| stm1i 32875 | State of component of unit... |
| stm1ri 32876 | State of component of unit... |
| stm1addi 32877 | Sum of states whose meet i... |
| staddi 32878 | If the sum of 2 states is ... |
| stm1add3i 32879 | Sum of states whose meet i... |
| stadd3i 32880 | If the sum of 3 states is ... |
| st0 32881 | The state of the zero subs... |
| strlem1 32882 | Lemma for strong state the... |
| strlem2 32883 | Lemma for strong state the... |
| strlem3a 32884 | Lemma for strong state the... |
| strlem3 32885 | Lemma for strong state the... |
| strlem4 32886 | Lemma for strong state the... |
| strlem5 32887 | Lemma for strong state the... |
| strlem6 32888 | Lemma for strong state the... |
| stri 32889 | Strong state theorem. The... |
| strb 32890 | Strong state theorem (bidi... |
| hstrlem2 32891 | Lemma for strong set of CH... |
| hstrlem3a 32892 | Lemma for strong set of CH... |
| hstrlem3 32893 | Lemma for strong set of CH... |
| hstrlem4 32894 | Lemma for strong set of CH... |
| hstrlem5 32895 | Lemma for strong set of CH... |
| hstrlem6 32896 | Lemma for strong set of CH... |
| hstri 32897 | Hilbert space admits a str... |
| hstrbi 32898 | Strong CH-state theorem (b... |
| largei 32899 | A Hilbert lattice admits a... |
| jplem1 32900 | Lemma for Jauch-Piron theo... |
| jplem2 32901 | Lemma for Jauch-Piron theo... |
| jpi 32902 | The function ` S ` , that ... |
| golem1 32903 | Lemma for Godowski's equat... |
| golem2 32904 | Lemma for Godowski's equat... |
| goeqi 32905 | Godowski's equation, shown... |
| stcltr1i 32906 | Property of a strong class... |
| stcltr2i 32907 | Property of a strong class... |
| stcltrlem1 32908 | Lemma for strong classical... |
| stcltrlem2 32909 | Lemma for strong classical... |
| stcltrthi 32910 | Theorem for classically st... |
| cvbr 32914 | Binary relation expressing... |
| cvbr2 32915 | Binary relation expressing... |
| cvcon3 32916 | Contraposition law for the... |
| cvpss 32917 | The covers relation implie... |
| cvnbtwn 32918 | The covers relation implie... |
| cvnbtwn2 32919 | The covers relation implie... |
| cvnbtwn3 32920 | The covers relation implie... |
| cvnbtwn4 32921 | The covers relation implie... |
| cvnsym 32922 | The covers relation is not... |
| cvnref 32923 | The covers relation is not... |
| cvntr 32924 | The covers relation is not... |
| spansncv2 32925 | Hilbert space has the cove... |
| mdbr 32926 | Binary relation expressing... |
| mdi 32927 | Consequence of the modular... |
| mdbr2 32928 | Binary relation expressing... |
| mdbr3 32929 | Binary relation expressing... |
| mdbr4 32930 | Binary relation expressing... |
| dmdbr 32931 | Binary relation expressing... |
| dmdmd 32932 | The dual modular pair prop... |
| mddmd 32933 | The modular pair property ... |
| dmdi 32934 | Consequence of the dual mo... |
| dmdbr2 32935 | Binary relation expressing... |
| dmdi2 32936 | Consequence of the dual mo... |
| dmdbr3 32937 | Binary relation expressing... |
| dmdbr4 32938 | Binary relation expressing... |
| dmdi4 32939 | Consequence of the dual mo... |
| dmdbr5 32940 | Binary relation expressing... |
| mddmd2 32941 | Relationship between modul... |
| mdsl0 32942 | A sublattice condition tha... |
| ssmd1 32943 | Ordering implies the modul... |
| ssmd2 32944 | Ordering implies the modul... |
| ssdmd1 32945 | Ordering implies the dual ... |
| ssdmd2 32946 | Ordering implies the dual ... |
| dmdsl3 32947 | Sublattice mapping for a d... |
| mdsl3 32948 | Sublattice mapping for a m... |
| mdslle1i 32949 | Order preservation of the ... |
| mdslle2i 32950 | Order preservation of the ... |
| mdslj1i 32951 | Join preservation of the o... |
| mdslj2i 32952 | Meet preservation of the r... |
| mdsl1i 32953 | If the modular pair proper... |
| mdsl2i 32954 | If the modular pair proper... |
| mdsl2bi 32955 | If the modular pair proper... |
| cvmdi 32956 | The covering property impl... |
| mdslmd1lem1 32957 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32958 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32959 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32960 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32961 | Preservation of the modula... |
| mdslmd2i 32962 | Preservation of the modula... |
| mdsldmd1i 32963 | Preservation of the dual m... |
| mdslmd3i 32964 | Modular pair conditions th... |
| mdslmd4i 32965 | Modular pair condition tha... |
| csmdsymi 32966 | Cross-symmetry implies M-s... |
| mdexchi 32967 | An exchange lemma for modu... |
| cvmd 32968 | The covering property impl... |
| cvdmd 32969 | The covering property impl... |
| ela 32971 | Atoms in a Hilbert lattice... |
| elat2 32972 | Expanded membership relati... |
| elatcv0 32973 | A Hilbert lattice element ... |
| atcv0 32974 | An atom covers the zero su... |
| atssch 32975 | Atoms are a subset of the ... |
| atelch 32976 | An atom is a Hilbert latti... |
| atne0 32977 | An atom is not the Hilbert... |
| atss 32978 | A lattice element smaller ... |
| atsseq 32979 | Two atoms in a subset rela... |
| atcveq0 32980 | A Hilbert lattice element ... |
| h1da 32981 | A 1-dimensional subspace i... |
| spansna 32982 | The span of the singleton ... |
| sh1dle 32983 | A 1-dimensional subspace i... |
| ch1dle 32984 | A 1-dimensional subspace i... |
| atom1d 32985 | The 1-dimensional subspace... |
| superpos 32986 | Superposition Principle. ... |
| chcv1 32987 | The Hilbert lattice has th... |
| chcv2 32988 | The Hilbert lattice has th... |
| chjatom 32989 | The join of a closed subsp... |
| shatomici 32990 | The lattice of Hilbert sub... |
| hatomici 32991 | The Hilbert lattice is ato... |
| hatomic 32992 | A Hilbert lattice is atomi... |
| shatomistici 32993 | The lattice of Hilbert sub... |
| hatomistici 32994 | ` CH ` is atomistic, i.e. ... |
| chpssati 32995 | Two Hilbert lattice elemen... |
| chrelati 32996 | The Hilbert lattice is rel... |
| chrelat2i 32997 | A consequence of relative ... |
| cvati 32998 | If a Hilbert lattice eleme... |
| cvbr4i 32999 | An alternate way to expres... |
| cvexchlem 33000 | Lemma for ~ cvexchi . (Co... |
| cvexchi 33001 | The Hilbert lattice satisf... |
| chrelat2 33002 | A consequence of relative ... |
| chrelat3 33003 | A consequence of relative ... |
| chrelat3i 33004 | A consequence of the relat... |
| chrelat4i 33005 | A consequence of relative ... |
| cvexch 33006 | The Hilbert lattice satisf... |
| cvp 33007 | The Hilbert lattice satisf... |
| atnssm0 33008 | The meet of a Hilbert latt... |
| atnemeq0 33009 | The meet of distinct atoms... |
| atssma 33010 | The meet with an atom's su... |
| atcv0eq 33011 | Two atoms covering the zer... |
| atcv1 33012 | Two atoms covering the zer... |
| atexch 33013 | The Hilbert lattice satisf... |
| atomli 33014 | An assertion holding in at... |
| atoml2i 33015 | An assertion holding in at... |
| atordi 33016 | An ordering law for a Hilb... |
| atcvatlem 33017 | Lemma for ~ atcvati . (Co... |
| atcvati 33018 | A nonzero Hilbert lattice ... |
| atcvat2i 33019 | A Hilbert lattice element ... |
| atord 33020 | An ordering law for a Hilb... |
| atcvat2 33021 | A Hilbert lattice element ... |
| chirredlem1 33022 | Lemma for ~ chirredi . (C... |
| chirredlem2 33023 | Lemma for ~ chirredi . (C... |
| chirredlem3 33024 | Lemma for ~ chirredi . (C... |
| chirredlem4 33025 | Lemma for ~ chirredi . (C... |
| chirredi 33026 | The Hilbert lattice is irr... |
| chirred 33027 | The Hilbert lattice is irr... |
| atcvat3i 33028 | A condition implying that ... |
| atcvat4i 33029 | A condition implying exist... |
| atdmd 33030 | Two Hilbert lattice elemen... |
| atmd 33031 | Two Hilbert lattice elemen... |
| atmd2 33032 | Two Hilbert lattice elemen... |
| atabsi 33033 | Absorption of an incompara... |
| atabs2i 33034 | Absorption of an incompara... |
| mdsymlem1 33035 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 33036 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 33037 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 33038 | Lemma for ~ mdsymi . This... |
| mdsymlem5 33039 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 33040 | Lemma for ~ mdsymi . This... |
| mdsymlem7 33041 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 33042 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 33043 | M-symmetry of the Hilbert ... |
| mdsym 33044 | M-symmetry of the Hilbert ... |
| dmdsym 33045 | Dual M-symmetry of the Hil... |
| atdmd2 33046 | Two Hilbert lattice elemen... |
| sumdmdii 33047 | If the subspace sum of two... |
| cmmdi 33048 | Commuting subspaces form a... |
| cmdmdi 33049 | Commuting subspaces form a... |
| sumdmdlem 33050 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 33051 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 33052 | The subspace sum of two Hi... |
| dmdbr4ati 33053 | Dual modular pair property... |
| dmdbr5ati 33054 | Dual modular pair property... |
| dmdbr6ati 33055 | Dual modular pair property... |
| dmdbr7ati 33056 | Dual modular pair property... |
| mdoc1i 33057 | Orthocomplements form a mo... |
| mdoc2i 33058 | Orthocomplements form a mo... |
| dmdoc1i 33059 | Orthocomplements form a du... |
| dmdoc2i 33060 | Orthocomplements form a du... |
| mdcompli 33061 | A condition equivalent to ... |
| dmdcompli 33062 | A condition equivalent to ... |
| mddmdin0i 33063 | If dual modular implies mo... |
| cdjreui 33064 | A member of the sum of dis... |
| cdj1i 33065 | Two ways to express " ` A ... |
| cdj3lem1 33066 | A property of " ` A ` and ... |
| cdj3lem2 33067 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 33068 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 33069 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 33070 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 33071 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 33072 | Lemma for ~ cdj3i . The s... |
| cdj3i 33073 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 33074 | (_This theorem is a dummy ... |
| tgacne 33075 | Non-collinearity of triang... |
| tgbcne 33076 | Non-collinearity of triang... |
| cgrahlb 33077 | Two null angles are congru... |
| inagbtwn 33078 | A flat sub-angle forces th... |
| inagcgrasubhl 33079 | In-angle transport, degene... |
| inagcgrasubbtwn 33080 | In-angle transport, degene... |
| inagcgrabtwn 33081 | In-angle transport, degene... |
| iscgrag 33082 | Ray normalization: an angl... |
| cgrgptxfr 33083 | Transfer a segment point a... |
| inagcgratrgmain 33084 | In-angle transport, the no... |
| inagcgratrgw 33085 | ~ inagcgratrgmain with the... |
| inagcgratrgxfr 33086 | In-angle transport with th... |
| inagcgratrgncol 33087 | In-angle transport once th... |
| inagcgratrgn 33088 | In-angle transport, non-co... |
| inaglea 33089 | If ` C ` lies inside angle... |
| cgrainag 33090 | Target-side in-angle trans... |
| leawittrg 33091 | Target-side ` leA ` witnes... |
| leacgra2 33092 | Target-side ` leA ` transp... |
| leacgra1 33093 | Source-side ` leA ` transp... |
| leacgra 33094 | GeoCoq l11__30 analogue: `... |
| tgmircgrg 33095 | Reflecting ` B ` through t... |
| tgmircgra 33096 | The angle congruence the e... |
| tgextamirn 33097 | The midpoint-reflected poi... |
| tgextancd 33098 | The target-side non-collin... |
| tgextahpl 33099 | The generic open half-plan... |
| tgextalem1inag 33100 | The point obtained by refl... |
| tgextale 33101 | The non-strict exterior-an... |
| tgextamirnhl 33102 | The midpoint-reflected poi... |
| tgextahp 33103 | The reflected point and ` ... |
| tgextancgra 33104 | The exterior angle is NOT ... |
| tgextaoa 33105 | The two exterior angles at... |
| tgextalt1 33106 | ~ tgexta 's FIRST conjunct... |
| tgextalt2 33107 | ~ tgexta 's SECOND conjunc... |
| tgexta 33108 | The exterior-angle theorem... |
| tgextan 33109 | ~ tgexta 's strictness can... |
| tgaasncol 33110 | Slide a non-collinearity a... |
| tgaasexc 33111 | The contradiction branch o... |
| tgaasne 33112 | Two DISTINCT points ` F ` ... |
| tgaascgra 33113 | The comparison triangle of... |
| tgaas1h 33114 | ~ tgaas1 with the construc... |
| tgaas1 33115 | Under the AAS hypotheses, ... |
| tgaas 33116 | Angle-angle-side: two tria... |
| sa-abvi 33117 | A theorem about the univer... |
| xfree 33118 | A partial converse to ~ 19... |
| xfree2 33119 | A partial converse to ~ 19... |
| addltmulALT 33120 | A proof readability experi... |
| ad11antr 33121 | Deduction adding 11 conjun... |
| simp-12l 33122 | Simplification of a conjun... |
| simp-12r 33123 | Simplification of a conjun... |
| an52ds 33124 | Inference exchanging the l... |
| an62ds 33125 | Inference exchanging the l... |
| an72ds 33126 | Inference exchanging the l... |
| an82ds 33127 | Inference exchanging the l... |
| syl22anbrc 33128 | Syllogism inference. (Con... |
| or3di 33129 | Distributive law for disju... |
| or3dir 33130 | Distributive law for disju... |
| 3o1cs 33131 | Deduction eliminating disj... |
| 3o2cs 33132 | Deduction eliminating disj... |
| 3o3cs 33133 | Deduction eliminating disj... |
| sbc2iedf 33134 | Conversion of implicit sub... |
| rspc2daf 33135 | Double restricted speciali... |
| ralcom4f 33136 | Commutation of restricted ... |
| rexcom4f 33137 | Commutation of restricted ... |
| 19.9d2rf 33138 | A deduction version of one... |
| 19.9d2r 33139 | A deduction version of one... |
| r19.29ffa 33140 | A commonly used pattern ba... |
| reu6dv 33141 | A condition which implies ... |
| eqtrb 33142 | A transposition of equalit... |
| eqelbid 33143 | A variable elimination law... |
| opsbc2ie 33144 | Conversion of implicit sub... |
| opreu2reuALT 33145 | Correspondence between uni... |
| 2reucom 33148 | Double restricted existent... |
| 2reu2rex1 33149 | Double restricted existent... |
| 2reureurex 33150 | Double restricted existent... |
| 2reu2reu2 33151 | Double restricted existent... |
| opreu2reu1 33152 | Equivalent definition of t... |
| sq2reunnltb 33153 | There exists a unique deco... |
| addsqnot2reu 33154 | For each complex number ` ... |
| sbceqbidf 33155 | Equality theorem for class... |
| sbcies 33156 | A special version of class... |
| mo5f 33157 | Alternate definition of "a... |
| nmo 33158 | Negation of "at most one".... |
| reuxfrdf 33159 | Transfer existential uniqu... |
| rexunirn 33160 | Restricted existential qua... |
| rmoxfrd 33161 | Transfer "at most one" res... |
| rmoun 33162 | "At most one" restricted e... |
| rmounid 33163 | A case where an "at most o... |
| riotaeqbidva 33164 | Equivalent wff's yield equ... |
| dmrab 33165 | Domain of a restricted cla... |
| difrab2 33166 | Difference of two restrict... |
| rabexgfGS 33167 | Separation Scheme in terms... |
| rabsnel 33168 | Truth implied by equality ... |
| rabsspr 33169 | Conditions for a restricte... |
| rabsstp 33170 | Conditions for a restricte... |
| 3unrab 33171 | Union of three restricted ... |
| foresf1o 33172 | From a surjective function... |
| rabfodom 33173 | Domination relation for re... |
| rabrexfi 33174 | Conditions for a class abs... |
| abrexdomjm 33175 | An indexed set is dominate... |
| abrexdom2jm 33176 | An indexed set is dominate... |
| abrexexd 33177 | Existence of a class abstr... |
| elabreximd 33178 | Class substitution in an i... |
| elabreximdv 33179 | Class substitution in an i... |
| abrexss 33180 | A necessary condition for ... |
| nelun 33181 | Negated membership for a u... |
| snsssng 33182 | If a singleton is a subset... |
| n0nsnel 33183 | If a class with one elemen... |
| inin 33184 | Intersection with an inter... |
| difininv 33185 | Condition for the intersec... |
| difeq 33186 | Rewriting an equation with... |
| eqdif 33187 | If both set differences of... |
| indifbi 33188 | Two ways to express equali... |
| diffib 33189 | Case where ~ diffi is a bi... |
| difxp1ss 33190 | Difference law for Cartesi... |
| difxp2ss 33191 | Difference law for Cartesi... |
| indifundif 33192 | A remarkable equation with... |
| elpwincl1 33193 | Closure of intersection wi... |
| elpwdifcl 33194 | Closure of class differenc... |
| elpwiuncl 33195 | Closure of indexed union w... |
| elpreq 33196 | Equality wihin a pair. (C... |
| prssad 33197 | If a pair is a subset of a... |
| prssbd 33198 | If a pair is a subset of a... |
| nelpr 33199 | A set ` A ` not in a pair ... |
| inpr0 33200 | Rewrite an empty intersect... |
| neldifpr1 33201 | The first element of a pai... |
| neldifpr2 33202 | The second element of a pa... |
| unidifsnel 33203 | The other element of a pai... |
| unidifsnne 33204 | The other element of a pai... |
| tpssg 33205 | An unordered triple of ele... |
| tpssd 33206 | Deduction version of tpssi... |
| tpssad 33207 | If an ordered triple is a ... |
| tpssbd 33208 | If an ordered triple is a ... |
| tpsscd 33209 | If an ordered triple is a ... |
| ifeqeqx 33210 | An equality theorem tailor... |
| elimifd 33211 | Elimination of a condition... |
| elim2if 33212 | Elimination of two conditi... |
| elim2ifim 33213 | Elimination of two conditi... |
| ifeq3da 33214 | Given an expression ` C ` ... |
| ifnetrue 33215 | Deduce truth from a condit... |
| ifnefals 33216 | Deduce falsehood from a co... |
| ifnebib 33217 | The converse of ~ ifbi hol... |
| ififcom 33218 | Commute two nested conditi... |
| uniinn0 33219 | Sufficient and necessary c... |
| difuncomp 33220 | Express a class difference... |
| elpwunicl 33221 | Closure of a set union wit... |
| cbviunf 33222 | Rule used to change the bo... |
| iuneq12daf 33223 | Equality deduction for ind... |
| iunin1f 33224 | Indexed union of intersect... |
| ssiun3 33225 | Subset equivalence for an ... |
| ssiun2sf 33226 | Subset relationship for an... |
| iuninc 33227 | The union of an increasing... |
| iundifdifd 33228 | The intersection of a set ... |
| iundifdif 33229 | The intersection of a set ... |
| iunrdx 33230 | Re-index an indexed union.... |
| iunrnmptss 33231 | A subset relation for an i... |
| iunxunsn 33232 | Appending a set to an inde... |
| iunxunpr 33233 | Appending two sets to an i... |
| iunxpssiun1 33234 | Provide an upper bound for... |
| iinabrex 33235 | Rewriting an indexed inter... |
| disjnf 33236 | In case ` x ` is not free ... |
| cbvdisjf 33237 | Change bound variables in ... |
| disjss1f 33238 | A subset of a disjoint col... |
| disjeq1f 33239 | Equality theorem for disjo... |
| disjxun0 33240 | Simplify a disjoint union.... |
| disjdifprg 33241 | A trivial partition into a... |
| disjdifprg2 33242 | A trivial partition of a s... |
| disji2f 33243 | Property of a disjoint col... |
| disjif 33244 | Property of a disjoint col... |
| disjorf 33245 | Two ways to say that a col... |
| disjorsf 33246 | Two ways to say that a col... |
| disjif2 33247 | Property of a disjoint col... |
| disjabrex 33248 | Rewriting a disjoint colle... |
| disjabrexf 33249 | Rewriting a disjoint colle... |
| disjpreima 33250 | A preimage of a disjoint s... |
| disjrnmpt 33251 | Rewriting a disjoint colle... |
| disjin 33252 | If a collection is disjoin... |
| disjin2 33253 | If a collection is disjoin... |
| disjxpin 33254 | Derive a disjunction over ... |
| iundisjf 33255 | Rewrite a countable union ... |
| iundisj2f 33256 | A disjoint union is disjoi... |
| disjrdx 33257 | Re-index a disjunct collec... |
| disjex 33258 | Two ways to say that two c... |
| disjexc 33259 | A variant of ~ disjex , ap... |
| disjunsn 33260 | Append an element to a dis... |
| disjun0 33261 | Adding the empty element p... |
| disjiunel 33262 | A set of elements B of a d... |
| disjuniel 33263 | A set of elements B of a d... |
| xpdisjres 33264 | Restriction of a constant ... |
| opeldifid 33265 | Ordered pair elementhood o... |
| difres 33266 | Case when class difference... |
| imadifxp 33267 | Image of the difference wi... |
| relfi 33268 | A relation (set) is finite... |
| 0res 33269 | Restriction of the empty f... |
| fcoinver 33270 | Build an equivalence relat... |
| fcoinvbr 33271 | Binary relation for the eq... |
| brabgaf 33272 | The law of concretion for ... |
| brelg 33273 | Two things in a binary rel... |
| br8d 33274 | Substitution for an eight-... |
| fnfvor 33275 | Relation between two funct... |
| ofrco 33276 | Function relation between ... |
| opabdm 33277 | Domain of an ordered-pair ... |
| opabrn 33278 | Range of an ordered-pair c... |
| opabssi 33279 | Sufficient condition for a... |
| opabid2ss 33280 | One direction of ~ opabid2... |
| ssrelf 33281 | A subclass relationship de... |
| eqrelrd2 33282 | A version of ~ eqrelrdv2 w... |
| erbr3b 33283 | Biconditional for equivale... |
| iunsnima 33284 | Image of a singleton by an... |
| iunsnima2 33285 | Version of ~ iunsnima with... |
| fconst7v 33286 | An alternative way to expr... |
| constcof 33287 | Composition with a constan... |
| ac6sf2 33288 | Alternate version of ~ ac6... |
| ac6mapd 33289 | Axiom of choice equivalent... |
| fnresin 33290 | Restriction of a function ... |
| fresunsn 33291 | Recover the original funct... |
| f1o3d 33292 | Describe an implicit one-t... |
| eldmne0 33293 | A function of nonempty dom... |
| f1rnen 33294 | Equinumerosity of the rang... |
| f1oeq3dd 33295 | Equality deduction for one... |
| rinvf1o 33296 | Sufficient conditions for ... |
| fresf1o 33297 | Conditions for a restricti... |
| nfpconfp 33298 | The set of fixed points of... |
| fmptco1f1o 33299 | The action of composing (t... |
| cofmpt2 33300 | Express composition of a m... |
| f1mptrn 33301 | Express injection for a ma... |
| dfimafnf 33302 | Alternate definition of th... |
| funimass4f 33303 | Membership relation for th... |
| suppss2f 33304 | Show that the support of a... |
| ofrn 33305 | The range of the function ... |
| ofrn2 33306 | The range of the function ... |
| off2 33307 | The function operation pro... |
| ofresid 33308 | Applying an operation rest... |
| unipreima 33309 | Preimage of a class union.... |
| opfv 33310 | Value of a function produc... |
| xppreima 33311 | The preimage of a Cartesia... |
| 2ndimaxp 33312 | Image of a cartesian produ... |
| dmdju 33313 | Domain of a disjoint union... |
| djussxp2 33314 | Stronger version of ~ djus... |
| 2ndresdju 33315 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 33316 | The ` 2nd ` function restr... |
| xppreima2 33317 | The preimage of a Cartesia... |
| abfmpunirn 33318 | Membership in a union of a... |
| rabfmpunirn 33319 | Membership in a union of a... |
| abfmpeld 33320 | Membership in an element o... |
| abfmpel 33321 | Membership in an element o... |
| fmptdf2 33322 | Domain and codomain of the... |
| fmptcof2 33323 | Composition of two functio... |
| fcomptf 33324 | Express composition of two... |
| acunirnmpt 33325 | Axiom of choice for the un... |
| acunirnmpt2 33326 | Axiom of choice for the un... |
| acunirnmpt2f 33327 | Axiom of choice for the un... |
| aciunf1lem 33328 | Choice in an index union. ... |
| aciunf1 33329 | Choice in an index union. ... |
| ofoprabco 33330 | Function operation as a co... |
| ofpreima 33331 | Express the preimage of a ... |
| ofpreima2 33332 | Express the preimage of a ... |
| funcnv5mpt 33333 | Two ways to say that a fun... |
| funcnv4mpt 33334 | Two ways to say that a fun... |
| preimane 33335 | Different elements have di... |
| fnpreimac 33336 | Choose a set ` x ` contain... |
| fgreu 33337 | Exactly one point of a fun... |
| fcnvgreu 33338 | If the converse of a relat... |
| mptssALT 33339 | Deduce subset relation of ... |
| dfcnv2 33340 | Alternative definition of ... |
| partfun2 33341 | Rewrite a function defined... |
| rnressnsn 33342 | The range of a restriction... |
| mpomptxf 33343 | Express a two-argument fun... |
| of0r 33344 | Function operation with th... |
| suppovss 33345 | A bound for the support of... |
| elsuppfnd 33346 | Deduce membership in the s... |
| fisuppov1 33347 | Formula building theorem f... |
| suppun2 33348 | The support of a union is ... |
| fdifsupp 33349 | Express the support of a f... |
| suppiniseg 33350 | Relation between the suppo... |
| fsuppinisegfi 33351 | The initial segment ` ( ``... |
| fressupp 33352 | The restriction of a funct... |
| fdifsuppconst 33353 | A function is a zero const... |
| ressupprn 33354 | The range of a function re... |
| supppreima 33355 | Express the support of a f... |
| fsupprnfi 33356 | Finite support implies fin... |
| mptiffisupp 33357 | Conditions for a mapping f... |
| cosnopne 33358 | Composition of two ordered... |
| cosnop 33359 | Composition of two ordered... |
| cnvprop 33360 | Converse of a pair of orde... |
| brprop 33361 | Binary relation for a pair... |
| mptprop 33362 | Rewrite pairs of ordered p... |
| coprprop 33363 | Composition of two pairs o... |
| fmptunsnop 33364 | Two ways to express a func... |
| gtiso 33365 | Two ways to write a strict... |
| isoun 33366 | Infer an isomorphism from ... |
| disjdsct 33367 | A disjoint collection is d... |
| df1stres 33368 | Definition for a restricti... |
| df2ndres 33369 | Definition for a restricti... |
| 1stpreimas 33370 | The preimage of a singleto... |
| 1stpreima 33371 | The preimage by ` 1st ` is... |
| 2ndpreima 33372 | The preimage by ` 2nd ` is... |
| curry2ima 33373 | The image of a curried fun... |
| preiman0 33374 | The preimage of a nonempty... |
| intimafv 33375 | The intersection of an ima... |
| snct 33376 | A singleton is countable. ... |
| prct 33377 | An unordered pair is count... |
| mpocti 33378 | An operation is countable ... |
| mptctf 33379 | A countable mapping set is... |
| abrexctf 33380 | An image set of a countabl... |
| padct 33381 | Index a countable set with... |
| fcobij 33382 | Composing functions with a... |
| fcobijfs 33383 | Composing finitely support... |
| fcobijfs2 33384 | Composing finitely support... |
| suppss3 33385 | Deduce a function's suppor... |
| fsuppcurry1 33386 | Finite support of a currie... |
| fsuppcurry2 33387 | Finite support of a currie... |
| offinsupp1 33388 | Finite support for a funct... |
| ffs2 33389 | Rewrite a function's suppo... |
| ffsrn 33390 | The range of a finitely su... |
| cocnvf1o 33391 | Composing with the inverse... |
| resf1o 33392 | Restriction of functions t... |
| maprnin 33393 | Restricting the range of t... |
| fpwrelmapffslem 33394 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 33395 | Define a canonical mapping... |
| fpwrelmapffs 33396 | Define a canonical mapping... |
| sgnval2 33397 | Value of the signum of a r... |
| creq0 33398 | The real representation of... |
| 1nei 33399 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 33400 | An integer unit times itse... |
| nnmulge 33401 | Multiplying by a positive ... |
| submuladdd 33402 | The product of a differenc... |
| binom2subadd 33403 | The difference of the squa... |
| cjsubd 33404 | Complex conjugate distribu... |
| re0cj 33405 | The conjugate of a pure im... |
| receqid 33406 | Real numbers equal to thei... |
| pythagreim 33407 | A simplified version of th... |
| efiargd 33408 | The exponential of the "ar... |
| arginv 33409 | The argument of the invers... |
| argcj 33410 | The argument of the conjug... |
| quad3d 33411 | Variant of quadratic equat... |
| lt2addrd 33412 | If the right-hand side of ... |
| nn0mnfxrd 33413 | Nonnegative integers or mi... |
| xrlelttric 33414 | Trichotomy law for extende... |
| xaddeq0 33415 | Two extended reals which a... |
| rexmul2 33416 | If the result ` A ` of an ... |
| xrinfm 33417 | The extended real numbers ... |
| le2halvesd 33418 | A sum is less than the who... |
| xraddge02 33419 | A number is less than or e... |
| xrge0addge 33420 | A number is less than or e... |
| xlt2addrd 33421 | If the right-hand side of ... |
| xrge0infss 33422 | Any subset of nonnegative ... |
| xrge0infssd 33423 | Inequality deduction for i... |
| xrge0addcld 33424 | Nonnegative extended reals... |
| xrge0subcld 33425 | Condition for closure of n... |
| infxrge0lb 33426 | A member of a set of nonne... |
| infxrge0glb 33427 | The infimum of a set of no... |
| infxrge0gelb 33428 | The infimum of a set of no... |
| xrofsup 33429 | The supremum is preserved ... |
| supxrnemnf 33430 | The supremum of a nonempty... |
| xnn0gt0 33431 | Nonzero extended nonnegati... |
| xnn01gt 33432 | An extended nonnegative in... |
| nn0xmulclb 33433 | Finite multiplication in t... |
| xnn0nn0d 33434 | Conditions for an extended... |
| xnn0nnd 33435 | Conditions for an extended... |
| joiniooico 33436 | Disjoint joining an open i... |
| ubico 33437 | A right-open interval does... |
| xeqlelt 33438 | Equality in terms of 'less... |
| eliccelico 33439 | Relate elementhood to a cl... |
| elicoelioo 33440 | Relate elementhood to a cl... |
| iocinioc2 33441 | Intersection between two o... |
| xrdifh 33442 | Class difference of a half... |
| iocinif 33443 | Relate intersection of two... |
| difioo 33444 | The difference between two... |
| difico 33445 | The difference between two... |
| uzssico 33446 | Upper integer sets are a s... |
| fz2ssnn0 33447 | A finite set of sequential... |
| nndiffz1 33448 | Upper set of the positive ... |
| ssnnssfz 33449 | For any finite subset of `... |
| fzm1ne1 33450 | Elementhood of an integer ... |
| fzspl 33451 | Split the last element of ... |
| fzdif2 33452 | Split the last element of ... |
| fzodif2 33453 | Split the last element of ... |
| fzodif1 33454 | Set difference of two half... |
| fzsplit3 33455 | Split a finite interval of... |
| nn0diffz0 33456 | Upper set of the nonnegati... |
| bcm1n 33457 | The proportion of one bino... |
| iundisjfi 33458 | Rewrite a countable union ... |
| iundisj2fi 33459 | A disjoint union is disjoi... |
| iundisjcnt 33460 | Rewrite a countable union ... |
| iundisj2cnt 33461 | A countable disjoint union... |
| f1ocnt 33462 | Given a countable set ` A ... |
| fz1nnct 33463 | NN and integer ranges star... |
| fz1nntr 33464 | NN and integer ranges star... |
| fzo0opth 33465 | Equality for a half open i... |
| nn0difffzod 33466 | A nonnegative integer that... |
| suppssnn0 33467 | Show that the support of a... |
| hashunif 33468 | The cardinality of a disjo... |
| hashxpe 33469 | The size of the Cartesian ... |
| hashgt1 33470 | Restate "set contains at l... |
| hashne0 33471 | Deduce that the size of a ... |
| hashimaf1 33472 | Taking the image of a set ... |
| elq2 33473 | Elementhood in the rationa... |
| znumd 33474 | Numerator of an integer. ... |
| zdend 33475 | Denominator of an integer.... |
| numdenneg 33476 | Numerator and denominator ... |
| divnumden2 33477 | Calculate the reduced form... |
| expgt0b 33478 | A real number ` A ` raised... |
| nn0split01 33479 | Split 0 and 1 from the non... |
| nn0disj01 33480 | The pair ` { 0 , 1 } ` doe... |
| nnindf 33481 | Principle of Mathematical ... |
| nn0min 33482 | Extracting the minimum pos... |
| subne0nn 33483 | A nonnegative difference i... |
| ltesubnnd 33484 | Subtracting an integer num... |
| fprodeq02 33485 | If one of the factors is z... |
| fprodex01 33486 | A product of factors equal... |
| prodpr 33487 | A product over a pair is t... |
| prodtp 33488 | A product over a triple is... |
| fsumub 33489 | An upper bound for a term ... |
| fsumiunle 33490 | Upper bound for a sum of n... |
| dfdec100 33491 | Split the hundreds from a ... |
| sgnsgn 33492 | Signum is idempotent. (Co... |
| sgnmulsgp 33493 | If two real numbers are of... |
| nexple 33494 | A lower bound for an expon... |
| 2exple2exp 33495 | If a nonnegative integer `... |
| expevenpos 33496 | Even powers are positive. ... |
| oexpled 33497 | Odd power monomials are mo... |
| indsumin 33498 | Finite sum of a product wi... |
| prodindf 33499 | The product of indicators ... |
| indsn 33500 | The indicator function of ... |
| indf1o 33501 | The bijection between a po... |
| indpreima 33502 | A function with range ` { ... |
| indf1ofs 33503 | The bijection between fini... |
| indsupp 33504 | The support of the indicat... |
| indfsd 33505 | The indicator function of ... |
| indfsid 33506 | Conditions for a function ... |
| dp2eq1 33509 | Equality theorem for the d... |
| dp2eq2 33510 | Equality theorem for the d... |
| dp2eq1i 33511 | Equality theorem for the d... |
| dp2eq2i 33512 | Equality theorem for the d... |
| dp2eq12i 33513 | Equality theorem for the d... |
| dp20u 33514 | Add a zero in the tenths (... |
| dp20h 33515 | Add a zero in the unit pla... |
| dp2cl 33516 | Closure for the decimal fr... |
| dp2clq 33517 | Closure for a decimal frac... |
| rpdp2cl 33518 | Closure for a decimal frac... |
| rpdp2cl2 33519 | Closure for a decimal frac... |
| dp2lt10 33520 | Decimal fraction builds re... |
| dp2lt 33521 | Comparing two decimal frac... |
| dp2ltsuc 33522 | Comparing a decimal fracti... |
| dp2ltc 33523 | Comparing two decimal expa... |
| dpval 33526 | Define the value of the de... |
| dpcl 33527 | Prove that the closure of ... |
| dpfrac1 33528 | Prove a simple equivalence... |
| dpval2 33529 | Value of the decimal point... |
| dpval3 33530 | Value of the decimal point... |
| dpmul10 33531 | Multiply by 10 a decimal e... |
| decdiv10 33532 | Divide a decimal number by... |
| dpmul100 33533 | Multiply by 100 a decimal ... |
| dp3mul10 33534 | Multiply by 10 a decimal e... |
| dpmul1000 33535 | Multiply by 1000 a decimal... |
| dpval3rp 33536 | Value of the decimal point... |
| dp0u 33537 | Add a zero in the tenths p... |
| dp0h 33538 | Remove a zero in the units... |
| rpdpcl 33539 | Closure of the decimal poi... |
| dplt 33540 | Comparing two decimal expa... |
| dplti 33541 | Comparing a decimal expans... |
| dpgti 33542 | Comparing a decimal expans... |
| dpltc 33543 | Comparing two decimal inte... |
| dpexpp1 33544 | Add one zero to the mantis... |
| 0dp2dp 33545 | Multiply by 10 a decimal e... |
| dpadd2 33546 | Addition with one decimal,... |
| dpadd 33547 | Addition with one decimal.... |
| dpadd3 33548 | Addition with two decimals... |
| dpmul 33549 | Multiplication with one de... |
| dpmul4 33550 | An upper bound to multipli... |
| threehalves 33551 | Example theorem demonstrat... |
| 1mhdrd 33552 | Example theorem demonstrat... |
| xdivval 33555 | Value of division: the (un... |
| xrecex 33556 | Existence of reciprocal of... |
| xmulcand 33557 | Cancellation law for exten... |
| xreceu 33558 | Existential uniqueness of ... |
| xdivcld 33559 | Closure law for the extend... |
| xdivcl 33560 | Closure law for the extend... |
| xdivmul 33561 | Relationship between divis... |
| rexdiv 33562 | The extended real division... |
| xdivrec 33563 | Relationship between divis... |
| xdivid 33564 | A number divided by itself... |
| xdiv0 33565 | Division into zero is zero... |
| xdiv0rp 33566 | Division into zero is zero... |
| eliccioo 33567 | Membership in a closed int... |
| elxrge02 33568 | Elementhood in the set of ... |
| xdivpnfrp 33569 | Plus infinity divided by a... |
| rpxdivcld 33570 | Closure law for extended d... |
| xrpxdivcld 33571 | Closure law for extended d... |
| wrdres 33572 | Condition for the restrict... |
| wrdsplex 33573 | Existence of a split of a ... |
| wrdfsupp 33574 | A word has finite support.... |
| wrdpmcl 33575 | Closure of a word with per... |
| pfx1s2 33576 | The prefix of length 1 of ... |
| pfxrn2 33577 | The range of a prefix of a... |
| pfxrn3 33578 | Express the range of a pre... |
| pfxf1 33579 | Condition for a prefix to ... |
| s2f1 33580 | Conditions for a length 2 ... |
| s3f1 33581 | Conditions for a length 3 ... |
| s3clhash 33582 | Closure of the words of le... |
| pfxlsw2ccat 33583 | Reconstruct a word from it... |
| ccatws1f1o 33584 | Conditions for the concate... |
| ccatws1f1olast 33585 | Two ways to reorder symbol... |
| wrdt2ind 33586 | Perform an induction over ... |
| swrdrn2 33587 | The range of a subword is ... |
| swrdrndisj 33588 | Condition for the range of... |
| splfv3 33589 | Symbols to the right of a ... |
| 1cshid 33590 | Cyclically shifting a sing... |
| cshw1s2 33591 | Cyclically shifting a leng... |
| cshwrnid 33592 | Cyclically shifting a word... |
| cshf1o 33593 | Condition for the cyclic s... |
| ressplusf 33594 | The group operation functi... |
| ressnm 33595 | The norm in a restricted s... |
| abvpropd2 33596 | Weaker version of ~ abvpro... |
| ressprs 33597 | The restriction of a prose... |
| posrasymb 33598 | A poset ordering is asymme... |
| odutos 33599 | Being a toset is a self-du... |
| tlt2 33600 | In a Toset, two elements m... |
| tlt3 33601 | In a Toset, two elements m... |
| trleile 33602 | In a Toset, two elements m... |
| toslublem 33603 | Lemma for ~ toslub and ~ x... |
| toslub 33604 | In a toset, the lowest upp... |
| tosglblem 33605 | Lemma for ~ tosglb and ~ x... |
| tosglb 33606 | Same theorem as ~ toslub ,... |
| clatp0cl 33607 | The poset zero of a comple... |
| clatp1cl 33608 | The poset one of a complet... |
| mntoval 33613 | Operation value of the mon... |
| ismnt 33614 | Express the statement " ` ... |
| ismntd 33615 | Property of being a monoto... |
| mntf 33616 | A monotone function is a f... |
| mgcoval 33617 | Operation value of the mon... |
| mgcval 33618 | Monotone Galois connection... |
| mgcf1 33619 | The lower adjoint ` F ` of... |
| mgcf2 33620 | The upper adjoint ` G ` of... |
| mgccole1 33621 | An inequality for the kern... |
| mgccole2 33622 | Inequality for the closure... |
| mgcmnt1 33623 | The lower adjoint ` F ` of... |
| mgcmnt2 33624 | The upper adjoint ` G ` of... |
| mgcmntco 33625 | A Galois connection like s... |
| dfmgc2lem 33626 | Lemma for dfmgc2, backward... |
| dfmgc2 33627 | Alternate definition of th... |
| mgcmnt1d 33628 | Galois connection implies ... |
| mgcmnt2d 33629 | Galois connection implies ... |
| mgccnv 33630 | The inverse Galois connect... |
| pwrssmgc 33631 | Given a function ` F ` , e... |
| mgcf1olem1 33632 | Property of a Galois conne... |
| mgcf1olem2 33633 | Property of a Galois conne... |
| mgcf1o 33634 | Given a Galois connection,... |
| xrs0 33637 | The zero of the extended r... |
| xrslt 33638 | The "strictly less than" r... |
| xrsinvgval 33639 | The inversion operation in... |
| xrsmulgzz 33640 | The "multiple" function in... |
| xrstos 33641 | The extended real numbers ... |
| xrsclat 33642 | The extended real numbers ... |
| xrsp0 33643 | The poset 0 of the extende... |
| xrsp1 33644 | The poset 1 of the extende... |
| xrge00 33645 | The zero of the extended n... |
| xrge0mulgnn0 33646 | The group multiple functio... |
| xrge0addass 33647 | Associativity of extended ... |
| xrge0addgt0 33648 | The sum of nonnegative and... |
| xrge0adddir 33649 | Right-distributivity of ex... |
| xrge0adddi 33650 | Left-distributivity of ext... |
| xrge0npcan 33651 | Extended nonnegative real ... |
| fsumrp0cl 33652 | Closure of a finite sum of... |
| mndcld 33653 | Closure of the operation o... |
| mndassd 33654 | A monoid operation is asso... |
| mndlrinv 33655 | In a monoid, if an element... |
| mndlrinvb 33656 | In a monoid, if an element... |
| mndlactf1 33657 | If an element ` X ` of a m... |
| mndlactfo 33658 | An element ` X ` of a mono... |
| mndractf1 33659 | If an element ` X ` of a m... |
| mndractfo 33660 | An element ` X ` of a mono... |
| mndlactf1o 33661 | An element ` X ` of a mono... |
| mndractf1o 33662 | An element ` X ` of a mono... |
| cmn4d 33663 | Commutative/associative la... |
| cmn246135 33664 | Rearrange terms in a commu... |
| cmn145236 33665 | Rearrange terms in a commu... |
| abliso 33666 | The image of an Abelian gr... |
| lmhmghmd 33667 | A module homomorphism is a... |
| mhmimasplusg 33668 | Value of the operation of ... |
| lmhmimasvsca 33669 | Value of the scalar produc... |
| grpidcld 33670 | The identity element of a ... |
| grpinvinvd 33671 | Double inverse law for gro... |
| grpsubcld 33672 | Closure of group subtracti... |
| subgsubcld 33673 | A subgroup is closed under... |
| subgmulgcld 33674 | Closure of the group multi... |
| ressmulgnn0d 33675 | Values for the group multi... |
| ablcomd 33676 | An abelian group operation... |
| gsumsubg 33677 | The group sum in a subgrou... |
| gsumsra 33678 | The group sum in a subring... |
| gsummpt2co 33679 | Split a finite sum into a ... |
| gsummpt2d 33680 | Express a finite sum over ... |
| lmodvslmhm 33681 | Scalar multiplication in a... |
| gsumvsmul1 33682 | Pull a scalar multiplicati... |
| gsummptres 33683 | Extend a finite group sum ... |
| gsummptres2 33684 | Extend a finite group sum ... |
| gsummptfsres 33685 | Extend a finitely supporte... |
| gsummptf1od 33686 | Re-index a finite group su... |
| gsummptrev 33687 | Revert ordering in a group... |
| gsummptp1 33688 | Reindex a zero-based sum a... |
| gsummptfzsplitra 33689 | Split a group sum expresse... |
| gsummptfzsplitla 33690 | Split a group sum expresse... |
| gsummptfsf1o 33691 | Re-index a finite group su... |
| gsumfs2d 33692 | Express a finite sum over ... |
| gsumzresunsn 33693 | Append an element to a fin... |
| gsumpart 33694 | Express a group sum as a d... |
| gsumtp 33695 | Group sum of an unordered ... |
| gsumzrsum 33696 | Relate a group sum on ` ZZ... |
| gsummulgc2 33697 | A finite group sum multipl... |
| gsumhashmul 33698 | Express a group sum by gro... |
| gsummulsubdishift1 33699 | Distribute a subtraction o... |
| gsummulsubdishift2 33700 | Distribute a subtraction o... |
| gsummulsubdishift1s 33701 | Distribute a subtraction o... |
| gsummulsubdishift2s 33702 | Distribute a subtraction o... |
| suppgsumssiun 33703 | The support of a function ... |
| xrge0tsmsd 33704 | Any finite or infinite sum... |
| xrge0tsmsbi 33705 | Any limit of a finite or i... |
| xrge0tsmseq 33706 | Any limit of a finite or i... |
| gsumwun 33707 | In a commutative ring, a g... |
| gsumwrd2dccatlem 33708 | Lemma for ~ gsumwrd2dccat ... |
| gsumwrd2dccat 33709 | Rewrite a sum ranging over... |
| cntzun 33710 | The centralizer of a union... |
| cntzsnid 33711 | The centralizer of the ide... |
| cntrcrng 33712 | The center of a ring is a ... |
| symgfcoeu 33713 | Uniqueness property of per... |
| symgcom 33714 | Two permutations ` X ` and... |
| symgcom2 33715 | Two permutations ` X ` and... |
| symgcntz 33716 | All elements of a (finite)... |
| odpmco 33717 | The composition of two odd... |
| symgsubg 33718 | The value of the group sub... |
| pmtrprfv2 33719 | In a transposition of two ... |
| pmtrcnel 33720 | Composing a permutation ` ... |
| pmtrcnel2 33721 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33722 | Composing a permutation ` ... |
| fzo0pmtrlast 33723 | Reorder a half-open intege... |
| wrdpmtrlast 33724 | Reorder a word, so that th... |
| pmtridf1o 33725 | Transpositions of ` X ` an... |
| pmtridfv1 33726 | Value at X of the transpos... |
| pmtridfv2 33727 | Value at Y of the transpos... |
| psgnid 33728 | Permutation sign of the id... |
| psgndmfi 33729 | For a finite base set, the... |
| pmtrto1cl 33730 | Useful lemma for the follo... |
| psgnfzto1stlem 33731 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33732 | Value of our permutation `... |
| fzto1st1 33733 | Special case where the per... |
| fzto1st 33734 | The function moving one el... |
| fzto1stinvn 33735 | Value of the inverse of ou... |
| psgnfzto1st 33736 | The permutation sign for m... |
| tocycval 33739 | Value of the cycle builder... |
| tocycfv 33740 | Function value of a permut... |
| tocycfvres1 33741 | A cyclic permutation is a ... |
| tocycfvres2 33742 | A cyclic permutation is th... |
| cycpmfvlem 33743 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33744 | Value of a cycle function ... |
| cycpmfv2 33745 | Value of a cycle function ... |
| cycpmfv3 33746 | Values outside of the orbi... |
| cycpmcl 33747 | Cyclic permutations are pe... |
| tocycf 33748 | The permutation cycle buil... |
| tocyc01 33749 | Permutation cycles built f... |
| cycpm2tr 33750 | A cyclic permutation of 2 ... |
| cycpm2cl 33751 | Closure for the 2-cycles. ... |
| cyc2fv1 33752 | Function value of a 2-cycl... |
| cyc2fv2 33753 | Function value of a 2-cycl... |
| trsp2cyc 33754 | Exhibit the word a transpo... |
| cycpmco2f1 33755 | The word U used in ~ cycpm... |
| cycpmco2rn 33756 | The orbit of the compositi... |
| cycpmco2lem1 33757 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33758 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33759 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33760 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33761 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33762 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33763 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33764 | The composition of a cycli... |
| cyc2fvx 33765 | Function value of a 2-cycl... |
| cycpm3cl 33766 | Closure of the 3-cycles in... |
| cycpm3cl2 33767 | Closure of the 3-cycles in... |
| cyc3fv1 33768 | Function value of a 3-cycl... |
| cyc3fv2 33769 | Function value of a 3-cycl... |
| cyc3fv3 33770 | Function value of a 3-cycl... |
| cyc3co2 33771 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33772 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33773 | A formula for computing co... |
| cycpmrn 33774 | The range of the word used... |
| tocyccntz 33775 | All elements of a (finite)... |
| evpmval 33776 | Value of the set of even p... |
| cnmsgn0g 33777 | The neutral element of the... |
| evpmsubg 33778 | The alternating group is a... |
| evpmid 33779 | The identity is an even pe... |
| altgnsg 33780 | The alternating group ` ( ... |
| cyc3evpm 33781 | 3-Cycles are even permutat... |
| cyc3genpmlem 33782 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33783 | The alternating group ` A ... |
| cycpmgcl 33784 | Cyclic permutations are pe... |
| cycpmconjslem1 33785 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33786 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33787 | All cycles of the same len... |
| cyc3conja 33788 | All 3-cycles are conjugate... |
| sgnsv 33791 | The sign mapping. (Contri... |
| sgnsval 33792 | The sign value. (Contribu... |
| sgnsf 33793 | The sign function. (Contr... |
| fxpval 33796 | Value of the set of fixed ... |
| fxpss 33797 | The set of fixed points is... |
| fxpgaval 33798 | Value of the set of fixed ... |
| isfxp 33799 | Property of being a fixed ... |
| fxpgaeq 33800 | A fixed point ` X ` is inv... |
| conjga 33801 | Group conjugation induces ... |
| cntrval2 33802 | Express the center ` Z ` o... |
| fxpsubm 33803 | Provided the group action ... |
| fxpsubg 33804 | The fixed points of a grou... |
| fxpsubrg 33805 | The fixed points of a grou... |
| fxpsdrg 33806 | The fixed points of a grou... |
| inftmrel 33811 | The infinitesimal relation... |
| isinftm 33812 | Express ` x ` is infinites... |
| isarchi 33813 | Express the predicate " ` ... |
| pnfinf 33814 | Plus infinity is an infini... |
| xrnarchi 33815 | The completed real line is... |
| isarchi2 33816 | Alternative way to express... |
| submarchi 33817 | A submonoid is archimedean... |
| isarchi3 33818 | This is the usual definiti... |
| archirng 33819 | Property of Archimedean or... |
| archirngz 33820 | Property of Archimedean le... |
| archiexdiv 33821 | In an Archimedean group, g... |
| archiabllem1a 33822 | Lemma for ~ archiabl : In... |
| archiabllem1b 33823 | Lemma for ~ archiabl . (C... |
| archiabllem1 33824 | Archimedean ordered groups... |
| archiabllem2a 33825 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33826 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33827 | Lemma for ~ archiabl . (C... |
| archiabllem2 33828 | Archimedean ordered groups... |
| archiabl 33829 | Archimedean left- and righ... |
| isarchiofld 33830 | Axiom of Archimedes : a ch... |
| isslmd 33833 | The predicate "is a semimo... |
| slmdlema 33834 | Lemma for properties of a ... |
| lmodslmd 33835 | Left semimodules generaliz... |
| slmdcmn 33836 | A semimodule is a commutat... |
| slmdmnd 33837 | A semimodule is a monoid. ... |
| slmdsrg 33838 | The scalar component of a ... |
| slmdbn0 33839 | The base set of a semimodu... |
| slmdacl 33840 | Closure of ring addition f... |
| slmdmcl 33841 | Closure of ring multiplica... |
| slmdsn0 33842 | The set of scalars in a se... |
| slmdvacl 33843 | Closure of vector addition... |
| slmdass 33844 | Semiring left module vecto... |
| slmdvscl 33845 | Closure of scalar product ... |
| slmdvsdi 33846 | Distributive law for scala... |
| slmdvsdir 33847 | Distributive law for scala... |
| slmdvsass 33848 | Associative law for scalar... |
| slmd0cl 33849 | The ring zero in a semimod... |
| slmd1cl 33850 | The ring unity in a semiri... |
| slmdvs1 33851 | Scalar product with ring u... |
| slmd0vcl 33852 | The zero vector is a vecto... |
| slmd0vlid 33853 | Left identity law for the ... |
| slmd0vrid 33854 | Right identity law for the... |
| slmd0vs 33855 | Zero times a vector is the... |
| slmdvs0 33856 | Anything times the zero ve... |
| gsumvsca1 33857 | Scalar product of a finite... |
| gsumvsca2 33858 | Scalar product of a finite... |
| prmsimpcyc 33859 | A group of prime order is ... |
| ringrngd 33860 | A unital ring is a non-uni... |
| urpropd 33861 | Sufficient condition for r... |
| subrgmcld 33862 | A subring is closed under ... |
| ress1r 33863 | ` 1r ` is unaffected by re... |
| ringm1expp1 33864 | Ring exponentiation of min... |
| ringinvval 33865 | The ring inverse expressed... |
| dvrcan5 33866 | Cancellation law for commo... |
| subrgchr 33867 | If ` A ` is a subring of `... |
| rmfsupp2 33868 | A mapping of a multiplicat... |
| unitnz 33869 | In a nonzero ring, a unit ... |
| isunit2 33870 | Alternate definition of be... |
| isunit3 33871 | Alternate definition of be... |
| isunitc 33872 | Characterize units in a co... |
| elrgspnlem1 33873 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem2 33874 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem3 33875 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem4 33876 | Lemma for ~ elrgspn . (Co... |
| elrgspn 33877 | Membership in the subring ... |
| elrgspnsubrunlem1 33878 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrunlem2 33879 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrun 33880 | Membership in the ring spa... |
| irrednzr 33881 | A ring with an irreducible... |
| 0ringsubrg 33882 | A subring of a zero ring i... |
| 0ringcring 33883 | The zero ring is commutati... |
| reldmrloc 33888 | Ring localization is a pro... |
| erlval 33889 | Value of the ring localiza... |
| rlocval 33890 | Expand the value of the ri... |
| erlcl1 33891 | Closure for the ring local... |
| erlcl2 33892 | Closure for the ring local... |
| erldi 33893 | Main property of the ring ... |
| erlbrd 33894 | Deduce the ring localizati... |
| erlbr2d 33895 | Deduce the ring localizati... |
| erler 33896 | The relation used to build... |
| erld2 33897 | Main property of the ring ... |
| elrlocbasi 33898 | Membership in the basis of... |
| rlocbas 33899 | The base set of a ring loc... |
| rlocaddval 33900 | Value of the addition in t... |
| rlocmulval 33901 | Value of the addition in t... |
| rloccring 33902 | The ring localization ` L ... |
| rloc0g 33903 | The zero of a ring localiz... |
| rloc1r 33904 | The multiplicative identit... |
| rlocf1 33905 | The embedding ` F ` of a r... |
| rlocinvunit 33906 | In the localization of a r... |
| rlocisunit 33907 | Characterize the units of ... |
| domnmuln0rd 33908 | In a domain, factors of a ... |
| domnprodn0 33909 | In a domain, a finite prod... |
| domnprodeq0 33910 | A product over a domain is... |
| domnpropd 33911 | If two structures have the... |
| idompropd 33912 | If two structures have the... |
| idomrcan 33913 | Right-cancellation law for... |
| 1rrg 33914 | The multiplicative identit... |
| rrgsubm 33915 | The left regular elements ... |
| subrdom 33916 | A subring of a domain is a... |
| subridom 33917 | A subring of an integral d... |
| subrfld 33918 | A subring of a field is an... |
| ricnzr1 33919 | A ring isomorphism maps a ... |
| ricdomn1 33920 | A ring isomorphism maps a ... |
| ricdomn 33921 | A ring is a domain if and ... |
| eufndx 33924 | Index value of the Euclide... |
| eufid 33925 | Utility theorem: index-ind... |
| rndrhmcl 33928 | The image of a division ri... |
| qfld 33929 | The field of rational numb... |
| subsdrg 33930 | A subring of a sub-divisio... |
| sdrgdvcl 33931 | A sub-division-ring is clo... |
| sdrginvcl 33932 | A sub-division-ring is clo... |
| primefldchr 33933 | The characteristic of a pr... |
| fracval 33936 | Value of the field of frac... |
| fracbas 33937 | The base of the field of f... |
| fracerl 33938 | Rewrite the ring localizat... |
| fracf1 33939 | The embedding of a commuta... |
| fracfld 33940 | The field of fractions of ... |
| idomsubr 33941 | Every integral domain is i... |
| fldgenval 33944 | Value of the field generat... |
| fldgenssid 33945 | The field generated by a s... |
| fldgensdrg 33946 | A generated subfield is a ... |
| fldgenssv 33947 | A generated subfield is a ... |
| fldgenss 33948 | Generated subfields preser... |
| fldgenidfld 33949 | The subfield generated by ... |
| fldgenssp 33950 | The field generated by a s... |
| fldgenid 33951 | The subfield of a field ` ... |
| fldgenfld 33952 | A generated subfield is a ... |
| primefldgen1 33953 | The prime field of a divis... |
| 1fldgenq 33954 | The field of rational numb... |
| rhmdvd 33955 | A ring homomorphism preser... |
| kerunit 33956 | If a unit element lies in ... |
| reldmresv 33959 | The scalar restriction is ... |
| resvval 33960 | Value of structure restric... |
| resvid2 33961 | General behavior of trivia... |
| resvval2 33962 | Value of nontrivial struct... |
| resvsca 33963 | Base set of a structure re... |
| resvlem 33964 | Other elements of a scalar... |
| resvbas 33965 | ` Base ` is unaffected by ... |
| resvplusg 33966 | ` +g ` is unaffected by sc... |
| resvvsca 33967 | ` .s ` is unaffected by sc... |
| resvmulr 33968 | ` .r ` is unaffected by sc... |
| resv0g 33969 | ` 0g ` is unaffected by sc... |
| resv1r 33970 | ` 1r ` is unaffected by sc... |
| resvcmn 33971 | Scalar restriction preserv... |
| gzcrng 33972 | The gaussian integers form... |
| cnfldfld 33973 | The complex numbers form a... |
| reofld 33974 | The real numbers form an o... |
| nn0omnd 33975 | The nonnegative integers f... |
| gsumind 33976 | The group sum of an indica... |
| rearchi 33977 | The field of the real numb... |
| nn0archi 33978 | The monoid of the nonnegat... |
| xrge0slmod 33979 | The extended nonnegative r... |
| qusker 33980 | The kernel of a quotient m... |
| eqgvscpbl 33981 | The left coset equivalence... |
| qusvscpbl 33982 | The quotient map distribut... |
| qusvsval 33983 | Value of the scalar multip... |
| imaslmod 33984 | The image structure of a l... |
| imasmhm 33985 | Given a function ` F ` wit... |
| imasghm 33986 | Given a function ` F ` wit... |
| imasrhm 33987 | Given a function ` F ` wit... |
| imaslmhm 33988 | Given a function ` F ` wit... |
| quslmod 33989 | If ` G ` is a submodule in... |
| quslmhm 33990 | If ` G ` is a submodule of... |
| quslvec 33991 | If ` S ` is a vector subsp... |
| znfermltl 33992 | Fermat's little theorem in... |
| islinds5 33993 | A set is linearly independ... |
| ellspds 33994 | Variation on ~ ellspd . (... |
| 0ellsp 33995 | Zero is in all spans. (Co... |
| 0nellinds 33996 | The group identity cannot ... |
| elrsp 33997 | Write the elements of a ri... |
| ellpi 33998 | Elementhood in a left prin... |
| lpirlidllpi 33999 | In a principal ideal ring,... |
| rspidlid 34000 | The ideal span of an ideal... |
| rsp2idlid 34001 | The ideal span of a two-si... |
| lbslsp 34002 | Any element of a left modu... |
| lindssn 34003 | Any singleton of a nonzero... |
| lindflbs 34004 | Conditions for an independ... |
| islbs5 34005 | An equivalent formulation ... |
| linds2eq 34006 | Deduce equality of element... |
| lindfpropd 34007 | Property deduction for lin... |
| lindspropd 34008 | Property deduction for lin... |
| dvdsruassoi 34009 | If two elements ` X ` and ... |
| dvdsruasso 34010 | Two elements ` X ` and ` Y... |
| dvdsruasso2 34011 | A reformulation of ~ dvdsr... |
| dvdsrspss 34012 | In a ring, an element ` X ... |
| rspsnasso 34013 | Two elements ` X ` and ` Y... |
| unitprodclb 34014 | A finite product is a unit... |
| elgrplsmsn 34015 | Membership in a sumset wit... |
| lsmsnorb 34016 | The sumset of a group with... |
| lsmsnorb2 34017 | The sumset of a single ele... |
| ringlsmss 34018 | Closure of the product of ... |
| ringlsmss1 34019 | The product of an ideal ` ... |
| ringlsmss2 34020 | The product with an ideal ... |
| lsmsnpridl 34021 | The product of the ring wi... |
| lsmsnidl 34022 | The product of the ring wi... |
| lsmssass 34023 | Group sum is associative, ... |
| grplsm0l 34024 | Sumset with the identity s... |
| grplsmid 34025 | The direct sum of an eleme... |
| quslsm 34026 | Express the image by the q... |
| qusbas2 34027 | Alternate definition of th... |
| qus0g 34028 | The identity element of a ... |
| qusima 34029 | The image of a subgroup by... |
| qusrn 34030 | The natural map from eleme... |
| nsgqus0 34031 | A normal subgroup ` N ` is... |
| nsgmgclem 34032 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 34033 | There is a monotone Galois... |
| nsgqusf1olem1 34034 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 34035 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 34036 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 34037 | The canonical projection h... |
| lmhmqusker 34038 | A surjective module homomo... |
| lmicqusker 34039 | The image ` H ` of a modul... |
| intlidl 34040 | The intersection of a none... |
| inlidl 34041 | The intersection of two id... |
| pidlnzb 34042 | A principal ideal is nonze... |
| lidlunitel 34043 | If an ideal ` I ` contains... |
| unitpidl1 34044 | The ideal ` I ` generated ... |
| rhmquskerlem 34045 | The mapping ` J ` induced ... |
| rhmqusker 34046 | A surjective ring homomorp... |
| ricqusker 34047 | The image ` H ` of a ring ... |
| elrspunidl 34048 | Elementhood in the span of... |
| elrspunsn 34049 | Membership to the span of ... |
| lidlincl 34050 | Ideals are closed under in... |
| idlinsubrg 34051 | The intersection between a... |
| rhmimaidl 34052 | The image of an ideal ` I ... |
| drngidlhash 34053 | A ring is a division ring ... |
| mxidlval 34056 | The set of maximal ideals ... |
| ismxidl 34057 | The predicate "is a maxima... |
| mxidlidl 34058 | A maximal ideal is an idea... |
| mxidlnr 34059 | A maximal ideal is proper.... |
| mxidlmax 34060 | A maximal ideal is a maxim... |
| mxidln1 34061 | One is not contained in an... |
| mxidlnzr 34062 | A ring with a maximal idea... |
| mxidlmaxv 34063 | An ideal ` I ` strictly co... |
| crngmxidl 34064 | In a commutative ring, max... |
| mxidlprm 34065 | Every maximal ideal is pri... |
| mxidlirredi 34066 | In an integral domain, the... |
| mxidlirred 34067 | In a principal ideal domai... |
| ssmxidllem 34068 | The set ` P ` used in the ... |
| ssmxidl 34069 | Let ` R ` be a ring, and l... |
| drng0mxidl 34070 | In a division ring, the ze... |
| drngmxidl 34071 | The zero ideal is the only... |
| drngmxidlr 34072 | If a ring's only maximal i... |
| krull 34073 | Krull's theorem: Any nonz... |
| mxidlnzrb 34074 | A ring is nonzero if and o... |
| krullndrng 34075 | Krull's theorem for non-di... |
| opprabs 34076 | The opposite ring of the o... |
| oppreqg 34077 | Group coset equivalence re... |
| opprnsg 34078 | Normal subgroups of the op... |
| opprlidlabs 34079 | The ideals of the opposite... |
| oppr2idl 34080 | Two sided ideal of the opp... |
| opprmxidlabs 34081 | The maximal ideal of the o... |
| opprqusbas 34082 | The base of the quotient o... |
| opprqusplusg 34083 | The group operation of the... |
| opprqus0g 34084 | The group identity element... |
| opprqusmulr 34085 | The multiplication operati... |
| opprqus1r 34086 | The ring unity of the quot... |
| opprqusdrng 34087 | The quotient of the opposi... |
| qsdrngilem 34088 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 34089 | A quotient by a maximal le... |
| qsdrnglem2 34090 | Lemma for ~ qsdrng . (Con... |
| qsdrng 34091 | An ideal ` M ` is both lef... |
| qsfld 34092 | An ideal ` M ` in the comm... |
| mxidlprmALT 34093 | Every maximal ideal is pri... |
| drnglring 34094 | A division ring is a local... |
| dflring2 34095 | Alternate definition of a ... |
| dflringlem 34096 | Lemma for ~ dflring3 . If... |
| dflringlem2 34097 | Lemma for ~ dflring3 . In... |
| dflringlem3 34098 | Lemma for ~ dflring3 . In... |
| dflring3 34099 | Alternate definition of a ... |
| dflring4 34100 | Alternate definition of a ... |
| fldlring 34101 | A field is a local ring. ... |
| idlsrgstr 34104 | A constructed semiring of ... |
| idlsrgval 34105 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 34106 | Base of the ideals of a ri... |
| idlsrgplusg 34107 | Additive operation of the ... |
| idlsrg0g 34108 | The zero ideal is the addi... |
| idlsrgmulr 34109 | Multiplicative operation o... |
| idlsrgtset 34110 | Topology component of the ... |
| idlsrgmulrval 34111 | Value of the ring multipli... |
| idlsrgmulrcl 34112 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 34113 | In a commutative ring, the... |
| idlsrgmulrss2 34114 | The product of two ideals ... |
| idlsrgmulrssin 34115 | In a commutative ring, the... |
| idlsrgmnd 34116 | The ideals of a ring form ... |
| idlsrgcmnd 34117 | The ideals of a ring form ... |
| rprmval 34118 | The prime elements of a ri... |
| isrprm 34119 | Property for ` P ` to be a... |
| rprmcl 34120 | A ring prime is an element... |
| rprmdvds 34121 | If a ring prime ` Q ` divi... |
| rprmnz 34122 | A ring prime is nonzero. ... |
| rprmnunit 34123 | A ring prime is not a unit... |
| rsprprmprmidl 34124 | In a commutative ring, ide... |
| rsprprmprmidlb 34125 | An ideal generated by a si... |
| rprmndvdsr1 34126 | A ring prime element does ... |
| rprmasso 34127 | In an integral domain, the... |
| rprmasso2 34128 | In an integral domain, if ... |
| rprmasso3 34129 | In an integral domain, if ... |
| unitmulrprm 34130 | A ring unit multiplied by ... |
| rprmndvdsru 34131 | A ring prime element does ... |
| rprmirredlem 34132 | Lemma for ~ rprmirred . (... |
| rprmirred 34133 | In an integral domain, rin... |
| rprmirredb 34134 | In a principal ideal domai... |
| rprmdvdspow 34135 | If a prime element divides... |
| rprmdvdsprod 34136 | If a prime element ` Q ` d... |
| 1arithidomlem1 34137 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 34138 | Lemma for ~ 1arithidom : i... |
| 1arithidom 34139 | Uniqueness of prime factor... |
| isufd 34142 | The property of being a Un... |
| ufdprmidl 34143 | In a unique factorization ... |
| ufdidom 34144 | A nonzero unique factoriza... |
| pidufd 34145 | Every principal ideal doma... |
| 1arithufdlem1 34146 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 34147 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 34148 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 34149 | Lemma for ~ 1arithufd . N... |
| 1arithufd 34150 | Existence of a factorizati... |
| dfufd2lem 34151 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 34152 | Alternative definition of ... |
| zringidom 34153 | The ring of integers is an... |
| zringpid 34154 | The ring of integers is a ... |
| dfprm3 34155 | The (positive) prime eleme... |
| zringfrac 34156 | The field of fractions of ... |
| assaassd 34157 | Left-associative property ... |
| assaassrd 34158 | Right-associative property... |
| 0ringmon1p 34159 | There are no monic polynom... |
| fply1 34160 | Conditions for a function ... |
| ply1lvec 34161 | In a division ring, the un... |
| evls1fn 34162 | Functionality of the subri... |
| evls1dm 34163 | The domain of the subring ... |
| evls1fvf 34164 | The subring evaluation fun... |
| evl1fvf 34165 | The univariate polynomial ... |
| evl1fpws 34166 | Evaluation of a univariate... |
| ressply1evls1 34167 | Subring evaluation of a un... |
| ressdeg1 34168 | The degree of a univariate... |
| ressply10g 34169 | A restricted polynomial al... |
| ressply1mon1p 34170 | The monic polynomials of a... |
| ressply1invg 34171 | An element of a restricted... |
| ressply1sub 34172 | A restricted polynomial al... |
| ressasclcl 34173 | Closure of the univariate ... |
| evls1subd 34174 | Univariate polynomial eval... |
| deg1le0eq0 34175 | A polynomial with nonposit... |
| ply1asclunit 34176 | A nonzero scalar polynomia... |
| ply1unit 34177 | In a field ` F ` , a polyn... |
| evl1deg1 34178 | Evaluation of a univariate... |
| evl1deg2 34179 | Evaluation of a univariate... |
| evl1deg3 34180 | Evaluation of a univariate... |
| evls1monply1 34181 | Subring evaluation of a sc... |
| ply1dg1rt 34182 | Express the root ` - B / A... |
| ply1dg1rtn0 34183 | Polynomials of degree 1 ov... |
| ply1mulrtss 34184 | The roots of a factor ` F ... |
| deg1prod 34185 | Degree of a product of pol... |
| ply1dg3rt0irred 34186 | If a cubic polynomial over... |
| m1pmeq 34187 | If two monic polynomials `... |
| ply1fermltl 34188 | Fermat's little theorem fo... |
| coe1mon 34189 | Coefficient vector of a mo... |
| ply1moneq 34190 | Two monomials are equal if... |
| ply1coedeg 34191 | Decompose a univariate pol... |
| coe1zfv 34192 | The coefficients of the ze... |
| coe1vr1 34193 | Polynomial coefficient of ... |
| deg1vr 34194 | The degree of the variable... |
| vr1nz 34195 | A univariate polynomial va... |
| ply1degltel 34196 | Characterize elementhood i... |
| ply1degleel 34197 | Characterize elementhood i... |
| ply1degltlss 34198 | The space ` S ` of the uni... |
| gsummoncoe1fzo 34199 | A coefficient of the polyn... |
| gsummoncoe1fz 34200 | A coefficient of the polyn... |
| ply1gsumz 34201 | If a polynomial given as a... |
| deg1addlt 34202 | If both factors have degre... |
| ig1pnunit 34203 | The polynomial ideal gener... |
| ig1pmindeg 34204 | The polynomial ideal gener... |
| q1pdir 34205 | Distribution of univariate... |
| q1pvsca 34206 | Scalar multiplication prop... |
| r1pvsca 34207 | Scalar multiplication prop... |
| r1p0 34208 | Polynomial remainder opera... |
| r1pcyc 34209 | The polynomial remainder o... |
| r1padd1 34210 | Addition property of the p... |
| r1plmhm 34211 | The univariate polynomial ... |
| r1pquslmic 34212 | The univariate polynomial ... |
| psrbasfsupp 34213 | Rewrite a finite support f... |
| psrnzr 34214 | The ring of power series o... |
| mplnzr 34215 | The multivariate polynomia... |
| 0mplrim 34216 | Build a ring isomorphism b... |
| 0mplric 34217 | Multivariate polynomials w... |
| mplasclco 34218 | Case where composing an al... |
| selvascl 34219 | The "variable selection" f... |
| selvply1rhmlema 34220 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlemb 34221 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem1 34222 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem2 34223 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem3 34224 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem4 34225 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem5 34226 | Lemma for ~ selvply1rhm . ... |
| selvply1rhm 34227 | Build a ring homomorphism ... |
| selvply1rhm0 34228 | The ring homomorphism ` H ... |
| mplidomlem 34229 | Lemma for ~ mplidom . (Co... |
| mplidom 34230 | The multivariate polynomia... |
| extvval 34233 | Value of the "variable ext... |
| extvfval 34234 | The "variable extension" f... |
| extvfv 34235 | The "variable extension" f... |
| extvfvv 34236 | The "variable extension" f... |
| extvfvvcl 34237 | Closure for the "variable ... |
| extvfvcl 34238 | Closure for the "variable ... |
| extvfvalf 34239 | The "variable extension" f... |
| mvrvalind 34240 | Value of the generating el... |
| mplmulmvr 34241 | Multiply a polynomial ` F ... |
| evlscaval 34242 | Polynomial evaluation for ... |
| evlvarval 34243 | Polynomial evaluation buil... |
| evlextv 34244 | Evaluating a variable-exte... |
| mplvrpmlem 34245 | Lemma for ~ mplvrpmga and ... |
| mplvrpmfgalem 34246 | Permuting variables in a m... |
| mplvrpmga 34247 | The action of permuting va... |
| mplvrpmmhm 34248 | The action of permuting va... |
| mplvrpmrhm 34249 | The action of permuting va... |
| psrgsum 34250 | Finite commutative sums of... |
| psrmon 34251 | A monomial is a power seri... |
| psrmonmul 34252 | The product of two power s... |
| psrmonmul2 34253 | The product of two power s... |
| psrmonprod 34254 | Finite product of bags of ... |
| mplgsum 34255 | Finite commutative sums of... |
| mplmonprod 34256 | Finite product of monomial... |
| splyval 34261 | The symmetric polynomials ... |
| splysubrg 34262 | The symmetric polynomials ... |
| issply 34263 | Conditions for being a sym... |
| esplyval 34264 | The elementary polynomials... |
| esplyfval 34265 | The ` K ` -th elementary p... |
| esplyfval0 34266 | The ` 0 ` -th elementary s... |
| esplyfval2 34267 | When ` K ` is out-of-bound... |
| esplylem 34268 | Lemma for ~ esplyfv and ot... |
| esplympl 34269 | Elementary symmetric polyn... |
| esplymhp 34270 | The ` K ` -th elementary s... |
| esplyfv1 34271 | Coefficient for the ` K ` ... |
| esplyfv 34272 | Coefficient for the ` K ` ... |
| esplysply 34273 | The ` K ` -th elementary s... |
| esplyfval3 34274 | Alternate expression for t... |
| esplyfval1 34275 | The first elementary symme... |
| esplyfvaln 34276 | The last elementary symmet... |
| esplyind 34277 | A recursive formula for th... |
| esplyindfv 34278 | A recursive formula for th... |
| esplyfvn 34279 | Express the last elementar... |
| vietadeg1 34280 | The degree of a product of... |
| vietalem 34281 | Lemma for ~ vieta : induct... |
| vieta 34282 | Vieta's Formulas: Coeffic... |
| sra1r 34283 | The unity element of a sub... |
| sradrng 34284 | Condition for a subring al... |
| sraidom 34285 | Condition for a subring al... |
| srasubrg 34286 | A subring of the original ... |
| sralvec 34287 | Given a sub division ring ... |
| srafldlvec 34288 | Given a subfield ` F ` of ... |
| resssra 34289 | The subring algebra of a r... |
| lsssra 34290 | A subring is a subspace of... |
| srapwov 34291 | The "power" operation on a... |
| drgext0g 34292 | The additive neutral eleme... |
| drgextvsca 34293 | The scalar multiplication ... |
| drgext0gsca 34294 | The additive neutral eleme... |
| drgextsubrg 34295 | The scalar field is a subr... |
| drgextlsp 34296 | The scalar field is a subs... |
| drgextgsum 34297 | Group sum in a division ri... |
| lvecdimfi 34298 | Finite version of ~ lvecdi... |
| exsslsb 34299 | Any finite generating set ... |
| lbslelsp 34300 | The size of a basis ` X ` ... |
| dimval 34303 | The dimension of a vector ... |
| dimvalfi 34304 | The dimension of a vector ... |
| dimcl 34305 | Closure of the vector spac... |
| lmimdim 34306 | Module isomorphisms preser... |
| lmicdim 34307 | Module isomorphisms preser... |
| lvecdim0i 34308 | A vector space of dimensio... |
| lvecdim0 34309 | A vector space of dimensio... |
| lssdimle 34310 | The dimension of a linear ... |
| dimpropd 34311 | If two structures have the... |
| rlmdim 34312 | The left vector space indu... |
| frlmdim 34313 | Dimension of a free left m... |
| tnglvec 34314 | Augmenting a structure wit... |
| tngdim 34315 | Dimension of a left vector... |
| rrxdim 34316 | Dimension of the generaliz... |
| matdim 34317 | Dimension of the space of ... |
| lbslsat 34318 | A nonzero vector ` X ` is ... |
| lsatdim 34319 | A line, spanned by a nonze... |
| drngdimgt0 34320 | The dimension of a vector ... |
| lmhmlvec2 34321 | A homomorphism of left vec... |
| kerlmhm 34322 | The kernel of a vector spa... |
| imlmhm 34323 | The image of a vector spac... |
| ply1degltdimlem 34324 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 34325 | The space ` S ` of the uni... |
| lindsunlem 34326 | Lemma for ~ lindsun . (Co... |
| lindsun 34327 | Condition for the union of... |
| lbsdiflsp0 34328 | The linear spans of two di... |
| dimkerim 34329 | Given a linear map ` F ` b... |
| qusdimsum 34330 | Let ` W ` be a vector spac... |
| fedgmullem1 34331 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 34332 | Lemma for ~ fedgmul . (Co... |
| fedgmul 34333 | The multiplicativity formu... |
| dimlssid 34334 | If the dimension of a line... |
| lvecendof1f1o 34335 | If an endomorphism ` U ` o... |
| lactlmhm 34336 | In an associative algebra ... |
| assalactf1o 34337 | In an associative algebra ... |
| assarrginv 34338 | If an element ` X ` of an ... |
| assafld 34339 | If an algebra ` A ` of fin... |
| relfldext 34346 | The field extension is a r... |
| brfldext 34347 | The field extension relati... |
| ccfldextrr 34348 | The field of the complex n... |
| fldextfld1 34349 | A field extension is only ... |
| fldextfld2 34350 | A field extension is only ... |
| fldextsubrg 34351 | Field extension implies a ... |
| sdrgfldext 34352 | A field ` E ` and any sub-... |
| fldextress 34353 | Field extension implies a ... |
| brfinext 34354 | The finite field extension... |
| extdgval 34355 | Value of the field extensi... |
| fldextsdrg 34356 | Deduce sub-division-ring f... |
| fldextsralvec 34357 | The subring algebra associ... |
| extdgcl 34358 | Closure of the field exten... |
| extdggt0 34359 | Degrees of field extension... |
| fldexttr 34360 | Field extension is a trans... |
| fldextid 34361 | The field extension relati... |
| extdgid 34362 | A trivial field extension ... |
| fldsdrgfldext 34363 | A sub-division-ring of a f... |
| fldsdrgfldext2 34364 | A sub-sub-division-ring of... |
| extdgmul 34365 | The multiplicativity formu... |
| finextfldext 34366 | A finite field extension i... |
| finexttrb 34367 | The extension ` E ` of ` K... |
| extdg1id 34368 | If the degree of the exten... |
| extdg1b 34369 | The degree of the extensio... |
| fldgenfldext 34370 | A subfield ` F ` extended ... |
| fldextchr 34371 | The characteristic of a su... |
| evls1fldgencl 34372 | Closure of the subring pol... |
| ccfldsrarelvec 34373 | The subring algebra of the... |
| ccfldextdgrr 34374 | The degree of the field ex... |
| fldextrspunlsplem 34375 | Lemma for ~ fldextrspunlsp... |
| fldextrspunlsp 34376 | Lemma for ~ fldextrspunfld... |
| fldextrspunlem1 34377 | Lemma for ~ fldextrspunfld... |
| fldextrspunfld 34378 | The ring generated by the ... |
| fldextrspunlem2 34379 | Part of the proof of Propo... |
| fldextrspundgle 34380 | Inequality involving the d... |
| fldextrspundglemul 34381 | Given two field extensions... |
| fldextrspundgdvdslem 34382 | Lemma for ~ fldextrspundgd... |
| fldextrspundgdvds 34383 | Given two finite extension... |
| fldext2rspun 34384 | Given two field extensions... |
| irngval 34387 | The elements of a field ` ... |
| elirng 34388 | Property for an element ` ... |
| irngss 34389 | All elements of a subring ... |
| irngssv 34390 | An integral element is an ... |
| 0ringirng 34391 | A zero ring ` R ` has no i... |
| irngnzply1lem 34392 | In the case of a field ` E... |
| irngnzply1 34393 | In the case of a field ` E... |
| extdgfialglem1 34394 | Lemma for ~ extdgfialg . ... |
| extdgfialglem2 34395 | Lemma for ~ extdgfialg . ... |
| extdgfialg 34396 | A finite field extension `... |
| bralgext 34399 | Express the fact that a fi... |
| finextalg 34400 | A finite field extension i... |
| ply1annidllem 34403 | Write the set ` Q ` of pol... |
| ply1annidl 34404 | The set ` Q ` of polynomia... |
| ply1annnr 34405 | The set ` Q ` of polynomia... |
| ply1annig1p 34406 | The ideal ` Q ` of polynom... |
| minplyval 34407 | Expand the value of the mi... |
| minplycl 34408 | The minimal polynomial is ... |
| ply1annprmidl 34409 | The set ` Q ` of polynomia... |
| minplymindeg 34410 | The minimal polynomial of ... |
| minplyann 34411 | The minimal polynomial for... |
| minplyirredlem 34412 | Lemma for ~ minplyirred . ... |
| minplyirred 34413 | A nonzero minimal polynomi... |
| irngnminplynz 34414 | Integral elements have non... |
| minplym1p 34415 | A minimal polynomial is mo... |
| minplynzm1p 34416 | If a minimal polynomial is... |
| minplyelirng 34417 | If the minimal polynomial ... |
| irredminply 34418 | An irreducible, monic, ann... |
| algextdeglem1 34419 | Lemma for ~ algextdeg . (... |
| algextdeglem2 34420 | Lemma for ~ algextdeg . B... |
| algextdeglem3 34421 | Lemma for ~ algextdeg . T... |
| algextdeglem4 34422 | Lemma for ~ algextdeg . B... |
| algextdeglem5 34423 | Lemma for ~ algextdeg . T... |
| algextdeglem6 34424 | Lemma for ~ algextdeg . B... |
| algextdeglem7 34425 | Lemma for ~ algextdeg . T... |
| algextdeglem8 34426 | Lemma for ~ algextdeg . T... |
| algextdeg 34427 | The degree of an algebraic... |
| rtelextdg2lem 34428 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 34429 | If an element ` X ` is a s... |
| fldext2chn 34430 | In a non-empty chain ` T `... |
| constrrtll 34433 | In the construction of con... |
| constrrtlc1 34434 | In the construction of con... |
| constrrtlc2 34435 | In the construction of con... |
| constrrtcclem 34436 | In the construction of con... |
| constrrtcc 34437 | In the construction of con... |
| isconstr 34438 | Property of being a constr... |
| constr0 34439 | The first step of the cons... |
| constrsuc 34440 | Membership in the successo... |
| constrlim 34441 | Limit step of the construc... |
| constrsscn 34442 | Closure of the constructib... |
| constrsslem 34443 | Lemma for ~ constrss . Th... |
| constr01 34444 | ` 0 ` and ` 1 ` are in all... |
| constrss 34445 | Constructed points are in ... |
| constrmon 34446 | The construction of constr... |
| constrconj 34447 | If a point ` X ` of the co... |
| constrfin 34448 | Each step of the construct... |
| constrelextdg2 34449 | If the ` N ` -th step ` ( ... |
| constrextdg2lem 34450 | Lemma for ~ constrextdg2 .... |
| constrextdg2 34451 | Any step ` ( C `` N ) ` of... |
| constrext2chnlem 34452 | Lemma for ~ constrext2chn ... |
| constrfiss 34453 | For any finite set ` A ` o... |
| constrllcllem 34454 | Constructible numbers are ... |
| constrlccllem 34455 | Constructible numbers are ... |
| constrcccllem 34456 | Constructible numbers are ... |
| constrcbvlem 34457 | Technical lemma for elimin... |
| constrllcl 34458 | Constructible numbers are ... |
| constrlccl 34459 | Constructible numbers are ... |
| constrcccl 34460 | Constructible numbers are ... |
| constrext2chn 34461 | If a constructible number ... |
| constrcn 34462 | Constructible numbers are ... |
| nn0constr 34463 | Nonnegative integers are c... |
| constraddcl 34464 | Constructive numbers are c... |
| constrnegcl 34465 | Constructible numbers are ... |
| zconstr 34466 | Integers are constructible... |
| constrdircl 34467 | Constructible numbers are ... |
| iconstr 34468 | The imaginary unit ` _i ` ... |
| constrremulcl 34469 | If two real numbers ` X ` ... |
| constrcjcl 34470 | Constructible numbers are ... |
| constrrecl 34471 | Constructible numbers are ... |
| constrimcl 34472 | Constructible numbers are ... |
| constrmulcl 34473 | Constructible numbers are ... |
| constrreinvcl 34474 | If a real number ` X ` is ... |
| constrinvcl 34475 | Constructible numbers are ... |
| constrcon 34476 | Contradiction of construct... |
| constrsdrg 34477 | Constructible numbers form... |
| constrfld 34478 | The constructible numbers ... |
| constrresqrtcl 34479 | If a positive real number ... |
| constrabscl 34480 | Constructible numbers are ... |
| constrsqrtcl 34481 | Constructible numbers are ... |
| 2sqr3minply 34482 | The polynomial ` ( ( X ^ 3... |
| 2sqr3nconstr 34483 | Doubling the cube is an im... |
| cos9thpiminplylem1 34484 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem2 34485 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem3 34486 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem4 34487 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem5 34488 | The constructed complex nu... |
| cos9thpiminplylem6 34489 | Evaluation of the polynomi... |
| cos9thpiminply 34490 | The polynomial ` ( ( X ^ 3... |
| cos9thpinconstrlem1 34491 | The complex number ` O ` ,... |
| cos9thpinconstrlem2 34492 | The complex number ` A ` i... |
| cos9thpinconstr 34493 | Trisecting an angle is an ... |
| trisecnconstr 34494 | Not all angles can be tris... |
| smatfval 34497 | Value of the submatrix. (... |
| smatrcl 34498 | Closure of the rectangular... |
| smatlem 34499 | Lemma for the next theorem... |
| smattl 34500 | Entries of a submatrix, to... |
| smattr 34501 | Entries of a submatrix, to... |
| smatbl 34502 | Entries of a submatrix, bo... |
| smatbr 34503 | Entries of a submatrix, bo... |
| smatcl 34504 | Closure of the square subm... |
| matmpo 34505 | Write a square matrix as a... |
| 1smat1 34506 | The submatrix of the ident... |
| submat1n 34507 | One case where the submatr... |
| submatres 34508 | Special case where the sub... |
| submateqlem1 34509 | Lemma for ~ submateq . (C... |
| submateqlem2 34510 | Lemma for ~ submateq . (C... |
| submateq 34511 | Sufficient condition for t... |
| submatminr1 34512 | If we take a submatrix by ... |
| lmatval 34515 | Value of the literal matri... |
| lmatfval 34516 | Entries of a literal matri... |
| lmatfvlem 34517 | Useful lemma to extract li... |
| lmatcl 34518 | Closure of the literal mat... |
| lmat22lem 34519 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 34520 | Entry of a 2x2 literal mat... |
| lmat22e12 34521 | Entry of a 2x2 literal mat... |
| lmat22e21 34522 | Entry of a 2x2 literal mat... |
| lmat22e22 34523 | Entry of a 2x2 literal mat... |
| lmat22det 34524 | The determinant of a liter... |
| mdetpmtr1 34525 | The determinant of a matri... |
| mdetpmtr2 34526 | The determinant of a matri... |
| mdetpmtr12 34527 | The determinant of a matri... |
| mdetlap1 34528 | A Laplace expansion of the... |
| madjusmdetlem1 34529 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 34530 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 34531 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 34532 | Lemma for ~ madjusmdet . ... |
| madjusmdet 34533 | Express the cofactor of th... |
| mdetlap 34534 | Laplace expansion of the d... |
| ist0cld 34535 | The predicate "is a T_0 sp... |
| txomap 34536 | Given two open maps ` F ` ... |
| qtopt1 34537 | If every equivalence class... |
| qtophaus 34538 | If an open map's graph in ... |
| circtopn 34539 | The topology of the unit c... |
| circcn 34540 | The function gluing the re... |
| reff 34541 | For any cover refinement, ... |
| locfinreflem 34542 | A locally finite refinemen... |
| locfinref 34543 | A locally finite refinemen... |
| iscref 34546 | The property that every op... |
| crefeq 34547 | Equality theorem for the "... |
| creftop 34548 | A space where every open c... |
| crefi 34549 | The property that every op... |
| crefdf 34550 | A formulation of ~ crefi e... |
| crefss 34551 | The "every open cover has ... |
| cmpcref 34552 | Equivalent definition of c... |
| cmpfiref 34553 | Every open cover of a Comp... |
| ldlfcntref 34556 | Every open cover of a Lind... |
| ispcmp 34559 | The predicate "is a paraco... |
| cmppcmp 34560 | Every compact space is par... |
| dispcmp 34561 | Every discrete space is pa... |
| pcmplfin 34562 | Given a paracompact topolo... |
| pcmplfinf 34563 | Given a paracompact topolo... |
| rspecval 34566 | Value of the spectrum of t... |
| rspecbas 34567 | The prime ideals form the ... |
| rspectset 34568 | Topology component of the ... |
| rspectopn 34569 | The topology component of ... |
| zarcls0 34570 | The closure of the identit... |
| zarcls1 34571 | The unit ideal ` B ` is th... |
| zarclsun 34572 | The union of two closed se... |
| zarclsiin 34573 | In a Zariski topology, the... |
| zarclsint 34574 | The intersection of a fami... |
| zarclssn 34575 | The closed points of Zaris... |
| zarcls 34576 | The open sets of the Zaris... |
| zartopn 34577 | The Zariski topology is a ... |
| zartop 34578 | The Zariski topology is a ... |
| zartopon 34579 | The points of the Zariski ... |
| zar0ring 34580 | The Zariski Topology of th... |
| zart0 34581 | The Zariski topology is T_... |
| zarmxt1 34582 | The Zariski topology restr... |
| zarcmplem 34583 | Lemma for ~ zarcmp . (Con... |
| zarcmp 34584 | The Zariski topology is co... |
| rspectps 34585 | The spectrum of a ring ` R... |
| rhmpreimacnlem 34586 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 34587 | The function mapping a pri... |
| metidval 34592 | Value of the metric identi... |
| metidss 34593 | As a relation, the metric ... |
| metidv 34594 | ` A ` and ` B ` identify b... |
| metideq 34595 | Basic property of the metr... |
| metider 34596 | The metric identification ... |
| pstmval 34597 | Value of the metric induce... |
| pstmfval 34598 | Function value of the metr... |
| pstmxmet 34599 | The metric induced by a ps... |
| hauseqcn 34600 | In a Hausdorff topology, t... |
| elunitge0 34601 | An element of the closed u... |
| unitssxrge0 34602 | The closed unit interval i... |
| unitdivcld 34603 | Necessary conditions for a... |
| iistmd 34604 | The closed unit interval f... |
| unicls 34605 | The union of the closed se... |
| tpr2tp 34606 | The usual topology on ` ( ... |
| tpr2uni 34607 | The usual topology on ` ( ... |
| xpinpreima 34608 | Rewrite the cartesian prod... |
| xpinpreima2 34609 | Rewrite the cartesian prod... |
| sqsscirc1 34610 | The complex square of side... |
| sqsscirc2 34611 | The complex square of side... |
| cnre2csqlem 34612 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 34613 | Image of a centered square... |
| tpr2rico 34614 | For any point of an open s... |
| cnvordtrestixx 34615 | The restriction of the 'gr... |
| prsdm 34616 | Domain of the relation of ... |
| prsrn 34617 | Range of the relation of a... |
| prsss 34618 | Relation of a subproset. ... |
| prsssdm 34619 | Domain of a subproset rela... |
| ordtprsval 34620 | Value of the order topolog... |
| ordtprsuni 34621 | Value of the order topolog... |
| ordtcnvNEW 34622 | The order dual generates t... |
| ordtrestNEW 34623 | The subspace topology of a... |
| ordtrest2NEWlem 34624 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 34625 | An interval-closed set ` A... |
| ordtconnlem1 34626 | Connectedness in the order... |
| ordtconn 34627 | Connectedness in the order... |
| mndpluscn 34628 | A mapping that is both a h... |
| mhmhmeotmd 34629 | Deduce a Topological Monoi... |
| rmulccn 34630 | Multiplication by a real c... |
| raddcn 34631 | Addition in the real numbe... |
| xrmulc1cn 34632 | The operation multiplying ... |
| fmcncfil 34633 | The image of a Cauchy filt... |
| xrge0hmph 34634 | The extended nonnegative r... |
| xrge0iifcnv 34635 | Define a bijection from ` ... |
| xrge0iifcv 34636 | The defined function's val... |
| xrge0iifiso 34637 | The defined bijection from... |
| xrge0iifhmeo 34638 | Expose a homeomorphism fro... |
| xrge0iifhom 34639 | The defined function from ... |
| xrge0iif1 34640 | Condition for the defined ... |
| xrge0iifmhm 34641 | The defined function from ... |
| xrge0pluscn 34642 | The addition operation of ... |
| xrge0mulc1cn 34643 | The operation multiplying ... |
| xrge0tps 34644 | The extended nonnegative r... |
| xrge0topn 34645 | The topology of the extend... |
| xrge0haus 34646 | The topology of the extend... |
| xrge0tmd 34647 | The extended nonnegative r... |
| xrge0tmdALT 34648 | Alternate proof of ~ xrge0... |
| lmlim 34649 | Relate a limit in a given ... |
| lmlimxrge0 34650 | Relate a limit in the nonn... |
| rge0scvg 34651 | Implication of convergence... |
| fsumcvg4 34652 | A serie with finite suppor... |
| pnfneige0 34653 | A neighborhood of ` +oo ` ... |
| lmxrge0 34654 | Express "sequence ` F ` co... |
| lmdvg 34655 | If a monotonic sequence of... |
| lmdvglim 34656 | If a monotonic real number... |
| pl1cn 34657 | A univariate polynomial is... |
| zringnm 34660 | The norm (function) for a ... |
| zzsnm 34661 | The norm of the ring of th... |
| zlm0 34662 | Zero of a ` ZZ ` -module. ... |
| zlm1 34663 | Unity element of a ` ZZ ` ... |
| zlmds 34664 | Distance in a ` ZZ ` -modu... |
| zlmtset 34665 | Topology in a ` ZZ ` -modu... |
| zlmnm 34666 | Norm of a ` ZZ ` -module (... |
| zhmnrg 34667 | The ` ZZ ` -module built f... |
| nmmulg 34668 | The norm of a group produc... |
| zrhnm 34669 | The norm of the image by `... |
| cnzh 34670 | The ` ZZ ` -module of ` CC... |
| rezh 34671 | The ` ZZ ` -module of ` RR... |
| qqhval 34674 | Value of the canonical hom... |
| zrhf1ker 34675 | The kernel of the homomorp... |
| zrhchr 34676 | The kernel of the homomorp... |
| zrhker 34677 | The kernel of the homomorp... |
| zrhunitpreima 34678 | The preimage by ` ZRHom ` ... |
| elzrhunit 34679 | Condition for the image by... |
| zrhneg 34680 | The canonical homomorphism... |
| zrhcntr 34681 | The canonical representati... |
| elzdif0 34682 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 34683 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 34684 | Value of the canonical hom... |
| qqhvval 34685 | Value of the canonical hom... |
| qqh0 34686 | The image of ` 0 ` by the ... |
| qqh1 34687 | The image of ` 1 ` by the ... |
| qqhf 34688 | ` QQHom ` as a function. ... |
| qqhvq 34689 | The image of a quotient by... |
| qqhghm 34690 | The ` QQHom ` homomorphism... |
| qqhrhm 34691 | The ` QQHom ` homomorphism... |
| qqhnm 34692 | The norm of the image by `... |
| qqhcn 34693 | The ` QQHom ` homomorphism... |
| qqhucn 34694 | The ` QQHom ` homomorphism... |
| rrhval 34698 | Value of the canonical hom... |
| rrhcn 34699 | If the topology of ` R ` i... |
| rrhf 34700 | If the topology of ` R ` i... |
| isrrext 34702 | Express the property " ` R... |
| rrextnrg 34703 | An extension of ` RR ` is ... |
| rrextdrg 34704 | An extension of ` RR ` is ... |
| rrextnlm 34705 | The norm of an extension o... |
| rrextchr 34706 | The ring characteristic of... |
| rrextcusp 34707 | An extension of ` RR ` is ... |
| rrexttps 34708 | An extension of ` RR ` is ... |
| rrexthaus 34709 | The topology of an extensi... |
| rrextust 34710 | The uniformity of an exten... |
| rerrext 34711 | The field of the real numb... |
| cnrrext 34712 | The field of the complex n... |
| qqtopn 34713 | The topology of the field ... |
| rrhfe 34714 | If ` R ` is an extension o... |
| rrhcne 34715 | If ` R ` is an extension o... |
| rrhqima 34716 | The ` RRHom ` homomorphism... |
| rrh0 34717 | The image of ` 0 ` by the ... |
| xrhval 34720 | The value of the embedding... |
| zrhre 34721 | The ` ZRHom ` homomorphism... |
| qqhre 34722 | The ` QQHom ` homomorphism... |
| rrhre 34723 | The ` RRHom ` homomorphism... |
| relmntop 34726 | Manifold is a relation. (... |
| ismntoplly 34727 | Property of being a manifo... |
| ismntop 34728 | Property of being a manifo... |
| esumex 34731 | An extended sum is a set b... |
| esumcl 34732 | Closure for extended sum i... |
| esumeq12dvaf 34733 | Equality deduction for ext... |
| esumeq12dva 34734 | Equality deduction for ext... |
| esumeq12d 34735 | Equality deduction for ext... |
| esumeq1 34736 | Equality theorem for an ex... |
| esumeq1d 34737 | Equality theorem for an ex... |
| esumeq2 34738 | Equality theorem for exten... |
| esumeq2d 34739 | Equality deduction for ext... |
| esumeq2dv 34740 | Equality deduction for ext... |
| esumeq2sdv 34741 | Equality deduction for ext... |
| nfesum1 34742 | Bound-variable hypothesis ... |
| nfesum2 34743 | Bound-variable hypothesis ... |
| cbvesum 34744 | Change bound variable in a... |
| cbvesumv 34745 | Change bound variable in a... |
| esumid 34746 | Identify the extended sum ... |
| esumgsum 34747 | A finite extended sum is t... |
| esumval 34748 | Develop the value of the e... |
| esumel 34749 | The extended sum is a limi... |
| esumnul 34750 | Extended sum over the empt... |
| esum0 34751 | Extended sum of zero. (Co... |
| esumf1o 34752 | Re-index an extended sum u... |
| esumc 34753 | Convert from the collectio... |
| esumrnmpt 34754 | Rewrite an extended sum in... |
| esumsplit 34755 | Split an extended sum into... |
| esummono 34756 | Extended sum is monotonic.... |
| esumpad 34757 | Extend an extended sum by ... |
| esumpad2 34758 | Remove zeroes from an exte... |
| esumadd 34759 | Addition of infinite sums.... |
| esumle 34760 | If all of the terms of an ... |
| gsumesum 34761 | Relate a group sum on ` ( ... |
| esumlub 34762 | The extended sum is the lo... |
| esumaddf 34763 | Addition of infinite sums.... |
| esumlef 34764 | If all of the terms of an ... |
| esumcst 34765 | The extended sum of a cons... |
| esumsnf 34766 | The extended sum of a sing... |
| esumsn 34767 | The extended sum of a sing... |
| esumpr 34768 | Extended sum over a pair. ... |
| esumpr2 34769 | Extended sum over a pair, ... |
| esumrnmpt2 34770 | Rewrite an extended sum in... |
| esumfzf 34771 | Formulating a partial exte... |
| esumfsup 34772 | Formulating an extended su... |
| esumfsupre 34773 | Formulating an extended su... |
| esumss 34774 | Change the index set to a ... |
| esumpinfval 34775 | The value of the extended ... |
| esumpfinvallem 34776 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34777 | The value of the extended ... |
| esumpfinvalf 34778 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34779 | The value of the extended ... |
| esumpcvgval 34780 | The value of the extended ... |
| esumpmono 34781 | The partial sums in an ext... |
| esumcocn 34782 | Lemma for ~ esummulc2 and ... |
| esummulc1 34783 | An extended sum multiplied... |
| esummulc2 34784 | An extended sum multiplied... |
| esumdivc 34785 | An extended sum divided by... |
| hashf2 34786 | Lemma for ~ hasheuni . (C... |
| hasheuni 34787 | The cardinality of a disjo... |
| esumcvg 34788 | The sequence of partial su... |
| esumcvg2 34789 | Simpler version of ~ esumc... |
| esumcvgsum 34790 | The value of the extended ... |
| esumsup 34791 | Express an extended sum as... |
| esumgect 34792 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34793 | All terms of a converging ... |
| esum2dlem 34794 | Lemma for ~ esum2d (finite... |
| esum2d 34795 | Write a double extended su... |
| esumiun 34796 | Sum over a nonnecessarily ... |
| ofceq 34799 | Equality theorem for funct... |
| ofcfval 34800 | Value of an operation appl... |
| ofcval 34801 | Evaluate a function/consta... |
| ofcfn 34802 | The function operation pro... |
| ofcfeqd2 34803 | Equality theorem for funct... |
| ofcfval3 34804 | General value of ` ( F oFC... |
| ofcf 34805 | The function/constant oper... |
| ofcfval2 34806 | The function operation exp... |
| ofcfval4 34807 | The function/constant oper... |
| ofcc 34808 | Left operation by a consta... |
| ofcof 34809 | Relate function operation ... |
| sigaex 34812 | Lemma for ~ issiga and ~ i... |
| sigaval 34813 | The set of sigma-algebra w... |
| issiga 34814 | An alternative definition ... |
| isrnsiga 34815 | The property of being a si... |
| 0elsiga 34816 | A sigma-algebra contains t... |
| baselsiga 34817 | A sigma-algebra contains i... |
| sigasspw 34818 | A sigma-algebra is a set o... |
| sigaclcu 34819 | A sigma-algebra is closed ... |
| sigaclcuni 34820 | A sigma-algebra is closed ... |
| sigaclfu 34821 | A sigma-algebra is closed ... |
| sigaclcu2 34822 | A sigma-algebra is closed ... |
| sigaclfu2 34823 | A sigma-algebra is closed ... |
| sigaclcu3 34824 | A sigma-algebra is closed ... |
| issgon 34825 | Property of being a sigma-... |
| sgon 34826 | A sigma-algebra is a sigma... |
| elsigass 34827 | An element of a sigma-alge... |
| elrnsiga 34828 | Dropping the base informat... |
| isrnsigau 34829 | The property of being a si... |
| unielsiga 34830 | A sigma-algebra contains i... |
| dmvlsiga 34831 | Lebesgue-measurable subset... |
| pwsiga 34832 | Any power set forms a sigm... |
| prsiga 34833 | The smallest possible sigm... |
| sigaclci 34834 | A sigma-algebra is closed ... |
| difunielsiga 34835 | A sigma-algebra is closed ... |
| unelsiga 34836 | A sigma-algebra is closed ... |
| difelsiga 34837 | A sigma-algebra is closed ... |
| inelsiga 34838 | A sigma-algebra is closed ... |
| sigainb 34839 | Building a sigma-algebra f... |
| insiga 34840 | The intersection of a coll... |
| sigagenval 34843 | Value of the generated sig... |
| sigagensiga 34844 | A generated sigma-algebra ... |
| sgsiga 34845 | A generated sigma-algebra ... |
| unisg 34846 | The sigma-algebra generate... |
| dmsigagen 34847 | A sigma-algebra can be gen... |
| sssigagen 34848 | A set is a subset of the s... |
| sssigagen2 34849 | A subset of the generating... |
| elsigagen 34850 | Any element of a set is al... |
| elsigagen2 34851 | Any countable union of ele... |
| sigagenss 34852 | The generated sigma-algebr... |
| sigagenss2 34853 | Sufficient condition for i... |
| sigagenid 34854 | The sigma-algebra generate... |
| ispisys 34855 | The property of being a pi... |
| ispisys2 34856 | The property of being a pi... |
| inelpisys 34857 | Pi-systems are closed unde... |
| sigapisys 34858 | All sigma-algebras are pi-... |
| isldsys 34859 | The property of being a la... |
| pwldsys 34860 | The power set of the unive... |
| unelldsys 34861 | Lambda-systems are closed ... |
| sigaldsys 34862 | All sigma-algebras are lam... |
| ldsysgenld 34863 | The intersection of all la... |
| sigapildsyslem 34864 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34865 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34866 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34867 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34868 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34869 | The lambda system ` E ` ge... |
| dynkin 34870 | Dynkin's lambda-pi theorem... |
| isros 34871 | The property of being a ri... |
| rossspw 34872 | A ring of sets is a collec... |
| 0elros 34873 | A ring of sets contains th... |
| unelros 34874 | A ring of sets is closed u... |
| difelros 34875 | A ring of sets is closed u... |
| inelros 34876 | A ring of sets is closed u... |
| fiunelros 34877 | A ring of sets is closed u... |
| issros 34878 | The property of being a se... |
| srossspw 34879 | A semiring of sets is a co... |
| 0elsros 34880 | A semiring of sets contain... |
| inelsros 34881 | A semiring of sets is clos... |
| diffiunisros 34882 | In semiring of sets, compl... |
| rossros 34883 | Rings of sets are semiring... |
| brsiga 34886 | The Borel Algebra on real ... |
| brsigarn 34887 | The Borel Algebra is a sig... |
| brsigasspwrn 34888 | The Borel Algebra is a set... |
| unibrsiga 34889 | The union of the Borel Alg... |
| cldssbrsiga 34890 | A Borel Algebra contains a... |
| sxval 34893 | Value of the product sigma... |
| sxsiga 34894 | A product sigma-algebra is... |
| sxsigon 34895 | A product sigma-algebra is... |
| sxuni 34896 | The base set of a product ... |
| elsx 34897 | The cartesian product of t... |
| measbase 34900 | The base set of a measure ... |
| measval 34901 | The value of the ` measure... |
| ismeas 34902 | The property of being a me... |
| isrnmeas 34903 | The property of being a me... |
| dmmeas 34904 | The domain of a measure is... |
| measbasedom 34905 | The base set of a measure ... |
| measfrge0 34906 | A measure is a function ov... |
| measfn 34907 | A measure is a function on... |
| measvxrge0 34908 | The values of a measure ar... |
| measvnul 34909 | The measure of the empty s... |
| measge0 34910 | A measure is nonnegative. ... |
| measle0 34911 | If the measure of a given ... |
| measvun 34912 | The measure of a countable... |
| measxun2 34913 | The measure the union of t... |
| measun 34914 | The measure the union of t... |
| measvunilem 34915 | Lemma for ~ measvuni . (C... |
| measvunilem0 34916 | Lemma for ~ measvuni . (C... |
| measvuni 34917 | The measure of a countable... |
| measssd 34918 | A measure is monotone with... |
| measunl 34919 | A measure is sub-additive ... |
| measiuns 34920 | The measure of the union o... |
| measiun 34921 | A measure is sub-additive.... |
| meascnbl 34922 | A measure is continuous fr... |
| measinblem 34923 | Lemma for ~ measinb . (Co... |
| measinb 34924 | Building a measure restric... |
| measres 34925 | Building a measure restric... |
| measinb2 34926 | Building a measure restric... |
| measdivcst 34927 | Division of a measure by a... |
| measdivcstALTV 34928 | Alternate version of ~ mea... |
| cntmeas 34929 | The Counting measure is a ... |
| pwcntmeas 34930 | The counting measure is a ... |
| cntnevol 34931 | Counting and Lebesgue meas... |
| voliune 34932 | The Lebesgue measure funct... |
| volfiniune 34933 | The Lebesgue measure funct... |
| volmeas 34934 | The Lebesgue measure is a ... |
| ddeval1 34937 | Value of the delta measure... |
| ddeval0 34938 | Value of the delta measure... |
| ddemeas 34939 | The Dirac delta measure is... |
| relae 34943 | 'almost everywhere' is a r... |
| brae 34944 | 'almost everywhere' relati... |
| braew 34945 | 'almost everywhere' relati... |
| truae 34946 | A truth holds almost every... |
| aean 34947 | A conjunction holds almost... |
| faeval 34949 | Value of the 'almost every... |
| relfae 34950 | The 'almost everywhere' bu... |
| brfae 34951 | 'almost everywhere' relati... |
| ismbfm 34954 | The predicate " ` F ` is a... |
| elunirnmbfm 34955 | The property of being a me... |
| mbfmfun 34956 | A measurable function is a... |
| mbfmf 34957 | A measurable function as a... |
| mbfmcnvima 34958 | The preimage by a measurab... |
| isanmbfm 34959 | The predicate to be a meas... |
| mbfmbfm 34960 | A measurable function to a... |
| mbfmcst 34961 | A constant function is mea... |
| 1stmbfm 34962 | The first projection map i... |
| 2ndmbfm 34963 | The second projection map ... |
| imambfm 34964 | If the sigma-algebra in th... |
| cnmbfm 34965 | A continuous function is m... |
| mbfmco 34966 | The composition of two mea... |
| mbfmco2 34967 | The pair building of two m... |
| mbfmvolf 34968 | Measurable functions with ... |
| elmbfmvol2 34969 | Measurable functions with ... |
| mbfmcnt 34970 | All functions are measurab... |
| br2base 34971 | The base set for the gener... |
| dya2ub 34972 | An upper bound for a dyadi... |
| sxbrsigalem0 34973 | The closed half-spaces of ... |
| sxbrsigalem3 34974 | The sigma-algebra generate... |
| dya2iocival 34975 | The function ` I ` returns... |
| dya2iocress 34976 | Dyadic intervals are subse... |
| dya2iocbrsiga 34977 | Dyadic intervals are Borel... |
| dya2icobrsiga 34978 | Dyadic intervals are Borel... |
| dya2icoseg 34979 | For any point and any clos... |
| dya2icoseg2 34980 | For any point and any open... |
| dya2iocrfn 34981 | The function returning dya... |
| dya2iocct 34982 | The dyadic rectangle set i... |
| dya2iocnrect 34983 | For any point of an open r... |
| dya2iocnei 34984 | For any point of an open s... |
| dya2iocuni 34985 | Every open set of ` ( RR X... |
| dya2iocucvr 34986 | The dyadic rectangular set... |
| sxbrsigalem1 34987 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34988 | The sigma-algebra generate... |
| sxbrsigalem4 34989 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34990 | First direction for ~ sxbr... |
| sxbrsigalem6 34991 | First direction for ~ sxbr... |
| sxbrsiga 34992 | The product sigma-algebra ... |
| omsval 34995 | Value of the function mapp... |
| omsfval 34996 | Value of the outer measure... |
| omscl 34997 | A closure lemma for the co... |
| omsf 34998 | A constructed outer measur... |
| oms0 34999 | A constructed outer measur... |
| omsmon 35000 | A constructed outer measur... |
| omssubaddlem 35001 | For any small margin ` E `... |
| omssubadd 35002 | A constructed outer measur... |
| carsgval 35005 | Value of the Caratheodory ... |
| carsgcl 35006 | Closure of the Caratheodor... |
| elcarsg 35007 | Property of being a Carath... |
| baselcarsg 35008 | The universe set, ` O ` , ... |
| 0elcarsg 35009 | The empty set is Caratheod... |
| carsguni 35010 | The union of all Caratheod... |
| elcarsgss 35011 | Caratheodory measurable se... |
| difelcarsg 35012 | The Caratheodory measurabl... |
| inelcarsg 35013 | The Caratheodory measurabl... |
| unelcarsg 35014 | The Caratheodory-measurabl... |
| difelcarsg2 35015 | The Caratheodory-measurabl... |
| carsgmon 35016 | Utility lemma: Apply mono... |
| carsgsigalem 35017 | Lemma for the following th... |
| fiunelcarsg 35018 | The Caratheodory measurabl... |
| carsgclctunlem1 35019 | Lemma for ~ carsgclctun . ... |
| carsggect 35020 | The outer measure is count... |
| carsgclctunlem2 35021 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 35022 | Lemma for ~ carsgclctun . ... |
| carsgclctun 35023 | The Caratheodory measurabl... |
| carsgsiga 35024 | The Caratheodory measurabl... |
| omsmeas 35025 | The restriction of a const... |
| pmeasmono 35026 | This theorem's hypotheses ... |
| pmeasadd 35027 | A premeasure on a ring of ... |
| itgeq12dv 35028 | Equality theorem for an in... |
| sitgval 35034 | Value of the simple functi... |
| issibf 35035 | The predicate " ` F ` is a... |
| sibf0 35036 | The constant zero function... |
| sibfmbl 35037 | A simple function is measu... |
| sibff 35038 | A simple function is a fun... |
| sibfrn 35039 | A simple function has fini... |
| sibfima 35040 | Any preimage of a singleto... |
| sibfinima 35041 | The measure of the interse... |
| sibfof 35042 | Applying function operatio... |
| sitgfval 35043 | Value of the Bochner integ... |
| sitgclg 35044 | Closure of the Bochner int... |
| sitgclbn 35045 | Closure of the Bochner int... |
| sitgclcn 35046 | Closure of the Bochner int... |
| sitgclre 35047 | Closure of the Bochner int... |
| sitg0 35048 | The integral of the consta... |
| sitgf 35049 | The integral for simple fu... |
| sitgaddlemb 35050 | Lemma for * sitgadd . (Co... |
| sitmval 35051 | Value of the simple functi... |
| sitmfval 35052 | Value of the integral dist... |
| sitmcl 35053 | Closure of the integral di... |
| sitmf 35054 | The integral metric as a f... |
| oddpwdc 35056 | Lemma for ~ eulerpart . T... |
| oddpwdcv 35057 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 35058 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 35059 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 35060 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 35061 | Lemma for ~ eulerpart . (... |
| eulerpartlems 35062 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 35063 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 35064 | Lemma for ~ eulerpart . (... |
| eulerpartleme 35065 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 35066 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 35067 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 35068 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 35069 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 35070 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 35071 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 35072 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 35073 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 35074 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 35075 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 35076 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 35077 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 35078 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 35079 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 35080 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 35081 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 35082 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 35083 | Lemma for ~ eulerpart . (... |
| eulerpart 35084 | Euler's theorem on partiti... |
| subiwrd 35087 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 35088 | Length of a subword of an ... |
| iwrdsplit 35089 | Lemma for ~ sseqp1 . (Con... |
| sseqval 35090 | Value of the strong sequen... |
| sseqfv1 35091 | Value of the strong sequen... |
| sseqfn 35092 | A strong recursive sequenc... |
| sseqmw 35093 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 35094 | A strong recursive sequenc... |
| sseqfres 35095 | The first elements in the ... |
| sseqfv2 35096 | Value of the strong sequen... |
| sseqp1 35097 | Value of the strong sequen... |
| fiblem 35100 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 35101 | Value of the Fibonacci seq... |
| fib1 35102 | Value of the Fibonacci seq... |
| fibp1 35103 | Value of the Fibonacci seq... |
| fib2 35104 | Value of the Fibonacci seq... |
| fib3 35105 | Value of the Fibonacci seq... |
| fib4 35106 | Value of the Fibonacci seq... |
| fib5 35107 | Value of the Fibonacci seq... |
| fib6 35108 | Value of the Fibonacci seq... |
| elprob 35111 | The property of being a pr... |
| domprobmeas 35112 | A probability measure is a... |
| domprobsiga 35113 | The domain of a probabilit... |
| probtot 35114 | The probability of the uni... |
| prob01 35115 | A probability is an elemen... |
| probnul 35116 | The probability of the emp... |
| unveldomd 35117 | The universe is an element... |
| unveldom 35118 | The universe is an element... |
| nuleldmp 35119 | The empty set is an elemen... |
| probcun 35120 | The probability of the uni... |
| probun 35121 | The probability of the uni... |
| probdif 35122 | The probability of the dif... |
| probinc 35123 | A probability law is incre... |
| probdsb 35124 | The probability of the com... |
| probmeasd 35125 | A probability measure is a... |
| probvalrnd 35126 | The value of a probability... |
| probtotrnd 35127 | The probability of the uni... |
| totprobd 35128 | Law of total probability, ... |
| totprob 35129 | Law of total probability. ... |
| probfinmeasb 35130 | Build a probability measur... |
| probfinmeasbALTV 35131 | Alternate version of ~ pro... |
| probmeasb 35132 | Build a probability from a... |
| cndprobval 35135 | The value of the condition... |
| cndprobin 35136 | An identity linking condit... |
| cndprob01 35137 | The conditional probabilit... |
| cndprobtot 35138 | The conditional probabilit... |
| cndprobnul 35139 | The conditional probabilit... |
| cndprobprob 35140 | The conditional probabilit... |
| bayesth 35141 | Bayes Theorem. (Contribut... |
| rrvmbfm 35144 | A real-valued random varia... |
| isrrvv 35145 | Elementhood to the set of ... |
| rrvvf 35146 | A real-valued random varia... |
| rrvfn 35147 | A real-valued random varia... |
| rrvdm 35148 | The domain of a random var... |
| rrvrnss 35149 | The range of a random vari... |
| rrvf2 35150 | A real-valued random varia... |
| rrvdmss 35151 | The domain of a random var... |
| rrvfinvima 35152 | For a real-value random va... |
| 0rrv 35153 | The constant function equa... |
| rrvadd 35154 | The sum of two random vari... |
| rrvmulc 35155 | A random variable multipli... |
| rrvsum 35156 | An indexed sum of random v... |
| boolesineq 35157 | Boole's inequality (union ... |
| orvcval 35160 | Value of the preimage mapp... |
| orvcval2 35161 | Another way to express the... |
| elorvc 35162 | Elementhood of a preimage.... |
| orvcval4 35163 | The value of the preimage ... |
| orvcoel 35164 | If the relation produces o... |
| orvccel 35165 | If the relation produces c... |
| elorrvc 35166 | Elementhood of a preimage ... |
| orrvcval4 35167 | The value of the preimage ... |
| orrvcoel 35168 | If the relation produces o... |
| orrvccel 35169 | If the relation produces c... |
| orvcgteel 35170 | Preimage maps produced by ... |
| orvcelval 35171 | Preimage maps produced by ... |
| orvcelel 35172 | Preimage maps produced by ... |
| dstrvval 35173 | The value of the distribut... |
| dstrvprob 35174 | The distribution of a rand... |
| orvclteel 35175 | Preimage maps produced by ... |
| dstfrvel 35176 | Elementhood of preimage ma... |
| dstfrvunirn 35177 | The limit of all preimage ... |
| orvclteinc 35178 | Preimage maps produced by ... |
| dstfrvinc 35179 | A cumulative distribution ... |
| dstfrvclim1 35180 | The limit of the cumulativ... |
| coinfliplem 35181 | Division in the extended r... |
| coinflipprob 35182 | The ` P ` we defined for c... |
| coinflipspace 35183 | The space of our coin-flip... |
| coinflipuniv 35184 | The universe of our coin-f... |
| coinfliprv 35185 | The ` X ` we defined for c... |
| coinflippv 35186 | The probability of heads i... |
| coinflippvt 35187 | The probability of tails i... |
| ballotlemoex 35188 | ` O ` is a set. (Contribu... |
| ballotlem1 35189 | The size of the universe i... |
| ballotlemelo 35190 | Elementhood in ` O ` . (C... |
| ballotlem2 35191 | The probability that the f... |
| ballotlemfval 35192 | The value of ` F ` . (Con... |
| ballotlemfelz 35193 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 35194 | If the ` J ` th ballot is ... |
| ballotlemfc0 35195 | ` F ` takes value 0 betwee... |
| ballotlemfcc 35196 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 35197 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 35198 | ` ( F `` C ) ` always star... |
| ballotleme 35199 | Elements of ` E ` . (Cont... |
| ballotlemodife 35200 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 35201 | If the first pick is a vot... |
| ballotlem5 35202 | If A is not ahead througho... |
| ballotlemi 35203 | Value of ` I ` for a given... |
| ballotlemiex 35204 | Properties of ` ( I `` C )... |
| ballotlemi1 35205 | The first tie cannot be re... |
| ballotlemii 35206 | The first tie cannot be re... |
| ballotlemsup 35207 | The set of zeroes of ` F `... |
| ballotlemimin 35208 | ` ( I `` C ) ` is the firs... |
| ballotlemic 35209 | If the first vote is for B... |
| ballotlem1c 35210 | If the first vote is for A... |
| ballotlemsval 35211 | Value of ` S ` . (Contrib... |
| ballotlemsv 35212 | Value of ` S ` evaluated a... |
| ballotlemsgt1 35213 | ` S ` maps values less tha... |
| ballotlemsdom 35214 | Domain of ` S ` for a give... |
| ballotlemsel1i 35215 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 35216 | The defined ` S ` is a bij... |
| ballotlemsi 35217 | The image by ` S ` of the ... |
| ballotlemsima 35218 | The image by ` S ` of an i... |
| ballotlemieq 35219 | If two countings share the... |
| ballotlemrval 35220 | Value of ` R ` . (Contrib... |
| ballotlemscr 35221 | The image of ` ( R `` C ) ... |
| ballotlemrv 35222 | Value of ` R ` evaluated a... |
| ballotlemrv1 35223 | Value of ` R ` before the ... |
| ballotlemrv2 35224 | Value of ` R ` after the t... |
| ballotlemro 35225 | Range of ` R ` is included... |
| ballotlemgval 35226 | Expand the value of ` .^ `... |
| ballotlemgun 35227 | A property of the defined ... |
| ballotlemfg 35228 | Express the value of ` ( F... |
| ballotlemfrc 35229 | Express the value of ` ( F... |
| ballotlemfrci 35230 | Reverse counting preserves... |
| ballotlemfrceq 35231 | Value of ` F ` for a rever... |
| ballotlemfrcn0 35232 | Value of ` F ` for a rever... |
| ballotlemrc 35233 | Range of ` R ` . (Contrib... |
| ballotlemirc 35234 | Applying ` R ` does not ch... |
| ballotlemrinv0 35235 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 35236 | ` R ` is its own inverse :... |
| ballotlem1ri 35237 | When the vote on the first... |
| ballotlem7 35238 | ` R ` is a bijection betwe... |
| ballotlem8 35239 | There are as many counting... |
| ballotth 35240 | Bertrand's ballot problem ... |
| fzssfzo 35241 | Condition for an integer i... |
| gsumncl 35242 | Closure of a group sum in ... |
| gsumnunsn 35243 | Closure of a group sum in ... |
| ccatmulgnn0dir 35244 | Concatenation of words fol... |
| ofcccat 35245 | Letterwise operations on w... |
| ofcs1 35246 | Letterwise operations on a... |
| ofcs2 35247 | Letterwise operations on a... |
| plyrecld 35248 | Closure of a polynomial wi... |
| signsplypnf 35249 | The quotient of a polynomi... |
| signsply0 35250 | Lemma for the rule of sign... |
| signspval 35251 | The value of the skipping ... |
| signsw0glem 35252 | Neutral element property o... |
| signswbase 35253 | The base of ` W ` is the u... |
| signswplusg 35254 | The operation of ` W ` . ... |
| signsw0g 35255 | The neutral element of ` W... |
| signswmnd 35256 | ` W ` is a monoid structur... |
| signswrid 35257 | The zero-skipping operatio... |
| signswlid 35258 | The zero-skipping operatio... |
| signswn0 35259 | The zero-skipping operatio... |
| signswch 35260 | The zero-skipping operatio... |
| signslema 35261 | Computational part of ~~? ... |
| signstfv 35262 | Value of the zero-skipping... |
| signstfval 35263 | Value of the zero-skipping... |
| signstcl 35264 | Closure of the zero skippi... |
| signstf 35265 | The zero skipping sign wor... |
| signstlen 35266 | Length of the zero skippin... |
| signstf0 35267 | Sign of a single letter wo... |
| signstfvn 35268 | Zero-skipping sign in a wo... |
| signsvtn0 35269 | If the last letter is nonz... |
| signstfvp 35270 | Zero-skipping sign in a wo... |
| signstfvneq0 35271 | In case the first letter i... |
| signstfvcl 35272 | Closure of the zero skippi... |
| signstfvc 35273 | Zero-skipping sign in a wo... |
| signstres 35274 | Restriction of a zero skip... |
| signstfveq0a 35275 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 35276 | In case the last letter is... |
| signsvvfval 35277 | The value of ` V ` , which... |
| signsvvf 35278 | ` V ` is a function. (Con... |
| signsvf0 35279 | There is no change of sign... |
| signsvf1 35280 | In a single-letter word, w... |
| signsvfn 35281 | Number of changes in a wor... |
| signsvtp 35282 | Adding a letter of the sam... |
| signsvtn 35283 | Adding a letter of a diffe... |
| signsvfpn 35284 | Adding a letter of the sam... |
| signsvfnn 35285 | Adding a letter of a diffe... |
| signlem0 35286 | Adding a zero as the highe... |
| signshf 35287 | ` H ` , corresponding to t... |
| signshwrd 35288 | ` H ` , corresponding to t... |
| signshlen 35289 | Length of ` H ` , correspo... |
| signshnz 35290 | ` H ` is not the empty wor... |
| iblidicc 35291 | The identity function is i... |
| rpsqrtcn 35292 | Continuity of the real pos... |
| divsqrtid 35293 | A real number divided by i... |
| cxpcncf1 35294 | The power function on comp... |
| efmul2picn 35295 | Multiplying by ` ( _i x. (... |
| fct2relem 35296 | Lemma for ~ ftc2re . (Con... |
| ftc2re 35297 | The Fundamental Theorem of... |
| fdvposlt 35298 | Functions with a positive ... |
| fdvneggt 35299 | Functions with a negative ... |
| fdvposle 35300 | Functions with a nonnegati... |
| fdvnegge 35301 | Functions with a nonpositi... |
| prodfzo03 35302 | A product of three factors... |
| actfunsnf1o 35303 | The action ` F ` of extend... |
| actfunsnrndisj 35304 | The action ` F ` of extend... |
| itgexpif 35305 | The basis for the circle m... |
| fsum2dsub 35306 | Lemma for ~ breprexp - Re-... |
| reprval 35309 | Value of the representatio... |
| repr0 35310 | There is exactly one repre... |
| reprf 35311 | Members of the representat... |
| reprsum 35312 | Sums of values of the memb... |
| reprle 35313 | Upper bound to the terms i... |
| reprsuc 35314 | Express the representation... |
| reprfi 35315 | Bounded representations ar... |
| reprss 35316 | Representations with terms... |
| reprinrn 35317 | Representations with term ... |
| reprlt 35318 | There are no representatio... |
| hashreprin 35319 | Express a sum of represent... |
| reprgt 35320 | There are no representatio... |
| reprinfz1 35321 | For the representation of ... |
| reprfi2 35322 | Corollary of ~ reprinfz1 .... |
| reprfz1 35323 | Corollary of ~ reprinfz1 .... |
| hashrepr 35324 | Develop the number of repr... |
| reprpmtf1o 35325 | Transposing ` 0 ` and ` X ... |
| reprdifc 35326 | Express the representation... |
| chpvalz 35327 | Value of the second Chebys... |
| chtvalz 35328 | Value of the Chebyshev fun... |
| breprexplema 35329 | Lemma for ~ breprexp (indu... |
| breprexplemb 35330 | Lemma for ~ breprexp (clos... |
| breprexplemc 35331 | Lemma for ~ breprexp (indu... |
| breprexp 35332 | Express the ` S ` th power... |
| breprexpnat 35333 | Express the ` S ` th power... |
| vtsval 35336 | Value of the Vinogradov tr... |
| vtscl 35337 | Closure of the Vinogradov ... |
| vtsprod 35338 | Express the Vinogradov tri... |
| circlemeth 35339 | The Hardy, Littlewood and ... |
| circlemethnat 35340 | The Hardy, Littlewood and ... |
| circlevma 35341 | The Circle Method, where t... |
| circlemethhgt 35342 | The circle method, where t... |
| hgt750lemc 35346 | An upper bound to the summ... |
| hgt750lemd 35347 | An upper bound to the summ... |
| hgt749d 35348 | A deduction version of ~ a... |
| logdivsqrle 35349 | Conditions for ` ( ( log `... |
| hgt750lem 35350 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 35351 | Decimal multiplication gal... |
| hgt750lemf 35352 | Lemma for the statement 7.... |
| hgt750lemg 35353 | Lemma for the statement 7.... |
| oddprm2 35354 | Two ways to write the set ... |
| hgt750lemb 35355 | An upper bound on the cont... |
| hgt750lema 35356 | An upper bound on the cont... |
| hgt750leme 35357 | An upper bound on the cont... |
| tgoldbachgnn 35358 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 35359 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 35360 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 35361 | Odd integers greater than ... |
| tgoldbachgt 35362 | Odd integers greater than ... |
| istrkg2d 35365 | Property of fulfilling dim... |
| axtglowdim2ALTV 35366 | Alternate version of ~ axt... |
| axtgupdim2ALTV 35367 | Alternate version of ~ axt... |
| cgranbtwn 35368 | Null angle implies between... |
| btwnlng13 35369 | If ` Z ` is between ` X ` ... |
| morleylemrneab 35370 | Lemma for morley . (Contr... |
| afsval 35373 | Value of the AFS relation ... |
| brafs 35374 | Binary relation form of th... |
| tg5segofs 35375 | Rephrase ~ axtg5seg using ... |
| lpadval 35378 | Value of the ` leftpad ` f... |
| lpadlem1 35379 | Lemma for the ` leftpad ` ... |
| lpadlem3 35380 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 35381 | Length of a left-padded wo... |
| lpadlem2 35382 | Lemma for the ` leftpad ` ... |
| lpadlen2 35383 | Length of a left-padded wo... |
| lpadmax 35384 | Length of a left-padded wo... |
| lpadleft 35385 | The contents of prefix of ... |
| lpadright 35386 | The suffix of a left-padde... |
| bnj170 35399 | ` /\ ` -manipulation. (Co... |
| bnj240 35400 | ` /\ ` -manipulation. (Co... |
| bnj248 35401 | ` /\ ` -manipulation. (Co... |
| bnj250 35402 | ` /\ ` -manipulation. (Co... |
| bnj251 35403 | ` /\ ` -manipulation. (Co... |
| bnj252 35404 | ` /\ ` -manipulation. (Co... |
| bnj253 35405 | ` /\ ` -manipulation. (Co... |
| bnj255 35406 | ` /\ ` -manipulation. (Co... |
| bnj256 35407 | ` /\ ` -manipulation. (Co... |
| bnj257 35408 | ` /\ ` -manipulation. (Co... |
| bnj258 35409 | ` /\ ` -manipulation. (Co... |
| bnj268 35410 | ` /\ ` -manipulation. (Co... |
| bnj290 35411 | ` /\ ` -manipulation. (Co... |
| bnj291 35412 | ` /\ ` -manipulation. (Co... |
| bnj312 35413 | ` /\ ` -manipulation. (Co... |
| bnj334 35414 | ` /\ ` -manipulation. (Co... |
| bnj345 35415 | ` /\ ` -manipulation. (Co... |
| bnj422 35416 | ` /\ ` -manipulation. (Co... |
| bnj432 35417 | ` /\ ` -manipulation. (Co... |
| bnj446 35418 | ` /\ ` -manipulation. (Co... |
| bnj23 35419 | First-order logic and set ... |
| bnj31 35420 | First-order logic and set ... |
| bnj62 35421 | First-order logic and set ... |
| bnj89 35422 | First-order logic and set ... |
| bnj90 35423 | First-order logic and set ... |
| bnj101 35424 | First-order logic and set ... |
| bnj105 35425 | First-order logic and set ... |
| bnj115 35426 | First-order logic and set ... |
| bnj132 35427 | First-order logic and set ... |
| bnj133 35428 | First-order logic and set ... |
| bnj156 35429 | First-order logic and set ... |
| bnj158 35430 | First-order logic and set ... |
| bnj168 35431 | First-order logic and set ... |
| bnj206 35432 | First-order logic and set ... |
| bnj216 35433 | First-order logic and set ... |
| bnj219 35434 | First-order logic and set ... |
| bnj226 35435 | First-order logic and set ... |
| bnj228 35436 | First-order logic and set ... |
| bnj519 35437 | First-order logic and set ... |
| bnj524 35438 | First-order logic and set ... |
| bnj525 35439 | First-order logic and set ... |
| bnj534 35440 | First-order logic and set ... |
| bnj538 35441 | First-order logic and set ... |
| bnj529 35442 | First-order logic and set ... |
| bnj551 35443 | First-order logic and set ... |
| bnj563 35444 | First-order logic and set ... |
| bnj564 35445 | First-order logic and set ... |
| bnj593 35446 | First-order logic and set ... |
| bnj596 35447 | First-order logic and set ... |
| bnj610 35448 | Pass from equality ( ` x =... |
| bnj642 35449 | ` /\ ` -manipulation. (Co... |
| bnj643 35450 | ` /\ ` -manipulation. (Co... |
| bnj645 35451 | ` /\ ` -manipulation. (Co... |
| bnj658 35452 | ` /\ ` -manipulation. (Co... |
| bnj667 35453 | ` /\ ` -manipulation. (Co... |
| bnj705 35454 | ` /\ ` -manipulation. (Co... |
| bnj706 35455 | ` /\ ` -manipulation. (Co... |
| bnj707 35456 | ` /\ ` -manipulation. (Co... |
| bnj708 35457 | ` /\ ` -manipulation. (Co... |
| bnj721 35458 | ` /\ ` -manipulation. (Co... |
| bnj832 35459 | ` /\ ` -manipulation. (Co... |
| bnj835 35460 | ` /\ ` -manipulation. (Co... |
| bnj836 35461 | ` /\ ` -manipulation. (Co... |
| bnj837 35462 | ` /\ ` -manipulation. (Co... |
| bnj769 35463 | ` /\ ` -manipulation. (Co... |
| bnj770 35464 | ` /\ ` -manipulation. (Co... |
| bnj771 35465 | ` /\ ` -manipulation. (Co... |
| bnj887 35466 | ` /\ ` -manipulation. (Co... |
| bnj918 35467 | First-order logic and set ... |
| bnj919 35468 | First-order logic and set ... |
| bnj923 35469 | First-order logic and set ... |
| bnj927 35470 | First-order logic and set ... |
| bnj931 35471 | First-order logic and set ... |
| bnj937 35472 | First-order logic and set ... |
| bnj941 35473 | First-order logic and set ... |
| bnj945 35474 | Technical lemma for ~ bnj6... |
| bnj946 35475 | First-order logic and set ... |
| bnj951 35476 | ` /\ ` -manipulation. (Co... |
| bnj956 35477 | First-order logic and set ... |
| bnj976 35478 | First-order logic and set ... |
| bnj982 35479 | First-order logic and set ... |
| bnj1019 35480 | First-order logic and set ... |
| bnj1023 35481 | First-order logic and set ... |
| bnj1095 35482 | First-order logic and set ... |
| bnj1096 35483 | First-order logic and set ... |
| bnj1098 35484 | First-order logic and set ... |
| bnj1101 35485 | First-order logic and set ... |
| bnj1113 35486 | First-order logic and set ... |
| bnj1109 35487 | First-order logic and set ... |
| bnj1131 35488 | First-order logic and set ... |
| bnj1138 35489 | First-order logic and set ... |
| bnj1143 35490 | First-order logic and set ... |
| bnj1146 35491 | First-order logic and set ... |
| bnj1149 35492 | First-order logic and set ... |
| bnj1185 35493 | First-order logic and set ... |
| bnj1196 35494 | First-order logic and set ... |
| bnj1198 35495 | First-order logic and set ... |
| bnj1209 35496 | First-order logic and set ... |
| bnj1211 35497 | First-order logic and set ... |
| bnj1213 35498 | First-order logic and set ... |
| bnj1212 35499 | First-order logic and set ... |
| bnj1219 35500 | First-order logic and set ... |
| bnj1224 35501 | First-order logic and set ... |
| bnj1230 35502 | First-order logic and set ... |
| bnj1232 35503 | First-order logic and set ... |
| bnj1235 35504 | First-order logic and set ... |
| bnj1239 35505 | First-order logic and set ... |
| bnj1238 35506 | First-order logic and set ... |
| bnj1241 35507 | First-order logic and set ... |
| bnj1247 35508 | First-order logic and set ... |
| bnj1254 35509 | First-order logic and set ... |
| bnj1262 35510 | First-order logic and set ... |
| bnj1266 35511 | First-order logic and set ... |
| bnj1265 35512 | First-order logic and set ... |
| bnj1275 35513 | First-order logic and set ... |
| bnj1276 35514 | First-order logic and set ... |
| bnj1292 35515 | First-order logic and set ... |
| bnj1293 35516 | First-order logic and set ... |
| bnj1294 35517 | First-order logic and set ... |
| bnj1299 35518 | First-order logic and set ... |
| bnj1304 35519 | First-order logic and set ... |
| bnj1316 35520 | First-order logic and set ... |
| bnj1317 35521 | First-order logic and set ... |
| bnj1322 35522 | First-order logic and set ... |
| bnj1340 35523 | First-order logic and set ... |
| bnj1345 35524 | First-order logic and set ... |
| bnj1350 35525 | First-order logic and set ... |
| bnj1351 35526 | First-order logic and set ... |
| bnj1352 35527 | First-order logic and set ... |
| bnj1361 35528 | First-order logic and set ... |
| bnj1366 35529 | First-order logic and set ... |
| bnj1379 35530 | First-order logic and set ... |
| bnj1383 35531 | First-order logic and set ... |
| bnj1385 35532 | First-order logic and set ... |
| bnj1386 35533 | First-order logic and set ... |
| bnj1397 35534 | First-order logic and set ... |
| bnj1400 35535 | First-order logic and set ... |
| bnj1405 35536 | First-order logic and set ... |
| bnj1422 35537 | First-order logic and set ... |
| bnj1424 35538 | First-order logic and set ... |
| bnj1436 35539 | First-order logic and set ... |
| bnj1441 35540 | First-order logic and set ... |
| bnj1441g 35541 | First-order logic and set ... |
| bnj1454 35542 | First-order logic and set ... |
| bnj1459 35543 | First-order logic and set ... |
| bnj1464 35544 | Conversion of implicit sub... |
| bnj1465 35545 | First-order logic and set ... |
| bnj1468 35546 | Conversion of implicit sub... |
| bnj1476 35547 | First-order logic and set ... |
| bnj1502 35548 | First-order logic and set ... |
| bnj1503 35549 | First-order logic and set ... |
| bnj1517 35550 | First-order logic and set ... |
| bnj1521 35551 | First-order logic and set ... |
| bnj1533 35552 | First-order logic and set ... |
| bnj1534 35553 | First-order logic and set ... |
| bnj1536 35554 | First-order logic and set ... |
| bnj1538 35555 | First-order logic and set ... |
| bnj1541 35556 | First-order logic and set ... |
| bnj1542 35557 | First-order logic and set ... |
| bnj110 35558 | Well-founded induction res... |
| bnj157 35559 | Well-founded induction res... |
| bnj66 35560 | Technical lemma for ~ bnj6... |
| bnj91 35561 | First-order logic and set ... |
| bnj92 35562 | First-order logic and set ... |
| bnj93 35563 | Technical lemma for ~ bnj9... |
| bnj95 35564 | Technical lemma for ~ bnj1... |
| bnj96 35565 | Technical lemma for ~ bnj1... |
| bnj97 35566 | Technical lemma for ~ bnj1... |
| bnj98 35567 | Technical lemma for ~ bnj1... |
| bnj106 35568 | First-order logic and set ... |
| bnj118 35569 | First-order logic and set ... |
| bnj121 35570 | First-order logic and set ... |
| bnj124 35571 | Technical lemma for ~ bnj1... |
| bnj125 35572 | Technical lemma for ~ bnj1... |
| bnj126 35573 | Technical lemma for ~ bnj1... |
| bnj130 35574 | Technical lemma for ~ bnj1... |
| bnj149 35575 | Technical lemma for ~ bnj1... |
| bnj150 35576 | Technical lemma for ~ bnj1... |
| bnj151 35577 | Technical lemma for ~ bnj1... |
| bnj154 35578 | Technical lemma for ~ bnj1... |
| bnj155 35579 | Technical lemma for ~ bnj1... |
| bnj153 35580 | Technical lemma for ~ bnj8... |
| bnj207 35581 | Technical lemma for ~ bnj8... |
| bnj213 35582 | First-order logic and set ... |
| bnj222 35583 | Technical lemma for ~ bnj2... |
| bnj229 35584 | Technical lemma for ~ bnj5... |
| bnj517 35585 | Technical lemma for ~ bnj5... |
| bnj518 35586 | Technical lemma for ~ bnj8... |
| bnj523 35587 | Technical lemma for ~ bnj8... |
| bnj526 35588 | Technical lemma for ~ bnj8... |
| bnj528 35589 | Technical lemma for ~ bnj8... |
| bnj535 35590 | Technical lemma for ~ bnj8... |
| bnj539 35591 | Technical lemma for ~ bnj8... |
| bnj540 35592 | Technical lemma for ~ bnj8... |
| bnj543 35593 | Technical lemma for ~ bnj8... |
| bnj544 35594 | Technical lemma for ~ bnj8... |
| bnj545 35595 | Technical lemma for ~ bnj8... |
| bnj546 35596 | Technical lemma for ~ bnj8... |
| bnj548 35597 | Technical lemma for ~ bnj8... |
| bnj553 35598 | Technical lemma for ~ bnj8... |
| bnj554 35599 | Technical lemma for ~ bnj8... |
| bnj556 35600 | Technical lemma for ~ bnj8... |
| bnj557 35601 | Technical lemma for ~ bnj8... |
| bnj558 35602 | Technical lemma for ~ bnj8... |
| bnj561 35603 | Technical lemma for ~ bnj8... |
| bnj562 35604 | Technical lemma for ~ bnj8... |
| bnj570 35605 | Technical lemma for ~ bnj8... |
| bnj571 35606 | Technical lemma for ~ bnj8... |
| bnj605 35607 | Technical lemma. This lem... |
| bnj581 35608 | Technical lemma for ~ bnj5... |
| bnj589 35609 | Technical lemma for ~ bnj8... |
| bnj590 35610 | Technical lemma for ~ bnj8... |
| bnj591 35611 | Technical lemma for ~ bnj8... |
| bnj594 35612 | Technical lemma for ~ bnj8... |
| bnj580 35613 | Technical lemma for ~ bnj5... |
| bnj579 35614 | Technical lemma for ~ bnj8... |
| bnj602 35615 | Equality theorem for the `... |
| bnj607 35616 | Technical lemma for ~ bnj8... |
| bnj609 35617 | Technical lemma for ~ bnj8... |
| bnj611 35618 | Technical lemma for ~ bnj8... |
| bnj600 35619 | Technical lemma for ~ bnj8... |
| bnj601 35620 | Technical lemma for ~ bnj8... |
| bnj852 35621 | Technical lemma for ~ bnj6... |
| bnj864 35622 | Technical lemma for ~ bnj6... |
| bnj865 35623 | Technical lemma for ~ bnj6... |
| bnj873 35624 | Technical lemma for ~ bnj6... |
| bnj849 35625 | Technical lemma for ~ bnj6... |
| bnj882 35626 | Definition (using hypothes... |
| bnj18eq1 35627 | Equality theorem for trans... |
| bnj893 35628 | Property of ` _trCl ` . U... |
| bnj900 35629 | Technical lemma for ~ bnj6... |
| bnj906 35630 | Property of ` _trCl ` . (... |
| bnj908 35631 | Technical lemma for ~ bnj6... |
| bnj911 35632 | Technical lemma for ~ bnj6... |
| bnj916 35633 | Technical lemma for ~ bnj6... |
| bnj917 35634 | Technical lemma for ~ bnj6... |
| bnj934 35635 | Technical lemma for ~ bnj6... |
| bnj929 35636 | Technical lemma for ~ bnj6... |
| bnj938 35637 | Technical lemma for ~ bnj6... |
| bnj944 35638 | Technical lemma for ~ bnj6... |
| bnj953 35639 | Technical lemma for ~ bnj6... |
| bnj958 35640 | Technical lemma for ~ bnj6... |
| bnj1000 35641 | Technical lemma for ~ bnj8... |
| bnj965 35642 | Technical lemma for ~ bnj8... |
| bnj964 35643 | Technical lemma for ~ bnj6... |
| bnj966 35644 | Technical lemma for ~ bnj6... |
| bnj967 35645 | Technical lemma for ~ bnj6... |
| bnj969 35646 | Technical lemma for ~ bnj6... |
| bnj970 35647 | Technical lemma for ~ bnj6... |
| bnj910 35648 | Technical lemma for ~ bnj6... |
| bnj978 35649 | Technical lemma for ~ bnj6... |
| bnj981 35650 | Technical lemma for ~ bnj6... |
| bnj983 35651 | Technical lemma for ~ bnj6... |
| bnj984 35652 | Technical lemma for ~ bnj6... |
| bnj985v 35653 | Version of ~ bnj985 with a... |
| bnj985 35654 | Technical lemma for ~ bnj6... |
| bnj986 35655 | Technical lemma for ~ bnj6... |
| bnj996 35656 | Technical lemma for ~ bnj6... |
| bnj998 35657 | Technical lemma for ~ bnj6... |
| bnj999 35658 | Technical lemma for ~ bnj6... |
| bnj1001 35659 | Technical lemma for ~ bnj6... |
| bnj1006 35660 | Technical lemma for ~ bnj6... |
| bnj1014 35661 | Technical lemma for ~ bnj6... |
| bnj1015 35662 | Technical lemma for ~ bnj6... |
| bnj1018g 35663 | Version of ~ bnj1018 with ... |
| bnj1018 35664 | Technical lemma for ~ bnj6... |
| bnj1020 35665 | Technical lemma for ~ bnj6... |
| bnj1021 35666 | Technical lemma for ~ bnj6... |
| bnj907 35667 | Technical lemma for ~ bnj6... |
| bnj1029 35668 | Property of ` _trCl ` . (... |
| bnj1033 35669 | Technical lemma for ~ bnj6... |
| bnj1034 35670 | Technical lemma for ~ bnj6... |
| bnj1039 35671 | Technical lemma for ~ bnj6... |
| bnj1040 35672 | Technical lemma for ~ bnj6... |
| bnj1047 35673 | Technical lemma for ~ bnj6... |
| bnj1049 35674 | Technical lemma for ~ bnj6... |
| bnj1052 35675 | Technical lemma for ~ bnj6... |
| bnj1053 35676 | Technical lemma for ~ bnj6... |
| bnj1071 35677 | Technical lemma for ~ bnj6... |
| bnj1083 35678 | Technical lemma for ~ bnj6... |
| bnj1090 35679 | Technical lemma for ~ bnj6... |
| bnj1093 35680 | Technical lemma for ~ bnj6... |
| bnj1097 35681 | Technical lemma for ~ bnj6... |
| bnj1110 35682 | Technical lemma for ~ bnj6... |
| bnj1112 35683 | Technical lemma for ~ bnj6... |
| bnj1118 35684 | Technical lemma for ~ bnj6... |
| bnj1121 35685 | Technical lemma for ~ bnj6... |
| bnj1123 35686 | Technical lemma for ~ bnj6... |
| bnj1030 35687 | Technical lemma for ~ bnj6... |
| bnj1124 35688 | Property of ` _trCl ` . (... |
| bnj1133 35689 | Technical lemma for ~ bnj6... |
| bnj1128 35690 | Technical lemma for ~ bnj6... |
| bnj1127 35691 | Property of ` _trCl ` . (... |
| bnj1125 35692 | Property of ` _trCl ` . (... |
| bnj1145 35693 | Technical lemma for ~ bnj6... |
| bnj1147 35694 | Property of ` _trCl ` . (... |
| bnj1137 35695 | Property of ` _trCl ` . (... |
| bnj1148 35696 | Property of ` _pred ` . (... |
| bnj1136 35697 | Technical lemma for ~ bnj6... |
| bnj1152 35698 | Technical lemma for ~ bnj6... |
| bnj1154 35699 | Property of ` Fr ` . (Con... |
| bnj1171 35700 | Technical lemma for ~ bnj6... |
| bnj1172 35701 | Technical lemma for ~ bnj6... |
| bnj1173 35702 | Technical lemma for ~ bnj6... |
| bnj1174 35703 | Technical lemma for ~ bnj6... |
| bnj1175 35704 | Technical lemma for ~ bnj6... |
| bnj1176 35705 | Technical lemma for ~ bnj6... |
| bnj1177 35706 | Technical lemma for ~ bnj6... |
| bnj1186 35707 | Technical lemma for ~ bnj6... |
| bnj1190 35708 | Technical lemma for ~ bnj6... |
| bnj1189 35709 | Technical lemma for ~ bnj6... |
| bnj69 35710 | Existence of a minimal ele... |
| bnj1228 35711 | Existence of a minimal ele... |
| bnj1204 35712 | Well-founded induction. T... |
| bnj1234 35713 | Technical lemma for ~ bnj6... |
| bnj1245 35714 | Technical lemma for ~ bnj6... |
| bnj1256 35715 | Technical lemma for ~ bnj6... |
| bnj1259 35716 | Technical lemma for ~ bnj6... |
| bnj1253 35717 | Technical lemma for ~ bnj6... |
| bnj1279 35718 | Technical lemma for ~ bnj6... |
| bnj1286 35719 | Technical lemma for ~ bnj6... |
| bnj1280 35720 | Technical lemma for ~ bnj6... |
| bnj1296 35721 | Technical lemma for ~ bnj6... |
| bnj1309 35722 | Technical lemma for ~ bnj6... |
| bnj1307 35723 | Technical lemma for ~ bnj6... |
| bnj1311 35724 | Technical lemma for ~ bnj6... |
| bnj1318 35725 | Technical lemma for ~ bnj6... |
| bnj1326 35726 | Technical lemma for ~ bnj6... |
| bnj1321 35727 | Technical lemma for ~ bnj6... |
| bnj1364 35728 | Property of ` _FrSe ` . (... |
| bnj1371 35729 | Technical lemma for ~ bnj6... |
| bnj1373 35730 | Technical lemma for ~ bnj6... |
| bnj1374 35731 | Technical lemma for ~ bnj6... |
| bnj1384 35732 | Technical lemma for ~ bnj6... |
| bnj1388 35733 | Technical lemma for ~ bnj6... |
| bnj1398 35734 | Technical lemma for ~ bnj6... |
| bnj1413 35735 | Property of ` _trCl ` . (... |
| bnj1408 35736 | Technical lemma for ~ bnj1... |
| bnj1414 35737 | Property of ` _trCl ` . (... |
| bnj1415 35738 | Technical lemma for ~ bnj6... |
| bnj1416 35739 | Technical lemma for ~ bnj6... |
| bnj1418 35740 | Property of ` _pred ` . (... |
| bnj1417 35741 | Technical lemma for ~ bnj6... |
| bnj1421 35742 | Technical lemma for ~ bnj6... |
| bnj1444 35743 | Technical lemma for ~ bnj6... |
| bnj1445 35744 | Technical lemma for ~ bnj6... |
| bnj1446 35745 | Technical lemma for ~ bnj6... |
| bnj1447 35746 | Technical lemma for ~ bnj6... |
| bnj1448 35747 | Technical lemma for ~ bnj6... |
| bnj1449 35748 | Technical lemma for ~ bnj6... |
| bnj1442 35749 | Technical lemma for ~ bnj6... |
| bnj1450 35750 | Technical lemma for ~ bnj6... |
| bnj1423 35751 | Technical lemma for ~ bnj6... |
| bnj1452 35752 | Technical lemma for ~ bnj6... |
| bnj1466 35753 | Technical lemma for ~ bnj6... |
| bnj1467 35754 | Technical lemma for ~ bnj6... |
| bnj1463 35755 | Technical lemma for ~ bnj6... |
| bnj1489 35756 | Technical lemma for ~ bnj6... |
| bnj1491 35757 | Technical lemma for ~ bnj6... |
| bnj1312 35758 | Technical lemma for ~ bnj6... |
| bnj1493 35759 | Technical lemma for ~ bnj6... |
| bnj1497 35760 | Technical lemma for ~ bnj6... |
| bnj1498 35761 | Technical lemma for ~ bnj6... |
| bnj60 35762 | Well-founded recursion, pa... |
| bnj1514 35763 | Technical lemma for ~ bnj1... |
| bnj1518 35764 | Technical lemma for ~ bnj1... |
| bnj1519 35765 | Technical lemma for ~ bnj1... |
| bnj1520 35766 | Technical lemma for ~ bnj1... |
| bnj1501 35767 | Technical lemma for ~ bnj1... |
| bnj1500 35768 | Well-founded recursion, pa... |
| bnj1525 35769 | Technical lemma for ~ bnj1... |
| bnj1529 35770 | Technical lemma for ~ bnj1... |
| bnj1523 35771 | Technical lemma for ~ bnj1... |
| bnj1522 35772 | Well-founded recursion, pa... |
| nfan1c 35773 | Variant of ~ nfan and comm... |
| cbvex1v 35774 | Rule used to change bound ... |
| dvelimalcased 35775 | Eliminate a disjoint varia... |
| dvelimalcasei 35776 | Eliminate a disjoint varia... |
| dvelimexcased 35777 | Eliminate a disjoint varia... |
| dvelimexcasei 35778 | Eliminate a disjoint varia... |
| reu6f 35779 | Version of ~ reu6 with dis... |
| rmo6 35780 | Alternate version of ~ rmo... |
| inv2 35781 | The intersection of the un... |
| jaeqifi 35782 | Inference combining four e... |
| exdifsn 35783 | There exists an element in... |
| axnulALT2 35784 | Alternate proof of ~ axnul... |
| f1preimaex 35785 | If the image under a one-t... |
| fnfvintima 35786 | Condition for a function v... |
| fwfffun 35787 | If ` ps ` is function-like... |
| fwfffunres 35788 | If ` ps ` is function-like... |
| ontr1i 35789 | Transitive law for ordinal... |
| eqsucidi 35790 | A set belongs to its succe... |
| suceldifsn0 35791 | A successor that is a memb... |
| ordunmax 35792 | The union of two ordinals ... |
| ordprcon 35793 | If an ordinal class is not... |
| xoromon 35794 | ` _om ` is either an ordin... |
| ot1st2nd 35795 | Reconstruction of a member... |
| omeueq 35796 | The division algorithm for... |
| mpofvd 35797 | Value of a two-argument fu... |
| mpofvopd 35798 | Value of a two-argument fu... |
| mpofvop2d 35799 | Value of a two-argument fu... |
| soinfdom 35800 | A strict order relation on... |
| abweex 35801 | The class of well-orders o... |
| ordtypeon 35802 | A proper class with a set-... |
| onf1we 35803 | If ` F ` maps ` A ` one-to... |
| fnrelpredd 35804 | A function that preserves ... |
| cardpred 35805 | The cardinality function p... |
| nummin 35806 | Every nonempty class of nu... |
| 1enumen 35807 | The Fundamental Theorem of... |
| 1enumcard 35808 | The Fundamental Theorem of... |
| r11 35809 | Value of the cumulative hi... |
| r12 35810 | Value of the cumulative hi... |
| onrankid 35811 | The rank of an ordinal num... |
| rankfilimb 35812 | The rank of a finite well-... |
| r1filim 35813 | A finite set appears in th... |
| r1ssel 35814 | A set is a subset of the v... |
| axnulALT3 35815 | Alternate proof of ~ axnul... |
| axprALT2 35816 | Alternate proof of ~ axpr ... |
| rankfo 35817 | The rank function maps the... |
| rankfn 35818 | The rank function is a fun... |
| trssfihf 35819 | If every element in a tran... |
| eqhf 35820 | The class of all hereditar... |
| werankwe 35821 | Construct a well-order giv... |
| werankwe2 35822 | Construct a well-order giv... |
| rankse 35823 | Construct a relation that ... |
| scotteqi 35824 | Equality theorem for the S... |
| elscott 35825 | Membership in a Scott's tr... |
| dfscott2 35826 | Alternate definition of a ... |
| dfscott3 35827 | Alternate definition of a ... |
| elscott2 35828 | Membership in a Scott's tr... |
| elscottrank 35829 | The rank of an element in ... |
| elscottrankeq 35830 | Elements in a Scott's tric... |
| elscottrankss 35831 | Relationship between the r... |
| scottrankeqel 35832 | If a member of the input c... |
| nelscottrankgt 35833 | If a member of the input c... |
| scottsn 35834 | Applying Scott's trick to ... |
| scott0bOLD 35835 | Obsolete version of ~ scot... |
| elscottn0 35836 | If the input class has ele... |
| 0elscott 35837 | If the empty set is an ele... |
| rankscott 35838 | The rank of a nonempty Sco... |
| rankscottu 35839 | An upper bound on the rank... |
| scottssr1 35840 | Relationship between a Sco... |
| scottidm 35841 | The Scott operation is ide... |
| trscott 35842 | The Scott's trick set of a... |
| erscottd 35843 | Applying the Scott operati... |
| ecscottd 35844 | Applying the Scott operati... |
| dfac15 35845 | Axiom of Choice equivalent... |
| dfac16 35846 | Axiom of Choice equivalent... |
| acnum 35847 | The Axiom of Choice implie... |
| weexenwe 35848 | If a well-ordering of an i... |
| infinfnum 35849 | Equivalence between two in... |
| rncardr1prc 35850 | The Axiom of Choice implie... |
| acwer1prclem 35851 | Lemma for ~ acwer1prc . (... |
| acwer1prc 35852 | The class of all well-orde... |
| 5on 35863 | Ordinal 5 is an ordinal nu... |
| 6on 35864 | Ordinal 6 is an ordinal nu... |
| 7on 35865 | Ordinal 7 is an ordinal nu... |
| 8on 35866 | Ordinal 8 is an ordinal nu... |
| 9on 35867 | Ordinal 9 is an ordinal nu... |
| 5onn 35868 | The ordinal 5 is a natural... |
| 6onn 35869 | The ordinal 6 is a natural... |
| 7onn 35870 | The ordinal 7 is a natural... |
| 8onn 35871 | The ordinal 8 is a natural... |
| 9onn 35872 | The ordinal 9 is a natural... |
| 8olt9o 35873 | Ordinal 8 is less than ord... |
| 7olt9o 35874 | Ordinal 7 is less than ord... |
| 6olt9o 35875 | Ordinal 6 is less than ord... |
| 5olt9o 35876 | Ordinal 5 is less than ord... |
| 4olt9o 35877 | Ordinal 4 is less than ord... |
| 3olt9o 35878 | Ordinal 3 is less than ord... |
| 2olt9o 35879 | Ordinal 2 is less than ord... |
| 1olt9o 35880 | Ordinal 1 is less than ord... |
| prcinf 35881 | Any proper class is litera... |
| fineqvrep 35882 | If all sets are finite, th... |
| fineqvpow 35883 | If all sets are finite, th... |
| fineqvac 35884 | If all sets are finite, th... |
| fineqvacALT 35885 | Shorter proof of ~ fineqva... |
| fineqvomon 35886 | If all sets are finite, th... |
| fineqvomonb 35887 | All sets are finite iff al... |
| omprcomonb 35888 | The class of all finite or... |
| fineqvnttrclselem1 35889 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem2 35890 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem3 35891 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclse 35892 | A counterexample demonstra... |
| fineqvinfep 35893 | A counterexample demonstra... |
| axreg 35895 | Derivation of ~ ax-reg fro... |
| axregscl 35896 | A version of ~ ax-regs wit... |
| axregszf 35897 | Derivation of ~ zfregs usi... |
| setindregs 35898 | Set (epsilon) induction. ... |
| setinds2regs 35899 | Principle of set induction... |
| noinfepfnregs 35900 | There are no infinite desc... |
| noinfepregs 35901 | There are no infinite desc... |
| tz9.1regs 35902 | Every set has a transitive... |
| unir1regs 35903 | The cumulative hierarchy o... |
| trssfihfregs 35904 | If every element in a tran... |
| eqhfregs 35905 | The class of all hereditar... |
| fineqvhfbregs 35906 | All sets are finite iff al... |
| axregs 35907 | Derivation of ~ ax-regs fr... |
| axsepg2 35908 | A generalization of ~ ax-s... |
| axsepg3 35909 | A generalization of ~ ax-s... |
| axsepg3ALT 35910 | Alternate proof of ~ axsep... |
| axsepg4 35911 | A generalization of ~ ax-s... |
| axsepg5 35912 | A generalization of ~ ax-s... |
| axnulg 35913 | A generalization of ~ ax-n... |
| axpowg 35914 | A generalization of ~ ax-p... |
| axpowg2 35915 | A generalization of ~ ax-p... |
| axpowg3 35916 | A generalization of ~ ax-p... |
| kardfn 35919 | The ` kard ` class is a fu... |
| kardval 35920 | The value of the ` kard ` ... |
| kardval2 35921 | The value of the ` kard ` ... |
| kard0 35922 | The ` kard ` cardinality o... |
| elkarden 35923 | Any member of the ` kard `... |
| kardeq0 35924 | Applying ` kard ` to a cla... |
| kardeng 35925 | Two sets are equinumerous ... |
| kardenir 35926 | If two sets are equinumero... |
| kard0b 35927 | The empty set is the only ... |
| kardsn 35928 | A singleton has cardinalit... |
| karddom 35929 | One set dominates another ... |
| kardsdom 35930 | One set strictly dominates... |
| kardexen 35931 | One set is equinumerous to... |
| kardcard2a 35932 | If two sets have equal non... |
| kardcard2b 35933 | If two sets have equal ` k... |
| kardcard2 35934 | Two numerable sets have eq... |
| ackardcard 35935 | The Axiom of Choice implie... |
| kardcard 35936 | Two sets have equal ` kard... |
| kardnnfi 35937 | The ` kard ` cardinal numb... |
| kardfi 35938 | The ` kard ` cardinal numb... |
| rankkardu 35939 | An upper bound on the rank... |
| 1enumkard 35940 | The Fundamental Theorem of... |
| gdlop1fv 35963 | The value of the first G&o... |
| gdlop2fv 35964 | The value of the second G&... |
| gdlop3fv 35965 | The value of the third G&o... |
| gdlop4fv 35966 | The value of the fourth G&... |
| gdlop5fv 35967 | The value of the fifth G&o... |
| gdlop6fv 35968 | The value of the sixth G&o... |
| gdlop7fv 35969 | The value of the seventh G... |
| gdlop8fv 35970 | The value of the eighth G&... |
| gdlopcfvlem 35971 | Lemma for ~ gdlopcfv2 thro... |
| gdlopcfv1 35972 | The value of the combined ... |
| gdlopcfv2 35973 | The value of the combined ... |
| gdlopcfv3 35974 | The value of the combined ... |
| gdlopcfv4 35975 | The value of the combined ... |
| gdlopcfv5 35976 | The value of the combined ... |
| gdlopcfv6 35977 | The value of the combined ... |
| gdlopcfv7 35978 | The value of the combined ... |
| gdlopcfv8 35979 | The value of the combined ... |
| lexoweon 35998 | Lexicographical order is a... |
| brlexod 35999 | Condition for two ordered ... |
| r0weseon 36000 | ` _R0 ` is a set-like well... |
| brr0d 36001 | Condition for two ordered ... |
| dmr0ss 36002 | The domain of ` _R0 ` is a... |
| r0maxss 36003 | If two ordered pairs are r... |
| j0iso 36004 | ` _J0 ` is the ` _R0 ` ord... |
| j00smo 36005 | ` _J0 ` is strictly monoto... |
| maxssj0 36006 | The value of ` _J0 ` is at... |
| j0ltalephlem 36007 | Lemma for ~ j0ltaleph . (... |
| j0ltaleph 36008 | The value of ` _J0 ` is le... |
| dfj2 36009 | Alternate definition of ` ... |
| jf1 36010 | ` _J ` is a one-to-one fun... |
| jf1o 36011 | ` _J ` is a bijection from... |
| dfk12 36012 | Alternate definition of ` ... |
| dfk22 36013 | Alternate definition of ` ... |
| dfk32 36014 | Alternate definition of ` ... |
| k1on 36015 | Values of the ` _K1 ` func... |
| k2on 36016 | Values of the ` _K2 ` func... |
| k39o 36017 | Values of the ` _K3 ` func... |
| jk1k2k3 36018 | Identity law for the ` _J ... |
| j0le 36019 | An upper bound on the valu... |
| k3n0j0lt 36020 | A strict upper bound on th... |
| k1le 36021 | The value of ` _K1 ` is bo... |
| k2le 36022 | The value of ` _K2 ` is bo... |
| k3n0k1lt 36023 | The value of ` _K1 ` is st... |
| k3n0k2lt 36024 | The value of ` _K2 ` is st... |
| jltalephlem 36025 | Lemma for ~ jltaleph . (C... |
| jltaleph 36026 | The value of ` _J ` is les... |
| fnlfvk30 36027 | The value of ` _FL ` when ... |
| fnlfvk3n0 36028 | The value of ` _FL ` when ... |
| gblacfnacd 36029 | If ` G ` is a global choic... |
| gblacfnac2d 36030 | If ` G ` is a global choic... |
| onvf1odlem1 36031 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem2 36032 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem3 36033 | Lemma for ~ onvf1od . The... |
| onvf1odlem4 36034 | Lemma for ~ onvf1od . If ... |
| onvf1od 36035 | If ` G ` is a global choic... |
| vonf1wev 36036 | If ` F ` maps the universe... |
| vonf1owev 36037 | If ` F ` is a bijection fr... |
| vonf1owevOLD 36038 | Obsolete version of ~ vonf... |
| wevgblacfn 36039 | If ` R ` is a well-orderin... |
| vonf1osev 36040 | If ` F ` is a bijection fr... |
| wevsev 36041 | If ` R ` is a well-orderin... |
| wevonprcf1o 36042 | If ` R ` is a set-like wel... |
| vonf1oonf1 36043 | If ` F ` is a bijection fr... |
| vonf1onprcf1ac 36044 | If ` F ` maps the universe... |
| onprcf1acwevdlem1 36045 | Lemma for ~ onprcf1acwevd ... |
| onprcf1acwevdlem2 36046 | Lemma for ~ onprcf1acwevd ... |
| onprcf1acwevd 36047 | If ` F ` maps the ordinals... |
| prconf1acwevd 36048 | If ` F ` maps the proper c... |
| vonf1oonfo 36049 | If ` F ` is a bijection fr... |
| onvfowev 36050 | If ` F ` maps the ordinals... |
| onf1onf1or 36051 | If ` F ` maps ` A ` one-to... |
| onf1oracwevd 36052 | If ` F ` maps the proper c... |
| onprcf1of1orad 36053 | If ` F ` is a bijection fr... |
| onprcf1of1orbd 36054 | If ` F ` is a bijection fr... |
| onprcf1of1ord 36055 | If ` F ` is a bijection fr... |
| f1oronf1or 36056 | If ` F ` maps ` A ` one-to... |
| zltp1ne 36057 | Integer ordering relation.... |
| nnltp1ne 36058 | Positive integer ordering ... |
| nn0ltp1ne 36059 | Nonnegative integer orderi... |
| fisshasheq 36060 | A finite set is equal to i... |
| 1enum 36061 | The Fundamental Theorem of... |
| cplgredgex 36062 | Any two (distinct) vertice... |
| cusgredgex 36063 | Any two (distinct) vertice... |
| cusgredgex2 36064 | Any two distinct vertices ... |
| revwlkb 36065 | Two words represent a walk... |
| usgrgt2cycl 36066 | A non-trivial cycle in a s... |
| usgrcyclgt2v 36067 | A simple graph with a non-... |
| cusgr3cyclex 36068 | Every complete simple grap... |
| 2cycl2d 36069 | Construction of a 2-cycle ... |
| acycgr0v 36070 | A null graph (with no vert... |
| acycgr1v 36071 | A multigraph with one vert... |
| acycgr2v 36072 | A simple graph with two ve... |
| prclisacycgr 36073 | A proper class (representi... |
| acycgrislfgr 36074 | An acyclic hypergraph is a... |
| upgracycumgr 36075 | An acyclic pseudograph is ... |
| umgracycusgr 36076 | An acyclic multigraph is a... |
| upgracycusgr 36077 | An acyclic pseudograph is ... |
| cusgracyclt3v 36078 | A complete simple graph is... |
| pthacycspth 36079 | A path in an acyclic graph... |
| acycgrsubgr 36080 | The subgraph of an acyclic... |
| srepdflem1 36085 | Lemma for ~ srepdf . (Con... |
| srepdflem2 36086 | Lemma for ~ srepdf . (Con... |
| srepdf 36087 | Derive the strong version ... |
| smrepsrep 36088 | Derive the strong version ... |
| vsmext 36089 | The universe is a standard... |
| vsmrep 36090 | The universe is a standard... |
| quartfull 36097 | The quartic equation, writ... |
| deranglem 36098 | Lemma for derangements. (... |
| derangval 36099 | Define the derangement fun... |
| derangf 36100 | The derangement number is ... |
| derang0 36101 | The derangement number of ... |
| derangsn 36102 | The derangement number of ... |
| derangenlem 36103 | One half of ~ derangen . ... |
| derangen 36104 | The derangement number is ... |
| subfacval 36105 | The subfactorial is define... |
| derangen2 36106 | Write the derangement numb... |
| subfacf 36107 | The subfactorial is a func... |
| subfaclefac 36108 | The subfactorial is less t... |
| subfac0 36109 | The subfactorial at zero. ... |
| subfac1 36110 | The subfactorial at one. ... |
| subfacp1lem1 36111 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 36112 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 36113 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 36114 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 36115 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 36116 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 36117 | Lemma for ~ subfacp1 . By... |
| subfacp1 36118 | A two-term recurrence for ... |
| subfacval2 36119 | A closed-form expression f... |
| subfaclim 36120 | The subfactorial converges... |
| subfacval3 36121 | Another closed form expres... |
| derangfmla 36122 | The derangements formula, ... |
| erdszelem1 36123 | Lemma for ~ erdsze . (Con... |
| erdszelem2 36124 | Lemma for ~ erdsze . (Con... |
| erdszelem3 36125 | Lemma for ~ erdsze . (Con... |
| erdszelem4 36126 | Lemma for ~ erdsze . (Con... |
| erdszelem5 36127 | Lemma for ~ erdsze . (Con... |
| erdszelem6 36128 | Lemma for ~ erdsze . (Con... |
| erdszelem7 36129 | Lemma for ~ erdsze . (Con... |
| erdszelem8 36130 | Lemma for ~ erdsze . (Con... |
| erdszelem9 36131 | Lemma for ~ erdsze . (Con... |
| erdszelem10 36132 | Lemma for ~ erdsze . (Con... |
| erdszelem11 36133 | Lemma for ~ erdsze . (Con... |
| erdsze 36134 | The Erdős-Szekeres th... |
| erdsze2lem1 36135 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 36136 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 36137 | Generalize the statement o... |
| kur14lem1 36138 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 36139 | Lemma for ~ kur14 . Write... |
| kur14lem3 36140 | Lemma for ~ kur14 . A clo... |
| kur14lem4 36141 | Lemma for ~ kur14 . Compl... |
| kur14lem5 36142 | Lemma for ~ kur14 . Closu... |
| kur14lem6 36143 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 36144 | Lemma for ~ kur14 : main p... |
| kur14lem8 36145 | Lemma for ~ kur14 . Show ... |
| kur14lem9 36146 | Lemma for ~ kur14 . Since... |
| kur14lem10 36147 | Lemma for ~ kur14 . Disch... |
| kur14 36148 | Kuratowski's closure-compl... |
| ispconn 36155 | The property of being a pa... |
| pconncn 36156 | The property of being a pa... |
| pconntop 36157 | A simply connected space i... |
| issconn 36158 | The property of being a si... |
| sconnpconn 36159 | A simply connected space i... |
| sconntop 36160 | A simply connected space i... |
| sconnpht 36161 | A closed path in a simply ... |
| cnpconn 36162 | An image of a path-connect... |
| pconnconn 36163 | A path-connected space is ... |
| txpconn 36164 | The topological product of... |
| ptpconn 36165 | The topological product of... |
| indispconn 36166 | The indiscrete topology (o... |
| connpconn 36167 | A connected and locally pa... |
| qtoppconn 36168 | A quotient of a path-conne... |
| pconnpi1 36169 | All fundamental groups in ... |
| sconnpht2 36170 | Any two paths in a simply ... |
| sconnpi1 36171 | A path-connected topologic... |
| txsconnlem 36172 | Lemma for ~ txsconn . (Co... |
| txsconn 36173 | The topological product of... |
| cvxpconn 36174 | A convex subset of the com... |
| cvxsconn 36175 | A convex subset of the com... |
| blsconn 36176 | An open ball in the comple... |
| cnllysconn 36177 | The topology of the comple... |
| resconn 36178 | A subset of ` RR ` is simp... |
| ioosconn 36179 | An open interval is simply... |
| iccsconn 36180 | A closed interval is simpl... |
| retopsconn 36181 | The real numbers are simpl... |
| iccllysconn 36182 | A closed interval is local... |
| rellysconn 36183 | The real numbers are local... |
| iisconn 36184 | The unit interval is simpl... |
| iillysconn 36185 | The unit interval is local... |
| iinllyconn 36186 | The unit interval is local... |
| fncvm 36189 | Lemma for covering maps. ... |
| cvmscbv 36190 | Change bound variables in ... |
| iscvm 36191 | The property of being a co... |
| cvmtop1 36192 | Reverse closure for a cove... |
| cvmtop2 36193 | Reverse closure for a cove... |
| cvmcn 36194 | A covering map is a contin... |
| cvmcov 36195 | Property of a covering map... |
| cvmsrcl 36196 | Reverse closure for an eve... |
| cvmsi 36197 | One direction of ~ cvmsval... |
| cvmsval 36198 | Elementhood in the set ` S... |
| cvmsss 36199 | An even covering is a subs... |
| cvmsn0 36200 | An even covering is nonemp... |
| cvmsuni 36201 | An even covering of ` U ` ... |
| cvmsdisj 36202 | An even covering of ` U ` ... |
| cvmshmeo 36203 | Every element of an even c... |
| cvmsf1o 36204 | ` F ` , localized to an el... |
| cvmscld 36205 | The sets of an even coveri... |
| cvmsss2 36206 | An open subset of an evenl... |
| cvmcov2 36207 | The covering map property ... |
| cvmseu 36208 | Every element in ` U. T ` ... |
| cvmsiota 36209 | Identify the unique elemen... |
| cvmopnlem 36210 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 36211 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 36212 | A covering map is an open ... |
| cvmliftmolem1 36213 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 36214 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 36215 | A lift of a continuous fun... |
| cvmliftmo 36216 | A lift of a continuous fun... |
| cvmliftlem1 36217 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 36218 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 36219 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 36220 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 36221 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 36222 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 36223 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 36224 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 36225 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 36226 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 36227 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 36228 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 36229 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 36230 | Lemma for ~ cvmlift . Dis... |
| cvmlift 36231 | One of the important prope... |
| cvmfo 36232 | A covering map is an onto ... |
| cvmliftiota 36233 | Write out a function ` H `... |
| cvmlift2lem1 36234 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 36235 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 36236 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 36237 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 36238 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 36239 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 36240 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 36241 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 36242 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 36243 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 36244 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 36245 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 36246 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 36247 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 36248 | A two-dimensional version ... |
| cvmliftphtlem 36249 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 36250 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 36251 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 36252 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 36253 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 36254 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 36255 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 36256 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 36257 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 36258 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 36259 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 36260 | A general version of ~ cvm... |
| snmlff 36261 | The function ` F ` from ~ ... |
| snmlfval 36262 | The function ` F ` from ~ ... |
| snmlval 36263 | The property " ` A ` is si... |
| snmlflim 36264 | If ` A ` is simply normal,... |
| goel 36279 | A "Godel-set of membership... |
| goelel3xp 36280 | A "Godel-set of membership... |
| goeleq12bg 36281 | Two "Godel-set of membersh... |
| gonafv 36282 | The "Godel-set for the She... |
| goaleq12d 36283 | Equality of the "Godel-set... |
| gonanegoal 36284 | The Godel-set for the Shef... |
| satf 36285 | The satisfaction predicate... |
| satfsucom 36286 | The satisfaction predicate... |
| satfn 36287 | The satisfaction predicate... |
| satom 36288 | The satisfaction predicate... |
| satfvsucom 36289 | The satisfaction predicate... |
| satfv0 36290 | The value of the satisfact... |
| satfvsuclem1 36291 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 36292 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 36293 | The value of the satisfact... |
| satfv1lem 36294 | Lemma for ~ satfv1 . (Con... |
| satfv1 36295 | The value of the satisfact... |
| satfsschain 36296 | The binary relation of a s... |
| satfvsucsuc 36297 | The satisfaction predicate... |
| satfbrsuc 36298 | The binary relation of a s... |
| satfrel 36299 | The value of the satisfact... |
| satfdmlem 36300 | Lemma for ~ satfdm . (Con... |
| satfdm 36301 | The domain of the satisfac... |
| satfrnmapom 36302 | The range of the satisfact... |
| satfv0fun 36303 | The value of the satisfact... |
| satf0 36304 | The satisfaction predicate... |
| satf0sucom 36305 | The satisfaction predicate... |
| satf00 36306 | The value of the satisfact... |
| satf0suclem 36307 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 36308 | The value of the satisfact... |
| satf0op 36309 | An element of a value of t... |
| satf0n0 36310 | The value of the satisfact... |
| sat1el2xp 36311 | The first component of an ... |
| fmlafv 36312 | The valid Godel formulas o... |
| fmla 36313 | The set of all valid Godel... |
| fmla0 36314 | The valid Godel formulas o... |
| fmla0xp 36315 | The valid Godel formulas o... |
| fmlasuc0 36316 | The valid Godel formulas o... |
| fmlafvel 36317 | A class is a valid Godel f... |
| fmlasuc 36318 | The valid Godel formulas o... |
| fmla1 36319 | The valid Godel formulas o... |
| isfmlasuc 36320 | The characterization of a ... |
| fmlasssuc 36321 | The Godel formulas of heig... |
| fmlaomn0 36322 | The empty set is not a God... |
| fmlan0 36323 | The empty set is not a God... |
| gonan0 36324 | The "Godel-set of NAND" is... |
| goaln0 36325 | The "Godel-set of universa... |
| gonarlem 36326 | Lemma for ~ gonar (inducti... |
| gonar 36327 | If the "Godel-set of NAND"... |
| goalrlem 36328 | Lemma for ~ goalr (inducti... |
| goalr 36329 | If the "Godel-set of unive... |
| fmla0disjsuc 36330 | The set of valid Godel for... |
| fmlasucdisj 36331 | The valid Godel formulas o... |
| satfdmfmla 36332 | The domain of the satisfac... |
| satffunlem 36333 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 36334 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 36335 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 36336 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 36337 | The domain of an ordered p... |
| satffunlem2lem2 36338 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 36339 | Lemma 1 for ~ satffun : in... |
| satffunlem2 36340 | Lemma 2 for ~ satffun : in... |
| satffun 36341 | The value of the satisfact... |
| satff 36342 | The satisfaction predicate... |
| satfun 36343 | The satisfaction predicate... |
| satfvel 36344 | An element of the value of... |
| satfv0fvfmla0 36345 | The value of the satisfact... |
| satefv 36346 | The simplified satisfactio... |
| sate0 36347 | The simplified satisfactio... |
| satef 36348 | The simplified satisfactio... |
| sate0fv0 36349 | A simplified satisfaction ... |
| satefvfmla0 36350 | The simplified satisfactio... |
| sategoelfvb 36351 | Characterization of a valu... |
| sategoelfv 36352 | Condition of a valuation `... |
| ex-sategoelel 36353 | Example of a valuation of ... |
| ex-sategoel 36354 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 36355 | The value of the satisfact... |
| 2goelgoanfmla1 36356 | Two Godel-sets of membersh... |
| satefvfmla1 36357 | The simplified satisfactio... |
| ex-sategoelelomsuc 36358 | Example of a valuation of ... |
| ex-sategoelel12 36359 | Example of a valuation of ... |
| prv 36360 | The "proves" relation on a... |
| elnanelprv 36361 | The wff ` ( A e. B -/\ B e... |
| prv0 36362 | Every wff encoded as ` U `... |
| prv1n 36363 | No wff encoded as a Godel-... |
| mvtval 36432 | The set of variable typeco... |
| mrexval 36433 | The set of "raw expression... |
| mexval 36434 | The set of expressions, wh... |
| mexval2 36435 | The set of expressions, wh... |
| mdvval 36436 | The set of disjoint variab... |
| mvrsval 36437 | The set of variables in an... |
| mvrsfpw 36438 | The set of variables in an... |
| mrsubffval 36439 | The substitution of some v... |
| mrsubfval 36440 | The substitution of some v... |
| mrsubval 36441 | The substitution of some v... |
| mrsubcv 36442 | The value of a substituted... |
| mrsubvr 36443 | The value of a substituted... |
| mrsubff 36444 | A substitution is a functi... |
| mrsubrn 36445 | Although it is defined for... |
| mrsubff1 36446 | When restricted to complet... |
| mrsubff1o 36447 | When restricted to complet... |
| mrsub0 36448 | The value of the substitut... |
| mrsubf 36449 | A substitution is a functi... |
| mrsubccat 36450 | Substitution distributes o... |
| mrsubcn 36451 | A substitution does not ch... |
| elmrsubrn 36452 | Characterization of the su... |
| mrsubco 36453 | The composition of two sub... |
| mrsubvrs 36454 | The set of variables in a ... |
| msubffval 36455 | A substitution applied to ... |
| msubfval 36456 | A substitution applied to ... |
| msubval 36457 | A substitution applied to ... |
| msubrsub 36458 | A substitution applied to ... |
| msubty 36459 | The type of a substituted ... |
| elmsubrn 36460 | Characterization of substi... |
| msubrn 36461 | Although it is defined for... |
| msubff 36462 | A substitution is a functi... |
| msubco 36463 | The composition of two sub... |
| msubf 36464 | A substitution is a functi... |
| mvhfval 36465 | Value of the function mapp... |
| mvhval 36466 | Value of the function mapp... |
| mpstval 36467 | A pre-statement is an orde... |
| elmpst 36468 | Property of being a pre-st... |
| msrfval 36469 | Value of the reduct of a p... |
| msrval 36470 | Value of the reduct of a p... |
| mpstssv 36471 | A pre-statement is an orde... |
| mpst123 36472 | Decompose a pre-statement ... |
| mpstrcl 36473 | The elements of a pre-stat... |
| msrf 36474 | The reduct of a pre-statem... |
| msrrcl 36475 | If ` X ` and ` Y ` have th... |
| mstaval 36476 | Value of the set of statem... |
| msrid 36477 | The reduct of a statement ... |
| msrfo 36478 | The reduct of a pre-statem... |
| mstapst 36479 | A statement is a pre-state... |
| elmsta 36480 | Property of being a statem... |
| ismfs 36481 | A formal system is a tuple... |
| mfsdisj 36482 | The constants and variable... |
| mtyf2 36483 | The type function maps var... |
| mtyf 36484 | The type function maps var... |
| mvtss 36485 | The set of variable typeco... |
| maxsta 36486 | An axiom is a statement. ... |
| mvtinf 36487 | Each variable typecode has... |
| msubff1 36488 | When restricted to complet... |
| msubff1o 36489 | When restricted to complet... |
| mvhf 36490 | The function mapping varia... |
| mvhf1 36491 | The function mapping varia... |
| msubvrs 36492 | The set of variables in a ... |
| mclsrcl 36493 | Reverse closure for the cl... |
| mclsssvlem 36494 | Lemma for ~ mclsssv . (Co... |
| mclsval 36495 | The function mapping varia... |
| mclsssv 36496 | The closure of a set of ex... |
| ssmclslem 36497 | Lemma for ~ ssmcls . (Con... |
| vhmcls 36498 | All variable hypotheses ar... |
| ssmcls 36499 | The original expressions a... |
| ss2mcls 36500 | The closure is monotonic u... |
| mclsax 36501 | The closure is closed unde... |
| mclsind 36502 | Induction theorem for clos... |
| mppspstlem 36503 | Lemma for ~ mppspst . (Co... |
| mppsval 36504 | Definition of a provable p... |
| elmpps 36505 | Definition of a provable p... |
| mppspst 36506 | A provable pre-statement i... |
| mthmval 36507 | A theorem is a pre-stateme... |
| elmthm 36508 | A theorem is a pre-stateme... |
| mthmi 36509 | A statement whose reduct i... |
| mthmsta 36510 | A theorem is a pre-stateme... |
| mppsthm 36511 | A provable pre-statement i... |
| mthmblem 36512 | Lemma for ~ mthmb . (Cont... |
| mthmb 36513 | If two statements have the... |
| mthmpps 36514 | Given a theorem, there is ... |
| mclsppslem 36515 | The closure is closed unde... |
| mclspps 36516 | The closure is closed unde... |
| rexxfr3d 36570 | Transfer existential quant... |
| rexxfr3dALT 36571 | Longer proof of ~ rexxfr3d... |
| rspssbasd 36572 | The span of a set of ring ... |
| ellcsrspsn 36573 | Membership in a left coset... |
| ply1divalg3 36574 | Uniqueness of polynomial r... |
| r1peuqusdeg1 36575 | Uniqueness of polynomial r... |
| problem1 36597 | Practice problem 1. Clues... |
| problem2 36598 | Practice problem 2. Clues... |
| problem3 36599 | Practice problem 3. Clues... |
| problem4 36600 | Practice problem 4. Clues... |
| problem5 36601 | Practice problem 5. Clues... |
| quad3 36602 | Variant of quadratic equat... |
| climuzcnv 36603 | Utility lemma to convert b... |
| sinccvglem 36604 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 36605 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 36606 | The circumference of a cir... |
| elfzm12 36607 | Membership in a curtailed ... |
| nn0seqcvg 36608 | A strictly-decreasing nonn... |
| lediv2aALT 36609 | Division of both sides of ... |
| abs2sqlei 36610 | The absolute values of two... |
| abs2sqlti 36611 | The absolute values of two... |
| abs2sqle 36612 | The absolute values of two... |
| abs2sqlt 36613 | The absolute values of two... |
| abs2difi 36614 | Difference of absolute val... |
| abs2difabsi 36615 | Absolute value of differen... |
| 2thALT 36616 | Alternate proof of ~ 2th .... |
| orbi2iALT 36617 | Alternate proof of ~ orbi2... |
| pm3.48ALT 36618 | Alternate proof of ~ pm3.4... |
| 3jcadALT 36619 | Alternate proof of ~ 3jcad... |
| currybi 36620 | Biconditional version of C... |
| antnest 36621 | Suppose ` ph ` , ` ps ` ar... |
| antnestlaw3lem 36622 | Lemma for ~ antnestlaw3 . ... |
| antnestlaw1 36623 | A law of nested antecedent... |
| antnestlaw2 36624 | A law of nested antecedent... |
| antnestlaw3 36625 | A law of nested antecedent... |
| antnestALT 36626 | Alternative proof of ~ ant... |
| mirimanoffr 36627 | Relative version of ~ miri... |
| mirimanoff 36628 | Mirimanoff's paradox: The... |
| mirimanofffr 36629 | Relative version of ~ miri... |
| mirimanofff 36630 | Mirimanoff's paradox ~ mir... |
| onprcr 36631 | Relative version of ~ onpr... |
| onprcALT 36632 | Alternate proof of ~ onprc... |
| cardprcr 36633 | Relative version of ~ card... |
| cardprcALT 36634 | Alternate proof of ~ cardp... |
| elpssex 36635 | A class has an element jus... |
| elpssmo 36636 | Uniqueness of elements is ... |
| elpsseu 36637 | A class has a unique eleme... |
| axextprim 36644 | ~ ax-ext without distinct ... |
| axrepprim 36645 | ~ ax-rep without distinct ... |
| axunprim 36646 | ~ ax-un without distinct v... |
| axpowprim 36647 | ~ ax-pow without distinct ... |
| axregprim 36648 | ~ ax-reg without distinct ... |
| axinfprim 36649 | ~ ax-inf without distinct ... |
| axacprim 36650 | ~ ax-ac without distinct v... |
| untelirr 36651 | We call a class "untanged"... |
| untuni 36652 | The union of a class is un... |
| untsucf 36653 | If a class is untangled, t... |
| unt0 36654 | The empty set is untangled... |
| untint 36655 | If there is an untangled e... |
| efrunt 36656 | If ` A ` is well-founded b... |
| untangtr 36657 | A transitive class is unta... |
| 3jaodd 36658 | Double deduction form of ~... |
| 3orit 36659 | Closed form of ~ 3ori . (... |
| biimpexp 36660 | A biconditional in the ant... |
| nepss 36661 | Two classes are unequal if... |
| 3ccased 36662 | Triple disjunction form of... |
| dfso3 36663 | Expansion of the definitio... |
| brtpid1 36664 | A binary relation involvin... |
| brtpid2 36665 | A binary relation involvin... |
| brtpid3 36666 | A binary relation involvin... |
| iota5f 36667 | A method for computing iot... |
| jath 36668 | Closed form of ~ ja . Pro... |
| xpab 36669 | Cartesian product of two c... |
| nnuni 36670 | The union of a finite ordi... |
| sqdivzi 36671 | Distribution of square ove... |
| supfz 36672 | The supremum of a finite s... |
| inffz 36673 | The infimum of a finite se... |
| fz0n 36674 | The sequence ` ( 0 ... ( N... |
| shftvalg 36675 | Value of a sequence shifte... |
| divcnvlin 36676 | Limit of the ratio of two ... |
| climlec3 36677 | Comparison of a constant t... |
| iexpire 36678 | ` _i ` raised to itself is... |
| bcneg1 36679 | The binomial coefficient o... |
| bcm1nt 36680 | The proportion of one bino... |
| bcprod 36681 | A product identity for bin... |
| bccolsum 36682 | A column-sum rule for bino... |
| iprodefisumlem 36683 | Lemma for ~ iprodefisum . ... |
| iprodefisum 36684 | Applying the exponential f... |
| iprodgam 36685 | An infinite product versio... |
| faclimlem1 36686 | Lemma for ~ faclim . Clos... |
| faclimlem2 36687 | Lemma for ~ faclim . Show... |
| faclimlem3 36688 | Lemma for ~ faclim . Alge... |
| faclim 36689 | An infinite product expres... |
| iprodfac 36690 | An infinite product expres... |
| faclim2 36691 | Another factorial limit du... |
| gcd32 36692 | Swap the second and third ... |
| gcdabsorb 36693 | Absorption law for gcd. (... |
| dftr6 36694 | A potential definition of ... |
| coep 36695 | Composition with the membe... |
| coepr 36696 | Composition with the conve... |
| dffr5 36697 | A quantifier-free definiti... |
| dfso2 36698 | Quantifier-free definition... |
| br8 36699 | Substitution for an eight-... |
| br6 36700 | Substitution for a six-pla... |
| br4 36701 | Substitution for a four-pl... |
| cnvco1 36702 | Another distributive law o... |
| cnvco2 36703 | Another distributive law o... |
| eldm3 36704 | Quantifier-free definition... |
| elrn3 36705 | Quantifier-free definition... |
| pocnv 36706 | The converse of a partial ... |
| socnv 36707 | The converse of a strict o... |
| elintfv 36708 | Membership in an intersect... |
| funpsstri 36709 | A condition for subset tri... |
| fundmpss 36710 | If a class ` F ` is a prop... |
| funsseq 36711 | Given two functions with e... |
| funbreq 36712 | An equality condition for ... |
| br1steq 36713 | Uniqueness condition for t... |
| br2ndeq 36714 | Uniqueness condition for t... |
| dfdm5 36715 | Definition of domain in te... |
| dfrn5 36716 | Definition of range in ter... |
| opelco3 36717 | Alternate way of saying th... |
| elima4 36718 | Quantifier-free expression... |
| fv1stcnv 36719 | The value of the converse ... |
| fv2ndcnv 36720 | The value of the converse ... |
| elpotr 36721 | A class of transitive sets... |
| dford5reg 36722 | Given ~ ax-reg , an ordina... |
| dfon2lem1 36723 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 36724 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 36725 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 36726 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 36727 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 36728 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 36729 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 36730 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 36731 | Lemma for ~ dfon2 . A cla... |
| dfon2 36732 | ` On ` consists of all set... |
| rdgprc0 36733 | The value of the recursive... |
| rdgprc 36734 | The value of the recursive... |
| dfrdg2 36735 | Alternate definition of th... |
| dfrdg3 36736 | Generalization of ~ dfrdg2... |
| axextdfeq 36737 | A version of ~ ax-ext for ... |
| ax8dfeq 36738 | A version of ~ ax-8 for us... |
| axextdist 36739 | ~ ax-ext with distinctors ... |
| axextbdist 36740 | ~ axextb with distinctors ... |
| 19.12b 36741 | Version of ~ 19.12vv with ... |
| exnel 36742 | There is always a set not ... |
| distel 36743 | Distinctors in terms of me... |
| axextndbi 36744 | ~ axextnd as a bicondition... |
| hbntg 36745 | A more general form of ~ h... |
| hbimtg 36746 | A more general and closed ... |
| hbaltg 36747 | A more general and closed ... |
| hbng 36748 | A more general form of ~ h... |
| hbimg 36749 | A more general form of ~ h... |
| wsuceq123 36754 | Equality theorem for well-... |
| wsuceq1 36755 | Equality theorem for well-... |
| wsuceq2 36756 | Equality theorem for well-... |
| wsuceq3 36757 | Equality theorem for well-... |
| nfwsuc 36758 | Bound-variable hypothesis ... |
| wlimeq12 36759 | Equality theorem for the l... |
| wlimeq1 36760 | Equality theorem for the l... |
| wlimeq2 36761 | Equality theorem for the l... |
| nfwlim 36762 | Bound-variable hypothesis ... |
| elwlim 36763 | Membership in the limit cl... |
| wzel 36764 | The zero of a well-founded... |
| wsuclem 36765 | Lemma for the supremum pro... |
| wsucex 36766 | Existence theorem for well... |
| wsuccl 36767 | If ` X ` is a set with an ... |
| wsuclb 36768 | A well-founded successor i... |
| wlimss 36769 | The class of limit points ... |
| txpss3v 36818 | A tail Cartesian product i... |
| txprel 36819 | A tail Cartesian product i... |
| brtxp 36820 | Characterize a ternary rel... |
| brtxp2 36821 | The binary relation over a... |
| dfpprod2 36822 | Expanded definition of par... |
| pprodcnveq 36823 | A converse law for paralle... |
| pprodss4v 36824 | The parallel product is a ... |
| brpprod 36825 | Characterize a quaternary ... |
| brpprod3a 36826 | Condition for parallel pro... |
| brpprod3b 36827 | Condition for parallel pro... |
| relsset 36828 | The subset class is a bina... |
| brsset 36829 | For sets, the ` SSet ` bin... |
| idsset 36830 | ` _I ` is equal to the int... |
| eltrans 36831 | Membership in the class of... |
| dfon3 36832 | A quantifier-free definiti... |
| dfon4 36833 | Another quantifier-free de... |
| brtxpsd 36834 | Expansion of a common form... |
| brtxpsd2 36835 | Another common abbreviatio... |
| brtxpsd3 36836 | A third common abbreviatio... |
| relbigcup 36837 | The ` Bigcup ` relationshi... |
| brbigcup 36838 | Binary relation over ` Big... |
| dfbigcup2 36839 | ` Bigcup ` using maps-to n... |
| fobigcup 36840 | ` Bigcup ` maps the univer... |
| fnbigcup 36841 | ` Bigcup ` is a function o... |
| fvbigcup 36842 | For sets, ` Bigcup ` yield... |
| elfix 36843 | Membership in the fixpoint... |
| elfix2 36844 | Alternative membership in ... |
| dffix2 36845 | The fixpoints of a class i... |
| fixssdm 36846 | The fixpoints of a class a... |
| fixssrn 36847 | The fixpoints of a class a... |
| fixcnv 36848 | The fixpoints of a class a... |
| fixun 36849 | The fixpoint operator dist... |
| ellimits 36850 | Membership in the class of... |
| limitssson 36851 | The class of all limit ord... |
| dfom5b 36852 | A quantifier-free definiti... |
| sscoid 36853 | A condition for subset and... |
| dffun10 36854 | Another potential definiti... |
| elfuns 36855 | Membership in the class of... |
| elfunsg 36856 | Closed form of ~ elfuns . ... |
| brsingle 36857 | The binary relation form o... |
| elsingles 36858 | Membership in the class of... |
| fnsingle 36859 | The singleton relationship... |
| fvsingle 36860 | The value of the singleton... |
| dfsingles2 36861 | Alternate definition of th... |
| snelsingles 36862 | A singleton is a member of... |
| dfiota3 36863 | A definition of iota using... |
| dffv5 36864 | Another quantifier-free de... |
| unisnif 36865 | Express union of singleton... |
| brimage 36866 | Binary relation form of th... |
| brimageg 36867 | Closed form of ~ brimage .... |
| funimage 36868 | ` Image A ` is a function.... |
| fnimage 36869 | ` Image R ` is a function ... |
| imageval 36870 | The image functor in maps-... |
| fvimage 36871 | Value of the image functor... |
| brcart 36872 | Binary relation form of th... |
| brdomain 36873 | Binary relation form of th... |
| brrange 36874 | Binary relation form of th... |
| brdomaing 36875 | Closed form of ~ brdomain ... |
| brrangeg 36876 | Closed form of ~ brrange .... |
| brimg 36877 | Binary relation form of th... |
| brapply 36878 | Binary relation form of th... |
| brcup 36879 | Binary relation form of th... |
| brcap 36880 | Binary relation form of th... |
| lemsuccf 36881 | Lemma for unfolding differ... |
| brsuccf 36882 | Binary relation form of th... |
| dfsuccf2 36883 | Alternate definition of Sc... |
| funpartlem 36884 | Lemma for ~ funpartfun . ... |
| funpartfun 36885 | The functional part of ` F... |
| funpartss 36886 | The functional part of ` F... |
| funpartfv 36887 | The function value of the ... |
| fullfunfnv 36888 | The full functional part o... |
| fullfunfv 36889 | The function value of the ... |
| brfullfun 36890 | A binary relation form con... |
| brrestrict 36891 | Binary relation form of th... |
| dfrecs2 36892 | A quantifier-free definiti... |
| dfrdg4 36893 | A quantifier-free definiti... |
| dfint3 36894 | Quantifier-free definition... |
| imagesset 36895 | The Image functor applied ... |
| brub 36896 | Binary relation form of th... |
| brlb 36897 | Binary relation form of th... |
| dffr7 36898 | Alternate quantifier-free ... |
| altopex 36903 | Alternative ordered pairs ... |
| altopthsn 36904 | Two alternate ordered pair... |
| altopeq12 36905 | Equality for alternate ord... |
| altopeq1 36906 | Equality for alternate ord... |
| altopeq2 36907 | Equality for alternate ord... |
| altopth1 36908 | Equality of the first memb... |
| altopth2 36909 | Equality of the second mem... |
| altopthg 36910 | Alternate ordered pair the... |
| altopthbg 36911 | Alternate ordered pair the... |
| altopth 36912 | The alternate ordered pair... |
| altopthb 36913 | Alternate ordered pair the... |
| altopthc 36914 | Alternate ordered pair the... |
| altopthd 36915 | Alternate ordered pair the... |
| altxpeq1 36916 | Equality for alternate Car... |
| altxpeq2 36917 | Equality for alternate Car... |
| elaltxp 36918 | Membership in alternate Ca... |
| altopelaltxp 36919 | Alternate ordered pair mem... |
| altxpsspw 36920 | An inclusion rule for alte... |
| altxpexg 36921 | The alternate Cartesian pr... |
| rankaltopb 36922 | Compute the rank of an alt... |
| nfaltop 36923 | Bound-variable hypothesis ... |
| sbcaltop 36924 | Distribution of class subs... |
| cgrrflx2d 36927 | Deduction form of ~ axcgrr... |
| cgrtr4d 36928 | Deduction form of ~ axcgrt... |
| cgrtr4and 36929 | Deduction form of ~ axcgrt... |
| cgrrflx 36930 | Reflexivity law for congru... |
| cgrrflxd 36931 | Deduction form of ~ cgrrfl... |
| cgrcomim 36932 | Congruence commutes on the... |
| cgrcom 36933 | Congruence commutes betwee... |
| cgrcomand 36934 | Deduction form of ~ cgrcom... |
| cgrtr 36935 | Transitivity law for congr... |
| cgrtrand 36936 | Deduction form of ~ cgrtr ... |
| cgrtr3 36937 | Transitivity law for congr... |
| cgrtr3and 36938 | Deduction form of ~ cgrtr3... |
| cgrcoml 36939 | Congruence commutes on the... |
| cgrcomr 36940 | Congruence commutes on the... |
| cgrcomlr 36941 | Congruence commutes on bot... |
| cgrcomland 36942 | Deduction form of ~ cgrcom... |
| cgrcomrand 36943 | Deduction form of ~ cgrcom... |
| cgrcomlrand 36944 | Deduction form of ~ cgrcom... |
| cgrtriv 36945 | Degenerate segments are co... |
| cgrid2 36946 | Identity law for congruenc... |
| cgrdegen 36947 | Two congruent segments are... |
| brofs 36948 | Binary relation form of th... |
| 5segofs 36949 | Rephrase ~ ax5seg using th... |
| ofscom 36950 | The outer five segment pre... |
| cgrextend 36951 | Link congruence over a pai... |
| cgrextendand 36952 | Deduction form of ~ cgrext... |
| segconeq 36953 | Two points that satisfy th... |
| segconeu 36954 | Existential uniqueness ver... |
| btwntriv2 36955 | Betweenness always holds f... |
| btwncomim 36956 | Betweenness commutes. Imp... |
| btwncom 36957 | Betweenness commutes. (Co... |
| btwncomand 36958 | Deduction form of ~ btwnco... |
| btwntriv1 36959 | Betweenness always holds f... |
| btwnswapid 36960 | If you can swap the first ... |
| btwnswapid2 36961 | If you can swap arguments ... |
| btwnintr 36962 | Inner transitivity law for... |
| btwnexch3 36963 | Exchange the first endpoin... |
| btwnexch3and 36964 | Deduction form of ~ btwnex... |
| btwnouttr2 36965 | Outer transitivity law for... |
| btwnexch2 36966 | Exchange the outer point o... |
| btwnouttr 36967 | Outer transitivity law for... |
| btwnexch 36968 | Outer transitivity law for... |
| btwnexchand 36969 | Deduction form of ~ btwnex... |
| btwndiff 36970 | There is always a ` c ` di... |
| trisegint 36971 | A line segment between two... |
| funtransport 36974 | The ` TransportTo ` relati... |
| fvtransport 36975 | Calculate the value of the... |
| transportcl 36976 | Closure law for segment tr... |
| transportprops 36977 | Calculate the defining pro... |
| brifs 36986 | Binary relation form of th... |
| ifscgr 36987 | Inner five segment congrue... |
| cgrsub 36988 | Removing identical parts f... |
| brcgr3 36989 | Binary relation form of th... |
| cgr3permute3 36990 | Permutation law for three-... |
| cgr3permute1 36991 | Permutation law for three-... |
| cgr3permute2 36992 | Permutation law for three-... |
| cgr3permute4 36993 | Permutation law for three-... |
| cgr3permute5 36994 | Permutation law for three-... |
| cgr3tr4 36995 | Transitivity law for three... |
| cgr3com 36996 | Commutativity law for thre... |
| cgr3rflx 36997 | Identity law for three-pla... |
| cgrxfr 36998 | A line segment can be divi... |
| btwnxfr 36999 | A condition for extending ... |
| colinrel 37000 | Colinearity is a relations... |
| brcolinear2 37001 | Alternate colinearity bina... |
| brcolinear 37002 | The binary relation form o... |
| colinearex 37003 | The colinear predicate exi... |
| colineardim1 37004 | If ` A ` is colinear with ... |
| colinearperm1 37005 | Permutation law for coline... |
| colinearperm3 37006 | Permutation law for coline... |
| colinearperm2 37007 | Permutation law for coline... |
| colinearperm4 37008 | Permutation law for coline... |
| colinearperm5 37009 | Permutation law for coline... |
| colineartriv1 37010 | Trivial case of colinearit... |
| colineartriv2 37011 | Trivial case of colinearit... |
| btwncolinear1 37012 | Betweenness implies coline... |
| btwncolinear2 37013 | Betweenness implies coline... |
| btwncolinear3 37014 | Betweenness implies coline... |
| btwncolinear4 37015 | Betweenness implies coline... |
| btwncolinear5 37016 | Betweenness implies coline... |
| btwncolinear6 37017 | Betweenness implies coline... |
| colinearxfr 37018 | Transfer law for colineari... |
| lineext 37019 | Extend a line with a missi... |
| brofs2 37020 | Change some conditions for... |
| brifs2 37021 | Change some conditions for... |
| brfs 37022 | Binary relation form of th... |
| fscgr 37023 | Congruence law for the gen... |
| linecgr 37024 | Congruence rule for lines.... |
| linecgrand 37025 | Deduction form of ~ linecg... |
| lineid 37026 | Identity law for points on... |
| idinside 37027 | Law for finding a point in... |
| endofsegid 37028 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 37029 | Deduction form of ~ endofs... |
| btwnconn1lem1 37030 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 37031 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 37032 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 37033 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 37034 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 37035 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 37036 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 37037 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 37038 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 37039 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 37040 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 37041 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 37042 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 37043 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 37044 | Connectitivy law for betwe... |
| btwnconn2 37045 | Another connectivity law f... |
| btwnconn3 37046 | Inner connectivity law for... |
| midofsegid 37047 | If two points fall in the ... |
| segcon2 37048 | Generalization of ~ axsegc... |
| brsegle 37051 | Binary relation form of th... |
| brsegle2 37052 | Alternate characterization... |
| seglecgr12im 37053 | Substitution law for segme... |
| seglecgr12 37054 | Substitution law for segme... |
| seglerflx 37055 | Segment comparison is refl... |
| seglemin 37056 | Any segment is at least as... |
| segletr 37057 | Segment less than is trans... |
| segleantisym 37058 | Antisymmetry law for segme... |
| seglelin 37059 | Linearity law for segment ... |
| btwnsegle 37060 | If ` B ` falls between ` A... |
| colinbtwnle 37061 | Given three colinear point... |
| broutsideof 37064 | Binary relation form of ` ... |
| broutsideof2 37065 | Alternate form of ` Outsid... |
| outsidene1 37066 | Outsideness implies inequa... |
| outsidene2 37067 | Outsideness implies inequa... |
| btwnoutside 37068 | A principle linking outsid... |
| broutsideof3 37069 | Characterization of outsid... |
| outsideofrflx 37070 | Reflexivity of outsideness... |
| outsideofcom 37071 | Commutativity law for outs... |
| outsideoftr 37072 | Transitivity law for outsi... |
| outsideofeq 37073 | Uniqueness law for ` Outsi... |
| outsideofeu 37074 | Given a nondegenerate ray,... |
| outsidele 37075 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 37076 | Outside of implies colinea... |
| funray 37083 | Show that the ` Ray ` rela... |
| fvray 37084 | Calculate the value of the... |
| funline 37085 | Show that the ` Line ` rel... |
| linedegen 37086 | When ` Line ` is applied w... |
| fvline 37087 | Calculate the value of the... |
| liness 37088 | A line is a subset of the ... |
| fvline2 37089 | Alternate definition of a ... |
| lineunray 37090 | A line is composed of a po... |
| lineelsb2 37091 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 37092 | Reflexivity law for line m... |
| linecom 37093 | Commutativity law for line... |
| linerflx2 37094 | Reflexivity law for line m... |
| ellines 37095 | Membership in the set of a... |
| linethru 37096 | If ` A ` is a line contain... |
| hilbert1.1 37097 | There is a line through an... |
| hilbert1.2 37098 | There is at most one line ... |
| linethrueu 37099 | There is a unique line goi... |
| lineintmo 37100 | Two distinct lines interse... |
| fwddifval 37105 | Calculate the value of the... |
| fwddifnval 37106 | The value of the forward d... |
| fwddifn0 37107 | The value of the n-iterate... |
| fwddifnp1 37108 | The value of the n-iterate... |
| rank0 37109 | The rank of the empty set ... |
| rankeq1o 37110 | The only set with rank ` 1... |
| hftr 37111 | The class of all hereditar... |
| hfext 37112 | Extensionality for HF sets... |
| hfninf 37113 | ` _om ` is not hereditaril... |
| nmulfn 37116 | Natural multiplication is ... |
| nmulprop 37117 | Show closure and value of ... |
| nmulcl 37118 | Closure law for natural mu... |
| nmulval 37119 | Show the value of natural ... |
| nmulcld 37120 | Closure law for natural mu... |
| nmulcom 37121 | Natural multiplication is ... |
| nmulr0 37122 | Natural multiplication by ... |
| nmull0 37123 | Natural multiplication by ... |
| nmulrid 37124 | Identity law for natural m... |
| nmullid 37125 | Identity law for natural m... |
| onelond 37126 | An element of an ordinal n... |
| ontr2d 37127 | Transitive law for ordinal... |
| onelssd 37128 | An element of an ordinal n... |
| nmulr0d 37129 | Natural multiplication by ... |
| nmull0d 37130 | Natural multiplication by ... |
| nmulridd 37131 | Identity law for natural m... |
| nmullidd 37132 | Identity law for natural m... |
| nmulcomd 37133 | Natural multiplication com... |
| naddridd 37134 | Identity law for natural a... |
| naddlidd 37135 | Identity law for natural a... |
| naddcomd 37136 | Natural addition commutes.... |
| naddassd 37137 | Natural addition associate... |
| nadd32d 37138 | Commutative/associative la... |
| nmuladdel 37139 | Ordering relationship for ... |
| nmuladdss 37140 | Ordering relationship for ... |
| nmulss1 37141 | Natural multiplication pre... |
| nmulel1 37142 | Natural multiplication by ... |
| ltnmul 37143 | Characterize less-than a n... |
| nmulle 37144 | A condition for bounding a... |
| ltnadd 37145 | Condition for bounding a n... |
| naddle 37146 | Condition for bounding nat... |
| nadddilem1 37147 | Lemma for ~ nadddi . Prov... |
| nadddilem2 37148 | Lemma for ~ nadddi . Prov... |
| nadddilem3 37149 | Lemma for ~ nadddi . Prov... |
| nadddilem4 37150 | Lemma for ~ nadddi . Prov... |
| nadddi 37151 | Natural multiplication dis... |
| nadddid 37152 | Natural multiplication dis... |
| nadddird 37153 | Natural multiplication dis... |
| rmoeqi 37154 | Equality inference for res... |
| rmoeqbii 37155 | Equality inference for res... |
| reueqi 37156 | Equality inference for res... |
| reueqbii 37157 | Equality inference for res... |
| sbceqbii 37158 | Formula-building inference... |
| disjeq1i 37159 | Equality theorem for disjo... |
| disjeq12i 37160 | Equality theorem for disjo... |
| rabeqbii 37161 | Equality theorem for restr... |
| iuneq12i 37162 | Equality theorem for index... |
| iineq1i 37163 | Equality theorem for index... |
| iineq12i 37164 | Equality theorem for index... |
| riotaeqbii 37165 | Equivalent wff's and equal... |
| riotaeqi 37166 | Equal domains yield equal ... |
| ixpeq1i 37167 | Equality inference for inf... |
| ixpeq12i 37168 | Equality inference for inf... |
| sumeq2si 37169 | Equality inference for sum... |
| sumeq12si 37170 | Equality inference for sum... |
| prodeq2si 37171 | Equality inference for pro... |
| prodeq12si 37172 | Equality inference for pro... |
| itgeq12i 37173 | Equality inference for an ... |
| itgeq1i 37174 | Equality inference for an ... |
| itgeq2i 37175 | Equality inference for an ... |
| ditgeq123i 37176 | Equality inference for the... |
| ditgeq12i 37177 | Equality inference for the... |
| ditgeq3i 37178 | Equality inference for the... |
| rmoeqdv 37179 | Formula-building rule for ... |
| rmoeqbidv 37180 | Formula-building rule for ... |
| sbequbidv 37181 | Deduction substituting bot... |
| disjeq12dv 37182 | Equality theorem for disjo... |
| ixpeq12dv 37183 | Equality theorem for infin... |
| sumeq12sdv 37184 | Equality deduction for sum... |
| prodeq12sdv 37185 | Equality deduction for pro... |
| itgeq12sdv 37186 | Equality theorem for an in... |
| itgeq2sdv 37187 | Equality theorem for an in... |
| ditgeq123dv 37188 | Equality theorem for the d... |
| ditgeq12d 37189 | Equality theorem for the d... |
| ditgeq3sdv 37190 | Equality theorem for the d... |
| in-ax8 37191 | A proof of ~ ax-8 that doe... |
| ss-ax8 37192 | A proof of ~ ax-8 that doe... |
| cbvralvw2 37193 | Change bound variable and ... |
| cbvrexvw2 37194 | Change bound variable and ... |
| cbvrmovw2 37195 | Change bound variable and ... |
| cbvreuvw2 37196 | Change bound variable and ... |
| cbvsbcvw2 37197 | Change bound variable of a... |
| cbvcsbvw2 37198 | Change bound variable of a... |
| cbviunvw2 37199 | Change bound variable and ... |
| cbviinvw2 37200 | Change bound variable and ... |
| cbvmptvw2 37201 | Change bound variable and ... |
| cbvdisjvw2 37202 | Change bound variable and ... |
| cbvriotavw2 37203 | Change bound variable and ... |
| cbvoprab1vw 37204 | Change the first bound var... |
| cbvoprab2vw 37205 | Change the second bound va... |
| cbvoprab123vw 37206 | Change all bound variables... |
| cbvoprab23vw 37207 | Change the second and thir... |
| cbvoprab13vw 37208 | Change the first and third... |
| cbvmpovw2 37209 | Change bound variables and... |
| cbvmpo1vw2 37210 | Change domains and the fir... |
| cbvmpo2vw2 37211 | Change domains and the sec... |
| cbvixpvw2 37212 | Change bound variable and ... |
| cbvsumvw2 37213 | Change bound variable and ... |
| cbvprodvw2 37214 | Change bound variable and ... |
| cbvitgvw2 37215 | Change bound variable and ... |
| cbvditgvw2 37216 | Change bound variable and ... |
| cbvmodavw 37217 | Change bound variable in t... |
| cbveudavw 37218 | Change bound variable in t... |
| cbvrmodavw 37219 | Change bound variable in t... |
| cbvreudavw 37220 | Change bound variable in t... |
| cbvsbdavw 37221 | Change bound variable in p... |
| cbvsbdavw2 37222 | Change bound variable in p... |
| cbvabdavw 37223 | Change bound variable in c... |
| cbvsbcdavw 37224 | Change bound variable of a... |
| cbvsbcdavw2 37225 | Change bound variable of a... |
| cbvcsbdavw 37226 | Change bound variable of a... |
| cbvcsbdavw2 37227 | Change bound variable of a... |
| cbvrabdavw 37228 | Change bound variable in r... |
| cbviundavw 37229 | Change bound variable in i... |
| cbviindavw 37230 | Change bound variable in i... |
| cbvopab1davw 37231 | Change the first bound var... |
| cbvopab2davw 37232 | Change the second bound va... |
| cbvopabdavw 37233 | Change bound variables in ... |
| cbvmptdavw 37234 | Change bound variable in a... |
| cbvdisjdavw 37235 | Change bound variable in a... |
| cbviotadavw 37236 | Change bound variable in a... |
| cbvriotadavw 37237 | Change bound variable in a... |
| cbvoprab1davw 37238 | Change the first bound var... |
| cbvoprab2davw 37239 | Change the second bound va... |
| cbvoprab3davw 37240 | Change the third bound var... |
| cbvoprab123davw 37241 | Change all bound variables... |
| cbvoprab12davw 37242 | Change the first and secon... |
| cbvoprab23davw 37243 | Change the second and thir... |
| cbvoprab13davw 37244 | Change the first and third... |
| cbvixpdavw 37245 | Change bound variable in a... |
| cbvsumdavw 37246 | Change bound variable in a... |
| cbvproddavw 37247 | Change bound variable in a... |
| cbvitgdavw 37248 | Change bound variable in a... |
| cbvditgdavw 37249 | Change bound variable in a... |
| cbvrmodavw2 37250 | Change bound variable and ... |
| cbvreudavw2 37251 | Change bound variable and ... |
| cbvrabdavw2 37252 | Change bound variable and ... |
| cbviundavw2 37253 | Change bound variable and ... |
| cbviindavw2 37254 | Change bound variable and ... |
| cbvmptdavw2 37255 | Change bound variable and ... |
| cbvdisjdavw2 37256 | Change bound variable and ... |
| cbvriotadavw2 37257 | Change bound variable and ... |
| cbvmpodavw2 37258 | Change bound variable and ... |
| cbvmpo1davw2 37259 | Change first bound variabl... |
| cbvmpo2davw2 37260 | Change second bound variab... |
| cbvixpdavw2 37261 | Change bound variable and ... |
| cbvsumdavw2 37262 | Change bound variable and ... |
| cbvproddavw2 37263 | Change bound variable and ... |
| cbvitgdavw2 37264 | Change bound variable and ... |
| cbvditgdavw2 37265 | Change bound variable and ... |
| mpomulnzcnf 37266 | Multiplication maps nonzer... |
| a1i14 37267 | Add two antecedents to a w... |
| a1i24 37268 | Add two antecedents to a w... |
| exp5d 37269 | An exportation inference. ... |
| exp5g 37270 | An exportation inference. ... |
| exp5k 37271 | An exportation inference. ... |
| exp56 37272 | An exportation inference. ... |
| exp58 37273 | An exportation inference. ... |
| exp510 37274 | An exportation inference. ... |
| exp511 37275 | An exportation inference. ... |
| exp512 37276 | An exportation inference. ... |
| 3com12d 37277 | Commutation in consequent.... |
| imp5p 37278 | A triple importation infer... |
| imp5q 37279 | A triple importation infer... |
| subtr 37280 | Transitivity of implicit s... |
| subtr2 37281 | Transitivity of implicit s... |
| trer 37282 | A relation intersected wit... |
| elicc3 37283 | An equivalent membership c... |
| finminlem 37284 | A useful lemma about finit... |
| gtinf 37285 | Any number greater than an... |
| opnrebl 37286 | A set is open in the stand... |
| opnrebl2 37287 | A set is open in the stand... |
| nn0prpwlem 37288 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 37289 | Two nonnegative integers a... |
| topbnd 37290 | Two equivalent expressions... |
| opnbnd 37291 | A set is open iff it is di... |
| cldbnd 37292 | A set is closed iff it con... |
| ntruni 37293 | A union of interiors is a ... |
| clsun 37294 | A pairwise union of closur... |
| clsint2 37295 | The closure of an intersec... |
| opnregcld 37296 | A set is regularly closed ... |
| cldregopn 37297 | A set if regularly open if... |
| neiin 37298 | Two neighborhoods intersec... |
| hmeoclda 37299 | Homeomorphisms preserve cl... |
| hmeocldb 37300 | Homeomorphisms preserve cl... |
| ivthALT 37301 | An alternate proof of the ... |
| fnerel 37304 | Fineness is a relation. (... |
| isfne 37305 | The predicate " ` B ` is f... |
| isfne4 37306 | The predicate " ` B ` is f... |
| isfne4b 37307 | A condition for a topology... |
| isfne2 37308 | The predicate " ` B ` is f... |
| isfne3 37309 | The predicate " ` B ` is f... |
| fnebas 37310 | A finer cover covers the s... |
| fnetg 37311 | A finer cover generates a ... |
| fnessex 37312 | If ` B ` is finer than ` A... |
| fneuni 37313 | If ` B ` is finer than ` A... |
| fneint 37314 | If a cover is finer than a... |
| fness 37315 | A cover is finer than its ... |
| fneref 37316 | Reflexivity of the finenes... |
| fnetr 37317 | Transitivity of the finene... |
| fneval 37318 | Two covers are finer than ... |
| fneer 37319 | Fineness intersected with ... |
| topfne 37320 | Fineness for covers corres... |
| topfneec 37321 | A cover is equivalent to a... |
| topfneec2 37322 | A topology is precisely id... |
| fnessref 37323 | A cover is finer iff it ha... |
| refssfne 37324 | A cover is a refinement if... |
| neibastop1 37325 | A collection of neighborho... |
| neibastop2lem 37326 | Lemma for ~ neibastop2 . ... |
| neibastop2 37327 | In the topology generated ... |
| neibastop3 37328 | The topology generated by ... |
| topmtcl 37329 | The meet of a collection o... |
| topmeet 37330 | Two equivalent formulation... |
| topjoin 37331 | Two equivalent formulation... |
| fnemeet1 37332 | The meet of a collection o... |
| fnemeet2 37333 | The meet of equivalence cl... |
| fnejoin1 37334 | Join of equivalence classe... |
| fnejoin2 37335 | Join of equivalence classe... |
| fgmin 37336 | Minimality property of a g... |
| neifg 37337 | The neighborhood filter of... |
| tailfval 37338 | The tail function for a di... |
| tailval 37339 | The tail of an element in ... |
| eltail 37340 | An element of a tail. (Co... |
| tailf 37341 | The tail function of a dir... |
| tailini 37342 | A tail contains its initia... |
| tailfb 37343 | The collection of tails of... |
| filnetlem1 37344 | Lemma for ~ filnet . Chan... |
| filnetlem2 37345 | Lemma for ~ filnet . The ... |
| filnetlem3 37346 | Lemma for ~ filnet . (Con... |
| filnetlem4 37347 | Lemma for ~ filnet . (Con... |
| filnet 37348 | A filter has the same conv... |
| tb-ax1 37349 | The first of three axioms ... |
| tb-ax2 37350 | The second of three axioms... |
| tb-ax3 37351 | The third of three axioms ... |
| tbsyl 37352 | The weak syllogism from Ta... |
| re1ax2lem 37353 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 37354 | ~ ax-2 rederived from the ... |
| naim1 37355 | Constructor theorem for ` ... |
| naim2 37356 | Constructor theorem for ` ... |
| naim1i 37357 | Constructor rule for ` -/\... |
| naim2i 37358 | Constructor rule for ` -/\... |
| naim12i 37359 | Constructor rule for ` -/\... |
| nabi1i 37360 | Constructor rule for ` -/\... |
| nabi2i 37361 | Constructor rule for ` -/\... |
| nabi12i 37362 | Constructor rule for ` -/\... |
| df3nandALT1 37365 | The double nand expressed ... |
| df3nandALT2 37366 | The double nand expressed ... |
| andnand1 37367 | Double and in terms of dou... |
| imnand2 37368 | An ` -> ` nand relation. ... |
| nalfal 37369 | Not all sets hold ` F. ` a... |
| nexntru 37370 | There does not exist a set... |
| nexfal 37371 | There does not exist a set... |
| neufal 37372 | There does not exist exact... |
| neutru 37373 | There does not exist exact... |
| nmotru 37374 | There does not exist at mo... |
| mofal 37375 | There exist at most one se... |
| nrmo 37376 | "At most one" restricted e... |
| meran1 37377 | A single axiom for proposi... |
| meran2 37378 | A single axiom for proposi... |
| meran3 37379 | A single axiom for proposi... |
| waj-ax 37380 | A single axiom for proposi... |
| lukshef-ax2 37381 | A single axiom for proposi... |
| arg-ax 37382 | A single axiom for proposi... |
| negsym1 37383 | In the paper "On Variable ... |
| imsym1 37384 | A symmetry with ` -> ` . ... |
| bisym1 37385 | A symmetry with ` <-> ` . ... |
| consym1 37386 | A symmetry with ` /\ ` . ... |
| dissym1 37387 | A symmetry with ` \/ ` . ... |
| nandsym1 37388 | A symmetry with ` -/\ ` . ... |
| unisym1 37389 | A symmetry with ` A. ` . ... |
| exisym1 37390 | A symmetry with ` E. ` . ... |
| unqsym1 37391 | A symmetry with ` E! ` . ... |
| amosym1 37392 | A symmetry with ` E* ` . ... |
| subsym1 37393 | A symmetry with ` [ x / y ... |
| ontopbas 37394 | An ordinal number is a top... |
| onsstopbas 37395 | The class of ordinal numbe... |
| onpsstopbas 37396 | The class of ordinal numbe... |
| ontgval 37397 | The topology generated fro... |
| ontgsucval 37398 | The topology generated fro... |
| onsuctop 37399 | A successor ordinal number... |
| onsuctopon 37400 | One of the topologies on a... |
| ordtoplem 37401 | Membership of the class of... |
| ordtop 37402 | An ordinal is a topology i... |
| onsucconni 37403 | A successor ordinal number... |
| onsucconn 37404 | A successor ordinal number... |
| ordtopconn 37405 | An ordinal topology is con... |
| onintopssconn 37406 | An ordinal topology is con... |
| onsuct0 37407 | A successor ordinal number... |
| ordtopt0 37408 | An ordinal topology is T_0... |
| onsucsuccmpi 37409 | The successor of a success... |
| onsucsuccmp 37410 | The successor of a success... |
| limsucncmpi 37411 | The successor of a limit o... |
| limsucncmp 37412 | The successor of a limit o... |
| ordcmp 37413 | An ordinal topology is com... |
| ssoninhaus 37414 | The ordinal topologies ` 1... |
| onint1 37415 | The ordinal T_1 spaces are... |
| oninhaus 37416 | The ordinal Hausdorff spac... |
| fveleq 37417 | Please add description her... |
| findfvcl 37418 | Please add description her... |
| findreccl 37419 | Please add description her... |
| findabrcl 37420 | Please add description her... |
| nnssi2 37421 | Convert a theorem for real... |
| nnssi3 37422 | Convert a theorem for real... |
| nndivsub 37423 | Please add description her... |
| nndivlub 37424 | A factor of a positive int... |
| ee7.2aOLD 37427 | Lemma for Euclid's Element... |
| weiunval 37428 | Value of the relation cons... |
| weiunlem 37429 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 37430 | Lemma for ~ weiunfr . (Co... |
| weiunpo 37431 | A partial ordering on an i... |
| weiunso 37432 | A strict ordering on an in... |
| weiunfr 37433 | A well-founded relation on... |
| weiunse 37434 | The relation constructed i... |
| weiunwe 37435 | A well-ordering on an inde... |
| numiunnum 37436 | An indexed union of sets i... |
| axtco 37437 | Axiom of Transitive Contai... |
| axtco1 37439 | Strong form of the Axiom o... |
| axtco2 37440 | Weak form of the Axiom of ... |
| axtco1from2 37441 | Strong form ~ axtco1 of th... |
| axtco1g 37442 | Strong form of the Axiom o... |
| axtco2g 37443 | Weak form of the Axiom of ... |
| axtcond 37444 | A version of the Axiom of ... |
| axuntco 37445 | Derivation of ~ ax-un from... |
| axnulregtco 37446 | Derivation of ~ ax-nul fro... |
| elALTtco 37447 | Derivation of ~ el from ~ ... |
| tz9.1ctco 37448 | Version of ~ tz9.1c derive... |
| tz9.1tco 37449 | Version of ~ tz9.1 derived... |
| tr0elw 37450 | Every nonempty transitive ... |
| tr0el 37451 | Every nonempty transitive ... |
| ttceq 37454 | Equality theorem for trans... |
| ttceqi 37455 | Equality inference for tra... |
| ttceqd 37456 | Equality deduction for tra... |
| nfttc 37457 | Bound-variable hypothesis ... |
| ttcid 37458 | The transitive closure con... |
| ttctr 37459 | The transitive closure of ... |
| ttctr2 37460 | The transitive closure of ... |
| ttctr3 37461 | The transitive closure of ... |
| ttcmin 37462 | The transitive closure of ... |
| ttcexrg 37463 | If the transitive closure ... |
| ttcss 37464 | A transitive closure conta... |
| ttcss2 37465 | The subclass relationship ... |
| ttcel 37466 | A transitive closure conta... |
| ttcel2 37467 | Elements turn into subclas... |
| ttctrid 37468 | The transitive closure of ... |
| ttcidm 37469 | The transitive closure ope... |
| ssttctr 37470 | Transitivity of ` A C_ TC+... |
| elttctr 37471 | Transitivity of ` A e. TC+... |
| dfttc2g 37472 | A shorter expression for t... |
| ttc0 37473 | The transitive closure of ... |
| ttc00 37474 | A class has an empty trans... |
| csbttc 37475 | Distribute proper substitu... |
| ttcuniun 37476 | Relationship between ` TC+... |
| ttciunun 37477 | Relationship between ` TC+... |
| ttcun 37478 | Distribute union of two cl... |
| ttcuni 37479 | Distribute union of a clas... |
| ttciun 37480 | Distribute indexed union t... |
| ttcpwss 37481 | The transitive closure of ... |
| ttcsnssg 37482 | The transitive closure is ... |
| ttcsnidg 37483 | The singleton transitive c... |
| ttcsnmin 37484 | The singleton transitive c... |
| ttcsng 37485 | Relationship between ` TC+... |
| ttcsnexg 37486 | If the transitive closure ... |
| ttcsnexbig 37487 | The transitive closure of ... |
| ttcsntrsucg 37488 | The singleton transitive c... |
| dfttc3gw 37489 | If the transitive closure ... |
| ttcwf 37490 | A set is well-founded iff ... |
| ttcwf2 37491 | If a transitive closure cl... |
| ttcwf3 37492 | The sets whose transitive ... |
| ttc0elw 37493 | If a transitive closure is... |
| dfttc4lem1 37494 | Lemma for ~ dfttc4 . (Con... |
| dfttc4lem2 37495 | Lemma for ~ dfttc4 . (Con... |
| dfttc4 37496 | An alternative expression ... |
| elttcirr 37497 | Irreflexivity of ` A e. TC... |
| ttcexg 37498 | The transitive closure of ... |
| ttcexbi 37499 | A class is a set iff its t... |
| dfttc3g 37500 | The transitive closure of ... |
| ttc0el 37501 | A transitive closure conta... |
| mh-setind 37502 | Principle of set induction... |
| mh-setindnd 37503 | A version of ~ mh-setind w... |
| regsfromregtco 37504 | Derivation of ~ ax-regs fr... |
| regsfromsetind 37505 | Derivation of ~ ax-regs fr... |
| regsfromunir1 37506 | Derivation of ~ ax-regs fr... |
| mh-inf3f1 37507 | A variant of ~ inf3 . If ... |
| mh-inf3sn 37508 | Version of ~ inf3 for the ... |
| mh-prprimbi 37509 | Shortest possible version ... |
| mh-unprimbi 37510 | Shortest possible version ... |
| mh-regprimbi 37511 | Shortest possible version ... |
| mh-infprim1bi 37512 | Shortest possible axiom of... |
| mh-infprim2bi 37513 | Shortest possible axiom of... |
| mh-infprim3bi 37514 | An axiom of infinity in pr... |
| dnival 37515 | Value of the "distance to ... |
| dnicld1 37516 | Closure theorem for the "d... |
| dnicld2 37517 | Closure theorem for the "d... |
| dnif 37518 | The "distance to nearest i... |
| dnizeq0 37519 | The distance to nearest in... |
| dnizphlfeqhlf 37520 | The distance to nearest in... |
| rddif2 37521 | Variant of ~ rddif . (Con... |
| dnibndlem1 37522 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 37523 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 37524 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 37525 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 37526 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 37527 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 37528 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 37529 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 37530 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 37531 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 37532 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 37533 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 37534 | Lemma for ~ dnibnd . (Con... |
| dnibnd 37535 | The "distance to nearest i... |
| dnicn 37536 | The "distance to nearest i... |
| knoppcnlem1 37537 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 37538 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 37539 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 37540 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 37541 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 37542 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 37543 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 37544 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 37545 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 37546 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 37547 | Lemma for ~ knoppcn . (Co... |
| knoppcn 37548 | The continuous nowhere dif... |
| knoppcld 37549 | Closure theorem for Knopp'... |
| unblimceq0lem 37550 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 37551 | If ` F ` is unbounded near... |
| unbdqndv1 37552 | If the difference quotient... |
| unbdqndv2lem1 37553 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 37554 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 37555 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 37556 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 37557 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 37558 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 37559 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 37560 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 37561 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 37562 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 37563 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 37564 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 37565 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 37566 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 37567 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 37568 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 37569 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 37570 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 37571 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 37572 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 37573 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 37574 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 37575 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 37576 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 37577 | Lemma for ~ knoppndv . (C... |
| knoppndv 37578 | The continuous nowhere dif... |
| knoppf 37579 | Knopp's function is a func... |
| knoppcn2 37580 | Variant of ~ knoppcn with ... |
| cnndvlem1 37581 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 37582 | Lemma for ~ cnndv . (Cont... |
| cnndv 37583 | There exists a continuous ... |
| bj-mp2c 37584 | A double _modus ponens_ in... |
| bj-mp2d 37585 | A double _modus ponens_ in... |
| bj-0 37586 | A syntactic theorem. See ... |
| bj-1 37587 | In this proof, the use of ... |
| bj-poni 37588 | Inference associated with ... |
| bj-nnclav 37589 | When ` F. ` is substituted... |
| bj-nnclavi 37590 | Inference associated with ... |
| bj-nnclavc 37591 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 37592 | Inference associated with ... |
| bj-jarrii 37593 | Inference associated with ... |
| bj-imim21 37594 | The propositional function... |
| bj-imim21i 37595 | The propositional function... |
| bj-imim11 37596 | The propositional function... |
| bj-imim11i 37597 | The propositional function... |
| bj-peircestab 37598 | Over minimal implicational... |
| bj-stabpeirce 37599 | This minimal implicational... |
| bj-bisimpl 37600 | Implication from equivalen... |
| bj-bisimpr 37601 | Implication from equivalen... |
| bj-syl66ib 37602 | A mixed syllogism inferenc... |
| bj-orim2 37603 | Proof of ~ orim2 from the ... |
| bj-currypeirce 37604 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 37605 | Peirce's axiom ~ peirce im... |
| bj-animbi 37606 | Conjunction in terms of im... |
| bj-currypara 37607 | Curry's paradox. The proo... |
| bj-con2com 37608 | A commuted form of the con... |
| bj-con2comi 37609 | Inference associated with ... |
| bj-nimn 37610 | If a formula is true, then... |
| bj-nimni 37611 | Inference associated with ... |
| bj-peircei 37612 | Inference associated with ... |
| bj-looinvi 37613 | Inference associated with ... |
| bj-looinvii 37614 | Inference associated with ... |
| bj-mt2bi 37615 | Version of ~ mt2 where the... |
| bj-fal 37616 | Shortening of ~ fal using ... |
| bj-ntrufal 37617 | The negation of a theorem ... |
| bj-dfnul2 37618 | Alternate definition of th... |
| bj-jaoi1 37619 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 37620 | Shortens ~ consensus (110>... |
| bj-dfbi4 37621 | Alternate definition of th... |
| bj-dfbi5 37622 | Alternate definition of th... |
| bj-dfbi6 37623 | Alternate definition of th... |
| bj-bijust0ALT 37624 | Alternate proof of ~ bijus... |
| bj-bijust00 37625 | A self-implication does no... |
| bj-consensus 37626 | Version of ~ consensus exp... |
| bj-consensusALT 37627 | Alternate proof of ~ bj-co... |
| bj-df-ifc 37628 | Candidate definition for t... |
| bj-dfif 37629 | Alternate definition of th... |
| bj-ififc 37630 | A biconditional connecting... |
| bj-imbi12 37631 | Uncurried (imported) form ... |
| bj-falor 37632 | Dual of ~ truan (which has... |
| bj-falor2 37633 | Dual of ~ truan . (Contri... |
| bj-bibibi 37634 | A property of the bicondit... |
| bj-imn3ani 37635 | Duplication of ~ bnj1224 .... |
| bj-andnotim 37636 | Two ways of expressing a c... |
| bj-bi3ant 37637 | This used to be in the mai... |
| bj-bisym 37638 | This used to be in the mai... |
| bj-bixor 37639 | Equivalence of two ternary... |
| bj-axdd2 37640 | This implication, proved u... |
| bj-axd2d 37641 | This implication, proved u... |
| bj-axtd 37642 | This implication, proved f... |
| bj-gl4 37643 | In a normal modal logic, t... |
| bj-axc4 37644 | Over minimal calculus, the... |
| prvlem1 37649 | An elementary property of ... |
| prvlem2 37650 | An elementary property of ... |
| bj-babygodel 37651 | See the section header com... |
| bj-babylob 37652 | See the section header com... |
| bj-godellob 37653 | Proof of Gödel's theo... |
| bj-exexalal 37654 | A lemma for changing bound... |
| bj-genr 37655 | Generalization rule on the... |
| bj-genl 37656 | Generalization rule on the... |
| bj-genan 37657 | Generalization rule on a c... |
| bj-mpgs 37658 | From a closed form theorem... |
| bj-almp 37659 | A quantified form of ~ ax-... |
| bj-sylggt 37660 | Stronger form of ~ sylgt ,... |
| bj-alrimg 37661 | The general form of the *a... |
| bj-sylgt2 37662 | Uncurried (imported) form ... |
| bj-nexdh 37663 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 37664 | Uncurried (imported) form ... |
| bj-alimii 37665 | Inference associated with ... |
| bj-ala1i 37666 | Add an antecedent in a uni... |
| bj-almpi 37667 | A quantified form of ~ mpi... |
| bj-almpig 37668 | A partially quantified for... |
| bj-alsyl 37669 | Syllogism under the univer... |
| bj-2alim 37670 | Closed form of ~ 2alimi . ... |
| bj-alimdh 37671 | General instance of ~ alim... |
| bj-alrimdh 37672 | Deduction form of Theorem ... |
| bj-alrimd 37673 | A slightly more general ~ ... |
| bj-exa1i 37674 | Add an antecedent in an ex... |
| bj-alanim 37675 | Closed form of ~ alanimi .... |
| bj-2albi 37676 | Closed form of ~ 2albii . ... |
| bj-notalbii 37677 | Equivalence of universal q... |
| bj-2exim 37678 | Closed form of ~ 2eximi . ... |
| bj-2exbi 37679 | Closed form of ~ 2exbii . ... |
| bj-3exbi 37680 | Closed form of ~ 3exbii . ... |
| bj-sylget 37681 | Dual statement of ~ sylgt ... |
| bj-sylget2 37682 | Uncurried (imported) form ... |
| bj-exlimg 37683 | The general form of the *e... |
| bj-sylge 37684 | Dual statement of ~ sylg (... |
| bj-exlimd 37685 | A slightly more general ~ ... |
| bj-nfimexal 37686 | A weak from of nonfreeness... |
| bj-exim 37687 | Theorem 19.22 of [Margaris... |
| bj-alexim 37688 | Closed form of ~ aleximi .... |
| bj-aleximiALT 37689 | Alternate proof of ~ alexi... |
| bj-hbxfrbi 37690 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 37691 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 37692 | Distribute quantifiers ove... |
| bj-exalimi 37693 | An inference for distribut... |
| bj-eximcom 37694 | A commuted form of ~ exim ... |
| bj-exalims 37695 | Distributing quantifiers o... |
| bj-exalimsi 37696 | An inference for distribut... |
| bj-axdd2ALT 37697 | Alternate proof of ~ bj-ax... |
| bj-ax12ig 37698 | A lemma used to prove a we... |
| bj-ax12i 37699 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 37700 | Closed form of ~ nfim and ... |
| bj-spimnfe 37701 | A universal specification ... |
| bj-spimenfa 37702 | An existential generalizat... |
| bj-spim 37703 | A lemma for universal spec... |
| bj-spime 37704 | A lemma for existential ge... |
| bj-cbvalimd0 37705 | A lemma for alpha-renaming... |
| bj-cbvalimdlem 37706 | A lemma for alpha-renaming... |
| bj-cbveximdlem 37707 | A lemma for alpha-renaming... |
| bj-cbvalimd 37708 | A lemma for alpha-renaming... |
| bj-cbveximd 37709 | A lemma for alpha-renaming... |
| bj-cbvalimdv 37710 | A lemma for alpha-renaming... |
| bj-cbveximdv 37711 | A lemma for alpha-renaming... |
| bj-spvw 37712 | Version of ~ spvw and ~ 19... |
| bj-spvew 37713 | Version of ~ 19.8v and ~ 1... |
| bj-alextruim 37714 | An equivalent expression f... |
| bj-exextruan 37715 | An equivalent expression f... |
| bj-cbvalvv 37716 | Universally quantifying ov... |
| bj-cbvexvv 37717 | Existentially quantifying ... |
| bj-cbvaw 37718 | Universally quantifying ov... |
| bj-cbvew 37719 | Existentially quantifying ... |
| bj-cbveaw 37720 | Universally quantifying ov... |
| bj-cbvaew 37721 | Exixtentially quantifying ... |
| bj-ax12wlem 37722 | A lemma used to prove a we... |
| bj-cbval 37723 | Changing a bound variable ... |
| bj-cbvex 37724 | Changing a bound variable ... |
| bj-df-sb 37727 | Proposed definition to rep... |
| bj-sbcex 37728 | Proof of ~ sbcex when taki... |
| bj-dfsbc 37729 | Proof of ~ df-sbc when tak... |
| bj-ssbeq 37730 | Substitution in an equalit... |
| bj-ssblem1 37731 | A lemma for the definiens ... |
| bj-ssblem2 37732 | An instance of ~ ax-11 pro... |
| bj-ax12v 37733 | A weaker form of ~ ax-12 a... |
| bj-ax12 37734 | Remove a DV condition from... |
| bj-ax12ssb 37735 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 37736 | Special case of ~ 19.41 pr... |
| bj-equsexval 37737 | Special case of ~ equsexv ... |
| bj-subst 37738 | Proof of ~ sbalex from cor... |
| bj-ssbid2 37739 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 37740 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 37741 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 37742 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 37743 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 37744 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 37745 | Proof of ~ ax6e (hence ~ a... |
| bj-spim0 37746 | A universal specialization... |
| bj-spimvwt 37747 | Closed form of ~ spimvw . ... |
| bj-spnfw 37748 | Theorem close to a closed ... |
| bj-cbvexiw 37749 | Change bound variable. Th... |
| bj-cbvexivw 37750 | Change bound variable. Th... |
| bj-modald 37751 | A short form of the axiom ... |
| bj-denot 37752 | A weakening of ~ ax-6 and ... |
| bj-eqs 37753 | A lemma for substitutions,... |
| bj-cbvexw 37754 | Change bound variable. Th... |
| bj-ax12w 37755 | The general statement that... |
| bj-ax89 37756 | A theorem which could be u... |
| bj-cleljusti 37757 | One direction of ~ cleljus... |
| bj-alcomexcom 37758 | Commutation of two existen... |
| bj-hbald 37759 | General statement that ~ h... |
| bj-hbalt 37760 | Closed form of (general in... |
| bj-hbal 37761 | More general instance of ~... |
| axc11n11 37762 | Proof of ~ axc11n from { ~... |
| axc11n11r 37763 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 37764 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 37765 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 37766 | Alternate proof of ~ bj-ax... |
| bj-sb 37767 | A weak variant of ~ sbid2 ... |
| bj-modalbe 37768 | The predicate-calculus ver... |
| bj-spst 37769 | Closed form of ~ sps . On... |
| bj-19.21bit 37770 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 37771 | Closed form of ~ 19.23bi .... |
| bj-nexrt 37772 | Closed form of ~ nexr . C... |
| bj-alrim 37773 | Closed form of ~ alrimi . ... |
| bj-alrim2 37774 | Uncurried (imported) form ... |
| bj-nfdt0 37775 | A theorem close to a close... |
| bj-nfdt 37776 | Closed form of ~ nf5d and ... |
| bj-nexdt 37777 | Closed form of ~ nexd . (... |
| bj-nexdvt 37778 | Closed form of ~ nexdv . ... |
| bj-alexbiex 37779 | Adding a second quantifier... |
| bj-exexbiex 37780 | Adding a second quantifier... |
| bj-alalbial 37781 | Adding a second quantifier... |
| bj-exalbial 37782 | Adding a second quantifier... |
| bj-19.9htbi 37783 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 37784 | Strengthening ~ hbnt by re... |
| bj-biexal1 37785 | A general FOL biconditiona... |
| bj-biexal2 37786 | When ` ph ` is substituted... |
| bj-biexal3 37787 | When ` ph ` is substituted... |
| bj-bialal 37788 | When ` ph ` is substituted... |
| bj-biexex 37789 | When ` ph ` is substituted... |
| bj-hbexd 37790 | A more general instance of... |
| bj-hbext 37791 | Closed form of ~ bj-hbex a... |
| bj-hbex 37792 | A more general instance of... |
| bj-nfalt 37793 | Closed form of ~ nfal . (... |
| bj-nfext 37794 | Closed form of ~ nfex . (... |
| bj-eeanvw 37795 | Version of ~ exdistrv with... |
| bj-modal4 37796 | First-order logic form of ... |
| bj-modal4e 37797 | First-order logic form of ... |
| bj-modalb 37798 | A short form of the axiom ... |
| bj-wnf1 37799 | When ` ph ` is substituted... |
| bj-wnf2 37800 | When ` ph ` is substituted... |
| bj-wnfanf 37801 | When ` ph ` is substituted... |
| bj-wnfenf 37802 | When ` ph ` is substituted... |
| bj-19.12 37803 | See ~ 19.12 . Could be la... |
| bj-substax12 37804 | Equivalent form of the axi... |
| bj-substw 37805 | Weak form of the LHS of ~ ... |
| bj-nnfa 37808 | Nonfreeness implies the eq... |
| bj-nnfad 37809 | Nonfreeness implies the eq... |
| bj-nnfai 37810 | Nonfreeness implies the eq... |
| bj-nnfe 37811 | Nonfreeness implies the eq... |
| bj-nnfed 37812 | Nonfreeness implies the eq... |
| bj-nnfei 37813 | Nonfreeness implies the eq... |
| bj-nnfea 37814 | Nonfreeness implies the eq... |
| bj-nnfead 37815 | Nonfreeness implies the eq... |
| bj-nnfeai 37816 | Nonfreeness implies the eq... |
| bj-alnnf 37817 | In deduction-style proofs,... |
| bj-alnnf2 37818 | If a proposition holds, th... |
| bj-dfnnf2 37819 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 37820 | New nonfreeness implies ol... |
| bj-nnfim1 37821 | A consequence of nonfreene... |
| bj-nnfim2 37822 | A consequence of nonfreene... |
| bj-nnftht 37823 | A variable is nonfree in a... |
| bj-nnfth 37824 | A variable is nonfree in a... |
| bj-nnf-alrim 37825 | Proof of the closed form o... |
| bj-stdpc5t 37826 | Alias of ~ bj-nnf-alrim fo... |
| bj-nnfbi 37827 | If two formulas are equiva... |
| bj-nnfbd0 37828 | If two formulas are equiva... |
| bj-nnfbii 37829 | If two formulas are equiva... |
| bj-nnfnt 37830 | A variable is nonfree in a... |
| bj-nnfnth 37831 | A variable is nonfree in t... |
| bj-nnfim 37832 | Nonfreeness in the anteced... |
| bj-nnfimd 37833 | Nonfreeness in the anteced... |
| bj-nnfan 37834 | Nonfreeness in both conjun... |
| bj-nnfand 37835 | Nonfreeness in both conjun... |
| bj-nnfor 37836 | Nonfreeness in both disjun... |
| bj-nnford 37837 | Nonfreeness in both disjun... |
| bj-nnfbit 37838 | Nonfreeness in both sides ... |
| bj-nnfbid 37839 | Nonfreeness in both sides ... |
| bj-nnf-exlim 37840 | Proof of the closed form o... |
| bj-19.21t 37841 | Statement ~ 19.21t proved ... |
| bj-19.23t 37842 | Statement ~ 19.23t proved ... |
| bj-19.36im 37843 | One direction of ~ 19.36 f... |
| bj-19.37im 37844 | One direction of ~ 19.37 f... |
| bj-19.42t 37845 | Closed form of ~ 19.42 fro... |
| bj-19.41t 37846 | Closed form of ~ 19.41 fro... |
| bj-pm11.53vw 37847 | Version of ~ pm11.53v with... |
| bj-nnfv 37848 | A non-occurring variable i... |
| bj-nnfbd 37849 | If two formulas are equiva... |
| bj-pm11.53a 37850 | A variant of ~ pm11.53v . ... |
| bj-equsvt 37851 | A variant of ~ equsv . (C... |
| bj-equsalvwd 37852 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 37853 | Variant of ~ equsexvw . (... |
| bj-nnf-spim 37854 | A universal specialization... |
| bj-nnf-spime 37855 | An existential generalizat... |
| bj-nnf-cbvaliv 37856 | The only DV conditions are... |
| bj-sbievwd 37857 | Variant of ~ sbievw . (Co... |
| bj-sbft 37858 | Version of ~ sbft using ` ... |
| bj-nnf-cbvali 37859 | Compared with ~ bj-nnf-cbv... |
| bj-nnf-cbval 37860 | Compared with ~ cbvalv1 , ... |
| bj-dfnnf3 37861 | Alternate definition of no... |
| bj-nfnnfTEMP 37862 | New nonfreeness is equival... |
| bj-wnfnf 37863 | When ` ph ` is substituted... |
| bj-nnfa1 37864 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 37865 | See ~ nfe1 . (Contributed... |
| bj-nnflemaa 37866 | One of four lemmas for non... |
| bj-nnflemee 37867 | One of four lemmas for non... |
| bj-nnflemae 37868 | One of four lemmas for non... |
| bj-nnflemea 37869 | One of four lemmas for non... |
| bj-nnfalt 37870 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 37871 | See ~ nfex and ~ bj-nfext ... |
| bj-pm11.53v 37872 | Version of ~ pm11.53v with... |
| bj-axc10 37873 | Alternate proof of ~ axc10... |
| bj-alequex 37874 | A fol lemma. See ~ aleque... |
| bj-spimt2 37875 | A step in the proof of ~ s... |
| bj-cbv3ta 37876 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 37877 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 37878 | A theorem close to a close... |
| bj-hbsb3 37879 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 37880 | A theorem close to a close... |
| bj-nfs1t2 37881 | A theorem close to a close... |
| bj-nfs1 37882 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 37883 | Version of ~ axc10 with a ... |
| bj-spimtv 37884 | Version of ~ spimt with a ... |
| bj-cbv3hv2 37885 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 37886 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 37887 | Version of ~ cbv2h with a ... |
| bj-cbv2v 37888 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 37889 | Version of ~ cbvald with a... |
| bj-cbvexdv 37890 | Version of ~ cbvexd with a... |
| bj-cbval2vv 37891 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 37892 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 37893 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 37894 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 37895 | Version of ~ cbvex4v with ... |
| bj-equsalhv 37896 | Version of ~ equsalh with ... |
| bj-axc11nv 37897 | Version of ~ axc11n with a... |
| bj-aecomsv 37898 | Version of ~ aecoms with a... |
| bj-axc11v 37899 | Version of ~ axc11 with a ... |
| bj-drnf2v 37900 | Version of ~ drnf2 with a ... |
| bj-equs45fv 37901 | Version of ~ equs45f with ... |
| bj-hbs1 37902 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 37903 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 37904 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 37905 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 37906 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 37907 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 37908 | Biconditional version of a... |
| bj-hbaeb 37909 | Biconditional version of ~... |
| bj-hbnaeb 37910 | Biconditional version of ~... |
| bj-dvv 37911 | A special instance of ~ bj... |
| bj-equsal1t 37912 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 37913 | Inference associated with ... |
| bj-equsal1 37914 | One direction of ~ equsal ... |
| bj-equsal2 37915 | One direction of ~ equsal ... |
| bj-equsal 37916 | Shorter proof of ~ equsal ... |
| stdpc5t 37917 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 37918 | More direct proof of ~ std... |
| 2stdpc5 37919 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 37920 | Proof of ~ 19.21t from ~ s... |
| exlimii 37921 | Inference associated with ... |
| ax11-pm 37922 | Proof of ~ ax-11 similar t... |
| ax6er 37923 | Commuted form of ~ ax6e . ... |
| exlimiieq1 37924 | Inferring a theorem when i... |
| exlimiieq2 37925 | Inferring a theorem when i... |
| ax11-pm2 37926 | Proof of ~ ax-11 from the ... |
| bj-sbsb 37927 | Biconditional showing two ... |
| bj-dfsb2 37928 | Alternate (dual) definitio... |
| bj-sbf3 37929 | Substitution has no effect... |
| bj-sbf4 37930 | Substitution has no effect... |
| bj-eu3f 37931 | Version of ~ eu3v where th... |
| bj-sblem1 37932 | Lemma for substitution. (... |
| bj-sblem2 37933 | Lemma for substitution. (... |
| bj-sblem 37934 | Lemma for substitution. (... |
| bj-sbievw1 37935 | Lemma for substitution. (... |
| bj-sbievw2 37936 | Lemma for substitution. (... |
| bj-sbievw 37937 | Lemma for substitution. C... |
| bj-sbievv 37938 | Version of ~ sbie with a s... |
| bj-moeub 37939 | Uniqueness is equivalent t... |
| bj-sbidmOLD 37940 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 37941 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 37942 | Curried (exported) form of... |
| bj-dvelimv 37943 | A version of ~ dvelim usin... |
| bj-nfeel2 37944 | Nonfreeness in a membershi... |
| bj-axc14nf 37945 | Proof of a version of ~ ax... |
| bj-axc14 37946 | Alternate proof of ~ axc14... |
| mobidvALT 37947 | Alternate proof of ~ mobid... |
| sbn1ALT 37948 | Alternate proof of ~ sbn1 ... |
| eliminable1 37949 | A theorem used to prove th... |
| eliminable2a 37950 | A theorem used to prove th... |
| eliminable2b 37951 | A theorem used to prove th... |
| eliminable2c 37952 | A theorem used to prove th... |
| eliminable3a 37953 | A theorem used to prove th... |
| eliminable3b 37954 | A theorem used to prove th... |
| eliminable-velab 37955 | A theorem used to prove th... |
| eliminable-veqab 37956 | A theorem used to prove th... |
| eliminable-abeqv 37957 | A theorem used to prove th... |
| eliminable-abeqab 37958 | A theorem used to prove th... |
| eliminable-abelv 37959 | A theorem used to prove th... |
| eliminable-abelab 37960 | A theorem used to prove th... |
| bj-denoteslem 37961 | Duplicate of ~ issettru an... |
| bj-denotesALTV 37962 | Moved to main as ~ iseqset... |
| bj-issettruALTV 37963 | Moved to main as ~ issettr... |
| bj-elabtru 37964 | This is as close as we can... |
| bj-issetwt 37965 | Closed form of ~ bj-issetw... |
| bj-issetw 37966 | The closest one can get to... |
| bj-issetiv 37967 | Version of ~ bj-isseti wit... |
| bj-isseti 37968 | Version of ~ isseti with a... |
| bj-ralvw 37969 | A weak version of ~ ralv n... |
| bj-rexvw 37970 | A weak version of ~ rexv n... |
| bj-rababw 37971 | A weak version of ~ rabab ... |
| bj-rexcom4bv 37972 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 37973 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 37974 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 37975 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 37976 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 37977 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 37978 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 37979 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 37980 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 37981 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 37982 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 37983 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 37984 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 37985 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 37986 | Remove from ~ spcimdv depe... |
| elelb 37987 | Equivalence between two co... |
| bj-pwvrelb 37988 | Characterization of the el... |
| bj-nfcsym 37989 | The nonfreeness quantifier... |
| bj-sbeqALT 37990 | Substitution in an equalit... |
| bj-sbeq 37991 | Distribute proper substitu... |
| bj-sbceqgALT 37992 | Distribute proper substitu... |
| bj-csbsnlem 37993 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 37994 | Substitution in a singleto... |
| bj-sbel1 37995 | Version of ~ sbcel1g when ... |
| bj-abv 37996 | The class of sets verifyin... |
| bj-abvALT 37997 | Alternate version of ~ bj-... |
| bj-ab0 37998 | The class of sets verifyin... |
| bj-abf 37999 | Shorter proof of ~ abf (wh... |
| bj-csbprc 38000 | More direct proof of ~ csb... |
| bj-exlimvmpi 38001 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 38002 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 38003 | Lemma for theorems of the ... |
| bj-exlimmpbir 38004 | Lemma for theorems of the ... |
| bj-vtoclf 38005 | Remove dependency on ~ ax-... |
| bj-vtocl 38006 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 38007 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 38008 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 38009 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 38010 | A version of ~ vtoclg with... |
| bj-rabeqbid 38011 | Version of ~ rabeqbidv wit... |
| bj-seex 38012 | Version of ~ seex with a d... |
| bj-nfcf 38013 | Version of ~ df-nfc with a... |
| bj-sepg 38014 | Version of ~ sepg which do... |
| bj-inex1gALT 38015 | Proof of ~ inex1g from ~ s... |
| bj-elabd2ALT 38016 | Alternate proof of ~ elabd... |
| bj-unrab 38017 | Generalization of ~ unrab ... |
| bj-inrab 38018 | Generalization of ~ inrab ... |
| bj-inrab2 38019 | Shorter proof of ~ inrab .... |
| bj-inrab3 38020 | Generalization of ~ dfrab3... |
| bj-rabtr 38021 | Restricted class abstracti... |
| bj-rabtrALT 38022 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 38023 | Proof of ~ bj-rabtr found ... |
| bj-gabss 38026 | Inclusion of generalized c... |
| bj-gabssd 38027 | Inclusion of generalized c... |
| bj-gabeqd 38028 | Equality of generalized cl... |
| bj-gabeqis 38029 | Equality of generalized cl... |
| bj-elgab 38030 | Elements of a generalized ... |
| bj-gabima 38031 | Generalized class abstract... |
| bj-ru1 38034 | A version of Russell's par... |
| bj-ru 38035 | Remove dependency on ~ ax-... |
| currysetlem 38036 | Lemma for ~ currysetlem , ... |
| curryset 38037 | Curry's paradox in set the... |
| currysetlem1 38038 | Lemma for ~ currysetALT . ... |
| currysetlem2 38039 | Lemma for ~ currysetALT . ... |
| currysetlem3 38040 | Lemma for ~ currysetALT . ... |
| currysetALT 38041 | Alternate proof of ~ curry... |
| bj-n0i 38042 | Inference associated with ... |
| bj-disjsn01 38043 | Disjointness of the single... |
| bj-0nel1 38044 | The empty set does not bel... |
| bj-1nel0 38045 | ` 1o ` does not belong to ... |
| bj-xpimasn 38046 | The image of a singleton, ... |
| bj-xpima1sn 38047 | The image of a singleton b... |
| bj-xpima1snALT 38048 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 38049 | The image of a singleton b... |
| bj-xpnzex 38050 | If the first factor of a p... |
| bj-xpexg2 38051 | Curried (exported) form of... |
| bj-xpnzexb 38052 | If the first factor of a p... |
| bj-cleq 38053 | Substitution property for ... |
| bj-snsetex 38054 | The class of sets "whose s... |
| bj-clexab 38055 | Sethood of certain classes... |
| bj-sngleq 38058 | Substitution property for ... |
| bj-elsngl 38059 | Characterization of the el... |
| bj-snglc 38060 | Characterization of the el... |
| bj-snglss 38061 | The singletonization of a ... |
| bj-0nelsngl 38062 | The empty set is not a mem... |
| bj-snglinv 38063 | Inverse of singletonizatio... |
| bj-snglex 38064 | A class is a set if and on... |
| bj-tageq 38067 | Substitution property for ... |
| bj-eltag 38068 | Characterization of the el... |
| bj-0eltag 38069 | The empty set belongs to t... |
| bj-tagn0 38070 | The tagging of a class is ... |
| bj-tagss 38071 | The tagging of a class is ... |
| bj-snglsstag 38072 | The singletonization is in... |
| bj-sngltagi 38073 | The singletonization is in... |
| bj-sngltag 38074 | The singletonization and t... |
| bj-tagci 38075 | Characterization of the el... |
| bj-tagcg 38076 | Characterization of the el... |
| bj-taginv 38077 | Inverse of tagging. (Cont... |
| bj-tagex 38078 | A class is a set if and on... |
| bj-xtageq 38079 | The products of a given cl... |
| bj-xtagex 38080 | The product of a set and t... |
| bj-projeq 38083 | Substitution property for ... |
| bj-projeq2 38084 | Substitution property for ... |
| bj-projun 38085 | The class projection on a ... |
| bj-projex 38086 | Sethood of the class proje... |
| bj-projval 38087 | Value of the class project... |
| bj-1upleq 38090 | Substitution property for ... |
| bj-pr1eq 38093 | Substitution property for ... |
| bj-pr1un 38094 | The first projection prese... |
| bj-pr1val 38095 | Value of the first project... |
| bj-pr11val 38096 | Value of the first project... |
| bj-pr1ex 38097 | Sethood of the first proje... |
| bj-1uplth 38098 | The characteristic propert... |
| bj-1uplex 38099 | A monuple is a set if and ... |
| bj-1upln0 38100 | A monuple is nonempty. (C... |
| bj-2upleq 38103 | Substitution property for ... |
| bj-pr21val 38104 | Value of the first project... |
| bj-pr2eq 38107 | Substitution property for ... |
| bj-pr2un 38108 | The second projection pres... |
| bj-pr2val 38109 | Value of the second projec... |
| bj-pr22val 38110 | Value of the second projec... |
| bj-pr2ex 38111 | Sethood of the second proj... |
| bj-2uplth 38112 | The characteristic propert... |
| bj-2uplex 38113 | A couple is a set if and o... |
| bj-2upln0 38114 | A couple is nonempty. (Co... |
| bj-2upln1upl 38115 | A couple is never equal to... |
| bj-rcleqf 38116 | Relative version of ~ cleq... |
| bj-rcleq 38117 | Relative version of ~ dfcl... |
| bj-reabeq 38118 | Relative form of ~ eqabb .... |
| bj-disj2r 38119 | Relative version of ~ ssdi... |
| bj-sscon 38120 | Contraposition law for rel... |
| bj-abex 38121 | Two ways of stating that t... |
| bj-clex 38122 | Two ways of stating that a... |
| bj-axsn 38123 | Two ways of stating the ax... |
| bj-snexg 38125 | A singleton built on a set... |
| bj-snex 38126 | A singleton is a set. See... |
| bj-axbun 38127 | Two ways of stating the ax... |
| bj-unexg 38129 | Existence of binary unions... |
| bj-prexg 38130 | Existence of unordered pai... |
| bj-prex 38131 | Existence of unordered pai... |
| bj-axadj 38132 | Two ways of stating the ax... |
| bj-adjg1 38134 | Existence of the result of... |
| bj-snfromadj 38135 | Singleton from adjunction ... |
| bj-prfromadj 38136 | Unordered pair from adjunc... |
| bj-adjfrombun 38137 | Adjunction from singleton ... |
| bj-elinxp 38138 | Membership in an intersect... |
| resssres 38139 | A restricted class is incl... |
| bj-elres 38140 | Membership in a restrictio... |
| bj-dfres2 38141 | Alternate definition of th... |
| bj-resss2 38142 | The restriction of a class... |
| relpart 38143 | Two ways of expressing the... |
| bj-dfrel3 38144 | Alternate definition of re... |
| relssresv 38145 | A relation is included in ... |
| bj-dfrel4v 38146 | A relation can be expresse... |
| bj-dfrel2 38147 | Alternate definition of re... |
| bj-cnvcnv2 38148 | The double converse of a c... |
| bj-cnvcnv 38149 | The double converse of a c... |
| elco 38150 | Membership in a compositio... |
| coi1res 38151 | Precomposing with the iden... |
| bj-coi1 38152 | The identity relation is a... |
| eleq2w2ALT 38153 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 38154 | Alternate proof of ~ clel3... |
| bj-pw0ALT 38155 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 38156 | Quantitative version of ~ ... |
| bj-unirel 38157 | Quantitative version of ~ ... |
| bj-elpwg 38158 | If the intersection of two... |
| bj-velpwALT 38159 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 38160 | Alternate proof of ~ elpwg... |
| bj-vjust 38161 | Justification theorem for ... |
| bj-nul 38162 | Two formulations of the ax... |
| bj-nuliota 38163 | Definition of the empty se... |
| bj-nuliotaALT 38164 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 38165 | Alternate proof of ~ vtocl... |
| bj-elsn12g 38166 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 38167 | Biconditional version of ~... |
| bj-pwcfsdom 38168 | Remove hypothesis from ~ p... |
| bj-grur1 38169 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 38170 | The extension of a predica... |
| bj-dfid2ALT 38171 | Alternate version of ~ dfi... |
| bj-0nelopab 38172 | The empty set is not an el... |
| bj-brrelex12ALT 38173 | Two classes related by a b... |
| bj-epelg 38174 | The membership relation an... |
| bj-epelb 38175 | Two classes are related by... |
| bj-nsnid 38176 | A set does not contain the... |
| bj-rdg0gALT 38177 | Alternate proof of ~ rdg0g... |
| bj-vn0ALT 38178 | Alternate proof of ~ vn0 w... |
| bj-axnul 38179 | Over the base theory ~ ax-... |
| bj-rep 38180 | Version of the axiom of re... |
| bj-axseprep 38181 | Axiom of separation (unive... |
| bj-axreprepsep 38182 | Strong axiom of replacemen... |
| bj-evaleq 38183 | Equality theorem for the `... |
| bj-evalfun 38184 | The evaluation at a class ... |
| bj-evalfn 38185 | The evaluation at a class ... |
| bj-evalf 38186 | The evaluation at a class ... |
| bj-evalval 38187 | Value of the evaluation at... |
| bj-evalid 38188 | The evaluation at a set of... |
| bj-ndxarg 38189 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 38190 | Closed general form of ~ s... |
| bj-rest00 38193 | An elementwise intersectio... |
| bj-restsn 38194 | An elementwise intersectio... |
| bj-restsnss 38195 | Special case of ~ bj-rests... |
| bj-restsnss2 38196 | Special case of ~ bj-rests... |
| bj-restsn0 38197 | An elementwise intersectio... |
| bj-restsn10 38198 | Special case of ~ bj-rests... |
| bj-restsnid 38199 | The elementwise intersecti... |
| bj-rest10 38200 | An elementwise intersectio... |
| bj-rest10b 38201 | Alternate version of ~ bj-... |
| bj-restn0 38202 | An elementwise intersectio... |
| bj-restn0b 38203 | Alternate version of ~ bj-... |
| bj-restpw 38204 | The elementwise intersecti... |
| bj-rest0 38205 | An elementwise intersectio... |
| bj-restb 38206 | An elementwise intersectio... |
| bj-restv 38207 | An elementwise intersectio... |
| bj-resta 38208 | An elementwise intersectio... |
| bj-restuni 38209 | The union of an elementwis... |
| bj-restuni2 38210 | The union of an elementwis... |
| bj-restreg 38211 | A reformulation of the axi... |
| bj-raldifsn 38212 | All elements of a set sati... |
| bj-0int 38213 | If ` A ` is a collection o... |
| bj-mooreset 38214 | A Moore collection is a se... |
| bj-ismoore 38217 | Characterization of Moore ... |
| bj-ismoored0 38218 | Necessary condition to be ... |
| bj-ismoored 38219 | Necessary condition to be ... |
| bj-ismoored2 38220 | Necessary condition to be ... |
| bj-ismooredr 38221 | Sufficient condition to be... |
| bj-ismooredr2 38222 | Sufficient condition to be... |
| bj-discrmoore 38223 | The powerclass ` ~P A ` is... |
| bj-0nmoore 38224 | The empty set is not a Moo... |
| bj-snmoore 38225 | A singleton is a Moore col... |
| bj-snmooreb 38226 | A singleton is a Moore col... |
| bj-prmoore 38227 | A pair formed of two neste... |
| bj-pr0moore 38228 | A pair containing the empt... |
| bj-0nelmpt 38229 | The empty set is not an el... |
| bj-mptval 38230 | Value of a function given ... |
| bj-dfmpoa 38231 | An equivalent definition o... |
| bj-mpomptALT 38232 | Alternate proof of ~ mpomp... |
| setsstrset 38247 | Relation between ~ df-sets... |
| bj-nfald 38248 | Variant of ~ nfald . (Con... |
| bj-nfexd 38249 | Variant of ~ nfexd . (Con... |
| cgsex2gd 38250 | Implicit substitution infe... |
| copsex2gd 38251 | Implicit substitution infe... |
| copsex2d 38252 | Implicit substitution dedu... |
| copsex2b 38253 | Biconditional form of ~ co... |
| opelopabd 38254 | Membership of an ordered p... |
| opelopabb 38255 | Membership of an ordered p... |
| opelopabbv 38256 | Membership of an ordered p... |
| bj-opelrelex 38257 | The coordinates of an orde... |
| bj-opelresdm 38258 | If an ordered pair is in a... |
| bj-brresdm 38259 | If two classes are related... |
| brabd0 38260 | Expressing that two sets a... |
| brabd 38261 | Expressing that two sets a... |
| bj-brab2a1 38262 | "Unbounded" version of ~ b... |
| bj-opabssvv 38263 | A variant of ~ relopabiv (... |
| bj-funidres 38264 | The restricted identity re... |
| bj-opelidb 38265 | Characterization of the or... |
| bj-opelidb1 38266 | Characterization of the or... |
| bj-inexeqex 38267 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 38268 | If the intersection of two... |
| bj-opelid 38269 | Characterization of the or... |
| bj-ideqg 38270 | Characterization of the cl... |
| bj-ideqgALT 38271 | Alternate proof of ~ bj-id... |
| bj-ideqb 38272 | Characterization of classe... |
| bj-idres 38273 | Alternate expression for t... |
| bj-opelidres 38274 | Characterization of the or... |
| bj-idreseq 38275 | Sufficient condition for t... |
| bj-idreseqb 38276 | Characterization for two c... |
| bj-ideqg1 38277 | For sets, the identity rel... |
| bj-ideqg1ALT 38278 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 38279 | Characterization of the co... |
| bj-elid3 38280 | Characterization of the co... |
| bj-elid4 38281 | Characterization of the el... |
| bj-elid5 38282 | Characterization of the el... |
| bj-elid6 38283 | Characterization of the el... |
| bj-elid7 38284 | Characterization of the el... |
| bj-diagval 38287 | Value of the functionalize... |
| bj-diagval2 38288 | Value of the functionalize... |
| bj-eldiag 38289 | Characterization of the el... |
| bj-eldiag2 38290 | Characterization of the el... |
| bj-imdirvallem 38293 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 38294 | Value of the functionalize... |
| bj-imdirval2lem 38295 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 38296 | Value of the functionalize... |
| bj-imdirval3 38297 | Value of the functionalize... |
| bj-imdiridlem 38298 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 38299 | Functorial property of the... |
| bj-opelopabid 38300 | Membership in an ordered-p... |
| bj-opabco 38301 | Composition of ordered-pai... |
| bj-xpcossxp 38302 | The composition of two Car... |
| bj-imdirco 38303 | Functorial property of the... |
| bj-iminvval 38306 | Value of the functionalize... |
| bj-iminvval2 38307 | Value of the functionalize... |
| bj-iminvid 38308 | Functorial property of the... |
| bj-inftyexpitaufo 38315 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 38318 | An element of the circle a... |
| bj-inftyexpiinv 38321 | Utility theorem for the in... |
| bj-inftyexpiinj 38322 | Injectivity of the paramet... |
| bj-inftyexpidisj 38323 | An element of the circle a... |
| bj-ccinftydisj 38326 | The circle at infinity is ... |
| bj-elccinfty 38327 | A lemma for infinite exten... |
| bj-ccssccbar 38330 | Complex numbers are extend... |
| bj-ccinftyssccbar 38331 | Infinite extended complex ... |
| bj-pinftyccb 38334 | The class ` pinfty ` is an... |
| bj-pinftynrr 38335 | The extended complex numbe... |
| bj-minftyccb 38338 | The class ` minfty ` is an... |
| bj-minftynrr 38339 | The extended complex numbe... |
| bj-pinftynminfty 38340 | The extended complex numbe... |
| bj-rrhatsscchat 38349 | The real projective line i... |
| bj-imafv 38364 | If the direct image of a s... |
| bj-funun 38365 | Value of a function expres... |
| bj-fununsn1 38366 | Value of a function expres... |
| bj-fununsn2 38367 | Value of a function expres... |
| bj-fvsnun1 38368 | The value of a function wi... |
| bj-fvsnun2 38369 | The value of a function wi... |
| bj-fvmptunsn1 38370 | Value of a function expres... |
| bj-fvmptunsn2 38371 | Value of a function expres... |
| bj-iomnnom 38372 | The canonical bijection fr... |
| bj-smgrpssmgm 38381 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 38382 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 38383 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 38384 | Monoids are semigroups (el... |
| bj-cmnssmnd 38385 | Commutative monoids are mo... |
| bj-cmnssmndel 38386 | Commutative monoids are mo... |
| bj-grpssmnd 38387 | Groups are monoids. (Cont... |
| bj-grpssmndel 38388 | Groups are monoids (elemen... |
| bj-ablssgrp 38389 | Abelian groups are groups.... |
| bj-ablssgrpel 38390 | Abelian groups are groups ... |
| bj-ablsscmn 38391 | Abelian groups are commuta... |
| bj-ablsscmnel 38392 | Abelian groups are commuta... |
| bj-modssabl 38393 | (The additive groups of) m... |
| bj-vecssmod 38394 | Vector spaces are modules.... |
| bj-vecssmodel 38395 | Vector spaces are modules ... |
| bj-finsumval0 38398 | Value of a finite sum. (C... |
| bj-fvimacnv0 38399 | Variant of ~ fvimacnv wher... |
| bj-isvec 38400 | The predicate "is a vector... |
| bj-fldssdrng 38401 | Fields are division rings.... |
| bj-flddrng 38402 | Fields are division rings ... |
| bj-rrdrg 38403 | The field of real numbers ... |
| bj-isclm 38404 | The predicate "is a subcom... |
| bj-isrvec 38407 | The predicate "is a real v... |
| bj-rvecmod 38408 | Real vector spaces are mod... |
| bj-rvecssmod 38409 | Real vector spaces are mod... |
| bj-rvecrr 38410 | The field of scalars of a ... |
| bj-isrvecd 38411 | The predicate "is a real v... |
| bj-rvecvec 38412 | Real vector spaces are vec... |
| bj-isrvec2 38413 | The predicate "is a real v... |
| bj-rvecssvec 38414 | Real vector spaces are vec... |
| bj-rveccmod 38415 | Real vector spaces are sub... |
| bj-rvecsscmod 38416 | Real vector spaces are sub... |
| bj-rvecsscvec 38417 | Real vector spaces are sub... |
| bj-rveccvec 38418 | Real vector spaces are sub... |
| bj-rvecssabl 38419 | (The additive groups of) r... |
| bj-rvecabl 38420 | (The additive groups of) r... |
| bj-subcom 38421 | A consequence of commutati... |
| bj-lineqi 38422 | Solution of a (scalar) lin... |
| bj-bary1lem 38423 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 38424 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 38425 | Barycentric coordinates in... |
| bj-endval 38428 | Value of the monoid of end... |
| bj-endbase 38429 | Base set of the monoid of ... |
| bj-endcomp 38430 | Composition law of the mon... |
| bj-endmnd 38431 | The monoid of endomorphism... |
| taupilem3 38432 | Lemma for tau-related theo... |
| taupilemrplb 38433 | A set of positive reals ha... |
| taupilem1 38434 | Lemma for ~ taupi . A pos... |
| taupilem2 38435 | Lemma for ~ taupi . The s... |
| taupi 38436 | Relationship between ` _ta... |
| dfgcd3 38437 | Alternate definition of th... |
| irrdifflemf 38438 | Lemma for ~ irrdiff . The... |
| irrdiff 38439 | The irrationals are exactl... |
| qdiff 38440 | The rationals are exactly ... |
| qdiffALT 38441 | Alternate proof of ~ qdiff... |
| iccioo01 38442 | The closed unit interval i... |
| csbrecsg 38443 | Move class substitution in... |
| csbrdgg 38444 | Move class substitution in... |
| csboprabg 38445 | Move class substitution in... |
| csbmpo123 38446 | Move class substitution in... |
| con1bii2 38447 | A contraposition inference... |
| con2bii2 38448 | A contraposition inference... |
| vtoclefex 38449 | Implicit substitution of a... |
| rnmptsn 38450 | The range of a function ma... |
| f1omptsnlem 38451 | This is the core of the pr... |
| f1omptsn 38452 | A function mapping to sing... |
| mptsnunlem 38453 | This is the core of the pr... |
| mptsnun 38454 | A class ` B ` is equal to ... |
| dissneqlem 38455 | This is the core of the pr... |
| dissneq 38456 | Any topology that contains... |
| exlimim 38457 | Closed form of ~ exlimimd ... |
| exlimimd 38458 | Existential elimination ru... |
| exellim 38459 | Closed form of ~ exellimdd... |
| exellimddv 38460 | Eliminate an antecedent wh... |
| topdifinfindis 38461 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 38462 | This is the core of the pr... |
| topdifinffin 38463 | Part of Exercise 3 of [Mun... |
| topdifinf 38464 | Part of Exercise 3 of [Mun... |
| topdifinfeq 38465 | Two different ways of defi... |
| icorempo 38466 | Closed-below, open-above i... |
| icoreresf 38467 | Closed-below, open-above i... |
| icoreval 38468 | Value of the closed-below,... |
| icoreelrnab 38469 | Elementhood in the set of ... |
| isbasisrelowllem1 38470 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 38471 | Lemma for ~ isbasisrelowl ... |
| icoreclin 38472 | The set of closed-below, o... |
| isbasisrelowl 38473 | The set of all closed-belo... |
| icoreunrn 38474 | The union of all closed-be... |
| istoprelowl 38475 | The set of all closed-belo... |
| icoreelrn 38476 | A class abstraction which ... |
| iooelexlt 38477 | An element of an open inte... |
| relowlssretop 38478 | The lower limit topology o... |
| relowlpssretop 38479 | The lower limit topology o... |
| sucneqond 38480 | Inequality of an ordinal s... |
| sucneqoni 38481 | Inequality of an ordinal s... |
| onsucuni3 38482 | If an ordinal number has a... |
| 1oequni2o 38483 | The ordinal number ` 1o ` ... |
| rdgsucuni 38484 | If an ordinal number has a... |
| rdgeqoa 38485 | If a recursive function wi... |
| elxp8 38486 | Membership in a Cartesian ... |
| cbveud 38487 | Deduction used to change b... |
| cbvreud 38488 | Deduction used to change b... |
| difunieq 38489 | The difference of unions i... |
| inunissunidif 38490 | Theorem about subsets of t... |
| rdgellim 38491 | Elementhood in a recursive... |
| rdglimss 38492 | A recursive definition at ... |
| rdgssun 38493 | In a recursive definition ... |
| exrecfnlem 38494 | Lemma for ~ exrecfn . (Co... |
| exrecfn 38495 | Theorem about the existenc... |
| exrecfnpw 38496 | For any base set, a set wh... |
| finorwe 38497 | If the Axiom of Infinity i... |
| dffinxpf 38500 | This theorem is the same a... |
| finxpeq1 38501 | Equality theorem for Carte... |
| finxpeq2 38502 | Equality theorem for Carte... |
| csbfinxpg 38503 | Distribute proper substitu... |
| finxpreclem1 38504 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 38505 | Lemma for ` ^^ ` recursion... |
| finxp0 38506 | The value of Cartesian exp... |
| finxp1o 38507 | The value of Cartesian exp... |
| finxpreclem3 38508 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 38509 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 38510 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 38511 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 38512 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 38513 | The value of Cartesian exp... |
| finxp2o 38514 | The value of Cartesian exp... |
| finxp3o 38515 | The value of Cartesian exp... |
| finxpnom 38516 | Cartesian exponentiation w... |
| finxp00 38517 | Cartesian exponentiation o... |
| iunctb2 38518 | Using the axiom of countab... |
| domalom 38519 | A class which dominates ev... |
| isinf2 38520 | The converse of ~ isinf . ... |
| ctbssinf 38521 | Using the axiom of choice,... |
| ralssiun 38522 | The index set of an indexe... |
| nlpineqsn 38523 | For every point ` p ` of a... |
| nlpfvineqsn 38524 | Given a subset ` A ` of ` ... |
| fvineqsnf1 38525 | A theorem about functions ... |
| fvineqsneu 38526 | A theorem about functions ... |
| fvineqsneq 38527 | A theorem about functions ... |
| pibp16 38528 | Property P000016 of pi-bas... |
| pibp19 38529 | Property P000019 of pi-bas... |
| pibp21 38530 | Property P000021 of pi-bas... |
| pibt1 38531 | Theorem T000001 of pi-base... |
| pibt2 38532 | Theorem T000002 of pi-base... |
| wl-section-prop 38533 | Intuitionistic logic is no... |
| wl-section-boot 38537 | In this section, I provide... |
| wl-luk-imim1i 38538 | Inference adding common co... |
| wl-luk-syl 38539 | An inference version of th... |
| wl-luk-imtrid 38540 | A syllogism rule of infere... |
| wl-luk-pm2.18d 38541 | Deduction based on reducti... |
| wl-luk-con4i 38542 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 38543 | Inference rule. Copy of ~... |
| wl-luk-a1i 38544 | Inference rule. Copy of ~... |
| wl-luk-mpi 38545 | A nested _modus ponens_ in... |
| wl-luk-imim2i 38546 | Inference adding common an... |
| wl-luk-imtrdi 38547 | A syllogism rule of infere... |
| wl-luk-ax3 38548 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 38549 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 38550 | This theorem, called "Asse... |
| wl-luk-com12 38551 | Inference that swaps (comm... |
| wl-luk-pm2.21 38552 | From a wff and its negatio... |
| wl-luk-con1i 38553 | A contraposition inference... |
| wl-luk-ja 38554 | Inference joining the ante... |
| wl-luk-imim2 38555 | A closed form of syllogism... |
| wl-luk-a1d 38556 | Deduction introducing an e... |
| wl-luk-ax2 38557 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 38558 | Principle of identity. Th... |
| wl-luk-notnotr 38559 | Converse of double negatio... |
| wl-luk-pm2.04 38560 | Swap antecedents. Theorem... |
| wl-section-impchain 38561 | An implication like ` ( ps... |
| wl-impchain-mp-x 38562 | This series of theorems pr... |
| wl-impchain-mp-0 38563 | This theorem is the start ... |
| wl-impchain-mp-1 38564 | This theorem is in fact a ... |
| wl-impchain-mp-2 38565 | This theorem is in fact a ... |
| wl-impchain-com-1.x 38566 | It is often convenient to ... |
| wl-impchain-com-1.1 38567 | A degenerate form of antec... |
| wl-impchain-com-1.2 38568 | This theorem is in fact a ... |
| wl-impchain-com-1.3 38569 | This theorem is in fact a ... |
| wl-impchain-com-1.4 38570 | This theorem is in fact a ... |
| wl-impchain-com-n.m 38571 | This series of theorems al... |
| wl-impchain-com-2.3 38572 | This theorem is in fact a ... |
| wl-impchain-com-2.4 38573 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 38574 | This theorem is in fact a ... |
| wl-impchain-a1-x 38575 | If an implication chain is... |
| wl-impchain-a1-1 38576 | Inference rule, a copy of ... |
| wl-impchain-a1-2 38577 | Inference rule, a copy of ... |
| wl-impchain-a1-3 38578 | Inference rule, a copy of ... |
| wl-ifp-ncond1 38579 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 38580 | If one case of an ` if- ` ... |
| wl-ifpimpr 38581 | If one case of an ` if- ` ... |
| wl-ifp4impr 38582 | If one case of an ` if- ` ... |
| wl-df-3xor 38583 | Alternative definition of ... |
| wl-df3xor2 38584 | Alternative definition of ... |
| wl-df3xor3 38585 | Alternative form of ~ wl-d... |
| wl-3xortru 38586 | If the first input is true... |
| wl-3xorfal 38587 | If the first input is fals... |
| wl-3xorbi 38588 | Triple xor can be replaced... |
| wl-3xorbi2 38589 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 38590 | Equivalence theorem for tr... |
| wl-3xorbi123i 38591 | Equivalence theorem for tr... |
| wl-3xorrot 38592 | Rotation law for triple xo... |
| wl-3xorcoma 38593 | Commutative law for triple... |
| wl-3xorcomb 38594 | Commutative law for triple... |
| wl-3xornot1 38595 | Flipping the first input f... |
| wl-3xornot 38596 | Triple xor distributes ove... |
| wl-1xor 38597 | In the recursive scheme ... |
| wl-2xor 38598 | In the recursive scheme ... |
| wl-df-3mintru2 38599 | Alternative definition of ... |
| wl-df2-3mintru2 38600 | The adder carry in disjunc... |
| wl-df3-3mintru2 38601 | The adder carry in conjunc... |
| wl-df4-3mintru2 38602 | An alternative definition ... |
| wl-1mintru1 38603 | Using the recursion formul... |
| wl-1mintru2 38604 | Using the recursion formul... |
| wl-2mintru1 38605 | Using the recursion formul... |
| wl-2mintru2 38606 | Using the recursion formul... |
| wl-df3maxtru1 38607 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 38609 | A version of ~ ax-wl-13v w... |
| wl-cleq-0 38610 |
Disclaimer: |
| wl-cleq-1 38611 |
Disclaimer: |
| wl-cleq-2 38612 |
Disclaimer: |
| wl-cleq-3 38613 |
Disclaimer: |
| wl-cleq-4 38614 |
Disclaimer: |
| wl-cleq-5 38615 |
Disclaimer: |
| wl-cleq-6 38616 |
Disclaimer: |
| wl-df-clab 38619 | Disclaimer: The material ... |
| wl-isseteq 38620 | A class equal to a set var... |
| wl-ax12v2cl 38621 | The class version of ~ ax1... |
| wl-df.clab 38622 | Define class abstractions,... |
| wl-df.cleq 38623 | Define the equality connec... |
| wl-dfcleq.basic 38624 | This theorem is a conserva... |
| wl-dfcleq.just 38625 | The hypotheses added to th... |
| wl-df.clel 38626 | Define the membership conn... |
| wl-dfclel.basic 38627 | This theorem gives a conse... |
| wl-dfclel.just 38628 | Add a hypothesis to ~ wl-d... |
| wl-dfcleq 38629 | The defining characterizat... |
| wl-dfclel 38630 | The defining characterizat... |
| wl-mps 38631 | Replacing a nested consequ... |
| wl-syls1 38632 | Replacing a nested consequ... |
| wl-syls2 38633 | Replacing a nested anteced... |
| wl-embant 38634 | A true wff can always be a... |
| wl-orel12 38635 | In a conjunctive normal fo... |
| wl-cases2-dnf 38636 | A particular instance of ~... |
| wl-cbvmotv 38637 | Change bound variable. Us... |
| wl-moteq 38638 | Change bound variable. Us... |
| wl-motae 38639 | Change bound variable. Us... |
| wl-moae 38640 | Two ways to express "at mo... |
| wl-euae 38641 | Two ways to express "exact... |
| wl-nax6im 38642 | The following series of th... |
| wl-hbae1 38643 | This specialization of ~ h... |
| wl-naevhba1v 38644 | An instance of ~ hbn1w app... |
| wl-spae 38645 | Prove an instance of ~ sp ... |
| wl-speqv 38646 | Under the assumption ` -. ... |
| wl-19.8eqv 38647 | Under the assumption ` -. ... |
| wl-19.2reqv 38648 | Under the assumption ` -. ... |
| wl-nfalv 38649 | If ` x ` is not present in... |
| wl-nfimf1 38650 | An antecedent is irrelevan... |
| wl-nfae1 38651 | Unlike ~ nfae , this speci... |
| wl-nfnae1 38652 | Unlike ~ nfnae , this spec... |
| wl-aetr 38653 | A transitive law for varia... |
| wl-axc11r 38654 | Same as ~ axc11r , but usi... |
| wl-dral1d 38655 | A version of ~ dral1 with ... |
| wl-cbvalnaed 38656 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 38657 | A more general version of ... |
| wl-exeq 38658 | The semantics of ` E. x y ... |
| wl-aleq 38659 | The semantics of ` A. x y ... |
| wl-nfeqfb 38660 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 38661 | If ` y ` is not free in ` ... |
| wl-equsalvw 38662 | Version of ~ equsalv with ... |
| wl-equsald 38663 | Deduction version of ~ equ... |
| wl-equsaldv 38664 | Deduction version of ~ equ... |
| wl-equsal 38665 | A useful equivalence relat... |
| wl-equsal1t 38666 | The expression ` x = y ` i... |
| wl-equsalcom 38667 | This simple equivalence ea... |
| wl-equsal1i 38668 | The antecedent ` x = y ` i... |
| wl-sbid2ft 38669 | A more general version of ... |
| wl-cbvalsbi 38670 | Change bounded variables i... |
| wl-sbrimt 38671 | Substitution with a variab... |
| wl-sblimt 38672 | Substitution with a variab... |
| wl-sb9v 38673 | Commutation of quantificat... |
| wl-sb8ft 38674 | Substitution of variable i... |
| wl-sb8eft 38675 | Substitution of variable i... |
| wl-sb8t 38676 | Substitution of variable i... |
| wl-sb8et 38677 | Substitution of variable i... |
| wl-sbhbt 38678 | Closed form of ~ sbhb . C... |
| wl-sbnf1 38679 | Two ways expressing that `... |
| wl-equsb3 38680 | ~ equsb3 with a distinctor... |
| wl-equsb4 38681 | Substitution applied to an... |
| wl-2sb6d 38682 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 38683 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 38684 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 38685 | Version of ~ sbcom2 with a... |
| wl-sbalnae 38686 | A theorem used in eliminat... |
| wl-sbal1 38687 | A theorem used in eliminat... |
| wl-sbal2 38688 | Move quantifier in and out... |
| wl-2spsbbi 38689 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 38690 | This theorem provides a ba... |
| wl-lem-nexmo 38691 | This theorem provides a ba... |
| wl-lem-moexsb 38692 | The antecedent ` A. x ( ph... |
| wl-alanbii 38693 | This theorem extends ~ ala... |
| wl-mo2df 38694 | Version of ~ mof with a co... |
| wl-mo2tf 38695 | Closed form of ~ mof with ... |
| wl-eudf 38696 | Version of ~ eu6 with a co... |
| wl-eutf 38697 | Closed form of ~ eu6 with ... |
| wl-euequf 38698 | ~ euequ proved with a dist... |
| wl-mo2t 38699 | Closed form of ~ mof . (C... |
| wl-mo3t 38700 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 38701 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 38702 | Substitution of variable i... |
| wl-sb8eutv 38703 | Substitution of variable i... |
| wl-sb8mot 38704 | Substitution of variable i... |
| wl-sb8motv 38705 | Substitution of variable i... |
| wl-issetft 38706 | A closed form of ~ issetf ... |
| wl-axc11rc11 38707 | Proving ~ axc11r from ~ ax... |
| wl-clabv 38708 | Variant of ~ df-clab , whe... |
| wl-dfclab 38709 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 38710 | Using class abstraction in... |
| wl-clabt 38711 | Using class abstraction in... |
| wl-eujustlem1 38712 | Version of ~ cbvexvw with ... |
| rabiun 38713 | Abstraction restricted to ... |
| iundif1 38714 | Indexed union of class dif... |
| imadifss 38715 | The difference of images i... |
| unceq 38716 | Equality theorem for uncur... |
| curunc 38717 | Currying of uncurrying. (... |
| unccur 38718 | Uncurrying of currying. (... |
| phpreu 38719 | Theorem related to pigeonh... |
| finixpnum 38720 | A finite Cartesian product... |
| fin2solem 38721 | Lemma for ~ fin2so . (Con... |
| fin2so 38722 | Any totally ordered Tarski... |
| ltflcei 38723 | Theorem to move the floor ... |
| leceifl 38724 | Theorem to move the floor ... |
| sin2h 38725 | Half-angle rule for sine. ... |
| cos2h 38726 | Half-angle rule for cosine... |
| tan2h 38727 | Half-angle rule for tangen... |
| lindsadd 38728 | In a vector space, the uni... |
| ptrest 38729 | Expressing a restriction o... |
| ptrecube 38730 | Any point in an open set o... |
| poimirlem1 38731 | Lemma for ~ poimir - the v... |
| poimirlem2 38732 | Lemma for ~ poimir - conse... |
| poimirlem3 38733 | Lemma for ~ poimir to add ... |
| poimirlem4 38734 | Lemma for ~ poimir connect... |
| poimirlem5 38735 | Lemma for ~ poimir to esta... |
| poimirlem6 38736 | Lemma for ~ poimir establi... |
| poimirlem7 38737 | Lemma for ~ poimir , simil... |
| poimirlem8 38738 | Lemma for ~ poimir , estab... |
| poimirlem9 38739 | Lemma for ~ poimir , estab... |
| poimirlem10 38740 | Lemma for ~ poimir establi... |
| poimirlem11 38741 | Lemma for ~ poimir connect... |
| poimirlem12 38742 | Lemma for ~ poimir connect... |
| poimirlem13 38743 | Lemma for ~ poimir - for a... |
| poimirlem14 38744 | Lemma for ~ poimir - for a... |
| poimirlem15 38745 | Lemma for ~ poimir , that ... |
| poimirlem16 38746 | Lemma for ~ poimir establi... |
| poimirlem17 38747 | Lemma for ~ poimir establi... |
| poimirlem18 38748 | Lemma for ~ poimir stating... |
| poimirlem19 38749 | Lemma for ~ poimir establi... |
| poimirlem20 38750 | Lemma for ~ poimir establi... |
| poimirlem21 38751 | Lemma for ~ poimir stating... |
| poimirlem22 38752 | Lemma for ~ poimir , that ... |
| poimirlem23 38753 | Lemma for ~ poimir , two w... |
| poimirlem24 38754 | Lemma for ~ poimir , two w... |
| poimirlem25 38755 | Lemma for ~ poimir stating... |
| poimirlem26 38756 | Lemma for ~ poimir showing... |
| poimirlem27 38757 | Lemma for ~ poimir showing... |
| poimirlem28 38758 | Lemma for ~ poimir , a var... |
| poimirlem29 38759 | Lemma for ~ poimir connect... |
| poimirlem30 38760 | Lemma for ~ poimir combini... |
| poimirlem31 38761 | Lemma for ~ poimir , assig... |
| poimirlem32 38762 | Lemma for ~ poimir , combi... |
| poimir 38763 | Poincare-Miranda theorem. ... |
| broucube 38764 | Brouwer - or as Kulpa call... |
| heicant 38765 | Heine-Cantor theorem: a co... |
| opnmbllem0 38766 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 38767 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 38768 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 38769 | The difference between two... |
| mblfinlem4 38770 | Backward direction of ~ is... |
| ismblfin 38771 | Measurability in terms of ... |
| ovoliunnfl 38772 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 38773 | Demonstration of ~ ovoliun... |
| voliunnfl 38774 | ~ voliun is incompatible w... |
| volsupnfl 38775 | ~ volsup is incompatible w... |
| mbfresfi 38776 | Measurability of a piecewi... |
| mbfposadd 38777 | If the sum of two measurab... |
| cnambfre 38778 | A real-valued, a.e. contin... |
| dvtanlem 38779 | Lemma for ~ dvtan - the do... |
| dvtan 38780 | Derivative of tangent. (C... |
| itg2addnclem 38781 | An alternate expression fo... |
| itg2addnclem2 38782 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 38783 | Lemma incomprehensible in ... |
| itg2addnc 38784 | Alternate proof of ~ itg2a... |
| itg2gt0cn 38785 | ~ itg2gt0 holds on functio... |
| ibladdnclem 38786 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 38787 | Choice-free analogue of ~ ... |
| itgaddnclem1 38788 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 38789 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 38790 | Choice-free analogue of ~ ... |
| iblsubnc 38791 | Choice-free analogue of ~ ... |
| itgsubnc 38792 | Choice-free analogue of ~ ... |
| iblabsnclem 38793 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 38794 | Choice-free analogue of ~ ... |
| iblmulc2nc 38795 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 38796 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 38797 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 38798 | Choice-free analogue of ~ ... |
| itgabsnc 38799 | Choice-free analogue of ~ ... |
| itggt0cn 38800 | ~ itggt0 holds for continu... |
| ftc1cnnclem 38801 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 38802 | Choice-free proof of ~ ftc... |
| ftc1anclem1 38803 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 38804 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 38805 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 38806 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 38807 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 38808 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 38809 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 38810 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 38811 | ~ ftc1a holds for function... |
| ftc2nc 38812 | Choice-free proof of ~ ftc... |
| asindmre 38813 | Real part of domain of dif... |
| dvasin 38814 | Derivative of arcsine. (C... |
| dvacos 38815 | Derivative of arccosine. ... |
| dvreasin 38816 | Real derivative of arcsine... |
| dvreacos 38817 | Real derivative of arccosi... |
| areacirclem1 38818 | Antiderivative of cross-se... |
| areacirclem2 38819 | Endpoint-inclusive continu... |
| areacirclem3 38820 | Integrability of cross-sec... |
| areacirclem4 38821 | Endpoint-inclusive continu... |
| areacirclem5 38822 | Finding the cross-section ... |
| areacirc 38823 | The area of a circle of ra... |
| findcard4 38824 | Schema for strong inductio... |
| dfproplem 38833 | Given a set A, the set of ... |
| varprop 38834 | Variables encoded as natur... |
| negprop 38835 | The negation of a sentence... |
| impprop 38836 | The implication between tw... |
| dfprop1 38837 | The set of variables encod... |
| dfprop2 38838 | Every sentence of proposit... |
| dfprop 38839 | The set of sentences of pr... |
| unirep 38840 | Define a quantity whose de... |
| cover2 38841 | Two ways of expressing the... |
| cover2g 38842 | Two ways of expressing the... |
| brabg2 38843 | Relation by a binary relat... |
| opelopab3 38844 | Ordered pair membership in... |
| cocanfo 38845 | Cancellation of a surjecti... |
| brresi2 38846 | Restriction of a binary re... |
| fnopabeqd 38847 | Equality deduction for fun... |
| fvopabf4g 38848 | Function value of an opera... |
| fnopabco 38849 | Composition of a function ... |
| opropabco 38850 | Composition of an operator... |
| cocnv 38851 | Composition with a functio... |
| f1ocan1fv 38852 | Cancel a composition by a ... |
| f1ocan2fv 38853 | Cancel a composition by th... |
| inixp 38854 | Intersection of Cartesian ... |
| upixp 38855 | Universal property of the ... |
| abrexdom 38856 | An indexed set is dominate... |
| abrexdom2 38857 | An indexed set is dominate... |
| ac6gf 38858 | Axiom of Choice. (Contrib... |
| indexa 38859 | If for every element of an... |
| indexdom 38860 | If for every element of an... |
| frinfm 38861 | A subset of a well-founded... |
| welb 38862 | A nonempty subset of a wel... |
| supex2g 38863 | Existence of supremum. (C... |
| supclt 38864 | Closure of supremum. (Con... |
| supubt 38865 | Upper bound property of su... |
| filbcmb 38866 | Combine a finite set of lo... |
| fzmul 38867 | Membership of a product in... |
| sdclem2 38868 | Lemma for ~ sdc . (Contri... |
| sdclem1 38869 | Lemma for ~ sdc . (Contri... |
| sdc 38870 | Strong dependent choice. ... |
| fdc 38871 | Finite version of dependen... |
| fdc1 38872 | Variant of ~ fdc with no s... |
| seqpo 38873 | Two ways to say that a seq... |
| incsequz 38874 | An increasing sequence of ... |
| incsequz2 38875 | An increasing sequence of ... |
| nnubfi 38876 | A bounded above set of pos... |
| nninfnub 38877 | An infinite set of positiv... |
| subspopn 38878 | An open set is open in the... |
| neificl 38879 | Neighborhoods are closed u... |
| lpss2 38880 | Limit points of a subset a... |
| metf1o 38881 | Use a bijection with a met... |
| blssp 38882 | A ball in the subspace met... |
| mettrifi 38883 | Generalized triangle inequ... |
| lmclim2 38884 | A sequence in a metric spa... |
| geomcau 38885 | If the distance between co... |
| caures 38886 | The restriction of a Cauch... |
| caushft 38887 | A shifted Cauchy sequence ... |
| constcncf 38888 | A constant function is a c... |
| cnres2 38889 | The restriction of a conti... |
| cnresima 38890 | A continuous function is c... |
| cncfres 38891 | A continuous function on c... |
| istotbnd 38895 | The predicate "is a totall... |
| istotbnd2 38896 | The predicate "is a totall... |
| istotbnd3 38897 | A metric space is totally ... |
| totbndmet 38898 | The predicate "totally bou... |
| 0totbnd 38899 | The metric (there is only ... |
| sstotbnd2 38900 | Condition for a subset of ... |
| sstotbnd 38901 | Condition for a subset of ... |
| sstotbnd3 38902 | Use a net that is not nece... |
| totbndss 38903 | A subset of a totally boun... |
| equivtotbnd 38904 | If the metric ` M ` is "st... |
| isbnd 38906 | The predicate "is a bounde... |
| bndmet 38907 | A bounded metric space is ... |
| isbndx 38908 | A "bounded extended metric... |
| isbnd2 38909 | The predicate "is a bounde... |
| isbnd3 38910 | A metric space is bounded ... |
| isbnd3b 38911 | A metric space is bounded ... |
| bndss 38912 | A subset of a bounded metr... |
| blbnd 38913 | A ball is bounded. (Contr... |
| ssbnd 38914 | A subset of a metric space... |
| totbndbnd 38915 | A totally bounded metric s... |
| equivbnd 38916 | If the metric ` M ` is "st... |
| bnd2lem 38917 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 38918 | If balls are totally bound... |
| prdsbnd 38919 | The product metric over fi... |
| prdstotbnd 38920 | The product metric over fi... |
| prdsbnd2 38921 | If balls are totally bound... |
| cntotbnd 38922 | A subset of the complex nu... |
| cnpwstotbnd 38923 | A subset of ` A ^ I ` , wh... |
| ismtyval 38926 | The set of isometries betw... |
| isismty 38927 | The condition "is an isome... |
| ismtycnv 38928 | The inverse of an isometry... |
| ismtyima 38929 | The image of a ball under ... |
| ismtyhmeolem 38930 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 38931 | An isometry is a homeomorp... |
| ismtybndlem 38932 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 38933 | Isometries preserve bounde... |
| ismtyres 38934 | A restriction of an isomet... |
| heibor1lem 38935 | Lemma for ~ heibor1 . A c... |
| heibor1 38936 | One half of ~ heibor , tha... |
| heiborlem1 38937 | Lemma for ~ heibor . We w... |
| heiborlem2 38938 | Lemma for ~ heibor . Subs... |
| heiborlem3 38939 | Lemma for ~ heibor . Usin... |
| heiborlem4 38940 | Lemma for ~ heibor . Usin... |
| heiborlem5 38941 | Lemma for ~ heibor . The ... |
| heiborlem6 38942 | Lemma for ~ heibor . Sinc... |
| heiborlem7 38943 | Lemma for ~ heibor . Sinc... |
| heiborlem8 38944 | Lemma for ~ heibor . The ... |
| heiborlem9 38945 | Lemma for ~ heibor . Disc... |
| heiborlem10 38946 | Lemma for ~ heibor . The ... |
| heibor 38947 | Generalized Heine-Borel Th... |
| bfplem1 38948 | Lemma for ~ bfp . The seq... |
| bfplem2 38949 | Lemma for ~ bfp . Using t... |
| bfp 38950 | Banach fixed point theorem... |
| rrnval 38953 | The n-dimensional Euclidea... |
| rrnmval 38954 | The value of the Euclidean... |
| rrnmet 38955 | Euclidean space is a metri... |
| rrndstprj1 38956 | The distance between two p... |
| rrndstprj2 38957 | Bound on the distance betw... |
| rrncmslem 38958 | Lemma for ~ rrncms . (Con... |
| rrncms 38959 | Euclidean space is complet... |
| repwsmet 38960 | The supremum metric on ` R... |
| rrnequiv 38961 | The supremum metric on ` R... |
| rrntotbnd 38962 | A set in Euclidean space i... |
| rrnheibor 38963 | Heine-Borel theorem for Eu... |
| ismrer1 38964 | An isometry between ` RR `... |
| reheibor 38965 | Heine-Borel theorem for re... |
| iccbnd 38966 | A closed interval in ` RR ... |
| icccmpALT 38967 | A closed interval in ` RR ... |
| isass 38972 | The predicate "is an assoc... |
| isexid 38973 | The predicate ` G ` has a ... |
| ismgmOLD 38976 | Obsolete version of ~ ismg... |
| clmgmOLD 38977 | Obsolete version of ~ mgmc... |
| opidonOLD 38978 | Obsolete version of ~ mgmi... |
| rngopidOLD 38979 | Obsolete version of ~ mgmi... |
| opidon2OLD 38980 | Obsolete version of ~ mgmf... |
| isexid2 38981 | Obsolete theorem. If ` G ... |
| exidu1 38982 | Obsolete theorem, use ~ mg... |
| idrval 38983 | Obsolete theorem, use ~ id... |
| iorlid 38984 | Obsolete theorem, use ~ mg... |
| cmpidelt 38985 | Obsolete theorem, use ~ mg... |
| smgrpismgmOLD 38988 | Obsolete version of ~ sgrp... |
| issmgrpOLD 38989 | Obsolete version of ~ issg... |
| smgrpmgm 38990 | Obsolete theorem, use ~ sg... |
| smgrpassOLD 38991 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 38994 | Obsolete version of ~ mnds... |
| mndoisexid 38995 | Obsolete theorem, use ~ mn... |
| mndoismgmOLD 38996 | Obsolete version of ~ mndm... |
| mndomgmid 38997 | Obsolete theorem, use ~ mn... |
| ismndo 38998 | Obsolete theorem, use ~ is... |
| ismndo1 38999 | Obsolete theorem, use ~ is... |
| ismndo2 39000 | Obsolete theorem, use ~ is... |
| grpomndo 39001 | Obsolete theorem, use ~ gr... |
| exidcl 39002 | Obsolete theorem, use ~ mg... |
| exidreslem 39003 | Obsolete theorem, use ~ 0g... |
| exidres 39004 | Obsolete theorem, use ~ id... |
| exidresid 39005 | Obsolete theorem, use ~ id... |
| ablo4pnp 39006 | Obsolete theorem, use ~ ab... |
| grpoeqdivid 39007 | Obsolete theorem, use ~ gr... |
| grposnOLD 39008 | The group operation for th... |
| elghomlem1OLD 39011 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 39012 | Obsolete as of 15-Mar-2020... |
| elghomOLD 39013 | Obsolete version of ~ isgh... |
| ghomlinOLD 39014 | Obsolete version of ~ ghml... |
| ghomidOLD 39015 | Obsolete version of ~ ghmi... |
| ghomf 39016 | Obsolete theorem, use ~ gh... |
| ghomco 39017 | Obsolete theorem, use ~ gh... |
| ghomdiv 39018 | Obsolete theorem, use ~ gh... |
| grpokerinj 39019 | Obsolete theorem, use ~ ke... |
| relrngo 39022 | Obsolete theorem. The cla... |
| isrngo 39023 | Obsolete theorem, use ~ df... |
| isrngod 39024 | Obsolete theorem, use ~ is... |
| rngoi 39025 | Obsolete theorem, use ~ df... |
| rngosm 39026 | Obsolete theorem. Functio... |
| rngocl 39027 | Obsolete theorem, use ~ ri... |
| rngoid 39028 | Obsolete theorem, use ~ ri... |
| rngoideu 39029 | Obsolete theorem, use ~ ri... |
| rngodi 39030 | Obsolete theorem, use ~ ri... |
| rngodir 39031 | Obsolete theorem, use ~ ri... |
| rngoass 39032 | Obsolete theorem, use ~ ri... |
| rngo2 39033 | Obsolete theorem, use ~ ri... |
| rngoablo 39034 | Obsolete theorem, use ~ ri... |
| rngoablo2 39035 | Obsolete theorem, use ~ ri... |
| rngogrpo 39036 | Obsolete theorem, use ~ ri... |
| rngone0 39037 | Obsolete theorem, use ~ ri... |
| rngogcl 39038 | Obsolete theorem, use ~ ri... |
| rngocom 39039 | Obsolete theorem, use ~ ri... |
| rngoaass 39040 | Obsolete theorem, use ~ ri... |
| rngoa32 39041 | Obsolete theorem, use ~ ri... |
| rngoa4 39042 | Obsolete theorem, use ~ ri... |
| rngorcan 39043 | Obsolete theorem, use ~ ri... |
| rngolcan 39044 | Obsolete theorem, use ~ ri... |
| rngo0cl 39045 | Obsolete theorem, use ~ ri... |
| rngo0rid 39046 | Obsolete theorem, use ~ ri... |
| rngo0lid 39047 | Obsolete theorem, use ~ ri... |
| rngolz 39048 | Obsolete theorem, use ~ ri... |
| rngorz 39049 | Obsolete theorem, use ~ ri... |
| rngosn3 39050 | Obsolete as of 25-Jan-2020... |
| rngosn4 39051 | Obsolete as of 25-Jan-2020... |
| rngosn6 39052 | Obsolete as of 25-Jan-2020... |
| rngonegcl 39053 | Obsolete theorem, use ~ ri... |
| rngoaddneg1 39054 | Obsolete theorem, use ~ ri... |
| rngoaddneg2 39055 | Obsolete theorem, use ~ ri... |
| rngosub 39056 | Obsolete theorem, use ~ ri... |
| rngmgmbs4 39057 | Obsolete theorem. The ran... |
| rngodm1dm2 39058 | Obsolete theorem. In a un... |
| rngorn1 39059 | Obsolete theorem. In a un... |
| rngorn1eq 39060 | Obsolete theorem. In a un... |
| rngomndo 39061 | Obsolete theorem, use ~ ri... |
| rngoidmlem 39062 | Obsolete theorem, use ~ ri... |
| rngolidm 39063 | Obsolete theorem, use ~ ri... |
| rngoridm 39064 | Obsolete theorem, use ~ ri... |
| rngo1cl 39065 | Obsolete theorem, use ~ ri... |
| rngoueqz 39066 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 39067 | Obsolete theorem, use ~ ri... |
| rngonegmn1r 39068 | Obsolete theorem, use ~ ri... |
| rngoneglmul 39069 | Obsolete theorem, use ~ ri... |
| rngonegrmul 39070 | Obsolete theorem, use ~ ri... |
| rngosubdi 39071 | Obsolete theorem, use ~ ri... |
| rngosubdir 39072 | Obsolete theorem, use ~ ri... |
| zerdivemp1x 39073 | Obsolete theorem, use ~ ri... |
| isdivrngo 39076 | Obsolete theorem, use ~ is... |
| drngoi 39077 | Obsolete theorem, use ~ dr... |
| gidsn 39078 | Obsolete as of 23-Jan-2020... |
| zrdivrng 39079 | Obsolete theorem, use ~ zr... |
| dvrunz 39080 | Obsolete theorem, use ~ dr... |
| isgrpda 39081 | Obsolete theorem, use ~ is... |
| isdrngo1 39082 | Obsolete theorem, use ~ is... |
| divrngcl 39083 | Obsolete theorem, use ~ dr... |
| isdrngo2 39084 | Obsolete theorem, use ~ is... |
| isdrngo3 39085 | Obsolete theorem, use ~ is... |
| rngohomval 39090 | Obsolete theorem, use ~ rh... |
| isrngohom 39091 | Obsolete theorem, use ~ is... |
| rngohomf 39092 | Obsolete theorem, use ~ rh... |
| rngohomcl 39093 | Obsolete theorem, use ~ rh... |
| rngohom1 39094 | Obsolete theorem, use ~ rh... |
| rngohomadd 39095 | Obsolete theorem, use ~ rh... |
| rngohommul 39096 | Obsolete theorem, use ~ rh... |
| rngogrphom 39097 | Obsolete theorem, use ~ rh... |
| rngohom0 39098 | Obsolete theorem, use ~ rh... |
| rngohomsub 39099 | Obsolete theorem, use ~ rh... |
| rngohomco 39100 | Obsolete theorem, use ~ rh... |
| rngokerinj 39101 | Obsolete theorem, use ~ rh... |
| rngoisoval 39103 | Obsolete theorem, use ~ ri... |
| isrngoiso 39104 | Obsolete theorem, use ~ is... |
| rngoiso1o 39105 | Obsolete theorem, use ~ ri... |
| rngoisohom 39106 | Obsolete theorem, use ~ ri... |
| rngoisocnv 39107 | Obsolete theorem, use ~ ri... |
| rngoisoco 39108 | Obsolete theorem, use ~ ri... |
| isriscg 39110 | Obsolete theorem, use ~ is... |
| isrisc 39111 | Obsolete theorem, use ~ is... |
| risc 39112 | Obsolete theorem, use ~ br... |
| risci 39113 | Obsolete theorem, use ~ br... |
| riscer 39114 | Obsolete theorem, use ~ ri... |
| iscom2 39121 | Obsolete theorem, used (as... |
| iscrngo 39122 | Obsolete theorem, use ~ is... |
| iscrngo2 39123 | Obsolete theorem, use ~ is... |
| iscringd 39124 | Obsolete theorem, use ~ is... |
| flddivrng 39125 | Obsolete theorem, use ~ fl... |
| crngorngo 39126 | Obsolete theorem, use ~ cr... |
| crngocom 39127 | Obsolete theorem, use ~ cr... |
| crngm23 39128 | Obsolete theorem, use ~ cr... |
| crngm4 39129 | Obsolete theorem, use ~ cr... |
| fldcrngo 39130 | Obsolete theorem, use ~ fl... |
| isfld2 39131 | Obsolete theorem, use ~ is... |
| crngohomfo 39132 | Obsolete theorem, use ~ cr... |
| idlval 39139 | Obsolete theorem, use ~ 2i... |
| isidl 39140 | Obsolete theorem, use ~ df... |
| isidlc 39141 | Obsolete theorem, use ~ df... |
| idlss 39142 | Obsolete theorem, use ~ 2i... |
| idlcl 39143 | Obsolete theorem, use ~ 2i... |
| idl0cl 39144 | Obsolete theorem, use ~ ri... |
| idladdcl 39145 | Obsolete theorem, use ~ 2i... |
| idllmulcl 39146 | Obsolete theorem, use ~ 2i... |
| idlrmulcl 39147 | Obsolete theorem, use ~ 2i... |
| idlnegcl 39148 | Obsolete theorem, use ~ 2i... |
| idlsubcl 39149 | Obsolete theorem, use ~ 2i... |
| rngoidl 39150 | Obsolete theorem, use ~ 2i... |
| 0idl 39151 | Obsolete theorem, use ~ 2i... |
| 1idl 39152 | Obsolete theorem, use ~ 2i... |
| 0rngo 39153 | Obsolete theorem, use ~ 0r... |
| divrngidl 39154 | Obsolete theorem, use ~ dr... |
| intidl 39155 | Obsolete theorem, use ~ in... |
| inidl 39156 | Obsolete theorem, use ~ in... |
| unichnidl 39157 | Obsolete theorem, use ~ un... |
| keridl 39158 | Obsolete theorem, use ~ ke... |
| pridlval 39159 | Obsolete theorem, use ~ pr... |
| ispridl 39160 | Obsolete theorem, use ~ is... |
| pridlidl 39161 | Obsolete theorem, use ~ pr... |
| pridlnr 39162 | Obsolete theorem, use ~ pr... |
| pridl 39163 | Obsolete theorem, use ~ is... |
| ispridl2 39164 | Obsolete theorem, use ~ pr... |
| maxidlval 39165 | Obsolete theorem, use ~ mx... |
| ismaxidl 39166 | Obsolete theorem, use ~ is... |
| maxidlidl 39167 | Obsolete theorem, use ~ mx... |
| maxidlnr 39168 | Obsolete theorem, use ~ mx... |
| maxidlmax 39169 | Obsolete theorem, use ~ mx... |
| maxidln1 39170 | Obsolete theorem, use ~ mx... |
| maxidln0 39171 | Obsolete theorem, use ~ mx... |
| isprrngo 39176 | Obsolete theorem, use ~ is... |
| prrngorngo 39177 | Obsolete theorem, use ~ pr... |
| smprngopr 39178 | Obsolete theorem, use ~ sm... |
| divrngpr 39179 | Obsolete theorem, use ~ dr... |
| isdmn 39180 | Obsolete theorem, use ~ is... |
| isdmn2 39181 | Obsolete theorem, use ~ is... |
| dmncrng 39182 | Obsolete theorem, use ~ id... |
| dmnrngo 39183 | Obsolete theorem, use ~ id... |
| flddmn 39184 | Obsolete theorem, use ~ fl... |
| igenval 39187 | Obsolete theorem, use ~ rs... |
| igenss 39188 | Obsolete theorem, use ~ rs... |
| igenidl 39189 | Obsolete theorem, use ~ rs... |
| igenmin 39190 | Obsolete theorem, use ~ rs... |
| igenidl2 39191 | Obsolete theorem, use ~ rs... |
| igenval2 39192 | Obsolete theorem, use ~ rs... |
| prnc 39193 | Obsolete theorem, use ~ rs... |
| isfldidl 39194 | Obsolete theorem, use ~ is... |
| isfldidl2 39195 | Obsolete theorem, use ~ is... |
| ispridlc 39196 | Obsolete theorem, use ~ is... |
| pridlc 39197 | Obsolete theorem, use ~ pr... |
| pridlc2 39198 | Obsolete theorem, use ~ pr... |
| pridlc3 39199 | Obsolete theorem, use ~ cm... |
| isdmn3 39200 | Obsolete theorem, use ~ is... |
| dmnnzd 39201 | Obsolete theorem, use ~ id... |
| dmncan1 39202 | Obsolete theorem, use ~ id... |
| dmncan2 39203 | Obsolete theorem, use ~ id... |
| efald2 39204 | A proof by contradiction. ... |
| notbinot1 39205 | Simplification rule of neg... |
| bicontr 39206 | Biconditional of its own n... |
| impor 39207 | An equivalent formula for ... |
| orfa 39208 | The falsum ` F. ` can be r... |
| notbinot2 39209 | Commutation rule between n... |
| biimpor 39210 | A rewriting rule for bicon... |
| orfa1 39211 | Add a contradicting disjun... |
| orfa2 39212 | Remove a contradicting dis... |
| bifald 39213 | Infer the equivalence to a... |
| cnf1dd 39214 | A lemma for Conjunctive No... |
| cnf2dd 39215 | A lemma for Conjunctive No... |
| cnfn1dd 39216 | A lemma for Conjunctive No... |
| cnfn2dd 39217 | A lemma for Conjunctive No... |
| or32dd 39218 | A rearrangement of disjunc... |
| notornotel1 39219 | A lemma for not-or-not eli... |
| notornotel2 39220 | A lemma for not-or-not eli... |
| contrd 39221 | A proof by contradiction, ... |
| an12i 39222 | An inference from commutin... |
| exmid2 39223 | An excluded middle law. (... |
| selconj 39224 | An inference for selecting... |
| truconj 39225 | Add true as a conjunct. (... |
| orel 39226 | An inference for disjuncti... |
| negel 39227 | An inference for negation ... |
| botel 39228 | An inference for bottom el... |
| tradd 39229 | Add top ad a conjunct. (C... |
| gm-sbtru 39230 | Substitution does not chan... |
| sbfal 39231 | Substitution does not chan... |
| sbcani 39232 | Distribution of class subs... |
| sbcori 39233 | Distribution of class subs... |
| sbcimi 39234 | Distribution of class subs... |
| sbcni 39235 | Move class substitution in... |
| sbali 39236 | Discard class substitution... |
| sbexi 39237 | Discard class substitution... |
| sbcalf 39238 | Move universal quantifier ... |
| sbcexf 39239 | Move existential quantifie... |
| sbcalfi 39240 | Move universal quantifier ... |
| sbcexfi 39241 | Move existential quantifie... |
| spsbcdi 39242 | A lemma for eliminating a ... |
| alrimii 39243 | A lemma for introducing a ... |
| spesbcdi 39244 | A lemma for introducing an... |
| exlimddvf 39245 | A lemma for eliminating an... |
| exlimddvfi 39246 | A lemma for eliminating an... |
| sbceq1ddi 39247 | A lemma for eliminating in... |
| sbccom2lem 39248 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 39249 | Commutative law for double... |
| sbccom2f 39250 | Commutative law for double... |
| sbccom2fi 39251 | Commutative law for double... |
| csbcom2fi 39252 | Commutative law for double... |
| fald 39253 | Refutation of falsity, in ... |
| tsim1 39254 | A Tseitin axiom for logica... |
| tsim2 39255 | A Tseitin axiom for logica... |
| tsim3 39256 | A Tseitin axiom for logica... |
| tsbi1 39257 | A Tseitin axiom for logica... |
| tsbi2 39258 | A Tseitin axiom for logica... |
| tsbi3 39259 | A Tseitin axiom for logica... |
| tsbi4 39260 | A Tseitin axiom for logica... |
| tsxo1 39261 | A Tseitin axiom for logica... |
| tsxo2 39262 | A Tseitin axiom for logica... |
| tsxo3 39263 | A Tseitin axiom for logica... |
| tsxo4 39264 | A Tseitin axiom for logica... |
| tsan1 39265 | A Tseitin axiom for logica... |
| tsan2 39266 | A Tseitin axiom for logica... |
| tsan3 39267 | A Tseitin axiom for logica... |
| tsna1 39268 | A Tseitin axiom for logica... |
| tsna2 39269 | A Tseitin axiom for logica... |
| tsna3 39270 | A Tseitin axiom for logica... |
| tsor1 39271 | A Tseitin axiom for logica... |
| tsor2 39272 | A Tseitin axiom for logica... |
| tsor3 39273 | A Tseitin axiom for logica... |
| ts3an1 39274 | A Tseitin axiom for triple... |
| ts3an2 39275 | A Tseitin axiom for triple... |
| ts3an3 39276 | A Tseitin axiom for triple... |
| ts3or1 39277 | A Tseitin axiom for triple... |
| ts3or2 39278 | A Tseitin axiom for triple... |
| ts3or3 39279 | A Tseitin axiom for triple... |
| iuneq2f 39280 | Equality deduction for ind... |
| rabeq12f 39281 | Equality deduction for res... |
| csbeq12 39282 | Equality deduction for sub... |
| sbeqi 39283 | Equality deduction for sub... |
| ralbi12f 39284 | Equality deduction for res... |
| oprabbi 39285 | Equality deduction for cla... |
| mpobi123f 39286 | Equality deduction for map... |
| iuneq12f 39287 | Equality deduction for ind... |
| iineq12f 39288 | Equality deduction for ind... |
| opabbi 39289 | Equality deduction for cla... |
| mptbi12f 39290 | Equality deduction for map... |
| orcomdd 39291 | Commutativity of logic dis... |
| scottexf 39292 | A version of ~ scottex wit... |
| scott0f 39293 | A version of ~ scott0b wit... |
| scottn0f 39294 | A version of ~ scott0f wit... |
| ac6s3f 39295 | Generalization of the Axio... |
| ac6s6 39296 | Generalization of the Axio... |
| ac6s6f 39297 | Generalization of the Axio... |
| el2v1 39353 | New way ( ~ elv , and the ... |
| el3v1 39354 | New way ( ~ elv , and the ... |
| el3v2 39355 | New way ( ~ elv , and the ... |
| el3v12 39356 | New way ( ~ elv , and the ... |
| el3v13 39357 | New way ( ~ elv , and the ... |
| el3v23 39358 | New way ( ~ elv , and the ... |
| anan 39359 | Multiple commutations in c... |
| triantru3 39360 | A wff is equivalent to its... |
| biorfd 39361 | A wff is equivalent to its... |
| eqbrtr 39362 | Substitution of equal clas... |
| eqbrb 39363 | Substitution of equal clas... |
| eqeltr 39364 | Substitution of equal clas... |
| eqelb 39365 | Substitution of equal clas... |
| eqeqan2d 39366 | Implication of introducing... |
| disjresin 39367 | The restriction to a disjo... |
| disjresdisj 39368 | The intersection of restri... |
| disjresdif 39369 | The difference between res... |
| disjresundif 39370 | Lemma for ~ ressucdifsn2 .... |
| inres2 39371 | Two ways of expressing the... |
| coideq 39372 | Equality theorem for compo... |
| nexmo1 39373 | If there is no case where ... |
| eqab2 39374 | Implication of a class abs... |
| r2alan 39375 | Double restricted universa... |
| ssrabi 39376 | Inference of restricted ab... |
| rabimbieq 39377 | Restricted equivalent wff'... |
| abeqin 39378 | Intersection with class ab... |
| abeqinbi 39379 | Intersection with class ab... |
| eqrabi 39380 | Class element of a restric... |
| rabeqel 39381 | Class element of a restric... |
| eqrelf 39382 | The equality connective be... |
| br1cnvinxp 39383 | Binary relation on the con... |
| releleccnv 39384 | Elementhood in a converse ... |
| releccnveq 39385 | Equality of converse ` R `... |
| xpv 39386 | Cartesian product of a cla... |
| vxp 39387 | Cartesian product of the u... |
| opelvvdif 39388 | Negated elementhood of ord... |
| vvdifopab 39389 | Ordered-pair class abstrac... |
| brvdif 39390 | Binary relation with unive... |
| brvdif2 39391 | Binary relation with unive... |
| brvvdif 39392 | Binary relation with the c... |
| brvbrvvdif 39393 | Binary relation with the c... |
| brcnvep 39394 | The converse of the binary... |
| elecALTV 39395 | Elementhood in the ` R ` -... |
| brcnvepres 39396 | Restricted converse epsilo... |
| brres2 39397 | Binary relation on a restr... |
| br1cnvres 39398 | Binary relation on the con... |
| elec1cnvres 39399 | Elementhood in the convers... |
| ec1cnvres 39400 | Converse restricted coset ... |
| eldmres 39401 | Elementhood in the domain ... |
| elrnres 39402 | Element of the range of a ... |
| eldmressnALTV 39403 | Element of the domain of a... |
| elrnressn 39404 | Element of the range of a ... |
| eldm4 39405 | Elementhood in a domain. ... |
| eldmres2 39406 | Elementhood in the domain ... |
| eldmres3 39407 | Elementhood in the domain ... |
| eceq1i 39408 | Equality theorem for ` C `... |
| ecres 39409 | Restricted coset of ` B ` ... |
| eccnvepres 39410 | Restricted converse epsilo... |
| eleccnvep 39411 | Elementhood in the convers... |
| eccnvep 39412 | The converse epsilon coset... |
| extep 39413 | Property of epsilon relati... |
| disjeccnvep 39414 | Property of the epsilon re... |
| eccnvepres2 39415 | The restricted converse ep... |
| eccnvepres3 39416 | Condition for a restricted... |
| eldmqsres 39417 | Elementhood in a restricte... |
| eldmqsres2 39418 | Elementhood in a restricte... |
| qsss1 39419 | Subclass theorem for quoti... |
| qseq1i 39420 | Equality theorem for quoti... |
| brinxprnres 39421 | Binary relation on a restr... |
| inxprnres 39422 | Restriction of a class as ... |
| dfres4 39423 | Alternate definition of th... |
| exan3 39424 | Equivalent expressions wit... |
| exanres 39425 | Equivalent expressions wit... |
| exanres3 39426 | Equivalent expressions wit... |
| exanres2 39427 | Equivalent expressions wit... |
| cnvepres 39428 | Restricted converse epsilo... |
| eqrel2 39429 | Equality of relations. (C... |
| rncnv 39430 | Range of converse is the d... |
| dfdm6 39431 | Alternate definition of do... |
| dfrn6 39432 | Alternate definition of ra... |
| rncnvepres 39433 | The range of the restricte... |
| dmecd 39434 | Equality of the coset of `... |
| dmec2d 39435 | Equality of the coset of `... |
| brid 39436 | Property of the identity b... |
| ideq2 39437 | For sets, the identity bin... |
| idresssidinxp 39438 | Condition for the identity... |
| idreseqidinxp 39439 | Condition for the identity... |
| extid 39440 | Property of identity relat... |
| inxpss 39441 | Two ways to say that an in... |
| idinxpss 39442 | Two ways to say that an in... |
| ref5 39443 | Two ways to say that an in... |
| inxpss3 39444 | Two ways to say that an in... |
| inxpss2 39445 | Two ways to say that inter... |
| inxpssidinxp 39446 | Two ways to say that inter... |
| idinxpssinxp 39447 | Two ways to say that inter... |
| idinxpssinxp2 39448 | Identity intersection with... |
| idinxpssinxp3 39449 | Identity intersection with... |
| idinxpssinxp4 39450 | Identity intersection with... |
| relcnveq3 39451 | Two ways of saying a relat... |
| relcnveq 39452 | Two ways of saying a relat... |
| relcnveq2 39453 | Two ways of saying a relat... |
| relcnveq4 39454 | Two ways of saying a relat... |
| qsresid 39455 | Simplification of a specia... |
| n0elqs 39456 | Two ways of expressing tha... |
| n0elqs2 39457 | Two ways of expressing tha... |
| rnresequniqs 39458 | The range of a restriction... |
| n0el2 39459 | Two ways of expressing tha... |
| cnvepresex 39460 | Sethood condition for the ... |
| cnvepima 39461 | The image of converse epsi... |
| inex3 39462 | Sufficient condition for t... |
| inxpex 39463 | Sufficient condition for a... |
| eqres 39464 | Converting a class constan... |
| brrabga 39465 | The law of concretion for ... |
| brcnvrabga 39466 | The law of concretion for ... |
| opideq 39467 | Equality conditions for or... |
| iss2 39468 | A subclass of the identity... |
| eldmcnv 39469 | Elementhood in a domain of... |
| dfrel5 39470 | Alternate definition of th... |
| dfrel6 39471 | Alternate definition of th... |
| cnvresrn 39472 | Converse restricted to ran... |
| relssinxpdmrn 39473 | Subset of restriction, spe... |
| cnvref4 39474 | Two ways to say that a rel... |
| cnvref5 39475 | Two ways to say that a rel... |
| ecin0 39476 | Two ways of saying that th... |
| ecinn0 39477 | Two ways of saying that th... |
| ineleq 39478 | Equivalence of restricted ... |
| inecmo 39479 | Equivalence of a double re... |
| inecmo2 39480 | Equivalence of a double re... |
| ineccnvmo 39481 | Equivalence of a double re... |
| alrmomorn 39482 | Equivalence of an "at most... |
| alrmomodm 39483 | Equivalence of an "at most... |
| ralmo 39484 | "At most one" can be restr... |
| ralrnmo 39485 | On the range, "at most one... |
| dmqsex 39486 | Sethood of the domain quot... |
| raldmqsmo 39487 | On the quotient carrier, "... |
| ralrmo3 39488 | Pull a restricted universa... |
| raldmqseu 39489 | Equivalence between "exact... |
| rsp3 39490 | From a restricted universa... |
| rsp3eq 39491 | From a restricted universa... |
| ineccnvmo2 39492 | Equivalence of a double un... |
| inecmo3 39493 | Equivalence of a double un... |
| moeu2 39494 | Uniqueness is equivalent t... |
| mopickr 39495 | "At most one" picks a vari... |
| moantr 39496 | Sufficient condition for t... |
| brabidgaw 39497 | The law of concretion for ... |
| brabidga 39498 | The law of concretion for ... |
| inxp2 39499 | Intersection with a Cartes... |
| opabf 39500 | A class abstraction of a c... |
| ec0 39501 | The empty-coset of a class... |
| brcnvin 39502 | Intersection with a conver... |
| ssdmral 39503 | Subclass of a domain. (Co... |
| xrnss3v 39505 | A range Cartesian product ... |
| xrnrel 39506 | A range Cartesian product ... |
| brxrn 39507 | Characterize a ternary rel... |
| brxrn2 39508 | A characterization of the ... |
| dfxrn2 39509 | Alternate definition of th... |
| brxrncnvep 39510 | The range product with con... |
| dmxrn 39511 | Domain of the range produc... |
| dmcnvep 39512 | Domain of converse epsilon... |
| dmxrncnvep 39513 | Domain of the range produc... |
| dmcnvepres 39514 | Domain of the restricted c... |
| dmuncnvepres 39515 | Domain of the union with t... |
| dmxrnuncnvepres 39516 | Domain of the combined rel... |
| ecun 39517 | The union coset of ` A ` .... |
| ecunres 39518 | The restricted union coset... |
| ecuncnvepres 39519 | The restricted union with ... |
| xrneq1 39520 | Equality theorem for the r... |
| xrneq1i 39521 | Equality theorem for the r... |
| xrneq1d 39522 | Equality theorem for the r... |
| xrneq2 39523 | Equality theorem for the r... |
| xrneq2i 39524 | Equality theorem for the r... |
| xrneq2d 39525 | Equality theorem for the r... |
| xrneq12 39526 | Equality theorem for the r... |
| xrneq12i 39527 | Equality theorem for the r... |
| xrneq12d 39528 | Equality theorem for the r... |
| elecxrn 39529 | Elementhood in the ` ( R |... |
| ecxrn 39530 | The ` ( R |X. S ) ` -coset... |
| relecxrn 39531 | The ` ( R |X. S ) ` -coset... |
| ecxrn2 39532 | The ` ( R |X. S ) ` -coset... |
| ecxrncnvep 39533 | The ` ( R |X. ``' _E ) ` -... |
| ecxrncnvep2 39534 | The ` ( R |X. ``' _E ) ` -... |
| disjressuc2 39535 | Double restricted quantifi... |
| disjecxrn 39536 | Two ways of saying that ` ... |
| disjecxrncnvep 39537 | Two ways of saying that co... |
| disjsuc2 39538 | Double restricted quantifi... |
| xrninxp 39539 | Intersection of a range Ca... |
| xrninxp2 39540 | Intersection of a range Ca... |
| xrninxpex 39541 | Sufficient condition for t... |
| inxpxrn 39542 | Two ways to express the in... |
| br1cnvxrn2 39543 | The converse of a binary r... |
| elec1cnvxrn2 39544 | Elementhood in the convers... |
| rnxrn 39545 | Range of the range Cartesi... |
| rnxrnres 39546 | Range of a range Cartesian... |
| rnxrncnvepres 39547 | Range of a range Cartesian... |
| rnxrnidres 39548 | Range of a range Cartesian... |
| xrnres 39549 | Two ways to express restri... |
| xrnres2 39550 | Two ways to express restri... |
| xrnres3 39551 | Two ways to express restri... |
| xrnres4 39552 | Two ways to express restri... |
| xrnresex 39553 | Sufficient condition for a... |
| xrnidresex 39554 | Sufficient condition for a... |
| xrncnvepresex 39555 | Sufficient condition for a... |
| dmxrncnvepres 39556 | Domain of the range produc... |
| dmxrncnvepres2 39557 | Domain of the range produc... |
| eldmxrncnvepres 39558 | Element of the domain of t... |
| eldmxrncnvepres2 39559 | Element of the domain of t... |
| eceldmqsxrncnvepres 39560 | An ` ( R |X. ( ``' _E |`` ... |
| eceldmqsxrncnvepres2 39561 | An ` ( R |X. ( ``' _E |`` ... |
| brin2 39562 | Binary relation on an inte... |
| brin3 39563 | Binary relation on an inte... |
| elrels2 39565 | The element of the relatio... |
| elrelsrel 39566 | The element of the relatio... |
| elrelsrelim 39567 | The element of the relatio... |
| elrels5 39568 | Equivalent expressions for... |
| elrels6 39569 | Equivalent expressions for... |
| dfqmap2 39571 | Alternate definition of th... |
| dfqmap3 39572 | Alternate definition of th... |
| ecqmap 39573 | ` QMap ` fibers are single... |
| ecqmap2 39574 | Fiber of ` QMap ` equals s... |
| qmapex 39575 | Quotient map exists if ` R... |
| relqmap 39576 | Quotient map is a relation... |
| dmqmap 39577 | ` QMap ` preserves the dom... |
| rnqmap 39578 | The range of the quotient ... |
| dfadjliftmap 39580 | Alternate (expanded) defin... |
| dfadjliftmap2 39581 | Alternate definition of th... |
| blockadjliftmap 39582 | A "two-stage" construction... |
| dfblockliftmap 39584 | Alternate definition of th... |
| dfblockliftmap2 39585 | Alternate definition of th... |
| dfsucmap3 39587 | Alternate definition of th... |
| dfsucmap2 39588 | Alternate definition of th... |
| dfsucmap4 39589 | Alternate definition of th... |
| brsucmap 39590 | Binary relation form of th... |
| relsucmap 39591 | The successor map is a rel... |
| dmsucmap 39592 | The domain of the successo... |
| dfsuccl2 39594 | Alternate definition of th... |
| mopre 39595 | There is at most one prede... |
| exeupre2 39596 | Whenever a predecessor exi... |
| dfsuccl3 39597 | Alternate definition of th... |
| dfsuccl4 39598 | Alternate definition that ... |
| dfpre 39600 | Alternate definition of th... |
| dfpre2 39601 | Alternate definition of th... |
| dfpre3 39602 | Alternate definition of th... |
| dfpred4 39603 | Alternate definition of th... |
| dfpre4 39604 | Alternate definition of th... |
| shiftstableeq2 39607 | Equality theorem for shift... |
| suceqsneq 39608 | One-to-one relationship be... |
| sucdifsn2 39609 | Absorption of union with a... |
| sucdifsn 39610 | The difference between the... |
| ressucdifsn2 39611 | The difference between res... |
| ressucdifsn 39612 | The difference between res... |
| sucmapsuc 39613 | A set is succeeded by its ... |
| sucmapleftuniq 39614 | Left uniqueness of the suc... |
| exeupre 39615 | Whenever a predecessor exi... |
| preex 39616 | The successor-predecessor ... |
| eupre2 39617 | Unique predecessor exists ... |
| eupre 39618 | Unique predecessor exists ... |
| presucmap 39619 | ` pre ` is really a predec... |
| preuniqval 39620 | Uniqueness/canonicity of `... |
| sucpre 39621 | ` suc ` is a right-inverse... |
| presuc 39622 | ` pre ` is a left-inverse ... |
| press 39623 | Predecessor is a subset of... |
| preel 39624 | Predecessor is a subset of... |
| dfcoss2 39627 | Alternate definition of th... |
| dfcoss3 39628 | Alternate definition of th... |
| dfcoss4 39629 | Alternate definition of th... |
| cosscnv 39630 | Class of cosets by the con... |
| coss1cnvres 39631 | Class of cosets by the con... |
| coss2cnvepres 39632 | Special case of ~ coss1cnv... |
| cossex 39633 | If ` A ` is a set then the... |
| cosscnvex 39634 | If ` A ` is a set then the... |
| 1cosscnvepresex 39635 | Sufficient condition for a... |
| 1cossxrncnvepresex 39636 | Sufficient condition for a... |
| relcoss 39637 | Cosets by ` R ` is a relat... |
| relcoels 39638 | Coelements on ` A ` is a r... |
| cossss 39639 | Subclass theorem for the c... |
| cosseq 39640 | Equality theorem for the c... |
| cosseqi 39641 | Equality theorem for the c... |
| cosseqd 39642 | Equality theorem for the c... |
| 1cossres 39643 | The class of cosets by a r... |
| dfcoels 39644 | Alternate definition of th... |
| brcoss 39645 | ` A ` and ` B ` are cosets... |
| brcoss2 39646 | Alternate form of the ` A ... |
| brcoss3 39647 | Alternate form of the ` A ... |
| brcosscnvcoss 39648 | For sets, the ` A ` and ` ... |
| brcoels 39649 | ` B ` and ` C ` are coelem... |
| cocossss 39650 | Two ways of saying that co... |
| cnvcosseq 39651 | The converse of cosets by ... |
| br2coss 39652 | Cosets by ` ,~ R ` binary ... |
| br1cossres 39653 | ` B ` and ` C ` are cosets... |
| br1cossres2 39654 | ` B ` and ` C ` are cosets... |
| brressn 39655 | Binary relation on a restr... |
| ressn2 39656 | A class ' R ' restricted t... |
| refressn 39657 | Any class ' R ' restricted... |
| antisymressn 39658 | Every class ' R ' restrict... |
| trressn 39659 | Any class ' R ' restricted... |
| relbrcoss 39660 | ` A ` and ` B ` are cosets... |
| br1cossinres 39661 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 39662 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 39663 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 39664 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 39665 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 39666 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 39667 | The domain of cosets is th... |
| dmcoss2 39668 | The domain of cosets is th... |
| rncossdmcoss 39669 | The range of cosets is the... |
| dm1cosscnvepres 39670 | The domain of cosets of th... |
| dmcoels 39671 | The domain of coelements i... |
| eldmcoss 39672 | Elementhood in the domain ... |
| eldmcoss2 39673 | Elementhood in the domain ... |
| eldm1cossres 39674 | Elementhood in the domain ... |
| eldm1cossres2 39675 | Elementhood in the domain ... |
| refrelcosslem 39676 | Lemma for the left side of... |
| refrelcoss3 39677 | The class of cosets by ` R... |
| refrelcoss2 39678 | The class of cosets by ` R... |
| symrelcoss3 39679 | The class of cosets by ` R... |
| symrelcoss2 39680 | The class of cosets by ` R... |
| cossssid 39681 | Equivalent expressions for... |
| cossssid2 39682 | Equivalent expressions for... |
| cossssid3 39683 | Equivalent expressions for... |
| cossssid4 39684 | Equivalent expressions for... |
| cossssid5 39685 | Equivalent expressions for... |
| brcosscnv 39686 | ` A ` and ` B ` are cosets... |
| brcosscnv2 39687 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 39688 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 39689 | Cosets by the converse ran... |
| cosscnvssid3 39690 | Equivalent expressions for... |
| cosscnvssid4 39691 | Equivalent expressions for... |
| cosscnvssid5 39692 | Equivalent expressions for... |
| coss0 39693 | Cosets by the empty set ar... |
| cossid 39694 | Cosets by the identity rel... |
| cosscnvid 39695 | Cosets by the converse ide... |
| trcoss 39696 | Sufficient condition for t... |
| eleccossin 39697 | Two ways of saying that th... |
| trcoss2 39698 | Equivalent expressions for... |
| cosselrels 39699 | Cosets of sets are element... |
| cnvelrels 39700 | The converse of a set is a... |
| cosscnvelrels 39701 | Cosets of converse sets ar... |
| dfssr2 39703 | Alternate definition of th... |
| relssr 39704 | The subset relation is a r... |
| brssr 39705 | The subset relation and su... |
| brssrid 39706 | Any set is a subset of its... |
| issetssr 39707 | Two ways of expressing set... |
| brssrres 39708 | Restricted subset binary r... |
| br1cnvssrres 39709 | Restricted converse subset... |
| brcnvssr 39710 | The converse of a subset r... |
| brcnvssrid 39711 | Any set is a converse subs... |
| br1cossxrncnvssrres 39712 | ` <. B , C >. ` and ` <. D... |
| extssr 39713 | Property of subset relatio... |
| dfrefrels2 39717 | Alternate definition of th... |
| dfrefrels3 39718 | Alternate definition of th... |
| dfrefrel2 39719 | Alternate definition of th... |
| dfrefrel3 39720 | Alternate definition of th... |
| dfrefrel5 39721 | Alternate definition of th... |
| elrefrels2 39722 | Element of the class of re... |
| elrefrels3 39723 | Element of the class of re... |
| elrefrelsrel 39724 | For sets, being an element... |
| refreleq 39725 | Equality theorem for refle... |
| refrelid 39726 | Identity relation is refle... |
| refrelcoss 39727 | The class of cosets by ` R... |
| refrelressn 39728 | Any class ' R ' restricted... |
| dfcnvrefrels2 39732 | Alternate definition of th... |
| dfcnvrefrels3 39733 | Alternate definition of th... |
| dfcnvrefrel2 39734 | Alternate definition of th... |
| dfcnvrefrel3 39735 | Alternate definition of th... |
| dfcnvrefrel4 39736 | Alternate definition of th... |
| dfcnvrefrel5 39737 | Alternate definition of th... |
| elcnvrefrels2 39738 | Element of the class of co... |
| elcnvrefrels3 39739 | Element of the class of co... |
| elcnvrefrelsrel 39740 | For sets, being an element... |
| cnvrefrelcoss2 39741 | Necessary and sufficient c... |
| cosselcnvrefrels2 39742 | Necessary and sufficient c... |
| cosselcnvrefrels3 39743 | Necessary and sufficient c... |
| cosselcnvrefrels4 39744 | Necessary and sufficient c... |
| cosselcnvrefrels5 39745 | Necessary and sufficient c... |
| dfsymrels2 39749 | Alternate definition of th... |
| dfsymrels3 39750 | Alternate definition of th... |
| elrelscnveq3 39751 | Two ways of saying a relat... |
| elrelscnveq 39752 | Two ways of saying a relat... |
| elrelscnveq2 39753 | Two ways of saying a relat... |
| elrelscnveq4 39754 | Two ways of saying a relat... |
| dfsymrels4 39755 | Alternate definition of th... |
| dfsymrels5 39756 | Alternate definition of th... |
| dfsymrel2 39757 | Alternate definition of th... |
| dfsymrel3 39758 | Alternate definition of th... |
| dfsymrel4 39759 | Alternate definition of th... |
| dfsymrel5 39760 | Alternate definition of th... |
| elsymrels2 39761 | Element of the class of sy... |
| elsymrels3 39762 | Element of the class of sy... |
| elsymrels4 39763 | Element of the class of sy... |
| elsymrels5 39764 | Element of the class of sy... |
| elsymrelsrel 39765 | For sets, being an element... |
| symreleq 39766 | Equality theorem for symme... |
| symrelim 39767 | Symmetric relation implies... |
| symrelcoss 39768 | The class of cosets by ` R... |
| idsymrel 39769 | The identity relation is s... |
| epnsymrel 39770 | The membership (epsilon) r... |
| symrefref2 39771 | Symmetry is a sufficient c... |
| symrefref3 39772 | Symmetry is a sufficient c... |
| refsymrels2 39773 | Elements of the class of r... |
| refsymrels3 39774 | Elements of the class of r... |
| refsymrel2 39775 | A relation which is reflex... |
| refsymrel3 39776 | A relation which is reflex... |
| elrefsymrels2 39777 | Elements of the class of r... |
| elrefsymrels3 39778 | Elements of the class of r... |
| elrefsymrelsrel 39779 | For sets, being an element... |
| dftrrels2 39783 | Alternate definition of th... |
| dftrrels3 39784 | Alternate definition of th... |
| dftrrel2 39785 | Alternate definition of th... |
| dftrrel3 39786 | Alternate definition of th... |
| eltrrels2 39787 | Element of the class of tr... |
| eltrrels3 39788 | Element of the class of tr... |
| eltrrelsrel 39789 | For sets, being an element... |
| trreleq 39790 | Equality theorem for the t... |
| trrelressn 39791 | Any class ' R ' restricted... |
| dfeqvrels2 39796 | Alternate definition of th... |
| dfeqvrels3 39797 | Alternate definition of th... |
| dfeqvrel2 39798 | Alternate definition of th... |
| dfeqvrel3 39799 | Alternate definition of th... |
| eleqvrels2 39800 | Element of the class of eq... |
| eleqvrels3 39801 | Element of the class of eq... |
| eleqvrelsrel 39802 | For sets, being an element... |
| elcoeleqvrels 39803 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 39804 | For sets, being an element... |
| eqvrelrel 39805 | An equivalence relation is... |
| eqvrelrefrel 39806 | An equivalence relation is... |
| eqvrelsymrel 39807 | An equivalence relation is... |
| eqvreltrrel 39808 | An equivalence relation is... |
| eqvrelim 39809 | Equivalence relation impli... |
| eqvreleq 39810 | Equality theorem for equiv... |
| eqvreleqi 39811 | Equality theorem for equiv... |
| eqvreleqd 39812 | Equality theorem for equiv... |
| eqvrelsym 39813 | An equivalence relation is... |
| eqvrelsymb 39814 | An equivalence relation is... |
| eqvreltr 39815 | An equivalence relation is... |
| eqvreltrd 39816 | A transitivity relation fo... |
| eqvreltr4d 39817 | A transitivity relation fo... |
| eqvrelref 39818 | An equivalence relation is... |
| eqvrelth 39819 | Basic property of equivale... |
| eqvrelcl 39820 | Elementhood in the field o... |
| eqvrelthi 39821 | Basic property of equivale... |
| eqvreldisj 39822 | Equivalence classes do not... |
| qsdisjALTV 39823 | Elements of a quotient set... |
| eqvrelqsel 39824 | If an element of a quotien... |
| eqvrelcoss 39825 | Two ways to express equiva... |
| eqvrelcoss3 39826 | Two ways to express equiva... |
| eqvrelcoss2 39827 | Two ways to express equiva... |
| eqvrelcoss4 39828 | Two ways to express equiva... |
| dfcoeleqvrels 39829 | Alternate definition of th... |
| dfcoeleqvrel 39830 | Alternate definition of th... |
| brredunds 39834 | Binary relation on the cla... |
| brredundsredund 39835 | For sets, binary relation ... |
| redundss3 39836 | Implication of redundancy ... |
| redundeq1 39837 | Equivalence of redundancy ... |
| redundpim3 39838 | Implication of redundancy ... |
| redundpbi1 39839 | Equivalence of redundancy ... |
| refrelsredund4 39840 | The naive version of the c... |
| refrelsredund2 39841 | The naive version of the c... |
| refrelsredund3 39842 | The naive version of the c... |
| refrelredund4 39843 | The naive version of the d... |
| refrelredund2 39844 | The naive version of the d... |
| refrelredund3 39845 | The naive version of the d... |
| dmqseq 39848 | Equality theorem for domai... |
| dmqseqi 39849 | Equality theorem for domai... |
| dmqseqd 39850 | Equality theorem for domai... |
| dmqseqeq1 39851 | Equality theorem for domai... |
| dmqseqeq1i 39852 | Equality theorem for domai... |
| dmqseqeq1d 39853 | Equality theorem for domai... |
| brdmqss 39854 | The domain quotient binary... |
| brdmqssqs 39855 | If ` A ` and ` R ` are set... |
| n0eldmqs 39856 | The empty set is not an el... |
| qseq 39857 | The quotient set equal to ... |
| n0eldmqseq 39858 | The empty set is not an el... |
| n0elim 39859 | Implication of that the em... |
| n0el3 39860 | Two ways of expressing tha... |
| cnvepresdmqss 39861 | The domain quotient binary... |
| cnvepresdmqs 39862 | The domain quotient predic... |
| unidmqs 39863 | The range of a relation is... |
| unidmqseq 39864 | The union of the domain qu... |
| dmqseqim 39865 | If the domain quotient of ... |
| dmqseqim2 39866 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 39867 | Elementhood in the domain ... |
| eldmqs1cossres 39868 | Elementhood in the domain ... |
| releldmqscoss 39869 | Elementhood in the domain ... |
| dmqscoelseq 39870 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 39871 | Two ways to express the eq... |
| brers 39876 | Binary equivalence relatio... |
| dferALTV2 39877 | Equivalence relation with ... |
| erALTVeq1 39878 | Equality theorem for equiv... |
| erALTVeq1i 39879 | Equality theorem for equiv... |
| erALTVeq1d 39880 | Equality theorem for equiv... |
| dfcomember 39881 | Alternate definition of th... |
| dfcomember2 39882 | Alternate definition of th... |
| dfcomember3 39883 | Alternate definition of th... |
| eqvreldmqs 39884 | Two ways to express comemb... |
| eqvreldmqs2 39885 | Two ways to express comemb... |
| brerser 39886 | Binary equivalence relatio... |
| erimeq2 39887 | Equivalence relation on it... |
| erimeq 39888 | Equivalence relation on it... |
| dffunsALTV 39892 | Alternate definition of th... |
| dffunsALTV2 39893 | Alternate definition of th... |
| dffunsALTV3 39894 | Alternate definition of th... |
| dffunsALTV4 39895 | Alternate definition of th... |
| dffunsALTV5 39896 | Alternate definition of th... |
| dffunALTV2 39897 | Alternate definition of th... |
| dffunALTV3 39898 | Alternate definition of th... |
| dffunALTV4 39899 | Alternate definition of th... |
| dffunALTV5 39900 | Alternate definition of th... |
| elfunsALTV 39901 | Elementhood in the class o... |
| elfunsALTV2 39902 | Elementhood in the class o... |
| elfunsALTV3 39903 | Elementhood in the class o... |
| elfunsALTV4 39904 | Elementhood in the class o... |
| elfunsALTV5 39905 | Elementhood in the class o... |
| elfunsALTVfunALTV 39906 | The element of the class o... |
| funALTVfun 39907 | Our definition of the func... |
| funALTVss 39908 | Subclass theorem for funct... |
| funALTVeq 39909 | Equality theorem for funct... |
| funALTVeqi 39910 | Equality inference for the... |
| funALTVeqd 39911 | Equality deduction for the... |
| dfdisjs 39917 | Alternate definition of th... |
| dfdisjs2 39918 | Alternate definition of th... |
| dfdisjs3 39919 | Alternate definition of th... |
| dfdisjs4 39920 | Alternate definition of th... |
| dfdisjs5 39921 | Alternate definition of th... |
| dfdisjALTV 39922 | Alternate definition of th... |
| dfdisjALTV2 39923 | Alternate definition of th... |
| dfdisjALTV3 39924 | Alternate definition of th... |
| dfdisjALTV4 39925 | Alternate definition of th... |
| dfdisjALTV5 39926 | Alternate definition of th... |
| dfdisjALTV5a 39927 | Alternate definition of th... |
| disjimeceqim 39928 | ` Disj ` implies coset-equ... |
| disjimeceqim2 39929 | ` Disj ` implies injectivi... |
| disjimeceqbi 39930 | ` Disj ` gives bicondition... |
| disjimeceqbi2 39931 | Injectivity of the block c... |
| disjimrmoeqec 39932 | Under ` Disj ` , every blo... |
| disjimdmqseq 39933 | Disjointness implies uniqu... |
| dfeldisj2 39934 | Alternate definition of th... |
| dfeldisj3 39935 | Alternate definition of th... |
| dfeldisj4 39936 | Alternate definition of th... |
| dfeldisj5 39937 | Alternate definition of th... |
| dfeldisj5a 39938 | Alternate definition of th... |
| eldisjim3 39939 | ` ElDisj ` elimination (tw... |
| eldisjdmqsim2 39940 | ElDisj of quotient implies... |
| eldisjdmqsim 39941 | Shared output implies equa... |
| suceldisj 39942 | Disjointness of successor ... |
| eldisjs 39943 | Elementhood in the class o... |
| eldisjs2 39944 | Elementhood in the class o... |
| eldisjs3 39945 | Elementhood in the class o... |
| eldisjs4 39946 | Elementhood in the class o... |
| eldisjs5 39947 | Elementhood in the class o... |
| eldisjsdisj 39948 | The element of the class o... |
| qmapeldisjs 39949 | When ` R ` is a set (e.g.,... |
| disjqmap2 39950 | Disjointness of ` QMap ` e... |
| disjqmap 39951 | Disjointness of ` QMap ` e... |
| eleldisjs 39952 | Elementhood in the disjoin... |
| eleldisjseldisj 39953 | The element of the disjoin... |
| disjrel 39954 | Disjoint relation is a rel... |
| disjss 39955 | Subclass theorem for disjo... |
| disjssi 39956 | Subclass theorem for disjo... |
| disjssd 39957 | Subclass theorem for disjo... |
| disjeq 39958 | Equality theorem for disjo... |
| disjeqi 39959 | Equality theorem for disjo... |
| disjeqd 39960 | Equality theorem for disjo... |
| disjdmqseqeq1 39961 | Lemma for the equality the... |
| eldisjss 39962 | Subclass theorem for disjo... |
| eldisjssi 39963 | Subclass theorem for disjo... |
| eldisjssd 39964 | Subclass theorem for disjo... |
| eldisjeq 39965 | Equality theorem for disjo... |
| eldisjeqi 39966 | Equality theorem for disjo... |
| eldisjeqd 39967 | Equality theorem for disjo... |
| disjres 39968 | Disjoint restriction. (Co... |
| eldisjn0elb 39969 | Two forms of disjoint elem... |
| disjxrn 39970 | Two ways of saying that a ... |
| disjxrnres5 39971 | Disjoint range Cartesian p... |
| disjorimxrn 39972 | Disjointness condition for... |
| disjimxrn 39973 | Disjointness condition for... |
| disjimres 39974 | Disjointness condition for... |
| disjimin 39975 | Disjointness condition for... |
| disjiminres 39976 | Disjointness condition for... |
| disjimxrnres 39977 | Disjointness condition for... |
| disjALTV0 39978 | The null class is disjoint... |
| disjALTVid 39979 | The class of identity rela... |
| disjALTVidres 39980 | The class of identity rela... |
| disjALTVinidres 39981 | The intersection with rest... |
| disjALTVxrnidres 39982 | The class of range Cartesi... |
| disjsuc 39983 | Disjoint range Cartesian p... |
| qmapeldisjsim 39984 | Injectivity of coset map f... |
| qmapeldisjsbi 39985 | Injectivity of coset map f... |
| rnqmapeleldisjsim 39986 | Element-disjointness of th... |
| dfantisymrel4 39988 | Alternate definition of th... |
| dfantisymrel5 39989 | Alternate definition of th... |
| antisymrelres 39990 | (Contributed by Peter Mazs... |
| antisymrelressn 39991 | (Contributed by Peter Mazs... |
| dfpart2 39996 | Alternate definition of th... |
| dfmembpart2 39997 | Alternate definition of th... |
| brparts 39998 | Binary partitions relation... |
| brparts2 39999 | Binary partitions relation... |
| brpartspart 40000 | Binary partition and the p... |
| parteq1 40001 | Equality theorem for parti... |
| parteq2 40002 | Equality theorem for parti... |
| parteq12 40003 | Equality theorem for parti... |
| parteq1i 40004 | Equality theorem for parti... |
| parteq1d 40005 | Equality theorem for parti... |
| partsuc2 40006 | Property of the partition.... |
| partsuc 40007 | Property of the partition.... |
| disjim 40008 | The "Divide et Aequivalere... |
| disjimi 40009 | Every disjoint relation ge... |
| detlem 40010 | If a relation is disjoint,... |
| eldisjim 40011 | If the elements of ` A ` a... |
| eldisjim2 40012 | Alternate form of ~ eldisj... |
| eqvrel0 40013 | The null class is an equiv... |
| det0 40014 | The cosets by the null cla... |
| eqvrelcoss0 40015 | The cosets by the null cla... |
| eqvrelid 40016 | The identity relation is a... |
| eqvrel1cossidres 40017 | The cosets by a restricted... |
| eqvrel1cossinidres 40018 | The cosets by an intersect... |
| eqvrel1cossxrnidres 40019 | The cosets by a range Cart... |
| detid 40020 | The cosets by the identity... |
| eqvrelcossid 40021 | The cosets by the identity... |
| detidres 40022 | The cosets by the restrict... |
| detinidres 40023 | The cosets by the intersec... |
| detxrnidres 40024 | The cosets by the range Ca... |
| disjlem14 40025 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 40026 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 40027 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 40028 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 40029 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 40030 | Lemma for ~ disjdmqseq via... |
| disjdmqs 40031 | If a relation is disjoint,... |
| disjdmqseq 40032 | If a relation is disjoint,... |
| eldisjn0el 40033 | Special case of ~ disjdmqs... |
| partim2 40034 | Disjoint relation on its n... |
| partim 40035 | Partition implies equivale... |
| partimeq 40036 | Partition implies that the... |
| eldisjlem19 40037 | Special case of ~ disjlem1... |
| membpartlem19 40038 | Together with ~ disjlem19 ... |
| petlem 40039 | If you can prove that the ... |
| petlemi 40040 | If you can prove disjointn... |
| pet02 40041 | Class ` A ` is a partition... |
| pet0 40042 | Class ` A ` is a partition... |
| petid2 40043 | Class ` A ` is a partition... |
| petid 40044 | A class is a partition by ... |
| petidres2 40045 | Class ` A ` is a partition... |
| petidres 40046 | A class is a partition by ... |
| petinidres2 40047 | Class ` A ` is a partition... |
| petinidres 40048 | A class is a partition by ... |
| petxrnidres2 40049 | Class ` A ` is a partition... |
| petxrnidres 40050 | A class is a partition by ... |
| eqvreldisj1 40051 | The elements of the quotie... |
| eqvreldisj2 40052 | The elements of the quotie... |
| eqvreldisj3 40053 | The elements of the quotie... |
| eqvreldisj4 40054 | Intersection with the conv... |
| eqvreldisj5 40055 | Range Cartesian product wi... |
| eqvrelqseqdisj2 40056 | Implication of ~ eqvreldis... |
| disjimeldisjdmqs 40057 | ` Disj ` implies element-d... |
| eldisjsim1 40058 | An element of the class of... |
| eldisjsim2 40059 | An element of the class of... |
| disjsssrels 40060 | The class of disjoint rela... |
| eldisjsim3 40061 | ` Disjs ` implies element-... |
| eldisjsim4 40062 | ` Disjs ` implies element-... |
| eldisjsim5 40063 | ` Disjs ` is closed under ... |
| eldisjs6 40064 | Elementhood in the class o... |
| eldisjs7 40065 | Elementhood in the class o... |
| dfdisjs6 40066 | Alternate definition of th... |
| dfdisjs7 40067 | Alternate definition of th... |
| fences3 40068 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 40069 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 40070 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 40071 | Lemma for the Partition-Eq... |
| mainer 40072 | The Main Theorem of Equiva... |
| partimcomember 40073 | Partition with general ` R... |
| mpet3 40074 | Member Partition-Equivalen... |
| cpet2 40075 | The conventional form of t... |
| cpet 40076 | The conventional form of M... |
| mpet 40077 | Member Partition-Equivalen... |
| mpet2 40078 | Member Partition-Equivalen... |
| mpets2 40079 | Member Partition-Equivalen... |
| mpets 40080 | Member Partition-Equivalen... |
| mainpart 40081 | Partition with general ` R... |
| fences 40082 | The Theorem of Fences by E... |
| fences2 40083 | The Theorem of Fences by E... |
| mainer2 40084 | The Main Theorem of Equiva... |
| mainerim 40085 | Every equivalence relation... |
| petincnvepres2 40086 | A partition-equivalence th... |
| petincnvepres 40087 | The shortest form of a par... |
| pet2 40088 | Partition-Equivalence Theo... |
| pet 40089 | Partition-Equivalence Theo... |
| pets 40090 | Partition-Equivalence Theo... |
| dmqsblocks 40091 | If the ~ pet span ` ( R |X... |
| dfpetparts2 40096 | Alternate definition of ` ... |
| dfpet2parts2 40097 | Grade stability applied to... |
| dfpeters2 40098 | Alternate definition of ` ... |
| typesafepets 40099 | Type-safe ~ pets scheme. ... |
| petseq 40100 | Generalized partition-equi... |
| pets2eq 40101 | Grade-stable generalized p... |
| prtlem60 40102 | Lemma for ~ prter3 . (Con... |
| bicomdd 40103 | Commute two sides of a bic... |
| jca2r 40104 | Inference conjoining the c... |
| jca3 40105 | Inference conjoining the c... |
| prtlem70 40106 | Lemma for ~ prter3 : a rea... |
| ibdr 40107 | Reverse of ~ ibd . (Contr... |
| prtlem100 40108 | Lemma for ~ prter3 . (Con... |
| prtlem5 40109 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 40110 | Lemma for ~ prter2 . (Con... |
| brabsb2 40111 | A closed form of ~ brabsb ... |
| eqbrrdv2 40112 | Other version of ~ eqbrrdi... |
| prtlem9 40113 | Lemma for ~ prter3 . (Con... |
| prtlem10 40114 | Lemma for ~ prter3 . (Con... |
| prtlem11 40115 | Lemma for ~ prter2 . (Con... |
| prtlem12 40116 | Lemma for ~ prtex and ~ pr... |
| prtlem13 40117 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 40118 | Lemma for ~ prtex , ~ prte... |
| prtlem400 40119 | Lemma for ~ prter2 and als... |
| erprt 40122 | The quotient set of an equ... |
| prtlem14 40123 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 40124 | Lemma for ~ prter1 and ~ p... |
| prtlem17 40125 | Lemma for ~ prter2 . (Con... |
| prtlem18 40126 | Lemma for ~ prter2 . (Con... |
| prtlem19 40127 | Lemma for ~ prter2 . (Con... |
| prter1 40128 | Every partition generates ... |
| prtex 40129 | The equivalence relation g... |
| prter2 40130 | The quotient set of the eq... |
| prter3 40131 | For every partition there ... |
| axc5 40142 | This theorem repeats ~ sp ... |
| ax4fromc4 40143 | Rederivation of Axiom ~ ax... |
| ax10fromc7 40144 | Rederivation of Axiom ~ ax... |
| ax6fromc10 40145 | Rederivation of Axiom ~ ax... |
| hba1-o 40146 | The setvar ` x ` is not fr... |
| axc4i-o 40147 | Inference version of ~ ax-... |
| equid1 40148 | Proof of ~ equid from our ... |
| equcomi1 40149 | Proof of ~ equcomi from ~ ... |
| aecom-o 40150 | Commutation law for identi... |
| aecoms-o 40151 | A commutation rule for ide... |
| hbae-o 40152 | All variables are effectiv... |
| dral1-o 40153 | Formula-building lemma for... |
| ax12fromc15 40154 | Rederivation of Axiom ~ ax... |
| ax13fromc9 40155 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 40156 | Axiom to quantify a variab... |
| sps-o 40157 | Generalization of antecede... |
| hbequid 40158 | Bound-variable hypothesis ... |
| nfequid-o 40159 | Bound-variable hypothesis ... |
| axc5c7 40160 | Proof of a single axiom th... |
| axc5c7toc5 40161 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 40162 | Rederivation of ~ ax-c7 fr... |
| axc711 40163 | Proof of a single axiom th... |
| nfa1-o 40164 | ` x ` is not free in ` A. ... |
| axc711toc7 40165 | Rederivation of ~ ax-c7 fr... |
| axc711to11 40166 | Rederivation of ~ ax-11 fr... |
| axc5c711 40167 | Proof of a single axiom th... |
| axc5c711toc5 40168 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 40169 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 40170 | Rederivation of ~ ax-11 fr... |
| equidqe 40171 | ~ equid with existential q... |
| axc5sp1 40172 | A special case of ~ ax-c5 ... |
| equidq 40173 | ~ equid with universal qua... |
| equid1ALT 40174 | Alternate proof of ~ equid... |
| axc11nfromc11 40175 | Rederivation of ~ ax-c11n ... |
| naecoms-o 40176 | A commutation rule for dis... |
| hbnae-o 40177 | All variables are effectiv... |
| dvelimf-o 40178 | Proof of ~ dvelimh that us... |
| dral2-o 40179 | Formula-building lemma for... |
| aev-o 40180 | A "distinctor elimination"... |
| ax5eq 40181 | Theorem to add distinct qu... |
| dveeq2-o 40182 | Quantifier introduction wh... |
| axc16g-o 40183 | A generalization of Axiom ... |
| dveeq1-o 40184 | Quantifier introduction wh... |
| dveeq1-o16 40185 | Version of ~ dveeq1 using ... |
| ax5el 40186 | Theorem to add distinct qu... |
| axc11n-16 40187 | This theorem shows that, g... |
| dveel2ALT 40188 | Alternate proof of ~ dveel... |
| ax12f 40189 | Basis step for constructin... |
| ax12eq 40190 | Basis step for constructin... |
| ax12el 40191 | Basis step for constructin... |
| ax12indn 40192 | Induction step for constru... |
| ax12indi 40193 | Induction step for constru... |
| ax12indalem 40194 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 40195 | Alternate proof of ~ ax12i... |
| ax12inda2 40196 | Induction step for constru... |
| ax12inda 40197 | Induction step for constru... |
| ax12v2-o 40198 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 40199 | Derive ~ ax-c15 from a hyp... |
| axc11-o 40200 | Show that ~ ax-c11 can be ... |
| fsumshftd 40201 | Index shift of a finite su... |
| riotaclbgBAD 40203 | Closure of restricted iota... |
| riotaclbBAD 40204 | Closure of restricted iota... |
| riotasvd 40205 | Deduction version of ~ rio... |
| riotasv2d 40206 | Value of description binde... |
| riotasv2s 40207 | The value of description b... |
| riotasv 40208 | Value of description binde... |
| riotasv3d 40209 | A property ` ch ` holding ... |
| elimhyps 40210 | A version of ~ elimhyp usi... |
| dedths 40211 | A version of weak deductio... |
| renegclALT 40212 | Closure law for negative o... |
| elimhyps2 40213 | Generalization of ~ elimhy... |
| dedths2 40214 | Generalization of ~ dedths... |
| nfcxfrdf 40215 | A utility lemma to transfe... |
| nfded 40216 | A deduction theorem that c... |
| nfded2 40217 | A deduction theorem that c... |
| nfunidALT2 40218 | Deduction version of ~ nfu... |
| nfunidALT 40219 | Deduction version of ~ nfu... |
| nfopdALT 40220 | Deduction version of bound... |
| cnaddcom 40221 | Recover the commutative la... |
| toycom 40222 | Show the commutative law f... |
| lshpset 40227 | The set of all hyperplanes... |
| islshp 40228 | The predicate "is a hyperp... |
| islshpsm 40229 | Hyperplane properties expr... |
| lshplss 40230 | A hyperplane is a subspace... |
| lshpne 40231 | A hyperplane is not equal ... |
| lshpnel 40232 | A hyperplane's generating ... |
| lshpnelb 40233 | The subspace sum of a hype... |
| lshpnel2N 40234 | Condition that determines ... |
| lshpne0 40235 | The member of the span in ... |
| lshpdisj 40236 | A hyperplane and the span ... |
| lshpcmp 40237 | If two hyperplanes are com... |
| lshpinN 40238 | The intersection of two di... |
| lsatset 40239 | The set of all 1-dim subsp... |
| islsat 40240 | The predicate "is a 1-dim ... |
| lsatlspsn2 40241 | The span of a nonzero sing... |
| lsatlspsn 40242 | The span of a nonzero sing... |
| islsati 40243 | A 1-dim subspace (atom) (o... |
| lsateln0 40244 | A 1-dim subspace (atom) (o... |
| lsatlss 40245 | The set of 1-dim subspaces... |
| lsatlssel 40246 | An atom is a subspace. (C... |
| lsatssv 40247 | An atom is a set of vector... |
| lsatn0 40248 | A 1-dim subspace (atom) of... |
| lsatspn0 40249 | The span of a vector is an... |
| lsator0sp 40250 | The span of a vector is ei... |
| lsatssn0 40251 | A subspace (or any class) ... |
| lsatcmp 40252 | If two atoms are comparabl... |
| lsatcmp2 40253 | If an atom is included in ... |
| lsatel 40254 | A nonzero vector in an ato... |
| lsatelbN 40255 | A nonzero vector in an ato... |
| lsat2el 40256 | Two atoms sharing a nonzer... |
| lsmsat 40257 | Convert comparison of atom... |
| lsatfixedN 40258 | Show equality with the spa... |
| lsmsatcv 40259 | Subspace sum has the cover... |
| lssatomic 40260 | The lattice of subspaces i... |
| lssats 40261 | The lattice of subspaces i... |
| lpssat 40262 | Two subspaces in a proper ... |
| lrelat 40263 | Subspaces are relatively a... |
| lssatle 40264 | The ordering of two subspa... |
| lssat 40265 | Two subspaces in a proper ... |
| islshpat 40266 | Hyperplane properties expr... |
| lcvfbr 40269 | The covers relation for a ... |
| lcvbr 40270 | The covers relation for a ... |
| lcvbr2 40271 | The covers relation for a ... |
| lcvbr3 40272 | The covers relation for a ... |
| lcvpss 40273 | The covers relation implie... |
| lcvnbtwn 40274 | The covers relation implie... |
| lcvntr 40275 | The covers relation is not... |
| lcvnbtwn2 40276 | The covers relation implie... |
| lcvnbtwn3 40277 | The covers relation implie... |
| lsmcv2 40278 | Subspace sum has the cover... |
| lcvat 40279 | If a subspace covers anoth... |
| lsatcv0 40280 | An atom covers the zero su... |
| lsatcveq0 40281 | A subspace covered by an a... |
| lsat0cv 40282 | A subspace is an atom iff ... |
| lcvexchlem1 40283 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 40284 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 40285 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 40286 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 40287 | Lemma for ~ lcvexch . (Co... |
| lcvexch 40288 | Subspaces satisfy the exch... |
| lcvp 40289 | Covering property of Defin... |
| lcv1 40290 | Covering property of a sub... |
| lcv2 40291 | Covering property of a sub... |
| lsatexch 40292 | The atom exchange property... |
| lsatnle 40293 | The meet of a subspace and... |
| lsatnem0 40294 | The meet of distinct atoms... |
| lsatexch1 40295 | The atom exch1ange propert... |
| lsatcv0eq 40296 | If the sum of two atoms co... |
| lsatcv1 40297 | Two atoms covering the zer... |
| lsatcvatlem 40298 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 40299 | A nonzero subspace less th... |
| lsatcvat2 40300 | A subspace covered by the ... |
| lsatcvat3 40301 | A condition implying that ... |
| islshpcv 40302 | Hyperplane properties expr... |
| l1cvpat 40303 | A subspace covered by the ... |
| l1cvat 40304 | Create an atom under an el... |
| lshpat 40305 | Create an atom under a hyp... |
| lflset 40308 | The set of linear function... |
| islfl 40309 | The predicate "is a linear... |
| lfli 40310 | Property of a linear funct... |
| islfld 40311 | Properties that determine ... |
| lflf 40312 | A linear functional is a f... |
| lflcl 40313 | A linear functional value ... |
| lfl0 40314 | A linear functional is zer... |
| lfladd 40315 | Property of a linear funct... |
| lflsub 40316 | Property of a linear funct... |
| lflmul 40317 | Property of a linear funct... |
| lfl0f 40318 | The zero function is a fun... |
| lfl1 40319 | A nonzero functional has a... |
| lfladdcl 40320 | Closure of addition of two... |
| lfladdcom 40321 | Commutativity of functiona... |
| lfladdass 40322 | Associativity of functiona... |
| lfladd0l 40323 | Functional addition with t... |
| lflnegcl 40324 | Closure of the negative of... |
| lflnegl 40325 | A functional plus its nega... |
| lflvscl 40326 | Closure of a scalar produc... |
| lflvsdi1 40327 | Distributive law for (righ... |
| lflvsdi2 40328 | Reverse distributive law f... |
| lflvsdi2a 40329 | Reverse distributive law f... |
| lflvsass 40330 | Associative law for (right... |
| lfl0sc 40331 | The (right vector space) s... |
| lflsc0N 40332 | The scalar product with th... |
| lfl1sc 40333 | The (right vector space) s... |
| lkrfval 40336 | The kernel of a functional... |
| lkrval 40337 | Value of the kernel of a f... |
| ellkr 40338 | Membership in the kernel o... |
| lkrval2 40339 | Value of the kernel of a f... |
| ellkr2 40340 | Membership in the kernel o... |
| lkrcl 40341 | A member of the kernel of ... |
| lkrf0 40342 | The value of a functional ... |
| lkr0f 40343 | The kernel of the zero fun... |
| lkrlss 40344 | The kernel of a linear fun... |
| lkrssv 40345 | The kernel of a linear fun... |
| lkrsc 40346 | The kernel of a nonzero sc... |
| lkrscss 40347 | The kernel of a scalar pro... |
| eqlkr 40348 | Two functionals with the s... |
| eqlkr2 40349 | Two functionals with the s... |
| eqlkr3 40350 | Two functionals with the s... |
| lkrlsp 40351 | The subspace sum of a kern... |
| lkrlsp2 40352 | The subspace sum of a kern... |
| lkrlsp3 40353 | The subspace sum of a kern... |
| lkrshp 40354 | The kernel of a nonzero fu... |
| lkrshp3 40355 | The kernels of nonzero fun... |
| lkrshpor 40356 | The kernel of a functional... |
| lkrshp4 40357 | A kernel is a hyperplane i... |
| lshpsmreu 40358 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 40359 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 40360 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 40361 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 40362 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 40363 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 40364 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 40365 | The set ` G ` defined by h... |
| lshpkr 40366 | The kernel of functional `... |
| lshpkrex 40367 | There exists a functional ... |
| lshpset2N 40368 | The set of all hyperplanes... |
| islshpkrN 40369 | The predicate "is a hyperp... |
| lfl1dim 40370 | Equivalent expressions for... |
| lfl1dim2N 40371 | Equivalent expressions for... |
| ldualset 40374 | Define the (left) dual of ... |
| ldualvbase 40375 | The vectors of a dual spac... |
| ldualelvbase 40376 | Utility theorem for conver... |
| ldualfvadd 40377 | Vector addition in the dua... |
| ldualvadd 40378 | Vector addition in the dua... |
| ldualvaddcl 40379 | The value of vector additi... |
| ldualvaddval 40380 | The value of the value of ... |
| ldualsca 40381 | The ring of scalars of the... |
| ldualsbase 40382 | Base set of scalar ring fo... |
| ldualsaddN 40383 | Scalar addition for the du... |
| ldualsmul 40384 | Scalar multiplication for ... |
| ldualfvs 40385 | Scalar product operation f... |
| ldualvs 40386 | Scalar product operation v... |
| ldualvsval 40387 | Value of scalar product op... |
| ldualvscl 40388 | The scalar product operati... |
| ldualvaddcom 40389 | Commutative law for vector... |
| ldualvsass 40390 | Associative law for scalar... |
| ldualvsass2 40391 | Associative law for scalar... |
| ldualvsdi1 40392 | Distributive law for scala... |
| ldualvsdi2 40393 | Reverse distributive law f... |
| ldualgrplem 40394 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 40395 | The dual of a vector space... |
| ldual0 40396 | The zero scalar of the dua... |
| ldual1 40397 | The unit scalar of the dua... |
| ldualneg 40398 | The negative of a scalar o... |
| ldual0v 40399 | The zero vector of the dua... |
| ldual0vcl 40400 | The dual zero vector is a ... |
| lduallmodlem 40401 | Lemma for ~ lduallmod . (... |
| lduallmod 40402 | The dual of a left module ... |
| lduallvec 40403 | The dual of a left vector ... |
| ldualvsub 40404 | The value of vector subtra... |
| ldualvsubcl 40405 | Closure of vector subtract... |
| ldualvsubval 40406 | The value of the value of ... |
| ldualssvscl 40407 | Closure of scalar product ... |
| ldualssvsubcl 40408 | Closure of vector subtract... |
| ldual0vs 40409 | Scalar zero times a functi... |
| lkr0f2 40410 | The kernel of the zero fun... |
| lduallkr3 40411 | The kernels of nonzero fun... |
| lkrpssN 40412 | Proper subset relation bet... |
| lkrin 40413 | Intersection of the kernel... |
| eqlkr4 40414 | Two functionals with the s... |
| ldual1dim 40415 | Equivalent expressions for... |
| ldualkrsc 40416 | The kernel of a nonzero sc... |
| lkrss 40417 | The kernel of a scalar pro... |
| lkrss2N 40418 | Two functionals with kerne... |
| lkreqN 40419 | Proportional functionals h... |
| lkrlspeqN 40420 | Condition for colinear fun... |
| isopos 40429 | The predicate "is an ortho... |
| opposet 40430 | Every orthoposet is a pose... |
| oposlem 40431 | Lemma for orthoposet prope... |
| op01dm 40432 | Conditions necessary for z... |
| op0cl 40433 | An orthoposet has a zero e... |
| op1cl 40434 | An orthoposet has a unity ... |
| op0le 40435 | Orthoposet zero is less th... |
| ople0 40436 | An element less than or eq... |
| opnlen0 40437 | An element not less than a... |
| lub0N 40438 | The least upper bound of t... |
| opltn0 40439 | A lattice element greater ... |
| ople1 40440 | Any element is less than t... |
| op1le 40441 | If the orthoposet unity is... |
| glb0N 40442 | The greatest lower bound o... |
| opoccl 40443 | Closure of orthocomplement... |
| opococ 40444 | Double negative law for or... |
| opcon3b 40445 | Contraposition law for ort... |
| opcon2b 40446 | Orthocomplement contraposi... |
| opcon1b 40447 | Orthocomplement contraposi... |
| oplecon3 40448 | Contraposition law for ort... |
| oplecon3b 40449 | Contraposition law for ort... |
| oplecon1b 40450 | Contraposition law for str... |
| opoc1 40451 | Orthocomplement of orthopo... |
| opoc0 40452 | Orthocomplement of orthopo... |
| opltcon3b 40453 | Contraposition law for str... |
| opltcon1b 40454 | Contraposition law for str... |
| opltcon2b 40455 | Contraposition law for str... |
| opexmid 40456 | Law of excluded middle for... |
| opnoncon 40457 | Law of contradiction for o... |
| riotaocN 40458 | The orthocomplement of the... |
| cmtfvalN 40459 | Value of commutes relation... |
| cmtvalN 40460 | Equivalence for commutes r... |
| isolat 40461 | The predicate "is an ortho... |
| ollat 40462 | An ortholattice is a latti... |
| olop 40463 | An ortholattice is an orth... |
| olposN 40464 | An ortholattice is a poset... |
| isolatiN 40465 | Properties that determine ... |
| oldmm1 40466 | De Morgan's law for meet i... |
| oldmm2 40467 | De Morgan's law for meet i... |
| oldmm3N 40468 | De Morgan's law for meet i... |
| oldmm4 40469 | De Morgan's law for meet i... |
| oldmj1 40470 | De Morgan's law for join i... |
| oldmj2 40471 | De Morgan's law for join i... |
| oldmj3 40472 | De Morgan's law for join i... |
| oldmj4 40473 | De Morgan's law for join i... |
| olj01 40474 | An ortholattice element jo... |
| olj02 40475 | An ortholattice element jo... |
| olm11 40476 | The meet of an ortholattic... |
| olm12 40477 | The meet of an ortholattic... |
| latmassOLD 40478 | Ortholattice meet is assoc... |
| latm12 40479 | A rearrangement of lattice... |
| latm32 40480 | A rearrangement of lattice... |
| latmrot 40481 | Rotate lattice meet of 3 c... |
| latm4 40482 | Rearrangement of lattice m... |
| latmmdiN 40483 | Lattice meet distributes o... |
| latmmdir 40484 | Lattice meet distributes o... |
| olm01 40485 | Meet with lattice zero is ... |
| olm02 40486 | Meet with lattice zero is ... |
| isoml 40487 | The predicate "is an ortho... |
| isomliN 40488 | Properties that determine ... |
| omlol 40489 | An orthomodular lattice is... |
| omlop 40490 | An orthomodular lattice is... |
| omllat 40491 | An orthomodular lattice is... |
| omllaw 40492 | The orthomodular law. (Co... |
| omllaw2N 40493 | Variation of orthomodular ... |
| omllaw3 40494 | Orthomodular law equivalen... |
| omllaw4 40495 | Orthomodular law equivalen... |
| omllaw5N 40496 | The orthomodular law. Rem... |
| cmtcomlemN 40497 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 40498 | Commutation is symmetric. ... |
| cmt2N 40499 | Commutation with orthocomp... |
| cmt3N 40500 | Commutation with orthocomp... |
| cmt4N 40501 | Commutation with orthocomp... |
| cmtbr2N 40502 | Alternate definition of th... |
| cmtbr3N 40503 | Alternate definition for t... |
| cmtbr4N 40504 | Alternate definition for t... |
| lecmtN 40505 | Ordered elements commute. ... |
| cmtidN 40506 | Any element commutes with ... |
| omlfh1N 40507 | Foulis-Holland Theorem, pa... |
| omlfh3N 40508 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 40509 | Analogue of modular law ~ ... |
| omlspjN 40510 | Contraction of a Sasaki pr... |
| cvrfval 40517 | Value of covers relation "... |
| cvrval 40518 | Binary relation expressing... |
| cvrlt 40519 | The covers relation implie... |
| cvrnbtwn 40520 | There is no element betwee... |
| ncvr1 40521 | No element covers the latt... |
| cvrletrN 40522 | Property of an element abo... |
| cvrval2 40523 | Binary relation expressing... |
| cvrnbtwn2 40524 | The covers relation implie... |
| cvrnbtwn3 40525 | The covers relation implie... |
| cvrcon3b 40526 | Contraposition law for the... |
| cvrle 40527 | The covers relation implie... |
| cvrnbtwn4 40528 | The covers relation implie... |
| cvrnle 40529 | The covers relation implie... |
| cvrne 40530 | The covers relation implie... |
| cvrnrefN 40531 | The covers relation is not... |
| cvrcmp 40532 | If two lattice elements th... |
| cvrcmp2 40533 | If two lattice elements co... |
| pats 40534 | The set of atoms in a pose... |
| isat 40535 | The predicate "is an atom"... |
| isat2 40536 | The predicate "is an atom"... |
| atcvr0 40537 | An atom covers zero. ( ~ ... |
| atbase 40538 | An atom is a member of the... |
| atssbase 40539 | The set of atoms is a subs... |
| 0ltat 40540 | An atom is greater than ze... |
| leatb 40541 | A poset element less than ... |
| leat 40542 | A poset element less than ... |
| leat2 40543 | A nonzero poset element le... |
| leat3 40544 | A poset element less than ... |
| meetat 40545 | The meet of any element wi... |
| meetat2 40546 | The meet of any element wi... |
| isatl 40548 | The predicate "is an atomi... |
| atllat 40549 | An atomic lattice is a lat... |
| atlpos 40550 | An atomic lattice is a pos... |
| atl0dm 40551 | Condition necessary for ze... |
| atl0cl 40552 | An atomic lattice has a ze... |
| atl0le 40553 | Orthoposet zero is less th... |
| atlle0 40554 | An element less than or eq... |
| atlltn0 40555 | A lattice element greater ... |
| isat3 40556 | The predicate "is an atom"... |
| atn0 40557 | An atom is not zero. ( ~ ... |
| atnle0 40558 | An atom is not less than o... |
| atlen0 40559 | A lattice element is nonze... |
| atcmp 40560 | If two atoms are comparabl... |
| atncmp 40561 | Frequently-used variation ... |
| atnlt 40562 | Two atoms cannot satisfy t... |
| atcvreq0 40563 | An element covered by an a... |
| atncvrN 40564 | Two atoms cannot satisfy t... |
| atlex 40565 | Every nonzero element of a... |
| atnle 40566 | Two ways of expressing "an... |
| atnem0 40567 | The meet of distinct atoms... |
| atlatmstc 40568 | An atomic, complete, ortho... |
| atlatle 40569 | The ordering of two Hilber... |
| atlrelat1 40570 | An atomistic lattice with ... |
| iscvlat 40572 | The predicate "is an atomi... |
| iscvlat2N 40573 | The predicate "is an atomi... |
| cvlatl 40574 | An atomic lattice with the... |
| cvllat 40575 | An atomic lattice with the... |
| cvlposN 40576 | An atomic lattice with the... |
| cvlexch1 40577 | An atomic covering lattice... |
| cvlexch2 40578 | An atomic covering lattice... |
| cvlexchb1 40579 | An atomic covering lattice... |
| cvlexchb2 40580 | An atomic covering lattice... |
| cvlexch3 40581 | An atomic covering lattice... |
| cvlexch4N 40582 | An atomic covering lattice... |
| cvlatexchb1 40583 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 40584 | A version of ~ cvlexchb2 f... |
| cvlatexch1 40585 | Atom exchange property. (... |
| cvlatexch2 40586 | Atom exchange property. (... |
| cvlatexch3 40587 | Atom exchange property. (... |
| cvlcvr1 40588 | The covering property. Pr... |
| cvlcvrp 40589 | A Hilbert lattice satisfie... |
| cvlatcvr1 40590 | An atom is covered by its ... |
| cvlatcvr2 40591 | An atom is covered by its ... |
| cvlsupr2 40592 | Two equivalent ways of exp... |
| cvlsupr3 40593 | Two equivalent ways of exp... |
| cvlsupr4 40594 | Consequence of superpositi... |
| cvlsupr5 40595 | Consequence of superpositi... |
| cvlsupr6 40596 | Consequence of superpositi... |
| cvlsupr7 40597 | Consequence of superpositi... |
| cvlsupr8 40598 | Consequence of superpositi... |
| ishlat1 40601 | The predicate "is a Hilber... |
| ishlat2 40602 | The predicate "is a Hilber... |
| ishlat3N 40603 | The predicate "is a Hilber... |
| ishlatiN 40604 | Properties that determine ... |
| hlomcmcv 40605 | A Hilbert lattice is ortho... |
| hloml 40606 | A Hilbert lattice is ortho... |
| hlclat 40607 | A Hilbert lattice is compl... |
| hlcvl 40608 | A Hilbert lattice is an at... |
| hlatl 40609 | A Hilbert lattice is atomi... |
| hlol 40610 | A Hilbert lattice is an or... |
| hlop 40611 | A Hilbert lattice is an or... |
| hllat 40612 | A Hilbert lattice is a lat... |
| hllatd 40613 | Deduction form of ~ hllat ... |
| hlomcmat 40614 | A Hilbert lattice is ortho... |
| hlpos 40615 | A Hilbert lattice is a pos... |
| hlatjcl 40616 | Closure of join operation.... |
| hlatjcom 40617 | Commutatitivity of join op... |
| hlatjidm 40618 | Idempotence of join operat... |
| hlatjass 40619 | Lattice join is associativ... |
| hlatj12 40620 | Swap 1st and 2nd members o... |
| hlatj32 40621 | Swap 2nd and 3rd members o... |
| hlatjrot 40622 | Rotate lattice join of 3 c... |
| hlatj4 40623 | Rearrangement of lattice j... |
| hlatlej1 40624 | A join's first argument is... |
| hlatlej2 40625 | A join's second argument i... |
| glbconN 40626 | De Morgan's law for GLB an... |
| glbconxN 40627 | De Morgan's law for GLB an... |
| atnlej1 40628 | If an atom is not less tha... |
| atnlej2 40629 | If an atom is not less tha... |
| hlsuprexch 40630 | A Hilbert lattice has the ... |
| hlexch1 40631 | A Hilbert lattice has the ... |
| hlexch2 40632 | A Hilbert lattice has the ... |
| hlexchb1 40633 | A Hilbert lattice has the ... |
| hlexchb2 40634 | A Hilbert lattice has the ... |
| hlsupr 40635 | A Hilbert lattice has the ... |
| hlsupr2 40636 | A Hilbert lattice has the ... |
| hlhgt4 40637 | A Hilbert lattice has a he... |
| hlhgt2 40638 | A Hilbert lattice has a he... |
| hl0lt1N 40639 | Lattice 0 is less than lat... |
| hlexch3 40640 | A Hilbert lattice has the ... |
| hlexch4N 40641 | A Hilbert lattice has the ... |
| hlatexchb1 40642 | A version of ~ hlexchb1 fo... |
| hlatexchb2 40643 | A version of ~ hlexchb2 fo... |
| hlatexch1 40644 | Atom exchange property. (... |
| hlatexch2 40645 | Atom exchange property. (... |
| hlatmstcOLDN 40646 | An atomic, complete, ortho... |
| hlatle 40647 | The ordering of two Hilber... |
| hlateq 40648 | The equality of two Hilber... |
| hlrelat1 40649 | An atomistic lattice with ... |
| hlrelat5N 40650 | An atomistic lattice with ... |
| hlrelat 40651 | A Hilbert lattice is relat... |
| hlrelat2 40652 | A consequence of relative ... |
| exatleN 40653 | A condition for an atom to... |
| hl2at 40654 | A Hilbert lattice has at l... |
| atex 40655 | At least one atom exists. ... |
| intnatN 40656 | If the intersection with a... |
| 2llnne2N 40657 | Condition implying that tw... |
| 2llnneN 40658 | Condition implying that tw... |
| cvr1 40659 | A Hilbert lattice has the ... |
| cvr2N 40660 | Less-than and covers equiv... |
| hlrelat3 40661 | The Hilbert lattice is rel... |
| cvrval3 40662 | Binary relation expressing... |
| cvrval4N 40663 | Binary relation expressing... |
| cvrval5 40664 | Binary relation expressing... |
| cvrp 40665 | A Hilbert lattice satisfie... |
| atcvr1 40666 | An atom is covered by its ... |
| atcvr2 40667 | An atom is covered by its ... |
| cvrexchlem 40668 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 40669 | A Hilbert lattice satisfie... |
| cvratlem 40670 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 40671 | A nonzero Hilbert lattice ... |
| ltltncvr 40672 | A chained strong ordering ... |
| ltcvrntr 40673 | Non-transitive condition f... |
| cvrntr 40674 | The covers relation is not... |
| atcvr0eq 40675 | The covers relation is not... |
| lnnat 40676 | A line (the join of two di... |
| atcvrj0 40677 | Two atoms covering the zer... |
| cvrat2 40678 | A Hilbert lattice element ... |
| atcvrneN 40679 | Inequality derived from at... |
| atcvrj1 40680 | Condition for an atom to b... |
| atcvrj2b 40681 | Condition for an atom to b... |
| atcvrj2 40682 | Condition for an atom to b... |
| atleneN 40683 | Inequality derived from at... |
| atltcvr 40684 | An equivalence of less-tha... |
| atle 40685 | Any nonzero element has an... |
| atlt 40686 | Two atoms are unequal iff ... |
| atlelt 40687 | Transfer less-than relatio... |
| 2atlt 40688 | Given an atom less than an... |
| atexchcvrN 40689 | Atom exchange property. V... |
| atexchltN 40690 | Atom exchange property. V... |
| cvrat3 40691 | A condition implying that ... |
| cvrat4 40692 | A condition implying exist... |
| cvrat42 40693 | Commuted version of ~ cvra... |
| 2atjm 40694 | The meet of a line (expres... |
| atbtwn 40695 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 40696 | There exists a 3rd atom ` ... |
| atbtwnex 40697 | Given atoms ` P ` in ` X `... |
| 3noncolr2 40698 | Two ways to express 3 non-... |
| 3noncolr1N 40699 | Two ways to express 3 non-... |
| hlatcon3 40700 | Atom exchange combined wit... |
| hlatcon2 40701 | Atom exchange combined wit... |
| 4noncolr3 40702 | A way to express 4 non-col... |
| 4noncolr2 40703 | A way to express 4 non-col... |
| 4noncolr1 40704 | A way to express 4 non-col... |
| athgt 40705 | A Hilbert lattice, whose h... |
| 3dim0 40706 | There exists a 3-dimension... |
| 3dimlem1 40707 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 40708 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 40709 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 40710 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 40711 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 40712 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 40713 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 40714 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 40715 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 40716 | Construct a 3-dimensional ... |
| 3dim2 40717 | Construct 2 new layers on ... |
| 3dim3 40718 | Construct a new layer on t... |
| 2dim 40719 | Generate a height-3 elemen... |
| 1dimN 40720 | An atom is covered by a he... |
| 1cvrco 40721 | The orthocomplement of an ... |
| 1cvratex 40722 | There exists an atom less ... |
| 1cvratlt 40723 | An atom less than or equal... |
| 1cvrjat 40724 | An element covered by the ... |
| 1cvrat 40725 | Create an atom under an el... |
| ps-1 40726 | The join of two atoms ` R ... |
| ps-2 40727 | Lattice analogue for the p... |
| 2atjlej 40728 | Two atoms are different if... |
| hlatexch3N 40729 | Rearrange join of atoms in... |
| hlatexch4 40730 | Exchange 2 atoms. (Contri... |
| ps-2b 40731 | Variation of projective ge... |
| 3atlem1 40732 | Lemma for ~ 3at . (Contri... |
| 3atlem2 40733 | Lemma for ~ 3at . (Contri... |
| 3atlem3 40734 | Lemma for ~ 3at . (Contri... |
| 3atlem4 40735 | Lemma for ~ 3at . (Contri... |
| 3atlem5 40736 | Lemma for ~ 3at . (Contri... |
| 3atlem6 40737 | Lemma for ~ 3at . (Contri... |
| 3atlem7 40738 | Lemma for ~ 3at . (Contri... |
| 3at 40739 | Any three non-colinear ato... |
| llnset 40754 | The set of lattice lines i... |
| islln 40755 | The predicate "is a lattic... |
| islln4 40756 | The predicate "is a lattic... |
| llni 40757 | Condition implying a latti... |
| llnbase 40758 | A lattice line is a lattic... |
| islln3 40759 | The predicate "is a lattic... |
| islln2 40760 | The predicate "is a lattic... |
| llni2 40761 | The join of two different ... |
| llnnleat 40762 | An atom cannot majorize a ... |
| llnneat 40763 | A lattice line is not an a... |
| 2atneat 40764 | The join of two distinct a... |
| llnn0 40765 | A lattice line is nonzero.... |
| islln2a 40766 | The predicate "is a lattic... |
| llnle 40767 | Any element greater than 0... |
| atcvrlln2 40768 | An atom under a line is co... |
| atcvrlln 40769 | An element covering an ato... |
| llnexatN 40770 | Given an atom on a line, t... |
| llncmp 40771 | If two lattice lines are c... |
| llnnlt 40772 | Two lattice lines cannot s... |
| 2llnmat 40773 | Two intersecting lines int... |
| 2at0mat0 40774 | Special case of ~ 2atmat0 ... |
| 2atmat0 40775 | The meet of two unequal li... |
| 2atm 40776 | An atom majorized by two d... |
| ps-2c 40777 | Variation of projective ge... |
| lplnset 40778 | The set of lattice planes ... |
| islpln 40779 | The predicate "is a lattic... |
| islpln4 40780 | The predicate "is a lattic... |
| lplni 40781 | Condition implying a latti... |
| islpln3 40782 | The predicate "is a lattic... |
| lplnbase 40783 | A lattice plane is a latti... |
| islpln5 40784 | The predicate "is a lattic... |
| islpln2 40785 | The predicate "is a lattic... |
| lplni2 40786 | The join of 3 different at... |
| lvolex3N 40787 | There is an atom outside o... |
| llnmlplnN 40788 | The intersection of a line... |
| lplnle 40789 | Any element greater than 0... |
| lplnnle2at 40790 | A lattice line (or atom) c... |
| lplnnleat 40791 | A lattice plane cannot maj... |
| lplnnlelln 40792 | A lattice plane is not les... |
| 2atnelpln 40793 | The join of two atoms is n... |
| lplnneat 40794 | No lattice plane is an ato... |
| lplnnelln 40795 | No lattice plane is a latt... |
| lplnn0N 40796 | A lattice plane is nonzero... |
| islpln2a 40797 | The predicate "is a lattic... |
| islpln2ah 40798 | The predicate "is a lattic... |
| lplnriaN 40799 | Property of a lattice plan... |
| lplnribN 40800 | Property of a lattice plan... |
| lplnric 40801 | Property of a lattice plan... |
| lplnri1 40802 | Property of a lattice plan... |
| lplnri2N 40803 | Property of a lattice plan... |
| lplnri3N 40804 | Property of a lattice plan... |
| lplnllnneN 40805 | Two lattice lines defined ... |
| llncvrlpln2 40806 | A lattice line under a lat... |
| llncvrlpln 40807 | An element covering a latt... |
| 2lplnmN 40808 | If the join of two lattice... |
| 2llnmj 40809 | The meet of two lattice li... |
| 2atmat 40810 | The meet of two intersecti... |
| lplncmp 40811 | If two lattice planes are ... |
| lplnexatN 40812 | Given a lattice line on a ... |
| lplnexllnN 40813 | Given an atom on a lattice... |
| lplnnlt 40814 | Two lattice planes cannot ... |
| 2llnjaN 40815 | The join of two different ... |
| 2llnjN 40816 | The join of two different ... |
| 2llnm2N 40817 | The meet of two different ... |
| 2llnm3N 40818 | Two lattice lines in a lat... |
| 2llnm4 40819 | Two lattice lines that maj... |
| 2llnmeqat 40820 | An atom equals the interse... |
| lvolset 40821 | The set of 3-dim lattice v... |
| islvol 40822 | The predicate "is a 3-dim ... |
| islvol4 40823 | The predicate "is a 3-dim ... |
| lvoli 40824 | Condition implying a 3-dim... |
| islvol3 40825 | The predicate "is a 3-dim ... |
| lvoli3 40826 | Condition implying a 3-dim... |
| lvolbase 40827 | A 3-dim lattice volume is ... |
| islvol5 40828 | The predicate "is a 3-dim ... |
| islvol2 40829 | The predicate "is a 3-dim ... |
| lvoli2 40830 | The join of 4 different at... |
| lvolnle3at 40831 | A lattice plane (or lattic... |
| lvolnleat 40832 | An atom cannot majorize a ... |
| lvolnlelln 40833 | A lattice line cannot majo... |
| lvolnlelpln 40834 | A lattice plane cannot maj... |
| 3atnelvolN 40835 | The join of 3 atoms is not... |
| 2atnelvolN 40836 | The join of two atoms is n... |
| lvolneatN 40837 | No lattice volume is an at... |
| lvolnelln 40838 | No lattice volume is a lat... |
| lvolnelpln 40839 | No lattice volume is a lat... |
| lvoln0N 40840 | A lattice volume is nonzer... |
| islvol2aN 40841 | The predicate "is a lattic... |
| 4atlem0a 40842 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 40843 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 40844 | Lemma for ~ 4at . (Contri... |
| 4atlem3 40845 | Lemma for ~ 4at . Break i... |
| 4atlem3a 40846 | Lemma for ~ 4at . Break i... |
| 4atlem3b 40847 | Lemma for ~ 4at . Break i... |
| 4atlem4a 40848 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 40849 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 40850 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 40851 | Lemma for ~ 4at . Frequen... |
| 4atlem9 40852 | Lemma for ~ 4at . Substit... |
| 4atlem10a 40853 | Lemma for ~ 4at . Substit... |
| 4atlem10b 40854 | Lemma for ~ 4at . Substit... |
| 4atlem10 40855 | Lemma for ~ 4at . Combine... |
| 4atlem11a 40856 | Lemma for ~ 4at . Substit... |
| 4atlem11b 40857 | Lemma for ~ 4at . Substit... |
| 4atlem11 40858 | Lemma for ~ 4at . Combine... |
| 4atlem12a 40859 | Lemma for ~ 4at . Substit... |
| 4atlem12b 40860 | Lemma for ~ 4at . Substit... |
| 4atlem12 40861 | Lemma for ~ 4at . Combine... |
| 4at 40862 | Four atoms determine a lat... |
| 4at2 40863 | Four atoms determine a lat... |
| lplncvrlvol2 40864 | A lattice line under a lat... |
| lplncvrlvol 40865 | An element covering a latt... |
| lvolcmp 40866 | If two lattice planes are ... |
| lvolnltN 40867 | Two lattice volumes cannot... |
| 2lplnja 40868 | The join of two different ... |
| 2lplnj 40869 | The join of two different ... |
| 2lplnm2N 40870 | The meet of two different ... |
| 2lplnmj 40871 | The meet of two lattice pl... |
| dalemkehl 40872 | Lemma for ~ dath . Freque... |
| dalemkelat 40873 | Lemma for ~ dath . Freque... |
| dalemkeop 40874 | Lemma for ~ dath . Freque... |
| dalempea 40875 | Lemma for ~ dath . Freque... |
| dalemqea 40876 | Lemma for ~ dath . Freque... |
| dalemrea 40877 | Lemma for ~ dath . Freque... |
| dalemsea 40878 | Lemma for ~ dath . Freque... |
| dalemtea 40879 | Lemma for ~ dath . Freque... |
| dalemuea 40880 | Lemma for ~ dath . Freque... |
| dalemyeo 40881 | Lemma for ~ dath . Freque... |
| dalemzeo 40882 | Lemma for ~ dath . Freque... |
| dalemclpjs 40883 | Lemma for ~ dath . Freque... |
| dalemclqjt 40884 | Lemma for ~ dath . Freque... |
| dalemclrju 40885 | Lemma for ~ dath . Freque... |
| dalem-clpjq 40886 | Lemma for ~ dath . Freque... |
| dalemceb 40887 | Lemma for ~ dath . Freque... |
| dalempeb 40888 | Lemma for ~ dath . Freque... |
| dalemqeb 40889 | Lemma for ~ dath . Freque... |
| dalemreb 40890 | Lemma for ~ dath . Freque... |
| dalemseb 40891 | Lemma for ~ dath . Freque... |
| dalemteb 40892 | Lemma for ~ dath . Freque... |
| dalemueb 40893 | Lemma for ~ dath . Freque... |
| dalempjqeb 40894 | Lemma for ~ dath . Freque... |
| dalemsjteb 40895 | Lemma for ~ dath . Freque... |
| dalemtjueb 40896 | Lemma for ~ dath . Freque... |
| dalemqrprot 40897 | Lemma for ~ dath . Freque... |
| dalemyeb 40898 | Lemma for ~ dath . Freque... |
| dalemcnes 40899 | Lemma for ~ dath . Freque... |
| dalempnes 40900 | Lemma for ~ dath . Freque... |
| dalemqnet 40901 | Lemma for ~ dath . Freque... |
| dalempjsen 40902 | Lemma for ~ dath . Freque... |
| dalemply 40903 | Lemma for ~ dath . Freque... |
| dalemsly 40904 | Lemma for ~ dath . Freque... |
| dalemswapyz 40905 | Lemma for ~ dath . Swap t... |
| dalemrot 40906 | Lemma for ~ dath . Rotate... |
| dalemrotyz 40907 | Lemma for ~ dath . Rotate... |
| dalem1 40908 | Lemma for ~ dath . Show t... |
| dalemcea 40909 | Lemma for ~ dath . Freque... |
| dalem2 40910 | Lemma for ~ dath . Show t... |
| dalemdea 40911 | Lemma for ~ dath . Freque... |
| dalemeea 40912 | Lemma for ~ dath . Freque... |
| dalem3 40913 | Lemma for ~ dalemdnee . (... |
| dalem4 40914 | Lemma for ~ dalemdnee . (... |
| dalemdnee 40915 | Lemma for ~ dath . Axis o... |
| dalem5 40916 | Lemma for ~ dath . Atom `... |
| dalem6 40917 | Lemma for ~ dath . Analog... |
| dalem7 40918 | Lemma for ~ dath . Analog... |
| dalem8 40919 | Lemma for ~ dath . Plane ... |
| dalem-cly 40920 | Lemma for ~ dalem9 . Cent... |
| dalem9 40921 | Lemma for ~ dath . Since ... |
| dalem10 40922 | Lemma for ~ dath . Atom `... |
| dalem11 40923 | Lemma for ~ dath . Analog... |
| dalem12 40924 | Lemma for ~ dath . Analog... |
| dalem13 40925 | Lemma for ~ dalem14 . (Co... |
| dalem14 40926 | Lemma for ~ dath . Planes... |
| dalem15 40927 | Lemma for ~ dath . The ax... |
| dalem16 40928 | Lemma for ~ dath . The at... |
| dalem17 40929 | Lemma for ~ dath . When p... |
| dalem18 40930 | Lemma for ~ dath . Show t... |
| dalem19 40931 | Lemma for ~ dath . Show t... |
| dalemccea 40932 | Lemma for ~ dath . Freque... |
| dalemddea 40933 | Lemma for ~ dath . Freque... |
| dalem-ccly 40934 | Lemma for ~ dath . Freque... |
| dalem-ddly 40935 | Lemma for ~ dath . Freque... |
| dalemccnedd 40936 | Lemma for ~ dath . Freque... |
| dalemclccjdd 40937 | Lemma for ~ dath . Freque... |
| dalemcceb 40938 | Lemma for ~ dath . Freque... |
| dalemswapyzps 40939 | Lemma for ~ dath . Swap t... |
| dalemrotps 40940 | Lemma for ~ dath . Rotate... |
| dalemcjden 40941 | Lemma for ~ dath . Show t... |
| dalem20 40942 | Lemma for ~ dath . Show t... |
| dalem21 40943 | Lemma for ~ dath . Show t... |
| dalem22 40944 | Lemma for ~ dath . Show t... |
| dalem23 40945 | Lemma for ~ dath . Show t... |
| dalem24 40946 | Lemma for ~ dath . Show t... |
| dalem25 40947 | Lemma for ~ dath . Show t... |
| dalem27 40948 | Lemma for ~ dath . Show t... |
| dalem28 40949 | Lemma for ~ dath . Lemma ... |
| dalem29 40950 | Lemma for ~ dath . Analog... |
| dalem30 40951 | Lemma for ~ dath . Analog... |
| dalem31N 40952 | Lemma for ~ dath . Analog... |
| dalem32 40953 | Lemma for ~ dath . Analog... |
| dalem33 40954 | Lemma for ~ dath . Analog... |
| dalem34 40955 | Lemma for ~ dath . Analog... |
| dalem35 40956 | Lemma for ~ dath . Analog... |
| dalem36 40957 | Lemma for ~ dath . Analog... |
| dalem37 40958 | Lemma for ~ dath . Analog... |
| dalem38 40959 | Lemma for ~ dath . Plane ... |
| dalem39 40960 | Lemma for ~ dath . Auxili... |
| dalem40 40961 | Lemma for ~ dath . Analog... |
| dalem41 40962 | Lemma for ~ dath . (Contr... |
| dalem42 40963 | Lemma for ~ dath . Auxili... |
| dalem43 40964 | Lemma for ~ dath . Planes... |
| dalem44 40965 | Lemma for ~ dath . Dummy ... |
| dalem45 40966 | Lemma for ~ dath . Dummy ... |
| dalem46 40967 | Lemma for ~ dath . Analog... |
| dalem47 40968 | Lemma for ~ dath . Analog... |
| dalem48 40969 | Lemma for ~ dath . Analog... |
| dalem49 40970 | Lemma for ~ dath . Analog... |
| dalem50 40971 | Lemma for ~ dath . Analog... |
| dalem51 40972 | Lemma for ~ dath . Constr... |
| dalem52 40973 | Lemma for ~ dath . Lines ... |
| dalem53 40974 | Lemma for ~ dath . The au... |
| dalem54 40975 | Lemma for ~ dath . Line `... |
| dalem55 40976 | Lemma for ~ dath . Lines ... |
| dalem56 40977 | Lemma for ~ dath . Analog... |
| dalem57 40978 | Lemma for ~ dath . Axis o... |
| dalem58 40979 | Lemma for ~ dath . Analog... |
| dalem59 40980 | Lemma for ~ dath . Analog... |
| dalem60 40981 | Lemma for ~ dath . ` B ` i... |
| dalem61 40982 | Lemma for ~ dath . Show t... |
| dalem62 40983 | Lemma for ~ dath . Elimin... |
| dalem63 40984 | Lemma for ~ dath . Combin... |
| dath 40985 | Desargues's theorem of pro... |
| dath2 40986 | Version of Desargues's the... |
| lineset 40987 | The set of lines in a Hilb... |
| isline 40988 | The predicate "is a line".... |
| islinei 40989 | Condition implying "is a l... |
| pointsetN 40990 | The set of points in a Hil... |
| ispointN 40991 | The predicate "is a point"... |
| atpointN 40992 | The singleton of an atom i... |
| psubspset 40993 | The set of projective subs... |
| ispsubsp 40994 | The predicate "is a projec... |
| ispsubsp2 40995 | The predicate "is a projec... |
| psubspi 40996 | Property of a projective s... |
| psubspi2N 40997 | Property of a projective s... |
| 0psubN 40998 | The empty set is a project... |
| snatpsubN 40999 | The singleton of an atom i... |
| pointpsubN 41000 | A point (singleton of an a... |
| linepsubN 41001 | A line is a projective sub... |
| atpsubN 41002 | The set of all atoms is a ... |
| psubssat 41003 | A projective subspace cons... |
| psubatN 41004 | A member of a projective s... |
| pmapfval 41005 | The projective map of a Hi... |
| pmapval 41006 | Value of the projective ma... |
| elpmap 41007 | Member of a projective map... |
| pmapssat 41008 | The projective map of a Hi... |
| pmapssbaN 41009 | A weakening of ~ pmapssat ... |
| pmaple 41010 | The projective map of a Hi... |
| pmap11 41011 | The projective map of a Hi... |
| pmapat 41012 | The projective map of an a... |
| elpmapat 41013 | Member of the projective m... |
| pmap0 41014 | Value of the projective ma... |
| pmapeq0 41015 | A projective map value is ... |
| pmap1N 41016 | Value of the projective ma... |
| pmapsub 41017 | The projective map of a Hi... |
| pmapglbx 41018 | The projective map of the ... |
| pmapglb 41019 | The projective map of the ... |
| pmapglb2N 41020 | The projective map of the ... |
| pmapglb2xN 41021 | The projective map of the ... |
| pmapmeet 41022 | The projective map of a me... |
| isline2 41023 | Definition of line in term... |
| linepmap 41024 | A line described with a pr... |
| isline3 41025 | Definition of line in term... |
| isline4N 41026 | Definition of line in term... |
| lneq2at 41027 | A line equals the join of ... |
| lnatexN 41028 | There is an atom in a line... |
| lnjatN 41029 | Given an atom in a line, t... |
| lncvrelatN 41030 | A lattice element covered ... |
| lncvrat 41031 | A line covers the atoms it... |
| lncmp 41032 | If two lines are comparabl... |
| 2lnat 41033 | Two intersecting lines int... |
| 2atm2atN 41034 | Two joins with a common at... |
| 2llnma1b 41035 | Generalization of ~ 2llnma... |
| 2llnma1 41036 | Two different intersecting... |
| 2llnma3r 41037 | Two different intersecting... |
| 2llnma2 41038 | Two different intersecting... |
| 2llnma2rN 41039 | Two different intersecting... |
| cdlema1N 41040 | A condition for required f... |
| cdlema2N 41041 | A condition for required f... |
| cdlemblem 41042 | Lemma for ~ cdlemb . (Con... |
| cdlemb 41043 | Given two atoms not less t... |
| paddfval 41046 | Projective subspace sum op... |
| paddval 41047 | Projective subspace sum op... |
| elpadd 41048 | Member of a projective sub... |
| elpaddn0 41049 | Member of projective subsp... |
| paddvaln0N 41050 | Projective subspace sum op... |
| elpaddri 41051 | Condition implying members... |
| elpaddatriN 41052 | Condition implying members... |
| elpaddat 41053 | Membership in a projective... |
| elpaddatiN 41054 | Consequence of membership ... |
| elpadd2at 41055 | Membership in a projective... |
| elpadd2at2 41056 | Membership in a projective... |
| paddunssN 41057 | Projective subspace sum in... |
| elpadd0 41058 | Member of projective subsp... |
| paddval0 41059 | Projective subspace sum wi... |
| padd01 41060 | Projective subspace sum wi... |
| padd02 41061 | Projective subspace sum wi... |
| paddcom 41062 | Projective subspace sum co... |
| paddssat 41063 | A projective subspace sum ... |
| sspadd1 41064 | A projective subspace sum ... |
| sspadd2 41065 | A projective subspace sum ... |
| paddss1 41066 | Subset law for projective ... |
| paddss2 41067 | Subset law for projective ... |
| paddss12 41068 | Subset law for projective ... |
| paddasslem1 41069 | Lemma for ~ paddass . (Co... |
| paddasslem2 41070 | Lemma for ~ paddass . (Co... |
| paddasslem3 41071 | Lemma for ~ paddass . Res... |
| paddasslem4 41072 | Lemma for ~ paddass . Com... |
| paddasslem5 41073 | Lemma for ~ paddass . Sho... |
| paddasslem6 41074 | Lemma for ~ paddass . (Co... |
| paddasslem7 41075 | Lemma for ~ paddass . Com... |
| paddasslem8 41076 | Lemma for ~ paddass . (Co... |
| paddasslem9 41077 | Lemma for ~ paddass . Com... |
| paddasslem10 41078 | Lemma for ~ paddass . Use... |
| paddasslem11 41079 | Lemma for ~ paddass . The... |
| paddasslem12 41080 | Lemma for ~ paddass . The... |
| paddasslem13 41081 | Lemma for ~ paddass . The... |
| paddasslem14 41082 | Lemma for ~ paddass . Rem... |
| paddasslem15 41083 | Lemma for ~ paddass . Use... |
| paddasslem16 41084 | Lemma for ~ paddass . Use... |
| paddasslem17 41085 | Lemma for ~ paddass . The... |
| paddasslem18 41086 | Lemma for ~ paddass . Com... |
| paddass 41087 | Projective subspace sum is... |
| padd12N 41088 | Commutative/associative la... |
| padd4N 41089 | Rearrangement of 4 terms i... |
| paddidm 41090 | Projective subspace sum is... |
| paddclN 41091 | The projective sum of two ... |
| paddssw1 41092 | Subset law for projective ... |
| paddssw2 41093 | Subset law for projective ... |
| paddss 41094 | Subset law for projective ... |
| pmodlem1 41095 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 41096 | Lemma for ~ pmod1i . (Con... |
| pmod1i 41097 | The modular law holds in a... |
| pmod2iN 41098 | Dual of the modular law. ... |
| pmodN 41099 | The modular law for projec... |
| pmodl42N 41100 | Lemma derived from modular... |
| pmapjoin 41101 | The projective map of the ... |
| pmapjat1 41102 | The projective map of the ... |
| pmapjat2 41103 | The projective map of the ... |
| pmapjlln1 41104 | The projective map of the ... |
| hlmod1i 41105 | A version of the modular l... |
| atmod1i1 41106 | Version of modular law ~ p... |
| atmod1i1m 41107 | Version of modular law ~ p... |
| atmod1i2 41108 | Version of modular law ~ p... |
| llnmod1i2 41109 | Version of modular law ~ p... |
| atmod2i1 41110 | Version of modular law ~ p... |
| atmod2i2 41111 | Version of modular law ~ p... |
| llnmod2i2 41112 | Version of modular law ~ p... |
| atmod3i1 41113 | Version of modular law tha... |
| atmod3i2 41114 | Version of modular law tha... |
| atmod4i1 41115 | Version of modular law tha... |
| atmod4i2 41116 | Version of modular law tha... |
| llnexchb2lem 41117 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 41118 | Line exchange property (co... |
| llnexch2N 41119 | Line exchange property (co... |
| dalawlem1 41120 | Lemma for ~ dalaw . Speci... |
| dalawlem2 41121 | Lemma for ~ dalaw . Utili... |
| dalawlem3 41122 | Lemma for ~ dalaw . First... |
| dalawlem4 41123 | Lemma for ~ dalaw . Secon... |
| dalawlem5 41124 | Lemma for ~ dalaw . Speci... |
| dalawlem6 41125 | Lemma for ~ dalaw . First... |
| dalawlem7 41126 | Lemma for ~ dalaw . Secon... |
| dalawlem8 41127 | Lemma for ~ dalaw . Speci... |
| dalawlem9 41128 | Lemma for ~ dalaw . Speci... |
| dalawlem10 41129 | Lemma for ~ dalaw . Combi... |
| dalawlem11 41130 | Lemma for ~ dalaw . First... |
| dalawlem12 41131 | Lemma for ~ dalaw . Secon... |
| dalawlem13 41132 | Lemma for ~ dalaw . Speci... |
| dalawlem14 41133 | Lemma for ~ dalaw . Combi... |
| dalawlem15 41134 | Lemma for ~ dalaw . Swap ... |
| dalaw 41135 | Desargues's law, derived f... |
| pclfvalN 41138 | The projective subspace cl... |
| pclvalN 41139 | Value of the projective su... |
| pclclN 41140 | Closure of the projective ... |
| elpclN 41141 | Membership in the projecti... |
| elpcliN 41142 | Implication of membership ... |
| pclssN 41143 | Ordering is preserved by s... |
| pclssidN 41144 | A set of atoms is included... |
| pclidN 41145 | The projective subspace cl... |
| pclbtwnN 41146 | A projective subspace sand... |
| pclunN 41147 | The projective subspace cl... |
| pclun2N 41148 | The projective subspace cl... |
| pclfinN 41149 | The projective subspace cl... |
| pclcmpatN 41150 | The set of projective subs... |
| polfvalN 41153 | The projective subspace po... |
| polvalN 41154 | Value of the projective su... |
| polval2N 41155 | Alternate expression for v... |
| polsubN 41156 | The polarity of a set of a... |
| polssatN 41157 | The polarity of a set of a... |
| pol0N 41158 | The polarity of the empty ... |
| pol1N 41159 | The polarity of the whole ... |
| 2pol0N 41160 | The closed subspace closur... |
| polpmapN 41161 | The polarity of a projecti... |
| 2polpmapN 41162 | Double polarity of a proje... |
| 2polvalN 41163 | Value of double polarity. ... |
| 2polssN 41164 | A set of atoms is a subset... |
| 3polN 41165 | Triple polarity cancels to... |
| polcon3N 41166 | Contraposition law for pol... |
| 2polcon4bN 41167 | Contraposition law for pol... |
| polcon2N 41168 | Contraposition law for pol... |
| polcon2bN 41169 | Contraposition law for pol... |
| pclss2polN 41170 | The projective subspace cl... |
| pcl0N 41171 | The projective subspace cl... |
| pcl0bN 41172 | The projective subspace cl... |
| pmaplubN 41173 | The LUB of a projective ma... |
| sspmaplubN 41174 | A set of atoms is a subset... |
| 2pmaplubN 41175 | Double projective map of a... |
| paddunN 41176 | The closure of the project... |
| poldmj1N 41177 | De Morgan's law for polari... |
| pmapj2N 41178 | The projective map of the ... |
| pmapocjN 41179 | The projective map of the ... |
| polatN 41180 | The polarity of the single... |
| 2polatN 41181 | Double polarity of the sin... |
| pnonsingN 41182 | The intersection of a set ... |
| psubclsetN 41185 | The set of closed projecti... |
| ispsubclN 41186 | The predicate "is a closed... |
| psubcliN 41187 | Property of a closed proje... |
| psubcli2N 41188 | Property of a closed proje... |
| psubclsubN 41189 | A closed projective subspa... |
| psubclssatN 41190 | A closed projective subspa... |
| pmapidclN 41191 | Projective map of the LUB ... |
| 0psubclN 41192 | The empty set is a closed ... |
| 1psubclN 41193 | The set of all atoms is a ... |
| atpsubclN 41194 | A point (singleton of an a... |
| pmapsubclN 41195 | A projective map value is ... |
| ispsubcl2N 41196 | Alternate predicate for "i... |
| psubclinN 41197 | The intersection of two cl... |
| paddatclN 41198 | The projective sum of a cl... |
| pclfinclN 41199 | The projective subspace cl... |
| linepsubclN 41200 | A line is a closed project... |
| polsubclN 41201 | A polarity is a closed pro... |
| poml4N 41202 | Orthomodular law for proje... |
| poml5N 41203 | Orthomodular law for proje... |
| poml6N 41204 | Orthomodular law for proje... |
| osumcllem1N 41205 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 41206 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 41207 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 41208 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 41209 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 41210 | Lemma for ~ osumclN . Use... |
| osumcllem7N 41211 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 41212 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 41213 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 41214 | Lemma for ~ osumclN . Con... |
| osumcllem11N 41215 | Lemma for ~ osumclN . (Co... |
| osumclN 41216 | Closure of orthogonal sum.... |
| pmapojoinN 41217 | For orthogonal elements, p... |
| pexmidN 41218 | Excluded middle law for cl... |
| pexmidlem1N 41219 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 41220 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 41221 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 41222 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 41223 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 41224 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 41225 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 41226 | Lemma for ~ pexmidN . The... |
| pexmidALTN 41227 | Excluded middle law for cl... |
| pl42lem1N 41228 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 41229 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 41230 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 41231 | Lemma for ~ pl42N . (Cont... |
| pl42N 41232 | Law holding in a Hilbert l... |
| watfvalN 41241 | The W atoms function. (Co... |
| watvalN 41242 | Value of the W atoms funct... |
| iswatN 41243 | The predicate "is a W atom... |
| lhpset 41244 | The set of co-atoms (latti... |
| islhp 41245 | The predicate "is a co-ato... |
| islhp2 41246 | The predicate "is a co-ato... |
| lhpbase 41247 | A co-atom is a member of t... |
| lhp1cvr 41248 | The lattice unity covers a... |
| lhplt 41249 | An atom under a co-atom is... |
| lhp2lt 41250 | The join of two atoms unde... |
| lhpexlt 41251 | There exists an atom less ... |
| lhp0lt 41252 | A co-atom is greater than ... |
| lhpn0 41253 | A co-atom is nonzero. TOD... |
| lhpexle 41254 | There exists an atom under... |
| lhpexnle 41255 | There exists an atom not u... |
| lhpexle1lem 41256 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 41257 | There exists an atom under... |
| lhpexle2lem 41258 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 41259 | There exists atom under a ... |
| lhpexle3lem 41260 | There exists atom under a ... |
| lhpexle3 41261 | There exists atom under a ... |
| lhpex2leN 41262 | There exist at least two d... |
| lhpoc 41263 | The orthocomplement of a c... |
| lhpoc2N 41264 | The orthocomplement of an ... |
| lhpocnle 41265 | The orthocomplement of a c... |
| lhpocat 41266 | The orthocomplement of a c... |
| lhpocnel 41267 | The orthocomplement of a c... |
| lhpocnel2 41268 | The orthocomplement of a c... |
| lhpjat1 41269 | The join of a co-atom (hyp... |
| lhpjat2 41270 | The join of a co-atom (hyp... |
| lhpj1 41271 | The join of a co-atom (hyp... |
| lhpmcvr 41272 | The meet of a lattice hype... |
| lhpmcvr2 41273 | Alternate way to express t... |
| lhpmcvr3 41274 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 41275 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 41276 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 41277 | Specialization of ~ lhpmcv... |
| lhpm0atN 41278 | If the meet of a lattice h... |
| lhpmat 41279 | An element covered by the ... |
| lhpmatb 41280 | An element covered by the ... |
| lhp2at0 41281 | Join and meet with differe... |
| lhp2atnle 41282 | Inequality for 2 different... |
| lhp2atne 41283 | Inequality for joins with ... |
| lhp2at0nle 41284 | Inequality for 2 different... |
| lhp2at0ne 41285 | Inequality for joins with ... |
| lhpelim 41286 | Eliminate an atom not unde... |
| lhpmod2i2 41287 | Modular law for hyperplane... |
| lhpmod6i1 41288 | Modular law for hyperplane... |
| lhprelat3N 41289 | The Hilbert lattice is rel... |
| cdlemb2 41290 | Given two atoms not under ... |
| lhple 41291 | Property of a lattice elem... |
| lhpat 41292 | Create an atom under a co-... |
| lhpat4N 41293 | Property of an atom under ... |
| lhpat2 41294 | Create an atom under a co-... |
| lhpat3 41295 | There is only one atom und... |
| 4atexlemk 41296 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 41297 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 41298 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 41299 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 41300 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 41301 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 41302 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 41303 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 41304 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 41305 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 41306 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 41307 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 41308 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 41309 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 41310 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 41311 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 41312 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 41313 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 41314 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 41315 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 41316 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 41317 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 41318 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 41319 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 41320 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 41321 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 41322 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 41323 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 41324 | Whenever there are at leas... |
| 4atex 41325 | Whenever there are at leas... |
| 4atex2 41326 | More general version of ~ ... |
| 4atex2-0aOLDN 41327 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 41328 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 41329 | Same as ~ 4atex2 except th... |
| 4atex3 41330 | More general version of ~ ... |
| lautset 41331 | The set of lattice automor... |
| islaut 41332 | The predicate "is a lattic... |
| lautle 41333 | Less-than or equal propert... |
| laut1o 41334 | A lattice automorphism is ... |
| laut11 41335 | One-to-one property of a l... |
| lautcl 41336 | A lattice automorphism val... |
| lautcnvclN 41337 | Reverse closure of a latti... |
| lautcnvle 41338 | Less-than or equal propert... |
| lautcnv 41339 | The converse of a lattice ... |
| lautlt 41340 | Less-than property of a la... |
| lautcvr 41341 | Covering property of a lat... |
| lautj 41342 | Meet property of a lattice... |
| lautm 41343 | Meet property of a lattice... |
| lauteq 41344 | A lattice automorphism arg... |
| idlaut 41345 | The identity function is a... |
| lautco 41346 | The composition of two lat... |
| pautsetN 41347 | The set of projective auto... |
| ispautN 41348 | The predicate "is a projec... |
| ldilfset 41357 | The mapping from fiducial ... |
| ldilset 41358 | The set of lattice dilatio... |
| isldil 41359 | The predicate "is a lattic... |
| ldillaut 41360 | A lattice dilation is an a... |
| ldil1o 41361 | A lattice dilation is a on... |
| ldilval 41362 | Value of a lattice dilatio... |
| idldil 41363 | The identity function is a... |
| ldilcnv 41364 | The converse of a lattice ... |
| ldilco 41365 | The composition of two lat... |
| ltrnfset 41366 | The set of all lattice tra... |
| ltrnset 41367 | The set of lattice transla... |
| isltrn 41368 | The predicate "is a lattic... |
| isltrn2N 41369 | The predicate "is a lattic... |
| ltrnu 41370 | Uniqueness property of a l... |
| ltrnldil 41371 | A lattice translation is a... |
| ltrnlaut 41372 | A lattice translation is a... |
| ltrn1o 41373 | A lattice translation is a... |
| ltrncl 41374 | Closure of a lattice trans... |
| ltrn11 41375 | One-to-one property of a l... |
| ltrncnvnid 41376 | If a translation is differ... |
| ltrncoidN 41377 | Two translations are equal... |
| ltrnle 41378 | Less-than or equal propert... |
| ltrncnvleN 41379 | Less-than or equal propert... |
| ltrnm 41380 | Lattice translation of a m... |
| ltrnj 41381 | Lattice translation of a m... |
| ltrncvr 41382 | Covering property of a lat... |
| ltrnval1 41383 | Value of a lattice transla... |
| ltrnid 41384 | A lattice translation is t... |
| ltrnnid 41385 | If a lattice translation i... |
| ltrnatb 41386 | The lattice translation of... |
| ltrncnvatb 41387 | The converse of the lattic... |
| ltrnel 41388 | The lattice translation of... |
| ltrnat 41389 | The lattice translation of... |
| ltrncnvat 41390 | The converse of the lattic... |
| ltrncnvel 41391 | The converse of the lattic... |
| ltrncoelN 41392 | Composition of lattice tra... |
| ltrncoat 41393 | Composition of lattice tra... |
| ltrncoval 41394 | Two ways to express value ... |
| ltrncnv 41395 | The converse of a lattice ... |
| ltrn11at 41396 | Frequently used one-to-one... |
| ltrneq2 41397 | The equality of two transl... |
| ltrneq 41398 | The equality of two transl... |
| idltrn 41399 | The identity function is a... |
| ltrnmw 41400 | Property of lattice transl... |
| dilfsetN 41401 | The mapping from fiducial ... |
| dilsetN 41402 | The set of dilations for a... |
| isdilN 41403 | The predicate "is a dilati... |
| trnfsetN 41404 | The mapping from fiducial ... |
| trnsetN 41405 | The set of translations fo... |
| istrnN 41406 | The predicate "is a transl... |
| trlfset 41409 | The set of all traces of l... |
| trlset 41410 | The set of traces of latti... |
| trlval 41411 | The value of the trace of ... |
| trlval2 41412 | The value of the trace of ... |
| trlcl 41413 | Closure of the trace of a ... |
| trlcnv 41414 | The trace of the converse ... |
| trljat1 41415 | The value of a translation... |
| trljat2 41416 | The value of a translation... |
| trljat3 41417 | The value of a translation... |
| trlat 41418 | If an atom differs from it... |
| trl0 41419 | If an atom not under the f... |
| trlator0 41420 | The trace of a lattice tra... |
| trlatn0 41421 | The trace of a lattice tra... |
| trlnidat 41422 | The trace of a lattice tra... |
| ltrnnidn 41423 | If a lattice translation i... |
| ltrnideq 41424 | Property of the identity l... |
| trlid0 41425 | The trace of the identity ... |
| trlnidatb 41426 | A lattice translation is n... |
| trlid0b 41427 | A lattice translation is t... |
| trlnid 41428 | Different translations wit... |
| ltrn2ateq 41429 | Property of the equality o... |
| ltrnateq 41430 | If any atom (under ` W ` )... |
| ltrnatneq 41431 | If any atom (under ` W ` )... |
| ltrnatlw 41432 | If the value of an atom eq... |
| trlle 41433 | The trace of a lattice tra... |
| trlne 41434 | The trace of a lattice tra... |
| trlnle 41435 | The atom not under the fid... |
| trlval3 41436 | The value of the trace of ... |
| trlval4 41437 | The value of the trace of ... |
| trlval5 41438 | The value of the trace of ... |
| arglem1N 41439 | Lemma for Desargues's law.... |
| cdlemc1 41440 | Part of proof of Lemma C i... |
| cdlemc2 41441 | Part of proof of Lemma C i... |
| cdlemc3 41442 | Part of proof of Lemma C i... |
| cdlemc4 41443 | Part of proof of Lemma C i... |
| cdlemc5 41444 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 41445 | Lemma for ~ cdlemc . (Con... |
| cdlemc 41446 | Lemma C in [Crawley] p. 11... |
| cdlemd1 41447 | Part of proof of Lemma D i... |
| cdlemd2 41448 | Part of proof of Lemma D i... |
| cdlemd3 41449 | Part of proof of Lemma D i... |
| cdlemd4 41450 | Part of proof of Lemma D i... |
| cdlemd5 41451 | Part of proof of Lemma D i... |
| cdlemd6 41452 | Part of proof of Lemma D i... |
| cdlemd7 41453 | Part of proof of Lemma D i... |
| cdlemd8 41454 | Part of proof of Lemma D i... |
| cdlemd9 41455 | Part of proof of Lemma D i... |
| cdlemd 41456 | If two translations agree ... |
| ltrneq3 41457 | Two translations agree at ... |
| cdleme00a 41458 | Part of proof of Lemma E i... |
| cdleme0aa 41459 | Part of proof of Lemma E i... |
| cdleme0a 41460 | Part of proof of Lemma E i... |
| cdleme0b 41461 | Part of proof of Lemma E i... |
| cdleme0c 41462 | Part of proof of Lemma E i... |
| cdleme0cp 41463 | Part of proof of Lemma E i... |
| cdleme0cq 41464 | Part of proof of Lemma E i... |
| cdleme0dN 41465 | Part of proof of Lemma E i... |
| cdleme0e 41466 | Part of proof of Lemma E i... |
| cdleme0fN 41467 | Part of proof of Lemma E i... |
| cdleme0gN 41468 | Part of proof of Lemma E i... |
| cdlemeulpq 41469 | Part of proof of Lemma E i... |
| cdleme01N 41470 | Part of proof of Lemma E i... |
| cdleme02N 41471 | Part of proof of Lemma E i... |
| cdleme0ex1N 41472 | Part of proof of Lemma E i... |
| cdleme0ex2N 41473 | Part of proof of Lemma E i... |
| cdleme0moN 41474 | Part of proof of Lemma E i... |
| cdleme1b 41475 | Part of proof of Lemma E i... |
| cdleme1 41476 | Part of proof of Lemma E i... |
| cdleme2 41477 | Part of proof of Lemma E i... |
| cdleme3b 41478 | Part of proof of Lemma E i... |
| cdleme3c 41479 | Part of proof of Lemma E i... |
| cdleme3d 41480 | Part of proof of Lemma E i... |
| cdleme3e 41481 | Part of proof of Lemma E i... |
| cdleme3fN 41482 | Part of proof of Lemma E i... |
| cdleme3g 41483 | Part of proof of Lemma E i... |
| cdleme3h 41484 | Part of proof of Lemma E i... |
| cdleme3fa 41485 | Part of proof of Lemma E i... |
| cdleme3 41486 | Part of proof of Lemma E i... |
| cdleme4 41487 | Part of proof of Lemma E i... |
| cdleme4a 41488 | Part of proof of Lemma E i... |
| cdleme5 41489 | Part of proof of Lemma E i... |
| cdleme6 41490 | Part of proof of Lemma E i... |
| cdleme7aa 41491 | Part of proof of Lemma E i... |
| cdleme7a 41492 | Part of proof of Lemma E i... |
| cdleme7b 41493 | Part of proof of Lemma E i... |
| cdleme7c 41494 | Part of proof of Lemma E i... |
| cdleme7d 41495 | Part of proof of Lemma E i... |
| cdleme7e 41496 | Part of proof of Lemma E i... |
| cdleme7ga 41497 | Part of proof of Lemma E i... |
| cdleme7 41498 | Part of proof of Lemma E i... |
| cdleme8 41499 | Part of proof of Lemma E i... |
| cdleme9a 41500 | Part of proof of Lemma E i... |
| cdleme9b 41501 | Utility lemma for Lemma E ... |
| cdleme9 41502 | Part of proof of Lemma E i... |
| cdleme10 41503 | Part of proof of Lemma E i... |
| cdleme8tN 41504 | Part of proof of Lemma E i... |
| cdleme9taN 41505 | Part of proof of Lemma E i... |
| cdleme9tN 41506 | Part of proof of Lemma E i... |
| cdleme10tN 41507 | Part of proof of Lemma E i... |
| cdleme16aN 41508 | Part of proof of Lemma E i... |
| cdleme11a 41509 | Part of proof of Lemma E i... |
| cdleme11c 41510 | Part of proof of Lemma E i... |
| cdleme11dN 41511 | Part of proof of Lemma E i... |
| cdleme11e 41512 | Part of proof of Lemma E i... |
| cdleme11fN 41513 | Part of proof of Lemma E i... |
| cdleme11g 41514 | Part of proof of Lemma E i... |
| cdleme11h 41515 | Part of proof of Lemma E i... |
| cdleme11j 41516 | Part of proof of Lemma E i... |
| cdleme11k 41517 | Part of proof of Lemma E i... |
| cdleme11l 41518 | Part of proof of Lemma E i... |
| cdleme11 41519 | Part of proof of Lemma E i... |
| cdleme12 41520 | Part of proof of Lemma E i... |
| cdleme13 41521 | Part of proof of Lemma E i... |
| cdleme14 41522 | Part of proof of Lemma E i... |
| cdleme15a 41523 | Part of proof of Lemma E i... |
| cdleme15b 41524 | Part of proof of Lemma E i... |
| cdleme15c 41525 | Part of proof of Lemma E i... |
| cdleme15d 41526 | Part of proof of Lemma E i... |
| cdleme15 41527 | Part of proof of Lemma E i... |
| cdleme16b 41528 | Part of proof of Lemma E i... |
| cdleme16c 41529 | Part of proof of Lemma E i... |
| cdleme16d 41530 | Part of proof of Lemma E i... |
| cdleme16e 41531 | Part of proof of Lemma E i... |
| cdleme16f 41532 | Part of proof of Lemma E i... |
| cdleme16g 41533 | Part of proof of Lemma E i... |
| cdleme16 41534 | Part of proof of Lemma E i... |
| cdleme17a 41535 | Part of proof of Lemma E i... |
| cdleme17b 41536 | Lemma leading to ~ cdleme1... |
| cdleme17c 41537 | Part of proof of Lemma E i... |
| cdleme17d1 41538 | Part of proof of Lemma E i... |
| cdleme0nex 41539 | Part of proof of Lemma E i... |
| cdleme18a 41540 | Part of proof of Lemma E i... |
| cdleme18b 41541 | Part of proof of Lemma E i... |
| cdleme18c 41542 | Part of proof of Lemma E i... |
| cdleme22gb 41543 | Utility lemma for Lemma E ... |
| cdleme18d 41544 | Part of proof of Lemma E i... |
| cdlemesner 41545 | Part of proof of Lemma E i... |
| cdlemedb 41546 | Part of proof of Lemma E i... |
| cdlemeda 41547 | Part of proof of Lemma E i... |
| cdlemednpq 41548 | Part of proof of Lemma E i... |
| cdlemednuN 41549 | Part of proof of Lemma E i... |
| cdleme20zN 41550 | Part of proof of Lemma E i... |
| cdleme20y 41551 | Part of proof of Lemma E i... |
| cdleme19a 41552 | Part of proof of Lemma E i... |
| cdleme19b 41553 | Part of proof of Lemma E i... |
| cdleme19c 41554 | Part of proof of Lemma E i... |
| cdleme19d 41555 | Part of proof of Lemma E i... |
| cdleme19e 41556 | Part of proof of Lemma E i... |
| cdleme19f 41557 | Part of proof of Lemma E i... |
| cdleme20aN 41558 | Part of proof of Lemma E i... |
| cdleme20bN 41559 | Part of proof of Lemma E i... |
| cdleme20c 41560 | Part of proof of Lemma E i... |
| cdleme20d 41561 | Part of proof of Lemma E i... |
| cdleme20e 41562 | Part of proof of Lemma E i... |
| cdleme20f 41563 | Part of proof of Lemma E i... |
| cdleme20g 41564 | Part of proof of Lemma E i... |
| cdleme20h 41565 | Part of proof of Lemma E i... |
| cdleme20i 41566 | Part of proof of Lemma E i... |
| cdleme20j 41567 | Part of proof of Lemma E i... |
| cdleme20k 41568 | Part of proof of Lemma E i... |
| cdleme20l1 41569 | Part of proof of Lemma E i... |
| cdleme20l2 41570 | Part of proof of Lemma E i... |
| cdleme20l 41571 | Part of proof of Lemma E i... |
| cdleme20m 41572 | Part of proof of Lemma E i... |
| cdleme20 41573 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 41574 | Part of proof of Lemma E i... |
| cdleme21b 41575 | Part of proof of Lemma E i... |
| cdleme21c 41576 | Part of proof of Lemma E i... |
| cdleme21at 41577 | Part of proof of Lemma E i... |
| cdleme21ct 41578 | Part of proof of Lemma E i... |
| cdleme21d 41579 | Part of proof of Lemma E i... |
| cdleme21e 41580 | Part of proof of Lemma E i... |
| cdleme21f 41581 | Part of proof of Lemma E i... |
| cdleme21g 41582 | Part of proof of Lemma E i... |
| cdleme21h 41583 | Part of proof of Lemma E i... |
| cdleme21i 41584 | Part of proof of Lemma E i... |
| cdleme21j 41585 | Combine ~ cdleme20 and ~ c... |
| cdleme21 41586 | Part of proof of Lemma E i... |
| cdleme21k 41587 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 41588 | Part of proof of Lemma E i... |
| cdleme22a 41589 | Part of proof of Lemma E i... |
| cdleme22b 41590 | Part of proof of Lemma E i... |
| cdleme22cN 41591 | Part of proof of Lemma E i... |
| cdleme22d 41592 | Part of proof of Lemma E i... |
| cdleme22e 41593 | Part of proof of Lemma E i... |
| cdleme22eALTN 41594 | Part of proof of Lemma E i... |
| cdleme22f 41595 | Part of proof of Lemma E i... |
| cdleme22f2 41596 | Part of proof of Lemma E i... |
| cdleme22g 41597 | Part of proof of Lemma E i... |
| cdleme23a 41598 | Part of proof of Lemma E i... |
| cdleme23b 41599 | Part of proof of Lemma E i... |
| cdleme23c 41600 | Part of proof of Lemma E i... |
| cdleme24 41601 | Quantified version of ~ cd... |
| cdleme25a 41602 | Lemma for ~ cdleme25b . (... |
| cdleme25b 41603 | Transform ~ cdleme24 . TO... |
| cdleme25c 41604 | Transform ~ cdleme25b . (... |
| cdleme25dN 41605 | Transform ~ cdleme25c . (... |
| cdleme25cl 41606 | Show closure of the unique... |
| cdleme25cv 41607 | Change bound variables in ... |
| cdleme26e 41608 | Part of proof of Lemma E i... |
| cdleme26ee 41609 | Part of proof of Lemma E i... |
| cdleme26eALTN 41610 | Part of proof of Lemma E i... |
| cdleme26fALTN 41611 | Part of proof of Lemma E i... |
| cdleme26f 41612 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 41613 | Part of proof of Lemma E i... |
| cdleme26f2 41614 | Part of proof of Lemma E i... |
| cdleme27cl 41615 | Part of proof of Lemma E i... |
| cdleme27a 41616 | Part of proof of Lemma E i... |
| cdleme27b 41617 | Lemma for ~ cdleme27N . (... |
| cdleme27N 41618 | Part of proof of Lemma E i... |
| cdleme28a 41619 | Lemma for ~ cdleme25b . T... |
| cdleme28b 41620 | Lemma for ~ cdleme25b . T... |
| cdleme28c 41621 | Part of proof of Lemma E i... |
| cdleme28 41622 | Quantified version of ~ cd... |
| cdleme29ex 41623 | Lemma for ~ cdleme29b . (... |
| cdleme29b 41624 | Transform ~ cdleme28 . (C... |
| cdleme29c 41625 | Transform ~ cdleme28b . (... |
| cdleme29cl 41626 | Show closure of the unique... |
| cdleme30a 41627 | Part of proof of Lemma E i... |
| cdleme31so 41628 | Part of proof of Lemma E i... |
| cdleme31sn 41629 | Part of proof of Lemma E i... |
| cdleme31sn1 41630 | Part of proof of Lemma E i... |
| cdleme31se 41631 | Part of proof of Lemma D i... |
| cdleme31se2 41632 | Part of proof of Lemma D i... |
| cdleme31sc 41633 | Part of proof of Lemma E i... |
| cdleme31sde 41634 | Part of proof of Lemma D i... |
| cdleme31snd 41635 | Part of proof of Lemma D i... |
| cdleme31sdnN 41636 | Part of proof of Lemma E i... |
| cdleme31sn1c 41637 | Part of proof of Lemma E i... |
| cdleme31sn2 41638 | Part of proof of Lemma E i... |
| cdleme31fv 41639 | Part of proof of Lemma E i... |
| cdleme31fv1 41640 | Part of proof of Lemma E i... |
| cdleme31fv1s 41641 | Part of proof of Lemma E i... |
| cdleme31fv2 41642 | Part of proof of Lemma E i... |
| cdleme31id 41643 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 41644 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 41645 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 41646 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 41647 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 41648 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 41649 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 41650 | Part of proof of Lemma E i... |
| cdlemefr29exN 41651 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 41652 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 41653 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 41654 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 41655 | TODO fix comment. (Contri... |
| cdlemefr29clN 41656 | Show closure of the unique... |
| cdleme43frv1snN 41657 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 41658 | Part of proof of Lemma E i... |
| cdlemefr32fva1 41659 | Part of proof of Lemma E i... |
| cdlemefr31fv1 41660 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 41661 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 41662 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 41663 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 41664 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 41665 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 41666 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 41667 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 41668 | Show closure of the unique... |
| cdleme43fsv1snlem 41669 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 41670 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 41671 | Part of proof of Lemma E i... |
| cdlemefs32fva1 41672 | Part of proof of Lemma E i... |
| cdlemefs31fv1 41673 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 41674 | Value of f(r) when r is an... |
| cdlemefs44 41675 | Value of f_s(r) when r is ... |
| cdlemefr45 41676 | Value of f(r) when r is an... |
| cdlemefr45e 41677 | Explicit expansion of ~ cd... |
| cdlemefs45 41678 | Value of f_s(r) when r is ... |
| cdlemefs45ee 41679 | Explicit expansion of ~ cd... |
| cdlemefs45eN 41680 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 41681 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 41682 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 41683 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 41684 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 41685 | Show closure of ` [_ R / s... |
| cdleme32fva 41686 | Part of proof of Lemma D i... |
| cdleme32fva1 41687 | Part of proof of Lemma D i... |
| cdleme32fvaw 41688 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 41689 | Part of proof of Lemma D i... |
| cdleme32a 41690 | Part of proof of Lemma D i... |
| cdleme32b 41691 | Part of proof of Lemma D i... |
| cdleme32c 41692 | Part of proof of Lemma D i... |
| cdleme32d 41693 | Part of proof of Lemma D i... |
| cdleme32e 41694 | Part of proof of Lemma D i... |
| cdleme32f 41695 | Part of proof of Lemma D i... |
| cdleme32le 41696 | Part of proof of Lemma D i... |
| cdleme35a 41697 | Part of proof of Lemma E i... |
| cdleme35fnpq 41698 | Part of proof of Lemma E i... |
| cdleme35b 41699 | Part of proof of Lemma E i... |
| cdleme35c 41700 | Part of proof of Lemma E i... |
| cdleme35d 41701 | Part of proof of Lemma E i... |
| cdleme35e 41702 | Part of proof of Lemma E i... |
| cdleme35f 41703 | Part of proof of Lemma E i... |
| cdleme35g 41704 | Part of proof of Lemma E i... |
| cdleme35h 41705 | Part of proof of Lemma E i... |
| cdleme35h2 41706 | Part of proof of Lemma E i... |
| cdleme35sn2aw 41707 | Part of proof of Lemma E i... |
| cdleme35sn3a 41708 | Part of proof of Lemma E i... |
| cdleme36a 41709 | Part of proof of Lemma E i... |
| cdleme36m 41710 | Part of proof of Lemma E i... |
| cdleme37m 41711 | Part of proof of Lemma E i... |
| cdleme38m 41712 | Part of proof of Lemma E i... |
| cdleme38n 41713 | Part of proof of Lemma E i... |
| cdleme39a 41714 | Part of proof of Lemma E i... |
| cdleme39n 41715 | Part of proof of Lemma E i... |
| cdleme40m 41716 | Part of proof of Lemma E i... |
| cdleme40n 41717 | Part of proof of Lemma E i... |
| cdleme40v 41718 | Part of proof of Lemma E i... |
| cdleme40w 41719 | Part of proof of Lemma E i... |
| cdleme42a 41720 | Part of proof of Lemma E i... |
| cdleme42c 41721 | Part of proof of Lemma E i... |
| cdleme42d 41722 | Part of proof of Lemma E i... |
| cdleme41sn3aw 41723 | Part of proof of Lemma E i... |
| cdleme41sn4aw 41724 | Part of proof of Lemma E i... |
| cdleme41snaw 41725 | Part of proof of Lemma E i... |
| cdleme41fva11 41726 | Part of proof of Lemma E i... |
| cdleme42b 41727 | Part of proof of Lemma E i... |
| cdleme42e 41728 | Part of proof of Lemma E i... |
| cdleme42f 41729 | Part of proof of Lemma E i... |
| cdleme42g 41730 | Part of proof of Lemma E i... |
| cdleme42h 41731 | Part of proof of Lemma E i... |
| cdleme42i 41732 | Part of proof of Lemma E i... |
| cdleme42k 41733 | Part of proof of Lemma E i... |
| cdleme42ke 41734 | Part of proof of Lemma E i... |
| cdleme42keg 41735 | Part of proof of Lemma E i... |
| cdleme42mN 41736 | Part of proof of Lemma E i... |
| cdleme42mgN 41737 | Part of proof of Lemma E i... |
| cdleme43aN 41738 | Part of proof of Lemma E i... |
| cdleme43bN 41739 | Lemma for Lemma E in [Craw... |
| cdleme43cN 41740 | Part of proof of Lemma E i... |
| cdleme43dN 41741 | Part of proof of Lemma E i... |
| cdleme46f2g2 41742 | Conversion for ` G ` to re... |
| cdleme46f2g1 41743 | Conversion for ` G ` to re... |
| cdleme17d2 41744 | Part of proof of Lemma E i... |
| cdleme17d3 41745 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 41746 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 41747 | Part of proof of Lemma E i... |
| cdleme48fv 41748 | Part of proof of Lemma D i... |
| cdleme48fvg 41749 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 41750 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 41751 | TODO: fix comment. TODO: ... |
| cdleme48b 41752 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 41753 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 41754 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 41755 | TODO: fix comment. (Contr... |
| cdleme4gfv 41756 | Part of proof of Lemma D i... |
| cdlemeg47b 41757 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 41758 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 41759 | Value of g_s(r) when r is ... |
| cdlemeg49le 41760 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 41761 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 41762 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 41763 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 41764 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 41765 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 41766 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 41767 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 41768 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 41769 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 41770 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 41771 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 41772 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 41773 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 41774 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 41775 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 41776 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 41777 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 41778 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 41779 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 41780 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 41781 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 41782 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 41783 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 41784 | TODO: fix comment. (Contr... |
| cdleme48gfv1 41785 | TODO: fix comment. (Contr... |
| cdleme48gfv 41786 | TODO: fix comment. (Contr... |
| cdleme48fgv 41787 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 41788 | Part of proof of Lemma D i... |
| cdleme50lebi 41789 | Part of proof of Lemma D i... |
| cdleme50eq 41790 | Part of proof of Lemma D i... |
| cdleme50f 41791 | Part of proof of Lemma D i... |
| cdleme50f1 41792 | Part of proof of Lemma D i... |
| cdleme50rnlem 41793 | Part of proof of Lemma D i... |
| cdleme50rn 41794 | Part of proof of Lemma D i... |
| cdleme50f1o 41795 | Part of proof of Lemma D i... |
| cdleme50laut 41796 | Part of proof of Lemma D i... |
| cdleme50ldil 41797 | Part of proof of Lemma D i... |
| cdleme50trn1 41798 | Part of proof that ` F ` i... |
| cdleme50trn2a 41799 | Part of proof that ` F ` i... |
| cdleme50trn2 41800 | Part of proof that ` F ` i... |
| cdleme50trn12 41801 | Part of proof that ` F ` i... |
| cdleme50trn3 41802 | Part of proof that ` F ` i... |
| cdleme50trn123 41803 | Part of proof that ` F ` i... |
| cdleme51finvfvN 41804 | Part of proof of Lemma E i... |
| cdleme51finvN 41805 | Part of proof of Lemma E i... |
| cdleme50ltrn 41806 | Part of proof of Lemma E i... |
| cdleme51finvtrN 41807 | Part of proof of Lemma E i... |
| cdleme50ex 41808 | Part of Lemma E in [Crawle... |
| cdleme 41809 | Lemma E in [Crawley] p. 11... |
| cdlemf1 41810 | Part of Lemma F in [Crawle... |
| cdlemf2 41811 | Part of Lemma F in [Crawle... |
| cdlemf 41812 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 41813 | ~ cdlemf with additional c... |
| cdlemftr3 41814 | Special case of ~ cdlemf s... |
| cdlemftr2 41815 | Special case of ~ cdlemf s... |
| cdlemftr1 41816 | Part of proof of Lemma G o... |
| cdlemftr0 41817 | Special case of ~ cdlemf s... |
| trlord 41818 | The ordering of two Hilber... |
| cdlemg1a 41819 | Shorter expression for ` G... |
| cdlemg1b2 41820 | This theorem can be used t... |
| cdlemg1idlemN 41821 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 41822 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 41823 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 41824 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 41825 | This theorem can be used t... |
| cdlemg1idN 41826 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 41827 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 41828 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 41829 | Version of ~ cdleme51finvt... |
| ltrniotaval 41830 | Value of the unique transl... |
| ltrniotacnvval 41831 | Converse value of the uniq... |
| ltrniotaidvalN 41832 | Value of the unique transl... |
| ltrniotavalbN 41833 | Value of the unique transl... |
| cdlemeiota 41834 | A translation is uniquely ... |
| cdlemg1ci2 41835 | Any function of the form o... |
| cdlemg1cN 41836 | Any translation belongs to... |
| cdlemg1cex 41837 | Any translation is one of ... |
| cdlemg2cN 41838 | Any translation belongs to... |
| cdlemg2dN 41839 | This theorem can be used t... |
| cdlemg2cex 41840 | Any translation is one of ... |
| cdlemg2ce 41841 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 41842 | Part of proof of Lemma E i... |
| cdlemg2fvlem 41843 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 41844 | ~ cdleme42keg with simpler... |
| cdlemg2idN 41845 | Version of ~ cdleme31id wi... |
| cdlemg3a 41846 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 41847 | TODO: Replace this with ~... |
| cdlemg2fv 41848 | Value of a translation in ... |
| cdlemg2fv2 41849 | Value of a translation in ... |
| cdlemg2k 41850 | ~ cdleme42keg with simpler... |
| cdlemg2kq 41851 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 41852 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 41853 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 41854 | TODO: Is there a simpler ... |
| cdlemb3 41855 | Given two atoms not under ... |
| cdlemg7fvbwN 41856 | Properties of a translatio... |
| cdlemg4a 41857 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 41858 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 41859 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 41860 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 41861 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 41862 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 41863 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 41864 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 41865 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 41866 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 41867 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 41868 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 41869 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 41870 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 41871 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 41872 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 41873 | Value of a translation com... |
| cdlemg7aN 41874 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 41875 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 41876 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 41877 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 41878 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 41879 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 41880 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 41881 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 41882 | The triples ` <. P , ( F `... |
| cdlemg9 41883 | The triples ` <. P , ( F `... |
| cdlemg10b 41884 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 41885 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 41886 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 41887 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 41888 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 41889 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 41890 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 41891 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 41892 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 41893 | The triples ` <. P , ( F `... |
| cdlemg12c 41894 | The triples ` <. P , ( F `... |
| cdlemg12d 41895 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 41896 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 41897 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 41898 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 41899 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 41900 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 41901 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 41902 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 41903 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 41904 | Eliminate the ` ( F `` P )... |
| cdlemg15 41905 | Eliminate the ` ( (... |
| cdlemg16 41906 | Part of proof of Lemma G o... |
| cdlemg16ALTN 41907 | This version of ~ cdlemg16... |
| cdlemg16z 41908 | Eliminate ` ( ( F `... |
| cdlemg16zz 41909 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 41910 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 41911 | Part of proof of Lemma G i... |
| cdlemg17dN 41912 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 41913 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 41914 | TODO: fix comment. (Contr... |
| cdlemg17f 41915 | TODO: fix comment. (Contr... |
| cdlemg17g 41916 | TODO: fix comment. (Contr... |
| cdlemg17h 41917 | TODO: fix comment. (Contr... |
| cdlemg17i 41918 | TODO: fix comment. (Contr... |
| cdlemg17ir 41919 | TODO: fix comment. (Contr... |
| cdlemg17j 41920 | TODO: fix comment. (Contr... |
| cdlemg17pq 41921 | Utility theorem for swappi... |
| cdlemg17bq 41922 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 41923 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 41924 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 41925 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 41926 | Part of Lemma G of [Crawle... |
| cdlemg18a 41927 | Show two lines are differe... |
| cdlemg18b 41928 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 41929 | Show two lines intersect a... |
| cdlemg18d 41930 | Show two lines intersect a... |
| cdlemg18 41931 | Show two lines intersect a... |
| cdlemg19a 41932 | Show two lines intersect a... |
| cdlemg19 41933 | Show two lines intersect a... |
| cdlemg20 41934 | Show two lines intersect a... |
| cdlemg21 41935 | Version of cdlemg19 with `... |
| cdlemg22 41936 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 41937 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 41938 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 41939 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 41940 | ~ cdlemg16zz restated for ... |
| cdlemg27a 41941 | For use with case when ` (... |
| cdlemg28a 41942 | Part of proof of Lemma G o... |
| cdlemg31b0N 41943 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 41944 | TODO: Fix comment. (Cont... |
| cdlemg27b 41945 | TODO: Fix comment. (Cont... |
| cdlemg31a 41946 | TODO: fix comment. (Contr... |
| cdlemg31b 41947 | TODO: fix comment. (Contr... |
| cdlemg31c 41948 | Show that when ` N ` is an... |
| cdlemg31d 41949 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 41950 | TODO: Fix comment. (Cont... |
| cdlemg33c0 41951 | TODO: Fix comment. (Cont... |
| cdlemg28b 41952 | Part of proof of Lemma G o... |
| cdlemg28 41953 | Part of proof of Lemma G o... |
| cdlemg29 41954 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 41955 | TODO: Fix comment. (Cont... |
| cdlemg33b 41956 | TODO: Fix comment. (Cont... |
| cdlemg33c 41957 | TODO: Fix comment. (Cont... |
| cdlemg33d 41958 | TODO: Fix comment. (Cont... |
| cdlemg33e 41959 | TODO: Fix comment. (Cont... |
| cdlemg33 41960 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 41961 | Use cdlemg33 to eliminate ... |
| cdlemg35 41962 | TODO: Fix comment. TODO:... |
| cdlemg36 41963 | Use cdlemg35 to eliminate ... |
| cdlemg38 41964 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 41965 | Eliminate ` =/= ` conditio... |
| cdlemg40 41966 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 41967 | Convert ~ cdlemg40 to func... |
| ltrnco 41968 | The composition of two tra... |
| trlcocnv 41969 | Swap the arguments of the ... |
| trlcoabs 41970 | Absorption into a composit... |
| trlcoabs2N 41971 | Absorption of the trace of... |
| trlcoat 41972 | The trace of a composition... |
| trlcocnvat 41973 | Commonly used special case... |
| trlconid 41974 | The composition of two dif... |
| trlcolem 41975 | Lemma for ~ trlco . (Cont... |
| trlco 41976 | The trace of a composition... |
| trlcone 41977 | If two translations have d... |
| cdlemg42 41978 | Part of proof of Lemma G o... |
| cdlemg43 41979 | Part of proof of Lemma G o... |
| cdlemg44a 41980 | Part of proof of Lemma G o... |
| cdlemg44b 41981 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 41982 | Part of proof of Lemma G o... |
| cdlemg47a 41983 | TODO: fix comment. TODO: ... |
| cdlemg46 41984 | Part of proof of Lemma G o... |
| cdlemg47 41985 | Part of proof of Lemma G o... |
| cdlemg48 41986 | Eliminate ` h ` from ~ cdl... |
| ltrncom 41987 | Composition is commutative... |
| ltrnco4 41988 | Rearrange a composition of... |
| trljco 41989 | Trace joined with trace of... |
| trljco2 41990 | Trace joined with trace of... |
| tgrpfset 41993 | The translation group maps... |
| tgrpset 41994 | The translation group for ... |
| tgrpbase 41995 | The base set of the transl... |
| tgrpopr 41996 | The group operation of the... |
| tgrpov 41997 | The group operation value ... |
| tgrpgrplem 41998 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 41999 | The translation group is a... |
| tgrpabl 42000 | The translation group is a... |
| tendofset 42007 | The set of all trace-prese... |
| tendoset 42008 | The set of trace-preservin... |
| istendo 42009 | The predicate "is a trace-... |
| tendotp 42010 | Trace-preserving property ... |
| istendod 42011 | Deduce the predicate "is a... |
| tendof 42012 | Functionality of a trace-p... |
| tendoeq1 42013 | Condition determining equa... |
| tendovalco 42014 | Value of composition of tr... |
| tendocoval 42015 | Value of composition of en... |
| tendocl 42016 | Closure of a trace-preserv... |
| tendoco2 42017 | Distribution of compositio... |
| tendoidcl 42018 | The identity is a trace-pr... |
| tendo1mul 42019 | Multiplicative identity mu... |
| tendo1mulr 42020 | Multiplicative identity mu... |
| tendococl 42021 | The composition of two tra... |
| tendoid 42022 | The identity value of a tr... |
| tendoeq2 42023 | Condition determining equa... |
| tendoplcbv 42024 | Define sum operation for t... |
| tendopl 42025 | Value of endomorphism sum ... |
| tendopl2 42026 | Value of result of endomor... |
| tendoplcl2 42027 | Value of result of endomor... |
| tendoplco2 42028 | Value of result of endomor... |
| tendopltp 42029 | Trace-preserving property ... |
| tendoplcl 42030 | Endomorphism sum is a trac... |
| tendoplcom 42031 | The endomorphism sum opera... |
| tendoplass 42032 | The endomorphism sum opera... |
| tendodi1 42033 | Endomorphism composition d... |
| tendodi2 42034 | Endomorphism composition d... |
| tendo0cbv 42035 | Define additive identity f... |
| tendo02 42036 | Value of additive identity... |
| tendo0co2 42037 | The additive identity trac... |
| tendo0tp 42038 | Trace-preserving property ... |
| tendo0cl 42039 | The additive identity is a... |
| tendo0pl 42040 | Property of the additive i... |
| tendo0plr 42041 | Property of the additive i... |
| tendoicbv 42042 | Define inverse function fo... |
| tendoi 42043 | Value of inverse endomorph... |
| tendoi2 42044 | Value of additive inverse ... |
| tendoicl 42045 | Closure of the additive in... |
| tendoipl 42046 | Property of the additive i... |
| tendoipl2 42047 | Property of the additive i... |
| erngfset 42048 | The division rings on trac... |
| erngset 42049 | The division ring on trace... |
| erngbase 42050 | The base set of the divisi... |
| erngfplus 42051 | Ring addition operation. ... |
| erngplus 42052 | Ring addition operation. ... |
| erngplus2 42053 | Ring addition operation. ... |
| erngfmul 42054 | Ring multiplication operat... |
| erngmul 42055 | Ring addition operation. ... |
| erngfset-rN 42056 | The division rings on trac... |
| erngset-rN 42057 | The division ring on trace... |
| erngbase-rN 42058 | The base set of the divisi... |
| erngfplus-rN 42059 | Ring addition operation. ... |
| erngplus-rN 42060 | Ring addition operation. ... |
| erngplus2-rN 42061 | Ring addition operation. ... |
| erngfmul-rN 42062 | Ring multiplication operat... |
| erngmul-rN 42063 | Ring addition operation. ... |
| cdlemh1 42064 | Part of proof of Lemma H o... |
| cdlemh2 42065 | Part of proof of Lemma H o... |
| cdlemh 42066 | Lemma H of [Crawley] p. 11... |
| cdlemi1 42067 | Part of proof of Lemma I o... |
| cdlemi2 42068 | Part of proof of Lemma I o... |
| cdlemi 42069 | Lemma I of [Crawley] p. 11... |
| cdlemj1 42070 | Part of proof of Lemma J o... |
| cdlemj2 42071 | Part of proof of Lemma J o... |
| cdlemj3 42072 | Part of proof of Lemma J o... |
| tendocan 42073 | Cancellation law: if the v... |
| tendoid0 42074 | A trace-preserving endomor... |
| tendo0mul 42075 | Additive identity multipli... |
| tendo0mulr 42076 | Additive identity multipli... |
| tendo1ne0 42077 | The identity (unity) is no... |
| tendoconid 42078 | The composition (product) ... |
| tendotr 42079 | The trace of the value of ... |
| cdlemk1 42080 | Part of proof of Lemma K o... |
| cdlemk2 42081 | Part of proof of Lemma K o... |
| cdlemk3 42082 | Part of proof of Lemma K o... |
| cdlemk4 42083 | Part of proof of Lemma K o... |
| cdlemk5a 42084 | Part of proof of Lemma K o... |
| cdlemk5 42085 | Part of proof of Lemma K o... |
| cdlemk6 42086 | Part of proof of Lemma K o... |
| cdlemk8 42087 | Part of proof of Lemma K o... |
| cdlemk9 42088 | Part of proof of Lemma K o... |
| cdlemk9bN 42089 | Part of proof of Lemma K o... |
| cdlemki 42090 | Part of proof of Lemma K o... |
| cdlemkvcl 42091 | Part of proof of Lemma K o... |
| cdlemk10 42092 | Part of proof of Lemma K o... |
| cdlemksv 42093 | Part of proof of Lemma K o... |
| cdlemksel 42094 | Part of proof of Lemma K o... |
| cdlemksat 42095 | Part of proof of Lemma K o... |
| cdlemksv2 42096 | Part of proof of Lemma K o... |
| cdlemk7 42097 | Part of proof of Lemma K o... |
| cdlemk11 42098 | Part of proof of Lemma K o... |
| cdlemk12 42099 | Part of proof of Lemma K o... |
| cdlemkoatnle 42100 | Utility lemma. (Contribut... |
| cdlemk13 42101 | Part of proof of Lemma K o... |
| cdlemkole 42102 | Utility lemma. (Contribut... |
| cdlemk14 42103 | Part of proof of Lemma K o... |
| cdlemk15 42104 | Part of proof of Lemma K o... |
| cdlemk16a 42105 | Part of proof of Lemma K o... |
| cdlemk16 42106 | Part of proof of Lemma K o... |
| cdlemk17 42107 | Part of proof of Lemma K o... |
| cdlemk1u 42108 | Part of proof of Lemma K o... |
| cdlemk5auN 42109 | Part of proof of Lemma K o... |
| cdlemk5u 42110 | Part of proof of Lemma K o... |
| cdlemk6u 42111 | Part of proof of Lemma K o... |
| cdlemkj 42112 | Part of proof of Lemma K o... |
| cdlemkuvN 42113 | Part of proof of Lemma K o... |
| cdlemkuel 42114 | Part of proof of Lemma K o... |
| cdlemkuat 42115 | Part of proof of Lemma K o... |
| cdlemkuv2 42116 | Part of proof of Lemma K o... |
| cdlemk18 42117 | Part of proof of Lemma K o... |
| cdlemk19 42118 | Part of proof of Lemma K o... |
| cdlemk7u 42119 | Part of proof of Lemma K o... |
| cdlemk11u 42120 | Part of proof of Lemma K o... |
| cdlemk12u 42121 | Part of proof of Lemma K o... |
| cdlemk21N 42122 | Part of proof of Lemma K o... |
| cdlemk20 42123 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 42124 | Utility lemma. (Contribut... |
| cdlemk13-2N 42125 | Part of proof of Lemma K o... |
| cdlemkole-2N 42126 | Utility lemma. (Contribut... |
| cdlemk14-2N 42127 | Part of proof of Lemma K o... |
| cdlemk15-2N 42128 | Part of proof of Lemma K o... |
| cdlemk16-2N 42129 | Part of proof of Lemma K o... |
| cdlemk17-2N 42130 | Part of proof of Lemma K o... |
| cdlemkj-2N 42131 | Part of proof of Lemma K o... |
| cdlemkuv-2N 42132 | Part of proof of Lemma K o... |
| cdlemkuel-2N 42133 | Part of proof of Lemma K o... |
| cdlemkuv2-2 42134 | Part of proof of Lemma K o... |
| cdlemk18-2N 42135 | Part of proof of Lemma K o... |
| cdlemk19-2N 42136 | Part of proof of Lemma K o... |
| cdlemk7u-2N 42137 | Part of proof of Lemma K o... |
| cdlemk11u-2N 42138 | Part of proof of Lemma K o... |
| cdlemk12u-2N 42139 | Part of proof of Lemma K o... |
| cdlemk21-2N 42140 | Part of proof of Lemma K o... |
| cdlemk20-2N 42141 | Part of proof of Lemma K o... |
| cdlemk22 42142 | Part of proof of Lemma K o... |
| cdlemk30 42143 | Part of proof of Lemma K o... |
| cdlemkuu 42144 | Convert between function a... |
| cdlemk31 42145 | Part of proof of Lemma K o... |
| cdlemk32 42146 | Part of proof of Lemma K o... |
| cdlemkuel-3 42147 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 42148 | Part of proof of Lemma K o... |
| cdlemk18-3N 42149 | Part of proof of Lemma K o... |
| cdlemk22-3 42150 | Part of proof of Lemma K o... |
| cdlemk23-3 42151 | Part of proof of Lemma K o... |
| cdlemk24-3 42152 | Part of proof of Lemma K o... |
| cdlemk25-3 42153 | Part of proof of Lemma K o... |
| cdlemk26b-3 42154 | Part of proof of Lemma K o... |
| cdlemk26-3 42155 | Part of proof of Lemma K o... |
| cdlemk27-3 42156 | Part of proof of Lemma K o... |
| cdlemk28-3 42157 | Part of proof of Lemma K o... |
| cdlemk33N 42158 | Part of proof of Lemma K o... |
| cdlemk34 42159 | Part of proof of Lemma K o... |
| cdlemk29-3 42160 | Part of proof of Lemma K o... |
| cdlemk35 42161 | Part of proof of Lemma K o... |
| cdlemk36 42162 | Part of proof of Lemma K o... |
| cdlemk37 42163 | Part of proof of Lemma K o... |
| cdlemk38 42164 | Part of proof of Lemma K o... |
| cdlemk39 42165 | Part of proof of Lemma K o... |
| cdlemk40 42166 | TODO: fix comment. (Contr... |
| cdlemk40t 42167 | TODO: fix comment. (Contr... |
| cdlemk40f 42168 | TODO: fix comment. (Contr... |
| cdlemk41 42169 | Part of proof of Lemma K o... |
| cdlemkfid1N 42170 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 42171 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 42172 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 42173 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 42174 | TODO: is this useful or sh... |
| cdlemky 42175 | Part of proof of Lemma K o... |
| cdlemkyu 42176 | Convert between function a... |
| cdlemkyuu 42177 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 42178 | Part of proof of Lemma K o... |
| cdlemk19ylem 42179 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 42180 | Part of proof of Lemma K o... |
| cdlemk19y 42181 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 42182 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 42183 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 42184 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 42185 | The value of the tau funct... |
| cdlemk35s 42186 | Substitution version of ~ ... |
| cdlemk35s-id 42187 | Substitution version of ~ ... |
| cdlemk39s 42188 | Substitution version of ~ ... |
| cdlemk39s-id 42189 | Substitution version of ~ ... |
| cdlemk42 42190 | Part of proof of Lemma K o... |
| cdlemk19xlem 42191 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 42192 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 42193 | Part of proof of Lemma K o... |
| cdlemk11tc 42194 | Part of proof of Lemma K o... |
| cdlemk11t 42195 | Part of proof of Lemma K o... |
| cdlemk45 42196 | Part of proof of Lemma K o... |
| cdlemk46 42197 | Part of proof of Lemma K o... |
| cdlemk47 42198 | Part of proof of Lemma K o... |
| cdlemk48 42199 | Part of proof of Lemma K o... |
| cdlemk49 42200 | Part of proof of Lemma K o... |
| cdlemk50 42201 | Part of proof of Lemma K o... |
| cdlemk51 42202 | Part of proof of Lemma K o... |
| cdlemk52 42203 | Part of proof of Lemma K o... |
| cdlemk53a 42204 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 42205 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 42206 | Part of proof of Lemma K o... |
| cdlemk54 42207 | Part of proof of Lemma K o... |
| cdlemk55a 42208 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 42209 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 42210 | Part of proof of Lemma K o... |
| cdlemkyyN 42211 | Part of proof of Lemma K o... |
| cdlemk43N 42212 | Part of proof of Lemma K o... |
| cdlemk35u 42213 | Substitution version of ~ ... |
| cdlemk55u1 42214 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 42215 | Part of proof of Lemma K o... |
| cdlemk39u1 42216 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 42217 | Part of proof of Lemma K o... |
| cdlemk19u1 42218 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 42219 | Part of Lemma K of [Crawle... |
| cdlemk56 42220 | Part of Lemma K of [Crawle... |
| cdlemk19w 42221 | Use a fixed element to eli... |
| cdlemk56w 42222 | Use a fixed element to eli... |
| cdlemk 42223 | Lemma K of [Crawley] p. 11... |
| tendoex 42224 | Generalization of Lemma K ... |
| cdleml1N 42225 | Part of proof of Lemma L o... |
| cdleml2N 42226 | Part of proof of Lemma L o... |
| cdleml3N 42227 | Part of proof of Lemma L o... |
| cdleml4N 42228 | Part of proof of Lemma L o... |
| cdleml5N 42229 | Part of proof of Lemma L o... |
| cdleml6 42230 | Part of proof of Lemma L o... |
| cdleml7 42231 | Part of proof of Lemma L o... |
| cdleml8 42232 | Part of proof of Lemma L o... |
| cdleml9 42233 | Part of proof of Lemma L o... |
| dva1dim 42234 | Two expressions for the 1-... |
| dvhb1dimN 42235 | Two expressions for the 1-... |
| erng1lem 42236 | Value of the endomorphism ... |
| erngdvlem1 42237 | Lemma for ~ eringring . (... |
| erngdvlem2N 42238 | Lemma for ~ eringring . (... |
| erngdvlem3 42239 | Lemma for ~ eringring . (... |
| erngdvlem4 42240 | Lemma for ~ erngdv . (Con... |
| eringring 42241 | An endomorphism ring is a ... |
| erngdv 42242 | An endomorphism ring is a ... |
| erng0g 42243 | The division ring zero of ... |
| erng1r 42244 | The division ring unity of... |
| erngdvlem1-rN 42245 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 42246 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 42247 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 42248 | Lemma for ~ erngdv . (Con... |
| erngring-rN 42249 | An endomorphism ring is a ... |
| erngdv-rN 42250 | An endomorphism ring is a ... |
| dvafset 42253 | The constructed partial ve... |
| dvaset 42254 | The constructed partial ve... |
| dvasca 42255 | The ring base set of the c... |
| dvabase 42256 | The ring base set of the c... |
| dvafplusg 42257 | Ring addition operation fo... |
| dvaplusg 42258 | Ring addition operation fo... |
| dvaplusgv 42259 | Ring addition operation fo... |
| dvafmulr 42260 | Ring multiplication operat... |
| dvamulr 42261 | Ring multiplication operat... |
| dvavbase 42262 | The vectors (vector base s... |
| dvafvadd 42263 | The vector sum operation f... |
| dvavadd 42264 | Ring addition operation fo... |
| dvafvsca 42265 | Ring addition operation fo... |
| dvavsca 42266 | Ring addition operation fo... |
| tendospcl 42267 | Closure of endomorphism sc... |
| tendospass 42268 | Associative law for endomo... |
| tendospdi1 42269 | Forward distributive law f... |
| tendocnv 42270 | Converse of a trace-preser... |
| tendospdi2 42271 | Reverse distributive law f... |
| tendospcanN 42272 | Cancellation law for trace... |
| dvaabl 42273 | The constructed partial ve... |
| dvalveclem 42274 | Lemma for ~ dvalvec . (Co... |
| dvalvec 42275 | The constructed partial ve... |
| dva0g 42276 | The zero vector of partial... |
| diaffval 42279 | The partial isomorphism A ... |
| diafval 42280 | The partial isomorphism A ... |
| diaval 42281 | The partial isomorphism A ... |
| diaelval 42282 | Member of the partial isom... |
| diafn 42283 | Functionality and domain o... |
| diadm 42284 | Domain of the partial isom... |
| diaeldm 42285 | Member of domain of the pa... |
| diadmclN 42286 | A member of domain of the ... |
| diadmleN 42287 | A member of domain of the ... |
| dian0 42288 | The value of the partial i... |
| dia0eldmN 42289 | The lattice zero belongs t... |
| dia1eldmN 42290 | The fiducial hyperplane (t... |
| diass 42291 | The value of the partial i... |
| diael 42292 | A member of the value of t... |
| diatrl 42293 | Trace of a member of the p... |
| diaelrnN 42294 | Any value of the partial i... |
| dialss 42295 | The value of partial isomo... |
| diaord 42296 | The partial isomorphism A ... |
| dia11N 42297 | The partial isomorphism A ... |
| diaf11N 42298 | The partial isomorphism A ... |
| diaclN 42299 | Closure of partial isomorp... |
| diacnvclN 42300 | Closure of partial isomorp... |
| dia0 42301 | The value of the partial i... |
| dia1N 42302 | The value of the partial i... |
| dia1elN 42303 | The largest subspace in th... |
| diaglbN 42304 | Partial isomorphism A of a... |
| diameetN 42305 | Partial isomorphism A of a... |
| diainN 42306 | Inverse partial isomorphis... |
| diaintclN 42307 | The intersection of partia... |
| diasslssN 42308 | The partial isomorphism A ... |
| diassdvaN 42309 | The partial isomorphism A ... |
| dia1dim 42310 | Two expressions for the 1-... |
| dia1dim2 42311 | Two expressions for a 1-di... |
| dia1dimid 42312 | A vector (translation) bel... |
| dia2dimlem1 42313 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 42314 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 42315 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 42316 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 42317 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 42318 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 42319 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 42320 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 42321 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 42322 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 42323 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 42324 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 42325 | Lemma for ~ dia2dim . Eli... |
| dia2dim 42326 | A two-dimensional subspace... |
| dvhfset 42329 | The constructed full vecto... |
| dvhset 42330 | The constructed full vecto... |
| dvhsca 42331 | The ring of scalars of the... |
| dvhbase 42332 | The ring base set of the c... |
| dvhfplusr 42333 | Ring addition operation fo... |
| dvhfmulr 42334 | Ring multiplication operat... |
| dvhmulr 42335 | Ring multiplication operat... |
| dvhvbase 42336 | The vectors (vector base s... |
| dvhelvbasei 42337 | Vector membership in the c... |
| dvhvaddcbv 42338 | Change bound variables to ... |
| dvhvaddval 42339 | The vector sum operation f... |
| dvhfvadd 42340 | The vector sum operation f... |
| dvhvadd 42341 | The vector sum operation f... |
| dvhopvadd 42342 | The vector sum operation f... |
| dvhopvadd2 42343 | The vector sum operation f... |
| dvhvaddcl 42344 | Closure of the vector sum ... |
| dvhvaddcomN 42345 | Commutativity of vector su... |
| dvhvaddass 42346 | Associativity of vector su... |
| dvhvscacbv 42347 | Change bound variables to ... |
| dvhvscaval 42348 | The scalar product operati... |
| dvhfvsca 42349 | Scalar product operation f... |
| dvhvsca 42350 | Scalar product operation f... |
| dvhopvsca 42351 | Scalar product operation f... |
| dvhvscacl 42352 | Closure of the scalar prod... |
| tendoinvcl 42353 | Closure of multiplicative ... |
| tendolinv 42354 | Left multiplicative invers... |
| tendorinv 42355 | Right multiplicative inver... |
| dvhgrp 42356 | The full vector space ` U ... |
| dvhlveclem 42357 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 42358 | The full vector space ` U ... |
| dvhlmod 42359 | The full vector space ` U ... |
| dvh0g 42360 | The zero vector of vector ... |
| dvheveccl 42361 | Properties of a unit vecto... |
| dvhopclN 42362 | Closure of a ` DVecH ` vec... |
| dvhopaddN 42363 | Sum of ` DVecH ` vectors e... |
| dvhopspN 42364 | Scalar product of ` DVecH ... |
| dvhopN 42365 | Decompose a ` DVecH ` vect... |
| dvhopellsm 42366 | Ordered pair membership in... |
| cdlemm10N 42367 | The image of the map ` G `... |
| docaffvalN 42370 | Subspace orthocomplement f... |
| docafvalN 42371 | Subspace orthocomplement f... |
| docavalN 42372 | Subspace orthocomplement f... |
| docaclN 42373 | Closure of subspace orthoc... |
| diaocN 42374 | Value of partial isomorphi... |
| doca2N 42375 | Double orthocomplement of ... |
| doca3N 42376 | Double orthocomplement of ... |
| dvadiaN 42377 | Any closed subspace is a m... |
| diarnN 42378 | Partial isomorphism A maps... |
| diaf1oN 42379 | The partial isomorphism A ... |
| djaffvalN 42382 | Subspace join for ` DVecA ... |
| djafvalN 42383 | Subspace join for ` DVecA ... |
| djavalN 42384 | Subspace join for ` DVecA ... |
| djaclN 42385 | Closure of subspace join f... |
| djajN 42386 | Transfer lattice join to `... |
| dibffval 42389 | The partial isomorphism B ... |
| dibfval 42390 | The partial isomorphism B ... |
| dibval 42391 | The partial isomorphism B ... |
| dibopelvalN 42392 | Member of the partial isom... |
| dibval2 42393 | Value of the partial isomo... |
| dibopelval2 42394 | Member of the partial isom... |
| dibval3N 42395 | Value of the partial isomo... |
| dibelval3 42396 | Member of the partial isom... |
| dibopelval3 42397 | Member of the partial isom... |
| dibelval1st 42398 | Membership in value of the... |
| dibelval1st1 42399 | Membership in value of the... |
| dibelval1st2N 42400 | Membership in value of the... |
| dibelval2nd 42401 | Membership in value of the... |
| dibn0 42402 | The value of the partial i... |
| dibfna 42403 | Functionality and domain o... |
| dibdiadm 42404 | Domain of the partial isom... |
| dibfnN 42405 | Functionality and domain o... |
| dibdmN 42406 | Domain of the partial isom... |
| dibeldmN 42407 | Member of domain of the pa... |
| dibord 42408 | The isomorphism B for a la... |
| dib11N 42409 | The isomorphism B for a la... |
| dibf11N 42410 | The partial isomorphism A ... |
| dibclN 42411 | Closure of partial isomorp... |
| dibvalrel 42412 | The value of partial isomo... |
| dib0 42413 | The value of partial isomo... |
| dib1dim 42414 | Two expressions for the 1-... |
| dibglbN 42415 | Partial isomorphism B of a... |
| dibintclN 42416 | The intersection of partia... |
| dib1dim2 42417 | Two expressions for a 1-di... |
| dibss 42418 | The partial isomorphism B ... |
| diblss 42419 | The value of partial isomo... |
| diblsmopel 42420 | Membership in subspace sum... |
| dicffval 42423 | The partial isomorphism C ... |
| dicfval 42424 | The partial isomorphism C ... |
| dicval 42425 | The partial isomorphism C ... |
| dicopelval 42426 | Membership in value of the... |
| dicelvalN 42427 | Membership in value of the... |
| dicval2 42428 | The partial isomorphism C ... |
| dicelval3 42429 | Member of the partial isom... |
| dicopelval2 42430 | Membership in value of the... |
| dicelval2N 42431 | Membership in value of the... |
| dicfnN 42432 | Functionality and domain o... |
| dicdmN 42433 | Domain of the partial isom... |
| dicvalrelN 42434 | The value of partial isomo... |
| dicssdvh 42435 | The partial isomorphism C ... |
| dicelval1sta 42436 | Membership in value of the... |
| dicelval1stN 42437 | Membership in value of the... |
| dicelval2nd 42438 | Membership in value of the... |
| dicvaddcl 42439 | Membership in value of the... |
| dicvscacl 42440 | Membership in value of the... |
| dicn0 42441 | The value of the partial i... |
| diclss 42442 | The value of partial isomo... |
| diclspsn 42443 | The value of isomorphism C... |
| cdlemn2 42444 | Part of proof of Lemma N o... |
| cdlemn2a 42445 | Part of proof of Lemma N o... |
| cdlemn3 42446 | Part of proof of Lemma N o... |
| cdlemn4 42447 | Part of proof of Lemma N o... |
| cdlemn4a 42448 | Part of proof of Lemma N o... |
| cdlemn5pre 42449 | Part of proof of Lemma N o... |
| cdlemn5 42450 | Part of proof of Lemma N o... |
| cdlemn6 42451 | Part of proof of Lemma N o... |
| cdlemn7 42452 | Part of proof of Lemma N o... |
| cdlemn8 42453 | Part of proof of Lemma N o... |
| cdlemn9 42454 | Part of proof of Lemma N o... |
| cdlemn10 42455 | Part of proof of Lemma N o... |
| cdlemn11a 42456 | Part of proof of Lemma N o... |
| cdlemn11b 42457 | Part of proof of Lemma N o... |
| cdlemn11c 42458 | Part of proof of Lemma N o... |
| cdlemn11pre 42459 | Part of proof of Lemma N o... |
| cdlemn11 42460 | Part of proof of Lemma N o... |
| cdlemn 42461 | Lemma N of [Crawley] p. 12... |
| dihordlem6 42462 | Part of proof of Lemma N o... |
| dihordlem7 42463 | Part of proof of Lemma N o... |
| dihordlem7b 42464 | Part of proof of Lemma N o... |
| dihjustlem 42465 | Part of proof after Lemma ... |
| dihjust 42466 | Part of proof after Lemma ... |
| dihord1 42467 | Part of proof after Lemma ... |
| dihord2a 42468 | Part of proof after Lemma ... |
| dihord2b 42469 | Part of proof after Lemma ... |
| dihord2cN 42470 | Part of proof after Lemma ... |
| dihord11b 42471 | Part of proof after Lemma ... |
| dihord10 42472 | Part of proof after Lemma ... |
| dihord11c 42473 | Part of proof after Lemma ... |
| dihord2pre 42474 | Part of proof after Lemma ... |
| dihord2pre2 42475 | Part of proof after Lemma ... |
| dihord2 42476 | Part of proof after Lemma ... |
| dihffval 42479 | The isomorphism H for a la... |
| dihfval 42480 | Isomorphism H for a lattic... |
| dihval 42481 | Value of isomorphism H for... |
| dihvalc 42482 | Value of isomorphism H for... |
| dihlsscpre 42483 | Closure of isomorphism H f... |
| dihvalcqpre 42484 | Value of isomorphism H for... |
| dihvalcq 42485 | Value of isomorphism H for... |
| dihvalb 42486 | Value of isomorphism H for... |
| dihopelvalbN 42487 | Ordered pair member of the... |
| dihvalcqat 42488 | Value of isomorphism H for... |
| dih1dimb 42489 | Two expressions for a 1-di... |
| dih1dimb2 42490 | Isomorphism H at an atom u... |
| dih1dimc 42491 | Isomorphism H at an atom n... |
| dib2dim 42492 | Extend ~ dia2dim to partia... |
| dih2dimb 42493 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 42494 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 42495 | Ordered pair member of the... |
| dihvalcq2 42496 | Value of isomorphism H for... |
| dihopelvalcpre 42497 | Member of value of isomorp... |
| dihopelvalc 42498 | Member of value of isomorp... |
| dihlss 42499 | The value of isomorphism H... |
| dihss 42500 | The value of isomorphism H... |
| dihssxp 42501 | An isomorphism H value is ... |
| dihopcl 42502 | Closure of an ordered pair... |
| xihopellsmN 42503 | Ordered pair membership in... |
| dihopellsm 42504 | Ordered pair membership in... |
| dihord6apre 42505 | Part of proof that isomorp... |
| dihord3 42506 | The isomorphism H for a la... |
| dihord4 42507 | The isomorphism H for a la... |
| dihord5b 42508 | Part of proof that isomorp... |
| dihord6b 42509 | Part of proof that isomorp... |
| dihord6a 42510 | Part of proof that isomorp... |
| dihord5apre 42511 | Part of proof that isomorp... |
| dihord5a 42512 | Part of proof that isomorp... |
| dihord 42513 | The isomorphism H is order... |
| dih11 42514 | The isomorphism H is one-t... |
| dihf11lem 42515 | Functionality of the isomo... |
| dihf11 42516 | The isomorphism H for a la... |
| dihfn 42517 | Functionality and domain o... |
| dihdm 42518 | Domain of isomorphism H. (... |
| dihcl 42519 | Closure of isomorphism H. ... |
| dihcnvcl 42520 | Closure of isomorphism H c... |
| dihcnvid1 42521 | The converse isomorphism o... |
| dihcnvid2 42522 | The isomorphism of a conve... |
| dihcnvord 42523 | Ordering property for conv... |
| dihcnv11 42524 | The converse of isomorphis... |
| dihsslss 42525 | The isomorphism H maps to ... |
| dihrnlss 42526 | The isomorphism H maps to ... |
| dihrnss 42527 | The isomorphism H maps to ... |
| dihvalrel 42528 | The value of isomorphism H... |
| dih0 42529 | The value of isomorphism H... |
| dih0bN 42530 | A lattice element is zero ... |
| dih0vbN 42531 | A vector is zero iff its s... |
| dih0cnv 42532 | The isomorphism H converse... |
| dih0rn 42533 | The zero subspace belongs ... |
| dih0sb 42534 | A subspace is zero iff the... |
| dih1 42535 | The value of isomorphism H... |
| dih1rn 42536 | The full vector space belo... |
| dih1cnv 42537 | The isomorphism H converse... |
| dihwN 42538 | Value of isomorphism H at ... |
| dihmeetlem1N 42539 | Isomorphism H of a conjunc... |
| dihglblem5apreN 42540 | A conjunction property of ... |
| dihglblem5aN 42541 | A conjunction property of ... |
| dihglblem2aN 42542 | Lemma for isomorphism H of... |
| dihglblem2N 42543 | The GLB of a set of lattic... |
| dihglblem3N 42544 | Isomorphism H of a lattice... |
| dihglblem3aN 42545 | Isomorphism H of a lattice... |
| dihglblem4 42546 | Isomorphism H of a lattice... |
| dihglblem5 42547 | Isomorphism H of a lattice... |
| dihmeetlem2N 42548 | Isomorphism H of a conjunc... |
| dihglbcpreN 42549 | Isomorphism H of a lattice... |
| dihglbcN 42550 | Isomorphism H of a lattice... |
| dihmeetcN 42551 | Isomorphism H of a lattice... |
| dihmeetbN 42552 | Isomorphism H of a lattice... |
| dihmeetbclemN 42553 | Lemma for isomorphism H of... |
| dihmeetlem3N 42554 | Lemma for isomorphism H of... |
| dihmeetlem4preN 42555 | Lemma for isomorphism H of... |
| dihmeetlem4N 42556 | Lemma for isomorphism H of... |
| dihmeetlem5 42557 | Part of proof that isomorp... |
| dihmeetlem6 42558 | Lemma for isomorphism H of... |
| dihmeetlem7N 42559 | Lemma for isomorphism H of... |
| dihjatc1 42560 | Lemma for isomorphism H of... |
| dihjatc2N 42561 | Isomorphism H of join with... |
| dihjatc3 42562 | Isomorphism H of join with... |
| dihmeetlem8N 42563 | Lemma for isomorphism H of... |
| dihmeetlem9N 42564 | Lemma for isomorphism H of... |
| dihmeetlem10N 42565 | Lemma for isomorphism H of... |
| dihmeetlem11N 42566 | Lemma for isomorphism H of... |
| dihmeetlem12N 42567 | Lemma for isomorphism H of... |
| dihmeetlem13N 42568 | Lemma for isomorphism H of... |
| dihmeetlem14N 42569 | Lemma for isomorphism H of... |
| dihmeetlem15N 42570 | Lemma for isomorphism H of... |
| dihmeetlem16N 42571 | Lemma for isomorphism H of... |
| dihmeetlem17N 42572 | Lemma for isomorphism H of... |
| dihmeetlem18N 42573 | Lemma for isomorphism H of... |
| dihmeetlem19N 42574 | Lemma for isomorphism H of... |
| dihmeetlem20N 42575 | Lemma for isomorphism H of... |
| dihmeetALTN 42576 | Isomorphism H of a lattice... |
| dih1dimatlem0 42577 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 42578 | Lemma for ~ dih1dimat . (... |
| dih1dimat 42579 | Any 1-dimensional subspace... |
| dihlsprn 42580 | The span of a vector belon... |
| dihlspsnssN 42581 | A subspace included in a 1... |
| dihlspsnat 42582 | The inverse isomorphism H ... |
| dihatlat 42583 | The isomorphism H of an at... |
| dihat 42584 | There exists at least one ... |
| dihpN 42585 | The value of isomorphism H... |
| dihlatat 42586 | The reverse isomorphism H ... |
| dihatexv 42587 | There is a nonzero vector ... |
| dihatexv2 42588 | There is a nonzero vector ... |
| dihglblem6 42589 | Isomorphism H of a lattice... |
| dihglb 42590 | Isomorphism H of a lattice... |
| dihglb2 42591 | Isomorphism H of a lattice... |
| dihmeet 42592 | Isomorphism H of a lattice... |
| dihintcl 42593 | The intersection of closed... |
| dihmeetcl 42594 | Closure of closed subspace... |
| dihmeet2 42595 | Reverse isomorphism H of a... |
| dochffval 42598 | Subspace orthocomplement f... |
| dochfval 42599 | Subspace orthocomplement f... |
| dochval 42600 | Subspace orthocomplement f... |
| dochval2 42601 | Subspace orthocomplement f... |
| dochcl 42602 | Closure of subspace orthoc... |
| dochlss 42603 | A subspace orthocomplement... |
| dochssv 42604 | A subspace orthocomplement... |
| dochfN 42605 | Domain and codomain of the... |
| dochvalr 42606 | Orthocomplement of a close... |
| doch0 42607 | Orthocomplement of the zer... |
| doch1 42608 | Orthocomplement of the uni... |
| dochoc0 42609 | The zero subspace is close... |
| dochoc1 42610 | The unit subspace (all vec... |
| dochvalr2 42611 | Orthocomplement of a close... |
| dochvalr3 42612 | Orthocomplement of a close... |
| doch2val2 42613 | Double orthocomplement for... |
| dochss 42614 | Subset law for orthocomple... |
| dochocss 42615 | Double negative law for or... |
| dochoc 42616 | Double negative law for or... |
| dochsscl 42617 | If a set of vectors is inc... |
| dochoccl 42618 | A set of vectors is closed... |
| dochord 42619 | Ordering law for orthocomp... |
| dochord2N 42620 | Ordering law for orthocomp... |
| dochord3 42621 | Ordering law for orthocomp... |
| doch11 42622 | Orthocomplement is one-to-... |
| dochsordN 42623 | Strict ordering law for or... |
| dochn0nv 42624 | An orthocomplement is nonz... |
| dihoml4c 42625 | Version of ~ dihoml4 with ... |
| dihoml4 42626 | Orthomodular law for const... |
| dochspss 42627 | The span of a set of vecto... |
| dochocsp 42628 | The span of an orthocomple... |
| dochspocN 42629 | The span of an orthocomple... |
| dochocsn 42630 | The double orthocomplement... |
| dochsncom 42631 | Swap vectors in an orthoco... |
| dochsat 42632 | The double orthocomplement... |
| dochshpncl 42633 | If a hyperplane is not clo... |
| dochlkr 42634 | Equivalent conditions for ... |
| dochkrshp 42635 | The closure of a kernel is... |
| dochkrshp2 42636 | Properties of the closure ... |
| dochkrshp3 42637 | Properties of the closure ... |
| dochkrshp4 42638 | Properties of the closure ... |
| dochdmj1 42639 | De Morgan-like law for sub... |
| dochnoncon 42640 | Law of noncontradiction. ... |
| dochnel2 42641 | A nonzero member of a subs... |
| dochnel 42642 | A nonzero vector doesn't b... |
| djhffval 42645 | Subspace join for ` DVecH ... |
| djhfval 42646 | Subspace join for ` DVecH ... |
| djhval 42647 | Subspace join for ` DVecH ... |
| djhval2 42648 | Value of subspace join for... |
| djhcl 42649 | Closure of subspace join f... |
| djhlj 42650 | Transfer lattice join to `... |
| djhljjN 42651 | Lattice join in terms of `... |
| djhjlj 42652 | ` DVecH ` vector space clo... |
| djhj 42653 | ` DVecH ` vector space clo... |
| djhcom 42654 | Subspace join is commutati... |
| djhspss 42655 | Subspace span of union is ... |
| djhsumss 42656 | Subspace sum is a subset o... |
| dihsumssj 42657 | The subspace sum of two is... |
| djhunssN 42658 | Subspace union is a subset... |
| dochdmm1 42659 | De Morgan-like law for clo... |
| djhexmid 42660 | Excluded middle property o... |
| djh01 42661 | Closed subspace join with ... |
| djh02 42662 | Closed subspace join with ... |
| djhlsmcl 42663 | A closed subspace sum equa... |
| djhcvat42 42664 | A covering property. ( ~ ... |
| dihjatb 42665 | Isomorphism H of lattice j... |
| dihjatc 42666 | Isomorphism H of lattice j... |
| dihjatcclem1 42667 | Lemma for isomorphism H of... |
| dihjatcclem2 42668 | Lemma for isomorphism H of... |
| dihjatcclem3 42669 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 42670 | Lemma for isomorphism H of... |
| dihjatcc 42671 | Isomorphism H of lattice j... |
| dihjat 42672 | Isomorphism H of lattice j... |
| dihprrnlem1N 42673 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 42674 | Lemma for ~ dihprrn . (Co... |
| dihprrn 42675 | The span of a vector pair ... |
| djhlsmat 42676 | The sum of two subspace at... |
| dihjat1lem 42677 | Subspace sum of a closed s... |
| dihjat1 42678 | Subspace sum of a closed s... |
| dihsmsprn 42679 | Subspace sum of a closed s... |
| dihjat2 42680 | The subspace sum of a clos... |
| dihjat3 42681 | Isomorphism H of lattice j... |
| dihjat4 42682 | Transfer the subspace sum ... |
| dihjat6 42683 | Transfer the subspace sum ... |
| dihsmsnrn 42684 | The subspace sum of two si... |
| dihsmatrn 42685 | The subspace sum of a clos... |
| dihjat5N 42686 | Transfer lattice join with... |
| dvh4dimat 42687 | There is an atom that is o... |
| dvh3dimatN 42688 | There is an atom that is o... |
| dvh2dimatN 42689 | Given an atom, there exist... |
| dvh1dimat 42690 | There exists an atom. (Co... |
| dvh1dim 42691 | There exists a nonzero vec... |
| dvh4dimlem 42692 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 42693 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 42694 | There is a vector that is ... |
| dvh3dim 42695 | There is a vector that is ... |
| dvh4dimN 42696 | There is a vector that is ... |
| dvh3dim2 42697 | There is a vector that is ... |
| dvh3dim3N 42698 | There is a vector that is ... |
| dochsnnz 42699 | The orthocomplement of a s... |
| dochsatshp 42700 | The orthocomplement of a s... |
| dochsatshpb 42701 | The orthocomplement of a s... |
| dochsnshp 42702 | The orthocomplement of a n... |
| dochshpsat 42703 | A hyperplane is closed iff... |
| dochkrsat 42704 | The orthocomplement of a k... |
| dochkrsat2 42705 | The orthocomplement of a k... |
| dochsat0 42706 | The orthocomplement of a k... |
| dochkrsm 42707 | The subspace sum of a clos... |
| dochexmidat 42708 | Special case of excluded m... |
| dochexmidlem1 42709 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 42710 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 42711 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 42712 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 42713 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 42714 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 42715 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 42716 | Lemma for ~ dochexmid . T... |
| dochexmid 42717 | Excluded middle law for cl... |
| dochsnkrlem1 42718 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 42719 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 42720 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 42721 | A (closed) kernel expresse... |
| dochsnkr2 42722 | Kernel of the explicit fun... |
| dochsnkr2cl 42723 | The ` X ` determining func... |
| dochflcl 42724 | Closure of the explicit fu... |
| dochfl1 42725 | The value of the explicit ... |
| dochfln0 42726 | The value of a functional ... |
| dochkr1 42727 | A nonzero functional has a... |
| dochkr1OLDN 42728 | A nonzero functional has a... |
| lpolsetN 42731 | The set of polarities of a... |
| islpolN 42732 | The predicate "is a polari... |
| islpoldN 42733 | Properties that determine ... |
| lpolfN 42734 | Functionality of a polarit... |
| lpolvN 42735 | The polarity of the whole ... |
| lpolconN 42736 | Contraposition property of... |
| lpolsatN 42737 | The polarity of an atomic ... |
| lpolpolsatN 42738 | Property of a polarity. (... |
| dochpolN 42739 | The subspace orthocompleme... |
| lcfl1lem 42740 | Property of a functional w... |
| lcfl1 42741 | Property of a functional w... |
| lcfl2 42742 | Property of a functional w... |
| lcfl3 42743 | Property of a functional w... |
| lcfl4N 42744 | Property of a functional w... |
| lcfl5 42745 | Property of a functional w... |
| lcfl5a 42746 | Property of a functional w... |
| lcfl6lem 42747 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 42748 | Lemma for ~ lcfl7N . If t... |
| lcfl6 42749 | Property of a functional w... |
| lcfl7N 42750 | Property of a functional w... |
| lcfl8 42751 | Property of a functional w... |
| lcfl8a 42752 | Property of a functional w... |
| lcfl8b 42753 | Property of a nonzero func... |
| lcfl9a 42754 | Property implying that a f... |
| lclkrlem1 42755 | The set of functionals hav... |
| lclkrlem2a 42756 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 42757 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 42758 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 42759 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 42760 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 42761 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 42762 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 42763 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 42764 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 42765 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 42766 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 42767 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 42768 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 42769 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 42770 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 42771 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 42772 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 42773 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 42774 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 42775 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 42776 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 42777 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 42778 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 42779 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 42780 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 42781 | The set of functionals hav... |
| lclkr 42782 | The set of functionals wit... |
| lcfls1lem 42783 | Property of a functional w... |
| lcfls1N 42784 | Property of a functional w... |
| lcfls1c 42785 | Property of a functional w... |
| lclkrslem1 42786 | The set of functionals hav... |
| lclkrslem2 42787 | The set of functionals hav... |
| lclkrs 42788 | The set of functionals hav... |
| lclkrs2 42789 | The set of functionals wit... |
| lcfrvalsnN 42790 | Reconstruction from the du... |
| lcfrlem1 42791 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 42792 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 42793 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 42794 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 42795 | Lemma for ~ lcfr . The se... |
| lcfrlem6 42796 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 42797 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 42798 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 42799 | Lemma for ~ lcf1o . (This... |
| lcf1o 42800 | Define a function ` J ` th... |
| lcfrlem10 42801 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 42802 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 42803 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 42804 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 42805 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 42806 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 42807 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 42808 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 42809 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 42810 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 42811 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 42812 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 42813 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 42814 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 42815 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 42816 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 42817 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 42818 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 42819 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 42820 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 42821 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 42822 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 42823 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 42824 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 42825 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 42826 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 42827 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 42828 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 42829 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 42830 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 42831 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 42832 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 42833 | Lemma for ~ lcfr . Elimin... |
| lcfr 42834 | Reconstruction of a subspa... |
| lcdfval 42837 | Dual vector space of funct... |
| lcdval 42838 | Dual vector space of funct... |
| lcdval2 42839 | Dual vector space of funct... |
| lcdlvec 42840 | The dual vector space of f... |
| lcdlmod 42841 | The dual vector space of f... |
| lcdvbase 42842 | Vector base set of a dual ... |
| lcdvbasess 42843 | The vector base set of the... |
| lcdvbaselfl 42844 | A vector in the base set o... |
| lcdvbasecl 42845 | Closure of the value of a ... |
| lcdvadd 42846 | Vector addition for the cl... |
| lcdvaddval 42847 | The value of the value of ... |
| lcdsca 42848 | The ring of scalars of the... |
| lcdsbase 42849 | Base set of scalar ring fo... |
| lcdsadd 42850 | Scalar addition for the cl... |
| lcdsmul 42851 | Scalar multiplication for ... |
| lcdvs 42852 | Scalar product for the clo... |
| lcdvsval 42853 | Value of scalar product op... |
| lcdvscl 42854 | The scalar product operati... |
| lcdlssvscl 42855 | Closure of scalar product ... |
| lcdvsass 42856 | Associative law for scalar... |
| lcd0 42857 | The zero scalar of the clo... |
| lcd1 42858 | The unit scalar of the clo... |
| lcdneg 42859 | The unit scalar of the clo... |
| lcd0v 42860 | The zero functional in the... |
| lcd0v2 42861 | The zero functional in the... |
| lcd0vvalN 42862 | Value of the zero function... |
| lcd0vcl 42863 | Closure of the zero functi... |
| lcd0vs 42864 | A scalar zero times a func... |
| lcdvs0N 42865 | A scalar times the zero fu... |
| lcdvsub 42866 | The value of vector subtra... |
| lcdvsubval 42867 | The value of the value of ... |
| lcdlss 42868 | Subspaces of a dual vector... |
| lcdlss2N 42869 | Subspaces of a dual vector... |
| lcdlsp 42870 | Span in the set of functio... |
| lcdlkreqN 42871 | Colinear functionals have ... |
| lcdlkreq2N 42872 | Colinear functionals have ... |
| mapdffval 42875 | Projectivity from vector s... |
| mapdfval 42876 | Projectivity from vector s... |
| mapdval 42877 | Value of projectivity from... |
| mapdvalc 42878 | Value of projectivity from... |
| mapdval2N 42879 | Value of projectivity from... |
| mapdval3N 42880 | Value of projectivity from... |
| mapdval4N 42881 | Value of projectivity from... |
| mapdval5N 42882 | Value of projectivity from... |
| mapdordlem1a 42883 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 42884 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 42885 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 42886 | Lemma for ~ mapdord . Ord... |
| mapdord 42887 | Ordering property of the m... |
| mapd11 42888 | The map defined by ~ df-ma... |
| mapddlssN 42889 | The mapping of a subspace ... |
| mapdsn 42890 | Value of the map defined b... |
| mapdsn2 42891 | Value of the map defined b... |
| mapdsn3 42892 | Value of the map defined b... |
| mapd1dim2lem1N 42893 | Value of the map defined b... |
| mapdrvallem2 42894 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 42895 | Lemma for ~ mapdrval . (C... |
| mapdrval 42896 | Given a dual subspace ` R ... |
| mapd1o 42897 | The map defined by ~ df-ma... |
| mapdrn 42898 | Range of the map defined b... |
| mapdunirnN 42899 | Union of the range of the ... |
| mapdrn2 42900 | Range of the map defined b... |
| mapdcnvcl 42901 | Closure of the converse of... |
| mapdcl 42902 | Closure the value of the m... |
| mapdcnvid1N 42903 | Converse of the value of t... |
| mapdsord 42904 | Strong ordering property o... |
| mapdcl2 42905 | The mapping of a subspace ... |
| mapdcnvid2 42906 | Value of the converse of t... |
| mapdcnvordN 42907 | Ordering property of the c... |
| mapdcnv11N 42908 | The converse of the map de... |
| mapdcv 42909 | Covering property of the c... |
| mapdincl 42910 | Closure of dual subspace i... |
| mapdin 42911 | Subspace intersection is p... |
| mapdlsmcl 42912 | Closure of dual subspace s... |
| mapdlsm 42913 | Subspace sum is preserved ... |
| mapd0 42914 | Projectivity map of the ze... |
| mapdcnvatN 42915 | Atoms are preserved by the... |
| mapdat 42916 | Atoms are preserved by the... |
| mapdspex 42917 | The map of a span equals t... |
| mapdn0 42918 | Transfer nonzero property ... |
| mapdncol 42919 | Transfer non-colinearity f... |
| mapdindp 42920 | Transfer (part of) vector ... |
| mapdpglem1 42921 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 42922 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 42923 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 42924 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 42925 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 42926 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 42927 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 42928 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 42929 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 42930 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 42931 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 42932 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 42933 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 42934 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 42935 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 42936 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 42937 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 42938 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 42939 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 42940 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 42941 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 42942 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 42943 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 42944 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 42945 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 42946 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 42947 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 42948 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 42949 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 42950 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 42951 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 42952 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 42953 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 42954 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 42955 | Part 1 of proof of the fir... |
| baerlem3lem1 42956 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 42957 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 42958 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 42959 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 42960 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 42961 | Lemma for ~ baerlem5b . (... |
| baerlem3 42962 | An equality that holds whe... |
| baerlem5a 42963 | An equality that holds whe... |
| baerlem5b 42964 | An equality that holds whe... |
| baerlem5amN 42965 | An equality that holds whe... |
| baerlem5bmN 42966 | An equality that holds whe... |
| baerlem5abmN 42967 | An equality that holds whe... |
| mapdindp0 42968 | Vector independence lemma.... |
| mapdindp1 42969 | Vector independence lemma.... |
| mapdindp2 42970 | Vector independence lemma.... |
| mapdindp3 42971 | Vector independence lemma.... |
| mapdindp4 42972 | Vector independence lemma.... |
| mapdhval 42973 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 42974 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 42975 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 42976 | Lemmma for ~~? mapdh . (C... |
| mapdheq 42977 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 42978 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 42979 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 42980 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 42981 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 42982 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 42983 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 42984 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 42985 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 42986 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 42987 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 42988 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 42989 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 42990 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 42991 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 42992 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 42993 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 42994 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 42995 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 42996 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 42997 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 42998 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 42999 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 43000 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 43001 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 43002 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 43003 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 43004 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 43007 | Map from nonzero vectors t... |
| hvmapfval 43008 | Map from nonzero vectors t... |
| hvmapval 43009 | Value of map from nonzero ... |
| hvmapvalvalN 43010 | Value of value of map (i.e... |
| hvmapidN 43011 | The value of the vector to... |
| hvmap1o 43012 | The vector to functional m... |
| hvmapclN 43013 | Closure of the vector to f... |
| hvmap1o2 43014 | The vector to functional m... |
| hvmapcl2 43015 | Closure of the vector to f... |
| hvmaplfl 43016 | The vector to functional m... |
| hvmaplkr 43017 | Kernel of the vector to fu... |
| mapdhvmap 43018 | Relationship between ` map... |
| lspindp5 43019 | Obtain an independent vect... |
| hdmaplem1 43020 | Lemma to convert a frequen... |
| hdmaplem2N 43021 | Lemma to convert a frequen... |
| hdmaplem3 43022 | Lemma to convert a frequen... |
| hdmaplem4 43023 | Lemma to convert a frequen... |
| mapdh8a 43024 | Part of Part (8) in [Baer]... |
| mapdh8aa 43025 | Part of Part (8) in [Baer]... |
| mapdh8ab 43026 | Part of Part (8) in [Baer]... |
| mapdh8ac 43027 | Part of Part (8) in [Baer]... |
| mapdh8ad 43028 | Part of Part (8) in [Baer]... |
| mapdh8b 43029 | Part of Part (8) in [Baer]... |
| mapdh8c 43030 | Part of Part (8) in [Baer]... |
| mapdh8d0N 43031 | Part of Part (8) in [Baer]... |
| mapdh8d 43032 | Part of Part (8) in [Baer]... |
| mapdh8e 43033 | Part of Part (8) in [Baer]... |
| mapdh8g 43034 | Part of Part (8) in [Baer]... |
| mapdh8i 43035 | Part of Part (8) in [Baer]... |
| mapdh8j 43036 | Part of Part (8) in [Baer]... |
| mapdh8 43037 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 43038 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 43039 | Lemma for part (9) in [Bae... |
| hdmap1ffval 43044 | Preliminary map from vecto... |
| hdmap1fval 43045 | Preliminary map from vecto... |
| hdmap1vallem 43046 | Value of preliminary map f... |
| hdmap1val 43047 | Value of preliminary map f... |
| hdmap1val0 43048 | Value of preliminary map f... |
| hdmap1val2 43049 | Value of preliminary map f... |
| hdmap1eq 43050 | The defining equation for ... |
| hdmap1cbv 43051 | Frequently used lemma to c... |
| hdmap1valc 43052 | Connect the value of the p... |
| hdmap1cl 43053 | Convert closure theorem ~ ... |
| hdmap1eq2 43054 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 43055 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 43056 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 43057 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 43058 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 43059 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 43060 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 43061 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 43062 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 43063 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 43064 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 43065 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 43066 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 43067 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 43068 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 43069 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 43070 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 43071 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 43072 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 43073 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 43074 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 43075 | Map from vectors to functi... |
| hdmapfval 43076 | Map from vectors to functi... |
| hdmapval 43077 | Value of map from vectors ... |
| hdmapfnN 43078 | Functionality of map from ... |
| hdmapcl 43079 | Closure of map from vector... |
| hdmapval2lem 43080 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 43081 | Value of map from vectors ... |
| hdmapval0 43082 | Value of map from vectors ... |
| hdmapeveclem 43083 | Lemma for ~ hdmapevec . T... |
| hdmapevec 43084 | Value of map from vectors ... |
| hdmapevec2 43085 | The inner product of the r... |
| hdmapval3lemN 43086 | Value of map from vectors ... |
| hdmapval3N 43087 | Value of map from vectors ... |
| hdmap10lem 43088 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 43089 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 43090 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 43091 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 43092 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 43093 | Part of proof of part 12 i... |
| hdmapnzcl 43094 | Nonzero vector closure of ... |
| hdmapneg 43095 | Part of proof of part 12 i... |
| hdmapsub 43096 | Part of proof of part 12 i... |
| hdmap11 43097 | Part of proof of part 12 i... |
| hdmaprnlem1N 43098 | Part of proof of part 12 i... |
| hdmaprnlem3N 43099 | Part of proof of part 12 i... |
| hdmaprnlem3uN 43100 | Part of proof of part 12 i... |
| hdmaprnlem4tN 43101 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 43102 | Part of proof of part 12 i... |
| hdmaprnlem6N 43103 | Part of proof of part 12 i... |
| hdmaprnlem7N 43104 | Part of proof of part 12 i... |
| hdmaprnlem8N 43105 | Part of proof of part 12 i... |
| hdmaprnlem9N 43106 | Part of proof of part 12 i... |
| hdmaprnlem3eN 43107 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 43108 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 43109 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 43110 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 43111 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 43112 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 43113 | Part of proof of part 12 i... |
| hdmapf1oN 43114 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 43115 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 43116 | Prior to part 14 in [Baer]... |
| hdmap14lem1 43117 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 43118 | Prior to part 14 in [Baer]... |
| hdmap14lem3 43119 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 43120 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 43121 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 43122 | Case where ` F ` is zero. ... |
| hdmap14lem7 43123 | Combine cases of ` F ` . ... |
| hdmap14lem8 43124 | Part of proof of part 14 i... |
| hdmap14lem9 43125 | Part of proof of part 14 i... |
| hdmap14lem10 43126 | Part of proof of part 14 i... |
| hdmap14lem11 43127 | Part of proof of part 14 i... |
| hdmap14lem12 43128 | Lemma for proof of part 14... |
| hdmap14lem13 43129 | Lemma for proof of part 14... |
| hdmap14lem14 43130 | Part of proof of part 14 i... |
| hdmap14lem15 43131 | Part of proof of part 14 i... |
| hgmapffval 43134 | Map from the scalar divisi... |
| hgmapfval 43135 | Map from the scalar divisi... |
| hgmapval 43136 | Value of map from the scal... |
| hgmapfnN 43137 | Functionality of scalar si... |
| hgmapcl 43138 | Closure of scalar sigma ma... |
| hgmapdcl 43139 | Closure of the vector spac... |
| hgmapvs 43140 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 43141 | Value of the scalar sigma ... |
| hgmapval1 43142 | Value of the scalar sigma ... |
| hgmapadd 43143 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 43144 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 43145 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 43146 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 43147 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 43148 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 43149 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 43150 | Part of proof of part 16 i... |
| hgmap11 43151 | The scalar sigma map is on... |
| hgmapf1oN 43152 | The scalar sigma map is a ... |
| hgmapeq0 43153 | The scalar sigma map is ze... |
| hdmapipcl 43154 | The inner product (Hermiti... |
| hdmapln1 43155 | Linearity property that wi... |
| hdmaplna1 43156 | Additive property of first... |
| hdmaplns1 43157 | Subtraction property of fi... |
| hdmaplnm1 43158 | Multiplicative property of... |
| hdmaplna2 43159 | Additive property of secon... |
| hdmapglnm2 43160 | g-linear property of secon... |
| hdmapgln2 43161 | g-linear property that wil... |
| hdmaplkr 43162 | Kernel of the vector to du... |
| hdmapellkr 43163 | Membership in the kernel (... |
| hdmapip0 43164 | Zero property that will be... |
| hdmapip1 43165 | Construct a proportional v... |
| hdmapip0com 43166 | Commutation property of Ba... |
| hdmapinvlem1 43167 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 43168 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 43169 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 43170 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 43171 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 43172 | Involution property of sca... |
| hgmapvvlem2 43173 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 43174 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 43175 | Value of a double involuti... |
| hdmapglem7a 43176 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 43177 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 43178 | Lemma for ~ hdmapg . Line... |
| hdmapg 43179 | Apply the scalar sigma fun... |
| hdmapoc 43180 | Express our constructed or... |
| hlhilset 43183 | The final Hilbert space co... |
| hlhilsca 43184 | The scalar of the final co... |
| hlhilbase 43185 | The base set of the final ... |
| hlhilplus 43186 | The vector addition for th... |
| hlhilslem 43187 | Lemma for ~ hlhilsbase etc... |
| hlhilsbase 43188 | The scalar base set of the... |
| hlhilsplus 43189 | Scalar addition for the fi... |
| hlhilsmul 43190 | Scalar multiplication for ... |
| hlhilsbase2 43191 | The scalar base set of the... |
| hlhilsplus2 43192 | Scalar addition for the fi... |
| hlhilsmul2 43193 | Scalar multiplication for ... |
| hlhils0 43194 | The scalar ring zero for t... |
| hlhils1N 43195 | The scalar ring unity for ... |
| hlhilvsca 43196 | The scalar product for the... |
| hlhilip 43197 | Inner product operation fo... |
| hlhilipval 43198 | Value of inner product ope... |
| hlhilnvl 43199 | The involution operation o... |
| hlhillvec 43200 | The final constructed Hilb... |
| hlhildrng 43201 | The star division ring for... |
| hlhilsrnglem 43202 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 43203 | The star division ring for... |
| hlhil0 43204 | The zero vector for the fi... |
| hlhillsm 43205 | The vector sum operation f... |
| hlhilocv 43206 | The orthocomplement for th... |
| hlhillcs 43207 | The closed subspaces of th... |
| hlhilphllem 43208 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 43209 | Lemma for ~ hlhil . (Cont... |
| hlathil 43210 | Construction of a Hilbert ... |
| iscsrg 43213 | A commutative semiring is ... |
| rhmzrhval 43214 | Evaluation of integers acr... |
| zndvdchrrhm 43215 | Construction of a ring hom... |
| relogbcld 43216 | Closure of the general log... |
| relogbexpd 43217 | Identity law for general l... |
| relogbzexpd 43218 | Power law for the general ... |
| logblebd 43219 | The general logarithm is m... |
| uzindd 43220 | Induction on the upper int... |
| fzadd2d 43221 | Membership of a sum in a f... |
| fzne2d 43222 | Elementhood in a finite se... |
| eqfnfv2d2 43223 | Equality of functions is d... |
| fzsplitnd 43224 | Split a finite interval of... |
| fzsplitnr 43225 | Split a finite interval of... |
| addassnni 43226 | Associative law for additi... |
| addcomnni 43227 | Commutative law for additi... |
| mulassnni 43228 | Associative law for multip... |
| mulcomnni 43229 | Commutative law for multip... |
| gcdcomnni 43230 | Commutative law for gcd. ... |
| gcdnegnni 43231 | Negation invariance for gc... |
| neggcdnni 43232 | Negation invariance for gc... |
| bccl2d 43233 | Closure of the binomial co... |
| recbothd 43234 | Take reciprocal on both si... |
| gcdmultiplei 43235 | The GCD of a multiple of a... |
| gcdaddmzz2nni 43236 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 43237 | Adding a multiple of one o... |
| gcdnncli 43238 | Closure of the gcd operato... |
| muldvds1d 43239 | If a product divides an in... |
| muldvds2d 43240 | If a product divides an in... |
| nndivdvdsd 43241 | A positive integer divides... |
| nnproddivdvdsd 43242 | A product of natural numbe... |
| coprmdvds2d 43243 | If an integer is divisible... |
| imadomfi 43244 | An image of a function und... |
| 12gcd5e1 43245 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 43246 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 43247 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 43248 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 43249 | Calculate the least common... |
| 12lcm5e60 43250 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 43251 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 43252 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 43253 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 43254 | Useful equation to calcula... |
| lcm1un 43255 | Least common multiple of n... |
| lcm2un 43256 | Least common multiple of n... |
| lcm3un 43257 | Least common multiple of n... |
| lcm4un 43258 | Least common multiple of n... |
| lcm5un 43259 | Least common multiple of n... |
| lcm6un 43260 | Least common multiple of n... |
| lcm7un 43261 | Least common multiple of n... |
| lcm8un 43262 | Least common multiple of n... |
| 3factsumint1 43263 | Move constants out of inte... |
| 3factsumint2 43264 | Move constants out of inte... |
| 3factsumint3 43265 | Move constants out of inte... |
| 3factsumint4 43266 | Move constants out of inte... |
| 3factsumint 43267 | Helpful equation for lcm i... |
| resopunitintvd 43268 | Restrict continuous functi... |
| resclunitintvd 43269 | Restrict continuous functi... |
| resdvopclptsd 43270 | Restrict derivative on uni... |
| lcmineqlem1 43271 | Part of lcm inequality lem... |
| lcmineqlem2 43272 | Part of lcm inequality lem... |
| lcmineqlem3 43273 | Part of lcm inequality lem... |
| lcmineqlem4 43274 | Part of lcm inequality lem... |
| lcmineqlem5 43275 | Technical lemma for recipr... |
| lcmineqlem6 43276 | Part of lcm inequality lem... |
| lcmineqlem7 43277 | Derivative of 1-x for chai... |
| lcmineqlem8 43278 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 43279 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 43280 | Induction step of ~ lcmine... |
| lcmineqlem11 43281 | Induction step, continuati... |
| lcmineqlem12 43282 | Base case for induction. ... |
| lcmineqlem13 43283 | Induction proof for lcm in... |
| lcmineqlem14 43284 | Technical lemma for inequa... |
| lcmineqlem15 43285 | F times the least common m... |
| lcmineqlem16 43286 | Technical divisibility lem... |
| lcmineqlem17 43287 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 43288 | Technical lemma to shift f... |
| lcmineqlem19 43289 | Dividing implies inequalit... |
| lcmineqlem20 43290 | Inequality for lcm lemma. ... |
| lcmineqlem21 43291 | The lcm inequality lemma w... |
| lcmineqlem22 43292 | The lcm inequality lemma w... |
| lcmineqlem23 43293 | Penultimate step to the lc... |
| lcmineqlem 43294 | The least common multiple ... |
| 3exp7 43295 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 43296 | First inequality in inequa... |
| 3lexlogpow5ineq2 43297 | Second inequality in inequ... |
| 3lexlogpow5ineq4 43298 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 43299 | Combined inequality chain ... |
| 3lexlogpow2ineq1 43300 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 43301 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 43302 | Result for bound in AKS in... |
| intlewftc 43303 | Inequality inference by in... |
| aks4d1lem1 43304 | Technical lemma to reduce ... |
| aks4d1p1p1 43305 | Exponential law for finite... |
| dvrelog2 43306 | The derivative of the loga... |
| dvrelog3 43307 | The derivative of the loga... |
| dvrelog2b 43308 | Derivative of the binary l... |
| 0nonelalab 43309 | Technical lemma for open i... |
| dvrelogpow2b 43310 | Derivative of the power of... |
| aks4d1p1p3 43311 | Bound of a ceiling of the ... |
| aks4d1p1p2 43312 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 43313 | Technical step for inequal... |
| dvle2 43314 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 43315 | Inequality lift to differe... |
| aks4d1p1p7 43316 | Bound of intermediary of i... |
| aks4d1p1p5 43317 | Show inequality for existe... |
| aks4d1p1 43318 | Show inequality for existe... |
| aks4d1p2 43319 | Technical lemma for existe... |
| aks4d1p3 43320 | There exists a small enoug... |
| aks4d1p4 43321 | There exists a small enoug... |
| aks4d1p5 43322 | Show that ` N ` and ` R ` ... |
| aks4d1p6 43323 | The maximal prime power ex... |
| aks4d1p7d1 43324 | Technical step in AKS lemm... |
| aks4d1p7 43325 | Technical step in AKS lemm... |
| aks4d1p8d1 43326 | If a prime divides one num... |
| aks4d1p8d2 43327 | Any prime power dividing a... |
| aks4d1p8d3 43328 | The remainder of a divisio... |
| aks4d1p8 43329 | Show that ` N ` and ` R ` ... |
| aks4d1p9 43330 | Show that the order is bou... |
| aks4d1 43331 | Lemma 4.1 from ~ https://w... |
| fldhmf1 43332 | A field homomorphism is in... |
| isprimroot 43335 | The value of a primitive r... |
| isprimroot2 43336 | Alternative way of creatin... |
| mndmolinv 43337 | An element of a monoid tha... |
| linvh 43338 | If an element has a unique... |
| primrootsunit1 43339 | Primitive roots have left ... |
| primrootsunit 43340 | Primitive roots have left ... |
| primrootscoprmpow 43341 | Coprime powers of primitiv... |
| posbezout 43342 | Bezout's identity restrict... |
| primrootscoprf 43343 | Coprime powers of primitiv... |
| primrootscoprbij 43344 | A bijection between coprim... |
| primrootscoprbij2 43345 | A bijection between coprim... |
| remexz 43346 | Division with rest. (Cont... |
| primrootlekpowne0 43347 | There is no smaller power ... |
| primrootspoweq0 43348 | The power of a ` R ` -th p... |
| aks6d1c1p1 43349 | Definition of the introspe... |
| aks6d1c1p1rcl 43350 | Reverse closure of the int... |
| aks6d1c1p2 43351 | ` P ` and linear factors a... |
| aks6d1c1p3 43352 | In a field with a Frobeniu... |
| aks6d1c1p4 43353 | The product of polynomials... |
| aks6d1c1p5 43354 | The product of exponents i... |
| aks6d1c1p7 43355 | ` X ` is introspective to ... |
| aks6d1c1p6 43356 | If a polynomials ` F ` is ... |
| aks6d1c1p8 43357 | If a number ` E ` is intro... |
| aks6d1c1 43358 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 43359 | Polynomial evaluation buil... |
| aks6d1c2p1 43360 | In the AKS-theorem the sub... |
| aks6d1c2p2 43361 | Injective condition for co... |
| hashscontpowcl 43362 | Closure of E for ~ https:/... |
| hashscontpow1 43363 | Helper lemma for to prove ... |
| hashscontpow 43364 | If a set contains all ` N ... |
| aks6d1c3 43365 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 43366 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 43367 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 43368 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 43369 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 43370 | If the number of elements ... |
| hashnexinjle 43371 | If the number of elements ... |
| aks6d1c2 43372 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 43373 | Special case related to ~ ... |
| idomnnzpownz 43374 | A nonzero power in an inte... |
| idomnnzgmulnz 43375 | A finite product of nonzer... |
| ringexp0nn 43376 | Zero to the power of a pos... |
| aks6d1c5lem0 43377 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 43378 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 43379 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 43380 | Lemma for Claim 5, contrad... |
| aks6d1c5 43381 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 43382 | Degree multiplication is a... |
| deg1pow 43383 | Exact degree of a power of... |
| 5bc2eq10 43384 | The value of 5 choose 2. ... |
| facp2 43385 | The factorial of a success... |
| 2np3bcnp1 43386 | Part of induction step for... |
| 2ap1caineq 43387 | Inequality for Theorem 6.6... |
| sticksstones1 43388 | Different strictly monoton... |
| sticksstones2 43389 | The range function on stri... |
| sticksstones3 43390 | The range function on stri... |
| sticksstones4 43391 | Equinumerosity lemma for s... |
| sticksstones5 43392 | Count the number of strict... |
| sticksstones6 43393 | Function induces an order ... |
| sticksstones7 43394 | Closure property of sticks... |
| sticksstones8 43395 | Establish mapping between ... |
| sticksstones9 43396 | Establish mapping between ... |
| sticksstones10 43397 | Establish mapping between ... |
| sticksstones11 43398 | Establish bijective mappin... |
| sticksstones12a 43399 | Establish bijective mappin... |
| sticksstones12 43400 | Establish bijective mappin... |
| sticksstones13 43401 | Establish bijective mappin... |
| sticksstones14 43402 | Sticks and stones with def... |
| sticksstones15 43403 | Sticks and stones with alm... |
| sticksstones16 43404 | Sticks and stones with col... |
| sticksstones17 43405 | Extend sticks and stones t... |
| sticksstones18 43406 | Extend sticks and stones t... |
| sticksstones19 43407 | Extend sticks and stones t... |
| sticksstones20 43408 | Lift sticks and stones to ... |
| sticksstones21 43409 | Lift sticks and stones to ... |
| sticksstones22 43410 | Non-exhaustive sticks and ... |
| sticksstones23 43411 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 43412 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 43413 | Every primitive root is ro... |
| aks6d1c6lem3 43414 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 43415 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 43416 | Lemma to construct the map... |
| aks6d1c6isolem2 43417 | Lemma to construct the gro... |
| aks6d1c6isolem3 43418 | The preimage of a map send... |
| aks6d1c6lem5 43419 | Eliminate the size hypothe... |
| bcled 43420 | Inequality for binomial co... |
| bcle2d 43421 | Inequality for binomial co... |
| aks6d1c7lem1 43422 | The last set of inequaliti... |
| aks6d1c7lem2 43423 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 43424 | Remove lots of hypotheses ... |
| aks6d1c7lem4 43425 | In the AKS algorithm there... |
| aks6d1c7 43426 | ` N ` is a prime power if ... |
| rhmqusspan 43427 | Ring homomorphism out of a... |
| aks5lem1 43428 | Section 5 of ~ https://www... |
| aks5lem2 43429 | Lemma for section 5 ~ http... |
| ply1asclzrhval 43430 | Transfer results from alge... |
| aks5lem3a 43431 | Lemma for AKS section 5. ... |
| aks5lem4a 43432 | Lemma for AKS section 5, r... |
| aks5lem5a 43433 | Lemma for AKS, section 5, ... |
| aks5lem6 43434 | Connect results of section... |
| indstrd 43435 | Strong induction, deductio... |
| grpods 43436 | Relate sums of elements of... |
| unitscyglem1 43437 | Lemma for unitscyg . (Con... |
| unitscyglem2 43438 | Lemma for unitscyg . (Con... |
| unitscyglem3 43439 | Lemma for unitscyg . (Con... |
| unitscyglem4 43440 | Lemma for unitscyg . (Con... |
| unitscyglem5 43441 | Lemma for unitscyg . (Con... |
| aks5lem7 43442 | Lemma for aks5. We clean ... |
| aks5lem8 43443 | Lemma for aks5. Clean up ... |
| exfinfldd 43445 | For any prime ` P ` and an... |
| aks5 43446 | The AKS Primality test, gi... |
| quadfac 43447 | The solution of a quadrati... |
| 25or6to4 43448 | Question 67 of 68 from a l... |
| jarrii 43449 | Inference associated with ... |
| intnanrt 43450 | Introduction of conjunct i... |
| ioin9i8 43451 | Miscellaneous inference cr... |
| jaodd 43452 | Double deduction form of ~... |
| syl3an12 43453 | A double syllogism inferen... |
| exbiii 43454 | Inference associated with ... |
| cbvex2ivw 43455 | One direction of ~ cbvex2v... |
| sbtd 43456 | A true statement is true u... |
| sbor2 43457 | One direction of ~ sbor , ... |
| sbalexi 43458 | Inference form of ~ sbalex... |
| nfalh 43459 | Version of ~ nfal with an ... |
| nfe2 43460 | An inner existential quant... |
| nfale2 43461 | An inner existential quant... |
| 19.9dev 43462 | ~ 19.9d in the case of an ... |
| 3rspcedvd 43463 | Triple application of ~ rs... |
| sn-axrep5v 43464 | A condensed form of ~ axre... |
| sn-axprlem3 43465 | ~ axprlem3 using only Tars... |
| sn-exelALT 43466 | Alternate proof of ~ exel ... |
| ssabdv 43467 | Deduction of abstraction s... |
| sn-iotalem 43468 | An unused lemma showing th... |
| sn-iotalemcor 43469 | Corollary of ~ sn-iotalem ... |
| abbi1sn 43470 | Originally part of ~ uniab... |
| brif2 43471 | Move a relation inside and... |
| brif12 43472 | Move a relation inside and... |
| pssexg 43473 | The proper subset of a set... |
| pssn0 43474 | A proper superset is nonem... |
| psspwb 43475 | Classes are proper subclas... |
| xppss12 43476 | Proper subset theorem for ... |
| elpwbi 43477 | Membership in a power set,... |
| imaopab 43478 | The image of a class of or... |
| eqresfnbd 43479 | Property of being the rest... |
| fmpocos 43480 | Composition of two functio... |
| ovmpogad 43481 | Value of an operation give... |
| ofun 43482 | A function operation of un... |
| dfqs3 43483 | Alternate definition of qu... |
| qseq12d 43484 | Equality theorem for quoti... |
| qsalrel 43485 | The quotient set is equal ... |
| supinf 43486 | The supremum is the infimu... |
| mapcod 43487 | Compose two mappings. (Co... |
| fisdomnn 43488 | A finite set is dominated ... |
| ltex 43489 | The less-than relation is ... |
| leex 43490 | The less-than-or-equal-to ... |
| subex 43491 | The subtraction operation ... |
| absex 43492 | The absolute value functio... |
| cjex 43493 | The conjugate function is ... |
| fzosumm1 43494 | Separate out the last term... |
| ccatcan2d 43495 | Cancellation law for conca... |
| c0exALT 43496 | Alternate proof of ~ c0ex ... |
| 0cnALT3 43497 | Alternate proof of ~ 0cn u... |
| elre0re 43498 | Specialized version of ~ 0... |
| lttrii 43499 | 'Less than' is transitive.... |
| remulcan2d 43500 | ~ mulcan2d for real number... |
| readdridaddlidd 43501 | Given some real number ` B... |
| 4p4e8ALT 43502 | A shorter proof of ~ 4p4e8... |
| 1p3e4 43503 | 1 + 3 = 4. (Contributed b... |
| 1p4e5 43504 | 1 + 4 = 5. (Contributed b... |
| 1p5e6 43505 | 1 + 5 = 6. (Contributed b... |
| 1p6e7 43506 | 1 + 6 = 7. (Contributed b... |
| 1p7e8 43507 | 1 + 7 = 8. (Contributed b... |
| 1p8e9 43508 | 1 + 8 = 9. (Contributed b... |
| 2p3e5 43509 | 2 + 3 = 5. (Contributed b... |
| 2p4e6 43510 | 2 + 4 = 6. (Contributed b... |
| 2p5e7 43511 | 2 + 5 = 7. (Contributed b... |
| 2p6e8 43512 | 2 + 6 = 8. (Contributed b... |
| 2p7e9 43513 | 2 + 7 = 9. (Contributed b... |
| 3p4e7 43514 | 3 + 4 = 7. (Contributed b... |
| 3p5e8 43515 | 3 + 5 = 8. (Contributed b... |
| 3p6e9 43516 | 3 + 6 = 9. (Contributed b... |
| 4p5e9 43517 | 4 + 5 = 9. (Contributed b... |
| 5ne0 43518 | The number 5 is nonzero. ... |
| 6ne0 43519 | The number 6 is nonzero. ... |
| 7ne0 43520 | The number 7 is nonzero. ... |
| 8ne0 43521 | The number 8 is nonzero. ... |
| 9ne0 43522 | The number 9 is nonzero. ... |
| sn-1ne2 43523 | A proof of ~ 1ne2 without ... |
| nnn1suc 43524 | A positive integer that is... |
| readdrcl2d 43525 | Reverse closure for additi... |
| raddswap12d 43526 | Swap the first two variabl... |
| lsubrotld 43527 | Rotate the variables left ... |
| rsubrotld 43528 | Rotate the variables left ... |
| lsubswap23d 43529 | Swap the second and third ... |
| addsubeq4com 43530 | Relation between sums and ... |
| sqsumi 43531 | A sum squared. (Contribut... |
| negn0nposznnd 43532 | Lemma for ~ dffltz . (Con... |
| sqmid3api 43533 | Value of the square of the... |
| decaddcom 43534 | Commute ones place in addi... |
| sqn5i 43535 | The square of a number end... |
| sqn5ii 43536 | The square of a number end... |
| decpmulnc 43537 | Partial products algorithm... |
| decpmul 43538 | Partial products algorithm... |
| sqdeccom12 43539 | The square of a number in ... |
| sq3deccom12 43540 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 43541 | 4 times 5 equals 20. (Con... |
| 3rdpwhole 43542 | A third of a number plus t... |
| sq4 43543 | The square of 4 is 16. (C... |
| sq5 43544 | The square of 5 is 25. (C... |
| sq6 43545 | The square of 6 is 36. (C... |
| sq7 43546 | The square of 7 is 49. (C... |
| sq8 43547 | The square of 8 is 64. (C... |
| sq9 43548 | The square of 9 is 81. (C... |
| rpsscn 43549 | The positive reals are a s... |
| 4rp 43550 | 4 is a positive real. (Co... |
| 6rp 43551 | 6 is a positive real. (Co... |
| 7rp 43552 | 7 is a positive real. (Co... |
| 8rp 43553 | 8 is a positive real. (Co... |
| 9rp 43554 | 9 is a positive real. (Co... |
| 235t711 43555 | Calculate a product by lon... |
| ex-decpmul 43556 | Example usage of ~ decpmul... |
| eluzp1 43557 | Membership in a successor ... |
| sn-eluzp1l 43558 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 43559 | The sum of ` N ` constant ... |
| fz1sump1 43560 | Add one more term to a sum... |
| oddnumth 43561 | The Odd Number Theorem. T... |
| nicomachus 43562 | Nicomachus's Theorem. The... |
| sumcubes 43563 | The sum of the first ` N `... |
| ine1 43564 | ` _i ` is not 1. (Contrib... |
| 0tie0 43565 | 0 times ` _i ` equals 0. ... |
| it1ei 43566 | ` _i ` times 1 equals ` _i... |
| 1tiei 43567 | 1 times ` _i ` equals ` _i... |
| itrere 43568 | ` _i ` times a real is rea... |
| retire 43569 | A real times ` _i ` is rea... |
| iocioodisjd 43570 | Adjacent intervals where t... |
| rpabsid 43571 | A positive real is its own... |
| oexpreposd 43572 | Lemma for ~ dffltz . For ... |
| explt1d 43573 | A nonnegative real number ... |
| expeq1d 43574 | A nonnegative real number ... |
| expeqidd 43575 | A nonnegative real number ... |
| exp11d 43576 | ~ exp11nnd for nonzero int... |
| 0dvds0 43577 | 0 divides 0. (Contributed... |
| absdvdsabsb 43578 | Divisibility is invariant ... |
| dvdsexpnn0 43579 | ~ dvdsexpnn generalized to... |
| dvdsexpb 43580 | ~ dvdssq generalized to po... |
| zdivgd 43581 | Two ways to express " ` N ... |
| efsubd 43582 | Difference of exponents la... |
| ef11d 43583 | General condition for the ... |
| logccne0d 43584 | The logarithm isn't 0 if i... |
| cxp112d 43585 | General condition for comp... |
| cxp111d 43586 | General condition for comp... |
| cxpi11d 43587 | ` _i ` to the powers of ` ... |
| logne0d 43588 | Deduction form of ~ logne0... |
| rxp112d 43589 | Real exponentiation is one... |
| log11d 43590 | The natural logarithm is o... |
| rplog11d 43591 | The natural logarithm is o... |
| rxp11d 43592 | Real exponentiation is one... |
| tanhalfpim 43593 | The tangent of ` _pi / 2 `... |
| sinpim 43594 | Sine of a number subtracte... |
| cospim 43595 | Cosine of a number subtrac... |
| tan3rdpi 43596 | The tangent of ` _pi / 3 `... |
| sin2t3rdpi 43597 | The sine of ` 2 x. ( _pi /... |
| cos2t3rdpi 43598 | The cosine of ` 2 x. ( _pi... |
| sin4t3rdpi 43599 | The sine of ` 4 x. ( _pi /... |
| cos4t3rdpi 43600 | The cosine of ` 4 x. ( _pi... |
| asin1half 43601 | The arcsine of ` 1 / 2 ` i... |
| acos1half 43602 | The arccosine of ` 1 / 2 `... |
| dvun 43603 | Condition for the union of... |
| redvmptabs 43604 | The derivative of the abso... |
| readvrec2 43605 | The antiderivative of 1/x ... |
| readvrec 43606 | For real numbers, the anti... |
| resuppsinopn 43607 | The support of sin ( ~ df-... |
| readvcot 43608 | Real antiderivative of cot... |
| resubval 43611 | Value of real subtraction,... |
| renegeulemv 43612 | Lemma for ~ renegeu and si... |
| renegeulem 43613 | Lemma for ~ renegeu and si... |
| renegeu 43614 | Existential uniqueness of ... |
| rernegcl 43615 | Closure law for negative r... |
| renegadd 43616 | Relationship between real ... |
| renegid 43617 | Addition of a real number ... |
| reneg0addlid 43618 | Negative zero is a left ad... |
| resubeulem1 43619 | Lemma for ~ resubeu . A v... |
| resubeulem2 43620 | Lemma for ~ resubeu . A v... |
| resubeu 43621 | Existential uniqueness of ... |
| rersubcl 43622 | Closure for real subtracti... |
| resubadd 43623 | Relation between real subt... |
| resubaddd 43624 | Relationship between subtr... |
| resubf 43625 | Real subtraction is an ope... |
| repncan2 43626 | Addition and subtraction o... |
| repncan3 43627 | Addition and subtraction o... |
| readdsub 43628 | Law for addition and subtr... |
| reladdrsub 43629 | Move LHS of a sum into RHS... |
| reltsub1 43630 | Subtraction from both side... |
| reltsubadd2 43631 | 'Less than' relationship b... |
| resubcan2 43632 | Cancellation law for real ... |
| resubsub4 43633 | Law for double subtraction... |
| rennncan2 43634 | Cancellation law for real ... |
| renpncan3 43635 | Cancellation law for real ... |
| repnpcan 43636 | Cancellation law for addit... |
| reppncan 43637 | Cancellation law for mixed... |
| resubidaddlidlem 43638 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 43639 | Any real number subtracted... |
| resubdi 43640 | Distribution of multiplica... |
| re1m1e0m0 43641 | Equality of two left-addit... |
| sn-00idlem1 43642 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 43643 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 43644 | Lemma for ~ sn-00id . (Co... |
| sn-00id 43645 | ~ 00id proven without ~ ax... |
| re0m0e0 43646 | Real number version of ~ 0... |
| readdlid 43647 | Real number version of ~ a... |
| sn-addlid 43648 | ~ addlid without ~ ax-mulc... |
| remul02 43649 | Real number version of ~ m... |
| sn-0ne2 43650 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 43651 | Real number version of ~ m... |
| sn-remul0ord 43652 | A product is zero iff one ... |
| resubid 43653 | Subtraction of a real numb... |
| readdrid 43654 | Real number version of ~ a... |
| resubid1 43655 | Real number version of ~ s... |
| renegneg 43656 | A real number is equal to ... |
| readdcan2 43657 | Commuted version of ~ read... |
| renegid2 43658 | Commuted version of ~ rene... |
| remulneg2d 43659 | Product with negative is n... |
| sn-it0e0 43660 | Proof of ~ it0e0 without ~... |
| sn-negex12 43661 | A combination of ~ cnegex ... |
| sn-negex 43662 | Proof of ~ cnegex without ... |
| sn-negex2 43663 | Proof of ~ cnegex2 without... |
| sn-addcand 43664 | ~ addcand without ~ ax-mul... |
| sn-addrid 43665 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 43666 | ~ addcan2d without ~ ax-mu... |
| reixi 43667 | ~ ixi without ~ ax-mulcom ... |
| rei4 43668 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 43669 | A number that sums to itse... |
| sn-mul01 43670 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 43671 | ~ negeu without ~ ax-mulco... |
| sn-subcl 43672 | ~ subcl without ~ ax-mulco... |
| sn-subf 43673 | ~ subf without ~ ax-mulcom... |
| resubeqsub 43674 | Equivalence between real s... |
| subresre 43675 | Subtraction restricted to ... |
| addinvcom 43676 | A number commutes with its... |
| remulinvcom 43677 | A left multiplicative inve... |
| remullid 43678 | Commuted version of ~ ax-1... |
| sn-1ticom 43679 | Lemma for ~ sn-mullid and ... |
| sn-mullid 43680 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 43681 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 43682 | The multiplicative inverse... |
| remulcand 43683 | Commuted version of ~ remu... |
| redivvald 43686 | Value of real division, wh... |
| rediveud 43687 | Existential uniqueness of ... |
| sn-redivcld 43688 | Closure law for real divis... |
| redivmuld 43689 | Relationship between divis... |
| redivmul2d 43690 | Relationship between divis... |
| redivcan2d 43691 | A cancellation law for div... |
| redivcan3d 43692 | A cancellation law for div... |
| rediveq0d 43693 | A ratio is zero iff the nu... |
| redivne0bd 43694 | The ratio of nonzero numbe... |
| rediveq1d 43695 | Equality in terms of unit ... |
| sn-rediv1d 43696 | A number divided by 1 is i... |
| sn-rediv0d 43697 | Division into zero is zero... |
| sn-redividd 43698 | A number divided by itself... |
| sn-rereccld 43699 | Closure law for reciprocal... |
| rerecne0d 43700 | The reciprocal of a nonzer... |
| rerecidd 43701 | Multiplication of a number... |
| rerecid2d 43702 | Multiplication of a number... |
| rerecrecd 43703 | A number is equal to the r... |
| redivrec2d 43704 | Relationship between divis... |
| rediv23d 43705 | A "commutative"/associativ... |
| redivdird 43706 | Distribution of division o... |
| rediv11d 43707 | One-to-one relationship fo... |
| sn-0tie0 43708 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 43709 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 43710 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 43711 | ~ ltaddneg without ~ ax-mu... |
| reposdif 43712 | Comparison of two numbers ... |
| relt0neg1 43713 | Comparison of a real and i... |
| relt0neg2 43714 | Comparison of a real and i... |
| sn-addlt0d 43715 | The sum of negative number... |
| sn-addgt0d 43716 | The sum of positive number... |
| sn-nnne0 43717 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 43718 | ~ elznn0nn restated using ... |
| nn0addcom 43719 | Addition is commutative fo... |
| zaddcomlem 43720 | Lemma for ~ zaddcom . (Co... |
| zaddcom 43721 | Addition is commutative fo... |
| renegmulnnass 43722 | Move multiplication by a n... |
| nn0mulcom 43723 | Multiplication is commutat... |
| zmulcomlem 43724 | Lemma for ~ zmulcom . (Co... |
| zmulcom 43725 | Multiplication is commutat... |
| mulgt0con1dlem 43726 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 43727 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 43728 | Lemma for ~ mulgt0b1d and ... |
| mulgt0b1d 43729 | Biconditional, deductive f... |
| sn-ltmul2d 43730 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 43731 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 43732 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 43733 | ~ ltp1 without ~ ax-mulcom... |
| sn-recgt0d 43734 | The reciprocal of a positi... |
| mulgt0b2d 43735 | Biconditional, deductive f... |
| sn-mulgt1d 43736 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 43737 | Negative one is a negative... |
| sn-reclt0d 43738 | The reciprocal of a negati... |
| mulltgt0d 43739 | Negative times positive is... |
| mullt0b1d 43740 | When the first term is neg... |
| mullt0b2d 43741 | When the second term is ne... |
| sn-mullt0d 43742 | The product of two negativ... |
| sn-msqgt0d 43743 | A nonzero square is positi... |
| sn-inelr 43744 | ~ inelr without ~ ax-mulco... |
| sn-itrere 43745 | ` _i ` times a real is rea... |
| sn-retire 43746 | Commuted version of ~ sn-i... |
| cnreeu 43747 | The reals in the expressio... |
| sn-sup2 43748 | ~ sup2 with exactly the sa... |
| sn-sup3d 43749 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 43750 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 43751 | ~ suprubd without ~ ax-mul... |
| sn-base0 43752 | Avoid axioms in ~ base0 by... |
| nelsubginvcld 43753 | The inverse of a non-subgr... |
| nelsubgcld 43754 | A non-subgroup-member plus... |
| nelsubgsubcld 43755 | A non-subgroup-member minu... |
| rnasclg 43756 | The set of injected scalar... |
| reldmfrlm 43757 | The domain of the free mod... |
| frlmbasfn 43758 | Elements of the free modul... |
| frlmvscl 43759 | Closure of scalar multipli... |
| frlmfielbas 43760 | The vectors of a finite fr... |
| frlmfzwrd 43761 | A vector of a module with ... |
| frlmfzowrd 43762 | A vector of a module with ... |
| frlmfzolen 43763 | The dimension of a vector ... |
| frlmfzowrdb 43764 | The vectors of a module wi... |
| frlmfzoccat 43765 | The concatenation of two v... |
| frlmvscadiccat 43766 | Scalar multiplication dist... |
| grpasscan2d 43767 | An associative cancellatio... |
| grpcominv1 43768 | If two elements commute, t... |
| grpcominv2 43769 | If two elements commute, t... |
| finsubmsubg 43770 | A submonoid of a finite gr... |
| opprmndb 43771 | A class is a monoid if and... |
| opprgrpb 43772 | A class is a group if and ... |
| opprablb 43773 | A class is an Abelian grou... |
| imacrhmcl 43774 | The image of a commutative... |
| riccrng1 43775 | Ring isomorphism preserves... |
| riccrng 43776 | A ring is commutative if a... |
| domnexpgn0cl 43777 | In a domain, a (nonnegativ... |
| drnginvrn0d 43778 | A multiplicative inverse i... |
| drngmullcan 43779 | Cancellation of a nonzero ... |
| drngmulrcan 43780 | Cancellation of a nonzero ... |
| drnginvmuld 43781 | Inverse of a nonzero produ... |
| lmod0vsd 43782 | Zero times a vector is the... |
| lmodvs0d 43783 | A scalar times the zero ve... |
| lmodvsassd 43784 | In a module, scalar multip... |
| ricdrng1 43785 | A ring isomorphic to a div... |
| ricdrng 43786 | A ring is a division ring ... |
| ricfld 43787 | A ring is a field if and o... |
| asclf1 43788 | Two ways of saying the sca... |
| abvexp 43789 | Move exponentiation in and... |
| fimgmcyclem 43790 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 43791 | Version of ~ odcl2 for fin... |
| fidomncyc 43792 | Version of ~ odcl2 for mul... |
| fiabv 43793 | In a finite domain (a fini... |
| lvecgrp 43794 | A vector space is a group.... |
| lvecring 43795 | The scalar component of a ... |
| frlm0vald 43796 | All coordinates of the zer... |
| frlmsnic 43797 | Given a free module with a... |
| uvccl 43798 | A unit vector is a vector.... |
| uvcn0 43799 | A unit vector is nonzero. ... |
| psrmnd 43800 | The ring of power series i... |
| mhmcopsr 43801 | The composition of a monoi... |
| mhmcoaddpsr 43802 | Show that the ring homomor... |
| rhmcomulpsr 43803 | Show that the ring homomor... |
| rhmpsr 43804 | Provide a ring homomorphis... |
| rhmpsr1 43805 | Provide a ring homomorphis... |
| evl0 43806 | The zero polynomial evalua... |
| evlsbagval 43807 | Polynomial evaluation buil... |
| evlvvvallem 43808 | Lemma for theorems using ~... |
| evlselvlem 43809 | Lemma for ~ evlselv . Use... |
| evlselv 43810 | Evaluating a selection of ... |
| fsuppind 43811 | Induction on functions ` F... |
| fsuppssindlem1 43812 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 43813 | Lemma for ~ fsuppssind . ... |
| fsuppssind 43814 | Induction on functions ` F... |
| mhpind 43815 | The homogeneous polynomial... |
| evlsmhpvvval 43816 | Give a formula for the eva... |
| mhphflem 43817 | Lemma for ~ mhphf . Add s... |
| mhphf 43818 | A homogeneous polynomial d... |
| mhphf2 43819 | A homogeneous polynomial d... |
| mhphf3 43820 | A homogeneous polynomial d... |
| mhphf4 43821 | A homogeneous polynomial d... |
| prjspval 43824 | Value of the projective sp... |
| prjsprel 43825 | Utility theorem regarding ... |
| prjspertr 43826 | The relation in ` PrjSp ` ... |
| prjsperref 43827 | The relation in ` PrjSp ` ... |
| prjspersym 43828 | The relation in ` PrjSp ` ... |
| prjsper 43829 | The relation used to defin... |
| prjspreln0 43830 | Two nonzero vectors are eq... |
| prjspvs 43831 | A nonzero multiple of a ve... |
| prjsprellsp 43832 | Two vectors are equivalent... |
| prjspeclsp 43833 | The vectors equivalent to ... |
| prjspval2 43834 | Alternate definition of pr... |
| prjspnval 43837 | Value of the n-dimensional... |
| prjspnerlem 43838 | A lemma showing that the e... |
| prjspnval2 43839 | Value of the n-dimensional... |
| prjspner 43840 | The relation used to defin... |
| prjspnvs 43841 | A nonzero multiple of a ve... |
| prjspnssbas 43842 | A projective space is a se... |
| elprjspnss 43843 | A point of a projective sp... |
| prjspnn0 43844 | A projective point is none... |
| frlmnzcoordex 43845 | Every nonzero vector of a ... |
| frlmnzcoordval 43846 | Value of ` J ` , a functio... |
| frlmnzcoordinf 43847 | The result ` ( J `` V ) ` ... |
| frlmnzcoordcl 43848 | Closure of the first nonze... |
| frlmnzcoordcl2 43849 | The first nonzero coordina... |
| frlmnzcoordn0 43850 | The function ` J ` gives a... |
| frlmnzcoordsca 43851 | In a free module, scaling ... |
| prjspnnormval 43852 | Value of ` F ` , a functio... |
| prjspnequivnorm 43853 | In a free module, a nonzer... |
| prjspnnorm 43854 | In a free module, two nonz... |
| 0prjspnlem 43855 | Lemma for ~ 0prjspn . The... |
| 0prjspnrel 43856 | In a zero-dimensional proj... |
| 0prjspn 43857 | A zero-dimensional project... |
| prjcrvfval 43860 | Value of the projective cu... |
| prjcrvval 43861 | Value of the projective cu... |
| prjcrv0 43862 | The "curve" (zero set) cor... |
| dffltz 43863 | Fermat's Last Theorem (FLT... |
| fltmul 43864 | A counterexample to FLT st... |
| fltltc 43865 | ` ( C ^ N ) ` is the large... |
| fltnltalem 43866 | Lemma for ~ fltnlta . A l... |
| fltnlta 43867 | In a Fermat counterexample... |
| iddii 43868 | Version of ~ a1ii with the... |
| bicomdALT 43869 | Alternate proof of ~ bicom... |
| alan 43870 | Alias for ~ 19.26 for easi... |
| exor 43871 | Alias for ~ 19.43 for easi... |
| rexor 43872 | Alias for ~ r19.43 for eas... |
| ruvALT 43873 | Alternate proof of ~ ruv w... |
| sn-wcdeq 43874 | Alternative to ~ wcdeq and... |
| sq45 43875 | 45 squared is 2025. (Cont... |
| sum9cubes 43876 | The sum of the first nine ... |
| sn-isghm 43877 | Longer proof of ~ isghm , ... |
| aprilfools2025 43878 | An abuse of notation. (Co... |
| nfa1w 43879 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 43880 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 43881 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 43882 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 43883 | Replace ~ ax-10 , ~ ax-11 ... |
| cu3addd 43884 | Cube of sum of three numbe... |
| negexpidd 43885 | The sum of a real number t... |
| rexlimdv3d 43886 | An extended version of ~ r... |
| 3cubeslem1 43887 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 43888 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 43889 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 43890 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 43891 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 43892 | Lemma for ~ 3cubes . This... |
| 3cubes 43893 | Every rational number is a... |
| omexcn 43894 | The axioms of the complex ... |
| rntrclfvOAI 43895 | The range of the transitiv... |
| moxfr 43896 | Transfer at-most-one betwe... |
| imaiinfv 43897 | Indexed intersection of an... |
| elrfi 43898 | Elementhood in a set of re... |
| elrfirn 43899 | Elementhood in a set of re... |
| elrfirn2 43900 | Elementhood in a set of re... |
| cmpfiiin 43901 | In a compact topology, a s... |
| ismrcd1 43902 | Any function from the subs... |
| ismrcd2 43903 | Second half of ~ ismrcd1 .... |
| istopclsd 43904 | A closure function which s... |
| ismrc 43905 | A function is a Moore clos... |
| isnacs 43908 | Expand definition of Noeth... |
| nacsfg 43909 | In a Noetherian-type closu... |
| isnacs2 43910 | Express Noetherian-type cl... |
| mrefg2 43911 | Slight variation on finite... |
| mrefg3 43912 | Slight variation on finite... |
| nacsacs 43913 | A closure system of Noethe... |
| isnacs3 43914 | A choice-free order equiva... |
| incssnn0 43915 | Transitivity induction of ... |
| nacsfix 43916 | An increasing sequence of ... |
| constmap 43917 | A constant (represented wi... |
| mapco2g 43918 | Renaming indices in a tupl... |
| mapco2 43919 | Post-composition (renaming... |
| mapfzcons 43920 | Extending a one-based mapp... |
| mapfzcons1 43921 | Recover prefix mapping fro... |
| mapfzcons1cl 43922 | A nonempty mapping has a p... |
| mapfzcons2 43923 | Recover added element from... |
| mptfcl 43924 | Interpret range of a maps-... |
| mzpclval 43929 | Substitution lemma for ` m... |
| elmzpcl 43930 | Double substitution lemma ... |
| mzpclall 43931 | The set of all functions w... |
| mzpcln0 43932 | Corollary of ~ mzpclall : ... |
| mzpcl1 43933 | Defining property 1 of a p... |
| mzpcl2 43934 | Defining property 2 of a p... |
| mzpcl34 43935 | Defining properties 3 and ... |
| mzpval 43936 | Value of the ` mzPoly ` fu... |
| dmmzp 43937 | ` mzPoly ` is defined for ... |
| mzpincl 43938 | Polynomial closedness is a... |
| mzpconst 43939 | Constant functions are pol... |
| mzpf 43940 | A polynomial function is a... |
| mzpproj 43941 | A projection function is p... |
| mzpadd 43942 | The pointwise sum of two p... |
| mzpmul 43943 | The pointwise product of t... |
| mzpconstmpt 43944 | A constant function expres... |
| mzpaddmpt 43945 | Sum of polynomial function... |
| mzpmulmpt 43946 | Product of polynomial func... |
| mzpsubmpt 43947 | The difference of two poly... |
| mzpnegmpt 43948 | Negation of a polynomial f... |
| mzpexpmpt 43949 | Raise a polynomial functio... |
| mzpindd 43950 | "Structural" induction to ... |
| mzpmfp 43951 | Relationship between multi... |
| mzpsubst 43952 | Substituting polynomials f... |
| mzprename 43953 | Simplified version of ~ mz... |
| mzpresrename 43954 | A polynomial is a polynomi... |
| mzpcompact2lem 43955 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 43956 | Polynomials are finitary o... |
| coeq0i 43957 | ~ coeq0 but without explic... |
| fzsplit1nn0 43958 | Split a finite 1-based set... |
| eldiophb 43961 | Initial expression of Diop... |
| eldioph 43962 | Condition for a set to be ... |
| diophrw 43963 | Renaming and adding unused... |
| eldioph2lem1 43964 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 43965 | Lemma for ~ eldioph2 . Co... |
| eldioph2 43966 | Construct a Diophantine se... |
| eldioph2b 43967 | While Diophantine sets wer... |
| eldiophelnn0 43968 | Remove antecedent on ` B `... |
| eldioph3b 43969 | Define Diophantine sets in... |
| eldioph3 43970 | Inference version of ~ eld... |
| ellz1 43971 | Membership in a lower set ... |
| lzunuz 43972 | The union of a lower set o... |
| fz1eqin 43973 | Express a one-based finite... |
| lzenom 43974 | Lower integers are countab... |
| elmapresaunres2 43975 | ~ fresaunres2 transposed t... |
| diophin 43976 | If two sets are Diophantin... |
| diophun 43977 | If two sets are Diophantin... |
| eldiophss 43978 | Diophantine sets are sets ... |
| diophrex 43979 | Projecting a Diophantine s... |
| eq0rabdioph 43980 | This is the first of a num... |
| eqrabdioph 43981 | Diophantine set builder fo... |
| 0dioph 43982 | The empty set is Diophanti... |
| vdioph 43983 | The "universal" set (as la... |
| anrabdioph 43984 | Diophantine set builder fo... |
| orrabdioph 43985 | Diophantine set builder fo... |
| 3anrabdioph 43986 | Diophantine set builder fo... |
| 3orrabdioph 43987 | Diophantine set builder fo... |
| 2sbcrex 43988 | Exchange an existential qu... |
| sbc2rex 43989 | Exchange a substitution wi... |
| sbc4rex 43990 | Exchange a substitution wi... |
| sbcrot3 43991 | Rotate a sequence of three... |
| sbcrot5 43992 | Rotate a sequence of five ... |
| sbccomieg 43993 | Commute two explicit subst... |
| rexrabdioph 43994 | Diophantine set builder fo... |
| rexfrabdioph 43995 | Diophantine set builder fo... |
| 2rexfrabdioph 43996 | Diophantine set builder fo... |
| 3rexfrabdioph 43997 | Diophantine set builder fo... |
| 4rexfrabdioph 43998 | Diophantine set builder fo... |
| 6rexfrabdioph 43999 | Diophantine set builder fo... |
| 7rexfrabdioph 44000 | Diophantine set builder fo... |
| rabdiophlem1 44001 | Lemma for arithmetic dioph... |
| rabdiophlem2 44002 | Lemma for arithmetic dioph... |
| elnn0rabdioph 44003 | Diophantine set builder fo... |
| rexzrexnn0 44004 | Rewrite an existential qua... |
| lerabdioph 44005 | Diophantine set builder fo... |
| eluzrabdioph 44006 | Diophantine set builder fo... |
| elnnrabdioph 44007 | Diophantine set builder fo... |
| ltrabdioph 44008 | Diophantine set builder fo... |
| nerabdioph 44009 | Diophantine set builder fo... |
| dvdsrabdioph 44010 | Divisibility is a Diophant... |
| eldioph4b 44011 | Membership in ` Dioph ` ex... |
| eldioph4i 44012 | Forward-only version of ~ ... |
| diophren 44013 | Change variables in a Diop... |
| rabrenfdioph 44014 | Change variable numbers in... |
| rabren3dioph 44015 | Change variable numbers in... |
| fphpd 44016 | Pigeonhole principle expre... |
| fphpdo 44017 | Pigeonhole principle for s... |
| ctbnfien 44018 | An infinite subset of a co... |
| fiphp3d 44019 | Infinite pigeonhole princi... |
| rencldnfilem 44020 | Lemma for ~ rencldnfi . (... |
| rencldnfi 44021 | A set of real numbers whic... |
| irrapxlem1 44022 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 44023 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 44024 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 44025 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 44026 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 44027 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 44028 | Dirichlet's approximation ... |
| pellexlem1 44029 | Lemma for ~ pellex . Arit... |
| pellexlem2 44030 | Lemma for ~ pellex . Arit... |
| pellexlem3 44031 | Lemma for ~ pellex . To e... |
| pellexlem4 44032 | Lemma for ~ pellex . Invo... |
| pellexlem5 44033 | Lemma for ~ pellex . Invo... |
| pellexlem6 44034 | Lemma for ~ pellex . Doin... |
| pellex 44035 | Every Pell equation has a ... |
| pell1qrval 44046 | Value of the set of first-... |
| elpell1qr 44047 | Membership in a first-quad... |
| pell14qrval 44048 | Value of the set of positi... |
| elpell14qr 44049 | Membership in the set of p... |
| pell1234qrval 44050 | Value of the set of genera... |
| elpell1234qr 44051 | Membership in the set of g... |
| pell1234qrre 44052 | General Pell solutions are... |
| pell1234qrne0 44053 | No solution to a Pell equa... |
| pell1234qrreccl 44054 | General solutions of the P... |
| pell1234qrmulcl 44055 | General solutions of the P... |
| pell14qrss1234 44056 | A positive Pell solution i... |
| pell14qrre 44057 | A positive Pell solution i... |
| pell14qrne0 44058 | A positive Pell solution i... |
| pell14qrgt0 44059 | A positive Pell solution i... |
| pell14qrrp 44060 | A positive Pell solution i... |
| pell1234qrdich 44061 | A general Pell solution is... |
| elpell14qr2 44062 | A number is a positive Pel... |
| pell14qrmulcl 44063 | Positive Pell solutions ar... |
| pell14qrreccl 44064 | Positive Pell solutions ar... |
| pell14qrdivcl 44065 | Positive Pell solutions ar... |
| pell14qrexpclnn0 44066 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 44067 | Positive Pell solutions ar... |
| pell1qrss14 44068 | First-quadrant Pell soluti... |
| pell14qrdich 44069 | A positive Pell solution i... |
| pell1qrge1 44070 | A Pell solution in the fir... |
| pell1qr1 44071 | 1 is a Pell solution and i... |
| elpell1qr2 44072 | The first quadrant solutio... |
| pell1qrgaplem 44073 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 44074 | First-quadrant Pell soluti... |
| pell14qrgap 44075 | Positive Pell solutions ar... |
| pell14qrgapw 44076 | Positive Pell solutions ar... |
| pellqrexplicit 44077 | Condition for a calculated... |
| infmrgelbi 44078 | Any lower bound of a nonem... |
| pellqrex 44079 | There is a nontrivial solu... |
| pellfundval 44080 | Value of the fundamental s... |
| pellfundre 44081 | The fundamental solution o... |
| pellfundge 44082 | Lower bound on the fundame... |
| pellfundgt1 44083 | Weak lower bound on the Pe... |
| pellfundlb 44084 | A nontrivial first quadran... |
| pellfundglb 44085 | If a real is larger than t... |
| pellfundex 44086 | The fundamental solution a... |
| pellfund14gap 44087 | There are no solutions bet... |
| pellfundrp 44088 | The fundamental Pell solut... |
| pellfundne1 44089 | The fundamental Pell solut... |
| reglogcl 44090 | General logarithm is a rea... |
| reglogltb 44091 | General logarithm preserve... |
| reglogleb 44092 | General logarithm preserve... |
| reglogmul 44093 | Multiplication law for gen... |
| reglogexp 44094 | Power law for general log.... |
| reglogbas 44095 | General log of the base is... |
| reglog1 44096 | General log of 1 is 0. (C... |
| reglogexpbas 44097 | General log of a power of ... |
| pellfund14 44098 | Every positive Pell soluti... |
| pellfund14b 44099 | The positive Pell solution... |
| rmxfval 44104 | Value of the X sequence. ... |
| rmyfval 44105 | Value of the Y sequence. ... |
| rmspecsqrtnq 44106 | The discriminant used to d... |
| rmspecnonsq 44107 | The discriminant used to d... |
| qirropth 44108 | This lemma implements the ... |
| rmspecfund 44109 | The base of exponent used ... |
| rmxyelqirr 44110 | The solutions used to cons... |
| rmxypairf1o 44111 | The function used to extra... |
| rmxyelxp 44112 | Lemma for ~ frmx and ~ frm... |
| frmx 44113 | The X sequence is a nonneg... |
| frmy 44114 | The Y sequence is an integ... |
| rmxyval 44115 | Main definition of the X a... |
| rmspecpos 44116 | The discriminant used to d... |
| rmxycomplete 44117 | The X and Y sequences take... |
| rmxynorm 44118 | The X and Y sequences defi... |
| rmbaserp 44119 | The base of exponentiation... |
| rmxyneg 44120 | Negation law for X and Y s... |
| rmxyadd 44121 | Addition formula for X and... |
| rmxy1 44122 | Value of the X and Y seque... |
| rmxy0 44123 | Value of the X and Y seque... |
| rmxneg 44124 | Negation law (even functio... |
| rmx0 44125 | Value of X sequence at 0. ... |
| rmx1 44126 | Value of X sequence at 1. ... |
| rmxadd 44127 | Addition formula for X seq... |
| rmyneg 44128 | Negation formula for Y seq... |
| rmy0 44129 | Value of Y sequence at 0. ... |
| rmy1 44130 | Value of Y sequence at 1. ... |
| rmyadd 44131 | Addition formula for Y seq... |
| rmxp1 44132 | Special addition-of-1 form... |
| rmyp1 44133 | Special addition of 1 form... |
| rmxm1 44134 | Subtraction of 1 formula f... |
| rmym1 44135 | Subtraction of 1 formula f... |
| rmxluc 44136 | The X sequence is a Lucas ... |
| rmyluc 44137 | The Y sequence is a Lucas ... |
| rmyluc2 44138 | Lucas sequence property of... |
| rmxdbl 44139 | "Double-angle formula" for... |
| rmydbl 44140 | "Double-angle formula" for... |
| monotuz 44141 | A function defined on an u... |
| monotoddzzfi 44142 | A function which is odd an... |
| monotoddzz 44143 | A function (given implicit... |
| oddcomabszz 44144 | An odd function which take... |
| 2nn0ind 44145 | Induction on nonnegative i... |
| zindbi 44146 | Inductively transfer a pro... |
| rmxypos 44147 | For all nonnegative indice... |
| ltrmynn0 44148 | The Y-sequence is strictly... |
| ltrmxnn0 44149 | The X-sequence is strictly... |
| lermxnn0 44150 | The X-sequence is monotoni... |
| rmxnn 44151 | The X-sequence is defined ... |
| ltrmy 44152 | The Y-sequence is strictly... |
| rmyeq0 44153 | Y is zero only at zero. (... |
| rmyeq 44154 | Y is one-to-one. (Contrib... |
| lermy 44155 | Y is monotonic (non-strict... |
| rmynn 44156 | ` rmY ` is positive for po... |
| rmynn0 44157 | ` rmY ` is nonnegative for... |
| rmyabs 44158 | ` rmY ` commutes with ` ab... |
| jm2.24nn 44159 | X(n) is strictly greater t... |
| jm2.17a 44160 | First half of lemma 2.17 o... |
| jm2.17b 44161 | Weak form of the second ha... |
| jm2.17c 44162 | Second half of lemma 2.17 ... |
| jm2.24 44163 | Lemma 2.24 of [JonesMatija... |
| rmygeid 44164 | Y(n) increases faster than... |
| congtr 44165 | A wff of the form ` A || (... |
| congadd 44166 | If two pairs of numbers ar... |
| congmul 44167 | If two pairs of numbers ar... |
| congsym 44168 | Congruence mod ` A ` is a ... |
| congneg 44169 | If two integers are congru... |
| congsub 44170 | If two pairs of numbers ar... |
| congid 44171 | Every integer is congruent... |
| mzpcong 44172 | Polynomials commute with c... |
| congrep 44173 | Every integer is congruent... |
| congabseq 44174 | If two integers are congru... |
| acongid 44175 | A wff like that in this th... |
| acongsym 44176 | Symmetry of alternating co... |
| acongneg2 44177 | Negate right side of alter... |
| acongtr 44178 | Transitivity of alternatin... |
| acongeq12d 44179 | Substitution deduction for... |
| acongrep 44180 | Every integer is alternati... |
| fzmaxdif 44181 | Bound on the difference be... |
| fzneg 44182 | Reflection of a finite ran... |
| acongeq 44183 | Two numbers in the fundame... |
| dvdsacongtr 44184 | Alternating congruence pas... |
| coprmdvdsb 44185 | Multiplication by a coprim... |
| modabsdifz 44186 | Divisibility in terms of m... |
| dvdsabsmod0 44187 | Divisibility in terms of m... |
| jm2.18 44188 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 44189 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 44190 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 44191 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 44192 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 44193 | Lemma 2.19 of [JonesMatija... |
| jm2.21 44194 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 44195 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 44196 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 44197 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 44198 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 44199 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 44200 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 44201 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 44202 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 44203 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 44204 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 44205 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 44206 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 44207 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 44208 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 44209 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 44210 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 44211 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 44212 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 44213 | Lemma for ~ rmydioph . Us... |
| rmydioph 44214 | ~ jm2.27 restated in terms... |
| rmxdiophlem 44215 | X can be expressed in term... |
| rmxdioph 44216 | X is a Diophantine functio... |
| jm3.1lem1 44217 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 44218 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 44219 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 44220 | Diophantine expression for... |
| expdiophlem1 44221 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 44222 | Lemma for ~ expdioph . Ex... |
| expdioph 44223 | The exponential function i... |
| setindtr 44224 | Set induction for sets con... |
| setindtrs 44225 | Set induction scheme witho... |
| dford3lem1 44226 | Lemma for ~ dford3 . (Con... |
| dford3lem2 44227 | Lemma for ~ dford3 . (Con... |
| dford3 44228 | Ordinals are precisely the... |
| dford4 44229 | ~ dford3 expressed in prim... |
| wopprc 44230 | Unrelated: Wiener pairs t... |
| rpnnen3lem 44231 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 44232 | Dedekind cut injection of ... |
| axac10 44233 | Characterization of choice... |
| harinf 44234 | The Hartogs number of an i... |
| wdom2d2 44235 | Deduction for weak dominan... |
| ttac 44236 | Tarski's theorem about cho... |
| pw2f1ocnv 44237 | Define a bijection between... |
| pw2f1o2 44238 | Define a bijection between... |
| pw2f1o2val 44239 | Function value of the ~ pw... |
| pw2f1o2val2 44240 | Membership in a mapped set... |
| limsuc2 44241 | Limit ordinals in the sens... |
| wepwsolem 44242 | Transfer an ordering on ch... |
| wepwso 44243 | A well-ordering induces a ... |
| dnnumch1 44244 | Define an enumeration of a... |
| dnnumch2 44245 | Define an enumeration (wea... |
| dnnumch3lem 44246 | Value of the ordinal injec... |
| dnnumch3 44247 | Define an injection from a... |
| dnwech 44248 | Define a well-ordering fro... |
| aomclem1 44249 | Lemma for ~ dfac11 . This... |
| aomclem2 44250 | Lemma for ~ dfac11 . Succ... |
| aomclem3 44251 | Lemma for ~ dfac11 . Succ... |
| aomclem4 44252 | Lemma for ~ dfac11 . Limi... |
| aomclem5 44253 | Lemma for ~ dfac11 . Comb... |
| aomclem6 44254 | Lemma for ~ dfac11 . Tran... |
| aomclem7 44255 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 44256 | Lemma for ~ dfac11 . Perf... |
| dfac11 44257 | The right-hand side of thi... |
| kelac1 44258 | Kelley's choice, basic for... |
| kelac2lem 44259 | Lemma for ~ kelac2 and ~ d... |
| kelac2 44260 | Kelley's choice, most comm... |
| dfac21 44261 | Tychonoff's theorem is a c... |
| islmodfg 44264 | Property of a finitely gen... |
| islssfg 44265 | Property of a finitely gen... |
| islssfg2 44266 | Property of a finitely gen... |
| islssfgi 44267 | Finitely spanned subspaces... |
| fglmod 44268 | Finitely generated left mo... |
| lsmfgcl 44269 | The sum of two finitely ge... |
| islnm 44272 | Property of being a Noethe... |
| islnm2 44273 | Property of being a Noethe... |
| lnmlmod 44274 | A Noetherian left module i... |
| lnmlssfg 44275 | A submodule of Noetherian ... |
| lnmlsslnm 44276 | All submodules of a Noethe... |
| lnmfg 44277 | A Noetherian left module i... |
| kercvrlsm 44278 | The domain of a linear fun... |
| lmhmfgima 44279 | A homomorphism maps finite... |
| lnmepi 44280 | Epimorphic images of Noeth... |
| lmhmfgsplit 44281 | If the kernel and range of... |
| lmhmlnmsplit 44282 | If the kernel and range of... |
| lnmlmic 44283 | Noetherian is an invariant... |
| pwssplit4 44284 | Splitting for structure po... |
| filnm 44285 | Finite left modules are No... |
| pwslnmlem0 44286 | Zeroeth powers are Noether... |
| pwslnmlem1 44287 | First powers are Noetheria... |
| pwslnmlem2 44288 | A sum of powers is Noether... |
| pwslnm 44289 | Finite powers of Noetheria... |
| unxpwdom3 44290 | Weaker version of ~ unxpwd... |
| pwfi2f1o 44291 | The ~ pw2f1o bijection rel... |
| pwfi2en 44292 | Finitely supported indicat... |
| frlmpwfi 44293 | Formal linear combinations... |
| gicabl 44294 | Being Abelian is a group i... |
| isnumbasgrplem1 44295 | A set which is equipollent... |
| harn0 44296 | The Hartogs number of a se... |
| numinfctb 44297 | A numerable infinite set c... |
| isnumbasgrplem2 44298 | If the (to be thought of a... |
| isnumbasgrplem3 44299 | Every nonempty numerable s... |
| isnumbasabl 44300 | A set is numerable iff it ... |
| isnumbasgrp 44301 | A set is numerable iff it ... |
| dfacbasgrp 44302 | A choice equivalent in abs... |
| islnr 44305 | Property of a left-Noether... |
| lnrring 44306 | Left-Noetherian rings are ... |
| lnrlnm 44307 | Left-Noetherian rings have... |
| islnr2 44308 | Property of being a left-N... |
| islnr3 44309 | Relate left-Noetherian rin... |
| lnr2i 44310 | Given an ideal in a left-N... |
| lpirlnr 44311 | Left principal ideal rings... |
| lnrfrlm 44312 | Finite-dimensional free mo... |
| lnrfg 44313 | Finitely-generated modules... |
| lnrfgtr 44314 | A submodule of a finitely ... |
| hbtlem1 44317 | Value of the leading coeff... |
| hbtlem2 44318 | Leading coefficient ideals... |
| hbtlem7 44319 | Functionality of leading c... |
| hbtlem4 44320 | The leading ideal function... |
| hbtlem3 44321 | The leading ideal function... |
| hbtlem5 44322 | The leading ideal function... |
| hbtlem6 44323 | There is a finite set of p... |
| hbt 44324 | The Hilbert Basis Theorem ... |
| dgrsub2 44329 | Subtracting two polynomial... |
| elmnc 44330 | Property of a monic polyno... |
| mncply 44331 | A monic polynomial is a po... |
| mnccoe 44332 | A monic polynomial has lea... |
| mncn0 44333 | A monic polynomial is not ... |
| dgraaval 44338 | Value of the degree functi... |
| dgraalem 44339 | Properties of the degree o... |
| dgraacl 44340 | Closure of the degree func... |
| dgraaf 44341 | Degree function on algebra... |
| dgraaub 44342 | Upper bound on degree of a... |
| dgraa0p 44343 | A rational polynomial of d... |
| mpaaeu 44344 | An algebraic number has ex... |
| mpaaval 44345 | Value of the minimal polyn... |
| mpaalem 44346 | Properties of the minimal ... |
| mpaacl 44347 | Minimal polynomial is a po... |
| mpaadgr 44348 | Minimal polynomial has deg... |
| mpaaroot 44349 | The minimal polynomial of ... |
| mpaamn 44350 | Minimal polynomial is moni... |
| itgoval 44355 | Value of the integral-over... |
| aaitgo 44356 | The standard algebraic num... |
| itgoss 44357 | An integral element is int... |
| itgocn 44358 | All integral elements are ... |
| cnsrexpcl 44359 | Exponentiation is closed i... |
| fsumcnsrcl 44360 | Finite sums are closed in ... |
| cnsrplycl 44361 | Polynomials are closed in ... |
| rgspnid 44362 | The span of a subring is i... |
| rngunsnply 44363 | Adjoining one element to a... |
| flcidc 44364 | Finite linear combinations... |
| algstr 44367 | Lemma to shorten proofs of... |
| algbase 44368 | The base set of a construc... |
| algaddg 44369 | The additive operation of ... |
| algmulr 44370 | The multiplicative operati... |
| algsca 44371 | The set of scalars of a co... |
| algvsca 44372 | The scalar product operati... |
| mendval 44373 | Value of the module endomo... |
| mendbas 44374 | Base set of the module end... |
| mendplusgfval 44375 | Addition in the module end... |
| mendplusg 44376 | A specific addition in the... |
| mendmulrfval 44377 | Multiplication in the modu... |
| mendmulr 44378 | A specific multiplication ... |
| mendsca 44379 | The module endomorphism al... |
| mendvscafval 44380 | Scalar multiplication in t... |
| mendvsca 44381 | A specific scalar multipli... |
| mendring 44382 | The module endomorphism al... |
| mendlmod 44383 | The module endomorphism al... |
| mendassa 44384 | The module endomorphism al... |
| idomodle 44385 | Limit on the number of ` N... |
| fiuneneq 44386 | Two finite sets of equal s... |
| idomsubgmo 44387 | The units of an integral d... |
| proot1mul 44388 | Any primitive ` N ` -th ro... |
| proot1hash 44389 | If an integral domain has ... |
| proot1ex 44390 | The complex field has prim... |
| mon1psubm 44393 | Monic polynomials are a mu... |
| deg1mhm 44394 | Homomorphic property of th... |
| cytpfn 44395 | Functionality of the cyclo... |
| cytpval 44396 | Substitutions for the Nth ... |
| fgraphopab 44397 | Express a function as a su... |
| fgraphxp 44398 | Express a function as a su... |
| hausgraph 44399 | The graph of a continuous ... |
| r1sssucd 44404 | Deductive form of ~ r1sssu... |
| iocunico 44405 | Split an open interval int... |
| iocinico 44406 | The intersection of two se... |
| iocmbl 44407 | An open-below, closed-abov... |
| cnioobibld 44408 | A bounded, continuous func... |
| arearect 44409 | The area of a rectangle wh... |
| areaquad 44410 | The area of a quadrilatera... |
| uniel 44411 | Two ways to say a union is... |
| unielss 44412 | Two ways to say the union ... |
| unielid 44413 | Two ways to say the union ... |
| ssunib 44414 | Two ways to say a class is... |
| rp-intrabeq 44415 | Equality theorem for supre... |
| rp-unirabeq 44416 | Equality theorem for infim... |
| onmaxnelsup 44417 | Two ways to say the maximu... |
| onsupneqmaxlim0 44418 | If the supremum of a class... |
| onsupcl2 44419 | The supremum of a set of o... |
| onuniintrab 44420 | The union of a set of ordi... |
| onintunirab 44421 | The intersection of a non-... |
| onsupnmax 44422 | If the union of a class of... |
| onsupuni 44423 | The supremum of a set of o... |
| onsupuni2 44424 | The supremum of a set of o... |
| onsupintrab 44425 | The supremum of a set of o... |
| onsupintrab2 44426 | The supremum of a set of o... |
| onsupcl3 44427 | The supremum of a set of o... |
| onsupex3 44428 | The supremum of a set of o... |
| onuniintrab2 44429 | The union of a set of ordi... |
| oninfint 44430 | The infimum of a non-empty... |
| oninfunirab 44431 | The infimum of a non-empty... |
| oninfcl2 44432 | The infimum of a non-empty... |
| onsupmaxb 44433 | The union of a class of or... |
| onexgt 44434 | For any ordinal, there is ... |
| onexomgt 44435 | For any ordinal, there is ... |
| omlimcl2 44436 | The product of a limit ord... |
| onexlimgt 44437 | For any ordinal, there is ... |
| onexoegt 44438 | For any ordinal, there is ... |
| oninfex2 44439 | The infimum of a non-empty... |
| onsupeqmax 44440 | Condition when the supremu... |
| onsupeqnmax 44441 | Condition when the supremu... |
| onsuplub 44442 | The supremum of a set of o... |
| onsupnub 44443 | An upper bound of a set of... |
| onfisupcl 44444 | Sufficient condition when ... |
| onelord 44445 | Every element of a ordinal... |
| onepsuc 44446 | Every ordinal is less than... |
| epsoon 44447 | The ordinals are strictly ... |
| epirron 44448 | The strict order on the or... |
| oneptr 44449 | The strict order on the or... |
| oneltr 44450 | The elementhood relation o... |
| oneptri 44451 | The strict, complete (line... |
| ordeldif 44452 | Membership in the differen... |
| ordeldifsucon 44453 | Membership in the differen... |
| ordeldif1o 44454 | Membership in the differen... |
| ordne0gt0 44455 | Ordinal zero is less than ... |
| ondif1i 44456 | Ordinal zero is less than ... |
| onsucelab 44457 | The successor of every ord... |
| dflim6 44458 | A limit ordinal is a nonze... |
| limnsuc 44459 | A limit ordinal is not an ... |
| onsucss 44460 | If one ordinal is less tha... |
| ordnexbtwnsuc 44461 | For any distinct pair of o... |
| orddif0suc 44462 | For any distinct pair of o... |
| onsucf1lem 44463 | For ordinals, the successo... |
| onsucf1olem 44464 | The successor operation is... |
| onsucrn 44465 | The successor operation is... |
| onsucf1o 44466 | The successor operation is... |
| dflim7 44467 | A limit ordinal is a nonze... |
| onov0suclim 44468 | Compactly express rules fo... |
| oa0suclim 44469 | Closed form expression of ... |
| om0suclim 44470 | Closed form expression of ... |
| oe0suclim 44471 | Closed form expression of ... |
| oaomoecl 44472 | The operations of addition... |
| onsupsucismax 44473 | If the union of a set of o... |
| onsssupeqcond 44474 | If for every element of a ... |
| limexissup 44475 | An ordinal which is a limi... |
| limiun 44476 | A limit ordinal is the uni... |
| limexissupab 44477 | An ordinal which is a limi... |
| om1om1r 44478 | Ordinal one is both a left... |
| oe0rif 44479 | Ordinal zero raised to any... |
| oasubex 44480 | While subtraction can't be... |
| nnamecl 44481 | Natural numbers are closed... |
| onsucwordi 44482 | The successor operation pr... |
| oalim2cl 44483 | The ordinal sum of any ord... |
| oaltublim 44484 | Given ` C ` is a limit ord... |
| oaordi3 44485 | Ordinal addition of the sa... |
| oaord3 44486 | When the same ordinal is a... |
| 1oaomeqom 44487 | Ordinal one plus omega is ... |
| oaabsb 44488 | The right addend absorbs t... |
| oaordnrex 44489 | When omega is added on the... |
| oaordnr 44490 | When the same ordinal is a... |
| omge1 44491 | Any nonzero ordinal produc... |
| omge2 44492 | Any nonzero ordinal produc... |
| omlim2 44493 | The nonzero product with a... |
| omord2lim 44494 | Given a limit ordinal, the... |
| omord2i 44495 | Ordinal multiplication of ... |
| omord2com 44496 | When the same nonzero ordi... |
| 2omomeqom 44497 | Ordinal two times omega is... |
| omnord1ex 44498 | When omega is multiplied o... |
| omnord1 44499 | When the same nonzero ordi... |
| oege1 44500 | Any nonzero ordinal power ... |
| oege2 44501 | Any power of an ordinal at... |
| rp-oelim2 44502 | The power of an ordinal at... |
| oeord2lim 44503 | Given a limit ordinal, the... |
| oeord2i 44504 | Ordinal exponentiation of ... |
| oeord2com 44505 | When the same base at leas... |
| nnoeomeqom 44506 | Any natural number at leas... |
| df3o2 44507 | Ordinal 3 is the unordered... |
| df3o3 44508 | Ordinal 3, fully expanded.... |
| oenord1ex 44509 | When ordinals two and thre... |
| oenord1 44510 | When two ordinals (both at... |
| oaomoencom 44511 | Ordinal addition, multipli... |
| oenassex 44512 | Ordinal two raised to two ... |
| oenass 44513 | Ordinal exponentiation is ... |
| cantnftermord 44514 | For terms of the form of a... |
| cantnfub 44515 | Given a finite number of t... |
| cantnfub2 44516 | Given a finite number of t... |
| bropabg 44517 | Equivalence for two classe... |
| cantnfresb 44518 | A Cantor normal form which... |
| cantnf2 44519 | For every ordinal, ` A ` ,... |
| oawordex2 44520 | If ` C ` is between ` A ` ... |
| nnawordexg 44521 | If an ordinal, ` B ` , is ... |
| succlg 44522 | Closure law for ordinal su... |
| dflim5 44523 | A limit ordinal is either ... |
| oacl2g 44524 | Closure law for ordinal ad... |
| onmcl 44525 | If an ordinal is less than... |
| omabs2 44526 | Ordinal multiplication by ... |
| omcl2 44527 | Closure law for ordinal mu... |
| omcl3g 44528 | Closure law for ordinal mu... |
| ordsssucb 44529 | An ordinal number is less ... |
| tfsconcatlem 44530 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 44531 | The concatenation of two t... |
| tfsconcatfn 44532 | The concatenation of two t... |
| tfsconcatfv1 44533 | An early value of the conc... |
| tfsconcatfv2 44534 | A latter value of the conc... |
| tfsconcatfv 44535 | The value of the concatena... |
| tfsconcatrn 44536 | The range of the concatena... |
| tfsconcatfo 44537 | The concatenation of two t... |
| tfsconcatb0 44538 | The concatentation with th... |
| tfsconcat0i 44539 | The concatentation with th... |
| tfsconcat0b 44540 | The concatentation with th... |
| tfsconcat00 44541 | The concatentation of two ... |
| tfsconcatrev 44542 | If the domain of a transfi... |
| tfsconcatrnss12 44543 | The range of the concatena... |
| tfsconcatrnss 44544 | The concatenation of trans... |
| tfsconcatrnsson 44545 | The concatenation of trans... |
| tfsnfin 44546 | A transfinite sequence is ... |
| rp-tfslim 44547 | The limit of a sequence of... |
| ofoafg 44548 | Addition operator for func... |
| ofoaf 44549 | Addition operator for func... |
| ofoafo 44550 | Addition operator for func... |
| ofoacl 44551 | Closure law for component ... |
| ofoaid1 44552 | Identity law for component... |
| ofoaid2 44553 | Identity law for component... |
| ofoaass 44554 | Component-wise addition of... |
| ofoacom 44555 | Component-wise addition of... |
| naddcnff 44556 | Addition operator for Cant... |
| naddcnffn 44557 | Addition operator for Cant... |
| naddcnffo 44558 | Addition of Cantor normal ... |
| naddcnfcl 44559 | Closure law for component-... |
| naddcnfcom 44560 | Component-wise ordinal add... |
| naddcnfid1 44561 | Identity law for component... |
| naddcnfid2 44562 | Identity law for component... |
| naddcnfass 44563 | Component-wise addition of... |
| onsucunifi 44564 | The successor to the union... |
| sucunisn 44565 | The successor to the union... |
| onsucunipr 44566 | The successor to the union... |
| onsucunitp 44567 | The successor to the union... |
| oaun3lem1 44568 | The class of all ordinal s... |
| oaun3lem2 44569 | The class of all ordinal s... |
| oaun3lem3 44570 | The class of all ordinal s... |
| oaun3lem4 44571 | The class of all ordinal s... |
| rp-abid 44572 | Two ways to express a clas... |
| oadif1lem 44573 | Express the set difference... |
| oadif1 44574 | Express the set difference... |
| oaun2 44575 | Ordinal addition as a unio... |
| oaun3 44576 | Ordinal addition as a unio... |
| naddov4 44577 | Alternate expression for n... |
| nadd2rabtr 44578 | The set of ordinals which ... |
| nadd2rabord 44579 | The set of ordinals which ... |
| nadd2rabex 44580 | The class of ordinals whic... |
| nadd2rabon 44581 | The set of ordinals which ... |
| nadd1rabtr 44582 | The set of ordinals which ... |
| nadd1rabord 44583 | The set of ordinals which ... |
| nadd1rabex 44584 | The class of ordinals whic... |
| nadd1rabon 44585 | The set of ordinals which ... |
| nadd1suc 44586 | Natural addition with 1 is... |
| naddass1 44587 | Natural addition of ordina... |
| naddgeoa 44588 | Natural addition results i... |
| naddonnn 44589 | Natural addition with a na... |
| naddwordnexlem0 44590 | When ` A ` is the sum of a... |
| naddwordnexlem1 44591 | When ` A ` is the sum of a... |
| naddwordnexlem2 44592 | When ` A ` is the sum of a... |
| naddwordnexlem3 44593 | When ` A ` is the sum of a... |
| oawordex3 44594 | When ` A ` is the sum of a... |
| naddwordnexlem4 44595 | When ` A ` is the sum of a... |
| ordsssucim 44596 | If an ordinal is less than... |
| insucid 44597 | The intersection of a clas... |
| oaltom 44598 | Multiplication eventually ... |
| oe2 44599 | Two ways to square an ordi... |
| omltoe 44600 | Exponentiation eventually ... |
| abeqabi 44601 | Generalized condition for ... |
| abpr 44602 | Condition for a class abst... |
| abtp 44603 | Condition for a class abst... |
| ralopabb 44604 | Restricted universal quant... |
| fpwfvss 44605 | Functions into a powerset ... |
| sdomne0 44606 | A class that strictly domi... |
| sdomne0d 44607 | A class that strictly domi... |
| safesnsupfiss 44608 | If ` B ` is a finite subse... |
| safesnsupfiub 44609 | If ` B ` is a finite subse... |
| safesnsupfidom1o 44610 | If ` B ` is a finite subse... |
| safesnsupfilb 44611 | If ` B ` is a finite subse... |
| isoeq145d 44612 | Equality deduction for iso... |
| resisoeq45d 44613 | Equality deduction for equ... |
| negslem1 44614 | An equivalence between ide... |
| nvocnvb 44615 | Equivalence to saying the ... |
| rp-brsslt 44616 | Binary relation form of a ... |
| nla0002 44617 | Extending a linear order t... |
| nla0003 44618 | Extending a linear order t... |
| nla0001 44619 | Extending a linear order t... |
| faosnf0.11b 44620 | ` B ` is called a non-limi... |
| dfno2 44621 | A surreal number, in the f... |
| onnoxpg 44622 | Every ordinal maps to a su... |
| onnobdayg 44623 | Every ordinal maps to a su... |
| bdaybndex 44624 | Bounds formed from the bir... |
| bdaybndbday 44625 | Bounds formed from the bir... |
| onnoxp 44626 | Every ordinal maps to a su... |
| onnoxpi 44627 | Every ordinal maps to a su... |
| 0fno 44628 | Ordinal zero maps to a sur... |
| 1fno 44629 | Ordinal one maps to a surr... |
| 2fno 44630 | Ordinal two maps to a surr... |
| 3fno 44631 | Ordinal three maps to a su... |
| 4fno 44632 | Ordinal four maps to a sur... |
| fnimafnex 44633 | The functional image of a ... |
| nlimsuc 44634 | A successor is not a limit... |
| nlim1NEW 44635 | 1 is not a limit ordinal. ... |
| nlim2NEW 44636 | 2 is not a limit ordinal. ... |
| nlim3 44637 | 3 is not a limit ordinal. ... |
| nlim4 44638 | 4 is not a limit ordinal. ... |
| oa1un 44639 | Given ` A e. On ` , let ` ... |
| oa1cl 44640 | ` A +o 1o ` is in ` On ` .... |
| 0finon 44641 | 0 is a finite ordinal. Se... |
| 1finon 44642 | 1 is a finite ordinal. Se... |
| 2finon 44643 | 2 is a finite ordinal. Se... |
| 3finon 44644 | 3 is a finite ordinal. Se... |
| 4finon 44645 | 4 is a finite ordinal. Se... |
| finona1cl 44646 | The finite ordinals are cl... |
| finonex 44647 | The finite ordinals are a ... |
| fzunt 44648 | Union of two adjacent fini... |
| fzuntd 44649 | Union of two adjacent fini... |
| fzunt1d 44650 | Union of two overlapping f... |
| fzuntgd 44651 | Union of two adjacent or o... |
| ifpan123g 44652 | Conjunction of conditional... |
| ifpan23 44653 | Conjunction of conditional... |
| ifpdfor2 44654 | Define or in terms of cond... |
| ifporcor 44655 | Corollary of commutation o... |
| ifpdfan2 44656 | Define and with conditiona... |
| ifpancor 44657 | Corollary of commutation o... |
| ifpdfor 44658 | Define or in terms of cond... |
| ifpdfan 44659 | Define and with conditiona... |
| ifpbi2 44660 | Equivalence theorem for co... |
| ifpbi3 44661 | Equivalence theorem for co... |
| ifpim1 44662 | Restate implication as con... |
| ifpnot 44663 | Restate negated wff as con... |
| ifpid2 44664 | Restate wff as conditional... |
| ifpim2 44665 | Restate implication as con... |
| ifpbi23 44666 | Equivalence theorem for co... |
| ifpbiidcor 44667 | Restatement of ~ biid . (... |
| ifpbicor 44668 | Corollary of commutation o... |
| ifpxorcor 44669 | Corollary of commutation o... |
| ifpbi1 44670 | Equivalence theorem for co... |
| ifpnot23 44671 | Negation of conditional lo... |
| ifpnotnotb 44672 | Factor conditional logic o... |
| ifpnorcor 44673 | Corollary of commutation o... |
| ifpnancor 44674 | Corollary of commutation o... |
| ifpnot23b 44675 | Negation of conditional lo... |
| ifpbiidcor2 44676 | Restatement of ~ biid . (... |
| ifpnot23c 44677 | Negation of conditional lo... |
| ifpnot23d 44678 | Negation of conditional lo... |
| ifpdfnan 44679 | Define nand as conditional... |
| ifpdfxor 44680 | Define xor as conditional ... |
| ifpbi12 44681 | Equivalence theorem for co... |
| ifpbi13 44682 | Equivalence theorem for co... |
| ifpbi123 44683 | Equivalence theorem for co... |
| ifpidg 44684 | Restate wff as conditional... |
| ifpid3g 44685 | Restate wff as conditional... |
| ifpid2g 44686 | Restate wff as conditional... |
| ifpid1g 44687 | Restate wff as conditional... |
| ifpim23g 44688 | Restate implication as con... |
| ifpim3 44689 | Restate implication as con... |
| ifpnim1 44690 | Restate negated implicatio... |
| ifpim4 44691 | Restate implication as con... |
| ifpnim2 44692 | Restate negated implicatio... |
| ifpim123g 44693 | Implication of conditional... |
| ifpim1g 44694 | Implication of conditional... |
| ifp1bi 44695 | Substitute the first eleme... |
| ifpbi1b 44696 | When the first variable is... |
| ifpimimb 44697 | Factor conditional logic o... |
| ifpororb 44698 | Factor conditional logic o... |
| ifpananb 44699 | Factor conditional logic o... |
| ifpnannanb 44700 | Factor conditional logic o... |
| ifpor123g 44701 | Disjunction of conditional... |
| ifpimim 44702 | Consequnce of implication.... |
| ifpbibib 44703 | Factor conditional logic o... |
| ifpxorxorb 44704 | Factor conditional logic o... |
| rp-fakeimass 44705 | A special case where impli... |
| rp-fakeanorass 44706 | A special case where a mix... |
| rp-fakeoranass 44707 | A special case where a mix... |
| rp-fakeinunass 44708 | A special case where a mix... |
| rp-fakeuninass 44709 | A special case where a mix... |
| rp-isfinite5 44710 | A set is said to be finite... |
| rp-isfinite6 44711 | A set is said to be finite... |
| intabssd 44712 | When for each element ` y ... |
| eu0 44713 | There is only one empty se... |
| epelon2 44714 | Over the ordinal numbers, ... |
| ontric3g 44715 | For all ` x , y e. On ` , ... |
| dfsucon 44716 | ` A ` is called a successo... |
| snen1g 44717 | A singleton is equinumerou... |
| snen1el 44718 | A singleton is equinumerou... |
| sn1dom 44719 | A singleton is dominated b... |
| pr2dom 44720 | An unordered pair is domin... |
| tr3dom 44721 | An unordered triple is dom... |
| ensucne0 44722 | A class equinumerous to a ... |
| ensucne0OLD 44723 | A class equinumerous to a ... |
| dfom6 44724 | Let ` _om ` be defined to ... |
| infordmin 44725 | ` _om ` is the smallest in... |
| iscard4 44726 | Two ways to express the pr... |
| minregex 44727 | Given any cardinal number ... |
| minregex2 44728 | Given any cardinal number ... |
| iscard5 44729 | Two ways to express the pr... |
| elrncard 44730 | Let us define a cardinal n... |
| harval3 44731 | ` ( har `` A ) ` is the le... |
| harval3on 44732 | For any ordinal number ` A... |
| omssrncard 44733 | All natural numbers are ca... |
| 0iscard 44734 | 0 is a cardinal number. (... |
| 1iscard 44735 | 1 is a cardinal number. (... |
| omiscard 44736 | ` _om ` is a cardinal numb... |
| sucomisnotcard 44737 | ` _om +o 1o ` is not a car... |
| nna1iscard 44738 | For any natural number, th... |
| har2o 44739 | The least cardinal greater... |
| en2pr 44740 | A class is equinumerous to... |
| pr2cv 44741 | If an unordered pair is eq... |
| pr2el1 44742 | If an unordered pair is eq... |
| pr2cv1 44743 | If an unordered pair is eq... |
| pr2el2 44744 | If an unordered pair is eq... |
| pr2cv2 44745 | If an unordered pair is eq... |
| pren2 44746 | An unordered pair is equin... |
| pr2eldif1 44747 | If an unordered pair is eq... |
| pr2eldif2 44748 | If an unordered pair is eq... |
| pren2d 44749 | A pair of two distinct set... |
| aleph1min 44750 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 44751 | ` aleph ` is a strictly or... |
| alephiso3 44752 | ` aleph ` is a strictly or... |
| pwelg 44753 | The powerclass is an eleme... |
| pwinfig 44754 | The powerclass of an infin... |
| pwinfi2 44755 | The powerclass of an infin... |
| pwinfi3 44756 | The powerclass of an infin... |
| pwinfi 44757 | The powerclass of an infin... |
| fipjust 44758 | A definition of the finite... |
| cllem0 44759 | The class of all sets with... |
| superficl 44760 | The class of all supersets... |
| superuncl 44761 | The class of all supersets... |
| ssficl 44762 | The class of all subsets o... |
| ssuncl 44763 | The class of all subsets o... |
| ssdifcl 44764 | The class of all subsets o... |
| sssymdifcl 44765 | The class of all subsets o... |
| fiinfi 44766 | If two classes have the fi... |
| rababg 44767 | Condition when restricted ... |
| elinintab 44768 | Two ways of saying a set i... |
| elmapintrab 44769 | Two ways to say a set is a... |
| elinintrab 44770 | Two ways of saying a set i... |
| inintabss 44771 | Upper bound on intersectio... |
| inintabd 44772 | Value of the intersection ... |
| xpinintabd 44773 | Value of the intersection ... |
| relintabex 44774 | If the intersection of a c... |
| elcnvcnvintab 44775 | Two ways of saying a set i... |
| relintab 44776 | Value of the intersection ... |
| nonrel 44777 | A non-relation is equal to... |
| elnonrel 44778 | Only an ordered pair where... |
| relnonrel 44779 | The non-relation part of a... |
| cnvnonrel 44780 | The converse of the non-re... |
| brnonrel 44781 | A non-relation cannot rela... |
| dmnonrel 44782 | The domain of the non-rela... |
| rnnonrel 44783 | The range of the non-relat... |
| resnonrel 44784 | A restriction of the non-r... |
| imanonrel 44785 | An image under the non-rel... |
| cononrel1 44786 | Composition with the non-r... |
| cononrel2 44787 | Composition with the non-r... |
| elmapintab 44788 | Two ways to say a set is a... |
| fvnonrel 44789 | The function value of any ... |
| elinlem 44790 | Two ways to say a set is a... |
| elcnvcnvlem 44791 | Two ways to say a set is a... |
| cnvcnvintabd 44792 | Value of the relationship ... |
| elcnvlem 44793 | Two ways to say a set is a... |
| elcnvintab 44794 | Two ways of saying a set i... |
| cnvintabd 44795 | Value of the converse of t... |
| undmrnresiss 44796 | Two ways of saying the ide... |
| reflexg 44797 | Two ways of saying a relat... |
| cnvssco 44798 | A condition weaker than re... |
| refimssco 44799 | Reflexive relations are su... |
| cleq2lem 44800 | Equality implies bijection... |
| cbvcllem 44801 | Change of bound variable i... |
| clublem 44802 | If a superset ` Y ` of ` X... |
| clss2lem 44803 | The closure of a property ... |
| dfid7 44804 | Definition of identity rel... |
| mptrcllem 44805 | Show two versions of a clo... |
| cotrintab 44806 | The intersection of a clas... |
| rclexi 44807 | The reflexive closure of a... |
| rtrclexlem 44808 | Existence of relation impl... |
| rtrclex 44809 | The reflexive-transitive c... |
| trclubgNEW 44810 | If a relation exists then ... |
| trclubNEW 44811 | If a relation exists then ... |
| trclexi 44812 | The transitive closure of ... |
| rtrclexi 44813 | The reflexive-transitive c... |
| clrellem 44814 | When the property ` ps ` h... |
| clcnvlem 44815 | When ` A ` , an upper boun... |
| cnvtrucl0 44816 | The converse of the trivia... |
| cnvrcl0 44817 | The converse of the reflex... |
| cnvtrcl0 44818 | The converse of the transi... |
| dmtrcl 44819 | The domain of the transiti... |
| rntrcl 44820 | The range of the transitiv... |
| dfrtrcl5 44821 | Definition of reflexive-tr... |
| trcleq2lemRP 44822 | Equality implies bijection... |
| sqrtcvallem1 44823 | Two ways of saying a compl... |
| reabsifneg 44824 | Alternate expression for t... |
| reabsifnpos 44825 | Alternate expression for t... |
| reabsifpos 44826 | Alternate expression for t... |
| reabsifnneg 44827 | Alternate expression for t... |
| reabssgn 44828 | Alternate expression for t... |
| sqrtcvallem2 44829 | Equivalent to saying that ... |
| sqrtcvallem3 44830 | Equivalent to saying that ... |
| sqrtcvallem4 44831 | Equivalent to saying that ... |
| sqrtcvallem5 44832 | Equivalent to saying that ... |
| sqrtcval 44833 | Explicit formula for the c... |
| sqrtcval2 44834 | Explicit formula for the c... |
| resqrtval 44835 | Real part of the complex s... |
| imsqrtval 44836 | Imaginary part of the comp... |
| resqrtvalex 44837 | Example for ~ resqrtval . ... |
| imsqrtvalex 44838 | Example for ~ imsqrtval . ... |
| al3im 44839 | Version of ~ ax-4 for a ne... |
| intima0 44840 | Two ways of expressing the... |
| elimaint 44841 | Element of image of inters... |
| cnviun 44842 | Converse of indexed union.... |
| imaiun1 44843 | The image of an indexed un... |
| coiun1 44844 | Composition with an indexe... |
| elintima 44845 | Element of intersection of... |
| intimass 44846 | The image under the inters... |
| intimass2 44847 | The image under the inters... |
| intimag 44848 | Requirement for the image ... |
| intimasn 44849 | Two ways to express the im... |
| intimasn2 44850 | Two ways to express the im... |
| ss2iundf 44851 | Subclass theorem for index... |
| ss2iundv 44852 | Subclass theorem for index... |
| cbviuneq12df 44853 | Rule used to change the bo... |
| cbviuneq12dv 44854 | Rule used to change the bo... |
| conrel1d 44855 | Deduction about compositio... |
| conrel2d 44856 | Deduction about compositio... |
| trrelind 44857 | The intersection of transi... |
| xpintrreld 44858 | The intersection of a tran... |
| restrreld 44859 | The restriction of a trans... |
| trrelsuperreldg 44860 | Concrete construction of a... |
| trficl 44861 | The class of all transitiv... |
| cnvtrrel 44862 | The converse of a transiti... |
| trrelsuperrel2dg 44863 | Concrete construction of a... |
| dfrcl2 44866 | Reflexive closure of a rel... |
| dfrcl3 44867 | Reflexive closure of a rel... |
| dfrcl4 44868 | Reflexive closure of a rel... |
| relexp2 44869 | A set operated on by the r... |
| relexpnul 44870 | If the domain and range of... |
| eliunov2 44871 | Membership in the indexed ... |
| eltrclrec 44872 | Membership in the indexed ... |
| elrtrclrec 44873 | Membership in the indexed ... |
| briunov2 44874 | Two classes related by the... |
| brmptiunrelexpd 44875 | If two elements are connec... |
| fvmptiunrelexplb0d 44876 | If the indexed union range... |
| fvmptiunrelexplb0da 44877 | If the indexed union range... |
| fvmptiunrelexplb1d 44878 | If the indexed union range... |
| brfvid 44879 | If two elements are connec... |
| brfvidRP 44880 | If two elements are connec... |
| fvilbd 44881 | A set is a subset of its i... |
| fvilbdRP 44882 | A set is a subset of its i... |
| brfvrcld 44883 | If two elements are connec... |
| brfvrcld2 44884 | If two elements are connec... |
| fvrcllb0d 44885 | A restriction of the ident... |
| fvrcllb0da 44886 | A restriction of the ident... |
| fvrcllb1d 44887 | A set is a subset of its i... |
| brtrclrec 44888 | Two classes related by the... |
| brrtrclrec 44889 | Two classes related by the... |
| briunov2uz 44890 | Two classes related by the... |
| eliunov2uz 44891 | Membership in the indexed ... |
| ov2ssiunov2 44892 | Any particular operator va... |
| relexp0eq 44893 | The zeroth power of relati... |
| iunrelexp0 44894 | Simplification of zeroth p... |
| relexpxpnnidm 44895 | Any positive power of a Ca... |
| relexpiidm 44896 | Any power of any restricti... |
| relexpss1d 44897 | The relational power of a ... |
| comptiunov2i 44898 | The composition two indexe... |
| corclrcl 44899 | The reflexive closure is i... |
| iunrelexpmin1 44900 | The indexed union of relat... |
| relexpmulnn 44901 | With exponents limited to ... |
| relexpmulg 44902 | With ordered exponents, th... |
| trclrelexplem 44903 | The union of relational po... |
| iunrelexpmin2 44904 | The indexed union of relat... |
| relexp01min 44905 | With exponents limited to ... |
| relexp1idm 44906 | Repeated raising a relatio... |
| relexp0idm 44907 | Repeated raising a relatio... |
| relexp0a 44908 | Absorption law for zeroth ... |
| relexpxpmin 44909 | The composition of powers ... |
| relexpaddss 44910 | The composition of two pow... |
| iunrelexpuztr 44911 | The indexed union of relat... |
| dftrcl3 44912 | Transitive closure of a re... |
| brfvtrcld 44913 | If two elements are connec... |
| fvtrcllb1d 44914 | A set is a subset of its i... |
| trclfvcom 44915 | The transitive closure of ... |
| cnvtrclfv 44916 | The converse of the transi... |
| cotrcltrcl 44917 | The transitive closure is ... |
| trclimalb2 44918 | Lower bound for image unde... |
| brtrclfv2 44919 | Two ways to indicate two e... |
| trclfvdecomr 44920 | The transitive closure of ... |
| trclfvdecoml 44921 | The transitive closure of ... |
| dmtrclfvRP 44922 | The domain of the transiti... |
| rntrclfvRP 44923 | The range of the transitiv... |
| rntrclfv 44924 | The range of the transitiv... |
| dfrtrcl3 44925 | Reflexive-transitive closu... |
| brfvrtrcld 44926 | If two elements are connec... |
| fvrtrcllb0d 44927 | A restriction of the ident... |
| fvrtrcllb0da 44928 | A restriction of the ident... |
| fvrtrcllb1d 44929 | A set is a subset of its i... |
| dfrtrcl4 44930 | Reflexive-transitive closu... |
| corcltrcl 44931 | The composition of the ref... |
| cortrcltrcl 44932 | Composition with the refle... |
| corclrtrcl 44933 | Composition with the refle... |
| cotrclrcl 44934 | The composition of the ref... |
| cortrclrcl 44935 | Composition with the refle... |
| cotrclrtrcl 44936 | Composition with the refle... |
| cortrclrtrcl 44937 | The reflexive-transitive c... |
| frege77d 44938 | If the images of both ` { ... |
| frege81d 44939 | If the image of ` U ` is a... |
| frege83d 44940 | If the image of the union ... |
| frege96d 44941 | If ` C ` follows ` A ` in ... |
| frege87d 44942 | If the images of both ` { ... |
| frege91d 44943 | If ` B ` follows ` A ` in ... |
| frege97d 44944 | If ` A ` contains all elem... |
| frege98d 44945 | If ` C ` follows ` A ` and... |
| frege102d 44946 | If either ` A ` and ` C ` ... |
| frege106d 44947 | If ` B ` follows ` A ` in ... |
| frege108d 44948 | If either ` A ` and ` C ` ... |
| frege109d 44949 | If ` A ` contains all elem... |
| frege114d 44950 | If either ` R ` relates ` ... |
| frege111d 44951 | If either ` A ` and ` C ` ... |
| frege122d 44952 | If ` F ` is a function, ` ... |
| frege124d 44953 | If ` F ` is a function, ` ... |
| frege126d 44954 | If ` F ` is a function, ` ... |
| frege129d 44955 | If ` F ` is a function and... |
| frege131d 44956 | If ` F ` is a function and... |
| frege133d 44957 | If ` F ` is a function and... |
| dfxor4 44958 | Express exclusive-or in te... |
| dfxor5 44959 | Express exclusive-or in te... |
| df3or2 44960 | Express triple-or in terms... |
| df3an2 44961 | Express triple-and in term... |
| nev 44962 | Express that not every set... |
| 0pssin 44963 | Express that an intersecti... |
| dfhe2 44966 | The property of relation `... |
| dfhe3 44967 | The property of relation `... |
| heeq12 44968 | Equality law for relations... |
| heeq1 44969 | Equality law for relations... |
| heeq2 44970 | Equality law for relations... |
| sbcheg 44971 | Distribute proper substitu... |
| hess 44972 | Subclass law for relations... |
| xphe 44973 | Any Cartesian product is h... |
| 0he 44974 | The empty relation is here... |
| 0heALT 44975 | The empty relation is here... |
| he0 44976 | Any relation is hereditary... |
| unhe1 44977 | The union of two relations... |
| snhesn 44978 | Any singleton is hereditar... |
| idhe 44979 | The identity relation is h... |
| psshepw 44980 | The relation between sets ... |
| sshepw 44981 | The relation between sets ... |
| rp-simp2-frege 44984 | Simplification of triple c... |
| rp-simp2 44985 | Simplification of triple c... |
| rp-frege3g 44986 | Add antecedent to ~ ax-fre... |
| frege3 44987 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 44988 | Double-use of ~ ax-frege2 ... |
| rp-frege24 44989 | Introducing an embedded an... |
| rp-frege4g 44990 | Deduction related to distr... |
| frege4 44991 | Special case of closed for... |
| frege5 44992 | A closed form of ~ syl . ... |
| rp-7frege 44993 | Distribute antecedent and ... |
| rp-4frege 44994 | Elimination of a nested an... |
| rp-6frege 44995 | Elimination of a nested an... |
| rp-8frege 44996 | Eliminate antecedent when ... |
| rp-frege25 44997 | Closed form for ~ a1dd . ... |
| frege6 44998 | A closed form of ~ imim2d ... |
| axfrege8 44999 | Swap antecedents. Identic... |
| frege7 45000 | A closed form of ~ syl6 . ... |
| frege26 45002 | Identical to ~ idd . Prop... |
| frege27 45003 | We cannot (at the same tim... |
| frege9 45004 | Closed form of ~ syl with ... |
| frege12 45005 | A closed form of ~ com23 .... |
| frege11 45006 | Elimination of a nested an... |
| frege24 45007 | Closed form for ~ a1d . D... |
| frege16 45008 | A closed form of ~ com34 .... |
| frege25 45009 | Closed form for ~ a1dd . ... |
| frege18 45010 | Closed form of a syllogism... |
| frege22 45011 | A closed form of ~ com45 .... |
| frege10 45012 | Result commuting anteceden... |
| frege17 45013 | A closed form of ~ com3l .... |
| frege13 45014 | A closed form of ~ com3r .... |
| frege14 45015 | Closed form of a deduction... |
| frege19 45016 | A closed form of ~ syl6 . ... |
| frege23 45017 | Syllogism followed by rota... |
| frege15 45018 | A closed form of ~ com4r .... |
| frege21 45019 | Replace antecedent in ante... |
| frege20 45020 | A closed form of ~ syl8 . ... |
| axfrege28 45021 | Contraposition. Identical... |
| frege29 45023 | Closed form of ~ con3d . ... |
| frege30 45024 | Commuted, closed form of ~... |
| axfrege31 45025 | Identical to ~ notnotr . ... |
| frege32 45027 | Deduce ~ con1 from ~ con3 ... |
| frege33 45028 | If ` ph ` or ` ps ` takes ... |
| frege34 45029 | If as a consequence of the... |
| frege35 45030 | Commuted, closed form of ~... |
| frege36 45031 | The case in which ` ps ` i... |
| frege37 45032 | If ` ch ` is a necessary c... |
| frege38 45033 | Identical to ~ pm2.21 . P... |
| frege39 45034 | Syllogism between ~ pm2.18... |
| frege40 45035 | Anything implies ~ pm2.18 ... |
| axfrege41 45036 | Identical to ~ notnot . A... |
| frege42 45038 | Not not ~ id . Propositio... |
| frege43 45039 | If there is a choice only ... |
| frege44 45040 | Similar to a commuted ~ pm... |
| frege45 45041 | Deduce ~ pm2.6 from ~ con1... |
| frege46 45042 | If ` ps ` holds when ` ph ... |
| frege47 45043 | Deduce consequence follows... |
| frege48 45044 | Closed form of syllogism w... |
| frege49 45045 | Closed form of deduction w... |
| frege50 45046 | Closed form of ~ jaoi . P... |
| frege51 45047 | Compare with ~ jaod . Pro... |
| axfrege52a 45048 | Justification for ~ ax-fre... |
| frege52aid 45050 | The case when the content ... |
| frege53aid 45051 | Specialization of ~ frege5... |
| frege53a 45052 | Lemma for ~ frege55a . Pr... |
| axfrege54a 45053 | Justification for ~ ax-fre... |
| frege54cor0a 45055 | Synonym for logical equiva... |
| frege54cor1a 45056 | Reflexive equality. (Cont... |
| frege55aid 45057 | Lemma for ~ frege57aid . ... |
| frege55lem1a 45058 | Necessary deduction regard... |
| frege55lem2a 45059 | Core proof of Proposition ... |
| frege55a 45060 | Proposition 55 of [Frege18... |
| frege55cor1a 45061 | Proposition 55 of [Frege18... |
| frege56aid 45062 | Lemma for ~ frege57aid . ... |
| frege56a 45063 | Proposition 56 of [Frege18... |
| frege57aid 45064 | This is the all important ... |
| frege57a 45065 | Analogue of ~ frege57aid .... |
| axfrege58a 45066 | Identical to ~ anifp . Ju... |
| frege58acor 45068 | Lemma for ~ frege59a . (C... |
| frege59a 45069 | A kind of Aristotelian inf... |
| frege60a 45070 | Swap antecedents of ~ ax-f... |
| frege61a 45071 | Lemma for ~ frege65a . Pr... |
| frege62a 45072 | A kind of Aristotelian inf... |
| frege63a 45073 | Proposition 63 of [Frege18... |
| frege64a 45074 | Lemma for ~ frege65a . Pr... |
| frege65a 45075 | A kind of Aristotelian inf... |
| frege66a 45076 | Swap antecedents of ~ freg... |
| frege67a 45077 | Lemma for ~ frege68a . Pr... |
| frege68a 45078 | Combination of applying a ... |
| axfrege52c 45079 | Justification for ~ ax-fre... |
| frege52b 45081 | The case when the content ... |
| frege53b 45082 | Lemma for frege102 (via ~ ... |
| axfrege54c 45083 | Reflexive equality of clas... |
| frege54b 45085 | Reflexive equality of sets... |
| frege54cor1b 45086 | Reflexive equality. (Cont... |
| frege55lem1b 45087 | Necessary deduction regard... |
| frege55lem2b 45088 | Lemma for ~ frege55b . Co... |
| frege55b 45089 | Lemma for ~ frege57b . Pr... |
| frege56b 45090 | Lemma for ~ frege57b . Pr... |
| frege57b 45091 | Analogue of ~ frege57aid .... |
| axfrege58b 45092 | If ` A. x ph ` is affirmed... |
| frege58bid 45094 | If ` A. x ph ` is affirmed... |
| frege58bcor 45095 | Lemma for ~ frege59b . (C... |
| frege59b 45096 | A kind of Aristotelian inf... |
| frege60b 45097 | Swap antecedents of ~ ax-f... |
| frege61b 45098 | Lemma for ~ frege65b . Pr... |
| frege62b 45099 | A kind of Aristotelian inf... |
| frege63b 45100 | Lemma for ~ frege91 . Pro... |
| frege64b 45101 | Lemma for ~ frege65b . Pr... |
| frege65b 45102 | A kind of Aristotelian inf... |
| frege66b 45103 | Swap antecedents of ~ freg... |
| frege67b 45104 | Lemma for ~ frege68b . Pr... |
| frege68b 45105 | Combination of applying a ... |
| frege53c 45106 | Proposition 53 of [Frege18... |
| frege54cor1c 45107 | Reflexive equality. (Cont... |
| frege55lem1c 45108 | Necessary deduction regard... |
| frege55lem2c 45109 | Core proof of Proposition ... |
| frege55c 45110 | Proposition 55 of [Frege18... |
| frege56c 45111 | Lemma for ~ frege57c . Pr... |
| frege57c 45112 | Swap order of implication ... |
| frege58c 45113 | Principle related to ~ sp ... |
| frege59c 45114 | A kind of Aristotelian inf... |
| frege60c 45115 | Swap antecedents of ~ freg... |
| frege61c 45116 | Lemma for ~ frege65c . Pr... |
| frege62c 45117 | A kind of Aristotelian inf... |
| frege63c 45118 | Analogue of ~ frege63b . ... |
| frege64c 45119 | Lemma for ~ frege65c . Pr... |
| frege65c 45120 | A kind of Aristotelian inf... |
| frege66c 45121 | Swap antecedents of ~ freg... |
| frege67c 45122 | Lemma for ~ frege68c . Pr... |
| frege68c 45123 | Combination of applying a ... |
| dffrege69 45124 | If from the proposition th... |
| frege70 45125 | Lemma for ~ frege72 . Pro... |
| frege71 45126 | Lemma for ~ frege72 . Pro... |
| frege72 45127 | If property ` A ` is hered... |
| frege73 45128 | Lemma for ~ frege87 . Pro... |
| frege74 45129 | If ` X ` has a property ` ... |
| frege75 45130 | If from the proposition th... |
| dffrege76 45131 | If from the two propositio... |
| frege77 45132 | If ` Y ` follows ` X ` in ... |
| frege78 45133 | Commuted form of ~ frege77... |
| frege79 45134 | Distributed form of ~ freg... |
| frege80 45135 | Add additional condition t... |
| frege81 45136 | If ` X ` has a property ` ... |
| frege82 45137 | Closed-form deduction base... |
| frege83 45138 | Apply commuted form of ~ f... |
| frege84 45139 | Commuted form of ~ frege81... |
| frege85 45140 | Commuted form of ~ frege77... |
| frege86 45141 | Conclusion about element o... |
| frege87 45142 | If ` Z ` is a result of an... |
| frege88 45143 | Commuted form of ~ frege87... |
| frege89 45144 | One direction of ~ dffrege... |
| frege90 45145 | Add antecedent to ~ frege8... |
| frege91 45146 | Every result of an applica... |
| frege92 45147 | Inference from ~ frege91 .... |
| frege93 45148 | Necessary condition for tw... |
| frege94 45149 | Looking one past a pair re... |
| frege95 45150 | Looking one past a pair re... |
| frege96 45151 | Every result of an applica... |
| frege97 45152 | The property of following ... |
| frege98 45153 | If ` Y ` follows ` X ` and... |
| dffrege99 45154 | If ` Z ` is identical with... |
| frege100 45155 | One direction of ~ dffrege... |
| frege101 45156 | Lemma for ~ frege102 . Pr... |
| frege102 45157 | If ` Z ` belongs to the ` ... |
| frege103 45158 | Proposition 103 of [Frege1... |
| frege104 45159 | Proposition 104 of [Frege1... |
| frege105 45160 | Proposition 105 of [Frege1... |
| frege106 45161 | Whatever follows ` X ` in ... |
| frege107 45162 | Proposition 107 of [Frege1... |
| frege108 45163 | If ` Y ` belongs to the ` ... |
| frege109 45164 | The property of belonging ... |
| frege110 45165 | Proposition 110 of [Frege1... |
| frege111 45166 | If ` Y ` belongs to the ` ... |
| frege112 45167 | Identity implies belonging... |
| frege113 45168 | Proposition 113 of [Frege1... |
| frege114 45169 | If ` X ` belongs to the ` ... |
| dffrege115 45170 | If from the circumstance t... |
| frege116 45171 | One direction of ~ dffrege... |
| frege117 45172 | Lemma for ~ frege118 . Pr... |
| frege118 45173 | Simplified application of ... |
| frege119 45174 | Lemma for ~ frege120 . Pr... |
| frege120 45175 | Simplified application of ... |
| frege121 45176 | Lemma for ~ frege122 . Pr... |
| frege122 45177 | If ` X ` is a result of an... |
| frege123 45178 | Lemma for ~ frege124 . Pr... |
| frege124 45179 | If ` X ` is a result of an... |
| frege125 45180 | Lemma for ~ frege126 . Pr... |
| frege126 45181 | If ` M ` follows ` Y ` in ... |
| frege127 45182 | Communte antecedents of ~ ... |
| frege128 45183 | Lemma for ~ frege129 . Pr... |
| frege129 45184 | If the procedure ` R ` is ... |
| frege130 45185 | Lemma for ~ frege131 . Pr... |
| frege131 45186 | If the procedure ` R ` is ... |
| frege132 45187 | Lemma for ~ frege133 . Pr... |
| frege133 45188 | If the procedure ` R ` is ... |
| enrelmap 45189 | The set of all possible re... |
| enrelmapr 45190 | The set of all possible re... |
| enmappw 45191 | The set of all mappings fr... |
| enmappwid 45192 | The set of all mappings fr... |
| rfovd 45193 | Value of the operator, ` (... |
| rfovfvd 45194 | Value of the operator, ` (... |
| rfovfvfvd 45195 | Value of the operator, ` (... |
| rfovcnvf1od 45196 | Properties of the operator... |
| rfovcnvd 45197 | Value of the converse of t... |
| rfovf1od 45198 | The value of the operator,... |
| rfovcnvfvd 45199 | Value of the converse of t... |
| fsovd 45200 | Value of the operator, ` (... |
| fsovrfovd 45201 | The operator which gives a... |
| fsovfvd 45202 | Value of the operator, ` (... |
| fsovfvfvd 45203 | Value of the operator, ` (... |
| fsovfd 45204 | The operator, ` ( A O B ) ... |
| fsovcnvlem 45205 | The ` O ` operator, which ... |
| fsovcnvd 45206 | The value of the converse ... |
| fsovcnvfvd 45207 | The value of the converse ... |
| fsovf1od 45208 | The value of ` ( A O B ) `... |
| dssmapfvd 45209 | Value of the duality opera... |
| dssmapfv2d 45210 | Value of the duality opera... |
| dssmapfv3d 45211 | Value of the duality opera... |
| dssmapnvod 45212 | For any base set ` B ` the... |
| dssmapf1od 45213 | For any base set ` B ` the... |
| dssmap2d 45214 | For any base set ` B ` the... |
| or3or 45215 | Decompose disjunction into... |
| andi3or 45216 | Distribute over triple dis... |
| uneqsn 45217 | If a union of classes is e... |
| brfvimex 45218 | If a binary relation holds... |
| brovmptimex 45219 | If a binary relation holds... |
| brovmptimex1 45220 | If a binary relation holds... |
| brovmptimex2 45221 | If a binary relation holds... |
| brcoffn 45222 | Conditions allowing the de... |
| brcofffn 45223 | Conditions allowing the de... |
| brco2f1o 45224 | Conditions allowing the de... |
| brco3f1o 45225 | Conditions allowing the de... |
| ntrclsbex 45226 | If (pseudo-)interior and (... |
| ntrclsrcomplex 45227 | The relative complement of... |
| neik0imk0p 45228 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 45229 | If an interior function is... |
| ntrkbimka 45230 | If the interiors of disjoi... |
| ntrk0kbimka 45231 | If the interiors of disjoi... |
| clsk3nimkb 45232 | If the base set is not emp... |
| clsk1indlem0 45233 | The ansatz closure functio... |
| clsk1indlem2 45234 | The ansatz closure functio... |
| clsk1indlem3 45235 | The ansatz closure functio... |
| clsk1indlem4 45236 | The ansatz closure functio... |
| clsk1indlem1 45237 | The ansatz closure functio... |
| clsk1independent 45238 | For generalized closure fu... |
| neik0pk1imk0 45239 | Kuratowski's K0' and K1 ax... |
| isotone1 45240 | Two different ways to say ... |
| isotone2 45241 | Two different ways to say ... |
| ntrk1k3eqk13 45242 | An interior function is bo... |
| ntrclsf1o 45243 | If (pseudo-)interior and (... |
| ntrclsnvobr 45244 | If (pseudo-)interior and (... |
| ntrclsiex 45245 | If (pseudo-)interior and (... |
| ntrclskex 45246 | If (pseudo-)interior and (... |
| ntrclsfv1 45247 | If (pseudo-)interior and (... |
| ntrclsfv2 45248 | If (pseudo-)interior and (... |
| ntrclselnel1 45249 | If (pseudo-)interior and (... |
| ntrclselnel2 45250 | If (pseudo-)interior and (... |
| ntrclsfv 45251 | The value of the interior ... |
| ntrclsfveq1 45252 | If interior and closure fu... |
| ntrclsfveq2 45253 | If interior and closure fu... |
| ntrclsfveq 45254 | If interior and closure fu... |
| ntrclsss 45255 | If interior and closure fu... |
| ntrclsneine0lem 45256 | If (pseudo-)interior and (... |
| ntrclsneine0 45257 | If (pseudo-)interior and (... |
| ntrclscls00 45258 | If (pseudo-)interior and (... |
| ntrclsiso 45259 | If (pseudo-)interior and (... |
| ntrclsk2 45260 | An interior function is co... |
| ntrclskb 45261 | The interiors of disjoint ... |
| ntrclsk3 45262 | The intersection of interi... |
| ntrclsk13 45263 | The interior of the inters... |
| ntrclsk4 45264 | Idempotence of the interio... |
| ntrneibex 45265 | If (pseudo-)interior and (... |
| ntrneircomplex 45266 | The relative complement of... |
| ntrneif1o 45267 | If (pseudo-)interior and (... |
| ntrneiiex 45268 | If (pseudo-)interior and (... |
| ntrneinex 45269 | If (pseudo-)interior and (... |
| ntrneicnv 45270 | If (pseudo-)interior and (... |
| ntrneifv1 45271 | If (pseudo-)interior and (... |
| ntrneifv2 45272 | If (pseudo-)interior and (... |
| ntrneiel 45273 | If (pseudo-)interior and (... |
| ntrneifv3 45274 | The value of the neighbors... |
| ntrneineine0lem 45275 | If (pseudo-)interior and (... |
| ntrneineine1lem 45276 | If (pseudo-)interior and (... |
| ntrneifv4 45277 | The value of the interior ... |
| ntrneiel2 45278 | Membership in iterated int... |
| ntrneineine0 45279 | If (pseudo-)interior and (... |
| ntrneineine1 45280 | If (pseudo-)interior and (... |
| ntrneicls00 45281 | If (pseudo-)interior and (... |
| ntrneicls11 45282 | If (pseudo-)interior and (... |
| ntrneiiso 45283 | If (pseudo-)interior and (... |
| ntrneik2 45284 | An interior function is co... |
| ntrneix2 45285 | An interior (closure) func... |
| ntrneikb 45286 | The interiors of disjoint ... |
| ntrneixb 45287 | The interiors (closures) o... |
| ntrneik3 45288 | The intersection of interi... |
| ntrneix3 45289 | The closure of the union o... |
| ntrneik13 45290 | The interior of the inters... |
| ntrneix13 45291 | The closure of the union o... |
| ntrneik4w 45292 | Idempotence of the interio... |
| ntrneik4 45293 | Idempotence of the interio... |
| clsneibex 45294 | If (pseudo-)closure and (p... |
| clsneircomplex 45295 | The relative complement of... |
| clsneif1o 45296 | If a (pseudo-)closure func... |
| clsneicnv 45297 | If a (pseudo-)closure func... |
| clsneikex 45298 | If closure and neighborhoo... |
| clsneinex 45299 | If closure and neighborhoo... |
| clsneiel1 45300 | If a (pseudo-)closure func... |
| clsneiel2 45301 | If a (pseudo-)closure func... |
| clsneifv3 45302 | Value of the neighborhoods... |
| clsneifv4 45303 | Value of the closure (inte... |
| neicvgbex 45304 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 45305 | The relative complement of... |
| neicvgf1o 45306 | If neighborhood and conver... |
| neicvgnvo 45307 | If neighborhood and conver... |
| neicvgnvor 45308 | If neighborhood and conver... |
| neicvgmex 45309 | If the neighborhoods and c... |
| neicvgnex 45310 | If the neighborhoods and c... |
| neicvgel1 45311 | A subset being an element ... |
| neicvgel2 45312 | The complement of a subset... |
| neicvgfv 45313 | The value of the neighborh... |
| ntrrn 45314 | The range of the interior ... |
| ntrf 45315 | The interior function of a... |
| ntrf2 45316 | The interior function is a... |
| ntrelmap 45317 | The interior function is a... |
| clsf2 45318 | The closure function is a ... |
| clselmap 45319 | The closure function is a ... |
| dssmapntrcls 45320 | The interior and closure o... |
| dssmapclsntr 45321 | The closure and interior o... |
| gneispa 45322 | Each point ` p ` of the ne... |
| gneispb 45323 | Given a neighborhood ` N `... |
| gneispace2 45324 | The predicate that ` F ` i... |
| gneispace3 45325 | The predicate that ` F ` i... |
| gneispace 45326 | The predicate that ` F ` i... |
| gneispacef 45327 | A generic neighborhood spa... |
| gneispacef2 45328 | A generic neighborhood spa... |
| gneispacefun 45329 | A generic neighborhood spa... |
| gneispacern 45330 | A generic neighborhood spa... |
| gneispacern2 45331 | A generic neighborhood spa... |
| gneispace0nelrn 45332 | A generic neighborhood spa... |
| gneispace0nelrn2 45333 | A generic neighborhood spa... |
| gneispace0nelrn3 45334 | A generic neighborhood spa... |
| gneispaceel 45335 | Every neighborhood of a po... |
| gneispaceel2 45336 | Every neighborhood of a po... |
| gneispacess 45337 | All supersets of a neighbo... |
| gneispacess2 45338 | All supersets of a neighbo... |
| k0004lem1 45339 | Application of ~ ssin to r... |
| k0004lem2 45340 | A mapping with a particula... |
| k0004lem3 45341 | When the value of a mappin... |
| k0004val 45342 | The topological simplex of... |
| k0004ss1 45343 | The topological simplex of... |
| k0004ss2 45344 | The topological simplex of... |
| k0004ss3 45345 | The topological simplex of... |
| k0004val0 45346 | The topological simplex of... |
| inductionexd 45347 | Simple induction example. ... |
| wwlemuld 45348 | Natural deduction form of ... |
| leeq1d 45349 | Specialization of ~ breq1d... |
| leeq2d 45350 | Specialization of ~ breq2d... |
| absmulrposd 45351 | Specialization of absmuld ... |
| imadisjld 45352 | Natural dduction form of o... |
| wnefimgd 45353 | The image of a mapping fro... |
| fco2d 45354 | Natural deduction form of ... |
| wfximgfd 45355 | The value of a function on... |
| extoimad 45356 | If |f(x)| <= C for all x t... |
| imo72b2lem0 45357 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 45358 | Natural deduction form of ... |
| imo72b2lem2 45359 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 45360 | Natural deduction form of ... |
| imo72b2lem1 45361 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 45362 | 'Less than or equal to' re... |
| lemuldiv4d 45363 | 'Less than or equal to' re... |
| imo72b2 45364 | IMO 1972 B2. (14th Intern... |
| int-addcomd 45365 | AdditionCommutativity gene... |
| int-addassocd 45366 | AdditionAssociativity gene... |
| int-addsimpd 45367 | AdditionSimplification gen... |
| int-mulcomd 45368 | MultiplicationCommutativit... |
| int-mulassocd 45369 | MultiplicationAssociativit... |
| int-mulsimpd 45370 | MultiplicationSimplificati... |
| int-leftdistd 45371 | AdditionMultiplicationLeft... |
| int-rightdistd 45372 | AdditionMultiplicationRigh... |
| int-sqdefd 45373 | SquareDefinition generator... |
| int-mul11d 45374 | First MultiplicationOne ge... |
| int-mul12d 45375 | Second MultiplicationOne g... |
| int-add01d 45376 | First AdditionZero generat... |
| int-add02d 45377 | Second AdditionZero genera... |
| int-sqgeq0d 45378 | SquareGEQZero generator ru... |
| int-eqprincd 45379 | PrincipleOfEquality genera... |
| int-eqtransd 45380 | EqualityTransitivity gener... |
| int-eqmvtd 45381 | EquMoveTerm generator rule... |
| int-eqineqd 45382 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 45383 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 45384 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 45385 | SecondPrincipleOfInequalit... |
| int-ineqtransd 45386 | InequalityTransitivity gen... |
| unitadd 45387 | Theorem used in conjunctio... |
| gsumws3 45388 | Valuation of a length 3 wo... |
| gsumws4 45389 | Valuation of a length 4 wo... |
| amgm2d 45390 | Arithmetic-geometric mean ... |
| amgm3d 45391 | Arithmetic-geometric mean ... |
| amgm4d 45392 | Arithmetic-geometric mean ... |
| spALT 45393 | ~ sp can be proven from th... |
| rr-spce 45394 | Prove an existential. (Co... |
| rexlimdvaacbv 45395 | Unpack a restricted existe... |
| rexlimddvcbvw 45396 | Unpack a restricted existe... |
| rexlimddvcbv 45397 | Unpack a restricted existe... |
| rr-elrnmpt3d 45398 | Elementhood in an image se... |
| rr-phpd 45399 | Equivalent of ~ php withou... |
| tfindsd 45400 | Deduction associated with ... |
| mnringvald 45403 | Value of the monoid ring f... |
| mnringnmulrd 45404 | Components of a monoid rin... |
| mnringbased 45405 | The base set of a monoid r... |
| mnringbaserd 45406 | The base set of a monoid r... |
| mnringelbased 45407 | Membership in the base set... |
| mnringbasefd 45408 | Elements of a monoid ring ... |
| mnringbasefsuppd 45409 | Elements of a monoid ring ... |
| mnringaddgd 45410 | The additive operation of ... |
| mnring0gd 45411 | The additive identity of a... |
| mnring0g2d 45412 | The additive identity of a... |
| mnringmulrd 45413 | The ring product of a mono... |
| mnringscad 45414 | The scalar ring of a monoi... |
| mnringvscad 45415 | The scalar product of a mo... |
| mnringlmodd 45416 | Monoid rings are left modu... |
| mnringmulrvald 45417 | Value of multiplication in... |
| mnringmulrcld 45418 | Monoid rings are closed un... |
| gru0eld 45419 | A nonempty Grothendieck un... |
| grusucd 45420 | Grothendieck universes are... |
| r1rankcld 45421 | Any rank of the cumulative... |
| grur1cld 45422 | Grothendieck universes are... |
| grurankcld 45423 | Grothendieck universes are... |
| grurankrcld 45424 | If a Grothendieck universe... |
| gruscottcld 45425 | If a Grothendieck universe... |
| dfcoll2 45428 | Alternate definition of th... |
| colleq12d 45429 | Equality theorem for the c... |
| colleq1 45430 | Equality theorem for the c... |
| colleq2 45431 | Equality theorem for the c... |
| nfcoll 45432 | Bound-variable hypothesis ... |
| collexd 45433 | The output of the collecti... |
| cpcolld 45434 | Property of the collection... |
| cpcoll2d 45435 | ~ cpcolld with an extra ex... |
| grucollcld 45436 | A Grothendieck universe co... |
| ismnu 45437 | The hypothesis of this the... |
| mnuop123d 45438 | Operations of a minimal un... |
| mnussd 45439 | Minimal universes are clos... |
| mnuss2d 45440 | ~ mnussd with arguments pr... |
| mnu0eld 45441 | A nonempty minimal univers... |
| mnuop23d 45442 | Second and third operation... |
| mnupwd 45443 | Minimal universes are clos... |
| mnusnd 45444 | Minimal universes are clos... |
| mnuprssd 45445 | A minimal universe contain... |
| mnuprss2d 45446 | Special case of ~ mnuprssd... |
| mnuop3d 45447 | Third operation of a minim... |
| mnuprdlem1 45448 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 45449 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 45450 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 45451 | Lemma for ~ mnuprd . Gene... |
| mnuprd 45452 | Minimal universes are clos... |
| mnuunid 45453 | Minimal universes are clos... |
| mnuund 45454 | Minimal universes are clos... |
| mnutrcld 45455 | Minimal universes contain ... |
| mnutrd 45456 | Minimal universes are tran... |
| mnurndlem1 45457 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 45458 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 45459 | Minimal universes contain ... |
| mnugrud 45460 | Minimal universes are Grot... |
| grumnudlem 45461 | Lemma for ~ grumnud . (Co... |
| grumnud 45462 | Grothendieck universes are... |
| grumnueq 45463 | The class of Grothendieck ... |
| expandan 45464 | Expand conjunction to prim... |
| expandexn 45465 | Expand an existential quan... |
| expandral 45466 | Expand a restricted univer... |
| expandrexn 45467 | Expand a restricted existe... |
| expandrex 45468 | Expand a restricted existe... |
| expanduniss 45469 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 45470 | Express the predicate on `... |
| rr-grothprimbi 45471 | Express "every set is cont... |
| inagrud 45472 | Inaccessible levels of the... |
| inaex 45473 | Assuming the Tarski-Grothe... |
| gruex 45474 | Assuming the Tarski-Grothe... |
| rr-groth 45475 | An equivalent of ~ ax-grot... |
| rr-grothprim 45476 | An equivalent of ~ ax-grot... |
| ismnushort 45477 | Express the predicate on `... |
| dfuniv2 45478 | Alternative definition of ... |
| rr-grothshortbi 45479 | Express "every set is cont... |
| rr-grothshort 45480 | A shorter equivalent of ~ ... |
| nanorxor 45481 | 'nand' is equivalent to th... |
| undisjrab 45482 | Union of two disjoint rest... |
| iso0 45483 | The empty set is an ` R , ... |
| ssrecnpr 45484 | ` RR ` is a subset of both... |
| seff 45485 | Let set ` S ` be the real ... |
| sblpnf 45486 | The infinity ball in the a... |
| prmunb2 45487 | The primes are unbounded. ... |
| dvgrat 45488 | Ratio test for divergence ... |
| cvgdvgrat 45489 | Ratio test for convergence... |
| radcnvrat 45490 | Let ` L ` be the limit, if... |
| reldvds 45491 | The divides relation is in... |
| nznngen 45492 | All positive integers in t... |
| nzss 45493 | The set of multiples of _m... |
| nzin 45494 | The intersection of the se... |
| nzprmdif 45495 | Subtract one prime's multi... |
| hashnzfz 45496 | Special case of ~ hashdvds... |
| hashnzfz2 45497 | Special case of ~ hashnzfz... |
| hashnzfzclim 45498 | As the upper bound ` K ` o... |
| caofcan 45499 | Transfer a cancellation la... |
| ofsubid 45500 | Function analogue of ~ sub... |
| ofmul12 45501 | Function analogue of ~ mul... |
| ofdivrec 45502 | Function analogue of ~ div... |
| ofdivcan4 45503 | Function analogue of ~ div... |
| ofdivdiv2 45504 | Function analogue of ~ div... |
| lhe4.4ex1a 45505 | Example of the Fundamental... |
| dvsconst 45506 | Derivative of a constant f... |
| dvsid 45507 | Derivative of the identity... |
| dvsef 45508 | Derivative of the exponent... |
| expgrowthi 45509 | Exponential growth and dec... |
| dvconstbi 45510 | The derivative of a functi... |
| expgrowth 45511 | Exponential growth and dec... |
| bccval 45514 | Value of the generalized b... |
| bcccl 45515 | Closure of the generalized... |
| bcc0 45516 | The generalized binomial c... |
| bccp1k 45517 | Generalized binomial coeff... |
| bccm1k 45518 | Generalized binomial coeff... |
| bccn0 45519 | Generalized binomial coeff... |
| bccn1 45520 | Generalized binomial coeff... |
| bccbc 45521 | The binomial coefficient a... |
| uzmptshftfval 45522 | When ` F ` is a maps-to fu... |
| dvradcnv2 45523 | The radius of convergence ... |
| binomcxplemwb 45524 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 45525 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 45526 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 45527 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 45528 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 45529 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 45530 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 45531 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 45532 | Lemma for ~ binomcxp . Wh... |
| binomcxp 45533 | Generalize the binomial th... |
| pm10.12 45534 | Theorem *10.12 in [Whitehe... |
| pm10.14 45535 | Theorem *10.14 in [Whitehe... |
| pm10.251 45536 | Theorem *10.251 in [Whiteh... |
| pm10.252 45537 | Theorem *10.252 in [Whiteh... |
| pm10.253 45538 | Theorem *10.253 in [Whiteh... |
| albitr 45539 | Theorem *10.301 in [Whiteh... |
| pm10.42 45540 | Theorem *10.42 in [Whitehe... |
| pm10.52 45541 | Theorem *10.52 in [Whitehe... |
| pm10.53 45542 | Theorem *10.53 in [Whitehe... |
| pm10.541 45543 | Theorem *10.541 in [Whiteh... |
| pm10.542 45544 | Theorem *10.542 in [Whiteh... |
| pm10.55 45545 | Theorem *10.55 in [Whitehe... |
| pm10.56 45546 | Theorem *10.56 in [Whitehe... |
| pm10.57 45547 | Theorem *10.57 in [Whitehe... |
| 2alanimi 45548 | Removes two universal quan... |
| 2al2imi 45549 | Removes two universal quan... |
| pm11.11 45550 | Theorem *11.11 in [Whitehe... |
| pm11.12 45551 | Theorem *11.12 in [Whitehe... |
| 19.21vv 45552 | Compare Theorem *11.3 in [... |
| 2alim 45553 | Theorem *11.32 in [Whitehe... |
| 2albi 45554 | Theorem *11.33 in [Whitehe... |
| 2exim 45555 | Theorem *11.34 in [Whitehe... |
| 2exbi 45556 | Theorem *11.341 in [Whiteh... |
| spsbce-2 45557 | Theorem *11.36 in [Whitehe... |
| 19.33-2 45558 | Theorem *11.421 in [Whiteh... |
| 19.36vv 45559 | Theorem *11.43 in [Whitehe... |
| 19.31vv 45560 | Theorem *11.44 in [Whitehe... |
| 19.37vv 45561 | Theorem *11.46 in [Whitehe... |
| 19.28vv 45562 | Theorem *11.47 in [Whitehe... |
| pm11.52 45563 | Theorem *11.52 in [Whitehe... |
| aaanv 45564 | Theorem *11.56 in [Whitehe... |
| pm11.57 45565 | Theorem *11.57 in [Whitehe... |
| pm11.58 45566 | Theorem *11.58 in [Whitehe... |
| pm11.59 45567 | Theorem *11.59 in [Whitehe... |
| pm11.6 45568 | Theorem *11.6 in [Whitehea... |
| pm11.61 45569 | Theorem *11.61 in [Whitehe... |
| pm11.62 45570 | Theorem *11.62 in [Whitehe... |
| pm11.63 45571 | Theorem *11.63 in [Whitehe... |
| pm11.7 45572 | Theorem *11.7 in [Whitehea... |
| pm11.71 45573 | Theorem *11.71 in [Whitehe... |
| sbeqal1 45574 | If ` x = y ` always implie... |
| sbeqal1i 45575 | Suppose you know ` x = y `... |
| sbeqal2i 45576 | If ` x = y ` implies ` x =... |
| axc5c4c711 45577 | Proof of a theorem that ca... |
| axc5c4c711toc5 45578 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 45579 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 45580 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 45581 | Rederivation of ~ ax-11 fr... |
| axc11next 45582 | This theorem shows that, g... |
| pm13.13a 45583 | One result of theorem *13.... |
| pm13.13b 45584 | Theorem *13.13 in [Whitehe... |
| pm13.14 45585 | Theorem *13.14 in [Whitehe... |
| pm13.192 45586 | Theorem *13.192 in [Whiteh... |
| pm13.193 45587 | Theorem *13.193 in [Whiteh... |
| pm13.194 45588 | Theorem *13.194 in [Whiteh... |
| pm13.195 45589 | Theorem *13.195 in [Whiteh... |
| pm13.196a 45590 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 45591 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 45592 | Theorem *13.22 in [Whitehe... |
| iotaexeu 45593 | The iota class exists. Th... |
| iotasbc 45594 | Definition *14.01 in [Whit... |
| iotasbc2 45595 | Theorem *14.111 in [Whiteh... |
| pm14.12 45596 | Theorem *14.12 in [Whitehe... |
| pm14.122a 45597 | Theorem *14.122 in [Whiteh... |
| pm14.122b 45598 | Theorem *14.122 in [Whiteh... |
| pm14.122c 45599 | Theorem *14.122 in [Whiteh... |
| pm14.123a 45600 | Theorem *14.123 in [Whiteh... |
| pm14.123b 45601 | Theorem *14.123 in [Whiteh... |
| pm14.123c 45602 | Theorem *14.123 in [Whiteh... |
| pm14.18 45603 | Theorem *14.18 in [Whitehe... |
| iotaequ 45604 | Theorem *14.2 in [Whitehea... |
| iotavalb 45605 | Theorem *14.202 in [Whiteh... |
| iotasbc5 45606 | Theorem *14.205 in [Whiteh... |
| pm14.24 45607 | Theorem *14.24 in [Whitehe... |
| iotavalsb 45608 | Theorem *14.242 in [Whiteh... |
| sbiota1 45609 | Theorem *14.25 in [Whitehe... |
| sbaniota 45610 | Theorem *14.26 in [Whitehe... |
| iotasbcq 45611 | Theorem *14.272 in [Whiteh... |
| elnev 45612 | Any set that contains one ... |
| rusbcALT 45613 | A version of Russell's par... |
| compeq 45614 | Equality between two ways ... |
| compne 45615 | The complement of ` A ` is... |
| compab 45616 | Two ways of saying "the co... |
| conss2 45617 | Contrapositive law for sub... |
| conss1 45618 | Contrapositive law for sub... |
| ralbidar 45619 | More general form of ~ ral... |
| rexbidar 45620 | More general form of ~ rex... |
| dropab1 45621 | Theorem to aid use of the ... |
| dropab2 45622 | Theorem to aid use of the ... |
| ipo0 45623 | If the identity relation p... |
| ifr0 45624 | A class that is founded by... |
| fvsb 45625 | Explicit substitution of a... |
| fveqsb 45626 | Implicit substitution of a... |
| xpexb 45627 | A Cartesian product exists... |
| trelpss 45628 | An element of a transitive... |
| addcomgi 45629 | Generalization of commutat... |
| addrval 45639 | Value of the operation of ... |
| subrval 45640 | Value of the operation of ... |
| mulvval 45641 | Value of the operation of ... |
| addrfv 45642 | Vector addition at a value... |
| subrfv 45643 | Vector subtraction at a va... |
| mulvfv 45644 | Scalar multiplication at a... |
| addrfn 45645 | Vector addition produces a... |
| subrfn 45646 | Vector subtraction produce... |
| mulvfn 45647 | Scalar multiplication prod... |
| addrcom 45648 | Vector addition is commuta... |
| idiALT 45652 | Placeholder for ~ idi . T... |
| exbir 45653 | Exportation implication al... |
| 3impexpbicom 45654 | Version of ~ 3impexp where... |
| 3impexpbicomi 45655 | Inference associated with ... |
| bi1imp 45656 | Importation inference simi... |
| bi2imp 45657 | Importation inference simi... |
| bi3impb 45658 | Similar to ~ 3impb with im... |
| bi3impa 45659 | Similar to ~ 3impa with im... |
| bi23impib 45660 | ~ 3impib with the inner im... |
| bi13impib 45661 | ~ 3impib with the outer im... |
| bi123impib 45662 | ~ 3impib with the implicat... |
| bi13impia 45663 | ~ 3impia with the outer im... |
| bi123impia 45664 | ~ 3impia with the implicat... |
| bi33imp12 45665 | ~ 3imp with innermost impl... |
| bi13imp23 45666 | ~ 3imp with outermost impl... |
| bi13imp2 45667 | Similar to ~ 3imp except t... |
| bi12imp3 45668 | Similar to ~ 3imp except a... |
| bi23imp1 45669 | Similar to ~ 3imp except a... |
| bi123imp0 45670 | Similar to ~ 3imp except a... |
| 4animp1 45671 | A single hypothesis unific... |
| 4an31 45672 | A rearrangement of conjunc... |
| 4an4132 45673 | A rearrangement of conjunc... |
| expcomdg 45674 | Biconditional form of ~ ex... |
| iidn3 45675 | ~ idn3 without virtual ded... |
| ee222 45676 | ~ e222 without virtual ded... |
| ee3bir 45677 | Right-biconditional form o... |
| ee13 45678 | ~ e13 without virtual dedu... |
| ee121 45679 | ~ e121 without virtual ded... |
| ee122 45680 | ~ e122 without virtual ded... |
| ee333 45681 | ~ e333 without virtual ded... |
| ee323 45682 | ~ e323 without virtual ded... |
| 3ornot23 45683 | If the second and third di... |
| orbi1r 45684 | ~ orbi1 with order of disj... |
| 3orbi123 45685 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 45686 | Closed form of ~ syl5 . D... |
| impexpd 45687 | The following User's Proof... |
| com3rgbi 45688 | The following User's Proof... |
| impexpdcom 45689 | The following User's Proof... |
| ee1111 45690 | Non-virtual deduction form... |
| pm2.43bgbi 45691 | Logical equivalence of a 2... |
| pm2.43cbi 45692 | Logical equivalence of a 3... |
| ee233 45693 | Non-virtual deduction form... |
| imbi13 45694 | Join three logical equival... |
| ee33 45695 | Non-virtual deduction form... |
| con5 45696 | Biconditional contrapositi... |
| con5i 45697 | Inference form of ~ con5 .... |
| exlimexi 45698 | Inference similar to Theor... |
| sb5ALT 45699 | Equivalence for substituti... |
| eexinst01 45700 | ~ exinst01 without virtual... |
| eexinst11 45701 | ~ exinst11 without virtual... |
| vk15.4j 45702 | Excercise 4j of Unit 15 of... |
| notnotrALT 45703 | Converse of double negatio... |
| con3ALT2 45704 | Contraposition. Alternate... |
| ssralv2 45705 | Quantification restricted ... |
| sbc3or 45706 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 45707 | Closed form of ~ alrimi wi... |
| rspsbc2 45708 | ~ rspsbc with two quantify... |
| sbcoreleleq 45709 | Substitution of a setvar v... |
| tratrb 45710 | If a class is transitive a... |
| ordelordALT 45711 | An element of an ordinal c... |
| sbcim2g 45712 | Distribution of class subs... |
| sbcbi 45713 | Implication form of ~ sbcb... |
| trsbc 45714 | Formula-building inference... |
| truniALT 45715 | The union of a class of tr... |
| onfrALTlem5 45716 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 45717 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 45718 | Lemma for ~ onfrALT . (Co... |
| ggen31 45719 | ~ gen31 without virtual de... |
| onfrALTlem2 45720 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 45721 | A theorem pertaining to th... |
| onfrALTlem1 45722 | Lemma for ~ onfrALT . (Co... |
| onfrALT 45723 | The membership relation is... |
| 19.41rg 45724 | Closed form of right-to-le... |
| opelopab4 45725 | Ordered pair membership in... |
| 2pm13.193 45726 | ~ pm13.193 for two variabl... |
| hbntal 45727 | A closed form of ~ hbn . ~... |
| hbimpg 45728 | A closed form of ~ hbim . ... |
| hbalg 45729 | Closed form of ~ hbal . D... |
| hbexg 45730 | Closed form of ~ nfex . D... |
| ax6e2eq 45731 | Alternate form of ~ ax6e f... |
| ax6e2nd 45732 | If at least two sets exist... |
| ax6e2ndeq 45733 | "At least two sets exist" ... |
| 2sb5nd 45734 | Equivalence for double sub... |
| 2uasbanh 45735 | Distribute the unabbreviat... |
| 2uasban 45736 | Distribute the unabbreviat... |
| e2ebind 45737 | Absorption of an existenti... |
| elpwgded 45738 | ~ elpwgdedVD in convention... |
| trelded 45739 | Deduction form of ~ trel .... |
| jaoded 45740 | Deduction form of ~ jao . ... |
| sbtT 45741 | A substitution into a theo... |
| not12an2impnot1 45742 | If a double conjunction is... |
| in1 45745 | Inference form of ~ df-vd1... |
| iin1 45746 | ~ in1 without virtual dedu... |
| dfvd1ir 45747 | Inference form of ~ df-vd1... |
| idn1 45748 | Virtual deduction identity... |
| dfvd1imp 45749 | Left-to-right part of defi... |
| dfvd1impr 45750 | Right-to-left part of defi... |
| dfvd2 45753 | Definition of a 2-hypothes... |
| dfvd2an 45756 | Definition of a 2-hypothes... |
| dfvd2ani 45757 | Inference form of ~ dfvd2a... |
| dfvd2anir 45758 | Right-to-left inference fo... |
| dfvd2i 45759 | Inference form of ~ dfvd2 ... |
| dfvd2ir 45760 | Right-to-left inference fo... |
| dfvd3 45765 | Definition of a 3-hypothes... |
| dfvd3i 45766 | Inference form of ~ dfvd3 ... |
| dfvd3ir 45767 | Right-to-left inference fo... |
| dfvd3an 45768 | Definition of a 3-hypothes... |
| dfvd3ani 45769 | Inference form of ~ dfvd3a... |
| dfvd3anir 45770 | Right-to-left inference fo... |
| vd01 45771 | A virtual hypothesis virtu... |
| vd02 45772 | Two virtual hypotheses vir... |
| vd03 45773 | A theorem is virtually inf... |
| vd12 45774 | A virtual deduction with 1... |
| vd13 45775 | A virtual deduction with 1... |
| vd23 45776 | A virtual deduction with 2... |
| dfvd2imp 45777 | The virtual deduction form... |
| dfvd2impr 45778 | A 2-antecedent nested impl... |
| in2 45779 | The virtual deduction intr... |
| int2 45780 | The virtual deduction intr... |
| iin2 45781 | ~ in2 without virtual dedu... |
| in2an 45782 | The virtual deduction intr... |
| in3 45783 | The virtual deduction intr... |
| iin3 45784 | ~ in3 without virtual dedu... |
| in3an 45785 | The virtual deduction intr... |
| int3 45786 | The virtual deduction intr... |
| idn2 45787 | Virtual deduction identity... |
| iden2 45788 | Virtual deduction identity... |
| idn3 45789 | Virtual deduction identity... |
| gen11 45790 | Virtual deduction generali... |
| gen11nv 45791 | Virtual deduction generali... |
| gen12 45792 | Virtual deduction generali... |
| gen21 45793 | Virtual deduction generali... |
| gen21nv 45794 | Virtual deduction form of ... |
| gen31 45795 | Virtual deduction generali... |
| gen22 45796 | Virtual deduction generali... |
| ggen22 45797 | ~ gen22 without virtual de... |
| exinst 45798 | Existential Instantiation.... |
| exinst01 45799 | Existential Instantiation.... |
| exinst11 45800 | Existential Instantiation.... |
| e1a 45801 | A Virtual deduction elimin... |
| el1 45802 | A Virtual deduction elimin... |
| e1bi 45803 | Biconditional form of ~ e1... |
| e1bir 45804 | Right biconditional form o... |
| e2 45805 | A virtual deduction elimin... |
| e2bi 45806 | Biconditional form of ~ e2... |
| e2bir 45807 | Right biconditional form o... |
| ee223 45808 | ~ e223 without virtual ded... |
| e223 45809 | A virtual deduction elimin... |
| e222 45810 | A virtual deduction elimin... |
| e220 45811 | A virtual deduction elimin... |
| ee220 45812 | ~ e220 without virtual ded... |
| e202 45813 | A virtual deduction elimin... |
| ee202 45814 | ~ e202 without virtual ded... |
| e022 45815 | A virtual deduction elimin... |
| ee022 45816 | ~ e022 without virtual ded... |
| e002 45817 | A virtual deduction elimin... |
| ee002 45818 | ~ e002 without virtual ded... |
| e020 45819 | A virtual deduction elimin... |
| ee020 45820 | ~ e020 without virtual ded... |
| e200 45821 | A virtual deduction elimin... |
| ee200 45822 | ~ e200 without virtual ded... |
| e221 45823 | A virtual deduction elimin... |
| ee221 45824 | ~ e221 without virtual ded... |
| e212 45825 | A virtual deduction elimin... |
| ee212 45826 | ~ e212 without virtual ded... |
| e122 45827 | A virtual deduction elimin... |
| e112 45828 | A virtual deduction elimin... |
| ee112 45829 | ~ e112 without virtual ded... |
| e121 45830 | A virtual deduction elimin... |
| e211 45831 | A virtual deduction elimin... |
| ee211 45832 | ~ e211 without virtual ded... |
| e210 45833 | A virtual deduction elimin... |
| ee210 45834 | ~ e210 without virtual ded... |
| e201 45835 | A virtual deduction elimin... |
| ee201 45836 | ~ e201 without virtual ded... |
| e120 45837 | A virtual deduction elimin... |
| ee120 45838 | Virtual deduction rule ~ e... |
| e021 45839 | A virtual deduction elimin... |
| ee021 45840 | ~ e021 without virtual ded... |
| e012 45841 | A virtual deduction elimin... |
| ee012 45842 | ~ e012 without virtual ded... |
| e102 45843 | A virtual deduction elimin... |
| ee102 45844 | ~ e102 without virtual ded... |
| e22 45845 | A virtual deduction elimin... |
| e22an 45846 | Conjunction form of ~ e22 ... |
| ee22an 45847 | ~ e22an without virtual de... |
| e111 45848 | A virtual deduction elimin... |
| e1111 45849 | A virtual deduction elimin... |
| e110 45850 | A virtual deduction elimin... |
| ee110 45851 | ~ e110 without virtual ded... |
| e101 45852 | A virtual deduction elimin... |
| ee101 45853 | ~ e101 without virtual ded... |
| e011 45854 | A virtual deduction elimin... |
| ee011 45855 | ~ e011 without virtual ded... |
| e100 45856 | A virtual deduction elimin... |
| ee100 45857 | ~ e100 without virtual ded... |
| e010 45858 | A virtual deduction elimin... |
| ee010 45859 | ~ e010 without virtual ded... |
| e001 45860 | A virtual deduction elimin... |
| ee001 45861 | ~ e001 without virtual ded... |
| e11 45862 | A virtual deduction elimin... |
| e11an 45863 | Conjunction form of ~ e11 ... |
| ee11an 45864 | ~ e11an without virtual de... |
| e01 45865 | A virtual deduction elimin... |
| e01an 45866 | Conjunction form of ~ e01 ... |
| ee01an 45867 | ~ e01an without virtual de... |
| e10 45868 | A virtual deduction elimin... |
| e10an 45869 | Conjunction form of ~ e10 ... |
| ee10an 45870 | ~ e10an without virtual de... |
| e02 45871 | A virtual deduction elimin... |
| e02an 45872 | Conjunction form of ~ e02 ... |
| ee02an 45873 | ~ e02an without virtual de... |
| eel021old 45874 | ~ el021old without virtual... |
| el021old 45875 | A virtual deduction elimin... |
| eel000cT 45876 | An elimination deduction. ... |
| eel0TT 45877 | An elimination deduction. ... |
| eelT00 45878 | An elimination deduction. ... |
| eelTTT 45879 | An elimination deduction. ... |
| eelT11 45880 | An elimination deduction. ... |
| eelT1 45881 | Syllogism inference combin... |
| eelT12 45882 | An elimination deduction. ... |
| eelTT1 45883 | An elimination deduction. ... |
| eelT01 45884 | An elimination deduction. ... |
| eel0T1 45885 | An elimination deduction. ... |
| eel12131 45886 | An elimination deduction. ... |
| eel2131 45887 | ~ syl2an with antecedents ... |
| eel3132 45888 | ~ syl2an with antecedents ... |
| eel0321old 45889 | ~ el0321old without virtua... |
| el0321old 45890 | A virtual deduction elimin... |
| eel2122old 45891 | ~ el2122old without virtua... |
| el2122old 45892 | A virtual deduction elimin... |
| eel0000 45893 | Elimination rule similar t... |
| eel00001 45894 | An elimination deduction. ... |
| eel00000 45895 | Elimination rule similar ~... |
| eel11111 45896 | Five-hypothesis eliminatio... |
| e12 45897 | A virtual deduction elimin... |
| e12an 45898 | Conjunction form of ~ e12 ... |
| el12 45899 | Virtual deduction form of ... |
| e20 45900 | A virtual deduction elimin... |
| e20an 45901 | Conjunction form of ~ e20 ... |
| ee20an 45902 | ~ e20an without virtual de... |
| e21 45903 | A virtual deduction elimin... |
| e21an 45904 | Conjunction form of ~ e21 ... |
| ee21an 45905 | ~ e21an without virtual de... |
| e333 45906 | A virtual deduction elimin... |
| e33 45907 | A virtual deduction elimin... |
| e33an 45908 | Conjunction form of ~ e33 ... |
| ee33an 45909 | ~ e33an without virtual de... |
| e3 45910 | Meta-connective form of ~ ... |
| e3bi 45911 | Biconditional form of ~ e3... |
| e3bir 45912 | Right biconditional form o... |
| e03 45913 | A virtual deduction elimin... |
| ee03 45914 | ~ e03 without virtual dedu... |
| e03an 45915 | Conjunction form of ~ e03 ... |
| ee03an 45916 | Conjunction form of ~ ee03... |
| e30 45917 | A virtual deduction elimin... |
| ee30 45918 | ~ e30 without virtual dedu... |
| e30an 45919 | A virtual deduction elimin... |
| ee30an 45920 | Conjunction form of ~ ee30... |
| e13 45921 | A virtual deduction elimin... |
| e13an 45922 | A virtual deduction elimin... |
| ee13an 45923 | ~ e13an without virtual de... |
| e31 45924 | A virtual deduction elimin... |
| ee31 45925 | ~ e31 without virtual dedu... |
| e31an 45926 | A virtual deduction elimin... |
| ee31an 45927 | ~ e31an without virtual de... |
| e23 45928 | A virtual deduction elimin... |
| e23an 45929 | A virtual deduction elimin... |
| ee23an 45930 | ~ e23an without virtual de... |
| e32 45931 | A virtual deduction elimin... |
| ee32 45932 | ~ e32 without virtual dedu... |
| e32an 45933 | A virtual deduction elimin... |
| ee32an 45934 | ~ e33an without virtual de... |
| e123 45935 | A virtual deduction elimin... |
| ee123 45936 | ~ e123 without virtual ded... |
| el123 45937 | A virtual deduction elimin... |
| e233 45938 | A virtual deduction elimin... |
| e323 45939 | A virtual deduction elimin... |
| e000 45940 | A virtual deduction elimin... |
| e00 45941 | Elimination rule identical... |
| e00an 45942 | Elimination rule identical... |
| eel00cT 45943 | An elimination deduction. ... |
| eelTT 45944 | An elimination deduction. ... |
| e0a 45945 | Elimination rule identical... |
| eelT 45946 | An elimination deduction. ... |
| eel0cT 45947 | An elimination deduction. ... |
| eelT0 45948 | An elimination deduction. ... |
| e0bi 45949 | Elimination rule identical... |
| e0bir 45950 | Elimination rule identical... |
| uun0.1 45951 | Convention notation form o... |
| un0.1 45952 | ` T. ` is the constant tru... |
| uunT1 45953 | A deduction unionizing a n... |
| uunT1p1 45954 | A deduction unionizing a n... |
| uunT21 45955 | A deduction unionizing a n... |
| uun121 45956 | A deduction unionizing a n... |
| uun121p1 45957 | A deduction unionizing a n... |
| uun132 45958 | A deduction unionizing a n... |
| uun132p1 45959 | A deduction unionizing a n... |
| anabss7p1 45960 | A deduction unionizing a n... |
| un10 45961 | A unionizing deduction. (... |
| un01 45962 | A unionizing deduction. (... |
| un2122 45963 | A deduction unionizing a n... |
| uun2131 45964 | A deduction unionizing a n... |
| uun2131p1 45965 | A deduction unionizing a n... |
| uunTT1 45966 | A deduction unionizing a n... |
| uunTT1p1 45967 | A deduction unionizing a n... |
| uunTT1p2 45968 | A deduction unionizing a n... |
| uunT11 45969 | A deduction unionizing a n... |
| uunT11p1 45970 | A deduction unionizing a n... |
| uunT11p2 45971 | A deduction unionizing a n... |
| uunT12 45972 | A deduction unionizing a n... |
| uunT12p1 45973 | A deduction unionizing a n... |
| uunT12p2 45974 | A deduction unionizing a n... |
| uunT12p3 45975 | A deduction unionizing a n... |
| uunT12p4 45976 | A deduction unionizing a n... |
| uunT12p5 45977 | A deduction unionizing a n... |
| uun111 45978 | A deduction unionizing a n... |
| 3anidm12p1 45979 | A deduction unionizing a n... |
| 3anidm12p2 45980 | A deduction unionizing a n... |
| uun123 45981 | A deduction unionizing a n... |
| uun123p1 45982 | A deduction unionizing a n... |
| uun123p2 45983 | A deduction unionizing a n... |
| uun123p3 45984 | A deduction unionizing a n... |
| uun123p4 45985 | A deduction unionizing a n... |
| uun2221 45986 | A deduction unionizing a n... |
| uun2221p1 45987 | A deduction unionizing a n... |
| uun2221p2 45988 | A deduction unionizing a n... |
| 3impdirp1 45989 | A deduction unionizing a n... |
| 3impcombi 45990 | A 1-hypothesis proposition... |
| trsspwALT 45991 | Virtual deduction proof of... |
| trsspwALT2 45992 | Virtual deduction proof of... |
| trsspwALT3 45993 | Short predicate calculus p... |
| sspwtr 45994 | Virtual deduction proof of... |
| sspwtrALT 45995 | Virtual deduction proof of... |
| sspwtrALT2 45996 | Short predicate calculus p... |
| pwtrVD 45997 | Virtual deduction proof of... |
| pwtrrVD 45998 | Virtual deduction proof of... |
| suctrALT 45999 | The successor of a transit... |
| snssiALTVD 46000 | Virtual deduction proof of... |
| snssiALT 46001 | If a class is an element o... |
| snsslVD 46002 | Virtual deduction proof of... |
| snssl 46003 | If a singleton is a subcla... |
| snelpwrVD 46004 | Virtual deduction proof of... |
| unipwrVD 46005 | Virtual deduction proof of... |
| unipwr 46006 | A class is a subclass of t... |
| sstrALT2VD 46007 | Virtual deduction proof of... |
| sstrALT2 46008 | Virtual deduction proof of... |
| suctrALT2VD 46009 | Virtual deduction proof of... |
| suctrALT2 46010 | Virtual deduction proof of... |
| elex2VD 46011 | Virtual deduction proof of... |
| elex22VD 46012 | Virtual deduction proof of... |
| eqsbc2VD 46013 | Virtual deduction proof of... |
| zfregs2VD 46014 | Virtual deduction proof of... |
| tpid3gVD 46015 | Virtual deduction proof of... |
| en3lplem1VD 46016 | Virtual deduction proof of... |
| en3lplem2VD 46017 | Virtual deduction proof of... |
| en3lpVD 46018 | Virtual deduction proof of... |
| simplbi2VD 46019 | Virtual deduction proof of... |
| 3ornot23VD 46020 | Virtual deduction proof of... |
| orbi1rVD 46021 | Virtual deduction proof of... |
| bitr3VD 46022 | Virtual deduction proof of... |
| 3orbi123VD 46023 | Virtual deduction proof of... |
| sbc3orgVD 46024 | Virtual deduction proof of... |
| 19.21a3con13vVD 46025 | Virtual deduction proof of... |
| exbirVD 46026 | Virtual deduction proof of... |
| exbiriVD 46027 | Virtual deduction proof of... |
| rspsbc2VD 46028 | Virtual deduction proof of... |
| 3impexpVD 46029 | Virtual deduction proof of... |
| 3impexpbicomVD 46030 | Virtual deduction proof of... |
| 3impexpbicomiVD 46031 | Virtual deduction proof of... |
| sbcoreleleqVD 46032 | Virtual deduction proof of... |
| hbra2VD 46033 | Virtual deduction proof of... |
| tratrbVD 46034 | Virtual deduction proof of... |
| al2imVD 46035 | Virtual deduction proof of... |
| syl5impVD 46036 | Virtual deduction proof of... |
| idiVD 46037 | Virtual deduction proof of... |
| ancomstVD 46038 | Closed form of ~ ancoms . ... |
| ssralv2VD 46039 | Quantification restricted ... |
| ordelordALTVD 46040 | An element of an ordinal c... |
| equncomVD 46041 | If a class equals the unio... |
| equncomiVD 46042 | Inference form of ~ equnco... |
| sucidALTVD 46043 | A set belongs to its succe... |
| sucidALT 46044 | A set belongs to its succe... |
| sucidVD 46045 | A set belongs to its succe... |
| imbi12VD 46046 | Implication form of ~ imbi... |
| imbi13VD 46047 | Join three logical equival... |
| sbcim2gVD 46048 | Distribution of class subs... |
| sbcbiVD 46049 | Implication form of ~ sbcb... |
| trsbcVD 46050 | Formula-building inference... |
| truniALTVD 46051 | The union of a class of tr... |
| ee33VD 46052 | Non-virtual deduction form... |
| trintALTVD 46053 | The intersection of a clas... |
| trintALT 46054 | The intersection of a clas... |
| undif3VD 46055 | The first equality of Exer... |
| sbcssgVD 46056 | Virtual deduction proof of... |
| csbingVD 46057 | Virtual deduction proof of... |
| onfrALTlem5VD 46058 | Virtual deduction proof of... |
| onfrALTlem4VD 46059 | Virtual deduction proof of... |
| onfrALTlem3VD 46060 | Virtual deduction proof of... |
| simplbi2comtVD 46061 | Virtual deduction proof of... |
| onfrALTlem2VD 46062 | Virtual deduction proof of... |
| onfrALTlem1VD 46063 | Virtual deduction proof of... |
| onfrALTVD 46064 | Virtual deduction proof of... |
| csbeq2gVD 46065 | Virtual deduction proof of... |
| csbsngVD 46066 | Virtual deduction proof of... |
| csbxpgVD 46067 | Virtual deduction proof of... |
| csbresgVD 46068 | Virtual deduction proof of... |
| csbrngVD 46069 | Virtual deduction proof of... |
| csbima12gALTVD 46070 | Virtual deduction proof of... |
| csbunigVD 46071 | Virtual deduction proof of... |
| csbfv12gALTVD 46072 | Virtual deduction proof of... |
| con5VD 46073 | Virtual deduction proof of... |
| relopabVD 46074 | Virtual deduction proof of... |
| 19.41rgVD 46075 | Virtual deduction proof of... |
| 2pm13.193VD 46076 | Virtual deduction proof of... |
| hbimpgVD 46077 | Virtual deduction proof of... |
| hbalgVD 46078 | Virtual deduction proof of... |
| hbexgVD 46079 | Virtual deduction proof of... |
| ax6e2eqVD 46080 | The following User's Proof... |
| ax6e2ndVD 46081 | The following User's Proof... |
| ax6e2ndeqVD 46082 | The following User's Proof... |
| 2sb5ndVD 46083 | The following User's Proof... |
| 2uasbanhVD 46084 | The following User's Proof... |
| e2ebindVD 46085 | The following User's Proof... |
| sb5ALTVD 46086 | The following User's Proof... |
| vk15.4jVD 46087 | The following User's Proof... |
| notnotrALTVD 46088 | The following User's Proof... |
| con3ALTVD 46089 | The following User's Proof... |
| elpwgdedVD 46090 | Membership in a power clas... |
| sspwimp 46091 | If a class is a subclass o... |
| sspwimpVD 46092 | The following User's Proof... |
| sspwimpcf 46093 | If a class is a subclass o... |
| sspwimpcfVD 46094 | The following User's Proof... |
| suctrALTcf 46095 | The successor of a transit... |
| suctrALTcfVD 46096 | The following User's Proof... |
| suctrALT3 46097 | The successor of a transit... |
| sspwimpALT 46098 | If a class is a subclass o... |
| unisnALT 46099 | A set equals the union of ... |
| notnotrALT2 46100 | Converse of double negatio... |
| sspwimpALT2 46101 | If a class is a subclass o... |
| e2ebindALT 46102 | Absorption of an existenti... |
| ax6e2ndALT 46103 | If at least two sets exist... |
| ax6e2ndeqALT 46104 | "At least two sets exist" ... |
| 2sb5ndALT 46105 | Equivalence for double sub... |
| chordthmALT 46106 | The intersecting chords th... |
| isosctrlem1ALT 46107 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 46108 | The indexed union of conne... |
| iunconnALT 46109 | The indexed union of conne... |
| sineq0ALT 46110 | A complex number whose sin... |
| rspesbcd 46111 | Restricted quantifier vers... |
| rext0 46112 | Nonempty existential quant... |
| relpropg 46113 | A pair of ordered pairs is... |
| cocanss2 46114 | Cancellation law for compo... |
| cocan2g 46115 | Cancellation law for compo... |
| cocanss1 46116 | Cancellation law for compo... |
| cocan1g 46117 | Cancellation law for compo... |
| mpoexwd 46118 | Deduction version of ~ mpo... |
| dmstructnn 46119 | The domain of an extensibl... |
| dmstructfi 46120 | The domain of an extensibl... |
| rnstructfi 46121 | The range of an extensible... |
| imasgimw 46122 | A relabeling of the elemen... |
| grpgimcanon 46123 | The result of "canonicaliz... |
| dfbi1ALTa 46124 | Version of ~ dfbi1ALT usin... |
| simprimi 46125 | Inference associated with ... |
| dfbi1ALTb 46126 | Further shorten ~ dfbi1ALT... |
| relpeq1 46129 | Equality theorem for relat... |
| relpeq2 46130 | Equality theorem for relat... |
| relpeq3 46131 | Equality theorem for relat... |
| relpeq4 46132 | Equality theorem for relat... |
| relpeq5 46133 | Equality theorem for relat... |
| nfrelp 46134 | Bound-variable hypothesis ... |
| relpf 46135 | A relation-preserving func... |
| relprel 46136 | A relation-preserving func... |
| relpmin 46137 | A preimage of a minimal el... |
| relpfrlem 46138 | Lemma for ~ relpfr . Prov... |
| relpfr 46139 | If the image of a set unde... |
| orbitex 46140 | Orbits exist. Given a set... |
| orbitinit 46141 | A set is contained in its ... |
| orbitcl 46142 | The orbit under a function... |
| orbitclmpt 46143 | Version of ~ orbitcl using... |
| trwf 46144 | The class of well-founded ... |
| rankrelp 46145 | The rank function preserve... |
| wffr 46146 | The class of well-founded ... |
| trfr 46147 | A transitive class well-fo... |
| tcfr 46148 | A set is well-founded if a... |
| xpwf 46149 | The Cartesian product of t... |
| dmwf 46150 | The domain of a well-found... |
| rnwf 46151 | The range of a well-founde... |
| relwf 46152 | A relation is a well-found... |
| ralabso 46153 | Simplification of restrict... |
| rexabso 46154 | Simplification of restrict... |
| ralabsod 46155 | Deduction form of ~ ralabs... |
| rexabsod 46156 | Deduction form of ~ rexabs... |
| ralabsobidv 46157 | Formula-building lemma for... |
| rexabsobidv 46158 | Formula-building lemma for... |
| ssabso 46159 | The notion " ` x ` is a su... |
| disjabso 46160 | Disjointness is absolute f... |
| n0abso 46161 | Nonemptiness is absolute f... |
| traxext 46162 | A transitive class models ... |
| modelaxreplem1 46163 | Lemma for ~ modelaxrep . ... |
| modelaxreplem2 46164 | Lemma for ~ modelaxrep . ... |
| modelaxreplem3 46165 | Lemma for ~ modelaxrep . ... |
| modelaxrep 46166 | Conditions which guarantee... |
| ssclaxsep 46167 | A class that is closed und... |
| 0elaxnul 46168 | A class that contains the ... |
| pwclaxpow 46169 | Suppose ` M ` is a transit... |
| prclaxpr 46170 | A class that is closed und... |
| uniclaxun 46171 | A class that is closed und... |
| sswfaxreg 46172 | A subclass of the class of... |
| omssaxinf2 46173 | A class that contains all ... |
| omelaxinf2 46174 | A transitive class that co... |
| dfac5prim 46175 | ~ dfac5 expanded into prim... |
| ac8prim 46176 | ~ ac8 expanded into primit... |
| modelac8prim 46177 | If ` M ` is a transitive c... |
| wfaxext 46178 | The class of well-founded ... |
| wfaxrep 46179 | The class of well-founded ... |
| wfaxsep 46180 | The class of well-founded ... |
| wfaxnul 46181 | The class of well-founded ... |
| wfaxpow 46182 | The class of well-founded ... |
| wfaxpr 46183 | The class of well-founded ... |
| wfaxun 46184 | The class of well-founded ... |
| wfaxreg 46185 | The class of well-founded ... |
| wfaxinf2 46186 | The class of well-founded ... |
| wfac8prim 46187 | The class of well-founded ... |
| brpermmodel 46188 | The membership relation in... |
| brpermmodelcnv 46189 | Ordinary membership expres... |
| permaxext 46190 | The Axiom of Extensionalit... |
| permaxrep 46191 | The Axiom of Replacement ~... |
| permaxsep 46192 | The Axiom of Separation ~ ... |
| permaxnul 46193 | The Null Set Axiom ~ ax-nu... |
| permaxpow 46194 | The Axiom of Power Sets ~ ... |
| permaxpr 46195 | The Axiom of Pairing ~ ax-... |
| permaxun 46196 | The Axiom of Union ~ ax-un... |
| permaxinf2lem 46197 | Lemma for ~ permaxinf2 . ... |
| permaxinf2 46198 | The Axiom of Infinity ~ ax... |
| permac8prim 46199 | The Axiom of Choice ~ ac8p... |
| nregmodelf1o 46200 | Define a permutation ` F `... |
| nregmodellem 46201 | Lemma for ~ nregmodel . (... |
| nregmodel 46202 | The Axiom of Regularity ~ ... |
| nregmodelaxext 46203 | The Axiom of Extensionalit... |
| hashnna 46204 | The ` # ` function on ` _o... |
| hashnnsuc 46205 | The ` # ` function on ` _o... |
| hashnnm 46206 | The ` # ` function on ` _o... |
| hashnnlt 46207 | The ` # ` function on ` _o... |
| hashnnltb 46208 | The ` # ` function on ` _o... |
| hashomf1o 46209 | The ` # ` function yields ... |
| hashomiso 46210 | The ` # ` function yields ... |
| hfpr 46211 | A pair whose members are h... |
| hfxp 46212 | The Cartesian product of t... |
| hfdm 46213 | The domain of a hereditari... |
| hfrn 46214 | The range of a hereditaril... |
| hfrel 46215 | A relation is a hereditari... |
| omhf 46216 | Finite ordinals are heredi... |
| omsshf 46217 | The class of finite ordina... |
| ackbij3 46220 | The Ackermann bijection is... |
| hfstructstruct 46223 | An HFStruct is an extensib... |
| hfstructfun 46224 | An HFStruct is a function.... |
| rnhfstructsshf 46225 | The range of an HFStruct c... |
| ishfstruct 46226 | Conditions that imply that... |
| rnhfstructhf 46227 | The range of an HFStruct i... |
| hfstructhf 46228 | A converted HFStruct is a ... |
| hfstructcan 46229 | Two HFStructs are equal if... |
| grpcanonhfstruct 46230 | A group defined on a hered... |
| fingrphf 46231 | Every finite group is isom... |
| hfstructwe 46234 |
The relation ` |
| evth2f 46237 | A version of ~ evth2 using... |
| elunif 46238 | A version of ~ eluni using... |
| rzalf 46239 | A version of ~ rzal using ... |
| fvelrnbf 46240 | A version of ~ fvelrnb usi... |
| rfcnpre1 46241 | If F is a continuous funct... |
| ubelsupr 46242 | If U belongs to A and U is... |
| fsumcnf 46243 | A finite sum of functions ... |
| mulltgt0 46244 | The product of a negative ... |
| rspcegf 46245 | A version of ~ rspcev usin... |
| rabexgf 46246 | A version of ~ rabexg usin... |
| fcnre 46247 | A function continuous with... |
| sumsnd 46248 | A sum of a singleton is th... |
| evthf 46249 | A version of ~ evth using ... |
| cnfex 46250 | The class of continuous fu... |
| fnchoice 46251 | For a finite set, a choice... |
| refsumcn 46252 | A finite sum of continuous... |
| rfcnpre2 46253 | If ` F ` is a continuous f... |
| cncmpmax 46254 | When the hypothesis for th... |
| rfcnpre3 46255 | If F is a continuous funct... |
| rfcnpre4 46256 | If F is a continuous funct... |
| sumpair 46257 | Sum of two distinct comple... |
| rfcnnnub 46258 | Given a real continuous fu... |
| refsum2cnlem1 46259 | This is the core Lemma for... |
| refsum2cn 46260 | The sum of two continuus r... |
| adantlllr 46261 | Deduction adding a conjunc... |
| 3adantlr3 46262 | Deduction adding a conjunc... |
| 3adantll2 46263 | Deduction adding a conjunc... |
| 3adantll3 46264 | Deduction adding a conjunc... |
| ssnel 46265 | If not element of a set, t... |
| sncldre 46266 | A singleton is closed w.r.... |
| n0p 46267 | A polynomial with a nonzer... |
| pm2.65ni 46268 | Inference rule for proof b... |
| iuneq2df 46269 | Equality deduction for ind... |
| nnfoctb 46270 | There exists a mapping fro... |
| elpwinss 46271 | An element of the powerset... |
| unidmex 46272 | If ` F ` is a set, then ` ... |
| ndisj2 46273 | A non-disjointness conditi... |
| zenom 46274 | The set of integer numbers... |
| uzwo4 46275 | Well-ordering principle: a... |
| unisn0 46276 | The union of the singleton... |
| ssin0 46277 | If two classes are disjoin... |
| inabs3 46278 | Absorption law for interse... |
| pwpwuni 46279 | Relationship between power... |
| disjiun2 46280 | In a disjoint collection, ... |
| 0pwfi 46281 | The empty set is in any po... |
| ssinss2d 46282 | Intersection preserves sub... |
| zct 46283 | The set of integer numbers... |
| pwfin0 46284 | A finite set always belong... |
| uzct 46285 | An upper integer set is co... |
| iunxsnf 46286 | A singleton index picks ou... |
| fiiuncl 46287 | If a set is closed under t... |
| iunp1 46288 | The addition of the next s... |
| fiunicl 46289 | If a set is closed under t... |
| ixpeq2d 46290 | Equality theorem for infin... |
| disjxp1 46291 | The sets of a cartesian pr... |
| disjsnxp 46292 | The sets in the cartesian ... |
| eliind 46293 | Membership in indexed inte... |
| rspcef 46294 | Restricted existential spe... |
| ixpssmapc 46295 | An infinite Cartesian prod... |
| elintd 46296 | Membership in class inters... |
| ssdf 46297 | A sufficient condition for... |
| brneqtrd 46298 | Substitution of equal clas... |
| ssnct 46299 | A set containing an uncoun... |
| ssuniint 46300 | Sufficient condition for b... |
| elintdv 46301 | Membership in class inters... |
| ssd 46302 | A sufficient condition for... |
| ralimralim 46303 | Introducing any antecedent... |
| snelmap 46304 | Membership of the element ... |
| xrnmnfpnf 46305 | An extended real that is n... |
| iuneq1i 46306 | Equality theorem for index... |
| ssinc 46307 | Inclusion relation for a m... |
| ssdec 46308 | Inclusion relation for a m... |
| elixpconstg 46309 | Membership in an infinite ... |
| iineq1d 46310 | Equality theorem for index... |
| metpsmet 46311 | A metric is a pseudometric... |
| ixpssixp 46312 | Subclass theorem for infin... |
| ballss3 46313 | A sufficient condition for... |
| iunincfi 46314 | Given a sequence of increa... |
| nsstr 46315 | If it's not a subclass, it... |
| rexanuz3 46316 | Combine two different uppe... |
| cbvmpo2 46317 | Rule to change the second ... |
| cbvmpo1 46318 | Rule to change the first b... |
| eliuniin 46319 | Indexed union of indexed i... |
| ssabf 46320 | Subclass of a class abstra... |
| pssnssi 46321 | A proper subclass does not... |
| rabidim2 46322 | Membership in a restricted... |
| eluni2f 46323 | Membership in class union.... |
| eliin2f 46324 | Membership in indexed inte... |
| nssd 46325 | Negation of subclass relat... |
| iineq12dv 46326 | Equality deduction for ind... |
| supxrcld 46327 | The supremum of an arbitra... |
| elrestd 46328 | A sufficient condition for... |
| eliuniincex 46329 | Counterexample to show tha... |
| eliincex 46330 | Counterexample to show tha... |
| eliinid 46331 | Membership in an indexed i... |
| abssf 46332 | Class abstraction in a sub... |
| supxrubd 46333 | A member of a set of exten... |
| ssrabf 46334 | Subclass of a restricted c... |
| ssrabdf 46335 | Subclass of a restricted c... |
| eliin2 46336 | Membership in indexed inte... |
| ssrab2f 46337 | Subclass relation for a re... |
| restuni3 46338 | The underlying set of a su... |
| rabssf 46339 | Restricted class abstracti... |
| eliuniin2 46340 | Indexed union of indexed i... |
| restuni4 46341 | The underlying set of a su... |
| restuni6 46342 | The underlying set of a su... |
| restuni5 46343 | The underlying set of a su... |
| unirestss 46344 | The union of an elementwis... |
| iniin1 46345 | Indexed intersection of in... |
| iniin2 46346 | Indexed intersection of in... |
| cbvrabv2 46347 | A more general version of ... |
| cbvrabv2w 46348 | A more general version of ... |
| iinssiin 46349 | Subset implication for an ... |
| eliind2 46350 | Membership in indexed inte... |
| iinssd 46351 | Subset implication for an ... |
| rabbida2 46352 | Equivalent wff's yield equ... |
| iinexd 46353 | The existence of an indexe... |
| rabexf 46354 | Separation Scheme in terms... |
| rabbida3 46355 | Equivalent wff's yield equ... |
| r19.36vf 46356 | Restricted quantifier vers... |
| raleqd 46357 | Equality deduction for res... |
| iinssf 46358 | Subset implication for an ... |
| iinssdf 46359 | Subset implication for an ... |
| resabs2i 46360 | Absorption law for restric... |
| ssdf2 46361 | A sufficient condition for... |
| rabssd 46362 | Restricted class abstracti... |
| rexnegd 46363 | Minus a real number. (Con... |
| rexlimd3 46364 | * Inference from Theorem 1... |
| nel1nelini 46365 | Membership in an intersect... |
| nel2nelini 46366 | Membership in an intersect... |
| eliunid 46367 | Membership in indexed unio... |
| reximdd 46368 | Deduction from Theorem 19.... |
| inopnd 46369 | The intersection of two op... |
| ss2rabdf 46370 | Deduction of restricted ab... |
| restopn3 46371 | If ` A ` is open, then ` A... |
| restopnssd 46372 | A topology restricted to a... |
| restsubel 46373 | A subset belongs in the sp... |
| toprestsubel 46374 | A subset is open in the to... |
| rabidd 46375 | An "identity" law of concr... |
| iunssdf 46376 | Subset theorem for an inde... |
| iinss2d 46377 | Subset implication for an ... |
| r19.3rzf 46378 | Restricted quantification ... |
| r19.28zf 46379 | Restricted quantifier vers... |
| iindif2f 46380 | Indexed intersection of cl... |
| ralfal 46381 | Two ways of expressing emp... |
| archd 46382 | Archimedean property of re... |
| nimnbi 46383 | If an implication is false... |
| nimnbi2 46384 | If an implication is false... |
| notbicom 46385 | Commutative law for the ne... |
| rexeqif 46386 | Equality inference for res... |
| rspced 46387 | Restricted existential spe... |
| fnresdmss 46388 | A function does not change... |
| fmptsnxp 46389 | Maps-to notation and Carte... |
| fvmpt2bd 46390 | Value of a function given ... |
| rnmptfi 46391 | The range of a function wi... |
| fresin2 46392 | Restriction of a function ... |
| ffi 46393 | A function with finite dom... |
| suprnmpt 46394 | An explicit bound for the ... |
| rnffi 46395 | The range of a function wi... |
| mptelpm 46396 | A function in maps-to nota... |
| rnmptpr 46397 | Range of a function define... |
| resmpti 46398 | Restriction of the mapping... |
| founiiun 46399 | Union expressed as an inde... |
| rnresun 46400 | Distribution law for range... |
| elrnmptf 46401 | The range of a function in... |
| rnmptssrn 46402 | Inclusion relation for two... |
| disjf1 46403 | A 1 to 1 mapping built fro... |
| rnsnf 46404 | The range of a function wh... |
| wessf1ornlem 46405 | Given a function ` F ` on ... |
| wessf1orn 46406 | Given a function ` F ` on ... |
| nelrnres 46407 | If ` A ` is not in the ran... |
| disjrnmpt2 46408 | Disjointness of the range ... |
| elrnmpt1sf 46409 | Elementhood in an image se... |
| founiiun0 46410 | Union expressed as an inde... |
| disjf1o 46411 | A bijection built from dis... |
| disjinfi 46412 | Only a finite number of di... |
| fvovco 46413 | Value of the composition o... |
| ssnnf1octb 46414 | There exists a bijection b... |
| nnf1oxpnn 46415 | There is a bijection betwe... |
| projf1o 46416 | A biijection from a set to... |
| fvmap 46417 | Function value for a membe... |
| fvixp2 46418 | Projection of a factor of ... |
| choicefi 46419 | For a finite set, a choice... |
| mpct 46420 | The exponentiation of a co... |
| cnmetcoval 46421 | Value of the distance func... |
| fcomptss 46422 | Express composition of two... |
| elmapsnd 46423 | Membership in a set expone... |
| mapss2 46424 | Subset inheritance for set... |
| difmap 46425 | Difference of two sets exp... |
| unirnmap 46426 | Given a subset of a set ex... |
| inmap 46427 | Intersection of two sets e... |
| fcoss 46428 | Composition of two mapping... |
| fsneqrn 46429 | Equality condition for two... |
| difmapsn 46430 | Difference of two sets exp... |
| mapssbi 46431 | Subset inheritance for set... |
| unirnmapsn 46432 | Equality theorem for a sub... |
| iunmapss 46433 | The indexed union of set e... |
| ssmapsn 46434 | A subset ` C ` of a set ex... |
| iunmapsn 46435 | The indexed union of set e... |
| absfico 46436 | Mapping domain and codomai... |
| icof 46437 | The set of left-closed rig... |
| elpmrn 46438 | The range of a partial fun... |
| imaexi 46439 | The image of a set is a se... |
| axccdom 46440 | Relax the constraint on ax... |
| dmmptdff 46441 | The domain of the mapping ... |
| dmmptdf 46442 | The domain of the mapping ... |
| elpmi2 46443 | The domain of a partial fu... |
| dmrelrnrel 46444 | A relation preserving func... |
| elrnmpoid 46445 | Membership in the range of... |
| axccd 46446 | An alternative version of ... |
| axccd2 46447 | An alternative version of ... |
| feqresmptf 46448 | Express a restricted funct... |
| dmmptssf 46449 | The domain of a mapping is... |
| dmmptdf2 46450 | The domain of the mapping ... |
| dmuz 46451 | Domain of the upper intege... |
| fmptd2f 46452 | Domain and codomain of the... |
| mpteq1df 46453 | An equality theorem for th... |
| mptexf 46454 | If the domain of a functio... |
| fvmpt4 46455 | Value of a function given ... |
| fmptf 46456 | Functionality of the mappi... |
| resimass 46457 | The image of a restriction... |
| mptssid 46458 | The mapping operation expr... |
| mptfnd 46459 | The maps-to notation defin... |
| rnmptlb 46460 | Boundness below of the ran... |
| rnmptbddlem 46461 | Boundness of the range of ... |
| rnmptbdd 46462 | Boundness of the range of ... |
| funimaeq 46463 | Membership relation for th... |
| rnmptssf 46464 | The range of a function gi... |
| rnmptbd2lem 46465 | Boundness below of the ran... |
| rnmptbd2 46466 | Boundness below of the ran... |
| infnsuprnmpt 46467 | The indexed infimum of rea... |
| suprclrnmpt 46468 | Closure of the indexed sup... |
| suprubrnmpt2 46469 | A member of a nonempty ind... |
| suprubrnmpt 46470 | A member of a nonempty ind... |
| rnmptssdf 46471 | The range of a function gi... |
| rnmptbdlem 46472 | Boundness above of the ran... |
| rnmptbd 46473 | Boundness above of the ran... |
| rnmptss2 46474 | The range of a function gi... |
| elmptima 46475 | The image of a function in... |
| ralrnmpt3 46476 | A restricted quantifier ov... |
| rnmptssbi 46477 | The range of a function gi... |
| imass2d 46478 | Subset theorem for image. ... |
| imassmpt 46479 | Membership relation for th... |
| fpmd 46480 | A total function is a part... |
| fconst7 46481 | An alternative way to expr... |
| fnmptif 46482 | Functionality and domain o... |
| dmmptif 46483 | Domain of the mapping oper... |
| mpteq2dfa 46484 | Slightly more general equa... |
| dmmpt1 46485 | The domain of the mapping ... |
| fmptff 46486 | Functionality of the mappi... |
| fvmptelcdmf 46487 | The value of a function at... |
| fmptdff 46488 | A version of ~ fmptd using... |
| fvmpt2df 46489 | Deduction version of ~ fvm... |
| rn1st 46490 | The range of a function wi... |
| rnmptssff 46491 | The range of a function gi... |
| rnmptssdff 46492 | The range of a function gi... |
| fvmpt4d 46493 | Value of a function given ... |
| sub2times 46494 | Subtracting from a number,... |
| nnxrd 46495 | A natural number is an ext... |
| nnxr 46496 | A natural number is an ext... |
| abssubrp 46497 | The distance of two distin... |
| elfzfzo 46498 | Relationship between membe... |
| oddfl 46499 | Odd number representation ... |
| abscosbd 46500 | Bound for the absolute val... |
| mul13d 46501 | Commutative/associative la... |
| negpilt0 46502 | Negative ` _pi ` is negati... |
| dstregt0 46503 | A complex number ` A ` tha... |
| subadd4b 46504 | Rearrangement of 4 terms i... |
| xrlttri5d 46505 | Not equal and not larger i... |
| zltlesub 46506 | If an integer ` N ` is les... |
| divlt0gt0d 46507 | The ratio of a negative nu... |
| subsub23d 46508 | Swap subtrahend and result... |
| 2timesgt 46509 | Double of a positive real ... |
| reopn 46510 | The reals are open with re... |
| sub31 46511 | Swap the first and third t... |
| nnne1ge2 46512 | A positive integer which i... |
| lefldiveq 46513 | A closed enough, smaller r... |
| negsubdi3d 46514 | Distribution of negative o... |
| ltdiv2dd 46515 | Division of a positive num... |
| abssinbd 46516 | Bound for the absolute val... |
| halffl 46517 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 46518 | Ordering relation for a st... |
| hashssle 46519 | The size of a subset of a ... |
| lttri5d 46520 | Not equal and not larger i... |
| fzisoeu 46521 | A finite ordered set has a... |
| lt3addmuld 46522 | If three real numbers are ... |
| absnpncan2d 46523 | Triangular inequality, com... |
| fperiodmullem 46524 | A function with period ` T... |
| fperiodmul 46525 | A function with period T i... |
| upbdrech 46526 | Choice of an upper bound f... |
| lt4addmuld 46527 | If four real numbers are l... |
| absnpncan3d 46528 | Triangular inequality, com... |
| upbdrech2 46529 | Choice of an upper bound f... |
| ssfiunibd 46530 | A finite union of bounded ... |
| fzdifsuc2 46531 | Remove a successor from th... |
| fzsscn 46532 | A finite sequence of integ... |
| divcan8d 46533 | A cancellation law for div... |
| dmmcand 46534 | Cancellation law for divis... |
| bccld 46535 | A binomial coefficient, in... |
| fzssnn0 46536 | A finite set of sequential... |
| xreqle 46537 | Equality implies 'less tha... |
| xaddlidd 46538 | ` 0 ` is a left identity f... |
| xadd0ge 46539 | A number is less than or e... |
| xrleneltd 46540 | 'Less than or equal to' an... |
| xaddcomd 46541 | The extended real addition... |
| supxrre3 46542 | The supremum of a nonempty... |
| uzfissfz 46543 | For any finite subset of t... |
| xleadd2d 46544 | Addition of extended reals... |
| suprltrp 46545 | The supremum of a nonempty... |
| xleadd1d 46546 | Addition of extended reals... |
| xreqled 46547 | Equality implies 'less tha... |
| xrgepnfd 46548 | An extended real greater t... |
| xrge0nemnfd 46549 | A nonnegative extended rea... |
| supxrgere 46550 | If a real number can be ap... |
| iuneqfzuzlem 46551 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 46552 | If two unions indexed by u... |
| xle2addd 46553 | Adding both side of two in... |
| supxrgelem 46554 | If an extended real number... |
| supxrge 46555 | If an extended real number... |
| suplesup 46556 | If any element of ` A ` ca... |
| infxrglb 46557 | The infimum of a set of ex... |
| xadd0ge2 46558 | A number is less than or e... |
| nepnfltpnf 46559 | An extended real that is n... |
| ltadd12dd 46560 | Addition to both sides of ... |
| nemnftgtmnft 46561 | An extended real that is n... |
| xrgtso 46562 | 'Greater than' is a strict... |
| rpex 46563 | The positive reals form a ... |
| xrge0ge0 46564 | A nonnegative extended rea... |
| xrssre 46565 | A subset of extended reals... |
| ssuzfz 46566 | A finite subset of the upp... |
| absfun 46567 | The absolute value is a fu... |
| infrpge 46568 | The infimum of a nonempty,... |
| xrlexaddrp 46569 | If an extended real number... |
| supsubc 46570 | The supremum function dist... |
| xralrple2 46571 | Show that ` A ` is less th... |
| nnuzdisj 46572 | The first ` N ` elements o... |
| ltdivgt1 46573 | Divsion by a number greate... |
| xrltned 46574 | 'Less than' implies not eq... |
| nnsplit 46575 | Express the set of positiv... |
| divdiv3d 46576 | Division into a fraction. ... |
| abslt2sqd 46577 | Comparison of the square o... |
| qenom 46578 | The set of rational number... |
| qct 46579 | The set of rational number... |
| lenlteq 46580 | 'less than or equal to' bu... |
| xrred 46581 | An extended real that is n... |
| rr2sscn2 46582 | The cartesian square of ` ... |
| infxr 46583 | The infimum of a set of ex... |
| infxrunb2 46584 | The infimum of an unbounde... |
| infxrbnd2 46585 | The infimum of a bounded-b... |
| infleinflem1 46586 | Lemma for ~ infleinf , cas... |
| infleinflem2 46587 | Lemma for ~ infleinf , whe... |
| infleinf 46588 | If any element of ` B ` ca... |
| xralrple4 46589 | Show that ` A ` is less th... |
| xralrple3 46590 | Show that ` A ` is less th... |
| eluzelzd 46591 | A member of an upper set o... |
| suplesup2 46592 | If any element of ` A ` is... |
| recnnltrp 46593 | ` N ` is a natural number ... |
| nnn0 46594 | The set of positive intege... |
| fzct 46595 | A finite set of sequential... |
| rpgtrecnn 46596 | Any positive real number i... |
| fzossuz 46597 | A half-open integer interv... |
| infxrrefi 46598 | The real and extended real... |
| xrralrecnnle 46599 | Show that ` A ` is less th... |
| fzoct 46600 | A finite set of sequential... |
| frexr 46601 | A function taking real val... |
| nnrecrp 46602 | The reciprocal of a positi... |
| reclt0d 46603 | The reciprocal of a negati... |
| lt0neg1dd 46604 | If a number is negative, i... |
| infxrcld 46605 | The infimum of an arbitrar... |
| xrralrecnnge 46606 | Show that ` A ` is less th... |
| reclt0 46607 | The reciprocal of a negati... |
| ltmulneg 46608 | Multiplying by a negative ... |
| allbutfi 46609 | For all but finitely many.... |
| ltdiv23neg 46610 | Swap denominator with othe... |
| xreqnltd 46611 | A consequence of trichotom... |
| mnfnre2 46612 | Minus infinity is not a re... |
| zssxr 46613 | The integers are a subset ... |
| fisupclrnmpt 46614 | A nonempty finite indexed ... |
| supxrunb3 46615 | The supremum of an unbound... |
| fimaxre4 46616 | A nonempty finite set of r... |
| ren0 46617 | The set of reals is nonemp... |
| eluzelz2 46618 | A member of an upper set o... |
| resabs2d 46619 | Absorption law for restric... |
| uzid2 46620 | Membership of the least me... |
| supxrleubrnmpt 46621 | The supremum of a nonempty... |
| uzssre2 46622 | An upper set of integers i... |
| uzssd 46623 | Subset relationship for tw... |
| eluzd 46624 | Membership in an upper set... |
| infxrlbrnmpt2 46625 | A member of a nonempty ind... |
| xrre4 46626 | An extended real is real i... |
| uz0 46627 | The upper integers functio... |
| eluzelz2d 46628 | A member of an upper set o... |
| infleinf2 46629 | If any element in ` B ` is... |
| unb2ltle 46630 | "Unbounded below" expresse... |
| uzidd2 46631 | Membership of the least me... |
| uzssd2 46632 | Subset relationship for tw... |
| rexabslelem 46633 | An indexed set of absolute... |
| rexabsle 46634 | An indexed set of absolute... |
| allbutfiinf 46635 | Given a "for all but finit... |
| supxrrernmpt 46636 | The real and extended real... |
| suprleubrnmpt 46637 | The supremum of a nonempty... |
| infrnmptle 46638 | An indexed infimum of exte... |
| infxrunb3 46639 | The infimum of an unbounde... |
| uzn0d 46640 | The upper integers are all... |
| uzssd3 46641 | Subset relationship for tw... |
| rexabsle2 46642 | An indexed set of absolute... |
| infxrunb3rnmpt 46643 | The infimum of an unbounde... |
| supxrre3rnmpt 46644 | The indexed supremum of a ... |
| uzublem 46645 | A set of reals, indexed by... |
| uzub 46646 | A set of reals, indexed by... |
| ssrexr 46647 | A subset of the reals is a... |
| supxrmnf2 46648 | Removing minus infinity fr... |
| supxrcli 46649 | The supremum of an arbitra... |
| uzid3 46650 | Membership of the least me... |
| infxrlesupxr 46651 | The supremum of a nonempty... |
| xnegeqd 46652 | Equality of two extended n... |
| xnegrecl 46653 | The extended real negative... |
| xnegnegi 46654 | Extended real version of ~... |
| xnegeqi 46655 | Equality of two extended n... |
| nfxnegd 46656 | Deduction version of ~ nfx... |
| xnegnegd 46657 | Extended real version of ~... |
| uzred 46658 | An upper integer is a real... |
| xnegcli 46659 | Closure of extended real n... |
| supminfrnmpt 46660 | The indexed supremum of a ... |
| infxrpnf 46661 | Adding plus infinity to a ... |
| infxrrnmptcl 46662 | The infimum of an arbitrar... |
| leneg2d 46663 | Negative of one side of 'l... |
| supxrltinfxr 46664 | The supremum of the empty ... |
| max1d 46665 | A number is less than or e... |
| supxrleubrnmptf 46666 | The supremum of a nonempty... |
| nleltd 46667 | 'Not less than or equal to... |
| zxrd 46668 | An integer is an extended ... |
| infxrgelbrnmpt 46669 | The infimum of an indexed ... |
| rphalfltd 46670 | Half of a positive real is... |
| uzssz2 46671 | An upper set of integers i... |
| leneg3d 46672 | Negative of one side of 'l... |
| max2d 46673 | A number is less than or e... |
| uzn0bi 46674 | The upper integers functio... |
| xnegrecl2 46675 | If the extended real negat... |
| nfxneg 46676 | Bound-variable hypothesis ... |
| uzxrd 46677 | An upper integer is an ext... |
| infxrpnf2 46678 | Removing plus infinity fro... |
| supminfxr 46679 | The extended real suprema ... |
| infrpgernmpt 46680 | The infimum of a nonempty,... |
| xnegre 46681 | An extended real is real i... |
| xnegrecl2d 46682 | If the extended real negat... |
| uzxr 46683 | An upper integer is an ext... |
| supminfxr2 46684 | The extended real suprema ... |
| xnegred 46685 | An extended real is real i... |
| supminfxrrnmpt 46686 | The indexed supremum of a ... |
| min1d 46687 | The minimum of two numbers... |
| min2d 46688 | The minimum of two numbers... |
| xrnpnfmnf 46689 | An extended real that is n... |
| uzsscn 46690 | An upper set of integers i... |
| absimnre 46691 | The absolute value of the ... |
| uzsscn2 46692 | An upper set of integers i... |
| xrtgcntopre 46693 | The standard topologies on... |
| absimlere 46694 | The absolute value of the ... |
| rpssxr 46695 | The positive reals are a s... |
| monoordxrv 46696 | Ordering relation for a mo... |
| monoordxr 46697 | Ordering relation for a mo... |
| monoord2xrv 46698 | Ordering relation for a mo... |
| monoord2xr 46699 | Ordering relation for a mo... |
| xrpnf 46700 | An extended real is plus i... |
| xlenegcon1 46701 | Extended real version of ~... |
| xlenegcon2 46702 | Extended real version of ~... |
| pimxrneun 46703 | The preimage of a set of e... |
| caucvgbf 46704 | A function is convergent i... |
| cvgcau 46705 | A convergent function is C... |
| cvgcaule 46706 | A convergent function is C... |
| rexanuz2nf 46707 | A simple counterexample re... |
| gtnelioc 46708 | A real number larger than ... |
| ioossioc 46709 | An open interval is a subs... |
| ioondisj2 46710 | A condition for two open i... |
| ioondisj1 46711 | A condition for two open i... |
| ioogtlb 46712 | An element of a closed int... |
| evthiccabs 46713 | Extreme Value Theorem on y... |
| ltnelicc 46714 | A real number smaller than... |
| eliood 46715 | Membership in an open real... |
| iooabslt 46716 | An upper bound for the dis... |
| gtnelicc 46717 | A real number greater than... |
| iooinlbub 46718 | An open interval has empty... |
| iocgtlb 46719 | An element of a left-open ... |
| iocleub 46720 | An element of a left-open ... |
| eliccd 46721 | Membership in a closed rea... |
| eliccre 46722 | A member of a closed inter... |
| eliooshift 46723 | Element of an open interva... |
| eliocd 46724 | Membership in a left-open ... |
| icoltub 46725 | An element of a left-close... |
| eliocre 46726 | A member of a left-open ri... |
| iooltub 46727 | An element of an open inte... |
| ioontr 46728 | The interior of an interva... |
| snunioo1 46729 | The closure of one end of ... |
| lbioc 46730 | A left-open right-closed i... |
| ioomidp 46731 | The midpoint is an element... |
| iccdifioo 46732 | If the open inverval is re... |
| iccdifprioo 46733 | An open interval is the cl... |
| ioossioobi 46734 | Biconditional form of ~ io... |
| iccshift 46735 | A closed interval shifted ... |
| iccsuble 46736 | An upper bound to the dist... |
| iocopn 46737 | A left-open right-closed i... |
| eliccelioc 46738 | Membership in a closed int... |
| iooshift 46739 | An open interval shifted b... |
| iccintsng 46740 | Intersection of two adiace... |
| icoiccdif 46741 | Left-closed right-open int... |
| icoopn 46742 | A left-closed right-open i... |
| icoub 46743 | A left-closed, right-open ... |
| eliccxrd 46744 | Membership in a closed rea... |
| pnfel0pnf 46745 | ` +oo ` is a nonnegative e... |
| eliccnelico 46746 | An element of a closed int... |
| eliccelicod 46747 | A member of a closed inter... |
| ge0xrre 46748 | A nonnegative extended rea... |
| ge0lere 46749 | A nonnegative extended Rea... |
| elicores 46750 | Membership in a left-close... |
| inficc 46751 | The infimum of a nonempty ... |
| qinioo 46752 | The rational numbers are d... |
| lenelioc 46753 | A real number smaller than... |
| ioonct 46754 | A nonempty open interval i... |
| xrgtnelicc 46755 | A real number greater than... |
| iccdificc 46756 | The difference of two clos... |
| iocnct 46757 | A nonempty left-open, righ... |
| iccnct 46758 | A closed interval, with mo... |
| iooiinicc 46759 | A closed interval expresse... |
| iccgelbd 46760 | An element of a closed int... |
| iooltubd 46761 | An element of an open inte... |
| icoltubd 46762 | An element of a left-close... |
| qelioo 46763 | The rational numbers are d... |
| tgqioo2 46764 | Every open set of reals is... |
| iccleubd 46765 | An element of a closed int... |
| elioored 46766 | A member of an open interv... |
| ioogtlbd 46767 | An element of a closed int... |
| ioofun 46768 | ` (,) ` is a function. (C... |
| icomnfinre 46769 | A left-closed, right-open,... |
| sqrlearg 46770 | The square compared with i... |
| ressiocsup 46771 | If the supremum belongs to... |
| ressioosup 46772 | If the supremum does not b... |
| iooiinioc 46773 | A left-open, right-closed ... |
| ressiooinf 46774 | If the infimum does not be... |
| iocleubd 46775 | An element of a left-open ... |
| uzinico 46776 | An upper interval of integ... |
| preimaiocmnf 46777 | Preimage of a right-closed... |
| uzinico2 46778 | An upper interval of integ... |
| uzinico3 46779 | An upper interval of integ... |
| dmico 46780 | The domain of the closed-b... |
| ndmico 46781 | The closed-below, open-abo... |
| uzubioo 46782 | The upper integers are unb... |
| uzubico 46783 | The upper integers are unb... |
| uzubioo2 46784 | The upper integers are unb... |
| uzubico2 46785 | The upper integers are unb... |
| iocgtlbd 46786 | An element of a left-open ... |
| xrtgioo2 46787 | The topology on the extend... |
| fsummulc1f 46788 | Closure of a finite sum of... |
| fsumnncl 46789 | Closure of a nonempty, fin... |
| fsumge0cl 46790 | The finite sum of nonnegat... |
| fsumf1of 46791 | Re-index a finite sum usin... |
| fsumiunss 46792 | Sum over a disjoint indexe... |
| fsumreclf 46793 | Closure of a finite sum of... |
| fsumlessf 46794 | A shorter sum of nonnegati... |
| fsumsupp0 46795 | Finite sum of function val... |
| fsumsermpt 46796 | A finite sum expressed in ... |
| fmul01 46797 | Multiplying a finite numbe... |
| fmulcl 46798 | If ' Y ' is closed under t... |
| fmuldfeqlem1 46799 | induction step for the pro... |
| fmuldfeq 46800 | X and Z are two equivalent... |
| fmul01lt1lem1 46801 | Given a finite multiplicat... |
| fmul01lt1lem2 46802 | Given a finite multiplicat... |
| fmul01lt1 46803 | Given a finite multiplicat... |
| cncfmptss 46804 | A continuous complex funct... |
| rrpsscn 46805 | The positive reals are a s... |
| mulc1cncfg 46806 | A version of ~ mulc1cncf u... |
| infrglb 46807 | The infimum of a nonempty ... |
| expcnfg 46808 | If ` F ` is a complex cont... |
| prodeq2ad 46809 | Equality deduction for pro... |
| fprodsplit1 46810 | Separate out a term in a f... |
| fprodexp 46811 | Positive integer exponenti... |
| fprodabs2 46812 | The absolute value of a fi... |
| fprod0 46813 | A finite product with a ze... |
| mccllem 46814 | * Induction step for ~ mcc... |
| mccl 46815 | A multinomial coefficient,... |
| fprodcnlem 46816 | A finite product of functi... |
| fprodcn 46817 | A finite product of functi... |
| clim1fr1 46818 | A class of sequences of fr... |
| isumneg 46819 | Negation of a converging s... |
| climrec 46820 | Limit of the reciprocal of... |
| climmulf 46821 | A version of ~ climmul usi... |
| climexp 46822 | The limit of natural power... |
| climinf 46823 | A bounded monotonic noninc... |
| climsuselem1 46824 | The subsequence index ` I ... |
| climsuse 46825 | A subsequence ` G ` of a c... |
| climrecf 46826 | A version of ~ climrec usi... |
| climneg 46827 | Complex limit of the negat... |
| climinff 46828 | A version of ~ climinf usi... |
| climdivf 46829 | Limit of the ratio of two ... |
| climreeq 46830 | If ` F ` is a real functio... |
| ellimciota 46831 | An explicit value for the ... |
| climaddf 46832 | A version of ~ climadd usi... |
| mullimc 46833 | Limit of the product of tw... |
| ellimcabssub0 46834 | An equivalent condition fo... |
| limcdm0 46835 | If a function has empty do... |
| islptre 46836 | An equivalence condition f... |
| limccog 46837 | Limit of the composition o... |
| limciccioolb 46838 | The limit of a function at... |
| climf 46839 | Express the predicate: Th... |
| mullimcf 46840 | Limit of the multiplicatio... |
| constlimc 46841 | Limit of constant function... |
| rexlim2d 46842 | Inference removing two res... |
| idlimc 46843 | Limit of the identity func... |
| divcnvg 46844 | The sequence of reciprocal... |
| limcperiod 46845 | If ` F ` is a periodic fun... |
| limcrecl 46846 | If ` F ` is a real-valued ... |
| sumnnodd 46847 | A series indexed by ` NN `... |
| lptioo2 46848 | The upper bound of an open... |
| lptioo1 46849 | The lower bound of an open... |
| limcmptdm 46850 | The domain of a maps-to fu... |
| clim2f 46851 | Express the predicate: Th... |
| limcicciooub 46852 | The limit of a function at... |
| ltmod 46853 | A sufficient condition for... |
| islpcn 46854 | A characterization for a l... |
| lptre2pt 46855 | If a set in the real line ... |
| limsupre 46856 | If a sequence is bounded, ... |
| limcresiooub 46857 | The left limit doesn't cha... |
| limcresioolb 46858 | The right limit doesn't ch... |
| limcleqr 46859 | If the left and the right ... |
| lptioo2cn 46860 | The upper bound of an open... |
| lptioo1cn 46861 | The lower bound of an open... |
| neglimc 46862 | Limit of the negative func... |
| addlimc 46863 | Sum of two limits. (Contr... |
| 0ellimcdiv 46864 | If the numerator converges... |
| clim2cf 46865 | Express the predicate ` F ... |
| limclner 46866 | For a limit point, both fr... |
| sublimc 46867 | Subtraction of two limits.... |
| reclimc 46868 | Limit of the reciprocal of... |
| clim0cf 46869 | Express the predicate ` F ... |
| limclr 46870 | For a limit point, both fr... |
| divlimc 46871 | Limit of the quotient of t... |
| expfac 46872 | Factorial grows faster tha... |
| climconstmpt 46873 | A constant sequence conver... |
| climresmpt 46874 | A function restricted to u... |
| climsubmpt 46875 | Limit of the difference of... |
| climsubc2mpt 46876 | Limit of the difference of... |
| climsubc1mpt 46877 | Limit of the difference of... |
| fnlimfv 46878 | The value of the limit fun... |
| climreclf 46879 | The limit of a convergent ... |
| climeldmeq 46880 | Two functions that are eve... |
| climf2 46881 | Express the predicate: Th... |
| fnlimcnv 46882 | The sequence of function v... |
| climeldmeqmpt 46883 | Two functions that are eve... |
| climfveq 46884 | Two functions that are eve... |
| clim2f2 46885 | Express the predicate: Th... |
| climfveqmpt 46886 | Two functions that are eve... |
| climd 46887 | Express the predicate: Th... |
| clim2d 46888 | The limit of complex numbe... |
| fnlimfvre 46889 | The limit function of real... |
| allbutfifvre 46890 | Given a sequence of real-v... |
| climleltrp 46891 | The limit of complex numbe... |
| fnlimfvre2 46892 | The limit function of real... |
| fnlimf 46893 | The limit function of real... |
| fnlimabslt 46894 | A sequence of function val... |
| climfveqf 46895 | Two functions that are eve... |
| climmptf 46896 | Exhibit a function ` G ` w... |
| climfveqmpt3 46897 | Two functions that are eve... |
| climeldmeqf 46898 | Two functions that are eve... |
| climreclmpt 46899 | The limit of B convergent ... |
| limsupref 46900 | If a sequence is bounded, ... |
| limsupbnd1f 46901 | If a sequence is eventuall... |
| climbddf 46902 | A converging sequence of c... |
| climeqf 46903 | Two functions that are eve... |
| climeldmeqmpt3 46904 | Two functions that are eve... |
| limsupcld 46905 | Closure of the superior li... |
| climfv 46906 | The limit of a convergent ... |
| limsupval3 46907 | The superior limit of an i... |
| climfveqmpt2 46908 | Two functions that are eve... |
| limsup0 46909 | The superior limit of the ... |
| climeldmeqmpt2 46910 | Two functions that are eve... |
| limsupresre 46911 | The supremum limit of a fu... |
| climeqmpt 46912 | Two functions that are eve... |
| climfvd 46913 | The limit of a convergent ... |
| limsuplesup 46914 | An upper bound for the sup... |
| limsupresico 46915 | The superior limit doesn't... |
| limsuppnfdlem 46916 | If the restriction of a fu... |
| limsuppnfd 46917 | If the restriction of a fu... |
| limsupresuz 46918 | If the real part of the do... |
| limsupub 46919 | If the limsup is not ` +oo... |
| limsupres 46920 | The superior limit of a re... |
| climinf2lem 46921 | A convergent, nonincreasin... |
| climinf2 46922 | A convergent, nonincreasin... |
| limsupvaluz 46923 | The superior limit, when t... |
| limsupresuz2 46924 | If the domain of a functio... |
| limsuppnflem 46925 | If the restriction of a fu... |
| limsuppnf 46926 | If the restriction of a fu... |
| limsupubuzlem 46927 | If the limsup is not ` +oo... |
| limsupubuz 46928 | For a real-valued function... |
| climinf2mpt 46929 | A bounded below, monotonic... |
| climinfmpt 46930 | A bounded below, monotonic... |
| climinf3 46931 | A convergent, nonincreasin... |
| limsupvaluzmpt 46932 | The superior limit, when t... |
| limsupequzmpt2 46933 | Two functions that are eve... |
| limsupubuzmpt 46934 | If the limsup is not ` +oo... |
| limsupmnflem 46935 | The superior limit of a fu... |
| limsupmnf 46936 | The superior limit of a fu... |
| limsupequzlem 46937 | Two functions that are eve... |
| limsupequz 46938 | Two functions that are eve... |
| limsupre2lem 46939 | Given a function on the ex... |
| limsupre2 46940 | Given a function on the ex... |
| limsupmnfuzlem 46941 | The superior limit of a fu... |
| limsupmnfuz 46942 | The superior limit of a fu... |
| limsupequzmptlem 46943 | Two functions that are eve... |
| limsupequzmpt 46944 | Two functions that are eve... |
| limsupre2mpt 46945 | Given a function on the ex... |
| limsupequzmptf 46946 | Two functions that are eve... |
| limsupre3lem 46947 | Given a function on the ex... |
| limsupre3 46948 | Given a function on the ex... |
| limsupre3mpt 46949 | Given a function on the ex... |
| limsupre3uzlem 46950 | Given a function on the ex... |
| limsupre3uz 46951 | Given a function on the ex... |
| limsupreuz 46952 | Given a function on the re... |
| limsupvaluz2 46953 | The superior limit, when t... |
| limsupreuzmpt 46954 | Given a function on the re... |
| supcnvlimsup 46955 | If a function on a set of ... |
| supcnvlimsupmpt 46956 | If a function on a set of ... |
| 0cnv 46957 | If ` (/) ` is a complex nu... |
| climuzlem 46958 | Express the predicate: Th... |
| climuz 46959 | Express the predicate: Th... |
| lmbr3v 46960 | Express the binary relatio... |
| climisp 46961 | If a sequence converges to... |
| lmbr3 46962 | Express the binary relatio... |
| climrescn 46963 | A sequence converging w.r.... |
| climxrrelem 46964 | If a sequence ranging over... |
| climxrre 46965 | If a sequence ranging over... |
| limsuplt2 46968 | The defining property of t... |
| liminfgord 46969 | Ordering property of the i... |
| limsupvald 46970 | The superior limit of a se... |
| limsupresicompt 46971 | The superior limit doesn't... |
| limsupcli 46972 | Closure of the superior li... |
| liminfgf 46973 | Closure of the inferior li... |
| liminfval 46974 | The inferior limit of a se... |
| climlimsup 46975 | A sequence of real numbers... |
| limsupge 46976 | The defining property of t... |
| liminfgval 46977 | Value of the inferior limi... |
| liminfcl 46978 | Closure of the inferior li... |
| liminfvald 46979 | The inferior limit of a se... |
| liminfval5 46980 | The inferior limit of an i... |
| limsupresxr 46981 | The superior limit of a fu... |
| liminfresxr 46982 | The inferior limit of a fu... |
| liminfval2 46983 | The superior limit, relati... |
| climlimsupcex 46984 | Counterexample for ~ climl... |
| liminfcld 46985 | Closure of the inferior li... |
| liminfresico 46986 | The inferior limit doesn't... |
| limsup10exlem 46987 | The range of the given fun... |
| limsup10ex 46988 | The superior limit of a fu... |
| liminf10ex 46989 | The inferior limit of a fu... |
| liminflelimsuplem 46990 | The superior limit is grea... |
| liminflelimsup 46991 | The superior limit is grea... |
| limsupgtlem 46992 | For any positive real, the... |
| limsupgt 46993 | Given a sequence of real n... |
| liminfresre 46994 | The inferior limit of a fu... |
| liminfresicompt 46995 | The inferior limit doesn't... |
| liminfltlimsupex 46996 | An example where the ` lim... |
| liminfgelimsup 46997 | The inferior limit is grea... |
| liminfvalxr 46998 | Alternate definition of ` ... |
| liminfresuz 46999 | If the real part of the do... |
| liminflelimsupuz 47000 | The superior limit is grea... |
| liminfvalxrmpt 47001 | Alternate definition of ` ... |
| liminfresuz2 47002 | If the domain of a functio... |
| liminfgelimsupuz 47003 | The inferior limit is grea... |
| liminfval4 47004 | Alternate definition of ` ... |
| liminfval3 47005 | Alternate definition of ` ... |
| liminfequzmpt2 47006 | Two functions that are eve... |
| liminfvaluz 47007 | Alternate definition of ` ... |
| liminf0 47008 | The inferior limit of the ... |
| limsupval4 47009 | Alternate definition of ` ... |
| liminfvaluz2 47010 | Alternate definition of ` ... |
| liminfvaluz3 47011 | Alternate definition of ` ... |
| liminflelimsupcex 47012 | A counterexample for ~ lim... |
| limsupvaluz3 47013 | Alternate definition of ` ... |
| liminfvaluz4 47014 | Alternate definition of ` ... |
| limsupvaluz4 47015 | Alternate definition of ` ... |
| climliminflimsupd 47016 | If a sequence of real numb... |
| liminfreuzlem 47017 | Given a function on the re... |
| liminfreuz 47018 | Given a function on the re... |
| liminfltlem 47019 | Given a sequence of real n... |
| liminflt 47020 | Given a sequence of real n... |
| climliminf 47021 | A sequence of real numbers... |
| liminflimsupclim 47022 | A sequence of real numbers... |
| climliminflimsup 47023 | A sequence of real numbers... |
| climliminflimsup2 47024 | A sequence of real numbers... |
| climliminflimsup3 47025 | A sequence of real numbers... |
| climliminflimsup4 47026 | A sequence of real numbers... |
| limsupub2 47027 | A extended real valued fun... |
| limsupubuz2 47028 | A sequence with values in ... |
| xlimpnfxnegmnf 47029 | A sequence converges to ` ... |
| liminflbuz2 47030 | A sequence with values in ... |
| liminfpnfuz 47031 | The inferior limit of a fu... |
| liminflimsupxrre 47032 | A sequence with values in ... |
| xlimrel 47035 | The limit on extended real... |
| xlimres 47036 | A function converges iff i... |
| xlimcl 47037 | The limit of a sequence of... |
| rexlimddv2 47038 | Restricted existential eli... |
| xlimclim 47039 | Given a sequence of reals,... |
| xlimconst 47040 | A constant sequence conver... |
| climxlim 47041 | A converging sequence in t... |
| xlimbr 47042 | Express the binary relatio... |
| fuzxrpmcn 47043 | A function mapping from an... |
| cnrefiisplem 47044 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 47045 | A non-real, complex number... |
| xlimxrre 47046 | If a sequence ranging over... |
| xlimmnfvlem1 47047 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 47048 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 47049 | A function converges to mi... |
| xlimconst2 47050 | A sequence that eventually... |
| xlimpnfvlem1 47051 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 47052 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 47053 | A function converges to pl... |
| xlimclim2lem 47054 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 47055 | Given a sequence of extend... |
| xlimmnf 47056 | A function converges to mi... |
| xlimpnf 47057 | A function converges to pl... |
| xlimmnfmpt 47058 | A function converges to pl... |
| xlimpnfmpt 47059 | A function converges to pl... |
| climxlim2lem 47060 | In this lemma for ~ climxl... |
| climxlim2 47061 | A sequence of extended rea... |
| dfxlim2v 47062 | An alternative definition ... |
| dfxlim2 47063 | An alternative definition ... |
| climresd 47064 | A function restricted to u... |
| climresdm 47065 | A real function converges ... |
| dmclimxlim 47066 | A real valued sequence tha... |
| xlimmnflimsup2 47067 | A sequence of extended rea... |
| xlimuni 47068 | An infinite sequence conve... |
| xlimclimdm 47069 | A sequence of extended rea... |
| xlimfun 47070 | The convergence relation o... |
| xlimmnflimsup 47071 | If a sequence of extended ... |
| xlimdm 47072 | Two ways to express that a... |
| xlimpnfxnegmnf2 47073 | A sequence converges to ` ... |
| xlimresdm 47074 | A function converges in th... |
| xlimpnfliminf 47075 | If a sequence of extended ... |
| xlimpnfliminf2 47076 | A sequence of extended rea... |
| xlimliminflimsup 47077 | A sequence of extended rea... |
| xlimlimsupleliminf 47078 | A sequence of extended rea... |
| coseq0 47079 | A complex number whose cos... |
| sinmulcos 47080 | Multiplication formula for... |
| coskpi2 47081 | The cosine of an integer m... |
| cosnegpi 47082 | The cosine of negative ` _... |
| sinaover2ne0 47083 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 47084 | The cosine of an integer m... |
| mulcncff 47085 | The multiplication of two ... |
| cncfmptssg 47086 | A continuous complex funct... |
| constcncfg 47087 | A constant function is a c... |
| idcncfg 47088 | The identity function is a... |
| cncfshift 47089 | A periodic continuous func... |
| resincncf 47090 | ` sin ` restricted to real... |
| addccncf2 47091 | Adding a constant is a con... |
| 0cnf 47092 | The empty set is a continu... |
| fsumcncf 47093 | The finite sum of continuo... |
| cncfperiod 47094 | A periodic continuous func... |
| subcncff 47095 | The subtraction of two con... |
| negcncfg 47096 | The opposite of a continuo... |
| cnfdmsn 47097 | A function with a singleto... |
| cncfcompt 47098 | Composition of continuous ... |
| addcncff 47099 | The sum of two continuous ... |
| ioccncflimc 47100 | Limit at the upper bound o... |
| cncfuni 47101 | A complex function on a su... |
| icccncfext 47102 | A continuous function on a... |
| cncficcgt0 47103 | A the absolute value of a ... |
| icocncflimc 47104 | Limit at the lower bound, ... |
| cncfdmsn 47105 | A complex function with a ... |
| divcncff 47106 | The quotient of two contin... |
| cncfshiftioo 47107 | A periodic continuous func... |
| cncfiooicclem1 47108 | A continuous function ` F ... |
| cncfiooicc 47109 | A continuous function ` F ... |
| cncfiooiccre 47110 | A continuous function ` F ... |
| cncfioobdlem 47111 | ` G ` actually extends ` F... |
| cncfioobd 47112 | A continuous function ` F ... |
| jumpncnp 47113 | Jump discontinuity or disc... |
| cxpcncf2 47114 | The complex power function... |
| fprodcncf 47115 | The finite product of cont... |
| add1cncf 47116 | Addition to a constant is ... |
| add2cncf 47117 | Addition to a constant is ... |
| sub1cncfd 47118 | Subtracting a constant is ... |
| sub2cncfd 47119 | Subtraction from a constan... |
| fprodsub2cncf 47120 | ` F ` is continuous. (Con... |
| fprodadd2cncf 47121 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 47122 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 47123 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 47124 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 47125 | The sequence ` S ` of fini... |
| dvsinexp 47126 | The derivative of sin^N . ... |
| dvcosre 47127 | The real derivative of the... |
| dvsinax 47128 | Derivative exercise: the d... |
| dvsubf 47129 | The subtraction rule for e... |
| dvmptconst 47130 | Function-builder for deriv... |
| dvcnre 47131 | From complex differentiati... |
| dvmptidg 47132 | Function-builder for deriv... |
| dvresntr 47133 | Function-builder for deriv... |
| fperdvper 47134 | The derivative of a period... |
| dvasinbx 47135 | Derivative exercise: the d... |
| dvresioo 47136 | Restriction of a derivativ... |
| dvdivf 47137 | The quotient rule for ever... |
| dvdivbd 47138 | A sufficient condition for... |
| dvsubcncf 47139 | A sufficient condition for... |
| dvmulcncf 47140 | A sufficient condition for... |
| dvcosax 47141 | Derivative exercise: the d... |
| dvdivcncf 47142 | A sufficient condition for... |
| dvbdfbdioolem1 47143 | Given a function with boun... |
| dvbdfbdioolem2 47144 | A function on an open inte... |
| dvbdfbdioo 47145 | A function on an open inte... |
| ioodvbdlimc1lem1 47146 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 47147 | Limit at the lower bound o... |
| ioodvbdlimc1 47148 | A real function with bound... |
| ioodvbdlimc2lem 47149 | Limit at the upper bound o... |
| ioodvbdlimc2 47150 | A real function with bound... |
| dvdmsscn 47151 | ` X ` is a subset of ` CC ... |
| dvmptmulf 47152 | Function-builder for deriv... |
| dvnmptdivc 47153 | Function-builder for itera... |
| dvdsn1add 47154 | If ` K ` divides ` N ` but... |
| dvxpaek 47155 | Derivative of the polynomi... |
| dvnmptconst 47156 | The ` N ` -th derivative o... |
| dvnxpaek 47157 | The ` n ` -th derivative o... |
| dvnmul 47158 | Function-builder for the `... |
| dvmptfprodlem 47159 | Induction step for ~ dvmpt... |
| dvmptfprod 47160 | Function-builder for deriv... |
| dvnprodlem1 47161 | ` D ` is bijective. (Cont... |
| dvnprodlem2 47162 | Induction step for ~ dvnpr... |
| dvnprodlem3 47163 | The multinomial formula fo... |
| dvnprod 47164 | The multinomial formula fo... |
| itgsin0pilem1 47165 | Calculation of the integra... |
| ibliccsinexp 47166 | sin^n on a closed interval... |
| itgsin0pi 47167 | Calculation of the integra... |
| iblioosinexp 47168 | sin^n on an open integral ... |
| itgsinexplem1 47169 | Integration by parts is ap... |
| itgsinexp 47170 | A recursive formula for th... |
| iblconstmpt 47171 | A constant function is int... |
| itgeq1d 47172 | Equality theorem for an in... |
| mbfres2cn 47173 | Measurability of a piecewi... |
| vol0 47174 | The measure of the empty s... |
| ditgeqiooicc 47175 | A function ` F ` on an ope... |
| volge0 47176 | The volume of a set is alw... |
| cnbdibl 47177 | A continuous bounded funct... |
| snmbl 47178 | A singleton is measurable.... |
| ditgeq3d 47179 | Equality theorem for the d... |
| iblempty 47180 | The empty function is inte... |
| iblsplit 47181 | The union of two integrabl... |
| volsn 47182 | A singleton has 0 Lebesgue... |
| itgvol0 47183 | If the domani is negligibl... |
| itgcoscmulx 47184 | Exercise: the integral of ... |
| iblsplitf 47185 | A version of ~ iblsplit us... |
| ibliooicc 47186 | If a function is integrabl... |
| volioc 47187 | The measure of a left-open... |
| iblspltprt 47188 | If a function is integrabl... |
| itgsincmulx 47189 | Exercise: the integral of ... |
| itgsubsticclem 47190 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 47191 | Integration by u-substitut... |
| itgioocnicc 47192 | The integral of a piecewis... |
| iblcncfioo 47193 | A continuous function ` F ... |
| itgspltprt 47194 | The ` S. ` integral splits... |
| itgiccshift 47195 | The integral of a function... |
| itgperiod 47196 | The integral of a periodic... |
| itgsbtaddcnst 47197 | Integral substitution, add... |
| volico 47198 | The measure of left-closed... |
| sublevolico 47199 | The Lebesgue measure of a ... |
| dmvolss 47200 | Lebesgue measurable sets a... |
| ismbl3 47201 | The predicate " ` A ` is L... |
| volioof 47202 | The function that assigns ... |
| ovolsplit 47203 | The Lebesgue outer measure... |
| fvvolioof 47204 | The function value of the ... |
| volioore 47205 | The measure of an open int... |
| fvvolicof 47206 | The function value of the ... |
| voliooico 47207 | An open interval and a lef... |
| ismbl4 47208 | The predicate " ` A ` is L... |
| volioofmpt 47209 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 47210 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 47211 | The Lebesgue measure of op... |
| volicofmpt 47212 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 47213 | The Lebesgue measure of a ... |
| voliccico 47214 | A closed interval and a le... |
| mbfdmssre 47215 | The domain of a measurable... |
| stoweidlem1 47216 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 47217 | lemma for ~ stoweid : here... |
| stoweidlem3 47218 | Lemma for ~ stoweid : if `... |
| stoweidlem4 47219 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 47220 | There exists a δ as ... |
| stoweidlem6 47221 | Lemma for ~ stoweid : two ... |
| stoweidlem7 47222 | This lemma is used to prov... |
| stoweidlem8 47223 | Lemma for ~ stoweid : two ... |
| stoweidlem9 47224 | Lemma for ~ stoweid : here... |
| stoweidlem10 47225 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 47226 | This lemma is used to prov... |
| stoweidlem12 47227 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 47228 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 47229 | There exists a ` k ` as in... |
| stoweidlem15 47230 | This lemma is used to prov... |
| stoweidlem16 47231 | Lemma for ~ stoweid . The... |
| stoweidlem17 47232 | This lemma proves that the... |
| stoweidlem18 47233 | This theorem proves Lemma ... |
| stoweidlem19 47234 | If a set of real functions... |
| stoweidlem20 47235 | If a set A of real functio... |
| stoweidlem21 47236 | Once the Stone Weierstrass... |
| stoweidlem22 47237 | If a set of real functions... |
| stoweidlem23 47238 | This lemma is used to prov... |
| stoweidlem24 47239 | This lemma proves that for... |
| stoweidlem25 47240 | This lemma proves that for... |
| stoweidlem26 47241 | This lemma is used to prov... |
| stoweidlem27 47242 | This lemma is used to prov... |
| stoweidlem28 47243 | There exists a δ as ... |
| stoweidlem29 47244 | When the hypothesis for th... |
| stoweidlem30 47245 | This lemma is used to prov... |
| stoweidlem31 47246 | This lemma is used to prov... |
| stoweidlem32 47247 | If a set A of real functio... |
| stoweidlem33 47248 | If a set of real functions... |
| stoweidlem34 47249 | This lemma proves that for... |
| stoweidlem35 47250 | This lemma is used to prov... |
| stoweidlem36 47251 | This lemma is used to prov... |
| stoweidlem37 47252 | This lemma is used to prov... |
| stoweidlem38 47253 | This lemma is used to prov... |
| stoweidlem39 47254 | This lemma is used to prov... |
| stoweidlem40 47255 | This lemma proves that q_n... |
| stoweidlem41 47256 | This lemma is used to prov... |
| stoweidlem42 47257 | This lemma is used to prov... |
| stoweidlem43 47258 | This lemma is used to prov... |
| stoweidlem44 47259 | This lemma is used to prov... |
| stoweidlem45 47260 | This lemma proves that, gi... |
| stoweidlem46 47261 | This lemma proves that set... |
| stoweidlem47 47262 | Subtracting a constant fro... |
| stoweidlem48 47263 | This lemma is used to prov... |
| stoweidlem49 47264 | There exists a function q_... |
| stoweidlem50 47265 | This lemma proves that set... |
| stoweidlem51 47266 | There exists a function x ... |
| stoweidlem52 47267 | There exists a neighborhoo... |
| stoweidlem53 47268 | This lemma is used to prov... |
| stoweidlem54 47269 | There exists a function ` ... |
| stoweidlem55 47270 | This lemma proves the exis... |
| stoweidlem56 47271 | This theorem proves Lemma ... |
| stoweidlem57 47272 | There exists a function x ... |
| stoweidlem58 47273 | This theorem proves Lemma ... |
| stoweidlem59 47274 | This lemma proves that the... |
| stoweidlem60 47275 | This lemma proves that the... |
| stoweidlem61 47276 | This lemma proves that the... |
| stoweidlem62 47277 | This theorem proves the St... |
| stoweid 47278 | This theorem proves the St... |
| stowei 47279 | This theorem proves the St... |
| wallispilem1 47280 | ` I ` is monotone: increas... |
| wallispilem2 47281 | A first set of properties ... |
| wallispilem3 47282 | I maps to real values. (C... |
| wallispilem4 47283 | ` F ` maps to explicit exp... |
| wallispilem5 47284 | The sequence ` H ` converg... |
| wallispi 47285 | Wallis' formula for π :... |
| wallispi2lem1 47286 | An intermediate step betwe... |
| wallispi2lem2 47287 | Two expressions are proven... |
| wallispi2 47288 | An alternative version of ... |
| stirlinglem1 47289 | A simple limit of fraction... |
| stirlinglem2 47290 | ` A ` maps to positive rea... |
| stirlinglem3 47291 | Long but simple algebraic ... |
| stirlinglem4 47292 | Algebraic manipulation of ... |
| stirlinglem5 47293 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 47294 | A series that converges to... |
| stirlinglem7 47295 | Algebraic manipulation of ... |
| stirlinglem8 47296 | If ` A ` converges to ` C ... |
| stirlinglem9 47297 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 47298 | A bound for any B(N)-B(N +... |
| stirlinglem11 47299 | ` B ` is decreasing. (Con... |
| stirlinglem12 47300 | The sequence ` B ` is boun... |
| stirlinglem13 47301 | ` B ` is decreasing and ha... |
| stirlinglem14 47302 | The sequence ` A ` converg... |
| stirlinglem15 47303 | The Stirling's formula is ... |
| stirling 47304 | Stirling's approximation f... |
| stirlingr 47305 | Stirling's approximation f... |
| dirkerval 47306 | The N_th Dirichlet kernel.... |
| dirker2re 47307 | The Dirichlet kernel value... |
| dirkerdenne0 47308 | The Dirichlet kernel denom... |
| dirkerval2 47309 | The N_th Dirichlet kernel ... |
| dirkerre 47310 | The Dirichlet kernel at an... |
| dirkerper 47311 | the Dirichlet kernel has p... |
| dirkerf 47312 | For any natural number ` N... |
| dirkertrigeqlem1 47313 | Sum of an even number of a... |
| dirkertrigeqlem2 47314 | Trigonometric equality lem... |
| dirkertrigeqlem3 47315 | Trigonometric equality lem... |
| dirkertrigeq 47316 | Trigonometric equality for... |
| dirkeritg 47317 | The definite integral of t... |
| dirkercncflem1 47318 | If ` Y ` is a multiple of ... |
| dirkercncflem2 47319 | Lemma used to prove that t... |
| dirkercncflem3 47320 | The Dirichlet kernel is co... |
| dirkercncflem4 47321 | The Dirichlet kernel is co... |
| dirkercncf 47322 | For any natural number ` N... |
| fourierdlem1 47323 | A partition interval is a ... |
| fourierdlem2 47324 | Membership in a partition.... |
| fourierdlem3 47325 | Membership in a partition.... |
| fourierdlem4 47326 | ` E ` is a function that m... |
| fourierdlem5 47327 | ` S ` is a function. (Con... |
| fourierdlem6 47328 | ` X ` is in the periodic p... |
| fourierdlem7 47329 | The difference between the... |
| fourierdlem8 47330 | A partition interval is a ... |
| fourierdlem9 47331 | ` H ` is a complex functio... |
| fourierdlem10 47332 | Condition on the bounds of... |
| fourierdlem11 47333 | If there is a partition, t... |
| fourierdlem12 47334 | A point of a partition is ... |
| fourierdlem13 47335 | Value of ` V ` in terms of... |
| fourierdlem14 47336 | Given the partition ` V ` ... |
| fourierdlem15 47337 | The range of the partition... |
| fourierdlem16 47338 | The coefficients of the fo... |
| fourierdlem17 47339 | The defined ` L ` is actua... |
| fourierdlem18 47340 | The function ` S ` is cont... |
| fourierdlem19 47341 | If two elements of ` D ` h... |
| fourierdlem20 47342 | Every interval in the part... |
| fourierdlem21 47343 | The coefficients of the fo... |
| fourierdlem22 47344 | The coefficients of the fo... |
| fourierdlem23 47345 | If ` F ` is continuous and... |
| fourierdlem24 47346 | A sufficient condition for... |
| fourierdlem25 47347 | If ` C ` is not in the ran... |
| fourierdlem26 47348 | Periodic image of a point ... |
| fourierdlem27 47349 | A partition open interval ... |
| fourierdlem28 47350 | Derivative of ` ( F `` ( X... |
| fourierdlem29 47351 | Explicit function value fo... |
| fourierdlem30 47352 | Sum of three small pieces ... |
| fourierdlem31 47353 | If ` A ` is finite and for... |
| fourierdlem32 47354 | Limit of a continuous func... |
| fourierdlem33 47355 | Limit of a continuous func... |
| fourierdlem34 47356 | A partition is one to one.... |
| fourierdlem35 47357 | There is a single point in... |
| fourierdlem36 47358 | ` F ` is an isomorphism. ... |
| fourierdlem37 47359 | ` I ` is a function that m... |
| fourierdlem38 47360 | The function ` F ` is cont... |
| fourierdlem39 47361 | Integration by parts of ... |
| fourierdlem40 47362 | ` H ` is a continuous func... |
| fourierdlem41 47363 | Lemma used to prove that e... |
| fourierdlem42 47364 | The set of points in a mov... |
| fourierdlem43 47365 | ` K ` is a real function. ... |
| fourierdlem44 47366 | A condition for having ` (... |
| fourierdlem46 47367 | The function ` F ` has a l... |
| fourierdlem47 47368 | For ` r ` large enough, th... |
| fourierdlem48 47369 | The given periodic functio... |
| fourierdlem49 47370 | The given periodic functio... |
| fourierdlem50 47371 | Continuity of ` O ` and it... |
| fourierdlem51 47372 | ` X ` is in the periodic p... |
| fourierdlem52 47373 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 47374 | The limit of ` F ( s ) ` a... |
| fourierdlem54 47375 | Given a partition ` Q ` an... |
| fourierdlem55 47376 | ` U ` is a real function. ... |
| fourierdlem56 47377 | Derivative of the ` K ` fu... |
| fourierdlem57 47378 | The derivative of ` O ` . ... |
| fourierdlem58 47379 | The derivative of ` K ` is... |
| fourierdlem59 47380 | The derivative of ` H ` is... |
| fourierdlem60 47381 | Given a differentiable fun... |
| fourierdlem61 47382 | Given a differentiable fun... |
| fourierdlem62 47383 | The function ` K ` is cont... |
| fourierdlem63 47384 | The upper bound of interva... |
| fourierdlem64 47385 | The partition ` V ` is fin... |
| fourierdlem65 47386 | The distance of two adjace... |
| fourierdlem66 47387 | Value of the ` G ` functio... |
| fourierdlem67 47388 | ` G ` is a function. (Con... |
| fourierdlem68 47389 | The derivative of ` O ` is... |
| fourierdlem69 47390 | A piecewise continuous fun... |
| fourierdlem70 47391 | A piecewise continuous fun... |
| fourierdlem71 47392 | A periodic piecewise conti... |
| fourierdlem72 47393 | The derivative of ` O ` is... |
| fourierdlem73 47394 | A version of the Riemann L... |
| fourierdlem74 47395 | Given a piecewise smooth f... |
| fourierdlem75 47396 | Given a piecewise smooth f... |
| fourierdlem76 47397 | Continuity of ` O ` and it... |
| fourierdlem77 47398 | If ` H ` is bounded, then ... |
| fourierdlem78 47399 | ` G ` is continuous when r... |
| fourierdlem79 47400 | ` E ` projects every inter... |
| fourierdlem80 47401 | The derivative of ` O ` is... |
| fourierdlem81 47402 | The integral of a piecewis... |
| fourierdlem82 47403 | Integral by substitution, ... |
| fourierdlem83 47404 | The fourier partial sum fo... |
| fourierdlem84 47405 | If ` F ` is piecewise cont... |
| fourierdlem85 47406 | Limit of the function ` G ... |
| fourierdlem86 47407 | Continuity of ` O ` and it... |
| fourierdlem87 47408 | The integral of ` G ` goes... |
| fourierdlem88 47409 | Given a piecewise continuo... |
| fourierdlem89 47410 | Given a piecewise continuo... |
| fourierdlem90 47411 | Given a piecewise continuo... |
| fourierdlem91 47412 | Given a piecewise continuo... |
| fourierdlem92 47413 | The integral of a piecewis... |
| fourierdlem93 47414 | Integral by substitution (... |
| fourierdlem94 47415 | For a piecewise smooth fun... |
| fourierdlem95 47416 | Algebraic manipulation of ... |
| fourierdlem96 47417 | limit for ` F ` at the low... |
| fourierdlem97 47418 | ` F ` is continuous on the... |
| fourierdlem98 47419 | ` F ` is continuous on the... |
| fourierdlem99 47420 | limit for ` F ` at the upp... |
| fourierdlem100 47421 | A piecewise continuous fun... |
| fourierdlem101 47422 | Integral by substitution f... |
| fourierdlem102 47423 | For a piecewise smooth fun... |
| fourierdlem103 47424 | The half lower part of the... |
| fourierdlem104 47425 | The half upper part of the... |
| fourierdlem105 47426 | A piecewise continuous fun... |
| fourierdlem106 47427 | For a piecewise smooth fun... |
| fourierdlem107 47428 | The integral of a piecewis... |
| fourierdlem108 47429 | The integral of a piecewis... |
| fourierdlem109 47430 | The integral of a piecewis... |
| fourierdlem110 47431 | The integral of a piecewis... |
| fourierdlem111 47432 | The fourier partial sum fo... |
| fourierdlem112 47433 | Here abbreviations (local ... |
| fourierdlem113 47434 | Fourier series convergence... |
| fourierdlem114 47435 | Fourier series convergence... |
| fourierdlem115 47436 | Fourier serier convergence... |
| fourierd 47437 | Fourier series convergence... |
| fourierclimd 47438 | Fourier series convergence... |
| fourierclim 47439 | Fourier series convergence... |
| fourier 47440 | Fourier series convergence... |
| fouriercnp 47441 | If ` F ` is continuous at ... |
| fourier2 47442 | Fourier series convergence... |
| sqwvfoura 47443 | Fourier coefficients for t... |
| sqwvfourb 47444 | Fourier series ` B ` coeff... |
| fourierswlem 47445 | The Fourier series for the... |
| fouriersw 47446 | Fourier series convergence... |
| fouriercn 47447 | If the derivative of ` F `... |
| elaa2lem 47448 | Elementhood in the set of ... |
| elaa2 47449 | Elementhood in the set of ... |
| etransclem1 47450 | ` H ` is a function. (Con... |
| etransclem2 47451 | Derivative of ` G ` . (Co... |
| etransclem3 47452 | The given ` if ` term is a... |
| etransclem4 47453 | ` F ` expressed as a finit... |
| etransclem5 47454 | A change of bound variable... |
| etransclem6 47455 | A change of bound variable... |
| etransclem7 47456 | The given product is an in... |
| etransclem8 47457 | ` F ` is a function. (Con... |
| etransclem9 47458 | If ` K ` divides ` N ` but... |
| etransclem10 47459 | The given ` if ` term is a... |
| etransclem11 47460 | A change of bound variable... |
| etransclem12 47461 | ` C ` applied to ` N ` . ... |
| etransclem13 47462 | ` F ` applied to ` Y ` . ... |
| etransclem14 47463 | Value of the term ` T ` , ... |
| etransclem15 47464 | Value of the term ` T ` , ... |
| etransclem16 47465 | Every element in the range... |
| etransclem17 47466 | The ` N ` -th derivative o... |
| etransclem18 47467 | The given function is inte... |
| etransclem19 47468 | The ` N ` -th derivative o... |
| etransclem20 47469 | ` H ` is smooth. (Contrib... |
| etransclem21 47470 | The ` N ` -th derivative o... |
| etransclem22 47471 | The ` N ` -th derivative o... |
| etransclem23 47472 | This is the claim proof in... |
| etransclem24 47473 | ` P ` divides the I -th de... |
| etransclem25 47474 | ` P ` factorial divides th... |
| etransclem26 47475 | Every term in the sum of t... |
| etransclem27 47476 | The ` N ` -th derivative o... |
| etransclem28 47477 | ` ( P - 1 ) ` factorial di... |
| etransclem29 47478 | The ` N ` -th derivative o... |
| etransclem30 47479 | The ` N ` -th derivative o... |
| etransclem31 47480 | The ` N ` -th derivative o... |
| etransclem32 47481 | This is the proof for the ... |
| etransclem33 47482 | ` F ` is smooth. (Contrib... |
| etransclem34 47483 | The ` N ` -th derivative o... |
| etransclem35 47484 | ` P ` does not divide the ... |
| etransclem36 47485 | The ` N ` -th derivative o... |
| etransclem37 47486 | ` ( P - 1 ) ` factorial di... |
| etransclem38 47487 | ` P ` divides the I -th de... |
| etransclem39 47488 | ` G ` is a function. (Con... |
| etransclem40 47489 | The ` N ` -th derivative o... |
| etransclem41 47490 | ` P ` does not divide the ... |
| etransclem42 47491 | The ` N ` -th derivative o... |
| etransclem43 47492 | ` G ` is a continuous func... |
| etransclem44 47493 | The given finite sum is no... |
| etransclem45 47494 | ` K ` is an integer. (Con... |
| etransclem46 47495 | This is the proof for equa... |
| etransclem47 47496 | ` _e ` is transcendental. ... |
| etransclem48 47497 | ` _e ` is transcendental. ... |
| etransc 47498 | ` _e ` is transcendental. ... |
| rrxtopn 47499 | The topology of the genera... |
| rrxngp 47500 | Generalized Euclidean real... |
| rrxtps 47501 | Generalized Euclidean real... |
| rrxtopnfi 47502 | The topology of the n-dime... |
| rrxtopon 47503 | The topology on generalize... |
| rrxtop 47504 | The topology on generalize... |
| rrndistlt 47505 | Given two points in the sp... |
| rrxtoponfi 47506 | The topology on n-dimensio... |
| rrxunitopnfi 47507 | The base set of the standa... |
| rrxtopn0 47508 | The topology of the zero-d... |
| qndenserrnbllem 47509 | n-dimensional rational num... |
| qndenserrnbl 47510 | n-dimensional rational num... |
| rrxtopn0b 47511 | The topology of the zero-d... |
| qndenserrnopnlem 47512 | n-dimensional rational num... |
| qndenserrnopn 47513 | n-dimensional rational num... |
| qndenserrn 47514 | n-dimensional rational num... |
| rrxsnicc 47515 | A multidimensional singlet... |
| rrnprjdstle 47516 | The distance between two p... |
| rrndsmet 47517 | ` D ` is a metric for the ... |
| rrndsxmet 47518 | ` D ` is an extended metri... |
| ioorrnopnlem 47519 | The a point in an indexed ... |
| ioorrnopn 47520 | The indexed product of ope... |
| ioorrnopnxrlem 47521 | Given a point ` F ` that b... |
| ioorrnopnxr 47522 | The indexed product of ope... |
| issal 47529 | Express the predicate " ` ... |
| pwsal 47530 | The power set of a given s... |
| salunicl 47531 | SAlg sigma-algebra is clos... |
| saluncl 47532 | The union of two sets in a... |
| prsal 47533 | The pair of the empty set ... |
| saldifcl 47534 | The complement of an eleme... |
| 0sal 47535 | The empty set belongs to e... |
| salgenval 47536 | The sigma-algebra generate... |
| saliunclf 47537 | SAlg sigma-algebra is clos... |
| saliuncl 47538 | SAlg sigma-algebra is clos... |
| salincl 47539 | The intersection of two se... |
| saluni 47540 | A set is an element of any... |
| saliinclf 47541 | SAlg sigma-algebra is clos... |
| saliincl 47542 | SAlg sigma-algebra is clos... |
| saldifcl2 47543 | The difference of two elem... |
| intsaluni 47544 | The union of an arbitrary ... |
| intsal 47545 | The arbitrary intersection... |
| salgenn0 47546 | The set used in the defini... |
| salgencl 47547 | ` SalGen ` actually genera... |
| issald 47548 | Sufficient condition to pr... |
| salexct 47549 | An example of nontrivial s... |
| sssalgen 47550 | A set is a subset of the s... |
| salgenss 47551 | The sigma-algebra generate... |
| salgenuni 47552 | The base set of the sigma-... |
| issalgend 47553 | One side of ~ dfsalgen2 . ... |
| salexct2 47554 | An example of a subset tha... |
| unisalgen 47555 | The union of a set belongs... |
| dfsalgen2 47556 | Alternate characterization... |
| salexct3 47557 | An example of a sigma-alge... |
| salgencntex 47558 | This counterexample shows ... |
| salgensscntex 47559 | This counterexample shows ... |
| issalnnd 47560 | Sufficient condition to pr... |
| dmvolsal 47561 | Lebesgue measurable sets f... |
| saldifcld 47562 | The complement of an eleme... |
| saluncld 47563 | The union of two sets in a... |
| salgencld 47564 | ` SalGen ` actually genera... |
| 0sald 47565 | The empty set belongs to e... |
| iooborel 47566 | An open interval is a Bore... |
| salincld 47567 | The intersection of two se... |
| salunid 47568 | A set is an element of any... |
| unisalgen2 47569 | The union of a set belongs... |
| bor1sal 47570 | The Borel sigma-algebra on... |
| iocborel 47571 | A left-open, right-closed ... |
| subsaliuncllem 47572 | A subspace sigma-algebra i... |
| subsaliuncl 47573 | A subspace sigma-algebra i... |
| subsalsal 47574 | A subspace sigma-algebra i... |
| subsaluni 47575 | A set belongs to the subsp... |
| salrestss 47576 | A sigma-algebra restricted... |
| sge0rnre 47579 | When ` sum^ ` is applied t... |
| fge0icoicc 47580 | If ` F ` maps to nonnegati... |
| sge0val 47581 | The value of the sum of no... |
| fge0npnf 47582 | If ` F ` maps to nonnegati... |
| sge0rnn0 47583 | The range used in the defi... |
| sge0vald 47584 | The value of the sum of no... |
| fge0iccico 47585 | A range of nonnegative ext... |
| gsumge0cl 47586 | Closure of group sum, for ... |
| sge0reval 47587 | Value of the sum of nonneg... |
| sge0pnfval 47588 | If a term in the sum of no... |
| fge0iccre 47589 | A range of nonnegative ext... |
| sge0z 47590 | Any nonnegative extended s... |
| sge00 47591 | The sum of nonnegative ext... |
| fsumlesge0 47592 | Every finite subsum of non... |
| sge0revalmpt 47593 | Value of the sum of nonneg... |
| sge0sn 47594 | A sum of a nonnegative ext... |
| sge0tsms 47595 | ` sum^ ` applied to a nonn... |
| sge0cl 47596 | The arbitrary sum of nonne... |
| sge0f1o 47597 | Re-index a nonnegative ext... |
| sge0snmpt 47598 | A sum of a nonnegative ext... |
| sge0ge0 47599 | The sum of nonnegative ext... |
| sge0xrcl 47600 | The arbitrary sum of nonne... |
| sge0repnf 47601 | The of nonnegative extende... |
| sge0fsum 47602 | The arbitrary sum of a fin... |
| sge0rern 47603 | If the sum of nonnegative ... |
| sge0supre 47604 | If the arbitrary sum of no... |
| sge0fsummpt 47605 | The arbitrary sum of a fin... |
| sge0sup 47606 | The arbitrary sum of nonne... |
| sge0less 47607 | A shorter sum of nonnegati... |
| sge0rnbnd 47608 | The range used in the defi... |
| sge0pr 47609 | Sum of a pair of nonnegati... |
| sge0gerp 47610 | The arbitrary sum of nonne... |
| sge0pnffigt 47611 | If the sum of nonnegative ... |
| sge0ssre 47612 | If a sum of nonnegative ex... |
| sge0lefi 47613 | A sum of nonnegative exten... |
| sge0lessmpt 47614 | A shorter sum of nonnegati... |
| sge0ltfirp 47615 | If the sum of nonnegative ... |
| sge0prle 47616 | The sum of a pair of nonne... |
| sge0gerpmpt 47617 | The arbitrary sum of nonne... |
| sge0resrnlem 47618 | The sum of nonnegative ext... |
| sge0resrn 47619 | The sum of nonnegative ext... |
| sge0ssrempt 47620 | If a sum of nonnegative ex... |
| sge0resplit 47621 | ` sum^ ` splits into two p... |
| sge0le 47622 | If all of the terms of sum... |
| sge0ltfirpmpt 47623 | If the extended sum of non... |
| sge0split 47624 | Split a sum of nonnegative... |
| sge0lempt 47625 | If all of the terms of sum... |
| sge0splitmpt 47626 | Split a sum of nonnegative... |
| sge0ss 47627 | Change the index set to a ... |
| sge0iunmptlemfi 47628 | Sum of nonnegative extende... |
| sge0p1 47629 | The addition of the next t... |
| sge0iunmptlemre 47630 | Sum of nonnegative extende... |
| sge0fodjrnlem 47631 | Re-index a nonnegative ext... |
| sge0fodjrn 47632 | Re-index a nonnegative ext... |
| sge0iunmpt 47633 | Sum of nonnegative extende... |
| sge0iun 47634 | Sum of nonnegative extende... |
| sge0nemnf 47635 | The generalized sum of non... |
| sge0rpcpnf 47636 | The sum of an infinite num... |
| sge0rernmpt 47637 | If the sum of nonnegative ... |
| sge0lefimpt 47638 | A sum of nonnegative exten... |
| nn0ssge0 47639 | Nonnegative integers are n... |
| sge0clmpt 47640 | The generalized sum of non... |
| sge0ltfirpmpt2 47641 | If the extended sum of non... |
| sge0isum 47642 | If a series of nonnegative... |
| sge0xrclmpt 47643 | The generalized sum of non... |
| sge0xp 47644 | Combine two generalized su... |
| sge0isummpt 47645 | If a series of nonnegative... |
| sge0ad2en 47646 | The value of the infinite ... |
| sge0isummpt2 47647 | If a series of nonnegative... |
| sge0xaddlem1 47648 | The extended addition of t... |
| sge0xaddlem2 47649 | The extended addition of t... |
| sge0xadd 47650 | The extended addition of t... |
| sge0fsummptf 47651 | The generalized sum of a f... |
| sge0snmptf 47652 | A sum of a nonnegative ext... |
| sge0ge0mpt 47653 | The sum of nonnegative ext... |
| sge0repnfmpt 47654 | The of nonnegative extende... |
| sge0pnffigtmpt 47655 | If the generalized sum of ... |
| sge0splitsn 47656 | Separate out a term in a g... |
| sge0pnffsumgt 47657 | If the sum of nonnegative ... |
| sge0gtfsumgt 47658 | If the generalized sum of ... |
| sge0uzfsumgt 47659 | If a real number is smalle... |
| sge0pnfmpt 47660 | If a term in the sum of no... |
| sge0seq 47661 | A series of nonnegative re... |
| sge0reuz 47662 | Value of the generalized s... |
| sge0reuzb 47663 | Value of the generalized s... |
| ismea 47666 | Express the predicate " ` ... |
| dmmeasal 47667 | The domain of a measure is... |
| meaf 47668 | A measure is a function th... |
| mea0 47669 | The measure of the empty s... |
| nnfoctbdjlem 47670 | There exists a mapping fro... |
| nnfoctbdj 47671 | There exists a mapping fro... |
| meadjuni 47672 | The measure of the disjoin... |
| meacl 47673 | The measure of a set is a ... |
| iundjiunlem 47674 | The sets in the sequence `... |
| iundjiun 47675 | Given a sequence ` E ` of ... |
| meaxrcl 47676 | The measure of a set is an... |
| meadjun 47677 | The measure of the union o... |
| meassle 47678 | The measure of a set is gr... |
| meaunle 47679 | The measure of the union o... |
| meadjiunlem 47680 | The sum of nonnegative ext... |
| meadjiun 47681 | The measure of the disjoin... |
| ismeannd 47682 | Sufficient condition to pr... |
| meaiunlelem 47683 | The measure of the union o... |
| meaiunle 47684 | The measure of the union o... |
| psmeasurelem 47685 | ` M ` applied to a disjoin... |
| psmeasure 47686 | Point supported measure, R... |
| voliunsge0lem 47687 | The Lebesgue measure funct... |
| voliunsge0 47688 | The Lebesgue measure funct... |
| volmea 47689 | The Lebesgue measure on th... |
| meage0 47690 | If the measure of a measur... |
| meadjunre 47691 | The measure of the union o... |
| meassre 47692 | If the measure of a measur... |
| meale0eq0 47693 | A measure that is less tha... |
| meadif 47694 | The measure of the differe... |
| meaiuninclem 47695 | Measures are continuous fr... |
| meaiuninc 47696 | Measures are continuous fr... |
| meaiuninc2 47697 | Measures are continuous fr... |
| meaiunincf 47698 | Measures are continuous fr... |
| meaiuninc3v 47699 | Measures are continuous fr... |
| meaiuninc3 47700 | Measures are continuous fr... |
| meaiininclem 47701 | Measures are continuous fr... |
| meaiininc 47702 | Measures are continuous fr... |
| meaiininc2 47703 | Measures are continuous fr... |
| caragenval 47708 | The sigma-algebra generate... |
| isome 47709 | Express the predicate " ` ... |
| caragenel 47710 | Membership in the Caratheo... |
| omef 47711 | An outer measure is a func... |
| ome0 47712 | The outer measure of the e... |
| omessle 47713 | The outer measure of a set... |
| omedm 47714 | The domain of an outer mea... |
| caragensplit 47715 | If ` E ` is in the set gen... |
| caragenelss 47716 | An element of the Caratheo... |
| carageneld 47717 | Membership in the Caratheo... |
| omecl 47718 | The outer measure of a set... |
| caragenss 47719 | The sigma-algebra generate... |
| omeunile 47720 | The outer measure of the u... |
| caragen0 47721 | The empty set belongs to a... |
| omexrcl 47722 | The outer measure of a set... |
| caragenunidm 47723 | The base set of an outer m... |
| caragensspw 47724 | The sigma-algebra generate... |
| omessre 47725 | If the outer measure of a ... |
| caragenuni 47726 | The base set of the sigma-... |
| caragenuncllem 47727 | The Caratheodory's constru... |
| caragenuncl 47728 | The Caratheodory's constru... |
| caragendifcl 47729 | The Caratheodory's constru... |
| caragenfiiuncl 47730 | The Caratheodory's constru... |
| omeunle 47731 | The outer measure of the u... |
| omeiunle 47732 | The outer measure of the i... |
| omelesplit 47733 | The outer measure of a set... |
| omeiunltfirp 47734 | If the outer measure of a ... |
| omeiunlempt 47735 | The outer measure of the i... |
| carageniuncllem1 47736 | The outer measure of ` A i... |
| carageniuncllem2 47737 | The Caratheodory's constru... |
| carageniuncl 47738 | The Caratheodory's constru... |
| caragenunicl 47739 | The Caratheodory's constru... |
| caragensal 47740 | Caratheodory's method gene... |
| caratheodorylem1 47741 | Lemma used to prove that C... |
| caratheodorylem2 47742 | Caratheodory's constructio... |
| caratheodory 47743 | Caratheodory's constructio... |
| 0ome 47744 | The map that assigns 0 to ... |
| isomenndlem 47745 | ` O ` is sub-additive w.r.... |
| isomennd 47746 | Sufficient condition to pr... |
| caragenel2d 47747 | Membership in the Caratheo... |
| omege0 47748 | If the outer measure of a ... |
| omess0 47749 | If the outer measure of a ... |
| caragencmpl 47750 | A measure built with the C... |
| vonval 47755 | Value of the Lebesgue meas... |
| ovnval 47756 | Value of the Lebesgue oute... |
| elhoi 47757 | Membership in a multidimen... |
| icoresmbl 47758 | A closed-below, open-above... |
| hoissre 47759 | The projection of a half-o... |
| ovnval2 47760 | Value of the Lebesgue oute... |
| volicorecl 47761 | The Lebesgue measure of a ... |
| hoiprodcl 47762 | The pre-measure of half-op... |
| hoicvr 47763 | ` I ` is a countable set o... |
| hoissrrn 47764 | A half-open interval is a ... |
| ovn0val 47765 | The Lebesgue outer measure... |
| ovnn0val 47766 | The value of a (multidimen... |
| ovnval2b 47767 | Value of the Lebesgue oute... |
| volicorescl 47768 | The Lebesgue measure of a ... |
| ovnprodcl 47769 | The product used in the de... |
| hoiprodcl2 47770 | The pre-measure of half-op... |
| hoicvrrex 47771 | Any subset of the multidim... |
| ovnsupge0 47772 | The set used in the defini... |
| ovnlecvr 47773 | Given a subset of multidim... |
| ovnpnfelsup 47774 | ` +oo ` is an element of t... |
| ovnsslelem 47775 | The (multidimensional, non... |
| ovnssle 47776 | The (multidimensional) Leb... |
| ovnlerp 47777 | The Lebesgue outer measure... |
| ovnf 47778 | The Lebesgue outer measure... |
| ovncvrrp 47779 | The Lebesgue outer measure... |
| ovn0lem 47780 | For any finite dimension, ... |
| ovn0 47781 | For any finite dimension, ... |
| ovncl 47782 | The Lebesgue outer measure... |
| ovn02 47783 | For the zero-dimensional s... |
| ovnxrcl 47784 | The Lebesgue outer measure... |
| ovnsubaddlem1 47785 | The Lebesgue outer measure... |
| ovnsubaddlem2 47786 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 47787 | ` ( voln* `` X ) ` is suba... |
| ovnome 47788 | ` ( voln* `` X ) ` is an o... |
| vonmea 47789 | ` ( voln `` X ) ` is a mea... |
| volicon0 47790 | The measure of a nonempty ... |
| hsphoif 47791 | ` H ` is a function (that ... |
| hoidmvval 47792 | The dimensional volume of ... |
| hoissrrn2 47793 | A half-open interval is a ... |
| hsphoival 47794 | ` H ` is a function (that ... |
| hoiprodcl3 47795 | The pre-measure of half-op... |
| volicore 47796 | The Lebesgue measure of a ... |
| hoidmvcl 47797 | The dimensional volume of ... |
| hoidmv0val 47798 | The dimensional volume of ... |
| hoidmvn0val 47799 | The dimensional volume of ... |
| hsphoidmvle2 47800 | The dimensional volume of ... |
| hsphoidmvle 47801 | The dimensional volume of ... |
| hoidmvval0 47802 | The dimensional volume of ... |
| hoiprodp1 47803 | The dimensional volume of ... |
| sge0hsphoire 47804 | If the generalized sum of ... |
| hoidmvval0b 47805 | The dimensional volume of ... |
| hoidmv1lelem1 47806 | The supremum of ` U ` belo... |
| hoidmv1lelem2 47807 | This is the contradiction ... |
| hoidmv1lelem3 47808 | The dimensional volume of ... |
| hoidmv1le 47809 | The dimensional volume of ... |
| hoidmvlelem1 47810 | The supremum of ` U ` belo... |
| hoidmvlelem2 47811 | This is the contradiction ... |
| hoidmvlelem3 47812 | This is the contradiction ... |
| hoidmvlelem4 47813 | The dimensional volume of ... |
| hoidmvlelem5 47814 | The dimensional volume of ... |
| hoidmvle 47815 | The dimensional volume of ... |
| ovnhoilem1 47816 | The Lebesgue outer measure... |
| ovnhoilem2 47817 | The Lebesgue outer measure... |
| ovnhoi 47818 | The Lebesgue outer measure... |
| dmovn 47819 | The domain of the Lebesgue... |
| hoicoto2 47820 | The half-open interval exp... |
| dmvon 47821 | Lebesgue measurable n-dime... |
| hoi2toco 47822 | The half-open interval exp... |
| hoidifhspval 47823 | ` D ` is a function that r... |
| hspval 47824 | The value of the half-spac... |
| ovnlecvr2 47825 | Given a subset of multidim... |
| ovncvr2 47826 | ` B ` and ` T ` are the le... |
| dmovnsal 47827 | The domain of the Lebesgue... |
| unidmovn 47828 | Base set of the n-dimensio... |
| rrnmbl 47829 | The set of n-dimensional R... |
| hoidifhspval2 47830 | ` D ` is a function that r... |
| hspdifhsp 47831 | A n-dimensional half-open ... |
| unidmvon 47832 | Base set of the n-dimensio... |
| hoidifhspf 47833 | ` D ` is a function that r... |
| hoidifhspval3 47834 | ` D ` is a function that r... |
| hoidifhspdmvle 47835 | The dimensional volume of ... |
| voncmpl 47836 | The Lebesgue measure is co... |
| hoiqssbllem1 47837 | The center of the n-dimens... |
| hoiqssbllem2 47838 | The center of the n-dimens... |
| hoiqssbllem3 47839 | A n-dimensional ball conta... |
| hoiqssbl 47840 | A n-dimensional ball conta... |
| hspmbllem1 47841 | Any half-space of the n-di... |
| hspmbllem2 47842 | Any half-space of the n-di... |
| hspmbllem3 47843 | Any half-space of the n-di... |
| hspmbl 47844 | Any half-space of the n-di... |
| hoimbllem 47845 | Any n-dimensional half-ope... |
| hoimbl 47846 | Any n-dimensional half-ope... |
| opnvonmbllem1 47847 | The half-open interval exp... |
| opnvonmbllem2 47848 | An open subset of the n-di... |
| opnvonmbl 47849 | An open subset of the n-di... |
| opnssborel 47850 | Open sets of a generalized... |
| borelmbl 47851 | All Borel subsets of the n... |
| volicorege0 47852 | The Lebesgue measure of a ... |
| isvonmbl 47853 | The predicate " ` A ` is m... |
| mblvon 47854 | The n-dimensional Lebesgue... |
| vonmblss 47855 | n-dimensional Lebesgue mea... |
| volico2 47856 | The measure of left-closed... |
| vonmblss2 47857 | n-dimensional Lebesgue mea... |
| ovolval2lem 47858 | The value of the Lebesgue ... |
| ovolval2 47859 | The value of the Lebesgue ... |
| ovnsubadd2lem 47860 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 47861 | ` ( voln* `` X ) ` is suba... |
| ovolval3 47862 | The value of the Lebesgue ... |
| ovnsplit 47863 | The n-dimensional Lebesgue... |
| ovolval4lem1 47864 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 47865 | The value of the Lebesgue ... |
| ovolval4 47866 | The value of the Lebesgue ... |
| ovolval5lem1 47867 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 47868 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 47869 | The value of the Lebesgue ... |
| ovolval5 47870 | The value of the Lebesgue ... |
| ovnovollem1 47871 | if ` F ` is a cover of ` B... |
| ovnovollem2 47872 | if ` I ` is a cover of ` (... |
| ovnovollem3 47873 | The 1-dimensional Lebesgue... |
| ovnovol 47874 | The 1-dimensional Lebesgue... |
| vonvolmbllem 47875 | If a subset ` B ` of real ... |
| vonvolmbl 47876 | A subset of Real numbers i... |
| vonvol 47877 | The 1-dimensional Lebesgue... |
| vonvolmbl2 47878 | A subset ` X ` of the spac... |
| vonvol2 47879 | The 1-dimensional Lebesgue... |
| hoimbl2 47880 | Any n-dimensional half-ope... |
| voncl 47881 | The Lebesgue measure of a ... |
| vonhoi 47882 | The Lebesgue outer measure... |
| vonxrcl 47883 | The Lebesgue measure of a ... |
| ioosshoi 47884 | A n-dimensional open inter... |
| vonn0hoi 47885 | The Lebesgue outer measure... |
| von0val 47886 | The Lebesgue measure (for ... |
| vonhoire 47887 | The Lebesgue measure of a ... |
| iinhoiicclem 47888 | A n-dimensional closed int... |
| iinhoiicc 47889 | A n-dimensional closed int... |
| iunhoiioolem 47890 | A n-dimensional open inter... |
| iunhoiioo 47891 | A n-dimensional open inter... |
| ioovonmbl 47892 | Any n-dimensional open int... |
| iccvonmbllem 47893 | Any n-dimensional closed i... |
| iccvonmbl 47894 | Any n-dimensional closed i... |
| vonioolem1 47895 | The sequence of the measur... |
| vonioolem2 47896 | The n-dimensional Lebesgue... |
| vonioo 47897 | The n-dimensional Lebesgue... |
| vonicclem1 47898 | The sequence of the measur... |
| vonicclem2 47899 | The n-dimensional Lebesgue... |
| vonicc 47900 | The n-dimensional Lebesgue... |
| snvonmbl 47901 | A n-dimensional singleton ... |
| vonn0ioo 47902 | The n-dimensional Lebesgue... |
| vonn0icc 47903 | The n-dimensional Lebesgue... |
| ctvonmbl 47904 | Any n-dimensional countabl... |
| vonn0ioo2 47905 | The n-dimensional Lebesgue... |
| vonsn 47906 | The n-dimensional Lebesgue... |
| vonn0icc2 47907 | The n-dimensional Lebesgue... |
| vonct 47908 | The n-dimensional Lebesgue... |
| vitali2 47909 | There are non-measurable s... |
| pimltmnf2f 47912 | Given a real-valued functi... |
| pimltmnf2 47913 | Given a real-valued functi... |
| preimagelt 47914 | The preimage of a right-op... |
| preimalegt 47915 | The preimage of a left-ope... |
| pimconstlt0 47916 | Given a constant function,... |
| pimconstlt1 47917 | Given a constant function,... |
| pimltpnff 47918 | Given a real-valued functi... |
| pimltpnf 47919 | Given a real-valued functi... |
| pimgtpnf2f 47920 | Given a real-valued functi... |
| pimgtpnf2 47921 | Given a real-valued functi... |
| salpreimagelt 47922 | If all the preimages of le... |
| pimrecltpos 47923 | The preimage of an unbound... |
| salpreimalegt 47924 | If all the preimages of ri... |
| pimiooltgt 47925 | The preimage of an open in... |
| preimaicomnf 47926 | Preimage of an open interv... |
| pimltpnf2f 47927 | Given a real-valued functi... |
| pimltpnf2 47928 | Given a real-valued functi... |
| pimgtmnf2 47929 | Given a real-valued functi... |
| pimdecfgtioc 47930 | Given a nonincreasing func... |
| pimincfltioc 47931 | Given a nondecreasing func... |
| pimdecfgtioo 47932 | Given a nondecreasing func... |
| pimincfltioo 47933 | Given a nondecreasing func... |
| preimaioomnf 47934 | Preimage of an open interv... |
| preimageiingt 47935 | A preimage of a left-close... |
| preimaleiinlt 47936 | A preimage of a left-open,... |
| pimgtmnff 47937 | Given a real-valued functi... |
| pimgtmnf 47938 | Given a real-valued functi... |
| pimrecltneg 47939 | The preimage of an unbound... |
| salpreimagtge 47940 | If all the preimages of le... |
| salpreimaltle 47941 | If all the preimages of ri... |
| issmflem 47942 | The predicate " ` F ` is a... |
| issmf 47943 | The predicate " ` F ` is a... |
| salpreimalelt 47944 | If all the preimages of ri... |
| salpreimagtlt 47945 | If all the preimages of le... |
| smfpreimalt 47946 | Given a function measurabl... |
| smff 47947 | A function measurable w.r.... |
| smfdmss 47948 | The domain of a function m... |
| issmff 47949 | The predicate " ` F ` is a... |
| issmfd 47950 | A sufficient condition for... |
| smfpreimaltf 47951 | Given a function measurabl... |
| issmfdf 47952 | A sufficient condition for... |
| sssmf 47953 | The restriction of a sigma... |
| mbfresmf 47954 | A real-valued measurable f... |
| cnfsmf 47955 | A continuous function is m... |
| incsmflem 47956 | A nondecreasing function i... |
| incsmf 47957 | A real-valued, nondecreasi... |
| smfsssmf 47958 | If a function is measurabl... |
| issmflelem 47959 | The predicate " ` F ` is a... |
| issmfle 47960 | The predicate " ` F ` is a... |
| smfpimltmpt 47961 | Given a function measurabl... |
| smfpimltxr 47962 | Given a function measurabl... |
| issmfdmpt 47963 | A sufficient condition for... |
| smfconst 47964 | Given a sigma-algebra over... |
| sssmfmpt 47965 | The restriction of a sigma... |
| cnfrrnsmf 47966 | A function, continuous fro... |
| smfid 47967 | The identity function is B... |
| bormflebmf 47968 | A Borel measurable functio... |
| smfpreimale 47969 | Given a function measurabl... |
| issmfgtlem 47970 | The predicate " ` F ` is a... |
| issmfgt 47971 | The predicate " ` F ` is a... |
| issmfled 47972 | A sufficient condition for... |
| smfpimltxrmptf 47973 | Given a function measurabl... |
| smfpimltxrmpt 47974 | Given a function measurabl... |
| smfmbfcex 47975 | A constant function, with ... |
| issmfgtd 47976 | A sufficient condition for... |
| smfpreimagt 47977 | Given a function measurabl... |
| smfaddlem1 47978 | Given the sum of two funct... |
| smfaddlem2 47979 | The sum of two sigma-measu... |
| smfadd 47980 | The sum of two sigma-measu... |
| decsmflem 47981 | A nonincreasing function i... |
| decsmf 47982 | A real-valued, nonincreasi... |
| smfpreimagtf 47983 | Given a function measurabl... |
| issmfgelem 47984 | The predicate " ` F ` is a... |
| issmfge 47985 | The predicate " ` F ` is a... |
| smflimlem1 47986 | Lemma for the proof that t... |
| smflimlem2 47987 | Lemma for the proof that t... |
| smflimlem3 47988 | The limit of sigma-measura... |
| smflimlem4 47989 | Lemma for the proof that t... |
| smflimlem5 47990 | Lemma for the proof that t... |
| smflimlem6 47991 | Lemma for the proof that t... |
| smflim 47992 | The limit of sigma-measura... |
| nsssmfmbflem 47993 | The sigma-measurable funct... |
| nsssmfmbf 47994 | The sigma-measurable funct... |
| smfpimgtxr 47995 | Given a function measurabl... |
| smfpimgtmpt 47996 | Given a function measurabl... |
| smfpreimage 47997 | Given a function measurabl... |
| mbfpsssmf 47998 | Real-valued measurable fun... |
| smfpimgtxrmptf 47999 | Given a function measurabl... |
| smfpimgtxrmpt 48000 | Given a function measurabl... |
| smfpimioompt 48001 | Given a function measurabl... |
| smfpimioo 48002 | Given a function measurabl... |
| smfresal 48003 | Given a sigma-measurable f... |
| smfrec 48004 | The reciprocal of a sigma-... |
| smfres 48005 | The restriction of sigma-m... |
| smfmullem1 48006 | The multiplication of two ... |
| smfmullem2 48007 | The multiplication of two ... |
| smfmullem3 48008 | The multiplication of two ... |
| smfmullem4 48009 | The multiplication of two ... |
| smfmul 48010 | The multiplication of two ... |
| smfmulc1 48011 | A sigma-measurable functio... |
| smfdiv 48012 | The fraction of two sigma-... |
| smfpimbor1lem1 48013 | Every open set belongs to ... |
| smfpimbor1lem2 48014 | Given a sigma-measurable f... |
| smfpimbor1 48015 | Given a sigma-measurable f... |
| smf2id 48016 | Twice the identity functio... |
| smfco 48017 | The composition of a Borel... |
| smfneg 48018 | The negative of a sigma-me... |
| smffmptf 48019 | A function measurable w.r.... |
| smffmpt 48020 | A function measurable w.r.... |
| smflim2 48021 | The limit of a sequence of... |
| smfpimcclem 48022 | Lemma for ~ smfpimcc given... |
| smfpimcc 48023 | Given a countable set of s... |
| issmfle2d 48024 | A sufficient condition for... |
| smflimmpt 48025 | The limit of a sequence of... |
| smfsuplem1 48026 | The supremum of a countabl... |
| smfsuplem2 48027 | The supremum of a countabl... |
| smfsuplem3 48028 | The supremum of a countabl... |
| smfsup 48029 | The supremum of a countabl... |
| smfsupmpt 48030 | The supremum of a countabl... |
| smfsupxr 48031 | The supremum of a countabl... |
| smfinflem 48032 | The infimum of a countable... |
| smfinf 48033 | The infimum of a countable... |
| smfinfmpt 48034 | The infimum of a countable... |
| smflimsuplem1 48035 | If ` H ` converges, the ` ... |
| smflimsuplem2 48036 | The superior limit of a se... |
| smflimsuplem3 48037 | The limit of the ` ( H `` ... |
| smflimsuplem4 48038 | If ` H ` converges, the ` ... |
| smflimsuplem5 48039 | ` H ` converges to the sup... |
| smflimsuplem6 48040 | The superior limit of a se... |
| smflimsuplem7 48041 | The superior limit of a se... |
| smflimsuplem8 48042 | The superior limit of a se... |
| smflimsup 48043 | The superior limit of a se... |
| smflimsupmpt 48044 | The superior limit of a se... |
| smfliminflem 48045 | The inferior limit of a co... |
| smfliminf 48046 | The inferior limit of a co... |
| smfliminfmpt 48047 | The inferior limit of a co... |
| adddmmbl 48048 | If two functions have doma... |
| adddmmbl2 48049 | If two functions have doma... |
| muldmmbl 48050 | If two functions have doma... |
| muldmmbl2 48051 | If two functions have doma... |
| smfdmmblpimne 48052 | If a measurable function w... |
| smfdivdmmbl 48053 | If a functions and a sigma... |
| smfpimne 48054 | Given a function measurabl... |
| smfpimne2 48055 | Given a function measurabl... |
| smfdivdmmbl2 48056 | If a functions and a sigma... |
| fsupdm 48057 | The domain of the sup func... |
| fsupdm2 48058 | The domain of the sup func... |
| smfsupdmmbllem 48059 | If a countable set of sigm... |
| smfsupdmmbl 48060 | If a countable set of sigm... |
| finfdm 48061 | The domain of the inf func... |
| finfdm2 48062 | The domain of the inf func... |
| smfinfdmmbllem 48063 | If a countable set of sigm... |
| smfinfdmmbl 48064 | If a countable set of sigm... |
| sigarval 48065 | Define the signed area by ... |
| sigarim 48066 | Signed area takes value in... |
| sigarac 48067 | Signed area is anticommuta... |
| sigaraf 48068 | Signed area is additive by... |
| sigarmf 48069 | Signed area is additive (w... |
| sigaras 48070 | Signed area is additive by... |
| sigarms 48071 | Signed area is additive (w... |
| sigarls 48072 | Signed area is linear by t... |
| sigarid 48073 | Signed area of a flat para... |
| sigarexp 48074 | Expand the signed area for... |
| sigarperm 48075 | Signed area ` ( A - C ) G ... |
| sigardiv 48076 | If signed area between vec... |
| sigarimcd 48077 | Signed area takes value in... |
| sigariz 48078 | If signed area is zero, th... |
| sigarcol 48079 | Given three points ` A ` ,... |
| sharhght 48080 | Let ` A B C ` be a triangl... |
| sigaradd 48081 | Subtracting (double) area ... |
| cevathlem1 48082 | Ceva's theorem first lemma... |
| cevathlem2 48083 | Ceva's theorem second lemm... |
| cevath 48084 | Ceva's theorem. Let ` A B... |
| simpcntrab 48085 | The center of a simple gro... |
| et-ltneverrefl 48086 | Less-than class is never r... |
| et-equeucl 48087 | Alternative proof that equ... |
| et-sqrtnegnre 48088 | The square root of a negat... |
| quantgodel 48089 | There can be no formula as... |
| quantgodelALT 48090 | There can be no formula as... |
| ormklocald 48091 | If elements of a certain s... |
| ormkglobd 48092 | If all adjacent elements o... |
| chnsubseqword 48093 | A subsequence of a chain i... |
| chnsubseqwl 48094 | A subsequence of a chain h... |
| chnsubseq 48095 | An order-preserving subseq... |
| chnsuslle 48096 | Length of a subsequence is... |
| chnerlem1 48097 | In a chain constructed on ... |
| chnerlem2 48098 | Lemma for ~ chner where th... |
| chnerlem3 48099 | Lemma for ~ chner - tricho... |
| chner 48100 | Any two elements are equiv... |
| wrddin 48101 | A word in two alphabets is... |
| wrddrin 48102 | A word whose alphabet is i... |
| wrddin2 48103 | Distribution of word class... |
| wrddun 48104 | Words in either of two alp... |
| wrddun2 48105 | Superadditivity of word co... |
| chndin 48106 | A chain in two alphabets a... |
| chndrin 48107 | A chain whose alphabet is ... |
| chndin2 48108 | Distribution of chain clas... |
| chndun 48109 | Chains in either of two al... |
| chndun2 48110 | Superaddivity of chain con... |
| chnrin 48111 | Satisfying two chain relat... |
| chnrrin 48112 | A chain of elements satisf... |
| chnrin2 48113 | Distribution of chain clas... |
| chnrun 48114 | Satisfying either of two c... |
| chnrun2 48115 | Superadditivity of chain c... |
| evenwodadd 48116 | If an integer is multiplie... |
| squeezedltsq 48117 | If a real value is squeeze... |
| sqrtnnaa 48118 | Square root of a natural n... |
| sqrtnzqaa 48119 | Square root of a nonzero r... |
| sqrtqaa 48120 | Square root of a rational ... |
| numtowerdt 48121 | Certain number sets and fi... |
| sin3t 48122 | Triple-angle formula for s... |
| cos3t 48123 | Triple-angle formula for c... |
| sin5tlem1 48124 | Lemma 1 for quintupled ang... |
| sin5tlem2 48125 | Lemma 2 for quintupled ang... |
| sin5tlem3 48126 | Lemma 3 for quintupled ang... |
| sin5tlem4 48127 | Lemma 4 for quintupled ang... |
| sin5tlem5 48128 | Lemma 5 for quintupled ang... |
| sin5t 48129 | Five-times-angle formula f... |
| cos5t 48130 | Five-times-angle formula f... |
| cos5teq 48131 | Five-times-angle formula f... |
| goldpolyfactor 48132 | Factorization of a polynom... |
| goldrarr 48133 | The golden ratio is a real... |
| goldrasin 48134 | Alternative trigonometric ... |
| goldrapos 48135 | Golden ratio is positive. ... |
| goldrarp 48136 | The golden ratio is a posi... |
| goldracos5teq 48137 | Lemma 1 for determining th... |
| goldratmolem2 48138 | Lemma 2 for determining th... |
| goldratmolem3 48139 | Lemma 3 for determining th... |
| goldratmolem4 48140 | Lemma 4 for determining th... |
| goldratval 48141 | Value of the golden ratio.... |
| lambert0 48142 | A value of Lambert W (prod... |
| lamberte 48143 | A value of Lambert W (prod... |
| cjnpoly 48144 | Complex conjugation operat... |
| tannpoly 48145 | The tangent function is no... |
| sinnpoly 48146 | Sine function is not a pol... |
| sqrtrrnpoly 48147 | Real square root is not a ... |
| sqrtnpoly 48148 | Square root function is no... |
| tmachlem-extapes 48149 | The class of all tapes is ... |
| tmachlem-finscan 48150 | Execution on any tape only... |
| tmachlem-agreeself 48151 | Any tape belongs to its ow... |
| tmachlem-agreeprod 48152 | Agreement set can be writt... |
| tmachlem-tpcomp 48153 | Product (discrete) topolog... |
| tmachlem-tpbase 48154 | The base set of product to... |
| tmachlem-tpitem 48155 | Topology lemma. (Contribu... |
| tmachlem-tpopen 48156 | Agreement sets are open in... |
| tmachlem-tpopen2 48157 | Variable-renaming lemma co... |
| tmachlem-uassst 48158 | Union of all agreement set... |
| tmachlem-exlargecover 48159 | Product topology of tapes ... |
| tmachlem-extpcover 48160 | Product topology of tapes ... |
| tmachlem-exagreecover 48161 | Particular properties of t... |
| tmachlem-agreesn 48162 | Scans for all tapes of a s... |
| tmachlem-agreefin 48163 | Any agreement set has a fi... |
| tmachlem-franscan 48164 | There is a finite number o... |
| tmachlem-fssscan 48165 | Any scan set is finite. (... |
| tmachfullfin 48166 | Folk theorem. For any alg... |
| hirstL-ax3 48167 | The third axiom of a syste... |
| ax3h 48168 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 48169 | A closed form showing (a i... |
| aibandbiaiaiffb 48170 | A closed form showing (a i... |
| notatnand 48171 | Do not use. Use intnanr i... |
| aistia 48172 | Given a is equivalent to `... |
| aisfina 48173 | Given a is equivalent to `... |
| bothtbothsame 48174 | Given both a, b are equiva... |
| bothfbothsame 48175 | Given both a, b are equiva... |
| aiffbbtat 48176 | Given a is equivalent to b... |
| aisbbisfaisf 48177 | Given a is equivalent to b... |
| axorbtnotaiffb 48178 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 48179 | Given a is equivalent to (... |
| axorbciffatcxorb 48180 | Given a is equivalent to (... |
| aibnbna 48181 | Given a implies b, (not b)... |
| aibnbaif 48182 | Given a implies b, not b, ... |
| aiffbtbat 48183 | Given a is equivalent to b... |
| astbstanbst 48184 | Given a is equivalent to T... |
| aistbistaandb 48185 | Given a is equivalent to T... |
| aisbnaxb 48186 | Given a is equivalent to b... |
| atbiffatnnb 48187 | If a implies b, then a imp... |
| bisaiaisb 48188 | Application of bicom1 with... |
| atbiffatnnbalt 48189 | If a implies b, then a imp... |
| abnotbtaxb 48190 | Assuming a, not b, there e... |
| abnotataxb 48191 | Assuming not a, b, there e... |
| conimpf 48192 | Assuming a, not b, and a i... |
| conimpfalt 48193 | Assuming a, not b, and a i... |
| aistbisfiaxb 48194 | Given a is equivalent to T... |
| aisfbistiaxb 48195 | Given a is equivalent to F... |
| aifftbifffaibif 48196 | Given a is equivalent to T... |
| aifftbifffaibifff 48197 | Given a is equivalent to T... |
| atnaiana 48198 | Given a, it is not the cas... |
| ainaiaandna 48199 | Given a, a implies it is n... |
| abcdta 48200 | Given (((a and b) and c) a... |
| abcdtb 48201 | Given (((a and b) and c) a... |
| abcdtc 48202 | Given (((a and b) and c) a... |
| abcdtd 48203 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 48204 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 48205 | Operands in a biconditiona... |
| nabctnabc 48206 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 48207 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 48208 | From one negated implicati... |
| twonotinotbothi 48209 | From these two negated imp... |
| clifte 48210 | show d is the same as an i... |
| cliftet 48211 | show d is the same as an i... |
| clifteta 48212 | show d is the same as an i... |
| cliftetb 48213 | show d is the same as an i... |
| confun 48214 | Given the hypotheses there... |
| confun2 48215 | Confun simplified to two p... |
| confun3 48216 | Confun's more complex form... |
| confun4 48217 | An attempt at derivative. ... |
| confun5 48218 | An attempt at derivative. ... |
| plcofph 48219 | Given, a,b and a "definiti... |
| pldofph 48220 | Given, a,b c, d, "definiti... |
| plvcofph 48221 | Given, a,b,d, and "definit... |
| plvcofphax 48222 | Given, a,b,d, and "definit... |
| plvofpos 48223 | rh is derivable because ON... |
| mdandyv0 48224 | Given the equivalences set... |
| mdandyv1 48225 | Given the equivalences set... |
| mdandyv2 48226 | Given the equivalences set... |
| mdandyv3 48227 | Given the equivalences set... |
| mdandyv4 48228 | Given the equivalences set... |
| mdandyv5 48229 | Given the equivalences set... |
| mdandyv6 48230 | Given the equivalences set... |
| mdandyv7 48231 | Given the equivalences set... |
| mdandyv8 48232 | Given the equivalences set... |
| mdandyv9 48233 | Given the equivalences set... |
| mdandyv10 48234 | Given the equivalences set... |
| mdandyv11 48235 | Given the equivalences set... |
| mdandyv12 48236 | Given the equivalences set... |
| mdandyv13 48237 | Given the equivalences set... |
| mdandyv14 48238 | Given the equivalences set... |
| mdandyv15 48239 | Given the equivalences set... |
| mdandyvr0 48240 | Given the equivalences set... |
| mdandyvr1 48241 | Given the equivalences set... |
| mdandyvr2 48242 | Given the equivalences set... |
| mdandyvr3 48243 | Given the equivalences set... |
| mdandyvr4 48244 | Given the equivalences set... |
| mdandyvr5 48245 | Given the equivalences set... |
| mdandyvr6 48246 | Given the equivalences set... |
| mdandyvr7 48247 | Given the equivalences set... |
| mdandyvr8 48248 | Given the equivalences set... |
| mdandyvr9 48249 | Given the equivalences set... |
| mdandyvr10 48250 | Given the equivalences set... |
| mdandyvr11 48251 | Given the equivalences set... |
| mdandyvr12 48252 | Given the equivalences set... |
| mdandyvr13 48253 | Given the equivalences set... |
| mdandyvr14 48254 | Given the equivalences set... |
| mdandyvr15 48255 | Given the equivalences set... |
| mdandyvrx0 48256 | Given the exclusivities se... |
| mdandyvrx1 48257 | Given the exclusivities se... |
| mdandyvrx2 48258 | Given the exclusivities se... |
| mdandyvrx3 48259 | Given the exclusivities se... |
| mdandyvrx4 48260 | Given the exclusivities se... |
| mdandyvrx5 48261 | Given the exclusivities se... |
| mdandyvrx6 48262 | Given the exclusivities se... |
| mdandyvrx7 48263 | Given the exclusivities se... |
| mdandyvrx8 48264 | Given the exclusivities se... |
| mdandyvrx9 48265 | Given the exclusivities se... |
| mdandyvrx10 48266 | Given the exclusivities se... |
| mdandyvrx11 48267 | Given the exclusivities se... |
| mdandyvrx12 48268 | Given the exclusivities se... |
| mdandyvrx13 48269 | Given the exclusivities se... |
| mdandyvrx14 48270 | Given the exclusivities se... |
| mdandyvrx15 48271 | Given the exclusivities se... |
| H15NH16TH15IH16 48272 | Given 15 hypotheses and a ... |
| dandysum2p2e4 48273 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 48274 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 48275 | Replacement of a nested an... |
| adh-minim 48276 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 48277 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 48278 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 48279 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 48280 | Fourth lemma for the deriv... |
| adh-minim-ax1 48281 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 48282 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 48283 | Sixth lemma for the deriva... |
| adh-minim-ax2c 48284 | Derivation of a commuted f... |
| adh-minim-ax2 48285 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 48286 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 48287 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 48288 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 48289 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 48290 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 48291 | Third lemma for the deriva... |
| adh-minimp-sylsimp 48292 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 48293 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 48294 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 48295 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 48296 | Fourth lemma for the deriv... |
| adh-minimp-ax2 48297 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 48298 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 48299 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 48300 | If a class with one elemen... |
| eusnsn 48301 | There is a unique element ... |
| absnsb 48302 | If the class abstraction `... |
| euabsneu 48303 | Another way to express exi... |
| elprneb 48304 | An element of a proper uno... |
| oppr 48305 | Equality for ordered pairs... |
| opprb 48306 | Equality for unordered pai... |
| or2expropbilem1 48307 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 48308 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 48309 | If two classes are strictl... |
| eubrv 48310 | If there is a unique set w... |
| eubrdm 48311 | If there is a unique set w... |
| eldmressn 48312 | Element of the domain of a... |
| iota0def 48313 | Example for a defined iota... |
| iota0ndef 48314 | Example for an undefined i... |
| fveqvfvv 48315 | If a function's value at a... |
| fnresfnco 48316 | Composition of two functio... |
| funcoressn 48317 | A composition restricted t... |
| funressnfv 48318 | A restriction to a singlet... |
| funressndmfvrn 48319 | The value of a function ` ... |
| funressnvmo 48320 | A function restricted to a... |
| funressnmo 48321 | A function restricted to a... |
| funressneu 48322 | There is exactly one value... |
| fresfo 48323 | Conditions for a restricti... |
| fsetsniunop 48324 | The class of all functions... |
| fsetabsnop 48325 | The class of all functions... |
| fsetsnf 48326 | The mapping of an element ... |
| fsetsnf1 48327 | The mapping of an element ... |
| fsetsnfo 48328 | The mapping of an element ... |
| fsetsnf1o 48329 | The mapping of an element ... |
| fsetsnprcnex 48330 | The class of all functions... |
| cfsetssfset 48331 | The class of constant func... |
| cfsetsnfsetfv 48332 | The function value of the ... |
| cfsetsnfsetf 48333 | The mapping of the class o... |
| cfsetsnfsetf1 48334 | The mapping of the class o... |
| cfsetsnfsetfo 48335 | The mapping of the class o... |
| cfsetsnfsetf1o 48336 | The mapping of the class o... |
| fsetprcnexALT 48337 | First version of proof for... |
| fcoreslem1 48338 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 48339 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 48340 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 48341 | Lemma 4 for ~ fcores . (C... |
| fcores 48342 | Every composite function `... |
| fcoresf1lem 48343 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 48344 | If a composition is inject... |
| fcoresf1b 48345 | A composition is injective... |
| fcoresfo 48346 | If a composition is surjec... |
| fcoresfob 48347 | A composition is surjectiv... |
| fcoresf1ob 48348 | A composition is bijective... |
| f1cof1blem 48349 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 48350 | The composition of three b... |
| 3f1oss2 48351 | The composition of three b... |
| f1cof1b 48352 | If the range of ` F ` equa... |
| funfocofob 48353 | If the domain of a functio... |
| fnfocofob 48354 | If the domain of a functio... |
| focofob 48355 | If the domain of a functio... |
| f1ocof1ob 48356 | If the range of ` F ` equa... |
| f1ocof1ob2 48357 | If the range of ` F ` equa... |
| aiotajust 48359 | Soundness justification th... |
| dfaiota2 48361 | Alternate definition of th... |
| reuabaiotaiota 48362 | The iota and the alternate... |
| reuaiotaiota 48363 | The iota and the alternate... |
| aiotaexb 48364 | The alternate iota over a ... |
| aiotavb 48365 | The alternate iota over a ... |
| aiotaint 48366 | This is to ~ df-aiota what... |
| dfaiota3 48367 | Alternate definition of ` ... |
| iotan0aiotaex 48368 | If the iota over a wff ` p... |
| aiotaexaiotaiota 48369 | The alternate iota over a ... |
| aiotaval 48370 | Theorem 8.19 in [Quine] p.... |
| aiota0def 48371 | Example for a defined alte... |
| aiota0ndef 48372 | Example for an undefined a... |
| aiotaexim 48373 | Representation of "the uni... |
| r19.32 48374 | Theorem 19.32 of [Margaris... |
| rexsb 48375 | An equivalent expression f... |
| rexrsb 48376 | An equivalent expression f... |
| 2rexsb 48377 | An equivalent expression f... |
| 2rexrsb 48378 | An equivalent expression f... |
| cbvral2 48379 | Change bound variables of ... |
| cbvrex2 48380 | Change bound variables of ... |
| ralndv1 48381 | Example for a theorem abou... |
| ralndv2 48382 | Second example for a theor... |
| reuf1odnf 48383 | There is exactly one eleme... |
| reuf1od 48384 | There is exactly one eleme... |
| euoreqb 48385 | There is a set which is eq... |
| 2reu3 48386 | Double restricted existent... |
| 2reu7 48387 | Two equivalent expressions... |
| 2reu8 48388 | Two equivalent expressions... |
| 2reu8i 48389 | Implication of a double re... |
| 2reuimp0 48390 | Implication of a double re... |
| 2reuimp 48391 | Implication of a double re... |
| ralbinrald 48398 | Elemination of a restricte... |
| alneu 48399 | If a statement holds for a... |
| eu2ndop1stv 48400 | If there is a unique secon... |
| dfateq12d 48401 | Equality deduction for "de... |
| nfdfat 48402 | Bound-variable hypothesis ... |
| dfdfat2 48403 | Alternate definition of th... |
| fundmdfat 48404 | A function is defined at a... |
| dfatprc 48405 | A function is not defined ... |
| dfatelrn 48406 | The value of a function ` ... |
| dfafv2 48407 | Alternative definition of ... |
| afveq12d 48408 | Equality deduction for fun... |
| afveq1 48409 | Equality theorem for funct... |
| afveq2 48410 | Equality theorem for funct... |
| nfafv 48411 | Bound-variable hypothesis ... |
| csbafv12g 48412 | Move class substitution in... |
| afvfundmfveq 48413 | If a class is a function r... |
| afvnfundmuv 48414 | If a set is not in the dom... |
| ndmafv 48415 | The value of a class outsi... |
| afvvdm 48416 | If the function value of a... |
| nfunsnafv 48417 | If the restriction of a cl... |
| afvvfunressn 48418 | If the function value of a... |
| afvprc 48419 | A function's value at a pr... |
| afvvv 48420 | If a function's value at a... |
| afvpcfv0 48421 | If the value of the altern... |
| afvnufveq 48422 | The value of the alternati... |
| afvvfveq 48423 | The value of the alternati... |
| afv0fv0 48424 | If the value of the altern... |
| afvfvn0fveq 48425 | If the function's value at... |
| afv0nbfvbi 48426 | The function's value at an... |
| afvfv0bi 48427 | The function's value at an... |
| afveu 48428 | The value of a function at... |
| fnbrafvb 48429 | Equivalence of function va... |
| fnopafvb 48430 | Equivalence of function va... |
| funbrafvb 48431 | Equivalence of function va... |
| funopafvb 48432 | Equivalence of function va... |
| funbrafv 48433 | The second argument of a b... |
| funbrafv2b 48434 | Function value in terms of... |
| dfafn5a 48435 | Representation of a functi... |
| dfafn5b 48436 | Representation of a functi... |
| fnrnafv 48437 | The range of a function ex... |
| afvelrnb 48438 | A member of a function's r... |
| afvelrnb0 48439 | A member of a function's r... |
| dfaimafn 48440 | Alternate definition of th... |
| dfaimafn2 48441 | Alternate definition of th... |
| afvelima 48442 | Function value in an image... |
| afvelrn 48443 | A function's value belongs... |
| fnafvelrn 48444 | A function's value belongs... |
| fafvelcdm 48445 | A function's value belongs... |
| ffnafv 48446 | A function maps to a class... |
| afvres 48447 | The value of a restricted ... |
| tz6.12-afv 48448 | Function value. Theorem 6... |
| tz6.12-1-afv 48449 | Function value (Theorem 6.... |
| dmfcoafv 48450 | Domains of a function comp... |
| afvco2 48451 | Value of a function compos... |
| rlimdmafv 48452 | Two ways to express that a... |
| aoveq123d 48453 | Equality deduction for ope... |
| nfaov 48454 | Bound-variable hypothesis ... |
| csbaovg 48455 | Move class substitution in... |
| aovfundmoveq 48456 | If a class is a function r... |
| aovnfundmuv 48457 | If an ordered pair is not ... |
| ndmaov 48458 | The value of an operation ... |
| ndmaovg 48459 | The value of an operation ... |
| aovvdm 48460 | If the operation value of ... |
| nfunsnaov 48461 | If the restriction of a cl... |
| aovvfunressn 48462 | If the operation value of ... |
| aovprc 48463 | The value of an operation ... |
| aovrcl 48464 | Reverse closure for an ope... |
| aovpcov0 48465 | If the alternative value o... |
| aovnuoveq 48466 | The alternative value of t... |
| aovvoveq 48467 | The alternative value of t... |
| aov0ov0 48468 | If the alternative value o... |
| aovovn0oveq 48469 | If the operation's value a... |
| aov0nbovbi 48470 | The operation's value on a... |
| aovov0bi 48471 | The operation's value on a... |
| rspceaov 48472 | A frequently used special ... |
| fnotaovb 48473 | Equivalence of operation v... |
| ffnaov 48474 | An operation maps to a cla... |
| faovcl 48475 | Closure law for an operati... |
| aovmpt4g 48476 | Value of a function given ... |
| aoprssdm 48477 | Domain of closure of an op... |
| ndmaovcl 48478 | The "closure" of an operat... |
| ndmaovrcl 48479 | Reverse closure law, in co... |
| ndmaovcom 48480 | Any operation is commutati... |
| ndmaovass 48481 | Any operation is associati... |
| ndmaovdistr 48482 | Any operation is distribut... |
| dfatafv2iota 48485 | If a function is defined a... |
| ndfatafv2 48486 | The alternate function val... |
| ndfatafv2undef 48487 | The alternate function val... |
| dfatafv2ex 48488 | The alternate function val... |
| afv2ex 48489 | The alternate function val... |
| afv2eq12d 48490 | Equality deduction for fun... |
| afv2eq1 48491 | Equality theorem for funct... |
| afv2eq2 48492 | Equality theorem for funct... |
| nfafv2 48493 | Bound-variable hypothesis ... |
| csbafv212g 48494 | Move class substitution in... |
| fexafv2ex 48495 | The alternate function val... |
| ndfatafv2nrn 48496 | The alternate function val... |
| ndmafv2nrn 48497 | The value of a class outsi... |
| funressndmafv2rn 48498 | The alternate function val... |
| afv2ndefb 48499 | Two ways to say that an al... |
| nfunsnafv2 48500 | If the restriction of a cl... |
| afv2prc 48501 | A function's value at a pr... |
| dfatafv2rnb 48502 | The alternate function val... |
| afv2orxorb 48503 | If a set is in the range o... |
| dmafv2rnb 48504 | The alternate function val... |
| fundmafv2rnb 48505 | The alternate function val... |
| afv2elrn 48506 | An alternate function valu... |
| afv20defat 48507 | If the alternate function ... |
| fnafv2elrn 48508 | An alternate function valu... |
| fafv2elcdm 48509 | An alternate function valu... |
| fafv2elrnb 48510 | An alternate function valu... |
| fcdmvafv2v 48511 | If the codomain of a funct... |
| tz6.12-2-afv2 48512 | Function value when ` F ` ... |
| afv2eu 48513 | The value of a function at... |
| afv2res 48514 | The value of a restricted ... |
| tz6.12-afv2 48515 | Function value (Theorem 6.... |
| tz6.12-1-afv2 48516 | Function value (Theorem 6.... |
| tz6.12c-afv2 48517 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 48518 | Corollary of Theorem 6.12(... |
| funressnbrafv2 48519 | The second argument of a b... |
| dfatbrafv2b 48520 | Equivalence of function va... |
| dfatopafv2b 48521 | Equivalence of function va... |
| funbrafv2 48522 | The second argument of a b... |
| fnbrafv2b 48523 | Equivalence of function va... |
| fnopafv2b 48524 | Equivalence of function va... |
| funbrafv22b 48525 | Equivalence of function va... |
| funopafv2b 48526 | Equivalence of function va... |
| dfatsnafv2 48527 | Singleton of function valu... |
| dfafv23 48528 | A definition of function v... |
| dfatdmfcoafv2 48529 | Domain of a function compo... |
| dfatcolem 48530 | Lemma for ~ dfatco . (Con... |
| dfatco 48531 | The predicate "defined at"... |
| afv2co2 48532 | Value of a function compos... |
| rlimdmafv2 48533 | Two ways to express that a... |
| dfafv22 48534 | Alternate definition of ` ... |
| afv2ndeffv0 48535 | If the alternate function ... |
| dfatafv2eqfv 48536 | If a function is defined a... |
| afv2rnfveq 48537 | If the alternate function ... |
| afv20fv0 48538 | If the alternate function ... |
| afv2fvn0fveq 48539 | If the function's value at... |
| afv2fv0 48540 | If the function's value at... |
| afv2fv0b 48541 | The function's value at an... |
| afv2fv0xorb 48542 | If a set is in the range o... |
| an4com24 48543 | Rearrangement of 4 conjunc... |
| 3an4ancom24 48544 | Commutative law for a conj... |
| 4an21 48545 | Rearrangement of 4 conjunc... |
| dfnelbr2 48548 | Alternate definition of th... |
| nelbr 48549 | The binary relation of a s... |
| nelbrim 48550 | If a set is related to ano... |
| nelbrnel 48551 | A set is related to anothe... |
| nelbrnelim 48552 | If a set is related to ano... |
| ralralimp 48553 | Selecting one of two alter... |
| otiunsndisjX 48554 | The union of singletons co... |
| fvifeq 48555 | Equality of function value... |
| rnfdmpr 48556 | The range of a one-to-one ... |
| imarnf1pr 48557 | The image of the range of ... |
| funop1 48558 | A function is an ordered p... |
| fun2dmnopgexmpl 48559 | A function with a domain c... |
| opabresex0d 48560 | A collection of ordered pa... |
| opabbrfex0d 48561 | A collection of ordered pa... |
| opabresexd 48562 | A collection of ordered pa... |
| opabbrfexd 48563 | A collection of ordered pa... |
| f1oresf1orab 48564 | Build a bijection by restr... |
| f1oresf1o 48565 | Build a bijection by restr... |
| f1oresf1o2 48566 | Build a bijection by restr... |
| fvmptrab 48567 | Value of a function mappin... |
| fvmptrabdm 48568 | Value of a function mappin... |
| cnambpcma 48569 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 48570 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 48571 | The sum of two complex num... |
| leaddsuble 48572 | Addition and subtraction o... |
| 2leaddle2 48573 | If two real numbers are le... |
| ltnltne 48574 | Variant of trichotomy law ... |
| p1lep2 48575 | A real number increasd by ... |
| ltsubsubaddltsub 48576 | If the result of subtracti... |
| zm1nn 48577 | An integer minus 1 is posi... |
| readdcnnred 48578 | The sum of a real number a... |
| resubcnnred 48579 | The difference of a real n... |
| recnmulnred 48580 | The product of a real numb... |
| cndivrenred 48581 | The quotient of an imagina... |
| sqrtnegnre 48582 | The square root of a negat... |
| nn0resubcl 48583 | Closure law for subtractio... |
| zgeltp1eq 48584 | If an integer is between a... |
| 1t10e1p1e11 48585 | 11 is 1 times 10 to the po... |
| deccarry 48586 | Add 1 to a 2 digit number ... |
| eluzge0nn0 48587 | If an integer is greater t... |
| nltle2tri 48588 | Negated extended trichotom... |
| ssfz12 48589 | Subset relationship for fi... |
| elfz2z 48590 | Membership of an integer i... |
| 2elfz3nn0 48591 | If there are two elements ... |
| fz0addcom 48592 | The addition of two member... |
| 2elfz2melfz 48593 | If the sum of two integers... |
| fz0addge0 48594 | The sum of two integers in... |
| elfzlble 48595 | Membership of an integer i... |
| elfzelfzlble 48596 | Membership of an element o... |
| elfz2nn 48597 | A member of a finite set o... |
| fzopred 48598 | Join a predecessor to the ... |
| fzopredsuc 48599 | Join a predecessor and a s... |
| 1fzopredsuc 48600 | Join 0 and a successor to ... |
| el1fzopredsuc 48601 | An element of an open inte... |
| subsubelfzo0 48602 | Subtracting a difference f... |
| 2ffzoeq 48603 | Two functions over a half-... |
| elfzo2nn 48604 | A member of a half-open ra... |
| nnmul2 48605 | If one factor of a product... |
| nnmul2b 48606 | A factor of a product of i... |
| 2ltceilhalf 48607 | The ceiling of half of an ... |
| ceilhalfgt1 48608 | The ceiling of half of an ... |
| ceilhalfelfzo1 48609 | A positive integer less th... |
| gpgedgvtx1lem 48610 | Lemma for ~ gpgedgvtx1 . ... |
| 2tceilhalfelfzo1 48611 | Two times a positive integ... |
| ceilbi 48612 | A condition equivalent to ... |
| ceilhalf1 48613 | The ceiling of one half is... |
| rehalfge1 48614 | Half of a real number grea... |
| ceilhalfnn 48615 | The ceiling of half of a p... |
| 1elfzo1ceilhalf1 48616 | 1 is in the half-open inte... |
| nnge2recfl0 48617 | The floor of the reciproca... |
| flmrecm1 48618 | The floor of an integer mi... |
| fldivmod 48619 | Expressing the floor of a ... |
| ceildivmod 48620 | Expressing the ceiling of ... |
| ceil5half3 48621 | The ceiling of half of 5 i... |
| submodaddmod 48622 | Subtraction and addition m... |
| difltmodne 48623 | Two nonnegative integers a... |
| zplusmodne 48624 | A nonnegative integer is n... |
| addmodne 48625 | The sum of a nonnegative i... |
| plusmod5ne 48626 | A nonnegative integer is n... |
| zp1modne 48627 | An integer is not itself p... |
| p1modne 48628 | A nonnegative integer is n... |
| m1modne 48629 | A nonnegative integer is n... |
| minusmod5ne 48630 | A nonnegative integer is n... |
| submodlt 48631 | The difference of an eleme... |
| submodneaddmod 48632 | An integer minus ` B ` is ... |
| m1modnep2mod 48633 | A nonnegative integer minu... |
| minusmodnep2tmod 48634 | A nonnegative integer minu... |
| m1mod0mod1 48635 | An integer decreased by 1 ... |
| elmod2 48636 | An integer modulo 2 is eit... |
| mod0mul 48637 | If an integer is 0 modulo ... |
| modn0mul 48638 | If an integer is not 0 mod... |
| m1modmmod 48639 | An integer decreased by 1 ... |
| difmodm1lt 48640 | The difference between an ... |
| 8mod5e3 48641 | 8 modulo 5 is 3. (Contrib... |
| modmkpkne 48642 | If an integer minus a cons... |
| modmknepk 48643 | A nonnegative integer less... |
| modlt0b 48644 | An integer with an absolut... |
| mod2addne 48645 | The sums of a nonnegative ... |
| modm1nep1 48646 | A nonnegative integer less... |
| modm2nep1 48647 | A nonnegative integer less... |
| modp2nep1 48648 | A nonnegative integer less... |
| modm1nep2 48649 | A nonnegative integer less... |
| modm1nem2 48650 | A nonnegative integer less... |
| modm1p1ne 48651 | If an integer minus one eq... |
| smonoord 48652 | Ordering relation for a st... |
| 2timesltsq 48653 | Two times an integer great... |
| 2timesltsqm1 48654 | Two times an integer great... |
| fsummsndifre 48655 | A finite sum with one of i... |
| fsumsplitsndif 48656 | Separate out a term in a f... |
| fsummmodsndifre 48657 | A finite sum of summands m... |
| fsummmodsnunz 48658 | A finite sum of summands m... |
| nndivides2 48659 | Definition of the divides ... |
| facnn0dvdsfac 48660 | The factorial of a nonnega... |
| muldvdsfacgt 48661 | The product of two differe... |
| muldvdsfacm1 48662 | The product of two differe... |
| setsidel 48663 | The injected slot is an el... |
| setsnidel 48664 | The injected slot is an el... |
| setsv 48665 | The value of the structure... |
| preimafvsnel 48666 | The preimage of a function... |
| preimafvn0 48667 | The preimage of a function... |
| uniimafveqt 48668 | The union of the image of ... |
| uniimaprimaeqfv 48669 | The union of the image of ... |
| setpreimafvex 48670 | The class ` P ` of all pre... |
| elsetpreimafvb 48671 | The characterization of an... |
| elsetpreimafv 48672 | An element of the class ` ... |
| elsetpreimafvssdm 48673 | An element of the class ` ... |
| fvelsetpreimafv 48674 | There is an element in a p... |
| preimafvelsetpreimafv 48675 | The preimage of a function... |
| preimafvsspwdm 48676 | The class ` P ` of all pre... |
| 0nelsetpreimafv 48677 | The empty set is not an el... |
| elsetpreimafvbi 48678 | An element of the preimage... |
| elsetpreimafveqfv 48679 | The elements of the preima... |
| eqfvelsetpreimafv 48680 | If an element of the domai... |
| elsetpreimafvrab 48681 | An element of the preimage... |
| imaelsetpreimafv 48682 | The image of an element of... |
| uniimaelsetpreimafv 48683 | The union of the image of ... |
| elsetpreimafveq 48684 | If two preimages of functi... |
| fundcmpsurinjlem1 48685 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 48686 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 48687 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 48688 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 48689 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 48690 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 48691 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 48692 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 48693 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 48694 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 48695 | Every function ` F : A -->... |
| fundcmpsurinj 48696 | Every function ` F : A -->... |
| fundcmpsurbijinj 48697 | Every function ` F : A -->... |
| fundcmpsurinjimaid 48698 | Every function ` F : A -->... |
| fundcmpsurinjALT 48699 | Alternate proof of ~ fundc... |
| iccpval 48702 | Partition consisting of a ... |
| iccpart 48703 | A special partition. Corr... |
| iccpartimp 48704 | Implications for a class b... |
| iccpartres 48705 | The restriction of a parti... |
| iccpartxr 48706 | If there is a partition, t... |
| iccpartgtprec 48707 | If there is a partition, t... |
| iccpartipre 48708 | If there is a partition, t... |
| iccpartiltu 48709 | If there is a partition, t... |
| iccpartigtl 48710 | If there is a partition, t... |
| iccpartlt 48711 | If there is a partition, t... |
| iccpartltu 48712 | If there is a partition, t... |
| iccpartgtl 48713 | If there is a partition, t... |
| iccpartgt 48714 | If there is a partition, t... |
| iccpartleu 48715 | If there is a partition, t... |
| iccpartgel 48716 | If there is a partition, t... |
| iccpartrn 48717 | If there is a partition, t... |
| iccpartf 48718 | The range of the partition... |
| iccpartel 48719 | If there is a partition, t... |
| iccelpart 48720 | An element of any partitio... |
| iccpartiun 48721 | A half-open interval of ex... |
| icceuelpartlem 48722 | Lemma for ~ icceuelpart . ... |
| icceuelpart 48723 | An element of a partitione... |
| iccpartdisj 48724 | The segments of a partitio... |
| iccpartnel 48725 | A point of a partition is ... |
| fargshiftfv 48726 | If a class is a function, ... |
| fargshiftf 48727 | If a class is a function, ... |
| fargshiftf1 48728 | If a function is 1-1, then... |
| fargshiftfo 48729 | If a function is onto, the... |
| fargshiftfva 48730 | The values of a shifted fu... |
| lswn0 48731 | The last symbol of a nonem... |
| nfich1 48734 | The first interchangeable ... |
| nfich2 48735 | The second interchangeable... |
| ichv 48736 | Setvar variables are inter... |
| ichf 48737 | Setvar variables are inter... |
| ichid 48738 | A setvar variable is alway... |
| icht 48739 | A theorem is interchangeab... |
| ichbidv 48740 | Formula building rule for ... |
| ichcircshi 48741 | The setvar variables are i... |
| ichan 48742 | If two setvar variables ar... |
| ichn 48743 | Negation does not affect i... |
| ichim 48744 | Formula building rule for ... |
| dfich2 48745 | Alternate definition of th... |
| ichcom 48746 | The interchangeability of ... |
| ichbi12i 48747 | Equivalence for interchang... |
| icheqid 48748 | In an equality for the sam... |
| icheq 48749 | In an equality of setvar v... |
| ichnfimlem 48750 | Lemma for ~ ichnfim : A s... |
| ichnfim 48751 | If in an interchangeabilit... |
| ichnfb 48752 | If ` x ` and ` y ` are int... |
| ichal 48753 | Move a universal quantifie... |
| ich2al 48754 | Two setvar variables are a... |
| ich2ex 48755 | Two setvar variables are a... |
| ichexmpl1 48756 | Example for interchangeabl... |
| ichexmpl2 48757 | Example for interchangeabl... |
| ich2exprop 48758 | If the setvar variables ar... |
| ichnreuop 48759 | If the setvar variables ar... |
| ichreuopeq 48760 | If the setvar variables ar... |
| sprid 48761 | Two identical representati... |
| elsprel 48762 | An unordered pair is an el... |
| spr0nelg 48763 | The empty set is not an el... |
| sprval 48766 | The set of all unordered p... |
| sprvalpw 48767 | The set of all unordered p... |
| sprssspr 48768 | The set of all unordered p... |
| spr0el 48769 | The empty set is not an un... |
| sprvalpwn0 48770 | The set of all unordered p... |
| sprel 48771 | An element of the set of a... |
| prssspr 48772 | An element of a subset of ... |
| prelspr 48773 | An unordered pair of eleme... |
| prsprel 48774 | The elements of a pair fro... |
| prsssprel 48775 | The elements of a pair fro... |
| sprvalpwle2 48776 | The set of all unordered p... |
| sprsymrelfvlem 48777 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 48778 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 48779 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 48780 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 48781 | The value of the function ... |
| sprsymrelf 48782 | The mapping ` F ` is a fun... |
| sprsymrelf1 48783 | The mapping ` F ` is a one... |
| sprsymrelfo 48784 | The mapping ` F ` is a fun... |
| sprsymrelf1o 48785 | The mapping ` F ` is a bij... |
| sprbisymrel 48786 | There is a bijection betwe... |
| sprsymrelen 48787 | The class ` P ` of subsets... |
| prpair 48788 | Characterization of a prop... |
| prproropf1olem0 48789 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 48790 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 48791 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 48792 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 48793 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 48794 | There is a bijection betwe... |
| prproropen 48795 | The set of proper pairs an... |
| prproropreud 48796 | There is exactly one order... |
| pairreueq 48797 | Two equivalent representat... |
| paireqne 48798 | Two sets are not equal iff... |
| prprval 48801 | The set of all proper unor... |
| prprvalpw 48802 | The set of all proper unor... |
| prprelb 48803 | An element of the set of a... |
| prprelprb 48804 | A set is an element of the... |
| prprspr2 48805 | The set of all proper unor... |
| prprsprreu 48806 | There is a unique proper u... |
| prprreueq 48807 | There is a unique proper u... |
| sbcpr 48808 | The proper substitution of... |
| reupr 48809 | There is a unique unordere... |
| reuprpr 48810 | There is a unique proper u... |
| poprelb 48811 | Equality for unordered pai... |
| 2exopprim 48812 | The existence of an ordere... |
| reuopreuprim 48813 | There is a unique unordere... |
| nprmmul1 48814 | Special factorization of a... |
| nprmmul2 48815 | Special factorization of a... |
| nprmmul3 48816 | Special factorization of a... |
| fmtno 48819 | The ` N ` th Fermat number... |
| fmtnoge3 48820 | Each Fermat number is grea... |
| fmtnonn 48821 | Each Fermat number is a po... |
| fmtnom1nn 48822 | A Fermat number minus one ... |
| fmtnoodd 48823 | Each Fermat number is odd.... |
| fmtnorn 48824 | A Fermat number is a funct... |
| fmtnof1 48825 | The enumeration of the Fer... |
| fmtnoinf 48826 | The set of Fermat numbers ... |
| fmtnorec1 48827 | The first recurrence relat... |
| sqrtpwpw2p 48828 | The floor of the square ro... |
| fmtnosqrt 48829 | The floor of the square ro... |
| fmtno0 48830 | The ` 0 ` th Fermat number... |
| fmtno1 48831 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 48832 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 48833 | The second recurrence rela... |
| fmtnodvds 48834 | Any Fermat number divides ... |
| goldbachthlem1 48835 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 48836 | Lemma 2 for ~ goldbachth .... |
| goldbachth 48837 | Goldbach's theorem: Two d... |
| fmtnorec3 48838 | The third recurrence relat... |
| fmtnorec4 48839 | The fourth recurrence rela... |
| fmtno2 48840 | The ` 2 ` nd Fermat number... |
| fmtno3 48841 | The ` 3 ` rd Fermat number... |
| fmtno4 48842 | The ` 4 ` th Fermat number... |
| fmtno5lem1 48843 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 48844 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 48845 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 48846 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 48847 | The ` 5 ` th Fermat number... |
| fmtno0prm 48848 | The ` 0 ` th Fermat number... |
| fmtno1prm 48849 | The ` 1 ` st Fermat number... |
| fmtno2prm 48850 | The ` 2 ` nd Fermat number... |
| 257prm 48851 | 257 is a prime number (the... |
| fmtno3prm 48852 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 48853 | Any power of two is coprim... |
| fmtnoprmfac1lem 48854 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 48855 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 48856 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 48857 | Divisor of Fermat number (... |
| fmtnofac2lem 48858 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 48859 | Divisor of Fermat number (... |
| fmtnofac1 48860 | Divisor of Fermat number (... |
| fmtno4sqrt 48861 | The floor of the square ro... |
| fmtno4prmfac 48862 | If P was a (prime) factor ... |
| fmtno4prmfac193 48863 | If P was a (prime) factor ... |
| fmtno4nprmfac193 48864 | 193 is not a (prime) facto... |
| fmtno4prm 48865 | The ` 4 `-th Fermat number... |
| 65537prm 48866 | 65537 is a prime number (t... |
| fmtnofz04prm 48867 | The first five Fermat numb... |
| fmtnole4prm 48868 | The first five Fermat numb... |
| fmtno5faclem1 48869 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 48870 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 48871 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 48872 | The factorization of the `... |
| fmtno5nprm 48873 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 48874 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 48875 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 48876 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 48877 | The mapping of a Fermat nu... |
| prminf2 48878 | The set of prime numbers i... |
| 2pwp1prm 48879 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 48880 | Every prime number of the ... |
| m2prm 48881 | The second Mersenne number... |
| m3prm 48882 | The third Mersenne number ... |
| flsqrt 48883 | A condition equivalent to ... |
| flsqrt5 48884 | The floor of the square ro... |
| 3ndvds4 48885 | 3 does not divide 4. (Con... |
| 139prmALT 48886 | 139 is a prime number. In... |
| 31prm 48887 | 31 is a prime number. In ... |
| m5prm 48888 | The fifth Mersenne number ... |
| 127prm 48889 | 127 is a prime number. (C... |
| m7prm 48890 | The seventh Mersenne numbe... |
| m11nprm 48891 | The eleventh Mersenne numb... |
| mod42tp1mod8 48892 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 48893 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 48894 | If ` P ` is a Sophie Germa... |
| lighneallem1 48895 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 48896 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 48897 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 48898 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 48899 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 48900 | Lemma 3 for ~ lighneal . ... |
| lighneal 48901 | If a power of a prime ` P ... |
| modexp2m1d 48902 | The square of an integer w... |
| proththdlem 48903 | Lemma for ~ proththd . (C... |
| proththd 48904 | Proth's theorem (1878). I... |
| 5tcu2e40 48905 | 5 times the cube of 2 is 4... |
| 3exp4mod41 48906 | 3 to the fourth power is -... |
| 41prothprmlem1 48907 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 48908 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 48909 | 41 is a _Proth prime_. (C... |
| nprmdvdsfacm1lem1 48910 | Lemma 1 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem2 48911 | Lemma 2 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem3 48912 | Lemma 3 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem4 48913 | Lemma 4 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1 48914 | A non-prime integer greate... |
| ppivalnnprm 48915 | Value of a term of the pri... |
| ppivalnnnprmge6 48916 | Value of a term of the pri... |
| ppivalnn4 48917 | Value of the term of the p... |
| ppivalnnnprm 48918 | Value of a term of the pri... |
| indprm 48919 | An indicator function for ... |
| indprmfz 48920 | An indicator function for ... |
| ppi1sum 48921 | Value of the prime-countin... |
| ppivalnn 48922 | Value of the prime-countin... |
| quad1 48923 | A condition for a quadrati... |
| requad01 48924 | A condition for a quadrati... |
| requad1 48925 | A condition for a quadrati... |
| requad2 48926 | A condition for a quadrati... |
| iseven 48931 | The predicate "is an even ... |
| isodd 48932 | The predicate "is an odd n... |
| evenz 48933 | An even number is an integ... |
| oddz 48934 | An odd number is an intege... |
| evendiv2z 48935 | The result of dividing an ... |
| oddp1div2z 48936 | The result of dividing an ... |
| oddm1div2z 48937 | The result of dividing an ... |
| isodd2 48938 | The predicate "is an odd n... |
| dfodd2 48939 | Alternate definition for o... |
| dfodd6 48940 | Alternate definition for o... |
| dfeven4 48941 | Alternate definition for e... |
| evenm1odd 48942 | The predecessor of an even... |
| evenp1odd 48943 | The successor of an even n... |
| oddp1eveni 48944 | The successor of an odd nu... |
| oddm1eveni 48945 | The predecessor of an odd ... |
| evennodd 48946 | An even number is not an o... |
| oddneven 48947 | An odd number is not an ev... |
| enege 48948 | The negative of an even nu... |
| onego 48949 | The negative of an odd num... |
| m1expevenALTV 48950 | Exponentiation of -1 by an... |
| m1expoddALTV 48951 | Exponentiation of -1 by an... |
| dfeven2 48952 | Alternate definition for e... |
| dfodd3 48953 | Alternate definition for o... |
| iseven2 48954 | The predicate "is an even ... |
| isodd3 48955 | The predicate "is an odd n... |
| 2dvdseven 48956 | 2 divides an even number. ... |
| m2even 48957 | A multiple of 2 is an even... |
| 2ndvdsodd 48958 | 2 does not divide an odd n... |
| 2dvdsoddp1 48959 | 2 divides an odd number in... |
| 2dvdsoddm1 48960 | 2 divides an odd number de... |
| dfeven3 48961 | Alternate definition for e... |
| dfodd4 48962 | Alternate definition for o... |
| dfodd5 48963 | Alternate definition for o... |
| zefldiv2ALTV 48964 | The floor of an even numbe... |
| zofldiv2ALTV 48965 | The floor of an odd number... |
| oddflALTV 48966 | Odd number representation ... |
| iseven5 48967 | The predicate "is an even ... |
| isodd7 48968 | The predicate "is an odd n... |
| dfeven5 48969 | Alternate definition for e... |
| dfodd7 48970 | Alternate definition for o... |
| gcd2odd1 48971 | The greatest common diviso... |
| zneoALTV 48972 | No even integer equals an ... |
| zeoALTV 48973 | An integer is even or odd.... |
| zeo2ALTV 48974 | An integer is even or odd ... |
| nneoALTV 48975 | A positive integer is even... |
| nneoiALTV 48976 | A positive integer is even... |
| odd2np1ALTV 48977 | An integer is odd iff it i... |
| oddm1evenALTV 48978 | An integer is odd iff its ... |
| oddp1evenALTV 48979 | An integer is odd iff its ... |
| oexpnegALTV 48980 | The exponential of the neg... |
| oexpnegnz 48981 | The exponential of the neg... |
| bits0ALTV 48982 | Value of the zeroth bit. ... |
| bits0eALTV 48983 | The zeroth bit of an even ... |
| bits0oALTV 48984 | The zeroth bit of an odd n... |
| divgcdoddALTV 48985 | Either ` A / ( A gcd B ) `... |
| opoeALTV 48986 | The sum of two odds is eve... |
| opeoALTV 48987 | The sum of an odd and an e... |
| omoeALTV 48988 | The difference of two odds... |
| omeoALTV 48989 | The difference of an odd a... |
| oddprmALTV 48990 | A prime not equal to ` 2 `... |
| 0evenALTV 48991 | 0 is an even number. (Con... |
| 0noddALTV 48992 | 0 is not an odd number. (... |
| 1oddALTV 48993 | 1 is an odd number. (Cont... |
| 1nevenALTV 48994 | 1 is not an even number. ... |
| 2evenALTV 48995 | 2 is an even number. (Con... |
| 2noddALTV 48996 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 48997 | An odd nonnegative integer... |
| nnoALTV 48998 | An alternate characterizat... |
| nn0oALTV 48999 | An alternate characterizat... |
| nn0e 49000 | An alternate characterizat... |
| nneven 49001 | An alternate characterizat... |
| nn0onn0exALTV 49002 | For each odd nonnegative i... |
| nn0enn0exALTV 49003 | For each even nonnegative ... |
| nnennexALTV 49004 | For each even positive int... |
| nnpw2evenALTV 49005 | 2 to the power of a positi... |
| epoo 49006 | The sum of an even and an ... |
| emoo 49007 | The difference of an even ... |
| epee 49008 | The sum of two even number... |
| emee 49009 | The difference of two even... |
| evensumeven 49010 | If a summand is even, the ... |
| 3odd 49011 | 3 is an odd number. (Cont... |
| 4even 49012 | 4 is an even number. (Con... |
| 5odd 49013 | 5 is an odd number. (Cont... |
| 6even 49014 | 6 is an even number. (Con... |
| 7odd 49015 | 7 is an odd number. (Cont... |
| 8even 49016 | 8 is an even number. (Con... |
| evenprm2 49017 | A prime number is even iff... |
| oddprmne2 49018 | Every prime number not bei... |
| oddprmuzge3 49019 | A prime number which is od... |
| evenltle 49020 | If an even number is great... |
| odd2prm2 49021 | If an odd number is the su... |
| even3prm2 49022 | If an even number is the s... |
| mogoldbblem 49023 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 49024 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 49025 | Lemma for ~ perfectALTV . ... |
| perfectALTV 49026 | The Euclid-Euler theorem, ... |
| fppr 49029 | The set of Fermat pseudopr... |
| fpprmod 49030 | The set of Fermat pseudopr... |
| fpprel 49031 | A Fermat pseudoprime to th... |
| fpprbasnn 49032 | The base of a Fermat pseud... |
| fpprnn 49033 | A Fermat pseudoprime to th... |
| fppr2odd 49034 | A Fermat pseudoprime to th... |
| 11t31e341 49035 | 341 is the product of 11 a... |
| 2exp340mod341 49036 | Eight to the eighth power ... |
| 341fppr2 49037 | 341 is the (smallest) _Pou... |
| 4fppr1 49038 | 4 is the (smallest) Fermat... |
| 8exp8mod9 49039 | Eight to the eighth power ... |
| 9fppr8 49040 | 9 is the (smallest) Fermat... |
| dfwppr 49041 | Alternate definition of a ... |
| fpprwppr 49042 | A Fermat pseudoprime to th... |
| fpprwpprb 49043 | An integer ` X ` which is ... |
| fpprel2 49044 | An alternate definition fo... |
| nfermltl8rev 49045 | Fermat's little theorem wi... |
| nfermltl2rev 49046 | Fermat's little theorem wi... |
| nfermltlrev 49047 | Fermat's little theorem re... |
| isgbe 49054 | The predicate "is an even ... |
| isgbow 49055 | The predicate "is a weak o... |
| isgbo 49056 | The predicate "is an odd G... |
| gbeeven 49057 | An even Goldbach number is... |
| gbowodd 49058 | A weak odd Goldbach number... |
| gbogbow 49059 | A (strong) odd Goldbach nu... |
| gboodd 49060 | An odd Goldbach number is ... |
| gbepos 49061 | Any even Goldbach number i... |
| gbowpos 49062 | Any weak odd Goldbach numb... |
| gbopos 49063 | Any odd Goldbach number is... |
| gbegt5 49064 | Any even Goldbach number i... |
| gbowgt5 49065 | Any weak odd Goldbach numb... |
| gbowge7 49066 | Any weak odd Goldbach numb... |
| gboge9 49067 | Any odd Goldbach number is... |
| gbege6 49068 | Any even Goldbach number i... |
| gbpart6 49069 | The Goldbach partition of ... |
| gbpart7 49070 | The (weak) Goldbach partit... |
| gbpart8 49071 | The Goldbach partition of ... |
| gbpart9 49072 | The (strong) Goldbach part... |
| gbpart11 49073 | The (strong) Goldbach part... |
| 6gbe 49074 | 6 is an even Goldbach numb... |
| 7gbow 49075 | 7 is a weak odd Goldbach n... |
| 8gbe 49076 | 8 is an even Goldbach numb... |
| 9gbo 49077 | 9 is an odd Goldbach numbe... |
| 11gbo 49078 | 11 is an odd Goldbach numb... |
| stgoldbwt 49079 | If the strong ternary Gold... |
| sbgoldbwt 49080 | If the strong binary Goldb... |
| sbgoldbst 49081 | If the strong binary Goldb... |
| sbgoldbaltlem1 49082 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 49083 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 49084 | An alternate (related to t... |
| sbgoldbb 49085 | If the strong binary Goldb... |
| sgoldbeven3prm 49086 | If the binary Goldbach con... |
| sbgoldbm 49087 | If the strong binary Goldb... |
| mogoldbb 49088 | If the modern version of t... |
| sbgoldbmb 49089 | The strong binary Goldbach... |
| sbgoldbo 49090 | If the strong binary Goldb... |
| nnsum3primes4 49091 | 4 is the sum of at most 3 ... |
| nnsum4primes4 49092 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 49093 | Every prime is "the sum of... |
| nnsum4primesprm 49094 | Every prime is "the sum of... |
| nnsum3primesgbe 49095 | Any even Goldbach number i... |
| nnsum4primesgbe 49096 | Any even Goldbach number i... |
| nnsum3primesle9 49097 | Every integer greater than... |
| nnsum4primesle9 49098 | Every integer greater than... |
| nnsum4primesodd 49099 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 49100 | If the (strong) ternary Go... |
| evengpop3 49101 | If the (weak) ternary Gold... |
| evengpoap3 49102 | If the (strong) ternary Go... |
| nnsum4primeseven 49103 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 49104 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 49105 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 49106 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 49107 | If the binary Goldbach con... |
| bgoldbtbndlem1 49108 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 49109 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 49110 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 49111 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 49112 | If the binary Goldbach con... |
| tgoldbachgtALTV 49115 | Variant of Thierry Arnoux'... |
| bgoldbachlt 49116 | The binary Goldbach conjec... |
| tgblthelfgott 49118 | The ternary Goldbach conje... |
| tgoldbachlt 49119 | The ternary Goldbach conje... |
| tgoldbach 49120 | The ternary Goldbach conje... |
| clnbgrprc0 49123 | The closed neighborhood is... |
| clnbgrcl 49124 | If a class ` X ` has at le... |
| clnbgrval 49125 | The closed neighborhood of... |
| dfclnbgr2 49126 | Alternate definition of th... |
| dfclnbgr4 49127 | Alternate definition of th... |
| elclnbgrelnbgr 49128 | An element of the closed n... |
| dfclnbgr3 49129 | Alternate definition of th... |
| clnbgrnvtx0 49130 | If a class ` X ` is not a ... |
| clnbgrel 49131 | Characterization of a memb... |
| clnbgrvtxel 49132 | Every vertex ` K ` is a me... |
| clnbgrisvtx 49133 | Every member ` N ` of the ... |
| clnbgrssvtx 49134 | The closed neighborhood of... |
| clnbgrn0 49135 | The closed neighborhood of... |
| clnbupgr 49136 | The closed neighborhood of... |
| clnbupgrel 49137 | A member of the closed nei... |
| clnbupgreli 49138 | A member of the closed nei... |
| clnbgr0vtx 49139 | In a null graph (with no v... |
| clnbgr0edg 49140 | In an empty graph (with no... |
| clnbgrsym 49141 | In a graph, the closed nei... |
| predgclnbgrel 49142 | If a (not necessarily prop... |
| clnbgredg 49143 | A vertex connected by an e... |
| clnbgrssedg 49144 | The vertices connected by ... |
| edgusgrclnbfin 49145 | The size of the closed nei... |
| clnbusgrfi 49146 | The closed neighborhood of... |
| clnbfiusgrfi 49147 | The closed neighborhood of... |
| clnbgrlevtx 49148 | The size of the closed nei... |
| dfsclnbgr2 49149 | Alternate definition of th... |
| sclnbgrel 49150 | Characterization of a memb... |
| sclnbgrelself 49151 | A vertex ` N ` is a member... |
| sclnbgrisvtx 49152 | Every member ` X ` of the ... |
| dfclnbgr5 49153 | Alternate definition of th... |
| dfnbgr5 49154 | Alternate definition of th... |
| dfnbgrss 49155 | Subset chain for different... |
| dfvopnbgr2 49156 | Alternate definition of th... |
| vopnbgrel 49157 | Characterization of a memb... |
| vopnbgrelself 49158 | A vertex ` N ` is a member... |
| dfclnbgr6 49159 | Alternate definition of th... |
| dfnbgr6 49160 | Alternate definition of th... |
| dfsclnbgr6 49161 | Alternate definition of a ... |
| dfnbgrss2 49162 | Subset chain for different... |
| isisubgr 49165 | The subgraph induced by a ... |
| isubgriedg 49166 | The edges of an induced su... |
| isubgrvtxuhgr 49167 | The subgraph induced by th... |
| isubgredgss 49168 | The edges of an induced su... |
| isubgredg 49169 | An edge of an induced subg... |
| isubgrvtx 49170 | The vertices of an induced... |
| isubgruhgr 49171 | An induced subgraph of a h... |
| isubgrsubgr 49172 | An induced subgraph of a h... |
| isubgrupgr 49173 | An induced subgraph of a p... |
| isubgrumgr 49174 | An induced subgraph of a m... |
| isubgrusgr 49175 | An induced subgraph of a s... |
| isubgr0uhgr 49176 | The subgraph induced by an... |
| grimfn 49182 | The graph isomorphism func... |
| grimdmrel 49183 | The domain of the graph is... |
| isgrim 49185 | An isomorphism of graphs i... |
| grimprop 49186 | Properties of an isomorphi... |
| grimf1o 49187 | An isomorphism of graphs i... |
| grimidvtxedg 49188 | The identity relation rest... |
| grimid 49189 | The identity relation rest... |
| grimuhgr 49190 | If there is a graph isomor... |
| grimcnv 49191 | The converse of a graph is... |
| grimco 49192 | The composition of graph i... |
| uhgrimedgi 49193 | An isomorphism between gra... |
| uhgrimedg 49194 | An isomorphism between gra... |
| uhgrimprop 49195 | An isomorphism between hyp... |
| isuspgrim0lem 49196 | An isomorphism of simple p... |
| isuspgrim0 49197 | An isomorphism of simple p... |
| isuspgrimlem 49198 | Lemma for ~ isuspgrim . (... |
| isuspgrim 49199 | A class is an isomorphism ... |
| upgrimwlklem1 49200 | Lemma 1 for ~ upgrimwlk an... |
| upgrimwlklem2 49201 | Lemma 2 for ~ upgrimwlk . ... |
| upgrimwlklem3 49202 | Lemma 3 for ~ upgrimwlk . ... |
| upgrimwlklem4 49203 | Lemma 4 for ~ upgrimwlk . ... |
| upgrimwlklem5 49204 | Lemma 5 for ~ upgrimwlk . ... |
| upgrimwlk 49205 | Graph isomorphisms between... |
| upgrimwlklen 49206 | Graph isomorphisms between... |
| upgrimtrlslem1 49207 | Lemma 1 for ~ upgrimtrls .... |
| upgrimtrlslem2 49208 | Lemma 2 for ~ upgrimtrls .... |
| upgrimtrls 49209 | Graph isomorphisms between... |
| upgrimpthslem1 49210 | Lemma 1 for ~ upgrimpths .... |
| upgrimpthslem2 49211 | Lemma 2 for ~ upgrimpths .... |
| upgrimpths 49212 | Graph isomorphisms between... |
| upgrimspths 49213 | Graph isomorphisms between... |
| upgrimcycls 49214 | Graph isomorphisms between... |
| brgric 49215 | The relation "is isomorphi... |
| brgrici 49216 | Prove that two graphs are ... |
| gricrcl 49217 | Reverse closure of the "is... |
| dfgric2 49218 | Alternate, explicit defini... |
| gricbri 49219 | Implications of two graphs... |
| gricushgr 49220 | The "is isomorphic to" rel... |
| gricuspgr 49221 | The "is isomorphic to" rel... |
| gricrel 49222 | The "is isomorphic to" rel... |
| gricref 49223 | Graph isomorphism is refle... |
| gricsym 49224 | Graph isomorphism is symme... |
| gricsymb 49225 | Graph isomorphism is symme... |
| grictr 49226 | Graph isomorphism is trans... |
| gricer 49227 | Isomorphism is an equivale... |
| gricen 49228 | Isomorphic graphs have equ... |
| opstrgric 49229 | A graph represented as an ... |
| ushggricedg 49230 | A simple hypergraph (with ... |
| cycldlenngric 49231 | Two simple pseudographs ar... |
| isubgrgrim 49232 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 49233 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 49234 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 49235 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 49236 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 49237 | Graph isomorphisms between... |
| grimedg 49238 | For two isomorphic graphs,... |
| grimedgi 49239 | Graph isomorphisms map edg... |
| grtriproplem 49242 | Lemma for ~ grtriprop . (... |
| grtri 49243 | The triangles in a graph. ... |
| grtriprop 49244 | The properties of a triang... |
| grtrif1o 49245 | Any bijection onto a trian... |
| isgrtri 49246 | A triangle in a graph. (C... |
| grtrissvtx 49247 | A triangle is a subset of ... |
| grtriclwlk3 49248 | A triangle induces a close... |
| cycl3grtrilem 49249 | Lemma for ~ cycl3grtri . ... |
| cycl3grtri 49250 | The vertices of a cycle of... |
| grtrimap 49251 | Conditions for mapping tri... |
| grimgrtri 49252 | Graph isomorphisms map tri... |
| usgrgrtrirex 49253 | Conditions for a simple gr... |
| stgrfv 49256 | The star graph S_N. (Contr... |
| stgrvtx 49257 | The vertices of the star g... |
| stgriedg 49258 | The indexed edges of the s... |
| stgredg 49259 | The edges of the star grap... |
| stgredgel 49260 | An edge of the star graph ... |
| stgredgiun 49261 | The edges of the star grap... |
| stgrusgra 49262 | The star graph S_N is a si... |
| stgr0 49263 | The star graph S_0 consist... |
| stgr1 49264 | The star graph S_1 consist... |
| stgrvtx0 49265 | The center ("internal node... |
| stgrorder 49266 | The order of a star graph ... |
| stgrnbgr0 49267 | All vertices of a star gra... |
| stgrclnbgr0 49268 | All vertices of a star gra... |
| isubgr3stgrlem1 49269 | Lemma 1 for ~ isubgr3stgr ... |
| isubgr3stgrlem2 49270 | Lemma 2 for ~ isubgr3stgr ... |
| isubgr3stgrlem3 49271 | Lemma 3 for ~ isubgr3stgr ... |
| isubgr3stgrlem4 49272 | Lemma 4 for ~ isubgr3stgr ... |
| isubgr3stgrlem5 49273 | Lemma 5 for ~ isubgr3stgr ... |
| isubgr3stgrlem6 49274 | Lemma 6 for ~ isubgr3stgr ... |
| isubgr3stgrlem7 49275 | Lemma 7 for ~ isubgr3stgr ... |
| isubgr3stgrlem8 49276 | Lemma 8 for ~ isubgr3stgr ... |
| isubgr3stgrlem9 49277 | Lemma 9 for ~ isubgr3stgr ... |
| isubgr3stgr 49278 | If a vertex of a simple gr... |
| grlimfn 49282 | The graph local isomorphis... |
| grlimdmrel 49283 | The domain of the graph lo... |
| isgrlim 49285 | A local isomorphism of gra... |
| isgrlim2 49286 | A local isomorphism of gra... |
| grlimprop 49287 | Properties of a local isom... |
| grlimf1o 49288 | A local isomorphism of gra... |
| grlimprop2 49289 | Properties of a local isom... |
| uhgrimgrlim 49290 | An isomorphism of hypergra... |
| uspgrlimlem1 49291 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 49292 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 49293 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 49294 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 49295 | A local isomorphism of sim... |
| usgrlimprop 49296 | Properties of a local isom... |
| clnbgrvtxedg 49297 | An edge ` E ` containing a... |
| grlimedgclnbgr 49298 | For two locally isomorphic... |
| grlimprclnbgr 49299 | For two locally isomorphic... |
| grlimprclnbgredg 49300 | For two locally isomorphic... |
| grlimpredg 49301 | For two locally isomorphic... |
| grlimprclnbgrvtx 49302 | For two locally isomorphic... |
| grlimgredgex 49303 | Local isomorphisms between... |
| grlimgrtrilem1 49304 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 49305 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 49306 | If one of two locally isom... |
| brgrlic 49307 | The relation "is locally i... |
| brgrilci 49308 | Prove that two graphs are ... |
| grlicrel 49309 | The "is locally isomorphic... |
| grlicrcl 49310 | Reverse closure of the "is... |
| dfgrlic2 49311 | Alternate, explicit defini... |
| grilcbri 49312 | Implications of two graphs... |
| dfgrlic3 49313 | Alternate, explicit defini... |
| grilcbri2 49314 | Implications of two graphs... |
| grlicref 49315 | Graph local isomorphism is... |
| grlicsym 49316 | Graph local isomorphism is... |
| grlicsymb 49317 | Graph local isomorphism is... |
| grlictr 49318 | Graph local isomorphism is... |
| grlicer 49319 | Local isomorphism is an eq... |
| grlicen 49320 | Locally isomorphic graphs ... |
| gricgrlic 49321 | Isomorphic hypergraphs are... |
| clnbgr3stgrgrlim 49322 | If all (closed) neighborho... |
| clnbgr3stgrgrlic 49323 | If all (closed) neighborho... |
| usgrexmpl1lem 49324 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 49325 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 49326 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 49327 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 49328 | ` G ` contains a triangle ... |
| usgrexmpl2lem 49329 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 49330 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 49331 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 49332 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 49333 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 49334 | The neighborhood of the fi... |
| usgrexmpl2nb1 49335 | The neighborhood of the se... |
| usgrexmpl2nb2 49336 | The neighborhood of the th... |
| usgrexmpl2nb3 49337 | The neighborhood of the fo... |
| usgrexmpl2nb4 49338 | The neighborhood of the fi... |
| usgrexmpl2nb5 49339 | The neighborhood of the si... |
| usgrexmpl2trifr 49340 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 49341 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 49342 | The graphs ` H ` and ` G `... |
| gpgov 49345 | The generalized Petersen g... |
| gpgvtx 49346 | The vertices of the genera... |
| gpgiedg 49347 | The indexed edges of the g... |
| gpgedg 49348 | The edges of the generaliz... |
| gpgiedgdmellem 49349 | Lemma for ~ gpgiedgdmel an... |
| gpgvtxel 49350 | A vertex in a generalized ... |
| gpgvtxel2 49351 | The second component of a ... |
| gpgiedgdmel 49352 | An index of edges of the g... |
| gpgedgel 49353 | An edge in a generalized P... |
| gpgprismgriedgdmel 49354 | An index of edges of the g... |
| gpgprismgriedgdmss 49355 | A subset of the index of e... |
| gpgvtx0 49356 | The outside vertices in a ... |
| gpgvtx1 49357 | The inside vertices in a g... |
| opgpgvtx 49358 | A vertex in a generalized ... |
| gpgusgralem 49359 | Lemma for ~ gpgusgra . (C... |
| gpgusgra 49360 | The generalized Petersen g... |
| gpgprismgrusgra 49361 | The generalized Petersen g... |
| gpgorder 49362 | The order of the generaliz... |
| gpg5order 49363 | The order of a generalized... |
| gpgedgvtx0 49364 | The edges starting at an o... |
| gpgedgvtx1 49365 | The edges starting at an i... |
| gpgvtxedg0 49366 | The edges starting at an o... |
| gpgvtxedg1 49367 | The edges starting at an i... |
| gpgedgiov 49368 | The edges of the generaliz... |
| gpgedg2ov 49369 | The edges of the generaliz... |
| gpgedg2iv 49370 | The edges of the generaliz... |
| gpg5nbgrvtx03starlem1 49371 | Lemma 1 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem2 49372 | Lemma 2 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem3 49373 | Lemma 3 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx13starlem1 49374 | Lemma 1 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem2 49375 | Lemma 2 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem3 49376 | Lemma 3 for ~ gpg5nbgr3sta... |
| gpgnbgrvtx0 49377 | The (open) neighborhood of... |
| gpgnbgrvtx1 49378 | The (open) neighborhood of... |
| gpg3nbgrvtx0 49379 | In a generalized Petersen ... |
| gpg3nbgrvtx0ALT 49380 | In a generalized Petersen ... |
| gpg3nbgrvtx1 49381 | In a generalized Petersen ... |
| gpgcubic 49382 | Every generalized Petersen... |
| gpg5nbgrvtx03star 49383 | In a generalized Petersen ... |
| gpg5nbgr3star 49384 | In a generalized Petersen ... |
| gpgvtxdg3 49385 | Every vertex in a generali... |
| gpg3kgrtriexlem1 49386 | Lemma 1 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem2 49387 | Lemma 2 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem3 49388 | Lemma 3 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem4 49389 | Lemma 4 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem5 49390 | Lemma 5 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem6 49391 | Lemma 6 for ~ gpg3kgrtriex... |
| gpg3kgrtriex 49392 | All generalized Petersen g... |
| gpg5gricstgr3 49393 | Each closed neighborhood i... |
| pglem 49394 | Lemma for theorems about P... |
| pgjsgr 49395 | A Petersen graph is a simp... |
| gpg5grlim 49396 | A local isomorphism betwee... |
| gpg5grlic 49397 | The two generalized Peters... |
| gpgprismgr4cycllem1 49398 | Lemma 1 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem2 49399 | Lemma 2 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem3 49400 | Lemma 3 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem4 49401 | Lemma 4 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem5 49402 | Lemma 5 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem6 49403 | Lemma 6 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem7 49404 | Lemma 7 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem8 49405 | Lemma 8 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem9 49406 | Lemma 9 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem10 49407 | Lemma 10 for ~ gpgprismgr4... |
| gpgprismgr4cycllem11 49408 | Lemma 11 for ~ gpgprismgr4... |
| gpgprismgr4cycl0 49409 | The generalized Petersen g... |
| gpgprismgr4cyclex 49410 | The generalized Petersen g... |
| pgnioedg1 49411 | An inside and an outside v... |
| pgnioedg2 49412 | An inside and an outside v... |
| pgnioedg3 49413 | An inside and an outside v... |
| pgnioedg4 49414 | An inside and an outside v... |
| pgnioedg5 49415 | An inside and an outside v... |
| pgnbgreunbgrlem1 49416 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem1 49417 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem2 49418 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem3 49419 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2 49420 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem3 49421 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem4 49422 | Lemma 4 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem1 49423 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem2 49424 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem3 49425 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5 49426 | Lemma 5 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem6 49427 | Lemma 6 for ~ pgnbgreunbgr... |
| pgnbgreunbgr 49428 | In a Petersen graph, two d... |
| pgn4cyclex 49429 | A cycle in a Petersen grap... |
| pg4cyclnex 49430 | In the Petersen graph G(5,... |
| gpg5ngric 49431 | The two generalized Peters... |
| lgricngricex 49432 | There are two different lo... |
| gpg5edgnedg 49433 | Two consecutive (according... |
| grlimedgnedg 49434 | In general, the image of a... |
| 1hegrlfgr 49435 | A graph ` G ` with one hyp... |
| upwlksfval 49438 | The set of simple walks (i... |
| isupwlk 49439 | Properties of a pair of fu... |
| isupwlkg 49440 | Generalization of ~ isupwl... |
| upwlkbprop 49441 | Basic properties of a simp... |
| upwlkwlk 49442 | A simple walk is a walk. ... |
| upgrwlkupwlk 49443 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 49444 | In a pseudograph, the defi... |
| upgrisupwlkALT 49445 | Alternate proof of ~ upgri... |
| upgredgssspr 49446 | The set of edges of a pseu... |
| uspgropssxp 49447 | The set ` G ` of "simple p... |
| uspgrsprfv 49448 | The value of the function ... |
| uspgrsprf 49449 | The mapping ` F ` is a fun... |
| uspgrsprf1 49450 | The mapping ` F ` is a one... |
| uspgrsprfo 49451 | The mapping ` F ` is a fun... |
| uspgrsprf1o 49452 | The mapping ` F ` is a bij... |
| uspgrex 49453 | The class ` G ` of all "si... |
| uspgrbispr 49454 | There is a bijection betwe... |
| uspgrspren 49455 | The set ` G ` of the "simp... |
| uspgrymrelen 49456 | The set ` G ` of the "simp... |
| uspgrbisymrel 49457 | There is a bijection betwe... |
| uspgrbisymrelALT 49458 | Alternate proof of ~ uspgr... |
| ovn0dmfun 49459 | If a class operation value... |
| xpsnopab 49460 | A Cartesian product with a... |
| xpiun 49461 | A Cartesian product expres... |
| fnxpdmdm 49462 | The domain of the domain o... |
| cnfldsrngbas 49463 | The base set of a subring ... |
| cnfldsrngadd 49464 | The group addition operati... |
| cnfldsrngmul 49465 | The ring multiplication op... |
| plusfreseq 49466 | If the empty set is not co... |
| mgmplusfreseq 49467 | If the empty set is not co... |
| 0mgm 49468 | A set with an empty base s... |
| opmpoismgm 49469 | A structure with a group a... |
| copissgrp 49470 | A structure with a constan... |
| copisnmnd 49471 | A structure with a constan... |
| 0nodd 49472 | 0 is not an odd integer. ... |
| 1odd 49473 | 1 is an odd integer. (Con... |
| 2nodd 49474 | 2 is not an odd integer. ... |
| oddibas 49475 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 49476 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 49477 | The structure of all odd i... |
| nnsgrpmgm 49478 | The structure of positive ... |
| nnsgrp 49479 | The structure of positive ... |
| nnsgrpnmnd 49480 | The structure of positive ... |
| nn0mnd 49481 | The set of nonnegative int... |
| gsumsplit2f 49482 | Split a group sum into two... |
| gsumdifsndf 49483 | Extract a summand from a f... |
| gsumfsupp 49484 | A group sum of a family ca... |
| iscllaw 49491 | The predicate "is a closed... |
| iscomlaw 49492 | The predicate "is a commut... |
| clcllaw 49493 | Closure of a closed operat... |
| isasslaw 49494 | The predicate "is an assoc... |
| asslawass 49495 | Associativity of an associ... |
| mgmplusgiopALT 49496 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 49497 | Slot 2 (group operation) o... |
| intopval 49504 | The internal (binary) oper... |
| intop 49505 | An internal (binary) opera... |
| clintopval 49506 | The closed (internal binar... |
| assintopval 49507 | The associative (closed in... |
| assintopmap 49508 | The associative (closed in... |
| isclintop 49509 | The predicate "is a closed... |
| clintop 49510 | A closed (internal binary)... |
| assintop 49511 | An associative (closed int... |
| isassintop 49512 | The predicate "is an assoc... |
| clintopcllaw 49513 | The closure law holds for ... |
| assintopcllaw 49514 | The closure low holds for ... |
| assintopasslaw 49515 | The associative low holds ... |
| assintopass 49516 | An associative (closed int... |
| ismgmALT 49525 | The predicate "is a magma"... |
| iscmgmALT 49526 | The predicate "is a commut... |
| issgrpALT 49527 | The predicate "is a semigr... |
| iscsgrpALT 49528 | The predicate "is a commut... |
| mgm2mgm 49529 | Equivalence of the two def... |
| sgrp2sgrp 49530 | Equivalence of the two def... |
| lmod0rng 49531 | If the scalar ring of a mo... |
| nzrneg1ne0 49532 | The additive inverse of th... |
| lidldomn1 49533 | If a (left) ideal (which i... |
| lidlabl 49534 | A (left) ideal of a ring i... |
| lidlrng 49535 | A (left) ideal of a ring i... |
| zlidlring 49536 | The zero (left) ideal of a... |
| uzlidlring 49537 | Only the zero (left) ideal... |
| lidldomnnring 49538 | A (left) ideal of a domain... |
| 0even 49539 | 0 is an even integer. (Co... |
| 1neven 49540 | 1 is not an even integer. ... |
| 2even 49541 | 2 is an even integer. (Co... |
| 2zlidl 49542 | The even integers are a (l... |
| 2zrng 49543 | The ring of integers restr... |
| 2zrngbas 49544 | The base set of R is the s... |
| 2zrngadd 49545 | The group addition operati... |
| 2zrng0 49546 | The additive identity of R... |
| 2zrngamgm 49547 | R is an (additive) magma. ... |
| 2zrngasgrp 49548 | R is an (additive) semigro... |
| 2zrngamnd 49549 | R is an (additive) monoid.... |
| 2zrngacmnd 49550 | R is a commutative (additi... |
| 2zrngagrp 49551 | R is an (additive) group. ... |
| 2zrngaabl 49552 | R is an (additive) abelian... |
| 2zrngmul 49553 | The ring multiplication op... |
| 2zrngmmgm 49554 | R is a (multiplicative) ma... |
| 2zrngmsgrp 49555 | R is a (multiplicative) se... |
| 2zrngALT 49556 | The ring of integers restr... |
| 2zrngnmlid 49557 | R has no multiplicative (l... |
| 2zrngnmrid 49558 | R has no multiplicative (r... |
| 2zrngnmlid2 49559 | R has no multiplicative (l... |
| 2zrngnring 49560 | R is not a unital ring. (... |
| cznrnglem 49561 | Lemma for ~ cznrng : The ... |
| cznabel 49562 | The ring constructed from ... |
| cznrng 49563 | The ring constructed from ... |
| cznnring 49564 | The ring constructed from ... |
| rngcvalALTV 49567 | Value of the category of n... |
| rngcbasALTV 49568 | Set of objects of the cate... |
| rngchomfvalALTV 49569 | Set of arrows of the categ... |
| rngchomALTV 49570 | Set of arrows of the categ... |
| elrngchomALTV 49571 | A morphism of non-unital r... |
| rngccofvalALTV 49572 | Composition in the categor... |
| rngccoALTV 49573 | Composition in the categor... |
| rngccatidALTV 49574 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 49575 | The category of non-unital... |
| rngcidALTV 49576 | The identity arrow in the ... |
| rngcsectALTV 49577 | A section in the category ... |
| rngcinvALTV 49578 | An inverse in the category... |
| rngcisoALTV 49579 | An isomorphism in the cate... |
| rngchomffvalALTV 49580 | The value of the functiona... |
| rngchomrnghmresALTV 49581 | The value of the functiona... |
| rngcrescrhmALTV 49582 | The category of non-unital... |
| rhmsubcALTVlem1 49583 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 49584 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 49585 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 49586 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 49587 | According to ~ df-subc , t... |
| rhmsubcALTVcat 49588 | The restriction of the cat... |
| ringcvalALTV 49591 | Value of the category of r... |
| funcringcsetcALTV2lem1 49592 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 49593 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 49594 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 49595 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 49596 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 49597 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 49598 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 49599 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 49600 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 49601 | The "natural forgetful fun... |
| ringcbasALTV 49602 | Set of objects of the cate... |
| ringchomfvalALTV 49603 | Set of arrows of the categ... |
| ringchomALTV 49604 | Set of arrows of the categ... |
| elringchomALTV 49605 | A morphism of rings is a f... |
| ringccofvalALTV 49606 | Composition in the categor... |
| ringccoALTV 49607 | Composition in the categor... |
| ringccatidALTV 49608 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 49609 | The category of rings is a... |
| ringcidALTV 49610 | The identity arrow in the ... |
| ringcsectALTV 49611 | A section in the category ... |
| ringcinvALTV 49612 | An inverse in the category... |
| ringcisoALTV 49613 | An isomorphism in the cate... |
| ringcbasbasALTV 49614 | An element of the base set... |
| funcringcsetclem1ALTV 49615 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 49616 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 49617 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 49618 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 49619 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 49620 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 49621 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 49622 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 49623 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 49624 | The "natural forgetful fun... |
| srhmsubcALTVlem1 49625 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 49626 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 49627 | According to ~ df-subc , t... |
| sringcatALTV 49628 | The restriction of the cat... |
| crhmsubcALTV 49629 | According to ~ df-subc , t... |
| cringcatALTV 49630 | The restriction of the cat... |
| drhmsubcALTV 49631 | According to ~ df-subc , t... |
| drngcatALTV 49632 | The restriction of the cat... |
| fldcatALTV 49633 | The restriction of the cat... |
| fldcALTV 49634 | The restriction of the cat... |
| fldhmsubcALTV 49635 | According to ~ df-subc , t... |
| isprmrng 49638 | The predicate "is a prime ... |
| prmringnzring 49639 | A prime ring is a nonzero ... |
| prmrngring 49640 | A prime ring is a ring. (... |
| smprngprmrng 49641 | A simple ring (a nonzero r... |
| drngprmrng 49642 | A division ring is a prime... |
| crngprmringidom 49643 | A commutative ring is a pr... |
| crngprmringdom 49644 | A commutative ring is a pr... |
| dfidom2 49645 | Alternate definition of th... |
| isidom2 49646 | The predicate "is an integ... |
| isidom3 49647 | The predicate "is a domain... |
| idomnzd 49648 | A domain has no zero-divis... |
| idomcanl 49649 | Cancellation law for domai... |
| idomcanr 49650 | Cancellation law for domai... |
| eliunxp2 49651 | Membership in a union of C... |
| mpomptx2 49652 | Express a two-argument fun... |
| cbvmpox2 49653 | Rule to change the bound v... |
| dmmpossx2 49654 | The domain of a mapping is... |
| mpoexxg2 49655 | Existence of an operation ... |
| ovmpordxf 49656 | Value of an operation give... |
| ovmpordx 49657 | Value of an operation give... |
| ovmpox2 49658 | The value of an operation ... |
| fdmdifeqresdif 49659 | The restriction of a condi... |
| ofaddmndmap 49660 | The function operation app... |
| mapsnop 49661 | A singleton of an ordered ... |
| fprmappr 49662 | A function with a domain o... |
| mapprop 49663 | An unordered pair containi... |
| ztprmneprm 49664 | A prime is not an integer ... |
| 2t6m3t4e0 49665 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 49666 | For any finite subset of `... |
| nn0sumltlt 49667 | If the sum of two nonnegat... |
| bcpascm1 49668 | Pascal's rule for the bino... |
| altgsumbc 49669 | The sum of binomial coeffi... |
| altgsumbcALT 49670 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 49671 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 49672 | An element of the (base se... |
| zlmodzxz0 49673 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 49674 | The scalar multiplication ... |
| zlmodzxzadd 49675 | The addition of the ` ZZ `... |
| zlmodzxzsubm 49676 | The subtraction of the ` Z... |
| zlmodzxzsub 49677 | The subtraction of the ` Z... |
| mgpsumunsn 49678 | Extract a summand/factor f... |
| mgpsumz 49679 | If the group sum for the m... |
| mgpsumn 49680 | If the group sum for the m... |
| exple2lt6 49681 | A nonnegative integer to t... |
| pgrple2abl 49682 | Every symmetric group on a... |
| pgrpgt2nabl 49683 | Every symmetric group on a... |
| invginvrid 49684 | Identity for a multiplicat... |
| rmsupp0 49685 | The support of a mapping o... |
| domnmsuppn0 49686 | The support of a mapping o... |
| rmsuppss 49687 | The support of a mapping o... |
| scmsuppss 49688 | The support of a mapping o... |
| rmsuppfi 49689 | The support of a mapping o... |
| rmfsupp 49690 | A mapping of a multiplicat... |
| scmsuppfi 49691 | The support of a mapping o... |
| scmfsupp 49692 | A mapping of a scalar mult... |
| suppmptcfin 49693 | The support of a mapping w... |
| mptcfsupp 49694 | A mapping with value 0 exc... |
| fsuppmptdmf 49695 | A mapping with a finite do... |
| lmodvsmdi 49696 | Multiple distributive law ... |
| gsumlsscl 49697 | Closure of a group sum in ... |
| assaascl0 49698 | The scalar 0 embedded into... |
| assaascl1 49699 | The scalar 1 embedded into... |
| ply1vr1smo 49700 | The variable in a polynomi... |
| ply1sclrmsm 49701 | The ring multiplication of... |
| coe1sclmulval 49702 | The value of the coefficie... |
| ply1mulgsumlem1 49703 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 49704 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 49705 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 49706 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 49707 | The product of two polynom... |
| evl1at0 49708 | Polynomial evaluation for ... |
| evl1at1 49709 | Polynomial evaluation for ... |
| linply1 49710 | A term of the form ` x - C... |
| lineval 49711 | A term of the form ` x - C... |
| linevalexample 49712 | The polynomial ` x - 3 ` o... |
| dmatALTval 49717 | The algebra of ` N ` x ` N... |
| dmatALTbas 49718 | The base set of the algebr... |
| dmatALTbasel 49719 | An element of the base set... |
| dmatbas 49720 | The set of all ` N ` x ` N... |
| lincop 49725 | A linear combination as op... |
| lincval 49726 | The value of a linear comb... |
| dflinc2 49727 | Alternative definition of ... |
| lcoop 49728 | A linear combination as op... |
| lcoval 49729 | The value of a linear comb... |
| lincfsuppcl 49730 | A linear combination of ve... |
| linccl 49731 | A linear combination of ve... |
| lincval0 49732 | The value of an empty line... |
| lincvalsng 49733 | The linear combination ove... |
| lincvalsn 49734 | The linear combination ove... |
| lincvalpr 49735 | The linear combination ove... |
| lincval1 49736 | The linear combination ove... |
| lcosn0 49737 | Properties of a linear com... |
| lincvalsc0 49738 | The linear combination whe... |
| lcoc0 49739 | Properties of a linear com... |
| linc0scn0 49740 | If a set contains the zero... |
| lincdifsn 49741 | A vector is a linear combi... |
| linc1 49742 | A vector is a linear combi... |
| lincellss 49743 | A linear combination of a ... |
| lco0 49744 | The set of empty linear co... |
| lcoel0 49745 | The zero vector is always ... |
| lincsum 49746 | The sum of two linear comb... |
| lincscm 49747 | A linear combinations mult... |
| lincsumcl 49748 | The sum of two linear comb... |
| lincscmcl 49749 | The multiplication of a li... |
| lincsumscmcl 49750 | The sum of a linear combin... |
| lincolss 49751 | According to the statement... |
| ellcoellss 49752 | Every linear combination o... |
| lcoss 49753 | A set of vectors of a modu... |
| lspsslco 49754 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 49755 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 49756 | Equivalence of a _span_ of... |
| rellininds 49760 | The class defining the rel... |
| linindsv 49762 | The classes of the module ... |
| islininds 49763 | The property of being a li... |
| linindsi 49764 | The implications of being ... |
| linindslinci 49765 | The implications of being ... |
| islinindfis 49766 | The property of being a li... |
| islinindfiss 49767 | The property of being a li... |
| linindscl 49768 | A linearly independent set... |
| lindepsnlininds 49769 | A linearly dependent subse... |
| islindeps 49770 | The property of being a li... |
| lincext1 49771 | Property 1 of an extension... |
| lincext2 49772 | Property 2 of an extension... |
| lincext3 49773 | Property 3 of an extension... |
| lindslinindsimp1 49774 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 49775 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 49776 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 49777 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 49778 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 49779 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 49780 | Implication 2 for ~ lindsl... |
| lindslininds 49781 | Equivalence of definitions... |
| linds0 49782 | The empty set is always a ... |
| el0ldep 49783 | A set containing the zero ... |
| el0ldepsnzr 49784 | A set containing the zero ... |
| lindsrng01 49785 | Any subset of a module is ... |
| lindszr 49786 | Any subset of a module ove... |
| snlindsntorlem 49787 | Lemma for ~ snlindsntor . ... |
| snlindsntor 49788 | A singleton is linearly in... |
| ldepsprlem 49789 | Lemma for ~ ldepspr . (Co... |
| ldepspr 49790 | If a vector is a scalar mu... |
| lincresunit3lem3 49791 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 49792 | Lemma 1 for properties of ... |
| lincresunitlem2 49793 | Lemma for properties of a ... |
| lincresunit1 49794 | Property 1 of a specially ... |
| lincresunit2 49795 | Property 2 of a specially ... |
| lincresunit3lem1 49796 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 49797 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 49798 | Property 3 of a specially ... |
| lincreslvec3 49799 | Property 3 of a specially ... |
| islindeps2 49800 | Conditions for being a lin... |
| islininds2 49801 | Implication of being a lin... |
| isldepslvec2 49802 | Alternative definition of ... |
| lindssnlvec 49803 | A singleton not containing... |
| lmod1lem1 49804 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 49805 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 49806 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 49807 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 49808 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 49809 | The (smallest) structure r... |
| lmod1zr 49810 | The (smallest) structure r... |
| lmod1zrnlvec 49811 | There is a (left) module (... |
| lmodn0 49812 | Left modules exist. (Cont... |
| zlmodzxzequa 49813 | Example of an equation wit... |
| zlmodzxznm 49814 | Example of a linearly depe... |
| zlmodzxzldeplem 49815 | A and B are not equal. (C... |
| zlmodzxzequap 49816 | Example of an equation wit... |
| zlmodzxzldeplem1 49817 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 49818 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 49819 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 49820 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 49821 | { A , B } is a linearly de... |
| ldepsnlinclem1 49822 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 49823 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 49824 | The class of all (left) ve... |
| ldepsnlinc 49825 | The reverse implication of... |
| ldepslinc 49826 | For (left) vector spaces, ... |
| suppdm 49827 | If the range of a function... |
| eluz2cnn0n1 49828 | An integer greater than 1 ... |
| divge1b 49829 | The ratio of a real number... |
| divgt1b 49830 | The ratio of a real number... |
| ltsubaddb 49831 | Equivalence for the "less ... |
| ltsubsubb 49832 | Equivalence for the "less ... |
| ltsubadd2b 49833 | Equivalence for the "less ... |
| divsub1dir 49834 | Distribution of division o... |
| expnegico01 49835 | An integer greater than 1 ... |
| elfzolborelfzop1 49836 | An element of a half-open ... |
| pw2m1lepw2m1 49837 | 2 to the power of a positi... |
| zgtp1leeq 49838 | If an integer is between a... |
| flsubz 49839 | An integer can be moved in... |
| nn0onn0ex 49840 | For each odd nonnegative i... |
| nn0enn0ex 49841 | For each even nonnegative ... |
| nnennex 49842 | For each even positive int... |
| nneop 49843 | A positive integer is even... |
| nneom 49844 | A positive integer is even... |
| nn0eo 49845 | A nonnegative integer is e... |
| nnpw2even 49846 | 2 to the power of a positi... |
| zefldiv2 49847 | The floor of an even integ... |
| zofldiv2 49848 | The floor of an odd intege... |
| nn0ofldiv2 49849 | The floor of an odd nonneg... |
| flnn0div2ge 49850 | The floor of a positive in... |
| flnn0ohalf 49851 | The floor of the half of a... |
| logcxp0 49852 | Logarithm of a complex pow... |
| regt1loggt0 49853 | The natural logarithm for ... |
| fdivval 49856 | The quotient of two functi... |
| fdivmpt 49857 | The quotient of two functi... |
| fdivmptf 49858 | The quotient of two functi... |
| refdivmptf 49859 | The quotient of two functi... |
| fdivpm 49860 | The quotient of two functi... |
| refdivpm 49861 | The quotient of two functi... |
| fdivmptfv 49862 | The function value of a qu... |
| refdivmptfv 49863 | The function value of a qu... |
| bigoval 49866 | Set of functions of order ... |
| elbigofrcl 49867 | Reverse closure of the "bi... |
| elbigo 49868 | Properties of a function o... |
| elbigo2 49869 | Properties of a function o... |
| elbigo2r 49870 | Sufficient condition for a... |
| elbigof 49871 | A function of order G(x) i... |
| elbigodm 49872 | The domain of a function o... |
| elbigoimp 49873 | The defining property of a... |
| elbigolo1 49874 | A function (into the posit... |
| rege1logbrege0 49875 | The general logarithm, wit... |
| rege1logbzge0 49876 | The general logarithm, wit... |
| fllogbd 49877 | A real number is between t... |
| relogbmulbexp 49878 | The logarithm of the produ... |
| relogbdivb 49879 | The logarithm of the quoti... |
| logbge0b 49880 | The logarithm of a number ... |
| logblt1b 49881 | The logarithm of a number ... |
| fldivexpfllog2 49882 | The floor of a positive re... |
| nnlog2ge0lt1 49883 | A positive integer is 1 if... |
| logbpw2m1 49884 | The floor of the binary lo... |
| fllog2 49885 | The floor of the binary lo... |
| blenval 49888 | The binary length of an in... |
| blen0 49889 | The binary length of 0. (... |
| blenn0 49890 | The binary length of a "nu... |
| blenre 49891 | The binary length of a pos... |
| blennn 49892 | The binary length of a pos... |
| blennnelnn 49893 | The binary length of a pos... |
| blennn0elnn 49894 | The binary length of a non... |
| blenpw2 49895 | The binary length of a pow... |
| blenpw2m1 49896 | The binary length of a pow... |
| nnpw2blen 49897 | A positive integer is betw... |
| nnpw2blenfzo 49898 | A positive integer is betw... |
| nnpw2blenfzo2 49899 | A positive integer is eith... |
| nnpw2pmod 49900 | Every positive integer can... |
| blen1 49901 | The binary length of 1. (... |
| blen2 49902 | The binary length of 2. (... |
| nnpw2p 49903 | Every positive integer can... |
| nnpw2pb 49904 | A number is a positive int... |
| blen1b 49905 | The binary length of a non... |
| blennnt2 49906 | The binary length of a pos... |
| nnolog2flm1 49907 | The floor of the binary lo... |
| blennn0em1 49908 | The binary length of the h... |
| blennngt2o2 49909 | The binary length of an od... |
| blengt1fldiv2p1 49910 | The binary length of an in... |
| blennn0e2 49911 | The binary length of an ev... |
| digfval 49914 | Operation to obtain the ` ... |
| digval 49915 | The ` K ` th digit of a no... |
| digvalnn0 49916 | The ` K ` th digit of a no... |
| nn0digval 49917 | The ` K ` th digit of a no... |
| dignn0fr 49918 | The digits of the fraction... |
| dignn0ldlem 49919 | Lemma for ~ dignnld . (Co... |
| dignnld 49920 | The leading digits of a po... |
| dig2nn0ld 49921 | The leading digits of a po... |
| dig2nn1st 49922 | The first (relevant) digit... |
| dig0 49923 | All digits of 0 are 0. (C... |
| digexp 49924 | The ` K ` th digit of a po... |
| dig1 49925 | All but one digits of 1 ar... |
| 0dig1 49926 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 49927 | The integers 0 and 1 corre... |
| dig2nn0 49928 | A digit of a nonnegative i... |
| 0dig2nn0e 49929 | The last bit of an even in... |
| 0dig2nn0o 49930 | The last bit of an odd int... |
| dig2bits 49931 | The ` K ` th digit of a no... |
| dignn0flhalflem1 49932 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 49933 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 49934 | The digits of the half of ... |
| dignn0flhalf 49935 | The digits of the rounded ... |
| nn0sumshdiglemA 49936 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 49937 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 49938 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 49939 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 49940 | A nonnegative integer can ... |
| nn0mulfsum 49941 | Trivial algorithm to calcu... |
| nn0mullong 49942 | Standard algorithm (also k... |
| naryfval 49945 | The set of the n-ary (endo... |
| naryfvalixp 49946 | The set of the n-ary (endo... |
| naryfvalel 49947 | An n-ary (endo)function on... |
| naryrcl 49948 | Reverse closure for n-ary ... |
| naryfvalelfv 49949 | The value of an n-ary (end... |
| naryfvalelwrdf 49950 | An n-ary (endo)function on... |
| 0aryfvalel 49951 | A nullary (endo)function o... |
| 0aryfvalelfv 49952 | The value of a nullary (en... |
| 1aryfvalel 49953 | A unary (endo)function on ... |
| fv1arycl 49954 | Closure of a unary (endo)f... |
| 1arympt1 49955 | A unary (endo)function in ... |
| 1arympt1fv 49956 | The value of a unary (endo... |
| 1arymaptfv 49957 | The value of the mapping o... |
| 1arymaptf 49958 | The mapping of unary (endo... |
| 1arymaptf1 49959 | The mapping of unary (endo... |
| 1arymaptfo 49960 | The mapping of unary (endo... |
| 1arymaptf1o 49961 | The mapping of unary (endo... |
| 1aryenef 49962 | The set of unary (endo)fun... |
| 1aryenefmnd 49963 | The set of unary (endo)fun... |
| 2aryfvalel 49964 | A binary (endo)function on... |
| fv2arycl 49965 | Closure of a binary (endo)... |
| 2arympt 49966 | A binary (endo)function in... |
| 2arymptfv 49967 | The value of a binary (end... |
| 2arymaptfv 49968 | The value of the mapping o... |
| 2arymaptf 49969 | The mapping of binary (end... |
| 2arymaptf1 49970 | The mapping of binary (end... |
| 2arymaptfo 49971 | The mapping of binary (end... |
| 2arymaptf1o 49972 | The mapping of binary (end... |
| 2aryenef 49973 | The set of binary (endo)fu... |
| itcoval 49978 | The value of the function ... |
| itcoval0 49979 | A function iterated zero t... |
| itcoval1 49980 | A function iterated once. ... |
| itcoval2 49981 | A function iterated twice.... |
| itcoval3 49982 | A function iterated three ... |
| itcoval0mpt 49983 | A mapping iterated zero ti... |
| itcovalsuc 49984 | The value of the function ... |
| itcovalsucov 49985 | The value of the function ... |
| itcovalendof 49986 | The n-th iterate of an end... |
| itcovalpclem1 49987 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 49988 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 49989 | The value of the function ... |
| itcovalt2lem2lem1 49990 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 49991 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 49992 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 49993 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 49994 | The value of the function ... |
| ackvalsuc1mpt 49995 | The Ackermann function at ... |
| ackvalsuc1 49996 | The Ackermann function at ... |
| ackval0 49997 | The Ackermann function at ... |
| ackval1 49998 | The Ackermann function at ... |
| ackval2 49999 | The Ackermann function at ... |
| ackval3 50000 | The Ackermann function at ... |
| ackendofnn0 50001 | The Ackermann function at ... |
| ackfnnn0 50002 | The Ackermann function at ... |
| ackval0val 50003 | The Ackermann function at ... |
| ackvalsuc0val 50004 | The Ackermann function at ... |
| ackvalsucsucval 50005 | The Ackermann function at ... |
| ackval0012 50006 | The Ackermann function at ... |
| ackval1012 50007 | The Ackermann function at ... |
| ackval2012 50008 | The Ackermann function at ... |
| ackval3012 50009 | The Ackermann function at ... |
| ackval40 50010 | The Ackermann function at ... |
| ackval41a 50011 | The Ackermann function at ... |
| ackval41 50012 | The Ackermann function at ... |
| ackval42 50013 | The Ackermann function at ... |
| ackval42a 50014 | The Ackermann function at ... |
| ackval50 50015 | The Ackermann function at ... |
| fv1prop 50016 | The function value of unor... |
| fv2prop 50017 | The function value of unor... |
| submuladdmuld 50018 | Transformation of a sum of... |
| affinecomb1 50019 | Combination of two real af... |
| affinecomb2 50020 | Combination of two real af... |
| affineid 50021 | Identity of an affine comb... |
| 1subrec1sub 50022 | Subtract the reciprocal of... |
| resum2sqcl 50023 | The sum of two squares of ... |
| resum2sqgt0 50024 | The sum of the square of a... |
| resum2sqrp 50025 | The sum of the square of a... |
| resum2sqorgt0 50026 | The sum of the square of t... |
| reorelicc 50027 | Membership in and outside ... |
| rrx2pxel 50028 | The x-coordinate of a poin... |
| rrx2pyel 50029 | The y-coordinate of a poin... |
| prelrrx2 50030 | A doubleton word of real n... |
| prelrrx2bd 50031 | Equivalent condition for `... |
| prelrrx2b 50032 | Equivalent condition for b... |
| rrx2pnecoorneor 50033 | If two different points ` ... |
| rrx2pnedifcoorneor 50034 | If two different points ` ... |
| rrx2pnedifcoorneorr 50035 | If two different points ` ... |
| rrx2xpref1o 50036 | There is a bijection betwe... |
| rrx2xpreen 50037 | The set of points in the t... |
| rrx2plord 50038 | The lexicographical orderi... |
| rrx2plord1 50039 | The lexicographical orderi... |
| rrx2plord2 50040 | The lexicographical orderi... |
| rrx2plordisom 50041 | The set of points in the t... |
| rrx2plordso 50042 | The lexicographical orderi... |
| ehl2eudisval0 50043 | The Euclidean distance of ... |
| ehl2eudis0lt 50044 | An upper bound of the Eucl... |
| lines 50049 | The lines passing through ... |
| line 50050 | The line passing through t... |
| rrxlines 50051 | Definition of lines passin... |
| rrxline 50052 | The line passing through t... |
| rrxlinesc 50053 | Definition of lines passin... |
| rrxlinec 50054 | The line passing through t... |
| eenglngeehlnmlem1 50055 | Lemma 1 for - eenglngeehln... |
| eenglngeehlnmlem2 50056 | Lemma 2 for - eenglngeehln... |
| rrx2line 50057 | The line passing through t... |
| rrx2vlinest 50058 | The vertical line passing ... |
| rrx2linest 50059 | The line passing through t... |
| rrx2linesl 50060 | The line passing through t... |
| rrx2linest2 50061 | The line passing through t... |
| elrrx2linest2 50062 | The line passing through t... |
| spheres 50063 | The spheres for given cent... |
| sphere 50064 | A sphere with center ` X `... |
| rrxsphere 50065 | The sphere with center ` M... |
| 2sphere 50066 | The sphere with center ` M... |
| 2sphere0 50067 | The sphere around the orig... |
| line2ylem 50068 | Lemma for ~ line2y . This... |
| line2 50069 | Example for a line ` G ` p... |
| line2xlem 50070 | Lemma for ~ line2x . This... |
| line2x 50071 | Example for a horizontal l... |
| line2y 50072 | Example for a vertical lin... |
| itsclc0lem1 50073 | Lemma for theorems about i... |
| itsclc0lem2 50074 | Lemma for theorems about i... |
| itsclc0lem3 50075 | Lemma for theorems about i... |
| itscnhlc0yqe 50076 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 50077 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 50078 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 50079 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 50080 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 50081 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 50082 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 50083 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 50084 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 50085 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 50086 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 50087 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 50088 | The intersection points of... |
| itsclc0b 50089 | The intersection points of... |
| itsclinecirc0 50090 | The intersection points of... |
| itsclinecirc0b 50091 | The intersection points of... |
| itsclinecirc0in 50092 | The intersection points of... |
| itsclquadb 50093 | Quadratic equation for the... |
| itsclquadeu 50094 | Quadratic equation for the... |
| 2itscplem1 50095 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 50096 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 50097 | Lemma D for ~ 2itscp . (C... |
| 2itscp 50098 | A condition for a quadrati... |
| itscnhlinecirc02plem1 50099 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 50100 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 50101 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 50102 | Intersection of a nonhoriz... |
| inlinecirc02plem 50103 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 50104 | Intersection of a line wit... |
| inlinecirc02preu 50105 | Intersection of a line wit... |
| imbi12d2 50106 | Distribution of implicatio... |
| imbi12d2a 50107 | Variant of ~ imbi12d2 . (... |
| imbi12d3 50108 | Variant of ~ imbi12d2 . (... |
| pm5.32rda 50109 | Distribution of implicatio... |
| pm5.32dar 50110 | Reverse distribution of im... |
| exp12bd 50111 | The import-export theorem ... |
| mpbiran3d 50112 | Equivalence with a conjunc... |
| mpbiran4d 50113 | Equivalence with a conjunc... |
| dtrucor3 50114 | An example of how ~ ax-5 w... |
| ralbidb 50115 | Formula-building rule for ... |
| ralbidc 50116 | Formula-building rule for ... |
| r19.41dv 50117 | A complex deduction form o... |
| rmotru 50118 | Two ways of expressing "at... |
| reutru 50119 | Two ways of expressing "ex... |
| reutruALT 50120 | Alternate proof of ~ reutr... |
| reueqbidva 50121 | Formula-building rule for ... |
| reuxfr1dd 50122 | Transfer existential uniqu... |
| ssdisjd 50123 | Subset preserves disjointn... |
| ssdisjdr 50124 | Subset preserves disjointn... |
| disjdifb 50125 | Relative complement is ant... |
| predisj 50126 | Preimages of disjoint sets... |
| elif0 50127 | Membership in a conditiona... |
| vsn 50128 | The singleton of the unive... |
| mosn 50129 | "At most one" element in a... |
| mo0 50130 | "At most one" element in a... |
| mosssn 50131 | "At most one" element in a... |
| mo0sn 50132 | Two ways of expressing "at... |
| mosssn2 50133 | Two ways of expressing "at... |
| unilbss 50134 | Superclass of the greatest... |
| iuneq0 50135 | An indexed union is empty ... |
| iineq0 50136 | An indexed intersection is... |
| iunlub 50137 | The indexed union is the t... |
| iinglb 50138 | The indexed intersection i... |
| iuneqconst2 50139 | Indexed union of identical... |
| iineqconst2 50140 | Indexed intersection of id... |
| inpw 50141 | Two ways of expressing a c... |
| opth1neg 50142 | Two ordered pairs are not ... |
| opth2neg 50143 | Two ordered pairs are not ... |
| brab2dd 50144 | Expressing that two sets a... |
| brab2ddw 50145 | Expressing that two sets a... |
| brab2ddw2 50146 | Expressing that two sets a... |
| iinxp 50147 | Indexed intersection of Ca... |
| intxpd 50148 | Intersection of Cartesian ... |
| coxp 50149 | Composition with a Cartesi... |
| cosn 50150 | Composition with an ordere... |
| cosni 50151 | Composition with an ordere... |
| inisegn0a 50152 | The inverse image of a sin... |
| dmrnxp 50153 | A Cartesian product is the... |
| mof0 50154 | There is at most one funct... |
| mof02 50155 | A variant of ~ mof0 . (Co... |
| mof0ALT 50156 | Alternate proof of ~ mof0 ... |
| eufsnlem 50157 | There is exactly one funct... |
| eufsn 50158 | There is exactly one funct... |
| eufsn2 50159 | There is exactly one funct... |
| mofsn 50160 | There is at most one funct... |
| mofsn2 50161 | There is at most one funct... |
| mofsssn 50162 | There is at most one funct... |
| mofmo 50163 | There is at most one funct... |
| mofeu 50164 | The uniqueness of a functi... |
| elfvne0 50165 | If a function value has a ... |
| fdomne0 50166 | A function with non-empty ... |
| f1sn2g 50167 | A function that maps a sin... |
| f102g 50168 | A function that maps the e... |
| f1mo 50169 | A function that maps a set... |
| f002 50170 | A function with an empty c... |
| map0cor 50171 | A function exists iff an e... |
| ffvbr 50172 | Relation with function val... |
| xpco2 50173 | Composition of a Cartesian... |
| ovsng 50174 | The operation value of a s... |
| ovsng2 50175 | The operation value of a s... |
| ovsn 50176 | The operation value of a s... |
| ovsn2 50177 | The operation value of a s... |
| ovconstbrd 50178 | Two ways of expressing ` A... |
| ovconstbrn0d 50179 | Two ways of expressing ` A... |
| elovconstbrd 50180 | Two ways of expressing ` A... |
| ovmpt4d 50181 | Deduction version of ~ ovm... |
| eqfnovd 50182 | Deduction for equality of ... |
| fonex 50183 | The domain of a surjection... |
| eloprab1st2nd 50184 | Reconstruction of a nested... |
| elovmpoeq 50185 | If a value of an operation... |
| fnmpompt 50186 | Characterization of the ou... |
| ovmpomptd 50187 | The outer value of an iter... |
| fvmpomptd 50188 | At an element of its inner... |
| resinsnlem 50189 | Lemma for ~ resinsnALT . ... |
| resinsn 50190 | Restriction to the interse... |
| resinsnALT 50191 | Restriction to the interse... |
| dftpos5 50192 | Alternate definition of ` ... |
| dftpos6 50193 | Alternate definition of ` ... |
| dmtposss 50194 | The domain of ` tpos F ` i... |
| tposres0 50195 | The transposition of a set... |
| tposresg 50196 | The transposition restrict... |
| tposrescnv 50197 | The transposition restrict... |
| tposres2 50198 | The transposition restrict... |
| tposres3 50199 | The transposition restrict... |
| tposres 50200 | The transposition restrict... |
| tposresxp 50201 | The transposition restrict... |
| tposf1o 50202 | Condition of a bijective t... |
| tposid 50203 | Swap an ordered pair. (Co... |
| tposidres 50204 | Swap an ordered pair. (Co... |
| tposidf1o 50205 | The swap function, or the ... |
| tposideq 50206 | Two ways of expressing the... |
| tposideq2 50207 | Two ways of expressing the... |
| ixpv 50208 | Infinite Cartesian product... |
| elixpmpt 50209 | Membership in an indexed C... |
| fvconst0ci 50210 | A constant function's valu... |
| fvconstdomi 50211 | A constant function's valu... |
| f1omo 50212 | There is at most one eleme... |
| f1omoOLD 50213 | Obsolete version of ~ f1om... |
| f1omoALT 50214 | There is at most one eleme... |
| iccin 50215 | Intersection of two closed... |
| iccdisj2 50216 | If the upper bound of one ... |
| iccdisj 50217 | If the upper bound of one ... |
| slotresfo 50218 | The condition of a structu... |
| mreuniss 50219 | The union of a collection ... |
| clduni 50220 | The union of closed sets i... |
| opncldbid 50221 | Conditions on open sets ar... |
| opndisj 50222 | Two ways of saying that tw... |
| clddisj 50223 | Two ways of saying that tw... |
| neircl 50224 | Reverse closure of the nei... |
| opnneilem 50225 | Lemma factoring out common... |
| opnneir 50226 | If something is true for a... |
| opnneirv 50227 | A variant of ~ opnneir wit... |
| opnneilv 50228 | The converse of ~ opnneir ... |
| opnneil 50229 | A variant of ~ opnneilv . ... |
| opnneibid 50230 | The equivalence between ne... |
| opnneibid2 50231 | The equivalence between ne... |
| restcls2lem 50232 | A closed set in a subspace... |
| restcls2 50233 | A closed set in a subspace... |
| restclsseplem 50234 | Lemma for ~ restclssep . ... |
| restclssep 50235 | Two disjoint closed sets i... |
| cnneiima 50236 | Given a continuous functio... |
| iooii 50237 | Open intervals are open se... |
| icccldii 50238 | Closed intervals are close... |
| i0oii 50239 | ` ( 0 [,) A ) ` is open in... |
| io1ii 50240 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 50241 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 50242 | Open neighborhood and neig... |
| sepnsepo 50243 | Open neighborhood and neig... |
| sepdisj 50244 | Separated sets are disjoin... |
| seposep 50245 | If two sets are separated ... |
| sepcsepo 50246 | If two sets are separated ... |
| sepfsepc 50247 | If two sets are separated ... |
| seppsepf 50248 | If two sets are precisely ... |
| seppcld 50249 | If two sets are precisely ... |
| isnrm4 50250 | A topological space is nor... |
| dfnrm2 50251 | A topological space is nor... |
| dfnrm3 50252 | A topological space is nor... |
| iscnrm3lem1 50253 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 50254 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem4 50255 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 50256 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 50257 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 50258 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 50259 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 50260 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 50261 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 50262 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 50263 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 50264 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 50265 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 50266 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 50267 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 50268 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 50269 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 50270 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 50271 | A completely normal topolo... |
| iscnrm3v 50272 | A topology is completely n... |
| iscnrm4 50273 | A completely normal topolo... |
| isprsd 50274 | Property of being a preord... |
| lubeldm2 50275 | Member of the domain of th... |
| glbeldm2 50276 | Member of the domain of th... |
| lubeldm2d 50277 | Member of the domain of th... |
| glbeldm2d 50278 | Member of the domain of th... |
| lubsscl 50279 | If a subset of ` S ` conta... |
| glbsscl 50280 | If a subset of ` S ` conta... |
| lubprlem 50281 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 50282 | The set of two comparable ... |
| lubpr 50283 | The LUB of the set of two ... |
| glbprlem 50284 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 50285 | The set of two comparable ... |
| glbpr 50286 | The GLB of the set of two ... |
| joindm2 50287 | The join of any two elemen... |
| joindm3 50288 | The join of any two elemen... |
| meetdm2 50289 | The meet of any two elemen... |
| meetdm3 50290 | The meet of any two elemen... |
| posjidm 50291 | Poset join is idempotent. ... |
| posmidm 50292 | Poset meet is idempotent. ... |
| resiposbas 50293 | Construct a poset ( ~ resi... |
| resipos 50294 | A set equipped with an ord... |
| exbaspos 50295 | There exists a poset for a... |
| exbasprs 50296 | There exists a preordered ... |
| basresposfo 50297 | The base function restrict... |
| basresprsfo 50298 | The base function restrict... |
| posnex 50299 | The class of posets is a p... |
| prsnex 50300 | The class of preordered se... |
| toslat 50301 | A toset is a lattice. (Co... |
| isclatd 50302 | The predicate "is a comple... |
| intubeu 50303 | Existential uniqueness of ... |
| unilbeu 50304 | Existential uniqueness of ... |
| ipolublem 50305 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 50306 | The domain of the LUB of t... |
| ipolub 50307 | The LUB of the inclusion p... |
| ipoglblem 50308 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 50309 | The domain of the GLB of t... |
| ipoglb 50310 | The GLB of the inclusion p... |
| ipolub0 50311 | The LUB of the empty set i... |
| ipolub00 50312 | The LUB of the empty set i... |
| ipoglb0 50313 | The GLB of the empty set i... |
| mrelatlubALT 50314 | Least upper bounds in a Mo... |
| mrelatglbALT 50315 | Greatest lower bounds in a... |
| mreclat 50316 | A Moore space is a complet... |
| topclat 50317 | A topology is a complete l... |
| toplatglb0 50318 | The empty intersection in ... |
| toplatlub 50319 | Least upper bounds in a to... |
| toplatglb 50320 | Greatest lower bounds in a... |
| toplatjoin 50321 | Joins in a topology are re... |
| toplatmeet 50322 | Meets in a topology are re... |
| topdlat 50323 | A topology is a distributi... |
| elmgpcntrd 50324 | The center of a ring. (Co... |
| asclelbasALT 50325 | Alternate proof of ~ ascle... |
| asclcntr 50326 | The algebra scalar lifting... |
| asclcom 50327 | Scalars are commutative af... |
| homf0 50328 | The base is empty iff the ... |
| catprslem 50329 | Lemma for ~ catprs . (Con... |
| catprs 50330 | A preorder can be extracte... |
| catprs2 50331 | A category equipped with t... |
| catprsc 50332 | A construction of the preo... |
| catprsc2 50333 | An alternate construction ... |
| endmndlem 50334 | A diagonal hom-set in a ca... |
| oppccatb 50335 | An opposite category is a ... |
| oppcmndclem 50336 | Lemma for ~ oppcmndc . Ev... |
| oppcendc 50337 | The opposite category of a... |
| oppcmndc 50338 | The opposite category of a... |
| idmon 50339 | An identity arrow, or an i... |
| idepi 50340 | An identity arrow, or an i... |
| sectrcl 50341 | Reverse closure for sectio... |
| sectrcl2 50342 | Reverse closure for sectio... |
| invrcl 50343 | Reverse closure for invers... |
| invrcl2 50344 | Reverse closure for invers... |
| isinv2 50345 | The property " ` F ` is an... |
| isisod 50346 | The predicate "is an isomo... |
| upeu2lem 50347 | Lemma for ~ upeu2 . There... |
| sectfn 50348 | The function value of the ... |
| invfn 50349 | The function value of the ... |
| isofnALT 50350 | The function value of the ... |
| isofval2 50351 | Function value of the func... |
| isorcl 50352 | Reverse closure for isomor... |
| isorcl2 50353 | Reverse closure for isomor... |
| isoval2 50354 | The isomorphisms are the d... |
| sectpropdlem 50355 | Lemma for ~ sectpropd . (... |
| sectpropd 50356 | Two structures with the sa... |
| invpropdlem 50357 | Lemma for ~ invpropd . (C... |
| invpropd 50358 | Two structures with the sa... |
| isopropdlem 50359 | Lemma for ~ isopropd . (C... |
| isopropd 50360 | Two structures with the sa... |
| cicfn 50361 | ` ~=c ` is a function on `... |
| cicrcl2 50362 | Isomorphism implies the st... |
| oppccic 50363 | Isomorphic objects are iso... |
| relcic 50364 | The set of isomorphic obje... |
| cicerALT 50365 | Isomorphism is an equivale... |
| cic1st2nd 50366 | Reconstruction of a pair o... |
| cic1st2ndbr 50367 | Rewrite the predicate of i... |
| cicpropdlem 50368 | Lemma for ~ cicpropd . (C... |
| cicpropd 50369 | Two structures with the sa... |
| oppccicb 50370 | Isomorphic objects are iso... |
| oppcciceq 50371 | The opposite category has ... |
| dmdm 50372 | The double domain of a fun... |
| iinfssclem1 50373 | Lemma for ~ iinfssc . (Co... |
| iinfssclem2 50374 | Lemma for ~ iinfssc . (Co... |
| iinfssclem3 50375 | Lemma for ~ iinfssc . (Co... |
| iinfssc 50376 | Indexed intersection of su... |
| iinfsubc 50377 | Indexed intersection of su... |
| iinfprg 50378 | Indexed intersection of fu... |
| infsubc 50379 | The intersection of two su... |
| infsubc2 50380 | The intersection of two su... |
| infsubc2d 50381 | The intersection of two su... |
| discsubclem 50382 | Lemma for ~ discsubc . (C... |
| discsubc 50383 | A discrete category, whose... |
| iinfconstbaslem 50384 | Lemma for ~ iinfconstbas .... |
| iinfconstbas 50385 | The discrete category is t... |
| nelsubclem 50386 | Lemma for ~ nelsubc . (Co... |
| nelsubc 50387 | An empty "hom-set" for non... |
| nelsubc2 50388 | An empty "hom-set" for non... |
| nelsubc3lem 50389 | Lemma for ~ nelsubc3 . (C... |
| nelsubc3 50390 | Remark 4.2(2) of [Adamek] ... |
| ssccatid 50391 | A category ` C ` restricte... |
| resccatlem 50392 | Lemma for ~ resccat . (Co... |
| resccat 50393 | A class ` C ` restricted b... |
| reldmfunc 50394 | The domain of ` Func ` is ... |
| func1st2nd 50395 | Rewrite the functor predic... |
| func1st 50396 | Extract the first member o... |
| func2nd 50397 | Extract the second member ... |
| funcrcl2 50398 | Reverse closure for a func... |
| funcrcl3 50399 | Reverse closure for a func... |
| funcf2lem 50400 | A utility theorem for prov... |
| funcf2lem2 50401 | A utility theorem for prov... |
| ixpovf 50402 | A member of an indexed pro... |
| 0funcglem 50403 | Lemma for ~ 0funcg . (Con... |
| 0funcg2 50404 | The functor from the empty... |
| 0funcg 50405 | The functor from the empty... |
| 0funclem 50406 | Lemma for ~ 0funcALT . (C... |
| 0func 50407 | The functor from the empty... |
| 0funcALT 50408 | Alternate proof of ~ 0func... |
| func0g 50409 | The source category of a f... |
| func0g2 50410 | The source category of a f... |
| initc 50411 | Sets with empty base are t... |
| cofu1st2nd 50412 | Rewrite the functor compos... |
| rescofuf 50413 | The restriction of functor... |
| cofu1a 50414 | Value of the object part o... |
| cofu2a 50415 | Value of the morphism part... |
| cofucla 50416 | The composition of two fun... |
| cofu1stfix 50417 | Post-composition by an obj... |
| funchomf 50418 | Source categories of a fun... |
| idfurcl 50419 | Reverse closure for an ide... |
| idfu1stf1o 50420 | The identity functor/inclu... |
| idfu1stalem 50421 | Lemma for ~ idfu1sta . (C... |
| idfu1sta 50422 | Value of the object part o... |
| idfu1a 50423 | Value of the object part o... |
| idfu2nda 50424 | Value of the morphism part... |
| imasubclem1 50425 | Lemma for ~ imasubc . (Co... |
| imasubclem2 50426 | Lemma for ~ imasubc . (Co... |
| imasubclem3 50427 | Lemma for ~ imasubc . (Co... |
| imaf1homlem 50428 | Lemma for ~ imaf1hom and o... |
| imaf1hom 50429 | The hom-set of an image of... |
| imaidfu2lem 50430 | Lemma for ~ imaidfu2 . (C... |
| imaidfu 50431 | The image of the identity ... |
| imaidfu2 50432 | The image of the identity ... |
| cofid1a 50433 | Express the object part of... |
| cofid2a 50434 | Express the morphism part ... |
| cofid1 50435 | Express the object part of... |
| cofid2 50436 | Express the morphism part ... |
| cofidvala 50437 | The property " ` F ` is a ... |
| cofidf2a 50438 | If " ` F ` is a section of... |
| cofidf1a 50439 | If " ` F ` is a section of... |
| cofidval 50440 | The property " ` <. F , G ... |
| cofidf2 50441 | If " ` F ` is a section of... |
| cofidf1 50442 | If " ` <. F , G >. ` is a ... |
| oppffn 50445 | ` oppFunc ` is a function ... |
| reldmoppf 50446 | The domain of ` oppFunc ` ... |
| oppfvalg 50447 | Value of the opposite func... |
| oppfrcllem 50448 | Lemma for ~ oppfrcl . (Co... |
| oppfrcl 50449 | If an opposite functor of ... |
| oppfrcl2 50450 | If an opposite functor of ... |
| oppfrcl3 50451 | If an opposite functor of ... |
| oppf1st2nd 50452 | Rewrite the opposite funct... |
| 2oppf 50453 | The double opposite functo... |
| eloppf 50454 | The pre-image of a non-emp... |
| eloppf2 50455 | Both components of a pre-i... |
| oppfvallem 50456 | Lemma for ~ oppfval . (Co... |
| oppfval 50457 | Value of the opposite func... |
| oppfval2 50458 | Value of the opposite func... |
| oppfval3 50459 | Value of the opposite func... |
| oppf1 50460 | Value of the object part o... |
| oppf2 50461 | Value of the morphism part... |
| oppfoppc 50462 | The opposite functor is a ... |
| oppfoppc2 50463 | The opposite functor is a ... |
| funcoppc2 50464 | A functor on opposite cate... |
| funcoppc4 50465 | A functor on opposite cate... |
| funcoppc5 50466 | A functor on opposite cate... |
| 2oppffunc 50467 | The opposite functor of an... |
| funcoppc3 50468 | A functor on opposite cate... |
| oppff1 50469 | The operation generating o... |
| oppff1o 50470 | The operation generating o... |
| cofuoppf 50471 | Composition of opposite fu... |
| imasubc 50472 | An image of a full functor... |
| imasubc2 50473 | An image of a full functor... |
| imassc 50474 | An image of a functor sati... |
| imaid 50475 | An image of a functor pres... |
| imaf1co 50476 | An image of a functor whos... |
| imasubc3 50477 | An image of a functor inje... |
| fthcomf 50478 | Source categories of a fai... |
| idfth 50479 | The inclusion functor is a... |
| idemb 50480 | The inclusion functor is a... |
| idsubc 50481 | The source category of an ... |
| idfullsubc 50482 | The source category of an ... |
| cofidfth 50483 | If " ` F ` is a section of... |
| fulloppf 50484 | The opposite functor of a ... |
| fthoppf 50485 | The opposite functor of a ... |
| ffthoppf 50486 | The opposite functor of a ... |
| upciclem1 50487 | Lemma for ~ upcic , ~ upeu... |
| upciclem2 50488 | Lemma for ~ upciclem3 and ... |
| upciclem3 50489 | Lemma for ~ upciclem4 . (... |
| upciclem4 50490 | Lemma for ~ upcic and ~ up... |
| upcic 50491 | A universal property defin... |
| upeu 50492 | A universal property defin... |
| upeu2 50493 | Generate new universal mor... |
| reldmup 50496 | The domain of ` UP ` is a ... |
| upfval 50497 | Function value of the clas... |
| upfval2 50498 | Function value of the clas... |
| upfval3 50499 | Function value of the clas... |
| isuplem 50500 | Lemma for ~ isup and other... |
| isup 50501 | The predicate "is a univer... |
| uppropd 50502 | If two categories have the... |
| reldmup2 50503 | The domain of ` ( D UP E )... |
| relup 50504 | The set of universal pairs... |
| uprcl 50505 | Reverse closure for the cl... |
| up1st2nd 50506 | Rewrite the universal prop... |
| up1st2ndr 50507 | Combine separated parts in... |
| up1st2ndb 50508 | Combine/separate parts in ... |
| up1st2nd2 50509 | Rewrite the universal prop... |
| uprcl2 50510 | Reverse closure for the cl... |
| uprcl3 50511 | Reverse closure for the cl... |
| uprcl4 50512 | Reverse closure for the cl... |
| uprcl5 50513 | Reverse closure for the cl... |
| uobrcl 50514 | Reverse closure for univer... |
| isup2 50515 | The universal property of ... |
| upeu3 50516 | The universal pair ` <. X ... |
| upeu4 50517 | Generate a new universal m... |
| uptposlem 50518 | Lemma for ~ uptpos . (Con... |
| uptpos 50519 | Rewrite the predicate of u... |
| oppcuprcl4 50520 | Reverse closure for the cl... |
| oppcuprcl3 50521 | Reverse closure for the cl... |
| oppcuprcl5 50522 | Reverse closure for the cl... |
| oppcuprcl2 50523 | Reverse closure for the cl... |
| uprcl2a 50524 | Reverse closure for the cl... |
| oppfuprcl 50525 | Reverse closure for the cl... |
| oppfuprcl2 50526 | Reverse closure for the cl... |
| oppcup3lem 50527 | Lemma for ~ oppcup3 . (Co... |
| oppcup 50528 | The universal pair ` <. X ... |
| oppcup2 50529 | The universal property for... |
| oppcup3 50530 | The universal property for... |
| uptrlem1 50531 | Lemma for ~ uptr . (Contr... |
| uptrlem2 50532 | Lemma for ~ uptr . (Contr... |
| uptrlem3 50533 | Lemma for ~ uptr . (Contr... |
| uptr 50534 | Universal property and ful... |
| uptri 50535 | Universal property and ful... |
| uptra 50536 | Universal property and ful... |
| uptrar 50537 | Universal property and ful... |
| uptrai 50538 | Universal property and ful... |
| uobffth 50539 | A fully faithful functor g... |
| uobeqw 50540 | If a full functor (in fact... |
| uobeq 50541 | If a full functor (in fact... |
| uptr2 50542 | Universal property and ful... |
| uptr2a 50543 | Universal property and ful... |
| isnatd 50544 | Property of being a natura... |
| natrcl2 50545 | Reverse closure for a natu... |
| natrcl3 50546 | Reverse closure for a natu... |
| isnat3 50547 | A natural transformation i... |
| natfn2 50548 | A natural transformation i... |
| nateq 50549 | Two natural transformation... |
| natideq 50550 | A natural transformation i... |
| natcoeq 50551 | A natural transformation e... |
| fucidcobi 50552 | Componentwise functor laws... |
| catbas 50553 | The base of the category s... |
| cathomfval 50554 | The hom-sets of the catego... |
| catcofval 50555 | Composition of the categor... |
| natoppf 50556 | A natural transformation i... |
| natoppf2 50557 | A natural transformation i... |
| natoppfb 50558 | A natural transformation i... |
| initoo2 50559 | An initial object is an ob... |
| termoo2 50560 | A terminal object is an ob... |
| zeroo2 50561 | A zero object is an object... |
| oppcinito 50562 | Initial objects are termin... |
| oppctermo 50563 | Terminal objects are initi... |
| oppczeroo 50564 | Zero objects are zero in t... |
| termoeu2 50565 | Terminal objects are essen... |
| initopropdlemlem 50566 | Lemma for ~ initopropdlem ... |
| initopropdlem 50567 | Lemma for ~ initopropd . ... |
| termopropdlem 50568 | Lemma for ~ termopropd . ... |
| zeroopropdlem 50569 | Lemma for ~ zeroopropd . ... |
| initopropd 50570 | Two structures with the sa... |
| termopropd 50571 | Two structures with the sa... |
| zeroopropd 50572 | Two structures with the sa... |
| reldmxpc 50573 | The binary product of cate... |
| reldmxpcALT 50574 | Alternate proof of ~ reldm... |
| elxpcbasex1 50575 | A non-empty base set of th... |
| elxpcbasex1ALT 50576 | Alternate proof of ~ elxpc... |
| elxpcbasex2 50577 | A non-empty base set of th... |
| elxpcbasex2ALT 50578 | Alternate proof of ~ elxpc... |
| xpcfucbas 50579 | The base set of the produc... |
| xpcfuchomfval 50580 | Set of morphisms of the bi... |
| xpcfuchom 50581 | Set of morphisms of the bi... |
| xpcfuchom2 50582 | Value of the set of morphi... |
| xpcfucco2 50583 | Value of composition in th... |
| xpcfuccocl 50584 | The composition of two nat... |
| xpcfucco3 50585 | Value of composition in th... |
| dfswapf2 50588 | Alternate definition of ` ... |
| swapfval 50589 | Value of the swap functor.... |
| swapfelvv 50590 | A swap functor is an order... |
| swapf2fvala 50591 | The morphism part of the s... |
| swapf2fval 50592 | The morphism part of the s... |
| swapf1vala 50593 | The object part of the swa... |
| swapf1val 50594 | The object part of the swa... |
| swapf2fn 50595 | The morphism part of the s... |
| swapf1a 50596 | The object part of the swa... |
| swapf2vala 50597 | The morphism part of the s... |
| swapf2a 50598 | The morphism part of the s... |
| swapf1 50599 | The object part of the swa... |
| swapf2val 50600 | The morphism part of the s... |
| swapf2 50601 | The morphism part of the s... |
| swapf1f1o 50602 | The object part of the swa... |
| swapf2f1o 50603 | The morphism part of the s... |
| swapf2f1oa 50604 | The morphism part of the s... |
| swapf2f1oaALT 50605 | Alternate proof of ~ swapf... |
| swapfid 50606 | Each identity morphism in ... |
| swapfida 50607 | Each identity morphism in ... |
| swapfcoa 50608 | Composition in the source ... |
| swapffunc 50609 | The swap functor is a func... |
| swapfffth 50610 | The swap functor is a full... |
| swapffunca 50611 | The swap functor is a func... |
| swapfiso 50612 | The swap functor is an iso... |
| swapciso 50613 | The product category is ca... |
| oppc1stflem 50614 | A utility theorem for prov... |
| oppc1stf 50615 | The opposite functor of th... |
| oppc2ndf 50616 | The opposite functor of th... |
| 1stfpropd 50617 | If two categories have the... |
| 2ndfpropd 50618 | If two categories have the... |
| diagpropd 50619 | If two categories have the... |
| cofuswapfcl 50620 | The bifunctor pre-composed... |
| cofuswapf1 50621 | The object part of a bifun... |
| cofuswapf2 50622 | The morphism part of a bif... |
| tposcurf1cl 50623 | The partially evaluated tr... |
| tposcurf11 50624 | Value of the double evalua... |
| tposcurf12 50625 | The partially evaluated tr... |
| tposcurf1 50626 | Value of the object part o... |
| tposcurf2 50627 | Value of the transposed cu... |
| tposcurf2val 50628 | Value of a component of th... |
| tposcurf2cl 50629 | The transposed curry funct... |
| tposcurfcl 50630 | The transposed curry funct... |
| diag1 50631 | The constant functor of ` ... |
| diag1a 50632 | The constant functor of ` ... |
| diag1f1lem 50633 | The object part of the dia... |
| diag1f1 50634 | The object part of the dia... |
| diag2f1lem 50635 | Lemma for ~ diag2f1 . The... |
| diag2f1 50636 | If ` B ` is non-empty, the... |
| fucofulem1 50637 | Lemma for proving functor ... |
| fucofulem2 50638 | Lemma for proving functor ... |
| fuco2el 50639 | Equivalence of product fun... |
| fuco2eld 50640 | Equivalence of product fun... |
| fuco2eld2 50641 | Equivalence of product fun... |
| fuco2eld3 50642 | Equivalence of product fun... |
| fucofvalg 50645 | Value of the function givi... |
| fucofval 50646 | Value of the function givi... |
| fucoelvv 50647 | A functor composition bifu... |
| fuco1 50648 | The object part of the fun... |
| fucof1 50649 | The object part of the fun... |
| fuco2 50650 | The morphism part of the f... |
| fucofn2 50651 | The morphism part of the f... |
| fucofvalne 50652 | Value of the function givi... |
| fuco11 50653 | The object part of the fun... |
| fuco11cl 50654 | The object part of the fun... |
| fuco11a 50655 | The object part of the fun... |
| fuco112 50656 | The object part of the fun... |
| fuco111 50657 | The object part of the fun... |
| fuco111x 50658 | The object part of the fun... |
| fuco112x 50659 | The object part of the fun... |
| fuco112xa 50660 | The object part of the fun... |
| fuco11id 50661 | The identity morphism of t... |
| fuco11idx 50662 | The identity morphism of t... |
| fuco21 50663 | The morphism part of the f... |
| fuco11b 50664 | The object part of the fun... |
| fuco11bALT 50665 | Alternate proof of ~ fuco1... |
| fuco22 50666 | The morphism part of the f... |
| fucofn22 50667 | The morphism part of the f... |
| fuco23 50668 | The morphism part of the f... |
| fuco22natlem1 50669 | Lemma for ~ fuco22nat . T... |
| fuco22natlem2 50670 | Lemma for ~ fuco22nat . T... |
| fuco22natlem3 50671 | Combine ~ fuco22natlem2 wi... |
| fuco22natlem 50672 | The composed natural trans... |
| fuco22nat 50673 | The composed natural trans... |
| fucof21 50674 | The morphism part of the f... |
| fucoid 50675 | Each identity morphism in ... |
| fucoid2 50676 | Each identity morphism in ... |
| fuco22a 50677 | The morphism part of the f... |
| fuco23alem 50678 | The naturality property ( ... |
| fuco23a 50679 | The morphism part of the f... |
| fucocolem1 50680 | Lemma for ~ fucoco . Asso... |
| fucocolem2 50681 | Lemma for ~ fucoco . The ... |
| fucocolem3 50682 | Lemma for ~ fucoco . The ... |
| fucocolem4 50683 | Lemma for ~ fucoco . The ... |
| fucoco 50684 | Composition in the source ... |
| fucoco2 50685 | Composition in the source ... |
| fucofunc 50686 | The functor composition bi... |
| fucofunca 50687 | The functor composition bi... |
| fucolid 50688 | Post-compose a natural tra... |
| fucorid 50689 | Pre-composing a natural tr... |
| fucorid2 50690 | Pre-composing a natural tr... |
| fucoridid 50691 | Horizontal composition wit... |
| fucoisfunc 50692 | A characterization of func... |
| fucoisfunc2lem1 50693 | Lemma for ~ fucoisfunc2lem... |
| fucoisfunc2lem2 50694 | Lemma for ~ fucoisfunc2 . ... |
| fucoisfunc2 50695 | Full componentwise charact... |
| fucoaltfunc 50696 | Post-compose functor compo... |
| fucoaltvallem 50697 | Lemma for ~ fucoaltval . ... |
| fucoaltval 50698 | At a pair consisting of a ... |
| fucoaltnelem 50699 | Lemma for ~ fucoaltne . (... |
| fucoaltne 50700 | A changed natural transfor... |
| postcofval 50701 | Value of the post-composit... |
| postcofcl 50702 | The post-composition funct... |
| postcof1st 50703 | The object part of the exp... |
| postcof2nd 50704 | The morphism part of the e... |
| postcof2 50705 | A component of the action ... |
| precofvallem 50706 | Lemma for ~ precofval to e... |
| precofval 50707 | Value of the pre-compositi... |
| precofvalALT 50708 | Alternate proof of ~ preco... |
| precofval2 50709 | Value of the pre-compositi... |
| precofcl 50710 | The pre-composition functo... |
| precofval3 50711 | Value of the pre-compositi... |
| precoffunc 50712 | The pre-composition functo... |
| reldmprcof 50715 | The domain of ` -o.F ` is ... |
| prcofvalg 50716 | Value of the pre-compositi... |
| prcofvala 50717 | Value of the pre-compositi... |
| prcofval 50718 | Value of the pre-compositi... |
| prcofpropd 50719 | If the categories have the... |
| prcofelvv 50720 | The pre-composition functo... |
| reldmprcof1 50721 | The domain of the object p... |
| reldmprcof2 50722 | The domain of the morphism... |
| prcoftposcurfuco 50723 | The pre-composition functo... |
| prcoftposcurfucoa 50724 | The pre-composition functo... |
| prcoffunc 50725 | The pre-composition functo... |
| prcoffunca 50726 | The pre-composition functo... |
| prcoffunca2 50727 | The pre-composition functo... |
| prcof1 50728 | The object part of the pre... |
| prcof2a 50729 | The morphism part of the p... |
| prcof2 50730 | The morphism part of the p... |
| prcof21a 50731 | The morphism part of the p... |
| prcof22a 50732 | The morphism part of the p... |
| prcofdiag1 50733 | A constant functor pre-com... |
| prcofdiag 50734 | A diagonal functor post-co... |
| catcrcl 50735 | Reverse closure for the ca... |
| catcrcl2 50736 | Reverse closure for the ca... |
| elcatchom 50737 | A morphism of the category... |
| catcsect 50738 | The property " ` F ` is a ... |
| catcinv 50739 | The property " ` F ` is an... |
| catcisoi 50740 | A functor is an isomorphis... |
| uobeq2 50741 | If a full functor (in fact... |
| uobeq3 50742 | An isomorphism between cat... |
| opf11 50743 | The object part of the op ... |
| opf12 50744 | The object part of the op ... |
| opf2fval 50745 | The morphism part of the o... |
| opf2 50746 | The morphism part of the o... |
| fucoppclem 50747 | Lemma for ~ fucoppc . (Co... |
| fucoppcid 50748 | The opposite category of f... |
| fucoppcco 50749 | The opposite category of f... |
| fucoppc 50750 | The isomorphism from the o... |
| fucoppcffth 50751 | A fully faithful functor f... |
| fucoppcfunc 50752 | A functor from the opposit... |
| fucoppccic 50753 | The opposite category of f... |
| oppfdiag1 50754 | A constant functor for opp... |
| oppfdiag1a 50755 | A constant functor for opp... |
| oppfdiag 50756 | A diagonal functor for opp... |
| isthinc 50759 | The predicate "is a thin c... |
| isthinc2 50760 | A thin category is a categ... |
| isthinc3 50761 | A thin category is a categ... |
| thinccat 50762 | A thin category is a categ... |
| thinccatd 50763 | A thin category is a categ... |
| thincssc 50764 | A thin category is a categ... |
| isthincd2lem1 50765 | Lemma for ~ isthincd2 and ... |
| thincmo2 50766 | Morphisms in the same hom-... |
| thinchom 50767 | A non-empty hom-set of a t... |
| thincmo 50768 | There is at most one morph... |
| thincmoALT 50769 | Alternate proof of ~ thinc... |
| thincmod 50770 | At most one morphism in ea... |
| thincn0eu 50771 | In a thin category, a hom-... |
| thincid 50772 | In a thin category, a morp... |
| thincmon 50773 | In a thin category, all mo... |
| thincepi 50774 | In a thin category, all mo... |
| isthincd2lem2 50775 | Lemma for ~ isthincd2 . (... |
| isthincd 50776 | The predicate "is a thin c... |
| isthincd2 50777 | The predicate " ` C ` is a... |
| oppcthin 50778 | The opposite category of a... |
| oppcthinco 50779 | If the opposite category o... |
| oppcthinendc 50780 | The opposite category of a... |
| oppcthinendcALT 50781 | Alternate proof of ~ oppct... |
| thincpropd 50782 | Two structures with the sa... |
| subthinc 50783 | A subcategory of a thin ca... |
| functhinclem1 50784 | Lemma for ~ functhinc . G... |
| functhinclem2 50785 | Lemma for ~ functhinc . (... |
| functhinclem3 50786 | Lemma for ~ functhinc . T... |
| functhinclem4 50787 | Lemma for ~ functhinc . O... |
| functhinc 50788 | A functor to a thin catego... |
| functhincfun 50789 | A functor to a thin catego... |
| fullthinc 50790 | A functor to a thin catego... |
| fullthinc2 50791 | A full functor to a thin c... |
| thincfth 50792 | A functor from a thin cate... |
| thincciso 50793 | Two thin categories are is... |
| thinccisod 50794 | Two thin categories are is... |
| thincciso2 50795 | Categories isomorphic to a... |
| thincciso3 50796 | Categories isomorphic to a... |
| thincciso4 50797 | Two isomorphic categories ... |
| 0thincg 50798 | Any structure with an empt... |
| 0thinc 50799 | The empty category (see ~ ... |
| indcthing 50800 | An indiscrete category, i.... |
| discthing 50801 | A discrete category, i.e.,... |
| indthinc 50802 | An indiscrete category in ... |
| indthincALT 50803 | An alternate proof of ~ in... |
| prsthinc 50804 | Preordered sets as categor... |
| setcthin 50805 | A category of sets all of ... |
| setc2othin 50806 | The category ` ( SetCat ``... |
| thincsect 50807 | In a thin category, one mo... |
| thincsect2 50808 | In a thin category, ` F ` ... |
| thincinv 50809 | In a thin category, ` F ` ... |
| thinciso 50810 | In a thin category, ` F : ... |
| thinccic 50811 | In a thin category, two ob... |
| istermc 50814 | The predicate "is a termin... |
| istermc2 50815 | The predicate "is a termin... |
| istermc3 50816 | The predicate "is a termin... |
| termcthin 50817 | A terminal category is a t... |
| termcthind 50818 | A terminal category is a t... |
| termccatd 50819 | A terminal category is a c... |
| termcbas 50820 | The base of a terminal cat... |
| termco 50821 | The object of a terminal c... |
| termcbas2 50822 | The base of a terminal cat... |
| termcbasmo 50823 | Two objects in a terminal ... |
| termchomn0 50824 | All hom-sets of a terminal... |
| termchommo 50825 | All morphisms of a termina... |
| termcid 50826 | The morphism of a terminal... |
| termcid2 50827 | The morphism of a terminal... |
| termchom 50828 | The hom-set of a terminal ... |
| termchom2 50829 | The hom-set of a terminal ... |
| setcsnterm 50830 | The category of one set, e... |
| setc1oterm 50831 | The category ` ( SetCat ``... |
| setc1obas 50832 | The base of the trivial ca... |
| setc1ohomfval 50833 | Set of morphisms of the tr... |
| setc1ocofval 50834 | Composition in the trivial... |
| setc1oid 50835 | The identity morphism of t... |
| funcsetc1ocl 50836 | The functor to the trivial... |
| funcsetc1o 50837 | Value of the functor to th... |
| isinito2lem 50838 | The predicate "is an initi... |
| isinito2 50839 | The predicate "is an initi... |
| isinito3 50840 | The predicate "is an initi... |
| dfinito4 50841 | An alternate definition of... |
| dftermo4 50842 | An alternate definition of... |
| termcpropd 50843 | Two structures with the sa... |
| oppctermhom 50844 | The opposite category of a... |
| oppctermco 50845 | The opposite category of a... |
| oppcterm 50846 | The opposite category of a... |
| functermclem 50847 | Lemma for ~ functermc . (... |
| functermc 50848 | Functor to a terminal cate... |
| functermc2 50849 | Functor to a terminal cate... |
| functermceu 50850 | There exists a unique func... |
| fulltermc 50851 | A functor to a terminal ca... |
| fulltermc2 50852 | Given a full functor to a ... |
| termcterm 50853 | A terminal category is a t... |
| termcterm2 50854 | A terminal object of the c... |
| termcterm3 50855 | In the category of small c... |
| termcciso 50856 | A category is isomorphic t... |
| termccisoeu 50857 | The isomorphism between te... |
| termc2 50858 | If there exists a unique f... |
| termc 50859 | Alternate definition of ` ... |
| dftermc2 50860 | Alternate definition of ` ... |
| eufunclem 50861 | If there exists a unique f... |
| eufunc 50862 | If there exists a unique f... |
| idfudiag1lem 50863 | Lemma for ~ idfudiag1bas a... |
| idfudiag1bas 50864 | If the identity functor of... |
| idfudiag1 50865 | If the identity functor of... |
| euendfunc 50866 | If there exists a unique e... |
| euendfunc2 50867 | If there exists a unique e... |
| termcarweu 50868 | There exists a unique disj... |
| arweuthinc 50869 | If a structure has a uniqu... |
| arweutermc 50870 | If a structure has a uniqu... |
| dftermc3 50871 | Alternate definition of ` ... |
| termcfuncval 50872 | The value of a functor fro... |
| diag1f1olem 50873 | To any functor from a term... |
| diag1f1o 50874 | The object part of the dia... |
| termcfunc1eq 50875 | Functors with terminal sou... |
| termccofix 50876 | Post-composition by an obj... |
| termcpostfix 50877 | Post-composition by an obj... |
| termcnatval 50878 | Value of natural transform... |
| diag2f1olem 50879 | Lemma for ~ diag2f1o . (C... |
| diag2f1o 50880 | If ` D ` is terminal, the ... |
| diagffth 50881 | The diagonal functor is a ... |
| diagciso 50882 | The diagonal functor is an... |
| diagcic 50883 | Any category ` C ` is isom... |
| funcsn 50884 | The category of one functo... |
| fucterm 50885 | The category of functors t... |
| 0fucterm 50886 | The category of functors f... |
| termfucterm 50887 | All functors between two t... |
| cofuterm 50888 | Post-compose with a functo... |
| uobeqterm 50889 | Universal objects and term... |
| isinito4 50890 | The predicate "is an initi... |
| isinito4a 50891 | The predicate "is an initi... |
| prstcval 50894 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 50895 | Lemma for ~ prstcnid and ~... |
| prstcnid 50896 | Components other than ` Ho... |
| prstcbas 50897 | The base set is unchanged.... |
| prstcleval 50898 | Value of the less-than-or-... |
| prstcle 50899 | Value of the less-than-or-... |
| prstcocval 50900 | Orthocomplementation is un... |
| prstcoc 50901 | Orthocomplementation is un... |
| prstchomval 50902 | Hom-sets of the constructe... |
| prstcprs 50903 | The category is a preorder... |
| prstcthin 50904 | The preordered set is equi... |
| prstchom 50905 | Hom-sets of the constructe... |
| prstchom2 50906 | Hom-sets of the constructe... |
| prstchom2ALT 50907 | Hom-sets of the constructe... |
| oduoppcbas 50908 | The dual of a preordered s... |
| oduoppcciso 50909 | The dual of a preordered s... |
| postcpos 50910 | The converted category is ... |
| postcposALT 50911 | Alternate proof of ~ postc... |
| postc 50912 | The converted category is ... |
| discsntermlem 50913 | A singlegon is an element ... |
| basrestermcfolem 50914 | An element of the class of... |
| discbas 50915 | A discrete category (a cat... |
| discthin 50916 | A discrete category (a cat... |
| discsnterm 50917 | A discrete category (a cat... |
| basrestermcfo 50918 | The base function restrict... |
| termcnex 50919 | The class of all terminal ... |
| mndtcval 50922 | Value of the category buil... |
| mndtcbasval 50923 | The base set of the catego... |
| mndtcbaseu 50924 | The category built from a ... |
| mndtcob 50925 | Lemma for ~ mndtchom and ~... |
| mndtcobeq 50926 | Two objects in a category ... |
| mndtchom 50927 | The only hom-set of the ca... |
| mndtcco 50928 | The composition of the cat... |
| mndtcco2 50929 | The composition of the cat... |
| mndtccatid 50930 | Lemma for ~ mndtccat and ~... |
| mndtccat 50931 | The function value is a ca... |
| mndtcid 50932 | The identity morphism, or ... |
| oppgoppchom 50933 | The converted opposite mon... |
| oppgoppcco 50934 | The converted opposite mon... |
| oppgoppcid 50935 | The converted opposite mon... |
| grptcmon 50936 | All morphisms in a categor... |
| grptcepi 50937 | All morphisms in a categor... |
| 2arwcatlem1 50938 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem2 50939 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem3 50940 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem4 50941 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem5 50942 | Lemma for ~ 2arwcat . (Co... |
| 2arwcat 50943 | The condition for a struct... |
| incat 50944 | Constructing a category wi... |
| setc1onsubc 50945 | Construct a category with ... |
| cnelsubcw 50946 | A weaker version of ~ cnel... |
| cnelsubc 50947 | Remark 4.2(2) of [Adamek] ... |
| parhomval 50948 | At objects in the base, th... |
| parhomel 50949 | Membership in a hom-set of... |
| parhomcl2 50950 | Variant of ~ parhomcl . (... |
| parhomidval 50951 | On the diagonal, the paral... |
| parhomssun 50952 | Every value of the paralle... |
| parhomcl 50953 | All hom-sets of the parall... |
| parhomideq2 50954 | Variant of ~ parhomideq . ... |
| parhomideq 50955 | An endomorphism in the par... |
| parhomid2 50956 | Variant of ~ parhomid . (... |
| parhomid 50957 | The identity value belongs... |
| parhomvaln0 50958 | Evaluate an inhabited para... |
| parhomne2 50959 | Variant of ~ parhomne . (... |
| parhomne 50960 | A morphism between differe... |
| parhomss2 50961 | Variant of ~ parhomss . (... |
| parhomss 50962 | When every member of the p... |
| parhomeqor2 50963 | Variant of ~ parhomeqor . ... |
| parhomeqor 50964 | In a composable pair, at l... |
| parhomco2 50965 | Variant of ~ parhomco . (... |
| parhomco 50966 | When every member of the p... |
| parcoval 50967 | The composition in the spe... |
| parcatid2 50968 | Variant of ~ parcatid . (... |
| parcatid 50969 | Construct a category with ... |
| paridval 50970 | The identity at any object... |
| parcoval2 50971 | Composition in a category ... |
| par2catid 50972 | A concrete category with t... |
| par2cat 50973 | The explicitly given two-o... |
| par2str 50974 | The object set, hom operat... |
| parmapcl2 50975 | Variant of ~ parmapcl . (... |
| parmapcl 50976 | A map fixing the identity ... |
| parmapco2 50977 | Variant of ~ parmapco . (... |
| parmapco 50978 | An identity-fixing map pre... |
| parmapfunc2 50979 | Variant of ~ parmapfunc . ... |
| parmapfunc 50980 | A map fixing the identity ... |
| parswapval 50981 | Evaluate the transposition... |
| parswapprop 50982 | The concrete transposition... |
| parswapfunc 50983 | The explicit exchange of t... |
| par2hom 50984 | The four hom-sets of the e... |
| par2diag 50985 | Two constant functors into... |
| fuconuniqlem 50986 | Lemma for ~ fuconuniq . (... |
| fuconuniq 50987 | The arrow map ` A ` fixes ... |
| fuconuniqvalslem 50988 | Lemma for ~ fuconuniqvals ... |
| fuconuniqvals 50989 | At the same natural-transf... |
| fuconuniq2 50990 | The concrete parallel-arro... |
| fuconuniqex 50991 | The object function of fun... |
| fuconuniqex2 50992 | There are categories for w... |
| lanfn 50997 | ` Lan ` is a function on `... |
| ranfn 50998 | ` Ran ` is a function on `... |
| reldmlan 50999 | The domain of ` Lan ` is a... |
| reldmran 51000 | The domain of ` Ran ` is a... |
| lanfval 51001 | Value of the function gene... |
| ranfval 51002 | Value of the function gene... |
| lanpropd 51003 | If the categories have the... |
| ranpropd 51004 | If the categories have the... |
| reldmlan2 51005 | The domain of ` ( P Lan E ... |
| reldmran2 51006 | The domain of ` ( P Ran E ... |
| lanval 51007 | Value of the set of left K... |
| ranval 51008 | Value of the set of right ... |
| lanrcl 51009 | Reverse closure for left K... |
| ranrcl 51010 | Reverse closure for right ... |
| rellan 51011 | The set of left Kan extens... |
| relran 51012 | The set of right Kan exten... |
| islan 51013 | A left Kan extension is a ... |
| islan2 51014 | A left Kan extension is a ... |
| lanval2 51015 | The set of left Kan extens... |
| isran 51016 | A right Kan extension is a... |
| isran2 51017 | A right Kan extension is a... |
| ranval2 51018 | The set of right Kan exten... |
| ranval3 51019 | The set of right Kan exten... |
| lanrcl2 51020 | Reverse closure for left K... |
| lanrcl3 51021 | Reverse closure for left K... |
| lanrcl4 51022 | The first component of a l... |
| lanrcl5 51023 | The second component of a ... |
| ranrcl2 51024 | Reverse closure for right ... |
| ranrcl3 51025 | Reverse closure for right ... |
| ranrcl4lem 51026 | Lemma for ~ ranrcl4 and ~ ... |
| ranrcl4 51027 | The first component of a r... |
| ranrcl5 51028 | The second component of a ... |
| lanup 51029 | The universal property of ... |
| ranup 51030 | The universal property of ... |
| reldmlmd 51035 | The domain of ` Limit ` is... |
| reldmcmd 51036 | The domain of ` Colimit ` ... |
| lmdfval 51037 | Function value of ` Limit ... |
| cmdfval 51038 | Function value of ` Colimi... |
| lmdrcl 51039 | Reverse closure for a limi... |
| cmdrcl 51040 | Reverse closure for a coli... |
| reldmlmd2 51041 | The domain of ` ( C Limit ... |
| reldmcmd2 51042 | The domain of ` ( C Colimi... |
| lmdfval2 51043 | The set of limits of a dia... |
| cmdfval2 51044 | The set of colimits of a d... |
| lmdpropd 51045 | If the categories have the... |
| cmdpropd 51046 | If the categories have the... |
| rellmd 51047 | The set of limits of a dia... |
| relcmd 51048 | The set of colimits of a d... |
| concl 51049 | A natural transformation f... |
| coccl 51050 | A natural transformation t... |
| concom 51051 | A cone to a diagram commut... |
| coccom 51052 | A co-cone to a diagram com... |
| islmd 51053 | The universal property of ... |
| iscmd 51054 | The universal property of ... |
| lmddu 51055 | The duality of limits and ... |
| cmddu 51056 | The duality of limits and ... |
| initocmd 51057 | Initial objects are the ob... |
| termolmd 51058 | Terminal objects are the o... |
| lmdran 51059 | To each limit of a diagram... |
| cmdlan 51060 | To each colimit of a diagr... |
| nfintd 51061 | Bound-variable hypothesis ... |
| nfiund 51062 | Bound-variable hypothesis ... |
| nfiundg 51063 | Bound-variable hypothesis ... |
| iunord 51064 | The indexed union of a col... |
| iunordi 51065 | The indexed union of a col... |
| spd 51066 | Specialization deduction, ... |
| tfis2d 51067 | Transfinite Induction Sche... |
| setrecseq 51068 | Equality theorem for set r... |
| nfsetrecs 51069 | Bound-variable hypothesis ... |
| setrec2mpt 51070 | Version of ~ setrec2 where... |
| setis 51071 | Version of ~ setrec2 expre... |
| elsetrecslem 51072 | Lemma for ~ elsetrecs . A... |
| elsetrecs 51073 | A set ` A ` is an element ... |
| setrecsss 51074 | The ` setrecs ` operator r... |
| setrecsres 51075 | A recursively generated cl... |
| vsetrec 51076 | Construct ` _V ` using set... |
| 0setrec 51077 | If a function sends the em... |
| onsetreclem1 51078 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 51079 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 51080 | Lemma for ~ onsetrec . (C... |
| onsetrec 51081 | Construct ` On ` using set... |
| elpglem1 51084 | Lemma for ~ elpg . (Contr... |
| elpglem2 51085 | Lemma for ~ elpg . (Contr... |
| elpglem3 51086 | Lemma for ~ elpg . (Contr... |
| elpg 51087 | Membership in the class of... |
| pgindlem 51088 | Lemma for ~ pgind . (Cont... |
| pgindnf 51089 | Version of ~ pgind with ex... |
| pgind 51090 | Induction on partizan game... |
| sbidd 51091 | An identity theorem for su... |
| sbidd-misc 51092 | An identity theorem for su... |
| gte-lte 51097 | Simple relationship betwee... |
| gt-lt 51098 | Simple relationship betwee... |
| gte-lteh 51099 | Relationship between ` <_ ... |
| gt-lth 51100 | Relationship between ` < `... |
| ex-gt 51101 | Simple example of ` > ` , ... |
| ex-gte 51102 | Simple example of ` >_ ` ,... |
| sinhval-named 51109 | Value of the named sinh fu... |
| coshval-named 51110 | Value of the named cosh fu... |
| tanhval-named 51111 | Value of the named tanh fu... |
| sinh-conventional 51112 | Conventional definition of... |
| sinhpcosh 51113 | Prove that ` ( sinh `` A )... |
| secval 51120 | Value of the secant functi... |
| cscval 51121 | Value of the cosecant func... |
| cotval 51122 | Value of the cotangent fun... |
| seccl 51123 | The closure of the secant ... |
| csccl 51124 | The closure of the cosecan... |
| cotcl 51125 | The closure of the cotange... |
| reseccl 51126 | The closure of the secant ... |
| recsccl 51127 | The closure of the cosecan... |
| recotcl 51128 | The closure of the cotange... |
| recsec 51129 | The reciprocal of secant i... |
| reccsc 51130 | The reciprocal of cosecant... |
| reccot 51131 | The reciprocal of cotangen... |
| rectan 51132 | The reciprocal of tangent ... |
| sec0 51133 | The value of the secant fu... |
| onetansqsecsq 51134 | Prove the tangent squared ... |
| cotsqcscsq 51135 | Prove the tangent squared ... |
| dvsec 51136 | Derivative of the secant f... |
| dvcsc 51137 | Derivative of the cosecant... |
| dvcot 51138 | Derivative of the cotangen... |
| ifnmfalse 51139 | If A is not a member of B,... |
| logb2aval 51140 | Define the value of the ` ... |
| mvlraddi 51147 | Move the right term in a s... |
| assraddsubi 51148 | Associate RHS addition-sub... |
| joinlmuladdmuli 51149 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 51150 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 51151 | Join AB-CB into (A-C) on L... |
| mvlrmuld 51152 | Move the right term in a p... |
| mvlrmuli 51153 | Move the right term in a p... |
| i2linesi 51154 | Solve for the intersection... |
| i2linesd 51155 | Solve for the intersection... |
| alimp-surprise 51156 | Demonstrate that when usin... |
| alimp-no-surprise 51157 | There is no "surprise" in ... |
| empty-surprise 51158 | Demonstrate that when usin... |
| empty-surprise2 51159 | "Prove" that false is true... |
| eximp-surprise 51160 | Show what implication insi... |
| eximp-surprise2 51161 | Show that "there exists" w... |
| dfrals2 51166 | The bounded "all some" for... |
| alsd 51167 | Introduction rule: "all s... |
| ralsd 51168 | Introduction rule for "all... |
| als1d 51169 | Deduction rule: Given "al... |
| als2d 51170 | Deduction rule: Given "al... |
| rals1d 51171 | Deduction rule: Given "al... |
| rals2d 51172 | Deduction rule: Given "al... |
| ralsn0d 51173 | Deduction rule: Given "al... |
| alsex 51174 | The consequent of an "all ... |
| ralsex 51175 | The consequent of an "all ... |
| alsbii 51176 | Congruence: equivalents ma... |
| ralsbii 51177 | Congruence for "all some" ... |
| alsbid 51178 | Deduction form of ~ alsbii... |
| nfals 51179 | Bound-variable hypothesis ... |
| nfrals 51180 | Bound-variable hypothesis ... |
| cbvals 51181 | Rule used to change bound ... |
| als-no-surprise 51182 | Demonstrate that there is ... |
| rals-no-surprise 51183 | Demonstrate that there is ... |
| ralrals 51184 | If the universal part of a... |
| rexrals 51185 | If a member of ` A ` satis... |
| alsanmo 51186 | An "all some" statement co... |
| ralsanmo 51187 | An "all some" statement re... |
| alsralrex 51188 | The general "all some" qua... |
| alsraln0 51189 | The general "all some" qua... |
| ralals 51190 | If ` ph ` holds for every ... |
| rexals 51191 | If some ` x ` in ` A ` sat... |
| n0als 51192 | If ` A ` is not empty, the... |
| 2alsraln0 51193 | Nested general "all some" ... |
| 2alsraln0id 51194 | Nested general "all some" ... |
| dfralseu2 51199 | The bounded "all some one"... |
| alseuals 51200 | "All some one" implies "al... |
| ralseurals 51201 | "All some one" restricted ... |
| alseud 51202 | Introduction rule: "all s... |
| ralseud 51203 | Introduction rule for "all... |
| alseu1d 51204 | Deduction rule: Given "al... |
| alseu2d 51205 | Deduction rule: Given "al... |
| ralseu1d 51206 | Deduction rule: Given "al... |
| ralseu2d 51207 | Deduction rule: Given "al... |
| alseubii 51208 | Congruence: equivalents ma... |
| ralseubii 51209 | Congruence for "all some o... |
| nfalseu 51210 | Bound-variable hypothesis ... |
| nfralseu 51211 | Bound-variable hypothesis ... |
| dfalseu2 51212 | An "all some one" statemen... |
| alseueu 51213 | "The ` ph ` is ` ps ` " im... |
| alseu-no-surprise 51214 | Demonstrate that there is ... |
| 5m4e1 51215 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 51216 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 51217 | Resolution rule. This is ... |
| testable 51218 | In classical logic all wff... |
| aacllem 51219 | Lemma for other theorems a... |
| wrdf1d 51220 | A one-to-one word maps its... |
| 1ne3 51221 | ` 1 ` is not equal to ` 3 ... |
| 2ne3 51222 | ` 2 ` is not equal to ` 3 ... |
| 1elfz13 51223 | Membership of 1 in the int... |
| 2elfz13 51224 | Membership of 2 in the int... |
| 3elfz13 51225 | Membership of 3 in the int... |
| rr3fvcl 51226 | The components of a 3-dime... |
| rr3fv1cld 51227 | First component of a 3-dim... |
| rr3fv2cld 51228 | Second component of a 3-di... |
| rr3fv3cld 51229 | Third component of a 3-dim... |
| crosspval 51234 | Value of the cross product... |
| crosspcle1d 51235 | Closure of the first compo... |
| crosspcle2d 51236 | Closure of the second comp... |
| crosspcle3d 51237 | Closure of the third compo... |
| crosspclem 51238 | Lemma for ~ crosspcld . C... |
| crosspcld 51239 | Closure of the cross produ... |
| crosspv1d 51240 | Value of the first compone... |
| crosspv2d 51241 | Value of the second compon... |
| crosspv3d 51242 | Value of the third compone... |
| crosspdot0lem 51243 | Lemma for ~ crosspdotd . ... |
| crosspdotsumlem 51244 | Lemma for ~ crosspdotd . ... |
| crosspdotd 51245 | Value of the scalar triple... |
| crosspaltd 51246 | Antisymmetry of the cross ... |
| crossp3d 51247 | The vector triple product ... |
| nellindf 51250 | A nonzero coefficient vect... |
| veronesevald 51251 | Value of the Veronese map ... |
| veronesefvcl 51252 | Every coordinate of the Ve... |
| veronesev1lem 51253 | Lemma for ~ veronesevrowd ... |
| veronesev2lem 51254 | Lemma for ~ veronesevrowd ... |
| veronesev3lem 51255 | Lemma for ~ veronesevrowd ... |
| veronesev4lem 51256 | Lemma for ~ veronesevrowd ... |
| veronesev5lem 51257 | Lemma for ~ veronesevrowd ... |
| veronesev6lem 51258 | Lemma for ~ veronesevrowd ... |
| veronesevrowd 51259 | The Veronese map at a poin... |
| veronesematbasd 51260 | The matrix whose ` i ` -th... |
| veronesematrowd 51261 | Currying the Veronese matr... |
| veronesematrowexpd 51262 | Currying the Veronese matr... |
| veroquadgsumlem 51263 | Lemma for ~ veroquadmodzer... |
| veroquadmodzerod 51264 | The columns of the Verones... |
| veroquadnolindfd 51265 | A nonzero homogeneous quad... |
| veroquaddetzerod 51266 | The Veronese matrix of six... |
| amgmwlem 51267 | Weighted version of ~ amgm... |
| amgmlemALT 51268 | Alternate proof of ~ amgml... |
| amgmw2d 51269 | Weighted arithmetic-geomet... |
| young2d 51270 | Young's inequality for ` n... |
| Copyright terms: Public domain | W3C validator |