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 ... |
| mp1i 13 | Inference detaching an ant... |
| a2i 14 | Inference distributing an ... |
| mpd 15 | A modus ponens deduction. ... |
| imim2i 16 | Inference adding common an... |
| syl 17 | An inference version of th... |
| 3syl 18 | Inference chaining two syl... |
| 4syl 19 | Inference chaining three s... |
| mpi 20 | A nested modus ponens infe... |
| mpisyl 21 | A syllogism combined with ... |
| id 22 | Principle of identity. Th... |
| idALT 23 | Alternate proof of ~ id . ... |
| idd 24 | Principle of identity ~ id... |
| a1d 25 | Deduction introducing an e... |
| 2a1d 26 | Deduction introducing two ... |
| a1i13 27 | Add two antecedents to a w... |
| 2a1 28 | A double form of ~ ax-1 . ... |
| a2d 29 | Deduction distributing an ... |
| sylcom 30 | Syllogism inference with c... |
| syl5com 31 | Syllogism inference with c... |
| com12 32 | Inference that swaps (comm... |
| syl11 33 | A syllogism inference. Co... |
| syl5 34 | A syllogism rule of infere... |
| syl6 35 | A syllogism rule of infere... |
| syl56 36 | Combine ~ syl5 and ~ syl6 ... |
| syl6com 37 | Syllogism inference with c... |
| mpcom 38 | Modus ponens inference wit... |
| syli 39 | Syllogism inference with c... |
| syl2im 40 | Replace two antecedents. ... |
| syl2imc 41 | A commuted version of ~ sy... |
| pm2.27 42 | This theorem, sometimes ca... |
| mpdd 43 | A nested modus ponens dedu... |
| mpid 44 | A nested modus ponens dedu... |
| mpdi 45 | A nested modus ponens dedu... |
| mpii 46 | A doubly nested modus pone... |
| syld 47 | Syllogism deduction. Dedu... |
| syldc 48 | Syllogism deduction. Comm... |
| mp2d 49 | A double modus ponens dedu... |
| a1dd 50 | Double deduction introduci... |
| 2a1dd 51 | Double deduction introduci... |
| pm2.43i 52 | Inference absorbing redund... |
| pm2.43d 53 | Deduction absorbing redund... |
| pm2.43a 54 | Inference absorbing redund... |
| pm2.43b 55 | Inference absorbing redund... |
| pm2.43 56 | Absorption of redundant an... |
| imim2d 57 | Deduction adding nested an... |
| imim2 58 | A closed form of syllogism... |
| embantd 59 | Deduction embedding an ant... |
| 3syld 60 | Triple syllogism deduction... |
| sylsyld 61 | A double syllogism inferen... |
| imim12i 62 | Inference joining two impl... |
| imim1i 63 | Inference adding common co... |
| imim3i 64 | Inference adding three nes... |
| sylc 65 | A syllogism inference comb... |
| syl3c 66 | A syllogism inference comb... |
| syl6mpi 67 | A syllogism inference. (C... |
| mpsyl 68 | Modus ponens combined with... |
| mpsylsyld 69 | Modus ponens combined with... |
| syl6c 70 | Inference combining ~ syl6... |
| syl6ci 71 | A syllogism inference comb... |
| syldd 72 | Nested syllogism deduction... |
| syl5d 73 | A nested syllogism deducti... |
| syl7 74 | A syllogism rule of infere... |
| syl6d 75 | A nested syllogism deducti... |
| syl8 76 | A syllogism rule of infere... |
| syl9 77 | A nested syllogism inferen... |
| syl9r 78 | A nested syllogism inferen... |
| syl10 79 | A nested syllogism inferen... |
| a1ddd 80 | Triple deduction introduci... |
| imim12d 81 | Deduction combining antece... |
| imim1d 82 | Deduction adding nested co... |
| imim1 83 | A closed form of syllogism... |
| pm2.83 84 | Theorem *2.83 of [Whitehea... |
| peirceroll 85 | Over minimal implicational... |
| com23 86 | Commutation of antecedents... |
| com3r 87 | Commutation of antecedents... |
| com13 88 | Commutation of antecedents... |
| com3l 89 | Commutation of antecedents... |
| pm2.04 90 | Swap antecedents. Theorem... |
| com34 91 | Commutation of antecedents... |
| com4l 92 | Commutation of antecedents... |
| com4t 93 | Commutation of antecedents... |
| com4r 94 | Commutation of antecedents... |
| com24 95 | Commutation of antecedents... |
| com14 96 | Commutation of antecedents... |
| com45 97 | Commutation of antecedents... |
| com35 98 | Commutation of antecedents... |
| com25 99 | Commutation of antecedents... |
| com5l 100 | Commutation of antecedents... |
| com15 101 | Commutation of antecedents... |
| com52l 102 | Commutation of antecedents... |
| com52r 103 | Commutation of antecedents... |
| com5r 104 | Commutation of antecedents... |
| imim12 105 | Closed form of ~ imim12i a... |
| jarr 106 | Elimination of a nested an... |
| jarri 107 | Inference associated with ... |
| pm2.86d 108 | Deduction associated with ... |
| pm2.86 109 | Converse of Axiom ~ ax-2 .... |
| pm2.86i 110 | Inference associated with ... |
| loolin 111 | The Linearity Axiom of the... |
| loowoz 112 | An alternate for the Linea... |
| con4 113 | Alias for ~ ax-3 to be use... |
| con4i 114 | Inference associated with ... |
| con4d 115 | Deduction associated with ... |
| mt4 116 | The rule of modus tollens.... |
| mt4d 117 | Modus tollens deduction. ... |
| mt4i 118 | Modus tollens inference. ... |
| pm2.21i 119 | A contradiction implies an... |
| pm2.24ii 120 | A contradiction implies an... |
| pm2.21d 121 | A contradiction implies an... |
| pm2.21ddALT 122 | Alternate proof of ~ pm2.2... |
| pm2.21 123 | From a wff and its negatio... |
| pm2.24 124 | Theorem *2.24 of [Whitehea... |
| jarl 125 | Elimination of a nested an... |
| jarli 126 | Inference associated with ... |
| pm2.18d 127 | Deduction form of the Clav... |
| pm2.18 128 | Clavius law, or "consequen... |
| pm2.18i 129 | Inference associated with ... |
| notnotr 130 | Double negation eliminatio... |
| notnotri 131 | Inference associated with ... |
| notnotriALT 132 | Alternate proof of ~ notno... |
| notnotrd 133 | Deduction associated with ... |
| con2d 134 | A contraposition deduction... |
| con2 135 | Contraposition. Theorem *... |
| mt2d 136 | Modus tollens deduction. ... |
| mt2i 137 | Modus tollens inference. ... |
| nsyl3 138 | A negated syllogism infere... |
| con2i 139 | A contraposition inference... |
| nsyl 140 | A negated syllogism infere... |
| nsyl2 141 | A negated syllogism infere... |
| notnot 142 | Double negation introducti... |
| notnoti 143 | Inference associated with ... |
| notnotd 144 | Deduction associated with ... |
| con1d 145 | A contraposition deduction... |
| con1 146 | Contraposition. Theorem *... |
| con1i 147 | A contraposition inference... |
| mt3d 148 | Modus tollens deduction. ... |
| mt3i 149 | Modus tollens inference. ... |
| pm2.24i 150 | Inference associated with ... |
| pm2.24d 151 | Deduction form of ~ pm2.24... |
| con3d 152 | A contraposition deduction... |
| con3 153 | Contraposition. Theorem *... |
| con3i 154 | A contraposition inference... |
| con3rr3 155 | Rotate through consequent ... |
| nsyld 156 | A negated syllogism deduct... |
| nsyli 157 | A negated syllogism infere... |
| nsyl4 158 | A negated syllogism infere... |
| nsyl5 159 | A negated syllogism infere... |
| pm3.2im 160 | Theorem *3.2 of [Whitehead... |
| jc 161 | Deduction joining the cons... |
| jcn 162 | Theorem joining the conseq... |
| jcnd 163 | Deduction joining the cons... |
| impi 164 | An importation inference. ... |
| expi 165 | An exportation inference. ... |
| simprim 166 | Simplification. Similar t... |
| simplim 167 | Simplification. Similar t... |
| pm2.5g 168 | General instance of Theore... |
| pm2.5 169 | Theorem *2.5 of [Whitehead... |
| conax1 170 | Contrapositive of ~ ax-1 .... |
| conax1k 171 | Weakening of ~ conax1 . G... |
| pm2.51 172 | Theorem *2.51 of [Whitehea... |
| pm2.52 173 | Theorem *2.52 of [Whitehea... |
| pm2.521g 174 | A general instance of Theo... |
| pm2.521g2 175 | A general instance of Theo... |
| pm2.521 176 | Theorem *2.521 of [Whitehe... |
| expt 177 | Exportation theorem ~ pm3.... |
| exptOLD 178 | Obsolete version of ~ expt... |
| impt 179 | Importation theorem ~ pm3.... |
| pm2.61d 180 | Deduction eliminating an a... |
| pm2.61d1 181 | Inference eliminating an a... |
| pm2.61d2 182 | Inference eliminating an a... |
| pm2.61i 183 | Inference eliminating an a... |
| pm2.61ii 184 | Inference eliminating two ... |
| pm2.61nii 185 | Inference eliminating two ... |
| pm2.61iii 186 | Inference eliminating thre... |
| ja 187 | Inference joining the ante... |
| jad 188 | Deduction form of ~ ja . ... |
| pm2.01 189 | Weak Clavius law. If a fo... |
| pm2.01i 190 | Inference associated with ... |
| pm2.01d 191 | Deduction based on reducti... |
| pm2.6 192 | Theorem *2.6 of [Whitehead... |
| pm2.61 193 | Theorem *2.61 of [Whitehea... |
| pm2.65 194 | Theorem *2.65 of [Whitehea... |
| pm2.65i 195 | Inference for proof by con... |
| pm2.65iOLD 196 | Obsolete version of ~ pm2.... |
| pm2.21dd 197 | A contradiction implies an... |
| pm2.65d 198 | Deduction for proof by con... |
| mto 199 | The rule of modus tollens.... |
| mtod 200 | Modus tollens deduction. ... |
| mtoi 201 | Modus tollens inference. ... |
| mt2 202 | A rule similar to modus to... |
| mt3 203 | A rule similar to modus to... |
| peirce 204 | Peirce's axiom. A non-int... |
| looinv 205 | The Inversion Axiom of the... |
| bijust0 206 | A self-implication (see ~ ... |
| bijust 207 | Theorem used to justify th... |
| impbi 210 | Property of the biconditio... |
| impbii 211 | Infer an equivalence from ... |
| impbidd 212 | Deduce an equivalence from... |
| impbid21d 213 | Deduce an equivalence from... |
| impbid 214 | Deduce an equivalence from... |
| dfbi1 215 | Relate the biconditional c... |
| dfbi1ALT 216 | Alternate proof of ~ dfbi1... |
| biimp 217 | Property of the biconditio... |
| biimpi 218 | Infer an implication from ... |
| sylbi 219 | A mixed syllogism inferenc... |
| sylib 220 | A mixed syllogism inferenc... |
| sylbb 221 | A mixed syllogism inferenc... |
| biimpr 222 | Property of the biconditio... |
| bicom1 223 | Commutative law for the bi... |
| bicom 224 | Commutative law for the bi... |
| bicomd 225 | Commute two sides of a bic... |
| bicomi 226 | Inference from commutative... |
| impbid1 227 | Infer an equivalence from ... |
| impbid2 228 | Infer an equivalence from ... |
| impcon4bid 229 | A variation on ~ impbid wi... |
| biimpri 230 | Infer a converse implicati... |
| biimpd 231 | Deduce an implication from... |
| mpbi 232 | An inference from a bicond... |
| mpbir 233 | An inference from a bicond... |
| mpbid 234 | A deduction from a bicondi... |
| mpbii 235 | An inference from a nested... |
| sylibr 236 | A mixed syllogism inferenc... |
| sylbir 237 | A mixed syllogism inferenc... |
| sylbbr 238 | A mixed syllogism inferenc... |
| sylbb1 239 | A mixed syllogism inferenc... |
| sylbb2 240 | A mixed syllogism inferenc... |
| sylibd 241 | A syllogism deduction. (C... |
| sylbid 242 | A syllogism deduction. (C... |
| mpbidi 243 | A deduction from a bicondi... |
| biimtrid 244 | A mixed syllogism inferenc... |
| biimtrrid 245 | A mixed syllogism inferenc... |
| imbitrid 246 | A mixed syllogism inferenc... |
| syl5ibcom 247 | A mixed syllogism inferenc... |
| imbitrrid 248 | A mixed syllogism inferenc... |
| syl5ibrcom 249 | A mixed syllogism inferenc... |
| biimprd 250 | Deduce a converse implicat... |
| biimpcd 251 | Deduce a commuted implicat... |
| biimprcd 252 | Deduce a converse commuted... |
| imbitrdi 253 | A mixed syllogism inferenc... |
| imbitrrdi 254 | A mixed syllogism inferenc... |
| biimtrdi 255 | A mixed syllogism inferenc... |
| biimtrrdi 256 | A mixed syllogism inferenc... |
| syl7bi 257 | A mixed syllogism inferenc... |
| syl8ib 258 | A syllogism rule of infere... |
| mpbird 259 | A deduction from a bicondi... |
| mpbiri 260 | An inference from a nested... |
| sylibrd 261 | A syllogism deduction. (C... |
| sylbird 262 | A syllogism deduction. (C... |
| biid 263 | Principle of identity for ... |
| biidd 264 | Principle of identity with... |
| pm5.1im 265 | Two propositions are equiv... |
| 2th 266 | Two truths are equivalent.... |
| 2thd 267 | Two truths are equivalent.... |
| monothetic 268 | Two self-implications (see... |
| ibi 269 | Inference that converts a ... |
| ibir 270 | Inference that converts a ... |
| ibd 271 | Deduction that converts a ... |
| pm5.74 272 | Distribution of implicatio... |
| pm5.74i 273 | Distribution of implicatio... |
| pm5.74ri 274 | Distribution of implicatio... |
| pm5.74d 275 | Distribution of implicatio... |
| pm5.74rd 276 | Distribution of implicatio... |
| bitri 277 | An inference from transiti... |
| bitr2i 278 | An inference from transiti... |
| bitr3i 279 | An inference from transiti... |
| bitr4i 280 | An inference from transiti... |
| bitrd 281 | Deduction form of ~ bitri ... |
| bitr2d 282 | Deduction form of ~ bitr2i... |
| bitr3d 283 | Deduction form of ~ bitr3i... |
| bitr4d 284 | Deduction form of ~ bitr4i... |
| bitrid 285 | A syllogism inference from... |
| bitr2id 286 | A syllogism inference from... |
| bitr3id 287 | A syllogism inference from... |
| bitr3di 288 | A syllogism inference from... |
| bitrdi 289 | A syllogism inference from... |
| bitr2di 290 | A syllogism inference from... |
| bitr4di 291 | A syllogism inference from... |
| bitr4id 292 | A syllogism inference from... |
| 3imtr3i 293 | A mixed syllogism inferenc... |
| 3imtr4i 294 | A mixed syllogism inferenc... |
| 3imtr3d 295 | More general version of ~ ... |
| 3imtr4d 296 | More general version of ~ ... |
| 3imtr3g 297 | More general version of ~ ... |
| 3imtr4g 298 | More general version of ~ ... |
| 3bitri 299 | A chained inference from t... |
| 3bitrri 300 | A chained inference from t... |
| 3bitr2i 301 | A chained inference from t... |
| 3bitr2ri 302 | A chained inference from t... |
| 3bitr3i 303 | A chained inference from t... |
| 3bitr3ri 304 | A chained inference from t... |
| 3bitr4i 305 | A chained inference from t... |
| 3bitr4ri 306 | A chained inference from t... |
| 3bitrd 307 | Deduction from transitivit... |
| 3bitrrd 308 | Deduction from transitivit... |
| 3bitr2d 309 | Deduction from transitivit... |
| 3bitr2rd 310 | Deduction from transitivit... |
| 3bitr3d 311 | Deduction from transitivit... |
| 3bitr3rd 312 | Deduction from transitivit... |
| 3bitr4d 313 | Deduction from transitivit... |
| 3bitr4rd 314 | Deduction from transitivit... |
| 3bitr3g 315 | More general version of ~ ... |
| 3bitr4g 316 | More general version of ~ ... |
| notnotb 317 | Double negation. Theorem ... |
| con34b 318 | A biconditional form of co... |
| con4bid 319 | A contraposition deduction... |
| notbid 320 | Deduction negating both si... |
| notbi 321 | Contraposition. Theorem *... |
| notbii 322 | Negate both sides of a log... |
| con4bii 323 | A contraposition inference... |
| mtbi 324 | An inference from a bicond... |
| mtbir 325 | An inference from a bicond... |
| mtbid 326 | A deduction from a bicondi... |
| mtbird 327 | A deduction from a bicondi... |
| mtbii 328 | An inference from a bicond... |
| mtbiri 329 | An inference from a bicond... |
| sylnib 330 | A mixed syllogism inferenc... |
| sylnibr 331 | A mixed syllogism inferenc... |
| sylnbi 332 | A mixed syllogism inferenc... |
| sylnbir 333 | A mixed syllogism inferenc... |
| xchnxbi 334 | Replacement of a subexpres... |
| xchnxbir 335 | Replacement of a subexpres... |
| xchbinx 336 | Replacement of a subexpres... |
| xchbinxr 337 | Replacement of a subexpres... |
| imbi2i 338 | Introduce an antecedent to... |
| bibi2i 339 | Inference adding a bicondi... |
| bibi1i 340 | Inference adding a bicondi... |
| bibi12i 341 | The equivalence of two equ... |
| imbi2d 342 | Deduction adding an antece... |
| imbi1d 343 | Deduction adding a consequ... |
| bibi2d 344 | Deduction adding a bicondi... |
| bibi1d 345 | Deduction adding a bicondi... |
| imbi12d 346 | Deduction joining two equi... |
| bibi12d 347 | Deduction joining two equi... |
| imbi12 348 | Closed form of ~ imbi12i .... |
| imbi1 349 | Theorem *4.84 of [Whitehea... |
| imbi2 350 | Theorem *4.85 of [Whitehea... |
| imbi1i 351 | Introduce a consequent to ... |
| imbi12i 352 | Join two logical equivalen... |
| bibi1 353 | Theorem *4.86 of [Whitehea... |
| bitr3 354 | Closed nested implication ... |
| con2bi 355 | Contraposition. Theorem *... |
| con2bid 356 | A contraposition deduction... |
| con1bid 357 | A contraposition deduction... |
| con1bii 358 | A contraposition inference... |
| con2bii 359 | A contraposition inference... |
| con1b 360 | Contraposition. Bidirecti... |
| con2b 361 | Contraposition. Bidirecti... |
| biimt 362 | A wff is equivalent to its... |
| pm5.5 363 | Theorem *5.5 of [Whitehead... |
| a1bi 364 | Inference introducing a th... |
| mt2bi 365 | A false consequent falsifi... |
| mtt 366 | Modus-tollens-like theorem... |
| imnot 367 | If a proposition is false,... |
| pm5.501 368 | Theorem *5.501 of [Whitehe... |
| ibib 369 | Implication in terms of im... |
| ibibr 370 | Implication in terms of im... |
| tbt 371 | A wff is equivalent to its... |
| nbn2 372 | The negation of a wff is e... |
| bibif 373 | Transfer negation via an e... |
| nbn 374 | The negation of a wff is e... |
| nbn3 375 | Transfer falsehood via equ... |
| pm5.21im 376 | Two propositions are equiv... |
| 2false 377 | Two falsehoods are equival... |
| 2falsed 378 | Two falsehoods are equival... |
| pm5.21ni 379 | Two propositions implying ... |
| pm5.21nii 380 | Eliminate an antecedent im... |
| pm5.21ndd 381 | Eliminate an antecedent im... |
| bija 382 | Combine antecedents into a... |
| pm5.18 383 | Theorem *5.18 of [Whitehea... |
| xor3 384 | Two ways to express "exclu... |
| nbbn 385 | Move negation outside of b... |
| biass 386 | Associative law for the bi... |
| biluk 387 | Lukasiewicz's shortest axi... |
| pm5.19 388 | Theorem *5.19 of [Whitehea... |
| bi2.04 389 | Logical equivalence of com... |
| pm5.4 390 | Antecedent absorption impl... |
| imdi 391 | Distributive law for impli... |
| pm5.41 392 | Theorem *5.41 of [Whitehea... |
| imbibi 393 | The antecedent of one side... |
| pm4.8 394 | Theorem *4.8 of [Whitehead... |
| pm4.81 395 | A formula is equivalent to... |
| imim21b 396 | Simplify an implication be... |
| pm4.63 399 | Theorem *4.63 of [Whitehea... |
| pm4.67 400 | Theorem *4.67 of [Whitehea... |
| imnan 401 | Express an implication in ... |
| imnani 402 | Infer an implication from ... |
| iman 403 | Implication in terms of co... |
| pm3.24 404 | Law of noncontradiction. ... |
| annim 405 | Express a conjunction in t... |
| pm4.61 406 | Theorem *4.61 of [Whitehea... |
| pm4.65 407 | Theorem *4.65 of [Whitehea... |
| imp 408 | Importation inference. (C... |
| impcom 409 | Importation inference with... |
| con3dimp 410 | Variant of ~ con3d with im... |
| mpnanrd 411 | Eliminate the right side o... |
| impd 412 | Importation deduction. (C... |
| impcomd 413 | Importation deduction with... |
| ex 414 | Exportation inference. (T... |
| expcom 415 | Exportation inference with... |
| expdcom 416 | Commuted form of ~ expd . ... |
| expd 417 | Exportation deduction. (C... |
| expcomd 418 | Deduction form of ~ expcom... |
| imp31 419 | An importation inference. ... |
| imp32 420 | An importation inference. ... |
| exp31 421 | An exportation inference. ... |
| exp32 422 | An exportation inference. ... |
| imp4b 423 | An importation inference. ... |
| imp4a 424 | An importation inference. ... |
| imp4c 425 | An importation inference. ... |
| imp4d 426 | An importation inference. ... |
| imp41 427 | An importation inference. ... |
| imp42 428 | An importation inference. ... |
| imp43 429 | An importation inference. ... |
| imp44 430 | An importation inference. ... |
| imp45 431 | An importation inference. ... |
| exp4b 432 | An exportation inference. ... |
| exp4a 433 | An exportation inference. ... |
| exp4c 434 | An exportation inference. ... |
| exp4d 435 | An exportation inference. ... |
| exp41 436 | An exportation inference. ... |
| exp42 437 | An exportation inference. ... |
| exp43 438 | An exportation inference. ... |
| exp44 439 | An exportation inference. ... |
| exp45 440 | An exportation inference. ... |
| imp5d 441 | An importation inference. ... |
| imp5a 442 | An importation inference. ... |
| imp5g 443 | An importation inference. ... |
| imp55 444 | An importation inference. ... |
| imp511 445 | An importation inference. ... |
| exp5c 446 | An exportation inference. ... |
| exp5j 447 | An exportation inference. ... |
| exp5l 448 | An exportation inference. ... |
| exp53 449 | An exportation inference. ... |
| pm3.3 450 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 451 | Theorem *3.31 (Imp) of [Wh... |
| impexp 452 | Import-export theorem. Pa... |
| impancom 453 | Mixed importation/commutat... |
| expdimp 454 | A deduction version of exp... |
| expimpd 455 | Exportation followed by a ... |
| impr 456 | Import a wff into a right ... |
| impl 457 | Export a wff from a left c... |
| expr 458 | Export a wff from a right ... |
| expl 459 | Export a wff from a left c... |
| ancoms 460 | Inference commuting conjun... |
| pm3.22 461 | Theorem *3.22 of [Whitehea... |
| ancom 462 | Commutative law for conjun... |
| ancomd 463 | Commutation of conjuncts i... |
| biancomi 464 | Commuting conjunction in a... |
| biancomd 465 | Commuting conjunction in a... |
| ancomst 466 | Closed form of ~ ancoms . ... |
| ancomsd 467 | Deduction commuting conjun... |
| anasss 468 | Associative law for conjun... |
| anassrs 469 | Associative law for conjun... |
| anass 470 | Associative law for conjun... |
| pm3.2 471 | Join antecedents with conj... |
| pm3.2i 472 | Infer conjunction of premi... |
| pm3.21 473 | Join antecedents with conj... |
| pm3.43i 474 | Nested conjunction of ante... |
| pm3.43 475 | Theorem *3.43 (Comp) of [W... |
| dfbi2 476 | A theorem similar to the s... |
| dfbi 477 | Definition ~ df-bi rewritt... |
| biimpa 478 | Importation inference from... |
| biimpar 479 | Importation inference from... |
| biimpac 480 | Importation inference from... |
| biimparc 481 | Importation inference from... |
| adantr 482 | Inference adding a conjunc... |
| adantl 483 | Inference adding a conjunc... |
| simpl 484 | Elimination of a conjunct.... |
| simpli 485 | Inference eliminating a co... |
| simpr 486 | Elimination of a conjunct.... |
| simpri 487 | Inference eliminating a co... |
| intnan 488 | Introduction of conjunct i... |
| intnanr 489 | Introduction of conjunct i... |
| intnand 490 | Introduction of conjunct i... |
| intnanrd 491 | Introduction of conjunct i... |
| adantld 492 | Deduction adding a conjunc... |
| adantrd 493 | Deduction adding a conjunc... |
| pm3.41 494 | Theorem *3.41 of [Whitehea... |
| pm3.42 495 | Theorem *3.42 of [Whitehea... |
| simpld 496 | Deduction eliminating a co... |
| simprd 497 | Deduction eliminating a co... |
| simplbi 498 | Deduction eliminating a co... |
| simprbi 499 | Deduction eliminating a co... |
| simprbda 500 | Deduction eliminating a co... |
| simplbda 501 | Deduction eliminating a co... |
| simplbi2 502 | Deduction eliminating a co... |
| simplbi2comt 503 | Closed form of ~ simplbi2c... |
| simplbi2com 504 | A deduction eliminating a ... |
| birani 505 | Inference adding a conjunc... |
| bilani 506 | Inference adding a conjunc... |
| biranri 507 | Inference adding a conjunc... |
| bilanri 508 | Inference adding a conjunc... |
| simpl2im 509 | Implication from an elimin... |
| simplbiim 510 | Implication from an elimin... |
| impel 511 | An inference for implicati... |
| mpan9 512 | Modus ponens conjoining di... |
| sylan9 513 | Nested syllogism inference... |
| sylan9r 514 | Nested syllogism inference... |
| sylan9bb 515 | Nested syllogism inference... |
| sylan9bbr 516 | Nested syllogism inference... |
| jca 517 | Deduce conjunction of the ... |
| jcad 518 | Deduction conjoining the c... |
| jca2 519 | Inference conjoining the c... |
| jca31 520 | Join three consequents. (... |
| jca32 521 | Join three consequents. (... |
| jcai 522 | Deduction replacing implic... |
| jcab 523 | Distributive law for impli... |
| pm4.76 524 | Theorem *4.76 of [Whitehea... |
| jctil 525 | Inference conjoining a the... |
| jctir 526 | Inference conjoining a the... |
| jccir 527 | Inference conjoining a con... |
| jccil 528 | Inference conjoining a con... |
| jctl 529 | Inference conjoining a the... |
| jctr 530 | Inference conjoining a the... |
| jctild 531 | Deduction conjoining a the... |
| jctird 532 | Deduction conjoining a the... |
| iba 533 | Introduction of antecedent... |
| ibar 534 | Introduction of antecedent... |
| biantru 535 | A wff is equivalent to its... |
| biantrur 536 | A wff is equivalent to its... |
| biantrud 537 | A wff is equivalent to its... |
| biantrurd 538 | A wff is equivalent to its... |
| bianfi 539 | A wff conjoined with false... |
| bianfd 540 | A wff conjoined with false... |
| baib 541 | Move conjunction outside o... |
| baibr 542 | Move conjunction outside o... |
| rbaibr 543 | Move conjunction outside o... |
| rbaib 544 | Move conjunction outside o... |
| baibd 545 | Move conjunction outside o... |
| rbaibd 546 | Move conjunction outside o... |
| bianabs 547 | Absorb a hypothesis into t... |
| pm5.44 548 | Theorem *5.44 of [Whitehea... |
| pm5.42 549 | Theorem *5.42 of [Whitehea... |
| ancl 550 | Conjoin antecedent to left... |
| anclb 551 | Conjoin antecedent to left... |
| ancr 552 | Conjoin antecedent to righ... |
| ancrb 553 | Conjoin antecedent to righ... |
| ancli 554 | Deduction conjoining antec... |
| ancri 555 | Deduction conjoining antec... |
| ancld 556 | Deduction conjoining antec... |
| ancrd 557 | Deduction conjoining antec... |
| impac 558 | Importation with conjuncti... |
| anc2l 559 | Conjoin antecedent to left... |
| anc2r 560 | Conjoin antecedent to righ... |
| anc2li 561 | Deduction conjoining antec... |
| anc2ri 562 | Deduction conjoining antec... |
| pm4.71 563 | Implication in terms of bi... |
| pm4.71r 564 | Implication in terms of bi... |
| pm4.71i 565 | Inference converting an im... |
| pm4.71ri 566 | Inference converting an im... |
| pm4.71d 567 | Deduction converting an im... |
| pm4.71rd 568 | Deduction converting an im... |
| pm4.24 569 | Theorem *4.24 of [Whitehea... |
| anidm 570 | Idempotent law for conjunc... |
| anidmdbi 571 | Conjunction idempotence wi... |
| anidms 572 | Inference from idempotent ... |
| imdistan 573 | Distribution of implicatio... |
| imdistani 574 | Distribution of implicatio... |
| imdistanri 575 | Distribution of implicatio... |
| imdistand 576 | Distribution of implicatio... |
| imdistanda 577 | Distribution of implicatio... |
| pm5.3 578 | Theorem *5.3 of [Whitehead... |
| pm5.32 579 | Distribution of implicatio... |
| pm5.32i 580 | Distribution of implicatio... |
| pm5.32ri 581 | Distribution of implicatio... |
| bianim 582 | Exchanging conjunction in ... |
| pm5.32d 583 | Distribution of implicatio... |
| pm5.32rd 584 | Distribution of implicatio... |
| pm5.32da 585 | Distribution of implicatio... |
| bian1d 586 | Adding a superfluous conju... |
| sylan 587 | A syllogism inference. (C... |
| sylanb 588 | A syllogism inference. (C... |
| sylanbr 589 | A syllogism inference. (C... |
| sylanbrc 590 | Syllogism inference. (Con... |
| syl2anc 591 | Syllogism inference combin... |
| syl2anc2 592 | Double syllogism inference... |
| sylancl 593 | Syllogism inference combin... |
| sylancr 594 | Syllogism inference combin... |
| sylancom 595 | Syllogism inference with c... |
| sylanblc 596 | Syllogism inference combin... |
| sylanblrc 597 | Syllogism inference combin... |
| syldan 598 | A syllogism deduction with... |
| sylbida 599 | A syllogism deduction. (C... |
| sylan2 600 | A syllogism inference. (C... |
| sylan2b 601 | A syllogism inference. (C... |
| sylan2br 602 | A syllogism inference. (C... |
| syl2an 603 | A double syllogism inferen... |
| syl2anr 604 | A double syllogism inferen... |
| syl2anb 605 | A double syllogism inferen... |
| syl2anbr 606 | A double syllogism inferen... |
| sylancb 607 | A syllogism inference comb... |
| sylancbr 608 | A syllogism inference comb... |
| syldanl 609 | A syllogism deduction with... |
| syland 610 | A syllogism deduction. (C... |
| sylani 611 | A syllogism inference. (C... |
| sylan2d 612 | A syllogism deduction. (C... |
| sylan2i 613 | A syllogism inference. (C... |
| syl2ani 614 | A syllogism inference. (C... |
| syl2and 615 | A syllogism deduction. (C... |
| anim12d 616 | Conjoin antecedents and co... |
| anim12d1 617 | Variant of ~ anim12d where... |
| anim1d 618 | Add a conjunct to right of... |
| anim2d 619 | Add a conjunct to left of ... |
| anim12i 620 | Conjoin antecedents and co... |
| anim12ci 621 | Variant of ~ anim12i with ... |
| anim1i 622 | Introduce conjunct to both... |
| anim1ci 623 | Introduce conjunct to both... |
| anim2i 624 | Introduce conjunct to both... |
| anim12ii 625 | Conjoin antecedents and co... |
| anim12dan 626 | Conjoin antecedents and co... |
| im2anan9 627 | Deduction joining nested i... |
| im2anan9r 628 | Deduction joining nested i... |
| pm3.45 629 | Theorem *3.45 (Fact) of [W... |
| anbi2i 630 | Introduce a left conjunct ... |
| anbi1i 631 | Introduce a right conjunct... |
| anbi2ci 632 | Variant of ~ anbi2i with c... |
| anbi1ci 633 | Variant of ~ anbi1i with c... |
| bianbi 634 | Exchanging conjunction in ... |
| anbi12i 635 | Conjoin both sides of two ... |
| anbi12ci 636 | Variant of ~ anbi12i with ... |
| anbi2d 637 | Deduction adding a left co... |
| anbi1d 638 | Deduction adding a right c... |
| anbi12d 639 | Deduction joining two equi... |
| anbi1 640 | Introduce a right conjunct... |
| anbi2 641 | Introduce a left conjunct ... |
| anbi1cd 642 | Introduce a proposition as... |
| an2anr 643 | Double commutation in conj... |
| pm4.38 644 | Theorem *4.38 of [Whitehea... |
| bi2anan9 645 | Deduction joining two equi... |
| bi2anan9r 646 | Deduction joining two equi... |
| bi2bian9 647 | Deduction joining two bico... |
| anbiim 648 | Adding biconditional when ... |
| bianass 649 | An inference to merge two ... |
| bianassc 650 | An inference to merge two ... |
| an21 651 | Swap two conjuncts. (Cont... |
| an12 652 | Swap two conjuncts. Note ... |
| an32 653 | A rearrangement of conjunc... |
| an13 654 | A rearrangement of conjunc... |
| an31 655 | A rearrangement of conjunc... |
| an12s 656 | Swap two conjuncts in ante... |
| ancom2s 657 | Inference commuting a nest... |
| an13s 658 | Swap two conjuncts in ante... |
| an32s 659 | Swap two conjuncts in ante... |
| ancom1s 660 | Inference commuting a nest... |
| an31s 661 | Swap two conjuncts in ante... |
| anass1rs 662 | Commutative-associative la... |
| an4 663 | Rearrangement of 4 conjunc... |
| an42 664 | Rearrangement of 4 conjunc... |
| an43 665 | Rearrangement of 4 conjunc... |
| an3 666 | A rearrangement of conjunc... |
| an4s 667 | Inference rearranging 4 co... |
| an42s 668 | Inference rearranging 4 co... |
| anabs1 669 | Absorption into embedded c... |
| anabs5 670 | Absorption into embedded c... |
| anabs7 671 | Absorption into embedded c... |
| anabsan 672 | Absorption of antecedent w... |
| anabss1 673 | Absorption of antecedent i... |
| anabss4 674 | Absorption of antecedent i... |
| anabss5 675 | Absorption of antecedent i... |
| anabsi5 676 | Absorption of antecedent i... |
| anabsi6 677 | Absorption of antecedent i... |
| anabsi7 678 | Absorption of antecedent i... |
| anabsi8 679 | Absorption of antecedent i... |
| anabss7 680 | Absorption of antecedent i... |
| anabsan2 681 | Absorption of antecedent w... |
| anabss3 682 | Absorption of antecedent i... |
| anandi 683 | Distribution of conjunctio... |
| anandir 684 | Distribution of conjunctio... |
| anandis 685 | Inference that undistribut... |
| anandirs 686 | Inference that undistribut... |
| sylanl1 687 | A syllogism inference. (C... |
| sylanl2 688 | A syllogism inference. (C... |
| sylanr1 689 | A syllogism inference. (C... |
| sylanr2 690 | A syllogism inference. (C... |
| syl6an 691 | A syllogism deduction comb... |
| syl2an2r 692 | ~ syl2anr with antecedents... |
| syl2an2 693 | ~ syl2an with antecedents ... |
| mpdan 694 | An inference based on modu... |
| mpancom 695 | An inference based on modu... |
| mpidan 696 | A deduction which "stacks"... |
| mpan 697 | An inference based on modu... |
| mpan2 698 | An inference based on modu... |
| mp2an 699 | An inference based on modu... |
| mp4an 700 | An inference based on modu... |
| mpan2d 701 | A deduction based on modus... |
| mpand 702 | A deduction based on modus... |
| mpani 703 | An inference based on modu... |
| mpan2i 704 | An inference based on modu... |
| mp2ani 705 | An inference based on modu... |
| mp2and 706 | A deduction based on modus... |
| mpanl1 707 | An inference based on modu... |
| mpanl2 708 | An inference based on modu... |
| mpanl12 709 | An inference based on modu... |
| mpanr1 710 | An inference based on modu... |
| mpanr2 711 | An inference based on modu... |
| mpanr12 712 | An inference based on modu... |
| mpanlr1 713 | An inference based on modu... |
| mpbirand 714 | Detach truth from conjunct... |
| mpbiran2d 715 | Detach truth from conjunct... |
| mpbiran 716 | Detach truth from conjunct... |
| mpbiran2 717 | Detach truth from conjunct... |
| mpbir2an 718 | Detach a conjunction of tr... |
| mpbi2and 719 | Detach a conjunction of tr... |
| mpbir2and 720 | Detach a conjunction of tr... |
| adantll 721 | Deduction adding a conjunc... |
| adantlr 722 | Deduction adding a conjunc... |
| adantrl 723 | Deduction adding a conjunc... |
| adantrr 724 | Deduction adding a conjunc... |
| adantlll 725 | Deduction adding a conjunc... |
| adantllr 726 | Deduction adding a conjunc... |
| adantlrl 727 | Deduction adding a conjunc... |
| adantlrr 728 | Deduction adding a conjunc... |
| adantrll 729 | Deduction adding a conjunc... |
| adantrlr 730 | Deduction adding a conjunc... |
| adantrrl 731 | Deduction adding a conjunc... |
| adantrrr 732 | Deduction adding a conjunc... |
| ad2antrr 733 | Deduction adding two conju... |
| ad2antlr 734 | Deduction adding two conju... |
| ad2antrl 735 | Deduction adding two conju... |
| ad2antll 736 | Deduction adding conjuncts... |
| ad3antrrr 737 | Deduction adding three con... |
| ad3antlr 738 | Deduction adding three con... |
| ad4antr 739 | Deduction adding 4 conjunc... |
| ad4antlr 740 | Deduction adding 4 conjunc... |
| ad5antr 741 | Deduction adding 5 conjunc... |
| ad5antlr 742 | Deduction adding 5 conjunc... |
| ad6antr 743 | Deduction adding 6 conjunc... |
| ad6antlr 744 | Deduction adding 6 conjunc... |
| ad7antr 745 | Deduction adding 7 conjunc... |
| ad7antlr 746 | Deduction adding 7 conjunc... |
| ad8antr 747 | Deduction adding 8 conjunc... |
| ad8antlr 748 | Deduction adding 8 conjunc... |
| ad9antr 749 | Deduction adding 9 conjunc... |
| ad9antlr 750 | Deduction adding 9 conjunc... |
| ad10antr 751 | Deduction adding 10 conjun... |
| ad10antlr 752 | Deduction adding 10 conjun... |
| ad2ant2l 753 | Deduction adding two conju... |
| ad2ant2r 754 | Deduction adding two conju... |
| ad2ant2lr 755 | Deduction adding two conju... |
| ad2ant2rl 756 | Deduction adding two conju... |
| adantl3r 757 | Deduction adding 1 conjunc... |
| ad4ant13 758 | Deduction adding conjuncts... |
| ad4ant14 759 | Deduction adding conjuncts... |
| ad4ant23 760 | Deduction adding conjuncts... |
| ad4ant24 761 | Deduction adding conjuncts... |
| adantl4r 762 | Deduction adding 1 conjunc... |
| ad5ant13 763 | Deduction adding conjuncts... |
| ad5ant14 764 | Deduction adding conjuncts... |
| ad5ant15 765 | Deduction adding conjuncts... |
| ad5ant23 766 | Deduction adding conjuncts... |
| ad5ant24 767 | Deduction adding conjuncts... |
| ad5ant25 768 | Deduction adding conjuncts... |
| adantl5r 769 | Deduction adding 1 conjunc... |
| adantl6r 770 | Deduction adding 1 conjunc... |
| pm3.33 771 | Theorem *3.33 (Syll) of [W... |
| pm3.34 772 | Theorem *3.34 (Syll) of [W... |
| simpll 773 | Simplification of a conjun... |
| simplld 774 | Deduction form of ~ simpll... |
| simplr 775 | Simplification of a conjun... |
| simplrd 776 | Deduction eliminating a do... |
| simprl 777 | Simplification of a conjun... |
| simprld 778 | Deduction eliminating a do... |
| simprr 779 | Simplification of a conjun... |
| simprrd 780 | Deduction form of ~ simprr... |
| simplll 781 | Simplification of a conjun... |
| simpllr 782 | Simplification of a conjun... |
| simplrl 783 | Simplification of a conjun... |
| simplrr 784 | Simplification of a conjun... |
| simprll 785 | Simplification of a conjun... |
| simprlr 786 | Simplification of a conjun... |
| simprrl 787 | Simplification of a conjun... |
| simprrr 788 | Simplification of a conjun... |
| simp-4l 789 | Simplification of a conjun... |
| simp-4r 790 | Simplification of a conjun... |
| simp-5l 791 | Simplification of a conjun... |
| simp-5r 792 | Simplification of a conjun... |
| simp-6l 793 | Simplification of a conjun... |
| simp-6r 794 | Simplification of a conjun... |
| simp-7l 795 | Simplification of a conjun... |
| simp-7r 796 | Simplification of a conjun... |
| simp-8l 797 | Simplification of a conjun... |
| simp-8r 798 | Simplification of a conjun... |
| simp-9l 799 | Simplification of a conjun... |
| simp-9r 800 | Simplification of a conjun... |
| simp-10l 801 | Simplification of a conjun... |
| simp-10r 802 | Simplification of a conjun... |
| simp-11l 803 | Simplification of a conjun... |
| simp-11r 804 | Simplification of a conjun... |
| pm2.01da 805 | Deduction based on reducti... |
| pm2.18da 806 | Deduction based on reducti... |
| impbida 807 | Deduce an equivalence from... |
| pm5.21nd 808 | Eliminate an antecedent im... |
| pm3.35 809 | Conjunctive detachment. T... |
| pm5.74da 810 | Distribution of implicatio... |
| bitr 811 | Theorem *4.22 of [Whitehea... |
| biantr 812 | A transitive law of equiva... |
| pm4.14 813 | Theorem *4.14 of [Whitehea... |
| pm3.37 814 | Theorem *3.37 (Transp) of ... |
| anim12 815 | Conjoin antecedents and co... |
| pm3.4 816 | Conjunction implies implic... |
| exbiri 817 | Inference form of ~ exbir ... |
| pm2.61ian 818 | Elimination of an antecede... |
| pm2.61dan 819 | Elimination of an antecede... |
| pm2.61ddan 820 | Elimination of two anteced... |
| pm2.61dda 821 | Elimination of two anteced... |
| mtand 822 | A modus tollens deduction.... |
| pm2.65da 823 | Deduction for proof by con... |
| condan 824 | Proof by contradiction. (... |
| biadan 825 | An implication is equivale... |
| biadani 826 | Inference associated with ... |
| biadaniALT 827 | Alternate proof of ~ biada... |
| biadanii 828 | Inference associated with ... |
| biadanid 829 | Deduction associated with ... |
| pm5.1 830 | Two propositions are equiv... |
| pm5.21 831 | Two propositions are equiv... |
| pm5.35 832 | Theorem *5.35 of [Whitehea... |
| abai 833 | Introduce one conjunct as ... |
| pm4.45im 834 | Conjunction with implicati... |
| impimprbi 835 | An implication and its rev... |
| nan 836 | Theorem to move a conjunct... |
| pm5.31 837 | Theorem *5.31 of [Whitehea... |
| pm5.31r 838 | Variant of ~ pm5.31 . (Co... |
| pm4.15 839 | Theorem *4.15 of [Whitehea... |
| pm5.36 840 | Theorem *5.36 of [Whitehea... |
| annotanannot 841 | A conjunction with a negat... |
| pm5.33 842 | Theorem *5.33 of [Whitehea... |
| syl12anc 843 | Syllogism combined with co... |
| syl21anc 844 | Syllogism combined with co... |
| syl22anc 845 | Syllogism combined with co... |
| bibiad 846 | Eliminate an hypothesis ` ... |
| syl1111anc 847 | Four-hypothesis eliminatio... |
| syldbl2 848 | Stacked hypotheseis implie... |
| mpsyl4anc 849 | An elimination deduction. ... |
| pm4.87 850 | Theorem *4.87 of [Whitehea... |
| bimsc1 851 | Removal of conjunct from o... |
| a2and 852 | Deduction distributing a c... |
| animpimp2impd 853 | Deduction deriving nested ... |
| pm4.64 856 | Theorem *4.64 of [Whitehea... |
| pm4.66 857 | Theorem *4.66 of [Whitehea... |
| pm2.53 858 | Theorem *2.53 of [Whitehea... |
| pm2.54 859 | Theorem *2.54 of [Whitehea... |
| imor 860 | Implication in terms of di... |
| imori 861 | Infer disjunction from imp... |
| imorri 862 | Infer implication from dis... |
| pm4.62 863 | Theorem *4.62 of [Whitehea... |
| jaoi 864 | Inference disjoining the a... |
| jao1i 865 | Add a disjunct in the ante... |
| jaod 866 | Deduction disjoining the a... |
| mpjaod 867 | Eliminate a disjunction in... |
| ori 868 | Infer implication from dis... |
| orri 869 | Infer disjunction from imp... |
| orrd 870 | Deduce disjunction from im... |
| ord 871 | Deduce implication from di... |
| orci 872 | Deduction introducing a di... |
| olci 873 | Deduction introducing a di... |
| orc 874 | Introduction of a disjunct... |
| olc 875 | Introduction of a disjunct... |
| pm1.4 876 | Axiom *1.4 of [WhiteheadRu... |
| orcom 877 | Commutative law for disjun... |
| orcomd 878 | Commutation of disjuncts i... |
| orcoms 879 | Commutation of disjuncts i... |
| orcd 880 | Deduction introducing a di... |
| olcd 881 | Deduction introducing a di... |
| orcs 882 | Deduction eliminating disj... |
| olcs 883 | Deduction eliminating disj... |
| olcnd 884 | A lemma for Conjunctive No... |
| orcnd 885 | A lemma for Conjunctive No... |
| mtord 886 | A modus tollens deduction ... |
| pm3.2ni 887 | Infer negated disjunction ... |
| pm2.45 888 | Theorem *2.45 of [Whitehea... |
| pm2.46 889 | Theorem *2.46 of [Whitehea... |
| pm2.47 890 | Theorem *2.47 of [Whitehea... |
| pm2.48 891 | Theorem *2.48 of [Whitehea... |
| pm2.49 892 | Theorem *2.49 of [Whitehea... |
| norbi 893 | If neither of two proposit... |
| nbior 894 | If two propositions are no... |
| orel1 895 | Elimination of disjunction... |
| pm2.25 896 | Theorem *2.25 of [Whitehea... |
| orel2 897 | Elimination of disjunction... |
| pm2.67-2 898 | Slight generalization of T... |
| pm2.67 899 | Theorem *2.67 of [Whitehea... |
| curryax 900 | A non-intuitionistic posit... |
| exmid 901 | Law of excluded middle, al... |
| exmidd 902 | Law of excluded middle in ... |
| pm2.1 903 | Theorem *2.1 of [Whitehead... |
| pm2.13 904 | Theorem *2.13 of [Whitehea... |
| pm2.621 905 | Theorem *2.621 of [Whitehe... |
| pm2.62 906 | Theorem *2.62 of [Whitehea... |
| pm2.68 907 | Theorem *2.68 of [Whitehea... |
| dfor2 908 | Logical 'or' expressed in ... |
| pm2.07 909 | Theorem *2.07 of [Whitehea... |
| pm1.2 910 | Axiom *1.2 of [WhiteheadRu... |
| oridm 911 | Idempotent law for disjunc... |
| pm4.25 912 | Theorem *4.25 of [Whitehea... |
| pm2.4 913 | Theorem *2.4 of [Whitehead... |
| pm2.41 914 | Theorem *2.41 of [Whitehea... |
| orim12i 915 | Disjoin antecedents and co... |
| orim1i 916 | Introduce disjunct to both... |
| orim2i 917 | Introduce disjunct to both... |
| orim12dALT 918 | Alternate proof of ~ orim1... |
| orbi2i 919 | Inference adding a left di... |
| orbi1i 920 | Inference adding a right d... |
| orbi12i 921 | Infer the disjunction of t... |
| orbi2d 922 | Deduction adding a left di... |
| orbi1d 923 | Deduction adding a right d... |
| orbi1 924 | Theorem *4.37 of [Whitehea... |
| orbi12d 925 | Deduction joining two equi... |
| pm1.5 926 | Axiom *1.5 (Assoc) of [Whi... |
| or12 927 | Swap two disjuncts. (Cont... |
| orass 928 | Associative law for disjun... |
| pm2.31 929 | Theorem *2.31 of [Whitehea... |
| pm2.32 930 | Theorem *2.32 of [Whitehea... |
| pm2.3 931 | Theorem *2.3 of [Whitehead... |
| or32 932 | A rearrangement of disjunc... |
| or4 933 | Rearrangement of 4 disjunc... |
| or42 934 | Rearrangement of 4 disjunc... |
| orordi 935 | Distribution of disjunctio... |
| orordir 936 | Distribution of disjunctio... |
| orimdi 937 | Disjunction distributes ov... |
| pm2.76 938 | Theorem *2.76 of [Whitehea... |
| pm2.85 939 | Theorem *2.85 of [Whitehea... |
| pm2.75 940 | Theorem *2.75 of [Whitehea... |
| pm4.78 941 | Implication distributes ov... |
| biort 942 | A disjunction with a true ... |
| biorf 943 | A wff is equivalent to its... |
| biortn 944 | A wff is equivalent to its... |
| biorfi 945 | The dual of ~ biorf is not... |
| biorfri 946 | A wff is equivalent to its... |
| biorfriOLD 947 | Obsolete version of ~ bior... |
| pm2.26 948 | Theorem *2.26 of [Whitehea... |
| pm2.63 949 | Theorem *2.63 of [Whitehea... |
| pm2.64 950 | Theorem *2.64 of [Whitehea... |
| pm2.42 951 | Theorem *2.42 of [Whitehea... |
| pm5.11g 952 | A general instance of Theo... |
| pm5.11 953 | Theorem *5.11 of [Whitehea... |
| pm5.12 954 | Theorem *5.12 of [Whitehea... |
| pm5.14 955 | Theorem *5.14 of [Whitehea... |
| pm5.13 956 | Theorem *5.13 of [Whitehea... |
| pm5.55 957 | Theorem *5.55 of [Whitehea... |
| pm4.72 958 | Implication in terms of bi... |
| imimorb 959 | Simplify an implication be... |
| oibabs 960 | Absorption of disjunction ... |
| orbidi 961 | Disjunction distributes ov... |
| pm5.7 962 | Disjunction distributes ov... |
| jaao 963 | Inference conjoining and d... |
| jaoa 964 | Inference disjoining and c... |
| jaoian 965 | Inference disjoining the a... |
| jaodan 966 | Deduction disjoining the a... |
| mpjaodan 967 | Eliminate a disjunction in... |
| pm3.44 968 | Theorem *3.44 of [Whitehea... |
| jao 969 | Disjunction of antecedents... |
| jaob 970 | Disjunction of antecedents... |
| pm4.77 971 | Theorem *4.77 of [Whitehea... |
| pm3.48 972 | Theorem *3.48 of [Whitehea... |
| orim12d 973 | Disjoin antecedents and co... |
| orim1d 974 | Disjoin antecedents and co... |
| orim2d 975 | Disjoin antecedents and co... |
| orim2 976 | Axiom *1.6 (Sum) of [White... |
| pm2.38 977 | Theorem *2.38 of [Whitehea... |
| pm2.36 978 | Theorem *2.36 of [Whitehea... |
| pm2.37 979 | Theorem *2.37 of [Whitehea... |
| pm2.81 980 | Theorem *2.81 of [Whitehea... |
| pm2.8 981 | Theorem *2.8 of [Whitehead... |
| pm2.73 982 | Theorem *2.73 of [Whitehea... |
| pm2.74 983 | Theorem *2.74 of [Whitehea... |
| pm2.82 984 | Theorem *2.82 of [Whitehea... |
| pm4.39 985 | Theorem *4.39 of [Whitehea... |
| animorl 986 | Conjunction implies disjun... |
| animorr 987 | Conjunction implies disjun... |
| animorlr 988 | Conjunction implies disjun... |
| animorrl 989 | Conjunction implies disjun... |
| ianor 990 | Negated conjunction in ter... |
| anor 991 | Conjunction in terms of di... |
| ioran 992 | Negated disjunction in ter... |
| pm4.52 993 | Theorem *4.52 of [Whitehea... |
| pm4.53 994 | Theorem *4.53 of [Whitehea... |
| pm4.54 995 | Theorem *4.54 of [Whitehea... |
| pm4.55 996 | Theorem *4.55 of [Whitehea... |
| pm4.56 997 | Theorem *4.56 of [Whitehea... |
| oran 998 | Disjunction in terms of co... |
| pm4.57 999 | Theorem *4.57 of [Whitehea... |
| pm3.1 1000 | Theorem *3.1 of [Whitehead... |
| pm3.11 1001 | Theorem *3.11 of [Whitehea... |
| pm3.12 1002 | Theorem *3.12 of [Whitehea... |
| pm3.13 1003 | Theorem *3.13 of [Whitehea... |
| pm3.14 1004 | Theorem *3.14 of [Whitehea... |
| pm4.44 1005 | Theorem *4.44 of [Whitehea... |
| pm4.45 1006 | Theorem *4.45 of [Whitehea... |
| orabs 1007 | Absorption of redundant in... |
| oranabs 1008 | Absorb a disjunct into a c... |
| pm5.61 1009 | Theorem *5.61 of [Whitehea... |
| pm5.6 1010 | Conjunction in antecedent ... |
| orcanai 1011 | Change disjunction in cons... |
| pm4.79 1012 | Theorem *4.79 of [Whitehea... |
| pm5.53 1013 | Theorem *5.53 of [Whitehea... |
| ordi 1014 | Distributive law for disju... |
| ordir 1015 | Distributive law for disju... |
| andi 1016 | Distributive law for conju... |
| andir 1017 | Distributive law for conju... |
| orddi 1018 | Double distributive law fo... |
| anddi 1019 | Double distributive law fo... |
| pm5.17 1020 | Theorem *5.17 of [Whitehea... |
| pm5.15 1021 | Theorem *5.15 of [Whitehea... |
| pm5.16 1022 | Theorem *5.16 of [Whitehea... |
| xor 1023 | Two ways to express exclus... |
| nbi2 1024 | Two ways to express "exclu... |
| xordi 1025 | Conjunction distributes ov... |
| pm5.54 1026 | Theorem *5.54 of [Whitehea... |
| pm5.62 1027 | Theorem *5.62 of [Whitehea... |
| pm5.63 1028 | Theorem *5.63 of [Whitehea... |
| niabn 1029 | Miscellaneous inference re... |
| ninba 1030 | Miscellaneous inference re... |
| pm4.43 1031 | Theorem *4.43 of [Whitehea... |
| pm4.82 1032 | Theorem *4.82 of [Whitehea... |
| pm4.83 1033 | Theorem *4.83 of [Whitehea... |
| pclem6 1034 | Negation inferred from emb... |
| bigolden 1035 | Dijkstra-Scholten's Golden... |
| pm5.71 1036 | Theorem *5.71 of [Whitehea... |
| pm5.75 1037 | Theorem *5.75 of [Whitehea... |
| ecase2d 1038 | Deduction for elimination ... |
| ecase3 1039 | Inference for elimination ... |
| ecase 1040 | Inference for elimination ... |
| ecase3d 1041 | Deduction for elimination ... |
| ecased 1042 | Deduction for elimination ... |
| ecase3ad 1043 | Deduction for elimination ... |
| ccase 1044 | Inference for combining ca... |
| ccased 1045 | Deduction for combining ca... |
| ccase2 1046 | Inference for combining ca... |
| 4cases 1047 | Inference eliminating two ... |
| 4casesdan 1048 | Deduction eliminating two ... |
| cases 1049 | Case disjunction according... |
| dedlem0a 1050 | Lemma for an alternate ver... |
| dedlem0b 1051 | Lemma for an alternate ver... |
| dedlema 1052 | Lemma for weak deduction t... |
| dedlemb 1053 | Lemma for weak deduction t... |
| cases2 1054 | Case disjunction according... |
| cases2ALT 1055 | Alternate proof of ~ cases... |
| dfbi3 1056 | An alternate definition of... |
| pm5.24 1057 | Theorem *5.24 of [Whitehea... |
| 4exmid 1058 | The disjunction of the fou... |
| consensus 1059 | The consensus theorem. Th... |
| pm4.42 1060 | Theorem *4.42 of [Whitehea... |
| prlem1 1061 | A specialized lemma for se... |
| prlem2 1062 | A specialized lemma for se... |
| oplem1 1063 | A specialized lemma for se... |
| dn1 1064 | A single axiom for Boolean... |
| bianir 1065 | A closed form of ~ mpbir ,... |
| jaoi2 1066 | Inference removing a negat... |
| jaoi3 1067 | Inference separating a dis... |
| ornld 1068 | Selecting one statement fr... |
| dfifp2 1071 | Alternate definition of th... |
| dfifp3 1072 | Alternate definition of th... |
| dfifp4 1073 | Alternate definition of th... |
| dfifp5 1074 | Alternate definition of th... |
| dfifp6 1075 | Alternate definition of th... |
| dfifp7 1076 | Alternate definition of th... |
| ifpdfbi 1077 | Define the biconditional a... |
| anifp 1078 | The conditional operator i... |
| ifpor 1079 | The conditional operator i... |
| ifpn 1080 | Conditional operator for t... |
| ifptru 1081 | Value of the conditional o... |
| ifpfal 1082 | Value of the conditional o... |
| ifpid 1083 | Value of the conditional o... |
| casesifp 1084 | Version of ~ cases express... |
| ifpbi123d 1085 | Equivalence deduction for ... |
| ifpbi23d 1086 | Equivalence deduction for ... |
| ifpimpda 1087 | Separation of the values o... |
| 1fpid3 1088 | The value of the condition... |
| elimh 1089 | Hypothesis builder for the... |
| dedt 1090 | The weak deduction theorem... |
| con3ALT 1091 | Proof of ~ con3 from its a... |
| 3orass 1096 | Associative law for triple... |
| 3orel1 1097 | Partial elimination of a t... |
| 3orrot 1098 | Rotation law for triple di... |
| 3orcoma 1099 | Commutation law for triple... |
| 3orcomb 1100 | Commutation law for triple... |
| 3anass 1101 | Associative law for triple... |
| 3anan12 1102 | Convert triple conjunction... |
| 3anan32 1103 | Convert triple conjunction... |
| 3ancoma 1104 | Commutation law for triple... |
| 3ancomb 1105 | Commutation law for triple... |
| 3anrot 1106 | Rotation law for triple co... |
| 3anrev 1107 | Reversal law for triple co... |
| anandi3 1108 | Distribution of triple con... |
| anandi3r 1109 | Distribution of triple con... |
| 3anidm 1110 | Idempotent law for conjunc... |
| 3an4anass 1111 | Associative law for four c... |
| 3ioran 1112 | Negated triple disjunction... |
| 3ianor 1113 | Negated triple conjunction... |
| 3anor 1114 | Triple conjunction express... |
| 3oran 1115 | Triple disjunction in term... |
| 3impa 1116 | Importation from double to... |
| 3imp 1117 | Importation inference. (C... |
| 3imp31 1118 | The importation inference ... |
| 3imp231 1119 | Importation inference. (C... |
| 3imp21 1120 | The importation inference ... |
| 3impb 1121 | Importation from double to... |
| bi23imp13 1122 | ~ 3imp with middle implica... |
| 3impib 1123 | Importation to triple conj... |
| 3impia 1124 | Importation to triple conj... |
| 3expa 1125 | Exportation from triple to... |
| 3exp 1126 | Exportation inference. (C... |
| 3expb 1127 | Exportation from triple to... |
| 3expia 1128 | Exportation from triple co... |
| 3expib 1129 | Exportation from triple co... |
| 3com12 1130 | Commutation in antecedent.... |
| 3com13 1131 | Commutation in antecedent.... |
| 3comr 1132 | Commutation in antecedent.... |
| 3com23 1133 | Commutation in antecedent.... |
| 3coml 1134 | Commutation in antecedent.... |
| 3jca 1135 | Join consequents with conj... |
| 3jcad 1136 | Deduction conjoining the c... |
| 3adant1 1137 | Deduction adding a conjunc... |
| 3adant2 1138 | Deduction adding a conjunc... |
| 3adant3 1139 | Deduction adding a conjunc... |
| 3ad2ant1 1140 | Deduction adding conjuncts... |
| 3ad2ant2 1141 | Deduction adding conjuncts... |
| 3ad2ant3 1142 | Deduction adding conjuncts... |
| simp1 1143 | Simplification of triple c... |
| simp2 1144 | Simplification of triple c... |
| simp3 1145 | Simplification of triple c... |
| simp1i 1146 | Infer a conjunct from a tr... |
| simp2i 1147 | Infer a conjunct from a tr... |
| simp3i 1148 | Infer a conjunct from a tr... |
| simp1d 1149 | Deduce a conjunct from a t... |
| simp2d 1150 | Deduce a conjunct from a t... |
| simp3d 1151 | Deduce a conjunct from a t... |
| simp1bi 1152 | Deduce a conjunct from a t... |
| simp2bi 1153 | Deduce a conjunct from a t... |
| simp3bi 1154 | Deduce a conjunct from a t... |
| 3simpa 1155 | Simplification of triple c... |
| 3simpb 1156 | Simplification of triple c... |
| 3simpc 1157 | Simplification of triple c... |
| 3anim123i 1158 | Join antecedents and conse... |
| 3anim1i 1159 | Add two conjuncts to antec... |
| 3anim2i 1160 | Add two conjuncts to antec... |
| 3anim3i 1161 | Add two conjuncts to antec... |
| 3anbi123i 1162 | Join 3 biconditionals with... |
| 3orbi123i 1163 | Join 3 biconditionals with... |
| 3anbi1i 1164 | Inference adding two conju... |
| 3anbi2i 1165 | Inference adding two conju... |
| 3anbi3i 1166 | Inference adding two conju... |
| syl3an 1167 | A triple syllogism inferen... |
| syl3anb 1168 | A triple syllogism inferen... |
| syl3anbr 1169 | A triple syllogism inferen... |
| syl3an1 1170 | A syllogism inference. (C... |
| syl3an2 1171 | A syllogism inference. (C... |
| syl3an3 1172 | A syllogism inference. (C... |
| syl3an132 1173 | ~ syl2an with antecedents ... |
| 3adantl1 1174 | Deduction adding a conjunc... |
| 3adantl2 1175 | Deduction adding a conjunc... |
| 3adantl3 1176 | Deduction adding a conjunc... |
| 3adantr1 1177 | Deduction adding a conjunc... |
| 3adantr2 1178 | Deduction adding a conjunc... |
| 3adantr3 1179 | Deduction adding a conjunc... |
| ad4ant123 1180 | Deduction adding conjuncts... |
| ad4ant124 1181 | Deduction adding conjuncts... |
| ad4ant134 1182 | Deduction adding conjuncts... |
| ad4ant234 1183 | Deduction adding conjuncts... |
| 3adant1l 1184 | Deduction adding a conjunc... |
| 3adant1r 1185 | Deduction adding a conjunc... |
| 3adant2l 1186 | Deduction adding a conjunc... |
| 3adant2r 1187 | Deduction adding a conjunc... |
| 3adant3l 1188 | Deduction adding a conjunc... |
| 3adant3r 1189 | Deduction adding a conjunc... |
| 3adant3r1 1190 | Deduction adding a conjunc... |
| 3adant3r2 1191 | Deduction adding a conjunc... |
| 3adant3r3 1192 | Deduction adding a conjunc... |
| 3ad2antl1 1193 | Deduction adding conjuncts... |
| 3ad2antl2 1194 | Deduction adding conjuncts... |
| 3ad2antl3 1195 | Deduction adding conjuncts... |
| 3ad2antr1 1196 | Deduction adding conjuncts... |
| 3ad2antr2 1197 | Deduction adding conjuncts... |
| 3ad2antr3 1198 | Deduction adding conjuncts... |
| simpl1 1199 | Simplification of conjunct... |
| simpl2 1200 | Simplification of conjunct... |
| simpl3 1201 | Simplification of conjunct... |
| simpr1 1202 | Simplification of conjunct... |
| simpr2 1203 | Simplification of conjunct... |
| simpr3 1204 | Simplification of conjunct... |
| simp1l 1205 | Simplification of triple c... |
| simp1r 1206 | Simplification of triple c... |
| simp2l 1207 | Simplification of triple c... |
| simp2r 1208 | Simplification of triple c... |
| simp3l 1209 | Simplification of triple c... |
| simp3r 1210 | Simplification of triple c... |
| simp11 1211 | Simplification of doubly t... |
| simp12 1212 | Simplification of doubly t... |
| simp13 1213 | Simplification of doubly t... |
| simp21 1214 | Simplification of doubly t... |
| simp22 1215 | Simplification of doubly t... |
| simp23 1216 | Simplification of doubly t... |
| simp31 1217 | Simplification of doubly t... |
| simp32 1218 | Simplification of doubly t... |
| simp33 1219 | Simplification of doubly t... |
| simpll1 1220 | Simplification of conjunct... |
| simpll2 1221 | Simplification of conjunct... |
| simpll3 1222 | Simplification of conjunct... |
| simplr1 1223 | Simplification of conjunct... |
| simplr2 1224 | Simplification of conjunct... |
| simplr3 1225 | Simplification of conjunct... |
| simprl1 1226 | Simplification of conjunct... |
| simprl2 1227 | Simplification of conjunct... |
| simprl3 1228 | Simplification of conjunct... |
| simprr1 1229 | Simplification of conjunct... |
| simprr2 1230 | Simplification of conjunct... |
| simprr3 1231 | Simplification of conjunct... |
| simpl1l 1232 | Simplification of conjunct... |
| simpl1r 1233 | Simplification of conjunct... |
| simpl2l 1234 | Simplification of conjunct... |
| simpl2r 1235 | Simplification of conjunct... |
| simpl3l 1236 | Simplification of conjunct... |
| simpl3r 1237 | Simplification of conjunct... |
| simpr1l 1238 | Simplification of conjunct... |
| simpr1r 1239 | Simplification of conjunct... |
| simpr2l 1240 | Simplification of conjunct... |
| simpr2r 1241 | Simplification of conjunct... |
| simpr3l 1242 | Simplification of conjunct... |
| simpr3r 1243 | Simplification of conjunct... |
| simp1ll 1244 | Simplification of conjunct... |
| simp1lr 1245 | Simplification of conjunct... |
| simp1rl 1246 | Simplification of conjunct... |
| simp1rr 1247 | Simplification of conjunct... |
| simp2ll 1248 | Simplification of conjunct... |
| simp2lr 1249 | Simplification of conjunct... |
| simp2rl 1250 | Simplification of conjunct... |
| simp2rr 1251 | Simplification of conjunct... |
| simp3ll 1252 | Simplification of conjunct... |
| simp3lr 1253 | Simplification of conjunct... |
| simp3rl 1254 | Simplification of conjunct... |
| simp3rr 1255 | Simplification of conjunct... |
| simpl11 1256 | Simplification of conjunct... |
| simpl12 1257 | Simplification of conjunct... |
| simpl13 1258 | Simplification of conjunct... |
| simpl21 1259 | Simplification of conjunct... |
| simpl22 1260 | Simplification of conjunct... |
| simpl23 1261 | Simplification of conjunct... |
| simpl31 1262 | Simplification of conjunct... |
| simpl32 1263 | Simplification of conjunct... |
| simpl33 1264 | Simplification of conjunct... |
| simpr11 1265 | Simplification of conjunct... |
| simpr12 1266 | Simplification of conjunct... |
| simpr13 1267 | Simplification of conjunct... |
| simpr21 1268 | Simplification of conjunct... |
| simpr22 1269 | Simplification of conjunct... |
| simpr23 1270 | Simplification of conjunct... |
| simpr31 1271 | Simplification of conjunct... |
| simpr32 1272 | Simplification of conjunct... |
| simpr33 1273 | Simplification of conjunct... |
| simp1l1 1274 | Simplification of conjunct... |
| simp1l2 1275 | Simplification of conjunct... |
| simp1l3 1276 | Simplification of conjunct... |
| simp1r1 1277 | Simplification of conjunct... |
| simp1r2 1278 | Simplification of conjunct... |
| simp1r3 1279 | Simplification of conjunct... |
| simp2l1 1280 | Simplification of conjunct... |
| simp2l2 1281 | Simplification of conjunct... |
| simp2l3 1282 | Simplification of conjunct... |
| simp2r1 1283 | Simplification of conjunct... |
| simp2r2 1284 | Simplification of conjunct... |
| simp2r3 1285 | Simplification of conjunct... |
| simp3l1 1286 | Simplification of conjunct... |
| simp3l2 1287 | Simplification of conjunct... |
| simp3l3 1288 | Simplification of conjunct... |
| simp3r1 1289 | Simplification of conjunct... |
| simp3r2 1290 | Simplification of conjunct... |
| simp3r3 1291 | Simplification of conjunct... |
| simp11l 1292 | Simplification of conjunct... |
| simp11r 1293 | Simplification of conjunct... |
| simp12l 1294 | Simplification of conjunct... |
| simp12r 1295 | Simplification of conjunct... |
| simp13l 1296 | Simplification of conjunct... |
| simp13r 1297 | Simplification of conjunct... |
| simp21l 1298 | Simplification of conjunct... |
| simp21r 1299 | Simplification of conjunct... |
| simp22l 1300 | Simplification of conjunct... |
| simp22r 1301 | Simplification of conjunct... |
| simp23l 1302 | Simplification of conjunct... |
| simp23r 1303 | Simplification of conjunct... |
| simp31l 1304 | Simplification of conjunct... |
| simp31r 1305 | Simplification of conjunct... |
| simp32l 1306 | Simplification of conjunct... |
| simp32r 1307 | Simplification of conjunct... |
| simp33l 1308 | Simplification of conjunct... |
| simp33r 1309 | Simplification of conjunct... |
| simp111 1310 | Simplification of conjunct... |
| simp112 1311 | Simplification of conjunct... |
| simp113 1312 | Simplification of conjunct... |
| simp121 1313 | Simplification of conjunct... |
| simp122 1314 | Simplification of conjunct... |
| simp123 1315 | Simplification of conjunct... |
| simp131 1316 | Simplification of conjunct... |
| simp132 1317 | Simplification of conjunct... |
| simp133 1318 | Simplification of conjunct... |
| simp211 1319 | Simplification of conjunct... |
| simp212 1320 | Simplification of conjunct... |
| simp213 1321 | Simplification of conjunct... |
| simp221 1322 | Simplification of conjunct... |
| simp222 1323 | Simplification of conjunct... |
| simp223 1324 | Simplification of conjunct... |
| simp231 1325 | Simplification of conjunct... |
| simp232 1326 | Simplification of conjunct... |
| simp233 1327 | Simplification of conjunct... |
| simp311 1328 | Simplification of conjunct... |
| simp312 1329 | Simplification of conjunct... |
| simp313 1330 | Simplification of conjunct... |
| simp321 1331 | Simplification of conjunct... |
| simp322 1332 | Simplification of conjunct... |
| simp323 1333 | Simplification of conjunct... |
| simp331 1334 | Simplification of conjunct... |
| simp332 1335 | Simplification of conjunct... |
| simp333 1336 | Simplification of conjunct... |
| 3anibar 1337 | Remove a hypothesis from t... |
| 3mix1 1338 | Introduction in triple dis... |
| 3mix2 1339 | Introduction in triple dis... |
| 3mix3 1340 | Introduction in triple dis... |
| 3mix1i 1341 | Introduction in triple dis... |
| 3mix2i 1342 | Introduction in triple dis... |
| 3mix3i 1343 | Introduction in triple dis... |
| 3mix1d 1344 | Deduction introducing trip... |
| 3mix2d 1345 | Deduction introducing trip... |
| 3mix3d 1346 | Deduction introducing trip... |
| 3pm3.2i 1347 | Infer conjunction of premi... |
| pm3.2an3 1348 | Version of ~ pm3.2 for a t... |
| mpbir3an 1349 | Detach a conjunction of tr... |
| mpbir3and 1350 | Detach a conjunction of tr... |
| syl3anbrc 1351 | Syllogism inference. (Con... |
| syl21anbrc 1352 | Syllogism inference. (Con... |
| 3imp3i2an 1353 | An elimination deduction. ... |
| ex3 1354 | Apply ~ ex to a hypothesis... |
| 3imp1 1355 | Importation to left triple... |
| 3impd 1356 | Importation deduction for ... |
| 3imp2 1357 | Importation to right tripl... |
| 3impdi 1358 | Importation inference (und... |
| 3impdir 1359 | Importation inference (und... |
| 3exp1 1360 | Exportation from left trip... |
| 3expd 1361 | Exportation deduction for ... |
| 3exp2 1362 | Exportation from right tri... |
| exp5o 1363 | A triple exportation infer... |
| exp516 1364 | A triple exportation infer... |
| exp520 1365 | A triple exportation infer... |
| 3impexp 1366 | Version of ~ impexp for a ... |
| 3an1rs 1367 | Swap conjuncts. (Contribu... |
| 3anassrs 1368 | Associative law for conjun... |
| 4anpull2 1369 | An equivalence of two four... |
| ad5ant245 1370 | Deduction adding conjuncts... |
| ad5ant234 1371 | Deduction adding conjuncts... |
| ad5ant235 1372 | Deduction adding conjuncts... |
| ad5ant123 1373 | Deduction adding conjuncts... |
| ad5ant124 1374 | Deduction adding conjuncts... |
| ad5ant125 1375 | Deduction adding conjuncts... |
| ad5ant134 1376 | Deduction adding conjuncts... |
| ad5ant135 1377 | Deduction adding conjuncts... |
| ad5ant145 1378 | Deduction adding conjuncts... |
| ad5ant2345 1379 | Deduction adding conjuncts... |
| syl3anc 1380 | Syllogism combined with co... |
| syl13anc 1381 | Syllogism combined with co... |
| syl31anc 1382 | Syllogism combined with co... |
| syl112anc 1383 | Syllogism combined with co... |
| syl121anc 1384 | Syllogism combined with co... |
| syl211anc 1385 | Syllogism combined with co... |
| syl23anc 1386 | Syllogism combined with co... |
| syl32anc 1387 | Syllogism combined with co... |
| syl122anc 1388 | Syllogism combined with co... |
| syl212anc 1389 | Syllogism combined with co... |
| syl221anc 1390 | Syllogism combined with co... |
| syl113anc 1391 | Syllogism combined with co... |
| syl131anc 1392 | Syllogism combined with co... |
| syl311anc 1393 | Syllogism combined with co... |
| syl33anc 1394 | Syllogism combined with co... |
| syl222anc 1395 | Syllogism combined with co... |
| syl123anc 1396 | Syllogism combined with co... |
| syl132anc 1397 | Syllogism combined with co... |
| syl213anc 1398 | Syllogism combined with co... |
| syl231anc 1399 | Syllogism combined with co... |
| syl312anc 1400 | Syllogism combined with co... |
| syl321anc 1401 | Syllogism combined with co... |
| syl133anc 1402 | Syllogism combined with co... |
| syl313anc 1403 | Syllogism combined with co... |
| syl331anc 1404 | Syllogism combined with co... |
| syl223anc 1405 | Syllogism combined with co... |
| syl232anc 1406 | Syllogism combined with co... |
| syl322anc 1407 | Syllogism combined with co... |
| syl233anc 1408 | Syllogism combined with co... |
| syl323anc 1409 | Syllogism combined with co... |
| syl332anc 1410 | Syllogism combined with co... |
| syl333anc 1411 | A syllogism inference comb... |
| syl3an1b 1412 | A syllogism inference. (C... |
| syl3an2b 1413 | A syllogism inference. (C... |
| syl3an3b 1414 | A syllogism inference. (C... |
| syl3an1br 1415 | A syllogism inference. (C... |
| syl3an2br 1416 | A syllogism inference. (C... |
| syl3an3br 1417 | A syllogism inference. (C... |
| syld3an3 1418 | A syllogism inference. (C... |
| syld3an1 1419 | A syllogism inference. (C... |
| syld3an2 1420 | A syllogism inference. (C... |
| syl3anl1 1421 | A syllogism inference. (C... |
| syl3anl2 1422 | A syllogism inference. (C... |
| syl3anl3 1423 | A syllogism inference. (C... |
| syl3anl 1424 | A triple syllogism inferen... |
| syl3anr1 1425 | A syllogism inference. (C... |
| syl3anr2 1426 | A syllogism inference. (C... |
| syl3anr3 1427 | A syllogism inference. (C... |
| 3anidm12 1428 | Inference from idempotent ... |
| 3anidm13 1429 | Inference from idempotent ... |
| 3anidm23 1430 | Inference from idempotent ... |
| syl2an3an 1431 | ~ syl3an with antecedents ... |
| syl2an23an 1432 | Deduction related to ~ syl... |
| 3ori 1433 | Infer implication from tri... |
| 3jao 1434 | Disjunction of three antec... |
| 3jaob 1435 | Disjunction of three antec... |
| 3jaobOLD 1436 | Obsolete version of ~ 3jao... |
| 3jaoi 1437 | Disjunction of three antec... |
| 3jaod 1438 | Disjunction of three antec... |
| 3jaoian 1439 | Disjunction of three antec... |
| 3jaodan 1440 | Disjunction of three antec... |
| mpjao3dan 1441 | Eliminate a three-way disj... |
| 3jaao 1442 | Inference conjoining and d... |
| syl3an9b 1443 | Nested syllogism inference... |
| 3orbi123d 1444 | Deduction joining 3 equiva... |
| 3anbi123d 1445 | Deduction joining 3 equiva... |
| 3anbi12d 1446 | Deduction conjoining and a... |
| 3anbi13d 1447 | Deduction conjoining and a... |
| 3anbi23d 1448 | Deduction conjoining and a... |
| 3anbi1d 1449 | Deduction adding conjuncts... |
| 3anbi2d 1450 | Deduction adding conjuncts... |
| 3anbi3d 1451 | Deduction adding conjuncts... |
| 3anim123d 1452 | Deduction joining 3 implic... |
| 3orim123d 1453 | Deduction joining 3 implic... |
| an6 1454 | Rearrangement of 6 conjunc... |
| 3an6 1455 | Analogue of ~ an4 for trip... |
| 3or6 1456 | Analogue of ~ or4 for trip... |
| mp3an1 1457 | An inference based on modu... |
| mp3an2 1458 | An inference based on modu... |
| mp3an3 1459 | An inference based on modu... |
| mp3an12 1460 | An inference based on modu... |
| mp3an13 1461 | An inference based on modu... |
| mp3an23 1462 | An inference based on modu... |
| mp3an1i 1463 | An inference based on modu... |
| mp3anl1 1464 | An inference based on modu... |
| mp3anl2 1465 | An inference based on modu... |
| mp3anl3 1466 | An inference based on modu... |
| mp3anr1 1467 | An inference based on modu... |
| mp3anr2 1468 | An inference based on modu... |
| mp3anr3 1469 | An inference based on modu... |
| mp3an 1470 | An inference based on modu... |
| mpd3an3 1471 | An inference based on modu... |
| mpd3an23 1472 | An inference based on modu... |
| mp3and 1473 | A deduction based on modus... |
| mp3an12i 1474 | ~ mp3an with antecedents i... |
| mp3an2i 1475 | ~ mp3an with antecedents i... |
| mp3an3an 1476 | ~ mp3an with antecedents i... |
| mp3an2ani 1477 | An elimination deduction. ... |
| biimp3a 1478 | Infer implication from a l... |
| biimp3ar 1479 | Infer implication from a l... |
| 3anandis 1480 | Inference that undistribut... |
| 3anandirs 1481 | Inference that undistribut... |
| ecase23d 1482 | Deduction for elimination ... |
| 3ecase 1483 | Inference for elimination ... |
| 3bior1fd 1484 | A disjunction is equivalen... |
| 3bior1fand 1485 | A disjunction is equivalen... |
| 3bior2fd 1486 | A wff is equivalent to its... |
| 3biant1d 1487 | A conjunction is equivalen... |
| intn3an1d 1488 | Introduction of a triple c... |
| intn3an2d 1489 | Introduction of a triple c... |
| intn3an3d 1490 | Introduction of a triple c... |
| an3andi 1491 | Distribution of conjunctio... |
| an33rean 1492 | Rearrange a 9-fold conjunc... |
| 3orel2 1493 | Partial elimination of a t... |
| 3orel2OLD 1494 | Obsolete version of ~ 3ore... |
| 3orel3 1495 | Partial elimination of a t... |
| 3orel13 1496 | Elimination of two disjunc... |
| 3pm3.2ni 1497 | Triple negated disjunction... |
| an42ds 1498 | Inference exchanging the l... |
| nanan 1501 | Conjunction in terms of al... |
| dfnan2 1502 | Alternative denial in term... |
| nanor 1503 | Alternative denial in term... |
| nancom 1504 | Alternative denial is comm... |
| nannan 1505 | Nested alternative denials... |
| nanim 1506 | Implication in terms of al... |
| nannot 1507 | Negation in terms of alter... |
| nanbi 1508 | Biconditional in terms of ... |
| nanbi1 1509 | Introduce a right anti-con... |
| nanbi2 1510 | Introduce a left anti-conj... |
| nanbi12 1511 | Join two logical equivalen... |
| nanbi1i 1512 | Introduce a right anti-con... |
| nanbi2i 1513 | Introduce a left anti-conj... |
| nanbi12i 1514 | Join two logical equivalen... |
| nanbi1d 1515 | Introduce a right anti-con... |
| nanbi2d 1516 | Introduce a left anti-conj... |
| nanbi12d 1517 | Join two logical equivalen... |
| nanass 1518 | A characterization of when... |
| xnor 1521 | Two ways to write XNOR (ex... |
| xorcom 1522 | The connector ` \/_ ` is c... |
| xorass 1523 | The connector ` \/_ ` is a... |
| excxor 1524 | This tautology shows that ... |
| xor2 1525 | Two ways to express "exclu... |
| xoror 1526 | Exclusive disjunction impl... |
| xornan 1527 | Exclusive disjunction impl... |
| xornan2 1528 | XOR implies NAND (written ... |
| xorneg2 1529 | The connector ` \/_ ` is n... |
| xorneg1 1530 | The connector ` \/_ ` is n... |
| xorneg 1531 | The connector ` \/_ ` is u... |
| xorbi12i 1532 | Equality property for excl... |
| xorbi12d 1533 | Equality property for excl... |
| anxordi 1534 | Conjunction distributes ov... |
| xorexmid 1535 | Exclusive-or variant of th... |
| norcom 1538 | The connector ` -\/ ` is c... |
| nornot 1539 | ` -. ` is expressible via ... |
| noran 1540 | ` /\ ` is expressible via ... |
| noror 1541 | ` \/ ` is expressible via ... |
| norasslem1 1542 | This lemma shows the equiv... |
| norasslem2 1543 | This lemma specializes ~ b... |
| norasslem3 1544 | This lemma specializes ~ b... |
| norass 1545 | A characterization of when... |
| trujust 1550 | Soundness justification th... |
| tru 1552 | The truth value ` T. ` is ... |
| dftru2 1553 | An alternate definition of... |
| trut 1554 | A proposition is equivalen... |
| mptru 1555 | Eliminate ` T. ` as an ant... |
| tbtru 1556 | A proposition is equivalen... |
| bitru 1557 | A theorem is equivalent to... |
| trud 1558 | Anything implies ` T. ` . ... |
| truan 1559 | True can be removed from a... |
| fal 1562 | The truth value ` F. ` is ... |
| nbfal 1563 | The negation of a proposit... |
| bifal 1564 | A contradiction is equival... |
| falim 1565 | The truth value ` F. ` imp... |
| falimd 1566 | The truth value ` F. ` imp... |
| dfnot 1567 | Given falsum ` F. ` , we c... |
| inegd 1568 | Negation introduction rule... |
| efald 1569 | Deduction based on reducti... |
| pm2.21fal 1570 | If a wff and its negation ... |
| truimtru 1571 | A ` -> ` identity. (Contr... |
| truimfal 1572 | A ` -> ` identity. (Contr... |
| falimtru 1573 | A ` -> ` identity. (Contr... |
| falimfal 1574 | A ` -> ` identity. (Contr... |
| nottru 1575 | A ` -. ` identity. (Contr... |
| notfal 1576 | A ` -. ` identity. (Contr... |
| trubitru 1577 | A ` <-> ` identity. (Cont... |
| falbitru 1578 | A ` <-> ` identity. (Cont... |
| trubifal 1579 | A ` <-> ` identity. (Cont... |
| falbifal 1580 | A ` <-> ` identity. (Cont... |
| truantru 1581 | A ` /\ ` identity. (Contr... |
| truanfal 1582 | A ` /\ ` identity. (Contr... |
| falantru 1583 | A ` /\ ` identity. (Contr... |
| falanfal 1584 | A ` /\ ` identity. (Contr... |
| truortru 1585 | A ` \/ ` identity. (Contr... |
| truorfal 1586 | A ` \/ ` identity. (Contr... |
| falortru 1587 | A ` \/ ` identity. (Contr... |
| falorfal 1588 | A ` \/ ` identity. (Contr... |
| trunantru 1589 | A ` -/\ ` identity. (Cont... |
| trunanfal 1590 | A ` -/\ ` identity. (Cont... |
| falnantru 1591 | A ` -/\ ` identity. (Cont... |
| falnanfal 1592 | A ` -/\ ` identity. (Cont... |
| truxortru 1593 | A ` \/_ ` identity. (Cont... |
| truxorfal 1594 | A ` \/_ ` identity. (Cont... |
| falxortru 1595 | A ` \/_ ` identity. (Cont... |
| falxorfal 1596 | A ` \/_ ` identity. (Cont... |
| trunortru 1597 | A ` -\/ ` identity. (Cont... |
| trunorfal 1598 | A ` -\/ ` identity. (Cont... |
| falnortru 1599 | A ` -\/ ` identity. (Cont... |
| falnorfal 1600 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1603 | Equality theorem for the a... |
| hadbi123i 1604 | Equality theorem for the a... |
| hadass 1605 | Associative law for the ad... |
| hadbi 1606 | The adder sum is the same ... |
| hadcoma 1607 | Commutative law for the ad... |
| hadcomb 1608 | Commutative law for the ad... |
| hadrot 1609 | Rotation law for the adder... |
| hadnot 1610 | The adder sum distributes ... |
| had1 1611 | If the first input is true... |
| had0 1612 | If the first input is fals... |
| hadifp 1613 | The value of the adder sum... |
| cador 1616 | The adder carry in disjunc... |
| cadan 1617 | The adder carry in conjunc... |
| cadbi123d 1618 | Equality theorem for the a... |
| cadbi123i 1619 | Equality theorem for the a... |
| cadcoma 1620 | Commutative law for the ad... |
| cadcomb 1621 | Commutative law for the ad... |
| cadrot 1622 | Rotation law for the adder... |
| cadnot 1623 | The adder carry distribute... |
| cad11 1624 | If (at least) two inputs a... |
| cad1 1625 | If one input is true, then... |
| cad0 1626 | If one input is false, the... |
| cadifp 1627 | The value of the carry is,... |
| cadtru 1628 | The adder carry is true as... |
| minimp 1629 | A single axiom for minimal... |
| minimp-syllsimp 1630 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1631 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1632 | Derivation of a commuted f... |
| minimp-ax2 1633 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1634 | Derivation of ~ pm2.43 (al... |
| impsingle 1635 | The shortest single axiom ... |
| impsingle-step4 1636 | Derivation of impsingle-st... |
| impsingle-step8 1637 | Derivation of impsingle-st... |
| impsingle-ax1 1638 | Derivation of impsingle-ax... |
| impsingle-step15 1639 | Derivation of impsingle-st... |
| impsingle-step18 1640 | Derivation of impsingle-st... |
| impsingle-step19 1641 | Derivation of impsingle-st... |
| impsingle-step20 1642 | Derivation of impsingle-st... |
| impsingle-step21 1643 | Derivation of impsingle-st... |
| impsingle-step22 1644 | Derivation of impsingle-st... |
| impsingle-step25 1645 | Derivation of impsingle-st... |
| impsingle-imim1 1646 | Derivation of impsingle-im... |
| impsingle-peirce 1647 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1648 | Derivation of ~ ax-2 from ... |
| meredith 1649 | Carew Meredith's sole axio... |
| merlem1 1650 | Step 3 of Meredith's proof... |
| merlem2 1651 | Step 4 of Meredith's proof... |
| merlem3 1652 | Step 7 of Meredith's proof... |
| merlem4 1653 | Step 8 of Meredith's proof... |
| merlem5 1654 | Step 11 of Meredith's proo... |
| merlem6 1655 | Step 12 of Meredith's proo... |
| merlem7 1656 | Between steps 14 and 15 of... |
| merlem8 1657 | Step 15 of Meredith's proo... |
| merlem9 1658 | Step 18 of Meredith's proo... |
| merlem10 1659 | Step 19 of Meredith's proo... |
| merlem11 1660 | Step 20 of Meredith's proo... |
| merlem12 1661 | Step 28 of Meredith's proo... |
| merlem13 1662 | Step 35 of Meredith's proo... |
| luk-1 1663 | 1 of 3 axioms for proposit... |
| luk-2 1664 | 2 of 3 axioms for proposit... |
| luk-3 1665 | 3 of 3 axioms for proposit... |
| luklem1 1666 | Used to rederive standard ... |
| luklem2 1667 | Used to rederive standard ... |
| luklem3 1668 | Used to rederive standard ... |
| luklem4 1669 | Used to rederive standard ... |
| luklem5 1670 | Used to rederive standard ... |
| luklem6 1671 | Used to rederive standard ... |
| luklem7 1672 | Used to rederive standard ... |
| luklem8 1673 | Used to rederive standard ... |
| ax1 1674 | Standard propositional axi... |
| ax2 1675 | Standard propositional axi... |
| ax3 1676 | Standard propositional axi... |
| nic-dfim 1677 | This theorem "defines" imp... |
| nic-dfneg 1678 | This theorem "defines" neg... |
| nic-mp 1679 | Derive Nicod's rule of mod... |
| nic-mpALT 1680 | A direct proof of ~ nic-mp... |
| nic-ax 1681 | Nicod's axiom derived from... |
| nic-axALT 1682 | A direct proof of ~ nic-ax... |
| nic-imp 1683 | Inference for ~ nic-mp usi... |
| nic-idlem1 1684 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1685 | Lemma for ~ nic-id . Infe... |
| nic-id 1686 | Theorem ~ id expressed wit... |
| nic-swap 1687 | The connector ` -/\ ` is s... |
| nic-isw1 1688 | Inference version of ~ nic... |
| nic-isw2 1689 | Inference for swapping nes... |
| nic-iimp1 1690 | Inference version of ~ nic... |
| nic-iimp2 1691 | Inference version of ~ nic... |
| nic-idel 1692 | Inference to remove the tr... |
| nic-ich 1693 | Chained inference. (Contr... |
| nic-idbl 1694 | Double the terms. Since d... |
| nic-bijust 1695 | Biconditional justificatio... |
| nic-bi1 1696 | Inference to extract one s... |
| nic-bi2 1697 | Inference to extract the o... |
| nic-stdmp 1698 | Derive the standard modus ... |
| nic-luk1 1699 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1700 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1701 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1702 | This alternative axiom for... |
| lukshefth1 1703 | Lemma for ~ renicax . (Co... |
| lukshefth2 1704 | Lemma for ~ renicax . (Co... |
| renicax 1705 | A rederivation of ~ nic-ax... |
| tbw-bijust 1706 | Justification for ~ tbw-ne... |
| tbw-negdf 1707 | The definition of negation... |
| tbw-ax1 1708 | The first of four axioms i... |
| tbw-ax2 1709 | The second of four axioms ... |
| tbw-ax3 1710 | The third of four axioms i... |
| tbw-ax4 1711 | The fourth of four axioms ... |
| tbwsyl 1712 | Used to rederive the Lukas... |
| tbwlem1 1713 | Used to rederive the Lukas... |
| tbwlem2 1714 | Used to rederive the Lukas... |
| tbwlem3 1715 | Used to rederive the Lukas... |
| tbwlem4 1716 | Used to rederive the Lukas... |
| tbwlem5 1717 | Used to rederive the Lukas... |
| re1luk1 1718 | ~ luk-1 derived from the T... |
| re1luk2 1719 | ~ luk-2 derived from the T... |
| re1luk3 1720 | ~ luk-3 derived from the T... |
| merco1 1721 | A single axiom for proposi... |
| merco1lem1 1722 | Used to rederive the Tarsk... |
| retbwax4 1723 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1724 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1725 | Used to rederive the Tarsk... |
| merco1lem3 1726 | Used to rederive the Tarsk... |
| merco1lem4 1727 | Used to rederive the Tarsk... |
| merco1lem5 1728 | Used to rederive the Tarsk... |
| merco1lem6 1729 | Used to rederive the Tarsk... |
| merco1lem7 1730 | Used to rederive the Tarsk... |
| retbwax3 1731 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1732 | Used to rederive the Tarsk... |
| merco1lem9 1733 | Used to rederive the Tarsk... |
| merco1lem10 1734 | Used to rederive the Tarsk... |
| merco1lem11 1735 | Used to rederive the Tarsk... |
| merco1lem12 1736 | Used to rederive the Tarsk... |
| merco1lem13 1737 | Used to rederive the Tarsk... |
| merco1lem14 1738 | Used to rederive the Tarsk... |
| merco1lem15 1739 | Used to rederive the Tarsk... |
| merco1lem16 1740 | Used to rederive the Tarsk... |
| merco1lem17 1741 | Used to rederive the Tarsk... |
| merco1lem18 1742 | Used to rederive the Tarsk... |
| retbwax1 1743 | ~ tbw-ax1 rederived from ~... |
| merco2 1744 | A single axiom for proposi... |
| mercolem1 1745 | Used to rederive the Tarsk... |
| mercolem2 1746 | Used to rederive the Tarsk... |
| mercolem3 1747 | Used to rederive the Tarsk... |
| mercolem4 1748 | Used to rederive the Tarsk... |
| mercolem5 1749 | Used to rederive the Tarsk... |
| mercolem6 1750 | Used to rederive the Tarsk... |
| mercolem7 1751 | Used to rederive the Tarsk... |
| mercolem8 1752 | Used to rederive the Tarsk... |
| re1tbw1 1753 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1754 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1755 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1756 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1757 | Justification for ~ rb-imd... |
| rb-imdf 1758 | The definition of implicat... |
| anmp 1759 | Modus ponens for ` { \/ , ... |
| rb-ax1 1760 | The first of four axioms i... |
| rb-ax2 1761 | The second of four axioms ... |
| rb-ax3 1762 | The third of four axioms i... |
| rb-ax4 1763 | The fourth of four axioms ... |
| rbsyl 1764 | Used to rederive the Lukas... |
| rblem1 1765 | Used to rederive the Lukas... |
| rblem2 1766 | Used to rederive the Lukas... |
| rblem3 1767 | Used to rederive the Lukas... |
| rblem4 1768 | Used to rederive the Lukas... |
| rblem5 1769 | Used to rederive the Lukas... |
| rblem6 1770 | Used to rederive the Lukas... |
| rblem7 1771 | Used to rederive the Lukas... |
| re1axmp 1772 | ~ ax-mp derived from Russe... |
| re2luk1 1773 | ~ luk-1 derived from Russe... |
| re2luk2 1774 | ~ luk-2 derived from Russe... |
| re2luk3 1775 | ~ luk-3 derived from Russe... |
| mptnan 1776 | Modus ponendo tollens 1, o... |
| mptxor 1777 | Modus ponendo tollens 2, o... |
| mtpor 1778 | Modus tollendo ponens (inc... |
| mtpxor 1779 | Modus tollendo ponens (ori... |
| stoic1a 1780 | Stoic logic Thema 1 (part ... |
| stoic1b 1781 | Stoic logic Thema 1 (part ... |
| stoic2a 1782 | Stoic logic Thema 2 versio... |
| stoic2b 1783 | Stoic logic Thema 2 versio... |
| stoic3 1784 | Stoic logic Thema 3. Stat... |
| stoic4a 1785 | Stoic logic Thema 4 versio... |
| stoic4b 1786 | Stoic logic Thema 4 versio... |
| alnex 1789 | Universal quantification o... |
| eximal 1790 | An equivalence between an ... |
| nf2 1793 | Alternate definition of no... |
| nf3 1794 | Alternate definition of no... |
| nf4 1795 | Alternate definition of no... |
| nfi 1796 | Deduce that ` x ` is not f... |
| nfri 1797 | Consequence of the definit... |
| nfd 1798 | Deduce that ` x ` is not f... |
| nfrd 1799 | Consequence of the definit... |
| nftht 1800 | Closed form of ~ nfth . (... |
| nfntht 1801 | Closed form of ~ nfnth . ... |
| nfntht2 1802 | Closed form of ~ nfnth . ... |
| gen2 1804 | Generalization applied twi... |
| mpg 1805 | Modus ponens combined with... |
| mpgbi 1806 | Modus ponens on biconditio... |
| mpgbir 1807 | Modus ponens on biconditio... |
| nex 1808 | Generalization rule for ne... |
| nfth 1809 | No variable is (effectivel... |
| nfnth 1810 | No variable is (effectivel... |
| hbth 1811 | No variable is (effectivel... |
| nftru 1812 | The true constant has no f... |
| nffal 1813 | The false constant has no ... |
| sptruw 1814 | Version of ~ sp when ` ph ... |
| altru 1815 | For all sets, ` T. ` is tr... |
| alfal 1816 | For all sets, ` -. F. ` is... |
| alim 1818 | Restatement of Axiom ~ ax-... |
| alimi 1819 | Inference quantifying both... |
| 2alimi 1820 | Inference doubly quantifyi... |
| ala1 1821 | Add an antecedent in a uni... |
| al2im 1822 | Closed form of ~ al2imi . ... |
| al2imi 1823 | Inference quantifying ante... |
| alanimi 1824 | Variant of ~ al2imi with c... |
| alimdh 1825 | Deduction form of Theorem ... |
| albi 1826 | Theorem 19.15 of [Margaris... |
| albii 1827 | Inference adding universal... |
| 2albii 1828 | Inference adding two unive... |
| 3albii 1829 | Inference adding three uni... |
| sylgt 1830 | Closed form of ~ sylg . (... |
| sylg 1831 | A syllogism combined with ... |
| alrimih 1832 | Inference form of Theorem ... |
| hbxfrbi 1833 | A utility lemma to transfe... |
| alex 1834 | Universal quantifier in te... |
| exnal 1835 | Existential quantification... |
| 2nalexn 1836 | Part of theorem *11.5 in [... |
| 2exnaln 1837 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1838 | Theorem *11.25 in [Whitehe... |
| alimex 1839 | An equivalence between an ... |
| aleximi 1840 | A variant of ~ al2imi : in... |
| alexbii 1841 | Biconditional form of ~ al... |
| exim 1842 | Theorem 19.22 of [Margaris... |
| eximi 1843 | Inference adding existenti... |
| 2eximi 1844 | Inference adding two exist... |
| eximii 1845 | Inference associated with ... |
| exa1 1846 | Add an antecedent in an ex... |
| 19.38 1847 | Theorem 19.38 of [Margaris... |
| 19.38a 1848 | Under a nonfreeness hypoth... |
| 19.38b 1849 | Under a nonfreeness hypoth... |
| imnang 1850 | Quantified implication in ... |
| alinexa 1851 | A transformation of quanti... |
| exnalimn 1852 | Existential quantification... |
| alexn 1853 | A relationship between two... |
| 2exnexn 1854 | Theorem *11.51 in [Whitehe... |
| exbi 1855 | Theorem 19.18 of [Margaris... |
| exbii 1856 | Inference adding existenti... |
| 2exbii 1857 | Inference adding two exist... |
| 3exbii 1858 | Inference adding three exi... |
| nfbiit 1859 | Equivalence theorem for th... |
| nfbii 1860 | Equality theorem for the n... |
| nfxfr 1861 | A utility lemma to transfe... |
| nfxfrd 1862 | A utility lemma to transfe... |
| nfnbi 1863 | A variable is nonfree in a... |
| nfnt 1864 | If a variable is nonfree i... |
| nfn 1865 | Inference associated with ... |
| nfnd 1866 | Deduction associated with ... |
| exanali 1867 | A transformation of quanti... |
| 2exanali 1868 | Theorem *11.521 in [Whiteh... |
| exancom 1869 | Commutation of conjunction... |
| exan 1870 | Place a conjunct in the sc... |
| alrimdh 1871 | Deduction form of Theorem ... |
| eximdh 1872 | Deduction from Theorem 19.... |
| nexdh 1873 | Deduction for generalizati... |
| albidh 1874 | Formula-building rule for ... |
| exbidh 1875 | Formula-building rule for ... |
| exsimpl 1876 | Simplification of an exist... |
| exsimpr 1877 | Simplification of an exist... |
| 19.26 1878 | Theorem 19.26 of [Margaris... |
| 19.26-2 1879 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1880 | Theorem ~ 19.26 with tripl... |
| 19.29 1881 | Theorem 19.29 of [Margaris... |
| 19.29r 1882 | Variation of ~ 19.29 . (C... |
| 19.29r2 1883 | Variation of ~ 19.29r with... |
| 19.29x 1884 | Variation of ~ 19.29 with ... |
| 19.35 1885 | Theorem 19.35 of [Margaris... |
| 19.35i 1886 | Inference associated with ... |
| 19.35ri 1887 | Inference associated with ... |
| 19.25 1888 | Theorem 19.25 of [Margaris... |
| 19.30 1889 | Theorem 19.30 of [Margaris... |
| 19.43 1890 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1891 | Obsolete proof of ~ 19.43 ... |
| 19.33 1892 | Theorem 19.33 of [Margaris... |
| 19.33b 1893 | The antecedent provides a ... |
| 19.40 1894 | Theorem 19.40 of [Margaris... |
| 19.40-2 1895 | Theorem *11.42 in [Whitehe... |
| 19.40b 1896 | The antecedent provides a ... |
| albiim 1897 | Split a biconditional and ... |
| 2albiim 1898 | Split a biconditional and ... |
| exintrbi 1899 | Add/remove a conjunct in t... |
| exintr 1900 | Introduce a conjunct in th... |
| alsyl 1901 | Universally quantified and... |
| nfimd 1902 | If in a context ` x ` is n... |
| nfimt 1903 | Closed form of ~ nfim and ... |
| nfim 1904 | If ` x ` is not free in ` ... |
| nfand 1905 | If in a context ` x ` is n... |
| nf3and 1906 | Deduction form of bound-va... |
| nfan 1907 | If ` x ` is not free in ` ... |
| nfnan 1908 | If ` x ` is not free in ` ... |
| nf3an 1909 | If ` x ` is not free in ` ... |
| nfbid 1910 | If in a context ` x ` is n... |
| nfbi 1911 | If ` x ` is not free in ` ... |
| nfor 1912 | If ` x ` is not free in ` ... |
| nf3or 1913 | If ` x ` is not free in ` ... |
| empty 1914 | Two characterizations of t... |
| emptyex 1915 | On the empty domain, any e... |
| emptyal 1916 | On the empty domain, any u... |
| emptynf 1917 | On the empty domain, any v... |
| ax5d 1919 | Version of ~ ax-5 with ant... |
| ax5e 1920 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1921 | If a formula holds for som... |
| nfv 1922 | If ` x ` is not present in... |
| nfvd 1923 | ~ nfv with antecedent. Us... |
| alimdv 1924 | Deduction form of Theorem ... |
| eximdv 1925 | Deduction form of Theorem ... |
| 2alimdv 1926 | Deduction form of Theorem ... |
| 2eximdv 1927 | Deduction form of Theorem ... |
| albidv 1928 | Formula-building rule for ... |
| exbidv 1929 | Formula-building rule for ... |
| nfbidv 1930 | An equality theorem for no... |
| 2albidv 1931 | Formula-building rule for ... |
| 2exbidv 1932 | Formula-building rule for ... |
| 3exbidv 1933 | Formula-building rule for ... |
| 4exbidv 1934 | Formula-building rule for ... |
| alrimiv 1935 | Inference form of Theorem ... |
| alrimivv 1936 | Inference form of Theorem ... |
| alrimdv 1937 | Deduction form of Theorem ... |
| exlimiv 1938 | Inference form of Theorem ... |
| exlimiiv 1939 | Inference (Rule C) associa... |
| exlimivv 1940 | Inference form of Theorem ... |
| exlimdv 1941 | Deduction form of Theorem ... |
| exlimdvv 1942 | Deduction form of Theorem ... |
| exlimddv 1943 | Existential elimination ru... |
| nexdv 1944 | Deduction for generalizati... |
| 2ax5 1945 | Quantification of two vari... |
| stdpc5v 1946 | Version of ~ stdpc5 with a... |
| 19.21v 1947 | Version of ~ 19.21 with a ... |
| 19.32v 1948 | Version of ~ 19.32 with a ... |
| 19.31v 1949 | Version of ~ 19.31 with a ... |
| 19.23v 1950 | Version of ~ 19.23 with a ... |
| 19.23vv 1951 | Theorem ~ 19.23v extended ... |
| pm11.53v 1952 | Version of ~ pm11.53 with ... |
| 19.36imv 1953 | One direction of ~ 19.36v ... |
| 19.36iv 1954 | Inference associated with ... |
| 19.37imv 1955 | One direction of ~ 19.37v ... |
| 19.37iv 1956 | Inference associated with ... |
| 19.41v 1957 | Version of ~ 19.41 with a ... |
| 19.41vv 1958 | Version of ~ 19.41 with tw... |
| 19.41vvv 1959 | Version of ~ 19.41 with th... |
| 19.41vvvv 1960 | Version of ~ 19.41 with fo... |
| 19.42v 1961 | Version of ~ 19.42 with a ... |
| exdistr 1962 | Distribution of existentia... |
| exdistrv 1963 | Distribute a pair of exist... |
| 4exdistrv 1964 | Distribute two pairs of ex... |
| 19.42vv 1965 | Version of ~ 19.42 with tw... |
| exdistr2 1966 | Distribution of existentia... |
| 19.42vvv 1967 | Version of ~ 19.42 with th... |
| 3exdistr 1968 | Distribution of existentia... |
| 4exdistr 1969 | Distribution of existentia... |
| weq 1970 | Extend wff definition to i... |
| speimfw 1971 | Specialization, with addit... |
| speimfwALT 1972 | Alternate proof of ~ speim... |
| spimfw 1973 | Specialization, with addit... |
| ax12i 1974 | Inference that has ~ ax-12... |
| ax6v 1976 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1977 | At least one individual ex... |
| spimw 1978 | Specialization. Lemma 8 o... |
| spimew 1979 | Existential introduction, ... |
| speiv 1980 | Inference from existential... |
| speivw 1981 | Version of ~ spei with a d... |
| exgen 1982 | Rule of existential genera... |
| extru 1983 | There exists a variable su... |
| 19.2 1984 | Theorem 19.2 of [Margaris]... |
| 19.2d 1985 | Deduction associated with ... |
| 19.8w 1986 | Weak version of ~ 19.8a an... |
| spnfw 1987 | Weak version of ~ sp . Us... |
| spfalw 1988 | Version of ~ sp when ` ph ... |
| spvw 1989 | Version of ~ sp when ` x `... |
| 19.3v 1990 | Version of ~ 19.3 with a d... |
| 19.8v 1991 | Version of ~ 19.8a with a ... |
| 19.9v 1992 | Version of ~ 19.9 with a d... |
| spimevw 1993 | Existential introduction, ... |
| spimvw 1994 | A weak form of specializat... |
| spsv 1995 | Generalization of antecede... |
| spvv 1996 | Specialization, using impl... |
| chvarvv 1997 | Implicit substitution of `... |
| 19.39 1998 | Theorem 19.39 of [Margaris... |
| 19.24 1999 | Theorem 19.24 of [Margaris... |
| 19.34 2000 | Theorem 19.34 of [Margaris... |
| 19.36v 2001 | Version of ~ 19.36 with a ... |
| 19.12vvv 2002 | Version of ~ 19.12vv with ... |
| 19.27v 2003 | Version of ~ 19.27 with a ... |
| 19.28v 2004 | Version of ~ 19.28 with a ... |
| 19.37v 2005 | Version of ~ 19.37 with a ... |
| 19.44v 2006 | Version of ~ 19.44 with a ... |
| 19.45v 2007 | Version of ~ 19.45 with a ... |
| equs4v 2008 | Version of ~ equs4 with a ... |
| alequexv 2009 | Version of ~ equs4v with i... |
| exsbim 2010 | One direction of the equiv... |
| equsv 2011 | If a formula does not cont... |
| equsalvw 2012 | Version of ~ equsalv with ... |
| equsexvw 2013 | Version of ~ equsexv with ... |
| cbvaliw 2014 | Change bound variable. Us... |
| cbvalivw 2015 | Change bound variable. Us... |
| ax7v 2017 | Weakened version of ~ ax-7... |
| ax7v1 2018 | First of two weakened vers... |
| ax7v2 2019 | Second of two weakened ver... |
| equid 2020 | Identity law for equality.... |
| nfequid 2021 | Bound-variable hypothesis ... |
| equcomiv 2022 | Weaker form of ~ equcomi w... |
| ax6evr 2023 | A commuted form of ~ ax6ev... |
| ax7 2024 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2025 | Commutative law for equali... |
| equcom 2026 | Commutative law for equali... |
| equcomd 2027 | Deduction form of ~ equcom... |
| equcoms 2028 | An inference commuting equ... |
| equtr 2029 | A transitive law for equal... |
| equtrr 2030 | A transitive law for equal... |
| equeuclr 2031 | Commuted version of ~ eque... |
| equeucl 2032 | Equality is a left-Euclide... |
| equequ1 2033 | An equivalence law for equ... |
| equequ2 2034 | An equivalence law for equ... |
| equtr2 2035 | Equality is a left-Euclide... |
| stdpc6 2036 | One of the two equality ax... |
| equvinv 2037 | A variable introduction la... |
| equvinva 2038 | A modified version of the ... |
| equvelv 2039 | A biconditional form of ~ ... |
| ax13b 2040 | An equivalence between two... |
| spfw 2041 | Weak version of ~ sp . Us... |
| spw 2042 | Weak version of the specia... |
| cbvalw 2043 | Change bound variable. Us... |
| cbvalvw 2044 | Change bound variable. Us... |
| cbvexvw 2045 | Change bound variable. Us... |
| cbvaldvaw 2046 | Rule used to change the bo... |
| cbvexdvaw 2047 | Rule used to change the bo... |
| cbval2vw 2048 | Rule used to change bound ... |
| cbvex2vw 2049 | Rule used to change bound ... |
| cbvex4vw 2050 | Rule used to change bound ... |
| alcomimw 2051 | Weak version of ~ ax-11 . ... |
| excomimw 2052 | Weak version of ~ excomim ... |
| alcomw 2053 | Weak version of ~ alcom an... |
| excomw 2054 | Weak version of ~ excom an... |
| hbn1fw 2055 | Weak version of ~ ax-10 fr... |
| hbn1w 2056 | Weak version of ~ hbn1 . ... |
| hba1w 2057 | Weak version of ~ hba1 . ... |
| hbe1w 2058 | Weak version of ~ hbe1 . ... |
| hbalw 2059 | Weak version of ~ hbal . ... |
| 19.8aw 2060 | If a formula is true, then... |
| exexw 2061 | Existential quantification... |
| spaev 2062 | A special instance of ~ sp... |
| cbvaev 2063 | Change bound variable in a... |
| aevlem0 2064 | Lemma for ~ aevlem . Inst... |
| aevlem 2065 | Lemma for ~ aev and ~ axc1... |
| aeveq 2066 | The antecedent ` A. x x = ... |
| aev 2067 | A "distinctor elimination"... |
| aev2 2068 | A version of ~ aev with tw... |
| hbaev 2069 | All variables are effectiv... |
| naev 2070 | If some set variables can ... |
| naev2 2071 | Generalization of ~ hbnaev... |
| hbnaev 2072 | Any variable is free in ` ... |
| sbjust 2073 | Justification theorem for ... |
| dfsb 2076 | Simplify definition ~ df-s... |
| sbtlem 2077 | In the case of ~ sbt , the... |
| sbt 2078 | A substitution into a theo... |
| sbtru 2079 | The result of substituting... |
| stdpc4 2080 | The specialization axiom o... |
| sbtALT 2081 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2082 | A double specialization us... |
| sbi1 2083 | Distribute substitution ov... |
| spsbim 2084 | Distribute substitution ov... |
| spsbbi 2085 | Biconditional property for... |
| sbimi 2086 | Distribute substitution ov... |
| sb2imi 2087 | Distribute substitution ov... |
| sbbii 2088 | Infer substitution into bo... |
| 2sbbii 2089 | Infer double substitution ... |
| sbimdv 2090 | Deduction substituting bot... |
| sbbidv 2091 | Deduction substituting bot... |
| sban 2092 | Conjunction inside and out... |
| sb3an 2093 | Threefold conjunction insi... |
| spsbe 2094 | Existential generalization... |
| sbequ 2095 | Equality property for subs... |
| sbequi 2096 | An equality theorem for su... |
| sb6 2097 | Alternate definition of su... |
| 2sb6 2098 | Equivalence for double sub... |
| sb1v 2099 | One direction of ~ sb5 , p... |
| sbv 2100 | Substitution for a variabl... |
| sbcom4 2101 | Commutativity law for subs... |
| pm11.07 2102 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2103 | Substitution in an implica... |
| sbrimvwOLD 2104 | Obsolete version of ~ sbri... |
| sbbiiev 2105 | An equivalence of substitu... |
| sbievw 2106 | Conversion of implicit sub... |
| sbievwOLD 2107 | Obsolete version of ~ sbie... |
| sbiedvw 2108 | Conversion of implicit sub... |
| 2sbievw 2109 | Conversion of double impli... |
| sbcom3vv 2110 | Substituting ` y ` for ` x... |
| sbievw2 2111 | ~ sbievw applied twice, av... |
| sbco2vv 2112 | A composition law for subs... |
| cbvsbv 2113 | Change the bound variable ... |
| sbco4lem 2114 | Lemma for ~ sbco4 . It re... |
| sbco4 2115 | Two ways of exchanging two... |
| equsb3 2116 | Substitution in an equalit... |
| equsb3r 2117 | Substitution applied to th... |
| equsb1v 2118 | Substitution applied to an... |
| nsb 2119 | Any substitution in an alw... |
| sbn1 2120 | One direction of ~ sbn , u... |
| wel 2122 | Extend wff definition to i... |
| ax8v 2124 | Weakened version of ~ ax-8... |
| ax8v1 2125 | First of two weakened vers... |
| ax8v2 2126 | Second of two weakened ver... |
| ax8 2127 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2128 | An identity law for the no... |
| elsb1 2129 | Substitution for the first... |
| cleljust 2130 | When the class variables i... |
| ax9v 2132 | Weakened version of ~ ax-9... |
| ax9v1 2133 | First of two weakened vers... |
| ax9v2 2134 | Second of two weakened ver... |
| ax9 2135 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2136 | An identity law for the no... |
| elequ2g 2137 | A form of ~ elequ2 with a ... |
| elsb2 2138 | Substitution for the secon... |
| elequ12 2139 | An identity law for the no... |
| ru0 2140 | The FOL statement used in ... |
| ax6dgen 2141 | Tarski's system uses the w... |
| ax10w 2142 | Weak version of ~ ax-10 fr... |
| ax11w 2143 | Weak version of ~ ax-11 fr... |
| ax11dgen 2144 | Degenerate instance of ~ a... |
| ax12wlem 2145 | Lemma for weak version of ... |
| ax12w 2146 | Weak version of ~ ax-12 fr... |
| ax12dgen 2147 | Degenerate instance of ~ a... |
| ax12wdemo 2148 | Example of an application ... |
| ax13w 2149 | Weak version (principal in... |
| ax13dgen1 2150 | Degenerate instance of ~ a... |
| ax13dgen2 2151 | Degenerate instance of ~ a... |
| ax13dgen3 2152 | Degenerate instance of ~ a... |
| ax13dgen4 2153 | Degenerate instance of ~ a... |
| hbn1 2155 | Alias for ~ ax-10 to be us... |
| hbe1 2156 | The setvar ` x ` is not fr... |
| hbe1a 2157 | Dual statement of ~ hbe1 .... |
| nf5-1 2158 | One direction of ~ nf5 can... |
| nf5i 2159 | Deduce that ` x ` is not f... |
| nf5dh 2160 | Deduce that ` x ` is not f... |
| nf5dv 2161 | Apply the definition of no... |
| nfnaew 2162 | All variables are effectiv... |
| nfe1 2163 | The setvar ` x ` is not fr... |
| nfa1 2164 | The setvar ` x ` is not fr... |
| nfna1 2165 | A convenience theorem part... |
| nfia1 2166 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2167 | The setvar ` x ` is not fr... |
| modal5 2168 | The analogue in our predic... |
| nfs1v 2169 | The setvar ` x ` is not fr... |
| alcoms 2171 | Swap quantifiers in an ant... |
| alcom 2172 | Theorem 19.5 of [Margaris]... |
| alrot3 2173 | Theorem *11.21 in [Whitehe... |
| alrot4 2174 | Rotate four universal quan... |
| excom 2175 | Theorem 19.11 of [Margaris... |
| excomim 2176 | One direction of Theorem 1... |
| excom13 2177 | Swap 1st and 3rd existenti... |
| exrot3 2178 | Rotate existential quantif... |
| exrot4 2179 | Rotate existential quantif... |
| hbal 2180 | If ` x ` is not free in ` ... |
| hbald 2181 | Deduction form of bound-va... |
| sbal 2182 | Move universal quantifier ... |
| sbalv 2183 | Quantify with new variable... |
| hbsbw 2184 | If ` z ` is not free in ` ... |
| sbcom2 2185 | Commutativity law for subs... |
| sbco4lemOLD 2186 | Obsolete version of ~ sbco... |
| sbco4OLD 2187 | Obsolete version of ~ sbco... |
| nfa2 2188 | Lemma 24 of [Monk2] p. 114... |
| nfexhe 2189 | Version of ~ nfex with the... |
| nfexa2 2190 | An inner universal quantif... |
| ax12v 2192 | This is essentially Axiom ... |
| ax12v2 2193 | It is possible to remove a... |
| ax12ev2 2194 | Version of ~ ax12v2 rewrit... |
| 19.8a 2195 | If a wff is true, it is tr... |
| 19.8ad 2196 | If a wff is true, it is tr... |
| sp 2197 | Specialization. A univers... |
| spi 2198 | Inference rule of universa... |
| sps 2199 | Generalization of antecede... |
| 2sp 2200 | A double specialization (s... |
| spsd 2201 | Deduction generalizing ant... |
| 19.2g 2202 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2203 | Inference form of ~ 19.21 ... |
| 19.21bbi 2204 | Inference removing two uni... |
| 19.23bi 2205 | Inference form of Theorem ... |
| nexr 2206 | Inference associated with ... |
| qexmid 2207 | Quantified excluded middle... |
| nf5r 2208 | Consequence of the definit... |
| nf5ri 2209 | Consequence of the definit... |
| nf5rd 2210 | Consequence of the definit... |
| spimedv 2211 | Deduction version of ~ spi... |
| spimefv 2212 | Version of ~ spime with a ... |
| nfim1 2213 | A closed form of ~ nfim . ... |
| nfan1 2214 | A closed form of ~ nfan . ... |
| 19.3t 2215 | Closed form of ~ 19.3 and ... |
| 19.3 2216 | A wff may be quantified wi... |
| 19.9d 2217 | A deduction version of one... |
| 19.9t 2218 | Closed form of ~ 19.9 and ... |
| 19.9 2219 | A wff may be existentially... |
| 19.21t 2220 | Closed form of Theorem 19.... |
| 19.21 2221 | Theorem 19.21 of [Margaris... |
| stdpc5 2222 | An axiom scheme of standar... |
| 19.21-2 2223 | Version of ~ 19.21 with tw... |
| 19.23t 2224 | Closed form of Theorem 19.... |
| 19.23 2225 | Theorem 19.23 of [Margaris... |
| alimd 2226 | Deduction form of Theorem ... |
| alrimi 2227 | Inference form of Theorem ... |
| alrimdd 2228 | Deduction form of Theorem ... |
| alrimd 2229 | Deduction form of Theorem ... |
| eximd 2230 | Deduction form of Theorem ... |
| exlimi 2231 | Inference associated with ... |
| exlimd 2232 | Deduction form of Theorem ... |
| exlimimdd 2233 | Existential elimination ru... |
| exlimdd 2234 | Existential elimination ru... |
| nexd 2235 | Deduction for generalizati... |
| albid 2236 | Formula-building rule for ... |
| exbid 2237 | Formula-building rule for ... |
| nfbidf 2238 | An equality theorem for ef... |
| 19.16 2239 | Theorem 19.16 of [Margaris... |
| 19.17 2240 | Theorem 19.17 of [Margaris... |
| 19.27 2241 | Theorem 19.27 of [Margaris... |
| 19.28 2242 | Theorem 19.28 of [Margaris... |
| 19.19 2243 | Theorem 19.19 of [Margaris... |
| 19.36 2244 | Theorem 19.36 of [Margaris... |
| 19.36i 2245 | Inference associated with ... |
| 19.37 2246 | Theorem 19.37 of [Margaris... |
| 19.32 2247 | Theorem 19.32 of [Margaris... |
| 19.31 2248 | Theorem 19.31 of [Margaris... |
| 19.41 2249 | Theorem 19.41 of [Margaris... |
| 19.42 2250 | Theorem 19.42 of [Margaris... |
| 19.44 2251 | Theorem 19.44 of [Margaris... |
| 19.45 2252 | Theorem 19.45 of [Margaris... |
| spimfv 2253 | Specialization, using impl... |
| chvarfv 2254 | Implicit substitution of `... |
| cbv3v2 2255 | Version of ~ cbv3 with two... |
| sbalex 2256 | Equivalence of two ways to... |
| sbalexOLD 2257 | Obsolete version of ~ sbal... |
| sb4av 2258 | Version of ~ sb4a with a d... |
| sbimd 2259 | Deduction substituting bot... |
| sbbid 2260 | Deduction substituting bot... |
| 2sbbid 2261 | Deduction doubly substitut... |
| sbequ1 2262 | An equality theorem for su... |
| sbequ2 2263 | An equality theorem for su... |
| stdpc7 2264 | One of the two equality ax... |
| sbequ12 2265 | An equality theorem for su... |
| sbequ12r 2266 | An equality theorem for su... |
| sbelx 2267 | Elimination of substitutio... |
| sbequ12a 2268 | An equality theorem for su... |
| sbid 2269 | An identity theorem for su... |
| sbcov 2270 | A composition law for subs... |
| sbcovOLD 2271 | Obsolete version of ~ sbco... |
| sb6a 2272 | Equivalence for substituti... |
| sbid2vw 2273 | Reverting substitution yie... |
| axc16g 2274 | Generalization of ~ axc16 ... |
| axc16 2275 | Proof of older axiom ~ ax-... |
| axc16gb 2276 | Biconditional strengthenin... |
| axc16nf 2277 | If ~ dtru is false, then t... |
| axc11v 2278 | Version of ~ axc11 with a ... |
| axc11rv 2279 | Version of ~ axc11r with a... |
| drsb2 2280 | Formula-building lemma for... |
| equsalv 2281 | An equivalence related to ... |
| equsexv 2282 | An equivalence related to ... |
| sbft 2283 | Substitution has no effect... |
| sbf 2284 | Substitution for a variabl... |
| sbf2 2285 | Substitution has no effect... |
| sbh 2286 | Substitution for a variabl... |
| hbs1 2287 | The setvar ` x ` is not fr... |
| nfs1f 2288 | If ` x ` is not free in ` ... |
| sb5 2289 | Alternate definition of su... |
| equs5av 2290 | A property related to subs... |
| 2sb5 2291 | Equivalence for double sub... |
| dfsb7 2292 | An alternate definition of... |
| sbn 2293 | Negation inside and outsid... |
| sbex 2294 | Move existential quantifie... |
| nf5 2295 | Alternate definition of ~ ... |
| nf6 2296 | An alternate definition of... |
| nf5d 2297 | Deduce that ` x ` is not f... |
| nf5di 2298 | Since the converse holds b... |
| 19.9h 2299 | A wff may be existentially... |
| 19.21h 2300 | Theorem 19.21 of [Margaris... |
| 19.23h 2301 | Theorem 19.23 of [Margaris... |
| exlimih 2302 | Inference associated with ... |
| exlimdh 2303 | Deduction form of Theorem ... |
| equsalhw 2304 | Version of ~ equsalh with ... |
| equsexhv 2305 | An equivalence related to ... |
| hba1 2306 | The setvar ` x ` is not fr... |
| hbnt 2307 | Closed theorem version of ... |
| hbn 2308 | If ` x ` is not free in ` ... |
| hbnd 2309 | Deduction form of bound-va... |
| hbim1 2310 | A closed form of ~ hbim . ... |
| hbimd 2311 | Deduction form of bound-va... |
| hbim 2312 | If ` x ` is not free in ` ... |
| hban 2313 | If ` x ` is not free in ` ... |
| hb3an 2314 | If ` x ` is not free in ` ... |
| sbi2 2315 | Introduction of implicatio... |
| sbim 2316 | Implication inside and out... |
| sbrim 2317 | Substitution in an implica... |
| sblim 2318 | Substitution in an implica... |
| sbor 2319 | Disjunction inside and out... |
| sbbi 2320 | Equivalence inside and out... |
| sblbis 2321 | Introduce left bicondition... |
| sbrbis 2322 | Introduce right biconditio... |
| sbrbif 2323 | Introduce right biconditio... |
| sbnf 2324 | Move nonfree predicate in ... |
| sbiev 2325 | Conversion of implicit sub... |
| sbievOLD 2326 | Obsolete version of ~ sbie... |
| sbiedw 2327 | Conversion of implicit sub... |
| axc7 2328 | Show that the original axi... |
| axc7e 2329 | Abbreviated version of ~ a... |
| modal-b 2330 | The analogue in our predic... |
| 19.9ht 2331 | A closed version of ~ 19.9... |
| axc4 2332 | Show that the original axi... |
| axc4i 2333 | Inference version of ~ axc... |
| nfal 2334 | If ` x ` is not free in ` ... |
| nfex 2335 | If ` x ` is not free in ` ... |
| hbex 2336 | If ` x ` is not free in ` ... |
| nfnf 2337 | If ` x ` is not free in ` ... |
| 19.12 2338 | Theorem 19.12 of [Margaris... |
| nfald 2339 | Deduction form of ~ nfal .... |
| nfexd 2340 | If ` x ` is not free in ` ... |
| nfsbv 2341 | If ` z ` is not free in ` ... |
| sbco2v 2342 | A composition law for subs... |
| aaan 2343 | Distribute universal quant... |
| eeor 2344 | Distribute existential qua... |
| cbv3v 2345 | Rule used to change bound ... |
| cbv1v 2346 | Rule used to change bound ... |
| cbv2w 2347 | Rule used to change bound ... |
| cbvaldw 2348 | Deduction used to change b... |
| cbvexdw 2349 | Deduction used to change b... |
| cbv3hv 2350 | Rule used to change bound ... |
| cbvalv1 2351 | Rule used to change bound ... |
| cbvexv1 2352 | Rule used to change bound ... |
| cbval2v 2353 | Rule used to change bound ... |
| cbvex2v 2354 | Rule used to change bound ... |
| dvelimhw 2355 | Proof of ~ dvelimh without... |
| pm11.53 2356 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2357 | Special case of ~ 19.12 wh... |
| eean 2358 | Distribute existential qua... |
| eeanv 2359 | Distribute a pair of exist... |
| eeeanv 2360 | Distribute three existenti... |
| ee4anv 2361 | Distribute two pairs of ex... |
| ee4anvOLD 2362 | Obsolete version of ~ ee4a... |
| sb8v 2363 | Substitution of variable i... |
| sb8f 2364 | Substitution of variable i... |
| sb8ef 2365 | Substitution of variable i... |
| 2sb8ef 2366 | An equivalent expression f... |
| sb6rfv 2367 | Reversed substitution. Ve... |
| sbnf2 2368 | Two ways of expressing " `... |
| exsb 2369 | An equivalent expression f... |
| 2exsb 2370 | An equivalent expression f... |
| sbbib 2371 | Reversal of substitution. ... |
| sbbibvv 2372 | Reversal of substitution. ... |
| cbvsbvf 2373 | Change the bound variable ... |
| cleljustALT 2374 | Alternate proof of ~ clelj... |
| cleljustALT2 2375 | Alternate proof of ~ clelj... |
| equs5aALT 2376 | Alternate proof of ~ equs5... |
| equs5eALT 2377 | Alternate proof of ~ equs5... |
| axc11r 2378 | Same as ~ axc11 but with r... |
| dral1v 2379 | Formula-building lemma for... |
| drex1v 2380 | Formula-building lemma for... |
| drnf1v 2381 | Formula-building lemma for... |
| ax13v 2383 | A weaker version of ~ ax-1... |
| ax13lem1 2384 | A version of ~ ax13v with ... |
| ax13 2385 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2386 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2387 | An equation between setvar... |
| dveeq2 2388 | Quantifier introduction wh... |
| nfeqf1 2389 | An equation between setvar... |
| dveeq1 2390 | Quantifier introduction wh... |
| nfeqf 2391 | A variable is effectively ... |
| axc9 2392 | Derive set.mm's original ~... |
| ax6e 2393 | At least one individual ex... |
| ax6 2394 | Theorem showing that ~ ax-... |
| axc10 2395 | Show that the original axi... |
| spimt 2396 | Closed theorem form of ~ s... |
| spim 2397 | Specialization, using impl... |
| spimed 2398 | Deduction version of ~ spi... |
| spime 2399 | Existential introduction, ... |
| spimv 2400 | A version of ~ spim with a... |
| spimvALT 2401 | Alternate proof of ~ spimv... |
| spimev 2402 | Distinct-variable version ... |
| spv 2403 | Specialization, using impl... |
| spei 2404 | Inference from existential... |
| chvar 2405 | Implicit substitution of `... |
| chvarv 2406 | Implicit substitution of `... |
| cbv3 2407 | Rule used to change bound ... |
| cbval 2408 | Rule used to change bound ... |
| cbvex 2409 | Rule used to change bound ... |
| cbvalv 2410 | Rule used to change bound ... |
| cbvexv 2411 | Rule used to change bound ... |
| cbv1 2412 | Rule used to change bound ... |
| cbv2 2413 | Rule used to change bound ... |
| cbv3h 2414 | Rule used to change bound ... |
| cbv1h 2415 | Rule used to change bound ... |
| cbv2h 2416 | Rule used to change bound ... |
| cbvald 2417 | Deduction used to change b... |
| cbvexd 2418 | Deduction used to change b... |
| cbvaldva 2419 | Rule used to change the bo... |
| cbvexdva 2420 | Rule used to change the bo... |
| cbval2 2421 | Rule used to change bound ... |
| cbvex2 2422 | Rule used to change bound ... |
| cbval2vv 2423 | Rule used to change bound ... |
| cbvex2vv 2424 | Rule used to change bound ... |
| cbvex4v 2425 | Rule used to change bound ... |
| equs4 2426 | Lemma used in proofs of im... |
| equsal 2427 | An equivalence related to ... |
| equsex 2428 | An equivalence related to ... |
| equsexALT 2429 | Alternate proof of ~ equse... |
| equsalh 2430 | An equivalence related to ... |
| equsexh 2431 | An equivalence related to ... |
| axc15 2432 | Derivation of set.mm's ori... |
| ax12 2433 | Rederivation of Axiom ~ ax... |
| ax12b 2434 | A bidirectional version of... |
| ax13ALT 2435 | Alternate proof of ~ ax13 ... |
| axc11n 2436 | Derive set.mm's original ~... |
| aecom 2437 | Commutation law for identi... |
| aecoms 2438 | A commutation rule for ide... |
| naecoms 2439 | A commutation rule for dis... |
| axc11 2440 | Show that ~ ax-c11 can be ... |
| hbae 2441 | All variables are effectiv... |
| hbnae 2442 | All variables are effectiv... |
| nfae 2443 | All variables are effectiv... |
| nfnae 2444 | All variables are effectiv... |
| hbnaes 2445 | Rule that applies ~ hbnae ... |
| axc16i 2446 | Inference with ~ axc16 as ... |
| axc16nfALT 2447 | Alternate proof of ~ axc16... |
| dral2 2448 | Formula-building lemma for... |
| dral1 2449 | Formula-building lemma for... |
| dral1ALT 2450 | Alternate proof of ~ dral1... |
| drex1 2451 | Formula-building lemma for... |
| drex2 2452 | Formula-building lemma for... |
| drnf1 2453 | Formula-building lemma for... |
| drnf2 2454 | Formula-building lemma for... |
| nfald2 2455 | Variation on ~ nfald which... |
| nfexd2 2456 | Variation on ~ nfexd which... |
| exdistrf 2457 | Distribution of existentia... |
| dvelimf 2458 | Version of ~ dvelimv witho... |
| dvelimdf 2459 | Deduction form of ~ dvelim... |
| dvelimh 2460 | Version of ~ dvelim withou... |
| dvelim 2461 | This theorem can be used t... |
| dvelimv 2462 | Similar to ~ dvelim with f... |
| dvelimnf 2463 | Version of ~ dvelim using ... |
| dveeq2ALT 2464 | Alternate proof of ~ dveeq... |
| equvini 2465 | A variable introduction la... |
| equvel 2466 | A variable elimination law... |
| equs5a 2467 | A property related to subs... |
| equs5e 2468 | A property related to subs... |
| equs45f 2469 | Two ways of expressing sub... |
| equs5 2470 | Lemma used in proofs of su... |
| dveel1 2471 | Quantifier introduction wh... |
| dveel2 2472 | Quantifier introduction wh... |
| axc14 2473 | Axiom ~ ax-c14 is redundan... |
| sb6x 2474 | Equivalence involving subs... |
| sbequ5 2475 | Substitution does not chan... |
| sbequ6 2476 | Substitution does not chan... |
| sb5rf 2477 | Reversed substitution. Us... |
| sb6rf 2478 | Reversed substitution. Fo... |
| ax12vALT 2479 | Alternate proof of ~ ax12v... |
| 2ax6elem 2480 | We can always find values ... |
| 2ax6e 2481 | We can always find values ... |
| 2sb5rf 2482 | Reversed double substituti... |
| 2sb6rf 2483 | Reversed double substituti... |
| sbel2x 2484 | Elimination of double subs... |
| sb4b 2485 | Simplified definition of s... |
| sb3b 2486 | Simplified definition of s... |
| sb3 2487 | One direction of a simplif... |
| sb1 2488 | One direction of a simplif... |
| sb2 2489 | One direction of a simplif... |
| sb4a 2490 | A version of one implicati... |
| dfsb1 2491 | Alternate definition of su... |
| hbsb2 2492 | Bound-variable hypothesis ... |
| nfsb2 2493 | Bound-variable hypothesis ... |
| hbsb2a 2494 | Special case of a bound-va... |
| sb4e 2495 | One direction of a simplif... |
| hbsb2e 2496 | Special case of a bound-va... |
| hbsb3 2497 | If ` y ` is not free in ` ... |
| nfs1 2498 | If ` y ` is not free in ` ... |
| axc16ALT 2499 | Alternate proof of ~ axc16... |
| axc16gALT 2500 | Alternate proof of ~ axc16... |
| equsb1 2501 | Substitution applied to an... |
| equsb2 2502 | Substitution applied to an... |
| dfsb2 2503 | An alternate definition of... |
| dfsb3 2504 | An alternate definition of... |
| drsb1 2505 | Formula-building lemma for... |
| sb2ae 2506 | In the case of two success... |
| sb6f 2507 | Equivalence for substituti... |
| sb5f 2508 | Equivalence for substituti... |
| nfsb4t 2509 | A variable not free in a p... |
| nfsb4 2510 | A variable not free in a p... |
| sbequ8 2511 | Elimination of equality fr... |
| sbie 2512 | Conversion of implicit sub... |
| sbied 2513 | Conversion of implicit sub... |
| sbiedv 2514 | Conversion of implicit sub... |
| 2sbiev 2515 | Conversion of double impli... |
| sbcom3 2516 | Substituting ` y ` for ` x... |
| sbco 2517 | A composition law for subs... |
| sbid2 2518 | An identity law for substi... |
| sbid2v 2519 | An identity law for substi... |
| sbidm 2520 | An idempotent law for subs... |
| sbco2 2521 | A composition law for subs... |
| sbco2d 2522 | A composition law for subs... |
| sbco3 2523 | A composition law for subs... |
| sbcom 2524 | A commutativity law for su... |
| sbtrt 2525 | Partially closed form of ~... |
| sbtr 2526 | A partial converse to ~ sb... |
| sb8 2527 | Substitution of variable i... |
| sb8e 2528 | Substitution of variable i... |
| sb9 2529 | Commutation of quantificat... |
| sb9i 2530 | Commutation of quantificat... |
| sbhb 2531 | Two ways of expressing " `... |
| nfsbd 2532 | Deduction version of ~ nfs... |
| nfsb 2533 | If ` z ` is not free in ` ... |
| hbsb 2534 | If ` z ` is not free in ` ... |
| sb7f 2535 | This version of ~ dfsb7 do... |
| sb7h 2536 | This version of ~ dfsb7 do... |
| sb10f 2537 | Hao Wang's identity axiom ... |
| sbal1 2538 | Check out ~ sbal for a ver... |
| sbal2 2539 | Move quantifier in and out... |
| 2sb8e 2540 | An equivalent expression f... |
| dfmoeu 2541 | An elementary proof of ~ m... |
| dfeumo 2542 | An elementary proof showin... |
| mojust 2544 | Soundness justification th... |
| dfmo 2546 | Simplify definition ~ df-m... |
| nexmo 2547 | Nonexistence implies uniqu... |
| exmo 2548 | Any proposition holds for ... |
| moabs 2549 | Absorption of existence co... |
| moim 2550 | The at-most-one quantifier... |
| moimi 2551 | The at-most-one quantifier... |
| moimdv 2552 | The at-most-one quantifier... |
| mobi 2553 | Equivalence theorem for th... |
| mobii 2554 | Formula-building rule for ... |
| mobidv 2555 | Formula-building rule for ... |
| mobid 2556 | Formula-building rule for ... |
| moa1 2557 | If an implication holds fo... |
| moan 2558 | "At most one" is still the... |
| moani 2559 | "At most one" is still tru... |
| moor 2560 | "At most one" is still the... |
| mooran1 2561 | "At most one" imports disj... |
| mooran2 2562 | "At most one" exports disj... |
| nfmo1 2563 | Bound-variable hypothesis ... |
| nfmod2 2564 | Bound-variable hypothesis ... |
| nfmodv 2565 | Bound-variable hypothesis ... |
| nfmov 2566 | Bound-variable hypothesis ... |
| nfmod 2567 | Bound-variable hypothesis ... |
| nfmo 2568 | Bound-variable hypothesis ... |
| mof 2569 | Version of ~ df-mo with di... |
| mo3 2570 | Alternate definition of th... |
| mo 2571 | Equivalent definitions of ... |
| mo4 2572 | At-most-one quantifier exp... |
| mo4f 2573 | At-most-one quantifier exp... |
| eu3v 2576 | An alternate way to expres... |
| eujust 2577 | Soundness justification th... |
| eujustALT 2578 | Alternate proof of ~ eujus... |
| eu6lem 2579 | Lemma of ~ eu6im . A diss... |
| eu6 2580 | Alternate definition of th... |
| eu6im 2581 | One direction of ~ eu6 nee... |
| euf 2582 | Version of ~ eu6 with disj... |
| euex 2583 | Existential uniqueness imp... |
| eumo 2584 | Existential uniqueness imp... |
| eumoi 2585 | Uniqueness inferred from e... |
| exmoeub 2586 | Existence implies that uni... |
| exmoeu 2587 | Existence is equivalent to... |
| moeuex 2588 | Uniqueness implies that ex... |
| moeu 2589 | Uniqueness is equivalent t... |
| eubi 2590 | Equivalence theorem for th... |
| eubii 2591 | Introduce unique existenti... |
| eubidv 2592 | Formula-building rule for ... |
| eubid 2593 | Formula-building rule for ... |
| nfeu1ALT 2594 | Alternate version of ~ nfe... |
| nfeu1 2595 | Bound-variable hypothesis ... |
| nfeud2 2596 | Bound-variable hypothesis ... |
| nfeudw 2597 | Bound-variable hypothesis ... |
| nfeud 2598 | Bound-variable hypothesis ... |
| nfeuw 2599 | Bound-variable hypothesis ... |
| nfeu 2600 | Bound-variable hypothesis ... |
| dfeu 2601 | Rederive ~ df-eu from the ... |
| dfmo2 2602 | Rederive ~ df-mo from the ... |
| euequ 2603 | There exists a unique set ... |
| sb8eulem 2604 | Lemma. Factor out the com... |
| sb8euv 2605 | Variable substitution in u... |
| sb8eu 2606 | Variable substitution in u... |
| sb8mo 2607 | Variable substitution for ... |
| cbvmovw 2608 | Change bound variable. Us... |
| cbvmow 2609 | Rule used to change bound ... |
| cbvmo 2610 | Rule used to change bound ... |
| cbveuvw 2611 | Change bound variable. Us... |
| cbveuw 2612 | Version of ~ cbveu with a ... |
| cbveu 2613 | Rule used to change bound ... |
| cbveuALT 2614 | Alternative proof of ~ cbv... |
| eu2 2615 | An alternate way of defini... |
| eu1 2616 | An alternate way to expres... |
| euor 2617 | Introduce a disjunct into ... |
| euorv 2618 | Introduce a disjunct into ... |
| euor2 2619 | Introduce or eliminate a d... |
| sbmo 2620 | Substitution into an at-mo... |
| eu4 2621 | Uniqueness using implicit ... |
| euimmo 2622 | Existential uniqueness imp... |
| euim 2623 | Add unique existential qua... |
| moanimlem 2624 | Factor out the common proo... |
| moanimv 2625 | Introduction of a conjunct... |
| moanim 2626 | Introduction of a conjunct... |
| euan 2627 | Introduction of a conjunct... |
| moanmo 2628 | Nested at-most-one quantif... |
| moaneu 2629 | Nested at-most-one and uni... |
| euanv 2630 | Introduction of a conjunct... |
| mopick 2631 | "At most one" picks a vari... |
| moexexlem 2632 | Factor out the proof skele... |
| 2moexv 2633 | Double quantification with... |
| moexexvw 2634 | "At most one" double quant... |
| 2moswapv 2635 | A condition allowing to sw... |
| 2euswapv 2636 | A condition allowing to sw... |
| 2euexv 2637 | Double quantification with... |
| 2exeuv 2638 | Double existential uniquen... |
| eupick 2639 | Existential uniqueness "pi... |
| eupicka 2640 | Version of ~ eupick with c... |
| eupickb 2641 | Existential uniqueness "pi... |
| eupickbi 2642 | Theorem *14.26 in [Whitehe... |
| mopick2 2643 | "At most one" can show the... |
| moexex 2644 | "At most one" double quant... |
| moexexv 2645 | "At most one" double quant... |
| 2moex 2646 | Double quantification with... |
| 2euex 2647 | Double quantification with... |
| 2eumo 2648 | Nested unique existential ... |
| 2eu2ex 2649 | Double existential uniquen... |
| 2moswap 2650 | A condition allowing to sw... |
| 2euswap 2651 | A condition allowing to sw... |
| 2exeu 2652 | Double existential uniquen... |
| 2mo2 2653 | Two ways of expressing "th... |
| 2mo 2654 | Two ways of expressing "th... |
| 2mos 2655 | Double "there exists at mo... |
| 2eu1 2656 | Double existential uniquen... |
| 2eu1v 2657 | Double existential uniquen... |
| 2eu2 2658 | Double existential uniquen... |
| 2eu3 2659 | Double existential uniquen... |
| 2eu4 2660 | This theorem provides us w... |
| 2eu5 2661 | An alternate definition of... |
| 2eu6 2662 | Two equivalent expressions... |
| 2eu7 2663 | Two equivalent expressions... |
| 2eu8 2664 | Two equivalent expressions... |
| euae 2665 | Two ways to express "exact... |
| exists1 2666 | Two ways to express "exact... |
| exists2 2667 | A condition implying that ... |
| barbara 2668 | "Barbara", one of the fund... |
| celarent 2669 | "Celarent", one of the syl... |
| darii 2670 | "Darii", one of the syllog... |
| dariiALT 2671 | Alternate proof of ~ darii... |
| ferio 2672 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2673 | Lemma for ~ barbari and th... |
| barbari 2674 | "Barbari", one of the syll... |
| barbariALT 2675 | Alternate proof of ~ barba... |
| celaront 2676 | "Celaront", one of the syl... |
| cesare 2677 | "Cesare", one of the syllo... |
| camestres 2678 | "Camestres", one of the sy... |
| festino 2679 | "Festino", one of the syll... |
| festinoALT 2680 | Alternate proof of ~ festi... |
| baroco 2681 | "Baroco", one of the syllo... |
| barocoALT 2682 | Alternate proof of ~ festi... |
| cesaro 2683 | "Cesaro", one of the syllo... |
| camestros 2684 | "Camestros", one of the sy... |
| datisi 2685 | "Datisi", one of the syllo... |
| disamis 2686 | "Disamis", one of the syll... |
| ferison 2687 | "Ferison", one of the syll... |
| bocardo 2688 | "Bocardo", one of the syll... |
| darapti 2689 | "Darapti", one of the syll... |
| daraptiALT 2690 | Alternate proof of ~ darap... |
| felapton 2691 | "Felapton", one of the syl... |
| calemes 2692 | "Calemes", one of the syll... |
| dimatis 2693 | "Dimatis", one of the syll... |
| fresison 2694 | "Fresison", one of the syl... |
| calemos 2695 | "Calemos", one of the syll... |
| fesapo 2696 | "Fesapo", one of the syllo... |
| bamalip 2697 | "Bamalip", one of the syll... |
| axia1 2698 | Left 'and' elimination (in... |
| axia2 2699 | Right 'and' elimination (i... |
| axia3 2700 | 'And' introduction (intuit... |
| axin1 2701 | 'Not' introduction (intuit... |
| axin2 2702 | 'Not' elimination (intuiti... |
| axio 2703 | Definition of 'or' (intuit... |
| axi4 2704 | Specialization (intuitioni... |
| axi5r 2705 | Converse of ~ axc4 (intuit... |
| axial 2706 | The setvar ` x ` is not fr... |
| axie1 2707 | The setvar ` x ` is not fr... |
| axie2 2708 | A key property of existent... |
| axi9 2709 | Axiom of existence (intuit... |
| axi10 2710 | Axiom of Quantifier Substi... |
| axi12 2711 | Axiom of Quantifier Introd... |
| axbnd 2712 | Axiom of Bundling (intuiti... |
| axexte 2714 | The axiom of extensionalit... |
| axextg 2715 | A generalization of the ax... |
| axextb 2716 | A bidirectional version of... |
| axextmo 2717 | There exists at most one s... |
| nulmo 2718 | There exists at most one e... |
| eleq1ab 2721 | Extension (in the sense of... |
| cleljustab 2722 | Extension of ~ cleljust fr... |
| abid 2723 | Simplification of class ab... |
| vexwt 2724 | A standard theorem of pred... |
| vexw 2725 | If ` ph ` is a theorem, th... |
| vextru 2726 | Every setvar is a member o... |
| nfsab1 2727 | Bound-variable hypothesis ... |
| hbab1 2728 | Bound-variable hypothesis ... |
| hbab 2729 | Bound-variable hypothesis ... |
| hbabg 2730 | Bound-variable hypothesis ... |
| nfsab 2731 | Bound-variable hypothesis ... |
| nfsabg 2732 | Bound-variable hypothesis ... |
| dfcleq 2734 | The defining characterizat... |
| cvjust 2735 | Every set is a class. Pro... |
| ax9ALT 2736 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2737 | A weaker version of ~ eleq... |
| eqriv 2738 | Infer equality of classes ... |
| eqrdv 2739 | Deduce equality of classes... |
| eqrdav 2740 | Deduce equality of classes... |
| eqid 2741 | Law of identity (reflexivi... |
| eqidd 2742 | Class identity law with an... |
| eqeq1d 2743 | Deduction from equality to... |
| eqeq1dALT 2744 | Alternate proof of ~ eqeq1... |
| eqeq1 2745 | Equality implies equivalen... |
| eqeq1i 2746 | Inference from equality to... |
| eqcomd 2747 | Deduction from commutative... |
| eqcom 2748 | Commutative law for class ... |
| eqcoms 2749 | Inference applying commuta... |
| eqcomi 2750 | Inference from commutative... |
| neqcomd 2751 | Commute an inequality. (C... |
| eqeq2d 2752 | Deduction from equality to... |
| eqeq2 2753 | Equality implies equivalen... |
| eqeq2i 2754 | Inference from equality to... |
| eqeqan12d 2755 | A useful inference for sub... |
| eqeqan12rd 2756 | A useful inference for sub... |
| eqeq12d 2757 | A useful inference for sub... |
| eqeq12 2758 | Equality relationship amon... |
| eqeq12i 2759 | A useful inference for sub... |
| eqeqan12dALT 2760 | Alternate proof of ~ eqeqa... |
| eqtr 2761 | Transitive law for class e... |
| eqtr2 2762 | A transitive law for class... |
| eqtr3 2763 | A transitive law for class... |
| eqtri 2764 | An equality transitivity i... |
| eqtr2i 2765 | An equality transitivity i... |
| eqtr3i 2766 | An equality transitivity i... |
| eqtr4i 2767 | An equality transitivity i... |
| 3eqtri 2768 | An inference from three ch... |
| 3eqtrri 2769 | An inference from three ch... |
| 3eqtr2i 2770 | An inference from three ch... |
| 3eqtr2ri 2771 | An inference from three ch... |
| 3eqtr3i 2772 | An inference from three ch... |
| 3eqtr3ri 2773 | An inference from three ch... |
| 3eqtr4i 2774 | An inference from three ch... |
| 3eqtr4ri 2775 | An inference from three ch... |
| eqtrd 2776 | An equality transitivity d... |
| eqtr2d 2777 | An equality transitivity d... |
| eqtr3d 2778 | An equality transitivity e... |
| eqtr4d 2779 | An equality transitivity e... |
| 3eqtrd 2780 | A deduction from three cha... |
| 3eqtrrd 2781 | A deduction from three cha... |
| 3eqtr2d 2782 | A deduction from three cha... |
| 3eqtr2rd 2783 | A deduction from three cha... |
| 3eqtr3d 2784 | A deduction from three cha... |
| 3eqtr3rd 2785 | A deduction from three cha... |
| 3eqtr4d 2786 | A deduction from three cha... |
| 3eqtr4rd 2787 | A deduction from three cha... |
| eqtrid 2788 | An equality transitivity d... |
| eqtr2id 2789 | An equality transitivity d... |
| eqtr3id 2790 | An equality transitivity d... |
| eqtr3di 2791 | An equality transitivity d... |
| eqtrdi 2792 | An equality transitivity d... |
| eqtr2di 2793 | An equality transitivity d... |
| eqtr4di 2794 | An equality transitivity d... |
| eqtr4id 2795 | An equality transitivity d... |
| sylan9eq 2796 | An equality transitivity d... |
| sylan9req 2797 | An equality transitivity d... |
| sylan9eqr 2798 | An equality transitivity d... |
| 3eqtr3g 2799 | A chained equality inferen... |
| 3eqtr3a 2800 | A chained equality inferen... |
| 3eqtr4g 2801 | A chained equality inferen... |
| 3eqtr4a 2802 | A chained equality inferen... |
| eq2tri 2803 | A compound transitive infe... |
| iseqsetvlem 2804 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2805 | Alternate proof of ~ iseqs... |
| abbi 2806 | Equivalent formulas yield ... |
| abbidv 2807 | Equivalent wff's yield equ... |
| abbii 2808 | Equivalent wff's yield equ... |
| abbid 2809 | Equivalent wff's yield equ... |
| abbib 2810 | Equal class abstractions r... |
| cbvabv 2811 | Rule used to change bound ... |
| cbvabw 2812 | Rule used to change bound ... |
| cbvab 2813 | Rule used to change bound ... |
| eqabbw 2814 | Version of ~ eqabb using i... |
| eqabcbw 2815 | Version of ~ eqabcb using ... |
| dfclel 2817 | Characterization of the el... |
| elex2 2818 | If a class contains anothe... |
| issettru 2819 | Weak version of ~ isset . ... |
| iseqsetv-clel 2820 | Alternate proof of ~ iseqs... |
| issetlem 2821 | Lemma for ~ elisset and ~ ... |
| elissetv 2822 | An element of a class exis... |
| elisset 2823 | An element of a class exis... |
| eleq1w 2824 | Weaker version of ~ eleq1 ... |
| eleq2w 2825 | Weaker version of ~ eleq2 ... |
| eleq1d 2826 | Deduction from equality to... |
| eleq2d 2827 | Deduction from equality to... |
| eleq2dALT 2828 | Alternate proof of ~ eleq2... |
| eleq1 2829 | Equality implies equivalen... |
| eleq2 2830 | Equality implies equivalen... |
| eleq12 2831 | Equality implies equivalen... |
| eleq1i 2832 | Inference from equality to... |
| eleq2i 2833 | Inference from equality to... |
| eleq12i 2834 | Inference from equality to... |
| eleq12d 2835 | Deduction from equality to... |
| eleq1a 2836 | A transitive-type law rela... |
| eqeltri 2837 | Substitution of equal clas... |
| eqeltrri 2838 | Substitution of equal clas... |
| eleqtri 2839 | Substitution of equal clas... |
| eleqtrri 2840 | Substitution of equal clas... |
| eqeltrd 2841 | Substitution of equal clas... |
| eqeltrrd 2842 | Deduction that substitutes... |
| eleqtrd 2843 | Deduction that substitutes... |
| eleqtrrd 2844 | Deduction that substitutes... |
| eqeltrid 2845 | A membership and equality ... |
| eqeltrrid 2846 | A membership and equality ... |
| eleqtrid 2847 | A membership and equality ... |
| eleqtrrid 2848 | A membership and equality ... |
| eqeltrdi 2849 | A membership and equality ... |
| eqeltrrdi 2850 | A membership and equality ... |
| eleqtrdi 2851 | A membership and equality ... |
| eleqtrrdi 2852 | A membership and equality ... |
| 3eltr3i 2853 | Substitution of equal clas... |
| 3eltr4i 2854 | Substitution of equal clas... |
| 3eltr3d 2855 | Substitution of equal clas... |
| 3eltr4d 2856 | Substitution of equal clas... |
| 3eltr3g 2857 | Substitution of equal clas... |
| 3eltr4g 2858 | Substitution of equal clas... |
| eleq2s 2859 | Substitution of equal clas... |
| eqneltri 2860 | If a class is not an eleme... |
| eqneltrd 2861 | If a class is not an eleme... |
| eqneltrrd 2862 | If a class is not an eleme... |
| neleqtrd 2863 | If a class is not an eleme... |
| neleqtrrd 2864 | If a class is not an eleme... |
| nelneq 2865 | A way of showing two class... |
| nelneq2 2866 | A way of showing two class... |
| eqsb1 2867 | Substitution for the left-... |
| clelsb1 2868 | Substitution for the first... |
| clelsb2 2869 | Substitution for the secon... |
| cleqh 2870 | Establish equality between... |
| hbxfreq 2871 | A utility lemma to transfe... |
| hblem 2872 | Change the free variable o... |
| hblemg 2873 | Change the free variable o... |
| eqabdv 2874 | Deduction from a wff to a ... |
| eqabcdv 2875 | Deduction from a wff to a ... |
| eqabi 2876 | Equality of a class variab... |
| abid1 2877 | Every class is equal to a ... |
| abid2 2878 | A simplification of class ... |
| eqab 2879 | One direction of ~ eqabb i... |
| eqabb 2880 | Equality of a class variab... |
| eqabcb 2881 | Equality of a class variab... |
| eqabrd 2882 | Equality of a class variab... |
| eqabri 2883 | Equality of a class variab... |
| eqabcri 2884 | Equality of a class variab... |
| clelab 2885 | Membership of a class vari... |
| clabel 2886 | Membership of a class abst... |
| sbab 2887 | The right-hand side of the... |
| nfcjust 2889 | Justification theorem for ... |
| nfci 2891 | Deduce that a class ` A ` ... |
| nfcii 2892 | Deduce that a class ` A ` ... |
| nfcr 2893 | Consequence of the not-fre... |
| nfcrALT 2894 | Alternate version of ~ nfc... |
| nfcri 2895 | Consequence of the not-fre... |
| nfcd 2896 | Deduce that a class ` A ` ... |
| nfcrd 2897 | Consequence of the not-fre... |
| nfcrii 2898 | Consequence of the not-fre... |
| nfceqdf 2899 | An equality theorem for ef... |
| nfceqi 2900 | Equality theorem for class... |
| nfcxfr 2901 | A utility lemma to transfe... |
| nfcxfrd 2902 | A utility lemma to transfe... |
| nfcv 2903 | If ` x ` is disjoint from ... |
| nfcvd 2904 | If ` x ` is disjoint from ... |
| nfab1 2905 | Bound-variable hypothesis ... |
| nfnfc1 2906 | The setvar ` x ` is bound ... |
| clelsb1fw 2907 | Substitution for the first... |
| clelsb1f 2908 | Substitution for the first... |
| nfab 2909 | Bound-variable hypothesis ... |
| nfabg 2910 | Bound-variable hypothesis ... |
| nfaba1 2911 | Bound-variable hypothesis ... |
| nfaba1g 2912 | Bound-variable hypothesis ... |
| nfeqd 2913 | Hypothesis builder for equ... |
| nfeld 2914 | Hypothesis builder for ele... |
| nfnfc 2915 | Hypothesis builder for ` F... |
| nfeq 2916 | Hypothesis builder for equ... |
| nfel 2917 | Hypothesis builder for ele... |
| nfeq1 2918 | Hypothesis builder for equ... |
| nfel1 2919 | Hypothesis builder for ele... |
| nfeq2 2920 | Hypothesis builder for equ... |
| nfel2 2921 | Hypothesis builder for ele... |
| drnfc1 2922 | Formula-building lemma for... |
| drnfc2 2923 | Formula-building lemma for... |
| nfabdw 2924 | Bound-variable hypothesis ... |
| nfabd 2925 | Bound-variable hypothesis ... |
| nfabd2 2926 | Bound-variable hypothesis ... |
| dvelimdc 2927 | Deduction form of ~ dvelim... |
| dvelimc 2928 | Version of ~ dvelim for cl... |
| nfcvf 2929 | If ` x ` and ` y ` are dis... |
| nfcvf2 2930 | If ` x ` and ` y ` are dis... |
| cleqf 2931 | Establish equality between... |
| eqabf 2932 | Equality of a class variab... |
| abid2f 2933 | A simplification of class ... |
| abid2fOLD 2934 | Obsolete version of ~ abid... |
| sbabel 2935 | Theorem to move a substitu... |
| neii 2938 | Inference associated with ... |
| neir 2939 | Inference associated with ... |
| nne 2940 | Negation of inequality. (... |
| neneqd 2941 | Deduction eliminating ineq... |
| neneq 2942 | From inequality to non-equ... |
| neqned 2943 | If it is not the case that... |
| neqne 2944 | From non-equality to inequ... |
| neirr 2945 | No class is unequal to its... |
| exmidne 2946 | Excluded middle with equal... |
| eqneqall 2947 | A contradiction concerning... |
| nonconne 2948 | Law of noncontradiction wi... |
| necon3ad 2949 | Contrapositive law deducti... |
| necon3bd 2950 | Contrapositive law deducti... |
| necon2ad 2951 | Contrapositive inference f... |
| necon2bd 2952 | Contrapositive inference f... |
| necon1ad 2953 | Contrapositive deduction f... |
| necon1bd 2954 | Contrapositive deduction f... |
| necon4ad 2955 | Contrapositive inference f... |
| necon4bd 2956 | Contrapositive inference f... |
| necon3d 2957 | Contrapositive law deducti... |
| necon1d 2958 | Contrapositive law deducti... |
| necon2d 2959 | Contrapositive inference f... |
| necon4d 2960 | Contrapositive inference f... |
| necon3ai 2961 | Contrapositive inference f... |
| necon3bi 2962 | Contrapositive inference f... |
| necon1ai 2963 | Contrapositive inference f... |
| necon1bi 2964 | Contrapositive inference f... |
| necon2ai 2965 | Contrapositive inference f... |
| necon2bi 2966 | Contrapositive inference f... |
| necon4ai 2967 | Contrapositive inference f... |
| necon3i 2968 | Contrapositive inference f... |
| necon1i 2969 | Contrapositive inference f... |
| necon2i 2970 | Contrapositive inference f... |
| necon4i 2971 | Contrapositive inference f... |
| necon3abid 2972 | Deduction from equality to... |
| necon3bbid 2973 | Deduction from equality to... |
| necon1abid 2974 | Contrapositive deduction f... |
| necon1bbid 2975 | Contrapositive inference f... |
| necon4abid 2976 | Contrapositive law deducti... |
| necon4bbid 2977 | Contrapositive law deducti... |
| necon2abid 2978 | Contrapositive deduction f... |
| necon2bbid 2979 | Contrapositive deduction f... |
| necon3bid 2980 | Deduction from equality to... |
| necon4bid 2981 | Contrapositive law deducti... |
| necon3abii 2982 | Deduction from equality to... |
| necon3bbii 2983 | Deduction from equality to... |
| necon1abii 2984 | Contrapositive inference f... |
| necon1bbii 2985 | Contrapositive inference f... |
| necon2abii 2986 | Contrapositive inference f... |
| necon2bbii 2987 | Contrapositive inference f... |
| necon3bii 2988 | Inference from equality to... |
| necom 2989 | Commutation of inequality.... |
| necomi 2990 | Inference from commutative... |
| necomd 2991 | Deduction from commutative... |
| nesym 2992 | Characterization of inequa... |
| nesymi 2993 | Inference associated with ... |
| nesymir 2994 | Inference associated with ... |
| neeq1d 2995 | Deduction for inequality. ... |
| neeq2d 2996 | Deduction for inequality. ... |
| neeq12d 2997 | Deduction for inequality. ... |
| neeq1 2998 | Equality theorem for inequ... |
| neeq2 2999 | Equality theorem for inequ... |
| neeq1i 3000 | Inference for inequality. ... |
| neeq2i 3001 | Inference for inequality. ... |
| neeq12i 3002 | Inference for inequality. ... |
| eqnetrd 3003 | Substitution of equal clas... |
| eqnetrrd 3004 | Substitution of equal clas... |
| neeqtrd 3005 | Substitution of equal clas... |
| eqnetri 3006 | Substitution of equal clas... |
| eqnetrri 3007 | Substitution of equal clas... |
| neeqtri 3008 | Substitution of equal clas... |
| neeqtrri 3009 | Substitution of equal clas... |
| neeqtrrd 3010 | Substitution of equal clas... |
| eqnetrrid 3011 | A chained equality inferen... |
| 3netr3d 3012 | Substitution of equality i... |
| 3netr4d 3013 | Substitution of equality i... |
| 3netr3g 3014 | Substitution of equality i... |
| 3netr4g 3015 | Substitution of equality i... |
| nebi 3016 | Contraposition law for ine... |
| pm13.18 3017 | Theorem *13.18 in [Whitehe... |
| pm13.181 3018 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3019 | Inference eliminating an i... |
| pm2.21ddne 3020 | A contradiction implies an... |
| pm2.61ne 3021 | Deduction eliminating an i... |
| pm2.61dne 3022 | Deduction eliminating an i... |
| pm2.61dane 3023 | Deduction eliminating an i... |
| pm2.61da2ne 3024 | Deduction eliminating two ... |
| pm2.61da3ne 3025 | Deduction eliminating thre... |
| pm2.61iine 3026 | Equality version of ~ pm2.... |
| mteqand 3027 | A modus tollens deduction ... |
| neor 3028 | Logical OR with an equalit... |
| neanior 3029 | A De Morgan's law for ineq... |
| ne3anior 3030 | A De Morgan's law for ineq... |
| neorian 3031 | A De Morgan's law for ineq... |
| nemtbir 3032 | An inference from an inequ... |
| nelne1 3033 | Two classes are different ... |
| nelne2 3034 | Two classes are different ... |
| nelelne 3035 | Two classes are different ... |
| neneor 3036 | If two classes are differe... |
| nfne 3037 | Bound-variable hypothesis ... |
| nfned 3038 | Bound-variable hypothesis ... |
| nabbib 3039 | Not equivalent wff's corre... |
| neli 3042 | Inference associated with ... |
| nelir 3043 | Inference associated with ... |
| nelcon3d 3044 | Contrapositive law deducti... |
| neleq12d 3045 | Equality theorem for negat... |
| neleq1 3046 | Equality theorem for negat... |
| neleq2 3047 | Equality theorem for negat... |
| nfnel 3048 | Bound-variable hypothesis ... |
| nfneld 3049 | Bound-variable hypothesis ... |
| nnel 3050 | Negation of negated member... |
| elnelne1 3051 | Two classes are different ... |
| elnelne2 3052 | Two classes are different ... |
| pm2.24nel 3053 | A contradiction concerning... |
| pm2.61danel 3054 | Deduction eliminating an e... |
| rgen 3057 | Generalization rule for re... |
| ralel 3058 | All elements of a class ar... |
| rgenw 3059 | Generalization rule for re... |
| rgen2w 3060 | Generalization rule for re... |
| mprg 3061 | Modus ponens combined with... |
| mprgbir 3062 | Modus ponens on biconditio... |
| ralrid 3063 | Sufficient condition for t... |
| raln 3064 | Restricted universally qua... |
| ralnex 3067 | Relationship between restr... |
| dfrex2 3068 | Relationship between restr... |
| nrex 3069 | Inference adding restricte... |
| alral 3070 | Universal quantification i... |
| rexex 3071 | Restricted existence impli... |
| rextru 3072 | Two ways of expressing tha... |
| ralimi2 3073 | Inference quantifying both... |
| reximi2 3074 | Inference quantifying both... |
| ralimia 3075 | Inference quantifying both... |
| reximia 3076 | Inference quantifying both... |
| ralimiaa 3077 | Inference quantifying both... |
| ralimi 3078 | Inference quantifying both... |
| reximi 3079 | Inference quantifying both... |
| ral2imi 3080 | Inference quantifying ante... |
| ralim 3081 | Distribution of restricted... |
| rexim 3082 | Theorem 19.22 of [Margaris... |
| ralbii2 3083 | Inference adding different... |
| rexbii2 3084 | Inference adding different... |
| ralbiia 3085 | Inference adding restricte... |
| rexbiia 3086 | Inference adding restricte... |
| ralbii 3087 | Inference adding restricte... |
| rexbii 3088 | Inference adding restricte... |
| ralanid 3089 | Cancellation law for restr... |
| rexanid 3090 | Cancellation law for restr... |
| ralcom3 3091 | A commutation law for rest... |
| dfral2 3092 | Relationship between restr... |
| rexnal 3093 | Relationship between restr... |
| ralinexa 3094 | A transformation of restri... |
| rexanali 3095 | A transformation of restri... |
| ralbi 3096 | Distribute a restricted un... |
| rexbi 3097 | Distribute restricted quan... |
| ralrexbid 3098 | Formula-building rule for ... |
| r19.35 3099 | Restricted quantifier vers... |
| r19.26m 3100 | Version of ~ 19.26 and ~ r... |
| r19.26 3101 | Restricted quantifier vers... |
| r19.26-3 3102 | Version of ~ r19.26 with t... |
| ralbiim 3103 | Split a biconditional and ... |
| r19.29 3104 | Restricted quantifier vers... |
| r19.29r 3105 | Restricted quantifier vers... |
| r19.29imd 3106 | Theorem 19.29 of [Margaris... |
| r19.40 3107 | Restricted quantifier vers... |
| r19.30 3108 | Restricted quantifier vers... |
| r19.43 3109 | Restricted quantifier vers... |
| 3r19.43 3110 | Restricted quantifier vers... |
| 2ralimi 3111 | Inference quantifying both... |
| 3ralimi 3112 | Inference quantifying both... |
| 4ralimi 3113 | Inference quantifying both... |
| 5ralimi 3114 | Inference quantifying both... |
| 6ralimi 3115 | Inference quantifying both... |
| 2ralbii 3116 | Inference adding two restr... |
| 2rexbii 3117 | Inference adding two restr... |
| 3ralbii 3118 | Inference adding three res... |
| 4ralbii 3119 | Inference adding four rest... |
| 2ralbiim 3120 | Split a biconditional and ... |
| ralnex2 3121 | Relationship between two r... |
| ralnex3 3122 | Relationship between three... |
| rexnal2 3123 | Relationship between two r... |
| rexnal3 3124 | Relationship between three... |
| nrexralim 3125 | Negation of a complex pred... |
| r19.26-2 3126 | Restricted quantifier vers... |
| 2r19.29 3127 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3128 | Theorem 19.29 of [Margaris... |
| r2allem 3129 | Lemma factoring out common... |
| r2exlem 3130 | Lemma factoring out common... |
| hbralrimi 3131 | Inference from Theorem 19.... |
| ralrimiv 3132 | Inference from Theorem 19.... |
| ralrimiva 3133 | Inference from Theorem 19.... |
| rexlimiva 3134 | Inference from Theorem 19.... |
| rexlimiv 3135 | Inference from Theorem 19.... |
| nrexdv 3136 | Deduction adding restricte... |
| ralrimivw 3137 | Inference from Theorem 19.... |
| rexlimivw 3138 | Weaker version of ~ rexlim... |
| ralrimdv 3139 | Inference from Theorem 19.... |
| rexlimdv 3140 | Inference from Theorem 19.... |
| ralrimdva 3141 | Inference from Theorem 19.... |
| rexlimdva 3142 | Inference from Theorem 19.... |
| rexlimdvaa 3143 | Inference from Theorem 19.... |
| rexlimdva2 3144 | Inference from Theorem 19.... |
| r19.29an 3145 | A commonly used pattern in... |
| rexlimdv3a 3146 | Inference from Theorem 19.... |
| rexlimdvw 3147 | Inference from Theorem 19.... |
| rexlimddv 3148 | Restricted existential eli... |
| r19.29a 3149 | A commonly used pattern in... |
| ralimdv2 3150 | Inference quantifying both... |
| reximdv2 3151 | Deduction quantifying both... |
| reximdvai 3152 | Deduction quantifying both... |
| ralimdva 3153 | Deduction quantifying both... |
| reximdva 3154 | Deduction quantifying both... |
| ralimdv 3155 | Deduction quantifying both... |
| reximdv 3156 | Deduction from Theorem 19.... |
| reximddv 3157 | Deduction from Theorem 19.... |
| reximddv3 3158 | Deduction from Theorem 19.... |
| reximssdv 3159 | Derivation of a restricted... |
| ralbidv2 3160 | Formula-building rule for ... |
| rexbidv2 3161 | Formula-building rule for ... |
| ralbidva 3162 | Formula-building rule for ... |
| rexbidva 3163 | Formula-building rule for ... |
| ralbidv 3164 | Formula-building rule for ... |
| rexbidv 3165 | Formula-building rule for ... |
| r19.21v 3166 | Restricted quantifier vers... |
| r19.37v 3167 | Restricted quantifier vers... |
| r19.23v 3168 | Restricted quantifier vers... |
| r19.36v 3169 | Restricted quantifier vers... |
| r19.27v 3170 | Restricted quantitifer ver... |
| r19.41v 3171 | Restricted quantifier vers... |
| r19.28v 3172 | Restricted quantifier vers... |
| r19.42v 3173 | Restricted quantifier vers... |
| r19.32v 3174 | Restricted quantifier vers... |
| r19.45v 3175 | Restricted quantifier vers... |
| r19.44v 3176 | One direction of a restric... |
| r2al 3177 | Double restricted universa... |
| r2ex 3178 | Double restricted existent... |
| r3al 3179 | Triple restricted universa... |
| r3ex 3180 | Triple existential quantif... |
| rgen2 3181 | Generalization rule for re... |
| ralrimivv 3182 | Inference from Theorem 19.... |
| rexlimivv 3183 | Inference from Theorem 19.... |
| ralrimivva 3184 | Inference from Theorem 19.... |
| ralrimdvv 3185 | Inference from Theorem 19.... |
| rgen3 3186 | Generalization rule for re... |
| ralrimivvva 3187 | Inference from Theorem 19.... |
| ralimdvva 3188 | Deduction doubly quantifyi... |
| reximdvva 3189 | Deduction doubly quantifyi... |
| ralimdvv 3190 | Deduction doubly quantifyi... |
| ralimdvvOLD 3191 | Obsolete version of ~ rali... |
| ralimd4v 3192 | Deduction quadrupally quan... |
| ralimd4vOLD 3193 | Obsolete version of ~ rali... |
| ralimd6v 3194 | Deduction sextupally quant... |
| ralimd6vOLD 3195 | Obsolete version of ~ rali... |
| ralrimdvva 3196 | Inference from Theorem 19.... |
| rexlimdvv 3197 | Inference from Theorem 19.... |
| rexlimdvva 3198 | Inference from Theorem 19.... |
| rexlimdvvva 3199 | Inference from Theorem 19.... |
| reximddv2 3200 | Double deduction from Theo... |
| r19.29vva 3201 | A commonly used pattern ba... |
| 2rexbiia 3202 | Inference adding two restr... |
| 2ralbidva 3203 | Formula-building rule for ... |
| 2rexbidva 3204 | Formula-building rule for ... |
| 2ralbidv 3205 | Formula-building rule for ... |
| 2rexbidv 3206 | Formula-building rule for ... |
| rexralbidv 3207 | Formula-building rule for ... |
| 3ralbidv 3208 | Formula-building rule for ... |
| 4ralbidv 3209 | Formula-building rule for ... |
| 6ralbidv 3210 | Formula-building rule for ... |
| r19.41vv 3211 | Version of ~ r19.41v with ... |
| reeanlem 3212 | Lemma factoring out common... |
| reeanv 3213 | Rearrange restricted exist... |
| 3reeanv 3214 | Rearrange three restricted... |
| 2ralor 3215 | Distribute restricted univ... |
| risset 3216 | Two ways to say " ` A ` be... |
| nelb 3217 | A definition of ` -. A e. ... |
| rspw 3218 | Restricted specialization.... |
| cbvralvw 3219 | Change the bound variable ... |
| cbvrexvw 3220 | Change the bound variable ... |
| cbvraldva 3221 | Rule used to change the bo... |
| cbvrexdva 3222 | Rule used to change the bo... |
| cbvral2vw 3223 | Change bound variables of ... |
| cbvrex2vw 3224 | Change bound variables of ... |
| cbvral3vw 3225 | Change bound variables of ... |
| cbvral4vw 3226 | Change bound variables of ... |
| cbvral6vw 3227 | Change bound variables of ... |
| cbvral8vw 3228 | Change bound variables of ... |
| rsp 3229 | Restricted specialization.... |
| rspa 3230 | Restricted specialization.... |
| rspe 3231 | Restricted specialization.... |
| rspec 3232 | Specialization rule for re... |
| r19.21bi 3233 | Inference from Theorem 19.... |
| r19.21be 3234 | Inference from Theorem 19.... |
| r19.21t 3235 | Restricted quantifier vers... |
| r19.21 3236 | Restricted quantifier vers... |
| r19.23t 3237 | Closed theorem form of ~ r... |
| r19.23 3238 | Restricted quantifier vers... |
| ralrimi 3239 | Inference from Theorem 19.... |
| ralrimia 3240 | Inference from Theorem 19.... |
| rexlimi 3241 | Restricted quantifier vers... |
| ralimdaa 3242 | Deduction quantifying both... |
| reximdai 3243 | Deduction from Theorem 19.... |
| r19.37 3244 | Restricted quantifier vers... |
| r19.41 3245 | Restricted quantifier vers... |
| ralrimd 3246 | Inference from Theorem 19.... |
| rexlimd2 3247 | Version of ~ rexlimd with ... |
| rexlimd 3248 | Deduction form of ~ rexlim... |
| r19.29af2 3249 | A commonly used pattern ba... |
| r19.29af 3250 | A commonly used pattern ba... |
| reximd2a 3251 | Deduction quantifying both... |
| ralbida 3252 | Formula-building rule for ... |
| rexbida 3253 | Formula-building rule for ... |
| ralbid 3254 | Formula-building rule for ... |
| rexbid 3255 | Formula-building rule for ... |
| rexbidvALT 3256 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3257 | Alternate proof of ~ rexbi... |
| rsp2 3258 | Restricted specialization,... |
| rsp2e 3259 | Restricted specialization.... |
| rspec2 3260 | Specialization rule for re... |
| rspec3 3261 | Specialization rule for re... |
| r2alf 3262 | Double restricted universa... |
| r2exf 3263 | Double restricted existent... |
| 2ralbida 3264 | Formula-building rule for ... |
| nfra1 3265 | The setvar ` x ` is not fr... |
| nfre1 3266 | The setvar ` x ` is not fr... |
| ralcom4 3267 | Commutation of restricted ... |
| rexcom4 3268 | Commutation of restricted ... |
| ralcom 3269 | Commutation of restricted ... |
| rexcom 3270 | Commutation of restricted ... |
| rexcom4a 3271 | Specialized existential co... |
| ralrot3 3272 | Rotate three restricted un... |
| ralcom13 3273 | Swap first and third restr... |
| rexcom13 3274 | Swap first and third restr... |
| rexrot4 3275 | Rotate four restricted exi... |
| 2ex2rexrot 3276 | Rotate two existential qua... |
| nfra2w 3277 | Similar to Lemma 24 of [Mo... |
| hbra1 3278 | The setvar ` x ` is not fr... |
| ralcomf 3279 | Commutation of restricted ... |
| rexcomf 3280 | Commutation of restricted ... |
| cbvralfw 3281 | Rule used to change bound ... |
| cbvrexfw 3282 | Rule used to change bound ... |
| cbvralw 3283 | Rule used to change bound ... |
| cbvrexw 3284 | Rule used to change bound ... |
| hbral 3285 | Bound-variable hypothesis ... |
| nfraldw 3286 | Deduction version of ~ nfr... |
| nfrexdw 3287 | Deduction version of ~ nfr... |
| nfralw 3288 | Bound-variable hypothesis ... |
| nfrexw 3289 | Bound-variable hypothesis ... |
| r19.12 3290 | Restricted quantifier vers... |
| reean 3291 | Rearrange restricted exist... |
| cbvralsvw 3292 | Change bound variable by u... |
| cbvrexsvw 3293 | Change bound variable by u... |
| cbvralsvwOLD 3294 | Obsolete version of ~ cbvr... |
| rexeq 3295 | Equality theorem for restr... |
| raleq 3296 | Equality theorem for restr... |
| raleqi 3297 | Equality inference for res... |
| rexeqi 3298 | Equality inference for res... |
| raleqdv 3299 | Equality deduction for res... |
| rexeqdv 3300 | Equality deduction for res... |
| raleqtrdv 3301 | Substitution of equal clas... |
| rexeqtrdv 3302 | Substitution of equal clas... |
| raleqtrrdv 3303 | Substitution of equal clas... |
| rexeqtrrdv 3304 | Substitution of equal clas... |
| raleqbidva 3305 | Equality deduction for res... |
| rexeqbidva 3306 | Equality deduction for res... |
| raleqbidvv 3307 | Version of ~ raleqbidv wit... |
| rexeqbidvv 3308 | Version of ~ rexeqbidv wit... |
| raleqbi1dv 3309 | Equality deduction for res... |
| rexeqbi1dv 3310 | Equality deduction for res... |
| raleleq 3311 | All elements of a class ar... |
| raleleqOLD 3312 | Obsolete version of ~ rale... |
| raleqbii 3313 | Equality deduction for res... |
| rexeqbii 3314 | Equality deduction for res... |
| raleqbidv 3315 | Equality deduction for res... |
| rexeqbidv 3316 | Equality deduction for res... |
| cbvraldva2 3317 | Rule used to change the bo... |
| cbvrexdva2 3318 | Rule used to change the bo... |
| sbralie 3319 | Implicit to explicit subst... |
| sbralieALT 3320 | Alternative shorter proof ... |
| sbralieOLD 3321 | Obsolete version of ~ sbra... |
| raleqf 3322 | Equality theorem for restr... |
| rexeqf 3323 | Equality theorem for restr... |
| raleqbid 3324 | Equality deduction for res... |
| rexeqbid 3325 | Equality deduction for res... |
| cbvralf 3326 | Rule used to change bound ... |
| cbvrexf 3327 | Rule used to change bound ... |
| cbvral 3328 | Rule used to change bound ... |
| cbvrex 3329 | Rule used to change bound ... |
| cbvralv 3330 | Change the bound variable ... |
| cbvrexv 3331 | Change the bound variable ... |
| cbvralsv 3332 | Change bound variable by u... |
| cbvrexsv 3333 | Change bound variable by u... |
| cbvral2v 3334 | Change bound variables of ... |
| cbvrex2v 3335 | Change bound variables of ... |
| cbvral3v 3336 | Change bound variables of ... |
| rgen2a 3337 | Generalization rule for re... |
| nfrald 3338 | Deduction version of ~ nfr... |
| nfrexd 3339 | Deduction version of ~ nfr... |
| nfral 3340 | Bound-variable hypothesis ... |
| nfrex 3341 | Bound-variable hypothesis ... |
| nfra2 3342 | Similar to Lemma 24 of [Mo... |
| ralcom2 3343 | Commutation of restricted ... |
| reu5 3348 | Restricted uniqueness in t... |
| reurmo 3349 | Restricted existential uni... |
| reurex 3350 | Restricted unique existenc... |
| mormo 3351 | Unrestricted "at most one"... |
| rmobiia 3352 | Formula-building rule for ... |
| reubiia 3353 | Formula-building rule for ... |
| rmobii 3354 | Formula-building rule for ... |
| reubii 3355 | Formula-building rule for ... |
| rmoanid 3356 | Cancellation law for restr... |
| reuanid 3357 | Cancellation law for restr... |
| 2reu2rex 3358 | Double restricted existent... |
| rmobidva 3359 | Formula-building rule for ... |
| reubidva 3360 | Formula-building rule for ... |
| rmobidv 3361 | Formula-building rule for ... |
| reubidv 3362 | Formula-building rule for ... |
| reueubd 3363 | Restricted existential uni... |
| rmo5 3364 | Restricted "at most one" i... |
| nrexrmo 3365 | Nonexistence implies restr... |
| moel 3366 | "At most one" element in a... |
| cbvrmovw 3367 | Change the bound variable ... |
| cbvreuvw 3368 | Change the bound variable ... |
| rmobida 3369 | Formula-building rule for ... |
| reubida 3370 | Formula-building rule for ... |
| cbvrmow 3371 | Change the bound variable ... |
| cbvreuw 3372 | Change the bound variable ... |
| nfrmo1 3373 | The setvar ` x ` is not fr... |
| nfreu1 3374 | The setvar ` x ` is not fr... |
| nfrmow 3375 | Bound-variable hypothesis ... |
| nfreuw 3376 | Bound-variable hypothesis ... |
| rmoeq1 3377 | Equality theorem for restr... |
| reueq1 3378 | Equality theorem for restr... |
| rmoeqd 3379 | Equality deduction for res... |
| reueqd 3380 | Equality deduction for res... |
| reueqdv 3381 | Formula-building rule for ... |
| reueqbidv 3382 | Formula-building rule for ... |
| rmoeq1f 3383 | Equality theorem for restr... |
| reueq1f 3384 | Equality theorem for restr... |
| cbvreu 3385 | Change the bound variable ... |
| cbvrmo 3386 | Change the bound variable ... |
| cbvrmov 3387 | Change the bound variable ... |
| cbvreuv 3388 | Change the bound variable ... |
| nfrmod 3389 | Deduction version of ~ nfr... |
| nfreud 3390 | Deduction version of ~ nfr... |
| nfrmo 3391 | Bound-variable hypothesis ... |
| nfreu 3392 | Bound-variable hypothesis ... |
| rabbidva2 3395 | Equivalent wff's yield equ... |
| rabbia2 3396 | Equivalent wff's yield equ... |
| rabbiia 3397 | Equivalent formulas yield ... |
| rabbii 3398 | Equivalent wff's correspon... |
| rabbidva 3399 | Equivalent wff's yield equ... |
| rabbidv 3400 | Equivalent wff's yield equ... |
| rabbieq 3401 | Equivalent wff's correspon... |
| rabswap 3402 | Swap with a membership rel... |
| cbvrabv 3403 | Rule to change the bound v... |
| rabeqcda 3404 | When ` ps ` is always true... |
| rabeqc 3405 | A restricted class abstrac... |
| rabeqi 3406 | Equality theorem for restr... |
| rabeq 3407 | Equality theorem for restr... |
| rabeqdv 3408 | Equality of restricted cla... |
| rabeqbidva 3409 | Equality of restricted cla... |
| rabeqbidvaOLD 3410 | Obsolete version of ~ rabe... |
| rabeqbidv 3411 | Equality of restricted cla... |
| rabrabi 3412 | Abstract builder restricte... |
| nfrab1 3413 | The abstraction variable i... |
| rabid 3414 | An "identity" law of concr... |
| rabidim1 3415 | Membership in a restricted... |
| reqabi 3416 | Inference from equality of... |
| rabrab 3417 | Abstract builder restricte... |
| rabbida4 3418 | Version of ~ rabbidva2 wit... |
| rabbida 3419 | Equivalent wff's yield equ... |
| rabbid 3420 | Version of ~ rabbidv with ... |
| rabeqd 3421 | Deduction form of ~ rabeq ... |
| rabeqbida 3422 | Version of ~ rabeqbidva wi... |
| rabbi 3423 | Equivalent wff's correspon... |
| rabid2f 3424 | An "identity" law for rest... |
| rabid2im 3425 | One direction of ~ rabid2 ... |
| rabid2 3426 | An "identity" law for rest... |
| rabeqf 3427 | Equality theorem for restr... |
| cbvrabw 3428 | Rule to change the bound v... |
| cbvrabwOLD 3429 | Obsolete version of ~ cbvr... |
| nfrabw 3430 | A variable not free in a w... |
| nfrab 3431 | A variable not free in a w... |
| cbvrab 3432 | Rule to change the bound v... |
| vjust 3434 | Justification theorem for ... |
| dfv2 3436 | Alternate definition of th... |
| vex 3437 | All setvar variables are s... |
| elv 3438 | If a proposition is implie... |
| elvd 3439 | If a proposition is implie... |
| el2v 3440 | If a proposition is implie... |
| el3v 3441 | If a proposition is implie... |
| el3v3 3442 | If a proposition is implie... |
| eqv 3443 | The universe contains ever... |
| eqvf 3444 | The universe contains ever... |
| abv 3445 | The class of sets verifyin... |
| abvALT 3446 | Alternate proof of ~ abv ,... |
| isset 3447 | Two ways to express that "... |
| cbvexeqsetf 3448 | The expression ` E. x x = ... |
| issetft 3449 | Closed theorem form of ~ i... |
| issetf 3450 | A version of ~ isset that ... |
| isseti 3451 | A way to say " ` A ` is a ... |
| issetri 3452 | A way to say " ` A ` is a ... |
| eqvisset 3453 | A class equal to a variabl... |
| elex 3454 | If a class is a member of ... |
| elexi 3455 | If a class is a member of ... |
| elexd 3456 | If a class is a member of ... |
| elex22 3457 | If two classes each contai... |
| prcnel 3458 | A proper class doesn't bel... |
| ralv 3459 | A universal quantifier res... |
| rexv 3460 | An existential quantifier ... |
| reuv 3461 | A unique existential quant... |
| rmov 3462 | An at-most-one quantifier ... |
| rabab 3463 | A class abstraction restri... |
| rexcom4b 3464 | Specialized existential co... |
| ceqsal1t 3465 | One direction of ~ ceqsalt... |
| ceqsalt 3466 | Closed theorem version of ... |
| ceqsralt 3467 | Restricted quantifier vers... |
| ceqsalg 3468 | A representation of explic... |
| ceqsalgALT 3469 | Alternate proof of ~ ceqsa... |
| ceqsal 3470 | A representation of explic... |
| ceqsalALT 3471 | A representation of explic... |
| ceqsalv 3472 | A representation of explic... |
| ceqsralv 3473 | Restricted quantifier vers... |
| gencl 3474 | Implicit substitution for ... |
| 2gencl 3475 | Implicit substitution for ... |
| 3gencl 3476 | Implicit substitution for ... |
| cgsexg 3477 | Implicit substitution infe... |
| cgsex2g 3478 | Implicit substitution infe... |
| cgsex4g 3479 | An implicit substitution i... |
| ceqsex 3480 | Elimination of an existent... |
| ceqsexv 3481 | Elimination of an existent... |
| ceqsexv2d 3482 | Elimination of an existent... |
| ceqsexv2dOLD 3483 | Obsolete version of ~ ceqs... |
| ceqsex2 3484 | Elimination of two existen... |
| ceqsex2v 3485 | Elimination of two existen... |
| ceqsex3v 3486 | Elimination of three exist... |
| ceqsex4v 3487 | Elimination of four existe... |
| ceqsex6v 3488 | Elimination of six existen... |
| ceqsex8v 3489 | Elimination of eight exist... |
| gencbvex 3490 | Change of bound variable u... |
| gencbvex2 3491 | Restatement of ~ gencbvex ... |
| gencbval 3492 | Change of bound variable u... |
| sbhypf 3493 | Introduce an explicit subs... |
| spcimgft 3494 | Closed theorem form of ~ s... |
| spcimgfi1 3495 | A closed version of ~ spci... |
| spcimgfi1OLD 3496 | Obsolete version of ~ spci... |
| spcgft 3497 | A closed version of ~ spcg... |
| spcimgf 3498 | Rule of specialization, us... |
| spcimegf 3499 | Existential specialization... |
| vtoclgft 3500 | Closed theorem form of ~ v... |
| vtocleg 3501 | Implicit substitution of a... |
| vtoclg 3502 | Implicit substitution of a... |
| vtocle 3503 | Implicit substitution of a... |
| vtoclbg 3504 | Implicit substitution of a... |
| vtocl 3505 | Implicit substitution of a... |
| vtoclOLD 3506 | Obsolete version of ~ vtoc... |
| vtocldf 3507 | Implicit substitution of a... |
| vtocld 3508 | Implicit substitution of a... |
| vtocl2d 3509 | Implicit substitution of t... |
| vtoclef 3510 | Implicit substitution of a... |
| vtoclf 3511 | Implicit substitution of a... |
| vtocl2 3512 | Implicit substitution of c... |
| vtocl3 3513 | Implicit substitution of c... |
| vtoclb 3514 | Implicit substitution of a... |
| vtoclgf 3515 | Implicit substitution of a... |
| vtoclg1f 3516 | Version of ~ vtoclgf with ... |
| vtocl2gf 3517 | Implicit substitution of a... |
| vtocl3gf 3518 | Implicit substitution of a... |
| vtocl2g 3519 | Implicit substitution of 2... |
| vtocl3g 3520 | Implicit substitution of a... |
| vtoclgaf 3521 | Implicit substitution of a... |
| vtoclga 3522 | Implicit substitution of a... |
| vtocl2ga 3523 | Implicit substitution of 2... |
| vtocl2gaf 3524 | Implicit substitution of 2... |
| vtocl3gaf 3525 | Implicit substitution of 3... |
| vtocl3ga 3526 | Implicit substitution of 3... |
| vtocl4g 3527 | Implicit substitution of 4... |
| vtocl4ga 3528 | Implicit substitution of 4... |
| vtoclegft 3529 | Implicit substitution of a... |
| vtoclri 3530 | Implicit substitution of a... |
| spcgf 3531 | Rule of specialization, us... |
| spcegf 3532 | Existential specialization... |
| spcimdv 3533 | Restricted specialization,... |
| spcdv 3534 | Rule of specialization, us... |
| spcimedv 3535 | Restricted existential spe... |
| spcgv 3536 | Rule of specialization, us... |
| spcegv 3537 | Existential specialization... |
| spcedv 3538 | Existential specialization... |
| spc2egv 3539 | Existential specialization... |
| spc2gv 3540 | Specialization with two qu... |
| spc2ed 3541 | Existential specialization... |
| spc2d 3542 | Specialization with 2 quan... |
| spc3egv 3543 | Existential specialization... |
| spc3gv 3544 | Specialization with three ... |
| spcv 3545 | Rule of specialization, us... |
| spcev 3546 | Existential specialization... |
| spc2ev 3547 | Existential specialization... |
| rspct 3548 | A closed version of ~ rspc... |
| rspcdf 3549 | Restricted specialization,... |
| rspc 3550 | Restricted specialization,... |
| rspce 3551 | Restricted existential spe... |
| rspcimdv 3552 | Restricted specialization,... |
| rspcimedv 3553 | Restricted existential spe... |
| rspcdv 3554 | Restricted specialization,... |
| rspcedv 3555 | Restricted existential spe... |
| rspcebdv 3556 | Restricted existential spe... |
| rspcdv2 3557 | Restricted specialization,... |
| rspcv 3558 | Restricted specialization,... |
| rspccv 3559 | Restricted specialization,... |
| rspcva 3560 | Restricted specialization,... |
| rspccva 3561 | Restricted specialization,... |
| rspcev 3562 | Restricted existential spe... |
| rspcdva 3563 | Restricted specialization,... |
| rspcedvd 3564 | Restricted existential spe... |
| rspcedvdw 3565 | Version of ~ rspcedvd wher... |
| rspceb2dv 3566 | Restricted existential spe... |
| rspcime 3567 | Prove a restricted existen... |
| rspceaimv 3568 | Restricted existential spe... |
| rspcedeq1vd 3569 | Restricted existential spe... |
| rspcedeq2vd 3570 | Restricted existential spe... |
| rspc2 3571 | Restricted specialization ... |
| rspc2gv 3572 | Restricted specialization ... |
| rspc2v 3573 | 2-variable restricted spec... |
| rspc2va 3574 | 2-variable restricted spec... |
| rspc2ev 3575 | 2-variable restricted exis... |
| 2rspcedvdw 3576 | Double application of ~ rs... |
| rspc2dv 3577 | 2-variable restricted spec... |
| rspc3v 3578 | 3-variable restricted spec... |
| rspc3ev 3579 | 3-variable restricted exis... |
| 3rspcedvdw 3580 | Triple application of ~ rs... |
| rspc3dv 3581 | 3-variable restricted spec... |
| rspc4v 3582 | 4-variable restricted spec... |
| rspc6v 3583 | 6-variable restricted spec... |
| rspc8v 3584 | 8-variable restricted spec... |
| rspceeqv 3585 | Restricted existential spe... |
| ralxpxfr2d 3586 | Transfer a universal quant... |
| rexraleqim 3587 | Statement following from e... |
| eqvincg 3588 | A variable introduction la... |
| eqvinc 3589 | A variable introduction la... |
| eqvincf 3590 | A variable introduction la... |
| alexeqg 3591 | Two ways to express substi... |
| ceqex 3592 | Equality implies equivalen... |
| ceqsexg 3593 | A representation of explic... |
| ceqsexgv 3594 | Elimination of an existent... |
| ceqsrexv 3595 | Elimination of a restricte... |
| ceqsrexbv 3596 | Elimination of a restricte... |
| ceqsralbv 3597 | Elimination of a restricte... |
| ceqsrex2v 3598 | Elimination of a restricte... |
| clel2g 3599 | Alternate definition of me... |
| clel2 3600 | Alternate definition of me... |
| clel3g 3601 | Alternate definition of me... |
| clel3 3602 | Alternate definition of me... |
| clel4g 3603 | Alternate definition of me... |
| clel4 3604 | Alternate definition of me... |
| clel5 3605 | Alternate definition of cl... |
| pm13.183 3606 | Compare theorem *13.183 in... |
| rr19.3v 3607 | Restricted quantifier vers... |
| rr19.28v 3608 | Restricted quantifier vers... |
| elab6g 3609 | Membership in a class abst... |
| elabd2 3610 | Membership in a class abst... |
| elabd3 3611 | Membership in a class abst... |
| elabgt 3612 | Membership in a class abst... |
| elabgtOLD 3613 | Obsolete version of ~ elab... |
| elabgf 3614 | Membership in a class abst... |
| elabf 3615 | Membership in a class abst... |
| elabg 3616 | Membership in a class abst... |
| elabgw 3617 | Membership in a class abst... |
| elab2gw 3618 | Membership in a class abst... |
| elab 3619 | Membership in a class abst... |
| elab2g 3620 | Membership in a class abst... |
| elabd 3621 | Explicit demonstration the... |
| elab2 3622 | Membership in a class abst... |
| elab4g 3623 | Membership in a class abst... |
| elab3gf 3624 | Membership in a class abst... |
| elab3g 3625 | Membership in a class abst... |
| elab3 3626 | Membership in a class abst... |
| elrabi 3627 | Implication for the member... |
| elrabf 3628 | Membership in a restricted... |
| rabtru 3629 | Abstract builder using the... |
| elrab3t 3630 | Membership in a restricted... |
| elrab 3631 | Membership in a restricted... |
| elrab3 3632 | Membership in a restricted... |
| elrabd 3633 | Membership in a restricted... |
| elrab2 3634 | Membership in a restricted... |
| elrab2w 3635 | Membership in a restricted... |
| ralab 3636 | Universal quantification o... |
| ralrab 3637 | Universal quantification o... |
| rexab 3638 | Existential quantification... |
| rexrab 3639 | Existential quantification... |
| ralab2 3640 | Universal quantification o... |
| ralrab2 3641 | Universal quantification o... |
| rexab2 3642 | Existential quantification... |
| rexrab2 3643 | Existential quantification... |
| reurab 3644 | Restricted existential uni... |
| abidnf 3645 | Identity used to create cl... |
| dedhb 3646 | A deduction theorem for co... |
| class2seteq 3647 | Writing a set as a class a... |
| nelrdva 3648 | Deduce negative membership... |
| eqeu 3649 | A condition which implies ... |
| moeq 3650 | There exists at most one s... |
| eueq 3651 | A class is a set if and on... |
| eueqi 3652 | There exists a unique set ... |
| eueq2 3653 | Equality has existential u... |
| eueq3 3654 | Equality has existential u... |
| moeq3 3655 | "At most one" property of ... |
| mosub 3656 | "At most one" remains true... |
| mo2icl 3657 | Theorem for inferring "at ... |
| mob2 3658 | Consequence of "at most on... |
| moi2 3659 | Consequence of "at most on... |
| mob 3660 | Equality implied by "at mo... |
| moi 3661 | Equality implied by "at mo... |
| morex 3662 | Derive membership from uni... |
| euxfr2w 3663 | Transfer existential uniqu... |
| euxfrw 3664 | Transfer existential uniqu... |
| euxfr2 3665 | Transfer existential uniqu... |
| euxfr 3666 | Transfer existential uniqu... |
| euind 3667 | Existential uniqueness via... |
| reu2 3668 | A way to express restricte... |
| reu6 3669 | A way to express restricte... |
| reu3 3670 | A way to express restricte... |
| reu6i 3671 | A condition which implies ... |
| eqreu 3672 | A condition which implies ... |
| rmo4 3673 | Restricted "at most one" u... |
| reu4 3674 | Restricted uniqueness usin... |
| reu7 3675 | Restricted uniqueness usin... |
| reu8 3676 | Restricted uniqueness usin... |
| rmo3f 3677 | Restricted "at most one" u... |
| rmo4f 3678 | Restricted "at most one" u... |
| reu2eqd 3679 | Deduce equality from restr... |
| reueq 3680 | Equality has existential u... |
| rmoeq 3681 | Equality's restricted exis... |
| rmoan 3682 | Restricted "at most one" s... |
| rmoim 3683 | Restricted "at most one" i... |
| rmoimia 3684 | Restricted "at most one" i... |
| rmoimi 3685 | Restricted "at most one" i... |
| rmoimi2 3686 | Restricted "at most one" i... |
| 2reu5a 3687 | Double restricted existent... |
| reuimrmo 3688 | Restricted uniqueness impl... |
| 2reuswap 3689 | A condition allowing swap ... |
| 2reuswap2 3690 | A condition allowing swap ... |
| reuxfrd 3691 | Transfer existential uniqu... |
| reuxfr 3692 | Transfer existential uniqu... |
| reuxfr1d 3693 | Transfer existential uniqu... |
| reuxfr1ds 3694 | Transfer existential uniqu... |
| reuxfr1 3695 | Transfer existential uniqu... |
| reuind 3696 | Existential uniqueness via... |
| 2rmorex 3697 | Double restricted quantifi... |
| 2reu5lem1 3698 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3699 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3700 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3701 | Double restricted existent... |
| 2reurmo 3702 | Double restricted quantifi... |
| 2reurex 3703 | Double restricted quantifi... |
| 2rmoswap 3704 | A condition allowing to sw... |
| 2rexreu 3705 | Double restricted existent... |
| cdeqi 3708 | Deduce conditional equalit... |
| cdeqri 3709 | Property of conditional eq... |
| cdeqth 3710 | Deduce conditional equalit... |
| cdeqnot 3711 | Distribute conditional equ... |
| cdeqal 3712 | Distribute conditional equ... |
| cdeqab 3713 | Distribute conditional equ... |
| cdeqal1 3714 | Distribute conditional equ... |
| cdeqab1 3715 | Distribute conditional equ... |
| cdeqim 3716 | Distribute conditional equ... |
| cdeqcv 3717 | Conditional equality for s... |
| cdeqeq 3718 | Distribute conditional equ... |
| cdeqel 3719 | Distribute conditional equ... |
| nfcdeq 3720 | If we have a conditional e... |
| nfccdeq 3721 | Variation of ~ nfcdeq for ... |
| rru 3722 | Relative version of Russel... |
| ru 3723 | Russell's Paradox. Propos... |
| ruOLD 3724 | Obsolete version of ~ ru a... |
| dfsbcq 3727 | Proper substitution of a c... |
| dfsbcq2 3728 | This theorem, which is sim... |
| sbsbc 3729 | Show that ~ df-sb and ~ df... |
| sbceq1d 3730 | Equality theorem for class... |
| sbceq1dd 3731 | Equality theorem for class... |
| sbceqbid 3732 | Equality theorem for class... |
| sbc8g 3733 | This is the closest we can... |
| sbc2or 3734 | The disjunction of two equ... |
| sbcex 3735 | By our definition of prope... |
| sbceq1a 3736 | Equality theorem for class... |
| sbceq2a 3737 | Equality theorem for class... |
| spsbc 3738 | Specialization: if a formu... |
| spsbcd 3739 | Specialization: if a formu... |
| sbcth 3740 | A substitution into a theo... |
| sbcthdv 3741 | Deduction version of ~ sbc... |
| sbcid 3742 | An identity theorem for su... |
| nfsbc1d 3743 | Deduction version of ~ nfs... |
| nfsbc1 3744 | Bound-variable hypothesis ... |
| nfsbc1v 3745 | Bound-variable hypothesis ... |
| nfsbcdw 3746 | Deduction version of ~ nfs... |
| nfsbcw 3747 | Bound-variable hypothesis ... |
| sbccow 3748 | A composition law for clas... |
| nfsbcd 3749 | Deduction version of ~ nfs... |
| nfsbc 3750 | Bound-variable hypothesis ... |
| sbcco 3751 | A composition law for clas... |
| sbcco2 3752 | A composition law for clas... |
| sbc5 3753 | An equivalence for class s... |
| sbc5ALT 3754 | Alternate proof of ~ sbc5 ... |
| sbc6g 3755 | An equivalence for class s... |
| sbc6 3756 | An equivalence for class s... |
| sbc7 3757 | An equivalence for class s... |
| cbvsbcw 3758 | Change bound variables in ... |
| cbvsbcvw 3759 | Change the bound variable ... |
| cbvsbc 3760 | Change bound variables in ... |
| cbvsbcv 3761 | Change the bound variable ... |
| sbciegft 3762 | Conversion of implicit sub... |
| sbciegf 3763 | Conversion of implicit sub... |
| sbcieg 3764 | Conversion of implicit sub... |
| sbcie2g 3765 | Conversion of implicit sub... |
| sbcie 3766 | Conversion of implicit sub... |
| sbciedf 3767 | Conversion of implicit sub... |
| sbcied 3768 | Conversion of implicit sub... |
| sbcied2 3769 | Conversion of implicit sub... |
| elrabsf 3770 | Membership in a restricted... |
| eqsbc1 3771 | Substitution for the left-... |
| sbcng 3772 | Move negation in and out o... |
| sbcimg 3773 | Distribution of class subs... |
| sbcan 3774 | Distribution of class subs... |
| sbcor 3775 | Distribution of class subs... |
| sbcbig 3776 | Distribution of class subs... |
| sbcn1 3777 | Move negation in and out o... |
| sbcim1 3778 | Distribution of class subs... |
| sbcbid 3779 | Formula-building deduction... |
| sbcbidv 3780 | Formula-building deduction... |
| sbcbii 3781 | Formula-building inference... |
| sbcbi1 3782 | Distribution of class subs... |
| sbcbi2 3783 | Substituting into equivale... |
| sbcal 3784 | Move universal quantifier ... |
| sbcex2 3785 | Move existential quantifie... |
| sbceqal 3786 | Class version of one impli... |
| sbeqalb 3787 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3788 | Substitution for the right... |
| sbc3an 3789 | Distribution of class subs... |
| sbcel1v 3790 | Class substitution into a ... |
| sbcel2gv 3791 | Class substitution into a ... |
| sbcel21v 3792 | Class substitution into a ... |
| sbcimdv 3793 | Substitution analogue of T... |
| sbctt 3794 | Substitution for a variabl... |
| sbcgf 3795 | Substitution for a variabl... |
| sbc19.21g 3796 | Substitution for a variabl... |
| sbcg 3797 | Substitution for a variabl... |
| sbcgfi 3798 | Substitution for a variabl... |
| sbc2iegf 3799 | Conversion of implicit sub... |
| sbc2ie 3800 | Conversion of implicit sub... |
| sbc2iedv 3801 | Conversion of implicit sub... |
| sbc3ie 3802 | Conversion of implicit sub... |
| sbccomlem 3803 | Lemma for ~ sbccom . (Con... |
| sbccomlemOLD 3804 | Obsolete version of ~ sbcc... |
| sbccom 3805 | Commutative law for double... |
| sbcralt 3806 | Interchange class substitu... |
| sbcrext 3807 | Interchange class substitu... |
| sbcralg 3808 | Interchange class substitu... |
| sbcrex 3809 | Interchange class substitu... |
| sbcreu 3810 | Interchange class substitu... |
| reu8nf 3811 | Restricted uniqueness usin... |
| sbcabel 3812 | Interchange class substitu... |
| rspsbc 3813 | Restricted quantifier vers... |
| rspsbca 3814 | Restricted quantifier vers... |
| rspesbca 3815 | Existence form of ~ rspsbc... |
| spesbc 3816 | Existence form of ~ spsbc ... |
| spesbcd 3817 | form of ~ spsbc . (Contri... |
| sbcth2 3818 | A substitution into a theo... |
| ra4v 3819 | Version of ~ ra4 with a di... |
| ra4 3820 | Restricted quantifier vers... |
| rmo2 3821 | Alternate definition of re... |
| rmo2i 3822 | Condition implying restric... |
| rmo3 3823 | Restricted "at most one" u... |
| rmob 3824 | Consequence of "at most on... |
| rmoi 3825 | Consequence of "at most on... |
| rmob2 3826 | Consequence of "restricted... |
| rmoi2 3827 | Consequence of "restricted... |
| rmoanim 3828 | Introduction of a conjunct... |
| rmoanimALT 3829 | Alternate proof of ~ rmoan... |
| reuan 3830 | Introduction of a conjunct... |
| 2reu1 3831 | Double restricted existent... |
| 2reu2 3832 | Double restricted existent... |
| csb2 3835 | Alternate expression for t... |
| csbeq1 3836 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3837 | Equality deduction for pro... |
| csbeq2 3838 | Substituting into equivale... |
| csbeq2d 3839 | Formula-building deduction... |
| csbeq2dv 3840 | Formula-building deduction... |
| csbeq2i 3841 | Formula-building inference... |
| csbeq12dv 3842 | Formula-building inference... |
| cbvcsbw 3843 | Change bound variables in ... |
| cbvcsb 3844 | Change bound variables in ... |
| cbvcsbv 3845 | Change the bound variable ... |
| csbid 3846 | Analogue of ~ sbid for pro... |
| csbeq1a 3847 | Equality theorem for prope... |
| csbcow 3848 | Composition law for chaine... |
| csbco 3849 | Composition law for chaine... |
| csbtt 3850 | Substitution doesn't affec... |
| csbconstgf 3851 | Substitution doesn't affec... |
| csbconstg 3852 | Substitution doesn't affec... |
| csbgfi 3853 | Substitution for a variabl... |
| csbconstgi 3854 | The proper substitution of... |
| nfcsb1d 3855 | Bound-variable hypothesis ... |
| nfcsb1 3856 | Bound-variable hypothesis ... |
| nfcsb1v 3857 | Bound-variable hypothesis ... |
| nfcsbd 3858 | Deduction version of ~ nfc... |
| nfcsbw 3859 | Bound-variable hypothesis ... |
| nfcsb 3860 | Bound-variable hypothesis ... |
| csbhypf 3861 | Introduce an explicit subs... |
| csbiebt 3862 | Conversion of implicit sub... |
| csbiedf 3863 | Conversion of implicit sub... |
| csbieb 3864 | Bidirectional conversion b... |
| csbiebg 3865 | Bidirectional conversion b... |
| csbiegf 3866 | Conversion of implicit sub... |
| csbief 3867 | Conversion of implicit sub... |
| csbie 3868 | Conversion of implicit sub... |
| csbied 3869 | Conversion of implicit sub... |
| csbied2 3870 | Conversion of implicit sub... |
| csbie2t 3871 | Conversion of implicit sub... |
| csbie2 3872 | Conversion of implicit sub... |
| csbie2g 3873 | Conversion of implicit sub... |
| cbvrabcsfw 3874 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3875 | A more general version of ... |
| cbvrexcsf 3876 | A more general version of ... |
| cbvreucsf 3877 | A more general version of ... |
| cbvrabcsf 3878 | A more general version of ... |
| cbvralv2 3879 | Rule used to change the bo... |
| cbvrexv2 3880 | Rule used to change the bo... |
| rspc2vd 3881 | Deduction version of 2-var... |
| difjust 3887 | Soundness justification th... |
| unjust 3889 | Soundness justification th... |
| injust 3891 | Soundness justification th... |
| dfin5 3893 | Alternate definition for t... |
| dfdif2 3894 | Alternate definition of cl... |
| eldif 3895 | Expansion of membership in... |
| eldifd 3896 | If a class is in one class... |
| eldifad 3897 | If a class is in the diffe... |
| eldifbd 3898 | If a class is in the diffe... |
| elneeldif 3899 | The elements of a set diff... |
| velcomp 3900 | Characterization of setvar... |
| elin 3901 | Expansion of membership in... |
| dfss2 3903 | Alternate definition of th... |
| dfss 3904 | Variant of subclass defini... |
| dfss3 3906 | Alternate definition of su... |
| dfss6 3907 | Alternate definition of su... |
| dfssf 3908 | Equivalence for subclass r... |
| dfss3f 3909 | Equivalence for subclass r... |
| nfss 3910 | If ` x ` is not free in ` ... |
| ssel 3911 | Membership relationships f... |
| ssel2 3912 | Membership relationships f... |
| sseli 3913 | Membership implication fro... |
| sselii 3914 | Membership inference from ... |
| sselid 3915 | Membership inference from ... |
| sseld 3916 | Membership deduction from ... |
| sselda 3917 | Membership deduction from ... |
| sseldd 3918 | Membership inference from ... |
| ssneld 3919 | If a class is not in anoth... |
| ssneldd 3920 | If an element is not in a ... |
| ssriv 3921 | Inference based on subclas... |
| ssrd 3922 | Deduction based on subclas... |
| ssrdv 3923 | Deduction based on subclas... |
| sstr2 3924 | Transitivity of subclass r... |
| sstr 3925 | Transitivity of subclass r... |
| sstri 3926 | Subclass transitivity infe... |
| sstrd 3927 | Subclass transitivity dedu... |
| sstrid 3928 | Subclass transitivity dedu... |
| sstrdi 3929 | Subclass transitivity dedu... |
| sylan9ss 3930 | A subclass transitivity de... |
| sylan9ssr 3931 | A subclass transitivity de... |
| eqss 3932 | The subclass relationship ... |
| eqssi 3933 | Infer equality from two su... |
| eqssd 3934 | Equality deduction from tw... |
| sssseq 3935 | If a class is a subclass o... |
| eqrd 3936 | Deduce equality of classes... |
| eqri 3937 | Infer equality of classes ... |
| eqelssd 3938 | Equality deduction from su... |
| ssid 3939 | Any class is a subclass of... |
| ssidd 3940 | Weakening of ~ ssid . (Co... |
| ssv 3941 | Any class is a subclass of... |
| sseq1 3942 | Equality theorem for subcl... |
| sseq2 3943 | Equality theorem for the s... |
| sseq12 3944 | Equality theorem for the s... |
| sseq1i 3945 | An equality inference for ... |
| sseq2i 3946 | An equality inference for ... |
| sseq12i 3947 | An equality inference for ... |
| sseq1d 3948 | An equality deduction for ... |
| sseq2d 3949 | An equality deduction for ... |
| sseq12d 3950 | An equality deduction for ... |
| eqsstrd 3951 | Substitution of equality i... |
| eqsstrrd 3952 | Substitution of equality i... |
| sseqtrd 3953 | Substitution of equality i... |
| sseqtrrd 3954 | Substitution of equality i... |
| eqsstrid 3955 | A chained subclass and equ... |
| eqsstrrid 3956 | A chained subclass and equ... |
| sseqtrdi 3957 | A chained subclass and equ... |
| sseqtrrdi 3958 | A chained subclass and equ... |
| sseqtrid 3959 | Subclass transitivity dedu... |
| sseqtrrid 3960 | Subclass transitivity dedu... |
| eqsstrdi 3961 | A chained subclass and equ... |
| eqsstrrdi 3962 | A chained subclass and equ... |
| eqsstri 3963 | Substitution of equality i... |
| eqsstrri 3964 | Substitution of equality i... |
| sseqtri 3965 | Substitution of equality i... |
| sseqtrri 3966 | Substitution of equality i... |
| 3sstr3i 3967 | Substitution of equality i... |
| 3sstr4i 3968 | Substitution of equality i... |
| 3sstr3g 3969 | Substitution of equality i... |
| 3sstr4g 3970 | Substitution of equality i... |
| 3sstr3d 3971 | Substitution of equality i... |
| 3sstr4d 3972 | Substitution of equality i... |
| eqimssd 3973 | Equality implies inclusion... |
| eqimsscd 3974 | Equality implies inclusion... |
| eqimss 3975 | Equality implies inclusion... |
| eqimss2 3976 | Equality implies inclusion... |
| eqimssi 3977 | Infer subclass relationshi... |
| eqimss2i 3978 | Infer subclass relationshi... |
| nssne1 3979 | Two classes are different ... |
| nssne2 3980 | Two classes are different ... |
| nss 3981 | Negation of subclass relat... |
| nssrex 3982 | Negation of subclass relat... |
| nelss 3983 | Demonstrate by witnesses t... |
| ssrexf 3984 | Restricted existential qua... |
| ssrmof 3985 | "At most one" existential ... |
| ssralv 3986 | Quantification restricted ... |
| ssrexv 3987 | Existential quantification... |
| ss2ralv 3988 | Two quantifications restri... |
| ss2rexv 3989 | Two existential quantifica... |
| ralss 3990 | Restricted universal quant... |
| rexss 3991 | Restricted existential qua... |
| ralssOLD 3992 | Obsolete version of ~ rals... |
| rexssOLD 3993 | Obsolete version of ~ rexs... |
| ss2abim 3994 | Class abstractions in a su... |
| ss2ab 3995 | Class abstractions in a su... |
| abss 3996 | Class abstraction in a sub... |
| ssab 3997 | Subclass of a class abstra... |
| ssabral 3998 | The relation for a subclas... |
| ss2abdv 3999 | Deduction of abstraction s... |
| ss2abi 4000 | Inference of abstraction s... |
| abssdv 4001 | Deduction of abstraction s... |
| abssi 4002 | Inference of abstraction s... |
| ss2rab 4003 | Restricted abstraction cla... |
| rabss 4004 | Restricted class abstracti... |
| ssrab 4005 | Subclass of a restricted c... |
| ss2rabd 4006 | Subclass of a restricted c... |
| ssrabdv 4007 | Subclass of a restricted c... |
| rabssdv 4008 | Subclass of a restricted c... |
| ss2rabdv 4009 | Deduction of restricted ab... |
| ss2rabi 4010 | Inference of restricted ab... |
| rabss2 4011 | Subclass law for restricte... |
| rabss2OLD 4012 | Obsolete version of ~ rabs... |
| ssab2 4013 | Subclass relation for the ... |
| ssrab2 4014 | Subclass relation for a re... |
| rabss3d 4015 | Subclass law for restricte... |
| ssrab3 4016 | Subclass relation for a re... |
| rabssrabd 4017 | Subclass of a restricted c... |
| ssrabeq 4018 | If the restricting class o... |
| rabssab 4019 | A restricted class is a su... |
| eqrrabd 4020 | Deduce equality with a res... |
| uniiunlem 4021 | A subset relationship usef... |
| dfpss2 4022 | Alternate definition of pr... |
| dfpss3 4023 | Alternate definition of pr... |
| psseq1 4024 | Equality theorem for prope... |
| psseq2 4025 | Equality theorem for prope... |
| psseq1i 4026 | An equality inference for ... |
| psseq2i 4027 | An equality inference for ... |
| psseq12i 4028 | An equality inference for ... |
| psseq1d 4029 | An equality deduction for ... |
| psseq2d 4030 | An equality deduction for ... |
| psseq12d 4031 | An equality deduction for ... |
| pssss 4032 | A proper subclass is a sub... |
| pssne 4033 | Two classes in a proper su... |
| pssssd 4034 | Deduce subclass from prope... |
| pssned 4035 | Proper subclasses are uneq... |
| sspss 4036 | Subclass in terms of prope... |
| pssirr 4037 | Proper subclass is irrefle... |
| pssn2lp 4038 | Proper subclass has no 2-c... |
| sspsstri 4039 | Two ways of stating tricho... |
| ssnpss 4040 | Partial trichotomy law for... |
| psstr 4041 | Transitive law for proper ... |
| sspsstr 4042 | Transitive law for subclas... |
| psssstr 4043 | Transitive law for subclas... |
| psstrd 4044 | Proper subclass inclusion ... |
| sspsstrd 4045 | Transitivity involving sub... |
| psssstrd 4046 | Transitivity involving sub... |
| npss 4047 | A class is not a proper su... |
| ssnelpss 4048 | A subclass missing a membe... |
| ssnelpssd 4049 | Subclass inclusion with on... |
| ssexnelpss 4050 | If there is an element of ... |
| dfdif3 4051 | Alternate definition of cl... |
| dfdif3OLD 4052 | Obsolete version of ~ dfdi... |
| difeq1 4053 | Equality theorem for class... |
| difeq2 4054 | Equality theorem for class... |
| difeq12 4055 | Equality theorem for class... |
| difeq1i 4056 | Inference adding differenc... |
| difeq2i 4057 | Inference adding differenc... |
| difeq12i 4058 | Equality inference for cla... |
| difeq1d 4059 | Deduction adding differenc... |
| difeq2d 4060 | Deduction adding differenc... |
| difeq12d 4061 | Equality deduction for cla... |
| difeqri 4062 | Inference from membership ... |
| nfdif 4063 | Bound-variable hypothesis ... |
| eldifi 4064 | Implication of membership ... |
| eldifn 4065 | Implication of membership ... |
| elndif 4066 | A set does not belong to a... |
| neldif 4067 | Implication of membership ... |
| difdif 4068 | Double class difference. ... |
| difss 4069 | Subclass relationship for ... |
| difssd 4070 | A difference of two classe... |
| difss2 4071 | If a class is contained in... |
| difss2d 4072 | If a class is contained in... |
| ssdifss 4073 | Preservation of a subclass... |
| ddif 4074 | Double complement under un... |
| ssconb 4075 | Contraposition law for sub... |
| sscon 4076 | Contraposition law for sub... |
| ssdif 4077 | Difference law for subsets... |
| ssdifd 4078 | If ` A ` is contained in `... |
| sscond 4079 | If ` A ` is contained in `... |
| ssdifssd 4080 | If ` A ` is contained in `... |
| ssdif2d 4081 | If ` A ` is contained in `... |
| raldifb 4082 | Restricted universal quant... |
| rexdifi 4083 | Restricted existential qua... |
| complss 4084 | Complementation reverses i... |
| compleq 4085 | Two classes are equal if a... |
| elun 4086 | Expansion of membership in... |
| elunnel1 4087 | A member of a union that i... |
| elunnel2 4088 | A member of a union that i... |
| uneqri 4089 | Inference from membership ... |
| unidm 4090 | Idempotent law for union o... |
| uncom 4091 | Commutative law for union ... |
| equncom 4092 | If a class equals the unio... |
| equncomi 4093 | Inference form of ~ equnco... |
| uneq1 4094 | Equality theorem for the u... |
| uneq2 4095 | Equality theorem for the u... |
| uneq12 4096 | Equality theorem for the u... |
| uneq1i 4097 | Inference adding union to ... |
| uneq2i 4098 | Inference adding union to ... |
| uneq12i 4099 | Equality inference for the... |
| uneq1d 4100 | Deduction adding union to ... |
| uneq2d 4101 | Deduction adding union to ... |
| uneq12d 4102 | Equality deduction for the... |
| nfun 4103 | Bound-variable hypothesis ... |
| unass 4104 | Associative law for union ... |
| un12 4105 | A rearrangement of union. ... |
| un23 4106 | A rearrangement of union. ... |
| un4 4107 | A rearrangement of the uni... |
| unundi 4108 | Union distributes over its... |
| unundir 4109 | Union distributes over its... |
| ssun1 4110 | Subclass relationship for ... |
| ssun2 4111 | Subclass relationship for ... |
| ssun3 4112 | Subclass law for union of ... |
| ssun4 4113 | Subclass law for union of ... |
| elun1 4114 | Membership law for union o... |
| elun2 4115 | Membership law for union o... |
| elunant 4116 | A statement is true for ev... |
| unss1 4117 | Subclass law for union of ... |
| ssequn1 4118 | A relationship between sub... |
| unss2 4119 | Subclass law for union of ... |
| unss12 4120 | Subclass law for union of ... |
| ssequn2 4121 | A relationship between sub... |
| unss 4122 | The union of two subclasse... |
| unssi 4123 | An inference showing the u... |
| unssd 4124 | A deduction showing the un... |
| unssad 4125 | If ` ( A u. B ) ` is conta... |
| unssbd 4126 | If ` ( A u. B ) ` is conta... |
| ssun 4127 | A condition that implies i... |
| rexun 4128 | Restricted existential qua... |
| ralunb 4129 | Restricted quantification ... |
| ralun 4130 | Restricted quantification ... |
| elini 4131 | Membership in an intersect... |
| elind 4132 | Deduce membership in an in... |
| elinel1 4133 | Membership in an intersect... |
| elinel2 4134 | Membership in an intersect... |
| elin2 4135 | Membership in a class defi... |
| elin1d 4136 | Elementhood in the first s... |
| elin2d 4137 | Elementhood in the first s... |
| elin3 4138 | Membership in a class defi... |
| nel1nelin 4139 | Membership in an intersect... |
| nel2nelin 4140 | Membership in an intersect... |
| incom 4141 | Commutative law for inters... |
| ineqcom 4142 | Two ways of expressing tha... |
| ineqcomi 4143 | Two ways of expressing tha... |
| ineqri 4144 | Inference from membership ... |
| ineq1 4145 | Equality theorem for inter... |
| ineq2 4146 | Equality theorem for inter... |
| ineq12 4147 | Equality theorem for inter... |
| ineq1i 4148 | Equality inference for int... |
| ineq2i 4149 | Equality inference for int... |
| ineq12i 4150 | Equality inference for int... |
| ineq1d 4151 | Equality deduction for int... |
| ineq2d 4152 | Equality deduction for int... |
| ineq12d 4153 | Equality deduction for int... |
| ineqan12d 4154 | Equality deduction for int... |
| sseqin2 4155 | A relationship between sub... |
| nfin 4156 | Bound-variable hypothesis ... |
| rabbi2dva 4157 | Deduction from a wff to a ... |
| inidm 4158 | Idempotent law for interse... |
| inass 4159 | Associative law for inters... |
| in12 4160 | A rearrangement of interse... |
| in32 4161 | A rearrangement of interse... |
| in13 4162 | A rearrangement of interse... |
| in31 4163 | A rearrangement of interse... |
| inrot 4164 | Rotate the intersection of... |
| in4 4165 | Rearrangement of intersect... |
| inindi 4166 | Intersection distributes o... |
| inindir 4167 | Intersection distributes o... |
| inss1 4168 | The intersection of two cl... |
| inss2 4169 | The intersection of two cl... |
| ssin 4170 | Subclass of intersection. ... |
| ssini 4171 | An inference showing that ... |
| ssind 4172 | A deduction showing that a... |
| ssrin 4173 | Add right intersection to ... |
| sslin 4174 | Add left intersection to s... |
| ssrind 4175 | Add right intersection to ... |
| ss2in 4176 | Intersection of subclasses... |
| ssinss1 4177 | Intersection preserves sub... |
| ssinss1d 4178 | Intersection preserves sub... |
| inss 4179 | Inclusion of an intersecti... |
| ralin 4180 | Restricted universal quant... |
| rexin 4181 | Restricted existential qua... |
| dfss7 4182 | Alternate definition of su... |
| symdifcom 4185 | Symmetric difference commu... |
| symdifeq1 4186 | Equality theorem for symme... |
| symdifeq2 4187 | Equality theorem for symme... |
| nfsymdif 4188 | Hypothesis builder for sym... |
| elsymdif 4189 | Membership in a symmetric ... |
| dfsymdif4 4190 | Alternate definition of th... |
| elsymdifxor 4191 | Membership in a symmetric ... |
| dfsymdif2 4192 | Alternate definition of th... |
| symdifass 4193 | Symmetric difference is as... |
| difsssymdif 4194 | The symmetric difference c... |
| difsymssdifssd 4195 | If the symmetric differenc... |
| unabs 4196 | Absorption law for union. ... |
| inabs 4197 | Absorption law for interse... |
| nssinpss 4198 | Negation of subclass expre... |
| nsspssun 4199 | Negation of subclass expre... |
| dfss4 4200 | Subclass defined in terms ... |
| dfun2 4201 | An alternate definition of... |
| dfin2 4202 | An alternate definition of... |
| difin 4203 | Difference with intersecti... |
| ssdifim 4204 | Implication of a class dif... |
| ssdifsym 4205 | Symmetric class difference... |
| dfss5 4206 | Alternate definition of su... |
| dfun3 4207 | Union defined in terms of ... |
| dfin3 4208 | Intersection defined in te... |
| dfin4 4209 | Alternate definition of th... |
| invdif 4210 | Intersection with universa... |
| indif 4211 | Intersection with class di... |
| indif2 4212 | Bring an intersection in a... |
| indif1 4213 | Bring an intersection in a... |
| indifcom 4214 | Commutation law for inters... |
| indi 4215 | Distributive law for inter... |
| undi 4216 | Distributive law for union... |
| indir 4217 | Distributive law for inter... |
| undir 4218 | Distributive law for union... |
| unineq 4219 | Infer equality from equali... |
| uneqin 4220 | Equality of union and inte... |
| difundi 4221 | Distributive law for class... |
| difundir 4222 | Distributive law for class... |
| difindi 4223 | Distributive law for class... |
| difindir 4224 | Distributive law for class... |
| indifdi 4225 | Distribute intersection ov... |
| indifdir 4226 | Distribute intersection ov... |
| difdif2 4227 | Class difference by a clas... |
| undm 4228 | De Morgan's law for union.... |
| indm 4229 | De Morgan's law for inters... |
| difun1 4230 | A relationship involving d... |
| undif3 4231 | An equality involving clas... |
| difin2 4232 | Represent a class differen... |
| dif32 4233 | Swap second and third argu... |
| difabs 4234 | Absorption-like law for cl... |
| sscon34b 4235 | Relative complementation r... |
| rcompleq 4236 | Two subclasses are equal i... |
| dfsymdif3 4237 | Alternate definition of th... |
| unabw 4238 | Union of two class abstrac... |
| unab 4239 | Union of two class abstrac... |
| inab 4240 | Intersection of two class ... |
| difab 4241 | Difference of two class ab... |
| abanssl 4242 | A class abstraction with a... |
| abanssr 4243 | A class abstraction with a... |
| notabw 4244 | A class abstraction define... |
| notab 4245 | A class abstraction define... |
| unrab 4246 | Union of two restricted cl... |
| inrab 4247 | Intersection of two restri... |
| inrab2 4248 | Intersection with a restri... |
| difrab 4249 | Difference of two restrict... |
| dfrab3 4250 | Alternate definition of re... |
| dfrab2 4251 | Alternate definition of re... |
| rabdif 4252 | Move difference in and out... |
| notrab 4253 | Complementation of restric... |
| dfrab3ss 4254 | Restricted class abstracti... |
| rabun2 4255 | Abstraction restricted to ... |
| reuun2 4256 | Transfer uniqueness to a s... |
| reuss2 4257 | Transfer uniqueness to a s... |
| reuss 4258 | Transfer uniqueness to a s... |
| reuun1 4259 | Transfer uniqueness to a s... |
| reupick 4260 | Restricted uniqueness "pic... |
| reupick3 4261 | Restricted uniqueness "pic... |
| reupick2 4262 | Restricted uniqueness "pic... |
| euelss 4263 | Transfer uniqueness of an ... |
| dfnul4 4266 | Alternate definition of th... |
| dfnul2 4267 | Alternate definition of th... |
| dfnul3 4268 | Alternate definition of th... |
| noel 4269 | The empty set has no eleme... |
| nel02 4270 | The empty set has no eleme... |
| n0i 4271 | If a class has elements, t... |
| ne0i 4272 | If a class has elements, t... |
| ne0d 4273 | Deduction form of ~ ne0i .... |
| n0ii 4274 | If a class has elements, t... |
| ne0ii 4275 | If a class has elements, t... |
| vn0 4276 | The universal class is not... |
| vn0ALT 4277 | Alternate proof of ~ vn0 .... |
| eq0f 4278 | A class is equal to the em... |
| neq0f 4279 | A class is not empty if an... |
| n0f 4280 | A class is nonempty if and... |
| eq0 4281 | A class is equal to the em... |
| eq0ALT 4282 | Alternate proof of ~ eq0 .... |
| neq0 4283 | A class is not empty if an... |
| n0 4284 | A class is nonempty if and... |
| nel0 4285 | From the general negation ... |
| reximdva0 4286 | Restricted existence deduc... |
| rspn0 4287 | Specialization for restric... |
| n0rex 4288 | There is an element in a n... |
| ssn0rex 4289 | There is an element in a c... |
| n0moeu 4290 | A case of equivalence of "... |
| rex0 4291 | Vacuous restricted existen... |
| reu0 4292 | Vacuous restricted uniquen... |
| rmo0 4293 | Vacuous restricted at-most... |
| 0el 4294 | Membership of the empty se... |
| n0el 4295 | Negated membership of the ... |
| eqeuel 4296 | A condition which implies ... |
| ssdif0 4297 | Subclass expressed in term... |
| difn0 4298 | If the difference of two s... |
| pssdifn0 4299 | A proper subclass has a no... |
| pssdif 4300 | A proper subclass has a no... |
| ndisj 4301 | Express that an intersecti... |
| inn0f 4302 | A nonempty intersection. ... |
| inn0 4303 | A nonempty intersection. ... |
| difin0ss 4304 | Difference, intersection, ... |
| inssdif0 4305 | Intersection, subclass, an... |
| inindif 4306 | The intersection and class... |
| difid 4307 | The difference between a c... |
| difidALT 4308 | Alternate proof of ~ difid... |
| dif0 4309 | The difference between a c... |
| ab0w 4310 | The class of sets verifyin... |
| ab0 4311 | The class of sets verifyin... |
| ab0ALT 4312 | Alternate proof of ~ ab0 ,... |
| dfnf5 4313 | Characterization of nonfre... |
| ab0orv 4314 | The class abstraction defi... |
| ab0orvALT 4315 | Alternate proof of ~ ab0or... |
| abn0 4316 | Nonempty class abstraction... |
| rab0 4317 | Any restricted class abstr... |
| rabeq0w 4318 | Condition for a restricted... |
| rabeq0 4319 | Condition for a restricted... |
| rabn0 4320 | Nonempty restricted class ... |
| rabxm 4321 | Law of excluded middle, in... |
| rabnc 4322 | Law of noncontradiction, i... |
| elneldisj 4323 | The set of elements ` s ` ... |
| elnelun 4324 | The union of the set of el... |
| un0 4325 | The union of a class with ... |
| in0 4326 | The intersection of a clas... |
| 0un 4327 | The union of the empty set... |
| 0in 4328 | The intersection of the em... |
| inv1 4329 | The intersection of a clas... |
| unv 4330 | The union of a class with ... |
| 0ss 4331 | The null set is a subset o... |
| ss0b 4332 | Any subset of the empty se... |
| ss0 4333 | Any subset of the empty se... |
| sseq0 4334 | A subclass of an empty cla... |
| ssn0 4335 | A class with a nonempty su... |
| 0dif 4336 | The difference between the... |
| abf 4337 | A class abstraction determ... |
| eq0rdv 4338 | Deduction for equality to ... |
| eq0rdvALT 4339 | Alternate proof of ~ eq0rd... |
| csbprc 4340 | The proper substitution of... |
| csb0 4341 | The proper substitution of... |
| sbcel12 4342 | Distribute proper substitu... |
| sbceqg 4343 | Distribute proper substitu... |
| sbceqi 4344 | Distribution of class subs... |
| sbcnel12g 4345 | Distribute proper substitu... |
| sbcne12 4346 | Distribute proper substitu... |
| sbcel1g 4347 | Move proper substitution i... |
| sbceq1g 4348 | Move proper substitution t... |
| sbcel2 4349 | Move proper substitution i... |
| sbceq2g 4350 | Move proper substitution t... |
| csbcom 4351 | Commutative law for double... |
| sbcnestgfw 4352 | Nest the composition of tw... |
| csbnestgfw 4353 | Nest the composition of tw... |
| sbcnestgw 4354 | Nest the composition of tw... |
| csbnestgw 4355 | Nest the composition of tw... |
| sbcco3gw 4356 | Composition of two substit... |
| sbcnestgf 4357 | Nest the composition of tw... |
| csbnestgf 4358 | Nest the composition of tw... |
| sbcnestg 4359 | Nest the composition of tw... |
| csbnestg 4360 | Nest the composition of tw... |
| sbcco3g 4361 | Composition of two substit... |
| csbco3g 4362 | Composition of two class s... |
| csbnest1g 4363 | Nest the composition of tw... |
| csbidm 4364 | Idempotent law for class s... |
| csbvarg 4365 | The proper substitution of... |
| csbvargi 4366 | The proper substitution of... |
| sbccsb 4367 | Substitution into a wff ex... |
| sbccsb2 4368 | Substitution into a wff ex... |
| rspcsbela 4369 | Special case related to ~ ... |
| sbnfc2 4370 | Two ways of expressing " `... |
| csbab 4371 | Move substitution into a c... |
| csbun 4372 | Distribution of class subs... |
| csbin 4373 | Distribute proper substitu... |
| csbie2df 4374 | Conversion of implicit sub... |
| 2nreu 4375 | If there are two different... |
| un00 4376 | Two classes are empty iff ... |
| vss 4377 | Only the universal class h... |
| 0pss 4378 | The null set is a proper s... |
| npss0 4379 | No set is a proper subset ... |
| pssv 4380 | Any non-universal class is... |
| disj 4381 | Two ways of saying that tw... |
| disjr 4382 | Two ways of saying that tw... |
| disj1 4383 | Two ways of saying that tw... |
| reldisj 4384 | Two ways of saying that tw... |
| disj3 4385 | Two ways of saying that tw... |
| disjne 4386 | Members of disjoint sets a... |
| disjeq0 4387 | Two disjoint sets are equa... |
| disjel 4388 | A set can't belong to both... |
| disj2 4389 | Two ways of saying that tw... |
| disj4 4390 | Two ways of saying that tw... |
| ssdisj 4391 | Intersection with a subcla... |
| disjpss 4392 | A class is a proper subset... |
| undisj1 4393 | The union of disjoint clas... |
| undisj2 4394 | The union of disjoint clas... |
| ssindif0 4395 | Subclass expressed in term... |
| inelcm 4396 | The intersection of classe... |
| minel 4397 | A minimum element of a cla... |
| undif4 4398 | Distribute union over diff... |
| disjssun 4399 | Subset relation for disjoi... |
| vdif0 4400 | Universal class equality i... |
| difrab0eq 4401 | If the difference between ... |
| pssnel 4402 | A proper subclass has a me... |
| disjdif 4403 | A class and its relative c... |
| disjdifr 4404 | A class and its relative c... |
| difin0 4405 | The difference of a class ... |
| unvdif 4406 | The union of a class and i... |
| undif1 4407 | Absorption of difference b... |
| undif2 4408 | Absorption of difference b... |
| undifabs 4409 | Absorption of difference b... |
| inundif 4410 | The intersection and class... |
| disjdif2 4411 | The difference of a class ... |
| difun2 4412 | Absorption of union by dif... |
| undif 4413 | Union of complementary par... |
| undifr 4414 | Union of complementary par... |
| undif5 4415 | An equality involving clas... |
| ssdifin0 4416 | A subset of a difference d... |
| ssdifeq0 4417 | A class is a subclass of i... |
| ssundif 4418 | A condition equivalent to ... |
| difcom 4419 | Swap the arguments of a cl... |
| pssdifcom1 4420 | Two ways to express overla... |
| pssdifcom2 4421 | Two ways to express non-co... |
| difdifdir 4422 | Distributive law for class... |
| uneqdifeq 4423 | Two ways to say that ` A `... |
| raldifeq 4424 | Equality theorem for restr... |
| rzal 4425 | Vacuous quantification is ... |
| rzalALT 4426 | Alternate proof of ~ rzal ... |
| rexn0 4427 | Restricted existential qua... |
| ralf0 4428 | The quantification of a fa... |
| ral0 4429 | Vacuous universal quantifi... |
| r19.2z 4430 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4431 | A response to the notion t... |
| r19.3rz 4432 | Restricted quantification ... |
| r19.28z 4433 | Restricted quantifier vers... |
| r19.3rzv 4434 | Restricted quantification ... |
| r19.3rzvOLD 4435 | Obsolete version of ~ r19.... |
| r19.9rzv 4436 | Restricted quantification ... |
| r19.28zv 4437 | Restricted quantifier vers... |
| r19.37zv 4438 | Restricted quantifier vers... |
| r19.45zv 4439 | Restricted version of Theo... |
| r19.44zv 4440 | Restricted version of Theo... |
| r19.27z 4441 | Restricted quantifier vers... |
| r19.27zv 4442 | Restricted quantifier vers... |
| r19.36zv 4443 | Restricted quantifier vers... |
| ralnralall 4444 | A contradiction concerning... |
| falseral0 4445 | A false statement can only... |
| falseral0OLD 4446 | Obsolete version of ~ fals... |
| ralidmw 4447 | Idempotent law for restric... |
| ralidm 4448 | Idempotent law for restric... |
| raaan 4449 | Rearrange restricted quant... |
| raaanv 4450 | Rearrange restricted quant... |
| sbss 4451 | Set substitution into the ... |
| sbcssg 4452 | Distribute proper substitu... |
| raaan2 4453 | Rearrange restricted quant... |
| 2reu4lem 4454 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4455 | Definition of double restr... |
| csbdif 4456 | Distribution of class subs... |
| dfif2 4459 | An alternate definition of... |
| dfif6 4460 | An alternate definition of... |
| ifeq1 4461 | Equality theorem for condi... |
| ifeq2 4462 | Equality theorem for condi... |
| iftrue 4463 | Value of the conditional o... |
| iftruei 4464 | Inference associated with ... |
| iftrued 4465 | Value of the conditional o... |
| iffalse 4466 | Value of the conditional o... |
| iffalsei 4467 | Inference associated with ... |
| iffalsed 4468 | Value of the conditional o... |
| ifnefalse 4469 | When values are unequal, b... |
| iftrueb 4470 | When the branches are not ... |
| ifsb 4471 | Distribute a function over... |
| dfif3 4472 | Alternate definition of th... |
| dfif4 4473 | Alternate definition of th... |
| dfif5 4474 | Alternate definition of th... |
| ifssun 4475 | A conditional class is inc... |
| ifeq12 4476 | Equality theorem for condi... |
| ifeq1d 4477 | Equality deduction for con... |
| ifeq2d 4478 | Equality deduction for con... |
| ifeq12d 4479 | Equality deduction for con... |
| ifbi 4480 | Equivalence theorem for co... |
| ifbid 4481 | Equivalence deduction for ... |
| ifbieq1d 4482 | Equivalence/equality deduc... |
| ifbieq2i 4483 | Equivalence/equality infer... |
| ifbieq2d 4484 | Equivalence/equality deduc... |
| ifbieq12i 4485 | Equivalence deduction for ... |
| ifbieq12d 4486 | Equivalence deduction for ... |
| nfifd 4487 | Deduction form of ~ nfif .... |
| nfif 4488 | Bound-variable hypothesis ... |
| ifeq1da 4489 | Conditional equality. (Co... |
| ifeq2da 4490 | Conditional equality. (Co... |
| ifeq12da 4491 | Equivalence deduction for ... |
| ifbieq12d2 4492 | Equivalence deduction for ... |
| ifclda 4493 | Conditional closure. (Con... |
| ifeqda 4494 | Separation of the values o... |
| elimif 4495 | Elimination of a condition... |
| ifbothda 4496 | A wff ` th ` containing a ... |
| ifboth 4497 | A wff ` th ` containing a ... |
| ifid 4498 | Identical true and false a... |
| eqif 4499 | Expansion of an equality w... |
| ifval 4500 | Another expression of the ... |
| elif 4501 | Membership in a conditiona... |
| ifel 4502 | Membership of a conditiona... |
| ifcl 4503 | Membership (closure) of a ... |
| ifcld 4504 | Membership (closure) of a ... |
| ifcli 4505 | Inference associated with ... |
| ifexd 4506 | Existence of the condition... |
| ifexg 4507 | Existence of the condition... |
| ifex 4508 | Existence of the condition... |
| ifeqor 4509 | The possible values of a c... |
| ifnot 4510 | Negating the first argumen... |
| ifan 4511 | Rewrite a conjunction in a... |
| ifor 4512 | Rewrite a disjunction in a... |
| 2if2 4513 | Resolve two nested conditi... |
| ifcomnan 4514 | Commute the conditions in ... |
| csbif 4515 | Distribute proper substitu... |
| dedth 4516 | Weak deduction theorem tha... |
| dedth2h 4517 | Weak deduction theorem eli... |
| dedth3h 4518 | Weak deduction theorem eli... |
| dedth4h 4519 | Weak deduction theorem eli... |
| dedth2v 4520 | Weak deduction theorem for... |
| dedth3v 4521 | Weak deduction theorem for... |
| dedth4v 4522 | Weak deduction theorem for... |
| elimhyp 4523 | Eliminate a hypothesis con... |
| elimhyp2v 4524 | Eliminate a hypothesis con... |
| elimhyp3v 4525 | Eliminate a hypothesis con... |
| elimhyp4v 4526 | Eliminate a hypothesis con... |
| elimel 4527 | Eliminate a membership hyp... |
| elimdhyp 4528 | Version of ~ elimhyp where... |
| keephyp 4529 | Transform a hypothesis ` p... |
| keephyp2v 4530 | Keep a hypothesis containi... |
| keephyp3v 4531 | Keep a hypothesis containi... |
| pwjust 4533 | Soundness justification th... |
| elpwg 4535 | Membership in a power clas... |
| elpw 4536 | Membership in a power clas... |
| velpw 4537 | Setvar variable membership... |
| elpwd 4538 | Membership in a power clas... |
| elpwi 4539 | Subset relation implied by... |
| elpwb 4540 | Characterization of the el... |
| elpwid 4541 | An element of a power clas... |
| elelpwi 4542 | If ` A ` belongs to a part... |
| sspw 4543 | The powerclass preserves i... |
| sspwi 4544 | The powerclass preserves i... |
| sspwd 4545 | The powerclass preserves i... |
| pweq 4546 | Equality theorem for power... |
| pweqALT 4547 | Alternate proof of ~ pweq ... |
| pweqi 4548 | Equality inference for pow... |
| pweqd 4549 | Equality deduction for pow... |
| pwunss 4550 | The power class of the uni... |
| nfpw 4551 | Bound-variable hypothesis ... |
| pwidg 4552 | A set is an element of its... |
| pwidb 4553 | A class is an element of i... |
| pwid 4554 | A set is a member of its p... |
| pwss 4555 | Subclass relationship for ... |
| pwundif 4556 | Break up the power class o... |
| snjust 4557 | Soundness justification th... |
| sneq 4568 | Equality theorem for singl... |
| sneqi 4569 | Equality inference for sin... |
| sneqd 4570 | Equality deduction for sin... |
| dfsn2 4571 | Alternate definition of si... |
| elsng 4572 | There is exactly one eleme... |
| elsn 4573 | There is exactly one eleme... |
| velsn 4574 | There is only one element ... |
| elsni 4575 | There is at most one eleme... |
| elsnd 4576 | There is at most one eleme... |
| rabsneq 4577 | Equality of class abstract... |
| absn 4578 | Condition for a class abst... |
| dfpr2 4579 | Alternate definition of a ... |
| dfsn2ALT 4580 | Alternate definition of si... |
| elprg 4581 | A member of a pair of clas... |
| elpri 4582 | If a class is an element o... |
| elpr 4583 | A member of a pair of clas... |
| elpr2g 4584 | A member of a pair of sets... |
| elpr2 4585 | A member of a pair of sets... |
| elprn1 4586 | A member of an unordered p... |
| elprn2 4587 | A member of an unordered p... |
| nelpr2 4588 | If a class is not an eleme... |
| nelpr1 4589 | If a class is not an eleme... |
| nelpri 4590 | If an element doesn't matc... |
| prneli 4591 | If an element doesn't matc... |
| nelprd 4592 | If an element doesn't matc... |
| eldifpr 4593 | Membership in a set with t... |
| rexdifpr 4594 | Restricted existential qua... |
| snidg 4595 | A set is a member of its s... |
| snidb 4596 | A class is a set iff it is... |
| snid 4597 | A set is a member of its s... |
| vsnid 4598 | A setvar variable is a mem... |
| elsn2g 4599 | There is exactly one eleme... |
| elsn2 4600 | There is exactly one eleme... |
| nelsn 4601 | If a class is not equal to... |
| rabeqsn 4602 | Conditions for a restricte... |
| rabsssn 4603 | Conditions for a restricte... |
| rabeqsnd 4604 | Conditions for a restricte... |
| ralsnsg 4605 | Substitution expressed in ... |
| rexsns 4606 | Restricted existential qua... |
| rexsngf 4607 | Restricted existential qua... |
| ralsngf 4608 | Restricted universal quant... |
| reusngf 4609 | Restricted existential uni... |
| ralsng 4610 | Substitution expressed in ... |
| rexsng 4611 | Restricted existential qua... |
| reusng 4612 | Restricted existential uni... |
| 2ralsng 4613 | Substitution expressed in ... |
| rexreusng 4614 | Restricted existential uni... |
| exsnrex 4615 | There is a set being the e... |
| ralsn 4616 | Convert a universal quanti... |
| rexsn 4617 | Convert an existential qua... |
| elunsn 4618 | Elementhood in a union wit... |
| elpwunsn 4619 | Membership in an extension... |
| eqoreldif 4620 | An element of a set is eit... |
| eltpg 4621 | Members of an unordered tr... |
| eldiftp 4622 | Membership in a set with t... |
| eltpi 4623 | A member of an unordered t... |
| eltp 4624 | A member of an unordered t... |
| el7g 4625 | Members of a set with seve... |
| dftp2 4626 | Alternate definition of un... |
| nfpr 4627 | Bound-variable hypothesis ... |
| ifpr 4628 | Membership of a conditiona... |
| ralprgf 4629 | Convert a restricted unive... |
| rexprgf 4630 | Convert a restricted exist... |
| ralprg 4631 | Convert a restricted unive... |
| rexprg 4632 | Convert a restricted exist... |
| raltpg 4633 | Convert a restricted unive... |
| rextpg 4634 | Convert a restricted exist... |
| ralpr 4635 | Convert a restricted unive... |
| rexpr 4636 | Convert a restricted exist... |
| reuprg0 4637 | Convert a restricted exist... |
| reuprg 4638 | Convert a restricted exist... |
| reurexprg 4639 | Convert a restricted exist... |
| raltp 4640 | Convert a universal quanti... |
| rextp 4641 | Convert an existential qua... |
| nfsn 4642 | Bound-variable hypothesis ... |
| csbsng 4643 | Distribute proper substitu... |
| csbprg 4644 | Distribute proper substitu... |
| elinsn 4645 | If the intersection of two... |
| disjsn 4646 | Intersection with the sing... |
| disjsn2 4647 | Two distinct singletons ar... |
| disjpr2 4648 | Two completely distinct un... |
| disjprsn 4649 | The disjoint intersection ... |
| disjtpsn 4650 | The disjoint intersection ... |
| disjtp2 4651 | Two completely distinct un... |
| snprc 4652 | The singleton of a proper ... |
| snnzb 4653 | A singleton is nonempty if... |
| rmosn 4654 | A restricted at-most-one q... |
| r19.12sn 4655 | Special case of ~ r19.12 w... |
| rabsn 4656 | Condition where a restrict... |
| rabsnifsb 4657 | A restricted class abstrac... |
| rabsnif 4658 | A restricted class abstrac... |
| rabrsn 4659 | A restricted class abstrac... |
| euabsn2 4660 | Another way to express exi... |
| euabsn 4661 | Another way to express exi... |
| reusn 4662 | A way to express restricte... |
| absneu 4663 | Restricted existential uni... |
| rabsneu 4664 | Restricted existential uni... |
| eusn 4665 | Two ways to express " ` A ... |
| rabsnt 4666 | Truth implied by equality ... |
| prcom 4667 | Commutative law for unorde... |
| preq1 4668 | Equality theorem for unord... |
| preq2 4669 | Equality theorem for unord... |
| preq12 4670 | Equality theorem for unord... |
| preq1i 4671 | Equality inference for uno... |
| preq2i 4672 | Equality inference for uno... |
| preq12i 4673 | Equality inference for uno... |
| preq1d 4674 | Equality deduction for uno... |
| preq2d 4675 | Equality deduction for uno... |
| preq12d 4676 | Equality deduction for uno... |
| tpeq1 4677 | Equality theorem for unord... |
| tpeq2 4678 | Equality theorem for unord... |
| tpeq3 4679 | Equality theorem for unord... |
| tpeq1d 4680 | Equality theorem for unord... |
| tpeq2d 4681 | Equality theorem for unord... |
| tpeq3d 4682 | Equality theorem for unord... |
| tpeq123d 4683 | Equality theorem for unord... |
| tprot 4684 | Rotation of the elements o... |
| tpcoma 4685 | Swap 1st and 2nd members o... |
| tpcomb 4686 | Swap 2nd and 3rd members o... |
| tpass 4687 | Split off the first elemen... |
| qdass 4688 | Two ways to write an unord... |
| qdassr 4689 | Two ways to write an unord... |
| tpidm12 4690 | Unordered triple ` { A , A... |
| tpidm13 4691 | Unordered triple ` { A , B... |
| tpidm23 4692 | Unordered triple ` { A , B... |
| tpidm 4693 | Unordered triple ` { A , A... |
| tppreq3 4694 | An unordered triple is an ... |
| prid1g 4695 | An unordered pair contains... |
| prid2g 4696 | An unordered pair contains... |
| prid1 4697 | An unordered pair contains... |
| prid2 4698 | An unordered pair contains... |
| ifpprsnss 4699 | An unordered pair is a sin... |
| prprc1 4700 | A proper class vanishes in... |
| prprc2 4701 | A proper class vanishes in... |
| prprc 4702 | An unordered pair containi... |
| tpid1 4703 | One of the three elements ... |
| tpid1g 4704 | Closed theorem form of ~ t... |
| tpid2 4705 | One of the three elements ... |
| tpid2g 4706 | Closed theorem form of ~ t... |
| tpid3g 4707 | Closed theorem form of ~ t... |
| tpid3 4708 | One of the three elements ... |
| snnzg 4709 | The singleton of a set is ... |
| snn0d 4710 | The singleton of a set is ... |
| snnz 4711 | The singleton of a set is ... |
| prnz 4712 | A pair containing a set is... |
| prnzg 4713 | A pair containing a set is... |
| tpnz 4714 | An unordered triple contai... |
| tpnzd 4715 | An unordered triple contai... |
| raltpd 4716 | Convert a universal quanti... |
| snssb 4717 | Characterization of the in... |
| snssg 4718 | The singleton formed on a ... |
| snss 4719 | The singleton of an elemen... |
| snssi 4720 | The singleton of an elemen... |
| snssd 4721 | The singleton of an elemen... |
| eldifsn 4722 | Membership in a set with a... |
| eldifsnd 4723 | Membership in a set with a... |
| ssdifsn 4724 | Subset of a set with an el... |
| elpwdifsn 4725 | A subset of a set is an el... |
| eldifsni 4726 | Membership in a set with a... |
| eldifsnneq 4727 | An element of a difference... |
| neldifsn 4728 | The class ` A ` is not in ... |
| neldifsnd 4729 | The class ` A ` is not in ... |
| rexdifsn 4730 | Restricted existential qua... |
| raldifsni 4731 | Rearrangement of a propert... |
| raldifsnb 4732 | Restricted universal quant... |
| eldifvsn 4733 | A set is an element of the... |
| difsn 4734 | An element not in a set ca... |
| difprsnss 4735 | Removal of a singleton fro... |
| difprsn1 4736 | Removal of a singleton fro... |
| difprsn2 4737 | Removal of a singleton fro... |
| diftpsn3 4738 | Removal of a singleton fro... |
| difpr 4739 | Removing two elements as p... |
| tpprceq3 4740 | An unordered triple is an ... |
| tppreqb 4741 | An unordered triple is an ... |
| difsnb 4742 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4743 | ` ( B \ { A } ) ` is a pro... |
| difsnid 4744 | If we remove a single elem... |
| eldifeldifsn 4745 | An element of a difference... |
| pw0 4746 | Compute the power set of t... |
| pwpw0 4747 | Compute the power set of t... |
| snsspr1 4748 | A singleton is a subset of... |
| snsspr2 4749 | A singleton is a subset of... |
| snsstp1 4750 | A singleton is a subset of... |
| snsstp2 4751 | A singleton is a subset of... |
| snsstp3 4752 | A singleton is a subset of... |
| prssg 4753 | A pair of elements of a cl... |
| prss 4754 | A pair of elements of a cl... |
| prssi 4755 | A pair of elements of a cl... |
| prssd 4756 | Deduction version of ~ prs... |
| prsspwg 4757 | An unordered pair belongs ... |
| ssprss 4758 | A pair as subset of a pair... |
| ssprsseq 4759 | A proper pair is a subset ... |
| sssn 4760 | The subsets of a singleton... |
| ssunsn2 4761 | The property of being sand... |
| ssunsn 4762 | Possible values for a set ... |
| eqsn 4763 | Two ways to express that a... |
| eqsnd 4764 | Deduce that a set is a sin... |
| eqsndOLD 4765 | Obsolete version of ~ eqsn... |
| issn 4766 | A sufficient condition for... |
| n0snor2el 4767 | A nonempty set is either a... |
| ssunpr 4768 | Possible values for a set ... |
| sspr 4769 | The subsets of a pair. (C... |
| sstp 4770 | The subsets of an unordere... |
| tpss 4771 | An unordered triple of ele... |
| tpssi 4772 | An unordered triple of ele... |
| sneqrg 4773 | Closed form of ~ sneqr . ... |
| sneqr 4774 | If the singletons of two s... |
| snsssn 4775 | If a singleton is a subset... |
| mosneq 4776 | There exists at most one s... |
| sneqbg 4777 | Two singletons of sets are... |
| snsspw 4778 | The singleton of a class i... |
| prsspw 4779 | An unordered pair belongs ... |
| preq1b 4780 | Biconditional equality lem... |
| preq2b 4781 | Biconditional equality lem... |
| preqr1 4782 | Reverse equality lemma for... |
| preqr2 4783 | Reverse equality lemma for... |
| preq12b 4784 | Equality relationship for ... |
| opthpr 4785 | An unordered pair has the ... |
| preqr1g 4786 | Reverse equality lemma for... |
| preq12bg 4787 | Closed form of ~ preq12b .... |
| prneimg 4788 | Two pairs are not equal if... |
| prneimg2 4789 | Two pairs are not equal if... |
| prnebg 4790 | A (proper) pair is not equ... |
| pr1eqbg 4791 | A (proper) pair is equal t... |
| pr1nebg 4792 | A (proper) pair is not equ... |
| preqsnd 4793 | Equivalence for a pair equ... |
| prnesn 4794 | A proper unordered pair is... |
| prneprprc 4795 | A proper unordered pair is... |
| preqsn 4796 | Equivalence for a pair equ... |
| preq12nebg 4797 | Equality relationship for ... |
| prel12g 4798 | Equality of two unordered ... |
| opthprneg 4799 | An unordered pair has the ... |
| elpreqprlem 4800 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4801 | Equality and membership ru... |
| elpreqprb 4802 | A set is an element of an ... |
| elpr2elpr 4803 | For an element ` A ` of an... |
| dfopif 4804 | Rewrite ~ df-op using ` if... |
| dfopg 4805 | Value of the ordered pair ... |
| dfop 4806 | Value of an ordered pair w... |
| opeq1 4807 | Equality theorem for order... |
| opeq2 4808 | Equality theorem for order... |
| opeq12 4809 | Equality theorem for order... |
| opeq1i 4810 | Equality inference for ord... |
| opeq2i 4811 | Equality inference for ord... |
| opeq12i 4812 | Equality inference for ord... |
| opeq1d 4813 | Equality deduction for ord... |
| opeq2d 4814 | Equality deduction for ord... |
| opeq12d 4815 | Equality deduction for ord... |
| oteq1 4816 | Equality theorem for order... |
| oteq2 4817 | Equality theorem for order... |
| oteq3 4818 | Equality theorem for order... |
| oteq1d 4819 | Equality deduction for ord... |
| oteq2d 4820 | Equality deduction for ord... |
| oteq3d 4821 | Equality deduction for ord... |
| oteq123d 4822 | Equality deduction for ord... |
| nfop 4823 | Bound-variable hypothesis ... |
| nfopd 4824 | Deduction version of bound... |
| csbopg 4825 | Distribution of class subs... |
| opidg 4826 | The ordered pair ` <. A , ... |
| opid 4827 | The ordered pair ` <. A , ... |
| ralunsn 4828 | Restricted quantification ... |
| 2ralunsn 4829 | Double restricted quantifi... |
| opprc 4830 | Expansion of an ordered pa... |
| opprc1 4831 | Expansion of an ordered pa... |
| opprc2 4832 | Expansion of an ordered pa... |
| oprcl 4833 | If an ordered pair has an ... |
| pwsn 4834 | The power set of a singlet... |
| pwpr 4835 | The power set of an unorde... |
| pwtp 4836 | The power set of an unorde... |
| pwpwpw0 4837 | Compute the power set of t... |
| pwv 4838 | The power class of the uni... |
| prproe 4839 | For an element of a proper... |
| 3elpr2eq 4840 | If there are three element... |
| dfuni2 4843 | Alternate definition of cl... |
| eluni 4844 | Membership in class union.... |
| eluni2 4845 | Membership in class union.... |
| elunii 4846 | Membership in class union.... |
| nfunid 4847 | Deduction version of ~ nfu... |
| nfuni 4848 | Bound-variable hypothesis ... |
| uniss 4849 | Subclass relationship for ... |
| unissi 4850 | Subclass relationship for ... |
| unissd 4851 | Subclass relationship for ... |
| unieq 4852 | Equality theorem for class... |
| unieqi 4853 | Inference of equality of t... |
| unieqd 4854 | Deduction of equality of t... |
| eluniab 4855 | Membership in union of a c... |
| elunirab 4856 | Membership in union of a c... |
| uniprg 4857 | The union of a pair is the... |
| unipr 4858 | The union of a pair is the... |
| unisng 4859 | A set equals the union of ... |
| unisn 4860 | A set equals the union of ... |
| unisnv 4861 | A set equals the union of ... |
| unisn3 4862 | Union of a singleton in th... |
| dfnfc2 4863 | An alternative statement o... |
| uniun 4864 | The class union of the uni... |
| uniin 4865 | The class union of the int... |
| ssuni 4866 | Subclass relationship for ... |
| uni0b 4867 | The union of a set is empt... |
| uni0c 4868 | The union of a set is empt... |
| uni0 4869 | The union of the empty set... |
| uni0OLD 4870 | Obsolete version of ~ uni0... |
| csbuni 4871 | Distribute proper substitu... |
| elssuni 4872 | An element of a class is a... |
| unissel 4873 | Condition turning a subcla... |
| unissb 4874 | Relationship involving mem... |
| uniss2 4875 | A subclass condition on th... |
| unidif 4876 | If the difference ` A \ B ... |
| ssunieq 4877 | Relationship implying unio... |
| unimax 4878 | Any member of a class is t... |
| pwuni 4879 | A class is a subclass of t... |
| dfint2 4882 | Alternate definition of cl... |
| inteq 4883 | Equality law for intersect... |
| inteqi 4884 | Equality inference for cla... |
| inteqd 4885 | Equality deduction for cla... |
| elint 4886 | Membership in class inters... |
| elint2 4887 | Membership in class inters... |
| elintg 4888 | Membership in class inters... |
| elinti 4889 | Membership in class inters... |
| nfint 4890 | Bound-variable hypothesis ... |
| elintabg 4891 | Two ways of saying a set i... |
| elintab 4892 | Membership in the intersec... |
| elintrab 4893 | Membership in the intersec... |
| elintrabg 4894 | Membership in the intersec... |
| int0 4895 | The intersection of the em... |
| intss1 4896 | An element of a class incl... |
| ssint 4897 | Subclass of a class inters... |
| ssintab 4898 | Subclass of the intersecti... |
| ssintub 4899 | Subclass of the least uppe... |
| ssmin 4900 | Subclass of the minimum va... |
| intmin 4901 | Any member of a class is t... |
| intss 4902 | Intersection of subclasses... |
| intssuni 4903 | The intersection of a none... |
| ssintrab 4904 | Subclass of the intersecti... |
| unissint 4905 | If the union of a class is... |
| intssuni2 4906 | Subclass relationship for ... |
| intminss 4907 | Under subset ordering, the... |
| intmin2 4908 | Any set is the smallest of... |
| intmin3 4909 | Under subset ordering, the... |
| intmin4 4910 | Elimination of a conjunct ... |
| intab 4911 | The intersection of a spec... |
| int0el 4912 | The intersection of a clas... |
| intun 4913 | The class intersection of ... |
| intprg 4914 | The intersection of a pair... |
| intpr 4915 | The intersection of a pair... |
| intsng 4916 | Intersection of a singleto... |
| intsn 4917 | The intersection of a sing... |
| uniintsn 4918 | Two ways to express " ` A ... |
| uniintab 4919 | The union and the intersec... |
| intunsn 4920 | Theorem joining a singleto... |
| rint0 4921 | Relative intersection of a... |
| elrint 4922 | Membership in a restricted... |
| elrint2 4923 | Membership in a restricted... |
| eliun 4928 | Membership in indexed unio... |
| eliin 4929 | Membership in indexed inte... |
| eliuni 4930 | Membership in an indexed u... |
| eliund 4931 | Membership in indexed unio... |
| iuncom 4932 | Commutation of indexed uni... |
| iuncom4 4933 | Commutation of union with ... |
| iunconst 4934 | Indexed union of a constan... |
| iinconst 4935 | Indexed intersection of a ... |
| iuneqconst 4936 | Indexed union of identical... |
| iuniin 4937 | Law combining indexed unio... |
| iinssiun 4938 | An indexed intersection is... |
| iunss1 4939 | Subclass theorem for index... |
| iinss1 4940 | Subclass theorem for index... |
| iuneq1 4941 | Equality theorem for index... |
| iineq1 4942 | Equality theorem for index... |
| ss2iun 4943 | Subclass theorem for index... |
| iuneq2 4944 | Equality theorem for index... |
| iineq2 4945 | Equality theorem for index... |
| iuneq2i 4946 | Equality inference for ind... |
| iineq2i 4947 | Equality inference for ind... |
| iineq2d 4948 | Equality deduction for ind... |
| iuneq2dv 4949 | Equality deduction for ind... |
| iineq2dv 4950 | Equality deduction for ind... |
| iuneq12df 4951 | Equality deduction for ind... |
| iuneq1d 4952 | Equality theorem for index... |
| iuneq12dOLD 4953 | Obsolete version of ~ iune... |
| iuneq12d 4954 | Equality deduction for ind... |
| iuneq2d 4955 | Equality deduction for ind... |
| nfiun 4956 | Bound-variable hypothesis ... |
| nfiin 4957 | Bound-variable hypothesis ... |
| nfiung 4958 | Bound-variable hypothesis ... |
| nfiing 4959 | Bound-variable hypothesis ... |
| nfiu1 4960 | Bound-variable hypothesis ... |
| nfii1 4961 | Bound-variable hypothesis ... |
| dfiun2g 4962 | Alternate definition of in... |
| dfiin2g 4963 | Alternate definition of in... |
| dfiun2 4964 | Alternate definition of in... |
| dfiin2 4965 | Alternate definition of in... |
| dfiunv2 4966 | Define double indexed unio... |
| cbviun 4967 | Rule used to change the bo... |
| cbviin 4968 | Change bound variables in ... |
| cbviung 4969 | Rule used to change the bo... |
| cbviing 4970 | Change bound variables in ... |
| cbviunv 4971 | Rule used to change the bo... |
| cbviinv 4972 | Change bound variables in ... |
| cbviunvg 4973 | Rule used to change the bo... |
| cbviinvg 4974 | Change bound variables in ... |
| iunssf 4975 | Subset theorem for an inde... |
| iunssfOLD 4976 | Obsolete version of ~ iuns... |
| iunss 4977 | Subset theorem for an inde... |
| iunssOLD 4978 | Obsolete version of ~ iuns... |
| ssiun 4979 | Subset implication for an ... |
| ssiun2 4980 | Identity law for subset of... |
| ssiun2s 4981 | Subset relationship for an... |
| iunss2 4982 | A subclass condition on th... |
| iunssd 4983 | Subset theorem for an inde... |
| iunab 4984 | The indexed union of a cla... |
| iunrab 4985 | The indexed union of a res... |
| iunxdif2 4986 | Indexed union with a class... |
| ssiinf 4987 | Subset theorem for an inde... |
| ssiin 4988 | Subset theorem for an inde... |
| iinss 4989 | Subset implication for an ... |
| iinss2 4990 | An indexed intersection is... |
| uniiun 4991 | Class union in terms of in... |
| intiin 4992 | Class intersection in term... |
| iunid 4993 | An indexed union of single... |
| iun0 4994 | An indexed union of the em... |
| 0iun 4995 | An empty indexed union is ... |
| 0iin 4996 | An empty indexed intersect... |
| viin 4997 | Indexed intersection with ... |
| iunsn 4998 | Indexed union of a singlet... |
| iunn0 4999 | There is a nonempty class ... |
| iinab 5000 | Indexed intersection of a ... |
| iinrab 5001 | Indexed intersection of a ... |
| iinrab2 5002 | Indexed intersection of a ... |
| iunin2 5003 | Indexed union of intersect... |
| iunin1 5004 | Indexed union of intersect... |
| iinun2 5005 | Indexed intersection of un... |
| iundif2 5006 | Indexed union of class dif... |
| iindif1 5007 | Indexed intersection of cl... |
| 2iunin 5008 | Rearrange indexed unions o... |
| iindif2 5009 | Indexed intersection of cl... |
| iinin2 5010 | Indexed intersection of in... |
| iinin1 5011 | Indexed intersection of in... |
| iinvdif 5012 | The indexed intersection o... |
| elriin 5013 | Elementhood in a relative ... |
| riin0 5014 | Relative intersection of a... |
| riinn0 5015 | Relative intersection of a... |
| riinrab 5016 | Relative intersection of a... |
| symdif0 5017 | Symmetric difference with ... |
| symdifv 5018 | The symmetric difference w... |
| symdifid 5019 | The symmetric difference o... |
| iinxsng 5020 | A singleton index picks ou... |
| iinxprg 5021 | Indexed intersection with ... |
| iunxsng 5022 | A singleton index picks ou... |
| iunxsn 5023 | A singleton index picks ou... |
| iunxsngf 5024 | A singleton index picks ou... |
| iunun 5025 | Separate a union in an ind... |
| iunxun 5026 | Separate a union in the in... |
| iunxdif3 5027 | An indexed union where som... |
| iunxprg 5028 | A pair index picks out two... |
| iunxiun 5029 | Separate an indexed union ... |
| iinuni 5030 | A relationship involving u... |
| iununi 5031 | A relationship involving u... |
| sspwuni 5032 | Subclass relationship for ... |
| pwssb 5033 | Two ways to express a coll... |
| elpwpw 5034 | Characterization of the el... |
| pwpwab 5035 | The double power class wri... |
| pwpwssunieq 5036 | The class of sets whose un... |
| elpwuni 5037 | Relationship for power cla... |
| iinpw 5038 | The power class of an inte... |
| iunpwss 5039 | Inclusion of an indexed un... |
| intss2 5040 | A nonempty intersection of... |
| rintn0 5041 | Relative intersection of a... |
| dfdisj2 5044 | Alternate definition for d... |
| disjss2 5045 | If each element of a colle... |
| disjeq2 5046 | Equality theorem for disjo... |
| disjeq2dv 5047 | Equality deduction for dis... |
| disjss1 5048 | A subset of a disjoint col... |
| disjeq1 5049 | Equality theorem for disjo... |
| disjeq1d 5050 | Equality theorem for disjo... |
| disjeq12d 5051 | Equality theorem for disjo... |
| cbvdisj 5052 | Change bound variables in ... |
| cbvdisjv 5053 | Change bound variables in ... |
| nfdisjw 5054 | Bound-variable hypothesis ... |
| nfdisj 5055 | Bound-variable hypothesis ... |
| nfdisj1 5056 | Bound-variable hypothesis ... |
| disjor 5057 | Two ways to say that a col... |
| disjors 5058 | Two ways to say that a col... |
| disji2 5059 | Property of a disjoint col... |
| disji 5060 | Property of a disjoint col... |
| invdisj 5061 | If there is a function ` C... |
| invdisjrab 5062 | The restricted class abstr... |
| disjiun 5063 | A disjoint collection yiel... |
| disjord 5064 | Conditions for a collectio... |
| disjiunb 5065 | Two ways to say that a col... |
| disjiund 5066 | Conditions for a collectio... |
| sndisj 5067 | Any collection of singleto... |
| 0disj 5068 | Any collection of empty se... |
| disjxsn 5069 | A singleton collection is ... |
| disjx0 5070 | An empty collection is dis... |
| disjprg 5071 | A pair collection is disjo... |
| disjxiun 5072 | An indexed union of a disj... |
| disjxun 5073 | The union of two disjoint ... |
| disjss3 5074 | Expand a disjoint collecti... |
| breq 5077 | Equality theorem for binar... |
| breq1 5078 | Equality theorem for a bin... |
| breq2 5079 | Equality theorem for a bin... |
| breq12 5080 | Equality theorem for a bin... |
| breqi 5081 | Equality inference for bin... |
| breq1i 5082 | Equality inference for a b... |
| breq2i 5083 | Equality inference for a b... |
| breq12i 5084 | Equality inference for a b... |
| breq1d 5085 | Equality deduction for a b... |
| breqd 5086 | Equality deduction for a b... |
| breq2d 5087 | Equality deduction for a b... |
| breq12d 5088 | Equality deduction for a b... |
| breq123d 5089 | Equality deduction for a b... |
| breqdi 5090 | Equality deduction for a b... |
| breqan12d 5091 | Equality deduction for a b... |
| breqan12rd 5092 | Equality deduction for a b... |
| eqnbrtrd 5093 | Substitution of equal clas... |
| nbrne1 5094 | Two classes are different ... |
| nbrne2 5095 | Two classes are different ... |
| eqbrtri 5096 | Substitution of equal clas... |
| eqbrtrd 5097 | Substitution of equal clas... |
| eqbrtrri 5098 | Substitution of equal clas... |
| eqbrtrrd 5099 | Substitution of equal clas... |
| breqtri 5100 | Substitution of equal clas... |
| breqtrd 5101 | Substitution of equal clas... |
| breqtrri 5102 | Substitution of equal clas... |
| breqtrrd 5103 | Substitution of equal clas... |
| 3brtr3i 5104 | Substitution of equality i... |
| 3brtr4i 5105 | Substitution of equality i... |
| 3brtr3d 5106 | Substitution of equality i... |
| 3brtr4d 5107 | Substitution of equality i... |
| 3brtr3g 5108 | Substitution of equality i... |
| 3brtr4g 5109 | Substitution of equality i... |
| eqbrtrid 5110 | A chained equality inferen... |
| eqbrtrrid 5111 | A chained equality inferen... |
| breqtrid 5112 | A chained equality inferen... |
| breqtrrid 5113 | A chained equality inferen... |
| eqbrtrdi 5114 | A chained equality inferen... |
| eqbrtrrdi 5115 | A chained equality inferen... |
| breqtrdi 5116 | A chained equality inferen... |
| breqtrrdi 5117 | A chained equality inferen... |
| ssbrd 5118 | Deduction from a subclass ... |
| ssbr 5119 | Implication from a subclas... |
| ssbri 5120 | Inference from a subclass ... |
| nfbrd 5121 | Deduction version of bound... |
| nfbr 5122 | Bound-variable hypothesis ... |
| brab1 5123 | Relationship between a bin... |
| br0 5124 | The empty binary relation ... |
| brne0 5125 | If two sets are in a binar... |
| brun 5126 | The union of two binary re... |
| brin 5127 | The intersection of two re... |
| brdif 5128 | The difference of two bina... |
| sbcbr123 5129 | Move substitution in and o... |
| sbcbr 5130 | Move substitution in and o... |
| sbcbr12g 5131 | Move substitution in and o... |
| sbcbr1g 5132 | Move substitution in and o... |
| sbcbr2g 5133 | Move substitution in and o... |
| brsymdif 5134 | Characterization of the sy... |
| brralrspcev 5135 | Restricted existential spe... |
| brimralrspcev 5136 | Restricted existential spe... |
| opabss 5139 | The collection of ordered ... |
| opabbid 5140 | Equivalent wff's yield equ... |
| opabbidv 5141 | Equivalent wff's yield equ... |
| opabbii 5142 | Equivalent wff's yield equ... |
| nfopabd 5143 | Bound-variable hypothesis ... |
| nfopab 5144 | Bound-variable hypothesis ... |
| nfopab1 5145 | The first abstraction vari... |
| nfopab2 5146 | The second abstraction var... |
| cbvopab 5147 | Rule used to change bound ... |
| cbvopabv 5148 | Rule used to change bound ... |
| cbvopab1 5149 | Change first bound variabl... |
| cbvopab1g 5150 | Change first bound variabl... |
| cbvopab2 5151 | Change second bound variab... |
| cbvopab1s 5152 | Change first bound variabl... |
| cbvopab1v 5153 | Rule used to change the fi... |
| cbvopab2v 5154 | Rule used to change the se... |
| unopab 5155 | Union of two ordered pair ... |
| mpteq12da 5158 | An equality inference for ... |
| mpteq12df 5159 | An equality inference for ... |
| mpteq12f 5160 | An equality theorem for th... |
| mpteq12dva 5161 | An equality inference for ... |
| mpteq12dv 5162 | An equality inference for ... |
| mpteq12 5163 | An equality theorem for th... |
| mpteq1 5164 | An equality theorem for th... |
| mpteq1d 5165 | An equality theorem for th... |
| mpteq1i 5166 | An equality theorem for th... |
| mpteq2da 5167 | Slightly more general equa... |
| mpteq2dva 5168 | Slightly more general equa... |
| mpteq2dv 5169 | An equality inference for ... |
| mpteq2ia 5170 | An equality inference for ... |
| mpteq2i 5171 | An equality inference for ... |
| mpteq12i 5172 | An equality inference for ... |
| nfmpt 5173 | Bound-variable hypothesis ... |
| nfmpt1 5174 | Bound-variable hypothesis ... |
| cbvmptf 5175 | Rule to change the bound v... |
| cbvmptfg 5176 | Rule to change the bound v... |
| cbvmpt 5177 | Rule to change the bound v... |
| cbvmptg 5178 | Rule to change the bound v... |
| cbvmptv 5179 | Rule to change the bound v... |
| cbvmptvg 5180 | Rule to change the bound v... |
| mptv 5181 | Function with universal do... |
| dftr2 5184 | An alternate way of defini... |
| dftr2c 5185 | Variant of ~ dftr2 with co... |
| dftr5 5186 | An alternate way of defini... |
| dftr3 5187 | An alternate way of defini... |
| dftr4 5188 | An alternate way of defini... |
| treq 5189 | Equality theorem for the t... |
| trel 5190 | In a transitive class, the... |
| trel3 5191 | In a transitive class, the... |
| trss 5192 | An element of a transitive... |
| trun 5193 | The union of transitive cl... |
| trin 5194 | The intersection of transi... |
| tr0 5195 | The empty set is transitiv... |
| trv 5196 | The universe is transitive... |
| triun 5197 | An indexed union of a clas... |
| truni 5198 | The union of a class of tr... |
| triin 5199 | An indexed intersection of... |
| trint 5200 | The intersection of a clas... |
| trintss 5201 | Any nonempty transitive cl... |
| axrep1 5203 | The version of the Axiom o... |
| axreplem 5204 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5205 | Axiom of Replacement expre... |
| axrep3 5206 | Axiom of Replacement sligh... |
| axrep4v 5207 | Version of ~ axrep4 with a... |
| axrep4 5208 | A more traditional version... |
| axrep4OLD 5209 | Obsolete version of ~ axre... |
| axrep5 5210 | Axiom of Replacement (simi... |
| axrep6 5211 | A condensed form of ~ ax-r... |
| axrep6OLD 5212 | Obsolete version of ~ axre... |
| replem 5213 | A lemma for variants of th... |
| zfrep6 5214 | A version of the Axiom of ... |
| axrep6g 5215 | ~ axrep6 in class notation... |
| zfrepclf 5216 | An inference based on the ... |
| zfrep3cl 5217 | An inference based on the ... |
| zfrep4 5218 | A version of Replacement u... |
| axsepgfromrep 5219 | A more general version ~ a... |
| axsep 5220 | Axiom scheme of separation... |
| axsepg 5222 | A more general version of ... |
| zfauscl 5223 | Separation Scheme (Aussond... |
| sepexlem 5224 | Lemma for ~ sepex . Use ~... |
| sepex 5225 | Convert implication to equ... |
| sepexi 5226 | Convert implication to equ... |
| bm1.3iiOLD 5227 | Obsolete version of ~ sepe... |
| ax6vsep 5228 | Derive ~ ax6v (a weakened ... |
| axnulALT 5229 | Alternate proof of ~ axnul... |
| axnul 5230 | The Null Set Axiom of ZF s... |
| 0ex 5232 | The Null Set Axiom of ZF s... |
| al0ssb 5233 | The empty set is the uniqu... |
| sseliALT 5234 | Alternate proof of ~ sseli... |
| csbexg 5235 | The existence of proper su... |
| csbex 5236 | The existence of proper su... |
| unisn2 5237 | A version of ~ unisn witho... |
| exnelv 5238 | For any set ` x ` , there ... |
| nalset 5239 | No set contains all sets. ... |
| nalsetOLD 5240 | Obsolete version of ~ nals... |
| vneqv 5241 | The universal class is not... |
| vnex 5242 | The universal class does n... |
| vnexOLD 5243 | Obsolete proof of ~ vnex a... |
| nvel 5244 | The universal class does n... |
| vprc 5245 | The universal class is not... |
| vprcOLD 5246 | Obsolete proof of ~ vprc ,... |
| nvelOLD 5247 | Obsolete proof of ~ nvel ,... |
| inex1 5248 | Separation Scheme (Aussond... |
| inex2 5249 | Separation Scheme (Aussond... |
| inex1g 5250 | Closed-form, generalized S... |
| inex2g 5251 | Sufficient condition for a... |
| ssex 5252 | The subset of a set is als... |
| ssexi 5253 | The subset of a set is als... |
| ssexg 5254 | The subset of a set is als... |
| ssexd 5255 | A subclass of a set is a s... |
| abexd 5256 | Conditions for a class abs... |
| abex 5257 | Conditions for a class abs... |
| prcssprc 5258 | The superclass of a proper... |
| sselpwd 5259 | Elementhood to a power set... |
| difexg 5260 | Existence of a difference.... |
| difexi 5261 | Existence of a difference,... |
| difexd 5262 | Existence of a difference.... |
| zfausab 5263 | Separation Scheme (Aussond... |
| elpw2g 5264 | Membership in a power clas... |
| elpw2 5265 | Membership in a power clas... |
| elpwi2 5266 | Membership in a power clas... |
| rabelpw 5267 | A restricted class abstrac... |
| rabexg 5268 | Separation Scheme in terms... |
| rabexgOLD 5269 | Obsolete version of ~ rabe... |
| rabex 5270 | Separation Scheme in terms... |
| rabexd 5271 | Separation Scheme in terms... |
| rabex2 5272 | Separation Scheme in terms... |
| rab2ex 5273 | A class abstraction based ... |
| elssabg 5274 | Membership in a class abst... |
| intex 5275 | The intersection of a none... |
| intnex 5276 | If a class intersection is... |
| intexab 5277 | The intersection of a none... |
| intexrab 5278 | The intersection of a none... |
| iinexg 5279 | The existence of a class i... |
| intabs 5280 | Absorption of a redundant ... |
| inuni 5281 | The intersection of a unio... |
| axpweq 5282 | Two equivalent ways to exp... |
| pwnss 5283 | The power set of a set is ... |
| pwne 5284 | No set equals its power se... |
| difelpw 5285 | A difference is an element... |
| class2set 5286 | The class of elements of `... |
| 0elpw 5287 | Every power class contains... |
| pwne0 5288 | A power class is never emp... |
| 0nep0 5289 | The empty set and its powe... |
| 0inp0 5290 | Something cannot be equal ... |
| unidif0 5291 | The removal of the empty s... |
| unidif0OLD 5292 | Obsolete version of ~ unid... |
| eqsnuniex 5293 | If a class is equal to the... |
| iin0 5294 | An indexed intersection of... |
| notzfaus 5295 | In the Separation Scheme ~... |
| intv 5296 | The intersection of the un... |
| zfpow 5298 | Axiom of Power Sets expres... |
| axpow2 5299 | A variant of the Axiom of ... |
| axpow3 5300 | A variant of the Axiom of ... |
| elALT2 5301 | Alternate proof of ~ el us... |
| dtruALT2 5302 | Alternate proof of ~ dtru ... |
| dtrucor 5303 | Corollary of ~ dtru . Thi... |
| dtrucor2 5304 | The theorem form of the de... |
| dvdemo1 5305 | Demonstration of a theorem... |
| dvdemo2 5306 | Demonstration of a theorem... |
| nfnid 5307 | A setvar variable is not f... |
| nfcvb 5308 | The "distinctor" expressio... |
| vpwex 5309 | Power set axiom: the power... |
| pwexg 5310 | Power set axiom expressed ... |
| pwexd 5311 | Deduction version of the p... |
| pwex 5312 | Power set axiom expressed ... |
| pwel 5313 | Quantitative version of ~ ... |
| abssexg 5314 | Existence of a class of su... |
| snexALT 5315 | Alternate proof of ~ snex ... |
| p0ex 5316 | The power set of the empty... |
| p0exALT 5317 | Alternate proof of ~ p0ex ... |
| pp0ex 5318 | The power set of the power... |
| ord3ex 5319 | The ordinal number 3 is a ... |
| dtruALT 5320 | Alternate proof of ~ dtru ... |
| axc16b 5321 | This theorem shows that Ax... |
| eunex 5322 | Existential uniqueness imp... |
| eusv1 5323 | Two ways to express single... |
| eusvnf 5324 | Even if ` x ` is free in `... |
| eusvnfb 5325 | Two ways to say that ` A (... |
| eusv2i 5326 | Two ways to express single... |
| eusv2nf 5327 | Two ways to express single... |
| eusv2 5328 | Two ways to express single... |
| reusv1 5329 | Two ways to express single... |
| reusv2lem1 5330 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5331 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5332 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5333 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5334 | Lemma for ~ reusv2 . (Con... |
| reusv2 5335 | Two ways to express single... |
| reusv3i 5336 | Two ways of expressing exi... |
| reusv3 5337 | Two ways to express single... |
| eusv4 5338 | Two ways to express single... |
| alxfr 5339 | Transfer universal quantif... |
| ralxfrd 5340 | Transfer universal quantif... |
| rexxfrd 5341 | Transfer existential quant... |
| ralxfr2d 5342 | Transfer universal quantif... |
| rexxfr2d 5343 | Transfer existential quant... |
| ralxfrd2 5344 | Transfer universal quantif... |
| rexxfrd2 5345 | Transfer existence from a ... |
| ralxfr 5346 | Transfer universal quantif... |
| ralxfrALT 5347 | Alternate proof of ~ ralxf... |
| rexxfr 5348 | Transfer existence from a ... |
| rabxfrd 5349 | Membership in a restricted... |
| rabxfr 5350 | Membership in a restricted... |
| reuhypd 5351 | A theorem useful for elimi... |
| reuhyp 5352 | A theorem useful for elimi... |
| zfpair 5353 | The Axiom of Pairing of Ze... |
| axprALT 5354 | Alternate proof of ~ axpr ... |
| axprlem1 5355 | Lemma for ~ axpr . There ... |
| axprlem2 5356 | Lemma for ~ axpr . There ... |
| axprlem3 5357 | Lemma for ~ axpr . Elimin... |
| axprlem4 5358 | Lemma for ~ axpr . If an ... |
| axpr 5359 | Unabbreviated version of t... |
| axprlem1OLD 5360 | Obsolete version of ~ axpr... |
| axprlem3OLD 5361 | Obsolete version of ~ axpr... |
| axprlem4OLD 5362 | Obsolete version of ~ axpr... |
| axprlem5OLD 5363 | Obsolete version of ~ axpr... |
| axprOLD 5364 | Obsolete version of ~ axpr... |
| zfpair2 5366 | Derive the abbreviated ver... |
| vsnex 5367 | A singleton built on a set... |
| axprglem 5368 | Lemma for ~ axprg . (Cont... |
| axprg 5369 | Derive The Axiom of Pairin... |
| prex 5370 | The Axiom of Pairing using... |
| snex 5371 | A singleton is a set. The... |
| snexg 5372 | A singleton built on a set... |
| snexgALT 5373 | Alternate proof of ~ snexg... |
| snexOLD 5374 | Obsolete version of ~ snex... |
| prexOLD 5375 | Obsolete version of ~ prex... |
| exel 5376 | There exist two sets, one ... |
| exexneq 5377 | There exist two different ... |
| exneq 5378 | Given any set (the " ` y `... |
| dtru 5379 | Given any set (the " ` y `... |
| el 5380 | Any set is an element of s... |
| elOLD 5381 | Obsolete version of ~ el a... |
| sels 5382 | If a class is a set, then ... |
| selsALT 5383 | Alternate proof of ~ sels ... |
| elALT 5384 | Alternate proof of ~ el , ... |
| snelpwg 5385 | A singleton of a set is a ... |
| snelpwi 5386 | If a set is a member of a ... |
| snelpw 5387 | A singleton of a set is a ... |
| prelpw 5388 | An unordered pair of two s... |
| prelpwi 5389 | If two sets are members of... |
| rext 5390 | A theorem similar to exten... |
| sspwb 5391 | The powerclass constructio... |
| unipw 5392 | A class equals the union o... |
| univ 5393 | The union of the universe ... |
| pwtr 5394 | A class is transitive iff ... |
| ssextss 5395 | An extensionality-like pri... |
| ssext 5396 | An extensionality-like pri... |
| nssss 5397 | Negation of subclass relat... |
| pweqb 5398 | Classes are equal if and o... |
| intidg 5399 | The intersection of all se... |
| moabex 5400 | "At most one" existence im... |
| moabexOLD 5401 | Obsolete version of ~ moab... |
| rmorabex 5402 | Restricted "at most one" e... |
| euabex 5403 | The abstraction of a wff w... |
| nnullss 5404 | A nonempty class (even if ... |
| exss 5405 | Restricted existence in a ... |
| opex 5406 | An ordered pair of classes... |
| opexOLD 5407 | Obsolete version of ~ opex... |
| otex 5408 | An ordered triple of class... |
| elopg 5409 | Characterization of the el... |
| elop 5410 | Characterization of the el... |
| opi1 5411 | One of the two elements in... |
| opi2 5412 | One of the two elements of... |
| opeluu 5413 | Each member of an ordered ... |
| op1stb 5414 | Extract the first member o... |
| brv 5415 | Two classes are always in ... |
| opnz 5416 | An ordered pair is nonempt... |
| opnzi 5417 | An ordered pair is nonempt... |
| opth1 5418 | Equality of the first memb... |
| opth 5419 | The ordered pair theorem. ... |
| opthg 5420 | Ordered pair theorem. ` C ... |
| opth1g 5421 | Equality of the first memb... |
| opthg2 5422 | Ordered pair theorem. (Co... |
| opth2 5423 | Ordered pair theorem. (Co... |
| opthneg 5424 | Two ordered pairs are not ... |
| opthne 5425 | Two ordered pairs are not ... |
| otth2 5426 | Ordered triple theorem, wi... |
| otth 5427 | Ordered triple theorem. (... |
| otthg 5428 | Ordered triple theorem, cl... |
| otthne 5429 | Contrapositive of the orde... |
| eqvinop 5430 | A variable introduction la... |
| sbcop1 5431 | The proper substitution of... |
| sbcop 5432 | The proper substitution of... |
| copsexgw 5433 | Version of ~ copsexg with ... |
| copsexgwOLD 5434 | Obsolete version of ~ cops... |
| copsexg 5435 | Substitution of class ` A ... |
| copsex2t 5436 | Closed theorem form of ~ c... |
| copsex2g 5437 | Implicit substitution infe... |
| copsex2dv 5438 | Implicit substitution dedu... |
| copsex4g 5439 | An implicit substitution i... |
| 0nelop 5440 | A property of ordered pair... |
| opwo0id 5441 | An ordered pair is equal t... |
| opeqex 5442 | Equivalence of existence i... |
| oteqex2 5443 | Equivalence of existence i... |
| oteqex 5444 | Equivalence of existence i... |
| opcom 5445 | An ordered pair commutes i... |
| moop2 5446 | "At most one" property of ... |
| opeqsng 5447 | Equivalence for an ordered... |
| opeqsn 5448 | Equivalence for an ordered... |
| opeqpr 5449 | Equivalence for an ordered... |
| snopeqop 5450 | Equivalence for an ordered... |
| propeqop 5451 | Equivalence for an ordered... |
| propssopi 5452 | If a pair of ordered pairs... |
| snopeqopsnid 5453 | Equivalence for an ordered... |
| mosubopt 5454 | "At most one" remains true... |
| mosubop 5455 | "At most one" remains true... |
| euop2 5456 | Transfer existential uniqu... |
| euotd 5457 | Prove existential uniquene... |
| opthwiener 5458 | Justification theorem for ... |
| uniop 5459 | The union of an ordered pa... |
| uniopel 5460 | Ordered pair membership is... |
| opthhausdorff 5461 | Justification theorem for ... |
| opthhausdorff0 5462 | Justification theorem for ... |
| otsndisj 5463 | The singletons consisting ... |
| otiunsndisj 5464 | The union of singletons co... |
| iunopeqop 5465 | Implication of an ordered ... |
| iunopeqopOLD 5466 | Obsolete version of ~ iuno... |
| brsnop 5467 | Binary relation for an ord... |
| brtp 5468 | A necessary and sufficient... |
| opabidw 5469 | The law of concretion. Sp... |
| opabid 5470 | The law of concretion. Sp... |
| elopabw 5471 | Membership in a class abst... |
| elopab 5472 | Membership in a class abst... |
| rexopabb 5473 | Restricted existential qua... |
| vopelopabsb 5474 | The law of concretion in t... |
| opelopabsb 5475 | The law of concretion in t... |
| brabsb 5476 | The law of concretion in t... |
| opelopabt 5477 | Closed theorem form of ~ o... |
| opelopabga 5478 | The law of concretion. Th... |
| brabga 5479 | The law of concretion for ... |
| opelopab2a 5480 | Ordered pair membership in... |
| opelopaba 5481 | The law of concretion. Th... |
| braba 5482 | The law of concretion for ... |
| opelopabg 5483 | The law of concretion. Th... |
| brabg 5484 | The law of concretion for ... |
| opelopabgf 5485 | The law of concretion. Th... |
| opelopab2 5486 | Ordered pair membership in... |
| opelopab 5487 | The law of concretion. Th... |
| brab 5488 | The law of concretion for ... |
| opelopabaf 5489 | The law of concretion. Th... |
| opelopabf 5490 | The law of concretion. Th... |
| ssopab2 5491 | Equivalence of ordered pai... |
| ssopab2bw 5492 | Equivalence of ordered pai... |
| eqopab2bw 5493 | Equivalence of ordered pai... |
| ssopab2b 5494 | Equivalence of ordered pai... |
| ssopab2i 5495 | Inference of ordered pair ... |
| ssopab2dv 5496 | Inference of ordered pair ... |
| eqopab2b 5497 | Equivalence of ordered pai... |
| opabn0 5498 | Nonempty ordered pair clas... |
| opab0 5499 | Empty ordered pair class a... |
| csbopab 5500 | Move substitution into a c... |
| csbopabw 5501 | Move substitution into a c... |
| csbmpt12 5502 | Move substitution into a m... |
| csbmpt2 5503 | Move substitution into the... |
| iunopab 5504 | Move indexed union inside ... |
| elopabr 5505 | Membership in an ordered-p... |
| elopabran 5506 | Membership in an ordered-p... |
| rbropapd 5507 | Properties of a pair in an... |
| rbropap 5508 | Properties of a pair in a ... |
| 2rbropap 5509 | Properties of a pair in a ... |
| 0nelopab 5510 | The empty set is never an ... |
| brabv 5511 | If two classes are in a re... |
| pwin 5512 | The power class of the int... |
| pwssun 5513 | The power class of the uni... |
| pwun 5514 | The power class of the uni... |
| dfid4 5517 | The identity function expr... |
| dfid2 5518 | Alternate definition of th... |
| dfid3 5519 | A stronger version of ~ df... |
| epelg 5522 | The membership relation an... |
| epeli 5523 | The membership relation an... |
| epel 5524 | The membership relation an... |
| 0sn0ep 5525 | An example for the members... |
| epn0 5526 | The membership relation is... |
| poss 5531 | Subset theorem for the par... |
| poeq1 5532 | Equality theorem for parti... |
| poeq2 5533 | Equality theorem for parti... |
| poeq12d 5534 | Equality deduction for par... |
| nfpo 5535 | Bound-variable hypothesis ... |
| nfso 5536 | Bound-variable hypothesis ... |
| pocl 5537 | Characteristic properties ... |
| ispod 5538 | Sufficient conditions for ... |
| swopolem 5539 | Perform the substitutions ... |
| swopo 5540 | A strict weak order is a p... |
| poirr 5541 | A partial order is irrefle... |
| potr 5542 | A partial order is a trans... |
| po2nr 5543 | A partial order has no 2-c... |
| po3nr 5544 | A partial order has no 3-c... |
| po2ne 5545 | Two sets related by a part... |
| po0 5546 | Any relation is a partial ... |
| pofun 5547 | The inverse image of a par... |
| sopo 5548 | A strict linear order is a... |
| soss 5549 | Subset theorem for the str... |
| soeq1 5550 | Equality theorem for the s... |
| soeq2 5551 | Equality theorem for the s... |
| soeq12d 5552 | Equality deduction for tot... |
| sonr 5553 | A strict order relation is... |
| sotr 5554 | A strict order relation is... |
| sotrd 5555 | Transitivity law for stric... |
| solin 5556 | A strict order relation is... |
| so2nr 5557 | A strict order relation ha... |
| so3nr 5558 | A strict order relation ha... |
| sotric 5559 | A strict order relation sa... |
| sotrieq 5560 | Trichotomy law for strict ... |
| sotrieq2 5561 | Trichotomy law for strict ... |
| soasym 5562 | Asymmetry law for strict o... |
| sotr2 5563 | A transitivity relation. ... |
| issod 5564 | An irreflexive, transitive... |
| issoi 5565 | An irreflexive, transitive... |
| isso2i 5566 | Deduce strict ordering fro... |
| so0 5567 | Any relation is a strict o... |
| somo 5568 | A totally ordered set has ... |
| sotrine 5569 | Trichotomy law for strict ... |
| sotr3 5570 | Transitivity law for stric... |
| dffr6 5577 | Alternate definition of ~ ... |
| frd 5578 | A nonempty subset of an ` ... |
| fri 5579 | A nonempty subset of an ` ... |
| seex 5580 | The ` R ` -preimage of an ... |
| exse 5581 | Any relation on a set is s... |
| dffr2 5582 | Alternate definition of we... |
| dffr2ALT 5583 | Alternate proof of ~ dffr2... |
| frc 5584 | Property of well-founded r... |
| frss 5585 | Subset theorem for the wel... |
| sess1 5586 | Subset theorem for the set... |
| sess2 5587 | Subset theorem for the set... |
| freq1 5588 | Equality theorem for the w... |
| freq2 5589 | Equality theorem for the w... |
| freq12d 5590 | Equality deduction for wel... |
| seeq1 5591 | Equality theorem for the s... |
| seeq2 5592 | Equality theorem for the s... |
| seeq12d 5593 | Equality deduction for the... |
| nffr 5594 | Bound-variable hypothesis ... |
| nfse 5595 | Bound-variable hypothesis ... |
| nfwe 5596 | Bound-variable hypothesis ... |
| frirr 5597 | A well-founded relation is... |
| fr2nr 5598 | A well-founded relation ha... |
| fr0 5599 | Any relation is well-found... |
| frminex 5600 | If an element of a well-fo... |
| efrirr 5601 | A well-founded class does ... |
| efrn2lp 5602 | A well-founded class conta... |
| epse 5603 | The membership relation is... |
| tz7.2 5604 | Similar to Theorem 7.2 of ... |
| dfepfr 5605 | An alternate way of saying... |
| epfrc 5606 | A subset of a well-founded... |
| wess 5607 | Subset theorem for the wel... |
| weeq1 5608 | Equality theorem for the w... |
| weeq2 5609 | Equality theorem for the w... |
| weeq12d 5610 | Equality deduction for wel... |
| wefr 5611 | A well-ordering is well-fo... |
| weso 5612 | A well-ordering is a stric... |
| wecmpep 5613 | The elements of a class we... |
| wetrep 5614 | On a class well-ordered by... |
| wefrc 5615 | A nonempty subclass of a c... |
| we0 5616 | Any relation is a well-ord... |
| wereu 5617 | A nonempty subset of an ` ... |
| wereu2 5618 | A nonempty subclass of an ... |
| xpeq1 5635 | Equality theorem for Carte... |
| xpss12 5636 | Subset theorem for Cartesi... |
| xpss 5637 | A Cartesian product is inc... |
| inxpssres 5638 | Intersection with a Cartes... |
| relxp 5639 | A Cartesian product is a r... |
| xpss1 5640 | Subset relation for Cartes... |
| xpss2 5641 | Subset relation for Cartes... |
| xpeq2 5642 | Equality theorem for Carte... |
| elxpi 5643 | Membership in a Cartesian ... |
| elxp 5644 | Membership in a Cartesian ... |
| elxp2 5645 | Membership in a Cartesian ... |
| xpeq12 5646 | Equality theorem for Carte... |
| xpeq1i 5647 | Equality inference for Car... |
| xpeq2i 5648 | Equality inference for Car... |
| xpeq12i 5649 | Equality inference for Car... |
| xpeq1d 5650 | Equality deduction for Car... |
| xpeq2d 5651 | Equality deduction for Car... |
| xpeq12d 5652 | Equality deduction for Car... |
| sqxpeqd 5653 | Equality deduction for a C... |
| nfxp 5654 | Bound-variable hypothesis ... |
| 0nelxp 5655 | The empty set is not a mem... |
| 0nelelxp 5656 | A member of a Cartesian pr... |
| opelxp 5657 | Ordered pair membership in... |
| opelxpi 5658 | Ordered pair membership in... |
| opelxpii 5659 | Ordered pair membership in... |
| opelxpd 5660 | Ordered pair membership in... |
| opelvv 5661 | Ordered pair membership in... |
| opelvvg 5662 | Ordered pair membership in... |
| opelxp1 5663 | The first member of an ord... |
| opelxp2 5664 | The second member of an or... |
| otelxp 5665 | Ordered triple membership ... |
| otelxp1 5666 | The first member of an ord... |
| otel3xp 5667 | An ordered triple is an el... |
| opabssxpd 5668 | An ordered-pair class abst... |
| rabxp 5669 | Class abstraction restrict... |
| brxp 5670 | Binary relation on a Carte... |
| pwvrel 5671 | A set is a binary relation... |
| pwvabrel 5672 | The powerclass of the cart... |
| brrelex12 5673 | Two classes related by a b... |
| brrelex1 5674 | If two classes are related... |
| brrelex2 5675 | If two classes are related... |
| brrelex12i 5676 | Two classes that are relat... |
| brrelex1i 5677 | The first argument of a bi... |
| brrelex2i 5678 | The second argument of a b... |
| nprrel12 5679 | Proper classes are not rel... |
| nprrel 5680 | No proper class is related... |
| 0nelrel0 5681 | A binary relation does not... |
| 0nelrel 5682 | A binary relation does not... |
| fconstmpt 5683 | Representation of a consta... |
| vtoclr 5684 | Variable to class conversi... |
| opthprc 5685 | Justification theorem for ... |
| brel 5686 | Two things in a binary rel... |
| elxp3 5687 | Membership in a Cartesian ... |
| opeliunxp 5688 | Membership in a union of C... |
| opeliun2xp 5689 | Membership of an ordered p... |
| xpundi 5690 | Distributive law for Carte... |
| xpundir 5691 | Distributive law for Carte... |
| xpiundi 5692 | Distributive law for Carte... |
| xpiundir 5693 | Distributive law for Carte... |
| iunxpconst 5694 | Membership in a union of C... |
| xpun 5695 | The Cartesian product of t... |
| elvv 5696 | Membership in universal cl... |
| elvvv 5697 | Membership in universal cl... |
| elvvuni 5698 | An ordered pair contains i... |
| brinxp2 5699 | Intersection of binary rel... |
| brinxp 5700 | Intersection of binary rel... |
| opelinxp 5701 | Ordered pair element in an... |
| poinxp 5702 | Intersection of partial or... |
| soinxp 5703 | Intersection of total orde... |
| frinxp 5704 | Intersection of well-found... |
| seinxp 5705 | Intersection of set-like r... |
| weinxp 5706 | Intersection of well-order... |
| posn 5707 | Partial ordering of a sing... |
| sosn 5708 | Strict ordering on a singl... |
| frsn 5709 | Founded relation on a sing... |
| wesn 5710 | Well-ordering of a singlet... |
| elopaelxp 5711 | Membership in an ordered-p... |
| bropaex12 5712 | Two classes related by an ... |
| opabssxp 5713 | An abstraction relation is... |
| brab2a 5714 | The law of concretion for ... |
| optocl 5715 | Implicit substitution of c... |
| optoclOLD 5716 | Obsolete version of ~ opto... |
| 2optocl 5717 | Implicit substitution of c... |
| 3optocl 5718 | Implicit substitution of c... |
| opbrop 5719 | Ordered pair membership in... |
| 0xp 5720 | The Cartesian product with... |
| xp0 5721 | The Cartesian product with... |
| csbxp 5722 | Distribute proper substitu... |
| releq 5723 | Equality theorem for the r... |
| releqi 5724 | Equality inference for the... |
| releqd 5725 | Equality deduction for the... |
| nfrel 5726 | Bound-variable hypothesis ... |
| sbcrel 5727 | Distribute proper substitu... |
| relss 5728 | Subclass theorem for relat... |
| ssrel 5729 | A subclass relationship de... |
| eqrel 5730 | Extensionality principle f... |
| ssrel2 5731 | A subclass relationship de... |
| ssrel3 5732 | Subclass relation in anoth... |
| relssi 5733 | Inference from subclass pr... |
| relssdv 5734 | Deduction from subclass pr... |
| eqrelriv 5735 | Inference from extensional... |
| eqrelriiv 5736 | Inference from extensional... |
| eqbrriv 5737 | Inference from extensional... |
| eqrelrdv 5738 | Deduce equality of relatio... |
| eqbrrdv 5739 | Deduction from extensional... |
| eqbrrdiv 5740 | Deduction from extensional... |
| eqrelrdv2 5741 | A version of ~ eqrelrdv . ... |
| ssrelrel 5742 | A subclass relationship de... |
| eqrelrel 5743 | Extensionality principle f... |
| elrel 5744 | A member of a relation is ... |
| rel0 5745 | The empty set is a relatio... |
| nrelv 5746 | The universal class is not... |
| relsng 5747 | A singleton is a relation ... |
| relsnb 5748 | An at-most-singleton is a ... |
| relsnopg 5749 | A singleton of an ordered ... |
| relsn 5750 | A singleton is a relation ... |
| relsnop 5751 | A singleton of an ordered ... |
| copsex2gb 5752 | Implicit substitution infe... |
| copsex2ga 5753 | Implicit substitution infe... |
| elopaba 5754 | Membership in an ordered-p... |
| xpsspw 5755 | A Cartesian product is inc... |
| unixpss 5756 | The double class union of ... |
| relun 5757 | The union of two relations... |
| relin1 5758 | The intersection with a re... |
| relin2 5759 | The intersection with a re... |
| relinxp 5760 | Intersection with a Cartes... |
| reldif 5761 | A difference cutting down ... |
| reliun 5762 | An indexed union is a rela... |
| reliin 5763 | An indexed intersection is... |
| reluni 5764 | The union of a class is a ... |
| relint 5765 | The intersection of a clas... |
| relopabiv 5766 | A class of ordered pairs i... |
| relopabv 5767 | A class of ordered pairs i... |
| relopabi 5768 | A class of ordered pairs i... |
| relopabiALT 5769 | Alternate proof of ~ relop... |
| relopab 5770 | A class of ordered pairs i... |
| mptrel 5771 | The maps-to notation alway... |
| reli 5772 | The identity relation is a... |
| rele 5773 | The membership relation is... |
| opabid2 5774 | A relation expressed as an... |
| inopab 5775 | Intersection of two ordere... |
| difopab 5776 | Difference of two ordered-... |
| inxp 5777 | Intersection of two Cartes... |
| xpindi 5778 | Distributive law for Carte... |
| xpindir 5779 | Distributive law for Carte... |
| xpiindi 5780 | Distributive law for Carte... |
| xpriindi 5781 | Distributive law for Carte... |
| eliunxp 5782 | Membership in a union of C... |
| opeliunxp2 5783 | Membership in a union of C... |
| raliunxp 5784 | Write a double restricted ... |
| rexiunxp 5785 | Write a double restricted ... |
| ralxp 5786 | Universal quantification r... |
| rexxp 5787 | Existential quantification... |
| exopxfr 5788 | Transfer ordered-pair exis... |
| exopxfr2 5789 | Transfer ordered-pair exis... |
| djussxp 5790 | Disjoint union is a subset... |
| ralxpf 5791 | Version of ~ ralxp with bo... |
| rexxpf 5792 | Version of ~ rexxp with bo... |
| iunxpf 5793 | Indexed union on a Cartesi... |
| opabbi2dv 5794 | Deduce equality of a relat... |
| relop 5795 | A necessary and sufficient... |
| ideqg 5796 | For sets, the identity rel... |
| ideq 5797 | For sets, the identity rel... |
| ididg 5798 | A set is identical to itse... |
| issetid 5799 | Two ways of expressing set... |
| coss1 5800 | Subclass theorem for compo... |
| coss2 5801 | Subclass theorem for compo... |
| coeq1 5802 | Equality theorem for compo... |
| coeq2 5803 | Equality theorem for compo... |
| coeq1i 5804 | Equality inference for com... |
| coeq2i 5805 | Equality inference for com... |
| coeq1d 5806 | Equality deduction for com... |
| coeq2d 5807 | Equality deduction for com... |
| coeq12i 5808 | Equality inference for com... |
| coeq12d 5809 | Equality deduction for com... |
| nfco 5810 | Bound-variable hypothesis ... |
| brcog 5811 | Ordered pair membership in... |
| opelco2g 5812 | Ordered pair membership in... |
| brcogw 5813 | Ordered pair membership in... |
| eqbrrdva 5814 | Deduction from extensional... |
| brco 5815 | Binary relation on a compo... |
| opelco 5816 | Ordered pair membership in... |
| cnvss 5817 | Subset theorem for convers... |
| cnveq 5818 | Equality theorem for conve... |
| cnveqi 5819 | Equality inference for con... |
| cnveqd 5820 | Equality deduction for con... |
| elcnv 5821 | Membership in a converse r... |
| elcnv2 5822 | Membership in a converse r... |
| nfcnv 5823 | Bound-variable hypothesis ... |
| brcnvg 5824 | The converse of a binary r... |
| opelcnvg 5825 | Ordered-pair membership in... |
| opelcnv 5826 | Ordered-pair membership in... |
| brcnv 5827 | The converse of a binary r... |
| cnv0 5828 | The converse of the empty ... |
| cnv0OLD 5829 | Obsolete version of ~ cnv0... |
| cnvi 5830 | The converse of the identi... |
| csbcnv 5831 | Move class substitution in... |
| csbcnvOLD 5832 | Obsolete version of ~ csbc... |
| csbcnvgALTOLD 5833 | Obsolete version of ~ csbc... |
| cnvco 5834 | Distributive law of conver... |
| cnvuni 5835 | The converse of a class un... |
| dfdm3 5836 | Alternate definition of do... |
| dfrn2 5837 | Alternate definition of ra... |
| dfrn3 5838 | Alternate definition of ra... |
| elrn2g 5839 | Membership in a range. (C... |
| elrng 5840 | Membership in a range. (C... |
| elrn2 5841 | Membership in a range. (C... |
| elrn 5842 | Membership in a range. (C... |
| ssrelrn 5843 | If a relation is a subset ... |
| dfdm4 5844 | Alternate definition of do... |
| dfdmf 5845 | Definition of domain, usin... |
| csbdm 5846 | Distribute proper substitu... |
| eldmg 5847 | Domain membership. Theore... |
| eldm2g 5848 | Domain membership. Theore... |
| eldm 5849 | Membership in a domain. T... |
| eldm2 5850 | Membership in a domain. T... |
| dmss 5851 | Subset theorem for domain.... |
| dmeq 5852 | Equality theorem for domai... |
| dmeqi 5853 | Equality inference for dom... |
| dmeqd 5854 | Equality deduction for dom... |
| opeldmd 5855 | Membership of first of an ... |
| opeldm 5856 | Membership of first of an ... |
| breldm 5857 | Membership of first of a b... |
| breldmg 5858 | Membership of first of a b... |
| dmun 5859 | The domain of a union is t... |
| dmin 5860 | The domain of an intersect... |
| breldmd 5861 | Membership of first of a b... |
| dmiun 5862 | The domain of an indexed u... |
| dmuni 5863 | The domain of a union. Pa... |
| dmopab 5864 | The domain of a class of o... |
| dmopabelb 5865 | A set is an element of the... |
| dmopab2rex 5866 | The domain of an ordered p... |
| dmopabss 5867 | Upper bound for the domain... |
| dmopab3 5868 | The domain of a restricted... |
| dm0 5869 | The domain of the empty se... |
| dmi 5870 | The domain of the identity... |
| dmv 5871 | The domain of the universe... |
| dmep 5872 | The domain of the membersh... |
| dm0rn0 5873 | An empty domain is equival... |
| dm0rn0OLD 5874 | Obsolete version of ~ dm0r... |
| rn0 5875 | The range of the empty set... |
| rnep 5876 | The range of the membershi... |
| reldm0 5877 | A relation is empty iff it... |
| dmxp 5878 | The domain of a Cartesian ... |
| dmxpid 5879 | The domain of a Cartesian ... |
| dmxpin 5880 | The domain of the intersec... |
| xpid11 5881 | The Cartesian square is a ... |
| dmcnvcnv 5882 | The domain of the double c... |
| rncnvcnv 5883 | The range of the double co... |
| elreldm 5884 | The first member of an ord... |
| rneq 5885 | Equality theorem for range... |
| rneqi 5886 | Equality inference for ran... |
| rneqd 5887 | Equality deduction for ran... |
| rnss 5888 | Subset theorem for range. ... |
| rnssi 5889 | Subclass inference for ran... |
| brelrng 5890 | The second argument of a b... |
| brelrn 5891 | The second argument of a b... |
| opelrn 5892 | Membership of second membe... |
| releldm 5893 | The first argument of a bi... |
| relelrn 5894 | The second argument of a b... |
| releldmb 5895 | Membership in a domain. (... |
| relelrnb 5896 | Membership in a range. (C... |
| releldmi 5897 | The first argument of a bi... |
| relelrni 5898 | The second argument of a b... |
| dfrnf 5899 | Definition of range, using... |
| nfdm 5900 | Bound-variable hypothesis ... |
| nfrn 5901 | Bound-variable hypothesis ... |
| dmiin 5902 | Domain of an intersection.... |
| rnopab 5903 | The range of a class of or... |
| rnopabss 5904 | Upper bound for the range ... |
| rnopab3 5905 | The range of a restricted ... |
| rnmpt 5906 | The range of a function in... |
| elrnmpt 5907 | The range of a function in... |
| elrnmpt1s 5908 | Elementhood in an image se... |
| elrnmpt1 5909 | Elementhood in an image se... |
| elrnmptg 5910 | Membership in the range of... |
| elrnmpti 5911 | Membership in the range of... |
| elrnmptd 5912 | The range of a function in... |
| elrnmpt1d 5913 | Elementhood in an image se... |
| elrnmptdv 5914 | Elementhood in the range o... |
| elrnmpt2d 5915 | Elementhood in the range o... |
| nelrnmpt 5916 | Non-membership in the rang... |
| dfiun3g 5917 | Alternate definition of in... |
| dfiin3g 5918 | Alternate definition of in... |
| dfiun3 5919 | Alternate definition of in... |
| dfiin3 5920 | Alternate definition of in... |
| riinint 5921 | Express a relative indexed... |
| relrn0 5922 | A relation is empty iff it... |
| dmrnssfld 5923 | The domain and range of a ... |
| dmcoss 5924 | Domain of a composition. ... |
| dmcossOLD 5925 | Obsolete version of ~ dmco... |
| rncoss 5926 | Range of a composition. (... |
| dmcosseq 5927 | Domain of a composition. ... |
| dmcosseqOLD 5928 | Obsolete version of ~ dmco... |
| dmcosseqOLDOLD 5929 | Obsolete version of ~ dmco... |
| dmcoeq 5930 | Domain of a composition. ... |
| rncoeq 5931 | Range of a composition. (... |
| reseq1 5932 | Equality theorem for restr... |
| reseq2 5933 | Equality theorem for restr... |
| reseq1i 5934 | Equality inference for res... |
| reseq2i 5935 | Equality inference for res... |
| reseq12i 5936 | Equality inference for res... |
| reseq1d 5937 | Equality deduction for res... |
| reseq2d 5938 | Equality deduction for res... |
| reseq12d 5939 | Equality deduction for res... |
| nfres 5940 | Bound-variable hypothesis ... |
| csbres 5941 | Distribute proper substitu... |
| res0 5942 | A restriction to the empty... |
| dfres3 5943 | Alternate definition of re... |
| opelres 5944 | Ordered pair elementhood i... |
| brres 5945 | Binary relation on a restr... |
| opelresi 5946 | Ordered pair membership in... |
| brresi 5947 | Binary relation on a restr... |
| opres 5948 | Ordered pair membership in... |
| resieq 5949 | A restricted identity rela... |
| opelidres 5950 | ` <. A , A >. ` belongs to... |
| resres 5951 | The restriction of a restr... |
| resundi 5952 | Distributive law for restr... |
| resundir 5953 | Distributive law for restr... |
| resindi 5954 | Class restriction distribu... |
| resindir 5955 | Class restriction distribu... |
| inres 5956 | Move intersection into cla... |
| resdifcom 5957 | Commutative law for restri... |
| resiun1 5958 | Distribution of restrictio... |
| resiun2 5959 | Distribution of restrictio... |
| resss 5960 | A class includes its restr... |
| rescom 5961 | Commutative law for restri... |
| ssres 5962 | Subclass theorem for restr... |
| ssres2 5963 | Subclass theorem for restr... |
| relres 5964 | A restriction is a relatio... |
| resabs1 5965 | Absorption law for restric... |
| resabs1i 5966 | Absorption law for restric... |
| resabs1d 5967 | Absorption law for restric... |
| resabs2 5968 | Absorption law for restric... |
| residm 5969 | Idempotent law for restric... |
| dmresss 5970 | The domain of a restrictio... |
| dmres 5971 | The domain of a restrictio... |
| ssdmres 5972 | A domain restricted to a s... |
| dmresexg 5973 | The domain of a restrictio... |
| resima 5974 | A restriction to an image.... |
| resima2 5975 | Image under a restricted c... |
| rnresss 5976 | The range of a restriction... |
| xpssres 5977 | Restriction of a constant ... |
| elinxp 5978 | Membership in an intersect... |
| elres 5979 | Membership in a restrictio... |
| elsnres 5980 | Membership in restriction ... |
| relssres 5981 | Simplification law for res... |
| dmressnsn 5982 | The domain of a restrictio... |
| eldmressnsn 5983 | The element of the domain ... |
| eldmeldmressn 5984 | An element of the domain (... |
| resdm 5985 | A relation restricted to i... |
| resexg 5986 | The restriction of a set i... |
| resexd 5987 | The restriction of a set i... |
| resex 5988 | The restriction of a set i... |
| resindm 5989 | When restricting a relatio... |
| resdmdfsn 5990 | Restricting a relation to ... |
| reldisjun 5991 | Split a relation into two ... |
| relresdm1 5992 | Restriction of a disjoint ... |
| resopab 5993 | Restriction of a class abs... |
| iss 5994 | A subclass of the identity... |
| resopab2 5995 | Restriction of a class abs... |
| resmpt 5996 | Restriction of the mapping... |
| resmpt3 5997 | Unconditional restriction ... |
| resmptf 5998 | Restriction of the mapping... |
| resmptd 5999 | Restriction of the mapping... |
| dfres2 6000 | Alternate definition of th... |
| mptss 6001 | Sufficient condition for i... |
| elimampt 6002 | Membership in the image of... |
| elidinxp 6003 | Characterization of the el... |
| elidinxpid 6004 | Characterization of the el... |
| elrid 6005 | Characterization of the el... |
| idinxpres 6006 | The intersection of the id... |
| idinxpresid 6007 | The intersection of the id... |
| idssxp 6008 | A diagonal set as a subset... |
| opabresid 6009 | The restricted identity re... |
| mptresid 6010 | The restricted identity re... |
| dmresi 6011 | The domain of a restricted... |
| restidsing 6012 | Restriction of the identit... |
| iresn0n0 6013 | The identity function rest... |
| imaeq1 6014 | Equality theorem for image... |
| imaeq2 6015 | Equality theorem for image... |
| imaeq1i 6016 | Equality theorem for image... |
| imaeq2i 6017 | Equality theorem for image... |
| imaeq1d 6018 | Equality theorem for image... |
| imaeq2d 6019 | Equality theorem for image... |
| imaeq12d 6020 | Equality theorem for image... |
| dfima2 6021 | Alternate definition of im... |
| dfima3 6022 | Alternate definition of im... |
| elimag 6023 | Membership in an image. T... |
| elima 6024 | Membership in an image. T... |
| elima2 6025 | Membership in an image. T... |
| elima3 6026 | Membership in an image. T... |
| nfima 6027 | Bound-variable hypothesis ... |
| nfimad 6028 | Deduction version of bound... |
| imadmrn 6029 | The image of the domain of... |
| imassrn 6030 | The image of a class is a ... |
| mptima 6031 | Image of a function in map... |
| mptimass 6032 | Image of a function in map... |
| imai 6033 | Image under the identity r... |
| rnresi 6034 | The range of the restricte... |
| resiima 6035 | The image of a restriction... |
| ima0 6036 | Image of the empty set. T... |
| 0ima 6037 | Image under the empty rela... |
| csbima12 6038 | Move class substitution in... |
| imadisj 6039 | A class whose image under ... |
| imadisjlnd 6040 | Deduction form of one nega... |
| cnvimass 6041 | A preimage under any class... |
| cnvimarndm 6042 | The preimage of the range ... |
| imasng 6043 | The image of a singleton. ... |
| relimasn 6044 | The image of a singleton. ... |
| elrelimasn 6045 | Elementhood in the image o... |
| elimasng1 6046 | Membership in an image of ... |
| elimasn1 6047 | Membership in an image of ... |
| elimasng 6048 | Membership in an image of ... |
| elimasn 6049 | Membership in an image of ... |
| elimasni 6050 | Membership in an image of ... |
| args 6051 | Two ways to express the cl... |
| elinisegg 6052 | Membership in the inverse ... |
| eliniseg 6053 | Membership in the inverse ... |
| epin 6054 | Any set is equal to its pr... |
| epini 6055 | Any set is equal to its pr... |
| iniseg 6056 | An idiom that signifies an... |
| inisegn0 6057 | Nonemptiness of an initial... |
| dffr3 6058 | Alternate definition of we... |
| dfse2 6059 | Alternate definition of se... |
| imass1 6060 | Subset theorem for image. ... |
| imass2 6061 | Subset theorem for image. ... |
| ndmima 6062 | The image of a singleton o... |
| relcnv 6063 | A converse is a relation. ... |
| relbrcnvg 6064 | When ` R ` is a relation, ... |
| eliniseg2 6065 | Eliminate the class existe... |
| relbrcnv 6066 | When ` R ` is a relation, ... |
| relco 6067 | A composition is a relatio... |
| cotrg 6068 | Two ways of saying that th... |
| cotr 6069 | Two ways of saying a relat... |
| idrefALT 6070 | Alternate proof of ~ idref... |
| cnvsym 6071 | Two ways of saying a relat... |
| intasym 6072 | Two ways of saying a relat... |
| asymref 6073 | Two ways of saying a relat... |
| asymref2 6074 | Two ways of saying a relat... |
| intirr 6075 | Two ways of saying a relat... |
| brcodir 6076 | Two ways of saying that tw... |
| codir 6077 | Two ways of saying a relat... |
| qfto 6078 | A quantifier-free way of e... |
| xpidtr 6079 | A Cartesian square is a tr... |
| trin2 6080 | The intersection of two tr... |
| poirr2 6081 | A partial order is irrefle... |
| trinxp 6082 | The relation induced by a ... |
| soirri 6083 | A strict order relation is... |
| sotri 6084 | A strict order relation is... |
| son2lpi 6085 | A strict order relation ha... |
| sotri2 6086 | A transitivity relation. ... |
| sotri3 6087 | A transitivity relation. ... |
| poleloe 6088 | Express "less than or equa... |
| poltletr 6089 | Transitive law for general... |
| somin1 6090 | Property of a minimum in a... |
| somincom 6091 | Commutativity of minimum i... |
| somin2 6092 | Property of a minimum in a... |
| soltmin 6093 | Being less than a minimum,... |
| cnvopab 6094 | The converse of a class ab... |
| cnvopabOLD 6095 | Obsolete version of ~ cnvo... |
| mptcnv 6096 | The converse of a mapping ... |
| cnvun 6097 | The converse of a union is... |
| cnvdif 6098 | Distributive law for conve... |
| cnvin 6099 | Distributive law for conve... |
| rnun 6100 | Distributive law for range... |
| rnin 6101 | The range of an intersecti... |
| rniun 6102 | The range of an indexed un... |
| rnuni 6103 | The range of a union. Par... |
| imaundi 6104 | Distributive law for image... |
| imaundir 6105 | The image of a union. (Co... |
| imadifssran 6106 | Condition for the range of... |
| cnvimassrndm 6107 | The preimage of a superset... |
| dminss 6108 | An upper bound for interse... |
| imainss 6109 | An upper bound for interse... |
| inimass 6110 | The image of an intersecti... |
| inimasn 6111 | The intersection of the im... |
| cnvxp 6112 | The converse of a Cartesia... |
| xp0OLD 6113 | Obsolete version of ~ xp0 ... |
| xpnz 6114 | The Cartesian product of n... |
| xpeq0 6115 | At least one member of an ... |
| xpdisj1 6116 | Cartesian products with di... |
| xpdisj2 6117 | Cartesian products with di... |
| xpsndisj 6118 | Cartesian products with tw... |
| difxp 6119 | Difference of Cartesian pr... |
| difxp1 6120 | Difference law for Cartesi... |
| difxp2 6121 | Difference law for Cartesi... |
| djudisj 6122 | Disjoint unions with disjo... |
| xpdifid 6123 | The set of distinct couple... |
| resdisj 6124 | A double restriction to di... |
| rnxp 6125 | The range of a Cartesian p... |
| dmxpss 6126 | The domain of a Cartesian ... |
| rnxpss 6127 | The range of a Cartesian p... |
| rnxpid 6128 | The range of a Cartesian s... |
| ssxpb 6129 | A Cartesian product subcla... |
| xp11 6130 | The Cartesian product of n... |
| xpcan 6131 | Cancellation law for Carte... |
| xpcan2 6132 | Cancellation law for Carte... |
| ssrnres 6133 | Two ways to express surjec... |
| rninxp 6134 | Two ways to express surjec... |
| dminxp 6135 | Two ways to express totali... |
| imainrect 6136 | Image by a restricted and ... |
| xpima 6137 | Direct image by a Cartesia... |
| xpima1 6138 | Direct image by a Cartesia... |
| xpima2 6139 | Direct image by a Cartesia... |
| xpimasn 6140 | Direct image of a singleto... |
| sossfld 6141 | The base set of a strict o... |
| sofld 6142 | The base set of a nonempty... |
| cnvcnv3 6143 | The set of all ordered pai... |
| dfrel2 6144 | Alternate definition of re... |
| dfrel4v 6145 | A relation can be expresse... |
| dfrel4 6146 | A relation can be expresse... |
| cnvcnv 6147 | The double converse of a c... |
| cnvcnv2 6148 | The double converse of a c... |
| cnvcnvss 6149 | The double converse of a c... |
| cnvrescnv 6150 | Two ways to express the co... |
| cnveqb 6151 | Equality theorem for conve... |
| cnveq0 6152 | A relation empty iff its c... |
| dfrel3 6153 | Alternate definition of re... |
| elid 6154 | Characterization of the el... |
| dmresv 6155 | The domain of a universal ... |
| rnresv 6156 | The range of a universal r... |
| dfrn4 6157 | Range defined in terms of ... |
| csbrn 6158 | Distribute proper substitu... |
| rescnvcnv 6159 | The restriction of the dou... |
| cnvcnvres 6160 | The double converse of the... |
| imacnvcnv 6161 | The image of the double co... |
| dmsnn0 6162 | The domain of a singleton ... |
| rnsnn0 6163 | The range of a singleton i... |
| dmsn0 6164 | The domain of the singleto... |
| cnvsn0 6165 | The converse of the single... |
| dmsn0el 6166 | The domain of a singleton ... |
| relsn2 6167 | A singleton is a relation ... |
| dmsnopg 6168 | The domain of a singleton ... |
| dmsnopss 6169 | The domain of a singleton ... |
| dmpropg 6170 | The domain of an unordered... |
| dmsnop 6171 | The domain of a singleton ... |
| dmprop 6172 | The domain of an unordered... |
| dmtpop 6173 | The domain of an unordered... |
| cnvcnvsn 6174 | Double converse of a singl... |
| dmsnsnsn 6175 | The domain of the singleto... |
| rnsnopg 6176 | The range of a singleton o... |
| rnpropg 6177 | The range of a pair of ord... |
| cnvsng 6178 | Converse of a singleton of... |
| rnsnop 6179 | The range of a singleton o... |
| op1sta 6180 | Extract the first member o... |
| cnvsn 6181 | Converse of a singleton of... |
| op2ndb 6182 | Extract the second member ... |
| op2nda 6183 | Extract the second member ... |
| opswap 6184 | Swap the members of an ord... |
| cnvresima 6185 | An image under the convers... |
| resdm2 6186 | A class restricted to its ... |
| resdmres 6187 | Restriction to the domain ... |
| resresdm 6188 | A restriction by an arbitr... |
| imadmres 6189 | The image of the domain of... |
| resdmss 6190 | Subset relationship for th... |
| resdifdi 6191 | Distributive law for restr... |
| resdifdir 6192 | Distributive law for restr... |
| mptpreima 6193 | The preimage of a function... |
| mptiniseg 6194 | Converse singleton image o... |
| dmmpt 6195 | The domain of the mapping ... |
| dmmptss 6196 | The domain of a mapping is... |
| dmmptg 6197 | The domain of the mapping ... |
| rnmpt0f 6198 | The range of a function in... |
| rnmptn0 6199 | The range of a function in... |
| dfco2 6200 | Alternate definition of a ... |
| dfco2a 6201 | Generalization of ~ dfco2 ... |
| coundi 6202 | Class composition distribu... |
| coundir 6203 | Class composition distribu... |
| cores 6204 | Restricted first member of... |
| resco 6205 | Associative law for the re... |
| imaco 6206 | Image of the composition o... |
| rnco 6207 | The range of the compositi... |
| rncoOLD 6208 | Obsolete version of ~ rnco... |
| rnco2 6209 | The range of the compositi... |
| dmco 6210 | The domain of a compositio... |
| coeq0 6211 | A composition of two relat... |
| coiun 6212 | Composition with an indexe... |
| cocnvcnv1 6213 | A composition is not affec... |
| cocnvcnv2 6214 | A composition is not affec... |
| cores2 6215 | Absorption of a reverse (p... |
| co02 6216 | Composition with the empty... |
| co01 6217 | Composition with the empty... |
| coi1 6218 | Composition with the ident... |
| coi2 6219 | Composition with the ident... |
| coires1 6220 | Composition with a restric... |
| coass 6221 | Associative law for class ... |
| relcnvtrg 6222 | General form of ~ relcnvtr... |
| relcnvtr 6223 | A relation is transitive i... |
| relssdmrn 6224 | A relation is included in ... |
| resssxp 6225 | If the ` R ` -image of a c... |
| cnvssrndm 6226 | The converse is a subset o... |
| cossxp 6227 | Composition as a subset of... |
| relrelss 6228 | Two ways to describe the s... |
| unielrel 6229 | The membership relation fo... |
| relfld 6230 | The double union of a rela... |
| relresfld 6231 | Restriction of a relation ... |
| relcoi2 6232 | Composition with the ident... |
| relcoi1 6233 | Composition with the ident... |
| unidmrn 6234 | The double union of the co... |
| relcnvfld 6235 | if ` R ` is a relation, it... |
| dfdm2 6236 | Alternate definition of do... |
| unixp 6237 | The double class union of ... |
| unixp0 6238 | A Cartesian product is emp... |
| unixpid 6239 | Field of a Cartesian squar... |
| ressn 6240 | Restriction of a class to ... |
| cnviin 6241 | The converse of an interse... |
| cnvpo 6242 | The converse of a partial ... |
| cnvso 6243 | The converse of a strict o... |
| xpco 6244 | Composition of two Cartesi... |
| xpcoid 6245 | Composition of two Cartesi... |
| elsnxp 6246 | Membership in a Cartesian ... |
| reu3op 6247 | There is a unique ordered ... |
| reuop 6248 | There is a unique ordered ... |
| opreu2reurex 6249 | There is a unique ordered ... |
| opreu2reu 6250 | If there is a unique order... |
| dfpo2 6251 | Quantifier-free definition... |
| csbcog 6252 | Distribute proper substitu... |
| snres0 6253 | Condition for restriction ... |
| imaindm 6254 | The image is unaffected by... |
| predeq123 6257 | Equality theorem for the p... |
| predeq1 6258 | Equality theorem for the p... |
| predeq2 6259 | Equality theorem for the p... |
| predeq3 6260 | Equality theorem for the p... |
| nfpred 6261 | Bound-variable hypothesis ... |
| csbpredg 6262 | Move class substitution in... |
| predpredss 6263 | If ` A ` is a subset of ` ... |
| predss 6264 | The predecessor class of `... |
| sspred 6265 | Another subset/predecessor... |
| dfpred2 6266 | An alternate definition of... |
| dfpred3 6267 | An alternate definition of... |
| dfpred3g 6268 | An alternate definition of... |
| elpredgg 6269 | Membership in a predecesso... |
| elpredg 6270 | Membership in a predecesso... |
| elpredimg 6271 | Membership in a predecesso... |
| elpredim 6272 | Membership in a predecesso... |
| elpred 6273 | Membership in a predecesso... |
| predexg 6274 | The predecessor class exis... |
| dffr4 6275 | Alternate definition of we... |
| predel 6276 | Membership in the predeces... |
| predtrss 6277 | If ` R ` is transitive ove... |
| predpo 6278 | Property of the predecesso... |
| predso 6279 | Property of the predecesso... |
| setlikespec 6280 | If ` R ` is set-like in ` ... |
| predidm 6281 | Idempotent law for the pre... |
| predin 6282 | Intersection law for prede... |
| predun 6283 | Union law for predecessor ... |
| preddif 6284 | Difference law for predece... |
| predep 6285 | The predecessor under the ... |
| trpred 6286 | The class of predecessors ... |
| preddowncl 6287 | A property of classes that... |
| predpoirr 6288 | Given a partial ordering, ... |
| predfrirr 6289 | Given a well-founded relat... |
| pred0 6290 | The predecessor class over... |
| dfse3 6291 | Alternate definition of se... |
| predrelss 6292 | Subset carries from relati... |
| predprc 6293 | The predecessor of a prope... |
| predres 6294 | Predecessor class is unaff... |
| frpomin 6295 | Every nonempty (possibly p... |
| frpomin2 6296 | Every nonempty (possibly p... |
| frpoind 6297 | The principle of well-foun... |
| frpoinsg 6298 | Well-Founded Induction Sch... |
| frpoins2fg 6299 | Well-Founded Induction sch... |
| frpoins2g 6300 | Well-Founded Induction sch... |
| frpoins3g 6301 | Well-Founded Induction sch... |
| tz6.26 6302 | All nonempty subclasses of... |
| tz6.26i 6303 | All nonempty subclasses of... |
| wfi 6304 | The Principle of Well-Orde... |
| wfii 6305 | The Principle of Well-Orde... |
| wfisg 6306 | Well-Ordered Induction Sch... |
| wfis 6307 | Well-Ordered Induction Sch... |
| wfis2fg 6308 | Well-Ordered Induction Sch... |
| wfis2f 6309 | Well-Ordered Induction sch... |
| wfis2g 6310 | Well-Ordered Induction Sch... |
| wfis2 6311 | Well-Ordered Induction sch... |
| wfis3 6312 | Well-Ordered Induction sch... |
| ordeq 6321 | Equality theorem for the o... |
| elong 6322 | An ordinal number is an or... |
| elon 6323 | An ordinal number is an or... |
| eloni 6324 | An ordinal number has the ... |
| elon2 6325 | An ordinal number is an or... |
| limeq 6326 | Equality theorem for the l... |
| ordwe 6327 | Membership well-orders eve... |
| ordtr 6328 | An ordinal class is transi... |
| ordfr 6329 | Membership is well-founded... |
| ordelss 6330 | An element of an ordinal c... |
| trssord 6331 | A transitive subclass of a... |
| ordirr 6332 | No ordinal class is a memb... |
| nordeq 6333 | A member of an ordinal cla... |
| ordn2lp 6334 | An ordinal class cannot be... |
| tz7.5 6335 | A nonempty subclass of an ... |
| ordelord 6336 | An element of an ordinal c... |
| tron 6337 | The class of all ordinal n... |
| ordelon 6338 | An element of an ordinal c... |
| onelon 6339 | An element of an ordinal n... |
| tz7.7 6340 | A transitive class belongs... |
| ordelssne 6341 | For ordinal classes, membe... |
| ordelpss 6342 | For ordinal classes, membe... |
| ordsseleq 6343 | For ordinal classes, inclu... |
| ordin 6344 | The intersection of two or... |
| onin 6345 | The intersection of two or... |
| ordtri3or 6346 | A trichotomy law for ordin... |
| ordtri1 6347 | A trichotomy law for ordin... |
| ontri1 6348 | A trichotomy law for ordin... |
| ordtri2 6349 | A trichotomy law for ordin... |
| ordtri3 6350 | A trichotomy law for ordin... |
| ordtri4 6351 | A trichotomy law for ordin... |
| orddisj 6352 | An ordinal class and its s... |
| onfr 6353 | The ordinal class is well-... |
| onelpss 6354 | Relationship between membe... |
| onsseleq 6355 | Relationship between subse... |
| onelss 6356 | An element of an ordinal n... |
| oneltri 6357 | The elementhood relation o... |
| ordtr1 6358 | Transitive law for ordinal... |
| ordtr2 6359 | Transitive law for ordinal... |
| ordtr3 6360 | Transitive law for ordinal... |
| ontr1 6361 | Transitive law for ordinal... |
| ontr2 6362 | Transitive law for ordinal... |
| onelssex 6363 | Ordinal less than is equiv... |
| ordunidif 6364 | The union of an ordinal st... |
| ordintdif 6365 | If ` B ` is smaller than `... |
| onintss 6366 | If a property is true for ... |
| oneqmini 6367 | A way to show that an ordi... |
| ord0 6368 | The empty set is an ordina... |
| 0elon 6369 | The empty set is an ordina... |
| ord0eln0 6370 | A nonempty ordinal contain... |
| on0eln0 6371 | An ordinal number contains... |
| dflim2 6372 | An alternate definition of... |
| inton 6373 | The intersection of the cl... |
| nlim0 6374 | The empty set is not a lim... |
| limord 6375 | A limit ordinal is ordinal... |
| limuni 6376 | A limit ordinal is its own... |
| limuni2 6377 | The union of a limit ordin... |
| 0ellim 6378 | A limit ordinal contains t... |
| limelon 6379 | A limit ordinal class that... |
| onn0 6380 | The class of all ordinal n... |
| suceqd 6381 | Deduction associated with ... |
| suceq 6382 | Equality of successors. (... |
| elsuci 6383 | Membership in a successor.... |
| elsucg 6384 | Membership in a successor.... |
| elsuc2g 6385 | Variant of membership in a... |
| elsuc 6386 | Membership in a successor.... |
| elsuc2 6387 | Membership in a successor.... |
| nfsuc 6388 | Bound-variable hypothesis ... |
| elelsuc 6389 | Membership in a successor.... |
| sucel 6390 | Membership of a successor ... |
| suc0 6391 | The successor of the empty... |
| sucprc 6392 | A proper class is its own ... |
| unisucs 6393 | The union of the successor... |
| unisucg 6394 | A transitive class is equa... |
| unisuc 6395 | A transitive class is equa... |
| sssucid 6396 | A class is included in its... |
| sucidg 6397 | Part of Proposition 7.23 o... |
| sucid 6398 | A set belongs to its succe... |
| nsuceq0 6399 | No successor is empty. (C... |
| eqelsuc 6400 | A set belongs to the succe... |
| iunsuc 6401 | Inductive definition for t... |
| suctr 6402 | The successor of a transit... |
| trsuc 6403 | A set whose successor belo... |
| trsucss 6404 | A member of the successor ... |
| ordsssuc 6405 | An ordinal is a subset of ... |
| onsssuc 6406 | A subset of an ordinal num... |
| ordsssuc2 6407 | An ordinal subset of an or... |
| onmindif 6408 | When its successor is subt... |
| ordnbtwn 6409 | There is no set between an... |
| onnbtwn 6410 | There is no set between an... |
| sucssel 6411 | A set whose successor is a... |
| orddif 6412 | Ordinal derived from its s... |
| orduniss 6413 | An ordinal class includes ... |
| ordtri2or 6414 | A trichotomy law for ordin... |
| ordtri2or2 6415 | A trichotomy law for ordin... |
| ordtri2or3 6416 | A consequence of total ord... |
| ordelinel 6417 | The intersection of two or... |
| ordssun 6418 | Property of a subclass of ... |
| ordequn 6419 | The maximum (i.e. union) o... |
| ordun 6420 | The maximum (i.e., union) ... |
| onunel 6421 | The union of two ordinals ... |
| ordunisssuc 6422 | A subclass relationship fo... |
| suc11 6423 | The successor operation be... |
| onun2 6424 | The union of two ordinals ... |
| ontr 6425 | An ordinal number is a tra... |
| onunisuc 6426 | An ordinal number is equal... |
| onordi 6427 | An ordinal number is an or... |
| onirri 6428 | An ordinal number is not a... |
| oneli 6429 | A member of an ordinal num... |
| onelssi 6430 | A member of an ordinal num... |
| onssneli 6431 | An ordering law for ordina... |
| onssnel2i 6432 | An ordering law for ordina... |
| onelini 6433 | An element of an ordinal n... |
| oneluni 6434 | An ordinal number equals i... |
| onunisuci 6435 | An ordinal number is equal... |
| onsseli 6436 | Subset is equivalent to me... |
| onun2i 6437 | The union of two ordinal n... |
| unizlim 6438 | An ordinal equal to its ow... |
| on0eqel 6439 | An ordinal number either e... |
| snsn0non 6440 | The singleton of the singl... |
| onxpdisj 6441 | Ordinal numbers and ordere... |
| onnev 6442 | The class of ordinal numbe... |
| iotajust 6444 | Soundness justification th... |
| dfiota2 6446 | Alternate definition for d... |
| nfiota1 6447 | Bound-variable hypothesis ... |
| nfiotadw 6448 | Deduction version of ~ nfi... |
| nfiotaw 6449 | Bound-variable hypothesis ... |
| nfiotad 6450 | Deduction version of ~ nfi... |
| nfiota 6451 | Bound-variable hypothesis ... |
| cbviotaw 6452 | Change bound variables in ... |
| cbviotavw 6453 | Change bound variables in ... |
| cbviota 6454 | Change bound variables in ... |
| cbviotav 6455 | Change bound variables in ... |
| sb8iota 6456 | Variable substitution in d... |
| iotaeq 6457 | Equality theorem for descr... |
| iotabi 6458 | Equivalence theorem for de... |
| uniabio 6459 | Part of Theorem 8.17 in [Q... |
| iotaval2 6460 | Version of ~ iotaval using... |
| iotauni2 6461 | Version of ~ iotauni using... |
| iotanul2 6462 | Version of ~ iotanul using... |
| iotaval 6463 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6464 | The ` iota ` class is a su... |
| iotaex 6465 | Theorem 8.23 in [Quine] p.... |
| iotauni 6466 | Equivalence between two di... |
| iotaint 6467 | Equivalence between two di... |
| iota1 6468 | Property of iota. (Contri... |
| iotanul 6469 | Theorem 8.22 in [Quine] p.... |
| iota4 6470 | Theorem *14.22 in [Whitehe... |
| iota4an 6471 | Theorem *14.23 in [Whitehe... |
| iota5 6472 | A method for computing iot... |
| iotabidv 6473 | Formula-building deduction... |
| iotabii 6474 | Formula-building deduction... |
| iotacl 6475 | Membership law for descrip... |
| iota2df 6476 | A condition that allows to... |
| iota2d 6477 | A condition that allows to... |
| iota2 6478 | The unique element such th... |
| iotan0 6479 | Representation of "the uni... |
| sniota 6480 | A class abstraction with a... |
| dfiota4 6481 | The ` iota ` operation usi... |
| csbiota 6482 | Class substitution within ... |
| dffun2 6499 | Alternate definition of a ... |
| dffun6 6500 | Alternate definition of a ... |
| dffun3 6501 | Alternate definition of fu... |
| dffun4 6502 | Alternate definition of a ... |
| dffun5 6503 | Alternate definition of fu... |
| dffun6f 6504 | Definition of function, us... |
| funmo 6505 | A function has at most one... |
| funrel 6506 | A function is a relation. ... |
| 0nelfun 6507 | A function does not contai... |
| funss 6508 | Subclass theorem for funct... |
| funeq 6509 | Equality theorem for funct... |
| funeqi 6510 | Equality inference for the... |
| funeqd 6511 | Equality deduction for the... |
| nffun 6512 | Bound-variable hypothesis ... |
| sbcfung 6513 | Distribute proper substitu... |
| funeu 6514 | There is exactly one value... |
| funeu2 6515 | There is exactly one value... |
| dffun7 6516 | Alternate definition of a ... |
| dffun8 6517 | Alternate definition of a ... |
| dffun9 6518 | Alternate definition of a ... |
| funfn 6519 | A class is a function if a... |
| funfnd 6520 | A function is a function o... |
| funi 6521 | The identity relation is a... |
| nfunv 6522 | The universal class is not... |
| funopg 6523 | A Kuratowski ordered pair ... |
| funopab 6524 | A class of ordered pairs i... |
| funopabeq 6525 | A class of ordered pairs o... |
| funopab4 6526 | A class of ordered pairs o... |
| funmpt 6527 | A function in maps-to nota... |
| funmpt2 6528 | Functionality of a class g... |
| funco 6529 | The composition of two fun... |
| funresfunco 6530 | Composition of two functio... |
| funres 6531 | A restriction of a functio... |
| funresd 6532 | A restriction of a functio... |
| funssres 6533 | The restriction of a funct... |
| fun2ssres 6534 | Equality of restrictions o... |
| funun 6535 | The union of functions wit... |
| fununmo 6536 | If the union of classes is... |
| fununfun 6537 | If the union of classes is... |
| fundif 6538 | A function with removed el... |
| funcnvsn 6539 | The converse singleton of ... |
| funsng 6540 | A singleton of an ordered ... |
| fnsng 6541 | Functionality and domain o... |
| funsn 6542 | A singleton of an ordered ... |
| funprg 6543 | A set of two pairs is a fu... |
| funtpg 6544 | A set of three pairs is a ... |
| funpr 6545 | A function with a domain o... |
| funtp 6546 | A function with a domain o... |
| fnsn 6547 | Functionality and domain o... |
| fnprg 6548 | Function with a domain of ... |
| fntpg 6549 | Function with a domain of ... |
| fntp 6550 | A function with a domain o... |
| funcnvpr 6551 | The converse pair of order... |
| funcnvtp 6552 | The converse triple of ord... |
| funcnvqp 6553 | The converse quadruple of ... |
| fun0 6554 | The empty set is a functio... |
| funcnv0 6555 | The converse of the empty ... |
| funcnvcnv 6556 | The double converse of a f... |
| funcnv2 6557 | A simpler equivalence for ... |
| funcnv 6558 | The converse of a class is... |
| funcnv3 6559 | A condition showing a clas... |
| fun2cnv 6560 | The double converse of a c... |
| svrelfun 6561 | A single-valued relation i... |
| fncnv 6562 | Single-rootedness (see ~ f... |
| fun11 6563 | Two ways of stating that `... |
| fununi 6564 | The union of a chain (with... |
| funin 6565 | The intersection with a fu... |
| funres11 6566 | The restriction of a one-t... |
| funcnvres 6567 | The converse of a restrict... |
| cnvresid 6568 | Converse of a restricted i... |
| funcnvres2 6569 | The converse of a restrict... |
| funimacnv 6570 | The image of the preimage ... |
| funimass1 6571 | A kind of contraposition l... |
| funimass2 6572 | A kind of contraposition l... |
| imadif 6573 | The image of a difference ... |
| imain 6574 | The image of an intersecti... |
| f1imadifssran 6575 | Condition for the range of... |
| funimaexg 6576 | Axiom of Replacement using... |
| funimaex 6577 | The image of a set under a... |
| isarep1 6578 | Part of a study of the Axi... |
| isarep2 6579 | Part of a study of the Axi... |
| fneq1 6580 | Equality theorem for funct... |
| fneq2 6581 | Equality theorem for funct... |
| fneq1d 6582 | Equality deduction for fun... |
| fneq2d 6583 | Equality deduction for fun... |
| fneq12d 6584 | Equality deduction for fun... |
| fneq12 6585 | Equality theorem for funct... |
| fneq1i 6586 | Equality inference for fun... |
| fneq2i 6587 | Equality inference for fun... |
| nffn 6588 | Bound-variable hypothesis ... |
| fnfun 6589 | A function with domain is ... |
| fnfund 6590 | A function with domain is ... |
| fnrel 6591 | A function with domain is ... |
| fndm 6592 | The domain of a function. ... |
| fndmi 6593 | The domain of a function. ... |
| fndmd 6594 | The domain of a function. ... |
| funfni 6595 | Inference to convert a fun... |
| fndmu 6596 | A function has a unique do... |
| fnbr 6597 | The first argument of bina... |
| fnop 6598 | The first argument of an o... |
| fneu 6599 | There is exactly one value... |
| fneu2 6600 | There is exactly one value... |
| fnunres1 6601 | Restriction of a disjoint ... |
| fnunres2 6602 | Restriction of a disjoint ... |
| fnun 6603 | The union of two functions... |
| fnund 6604 | The union of two functions... |
| fnunop 6605 | Extension of a function wi... |
| fncofn 6606 | Composition of a function ... |
| fnco 6607 | Composition of two functio... |
| fnresdm 6608 | A function does not change... |
| fnresdisj 6609 | A function restricted to a... |
| 2elresin 6610 | Membership in two function... |
| fnssresb 6611 | Restriction of a function ... |
| fnssres 6612 | Restriction of a function ... |
| fnssresd 6613 | Restriction of a function ... |
| fnresin1 6614 | Restriction of a function'... |
| fnresin2 6615 | Restriction of a function'... |
| fnres 6616 | An equivalence for functio... |
| idfn 6617 | The identity relation is a... |
| fnresi 6618 | The restricted identity re... |
| fnima 6619 | The image of a function's ... |
| fn0 6620 | A function with empty doma... |
| fnimadisj 6621 | A class that is disjoint w... |
| fnimaeq0 6622 | Images under a function ne... |
| dfmpt3 6623 | Alternate definition for t... |
| mptfnf 6624 | The maps-to notation defin... |
| fnmptf 6625 | The maps-to notation defin... |
| fnopabg 6626 | Functionality and domain o... |
| fnopab 6627 | Functionality and domain o... |
| mptfng 6628 | The maps-to notation defin... |
| fnmpt 6629 | The maps-to notation defin... |
| fnmptd 6630 | The maps-to notation defin... |
| mpt0 6631 | A mapping operation with e... |
| fnmpti 6632 | Functionality and domain o... |
| dmmpti 6633 | Domain of the mapping oper... |
| dmmptd 6634 | The domain of the mapping ... |
| mptun 6635 | Union of mappings which ar... |
| partfun 6636 | Rewrite a function defined... |
| feq1 6637 | Equality theorem for funct... |
| feq2 6638 | Equality theorem for funct... |
| feq3 6639 | Equality theorem for funct... |
| feq23 6640 | Equality theorem for funct... |
| feq1d 6641 | Equality deduction for fun... |
| feq1dd 6642 | Equality deduction for fun... |
| feq2d 6643 | Equality deduction for fun... |
| feq3d 6644 | Equality deduction for fun... |
| feq2dd 6645 | Equality deduction for fun... |
| feq3dd 6646 | Equality deduction for fun... |
| feq12d 6647 | Equality deduction for fun... |
| feq123d 6648 | Equality deduction for fun... |
| feq123 6649 | Equality theorem for funct... |
| feq1i 6650 | Equality inference for fun... |
| feq2i 6651 | Equality inference for fun... |
| feq12i 6652 | Equality inference for fun... |
| feq23i 6653 | Equality inference for fun... |
| feq23d 6654 | Equality deduction for fun... |
| nff 6655 | Bound-variable hypothesis ... |
| sbcfng 6656 | Distribute proper substitu... |
| sbcfg 6657 | Distribute proper substitu... |
| elimf 6658 | Eliminate a mapping hypoth... |
| ffn 6659 | A mapping is a function wi... |
| ffnd 6660 | A mapping is a function wi... |
| dffn2 6661 | Any function is a mapping ... |
| ffun 6662 | A mapping is a function. ... |
| ffund 6663 | A mapping is a function, d... |
| frel 6664 | A mapping is a relation. ... |
| freld 6665 | A mapping is a relation. ... |
| frn 6666 | The range of a mapping. (... |
| frnd 6667 | Deduction form of ~ frn . ... |
| fdm 6668 | The domain of a mapping. ... |
| fdmd 6669 | Deduction form of ~ fdm . ... |
| fdmi 6670 | Inference associated with ... |
| dffn3 6671 | A function maps to its ran... |
| ffrn 6672 | A function maps to its ran... |
| ffrnb 6673 | Characterization of a func... |
| ffrnbd 6674 | A function maps to its ran... |
| fss 6675 | Expanding the codomain of ... |
| fssd 6676 | Expanding the codomain of ... |
| fssdmd 6677 | Expressing that a class is... |
| fssdm 6678 | Expressing that a class is... |
| fimass 6679 | The image of a class under... |
| fimassd 6680 | The image of a class is a ... |
| fimacnv 6681 | The preimage of the codoma... |
| fcof 6682 | Composition of a function ... |
| fco 6683 | Composition of two functio... |
| fcod 6684 | Composition of two mapping... |
| fco2 6685 | Functionality of a composi... |
| fssxp 6686 | A mapping is a class of or... |
| funssxp 6687 | Two ways of specifying a p... |
| ffdm 6688 | A mapping is a partial fun... |
| ffdmd 6689 | The domain of a function. ... |
| fdmrn 6690 | A different way to write `... |
| funcofd 6691 | Composition of two functio... |
| opelf 6692 | The members of an ordered ... |
| fun 6693 | The union of two functions... |
| fun2 6694 | The union of two functions... |
| fun2d 6695 | The union of functions wit... |
| fnfco 6696 | Composition of two functio... |
| fssres 6697 | Restriction of a function ... |
| fssresd 6698 | Restriction of a function ... |
| fssres2 6699 | Restriction of a restricte... |
| fresin 6700 | An identity for the mappin... |
| resasplit 6701 | If two functions agree on ... |
| fresaun 6702 | The union of two functions... |
| fresaunres2 6703 | From the union of two func... |
| fresaunres1 6704 | From the union of two func... |
| fcoi1 6705 | Composition of a mapping a... |
| fcoi2 6706 | Composition of restricted ... |
| feu 6707 | There is exactly one value... |
| fcnvres 6708 | The converse of a restrict... |
| fimacnvdisj 6709 | The preimage of a class di... |
| fint 6710 | Function into an intersect... |
| fin 6711 | Mapping into an intersecti... |
| f0 6712 | The empty function. (Cont... |
| f00 6713 | A class is a function with... |
| f0bi 6714 | A function with empty doma... |
| f0dom0 6715 | A function is empty iff it... |
| f0rn0 6716 | If there is no element in ... |
| fconst 6717 | A Cartesian product with a... |
| fconstg 6718 | A Cartesian product with a... |
| fnconstg 6719 | A Cartesian product with a... |
| fconst6g 6720 | Constant function with loo... |
| fconst6 6721 | A constant function as a m... |
| f1eq1 6722 | Equality theorem for one-t... |
| f1eq2 6723 | Equality theorem for one-t... |
| f1eq3 6724 | Equality theorem for one-t... |
| nff1 6725 | Bound-variable hypothesis ... |
| dff12 6726 | Alternate definition of a ... |
| f1f 6727 | A one-to-one mapping is a ... |
| f1fn 6728 | A one-to-one mapping is a ... |
| f1fun 6729 | A one-to-one mapping is a ... |
| f1rel 6730 | A one-to-one onto mapping ... |
| f1dm 6731 | The domain of a one-to-one... |
| f1ss 6732 | A function that is one-to-... |
| f1ssr 6733 | A function that is one-to-... |
| f1ssres 6734 | A function that is one-to-... |
| f1resf1 6735 | The restriction of an inje... |
| f1cnvcnv 6736 | Two ways to express that a... |
| f1cof1 6737 | Composition of two one-to-... |
| f1co 6738 | Composition of one-to-one ... |
| foeq1 6739 | Equality theorem for onto ... |
| foeq2 6740 | Equality theorem for onto ... |
| foeq3 6741 | Equality theorem for onto ... |
| nffo 6742 | Bound-variable hypothesis ... |
| fof 6743 | An onto mapping is a mappi... |
| fofun 6744 | An onto mapping is a funct... |
| fofn 6745 | An onto mapping is a funct... |
| forn 6746 | The codomain of an onto fu... |
| dffo2 6747 | Alternate definition of an... |
| foima 6748 | The image of the domain of... |
| dffn4 6749 | A function maps onto its r... |
| funforn 6750 | A function maps its domain... |
| fodmrnu 6751 | An onto function has uniqu... |
| fimadmfo 6752 | A function is a function o... |
| fores 6753 | Restriction of an onto fun... |
| fimadmfoALT 6754 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6755 | The preimage of the codoma... |
| focofo 6756 | Composition of onto functi... |
| foco 6757 | Composition of onto functi... |
| foconst 6758 | A nonzero constant functio... |
| f1oeq1 6759 | Equality theorem for one-t... |
| f1oeq2 6760 | Equality theorem for one-t... |
| f1oeq3 6761 | Equality theorem for one-t... |
| f1oeq23 6762 | Equality theorem for one-t... |
| f1eq123d 6763 | Equality deduction for one... |
| foeq123d 6764 | Equality deduction for ont... |
| f1oeq123d 6765 | Equality deduction for one... |
| f1oeq1d 6766 | Equality deduction for one... |
| f1oeq2d 6767 | Equality deduction for one... |
| f1oeq3d 6768 | Equality deduction for one... |
| nff1o 6769 | Bound-variable hypothesis ... |
| f1of1 6770 | A one-to-one onto mapping ... |
| f1of 6771 | A one-to-one onto mapping ... |
| f1ofn 6772 | A one-to-one onto mapping ... |
| f1ofun 6773 | A one-to-one onto mapping ... |
| f1orel 6774 | A one-to-one onto mapping ... |
| f1odm 6775 | The domain of a one-to-one... |
| dff1o2 6776 | Alternate definition of on... |
| dff1o3 6777 | Alternate definition of on... |
| f1ofo 6778 | A one-to-one onto function... |
| dff1o4 6779 | Alternate definition of on... |
| dff1o5 6780 | Alternate definition of on... |
| f1orn 6781 | A one-to-one function maps... |
| f1f1orn 6782 | A one-to-one function maps... |
| f1ocnv 6783 | The converse of a one-to-o... |
| f1ocnvb 6784 | A relation is a one-to-one... |
| f1ores 6785 | The restriction of a one-t... |
| f1orescnv 6786 | The converse of a one-to-o... |
| f1imacnv 6787 | Preimage of an image. (Co... |
| foimacnv 6788 | A reverse version of ~ f1i... |
| foun 6789 | The union of two onto func... |
| f1oun 6790 | The union of two one-to-on... |
| f1un 6791 | The union of two one-to-on... |
| resdif 6792 | The restriction of a one-t... |
| resin 6793 | The restriction of a one-t... |
| f1oco 6794 | Composition of one-to-one ... |
| f1cnv 6795 | The converse of an injecti... |
| funcocnv2 6796 | Composition with the conve... |
| fococnv2 6797 | The composition of an onto... |
| f1ococnv2 6798 | The composition of a one-t... |
| f1cocnv2 6799 | Composition of an injectiv... |
| f1ococnv1 6800 | The composition of a one-t... |
| f1cocnv1 6801 | Composition of an injectiv... |
| funcoeqres 6802 | Express a constraint on a ... |
| f1ssf1 6803 | A subset of an injective f... |
| f10 6804 | The empty set maps one-to-... |
| f10d 6805 | The empty set maps one-to-... |
| f1o00 6806 | One-to-one onto mapping of... |
| fo00 6807 | Onto mapping of the empty ... |
| f1o0 6808 | One-to-one onto mapping of... |
| f1oi 6809 | A restriction of the ident... |
| f1oiOLD 6810 | Obsolete version of ~ f1oi... |
| f1ovi 6811 | The identity relation is a... |
| f1osn 6812 | A singleton of an ordered ... |
| f1osng 6813 | A singleton of an ordered ... |
| f1sng 6814 | A singleton of an ordered ... |
| fsnd 6815 | A singleton of an ordered ... |
| f1oprswap 6816 | A two-element swap is a bi... |
| f1oprg 6817 | An unordered pair of order... |
| tz6.12-2 6818 | Function value when ` F ` ... |
| tz6.12-2OLD 6819 | Obsolete version of ~ tz6.... |
| fveu 6820 | The value of a function at... |
| brprcneu 6821 | If ` A ` is a proper class... |
| brprcneuALT 6822 | Alternate proof of ~ brprc... |
| fvprc 6823 | A function's value at a pr... |
| fvprcALT 6824 | Alternate proof of ~ fvprc... |
| rnfvprc 6825 | The range of a function va... |
| fv2 6826 | Alternate definition of fu... |
| dffv3 6827 | A definition of function v... |
| dffv4 6828 | The previous definition of... |
| elfv 6829 | Membership in a function v... |
| fveq1 6830 | Equality theorem for funct... |
| fveq2 6831 | Equality theorem for funct... |
| fveq1i 6832 | Equality inference for fun... |
| fveq1d 6833 | Equality deduction for fun... |
| fveq2i 6834 | Equality inference for fun... |
| fveq2d 6835 | Equality deduction for fun... |
| 2fveq3 6836 | Equality theorem for neste... |
| fveq12i 6837 | Equality deduction for fun... |
| fveq12d 6838 | Equality deduction for fun... |
| fveqeq2d 6839 | Equality deduction for fun... |
| fveqeq2 6840 | Equality deduction for fun... |
| nffv 6841 | Bound-variable hypothesis ... |
| nffvmpt1 6842 | Bound-variable hypothesis ... |
| nffvd 6843 | Deduction version of bound... |
| fvex 6844 | The value of a class exist... |
| fvexi 6845 | The value of a class exist... |
| fvexd 6846 | The value of a class exist... |
| fvif 6847 | Move a conditional outside... |
| iffv 6848 | Move a conditional outside... |
| fv3 6849 | Alternate definition of th... |
| fvres 6850 | The value of a restricted ... |
| fvresd 6851 | The value of a restricted ... |
| funssfv 6852 | The value of a member of t... |
| tz6.12c 6853 | Corollary of Theorem 6.12(... |
| tz6.12-1 6854 | Function value. Theorem 6... |
| tz6.12 6855 | Function value. Theorem 6... |
| tz6.12f 6856 | Function value, using boun... |
| tz6.12i 6857 | Corollary of Theorem 6.12(... |
| fvbr0 6858 | Two possibilities for the ... |
| fvrn0 6859 | A function value is a memb... |
| fvn0fvelrn 6860 | If the value of a function... |
| elfvunirn 6861 | A function value is a subs... |
| fvssunirn 6862 | The result of a function v... |
| ndmfv 6863 | The value of a class outsi... |
| ndmfvrcl 6864 | Reverse closure law for fu... |
| elfvdm 6865 | If a function value has a ... |
| elfvex 6866 | If a function value has a ... |
| elfvexd 6867 | If a function value has a ... |
| eliman0 6868 | A nonempty function value ... |
| nfvres 6869 | The value of a non-member ... |
| nfunsn 6870 | If the restriction of a cl... |
| fvfundmfvn0 6871 | If the "value of a class" ... |
| 0fv 6872 | Function value of the empt... |
| fv2prc 6873 | A function value of a func... |
| elfv2ex 6874 | If a function value of a f... |
| fveqres 6875 | Equal values imply equal v... |
| csbfv12 6876 | Move class substitution in... |
| csbfv2g 6877 | Move class substitution in... |
| csbfv 6878 | Substitution for a functio... |
| funbrfv 6879 | The second argument of a b... |
| funopfv 6880 | The second element in an o... |
| fnbrfvb 6881 | Equivalence of function va... |
| fnopfvb 6882 | Equivalence of function va... |
| fvelima2 6883 | Function value in an image... |
| funbrfvb 6884 | Equivalence of function va... |
| funopfvb 6885 | Equivalence of function va... |
| fnbrfvb2 6886 | Version of ~ fnbrfvb for f... |
| fdmeu 6887 | There is exactly one codom... |
| funbrfv2b 6888 | Function value in terms of... |
| dffn5 6889 | Representation of a functi... |
| fnrnfv 6890 | The range of a function ex... |
| fvelrnb 6891 | A member of a function's r... |
| foelcdmi 6892 | A member of a surjective f... |
| dfimafn 6893 | Alternate definition of th... |
| dfimafn2 6894 | Alternate definition of th... |
| funimass4 6895 | Membership relation for th... |
| fvelima 6896 | Function value in an image... |
| funimassd 6897 | Sufficient condition for t... |
| fvelimad 6898 | Function value in an image... |
| feqmptd 6899 | Deduction form of ~ dffn5 ... |
| feqresmpt 6900 | Express a restricted funct... |
| feqmptdf 6901 | Deduction form of ~ dffn5f... |
| dffn5f 6902 | Representation of a functi... |
| fvelimab 6903 | Function value in an image... |
| fvelimabd 6904 | Deduction form of ~ fvelim... |
| fimarab 6905 | Expressing the image of a ... |
| unima 6906 | Image of a union. (Contri... |
| fvi 6907 | The value of the identity ... |
| fviss 6908 | The value of the identity ... |
| fniinfv 6909 | The indexed intersection o... |
| fnsnfv 6910 | Singleton of function valu... |
| opabiotafun 6911 | Define a function whose va... |
| opabiotadm 6912 | Define a function whose va... |
| opabiota 6913 | Define a function whose va... |
| fnimapr 6914 | The image of a pair under ... |
| fnimatpd 6915 | The image of an unordered ... |
| ssimaex 6916 | The existence of a subimag... |
| ssimaexg 6917 | The existence of a subimag... |
| funfv 6918 | A simplified expression fo... |
| funfv2 6919 | The value of a function. ... |
| funfv2f 6920 | The value of a function. ... |
| fvun 6921 | Value of the union of two ... |
| fvun1 6922 | The value of a union when ... |
| fvun2 6923 | The value of a union when ... |
| fvun1d 6924 | The value of a union when ... |
| fvun2d 6925 | The value of a union when ... |
| dffv2 6926 | Alternate definition of fu... |
| dmfco 6927 | Domains of a function comp... |
| fvco2 6928 | Value of a function compos... |
| fvco 6929 | Value of a function compos... |
| fvcod 6930 | Value of a function compos... |
| fvco3 6931 | Value of a function compos... |
| fvco3d 6932 | Value of a function compos... |
| fvco4i 6933 | Conditions for a compositi... |
| fvopab3g 6934 | Value of a function given ... |
| fvopab3ig 6935 | Value of a function given ... |
| brfvopabrbr 6936 | The binary relation of a f... |
| fvmptg 6937 | Value of a function given ... |
| fvmpti 6938 | Value of a function given ... |
| fvmpt 6939 | Value of a function given ... |
| fvmpt2f 6940 | Value of a function given ... |
| funcnvmpt 6941 | Condition for a function i... |
| fvtresfn 6942 | Functionality of a tuple-r... |
| fvmpts 6943 | Value of a function given ... |
| fvmpt3 6944 | Value of a function given ... |
| fvmpt3i 6945 | Value of a function given ... |
| fvmptdf 6946 | Deduction version of ~ fvm... |
| fvmptd 6947 | Deduction version of ~ fvm... |
| fvmptd2 6948 | Deduction version of ~ fvm... |
| mptrcl 6949 | Reverse closure for a mapp... |
| fvmpt2i 6950 | Value of a function given ... |
| fvmpt2 6951 | Value of a function given ... |
| fvmptss 6952 | If all the values of the m... |
| fvmpt2d 6953 | Deduction version of ~ fvm... |
| fvmptex 6954 | Express a function ` F ` w... |
| fvmptd3f 6955 | Alternate deduction versio... |
| fvmptd2f 6956 | Alternate deduction versio... |
| fvmptdv 6957 | Alternate deduction versio... |
| fvmptdv2 6958 | Alternate deduction versio... |
| mpteqb 6959 | Bidirectional equality the... |
| fvmptt 6960 | Closed theorem form of ~ f... |
| fvmptf 6961 | Value of a function given ... |
| fvmptnf 6962 | The value of a function gi... |
| fvmptd3 6963 | Deduction version of ~ fvm... |
| fvmptd4 6964 | Deduction version of ~ fvm... |
| fvmptn 6965 | This somewhat non-intuitiv... |
| fvmptss2 6966 | A mapping always evaluates... |
| elfvmptrab1w 6967 | Implications for the value... |
| elfvmptrab1 6968 | Implications for the value... |
| elfvmptrab 6969 | Implications for the value... |
| fvopab4ndm 6970 | Value of a function given ... |
| fvmptndm 6971 | Value of a function given ... |
| fvmptrabfv 6972 | Value of a function mappin... |
| fvopab5 6973 | The value of a function th... |
| fvopab6 6974 | Value of a function given ... |
| eqfnfv 6975 | Equality of functions is d... |
| eqfnfv2 6976 | Equality of functions is d... |
| eqfnfv3 6977 | Derive equality of functio... |
| eqfnfvd 6978 | Deduction for equality of ... |
| eqfnfv2f 6979 | Equality of functions is d... |
| fsneq 6980 | Equality condition for two... |
| eqfunfv 6981 | Equality of functions is d... |
| eqfnun 6982 | Two functions on ` A u. B ... |
| fvreseq0 6983 | Equality of restricted fun... |
| fvreseq1 6984 | Equality of a function res... |
| fvreseq 6985 | Equality of restricted fun... |
| fnmptfvd 6986 | A function with a given do... |
| fndmdif 6987 | Two ways to express the lo... |
| fndmdifcom 6988 | The difference set between... |
| fndmdifeq0 6989 | The difference set of two ... |
| fndmin 6990 | Two ways to express the lo... |
| fneqeql 6991 | Two functions are equal if... |
| fneqeql2 6992 | Two functions are equal if... |
| fnreseql 6993 | Two functions are equal on... |
| chfnrn 6994 | The range of a choice func... |
| funfvop 6995 | Ordered pair with function... |
| funfvbrb 6996 | Two ways to say that ` A `... |
| fvimacnvi 6997 | A member of a preimage is ... |
| fvimacnv 6998 | The argument of a function... |
| funimass3 6999 | A kind of contraposition l... |
| funimass5 7000 | A subclass of a preimage i... |
| funconstss 7001 | Two ways of specifying tha... |
| fvimacnvALT 7002 | Alternate proof of ~ fvima... |
| elpreima 7003 | Membership in the preimage... |
| elpreimad 7004 | Membership in the preimage... |
| fniniseg 7005 | Membership in the preimage... |
| fncnvima2 7006 | Inverse images under funct... |
| fniniseg2 7007 | Inverse point images under... |
| unpreima 7008 | Preimage of a union. (Con... |
| inpreima 7009 | Preimage of an intersectio... |
| difpreima 7010 | Preimage of a difference. ... |
| respreima 7011 | The preimage of a restrict... |
| cnvimainrn 7012 | The preimage of the inters... |
| sspreima 7013 | The preimage of a subset i... |
| iinpreima 7014 | Preimage of an intersectio... |
| intpreima 7015 | Preimage of an intersectio... |
| fimacnvinrn 7016 | Taking the converse image ... |
| fimacnvinrn2 7017 | Taking the converse image ... |
| rescnvimafod 7018 | The restriction of a funct... |
| fvn0ssdmfun 7019 | If a class' function value... |
| fnopfv 7020 | Ordered pair with function... |
| fvelrn 7021 | A function's value belongs... |
| nelrnfvne 7022 | A function value cannot be... |
| fveqdmss 7023 | If the empty set is not co... |
| fveqressseq 7024 | If the empty set is not co... |
| fnfvelrn 7025 | A function's value belongs... |
| ffvelcdm 7026 | A function's value belongs... |
| fnfvelrnd 7027 | A function's value belongs... |
| ffvelcdmi 7028 | A function's value belongs... |
| ffvelcdmda 7029 | A function's value belongs... |
| ffvelcdmd 7030 | A function's value belongs... |
| feldmfvelcdm 7031 | A class is an element of t... |
| rexrn 7032 | Restricted existential qua... |
| ralrn 7033 | Restricted universal quant... |
| elrnrexdm 7034 | For any element in the ran... |
| elrnrexdmb 7035 | For any element in the ran... |
| eldmrexrn 7036 | For any element in the dom... |
| eldmrexrnb 7037 | For any element in the dom... |
| fvcofneq 7038 | The values of two function... |
| ralrnmptw 7039 | A restricted quantifier ov... |
| rexrnmptw 7040 | A restricted quantifier ov... |
| ralrnmpt 7041 | A restricted quantifier ov... |
| rexrnmpt 7042 | A restricted quantifier ov... |
| f0cli 7043 | Unconditional closure of a... |
| dff2 7044 | Alternate definition of a ... |
| dff3 7045 | Alternate definition of a ... |
| dff4 7046 | Alternate definition of a ... |
| dffo3 7047 | An onto mapping expressed ... |
| dffo4 7048 | Alternate definition of an... |
| dffo5 7049 | Alternate definition of an... |
| exfo 7050 | A relation equivalent to t... |
| dffo3f 7051 | An onto mapping expressed ... |
| foelrn 7052 | Property of a surjective f... |
| foelrnf 7053 | Property of a surjective f... |
| foco2 7054 | If a composition of two fu... |
| fmpt 7055 | Functionality of the mappi... |
| f1ompt 7056 | Express bijection for a ma... |
| fmpti 7057 | Functionality of the mappi... |
| fvmptelcdm 7058 | The value of a function at... |
| fmptd 7059 | Domain and codomain of the... |
| fmpttd 7060 | Version of ~ fmptd with in... |
| fmpt3d 7061 | Domain and codomain of the... |
| fmptdf 7062 | A version of ~ fmptd using... |
| fompt 7063 | Express being onto for a m... |
| ffnfv 7064 | A function maps to a class... |
| ffnfvf 7065 | A function maps to a class... |
| fnfvrnss 7066 | An upper bound for range d... |
| fcdmssb 7067 | A function is a function i... |
| rnmptss 7068 | The range of an operation ... |
| rnmptssd 7069 | The range of a function gi... |
| fmpt2d 7070 | Domain and codomain of the... |
| ffvresb 7071 | A necessary and sufficient... |
| fssrescdmd 7072 | Restriction of a function ... |
| f1oresrab 7073 | Build a bijection between ... |
| f1ossf1o 7074 | Restricting a bijection, w... |
| fmptco 7075 | Composition of two functio... |
| fmptcof 7076 | Version of ~ fmptco where ... |
| fmptcos 7077 | Composition of two functio... |
| cofmpt 7078 | Express composition of a m... |
| fcompt 7079 | Express composition of two... |
| fcoconst 7080 | Composition with a constan... |
| fsn 7081 | A function maps a singleto... |
| fsn2 7082 | A function that maps a sin... |
| fsng 7083 | A function maps a singleto... |
| fsn2g 7084 | A function that maps a sin... |
| xpsng 7085 | The Cartesian product of t... |
| xpprsng 7086 | The Cartesian product of a... |
| xpsn 7087 | The Cartesian product of t... |
| f1o2sn 7088 | A singleton consisting in ... |
| residpr 7089 | Restriction of the identit... |
| dfmpt 7090 | Alternate definition for t... |
| fnasrn 7091 | A function expressed as th... |
| idref 7092 | Two ways to state that a r... |
| funiun 7093 | A function is a union of s... |
| funopsn 7094 | If a function is an ordere... |
| funopsnOLD 7095 | Obsolete version of ~ funo... |
| funop 7096 | An ordered pair is a funct... |
| funopdmsn 7097 | The domain of a function w... |
| funsndifnop 7098 | A singleton of an ordered ... |
| funsneqopb 7099 | A singleton of an ordered ... |
| ressnop0 7100 | If ` A ` is not in ` C ` ,... |
| fpr 7101 | A function with a domain o... |
| fprg 7102 | A function with a domain o... |
| ftpg 7103 | A function with a domain o... |
| ftp 7104 | A function with a domain o... |
| fnressn 7105 | A function restricted to a... |
| funressn 7106 | A function restricted to a... |
| fressnfv 7107 | The value of a function re... |
| fvrnressn 7108 | If the value of a function... |
| fvressn 7109 | The value of a function re... |
| fvconst 7110 | The value of a constant fu... |
| fnsnr 7111 | If a class belongs to a fu... |
| fnsnbg 7112 | A function's domain is a s... |
| fnsnb 7113 | A function whose domain is... |
| fnsnbOLD 7114 | Obsolete version of ~ fnsn... |
| fmptsn 7115 | Express a singleton functi... |
| fmptsng 7116 | Express a singleton functi... |
| fmptsnd 7117 | Express a singleton functi... |
| fmptap 7118 | Append an additional value... |
| fmptapd 7119 | Append an additional value... |
| fmptpr 7120 | Express a pair function in... |
| fvresi 7121 | The value of a restricted ... |
| fninfp 7122 | Express the class of fixed... |
| fnelfp 7123 | Property of a fixed point ... |
| fndifnfp 7124 | Express the class of non-f... |
| fnelnfp 7125 | Property of a non-fixed po... |
| fnnfpeq0 7126 | A function is the identity... |
| fvunsn 7127 | Remove an ordered pair not... |
| fvsng 7128 | The value of a singleton o... |
| fvsn 7129 | The value of a singleton o... |
| fvsnun1 7130 | The value of a function wi... |
| fvsnun2 7131 | The value of a function wi... |
| fnsnsplit 7132 | Split a function into a si... |
| fsnunf 7133 | Adjoining a point to a fun... |
| fsnunf2 7134 | Adjoining a point to a pun... |
| fsnunfv 7135 | Recover the added point fr... |
| fsnunres 7136 | Recover the original funct... |
| funresdfunsn 7137 | Restricting a function to ... |
| fvpr1g 7138 | The value of a function wi... |
| fvpr2g 7139 | The value of a function wi... |
| fvpr1 7140 | The value of a function wi... |
| fvpr2 7141 | The value of a function wi... |
| fprb 7142 | A condition for functionho... |
| fvtp1 7143 | The first value of a funct... |
| fvtp2 7144 | The second value of a func... |
| fvtp3 7145 | The third value of a funct... |
| fvtp1g 7146 | The value of a function wi... |
| fvtp2g 7147 | The value of a function wi... |
| fvtp3g 7148 | The value of a function wi... |
| tpres 7149 | An unordered triple of ord... |
| fvconst2g 7150 | The value of a constant fu... |
| fconst2g 7151 | A constant function expres... |
| fvconst2 7152 | The value of a constant fu... |
| fconst2 7153 | A constant function expres... |
| fconst5 7154 | Two ways to express that a... |
| rnmptc 7155 | Range of a constant functi... |
| fnprb 7156 | A function whose domain ha... |
| fntpb 7157 | A function whose domain ha... |
| fnpr2g 7158 | A function whose domain ha... |
| fpr2g 7159 | A function that maps a pai... |
| fconstfv 7160 | A constant function expres... |
| fconst3 7161 | Two ways to express a cons... |
| fconst4 7162 | Two ways to express a cons... |
| resfunexg 7163 | The restriction of a funct... |
| resiexd 7164 | The restriction of the ide... |
| fnex 7165 | If the domain of a functio... |
| fnexd 7166 | If the domain of a functio... |
| funex 7167 | If the domain of a functio... |
| opabex 7168 | Existence of a function ex... |
| mptexg 7169 | If the domain of a functio... |
| mptexgf 7170 | If the domain of a functio... |
| mptex 7171 | If the domain of a functio... |
| mptexd 7172 | If the domain of a functio... |
| mptrabex 7173 | If the domain of a functio... |
| fex 7174 | If the domain of a mapping... |
| fexd 7175 | If the domain of a mapping... |
| mptfvmpt 7176 | A function in maps-to nota... |
| eufnfv 7177 | A function is uniquely det... |
| funfvima 7178 | A function's value in a pr... |
| funfvima2 7179 | A function's value in an i... |
| funfvima2d 7180 | A function's value in a pr... |
| fnfvima 7181 | The function value of an o... |
| fnfvimad 7182 | A function's value belongs... |
| resfvresima 7183 | The value of the function ... |
| funfvima3 7184 | A class including a functi... |
| ralima 7185 | Universal quantification u... |
| rexima 7186 | Existential quantification... |
| reximaOLD 7187 | Obsolete version of ~ rexi... |
| ralimaOLD 7188 | Obsolete version of ~ rali... |
| fvclss 7189 | Upper bound for the class ... |
| elabrex 7190 | Elementhood in an image se... |
| elabrexg 7191 | Elementhood in an image se... |
| abrexco 7192 | Composition of two image m... |
| imaiun 7193 | The image of an indexed un... |
| imauni 7194 | The image of a union is th... |
| fniunfv 7195 | The indexed union of a fun... |
| funiunfv 7196 | The indexed union of a fun... |
| funiunfvf 7197 | The indexed union of a fun... |
| eluniima 7198 | Membership in the union of... |
| elunirn 7199 | Membership in the union of... |
| elunirnALT 7200 | Alternate proof of ~ eluni... |
| fnunirn 7201 | Membership in a union of s... |
| dff13 7202 | A one-to-one function in t... |
| dff13f 7203 | A one-to-one function in t... |
| f1veqaeq 7204 | If the values of a one-to-... |
| f1cofveqaeq 7205 | If the values of a composi... |
| f1cofveqaeqALT 7206 | Alternate proof of ~ f1cof... |
| dff14i 7207 | A one-to-one function maps... |
| 2f1fvneq 7208 | If two one-to-one function... |
| f1mpt 7209 | Express injection for a ma... |
| f1fveq 7210 | Equality of function value... |
| f1elima 7211 | Membership in the image of... |
| f1imass 7212 | Taking images under a one-... |
| f1imaeq 7213 | Taking images under a one-... |
| f1imapss 7214 | Taking images under a one-... |
| fpropnf1 7215 | A function, given by an un... |
| f1dom3fv3dif 7216 | The function values for a ... |
| f1dom3el3dif 7217 | The codomain of a 1-1 func... |
| dff14a 7218 | A one-to-one function in t... |
| dff14b 7219 | A one-to-one function in t... |
| f1ounsn 7220 | Extension of a bijection b... |
| f12dfv 7221 | A one-to-one function with... |
| f13dfv 7222 | A one-to-one function with... |
| dff1o6 7223 | A one-to-one onto function... |
| f1ocnvfv1 7224 | The converse value of the ... |
| f1ocnvfv2 7225 | The value of the converse ... |
| f1ocnvfv 7226 | Relationship between the v... |
| f1ocnvfvb 7227 | Relationship between the v... |
| nvof1o 7228 | An involution is a bijecti... |
| nvocnv 7229 | The converse of an involut... |
| f1cdmsn 7230 | If a one-to-one function w... |
| fsnex 7231 | Relate a function with a s... |
| f1prex 7232 | Relate a one-to-one functi... |
| f1ocnvdm 7233 | The value of the converse ... |
| f1ocnvfvrneq 7234 | If the values of a one-to-... |
| fcof1 7235 | An application is injectiv... |
| fcofo 7236 | An application is surjecti... |
| cbvfo 7237 | Change bound variable betw... |
| cbvexfo 7238 | Change bound variable betw... |
| cocan1 7239 | An injection is left-cance... |
| cocan2 7240 | A surjection is right-canc... |
| fcof1oinvd 7241 | Show that a function is th... |
| fcof1od 7242 | A function is bijective if... |
| 2fcoidinvd 7243 | Show that a function is th... |
| fcof1o 7244 | Show that two functions ar... |
| 2fvcoidd 7245 | Show that the composition ... |
| 2fvidf1od 7246 | A function is bijective if... |
| 2fvidinvd 7247 | Show that two functions ar... |
| foeqcnvco 7248 | Condition for function equ... |
| f1eqcocnv 7249 | Condition for function equ... |
| fveqf1o 7250 | Given a bijection ` F ` , ... |
| f1ocoima 7251 | The composition of two bij... |
| nf1const 7252 | A constant function from a... |
| nf1oconst 7253 | A constant function from a... |
| f1ofvswap 7254 | Swapping two values in a b... |
| fvf1pr 7255 | Values of a one-to-one fun... |
| fliftrel 7256 | ` F ` , a function lift, i... |
| fliftel 7257 | Elementhood in the relatio... |
| fliftel1 7258 | Elementhood in the relatio... |
| fliftcnv 7259 | Converse of the relation `... |
| fliftfun 7260 | The function ` F ` is the ... |
| fliftfund 7261 | The function ` F ` is the ... |
| fliftfuns 7262 | The function ` F ` is the ... |
| fliftf 7263 | The domain and range of th... |
| fliftval 7264 | The value of the function ... |
| isoeq1 7265 | Equality theorem for isomo... |
| isoeq2 7266 | Equality theorem for isomo... |
| isoeq3 7267 | Equality theorem for isomo... |
| isoeq4 7268 | Equality theorem for isomo... |
| isoeq5 7269 | Equality theorem for isomo... |
| nfiso 7270 | Bound-variable hypothesis ... |
| isof1o 7271 | An isomorphism is a one-to... |
| isof1oidb 7272 | A function is a bijection ... |
| isof1oopb 7273 | A function is a bijection ... |
| isorel 7274 | An isomorphism connects bi... |
| soisores 7275 | Express the condition of i... |
| soisoi 7276 | Infer isomorphism from one... |
| isoid 7277 | Identity law for isomorphi... |
| isocnv 7278 | Converse law for isomorphi... |
| isocnv2 7279 | Converse law for isomorphi... |
| isocnv3 7280 | Complementation law for is... |
| isores2 7281 | An isomorphism from one we... |
| isores1 7282 | An isomorphism from one we... |
| isores3 7283 | Induced isomorphism on a s... |
| isotr 7284 | Composition (transitive) l... |
| isomin 7285 | Isomorphisms preserve mini... |
| isoini 7286 | Isomorphisms preserve init... |
| isoini2 7287 | Isomorphisms are isomorphi... |
| isofrlem 7288 | Lemma for ~ isofr . (Cont... |
| isoselem 7289 | Lemma for ~ isose . (Cont... |
| isofr 7290 | An isomorphism preserves w... |
| isose 7291 | An isomorphism preserves s... |
| isofr2 7292 | A weak form of ~ isofr tha... |
| isopolem 7293 | Lemma for ~ isopo . (Cont... |
| isopo 7294 | An isomorphism preserves t... |
| isosolem 7295 | Lemma for ~ isoso . (Cont... |
| isoso 7296 | An isomorphism preserves t... |
| isowe 7297 | An isomorphism preserves t... |
| isowe2 7298 | A weak form of ~ isowe tha... |
| f1oiso 7299 | Any one-to-one onto functi... |
| f1oiso2 7300 | Any one-to-one onto functi... |
| f1owe 7301 | Well-ordering of isomorphi... |
| weniso 7302 | A set-like well-ordering h... |
| weisoeq 7303 | Thus, there is at most one... |
| weisoeq2 7304 | Thus, there is at most one... |
| knatar 7305 | The Knaster-Tarski theorem... |
| fvresval 7306 | The value of a restricted ... |
| funeldmb 7307 | If ` (/) ` is not part of ... |
| eqfunresadj 7308 | Law for adjoining an eleme... |
| eqfunressuc 7309 | Law for equality of restri... |
| fnssintima 7310 | Condition for subset of an... |
| imaeqsexvOLD 7311 | Obsolete version of ~ rexi... |
| imaeqsalvOLD 7312 | Obsolete version of ~ rali... |
| fnimasnd 7313 | The image of a function by... |
| canth 7314 | No set ` A ` is equinumero... |
| ncanth 7315 | Cantor's theorem fails for... |
| riotaeqdv 7318 | Formula-building deduction... |
| riotabidv 7319 | Formula-building deduction... |
| riotaeqbidv 7320 | Equality deduction for res... |
| riotaex 7321 | Restricted iota is a set. ... |
| riotav 7322 | An iota restricted to the ... |
| riotauni 7323 | Restricted iota in terms o... |
| nfriota1 7324 | The abstraction variable i... |
| nfriotadw 7325 | Deduction version of ~ nfr... |
| cbvriotaw 7326 | Change bound variable in a... |
| cbvriotavw 7327 | Change bound variable in a... |
| nfriotad 7328 | Deduction version of ~ nfr... |
| nfriota 7329 | A variable not free in a w... |
| cbvriota 7330 | Change bound variable in a... |
| cbvriotav 7331 | Change bound variable in a... |
| csbriota 7332 | Interchange class substitu... |
| riotacl2 7333 | Membership law for "the un... |
| riotacl 7334 | Closure of restricted iota... |
| riotasbc 7335 | Substitution law for descr... |
| riotabidva 7336 | Equivalent wff's yield equ... |
| riotabiia 7337 | Equivalent wff's yield equ... |
| riota1 7338 | Property of restricted iot... |
| riota1a 7339 | Property of iota. (Contri... |
| riota2df 7340 | A deduction version of ~ r... |
| riota2f 7341 | This theorem shows a condi... |
| riota2 7342 | This theorem shows a condi... |
| riotaeqimp 7343 | If two restricted iota des... |
| riotaprop 7344 | Properties of a restricted... |
| riota5f 7345 | A method for computing res... |
| riota5 7346 | A method for computing res... |
| riotass2 7347 | Restriction of a unique el... |
| riotass 7348 | Restriction of a unique el... |
| moriotass 7349 | Restriction of a unique el... |
| snriota 7350 | A restricted class abstrac... |
| riotaxfrd 7351 | Change the variable ` x ` ... |
| eusvobj2 7352 | Specify the same property ... |
| eusvobj1 7353 | Specify the same object in... |
| f1ofveu 7354 | There is one domain elemen... |
| f1ocnvfv3 7355 | Value of the converse of a... |
| riotaund 7356 | Restricted iota equals the... |
| riotassuni 7357 | The restricted iota class ... |
| riotaclb 7358 | Bidirectional closure of r... |
| riotarab 7359 | Restricted iota of a restr... |
| oveq 7366 | Equality theorem for opera... |
| oveq1 7367 | Equality theorem for opera... |
| oveq2 7368 | Equality theorem for opera... |
| oveq12 7369 | Equality theorem for opera... |
| oveq1i 7370 | Equality inference for ope... |
| oveq2i 7371 | Equality inference for ope... |
| oveq12i 7372 | Equality inference for ope... |
| oveqi 7373 | Equality inference for ope... |
| oveq123i 7374 | Equality inference for ope... |
| oveq1d 7375 | Equality deduction for ope... |
| oveq2d 7376 | Equality deduction for ope... |
| oveqd 7377 | Equality deduction for ope... |
| oveq12d 7378 | Equality deduction for ope... |
| oveqan12d 7379 | Equality deduction for ope... |
| oveqan12rd 7380 | Equality deduction for ope... |
| oveq123d 7381 | Equality deduction for ope... |
| fvoveq1d 7382 | Equality deduction for nes... |
| fvoveq1 7383 | Equality theorem for neste... |
| ovanraleqv 7384 | Equality theorem for a con... |
| imbrov2fvoveq 7385 | Equality theorem for neste... |
| ovrspc2v 7386 | If an operation value is a... |
| oveqrspc2v 7387 | Restricted specialization ... |
| oveqdr 7388 | Equality of two operations... |
| nfovd 7389 | Deduction version of bound... |
| nfov 7390 | Bound-variable hypothesis ... |
| oprabidw 7391 | The law of concretion. Sp... |
| oprabid 7392 | The law of concretion. Sp... |
| ovex 7393 | The result of an operation... |
| ovexi 7394 | The result of an operation... |
| ovexd 7395 | The result of an operation... |
| ovssunirn 7396 | The result of an operation... |
| 0ov 7397 | Operation value of the emp... |
| ovprc 7398 | The value of an operation ... |
| ovprc1 7399 | The value of an operation ... |
| ovprc2 7400 | The value of an operation ... |
| ovrcl 7401 | Reverse closure for an ope... |
| elfvov1 7402 | Utility theorem: reverse c... |
| elfvov2 7403 | Utility theorem: reverse c... |
| csbov123 7404 | Move class substitution in... |
| csbov 7405 | Move class substitution in... |
| csbov12g 7406 | Move class substitution in... |
| csbov1g 7407 | Move class substitution in... |
| csbov2g 7408 | Move class substitution in... |
| rspceov 7409 | A frequently used special ... |
| elovimad 7410 | Elementhood of the image s... |
| fnbrovb 7411 | Value of a binary operatio... |
| fnotovb 7412 | Equivalence of operation v... |
| opabbrex 7413 | A collection of ordered pa... |
| opabresex2 7414 | Restrictions of a collecti... |
| fvmptopab 7415 | The function value of a ma... |
| f1opr 7416 | Condition for an operation... |
| brfvopab 7417 | The classes involved in a ... |
| dfoprab2 7418 | Class abstraction for oper... |
| reloprab 7419 | An operation class abstrac... |
| oprabv 7420 | If a pair and a class are ... |
| nfoprab1 7421 | The abstraction variables ... |
| nfoprab2 7422 | The abstraction variables ... |
| nfoprab3 7423 | The abstraction variables ... |
| nfoprab 7424 | Bound-variable hypothesis ... |
| oprabbid 7425 | Equivalent wff's yield equ... |
| oprabbidv 7426 | Equivalent wff's yield equ... |
| oprabbii 7427 | Equivalent wff's yield equ... |
| ssoprab2 7428 | Equivalence of ordered pai... |
| ssoprab2b 7429 | Equivalence of ordered pai... |
| eqoprab2bw 7430 | Equivalence of ordered pai... |
| eqoprab2b 7431 | Equivalence of ordered pai... |
| mpoeq123 7432 | An equality theorem for th... |
| mpoeq12 7433 | An equality theorem for th... |
| mpoeq123dva 7434 | An equality deduction for ... |
| mpoeq123dv 7435 | An equality deduction for ... |
| mpoeq123i 7436 | An equality inference for ... |
| mpoeq3dva 7437 | Slightly more general equa... |
| mpoeq3ia 7438 | An equality inference for ... |
| mpoeq3dv 7439 | An equality deduction for ... |
| nfmpo1 7440 | Bound-variable hypothesis ... |
| nfmpo2 7441 | Bound-variable hypothesis ... |
| nfmpo 7442 | Bound-variable hypothesis ... |
| 0mpo0 7443 | A mapping operation with e... |
| mpo0v 7444 | A mapping operation with e... |
| mpo0 7445 | A mapping operation with e... |
| oprab4 7446 | Two ways to state the doma... |
| cbvoprab1 7447 | Rule used to change first ... |
| cbvoprab2 7448 | Change the second bound va... |
| cbvoprab12 7449 | Rule used to change first ... |
| cbvoprab12v 7450 | Rule used to change first ... |
| cbvoprab3 7451 | Rule used to change the th... |
| cbvoprab3v 7452 | Rule used to change the th... |
| cbvmpox 7453 | Rule to change the bound v... |
| cbvmpo 7454 | Rule to change the bound v... |
| cbvmpov 7455 | Rule to change the bound v... |
| elimdelov 7456 | Eliminate a hypothesis whi... |
| brif1 7457 | Move a relation inside and... |
| ovif 7458 | Move a conditional outside... |
| ovif2 7459 | Move a conditional outside... |
| ovif12 7460 | Move a conditional outside... |
| ifov 7461 | Move a conditional outside... |
| ifmpt2v 7462 | Move a conditional inside ... |
| dmoprab 7463 | The domain of an operation... |
| dmoprabss 7464 | The domain of an operation... |
| rnoprab 7465 | The range of an operation ... |
| rnoprab2 7466 | The range of a restricted ... |
| reldmoprab 7467 | The domain of an operation... |
| oprabss 7468 | Structure of an operation ... |
| eloprabga 7469 | The law of concretion for ... |
| eloprabg 7470 | The law of concretion for ... |
| ssoprab2i 7471 | Inference of operation cla... |
| mpov 7472 | Operation with universal d... |
| mpomptx 7473 | Express a two-argument fun... |
| mpompt 7474 | Express a two-argument fun... |
| mpodifsnif 7475 | A mapping with two argumen... |
| mposnif 7476 | A mapping with two argumen... |
| fconstmpo 7477 | Representation of a consta... |
| resoprab 7478 | Restriction of an operatio... |
| resoprab2 7479 | Restriction of an operator... |
| resmpo 7480 | Restriction of the mapping... |
| funoprabg 7481 | "At most one" is a suffici... |
| funoprab 7482 | "At most one" is a suffici... |
| fnoprabg 7483 | Functionality and domain o... |
| mpofun 7484 | The maps-to notation for a... |
| fnoprab 7485 | Functionality and domain o... |
| ffnov 7486 | An operation maps to a cla... |
| fovcld 7487 | Closure law for an operati... |
| fovcl 7488 | Closure law for an operati... |
| eqfnov 7489 | Equality of two operations... |
| eqfnov2 7490 | Two operators with the sam... |
| fnov 7491 | Representation of a functi... |
| mpo2eqb 7492 | Bidirectional equality the... |
| rnmpo 7493 | The range of an operation ... |
| reldmmpo 7494 | The domain of an operation... |
| elrnmpog 7495 | Membership in the range of... |
| elrnmpo 7496 | Membership in the range of... |
| elimampo 7497 | Membership in the image of... |
| elrnmpores 7498 | Membership in the range of... |
| ralrnmpo 7499 | A restricted quantifier ov... |
| rexrnmpo 7500 | A restricted quantifier ov... |
| ovid 7501 | The value of an operation ... |
| ovidig 7502 | The value of an operation ... |
| ovidi 7503 | The value of an operation ... |
| ov 7504 | The value of an operation ... |
| ovigg 7505 | The value of an operation ... |
| ovig 7506 | The value of an operation ... |
| ovmpt4g 7507 | Value of a function given ... |
| ovmpos 7508 | Value of a function given ... |
| ov2gf 7509 | The value of an operation ... |
| ovmpodxf 7510 | Value of an operation give... |
| ovmpodx 7511 | Value of an operation give... |
| ovmpod 7512 | Value of an operation give... |
| ovmpox 7513 | The value of an operation ... |
| ovmpoga 7514 | Value of an operation give... |
| ovmpoa 7515 | Value of an operation give... |
| ovmpodf 7516 | Alternate deduction versio... |
| ovmpodv 7517 | Alternate deduction versio... |
| ovmpodv2 7518 | Alternate deduction versio... |
| ovmpog 7519 | Value of an operation give... |
| ovmpo 7520 | Value of an operation give... |
| ovmpot 7521 | The value of an operation ... |
| fvmpopr2d 7522 | Value of an operation give... |
| ov3 7523 | The value of an operation ... |
| ov6g 7524 | The value of an operation ... |
| ovg 7525 | The value of an operation ... |
| ovres 7526 | The value of a restricted ... |
| ovresd 7527 | Lemma for converting metri... |
| oprres 7528 | The restriction of an oper... |
| oprssov 7529 | The value of a member of t... |
| fovcdm 7530 | An operation's value belon... |
| fovcdmda 7531 | An operation's value belon... |
| fovcdmd 7532 | An operation's value belon... |
| fnrnov 7533 | The range of an operation ... |
| foov 7534 | An onto mapping of an oper... |
| fnovrn 7535 | An operation's value belon... |
| ovelrn 7536 | A member of an operation's... |
| funimassov 7537 | Membership relation for th... |
| ovelimab 7538 | Operation value in an imag... |
| ovima0 7539 | An operation value is a me... |
| ovconst2 7540 | The value of a constant op... |
| oprssdm 7541 | Domain of closure of an op... |
| nssdmovg 7542 | The value of an operation ... |
| ndmovg 7543 | The value of an operation ... |
| ndmov 7544 | The value of an operation ... |
| ndmovcl 7545 | The closure of an operatio... |
| ndmovrcl 7546 | Reverse closure law, when ... |
| ndmovcom 7547 | Any operation is commutati... |
| ndmovass 7548 | Any operation is associati... |
| ndmovdistr 7549 | Any operation is distribut... |
| ndmovord 7550 | Elimination of redundant a... |
| ndmovordi 7551 | Elimination of redundant a... |
| caovclg 7552 | Convert an operation closu... |
| caovcld 7553 | Convert an operation closu... |
| caovcl 7554 | Convert an operation closu... |
| caovcomg 7555 | Convert an operation commu... |
| caovcomd 7556 | Convert an operation commu... |
| caovcom 7557 | Convert an operation commu... |
| caovassg 7558 | Convert an operation assoc... |
| caovassd 7559 | Convert an operation assoc... |
| caovass 7560 | Convert an operation assoc... |
| caovcang 7561 | Convert an operation cance... |
| caovcand 7562 | Convert an operation cance... |
| caovcanrd 7563 | Commute the arguments of a... |
| caovcan 7564 | Convert an operation cance... |
| caovordig 7565 | Convert an operation order... |
| caovordid 7566 | Convert an operation order... |
| caovordg 7567 | Convert an operation order... |
| caovordd 7568 | Convert an operation order... |
| caovord2d 7569 | Operation ordering law wit... |
| caovord3d 7570 | Ordering law. (Contribute... |
| caovord 7571 | Convert an operation order... |
| caovord2 7572 | Operation ordering law wit... |
| caovord3 7573 | Ordering law. (Contribute... |
| caovdig 7574 | Convert an operation distr... |
| caovdid 7575 | Convert an operation distr... |
| caovdir2d 7576 | Convert an operation distr... |
| caovdirg 7577 | Convert an operation rever... |
| caovdird 7578 | Convert an operation distr... |
| caovdi 7579 | Convert an operation distr... |
| caov32d 7580 | Rearrange arguments in a c... |
| caov12d 7581 | Rearrange arguments in a c... |
| caov31d 7582 | Rearrange arguments in a c... |
| caov13d 7583 | Rearrange arguments in a c... |
| caov4d 7584 | Rearrange arguments in a c... |
| caov411d 7585 | Rearrange arguments in a c... |
| caov42d 7586 | Rearrange arguments in a c... |
| caov32 7587 | Rearrange arguments in a c... |
| caov12 7588 | Rearrange arguments in a c... |
| caov31 7589 | Rearrange arguments in a c... |
| caov13 7590 | Rearrange arguments in a c... |
| caov4 7591 | Rearrange arguments in a c... |
| caov411 7592 | Rearrange arguments in a c... |
| caov42 7593 | Rearrange arguments in a c... |
| caovdir 7594 | Reverse distributive law. ... |
| caovdilem 7595 | Lemma used by real number ... |
| caovlem2 7596 | Lemma used in real number ... |
| caovmo 7597 | Uniqueness of inverse elem... |
| imaeqexov 7598 | Substitute an operation va... |
| imaeqalov 7599 | Substitute an operation va... |
| mpondm0 7600 | The value of an operation ... |
| elmpocl 7601 | If a two-parameter class i... |
| elmpocl1 7602 | If a two-parameter class i... |
| elmpocl2 7603 | If a two-parameter class i... |
| elovmpod 7604 | Utility lemma for two-para... |
| elovmpo 7605 | Utility lemma for two-para... |
| elovmporab 7606 | Implications for the value... |
| elovmporab1w 7607 | Implications for the value... |
| elovmporab1 7608 | Implications for the value... |
| 2mpo0 7609 | If the operation value of ... |
| relmptopab 7610 | Any function to sets of or... |
| f1ocnvd 7611 | Describe an implicit one-t... |
| f1od 7612 | Describe an implicit one-t... |
| f1ocnv2d 7613 | Describe an implicit one-t... |
| f1o2d 7614 | Describe an implicit one-t... |
| f1opw2 7615 | A one-to-one mapping induc... |
| f1opw 7616 | A one-to-one mapping induc... |
| elovmpt3imp 7617 | If the value of a function... |
| ovmpt3rab1 7618 | The value of an operation ... |
| ovmpt3rabdm 7619 | If the value of a function... |
| elovmpt3rab1 7620 | Implications for the value... |
| elovmpt3rab 7621 | Implications for the value... |
| ofeqd 7626 | Equality theorem for funct... |
| ofeq 7627 | Equality theorem for funct... |
| ofreq 7628 | Equality theorem for funct... |
| ofexg 7629 | A function operation restr... |
| nfof 7630 | Hypothesis builder for fun... |
| nfofr 7631 | Hypothesis builder for fun... |
| ofrfvalg 7632 | Value of a relation applie... |
| offval 7633 | Value of an operation appl... |
| ofrfval 7634 | Value of a relation applie... |
| ofval 7635 | Evaluate a function operat... |
| ofrval 7636 | Exhibit a function relatio... |
| offn 7637 | The function operation pro... |
| offun 7638 | The function operation pro... |
| offval2f 7639 | The function operation exp... |
| ofmresval 7640 | Value of a restriction of ... |
| fnfvof 7641 | Function value of a pointw... |
| off 7642 | The function operation pro... |
| ofres 7643 | Restrict the operands of a... |
| offval2 7644 | The function operation exp... |
| ofrfval2 7645 | The function relation acti... |
| offvalfv 7646 | The function operation exp... |
| ofmpteq 7647 | Value of a pointwise opera... |
| coof 7648 | The composition of a _homo... |
| ofco 7649 | The composition of a funct... |
| offveq 7650 | Convert an identity of the... |
| offveqb 7651 | Equivalent expressions for... |
| ofc1 7652 | Left operation by a consta... |
| ofc2 7653 | Right operation by a const... |
| ofc12 7654 | Function operation on two ... |
| caofref 7655 | Transfer a reflexive law t... |
| caofinvl 7656 | Transfer a left inverse la... |
| caofid0l 7657 | Transfer a left identity l... |
| caofid0r 7658 | Transfer a right identity ... |
| caofid1 7659 | Transfer a right absorptio... |
| caofid2 7660 | Transfer a right absorptio... |
| caofcom 7661 | Transfer a commutative law... |
| caofidlcan 7662 | Transfer a cancellation/id... |
| caofrss 7663 | Transfer a relation subset... |
| caofass 7664 | Transfer an associative la... |
| caoftrn 7665 | Transfer a transitivity la... |
| caofdi 7666 | Transfer a distributive la... |
| caofdir 7667 | Transfer a reverse distrib... |
| caonncan 7668 | Transfer ~ nncan -shaped l... |
| relrpss 7671 | The proper subset relation... |
| brrpssg 7672 | The proper subset relation... |
| brrpss 7673 | The proper subset relation... |
| porpss 7674 | Every class is partially o... |
| sorpss 7675 | Express strict ordering un... |
| sorpssi 7676 | Property of a chain of set... |
| sorpssun 7677 | A chain of sets is closed ... |
| sorpssin 7678 | A chain of sets is closed ... |
| sorpssuni 7679 | In a chain of sets, a maxi... |
| sorpssint 7680 | In a chain of sets, a mini... |
| sorpsscmpl 7681 | The componentwise compleme... |
| zfun 7683 | Axiom of Union expressed w... |
| axun2 7684 | A variant of the Axiom of ... |
| uniex2 7685 | The Axiom of Union using t... |
| vuniex 7686 | The union of a setvar is a... |
| uniexg 7687 | The ZF Axiom of Union in c... |
| uniex 7688 | The Axiom of Union in clas... |
| uniexd 7689 | Deduction version of the Z... |
| unexg 7690 | The union of two sets is a... |
| unex 7691 | The union of two sets is a... |
| unexOLD 7692 | Obsolete version of ~ unex... |
| tpex 7693 | An unordered triple of cla... |
| unexb 7694 | Existence of union is equi... |
| unexbOLD 7695 | Obsolete version of ~ unex... |
| unexgOLD 7696 | Obsolete version of ~ unex... |
| xpexg 7697 | The Cartesian product of t... |
| xpexd 7698 | The Cartesian product of t... |
| 3xpexg 7699 | The Cartesian product of t... |
| xpex 7700 | The Cartesian product of t... |
| unexd 7701 | The union of two sets is a... |
| sqxpexg 7702 | The Cartesian square of a ... |
| abnexg 7703 | Sufficient condition for a... |
| abnex 7704 | Sufficient condition for a... |
| snnex 7705 | The class of all singleton... |
| pwnex 7706 | The class of all power set... |
| difex2 7707 | If the subtrahend of a cla... |
| difsnexi 7708 | If the difference of a cla... |
| uniuni 7709 | Expression for double unio... |
| uniexr 7710 | Converse of the Axiom of U... |
| uniexb 7711 | The Axiom of Union and its... |
| pwexr 7712 | Converse of the Axiom of P... |
| pwexb 7713 | The Axiom of Power Sets an... |
| elpwpwel 7714 | A class belongs to a doubl... |
| eldifpw 7715 | Membership in a power clas... |
| elpwun 7716 | Membership in the power cl... |
| pwuncl 7717 | Power classes are closed u... |
| iunpw 7718 | An indexed union of a powe... |
| fr3nr 7719 | A well-founded relation ha... |
| epne3 7720 | A well-founded class conta... |
| dfwe2 7721 | Alternate definition of we... |
| epweon 7722 | The membership relation we... |
| epweonALT 7723 | Alternate proof of ~ epweo... |
| ordon 7724 | The class of all ordinal n... |
| onprc 7725 | No set contains all ordina... |
| ssorduni 7726 | The union of a class of or... |
| ssonuni 7727 | The union of a set of ordi... |
| ssonunii 7728 | The union of a set of ordi... |
| ordeleqon 7729 | A way to express the ordin... |
| ordsson 7730 | Any ordinal class is a sub... |
| dford5 7731 | A class is ordinal iff it ... |
| onss 7732 | An ordinal number is a sub... |
| predon 7733 | The predecessor of an ordi... |
| ssonprc 7734 | Two ways of saying a class... |
| onuni 7735 | The union of an ordinal nu... |
| orduni 7736 | The union of an ordinal cl... |
| onint 7737 | The intersection (infimum)... |
| onint0 7738 | The intersection of a clas... |
| onssmin 7739 | A nonempty class of ordina... |
| onminesb 7740 | If a property is true for ... |
| onminsb 7741 | If a property is true for ... |
| oninton 7742 | The intersection of a none... |
| onintrab 7743 | The intersection of a clas... |
| onintrab2 7744 | An existence condition equ... |
| onnmin 7745 | No member of a set of ordi... |
| onnminsb 7746 | An ordinal number smaller ... |
| oneqmin 7747 | A way to show that an ordi... |
| uniordint 7748 | The union of a set of ordi... |
| onminex 7749 | If a wff is true for an or... |
| sucon 7750 | The class of all ordinal n... |
| sucexb 7751 | A successor exists iff its... |
| sucexg 7752 | The successor of a set is ... |
| sucex 7753 | The successor of a set is ... |
| onmindif2 7754 | The minimum of a class of ... |
| ordsuci 7755 | The successor of an ordina... |
| sucexeloni 7756 | If the successor of an ord... |
| onsuc 7757 | The successor of an ordina... |
| ordsuc 7758 | A class is ordinal if and ... |
| ordpwsuc 7759 | The collection of ordinals... |
| onpwsuc 7760 | The collection of ordinal ... |
| onsucb 7761 | A class is an ordinal numb... |
| ordsucss 7762 | The successor of an elemen... |
| onpsssuc 7763 | An ordinal number is a pro... |
| ordelsuc 7764 | A set belongs to an ordina... |
| onsucmin 7765 | The successor of an ordina... |
| ordsucelsuc 7766 | Membership is inherited by... |
| ordsucsssuc 7767 | The subclass relationship ... |
| ordsucuniel 7768 | Given an element ` A ` of ... |
| ordsucun 7769 | The successor of the maxim... |
| ordunpr 7770 | The maximum of two ordinal... |
| ordunel 7771 | The maximum of two ordinal... |
| onsucuni 7772 | A class of ordinal numbers... |
| ordsucuni 7773 | An ordinal class is a subc... |
| orduniorsuc 7774 | An ordinal class is either... |
| unon 7775 | The class of all ordinal n... |
| ordunisuc 7776 | An ordinal class is equal ... |
| orduniss2 7777 | The union of the ordinal s... |
| onsucuni2 7778 | A successor ordinal is the... |
| 0elsuc 7779 | The successor of an ordina... |
| limon 7780 | The class of ordinal numbe... |
| onuniorsuc 7781 | An ordinal number is eithe... |
| onssi 7782 | An ordinal number is a sub... |
| onsuci 7783 | The successor of an ordina... |
| onuninsuci 7784 | An ordinal is equal to its... |
| onsucssi 7785 | A set belongs to an ordina... |
| nlimsucg 7786 | A successor is not a limit... |
| orduninsuc 7787 | An ordinal class is equal ... |
| ordunisuc2 7788 | An ordinal equal to its un... |
| ordzsl 7789 | An ordinal is zero, a succ... |
| onzsl 7790 | An ordinal number is zero,... |
| dflim3 7791 | An alternate definition of... |
| dflim4 7792 | An alternate definition of... |
| limsuc 7793 | The successor of a member ... |
| limsssuc 7794 | A class includes a limit o... |
| nlimon 7795 | Two ways to express the cl... |
| limuni3 7796 | The union of a nonempty cl... |
| tfi 7797 | The Principle of Transfini... |
| tfisg 7798 | A closed form of ~ tfis . ... |
| tfis 7799 | Transfinite Induction Sche... |
| tfis2f 7800 | Transfinite Induction Sche... |
| tfis2 7801 | Transfinite Induction Sche... |
| tfis3 7802 | Transfinite Induction Sche... |
| tfisi 7803 | A transfinite induction sc... |
| tfinds 7804 | Principle of Transfinite I... |
| tfindsg 7805 | Transfinite Induction (inf... |
| tfindsg2 7806 | Transfinite Induction (inf... |
| tfindes 7807 | Transfinite Induction with... |
| tfinds2 7808 | Transfinite Induction (inf... |
| tfinds3 7809 | Principle of Transfinite I... |
| dfom2 7812 | An alternate definition of... |
| elom 7813 | Membership in omega. The ... |
| omsson 7814 | Omega is a subset of ` On ... |
| limomss 7815 | The class of natural numbe... |
| nnon 7816 | A natural number is an ord... |
| nnoni 7817 | A natural number is an ord... |
| nnord 7818 | A natural number is ordina... |
| trom 7819 | The class of finite ordina... |
| ordom 7820 | The class of finite ordina... |
| elnn 7821 | A member of a natural numb... |
| omon 7822 | The class of natural numbe... |
| omelon2 7823 | Omega is an ordinal number... |
| nnlim 7824 | A natural number is not a ... |
| omssnlim 7825 | The class of natural numbe... |
| limom 7826 | Omega is a limit ordinal. ... |
| peano2b 7827 | A class belongs to omega i... |
| nnsuc 7828 | A nonzero natural number i... |
| omsucne 7829 | A natural number is not th... |
| ssnlim 7830 | An ordinal subclass of non... |
| omsinds 7831 | Strong (or "total") induct... |
| omun 7832 | The union of two finite or... |
| peano1 7833 | Zero is a natural number. ... |
| peano2 7834 | The successor of any natur... |
| peano3 7835 | The successor of any natur... |
| peano4 7836 | Two natural numbers are eq... |
| peano5 7837 | The induction postulate: a... |
| nn0suc 7838 | A natural number is either... |
| find 7839 | The Principle of Finite In... |
| finds 7840 | Principle of Finite Induct... |
| findsg 7841 | Principle of Finite Induct... |
| finds2 7842 | Principle of Finite Induct... |
| finds1 7843 | Principle of Finite Induct... |
| findes 7844 | Finite induction with expl... |
| dmexg 7845 | The domain of a set is a s... |
| rnexg 7846 | The range of a set is a se... |
| dmexd 7847 | The domain of a set is a s... |
| fndmexd 7848 | If a function is a set, it... |
| dmfex 7849 | If a mapping is a set, its... |
| fndmexb 7850 | The domain of a function i... |
| fdmexb 7851 | The domain of a function i... |
| dmfexALT 7852 | Alternate proof of ~ dmfex... |
| dmex 7853 | The domain of a set is a s... |
| rnex 7854 | The range of a set is a se... |
| iprc 7855 | The identity function is a... |
| resiexg 7856 | The existence of a restric... |
| imaexg 7857 | The image of a set is a se... |
| imaex 7858 | The image of a set is a se... |
| rnexd 7859 | The range of a set is a se... |
| imaexd 7860 | The image of a set is a se... |
| exse2 7861 | Any set relation is set-li... |
| xpexr 7862 | If a Cartesian product is ... |
| xpexr2 7863 | If a nonempty Cartesian pr... |
| xpexcnv 7864 | A condition where the conv... |
| soex 7865 | If the relation in a stric... |
| elxp4 7866 | Membership in a Cartesian ... |
| elxp5 7867 | Membership in a Cartesian ... |
| cnvexg 7868 | The converse of a set is a... |
| cnvex 7869 | The converse of a set is a... |
| relcnvexb 7870 | A relation is a set iff it... |
| f1oexrnex 7871 | If the range of a 1-1 onto... |
| f1oexbi 7872 | There is a one-to-one onto... |
| coexg 7873 | The composition of two set... |
| coex 7874 | The composition of two set... |
| coexd 7875 | The composition of two set... |
| funcnvuni 7876 | The union of a chain (with... |
| fun11uni 7877 | The union of a chain (with... |
| resf1extb 7878 | Extension of an injection ... |
| resf1ext2b 7879 | Extension of an injection ... |
| fex2 7880 | A function with bounded do... |
| fabexd 7881 | Existence of a set of func... |
| fabexg 7882 | Existence of a set of func... |
| fabexgOLD 7883 | Obsolete version of ~ fabe... |
| fabex 7884 | Existence of a set of func... |
| mapex 7885 | The class of all functions... |
| f1oabexg 7886 | The class of all 1-1-onto ... |
| f1oabexgOLD 7887 | Obsolete version of ~ f1oa... |
| fiunlem 7888 | Lemma for ~ fiun and ~ f1i... |
| fiun 7889 | The union of a chain (with... |
| f1iun 7890 | The union of a chain (with... |
| fviunfun 7891 | The function value of an i... |
| ffoss 7892 | Relationship between a map... |
| f11o 7893 | Relationship between one-t... |
| resfunexgALT 7894 | Alternate proof of ~ resfu... |
| cofunexg 7895 | Existence of a composition... |
| cofunex2g 7896 | Existence of a composition... |
| fnexALT 7897 | Alternate proof of ~ fnex ... |
| funexw 7898 | Weak version of ~ funex th... |
| mptexw 7899 | Weak version of ~ mptex th... |
| funrnex 7900 | If the domain of a functio... |
| zfrep6OLD 7901 | Obsolete proof of ~ zfrep6... |
| focdmex 7902 | If the domain of an onto f... |
| f1dmex 7903 | If the codomain of a one-t... |
| f1ovv 7904 | The codomain/range of a 1-... |
| fvclex 7905 | Existence of the class of ... |
| fvresex 7906 | Existence of the class of ... |
| abrexexg 7907 | Existence of a class abstr... |
| abrexex 7908 | Existence of a class abstr... |
| iunexg 7909 | The existence of an indexe... |
| abrexex2g 7910 | Existence of an existentia... |
| opabex3d 7911 | Existence of an ordered pa... |
| opabex3rd 7912 | Existence of an ordered pa... |
| opabex3 7913 | Existence of an ordered pa... |
| iunex 7914 | The existence of an indexe... |
| abrexex2 7915 | Existence of an existentia... |
| abexssex 7916 | Existence of a class abstr... |
| abexex 7917 | A condition where a class ... |
| f1oweALT 7918 | Alternate proof of ~ f1owe... |
| wemoiso 7919 | Thus, there is at most one... |
| wemoiso2 7920 | Thus, there is at most one... |
| oprabexd 7921 | Existence of an operator a... |
| oprabex 7922 | Existence of an operation ... |
| oprabex3 7923 | Existence of an operation ... |
| oprabrexex2 7924 | Existence of an existentia... |
| ab2rexex 7925 | Existence of a class abstr... |
| ab2rexex2 7926 | Existence of an existentia... |
| xpexgALT 7927 | Alternate proof of ~ xpexg... |
| offval3 7928 | General value of ` ( F oF ... |
| offres 7929 | Pointwise combination comm... |
| ofmres 7930 | Equivalent expressions for... |
| ofmresex 7931 | Existence of a restriction... |
| mptcnfimad 7932 | The converse of a mapping ... |
| 1stval 7937 | The value of the function ... |
| 2ndval 7938 | The value of the function ... |
| 1stnpr 7939 | Value of the first-member ... |
| 2ndnpr 7940 | Value of the second-member... |
| 1st0 7941 | The value of the first-mem... |
| 2nd0 7942 | The value of the second-me... |
| op1st 7943 | Extract the first member o... |
| op2nd 7944 | Extract the second member ... |
| op1std 7945 | Extract the first member o... |
| op2ndd 7946 | Extract the second member ... |
| op1stg 7947 | Extract the first member o... |
| op2ndg 7948 | Extract the second member ... |
| ot1stg 7949 | Extract the first member o... |
| ot2ndg 7950 | Extract the second member ... |
| ot3rdg 7951 | Extract the third member o... |
| 1stval2 7952 | Alternate value of the fun... |
| 2ndval2 7953 | Alternate value of the fun... |
| oteqimp 7954 | The components of an order... |
| fo1st 7955 | The ` 1st ` function maps ... |
| fo2nd 7956 | The ` 2nd ` function maps ... |
| br1steqg 7957 | Uniqueness condition for t... |
| br2ndeqg 7958 | Uniqueness condition for t... |
| f1stres 7959 | Mapping of a restriction o... |
| f2ndres 7960 | Mapping of a restriction o... |
| fo1stres 7961 | Onto mapping of a restrict... |
| fo2ndres 7962 | Onto mapping of a restrict... |
| 1st2val 7963 | Value of an alternate defi... |
| 2nd2val 7964 | Value of an alternate defi... |
| 1stcof 7965 | Composition of the first m... |
| 2ndcof 7966 | Composition of the second ... |
| xp1st 7967 | Location of the first elem... |
| xp2nd 7968 | Location of the second ele... |
| elxp6 7969 | Membership in a Cartesian ... |
| elxp7 7970 | Membership in a Cartesian ... |
| eqopi 7971 | Equality with an ordered p... |
| xp2 7972 | Representation of Cartesia... |
| unielxp 7973 | The membership relation fo... |
| 1st2nd2 7974 | Reconstruction of a member... |
| 1st2ndb 7975 | Reconstruction of an order... |
| xpopth 7976 | An ordered pair theorem fo... |
| eqop 7977 | Two ways to express equali... |
| eqop2 7978 | Two ways to express equali... |
| op1steq 7979 | Two ways of expressing tha... |
| opreuopreu 7980 | There is a unique ordered ... |
| el2xptp 7981 | A member of a nested Carte... |
| el2xptp0 7982 | A member of a nested Carte... |
| el2xpss 7983 | Version of ~ elrel for tri... |
| 2nd1st 7984 | Swap the members of an ord... |
| 1st2nd 7985 | Reconstruction of a member... |
| 1stdm 7986 | The first ordered pair com... |
| 2ndrn 7987 | The second ordered pair co... |
| 1st2ndbr 7988 | Express an element of a re... |
| releldm2 7989 | Two ways of expressing mem... |
| reldm 7990 | An expression for the doma... |
| releldmdifi 7991 | One way of expressing memb... |
| funfv1st2nd 7992 | The function value for the... |
| funelss 7993 | If the first component of ... |
| funeldmdif 7994 | Two ways of expressing mem... |
| sbcopeq1a 7995 | Equality theorem for subst... |
| csbopeq1a 7996 | Equality theorem for subst... |
| sbcoteq1a 7997 | Equality theorem for subst... |
| dfopab2 7998 | A way to define an ordered... |
| dfoprab3s 7999 | A way to define an operati... |
| dfoprab3 8000 | Operation class abstractio... |
| dfoprab4 8001 | Operation class abstractio... |
| dfoprab4f 8002 | Operation class abstractio... |
| opabex2 8003 | Condition for an operation... |
| opabn1stprc 8004 | An ordered-pair class abst... |
| opiota 8005 | The property of a uniquely... |
| cnvoprab 8006 | The converse of a class ab... |
| dfxp3 8007 | Define the Cartesian produ... |
| elopabi 8008 | A consequence of membershi... |
| eloprabi 8009 | A consequence of membershi... |
| mpomptsx 8010 | Express a two-argument fun... |
| mpompts 8011 | Express a two-argument fun... |
| dmmpossx 8012 | The domain of a mapping is... |
| fmpox 8013 | Functionality, domain and ... |
| fmpo 8014 | Functionality, domain and ... |
| fnmpo 8015 | Functionality and domain o... |
| fnmpoi 8016 | Functionality and domain o... |
| dmmpo 8017 | Domain of a class given by... |
| ovmpoelrn 8018 | An operation's value belon... |
| dmmpoga 8019 | Domain of an operation giv... |
| dmmpog 8020 | Domain of an operation giv... |
| mpoexxg 8021 | Existence of an operation ... |
| mpoexg 8022 | Existence of an operation ... |
| mpoexga 8023 | If the domain of an operat... |
| mpoexw 8024 | Weak version of ~ mpoex th... |
| mpoex 8025 | If the domain of an operat... |
| mptmpoopabbrd 8026 | The operation value of a f... |
| mptmpoopabovd 8027 | The operation value of a f... |
| el2mpocsbcl 8028 | If the operation value of ... |
| el2mpocl 8029 | If the operation value of ... |
| fnmpoovd 8030 | A function with a Cartesia... |
| offval22 8031 | The function operation exp... |
| brovpreldm 8032 | If a binary relation holds... |
| bropopvvv 8033 | If a binary relation holds... |
| bropfvvvvlem 8034 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8035 | If a binary relation holds... |
| ovmptss 8036 | If all the values of the m... |
| relmpoopab 8037 | Any function to sets of or... |
| fmpoco 8038 | Composition of two functio... |
| oprabco 8039 | Composition of a function ... |
| oprab2co 8040 | Composition of operator ab... |
| df1st2 8041 | An alternate possible defi... |
| df2nd2 8042 | An alternate possible defi... |
| 1stconst 8043 | The mapping of a restricti... |
| 2ndconst 8044 | The mapping of a restricti... |
| dfmpo 8045 | Alternate definition for t... |
| mposn 8046 | An operation (in maps-to n... |
| curry1 8047 | Composition with ` ``' ( 2... |
| curry1val 8048 | The value of a curried fun... |
| curry1f 8049 | Functionality of a curried... |
| curry2 8050 | Composition with ` ``' ( 1... |
| curry2f 8051 | Functionality of a curried... |
| curry2val 8052 | The value of a curried fun... |
| cnvf1olem 8053 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8054 | Describe a function that m... |
| fparlem1 8055 | Lemma for ~ fpar . (Contr... |
| fparlem2 8056 | Lemma for ~ fpar . (Contr... |
| fparlem3 8057 | Lemma for ~ fpar . (Contr... |
| fparlem4 8058 | Lemma for ~ fpar . (Contr... |
| fpar 8059 | Merge two functions in par... |
| fsplit 8060 | A function that can be use... |
| fsplitfpar 8061 | Merge two functions with a... |
| offsplitfpar 8062 | Express the function opera... |
| f2ndf 8063 | The ` 2nd ` (second compon... |
| fo2ndf 8064 | The ` 2nd ` (second compon... |
| f1o2ndf1 8065 | The ` 2nd ` (second compon... |
| opco1 8066 | Value of an operation prec... |
| opco2 8067 | Value of an operation prec... |
| opco1i 8068 | Inference form of ~ opco1 ... |
| mpof1o2d 8069 | Sufficient condition for a... |
| frxp 8070 | A lexicographical ordering... |
| xporderlem 8071 | Lemma for lexicographical ... |
| poxp 8072 | A lexicographical ordering... |
| soxp 8073 | A lexicographical ordering... |
| wexp 8074 | A lexicographical ordering... |
| fnwelem 8075 | Lemma for ~ fnwe . (Contr... |
| fnwe 8076 | A variant on lexicographic... |
| fnse 8077 | Condition for the well-ord... |
| fvproj 8078 | Value of a function on ord... |
| fimaproj 8079 | Image of a cartesian produ... |
| ralxpes 8080 | A version of ~ ralxp with ... |
| ralxp3f 8081 | Restricted for all over a ... |
| ralxp3 8082 | Restricted for all over a ... |
| ralxp3es 8083 | Restricted for-all over a ... |
| frpoins3xpg 8084 | Special case of founded pa... |
| frpoins3xp3g 8085 | Special case of founded pa... |
| xpord2lem 8086 | Lemma for Cartesian produc... |
| poxp2 8087 | Another way of partially o... |
| frxp2 8088 | Another way of giving a we... |
| xpord2pred 8089 | Calculate the predecessor ... |
| sexp2 8090 | Condition for the relation... |
| xpord2indlem 8091 | Induction over the Cartesi... |
| xpord2ind 8092 | Induction over the Cartesi... |
| xpord3lem 8093 | Lemma for triple ordering.... |
| poxp3 8094 | Triple Cartesian product p... |
| frxp3 8095 | Give well-foundedness over... |
| xpord3pred 8096 | Calculate the predecsessor... |
| sexp3 8097 | Show that the triple order... |
| xpord3inddlem 8098 | Induction over the triple ... |
| xpord3indd 8099 | Induction over the triple ... |
| xpord3ind 8100 | Induction over the triple ... |
| orderseqlem 8101 | Lemma for ~ poseq and ~ so... |
| poseq 8102 | A partial ordering of ordi... |
| soseq 8103 | A linear ordering of ordin... |
| suppval 8106 | The value of the operation... |
| supp0prc 8107 | The support of a class is ... |
| suppvalbr 8108 | The value of the operation... |
| supp0 8109 | The support of the empty s... |
| suppval1 8110 | The value of the operation... |
| suppvalfng 8111 | The value of the operation... |
| suppvalfn 8112 | The value of the operation... |
| elsuppfng 8113 | An element of the support ... |
| elsuppfn 8114 | An element of the support ... |
| fvdifsupp 8115 | Function value is zero out... |
| cnvimadfsn 8116 | The support of functions "... |
| suppimacnvss 8117 | The support of functions "... |
| suppimacnv 8118 | Support sets of functions ... |
| fsuppeq 8119 | Two ways of writing the su... |
| fsuppeqg 8120 | Version of ~ fsuppeq avoid... |
| suppssdm 8121 | The support of a function ... |
| suppsnop 8122 | The support of a singleton... |
| snopsuppss 8123 | The support of a singleton... |
| fvn0elsupp 8124 | If the function value for ... |
| fvn0elsuppb 8125 | The function value for a g... |
| rexsupp 8126 | Existential quantification... |
| ressuppss 8127 | The support of the restric... |
| suppun 8128 | The support of a class/fun... |
| ressuppssdif 8129 | The support of the restric... |
| mptsuppdifd 8130 | The support of a function ... |
| mptsuppd 8131 | The support of a function ... |
| extmptsuppeq 8132 | The support of an extended... |
| suppfnss 8133 | The support of a function ... |
| funsssuppss 8134 | The support of a function ... |
| fnsuppres 8135 | Two ways to express restri... |
| fnsuppeq0 8136 | The support of a function ... |
| fczsupp0 8137 | The support of a constant ... |
| suppss 8138 | Show that the support of a... |
| suppssr 8139 | A function is zero outside... |
| suppssrg 8140 | A function is zero outside... |
| suppssov1 8141 | Formula building theorem f... |
| suppssov2 8142 | Formula building theorem f... |
| suppssof1 8143 | Formula building theorem f... |
| suppss2 8144 | Show that the support of a... |
| suppsssn 8145 | Show that the support of a... |
| suppssfv 8146 | Formula building theorem f... |
| suppofssd 8147 | Condition for the support ... |
| suppofss1d 8148 | Condition for the support ... |
| suppofss2d 8149 | Condition for the support ... |
| suppco 8150 | The support of the composi... |
| suppcoss 8151 | The support of the composi... |
| supp0cosupp0 8152 | The support of the composi... |
| imacosupp 8153 | The image of the support o... |
| opeliunxp2f 8154 | Membership in a union of C... |
| mpoxeldm 8155 | If there is an element of ... |
| mpoxneldm 8156 | If the first argument of a... |
| mpoxopn0yelv 8157 | If there is an element of ... |
| mpoxopynvov0g 8158 | If the second argument of ... |
| mpoxopxnop0 8159 | If the first argument of a... |
| mpoxopx0ov0 8160 | If the first argument of a... |
| mpoxopxprcov0 8161 | If the components of the f... |
| mpoxopynvov0 8162 | If the second argument of ... |
| mpoxopoveq 8163 | Value of an operation give... |
| mpoxopovel 8164 | Element of the value of an... |
| mpoxopoveqd 8165 | Value of an operation give... |
| brovex 8166 | A binary relation of the v... |
| brovmpoex 8167 | A binary relation of the v... |
| sprmpod 8168 | The extension of a binary ... |
| tposss 8171 | Subset theorem for transpo... |
| tposeq 8172 | Equality theorem for trans... |
| tposeqd 8173 | Equality theorem for trans... |
| tposssxp 8174 | The transposition is a sub... |
| reltpos 8175 | The transposition is a rel... |
| brtpos2 8176 | Value of the transposition... |
| brtpos0 8177 | The behavior of ` tpos ` w... |
| reldmtpos 8178 | Necessary and sufficient c... |
| brtpos 8179 | The transposition swaps ar... |
| ottpos 8180 | The transposition swaps th... |
| relbrtpos 8181 | The transposition swaps ar... |
| dmtpos 8182 | The domain of ` tpos F ` w... |
| rntpos 8183 | The range of ` tpos F ` wh... |
| tposexg 8184 | The transposition of a set... |
| ovtpos 8185 | The transposition swaps th... |
| tposfun 8186 | The transposition of a fun... |
| dftpos2 8187 | Alternate definition of ` ... |
| dftpos3 8188 | Alternate definition of ` ... |
| dftpos4 8189 | Alternate definition of ` ... |
| tpostpos 8190 | Value of the double transp... |
| tpostpos2 8191 | Value of the double transp... |
| tposfn2 8192 | The domain of a transposit... |
| tposfo2 8193 | Condition for a surjective... |
| tposf2 8194 | The domain and codomain of... |
| tposf12 8195 | Condition for an injective... |
| tposf1o2 8196 | Condition of a bijective t... |
| tposfo 8197 | The domain and codomain/ra... |
| tposf 8198 | The domain and codomain of... |
| tposfn 8199 | Functionality of a transpo... |
| tpos0 8200 | Transposition of the empty... |
| tposco 8201 | Transposition of a composi... |
| tpossym 8202 | Two ways to say a function... |
| tposeqi 8203 | Equality theorem for trans... |
| tposex 8204 | A transposition is a set. ... |
| nftpos 8205 | Hypothesis builder for tra... |
| tposoprab 8206 | Transposition of a class o... |
| tposmpo 8207 | Transposition of a two-arg... |
| tposconst 8208 | The transposition of a con... |
| mpocurryd 8213 | The currying of an operati... |
| mpocurryvald 8214 | The value of a curried ope... |
| fvmpocurryd 8215 | The value of the value of ... |
| pwuninel2 8218 | Proof of ~ pwuninel under ... |
| pwuninel 8219 | The powerclass of the unio... |
| undefval 8220 | Value of the undefined val... |
| undefnel2 8221 | The undefined value genera... |
| undefnel 8222 | The undefined value genera... |
| undefne0 8223 | The undefined value genera... |
| frecseq123 8226 | Equality theorem for the w... |
| nffrecs 8227 | Bound-variable hypothesis ... |
| csbfrecsg 8228 | Move class substitution in... |
| fpr3g 8229 | Functions defined by well-... |
| frrlem1 8230 | Lemma for well-founded rec... |
| frrlem2 8231 | Lemma for well-founded rec... |
| frrlem3 8232 | Lemma for well-founded rec... |
| frrlem4 8233 | Lemma for well-founded rec... |
| frrlem5 8234 | Lemma for well-founded rec... |
| frrlem6 8235 | Lemma for well-founded rec... |
| frrlem7 8236 | Lemma for well-founded rec... |
| frrlem8 8237 | Lemma for well-founded rec... |
| frrlem9 8238 | Lemma for well-founded rec... |
| frrlem10 8239 | Lemma for well-founded rec... |
| frrlem11 8240 | Lemma for well-founded rec... |
| frrlem12 8241 | Lemma for well-founded rec... |
| frrlem13 8242 | Lemma for well-founded rec... |
| frrlem14 8243 | Lemma for well-founded rec... |
| fprlem1 8244 | Lemma for well-founded rec... |
| fprlem2 8245 | Lemma for well-founded rec... |
| fpr2a 8246 | Weak version of ~ fpr2 whi... |
| fpr1 8247 | Law of well-founded recurs... |
| fpr2 8248 | Law of well-founded recurs... |
| fpr3 8249 | Law of well-founded recurs... |
| frrrel 8250 | Show without using the axi... |
| frrdmss 8251 | Show without using the axi... |
| frrdmcl 8252 | Show without using the axi... |
| fprfung 8253 | A "function" defined by we... |
| fprresex 8254 | The restriction of a funct... |
| wrecseq123 8257 | General equality theorem f... |
| nfwrecs 8258 | Bound-variable hypothesis ... |
| wrecseq1 8259 | Equality theorem for the w... |
| wrecseq2 8260 | Equality theorem for the w... |
| wrecseq3 8261 | Equality theorem for the w... |
| csbwrecsg 8262 | Move class substitution in... |
| wfr3g 8263 | Functions defined by well-... |
| wfrrel 8264 | The well-ordered recursion... |
| wfrdmss 8265 | The domain of the well-ord... |
| wfrdmcl 8266 | The predecessor class of a... |
| wfrfun 8267 | The "function" generated b... |
| wfrresex 8268 | Show without using the axi... |
| wfr2a 8269 | A weak version of ~ wfr2 w... |
| wfr1 8270 | The Principle of Well-Orde... |
| wfr2 8271 | The Principle of Well-Orde... |
| wfr3 8272 | The principle of Well-Orde... |
| iunon 8273 | The indexed union of a set... |
| iinon 8274 | The nonempty indexed inter... |
| onfununi 8275 | A property of functions on... |
| onovuni 8276 | A variant of ~ onfununi fo... |
| onoviun 8277 | A variant of ~ onovuni wit... |
| onnseq 8278 | There are no length ` _om ... |
| dfsmo2 8281 | Alternate definition of a ... |
| issmo 8282 | Conditions for which ` A `... |
| issmo2 8283 | Alternate definition of a ... |
| smoeq 8284 | Equality theorem for stric... |
| smodm 8285 | The domain of a strictly m... |
| smores 8286 | A strictly monotone functi... |
| smores3 8287 | A strictly monotone functi... |
| smores2 8288 | A strictly monotone ordina... |
| smodm2 8289 | The domain of a strictly m... |
| smofvon2 8290 | The function values of a s... |
| iordsmo 8291 | The identity relation rest... |
| smo0 8292 | The null set is a strictly... |
| smofvon 8293 | If ` B ` is a strictly mon... |
| smoel 8294 | If ` x ` is less than ` y ... |
| smoiun 8295 | The value of a strictly mo... |
| smoiso 8296 | If ` F ` is an isomorphism... |
| smoel2 8297 | A strictly monotone ordina... |
| smo11 8298 | A strictly monotone ordina... |
| smoord 8299 | A strictly monotone ordina... |
| smoword 8300 | A strictly monotone ordina... |
| smogt 8301 | A strictly monotone ordina... |
| smocdmdom 8302 | The codomain of a strictly... |
| smoiso2 8303 | The strictly monotone ordi... |
| dfrecs3 8306 | The old definition of tran... |
| recseq 8307 | Equality theorem for ` rec... |
| nfrecs 8308 | Bound-variable hypothesis ... |
| tfrlem1 8309 | A technical lemma for tran... |
| tfrlem3a 8310 | Lemma for transfinite recu... |
| tfrlem3 8311 | Lemma for transfinite recu... |
| tfrlem4 8312 | Lemma for transfinite recu... |
| tfrlem5 8313 | Lemma for transfinite recu... |
| recsfval 8314 | Lemma for transfinite recu... |
| tfrlem6 8315 | Lemma for transfinite recu... |
| tfrlem7 8316 | Lemma for transfinite recu... |
| tfrlem8 8317 | Lemma for transfinite recu... |
| tfrlem9 8318 | Lemma for transfinite recu... |
| tfrlem9a 8319 | Lemma for transfinite recu... |
| tfrlem10 8320 | Lemma for transfinite recu... |
| tfrlem11 8321 | Lemma for transfinite recu... |
| tfrlem12 8322 | Lemma for transfinite recu... |
| tfrlem13 8323 | Lemma for transfinite recu... |
| tfrlem14 8324 | Lemma for transfinite recu... |
| tfrlem15 8325 | Lemma for transfinite recu... |
| tfrlem16 8326 | Lemma for finite recursion... |
| tfr1a 8327 | A weak version of ~ tfr1 w... |
| tfr2a 8328 | A weak version of ~ tfr2 w... |
| tfr2b 8329 | Without assuming ~ ax-rep ... |
| tfr1 8330 | Principle of Transfinite R... |
| tfr2 8331 | Principle of Transfinite R... |
| tfr3 8332 | Principle of Transfinite R... |
| tfr1ALT 8333 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8334 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8335 | Alternate proof of ~ tfr3 ... |
| recsfnon 8336 | Strong transfinite recursi... |
| recsval 8337 | Strong transfinite recursi... |
| tz7.44lem1 8338 | The ordered pair abstracti... |
| tz7.44-1 8339 | The value of ` F ` at ` (/... |
| tz7.44-2 8340 | The value of ` F ` at a su... |
| tz7.44-3 8341 | The value of ` F ` at a li... |
| rdgeq1 8344 | Equality theorem for the r... |
| rdgeq2 8345 | Equality theorem for the r... |
| rdgeq12 8346 | Equality theorem for the r... |
| nfrdg 8347 | Bound-variable hypothesis ... |
| rdglem1 8348 | Lemma used with the recurs... |
| rdgfun 8349 | The recursive definition g... |
| rdgdmlim 8350 | The domain of the recursiv... |
| rdgfnon 8351 | The recursive definition g... |
| rdgvalg 8352 | Value of the recursive def... |
| rdgval 8353 | Value of the recursive def... |
| rdg0 8354 | The initial value of the r... |
| rdgseg 8355 | The initial segments of th... |
| rdgsucg 8356 | The value of the recursive... |
| rdgsuc 8357 | The value of the recursive... |
| rdglimg 8358 | The value of the recursive... |
| rdglim 8359 | The value of the recursive... |
| rdg0g 8360 | The initial value of the r... |
| rdgsucmptf 8361 | The value of the recursive... |
| rdgsucmptnf 8362 | The value of the recursive... |
| rdgsucmpt2 8363 | This version of ~ rdgsucmp... |
| rdgsucmpt 8364 | The value of the recursive... |
| rdglim2 8365 | The value of the recursive... |
| rdglim2a 8366 | The value of the recursive... |
| rdg0n 8367 | If ` A ` is a proper class... |
| frfnom 8368 | The function generated by ... |
| fr0g 8369 | The initial value resultin... |
| frsuc 8370 | The successor value result... |
| frsucmpt 8371 | The successor value result... |
| frsucmptn 8372 | The value of the finite re... |
| frsucmpt2 8373 | The successor value result... |
| tz7.48lem 8374 | A way of showing an ordina... |
| tz7.48-2 8375 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8376 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8377 | Proposition 7.48(3) of [Ta... |
| tz7.49 8378 | Proposition 7.49 of [Takeu... |
| tz7.49c 8379 | Corollary of Proposition 7... |
| seqomlem0 8382 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8383 | Lemma for ` seqom ` . The... |
| seqomlem2 8384 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8385 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8386 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8387 | Equality theorem for ` seq... |
| fnseqom 8388 | An index-aware recursive d... |
| seqom0g 8389 | Value of an index-aware re... |
| seqomsuc 8390 | Value of an index-aware re... |
| omsucelsucb 8391 | Membership is inherited by... |
| df1o2 8406 | Expanded value of the ordi... |
| df2o3 8407 | Expanded value of the ordi... |
| df2o2 8408 | Expanded value of the ordi... |
| 1oex 8409 | Ordinal 1 is a set. (Cont... |
| 2oex 8410 | ` 2o ` is a set. (Contrib... |
| 1on 8411 | Ordinal 1 is an ordinal nu... |
| 2on 8412 | Ordinal 2 is an ordinal nu... |
| 2on0 8413 | Ordinal two is not zero. ... |
| ord3 8414 | Ordinal 3 is an ordinal cl... |
| 3on 8415 | Ordinal 3 is an ordinal nu... |
| 4on 8416 | Ordinal 4 is an ordinal nu... |
| 1n0 8417 | Ordinal one is not equal t... |
| nlim1 8418 | 1 is not a limit ordinal. ... |
| nlim2 8419 | 2 is not a limit ordinal. ... |
| xp01disj 8420 | Cartesian products with th... |
| xp01disjl 8421 | Cartesian products with th... |
| ordgt0ge1 8422 | Two ways to express that a... |
| ordge1n0 8423 | An ordinal greater than or... |
| el1o 8424 | Membership in ordinal one.... |
| ord1eln01 8425 | An ordinal that is not 0 o... |
| ord2eln012 8426 | An ordinal that is not 0, ... |
| 1ellim 8427 | A limit ordinal contains 1... |
| 2ellim 8428 | A limit ordinal contains 2... |
| dif1o 8429 | Two ways to say that ` A `... |
| ondif1 8430 | Two ways to say that ` A `... |
| ondif2 8431 | Two ways to say that ` A `... |
| 2oconcl 8432 | Closure of the pair swappi... |
| 0lt1o 8433 | Ordinal zero is less than ... |
| dif20el 8434 | An ordinal greater than on... |
| 0we1 8435 | The empty set is a well-or... |
| brwitnlem 8436 | Lemma for relations which ... |
| fnoa 8437 | Functionality and domain o... |
| fnom 8438 | Functionality and domain o... |
| fnoe 8439 | Functionality and domain o... |
| oav 8440 | Value of ordinal addition.... |
| omv 8441 | Value of ordinal multiplic... |
| oe0lem 8442 | A helper lemma for ~ oe0 a... |
| oev 8443 | Value of ordinal exponenti... |
| oevn0 8444 | Value of ordinal exponenti... |
| oa0 8445 | Addition with zero. Propo... |
| om0 8446 | Ordinal multiplication wit... |
| oe0m 8447 | Value of zero raised to an... |
| om0x 8448 | Ordinal multiplication wit... |
| oe0m0 8449 | Ordinal exponentiation wit... |
| oe0m1 8450 | Ordinal exponentiation wit... |
| oe0 8451 | Ordinal exponentiation wit... |
| oev2 8452 | Alternate value of ordinal... |
| oasuc 8453 | Addition with successor. ... |
| oesuclem 8454 | Lemma for ~ oesuc . (Cont... |
| omsuc 8455 | Multiplication with succes... |
| oesuc 8456 | Ordinal exponentiation wit... |
| onasuc 8457 | Addition with successor. ... |
| onmsuc 8458 | Multiplication with succes... |
| onesuc 8459 | Exponentiation with a succ... |
| oa1suc 8460 | Addition with 1 is same as... |
| oalim 8461 | Ordinal addition with a li... |
| omlim 8462 | Ordinal multiplication wit... |
| oelim 8463 | Ordinal exponentiation wit... |
| oacl 8464 | Closure law for ordinal ad... |
| omcl 8465 | Closure law for ordinal mu... |
| oecl 8466 | Closure law for ordinal ex... |
| oa0r 8467 | Ordinal addition with zero... |
| om0r 8468 | Ordinal multiplication wit... |
| o1p1e2 8469 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8470 | 2 + 2 = 4 for ordinal numb... |
| om1 8471 | Ordinal multiplication wit... |
| om1r 8472 | Ordinal multiplication wit... |
| oe1 8473 | Ordinal exponentiation wit... |
| oe1m 8474 | Ordinal exponentiation wit... |
| oaordi 8475 | Ordering property of ordin... |
| oaord 8476 | Ordering property of ordin... |
| oacan 8477 | Left cancellation law for ... |
| oaword 8478 | Weak ordering property of ... |
| oawordri 8479 | Weak ordering property of ... |
| oaord1 8480 | An ordinal is less than it... |
| oaword1 8481 | An ordinal is less than or... |
| oaword2 8482 | An ordinal is less than or... |
| oawordeulem 8483 | Lemma for ~ oawordex . (C... |
| oawordeu 8484 | Existence theorem for weak... |
| oawordexr 8485 | Existence theorem for weak... |
| oawordex 8486 | Existence theorem for weak... |
| oaordex 8487 | Existence theorem for orde... |
| oa00 8488 | An ordinal sum is zero iff... |
| oalimcl 8489 | The ordinal sum with a lim... |
| oaass 8490 | Ordinal addition is associ... |
| oarec 8491 | Recursive definition of or... |
| oaf1o 8492 | Left addition by a constan... |
| oacomf1olem 8493 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8494 | Define a bijection from ` ... |
| omordi 8495 | Ordering property of ordin... |
| omord2 8496 | Ordering property of ordin... |
| omord 8497 | Ordering property of ordin... |
| omcan 8498 | Left cancellation law for ... |
| omword 8499 | Weak ordering property of ... |
| omwordi 8500 | Weak ordering property of ... |
| omwordri 8501 | Weak ordering property of ... |
| omword1 8502 | An ordinal is less than or... |
| omword2 8503 | An ordinal is less than or... |
| om00 8504 | The product of two ordinal... |
| om00el 8505 | The product of two nonzero... |
| omordlim 8506 | Ordering involving the pro... |
| omlimcl 8507 | The product of any nonzero... |
| odi 8508 | Distributive law for ordin... |
| omass 8509 | Multiplication of ordinal ... |
| oneo 8510 | If an ordinal number is ev... |
| omeulem1 8511 | Lemma for ~ omeu : existen... |
| omeulem2 8512 | Lemma for ~ omeu : uniquen... |
| omopth2 8513 | An ordered pair-like theor... |
| omeu 8514 | The division algorithm for... |
| om2 8515 | Two ways to double an ordi... |
| oen0 8516 | Ordinal exponentiation wit... |
| oeordi 8517 | Ordering law for ordinal e... |
| oeord 8518 | Ordering property of ordin... |
| oecan 8519 | Left cancellation law for ... |
| oeword 8520 | Weak ordering property of ... |
| oewordi 8521 | Weak ordering property of ... |
| oewordri 8522 | Weak ordering property of ... |
| oeworde 8523 | Ordinal exponentiation com... |
| oeordsuc 8524 | Ordering property of ordin... |
| oelim2 8525 | Ordinal exponentiation wit... |
| oeoalem 8526 | Lemma for ~ oeoa . (Contr... |
| oeoa 8527 | Sum of exponents law for o... |
| oeoelem 8528 | Lemma for ~ oeoe . (Contr... |
| oeoe 8529 | Product of exponents law f... |
| oelimcl 8530 | The ordinal exponential wi... |
| oeeulem 8531 | Lemma for ~ oeeu . (Contr... |
| oeeui 8532 | The division algorithm for... |
| oeeu 8533 | The division algorithm for... |
| nna0 8534 | Addition with zero. Theor... |
| nnm0 8535 | Multiplication with zero. ... |
| nnasuc 8536 | Addition with successor. ... |
| nnmsuc 8537 | Multiplication with succes... |
| nnesuc 8538 | Exponentiation with a succ... |
| nna0r 8539 | Addition to zero. Remark ... |
| nnm0r 8540 | Multiplication with zero. ... |
| nnacl 8541 | Closure of addition of nat... |
| nnmcl 8542 | Closure of multiplication ... |
| nnecl 8543 | Closure of exponentiation ... |
| nnacli 8544 | ` _om ` is closed under ad... |
| nnmcli 8545 | ` _om ` is closed under mu... |
| nnarcl 8546 | Reverse closure law for ad... |
| nnacom 8547 | Addition of natural number... |
| nnaordi 8548 | Ordering property of addit... |
| nnaord 8549 | Ordering property of addit... |
| nnaordr 8550 | Ordering property of addit... |
| nnawordi 8551 | Adding to both sides of an... |
| nnaass 8552 | Addition of natural number... |
| nndi 8553 | Distributive law for natur... |
| nnmass 8554 | Multiplication of natural ... |
| nnmsucr 8555 | Multiplication with succes... |
| nnmcom 8556 | Multiplication of natural ... |
| nnaword 8557 | Weak ordering property of ... |
| nnacan 8558 | Cancellation law for addit... |
| nnaword1 8559 | Weak ordering property of ... |
| nnaword2 8560 | Weak ordering property of ... |
| nnmordi 8561 | Ordering property of multi... |
| nnmord 8562 | Ordering property of multi... |
| nnmword 8563 | Weak ordering property of ... |
| nnmcan 8564 | Cancellation law for multi... |
| nnmwordi 8565 | Weak ordering property of ... |
| nnmwordri 8566 | Weak ordering property of ... |
| nnawordex 8567 | Equivalence for weak order... |
| nnaordex 8568 | Equivalence for ordering. ... |
| nnaordex2 8569 | Equivalence for ordering. ... |
| 1onn 8570 | The ordinal 1 is a natural... |
| 1onnALT 8571 | Shorter proof of ~ 1onn us... |
| 2onn 8572 | The ordinal 2 is a natural... |
| 2onnALT 8573 | Shorter proof of ~ 2onn us... |
| 3onn 8574 | The ordinal 3 is a natural... |
| 4onn 8575 | The ordinal 4 is a natural... |
| 1one2o 8576 | Ordinal one is not ordinal... |
| oaabslem 8577 | Lemma for ~ oaabs . (Cont... |
| oaabs 8578 | Ordinal addition absorbs a... |
| oaabs2 8579 | The absorption law ~ oaabs... |
| omabslem 8580 | Lemma for ~ omabs . (Cont... |
| omabs 8581 | Ordinal multiplication is ... |
| nnm1 8582 | Multiply an element of ` _... |
| nnm2 8583 | Multiply an element of ` _... |
| nn2m 8584 | Multiply an element of ` _... |
| nnneo 8585 | If a natural number is eve... |
| nneob 8586 | A natural number is even i... |
| omsmolem 8587 | Lemma for ~ omsmo . (Cont... |
| omsmo 8588 | A strictly monotonic ordin... |
| omopthlem1 8589 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8590 | Lemma for ~ omopthi . (Co... |
| omopthi 8591 | An ordered pair theorem fo... |
| omopth 8592 | An ordered pair theorem fo... |
| nnasmo 8593 | There is at most one left ... |
| eldifsucnn 8594 | Condition for membership i... |
| on2recsfn 8597 | Show that double recursion... |
| on2recsov 8598 | Calculate the value of the... |
| on2ind 8599 | Double induction over ordi... |
| on3ind 8600 | Triple induction over ordi... |
| coflton 8601 | Cofinality theorem for ord... |
| cofon1 8602 | Cofinality theorem for ord... |
| cofon2 8603 | Cofinality theorem for ord... |
| cofonr 8604 | Inverse cofinality law for... |
| naddfn 8605 | Natural addition is a func... |
| naddcllem 8606 | Lemma for ordinal addition... |
| naddcl 8607 | Closure law for natural ad... |
| naddov 8608 | The value of natural addit... |
| naddov2 8609 | Alternate expression for n... |
| naddov3 8610 | Alternate expression for n... |
| naddf 8611 | Function statement for nat... |
| naddcom 8612 | Natural addition commutes.... |
| naddrid 8613 | Ordinal zero is the additi... |
| naddlid 8614 | Ordinal zero is the additi... |
| naddssim 8615 | Ordinal less-than-or-equal... |
| naddelim 8616 | Ordinal less-than is prese... |
| naddel1 8617 | Ordinal less-than is not a... |
| naddel2 8618 | Ordinal less-than is not a... |
| naddss1 8619 | Ordinal less-than-or-equal... |
| naddss2 8620 | Ordinal less-than-or-equal... |
| naddword1 8621 | Weak-ordering principle fo... |
| naddword2 8622 | Weak-ordering principle fo... |
| naddunif 8623 | Uniformity theorem for nat... |
| naddasslem1 8624 | Lemma for ~ naddass . Exp... |
| naddasslem2 8625 | Lemma for ~ naddass . Exp... |
| naddass 8626 | Natural ordinal addition i... |
| nadd32 8627 | Commutative/associative la... |
| nadd4 8628 | Rearragement of terms in a... |
| nadd42 8629 | Rearragement of terms in a... |
| naddel12 8630 | Natural addition to both s... |
| naddsuc2 8631 | Natural addition with succ... |
| naddoa 8632 | Natural addition of a natu... |
| omnaddcl 8633 | The naturals are closed un... |
| dfer2 8638 | Alternate definition of eq... |
| dfec2 8640 | Alternate definition of ` ... |
| ecexg 8641 | An equivalence class modul... |
| ecexr 8642 | A nonempty equivalence cla... |
| dfqs2 8644 | Alternate definition of qu... |
| ereq1 8645 | Equality theorem for equiv... |
| ereq2 8646 | Equality theorem for equiv... |
| errel 8647 | An equivalence relation is... |
| erdm 8648 | The domain of an equivalen... |
| ercl 8649 | Elementhood in the field o... |
| ersym 8650 | An equivalence relation is... |
| ercl2 8651 | Elementhood in the field o... |
| ersymb 8652 | An equivalence relation is... |
| ertr 8653 | An equivalence relation is... |
| ertrd 8654 | A transitivity relation fo... |
| ertr2d 8655 | A transitivity relation fo... |
| ertr3d 8656 | A transitivity relation fo... |
| ertr4d 8657 | A transitivity relation fo... |
| erref 8658 | An equivalence relation is... |
| ercnv 8659 | The converse of an equival... |
| errn 8660 | The range and domain of an... |
| erssxp 8661 | An equivalence relation is... |
| erex 8662 | An equivalence relation is... |
| erexb 8663 | An equivalence relation is... |
| iserd 8664 | A reflexive, symmetric, tr... |
| iseri 8665 | A reflexive, symmetric, tr... |
| iseriALT 8666 | Alternate proof of ~ iseri... |
| brinxper 8667 | Conditions for a reflexive... |
| brdifun 8668 | Evaluate the incomparabili... |
| swoer 8669 | Incomparability under a st... |
| swoord1 8670 | The incomparability equiva... |
| swoord2 8671 | The incomparability equiva... |
| swoso 8672 | If the incomparability rel... |
| eqerlem 8673 | Lemma for ~ eqer . (Contr... |
| eqer 8674 | Equivalence relation invol... |
| ider 8675 | The identity relation is a... |
| 0er 8676 | The empty set is an equiva... |
| eceq1 8677 | Equality theorem for equiv... |
| eceq1d 8678 | Equality theorem for equiv... |
| eceq2 8679 | Equality theorem for equiv... |
| eceq2i 8680 | Equality theorem for the `... |
| eceq2d 8681 | Equality theorem for the `... |
| elecg 8682 | Membership in an equivalen... |
| ecref 8683 | All elements are in their ... |
| elec 8684 | Membership in an equivalen... |
| relelec 8685 | Membership in an equivalen... |
| elecres 8686 | Elementhood in the restric... |
| elecreseq 8687 | The restricted coset of ` ... |
| elecex 8688 | Condition for a coset to b... |
| ecss 8689 | An equivalence class is a ... |
| ecdmn0 8690 | A representative of a none... |
| ereldm 8691 | Equality of equivalence cl... |
| erth 8692 | Basic property of equivale... |
| erth2 8693 | Basic property of equivale... |
| erthi 8694 | Basic property of equivale... |
| erdisj 8695 | Equivalence classes do not... |
| ecidsn 8696 | An equivalence class modul... |
| qseq1 8697 | Equality theorem for quoti... |
| qseq2 8698 | Equality theorem for quoti... |
| qseq2i 8699 | Equality theorem for quoti... |
| qseq1d 8700 | Equality theorem for quoti... |
| qseq2d 8701 | Equality theorem for quoti... |
| qseq12 8702 | Equality theorem for quoti... |
| 0qs 8703 | Quotient set with the empt... |
| elqsg 8704 | Closed form of ~ elqs . (... |
| elqs 8705 | Membership in a quotient s... |
| elqsi 8706 | Membership in a quotient s... |
| elqsecl 8707 | Membership in a quotient s... |
| ecelqs 8708 | Membership of an equivalen... |
| ecelqsw 8709 | Membership of an equivalen... |
| ecelqsi 8710 | Membership of an equivalen... |
| ecopqsi 8711 | "Closure" law for equivale... |
| qsexg 8712 | A quotient set exists. (C... |
| qsex 8713 | A quotient set exists. (C... |
| uniqs 8714 | The union of a quotient se... |
| uniqsw 8715 | The union of a quotient se... |
| qsss 8716 | A quotient set is a set of... |
| uniqs2 8717 | The union of a quotient se... |
| snecg 8718 | The singleton of a coset i... |
| snec 8719 | The singleton of an equiva... |
| ecqs 8720 | Equivalence class in terms... |
| ecid 8721 | A set is equal to its cose... |
| qsid 8722 | A set is equal to its quot... |
| ectocld 8723 | Implicit substitution of c... |
| ectocl 8724 | Implicit substitution of c... |
| elqsn0 8725 | A quotient set does not co... |
| ecelqsdm 8726 | Membership of an equivalen... |
| ecelqsdmb 8727 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8728 | ` R ` -coset in its domain... |
| xpider 8729 | A Cartesian square is an e... |
| iiner 8730 | The intersection of a none... |
| riiner 8731 | The relative intersection ... |
| erinxp 8732 | A restricted equivalence r... |
| ecinxp 8733 | Restrict the relation in a... |
| qsinxp 8734 | Restrict the equivalence r... |
| qsdisj 8735 | Members of a quotient set ... |
| qsdisj2 8736 | A quotient set is a disjoi... |
| qsel 8737 | If an element of a quotien... |
| uniinqs 8738 | Class union distributes ov... |
| qliftlem 8739 | Lemma for theorems about a... |
| qliftrel 8740 | ` F ` , a function lift, i... |
| qliftel 8741 | Elementhood in the relatio... |
| qliftel1 8742 | Elementhood in the relatio... |
| qliftfun 8743 | The function ` F ` is the ... |
| qliftfund 8744 | The function ` F ` is the ... |
| qliftfuns 8745 | The function ` F ` is the ... |
| qliftf 8746 | The domain and codomain of... |
| qliftval 8747 | The value of the function ... |
| ecoptocl 8748 | Implicit substitution of c... |
| 2ecoptocl 8749 | Implicit substitution of c... |
| 3ecoptocl 8750 | Implicit substitution of c... |
| brecop 8751 | Binary relation on a quoti... |
| brecop2 8752 | Binary relation on a quoti... |
| eroveu 8753 | Lemma for ~ erov and ~ ero... |
| erovlem 8754 | Lemma for ~ erov and ~ ero... |
| erov 8755 | The value of an operation ... |
| eroprf 8756 | Functionality of an operat... |
| erov2 8757 | The value of an operation ... |
| eroprf2 8758 | Functionality of an operat... |
| ecopoveq 8759 | This is the first of sever... |
| ecopovsym 8760 | Assuming the operation ` F... |
| ecopovtrn 8761 | Assuming that operation ` ... |
| ecopover 8762 | Assuming that operation ` ... |
| eceqoveq 8763 | Equality of equivalence re... |
| ecovcom 8764 | Lemma used to transfer a c... |
| ecovass 8765 | Lemma used to transfer an ... |
| ecovdi 8766 | Lemma used to transfer a d... |
| mapprc 8771 | When ` A ` is a proper cla... |
| pmex 8772 | The class of all partial f... |
| mapexOLD 8773 | Obsolete version of ~ mape... |
| fnmap 8774 | Set exponentiation has a u... |
| fnpm 8775 | Partial function exponenti... |
| reldmmap 8776 | Set exponentiation is a we... |
| mapvalg 8777 | The value of set exponenti... |
| pmvalg 8778 | The value of the partial m... |
| mapval 8779 | The value of set exponenti... |
| elmapg 8780 | Membership relation for se... |
| elmapd 8781 | Deduction form of ~ elmapg... |
| elmapdd 8782 | Deduction associated with ... |
| mapdm0 8783 | The empty set is the only ... |
| elpmg 8784 | The predicate "is a partia... |
| elpm2g 8785 | The predicate "is a partia... |
| elpm2r 8786 | Sufficient condition for b... |
| elpmi 8787 | A partial function is a fu... |
| pmfun 8788 | A partial function is a fu... |
| elmapex 8789 | Eliminate antecedent for m... |
| elmapi 8790 | A mapping is a function, f... |
| mapfset 8791 | If ` B ` is a set, the val... |
| mapssfset 8792 | The value of the set expon... |
| mapfoss 8793 | The value of the set expon... |
| fsetsspwxp 8794 | The class of all functions... |
| fset0 8795 | The set of functions from ... |
| fsetdmprc0 8796 | The set of functions with ... |
| fsetex 8797 | The set of functions betwe... |
| f1setex 8798 | The set of injections betw... |
| fosetex 8799 | The set of surjections bet... |
| f1osetex 8800 | The set of bijections betw... |
| fsetfcdm 8801 | The class of functions wit... |
| fsetfocdm 8802 | The class of functions wit... |
| fsetprcnex 8803 | The class of all functions... |
| fsetcdmex 8804 | The class of all functions... |
| fsetexb 8805 | The class of all functions... |
| elmapfn 8806 | A mapping is a function wi... |
| elmapfun 8807 | A mapping is always a func... |
| elmapssres 8808 | A restricted mapping is a ... |
| elmapssresd 8809 | A restricted mapping is a ... |
| fpmg 8810 | A total function is a part... |
| pmss12g 8811 | Subset relation for the se... |
| pmresg 8812 | Elementhood of a restricte... |
| elmap 8813 | Membership relation for se... |
| mapval2 8814 | Alternate expression for t... |
| elpm 8815 | The predicate "is a partia... |
| elpm2 8816 | The predicate "is a partia... |
| fpm 8817 | A total function is a part... |
| mapsspm 8818 | Set exponentiation is a su... |
| pmsspw 8819 | Partial maps are a subset ... |
| mapsspw 8820 | Set exponentiation is a su... |
| mapfvd 8821 | The value of a function th... |
| elmapresaun 8822 | ~ fresaun transposed to ma... |
| fvmptmap 8823 | Special case of ~ fvmpt fo... |
| map0e 8824 | Set exponentiation with an... |
| map0b 8825 | Set exponentiation with an... |
| map0g 8826 | Set exponentiation is empt... |
| 0map0sn0 8827 | The set of mappings of the... |
| mapsnd 8828 | The value of set exponenti... |
| map0 8829 | Set exponentiation is empt... |
| mapsn 8830 | The value of set exponenti... |
| mapss 8831 | Subset inheritance for set... |
| fdiagfn 8832 | Functionality of the diago... |
| fvdiagfn 8833 | Functionality of the diago... |
| mapsnconst 8834 | Every singleton map is a c... |
| mapsncnv 8835 | Expression for the inverse... |
| mapsnf1o2 8836 | Explicit bijection between... |
| mapsnf1o3 8837 | Explicit bijection in the ... |
| ralxpmap 8838 | Quantification over functi... |
| dfixp 8841 | Eliminate the expression `... |
| ixpsnval 8842 | The value of an infinite C... |
| elixp2 8843 | Membership in an infinite ... |
| fvixp 8844 | Projection of a factor of ... |
| ixpfn 8845 | A nuple is a function. (C... |
| elixp 8846 | Membership in an infinite ... |
| elixpconst 8847 | Membership in an infinite ... |
| ixpconstg 8848 | Infinite Cartesian product... |
| ixpconst 8849 | Infinite Cartesian product... |
| ixpeq1 8850 | Equality theorem for infin... |
| ixpeq1d 8851 | Equality theorem for infin... |
| ss2ixp 8852 | Subclass theorem for infin... |
| ixpeq2 8853 | Equality theorem for infin... |
| ixpeq2dva 8854 | Equality theorem for infin... |
| ixpeq2dv 8855 | Equality theorem for infin... |
| cbvixp 8856 | Change bound variable in a... |
| cbvixpv 8857 | Change bound variable in a... |
| nfixpw 8858 | Bound-variable hypothesis ... |
| nfixp 8859 | Bound-variable hypothesis ... |
| nfixp1 8860 | The index variable in an i... |
| ixpprc 8861 | A cartesian product of pro... |
| ixpf 8862 | A member of an infinite Ca... |
| uniixp 8863 | The union of an infinite C... |
| ixpexg 8864 | The existence of an infini... |
| ixpin 8865 | The intersection of two in... |
| ixpiin 8866 | The indexed intersection o... |
| ixpint 8867 | The intersection of a coll... |
| ixp0x 8868 | An infinite Cartesian prod... |
| ixpssmap2g 8869 | An infinite Cartesian prod... |
| ixpssmapg 8870 | An infinite Cartesian prod... |
| 0elixp 8871 | Membership of the empty se... |
| ixpn0 8872 | The infinite Cartesian pro... |
| ixp0 8873 | The infinite Cartesian pro... |
| ixpssmap 8874 | An infinite Cartesian prod... |
| resixp 8875 | Restriction of an element ... |
| undifixp 8876 | Union of two projections o... |
| mptelixpg 8877 | Condition for an explicit ... |
| resixpfo 8878 | Restriction of elements of... |
| elixpsn 8879 | Membership in a class of s... |
| ixpsnf1o 8880 | A bijection between a clas... |
| mapsnf1o 8881 | A bijection between a set ... |
| boxriin 8882 | A rectangular subset of a ... |
| boxcutc 8883 | The relative complement of... |
| relen 8892 | Equinumerosity is a relati... |
| reldom 8893 | Dominance is a relation. ... |
| relsdom 8894 | Strict dominance is a rela... |
| encv 8895 | If two classes are equinum... |
| breng 8896 | Equinumerosity relation. ... |
| bren 8897 | Equinumerosity relation. ... |
| brdom2g 8898 | Dominance relation. This ... |
| brdomg 8899 | Dominance relation. (Cont... |
| brdomi 8900 | Dominance relation. (Cont... |
| brdom 8901 | Dominance relation. (Cont... |
| domen 8902 | Dominance in terms of equi... |
| domeng 8903 | Dominance in terms of equi... |
| ctex 8904 | A countable set is a set. ... |
| f1oen4g 8905 | The domain and range of a ... |
| f1dom4g 8906 | The domain of a one-to-one... |
| f1oen3g 8907 | The domain and range of a ... |
| f1dom3g 8908 | The domain of a one-to-one... |
| f1oen2g 8909 | The domain and range of a ... |
| f1dom2g 8910 | The domain of a one-to-one... |
| f1oeng 8911 | The domain and range of a ... |
| f1domg 8912 | The domain of a one-to-one... |
| f1oen 8913 | The domain and range of a ... |
| f1dom 8914 | The domain of a one-to-one... |
| brsdom 8915 | Strict dominance relation,... |
| isfi 8916 | Express " ` A ` is finite"... |
| enssdom 8917 | Equinumerosity implies dom... |
| enssdomOLD 8918 | Obsolete version of ~ enss... |
| dfdom2 8919 | Alternate definition of do... |
| endom 8920 | Equinumerosity implies dom... |
| sdomdom 8921 | Strict dominance implies d... |
| sdomnen 8922 | Strict dominance implies n... |
| brdom2 8923 | Dominance in terms of stri... |
| bren2 8924 | Equinumerosity expressed i... |
| enrefg 8925 | Equinumerosity is reflexiv... |
| enref 8926 | Equinumerosity is reflexiv... |
| eqeng 8927 | Equality implies equinumer... |
| domrefg 8928 | Dominance is reflexive. (... |
| en2d 8929 | Equinumerosity inference f... |
| en3d 8930 | Equinumerosity inference f... |
| en2i 8931 | Equinumerosity inference f... |
| en3i 8932 | Equinumerosity inference f... |
| dom2lem 8933 | A mapping (first hypothesi... |
| dom2d 8934 | A mapping (first hypothesi... |
| dom3d 8935 | A mapping (first hypothesi... |
| dom2 8936 | A mapping (first hypothesi... |
| dom3 8937 | A mapping (first hypothesi... |
| idssen 8938 | Equality implies equinumer... |
| domssl 8939 | If ` A ` is a subset of ` ... |
| domssr 8940 | If ` C ` is a superset of ... |
| ssdomg 8941 | A set dominates its subset... |
| ener 8942 | Equinumerosity is an equiv... |
| ensymb 8943 | Symmetry of equinumerosity... |
| ensym 8944 | Symmetry of equinumerosity... |
| ensymi 8945 | Symmetry of equinumerosity... |
| ensymd 8946 | Symmetry of equinumerosity... |
| entr 8947 | Transitivity of equinumero... |
| domtr 8948 | Transitivity of dominance ... |
| entri 8949 | A chained equinumerosity i... |
| entr2i 8950 | A chained equinumerosity i... |
| entr3i 8951 | A chained equinumerosity i... |
| entr4i 8952 | A chained equinumerosity i... |
| endomtr 8953 | Transitivity of equinumero... |
| domentr 8954 | Transitivity of dominance ... |
| f1imaeng 8955 | If a function is one-to-on... |
| f1imaen2g 8956 | If a function is one-to-on... |
| f1imaen3g 8957 | If a set function is one-t... |
| f1imaen 8958 | If a function is one-to-on... |
| en0 8959 | The empty set is equinumer... |
| en0ALT 8960 | Shorter proof of ~ en0 , d... |
| en0r 8961 | The empty set is equinumer... |
| ensn1 8962 | A singleton is equinumerou... |
| ensn1g 8963 | A singleton is equinumerou... |
| enpr1g 8964 | ` { A , A } ` has only one... |
| en1 8965 | A set is equinumerous to o... |
| en1b 8966 | A set is equinumerous to o... |
| reuen1 8967 | Two ways to express "exact... |
| euen1 8968 | Two ways to express "exact... |
| euen1b 8969 | Two ways to express " ` A ... |
| en1uniel 8970 | A singleton contains its s... |
| 2dom 8971 | A set that dominates ordin... |
| fundmen 8972 | A function is equinumerous... |
| fundmeng 8973 | A function is equinumerous... |
| cnven 8974 | A relational set is equinu... |
| cnvct 8975 | If a set is countable, so ... |
| fndmeng 8976 | A function is equinumerate... |
| mapsnend 8977 | Set exponentiation to a si... |
| mapsnen 8978 | Set exponentiation to a si... |
| snmapen 8979 | Set exponentiation: a sing... |
| snmapen1 8980 | Set exponentiation: a sing... |
| map1 8981 | Set exponentiation: ordina... |
| en2sn 8982 | Two singletons are equinum... |
| 0fi 8983 | The empty set is finite. ... |
| snfi 8984 | A singleton is finite. (C... |
| fiprc 8985 | The class of finite sets i... |
| unen 8986 | Equinumerosity of union of... |
| enrefnn 8987 | Equinumerosity is reflexiv... |
| en2prd 8988 | Two proper unordered pairs... |
| enpr2d 8989 | A pair with distinct eleme... |
| ssct 8990 | Any subset of a countable ... |
| difsnen 8991 | All decrements of a set ar... |
| domdifsn 8992 | Dominance over a set with ... |
| xpsnen 8993 | A set is equinumerous to i... |
| xpsneng 8994 | A set is equinumerous to i... |
| xp1en 8995 | One times a cardinal numbe... |
| endisj 8996 | Any two sets are equinumer... |
| undom 8997 | Dominance law for union. ... |
| xpcomf1o 8998 | The canonical bijection fr... |
| xpcomco 8999 | Composition with the bijec... |
| xpcomen 9000 | Commutative law for equinu... |
| xpcomeng 9001 | Commutative law for equinu... |
| xpsnen2g 9002 | A set is equinumerous to i... |
| xpassen 9003 | Associative law for equinu... |
| xpdom2 9004 | Dominance law for Cartesia... |
| xpdom2g 9005 | Dominance law for Cartesia... |
| xpdom1g 9006 | Dominance law for Cartesia... |
| xpdom3 9007 | A set is dominated by its ... |
| xpdom1 9008 | Dominance law for Cartesia... |
| domunsncan 9009 | A singleton cancellation l... |
| omxpenlem 9010 | Lemma for ~ omxpen . (Con... |
| omxpen 9011 | The cardinal and ordinal p... |
| omf1o 9012 | Construct an explicit bije... |
| pw2f1olem 9013 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9014 | The power set of a set is ... |
| pw2eng 9015 | The power set of a set is ... |
| pw2en 9016 | The power set of a set is ... |
| fopwdom 9017 | Covering implies injection... |
| enfixsn 9018 | Given two equipollent sets... |
| sbthlem1 9019 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9020 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9021 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9022 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9023 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9024 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9025 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9026 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9027 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9028 | Lemma for ~ sbth . (Contr... |
| sbth 9029 | Schroeder-Bernstein Theore... |
| sbthb 9030 | Schroeder-Bernstein Theore... |
| sbthcl 9031 | Schroeder-Bernstein Theore... |
| dfsdom2 9032 | Alternate definition of st... |
| brsdom2 9033 | Alternate definition of st... |
| sdomnsym 9034 | Strict dominance is asymme... |
| domnsym 9035 | Theorem 22(i) of [Suppes] ... |
| 0domg 9036 | Any set dominates the empt... |
| dom0 9037 | A set dominated by the emp... |
| 0sdomg 9038 | A set strictly dominates t... |
| 0dom 9039 | Any set dominates the empt... |
| 0sdom 9040 | A set strictly dominates t... |
| sdom0 9041 | The empty set does not str... |
| sdomdomtr 9042 | Transitivity of strict dom... |
| sdomentr 9043 | Transitivity of strict dom... |
| domsdomtr 9044 | Transitivity of dominance ... |
| ensdomtr 9045 | Transitivity of equinumero... |
| sdomirr 9046 | Strict dominance is irrefl... |
| sdomtr 9047 | Strict dominance is transi... |
| sdomn2lp 9048 | Strict dominance has no 2-... |
| enen1 9049 | Equality-like theorem for ... |
| enen2 9050 | Equality-like theorem for ... |
| domen1 9051 | Equality-like theorem for ... |
| domen2 9052 | Equality-like theorem for ... |
| sdomen1 9053 | Equality-like theorem for ... |
| sdomen2 9054 | Equality-like theorem for ... |
| domtriord 9055 | Dominance is trichotomous ... |
| sdomel 9056 | For ordinals, strict domin... |
| sdomdif 9057 | The difference of a set fr... |
| onsdominel 9058 | An ordinal with more eleme... |
| domunsn 9059 | Dominance over a set with ... |
| fodomr 9060 | There exists a mapping fro... |
| pwdom 9061 | Injection of sets implies ... |
| canth2 9062 | Cantor's Theorem. No set ... |
| canth2g 9063 | Cantor's theorem with the ... |
| 2pwuninel 9064 | The power set of the power... |
| 2pwne 9065 | No set equals the power se... |
| disjen 9066 | A stronger form of ~ pwuni... |
| disjenex 9067 | Existence version of ~ dis... |
| domss2 9068 | A corollary of ~ disjenex ... |
| domssex2 9069 | A corollary of ~ disjenex ... |
| domssex 9070 | Weakening of ~ domssex2 to... |
| xpf1o 9071 | Construct a bijection on a... |
| xpen 9072 | Equinumerosity law for Car... |
| mapen 9073 | Two set exponentiations ar... |
| mapdom1 9074 | Order-preserving property ... |
| mapxpen 9075 | Equinumerosity law for dou... |
| xpmapenlem 9076 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9077 | Equinumerosity law for set... |
| mapunen 9078 | Equinumerosity law for set... |
| map2xp 9079 | A cardinal power with expo... |
| mapdom2 9080 | Order-preserving property ... |
| mapdom3 9081 | Set exponentiation dominat... |
| pwen 9082 | If two sets are equinumero... |
| ssenen 9083 | Equinumerosity of equinume... |
| limenpsi 9084 | A limit ordinal is equinum... |
| limensuci 9085 | A limit ordinal is equinum... |
| limensuc 9086 | A limit ordinal is equinum... |
| infensuc 9087 | Any infinite ordinal is eq... |
| dif1enlem 9088 | Lemma for ~ rexdif1en and ... |
| rexdif1en 9089 | If a set is equinumerous t... |
| dif1en 9090 | If a set ` A ` is equinume... |
| dif1ennn 9091 | If a set ` A ` is equinume... |
| findcard 9092 | Schema for induction on th... |
| findcard2 9093 | Schema for induction on th... |
| findcard2s 9094 | Variation of ~ findcard2 r... |
| findcard2d 9095 | Deduction version of ~ fin... |
| nnfi 9096 | Natural numbers are finite... |
| pssnn 9097 | A proper subset of a natur... |
| ssnnfi 9098 | A subset of a natural numb... |
| unfi 9099 | The union of two finite se... |
| unfid 9100 | The union of two finite se... |
| ssfi 9101 | A subset of a finite set i... |
| ssfiALT 9102 | Shorter proof of ~ ssfi us... |
| diffi 9103 | If ` A ` is finite, ` ( A ... |
| cnvfi 9104 | If a set is finite, its co... |
| pwssfi 9105 | Every element of the power... |
| fnfi 9106 | A version of ~ fnex for fi... |
| f1oenfi 9107 | If the domain of a one-to-... |
| f1oenfirn 9108 | If the range of a one-to-o... |
| f1domfi 9109 | If the codomain of a one-t... |
| f1domfi2 9110 | If the domain of a one-to-... |
| enreffi 9111 | Equinumerosity is reflexiv... |
| ensymfib 9112 | Symmetry of equinumerosity... |
| entrfil 9113 | Transitivity of equinumero... |
| enfii 9114 | A set equinumerous to a fi... |
| enfi 9115 | Equinumerous sets have the... |
| enfiALT 9116 | Shorter proof of ~ enfi us... |
| domfi 9117 | A set dominated by a finit... |
| entrfi 9118 | Transitivity of equinumero... |
| entrfir 9119 | Transitivity of equinumero... |
| domtrfil 9120 | Transitivity of dominance ... |
| domtrfi 9121 | Transitivity of dominance ... |
| domtrfir 9122 | Transitivity of dominance ... |
| f1imaenfi 9123 | If a function is one-to-on... |
| ssdomfi 9124 | A finite set dominates its... |
| ssdomfi2 9125 | A set dominates its finite... |
| sbthfilem 9126 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9127 | Schroeder-Bernstein Theore... |
| domnsymfi 9128 | If a set dominates a finit... |
| sdomdomtrfi 9129 | Transitivity of strict dom... |
| domsdomtrfi 9130 | Transitivity of dominance ... |
| sucdom2 9131 | Strict dominance of a set ... |
| phplem1 9132 | Lemma for Pigeonhole Princ... |
| phplem2 9133 | Lemma for Pigeonhole Princ... |
| nneneq 9134 | Two equinumerous natural n... |
| php 9135 | Pigeonhole Principle. A n... |
| php2 9136 | Corollary of Pigeonhole Pr... |
| php3 9137 | Corollary of Pigeonhole Pr... |
| php4 9138 | Corollary of the Pigeonhol... |
| php5 9139 | Corollary of the Pigeonhol... |
| phpeqd 9140 | Corollary of the Pigeonhol... |
| nndomog 9141 | Cardinal ordering agrees w... |
| onomeneq 9142 | An ordinal number equinume... |
| onfin 9143 | An ordinal number is finit... |
| ordfin 9144 | A generalization of ~ onfi... |
| onfin2 9145 | A set is a natural number ... |
| nndomo 9146 | Cardinal ordering agrees w... |
| nnsdomo 9147 | Cardinal ordering agrees w... |
| sucdom 9148 | Strict dominance of a set ... |
| snnen2o 9149 | A singleton ` { A } ` is n... |
| 0sdom1dom 9150 | Strict dominance over 0 is... |
| 0sdom1domALT 9151 | Alternate proof of ~ 0sdom... |
| 1sdom2 9152 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9153 | Alternate proof of ~ 1sdom... |
| sdom1 9154 | A set has less than one me... |
| modom 9155 | Two ways to express "at mo... |
| modom2 9156 | Two ways to express "at mo... |
| rex2dom 9157 | A set that has at least 2 ... |
| 1sdom2dom 9158 | Strict dominance over 1 is... |
| 1sdom 9159 | A set that strictly domina... |
| unxpdomlem1 9160 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9161 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9162 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9163 | Cartesian product dominate... |
| unxpdom2 9164 | Corollary of ~ unxpdom . ... |
| sucxpdom 9165 | Cartesian product dominate... |
| pssinf 9166 | A set equinumerous to a pr... |
| fisseneq 9167 | A finite set is equal to i... |
| ominf 9168 | The set of natural numbers... |
| isinf 9169 | Any set that is not finite... |
| fineqvlem 9170 | Lemma for ~ fineqv . (Con... |
| fineqv 9171 | If the Axiom of Infinity i... |
| xpfir 9172 | The components of a nonemp... |
| ssfid 9173 | A subset of a finite set i... |
| infi 9174 | The intersection of two se... |
| rabfi 9175 | A restricted class built f... |
| finresfin 9176 | The restriction of a finit... |
| f1finf1o 9177 | Any injection from one fin... |
| nfielex 9178 | If a class is not finite, ... |
| en1eqsn 9179 | A set with one element is ... |
| en1eqsnbi 9180 | A set containing an elemen... |
| dif1ennnALT 9181 | Alternate proof of ~ dif1e... |
| enp1ilem 9182 | Lemma for uses of ~ enp1i ... |
| enp1i 9183 | Proof induction for ~ en2 ... |
| en2 9184 | A set equinumerous to ordi... |
| en3 9185 | A set equinumerous to ordi... |
| en4 9186 | A set equinumerous to ordi... |
| findcard3 9187 | Schema for strong inductio... |
| ac6sfi 9188 | A version of ~ ac6s for fi... |
| frfi 9189 | A partial order is well-fo... |
| fimax2g 9190 | A finite set has a maximum... |
| fimaxg 9191 | A finite set has a maximum... |
| fisupg 9192 | Lemma showing existence an... |
| wofi 9193 | A total order on a finite ... |
| ordunifi 9194 | The maximum of a finite co... |
| nnunifi 9195 | The union (supremum) of a ... |
| unblem1 9196 | Lemma for ~ unbnn . After... |
| unblem2 9197 | Lemma for ~ unbnn . The v... |
| unblem3 9198 | Lemma for ~ unbnn . The v... |
| unblem4 9199 | Lemma for ~ unbnn . The f... |
| unbnn 9200 | Any unbounded subset of na... |
| unbnn2 9201 | Version of ~ unbnn that do... |
| isfinite2 9202 | Any set strictly dominated... |
| nnsdomg 9203 | Omega strictly dominates a... |
| isfiniteg 9204 | A set is finite iff it is ... |
| infsdomnn 9205 | An infinite set strictly d... |
| infn0 9206 | An infinite set is not emp... |
| infn0ALT 9207 | Shorter proof of ~ infn0 u... |
| fin2inf 9208 | This (useless) theorem, wh... |
| unfilem1 9209 | Lemma for proving that the... |
| unfilem2 9210 | Lemma for proving that the... |
| unfilem3 9211 | Lemma for proving that the... |
| unfir 9212 | If a union is finite, the ... |
| unfib 9213 | A union is finite if and o... |
| unfi2 9214 | The union of two finite se... |
| difinf 9215 | An infinite set ` A ` minu... |
| fodomfi 9216 | An onto function implies d... |
| fofi 9217 | If an onto function has a ... |
| f1fi 9218 | If a 1-to-1 function has a... |
| imafi 9219 | Images of finite sets are ... |
| imafiOLD 9220 | Obsolete version of ~ imaf... |
| pwfir 9221 | If the power set of a set ... |
| pwfilem 9222 | Lemma for ~ pwfi . (Contr... |
| pwfi 9223 | The power set of a finite ... |
| xpfi 9224 | The Cartesian product of t... |
| 3xpfi 9225 | The Cartesian product of t... |
| domunfican 9226 | A finite set union cancell... |
| infcntss 9227 | Every infinite set has a d... |
| prfi 9228 | An unordered pair is finit... |
| prfiALT 9229 | Shorter proof of ~ prfi us... |
| tpfi 9230 | An unordered triple is fin... |
| fiint 9231 | Equivalent ways of stating... |
| fodomfir 9232 | There exists a mapping fro... |
| fodomfib 9233 | Equivalence of an onto map... |
| fodomfiOLD 9234 | Obsolete version of ~ fodo... |
| fodomfibOLD 9235 | Obsolete version of ~ fodo... |
| fofinf1o 9236 | Any surjection from one fi... |
| rneqdmfinf1o 9237 | Any function from a finite... |
| fidomdm 9238 | Any finite set dominates i... |
| dmfi 9239 | The domain of a finite set... |
| fundmfibi 9240 | A function is finite if an... |
| resfnfinfin 9241 | The restriction of a funct... |
| residfi 9242 | A restricted identity func... |
| cnvfiALT 9243 | Shorter proof of ~ cnvfi u... |
| rnfi 9244 | The range of a finite set ... |
| f1dmvrnfibi 9245 | A one-to-one function whos... |
| f1vrnfibi 9246 | A one-to-one function whic... |
| iunfi 9247 | The finite union of finite... |
| unifi 9248 | The finite union of finite... |
| unifi2 9249 | The finite union of finite... |
| infssuni 9250 | If an infinite set ` A ` i... |
| unirnffid 9251 | The union of the range of ... |
| mapfi 9252 | Set exponentiation of fini... |
| ixpfi 9253 | A Cartesian product of fin... |
| ixpfi2 9254 | A Cartesian product of fin... |
| mptfi 9255 | A finite mapping set is fi... |
| abrexfi 9256 | An image set from a finite... |
| cnvimamptfin 9257 | A preimage of a mapping wi... |
| elfpw 9258 | Membership in a class of f... |
| unifpw 9259 | A set is the union of its ... |
| f1opwfi 9260 | A one-to-one mapping induc... |
| fissuni 9261 | A finite subset of a union... |
| fipreima 9262 | Given a finite subset ` A ... |
| finsschain 9263 | A finite subset of the uni... |
| indexfi 9264 | If for every element of a ... |
| imafi2 9265 | The image by a finite set ... |
| unifi3 9266 | If a union is finite, then... |
| tfsnfin2 9267 | A transfinite sequence is ... |
| relfsupp 9270 | The property of a function... |
| relprcnfsupp 9271 | A proper class is never fi... |
| isfsupp 9272 | The property of a class to... |
| isfsuppd 9273 | Deduction form of ~ isfsup... |
| funisfsupp 9274 | The property of a function... |
| fsuppimp 9275 | Implications of a class be... |
| fsuppimpd 9276 | A finitely supported funct... |
| fsuppfund 9277 | A finitely supported funct... |
| fisuppfi 9278 | A function on a finite set... |
| fidmfisupp 9279 | A function with a finite d... |
| finnzfsuppd 9280 | If a function is zero outs... |
| fdmfisuppfi 9281 | The support of a function ... |
| fdmfifsupp 9282 | A function with a finite d... |
| fsuppmptdm 9283 | A mapping with a finite do... |
| fndmfisuppfi 9284 | The support of a function ... |
| fndmfifsupp 9285 | A function with a finite d... |
| suppeqfsuppbi 9286 | If two functions have the ... |
| suppssfifsupp 9287 | If the support of a functi... |
| fsuppsssupp 9288 | If the support of a functi... |
| fsuppsssuppgd 9289 | If the support of a functi... |
| fsuppss 9290 | A subset of a finitely sup... |
| fsuppssov1 9291 | Formula building theorem f... |
| fsuppxpfi 9292 | The cartesian product of t... |
| fczfsuppd 9293 | A constant function with v... |
| fsuppun 9294 | The union of two finitely ... |
| fsuppunfi 9295 | The union of the support o... |
| fsuppunbi 9296 | If the union of two classe... |
| 0fsupp 9297 | The empty set is a finitel... |
| snopfsupp 9298 | A singleton containing an ... |
| funsnfsupp 9299 | Finite support for a funct... |
| fsuppres 9300 | The restriction of a finit... |
| fmptssfisupp 9301 | The restriction of a mappi... |
| ressuppfi 9302 | If the support of the rest... |
| resfsupp 9303 | If the restriction of a fu... |
| resfifsupp 9304 | The restriction of a funct... |
| ffsuppbi 9305 | Two ways of saying that a ... |
| fsuppmptif 9306 | A function mapping an argu... |
| sniffsupp 9307 | A function mapping all but... |
| fsuppcolem 9308 | Lemma for ~ fsuppco . For... |
| fsuppco 9309 | The composition of a 1-1 f... |
| fsuppco2 9310 | The composition of a funct... |
| fsuppcor 9311 | The composition of a funct... |
| mapfienlem1 9312 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9313 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9314 | Lemma 3 for ~ mapfien . (... |
| mapfien 9315 | A bijection of the base se... |
| mapfien2 9316 | Equinumerousity relation f... |
| fival 9319 | The set of all the finite ... |
| elfi 9320 | Specific properties of an ... |
| elfi2 9321 | The empty intersection nee... |
| elfir 9322 | Sufficient condition for a... |
| intrnfi 9323 | Sufficient condition for t... |
| iinfi 9324 | An indexed intersection of... |
| inelfi 9325 | The intersection of two se... |
| ssfii 9326 | Any element of a set ` A `... |
| fi0 9327 | The set of finite intersec... |
| fieq0 9328 | A set is empty iff the cla... |
| fiin 9329 | The elements of ` ( fi `` ... |
| dffi2 9330 | The set of finite intersec... |
| fiss 9331 | Subset relationship for fu... |
| inficl 9332 | A set which is closed unde... |
| fipwuni 9333 | The set of finite intersec... |
| fisn 9334 | A singleton is closed unde... |
| fiuni 9335 | The union of the finite in... |
| fipwss 9336 | If a set is a family of su... |
| elfiun 9337 | A finite intersection of e... |
| dffi3 9338 | The set of finite intersec... |
| fifo 9339 | Describe a surjection from... |
| marypha1lem 9340 | Core induction for Philip ... |
| marypha1 9341 | (Philip) Hall's marriage t... |
| marypha2lem1 9342 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9343 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9344 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9345 | Lemma for ~ marypha2 . Pr... |
| marypha2 9346 | Version of ~ marypha1 usin... |
| dfsup2 9351 | Quantifier-free definition... |
| supeq1 9352 | Equality theorem for supre... |
| supeq1d 9353 | Equality deduction for sup... |
| supeq1i 9354 | Equality inference for sup... |
| supeq2 9355 | Equality theorem for supre... |
| supeq3 9356 | Equality theorem for supre... |
| supeq123d 9357 | Equality deduction for sup... |
| nfsup 9358 | Hypothesis builder for sup... |
| supmo 9359 | Any class ` B ` has at mos... |
| supexd 9360 | A supremum is a set. (Con... |
| supeu 9361 | A supremum is unique. Sim... |
| supval2 9362 | Alternate expression for t... |
| eqsup 9363 | Sufficient condition for a... |
| eqsupd 9364 | Sufficient condition for a... |
| supcl 9365 | A supremum belongs to its ... |
| supub 9366 | A supremum is an upper bou... |
| suplub 9367 | A supremum is the least up... |
| suplub2 9368 | Bidirectional form of ~ su... |
| supnub 9369 | An upper bound is not less... |
| supssd 9370 | Inequality deduction for s... |
| supex 9371 | A supremum is a set. (Con... |
| sup00 9372 | The supremum under an empt... |
| sup0riota 9373 | The supremum of an empty s... |
| sup0 9374 | The supremum of an empty s... |
| supmax 9375 | The greatest element of a ... |
| fisup2g 9376 | A finite set satisfies the... |
| fisupcl 9377 | A nonempty finite set cont... |
| supgtoreq 9378 | The supremum of a finite s... |
| suppr 9379 | The supremum of a pair. (... |
| supsn 9380 | The supremum of a singleto... |
| supisolem 9381 | Lemma for ~ supiso . (Con... |
| supisoex 9382 | Lemma for ~ supiso . (Con... |
| supiso 9383 | Image of a supremum under ... |
| infeq1 9384 | Equality theorem for infim... |
| infeq1d 9385 | Equality deduction for inf... |
| infeq1i 9386 | Equality inference for inf... |
| infeq2 9387 | Equality theorem for infim... |
| infeq3 9388 | Equality theorem for infim... |
| infeq123d 9389 | Equality deduction for inf... |
| nfinf 9390 | Hypothesis builder for inf... |
| infexd 9391 | An infimum is a set. (Con... |
| eqinf 9392 | Sufficient condition for a... |
| eqinfd 9393 | Sufficient condition for a... |
| infval 9394 | Alternate expression for t... |
| infcllem 9395 | Lemma for ~ infcl , ~ infl... |
| infcl 9396 | An infimum belongs to its ... |
| inflb 9397 | An infimum is a lower boun... |
| infglb 9398 | An infimum is the greatest... |
| infglbb 9399 | Bidirectional form of ~ in... |
| infnlb 9400 | A lower bound is not great... |
| infssd 9401 | Inequality deduction for i... |
| infex 9402 | An infimum is a set. (Con... |
| infmin 9403 | The smallest element of a ... |
| infmo 9404 | Any class ` B ` has at mos... |
| infeu 9405 | An infimum is unique. (Co... |
| fimin2g 9406 | A finite set has a minimum... |
| fiming 9407 | A finite set has a minimum... |
| fiinfg 9408 | Lemma showing existence an... |
| fiinf2g 9409 | A finite set satisfies the... |
| fiinfcl 9410 | A nonempty finite set cont... |
| infltoreq 9411 | The infimum of a finite se... |
| infpr 9412 | The infimum of a pair. (C... |
| infsupprpr 9413 | The infimum of a proper pa... |
| infsn 9414 | The infimum of a singleton... |
| inf00 9415 | The infimum regarding an e... |
| infempty 9416 | The infimum of an empty se... |
| infiso 9417 | Image of an infimum under ... |
| dfoi 9420 | Rewrite ~ df-oi with abbre... |
| oieq1 9421 | Equality theorem for ordin... |
| oieq2 9422 | Equality theorem for ordin... |
| nfoi 9423 | Hypothesis builder for ord... |
| ordiso2 9424 | Generalize ~ ordiso to pro... |
| ordiso 9425 | Order-isomorphic ordinal n... |
| ordtypecbv 9426 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9427 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9428 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9429 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9430 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9431 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9432 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9433 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9434 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9435 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9436 | Lemma for ~ ordtype . Usi... |
| oi0 9437 | Definition of the ordinal ... |
| oicl 9438 | The order type of the well... |
| oif 9439 | The order isomorphism of t... |
| oiiso2 9440 | The order isomorphism of t... |
| ordtype 9441 | For any set-like well-orde... |
| oiiniseg 9442 | ` ran F ` is an initial se... |
| ordtype2 9443 | For any set-like well-orde... |
| oiexg 9444 | The order isomorphism on a... |
| oion 9445 | The order type of the well... |
| oiiso 9446 | The order isomorphism of t... |
| oien 9447 | The order type of a well-o... |
| oieu 9448 | Uniqueness of the unique o... |
| oismo 9449 | When ` A ` is a subclass o... |
| oiid 9450 | The order type of an ordin... |
| hartogslem1 9451 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9452 | Lemma for ~ hartogs . (Co... |
| hartogs 9453 | The class of ordinals domi... |
| wofib 9454 | The only sets which are we... |
| wemaplem1 9455 | Value of the lexicographic... |
| wemaplem2 9456 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9457 | Lemma for ~ wemapso . Tra... |
| wemappo 9458 | Construct lexicographic or... |
| wemapsolem 9459 | Lemma for ~ wemapso . (Co... |
| wemapso 9460 | Construct lexicographic or... |
| wemapso2lem 9461 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9462 | An alternative to having a... |
| card2on 9463 | The alternate definition o... |
| card2inf 9464 | The alternate definition o... |
| harf 9467 | Functionality of the Harto... |
| harcl 9468 | Values of the Hartogs func... |
| harval 9469 | Function value of the Hart... |
| elharval 9470 | The Hartogs number of a se... |
| harndom 9471 | The Hartogs number of a se... |
| harword 9472 | Weak ordering property of ... |
| relwdom 9475 | Weak dominance is a relati... |
| brwdom 9476 | Property of weak dominance... |
| brwdomi 9477 | Property of weak dominance... |
| brwdomn0 9478 | Weak dominance over nonemp... |
| 0wdom 9479 | Any set weakly dominates t... |
| fowdom 9480 | An onto function implies w... |
| wdomref 9481 | Reflexivity of weak domina... |
| brwdom2 9482 | Alternate characterization... |
| domwdom 9483 | Weak dominance is implied ... |
| wdomtr 9484 | Transitivity of weak domin... |
| wdomen1 9485 | Equality-like theorem for ... |
| wdomen2 9486 | Equality-like theorem for ... |
| wdompwdom 9487 | Weak dominance strengthens... |
| canthwdom 9488 | Cantor's Theorem, stated u... |
| wdom2d 9489 | Deduce weak dominance from... |
| wdomd 9490 | Deduce weak dominance from... |
| brwdom3 9491 | Condition for weak dominan... |
| brwdom3i 9492 | Weak dominance implies exi... |
| unwdomg 9493 | Weak dominance of a (disjo... |
| xpwdomg 9494 | Weak dominance of a Cartes... |
| wdomima2g 9495 | A set is weakly dominant o... |
| wdomimag 9496 | A set is weakly dominant o... |
| unxpwdom2 9497 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9498 | If a Cartesian product is ... |
| ixpiunwdom 9499 | Describe an onto function ... |
| harwdom 9500 | The value of the Hartogs f... |
| axreg2 9502 | Axiom of Regularity expres... |
| zfregcl 9503 | The Axiom of Regularity wi... |
| zfregclOLD 9504 | Obsolete version of ~ zfre... |
| zfreg 9505 | The Axiom of Regularity us... |
| elirrv 9506 | The membership relation is... |
| elirrvOLD 9507 | Obsolete version of ~ elir... |
| elirrvOLDOLD 9508 | Obsolete version of ~ elir... |
| elirr 9509 | No class is a member of it... |
| elneq 9510 | A class is not equal to an... |
| nelaneq 9511 | A class is not an element ... |
| nelaneqOLD 9512 | Obsolete version of ~ nela... |
| nelaneqOLDOLD 9513 | Obsolete version of ~ nela... |
| epinid0 9514 | The membership relation an... |
| sucprcreg 9515 | A class is equal to its su... |
| sucprcregOLD 9516 | Obsolete version of ~ sucp... |
| ruv 9517 | The Russell class is equal... |
| ruALT 9518 | Alternate proof of ~ ru , ... |
| disjcsn 9519 | A class is disjoint from i... |
| zfregfr 9520 | The membership relation is... |
| elirrvALT 9521 | Alternate proof of ~ elirr... |
| en2lp 9522 | No class has 2-cycle membe... |
| elnanel 9523 | Two classes are not elemen... |
| cnvepnep 9524 | The membership (epsilon) r... |
| epnsym 9525 | The membership (epsilon) r... |
| elnotel 9526 | A class cannot be an eleme... |
| elnel 9527 | A class cannot be an eleme... |
| en3lplem1 9528 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9529 | Lemma for ~ en3lp . (Cont... |
| en3lp 9530 | No class has 3-cycle membe... |
| preleqg 9531 | Equality of two unordered ... |
| preleq 9532 | Equality of two unordered ... |
| preleqALT 9533 | Alternate proof of ~ prele... |
| opthreg 9534 | Theorem for alternate repr... |
| suc11reg 9535 | The successor operation be... |
| dford2 9536 | Assuming ~ ax-reg , an ord... |
| inf0 9537 | Existence of ` _om ` impli... |
| inf1 9538 | Variation of Axiom of Infi... |
| inf2 9539 | Variation of Axiom of Infi... |
| inf3lema 9540 | Lemma for our Axiom of Inf... |
| inf3lemb 9541 | Lemma for our Axiom of Inf... |
| inf3lemc 9542 | Lemma for our Axiom of Inf... |
| inf3lemd 9543 | Lemma for our Axiom of Inf... |
| inf3lem1 9544 | Lemma for our Axiom of Inf... |
| inf3lem2 9545 | Lemma for our Axiom of Inf... |
| inf3lem3 9546 | Lemma for our Axiom of Inf... |
| inf3lem4 9547 | Lemma for our Axiom of Inf... |
| inf3lem5 9548 | Lemma for our Axiom of Inf... |
| inf3lem6 9549 | Lemma for our Axiom of Inf... |
| inf3lem7 9550 | Lemma for our Axiom of Inf... |
| inf3 9551 | Our Axiom of Infinity ~ ax... |
| infeq5i 9552 | Half of ~ infeq5 . (Contr... |
| infeq5 9553 | The statement "there exist... |
| zfinf 9555 | Axiom of Infinity expresse... |
| axinf2 9556 | A standard version of Axio... |
| zfinf2 9558 | A standard version of the ... |
| omex 9559 | The existence of omega (th... |
| axinf 9560 | The first version of the A... |
| inf5 9561 | The statement "there exist... |
| omelon 9562 | Omega is an ordinal number... |
| dfom3 9563 | The class of natural numbe... |
| elom3 9564 | A simplification of ~ elom... |
| dfom4 9565 | A simplification of ~ df-o... |
| dfom5 9566 | ` _om ` is the smallest li... |
| oancom 9567 | Ordinal addition is not co... |
| isfinite 9568 | A set is finite iff it is ... |
| fict 9569 | A finite set is countable ... |
| nnsdom 9570 | A natural number is strict... |
| omenps 9571 | Omega is equinumerous to a... |
| omensuc 9572 | The set of natural numbers... |
| infdifsn 9573 | Removing a singleton from ... |
| infdiffi 9574 | Removing a finite set from... |
| unbnn3 9575 | Any unbounded subset of na... |
| noinfep 9576 | Using the Axiom of Regular... |
| cantnffval 9579 | The value of the Cantor no... |
| cantnfdm 9580 | The domain of the Cantor n... |
| cantnfvalf 9581 | Lemma for ~ cantnf . The ... |
| cantnfs 9582 | Elementhood in the set of ... |
| cantnfcl 9583 | Basic properties of the or... |
| cantnfval 9584 | The value of the Cantor no... |
| cantnfval2 9585 | Alternate expression for t... |
| cantnfsuc 9586 | The value of the recursive... |
| cantnfle 9587 | A lower bound on the ` CNF... |
| cantnflt 9588 | An upper bound on the part... |
| cantnflt2 9589 | An upper bound on the ` CN... |
| cantnff 9590 | The ` CNF ` function is a ... |
| cantnf0 9591 | The value of the zero func... |
| cantnfrescl 9592 | A function is finitely sup... |
| cantnfres 9593 | The ` CNF ` function respe... |
| cantnfp1lem1 9594 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9595 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9596 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9597 | If ` F ` is created by add... |
| oemapso 9598 | The relation ` T ` is a st... |
| oemapval 9599 | Value of the relation ` T ... |
| oemapvali 9600 | If ` F < G ` , then there ... |
| cantnflem1a 9601 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9602 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9603 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9604 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9605 | Lemma for ~ cantnf . This... |
| cantnflem2 9606 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9607 | Lemma for ~ cantnf . Here... |
| cantnflem4 9608 | Lemma for ~ cantnf . Comp... |
| cantnf 9609 | The Cantor Normal Form the... |
| oemapwe 9610 | The lexicographic order on... |
| cantnffval2 9611 | An alternate definition of... |
| cantnff1o 9612 | Simplify the isomorphism o... |
| wemapwe 9613 | Construct lexicographic or... |
| oef1o 9614 | A bijection of the base se... |
| cnfcomlem 9615 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9616 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9617 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9618 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9619 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9620 | Any infinite ordinal ` B `... |
| cnfcom3clem 9621 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9622 | Wrap the construction of ~... |
| ttrcleq 9625 | Equality theorem for trans... |
| nfttrcld 9626 | Bound variable hypothesis ... |
| nfttrcl 9627 | Bound variable hypothesis ... |
| relttrcl 9628 | The transitive closure of ... |
| brttrcl 9629 | Characterization of elemen... |
| brttrcl2 9630 | Characterization of elemen... |
| ssttrcl 9631 | If ` R ` is a relation, th... |
| ttrcltr 9632 | The transitive closure of ... |
| ttrclresv 9633 | The transitive closure of ... |
| ttrclco 9634 | Composition law for the tr... |
| cottrcl 9635 | Composition law for the tr... |
| ttrclss 9636 | If ` R ` is a subclass of ... |
| dmttrcl 9637 | The domain of a transitive... |
| rnttrcl 9638 | The range of a transitive ... |
| ttrclexg 9639 | If ` R ` is a set, then so... |
| dfttrcl2 9640 | When ` R ` is a set and a ... |
| ttrclselem1 9641 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9642 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9643 | If ` R ` is set-like over ... |
| trcl 9644 | For any set ` A ` , show t... |
| tz9.1 9645 | Every set has a transitive... |
| tz9.1c 9646 | Alternate expression for t... |
| epfrs 9647 | The strong form of the Axi... |
| zfregs 9648 | The strong form of the Axi... |
| zfregs2 9649 | Alternate strong form of t... |
| tcvalg 9652 | Value of the transitive cl... |
| tcid 9653 | Defining property of the t... |
| tctr 9654 | Defining property of the t... |
| tcmin 9655 | Defining property of the t... |
| tc2 9656 | A variant of the definitio... |
| tcsni 9657 | The transitive closure of ... |
| tcss 9658 | The transitive closure fun... |
| tcel 9659 | The transitive closure fun... |
| tcidm 9660 | The transitive closure fun... |
| tc0 9661 | The transitive closure of ... |
| tc00 9662 | The transitive closure is ... |
| setind 9663 | Set (epsilon) induction. ... |
| setind2 9664 | Set (epsilon) induction, s... |
| setinds 9665 | Principle of set induction... |
| setinds2f 9666 | ` _E ` induction schema, u... |
| setinds2 9667 | ` _E ` induction schema, u... |
| frmin 9668 | Every (possibly proper) su... |
| frind 9669 | A subclass of a well-found... |
| frinsg 9670 | Well-Founded Induction Sch... |
| frins 9671 | Well-Founded Induction Sch... |
| frins2f 9672 | Well-Founded Induction sch... |
| frins2 9673 | Well-Founded Induction sch... |
| frins3 9674 | Well-Founded Induction sch... |
| frr3g 9675 | Functions defined by well-... |
| frrlem15 9676 | Lemma for general well-fou... |
| frrlem16 9677 | Lemma for general well-fou... |
| frr1 9678 | Law of general well-founde... |
| frr2 9679 | Law of general well-founde... |
| frr3 9680 | Law of general well-founde... |
| r1funlim 9685 | The cumulative hierarchy o... |
| r1fnon 9686 | The cumulative hierarchy o... |
| r10 9687 | Value of the cumulative hi... |
| r1sucg 9688 | Value of the cumulative hi... |
| r1suc 9689 | Value of the cumulative hi... |
| r1limg 9690 | Value of the cumulative hi... |
| r1lim 9691 | Value of the cumulative hi... |
| r1fin 9692 | The first ` _om ` levels o... |
| r1sdom 9693 | Each stage in the cumulati... |
| r111 9694 | The cumulative hierarchy i... |
| r1tr 9695 | The cumulative hierarchy o... |
| r1tr2 9696 | The union of a cumulative ... |
| r1ordg 9697 | Ordering relation for the ... |
| r1ord3g 9698 | Ordering relation for the ... |
| r1ord 9699 | Ordering relation for the ... |
| r1ord2 9700 | Ordering relation for the ... |
| r1ord3 9701 | Ordering relation for the ... |
| r1sssuc 9702 | The value of the cumulativ... |
| r1pwss 9703 | Each set of the cumulative... |
| r1sscl 9704 | Each set of the cumulative... |
| r1val1 9705 | The value of the cumulativ... |
| tz9.12lem1 9706 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9707 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9708 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9709 | A set is well-founded if a... |
| tz9.13 9710 | Every set is well-founded,... |
| tz9.13g 9711 | Every set is well-founded,... |
| rankwflemb 9712 | Two ways of saying a set i... |
| rankf 9713 | The domain and codomain of... |
| rankon 9714 | The rank of a set is an or... |
| r1elwf 9715 | Any member of the cumulati... |
| rankvalb 9716 | Value of the rank function... |
| rankr1ai 9717 | One direction of ~ rankr1a... |
| rankvaln 9718 | Value of the rank function... |
| rankidb 9719 | Identity law for the rank ... |
| rankdmr1 9720 | A rank is a member of the ... |
| rankr1ag 9721 | A version of ~ rankr1a tha... |
| rankr1bg 9722 | A relationship between ran... |
| r1rankidb 9723 | Any set is a subset of the... |
| r1elssi 9724 | The range of the ` R1 ` fu... |
| r1elss 9725 | The range of the ` R1 ` fu... |
| pwwf 9726 | A power set is well-founde... |
| sswf 9727 | A subset of a well-founded... |
| snwf 9728 | A singleton is well-founde... |
| unwf 9729 | A binary union is well-fou... |
| prwf 9730 | An unordered pair is well-... |
| opwf 9731 | An ordered pair is well-fo... |
| unir1 9732 | The cumulative hierarchy o... |
| jech9.3 9733 | Every set belongs to some ... |
| rankwflem 9734 | Every set is well-founded,... |
| rankval 9735 | Value of the rank function... |
| rankvalg 9736 | Value of the rank function... |
| rankval2 9737 | Value of an alternate defi... |
| uniwf 9738 | A union is well-founded if... |
| rankr1clem 9739 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9740 | A relationship between the... |
| rankidn 9741 | A relationship between the... |
| rankpwi 9742 | The rank of a power set. ... |
| rankelb 9743 | The membership relation is... |
| wfelirr 9744 | A well-founded set is not ... |
| rankval3b 9745 | The value of the rank func... |
| ranksnb 9746 | The rank of a singleton. ... |
| rankonidlem 9747 | Lemma for ~ rankonid . (C... |
| rankonid 9748 | The rank of an ordinal num... |
| onwf 9749 | The ordinals are all well-... |
| onssr1 9750 | Initial segments of the or... |
| rankr1g 9751 | A relationship between the... |
| rankid 9752 | Identity law for the rank ... |
| rankr1 9753 | A relationship between the... |
| ssrankr1 9754 | A relationship between an ... |
| rankr1a 9755 | A relationship between ran... |
| r1val2 9756 | The value of the cumulativ... |
| r1val3 9757 | The value of the cumulativ... |
| rankel 9758 | The membership relation is... |
| rankval3 9759 | The value of the rank func... |
| bndrank 9760 | Any class whose elements h... |
| unbndrank 9761 | The elements of a proper c... |
| rankpw 9762 | The rank of a power set. ... |
| ranklim 9763 | The rank of a set belongs ... |
| r1pw 9764 | A stronger property of ` R... |
| r1pwALT 9765 | Alternate shorter proof of... |
| r1pwcl 9766 | The cumulative hierarchy o... |
| rankssb 9767 | The subset relation is inh... |
| rankss 9768 | The subset relation is inh... |
| rankunb 9769 | The rank of the union of t... |
| rankprb 9770 | The rank of an unordered p... |
| rankopb 9771 | The rank of an ordered pai... |
| rankuni2b 9772 | The value of the rank func... |
| ranksn 9773 | The rank of a singleton. ... |
| rankuni2 9774 | The rank of a union. Part... |
| rankun 9775 | The rank of the union of t... |
| rankpr 9776 | The rank of an unordered p... |
| rankop 9777 | The rank of an ordered pai... |
| r1rankid 9778 | Any set is a subset of the... |
| rankeq0b 9779 | A set is empty iff its ran... |
| rankeq0 9780 | A set is empty iff its ran... |
| rankr1id 9781 | The rank of the hierarchy ... |
| rankuni 9782 | The rank of a union. Part... |
| rankr1b 9783 | A relationship between ran... |
| ranksuc 9784 | The rank of a successor. ... |
| rankuniss 9785 | Upper bound of the rank of... |
| rankval4 9786 | The rank of a set is the s... |
| rankbnd 9787 | The rank of a set is bound... |
| rankbnd2 9788 | The rank of a set is bound... |
| rankc1 9789 | A relationship that can be... |
| rankc2 9790 | A relationship that can be... |
| rankelun 9791 | Rank membership is inherit... |
| rankelpr 9792 | Rank membership is inherit... |
| rankelop 9793 | Rank membership is inherit... |
| rankxpl 9794 | A lower bound on the rank ... |
| rankxpu 9795 | An upper bound on the rank... |
| rankfu 9796 | An upper bound on the rank... |
| rankmapu 9797 | An upper bound on the rank... |
| rankxplim 9798 | The rank of a Cartesian pr... |
| rankxplim2 9799 | If the rank of a Cartesian... |
| rankxplim3 9800 | The rank of a Cartesian pr... |
| rankxpsuc 9801 | The rank of a Cartesian pr... |
| tcwf 9802 | The transitive closure fun... |
| tcrank 9803 | This theorem expresses two... |
| scottex 9804 | Scott's trick collects all... |
| scott0 9805 | Scott's trick collects all... |
| scottexs 9806 | Theorem scheme version of ... |
| scott0s 9807 | Theorem scheme version of ... |
| cplem1 9808 | Lemma for the Collection P... |
| cplem2 9809 | Lemma for the Collection P... |
| cp 9810 | Collection Principle. Thi... |
| bnd 9811 | A very strong generalizati... |
| bnd2 9812 | A variant of the Boundedne... |
| kardex 9813 | The collection of all sets... |
| karden 9814 | If we allow the Axiom of R... |
| htalem 9815 | Lemma for defining an emul... |
| hta 9816 | A ZFC emulation of Hilbert... |
| djueq12 9823 | Equality theorem for disjo... |
| djueq1 9824 | Equality theorem for disjo... |
| djueq2 9825 | Equality theorem for disjo... |
| nfdju 9826 | Bound-variable hypothesis ... |
| djuex 9827 | The disjoint union of sets... |
| djuexb 9828 | The disjoint union of two ... |
| djulcl 9829 | Left closure of disjoint u... |
| djurcl 9830 | Right closure of disjoint ... |
| djulf1o 9831 | The left injection functio... |
| djurf1o 9832 | The right injection functi... |
| inlresf 9833 | The left injection restric... |
| inlresf1 9834 | The left injection restric... |
| inrresf 9835 | The right injection restri... |
| inrresf1 9836 | The right injection restri... |
| djuin 9837 | The images of any classes ... |
| djur 9838 | A member of a disjoint uni... |
| djuss 9839 | A disjoint union is a subc... |
| djuunxp 9840 | The union of a disjoint un... |
| djuexALT 9841 | Alternate proof of ~ djuex... |
| eldju1st 9842 | The first component of an ... |
| eldju2ndl 9843 | The second component of an... |
| eldju2ndr 9844 | The second component of an... |
| djuun 9845 | The disjoint union of two ... |
| 1stinl 9846 | The first component of the... |
| 2ndinl 9847 | The second component of th... |
| 1stinr 9848 | The first component of the... |
| 2ndinr 9849 | The second component of th... |
| updjudhf 9850 | The mapping of an element ... |
| updjudhcoinlf 9851 | The composition of the map... |
| updjudhcoinrg 9852 | The composition of the map... |
| updjud 9853 | Universal property of the ... |
| cardf2 9862 | The cardinality function i... |
| cardon 9863 | The cardinal number of a s... |
| isnum2 9864 | A way to express well-orde... |
| isnumi 9865 | A set equinumerous to an o... |
| ennum 9866 | Equinumerous sets are equi... |
| finnum 9867 | Every finite set is numera... |
| onenon 9868 | Every ordinal number is nu... |
| tskwe 9869 | A Tarski set is well-order... |
| xpnum 9870 | The cartesian product of n... |
| cardval3 9871 | An alternate definition of... |
| cardid2 9872 | Any numerable set is equin... |
| isnum3 9873 | A set is numerable iff it ... |
| oncardval 9874 | The value of the cardinal ... |
| oncardid 9875 | Any ordinal number is equi... |
| cardonle 9876 | The cardinal of an ordinal... |
| card0 9877 | The cardinality of the emp... |
| cardidm 9878 | The cardinality function i... |
| oncard 9879 | A set is a cardinal number... |
| ficardom 9880 | The cardinal number of a f... |
| ficardid 9881 | A finite set is equinumero... |
| cardnn 9882 | The cardinality of a natur... |
| cardnueq0 9883 | The empty set is the only ... |
| cardne 9884 | No member of a cardinal nu... |
| carden2a 9885 | If two sets have equal non... |
| carden2b 9886 | If two sets are equinumero... |
| card1 9887 | A set has cardinality one ... |
| cardsn 9888 | A singleton has cardinalit... |
| carddomi2 9889 | Two sets have the dominanc... |
| sdomsdomcardi 9890 | A set strictly dominates i... |
| cardlim 9891 | An infinite cardinal is a ... |
| cardsdomelir 9892 | A cardinal strictly domina... |
| cardsdomel 9893 | A cardinal strictly domina... |
| iscard 9894 | Two ways to express the pr... |
| iscard2 9895 | Two ways to express the pr... |
| carddom2 9896 | Two numerable sets have th... |
| harcard 9897 | The class of ordinal numbe... |
| cardprclem 9898 | Lemma for ~ cardprc . (Co... |
| cardprc 9899 | The class of all cardinal ... |
| carduni 9900 | The union of a set of card... |
| cardiun 9901 | The indexed union of a set... |
| cardennn 9902 | If ` A ` is equinumerous t... |
| cardsucinf 9903 | The cardinality of the suc... |
| cardsucnn 9904 | The cardinality of the suc... |
| cardom 9905 | The set of natural numbers... |
| carden2 9906 | Two numerable sets are equ... |
| cardsdom2 9907 | A numerable set is strictl... |
| domtri2 9908 | Trichotomy of dominance fo... |
| nnsdomel 9909 | Strict dominance and eleme... |
| cardval2 9910 | An alternate version of th... |
| isinffi 9911 | An infinite set contains s... |
| fidomtri 9912 | Trichotomy of dominance wi... |
| fidomtri2 9913 | Trichotomy of dominance wi... |
| harsdom 9914 | The Hartogs number of a we... |
| onsdom 9915 | Any well-orderable set is ... |
| harval2 9916 | An alternate expression fo... |
| harsucnn 9917 | The next cardinal after a ... |
| cardmin2 9918 | The smallest ordinal that ... |
| pm54.43lem 9919 | In Theorem *54.43 of [Whit... |
| pm54.43 9920 | Theorem *54.43 of [Whitehe... |
| enpr2 9921 | An unordered pair with dis... |
| pr2ne 9922 | If an unordered pair has t... |
| prdom2 9923 | An unordered pair has at m... |
| en2eqpr 9924 | Building a set with two el... |
| en2eleq 9925 | Express a set of pair card... |
| en2other2 9926 | Taking the other element t... |
| dif1card 9927 | The cardinality of a nonem... |
| leweon 9928 | Lexicographical order is a... |
| r0weon 9929 | A set-like well-ordering o... |
| infxpenlem 9930 | Lemma for ~ infxpen . (Co... |
| infxpen 9931 | Every infinite ordinal is ... |
| xpomen 9932 | The Cartesian product of o... |
| xpct 9933 | The cartesian product of t... |
| infxpidm2 9934 | Every infinite well-ordera... |
| infxpenc 9935 | A canonical version of ~ i... |
| infxpenc2lem1 9936 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9937 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9938 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9939 | Existence form of ~ infxpe... |
| iunmapdisj 9940 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9941 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9942 | Lemma for ~ fseqen . (Con... |
| fseqdom 9943 | One half of ~ fseqen . (C... |
| fseqen 9944 | A set that is equinumerous... |
| infpwfidom 9945 | The collection of finite s... |
| dfac8alem 9946 | Lemma for ~ dfac8a . If t... |
| dfac8a 9947 | Numeration theorem: every ... |
| dfac8b 9948 | The well-ordering theorem:... |
| dfac8clem 9949 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9950 | If the union of a set is w... |
| ac10ct 9951 | A proof of the well-orderi... |
| ween 9952 | A set is numerable iff it ... |
| ac5num 9953 | A version of ~ ac5b with t... |
| ondomen 9954 | If a set is dominated by a... |
| numdom 9955 | A set dominated by a numer... |
| ssnum 9956 | A subset of a numerable se... |
| onssnum 9957 | All subsets of the ordinal... |
| indcardi 9958 | Indirect strong induction ... |
| acnrcl 9959 | Reverse closure for the ch... |
| acneq 9960 | Equality theorem for the c... |
| isacn 9961 | The property of being a ch... |
| acni 9962 | The property of being a ch... |
| acni2 9963 | The property of being a ch... |
| acni3 9964 | The property of being a ch... |
| acnlem 9965 | Construct a mapping satisf... |
| numacn 9966 | A well-orderable set has c... |
| finacn 9967 | Every set has finite choic... |
| acndom 9968 | A set with long choice seq... |
| acnnum 9969 | A set ` X ` which has choi... |
| acnen 9970 | The class of choice sets o... |
| acndom2 9971 | A set smaller than one wit... |
| acnen2 9972 | The class of sets with cho... |
| fodomacn 9973 | A version of ~ fodom that ... |
| fodomnum 9974 | A version of ~ fodom that ... |
| fonum 9975 | A surjection maps numerabl... |
| numwdom 9976 | A surjection maps numerabl... |
| fodomfi2 9977 | Onto functions define domi... |
| wdomfil 9978 | Weak dominance agrees with... |
| infpwfien 9979 | Any infinite well-orderabl... |
| inffien 9980 | The set of finite intersec... |
| wdomnumr 9981 | Weak dominance agrees with... |
| alephfnon 9982 | The aleph function is a fu... |
| aleph0 9983 | The first infinite cardina... |
| alephlim 9984 | Value of the aleph functio... |
| alephsuc 9985 | Value of the aleph functio... |
| alephon 9986 | An aleph is an ordinal num... |
| alephcard 9987 | Every aleph is a cardinal ... |
| alephnbtwn 9988 | No cardinal can be sandwic... |
| alephnbtwn2 9989 | No set has equinumerosity ... |
| alephordilem1 9990 | Lemma for ~ alephordi . (... |
| alephordi 9991 | Strict ordering property o... |
| alephord 9992 | Ordering property of the a... |
| alephord2 9993 | Ordering property of the a... |
| alephord2i 9994 | Ordering property of the a... |
| alephord3 9995 | Ordering property of the a... |
| alephsucdom 9996 | A set dominated by an alep... |
| alephsuc2 9997 | An alternate representatio... |
| alephdom 9998 | Relationship between inclu... |
| alephgeom 9999 | Every aleph is greater tha... |
| alephislim 10000 | Every aleph is a limit ord... |
| aleph11 10001 | The aleph function is one-... |
| alephf1 10002 | The aleph function is a on... |
| alephsdom 10003 | If an ordinal is smaller t... |
| alephdom2 10004 | A dominated initial ordina... |
| alephle 10005 | The argument of the aleph ... |
| cardaleph 10006 | Given any transfinite card... |
| cardalephex 10007 | Every transfinite cardinal... |
| infenaleph 10008 | An infinite numerable set ... |
| isinfcard 10009 | Two ways to express the pr... |
| iscard3 10010 | Two ways to express the pr... |
| cardnum 10011 | Two ways to express the cl... |
| alephinit 10012 | An infinite initial ordina... |
| carduniima 10013 | The union of the image of ... |
| cardinfima 10014 | If a mapping to cardinals ... |
| alephiso 10015 | Aleph is an order isomorph... |
| alephprc 10016 | The class of all transfini... |
| alephsson 10017 | The class of transfinite c... |
| unialeph 10018 | The union of the class of ... |
| alephsmo 10019 | The aleph function is stri... |
| alephf1ALT 10020 | Alternate proof of ~ aleph... |
| alephfplem1 10021 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10022 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10023 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10024 | Lemma for ~ alephfp . (Co... |
| alephfp 10025 | The aleph function has a f... |
| alephfp2 10026 | The aleph function has at ... |
| alephval3 10027 | An alternate way to expres... |
| alephsucpw2 10028 | The power set of an aleph ... |
| mappwen 10029 | Power rule for cardinal ar... |
| finnisoeu 10030 | A finite totally ordered s... |
| iunfictbso 10031 | Countability of a countabl... |
| aceq1 10034 | Equivalence of two version... |
| aceq0 10035 | Equivalence of two version... |
| aceq2 10036 | Equivalence of two version... |
| aceq3lem 10037 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10038 | Equivalence of two version... |
| dfac4 10039 | Equivalence of two version... |
| dfac5lem1 10040 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10041 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10042 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10043 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10044 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4OLD 10045 | Obsolete version of ~ dfac... |
| dfac5 10046 | Equivalence of two version... |
| dfac2a 10047 | Our Axiom of Choice (in th... |
| dfac2b 10048 | Axiom of Choice (first for... |
| dfac2 10049 | Axiom of Choice (first for... |
| dfac7 10050 | Equivalence of the Axiom o... |
| dfac0 10051 | Equivalence of two version... |
| dfac1 10052 | Equivalence of two version... |
| dfac8 10053 | A proof of the equivalency... |
| dfac9 10054 | Equivalence of the axiom o... |
| dfac10 10055 | Axiom of Choice equivalent... |
| dfac10c 10056 | Axiom of Choice equivalent... |
| dfac10b 10057 | Axiom of Choice equivalent... |
| acacni 10058 | A choice equivalent: every... |
| dfacacn 10059 | A choice equivalent: every... |
| dfac13 10060 | The axiom of choice holds ... |
| dfac12lem1 10061 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10062 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10063 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10064 | The axiom of choice holds ... |
| dfac12k 10065 | Equivalence of ~ dfac12 an... |
| dfac12a 10066 | The axiom of choice holds ... |
| dfac12 10067 | The axiom of choice holds ... |
| kmlem1 10068 | Lemma for 5-quantifier AC ... |
| kmlem2 10069 | Lemma for 5-quantifier AC ... |
| kmlem3 10070 | Lemma for 5-quantifier AC ... |
| kmlem4 10071 | Lemma for 5-quantifier AC ... |
| kmlem5 10072 | Lemma for 5-quantifier AC ... |
| kmlem6 10073 | Lemma for 5-quantifier AC ... |
| kmlem7 10074 | Lemma for 5-quantifier AC ... |
| kmlem8 10075 | Lemma for 5-quantifier AC ... |
| kmlem9 10076 | Lemma for 5-quantifier AC ... |
| kmlem10 10077 | Lemma for 5-quantifier AC ... |
| kmlem11 10078 | Lemma for 5-quantifier AC ... |
| kmlem12 10079 | Lemma for 5-quantifier AC ... |
| kmlem13 10080 | Lemma for 5-quantifier AC ... |
| kmlem14 10081 | Lemma for 5-quantifier AC ... |
| kmlem15 10082 | Lemma for 5-quantifier AC ... |
| kmlem16 10083 | Lemma for 5-quantifier AC ... |
| dfackm 10084 | Equivalence of the Axiom o... |
| undjudom 10085 | Cardinal addition dominate... |
| endjudisj 10086 | Equinumerosity of a disjoi... |
| djuen 10087 | Disjoint unions of equinum... |
| djuenun 10088 | Disjoint union is equinume... |
| dju1en 10089 | Cardinal addition with car... |
| dju1dif 10090 | Adding and subtracting one... |
| dju1p1e2 10091 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10092 | Alternate proof of ~ dju1p... |
| dju0en 10093 | Cardinal addition with car... |
| xp2dju 10094 | Two times a cardinal numbe... |
| djucomen 10095 | Commutative law for cardin... |
| djuassen 10096 | Associative law for cardin... |
| xpdjuen 10097 | Cardinal multiplication di... |
| mapdjuen 10098 | Sum of exponents law for c... |
| pwdjuen 10099 | Sum of exponents law for c... |
| djudom1 10100 | Ordering law for cardinal ... |
| djudom2 10101 | Ordering law for cardinal ... |
| djudoml 10102 | A set is dominated by its ... |
| djuxpdom 10103 | Cartesian product dominate... |
| djufi 10104 | The disjoint union of two ... |
| cdainflem 10105 | Any partition of omega int... |
| djuinf 10106 | A set is infinite iff the ... |
| infdju1 10107 | An infinite set is equinum... |
| pwdju1 10108 | The sum of a powerset with... |
| pwdjuidm 10109 | If the natural numbers inj... |
| djulepw 10110 | If ` A ` is idempotent und... |
| onadju 10111 | The cardinal and ordinal s... |
| cardadju 10112 | The cardinal sum is equinu... |
| djunum 10113 | The disjoint union of two ... |
| unnum 10114 | The union of two numerable... |
| nnadju 10115 | The cardinal and ordinal s... |
| nnadjuALT 10116 | Shorter proof of ~ nnadju ... |
| ficardadju 10117 | The disjoint union of fini... |
| ficardun 10118 | The cardinality of the uni... |
| ficardun2 10119 | The cardinality of the uni... |
| pwsdompw 10120 | Lemma for ~ domtriom . Th... |
| unctb 10121 | The union of two countable... |
| infdjuabs 10122 | Absorption law for additio... |
| infunabs 10123 | An infinite set is equinum... |
| infdju 10124 | The sum of two cardinal nu... |
| infdif 10125 | The cardinality of an infi... |
| infdif2 10126 | Cardinality ordering for a... |
| infxpdom 10127 | Dominance law for multipli... |
| infxpabs 10128 | Absorption law for multipl... |
| infunsdom1 10129 | The union of two sets that... |
| infunsdom 10130 | The union of two sets that... |
| infxp 10131 | Absorption law for multipl... |
| pwdjudom 10132 | A property of dominance ov... |
| infpss 10133 | Every infinite set has an ... |
| infmap2 10134 | An exponentiation law for ... |
| ackbij2lem1 10135 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10136 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10137 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10138 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10139 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10140 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10141 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10142 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10143 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10144 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10145 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10146 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10147 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10148 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10149 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10150 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10151 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10152 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10153 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10154 | The Ackermann bijection, p... |
| ackbij1b 10155 | The Ackermann bijection, p... |
| ackbij2lem2 10156 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10157 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10158 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10159 | The Ackermann bijection, p... |
| r1om 10160 | The set of hereditarily fi... |
| fictb 10161 | A set is countable iff its... |
| cflem 10162 | A lemma used to simplify c... |
| cflemOLD 10163 | Obsolete version of ~ cfle... |
| cfval 10164 | Value of the cofinality fu... |
| cff 10165 | Cofinality is a function o... |
| cfub 10166 | An upper bound on cofinali... |
| cflm 10167 | Value of the cofinality fu... |
| cf0 10168 | Value of the cofinality fu... |
| cardcf 10169 | Cofinality is a cardinal n... |
| cflecard 10170 | Cofinality is bounded by t... |
| cfle 10171 | Cofinality is bounded by i... |
| cfon 10172 | The cofinality of any set ... |
| cfeq0 10173 | Only the ordinal zero has ... |
| cfsuc 10174 | Value of the cofinality fu... |
| cff1 10175 | There is always a map from... |
| cfflb 10176 | If there is a cofinal map ... |
| cfval2 10177 | Another expression for the... |
| coflim 10178 | A simpler expression for t... |
| cflim3 10179 | Another expression for the... |
| cflim2 10180 | The cofinality function is... |
| cfom 10181 | Value of the cofinality fu... |
| cfss 10182 | There is a cofinal subset ... |
| cfslb 10183 | Any cofinal subset of ` A ... |
| cfslbn 10184 | Any subset of ` A ` smalle... |
| cfslb2n 10185 | Any small collection of sm... |
| cofsmo 10186 | Any cofinal map implies th... |
| cfsmolem 10187 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10188 | The map in ~ cff1 can be a... |
| cfcoflem 10189 | Lemma for ~ cfcof , showin... |
| coftr 10190 | If there is a cofinal map ... |
| cfcof 10191 | If there is a cofinal map ... |
| cfidm 10192 | The cofinality function is... |
| alephsing 10193 | The cofinality of a limit ... |
| sornom 10194 | The range of a single-step... |
| isfin1a 10209 | Definition of a Ia-finite ... |
| fin1ai 10210 | Property of a Ia-finite se... |
| isfin2 10211 | Definition of a II-finite ... |
| fin2i 10212 | Property of a II-finite se... |
| isfin3 10213 | Definition of a III-finite... |
| isfin4 10214 | Definition of a IV-finite ... |
| fin4i 10215 | Infer that a set is IV-inf... |
| isfin5 10216 | Definition of a V-finite s... |
| isfin6 10217 | Definition of a VI-finite ... |
| isfin7 10218 | Definition of a VII-finite... |
| sdom2en01 10219 | A set with less than two e... |
| infpssrlem1 10220 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10221 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10222 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10223 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10224 | Lemma for ~ infpssr . (Co... |
| infpssr 10225 | Dedekind infinity implies ... |
| fin4en1 10226 | Dedekind finite is a cardi... |
| ssfin4 10227 | Dedekind finite sets have ... |
| domfin4 10228 | A set dominated by a Dedek... |
| ominf4 10229 | ` _om ` is Dedekind infini... |
| infpssALT 10230 | Alternate proof of ~ infps... |
| isfin4-2 10231 | Alternate definition of IV... |
| isfin4p1 10232 | Alternate definition of IV... |
| fin23lem7 10233 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10234 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10235 | A II-finite set contains m... |
| isfin2-2 10236 | ` Fin2 ` expressed in term... |
| ssfin2 10237 | A subset of a II-finite se... |
| enfin2i 10238 | II-finiteness is a cardina... |
| fin23lem24 10239 | Lemma for ~ fin23 . In a ... |
| fincssdom 10240 | In a chain of finite sets,... |
| fin23lem25 10241 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10242 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10243 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10244 | Lemma for ~ fin23 but coul... |
| fin23lem27 10245 | The mapping constructed in... |
| isfin3ds 10246 | Property of a III-finite s... |
| ssfin3ds 10247 | A subset of a III-finite s... |
| fin23lem12 10248 | The beginning of the proof... |
| fin23lem13 10249 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10250 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10251 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10252 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10253 | Lemma for ~ fin23 . The f... |
| fin23lem20 10254 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10255 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10256 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10257 | Lemma for ~ fin23 . The r... |
| fin23lem29 10258 | Lemma for ~ fin23 . The r... |
| fin23lem30 10259 | Lemma for ~ fin23 . The r... |
| fin23lem31 10260 | Lemma for ~ fin23 . The r... |
| fin23lem32 10261 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10262 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10263 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10264 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10265 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10266 | Lemma for ~ fin23 . The c... |
| fin23lem39 10267 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10268 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10269 | Lemma for ~ fin23 . A set... |
| isf32lem1 10270 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10271 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10272 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10273 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10274 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10275 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10276 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10277 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10278 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10279 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10280 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10281 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10282 | One half of ~ isfin3-2 . ... |
| isf33lem 10283 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10284 | Weakly Dedekind-infinite s... |
| isfin3-3 10285 | Weakly Dedekind-infinite s... |
| fin33i 10286 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10287 | Lemma for ~ compsscnv . (... |
| compsscnv 10288 | Complementation on a power... |
| isf34lem1 10289 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10290 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10291 | Complementation is an anti... |
| isf34lem3 10292 | Lemma for ~ isfin3-4 . (C... |
| compss 10293 | Express image under of the... |
| isf34lem4 10294 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10295 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10296 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10297 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10298 | Inference from ~ isfin3-4 ... |
| isfin3-4 10299 | Weakly Dedekind-infinite s... |
| fin11a 10300 | Every I-finite set is Ia-f... |
| enfin1ai 10301 | Ia-finiteness is a cardina... |
| isfin1-2 10302 | A set is finite in the usu... |
| isfin1-3 10303 | A set is I-finite iff ever... |
| isfin1-4 10304 | A set is I-finite iff ever... |
| dffin1-5 10305 | Compact quantifier-free ve... |
| fin23 10306 | Every II-finite set (every... |
| fin34 10307 | Every III-finite set is IV... |
| isfin5-2 10308 | Alternate definition of V-... |
| fin45 10309 | Every IV-finite set is V-f... |
| fin56 10310 | Every V-finite set is VI-f... |
| fin17 10311 | Every I-finite set is VII-... |
| fin67 10312 | Every VI-finite set is VII... |
| isfin7-2 10313 | A set is VII-finite iff it... |
| fin71num 10314 | A well-orderable set is VI... |
| dffin7-2 10315 | Class form of ~ isfin7-2 .... |
| dfacfin7 10316 | Axiom of Choice equivalent... |
| fin1a2lem1 10317 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10318 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10319 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10320 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10321 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10322 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10323 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10324 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10325 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10326 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10327 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10328 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10329 | Lemma for ~ fin1a2 . (Con... |
| fin12 10330 | Weak theorem which skips I... |
| fin1a2s 10331 | An II-infinite set can hav... |
| fin1a2 10332 | Every Ia-finite set is II-... |
| itunifval 10333 | Function value of iterated... |
| itunifn 10334 | Functionality of the itera... |
| ituni0 10335 | A zero-fold iterated union... |
| itunisuc 10336 | Successor iterated union. ... |
| itunitc1 10337 | Each union iterate is a me... |
| itunitc 10338 | The union of all union ite... |
| ituniiun 10339 | Unwrap an iterated union f... |
| hsmexlem7 10340 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10341 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10342 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10343 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10344 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10345 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10346 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10347 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10348 | Lemma for ~ hsmex . (Cont... |
| hsmex 10349 | The collection of heredita... |
| hsmex2 10350 | The set of hereditary size... |
| hsmex3 10351 | The set of hereditary size... |
| axcc2lem 10353 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10354 | A possibly more useful ver... |
| axcc3 10355 | A possibly more useful ver... |
| axcc4 10356 | A version of ~ axcc3 that ... |
| acncc 10357 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10358 | Relax the constraint on ~ ... |
| domtriomlem 10359 | Lemma for ~ domtriom . (C... |
| domtriom 10360 | Trichotomy of equinumerosi... |
| fin41 10361 | Under countable choice, th... |
| dominf 10362 | A nonempty set that is a s... |
| dcomex 10364 | The Axiom of Dependent Cho... |
| axdc2lem 10365 | Lemma for ~ axdc2 . We co... |
| axdc2 10366 | An apparent strengthening ... |
| axdc3lem 10367 | The class ` S ` of finite ... |
| axdc3lem2 10368 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10369 | Simple substitution lemma ... |
| axdc3lem4 10370 | Lemma for ~ axdc3 . We ha... |
| axdc3 10371 | Dependent Choice. Axiom D... |
| axdc4lem 10372 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10373 | A more general version of ... |
| axcclem 10374 | Lemma for ~ axcc . (Contr... |
| axcc 10375 | Although CC can be proven ... |
| zfac 10377 | Axiom of Choice expressed ... |
| ac2 10378 | Axiom of Choice equivalent... |
| ac3 10379 | Axiom of Choice using abbr... |
| axac3 10381 | This theorem asserts that ... |
| ackm 10382 | A remarkable equivalent to... |
| axac2 10383 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10384 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10385 | Apply a choice equivalent.... |
| cardeqv 10386 | All sets are well-orderabl... |
| numth3 10387 | All sets are well-orderabl... |
| numth2 10388 | Numeration theorem: any se... |
| numth 10389 | Numeration theorem: every ... |
| ac7 10390 | An Axiom of Choice equival... |
| ac7g 10391 | An Axiom of Choice equival... |
| ac4 10392 | Equivalent of Axiom of Cho... |
| ac4c 10393 | Equivalent of Axiom of Cho... |
| ac5 10394 | An Axiom of Choice equival... |
| ac5b 10395 | Equivalent of Axiom of Cho... |
| ac6num 10396 | A version of ~ ac6 which t... |
| ac6 10397 | Equivalent of Axiom of Cho... |
| ac6c4 10398 | Equivalent of Axiom of Cho... |
| ac6c5 10399 | Equivalent of Axiom of Cho... |
| ac9 10400 | An Axiom of Choice equival... |
| ac6s 10401 | Equivalent of Axiom of Cho... |
| ac6n 10402 | Equivalent of Axiom of Cho... |
| ac6s2 10403 | Generalization of the Axio... |
| ac6s3 10404 | Generalization of the Axio... |
| ac6sg 10405 | ~ ac6s with sethood as ant... |
| ac6sf 10406 | Version of ~ ac6 with boun... |
| ac6s4 10407 | Generalization of the Axio... |
| ac6s5 10408 | Generalization of the Axio... |
| ac8 10409 | An Axiom of Choice equival... |
| ac9s 10410 | An Axiom of Choice equival... |
| numthcor 10411 | Any set is strictly domina... |
| weth 10412 | Well-ordering theorem: any... |
| zorn2lem1 10413 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10414 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10415 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10416 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10417 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10418 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10419 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10420 | Zorn's Lemma of [Monk1] p.... |
| zorng 10421 | Zorn's Lemma. If the unio... |
| zornn0g 10422 | Variant of Zorn's lemma ~ ... |
| zorn2 10423 | Zorn's Lemma of [Monk1] p.... |
| zorn 10424 | Zorn's Lemma. If the unio... |
| zornn0 10425 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10426 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10427 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10428 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10429 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10430 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10431 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10432 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10433 | The Teichmüller-Tukey... |
| ttukeyg 10434 | The Teichmüller-Tukey... |
| ttukey 10435 | The Teichmüller-Tukey... |
| axdclem 10436 | Lemma for ~ axdc . (Contr... |
| axdclem2 10437 | Lemma for ~ axdc . Using ... |
| axdc 10438 | This theorem derives ~ ax-... |
| fodomg 10439 | An onto function implies d... |
| fodom 10440 | An onto function implies d... |
| dmct 10441 | The domain of a countable ... |
| rnct 10442 | The range of a countable s... |
| fodomb 10443 | Equivalence of an onto map... |
| wdomac 10444 | When assuming AC, weak and... |
| brdom3 10445 | Equivalence to a dominance... |
| brdom5 10446 | An equivalence to a domina... |
| brdom4 10447 | An equivalence to a domina... |
| brdom7disj 10448 | An equivalence to a domina... |
| brdom6disj 10449 | An equivalence to a domina... |
| fin71ac 10450 | Once we allow AC, the "str... |
| imadomg 10451 | An image of a function und... |
| fimact 10452 | The image by a function of... |
| fnrndomg 10453 | The range of a function is... |
| fnct 10454 | If the domain of a functio... |
| mptct 10455 | A countable mapping set is... |
| iunfo 10456 | Existence of an onto funct... |
| iundom2g 10457 | An upper bound for the car... |
| iundomg 10458 | An upper bound for the car... |
| iundom 10459 | An upper bound for the car... |
| unidom 10460 | An upper bound for the car... |
| uniimadom 10461 | An upper bound for the car... |
| uniimadomf 10462 | An upper bound for the car... |
| cardval 10463 | The value of the cardinal ... |
| cardid 10464 | Any set is equinumerous to... |
| cardidg 10465 | Any set is equinumerous to... |
| cardidd 10466 | Any set is equinumerous to... |
| cardf 10467 | The cardinality function i... |
| carden 10468 | Two sets are equinumerous ... |
| cardeq0 10469 | Only the empty set has car... |
| unsnen 10470 | Equinumerosity of a set wi... |
| carddom 10471 | Two sets have the dominanc... |
| cardsdom 10472 | Two sets have the strict d... |
| domtri 10473 | Trichotomy law for dominan... |
| entric 10474 | Trichotomy of equinumerosi... |
| entri2 10475 | Trichotomy of dominance an... |
| entri3 10476 | Trichotomy of dominance. ... |
| sdomsdomcard 10477 | A set strictly dominates i... |
| canth3 10478 | Cantor's theorem in terms ... |
| infxpidm 10479 | Every infinite class is eq... |
| ondomon 10480 | The class of ordinals domi... |
| cardmin 10481 | The smallest ordinal that ... |
| ficard 10482 | A set is finite iff its ca... |
| infinfg 10483 | Equivalence between two in... |
| infinf 10484 | Equivalence between two in... |
| unirnfdomd 10485 | The union of the range of ... |
| konigthlem 10486 | Lemma for ~ konigth . (Co... |
| konigth 10487 | Konig's Theorem. If ` m (... |
| alephsucpw 10488 | The power set of an aleph ... |
| aleph1 10489 | The set exponentiation of ... |
| alephval2 10490 | An alternate way to expres... |
| dominfac 10491 | A nonempty set that is a s... |
| iunctb 10492 | The countable union of cou... |
| unictb 10493 | The countable union of cou... |
| infmap 10494 | An exponentiation law for ... |
| alephadd 10495 | The sum of two alephs is t... |
| alephmul 10496 | The product of two alephs ... |
| alephexp1 10497 | An exponentiation law for ... |
| alephsuc3 10498 | An alternate representatio... |
| alephexp2 10499 | An expression equinumerous... |
| alephreg 10500 | A successor aleph is regul... |
| pwcfsdom 10501 | A corollary of Konig's The... |
| cfpwsdom 10502 | A corollary of Konig's The... |
| alephom 10503 | From ~ canth2 , we know th... |
| smobeth 10504 | The beth function is stric... |
| nd1 10505 | A lemma for proving condit... |
| nd2 10506 | A lemma for proving condit... |
| nd3 10507 | A lemma for proving condit... |
| nd4 10508 | A lemma for proving condit... |
| axextnd 10509 | A version of the Axiom of ... |
| axrepndlem1 10510 | Lemma for the Axiom of Rep... |
| axrepndlem2 10511 | Lemma for the Axiom of Rep... |
| axrepnd 10512 | A version of the Axiom of ... |
| axunndlem1 10513 | Lemma for the Axiom of Uni... |
| axunnd 10514 | A version of the Axiom of ... |
| axpowndlem1 10515 | Lemma for the Axiom of Pow... |
| axpowndlem2 10516 | Lemma for the Axiom of Pow... |
| axpowndlem3 10517 | Lemma for the Axiom of Pow... |
| axpowndlem4 10518 | Lemma for the Axiom of Pow... |
| axpownd 10519 | A version of the Axiom of ... |
| axregndlem1 10520 | Lemma for the Axiom of Reg... |
| axregndlem2 10521 | Lemma for the Axiom of Reg... |
| axregnd 10522 | A version of the Axiom of ... |
| axinfndlem1 10523 | Lemma for the Axiom of Inf... |
| axinfnd 10524 | A version of the Axiom of ... |
| axacndlem1 10525 | Lemma for the Axiom of Cho... |
| axacndlem2 10526 | Lemma for the Axiom of Cho... |
| axacndlem3 10527 | Lemma for the Axiom of Cho... |
| axacndlem4 10528 | Lemma for the Axiom of Cho... |
| axacndlem5 10529 | Lemma for the Axiom of Cho... |
| axacnd 10530 | A version of the Axiom of ... |
| zfcndext 10531 | Axiom of Extensionality ~ ... |
| zfcndrep 10532 | Axiom of Replacement ~ ax-... |
| zfcndun 10533 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10534 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10535 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10536 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10537 | Axiom of Choice ~ ax-ac , ... |
| elgch 10540 | Elementhood in the collect... |
| fingch 10541 | A finite set is a GCH-set.... |
| gchi 10542 | The only GCH-sets which ha... |
| gchen1 10543 | If ` A <_ B < ~P A ` , and... |
| gchen2 10544 | If ` A < B <_ ~P A ` , and... |
| gchor 10545 | If ` A <_ B <_ ~P A ` , an... |
| engch 10546 | The property of being a GC... |
| gchdomtri 10547 | Under certain conditions, ... |
| fpwwe2cbv 10548 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10549 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10550 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10551 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10552 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10553 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10554 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10555 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10556 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10557 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10558 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10559 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10560 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10561 | Given any function ` F ` f... |
| fpwwecbv 10562 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10563 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10564 | Given any function ` F ` f... |
| canth4 10565 | An "effective" form of Can... |
| canthnumlem 10566 | Lemma for ~ canthnum . (C... |
| canthnum 10567 | The set of well-orderable ... |
| canthwelem 10568 | Lemma for ~ canthwe . (Co... |
| canthwe 10569 | The set of well-orders of ... |
| canthp1lem1 10570 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10571 | Lemma for ~ canthp1 . (Co... |
| canthp1 10572 | A slightly stronger form o... |
| finngch 10573 | The exclusion of finite se... |
| gchdju1 10574 | An infinite GCH-set is ide... |
| gchinf 10575 | An infinite GCH-set is Ded... |
| pwfseqlem1 10576 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10577 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10578 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10579 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10580 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10581 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10582 | The powerset of a Dedekind... |
| pwxpndom2 10583 | The powerset of a Dedekind... |
| pwxpndom 10584 | The powerset of a Dedekind... |
| pwdjundom 10585 | The powerset of a Dedekind... |
| gchdjuidm 10586 | An infinite GCH-set is ide... |
| gchxpidm 10587 | An infinite GCH-set is ide... |
| gchpwdom 10588 | A relationship between dom... |
| gchaleph 10589 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10590 | If ` ( aleph `` A ) ` and ... |
| hargch 10591 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10592 | If ` ( aleph `` suc A ) ` ... |
| gch2 10593 | It is sufficient to requir... |
| gch3 10594 | An equivalent formulation ... |
| gch-kn 10595 | The equivalence of two ver... |
| gchaclem 10596 | Lemma for ~ gchac (obsolet... |
| gchhar 10597 | A "local" form of ~ gchac ... |
| gchacg 10598 | A "local" form of ~ gchac ... |
| gchac 10599 | The Generalized Continuum ... |
| elwina 10604 | Conditions of weak inacces... |
| elina 10605 | Conditions of strong inacc... |
| winaon 10606 | A weakly inaccessible card... |
| inawinalem 10607 | Lemma for ~ inawina . (Co... |
| inawina 10608 | Every strongly inaccessibl... |
| omina 10609 | ` _om ` is a strongly inac... |
| winacard 10610 | A weakly inaccessible card... |
| winainflem 10611 | A weakly inaccessible card... |
| winainf 10612 | A weakly inaccessible card... |
| winalim 10613 | A weakly inaccessible card... |
| winalim2 10614 | A nontrivial weakly inacce... |
| winafp 10615 | A nontrivial weakly inacce... |
| winafpi 10616 | This theorem, which states... |
| gchina 10617 | Assuming the GCH, weakly a... |
| iswun 10622 | Properties of a weak unive... |
| wuntr 10623 | A weak universe is transit... |
| wununi 10624 | A weak universe is closed ... |
| wunpw 10625 | A weak universe is closed ... |
| wunelss 10626 | The elements of a weak uni... |
| wunpr 10627 | A weak universe is closed ... |
| wunun 10628 | A weak universe is closed ... |
| wuntp 10629 | A weak universe is closed ... |
| wunss 10630 | A weak universe is closed ... |
| wunin 10631 | A weak universe is closed ... |
| wundif 10632 | A weak universe is closed ... |
| wunint 10633 | A weak universe is closed ... |
| wunsn 10634 | A weak universe is closed ... |
| wunsuc 10635 | A weak universe is closed ... |
| wun0 10636 | A weak universe contains t... |
| wunr1om 10637 | A weak universe is infinit... |
| wunom 10638 | A weak universe contains a... |
| wunfi 10639 | A weak universe contains a... |
| wunop 10640 | A weak universe is closed ... |
| wunot 10641 | A weak universe is closed ... |
| wunxp 10642 | A weak universe is closed ... |
| wunpm 10643 | A weak universe is closed ... |
| wunmap 10644 | A weak universe is closed ... |
| wunf 10645 | A weak universe is closed ... |
| wundm 10646 | A weak universe is closed ... |
| wunrn 10647 | A weak universe is closed ... |
| wuncnv 10648 | A weak universe is closed ... |
| wunres 10649 | A weak universe is closed ... |
| wunfv 10650 | A weak universe is closed ... |
| wunco 10651 | A weak universe is closed ... |
| wuntpos 10652 | A weak universe is closed ... |
| intwun 10653 | The intersection of a coll... |
| r1limwun 10654 | Each limit stage in the cu... |
| r1wunlim 10655 | The weak universes in the ... |
| wunex2 10656 | Construct a weak universe ... |
| wunex 10657 | Construct a weak universe ... |
| uniwun 10658 | Every set is contained in ... |
| wunex3 10659 | Construct a weak universe ... |
| wuncval 10660 | Value of the weak universe... |
| wuncid 10661 | The weak universe closure ... |
| wunccl 10662 | The weak universe closure ... |
| wuncss 10663 | The weak universe closure ... |
| wuncidm 10664 | The weak universe closure ... |
| wuncval2 10665 | Our earlier expression for... |
| eltskg 10668 | Properties of a Tarski cla... |
| eltsk2g 10669 | Properties of a Tarski cla... |
| tskpwss 10670 | First axiom of a Tarski cl... |
| tskpw 10671 | Second axiom of a Tarski c... |
| tsken 10672 | Third axiom of a Tarski cl... |
| 0tsk 10673 | The empty set is a (transi... |
| tsksdom 10674 | An element of a Tarski cla... |
| tskssel 10675 | A part of a Tarski class s... |
| tskss 10676 | The subsets of an element ... |
| tskin 10677 | The intersection of two el... |
| tsksn 10678 | A singleton of an element ... |
| tsktrss 10679 | A transitive element of a ... |
| tsksuc 10680 | If an element of a Tarski ... |
| tsk0 10681 | A nonempty Tarski class co... |
| tsk1 10682 | One is an element of a non... |
| tsk2 10683 | Two is an element of a non... |
| 2domtsk 10684 | If a Tarski class is not e... |
| tskr1om 10685 | A nonempty Tarski class is... |
| tskr1om2 10686 | A nonempty Tarski class co... |
| tskinf 10687 | A nonempty Tarski class is... |
| tskpr 10688 | If ` A ` and ` B ` are mem... |
| tskop 10689 | If ` A ` and ` B ` are mem... |
| tskxpss 10690 | A Cartesian product of two... |
| tskwe2 10691 | A Tarski class is well-ord... |
| inttsk 10692 | The intersection of a coll... |
| inar1 10693 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10694 | Alternate proof of ~ r1om ... |
| rankcf 10695 | Any set must be at least a... |
| inatsk 10696 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10697 | The set of hereditarily fi... |
| tskord 10698 | A Tarski class contains al... |
| tskcard 10699 | An even more direct relati... |
| r1tskina 10700 | There is a direct relation... |
| tskuni 10701 | The union of an element of... |
| tskwun 10702 | A nonempty transitive Tars... |
| tskint 10703 | The intersection of an ele... |
| tskun 10704 | The union of two elements ... |
| tskxp 10705 | The Cartesian product of t... |
| tskmap 10706 | Set exponentiation is an e... |
| tskurn 10707 | A transitive Tarski class ... |
| elgrug 10710 | Properties of a Grothendie... |
| grutr 10711 | A Grothendieck universe is... |
| gruelss 10712 | A Grothendieck universe is... |
| grupw 10713 | A Grothendieck universe co... |
| gruss 10714 | Any subset of an element o... |
| grupr 10715 | A Grothendieck universe co... |
| gruurn 10716 | A Grothendieck universe co... |
| gruiun 10717 | If ` B ( x ) ` is a family... |
| gruuni 10718 | A Grothendieck universe co... |
| grurn 10719 | A Grothendieck universe co... |
| gruima 10720 | A Grothendieck universe co... |
| gruel 10721 | Any element of an element ... |
| grusn 10722 | A Grothendieck universe co... |
| gruop 10723 | A Grothendieck universe co... |
| gruun 10724 | A Grothendieck universe co... |
| gruxp 10725 | A Grothendieck universe co... |
| grumap 10726 | A Grothendieck universe co... |
| gruixp 10727 | A Grothendieck universe co... |
| gruiin 10728 | A Grothendieck universe co... |
| gruf 10729 | A Grothendieck universe co... |
| gruen 10730 | A Grothendieck universe co... |
| gruwun 10731 | A nonempty Grothendieck un... |
| intgru 10732 | The intersection of a fami... |
| ingru 10733 | The intersection of a univ... |
| wfgru 10734 | The wellfounded part of a ... |
| grudomon 10735 | Each ordinal that is compa... |
| gruina 10736 | If a Grothendieck universe... |
| grur1a 10737 | A characterization of Grot... |
| grur1 10738 | A characterization of Grot... |
| grutsk1 10739 | Grothendieck universes are... |
| grutsk 10740 | Grothendieck universes are... |
| axgroth5 10742 | The Tarski-Grothendieck ax... |
| axgroth2 10743 | Alternate version of the T... |
| grothpw 10744 | Derive the Axiom of Power ... |
| grothpwex 10745 | Derive the Axiom of Power ... |
| axgroth6 10746 | The Tarski-Grothendieck ax... |
| grothomex 10747 | The Tarski-Grothendieck Ax... |
| grothac 10748 | The Tarski-Grothendieck Ax... |
| axgroth3 10749 | Alternate version of the T... |
| axgroth4 10750 | Alternate version of the T... |
| grothprimlem 10751 | Lemma for ~ grothprim . E... |
| grothprim 10752 | The Tarski-Grothendieck Ax... |
| grothtsk 10753 | The Tarski-Grothendieck Ax... |
| inaprc 10754 | An equivalent to the Tarsk... |
| tskmval 10757 | Value of our tarski map. ... |
| tskmid 10758 | The set ` A ` is an elemen... |
| tskmcl 10759 | A Tarski class that contai... |
| sstskm 10760 | Being a part of ` ( tarski... |
| eltskm 10761 | Belonging to ` ( tarskiMap... |
| elni 10794 | Membership in the class of... |
| elni2 10795 | Membership in the class of... |
| pinn 10796 | A positive integer is a na... |
| pion 10797 | A positive integer is an o... |
| piord 10798 | A positive integer is ordi... |
| niex 10799 | The class of positive inte... |
| 0npi 10800 | The empty set is not a pos... |
| 1pi 10801 | Ordinal 'one' is a positiv... |
| addpiord 10802 | Positive integer addition ... |
| mulpiord 10803 | Positive integer multiplic... |
| mulidpi 10804 | 1 is an identity element f... |
| ltpiord 10805 | Positive integer 'less tha... |
| ltsopi 10806 | Positive integer 'less tha... |
| ltrelpi 10807 | Positive integer 'less tha... |
| dmaddpi 10808 | Domain of addition on posi... |
| dmmulpi 10809 | Domain of multiplication o... |
| addclpi 10810 | Closure of addition of pos... |
| mulclpi 10811 | Closure of multiplication ... |
| addcompi 10812 | Addition of positive integ... |
| addasspi 10813 | Addition of positive integ... |
| mulcompi 10814 | Multiplication of positive... |
| mulasspi 10815 | Multiplication of positive... |
| distrpi 10816 | Multiplication of positive... |
| addcanpi 10817 | Addition cancellation law ... |
| mulcanpi 10818 | Multiplication cancellatio... |
| addnidpi 10819 | There is no identity eleme... |
| ltexpi 10820 | Ordering on positive integ... |
| ltapi 10821 | Ordering property of addit... |
| ltmpi 10822 | Ordering property of multi... |
| 1lt2pi 10823 | One is less than two (one ... |
| nlt1pi 10824 | No positive integer is les... |
| indpi 10825 | Principle of Finite Induct... |
| enqbreq 10837 | Equivalence relation for p... |
| enqbreq2 10838 | Equivalence relation for p... |
| enqer 10839 | The equivalence relation f... |
| enqex 10840 | The equivalence relation f... |
| nqex 10841 | The class of positive frac... |
| 0nnq 10842 | The empty set is not a pos... |
| elpqn 10843 | Each positive fraction is ... |
| ltrelnq 10844 | Positive fraction 'less th... |
| pinq 10845 | The representatives of pos... |
| 1nq 10846 | The positive fraction 'one... |
| nqereu 10847 | There is a unique element ... |
| nqerf 10848 | Corollary of ~ nqereu : th... |
| nqercl 10849 | Corollary of ~ nqereu : cl... |
| nqerrel 10850 | Any member of ` ( N. X. N.... |
| nqerid 10851 | Corollary of ~ nqereu : th... |
| enqeq 10852 | Corollary of ~ nqereu : if... |
| nqereq 10853 | The function ` /Q ` acts a... |
| addpipq2 10854 | Addition of positive fract... |
| addpipq 10855 | Addition of positive fract... |
| addpqnq 10856 | Addition of positive fract... |
| mulpipq2 10857 | Multiplication of positive... |
| mulpipq 10858 | Multiplication of positive... |
| mulpqnq 10859 | Multiplication of positive... |
| ordpipq 10860 | Ordering of positive fract... |
| ordpinq 10861 | Ordering of positive fract... |
| addpqf 10862 | Closure of addition on pos... |
| addclnq 10863 | Closure of addition on pos... |
| mulpqf 10864 | Closure of multiplication ... |
| mulclnq 10865 | Closure of multiplication ... |
| addnqf 10866 | Domain of addition on posi... |
| mulnqf 10867 | Domain of multiplication o... |
| addcompq 10868 | Addition of positive fract... |
| addcomnq 10869 | Addition of positive fract... |
| mulcompq 10870 | Multiplication of positive... |
| mulcomnq 10871 | Multiplication of positive... |
| adderpqlem 10872 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10873 | Lemma for ~ mulerpq . (Co... |
| adderpq 10874 | Addition is compatible wit... |
| mulerpq 10875 | Multiplication is compatib... |
| addassnq 10876 | Addition of positive fract... |
| mulassnq 10877 | Multiplication of positive... |
| mulcanenq 10878 | Lemma for distributive law... |
| distrnq 10879 | Multiplication of positive... |
| 1nqenq 10880 | The equivalence class of r... |
| mulidnq 10881 | Multiplication identity el... |
| recmulnq 10882 | Relationship between recip... |
| recidnq 10883 | A positive fraction times ... |
| recclnq 10884 | Closure law for positive f... |
| recrecnq 10885 | Reciprocal of reciprocal o... |
| dmrecnq 10886 | Domain of reciprocal on po... |
| ltsonq 10887 | 'Less than' is a strict or... |
| lterpq 10888 | Compatibility of ordering ... |
| ltanq 10889 | Ordering property of addit... |
| ltmnq 10890 | Ordering property of multi... |
| 1lt2nq 10891 | One is less than two (one ... |
| ltaddnq 10892 | The sum of two fractions i... |
| ltexnq 10893 | Ordering on positive fract... |
| halfnq 10894 | One-half of any positive f... |
| nsmallnq 10895 | The is no smallest positiv... |
| ltbtwnnq 10896 | There exists a number betw... |
| ltrnq 10897 | Ordering property of recip... |
| archnq 10898 | For any fraction, there is... |
| npex 10904 | The class of positive real... |
| elnp 10905 | Membership in positive rea... |
| elnpi 10906 | Membership in positive rea... |
| prn0 10907 | A positive real is not emp... |
| prpssnq 10908 | A positive real is a subse... |
| elprnq 10909 | A positive real is a set o... |
| 0npr 10910 | The empty set is not a pos... |
| prcdnq 10911 | A positive real is closed ... |
| prub 10912 | A positive fraction not in... |
| prnmax 10913 | A positive real has no lar... |
| npomex 10914 | A simplifying observation,... |
| prnmadd 10915 | A positive real has no lar... |
| ltrelpr 10916 | Positive real 'less than' ... |
| genpv 10917 | Value of general operation... |
| genpelv 10918 | Membership in value of gen... |
| genpprecl 10919 | Pre-closure law for genera... |
| genpdm 10920 | Domain of general operatio... |
| genpn0 10921 | The result of an operation... |
| genpss 10922 | The result of an operation... |
| genpnnp 10923 | The result of an operation... |
| genpcd 10924 | Downward closure of an ope... |
| genpnmax 10925 | An operation on positive r... |
| genpcl 10926 | Closure of an operation on... |
| genpass 10927 | Associativity of an operat... |
| plpv 10928 | Value of addition on posit... |
| mpv 10929 | Value of multiplication on... |
| dmplp 10930 | Domain of addition on posi... |
| dmmp 10931 | Domain of multiplication o... |
| nqpr 10932 | The canonical embedding of... |
| 1pr 10933 | The positive real number '... |
| addclprlem1 10934 | Lemma to prove downward cl... |
| addclprlem2 10935 | Lemma to prove downward cl... |
| addclpr 10936 | Closure of addition on pos... |
| mulclprlem 10937 | Lemma to prove downward cl... |
| mulclpr 10938 | Closure of multiplication ... |
| addcompr 10939 | Addition of positive reals... |
| addasspr 10940 | Addition of positive reals... |
| mulcompr 10941 | Multiplication of positive... |
| mulasspr 10942 | Multiplication of positive... |
| distrlem1pr 10943 | Lemma for distributive law... |
| distrlem4pr 10944 | Lemma for distributive law... |
| distrlem5pr 10945 | Lemma for distributive law... |
| distrpr 10946 | Multiplication of positive... |
| 1idpr 10947 | 1 is an identity element f... |
| ltprord 10948 | Positive real 'less than' ... |
| psslinpr 10949 | Proper subset is a linear ... |
| ltsopr 10950 | Positive real 'less than' ... |
| prlem934 10951 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10952 | The sum of two positive re... |
| ltaddpr2 10953 | The sum of two positive re... |
| ltexprlem1 10954 | Lemma for Proposition 9-3.... |
| ltexprlem2 10955 | Lemma for Proposition 9-3.... |
| ltexprlem3 10956 | Lemma for Proposition 9-3.... |
| ltexprlem4 10957 | Lemma for Proposition 9-3.... |
| ltexprlem5 10958 | Lemma for Proposition 9-3.... |
| ltexprlem6 10959 | Lemma for Proposition 9-3.... |
| ltexprlem7 10960 | Lemma for Proposition 9-3.... |
| ltexpri 10961 | Proposition 9-3.5(iv) of [... |
| ltaprlem 10962 | Lemma for Proposition 9-3.... |
| ltapr 10963 | Ordering property of addit... |
| addcanpr 10964 | Addition cancellation law ... |
| prlem936 10965 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 10966 | Lemma for Proposition 9-3.... |
| reclem3pr 10967 | Lemma for Proposition 9-3.... |
| reclem4pr 10968 | Lemma for Proposition 9-3.... |
| recexpr 10969 | The reciprocal of a positi... |
| suplem1pr 10970 | The union of a nonempty, b... |
| suplem2pr 10971 | The union of a set of posi... |
| supexpr 10972 | The union of a nonempty, b... |
| enrer 10981 | The equivalence relation f... |
| nrex1 10982 | The class of signed reals ... |
| enrbreq 10983 | Equivalence relation for s... |
| enreceq 10984 | Equivalence class equality... |
| enrex 10985 | The equivalence relation f... |
| ltrelsr 10986 | Signed real 'less than' is... |
| addcmpblnr 10987 | Lemma showing compatibilit... |
| mulcmpblnrlem 10988 | Lemma used in lemma showin... |
| mulcmpblnr 10989 | Lemma showing compatibilit... |
| prsrlem1 10990 | Decomposing signed reals i... |
| addsrmo 10991 | There is at most one resul... |
| mulsrmo 10992 | There is at most one resul... |
| addsrpr 10993 | Addition of signed reals i... |
| mulsrpr 10994 | Multiplication of signed r... |
| ltsrpr 10995 | Ordering of signed reals i... |
| gt0srpr 10996 | Greater than zero in terms... |
| 0nsr 10997 | The empty set is not a sig... |
| 0r 10998 | The constant ` 0R ` is a s... |
| 1sr 10999 | The constant ` 1R ` is a s... |
| m1r 11000 | The constant ` -1R ` is a ... |
| addclsr 11001 | Closure of addition on sig... |
| mulclsr 11002 | Closure of multiplication ... |
| dmaddsr 11003 | Domain of addition on sign... |
| dmmulsr 11004 | Domain of multiplication o... |
| addcomsr 11005 | Addition of signed reals i... |
| addasssr 11006 | Addition of signed reals i... |
| mulcomsr 11007 | Multiplication of signed r... |
| mulasssr 11008 | Multiplication of signed r... |
| distrsr 11009 | Multiplication of signed r... |
| m1p1sr 11010 | Minus one plus one is zero... |
| m1m1sr 11011 | Minus one times minus one ... |
| ltsosr 11012 | Signed real 'less than' is... |
| 0lt1sr 11013 | 0 is less than 1 for signe... |
| 1ne0sr 11014 | 1 and 0 are distinct for s... |
| 0idsr 11015 | The signed real number 0 i... |
| 1idsr 11016 | 1 is an identity element f... |
| 00sr 11017 | A signed real times 0 is 0... |
| ltasr 11018 | Ordering property of addit... |
| pn0sr 11019 | A signed real plus its neg... |
| negexsr 11020 | Existence of negative sign... |
| recexsrlem 11021 | The reciprocal of a positi... |
| addgt0sr 11022 | The sum of two positive si... |
| mulgt0sr 11023 | The product of two positiv... |
| sqgt0sr 11024 | The square of a nonzero si... |
| recexsr 11025 | The reciprocal of a nonzer... |
| mappsrpr 11026 | Mapping from positive sign... |
| ltpsrpr 11027 | Mapping of order from posi... |
| map2psrpr 11028 | Equivalence for positive s... |
| supsrlem 11029 | Lemma for supremum theorem... |
| supsr 11030 | A nonempty, bounded set of... |
| opelcn 11047 | Ordered pair membership in... |
| opelreal 11048 | Ordered pair membership in... |
| elreal 11049 | Membership in class of rea... |
| elreal2 11050 | Ordered pair membership in... |
| 0ncn 11051 | The empty set is not a com... |
| ltrelre 11052 | 'Less than' is a relation ... |
| addcnsr 11053 | Addition of complex number... |
| mulcnsr 11054 | Multiplication of complex ... |
| eqresr 11055 | Equality of real numbers i... |
| addresr 11056 | Addition of real numbers i... |
| mulresr 11057 | Multiplication of real num... |
| ltresr 11058 | Ordering of real subset of... |
| ltresr2 11059 | Ordering of real subset of... |
| dfcnqs 11060 | Technical trick to permit ... |
| addcnsrec 11061 | Technical trick to permit ... |
| mulcnsrec 11062 | Technical trick to permit ... |
| axaddf 11063 | Addition is an operation o... |
| axmulf 11064 | Multiplication is an opera... |
| axcnex 11065 | The complex numbers form a... |
| axresscn 11066 | The real numbers are a sub... |
| ax1cn 11067 | 1 is a complex number. Ax... |
| axicn 11068 | ` _i ` is a complex number... |
| axaddcl 11069 | Closure law for addition o... |
| axaddrcl 11070 | Closure law for addition i... |
| axmulcl 11071 | Closure law for multiplica... |
| axmulrcl 11072 | Closure law for multiplica... |
| axmulcom 11073 | Multiplication of complex ... |
| axaddass 11074 | Addition of complex number... |
| axmulass 11075 | Multiplication of complex ... |
| axdistr 11076 | Distributive law for compl... |
| axi2m1 11077 | i-squared equals -1 (expre... |
| ax1ne0 11078 | 1 and 0 are distinct. Axi... |
| ax1rid 11079 | ` 1 ` is an identity eleme... |
| axrnegex 11080 | Existence of negative of r... |
| axrrecex 11081 | Existence of reciprocal of... |
| axcnre 11082 | A complex number can be ex... |
| axpre-lttri 11083 | Ordering on reals satisfie... |
| axpre-lttrn 11084 | Ordering on reals is trans... |
| axpre-ltadd 11085 | Ordering property of addit... |
| axpre-mulgt0 11086 | The product of two positiv... |
| axpre-sup 11087 | A nonempty, bounded-above ... |
| wuncn 11088 | A weak universe containing... |
| cnex 11114 | Alias for ~ ax-cnex . See... |
| addcl 11115 | Alias for ~ ax-addcl , for... |
| readdcl 11116 | Alias for ~ ax-addrcl , fo... |
| mulcl 11117 | Alias for ~ ax-mulcl , for... |
| remulcl 11118 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11119 | Alias for ~ ax-mulcom , fo... |
| addass 11120 | Alias for ~ ax-addass , fo... |
| mulass 11121 | Alias for ~ ax-mulass , fo... |
| adddi 11122 | Alias for ~ ax-distr , for... |
| recn 11123 | A real number is a complex... |
| reex 11124 | The real numbers form a se... |
| reelprrecn 11125 | Reals are a subset of the ... |
| cnelprrecn 11126 | Complex numbers are a subs... |
| mpoaddf 11127 | Addition is an operation o... |
| mpomulf 11128 | Multiplication is an opera... |
| elimne0 11129 | Hypothesis for weak deduct... |
| adddir 11130 | Distributive law for compl... |
| 0cn 11131 | Zero is a complex number. ... |
| 0cnd 11132 | Zero is a complex number, ... |
| c0ex 11133 | Zero is a set. (Contribut... |
| 1cnd 11134 | One is a complex number, d... |
| 1ex 11135 | One is a set. (Contribute... |
| cnre 11136 | Alias for ~ ax-cnre , for ... |
| mulrid 11137 | The number 1 is an identit... |
| mullid 11138 | Identity law for multiplic... |
| 1re 11139 | The number 1 is real. Thi... |
| 1red 11140 | The number 1 is real, dedu... |
| 0re 11141 | The number 0 is real. Rem... |
| 0red 11142 | The number 0 is real, dedu... |
| pr01ssre 11143 | The pair ` { 0 , 1 } ` is ... |
| mulridi 11144 | Identity law for multiplic... |
| mullidi 11145 | Identity law for multiplic... |
| addcli 11146 | Closure law for addition. ... |
| mulcli 11147 | Closure law for multiplica... |
| mulcomi 11148 | Commutative law for multip... |
| mulcomli 11149 | Commutative law for multip... |
| addassi 11150 | Associative law for additi... |
| mulassi 11151 | Associative law for multip... |
| adddii 11152 | Distributive law (left-dis... |
| adddiri 11153 | Distributive law (right-di... |
| recni 11154 | A real number is a complex... |
| readdcli 11155 | Closure law for addition o... |
| remulcli 11156 | Closure law for multiplica... |
| mulridd 11157 | Identity law for multiplic... |
| mullidd 11158 | Identity law for multiplic... |
| addcld 11159 | Closure law for addition. ... |
| mulcld 11160 | Closure law for multiplica... |
| mulcomd 11161 | Commutative law for multip... |
| addassd 11162 | Associative law for additi... |
| mulassd 11163 | Associative law for multip... |
| adddid 11164 | Distributive law (left-dis... |
| adddird 11165 | Distributive law (right-di... |
| adddirp1d 11166 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11167 | Join AB+CB into (A+C) on L... |
| recnd 11168 | Deduction from real number... |
| readdcld 11169 | Closure law for addition o... |
| remulcld 11170 | Closure law for multiplica... |
| pnfnre 11181 | Plus infinity is not a rea... |
| pnfnre2 11182 | Plus infinity is not a rea... |
| mnfnre 11183 | Minus infinity is not a re... |
| ressxr 11184 | The standard reals are a s... |
| rexpssxrxp 11185 | The Cartesian product of s... |
| rexr 11186 | A standard real is an exte... |
| 0xr 11187 | Zero is an extended real. ... |
| renepnf 11188 | No (finite) real equals pl... |
| renemnf 11189 | No real equals minus infin... |
| rexrd 11190 | A standard real is an exte... |
| renepnfd 11191 | No (finite) real equals pl... |
| renemnfd 11192 | No real equals minus infin... |
| pnfex 11193 | Plus infinity exists. (Co... |
| pnfxr 11194 | Plus infinity belongs to t... |
| pnfnemnf 11195 | Plus and minus infinity ar... |
| mnfnepnf 11196 | Minus and plus infinity ar... |
| mnfxr 11197 | Minus infinity belongs to ... |
| rexri 11198 | A standard real is an exte... |
| 1xr 11199 | ` 1 ` is an extended real ... |
| renfdisj 11200 | The reals and the infiniti... |
| ltrelxr 11201 | "Less than" is a relation ... |
| ltrel 11202 | "Less than" is a relation.... |
| lerelxr 11203 | "Less than or equal to" is... |
| lerel 11204 | "Less than or equal to" is... |
| xrlenlt 11205 | "Less than or equal to" ex... |
| xrlenltd 11206 | "Less than or equal to" ex... |
| xrltnle 11207 | "Less than" expressed in t... |
| xrltnled 11208 | 'Less than' in terms of 'l... |
| xrnltled 11209 | "Not less than" implies "l... |
| ssxr 11210 | The three (non-exclusive) ... |
| ltxrlt 11211 | The standard less-than ` <... |
| axlttri 11212 | Ordering on reals satisfie... |
| axlttrn 11213 | Ordering on reals is trans... |
| axltadd 11214 | Ordering property of addit... |
| axmulgt0 11215 | The product of two positiv... |
| axsup 11216 | A nonempty, bounded-above ... |
| lttr 11217 | Alias for ~ axlttrn , for ... |
| mulgt0 11218 | The product of two positiv... |
| lenlt 11219 | 'Less than or equal to' ex... |
| ltnle 11220 | 'Less than' expressed in t... |
| ltso 11221 | 'Less than' is a strict or... |
| gtso 11222 | 'Greater than' is a strict... |
| lttri2 11223 | Consequence of trichotomy.... |
| lttri3 11224 | Trichotomy law for 'less t... |
| lttri4 11225 | Trichotomy law for 'less t... |
| letri3 11226 | Trichotomy law. (Contribu... |
| leloe 11227 | 'Less than or equal to' ex... |
| eqlelt 11228 | Equality in terms of 'less... |
| ltle 11229 | 'Less than' implies 'less ... |
| leltne 11230 | 'Less than or equal to' im... |
| lelttr 11231 | Transitive law. (Contribu... |
| leltletr 11232 | Transitive law, weaker for... |
| ltletr 11233 | Transitive law. (Contribu... |
| ltleletr 11234 | Transitive law, weaker for... |
| letr 11235 | Transitive law. (Contribu... |
| ltnr 11236 | 'Less than' is irreflexive... |
| leid 11237 | 'Less than or equal to' is... |
| ltne 11238 | 'Less than' implies not eq... |
| ltnsym 11239 | 'Less than' is not symmetr... |
| ltnsym2 11240 | 'Less than' is antisymmetr... |
| letric 11241 | Trichotomy law. (Contribu... |
| ltlen 11242 | 'Less than' expressed in t... |
| eqle 11243 | Equality implies 'less tha... |
| eqled 11244 | Equality implies 'less tha... |
| ltadd2 11245 | Addition to both sides of ... |
| ne0gt0 11246 | A nonzero nonnegative numb... |
| lecasei 11247 | Ordering elimination by ca... |
| lelttric 11248 | Trichotomy law. (Contribu... |
| ltlecasei 11249 | Ordering elimination by ca... |
| ltnri 11250 | 'Less than' is irreflexive... |
| eqlei 11251 | Equality implies 'less tha... |
| eqlei2 11252 | Equality implies 'less tha... |
| gtneii 11253 | 'Less than' implies not eq... |
| ltneii 11254 | 'Greater than' implies not... |
| lttri2i 11255 | Consequence of trichotomy.... |
| lttri3i 11256 | Consequence of trichotomy.... |
| letri3i 11257 | Consequence of trichotomy.... |
| leloei 11258 | 'Less than or equal to' in... |
| ltleni 11259 | 'Less than' expressed in t... |
| ltnsymi 11260 | 'Less than' is not symmetr... |
| lenlti 11261 | 'Less than or equal to' in... |
| ltnlei 11262 | 'Less than' in terms of 'l... |
| ltlei 11263 | 'Less than' implies 'less ... |
| ltleii 11264 | 'Less than' implies 'less ... |
| ltnei 11265 | 'Less than' implies not eq... |
| letrii 11266 | Trichotomy law for 'less t... |
| lttri 11267 | 'Less than' is transitive.... |
| lelttri 11268 | 'Less than or equal to', '... |
| ltletri 11269 | 'Less than', 'less than or... |
| letri 11270 | 'Less than or equal to' is... |
| le2tri3i 11271 | Extended trichotomy law fo... |
| ltadd2i 11272 | Addition to both sides of ... |
| mulgt0i 11273 | The product of two positiv... |
| mulgt0ii 11274 | The product of two positiv... |
| ltnrd 11275 | 'Less than' is irreflexive... |
| gtned 11276 | 'Less than' implies not eq... |
| ltned 11277 | 'Greater than' implies not... |
| ne0gt0d 11278 | A nonzero nonnegative numb... |
| lttrid 11279 | Ordering on reals satisfie... |
| lttri2d 11280 | Consequence of trichotomy.... |
| lttri3d 11281 | Consequence of trichotomy.... |
| lttri4d 11282 | Trichotomy law for 'less t... |
| letri3d 11283 | Consequence of trichotomy.... |
| leloed 11284 | 'Less than or equal to' in... |
| eqleltd 11285 | Equality in terms of 'less... |
| ltlend 11286 | 'Less than' expressed in t... |
| lenltd 11287 | 'Less than or equal to' in... |
| ltnled 11288 | 'Less than' in terms of 'l... |
| ltled 11289 | 'Less than' implies 'less ... |
| ltnsymd 11290 | 'Less than' implies 'less ... |
| nltled 11291 | 'Not less than ' implies '... |
| lensymd 11292 | 'Less than or equal to' im... |
| letrid 11293 | Trichotomy law for 'less t... |
| leltned 11294 | 'Less than or equal to' im... |
| leneltd 11295 | 'Less than or equal to' an... |
| mulgt0d 11296 | The product of two positiv... |
| ltadd2d 11297 | Addition to both sides of ... |
| letrd 11298 | Transitive law deduction f... |
| lelttrd 11299 | Transitive law deduction f... |
| ltadd2dd 11300 | Addition to both sides of ... |
| ltletrd 11301 | Transitive law deduction f... |
| lttrd 11302 | Transitive law deduction f... |
| lelttrdi 11303 | If a number is less than a... |
| dedekind 11304 | The Dedekind cut theorem. ... |
| dedekindle 11305 | The Dedekind cut theorem, ... |
| mul12 11306 | Commutative/associative la... |
| mul32 11307 | Commutative/associative la... |
| mul31 11308 | Commutative/associative la... |
| mul4 11309 | Rearrangement of 4 factors... |
| mul4r 11310 | Rearrangement of 4 factors... |
| muladd11 11311 | A simple product of sums e... |
| 1p1times 11312 | Two times a number. (Cont... |
| peano2cn 11313 | A theorem for complex numb... |
| peano2re 11314 | A theorem for reals analog... |
| readdcan 11315 | Cancellation law for addit... |
| 00id 11316 | ` 0 ` is its own additive ... |
| mul02lem1 11317 | Lemma for ~ mul02 . If an... |
| mul02lem2 11318 | Lemma for ~ mul02 . Zero ... |
| mul02 11319 | Multiplication by ` 0 ` . ... |
| mul01 11320 | Multiplication by ` 0 ` . ... |
| addrid 11321 | ` 0 ` is an additive ident... |
| cnegex 11322 | Existence of the negative ... |
| cnegex2 11323 | Existence of a left invers... |
| addlid 11324 | ` 0 ` is a left identity f... |
| addcan 11325 | Cancellation law for addit... |
| addcan2 11326 | Cancellation law for addit... |
| addcom 11327 | Addition commutes. This u... |
| addridi 11328 | ` 0 ` is an additive ident... |
| addlidi 11329 | ` 0 ` is a left identity f... |
| mul02i 11330 | Multiplication by 0. Theo... |
| mul01i 11331 | Multiplication by ` 0 ` . ... |
| addcomi 11332 | Addition commutes. Based ... |
| addcomli 11333 | Addition commutes. (Contr... |
| addcani 11334 | Cancellation law for addit... |
| addcan2i 11335 | Cancellation law for addit... |
| mul12i 11336 | Commutative/associative la... |
| mul32i 11337 | Commutative/associative la... |
| mul4i 11338 | Rearrangement of 4 factors... |
| mul02d 11339 | Multiplication by 0. Theo... |
| mul01d 11340 | Multiplication by ` 0 ` . ... |
| addridd 11341 | ` 0 ` is an additive ident... |
| addlidd 11342 | ` 0 ` is a left identity f... |
| addcomd 11343 | Addition commutes. Based ... |
| addcand 11344 | Cancellation law for addit... |
| addcan2d 11345 | Cancellation law for addit... |
| addcanad 11346 | Cancelling a term on the l... |
| addcan2ad 11347 | Cancelling a term on the r... |
| addneintrd 11348 | Introducing a term on the ... |
| addneintr2d 11349 | Introducing a term on the ... |
| mul12d 11350 | Commutative/associative la... |
| mul32d 11351 | Commutative/associative la... |
| mul31d 11352 | Commutative/associative la... |
| mul4d 11353 | Rearrangement of 4 factors... |
| muladd11r 11354 | A simple product of sums e... |
| comraddd 11355 | Commute RHS addition, in d... |
| comraddi 11356 | Commute RHS addition. See... |
| ltaddneg 11357 | Adding a negative number t... |
| ltaddnegr 11358 | Adding a negative number t... |
| add12 11359 | Commutative/associative la... |
| add32 11360 | Commutative/associative la... |
| add32r 11361 | Commutative/associative la... |
| add4 11362 | Rearrangement of 4 terms i... |
| add42 11363 | Rearrangement of 4 terms i... |
| add12i 11364 | Commutative/associative la... |
| add32i 11365 | Commutative/associative la... |
| add4i 11366 | Rearrangement of 4 terms i... |
| add42i 11367 | Rearrangement of 4 terms i... |
| add12d 11368 | Commutative/associative la... |
| add32d 11369 | Commutative/associative la... |
| add4d 11370 | Rearrangement of 4 terms i... |
| add42d 11371 | Rearrangement of 4 terms i... |
| 0cnALT 11376 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11377 | Alternate proof of ~ 0cnAL... |
| negeu 11378 | Existential uniqueness of ... |
| subval 11379 | Value of subtraction, whic... |
| negeq 11380 | Equality theorem for negat... |
| negeqi 11381 | Equality inference for neg... |
| negeqd 11382 | Equality deduction for neg... |
| nfnegd 11383 | Deduction version of ~ nfn... |
| nfneg 11384 | Bound-variable hypothesis ... |
| csbnegg 11385 | Move class substitution in... |
| negex 11386 | A negative is a set. (Con... |
| subcl 11387 | Closure law for subtractio... |
| negcl 11388 | Closure law for negative. ... |
| negicn 11389 | ` -u _i ` is a complex num... |
| subf 11390 | Subtraction is an operatio... |
| subadd 11391 | Relationship between subtr... |
| subadd2 11392 | Relationship between subtr... |
| subsub23 11393 | Swap subtrahend and result... |
| pncan 11394 | Cancellation law for subtr... |
| pncan2 11395 | Cancellation law for subtr... |
| pncan3 11396 | Subtraction and addition o... |
| npcan 11397 | Cancellation law for subtr... |
| addsubass 11398 | Associative-type law for a... |
| addsub 11399 | Law for addition and subtr... |
| subadd23 11400 | Commutative/associative la... |
| addsub12 11401 | Commutative/associative la... |
| 2addsub 11402 | Law for subtraction and ad... |
| addsubeq4 11403 | Relation between sums and ... |
| pncan3oi 11404 | Subtraction and addition o... |
| mvrraddi 11405 | Move the right term in a s... |
| mvrladdi 11406 | Move the left term in a su... |
| mvlladdi 11407 | Move the left term in a su... |
| subid 11408 | Subtraction of a number fr... |
| subid1 11409 | Identity law for subtracti... |
| npncan 11410 | Cancellation law for subtr... |
| nppcan 11411 | Cancellation law for subtr... |
| nnpcan 11412 | Cancellation law for subtr... |
| nppcan3 11413 | Cancellation law for subtr... |
| subcan2 11414 | Cancellation law for subtr... |
| subeq0 11415 | If the difference between ... |
| npncan2 11416 | Cancellation law for subtr... |
| subsub2 11417 | Law for double subtraction... |
| nncan 11418 | Cancellation law for subtr... |
| subsub 11419 | Law for double subtraction... |
| nppcan2 11420 | Cancellation law for subtr... |
| subsub3 11421 | Law for double subtraction... |
| subsub4 11422 | Law for double subtraction... |
| sub32 11423 | Swap the second and third ... |
| nnncan 11424 | Cancellation law for subtr... |
| nnncan1 11425 | Cancellation law for subtr... |
| nnncan2 11426 | Cancellation law for subtr... |
| npncan3 11427 | Cancellation law for subtr... |
| pnpcan 11428 | Cancellation law for mixed... |
| pnpcan2 11429 | Cancellation law for mixed... |
| pnncan 11430 | Cancellation law for mixed... |
| ppncan 11431 | Cancellation law for mixed... |
| addsub4 11432 | Rearrangement of 4 terms i... |
| subadd4 11433 | Rearrangement of 4 terms i... |
| sub4 11434 | Rearrangement of 4 terms i... |
| neg0 11435 | Minus 0 equals 0. (Contri... |
| negid 11436 | Addition of a number and i... |
| negsub 11437 | Relationship between subtr... |
| subneg 11438 | Relationship between subtr... |
| negneg 11439 | A number is equal to the n... |
| neg11 11440 | Negative is one-to-one. (... |
| negcon1 11441 | Negative contraposition la... |
| negcon2 11442 | Negative contraposition la... |
| negeq0 11443 | A number is zero iff its n... |
| subcan 11444 | Cancellation law for subtr... |
| negsubdi 11445 | Distribution of negative o... |
| negdi 11446 | Distribution of negative o... |
| negdi2 11447 | Distribution of negative o... |
| negsubdi2 11448 | Distribution of negative o... |
| neg2sub 11449 | Relationship between subtr... |
| renegcli 11450 | Closure law for negative o... |
| resubcli 11451 | Closure law for subtractio... |
| renegcl 11452 | Closure law for negative o... |
| resubcl 11453 | Closure law for subtractio... |
| negreb 11454 | The negative of a real is ... |
| peano2cnm 11455 | "Reverse" second Peano pos... |
| peano2rem 11456 | "Reverse" second Peano pos... |
| negcli 11457 | Closure law for negative. ... |
| negidi 11458 | Addition of a number and i... |
| negnegi 11459 | A number is equal to the n... |
| subidi 11460 | Subtraction of a number fr... |
| subid1i 11461 | Identity law for subtracti... |
| negne0bi 11462 | A number is nonzero iff it... |
| negrebi 11463 | The negative of a real is ... |
| negne0i 11464 | The negative of a nonzero ... |
| subcli 11465 | Closure law for subtractio... |
| pncan3i 11466 | Subtraction and addition o... |
| negsubi 11467 | Relationship between subtr... |
| subnegi 11468 | Relationship between subtr... |
| subeq0i 11469 | If the difference between ... |
| neg11i 11470 | Negative is one-to-one. (... |
| negcon1i 11471 | Negative contraposition la... |
| negcon2i 11472 | Negative contraposition la... |
| negdii 11473 | Distribution of negative o... |
| negsubdii 11474 | Distribution of negative o... |
| negsubdi2i 11475 | Distribution of negative o... |
| subaddi 11476 | Relationship between subtr... |
| subadd2i 11477 | Relationship between subtr... |
| subaddrii 11478 | Relationship between subtr... |
| subsub23i 11479 | Swap subtrahend and result... |
| addsubassi 11480 | Associative-type law for s... |
| addsubi 11481 | Law for subtraction and ad... |
| subcani 11482 | Cancellation law for subtr... |
| subcan2i 11483 | Cancellation law for subtr... |
| pnncani 11484 | Cancellation law for mixed... |
| addsub4i 11485 | Rearrangement of 4 terms i... |
| 0reALT 11486 | Alternate proof of ~ 0re .... |
| negcld 11487 | Closure law for negative. ... |
| subidd 11488 | Subtraction of a number fr... |
| subid1d 11489 | Identity law for subtracti... |
| negidd 11490 | Addition of a number and i... |
| negnegd 11491 | A number is equal to the n... |
| negeq0d 11492 | A number is zero iff its n... |
| negne0bd 11493 | A number is nonzero iff it... |
| negcon1d 11494 | Contraposition law for una... |
| negcon1ad 11495 | Contraposition law for una... |
| neg11ad 11496 | The negatives of two compl... |
| negned 11497 | If two complex numbers are... |
| negne0d 11498 | The negative of a nonzero ... |
| negrebd 11499 | The negative of a real is ... |
| subcld 11500 | Closure law for subtractio... |
| pncand 11501 | Cancellation law for subtr... |
| pncan2d 11502 | Cancellation law for subtr... |
| pncan3d 11503 | Subtraction and addition o... |
| npcand 11504 | Cancellation law for subtr... |
| nncand 11505 | Cancellation law for subtr... |
| negsubd 11506 | Relationship between subtr... |
| subnegd 11507 | Relationship between subtr... |
| subeq0d 11508 | If the difference between ... |
| subne0d 11509 | Two unequal numbers have n... |
| subeq0ad 11510 | The difference of two comp... |
| subne0ad 11511 | If the difference of two c... |
| neg11d 11512 | If the difference between ... |
| negdid 11513 | Distribution of negative o... |
| negdi2d 11514 | Distribution of negative o... |
| negsubdid 11515 | Distribution of negative o... |
| negsubdi2d 11516 | Distribution of negative o... |
| neg2subd 11517 | Relationship between subtr... |
| subaddd 11518 | Relationship between subtr... |
| subadd2d 11519 | Relationship between subtr... |
| addsubassd 11520 | Associative-type law for s... |
| addsubd 11521 | Law for subtraction and ad... |
| subadd23d 11522 | Commutative/associative la... |
| addsub12d 11523 | Commutative/associative la... |
| npncand 11524 | Cancellation law for subtr... |
| nppcand 11525 | Cancellation law for subtr... |
| nppcan2d 11526 | Cancellation law for subtr... |
| nppcan3d 11527 | Cancellation law for subtr... |
| subsubd 11528 | Law for double subtraction... |
| subsub2d 11529 | Law for double subtraction... |
| subsub3d 11530 | Law for double subtraction... |
| subsub4d 11531 | Law for double subtraction... |
| sub32d 11532 | Swap the second and third ... |
| nnncand 11533 | Cancellation law for subtr... |
| nnncan1d 11534 | Cancellation law for subtr... |
| nnncan2d 11535 | Cancellation law for subtr... |
| npncan3d 11536 | Cancellation law for subtr... |
| pnpcand 11537 | Cancellation law for mixed... |
| pnpcan2d 11538 | Cancellation law for mixed... |
| pnncand 11539 | Cancellation law for mixed... |
| ppncand 11540 | Cancellation law for mixed... |
| subcand 11541 | Cancellation law for subtr... |
| subcan2d 11542 | Cancellation law for subtr... |
| subcanad 11543 | Cancellation law for subtr... |
| subneintrd 11544 | Introducing subtraction on... |
| subcan2ad 11545 | Cancellation law for subtr... |
| subneintr2d 11546 | Introducing subtraction on... |
| addsub4d 11547 | Rearrangement of 4 terms i... |
| subadd4d 11548 | Rearrangement of 4 terms i... |
| sub4d 11549 | Rearrangement of 4 terms i... |
| 2addsubd 11550 | Law for subtraction and ad... |
| addsubeq4d 11551 | Relation between sums and ... |
| subsubadd23 11552 | Swap the second and the th... |
| addsubsub23 11553 | Swap the second and the th... |
| subeqxfrd 11554 | Transfer two terms of a su... |
| mvlraddd 11555 | Move the right term in a s... |
| mvlladdd 11556 | Move the left term in a su... |
| mvrraddd 11557 | Move the right term in a s... |
| mvrladdd 11558 | Move the left term in a su... |
| assraddsubd 11559 | Associate RHS addition-sub... |
| subaddeqd 11560 | Transfer two terms of a su... |
| addlsub 11561 | Left-subtraction: Subtrac... |
| addrsub 11562 | Right-subtraction: Subtra... |
| subexsub 11563 | A subtraction law: Exchan... |
| addid0 11564 | If adding a number to a an... |
| addn0nid 11565 | Adding a nonzero number to... |
| pnpncand 11566 | Addition/subtraction cance... |
| subeqrev 11567 | Reverse the order of subtr... |
| addeq0 11568 | Two complex numbers add up... |
| pncan1 11569 | Cancellation law for addit... |
| npcan1 11570 | Cancellation law for subtr... |
| subeq0bd 11571 | If two complex numbers are... |
| renegcld 11572 | Closure law for negative o... |
| resubcld 11573 | Closure law for subtractio... |
| negn0 11574 | The image under negation o... |
| negf1o 11575 | Negation is an isomorphism... |
| kcnktkm1cn 11576 | k times k minus 1 is a com... |
| muladd 11577 | Product of two sums. (Con... |
| subdi 11578 | Distribution of multiplica... |
| subdir 11579 | Distribution of multiplica... |
| ine0 11580 | The imaginary unit ` _i ` ... |
| mulneg1 11581 | Product with negative is n... |
| mulneg2 11582 | The product with a negativ... |
| mulneg12 11583 | Swap the negative sign in ... |
| mul2neg 11584 | Product of two negatives. ... |
| submul2 11585 | Convert a subtraction to a... |
| mulm1 11586 | Product with minus one is ... |
| addneg1mul 11587 | Addition with product with... |
| mulsub 11588 | Product of two differences... |
| mulsub2 11589 | Swap the order of subtract... |
| mulm1i 11590 | Product with minus one is ... |
| mulneg1i 11591 | Product with negative is n... |
| mulneg2i 11592 | Product with negative is n... |
| mul2negi 11593 | Product of two negatives. ... |
| subdii 11594 | Distribution of multiplica... |
| subdiri 11595 | Distribution of multiplica... |
| muladdi 11596 | Product of two sums. (Con... |
| mulm1d 11597 | Product with minus one is ... |
| mulneg1d 11598 | Product with negative is n... |
| mulneg2d 11599 | Product with negative is n... |
| mul2negd 11600 | Product of two negatives. ... |
| subdid 11601 | Distribution of multiplica... |
| subdird 11602 | Distribution of multiplica... |
| muladdd 11603 | Product of two sums. (Con... |
| mulsubd 11604 | Product of two differences... |
| muls1d 11605 | Multiplication by one minu... |
| mulsubfacd 11606 | Multiplication followed by... |
| addmulsub 11607 | The product of a sum and a... |
| subaddmulsub 11608 | The difference with a prod... |
| mulsubaddmulsub 11609 | A special difference of a ... |
| gt0ne0 11610 | Positive implies nonzero. ... |
| lt0ne0 11611 | A number which is less tha... |
| ltadd1 11612 | Addition to both sides of ... |
| leadd1 11613 | Addition to both sides of ... |
| leadd2 11614 | Addition to both sides of ... |
| ltsubadd 11615 | 'Less than' relationship b... |
| ltsubadd2 11616 | 'Less than' relationship b... |
| lesubadd 11617 | 'Less than or equal to' re... |
| lesubadd2 11618 | 'Less than or equal to' re... |
| ltaddsub 11619 | 'Less than' relationship b... |
| ltaddsub2 11620 | 'Less than' relationship b... |
| leaddsub 11621 | 'Less than or equal to' re... |
| leaddsub2 11622 | 'Less than or equal to' re... |
| suble 11623 | Swap subtrahends in an ine... |
| lesub 11624 | Swap subtrahends in an ine... |
| ltsub23 11625 | 'Less than' relationship b... |
| ltsub13 11626 | 'Less than' relationship b... |
| le2add 11627 | Adding both sides of two '... |
| ltleadd 11628 | Adding both sides of two o... |
| leltadd 11629 | Adding both sides of two o... |
| lt2add 11630 | Adding both sides of two '... |
| addgt0 11631 | The sum of 2 positive numb... |
| addgegt0 11632 | The sum of nonnegative and... |
| addgtge0 11633 | The sum of nonnegative and... |
| addge0 11634 | The sum of 2 nonnegative n... |
| ltaddpos 11635 | Adding a positive number t... |
| ltaddpos2 11636 | Adding a positive number t... |
| ltsubpos 11637 | Subtracting a positive num... |
| posdif 11638 | Comparison of two numbers ... |
| lesub1 11639 | Subtraction from both side... |
| lesub2 11640 | Subtraction of both sides ... |
| ltsub1 11641 | Subtraction from both side... |
| ltsub2 11642 | Subtraction of both sides ... |
| lt2sub 11643 | Subtracting both sides of ... |
| le2sub 11644 | Subtracting both sides of ... |
| ltneg 11645 | Negative of both sides of ... |
| ltnegcon1 11646 | Contraposition of negative... |
| ltnegcon2 11647 | Contraposition of negative... |
| leneg 11648 | Negative of both sides of ... |
| lenegcon1 11649 | Contraposition of negative... |
| lenegcon2 11650 | Contraposition of negative... |
| lt0neg1 11651 | Comparison of a number and... |
| lt0neg2 11652 | Comparison of a number and... |
| le0neg1 11653 | Comparison of a number and... |
| le0neg2 11654 | Comparison of a number and... |
| addge01 11655 | A number is less than or e... |
| addge02 11656 | A number is less than or e... |
| add20 11657 | Two nonnegative numbers ar... |
| subge0 11658 | Nonnegative subtraction. ... |
| suble0 11659 | Nonpositive subtraction. ... |
| leaddle0 11660 | The sum of a real number a... |
| subge02 11661 | Nonnegative subtraction. ... |
| lesub0 11662 | Lemma to show a nonnegativ... |
| mulge0 11663 | The product of two nonnega... |
| mullt0 11664 | The product of two negativ... |
| msqgt0 11665 | A nonzero square is positi... |
| msqge0 11666 | A square is nonnegative. ... |
| 0lt1 11667 | 0 is less than 1. Theorem... |
| 0le1 11668 | 0 is less than or equal to... |
| relin01 11669 | An interval law for less t... |
| ltordlem 11670 | Lemma for ~ ltord1 . (Con... |
| ltord1 11671 | Infer an ordering relation... |
| leord1 11672 | Infer an ordering relation... |
| eqord1 11673 | A strictly increasing real... |
| ltord2 11674 | Infer an ordering relation... |
| leord2 11675 | Infer an ordering relation... |
| eqord2 11676 | A strictly decreasing real... |
| wloglei 11677 | Form of ~ wlogle where bot... |
| wlogle 11678 | If the predicate ` ch ( x ... |
| leidi 11679 | 'Less than or equal to' is... |
| gt0ne0i 11680 | Positive means nonzero (us... |
| gt0ne0ii 11681 | Positive implies nonzero. ... |
| msqgt0i 11682 | A nonzero square is positi... |
| msqge0i 11683 | A square is nonnegative. ... |
| addgt0i 11684 | Addition of 2 positive num... |
| addge0i 11685 | Addition of 2 nonnegative ... |
| addgegt0i 11686 | Addition of nonnegative an... |
| addgt0ii 11687 | Addition of 2 positive num... |
| add20i 11688 | Two nonnegative numbers ar... |
| ltnegi 11689 | Negative of both sides of ... |
| lenegi 11690 | Negative of both sides of ... |
| ltnegcon2i 11691 | Contraposition of negative... |
| mulge0i 11692 | The product of two nonnega... |
| lesub0i 11693 | Lemma to show a nonnegativ... |
| ltaddposi 11694 | Adding a positive number t... |
| posdifi 11695 | Comparison of two numbers ... |
| ltnegcon1i 11696 | Contraposition of negative... |
| lenegcon1i 11697 | Contraposition of negative... |
| subge0i 11698 | Nonnegative subtraction. ... |
| ltadd1i 11699 | Addition to both sides of ... |
| leadd1i 11700 | Addition to both sides of ... |
| leadd2i 11701 | Addition to both sides of ... |
| ltsubaddi 11702 | 'Less than' relationship b... |
| lesubaddi 11703 | 'Less than or equal to' re... |
| ltsubadd2i 11704 | 'Less than' relationship b... |
| lesubadd2i 11705 | 'Less than or equal to' re... |
| ltaddsubi 11706 | 'Less than' relationship b... |
| lt2addi 11707 | Adding both side of two in... |
| le2addi 11708 | Adding both side of two in... |
| gt0ne0d 11709 | Positive implies nonzero. ... |
| lt0ne0d 11710 | Something less than zero i... |
| leidd 11711 | 'Less than or equal to' is... |
| msqgt0d 11712 | A nonzero square is positi... |
| msqge0d 11713 | A square is nonnegative. ... |
| lt0neg1d 11714 | Comparison of a number and... |
| lt0neg2d 11715 | Comparison of a number and... |
| le0neg1d 11716 | Comparison of a number and... |
| le0neg2d 11717 | Comparison of a number and... |
| addgegt0d 11718 | Addition of nonnegative an... |
| addgtge0d 11719 | Addition of positive and n... |
| addgt0d 11720 | Addition of 2 positive num... |
| addge0d 11721 | Addition of 2 nonnegative ... |
| mulge0d 11722 | The product of two nonnega... |
| ltnegd 11723 | Negative of both sides of ... |
| lenegd 11724 | Negative of both sides of ... |
| ltnegcon1d 11725 | Contraposition of negative... |
| ltnegcon2d 11726 | Contraposition of negative... |
| lenegcon1d 11727 | Contraposition of negative... |
| lenegcon2d 11728 | Contraposition of negative... |
| ltaddposd 11729 | Adding a positive number t... |
| ltaddpos2d 11730 | Adding a positive number t... |
| ltsubposd 11731 | Subtracting a positive num... |
| posdifd 11732 | Comparison of two numbers ... |
| addge01d 11733 | A number is less than or e... |
| addge02d 11734 | A number is less than or e... |
| subge0d 11735 | Nonnegative subtraction. ... |
| suble0d 11736 | Nonpositive subtraction. ... |
| subge02d 11737 | Nonnegative subtraction. ... |
| ltadd1d 11738 | Addition to both sides of ... |
| leadd1d 11739 | Addition to both sides of ... |
| leadd2d 11740 | Addition to both sides of ... |
| ltsubaddd 11741 | 'Less than' relationship b... |
| lesubaddd 11742 | 'Less than or equal to' re... |
| ltsubadd2d 11743 | 'Less than' relationship b... |
| lesubadd2d 11744 | 'Less than or equal to' re... |
| ltaddsubd 11745 | 'Less than' relationship b... |
| ltaddsub2d 11746 | 'Less than' relationship b... |
| leaddsub2d 11747 | 'Less than or equal to' re... |
| subled 11748 | Swap subtrahends in an ine... |
| lesubd 11749 | Swap subtrahends in an ine... |
| ltsub23d 11750 | 'Less than' relationship b... |
| ltsub13d 11751 | 'Less than' relationship b... |
| lesub1d 11752 | Subtraction from both side... |
| lesub2d 11753 | Subtraction of both sides ... |
| ltsub1d 11754 | Subtraction from both side... |
| ltsub2d 11755 | Subtraction of both sides ... |
| ltadd1dd 11756 | Addition to both sides of ... |
| ltsub1dd 11757 | Subtraction from both side... |
| ltsub2dd 11758 | Subtraction of both sides ... |
| leadd1dd 11759 | Addition to both sides of ... |
| leadd2dd 11760 | Addition to both sides of ... |
| lesub1dd 11761 | Subtraction from both side... |
| lesub2dd 11762 | Subtraction of both sides ... |
| lesub3d 11763 | The result of subtracting ... |
| le2addd 11764 | Adding both side of two in... |
| le2subd 11765 | Subtracting both sides of ... |
| ltleaddd 11766 | Adding both sides of two o... |
| leltaddd 11767 | Adding both sides of two o... |
| lt2addd 11768 | Adding both side of two in... |
| lt2subd 11769 | Subtracting both sides of ... |
| possumd 11770 | Condition for a positive s... |
| sublt0d 11771 | When a subtraction gives a... |
| ltaddsublt 11772 | Addition and subtraction o... |
| 1le1 11773 | One is less than or equal ... |
| ixi 11774 | ` _i ` times itself is min... |
| recextlem1 11775 | Lemma for ~ recex . (Cont... |
| recextlem2 11776 | Lemma for ~ recex . (Cont... |
| recex 11777 | Existence of reciprocal of... |
| mulcand 11778 | Cancellation law for multi... |
| mulcan2d 11779 | Cancellation law for multi... |
| mulcanad 11780 | Cancellation of a nonzero ... |
| mulcan2ad 11781 | Cancellation of a nonzero ... |
| mulcan 11782 | Cancellation law for multi... |
| mulcan2 11783 | Cancellation law for multi... |
| mulcani 11784 | Cancellation law for multi... |
| mul0or 11785 | If a product is zero, one ... |
| mulne0b 11786 | The product of two nonzero... |
| mulne0 11787 | The product of two nonzero... |
| mulne0i 11788 | The product of two nonzero... |
| muleqadd 11789 | Property of numbers whose ... |
| receu 11790 | Existential uniqueness of ... |
| mulnzcnf 11791 | Multiplication maps nonzer... |
| mul0ori 11792 | If a product is zero, one ... |
| mul0ord 11793 | If a product is zero, one ... |
| msq0i 11794 | A number is zero iff its s... |
| msq0d 11795 | A number is zero iff its s... |
| mulne0bd 11796 | The product of two nonzero... |
| mulne0d 11797 | The product of two nonzero... |
| mulcan1g 11798 | A generalized form of the ... |
| mulcan2g 11799 | A generalized form of the ... |
| mulne0bad 11800 | A factor of a nonzero comp... |
| mulne0bbd 11801 | A factor of a nonzero comp... |
| 1div0 11804 | You can't divide by zero, ... |
| 1div0OLD 11805 | Obsolete version of ~ 1div... |
| divval 11806 | Value of division: if ` A ... |
| divmul 11807 | Relationship between divis... |
| divmul2 11808 | Relationship between divis... |
| divmul3 11809 | Relationship between divis... |
| divcl 11810 | Closure law for division. ... |
| reccl 11811 | Closure law for reciprocal... |
| divcan2 11812 | A cancellation law for div... |
| divcan1 11813 | A cancellation law for div... |
| diveq0 11814 | A ratio is zero iff the nu... |
| divne0b 11815 | The ratio of nonzero numbe... |
| divne0 11816 | The ratio of nonzero numbe... |
| recne0 11817 | The reciprocal of a nonzer... |
| recid 11818 | Multiplication of a number... |
| recid2 11819 | Multiplication of a number... |
| divrec 11820 | Relationship between divis... |
| divrec2 11821 | Relationship between divis... |
| divass 11822 | An associative law for div... |
| div23 11823 | A commutative/associative ... |
| div32 11824 | A commutative/associative ... |
| div13 11825 | A commutative/associative ... |
| div12 11826 | A commutative/associative ... |
| divmulass 11827 | An associative law for div... |
| divmulasscom 11828 | An associative/commutative... |
| divdir 11829 | Distribution of division o... |
| divcan3 11830 | A cancellation law for div... |
| divcan4 11831 | A cancellation law for div... |
| div11 11832 | One-to-one relationship fo... |
| div11OLD 11833 | Obsolete version of ~ div1... |
| diveq1 11834 | Equality in terms of unit ... |
| divid 11835 | A number divided by itself... |
| dividOLD 11836 | Obsolete version of ~ divi... |
| div0 11837 | Division into zero is zero... |
| div0OLD 11838 | Obsolete version of ~ div0... |
| div1 11839 | A number divided by 1 is i... |
| 1div1e1 11840 | 1 divided by 1 is 1. (Con... |
| divneg 11841 | Move negative sign inside ... |
| muldivdir 11842 | Distribution of division o... |
| divsubdir 11843 | Distribution of division o... |
| muldivdid 11844 | Distribution of division o... |
| subdivcomb1 11845 | Bring a term in a subtract... |
| subdivcomb2 11846 | Bring a term in a subtract... |
| recrec 11847 | A number is equal to the r... |
| rec11 11848 | Reciprocal is one-to-one. ... |
| rec11r 11849 | Mutual reciprocals. (Cont... |
| divmuldiv 11850 | Multiplication of two rati... |
| divdivdiv 11851 | Division of two ratios. T... |
| divcan5 11852 | Cancellation of common fac... |
| divmul13 11853 | Swap the denominators in t... |
| divmul24 11854 | Swap the numerators in the... |
| divmuleq 11855 | Cross-multiply in an equal... |
| recdiv 11856 | The reciprocal of a ratio.... |
| divcan6 11857 | Cancellation of inverted f... |
| divdiv32 11858 | Swap denominators in a div... |
| divcan7 11859 | Cancel equal divisors in a... |
| dmdcan 11860 | Cancellation law for divis... |
| divdiv1 11861 | Division into a fraction. ... |
| divdiv2 11862 | Division by a fraction. (... |
| recdiv2 11863 | Division into a reciprocal... |
| ddcan 11864 | Cancellation in a double d... |
| divadddiv 11865 | Addition of two ratios. T... |
| divsubdiv 11866 | Subtraction of two ratios.... |
| conjmul 11867 | Two numbers whose reciproc... |
| rereccl 11868 | Closure law for reciprocal... |
| redivcl 11869 | Closure law for division o... |
| eqneg 11870 | A number equal to its nega... |
| eqnegd 11871 | A complex number equals it... |
| eqnegad 11872 | If a complex number equals... |
| div2neg 11873 | Quotient of two negatives.... |
| divneg2 11874 | Move negative sign inside ... |
| recclzi 11875 | Closure law for reciprocal... |
| recne0zi 11876 | The reciprocal of a nonzer... |
| recidzi 11877 | Multiplication of a number... |
| div1i 11878 | A number divided by 1 is i... |
| eqnegi 11879 | A number equal to its nega... |
| reccli 11880 | Closure law for reciprocal... |
| recidi 11881 | Multiplication of a number... |
| recreci 11882 | A number is equal to the r... |
| dividi 11883 | A number divided by itself... |
| div0i 11884 | Division into zero is zero... |
| divclzi 11885 | Closure law for division. ... |
| divcan1zi 11886 | A cancellation law for div... |
| divcan2zi 11887 | A cancellation law for div... |
| divreczi 11888 | Relationship between divis... |
| divcan3zi 11889 | A cancellation law for div... |
| divcan4zi 11890 | A cancellation law for div... |
| rec11i 11891 | Reciprocal is one-to-one. ... |
| divcli 11892 | Closure law for division. ... |
| divcan2i 11893 | A cancellation law for div... |
| divcan1i 11894 | A cancellation law for div... |
| divreci 11895 | Relationship between divis... |
| divcan3i 11896 | A cancellation law for div... |
| divcan4i 11897 | A cancellation law for div... |
| divne0i 11898 | The ratio of nonzero numbe... |
| rec11ii 11899 | Reciprocal is one-to-one. ... |
| divasszi 11900 | An associative law for div... |
| divmulzi 11901 | Relationship between divis... |
| divdirzi 11902 | Distribution of division o... |
| divdiv23zi 11903 | Swap denominators in a div... |
| divmuli 11904 | Relationship between divis... |
| divdiv32i 11905 | Swap denominators in a div... |
| divassi 11906 | An associative law for div... |
| divdiri 11907 | Distribution of division o... |
| div23i 11908 | A commutative/associative ... |
| div11i 11909 | One-to-one relationship fo... |
| divmuldivi 11910 | Multiplication of two rati... |
| divmul13i 11911 | Swap denominators of two r... |
| divadddivi 11912 | Addition of two ratios. T... |
| divdivdivi 11913 | Division of two ratios. T... |
| rerecclzi 11914 | Closure law for reciprocal... |
| rereccli 11915 | Closure law for reciprocal... |
| redivclzi 11916 | Closure law for division o... |
| redivcli 11917 | Closure law for division o... |
| div1d 11918 | A number divided by 1 is i... |
| reccld 11919 | Closure law for reciprocal... |
| recne0d 11920 | The reciprocal of a nonzer... |
| recidd 11921 | Multiplication of a number... |
| recid2d 11922 | Multiplication of a number... |
| recrecd 11923 | A number is equal to the r... |
| dividd 11924 | A number divided by itself... |
| div0d 11925 | Division into zero is zero... |
| divcld 11926 | Closure law for division. ... |
| divcan1d 11927 | A cancellation law for div... |
| divcan2d 11928 | A cancellation law for div... |
| divrecd 11929 | Relationship between divis... |
| divrec2d 11930 | Relationship between divis... |
| divcan3d 11931 | A cancellation law for div... |
| divcan4d 11932 | A cancellation law for div... |
| diveq0d 11933 | A ratio is zero iff the nu... |
| diveq1d 11934 | Equality in terms of unit ... |
| diveq1ad 11935 | The quotient of two comple... |
| diveq0ad 11936 | A fraction of complex numb... |
| divne1d 11937 | If two complex numbers are... |
| divne0bd 11938 | A ratio is zero iff the nu... |
| divnegd 11939 | Move negative sign inside ... |
| divneg2d 11940 | Move negative sign inside ... |
| div2negd 11941 | Quotient of two negatives.... |
| divne0d 11942 | The ratio of nonzero numbe... |
| recdivd 11943 | The reciprocal of a ratio.... |
| recdiv2d 11944 | Division into a reciprocal... |
| divcan6d 11945 | Cancellation of inverted f... |
| ddcand 11946 | Cancellation in a double d... |
| rec11d 11947 | Reciprocal is one-to-one. ... |
| divmuld 11948 | Relationship between divis... |
| div32d 11949 | A commutative/associative ... |
| div13d 11950 | A commutative/associative ... |
| divdiv32d 11951 | Swap denominators in a div... |
| divcan5d 11952 | Cancellation of common fac... |
| divcan5rd 11953 | Cancellation of common fac... |
| divcan7d 11954 | Cancel equal divisors in a... |
| dmdcand 11955 | Cancellation law for divis... |
| dmdcan2d 11956 | Cancellation law for divis... |
| divdiv1d 11957 | Division into a fraction. ... |
| divdiv2d 11958 | Division by a fraction. (... |
| divmul2d 11959 | Relationship between divis... |
| divmul3d 11960 | Relationship between divis... |
| divassd 11961 | An associative law for div... |
| div12d 11962 | A commutative/associative ... |
| div23d 11963 | A commutative/associative ... |
| divdird 11964 | Distribution of division o... |
| divsubdird 11965 | Distribution of division o... |
| div11d 11966 | One-to-one relationship fo... |
| divmuldivd 11967 | Multiplication of two rati... |
| divmul13d 11968 | Swap denominators of two r... |
| divmul24d 11969 | Swap the numerators in the... |
| divadddivd 11970 | Addition of two ratios. T... |
| divsubdivd 11971 | Subtraction of two ratios.... |
| divmuleqd 11972 | Cross-multiply in an equal... |
| divdivdivd 11973 | Division of two ratios. T... |
| diveq1bd 11974 | If two complex numbers are... |
| div2sub 11975 | Swap the order of subtract... |
| div2subd 11976 | Swap subtrahend and minuen... |
| rereccld 11977 | Closure law for reciprocal... |
| redivcld 11978 | Closure law for division o... |
| subrecd 11979 | Subtraction of reciprocals... |
| subrec 11980 | Subtraction of reciprocals... |
| subreci 11981 | Subtraction of reciprocals... |
| mvllmuld 11982 | Move the left term in a pr... |
| mvllmuli 11983 | Move the left term in a pr... |
| ldiv 11984 | Left-division. (Contribut... |
| rdiv 11985 | Right-division. (Contribu... |
| mdiv 11986 | A division law. (Contribu... |
| lineq 11987 | Solution of a (scalar) lin... |
| elimgt0 11988 | Hypothesis for weak deduct... |
| elimge0 11989 | Hypothesis for weak deduct... |
| ltp1 11990 | A number is less than itse... |
| lep1 11991 | A number is less than or e... |
| ltm1 11992 | A number minus 1 is less t... |
| lem1 11993 | A number minus 1 is less t... |
| letrp1 11994 | A transitive property of '... |
| p1le 11995 | A transitive property of p... |
| recgt0 11996 | The reciprocal of a positi... |
| prodgt0 11997 | Infer that a multiplicand ... |
| prodgt02 11998 | Infer that a multiplier is... |
| ltmul1a 11999 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12000 | Multiplication of both sid... |
| ltmul2 12001 | Multiplication of both sid... |
| lemul1 12002 | Multiplication of both sid... |
| lemul2 12003 | Multiplication of both sid... |
| lemul1a 12004 | Multiplication of both sid... |
| lemul2a 12005 | Multiplication of both sid... |
| ltmul12a 12006 | Comparison of product of t... |
| lemul12b 12007 | Comparison of product of t... |
| lemul12a 12008 | Comparison of product of t... |
| mulgt1OLD 12009 | Obsolete version of ~ mulg... |
| ltmulgt11 12010 | Multiplication by a number... |
| ltmulgt12 12011 | Multiplication by a number... |
| mulgt1 12012 | The product of two numbers... |
| lemulge11 12013 | Multiplication by a number... |
| lemulge12 12014 | Multiplication by a number... |
| ltdiv1 12015 | Division of both sides of ... |
| lediv1 12016 | Division of both sides of ... |
| gt0div 12017 | Division of a positive num... |
| ge0div 12018 | Division of a nonnegative ... |
| divgt0 12019 | The ratio of two positive ... |
| divge0 12020 | The ratio of nonnegative a... |
| mulge0b 12021 | A condition for multiplica... |
| mulle0b 12022 | A condition for multiplica... |
| mulsuble0b 12023 | A condition for multiplica... |
| ltmuldiv 12024 | 'Less than' relationship b... |
| ltmuldiv2 12025 | 'Less than' relationship b... |
| ltdivmul 12026 | 'Less than' relationship b... |
| ledivmul 12027 | 'Less than or equal to' re... |
| ltdivmul2 12028 | 'Less than' relationship b... |
| lt2mul2div 12029 | 'Less than' relationship b... |
| ledivmul2 12030 | 'Less than or equal to' re... |
| lemuldiv 12031 | 'Less than or equal' relat... |
| lemuldiv2 12032 | 'Less than or equal' relat... |
| ltrec 12033 | The reciprocal of both sid... |
| lerec 12034 | The reciprocal of both sid... |
| lt2msq1 12035 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12036 | Two nonnegative numbers co... |
| ltdiv2 12037 | Division of a positive num... |
| ltrec1 12038 | Reciprocal swap in a 'less... |
| lerec2 12039 | Reciprocal swap in a 'less... |
| ledivdiv 12040 | Invert ratios of positive ... |
| lediv2 12041 | Division of a positive num... |
| ltdiv23 12042 | Swap denominator with othe... |
| lediv23 12043 | Swap denominator with othe... |
| lediv12a 12044 | Comparison of ratio of two... |
| lediv2a 12045 | Division of both sides of ... |
| reclt1 12046 | The reciprocal of a positi... |
| recgt1 12047 | The reciprocal of a positi... |
| recgt1i 12048 | The reciprocal of a number... |
| recp1lt1 12049 | Construct a number less th... |
| recreclt 12050 | Given a positive number ` ... |
| le2msq 12051 | The square function on non... |
| msq11 12052 | The square of a nonnegativ... |
| ledivp1 12053 | "Less than or equal to" an... |
| squeeze0 12054 | If a nonnegative number is... |
| ltp1i 12055 | A number is less than itse... |
| recgt0i 12056 | The reciprocal of a positi... |
| recgt0ii 12057 | The reciprocal of a positi... |
| prodgt0i 12058 | Infer that a multiplicand ... |
| divgt0i 12059 | The ratio of two positive ... |
| divge0i 12060 | The ratio of nonnegative a... |
| ltreci 12061 | The reciprocal of both sid... |
| lereci 12062 | The reciprocal of both sid... |
| lt2msqi 12063 | The square function on non... |
| le2msqi 12064 | The square function on non... |
| msq11i 12065 | The square of a nonnegativ... |
| divgt0i2i 12066 | The ratio of two positive ... |
| ltrecii 12067 | The reciprocal of both sid... |
| divgt0ii 12068 | The ratio of two positive ... |
| ltmul1i 12069 | Multiplication of both sid... |
| ltdiv1i 12070 | Division of both sides of ... |
| ltmuldivi 12071 | 'Less than' relationship b... |
| ltmul2i 12072 | Multiplication of both sid... |
| lemul1i 12073 | Multiplication of both sid... |
| lemul2i 12074 | Multiplication of both sid... |
| ltdiv23i 12075 | Swap denominator with othe... |
| ledivp1i 12076 | "Less than or equal to" an... |
| ltdivp1i 12077 | Less-than and division rel... |
| ltdiv23ii 12078 | Swap denominator with othe... |
| ltmul1ii 12079 | Multiplication of both sid... |
| ltdiv1ii 12080 | Division of both sides of ... |
| ltp1d 12081 | A number is less than itse... |
| lep1d 12082 | A number is less than or e... |
| ltm1d 12083 | A number minus 1 is less t... |
| lem1d 12084 | A number minus 1 is less t... |
| recgt0d 12085 | The reciprocal of a positi... |
| divgt0d 12086 | The ratio of two positive ... |
| mulgt1d 12087 | The product of two numbers... |
| lemulge11d 12088 | Multiplication by a number... |
| lemulge12d 12089 | Multiplication by a number... |
| lemul1ad 12090 | Multiplication of both sid... |
| lemul2ad 12091 | Multiplication of both sid... |
| ltmul12ad 12092 | Comparison of product of t... |
| lemul12ad 12093 | Comparison of product of t... |
| lemul12bd 12094 | Comparison of product of t... |
| fimaxre 12095 | A finite set of real numbe... |
| fimaxre2 12096 | A nonempty finite set of r... |
| fimaxre3 12097 | A nonempty finite set of r... |
| fiminre 12098 | A nonempty finite set of r... |
| fiminre2 12099 | A nonempty finite set of r... |
| negfi 12100 | The negation of a finite s... |
| lbreu 12101 | If a set of reals contains... |
| lbcl 12102 | If a set of reals contains... |
| lble 12103 | If a set of reals contains... |
| lbinf 12104 | If a set of reals contains... |
| lbinfcl 12105 | If a set of reals contains... |
| lbinfle 12106 | If a set of reals contains... |
| sup2 12107 | A nonempty, bounded-above ... |
| sup3 12108 | A version of the completen... |
| infm3lem 12109 | Lemma for ~ infm3 . (Cont... |
| infm3 12110 | The completeness axiom for... |
| suprcl 12111 | Closure of supremum of a n... |
| suprub 12112 | A member of a nonempty bou... |
| suprubd 12113 | Natural deduction form of ... |
| suprcld 12114 | Natural deduction form of ... |
| suprlub 12115 | The supremum of a nonempty... |
| suprnub 12116 | An upper bound is not less... |
| suprleub 12117 | The supremum of a nonempty... |
| supaddc 12118 | The supremum function dist... |
| supadd 12119 | The supremum function dist... |
| supmul1 12120 | The supremum function dist... |
| supmullem1 12121 | Lemma for ~ supmul . (Con... |
| supmullem2 12122 | Lemma for ~ supmul . (Con... |
| supmul 12123 | The supremum function dist... |
| sup3ii 12124 | A version of the completen... |
| suprclii 12125 | Closure of supremum of a n... |
| suprubii 12126 | A member of a nonempty bou... |
| suprlubii 12127 | The supremum of a nonempty... |
| suprnubii 12128 | An upper bound is not less... |
| suprleubii 12129 | The supremum of a nonempty... |
| riotaneg 12130 | The negative of the unique... |
| negiso 12131 | Negation is an order anti-... |
| dfinfre 12132 | The infimum of a set of re... |
| infrecl 12133 | Closure of infimum of a no... |
| infrenegsup 12134 | The infimum of a set of re... |
| infregelb 12135 | Any lower bound of a nonem... |
| infrelb 12136 | If a nonempty set of real ... |
| infrefilb 12137 | The infimum of a finite se... |
| supfirege 12138 | The supremum of a finite s... |
| neg1cn 12139 | -1 is a complex number. (... |
| neg1rr 12140 | -1 is a real number. (Con... |
| neg1ne0 12141 | -1 is nonzero. (Contribut... |
| neg1lt0 12142 | -1 is less than 0. (Contr... |
| negneg1e1 12143 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12144 | The imaginary unit ` _i ` ... |
| rimul 12145 | A real number times the im... |
| cru 12146 | The representation of comp... |
| crne0 12147 | The real representation of... |
| creur 12148 | The real part of a complex... |
| creui 12149 | The imaginary part of a co... |
| cju 12150 | The complex conjugate of a... |
| ofsubeq0 12151 | Function analogue of ~ sub... |
| ofnegsub 12152 | Function analogue of ~ neg... |
| ofsubge0 12153 | Function analogue of ~ sub... |
| indv 12156 | Value of the indicator fun... |
| indval 12157 | Value of the indicator fun... |
| indval0 12158 | The indicator function gen... |
| indval2 12159 | Alternate value of the ind... |
| indf 12160 | An indicator function as a... |
| indfval 12161 | Value of the indicator fun... |
| fvindre 12162 | The range of the indicator... |
| ind1 12163 | Value of the indicator fun... |
| ind0 12164 | Value of the indicator fun... |
| ind1a 12165 | Value of the indicator fun... |
| indconst0 12166 | Indicator of the empty set... |
| indconst1 12167 | Indicator of the whole set... |
| indpi1 12168 | Preimage of the singleton ... |
| nnexALT 12171 | Alternate proof of ~ nnex ... |
| peano5nni 12172 | Peano's inductive postulat... |
| nnssre 12173 | The positive integers are ... |
| nnsscn 12174 | The positive integers are ... |
| nnex 12175 | The set of positive intege... |
| nnre 12176 | A positive integer is a re... |
| nncn 12177 | A positive integer is a co... |
| nnrei 12178 | A positive integer is a re... |
| nncni 12179 | A positive integer is a co... |
| 1nn 12180 | Peano postulate: 1 is a po... |
| peano2nn 12181 | Peano postulate: a success... |
| dfnn2 12182 | Alternate definition of th... |
| dfnn3 12183 | Alternate definition of th... |
| nnred 12184 | A positive integer is a re... |
| nncnd 12185 | A positive integer is a co... |
| peano2nnd 12186 | Peano postulate: a success... |
| nnind 12187 | Principle of Mathematical ... |
| nnindALT 12188 | Principle of Mathematical ... |
| nnindd 12189 | Principle of Mathematical ... |
| nn1m1nn 12190 | Every positive integer is ... |
| nn1suc 12191 | If a statement holds for 1... |
| nnaddcl 12192 | Closure of addition of pos... |
| nnmulcl 12193 | Closure of multiplication ... |
| nnmulcli 12194 | Closure of multiplication ... |
| nnadd1com 12195 | Addition with 1 is commuta... |
| nnaddcom 12196 | Addition is commutative fo... |
| nnaddcomli 12197 | Version of ~ addcomli for ... |
| nnmtmip 12198 | "Minus times minus is plus... |
| nn2ge 12199 | There exists a positive in... |
| nnge1 12200 | A positive integer is one ... |
| nngt1ne1 12201 | A positive integer is grea... |
| nnle1eq1 12202 | A positive integer is less... |
| nngt0 12203 | A positive integer is posi... |
| nnnlt1 12204 | A positive integer is not ... |
| nnnle0 12205 | A positive integer is not ... |
| nnne0 12206 | A positive integer is nonz... |
| nnneneg 12207 | No positive integer is equ... |
| 0nnn 12208 | Zero is not a positive int... |
| 0nnnALT 12209 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12210 | Alternate version of ~ nnn... |
| nngt0i 12211 | A positive integer is posi... |
| nnne0i 12212 | A positive integer is nonz... |
| nndivre 12213 | The quotient of a real and... |
| nnrecre 12214 | The reciprocal of a positi... |
| nnrecgt0 12215 | The reciprocal of a positi... |
| nnsub 12216 | Subtraction of positive in... |
| nnsubi 12217 | Subtraction of positive in... |
| nndiv 12218 | Two ways to express " ` A ... |
| nndivtr 12219 | Transitive property of div... |
| nnge1d 12220 | A positive integer is one ... |
| nngt0d 12221 | A positive integer is posi... |
| nnne0d 12222 | A positive integer is nonz... |
| nnrecred 12223 | The reciprocal of a positi... |
| nnaddcld 12224 | Closure of addition of pos... |
| nnmulcld 12225 | Closure of multiplication ... |
| nndivred 12226 | A positive integer is one ... |
| 1t1e1ALT 12227 | Alternate proof of ~ 1t1e1... |
| nnadddir 12228 | Right-distributivity for n... |
| nnmul1com 12229 | Multiplication with 1 is c... |
| nnmulcom 12230 | Multiplication is commutat... |
| 0ne1 12247 | Zero is different from one... |
| 1m1e0 12248 | One minus one equals zero.... |
| 2nn 12249 | 2 is a positive integer. ... |
| 2re 12250 | The number 2 is real. (Co... |
| 2cn 12251 | The number 2 is a complex ... |
| 2cnALT 12252 | Alternate proof of ~ 2cn .... |
| 2ex 12253 | The number 2 is a set. (C... |
| 2cnd 12254 | The number 2 is a complex ... |
| 3nn 12255 | 3 is a positive integer. ... |
| 3re 12256 | The number 3 is real. (Co... |
| 3cn 12257 | The number 3 is a complex ... |
| 3ex 12258 | The number 3 is a set. (C... |
| 4nn 12259 | 4 is a positive integer. ... |
| 4re 12260 | The number 4 is real. (Co... |
| 4cn 12261 | The number 4 is a complex ... |
| 5nn 12262 | 5 is a positive integer. ... |
| 5re 12263 | The number 5 is real. (Co... |
| 5cn 12264 | The number 5 is a complex ... |
| 6nn 12265 | 6 is a positive integer. ... |
| 6re 12266 | The number 6 is real. (Co... |
| 6cn 12267 | The number 6 is a complex ... |
| 7nn 12268 | 7 is a positive integer. ... |
| 7re 12269 | The number 7 is real. (Co... |
| 7cn 12270 | The number 7 is a complex ... |
| 8nn 12271 | 8 is a positive integer. ... |
| 8re 12272 | The number 8 is real. (Co... |
| 8cn 12273 | The number 8 is a complex ... |
| 9nn 12274 | 9 is a positive integer. ... |
| 9re 12275 | The number 9 is real. (Co... |
| 9cn 12276 | The number 9 is a complex ... |
| 0le0 12277 | Zero is nonnegative. (Con... |
| 0le2 12278 | The number 0 is less than ... |
| 2pos 12279 | The number 2 is positive. ... |
| 2ne0 12280 | The number 2 is nonzero. ... |
| 3pos 12281 | The number 3 is positive. ... |
| 3ne0 12282 | The number 3 is nonzero. ... |
| 4pos 12283 | The number 4 is positive. ... |
| 4ne0 12284 | The number 4 is nonzero. ... |
| 5pos 12285 | The number 5 is positive. ... |
| 6pos 12286 | The number 6 is positive. ... |
| 7pos 12287 | The number 7 is positive. ... |
| 8pos 12288 | The number 8 is positive. ... |
| 9pos 12289 | The number 9 is positive. ... |
| 1pneg1e0 12290 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12291 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12292 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12293 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12294 | Function value at ` N + 1 ... |
| 1p0e1 12295 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12296 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12297 | 2 - 1 = 1. The result is ... |
| 1e2m1 12298 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12299 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12300 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12301 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12302 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12303 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12304 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12305 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12306 | Two plus two equals four. ... |
| 2times 12307 | Two times a number. (Cont... |
| times2 12308 | A number times 2. (Contri... |
| 2timesi 12309 | Two times a number. (Cont... |
| times2i 12310 | A number times 2. (Contri... |
| 2txmxeqx 12311 | Two times a complex number... |
| 2div2e1 12312 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12313 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12314 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12315 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12316 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12317 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12318 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12319 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12320 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12321 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12322 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12323 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12324 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12325 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12326 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12327 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12328 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12329 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12330 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12331 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12332 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12333 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12334 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12335 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12336 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12337 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12338 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12339 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12340 | 2 times 0 equals 0. (Cont... |
| 4div2e2 12341 | One half of four is two. ... |
| 1lt2 12342 | 1 is less than 2. (Contri... |
| 2lt3 12343 | 2 is less than 3. (Contri... |
| 1lt3 12344 | 1 is less than 3. (Contri... |
| 3lt4 12345 | 3 is less than 4. (Contri... |
| 2lt4 12346 | 2 is less than 4. (Contri... |
| 1lt4 12347 | 1 is less than 4. (Contri... |
| 4lt5 12348 | 4 is less than 5. (Contri... |
| 3lt5 12349 | 3 is less than 5. (Contri... |
| 2lt5 12350 | 2 is less than 5. (Contri... |
| 1lt5 12351 | 1 is less than 5. (Contri... |
| 5lt6 12352 | 5 is less than 6. (Contri... |
| 4lt6 12353 | 4 is less than 6. (Contri... |
| 3lt6 12354 | 3 is less than 6. (Contri... |
| 2lt6 12355 | 2 is less than 6. (Contri... |
| 1lt6 12356 | 1 is less than 6. (Contri... |
| 6lt7 12357 | 6 is less than 7. (Contri... |
| 5lt7 12358 | 5 is less than 7. (Contri... |
| 4lt7 12359 | 4 is less than 7. (Contri... |
| 3lt7 12360 | 3 is less than 7. (Contri... |
| 2lt7 12361 | 2 is less than 7. (Contri... |
| 1lt7 12362 | 1 is less than 7. (Contri... |
| 7lt8 12363 | 7 is less than 8. (Contri... |
| 6lt8 12364 | 6 is less than 8. (Contri... |
| 5lt8 12365 | 5 is less than 8. (Contri... |
| 4lt8 12366 | 4 is less than 8. (Contri... |
| 3lt8 12367 | 3 is less than 8. (Contri... |
| 2lt8 12368 | 2 is less than 8. (Contri... |
| 1lt8 12369 | 1 is less than 8. (Contri... |
| 8lt9 12370 | 8 is less than 9. (Contri... |
| 7lt9 12371 | 7 is less than 9. (Contri... |
| 6lt9 12372 | 6 is less than 9. (Contri... |
| 5lt9 12373 | 5 is less than 9. (Contri... |
| 4lt9 12374 | 4 is less than 9. (Contri... |
| 3lt9 12375 | 3 is less than 9. (Contri... |
| 2lt9 12376 | 2 is less than 9. (Contri... |
| 1lt9 12377 | 1 is less than 9. (Contri... |
| 0ne2 12378 | 0 is not equal to 2. (Con... |
| 1ne2 12379 | 1 is not equal to 2. (Con... |
| 1le2 12380 | 1 is less than or equal to... |
| 2cnne0 12381 | 2 is a nonzero complex num... |
| 2rene0 12382 | 2 is a nonzero real number... |
| 1le3 12383 | 1 is less than or equal to... |
| neg1mulneg1e1 12384 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12385 | One-half is real. (Contri... |
| halfcn 12386 | One-half is a complex numb... |
| halfgt0 12387 | One-half is greater than z... |
| halfge0 12388 | One-half is not negative. ... |
| halflt1 12389 | One-half is less than one.... |
| 2halves 12390 | Two halves make a whole. ... |
| 1mhlfehlf 12391 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12392 | An eighth of four thirds i... |
| halfthird 12393 | Half minus a third. (Cont... |
| halfpm6th 12394 | One half plus or minus one... |
| it0e0 12395 | i times 0 equals 0. (Cont... |
| 2mulicn 12396 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12397 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12398 | Closure of half of a numbe... |
| rehalfcl 12399 | Real closure of half. (Co... |
| half0 12400 | Half of a number is zero i... |
| halfpos2 12401 | A number is positive iff i... |
| halfpos 12402 | A positive number is great... |
| halfnneg2 12403 | A number is nonnegative if... |
| halfaddsubcl 12404 | Closure of half-sum and ha... |
| halfaddsub 12405 | Sum and difference of half... |
| subhalfhalf 12406 | Subtracting the half of a ... |
| lt2halves 12407 | A sum is less than the who... |
| addltmul 12408 | Sum is less than product f... |
| nominpos 12409 | There is no smallest posit... |
| avglt1 12410 | Ordering property for aver... |
| avglt2 12411 | Ordering property for aver... |
| avgle1 12412 | Ordering property for aver... |
| avgle2 12413 | Ordering property for aver... |
| avgle 12414 | The average of two numbers... |
| 2timesd 12415 | Two times a number. (Cont... |
| times2d 12416 | A number times 2. (Contri... |
| halfcld 12417 | Closure of half of a numbe... |
| 2halvesd 12418 | Two halves make a whole. ... |
| rehalfcld 12419 | Real closure of half. (Co... |
| lt2halvesd 12420 | A sum is less than the who... |
| rehalfcli 12421 | Half a real number is real... |
| lt2addmuld 12422 | If two real numbers are le... |
| add1p1 12423 | Adding two times 1 to a nu... |
| sub1m1 12424 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12425 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12426 | A complex number increased... |
| div4p1lem1div2 12427 | An integer greater than 5,... |
| nnunb 12428 | The set of positive intege... |
| arch 12429 | Archimedean property of re... |
| nnrecl 12430 | There exists a positive in... |
| bndndx 12431 | A bounded real sequence ` ... |
| elnn0 12434 | Nonnegative integers expre... |
| nnssnn0 12435 | Positive naturals are a su... |
| nn0ssre 12436 | Nonnegative integers are a... |
| nn0sscn 12437 | Nonnegative integers are a... |
| nn0ex 12438 | The set of nonnegative int... |
| nnnn0 12439 | A positive integer is a no... |
| nnnn0i 12440 | A positive integer is a no... |
| nn0re 12441 | A nonnegative integer is a... |
| nn0cn 12442 | A nonnegative integer is a... |
| nn0rei 12443 | A nonnegative integer is a... |
| nn0cni 12444 | A nonnegative integer is a... |
| dfn2 12445 | The set of positive intege... |
| elnnne0 12446 | The positive integer prope... |
| 0nn0 12447 | 0 is a nonnegative integer... |
| 1nn0 12448 | 1 is a nonnegative integer... |
| 2nn0 12449 | 2 is a nonnegative integer... |
| 3nn0 12450 | 3 is a nonnegative integer... |
| 4nn0 12451 | 4 is a nonnegative integer... |
| 5nn0 12452 | 5 is a nonnegative integer... |
| 6nn0 12453 | 6 is a nonnegative integer... |
| 7nn0 12454 | 7 is a nonnegative integer... |
| 8nn0 12455 | 8 is a nonnegative integer... |
| 9nn0 12456 | 9 is a nonnegative integer... |
| nn0ge0 12457 | A nonnegative integer is g... |
| nn0nlt0 12458 | A nonnegative integer is n... |
| nn0ge0i 12459 | Nonnegative integers are n... |
| nn0le0eq0 12460 | A nonnegative integer is l... |
| nn0p1gt0 12461 | A nonnegative integer incr... |
| nnnn0addcl 12462 | A positive integer plus a ... |
| nn0nnaddcl 12463 | A nonnegative integer plus... |
| 0mnnnnn0 12464 | The result of subtracting ... |
| un0addcl 12465 | If ` S ` is closed under a... |
| un0mulcl 12466 | If ` S ` is closed under m... |
| nn0addcl 12467 | Closure of addition of non... |
| nn0mulcl 12468 | Closure of multiplication ... |
| nn0addcli 12469 | Closure of addition of non... |
| nn0mulcli 12470 | Closure of multiplication ... |
| nn0p1nn 12471 | A nonnegative integer plus... |
| peano2nn0 12472 | Second Peano postulate for... |
| nnm1nn0 12473 | A positive integer minus 1... |
| elnn0nn 12474 | The nonnegative integer pr... |
| elnnnn0 12475 | The positive integer prope... |
| elnnnn0b 12476 | The positive integer prope... |
| elnnnn0c 12477 | The positive integer prope... |
| nn0addge1 12478 | A number is less than or e... |
| nn0addge2 12479 | A number is less than or e... |
| nn0addge1i 12480 | A number is less than or e... |
| nn0addge2i 12481 | A number is less than or e... |
| nn0sub 12482 | Subtraction of nonnegative... |
| ltsubnn0 12483 | Subtracting a nonnegative ... |
| nn0negleid 12484 | A nonnegative integer is g... |
| difgtsumgt 12485 | If the difference of a rea... |
| nn0le2x 12486 | A nonnegative integer is l... |
| nn0le2xi 12487 | A nonnegative integer is l... |
| nn0lele2xi 12488 | 'Less than or equal to' im... |
| fcdmnn0supp 12489 | Two ways to write the supp... |
| fcdmnn0fsupp 12490 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12491 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12492 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12493 | A positive integer is a no... |
| nn0red 12494 | A nonnegative integer is a... |
| nn0cnd 12495 | A nonnegative integer is a... |
| nn0ge0d 12496 | A nonnegative integer is g... |
| nn0addcld 12497 | Closure of addition of non... |
| nn0mulcld 12498 | Closure of multiplication ... |
| nn0readdcl 12499 | Closure law for addition o... |
| nn0n0n1ge2 12500 | A nonnegative integer whic... |
| nn0n0n1ge2b 12501 | A nonnegative integer is n... |
| nn0ge2m1nn 12502 | If a nonnegative integer i... |
| nn0ge2m1nn0 12503 | If a nonnegative integer i... |
| nn0nndivcl 12504 | Closure law for dividing o... |
| elxnn0 12507 | An extended nonnegative in... |
| nn0ssxnn0 12508 | The standard nonnegative i... |
| nn0xnn0 12509 | A standard nonnegative int... |
| xnn0xr 12510 | An extended nonnegative in... |
| 0xnn0 12511 | Zero is an extended nonneg... |
| pnf0xnn0 12512 | Positive infinity is an ex... |
| nn0nepnf 12513 | No standard nonnegative in... |
| nn0xnn0d 12514 | A standard nonnegative int... |
| nn0nepnfd 12515 | No standard nonnegative in... |
| xnn0nemnf 12516 | No extended nonnegative in... |
| xnn0xrnemnf 12517 | The extended nonnegative i... |
| xnn0nnn0pnf 12518 | An extended nonnegative in... |
| elz 12521 | Membership in the set of i... |
| nnnegz 12522 | The negative of a positive... |
| zre 12523 | An integer is a real. (Co... |
| zcn 12524 | An integer is a complex nu... |
| zrei 12525 | An integer is a real numbe... |
| zssre 12526 | The integers are a subset ... |
| zsscn 12527 | The integers are a subset ... |
| zex 12528 | The set of integers exists... |
| elnnz 12529 | Positive integer property ... |
| 0z 12530 | Zero is an integer. (Cont... |
| 0zd 12531 | Zero is an integer, deduct... |
| elnn0z 12532 | Nonnegative integer proper... |
| elznn0nn 12533 | Integer property expressed... |
| elznn0 12534 | Integer property expressed... |
| elznn 12535 | Integer property expressed... |
| zle0orge1 12536 | There is no integer in the... |
| elz2 12537 | Membership in the set of i... |
| dfz2 12538 | Alternative definition of ... |
| zexALT 12539 | Alternate proof of ~ zex .... |
| nnz 12540 | A positive integer is an i... |
| nnssz 12541 | Positive integers are a su... |
| nn0ssz 12542 | Nonnegative integers are a... |
| nn0z 12543 | A nonnegative integer is a... |
| nn0zd 12544 | A nonnegative integer is a... |
| nnzd 12545 | A positive integer is an i... |
| nnzi 12546 | A positive integer is an i... |
| nn0zi 12547 | A nonnegative integer is a... |
| elnnz1 12548 | Positive integer property ... |
| znnnlt1 12549 | An integer is not a positi... |
| nnzrab 12550 | Positive integers expresse... |
| nn0zrab 12551 | Nonnegative integers expre... |
| 1z 12552 | One is an integer. (Contr... |
| 1zzd 12553 | One is an integer, deducti... |
| 2z 12554 | 2 is an integer. (Contrib... |
| 3z 12555 | 3 is an integer. (Contrib... |
| 4z 12556 | 4 is an integer. (Contrib... |
| znegcl 12557 | Closure law for negative i... |
| neg1z 12558 | -1 is an integer. (Contri... |
| znegclb 12559 | A complex number is an int... |
| nn0negz 12560 | The negative of a nonnegat... |
| nn0negzi 12561 | The negative of a nonnegat... |
| zaddcl 12562 | Closure of addition of int... |
| peano2z 12563 | Second Peano postulate gen... |
| zsubcl 12564 | Closure of subtraction of ... |
| peano2zm 12565 | "Reverse" second Peano pos... |
| zletr 12566 | Transitive law of ordering... |
| zrevaddcl 12567 | Reverse closure law for ad... |
| znnsub 12568 | The positive difference of... |
| znn0sub 12569 | The nonnegative difference... |
| nzadd 12570 | The sum of a real number n... |
| zmulcl 12571 | Closure of multiplication ... |
| zltp1le 12572 | Integer ordering relation.... |
| zleltp1 12573 | Integer ordering relation.... |
| zlem1lt 12574 | Integer ordering relation.... |
| zltlem1 12575 | Integer ordering relation.... |
| zltlem1d 12576 | Integer ordering relation,... |
| zltp1led 12577 | Integer ordering relation,... |
| zgt0ge1 12578 | An integer greater than ` ... |
| nnleltp1 12579 | Positive integer ordering ... |
| nnltp1le 12580 | Positive integer ordering ... |
| nnaddm1cl 12581 | Closure of addition of pos... |
| nn0ltp1le 12582 | Nonnegative integer orderi... |
| nn0leltp1 12583 | Nonnegative integer orderi... |
| nn0ltlem1 12584 | Nonnegative integer orderi... |
| nn0sub2 12585 | Subtraction of nonnegative... |
| nn0lt10b 12586 | A nonnegative integer less... |
| nn0lt2 12587 | A nonnegative integer less... |
| nn0le2is012 12588 | A nonnegative integer whic... |
| nn0lem1lt 12589 | Nonnegative integer orderi... |
| nnlem1lt 12590 | Positive integer ordering ... |
| nnltlem1 12591 | Positive integer ordering ... |
| nnm1ge0 12592 | A positive integer decreas... |
| nn0ge0div 12593 | Division of a nonnegative ... |
| zdiv 12594 | Two ways to express " ` M ... |
| zdivadd 12595 | Property of divisibility: ... |
| zdivmul 12596 | Property of divisibility: ... |
| zextle 12597 | An extensionality-like pro... |
| zextlt 12598 | An extensionality-like pro... |
| recnz 12599 | The reciprocal of a number... |
| btwnnz 12600 | A number between an intege... |
| gtndiv 12601 | A larger number does not d... |
| halfnz 12602 | One-half is not an integer... |
| 3halfnz 12603 | Three halves is not an int... |
| suprzcl 12604 | The supremum of a bounded-... |
| prime 12605 | Two ways to express " ` A ... |
| msqznn 12606 | The square of a nonzero in... |
| zneo 12607 | No even integer equals an ... |
| nneo 12608 | A positive integer is even... |
| nneoi 12609 | A positive integer is even... |
| zeo 12610 | An integer is even or odd.... |
| zeo2 12611 | An integer is even or odd ... |
| peano2uz2 12612 | Second Peano postulate for... |
| peano5uzi 12613 | Peano's inductive postulat... |
| peano5uzti 12614 | Peano's inductive postulat... |
| dfuzi 12615 | An expression for the uppe... |
| uzind 12616 | Induction on the upper int... |
| uzind2 12617 | Induction on the upper int... |
| uzind3 12618 | Induction on the upper int... |
| nn0ind 12619 | Principle of Mathematical ... |
| nn0indALT 12620 | Principle of Mathematical ... |
| nn0indd 12621 | Principle of Mathematical ... |
| fzind 12622 | Induction on the integers ... |
| fnn0ind 12623 | Induction on the integers ... |
| nn0ind-raph 12624 | Principle of Mathematical ... |
| zindd 12625 | Principle of Mathematical ... |
| fzindd 12626 | Induction on the integers ... |
| btwnz 12627 | Any real number can be san... |
| zred 12628 | An integer is a real numbe... |
| zcnd 12629 | An integer is a complex nu... |
| znegcld 12630 | Closure law for negative i... |
| peano2zd 12631 | Deduction from second Pean... |
| zaddcld 12632 | Closure of addition of int... |
| zsubcld 12633 | Closure of subtraction of ... |
| zmulcld 12634 | Closure of multiplication ... |
| znnn0nn 12635 | The negative of a negative... |
| zadd2cl 12636 | Increasing an integer by 2... |
| zriotaneg 12637 | The negative of the unique... |
| suprfinzcl 12638 | The supremum of a nonempty... |
| 9p1e10 12641 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12642 | Version of the definition ... |
| decex 12643 | A decimal number is a set.... |
| deceq1 12644 | Equality theorem for the d... |
| deceq2 12645 | Equality theorem for the d... |
| deceq1i 12646 | Equality theorem for the d... |
| deceq2i 12647 | Equality theorem for the d... |
| deceq12i 12648 | Equality theorem for the d... |
| numnncl 12649 | Closure for a numeral (wit... |
| num0u 12650 | Add a zero in the units pl... |
| num0h 12651 | Add a zero in the higher p... |
| numcl 12652 | Closure for a decimal inte... |
| numsuc 12653 | The successor of a decimal... |
| deccl 12654 | Closure for a numeral. (C... |
| 10nn 12655 | 10 is a positive integer. ... |
| 10pos 12656 | The number 10 is positive.... |
| 10nn0 12657 | 10 is a nonnegative intege... |
| 10re 12658 | The number 10 is real. (C... |
| decnncl 12659 | Closure for a numeral. (C... |
| dec0u 12660 | Add a zero in the units pl... |
| dec0h 12661 | Add a zero in the higher p... |
| numnncl2 12662 | Closure for a decimal inte... |
| decnncl2 12663 | Closure for a decimal inte... |
| numlt 12664 | Comparing two decimal inte... |
| numltc 12665 | Comparing two decimal inte... |
| le9lt10 12666 | A "decimal digit" (i.e. a ... |
| declt 12667 | Comparing two decimal inte... |
| decltc 12668 | Comparing two decimal inte... |
| declth 12669 | Comparing two decimal inte... |
| decsuc 12670 | The successor of a decimal... |
| 3declth 12671 | Comparing two decimal inte... |
| 3decltc 12672 | Comparing two decimal inte... |
| decle 12673 | Comparing two decimal inte... |
| decleh 12674 | Comparing two decimal inte... |
| declei 12675 | Comparing a digit to a dec... |
| numlti 12676 | Comparing a digit to a dec... |
| declti 12677 | Comparing a digit to a dec... |
| decltdi 12678 | Comparing a digit to a dec... |
| numsucc 12679 | The successor of a decimal... |
| decsucc 12680 | The successor of a decimal... |
| 1e0p1 12681 | The successor of zero. (C... |
| dec10p 12682 | Ten plus an integer. (Con... |
| numma 12683 | Perform a multiply-add of ... |
| nummac 12684 | Perform a multiply-add of ... |
| numma2c 12685 | Perform a multiply-add of ... |
| numadd 12686 | Add two decimal integers `... |
| numaddc 12687 | Add two decimal integers `... |
| nummul1c 12688 | The product of a decimal i... |
| nummul2c 12689 | The product of a decimal i... |
| decma 12690 | Perform a multiply-add of ... |
| decmac 12691 | Perform a multiply-add of ... |
| decma2c 12692 | Perform a multiply-add of ... |
| decadd 12693 | Add two numerals ` M ` and... |
| decaddc 12694 | Add two numerals ` M ` and... |
| decaddc2 12695 | Add two numerals ` M ` and... |
| decrmanc 12696 | Perform a multiply-add of ... |
| decrmac 12697 | Perform a multiply-add of ... |
| decaddm10 12698 | The sum of two multiples o... |
| decaddi 12699 | Add two numerals ` M ` and... |
| decaddci 12700 | Add two numerals ` M ` and... |
| decaddci2 12701 | Add two numerals ` M ` and... |
| decsubi 12702 | Difference between a numer... |
| decmul1 12703 | The product of a numeral w... |
| decmul1c 12704 | The product of a numeral w... |
| decmul2c 12705 | The product of a numeral w... |
| decmulnc 12706 | The product of a numeral w... |
| 11multnc 12707 | The product of 11 (as nume... |
| decmul10add 12708 | A multiplication of a numb... |
| 6p5lem 12709 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12710 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12711 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12712 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12713 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12714 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12715 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12716 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12717 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12718 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12719 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12720 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12721 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12722 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12723 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12724 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12725 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12726 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12727 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12728 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12729 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12730 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12731 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12732 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12733 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12734 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12735 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12736 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12737 | 4 times 3 equals 12. (Con... |
| 4t4e16 12738 | 4 times 4 equals 16. (Con... |
| 5t2e10 12739 | 5 times 2 equals 10. (Con... |
| 5t3e15 12740 | 5 times 3 equals 15. (Con... |
| 5t4e20 12741 | 5 times 4 equals 20. (Con... |
| 5t5e25 12742 | 5 times 5 equals 25. (Con... |
| 6t2e12 12743 | 6 times 2 equals 12. (Con... |
| 6t3e18 12744 | 6 times 3 equals 18. (Con... |
| 6t4e24 12745 | 6 times 4 equals 24. (Con... |
| 6t5e30 12746 | 6 times 5 equals 30. (Con... |
| 6t6e36 12747 | 6 times 6 equals 36. (Con... |
| 7t2e14 12748 | 7 times 2 equals 14. (Con... |
| 7t3e21 12749 | 7 times 3 equals 21. (Con... |
| 7t4e28 12750 | 7 times 4 equals 28. (Con... |
| 7t5e35 12751 | 7 times 5 equals 35. (Con... |
| 7t6e42 12752 | 7 times 6 equals 42. (Con... |
| 7t7e49 12753 | 7 times 7 equals 49. (Con... |
| 8t2e16 12754 | 8 times 2 equals 16. (Con... |
| 8t3e24 12755 | 8 times 3 equals 24. (Con... |
| 8t4e32 12756 | 8 times 4 equals 32. (Con... |
| 8t5e40 12757 | 8 times 5 equals 40. (Con... |
| 8t6e48 12758 | 8 times 6 equals 48. (Con... |
| 8t7e56 12759 | 8 times 7 equals 56. (Con... |
| 8t8e64 12760 | 8 times 8 equals 64. (Con... |
| 9t2e18 12761 | 9 times 2 equals 18. (Con... |
| 9t3e27 12762 | 9 times 3 equals 27. (Con... |
| 9t4e36 12763 | 9 times 4 equals 36. (Con... |
| 9t5e45 12764 | 9 times 5 equals 45. (Con... |
| 9t6e54 12765 | 9 times 6 equals 54. (Con... |
| 9t7e63 12766 | 9 times 7 equals 63. (Con... |
| 9t8e72 12767 | 9 times 8 equals 72. (Con... |
| 9t9e81 12768 | 9 times 9 equals 81. (Con... |
| 9t11e99 12769 | 9 times 11 equals 99. (Co... |
| 9lt10 12770 | 9 is less than 10. (Contr... |
| 8lt10 12771 | 8 is less than 10. (Contr... |
| 7lt10 12772 | 7 is less than 10. (Contr... |
| 6lt10 12773 | 6 is less than 10. (Contr... |
| 5lt10 12774 | 5 is less than 10. (Contr... |
| 4lt10 12775 | 4 is less than 10. (Contr... |
| 3lt10 12776 | 3 is less than 10. (Contr... |
| 2lt10 12777 | 2 is less than 10. (Contr... |
| 1lt10 12778 | 1 is less than 10. (Contr... |
| decbin0 12779 | Decompose base 4 into base... |
| decbin2 12780 | Decompose base 4 into base... |
| decbin3 12781 | Decompose base 4 into base... |
| 5recm6rec 12782 | One fifth minus one sixth.... |
| uzval 12785 | The value of the upper int... |
| uzf 12786 | The domain and codomain of... |
| eluz1 12787 | Membership in the upper se... |
| eluzel2 12788 | Implication of membership ... |
| eluz2 12789 | Membership in an upper set... |
| eluzmn 12790 | Membership in an earlier u... |
| eluz1i 12791 | Membership in an upper set... |
| eluzuzle 12792 | An integer in an upper set... |
| eluzelz 12793 | A member of an upper set o... |
| eluzelre 12794 | A member of an upper set o... |
| eluzelcn 12795 | A member of an upper set o... |
| eluzle 12796 | Implication of membership ... |
| eluz 12797 | Membership in an upper set... |
| uzid 12798 | Membership of the least me... |
| uzidd 12799 | Membership of the least me... |
| uzn0 12800 | The upper integers are all... |
| uztrn 12801 | Transitive law for sets of... |
| uztrn2 12802 | Transitive law for sets of... |
| uzneg 12803 | Contraposition law for upp... |
| uzssz 12804 | An upper set of integers i... |
| uzssre 12805 | An upper set of integers i... |
| uzss 12806 | Subset relationship for tw... |
| uztric 12807 | Totality of the ordering r... |
| uz11 12808 | The upper integers functio... |
| eluzp1m1 12809 | Membership in the next upp... |
| eluzp1l 12810 | Strict ordering implied by... |
| eluzp1p1 12811 | Membership in the next upp... |
| eluzadd 12812 | Membership in a later uppe... |
| eluzsub 12813 | Membership in an earlier u... |
| eluzaddi 12814 | Membership in a later uppe... |
| eluzsubi 12815 | Membership in an earlier u... |
| subeluzsub 12816 | Membership of a difference... |
| uzm1 12817 | Choices for an element of ... |
| uznn0sub 12818 | The nonnegative difference... |
| uzin 12819 | Intersection of two upper ... |
| uzp1 12820 | Choices for an element of ... |
| nn0uz 12821 | Nonnegative integers expre... |
| nnuz 12822 | Positive integers expresse... |
| elnnuz 12823 | A positive integer express... |
| elnn0uz 12824 | A nonnegative integer expr... |
| 1eluzge0 12825 | 1 is an integer greater th... |
| 2eluzge0 12826 | 2 is an integer greater th... |
| 2eluzge1 12827 | 2 is an integer greater th... |
| 5eluz3 12828 | 5 is an integer greater th... |
| uzuzle23 12829 | An integer greater than or... |
| uzuzle24 12830 | An integer greater than or... |
| uzuzle34 12831 | An integer greater than or... |
| uzuzle35 12832 | An integer greater than or... |
| eluz2nn 12833 | An integer greater than or... |
| eluz3nn 12834 | An integer greater than or... |
| eluz4nn 12835 | An integer greater than or... |
| eluz5nn 12836 | An integer greater than or... |
| eluzge2nn0 12837 | If an integer is greater t... |
| eluz2n0 12838 | An integer greater than or... |
| uz3m2nn 12839 | An integer greater than or... |
| uznnssnn 12840 | The upper integers startin... |
| raluz 12841 | Restricted universal quant... |
| raluz2 12842 | Restricted universal quant... |
| rexuz 12843 | Restricted existential qua... |
| rexuz2 12844 | Restricted existential qua... |
| 2rexuz 12845 | Double existential quantif... |
| peano2uz 12846 | Second Peano postulate for... |
| peano2uzs 12847 | Second Peano postulate for... |
| peano2uzr 12848 | Reversed second Peano axio... |
| uzaddcl 12849 | Addition closure law for a... |
| nn0pzuz 12850 | The sum of a nonnegative i... |
| uzind4 12851 | Induction on the upper set... |
| uzind4ALT 12852 | Induction on the upper set... |
| uzind4s 12853 | Induction on the upper set... |
| uzind4s2 12854 | Induction on the upper set... |
| uzind4i 12855 | Induction on the upper int... |
| uzwo 12856 | Well-ordering principle: a... |
| uzwo2 12857 | Well-ordering principle: a... |
| nnwo 12858 | Well-ordering principle: a... |
| nnwof 12859 | Well-ordering principle: a... |
| nnwos 12860 | Well-ordering principle: a... |
| indstr 12861 | Strong Mathematical Induct... |
| eluznn0 12862 | Membership in a nonnegativ... |
| eluznn 12863 | Membership in a positive u... |
| eluz2b1 12864 | Two ways to say "an intege... |
| eluz2gt1 12865 | An integer greater than or... |
| eluz2b2 12866 | Two ways to say "an intege... |
| eluz2b3 12867 | Two ways to say "an intege... |
| uz2m1nn 12868 | One less than an integer g... |
| 1nuz2 12869 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12870 | A positive integer is eith... |
| uz2mulcl 12871 | Closure of multiplication ... |
| indstr2 12872 | Strong Mathematical Induct... |
| uzinfi 12873 | Extract the lower bound of... |
| nninf 12874 | The infimum of the set of ... |
| nn0inf 12875 | The infimum of the set of ... |
| infssuzle 12876 | The infimum of a subset of... |
| infssuzcl 12877 | The infimum of a subset of... |
| ublbneg 12878 | The image under negation o... |
| eqreznegel 12879 | Two ways to express the im... |
| supminf 12880 | The supremum of a bounded-... |
| lbzbi 12881 | If a set of reals is bound... |
| zsupss 12882 | Any nonempty bounded subse... |
| suprzcl2 12883 | The supremum of a bounded-... |
| suprzub 12884 | The supremum of a bounded-... |
| uzsupss 12885 | Any bounded subset of an u... |
| nn01to3 12886 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12887 | Alternate proof of ~ nn0ge... |
| uzwo3 12888 | Well-ordering principle: a... |
| zmin 12889 | There is a unique smallest... |
| zmax 12890 | There is a unique largest ... |
| zbtwnre 12891 | There is a unique integer ... |
| rebtwnz 12892 | There is a unique greatest... |
| elq 12895 | Membership in the set of r... |
| qmulz 12896 | If ` A ` is rational, then... |
| znq 12897 | The ratio of an integer an... |
| qre 12898 | A rational number is a rea... |
| zq 12899 | An integer is a rational n... |
| qred 12900 | A rational number is a rea... |
| zssq 12901 | The integers are a subset ... |
| nn0ssq 12902 | The nonnegative integers a... |
| nnssq 12903 | The positive integers are ... |
| qssre 12904 | The rationals are a subset... |
| qsscn 12905 | The rationals are a subset... |
| qex 12906 | The set of rational number... |
| nnq 12907 | A positive integer is rati... |
| qcn 12908 | A rational number is a com... |
| qexALT 12909 | Alternate proof of ~ qex .... |
| qaddcl 12910 | Closure of addition of rat... |
| qnegcl 12911 | Closure law for the negati... |
| qmulcl 12912 | Closure of multiplication ... |
| qsubcl 12913 | Closure of subtraction of ... |
| qreccl 12914 | Closure of reciprocal of r... |
| qdivcl 12915 | Closure of division of rat... |
| qrevaddcl 12916 | Reverse closure law for ad... |
| nnrecq 12917 | The reciprocal of a positi... |
| irradd 12918 | The sum of an irrational n... |
| irrmul 12919 | The product of an irration... |
| elpq 12920 | A positive rational is the... |
| elpqb 12921 | A class is a positive rati... |
| rpnnen1lem2 12922 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12923 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12924 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12925 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12926 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12927 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12928 | One half of ~ rpnnen , whe... |
| reexALT 12929 | Alternate proof of ~ reex ... |
| cnref1o 12930 | There is a natural one-to-... |
| cnexALT 12931 | The set of complex numbers... |
| xrex 12932 | The set of extended reals ... |
| mpoaddex 12933 | The addition operation is ... |
| addex 12934 | The addition operation is ... |
| mpomulex 12935 | The multiplication operati... |
| mulex 12936 | The multiplication operati... |
| elrp 12939 | Membership in the set of p... |
| elrpii 12940 | Membership in the set of p... |
| 1rp 12941 | 1 is a positive real. (Co... |
| 2rp 12942 | 2 is a positive real. (Co... |
| 3rp 12943 | 3 is a positive real. (Co... |
| 5rp 12944 | 5 is a positive real. (Co... |
| rpssre 12945 | The positive reals are a s... |
| rpre 12946 | A positive real is a real.... |
| rpxr 12947 | A positive real is an exte... |
| rpcn 12948 | A positive real is a compl... |
| nnrp 12949 | A positive integer is a po... |
| rpgt0 12950 | A positive real is greater... |
| rpge0 12951 | A positive real is greater... |
| rpregt0 12952 | A positive real is a posit... |
| rprege0 12953 | A positive real is a nonne... |
| rpne0 12954 | A positive real is nonzero... |
| rprene0 12955 | A positive real is a nonze... |
| rpcnne0 12956 | A positive real is a nonze... |
| neglt 12957 | The negative of a positive... |
| rpcndif0 12958 | A positive real number is ... |
| ralrp 12959 | Quantification over positi... |
| rexrp 12960 | Quantification over positi... |
| rpaddcl 12961 | Closure law for addition o... |
| rpmulcl 12962 | Closure law for multiplica... |
| rpmtmip 12963 | "Minus times minus is plus... |
| rpdivcl 12964 | Closure law for division o... |
| rpreccl 12965 | Closure law for reciprocat... |
| rphalfcl 12966 | Closure law for half of a ... |
| rpgecl 12967 | A number greater than or e... |
| rphalflt 12968 | Half of a positive real is... |
| rerpdivcl 12969 | Closure law for division o... |
| ge0p1rp 12970 | A nonnegative number plus ... |
| rpneg 12971 | Either a nonzero real or i... |
| negelrp 12972 | Elementhood of a negation ... |
| negelrpd 12973 | The negation of a negative... |
| 0nrp 12974 | Zero is not a positive rea... |
| ltsubrp 12975 | Subtracting a positive rea... |
| ltaddrp 12976 | Adding a positive number t... |
| difrp 12977 | Two ways to say one number... |
| elrpd 12978 | Membership in the set of p... |
| nnrpd 12979 | A positive integer is a po... |
| zgt1rpn0n1 12980 | An integer greater than 1 ... |
| rpred 12981 | A positive real is a real.... |
| rpxrd 12982 | A positive real is an exte... |
| rpcnd 12983 | A positive real is a compl... |
| rpgt0d 12984 | A positive real is greater... |
| rpge0d 12985 | A positive real is greater... |
| rpne0d 12986 | A positive real is nonzero... |
| rpregt0d 12987 | A positive real is real an... |
| rprege0d 12988 | A positive real is real an... |
| rprene0d 12989 | A positive real is a nonze... |
| rpcnne0d 12990 | A positive real is a nonze... |
| rpreccld 12991 | Closure law for reciprocat... |
| rprecred 12992 | Closure law for reciprocat... |
| rphalfcld 12993 | Closure law for half of a ... |
| reclt1d 12994 | The reciprocal of a positi... |
| recgt1d 12995 | The reciprocal of a positi... |
| rpaddcld 12996 | Closure law for addition o... |
| rpmulcld 12997 | Closure law for multiplica... |
| rpdivcld 12998 | Closure law for division o... |
| ltrecd 12999 | The reciprocal of both sid... |
| lerecd 13000 | The reciprocal of both sid... |
| ltrec1d 13001 | Reciprocal swap in a 'less... |
| lerec2d 13002 | Reciprocal swap in a 'less... |
| lediv2ad 13003 | Division of both sides of ... |
| ltdiv2d 13004 | Division of a positive num... |
| lediv2d 13005 | Division of a positive num... |
| ledivdivd 13006 | Invert ratios of positive ... |
| divge1 13007 | The ratio of a number over... |
| divlt1lt 13008 | A real number divided by a... |
| divle1le 13009 | A real number divided by a... |
| ledivge1le 13010 | If a number is less than o... |
| ge0p1rpd 13011 | A nonnegative number plus ... |
| rerpdivcld 13012 | Closure law for division o... |
| ltsubrpd 13013 | Subtracting a positive rea... |
| ltaddrpd 13014 | Adding a positive number t... |
| ltaddrp2d 13015 | Adding a positive number t... |
| ltmulgt11d 13016 | Multiplication by a number... |
| ltmulgt12d 13017 | Multiplication by a number... |
| gt0divd 13018 | Division of a positive num... |
| ge0divd 13019 | Division of a nonnegative ... |
| rpgecld 13020 | A number greater than or e... |
| divge0d 13021 | The ratio of nonnegative a... |
| ltmul1d 13022 | The ratio of nonnegative a... |
| ltmul2d 13023 | Multiplication of both sid... |
| lemul1d 13024 | Multiplication of both sid... |
| lemul2d 13025 | Multiplication of both sid... |
| ltdiv1d 13026 | Division of both sides of ... |
| lediv1d 13027 | Division of both sides of ... |
| ltmuldivd 13028 | 'Less than' relationship b... |
| ltmuldiv2d 13029 | 'Less than' relationship b... |
| lemuldivd 13030 | 'Less than or equal to' re... |
| lemuldiv2d 13031 | 'Less than or equal to' re... |
| ltdivmuld 13032 | 'Less than' relationship b... |
| ltdivmul2d 13033 | 'Less than' relationship b... |
| ledivmuld 13034 | 'Less than or equal to' re... |
| ledivmul2d 13035 | 'Less than or equal to' re... |
| ltmul1dd 13036 | The ratio of nonnegative a... |
| ltmul2dd 13037 | Multiplication of both sid... |
| ltdiv1dd 13038 | Division of both sides of ... |
| lediv1dd 13039 | Division of both sides of ... |
| lediv12ad 13040 | Comparison of ratio of two... |
| mul2lt0rlt0 13041 | If the result of a multipl... |
| mul2lt0rgt0 13042 | If the result of a multipl... |
| mul2lt0llt0 13043 | If the result of a multipl... |
| mul2lt0lgt0 13044 | If the result of a multipl... |
| mul2lt0bi 13045 | If the result of a multipl... |
| prodge0rd 13046 | Infer that a multiplicand ... |
| prodge0ld 13047 | Infer that a multiplier is... |
| ltdiv23d 13048 | Swap denominator with othe... |
| lediv23d 13049 | Swap denominator with othe... |
| lt2mul2divd 13050 | The ratio of nonnegative a... |
| nnledivrp 13051 | Division of a positive int... |
| nn0ledivnn 13052 | Division of a nonnegative ... |
| addlelt 13053 | If the sum of a real numbe... |
| ge2halflem1 13054 | Half of an integer greater... |
| ltxr 13061 | The 'less than' binary rel... |
| elxr 13062 | Membership in the set of e... |
| xrnemnf 13063 | An extended real other tha... |
| xrnepnf 13064 | An extended real other tha... |
| xrltnr 13065 | The extended real 'less th... |
| ltpnf 13066 | Any (finite) real is less ... |
| ltpnfd 13067 | Any (finite) real is less ... |
| 0ltpnf 13068 | Zero is less than plus inf... |
| mnflt 13069 | Minus infinity is less tha... |
| mnfltd 13070 | Minus infinity is less tha... |
| mnflt0 13071 | Minus infinity is less tha... |
| mnfltpnf 13072 | Minus infinity is less tha... |
| mnfltxr 13073 | Minus infinity is less tha... |
| pnfnlt 13074 | No extended real is greate... |
| nltmnf 13075 | No extended real is less t... |
| pnfge 13076 | Plus infinity is an upper ... |
| pnfged 13077 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13078 | An extended nonnegative in... |
| 0lepnf 13079 | 0 less than or equal to po... |
| xnn0ge0 13080 | An extended nonnegative in... |
| mnfle 13081 | Minus infinity is less tha... |
| mnfled 13082 | Minus infinity is less tha... |
| xrltnsym 13083 | Ordering on the extended r... |
| xrltnsym2 13084 | 'Less than' is antisymmetr... |
| xrlttri 13085 | Ordering on the extended r... |
| xrlttr 13086 | Ordering on the extended r... |
| xrltso 13087 | 'Less than' is a strict or... |
| xrlttri2 13088 | Trichotomy law for 'less t... |
| xrlttri3 13089 | Trichotomy law for 'less t... |
| xrleloe 13090 | 'Less than or equal' expre... |
| xrleltne 13091 | 'Less than or equal to' im... |
| xrltlen 13092 | 'Less than' expressed in t... |
| dfle2 13093 | Alternative definition of ... |
| dflt2 13094 | Alternative definition of ... |
| xrltle 13095 | 'Less than' implies 'less ... |
| xrltled 13096 | 'Less than' implies 'less ... |
| xrleid 13097 | 'Less than or equal to' is... |
| xrleidd 13098 | 'Less than or equal to' is... |
| xrletri 13099 | Trichotomy law for extende... |
| xrletri3 13100 | Trichotomy law for extende... |
| xrletrid 13101 | Trichotomy law for extende... |
| xrlelttr 13102 | Transitive law for orderin... |
| xrltletr 13103 | Transitive law for orderin... |
| xrletr 13104 | Transitive law for orderin... |
| xrlttrd 13105 | Transitive law for orderin... |
| xrlelttrd 13106 | Transitive law for orderin... |
| xrltletrd 13107 | Transitive law for orderin... |
| xrletrd 13108 | Transitive law for orderin... |
| xrltne 13109 | 'Less than' implies not eq... |
| xrgtned 13110 | 'Greater than' implies not... |
| nltpnft 13111 | An extended real is not le... |
| xgepnf 13112 | An extended real which is ... |
| ngtmnft 13113 | An extended real is not gr... |
| xlemnf 13114 | An extended real which is ... |
| xrrebnd 13115 | An extended real is real i... |
| xrre 13116 | A way of proving that an e... |
| xrre2 13117 | An extended real between t... |
| xrre3 13118 | A way of proving that an e... |
| ge0gtmnf 13119 | A nonnegative extended rea... |
| ge0nemnf 13120 | A nonnegative extended rea... |
| xrrege0 13121 | A nonnegative extended rea... |
| xrmax1 13122 | An extended real is less t... |
| xrmax2 13123 | An extended real is less t... |
| xrmin1 13124 | The minimum of two extende... |
| xrmin2 13125 | The minimum of two extende... |
| xrmaxeq 13126 | The maximum of two extende... |
| xrmineq 13127 | The minimum of two extende... |
| xrmaxlt 13128 | Two ways of saying the max... |
| xrltmin 13129 | Two ways of saying an exte... |
| xrmaxle 13130 | Two ways of saying the max... |
| xrlemin 13131 | Two ways of saying a numbe... |
| max1 13132 | A number is less than or e... |
| max1ALT 13133 | A number is less than or e... |
| max2 13134 | A number is less than or e... |
| 2resupmax 13135 | The supremum of two real n... |
| min1 13136 | The minimum of two numbers... |
| min2 13137 | The minimum of two numbers... |
| maxle 13138 | Two ways of saying the max... |
| lemin 13139 | Two ways of saying a numbe... |
| maxlt 13140 | Two ways of saying the max... |
| ltmin 13141 | Two ways of saying a numbe... |
| lemaxle 13142 | A real number which is les... |
| max0sub 13143 | Decompose a real number in... |
| ifle 13144 | An if statement transforms... |
| z2ge 13145 | There exists an integer gr... |
| qbtwnre 13146 | The rational numbers are d... |
| qbtwnxr 13147 | The rational numbers are d... |
| qsqueeze 13148 | If a nonnegative real is l... |
| qextltlem 13149 | Lemma for ~ qextlt and qex... |
| qextlt 13150 | An extensionality-like pro... |
| qextle 13151 | An extensionality-like pro... |
| xralrple 13152 | Show that ` A ` is less th... |
| alrple 13153 | Show that ` A ` is less th... |
| xnegeq 13154 | Equality of two extended n... |
| xnegex 13155 | A negative extended real e... |
| xnegpnf 13156 | Minus ` +oo ` . Remark of... |
| xnegmnf 13157 | Minus ` -oo ` . Remark of... |
| rexneg 13158 | Minus a real number. Rema... |
| xneg0 13159 | The negative of zero. (Co... |
| xnegcl 13160 | Closure of extended real n... |
| xnegneg 13161 | Extended real version of ~... |
| xneg11 13162 | Extended real version of ~... |
| xltnegi 13163 | Forward direction of ~ xlt... |
| xltneg 13164 | Extended real version of ~... |
| xleneg 13165 | Extended real version of ~... |
| xlt0neg1 13166 | Extended real version of ~... |
| xlt0neg2 13167 | Extended real version of ~... |
| xle0neg1 13168 | Extended real version of ~... |
| xle0neg2 13169 | Extended real version of ~... |
| xaddval 13170 | Value of the extended real... |
| xaddf 13171 | The extended real addition... |
| xmulval 13172 | Value of the extended real... |
| xaddpnf1 13173 | Addition of positive infin... |
| xaddpnf2 13174 | Addition of positive infin... |
| xaddmnf1 13175 | Addition of negative infin... |
| xaddmnf2 13176 | Addition of negative infin... |
| pnfaddmnf 13177 | Addition of positive and n... |
| mnfaddpnf 13178 | Addition of negative and p... |
| rexadd 13179 | The extended real addition... |
| rexsub 13180 | Extended real subtraction ... |
| rexaddd 13181 | The extended real addition... |
| xnn0xaddcl 13182 | The extended nonnegative i... |
| xaddnemnf 13183 | Closure of extended real a... |
| xaddnepnf 13184 | Closure of extended real a... |
| xnegid 13185 | Extended real version of ~... |
| xaddcl 13186 | The extended real addition... |
| xaddcom 13187 | The extended real addition... |
| xaddrid 13188 | Extended real version of ~... |
| xaddlid 13189 | Extended real version of ~... |
| xaddridd 13190 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13191 | Extended nonnegative integ... |
| xnn0lenn0nn0 13192 | An extended nonnegative in... |
| xnn0le2is012 13193 | An extended nonnegative in... |
| xnn0xadd0 13194 | The sum of two extended no... |
| xnegdi 13195 | Extended real version of ~... |
| xaddass 13196 | Associativity of extended ... |
| xaddass2 13197 | Associativity of extended ... |
| xpncan 13198 | Extended real version of ~... |
| xnpcan 13199 | Extended real version of ~... |
| xleadd1a 13200 | Extended real version of ~... |
| xleadd2a 13201 | Commuted form of ~ xleadd1... |
| xleadd1 13202 | Weakened version of ~ xlea... |
| xltadd1 13203 | Extended real version of ~... |
| xltadd2 13204 | Extended real version of ~... |
| xaddge0 13205 | The sum of nonnegative ext... |
| xle2add 13206 | Extended real version of ~... |
| xlt2add 13207 | Extended real version of ~... |
| xsubge0 13208 | Extended real version of ~... |
| xposdif 13209 | Extended real version of ~... |
| xlesubadd 13210 | Under certain conditions, ... |
| xmullem 13211 | Lemma for ~ rexmul . (Con... |
| xmullem2 13212 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13213 | Extended real multiplicati... |
| xmul01 13214 | Extended real version of ~... |
| xmul02 13215 | Extended real version of ~... |
| xmulneg1 13216 | Extended real version of ~... |
| xmulneg2 13217 | Extended real version of ~... |
| rexmul 13218 | The extended real multipli... |
| xmulf 13219 | The extended real multipli... |
| xmulcl 13220 | Closure of extended real m... |
| xmulpnf1 13221 | Multiplication by plus inf... |
| xmulpnf2 13222 | Multiplication by plus inf... |
| xmulmnf1 13223 | Multiplication by minus in... |
| xmulmnf2 13224 | Multiplication by minus in... |
| xmulpnf1n 13225 | Multiplication by plus inf... |
| xmulrid 13226 | Extended real version of ~... |
| xmullid 13227 | Extended real version of ~... |
| xmulm1 13228 | Extended real version of ~... |
| xmulasslem2 13229 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13230 | Extended real version of ~... |
| xmulge0 13231 | Extended real version of ~... |
| xmulasslem 13232 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13233 | Lemma for ~ xmulass . (Co... |
| xmulass 13234 | Associativity of the exten... |
| xlemul1a 13235 | Extended real version of ~... |
| xlemul2a 13236 | Extended real version of ~... |
| xlemul1 13237 | Extended real version of ~... |
| xlemul2 13238 | Extended real version of ~... |
| xltmul1 13239 | Extended real version of ~... |
| xltmul2 13240 | Extended real version of ~... |
| xadddilem 13241 | Lemma for ~ xadddi . (Con... |
| xadddi 13242 | Distributive property for ... |
| xadddir 13243 | Commuted version of ~ xadd... |
| xadddi2 13244 | The assumption that the mu... |
| xadddi2r 13245 | Commuted version of ~ xadd... |
| x2times 13246 | Extended real version of ~... |
| xnegcld 13247 | Closure of extended real n... |
| xaddcld 13248 | The extended real addition... |
| xmulcld 13249 | Closure of extended real m... |
| xadd4d 13250 | Rearrangement of 4 terms i... |
| xnn0add4d 13251 | Rearrangement of 4 terms i... |
| xrsupexmnf 13252 | Adding minus infinity to a... |
| xrinfmexpnf 13253 | Adding plus infinity to a ... |
| xrsupsslem 13254 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13255 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13256 | Any subset of extended rea... |
| xrinfmss 13257 | Any subset of extended rea... |
| xrinfmss2 13258 | Any subset of extended rea... |
| xrub 13259 | By quantifying only over r... |
| supxr 13260 | The supremum of a set of e... |
| supxr2 13261 | The supremum of a set of e... |
| supxrcl 13262 | The supremum of an arbitra... |
| supxrun 13263 | The supremum of the union ... |
| supxrmnf 13264 | Adding minus infinity to a... |
| supxrpnf 13265 | The supremum of a set of e... |
| supxrunb1 13266 | The supremum of an unbound... |
| supxrunb2 13267 | The supremum of an unbound... |
| supxrbnd1 13268 | The supremum of a bounded-... |
| supxrbnd2 13269 | The supremum of a bounded-... |
| xrsup0 13270 | The supremum of an empty s... |
| supxrub 13271 | A member of a set of exten... |
| supxrlub 13272 | The supremum of a set of e... |
| supxrleub 13273 | The supremum of a set of e... |
| supxrre 13274 | The real and extended real... |
| supxrbnd 13275 | The supremum of a bounded-... |
| supxrgtmnf 13276 | The supremum of a nonempty... |
| supxrre1 13277 | The supremum of a nonempty... |
| supxrre2 13278 | The supremum of a nonempty... |
| supxrss 13279 | Smaller sets of extended r... |
| xrsupssd 13280 | Inequality deduction for s... |
| infxrcl 13281 | The infimum of an arbitrar... |
| infxrlb 13282 | A member of a set of exten... |
| infxrgelb 13283 | The infimum of a set of ex... |
| infxrre 13284 | The real and extended real... |
| infxrmnf 13285 | The infinimum of a set of ... |
| xrinf0 13286 | The infimum of the empty s... |
| infxrss 13287 | Larger sets of extended re... |
| reltre 13288 | For all real numbers there... |
| rpltrp 13289 | For all positive real numb... |
| reltxrnmnf 13290 | For all extended real numb... |
| infmremnf 13291 | The infimum of the reals i... |
| infmrp1 13292 | The infimum of the positiv... |
| ixxval 13301 | Value of the interval func... |
| elixx1 13302 | Membership in an interval ... |
| ixxf 13303 | The set of intervals of ex... |
| ixxex 13304 | The set of intervals of ex... |
| ixxssxr 13305 | The set of intervals of ex... |
| elixx3g 13306 | Membership in a set of ope... |
| ixxssixx 13307 | An interval is a subset of... |
| ixxdisj 13308 | Split an interval into dis... |
| ixxun 13309 | Split an interval into two... |
| ixxin 13310 | Intersection of two interv... |
| ixxss1 13311 | Subset relationship for in... |
| ixxss2 13312 | Subset relationship for in... |
| ixxss12 13313 | Subset relationship for in... |
| ixxub 13314 | Extract the upper bound of... |
| ixxlb 13315 | Extract the lower bound of... |
| iooex 13316 | The set of open intervals ... |
| iooval 13317 | Value of the open interval... |
| ioo0 13318 | An empty open interval of ... |
| ioon0 13319 | An open interval of extend... |
| ndmioo 13320 | The open interval function... |
| iooid 13321 | An open interval with iden... |
| elioo3g 13322 | Membership in a set of ope... |
| elioore 13323 | A member of an open interv... |
| lbioo 13324 | An open interval does not ... |
| ubioo 13325 | An open interval does not ... |
| iooval2 13326 | Value of the open interval... |
| iooin 13327 | Intersection of two open i... |
| iooss1 13328 | Subset relationship for op... |
| iooss2 13329 | Subset relationship for op... |
| iocval 13330 | Value of the open-below, c... |
| icoval 13331 | Value of the closed-below,... |
| iccval 13332 | Value of the closed interv... |
| elioo1 13333 | Membership in an open inte... |
| elioo2 13334 | Membership in an open inte... |
| elioc1 13335 | Membership in an open-belo... |
| elico1 13336 | Membership in a closed-bel... |
| elicc1 13337 | Membership in a closed int... |
| iccid 13338 | A closed interval with ide... |
| ico0 13339 | An empty open interval of ... |
| ioc0 13340 | An empty open interval of ... |
| icc0 13341 | An empty closed interval o... |
| dfrp2 13342 | Alternate definition of th... |
| elicod 13343 | Membership in a left-close... |
| icogelb 13344 | An element of a left-close... |
| icogelbd 13345 | An element of a left-close... |
| elicore 13346 | A member of a left-closed ... |
| ubioc1 13347 | The upper bound belongs to... |
| lbico1 13348 | The lower bound belongs to... |
| iccleub 13349 | An element of a closed int... |
| iccgelb 13350 | An element of a closed int... |
| elioo5 13351 | Membership in an open inte... |
| eliooxr 13352 | A nonempty open interval s... |
| eliooord 13353 | Ordering implied by a memb... |
| elioo4g 13354 | Membership in an open inte... |
| ioossre 13355 | An open interval is a set ... |
| ioosscn 13356 | An open interval is a set ... |
| elioc2 13357 | Membership in an open-belo... |
| elico2 13358 | Membership in a closed-bel... |
| elicc2 13359 | Membership in a closed rea... |
| elicc2i 13360 | Inference for membership i... |
| elicc4 13361 | Membership in a closed rea... |
| iccss 13362 | Condition for a closed int... |
| iccssioo 13363 | Condition for a closed int... |
| icossico 13364 | Condition for a closed-bel... |
| iccss2 13365 | Condition for a closed int... |
| iccssico 13366 | Condition for a closed int... |
| iccssioo2 13367 | Condition for a closed int... |
| iccssico2 13368 | Condition for a closed int... |
| icossico2d 13369 | Condition for a closed-bel... |
| ioomax 13370 | The open interval from min... |
| iccmax 13371 | The closed interval from m... |
| ioopos 13372 | The set of positive reals ... |
| ioorp 13373 | The set of positive reals ... |
| iooshf 13374 | Shift the arguments of the... |
| iocssre 13375 | A closed-above interval wi... |
| icossre 13376 | A closed-below interval wi... |
| iccssre 13377 | A closed real interval is ... |
| iccssxr 13378 | A closed interval is a set... |
| iocssxr 13379 | An open-below, closed-abov... |
| icossxr 13380 | A closed-below, open-above... |
| ioossicc 13381 | An open interval is a subs... |
| iccssred 13382 | A closed real interval is ... |
| eliccxr 13383 | A member of a closed inter... |
| icossicc 13384 | A closed-below, open-above... |
| iocssicc 13385 | A closed-above, open-below... |
| ioossico 13386 | An open interval is a subs... |
| iocssioo 13387 | Condition for a closed int... |
| icossioo 13388 | Condition for a closed int... |
| ioossioo 13389 | Condition for an open inte... |
| iccsupr 13390 | A nonempty subset of a clo... |
| elioopnf 13391 | Membership in an unbounded... |
| elioomnf 13392 | Membership in an unbounded... |
| elicopnf 13393 | Membership in a closed unb... |
| repos 13394 | Two ways of saying that a ... |
| ioof 13395 | The set of open intervals ... |
| iccf 13396 | The set of closed interval... |
| unirnioo 13397 | The union of the range of ... |
| dfioo2 13398 | Alternate definition of th... |
| ioorebas 13399 | Open intervals are element... |
| xrge0neqmnf 13400 | A nonnegative extended rea... |
| xrge0nre 13401 | An extended real which is ... |
| elrege0 13402 | The predicate "is a nonneg... |
| nn0rp0 13403 | A nonnegative integer is a... |
| rge0ssre 13404 | Nonnegative real numbers a... |
| elxrge0 13405 | Elementhood in the set of ... |
| 0e0icopnf 13406 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13407 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13408 | The nonnegative reals are ... |
| ge0mulcl 13409 | The nonnegative reals are ... |
| ge0xaddcl 13410 | The nonnegative reals are ... |
| ge0xmulcl 13411 | The nonnegative extended r... |
| lbicc2 13412 | The lower bound of a close... |
| ubicc2 13413 | The upper bound of a close... |
| elicc01 13414 | Membership in the closed r... |
| elunitrn 13415 | The closed unit interval i... |
| elunitcn 13416 | The closed unit interval i... |
| 0elunit 13417 | Zero is an element of the ... |
| 1elunit 13418 | One is an element of the c... |
| iooneg 13419 | Membership in a negated op... |
| iccneg 13420 | Membership in a negated cl... |
| icoshft 13421 | A shifted real is a member... |
| icoshftf1o 13422 | Shifting a closed-below, o... |
| icoun 13423 | The union of two adjacent ... |
| icodisj 13424 | Adjacent left-closed right... |
| ioounsn 13425 | The union of an open inter... |
| snunioo 13426 | The closure of one end of ... |
| snunico 13427 | The closure of the open en... |
| snunioc 13428 | The closure of the open en... |
| prunioo 13429 | The closure of an open rea... |
| ioodisj 13430 | If the upper bound of one ... |
| ioojoin 13431 | Join two open intervals to... |
| difreicc 13432 | The class difference of ` ... |
| iccsplit 13433 | Split a closed interval in... |
| iccshftr 13434 | Membership in a shifted in... |
| iccshftri 13435 | Membership in a shifted in... |
| iccshftl 13436 | Membership in a shifted in... |
| iccshftli 13437 | Membership in a shifted in... |
| iccdil 13438 | Membership in a dilated in... |
| iccdili 13439 | Membership in a dilated in... |
| icccntr 13440 | Membership in a contracted... |
| icccntri 13441 | Membership in a contracted... |
| divelunit 13442 | A condition for a ratio to... |
| lincmb01cmp 13443 | A linear combination of tw... |
| iccf1o 13444 | Describe a bijection from ... |
| iccen 13445 | Any nontrivial closed inte... |
| xov1plusxeqvd 13446 | A complex number ` X ` is ... |
| unitssre 13447 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13448 | The closed unit interval i... |
| supicc 13449 | Supremum of a bounded set ... |
| supiccub 13450 | The supremum of a bounded ... |
| supicclub 13451 | The supremum of a bounded ... |
| supicclub2 13452 | The supremum of a bounded ... |
| zltaddlt1le 13453 | The sum of an integer and ... |
| xnn0xrge0 13454 | An extended nonnegative in... |
| nnge2recico01 13455 | The reciprocal of an integ... |
| fzval 13458 | The value of a finite set ... |
| fzval2 13459 | An alternative way of expr... |
| fzf 13460 | Establish the domain and c... |
| elfz1 13461 | Membership in a finite set... |
| elfz 13462 | Membership in a finite set... |
| elfz2 13463 | Membership in a finite set... |
| elfzd 13464 | Membership in a finite set... |
| elfz5 13465 | Membership in a finite set... |
| elfz4 13466 | Membership in a finite set... |
| elfzuzb 13467 | Membership in a finite set... |
| eluzfz 13468 | Membership in a finite set... |
| elfzuz 13469 | A member of a finite set o... |
| elfzuz3 13470 | Membership in a finite set... |
| elfzel2 13471 | Membership in a finite set... |
| elfzel1 13472 | Membership in a finite set... |
| elfzelz 13473 | A member of a finite set o... |
| elfzelzd 13474 | A member of a finite set o... |
| fzssz 13475 | A finite sequence of integ... |
| elfzle1 13476 | A member of a finite set o... |
| elfzle2 13477 | A member of a finite set o... |
| elfzuz2 13478 | Implication of membership ... |
| elfzle3 13479 | Membership in a finite set... |
| eluzfz1 13480 | Membership in a finite set... |
| eluzfz2 13481 | Membership in a finite set... |
| eluzfz2b 13482 | Membership in a finite set... |
| elfz3 13483 | Membership in a finite set... |
| elfz1eq 13484 | Membership in a finite set... |
| elfzubelfz 13485 | If there is a member in a ... |
| peano2fzr 13486 | A Peano-postulate-like the... |
| fzn0 13487 | Properties of a finite int... |
| fz0 13488 | A finite set of sequential... |
| fzn 13489 | A finite set of sequential... |
| fzen 13490 | A shifted finite set of se... |
| fz1n 13491 | A 1-based finite set of se... |
| 0nelfz1 13492 | 0 is not an element of a f... |
| 0fz1 13493 | Two ways to say a finite 1... |
| fz10 13494 | There are no integers betw... |
| uzsubsubfz 13495 | Membership of an integer g... |
| uzsubsubfz1 13496 | Membership of an integer g... |
| ige3m2fz 13497 | Membership of an integer g... |
| fzsplit2 13498 | Split a finite interval of... |
| fzsplit 13499 | Split a finite interval of... |
| fzdisj 13500 | Condition for two finite i... |
| fz01en 13501 | 0-based and 1-based finite... |
| elfznn 13502 | A member of a finite set o... |
| elfz1end 13503 | A nonempty finite range of... |
| fz1ssnn 13504 | A finite set of positive i... |
| fznn0sub 13505 | Subtraction closure for a ... |
| fzmmmeqm 13506 | Subtracting the difference... |
| fzaddel 13507 | Membership of a sum in a f... |
| fzadd2 13508 | Membership of a sum in a f... |
| fzsubel 13509 | Membership of a difference... |
| fzopth 13510 | A finite set of sequential... |
| fzass4 13511 | Two ways to express a nond... |
| fzss1 13512 | Subset relationship for fi... |
| fzss2 13513 | Subset relationship for fi... |
| fzssuz 13514 | A finite set of sequential... |
| fzsn 13515 | A finite interval of integ... |
| fzssp1 13516 | Subset relationship for fi... |
| fzssnn 13517 | Finite sets of sequential ... |
| ssfzunsnext 13518 | A subset of a finite seque... |
| ssfzunsn 13519 | A subset of a finite seque... |
| fzsuc 13520 | Join a successor to the en... |
| fzpred 13521 | Join a predecessor to the ... |
| fzpreddisj 13522 | A finite set of sequential... |
| elfzp1 13523 | Append an element to a fin... |
| fzp1ss 13524 | Subset relationship for fi... |
| fzelp1 13525 | Membership in a set of seq... |
| fzp1elp1 13526 | Add one to an element of a... |
| fznatpl1 13527 | Shift membership in a fini... |
| fzpr 13528 | A finite interval of integ... |
| fztp 13529 | A finite interval of integ... |
| fz12pr 13530 | An integer range between 1... |
| fzsuc2 13531 | Join a successor to the en... |
| fzp1disj 13532 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13533 | Remove a successor from th... |
| fzprval 13534 | Two ways of defining the f... |
| fztpval 13535 | Two ways of defining the f... |
| fzrev 13536 | Reversal of start and end ... |
| fzrev2 13537 | Reversal of start and end ... |
| fzrev2i 13538 | Reversal of start and end ... |
| fzrev3 13539 | The "complement" of a memb... |
| fzrev3i 13540 | The "complement" of a memb... |
| fznn 13541 | Finite set of sequential i... |
| elfz1b 13542 | Membership in a 1-based fi... |
| elfz1uz 13543 | Membership in a 1-based fi... |
| elfzm11 13544 | Membership in a finite set... |
| uzsplit 13545 | Express an upper integer s... |
| uzdisj 13546 | The first ` N ` elements o... |
| fseq1p1m1 13547 | Add/remove an item to/from... |
| fseq1m1p1 13548 | Add/remove an item to/from... |
| fz1sbc 13549 | Quantification over a one-... |
| elfzp1b 13550 | An integer is a member of ... |
| elfzm1b 13551 | An integer is a member of ... |
| elfzp12 13552 | Options for membership in ... |
| fzne1 13553 | Elementhood in a finite se... |
| fzdif1 13554 | Split the first element of... |
| fz0dif1 13555 | Split the first element of... |
| fzm1 13556 | Choices for an element of ... |
| fzneuz 13557 | No finite set of sequentia... |
| fznuz 13558 | Disjointness of the upper ... |
| uznfz 13559 | Disjointness of the upper ... |
| fzp1nel 13560 | One plus the upper bound o... |
| fzrevral 13561 | Reversal of scanning order... |
| fzrevral2 13562 | Reversal of scanning order... |
| fzrevral3 13563 | Reversal of scanning order... |
| fzshftral 13564 | Shift the scanning order i... |
| ige2m1fz1 13565 | Membership of an integer g... |
| ige2m1fz 13566 | Membership in a 0-based fi... |
| elfz2nn0 13567 | Membership in a finite set... |
| fznn0 13568 | Characterization of a fini... |
| elfznn0 13569 | A member of a finite set o... |
| elfz3nn0 13570 | The upper bound of a nonem... |
| fz0ssnn0 13571 | Finite sets of sequential ... |
| fz1ssfz0 13572 | Subset relationship for fi... |
| 0elfz 13573 | 0 is an element of a finit... |
| nn0fz0 13574 | A nonnegative integer is a... |
| elfz0add 13575 | An element of a finite set... |
| fz0sn 13576 | An integer range from 0 to... |
| fz0tp 13577 | An integer range from 0 to... |
| fz0to3un2pr 13578 | An integer range from 0 to... |
| fz0to4untppr 13579 | An integer range from 0 to... |
| fz0to5un2tp 13580 | An integer range from 0 to... |
| elfz0ubfz0 13581 | An element of a finite set... |
| elfz0fzfz0 13582 | A member of a finite set o... |
| fz0fzelfz0 13583 | If a member of a finite se... |
| fznn0sub2 13584 | Subtraction closure for a ... |
| uzsubfz0 13585 | Membership of an integer g... |
| fz0fzdiffz0 13586 | The difference of an integ... |
| elfzmlbm 13587 | Subtracting the lower boun... |
| elfzmlbp 13588 | Subtracting the lower boun... |
| fzctr 13589 | Lemma for theorems about t... |
| difelfzle 13590 | The difference of two inte... |
| difelfznle 13591 | The difference of two inte... |
| nn0split 13592 | Express the set of nonnega... |
| nn0disj 13593 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13594 | A finite set of sequential... |
| fvffz0 13595 | The function value of a fu... |
| 1fv 13596 | A function on a singleton.... |
| 4fvwrd4 13597 | The first four function va... |
| 2ffzeq 13598 | Two functions over 0-based... |
| preduz 13599 | The value of the predecess... |
| prednn 13600 | The value of the predecess... |
| prednn0 13601 | The value of the predecess... |
| predfz 13602 | Calculate the predecessor ... |
| fzof 13605 | Functionality of the half-... |
| elfzoel1 13606 | Reverse closure for half-o... |
| elfzoel2 13607 | Reverse closure for half-o... |
| elfzoelz 13608 | Reverse closure for half-o... |
| fzoval 13609 | Value of the half-open int... |
| elfzo 13610 | Membership in a half-open ... |
| elfzo2 13611 | Membership in a half-open ... |
| elfzod 13612 | Membership in a half-open ... |
| elfzouz 13613 | Membership in a half-open ... |
| nelfzo 13614 | An integer not being a mem... |
| fzolb 13615 | The left endpoint of a hal... |
| fzolb2 13616 | The left endpoint of a hal... |
| elfzole1 13617 | A member in a half-open in... |
| elfzolt2 13618 | A member in a half-open in... |
| elfzolt3 13619 | Membership in a half-open ... |
| elfzolt2b 13620 | A member in a half-open in... |
| elfzolt3b 13621 | Membership in a half-open ... |
| elfzop1le2 13622 | A member in a half-open in... |
| fzonel 13623 | A half-open range does not... |
| elfzouz2 13624 | The upper bound of a half-... |
| elfzofz 13625 | A half-open range is conta... |
| elfzo3 13626 | Express membership in a ha... |
| fzon0 13627 | A half-open integer interv... |
| fzossfz 13628 | A half-open range is conta... |
| fzossz 13629 | A half-open integer interv... |
| fzon 13630 | A half-open set of sequent... |
| fzo0n 13631 | A half-open range of nonne... |
| fzonlt0 13632 | A half-open integer range ... |
| fzo0 13633 | Half-open sets with equal ... |
| fzonnsub 13634 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13635 | If ` M < N ` then ` N - M ... |
| fzoss1 13636 | Subset relationship for ha... |
| fzoss2 13637 | Subset relationship for ha... |
| fzossrbm1 13638 | Subset of a half-open rang... |
| fzo0ss1 13639 | Subset relationship for ha... |
| fzossnn0 13640 | A half-open integer range ... |
| fzospliti 13641 | One direction of splitting... |
| fzosplit 13642 | Split a half-open integer ... |
| fzodisj 13643 | Abutting half-open integer... |
| fzouzsplit 13644 | Split an upper integer set... |
| fzouzdisj 13645 | A half-open integer range ... |
| fzoun 13646 | A half-open integer range ... |
| fzodisjsn 13647 | A half-open integer range ... |
| prinfzo0 13648 | The intersection of a half... |
| lbfzo0 13649 | An integer is strictly gre... |
| elfzo0 13650 | Membership in a half-open ... |
| elfzo0z 13651 | Membership in a half-open ... |
| nn0p1elfzo 13652 | A nonnegative integer incr... |
| elfzo0le 13653 | A member in a half-open ra... |
| elfzolem1 13654 | A member in a half-open in... |
| elfzo0subge1 13655 | The difference of the uppe... |
| elfzo0suble 13656 | The difference of the uppe... |
| elfzonn0 13657 | A member of a half-open ra... |
| fzonmapblen 13658 | The result of subtracting ... |
| fzofzim 13659 | If a nonnegative integer i... |
| fz1fzo0m1 13660 | Translation of one between... |
| fzossnn 13661 | Half-open integer ranges s... |
| elfzo1 13662 | Membership in a half-open ... |
| fzo1lb 13663 | 1 is the left endpoint of ... |
| 1elfzo1 13664 | 1 is in a half-open range ... |
| fzo1fzo0n0 13665 | An integer between 1 and a... |
| fzo0n0 13666 | A half-open integer range ... |
| fzoaddel 13667 | Translate membership in a ... |
| fzo0addel 13668 | Translate membership in a ... |
| fzo0addelr 13669 | Translate membership in a ... |
| fzoaddel2 13670 | Translate membership in a ... |
| elfzoextl 13671 | Membership of an integer i... |
| elfzoext 13672 | Membership of an integer i... |
| elincfzoext 13673 | Membership of an increased... |
| fzosubel 13674 | Translate membership in a ... |
| fzosubel2 13675 | Membership in a translated... |
| fzosubel3 13676 | Membership in a translated... |
| eluzgtdifelfzo 13677 | Membership of the differen... |
| ige2m2fzo 13678 | Membership of an integer g... |
| fzocatel 13679 | Translate membership in a ... |
| ubmelfzo 13680 | If an integer in a 1-based... |
| elfzodifsumelfzo 13681 | If an integer is in a half... |
| elfzom1elp1fzo 13682 | Membership of an integer i... |
| elfzom1elfzo 13683 | Membership in a half-open ... |
| fzval3 13684 | Expressing a closed intege... |
| fz0add1fz1 13685 | Translate membership in a ... |
| fzosn 13686 | Expressing a singleton as ... |
| elfzomin 13687 | Membership of an integer i... |
| zpnn0elfzo 13688 | Membership of an integer i... |
| zpnn0elfzo1 13689 | Membership of an integer i... |
| fzosplitsnm1 13690 | Removing a singleton from ... |
| elfzonlteqm1 13691 | If an element of a half-op... |
| fzonn0p1 13692 | A nonnegative integer is a... |
| fzossfzop1 13693 | A half-open range of nonne... |
| fzonn0p1p1 13694 | If a nonnegative integer i... |
| elfzom1p1elfzo 13695 | Increasing an element of a... |
| fzo0ssnn0 13696 | Half-open integer ranges s... |
| fzo01 13697 | Expressing the singleton o... |
| fzo12sn 13698 | A 1-based half-open intege... |
| fzo13pr 13699 | A 1-based half-open intege... |
| fzo0to2pr 13700 | A half-open integer range ... |
| fz01pr 13701 | An integer range between 0... |
| fzo0to3tp 13702 | A half-open integer range ... |
| fzo0to42pr 13703 | A half-open integer range ... |
| fzo1to4tp 13704 | A half-open integer range ... |
| fzo0sn0fzo1 13705 | A half-open range of nonne... |
| elfzo0l 13706 | A member of a half-open ra... |
| fzoend 13707 | The endpoint of a half-ope... |
| fzo0end 13708 | The endpoint of a zero-bas... |
| ssfzo12 13709 | Subset relationship for ha... |
| ssfzoulel 13710 | If a half-open integer ran... |
| ssfzo12bi 13711 | Subset relationship for ha... |
| fzoopth 13712 | A half-open integer range ... |
| ubmelm1fzo 13713 | The result of subtracting ... |
| fzofzp1 13714 | If a point is in a half-op... |
| fzofzp1b 13715 | If a point is in a half-op... |
| elfzom1b 13716 | An integer is a member of ... |
| elfzom1elp1fzo1 13717 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13718 | Membership of a positive i... |
| elfzonelfzo 13719 | If an element of a half-op... |
| elfzodif0 13720 | If an integer ` M ` is in ... |
| fzonfzoufzol 13721 | If an element of a half-op... |
| elfzomelpfzo 13722 | An integer increased by an... |
| elfznelfzo 13723 | A value in a finite set of... |
| elfznelfzob 13724 | A value in a finite set of... |
| peano2fzor 13725 | A Peano-postulate-like the... |
| fzosplitsn 13726 | Extending a half-open rang... |
| fzosplitpr 13727 | Extending a half-open inte... |
| fzosplitprm1 13728 | Extending a half-open inte... |
| fzosplitsni 13729 | Membership in a half-open ... |
| fzisfzounsn 13730 | A finite interval of integ... |
| elfzr 13731 | A member of a finite inter... |
| elfzlmr 13732 | A member of a finite inter... |
| elfz0lmr 13733 | A member of a finite inter... |
| fzone1 13734 | Elementhood in a half-open... |
| fzom1ne1 13735 | Elementhood in a half-open... |
| fzostep1 13736 | Two possibilities for a nu... |
| fzoshftral 13737 | Shift the scanning order i... |
| fzind2 13738 | Induction on the integers ... |
| fvinim0ffz 13739 | The function values for th... |
| injresinjlem 13740 | Lemma for ~ injresinj . (... |
| injresinj 13741 | A function whose restricti... |
| subfzo0 13742 | The difference between two... |
| fvf1tp 13743 | Values of a one-to-one fun... |
| flval 13748 | Value of the floor (greate... |
| flcl 13749 | The floor (greatest intege... |
| reflcl 13750 | The floor (greatest intege... |
| fllelt 13751 | A basic property of the fl... |
| flcld 13752 | The floor (greatest intege... |
| flle 13753 | A basic property of the fl... |
| flltp1 13754 | A basic property of the fl... |
| fllep1 13755 | A basic property of the fl... |
| fraclt1 13756 | The fractional part of a r... |
| fracle1 13757 | The fractional part of a r... |
| fracge0 13758 | The fractional part of a r... |
| flge 13759 | The floor function value i... |
| fllt 13760 | The floor function value i... |
| flflp1 13761 | Move floor function betwee... |
| flid 13762 | An integer is its own floo... |
| flidm 13763 | The floor function is idem... |
| flidz 13764 | A real number equals its f... |
| flltnz 13765 | The floor of a non-integer... |
| flwordi 13766 | Ordering relation for the ... |
| flword2 13767 | Ordering relation for the ... |
| flval2 13768 | An alternate way to define... |
| flval3 13769 | An alternate way to define... |
| flbi 13770 | A condition equivalent to ... |
| flbi2 13771 | A condition equivalent to ... |
| adddivflid 13772 | The floor of a sum of an i... |
| ico01fl0 13773 | The floor of a real number... |
| flge0nn0 13774 | The floor of a number grea... |
| flge1nn 13775 | The floor of a number grea... |
| fldivnn0 13776 | The floor function of a di... |
| refldivcl 13777 | The floor function of a di... |
| divfl0 13778 | The floor of a fraction is... |
| fladdz 13779 | An integer can be moved in... |
| flzadd 13780 | An integer can be moved in... |
| flmulnn0 13781 | Move a nonnegative integer... |
| btwnzge0 13782 | A real bounded between an ... |
| 2tnp1ge0ge0 13783 | Two times an integer plus ... |
| flhalf 13784 | Ordering relation for the ... |
| fldivle 13785 | The floor function of a di... |
| fldivnn0le 13786 | The floor function of a di... |
| flltdivnn0lt 13787 | The floor function of a di... |
| ltdifltdiv 13788 | If the dividend of a divis... |
| fldiv4p1lem1div2 13789 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13790 | The floor of an integer gr... |
| fldiv4lem1div2 13791 | The floor of a positive in... |
| ceilval 13792 | The value of the ceiling f... |
| dfceil2 13793 | Alternative definition of ... |
| ceilval2 13794 | The value of the ceiling f... |
| ceicl 13795 | The ceiling function retur... |
| ceilcl 13796 | Closure of the ceiling fun... |
| ceilcld 13797 | Closure of the ceiling fun... |
| ceige 13798 | The ceiling of a real numb... |
| ceilge 13799 | The ceiling of a real numb... |
| ceilged 13800 | The ceiling of a real numb... |
| ceim1l 13801 | One less than the ceiling ... |
| ceilm1lt 13802 | One less than the ceiling ... |
| ceile 13803 | The ceiling of a real numb... |
| ceille 13804 | The ceiling of a real numb... |
| ceilid 13805 | An integer is its own ceil... |
| ceilidz 13806 | A real number equals its c... |
| flleceil 13807 | The floor of a real number... |
| fleqceilz 13808 | A real number is an intege... |
| quoremz 13809 | Quotient and remainder of ... |
| quoremnn0 13810 | Quotient and remainder of ... |
| quoremnn0ALT 13811 | Alternate proof of ~ quore... |
| intfrac2 13812 | Decompose a real into inte... |
| intfracq 13813 | Decompose a rational numbe... |
| fldiv 13814 | Cancellation of the embedd... |
| fldiv2 13815 | Cancellation of an embedde... |
| fznnfl 13816 | Finite set of sequential i... |
| uzsup 13817 | An upper set of integers i... |
| ioopnfsup 13818 | An upper set of reals is u... |
| icopnfsup 13819 | An upper set of reals is u... |
| rpsup 13820 | The positive reals are unb... |
| resup 13821 | The real numbers are unbou... |
| xrsup 13822 | The extended real numbers ... |
| modval 13825 | The value of the modulo op... |
| modvalr 13826 | The value of the modulo op... |
| modcl 13827 | Closure law for the modulo... |
| flpmodeq 13828 | Partition of a division in... |
| modcld 13829 | Closure law for the modulo... |
| mod0 13830 | ` A mod B ` is zero iff ` ... |
| mulmod0 13831 | The product of an integer ... |
| negmod0 13832 | ` A ` is divisible by ` B ... |
| modge0 13833 | The modulo operation is no... |
| modlt 13834 | The modulo operation is le... |
| modelico 13835 | Modular reduction produces... |
| moddiffl 13836 | Value of the modulo operat... |
| moddifz 13837 | The modulo operation diffe... |
| modfrac 13838 | The fractional part of a n... |
| flmod 13839 | The floor function express... |
| intfrac 13840 | Break a number into its in... |
| zmod10 13841 | An integer modulo 1 is 0. ... |
| zmod1congr 13842 | Two arbitrary integers are... |
| modmulnn 13843 | Move a positive integer in... |
| modvalp1 13844 | The value of the modulo op... |
| zmodcl 13845 | Closure law for the modulo... |
| zmodcld 13846 | Closure law for the modulo... |
| zmodfz 13847 | An integer mod ` B ` lies ... |
| zmodfzo 13848 | An integer mod ` B ` lies ... |
| zmodfzp1 13849 | An integer mod ` B ` lies ... |
| modid 13850 | Identity law for modulo. ... |
| modid0 13851 | A positive real number mod... |
| modid2 13852 | Identity law for modulo. ... |
| zmodid2 13853 | Identity law for modulo re... |
| zmodidfzo 13854 | Identity law for modulo re... |
| zmodidfzoimp 13855 | Identity law for modulo re... |
| 0mod 13856 | Special case: 0 modulo a p... |
| 1mod 13857 | Special case: 1 modulo a r... |
| modabs 13858 | Absorption law for modulo.... |
| modabs2 13859 | Absorption law for modulo.... |
| modcyc 13860 | The modulo operation is pe... |
| modcyc2 13861 | The modulo operation is pe... |
| modadd1 13862 | Addition property of the m... |
| modaddb 13863 | Addition property of the m... |
| modaddid 13864 | The sums of two nonnegativ... |
| modaddabs 13865 | Absorption law for modulo.... |
| modaddmod 13866 | The sum of a real number m... |
| muladdmodid 13867 | The sum of a positive real... |
| mulp1mod1 13868 | The product of an integer ... |
| muladdmod 13869 | A real number is the sum o... |
| modmuladd 13870 | Decomposition of an intege... |
| modmuladdim 13871 | Implication of a decomposi... |
| modmuladdnn0 13872 | Implication of a decomposi... |
| negmod 13873 | The negation of a number m... |
| m1modnnsub1 13874 | Minus one modulo a positiv... |
| m1modge3gt1 13875 | Minus one modulo an intege... |
| addmodid 13876 | The sum of a positive inte... |
| addmodidr 13877 | The sum of a positive inte... |
| modadd2mod 13878 | The sum of a real number m... |
| modm1p1mod0 13879 | If a real number modulo a ... |
| modltm1p1mod 13880 | If a real number modulo a ... |
| modmul1 13881 | Multiplication property of... |
| modmul12d 13882 | Multiplication property of... |
| modnegd 13883 | Negation property of the m... |
| modadd12d 13884 | Additive property of the m... |
| modsub12d 13885 | Subtraction property of th... |
| modsubmod 13886 | The difference of a real n... |
| modsubmodmod 13887 | The difference of a real n... |
| 2txmodxeq0 13888 | Two times a positive real ... |
| 2submod 13889 | If a real number is betwee... |
| modifeq2int 13890 | If a nonnegative integer i... |
| modaddmodup 13891 | The sum of an integer modu... |
| modaddmodlo 13892 | The sum of an integer modu... |
| modmulmod 13893 | The product of a real numb... |
| modmulmodr 13894 | The product of an integer ... |
| modaddmulmod 13895 | The sum of a real number a... |
| moddi 13896 | Distribute multiplication ... |
| modsubdir 13897 | Distribute the modulo oper... |
| modeqmodmin 13898 | A real number equals the d... |
| modirr 13899 | A number modulo an irratio... |
| modfzo0difsn 13900 | For a number within a half... |
| modsumfzodifsn 13901 | The sum of a number within... |
| modlteq 13902 | Two nonnegative integers l... |
| addmodlteq 13903 | Two nonnegative integers l... |
| om2uz0i 13904 | The mapping ` G ` is a one... |
| om2uzsuci 13905 | The value of ` G ` (see ~ ... |
| om2uzuzi 13906 | The value ` G ` (see ~ om2... |
| om2uzlti 13907 | Less-than relation for ` G... |
| om2uzlt2i 13908 | The mapping ` G ` (see ~ o... |
| om2uzrani 13909 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13910 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13911 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13912 | An alternative definition ... |
| om2uzrdg 13913 | A helper lemma for the val... |
| uzrdglem 13914 | A helper lemma for the val... |
| uzrdgfni 13915 | The recursive definition g... |
| uzrdg0i 13916 | Initial value of a recursi... |
| uzrdgsuci 13917 | Successor value of a recur... |
| ltweuz 13918 | ` < ` is a well-founded re... |
| ltwenn 13919 | Less than well-orders the ... |
| ltwefz 13920 | Less than well-orders a se... |
| uzenom 13921 | An upper integer set is de... |
| uzinf 13922 | An upper integer set is in... |
| nnnfi 13923 | The set of positive intege... |
| uzrdgxfr 13924 | Transfer the value of the ... |
| fzennn 13925 | The cardinality of a finit... |
| fzen2 13926 | The cardinality of a finit... |
| cardfz 13927 | The cardinality of a finit... |
| hashgf1o 13928 | ` G ` maps ` _om ` one-to-... |
| fzfi 13929 | A finite interval of integ... |
| fzfid 13930 | Commonly used special case... |
| fzofi 13931 | Half-open integer sets are... |
| fsequb 13932 | The values of a finite rea... |
| fsequb2 13933 | The values of a finite rea... |
| fseqsupcl 13934 | The values of a finite rea... |
| fseqsupubi 13935 | The values of a finite rea... |
| nn0ennn 13936 | The nonnegative integers a... |
| nnenom 13937 | The set of positive intege... |
| nnct 13938 | ` NN ` is countable. (Con... |
| uzindi 13939 | Indirect strong induction ... |
| axdc4uzlem 13940 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13941 | A version of ~ axdc4 that ... |
| ssnn0fi 13942 | A subset of the nonnegativ... |
| rabssnn0fi 13943 | A subset of the nonnegativ... |
| uzsinds 13944 | Strong (or "total") induct... |
| nnsinds 13945 | Strong (or "total") induct... |
| nn0sinds 13946 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13947 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13948 | If all functions of a fini... |
| fsuppmapnn0fiubex 13949 | If all functions of a fini... |
| fsuppmapnn0fiub0 13950 | If all functions of a fini... |
| suppssfz 13951 | Condition for a function o... |
| fsuppmapnn0ub 13952 | If a function over the non... |
| fsuppmapnn0fz 13953 | If a function over the non... |
| mptnn0fsupp 13954 | A mapping from the nonnega... |
| mptnn0fsuppd 13955 | A mapping from the nonnega... |
| mptnn0fsuppr 13956 | A finitely supported mappi... |
| f13idfv 13957 | A one-to-one function with... |
| seqex 13960 | Existence of the sequence ... |
| seqeq1 13961 | Equality theorem for the s... |
| seqeq2 13962 | Equality theorem for the s... |
| seqeq3 13963 | Equality theorem for the s... |
| seqeq1d 13964 | Equality deduction for the... |
| seqeq2d 13965 | Equality deduction for the... |
| seqeq3d 13966 | Equality deduction for the... |
| seqeq123d 13967 | Equality deduction for the... |
| nfseq 13968 | Hypothesis builder for the... |
| seqval 13969 | Value of the sequence buil... |
| seqfn 13970 | The sequence builder funct... |
| seq1 13971 | Value of the sequence buil... |
| seq1i 13972 | Value of the sequence buil... |
| seqp1 13973 | Value of the sequence buil... |
| seqexw 13974 | Weak version of ~ seqex th... |
| seqp1d 13975 | Value of the sequence buil... |
| seqm1 13976 | Value of the sequence buil... |
| seqcl2 13977 | Closure properties of the ... |
| seqf2 13978 | Range of the recursive seq... |
| seqcl 13979 | Closure properties of the ... |
| seqf 13980 | Range of the recursive seq... |
| seqfveq2 13981 | Equality of sequences. (C... |
| seqfeq2 13982 | Equality of sequences. (C... |
| seqfveq 13983 | Equality of sequences. (C... |
| seqfeq 13984 | Equality of sequences. (C... |
| seqshft2 13985 | Shifting the index set of ... |
| seqres 13986 | Restricting its characteri... |
| serf 13987 | An infinite series of comp... |
| serfre 13988 | An infinite series of real... |
| monoord 13989 | Ordering relation for a mo... |
| monoord2 13990 | Ordering relation for a mo... |
| sermono 13991 | The partial sums in an inf... |
| seqsplit 13992 | Split a sequence into two ... |
| seq1p 13993 | Removing the first term fr... |
| seqcaopr3 13994 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 13995 | The sum of two infinite se... |
| seqcaopr 13996 | The sum of two infinite se... |
| seqf1olem2a 13997 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 13998 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 13999 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14000 | Rearrange a sum via an arb... |
| seradd 14001 | The sum of two infinite se... |
| sersub 14002 | The difference of two infi... |
| seqid3 14003 | A sequence that consists e... |
| seqid 14004 | Discarding the first few t... |
| seqid2 14005 | The last few partial sums ... |
| seqhomo 14006 | Apply a homomorphism to a ... |
| seqz 14007 | If the operation ` .+ ` ha... |
| seqfeq4 14008 | Equality of series under d... |
| seqfeq3 14009 | Equality of series under d... |
| seqdistr 14010 | The distributive property ... |
| ser0 14011 | The value of the partial s... |
| ser0f 14012 | A zero-valued infinite ser... |
| serge0 14013 | A finite sum of nonnegativ... |
| serle 14014 | Comparison of partial sums... |
| ser1const 14015 | Value of the partial serie... |
| seqof 14016 | Distribute function operat... |
| seqof2 14017 | Distribute function operat... |
| expval 14020 | Value of exponentiation to... |
| expnnval 14021 | Value of exponentiation to... |
| exp0 14022 | Value of a complex number ... |
| 0exp0e1 14023 | The zeroth power of zero e... |
| exp1 14024 | Value of a complex number ... |
| expp1 14025 | Value of a complex number ... |
| expneg 14026 | Value of a complex number ... |
| expneg2 14027 | Value of a complex number ... |
| expn1 14028 | A complex number raised to... |
| expcllem 14029 | Lemma for proving nonnegat... |
| expcl2lem 14030 | Lemma for proving integer ... |
| nnexpcl 14031 | Closure of exponentiation ... |
| nn0expcl 14032 | Closure of exponentiation ... |
| zexpcl 14033 | Closure of exponentiation ... |
| qexpcl 14034 | Closure of exponentiation ... |
| reexpcl 14035 | Closure of exponentiation ... |
| expcl 14036 | Closure law for nonnegativ... |
| rpexpcl 14037 | Closure law for integer ex... |
| qexpclz 14038 | Closure of integer exponen... |
| reexpclz 14039 | Closure of integer exponen... |
| expclzlem 14040 | Lemma for ~ expclz . (Con... |
| expclz 14041 | Closure law for integer ex... |
| m1expcl2 14042 | Closure of integer exponen... |
| m1expcl 14043 | Closure of exponentiation ... |
| zexpcld 14044 | Closure of exponentiation ... |
| nn0expcli 14045 | Closure of exponentiation ... |
| nn0sqcl 14046 | The square of a nonnegativ... |
| expm1t 14047 | Exponentiation in terms of... |
| 1exp 14048 | Value of 1 raised to an in... |
| expeq0 14049 | A positive integer power i... |
| expne0 14050 | A positive integer power i... |
| expne0i 14051 | An integer power is nonzer... |
| expgt0 14052 | A positive real raised to ... |
| expnegz 14053 | Value of a nonzero complex... |
| 0exp 14054 | Value of zero raised to a ... |
| expge0 14055 | A nonnegative real raised ... |
| expge1 14056 | A real greater than or equ... |
| expgt1 14057 | A real greater than 1 rais... |
| mulexp 14058 | Nonnegative integer expone... |
| mulexpz 14059 | Integer exponentiation of ... |
| exprec 14060 | Integer exponentiation of ... |
| expadd 14061 | Sum of exponents law for n... |
| expaddzlem 14062 | Lemma for ~ expaddz . (Co... |
| expaddz 14063 | Sum of exponents law for i... |
| expmul 14064 | Product of exponents law f... |
| expmulz 14065 | Product of exponents law f... |
| m1expeven 14066 | Exponentiation of negative... |
| expsub 14067 | Exponent subtraction law f... |
| expp1z 14068 | Value of a nonzero complex... |
| expm1 14069 | Value of a nonzero complex... |
| expdiv 14070 | Nonnegative integer expone... |
| sqval 14071 | Value of the square of a c... |
| sqneg 14072 | The square of the negative... |
| sqnegd 14073 | The square of the negative... |
| sqsubswap 14074 | Swap the order of subtract... |
| sqcl 14075 | Closure of square. (Contr... |
| sqmul 14076 | Distribution of squaring o... |
| sqeq0 14077 | A complex number is zero i... |
| sqdiv 14078 | Distribution of squaring o... |
| sqdivid 14079 | The square of a nonzero co... |
| sqne0 14080 | A complex number is nonzer... |
| resqcl 14081 | Closure of squaring in rea... |
| resqcld 14082 | Closure of squaring in rea... |
| sqgt0 14083 | The square of a nonzero re... |
| sqn0rp 14084 | The square of a nonzero re... |
| nnsqcl 14085 | The positive naturals are ... |
| zsqcl 14086 | Integers are closed under ... |
| qsqcl 14087 | The square of a rational i... |
| sq11 14088 | The square function is one... |
| nn0sq11 14089 | The square function is one... |
| lt2sq 14090 | The square function is inc... |
| le2sq 14091 | The square function is non... |
| le2sq2 14092 | The square function is non... |
| sqge0 14093 | The square of a real is no... |
| sqge0d 14094 | The square of a real is no... |
| zsqcl2 14095 | The square of an integer i... |
| 0expd 14096 | Value of zero raised to a ... |
| exp0d 14097 | Value of a complex number ... |
| exp1d 14098 | Value of a complex number ... |
| expeq0d 14099 | If a positive integer powe... |
| sqvald 14100 | Value of square. Inferenc... |
| sqcld 14101 | Closure of square. (Contr... |
| sqeq0d 14102 | A number is zero iff its s... |
| expcld 14103 | Closure law for nonnegativ... |
| expp1d 14104 | Value of a complex number ... |
| expaddd 14105 | Sum of exponents law for n... |
| expmuld 14106 | Product of exponents law f... |
| sqrecd 14107 | Square of reciprocal is re... |
| expclzd 14108 | Closure law for integer ex... |
| expne0d 14109 | A nonnegative integer powe... |
| expnegd 14110 | Value of a nonzero complex... |
| exprecd 14111 | An integer power of a reci... |
| expp1zd 14112 | Value of a nonzero complex... |
| expm1d 14113 | Value of a nonzero complex... |
| expsubd 14114 | Exponent subtraction law f... |
| sqmuld 14115 | Distribution of squaring o... |
| sqdivd 14116 | Distribution of squaring o... |
| expdivd 14117 | Nonnegative integer expone... |
| mulexpd 14118 | Nonnegative integer expone... |
| znsqcld 14119 | The square of a nonzero in... |
| reexpcld 14120 | Closure of exponentiation ... |
| expge0d 14121 | A nonnegative real raised ... |
| expge1d 14122 | A real greater than or equ... |
| ltexp2a 14123 | Exponent ordering relation... |
| expmordi 14124 | Base ordering relationship... |
| rpexpmord 14125 | Base ordering relationship... |
| expcan 14126 | Cancellation law for integ... |
| ltexp2 14127 | Strict ordering law for ex... |
| leexp2 14128 | Ordering law for exponenti... |
| leexp2a 14129 | Weak ordering relationship... |
| ltexp2r 14130 | The integer powers of a fi... |
| leexp2r 14131 | Weak ordering relationship... |
| leexp1a 14132 | Weak base ordering relatio... |
| leexp1ad 14133 | Weak base ordering relatio... |
| exple1 14134 | A real between 0 and 1 inc... |
| expubnd 14135 | An upper bound on ` A ^ N ... |
| sumsqeq0 14136 | The sum of two squres of r... |
| sqvali 14137 | Value of square. Inferenc... |
| sqcli 14138 | Closure of square. (Contr... |
| sqeq0i 14139 | A complex number is zero i... |
| sqrecii 14140 | The square of a reciprocal... |
| sqmuli 14141 | Distribution of squaring o... |
| sqdivi 14142 | Distribution of squaring o... |
| resqcli 14143 | Closure of square in reals... |
| sqgt0i 14144 | The square of a nonzero re... |
| sqge0i 14145 | The square of a real is no... |
| lt2sqi 14146 | The square function on non... |
| le2sqi 14147 | The square function on non... |
| sq11i 14148 | The square function is one... |
| sq0 14149 | The square of 0 is 0. (Co... |
| sq0i 14150 | If a number is zero, then ... |
| sq0id 14151 | If a number is zero, then ... |
| sq1 14152 | The square of 1 is 1. (Co... |
| neg1sqe1 14153 | The square of ` -u 1 ` is ... |
| sq2 14154 | The square of 2 is 4. (Co... |
| sq3 14155 | The square of 3 is 9. (Co... |
| sq4e2t8 14156 | The square of 4 is 2 times... |
| cu2 14157 | The cube of 2 is 8. (Cont... |
| irec 14158 | The reciprocal of ` _i ` .... |
| i2 14159 | ` _i ` squared. (Contribu... |
| i3 14160 | ` _i ` cubed. (Contribute... |
| i4 14161 | ` _i ` to the fourth power... |
| nnlesq 14162 | A positive integer is less... |
| zzlesq 14163 | An integer is less than or... |
| iexpcyc 14164 | Taking ` _i ` to the ` K `... |
| expnass 14165 | A counterexample showing t... |
| sqlecan 14166 | Cancel one factor of a squ... |
| subsq 14167 | Factor the difference of t... |
| subsq2 14168 | Express the difference of ... |
| binom2i 14169 | The square of a binomial. ... |
| subsqi 14170 | Factor the difference of t... |
| sqeqori 14171 | The squares of two complex... |
| subsq0i 14172 | The two solutions to the d... |
| sqeqor 14173 | The squares of two complex... |
| binom2 14174 | The square of a binomial. ... |
| binom2d 14175 | Deduction form of ~ binom2... |
| binom21 14176 | Special case of ~ binom2 w... |
| binom2sub 14177 | Expand the square of a sub... |
| binom2sub1 14178 | Special case of ~ binom2su... |
| binom2subi 14179 | Expand the square of a sub... |
| mulbinom2 14180 | The square of a binomial w... |
| binom3 14181 | The cube of a binomial. (... |
| sq01 14182 | If a complex number equals... |
| zesq 14183 | An integer is even iff its... |
| nnesq 14184 | A positive integer is even... |
| crreczi 14185 | Reciprocal of a complex nu... |
| bernneq 14186 | Bernoulli's inequality, du... |
| bernneq2 14187 | Variation of Bernoulli's i... |
| bernneq3 14188 | A corollary of ~ bernneq .... |
| expnbnd 14189 | Exponentiation with a base... |
| expnlbnd 14190 | The reciprocal of exponent... |
| expnlbnd2 14191 | The reciprocal of exponent... |
| expmulnbnd 14192 | Exponentiation with a base... |
| digit2 14193 | Two ways to express the ` ... |
| digit1 14194 | Two ways to express the ` ... |
| modexp 14195 | Exponentiation property of... |
| discr1 14196 | A nonnegative quadratic fo... |
| discr 14197 | If a quadratic polynomial ... |
| expnngt1 14198 | If an integer power with a... |
| expnngt1b 14199 | An integer power with an i... |
| sqoddm1div8 14200 | A squared odd number minus... |
| nnsqcld 14201 | The naturals are closed un... |
| nnexpcld 14202 | Closure of exponentiation ... |
| nn0expcld 14203 | Closure of exponentiation ... |
| rpexpcld 14204 | Closure law for exponentia... |
| ltexp2rd 14205 | The power of a positive nu... |
| reexpclzd 14206 | Closure of exponentiation ... |
| sqgt0d 14207 | The square of a nonzero re... |
| ltexp2d 14208 | Ordering relationship for ... |
| leexp2d 14209 | Ordering law for exponenti... |
| expcand 14210 | Ordering relationship for ... |
| leexp2ad 14211 | Ordering relationship for ... |
| leexp2rd 14212 | Ordering relationship for ... |
| lt2sqd 14213 | The square function on non... |
| le2sqd 14214 | The square function on non... |
| sq11d 14215 | The square function is one... |
| ltexp1d 14216 | Elevating to a positive po... |
| ltexp1dd 14217 | Raising both sides of 'les... |
| exp11nnd 14218 | The function elevating non... |
| mulsubdivbinom2 14219 | The square of a binomial w... |
| muldivbinom2 14220 | The square of a binomial w... |
| sq10 14221 | The square of 10 is 100. ... |
| sq10e99m1 14222 | The square of 10 is 99 plu... |
| 3dec 14223 | A "decimal constructor" wh... |
| nn0le2msqi 14224 | The square function on non... |
| nn0opthlem1 14225 | A rather pretty lemma for ... |
| nn0opthlem2 14226 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14227 | An ordered pair theorem fo... |
| nn0opth2i 14228 | An ordered pair theorem fo... |
| nn0opth2 14229 | An ordered pair theorem fo... |
| facnn 14232 | Value of the factorial fun... |
| fac0 14233 | The factorial of 0. (Cont... |
| fac1 14234 | The factorial of 1. (Cont... |
| facp1 14235 | The factorial of a success... |
| fac2 14236 | The factorial of 2. (Cont... |
| fac3 14237 | The factorial of 3. (Cont... |
| fac4 14238 | The factorial of 4. (Cont... |
| facnn2 14239 | Value of the factorial fun... |
| faccl 14240 | Closure of the factorial f... |
| faccld 14241 | Closure of the factorial f... |
| facmapnn 14242 | The factorial function res... |
| facne0 14243 | The factorial function is ... |
| facdiv 14244 | A positive integer divides... |
| facndiv 14245 | No positive integer (great... |
| facwordi 14246 | Ordering property of facto... |
| faclbnd 14247 | A lower bound for the fact... |
| faclbnd2 14248 | A lower bound for the fact... |
| faclbnd3 14249 | A lower bound for the fact... |
| faclbnd4lem1 14250 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14251 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14252 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14253 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14254 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14255 | The factorial function gro... |
| faclbnd6 14256 | Geometric lower bound for ... |
| facubnd 14257 | An upper bound for the fac... |
| facavg 14258 | The product of two factori... |
| bcval 14261 | Value of the binomial coef... |
| bcval2 14262 | Value of the binomial coef... |
| bcval3 14263 | Value of the binomial coef... |
| bcval4 14264 | Value of the binomial coef... |
| bcrpcl 14265 | Closure of the binomial co... |
| bccmpl 14266 | "Complementing" its second... |
| bcn0 14267 | ` N ` choose 0 is 1. Rema... |
| bc0k 14268 | The binomial coefficient "... |
| bcnn 14269 | ` N ` choose ` N ` is 1. ... |
| bcn1 14270 | Binomial coefficient: ` N ... |
| bcnp1n 14271 | Binomial coefficient: ` N ... |
| bcm1k 14272 | The proportion of one bino... |
| bcp1n 14273 | The proportion of one bino... |
| bcp1nk 14274 | The proportion of one bino... |
| bcval5 14275 | Write out the top and bott... |
| bcn2 14276 | Binomial coefficient: ` N ... |
| bcp1m1 14277 | Compute the binomial coeff... |
| bcpasc 14278 | Pascal's rule for the bino... |
| bccl 14279 | A binomial coefficient, in... |
| bccl2 14280 | A binomial coefficient, in... |
| bcn2m1 14281 | Compute the binomial coeff... |
| bcn2p1 14282 | Compute the binomial coeff... |
| permnn 14283 | The number of permutations... |
| bcnm1 14284 | The binomial coefficient o... |
| 4bc3eq4 14285 | The value of four choose t... |
| 4bc2eq6 14286 | The value of four choose t... |
| hashkf 14289 | The finite part of the siz... |
| hashgval 14290 | The value of the ` # ` fun... |
| hashginv 14291 | The converse of ` G ` maps... |
| hashinf 14292 | The value of the ` # ` fun... |
| hashbnd 14293 | If ` A ` has size bounded ... |
| hashfxnn0 14294 | The size function is a fun... |
| hashf 14295 | The size function maps all... |
| hashxnn0 14296 | The value of the hash func... |
| hashresfn 14297 | Restriction of the domain ... |
| dmhashres 14298 | Restriction of the domain ... |
| hashnn0pnf 14299 | The value of the hash func... |
| hashnnn0genn0 14300 | If the size of a set is no... |
| hashnemnf 14301 | The size of a set is never... |
| hashv01gt1 14302 | The size of a set is eithe... |
| hashfz1 14303 | The set ` ( 1 ... N ) ` ha... |
| hashen 14304 | Two finite sets have the s... |
| hasheni 14305 | Equinumerous sets have the... |
| hasheqf1o 14306 | The size of two finite set... |
| fiinfnf1o 14307 | There is no bijection betw... |
| hasheqf1oi 14308 | The size of two sets is eq... |
| hashf1rn 14309 | The size of a finite set w... |
| hasheqf1od 14310 | The size of two sets is eq... |
| fz1eqb 14311 | Two possibly-empty 1-based... |
| hashcard 14312 | The size function of the c... |
| hashcl 14313 | Closure of the ` # ` funct... |
| hashxrcl 14314 | Extended real closure of t... |
| hashclb 14315 | Reverse closure of the ` #... |
| nfile 14316 | The size of any infinite s... |
| hashvnfin 14317 | A set of finite size is a ... |
| hashnfinnn0 14318 | The size of an infinite se... |
| isfinite4 14319 | A finite set is equinumero... |
| hasheq0 14320 | Two ways of saying a set i... |
| hashneq0 14321 | Two ways of saying a set i... |
| hashgt0n0 14322 | If the size of a set is gr... |
| hashnncl 14323 | Positive natural closure o... |
| hash0 14324 | The empty set has size zer... |
| hashelne0d 14325 | A set with an element has ... |
| hashsng 14326 | The size of a singleton. ... |
| hashen1 14327 | A set has size 1 if and on... |
| hash1elsn 14328 | A set of size 1 with a kno... |
| hashrabrsn 14329 | The size of a restricted c... |
| hashrabsn01 14330 | The size of a restricted c... |
| hashrabsn1 14331 | If the size of a restricte... |
| hashfn 14332 | A function is equinumerous... |
| fseq1hash 14333 | The value of the size func... |
| hashgadd 14334 | ` G ` maps ordinal additio... |
| hashgval2 14335 | A short expression for the... |
| hashdom 14336 | Dominance relation for the... |
| hashdomi 14337 | Non-strict order relation ... |
| hashsdom 14338 | Strict dominance relation ... |
| hashun 14339 | The size of the union of d... |
| hashun2 14340 | The size of the union of f... |
| hashun3 14341 | The size of the union of f... |
| hashinfxadd 14342 | The extended real addition... |
| hashunx 14343 | The size of the union of d... |
| hashge0 14344 | The cardinality of a set i... |
| hashgt0 14345 | The cardinality of a nonem... |
| hashge1 14346 | The cardinality of a nonem... |
| 1elfz0hash 14347 | 1 is an element of the fin... |
| hashnn0n0nn 14348 | If a nonnegative integer i... |
| hashunsng 14349 | The size of the union of a... |
| hashunsngx 14350 | The size of the union of a... |
| hashunsnggt 14351 | The size of a set is great... |
| hashprg 14352 | The size of an unordered p... |
| elprchashprn2 14353 | If one element of an unord... |
| hashprb 14354 | The size of an unordered p... |
| hashprdifel 14355 | The elements of an unorder... |
| prhash2ex 14356 | There is (at least) one se... |
| hashle00 14357 | If the size of a set is le... |
| hashgt0elex 14358 | If the size of a set is gr... |
| hashgt0elexb 14359 | The size of a set is great... |
| hashp1i 14360 | Size of a finite ordinal. ... |
| hash1 14361 | Size of a finite ordinal. ... |
| hash2 14362 | Size of a finite ordinal. ... |
| hash3 14363 | Size of a finite ordinal. ... |
| hash4 14364 | Size of a finite ordinal. ... |
| pr0hash2ex 14365 | There is (at least) one se... |
| hashss 14366 | The size of a subset is le... |
| prsshashgt1 14367 | The size of a superset of ... |
| hashin 14368 | The size of the intersecti... |
| hashssdif 14369 | The size of the difference... |
| hashdif 14370 | The size of the difference... |
| hashdifsn 14371 | The size of the difference... |
| hashdifpr 14372 | The size of the difference... |
| hashsn01 14373 | The size of a singleton is... |
| hashsnle1 14374 | The size of a singleton is... |
| hashsnlei 14375 | Get an upper bound on a co... |
| hash1snb 14376 | The size of a set is 1 if ... |
| euhash1 14377 | The size of a set is 1 in ... |
| hash1n0 14378 | If the size of a set is 1 ... |
| hashgt12el 14379 | In a set with more than on... |
| hashgt12el2 14380 | In a set with more than on... |
| hashgt23el 14381 | A set with more than two e... |
| hashunlei 14382 | Get an upper bound on a co... |
| hashsslei 14383 | Get an upper bound on a co... |
| hashfz 14384 | Value of the numeric cardi... |
| fzsdom2 14385 | Condition for finite range... |
| hashfzo 14386 | Cardinality of a half-open... |
| hashfzo0 14387 | Cardinality of a half-open... |
| hashfzp1 14388 | Value of the numeric cardi... |
| hashfz0 14389 | Value of the numeric cardi... |
| hashxplem 14390 | Lemma for ~ hashxp . (Con... |
| hashxp 14391 | The size of the Cartesian ... |
| hashmap 14392 | The size of the set expone... |
| hashpw 14393 | The size of the power set ... |
| hashfun 14394 | A finite set is a function... |
| hashres 14395 | The number of elements of ... |
| hashreshashfun 14396 | The number of elements of ... |
| hashimarn 14397 | The size of the image of a... |
| hashimarni 14398 | If the size of the image o... |
| hashfundm 14399 | The size of a set function... |
| hashf1dmrn 14400 | The size of the domain of ... |
| hashf1dmcdm 14401 | The size of the domain of ... |
| resunimafz0 14402 | TODO-AV: Revise using ` F... |
| fnfz0hash 14403 | The size of a function on ... |
| ffz0hash 14404 | The size of a function on ... |
| fnfz0hashnn0 14405 | The size of a function on ... |
| ffzo0hash 14406 | The size of a function on ... |
| fnfzo0hash 14407 | The size of a function on ... |
| fnfzo0hashnn0 14408 | The value of the size func... |
| hashbclem 14409 | Lemma for ~ hashbc : induc... |
| hashbc 14410 | The binomial coefficient c... |
| hashfacen 14411 | The number of bijections b... |
| hashf1lem1 14412 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14413 | Lemma for ~ hashf1 . (Con... |
| hashf1 14414 | The permutation number ` |... |
| hashfac 14415 | A factorial counts the num... |
| leiso 14416 | Two ways to write a strict... |
| leisorel 14417 | Version of ~ isorel for st... |
| fz1isolem 14418 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14419 | Any finite ordered set has... |
| ishashinf 14420 | Any set that is not finite... |
| seqcoll 14421 | The function ` F ` contain... |
| seqcoll2 14422 | The function ` F ` contain... |
| phphashd 14423 | Corollary of the Pigeonhol... |
| phphashrd 14424 | Corollary of the Pigeonhol... |
| hashprlei 14425 | An unordered pair has at m... |
| hash2pr 14426 | A set of size two is an un... |
| hash2prde 14427 | A set of size two is an un... |
| hash2exprb 14428 | A set of size two is an un... |
| hash2prb 14429 | A set of size two is a pro... |
| prprrab 14430 | The set of proper pairs of... |
| nehash2 14431 | The cardinality of a set w... |
| hash2prd 14432 | A set of size two is an un... |
| hash2pwpr 14433 | If the size of a subset of... |
| hashle2pr 14434 | A nonempty set of size les... |
| hashle2prv 14435 | A nonempty subset of a pow... |
| pr2pwpr 14436 | The set of subsets of a pa... |
| hashge2el2dif 14437 | A set with size at least 2... |
| hashge2el2difr 14438 | A set with at least 2 diff... |
| hashge2el2difb 14439 | A set has size at least 2 ... |
| hashdmpropge2 14440 | The size of the domain of ... |
| hashtplei 14441 | An unordered triple has at... |
| hashtpg 14442 | The size of an unordered t... |
| hash7g 14443 | The size of an unordered s... |
| hashge3el3dif 14444 | A set with size at least 3... |
| elss2prb 14445 | An element of the set of s... |
| hash2sspr 14446 | A subset of size two is an... |
| exprelprel 14447 | If there is an element of ... |
| hash3tr 14448 | A set of size three is an ... |
| hash1to3 14449 | If the size of a set is be... |
| hash3tpde 14450 | A set of size three is an ... |
| hash3tpexb 14451 | A set of size three is an ... |
| hash3tpb 14452 | A set of size three is a p... |
| tpf1ofv0 14453 | The value of a one-to-one ... |
| tpf1ofv1 14454 | The value of a one-to-one ... |
| tpf1ofv2 14455 | The value of a one-to-one ... |
| tpf 14456 | A function into a (proper)... |
| tpfo 14457 | A function onto a (proper)... |
| tpf1o 14458 | A bijection onto a (proper... |
| fundmge2nop0 14459 | A function with a domain c... |
| fundmge2nop 14460 | A function with a domain c... |
| fun2dmnop0 14461 | A function with a domain c... |
| fun2dmnop 14462 | A function with a domain c... |
| hashdifsnp1 14463 | If the size of a set is a ... |
| fi1uzind 14464 | Properties of an ordered p... |
| brfi1uzind 14465 | Properties of a binary rel... |
| brfi1ind 14466 | Properties of a binary rel... |
| brfi1indALT 14467 | Alternate proof of ~ brfi1... |
| opfi1uzind 14468 | Properties of an ordered p... |
| opfi1ind 14469 | Properties of an ordered p... |
| iswrd 14472 | Property of being a word o... |
| wrdval 14473 | Value of the set of words ... |
| iswrdi 14474 | A zero-based sequence is a... |
| wrdf 14475 | A word is a zero-based seq... |
| wrdfd 14476 | A word is a zero-based seq... |
| iswrdb 14477 | A word over an alphabet is... |
| wrddm 14478 | The indices of a word (i.e... |
| sswrd 14479 | The set of words respects ... |
| snopiswrd 14480 | A singleton of an ordered ... |
| wrdexg 14481 | The set of words over a se... |
| wrdexb 14482 | The set of words over a se... |
| wrdexi 14483 | The set of words over a se... |
| wrdsymbcl 14484 | A symbol within a word ove... |
| wrdfn 14485 | A word is a function with ... |
| wrdv 14486 | A word over an alphabet is... |
| wrdlndm 14487 | The length of a word is no... |
| iswrdsymb 14488 | An arbitrary word is a wor... |
| wrdfin 14489 | A word is a finite set. (... |
| lencl 14490 | The length of a word is a ... |
| lennncl 14491 | The length of a nonempty w... |
| wrdffz 14492 | A word is a function from ... |
| wrdeq 14493 | Equality theorem for the s... |
| wrdeqi 14494 | Equality theorem for the s... |
| iswrddm0 14495 | A function with empty doma... |
| wrd0 14496 | The empty set is a word (t... |
| 0wrd0 14497 | The empty word is the only... |
| ffz0iswrd 14498 | A sequence with zero-based... |
| wrdsymb 14499 | A word is a word over the ... |
| nfwrd 14500 | Hypothesis builder for ` W... |
| csbwrdg 14501 | Class substitution for the... |
| wrdnval 14502 | Words of a fixed length ar... |
| wrdmap 14503 | Words as a mapping. (Cont... |
| hashwrdn 14504 | If there is only a finite ... |
| wrdnfi 14505 | If there is only a finite ... |
| wrdsymb0 14506 | A symbol at a position "ou... |
| wrdlenge1n0 14507 | A word with length at leas... |
| len0nnbi 14508 | The length of a word is a ... |
| wrdlenge2n0 14509 | A word with length at leas... |
| wrdsymb1 14510 | The first symbol of a none... |
| wrdlen1 14511 | A word of length 1 starts ... |
| fstwrdne 14512 | The first symbol of a none... |
| fstwrdne0 14513 | The first symbol of a none... |
| eqwrd 14514 | Two words are equal iff th... |
| elovmpowrd 14515 | Implications for the value... |
| elovmptnn0wrd 14516 | Implications for the value... |
| wrdred1 14517 | A word truncated by a symb... |
| wrdred1hash 14518 | The length of a word trunc... |
| lsw 14521 | Extract the last symbol of... |
| lsw0 14522 | The last symbol of an empt... |
| lsw0g 14523 | The last symbol of an empt... |
| lsw1 14524 | The last symbol of a word ... |
| lswcl 14525 | Closure of the last symbol... |
| lswlgt0cl 14526 | The last symbol of a nonem... |
| ccatfn 14529 | The concatenation operator... |
| ccatfval 14530 | Value of the concatenation... |
| ccatcl 14531 | The concatenation of two w... |
| ccatlen 14532 | The length of a concatenat... |
| ccat0 14533 | The concatenation of two w... |
| ccatval1 14534 | Value of a symbol in the l... |
| ccatval2 14535 | Value of a symbol in the r... |
| ccatval3 14536 | Value of a symbol in the r... |
| elfzelfzccat 14537 | An element of a finite set... |
| ccatvalfn 14538 | The concatenation of two w... |
| ccatdmss 14539 | The domain of a concatenat... |
| ccatsymb 14540 | The symbol at a given posi... |
| ccatfv0 14541 | The first symbol of a conc... |
| ccatval1lsw 14542 | The last symbol of the lef... |
| ccatval21sw 14543 | The first symbol of the ri... |
| ccatlid 14544 | Concatenation of a word by... |
| ccatrid 14545 | Concatenation of a word by... |
| ccatass 14546 | Associative law for concat... |
| ccatrn 14547 | The range of a concatenate... |
| ccatidid 14548 | Concatenation of the empty... |
| lswccatn0lsw 14549 | The last symbol of a word ... |
| lswccat0lsw 14550 | The last symbol of a word ... |
| ccatalpha 14551 | A concatenation of two arb... |
| ccatrcl1 14552 | Reverse closure of a conca... |
| ids1 14555 | Identity function protecti... |
| s1val 14556 | Value of a singleton word.... |
| s1rn 14557 | The range of a singleton w... |
| s1eq 14558 | Equality theorem for a sin... |
| s1eqd 14559 | Equality theorem for a sin... |
| s1cl 14560 | A singleton word is a word... |
| s1cld 14561 | A singleton word is a word... |
| s1prc 14562 | Value of a singleton word ... |
| s1cli 14563 | A singleton word is a word... |
| s1len 14564 | Length of a singleton word... |
| s1nz 14565 | A singleton word is not th... |
| s1dm 14566 | The domain of a singleton ... |
| s1dmALT 14567 | Alternate version of ~ s1d... |
| s1fv 14568 | Sole symbol of a singleton... |
| lsws1 14569 | The last symbol of a singl... |
| eqs1 14570 | A word of length 1 is a si... |
| wrdl1exs1 14571 | A word of length 1 is a si... |
| wrdl1s1 14572 | A word of length 1 is a si... |
| s111 14573 | The singleton word functio... |
| ccatws1cl 14574 | The concatenation of a wor... |
| ccatws1clv 14575 | The concatenation of a wor... |
| ccat2s1cl 14576 | The concatenation of two s... |
| ccats1alpha 14577 | A concatenation of a word ... |
| ccatws1len 14578 | The length of the concaten... |
| ccatws1lenp1b 14579 | The length of a word is ` ... |
| wrdlenccats1lenm1 14580 | The length of a word is th... |
| ccat2s1len 14581 | The length of the concaten... |
| ccatw2s1cl 14582 | The concatenation of a wor... |
| ccatw2s1len 14583 | The length of the concaten... |
| ccats1val1 14584 | Value of a symbol in the l... |
| ccats1val2 14585 | Value of the symbol concat... |
| ccat1st1st 14586 | The first symbol of a word... |
| ccat2s1p1 14587 | Extract the first of two c... |
| ccat2s1p2 14588 | Extract the second of two ... |
| ccatw2s1ass 14589 | Associative law for a conc... |
| ccatws1n0 14590 | The concatenation of a wor... |
| ccatws1ls 14591 | The last symbol of the con... |
| lswccats1 14592 | The last symbol of a word ... |
| lswccats1fst 14593 | The last symbol of a nonem... |
| ccatw2s1p1 14594 | Extract the symbol of the ... |
| ccatw2s1p2 14595 | Extract the second of two ... |
| ccat2s1fvw 14596 | Extract a symbol of a word... |
| ccat2s1fst 14597 | The first symbol of the co... |
| swrdnznd 14600 | The value of a subword ope... |
| swrdval 14601 | Value of a subword. (Cont... |
| swrd00 14602 | A zero length substring. ... |
| swrdcl 14603 | Closure of the subword ext... |
| swrdval2 14604 | Value of the subword extra... |
| swrdlen 14605 | Length of an extracted sub... |
| swrdfv 14606 | A symbol in an extracted s... |
| swrdfv0 14607 | The first symbol in an ext... |
| swrdf 14608 | A subword of a word is a f... |
| swrdvalfn 14609 | Value of the subword extra... |
| swrdrn 14610 | The range of a subword of ... |
| swrdlend 14611 | The value of the subword e... |
| swrdnd 14612 | The value of the subword e... |
| swrdnd2 14613 | Value of the subword extra... |
| swrdnnn0nd 14614 | The value of a subword ope... |
| swrdnd0 14615 | The value of a subword ope... |
| swrd0 14616 | A subword of an empty set ... |
| swrdrlen 14617 | Length of a right-anchored... |
| swrdlen2 14618 | Length of an extracted sub... |
| swrdfv2 14619 | A symbol in an extracted s... |
| swrdwrdsymb 14620 | A subword is a word over t... |
| swrdsb0eq 14621 | Two subwords with the same... |
| swrdsbslen 14622 | Two subwords with the same... |
| swrdspsleq 14623 | Two words have a common su... |
| swrds1 14624 | Extract a single symbol fr... |
| swrdlsw 14625 | Extract the last single sy... |
| ccatswrd 14626 | Joining two adjacent subwo... |
| swrdccat2 14627 | Recover the right half of ... |
| pfxnndmnd 14630 | The value of a prefix oper... |
| pfxval 14631 | Value of a prefix operatio... |
| pfx00 14632 | The zero length prefix is ... |
| pfx0 14633 | A prefix of an empty set i... |
| pfxval0 14634 | Value of a prefix operatio... |
| pfxcl 14635 | Closure of the prefix extr... |
| pfxmpt 14636 | Value of the prefix extrac... |
| pfxres 14637 | Value of the prefix extrac... |
| pfxf 14638 | A prefix of a word is a fu... |
| pfxfn 14639 | Value of the prefix extrac... |
| pfxfv 14640 | A symbol in a prefix of a ... |
| pfxlen 14641 | Length of a prefix. (Cont... |
| pfxid 14642 | A word is a prefix of itse... |
| pfxrn 14643 | The range of a prefix of a... |
| pfxn0 14644 | A prefix consisting of at ... |
| pfxnd 14645 | The value of a prefix oper... |
| pfxnd0 14646 | The value of a prefix oper... |
| pfxwrdsymb 14647 | A prefix of a word is a wo... |
| addlenpfx 14648 | The sum of the lengths of ... |
| pfxfv0 14649 | The first symbol of a pref... |
| pfxtrcfv 14650 | A symbol in a word truncat... |
| pfxtrcfv0 14651 | The first symbol in a word... |
| pfxfvlsw 14652 | The last symbol in a nonem... |
| pfxeq 14653 | The prefixes of two words ... |
| pfxtrcfvl 14654 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14655 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14656 | Two (nonempty) words are e... |
| disjwrdpfx 14657 | Sets of words are disjoint... |
| ccatpfx 14658 | Concatenating a prefix wit... |
| pfxccat1 14659 | Recover the left half of a... |
| pfx1 14660 | The prefix of length one o... |
| swrdswrdlem 14661 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14662 | A subword of a subword is ... |
| pfxswrd 14663 | A prefix of a subword is a... |
| swrdpfx 14664 | A subword of a prefix is a... |
| pfxpfx 14665 | A prefix of a prefix is a ... |
| pfxpfxid 14666 | A prefix of a prefix with ... |
| pfxcctswrd 14667 | The concatenation of the p... |
| lenpfxcctswrd 14668 | The length of the concaten... |
| lenrevpfxcctswrd 14669 | The length of the concaten... |
| pfxlswccat 14670 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14671 | The last symbol of a word ... |
| ccats1pfxeqrex 14672 | There exists a symbol such... |
| ccatopth 14673 | An ~ opth -like theorem fo... |
| ccatopth2 14674 | An ~ opth -like theorem fo... |
| ccatlcan 14675 | Concatenation of words is ... |
| ccatrcan 14676 | Concatenation of words is ... |
| wrdeqs1cat 14677 | Decompose a nonempty word ... |
| cats1un 14678 | Express a word with an ext... |
| wrdind 14679 | Perform induction over the... |
| wrd2ind 14680 | Perform induction over the... |
| swrdccatfn 14681 | The subword of a concatena... |
| swrdccatin1 14682 | The subword of a concatena... |
| pfxccatin12lem4 14683 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14684 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14685 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14686 | The subword of a concatena... |
| pfxccatin12lem2c 14687 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14688 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14689 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14690 | The subword of a concatena... |
| pfxccat3 14691 | The subword of a concatena... |
| swrdccat 14692 | The subword of a concatena... |
| pfxccatpfx1 14693 | A prefix of a concatenatio... |
| pfxccatpfx2 14694 | A prefix of a concatenatio... |
| pfxccat3a 14695 | A prefix of a concatenatio... |
| swrdccat3blem 14696 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14697 | A suffix of a concatenatio... |
| pfxccatid 14698 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14699 | A word is a prefix of a wo... |
| swrdccatin1d 14700 | The subword of a concatena... |
| swrdccatin2d 14701 | The subword of a concatena... |
| pfxccatin12d 14702 | The subword of a concatena... |
| reuccatpfxs1lem 14703 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14704 | There is a unique word hav... |
| reuccatpfxs1v 14705 | There is a unique word hav... |
| splval 14708 | Value of the substring rep... |
| splcl 14709 | Closure of the substring r... |
| splid 14710 | Splicing a subword for the... |
| spllen 14711 | The length of a splice. (... |
| splfv1 14712 | Symbols to the left of a s... |
| splfv2a 14713 | Symbols within the replace... |
| splval2 14714 | Value of a splice, assumin... |
| revval 14717 | Value of the word reversin... |
| revcl 14718 | The reverse of a word is a... |
| revlen 14719 | The reverse of a word has ... |
| revfv 14720 | Reverse of a word at a poi... |
| rev0 14721 | The empty word is its own ... |
| revs1 14722 | Singleton words are their ... |
| revccat 14723 | Antiautomorphic property o... |
| revrev 14724 | Reversal is an involution ... |
| reps 14727 | Construct a function mappi... |
| repsundef 14728 | A function mapping a half-... |
| repsconst 14729 | Construct a function mappi... |
| repsf 14730 | The constructed function m... |
| repswsymb 14731 | The symbols of a "repeated... |
| repsw 14732 | A function mapping a half-... |
| repswlen 14733 | The length of a "repeated ... |
| repsw0 14734 | The "repeated symbol word"... |
| repsdf2 14735 | Alternative definition of ... |
| repswsymball 14736 | All the symbols of a "repe... |
| repswsymballbi 14737 | A word is a "repeated symb... |
| repswfsts 14738 | The first symbol of a none... |
| repswlsw 14739 | The last symbol of a nonem... |
| repsw1 14740 | The "repeated symbol word"... |
| repswswrd 14741 | A subword of a "repeated s... |
| repswpfx 14742 | A prefix of a repeated sym... |
| repswccat 14743 | The concatenation of two "... |
| repswrevw 14744 | The reverse of a "repeated... |
| cshfn 14747 | Perform a cyclical shift f... |
| cshword 14748 | Perform a cyclical shift f... |
| cshnz 14749 | A cyclical shift is the em... |
| 0csh0 14750 | Cyclically shifting an emp... |
| cshw0 14751 | A word cyclically shifted ... |
| cshwmodn 14752 | Cyclically shifting a word... |
| cshwsublen 14753 | Cyclically shifting a word... |
| cshwn 14754 | A word cyclically shifted ... |
| cshwcl 14755 | A cyclically shifted word ... |
| cshwlen 14756 | The length of a cyclically... |
| cshwf 14757 | A cyclically shifted word ... |
| cshwfn 14758 | A cyclically shifted word ... |
| cshwrn 14759 | The range of a cyclically ... |
| cshwidxmod 14760 | The symbol at a given inde... |
| cshwidxmodr 14761 | The symbol at a given inde... |
| cshwidx0mod 14762 | The symbol at index 0 of a... |
| cshwidx0 14763 | The symbol at index 0 of a... |
| cshwidxm1 14764 | The symbol at index ((n-N)... |
| cshwidxm 14765 | The symbol at index (n-N) ... |
| cshwidxn 14766 | The symbol at index (n-1) ... |
| cshf1 14767 | Cyclically shifting a word... |
| cshinj 14768 | If a word is injectiv (reg... |
| repswcshw 14769 | A cyclically shifted "repe... |
| 2cshw 14770 | Cyclically shifting a word... |
| 2cshwid 14771 | Cyclically shifting a word... |
| lswcshw 14772 | The last symbol of a word ... |
| 2cshwcom 14773 | Cyclically shifting a word... |
| cshwleneq 14774 | If the results of cyclical... |
| 3cshw 14775 | Cyclically shifting a word... |
| cshweqdif2 14776 | If cyclically shifting two... |
| cshweqdifid 14777 | If cyclically shifting a w... |
| cshweqrep 14778 | If cyclically shifting a w... |
| cshw1 14779 | If cyclically shifting a w... |
| cshw1repsw 14780 | If cyclically shifting a w... |
| cshwsexa 14781 | The class of (different!) ... |
| 2cshwcshw 14782 | If a word is a cyclically ... |
| scshwfzeqfzo 14783 | For a nonempty word the se... |
| cshwcshid 14784 | A cyclically shifted word ... |
| cshwcsh2id 14785 | A cyclically shifted word ... |
| cshimadifsn 14786 | The image of a cyclically ... |
| cshimadifsn0 14787 | The image of a cyclically ... |
| wrdco 14788 | Mapping a word by a functi... |
| lenco 14789 | Length of a mapped word is... |
| s1co 14790 | Mapping of a singleton wor... |
| revco 14791 | Mapping of words (i.e., a ... |
| ccatco 14792 | Mapping of words commutes ... |
| cshco 14793 | Mapping of words commutes ... |
| swrdco 14794 | Mapping of words commutes ... |
| pfxco 14795 | Mapping of words commutes ... |
| lswco 14796 | Mapping of (nonempty) word... |
| repsco 14797 | Mapping of words commutes ... |
| cats1cld 14812 | Closure of concatenation w... |
| cats1co 14813 | Closure of concatenation w... |
| cats1cli 14814 | Closure of concatenation w... |
| cats1fvn 14815 | The last symbol of a conca... |
| cats1fv 14816 | A symbol other than the la... |
| cats1len 14817 | The length of concatenatio... |
| cats1cat 14818 | Closure of concatenation w... |
| cats2cat 14819 | Closure of concatenation o... |
| s2eqd 14820 | Equality theorem for a dou... |
| s3eqd 14821 | Equality theorem for a len... |
| s4eqd 14822 | Equality theorem for a len... |
| s5eqd 14823 | Equality theorem for a len... |
| s6eqd 14824 | Equality theorem for a len... |
| s7eqd 14825 | Equality theorem for a len... |
| s8eqd 14826 | Equality theorem for a len... |
| s3eq2 14827 | Equality theorem for a len... |
| s2cld 14828 | A doubleton word is a word... |
| s3cld 14829 | A length 3 string is a wor... |
| s4cld 14830 | A length 4 string is a wor... |
| s5cld 14831 | A length 5 string is a wor... |
| s6cld 14832 | A length 6 string is a wor... |
| s7cld 14833 | A length 7 string is a wor... |
| s8cld 14834 | A length 8 string is a wor... |
| s2cl 14835 | A doubleton word is a word... |
| s3cl 14836 | A length 3 string is a wor... |
| s2cli 14837 | A doubleton word is a word... |
| s3cli 14838 | A length 3 string is a wor... |
| s4cli 14839 | A length 4 string is a wor... |
| s5cli 14840 | A length 5 string is a wor... |
| s6cli 14841 | A length 6 string is a wor... |
| s7cli 14842 | A length 7 string is a wor... |
| s8cli 14843 | A length 8 string is a wor... |
| s2fv0 14844 | Extract the first symbol f... |
| s2fv1 14845 | Extract the second symbol ... |
| s2len 14846 | The length of a doubleton ... |
| s2dm 14847 | The domain of a doubleton ... |
| s3fv0 14848 | Extract the first symbol f... |
| s3fv1 14849 | Extract the second symbol ... |
| s3fv2 14850 | Extract the third symbol f... |
| s3len 14851 | The length of a length 3 s... |
| s4fv0 14852 | Extract the first symbol f... |
| s4fv1 14853 | Extract the second symbol ... |
| s4fv2 14854 | Extract the third symbol f... |
| s4fv3 14855 | Extract the fourth symbol ... |
| s4len 14856 | The length of a length 4 s... |
| s5len 14857 | The length of a length 5 s... |
| s6len 14858 | The length of a length 6 s... |
| s7len 14859 | The length of a length 7 s... |
| s8len 14860 | The length of a length 8 s... |
| lsws2 14861 | The last symbol of a doubl... |
| lsws3 14862 | The last symbol of a 3 let... |
| lsws4 14863 | The last symbol of a 4 let... |
| s2prop 14864 | A length 2 word is an unor... |
| s2dmALT 14865 | Alternate version of ~ s2d... |
| s3tpop 14866 | A length 3 word is an unor... |
| s4prop 14867 | A length 4 word is a union... |
| s3fn 14868 | A length 3 word is a funct... |
| funcnvs1 14869 | The converse of a singleto... |
| funcnvs2 14870 | The converse of a length 2... |
| funcnvs3 14871 | The converse of a length 3... |
| funcnvs4 14872 | The converse of a length 4... |
| s2f1o 14873 | A length 2 word with mutua... |
| f1oun2prg 14874 | A union of unordered pairs... |
| s4f1o 14875 | A length 4 word with mutua... |
| s4dom 14876 | The domain of a length 4 w... |
| s2co 14877 | Mapping a doubleton word b... |
| s3co 14878 | Mapping a length 3 string ... |
| s0s1 14879 | Concatenation of fixed len... |
| s1s2 14880 | Concatenation of fixed len... |
| s1s3 14881 | Concatenation of fixed len... |
| s1s4 14882 | Concatenation of fixed len... |
| s1s5 14883 | Concatenation of fixed len... |
| s1s6 14884 | Concatenation of fixed len... |
| s1s7 14885 | Concatenation of fixed len... |
| s2s2 14886 | Concatenation of fixed len... |
| s4s2 14887 | Concatenation of fixed len... |
| s4s3 14888 | Concatenation of fixed len... |
| s4s4 14889 | Concatenation of fixed len... |
| s3s4 14890 | Concatenation of fixed len... |
| s2s5 14891 | Concatenation of fixed len... |
| s5s2 14892 | Concatenation of fixed len... |
| s2eq2s1eq 14893 | Two length 2 words are equ... |
| s2eq2seq 14894 | Two length 2 words are equ... |
| s3eqs2s1eq 14895 | Two length 3 words are equ... |
| s3eq3seq 14896 | Two length 3 words are equ... |
| swrds2 14897 | Extract two adjacent symbo... |
| swrds2m 14898 | Extract two adjacent symbo... |
| wrdlen2i 14899 | Implications of a word of ... |
| wrd2pr2op 14900 | A word of length two repre... |
| wrdlen2 14901 | A word of length two. (Co... |
| wrdlen2s2 14902 | A word of length two as do... |
| wrdl2exs2 14903 | A word of length two is a ... |
| pfx2 14904 | A prefix of length two. (... |
| wrd3tpop 14905 | A word of length three rep... |
| wrdlen3s3 14906 | A word of length three as ... |
| repsw2 14907 | The "repeated symbol word"... |
| repsw3 14908 | The "repeated symbol word"... |
| swrd2lsw 14909 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14910 | Two words of length at lea... |
| ccatw2s1ccatws2 14911 | The concatenation of a wor... |
| ccat2s1fvwALT 14912 | Alternate proof of ~ ccat2... |
| wwlktovf 14913 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14914 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14915 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14916 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14917 | There is a bijection betwe... |
| eqwrds3 14918 | A word is equal with a len... |
| wrdl3s3 14919 | A word of length 3 is a le... |
| s2rn 14920 | Range of a length 2 string... |
| s3rn 14921 | Range of a length 3 string... |
| s7rn 14922 | Range of a length 7 string... |
| s7f1o 14923 | A length 7 word with mutua... |
| s3sndisj 14924 | The singletons consisting ... |
| s3iunsndisj 14925 | The union of singletons co... |
| ofccat 14926 | Letterwise operations on w... |
| ofs1 14927 | Letterwise operations on a... |
| ofs2 14928 | Letterwise operations on a... |
| coss12d 14929 | Subset deduction for compo... |
| trrelssd 14930 | The composition of subclas... |
| xpcogend 14931 | The most interesting case ... |
| xpcoidgend 14932 | If two classes are not dis... |
| cotr2g 14933 | Two ways of saying that th... |
| cotr2 14934 | Two ways of saying a relat... |
| cotr3 14935 | Two ways of saying a relat... |
| coemptyd 14936 | Deduction about compositio... |
| xptrrel 14937 | The cross product is alway... |
| 0trrel 14938 | The empty class is a trans... |
| cleq1lem 14939 | Equality implies bijection... |
| cleq1 14940 | Equality of relations impl... |
| clsslem 14941 | The closure of a subclass ... |
| trcleq1 14946 | Equality of relations impl... |
| trclsslem 14947 | The transitive closure (as... |
| trcleq2lem 14948 | Equality implies bijection... |
| cvbtrcl 14949 | Change of bound variable i... |
| trcleq12lem 14950 | Equality implies bijection... |
| trclexlem 14951 | Existence of relation impl... |
| trclublem 14952 | If a relation exists then ... |
| trclubi 14953 | The Cartesian product of t... |
| trclubgi 14954 | The union with the Cartesi... |
| trclub 14955 | The Cartesian product of t... |
| trclubg 14956 | The union with the Cartesi... |
| trclfv 14957 | The transitive closure of ... |
| brintclab 14958 | Two ways to express a bina... |
| brtrclfv 14959 | Two ways of expressing the... |
| brcnvtrclfv 14960 | Two ways of expressing the... |
| brtrclfvcnv 14961 | Two ways of expressing the... |
| brcnvtrclfvcnv 14962 | Two ways of expressing the... |
| trclfvss 14963 | The transitive closure (as... |
| trclfvub 14964 | The transitive closure of ... |
| trclfvlb 14965 | The transitive closure of ... |
| trclfvcotr 14966 | The transitive closure of ... |
| trclfvlb2 14967 | The transitive closure of ... |
| trclfvlb3 14968 | The transitive closure of ... |
| cotrtrclfv 14969 | The transitive closure of ... |
| trclidm 14970 | The transitive closure of ... |
| trclun 14971 | Transitive closure of a un... |
| trclfvg 14972 | The value of the transitiv... |
| trclfvcotrg 14973 | The value of the transitiv... |
| reltrclfv 14974 | The transitive closure of ... |
| dmtrclfv 14975 | The domain of the transiti... |
| reldmrelexp 14978 | The domain of the repeated... |
| relexp0g 14979 | A relation composed zero t... |
| relexp0 14980 | A relation composed zero t... |
| relexp0d 14981 | A relation composed zero t... |
| relexpsucnnr 14982 | A reduction for relation e... |
| relexp1g 14983 | A relation composed once i... |
| dfid5 14984 | Identity relation is equal... |
| dfid6 14985 | Identity relation expresse... |
| relexp1d 14986 | A relation composed once i... |
| relexpsucnnl 14987 | A reduction for relation e... |
| relexpsucl 14988 | A reduction for relation e... |
| relexpsucr 14989 | A reduction for relation e... |
| relexpsucrd 14990 | A reduction for relation e... |
| relexpsucld 14991 | A reduction for relation e... |
| relexpcnv 14992 | Commutation of converse an... |
| relexpcnvd 14993 | Commutation of converse an... |
| relexp0rel 14994 | The exponentiation of a cl... |
| relexprelg 14995 | The exponentiation of a cl... |
| relexprel 14996 | The exponentiation of a re... |
| relexpreld 14997 | The exponentiation of a re... |
| relexpnndm 14998 | The domain of an exponenti... |
| relexpdmg 14999 | The domain of an exponenti... |
| relexpdm 15000 | The domain of an exponenti... |
| relexpdmd 15001 | The domain of an exponenti... |
| relexpnnrn 15002 | The range of an exponentia... |
| relexprng 15003 | The range of an exponentia... |
| relexprn 15004 | The range of an exponentia... |
| relexprnd 15005 | The range of an exponentia... |
| relexpfld 15006 | The field of an exponentia... |
| relexpfldd 15007 | The field of an exponentia... |
| relexpaddnn 15008 | Relation composition becom... |
| relexpuzrel 15009 | The exponentiation of a cl... |
| relexpaddg 15010 | Relation composition becom... |
| relexpaddd 15011 | Relation composition becom... |
| rtrclreclem1 15014 | The reflexive, transitive ... |
| dfrtrclrec2 15015 | If two elements are connec... |
| rtrclreclem2 15016 | The reflexive, transitive ... |
| rtrclreclem3 15017 | The reflexive, transitive ... |
| rtrclreclem4 15018 | The reflexive, transitive ... |
| dfrtrcl2 15019 | The two definitions ` t* `... |
| relexpindlem 15020 | Principle of transitive in... |
| relexpind 15021 | Principle of transitive in... |
| rtrclind 15022 | Principle of transitive in... |
| shftlem 15025 | Two ways to write a shifte... |
| shftuz 15026 | A shift of the upper integ... |
| shftfval 15027 | The value of the sequence ... |
| shftdm 15028 | Domain of a relation shift... |
| shftfib 15029 | Value of a fiber of the re... |
| shftfn 15030 | Functionality and domain o... |
| shftval 15031 | Value of a sequence shifte... |
| shftval2 15032 | Value of a sequence shifte... |
| shftval3 15033 | Value of a sequence shifte... |
| shftval4 15034 | Value of a sequence shifte... |
| shftval5 15035 | Value of a shifted sequenc... |
| shftf 15036 | Functionality of a shifted... |
| 2shfti 15037 | Composite shift operations... |
| shftidt2 15038 | Identity law for the shift... |
| shftidt 15039 | Identity law for the shift... |
| shftcan1 15040 | Cancellation law for the s... |
| shftcan2 15041 | Cancellation law for the s... |
| seqshft 15042 | Shifting the index set of ... |
| sgnval 15045 | Value of the signum functi... |
| sgn0 15046 | The signum of 0 is 0. (Co... |
| sgnp 15047 | The signum of a positive e... |
| sgnrrp 15048 | The signum of a positive r... |
| sgn1 15049 | The signum of 1 is 1. (Co... |
| sgnpnf 15050 | The signum of ` +oo ` is 1... |
| sgnn 15051 | The signum of a negative e... |
| sgnmnf 15052 | The signum of ` -oo ` is -... |
| cjval 15059 | The value of the conjugate... |
| cjth 15060 | The defining property of t... |
| cjf 15061 | Domain and codomain of the... |
| cjcl 15062 | The conjugate of a complex... |
| reval 15063 | The value of the real part... |
| imval 15064 | The value of the imaginary... |
| imre 15065 | The imaginary part of a co... |
| reim 15066 | The real part of a complex... |
| recl 15067 | The real part of a complex... |
| imcl 15068 | The imaginary part of a co... |
| ref 15069 | Domain and codomain of the... |
| imf 15070 | Domain and codomain of the... |
| crre 15071 | The real part of a complex... |
| crim 15072 | The real part of a complex... |
| replim 15073 | Reconstruct a complex numb... |
| remim 15074 | Value of the conjugate of ... |
| reim0 15075 | The imaginary part of a re... |
| reim0b 15076 | A number is real iff its i... |
| rereb 15077 | A number is real iff it eq... |
| mulre 15078 | A product with a nonzero r... |
| rere 15079 | A real number equals its r... |
| cjreb 15080 | A number is real iff it eq... |
| recj 15081 | Real part of a complex con... |
| reneg 15082 | Real part of negative. (C... |
| readd 15083 | Real part distributes over... |
| resub 15084 | Real part distributes over... |
| remullem 15085 | Lemma for ~ remul , ~ immu... |
| remul 15086 | Real part of a product. (... |
| remul2 15087 | Real part of a product. (... |
| rediv 15088 | Real part of a division. ... |
| imcj 15089 | Imaginary part of a comple... |
| imneg 15090 | The imaginary part of a ne... |
| imadd 15091 | Imaginary part distributes... |
| imsub 15092 | Imaginary part distributes... |
| immul 15093 | Imaginary part of a produc... |
| immul2 15094 | Imaginary part of a produc... |
| imdiv 15095 | Imaginary part of a divisi... |
| cjre 15096 | A real number equals its c... |
| cjcj 15097 | The conjugate of the conju... |
| cjadd 15098 | Complex conjugate distribu... |
| cjmul 15099 | Complex conjugate distribu... |
| ipcnval 15100 | Standard inner product on ... |
| cjmulrcl 15101 | A complex number times its... |
| cjmulval 15102 | A complex number times its... |
| cjmulge0 15103 | A complex number times its... |
| cjneg 15104 | Complex conjugate of negat... |
| addcj 15105 | A number plus its conjugat... |
| cjsub 15106 | Complex conjugate distribu... |
| cjexp 15107 | Complex conjugate of posit... |
| imval2 15108 | The imaginary part of a nu... |
| re0 15109 | The real part of zero. (C... |
| im0 15110 | The imaginary part of zero... |
| re1 15111 | The real part of one. (Co... |
| im1 15112 | The imaginary part of one.... |
| rei 15113 | The real part of ` _i ` . ... |
| imi 15114 | The imaginary part of ` _i... |
| cj0 15115 | The conjugate of zero. (C... |
| cji 15116 | The complex conjugate of t... |
| cjreim 15117 | The conjugate of a represe... |
| cjreim2 15118 | The conjugate of the repre... |
| cj11 15119 | Complex conjugate is a one... |
| cjne0 15120 | A number is nonzero iff it... |
| cjdiv 15121 | Complex conjugate distribu... |
| cnrecnv 15122 | The inverse to the canonic... |
| sqeqd 15123 | A deduction for showing tw... |
| recli 15124 | The real part of a complex... |
| imcli 15125 | The imaginary part of a co... |
| cjcli 15126 | Closure law for complex co... |
| replimi 15127 | Construct a complex number... |
| cjcji 15128 | The conjugate of the conju... |
| reim0bi 15129 | A number is real iff its i... |
| rerebi 15130 | A real number equals its r... |
| cjrebi 15131 | A number is real iff it eq... |
| recji 15132 | Real part of a complex con... |
| imcji 15133 | Imaginary part of a comple... |
| cjmulrcli 15134 | A complex number times its... |
| cjmulvali 15135 | A complex number times its... |
| cjmulge0i 15136 | A complex number times its... |
| renegi 15137 | Real part of negative. (C... |
| imnegi 15138 | Imaginary part of negative... |
| cjnegi 15139 | Complex conjugate of negat... |
| addcji 15140 | A number plus its conjugat... |
| readdi 15141 | Real part distributes over... |
| imaddi 15142 | Imaginary part distributes... |
| remuli 15143 | Real part of a product. (... |
| immuli 15144 | Imaginary part of a produc... |
| cjaddi 15145 | Complex conjugate distribu... |
| cjmuli 15146 | Complex conjugate distribu... |
| ipcni 15147 | Standard inner product on ... |
| cjdivi 15148 | Complex conjugate distribu... |
| crrei 15149 | The real part of a complex... |
| crimi 15150 | The imaginary part of a co... |
| recld 15151 | The real part of a complex... |
| imcld 15152 | The imaginary part of a co... |
| cjcld 15153 | Closure law for complex co... |
| replimd 15154 | Construct a complex number... |
| remimd 15155 | Value of the conjugate of ... |
| cjcjd 15156 | The conjugate of the conju... |
| reim0bd 15157 | A number is real iff its i... |
| rerebd 15158 | A real number equals its r... |
| cjrebd 15159 | A number is real iff it eq... |
| cjne0d 15160 | A number is nonzero iff it... |
| recjd 15161 | Real part of a complex con... |
| imcjd 15162 | Imaginary part of a comple... |
| cjmulrcld 15163 | A complex number times its... |
| cjmulvald 15164 | A complex number times its... |
| cjmulge0d 15165 | A complex number times its... |
| renegd 15166 | Real part of negative. (C... |
| imnegd 15167 | Imaginary part of negative... |
| cjnegd 15168 | Complex conjugate of negat... |
| addcjd 15169 | A number plus its conjugat... |
| cjexpd 15170 | Complex conjugate of posit... |
| readdd 15171 | Real part distributes over... |
| imaddd 15172 | Imaginary part distributes... |
| resubd 15173 | Real part distributes over... |
| imsubd 15174 | Imaginary part distributes... |
| remuld 15175 | Real part of a product. (... |
| immuld 15176 | Imaginary part of a produc... |
| cjaddd 15177 | Complex conjugate distribu... |
| cjmuld 15178 | Complex conjugate distribu... |
| ipcnd 15179 | Standard inner product on ... |
| cjdivd 15180 | Complex conjugate distribu... |
| rered 15181 | A real number equals its r... |
| reim0d 15182 | The imaginary part of a re... |
| cjred 15183 | A real number equals its c... |
| remul2d 15184 | Real part of a product. (... |
| immul2d 15185 | Imaginary part of a produc... |
| redivd 15186 | Real part of a division. ... |
| imdivd 15187 | Imaginary part of a divisi... |
| crred 15188 | The real part of a complex... |
| crimd 15189 | The imaginary part of a co... |
| sqrtval 15194 | Value of square root funct... |
| absval 15195 | The absolute value (modulu... |
| rennim 15196 | A real number does not lie... |
| cnpart 15197 | The specification of restr... |
| sqrt0 15198 | The square root of zero is... |
| 01sqrexlem1 15199 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15200 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15201 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15202 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15203 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15204 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15205 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15206 | Existence of a square root... |
| resqrex 15207 | Existence of a square root... |
| sqrmo 15208 | Uniqueness for the square ... |
| resqreu 15209 | Existence and uniqueness f... |
| resqrtcl 15210 | Closure of the square root... |
| resqrtthlem 15211 | Lemma for ~ resqrtth . (C... |
| resqrtth 15212 | Square root theorem over t... |
| remsqsqrt 15213 | Square of square root. (C... |
| sqrtge0 15214 | The square root function i... |
| sqrtgt0 15215 | The square root function i... |
| sqrtmul 15216 | Square root distributes ov... |
| sqrtle 15217 | Square root is monotonic. ... |
| sqrtlt 15218 | Square root is strictly mo... |
| sqrt11 15219 | The square root function i... |
| sqrt00 15220 | A square root is zero iff ... |
| rpsqrtcl 15221 | The square root of a posit... |
| sqrtdiv 15222 | Square root distributes ov... |
| sqrtneglem 15223 | The square root of a negat... |
| sqrtneg 15224 | The square root of a negat... |
| sqrtsq2 15225 | Relationship between squar... |
| sqrtsq 15226 | Square root of square. (C... |
| sqrtmsq 15227 | Square root of square. (C... |
| sqrt1 15228 | The square root of 1 is 1.... |
| sqrt4 15229 | The square root of 4 is 2.... |
| sqrt9 15230 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15231 | The square root of 2 is bo... |
| sqrtm1 15232 | The imaginary unit is the ... |
| nn0sqeq1 15233 | A natural number with squa... |
| absneg 15234 | Absolute value of the nega... |
| abscl 15235 | Real closure of absolute v... |
| abscj 15236 | The absolute value of a nu... |
| absvalsq 15237 | Square of value of absolut... |
| absvalsq2 15238 | Square of value of absolut... |
| sqabsadd 15239 | Square of absolute value o... |
| sqabssub 15240 | Square of absolute value o... |
| absval2 15241 | Value of absolute value fu... |
| abs0 15242 | The absolute value of 0. ... |
| absi 15243 | The absolute value of the ... |
| absge0 15244 | Absolute value is nonnegat... |
| absrpcl 15245 | The absolute value of a no... |
| abs00 15246 | The absolute value of a nu... |
| abs00ad 15247 | A complex number is zero i... |
| abs00bd 15248 | If a complex number is zer... |
| absreimsq 15249 | Square of the absolute val... |
| absreim 15250 | Absolute value of a number... |
| absmul 15251 | Absolute value distributes... |
| absdiv 15252 | Absolute value distributes... |
| absid 15253 | A nonnegative number is it... |
| abs1 15254 | The absolute value of one ... |
| absnid 15255 | For a negative number, its... |
| leabs 15256 | A real number is less than... |
| absor 15257 | The absolute value of a re... |
| absre 15258 | Absolute value of a real n... |
| absresq 15259 | Square of the absolute val... |
| absmod0 15260 | ` A ` is divisible by ` B ... |
| absexp 15261 | Absolute value of positive... |
| absexpz 15262 | Absolute value of integer ... |
| abssq 15263 | Square can be moved in and... |
| sqabs 15264 | The squares of two reals a... |
| absrele 15265 | The absolute value of a co... |
| absimle 15266 | The absolute value of a co... |
| max0add 15267 | The sum of the positive an... |
| absz 15268 | A real number is an intege... |
| nn0abscl 15269 | The absolute value of an i... |
| zabscl 15270 | The absolute value of an i... |
| zabs0b 15271 | An integer has an absolute... |
| abslt 15272 | Absolute value and 'less t... |
| absle 15273 | Absolute value and 'less t... |
| abssubne0 15274 | If the absolute value of a... |
| absdiflt 15275 | The absolute value of a di... |
| absdifle 15276 | The absolute value of a di... |
| elicc4abs 15277 | Membership in a symmetric ... |
| lenegsq 15278 | Comparison to a nonnegativ... |
| releabs 15279 | The real part of a number ... |
| recval 15280 | Reciprocal expressed with ... |
| absidm 15281 | The absolute value functio... |
| absgt0 15282 | The absolute value of a no... |
| nnabscl 15283 | The absolute value of a no... |
| abssub 15284 | Swapping order of subtract... |
| abssubge0 15285 | Absolute value of a nonneg... |
| abssuble0 15286 | Absolute value of a nonpos... |
| absmax 15287 | The maximum of two numbers... |
| abstri 15288 | Triangle inequality for ab... |
| abs3dif 15289 | Absolute value of differen... |
| abs2dif 15290 | Difference of absolute val... |
| abs2dif2 15291 | Difference of absolute val... |
| abs2difabs 15292 | Absolute value of differen... |
| abs1m 15293 | For any complex number, th... |
| recan 15294 | Cancellation law involving... |
| absf 15295 | Mapping domain and codomai... |
| abs3lem 15296 | Lemma involving absolute v... |
| abslem2 15297 | Lemma involving absolute v... |
| rddif 15298 | The difference between a r... |
| absrdbnd 15299 | Bound on the absolute valu... |
| fzomaxdiflem 15300 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15301 | A bound on the separation ... |
| uzin2 15302 | The upper integers are clo... |
| rexanuz 15303 | Combine two different uppe... |
| rexanre 15304 | Combine two different uppe... |
| rexfiuz 15305 | Combine finitely many diff... |
| rexuz3 15306 | Restrict the base of the u... |
| rexanuz2 15307 | Combine two different uppe... |
| r19.29uz 15308 | A version of ~ 19.29 for u... |
| r19.2uz 15309 | A version of ~ r19.2z for ... |
| rexuzre 15310 | Convert an upper real quan... |
| rexico 15311 | Restrict the base of an up... |
| cau3lem 15312 | Lemma for ~ cau3 . (Contr... |
| cau3 15313 | Convert between three-quan... |
| cau4 15314 | Change the base of a Cauch... |
| caubnd2 15315 | A Cauchy sequence of compl... |
| caubnd 15316 | A Cauchy sequence of compl... |
| sqreulem 15317 | Lemma for ~ sqreu : write ... |
| sqreu 15318 | Existence and uniqueness f... |
| sqrtcl 15319 | Closure of the square root... |
| sqrtthlem 15320 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15321 | Mapping domain and codomai... |
| sqrtth 15322 | Square root theorem over t... |
| sqrtrege0 15323 | The square root function m... |
| eqsqrtor 15324 | Solve an equation containi... |
| eqsqrtd 15325 | A deduction for showing th... |
| eqsqrt2d 15326 | A deduction for showing th... |
| amgm2 15327 | Arithmetic-geometric mean ... |
| sqrtthi 15328 | Square root theorem. Theo... |
| sqrtcli 15329 | The square root of a nonne... |
| sqrtgt0i 15330 | The square root of a posit... |
| sqrtmsqi 15331 | Square root of square. (C... |
| sqrtsqi 15332 | Square root of square. (C... |
| sqsqrti 15333 | Square of square root. (C... |
| sqrtge0i 15334 | The square root of a nonne... |
| absidi 15335 | A nonnegative number is it... |
| absnidi 15336 | A negative number is the n... |
| leabsi 15337 | A real number is less than... |
| absori 15338 | The absolute value of a re... |
| absrei 15339 | Absolute value of a real n... |
| sqrtpclii 15340 | The square root of a posit... |
| sqrtgt0ii 15341 | The square root of a posit... |
| sqrt11i 15342 | The square root function i... |
| sqrtmuli 15343 | Square root distributes ov... |
| sqrtmulii 15344 | Square root distributes ov... |
| sqrtmsq2i 15345 | Relationship between squar... |
| sqrtlei 15346 | Square root is monotonic. ... |
| sqrtlti 15347 | Square root is strictly mo... |
| abslti 15348 | Absolute value and 'less t... |
| abslei 15349 | Absolute value and 'less t... |
| cnsqrt00 15350 | A square root of a complex... |
| absvalsqi 15351 | Square of value of absolut... |
| absvalsq2i 15352 | Square of value of absolut... |
| abscli 15353 | Real closure of absolute v... |
| absge0i 15354 | Absolute value is nonnegat... |
| absval2i 15355 | Value of absolute value fu... |
| abs00i 15356 | The absolute value of a nu... |
| absgt0i 15357 | The absolute value of a no... |
| absnegi 15358 | Absolute value of negative... |
| abscji 15359 | The absolute value of a nu... |
| releabsi 15360 | The real part of a number ... |
| abssubi 15361 | Swapping order of subtract... |
| absmuli 15362 | Absolute value distributes... |
| sqabsaddi 15363 | Square of absolute value o... |
| sqabssubi 15364 | Square of absolute value o... |
| absdivzi 15365 | Absolute value distributes... |
| abstrii 15366 | Triangle inequality for ab... |
| abs3difi 15367 | Absolute value of differen... |
| abs3lemi 15368 | Lemma involving absolute v... |
| rpsqrtcld 15369 | The square root of a posit... |
| sqrtgt0d 15370 | The square root of a posit... |
| absnidd 15371 | A negative number is the n... |
| leabsd 15372 | A real number is less than... |
| absord 15373 | The absolute value of a re... |
| absred 15374 | Absolute value of a real n... |
| resqrtcld 15375 | The square root of a nonne... |
| sqrtmsqd 15376 | Square root of square. (C... |
| sqrtsqd 15377 | Square root of square. (C... |
| sqrtge0d 15378 | The square root of a nonne... |
| sqrtnegd 15379 | The square root of a negat... |
| absidd 15380 | A nonnegative number is it... |
| sqrtdivd 15381 | Square root distributes ov... |
| sqrtmuld 15382 | Square root distributes ov... |
| sqrtsq2d 15383 | Relationship between squar... |
| sqrtled 15384 | Square root is monotonic. ... |
| sqrtltd 15385 | Square root is strictly mo... |
| sqr11d 15386 | The square root function i... |
| nn0absid 15387 | A nonnegative integer is i... |
| nn0absidi 15388 | A nonnegative integer is i... |
| absltd 15389 | Absolute value and 'less t... |
| absled 15390 | Absolute value and 'less t... |
| abssubge0d 15391 | Absolute value of a nonneg... |
| abssuble0d 15392 | Absolute value of a nonpos... |
| absdifltd 15393 | The absolute value of a di... |
| absdifled 15394 | The absolute value of a di... |
| icodiamlt 15395 | Two elements in a half-ope... |
| abscld 15396 | Real closure of absolute v... |
| sqrtcld 15397 | Closure of the square root... |
| sqrtrege0d 15398 | The real part of the squar... |
| sqsqrtd 15399 | Square root theorem. Theo... |
| msqsqrtd 15400 | Square root theorem. Theo... |
| sqr00d 15401 | A square root is zero iff ... |
| absvalsqd 15402 | Square of value of absolut... |
| absvalsq2d 15403 | Square of value of absolut... |
| absge0d 15404 | Absolute value is nonnegat... |
| absval2d 15405 | Value of absolute value fu... |
| abs00d 15406 | The absolute value of a nu... |
| absne0d 15407 | The absolute value of a nu... |
| absrpcld 15408 | The absolute value of a no... |
| absnegd 15409 | Absolute value of negative... |
| abscjd 15410 | The absolute value of a nu... |
| releabsd 15411 | The real part of a number ... |
| absexpd 15412 | Absolute value of positive... |
| abssubd 15413 | Swapping order of subtract... |
| absmuld 15414 | Absolute value distributes... |
| absdivd 15415 | Absolute value distributes... |
| abstrid 15416 | Triangle inequality for ab... |
| abs2difd 15417 | Difference of absolute val... |
| abs2dif2d 15418 | Difference of absolute val... |
| abs2difabsd 15419 | Absolute value of differen... |
| abs3difd 15420 | Absolute value of differen... |
| abs3lemd 15421 | Lemma involving absolute v... |
| reusq0 15422 | A complex number is the sq... |
| bhmafibid1cn 15423 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15424 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15425 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15426 | The Brahmagupta-Fibonacci ... |
| limsupgord 15429 | Ordering property of the s... |
| limsupcl 15430 | Closure of the superior li... |
| limsupval 15431 | The superior limit of an i... |
| limsupgf 15432 | Closure of the superior li... |
| limsupgval 15433 | Value of the superior limi... |
| limsupgle 15434 | The defining property of t... |
| limsuple 15435 | The defining property of t... |
| limsuplt 15436 | The defining property of t... |
| limsupval2 15437 | The superior limit, relati... |
| limsupgre 15438 | If a sequence of real numb... |
| limsupbnd1 15439 | If a sequence is eventuall... |
| limsupbnd2 15440 | If a sequence is eventuall... |
| climrel 15449 | The limit relation is a re... |
| rlimrel 15450 | The limit relation is a re... |
| clim 15451 | Express the predicate: Th... |
| rlim 15452 | Express the predicate: Th... |
| rlim2 15453 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15454 | Use strictly less-than in ... |
| rlim3 15455 | Restrict the range of the ... |
| climcl 15456 | Closure of the limit of a ... |
| rlimpm 15457 | Closure of a function with... |
| rlimf 15458 | Closure of a function with... |
| rlimss 15459 | Domain closure of a functi... |
| rlimcl 15460 | Closure of the limit of a ... |
| clim2 15461 | Express the predicate: Th... |
| clim2c 15462 | Express the predicate ` F ... |
| clim0 15463 | Express the predicate ` F ... |
| clim0c 15464 | Express the predicate ` F ... |
| rlim0 15465 | Express the predicate ` B ... |
| rlim0lt 15466 | Use strictly less-than in ... |
| climi 15467 | Convergence of a sequence ... |
| climi2 15468 | Convergence of a sequence ... |
| climi0 15469 | Convergence of a sequence ... |
| rlimi 15470 | Convergence at infinity of... |
| rlimi2 15471 | Convergence at infinity of... |
| ello1 15472 | Elementhood in the set of ... |
| ello12 15473 | Elementhood in the set of ... |
| ello12r 15474 | Sufficient condition for e... |
| lo1f 15475 | An eventually upper bounde... |
| lo1dm 15476 | An eventually upper bounde... |
| lo1bdd 15477 | The defining property of a... |
| ello1mpt 15478 | Elementhood in the set of ... |
| ello1mpt2 15479 | Elementhood in the set of ... |
| ello1d 15480 | Sufficient condition for e... |
| lo1bdd2 15481 | If an eventually bounded f... |
| lo1bddrp 15482 | Refine ~ o1bdd2 to give a ... |
| elo1 15483 | Elementhood in the set of ... |
| elo12 15484 | Elementhood in the set of ... |
| elo12r 15485 | Sufficient condition for e... |
| o1f 15486 | An eventually bounded func... |
| o1dm 15487 | An eventually bounded func... |
| o1bdd 15488 | The defining property of a... |
| lo1o1 15489 | A function is eventually b... |
| lo1o12 15490 | A function is eventually b... |
| elo1mpt 15491 | Elementhood in the set of ... |
| elo1mpt2 15492 | Elementhood in the set of ... |
| elo1d 15493 | Sufficient condition for e... |
| o1lo1 15494 | A real function is eventua... |
| o1lo12 15495 | A lower bounded real funct... |
| o1lo1d 15496 | A real eventually bounded ... |
| icco1 15497 | Derive eventual boundednes... |
| o1bdd2 15498 | If an eventually bounded f... |
| o1bddrp 15499 | Refine ~ o1bdd2 to give a ... |
| climconst 15500 | An (eventually) constant s... |
| rlimconst 15501 | A constant sequence conver... |
| rlimclim1 15502 | Forward direction of ~ rli... |
| rlimclim 15503 | A sequence on an upper int... |
| climrlim2 15504 | Produce a real limit from ... |
| climconst2 15505 | A constant sequence conver... |
| climz 15506 | The zero sequence converge... |
| rlimuni 15507 | A real function whose doma... |
| rlimdm 15508 | Two ways to express that a... |
| climuni 15509 | An infinite sequence of co... |
| fclim 15510 | The limit relation is func... |
| climdm 15511 | Two ways to express that a... |
| climeu 15512 | An infinite sequence of co... |
| climreu 15513 | An infinite sequence of co... |
| climmo 15514 | An infinite sequence of co... |
| rlimres 15515 | The restriction of a funct... |
| lo1res 15516 | The restriction of an even... |
| o1res 15517 | The restriction of an even... |
| rlimres2 15518 | The restriction of a funct... |
| lo1res2 15519 | The restriction of a funct... |
| o1res2 15520 | The restriction of a funct... |
| lo1resb 15521 | The restriction of a funct... |
| rlimresb 15522 | The restriction of a funct... |
| o1resb 15523 | The restriction of a funct... |
| climeq 15524 | Two functions that are eve... |
| lo1eq 15525 | Two functions that are eve... |
| rlimeq 15526 | Two functions that are eve... |
| o1eq 15527 | Two functions that are eve... |
| climmpt 15528 | Exhibit a function ` G ` w... |
| 2clim 15529 | If two sequences converge ... |
| climmpt2 15530 | Relate an integer limit on... |
| climshftlem 15531 | A shifted function converg... |
| climres 15532 | A function restricted to u... |
| climshft 15533 | A shifted function converg... |
| serclim0 15534 | The zero series converges ... |
| rlimcld2 15535 | If ` D ` is a closed set i... |
| rlimrege0 15536 | The limit of a sequence of... |
| rlimrecl 15537 | The limit of a real sequen... |
| rlimge0 15538 | The limit of a sequence of... |
| climshft2 15539 | A shifted function converg... |
| climrecl 15540 | The limit of a convergent ... |
| climge0 15541 | A nonnegative sequence con... |
| climabs0 15542 | Convergence to zero of the... |
| o1co 15543 | Sufficient condition for t... |
| o1compt 15544 | Sufficient condition for t... |
| rlimcn1 15545 | Image of a limit under a c... |
| rlimcn1b 15546 | Image of a limit under a c... |
| rlimcn3 15547 | Image of a limit under a c... |
| rlimcn2 15548 | Image of a limit under a c... |
| climcn1 15549 | Image of a limit under a c... |
| climcn2 15550 | Image of a limit under a c... |
| addcn2 15551 | Complex number addition is... |
| subcn2 15552 | Complex number subtraction... |
| mulcn2 15553 | Complex number multiplicat... |
| reccn2 15554 | The reciprocal function is... |
| cn1lem 15555 | A sufficient condition for... |
| abscn2 15556 | The absolute value functio... |
| cjcn2 15557 | The complex conjugate func... |
| recn2 15558 | The real part function is ... |
| imcn2 15559 | The imaginary part functio... |
| climcn1lem 15560 | The limit of a continuous ... |
| climabs 15561 | Limit of the absolute valu... |
| climcj 15562 | Limit of the complex conju... |
| climre 15563 | Limit of the real part of ... |
| climim 15564 | Limit of the imaginary par... |
| rlimmptrcl 15565 | Reverse closure for a real... |
| rlimabs 15566 | Limit of the absolute valu... |
| rlimcj 15567 | Limit of the complex conju... |
| rlimre 15568 | Limit of the real part of ... |
| rlimim 15569 | Limit of the imaginary par... |
| o1of2 15570 | Show that a binary operati... |
| o1add 15571 | The sum of two eventually ... |
| o1mul 15572 | The product of two eventua... |
| o1sub 15573 | The difference of two even... |
| rlimo1 15574 | Any function with a finite... |
| rlimdmo1 15575 | A convergent function is e... |
| o1rlimmul 15576 | The product of an eventual... |
| o1const 15577 | A constant function is eve... |
| lo1const 15578 | A constant function is eve... |
| lo1mptrcl 15579 | Reverse closure for an eve... |
| o1mptrcl 15580 | Reverse closure for an eve... |
| o1add2 15581 | The sum of two eventually ... |
| o1mul2 15582 | The product of two eventua... |
| o1sub2 15583 | The product of two eventua... |
| lo1add 15584 | The sum of two eventually ... |
| lo1mul 15585 | The product of an eventual... |
| lo1mul2 15586 | The product of an eventual... |
| o1dif 15587 | If the difference of two f... |
| lo1sub 15588 | The difference of an event... |
| climadd 15589 | Limit of the sum of two co... |
| climmul 15590 | Limit of the product of tw... |
| climsub 15591 | Limit of the difference of... |
| climaddc1 15592 | Limit of a constant ` C ` ... |
| climaddc2 15593 | Limit of a constant ` C ` ... |
| climmulc2 15594 | Limit of a sequence multip... |
| climsubc1 15595 | Limit of a constant ` C ` ... |
| climsubc2 15596 | Limit of a constant ` C ` ... |
| climle 15597 | Comparison of the limits o... |
| climsqz 15598 | Convergence of a sequence ... |
| climsqz2 15599 | Convergence of a sequence ... |
| rlimadd 15600 | Limit of the sum of two co... |
| rlimsub 15601 | Limit of the difference of... |
| rlimmul 15602 | Limit of the product of tw... |
| rlimdiv 15603 | Limit of the quotient of t... |
| rlimneg 15604 | Limit of the negative of a... |
| rlimle 15605 | Comparison of the limits o... |
| rlimsqzlem 15606 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15607 | Convergence of a sequence ... |
| rlimsqz2 15608 | Convergence of a sequence ... |
| lo1le 15609 | Transfer eventual upper bo... |
| o1le 15610 | Transfer eventual boundedn... |
| rlimno1 15611 | A function whose inverse c... |
| clim2ser 15612 | The limit of an infinite s... |
| clim2ser2 15613 | The limit of an infinite s... |
| iserex 15614 | An infinite series converg... |
| isermulc2 15615 | Multiplication of an infin... |
| climlec2 15616 | Comparison of a constant t... |
| iserle 15617 | Comparison of the limits o... |
| iserge0 15618 | The limit of an infinite s... |
| climub 15619 | The limit of a monotonic s... |
| climserle 15620 | The partial sums of a conv... |
| isershft 15621 | Index shift of the limit o... |
| isercolllem1 15622 | Lemma for ~ isercoll . (C... |
| isercolllem2 15623 | Lemma for ~ isercoll . (C... |
| isercolllem3 15624 | Lemma for ~ isercoll . (C... |
| isercoll 15625 | Rearrange an infinite seri... |
| isercoll2 15626 | Generalize ~ isercoll so t... |
| climsup 15627 | A bounded monotonic sequen... |
| climcau 15628 | A converging sequence of c... |
| climbdd 15629 | A converging sequence of c... |
| caucvgrlem 15630 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15631 | A Cauchy sequence of real ... |
| caucvgrlem2 15632 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15633 | A Cauchy sequence of compl... |
| caurcvg 15634 | A Cauchy sequence of real ... |
| caurcvg2 15635 | A Cauchy sequence of real ... |
| caucvg 15636 | A Cauchy sequence of compl... |
| caucvgb 15637 | A function is convergent i... |
| serf0 15638 | If an infinite series conv... |
| iseraltlem1 15639 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15640 | Lemma for ~ iseralt . The... |
| iseraltlem3 15641 | Lemma for ~ iseralt . Fro... |
| iseralt 15642 | The alternating series tes... |
| sumex 15645 | A sum is a set. (Contribu... |
| sumeq1 15646 | Equality theorem for a sum... |
| nfsum1 15647 | Bound-variable hypothesis ... |
| nfsum 15648 | Bound-variable hypothesis ... |
| sumeq2w 15649 | Equality theorem for sum, ... |
| sumeq2ii 15650 | Equality theorem for sum, ... |
| sumeq2 15651 | Equality theorem for sum. ... |
| cbvsum 15652 | Change bound variable in a... |
| cbvsumv 15653 | Change bound variable in a... |
| sumeq1i 15654 | Equality inference for sum... |
| sumeq2i 15655 | Equality inference for sum... |
| sumeq12i 15656 | Equality inference for sum... |
| sumeq1d 15657 | Equality deduction for sum... |
| sumeq2d 15658 | Equality deduction for sum... |
| sumeq2dv 15659 | Equality deduction for sum... |
| sumeq2sdv 15660 | Equality deduction for sum... |
| sumeq2sdvOLD 15661 | Obsolete version of ~ sume... |
| 2sumeq2dv 15662 | Equality deduction for dou... |
| sumeq12dv 15663 | Equality deduction for sum... |
| sumeq12rdv 15664 | Equality deduction for sum... |
| sum2id 15665 | The second class argument ... |
| sumfc 15666 | A lemma to facilitate conv... |
| fz1f1o 15667 | A lemma for working with f... |
| sumrblem 15668 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15669 | The sequence of partial su... |
| sumrb 15670 | Rebase the starting point ... |
| summolem3 15671 | Lemma for ~ summo . (Cont... |
| summolem2a 15672 | Lemma for ~ summo . (Cont... |
| summolem2 15673 | Lemma for ~ summo . (Cont... |
| summo 15674 | A sum has at most one limi... |
| zsum 15675 | Series sum with index set ... |
| isum 15676 | Series sum with an upper i... |
| fsum 15677 | The value of a sum over a ... |
| sum0 15678 | Any sum over the empty set... |
| sumz 15679 | Any sum of zero over a sum... |
| fsumf1o 15680 | Re-index a finite sum usin... |
| sumss 15681 | Change the index set to a ... |
| fsumss 15682 | Change the index set to a ... |
| sumss2 15683 | Change the index set of a ... |
| fsumcvg2 15684 | The sequence of partial su... |
| fsumsers 15685 | Special case of series sum... |
| fsumcvg3 15686 | A finite sum is convergent... |
| fsumser 15687 | A finite sum expressed in ... |
| fsumcl2lem 15688 | - Lemma for finite sum clo... |
| fsumcllem 15689 | - Lemma for finite sum clo... |
| fsumcl 15690 | Closure of a finite sum of... |
| fsumrecl 15691 | Closure of a finite sum of... |
| fsumzcl 15692 | Closure of a finite sum of... |
| fsumnn0cl 15693 | Closure of a finite sum of... |
| fsumrpcl 15694 | Closure of a finite sum of... |
| fsumclf 15695 | Closure of a finite sum of... |
| fsumzcl2 15696 | A finite sum with integer ... |
| fsumadd 15697 | The sum of two finite sums... |
| fsumsplit 15698 | Split a sum into two parts... |
| fsumsplitf 15699 | Split a sum into two parts... |
| sumsnf 15700 | A sum of a singleton is th... |
| fsumsplitsn 15701 | Separate out a term in a f... |
| fsumsplit1 15702 | Separate out a term in a f... |
| sumsn 15703 | A sum of a singleton is th... |
| fsum1 15704 | The finite sum of ` A ( k ... |
| sumpr 15705 | A sum over a pair is the s... |
| sumtp 15706 | A sum over a triple is the... |
| sumsns 15707 | A sum of a singleton is th... |
| fsumm1 15708 | Separate out the last term... |
| fzosump1 15709 | Separate out the last term... |
| fsum1p 15710 | Separate out the first ter... |
| fsummsnunz 15711 | A finite sum all of whose ... |
| fsumsplitsnun 15712 | Separate out a term in a f... |
| fsump1 15713 | The addition of the next t... |
| isumclim 15714 | An infinite sum equals the... |
| isumclim2 15715 | A converging series conver... |
| isumclim3 15716 | The sequence of partial fi... |
| sumnul 15717 | The sum of a non-convergen... |
| isumcl 15718 | The sum of a converging in... |
| isummulc2 15719 | An infinite sum multiplied... |
| isummulc1 15720 | An infinite sum multiplied... |
| isumdivc 15721 | An infinite sum divided by... |
| isumrecl 15722 | The sum of a converging in... |
| isumge0 15723 | An infinite sum of nonnega... |
| isumadd 15724 | Addition of infinite sums.... |
| sumsplit 15725 | Split a sum into two parts... |
| fsump1i 15726 | Optimized version of ~ fsu... |
| fsum2dlem 15727 | Lemma for ~ fsum2d - induc... |
| fsum2d 15728 | Write a double sum as a su... |
| fsumxp 15729 | Combine two sums into a si... |
| fsumcnv 15730 | Transform a region of summ... |
| fsumcom2 15731 | Interchange order of summa... |
| fsumcom 15732 | Interchange order of summa... |
| fsum0diaglem 15733 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15734 | Two ways to express "the s... |
| mptfzshft 15735 | 1-1 onto function in maps-... |
| fsumrev 15736 | Reversal of a finite sum. ... |
| fsumshft 15737 | Index shift of a finite su... |
| fsumshftm 15738 | Negative index shift of a ... |
| fsumrev2 15739 | Reversal of a finite sum. ... |
| fsum0diag2 15740 | Two ways to express "the s... |
| fsummulc2 15741 | A finite sum multiplied by... |
| fsummulc1 15742 | A finite sum multiplied by... |
| fsumdivc 15743 | A finite sum divided by a ... |
| fsumneg 15744 | Negation of a finite sum. ... |
| fsumsub 15745 | Split a finite sum over a ... |
| fsum2mul 15746 | Separate the nested sum of... |
| fsumconst 15747 | The sum of constant terms ... |
| fsumconst1 15748 | The sum of 1 over a finite... |
| fsumdifsnconst 15749 | The sum of constant terms ... |
| modfsummodslem1 15750 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15751 | Induction step for ~ modfs... |
| modfsummod 15752 | A finite sum modulo a posi... |
| fsumge0 15753 | If all of the terms of a f... |
| fsumless 15754 | A shorter sum of nonnegati... |
| fsumge1 15755 | A sum of nonnegative numbe... |
| fsum00 15756 | A sum of nonnegative numbe... |
| fsumle 15757 | If all of the terms of fin... |
| fsumlt 15758 | If every term in one finit... |
| fsumabs 15759 | Generalized triangle inequ... |
| telfsumo 15760 | Sum of a telescoping serie... |
| telfsumo2 15761 | Sum of a telescoping serie... |
| telfsum 15762 | Sum of a telescoping serie... |
| telfsum2 15763 | Sum of a telescoping serie... |
| fsumparts 15764 | Summation by parts. (Cont... |
| fsumrelem 15765 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15766 | The real part of a sum. (... |
| fsumim 15767 | The imaginary part of a su... |
| fsumcj 15768 | The complex conjugate of a... |
| fsumrlim 15769 | Limit of a finite sum of c... |
| fsumo1 15770 | The finite sum of eventual... |
| o1fsum 15771 | If ` A ( k ) ` is O(1), th... |
| seqabs 15772 | Generalized triangle inequ... |
| iserabs 15773 | Generalized triangle inequ... |
| cvgcmp 15774 | A comparison test for conv... |
| cvgcmpub 15775 | An upper bound for the lim... |
| cvgcmpce 15776 | A comparison test for conv... |
| abscvgcvg 15777 | An absolutely convergent s... |
| climfsum 15778 | Limit of a finite sum of c... |
| fsumiun 15779 | Sum over a disjoint indexe... |
| hashiun 15780 | The cardinality of a disjo... |
| hash2iun 15781 | The cardinality of a neste... |
| hash2iun1dif1 15782 | The cardinality of a neste... |
| hashrabrex 15783 | The number of elements in ... |
| hashuni 15784 | The cardinality of a disjo... |
| qshash 15785 | The cardinality of a set w... |
| indsum 15786 | Finite sum of a product wi... |
| indsumhash 15787 | The finite sum of the indi... |
| ackbijnn 15788 | Translate the Ackermann bi... |
| binomlem 15789 | Lemma for ~ binom (binomia... |
| binom 15790 | The binomial theorem: ` ( ... |
| binom1p 15791 | Special case of the binomi... |
| binom11 15792 | Special case of the binomi... |
| binom1dif 15793 | A summation for the differ... |
| bcxmaslem1 15794 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15795 | Parallel summation (Christ... |
| incexclem 15796 | Lemma for ~ incexc . (Con... |
| incexc 15797 | The inclusion/exclusion pr... |
| incexc2 15798 | The inclusion/exclusion pr... |
| isumshft 15799 | Index shift of an infinite... |
| isumsplit 15800 | Split off the first ` N ` ... |
| isum1p 15801 | The infinite sum of a conv... |
| isumnn0nn 15802 | Sum from 0 to infinity in ... |
| isumrpcl 15803 | The infinite sum of positi... |
| isumle 15804 | Comparison of two infinite... |
| isumless 15805 | A finite sum of nonnegativ... |
| isumsup2 15806 | An infinite sum of nonnega... |
| isumsup 15807 | An infinite sum of nonnega... |
| isumltss 15808 | A partial sum of a series ... |
| climcndslem1 15809 | Lemma for ~ climcnds : bou... |
| climcndslem2 15810 | Lemma for ~ climcnds : bou... |
| climcnds 15811 | The Cauchy condensation te... |
| divrcnv 15812 | The sequence of reciprocal... |
| divcnv 15813 | The sequence of reciprocal... |
| flo1 15814 | The floor function satisfi... |
| divcnvshft 15815 | Limit of a ratio function.... |
| supcvg 15816 | Extract a sequence ` f ` i... |
| infcvgaux1i 15817 | Auxiliary theorem for appl... |
| infcvgaux2i 15818 | Auxiliary theorem for appl... |
| harmonic 15819 | The harmonic series ` H ` ... |
| arisum 15820 | Arithmetic series sum of t... |
| arisum2 15821 | Arithmetic series sum of t... |
| trireciplem 15822 | Lemma for ~ trirecip . Sh... |
| trirecip 15823 | The sum of the reciprocals... |
| expcnv 15824 | A sequence of powers of a ... |
| explecnv 15825 | A sequence of terms conver... |
| geoserg 15826 | The value of the finite ge... |
| geoser 15827 | The value of the finite ge... |
| pwdif 15828 | The difference of two numb... |
| pwm1geoser 15829 | The n-th power of a number... |
| geolim 15830 | The partial sums in the in... |
| geolim2 15831 | The partial sums in the ge... |
| georeclim 15832 | The limit of a geometric s... |
| geo2sum 15833 | The value of the finite ge... |
| geo2sum2 15834 | The value of the finite ge... |
| geo2lim 15835 | The value of the infinite ... |
| geomulcvg 15836 | The geometric series conve... |
| geoisum 15837 | The infinite sum of ` 1 + ... |
| geoisumr 15838 | The infinite sum of recipr... |
| geoisum1 15839 | The infinite sum of ` A ^ ... |
| geoisum1c 15840 | The infinite sum of ` A x.... |
| 0.999... 15841 | The recurring decimal 0.99... |
| geoihalfsum 15842 | Prove that the infinite ge... |
| cvgrat 15843 | Ratio test for convergence... |
| mertenslem1 15844 | Lemma for ~ mertens . (Co... |
| mertenslem2 15845 | Lemma for ~ mertens . (Co... |
| mertens 15846 | Mertens' theorem. If ` A ... |
| prodf 15847 | An infinite product of com... |
| clim2prod 15848 | The limit of an infinite p... |
| clim2div 15849 | The limit of an infinite p... |
| prodfmul 15850 | The product of two infinit... |
| prodf1 15851 | The value of the partial p... |
| prodf1f 15852 | A one-valued infinite prod... |
| prodfclim1 15853 | The constant one product c... |
| prodfn0 15854 | No term of a nonzero infin... |
| prodfrec 15855 | The reciprocal of an infin... |
| prodfdiv 15856 | The quotient of two infini... |
| ntrivcvg 15857 | A non-trivially converging... |
| ntrivcvgn0 15858 | A product that converges t... |
| ntrivcvgfvn0 15859 | Any value of a product seq... |
| ntrivcvgtail 15860 | A tail of a non-trivially ... |
| ntrivcvgmullem 15861 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15862 | The product of two non-tri... |
| prodex 15865 | A product is a set. (Cont... |
| prodeq1f 15866 | Equality theorem for a pro... |
| prodeq1 15867 | Equality theorem for a pro... |
| nfcprod1 15868 | Bound-variable hypothesis ... |
| nfcprod 15869 | Bound-variable hypothesis ... |
| prodeq2w 15870 | Equality theorem for produ... |
| prodeq2ii 15871 | Equality theorem for produ... |
| prodeq2 15872 | Equality theorem for produ... |
| cbvprod 15873 | Change bound variable in a... |
| cbvprodv 15874 | Change bound variable in a... |
| cbvprodi 15875 | Change bound variable in a... |
| prodeq1i 15876 | Equality inference for pro... |
| prodeq1iOLD 15877 | Obsolete version of ~ prod... |
| prodeq2i 15878 | Equality inference for pro... |
| prodeq12i 15879 | Equality inference for pro... |
| prodeq1d 15880 | Equality deduction for pro... |
| prodeq2d 15881 | Equality deduction for pro... |
| prodeq2dv 15882 | Equality deduction for pro... |
| prodeq2sdv 15883 | Equality deduction for pro... |
| prodeq2sdvOLD 15884 | Obsolete version of ~ prod... |
| 2cprodeq2dv 15885 | Equality deduction for dou... |
| prodeq12dv 15886 | Equality deduction for pro... |
| prodeq12rdv 15887 | Equality deduction for pro... |
| prod2id 15888 | The second class argument ... |
| prodrblem 15889 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15890 | The sequence of partial pr... |
| prodrblem2 15891 | Lemma for ~ prodrb . (Con... |
| prodrb 15892 | Rebase the starting point ... |
| prodmolem3 15893 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15894 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15895 | Lemma for ~ prodmo . (Con... |
| prodmo 15896 | A product has at most one ... |
| zprod 15897 | Series product with index ... |
| iprod 15898 | Series product with an upp... |
| zprodn0 15899 | Nonzero series product wit... |
| iprodn0 15900 | Nonzero series product wit... |
| fprod 15901 | The value of a product ove... |
| fprodntriv 15902 | A non-triviality lemma for... |
| prod0 15903 | A product over the empty s... |
| prod1 15904 | Any product of one over a ... |
| prodfc 15905 | A lemma to facilitate conv... |
| fprodf1o 15906 | Re-index a finite product ... |
| prodss 15907 | Change the index set to a ... |
| fprodss 15908 | Change the index set to a ... |
| fprodser 15909 | A finite product expressed... |
| fprodcl2lem 15910 | Finite product closure lem... |
| fprodcllem 15911 | Finite product closure lem... |
| fprodcl 15912 | Closure of a finite produc... |
| fprodrecl 15913 | Closure of a finite produc... |
| fprodzcl 15914 | Closure of a finite produc... |
| fprodnncl 15915 | Closure of a finite produc... |
| fprodrpcl 15916 | Closure of a finite produc... |
| fprodnn0cl 15917 | Closure of a finite produc... |
| fprodcllemf 15918 | Finite product closure lem... |
| fprodreclf 15919 | Closure of a finite produc... |
| fprodmul 15920 | The product of two finite ... |
| fproddiv 15921 | The quotient of two finite... |
| prodsn 15922 | A product of a singleton i... |
| fprod1 15923 | A finite product of only o... |
| prodsnf 15924 | A product of a singleton i... |
| climprod1 15925 | The limit of a product ove... |
| fprodsplit 15926 | Split a finite product int... |
| fprodm1 15927 | Separate out the last term... |
| fprod1p 15928 | Separate out the first ter... |
| fprodp1 15929 | Multiply in the last term ... |
| fprodm1s 15930 | Separate out the last term... |
| fprodp1s 15931 | Multiply in the last term ... |
| prodsns 15932 | A product of the singleton... |
| fprodfac 15933 | Factorial using product no... |
| fprodabs 15934 | The absolute value of a fi... |
| fprodeq0 15935 | Any finite product contain... |
| fprodshft 15936 | Shift the index of a finit... |
| fprodrev 15937 | Reversal of a finite produ... |
| fprodconst 15938 | The product of constant te... |
| fprodn0 15939 | A finite product of nonzer... |
| fprod2dlem 15940 | Lemma for ~ fprod2d - indu... |
| fprod2d 15941 | Write a double product as ... |
| fprodxp 15942 | Combine two products into ... |
| fprodcnv 15943 | Transform a product region... |
| fprodcom2 15944 | Interchange order of multi... |
| fprodcom 15945 | Interchange product order.... |
| fprod0diag 15946 | Two ways to express "the p... |
| fproddivf 15947 | The quotient of two finite... |
| fprodsplitf 15948 | Split a finite product int... |
| fprodsplitsn 15949 | Separate out a term in a f... |
| fprodsplit1f 15950 | Separate out a term in a f... |
| fprodn0f 15951 | A finite product of nonzer... |
| fprodclf 15952 | Closure of a finite produc... |
| fprodge0 15953 | If all the terms of a fini... |
| fprodeq0g 15954 | Any finite product contain... |
| fprodge1 15955 | If all of the terms of a f... |
| fprodle 15956 | If all the terms of two fi... |
| fprodmodd 15957 | If all factors of two fini... |
| iprodclim 15958 | An infinite product equals... |
| iprodclim2 15959 | A converging product conve... |
| iprodclim3 15960 | The sequence of partial fi... |
| iprodcl 15961 | The product of a non-trivi... |
| iprodrecl 15962 | The product of a non-trivi... |
| iprodmul 15963 | Multiplication of infinite... |
| risefacval 15968 | The value of the rising fa... |
| fallfacval 15969 | The value of the falling f... |
| risefacval2 15970 | One-based value of rising ... |
| fallfacval2 15971 | One-based value of falling... |
| fallfacval3 15972 | A product representation o... |
| risefaccllem 15973 | Lemma for rising factorial... |
| fallfaccllem 15974 | Lemma for falling factoria... |
| risefaccl 15975 | Closure law for rising fac... |
| fallfaccl 15976 | Closure law for falling fa... |
| rerisefaccl 15977 | Closure law for rising fac... |
| refallfaccl 15978 | Closure law for falling fa... |
| nnrisefaccl 15979 | Closure law for rising fac... |
| zrisefaccl 15980 | Closure law for rising fac... |
| zfallfaccl 15981 | Closure law for falling fa... |
| nn0risefaccl 15982 | Closure law for rising fac... |
| rprisefaccl 15983 | Closure law for rising fac... |
| risefallfac 15984 | A relationship between ris... |
| fallrisefac 15985 | A relationship between fal... |
| risefall0lem 15986 | Lemma for ~ risefac0 and ~... |
| risefac0 15987 | The value of the rising fa... |
| fallfac0 15988 | The value of the falling f... |
| risefacp1 15989 | The value of the rising fa... |
| fallfacp1 15990 | The value of the falling f... |
| risefacp1d 15991 | The value of the rising fa... |
| fallfacp1d 15992 | The value of the falling f... |
| risefac1 15993 | The value of rising factor... |
| fallfac1 15994 | The value of falling facto... |
| risefacfac 15995 | Relate rising factorial to... |
| fallfacfwd 15996 | The forward difference of ... |
| 0fallfac 15997 | The value of the zero fall... |
| 0risefac 15998 | The value of the zero risi... |
| binomfallfaclem1 15999 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16000 | Lemma for ~ binomfallfac .... |
| binomfallfac 16001 | A version of the binomial ... |
| binomrisefac 16002 | A version of the binomial ... |
| fallfacval4 16003 | Represent the falling fact... |
| bcfallfac 16004 | Binomial coefficient in te... |
| fallfacfac 16005 | Relate falling factorial t... |
| bpolylem 16008 | Lemma for ~ bpolyval . (C... |
| bpolyval 16009 | The value of the Bernoulli... |
| bpoly0 16010 | The value of the Bernoulli... |
| bpoly1 16011 | The value of the Bernoulli... |
| bpolycl 16012 | Closure law for Bernoulli ... |
| bpolysum 16013 | A sum for Bernoulli polyno... |
| bpolydiflem 16014 | Lemma for ~ bpolydif . (C... |
| bpolydif 16015 | Calculate the difference b... |
| fsumkthpow 16016 | A closed-form expression f... |
| bpoly2 16017 | The Bernoulli polynomials ... |
| bpoly3 16018 | The Bernoulli polynomials ... |
| bpoly4 16019 | The Bernoulli polynomials ... |
| fsumcube 16020 | Express the sum of cubes i... |
| eftcl 16033 | Closure of a term in the s... |
| reeftcl 16034 | The terms of the series ex... |
| eftabs 16035 | The absolute value of a te... |
| eftval 16036 | The value of a term in the... |
| efcllem 16037 | Lemma for ~ efcl . The se... |
| ef0lem 16038 | The series defining the ex... |
| efval 16039 | Value of the exponential f... |
| esum 16040 | Value of Euler's constant ... |
| eff 16041 | Domain and codomain of the... |
| efcl 16042 | Closure law for the expone... |
| efcld 16043 | Closure law for the expone... |
| efval2 16044 | Value of the exponential f... |
| efcvg 16045 | The series that defines th... |
| efcvgfsum 16046 | Exponential function conve... |
| reefcl 16047 | The exponential function i... |
| reefcld 16048 | The exponential function i... |
| ere 16049 | Euler's constant ` _e ` = ... |
| ege2le3 16050 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16051 | Value of the exponential f... |
| efcj 16052 | The exponential of a compl... |
| efaddlem 16053 | Lemma for ~ efadd (exponen... |
| efadd 16054 | Sum of exponents law for e... |
| fprodefsum 16055 | Move the exponential funct... |
| efcan 16056 | Cancellation law for expon... |
| efne0d 16057 | The exponential of a compl... |
| efne0 16058 | The exponential of a compl... |
| efne0OLD 16059 | Obsolete version of ~ efne... |
| efneg 16060 | The exponential of the opp... |
| eff2 16061 | The exponential function m... |
| efsub 16062 | Difference of exponents la... |
| efexp 16063 | The exponential of an inte... |
| efzval 16064 | Value of the exponential f... |
| efgt0 16065 | The exponential of a real ... |
| rpefcl 16066 | The exponential of a real ... |
| rpefcld 16067 | The exponential of a real ... |
| eftlcvg 16068 | The tail series of the exp... |
| eftlcl 16069 | Closure of the sum of an i... |
| reeftlcl 16070 | Closure of the sum of an i... |
| eftlub 16071 | An upper bound on the abso... |
| efsep 16072 | Separate out the next term... |
| effsumlt 16073 | The partial sums of the se... |
| eft0val 16074 | The value of the first ter... |
| ef4p 16075 | Separate out the first fou... |
| efgt1p2 16076 | The exponential of a posit... |
| efgt1p 16077 | The exponential of a posit... |
| efgt1 16078 | The exponential of a posit... |
| eflt 16079 | The exponential function o... |
| efle 16080 | The exponential function o... |
| reef11 16081 | The exponential function o... |
| reeff1 16082 | The exponential function m... |
| eflegeo 16083 | The exponential function o... |
| sinval 16084 | Value of the sine function... |
| cosval 16085 | Value of the cosine functi... |
| sinf 16086 | Domain and codomain of the... |
| cosf 16087 | Domain and codomain of the... |
| sincl 16088 | Closure of the sine functi... |
| coscl 16089 | Closure of the cosine func... |
| tanval 16090 | Value of the tangent funct... |
| tancl 16091 | The closure of the tangent... |
| sincld 16092 | Closure of the sine functi... |
| coscld 16093 | Closure of the cosine func... |
| tancld 16094 | Closure of the tangent fun... |
| tanval2 16095 | Express the tangent functi... |
| tanval3 16096 | Express the tangent functi... |
| resinval 16097 | The sine of a real number ... |
| recosval 16098 | The cosine of a real numbe... |
| efi4p 16099 | Separate out the first fou... |
| resin4p 16100 | Separate out the first fou... |
| recos4p 16101 | Separate out the first fou... |
| resincl 16102 | The sine of a real number ... |
| recoscl 16103 | The cosine of a real numbe... |
| retancl 16104 | The closure of the tangent... |
| resincld 16105 | Closure of the sine functi... |
| recoscld 16106 | Closure of the cosine func... |
| retancld 16107 | Closure of the tangent fun... |
| sinneg 16108 | The sine of a negative is ... |
| cosneg 16109 | The cosines of a number an... |
| tanneg 16110 | The tangent of a negative ... |
| sin0 16111 | Value of the sine function... |
| cos0 16112 | Value of the cosine functi... |
| tan0 16113 | The value of the tangent f... |
| efival 16114 | The exponential function i... |
| efmival 16115 | The exponential function i... |
| sinhval 16116 | Value of the hyperbolic si... |
| coshval 16117 | Value of the hyperbolic co... |
| resinhcl 16118 | The hyperbolic sine of a r... |
| rpcoshcl 16119 | The hyperbolic cosine of a... |
| recoshcl 16120 | The hyperbolic cosine of a... |
| retanhcl 16121 | The hyperbolic tangent of ... |
| tanhlt1 16122 | The hyperbolic tangent of ... |
| tanhbnd 16123 | The hyperbolic tangent of ... |
| efeul 16124 | Eulerian representation of... |
| efieq 16125 | The exponentials of two im... |
| sinadd 16126 | Addition formula for sine.... |
| cosadd 16127 | Addition formula for cosin... |
| tanaddlem 16128 | A useful intermediate step... |
| tanadd 16129 | Addition formula for tange... |
| sinsub 16130 | Sine of difference. (Cont... |
| cossub 16131 | Cosine of difference. (Co... |
| addsin 16132 | Sum of sines. (Contribute... |
| subsin 16133 | Difference of sines. (Con... |
| sinmul 16134 | Product of sines can be re... |
| cosmul 16135 | Product of cosines can be ... |
| addcos 16136 | Sum of cosines. (Contribu... |
| subcos 16137 | Difference of cosines. (C... |
| sincossq 16138 | Sine squared plus cosine s... |
| sin2t 16139 | Double-angle formula for s... |
| cos2t 16140 | Double-angle formula for c... |
| cos2tsin 16141 | Double-angle formula for c... |
| sinbnd 16142 | The sine of a real number ... |
| cosbnd 16143 | The cosine of a real numbe... |
| sinbnd2 16144 | The sine of a real number ... |
| cosbnd2 16145 | The cosine of a real numbe... |
| ef01bndlem 16146 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16147 | Bounds on the sine of a po... |
| cos01bnd 16148 | Bounds on the cosine of a ... |
| cos1bnd 16149 | Bounds on the cosine of 1.... |
| cos2bnd 16150 | Bounds on the cosine of 2.... |
| sinltx 16151 | The sine of a positive rea... |
| sin01gt0 16152 | The sine of a positive rea... |
| cos01gt0 16153 | The cosine of a positive r... |
| sin02gt0 16154 | The sine of a positive rea... |
| sincos1sgn 16155 | The signs of the sine and ... |
| sincos2sgn 16156 | The signs of the sine and ... |
| sin4lt0 16157 | The sine of 4 is negative.... |
| absefi 16158 | The absolute value of the ... |
| absef 16159 | The absolute value of the ... |
| absefib 16160 | A complex number is real i... |
| efieq1re 16161 | A number whose imaginary e... |
| demoivre 16162 | De Moivre's Formula. Proo... |
| demoivreALT 16163 | Alternate proof of ~ demoi... |
| eirrlem 16166 | Lemma for ~ eirr . (Contr... |
| eirr 16167 | ` _e ` is irrational. (Co... |
| egt2lt3 16168 | Euler's constant ` _e ` = ... |
| epos 16169 | Euler's constant ` _e ` is... |
| epr 16170 | Euler's constant ` _e ` is... |
| ene0 16171 | ` _e ` is not 0. (Contrib... |
| ene1 16172 | ` _e ` is not 1. (Contrib... |
| xpnnen 16173 | The Cartesian product of t... |
| znnen 16174 | The set of integers and th... |
| qnnen 16175 | The rational numbers are c... |
| rpnnen2lem1 16176 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16177 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16178 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16179 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16180 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16181 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16182 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16183 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16184 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16185 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16186 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16187 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16188 | The other half of ~ rpnnen... |
| rpnnen 16189 | The cardinality of the con... |
| rexpen 16190 | The real numbers are equin... |
| cpnnen 16191 | The complex numbers are eq... |
| rucALT 16192 | Alternate proof of ~ ruc .... |
| ruclem1 16193 | Lemma for ~ ruc (the reals... |
| ruclem2 16194 | Lemma for ~ ruc . Orderin... |
| ruclem3 16195 | Lemma for ~ ruc . The con... |
| ruclem4 16196 | Lemma for ~ ruc . Initial... |
| ruclem6 16197 | Lemma for ~ ruc . Domain ... |
| ruclem7 16198 | Lemma for ~ ruc . Success... |
| ruclem8 16199 | Lemma for ~ ruc . The int... |
| ruclem9 16200 | Lemma for ~ ruc . The fir... |
| ruclem10 16201 | Lemma for ~ ruc . Every f... |
| ruclem11 16202 | Lemma for ~ ruc . Closure... |
| ruclem12 16203 | Lemma for ~ ruc . The sup... |
| ruclem13 16204 | Lemma for ~ ruc . There i... |
| ruc 16205 | The set of positive intege... |
| resdomq 16206 | The set of rationals is st... |
| aleph1re 16207 | There are at least aleph-o... |
| aleph1irr 16208 | There are at least aleph-o... |
| cnso 16209 | The complex numbers can be... |
| sqrt2irrlem 16210 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16211 | The square root of 2 is ir... |
| sqrt2re 16212 | The square root of 2 exist... |
| sqrt2irr0 16213 | The square root of 2 is an... |
| nthruc 16214 | The sequence ` NN ` , ` ZZ... |
| nthruz 16215 | The sequence ` NN ` , ` NN... |
| divides 16218 | Define the divides relatio... |
| dvdsval2 16219 | One nonzero integer divide... |
| dvdsval3 16220 | One nonzero integer divide... |
| dvdszrcl 16221 | Reverse closure for the di... |
| dvdsmod0 16222 | If a positive integer divi... |
| p1modz1 16223 | If a number greater than 1... |
| dvdsmodexp 16224 | If a positive integer divi... |
| nndivdvds 16225 | Strong form of ~ dvdsval2 ... |
| nndivides 16226 | Definition of the divides ... |
| moddvds 16227 | Two ways to say ` A == B `... |
| modm1div 16228 | An integer greater than on... |
| addmulmodb 16229 | An integer plus a product ... |
| dvds0lem 16230 | A lemma to assist theorems... |
| dvds1lem 16231 | A lemma to assist theorems... |
| dvds2lem 16232 | A lemma to assist theorems... |
| iddvds 16233 | An integer divides itself.... |
| 1dvds 16234 | 1 divides any integer. Th... |
| dvds0 16235 | Any integer divides 0. Th... |
| negdvdsb 16236 | An integer divides another... |
| dvdsnegb 16237 | An integer divides another... |
| absdvdsb 16238 | An integer divides another... |
| dvdsabsb 16239 | An integer divides another... |
| 0dvds 16240 | Only 0 is divisible by 0. ... |
| dvdsmul1 16241 | An integer divides a multi... |
| dvdsmul2 16242 | An integer divides a multi... |
| iddvdsexp 16243 | An integer divides a posit... |
| muldvds1 16244 | If a product divides an in... |
| muldvds2 16245 | If a product divides an in... |
| dvdscmul 16246 | Multiplication by a consta... |
| dvdsmulc 16247 | Multiplication by a consta... |
| dvdscmulr 16248 | Cancellation law for the d... |
| dvdsmulcr 16249 | Cancellation law for the d... |
| summodnegmod 16250 | The sum of two integers mo... |
| difmod0 16251 | The difference of two inte... |
| modmulconst 16252 | Constant multiplication in... |
| dvds2ln 16253 | If an integer divides each... |
| dvds2add 16254 | If an integer divides each... |
| dvds2sub 16255 | If an integer divides each... |
| dvds2addd 16256 | Deduction form of ~ dvds2a... |
| dvds2subd 16257 | Deduction form of ~ dvds2s... |
| dvdstr 16258 | The divides relation is tr... |
| dvdstrd 16259 | The divides relation is tr... |
| dvdsmultr1 16260 | If an integer divides anot... |
| dvdsmultr1d 16261 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16262 | If an integer divides anot... |
| dvdsmultr2d 16263 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16264 | If an integer divides eith... |
| dvdssub2 16265 | If an integer divides a di... |
| dvdsadd 16266 | An integer divides another... |
| dvdsaddr 16267 | An integer divides another... |
| dvdssub 16268 | An integer divides another... |
| dvdssubr 16269 | An integer divides another... |
| dvdsadd2b 16270 | Adding a multiple of the b... |
| dvdsaddre2b 16271 | Adding a multiple of the b... |
| fsumdvds 16272 | If every term in a sum is ... |
| dvdslelem 16273 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16274 | The divisors of a positive... |
| dvdsleabs 16275 | The divisors of a nonzero ... |
| dvdsleabs2 16276 | Transfer divisibility to a... |
| dvdsabseq 16277 | If two integers divide eac... |
| dvdseq 16278 | If two nonnegative integer... |
| divconjdvds 16279 | If a nonzero integer ` M `... |
| dvdsdivcl 16280 | The complement of a diviso... |
| dvdsflip 16281 | An involution of the divis... |
| dvdsssfz1 16282 | The set of divisors of a n... |
| dvds1 16283 | The only nonnegative integ... |
| alzdvds 16284 | Only 0 is divisible by all... |
| dvdsext 16285 | Poset extensionality for d... |
| fzm1ndvds 16286 | No number between ` 1 ` an... |
| fzo0dvdseq 16287 | Zero is the only one of th... |
| fzocongeq 16288 | Two different elements of ... |
| addmodlteqALT 16289 | Two nonnegative integers l... |
| dvdsfac 16290 | A positive integer divides... |
| dvdsexp2im 16291 | If an integer divides anot... |
| dvdsexp 16292 | A power divides a power wi... |
| dvdsmod 16293 | Any number ` K ` whose mod... |
| mulmoddvds 16294 | If an integer is divisible... |
| 3dvds 16295 | A rule for divisibility by... |
| 3dvdsdec 16296 | A decimal number is divisi... |
| 3dvds2dec 16297 | A decimal number is divisi... |
| fprodfvdvdsd 16298 | A finite product of intege... |
| fproddvdsd 16299 | A finite product of intege... |
| evenelz 16300 | An even number is an integ... |
| zeo3 16301 | An integer is even or odd.... |
| zeo4 16302 | An integer is even or odd ... |
| zeneo 16303 | No even integer equals an ... |
| odd2np1lem 16304 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16305 | An integer is odd iff it i... |
| even2n 16306 | An integer is even iff it ... |
| oddm1even 16307 | An integer is odd iff its ... |
| oddp1even 16308 | An integer is odd iff its ... |
| oexpneg 16309 | The exponential of the neg... |
| mod2eq0even 16310 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16311 | An integer is 1 modulo 2 i... |
| oddnn02np1 16312 | A nonnegative integer is o... |
| oddge22np1 16313 | An integer greater than on... |
| evennn02n 16314 | A nonnegative integer is e... |
| evennn2n 16315 | A positive integer is even... |
| 2tp1odd 16316 | A number which is twice an... |
| mulsucdiv2z 16317 | An integer multiplied with... |
| sqoddm1div8z 16318 | A squared odd number minus... |
| 2teven 16319 | A number which is twice an... |
| zeo5 16320 | An integer is either even ... |
| evend2 16321 | An integer is even iff its... |
| oddp1d2 16322 | An integer is odd iff its ... |
| zob 16323 | Alternate characterization... |
| oddm1d2 16324 | An integer is odd iff its ... |
| ltoddhalfle 16325 | An integer is less than ha... |
| halfleoddlt 16326 | An integer is greater than... |
| opoe 16327 | The sum of two odds is eve... |
| omoe 16328 | The difference of two odds... |
| opeo 16329 | The sum of an odd and an e... |
| omeo 16330 | The difference of an odd a... |
| z0even 16331 | 2 divides 0. That means 0... |
| n2dvds1 16332 | 2 does not divide 1. That... |
| n2dvdsm1 16333 | 2 does not divide -1. Tha... |
| z2even 16334 | 2 divides 2. That means 2... |
| n2dvds3 16335 | 2 does not divide 3. That... |
| z4even 16336 | 2 divides 4. That means 4... |
| 4dvdseven 16337 | An integer which is divisi... |
| m1expe 16338 | Exponentiation of -1 by an... |
| m1expo 16339 | Exponentiation of -1 by an... |
| m1exp1 16340 | Exponentiation of negative... |
| nn0enne 16341 | A positive integer is an e... |
| nn0ehalf 16342 | The half of an even nonneg... |
| nnehalf 16343 | The half of an even positi... |
| nn0onn 16344 | An odd nonnegative integer... |
| nn0o1gt2 16345 | An odd nonnegative integer... |
| nno 16346 | An alternate characterizat... |
| nn0o 16347 | An alternate characterizat... |
| nn0ob 16348 | Alternate characterization... |
| nn0oddm1d2 16349 | A positive integer is odd ... |
| nnoddm1d2 16350 | A positive integer is odd ... |
| sumeven 16351 | If every term in a sum is ... |
| sumodd 16352 | If every term in a sum is ... |
| evensumodd 16353 | If every term in a sum wit... |
| oddsumodd 16354 | If every term in a sum wit... |
| pwp1fsum 16355 | The n-th power of a number... |
| oddpwp1fsum 16356 | An odd power of a number i... |
| divalglem0 16357 | Lemma for ~ divalg . (Con... |
| divalglem1 16358 | Lemma for ~ divalg . (Con... |
| divalglem2 16359 | Lemma for ~ divalg . (Con... |
| divalglem4 16360 | Lemma for ~ divalg . (Con... |
| divalglem5 16361 | Lemma for ~ divalg . (Con... |
| divalglem6 16362 | Lemma for ~ divalg . (Con... |
| divalglem7 16363 | Lemma for ~ divalg . (Con... |
| divalglem8 16364 | Lemma for ~ divalg . (Con... |
| divalglem9 16365 | Lemma for ~ divalg . (Con... |
| divalglem10 16366 | Lemma for ~ divalg . (Con... |
| divalg 16367 | The division algorithm (th... |
| divalgb 16368 | Express the division algor... |
| divalg2 16369 | The division algorithm (th... |
| divalgmod 16370 | The result of the ` mod ` ... |
| divalgmodcl 16371 | The result of the ` mod ` ... |
| modremain 16372 | The result of the modulo o... |
| ndvdssub 16373 | Corollary of the division ... |
| ndvdsadd 16374 | Corollary of the division ... |
| ndvdsp1 16375 | Special case of ~ ndvdsadd... |
| ndvdsi 16376 | A quick test for non-divis... |
| 5ndvds3 16377 | 5 does not divide 3. (Con... |
| 5ndvds6 16378 | 5 does not divide 6. (Con... |
| flodddiv4 16379 | The floor of an odd intege... |
| fldivndvdslt 16380 | The floor of an integer di... |
| flodddiv4lt 16381 | The floor of an odd number... |
| flodddiv4t2lthalf 16382 | The floor of an odd number... |
| bitsfval 16387 | Expand the definition of t... |
| bitsval 16388 | Expand the definition of t... |
| bitsval2 16389 | Expand the definition of t... |
| bitsss 16390 | The set of bits of an inte... |
| bitsf 16391 | The ` bits ` function is a... |
| bits0 16392 | Value of the zeroth bit. ... |
| bits0e 16393 | The zeroth bit of an even ... |
| bits0o 16394 | The zeroth bit of an odd n... |
| bitsp1 16395 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16396 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16397 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16398 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16399 | The bits of a number are a... |
| bitsmod 16400 | Truncating the bit sequenc... |
| bitsfi 16401 | Every number is associated... |
| bitscmp 16402 | The bit complement of ` N ... |
| 0bits 16403 | The bits of zero. (Contri... |
| m1bits 16404 | The bits of negative one. ... |
| bitsinv1lem 16405 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16406 | There is an explicit inver... |
| bitsinv2 16407 | There is an explicit inver... |
| bitsf1ocnv 16408 | The ` bits ` function rest... |
| bitsf1o 16409 | The ` bits ` function rest... |
| bitsf1 16410 | The ` bits ` function is a... |
| 2ebits 16411 | The bits of a power of two... |
| bitsinv 16412 | The inverse of the ` bits ... |
| bitsinvp1 16413 | Recursive definition of th... |
| sadadd2lem2 16414 | The core of the proof of ~... |
| sadfval 16416 | Define the addition of two... |
| sadcf 16417 | The carry sequence is a se... |
| sadc0 16418 | The initial element of the... |
| sadcp1 16419 | The carry sequence (which ... |
| sadval 16420 | The full adder sequence is... |
| sadcaddlem 16421 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16422 | Non-recursive definition o... |
| sadadd2lem 16423 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16424 | Sum of initial segments of... |
| sadadd3 16425 | Sum of initial segments of... |
| sadcl 16426 | The sum of two sequences i... |
| sadcom 16427 | The adder sequence functio... |
| saddisjlem 16428 | Lemma for ~ sadadd . (Con... |
| saddisj 16429 | The sum of disjoint sequen... |
| sadaddlem 16430 | Lemma for ~ sadadd . (Con... |
| sadadd 16431 | For sequences that corresp... |
| sadid1 16432 | The adder sequence functio... |
| sadid2 16433 | The adder sequence functio... |
| sadasslem 16434 | Lemma for ~ sadass . (Con... |
| sadass 16435 | Sequence addition is assoc... |
| sadeq 16436 | Any element of a sequence ... |
| bitsres 16437 | Restrict the bits of a num... |
| bitsuz 16438 | The bits of a number are a... |
| bitsshft 16439 | Shifting a bit sequence to... |
| smufval 16441 | The multiplication of two ... |
| smupf 16442 | The sequence of partial su... |
| smup0 16443 | The initial element of the... |
| smupp1 16444 | The initial element of the... |
| smuval 16445 | Define the addition of two... |
| smuval2 16446 | The partial sum sequence s... |
| smupvallem 16447 | If ` A ` only has elements... |
| smucl 16448 | The product of two sequenc... |
| smu01lem 16449 | Lemma for ~ smu01 and ~ sm... |
| smu01 16450 | Multiplication of a sequen... |
| smu02 16451 | Multiplication of a sequen... |
| smupval 16452 | Rewrite the elements of th... |
| smup1 16453 | Rewrite ~ smupp1 using onl... |
| smueqlem 16454 | Any element of a sequence ... |
| smueq 16455 | Any element of a sequence ... |
| smumullem 16456 | Lemma for ~ smumul . (Con... |
| smumul 16457 | For sequences that corresp... |
| gcdval 16460 | The value of the ` gcd ` o... |
| gcd0val 16461 | The value, by convention, ... |
| gcdn0val 16462 | The value of the ` gcd ` o... |
| gcdcllem1 16463 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16464 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16465 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16466 | Closure of the ` gcd ` ope... |
| gcddvds 16467 | The gcd of two integers di... |
| dvdslegcd 16468 | An integer which divides b... |
| nndvdslegcd 16469 | A positive integer which d... |
| gcdcl 16470 | Closure of the ` gcd ` ope... |
| gcdnncl 16471 | Closure of the ` gcd ` ope... |
| gcdcld 16472 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16473 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16474 | An integer is the product ... |
| divgcdz 16475 | An integer divided by the ... |
| gcdf 16476 | Domain and codomain of the... |
| gcdcom 16477 | The ` gcd ` operator is co... |
| gcdcomd 16478 | The ` gcd ` operator is co... |
| divgcdnn 16479 | A positive integer divided... |
| divgcdnnr 16480 | A positive integer divided... |
| gcdeq0 16481 | The gcd of two integers is... |
| gcdn0gt0 16482 | The gcd of two integers is... |
| gcd0id 16483 | The gcd of 0 and an intege... |
| gcdid0 16484 | The gcd of an integer and ... |
| nn0gcdid0 16485 | The gcd of a nonnegative i... |
| gcdneg 16486 | Negating one operand of th... |
| neggcd 16487 | Negating one operand of th... |
| gcdaddmlem 16488 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16489 | Adding a multiple of one o... |
| gcdadd 16490 | The GCD of two numbers is ... |
| gcdid 16491 | The gcd of a number and it... |
| gcd1 16492 | The gcd of a number with 1... |
| gcdabs1 16493 | ` gcd ` of the absolute va... |
| gcdabs2 16494 | ` gcd ` of the absolute va... |
| gcdabs 16495 | The gcd of two integers is... |
| modgcd 16496 | The gcd remains unchanged ... |
| 1gcd 16497 | The GCD of one and an inte... |
| gcdmultipled 16498 | The greatest common diviso... |
| gcdmultiplez 16499 | The GCD of a multiple of a... |
| gcdmultiple 16500 | The GCD of a multiple of a... |
| dvdsgcdidd 16501 | The greatest common diviso... |
| 6gcd4e2 16502 | The greatest common diviso... |
| bezoutlem1 16503 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16504 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16505 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16506 | Lemma for ~ bezout . (Con... |
| bezout 16507 | Bézout's identity: ... |
| dvdsgcd 16508 | An integer which divides e... |
| dvdsgcdb 16509 | Biconditional form of ~ dv... |
| dfgcd2 16510 | Alternate definition of th... |
| gcdass 16511 | Associative law for ` gcd ... |
| mulgcd 16512 | Distribute multiplication ... |
| absmulgcd 16513 | Distribute absolute value ... |
| mulgcdr 16514 | Reverse distribution law f... |
| gcddiv 16515 | Division law for GCD. (Con... |
| gcdzeq 16516 | A positive integer ` A ` i... |
| gcdeq 16517 | ` A ` is equal to its gcd ... |
| dvdssqim 16518 | Unidirectional form of ~ d... |
| dvdsexpim 16519 | If two numbers are divisib... |
| dvdsmulgcd 16520 | A divisibility equivalent ... |
| rpmulgcd 16521 | If ` K ` and ` M ` are rel... |
| rplpwr 16522 | If ` A ` and ` B ` are rel... |
| rprpwr 16523 | If ` A ` and ` B ` are rel... |
| rppwr 16524 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16525 | If ` A ` and ` B ` are rel... |
| sqgcd 16526 | Square distributes over gc... |
| expgcd 16527 | Exponentiation distributes... |
| nn0expgcd 16528 | Exponentiation distributes... |
| zexpgcd 16529 | Exponentiation distributes... |
| dvdssqlem 16530 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16531 | Two numbers are divisible ... |
| bezoutr 16532 | Partial converse to ~ bezo... |
| bezoutr1 16533 | Converse of ~ bezout for w... |
| nn0seqcvgd 16534 | A strictly-decreasing nonn... |
| seq1st 16535 | A sequence whose iteration... |
| algr0 16536 | The value of the algorithm... |
| algrf 16537 | An algorithm is a step fun... |
| algrp1 16538 | The value of the algorithm... |
| alginv 16539 | If ` I ` is an invariant o... |
| algcvg 16540 | One way to prove that an a... |
| algcvgblem 16541 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16542 | Two ways of expressing tha... |
| algcvga 16543 | The countdown function ` C... |
| algfx 16544 | If ` F ` reaches a fixed p... |
| eucalgval2 16545 | The value of the step func... |
| eucalgval 16546 | Euclid's Algorithm ~ eucal... |
| eucalgf 16547 | Domain and codomain of the... |
| eucalginv 16548 | The invariant of the step ... |
| eucalglt 16549 | The second member of the s... |
| eucalgcvga 16550 | Once Euclid's Algorithm ha... |
| eucalg 16551 | Euclid's Algorithm compute... |
| lcmval 16556 | Value of the ` lcm ` opera... |
| lcmcom 16557 | The ` lcm ` operator is co... |
| lcm0val 16558 | The value, by convention, ... |
| lcmn0val 16559 | The value of the ` lcm ` o... |
| lcmcllem 16560 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16561 | Closure of the ` lcm ` ope... |
| dvdslcm 16562 | The lcm of two integers is... |
| lcmledvds 16563 | A positive integer which b... |
| lcmeq0 16564 | The lcm of two integers is... |
| lcmcl 16565 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16566 | The greatest common diviso... |
| lcmneg 16567 | Negating one operand of th... |
| neglcm 16568 | Negating one operand of th... |
| lcmabs 16569 | The lcm of two integers is... |
| lcmgcdlem 16570 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16571 | The product of two numbers... |
| lcmdvds 16572 | The lcm of two integers di... |
| lcmid 16573 | The lcm of an integer and ... |
| lcm1 16574 | The lcm of an integer and ... |
| lcmgcdnn 16575 | The product of two positiv... |
| lcmgcdeq 16576 | Two integers' absolute val... |
| lcmdvdsb 16577 | Biconditional form of ~ lc... |
| lcmass 16578 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16579 | The least common multiple ... |
| 6lcm4e12 16580 | The least common multiple ... |
| absproddvds 16581 | The absolute value of the ... |
| absprodnn 16582 | The absolute value of the ... |
| fissn0dvds 16583 | For each finite subset of ... |
| fissn0dvdsn0 16584 | For each finite subset of ... |
| lcmfval 16585 | Value of the ` _lcm ` func... |
| lcmf0val 16586 | The value, by convention, ... |
| lcmfn0val 16587 | The value of the ` _lcm ` ... |
| lcmfnnval 16588 | The value of the ` _lcm ` ... |
| lcmfcllem 16589 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16590 | Closure of the ` _lcm ` fu... |
| lcmfpr 16591 | The value of the ` _lcm ` ... |
| lcmfcl 16592 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16593 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16594 | The least common multiple ... |
| dvdslcmf 16595 | The least common multiple ... |
| lcmfledvds 16596 | A positive integer which i... |
| lcmf 16597 | Characterization of the le... |
| lcmf0 16598 | The least common multiple ... |
| lcmfsn 16599 | The least common multiple ... |
| lcmftp 16600 | The least common multiple ... |
| lcmfunsnlem1 16601 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16602 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16603 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16604 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16605 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16606 | The least common multiple ... |
| lcmfdvdsb 16607 | Biconditional form of ~ lc... |
| lcmfunsn 16608 | The ` _lcm ` function for ... |
| lcmfun 16609 | The ` _lcm ` function for ... |
| lcmfass 16610 | Associative law for the ` ... |
| lcmf2a3a4e12 16611 | The least common multiple ... |
| lcmflefac 16612 | The least common multiple ... |
| coprmgcdb 16613 | Two positive integers are ... |
| ncoprmgcdne1b 16614 | Two positive integers are ... |
| ncoprmgcdgt1b 16615 | Two positive integers are ... |
| coprmdvds1 16616 | If two positive integers a... |
| coprmdvds 16617 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16618 | If an integer is divisible... |
| mulgcddvds 16619 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16620 | If ` M ` is relatively pri... |
| qredeq 16621 | Two equal reduced fraction... |
| qredeu 16622 | Every rational number has ... |
| rpmul 16623 | If ` K ` is relatively pri... |
| rpdvds 16624 | If ` K ` is relatively pri... |
| coprmprod 16625 | The product of the element... |
| coprmproddvdslem 16626 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16627 | If a positive integer is d... |
| congr 16628 | Definition of congruence b... |
| divgcdcoprm0 16629 | Integers divided by gcd ar... |
| divgcdcoprmex 16630 | Integers divided by gcd ar... |
| cncongr1 16631 | One direction of the bicon... |
| cncongr2 16632 | The other direction of the... |
| cncongr 16633 | Cancellability of Congruen... |
| cncongrcoprm 16634 | Corollary 1 of Cancellabil... |
| isprm 16637 | The predicate "is a prime ... |
| prmnn 16638 | A prime number is a positi... |
| prmz 16639 | A prime number is an integ... |
| prmssnn 16640 | The prime numbers are a su... |
| prmex 16641 | The set of prime numbers e... |
| 0nprm 16642 | 0 is not a prime number. ... |
| 1nprm 16643 | 1 is not a prime number. ... |
| 1idssfct 16644 | The positive divisors of a... |
| isprm2lem 16645 | Lemma for ~ isprm2 . (Con... |
| isprm2 16646 | The predicate "is a prime ... |
| isprm3 16647 | The predicate "is a prime ... |
| isprm4 16648 | The predicate "is a prime ... |
| prmind2 16649 | A variation on ~ prmind as... |
| prmind 16650 | Perform induction over the... |
| dvdsprime 16651 | If ` M ` divides a prime, ... |
| nprm 16652 | A product of two integers ... |
| nprmi 16653 | An inference for composite... |
| dvdsnprmd 16654 | If a number is divisible b... |
| prm2orodd 16655 | A prime number is either 2... |
| 2prm 16656 | 2 is a prime number. (Con... |
| 2mulprm 16657 | A multiple of two is prime... |
| 3prm 16658 | 3 is a prime number. (Con... |
| 4nprm 16659 | 4 is not a prime number. ... |
| prmuz2 16660 | A prime number is an integ... |
| prmssuz2 16661 | The primes are integers gr... |
| prmgt1 16662 | A prime number is an integ... |
| prmm2nn0 16663 | Subtracting 2 from a prime... |
| oddprmgt2 16664 | An odd prime is greater th... |
| oddprmge3 16665 | An odd prime is greater th... |
| ge2nprmge4 16666 | A composite integer greate... |
| sqnprm 16667 | A square is never prime. ... |
| dvdsprm 16668 | An integer greater than or... |
| exprmfct 16669 | Every integer greater than... |
| prmdvdsfz 16670 | Each integer greater than ... |
| nprmdvds1 16671 | No prime number divides 1.... |
| isprm5 16672 | One need only check prime ... |
| isprm7 16673 | One need only check prime ... |
| maxprmfct 16674 | The set of prime factors o... |
| divgcdodd 16675 | Either ` A / ( A gcd B ) `... |
| coprm 16676 | A prime number either divi... |
| prmrp 16677 | Unequal prime numbers are ... |
| euclemma 16678 | Euclid's lemma. A prime n... |
| isprm6 16679 | A number is prime iff it s... |
| prmdvdsexp 16680 | A prime divides a positive... |
| prmdvdsexpb 16681 | A prime divides a positive... |
| prmdvdsexpr 16682 | If a prime divides a nonne... |
| prmdvdssq 16683 | Condition for a prime divi... |
| prmexpb 16684 | Two positive prime powers ... |
| prmfac1 16685 | The factorial of a number ... |
| dvdszzq 16686 | Divisibility for an intege... |
| rpexp 16687 | If two numbers ` A ` and `... |
| rpexp1i 16688 | Relative primality passes ... |
| rpexp12i 16689 | Relative primality passes ... |
| prmndvdsfaclt 16690 | A prime number does not di... |
| prmdvdsbc 16691 | Condition for a prime numb... |
| prmdvdsncoprmbd 16692 | Two positive integers are ... |
| ncoprmlnprm 16693 | If two positive integers a... |
| cncongrprm 16694 | Corollary 2 of Cancellabil... |
| isevengcd2 16695 | The predicate "is an even ... |
| isoddgcd1 16696 | The predicate "is an odd n... |
| 3lcm2e6 16697 | The least common multiple ... |
| qnumval 16702 | Value of the canonical num... |
| qdenval 16703 | Value of the canonical den... |
| qnumdencl 16704 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16705 | The canonical numerator of... |
| qdencl 16706 | The canonical denominator ... |
| fnum 16707 | Canonical numerator define... |
| fden 16708 | Canonical denominator defi... |
| qnumdenbi 16709 | Two numbers are the canoni... |
| qnumdencoprm 16710 | The canonical representati... |
| qeqnumdivden 16711 | Recover a rational number ... |
| qmuldeneqnum 16712 | Multiplying a rational by ... |
| divnumden 16713 | Calculate the reduced form... |
| divdenle 16714 | Reducing a quotient never ... |
| qnumgt0 16715 | A rational is positive iff... |
| qgt0numnn 16716 | A rational is positive iff... |
| nn0gcdsq 16717 | Squaring commutes with GCD... |
| zgcdsq 16718 | ~ nn0gcdsq extended to int... |
| numdensq 16719 | Squaring a rational square... |
| numsq 16720 | Square commutes with canon... |
| densq 16721 | Square commutes with canon... |
| qden1elz 16722 | A rational is an integer i... |
| zsqrtelqelz 16723 | If an integer has a ration... |
| nonsq 16724 | Any integer strictly betwe... |
| numdenexp 16725 | Elevating a rational numbe... |
| numexp 16726 | Elevating to a nonnegative... |
| denexp 16727 | Elevating to a nonnegative... |
| phival 16732 | Value of the Euler ` phi `... |
| phicl2 16733 | Bounds and closure for the... |
| phicl 16734 | Closure for the value of t... |
| phibndlem 16735 | Lemma for ~ phibnd . (Con... |
| phibnd 16736 | A slightly tighter bound o... |
| phicld 16737 | Closure for the value of t... |
| phi1 16738 | Value of the Euler ` phi `... |
| dfphi2 16739 | Alternate definition of th... |
| hashdvds 16740 | The number of numbers in a... |
| phiprmpw 16741 | Value of the Euler ` phi `... |
| phiprm 16742 | Value of the Euler ` phi `... |
| crth 16743 | The Chinese Remainder Theo... |
| phimullem 16744 | Lemma for ~ phimul . (Con... |
| phimul 16745 | The Euler ` phi ` function... |
| eulerthlem1 16746 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16747 | Lemma for ~ eulerth . (Co... |
| eulerth 16748 | Euler's theorem, a general... |
| fermltl 16749 | Fermat's little theorem. ... |
| prmdiv 16750 | Show an explicit expressio... |
| prmdiveq 16751 | The modular inverse of ` A... |
| prmdivdiv 16752 | The (modular) inverse of t... |
| hashgcdlem 16753 | A correspondence between e... |
| dvdsfi 16754 | A natural number has finit... |
| hashgcdeq 16755 | Number of initial positive... |
| phisum 16756 | The divisor sum identity o... |
| odzval 16757 | Value of the order functio... |
| odzcllem 16758 | - Lemma for ~ odzcl , show... |
| odzcl 16759 | The order of a group eleme... |
| odzid 16760 | Any element raised to the ... |
| odzdvds 16761 | The only powers of ` A ` t... |
| odzphi 16762 | The order of any group ele... |
| modprm1div 16763 | A prime number divides an ... |
| m1dvdsndvds 16764 | If an integer minus 1 is d... |
| modprminv 16765 | Show an explicit expressio... |
| modprminveq 16766 | The modular inverse of ` A... |
| vfermltl 16767 | Variant of Fermat's little... |
| vfermltlALT 16768 | Alternate proof of ~ vferm... |
| powm2modprm 16769 | If an integer minus 1 is d... |
| reumodprminv 16770 | For any prime number and f... |
| modprm0 16771 | For two positive integers ... |
| nnnn0modprm0 16772 | For a positive integer and... |
| modprmn0modprm0 16773 | For an integer not being 0... |
| coprimeprodsq 16774 | If three numbers are copri... |
| coprimeprodsq2 16775 | If three numbers are copri... |
| oddprm 16776 | A prime not equal to ` 2 `... |
| nnoddn2prm 16777 | A prime not equal to ` 2 `... |
| oddn2prm 16778 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16779 | A number is a prime number... |
| prm23lt5 16780 | A prime less than 5 is eit... |
| prm23ge5 16781 | A prime is either 2 or 3 o... |
| pythagtriplem1 16782 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16783 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16784 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16785 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16786 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16787 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16788 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16789 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16790 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16791 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16792 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16793 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16794 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16795 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16796 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16797 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16798 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16799 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16800 | Parameterize the Pythagore... |
| iserodd 16801 | Collect the odd terms in a... |
| pclem 16804 | - Lemma for the prime powe... |
| pcprecl 16805 | Closure of the prime power... |
| pcprendvds 16806 | Non-divisibility property ... |
| pcprendvds2 16807 | Non-divisibility property ... |
| pcpre1 16808 | Value of the prime power p... |
| pcpremul 16809 | Multiplicative property of... |
| pcval 16810 | The value of the prime pow... |
| pceulem 16811 | Lemma for ~ pceu . (Contr... |
| pceu 16812 | Uniqueness for the prime p... |
| pczpre 16813 | Connect the prime count pr... |
| pczcl 16814 | Closure of the prime power... |
| pccl 16815 | Closure of the prime power... |
| pccld 16816 | Closure of the prime power... |
| pcmul 16817 | Multiplication property of... |
| pcdiv 16818 | Division property of the p... |
| pcqmul 16819 | Multiplication property of... |
| pc0 16820 | The value of the prime pow... |
| pc1 16821 | Value of the prime count f... |
| pcqcl 16822 | Closure of the general pri... |
| pcqdiv 16823 | Division property of the p... |
| pcrec 16824 | Prime power of a reciproca... |
| pcexp 16825 | Prime power of an exponent... |
| pcxnn0cl 16826 | Extended nonnegative integ... |
| pcxcl 16827 | Extended real closure of t... |
| pcge0 16828 | The prime count of an inte... |
| pczdvds 16829 | Defining property of the p... |
| pcdvds 16830 | Defining property of the p... |
| pczndvds 16831 | Defining property of the p... |
| pcndvds 16832 | Defining property of the p... |
| pczndvds2 16833 | The remainder after dividi... |
| pcndvds2 16834 | The remainder after dividi... |
| pcdvdsb 16835 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16836 | There are a positive numbe... |
| pceq0 16837 | There are zero powers of a... |
| pcidlem 16838 | The prime count of a prime... |
| pcid 16839 | The prime count of a prime... |
| pcneg 16840 | The prime count of a negat... |
| pcabs 16841 | The prime count of an abso... |
| pcdvdstr 16842 | The prime count increases ... |
| pcgcd1 16843 | The prime count of a GCD i... |
| pcgcd 16844 | The prime count of a GCD i... |
| pc2dvds 16845 | A characterization of divi... |
| pc11 16846 | The prime count function, ... |
| pcz 16847 | The prime count function c... |
| pcprmpw2 16848 | Self-referential expressio... |
| pcprmpw 16849 | Self-referential expressio... |
| dvdsprmpweq 16850 | If a positive integer divi... |
| dvdsprmpweqnn 16851 | If an integer greater than... |
| dvdsprmpweqle 16852 | If a positive integer divi... |
| difsqpwdvds 16853 | If the difference of two s... |
| pcaddlem 16854 | Lemma for ~ pcadd . The o... |
| pcadd 16855 | An inequality for the prim... |
| pcadd2 16856 | The inequality of ~ pcadd ... |
| pcmptcl 16857 | Closure for the prime powe... |
| pcmpt 16858 | Construct a function with ... |
| pcmpt2 16859 | Dividing two prime count m... |
| pcmptdvds 16860 | The partial products of th... |
| pcprod 16861 | The product of the primes ... |
| sumhash 16862 | The sum of 1 over a set is... |
| fldivp1 16863 | The difference between the... |
| pcfaclem 16864 | Lemma for ~ pcfac . (Cont... |
| pcfac 16865 | Calculate the prime count ... |
| pcbc 16866 | Calculate the prime count ... |
| qexpz 16867 | If a power of a rational n... |
| expnprm 16868 | A second or higher power o... |
| oddprmdvds 16869 | Every positive integer whi... |
| prmpwdvds 16870 | A relation involving divis... |
| pockthlem 16871 | Lemma for ~ pockthg . (Co... |
| pockthg 16872 | The generalized Pocklingto... |
| pockthi 16873 | Pocklington's theorem, whi... |
| unbenlem 16874 | Lemma for ~ unben . (Cont... |
| unben 16875 | An unbounded set of positi... |
| infpnlem1 16876 | Lemma for ~ infpn . The s... |
| infpnlem2 16877 | Lemma for ~ infpn . For a... |
| infpn 16878 | There exist infinitely man... |
| infpn2 16879 | There exist infinitely man... |
| prmunb 16880 | The primes are unbounded. ... |
| prminf 16881 | There are an infinite numb... |
| prmreclem1 16882 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16883 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16884 | Lemma for ~ prmrec . The ... |
| prmreclem4 16885 | Lemma for ~ prmrec . Show... |
| prmreclem5 16886 | Lemma for ~ prmrec . Here... |
| prmreclem6 16887 | Lemma for ~ prmrec . If t... |
| prmrec 16888 | The sum of the reciprocals... |
| 1arithlem1 16889 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16890 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16891 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16892 | Lemma for ~ 1arith . (Con... |
| 1arith 16893 | Fundamental theorem of ari... |
| 1arith2 16894 | Fundamental theorem of ari... |
| elgz 16897 | Elementhood in the gaussia... |
| gzcn 16898 | A gaussian integer is a co... |
| zgz 16899 | An integer is a gaussian i... |
| igz 16900 | ` _i ` is a gaussian integ... |
| gznegcl 16901 | The gaussian integers are ... |
| gzcjcl 16902 | The gaussian integers are ... |
| gzaddcl 16903 | The gaussian integers are ... |
| gzmulcl 16904 | The gaussian integers are ... |
| gzreim 16905 | Construct a gaussian integ... |
| gzsubcl 16906 | The gaussian integers are ... |
| gzabssqcl 16907 | The squared norm of a gaus... |
| 4sqlem5 16908 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16909 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16910 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16911 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16912 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16913 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16914 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16915 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16916 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16917 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16918 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16919 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16920 | Euler's four-square identi... |
| 4sqlem11 16921 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16922 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16923 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16924 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16925 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16926 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16927 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16928 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16929 | Lemma for ~ 4sq . The pro... |
| 4sq 16930 | Lagrange's four-square the... |
| vdwapfval 16937 | Define the arithmetic prog... |
| vdwapf 16938 | The arithmetic progression... |
| vdwapval 16939 | Value of the arithmetic pr... |
| vdwapun 16940 | Remove the first element o... |
| vdwapid1 16941 | The first element of an ar... |
| vdwap0 16942 | Value of a length-1 arithm... |
| vdwap1 16943 | Value of a length-1 arithm... |
| vdwmc 16944 | The predicate " The ` <. R... |
| vdwmc2 16945 | Expand out the definition ... |
| vdwpc 16946 | The predicate " The colori... |
| vdwlem1 16947 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16948 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16949 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16950 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16951 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16952 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16953 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16954 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16955 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16956 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16957 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16958 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16959 | Lemma for ~ vdw . Main in... |
| vdw 16960 | Van der Waerden's theorem.... |
| vdwnnlem1 16961 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16962 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16963 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16964 | Van der Waerden's theorem,... |
| ramtlecl 16966 | The set ` T ` of numbers w... |
| hashbcval 16968 | Value of the "binomial set... |
| hashbccl 16969 | The binomial set is a fini... |
| hashbcss 16970 | Subset relation for the bi... |
| hashbc0 16971 | The set of subsets of size... |
| hashbc2 16972 | The size of the binomial s... |
| 0hashbc 16973 | There are no subsets of th... |
| ramval 16974 | The value of the Ramsey nu... |
| ramcl2lem 16975 | Lemma for extended real cl... |
| ramtcl 16976 | The Ramsey number has the ... |
| ramtcl2 16977 | The Ramsey number is an in... |
| ramtub 16978 | The Ramsey number is a low... |
| ramub 16979 | The Ramsey number is a low... |
| ramub2 16980 | It is sufficient to check ... |
| rami 16981 | The defining property of a... |
| ramcl2 16982 | The Ramsey number is eithe... |
| ramxrcl 16983 | The Ramsey number is an ex... |
| ramubcl 16984 | If the Ramsey number is up... |
| ramlb 16985 | Establish a lower bound on... |
| 0ram 16986 | The Ramsey number when ` M... |
| 0ram2 16987 | The Ramsey number when ` M... |
| ram0 16988 | The Ramsey number when ` R... |
| 0ramcl 16989 | Lemma for ~ ramcl : Exist... |
| ramz2 16990 | The Ramsey number when ` F... |
| ramz 16991 | The Ramsey number when ` F... |
| ramub1lem1 16992 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16993 | Lemma for ~ ramub1 . (Con... |
| ramub1 16994 | Inductive step for Ramsey'... |
| ramcl 16995 | Ramsey's theorem: the Rams... |
| ramsey 16996 | Ramsey's theorem with the ... |
| prmoval 16999 | Value of the primorial fun... |
| prmocl 17000 | Closure of the primorial f... |
| prmone0 17001 | The primorial function is ... |
| prmo0 17002 | The primorial of 0. (Cont... |
| prmo1 17003 | The primorial of 1. (Cont... |
| prmop1 17004 | The primorial of a success... |
| prmonn2 17005 | Value of the primorial fun... |
| prmo2 17006 | The primorial of 2. (Cont... |
| prmo3 17007 | The primorial of 3. (Cont... |
| prmdvdsprmo 17008 | The primorial of a number ... |
| prmdvdsprmop 17009 | The primorial of a number ... |
| fvprmselelfz 17010 | The value of the prime sel... |
| fvprmselgcd1 17011 | The greatest common diviso... |
| prmolefac 17012 | The primorial of a positiv... |
| prmodvdslcmf 17013 | The primorial of a nonnega... |
| prmolelcmf 17014 | The primorial of a positiv... |
| prmgaplem1 17015 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17016 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17017 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17018 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17019 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17020 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17021 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17022 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17023 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17024 | Lemma for ~ prmgap . (Con... |
| prmgap 17025 | The prime gap theorem: for... |
| prmgaplcm 17026 | Alternate proof of ~ prmga... |
| prmgapprmolem 17027 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17028 | Alternate proof of ~ prmga... |
| dec2dvds 17029 | Divisibility by two is obv... |
| dec5dvds 17030 | Divisibility by five is ob... |
| dec5dvds2 17031 | Divisibility by five is ob... |
| dec5nprm 17032 | A decimal number greater t... |
| dec2nprm 17033 | A decimal number greater t... |
| modxai 17034 | Add exponents in a power m... |
| mod2xi 17035 | Double exponents in a powe... |
| modxp1i 17036 | Add one to an exponent in ... |
| mod2xnegi 17037 | Version of ~ mod2xi with a... |
| modsubi 17038 | Subtract from within a mod... |
| gcdi 17039 | Calculate a GCD via Euclid... |
| gcdmodi 17040 | Calculate a GCD via Euclid... |
| numexp0 17041 | Calculate an integer power... |
| numexp1 17042 | Calculate an integer power... |
| numexpp1 17043 | Calculate an integer power... |
| numexp2x 17044 | Double an integer power. ... |
| decsplit0b 17045 | Split a decimal number int... |
| decsplit0 17046 | Split a decimal number int... |
| decsplit1 17047 | Split a decimal number int... |
| decsplit 17048 | Split a decimal number int... |
| karatsuba 17049 | The Karatsuba multiplicati... |
| 2exp4 17050 | Two to the fourth power is... |
| 2exp5 17051 | Two to the fifth power is ... |
| 2exp6 17052 | Two to the sixth power is ... |
| 2exp7 17053 | Two to the seventh power i... |
| 2exp8 17054 | Two to the eighth power is... |
| 2exp11 17055 | Two to the eleventh power ... |
| 2exp16 17056 | Two to the sixteenth power... |
| 3exp3 17057 | Three to the third power i... |
| 2expltfac 17058 | The factorial grows faster... |
| cshwsidrepsw 17059 | If cyclically shifting a w... |
| cshwsidrepswmod0 17060 | If cyclically shifting a w... |
| cshwshashlem1 17061 | If cyclically shifting a w... |
| cshwshashlem2 17062 | If cyclically shifting a w... |
| cshwshashlem3 17063 | If cyclically shifting a w... |
| cshwsdisj 17064 | The singletons resulting b... |
| cshwsiun 17065 | The set of (different!) wo... |
| cshwsex 17066 | The class of (different!) ... |
| cshws0 17067 | The size of the set of (di... |
| cshwrepswhash1 17068 | The size of the set of (di... |
| cshwshashnsame 17069 | If a word (not consisting ... |
| cshwshash 17070 | If a word has a length bei... |
| prmlem0 17071 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17072 | A quick proof skeleton to ... |
| prmlem1 17073 | A quick proof skeleton to ... |
| 5prm 17074 | 5 is a prime number. (Con... |
| 6nprm 17075 | 6 is not a prime number. ... |
| 7prm 17076 | 7 is a prime number. (Con... |
| 8nprm 17077 | 8 is not a prime number. ... |
| 9nprm 17078 | 9 is not a prime number. ... |
| 10nprm 17079 | 10 is not a prime number. ... |
| 11prm 17080 | 11 is a prime number. (Co... |
| 13prm 17081 | 13 is a prime number. (Co... |
| 17prm 17082 | 17 is a prime number. (Co... |
| 19prm 17083 | 19 is a prime number. (Co... |
| 23prm 17084 | 23 is a prime number. (Co... |
| prmlem2 17085 | Our last proving session g... |
| 37prm 17086 | 37 is a prime number. (Co... |
| 43prm 17087 | 43 is a prime number. (Co... |
| 83prm 17088 | 83 is a prime number. (Co... |
| 139prm 17089 | 139 is a prime number. (C... |
| 163prm 17090 | 163 is a prime number. (C... |
| 317prm 17091 | 317 is a prime number. (C... |
| 631prm 17092 | 631 is a prime number. (C... |
| prmo4 17093 | The primorial of 4. (Cont... |
| prmo5 17094 | The primorial of 5. (Cont... |
| prmo6 17095 | The primorial of 6. (Cont... |
| 1259lem1 17096 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17097 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17098 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17099 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17100 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17101 | 1259 is a prime number. (... |
| 2503lem1 17102 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17103 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17104 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17105 | 2503 is a prime number. (... |
| 4001lem1 17106 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17107 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17108 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17109 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17110 | 4001 is a prime number. (... |
| brstruct 17113 | The structure relation is ... |
| isstruct2 17114 | The property of being a st... |
| structex 17115 | A structure is a set. (Co... |
| structn0fun 17116 | A structure without the em... |
| isstruct 17117 | The property of being a st... |
| structcnvcnv 17118 | Two ways to express the re... |
| structfung 17119 | The converse of the conver... |
| structfun 17120 | Convert between two kinds ... |
| structfn 17121 | Convert between two kinds ... |
| strleun 17122 | Combine two structures int... |
| strle1 17123 | Make a structure from a si... |
| strle2 17124 | Make a structure from a pa... |
| strle3 17125 | Make a structure from a tr... |
| sbcie2s 17126 | A special version of class... |
| sbcie3s 17127 | A special version of class... |
| reldmsets 17130 | The structure override ope... |
| setsvalg 17131 | Value of the structure rep... |
| setsval 17132 | Value of the structure rep... |
| fvsetsid 17133 | The value of the structure... |
| fsets 17134 | The structure replacement ... |
| setsdm 17135 | The domain of a structure ... |
| setsfun 17136 | A structure with replaceme... |
| setsfun0 17137 | A structure with replaceme... |
| setsn0fun 17138 | The value of the structure... |
| setsstruct2 17139 | An extensible structure wi... |
| setsexstruct2 17140 | An extensible structure wi... |
| setsstruct 17141 | An extensible structure wi... |
| wunsets 17142 | Closure of structure repla... |
| setsres 17143 | The structure replacement ... |
| setsabs 17144 | Replacing the same compone... |
| setscom 17145 | Different components can b... |
| sloteq 17148 | Equality theorem for the `... |
| slotfn 17149 | A slot is a function on se... |
| strfvnd 17150 | Deduction version of ~ str... |
| strfvn 17151 | Value of a structure compo... |
| strfvss 17152 | A structure component extr... |
| wunstr 17153 | Closure of a structure ind... |
| str0 17154 | All components of the empt... |
| strfvi 17155 | Structure slot extractors ... |
| fveqprc 17156 | Lemma for showing the equa... |
| oveqprc 17157 | Lemma for showing the equa... |
| wunndx 17160 | Closure of the index extra... |
| ndxarg 17161 | Get the numeric argument f... |
| ndxid 17162 | A structure component extr... |
| strndxid 17163 | The value of a structure c... |
| setsidvald 17164 | Value of the structure rep... |
| strfvd 17165 | Deduction version of ~ str... |
| strfv2d 17166 | Deduction version of ~ str... |
| strfv2 17167 | A variation on ~ strfv to ... |
| strfv 17168 | Extract a structure compon... |
| strfv3 17169 | Variant on ~ strfv for lar... |
| strssd 17170 | Deduction version of ~ str... |
| strss 17171 | Propagate component extrac... |
| setsid 17172 | Value of the structure rep... |
| setsnid 17173 | Value of the structure rep... |
| baseval 17176 | Value of the base set extr... |
| baseid 17177 | Utility theorem: index-ind... |
| basfn 17178 | The base set extractor is ... |
| base0 17179 | The base set of the empty ... |
| elbasfv 17180 | Utility theorem: reverse c... |
| elbasov 17181 | Utility theorem: reverse c... |
| strov2rcl 17182 | Partial reverse closure fo... |
| basendx 17183 | Index value of the base se... |
| basendxnn 17184 | The index value of the bas... |
| basndxelwund 17185 | The index of the base set ... |
| basprssdmsets 17186 | The pair of the base index... |
| opelstrbas 17187 | The base set of a structur... |
| 1strstr 17188 | A constructed one-slot str... |
| 1strbas 17189 | The base set of a construc... |
| 1strwunbndx 17190 | A constructed one-slot str... |
| 1strwun 17191 | A constructed one-slot str... |
| 2strstr 17192 | A constructed two-slot str... |
| 2strbas 17193 | The base set of a construc... |
| 2strop 17194 | The other slot of a constr... |
| reldmress 17197 | The structure restriction ... |
| ressval 17198 | Value of structure restric... |
| ressid2 17199 | General behavior of trivia... |
| ressval2 17200 | Value of nontrivial struct... |
| ressbas 17201 | Base set of a structure re... |
| ressbasssg 17202 | The base set of a restrict... |
| ressbas2 17203 | Base set of a structure re... |
| ressbasss 17204 | The base set of a restrict... |
| ressbasssOLD 17205 | Obsolete version of ~ ress... |
| ressbasss2 17206 | The base set of a restrict... |
| resseqnbas 17207 | The components of an exten... |
| ress0 17208 | All restrictions of the nu... |
| ressid 17209 | Behavior of trivial restri... |
| ressinbas 17210 | Restriction only cares abo... |
| ressval3d 17211 | Value of structure restric... |
| ressress 17212 | Restriction composition la... |
| ressabs 17213 | Restriction absorption law... |
| wunress 17214 | Closure of structure restr... |
| plusgndx 17241 | Index value of the ~ df-pl... |
| plusgid 17242 | Utility theorem: index-ind... |
| plusgndxnn 17243 | The index of the slot for ... |
| basendxltplusgndx 17244 | The index of the slot for ... |
| basendxnplusgndx 17245 | The slot for the base set ... |
| grpstr 17246 | A constructed group is a s... |
| grpbase 17247 | The base set of a construc... |
| grpplusg 17248 | The operation of a constru... |
| ressplusg 17249 | ` +g ` is unaffected by re... |
| grpbasex 17250 | The base of an explicitly ... |
| grpplusgx 17251 | The operation of an explic... |
| mulrndx 17252 | Index value of the ~ df-mu... |
| mulridx 17253 | Utility theorem: index-ind... |
| basendxnmulrndx 17254 | The slot for the base set ... |
| plusgndxnmulrndx 17255 | The slot for the group (ad... |
| rngstr 17256 | A constructed ring is a st... |
| rngbase 17257 | The base set of a construc... |
| rngplusg 17258 | The additive operation of ... |
| rngmulr 17259 | The multiplicative operati... |
| starvndx 17260 | Index value of the ~ df-st... |
| starvid 17261 | Utility theorem: index-ind... |
| starvndxnbasendx 17262 | The slot for the involutio... |
| starvndxnplusgndx 17263 | The slot for the involutio... |
| starvndxnmulrndx 17264 | The slot for the involutio... |
| ressmulr 17265 | ` .r ` is unaffected by re... |
| ressstarv 17266 | ` *r ` is unaffected by re... |
| srngstr 17267 | A constructed star ring is... |
| srngbase 17268 | The base set of a construc... |
| srngplusg 17269 | The addition operation of ... |
| srngmulr 17270 | The multiplication operati... |
| srnginvl 17271 | The involution function of... |
| scandx 17272 | Index value of the ~ df-sc... |
| scaid 17273 | Utility theorem: index-ind... |
| scandxnbasendx 17274 | The slot for the scalar is... |
| scandxnplusgndx 17275 | The slot for the scalar fi... |
| scandxnmulrndx 17276 | The slot for the scalar fi... |
| vscandx 17277 | Index value of the ~ df-vs... |
| vscaid 17278 | Utility theorem: index-ind... |
| vscandxnbasendx 17279 | The slot for the scalar pr... |
| vscandxnplusgndx 17280 | The slot for the scalar pr... |
| vscandxnmulrndx 17281 | The slot for the scalar pr... |
| vscandxnscandx 17282 | The slot for the scalar pr... |
| lmodstr 17283 | A constructed left module ... |
| lmodbase 17284 | The base set of a construc... |
| lmodplusg 17285 | The additive operation of ... |
| lmodsca 17286 | The set of scalars of a co... |
| lmodvsca 17287 | The scalar product operati... |
| ipndx 17288 | Index value of the ~ df-ip... |
| ipid 17289 | Utility theorem: index-ind... |
| ipndxnbasendx 17290 | The slot for the inner pro... |
| ipndxnplusgndx 17291 | The slot for the inner pro... |
| ipndxnmulrndx 17292 | The slot for the inner pro... |
| slotsdifipndx 17293 | The slot for the scalar is... |
| ipsstr 17294 | Lemma to shorten proofs of... |
| ipsbase 17295 | The base set of a construc... |
| ipsaddg 17296 | The additive operation of ... |
| ipsmulr 17297 | The multiplicative operati... |
| ipssca 17298 | The set of scalars of a co... |
| ipsvsca 17299 | The scalar product operati... |
| ipsip 17300 | The multiplicative operati... |
| resssca 17301 | ` Scalar ` is unaffected b... |
| ressvsca 17302 | ` .s ` is unaffected by re... |
| ressip 17303 | The inner product is unaff... |
| phlstr 17304 | A constructed pre-Hilbert ... |
| phlbase 17305 | The base set of a construc... |
| phlplusg 17306 | The additive operation of ... |
| phlsca 17307 | The ring of scalars of a c... |
| phlvsca 17308 | The scalar product operati... |
| phlip 17309 | The inner product (Hermiti... |
| tsetndx 17310 | Index value of the ~ df-ts... |
| tsetid 17311 | Utility theorem: index-ind... |
| tsetndxnn 17312 | The index of the slot for ... |
| basendxlttsetndx 17313 | The index of the slot for ... |
| tsetndxnbasendx 17314 | The slot for the topology ... |
| tsetndxnplusgndx 17315 | The slot for the topology ... |
| tsetndxnmulrndx 17316 | The slot for the topology ... |
| tsetndxnstarvndx 17317 | The slot for the topology ... |
| slotstnscsi 17318 | The slots ` Scalar ` , ` .... |
| topgrpstr 17319 | A constructed topological ... |
| topgrpbas 17320 | The base set of a construc... |
| topgrpplusg 17321 | The additive operation of ... |
| topgrptset 17322 | The topology of a construc... |
| resstset 17323 | ` TopSet ` is unaffected b... |
| plendx 17324 | Index value of the ~ df-pl... |
| pleid 17325 | Utility theorem: self-refe... |
| plendxnn 17326 | The index value of the ord... |
| basendxltplendx 17327 | The index value of the ` B... |
| plendxnbasendx 17328 | The slot for the order is ... |
| plendxnplusgndx 17329 | The slot for the "less tha... |
| plendxnmulrndx 17330 | The slot for the "less tha... |
| plendxnscandx 17331 | The slot for the "less tha... |
| plendxnvscandx 17332 | The slot for the "less tha... |
| slotsdifplendx 17333 | The index of the slot for ... |
| otpsstr 17334 | Functionality of a topolog... |
| otpsbas 17335 | The base set of a topologi... |
| otpstset 17336 | The open sets of a topolog... |
| otpsle 17337 | The order of a topological... |
| ressle 17338 | ` le ` is unaffected by re... |
| ocndx 17339 | Index value of the ~ df-oc... |
| ocid 17340 | Utility theorem: index-ind... |
| basendxnocndx 17341 | The slot for the orthocomp... |
| plendxnocndx 17342 | The slot for the orthocomp... |
| dsndx 17343 | Index value of the ~ df-ds... |
| dsid 17344 | Utility theorem: index-ind... |
| dsndxnn 17345 | The index of the slot for ... |
| basendxltdsndx 17346 | The index of the slot for ... |
| dsndxnbasendx 17347 | The slot for the distance ... |
| dsndxnplusgndx 17348 | The slot for the distance ... |
| dsndxnmulrndx 17349 | The slot for the distance ... |
| slotsdnscsi 17350 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17351 | The slot for the distance ... |
| slotsdifdsndx 17352 | The index of the slot for ... |
| unifndx 17353 | Index value of the ~ df-un... |
| unifid 17354 | Utility theorem: index-ind... |
| unifndxnn 17355 | The index of the slot for ... |
| basendxltunifndx 17356 | The index of the slot for ... |
| unifndxnbasendx 17357 | The slot for the uniform s... |
| unifndxntsetndx 17358 | The slot for the uniform s... |
| slotsdifunifndx 17359 | The index of the slot for ... |
| ressunif 17360 | ` UnifSet ` is unaffected ... |
| odrngstr 17361 | Functionality of an ordere... |
| odrngbas 17362 | The base set of an ordered... |
| odrngplusg 17363 | The addition operation of ... |
| odrngmulr 17364 | The multiplication operati... |
| odrngtset 17365 | The open sets of an ordere... |
| odrngle 17366 | The order of an ordered me... |
| odrngds 17367 | The metric of an ordered m... |
| ressds 17368 | ` dist ` is unaffected by ... |
| homndx 17369 | Index value of the ~ df-ho... |
| homid 17370 | Utility theorem: index-ind... |
| ccondx 17371 | Index value of the ~ df-cc... |
| ccoid 17372 | Utility theorem: index-ind... |
| slotsbhcdif 17373 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17374 | The index of the slot for ... |
| slotsdifocndx 17375 | The index of the slot for ... |
| resshom 17376 | ` Hom ` is unaffected by r... |
| ressco 17377 | ` comp ` is unaffected by ... |
| restfn 17382 | The subspace topology oper... |
| topnfn 17383 | The topology extractor fun... |
| restval 17384 | The subspace topology indu... |
| elrest 17385 | The predicate "is an open ... |
| elrestr 17386 | Sufficient condition for b... |
| 0rest 17387 | Value of the structure res... |
| restid2 17388 | The subspace topology over... |
| restsspw 17389 | The subspace topology is a... |
| firest 17390 | The finite intersections o... |
| restid 17391 | The subspace topology of t... |
| topnval 17392 | Value of the topology extr... |
| topnid 17393 | Value of the topology extr... |
| topnpropd 17394 | The topology extractor fun... |
| reldmprds 17406 | The structure product is a... |
| prdsbasex 17408 | Lemma for structure produc... |
| imasvalstr 17409 | An image structure value i... |
| prdsvalstr 17410 | Structure product value is... |
| prdsbaslem 17411 | Lemma for ~ prdsbas and si... |
| prdsvallem 17412 | Lemma for ~ prdsval . (Co... |
| prdsval 17413 | Value of the structure pro... |
| prdssca 17414 | Scalar ring of a structure... |
| prdsbas 17415 | Base set of a structure pr... |
| prdsplusg 17416 | Addition in a structure pr... |
| prdsmulr 17417 | Multiplication in a struct... |
| prdsvsca 17418 | Scalar multiplication in a... |
| prdsip 17419 | Inner product in a structu... |
| prdsle 17420 | Structure product weak ord... |
| prdsless 17421 | Closure of the order relat... |
| prdsds 17422 | Structure product distance... |
| prdsdsfn 17423 | Structure product distance... |
| prdstset 17424 | Structure product topology... |
| prdshom 17425 | Structure product hom-sets... |
| prdsco 17426 | Structure product composit... |
| prdsbas2 17427 | The base set of a structur... |
| prdsbasmpt 17428 | A constructed tuple is a p... |
| prdsbasfn 17429 | Points in the structure pr... |
| prdsbasprj 17430 | Each point in a structure ... |
| prdsplusgval 17431 | Value of a componentwise s... |
| prdsplusgfval 17432 | Value of a structure produ... |
| prdsmulrval 17433 | Value of a componentwise r... |
| prdsmulrfval 17434 | Value of a structure produ... |
| prdsleval 17435 | Value of the product order... |
| prdsdsval 17436 | Value of the metric in a s... |
| prdsvscaval 17437 | Scalar multiplication in a... |
| prdsvscafval 17438 | Scalar multiplication of a... |
| prdsbas3 17439 | The base set of an indexed... |
| prdsbasmpt2 17440 | A constructed tuple is a p... |
| prdsbascl 17441 | An element of the base has... |
| prdsdsval2 17442 | Value of the metric in a s... |
| prdsdsval3 17443 | Value of the metric in a s... |
| pwsval 17444 | Value of a structure power... |
| pwsbas 17445 | Base set of a structure po... |
| pwselbasb 17446 | Membership in the base set... |
| pwselbas 17447 | An element of a structure ... |
| pwselbasr 17448 | The reverse direction of ~... |
| pwsplusgval 17449 | Value of addition in a str... |
| pwsmulrval 17450 | Value of multiplication in... |
| pwsle 17451 | Ordering in a structure po... |
| pwsleval 17452 | Ordering in a structure po... |
| pwsvscafval 17453 | Scalar multiplication in a... |
| pwsvscaval 17454 | Scalar multiplication of a... |
| pwssca 17455 | The ring of scalars of a s... |
| pwsdiagel 17456 | Membership of diagonal ele... |
| pwssnf1o 17457 | Triviality of singleton po... |
| imasval 17470 | Value of an image structur... |
| imasbas 17471 | The base set of an image s... |
| imasds 17472 | The distance function of a... |
| imasdsfn 17473 | The distance function is a... |
| imasdsval 17474 | The distance function of a... |
| imasdsval2 17475 | The distance function of a... |
| imasplusg 17476 | The group operation in an ... |
| imasmulr 17477 | The ring multiplication in... |
| imassca 17478 | The scalar field of an ima... |
| imasvsca 17479 | The scalar multiplication ... |
| imasip 17480 | The inner product of an im... |
| imastset 17481 | The topology of an image s... |
| imasle 17482 | The ordering of an image s... |
| f1ocpbllem 17483 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17484 | An injection is compatible... |
| f1ovscpbl 17485 | An injection is compatible... |
| f1olecpbl 17486 | An injection is compatible... |
| imasaddfnlem 17487 | The image structure operat... |
| imasaddvallem 17488 | The operation of an image ... |
| imasaddflem 17489 | The image set operations a... |
| imasaddfn 17490 | The image structure's grou... |
| imasaddval 17491 | The value of an image stru... |
| imasaddf 17492 | The image structure's grou... |
| imasmulfn 17493 | The image structure's ring... |
| imasmulval 17494 | The value of an image stru... |
| imasmulf 17495 | The image structure's ring... |
| imasvscafn 17496 | The image structure's scal... |
| imasvscaval 17497 | The value of an image stru... |
| imasvscaf 17498 | The image structure's scal... |
| imasless 17499 | The order relation defined... |
| imasleval 17500 | The value of the image str... |
| qusval 17501 | Value of a quotient struct... |
| quslem 17502 | The function in ~ qusval i... |
| qusin 17503 | Restrict the equivalence r... |
| qusbas 17504 | Base set of a quotient str... |
| quss 17505 | The scalar field of a quot... |
| divsfval 17506 | Value of the function in ~... |
| ercpbllem 17507 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17508 | Translate the function com... |
| erlecpbl 17509 | Translate the relation com... |
| qusaddvallem 17510 | Value of an operation defi... |
| qusaddflem 17511 | The operation of a quotien... |
| qusaddval 17512 | The addition in a quotient... |
| qusaddf 17513 | The addition in a quotient... |
| qusmulval 17514 | The multiplication in a qu... |
| qusmulf 17515 | The multiplication in a qu... |
| fnpr2o 17516 | Function with a domain of ... |
| fnpr2ob 17517 | Biconditional version of ~... |
| fvpr0o 17518 | The value of a function wi... |
| fvpr1o 17519 | The value of a function wi... |
| fvprif 17520 | The value of the pair func... |
| xpsfrnel 17521 | Elementhood in the target ... |
| xpsfeq 17522 | A function on ` 2o ` is de... |
| xpsfrnel2 17523 | Elementhood in the target ... |
| xpscf 17524 | Equivalent condition for t... |
| xpsfval 17525 | The value of the function ... |
| xpsff1o 17526 | The function appearing in ... |
| xpsfrn 17527 | A short expression for the... |
| xpsff1o2 17528 | The function appearing in ... |
| xpsval 17529 | Value of the binary struct... |
| xpsrnbas 17530 | The indexed structure prod... |
| xpsbas 17531 | The base set of the binary... |
| xpsaddlem 17532 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17533 | Value of the addition oper... |
| xpsmul 17534 | Value of the multiplicatio... |
| xpssca 17535 | Value of the scalar field ... |
| xpsvsca 17536 | Value of the scalar multip... |
| xpsless 17537 | Closure of the ordering in... |
| xpsle 17538 | Value of the ordering in a... |
| ismre 17547 | Property of being a Moore ... |
| fnmre 17548 | The Moore collection gener... |
| mresspw 17549 | A Moore collection is a su... |
| mress 17550 | A Moore-closed subset is a... |
| mre1cl 17551 | In any Moore collection th... |
| mreintcl 17552 | A nonempty collection of c... |
| mreiincl 17553 | A nonempty indexed interse... |
| mrerintcl 17554 | The relative intersection ... |
| mreriincl 17555 | The relative intersection ... |
| mreincl 17556 | Two closed sets have a clo... |
| mreuni 17557 | Since the entire base set ... |
| mreunirn 17558 | Two ways to express the no... |
| ismred 17559 | Properties that determine ... |
| ismred2 17560 | Properties that determine ... |
| mremre 17561 | The Moore collections of s... |
| submre 17562 | The subcollection of a clo... |
| xrsle 17563 | The ordering of the extend... |
| xrge0le 17564 | The "less than or equal to... |
| xrsbas 17565 | The base set of the extend... |
| xrge0base 17566 | The base of the extended n... |
| mrcflem 17567 | The domain and codomain of... |
| fnmrc 17568 | Moore-closure is a well-be... |
| mrcfval 17569 | Value of the function expr... |
| mrcf 17570 | The Moore closure is a fun... |
| mrcval 17571 | Evaluation of the Moore cl... |
| mrccl 17572 | The Moore closure of a set... |
| mrcsncl 17573 | The Moore closure of a sin... |
| mrcid 17574 | The closure of a closed se... |
| mrcssv 17575 | The closure of a set is a ... |
| mrcidb 17576 | A set is closed iff it is ... |
| mrcss 17577 | Closure preserves subset o... |
| mrcssid 17578 | The closure of a set is a ... |
| mrcidb2 17579 | A set is closed iff it con... |
| mrcidm 17580 | The closure operation is i... |
| mrcsscl 17581 | The closure is the minimal... |
| mrcuni 17582 | Idempotence of closure und... |
| mrcun 17583 | Idempotence of closure und... |
| mrcssvd 17584 | The Moore closure of a set... |
| mrcssd 17585 | Moore closure preserves su... |
| mrcssidd 17586 | A set is contained in its ... |
| mrcidmd 17587 | Moore closure is idempoten... |
| mressmrcd 17588 | In a Moore system, if a se... |
| submrc 17589 | In a closure system which ... |
| mrieqvlemd 17590 | In a Moore system, if ` Y ... |
| mrisval 17591 | Value of the set of indepe... |
| ismri 17592 | Criterion for a set to be ... |
| ismri2 17593 | Criterion for a subset of ... |
| ismri2d 17594 | Criterion for a subset of ... |
| ismri2dd 17595 | Definition of independence... |
| mriss 17596 | An independent set of a Mo... |
| mrissd 17597 | An independent set of a Mo... |
| ismri2dad 17598 | Consequence of a set in a ... |
| mrieqvd 17599 | In a Moore system, a set i... |
| mrieqv2d 17600 | In a Moore system, a set i... |
| mrissmrcd 17601 | In a Moore system, if an i... |
| mrissmrid 17602 | In a Moore system, subsets... |
| mreexd 17603 | In a Moore system, the clo... |
| mreexmrid 17604 | In a Moore system whose cl... |
| mreexexlemd 17605 | This lemma is used to gene... |
| mreexexlem2d 17606 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17607 | Base case of the induction... |
| mreexexlem4d 17608 | Induction step of the indu... |
| mreexexd 17609 | Exchange-type theorem. In... |
| mreexdomd 17610 | In a Moore system whose cl... |
| mreexfidimd 17611 | In a Moore system whose cl... |
| isacs 17612 | A set is an algebraic clos... |
| acsmre 17613 | Algebraic closure systems ... |
| isacs2 17614 | In the definition of an al... |
| acsfiel 17615 | A set is closed in an alge... |
| acsfiel2 17616 | A set is closed in an alge... |
| acsmred 17617 | An algebraic closure syste... |
| isacs1i 17618 | A closure system determine... |
| mreacs 17619 | Algebraicity is a composab... |
| acsfn 17620 | Algebraicity of a conditio... |
| acsfn0 17621 | Algebraicity of a point cl... |
| acsfn1 17622 | Algebraicity of a one-argu... |
| acsfn1c 17623 | Algebraicity of a one-argu... |
| acsfn2 17624 | Algebraicity of a two-argu... |
| iscat 17633 | The predicate "is a catego... |
| iscatd 17634 | Properties that determine ... |
| catidex 17635 | Each object in a category ... |
| catideu 17636 | Each object in a category ... |
| cidfval 17637 | Each object in a category ... |
| cidval 17638 | Each object in a category ... |
| cidffn 17639 | The identity arrow constru... |
| cidfn 17640 | The identity arrow operato... |
| catidd 17641 | Deduce the identity arrow ... |
| iscatd2 17642 | Version of ~ iscatd with a... |
| catidcl 17643 | Each object in a category ... |
| catlid 17644 | Left identity property of ... |
| catrid 17645 | Right identity property of... |
| catcocl 17646 | Closure of a composition a... |
| catass 17647 | Associativity of compositi... |
| catcone0 17648 | Composition of non-empty h... |
| 0catg 17649 | Any structure with an empt... |
| 0cat 17650 | The empty set is a categor... |
| homffval 17651 | Value of the functionalize... |
| fnhomeqhomf 17652 | If the Hom-set operation i... |
| homfval 17653 | Value of the functionalize... |
| homffn 17654 | The functionalized Hom-set... |
| homfeq 17655 | Condition for two categori... |
| homfeqd 17656 | If two structures have the... |
| homfeqbas 17657 | Deduce equality of base se... |
| homfeqval 17658 | Value of the functionalize... |
| comfffval 17659 | Value of the functionalize... |
| comffval 17660 | Value of the functionalize... |
| comfval 17661 | Value of the functionalize... |
| comfffval2 17662 | Value of the functionalize... |
| comffval2 17663 | Value of the functionalize... |
| comfval2 17664 | Value of the functionalize... |
| comfffn 17665 | The functionalized composi... |
| comffn 17666 | The functionalized composi... |
| comfeq 17667 | Condition for two categori... |
| comfeqd 17668 | Condition for two categori... |
| comfeqval 17669 | Equality of two compositio... |
| catpropd 17670 | Two structures with the sa... |
| cidpropd 17671 | Two structures with the sa... |
| oppcval 17674 | Value of the opposite cate... |
| oppchomfval 17675 | Hom-sets of the opposite c... |
| oppchom 17676 | Hom-sets of the opposite c... |
| oppccofval 17677 | Composition in the opposit... |
| oppcco 17678 | Composition in the opposit... |
| oppcbas 17679 | Base set of an opposite ca... |
| oppccatid 17680 | Lemma for ~ oppccat . (Co... |
| oppchomf 17681 | Hom-sets of the opposite c... |
| oppcid 17682 | Identity function of an op... |
| oppccat 17683 | An opposite category is a ... |
| 2oppcbas 17684 | The double opposite catego... |
| 2oppchomf 17685 | The double opposite catego... |
| 2oppccomf 17686 | The double opposite catego... |
| oppchomfpropd 17687 | If two categories have the... |
| oppccomfpropd 17688 | If two categories have the... |
| oppccatf 17689 | ` oppCat ` restricted to `... |
| monfval 17694 | Definition of a monomorphi... |
| ismon 17695 | Definition of a monomorphi... |
| ismon2 17696 | Write out the monomorphism... |
| monhom 17697 | A monomorphism is a morphi... |
| moni 17698 | Property of a monomorphism... |
| monpropd 17699 | If two categories have the... |
| oppcmon 17700 | A monomorphism in the oppo... |
| oppcepi 17701 | An epimorphism in the oppo... |
| isepi 17702 | Definition of an epimorphi... |
| isepi2 17703 | Write out the epimorphism ... |
| epihom 17704 | An epimorphism is a morphi... |
| epii 17705 | Property of an epimorphism... |
| sectffval 17712 | Value of the section opera... |
| sectfval 17713 | Value of the section relat... |
| sectss 17714 | The section relation is a ... |
| issect 17715 | The property " ` F ` is a ... |
| issect2 17716 | Property of being a sectio... |
| sectcan 17717 | If ` G ` is a section of `... |
| sectco 17718 | Composition of two section... |
| isofval 17719 | Function value of the func... |
| invffval 17720 | Value of the inverse relat... |
| invfval 17721 | Value of the inverse relat... |
| isinv 17722 | Value of the inverse relat... |
| invss 17723 | The inverse relation is a ... |
| invsym 17724 | The inverse relation is sy... |
| invsym2 17725 | The inverse relation is sy... |
| invfun 17726 | The inverse relation is a ... |
| isoval 17727 | The isomorphisms are the d... |
| inviso1 17728 | If ` G ` is an inverse to ... |
| inviso2 17729 | If ` G ` is an inverse to ... |
| invf 17730 | The inverse relation is a ... |
| invf1o 17731 | The inverse relation is a ... |
| invinv 17732 | The inverse of the inverse... |
| invco 17733 | The composition of two iso... |
| dfiso2 17734 | Alternate definition of an... |
| dfiso3 17735 | Alternate definition of an... |
| inveq 17736 | If there are two inverses ... |
| isofn 17737 | The function value of the ... |
| isohom 17738 | An isomorphism is a homomo... |
| isoco 17739 | The composition of two iso... |
| oppcsect 17740 | A section in the opposite ... |
| oppcsect2 17741 | A section in the opposite ... |
| oppcinv 17742 | An inverse in the opposite... |
| oppciso 17743 | An isomorphism in the oppo... |
| sectmon 17744 | If ` F ` is a section of `... |
| monsect 17745 | If ` F ` is a monomorphism... |
| sectepi 17746 | If ` F ` is a section of `... |
| episect 17747 | If ` F ` is an epimorphism... |
| sectid 17748 | The identity is a section ... |
| invid 17749 | The inverse of the identit... |
| idiso 17750 | The identity is an isomorp... |
| idinv 17751 | The inverse of the identit... |
| invisoinvl 17752 | The inverse of an isomorph... |
| invisoinvr 17753 | The inverse of an isomorph... |
| invcoisoid 17754 | The inverse of an isomorph... |
| isocoinvid 17755 | The inverse of an isomorph... |
| rcaninv 17756 | Right cancellation of an i... |
| cicfval 17759 | The set of isomorphic obje... |
| brcic 17760 | The relation "is isomorphi... |
| cic 17761 | Objects ` X ` and ` Y ` in... |
| brcici 17762 | Prove that two objects are... |
| cicref 17763 | Isomorphism is reflexive. ... |
| ciclcl 17764 | Isomorphism implies the le... |
| cicrcl 17765 | Isomorphism implies the ri... |
| cicsym 17766 | Isomorphism is symmetric. ... |
| cictr 17767 | Isomorphism is transitive.... |
| cicer 17768 | Isomorphism is an equivale... |
| sscrel 17775 | The subcategory subset rel... |
| brssc 17776 | The subcategory subset rel... |
| sscpwex 17777 | An analogue of ~ pwex for ... |
| subcrcl 17778 | Reverse closure for the su... |
| sscfn1 17779 | The subcategory subset rel... |
| sscfn2 17780 | The subcategory subset rel... |
| ssclem 17781 | Lemma for ~ ssc1 and simil... |
| isssc 17782 | Value of the subcategory s... |
| ssc1 17783 | Infer subset relation on o... |
| ssc2 17784 | Infer subset relation on m... |
| sscres 17785 | Any function restricted to... |
| sscid 17786 | The subcategory subset rel... |
| ssctr 17787 | The subcategory subset rel... |
| ssceq 17788 | The subcategory subset rel... |
| rescval 17789 | Value of the category rest... |
| rescval2 17790 | Value of the category rest... |
| rescbas 17791 | Base set of the category r... |
| reschom 17792 | Hom-sets of the category r... |
| reschomf 17793 | Hom-sets of the category r... |
| rescco 17794 | Composition in the categor... |
| rescabs 17795 | Restriction absorption law... |
| rescabs2 17796 | Restriction absorption law... |
| issubc 17797 | Elementhood in the set of ... |
| issubc2 17798 | Elementhood in the set of ... |
| 0ssc 17799 | For any category ` C ` , t... |
| 0subcat 17800 | For any category ` C ` , t... |
| catsubcat 17801 | For any category ` C ` , `... |
| subcssc 17802 | An element in the set of s... |
| subcfn 17803 | An element in the set of s... |
| subcss1 17804 | The objects of a subcatego... |
| subcss2 17805 | The morphisms of a subcate... |
| subcidcl 17806 | The identity of the origin... |
| subccocl 17807 | A subcategory is closed un... |
| subccatid 17808 | A subcategory is a categor... |
| subcid 17809 | The identity in a subcateg... |
| subccat 17810 | A subcategory is a categor... |
| issubc3 17811 | Alternate definition of a ... |
| fullsubc 17812 | The full subcategory gener... |
| fullresc 17813 | The category formed by str... |
| resscat 17814 | A category restricted to a... |
| subsubc 17815 | A subcategory of a subcate... |
| relfunc 17824 | The set of functors is a r... |
| funcrcl 17825 | Reverse closure for a func... |
| isfunc 17826 | Value of the set of functo... |
| isfuncd 17827 | Deduce that an operation i... |
| funcf1 17828 | The object part of a funct... |
| funcixp 17829 | The morphism part of a fun... |
| funcf2 17830 | The morphism part of a fun... |
| funcfn2 17831 | The morphism part of a fun... |
| funcid 17832 | A functor maps each identi... |
| funcco 17833 | A functor maps composition... |
| funcsect 17834 | The image of a section und... |
| funcinv 17835 | The image of an inverse un... |
| funciso 17836 | The image of an isomorphis... |
| funcoppc 17837 | A functor on categories yi... |
| idfuval 17838 | Value of the identity func... |
| idfu2nd 17839 | Value of the morphism part... |
| idfu2 17840 | Value of the morphism part... |
| idfu1st 17841 | Value of the object part o... |
| idfu1 17842 | Value of the object part o... |
| idfucl 17843 | The identity functor is a ... |
| cofuval 17844 | Value of the composition o... |
| cofu1st 17845 | Value of the object part o... |
| cofu1 17846 | Value of the object part o... |
| cofu2nd 17847 | Value of the morphism part... |
| cofu2 17848 | Value of the morphism part... |
| cofuval2 17849 | Value of the composition o... |
| cofucl 17850 | The composition of two fun... |
| cofuass 17851 | Functor composition is ass... |
| cofulid 17852 | The identity functor is a ... |
| cofurid 17853 | The identity functor is a ... |
| resfval 17854 | Value of the functor restr... |
| resfval2 17855 | Value of the functor restr... |
| resf1st 17856 | Value of the functor restr... |
| resf2nd 17857 | Value of the functor restr... |
| funcres 17858 | A functor restricted to a ... |
| funcres2b 17859 | Condition for a functor to... |
| funcres2 17860 | A functor into a restricte... |
| idfusubc0 17861 | The identity functor for a... |
| idfusubc 17862 | The identity functor for a... |
| wunfunc 17863 | A weak universe is closed ... |
| funcpropd 17864 | If two categories have the... |
| funcres2c 17865 | Condition for a functor to... |
| fullfunc 17870 | A full functor is a functo... |
| fthfunc 17871 | A faithful functor is a fu... |
| relfull 17872 | The set of full functors i... |
| relfth 17873 | The set of faithful functo... |
| isfull 17874 | Value of the set of full f... |
| isfull2 17875 | Equivalent condition for a... |
| fullfo 17876 | The morphism map of a full... |
| fulli 17877 | The morphism map of a full... |
| isfth 17878 | Value of the set of faithf... |
| isfth2 17879 | Equivalent condition for a... |
| isffth2 17880 | A fully faithful functor i... |
| fthf1 17881 | The morphism map of a fait... |
| fthi 17882 | The morphism map of a fait... |
| ffthf1o 17883 | The morphism map of a full... |
| fullpropd 17884 | If two categories have the... |
| fthpropd 17885 | If two categories have the... |
| fulloppc 17886 | The opposite functor of a ... |
| fthoppc 17887 | The opposite functor of a ... |
| ffthoppc 17888 | The opposite functor of a ... |
| fthsect 17889 | A faithful functor reflect... |
| fthinv 17890 | A faithful functor reflect... |
| fthmon 17891 | A faithful functor reflect... |
| fthepi 17892 | A faithful functor reflect... |
| ffthiso 17893 | A fully faithful functor r... |
| fthres2b 17894 | Condition for a faithful f... |
| fthres2c 17895 | Condition for a faithful f... |
| fthres2 17896 | A faithful functor into a ... |
| idffth 17897 | The identity functor is a ... |
| cofull 17898 | The composition of two ful... |
| cofth 17899 | The composition of two fai... |
| coffth 17900 | The composition of two ful... |
| rescfth 17901 | The inclusion functor from... |
| ressffth 17902 | The inclusion functor from... |
| fullres2c 17903 | Condition for a full funct... |
| ffthres2c 17904 | Condition for a fully fait... |
| inclfusubc 17905 | The "inclusion functor" fr... |
| fnfuc 17910 | The ` FuncCat ` operation ... |
| natfval 17911 | Value of the function givi... |
| isnat 17912 | Property of being a natura... |
| isnat2 17913 | Property of being a natura... |
| natffn 17914 | The natural transformation... |
| natrcl 17915 | Reverse closure for a natu... |
| nat1st2nd 17916 | Rewrite the natural transf... |
| natixp 17917 | A natural transformation i... |
| natcl 17918 | A component of a natural t... |
| natfn 17919 | A natural transformation i... |
| nati 17920 | Naturality property of a n... |
| wunnat 17921 | A weak universe is closed ... |
| catstr 17922 | A category structure is a ... |
| fucval 17923 | Value of the functor categ... |
| fuccofval 17924 | Value of the functor categ... |
| fucbas 17925 | The objects of the functor... |
| fuchom 17926 | The morphisms in the funct... |
| fucco 17927 | Value of the composition o... |
| fuccoval 17928 | Value of the functor categ... |
| fuccocl 17929 | The composition of two nat... |
| fucidcl 17930 | The identity natural trans... |
| fuclid 17931 | Left identity of natural t... |
| fucrid 17932 | Right identity of natural ... |
| fucass 17933 | Associativity of natural t... |
| fuccatid 17934 | The functor category is a ... |
| fuccat 17935 | The functor category is a ... |
| fucid 17936 | The identity morphism in t... |
| fucsect 17937 | Two natural transformation... |
| fucinv 17938 | Two natural transformation... |
| invfuc 17939 | If ` V ( x ) ` is an inver... |
| fuciso 17940 | A natural transformation i... |
| natpropd 17941 | If two categories have the... |
| fucpropd 17942 | If two categories have the... |
| initofn 17949 | ` InitO ` is a function on... |
| termofn 17950 | ` TermO ` is a function on... |
| zeroofn 17951 | ` ZeroO ` is a function on... |
| initorcl 17952 | Reverse closure for an ini... |
| termorcl 17953 | Reverse closure for a term... |
| zeroorcl 17954 | Reverse closure for a zero... |
| initoval 17955 | The value of the initial o... |
| termoval 17956 | The value of the terminal ... |
| zerooval 17957 | The value of the zero obje... |
| isinito 17958 | The predicate "is an initi... |
| istermo 17959 | The predicate "is a termin... |
| iszeroo 17960 | The predicate "is a zero o... |
| isinitoi 17961 | Implication of a class bei... |
| istermoi 17962 | Implication of a class bei... |
| initoid 17963 | For an initial object, the... |
| termoid 17964 | For a terminal object, the... |
| dfinito2 17965 | An initial object is a ter... |
| dftermo2 17966 | A terminal object is an in... |
| dfinito3 17967 | An alternate definition of... |
| dftermo3 17968 | An alternate definition of... |
| initoo 17969 | An initial object is an ob... |
| termoo 17970 | A terminal object is an ob... |
| iszeroi 17971 | Implication of a class bei... |
| 2initoinv 17972 | Morphisms between two init... |
| initoeu1 17973 | Initial objects are essent... |
| initoeu1w 17974 | Initial objects are essent... |
| initoeu2lem0 17975 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17976 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17977 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17978 | Initial objects are essent... |
| 2termoinv 17979 | Morphisms between two term... |
| termoeu1 17980 | Terminal objects are essen... |
| termoeu1w 17981 | Terminal objects are essen... |
| homarcl 17990 | Reverse closure for an arr... |
| homafval 17991 | Value of the disjointified... |
| homaf 17992 | Functionality of the disjo... |
| homaval 17993 | Value of the disjointified... |
| elhoma 17994 | Value of the disjointified... |
| elhomai 17995 | Produce an arrow from a mo... |
| elhomai2 17996 | Produce an arrow from a mo... |
| homarcl2 17997 | Reverse closure for the do... |
| homarel 17998 | An arrow is an ordered pai... |
| homa1 17999 | The first component of an ... |
| homahom2 18000 | The second component of an... |
| homahom 18001 | The second component of an... |
| homadm 18002 | The domain of an arrow wit... |
| homacd 18003 | The codomain of an arrow w... |
| homadmcd 18004 | Decompose an arrow into do... |
| arwval 18005 | The set of arrows is the u... |
| arwrcl 18006 | The first component of an ... |
| arwhoma 18007 | An arrow is contained in t... |
| homarw 18008 | A hom-set is a subset of t... |
| arwdm 18009 | The domain of an arrow is ... |
| arwcd 18010 | The codomain of an arrow i... |
| dmaf 18011 | The domain function is a f... |
| cdaf 18012 | The codomain function is a... |
| arwhom 18013 | The second component of an... |
| arwdmcd 18014 | Decompose an arrow into do... |
| idafval 18019 | Value of the identity arro... |
| idaval 18020 | Value of the identity arro... |
| ida2 18021 | Morphism part of the ident... |
| idahom 18022 | Domain and codomain of the... |
| idadm 18023 | Domain of the identity arr... |
| idacd 18024 | Codomain of the identity a... |
| idaf 18025 | The identity arrow functio... |
| coafval 18026 | The value of the compositi... |
| eldmcoa 18027 | A pair ` <. G , F >. ` is ... |
| dmcoass 18028 | The domain of composition ... |
| homdmcoa 18029 | If ` F : X --> Y ` and ` G... |
| coaval 18030 | Value of composition for c... |
| coa2 18031 | The morphism part of arrow... |
| coahom 18032 | The composition of two com... |
| coapm 18033 | Composition of arrows is a... |
| arwlid 18034 | Left identity of a categor... |
| arwrid 18035 | Right identity of a catego... |
| arwass 18036 | Associativity of compositi... |
| setcval 18039 | Value of the category of s... |
| setcbas 18040 | Set of objects of the cate... |
| setchomfval 18041 | Set of arrows of the categ... |
| setchom 18042 | Set of arrows of the categ... |
| elsetchom 18043 | A morphism of sets is a fu... |
| setccofval 18044 | Composition in the categor... |
| setcco 18045 | Composition in the categor... |
| setccatid 18046 | Lemma for ~ setccat . (Co... |
| setccat 18047 | The category of sets is a ... |
| setcid 18048 | The identity arrow in the ... |
| setcmon 18049 | A monomorphism of sets is ... |
| setcepi 18050 | An epimorphism of sets is ... |
| setcsect 18051 | A section in the category ... |
| setcinv 18052 | An inverse in the category... |
| setciso 18053 | An isomorphism in the cate... |
| resssetc 18054 | The restriction of the cat... |
| funcsetcres2 18055 | A functor into a smaller c... |
| setc2obas 18056 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18057 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18058 | The category of sets in a ... |
| cat1 18059 | The definition of category... |
| catcval 18062 | Value of the category of c... |
| catcbas 18063 | Set of objects of the cate... |
| catchomfval 18064 | Set of arrows of the categ... |
| catchom 18065 | Set of arrows of the categ... |
| catccofval 18066 | Composition in the categor... |
| catcco 18067 | Composition in the categor... |
| catccatid 18068 | Lemma for ~ catccat . (Co... |
| catcid 18069 | The identity arrow in the ... |
| catccat 18070 | The category of categories... |
| resscatc 18071 | The restriction of the cat... |
| catcisolem 18072 | Lemma for ~ catciso . (Co... |
| catciso 18073 | A functor is an isomorphis... |
| catcbascl 18074 | An element of the base set... |
| catcslotelcl 18075 | A slot entry of an element... |
| catcbaselcl 18076 | The base set of an element... |
| catchomcl 18077 | The Hom-set of an element ... |
| catcccocl 18078 | The composition operation ... |
| catcoppccl 18079 | The category of categories... |
| catcfuccl 18080 | The category of categories... |
| fncnvimaeqv 18081 | The inverse images of the ... |
| bascnvimaeqv 18082 | The inverse image of the u... |
| estrcval 18085 | Value of the category of e... |
| estrcbas 18086 | Set of objects of the cate... |
| estrchomfval 18087 | Set of morphisms ("arrows"... |
| estrchom 18088 | The morphisms between exte... |
| elestrchom 18089 | A morphism between extensi... |
| estrccofval 18090 | Composition in the categor... |
| estrcco 18091 | Composition in the categor... |
| estrcbasbas 18092 | An element of the base set... |
| estrccatid 18093 | Lemma for ~ estrccat . (C... |
| estrccat 18094 | The category of extensible... |
| estrcid 18095 | The identity arrow in the ... |
| estrchomfn 18096 | The Hom-set operation in t... |
| estrchomfeqhom 18097 | The functionalized Hom-set... |
| estrreslem1 18098 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18099 | Lemma 2 for ~ estrres . (... |
| estrres 18100 | Any restriction of a categ... |
| funcestrcsetclem1 18101 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18102 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18103 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18104 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18105 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18106 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18107 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18108 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18109 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18110 | The "natural forgetful fun... |
| fthestrcsetc 18111 | The "natural forgetful fun... |
| fullestrcsetc 18112 | The "natural forgetful fun... |
| equivestrcsetc 18113 | The "natural forgetful fun... |
| setc1strwun 18114 | A constructed one-slot str... |
| funcsetcestrclem1 18115 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18116 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18117 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18118 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18119 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18120 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18121 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18122 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18123 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18124 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18125 | The "embedding functor" fr... |
| fthsetcestrc 18126 | The "embedding functor" fr... |
| fullsetcestrc 18127 | The "embedding functor" fr... |
| embedsetcestrc 18128 | The "embedding functor" fr... |
| fnxpc 18137 | The binary product of cate... |
| xpcval 18138 | Value of the binary produc... |
| xpcbas 18139 | Set of objects of the bina... |
| xpchomfval 18140 | Set of morphisms of the bi... |
| xpchom 18141 | Set of morphisms of the bi... |
| relxpchom 18142 | A hom-set in the binary pr... |
| xpccofval 18143 | Value of composition in th... |
| xpcco 18144 | Value of composition in th... |
| xpcco1st 18145 | Value of composition in th... |
| xpcco2nd 18146 | Value of composition in th... |
| xpchom2 18147 | Value of the set of morphi... |
| xpcco2 18148 | Value of composition in th... |
| xpccatid 18149 | The product of two categor... |
| xpcid 18150 | The identity morphism in t... |
| xpccat 18151 | The product of two categor... |
| 1stfval 18152 | Value of the first project... |
| 1stf1 18153 | Value of the first project... |
| 1stf2 18154 | Value of the first project... |
| 2ndfval 18155 | Value of the first project... |
| 2ndf1 18156 | Value of the first project... |
| 2ndf2 18157 | Value of the first project... |
| 1stfcl 18158 | The first projection funct... |
| 2ndfcl 18159 | The second projection func... |
| prfval 18160 | Value of the pairing funct... |
| prf1 18161 | Value of the pairing funct... |
| prf2fval 18162 | Value of the pairing funct... |
| prf2 18163 | Value of the pairing funct... |
| prfcl 18164 | The pairing of functors ` ... |
| prf1st 18165 | Cancellation of pairing wi... |
| prf2nd 18166 | Cancellation of pairing wi... |
| 1st2ndprf 18167 | Break a functor into a pro... |
| catcxpccl 18168 | The category of categories... |
| xpcpropd 18169 | If two categories have the... |
| evlfval 18178 | Value of the evaluation fu... |
| evlf2 18179 | Value of the evaluation fu... |
| evlf2val 18180 | Value of the evaluation na... |
| evlf1 18181 | Value of the evaluation fu... |
| evlfcllem 18182 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18183 | The evaluation functor is ... |
| curfval 18184 | Value of the curry functor... |
| curf1fval 18185 | Value of the object part o... |
| curf1 18186 | Value of the object part o... |
| curf11 18187 | Value of the double evalua... |
| curf12 18188 | The partially evaluated cu... |
| curf1cl 18189 | The partially evaluated cu... |
| curf2 18190 | Value of the curry functor... |
| curf2val 18191 | Value of a component of th... |
| curf2cl 18192 | The curry functor at a mor... |
| curfcl 18193 | The curry functor of a fun... |
| curfpropd 18194 | If two categories have the... |
| uncfval 18195 | Value of the uncurry funct... |
| uncfcl 18196 | The uncurry operation take... |
| uncf1 18197 | Value of the uncurry funct... |
| uncf2 18198 | Value of the uncurry funct... |
| curfuncf 18199 | Cancellation of curry with... |
| uncfcurf 18200 | Cancellation of uncurry wi... |
| diagval 18201 | Define the diagonal functo... |
| diagcl 18202 | The diagonal functor is a ... |
| diag1cl 18203 | The constant functor of ` ... |
| diag11 18204 | Value of the constant func... |
| diag12 18205 | Value of the constant func... |
| diag2 18206 | Value of the diagonal func... |
| diag2cl 18207 | The diagonal functor at a ... |
| curf2ndf 18208 | As shown in ~ diagval , th... |
| hofval 18213 | Value of the Hom functor, ... |
| hof1fval 18214 | The object part of the Hom... |
| hof1 18215 | The object part of the Hom... |
| hof2fval 18216 | The morphism part of the H... |
| hof2val 18217 | The morphism part of the H... |
| hof2 18218 | The morphism part of the H... |
| hofcllem 18219 | Lemma for ~ hofcl . (Cont... |
| hofcl 18220 | Closure of the Hom functor... |
| oppchofcl 18221 | Closure of the opposite Ho... |
| yonval 18222 | Value of the Yoneda embedd... |
| yoncl 18223 | The Yoneda embedding is a ... |
| yon1cl 18224 | The Yoneda embedding at an... |
| yon11 18225 | Value of the Yoneda embedd... |
| yon12 18226 | Value of the Yoneda embedd... |
| yon2 18227 | Value of the Yoneda embedd... |
| hofpropd 18228 | If two categories have the... |
| yonpropd 18229 | If two categories have the... |
| oppcyon 18230 | Value of the opposite Yone... |
| oyoncl 18231 | The opposite Yoneda embedd... |
| oyon1cl 18232 | The opposite Yoneda embedd... |
| yonedalem1 18233 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18234 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18235 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18236 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18237 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18238 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18239 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18240 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18241 | Lemma for ~ yoneda . (Con... |
| yonedainv 18242 | The Yoneda Lemma with expl... |
| yonffthlem 18243 | Lemma for ~ yonffth . (Co... |
| yoneda 18244 | The Yoneda Lemma. There i... |
| yonffth 18245 | The Yoneda Lemma. The Yon... |
| yoniso 18246 | If the codomain is recover... |
| oduval 18249 | Value of an order dual str... |
| oduleval 18250 | Value of the less-equal re... |
| oduleg 18251 | Truth of the less-equal re... |
| odubas 18252 | Base set of an order dual ... |
| isprs 18257 | Property of being a preord... |
| prslem 18258 | Lemma for ~ prsref and ~ p... |
| prsref 18259 | "Less than or equal to" is... |
| prstr 18260 | "Less than or equal to" is... |
| oduprs 18261 | Being a proset is a self-d... |
| isdrs 18262 | Property of being a direct... |
| drsdir 18263 | Direction of a directed se... |
| drsprs 18264 | A directed set is a proset... |
| drsbn0 18265 | The base of a directed set... |
| drsdirfi 18266 | Any _finite_ number of ele... |
| isdrs2 18267 | Directed sets may be defin... |
| ispos 18275 | The predicate "is a poset"... |
| ispos2 18276 | A poset is an antisymmetri... |
| posprs 18277 | A poset is a proset. (Con... |
| posi 18278 | Lemma for poset properties... |
| posref 18279 | A poset ordering is reflex... |
| posasymb 18280 | A poset ordering is asymme... |
| postr 18281 | A poset ordering is transi... |
| 0pos 18282 | Technical lemma to simplif... |
| isposd 18283 | Properties that determine ... |
| isposi 18284 | Properties that determine ... |
| isposix 18285 | Properties that determine ... |
| pospropd 18286 | Posethood is determined on... |
| odupos 18287 | Being a poset is a self-du... |
| oduposb 18288 | Being a poset is a self-du... |
| pltfval 18290 | Value of the less-than rel... |
| pltval 18291 | Less-than relation. ( ~ d... |
| pltle 18292 | "Less than" implies "less ... |
| pltne 18293 | The "less than" relation i... |
| pltirr 18294 | The "less than" relation i... |
| pleval2i 18295 | One direction of ~ pleval2... |
| pleval2 18296 | "Less than or equal to" in... |
| pltnle 18297 | "Less than" implies not co... |
| pltval3 18298 | Alternate expression for t... |
| pltnlt 18299 | The less-than relation imp... |
| pltn2lp 18300 | The less-than relation has... |
| plttr 18301 | The less-than relation is ... |
| pltletr 18302 | Transitive law for chained... |
| plelttr 18303 | Transitive law for chained... |
| pospo 18304 | Write a poset structure in... |
| lubfval 18309 | Value of the least upper b... |
| lubdm 18310 | Domain of the least upper ... |
| lubfun 18311 | The LUB is a function. (C... |
| lubeldm 18312 | Member of the domain of th... |
| lubelss 18313 | A member of the domain of ... |
| lubeu 18314 | Unique existence proper of... |
| lubval 18315 | Value of the least upper b... |
| lubcl 18316 | The least upper bound func... |
| lubprop 18317 | Properties of greatest low... |
| luble 18318 | The greatest lower bound i... |
| lublecllem 18319 | Lemma for ~ lublecl and ~ ... |
| lublecl 18320 | The set of all elements le... |
| lubid 18321 | The LUB of elements less t... |
| glbfval 18322 | Value of the greatest lowe... |
| glbdm 18323 | Domain of the greatest low... |
| glbfun 18324 | The GLB is a function. (C... |
| glbeldm 18325 | Member of the domain of th... |
| glbelss 18326 | A member of the domain of ... |
| glbeu 18327 | Unique existence proper of... |
| glbval 18328 | Value of the greatest lowe... |
| glbcl 18329 | The least upper bound func... |
| glbprop 18330 | Properties of greatest low... |
| glble 18331 | The greatest lower bound i... |
| joinfval 18332 | Value of join function for... |
| joinfval2 18333 | Value of join function for... |
| joindm 18334 | Domain of join function fo... |
| joindef 18335 | Two ways to say that a joi... |
| joinval 18336 | Join value. Since both si... |
| joincl 18337 | Closure of join of element... |
| joindmss 18338 | Subset property of domain ... |
| joinval2lem 18339 | Lemma for ~ joinval2 and ~... |
| joinval2 18340 | Value of join for a poset ... |
| joineu 18341 | Uniqueness of join of elem... |
| joinlem 18342 | Lemma for join properties.... |
| lejoin1 18343 | A join's first argument is... |
| lejoin2 18344 | A join's second argument i... |
| joinle 18345 | A join is less than or equ... |
| meetfval 18346 | Value of meet function for... |
| meetfval2 18347 | Value of meet function for... |
| meetdm 18348 | Domain of meet function fo... |
| meetdef 18349 | Two ways to say that a mee... |
| meetval 18350 | Meet value. Since both si... |
| meetcl 18351 | Closure of meet of element... |
| meetdmss 18352 | Subset property of domain ... |
| meetval2lem 18353 | Lemma for ~ meetval2 and ~... |
| meetval2 18354 | Value of meet for a poset ... |
| meeteu 18355 | Uniqueness of meet of elem... |
| meetlem 18356 | Lemma for meet properties.... |
| lemeet1 18357 | A meet's first argument is... |
| lemeet2 18358 | A meet's second argument i... |
| meetle 18359 | A meet is less than or equ... |
| joincomALT 18360 | The join of a poset is com... |
| joincom 18361 | The join of a poset is com... |
| meetcomALT 18362 | The meet of a poset is com... |
| meetcom 18363 | The meet of a poset is com... |
| join0 18364 | Lemma for ~ odumeet . (Co... |
| meet0 18365 | Lemma for ~ odujoin . (Co... |
| odulub 18366 | Least upper bounds in a du... |
| odujoin 18367 | Joins in a dual order are ... |
| oduglb 18368 | Greatest lower bounds in a... |
| odumeet 18369 | Meets in a dual order are ... |
| poslubmo 18370 | Least upper bounds in a po... |
| posglbmo 18371 | Greatest lower bounds in a... |
| poslubd 18372 | Properties which determine... |
| poslubdg 18373 | Properties which determine... |
| posglbdg 18374 | Properties which determine... |
| istos 18377 | The predicate "is a toset"... |
| tosso 18378 | Write the totally ordered ... |
| tospos 18379 | A Toset is a Poset. (Cont... |
| tleile 18380 | In a Toset, any two elemen... |
| tltnle 18381 | In a Toset, "less than" is... |
| p0val 18386 | Value of poset zero. (Con... |
| p1val 18387 | Value of poset zero. (Con... |
| p0le 18388 | Any element is less than o... |
| ple1 18389 | Any element is less than o... |
| resspos 18390 | The restriction of a Poset... |
| resstos 18391 | The restriction of a Toset... |
| islat 18394 | The predicate "is a lattic... |
| odulatb 18395 | Being a lattice is self-du... |
| odulat 18396 | Being a lattice is self-du... |
| latcl2 18397 | The join and meet of any t... |
| latlem 18398 | Lemma for lattice properti... |
| latpos 18399 | A lattice is a poset. (Co... |
| latjcl 18400 | Closure of join operation ... |
| latmcl 18401 | Closure of meet operation ... |
| latref 18402 | A lattice ordering is refl... |
| latasymb 18403 | A lattice ordering is asym... |
| latasym 18404 | A lattice ordering is asym... |
| lattr 18405 | A lattice ordering is tran... |
| latasymd 18406 | Deduce equality from latti... |
| lattrd 18407 | A lattice ordering is tran... |
| latjcom 18408 | The join of a lattice comm... |
| latlej1 18409 | A join's first argument is... |
| latlej2 18410 | A join's second argument i... |
| latjle12 18411 | A join is less than or equ... |
| latleeqj1 18412 | "Less than or equal to" in... |
| latleeqj2 18413 | "Less than or equal to" in... |
| latjlej1 18414 | Add join to both sides of ... |
| latjlej2 18415 | Add join to both sides of ... |
| latjlej12 18416 | Add join to both sides of ... |
| latnlej 18417 | An idiom to express that a... |
| latnlej1l 18418 | An idiom to express that a... |
| latnlej1r 18419 | An idiom to express that a... |
| latnlej2 18420 | An idiom to express that a... |
| latnlej2l 18421 | An idiom to express that a... |
| latnlej2r 18422 | An idiom to express that a... |
| latjidm 18423 | Lattice join is idempotent... |
| latmcom 18424 | The join of a lattice comm... |
| latmle1 18425 | A meet is less than or equ... |
| latmle2 18426 | A meet is less than or equ... |
| latlem12 18427 | An element is less than or... |
| latleeqm1 18428 | "Less than or equal to" in... |
| latleeqm2 18429 | "Less than or equal to" in... |
| latmlem1 18430 | Add meet to both sides of ... |
| latmlem2 18431 | Add meet to both sides of ... |
| latmlem12 18432 | Add join to both sides of ... |
| latnlemlt 18433 | Negation of "less than or ... |
| latnle 18434 | Equivalent expressions for... |
| latmidm 18435 | Lattice meet is idempotent... |
| latabs1 18436 | Lattice absorption law. F... |
| latabs2 18437 | Lattice absorption law. F... |
| latledi 18438 | An ortholattice is distrib... |
| latmlej11 18439 | Ordering of a meet and joi... |
| latmlej12 18440 | Ordering of a meet and joi... |
| latmlej21 18441 | Ordering of a meet and joi... |
| latmlej22 18442 | Ordering of a meet and joi... |
| lubsn 18443 | The least upper bound of a... |
| latjass 18444 | Lattice join is associativ... |
| latj12 18445 | Swap 1st and 2nd members o... |
| latj32 18446 | Swap 2nd and 3rd members o... |
| latj13 18447 | Swap 1st and 3rd members o... |
| latj31 18448 | Swap 2nd and 3rd members o... |
| latjrot 18449 | Rotate lattice join of 3 c... |
| latj4 18450 | Rearrangement of lattice j... |
| latj4rot 18451 | Rotate lattice join of 4 c... |
| latjjdi 18452 | Lattice join distributes o... |
| latjjdir 18453 | Lattice join distributes o... |
| mod1ile 18454 | The weak direction of the ... |
| mod2ile 18455 | The weak direction of the ... |
| latmass 18456 | Lattice meet is associativ... |
| latdisdlem 18457 | Lemma for ~ latdisd . (Co... |
| latdisd 18458 | In a lattice, joins distri... |
| isclat 18461 | The predicate "is a comple... |
| clatpos 18462 | A complete lattice is a po... |
| clatlem 18463 | Lemma for properties of a ... |
| clatlubcl 18464 | Any subset of the base set... |
| clatlubcl2 18465 | Any subset of the base set... |
| clatglbcl 18466 | Any subset of the base set... |
| clatglbcl2 18467 | Any subset of the base set... |
| oduclatb 18468 | Being a complete lattice i... |
| clatl 18469 | A complete lattice is a la... |
| isglbd 18470 | Properties that determine ... |
| lublem 18471 | Lemma for the least upper ... |
| lubub 18472 | The LUB of a complete latt... |
| lubl 18473 | The LUB of a complete latt... |
| lubss 18474 | Subset law for least upper... |
| lubel 18475 | An element of a set is les... |
| lubun 18476 | The LUB of a union. (Cont... |
| clatglb 18477 | Properties of greatest low... |
| clatglble 18478 | The greatest lower bound i... |
| clatleglb 18479 | Two ways of expressing "le... |
| clatglbss 18480 | Subset law for greatest lo... |
| isdlat 18483 | Property of being a distri... |
| dlatmjdi 18484 | In a distributive lattice,... |
| dlatl 18485 | A distributive lattice is ... |
| odudlatb 18486 | The dual of a distributive... |
| dlatjmdi 18487 | In a distributive lattice,... |
| ipostr 18490 | The structure of ~ df-ipo ... |
| ipoval 18491 | Value of the inclusion pos... |
| ipobas 18492 | Base set of the inclusion ... |
| ipolerval 18493 | Relation of the inclusion ... |
| ipotset 18494 | Topology of the inclusion ... |
| ipole 18495 | Weak order condition of th... |
| ipolt 18496 | Strict order condition of ... |
| ipopos 18497 | The inclusion poset on a f... |
| isipodrs 18498 | Condition for a family of ... |
| ipodrscl 18499 | Direction by inclusion as ... |
| ipodrsfi 18500 | Finite upper bound propert... |
| fpwipodrs 18501 | The finite subsets of any ... |
| ipodrsima 18502 | The monotone image of a di... |
| isacs3lem 18503 | An algebraic closure syste... |
| acsdrsel 18504 | An algebraic closure syste... |
| isacs4lem 18505 | In a closure system in whi... |
| isacs5lem 18506 | If closure commutes with d... |
| acsdrscl 18507 | In an algebraic closure sy... |
| acsficl 18508 | A closure in an algebraic ... |
| isacs5 18509 | A closure system is algebr... |
| isacs4 18510 | A closure system is algebr... |
| isacs3 18511 | A closure system is algebr... |
| acsficld 18512 | In an algebraic closure sy... |
| acsficl2d 18513 | In an algebraic closure sy... |
| acsfiindd 18514 | In an algebraic closure sy... |
| acsmapd 18515 | In an algebraic closure sy... |
| acsmap2d 18516 | In an algebraic closure sy... |
| acsinfd 18517 | In an algebraic closure sy... |
| acsdomd 18518 | In an algebraic closure sy... |
| acsinfdimd 18519 | In an algebraic closure sy... |
| acsexdimd 18520 | In an algebraic closure sy... |
| mrelatglb 18521 | Greatest lower bounds in a... |
| mrelatglb0 18522 | The empty intersection in ... |
| mrelatlub 18523 | Least upper bounds in a Mo... |
| mreclatBAD 18524 | A Moore space is a complet... |
| isps 18529 | The predicate "is a poset"... |
| psrel 18530 | A poset is a relation. (C... |
| psref2 18531 | A poset is antisymmetric a... |
| pstr2 18532 | A poset is transitive. (C... |
| pslem 18533 | Lemma for ~ psref and othe... |
| psdmrn 18534 | The domain and range of a ... |
| psref 18535 | A poset is reflexive. (Co... |
| psrn 18536 | The range of a poset equal... |
| psasym 18537 | A poset is antisymmetric. ... |
| pstr 18538 | A poset is transitive. (C... |
| cnvps 18539 | The converse of a poset is... |
| cnvpsb 18540 | The converse of a poset is... |
| psss 18541 | Any subset of a partially ... |
| psssdm2 18542 | Field of a subposet. (Con... |
| psssdm 18543 | Field of a subposet. (Con... |
| istsr 18544 | The predicate is a toset. ... |
| istsr2 18545 | The predicate is a toset. ... |
| tsrlin 18546 | A toset is a linear order.... |
| tsrlemax 18547 | Two ways of saying a numbe... |
| tsrps 18548 | A toset is a poset. (Cont... |
| cnvtsr 18549 | The converse of a toset is... |
| tsrss 18550 | Any subset of a totally or... |
| ledm 18551 | The domain of ` <_ ` is ` ... |
| lern 18552 | The range of ` <_ ` is ` R... |
| lefld 18553 | The field of the 'less or ... |
| letsr 18554 | The "less than or equal to... |
| isdir 18559 | A condition for a relation... |
| reldir 18560 | A direction is a relation.... |
| dirdm 18561 | A direction's domain is eq... |
| dirref 18562 | A direction is reflexive. ... |
| dirtr 18563 | A direction is transitive.... |
| dirge 18564 | For any two elements of a ... |
| tsrdir 18565 | A totally ordered set is a... |
| ischn 18568 | Property of being a chain.... |
| chnwrd 18569 | A chain is an ordered sequ... |
| chnltm1 18570 | Basic property of a chain.... |
| pfxchn 18571 | A prefix of a chain is sti... |
| nfchnd 18572 | Bound-variable hypothesis ... |
| chneq1 18573 | Equality theorem for chain... |
| chneq2 18574 | Equality theorem for chain... |
| chneq12 18575 | Equality theorem for chain... |
| chnrss 18576 | Chains under a relation ar... |
| chndss 18577 | Chains with an alphabet ar... |
| chnrdss 18578 | Subset theorem for chains.... |
| chnexg 18579 | Chains with a set given fo... |
| nulchn 18580 | Empty set is an increasing... |
| s1chn 18581 | A singleton word is always... |
| chnind 18582 | Induction over a chain. S... |
| chnub 18583 | In a chain, the last eleme... |
| chnlt 18584 | Compare any two elements i... |
| chnso 18585 | A chain induces a total or... |
| chnccats1 18586 | Extend a chain with a sing... |
| chnccat 18587 | Concatenate two chains. (... |
| chnrev 18588 | Reverse of a chain is chai... |
| chnflenfi 18589 | There is a finite number o... |
| chnf 18590 | A chain is a zero-based fi... |
| chnpof1 18591 | A chain under relation whi... |
| chnpoadomd 18592 | A chain under relation whi... |
| chnpolleha 18593 | A chain under relation whi... |
| chnpolfz 18594 | Provided that chain's rela... |
| chnfi 18595 | There is a finite number o... |
| chninf 18596 | There is an infinite numbe... |
| chnfibg 18597 | Given a partial order, the... |
| ex-chn1 18598 | Example: a doubleton of tw... |
| ex-chn2 18599 | Example: sequence <" ZZ NN... |
| ismgm 18604 | The predicate "is a magma"... |
| ismgmn0 18605 | The predicate "is a magma"... |
| mgmcl 18606 | Closure of the operation o... |
| isnmgm 18607 | A condition for a structur... |
| mgmsscl 18608 | If the base set of a magma... |
| plusffval 18609 | The group addition operati... |
| plusfval 18610 | The group addition operati... |
| plusfeq 18611 | If the addition operation ... |
| plusffn 18612 | The group addition operati... |
| mgmplusf 18613 | The group addition functio... |
| mgmpropd 18614 | If two structures have the... |
| ismgmd 18615 | Deduce a magma from its pr... |
| issstrmgm 18616 | Characterize a substructur... |
| intopsn 18617 | The internal operation for... |
| mgmb1mgm1 18618 | The only magma with a base... |
| mgm0 18619 | Any set with an empty base... |
| mgm0b 18620 | The structure with an empt... |
| mgm1 18621 | The structure with one ele... |
| opifismgm 18622 | A structure with a group a... |
| mgmidmo 18623 | A two-sided identity eleme... |
| grpidval 18624 | The value of the identity ... |
| grpidpropd 18625 | If two structures have the... |
| fn0g 18626 | The group zero extractor i... |
| 0g0 18627 | The identity element funct... |
| ismgmid 18628 | The identity element of a ... |
| mgmidcl 18629 | The identity element of a ... |
| mgmlrid 18630 | The identity element of a ... |
| ismgmid2 18631 | Show that a given element ... |
| lidrideqd 18632 | If there is a left and rig... |
| lidrididd 18633 | If there is a left and rig... |
| grpidd 18634 | Deduce the identity elemen... |
| mgmidsssn0 18635 | Property of the set of ide... |
| grpinvalem 18636 | Lemma for ~ grpinva . (Co... |
| grpinva 18637 | Deduce right inverse from ... |
| grprida 18638 | Deduce right identity from... |
| gsumvalx 18639 | Expand out the substitutio... |
| gsumval 18640 | Expand out the substitutio... |
| gsumpropd 18641 | The group sum depends only... |
| gsumpropd2lem 18642 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18643 | A stronger version of ~ gs... |
| gsummgmpropd 18644 | A stronger version of ~ gs... |
| gsumress 18645 | The group sum in a substru... |
| gsumval1 18646 | Value of the group sum ope... |
| gsum0 18647 | Value of the empty group s... |
| gsumval2a 18648 | Value of the group sum ope... |
| gsumval2 18649 | Value of the group sum ope... |
| gsumsplit1r 18650 | Splitting off the rightmos... |
| gsumprval 18651 | Value of the group sum ope... |
| gsumpr12val 18652 | Value of the group sum ope... |
| mgmhmrcl 18657 | Reverse closure of a magma... |
| submgmrcl 18658 | Reverse closure for submag... |
| ismgmhm 18659 | Property of a magma homomo... |
| mgmhmf 18660 | A magma homomorphism is a ... |
| mgmhmpropd 18661 | Magma homomorphism depends... |
| mgmhmlin 18662 | A magma homomorphism prese... |
| mgmhmf1o 18663 | A magma homomorphism is bi... |
| idmgmhm 18664 | The identity homomorphism ... |
| issubmgm 18665 | Expand definition of a sub... |
| issubmgm2 18666 | Submagmas are subsets that... |
| rabsubmgmd 18667 | Deduction for proving that... |
| submgmss 18668 | Submagmas are subsets of t... |
| submgmid 18669 | Every magma is trivially a... |
| submgmcl 18670 | Submagmas are closed under... |
| submgmmgm 18671 | Submagmas are themselves m... |
| submgmbas 18672 | The base set of a submagma... |
| subsubmgm 18673 | A submagma of a submagma i... |
| resmgmhm 18674 | Restriction of a magma hom... |
| resmgmhm2 18675 | One direction of ~ resmgmh... |
| resmgmhm2b 18676 | Restriction of the codomai... |
| mgmhmco 18677 | The composition of magma h... |
| mgmhmima 18678 | The homomorphic image of a... |
| mgmhmeql 18679 | The equalizer of two magma... |
| submgmacs 18680 | Submagmas are an algebraic... |
| issgrp 18683 | The predicate "is a semigr... |
| issgrpv 18684 | The predicate "is a semigr... |
| issgrpn0 18685 | The predicate "is a semigr... |
| isnsgrp 18686 | A condition for a structur... |
| sgrpmgm 18687 | A semigroup is a magma. (... |
| sgrpass 18688 | A semigroup operation is a... |
| sgrpcl 18689 | Closure of the operation o... |
| sgrp0 18690 | Any set with an empty base... |
| sgrp0b 18691 | The structure with an empt... |
| sgrp1 18692 | The structure with one ele... |
| issgrpd 18693 | Deduce a semigroup from it... |
| sgrppropd 18694 | If two structures are sets... |
| prdsplusgsgrpcl 18695 | Structure product pointwis... |
| prdssgrpd 18696 | The product of a family of... |
| ismnddef 18699 | The predicate "is a monoid... |
| ismnd 18700 | The predicate "is a monoid... |
| isnmnd 18701 | A condition for a structur... |
| sgrpidmnd 18702 | A semigroup with an identi... |
| mndsgrp 18703 | A monoid is a semigroup. ... |
| mndmgm 18704 | A monoid is a magma. (Con... |
| mndcl 18705 | Closure of the operation o... |
| mndass 18706 | A monoid operation is asso... |
| mndid 18707 | A monoid has a two-sided i... |
| mndideu 18708 | The two-sided identity ele... |
| mnd32g 18709 | Commutative/associative la... |
| mnd12g 18710 | Commutative/associative la... |
| mnd4g 18711 | Commutative/associative la... |
| mndidcl 18712 | The identity element of a ... |
| mndbn0 18713 | The base set of a monoid i... |
| hashfinmndnn 18714 | A finite monoid has positi... |
| mndplusf 18715 | The group addition operati... |
| mndlrid 18716 | A monoid's identity elemen... |
| mndlid 18717 | The identity element of a ... |
| mndrid 18718 | The identity element of a ... |
| ismndd 18719 | Deduce a monoid from its p... |
| mndpfo 18720 | The addition operation of ... |
| mndfo 18721 | The addition operation of ... |
| mndpropd 18722 | If two structures have the... |
| mndprop 18723 | If two structures have the... |
| issubmnd 18724 | Characterize a submonoid b... |
| ress0g 18725 | ` 0g ` is unaffected by re... |
| submnd0 18726 | The zero of a submonoid is... |
| mndinvmod 18727 | Uniqueness of an inverse e... |
| mndpsuppss 18728 | The support of a mapping o... |
| mndpsuppfi 18729 | The support of a mapping o... |
| mndpfsupp 18730 | A mapping of a scalar mult... |
| prdsplusgcl 18731 | Structure product pointwis... |
| prdsidlem 18732 | Characterization of identi... |
| prdsmndd 18733 | The product of a family of... |
| prds0g 18734 | The identity in a product ... |
| pwsmnd 18735 | The structure power of a m... |
| pws0g 18736 | The identity in a structur... |
| imasmnd2 18737 | The image structure of a m... |
| imasmnd 18738 | The image structure of a m... |
| imasmndf1 18739 | The image of a monoid unde... |
| xpsmnd 18740 | The binary product of mono... |
| xpsmnd0 18741 | The identity element of a ... |
| mnd1 18742 | The (smallest) structure r... |
| mnd1id 18743 | The singleton element of a... |
| ismhm 18748 | Property of a monoid homom... |
| ismhmd 18749 | Deduction version of ~ ism... |
| mhmrcl1 18750 | Reverse closure of a monoi... |
| mhmrcl2 18751 | Reverse closure of a monoi... |
| mhmf 18752 | A monoid homomorphism is a... |
| ismhm0 18753 | Property of a monoid homom... |
| mhmismgmhm 18754 | Each monoid homomorphism i... |
| mhmpropd 18755 | Monoid homomorphism depend... |
| mhmlin 18756 | A monoid homomorphism comm... |
| mhm0 18757 | A monoid homomorphism pres... |
| idmhm 18758 | The identity homomorphism ... |
| mhmf1o 18759 | A monoid homomorphism is b... |
| mndvcl 18760 | Tuple-wise additive closur... |
| mndvass 18761 | Tuple-wise associativity i... |
| mndvlid 18762 | Tuple-wise left identity i... |
| mndvrid 18763 | Tuple-wise right identity ... |
| mhmvlin 18764 | Tuple extension of monoid ... |
| submrcl 18765 | Reverse closure for submon... |
| issubm 18766 | Expand definition of a sub... |
| issubm2 18767 | Submonoids are subsets tha... |
| issubmndb 18768 | The submonoid predicate. ... |
| issubmd 18769 | Deduction for proving a su... |
| mndissubm 18770 | If the base set of a monoi... |
| resmndismnd 18771 | If the base set of a monoi... |
| submss 18772 | Submonoids are subsets of ... |
| submid 18773 | Every monoid is trivially ... |
| subm0cl 18774 | Submonoids contain zero. ... |
| submcl 18775 | Submonoids are closed unde... |
| submmnd 18776 | Submonoids are themselves ... |
| submbas 18777 | The base set of a submonoi... |
| subm0 18778 | Submonoids have the same i... |
| subsubm 18779 | A submonoid of a submonoid... |
| 0subm 18780 | The zero submonoid of an a... |
| insubm 18781 | The intersection of two su... |
| 0mhm 18782 | The constant zero linear f... |
| resmhm 18783 | Restriction of a monoid ho... |
| resmhm2 18784 | One direction of ~ resmhm2... |
| resmhm2b 18785 | Restriction of the codomai... |
| mhmco 18786 | The composition of monoid ... |
| mhmimalem 18787 | Lemma for ~ mhmima and sim... |
| mhmima 18788 | The homomorphic image of a... |
| mhmeql 18789 | The equalizer of two monoi... |
| submacs 18790 | Submonoids are an algebrai... |
| mndind 18791 | Induction in a monoid. In... |
| prdspjmhm 18792 | A projection from a produc... |
| pwspjmhm 18793 | A projection from a struct... |
| pwsdiagmhm 18794 | Diagonal monoid homomorphi... |
| pwsco1mhm 18795 | Right composition with a f... |
| pwsco2mhm 18796 | Left composition with a mo... |
| gsumvallem2 18797 | Lemma for properties of th... |
| gsumsubm 18798 | Evaluate a group sum in a ... |
| gsumz 18799 | Value of a group sum over ... |
| gsumwsubmcl 18800 | Closure of the composite i... |
| gsumws1 18801 | A singleton composite reco... |
| gsumwcl 18802 | Closure of the composite o... |
| gsumsgrpccat 18803 | Homomorphic property of no... |
| gsumccat 18804 | Homomorphic property of co... |
| gsumws2 18805 | Valuation of a pair in a m... |
| gsumccatsn 18806 | Homomorphic property of co... |
| gsumspl 18807 | The primary purpose of the... |
| gsumwmhm 18808 | Behavior of homomorphisms ... |
| gsumwspan 18809 | The submonoid generated by... |
| frmdval 18814 | Value of the free monoid c... |
| frmdbas 18815 | The base set of a free mon... |
| frmdelbas 18816 | An element of the base set... |
| frmdplusg 18817 | The monoid operation of a ... |
| frmdadd 18818 | Value of the monoid operat... |
| vrmdfval 18819 | The canonical injection fr... |
| vrmdval 18820 | The value of the generatin... |
| vrmdf 18821 | The mapping from the index... |
| frmdmnd 18822 | A free monoid is a monoid.... |
| frmd0 18823 | The identity of the free m... |
| frmdsssubm 18824 | The set of words taking va... |
| frmdgsum 18825 | Any word in a free monoid ... |
| frmdss2 18826 | A subset of generators is ... |
| frmdup1 18827 | Any assignment of the gene... |
| frmdup2 18828 | The evaluation map has the... |
| frmdup3lem 18829 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18830 | Universal property of the ... |
| efmnd 18833 | The monoid of endofunction... |
| efmndbas 18834 | The base set of the monoid... |
| efmndbasabf 18835 | The base set of the monoid... |
| elefmndbas 18836 | Two ways of saying a funct... |
| elefmndbas2 18837 | Two ways of saying a funct... |
| efmndbasf 18838 | Elements in the monoid of ... |
| efmndhash 18839 | The monoid of endofunction... |
| efmndbasfi 18840 | The monoid of endofunction... |
| efmndfv 18841 | The function value of an e... |
| efmndtset 18842 | The topology of the monoid... |
| efmndplusg 18843 | The group operation of a m... |
| efmndov 18844 | The value of the group ope... |
| efmndcl 18845 | The group operation of the... |
| efmndtopn 18846 | The topology of the monoid... |
| symggrplem 18847 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18848 | The monoid of endofunction... |
| efmndsgrp 18849 | The monoid of endofunction... |
| ielefmnd 18850 | The identity function rest... |
| efmndid 18851 | The identity function rest... |
| efmndmnd 18852 | The monoid of endofunction... |
| efmnd0nmnd 18853 | Even the monoid of endofun... |
| efmndbas0 18854 | The base set of the monoid... |
| efmnd1hash 18855 | The monoid of endofunction... |
| efmnd1bas 18856 | The monoid of endofunction... |
| efmnd2hash 18857 | The monoid of endofunction... |
| submefmnd 18858 | If the base set of a monoi... |
| sursubmefmnd 18859 | The set of surjective endo... |
| injsubmefmnd 18860 | The set of injective endof... |
| idressubmefmnd 18861 | The singleton containing o... |
| idresefmnd 18862 | The structure with the sin... |
| smndex1ibas 18863 | The modulo function ` I ` ... |
| smndex1iidm 18864 | The modulo function ` I ` ... |
| smndex1gbas 18865 | The constant functions ` (... |
| smndex1gbasOLD 18866 | Obsolete version of ~ smnd... |
| smndex1gid 18867 | The composition of a const... |
| smndex1gidOLD 18868 | Obsolete version of ~ smnd... |
| smndex1igid 18869 | The composition of the mod... |
| smndex1igidOLD 18870 | Obsolete version of ~ smnd... |
| smndex1basss 18871 | The modulo function ` I ` ... |
| smndex1bas 18872 | The base set of the monoid... |
| smndex1mgm 18873 | The monoid of endofunction... |
| smndex1sgrp 18874 | The monoid of endofunction... |
| smndex1mndlem 18875 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18876 | The monoid of endofunction... |
| smndex1id 18877 | The modulo function ` I ` ... |
| smndex1n0mnd 18878 | The identity of the monoid... |
| nsmndex1 18879 | The base set ` B ` of the ... |
| smndex2dbas 18880 | The doubling function ` D ... |
| smndex2dnrinv 18881 | The doubling function ` D ... |
| smndex2hbas 18882 | The halving functions ` H ... |
| smndex2dlinvh 18883 | The halving functions ` H ... |
| mgm2nsgrplem1 18884 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18885 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18886 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18887 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18888 | A small magma (with two el... |
| sgrp2nmndlem1 18889 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18890 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18891 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18892 | A small semigroup (with tw... |
| sgrp2rid2ex 18893 | A small semigroup (with tw... |
| sgrp2nmndlem4 18894 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18895 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18896 | A small semigroup (with tw... |
| mgmnsgrpex 18897 | There is a magma which is ... |
| sgrpnmndex 18898 | There is a semigroup which... |
| sgrpssmgm 18899 | The class of all semigroup... |
| mndsssgrp 18900 | The class of all monoids i... |
| pwmndgplus 18901 | The operation of the monoi... |
| pwmndid 18902 | The identity of the monoid... |
| pwmnd 18903 | The power set of a class `... |
| isgrp 18910 | The predicate "is a group"... |
| grpmnd 18911 | A group is a monoid. (Con... |
| grpcl 18912 | Closure of the operation o... |
| grpass 18913 | A group operation is assoc... |
| grpinvex 18914 | Every member of a group ha... |
| grpideu 18915 | The two-sided identity ele... |
| grpassd 18916 | A group operation is assoc... |
| grpmndd 18917 | A group is a monoid. (Con... |
| grpcld 18918 | Closure of the operation o... |
| grpplusf 18919 | The group addition operati... |
| grpplusfo 18920 | The group addition operati... |
| resgrpplusfrn 18921 | The underlying set of a gr... |
| grppropd 18922 | If two structures have the... |
| grpprop 18923 | If two structures have the... |
| grppropstr 18924 | Generalize a specific 2-el... |
| grpss 18925 | Show that a structure exte... |
| isgrpd2e 18926 | Deduce a group from its pr... |
| isgrpd2 18927 | Deduce a group from its pr... |
| isgrpde 18928 | Deduce a group from its pr... |
| isgrpd 18929 | Deduce a group from its pr... |
| isgrpi 18930 | Properties that determine ... |
| grpsgrp 18931 | A group is a semigroup. (... |
| grpmgmd 18932 | A group is a magma, deduct... |
| dfgrp2 18933 | Alternate definition of a ... |
| dfgrp2e 18934 | Alternate definition of a ... |
| isgrpix 18935 | Properties that determine ... |
| grpidcl 18936 | The identity element of a ... |
| grpbn0 18937 | The base set of a group is... |
| grplid 18938 | The identity element of a ... |
| grprid 18939 | The identity element of a ... |
| grplidd 18940 | The identity element of a ... |
| grpridd 18941 | The identity element of a ... |
| grpn0 18942 | A group is not empty. (Co... |
| hashfingrpnn 18943 | A finite group has positiv... |
| grprcan 18944 | Right cancellation law for... |
| grpinveu 18945 | The left inverse element o... |
| grpid 18946 | Two ways of saying that an... |
| isgrpid2 18947 | Properties showing that an... |
| grpidd2 18948 | Deduce the identity elemen... |
| grpinvfval 18949 | The inverse function of a ... |
| grpinvfvalALT 18950 | Shorter proof of ~ grpinvf... |
| grpinvval 18951 | The inverse of a group ele... |
| grpinvfn 18952 | Functionality of the group... |
| grpinvfvi 18953 | The group inverse function... |
| grpsubfval 18954 | Group subtraction (divisio... |
| grpsubfvalALT 18955 | Shorter proof of ~ grpsubf... |
| grpsubval 18956 | Group subtraction (divisio... |
| grpinvf 18957 | The group inversion operat... |
| grpinvcl 18958 | A group element's inverse ... |
| grpinvcld 18959 | A group element's inverse ... |
| grplinv 18960 | The left inverse of a grou... |
| grprinv 18961 | The right inverse of a gro... |
| grpinvid1 18962 | The inverse of a group ele... |
| grpinvid2 18963 | The inverse of a group ele... |
| isgrpinv 18964 | Properties showing that a ... |
| grplinvd 18965 | The left inverse of a grou... |
| grprinvd 18966 | The right inverse of a gro... |
| grplrinv 18967 | In a group, every member h... |
| grpidinv2 18968 | A group's properties using... |
| grpidinv 18969 | A group has a left and rig... |
| grpinvid 18970 | The inverse of the identit... |
| grplcan 18971 | Left cancellation law for ... |
| grpasscan1 18972 | An associative cancellatio... |
| grpasscan2 18973 | An associative cancellatio... |
| grpidrcan 18974 | If right adding an element... |
| grpidlcan 18975 | If left adding an element ... |
| grpinvinv 18976 | Double inverse law for gro... |
| grpinvcnv 18977 | The group inverse is its o... |
| grpinv11 18978 | The group inverse is one-t... |
| grpinv11OLD 18979 | Obsolete version of ~ grpi... |
| grpinvf1o 18980 | The group inverse is a one... |
| grpinvnz 18981 | The inverse of a nonzero g... |
| grpinvnzcl 18982 | The inverse of a nonzero g... |
| grpsubinv 18983 | Subtraction of an inverse.... |
| grplmulf1o 18984 | Left multiplication by a g... |
| grpraddf1o 18985 | Right addition by a group ... |
| grpinvpropd 18986 | If two structures have the... |
| grpidssd 18987 | If the base set of a group... |
| grpinvssd 18988 | If the base set of a group... |
| grpinvadd 18989 | The inverse of the group o... |
| grpsubf 18990 | Functionality of group sub... |
| grpsubcl 18991 | Closure of group subtracti... |
| grpsubrcan 18992 | Right cancellation law for... |
| grpinvsub 18993 | Inverse of a group subtrac... |
| grpinvval2 18994 | A ~ df-neg -like equation ... |
| grpsubid 18995 | Subtraction of a group ele... |
| grpsubid1 18996 | Subtraction of the identit... |
| grpsubeq0 18997 | If the difference between ... |
| grpsubadd0sub 18998 | Subtraction expressed as a... |
| grpsubadd 18999 | Relationship between group... |
| grpsubsub 19000 | Double group subtraction. ... |
| grpaddsubass 19001 | Associative-type law for g... |
| grppncan 19002 | Cancellation law for subtr... |
| grpnpcan 19003 | Cancellation law for subtr... |
| grpsubsub4 19004 | Double group subtraction (... |
| grppnpcan2 19005 | Cancellation law for mixed... |
| grpnpncan 19006 | Cancellation law for group... |
| grpnpncan0 19007 | Cancellation law for group... |
| grpnnncan2 19008 | Cancellation law for group... |
| dfgrp3lem 19009 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 19010 | Alternate definition of a ... |
| dfgrp3e 19011 | Alternate definition of a ... |
| grplactfval 19012 | The left group action of e... |
| grplactval 19013 | The value of the left grou... |
| grplactcnv 19014 | The left group action of e... |
| grplactf1o 19015 | The left group action of e... |
| grpsubpropd 19016 | Weak property deduction fo... |
| grpsubpropd2 19017 | Strong property deduction ... |
| grp1 19018 | The (smallest) structure r... |
| grp1inv 19019 | The inverse function of th... |
| prdsinvlem 19020 | Characterization of invers... |
| prdsgrpd 19021 | The product of a family of... |
| prdsinvgd 19022 | Negation in a product of g... |
| pwsgrp 19023 | A structure power of a gro... |
| pwsinvg 19024 | Negation in a group power.... |
| pwssub 19025 | Subtraction in a group pow... |
| imasgrp2 19026 | The image structure of a g... |
| imasgrp 19027 | The image structure of a g... |
| imasgrpf1 19028 | The image of a group under... |
| qusgrp2 19029 | Prove that a quotient stru... |
| xpsgrp 19030 | The binary product of grou... |
| xpsinv 19031 | Value of the negation oper... |
| xpsgrpsub 19032 | Value of the subtraction o... |
| mhmlem 19033 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19034 | A surjective monoid morphi... |
| mhmmnd 19035 | The image of a monoid ` G ... |
| mhmfmhm 19036 | The function fulfilling th... |
| ghmgrp 19037 | The image of a group ` G `... |
| mulgfval 19040 | Group multiple (exponentia... |
| mulgfvalALT 19041 | Shorter proof of ~ mulgfva... |
| mulgval 19042 | Value of the group multipl... |
| mulgfn 19043 | Functionality of the group... |
| mulgfvi 19044 | The group multiple operati... |
| mulg0 19045 | Group multiple (exponentia... |
| mulgnn 19046 | Group multiple (exponentia... |
| ressmulgnn 19047 | Values for the group multi... |
| ressmulgnn0 19048 | Values for the group multi... |
| ressmulgnnd 19049 | Values for the group multi... |
| mulgnngsum 19050 | Group multiple (exponentia... |
| mulgnn0gsum 19051 | Group multiple (exponentia... |
| mulg1 19052 | Group multiple (exponentia... |
| mulgnnp1 19053 | Group multiple (exponentia... |
| mulg2 19054 | Group multiple (exponentia... |
| mulgnegnn 19055 | Group multiple (exponentia... |
| mulgnn0p1 19056 | Group multiple (exponentia... |
| mulgnnsubcl 19057 | Closure of the group multi... |
| mulgnn0subcl 19058 | Closure of the group multi... |
| mulgsubcl 19059 | Closure of the group multi... |
| mulgnncl 19060 | Closure of the group multi... |
| mulgnn0cl 19061 | Closure of the group multi... |
| mulgcl 19062 | Closure of the group multi... |
| mulgneg 19063 | Group multiple (exponentia... |
| mulgnegneg 19064 | The inverse of a negative ... |
| mulgm1 19065 | Group multiple (exponentia... |
| mulgnn0cld 19066 | Closure of the group multi... |
| mulgcld 19067 | Deduction associated with ... |
| mulgaddcomlem 19068 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19069 | The group multiple operato... |
| mulginvcom 19070 | The group multiple operato... |
| mulginvinv 19071 | The group multiple operato... |
| mulgnn0z 19072 | A group multiple of the id... |
| mulgz 19073 | A group multiple of the id... |
| mulgnndir 19074 | Sum of group multiples, fo... |
| mulgnn0dir 19075 | Sum of group multiples, ge... |
| mulgdirlem 19076 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19077 | Sum of group multiples, ge... |
| mulgp1 19078 | Group multiple (exponentia... |
| mulgneg2 19079 | Group multiple (exponentia... |
| mulgnnass 19080 | Product of group multiples... |
| mulgnn0ass 19081 | Product of group multiples... |
| mulgass 19082 | Product of group multiples... |
| mulgassr 19083 | Reversed product of group ... |
| mulgmodid 19084 | Casting out multiples of t... |
| mulgsubdir 19085 | Distribution of group mult... |
| mhmmulg 19086 | A homomorphism of monoids ... |
| mulgpropd 19087 | Two structures with the sa... |
| submmulgcl 19088 | Closure of the group multi... |
| submmulg 19089 | A group multiple is the sa... |
| pwsmulg 19090 | Value of a group multiple ... |
| issubg 19097 | The subgroup predicate. (... |
| subgss 19098 | A subgroup is a subset. (... |
| subgid 19099 | A group is a subgroup of i... |
| subggrp 19100 | A subgroup is a group. (C... |
| subgbas 19101 | The base of the restricted... |
| subgrcl 19102 | Reverse closure for the su... |
| subg0 19103 | A subgroup of a group must... |
| subginv 19104 | The inverse of an element ... |
| subg0cl 19105 | The group identity is an e... |
| subginvcl 19106 | The inverse of an element ... |
| subgcl 19107 | A subgroup is closed under... |
| subgsubcl 19108 | A subgroup is closed under... |
| subgsub 19109 | The subtraction of element... |
| subgmulgcl 19110 | Closure of the group multi... |
| subgmulg 19111 | A group multiple is the sa... |
| issubg2 19112 | Characterize the subgroups... |
| issubgrpd2 19113 | Prove a subgroup by closur... |
| issubgrpd 19114 | Prove a subgroup by closur... |
| issubg3 19115 | A subgroup is a symmetric ... |
| issubg4 19116 | A subgroup is a nonempty s... |
| grpissubg 19117 | If the base set of a group... |
| resgrpisgrp 19118 | If the base set of a group... |
| subgsubm 19119 | A subgroup is a submonoid.... |
| subsubg 19120 | A subgroup of a subgroup i... |
| subgint 19121 | The intersection of a none... |
| 0subg 19122 | The zero subgroup of an ar... |
| trivsubgd 19123 | The only subgroup of a tri... |
| trivsubgsnd 19124 | The only subgroup of a tri... |
| isnsg 19125 | Property of being a normal... |
| isnsg2 19126 | Weaken the condition of ~ ... |
| nsgbi 19127 | Defining property of a nor... |
| nsgsubg 19128 | A normal subgroup is a sub... |
| nsgconj 19129 | The conjugation of an elem... |
| isnsg3 19130 | A subgroup is normal iff t... |
| subgacs 19131 | Subgroups are an algebraic... |
| nsgacs 19132 | Normal subgroups form an a... |
| elnmz 19133 | Elementhood in the normali... |
| nmzbi 19134 | Defining property of the n... |
| nmzsubg 19135 | The normalizer N_G(S) of a... |
| ssnmz 19136 | A subgroup is a subset of ... |
| isnsg4 19137 | A subgroup is normal iff i... |
| nmznsg 19138 | Any subgroup is a normal s... |
| 0nsg 19139 | The zero subgroup is norma... |
| nsgid 19140 | The whole group is a norma... |
| 0idnsgd 19141 | The whole group and the ze... |
| trivnsgd 19142 | The only normal subgroup o... |
| triv1nsgd 19143 | A trivial group has exactl... |
| 1nsgtrivd 19144 | A group with exactly one n... |
| releqg 19145 | The left coset equivalence... |
| eqgfval 19146 | Value of the subgroup left... |
| eqgval 19147 | Value of the subgroup left... |
| eqger 19148 | The subgroup coset equival... |
| eqglact 19149 | A left coset can be expres... |
| eqgid 19150 | The left coset containing ... |
| eqgen 19151 | Each coset is equipotent t... |
| eqgcpbl 19152 | The subgroup coset equival... |
| eqg0el 19153 | Equivalence class of a quo... |
| quselbas 19154 | Membership in the base set... |
| quseccl0 19155 | Closure of the quotient ma... |
| qusgrp 19156 | If ` Y ` is a normal subgr... |
| quseccl 19157 | Closure of the quotient ma... |
| qusadd 19158 | Value of the group operati... |
| qus0 19159 | Value of the group identit... |
| qusinv 19160 | Value of the group inverse... |
| qussub 19161 | Value of the group subtrac... |
| ecqusaddd 19162 | Addition of equivalence cl... |
| ecqusaddcl 19163 | Closure of the addition in... |
| lagsubg2 19164 | Lagrange's theorem for fin... |
| lagsubg 19165 | Lagrange's theorem for Gro... |
| eqg0subg 19166 | The coset equivalence rela... |
| eqg0subgecsn 19167 | The equivalence classes mo... |
| qus0subgbas 19168 | The base set of a quotient... |
| qus0subgadd 19169 | The addition in a quotient... |
| cycsubmel 19170 | Characterization of an ele... |
| cycsubmcl 19171 | The set of nonnegative int... |
| cycsubm 19172 | The set of nonnegative int... |
| cyccom 19173 | Condition for an operation... |
| cycsubmcom 19174 | The operation of a monoid ... |
| cycsubggend 19175 | The cyclic subgroup genera... |
| cycsubgcl 19176 | The set of integer powers ... |
| cycsubgss 19177 | The cyclic subgroup genera... |
| cycsubg 19178 | The cyclic group generated... |
| cycsubgcld 19179 | The cyclic subgroup genera... |
| cycsubg2 19180 | The subgroup generated by ... |
| cycsubg2cl 19181 | Any multiple of an element... |
| reldmghm 19184 | Lemma for group homomorphi... |
| isghm 19185 | Property of being a homomo... |
| isghmOLD 19186 | Obsolete version of ~ isgh... |
| isghm3 19187 | Property of a group homomo... |
| ghmgrp1 19188 | A group homomorphism is on... |
| ghmgrp2 19189 | A group homomorphism is on... |
| ghmf 19190 | A group homomorphism is a ... |
| ghmlin 19191 | A homomorphism of groups i... |
| ghmid 19192 | A homomorphism of groups p... |
| ghminv 19193 | A homomorphism of groups p... |
| ghmsub 19194 | Linearity of subtraction t... |
| isghmd 19195 | Deduction for a group homo... |
| ghmmhm 19196 | A group homomorphism is a ... |
| ghmmhmb 19197 | Group homomorphisms and mo... |
| ghmmulg 19198 | A group homomorphism prese... |
| ghmrn 19199 | The range of a homomorphis... |
| 0ghm 19200 | The constant zero linear f... |
| idghm 19201 | The identity homomorphism ... |
| resghm 19202 | Restriction of a homomorph... |
| resghm2 19203 | One direction of ~ resghm2... |
| resghm2b 19204 | Restriction of the codomai... |
| ghmghmrn 19205 | A group homomorphism from ... |
| ghmco 19206 | The composition of group h... |
| ghmima 19207 | The image of a subgroup un... |
| ghmpreima 19208 | The inverse image of a sub... |
| ghmeql 19209 | The equalizer of two group... |
| ghmnsgima 19210 | The image of a normal subg... |
| ghmnsgpreima 19211 | The inverse image of a nor... |
| ghmker 19212 | The kernel of a homomorphi... |
| ghmeqker 19213 | Two source points map to t... |
| pwsdiagghm 19214 | Diagonal homomorphism into... |
| f1ghm0to0 19215 | If a group homomorphism ` ... |
| ghmf1 19216 | Two ways of saying a group... |
| kerf1ghm 19217 | A group homomorphism ` F `... |
| ghmf1o 19218 | A bijective group homomorp... |
| conjghm 19219 | Conjugation is an automorp... |
| conjsubg 19220 | A conjugated subgroup is a... |
| conjsubgen 19221 | A conjugated subgroup is e... |
| conjnmz 19222 | A subgroup is unchanged un... |
| conjnmzb 19223 | Alternative condition for ... |
| conjnsg 19224 | A normal subgroup is uncha... |
| qusghm 19225 | If ` Y ` is a normal subgr... |
| ghmpropd 19226 | Group homomorphism depends... |
| gimfn 19231 | The group isomorphism func... |
| isgim 19232 | An isomorphism of groups i... |
| gimf1o 19233 | An isomorphism of groups i... |
| gimghm 19234 | An isomorphism of groups i... |
| isgim2 19235 | A group isomorphism is a h... |
| subggim 19236 | Behavior of subgroups unde... |
| gimcnv 19237 | The converse of a group is... |
| gimco 19238 | The composition of group i... |
| gim0to0 19239 | A group isomorphism maps t... |
| brgic 19240 | The relation "is isomorphi... |
| brgici 19241 | Prove isomorphic by an exp... |
| gicref 19242 | Isomorphism is reflexive. ... |
| giclcl 19243 | Isomorphism implies the le... |
| gicrcl 19244 | Isomorphism implies the ri... |
| gicsym 19245 | Isomorphism is symmetric. ... |
| gictr 19246 | Isomorphism is transitive.... |
| gicer 19247 | Isomorphism is an equivale... |
| gicen 19248 | Isomorphic groups have equ... |
| gicsubgen 19249 | A less trivial example of ... |
| ghmqusnsglem1 19250 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19251 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19252 | The mapping ` H ` induced ... |
| ghmquskerlem1 19253 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19254 | In the case of theorem ~ g... |
| ghmquskerlem2 19255 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19256 | The mapping ` H ` induced ... |
| ghmqusker 19257 | A surjective group homomor... |
| gicqusker 19258 | The image ` H ` of a group... |
| isga 19261 | The predicate "is a (left)... |
| gagrp 19262 | The left argument of a gro... |
| gaset 19263 | The right argument of a gr... |
| gagrpid 19264 | The identity of the group ... |
| gaf 19265 | The mapping of the group a... |
| gafo 19266 | A group action is onto its... |
| gaass 19267 | An "associative" property ... |
| ga0 19268 | The action of a group on t... |
| gaid 19269 | The trivial action of a gr... |
| subgga 19270 | A subgroup acts on its par... |
| gass 19271 | A subset of a group action... |
| gasubg 19272 | The restriction of a group... |
| gaid2 19273 | A group operation is a lef... |
| galcan 19274 | The action of a particular... |
| gacan 19275 | Group inverses cancel in a... |
| gapm 19276 | The action of a particular... |
| gaorb 19277 | The orbit equivalence rela... |
| gaorber 19278 | The orbit equivalence rela... |
| gastacl 19279 | The stabilizer subgroup in... |
| gastacos 19280 | Write the coset relation f... |
| orbstafun 19281 | Existence and uniqueness f... |
| orbstaval 19282 | Value of the function at a... |
| orbsta 19283 | The Orbit-Stabilizer theor... |
| orbsta2 19284 | Relation between the size ... |
| cntrval 19289 | Substitute definition of t... |
| cntzfval 19290 | First level substitution f... |
| cntzval 19291 | Definition substitution fo... |
| elcntz 19292 | Elementhood in the central... |
| cntzel 19293 | Membership in a centralize... |
| cntzsnval 19294 | Special substitution for t... |
| elcntzsn 19295 | Value of the centralizer o... |
| sscntz 19296 | A centralizer expression f... |
| cntzrcl 19297 | Reverse closure for elemen... |
| cntzssv 19298 | The centralizer is uncondi... |
| cntzi 19299 | Membership in a centralize... |
| elcntr 19300 | Elementhood in the center ... |
| cntrss 19301 | The center is a subset of ... |
| cntri 19302 | Defining property of the c... |
| resscntz 19303 | Centralizer in a substruct... |
| cntzsgrpcl 19304 | Centralizers are closed un... |
| cntz2ss 19305 | Centralizers reverse the s... |
| cntzrec 19306 | Reciprocity relationship f... |
| cntziinsn 19307 | Express any centralizer as... |
| cntzsubm 19308 | Centralizers in a monoid a... |
| cntzsubg 19309 | Centralizers in a group ar... |
| cntzidss 19310 | If the elements of ` S ` c... |
| cntzmhm 19311 | Centralizers in a monoid a... |
| cntzmhm2 19312 | Centralizers in a monoid a... |
| cntrsubgnsg 19313 | A central subgroup is norm... |
| cntrnsg 19314 | The center of a group is a... |
| oppgval 19317 | Value of the opposite grou... |
| oppgplusfval 19318 | Value of the addition oper... |
| oppgplus 19319 | Value of the addition oper... |
| setsplusg 19320 | The other components of an... |
| oppgbas 19321 | Base set of an opposite gr... |
| oppgtset 19322 | Topology of an opposite gr... |
| oppgtopn 19323 | Topology of an opposite gr... |
| oppgmnd 19324 | The opposite of a monoid i... |
| oppgmndb 19325 | Bidirectional form of ~ op... |
| oppgid 19326 | Zero in a monoid is a symm... |
| oppggrp 19327 | The opposite of a group is... |
| oppggrpb 19328 | Bidirectional form of ~ op... |
| oppginv 19329 | Inverses in a group are a ... |
| invoppggim 19330 | The inverse is an antiauto... |
| oppggic 19331 | Every group is (naturally)... |
| oppgsubm 19332 | Being a submonoid is a sym... |
| oppgsubg 19333 | Being a subgroup is a symm... |
| oppgcntz 19334 | A centralizer in a group i... |
| oppgcntr 19335 | The center of a group is t... |
| gsumwrev 19336 | A sum in an opposite monoi... |
| oppgle 19337 | less-than relation of an o... |
| oppglt 19338 | less-than relation of an o... |
| symgval 19341 | The value of the symmetric... |
| symgbas 19342 | The base set of the symmet... |
| elsymgbas2 19343 | Two ways of saying a funct... |
| elsymgbas 19344 | Two ways of saying a funct... |
| symgbasf1o 19345 | Elements in the symmetric ... |
| symgbasf 19346 | A permutation (element of ... |
| symgbasmap 19347 | A permutation (element of ... |
| symghash 19348 | The symmetric group on ` n... |
| symgbasfi 19349 | The symmetric group on a f... |
| symgfv 19350 | The function value of a pe... |
| symgfvne 19351 | The function values of a p... |
| symgressbas 19352 | The symmetric group on ` A... |
| symgplusg 19353 | The group operation of a s... |
| symgov 19354 | The value of the group ope... |
| symgcl 19355 | The group operation of the... |
| idresperm 19356 | The identity function rest... |
| symgmov1 19357 | For a permutation of a set... |
| symgmov2 19358 | For a permutation of a set... |
| symgbas0 19359 | The base set of the symmet... |
| symg1hash 19360 | The symmetric group on a s... |
| symg1bas 19361 | The symmetric group on a s... |
| symg2hash 19362 | The symmetric group on a (... |
| symg2bas 19363 | The symmetric group on a p... |
| 0symgefmndeq 19364 | The symmetric group on the... |
| snsymgefmndeq 19365 | The symmetric group on a s... |
| symgpssefmnd 19366 | For a set ` A ` with more ... |
| symgvalstruct 19367 | The value of the symmetric... |
| symgsubmefmnd 19368 | The symmetric group on a s... |
| symgtset 19369 | The topology of the symmet... |
| symggrp 19370 | The symmetric group on a s... |
| symgid 19371 | The group identity element... |
| symginv 19372 | The group inverse in the s... |
| symgsubmefmndALT 19373 | The symmetric group on a s... |
| galactghm 19374 | The currying of a group ac... |
| lactghmga 19375 | The converse of ~ galactgh... |
| symgtopn 19376 | The topology of the symmet... |
| symgga 19377 | The symmetric group induce... |
| pgrpsubgsymgbi 19378 | Every permutation group is... |
| pgrpsubgsymg 19379 | Every permutation group is... |
| idressubgsymg 19380 | The singleton containing o... |
| idrespermg 19381 | The structure with the sin... |
| cayleylem1 19382 | Lemma for ~ cayley . (Con... |
| cayleylem2 19383 | Lemma for ~ cayley . (Con... |
| cayley 19384 | Cayley's Theorem (construc... |
| cayleyth 19385 | Cayley's Theorem (existenc... |
| symgfix2 19386 | If a permutation does not ... |
| symgextf 19387 | The extension of a permuta... |
| symgextfv 19388 | The function value of the ... |
| symgextfve 19389 | The function value of the ... |
| symgextf1lem 19390 | Lemma for ~ symgextf1 . (... |
| symgextf1 19391 | The extension of a permuta... |
| symgextfo 19392 | The extension of a permuta... |
| symgextf1o 19393 | The extension of a permuta... |
| symgextsymg 19394 | The extension of a permuta... |
| symgextres 19395 | The restriction of the ext... |
| gsumccatsymgsn 19396 | Homomorphic property of co... |
| gsmsymgrfixlem1 19397 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19398 | The composition of permuta... |
| fvcosymgeq 19399 | The values of two composit... |
| gsmsymgreqlem1 19400 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19401 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19402 | Two combination of permuta... |
| symgfixelq 19403 | A permutation of a set fix... |
| symgfixels 19404 | The restriction of a permu... |
| symgfixelsi 19405 | The restriction of a permu... |
| symgfixf 19406 | The mapping of a permutati... |
| symgfixf1 19407 | The mapping of a permutati... |
| symgfixfolem1 19408 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19409 | The mapping of a permutati... |
| symgfixf1o 19410 | The mapping of a permutati... |
| f1omvdmvd 19413 | A permutation of any class... |
| f1omvdcnv 19414 | A permutation and its inve... |
| mvdco 19415 | Composing two permutations... |
| f1omvdconj 19416 | Conjugation of a permutati... |
| f1otrspeq 19417 | A transposition is charact... |
| f1omvdco2 19418 | If exactly one of two perm... |
| f1omvdco3 19419 | If a point is moved by exa... |
| pmtrfval 19420 | The function generating tr... |
| pmtrval 19421 | A generated transposition,... |
| pmtrfv 19422 | General value of mapping a... |
| pmtrprfv 19423 | In a transposition of two ... |
| pmtrprfv3 19424 | In a transposition of two ... |
| pmtrf 19425 | Functionality of a transpo... |
| pmtrmvd 19426 | A transposition moves prec... |
| pmtrrn 19427 | Transposing two points giv... |
| pmtrfrn 19428 | A transposition (as a kind... |
| pmtrffv 19429 | Mapping of a point under a... |
| pmtrrn2 19430 | For any transposition ther... |
| pmtrfinv 19431 | A transposition function i... |
| pmtrfmvdn0 19432 | A transposition moves at l... |
| pmtrff1o 19433 | A transposition function i... |
| pmtrfcnv 19434 | A transposition function i... |
| pmtrfb 19435 | An intrinsic characterizat... |
| pmtrfconj 19436 | Any conjugate of a transpo... |
| symgsssg 19437 | The symmetric group has su... |
| symgfisg 19438 | The symmetric group has a ... |
| symgtrf 19439 | Transpositions are element... |
| symggen 19440 | The span of the transposit... |
| symggen2 19441 | A finite permutation group... |
| symgtrinv 19442 | To invert a permutation re... |
| pmtr3ncomlem1 19443 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19444 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19445 | Transpositions over sets w... |
| pmtrdifellem1 19446 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19447 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19448 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19449 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19450 | A transposition of element... |
| pmtrdifwrdellem1 19451 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19452 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19453 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19454 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19455 | A sequence of transpositio... |
| pmtrdifwrdel2 19456 | A sequence of transpositio... |
| pmtrprfval 19457 | The transpositions on a pa... |
| pmtrprfvalrn 19458 | The range of the transposi... |
| psgnunilem1 19463 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19464 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19465 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19466 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19467 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19468 | Addition and subtraction o... |
| psgnuni 19469 | If the same permutation ca... |
| psgnfval 19470 | Function definition of the... |
| psgnfn 19471 | Functionality and domain o... |
| psgndmsubg 19472 | The finitary permutations ... |
| psgneldm 19473 | Property of being a finita... |
| psgneldm2 19474 | The finitary permutations ... |
| psgneldm2i 19475 | A sequence of transpositio... |
| psgneu 19476 | A finitary permutation has... |
| psgnval 19477 | Value of the permutation s... |
| psgnvali 19478 | A finitary permutation has... |
| psgnvalii 19479 | Any representation of a pe... |
| psgnpmtr 19480 | All transpositions are odd... |
| psgn0fv0 19481 | The permutation sign funct... |
| sygbasnfpfi 19482 | The class of non-fixed poi... |
| psgnfvalfi 19483 | Function definition of the... |
| psgnvalfi 19484 | Value of the permutation s... |
| psgnran 19485 | The range of the permutati... |
| gsmtrcl 19486 | The group sum of transposi... |
| psgnfitr 19487 | A permutation of a finite ... |
| psgnfieu 19488 | A permutation of a finite ... |
| pmtrsn 19489 | The value of the transposi... |
| psgnsn 19490 | The permutation sign funct... |
| psgnprfval 19491 | The permutation sign funct... |
| psgnprfval1 19492 | The permutation sign of th... |
| psgnprfval2 19493 | The permutation sign of th... |
| odfval 19502 | Value of the order functio... |
| odfvalALT 19503 | Shorter proof of ~ odfval ... |
| odval 19504 | Second substitution for th... |
| odlem1 19505 | The group element order is... |
| odcl 19506 | The order of a group eleme... |
| odf 19507 | Functionality of the group... |
| odid 19508 | Any element to the power o... |
| odlem2 19509 | Any positive annihilator o... |
| odmodnn0 19510 | Reduce the argument of a g... |
| mndodconglem 19511 | Lemma for ~ mndodcong . (... |
| mndodcong 19512 | If two multipliers are con... |
| mndodcongi 19513 | If two multipliers are con... |
| oddvdsnn0 19514 | The only multiples of ` A ... |
| odnncl 19515 | If a nonzero multiple of a... |
| odmod 19516 | Reduce the argument of a g... |
| oddvds 19517 | The only multiples of ` A ... |
| oddvdsi 19518 | Any group element is annih... |
| odcong 19519 | If two multipliers are con... |
| odeq 19520 | The ~ oddvds property uniq... |
| odval2 19521 | A non-conditional definiti... |
| odcld 19522 | The order of a group eleme... |
| odm1inv 19523 | The (order-1)th multiple o... |
| odmulgid 19524 | A relationship between the... |
| odmulg2 19525 | The order of a multiple di... |
| odmulg 19526 | Relationship between the o... |
| odmulgeq 19527 | A multiple of a point of f... |
| odbezout 19528 | If ` N ` is coprime to the... |
| od1 19529 | The order of the group ide... |
| odeq1 19530 | The group identity is the ... |
| odinv 19531 | The order of the inverse o... |
| odf1 19532 | The multiples of an elemen... |
| odinf 19533 | The multiples of an elemen... |
| dfod2 19534 | An alternative definition ... |
| odcl2 19535 | The order of an element of... |
| oddvds2 19536 | The order of an element of... |
| finodsubmsubg 19537 | A submonoid whose elements... |
| 0subgALT 19538 | A shorter proof of ~ 0subg... |
| submod 19539 | The order of an element is... |
| subgod 19540 | The order of an element is... |
| odsubdvds 19541 | The order of an element of... |
| odf1o1 19542 | An element with zero order... |
| odf1o2 19543 | An element with nonzero or... |
| odhash 19544 | An element of zero order g... |
| odhash2 19545 | If an element has nonzero ... |
| odhash3 19546 | An element which generates... |
| odngen 19547 | A cyclic subgroup of size ... |
| gexval 19548 | Value of the exponent of a... |
| gexlem1 19549 | The group element order is... |
| gexcl 19550 | The exponent of a group is... |
| gexid 19551 | Any element to the power o... |
| gexlem2 19552 | Any positive annihilator o... |
| gexdvdsi 19553 | Any group element is annih... |
| gexdvds 19554 | The only ` N ` that annihi... |
| gexdvds2 19555 | An integer divides the gro... |
| gexod 19556 | Any group element is annih... |
| gexcl3 19557 | If the order of every grou... |
| gexnnod 19558 | Every group element has fi... |
| gexcl2 19559 | The exponent of a finite g... |
| gexdvds3 19560 | The exponent of a finite g... |
| gex1 19561 | A group or monoid has expo... |
| ispgp 19562 | A group is a ` P ` -group ... |
| pgpprm 19563 | Reverse closure for the fi... |
| pgpgrp 19564 | Reverse closure for the se... |
| pgpfi1 19565 | A finite group with order ... |
| pgp0 19566 | The identity subgroup is a... |
| subgpgp 19567 | A subgroup of a p-group is... |
| sylow1lem1 19568 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19569 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19570 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19571 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19572 | Lemma for ~ sylow1 . Usin... |
| sylow1 19573 | Sylow's first theorem. If... |
| odcau 19574 | Cauchy's theorem for the o... |
| pgpfi 19575 | The converse to ~ pgpfi1 .... |
| pgpfi2 19576 | Alternate version of ~ pgp... |
| pgphash 19577 | The order of a p-group. (... |
| isslw 19578 | The property of being a Sy... |
| slwprm 19579 | Reverse closure for the fi... |
| slwsubg 19580 | A Sylow ` P ` -subgroup is... |
| slwispgp 19581 | Defining property of a Syl... |
| slwpss 19582 | A proper superset of a Syl... |
| slwpgp 19583 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19584 | Every ` P ` -subgroup is c... |
| slwn0 19585 | Every finite group contain... |
| subgslw 19586 | A Sylow subgroup that is c... |
| sylow2alem1 19587 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19588 | Lemma for ~ sylow2a . All... |
| sylow2a 19589 | A named lemma of Sylow's s... |
| sylow2blem1 19590 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19591 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19592 | Sylow's second theorem. P... |
| sylow2b 19593 | Sylow's second theorem. A... |
| slwhash 19594 | A sylow subgroup has cardi... |
| fislw 19595 | The sylow subgroups of a f... |
| sylow2 19596 | Sylow's second theorem. S... |
| sylow3lem1 19597 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19598 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19599 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19600 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19601 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19602 | Lemma for ~ sylow3 , secon... |
| sylow3 19603 | Sylow's third theorem. Th... |
| lsmfval 19608 | The subgroup sum function ... |
| lsmvalx 19609 | Subspace sum value (for a ... |
| lsmelvalx 19610 | Subspace sum membership (f... |
| lsmelvalix 19611 | Subspace sum membership (f... |
| oppglsm 19612 | The subspace sum operation... |
| lsmssv 19613 | Subgroup sum is a subset o... |
| lsmless1x 19614 | Subset implies subgroup su... |
| lsmless2x 19615 | Subset implies subgroup su... |
| lsmub1x 19616 | Subgroup sum is an upper b... |
| lsmub2x 19617 | Subgroup sum is an upper b... |
| lsmval 19618 | Subgroup sum value (for a ... |
| lsmelval 19619 | Subgroup sum membership (f... |
| lsmelvali 19620 | Subgroup sum membership (f... |
| lsmelvalm 19621 | Subgroup sum membership an... |
| lsmelvalmi 19622 | Membership of vector subtr... |
| lsmsubm 19623 | The sum of two commuting s... |
| lsmsubg 19624 | The sum of two commuting s... |
| lsmcom2 19625 | Subgroup sum commutes. (C... |
| smndlsmidm 19626 | The direct product is idem... |
| lsmub1 19627 | Subgroup sum is an upper b... |
| lsmub2 19628 | Subgroup sum is an upper b... |
| lsmunss 19629 | Union of subgroups is a su... |
| lsmless1 19630 | Subset implies subgroup su... |
| lsmless2 19631 | Subset implies subgroup su... |
| lsmless12 19632 | Subset implies subgroup su... |
| lsmidm 19633 | Subgroup sum is idempotent... |
| lsmlub 19634 | The least upper bound prop... |
| lsmss1 19635 | Subgroup sum with a subset... |
| lsmss1b 19636 | Subgroup sum with a subset... |
| lsmss2 19637 | Subgroup sum with a subset... |
| lsmss2b 19638 | Subgroup sum with a subset... |
| lsmass 19639 | Subgroup sum is associativ... |
| mndlsmidm 19640 | Subgroup sum is idempotent... |
| lsm01 19641 | Subgroup sum with the zero... |
| lsm02 19642 | Subgroup sum with the zero... |
| subglsm 19643 | The subgroup sum evaluated... |
| lssnle 19644 | Equivalent expressions for... |
| lsmmod 19645 | The modular law holds for ... |
| lsmmod2 19646 | Modular law dual for subgr... |
| lsmpropd 19647 | If two structures have the... |
| cntzrecd 19648 | Commute the "subgroups com... |
| lsmcntz 19649 | The "subgroups commute" pr... |
| lsmcntzr 19650 | The "subgroups commute" pr... |
| lsmdisj 19651 | Disjointness from a subgro... |
| lsmdisj2 19652 | Association of the disjoin... |
| lsmdisj3 19653 | Association of the disjoin... |
| lsmdisjr 19654 | Disjointness from a subgro... |
| lsmdisj2r 19655 | Association of the disjoin... |
| lsmdisj3r 19656 | Association of the disjoin... |
| lsmdisj2a 19657 | Association of the disjoin... |
| lsmdisj2b 19658 | Association of the disjoin... |
| lsmdisj3a 19659 | Association of the disjoin... |
| lsmdisj3b 19660 | Association of the disjoin... |
| subgdisj1 19661 | Vectors belonging to disjo... |
| subgdisj2 19662 | Vectors belonging to disjo... |
| subgdisjb 19663 | Vectors belonging to disjo... |
| pj1fval 19664 | The left projection functi... |
| pj1val 19665 | The left projection functi... |
| pj1eu 19666 | Uniqueness of a left proje... |
| pj1f 19667 | The left projection functi... |
| pj2f 19668 | The right projection funct... |
| pj1id 19669 | Any element of a direct su... |
| pj1eq 19670 | Any element of a direct su... |
| pj1lid 19671 | The left projection functi... |
| pj1rid 19672 | The left projection functi... |
| pj1ghm 19673 | The left projection functi... |
| pj1ghm2 19674 | The left projection functi... |
| lsmhash 19675 | The order of the direct pr... |
| efgmval 19682 | Value of the formal invers... |
| efgmf 19683 | The formal inverse operati... |
| efgmnvl 19684 | The inversion function on ... |
| efgrcl 19685 | Lemma for ~ efgval . (Con... |
| efglem 19686 | Lemma for ~ efgval . (Con... |
| efgval 19687 | Value of the free group co... |
| efger 19688 | Value of the free group co... |
| efgi 19689 | Value of the free group co... |
| efgi0 19690 | Value of the free group co... |
| efgi1 19691 | Value of the free group co... |
| efgtf 19692 | Value of the free group co... |
| efgtval 19693 | Value of the extension fun... |
| efgval2 19694 | Value of the free group co... |
| efgi2 19695 | Value of the free group co... |
| efgtlen 19696 | Value of the free group co... |
| efginvrel2 19697 | The inverse of the reverse... |
| efginvrel1 19698 | The inverse of the reverse... |
| efgsf 19699 | Value of the auxiliary fun... |
| efgsdm 19700 | Elementhood in the domain ... |
| efgsval 19701 | Value of the auxiliary fun... |
| efgsdmi 19702 | Property of the last link ... |
| efgsval2 19703 | Value of the auxiliary fun... |
| efgsrel 19704 | The start and end of any e... |
| efgs1 19705 | A singleton of an irreduci... |
| efgs1b 19706 | Every extension sequence e... |
| efgsp1 19707 | If ` F ` is an extension s... |
| efgsres 19708 | An initial segment of an e... |
| efgsfo 19709 | For any word, there is a s... |
| efgredlema 19710 | The reduced word that form... |
| efgredlemf 19711 | Lemma for ~ efgredleme . ... |
| efgredlemg 19712 | Lemma for ~ efgred . (Con... |
| efgredleme 19713 | Lemma for ~ efgred . (Con... |
| efgredlemd 19714 | The reduced word that form... |
| efgredlemc 19715 | The reduced word that form... |
| efgredlemb 19716 | The reduced word that form... |
| efgredlem 19717 | The reduced word that form... |
| efgred 19718 | The reduced word that form... |
| efgrelexlema 19719 | If two words ` A , B ` are... |
| efgrelexlemb 19720 | If two words ` A , B ` are... |
| efgrelex 19721 | If two words ` A , B ` are... |
| efgredeu 19722 | There is a unique reduced ... |
| efgred2 19723 | Two extension sequences ha... |
| efgcpbllema 19724 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19725 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19726 | Two extension sequences ha... |
| efgcpbl2 19727 | Two extension sequences ha... |
| frgpval 19728 | Value of the free group co... |
| frgpcpbl 19729 | Compatibility of the group... |
| frgp0 19730 | The free group is a group.... |
| frgpeccl 19731 | Closure of the quotient ma... |
| frgpgrp 19732 | The free group is a group.... |
| frgpadd 19733 | Addition in the free group... |
| frgpinv 19734 | The inverse of an element ... |
| frgpmhm 19735 | The "natural map" from wor... |
| vrgpfval 19736 | The canonical injection fr... |
| vrgpval 19737 | The value of the generatin... |
| vrgpf 19738 | The mapping from the index... |
| vrgpinv 19739 | The inverse of a generatin... |
| frgpuptf 19740 | Any assignment of the gene... |
| frgpuptinv 19741 | Any assignment of the gene... |
| frgpuplem 19742 | Any assignment of the gene... |
| frgpupf 19743 | Any assignment of the gene... |
| frgpupval 19744 | Any assignment of the gene... |
| frgpup1 19745 | Any assignment of the gene... |
| frgpup2 19746 | The evaluation map has the... |
| frgpup3lem 19747 | The evaluation map has the... |
| frgpup3 19748 | Universal property of the ... |
| 0frgp 19749 | The free group on zero gen... |
| isabl 19754 | The predicate "is an Abeli... |
| ablgrp 19755 | An Abelian group is a grou... |
| ablgrpd 19756 | An Abelian group is a grou... |
| ablcmn 19757 | An Abelian group is a comm... |
| ablcmnd 19758 | An Abelian group is a comm... |
| iscmn 19759 | The predicate "is a commut... |
| isabl2 19760 | The predicate "is an Abeli... |
| cmnpropd 19761 | If two structures have the... |
| ablpropd 19762 | If two structures have the... |
| ablprop 19763 | If two structures have the... |
| iscmnd 19764 | Properties that determine ... |
| isabld 19765 | Properties that determine ... |
| isabli 19766 | Properties that determine ... |
| cmnmnd 19767 | A commutative monoid is a ... |
| cmncom 19768 | A commutative monoid is co... |
| ablcom 19769 | An Abelian group operation... |
| cmn32 19770 | Commutative/associative la... |
| cmn4 19771 | Commutative/associative la... |
| cmn12 19772 | Commutative/associative la... |
| abl32 19773 | Commutative/associative la... |
| cmnmndd 19774 | A commutative monoid is a ... |
| cmnbascntr 19775 | The base set of a commutat... |
| rinvmod 19776 | Uniqueness of a right inve... |
| ablinvadd 19777 | The inverse of an Abelian ... |
| ablsub2inv 19778 | Abelian group subtraction ... |
| ablsubadd 19779 | Relationship between Abeli... |
| ablsub4 19780 | Commutative/associative su... |
| abladdsub4 19781 | Abelian group addition/sub... |
| abladdsub 19782 | Associative-type law for g... |
| ablsubadd23 19783 | Commutative/associative la... |
| ablsubaddsub 19784 | Double subtraction and add... |
| ablpncan2 19785 | Cancellation law for subtr... |
| ablpncan3 19786 | A cancellation law for Abe... |
| ablsubsub 19787 | Law for double subtraction... |
| ablsubsub4 19788 | Law for double subtraction... |
| ablpnpcan 19789 | Cancellation law for mixed... |
| ablnncan 19790 | Cancellation law for group... |
| ablsub32 19791 | Swap the second and third ... |
| ablnnncan 19792 | Cancellation law for group... |
| ablnnncan1 19793 | Cancellation law for group... |
| ablsubsub23 19794 | Swap subtrahend and result... |
| mulgnn0di 19795 | Group multiple of a sum, f... |
| mulgdi 19796 | Group multiple of a sum. ... |
| mulgmhm 19797 | The map from ` x ` to ` n ... |
| mulgghm 19798 | The map from ` x ` to ` n ... |
| mulgsubdi 19799 | Group multiple of a differ... |
| ghmfghm 19800 | The function fulfilling th... |
| ghmcmn 19801 | The image of a commutative... |
| ghmabl 19802 | The image of an abelian gr... |
| invghm 19803 | The inversion map is a gro... |
| eqgabl 19804 | Value of the subgroup cose... |
| qusecsub 19805 | Two subgroup cosets are eq... |
| subgabl 19806 | A subgroup of an abelian g... |
| subcmn 19807 | A submonoid of a commutati... |
| submcmn 19808 | A submonoid of a commutati... |
| submcmn2 19809 | A submonoid is commutative... |
| cntzcmn 19810 | The centralizer of any sub... |
| cntzcmnss 19811 | Any subset in a commutativ... |
| cntrcmnd 19812 | The center of a monoid is ... |
| cntrabl 19813 | The center of a group is a... |
| cntzspan 19814 | If the generators commute,... |
| cntzcmnf 19815 | Discharge the centralizer ... |
| ghmplusg 19816 | The pointwise sum of two l... |
| ablnsg 19817 | Every subgroup of an abeli... |
| odadd1 19818 | The order of a product in ... |
| odadd2 19819 | The order of a product in ... |
| odadd 19820 | The order of a product is ... |
| gex2abl 19821 | A group with exponent 2 (o... |
| gexexlem 19822 | Lemma for ~ gexex . (Cont... |
| gexex 19823 | In an abelian group with f... |
| torsubg 19824 | The set of all elements of... |
| oddvdssubg 19825 | The set of all elements wh... |
| lsmcomx 19826 | Subgroup sum commutes (ext... |
| ablcntzd 19827 | All subgroups in an abelia... |
| lsmcom 19828 | Subgroup sum commutes. (C... |
| lsmsubg2 19829 | The sum of two subgroups i... |
| lsm4 19830 | Commutative/associative la... |
| prdscmnd 19831 | The product of a family of... |
| prdsabld 19832 | The product of a family of... |
| pwscmn 19833 | The structure power on a c... |
| pwsabl 19834 | The structure power on an ... |
| qusabl 19835 | If ` Y ` is a subgroup of ... |
| abl1 19836 | The (smallest) structure r... |
| abln0 19837 | Abelian groups (and theref... |
| cnaddablx 19838 | The complex numbers are an... |
| cnaddabl 19839 | The complex numbers are an... |
| cnaddid 19840 | The group identity element... |
| cnaddinv 19841 | Value of the group inverse... |
| zaddablx 19842 | The integers are an Abelia... |
| frgpnabllem1 19843 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19844 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19845 | The free group on two or m... |
| imasabl 19846 | The image structure of an ... |
| iscyg 19849 | Definition of a cyclic gro... |
| iscyggen 19850 | The property of being a cy... |
| iscyggen2 19851 | The property of being a cy... |
| iscyg2 19852 | A cyclic group is a group ... |
| cyggeninv 19853 | The inverse of a cyclic ge... |
| cyggenod 19854 | An element is the generato... |
| cyggenod2 19855 | In an infinite cyclic grou... |
| iscyg3 19856 | Definition of a cyclic gro... |
| iscygd 19857 | Definition of a cyclic gro... |
| iscygodd 19858 | Show that a group with an ... |
| cycsubmcmn 19859 | The set of nonnegative int... |
| cyggrp 19860 | A cyclic group is a group.... |
| cygabl 19861 | A cyclic group is abelian.... |
| cygctb 19862 | A cyclic group is countabl... |
| 0cyg 19863 | The trivial group is cycli... |
| prmcyg 19864 | A group with prime order i... |
| lt6abl 19865 | A group with fewer than ` ... |
| ghmcyg 19866 | The image of a cyclic grou... |
| cyggex2 19867 | The exponent of a cyclic g... |
| cyggex 19868 | The exponent of a finite c... |
| cyggexb 19869 | A finite abelian group is ... |
| giccyg 19870 | Cyclicity is a group prope... |
| cycsubgcyg 19871 | The cyclic subgroup genera... |
| cycsubgcyg2 19872 | The cyclic subgroup genera... |
| gsumval3a 19873 | Value of the group sum ope... |
| gsumval3eu 19874 | The group sum as defined i... |
| gsumval3lem1 19875 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19876 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19877 | Value of the group sum ope... |
| gsumcllem 19878 | Lemma for ~ gsumcl and rel... |
| gsumzres 19879 | Extend a finite group sum ... |
| gsumzcl2 19880 | Closure of a finite group ... |
| gsumzcl 19881 | Closure of a finite group ... |
| gsumzf1o 19882 | Re-index a finite group su... |
| gsumres 19883 | Extend a finite group sum ... |
| gsumcl2 19884 | Closure of a finite group ... |
| gsumcl 19885 | Closure of a finite group ... |
| gsumf1o 19886 | Re-index a finite group su... |
| gsumreidx 19887 | Re-index a finite group su... |
| gsumzsubmcl 19888 | Closure of a group sum in ... |
| gsumsubmcl 19889 | Closure of a group sum in ... |
| gsumsubgcl 19890 | Closure of a group sum in ... |
| gsumzaddlem 19891 | The sum of two group sums.... |
| gsumzadd 19892 | The sum of two group sums.... |
| gsumadd 19893 | The sum of two group sums.... |
| gsummptfsadd 19894 | The sum of two group sums ... |
| gsummptfidmadd 19895 | The sum of two group sums ... |
| gsummptfidmadd2 19896 | The sum of two group sums ... |
| gsumzsplit 19897 | Split a group sum into two... |
| gsumsplit 19898 | Split a group sum into two... |
| gsumsplit2 19899 | Split a group sum into two... |
| gsummptfidmsplit 19900 | Split a group sum expresse... |
| gsummptfidmsplitres 19901 | Split a group sum expresse... |
| gsummptfzsplit 19902 | Split a group sum expresse... |
| gsummptfzsplitl 19903 | Split a group sum expresse... |
| gsumconst 19904 | Sum of a constant series. ... |
| gsumconstf 19905 | Sum of a constant series. ... |
| gsummptshft 19906 | Index shift of a finite gr... |
| gsumzmhm 19907 | Apply a group homomorphism... |
| gsummhm 19908 | Apply a group homomorphism... |
| gsummhm2 19909 | Apply a group homomorphism... |
| gsummptmhm 19910 | Apply a group homomorphism... |
| gsummulglem 19911 | Lemma for ~ gsummulg and ~... |
| gsummulg 19912 | Nonnegative multiple of a ... |
| gsummulgz 19913 | Integer multiple of a grou... |
| gsumzoppg 19914 | The opposite of a group su... |
| gsumzinv 19915 | Inverse of a group sum. (... |
| gsuminv 19916 | Inverse of a group sum. (... |
| gsummptfidminv 19917 | Inverse of a group sum exp... |
| gsumsub 19918 | The difference of two grou... |
| gsummptfssub 19919 | The difference of two grou... |
| gsummptfidmsub 19920 | The difference of two grou... |
| gsumsnfd 19921 | Group sum of a singleton, ... |
| gsumsnd 19922 | Group sum of a singleton, ... |
| gsumsnf 19923 | Group sum of a singleton, ... |
| gsumsn 19924 | Group sum of a singleton. ... |
| gsumpr 19925 | Group sum of a pair. (Con... |
| gsumzunsnd 19926 | Append an element to a fin... |
| gsumunsnfd 19927 | Append an element to a fin... |
| gsumunsnd 19928 | Append an element to a fin... |
| gsumunsnf 19929 | Append an element to a fin... |
| gsumunsn 19930 | Append an element to a fin... |
| gsumdifsnd 19931 | Extract a summand from a f... |
| gsumpt 19932 | Sum of a family that is no... |
| gsummptf1o 19933 | Re-index a finite group su... |
| gsummptun 19934 | Group sum of a disjoint un... |
| gsummpt1n0 19935 | If only one summand in a f... |
| gsummptif1n0 19936 | If only one summand in a f... |
| gsummptcl 19937 | Closure of a finite group ... |
| gsummptfif1o 19938 | Re-index a finite group su... |
| gsummptfzcl 19939 | Closure of a finite group ... |
| gsum2dlem1 19940 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19941 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19942 | Write a sum over a two-dim... |
| gsum2d2lem 19943 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19944 | Write a group sum over a t... |
| gsumcom2 19945 | Two-dimensional commutatio... |
| gsumxp 19946 | Write a group sum over a c... |
| gsumcom 19947 | Commute the arguments of a... |
| gsumcom3 19948 | A commutative law for fini... |
| gsumcom3fi 19949 | A commutative law for fini... |
| gsumxp2 19950 | Write a group sum over a c... |
| prdsgsum 19951 | Finite commutative sums in... |
| pwsgsum 19952 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19953 | Replacing a finitely suppo... |
| nn0gsumfz 19954 | Replacing a finitely suppo... |
| nn0gsumfz0 19955 | Replacing a finitely suppo... |
| gsummptnn0fz 19956 | A final group sum over a f... |
| gsummptnn0fzfv 19957 | A final group sum over a f... |
| telgsumfzslem 19958 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19959 | Telescoping group sum rang... |
| telgsumfz 19960 | Telescoping group sum rang... |
| telgsumfz0s 19961 | Telescoping finite group s... |
| telgsumfz0 19962 | Telescoping finite group s... |
| telgsums 19963 | Telescoping finitely suppo... |
| telgsum 19964 | Telescoping finitely suppo... |
| reldmdprd 19969 | The domain of the internal... |
| dmdprd 19970 | The domain of definition o... |
| dmdprdd 19971 | Show that a given family i... |
| dprddomprc 19972 | A family of subgroups inde... |
| dprddomcld 19973 | If a family of subgroups i... |
| dprdval0prc 19974 | The internal direct produc... |
| dprdval 19975 | The value of the internal ... |
| eldprd 19976 | A class ` A ` is an intern... |
| dprdgrp 19977 | Reverse closure for the in... |
| dprdf 19978 | The function ` S ` is a fa... |
| dprdf2 19979 | The function ` S ` is a fa... |
| dprdcntz 19980 | The function ` S ` is a fa... |
| dprddisj 19981 | The function ` S ` is a fa... |
| dprdw 19982 | The property of being a fi... |
| dprdwd 19983 | A mapping being a finitely... |
| dprdff 19984 | A finitely supported funct... |
| dprdfcl 19985 | A finitely supported funct... |
| dprdffsupp 19986 | A finitely supported funct... |
| dprdfcntz 19987 | A function on the elements... |
| dprdssv 19988 | The internal direct produc... |
| dprdfid 19989 | A function mapping all but... |
| eldprdi 19990 | The domain of definition o... |
| dprdfinv 19991 | Take the inverse of a grou... |
| dprdfadd 19992 | Take the sum of group sums... |
| dprdfsub 19993 | Take the difference of gro... |
| dprdfeq0 19994 | The zero function is the o... |
| dprdf11 19995 | Two group sums over a dire... |
| dprdsubg 19996 | The internal direct produc... |
| dprdub 19997 | Each factor is a subset of... |
| dprdlub 19998 | The direct product is smal... |
| dprdspan 19999 | The direct product is the ... |
| dprdres 20000 | Restriction of a direct pr... |
| dprdss 20001 | Create a direct product by... |
| dprdz 20002 | A family consisting entire... |
| dprd0 20003 | The empty family is an int... |
| dprdf1o 20004 | Rearrange the index set of... |
| dprdf1 20005 | Rearrange the index set of... |
| subgdmdprd 20006 | A direct product in a subg... |
| subgdprd 20007 | A direct product in a subg... |
| dprdsn 20008 | A singleton family is an i... |
| dmdprdsplitlem 20009 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 20010 | The function ` S ` is a fa... |
| dprddisj2 20011 | The function ` S ` is a fa... |
| dprd2dlem2 20012 | The direct product of a co... |
| dprd2dlem1 20013 | The direct product of a co... |
| dprd2da 20014 | The direct product of a co... |
| dprd2db 20015 | The direct product of a co... |
| dprd2d2 20016 | The direct product of a co... |
| dmdprdsplit2lem 20017 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 20018 | The direct product splits ... |
| dmdprdsplit 20019 | The direct product splits ... |
| dprdsplit 20020 | The direct product is the ... |
| dmdprdpr 20021 | A singleton family is an i... |
| dprdpr 20022 | A singleton family is an i... |
| dpjlem 20023 | Lemma for theorems about d... |
| dpjcntz 20024 | The two subgroups that app... |
| dpjdisj 20025 | The two subgroups that app... |
| dpjlsm 20026 | The two subgroups that app... |
| dpjfval 20027 | Value of the direct produc... |
| dpjval 20028 | Value of the direct produc... |
| dpjf 20029 | The ` X ` -th index projec... |
| dpjidcl 20030 | The key property of projec... |
| dpjeq 20031 | Decompose a group sum into... |
| dpjid 20032 | The key property of projec... |
| dpjlid 20033 | The ` X ` -th index projec... |
| dpjrid 20034 | The ` Y ` -th index projec... |
| dpjghm 20035 | The direct product is the ... |
| dpjghm2 20036 | The direct product is the ... |
| ablfacrplem 20037 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20038 | A finite abelian group who... |
| ablfacrp2 20039 | The factors ` K , L ` of ~... |
| ablfac1lem 20040 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20041 | The factors of ~ ablfac1b ... |
| ablfac1b 20042 | Any abelian group is the d... |
| ablfac1c 20043 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20044 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20045 | The factorization of ~ abl... |
| pgpfac1lem1 20046 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20047 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20048 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20049 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20050 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20051 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20052 | Factorization of a finite ... |
| pgpfaclem1 20053 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20054 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20055 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20056 | Full factorization of a fi... |
| ablfaclem1 20057 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20058 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20059 | Lemma for ~ ablfac . (Con... |
| ablfac 20060 | The Fundamental Theorem of... |
| ablfac2 20061 | Choose generators for each... |
| issimpg 20064 | The predicate "is a simple... |
| issimpgd 20065 | Deduce a simple group from... |
| simpggrp 20066 | A simple group is a group.... |
| simpggrpd 20067 | A simple group is a group.... |
| simpg2nsg 20068 | A simple group has two nor... |
| trivnsimpgd 20069 | Trivial groups are not sim... |
| simpgntrivd 20070 | Simple groups are nontrivi... |
| simpgnideld 20071 | A simple group contains a ... |
| simpgnsgd 20072 | The only normal subgroups ... |
| simpgnsgeqd 20073 | A normal subgroup of a sim... |
| 2nsgsimpgd 20074 | If any normal subgroup of ... |
| simpgnsgbid 20075 | A nontrivial group is simp... |
| ablsimpnosubgd 20076 | A subgroup of an abelian s... |
| ablsimpg1gend 20077 | An abelian simple group is... |
| ablsimpgcygd 20078 | An abelian simple group is... |
| ablsimpgfindlem1 20079 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20080 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20081 | Relationship between the o... |
| ablsimpgfind 20082 | An abelian simple group is... |
| fincygsubgd 20083 | The subgroup referenced in... |
| fincygsubgodd 20084 | Calculate the order of a s... |
| fincygsubgodexd 20085 | A finite cyclic group has ... |
| prmgrpsimpgd 20086 | A group of prime order is ... |
| ablsimpgprmd 20087 | An abelian simple group ha... |
| ablsimpgd 20088 | An abelian group is simple... |
| isomnd 20093 | A (left) ordered monoid is... |
| isogrp 20094 | A (left-)ordered group is ... |
| ogrpgrp 20095 | A left-ordered group is a ... |
| omndmnd 20096 | A left-ordered monoid is a... |
| omndtos 20097 | A left-ordered monoid is a... |
| omndadd 20098 | In an ordered monoid, the ... |
| omndaddr 20099 | In a right ordered monoid,... |
| omndadd2d 20100 | In a commutative left orde... |
| omndadd2rd 20101 | In a left- and right- orde... |
| submomnd 20102 | A submonoid of an ordered ... |
| omndmul2 20103 | In an ordered monoid, the ... |
| omndmul3 20104 | In an ordered monoid, the ... |
| omndmul 20105 | In a commutative ordered m... |
| ogrpinv0le 20106 | In an ordered group, the o... |
| ogrpsub 20107 | In an ordered group, the o... |
| ogrpaddlt 20108 | In an ordered group, stric... |
| ogrpaddltbi 20109 | In a right ordered group, ... |
| ogrpaddltrd 20110 | In a right ordered group, ... |
| ogrpaddltrbid 20111 | In a right ordered group, ... |
| ogrpsublt 20112 | In an ordered group, stric... |
| ogrpinv0lt 20113 | In an ordered group, the o... |
| ogrpinvlt 20114 | In an ordered group, the o... |
| gsumle 20115 | A finite sum in an ordered... |
| fnmgp 20118 | The multiplicative group o... |
| mgpval 20119 | Value of the multiplicatio... |
| mgpplusg 20120 | Value of the group operati... |
| mgpbas 20121 | Base set of the multiplica... |
| mgpsca 20122 | The multiplication monoid ... |
| mgptset 20123 | Topology component of the ... |
| mgptopn 20124 | Topology of the multiplica... |
| mgpds 20125 | Distance function of the m... |
| mgpress 20126 | Subgroup commutes with the... |
| prdsmgp 20127 | The multiplicative monoid ... |
| isrng 20130 | The predicate "is a non-un... |
| rngabl 20131 | A non-unital ring is an (a... |
| rngmgp 20132 | A non-unital ring is a sem... |
| rngmgpf 20133 | Restricted functionality o... |
| rnggrp 20134 | A non-unital ring is a (ad... |
| rngass 20135 | Associative law for the mu... |
| rngdi 20136 | Distributive law for the m... |
| rngdir 20137 | Distributive law for the m... |
| rngacl 20138 | Closure of the addition op... |
| rng0cl 20139 | The zero element of a non-... |
| rngcl 20140 | Closure of the multiplicat... |
| rnglz 20141 | The zero of a non-unital r... |
| rngrz 20142 | The zero of a non-unital r... |
| rngmneg1 20143 | Negation of a product in a... |
| rngmneg2 20144 | Negation of a product in a... |
| rngm2neg 20145 | Double negation of a produ... |
| rngansg 20146 | Every additive subgroup of... |
| rngsubdi 20147 | Ring multiplication distri... |
| rngsubdir 20148 | Ring multiplication distri... |
| isrngd 20149 | Properties that determine ... |
| rngpropd 20150 | If two structures have the... |
| prdsmulrngcl 20151 | Closure of the multiplicat... |
| prdsrngd 20152 | A product of non-unital ri... |
| imasrng 20153 | The image structure of a n... |
| imasrngf1 20154 | The image of a non-unital ... |
| xpsrngd 20155 | A product of two non-unita... |
| qusrng 20156 | The quotient structure of ... |
| ringidval 20159 | The value of the unity ele... |
| dfur2 20160 | The multiplicative identit... |
| ringurd 20161 | Deduce the unity element o... |
| issrg 20164 | The predicate "is a semiri... |
| srgcmn 20165 | A semiring is a commutativ... |
| srgmnd 20166 | A semiring is a monoid. (... |
| srgmgp 20167 | A semiring is a monoid und... |
| srgdilem 20168 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20169 | Closure of the multiplicat... |
| srgass 20170 | Associative law for the mu... |
| srgideu 20171 | The unity element of a sem... |
| srgfcl 20172 | Functionality of the multi... |
| srgdi 20173 | Distributive law for the m... |
| srgdir 20174 | Distributive law for the m... |
| srgidcl 20175 | The unity element of a sem... |
| srg0cl 20176 | The zero element of a semi... |
| srgidmlem 20177 | Lemma for ~ srglidm and ~ ... |
| srglidm 20178 | The unity element of a sem... |
| srgridm 20179 | The unity element of a sem... |
| issrgid 20180 | Properties showing that an... |
| srgacl 20181 | Closure of the addition op... |
| srgcom 20182 | Commutativity of the addit... |
| srgrz 20183 | The zero of a semiring is ... |
| srglz 20184 | The zero of a semiring is ... |
| srgisid 20185 | In a semiring, the only le... |
| o2timesd 20186 | An element of a ring-like ... |
| rglcom4d 20187 | Restricted commutativity o... |
| srgo2times 20188 | A semiring element plus it... |
| srgcom4lem 20189 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20190 | Restricted commutativity o... |
| srg1zr 20191 | The only semiring with a b... |
| srgen1zr 20192 | The only semiring with one... |
| srgmulgass 20193 | An associative property be... |
| srgpcomp 20194 | If two elements of a semir... |
| srgpcompp 20195 | If two elements of a semir... |
| srgpcomppsc 20196 | If two elements of a semir... |
| srglmhm 20197 | Left-multiplication in a s... |
| srgrmhm 20198 | Right-multiplication in a ... |
| srgsummulcr 20199 | A finite semiring sum mult... |
| sgsummulcl 20200 | A finite semiring sum mult... |
| srg1expzeq1 20201 | The exponentiation (by a n... |
| srgbinomlem1 20202 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20203 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20204 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20205 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20206 | Lemma for ~ srgbinom . In... |
| srgbinom 20207 | The binomial theorem for c... |
| csrgbinom 20208 | The binomial theorem for c... |
| isring 20213 | The predicate "is a (unita... |
| ringgrp 20214 | A ring is a group. (Contr... |
| ringmgp 20215 | A ring is a monoid under m... |
| iscrng 20216 | A commutative ring is a ri... |
| crngmgp 20217 | A commutative ring's multi... |
| ringgrpd 20218 | A ring is a group. (Contr... |
| ringmnd 20219 | A ring is a monoid under a... |
| ringmgm 20220 | A ring is a magma. (Contr... |
| crngring 20221 | A commutative ring is a ri... |
| crngringd 20222 | A commutative ring is a ri... |
| crnggrpd 20223 | A commutative ring is a gr... |
| mgpf 20224 | Restricted functionality o... |
| ringdilem 20225 | Properties of a unital rin... |
| ringcl 20226 | Closure of the multiplicat... |
| crngcom 20227 | A commutative ring's multi... |
| iscrng2 20228 | A commutative ring is a ri... |
| ringass 20229 | Associative law for multip... |
| ringideu 20230 | The unity element of a rin... |
| crngcomd 20231 | Multiplication is commutat... |
| crngbascntr 20232 | The base set of a commutat... |
| ringassd 20233 | Associative law for multip... |
| crng12d 20234 | Commutative/associative la... |
| crng32d 20235 | Commutative/associative la... |
| ringcld 20236 | Closure of the multiplicat... |
| ringdi 20237 | Distributive law for the m... |
| ringdir 20238 | Distributive law for the m... |
| ringdid 20239 | Distributive law for the m... |
| ringdird 20240 | Distributive law for the m... |
| ringidcl 20241 | The unity element of a rin... |
| ringidcld 20242 | The unity element of a rin... |
| ring0cl 20243 | The zero element of a ring... |
| ringidmlem 20244 | Lemma for ~ ringlidm and ~... |
| ringlidm 20245 | The unity element of a rin... |
| ringridm 20246 | The unity element of a rin... |
| isringid 20247 | Properties showing that an... |
| ringlidmd 20248 | The unity element of a rin... |
| ringridmd 20249 | The unity element of a rin... |
| ringid 20250 | The multiplication operati... |
| ringo2times 20251 | A ring element plus itself... |
| ringadd2 20252 | A ring element plus itself... |
| ringidss 20253 | A subset of the multiplica... |
| ringacl 20254 | Closure of the addition op... |
| ringcomlem 20255 | Lemma for ~ ringcom . Thi... |
| ringcom 20256 | Commutativity of the addit... |
| ringabl 20257 | A ring is an Abelian group... |
| ringcmn 20258 | A ring is a commutative mo... |
| ringabld 20259 | A ring is an Abelian group... |
| ringcmnd 20260 | A ring is a commutative mo... |
| ringrng 20261 | A unital ring is a non-uni... |
| ringssrng 20262 | The unital rings are non-u... |
| isringrng 20263 | The predicate "is a unital... |
| ringpropd 20264 | If two structures have the... |
| crngpropd 20265 | If two structures have the... |
| ringprop 20266 | If two structures have the... |
| isringd 20267 | Properties that determine ... |
| iscrngd 20268 | Properties that determine ... |
| ringlz 20269 | The zero of a unital ring ... |
| ringrz 20270 | The zero of a unital ring ... |
| ringlzd 20271 | The zero of a unital ring ... |
| ringrzd 20272 | The zero of a unital ring ... |
| ringsrg 20273 | Any ring is also a semirin... |
| ring1eq0 20274 | If one and zero are equal,... |
| ring1ne0 20275 | If a ring has at least two... |
| ringinvnz1ne0 20276 | In a unital ring, a left i... |
| ringinvnzdiv 20277 | In a unital ring, a left i... |
| ringnegl 20278 | Negation in a ring is the ... |
| ringnegr 20279 | Negation in a ring is the ... |
| ringmneg1 20280 | Negation of a product in a... |
| ringmneg2 20281 | Negation of a product in a... |
| ringm2neg 20282 | Double negation of a produ... |
| ringsubdi 20283 | Ring multiplication distri... |
| ringsubdir 20284 | Ring multiplication distri... |
| mulgass2 20285 | An associative property be... |
| ring1 20286 | The (smallest) structure r... |
| ringn0 20287 | Rings exist. (Contributed... |
| ringlghm 20288 | Left-multiplication in a r... |
| ringrghm 20289 | Right-multiplication in a ... |
| gsummulc1 20290 | A finite ring sum multipli... |
| gsummulc2 20291 | A finite ring sum multipli... |
| gsummgp0 20292 | If one factor in a finite ... |
| gsumdixp 20293 | Distribute a binary produc... |
| prdsmulrcl 20294 | A structure product of rin... |
| prdsringd 20295 | A product of rings is a ri... |
| prdscrngd 20296 | A product of commutative r... |
| prds1 20297 | Value of the ring unity in... |
| pwsring 20298 | A structure power of a rin... |
| pws1 20299 | Value of the ring unity in... |
| pwscrng 20300 | A structure power of a com... |
| pwsmgp 20301 | The multiplicative group o... |
| pwspjmhmmgpd 20302 | The projection given by ~ ... |
| pwsexpg 20303 | Value of a group exponenti... |
| pwsgprod 20304 | Finite products in a power... |
| imasring 20305 | The image structure of a r... |
| imasringf1 20306 | The image of a ring under ... |
| xpsringd 20307 | A product of two rings is ... |
| xpsring1d 20308 | The multiplicative identit... |
| qusring2 20309 | The quotient structure of ... |
| crngbinom 20310 | The binomial theorem for c... |
| opprval 20313 | Value of the opposite ring... |
| opprmulfval 20314 | Value of the multiplicatio... |
| opprmul 20315 | Value of the multiplicatio... |
| crngoppr 20316 | In a commutative ring, the... |
| opprlem 20317 | Lemma for ~ opprbas and ~ ... |
| opprbas 20318 | Base set of an opposite ri... |
| oppradd 20319 | Addition operation of an o... |
| opprrng 20320 | An opposite non-unital rin... |
| opprrngb 20321 | A class is a non-unital ri... |
| opprring 20322 | An opposite ring is a ring... |
| opprringb 20323 | Bidirectional form of ~ op... |
| oppr0 20324 | Additive identity of an op... |
| oppr1 20325 | Multiplicative identity of... |
| opprneg 20326 | The negative function in a... |
| opprsubg 20327 | Being a subgroup is a symm... |
| mulgass3 20328 | An associative property be... |
| reldvdsr 20335 | The divides relation is a ... |
| dvdsrval 20336 | Value of the divides relat... |
| dvdsr 20337 | Value of the divides relat... |
| dvdsr2 20338 | Value of the divides relat... |
| dvdsrmul 20339 | A left-multiple of ` X ` i... |
| dvdsrcl 20340 | Closure of a dividing elem... |
| dvdsrcl2 20341 | Closure of a dividing elem... |
| dvdsrid 20342 | An element in a (unital) r... |
| dvdsrtr 20343 | Divisibility is transitive... |
| dvdsrmul1 20344 | The divisibility relation ... |
| dvdsrneg 20345 | An element divides its neg... |
| dvdsr01 20346 | In a ring, zero is divisib... |
| dvdsr02 20347 | Only zero is divisible by ... |
| isunit 20348 | Property of being a unit o... |
| 1unit 20349 | The multiplicative identit... |
| unitcl 20350 | A unit is an element of th... |
| unitss 20351 | The set of units is contai... |
| opprunit 20352 | Being a unit is a symmetri... |
| crngunit 20353 | Property of being a unit i... |
| dvdsunit 20354 | A divisor of a unit is a u... |
| unitmulcl 20355 | The product of units is a ... |
| unitmulclb 20356 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20357 | The base set of the group ... |
| unitgrp 20358 | The group of units is a gr... |
| unitabl 20359 | The group of units of a co... |
| unitgrpid 20360 | The identity of the group ... |
| unitsubm 20361 | The group of units is a su... |
| invrfval 20364 | Multiplicative inverse fun... |
| unitinvcl 20365 | The inverse of a unit exis... |
| unitinvinv 20366 | The inverse of the inverse... |
| ringinvcl 20367 | The inverse of a unit is a... |
| unitlinv 20368 | A unit times its inverse i... |
| unitrinv 20369 | A unit times its inverse i... |
| 1rinv 20370 | The inverse of the ring un... |
| 0unit 20371 | The additive identity is a... |
| unitnegcl 20372 | The negative of a unit is ... |
| ringunitnzdiv 20373 | In a unitary ring, a unit ... |
| ring1nzdiv 20374 | In a unitary ring, the rin... |
| dvrfval 20377 | Division operation in a ri... |
| dvrval 20378 | Division operation in a ri... |
| dvrcl 20379 | Closure of division operat... |
| unitdvcl 20380 | The units are closed under... |
| dvrid 20381 | A ring element divided by ... |
| dvr1 20382 | A ring element divided by ... |
| dvrass 20383 | An associative law for div... |
| dvrcan1 20384 | A cancellation law for div... |
| dvrcan3 20385 | A cancellation law for div... |
| dvreq1 20386 | Equality in terms of ratio... |
| dvrdir 20387 | Distributive law for the d... |
| rdivmuldivd 20388 | Multiplication of two rati... |
| ringinvdv 20389 | Write the inverse function... |
| rngidpropd 20390 | The ring unity depends onl... |
| dvdsrpropd 20391 | The divisibility relation ... |
| unitpropd 20392 | The set of units depends o... |
| invrpropd 20393 | The ring inverse function ... |
| isirred 20394 | An irreducible element of ... |
| isnirred 20395 | The property of being a no... |
| isirred2 20396 | Expand out the class diffe... |
| opprirred 20397 | Irreducibility is symmetri... |
| irredn0 20398 | The additive identity is n... |
| irredcl 20399 | An irreducible element is ... |
| irrednu 20400 | An irreducible element is ... |
| irredn1 20401 | The multiplicative identit... |
| irredrmul 20402 | The product of an irreduci... |
| irredlmul 20403 | The product of a unit and ... |
| irredmul 20404 | If product of two elements... |
| irredneg 20405 | The negative of an irreduc... |
| irrednegb 20406 | An element is irreducible ... |
| rnghmrcl 20413 | Reverse closure of a non-u... |
| rnghmfn 20414 | The mapping of two non-uni... |
| rnghmval 20415 | The set of the non-unital ... |
| isrnghm 20416 | A function is a non-unital... |
| isrnghmmul 20417 | A function is a non-unital... |
| rnghmmgmhm 20418 | A non-unital ring homomorp... |
| rnghmval2 20419 | The non-unital ring homomo... |
| isrngim 20420 | An isomorphism of non-unit... |
| rngimrcl 20421 | Reverse closure for an iso... |
| rnghmghm 20422 | A non-unital ring homomorp... |
| rnghmf 20423 | A ring homomorphism is a f... |
| rnghmmul 20424 | A homomorphism of non-unit... |
| isrnghm2d 20425 | Demonstration of non-unita... |
| isrnghmd 20426 | Demonstration of non-unita... |
| rnghmf1o 20427 | A non-unital ring homomorp... |
| isrngim2 20428 | An isomorphism of non-unit... |
| rngimf1o 20429 | An isomorphism of non-unit... |
| rngimrnghm 20430 | An isomorphism of non-unit... |
| rngimcnv 20431 | The converse of an isomorp... |
| rnghmco 20432 | The composition of non-uni... |
| idrnghm 20433 | The identity homomorphism ... |
| c0mgm 20434 | The constant mapping to ze... |
| c0mhm 20435 | The constant mapping to ze... |
| c0ghm 20436 | The constant mapping to ze... |
| c0snmgmhm 20437 | The constant mapping to ze... |
| c0snmhm 20438 | The constant mapping to ze... |
| c0snghm 20439 | The constant mapping to ze... |
| rngisomfv1 20440 | If there is a non-unital r... |
| rngisom1 20441 | If there is a non-unital r... |
| rngisomring 20442 | If there is a non-unital r... |
| rngisomring1 20443 | If there is a non-unital r... |
| dfrhm2 20449 | The property of a ring hom... |
| rhmrcl1 20451 | Reverse closure of a ring ... |
| rhmrcl2 20452 | Reverse closure of a ring ... |
| isrhm 20453 | A function is a ring homom... |
| rhmmhm 20454 | A ring homomorphism is a h... |
| rhmisrnghm 20455 | Each unital ring homomorph... |
| rimrcl 20456 | Reverse closure for an iso... |
| isrim0 20457 | A ring isomorphism is a ho... |
| rhmghm 20458 | A ring homomorphism is an ... |
| rhmf 20459 | A ring homomorphism is a f... |
| rimcnv 20460 | The converse of a ring iso... |
| rhmmul 20461 | A homomorphism of rings pr... |
| isrhm2d 20462 | Demonstration of ring homo... |
| isrhmd 20463 | Demonstration of ring homo... |
| rhm1 20464 | Ring homomorphisms are req... |
| idrhm 20465 | The identity homomorphism ... |
| rhmf1o 20466 | A ring homomorphism is bij... |
| isrim 20467 | An isomorphism of rings is... |
| rimf1o 20468 | An isomorphism of rings is... |
| rimrhm 20469 | A ring isomorphism is a ho... |
| rimrcl1 20470 | Reverse closure of a ring ... |
| rimrcl2 20471 | Reverse closure of a ring ... |
| rimgim 20472 | An isomorphism of rings is... |
| rimisrngim 20473 | Each unital ring isomorphi... |
| rhmfn 20474 | The mapping of two rings t... |
| rhmval 20475 | The ring homomorphisms bet... |
| rhmco 20476 | The composition of ring ho... |
| pwsco1rhm 20477 | Right composition with a f... |
| pwsco2rhm 20478 | Left composition with a ri... |
| brric 20479 | The relation "is isomorphi... |
| brrici 20480 | Prove isomorphic by an exp... |
| ricsym 20481 | Ring isomorphism is symmet... |
| brric2 20482 | The relation "is isomorphi... |
| ricgic 20483 | If two rings are (ring) is... |
| rhmdvdsr 20484 | A ring homomorphism preser... |
| rhmopp 20485 | A ring homomorphism is als... |
| elrhmunit 20486 | Ring homomorphisms preserv... |
| rhmunitinv 20487 | Ring homomorphisms preserv... |
| isnzr 20490 | Property of a nonzero ring... |
| nzrnz 20491 | One and zero are different... |
| nzrring 20492 | A nonzero ring is a ring. ... |
| nzrringOLD 20493 | Obsolete version of ~ nzrr... |
| isnzr2 20494 | Equivalent characterizatio... |
| isnzr2hash 20495 | Equivalent characterizatio... |
| nzrpropd 20496 | If two structures have the... |
| opprnzrb 20497 | The opposite of a nonzero ... |
| opprnzr 20498 | The opposite of a nonzero ... |
| ringelnzr 20499 | A ring is nonzero if it ha... |
| nzrunit 20500 | A unit is nonzero in any n... |
| 0ringnnzr 20501 | A ring is a zero ring iff ... |
| 0ring 20502 | If a ring has only one ele... |
| 0ringdif 20503 | A zero ring is a ring whic... |
| 0ringbas 20504 | The base set of a zero rin... |
| 0ring01eq 20505 | In a ring with only one el... |
| 01eq0ring 20506 | If the zero and the identi... |
| 01eq0ringOLD 20507 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20508 | In a unital ring the zero ... |
| 0ring1eq0 20509 | In a zero ring, a ring whi... |
| c0rhm 20510 | The constant mapping to ze... |
| c0rnghm 20511 | The constant mapping to ze... |
| zrrnghm 20512 | The constant mapping to ze... |
| nrhmzr 20513 | There is no ring homomorph... |
| islring 20516 | The predicate "is a local ... |
| lringnzr 20517 | A local ring is a nonzero ... |
| lringring 20518 | A local ring is a ring. (... |
| lringnz 20519 | A local ring is a nonzero ... |
| lringuplu 20520 | If the sum of two elements... |
| issubrng 20523 | The subring of non-unital ... |
| subrngss 20524 | A subring is a subset. (C... |
| subrngid 20525 | Every non-unital ring is a... |
| subrngrng 20526 | A subring is a non-unital ... |
| subrngrcl 20527 | Reverse closure for a subr... |
| subrngsubg 20528 | A subring is a subgroup. ... |
| subrngringnsg 20529 | A subring is a normal subg... |
| subrngbas 20530 | Base set of a subring stru... |
| subrng0 20531 | A subring always has the s... |
| subrngacl 20532 | A subring is closed under ... |
| subrngmcl 20533 | A subring is closed under ... |
| issubrng2 20534 | Characterize the subrings ... |
| opprsubrng 20535 | Being a subring is a symme... |
| subrngint 20536 | The intersection of a none... |
| subrngin 20537 | The intersection of two su... |
| subrngmre 20538 | The subrings of a non-unit... |
| subsubrng 20539 | A subring of a subring is ... |
| subsubrng2 20540 | The set of subrings of a s... |
| rhmimasubrnglem 20541 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20542 | The homomorphic image of a... |
| cntzsubrng 20543 | Centralizers in a non-unit... |
| subrngpropd 20544 | If two structures have the... |
| issubrg 20547 | The subring predicate. (C... |
| subrgss 20548 | A subring is a subset. (C... |
| subrgid 20549 | Every ring is a subring of... |
| subrgring 20550 | A subring is a ring. (Con... |
| subrgcrng 20551 | A subring of a commutative... |
| subrgrcl 20552 | Reverse closure for a subr... |
| subrgsubg 20553 | A subring is a subgroup. ... |
| subrgsubrng 20554 | A subring of a unital ring... |
| subrg0 20555 | A subring always has the s... |
| subrg1cl 20556 | A subring contains the mul... |
| subrgbas 20557 | Base set of a subring stru... |
| subrg1 20558 | A subring always has the s... |
| subrgacl 20559 | A subring is closed under ... |
| subrgmcl 20560 | A subring is closed under ... |
| subrgsubm 20561 | A subring is a submonoid o... |
| subrgdvds 20562 | If an element divides anot... |
| subrguss 20563 | A unit of a subring is a u... |
| subrginv 20564 | A subring always has the s... |
| subrgdv 20565 | A subring always has the s... |
| subrgunit 20566 | An element of a ring is a ... |
| subrgugrp 20567 | The units of a subring for... |
| issubrg2 20568 | Characterize the subrings ... |
| opprsubrg 20569 | Being a subring is a symme... |
| subrgnzr 20570 | A subring of a nonzero rin... |
| subrgint 20571 | The intersection of a none... |
| subrgin 20572 | The intersection of two su... |
| subrgmre 20573 | The subrings of a ring are... |
| subsubrg 20574 | A subring of a subring is ... |
| subsubrg2 20575 | The set of subrings of a s... |
| issubrg3 20576 | A subring is an additive s... |
| resrhm 20577 | Restriction of a ring homo... |
| resrhm2b 20578 | Restriction of the codomai... |
| rhmeql 20579 | The equalizer of two ring ... |
| rhmima 20580 | The homomorphic image of a... |
| rnrhmsubrg 20581 | The range of a ring homomo... |
| cntzsubr 20582 | Centralizers in a ring are... |
| pwsdiagrhm 20583 | Diagonal homomorphism into... |
| subrgpropd 20584 | If two structures have the... |
| rhmpropd 20585 | Ring homomorphism depends ... |
| rgspnval 20588 | Value of the ring-span of ... |
| rgspncl 20589 | The ring-span of a set is ... |
| rgspnssid 20590 | The ring-span of a set con... |
| rgspnmin 20591 | The ring-span is contained... |
| rngcval 20594 | Value of the category of n... |
| rnghmresfn 20595 | The class of non-unital ri... |
| rnghmresel 20596 | An element of the non-unit... |
| rngcbas 20597 | Set of objects of the cate... |
| rngchomfval 20598 | Set of arrows of the categ... |
| rngchom 20599 | Set of arrows of the categ... |
| elrngchom 20600 | A morphism of non-unital r... |
| rngchomfeqhom 20601 | The functionalized Hom-set... |
| rngccofval 20602 | Composition in the categor... |
| rngcco 20603 | Composition in the categor... |
| dfrngc2 20604 | Alternate definition of th... |
| rnghmsscmap2 20605 | The non-unital ring homomo... |
| rnghmsscmap 20606 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20607 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20608 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20609 | The non-unital ring homomo... |
| rngccat 20610 | The category of non-unital... |
| rngcid 20611 | The identity arrow in the ... |
| rngcsect 20612 | A section in the category ... |
| rngcinv 20613 | An inverse in the category... |
| rngciso 20614 | An isomorphism in the cate... |
| rngcifuestrc 20615 | The "inclusion functor" fr... |
| funcrngcsetc 20616 | The "natural forgetful fun... |
| funcrngcsetcALT 20617 | Alternate proof of ~ funcr... |
| zrinitorngc 20618 | The zero ring is an initia... |
| zrtermorngc 20619 | The zero ring is a termina... |
| zrzeroorngc 20620 | The zero ring is a zero ob... |
| ringcval 20623 | Value of the category of u... |
| rhmresfn 20624 | The class of unital ring h... |
| rhmresel 20625 | An element of the unital r... |
| ringcbas 20626 | Set of objects of the cate... |
| ringchomfval 20627 | Set of arrows of the categ... |
| ringchom 20628 | Set of arrows of the categ... |
| elringchom 20629 | A morphism of unital rings... |
| ringchomfeqhom 20630 | The functionalized Hom-set... |
| ringccofval 20631 | Composition in the categor... |
| ringcco 20632 | Composition in the categor... |
| dfringc2 20633 | Alternate definition of th... |
| rhmsscmap2 20634 | The unital ring homomorphi... |
| rhmsscmap 20635 | The unital ring homomorphi... |
| rhmsubcsetclem1 20636 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20637 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20638 | The unital ring homomorphi... |
| ringccat 20639 | The category of unital rin... |
| ringcid 20640 | The identity arrow in the ... |
| rhmsscrnghm 20641 | The unital ring homomorphi... |
| rhmsubcrngclem1 20642 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20643 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20644 | The unital ring homomorphi... |
| rngcresringcat 20645 | The restriction of the cat... |
| ringcsect 20646 | A section in the category ... |
| ringcinv 20647 | An inverse in the category... |
| ringciso 20648 | An isomorphism in the cate... |
| ringcbasbas 20649 | An element of the base set... |
| funcringcsetc 20650 | The "natural forgetful fun... |
| zrtermoringc 20651 | The zero ring is a termina... |
| zrninitoringc 20652 | The zero ring is not an in... |
| srhmsubclem1 20653 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20654 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20655 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20656 | According to ~ df-subc , t... |
| sringcat 20657 | The restriction of the cat... |
| crhmsubc 20658 | According to ~ df-subc , t... |
| cringcat 20659 | The restriction of the cat... |
| rngcrescrhm 20660 | The category of non-unital... |
| rhmsubclem1 20661 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20662 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20663 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20664 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20665 | According to ~ df-subc , t... |
| rhmsubccat 20666 | The restriction of the cat... |
| rrgval 20673 | Value of the set or left-r... |
| isrrg 20674 | Membership in the set of l... |
| rrgeq0i 20675 | Property of a left-regular... |
| rrgeq0 20676 | Left-multiplication by a l... |
| rrgsupp 20677 | Left multiplication by a l... |
| rrgss 20678 | Left-regular elements are ... |
| unitrrg 20679 | Units are regular elements... |
| rrgnz 20680 | In a nonzero ring, the zer... |
| isdomn 20681 | Expand definition of a dom... |
| domnnzr 20682 | A domain is a nonzero ring... |
| domnring 20683 | A domain is a ring. (Cont... |
| domneq0 20684 | In a domain, a product is ... |
| domnmuln0 20685 | In a domain, a product of ... |
| isdomn5 20686 | The equivalence between th... |
| isdomn2 20687 | A ring is a domain iff all... |
| isdomn2OLD 20688 | Obsolete version of ~ isdo... |
| domnrrg 20689 | In a domain, a nonzero ele... |
| isdomn6 20690 | A ring is a domain iff the... |
| isdomn3 20691 | Nonzero elements form a mu... |
| isdomn4 20692 | A ring is a domain iff it ... |
| opprdomnb 20693 | A class is a domain if and... |
| opprdomn 20694 | The opposite of a domain i... |
| isdomn4r 20695 | A ring is a domain iff it ... |
| domnlcanb 20696 | Left-cancellation law for ... |
| domnlcan 20697 | Left-cancellation law for ... |
| domnrcanb 20698 | Right-cancellation law for... |
| domnrcan 20699 | Right-cancellation law for... |
| domneq0r 20700 | Right multiplication by a ... |
| isidom 20701 | An integral domain is a co... |
| idomdomd 20702 | An integral domain is a do... |
| idomcringd 20703 | An integral domain is a co... |
| idomringd 20704 | An integral domain is a ri... |
| isdrng 20709 | The predicate "is a divisi... |
| drngunit 20710 | Elementhood in the set of ... |
| drngui 20711 | The set of units of a divi... |
| drngring 20712 | A division ring is a ring.... |
| drngringd 20713 | A division ring is a ring.... |
| drnggrpd 20714 | A division ring is a group... |
| drnggrp 20715 | A division ring is a group... |
| isfld 20716 | A field is a commutative d... |
| flddrngd 20717 | A field is a division ring... |
| fldcrngd 20718 | A field is a commutative r... |
| isdrng2 20719 | A division ring can equiva... |
| drngprop 20720 | If two structures have the... |
| drngmgp 20721 | A division ring contains a... |
| drngid 20722 | A division ring's unity is... |
| drngunz 20723 | A division ring's unity is... |
| drngnzr 20724 | A division ring is a nonze... |
| drngdomn 20725 | A division ring is a domai... |
| drngmcl 20726 | The product of two nonzero... |
| drngmclOLD 20727 | Obsolete version of ~ drng... |
| drngid2 20728 | Properties showing that an... |
| drnginvrcl 20729 | Closure of the multiplicat... |
| drnginvrn0 20730 | The multiplicative inverse... |
| drnginvrcld 20731 | Closure of the multiplicat... |
| drnginvrl 20732 | Property of the multiplica... |
| drnginvrr 20733 | Property of the multiplica... |
| drnginvrld 20734 | Property of the multiplica... |
| drnginvrrd 20735 | Property of the multiplica... |
| drngmul0or 20736 | A product is zero iff one ... |
| drngmul0orOLD 20737 | Obsolete version of ~ drng... |
| drngmulne0 20738 | A product is nonzero iff b... |
| drngmuleq0 20739 | An element is zero iff its... |
| opprdrng 20740 | The opposite of a division... |
| isdrngd 20741 | Properties that characteri... |
| isdrngrd 20742 | Properties that characteri... |
| isdrngdOLD 20743 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20744 | Obsolete version of ~ isdr... |
| drngpropd 20745 | If two structures have the... |
| fldpropd 20746 | If two structures have the... |
| fldidom 20747 | A field is an integral dom... |
| fidomndrnglem 20748 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20749 | A finite domain is a divis... |
| fiidomfld 20750 | A finite integral domain i... |
| rng1nnzr 20751 | The (smallest) structure r... |
| ring1zr 20752 | The only (unital) ring wit... |
| rngen1zr 20753 | The only (unital) ring wit... |
| ringen1zr 20754 | The only unital ring with ... |
| rng1nfld 20755 | The zero ring is not a fie... |
| issubdrg 20756 | Characterize the subfields... |
| drhmsubc 20757 | According to ~ df-subc , t... |
| drngcat 20758 | The restriction of the cat... |
| fldcat 20759 | The restriction of the cat... |
| fldc 20760 | The restriction of the cat... |
| fldhmsubc 20761 | According to ~ df-subc , t... |
| issdrg 20764 | Property of a division sub... |
| sdrgrcl 20765 | Reverse closure for a sub-... |
| sdrgdrng 20766 | A sub-division-ring is a d... |
| sdrgsubrg 20767 | A sub-division-ring is a s... |
| sdrgid 20768 | Every division ring is a d... |
| sdrgss 20769 | A division subring is a su... |
| sdrgbas 20770 | Base set of a sub-division... |
| issdrg2 20771 | Property of a division sub... |
| sdrgunit 20772 | A unit of a sub-division-r... |
| imadrhmcl 20773 | The image of a (nontrivial... |
| fldsdrgfld 20774 | A sub-division-ring of a f... |
| acsfn1p 20775 | Construction of a closure ... |
| subrgacs 20776 | Closure property of subrin... |
| sdrgacs 20777 | Closure property of divisi... |
| cntzsdrg 20778 | Centralizers in division r... |
| subdrgint 20779 | The intersection of a none... |
| sdrgint 20780 | The intersection of a none... |
| primefld 20781 | The smallest sub division ... |
| primefld0cl 20782 | The prime field contains t... |
| primefld1cl 20783 | The prime field contains t... |
| abvfval 20786 | Value of the set of absolu... |
| isabv 20787 | Elementhood in the set of ... |
| isabvd 20788 | Properties that determine ... |
| abvrcl 20789 | Reverse closure for the ab... |
| abvfge0 20790 | An absolute value is a fun... |
| abvf 20791 | An absolute value is a fun... |
| abvcl 20792 | An absolute value is a fun... |
| abvge0 20793 | The absolute value of a nu... |
| abveq0 20794 | The value of an absolute v... |
| abvne0 20795 | The absolute value of a no... |
| abvgt0 20796 | The absolute value of a no... |
| abvmul 20797 | An absolute value distribu... |
| abvtri 20798 | An absolute value satisfie... |
| abv0 20799 | The absolute value of zero... |
| abv1z 20800 | The absolute value of one ... |
| abv1 20801 | The absolute value of one ... |
| abvneg 20802 | The absolute value of a ne... |
| abvsubtri 20803 | An absolute value satisfie... |
| abvrec 20804 | The absolute value distrib... |
| abvdiv 20805 | The absolute value distrib... |
| abvdom 20806 | Any ring with an absolute ... |
| abvres 20807 | The restriction of an abso... |
| abvtrivd 20808 | The trivial absolute value... |
| abvtrivg 20809 | The trivial absolute value... |
| abvtriv 20810 | The trivial absolute value... |
| abvpropd 20811 | If two structures have the... |
| abvn0b 20812 | Another characterization o... |
| staffval 20817 | The functionalization of t... |
| stafval 20818 | The functionalization of t... |
| staffn 20819 | The functionalization is e... |
| issrng 20820 | The predicate "is a star r... |
| srngrhm 20821 | The involution function in... |
| srngring 20822 | A star ring is a ring. (C... |
| srngcnv 20823 | The involution function in... |
| srngf1o 20824 | The involution function in... |
| srngcl 20825 | The involution function in... |
| srngnvl 20826 | The involution function in... |
| srngadd 20827 | The involution function in... |
| srngmul 20828 | The involution function in... |
| srng1 20829 | The conjugate of the ring ... |
| srng0 20830 | The conjugate of the ring ... |
| issrngd 20831 | Properties that determine ... |
| idsrngd 20832 | A commutative ring is a st... |
| isorng 20837 | An ordered ring is a ring ... |
| orngring 20838 | An ordered ring is a ring.... |
| orngogrp 20839 | An ordered ring is an orde... |
| isofld 20840 | An ordered field is a fiel... |
| orngmul 20841 | In an ordered ring, the or... |
| orngsqr 20842 | In an ordered ring, all sq... |
| ornglmulle 20843 | In an ordered ring, multip... |
| orngrmulle 20844 | In an ordered ring, multip... |
| ornglmullt 20845 | In an ordered ring, multip... |
| orngrmullt 20846 | In an ordered ring, multip... |
| orngmullt 20847 | In an ordered ring, the st... |
| ofldfld 20848 | An ordered field is a fiel... |
| ofldtos 20849 | An ordered field is a tota... |
| orng0le1 20850 | In an ordered ring, the ri... |
| ofldlt1 20851 | In an ordered field, the r... |
| suborng 20852 | Every subring of an ordere... |
| subofld 20853 | Every subfield of an order... |
| islmod 20858 | The predicate "is a left m... |
| lmodlema 20859 | Lemma for properties of a ... |
| islmodd 20860 | Properties that determine ... |
| lmodgrp 20861 | A left module is a group. ... |
| lmodring 20862 | The scalar component of a ... |
| lmodfgrp 20863 | The scalar component of a ... |
| lmodgrpd 20864 | A left module is a group. ... |
| lmodbn0 20865 | The base set of a left mod... |
| lmodacl 20866 | Closure of ring addition f... |
| lmodmcl 20867 | Closure of ring multiplica... |
| lmodsn0 20868 | The set of scalars in a le... |
| lmodvacl 20869 | Closure of vector addition... |
| lmodass 20870 | Left module vector sum is ... |
| lmodlcan 20871 | Left cancellation law for ... |
| lmodvscl 20872 | Closure of scalar product ... |
| lmodvscld 20873 | Closure of scalar product ... |
| scaffval 20874 | The scalar multiplication ... |
| scafval 20875 | The scalar multiplication ... |
| scafeq 20876 | If the scalar multiplicati... |
| scaffn 20877 | The scalar multiplication ... |
| lmodscaf 20878 | The scalar multiplication ... |
| lmodvsdi 20879 | Distributive law for scala... |
| lmodvsdir 20880 | Distributive law for scala... |
| lmodvsass 20881 | Associative law for scalar... |
| lmod0cl 20882 | The ring zero in a left mo... |
| lmod1cl 20883 | The ring unity in a left m... |
| lmodvs1 20884 | Scalar product with the ri... |
| lmod0vcl 20885 | The zero vector is a vecto... |
| lmod0vlid 20886 | Left identity law for the ... |
| lmod0vrid 20887 | Right identity law for the... |
| lmod0vid 20888 | Identity equivalent to the... |
| lmod0vs 20889 | Zero times a vector is the... |
| lmodvs0 20890 | Anything times the zero ve... |
| lmodvsmmulgdi 20891 | Distributive law for a gro... |
| lmodfopnelem1 20892 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20893 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20894 | The (functionalized) opera... |
| lcomf 20895 | A linear-combination sum i... |
| lcomfsupp 20896 | A linear-combination sum i... |
| lmodvnegcl 20897 | Closure of vector negative... |
| lmodvnegid 20898 | Addition of a vector with ... |
| lmodvneg1 20899 | Minus 1 times a vector is ... |
| lmodvsneg 20900 | Multiplication of a vector... |
| lmodvsubcl 20901 | Closure of vector subtract... |
| lmodcom 20902 | Left module vector sum is ... |
| lmodabl 20903 | A left module is an abelia... |
| lmodcmn 20904 | A left module is a commuta... |
| lmodnegadd 20905 | Distribute negation throug... |
| lmod4 20906 | Commutative/associative la... |
| lmodvsubadd 20907 | Relationship between vecto... |
| lmodvaddsub4 20908 | Vector addition/subtractio... |
| lmodvpncan 20909 | Addition/subtraction cance... |
| lmodvnpcan 20910 | Cancellation law for vecto... |
| lmodvsubval2 20911 | Value of vector subtractio... |
| lmodsubvs 20912 | Subtraction of a scalar pr... |
| lmodsubdi 20913 | Scalar multiplication dist... |
| lmodsubdir 20914 | Scalar multiplication dist... |
| lmodsubeq0 20915 | If the difference between ... |
| lmodsubid 20916 | Subtraction of a vector fr... |
| lmodvsghm 20917 | Scalar multiplication of t... |
| lmodprop2d 20918 | If two structures have the... |
| lmodpropd 20919 | If two structures have the... |
| gsumvsmul 20920 | Pull a scalar multiplicati... |
| mptscmfsupp0 20921 | A mapping to a scalar prod... |
| mptscmfsuppd 20922 | A function mapping to a sc... |
| rmodislmodlem 20923 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20924 | The right module ` R ` ind... |
| lssset 20927 | The set of all (not necess... |
| islss 20928 | The predicate "is a subspa... |
| islssd 20929 | Properties that determine ... |
| lssss 20930 | A subspace is a set of vec... |
| lssel 20931 | A subspace member is a vec... |
| lss1 20932 | The set of vectors in a le... |
| lssuni 20933 | The union of all subspaces... |
| lssn0 20934 | A subspace is not empty. ... |
| 00lss 20935 | The empty structure has no... |
| lsscl 20936 | Closure property of a subs... |
| lssvacl 20937 | Closure of vector addition... |
| lssvsubcl 20938 | Closure of vector subtract... |
| lssvancl1 20939 | Non-closure: if one vector... |
| lssvancl2 20940 | Non-closure: if one vector... |
| lss0cl 20941 | The zero vector belongs to... |
| lsssn0 20942 | The singleton of the zero ... |
| lss0ss 20943 | The zero subspace is inclu... |
| lssle0 20944 | No subspace is smaller tha... |
| lssne0 20945 | A nonzero subspace has a n... |
| lssvneln0 20946 | A vector ` X ` which doesn... |
| lssneln0 20947 | A vector ` X ` which doesn... |
| lssssr 20948 | Conclude subspace ordering... |
| lssvscl 20949 | Closure of scalar product ... |
| lssvnegcl 20950 | Closure of negative vector... |
| lsssubg 20951 | All subspaces are subgroup... |
| lsssssubg 20952 | All subspaces are subgroup... |
| islss3 20953 | A linear subspace of a mod... |
| lsslmod 20954 | A submodule is a module. ... |
| lsslss 20955 | The subspaces of a subspac... |
| islss4 20956 | A linear subspace is a sub... |
| lss1d 20957 | One-dimensional subspace (... |
| lssintcl 20958 | The intersection of a none... |
| lssincl 20959 | The intersection of two su... |
| lssmre 20960 | The subspaces of a module ... |
| lssacs 20961 | Submodules are an algebrai... |
| prdsvscacl 20962 | Pointwise scalar multiplic... |
| prdslmodd 20963 | The product of a family of... |
| pwslmod 20964 | A structure power of a lef... |
| lspfval 20967 | The span function for a le... |
| lspf 20968 | The span function on a lef... |
| lspval 20969 | The span of a set of vecto... |
| lspcl 20970 | The span of a set of vecto... |
| lspsncl 20971 | The span of a singleton is... |
| lspprcl 20972 | The span of a pair is a su... |
| lsptpcl 20973 | The span of an unordered t... |
| lspsnsubg 20974 | The span of a singleton is... |
| 00lsp 20975 | ~ fvco4i lemma for linear ... |
| lspid 20976 | The span of a subspace is ... |
| lspssv 20977 | A span is a set of vectors... |
| lspss 20978 | Span preserves subset orde... |
| lspssid 20979 | A set of vectors is a subs... |
| lspidm 20980 | The span of a set of vecto... |
| lspun 20981 | The span of union is the s... |
| lspssp 20982 | If a set of vectors is a s... |
| mrclsp 20983 | Moore closure generalizes ... |
| lspsnss 20984 | The span of the singleton ... |
| ellspsn3 20985 | A member of the span of th... |
| lspprss 20986 | The span of a pair of vect... |
| lspsnid 20987 | A vector belongs to the sp... |
| ellspsn6 20988 | Relationship between a vec... |
| ellspsn5b 20989 | Relationship between a vec... |
| ellspsn5 20990 | Relationship between a vec... |
| lspprid1 20991 | A member of a pair of vect... |
| lspprid2 20992 | A member of a pair of vect... |
| lspprvacl 20993 | The sum of two vectors bel... |
| lssats2 20994 | A way to express atomistic... |
| ellspsni 20995 | A scalar product with a ve... |
| lspsn 20996 | Span of the singleton of a... |
| ellspsn 20997 | Member of span of the sing... |
| lspsnvsi 20998 | Span of a scalar product o... |
| lspsnss2 20999 | Comparable spans of single... |
| lspsnneg 21000 | Negation does not change t... |
| lspsnsub 21001 | Swapping subtraction order... |
| lspsn0 21002 | Span of the singleton of t... |
| lsp0 21003 | Span of the empty set. (C... |
| lspuni0 21004 | Union of the span of the e... |
| lspun0 21005 | The span of a union with t... |
| lspsneq0 21006 | Span of the singleton is t... |
| lspsneq0b 21007 | Equal singleton spans impl... |
| lmodindp1 21008 | Two independent (non-colin... |
| lsslsp 21009 | Spans in submodules corres... |
| lss0v 21010 | The zero vector in a submo... |
| lsspropd 21011 | If two structures have the... |
| lsppropd 21012 | If two structures have the... |
| reldmlmhm 21019 | Lemma for module homomorph... |
| lmimfn 21020 | Lemma for module isomorphi... |
| islmhm 21021 | Property of being a homomo... |
| islmhm3 21022 | Property of a module homom... |
| lmhmlem 21023 | Non-quantified consequence... |
| lmhmsca 21024 | A homomorphism of left mod... |
| lmghm 21025 | A homomorphism of left mod... |
| lmhmlmod2 21026 | A homomorphism of left mod... |
| lmhmlmod1 21027 | A homomorphism of left mod... |
| lmhmf 21028 | A homomorphism of left mod... |
| lmhmlin 21029 | A homomorphism of left mod... |
| lmodvsinv 21030 | Multiplication of a vector... |
| lmodvsinv2 21031 | Multiplying a negated vect... |
| islmhm2 21032 | A one-equation proof of li... |
| islmhmd 21033 | Deduction for a module hom... |
| 0lmhm 21034 | The constant zero linear f... |
| idlmhm 21035 | The identity function on a... |
| invlmhm 21036 | The negative function on a... |
| lmhmco 21037 | The composition of two mod... |
| lmhmplusg 21038 | The pointwise sum of two l... |
| lmhmvsca 21039 | The pointwise scalar produ... |
| lmhmf1o 21040 | A bijective module homomor... |
| lmhmima 21041 | The image of a subspace un... |
| lmhmpreima 21042 | The inverse image of a sub... |
| lmhmlsp 21043 | Homomorphisms preserve spa... |
| lmhmrnlss 21044 | The range of a homomorphis... |
| lmhmkerlss 21045 | The kernel of a homomorphi... |
| reslmhm 21046 | Restriction of a homomorph... |
| reslmhm2 21047 | Expansion of the codomain ... |
| reslmhm2b 21048 | Expansion of the codomain ... |
| lmhmeql 21049 | The equalizer of two modul... |
| lspextmo 21050 | A linear function is compl... |
| pwsdiaglmhm 21051 | Diagonal homomorphism into... |
| pwssplit0 21052 | Splitting for structure po... |
| pwssplit1 21053 | Splitting for structure po... |
| pwssplit2 21054 | Splitting for structure po... |
| pwssplit3 21055 | Splitting for structure po... |
| islmim 21056 | An isomorphism of left mod... |
| lmimf1o 21057 | An isomorphism of left mod... |
| lmimlmhm 21058 | An isomorphism of modules ... |
| lmimgim 21059 | An isomorphism of modules ... |
| islmim2 21060 | An isomorphism of left mod... |
| lmimcnv 21061 | The converse of a bijectiv... |
| brlmic 21062 | The relation "is isomorphi... |
| brlmici 21063 | Prove isomorphic by an exp... |
| lmiclcl 21064 | Isomorphism implies the le... |
| lmicrcl 21065 | Isomorphism implies the ri... |
| lmicsym 21066 | Module isomorphism is symm... |
| lmhmpropd 21067 | Module homomorphism depend... |
| islbs 21070 | The predicate " ` B ` is a... |
| lbsss 21071 | A basis is a set of vector... |
| lbsel 21072 | An element of a basis is a... |
| lbssp 21073 | The span of a basis is the... |
| lbsind 21074 | A basis is linearly indepe... |
| lbsind2 21075 | A basis is linearly indepe... |
| lbspss 21076 | No proper subset of a basi... |
| lsmcl 21077 | The sum of two subspaces i... |
| lsmspsn 21078 | Member of subspace sum of ... |
| lsmelval2 21079 | Subspace sum membership in... |
| lsmsp 21080 | Subspace sum in terms of s... |
| lsmsp2 21081 | Subspace sum of spans of s... |
| lsmssspx 21082 | Subspace sum (in its exten... |
| lsmpr 21083 | The span of a pair of vect... |
| lsppreli 21084 | A vector expressed as a su... |
| lsmelpr 21085 | Two ways to say that a vec... |
| lsppr0 21086 | The span of a vector paire... |
| lsppr 21087 | Span of a pair of vectors.... |
| lspprel 21088 | Member of the span of a pa... |
| lspprabs 21089 | Absorption of vector sum i... |
| lspvadd 21090 | The span of a vector sum i... |
| lspsntri 21091 | Triangle-type inequality f... |
| lspsntrim 21092 | Triangle-type inequality f... |
| lbspropd 21093 | If two structures have the... |
| pj1lmhm 21094 | The left projection functi... |
| pj1lmhm2 21095 | The left projection functi... |
| islvec 21098 | The predicate "is a left v... |
| lvecdrng 21099 | The set of scalars of a le... |
| lveclmod 21100 | A left vector space is a l... |
| lveclmodd 21101 | A vector space is a left m... |
| lvecgrpd 21102 | A vector space is a group.... |
| lsslvec 21103 | A vector subspace is a vec... |
| lmhmlvec 21104 | The property for modules t... |
| lvecvs0or 21105 | If a scalar product is zer... |
| lvecvsn0 21106 | A scalar product is nonzer... |
| lssvs0or 21107 | If a scalar product belong... |
| lvecvscan 21108 | Cancellation law for scala... |
| lvecvscan2 21109 | Cancellation law for scala... |
| lvecinv 21110 | Invert coefficient of scal... |
| lspsnvs 21111 | A nonzero scalar product d... |
| lspsneleq 21112 | Membership relation that i... |
| lspsncmp 21113 | Comparable spans of nonzer... |
| lspsnne1 21114 | Two ways to express that v... |
| lspsnne2 21115 | Two ways to express that v... |
| lspsnnecom 21116 | Swap two vectors with diff... |
| lspabs2 21117 | Absorption law for span of... |
| lspabs3 21118 | Absorption law for span of... |
| lspsneq 21119 | Equal spans of singletons ... |
| lspsneu 21120 | Nonzero vectors with equal... |
| ellspsn4 21121 | A member of the span of th... |
| lspdisj 21122 | The span of a vector not i... |
| lspdisjb 21123 | A nonzero vector is not in... |
| lspdisj2 21124 | Unequal spans are disjoint... |
| lspfixed 21125 | Show membership in the spa... |
| lspexch 21126 | Exchange property for span... |
| lspexchn1 21127 | Exchange property for span... |
| lspexchn2 21128 | Exchange property for span... |
| lspindpi 21129 | Partial independence prope... |
| lspindp1 21130 | Alternate way to say 3 vec... |
| lspindp2l 21131 | Alternate way to say 3 vec... |
| lspindp2 21132 | Alternate way to say 3 vec... |
| lspindp3 21133 | Independence of 2 vectors ... |
| lspindp4 21134 | (Partial) independence of ... |
| lvecindp 21135 | Compute the ` X ` coeffici... |
| lvecindp2 21136 | Sums of independent vector... |
| lspsnsubn0 21137 | Unequal singleton spans im... |
| lsmcv 21138 | Subspace sum has the cover... |
| lspsolvlem 21139 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21140 | If ` X ` is in the span of... |
| lssacsex 21141 | In a vector space, subspac... |
| lspsnat 21142 | There is no subspace stric... |
| lspsncv0 21143 | The span of a singleton co... |
| lsppratlem1 21144 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21145 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21146 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21147 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21148 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21149 | Lemma for ~ lspprat . Neg... |
| lspprat 21150 | A proper subspace of the s... |
| islbs2 21151 | An equivalent formulation ... |
| islbs3 21152 | An equivalent formulation ... |
| lbsacsbs 21153 | Being a basis in a vector ... |
| lvecdim 21154 | The dimension theorem for ... |
| lbsextlem1 21155 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21156 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21157 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21158 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21159 | For any linearly independe... |
| lbsext 21160 | For any linearly independe... |
| lbsexg 21161 | Every vector space has a b... |
| lbsex 21162 | Every vector space has a b... |
| lvecprop2d 21163 | If two structures have the... |
| lvecpropd 21164 | If two structures have the... |
| sraval 21169 | Lemma for ~ srabase throug... |
| sralem 21170 | Lemma for ~ srabase and si... |
| srabase 21171 | Base set of a subring alge... |
| sraaddg 21172 | Additive operation of a su... |
| sramulr 21173 | Multiplicative operation o... |
| srasca 21174 | The set of scalars of a su... |
| sravsca 21175 | The scalar product operati... |
| sraip 21176 | The inner product operatio... |
| sratset 21177 | Topology component of a su... |
| sratopn 21178 | Topology component of a su... |
| srads 21179 | Distance function of a sub... |
| sraring 21180 | Condition for a subring al... |
| sralmod 21181 | The subring algebra is a l... |
| sralmod0 21182 | The subring module inherit... |
| issubrgd 21183 | Prove a subring by closure... |
| rlmfn 21184 | ` ringLMod ` is a function... |
| rlmval 21185 | Value of the ring module. ... |
| rlmval2 21186 | Value of the ring module e... |
| rlmbas 21187 | Base set of the ring modul... |
| rlmplusg 21188 | Vector addition in the rin... |
| rlm0 21189 | Zero vector in the ring mo... |
| rlmsub 21190 | Subtraction in the ring mo... |
| rlmmulr 21191 | Ring multiplication in the... |
| rlmsca 21192 | Scalars in the ring module... |
| rlmsca2 21193 | Scalars in the ring module... |
| rlmvsca 21194 | Scalar multiplication in t... |
| rlmtopn 21195 | Topology component of the ... |
| rlmds 21196 | Metric component of the ri... |
| rlmlmod 21197 | The ring module is a modul... |
| rlmlvec 21198 | The ring module over a div... |
| rlmlsm 21199 | Subgroup sum of the ring m... |
| rlmvneg 21200 | Vector negation in the rin... |
| rlmscaf 21201 | Functionalized scalar mult... |
| ixpsnbasval 21202 | The value of an infinite C... |
| lidlval 21207 | Value of the set of ring i... |
| rspval 21208 | Value of the ring span fun... |
| lidlss 21209 | An ideal is a subset of th... |
| lidlssbas 21210 | The base set of the restri... |
| lidlbas 21211 | A (left) ideal of a ring i... |
| islidl 21212 | Predicate of being a (left... |
| rnglidlmcl 21213 | A (left) ideal containing ... |
| rngridlmcl 21214 | A right ideal (which is a ... |
| dflidl2rng 21215 | Alternate (the usual textb... |
| isridlrng 21216 | A right ideal is a left id... |
| lidl0cl 21217 | An ideal contains 0. (Con... |
| lidlacl 21218 | An ideal is closed under a... |
| lidlnegcl 21219 | An ideal contains negative... |
| lidlsubg 21220 | An ideal is a subgroup of ... |
| lidlsubcl 21221 | An ideal is closed under s... |
| lidlmcl 21222 | An ideal is closed under l... |
| lidl1el 21223 | An ideal contains 1 iff it... |
| dflidl2 21224 | Alternate (the usual textb... |
| lidl0ALT 21225 | Alternate proof for ~ lidl... |
| rnglidl0 21226 | Every non-unital ring cont... |
| lidl0 21227 | Every ring contains a zero... |
| lidl1ALT 21228 | Alternate proof for ~ lidl... |
| rnglidl1 21229 | The base set of every non-... |
| lidl1 21230 | Every ring contains a unit... |
| lidlacs 21231 | The ideal system is an alg... |
| rspcl 21232 | The span of a set of ring ... |
| rspssid 21233 | The span of a set of ring ... |
| rsp1 21234 | The span of the identity e... |
| rsp0 21235 | The span of the zero eleme... |
| rspssp 21236 | The ideal span of a set of... |
| elrspsn 21237 | Membership in a principal ... |
| mrcrsp 21238 | Moore closure generalizes ... |
| lidlnz 21239 | A nonzero ideal contains a... |
| drngnidl 21240 | A division ring has only t... |
| lidlrsppropd 21241 | The left ideals and ring s... |
| rnglidlmmgm 21242 | The multiplicative group o... |
| rnglidlmsgrp 21243 | The multiplicative group o... |
| rnglidlrng 21244 | A (left) ideal of a non-un... |
| lidlnsg 21245 | An ideal is a normal subgr... |
| 2idlval 21248 | Definition of a two-sided ... |
| isridl 21249 | A right ideal is a left id... |
| 2idlelb 21250 | Membership in a two-sided ... |
| 2idllidld 21251 | A two-sided ideal is a lef... |
| 2idlridld 21252 | A two-sided ideal is a rig... |
| df2idl2rng 21253 | Alternate (the usual textb... |
| df2idl2 21254 | Alternate (the usual textb... |
| ridl0 21255 | Every ring contains a zero... |
| ridl1 21256 | Every ring contains a unit... |
| 2idl0 21257 | Every ring contains a zero... |
| 2idl1 21258 | Every ring contains a unit... |
| 2idlss 21259 | A two-sided ideal is a sub... |
| 2idlbas 21260 | The base set of a two-side... |
| 2idlelbas 21261 | The base set of a two-side... |
| rng2idlsubrng 21262 | A two-sided ideal of a non... |
| rng2idlnsg 21263 | A two-sided ideal of a non... |
| rng2idl0 21264 | The zero (additive identit... |
| rng2idlsubgsubrng 21265 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21266 | A two-sided ideal of a non... |
| rng2idlsubg0 21267 | The zero (additive identit... |
| 2idlcpblrng 21268 | The coset equivalence rela... |
| 2idlcpbl 21269 | The coset equivalence rela... |
| qus2idrng 21270 | The quotient of a non-unit... |
| qus1 21271 | The multiplicative identit... |
| qusring 21272 | If ` S ` is a two-sided id... |
| qusrhm 21273 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21274 | The preimage of an ideal b... |
| kerlidl 21275 | The kernel of a ring homom... |
| qusmul2idl 21276 | Value of the ring operatio... |
| crngridl 21277 | In a commutative ring, the... |
| crng2idl 21278 | In a commutative ring, a t... |
| qusmulrng 21279 | Value of the multiplicatio... |
| quscrng 21280 | The quotient of a commutat... |
| qusmulcrng 21281 | Value of the ring operatio... |
| rhmqusnsg 21282 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21283 | The ring unity of a two-si... |
| rngqiprngghmlem1 21284 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21285 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21286 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21287 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21288 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21289 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21290 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21291 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21292 | The base set of the produc... |
| rngqiprng 21293 | The product of the quotien... |
| rngqiprngimf 21294 | ` F ` is a function from (... |
| rngqiprngimfv 21295 | The value of the function ... |
| rngqiprngghm 21296 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21297 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21298 | ` F ` is a function from (... |
| rngqiprnglin 21299 | ` F ` is linear with respe... |
| rngqiprngho 21300 | ` F ` is a homomorphism of... |
| rngqiprngim 21301 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21302 | The unity of a two-sided i... |
| rngringbdlem1 21303 | In a unital ring, the quot... |
| rngringbdlem2 21304 | A non-unital ring is unita... |
| rngringbd 21305 | A non-unital ring is unita... |
| ring2idlqus 21306 | For every unital ring ther... |
| ring2idlqusb 21307 | A non-unital ring is unita... |
| rngqiprngfulem1 21308 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21309 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21310 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21311 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21312 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21313 | The ring unity of the prod... |
| rngqiprngfu 21314 | The function value of ` F ... |
| rngqiprngu 21315 | If a non-unital ring has a... |
| ring2idlqus1 21316 | If a non-unital ring has a... |
| lpival 21321 | Value of the set of princi... |
| islpidl 21322 | Property of being a princi... |
| lpi0 21323 | The zero ideal is always p... |
| lpi1 21324 | The unit ideal is always p... |
| islpir 21325 | Principal ideal rings are ... |
| lpiss 21326 | Principal ideals are a sub... |
| islpir2 21327 | Principal ideal rings are ... |
| lpirring 21328 | Principal ideal rings are ... |
| drnglpir 21329 | Division rings are princip... |
| rspsn 21330 | Membership in principal id... |
| lidldvgen 21331 | An element generates an id... |
| lpigen 21332 | An ideal is principal iff ... |
| cnfldstr 21353 | The field of complex numbe... |
| cnfldex 21354 | The field of complex numbe... |
| cnfldbas 21355 | The base set of the field ... |
| mpocnfldadd 21356 | The addition operation of ... |
| cnfldadd 21357 | The addition operation of ... |
| mpocnfldmul 21358 | The multiplication operati... |
| cnfldmul 21359 | The multiplication operati... |
| cnfldcj 21360 | The conjugation operation ... |
| cnfldtset 21361 | The topology component of ... |
| cnfldle 21362 | The ordering of the field ... |
| cnfldds 21363 | The metric of the field of... |
| cnfldunif 21364 | The uniform structure comp... |
| cnfldfun 21365 | The field of complex numbe... |
| cnfldfunALT 21366 | The field of complex numbe... |
| xrsstr 21367 | The extended real structur... |
| xrsex 21368 | The extended real structur... |
| xrsadd 21369 | The addition operation of ... |
| xrsmul 21370 | The multiplication operati... |
| xrstset 21371 | The topology component of ... |
| cncrng 21372 | The complex numbers form a... |
| cnring 21373 | The complex numbers form a... |
| xrsmcmn 21374 | The "multiplicative group"... |
| cnfld0 21375 | Zero is the zero element o... |
| cnfld1 21376 | One is the unity element o... |
| cnfldneg 21377 | The additive inverse in th... |
| cnfldplusf 21378 | The functionalized additio... |
| cnfldsub 21379 | The subtraction operator i... |
| cndrng 21380 | The complex numbers form a... |
| cnflddiv 21381 | The division operation in ... |
| cnfldinv 21382 | The multiplicative inverse... |
| cnfldmulg 21383 | The group multiple functio... |
| cnfldexp 21384 | The exponentiation operato... |
| cnsrng 21385 | The complex numbers form a... |
| xrsmgm 21386 | The "additive group" of th... |
| xrsnsgrp 21387 | The "additive group" of th... |
| xrsmgmdifsgrp 21388 | The "additive group" of th... |
| xrsds 21389 | The metric of the extended... |
| xrsdsval 21390 | The metric of the extended... |
| xrsdsreval 21391 | The metric of the extended... |
| xrsdsreclblem 21392 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21393 | The metric of the extended... |
| cnsubmlem 21394 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21395 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21396 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 21397 | Lemma for ~ resubdrg and f... |
| qsubdrg 21398 | The rational numbers form ... |
| zsubrg 21399 | The integers form a subrin... |
| gzsubrg 21400 | The gaussian integers form... |
| nn0subm 21401 | The nonnegative integers f... |
| rege0subm 21402 | The nonnegative reals form... |
| absabv 21403 | The regular absolute value... |
| zsssubrg 21404 | The integers are a subset ... |
| qsssubdrg 21405 | The rational numbers are a... |
| cnsubrg 21406 | There are no subrings of t... |
| cnmgpabl 21407 | The unit group of the comp... |
| cnmgpid 21408 | The group identity element... |
| cnmsubglem 21409 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21410 | The positive reals form a ... |
| gzrngunitlem 21411 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21412 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21413 | Relate a group sum on ` CC... |
| regsumfsum 21414 | Relate a group sum on ` ( ... |
| expmhm 21415 | Exponentiation is a monoid... |
| nn0srg 21416 | The nonnegative integers f... |
| rge0srg 21417 | The nonnegative real numbe... |
| xrge0plusg 21418 | The additive law of the ex... |
| xrs1mnd 21419 | The extended real numbers,... |
| xrs10 21420 | The zero of the extended r... |
| xrs1cmn 21421 | The extended real numbers ... |
| xrge0subm 21422 | The nonnegative extended r... |
| xrge0cmn 21423 | The nonnegative extended r... |
| xrge0omnd 21424 | The nonnegative extended r... |
| zringcrng 21427 | The ring of integers is a ... |
| zringring 21428 | The ring of integers is a ... |
| zringrng 21429 | The ring of integers is a ... |
| zringabl 21430 | The ring of integers is an... |
| zringgrp 21431 | The ring of integers is an... |
| zringbas 21432 | The integers are the base ... |
| zringplusg 21433 | The addition operation of ... |
| zringsub 21434 | The subtraction of element... |
| zringmulg 21435 | The multiplication (group ... |
| zringmulr 21436 | The multiplication operati... |
| zring0 21437 | The zero element of the ri... |
| zring1 21438 | The unity element of the r... |
| zringnzr 21439 | The ring of integers is a ... |
| dvdsrzring 21440 | Ring divisibility in the r... |
| zringlpirlem1 21441 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21442 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21443 | Lemma for ~ zringlpir . A... |
| zringinvg 21444 | The additive inverse of an... |
| zringunit 21445 | The units of ` ZZ ` are th... |
| zringlpir 21446 | The integers are a princip... |
| zringndrg 21447 | The integers are not a div... |
| zringcyg 21448 | The integers are a cyclic ... |
| zringsubgval 21449 | Subtraction in the ring of... |
| zringmpg 21450 | The multiplicative group o... |
| prmirredlem 21451 | A positive integer is irre... |
| dfprm2 21452 | The positive irreducible e... |
| prmirred 21453 | The irreducible elements o... |
| expghm 21454 | Exponentiation is a group ... |
| mulgghm2 21455 | The powers of a group elem... |
| mulgrhm 21456 | The powers of the element ... |
| mulgrhm2 21457 | The powers of the element ... |
| irinitoringc 21458 | The ring of integers is an... |
| nzerooringczr 21459 | There is no zero object in... |
| pzriprnglem1 21460 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21461 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21462 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21463 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21464 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21465 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21466 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21467 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21468 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21469 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21470 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21471 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21472 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21473 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21474 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21475 | The ring unity of the ring... |
| pzriprng 21476 | The non-unital ring ` ( ZZ... |
| pzriprng1 21477 | The ring unity of the ring... |
| zrhval 21486 | Define the unique homomorp... |
| zrhval2 21487 | Alternate value of the ` Z... |
| zrhmulg 21488 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21489 | The ` ZRHom ` homomorphism... |
| zrhrhm 21490 | The ` ZRHom ` homomorphism... |
| zrh1 21491 | Interpretation of 1 in a r... |
| zrh0 21492 | Interpretation of 0 in a r... |
| zrhpropd 21493 | The ` ZZ ` ring homomorphi... |
| zlmval 21494 | Augment an abelian group w... |
| zlmlem 21495 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21496 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21497 | Group operation of a ` ZZ ... |
| zlmmulr 21498 | Ring operation of a ` ZZ `... |
| zlmsca 21499 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21500 | Scalar multiplication oper... |
| zlmlmod 21501 | The ` ZZ ` -module operati... |
| chrval 21502 | Definition substitution of... |
| chrcl 21503 | Closure of the characteris... |
| chrid 21504 | The canonical ` ZZ ` ring ... |
| chrdvds 21505 | The ` ZZ ` ring homomorphi... |
| chrcong 21506 | If two integers are congru... |
| dvdschrmulg 21507 | In a ring, any multiple of... |
| fermltlchr 21508 | A generalization of Fermat... |
| chrnzr 21509 | Nonzero rings are precisel... |
| chrrhm 21510 | The characteristic restric... |
| domnchr 21511 | The characteristic of a do... |
| znlidl 21512 | The set ` n ZZ ` is an ide... |
| zncrng2 21513 | Making a commutative ring ... |
| znval 21514 | The value of the ` Z/nZ ` ... |
| znle 21515 | The value of the ` Z/nZ ` ... |
| znval2 21516 | Self-referential expressio... |
| znbaslem 21517 | Lemma for ~ znbas . (Cont... |
| znbas2 21518 | The base set of ` Z/nZ ` i... |
| znadd 21519 | The additive structure of ... |
| znmul 21520 | The multiplicative structu... |
| znzrh 21521 | The ` ZZ ` ring homomorphi... |
| znbas 21522 | The base set of ` Z/nZ ` s... |
| zncrng 21523 | ` Z/nZ ` is a commutative ... |
| znzrh2 21524 | The ` ZZ ` ring homomorphi... |
| znzrhval 21525 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21526 | The ` ZZ ` ring homomorphi... |
| zncyg 21527 | The group ` ZZ / n ZZ ` is... |
| zndvds 21528 | Express equality of equiva... |
| zndvds0 21529 | Special case of ~ zndvds w... |
| znf1o 21530 | The function ` F ` enumera... |
| zzngim 21531 | The ` ZZ ` ring homomorphi... |
| znle2 21532 | The ordering of the ` Z/nZ... |
| znleval 21533 | The ordering of the ` Z/nZ... |
| znleval2 21534 | The ordering of the ` Z/nZ... |
| zntoslem 21535 | Lemma for ~ zntos . (Cont... |
| zntos 21536 | The ` Z/nZ ` structure is ... |
| znhash 21537 | The ` Z/nZ ` structure has... |
| znfi 21538 | The ` Z/nZ ` structure is ... |
| znfld 21539 | The ` Z/nZ ` structure is ... |
| znidomb 21540 | The ` Z/nZ ` structure is ... |
| znchr 21541 | Cyclic rings are defined b... |
| znunit 21542 | The units of ` Z/nZ ` are ... |
| znunithash 21543 | The size of the unit group... |
| znrrg 21544 | The regular elements of ` ... |
| cygznlem1 21545 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21546 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21547 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21548 | A cyclic group with ` n ` ... |
| cygzn 21549 | A cyclic group with ` n ` ... |
| cygth 21550 | The "fundamental theorem o... |
| cyggic 21551 | Cyclic groups are isomorph... |
| frgpcyg 21552 | A free group is cyclic iff... |
| freshmansdream 21553 | For a prime number ` P ` ,... |
| frobrhm 21554 | In a commutative ring with... |
| ofldchr 21555 | The characteristic of an o... |
| cnmsgnsubg 21556 | The signs form a multiplic... |
| cnmsgnbas 21557 | The base set of the sign s... |
| cnmsgngrp 21558 | The group of signs under m... |
| psgnghm 21559 | The sign is a homomorphism... |
| psgnghm2 21560 | The sign is a homomorphism... |
| psgninv 21561 | The sign of a permutation ... |
| psgnco 21562 | Multiplicativity of the pe... |
| zrhpsgnmhm 21563 | Embedding of permutation s... |
| zrhpsgninv 21564 | The embedded sign of a per... |
| evpmss 21565 | Even permutations are perm... |
| psgnevpmb 21566 | A class is an even permuta... |
| psgnodpm 21567 | A permutation which is odd... |
| psgnevpm 21568 | A permutation which is eve... |
| psgnodpmr 21569 | If a permutation has sign ... |
| zrhpsgnevpm 21570 | The sign of an even permut... |
| zrhpsgnodpm 21571 | The sign of an odd permuta... |
| cofipsgn 21572 | Composition of any class `... |
| zrhpsgnelbas 21573 | Embedding of permutation s... |
| zrhcopsgnelbas 21574 | Embedding of permutation s... |
| evpmodpmf1o 21575 | The function for performin... |
| pmtrodpm 21576 | A transposition is an odd ... |
| psgnfix1 21577 | A permutation of a finite ... |
| psgnfix2 21578 | A permutation of a finite ... |
| psgndiflemB 21579 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21580 | Lemma 2 for ~ psgndif . (... |
| psgndif 21581 | Embedding of permutation s... |
| copsgndif 21582 | Embedding of permutation s... |
| rebase 21585 | The base of the field of r... |
| remulg 21586 | The multiplication (group ... |
| resubdrg 21587 | The real numbers form a di... |
| resubgval 21588 | Subtraction in the field o... |
| replusg 21589 | The addition operation of ... |
| remulr 21590 | The multiplication operati... |
| re0g 21591 | The zero element of the fi... |
| re1r 21592 | The unity element of the f... |
| rele2 21593 | The ordering relation of t... |
| relt 21594 | The ordering relation of t... |
| reds 21595 | The distance of the field ... |
| redvr 21596 | The division operation of ... |
| retos 21597 | The real numbers are a tot... |
| refld 21598 | The real numbers form a fi... |
| refldcj 21599 | The conjugation operation ... |
| resrng 21600 | The real numbers form a st... |
| regsumsupp 21601 | The group sum over the rea... |
| rzgrp 21602 | The quotient group ` RR / ... |
| isphl 21607 | The predicate "is a genera... |
| phllvec 21608 | A pre-Hilbert space is a l... |
| phllmod 21609 | A pre-Hilbert space is a l... |
| phlsrng 21610 | The scalar ring of a pre-H... |
| phllmhm 21611 | The inner product of a pre... |
| ipcl 21612 | Closure of the inner produ... |
| ipcj 21613 | Conjugate of an inner prod... |
| iporthcom 21614 | Orthogonality (meaning inn... |
| ip0l 21615 | Inner product with a zero ... |
| ip0r 21616 | Inner product with a zero ... |
| ipeq0 21617 | The inner product of a vec... |
| ipdir 21618 | Distributive law for inner... |
| ipdi 21619 | Distributive law for inner... |
| ip2di 21620 | Distributive law for inner... |
| ipsubdir 21621 | Distributive law for inner... |
| ipsubdi 21622 | Distributive law for inner... |
| ip2subdi 21623 | Distributive law for inner... |
| ipass 21624 | Associative law for inner ... |
| ipassr 21625 | "Associative" law for seco... |
| ipassr2 21626 | "Associative" law for inne... |
| ipffval 21627 | The inner product operatio... |
| ipfval 21628 | The inner product operatio... |
| ipfeq 21629 | If the inner product opera... |
| ipffn 21630 | The inner product operatio... |
| phlipf 21631 | The inner product operatio... |
| ip2eq 21632 | Two vectors are equal iff ... |
| isphld 21633 | Properties that determine ... |
| phlpropd 21634 | If two structures have the... |
| ssipeq 21635 | The inner product on a sub... |
| phssipval 21636 | The inner product on a sub... |
| phssip 21637 | The inner product (as a fu... |
| phlssphl 21638 | A subspace of an inner pro... |
| ocvfval 21645 | The orthocomplement operat... |
| ocvval 21646 | Value of the orthocompleme... |
| elocv 21647 | Elementhood in the orthoco... |
| ocvi 21648 | Property of a member of th... |
| ocvss 21649 | The orthocomplement of a s... |
| ocvocv 21650 | A set is contained in its ... |
| ocvlss 21651 | The orthocomplement of a s... |
| ocv2ss 21652 | Orthocomplements reverse s... |
| ocvin 21653 | An orthocomplement has tri... |
| ocvsscon 21654 | Two ways to say that ` S `... |
| ocvlsp 21655 | The orthocomplement of a l... |
| ocv0 21656 | The orthocomplement of the... |
| ocvz 21657 | The orthocomplement of the... |
| ocv1 21658 | The orthocomplement of the... |
| unocv 21659 | The orthocomplement of a u... |
| iunocv 21660 | The orthocomplement of an ... |
| cssval 21661 | The set of closed subspace... |
| iscss 21662 | The predicate "is a closed... |
| cssi 21663 | Property of a closed subsp... |
| cssss 21664 | A closed subspace is a sub... |
| iscss2 21665 | It is sufficient to prove ... |
| ocvcss 21666 | The orthocomplement of any... |
| cssincl 21667 | The zero subspace is a clo... |
| css0 21668 | The zero subspace is a clo... |
| css1 21669 | The whole space is a close... |
| csslss 21670 | A closed subspace of a pre... |
| lsmcss 21671 | A subset of a pre-Hilbert ... |
| cssmre 21672 | The closed subspaces of a ... |
| mrccss 21673 | The Moore closure correspo... |
| thlval 21674 | Value of the Hilbert latti... |
| thlbas 21675 | Base set of the Hilbert la... |
| thlle 21676 | Ordering on the Hilbert la... |
| thlleval 21677 | Ordering on the Hilbert la... |
| thloc 21678 | Orthocomplement on the Hil... |
| pjfval 21685 | The value of the projectio... |
| pjdm 21686 | A subspace is in the domai... |
| pjpm 21687 | The projection map is a pa... |
| pjfval2 21688 | Value of the projection ma... |
| pjval 21689 | Value of the projection ma... |
| pjdm2 21690 | A subspace is in the domai... |
| pjff 21691 | A projection is a linear o... |
| pjf 21692 | A projection is a function... |
| pjf2 21693 | A projection is a function... |
| pjfo 21694 | A projection is a surjecti... |
| pjcss 21695 | A projection subspace is a... |
| ocvpj 21696 | The orthocomplement of a p... |
| ishil 21697 | The predicate "is a Hilber... |
| ishil2 21698 | The predicate "is a Hilber... |
| isobs 21699 | The predicate "is an ortho... |
| obsip 21700 | The inner product of two e... |
| obsipid 21701 | A basis element has length... |
| obsrcl 21702 | Reverse closure for an ort... |
| obsss 21703 | An orthonormal basis is a ... |
| obsne0 21704 | A basis element is nonzero... |
| obsocv 21705 | An orthonormal basis has t... |
| obs2ocv 21706 | The double orthocomplement... |
| obselocv 21707 | A basis element is in the ... |
| obs2ss 21708 | A basis has no proper subs... |
| obslbs 21709 | An orthogonal basis is a l... |
| reldmdsmm 21712 | The direct sum is a well-b... |
| dsmmval 21713 | Value of the module direct... |
| dsmmbase 21714 | Base set of the module dir... |
| dsmmval2 21715 | Self-referential definitio... |
| dsmmbas2 21716 | Base set of the direct sum... |
| dsmmfi 21717 | For finite products, the d... |
| dsmmelbas 21718 | Membership in the finitely... |
| dsmm0cl 21719 | The all-zero vector is con... |
| dsmmacl 21720 | The finite hull is closed ... |
| prdsinvgd2 21721 | Negation of a single coord... |
| dsmmsubg 21722 | The finite hull of a produ... |
| dsmmlss 21723 | The finite hull of a produ... |
| dsmmlmod 21724 | The direct sum of a family... |
| frlmval 21727 | Value of the "free module"... |
| frlmlmod 21728 | The free module is a modul... |
| frlmpws 21729 | The free module as a restr... |
| frlmlss 21730 | The base set of the free m... |
| frlmpwsfi 21731 | The finite free module is ... |
| frlmsca 21732 | The ring of scalars of a f... |
| frlm0 21733 | Zero in a free module (rin... |
| frlmbas 21734 | Base set of the free modul... |
| frlmelbas 21735 | Membership in the base set... |
| frlmrcl 21736 | If a free module is inhabi... |
| frlmbasfsupp 21737 | Elements of the free modul... |
| frlmbasmap 21738 | Elements of the free modul... |
| frlmbasf 21739 | Elements of the free modul... |
| frlmlvec 21740 | The free module over a div... |
| frlmfibas 21741 | The base set of the finite... |
| elfrlmbasn0 21742 | If the dimension of a free... |
| frlmplusgval 21743 | Addition in a free module.... |
| frlmsubgval 21744 | Subtraction in a free modu... |
| frlmvscafval 21745 | Scalar multiplication in a... |
| frlmvplusgvalc 21746 | Coordinates of a sum with ... |
| frlmvscaval 21747 | Coordinates of a scalar mu... |
| frlmplusgvalb 21748 | Addition in a free module ... |
| frlmvscavalb 21749 | Scalar multiplication in a... |
| frlmvplusgscavalb 21750 | Addition combined with sca... |
| frlmgsum 21751 | Finite commutative sums in... |
| frlmsplit2 21752 | Restriction is homomorphic... |
| frlmsslss 21753 | A subset of a free module ... |
| frlmsslss2 21754 | A subset of a free module ... |
| frlmbas3 21755 | An element of the base set... |
| mpofrlmd 21756 | Elements of the free modul... |
| frlmip 21757 | The inner product of a fre... |
| frlmipval 21758 | The inner product of a fre... |
| frlmphllem 21759 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21760 | Conditions for a free modu... |
| uvcfval 21763 | Value of the unit-vector g... |
| uvcval 21764 | Value of a single unit vec... |
| uvcvval 21765 | Value of a unit vector coo... |
| uvcvvcl 21766 | A coordinate of a unit vec... |
| uvcvvcl2 21767 | A unit vector coordinate i... |
| uvcvv1 21768 | The unit vector is one at ... |
| uvcvv0 21769 | The unit vector is zero at... |
| uvcff 21770 | Domain and codomain of the... |
| uvcf1 21771 | In a nonzero ring, each un... |
| uvcresum 21772 | Any element of a free modu... |
| frlmssuvc1 21773 | A scalar multiple of a uni... |
| frlmssuvc2 21774 | A nonzero scalar multiple ... |
| frlmsslsp 21775 | A subset of a free module ... |
| frlmlbs 21776 | The unit vectors comprise ... |
| frlmup1 21777 | Any assignment of unit vec... |
| frlmup2 21778 | The evaluation map has the... |
| frlmup3 21779 | The range of such an evalu... |
| frlmup4 21780 | Universal property of the ... |
| ellspd 21781 | The elements of the span o... |
| elfilspd 21782 | Simplified version of ~ el... |
| rellindf 21787 | The independent-family pre... |
| islinds 21788 | Property of an independent... |
| linds1 21789 | An independent set of vect... |
| linds2 21790 | An independent set of vect... |
| islindf 21791 | Property of an independent... |
| islinds2 21792 | Expanded property of an in... |
| islindf2 21793 | Property of an independent... |
| lindff 21794 | Functional property of a l... |
| lindfind 21795 | A linearly independent fam... |
| lindsind 21796 | A linearly independent set... |
| lindfind2 21797 | In a linearly independent ... |
| lindsind2 21798 | In a linearly independent ... |
| lindff1 21799 | A linearly independent fam... |
| lindfrn 21800 | The range of an independen... |
| f1lindf 21801 | Rearranging and deleting e... |
| lindfres 21802 | Any restriction of an inde... |
| lindsss 21803 | Any subset of an independe... |
| f1linds 21804 | A family constructed from ... |
| islindf3 21805 | In a nonzero ring, indepen... |
| lindfmm 21806 | Linear independence of a f... |
| lindsmm 21807 | Linear independence of a s... |
| lindsmm2 21808 | The monomorphic image of a... |
| lsslindf 21809 | Linear independence is unc... |
| lsslinds 21810 | Linear independence is unc... |
| islbs4 21811 | A basis is an independent ... |
| lbslinds 21812 | A basis is independent. (... |
| islinds3 21813 | A subset is linearly indep... |
| islinds4 21814 | A set is independent in a ... |
| lmimlbs 21815 | The isomorphic image of a ... |
| lmiclbs 21816 | Having a basis is an isomo... |
| islindf4 21817 | A family is independent if... |
| islindf5 21818 | A family is independent if... |
| indlcim 21819 | An independent, spanning f... |
| lbslcic 21820 | A module with a basis is i... |
| lmisfree 21821 | A module has a basis iff i... |
| lvecisfrlm 21822 | Every vector space is isom... |
| lmimco 21823 | The composition of two iso... |
| lmictra 21824 | Module isomorphism is tran... |
| uvcf1o 21825 | In a nonzero ring, the map... |
| uvcendim 21826 | In a nonzero ring, the num... |
| frlmisfrlm 21827 | A free module is isomorphi... |
| frlmiscvec 21828 | Every free module is isomo... |
| isassa 21835 | The properties of an assoc... |
| assalem 21836 | The properties of an assoc... |
| assaass 21837 | Left-associative property ... |
| assaassr 21838 | Right-associative property... |
| assalmod 21839 | An associative algebra is ... |
| assaring 21840 | An associative algebra is ... |
| assasca 21841 | The scalars of an associat... |
| assa2ass 21842 | Left- and right-associativ... |
| assa2ass2 21843 | Left- and right-associativ... |
| isassad 21844 | Sufficient condition for b... |
| issubassa3 21845 | A subring that is also a s... |
| issubassa 21846 | The subalgebras of an asso... |
| sraassab 21847 | A subring algebra is an as... |
| sraassa 21848 | The subring algebra over a... |
| rlmassa 21849 | The ring module over a com... |
| assapropd 21850 | If two structures have the... |
| aspval 21851 | Value of the algebraic clo... |
| asplss 21852 | The algebraic span of a se... |
| aspid 21853 | The algebraic span of a su... |
| aspsubrg 21854 | The algebraic span of a se... |
| aspss 21855 | Span preserves subset orde... |
| aspssid 21856 | A set of vectors is a subs... |
| asclfval 21857 | Function value of the alge... |
| asclval 21858 | Value of a mapped algebra ... |
| asclfn 21859 | Unconditional functionalit... |
| asclf 21860 | The algebra scalar lifting... |
| asclghm 21861 | The algebra scalar lifting... |
| asclelbas 21862 | Lifted scalars are in the ... |
| ascl0 21863 | The scalar 0 embedded into... |
| ascl1 21864 | The scalar 1 embedded into... |
| asclmul1 21865 | Left multiplication by a l... |
| asclmul2 21866 | Right multiplication by a ... |
| ascldimul 21867 | The algebra scalar lifting... |
| asclinvg 21868 | The group inverse (negatio... |
| asclrhm 21869 | The algebra scalar lifting... |
| rnascl 21870 | The set of lifted scalars ... |
| issubassa2 21871 | A subring of a unital alge... |
| rnasclsubrg 21872 | The scalar multiples of th... |
| rnasclmulcl 21873 | (Vector) multiplication is... |
| rnasclassa 21874 | The scalar multiples of th... |
| ressascl 21875 | The lifting of scalars is ... |
| asclpropd 21876 | If two structures have the... |
| aspval2 21877 | The algebraic closure is t... |
| assamulgscmlem1 21878 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21879 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21880 | Exponentiation of a scalar... |
| asclmulg 21881 | Apply group multiplication... |
| zlmassa 21882 | The ` ZZ ` -module operati... |
| reldmpsr 21893 | The multivariate power ser... |
| psrval 21894 | Value of the multivariate ... |
| psrvalstr 21895 | The multivariate power ser... |
| psrbag 21896 | Elementhood in the set of ... |
| psrbagf 21897 | A finite bag is a function... |
| psrbagfsupp 21898 | Finite bags have finite su... |
| snifpsrbag 21899 | A bag containing one eleme... |
| fczpsrbag 21900 | The constant function equa... |
| psrbaglesupp 21901 | The support of a dominated... |
| psrbaglecl 21902 | The set of finite bags is ... |
| psrbagaddcl 21903 | The sum of two finite bags... |
| psrbagcon 21904 | The analogue of the statem... |
| psrbaglefi 21905 | There are finitely many ba... |
| psrbagconcl 21906 | The complement of a bag is... |
| psrbagleadd1 21907 | The analogue of " ` X <_ F... |
| psrbagconf1o 21908 | Bag complementation is a b... |
| psrbagres 21909 | Restrict a bag of variable... |
| gsumbagdiaglem 21910 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21911 | Two-dimensional commutatio... |
| psrass1lem 21912 | A group sum commutation us... |
| psrbas 21913 | The base set of the multiv... |
| psrelbas 21914 | An element of the set of p... |
| psrelbasfun 21915 | An element of the set of p... |
| psrplusg 21916 | The addition operation of ... |
| psradd 21917 | The addition operation of ... |
| psraddcl 21918 | Closure of the power serie... |
| rhmpsrlem1 21919 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21920 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21921 | The multiplication operati... |
| psrmulfval 21922 | The multiplication operati... |
| psrmulval 21923 | The multiplication operati... |
| psrmulcllem 21924 | Closure of the power serie... |
| psrmulcl 21925 | Closure of the power serie... |
| psrsca 21926 | The scalar field of the mu... |
| psrvscafval 21927 | The scalar multiplication ... |
| psrvsca 21928 | The scalar multiplication ... |
| psrvscaval 21929 | The scalar multiplication ... |
| psrvscacl 21930 | Closure of the power serie... |
| psr0cl 21931 | The zero element of the ri... |
| psr0lid 21932 | The zero element of the ri... |
| psrnegcl 21933 | The negative function in t... |
| psrlinv 21934 | The negative function in t... |
| psrgrp 21935 | The ring of power series i... |
| psr0 21936 | The zero element of the ri... |
| psrneg 21937 | The negative function of t... |
| psrlmod 21938 | The ring of power series i... |
| psr1cl 21939 | The identity element of th... |
| psrlidm 21940 | The identity element of th... |
| psrridm 21941 | The identity element of th... |
| psrass1 21942 | Associative identity for t... |
| psrdi 21943 | Distributive law for the r... |
| psrdir 21944 | Distributive law for the r... |
| psrass23l 21945 | Associative identity for t... |
| psrcom 21946 | Commutative law for the ri... |
| psrass23 21947 | Associative identities for... |
| psrring 21948 | The ring of power series i... |
| psr1 21949 | The identity element of th... |
| psrcrng 21950 | The ring of power series i... |
| psrassa 21951 | The ring of power series i... |
| resspsrbas 21952 | A restricted power series ... |
| resspsradd 21953 | A restricted power series ... |
| resspsrmul 21954 | A restricted power series ... |
| resspsrvsca 21955 | A restricted power series ... |
| subrgpsr 21956 | A subring of the base ring... |
| psrascl 21957 | Value of the scalar inject... |
| psrasclcl 21958 | A scalar is lifted into a ... |
| mvrfval 21959 | Value of the generating el... |
| mvrval 21960 | Value of the generating el... |
| mvrval2 21961 | Value of the generating el... |
| mvrid 21962 | The ` X i ` -th coefficien... |
| mvrf 21963 | The power series variable ... |
| mvrf1 21964 | The power series variable ... |
| mvrcl2 21965 | A power series variable is... |
| reldmmpl 21966 | The multivariate polynomia... |
| mplval 21967 | Value of the set of multiv... |
| mplbas 21968 | Base set of the set of mul... |
| mplelbas 21969 | Property of being a polyno... |
| mvrcl 21970 | A power series variable is... |
| mvrf2 21971 | The power series/polynomia... |
| mplrcl 21972 | Reverse closure for the po... |
| mplelsfi 21973 | A polynomial treated as a ... |
| mplval2 21974 | Self-referential expressio... |
| mplbasss 21975 | The set of polynomials is ... |
| mplelf 21976 | A polynomial is defined as... |
| mplsubglem 21977 | If ` A ` is an ideal of se... |
| mpllsslem 21978 | If ` A ` is an ideal of su... |
| mplsubglem2 21979 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21980 | The set of polynomials is ... |
| mpllss 21981 | The set of polynomials is ... |
| mplsubrglem 21982 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21983 | The set of polynomials is ... |
| mpl0 21984 | The zero polynomial. (Con... |
| mplplusg 21985 | Value of addition in a pol... |
| mplmulr 21986 | Value of multiplication in... |
| mpladd 21987 | The addition operation on ... |
| mplneg 21988 | The negative function on m... |
| mplmul 21989 | The multiplication operati... |
| mpl1 21990 | The identity element of th... |
| mplsca 21991 | The scalar field of a mult... |
| mplvsca2 21992 | The scalar multiplication ... |
| mplvsca 21993 | The scalar multiplication ... |
| mplvscaval 21994 | The scalar multiplication ... |
| mplgrp 21995 | The polynomial ring is a g... |
| mpllmod 21996 | The polynomial ring is a l... |
| mplring 21997 | The polynomial ring is a r... |
| mpllvec 21998 | The polynomial ring is a v... |
| mplcrng 21999 | The polynomial ring is a c... |
| mplassa 22000 | The polynomial ring is an ... |
| mplringd 22001 | The polynomial ring is a r... |
| mplcrngd 22002 | The polynomial ring is a c... |
| mpllmodd 22003 | The polynomial ring is a l... |
| mplascl0 22004 | The zero scalar as a polyn... |
| mplascl1 22005 | The one scalar as a polyno... |
| ressmplbas2 22006 | The base set of a restrict... |
| ressmplbas 22007 | A restricted polynomial al... |
| ressmpladd 22008 | A restricted polynomial al... |
| ressmplmul 22009 | A restricted polynomial al... |
| ressmplvsca 22010 | A restricted power series ... |
| subrgmpl 22011 | A subring of the base ring... |
| mplsubrgcl 22012 | An element of a polynomial... |
| subrgmvr 22013 | The variables in a subring... |
| subrgmvrf 22014 | The variables in a polynom... |
| mplmon 22015 | A monomial is a polynomial... |
| mplmonmul 22016 | The product of two monomia... |
| mplcoe1 22017 | Decompose a polynomial int... |
| mplcoe3 22018 | Decompose a monomial in on... |
| mplcoe5lem 22019 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 22020 | Decompose a monomial into ... |
| mplcoe2 22021 | Decompose a monomial into ... |
| mplbas2 22022 | An alternative expression ... |
| ltbval 22023 | Value of the well-order on... |
| ltbwe 22024 | The finite bag order is a ... |
| reldmopsr 22025 | Lemma for ordered power se... |
| opsrval 22026 | The value of the "ordered ... |
| opsrle 22027 | An alternative expression ... |
| opsrval2 22028 | Self-referential expressio... |
| opsrbaslem 22029 | Get a component of the ord... |
| opsrbas 22030 | The base set of the ordere... |
| opsrplusg 22031 | The addition operation of ... |
| opsrmulr 22032 | The multiplication operati... |
| opsrvsca 22033 | The scalar product operati... |
| opsrsca 22034 | The scalar ring of the ord... |
| opsrtoslem1 22035 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22036 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22037 | The ordered power series s... |
| opsrso 22038 | The ordered power series s... |
| opsrcrng 22039 | The ring of ordered power ... |
| opsrassa 22040 | The ring of ordered power ... |
| mplmon2 22041 | Express a scaled monomial.... |
| psrbag0 22042 | The empty bag is a bag. (... |
| psrbagsn 22043 | A singleton bag is a bag. ... |
| mplascl 22044 | Value of the scalar inject... |
| mplasclf 22045 | The scalar injection is a ... |
| subrgascl 22046 | The scalar injection funct... |
| subrgasclcl 22047 | The scalars in a polynomia... |
| mplmon2cl 22048 | A scaled monomial is a pol... |
| mplmon2mul 22049 | Product of scaled monomial... |
| mplind 22050 | Prove a property of polyno... |
| mplcoe4 22051 | Decompose a polynomial int... |
| evlslem4 22056 | The support of a tensor pr... |
| psrbagev1 22057 | A bag of multipliers provi... |
| psrbagev2 22058 | Closure of a sum using a b... |
| evlslem2 22059 | A linear function on the p... |
| evlslem3 22060 | Lemma for ~ evlseu . Poly... |
| evlslem6 22061 | Lemma for ~ evlseu . Fini... |
| evlslem1 22062 | Lemma for ~ evlseu , give ... |
| evlseu 22063 | For a given interpretation... |
| reldmevls 22064 | Well-behaved binary operat... |
| mpfrcl 22065 | Reverse closure for the se... |
| evlsval 22066 | Value of the polynomial ev... |
| evlsval2 22067 | Characterizing properties ... |
| evlsrhm 22068 | Polynomial evaluation is a... |
| evlsval3 22069 | Give a formula for the pol... |
| evlsvval 22070 | Give a formula for the eva... |
| evlsvvvallem 22071 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 22072 | Lemma for theorems using ~... |
| evlsvvval 22073 | Give a formula for the eva... |
| evlssca 22074 | Polynomial evaluation maps... |
| evlsvar 22075 | Polynomial evaluation maps... |
| evlsgsumadd 22076 | Polynomial evaluation maps... |
| evlsgsummul 22077 | Polynomial evaluation maps... |
| evlspw 22078 | Polynomial evaluation for ... |
| evlsvarpw 22079 | Polynomial evaluation for ... |
| evlval 22080 | Value of the simple/same r... |
| evlrhm 22081 | The simple evaluation map ... |
| evlcl 22082 | A polynomial over the ring... |
| evladdval 22083 | Polynomial evaluation buil... |
| evlmulval 22084 | Polynomial evaluation buil... |
| evlsscasrng 22085 | The evaluation of a scalar... |
| evlsca 22086 | Simple polynomial evaluati... |
| evlsvarsrng 22087 | The evaluation of the vari... |
| evlvar 22088 | Simple polynomial evaluati... |
| mpfconst 22089 | Constants are multivariate... |
| mpfproj 22090 | Projections are multivaria... |
| mpfsubrg 22091 | Polynomial functions are a... |
| mpff 22092 | Polynomial functions are f... |
| mpfaddcl 22093 | The sum of multivariate po... |
| mpfmulcl 22094 | The product of multivariat... |
| mpfind 22095 | Prove a property of polyno... |
| selvffval 22098 | Value of the "variable sel... |
| selvfval 22099 | Value of the "variable sel... |
| selvval 22100 | Value of the "variable sel... |
| mhmcompl 22101 | The composition of a monoi... |
| mplmapghm 22102 | The function ` H ` mapping... |
| mhmcoaddmpl 22103 | Show that the ring homomor... |
| rhmcomulmpl 22104 | Show that the ring homomor... |
| evlscl 22105 | A polynomial over the ring... |
| evlsscaval 22106 | Polynomial evaluation buil... |
| evlsvarval 22107 | Polynomial evaluation buil... |
| evlsexpval 22108 | Polynomial evaluation buil... |
| evlsaddval 22109 | Polynomial evaluation buil... |
| evlsmulval 22110 | Polynomial evaluation buil... |
| evlsmaprhm 22111 | The function ` F ` mapping... |
| evlsevl 22112 | Evaluation in a subring is... |
| evlvvval 22113 | Give a formula for the eva... |
| selvcllem1 22114 | ` T ` is an associative al... |
| selvcllem2 22115 | ` D ` is a ring homomorphi... |
| selvcllem3 22116 | The third argument passed ... |
| selvcllemh 22117 | Apply the third argument (... |
| selvcllem4 22118 | The fourth argument passed... |
| selvcllem5 22119 | The fifth argument passed ... |
| selvcl 22120 | Closure of the "variable s... |
| selvval2 22121 | Value of the "variable sel... |
| selvvvval 22122 | Recover the original polyn... |
| selvadd 22123 | The "variable selection" f... |
| selvmul 22124 | The "variable selection" f... |
| reldmmhp 22129 | The domain of the homogene... |
| mhpfval 22130 | Value of the "homogeneous ... |
| mhpval 22131 | Value of the "homogeneous ... |
| ismhp 22132 | Property of being a homoge... |
| ismhp2 22133 | Deduce a homogeneous polyn... |
| ismhp3 22134 | A polynomial is homogeneou... |
| mhprcl 22135 | Reverse closure for homoge... |
| mhpmpl 22136 | A homogeneous polynomial i... |
| mhpdeg 22137 | All nonzero terms of a hom... |
| mhp0cl 22138 | The zero polynomial is hom... |
| mhpsclcl 22139 | A scalar (or constant) pol... |
| mhpvarcl 22140 | A power series variable is... |
| mhpmulcl 22141 | A product of homogeneous p... |
| mhppwdeg 22142 | Degree of a homogeneous po... |
| mhpaddcl 22143 | Homogeneous polynomials ar... |
| mhpinvcl 22144 | Homogeneous polynomials ar... |
| mhpsubg 22145 | Homogeneous polynomials fo... |
| mhpvscacl 22146 | Homogeneous polynomials ar... |
| mhplss 22147 | Homogeneous polynomials fo... |
| psdffval 22149 | Value of the power series ... |
| psdfval 22150 | Give a map between power s... |
| psdval 22151 | Evaluate the partial deriv... |
| psdcoef 22152 | Coefficient of a term of t... |
| psdcl 22153 | The derivative of a power ... |
| psdmplcl 22154 | The derivative of a polyno... |
| psdadd 22155 | The derivative of a sum is... |
| psdvsca 22156 | The derivative of a scaled... |
| psdmullem 22157 | Lemma for ~ psdmul . Tran... |
| psdmul 22158 | Product rule for power ser... |
| psd1 22159 | The derivative of one is z... |
| psdascl 22160 | The derivative of a consta... |
| psdmvr 22161 | The partial derivative of ... |
| psdpw 22162 | Power rule for partial der... |
| psr1baslem 22174 | The set of finite bags on ... |
| psr1val 22175 | Value of the ring of univa... |
| psr1crng 22176 | The ring of univariate pow... |
| psr1assa 22177 | The ring of univariate pow... |
| psr1tos 22178 | The ordered power series s... |
| psr1bas2 22179 | The base set of the ring o... |
| psr1bas 22180 | The base set of the ring o... |
| vr1val 22181 | The value of the generator... |
| vr1cl2 22182 | The variable ` X ` is a me... |
| ply1val 22183 | The value of the set of un... |
| ply1bas 22184 | The value of the base set ... |
| ply1lss 22185 | Univariate polynomials for... |
| ply1subrg 22186 | Univariate polynomials for... |
| ply1crng 22187 | The ring of univariate pol... |
| ply1assa 22188 | The ring of univariate pol... |
| psr1bascl 22189 | A univariate power series ... |
| psr1basf 22190 | Univariate power series ba... |
| ply1basf 22191 | Univariate polynomial base... |
| ply1bascl 22192 | A univariate polynomial is... |
| ply1bascl2 22193 | A univariate polynomial is... |
| coe1fval 22194 | Value of the univariate po... |
| coe1fv 22195 | Value of an evaluated coef... |
| fvcoe1 22196 | Value of a multivariate co... |
| coe1fval3 22197 | Univariate power series co... |
| coe1f2 22198 | Functionality of univariat... |
| coe1fval2 22199 | Univariate polynomial coef... |
| coe1f 22200 | Functionality of univariat... |
| coe1fvalcl 22201 | A coefficient of a univari... |
| coe1sfi 22202 | Finite support of univaria... |
| coe1fsupp 22203 | The coefficient vector of ... |
| mptcoe1fsupp 22204 | A mapping involving coeffi... |
| coe1ae0 22205 | The coefficient vector of ... |
| vr1cl 22206 | The generator of a univari... |
| opsr0 22207 | Zero in the ordered power ... |
| opsr1 22208 | One in the ordered power s... |
| psr1plusg 22209 | Value of addition in a uni... |
| psr1vsca 22210 | Value of scalar multiplica... |
| psr1mulr 22211 | Value of multiplication in... |
| ply1plusg 22212 | Value of addition in a uni... |
| ply1vsca 22213 | Value of scalar multiplica... |
| ply1mulr 22214 | Value of multiplication in... |
| ply1ass23l 22215 | Associative identity with ... |
| ressply1bas2 22216 | The base set of a restrict... |
| ressply1bas 22217 | A restricted polynomial al... |
| ressply1add 22218 | A restricted polynomial al... |
| ressply1mul 22219 | A restricted polynomial al... |
| ressply1vsca 22220 | A restricted power series ... |
| subrgply1 22221 | A subring of the base ring... |
| gsumply1subr 22222 | Evaluate a group sum in a ... |
| psrbaspropd 22223 | Property deduction for pow... |
| psrplusgpropd 22224 | Property deduction for pow... |
| mplbaspropd 22225 | Property deduction for pol... |
| psropprmul 22226 | Reversing multiplication i... |
| ply1opprmul 22227 | Reversing multiplication i... |
| 00ply1bas 22228 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22229 | Protection compatibility o... |
| ply1plusgfvi 22230 | Protection compatibility o... |
| ply1baspropd 22231 | Property deduction for uni... |
| ply1plusgpropd 22232 | Property deduction for uni... |
| opsrring 22233 | Ordered power series form ... |
| opsrlmod 22234 | Ordered power series form ... |
| psr1ring 22235 | Univariate power series fo... |
| ply1ring 22236 | Univariate polynomials for... |
| psr1lmod 22237 | Univariate power series fo... |
| psr1sca 22238 | Scalars of a univariate po... |
| psr1sca2 22239 | Scalars of a univariate po... |
| ply1lmod 22240 | Univariate polynomials for... |
| ply1sca 22241 | Scalars of a univariate po... |
| ply1sca2 22242 | Scalars of a univariate po... |
| ply1ascl0 22243 | The zero scalar as a polyn... |
| ply1ascl1 22244 | The multiplicative identit... |
| ply1mpl0 22245 | The univariate polynomial ... |
| ply10s0 22246 | Zero times a univariate po... |
| ply1mpl1 22247 | The univariate polynomial ... |
| ply1ascl 22248 | The univariate polynomial ... |
| subrg1ascl 22249 | The scalar injection funct... |
| subrg1asclcl 22250 | The scalars in a polynomia... |
| subrgvr1 22251 | The variables in a subring... |
| subrgvr1cl 22252 | The variables in a polynom... |
| coe1z 22253 | The coefficient vector of ... |
| coe1add 22254 | The coefficient vector of ... |
| coe1addfv 22255 | A particular coefficient o... |
| coe1subfv 22256 | A particular coefficient o... |
| coe1mul2lem1 22257 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22258 | An equivalence for ~ coe1m... |
| coe1mul2 22259 | The coefficient vector of ... |
| coe1mul 22260 | The coefficient vector of ... |
| ply1moncl 22261 | Closure of the expression ... |
| ply1tmcl 22262 | Closure of the expression ... |
| coe1tm 22263 | Coefficient vector of a po... |
| coe1tmfv1 22264 | Nonzero coefficient of a p... |
| coe1tmfv2 22265 | Zero coefficient of a poly... |
| coe1tmmul2 22266 | Coefficient vector of a po... |
| coe1tmmul 22267 | Coefficient vector of a po... |
| coe1tmmul2fv 22268 | Function value of a right-... |
| coe1pwmul 22269 | Coefficient vector of a po... |
| coe1pwmulfv 22270 | Function value of a right-... |
| ply1scltm 22271 | A scalar is a term with ze... |
| coe1sclmul 22272 | Coefficient vector of a po... |
| coe1sclmulfv 22273 | A single coefficient of a ... |
| coe1sclmul2 22274 | Coefficient vector of a po... |
| ply1sclf 22275 | A scalar polynomial is a p... |
| ply1sclcl 22276 | The value of the algebra s... |
| coe1scl 22277 | Coefficient vector of a sc... |
| ply1sclid 22278 | Recover the base scalar fr... |
| ply1sclf1 22279 | The polynomial scalar func... |
| ply1scl0 22280 | The zero scalar is zero. ... |
| ply1scln0 22281 | Nonzero scalars create non... |
| ply1scl1 22282 | The one scalar is the unit... |
| coe1id 22283 | Coefficient vector of the ... |
| ply1idvr1 22284 | The identity of a polynomi... |
| ply1idvr1OLD 22285 | Obsolete version of ~ ply1... |
| cply1mul 22286 | The product of two constan... |
| ply1coefsupp 22287 | The decomposition of a uni... |
| ply1coe 22288 | Decompose a univariate pol... |
| eqcoe1ply1eq 22289 | Two polynomials over the s... |
| ply1coe1eq 22290 | Two polynomials over the s... |
| cply1coe0 22291 | All but the first coeffici... |
| cply1coe0bi 22292 | A polynomial is constant (... |
| coe1fzgsumdlem 22293 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22294 | Value of an evaluated coef... |
| ply1scleq 22295 | Equality of a constant pol... |
| ply1chr 22296 | The characteristic of a po... |
| gsumsmonply1 22297 | A finite group sum of scal... |
| gsummoncoe1 22298 | A coefficient of the polyn... |
| gsumply1eq 22299 | Two univariate polynomials... |
| lply1binom 22300 | The binomial theorem for l... |
| lply1binomsc 22301 | The binomial theorem for l... |
| ply1fermltlchr 22302 | Fermat's little theorem fo... |
| reldmevls1 22307 | Well-behaved binary operat... |
| ply1frcl 22308 | Reverse closure for the se... |
| evls1fval 22309 | Value of the univariate po... |
| evls1val 22310 | Value of the univariate po... |
| evls1rhmlem 22311 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22312 | Polynomial evaluation is a... |
| evls1sca 22313 | Univariate polynomial eval... |
| evls1gsumadd 22314 | Univariate polynomial eval... |
| evls1gsummul 22315 | Univariate polynomial eval... |
| evls1pw 22316 | Univariate polynomial eval... |
| evls1varpw 22317 | Univariate polynomial eval... |
| evl1fval 22318 | Value of the simple/same r... |
| evl1val 22319 | Value of the simple/same r... |
| evl1fval1lem 22320 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22321 | Value of the simple/same r... |
| evl1rhm 22322 | Polynomial evaluation is a... |
| fveval1fvcl 22323 | The function value of the ... |
| evl1sca 22324 | Polynomial evaluation maps... |
| evl1scad 22325 | Polynomial evaluation buil... |
| evl1var 22326 | Polynomial evaluation maps... |
| evl1vard 22327 | Polynomial evaluation buil... |
| evls1var 22328 | Univariate polynomial eval... |
| evls1scasrng 22329 | The evaluation of a scalar... |
| evls1varsrng 22330 | The evaluation of the vari... |
| evl1addd 22331 | Polynomial evaluation buil... |
| evl1subd 22332 | Polynomial evaluation buil... |
| evl1muld 22333 | Polynomial evaluation buil... |
| evl1vsd 22334 | Polynomial evaluation buil... |
| evl1expd 22335 | Polynomial evaluation buil... |
| pf1const 22336 | Constants are polynomial f... |
| pf1id 22337 | The identity is a polynomi... |
| pf1subrg 22338 | Polynomial functions are a... |
| pf1rcl 22339 | Reverse closure for the se... |
| pf1f 22340 | Polynomial functions are f... |
| mpfpf1 22341 | Convert a multivariate pol... |
| pf1mpf 22342 | Convert a univariate polyn... |
| pf1addcl 22343 | The sum of multivariate po... |
| pf1mulcl 22344 | The product of multivariat... |
| pf1ind 22345 | Prove a property of polyno... |
| evl1gsumdlem 22346 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22347 | Polynomial evaluation buil... |
| evl1gsumadd 22348 | Univariate polynomial eval... |
| evl1gsumaddval 22349 | Value of a univariate poly... |
| evl1gsummul 22350 | Univariate polynomial eval... |
| evl1varpw 22351 | Univariate polynomial eval... |
| evl1varpwval 22352 | Value of a univariate poly... |
| evl1scvarpw 22353 | Univariate polynomial eval... |
| evl1scvarpwval 22354 | Value of a univariate poly... |
| evl1gsummon 22355 | Value of a univariate poly... |
| evls1scafv 22356 | Value of the univariate po... |
| evls1expd 22357 | Univariate polynomial eval... |
| evls1varpwval 22358 | Univariate polynomial eval... |
| evls1fpws 22359 | Evaluation of a univariate... |
| ressply1evl 22360 | Evaluation of a univariate... |
| evls1addd 22361 | Univariate polynomial eval... |
| evls1muld 22362 | Univariate polynomial eval... |
| evls1vsca 22363 | Univariate polynomial eval... |
| asclply1subcl 22364 | Closure of the algebra sca... |
| evls1fvcl 22365 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22366 | The function ` F ` mapping... |
| evls1maplmhm 22367 | The function ` F ` mapping... |
| evls1maprnss 22368 | The function ` F ` mapping... |
| evl1maprhm 22369 | The function ` F ` mapping... |
| rhmmpl 22370 | Provide a ring homomorphis... |
| ply1vscl 22371 | Closure of scalar multipli... |
| mhmcoply1 22372 | The composition of a monoi... |
| rhmply1 22373 | Provide a ring homomorphis... |
| rhmply1vr1 22374 | A ring homomorphism betwee... |
| rhmply1vsca 22375 | Apply a ring homomorphism ... |
| rhmply1mon 22376 | Apply a ring homomorphism ... |
| mamufval 22379 | Functional value of the ma... |
| mamuval 22380 | Multiplication of two matr... |
| mamufv 22381 | A cell in the multiplicati... |
| mamudm 22382 | The domain of the matrix m... |
| mamufacex 22383 | Every solution of the equa... |
| mamures 22384 | Rows in a matrix product a... |
| grpvlinv 22385 | Tuple-wise left inverse in... |
| grpvrinv 22386 | Tuple-wise right inverse i... |
| ringvcl 22387 | Tuple-wise multiplication ... |
| mamucl 22388 | Operation closure of matri... |
| mamuass 22389 | Matrix multiplication is a... |
| mamudi 22390 | Matrix multiplication dist... |
| mamudir 22391 | Matrix multiplication dist... |
| mamuvs1 22392 | Matrix multiplication dist... |
| mamuvs2 22393 | Matrix multiplication dist... |
| matbas0pc 22396 | There is no matrix with a ... |
| matbas0 22397 | There is no matrix for a n... |
| matval 22398 | Value of the matrix algebr... |
| matrcl 22399 | Reverse closure for the ma... |
| matbas 22400 | The matrix ring has the sa... |
| matplusg 22401 | The matrix ring has the sa... |
| matsca 22402 | The matrix ring has the sa... |
| matvsca 22403 | The matrix ring has the sa... |
| mat0 22404 | The matrix ring has the sa... |
| matinvg 22405 | The matrix ring has the sa... |
| mat0op 22406 | Value of a zero matrix as ... |
| matsca2 22407 | The scalars of the matrix ... |
| matbas2 22408 | The base set of the matrix... |
| matbas2i 22409 | A matrix is a function. (... |
| matbas2d 22410 | The base set of the matrix... |
| eqmat 22411 | Two square matrices of the... |
| matecl 22412 | Each entry (according to W... |
| matecld 22413 | Each entry (according to W... |
| matplusg2 22414 | Addition in the matrix rin... |
| matvsca2 22415 | Scalar multiplication in t... |
| matlmod 22416 | The matrix ring is a linea... |
| matgrp 22417 | The matrix ring is a group... |
| matvscl 22418 | Closure of the scalar mult... |
| matsubg 22419 | The matrix ring has the sa... |
| matplusgcell 22420 | Addition in the matrix rin... |
| matsubgcell 22421 | Subtraction in the matrix ... |
| matinvgcell 22422 | Additive inversion in the ... |
| matvscacell 22423 | Scalar multiplication in t... |
| matgsum 22424 | Finite commutative sums in... |
| matmulr 22425 | Multiplication in the matr... |
| mamumat1cl 22426 | The identity matrix (as op... |
| mat1comp 22427 | The components of the iden... |
| mamulid 22428 | The identity matrix (as op... |
| mamurid 22429 | The identity matrix (as op... |
| matring 22430 | Existence of the matrix ri... |
| matassa 22431 | Existence of the matrix al... |
| matmulcell 22432 | Multiplication in the matr... |
| mpomatmul 22433 | Multiplication of two N x ... |
| mat1 22434 | Value of an identity matri... |
| mat1ov 22435 | Entries of an identity mat... |
| mat1bas 22436 | The identity matrix is a m... |
| matsc 22437 | The identity matrix multip... |
| ofco2 22438 | Distribution law for the f... |
| oftpos 22439 | The transposition of the v... |
| mattposcl 22440 | The transpose of a square ... |
| mattpostpos 22441 | The transpose of the trans... |
| mattposvs 22442 | The transposition of a mat... |
| mattpos1 22443 | The transposition of the i... |
| tposmap 22444 | The transposition of an I ... |
| mamutpos 22445 | Behavior of transposes in ... |
| mattposm 22446 | Multiplying two transposed... |
| matgsumcl 22447 | Closure of a group sum ove... |
| madetsumid 22448 | The identity summand in th... |
| matepmcl 22449 | Each entry of a matrix wit... |
| matepm2cl 22450 | Each entry of a matrix wit... |
| madetsmelbas 22451 | A summand of the determina... |
| madetsmelbas2 22452 | A summand of the determina... |
| mat0dimbas0 22453 | The empty set is the one a... |
| mat0dim0 22454 | The zero of the algebra of... |
| mat0dimid 22455 | The identity of the algebr... |
| mat0dimscm 22456 | The scalar multiplication ... |
| mat0dimcrng 22457 | The algebra of matrices wi... |
| mat1dimelbas 22458 | A matrix with dimension 1 ... |
| mat1dimbas 22459 | A matrix with dimension 1 ... |
| mat1dim0 22460 | The zero of the algebra of... |
| mat1dimid 22461 | The identity of the algebr... |
| mat1dimscm 22462 | The scalar multiplication ... |
| mat1dimmul 22463 | The ring multiplication in... |
| mat1dimcrng 22464 | The algebra of matrices wi... |
| mat1f1o 22465 | There is a 1-1 function fr... |
| mat1rhmval 22466 | The value of the ring homo... |
| mat1rhmelval 22467 | The value of the ring homo... |
| mat1rhmcl 22468 | The value of the ring homo... |
| mat1f 22469 | There is a function from a... |
| mat1ghm 22470 | There is a group homomorph... |
| mat1mhm 22471 | There is a monoid homomorp... |
| mat1rhm 22472 | There is a ring homomorphi... |
| mat1rngiso 22473 | There is a ring isomorphis... |
| mat1ric 22474 | A ring is isomorphic to th... |
| dmatval 22479 | The set of ` N ` x ` N ` d... |
| dmatel 22480 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22481 | An ` N ` x ` N ` diagonal ... |
| dmatid 22482 | The identity matrix is a d... |
| dmatelnd 22483 | An extradiagonal entry of ... |
| dmatmul 22484 | The product of two diagona... |
| dmatsubcl 22485 | The difference of two diag... |
| dmatsgrp 22486 | The set of diagonal matric... |
| dmatmulcl 22487 | The product of two diagona... |
| dmatsrng 22488 | The set of diagonal matric... |
| dmatcrng 22489 | The subring of diagonal ma... |
| dmatscmcl 22490 | The multiplication of a di... |
| scmatval 22491 | The set of ` N ` x ` N ` s... |
| scmatel 22492 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22493 | A scalar matrix can be exp... |
| scmatscmide 22494 | An entry of a scalar matri... |
| scmatscmiddistr 22495 | Distributive law for scala... |
| scmatmat 22496 | An ` N ` x ` N ` scalar ma... |
| scmate 22497 | An entry of an ` N ` x ` N... |
| scmatmats 22498 | The set of an ` N ` x ` N ... |
| scmateALT 22499 | Alternate proof of ~ scmat... |
| scmatscm 22500 | The multiplication of a ma... |
| scmatid 22501 | The identity matrix is a s... |
| scmatdmat 22502 | A scalar matrix is a diago... |
| scmataddcl 22503 | The sum of two scalar matr... |
| scmatsubcl 22504 | The difference of two scal... |
| scmatmulcl 22505 | The product of two scalar ... |
| scmatsgrp 22506 | The set of scalar matrices... |
| scmatsrng 22507 | The set of scalar matrices... |
| scmatcrng 22508 | The subring of scalar matr... |
| scmatsgrp1 22509 | The set of scalar matrices... |
| scmatsrng1 22510 | The set of scalar matrices... |
| smatvscl 22511 | Closure of the scalar mult... |
| scmatlss 22512 | The set of scalar matrices... |
| scmatstrbas 22513 | The set of scalar matrices... |
| scmatrhmval 22514 | The value of the ring homo... |
| scmatrhmcl 22515 | The value of the ring homo... |
| scmatf 22516 | There is a function from a... |
| scmatfo 22517 | There is a function from a... |
| scmatf1 22518 | There is a 1-1 function fr... |
| scmatf1o 22519 | There is a bijection betwe... |
| scmatghm 22520 | There is a group homomorph... |
| scmatmhm 22521 | There is a monoid homomorp... |
| scmatrhm 22522 | There is a ring homomorphi... |
| scmatrngiso 22523 | There is a ring isomorphis... |
| scmatric 22524 | A ring is isomorphic to ev... |
| mat0scmat 22525 | The empty matrix over a ri... |
| mat1scmat 22526 | A 1-dimensional matrix ove... |
| mvmulfval 22529 | Functional value of the ma... |
| mvmulval 22530 | Multiplication of a vector... |
| mvmulfv 22531 | A cell/element in the vect... |
| mavmulval 22532 | Multiplication of a vector... |
| mavmulfv 22533 | A cell/element in the vect... |
| mavmulcl 22534 | Multiplication of an NxN m... |
| 1mavmul 22535 | Multiplication of the iden... |
| mavmulass 22536 | Associativity of the multi... |
| mavmuldm 22537 | The domain of the matrix v... |
| mavmulsolcl 22538 | Every solution of the equa... |
| mavmul0 22539 | Multiplication of a 0-dime... |
| mavmul0g 22540 | The result of the 0-dimens... |
| mvmumamul1 22541 | The multiplication of an M... |
| mavmumamul1 22542 | The multiplication of an N... |
| marrepfval 22547 | First substitution for the... |
| marrepval0 22548 | Second substitution for th... |
| marrepval 22549 | Third substitution for the... |
| marrepeval 22550 | An entry of a matrix with ... |
| marrepcl 22551 | Closure of the row replace... |
| marepvfval 22552 | First substitution for the... |
| marepvval0 22553 | Second substitution for th... |
| marepvval 22554 | Third substitution for the... |
| marepveval 22555 | An entry of a matrix with ... |
| marepvcl 22556 | Closure of the column repl... |
| ma1repvcl 22557 | Closure of the column repl... |
| ma1repveval 22558 | An entry of an identity ma... |
| mulmarep1el 22559 | Element by element multipl... |
| mulmarep1gsum1 22560 | The sum of element by elem... |
| mulmarep1gsum2 22561 | The sum of element by elem... |
| 1marepvmarrepid 22562 | Replacing the ith row by 0... |
| submabas 22565 | Any subset of the index se... |
| submafval 22566 | First substitution for a s... |
| submaval0 22567 | Second substitution for a ... |
| submaval 22568 | Third substitution for a s... |
| submaeval 22569 | An entry of a submatrix of... |
| 1marepvsma1 22570 | The submatrix of the ident... |
| mdetfval 22573 | First substitution for the... |
| mdetleib 22574 | Full substitution of our d... |
| mdetleib2 22575 | Leibniz' formula can also ... |
| nfimdetndef 22576 | The determinant is not def... |
| mdetfval1 22577 | First substitution of an a... |
| mdetleib1 22578 | Full substitution of an al... |
| mdet0pr 22579 | The determinant function f... |
| mdet0f1o 22580 | The determinant function f... |
| mdet0fv0 22581 | The determinant of the emp... |
| mdetf 22582 | Functionality of the deter... |
| mdetcl 22583 | The determinant evaluates ... |
| m1detdiag 22584 | The determinant of a 1-dim... |
| mdetdiaglem 22585 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22586 | The determinant of a diago... |
| mdetdiagid 22587 | The determinant of a diago... |
| mdet1 22588 | The determinant of the ide... |
| mdetrlin 22589 | The determinant function i... |
| mdetrsca 22590 | The determinant function i... |
| mdetrsca2 22591 | The determinant function i... |
| mdetr0 22592 | The determinant of a matri... |
| mdet0 22593 | The determinant of the zer... |
| mdetrlin2 22594 | The determinant function i... |
| mdetralt 22595 | The determinant function i... |
| mdetralt2 22596 | The determinant function i... |
| mdetero 22597 | The determinant function i... |
| mdettpos 22598 | Determinant is invariant u... |
| mdetunilem1 22599 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22600 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22601 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22602 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22603 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22604 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22605 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22606 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22607 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22608 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22609 | According to the definitio... |
| mdetmul 22610 | Multiplicativity of the de... |
| m2detleiblem1 22611 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22612 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22613 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22614 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22615 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22616 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22617 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22618 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22623 | Lemma for ~ maducoeval2 . ... |
| madufval 22624 | First substitution for the... |
| maduval 22625 | Second substitution for th... |
| maducoeval 22626 | An entry of the adjunct (c... |
| maducoeval2 22627 | An entry of the adjunct (c... |
| maduf 22628 | Creating the adjunct of ma... |
| madutpos 22629 | The adjuct of a transposed... |
| madugsum 22630 | The determinant of a matri... |
| madurid 22631 | Multiplying a matrix with ... |
| madulid 22632 | Multiplying the adjunct of... |
| minmar1fval 22633 | First substitution for the... |
| minmar1val0 22634 | Second substitution for th... |
| minmar1val 22635 | Third substitution for the... |
| minmar1eval 22636 | An entry of a matrix for a... |
| minmar1marrep 22637 | The minor matrix is a spec... |
| minmar1cl 22638 | Closure of the row replace... |
| maducoevalmin1 22639 | The coefficients of an adj... |
| symgmatr01lem 22640 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22641 | Applying a permutation tha... |
| gsummatr01lem1 22642 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22643 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22644 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22645 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22646 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22647 | Replacing a row of a squar... |
| smadiadetlem0 22648 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22649 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22650 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22651 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22652 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22653 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22654 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22655 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22656 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22657 | The determinant of a subma... |
| smadiadetglem1 22658 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22659 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22660 | The determinant of a squar... |
| smadiadetg0 22661 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22662 | The determinant of a squar... |
| invrvald 22663 | If a matrix multiplied wit... |
| matinv 22664 | The inverse of a matrix is... |
| matunit 22665 | A matrix is a unit in the ... |
| slesolvec 22666 | Every solution of a system... |
| slesolinv 22667 | The solution of a system o... |
| slesolinvbi 22668 | The solution of a system o... |
| slesolex 22669 | Every system of linear equ... |
| cramerimplem1 22670 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22671 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22672 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22673 | One direction of Cramer's ... |
| cramerlem1 22674 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22675 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22676 | Lemma 3 for ~ cramer . (C... |
| cramer0 22677 | Special case of Cramer's r... |
| cramer 22678 | Cramer's rule. According ... |
| pmatring 22679 | The set of polynomial matr... |
| pmatlmod 22680 | The set of polynomial matr... |
| pmatassa 22681 | The set of polynomial matr... |
| pmat0op 22682 | The zero polynomial matrix... |
| pmat1op 22683 | The identity polynomial ma... |
| pmat1ovd 22684 | Entries of the identity po... |
| pmat0opsc 22685 | The zero polynomial matrix... |
| pmat1opsc 22686 | The identity polynomial ma... |
| pmat1ovscd 22687 | Entries of the identity po... |
| pmatcoe1fsupp 22688 | For a polynomial matrix th... |
| 1pmatscmul 22689 | The scalar product of the ... |
| cpmat 22696 | Value of the constructor o... |
| cpmatpmat 22697 | A constant polynomial matr... |
| cpmatel 22698 | Property of a constant pol... |
| cpmatelimp 22699 | Implication of a set being... |
| cpmatel2 22700 | Another property of a cons... |
| cpmatelimp2 22701 | Another implication of a s... |
| 1elcpmat 22702 | The identity of the ring o... |
| cpmatacl 22703 | The set of all constant po... |
| cpmatinvcl 22704 | The set of all constant po... |
| cpmatmcllem 22705 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22706 | The set of all constant po... |
| cpmatsubgpmat 22707 | The set of all constant po... |
| cpmatsrgpmat 22708 | The set of all constant po... |
| 0elcpmat 22709 | The zero of the ring of al... |
| mat2pmatfval 22710 | Value of the matrix transf... |
| mat2pmatval 22711 | The result of a matrix tra... |
| mat2pmatvalel 22712 | A (matrix) element of the ... |
| mat2pmatbas 22713 | The result of a matrix tra... |
| mat2pmatbas0 22714 | The result of a matrix tra... |
| mat2pmatf 22715 | The matrix transformation ... |
| mat2pmatf1 22716 | The matrix transformation ... |
| mat2pmatghm 22717 | The transformation of matr... |
| mat2pmatmul 22718 | The transformation of matr... |
| mat2pmat1 22719 | The transformation of the ... |
| mat2pmatmhm 22720 | The transformation of matr... |
| mat2pmatrhm 22721 | The transformation of matr... |
| mat2pmatlin 22722 | The transformation of matr... |
| 0mat2pmat 22723 | The transformed zero matri... |
| idmatidpmat 22724 | The transformed identity m... |
| d0mat2pmat 22725 | The transformed empty set ... |
| d1mat2pmat 22726 | The transformation of a ma... |
| mat2pmatscmxcl 22727 | A transformed matrix multi... |
| m2cpm 22728 | The result of a matrix tra... |
| m2cpmf 22729 | The matrix transformation ... |
| m2cpmf1 22730 | The matrix transformation ... |
| m2cpmghm 22731 | The transformation of matr... |
| m2cpmmhm 22732 | The transformation of matr... |
| m2cpmrhm 22733 | The transformation of matr... |
| m2pmfzmap 22734 | The transformed values of ... |
| m2pmfzgsumcl 22735 | Closure of the sum of scal... |
| cpm2mfval 22736 | Value of the inverse matri... |
| cpm2mval 22737 | The result of an inverse m... |
| cpm2mvalel 22738 | A (matrix) element of the ... |
| cpm2mf 22739 | The inverse matrix transfo... |
| m2cpminvid 22740 | The inverse transformation... |
| m2cpminvid2lem 22741 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22742 | The transformation applied... |
| m2cpmfo 22743 | The matrix transformation ... |
| m2cpmf1o 22744 | The matrix transformation ... |
| m2cpmrngiso 22745 | The transformation of matr... |
| matcpmric 22746 | The ring of matrices over ... |
| m2cpminv 22747 | The inverse matrix transfo... |
| m2cpminv0 22748 | The inverse matrix transfo... |
| decpmatval0 22751 | The matrix consisting of t... |
| decpmatval 22752 | The matrix consisting of t... |
| decpmate 22753 | An entry of the matrix con... |
| decpmatcl 22754 | Closure of the decompositi... |
| decpmataa0 22755 | The matrix consisting of t... |
| decpmatfsupp 22756 | The mapping to the matrice... |
| decpmatid 22757 | The matrix consisting of t... |
| decpmatmullem 22758 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22759 | The matrix consisting of t... |
| decpmatmulsumfsupp 22760 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22761 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22762 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22763 | Write a polynomial matrix ... |
| pmatcollpw2lem 22764 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22765 | Write a polynomial matrix ... |
| monmatcollpw 22766 | The matrix consisting of t... |
| pmatcollpwlem 22767 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22768 | Write a polynomial matrix ... |
| pmatcollpwfi 22769 | Write a polynomial matrix ... |
| pmatcollpw3lem 22770 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22771 | Write a polynomial matrix ... |
| pmatcollpw3fi 22772 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22773 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22774 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22775 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22776 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22777 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22778 | Write a scalar matrix over... |
| pm2mpf1lem 22781 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22782 | Value of the transformatio... |
| pm2mpfval 22783 | A polynomial matrix transf... |
| pm2mpcl 22784 | The transformation of poly... |
| pm2mpf 22785 | The transformation of poly... |
| pm2mpf1 22786 | The transformation of poly... |
| pm2mpcoe1 22787 | A coefficient of the polyn... |
| idpm2idmp 22788 | The transformation of the ... |
| mptcoe1matfsupp 22789 | The mapping extracting the... |
| mply1topmatcllem 22790 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22791 | A polynomial over matrices... |
| mply1topmatcl 22792 | A polynomial over matrices... |
| mp2pm2mplem1 22793 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22794 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22795 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22796 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22797 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22798 | A polynomial over matrices... |
| pm2mpghmlem2 22799 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22800 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22801 | The transformation of poly... |
| pm2mpf1o 22802 | The transformation of poly... |
| pm2mpghm 22803 | The transformation of poly... |
| pm2mpgrpiso 22804 | The transformation of poly... |
| pm2mpmhmlem1 22805 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22806 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22807 | The transformation of poly... |
| pm2mprhm 22808 | The transformation of poly... |
| pm2mprngiso 22809 | The transformation of poly... |
| pmmpric 22810 | The ring of polynomial mat... |
| monmat2matmon 22811 | The transformation of a po... |
| pm2mp 22812 | The transformation of a su... |
| chmatcl 22815 | Closure of the characteris... |
| chmatval 22816 | The entries of the charact... |
| chpmatfval 22817 | Value of the characteristi... |
| chpmatval 22818 | The characteristic polynom... |
| chpmatply1 22819 | The characteristic polynom... |
| chpmatval2 22820 | The characteristic polynom... |
| chpmat0d 22821 | The characteristic polynom... |
| chpmat1dlem 22822 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22823 | The characteristic polynom... |
| chpdmatlem0 22824 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22825 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22826 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22827 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22828 | The characteristic polynom... |
| chpscmat 22829 | The characteristic polynom... |
| chpscmat0 22830 | The characteristic polynom... |
| chpscmatgsumbin 22831 | The characteristic polynom... |
| chpscmatgsummon 22832 | The characteristic polynom... |
| chp0mat 22833 | The characteristic polynom... |
| chpidmat 22834 | The characteristic polynom... |
| chmaidscmat 22835 | The characteristic polynom... |
| fvmptnn04if 22836 | The function values of a m... |
| fvmptnn04ifa 22837 | The function value of a ma... |
| fvmptnn04ifb 22838 | The function value of a ma... |
| fvmptnn04ifc 22839 | The function value of a ma... |
| fvmptnn04ifd 22840 | The function value of a ma... |
| chfacfisf 22841 | The "characteristic factor... |
| chfacfisfcpmat 22842 | The "characteristic factor... |
| chfacffsupp 22843 | The "characteristic factor... |
| chfacfscmulcl 22844 | Closure of a scaled value ... |
| chfacfscmul0 22845 | A scaled value of the "cha... |
| chfacfscmulfsupp 22846 | A mapping of scaled values... |
| chfacfscmulgsum 22847 | Breaking up a sum of value... |
| chfacfpmmulcl 22848 | Closure of the value of th... |
| chfacfpmmul0 22849 | The value of the "characte... |
| chfacfpmmulfsupp 22850 | A mapping of values of the... |
| chfacfpmmulgsum 22851 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22852 | Breaking up a sum of value... |
| cayhamlem1 22853 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22854 | The right-hand fundamental... |
| cpmidgsum 22855 | Representation of the iden... |
| cpmidgsumm2pm 22856 | Representation of the iden... |
| cpmidpmatlem1 22857 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22858 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22859 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22860 | Representation of the iden... |
| cpmadugsumlemB 22861 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22862 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22863 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22864 | The product of the charact... |
| cpmadugsum 22865 | The product of the charact... |
| cpmidgsum2 22866 | Representation of the iden... |
| cpmidg2sum 22867 | Equality of two sums repre... |
| cpmadumatpolylem1 22868 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22869 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22870 | The product of the charact... |
| cayhamlem2 22871 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22872 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22873 | The coefficients of the ch... |
| cayhamlem3 22874 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22875 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22876 | The Cayley-Hamilton theore... |
| cayleyhamilton 22877 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22878 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22879 | The Cayley-Hamilton theore... |
| istopg 22882 | Express the predicate " ` ... |
| istop2g 22883 | Express the predicate " ` ... |
| uniopn 22884 | The union of a subset of a... |
| iunopn 22885 | The indexed union of a sub... |
| inopn 22886 | The intersection of two op... |
| fitop 22887 | A topology is closed under... |
| fiinopn 22888 | The intersection of a none... |
| iinopn 22889 | The intersection of a none... |
| unopn 22890 | The union of two open sets... |
| 0opn 22891 | The empty set is an open s... |
| 0ntop 22892 | The empty set is not a top... |
| topopn 22893 | The underlying set of a to... |
| eltopss 22894 | A member of a topology is ... |
| riinopn 22895 | A finite indexed relative ... |
| rintopn 22896 | A finite relative intersec... |
| istopon 22899 | Property of being a topolo... |
| topontop 22900 | A topology on a given base... |
| toponuni 22901 | The base set of a topology... |
| topontopi 22902 | A topology on a given base... |
| toponunii 22903 | The base set of a topology... |
| toptopon 22904 | Alternative definition of ... |
| toptopon2 22905 | A topology is the same thi... |
| topontopon 22906 | A topology on a set is a t... |
| funtopon 22907 | The class ` TopOn ` is a f... |
| toponrestid 22908 | Given a topology on a set,... |
| toponsspwpw 22909 | The set of topologies on a... |
| dmtopon 22910 | The domain of ` TopOn ` is... |
| fntopon 22911 | The class ` TopOn ` is a f... |
| toprntopon 22912 | A topology is the same thi... |
| toponmax 22913 | The base set of a topology... |
| toponss 22914 | A member of a topology is ... |
| toponcom 22915 | If ` K ` is a topology on ... |
| toponcomb 22916 | Biconditional form of ~ to... |
| topgele 22917 | The topologies over the sa... |
| topsn 22918 | The only topology on a sin... |
| istps 22921 | Express the predicate "is ... |
| istps2 22922 | Express the predicate "is ... |
| tpsuni 22923 | The base set of a topologi... |
| tpstop 22924 | The topology extractor on ... |
| tpspropd 22925 | A topological space depend... |
| tpsprop2d 22926 | A topological space depend... |
| topontopn 22927 | Express the predicate "is ... |
| tsettps 22928 | If the topology component ... |
| istpsi 22929 | Properties that determine ... |
| eltpsg 22930 | Properties that determine ... |
| eltpsi 22931 | Properties that determine ... |
| isbasisg 22934 | Express the predicate "the... |
| isbasis2g 22935 | Express the predicate "the... |
| isbasis3g 22936 | Express the predicate "the... |
| basis1 22937 | Property of a basis. (Con... |
| basis2 22938 | Property of a basis. (Con... |
| fiinbas 22939 | If a set is closed under f... |
| basdif0 22940 | A basis is not affected by... |
| baspartn 22941 | A disjoint system of sets ... |
| tgval 22942 | The topology generated by ... |
| tgval2 22943 | Definition of a topology g... |
| eltg 22944 | Membership in a topology g... |
| eltg2 22945 | Membership in a topology g... |
| eltg2b 22946 | Membership in a topology g... |
| eltg4i 22947 | An open set in a topology ... |
| eltg3i 22948 | The union of a set of basi... |
| eltg3 22949 | Membership in a topology g... |
| tgval3 22950 | Alternate expression for t... |
| tg1 22951 | Property of a member of a ... |
| tg2 22952 | Property of a member of a ... |
| bastg 22953 | A member of a basis is a s... |
| unitg 22954 | The topology generated by ... |
| tgss 22955 | Subset relation for genera... |
| tgcl 22956 | Show that a basis generate... |
| tgclb 22957 | The property ~ tgcl can be... |
| tgtopon 22958 | A basis generates a topolo... |
| topbas 22959 | A topology is its own basi... |
| tgtop 22960 | A topology is its own basi... |
| eltop 22961 | Membership in a topology, ... |
| eltop2 22962 | Membership in a topology. ... |
| eltop3 22963 | Membership in a topology. ... |
| fibas 22964 | A collection of finite int... |
| tgdom 22965 | A space has no more open s... |
| tgiun 22966 | The indexed union of a set... |
| tgidm 22967 | The topology generator fun... |
| bastop 22968 | Two ways to express that a... |
| tgtop11 22969 | The topology generation fu... |
| 0top 22970 | The singleton of the empty... |
| en1top 22971 | ` { (/) } ` is the only to... |
| en2top 22972 | If a topology has two elem... |
| tgss3 22973 | A criterion for determinin... |
| tgss2 22974 | A criterion for determinin... |
| basgen 22975 | Given a topology ` J ` , s... |
| basgen2 22976 | Given a topology ` J ` , s... |
| 2basgen 22977 | Conditions that determine ... |
| tgfiss 22978 | If a subbase is included i... |
| tgdif0 22979 | A generated topology is no... |
| bastop1 22980 | A subset of a topology is ... |
| bastop2 22981 | A version of ~ bastop1 tha... |
| distop 22982 | The discrete topology on a... |
| topnex 22983 | The class of all topologie... |
| distopon 22984 | The discrete topology on a... |
| sn0topon 22985 | The singleton of the empty... |
| sn0top 22986 | The singleton of the empty... |
| indislem 22987 | A lemma to eliminate some ... |
| indistopon 22988 | The indiscrete topology on... |
| indistop 22989 | The indiscrete topology on... |
| indisuni 22990 | The base set of the indisc... |
| fctop 22991 | The finite complement topo... |
| fctop2 22992 | The finite complement topo... |
| cctop 22993 | The countable complement t... |
| ppttop 22994 | The particular point topol... |
| pptbas 22995 | The particular point topol... |
| epttop 22996 | The excluded point topolog... |
| indistpsx 22997 | The indiscrete topology on... |
| indistps 22998 | The indiscrete topology on... |
| indistps2 22999 | The indiscrete topology on... |
| indistpsALT 23000 | The indiscrete topology on... |
| indistps2ALT 23001 | The indiscrete topology on... |
| distps 23002 | The discrete topology on a... |
| fncld 23009 | The closed-set generator i... |
| cldval 23010 | The set of closed sets of ... |
| ntrfval 23011 | The interior function on t... |
| clsfval 23012 | The closure function on th... |
| cldrcl 23013 | Reverse closure of the clo... |
| iscld 23014 | The predicate "the class `... |
| iscld2 23015 | A subset of the underlying... |
| cldss 23016 | A closed set is a subset o... |
| cldss2 23017 | The set of closed sets is ... |
| cldopn 23018 | The complement of a closed... |
| isopn2 23019 | A subset of the underlying... |
| opncld 23020 | The complement of an open ... |
| difopn 23021 | The difference of a closed... |
| topcld 23022 | The underlying set of a to... |
| ntrval 23023 | The interior of a subset o... |
| clsval 23024 | The closure of a subset of... |
| 0cld 23025 | The empty set is closed. ... |
| iincld 23026 | The indexed intersection o... |
| intcld 23027 | The intersection of a set ... |
| uncld 23028 | The union of two closed se... |
| cldcls 23029 | A closed subset equals its... |
| incld 23030 | The intersection of two cl... |
| riincld 23031 | An indexed relative inters... |
| iuncld 23032 | A finite indexed union of ... |
| unicld 23033 | A finite union of closed s... |
| clscld 23034 | The closure of a subset of... |
| clsf 23035 | The closure function is a ... |
| ntropn 23036 | The interior of a subset o... |
| clsval2 23037 | Express closure in terms o... |
| ntrval2 23038 | Interior expressed in term... |
| ntrdif 23039 | An interior of a complemen... |
| clsdif 23040 | A closure of a complement ... |
| clsss 23041 | Subset relationship for cl... |
| ntrss 23042 | Subset relationship for in... |
| sscls 23043 | A subset of a topology's u... |
| ntrss2 23044 | A subset includes its inte... |
| ssntr 23045 | An open subset of a set is... |
| clsss3 23046 | The closure of a subset of... |
| ntrss3 23047 | The interior of a subset o... |
| ntrin 23048 | A pairwise intersection of... |
| cmclsopn 23049 | The complement of a closur... |
| cmntrcld 23050 | The complement of an inter... |
| iscld3 23051 | A subset is closed iff it ... |
| iscld4 23052 | A subset is closed iff it ... |
| isopn3 23053 | A subset is open iff it eq... |
| clsidm 23054 | The closure operation is i... |
| ntridm 23055 | The interior operation is ... |
| clstop 23056 | The closure of a topology'... |
| ntrtop 23057 | The interior of a topology... |
| 0ntr 23058 | A subset with an empty int... |
| clsss2 23059 | If a subset is included in... |
| elcls 23060 | Membership in a closure. ... |
| elcls2 23061 | Membership in a closure. ... |
| clsndisj 23062 | Any open set containing a ... |
| ntrcls0 23063 | A subset whose closure has... |
| ntreq0 23064 | Two ways to say that a sub... |
| cldmre 23065 | The closed sets of a topol... |
| mrccls 23066 | Moore closure generalizes ... |
| cls0 23067 | The closure of the empty s... |
| ntr0 23068 | The interior of the empty ... |
| isopn3i 23069 | An open subset equals its ... |
| elcls3 23070 | Membership in a closure in... |
| opncldf1 23071 | A bijection useful for con... |
| opncldf2 23072 | The values of the open-clo... |
| opncldf3 23073 | The values of the converse... |
| isclo 23074 | A set ` A ` is clopen iff ... |
| isclo2 23075 | A set ` A ` is clopen iff ... |
| discld 23076 | The open sets of a discret... |
| sn0cld 23077 | The closed sets of the top... |
| indiscld 23078 | The closed sets of an indi... |
| mretopd 23079 | A Moore collection which i... |
| toponmre 23080 | The topologies over a give... |
| cldmreon 23081 | The closed sets of a topol... |
| iscldtop 23082 | A family is the closed set... |
| mreclatdemoBAD 23083 | The closed subspaces of a ... |
| neifval 23086 | Value of the neighborhood ... |
| neif 23087 | The neighborhood function ... |
| neiss2 23088 | A set with a neighborhood ... |
| neival 23089 | Value of the set of neighb... |
| isnei 23090 | The predicate "the class `... |
| neiint 23091 | An intuitive definition of... |
| isneip 23092 | The predicate "the class `... |
| neii1 23093 | A neighborhood is included... |
| neisspw 23094 | The neighborhoods of any s... |
| neii2 23095 | Property of a neighborhood... |
| neiss 23096 | Any neighborhood of a set ... |
| ssnei 23097 | A set is included in any o... |
| elnei 23098 | A point belongs to any of ... |
| 0nnei 23099 | The empty set is not a nei... |
| neips 23100 | A neighborhood of a set is... |
| opnneissb 23101 | An open set is a neighborh... |
| opnssneib 23102 | Any superset of an open se... |
| ssnei2 23103 | Any subset ` M ` of ` X ` ... |
| neindisj 23104 | Any neighborhood of an ele... |
| opnneiss 23105 | An open set is a neighborh... |
| opnneip 23106 | An open set is a neighborh... |
| opnnei 23107 | A set is open iff it is a ... |
| tpnei 23108 | The underlying set of a to... |
| neiuni 23109 | The union of the neighborh... |
| neindisj2 23110 | A point ` P ` belongs to t... |
| topssnei 23111 | A finer topology has more ... |
| innei 23112 | The intersection of two ne... |
| opnneiid 23113 | Only an open set is a neig... |
| neissex 23114 | For any neighborhood ` N `... |
| 0nei 23115 | The empty set is a neighbo... |
| neipeltop 23116 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23117 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23118 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23119 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23120 | If, to each element ` P ` ... |
| lpfval 23125 | The limit point function o... |
| lpval 23126 | The set of limit points of... |
| islp 23127 | The predicate "the class `... |
| lpsscls 23128 | The limit points of a subs... |
| lpss 23129 | The limit points of a subs... |
| lpdifsn 23130 | ` P ` is a limit point of ... |
| lpss3 23131 | Subset relationship for li... |
| islp2 23132 | The predicate " ` P ` is a... |
| islp3 23133 | The predicate " ` P ` is a... |
| maxlp 23134 | A point is a limit point o... |
| clslp 23135 | The closure of a subset of... |
| islpi 23136 | A point belonging to a set... |
| cldlp 23137 | A subset of a topological ... |
| isperf 23138 | Definition of a perfect sp... |
| isperf2 23139 | Definition of a perfect sp... |
| isperf3 23140 | A perfect space is a topol... |
| perflp 23141 | The limit points of a perf... |
| perfi 23142 | Property of a perfect spac... |
| perftop 23143 | A perfect space is a topol... |
| restrcl 23144 | Reverse closure for the su... |
| restbas 23145 | A subspace topology basis ... |
| tgrest 23146 | A subspace can be generate... |
| resttop 23147 | A subspace topology is a t... |
| resttopon 23148 | A subspace topology is a t... |
| restuni 23149 | The underlying set of a su... |
| stoig 23150 | The topological space buil... |
| restco 23151 | Composition of subspaces. ... |
| restabs 23152 | Equivalence of being a sub... |
| restin 23153 | When the subspace region i... |
| restuni2 23154 | The underlying set of a su... |
| resttopon2 23155 | The underlying set of a su... |
| rest0 23156 | The subspace topology indu... |
| restsn 23157 | The only subspace topology... |
| restsn2 23158 | The subspace topology indu... |
| restcld 23159 | A closed set of a subspace... |
| restcldi 23160 | A closed set is closed in ... |
| restcldr 23161 | A set which is closed in t... |
| restopnb 23162 | If ` B ` is an open subset... |
| ssrest 23163 | If ` K ` is a finer topolo... |
| restopn2 23164 | If ` A ` is open, then ` B... |
| restdis 23165 | A subspace of a discrete t... |
| restfpw 23166 | The restriction of the set... |
| neitr 23167 | The neighborhood of a trac... |
| restcls 23168 | A closure in a subspace to... |
| restntr 23169 | An interior in a subspace ... |
| restlp 23170 | The limit points of a subs... |
| restperf 23171 | Perfection of a subspace. ... |
| perfopn 23172 | An open subset of a perfec... |
| resstopn 23173 | The topology of a restrict... |
| resstps 23174 | A restricted topological s... |
| ordtbaslem 23175 | Lemma for ~ ordtbas . In ... |
| ordtval 23176 | Value of the order topolog... |
| ordtuni 23177 | Value of the order topolog... |
| ordtbas2 23178 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23179 | In a total order, the fini... |
| ordttopon 23180 | Value of the order topolog... |
| ordtopn1 23181 | An upward ray ` ( P , +oo ... |
| ordtopn2 23182 | A downward ray ` ( -oo , P... |
| ordtopn3 23183 | An open interval ` ( A , B... |
| ordtcld1 23184 | A downward ray ` ( -oo , P... |
| ordtcld2 23185 | An upward ray ` [ P , +oo ... |
| ordtcld3 23186 | A closed interval ` [ A , ... |
| ordttop 23187 | The order topology is a to... |
| ordtcnv 23188 | The order dual generates t... |
| ordtrest 23189 | The subspace topology of a... |
| ordtrest2lem 23190 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23191 | An interval-closed set ` A... |
| letopon 23192 | The topology of the extend... |
| letop 23193 | The topology of the extend... |
| letopuni 23194 | The topology of the extend... |
| xrstopn 23195 | The topology component of ... |
| xrstps 23196 | The extended real number s... |
| leordtvallem1 23197 | Lemma for ~ leordtval . (... |
| leordtvallem2 23198 | Lemma for ~ leordtval . (... |
| leordtval2 23199 | The topology of the extend... |
| leordtval 23200 | The topology of the extend... |
| iccordt 23201 | A closed interval is close... |
| iocpnfordt 23202 | An unbounded above open in... |
| icomnfordt 23203 | An unbounded above open in... |
| iooordt 23204 | An open interval is open i... |
| reordt 23205 | The real numbers are an op... |
| lecldbas 23206 | The set of closed interval... |
| pnfnei 23207 | A neighborhood of ` +oo ` ... |
| mnfnei 23208 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23209 | The restriction of the les... |
| ordtresticc 23210 | The restriction of the les... |
| lmrel 23217 | The topological space conv... |
| lmrcl 23218 | Reverse closure for the co... |
| lmfval 23219 | The relation "sequence ` f... |
| cnfval 23220 | The set of all continuous ... |
| cnpfval 23221 | The function mapping the p... |
| iscn 23222 | The predicate "the class `... |
| cnpval 23223 | The set of all functions f... |
| iscnp 23224 | The predicate "the class `... |
| iscn2 23225 | The predicate "the class `... |
| iscnp2 23226 | The predicate "the class `... |
| cntop1 23227 | Reverse closure for a cont... |
| cntop2 23228 | Reverse closure for a cont... |
| cnptop1 23229 | Reverse closure for a func... |
| cnptop2 23230 | Reverse closure for a func... |
| iscnp3 23231 | The predicate "the class `... |
| cnprcl 23232 | Reverse closure for a func... |
| cnf 23233 | A continuous function is a... |
| cnpf 23234 | A continuous function at p... |
| cnpcl 23235 | The value of a continuous ... |
| cnf2 23236 | A continuous function is a... |
| cnpf2 23237 | A continuous function at p... |
| cnprcl2 23238 | Reverse closure for a func... |
| tgcn 23239 | The continuity predicate w... |
| tgcnp 23240 | The "continuous at a point... |
| subbascn 23241 | The continuity predicate w... |
| ssidcn 23242 | The identity function is a... |
| cnpimaex 23243 | Property of a function con... |
| idcn 23244 | A restricted identity func... |
| lmbr 23245 | Express the binary relatio... |
| lmbr2 23246 | Express the binary relatio... |
| lmbrf 23247 | Express the binary relatio... |
| lmconst 23248 | A constant sequence conver... |
| lmcvg 23249 | Convergence property of a ... |
| iscnp4 23250 | The predicate "the class `... |
| cnpnei 23251 | A condition for continuity... |
| cnima 23252 | An open subset of the codo... |
| cnco 23253 | The composition of two con... |
| cnpco 23254 | The composition of a funct... |
| cnclima 23255 | A closed subset of the cod... |
| iscncl 23256 | A characterization of a co... |
| cncls2i 23257 | Property of the preimage o... |
| cnntri 23258 | Property of the preimage o... |
| cnclsi 23259 | Property of the image of a... |
| cncls2 23260 | Continuity in terms of clo... |
| cncls 23261 | Continuity in terms of clo... |
| cnntr 23262 | Continuity in terms of int... |
| cnss1 23263 | If the topology ` K ` is f... |
| cnss2 23264 | If the topology ` K ` is f... |
| cncnpi 23265 | A continuous function is c... |
| cnsscnp 23266 | The set of continuous func... |
| cncnp 23267 | A continuous function is c... |
| cncnp2 23268 | A continuous function is c... |
| cnnei 23269 | Continuity in terms of nei... |
| cnconst2 23270 | A constant function is con... |
| cnconst 23271 | A constant function is con... |
| cnrest 23272 | Continuity of a restrictio... |
| cnrest2 23273 | Equivalence of continuity ... |
| cnrest2r 23274 | Equivalence of continuity ... |
| cnpresti 23275 | One direction of ~ cnprest... |
| cnprest 23276 | Equivalence of continuity ... |
| cnprest2 23277 | Equivalence of point-conti... |
| cndis 23278 | Every function is continuo... |
| cnindis 23279 | Every function is continuo... |
| cnpdis 23280 | If ` A ` is an isolated po... |
| paste 23281 | Pasting lemma. If ` A ` a... |
| lmfpm 23282 | If ` F ` converges, then `... |
| lmfss 23283 | Inclusion of a function ha... |
| lmcl 23284 | Closure of a limit. (Cont... |
| lmss 23285 | Limit on a subspace. (Con... |
| sslm 23286 | A finer topology has fewer... |
| lmres 23287 | A function converges iff i... |
| lmff 23288 | If ` F ` converges, there ... |
| lmcls 23289 | Any convergent sequence of... |
| lmcld 23290 | Any convergent sequence of... |
| lmcnp 23291 | The image of a convergent ... |
| lmcn 23292 | The image of a convergent ... |
| ist0 23307 | The predicate "is a T_0 sp... |
| ist1 23308 | The predicate "is a T_1 sp... |
| ishaus 23309 | The predicate "is a Hausdo... |
| iscnrm 23310 | The property of being comp... |
| t0sep 23311 | Any two topologically indi... |
| t0dist 23312 | Any two distinct points in... |
| t1sncld 23313 | In a T_1 space, singletons... |
| t1ficld 23314 | In a T_1 space, finite set... |
| hausnei 23315 | Neighborhood property of a... |
| t0top 23316 | A T_0 space is a topologic... |
| t1top 23317 | A T_1 space is a topologic... |
| haustop 23318 | A Hausdorff space is a top... |
| isreg 23319 | The predicate "is a regula... |
| regtop 23320 | A regular space is a topol... |
| regsep 23321 | In a regular space, every ... |
| isnrm 23322 | The predicate "is a normal... |
| nrmtop 23323 | A normal space is a topolo... |
| cnrmtop 23324 | A completely normal space ... |
| iscnrm2 23325 | The property of being comp... |
| ispnrm 23326 | The property of being perf... |
| pnrmnrm 23327 | A perfectly normal space i... |
| pnrmtop 23328 | A perfectly normal space i... |
| pnrmcld 23329 | A closed set in a perfectl... |
| pnrmopn 23330 | An open set in a perfectly... |
| ist0-2 23331 | The predicate "is a T_0 sp... |
| ist0-3 23332 | The predicate "is a T_0 sp... |
| cnt0 23333 | The preimage of a T_0 topo... |
| ist1-2 23334 | An alternate characterizat... |
| t1t0 23335 | A T_1 space is a T_0 space... |
| ist1-3 23336 | A space is T_1 iff every p... |
| cnt1 23337 | The preimage of a T_1 topo... |
| ishaus2 23338 | Express the predicate " ` ... |
| haust1 23339 | A Hausdorff space is a T_1... |
| hausnei2 23340 | The Hausdorff condition st... |
| cnhaus 23341 | The preimage of a Hausdorf... |
| nrmsep3 23342 | In a normal space, given a... |
| nrmsep2 23343 | In a normal space, any two... |
| nrmsep 23344 | In a normal space, disjoin... |
| isnrm2 23345 | An alternate characterizat... |
| isnrm3 23346 | A topological space is nor... |
| cnrmi 23347 | A subspace of a completely... |
| cnrmnrm 23348 | A completely normal space ... |
| restcnrm 23349 | A subspace of a completely... |
| resthauslem 23350 | Lemma for ~ resthaus and s... |
| lpcls 23351 | The limit points of the cl... |
| perfcls 23352 | A subset of a perfect spac... |
| restt0 23353 | A subspace of a T_0 topolo... |
| restt1 23354 | A subspace of a T_1 topolo... |
| resthaus 23355 | A subspace of a Hausdorff ... |
| t1sep2 23356 | Any two points in a T_1 sp... |
| t1sep 23357 | Any two distinct points in... |
| sncld 23358 | A singleton is closed in a... |
| sshauslem 23359 | Lemma for ~ sshaus and sim... |
| sst0 23360 | A topology finer than a T_... |
| sst1 23361 | A topology finer than a T_... |
| sshaus 23362 | A topology finer than a Ha... |
| regsep2 23363 | In a regular space, a clos... |
| isreg2 23364 | A topological space is reg... |
| dnsconst 23365 | If a continuous mapping to... |
| ordtt1 23366 | The order topology is T_1 ... |
| lmmo 23367 | A sequence in a Hausdorff ... |
| lmfun 23368 | The convergence relation i... |
| dishaus 23369 | A discrete topology is Hau... |
| ordthauslem 23370 | Lemma for ~ ordthaus . (C... |
| ordthaus 23371 | The order topology of a to... |
| xrhaus 23372 | The topology of the extend... |
| iscmp 23375 | The predicate "is a compac... |
| cmpcov 23376 | An open cover of a compact... |
| cmpcov2 23377 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23378 | Combine ~ cmpcov with ~ ac... |
| cncmp 23379 | Compactness is respected b... |
| fincmp 23380 | A finite topology is compa... |
| 0cmp 23381 | The singleton of the empty... |
| cmptop 23382 | A compact topology is a to... |
| rncmp 23383 | The image of a compact set... |
| imacmp 23384 | The image of a compact set... |
| discmp 23385 | A discrete topology is com... |
| cmpsublem 23386 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23387 | Two equivalent ways of des... |
| tgcmp 23388 | A topology generated by a ... |
| cmpcld 23389 | A closed subset of a compa... |
| uncmp 23390 | The union of two compact s... |
| fiuncmp 23391 | A finite union of compact ... |
| sscmp 23392 | A subset of a compact topo... |
| hauscmplem 23393 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23394 | A compact subspace of a T2... |
| cmpfi 23395 | If a topology is compact a... |
| cmpfii 23396 | In a compact topology, a s... |
| bwth 23397 | The glorious Bolzano-Weier... |
| isconn 23400 | The predicate ` J ` is a c... |
| isconn2 23401 | The predicate ` J ` is a c... |
| connclo 23402 | The only nonempty clopen s... |
| conndisj 23403 | If a topology is connected... |
| conntop 23404 | A connected topology is a ... |
| indisconn 23405 | The indiscrete topology (o... |
| dfconn2 23406 | An alternate definition of... |
| connsuba 23407 | Connectedness for a subspa... |
| connsub 23408 | Two equivalent ways of say... |
| cnconn 23409 | Connectedness is respected... |
| nconnsubb 23410 | Disconnectedness for a sub... |
| connsubclo 23411 | If a clopen set meets a co... |
| connima 23412 | The image of a connected s... |
| conncn 23413 | A continuous function from... |
| iunconnlem 23414 | Lemma for ~ iunconn . (Co... |
| iunconn 23415 | The indexed union of conne... |
| unconn 23416 | The union of two connected... |
| clsconn 23417 | The closure of a connected... |
| conncompid 23418 | The connected component co... |
| conncompconn 23419 | The connected component co... |
| conncompss 23420 | The connected component co... |
| conncompcld 23421 | The connected component co... |
| conncompclo 23422 | The connected component co... |
| t1connperf 23423 | A connected T_1 space is p... |
| is1stc 23428 | The predicate "is a first-... |
| is1stc2 23429 | An equivalent way of sayin... |
| 1stctop 23430 | A first-countable topology... |
| 1stcclb 23431 | A property of points in a ... |
| 1stcfb 23432 | For any point ` A ` in a f... |
| is2ndc 23433 | The property of being seco... |
| 2ndctop 23434 | A second-countable topolog... |
| 2ndci 23435 | A countable basis generate... |
| 2ndcsb 23436 | Having a countable subbase... |
| 2ndcredom 23437 | A second-countable space h... |
| 2ndc1stc 23438 | A second-countable space i... |
| 1stcrestlem 23439 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23440 | A subspace of a first-coun... |
| 2ndcrest 23441 | A subspace of a second-cou... |
| 2ndcctbss 23442 | If a topology is second-co... |
| 2ndcdisj 23443 | Any disjoint family of ope... |
| 2ndcdisj2 23444 | Any disjoint collection of... |
| 2ndcomap 23445 | A surjective continuous op... |
| 2ndcsep 23446 | A second-countable topolog... |
| dis2ndc 23447 | A discrete space is second... |
| 1stcelcls 23448 | A point belongs to the clo... |
| 1stccnp 23449 | A mapping is continuous at... |
| 1stccn 23450 | A mapping ` X --> Y ` , wh... |
| islly 23455 | The property of being a lo... |
| isnlly 23456 | The property of being an n... |
| llyeq 23457 | Equality theorem for the `... |
| nllyeq 23458 | Equality theorem for the `... |
| llytop 23459 | A locally ` A ` space is a... |
| nllytop 23460 | A locally ` A ` space is a... |
| llyi 23461 | The property of a locally ... |
| nllyi 23462 | The property of an n-local... |
| nlly2i 23463 | Eliminate the neighborhood... |
| llynlly 23464 | A locally ` A ` space is n... |
| llyssnlly 23465 | A locally ` A ` space is n... |
| llyss 23466 | The "locally" predicate re... |
| nllyss 23467 | The "n-locally" predicate ... |
| subislly 23468 | The property of a subspace... |
| restnlly 23469 | If the property ` A ` pass... |
| restlly 23470 | If the property ` A ` pass... |
| islly2 23471 | An alternative expression ... |
| llyrest 23472 | An open subspace of a loca... |
| nllyrest 23473 | An open subspace of an n-l... |
| loclly 23474 | If ` A ` is a local proper... |
| llyidm 23475 | Idempotence of the "locall... |
| nllyidm 23476 | Idempotence of the "n-loca... |
| toplly 23477 | A topology is locally a to... |
| topnlly 23478 | A topology is n-locally a ... |
| hauslly 23479 | A Hausdorff space is local... |
| hausnlly 23480 | A Hausdorff space is n-loc... |
| hausllycmp 23481 | A compact Hausdorff space ... |
| cldllycmp 23482 | A closed subspace of a loc... |
| lly1stc 23483 | First-countability is a lo... |
| dislly 23484 | The discrete space ` ~P X ... |
| disllycmp 23485 | A discrete space is locall... |
| dis1stc 23486 | A discrete space is first-... |
| hausmapdom 23487 | If ` X ` is a first-counta... |
| hauspwdom 23488 | Simplify the cardinal ` A ... |
| refrel 23495 | Refinement is a relation. ... |
| isref 23496 | The property of being a re... |
| refbas 23497 | A refinement covers the sa... |
| refssex 23498 | Every set in a refinement ... |
| ssref 23499 | A subcover is a refinement... |
| refref 23500 | Reflexivity of refinement.... |
| reftr 23501 | Refinement is transitive. ... |
| refun0 23502 | Adding the empty set prese... |
| isptfin 23503 | The statement "is a point-... |
| islocfin 23504 | The statement "is a locall... |
| finptfin 23505 | A finite cover is a point-... |
| ptfinfin 23506 | A point covered by a point... |
| finlocfin 23507 | A finite cover of a topolo... |
| locfintop 23508 | A locally finite cover cov... |
| locfinbas 23509 | A locally finite cover mus... |
| locfinnei 23510 | A point covered by a local... |
| lfinpfin 23511 | A locally finite cover is ... |
| lfinun 23512 | Adding a finite set preser... |
| locfincmp 23513 | For a compact space, the l... |
| unisngl 23514 | Taking the union of the se... |
| dissnref 23515 | The set of singletons is a... |
| dissnlocfin 23516 | The set of singletons is l... |
| locfindis 23517 | The locally finite covers ... |
| locfincf 23518 | A locally finite cover in ... |
| comppfsc 23519 | A space where every open c... |
| kgenval 23522 | Value of the compact gener... |
| elkgen 23523 | Value of the compact gener... |
| kgeni 23524 | Property of the open sets ... |
| kgentopon 23525 | The compact generator gene... |
| kgenuni 23526 | The base set of the compac... |
| kgenftop 23527 | The compact generator gene... |
| kgenf 23528 | The compact generator is a... |
| kgentop 23529 | A compactly generated spac... |
| kgenss 23530 | The compact generator gene... |
| kgenhaus 23531 | The compact generator gene... |
| kgencmp 23532 | The compact generator topo... |
| kgencmp2 23533 | The compact generator topo... |
| kgenidm 23534 | The compact generator is i... |
| iskgen2 23535 | A space is compactly gener... |
| iskgen3 23536 | Derive the usual definitio... |
| llycmpkgen2 23537 | A locally compact space is... |
| cmpkgen 23538 | A compact space is compact... |
| llycmpkgen 23539 | A locally compact space is... |
| 1stckgenlem 23540 | The one-point compactifica... |
| 1stckgen 23541 | A first-countable space is... |
| kgen2ss 23542 | The compact generator pres... |
| kgencn 23543 | A function from a compactl... |
| kgencn2 23544 | A function ` F : J --> K `... |
| kgencn3 23545 | The set of continuous func... |
| kgen2cn 23546 | A continuous function is a... |
| txval 23551 | Value of the binary topolo... |
| txuni2 23552 | The underlying set of the ... |
| txbasex 23553 | The basis for the product ... |
| txbas 23554 | The set of Cartesian produ... |
| eltx 23555 | A set in a product is open... |
| txtop 23556 | The product of two topolog... |
| ptval 23557 | The value of the product t... |
| ptpjpre1 23558 | The preimage of a projecti... |
| elpt 23559 | Elementhood in the bases o... |
| elptr 23560 | A basic open set in the pr... |
| elptr2 23561 | A basic open set in the pr... |
| ptbasid 23562 | The base set of the produc... |
| ptuni2 23563 | The base set for the produ... |
| ptbasin 23564 | The basis for a product to... |
| ptbasin2 23565 | The basis for a product to... |
| ptbas 23566 | The basis for a product to... |
| ptpjpre2 23567 | The basis for a product to... |
| ptbasfi 23568 | The basis for the product ... |
| pttop 23569 | The product topology is a ... |
| ptopn 23570 | A basic open set in the pr... |
| ptopn2 23571 | A sub-basic open set in th... |
| xkotf 23572 | Functionality of function ... |
| xkobval 23573 | Alternative expression for... |
| xkoval 23574 | Value of the compact-open ... |
| xkotop 23575 | The compact-open topology ... |
| xkoopn 23576 | A basic open set of the co... |
| txtopi 23577 | The product of two topolog... |
| txtopon 23578 | The underlying set of the ... |
| txuni 23579 | The underlying set of the ... |
| txunii 23580 | The underlying set of the ... |
| ptuni 23581 | The base set for the produ... |
| ptunimpt 23582 | Base set of a product topo... |
| pttopon 23583 | The base set for the produ... |
| pttoponconst 23584 | The base set for a product... |
| ptuniconst 23585 | The base set for a product... |
| xkouni 23586 | The base set of the compac... |
| xkotopon 23587 | The base set of the compac... |
| ptval2 23588 | The value of the product t... |
| txopn 23589 | The product of two open se... |
| txcld 23590 | The product of two closed ... |
| txcls 23591 | Closure of a rectangle in ... |
| txss12 23592 | Subset property of the top... |
| txbasval 23593 | It is sufficient to consid... |
| neitx 23594 | The Cartesian product of t... |
| txcnpi 23595 | Continuity of a two-argume... |
| tx1cn 23596 | Continuity of the first pr... |
| tx2cn 23597 | Continuity of the second p... |
| ptpjcn 23598 | Continuity of a projection... |
| ptpjopn 23599 | The projection map is an o... |
| ptcld 23600 | A closed box in the produc... |
| ptcldmpt 23601 | A closed box in the produc... |
| ptclsg 23602 | The closure of a box in th... |
| ptcls 23603 | The closure of a box in th... |
| dfac14lem 23604 | Lemma for ~ dfac14 . By e... |
| dfac14 23605 | Theorem ~ ptcls is an equi... |
| xkoccn 23606 | The "constant function" fu... |
| txcnp 23607 | If two functions are conti... |
| ptcnplem 23608 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23609 | If every projection of a f... |
| upxp 23610 | Universal property of the ... |
| txcnmpt 23611 | A map into the product of ... |
| uptx 23612 | Universal property of the ... |
| txcn 23613 | A map into the product of ... |
| ptcn 23614 | If every projection of a f... |
| prdstopn 23615 | Topology of a structure pr... |
| prdstps 23616 | A structure product of top... |
| pwstps 23617 | A structure power of a top... |
| txrest 23618 | The subspace of a topologi... |
| txdis 23619 | The topological product of... |
| txindislem 23620 | Lemma for ~ txindis . (Co... |
| txindis 23621 | The topological product of... |
| txdis1cn 23622 | A function is jointly cont... |
| txlly 23623 | If the property ` A ` is p... |
| txnlly 23624 | If the property ` A ` is p... |
| pthaus 23625 | The product of a collectio... |
| ptrescn 23626 | Restriction is a continuou... |
| txtube 23627 | The "tube lemma". If ` X ... |
| txcmplem1 23628 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23629 | Lemma for ~ txcmp . (Cont... |
| txcmp 23630 | The topological product of... |
| txcmpb 23631 | The topological product of... |
| hausdiag 23632 | A topology is Hausdorff if... |
| hauseqlcld 23633 | In a Hausdorff topology, t... |
| txhaus 23634 | The topological product of... |
| txlm 23635 | Two sequences converge iff... |
| lmcn2 23636 | The image of a convergent ... |
| tx1stc 23637 | The topological product of... |
| tx2ndc 23638 | The topological product of... |
| txkgen 23639 | The topological product of... |
| xkohaus 23640 | If the codomain space is H... |
| xkoptsub 23641 | The compact-open topology ... |
| xkopt 23642 | The compact-open topology ... |
| xkopjcn 23643 | Continuity of a projection... |
| xkoco1cn 23644 | If ` F ` is a continuous f... |
| xkoco2cn 23645 | If ` F ` is a continuous f... |
| xkococnlem 23646 | Continuity of the composit... |
| xkococn 23647 | Continuity of the composit... |
| cnmptid 23648 | The identity function is c... |
| cnmptc 23649 | A constant function is con... |
| cnmpt11 23650 | The composition of continu... |
| cnmpt11f 23651 | The composition of continu... |
| cnmpt1t 23652 | The composition of continu... |
| cnmpt12f 23653 | The composition of continu... |
| cnmpt12 23654 | The composition of continu... |
| cnmpt1st 23655 | The projection onto the fi... |
| cnmpt2nd 23656 | The projection onto the se... |
| cnmpt2c 23657 | A constant function is con... |
| cnmpt21 23658 | The composition of continu... |
| cnmpt21f 23659 | The composition of continu... |
| cnmpt2t 23660 | The composition of continu... |
| cnmpt22 23661 | The composition of continu... |
| cnmpt22f 23662 | The composition of continu... |
| cnmpt1res 23663 | The restriction of a conti... |
| cnmpt2res 23664 | The restriction of a conti... |
| cnmptcom 23665 | The argument converse of a... |
| cnmptkc 23666 | The curried first projecti... |
| cnmptkp 23667 | The evaluation of the inne... |
| cnmptk1 23668 | The composition of a curri... |
| cnmpt1k 23669 | The composition of a one-a... |
| cnmptkk 23670 | The composition of two cur... |
| xkofvcn 23671 | Joint continuity of the fu... |
| cnmptk1p 23672 | The evaluation of a currie... |
| cnmptk2 23673 | The uncurrying of a currie... |
| xkoinjcn 23674 | Continuity of "injection",... |
| cnmpt2k 23675 | The currying of a two-argu... |
| txconn 23676 | The topological product of... |
| imasnopn 23677 | If a relation graph is ope... |
| imasncld 23678 | If a relation graph is clo... |
| imasncls 23679 | If a relation graph is clo... |
| qtopval 23682 | Value of the quotient topo... |
| qtopval2 23683 | Value of the quotient topo... |
| elqtop 23684 | Value of the quotient topo... |
| qtopres 23685 | The quotient topology is u... |
| qtoptop2 23686 | The quotient topology is a... |
| qtoptop 23687 | The quotient topology is a... |
| elqtop2 23688 | Value of the quotient topo... |
| qtopuni 23689 | The base set of the quotie... |
| elqtop3 23690 | Value of the quotient topo... |
| qtoptopon 23691 | The base set of the quotie... |
| qtopid 23692 | A quotient map is a contin... |
| idqtop 23693 | The quotient topology indu... |
| qtopcmplem 23694 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23695 | A quotient of a compact sp... |
| qtopconn 23696 | A quotient of a connected ... |
| qtopkgen 23697 | A quotient of a compactly ... |
| basqtop 23698 | An injection maps bases to... |
| tgqtop 23699 | An injection maps generate... |
| qtopcld 23700 | The property of being a cl... |
| qtopcn 23701 | Universal property of a qu... |
| qtopss 23702 | A surjective continuous fu... |
| qtopeu 23703 | Universal property of the ... |
| qtoprest 23704 | If ` A ` is a saturated op... |
| qtopomap 23705 | If ` F ` is a surjective c... |
| qtopcmap 23706 | If ` F ` is a surjective c... |
| imastopn 23707 | The topology of an image s... |
| imastps 23708 | The image of a topological... |
| qustps 23709 | A quotient structure is a ... |
| kqfval 23710 | Value of the function appe... |
| kqfeq 23711 | Two points in the Kolmogor... |
| kqffn 23712 | The topological indistingu... |
| kqval 23713 | Value of the quotient topo... |
| kqtopon 23714 | The Kolmogorov quotient is... |
| kqid 23715 | The topological indistingu... |
| ist0-4 23716 | The topological indistingu... |
| kqfvima 23717 | When the image set is open... |
| kqsat 23718 | Any open set is saturated ... |
| kqdisj 23719 | A version of ~ imain for t... |
| kqcldsat 23720 | Any closed set is saturate... |
| kqopn 23721 | The topological indistingu... |
| kqcld 23722 | The topological indistingu... |
| kqt0lem 23723 | Lemma for ~ kqt0 . (Contr... |
| isr0 23724 | The property " ` J ` is an... |
| r0cld 23725 | The analogue of the T_1 ax... |
| regr1lem 23726 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23727 | A Kolmogorov quotient of a... |
| kqreglem1 23728 | A Kolmogorov quotient of a... |
| kqreglem2 23729 | If the Kolmogorov quotient... |
| kqnrmlem1 23730 | A Kolmogorov quotient of a... |
| kqnrmlem2 23731 | If the Kolmogorov quotient... |
| kqtop 23732 | The Kolmogorov quotient is... |
| kqt0 23733 | The Kolmogorov quotient is... |
| kqf 23734 | The Kolmogorov quotient is... |
| r0sep 23735 | The separation property of... |
| nrmr0reg 23736 | A normal R_0 space is also... |
| regr1 23737 | A regular space is R_1, wh... |
| kqreg 23738 | The Kolmogorov quotient of... |
| kqnrm 23739 | The Kolmogorov quotient of... |
| hmeofn 23744 | The set of homeomorphisms ... |
| hmeofval 23745 | The set of all the homeomo... |
| ishmeo 23746 | The predicate F is a homeo... |
| hmeocn 23747 | A homeomorphism is continu... |
| hmeocnvcn 23748 | The converse of a homeomor... |
| hmeocnv 23749 | The converse of a homeomor... |
| hmeof1o2 23750 | A homeomorphism is a 1-1-o... |
| hmeof1o 23751 | A homeomorphism is a 1-1-o... |
| hmeoima 23752 | The image of an open set b... |
| hmeoopn 23753 | Homeomorphisms preserve op... |
| hmeocld 23754 | Homeomorphisms preserve cl... |
| hmeocls 23755 | Homeomorphisms preserve cl... |
| hmeontr 23756 | Homeomorphisms preserve in... |
| hmeoimaf1o 23757 | The function mapping open ... |
| hmeores 23758 | The restriction of a homeo... |
| hmeoco 23759 | The composite of two homeo... |
| idhmeo 23760 | The identity function is a... |
| hmeocnvb 23761 | The converse of a homeomor... |
| hmeoqtop 23762 | A homeomorphism is a quoti... |
| hmph 23763 | Express the predicate ` J ... |
| hmphi 23764 | If there is a homeomorphis... |
| hmphtop 23765 | Reverse closure for the ho... |
| hmphtop1 23766 | The relation "being homeom... |
| hmphtop2 23767 | The relation "being homeom... |
| hmphref 23768 | "Is homeomorphic to" is re... |
| hmphsym 23769 | "Is homeomorphic to" is sy... |
| hmphtr 23770 | "Is homeomorphic to" is tr... |
| hmpher 23771 | "Is homeomorphic to" is an... |
| hmphen 23772 | Homeomorphisms preserve th... |
| hmphsymb 23773 | "Is homeomorphic to" is sy... |
| haushmphlem 23774 | Lemma for ~ haushmph and s... |
| cmphmph 23775 | Compactness is a topologic... |
| connhmph 23776 | Connectedness is a topolog... |
| t0hmph 23777 | T_0 is a topological prope... |
| t1hmph 23778 | T_1 is a topological prope... |
| haushmph 23779 | Hausdorff-ness is a topolo... |
| reghmph 23780 | Regularity is a topologica... |
| nrmhmph 23781 | Normality is a topological... |
| hmph0 23782 | A topology homeomorphic to... |
| hmphdis 23783 | Homeomorphisms preserve to... |
| hmphindis 23784 | Homeomorphisms preserve to... |
| indishmph 23785 | Equinumerous sets equipped... |
| hmphen2 23786 | Homeomorphisms preserve th... |
| cmphaushmeo 23787 | A continuous bijection fro... |
| ordthmeolem 23788 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23789 | An order isomorphism is a ... |
| txhmeo 23790 | Lift a pair of homeomorphi... |
| txswaphmeolem 23791 | Show inverse for the "swap... |
| txswaphmeo 23792 | There is a homeomorphism f... |
| pt1hmeo 23793 | The canonical homeomorphis... |
| ptuncnv 23794 | Exhibit the converse funct... |
| ptunhmeo 23795 | Define a homeomorphism fro... |
| xpstopnlem1 23796 | The function ` F ` used in... |
| xpstps 23797 | A binary product of topolo... |
| xpstopnlem2 23798 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23799 | The topology on a binary p... |
| ptcmpfi 23800 | A topological product of f... |
| xkocnv 23801 | The inverse of the "curryi... |
| xkohmeo 23802 | The Exponential Law for to... |
| qtopf1 23803 | If a quotient map is injec... |
| qtophmeo 23804 | If two functions on a base... |
| t0kq 23805 | A topological space is T_0... |
| kqhmph 23806 | A topological space is T_0... |
| ist1-5lem 23807 | Lemma for ~ ist1-5 and sim... |
| t1r0 23808 | A T_1 space is R_0. That ... |
| ist1-5 23809 | A topological space is T_1... |
| ishaus3 23810 | A topological space is Hau... |
| nrmreg 23811 | A normal T_1 space is regu... |
| reghaus 23812 | A regular T_0 space is Hau... |
| nrmhaus 23813 | A T_1 normal space is Haus... |
| elmptrab 23814 | Membership in a one-parame... |
| elmptrab2 23815 | Membership in a one-parame... |
| isfbas 23816 | The predicate " ` F ` is a... |
| fbasne0 23817 | There are no empty filter ... |
| 0nelfb 23818 | No filter base contains th... |
| fbsspw 23819 | A filter base on a set is ... |
| fbelss 23820 | An element of the filter b... |
| fbdmn0 23821 | The domain of a filter bas... |
| isfbas2 23822 | The predicate " ` F ` is a... |
| fbasssin 23823 | A filter base contains sub... |
| fbssfi 23824 | A filter base contains sub... |
| fbssint 23825 | A filter base contains sub... |
| fbncp 23826 | A filter base does not con... |
| fbun 23827 | A necessary and sufficient... |
| fbfinnfr 23828 | No filter base containing ... |
| opnfbas 23829 | The collection of open sup... |
| trfbas2 23830 | Conditions for the trace o... |
| trfbas 23831 | Conditions for the trace o... |
| isfil 23834 | The predicate "is a filter... |
| filfbas 23835 | A filter is a filter base.... |
| 0nelfil 23836 | The empty set doesn't belo... |
| fileln0 23837 | An element of a filter is ... |
| filsspw 23838 | A filter is a subset of th... |
| filelss 23839 | An element of a filter is ... |
| filss 23840 | A filter is closed under t... |
| filin 23841 | A filter is closed under t... |
| filtop 23842 | The underlying set belongs... |
| isfil2 23843 | Derive the standard axioms... |
| isfildlem 23844 | Lemma for ~ isfild . (Con... |
| isfild 23845 | Sufficient condition for a... |
| filfi 23846 | A filter is closed under t... |
| filinn0 23847 | The intersection of two el... |
| filintn0 23848 | A filter has the finite in... |
| filn0 23849 | The empty set is not a fil... |
| infil 23850 | The intersection of two fi... |
| snfil 23851 | A singleton is a filter. ... |
| fbasweak 23852 | A filter base on any set i... |
| snfbas 23853 | Condition for a singleton ... |
| fsubbas 23854 | A condition for a set to g... |
| fbasfip 23855 | A filter base has the fini... |
| fbunfip 23856 | A helpful lemma for showin... |
| fgval 23857 | The filter generating clas... |
| elfg 23858 | A condition for elements o... |
| ssfg 23859 | A filter base is a subset ... |
| fgss 23860 | A bigger base generates a ... |
| fgss2 23861 | A condition for a filter t... |
| fgfil 23862 | A filter generates itself.... |
| elfilss 23863 | An element belongs to a fi... |
| filfinnfr 23864 | No filter containing a fin... |
| fgcl 23865 | A generated filter is a fi... |
| fgabs 23866 | Absorption law for filter ... |
| neifil 23867 | The neighborhoods of a non... |
| filunibas 23868 | Recover the base set from ... |
| filunirn 23869 | Two ways to express a filt... |
| filconn 23870 | A filter gives rise to a c... |
| fbasrn 23871 | Given a filter on a domain... |
| filuni 23872 | The union of a nonempty se... |
| trfil1 23873 | Conditions for the trace o... |
| trfil2 23874 | Conditions for the trace o... |
| trfil3 23875 | Conditions for the trace o... |
| trfilss 23876 | If ` A ` is a member of th... |
| fgtr 23877 | If ` A ` is a member of th... |
| trfg 23878 | The trace operation and th... |
| trnei 23879 | The trace, over a set ` A ... |
| cfinfil 23880 | Relative complements of th... |
| csdfil 23881 | The set of all elements wh... |
| supfil 23882 | The supersets of a nonempt... |
| zfbas 23883 | The set of upper sets of i... |
| uzrest 23884 | The restriction of the set... |
| uzfbas 23885 | The set of upper sets of i... |
| isufil 23890 | The property of being an u... |
| ufilfil 23891 | An ultrafilter is a filter... |
| ufilss 23892 | For any subset of the base... |
| ufilb 23893 | The complement is in an ul... |
| ufilmax 23894 | Any filter finer than an u... |
| isufil2 23895 | The maximal property of an... |
| ufprim 23896 | An ultrafilter is a prime ... |
| trufil 23897 | Conditions for the trace o... |
| filssufilg 23898 | A filter is contained in s... |
| filssufil 23899 | A filter is contained in s... |
| isufl 23900 | Define the (strong) ultraf... |
| ufli 23901 | Property of a set that sat... |
| numufl 23902 | Consequence of ~ filssufil... |
| fiufl 23903 | A finite set satisfies the... |
| acufl 23904 | The axiom of choice implie... |
| ssufl 23905 | If ` Y ` is a subset of ` ... |
| ufileu 23906 | If the ultrafilter contain... |
| filufint 23907 | A filter is equal to the i... |
| uffix 23908 | Lemma for ~ fixufil and ~ ... |
| fixufil 23909 | The condition describing a... |
| uffixfr 23910 | An ultrafilter is either f... |
| uffix2 23911 | A classification of fixed ... |
| uffixsn 23912 | The singleton of the gener... |
| ufildom1 23913 | An ultrafilter is generate... |
| uffinfix 23914 | An ultrafilter containing ... |
| cfinufil 23915 | An ultrafilter is free iff... |
| ufinffr 23916 | An infinite subset is cont... |
| ufilen 23917 | Any infinite set has an ul... |
| ufildr 23918 | An ultrafilter gives rise ... |
| fin1aufil 23919 | There are no definable fre... |
| fmval 23930 | Introduce a function that ... |
| fmfil 23931 | A mapping filter is a filt... |
| fmf 23932 | Pushing-forward via a func... |
| fmss 23933 | A finer filter produces a ... |
| elfm 23934 | An element of a mapping fi... |
| elfm2 23935 | An element of a mapping fi... |
| fmfg 23936 | The image filter of a filt... |
| elfm3 23937 | An alternate formulation o... |
| imaelfm 23938 | An image of a filter eleme... |
| rnelfmlem 23939 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23940 | A condition for a filter t... |
| fmfnfmlem1 23941 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23942 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23943 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23944 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23945 | A filter finer than an ima... |
| fmufil 23946 | An image filter of an ultr... |
| fmid 23947 | The filter map applied to ... |
| fmco 23948 | Composition of image filte... |
| ufldom 23949 | The ultrafilter lemma prop... |
| flimval 23950 | The set of limit points of... |
| elflim2 23951 | The predicate "is a limit ... |
| flimtop 23952 | Reverse closure for the li... |
| flimneiss 23953 | A filter contains the neig... |
| flimnei 23954 | A filter contains all of t... |
| flimelbas 23955 | A limit point of a filter ... |
| flimfil 23956 | Reverse closure for the li... |
| flimtopon 23957 | Reverse closure for the li... |
| elflim 23958 | The predicate "is a limit ... |
| flimss2 23959 | A limit point of a filter ... |
| flimss1 23960 | A limit point of a filter ... |
| neiflim 23961 | A point is a limit point o... |
| flimopn 23962 | The condition for being a ... |
| fbflim 23963 | A condition for a filter t... |
| fbflim2 23964 | A condition for a filter b... |
| flimclsi 23965 | The convergent points of a... |
| hausflimlem 23966 | If ` A ` and ` B ` are bot... |
| hausflimi 23967 | One direction of ~ hausfli... |
| hausflim 23968 | A condition for a topology... |
| flimcf 23969 | Fineness is properly chara... |
| flimrest 23970 | The set of limit points in... |
| flimclslem 23971 | Lemma for ~ flimcls . (Co... |
| flimcls 23972 | Closure in terms of filter... |
| flimsncls 23973 | If ` A ` is a limit point ... |
| hauspwpwf1 23974 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23975 | If ` X ` is a Hausdorff sp... |
| flffval 23976 | Given a topology and a fil... |
| flfval 23977 | Given a function from a fi... |
| flfnei 23978 | The property of being a li... |
| flfneii 23979 | A neighborhood of a limit ... |
| isflf 23980 | The property of being a li... |
| flfelbas 23981 | A limit point of a functio... |
| flffbas 23982 | Limit points of a function... |
| flftg 23983 | Limit points of a function... |
| hausflf 23984 | If a function has its valu... |
| hausflf2 23985 | If a convergent function h... |
| cnpflfi 23986 | Forward direction of ~ cnp... |
| cnpflf2 23987 | ` F ` is continuous at poi... |
| cnpflf 23988 | Continuity of a function a... |
| cnflf 23989 | A function is continuous i... |
| cnflf2 23990 | A function is continuous i... |
| flfcnp 23991 | A continuous function pres... |
| lmflf 23992 | The topological limit rela... |
| txflf 23993 | Two sequences converge in ... |
| flfcnp2 23994 | The image of a convergent ... |
| fclsval 23995 | The set of all cluster poi... |
| isfcls 23996 | A cluster point of a filte... |
| fclsfil 23997 | Reverse closure for the cl... |
| fclstop 23998 | Reverse closure for the cl... |
| fclstopon 23999 | Reverse closure for the cl... |
| isfcls2 24000 | A cluster point of a filte... |
| fclsopn 24001 | Write the cluster point co... |
| fclsopni 24002 | An open neighborhood of a ... |
| fclselbas 24003 | A cluster point is in the ... |
| fclsneii 24004 | A neighborhood of a cluste... |
| fclssscls 24005 | The set of cluster points ... |
| fclsnei 24006 | Cluster points in terms of... |
| supnfcls 24007 | The filter of supersets of... |
| fclsbas 24008 | Cluster points in terms of... |
| fclsss1 24009 | A finer topology has fewer... |
| fclsss2 24010 | A finer filter has fewer c... |
| fclsrest 24011 | The set of cluster points ... |
| fclscf 24012 | Characterization of finene... |
| flimfcls 24013 | A limit point is a cluster... |
| fclsfnflim 24014 | A filter clusters at a poi... |
| flimfnfcls 24015 | A filter converges to a po... |
| fclscmpi 24016 | Forward direction of ~ fcl... |
| fclscmp 24017 | A space is compact iff eve... |
| uffclsflim 24018 | The cluster points of an u... |
| ufilcmp 24019 | A space is compact iff eve... |
| fcfval 24020 | The set of cluster points ... |
| isfcf 24021 | The property of being a cl... |
| fcfnei 24022 | The property of being a cl... |
| fcfelbas 24023 | A cluster point of a funct... |
| fcfneii 24024 | A neighborhood of a cluste... |
| flfssfcf 24025 | A limit point of a functio... |
| uffcfflf 24026 | If the domain filter is an... |
| cnpfcfi 24027 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 24028 | A function ` F ` is contin... |
| cnfcf 24029 | Continuity of a function i... |
| flfcntr 24030 | A continuous function's va... |
| alexsublem 24031 | Lemma for ~ alexsub . (Co... |
| alexsub 24032 | The Alexander Subbase Theo... |
| alexsubb 24033 | Biconditional form of the ... |
| alexsubALTlem1 24034 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 24035 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 24036 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 24037 | Lemma for ~ alexsubALT . ... |
| alexsubALT 24038 | The Alexander Subbase Theo... |
| ptcmplem1 24039 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 24040 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 24041 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 24042 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 24043 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24044 | Tychonoff's theorem: The ... |
| ptcmp 24045 | Tychonoff's theorem: The ... |
| cnextval 24048 | The function applying cont... |
| cnextfval 24049 | The continuous extension o... |
| cnextrel 24050 | In the general case, a con... |
| cnextfun 24051 | If the target space is Hau... |
| cnextfvval 24052 | The value of the continuou... |
| cnextf 24053 | Extension by continuity. ... |
| cnextcn 24054 | Extension by continuity. ... |
| cnextfres1 24055 | ` F ` and its extension by... |
| cnextfres 24056 | ` F ` and its extension by... |
| istmd 24061 | The predicate "is a topolo... |
| tmdmnd 24062 | A topological monoid is a ... |
| tmdtps 24063 | A topological monoid is a ... |
| istgp 24064 | The predicate "is a topolo... |
| tgpgrp 24065 | A topological group is a g... |
| tgptmd 24066 | A topological group is a t... |
| tgptps 24067 | A topological group is a t... |
| tmdtopon 24068 | The topology of a topologi... |
| tgptopon 24069 | The topology of a topologi... |
| tmdcn 24070 | In a topological monoid, t... |
| tgpcn 24071 | In a topological group, th... |
| tgpinv 24072 | In a topological group, th... |
| grpinvhmeo 24073 | The inverse function in a ... |
| cnmpt1plusg 24074 | Continuity of the group su... |
| cnmpt2plusg 24075 | Continuity of the group su... |
| tmdcn2 24076 | Write out the definition o... |
| tgpsubcn 24077 | In a topological group, th... |
| istgp2 24078 | A group with a topology is... |
| tmdmulg 24079 | In a topological monoid, t... |
| tgpmulg 24080 | In a topological group, th... |
| tgpmulg2 24081 | In a topological monoid, t... |
| tmdgsum 24082 | In a topological monoid, t... |
| tmdgsum2 24083 | For any neighborhood ` U `... |
| oppgtmd 24084 | The opposite of a topologi... |
| oppgtgp 24085 | The opposite of a topologi... |
| distgp 24086 | Any group equipped with th... |
| indistgp 24087 | Any group equipped with th... |
| efmndtmd 24088 | The monoid of endofunction... |
| tmdlactcn 24089 | The left group action of e... |
| tgplacthmeo 24090 | The left group action of e... |
| submtmd 24091 | A submonoid of a topologic... |
| subgtgp 24092 | A subgroup of a topologica... |
| symgtgp 24093 | The symmetric group is a t... |
| subgntr 24094 | A subgroup of a topologica... |
| opnsubg 24095 | An open subgroup of a topo... |
| clssubg 24096 | The closure of a subgroup ... |
| clsnsg 24097 | The closure of a normal su... |
| cldsubg 24098 | A subgroup of finite index... |
| tgpconncompeqg 24099 | The connected component co... |
| tgpconncomp 24100 | The identity component, th... |
| tgpconncompss 24101 | The identity component is ... |
| ghmcnp 24102 | A group homomorphism on to... |
| snclseqg 24103 | The coset of the closure o... |
| tgphaus 24104 | A topological group is Hau... |
| tgpt1 24105 | Hausdorff and T1 are equiv... |
| tgpt0 24106 | Hausdorff and T0 are equiv... |
| qustgpopn 24107 | A quotient map in a topolo... |
| qustgplem 24108 | Lemma for ~ qustgp . (Con... |
| qustgp 24109 | The quotient of a topologi... |
| qustgphaus 24110 | The quotient of a topologi... |
| prdstmdd 24111 | The product of a family of... |
| prdstgpd 24112 | The product of a family of... |
| tsmsfbas 24115 | The collection of all sets... |
| tsmslem1 24116 | The finite partial sums of... |
| tsmsval2 24117 | Definition of the topologi... |
| tsmsval 24118 | Definition of the topologi... |
| tsmspropd 24119 | The group sum depends only... |
| eltsms 24120 | The property of being a su... |
| tsmsi 24121 | The property of being a su... |
| tsmscl 24122 | A sum in a topological gro... |
| haustsms 24123 | In a Hausdorff topological... |
| haustsms2 24124 | In a Hausdorff topological... |
| tsmscls 24125 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24126 | The convergent points of a... |
| tsmsid 24127 | If a sum is finite, the us... |
| haustsmsid 24128 | In a Hausdorff topological... |
| tsms0 24129 | The sum of zero is zero. ... |
| tsmssubm 24130 | Evaluate an infinite group... |
| tsmsres 24131 | Extend an infinite group s... |
| tsmsf1o 24132 | Re-index an infinite group... |
| tsmsmhm 24133 | Apply a continuous group h... |
| tsmsadd 24134 | The sum of two infinite gr... |
| tsmsinv 24135 | Inverse of an infinite gro... |
| tsmssub 24136 | The difference of two infi... |
| tgptsmscls 24137 | A sum in a topological gro... |
| tgptsmscld 24138 | The set of limit points to... |
| tsmssplit 24139 | Split a topological group ... |
| tsmsxplem1 24140 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24141 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24142 | Write a sum over a two-dim... |
| istrg 24151 | Express the predicate " ` ... |
| trgtmd 24152 | The multiplicative monoid ... |
| istdrg 24153 | Express the predicate " ` ... |
| tdrgunit 24154 | The unit group of a topolo... |
| trgtgp 24155 | A topological ring is a to... |
| trgtmd2 24156 | A topological ring is a to... |
| trgtps 24157 | A topological ring is a to... |
| trgring 24158 | A topological ring is a ri... |
| trggrp 24159 | A topological ring is a gr... |
| tdrgtrg 24160 | A topological division rin... |
| tdrgdrng 24161 | A topological division rin... |
| tdrgring 24162 | A topological division rin... |
| tdrgtmd 24163 | A topological division rin... |
| tdrgtps 24164 | A topological division rin... |
| istdrg2 24165 | A topological-ring divisio... |
| mulrcn 24166 | The functionalization of t... |
| invrcn2 24167 | The multiplicative inverse... |
| invrcn 24168 | The multiplicative inverse... |
| cnmpt1mulr 24169 | Continuity of ring multipl... |
| cnmpt2mulr 24170 | Continuity of ring multipl... |
| dvrcn 24171 | The division function is c... |
| istlm 24172 | The predicate " ` W ` is a... |
| vscacn 24173 | The scalar multiplication ... |
| tlmtmd 24174 | A topological module is a ... |
| tlmtps 24175 | A topological module is a ... |
| tlmlmod 24176 | A topological module is a ... |
| tlmtrg 24177 | The scalar ring of a topol... |
| tlmscatps 24178 | The scalar ring of a topol... |
| istvc 24179 | A topological vector space... |
| tvctdrg 24180 | The scalar field of a topo... |
| cnmpt1vsca 24181 | Continuity of scalar multi... |
| cnmpt2vsca 24182 | Continuity of scalar multi... |
| tlmtgp 24183 | A topological vector space... |
| tvctlm 24184 | A topological vector space... |
| tvclmod 24185 | A topological vector space... |
| tvclvec 24186 | A topological vector space... |
| ustfn 24189 | The defined uniform struct... |
| ustval 24190 | The class of all uniform s... |
| isust 24191 | The predicate " ` U ` is a... |
| ustssxp 24192 | Entourages are subsets of ... |
| ustssel 24193 | A uniform structure is upw... |
| ustbasel 24194 | The full set is always an ... |
| ustincl 24195 | A uniform structure is clo... |
| ustdiag 24196 | The diagonal set is includ... |
| ustinvel 24197 | If ` V ` is an entourage, ... |
| ustexhalf 24198 | For each entourage ` V ` t... |
| ustrel 24199 | The elements of uniform st... |
| ustfilxp 24200 | A uniform structure on a n... |
| ustne0 24201 | A uniform structure cannot... |
| ustssco 24202 | In an uniform structure, a... |
| ustexsym 24203 | In an uniform structure, f... |
| ustex2sym 24204 | In an uniform structure, f... |
| ustex3sym 24205 | In an uniform structure, f... |
| ustref 24206 | Any element of the base se... |
| ust0 24207 | The unique uniform structu... |
| ustn0 24208 | The empty set is not an un... |
| ustund 24209 | If two intersecting sets `... |
| ustelimasn 24210 | Any point ` A ` is near en... |
| ustneism 24211 | For a point ` A ` in ` X `... |
| ustbas2 24212 | Second direction for ~ ust... |
| ustuni 24213 | The set union of a uniform... |
| ustbas 24214 | Recover the base of an uni... |
| ustimasn 24215 | Lemma for ~ ustuqtop . (C... |
| trust 24216 | The trace of a uniform str... |
| utopval 24219 | The topology induced by a ... |
| elutop 24220 | Open sets in the topology ... |
| utoptop 24221 | The topology induced by a ... |
| utopbas 24222 | The base of the topology i... |
| utoptopon 24223 | Topology induced by a unif... |
| restutop 24224 | Restriction of a topology ... |
| restutopopn 24225 | The restriction of the top... |
| ustuqtoplem 24226 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24227 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24228 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24229 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24230 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24231 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24232 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24233 | For a given uniform struct... |
| utopsnneiplem 24234 | The neighborhoods of a poi... |
| utopsnneip 24235 | The neighborhoods of a poi... |
| utopsnnei 24236 | Images of singletons by en... |
| utop2nei 24237 | For any symmetrical entour... |
| utop3cls 24238 | Relation between a topolog... |
| utopreg 24239 | All Hausdorff uniform spac... |
| ussval 24246 | The uniform structure on u... |
| ussid 24247 | In case the base of the ` ... |
| isusp 24248 | The predicate ` W ` is a u... |
| ressuss 24249 | Value of the uniform struc... |
| ressust 24250 | The uniform structure of a... |
| ressusp 24251 | The restriction of a unifo... |
| tusval 24252 | The value of the uniform s... |
| tuslem 24253 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24254 | The base set of a construc... |
| tusunif 24255 | The uniform structure of a... |
| tususs 24256 | The uniform structure of a... |
| tustopn 24257 | The topology induced by a ... |
| tususp 24258 | A constructed uniform spac... |
| tustps 24259 | A constructed uniform spac... |
| uspreg 24260 | If a uniform space is Haus... |
| ucnval 24263 | The set of all uniformly c... |
| isucn 24264 | The predicate " ` F ` is a... |
| isucn2 24265 | The predicate " ` F ` is a... |
| ucnimalem 24266 | Reformulate the ` G ` func... |
| ucnima 24267 | An equivalent statement of... |
| ucnprima 24268 | The preimage by a uniforml... |
| iducn 24269 | The identity is uniformly ... |
| cstucnd 24270 | A constant function is uni... |
| ucncn 24271 | Uniform continuity implies... |
| iscfilu 24274 | The predicate " ` F ` is a... |
| cfilufbas 24275 | A Cauchy filter base is a ... |
| cfiluexsm 24276 | For a Cauchy filter base a... |
| fmucndlem 24277 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24278 | The image of a Cauchy filt... |
| cfilufg 24279 | The filter generated by a ... |
| trcfilu 24280 | Condition for the trace of... |
| cfiluweak 24281 | A Cauchy filter base is al... |
| neipcfilu 24282 | In an uniform space, a nei... |
| iscusp 24285 | The predicate " ` W ` is a... |
| cuspusp 24286 | A complete uniform space i... |
| cuspcvg 24287 | In a complete uniform spac... |
| iscusp2 24288 | The predicate " ` W ` is a... |
| cnextucn 24289 | Extension by continuity. ... |
| ucnextcn 24290 | Extension by continuity. ... |
| ispsmet 24291 | Express the predicate " ` ... |
| psmetdmdm 24292 | Recover the base set from ... |
| psmetf 24293 | The distance function of a... |
| psmetcl 24294 | Closure of the distance fu... |
| psmet0 24295 | The distance function of a... |
| psmettri2 24296 | Triangle inequality for th... |
| psmetsym 24297 | The distance function of a... |
| psmettri 24298 | Triangle inequality for th... |
| psmetge0 24299 | The distance function of a... |
| psmetxrge0 24300 | The distance function of a... |
| psmetres2 24301 | Restriction of a pseudomet... |
| psmetlecl 24302 | Real closure of an extende... |
| distspace 24303 | A set ` X ` together with ... |
| ismet 24310 | Express the predicate " ` ... |
| isxmet 24311 | Express the predicate " ` ... |
| ismeti 24312 | Properties that determine ... |
| isxmetd 24313 | Properties that determine ... |
| isxmet2d 24314 | It is safe to only require... |
| metflem 24315 | Lemma for ~ metf and other... |
| xmetf 24316 | Mapping of the distance fu... |
| metf 24317 | Mapping of the distance fu... |
| xmetcl 24318 | Closure of the distance fu... |
| metcl 24319 | Closure of the distance fu... |
| ismet2 24320 | An extended metric is a me... |
| metxmet 24321 | A metric is an extended me... |
| xmetdmdm 24322 | Recover the base set from ... |
| metdmdm 24323 | Recover the base set from ... |
| xmetunirn 24324 | Two ways to express an ext... |
| xmeteq0 24325 | The value of an extended m... |
| meteq0 24326 | The value of a metric is z... |
| xmettri2 24327 | Triangle inequality for th... |
| mettri2 24328 | Triangle inequality for th... |
| xmet0 24329 | The distance function of a... |
| met0 24330 | The distance function of a... |
| xmetge0 24331 | The distance function of a... |
| metge0 24332 | The distance function of a... |
| xmetlecl 24333 | Real closure of an extende... |
| xmetsym 24334 | The distance function of a... |
| xmetpsmet 24335 | An extended metric is a ps... |
| xmettpos 24336 | The distance function of a... |
| metsym 24337 | The distance function of a... |
| xmettri 24338 | Triangle inequality for th... |
| mettri 24339 | Triangle inequality for th... |
| xmettri3 24340 | Triangle inequality for th... |
| mettri3 24341 | Triangle inequality for th... |
| xmetrtri 24342 | One half of the reverse tr... |
| xmetrtri2 24343 | The reverse triangle inequ... |
| metrtri 24344 | Reverse triangle inequalit... |
| xmetgt0 24345 | The distance function of a... |
| metgt0 24346 | The distance function of a... |
| metn0 24347 | A metric space is nonempty... |
| xmetres2 24348 | Restriction of an extended... |
| metreslem 24349 | Lemma for ~ metres . (Con... |
| metres2 24350 | Lemma for ~ metres . (Con... |
| xmetres 24351 | A restriction of an extend... |
| metres 24352 | A restriction of a metric ... |
| 0met 24353 | The empty metric. (Contri... |
| prdsdsf 24354 | The product metric is a fu... |
| prdsxmetlem 24355 | The product metric is an e... |
| prdsxmet 24356 | The product metric is an e... |
| prdsmet 24357 | The product metric is a me... |
| ressprdsds 24358 | Restriction of a product m... |
| resspwsds 24359 | Restriction of a power met... |
| imasdsf1olem 24360 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24361 | The distance function is t... |
| imasf1oxmet 24362 | The image of an extended m... |
| imasf1omet 24363 | The image of a metric is a... |
| xpsdsfn 24364 | Closure of the metric in a... |
| xpsdsfn2 24365 | Closure of the metric in a... |
| xpsxmetlem 24366 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24367 | A product metric of extend... |
| xpsdsval 24368 | Value of the metric in a b... |
| xpsmet 24369 | The direct product of two ... |
| blfvalps 24370 | The value of the ball func... |
| blfval 24371 | The value of the ball func... |
| blvalps 24372 | The ball around a point ` ... |
| blval 24373 | The ball around a point ` ... |
| elblps 24374 | Membership in a ball. (Co... |
| elbl 24375 | Membership in a ball. (Co... |
| elbl2ps 24376 | Membership in a ball. (Co... |
| elbl2 24377 | Membership in a ball. (Co... |
| elbl3ps 24378 | Membership in a ball, with... |
| elbl3 24379 | Membership in a ball, with... |
| blcomps 24380 | Commute the arguments to t... |
| blcom 24381 | Commute the arguments to t... |
| xblpnfps 24382 | The infinity ball in an ex... |
| xblpnf 24383 | The infinity ball in an ex... |
| blpnf 24384 | The infinity ball in a sta... |
| bldisj 24385 | Two balls are disjoint if ... |
| blgt0 24386 | A nonempty ball implies th... |
| bl2in 24387 | Two balls are disjoint if ... |
| xblss2ps 24388 | One ball is contained in a... |
| xblss2 24389 | One ball is contained in a... |
| blss2ps 24390 | One ball is contained in a... |
| blss2 24391 | One ball is contained in a... |
| blhalf 24392 | A ball of radius ` R / 2 `... |
| blfps 24393 | Mapping of a ball. (Contr... |
| blf 24394 | Mapping of a ball. (Contr... |
| blrnps 24395 | Membership in the range of... |
| blrn 24396 | Membership in the range of... |
| xblcntrps 24397 | A ball contains its center... |
| xblcntr 24398 | A ball contains its center... |
| blcntrps 24399 | A ball contains its center... |
| blcntr 24400 | A ball contains its center... |
| xbln0 24401 | A ball is nonempty iff the... |
| bln0 24402 | A ball is not empty. (Con... |
| blelrnps 24403 | A ball belongs to the set ... |
| blelrn 24404 | A ball belongs to the set ... |
| blssm 24405 | A ball is a subset of the ... |
| unirnblps 24406 | The union of the set of ba... |
| unirnbl 24407 | The union of the set of ba... |
| blin 24408 | The intersection of two ba... |
| ssblps 24409 | The size of a ball increas... |
| ssbl 24410 | The size of a ball increas... |
| blssps 24411 | Any point ` P ` in a ball ... |
| blss 24412 | Any point ` P ` in a ball ... |
| blssexps 24413 | Two ways to express the ex... |
| blssex 24414 | Two ways to express the ex... |
| ssblex 24415 | A nested ball exists whose... |
| blin2 24416 | Given any two balls and a ... |
| blbas 24417 | The balls of a metric spac... |
| blres 24418 | A ball in a restricted met... |
| xmeterval 24419 | Value of the "finitely sep... |
| xmeter 24420 | The "finitely separated" r... |
| xmetec 24421 | The equivalence classes un... |
| blssec 24422 | A ball centered at ` P ` i... |
| blpnfctr 24423 | The infinity ball in an ex... |
| xmetresbl 24424 | An extended metric restric... |
| mopnval 24425 | An open set is a subset of... |
| mopntopon 24426 | The set of open sets of a ... |
| mopntop 24427 | The set of open sets of a ... |
| mopnuni 24428 | The union of all open sets... |
| elmopn 24429 | The defining property of a... |
| mopnfss 24430 | The family of open sets of... |
| mopnm 24431 | The base set of a metric s... |
| elmopn2 24432 | A defining property of an ... |
| mopnss 24433 | An open set of a metric sp... |
| isxms 24434 | Express the predicate " ` ... |
| isxms2 24435 | Express the predicate " ` ... |
| isms 24436 | Express the predicate " ` ... |
| isms2 24437 | Express the predicate " ` ... |
| xmstopn 24438 | The topology component of ... |
| mstopn 24439 | The topology component of ... |
| xmstps 24440 | An extended metric space i... |
| msxms 24441 | A metric space is an exten... |
| mstps 24442 | A metric space is a topolo... |
| xmsxmet 24443 | The distance function, sui... |
| msmet 24444 | The distance function, sui... |
| msf 24445 | The distance function of a... |
| xmsxmet2 24446 | The distance function, sui... |
| msmet2 24447 | The distance function, sui... |
| mscl 24448 | Closure of the distance fu... |
| xmscl 24449 | Closure of the distance fu... |
| xmsge0 24450 | The distance function in a... |
| xmseq0 24451 | The distance between two p... |
| xmssym 24452 | The distance function in a... |
| xmstri2 24453 | Triangle inequality for th... |
| mstri2 24454 | Triangle inequality for th... |
| xmstri 24455 | Triangle inequality for th... |
| mstri 24456 | Triangle inequality for th... |
| xmstri3 24457 | Triangle inequality for th... |
| mstri3 24458 | Triangle inequality for th... |
| msrtri 24459 | Reverse triangle inequalit... |
| xmspropd 24460 | Property deduction for an ... |
| mspropd 24461 | Property deduction for a m... |
| setsmsbas 24462 | The base set of a construc... |
| setsmsds 24463 | The distance function of a... |
| setsmstset 24464 | The topology of a construc... |
| setsmstopn 24465 | The topology of a construc... |
| setsxms 24466 | The constructed metric spa... |
| setsms 24467 | The constructed metric spa... |
| tmsval 24468 | For any metric there is an... |
| tmslem 24469 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24470 | The base set of a construc... |
| tmsds 24471 | The metric of a constructe... |
| tmstopn 24472 | The topology of a construc... |
| tmsxms 24473 | The constructed metric spa... |
| tmsms 24474 | The constructed metric spa... |
| imasf1obl 24475 | The image of a metric spac... |
| imasf1oxms 24476 | The image of a metric spac... |
| imasf1oms 24477 | The image of a metric spac... |
| prdsbl 24478 | A ball in the product metr... |
| mopni 24479 | An open set of a metric sp... |
| mopni2 24480 | An open set of a metric sp... |
| mopni3 24481 | An open set of a metric sp... |
| blssopn 24482 | The balls of a metric spac... |
| unimopn 24483 | The union of a collection ... |
| mopnin 24484 | The intersection of two op... |
| mopn0 24485 | The empty set is an open s... |
| rnblopn 24486 | A ball of a metric space i... |
| blopn 24487 | A ball of a metric space i... |
| neibl 24488 | The neighborhoods around a... |
| blnei 24489 | A ball around a point is a... |
| lpbl 24490 | Every ball around a limit ... |
| blsscls2 24491 | A smaller closed ball is c... |
| blcld 24492 | A "closed ball" in a metri... |
| blcls 24493 | The closure of an open bal... |
| blsscls 24494 | If two concentric balls ha... |
| metss 24495 | Two ways of saying that me... |
| metequiv 24496 | Two ways of saying that tw... |
| metequiv2 24497 | If there is a sequence of ... |
| metss2lem 24498 | Lemma for ~ metss2 . (Con... |
| metss2 24499 | If the metric ` D ` is "st... |
| comet 24500 | The composition of an exte... |
| stdbdmetval 24501 | Value of the standard boun... |
| stdbdxmet 24502 | The standard bounded metri... |
| stdbdmet 24503 | The standard bounded metri... |
| stdbdbl 24504 | The standard bounded metri... |
| stdbdmopn 24505 | The standard bounded metri... |
| mopnex 24506 | The topology generated by ... |
| methaus 24507 | The topology generated by ... |
| met1stc 24508 | The topology generated by ... |
| met2ndci 24509 | A separable metric space (... |
| met2ndc 24510 | A metric space is second-c... |
| metrest 24511 | Two alternate formulations... |
| ressxms 24512 | The restriction of a metri... |
| ressms 24513 | The restriction of a metri... |
| prdsmslem1 24514 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24515 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24516 | Lemma for ~ prdsxms . The... |
| prdsxms 24517 | The indexed product struct... |
| prdsms 24518 | The indexed product struct... |
| pwsxms 24519 | A power of an extended met... |
| pwsms 24520 | A power of a metric space ... |
| xpsxms 24521 | A binary product of metric... |
| xpsms 24522 | A binary product of metric... |
| tmsxps 24523 | Express the product of two... |
| tmsxpsmopn 24524 | Express the product of two... |
| tmsxpsval 24525 | Value of the product of tw... |
| tmsxpsval2 24526 | Value of the product of tw... |
| metcnp3 24527 | Two ways to express that `... |
| metcnp 24528 | Two ways to say a mapping ... |
| metcnp2 24529 | Two ways to say a mapping ... |
| metcn 24530 | Two ways to say a mapping ... |
| metcnpi 24531 | Epsilon-delta property of ... |
| metcnpi2 24532 | Epsilon-delta property of ... |
| metcnpi3 24533 | Epsilon-delta property of ... |
| txmetcnp 24534 | Continuity of a binary ope... |
| txmetcn 24535 | Continuity of a binary ope... |
| metuval 24536 | Value of the uniform struc... |
| metustel 24537 | Define a filter base ` F `... |
| metustss 24538 | Range of the elements of t... |
| metustrel 24539 | Elements of the filter bas... |
| metustto 24540 | Any two elements of the fi... |
| metustid 24541 | The identity diagonal is i... |
| metustsym 24542 | Elements of the filter bas... |
| metustexhalf 24543 | For any element ` A ` of t... |
| metustfbas 24544 | The filter base generated ... |
| metust 24545 | The uniform structure gene... |
| cfilucfil 24546 | Given a metric ` D ` and a... |
| metuust 24547 | The uniform structure gene... |
| cfilucfil2 24548 | Given a metric ` D ` and a... |
| blval2 24549 | The ball around a point ` ... |
| elbl4 24550 | Membership in a ball, alte... |
| metuel 24551 | Elementhood in the uniform... |
| metuel2 24552 | Elementhood in the uniform... |
| metustbl 24553 | The "section" image of an ... |
| psmetutop 24554 | The topology induced by a ... |
| xmetutop 24555 | The topology induced by a ... |
| xmsusp 24556 | If the uniform set of a me... |
| restmetu 24557 | The uniform structure gene... |
| metucn 24558 | Uniform continuity in metr... |
| dscmet 24559 | The discrete metric on any... |
| dscopn 24560 | The discrete metric genera... |
| nrmmetd 24561 | Show that a group norm gen... |
| abvmet 24562 | An absolute value ` F ` ge... |
| nmfval 24575 | The value of the norm func... |
| nmval 24576 | The value of the norm as t... |
| nmfval0 24577 | The value of the norm func... |
| nmfval2 24578 | The value of the norm func... |
| nmval2 24579 | The value of the norm on a... |
| nmf2 24580 | The norm on a metric group... |
| nmpropd 24581 | Weak property deduction fo... |
| nmpropd2 24582 | Strong property deduction ... |
| isngp 24583 | The property of being a no... |
| isngp2 24584 | The property of being a no... |
| isngp3 24585 | The property of being a no... |
| ngpgrp 24586 | A normed group is a group.... |
| ngpms 24587 | A normed group is a metric... |
| ngpxms 24588 | A normed group is an exten... |
| ngptps 24589 | A normed group is a topolo... |
| ngpmet 24590 | The (induced) metric of a ... |
| ngpds 24591 | Value of the distance func... |
| ngpdsr 24592 | Value of the distance func... |
| ngpds2 24593 | Write the distance between... |
| ngpds2r 24594 | Write the distance between... |
| ngpds3 24595 | Write the distance between... |
| ngpds3r 24596 | Write the distance between... |
| ngprcan 24597 | Cancel right addition insi... |
| ngplcan 24598 | Cancel left addition insid... |
| isngp4 24599 | Express the property of be... |
| ngpinvds 24600 | Two elements are the same ... |
| ngpsubcan 24601 | Cancel right subtraction i... |
| nmf 24602 | The norm on a normed group... |
| nmcl 24603 | The norm of a normed group... |
| nmge0 24604 | The norm of a normed group... |
| nmeq0 24605 | The identity is the only e... |
| nmne0 24606 | The norm of a nonzero elem... |
| nmrpcl 24607 | The norm of a nonzero elem... |
| nminv 24608 | The norm of a negated elem... |
| nmmtri 24609 | The triangle inequality fo... |
| nmsub 24610 | The norm of the difference... |
| nmrtri 24611 | Reverse triangle inequalit... |
| nm2dif 24612 | Inequality for the differe... |
| nmtri 24613 | The triangle inequality fo... |
| nmtri2 24614 | Triangle inequality for th... |
| ngpi 24615 | The properties of a normed... |
| nm0 24616 | Norm of the identity eleme... |
| nmgt0 24617 | The norm of a nonzero elem... |
| sgrim 24618 | The induced metric on a su... |
| sgrimval 24619 | The induced metric on a su... |
| subgnm 24620 | The norm in a subgroup. (... |
| subgnm2 24621 | A substructure assigns the... |
| subgngp 24622 | A normed group restricted ... |
| ngptgp 24623 | A normed abelian group is ... |
| ngppropd 24624 | Property deduction for a n... |
| reldmtng 24625 | The function ` toNrmGrp ` ... |
| tngval 24626 | Value of the function whic... |
| tnglem 24627 | Lemma for ~ tngbas and sim... |
| tngbas 24628 | The base set of a structur... |
| tngplusg 24629 | The group addition of a st... |
| tng0 24630 | The group identity of a st... |
| tngmulr 24631 | The ring multiplication of... |
| tngsca 24632 | The scalar ring of a struc... |
| tngvsca 24633 | The scalar multiplication ... |
| tngip 24634 | The inner product operatio... |
| tngds 24635 | The metric function of a s... |
| tngtset 24636 | The topology generated by ... |
| tngtopn 24637 | The topology generated by ... |
| tngnm 24638 | The topology generated by ... |
| tngngp2 24639 | A norm turns a group into ... |
| tngngpd 24640 | Derive the axioms for a no... |
| tngngp 24641 | Derive the axioms for a no... |
| tnggrpr 24642 | If a structure equipped wi... |
| tngngp3 24643 | Alternate definition of a ... |
| nrmtngdist 24644 | The augmentation of a norm... |
| nrmtngnrm 24645 | The augmentation of a norm... |
| tngngpim 24646 | The induced metric of a no... |
| isnrg 24647 | A normed ring is a ring wi... |
| nrgabv 24648 | The norm of a normed ring ... |
| nrgngp 24649 | A normed ring is a normed ... |
| nrgring 24650 | A normed ring is a ring. ... |
| nmmul 24651 | The norm of a product in a... |
| nrgdsdi 24652 | Distribute a distance calc... |
| nrgdsdir 24653 | Distribute a distance calc... |
| nm1 24654 | The norm of one in a nonze... |
| unitnmn0 24655 | The norm of a unit is nonz... |
| nminvr 24656 | The norm of an inverse in ... |
| nmdvr 24657 | The norm of a division in ... |
| nrgdomn 24658 | A nonzero normed ring is a... |
| nrgtgp 24659 | A normed ring is a topolog... |
| subrgnrg 24660 | A normed ring restricted t... |
| tngnrg 24661 | Given any absolute value o... |
| isnlm 24662 | A normed (left) module is ... |
| nmvs 24663 | Defining property of a nor... |
| nlmngp 24664 | A normed module is a norme... |
| nlmlmod 24665 | A normed module is a left ... |
| nlmnrg 24666 | The scalar component of a ... |
| nlmngp2 24667 | The scalar component of a ... |
| nlmdsdi 24668 | Distribute a distance calc... |
| nlmdsdir 24669 | Distribute a distance calc... |
| nlmmul0or 24670 | If a scalar product is zer... |
| sranlm 24671 | The subring algebra over a... |
| nlmvscnlem2 24672 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24673 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24674 | The scalar multiplication ... |
| rlmnlm 24675 | The ring module over a nor... |
| rlmnm 24676 | The norm function in the r... |
| nrgtrg 24677 | A normed ring is a topolog... |
| nrginvrcnlem 24678 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24679 | The ring inverse function ... |
| nrgtdrg 24680 | A normed division ring is ... |
| nlmtlm 24681 | A normed module is a topol... |
| isnvc 24682 | A normed vector space is j... |
| nvcnlm 24683 | A normed vector space is a... |
| nvclvec 24684 | A normed vector space is a... |
| nvclmod 24685 | A normed vector space is a... |
| isnvc2 24686 | A normed vector space is j... |
| nvctvc 24687 | A normed vector space is a... |
| lssnlm 24688 | A subspace of a normed mod... |
| lssnvc 24689 | A subspace of a normed vec... |
| rlmnvc 24690 | The ring module over a nor... |
| ngpocelbl 24691 | Membership of an off-cente... |
| nmoffn 24698 | The function producing ope... |
| reldmnghm 24699 | Lemma for normed group hom... |
| reldmnmhm 24700 | Lemma for module homomorph... |
| nmofval 24701 | Value of the operator norm... |
| nmoval 24702 | Value of the operator norm... |
| nmogelb 24703 | Property of the operator n... |
| nmolb 24704 | Any upper bound on the val... |
| nmolb2d 24705 | Any upper bound on the val... |
| nmof 24706 | The operator norm is a fun... |
| nmocl 24707 | The operator norm of an op... |
| nmoge0 24708 | The operator norm of an op... |
| nghmfval 24709 | A normed group homomorphis... |
| isnghm 24710 | A normed group homomorphis... |
| isnghm2 24711 | A normed group homomorphis... |
| isnghm3 24712 | A normed group homomorphis... |
| bddnghm 24713 | A bounded group homomorphi... |
| nghmcl 24714 | A normed group homomorphis... |
| nmoi 24715 | The operator norm achieves... |
| nmoix 24716 | The operator norm is a bou... |
| nmoi2 24717 | The operator norm is a bou... |
| nmoleub 24718 | The operator norm, defined... |
| nghmrcl1 24719 | Reverse closure for a norm... |
| nghmrcl2 24720 | Reverse closure for a norm... |
| nghmghm 24721 | A normed group homomorphis... |
| nmo0 24722 | The operator norm of the z... |
| nmoeq0 24723 | The operator norm is zero ... |
| nmoco 24724 | An upper bound on the oper... |
| nghmco 24725 | The composition of normed ... |
| nmotri 24726 | Triangle inequality for th... |
| nghmplusg 24727 | The sum of two bounded lin... |
| 0nghm 24728 | The zero operator is a nor... |
| nmoid 24729 | The operator norm of the i... |
| idnghm 24730 | The identity operator is a... |
| nmods 24731 | Upper bound for the distan... |
| nghmcn 24732 | A normed group homomorphis... |
| isnmhm 24733 | A normed module homomorphi... |
| nmhmrcl1 24734 | Reverse closure for a norm... |
| nmhmrcl2 24735 | Reverse closure for a norm... |
| nmhmlmhm 24736 | A normed module homomorphi... |
| nmhmnghm 24737 | A normed module homomorphi... |
| nmhmghm 24738 | A normed module homomorphi... |
| isnmhm2 24739 | A normed module homomorphi... |
| nmhmcl 24740 | A normed module homomorphi... |
| idnmhm 24741 | The identity operator is a... |
| 0nmhm 24742 | The zero operator is a bou... |
| nmhmco 24743 | The composition of bounded... |
| nmhmplusg 24744 | The sum of two bounded lin... |
| qtopbaslem 24745 | The set of open intervals ... |
| qtopbas 24746 | The set of open intervals ... |
| retopbas 24747 | A basis for the standard t... |
| retop 24748 | The standard topology on t... |
| uniretop 24749 | The underlying set of the ... |
| retopon 24750 | The standard topology on t... |
| retps 24751 | The standard topological s... |
| iooretop 24752 | Open intervals are open se... |
| icccld 24753 | Closed intervals are close... |
| icopnfcld 24754 | Right-unbounded closed int... |
| iocmnfcld 24755 | Left-unbounded closed inte... |
| qdensere 24756 | ` QQ ` is dense in the sta... |
| cnmetdval 24757 | Value of the distance func... |
| cnmet 24758 | The absolute value metric ... |
| cnxmet 24759 | The absolute value metric ... |
| cnbl0 24760 | Two ways to write the open... |
| cnblcld 24761 | Two ways to write the clos... |
| cnfldms 24762 | The complex number field i... |
| cnfldxms 24763 | The complex number field i... |
| cnfldtps 24764 | The complex number field i... |
| cnfldnm 24765 | The norm of the field of c... |
| cnngp 24766 | The complex numbers form a... |
| cnnrg 24767 | The complex numbers form a... |
| cnfldtopn 24768 | The topology of the comple... |
| cnfldtopon 24769 | The topology of the comple... |
| cnfldtop 24770 | The topology of the comple... |
| cnfldhaus 24771 | The topology of the comple... |
| unicntop 24772 | The underlying set of the ... |
| cnopn 24773 | The set of complex numbers... |
| cnn0opn 24774 | The set of nonzero complex... |
| zringnrg 24775 | The ring of integers is a ... |
| remetdval 24776 | Value of the distance func... |
| remet 24777 | The absolute value metric ... |
| rexmet 24778 | The absolute value metric ... |
| bl2ioo 24779 | A ball in terms of an open... |
| ioo2bl 24780 | An open interval of reals ... |
| ioo2blex 24781 | An open interval of reals ... |
| blssioo 24782 | The balls of the standard ... |
| tgioo 24783 | The topology generated by ... |
| qdensere2 24784 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24785 | An open ball in the comple... |
| rehaus 24786 | The standard topology on t... |
| tgqioo 24787 | The topology generated by ... |
| re2ndc 24788 | The standard topology on t... |
| resubmet 24789 | The subspace topology indu... |
| tgioo2 24790 | The standard topology on t... |
| rerest 24791 | The subspace topology indu... |
| tgioo4 24792 | The standard topology on t... |
| tgioo3 24793 | The standard topology on t... |
| xrtgioo 24794 | The topology on the extend... |
| xrrest 24795 | The subspace topology indu... |
| xrrest2 24796 | The subspace topology indu... |
| xrsxmet 24797 | The metric on the extended... |
| xrsdsre 24798 | The metric on the extended... |
| xrsblre 24799 | Any ball of the metric of ... |
| xrsmopn 24800 | The metric on the extended... |
| zcld 24801 | The integers are a closed ... |
| recld2 24802 | The real numbers are a clo... |
| zcld2 24803 | The integers are a closed ... |
| zdis 24804 | The integers are a discret... |
| sszcld 24805 | Every subset of the intege... |
| reperflem 24806 | A subset of the real numbe... |
| reperf 24807 | The real numbers are a per... |
| cnperf 24808 | The complex numbers are a ... |
| iccntr 24809 | The interior of a closed i... |
| icccmplem1 24810 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24811 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24812 | Lemma for ~ icccmp . (Con... |
| icccmp 24813 | A closed interval in ` RR ... |
| reconnlem1 24814 | Lemma for ~ reconn . Conn... |
| reconnlem2 24815 | Lemma for ~ reconn . (Con... |
| reconn 24816 | A subset of the reals is c... |
| retopconn 24817 | Corollary of ~ reconn . T... |
| iccconn 24818 | A closed interval is conne... |
| opnreen 24819 | Every nonempty open set is... |
| rectbntr0 24820 | A countable subset of the ... |
| xrge0gsumle 24821 | A finite sum in the nonneg... |
| xrge0tsms 24822 | Any finite or infinite sum... |
| xrge0tsms2 24823 | Any finite or infinite sum... |
| metdcnlem 24824 | The metric function of a m... |
| xmetdcn2 24825 | The metric function of an ... |
| xmetdcn 24826 | The metric function of an ... |
| metdcn2 24827 | The metric function of a m... |
| metdcn 24828 | The metric function of a m... |
| msdcn 24829 | The metric function of a m... |
| cnmpt1ds 24830 | Continuity of the metric f... |
| cnmpt2ds 24831 | Continuity of the metric f... |
| nmcn 24832 | The norm of a normed group... |
| ngnmcncn 24833 | The norm of a normed group... |
| abscn 24834 | The absolute value functio... |
| metdsval 24835 | Value of the "distance to ... |
| metdsf 24836 | The distance from a point ... |
| metdsge 24837 | The distance from the poin... |
| metds0 24838 | If a point is in a set, it... |
| metdstri 24839 | A generalization of the tr... |
| metdsle 24840 | The distance from a point ... |
| metdsre 24841 | The distance from a point ... |
| metdseq0 24842 | The distance from a point ... |
| metdscnlem 24843 | Lemma for ~ metdscn . (Co... |
| metdscn 24844 | The function ` F ` which g... |
| metdscn2 24845 | The function ` F ` which g... |
| metnrmlem1a 24846 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24847 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24848 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24849 | Lemma for ~ metnrm . (Con... |
| metnrm 24850 | A metric space is normal. ... |
| metreg 24851 | A metric space is regular.... |
| addcnlem 24852 | Lemma for ~ addcn , ~ subc... |
| addcn 24853 | Complex number addition is... |
| subcn 24854 | Complex number subtraction... |
| mulcn 24855 | Complex number multiplicat... |
| mpomulcn 24856 | Complex number multiplicat... |
| divcn 24857 | Complex number division is... |
| cnfldtgp 24858 | The complex numbers form a... |
| fsumcn 24859 | A finite sum of functions ... |
| fsum2cn 24860 | Version of ~ fsumcn for tw... |
| expcn 24861 | The power function on comp... |
| divccn 24862 | Division by a nonzero cons... |
| sqcn 24863 | The square function on com... |
| iitopon 24868 | The unit interval is a top... |
| iitop 24869 | The unit interval is a top... |
| iiuni 24870 | The base set of the unit i... |
| dfii2 24871 | Alternate definition of th... |
| dfii3 24872 | Alternate definition of th... |
| dfii4 24873 | Alternate definition of th... |
| dfii5 24874 | The unit interval expresse... |
| iicmp 24875 | The unit interval is compa... |
| iiconn 24876 | The unit interval is conne... |
| cncfval 24877 | The value of the continuou... |
| elcncf 24878 | Membership in the set of c... |
| elcncf2 24879 | Version of ~ elcncf with a... |
| cncfrss 24880 | Reverse closure of the con... |
| cncfrss2 24881 | Reverse closure of the con... |
| cncff 24882 | A continuous complex funct... |
| cncfi 24883 | Defining property of a con... |
| elcncf1di 24884 | Membership in the set of c... |
| elcncf1ii 24885 | Membership in the set of c... |
| rescncf 24886 | A continuous complex funct... |
| cncfcdm 24887 | Change the codomain of a c... |
| cncfss 24888 | The set of continuous func... |
| climcncf 24889 | Image of a limit under a c... |
| abscncf 24890 | Absolute value is continuo... |
| recncf 24891 | Real part is continuous. ... |
| imcncf 24892 | Imaginary part is continuo... |
| cjcncf 24893 | Complex conjugate is conti... |
| mulc1cncf 24894 | Multiplication by a consta... |
| divccncf 24895 | Division by a constant is ... |
| cncfco 24896 | The composition of two con... |
| cncfcompt2 24897 | Composition of continuous ... |
| cncfmet 24898 | Relate complex function co... |
| cncfcn 24899 | Relate complex function co... |
| cncfcn1 24900 | Relate complex function co... |
| cncfmptc 24901 | A constant function is a c... |
| cncfmptid 24902 | The identity function is a... |
| cncfmpt1f 24903 | Composition of continuous ... |
| cncfmpt2f 24904 | Composition of continuous ... |
| cncfmpt2ss 24905 | Composition of continuous ... |
| addccncf 24906 | Adding a constant is a con... |
| idcncf 24907 | The identity function is a... |
| sub1cncf 24908 | Subtracting a constant is ... |
| sub2cncf 24909 | Subtraction from a constan... |
| cdivcncf 24910 | Division with a constant n... |
| negcncf 24911 | The negative function is c... |
| negfcncf 24912 | The negative of a continuo... |
| abscncfALT 24913 | Absolute value is continuo... |
| cncfcnvcn 24914 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24915 | The power function on comp... |
| cnmptre 24916 | Lemma for ~ iirevcn and re... |
| cnmpopc 24917 | Piecewise definition of a ... |
| iirev 24918 | Reverse the unit interval.... |
| iirevcn 24919 | The reversion function is ... |
| iihalf1 24920 | Map the first half of ` II... |
| iihalf1cn 24921 | The first half function is... |
| iihalf2 24922 | Map the second half of ` I... |
| iihalf2cn 24923 | The second half function i... |
| elii1 24924 | Divide the unit interval i... |
| elii2 24925 | Divide the unit interval i... |
| iimulcl 24926 | The unit interval is close... |
| iimulcn 24927 | Multiplication is a contin... |
| icoopnst 24928 | A half-open interval start... |
| iocopnst 24929 | A half-open interval endin... |
| icchmeo 24930 | The natural bijection from... |
| icopnfcnv 24931 | Define a bijection from ` ... |
| icopnfhmeo 24932 | The defined bijection from... |
| iccpnfcnv 24933 | Define a bijection from ` ... |
| iccpnfhmeo 24934 | The defined bijection from... |
| xrhmeo 24935 | The bijection from ` [ -u ... |
| xrhmph 24936 | The extended reals are hom... |
| xrcmp 24937 | The topology of the extend... |
| xrconn 24938 | The topology of the extend... |
| icccvx 24939 | A linear combination of tw... |
| oprpiece1res1 24940 | Restriction to the first p... |
| oprpiece1res2 24941 | Restriction to the second ... |
| cnrehmeo 24942 | The canonical bijection fr... |
| cnheiborlem 24943 | Lemma for ~ cnheibor . (C... |
| cnheibor 24944 | Heine-Borel theorem for co... |
| cnllycmp 24945 | The topology on the comple... |
| rellycmp 24946 | The topology on the reals ... |
| bndth 24947 | The Boundedness Theorem. ... |
| evth 24948 | The Extreme Value Theorem.... |
| evth2 24949 | The Extreme Value Theorem,... |
| lebnumlem1 24950 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24951 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24952 | Lemma for ~ lebnum . By t... |
| lebnum 24953 | The Lebesgue number lemma,... |
| xlebnum 24954 | Generalize ~ lebnum to ext... |
| lebnumii 24955 | Specialize the Lebesgue nu... |
| ishtpy 24961 | Membership in the class of... |
| htpycn 24962 | A homotopy is a continuous... |
| htpyi 24963 | A homotopy evaluated at it... |
| ishtpyd 24964 | Deduction for membership i... |
| htpycom 24965 | Given a homotopy from ` F ... |
| htpyid 24966 | A homotopy from a function... |
| htpyco1 24967 | Compose a homotopy with a ... |
| htpyco2 24968 | Compose a homotopy with a ... |
| htpycc 24969 | Concatenate two homotopies... |
| isphtpy 24970 | Membership in the class of... |
| phtpyhtpy 24971 | A path homotopy is a homot... |
| phtpycn 24972 | A path homotopy is a conti... |
| phtpyi 24973 | Membership in the class of... |
| phtpy01 24974 | Two path-homotopic paths h... |
| isphtpyd 24975 | Deduction for membership i... |
| isphtpy2d 24976 | Deduction for membership i... |
| phtpycom 24977 | Given a homotopy from ` F ... |
| phtpyid 24978 | A homotopy from a path to ... |
| phtpyco2 24979 | Compose a path homotopy wi... |
| phtpycc 24980 | Concatenate two path homot... |
| phtpcrel 24982 | The path homotopy relation... |
| isphtpc 24983 | The relation "is path homo... |
| phtpcer 24984 | Path homotopy is an equiva... |
| phtpc01 24985 | Path homotopic paths have ... |
| reparphti 24986 | Lemma for ~ reparpht . (C... |
| reparpht 24987 | Reparametrization lemma. ... |
| phtpcco2 24988 | Compose a path homotopy wi... |
| pcofval 24999 | The value of the path conc... |
| pcoval 25000 | The concatenation of two p... |
| pcovalg 25001 | Evaluate the concatenation... |
| pcoval1 25002 | Evaluate the concatenation... |
| pco0 25003 | The starting point of a pa... |
| pco1 25004 | The ending point of a path... |
| pcoval2 25005 | Evaluate the concatenation... |
| pcocn 25006 | The concatenation of two p... |
| copco 25007 | The composition of a conca... |
| pcohtpylem 25008 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 25009 | Homotopy invariance of pat... |
| pcoptcl 25010 | A constant function is a p... |
| pcopt 25011 | Concatenation with a point... |
| pcopt2 25012 | Concatenation with a point... |
| pcoass 25013 | Order of concatenation doe... |
| pcorevcl 25014 | Closure for a reversed pat... |
| pcorevlem 25015 | Lemma for ~ pcorev . Prov... |
| pcorev 25016 | Concatenation with the rev... |
| pcorev2 25017 | Concatenation with the rev... |
| pcophtb 25018 | The path homotopy equivale... |
| om1val 25019 | The definition of the loop... |
| om1bas 25020 | The base set of the loop s... |
| om1elbas 25021 | Elementhood in the base se... |
| om1addcl 25022 | Closure of the group opera... |
| om1plusg 25023 | The group operation (which... |
| om1tset 25024 | The topology of the loop s... |
| om1opn 25025 | The topology of the loop s... |
| pi1val 25026 | The definition of the fund... |
| pi1bas 25027 | The base set of the fundam... |
| pi1blem 25028 | Lemma for ~ pi1buni . (Co... |
| pi1buni 25029 | Another way to write the l... |
| pi1bas2 25030 | The base set of the fundam... |
| pi1eluni 25031 | Elementhood in the base se... |
| pi1bas3 25032 | The base set of the fundam... |
| pi1cpbl 25033 | The group operation, loop ... |
| elpi1 25034 | The elements of the fundam... |
| elpi1i 25035 | The elements of the fundam... |
| pi1addf 25036 | The group operation of ` p... |
| pi1addval 25037 | The concatenation of two p... |
| pi1grplem 25038 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25039 | The fundamental group is a... |
| pi1id 25040 | The identity element of th... |
| pi1inv 25041 | An inverse in the fundamen... |
| pi1xfrf 25042 | Functionality of the loop ... |
| pi1xfrval 25043 | The value of the loop tran... |
| pi1xfr 25044 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25045 | Given a path ` F ` between... |
| pi1xfrcnv 25046 | Given a path ` F ` between... |
| pi1xfrgim 25047 | The mapping ` G ` between ... |
| pi1cof 25048 | Functionality of the loop ... |
| pi1coval 25049 | The value of the loop tran... |
| pi1coghm 25050 | The mapping ` G ` between ... |
| isclm 25053 | A subcomplex module is a l... |
| clmsca 25054 | The ring of scalars ` F ` ... |
| clmsubrg 25055 | The base set of the ring o... |
| clmlmod 25056 | A subcomplex module is a l... |
| clmgrp 25057 | A subcomplex module is an ... |
| clmabl 25058 | A subcomplex module is an ... |
| clmring 25059 | The scalar ring of a subco... |
| clmfgrp 25060 | The scalar ring of a subco... |
| clm0 25061 | The zero of the scalar rin... |
| clm1 25062 | The identity of the scalar... |
| clmadd 25063 | The addition of the scalar... |
| clmmul 25064 | The multiplication of the ... |
| clmcj 25065 | The conjugation of the sca... |
| isclmi 25066 | Reverse direction of ~ isc... |
| clmzss 25067 | The scalar ring of a subco... |
| clmsscn 25068 | The scalar ring of a subco... |
| clmsub 25069 | Subtraction in the scalar ... |
| clmneg 25070 | Negation in the scalar rin... |
| clmneg1 25071 | Minus one is in the scalar... |
| clmabs 25072 | Norm in the scalar ring of... |
| clmacl 25073 | Closure of ring addition f... |
| clmmcl 25074 | Closure of ring multiplica... |
| clmsubcl 25075 | Closure of ring subtractio... |
| lmhmclm 25076 | The domain of a linear ope... |
| clmvscl 25077 | Closure of scalar product ... |
| clmvsass 25078 | Associative law for scalar... |
| clmvscom 25079 | Commutative law for the sc... |
| clmvsdir 25080 | Distributive law for scala... |
| clmvsdi 25081 | Distributive law for scala... |
| clmvs1 25082 | Scalar product with ring u... |
| clmvs2 25083 | A vector plus itself is tw... |
| clm0vs 25084 | Zero times a vector is the... |
| clmopfne 25085 | The (functionalized) opera... |
| isclmp 25086 | The predicate "is a subcom... |
| isclmi0 25087 | Properties that determine ... |
| clmvneg1 25088 | Minus 1 times a vector is ... |
| clmvsneg 25089 | Multiplication of a vector... |
| clmmulg 25090 | The group multiple functio... |
| clmsubdir 25091 | Scalar multiplication dist... |
| clmpm1dir 25092 | Subtractive distributive l... |
| clmnegneg 25093 | Double negative of a vecto... |
| clmnegsubdi2 25094 | Distribution of negative o... |
| clmsub4 25095 | Rearrangement of 4 terms i... |
| clmvsrinv 25096 | A vector minus itself. (C... |
| clmvslinv 25097 | Minus a vector plus itself... |
| clmvsubval 25098 | Value of vector subtractio... |
| clmvsubval2 25099 | Value of vector subtractio... |
| clmvz 25100 | Two ways to express the ne... |
| zlmclm 25101 | The ` ZZ ` -module operati... |
| clmzlmvsca 25102 | The scalar product of a su... |
| nmoleub2lem 25103 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25104 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25105 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25106 | The operator norm is the s... |
| nmoleub2b 25107 | The operator norm is the s... |
| nmoleub3 25108 | The operator norm is the s... |
| nmhmcn 25109 | A linear operator over a n... |
| cmodscexp 25110 | The powers of ` _i ` belon... |
| cmodscmulexp 25111 | The scalar product of a ve... |
| cvslvec 25114 | A subcomplex vector space ... |
| cvsclm 25115 | A subcomplex vector space ... |
| iscvs 25116 | A subcomplex vector space ... |
| iscvsp 25117 | The predicate "is a subcom... |
| iscvsi 25118 | Properties that determine ... |
| cvsi 25119 | The properties of a subcom... |
| cvsunit 25120 | Unit group of the scalar r... |
| cvsdiv 25121 | Division of the scalar rin... |
| cvsdivcl 25122 | The scalar field of a subc... |
| cvsmuleqdivd 25123 | An equality involving rati... |
| cvsdiveqd 25124 | An equality involving rati... |
| cnlmodlem1 25125 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25126 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25127 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25128 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25129 | The set of complex numbers... |
| cnstrcvs 25130 | The set of complex numbers... |
| cnrbas 25131 | The set of complex numbers... |
| cnrlmod 25132 | The complex left module of... |
| cnrlvec 25133 | The complex left module of... |
| cncvs 25134 | The complex left module of... |
| recvs 25135 | The field of the real numb... |
| qcvs 25136 | The field of rational numb... |
| zclmncvs 25137 | The ring of integers as le... |
| isncvsngp 25138 | A normed subcomplex vector... |
| isncvsngpd 25139 | Properties that determine ... |
| ncvsi 25140 | The properties of a normed... |
| ncvsprp 25141 | Proportionality property o... |
| ncvsge0 25142 | The norm of a scalar produ... |
| ncvsm1 25143 | The norm of the opposite o... |
| ncvsdif 25144 | The norm of the difference... |
| ncvspi 25145 | The norm of a vector plus ... |
| ncvs1 25146 | From any nonzero vector of... |
| cnrnvc 25147 | The module of complex numb... |
| cnncvs 25148 | The module of complex numb... |
| cnnm 25149 | The norm of the normed sub... |
| ncvspds 25150 | Value of the distance func... |
| cnindmet 25151 | The metric induced on the ... |
| cnncvsaddassdemo 25152 | Derive the associative law... |
| cnncvsmulassdemo 25153 | Derive the associative law... |
| cnncvsabsnegdemo 25154 | Derive the absolute value ... |
| iscph 25159 | A subcomplex pre-Hilbert s... |
| cphphl 25160 | A subcomplex pre-Hilbert s... |
| cphnlm 25161 | A subcomplex pre-Hilbert s... |
| cphngp 25162 | A subcomplex pre-Hilbert s... |
| cphlmod 25163 | A subcomplex pre-Hilbert s... |
| cphlvec 25164 | A subcomplex pre-Hilbert s... |
| cphnvc 25165 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25166 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25167 | Lemma for ~ cphreccl . (C... |
| cphsca 25168 | A subcomplex pre-Hilbert s... |
| cphsubrg 25169 | The scalar field of a subc... |
| cphreccl 25170 | The scalar field of a subc... |
| cphdivcl 25171 | The scalar field of a subc... |
| cphcjcl 25172 | The scalar field of a subc... |
| cphsqrtcl 25173 | The scalar field of a subc... |
| cphabscl 25174 | The scalar field of a subc... |
| cphsqrtcl2 25175 | The scalar field of a subc... |
| cphsqrtcl3 25176 | If the scalar field of a s... |
| cphqss 25177 | The scalar field of a subc... |
| cphclm 25178 | A subcomplex pre-Hilbert s... |
| cphnmvs 25179 | Norm of a scalar product. ... |
| cphipcl 25180 | An inner product is a memb... |
| cphnmfval 25181 | The value of the norm in a... |
| cphnm 25182 | The square of the norm is ... |
| nmsq 25183 | The square of the norm is ... |
| cphnmf 25184 | The norm of a vector is a ... |
| cphnmcl 25185 | The norm of a vector is a ... |
| reipcl 25186 | An inner product of an ele... |
| ipge0 25187 | The inner product in a sub... |
| cphipcj 25188 | Conjugate of an inner prod... |
| cphipipcj 25189 | An inner product times its... |
| cphorthcom 25190 | Orthogonality (meaning inn... |
| cphip0l 25191 | Inner product with a zero ... |
| cphip0r 25192 | Inner product with a zero ... |
| cphipeq0 25193 | The inner product of a vec... |
| cphdir 25194 | Distributive law for inner... |
| cphdi 25195 | Distributive law for inner... |
| cph2di 25196 | Distributive law for inner... |
| cphsubdir 25197 | Distributive law for inner... |
| cphsubdi 25198 | Distributive law for inner... |
| cph2subdi 25199 | Distributive law for inner... |
| cphass 25200 | Associative law for inner ... |
| cphassr 25201 | "Associative" law for seco... |
| cph2ass 25202 | Move scalar multiplication... |
| cphassi 25203 | Associative law for the fi... |
| cphassir 25204 | "Associative" law for the ... |
| cphpyth 25205 | The pythagorean theorem fo... |
| tcphex 25206 | Lemma for ~ tcphbas and si... |
| tcphval 25207 | Define a function to augme... |
| tcphbas 25208 | The base set of a subcompl... |
| tchplusg 25209 | The addition operation of ... |
| tcphsub 25210 | The subtraction operation ... |
| tcphmulr 25211 | The ring operation of a su... |
| tcphsca 25212 | The scalar field of a subc... |
| tcphvsca 25213 | The scalar multiplication ... |
| tcphip 25214 | The inner product of a sub... |
| tcphtopn 25215 | The topology of a subcompl... |
| tcphphl 25216 | Augmentation of a subcompl... |
| tchnmfval 25217 | The norm of a subcomplex p... |
| tcphnmval 25218 | The norm of a subcomplex p... |
| cphtcphnm 25219 | The norm of a norm-augment... |
| tcphds 25220 | The distance of a pre-Hilb... |
| phclm 25221 | A pre-Hilbert space whose ... |
| tcphcphlem3 25222 | Lemma for ~ tcphcph : real... |
| ipcau2 25223 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25224 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25225 | Lemma for ~ tcphcph : homo... |
| tcphcph 25226 | The standard definition of... |
| ipcau 25227 | The Cauchy-Schwarz inequal... |
| nmparlem 25228 | Lemma for ~ nmpar . (Cont... |
| nmpar 25229 | A subcomplex pre-Hilbert s... |
| cphipval2 25230 | Value of the inner product... |
| 4cphipval2 25231 | Four times the inner produ... |
| cphipval 25232 | Value of the inner product... |
| ipcnlem2 25233 | The inner product operatio... |
| ipcnlem1 25234 | The inner product operatio... |
| ipcn 25235 | The inner product operatio... |
| cnmpt1ip 25236 | Continuity of inner produc... |
| cnmpt2ip 25237 | Continuity of inner produc... |
| csscld 25238 | A "closed subspace" in a s... |
| clsocv 25239 | The orthogonal complement ... |
| cphsscph 25240 | A subspace of a subcomplex... |
| lmmbr 25247 | Express the binary relatio... |
| lmmbr2 25248 | Express the binary relatio... |
| lmmbr3 25249 | Express the binary relatio... |
| lmmcvg 25250 | Convergence property of a ... |
| lmmbrf 25251 | Express the binary relatio... |
| lmnn 25252 | A condition that implies c... |
| cfilfval 25253 | The set of Cauchy filters ... |
| iscfil 25254 | The property of being a Ca... |
| iscfil2 25255 | The property of being a Ca... |
| cfilfil 25256 | A Cauchy filter is a filte... |
| cfili 25257 | Property of a Cauchy filte... |
| cfil3i 25258 | A Cauchy filter contains b... |
| cfilss 25259 | A filter finer than a Cauc... |
| fgcfil 25260 | The Cauchy filter conditio... |
| fmcfil 25261 | The Cauchy filter conditio... |
| iscfil3 25262 | A filter is Cauchy iff it ... |
| cfilfcls 25263 | Similar to ultrafilters ( ... |
| caufval 25264 | The set of Cauchy sequence... |
| iscau 25265 | Express the property " ` F... |
| iscau2 25266 | Express the property " ` F... |
| iscau3 25267 | Express the Cauchy sequenc... |
| iscau4 25268 | Express the property " ` F... |
| iscauf 25269 | Express the property " ` F... |
| caun0 25270 | A metric with a Cauchy seq... |
| caufpm 25271 | Inclusion of a Cauchy sequ... |
| caucfil 25272 | A Cauchy sequence predicat... |
| iscmet 25273 | The property " ` D ` is a ... |
| cmetcvg 25274 | The convergence of a Cauch... |
| cmetmet 25275 | A complete metric space is... |
| cmetmeti 25276 | A complete metric space is... |
| cmetcaulem 25277 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25278 | The convergence of a Cauch... |
| iscmet3lem3 25279 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25280 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25281 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25282 | The property " ` D ` is a ... |
| iscmet2 25283 | A metric ` D ` is complete... |
| cfilresi 25284 | A Cauchy filter on a metri... |
| cfilres 25285 | Cauchy filter on a metric ... |
| caussi 25286 | Cauchy sequence on a metri... |
| causs 25287 | Cauchy sequence on a metri... |
| equivcfil 25288 | If the metric ` D ` is "st... |
| equivcau 25289 | If the metric ` D ` is "st... |
| lmle 25290 | If the distance from each ... |
| nglmle 25291 | If the norm of each member... |
| lmclim 25292 | Relate a limit on the metr... |
| lmclimf 25293 | Relate a limit on the metr... |
| metelcls 25294 | A point belongs to the clo... |
| metcld 25295 | A subset of a metric space... |
| metcld2 25296 | A subset of a metric space... |
| caubl 25297 | Sufficient condition to en... |
| caublcls 25298 | The convergent point of a ... |
| metcnp4 25299 | Two ways to say a mapping ... |
| metcn4 25300 | Two ways to say a mapping ... |
| iscmet3i 25301 | Properties that determine ... |
| lmcau 25302 | Every convergent sequence ... |
| flimcfil 25303 | Every convergent filter in... |
| metsscmetcld 25304 | A complete subspace of a m... |
| cmetss 25305 | A subspace of a complete m... |
| equivcmet 25306 | If two metrics are strongl... |
| relcmpcmet 25307 | If ` D ` is a metric space... |
| cmpcmet 25308 | A compact metric space is ... |
| cfilucfil3 25309 | Given a metric ` D ` and a... |
| cfilucfil4 25310 | Given a metric ` D ` and a... |
| cncmet 25311 | The set of complex numbers... |
| recmet 25312 | The real numbers are a com... |
| bcthlem1 25313 | Lemma for ~ bcth . Substi... |
| bcthlem2 25314 | Lemma for ~ bcth . The ba... |
| bcthlem3 25315 | Lemma for ~ bcth . The li... |
| bcthlem4 25316 | Lemma for ~ bcth . Given ... |
| bcthlem5 25317 | Lemma for ~ bcth . The pr... |
| bcth 25318 | Baire's Category Theorem. ... |
| bcth2 25319 | Baire's Category Theorem, ... |
| bcth3 25320 | Baire's Category Theorem, ... |
| isbn 25327 | A Banach space is a normed... |
| bnsca 25328 | The scalar field of a Bana... |
| bnnvc 25329 | A Banach space is a normed... |
| bnnlm 25330 | A Banach space is a normed... |
| bnngp 25331 | A Banach space is a normed... |
| bnlmod 25332 | A Banach space is a left m... |
| bncms 25333 | A Banach space is a comple... |
| iscms 25334 | A complete metric space is... |
| cmscmet 25335 | The induced metric on a co... |
| bncmet 25336 | The induced metric on Bana... |
| cmsms 25337 | A complete metric space is... |
| cmspropd 25338 | Property deduction for a c... |
| cmssmscld 25339 | The restriction of a metri... |
| cmsss 25340 | The restriction of a compl... |
| lssbn 25341 | A subspace of a Banach spa... |
| cmetcusp1 25342 | If the uniform set of a co... |
| cmetcusp 25343 | The uniform space generate... |
| cncms 25344 | The field of complex numbe... |
| cnflduss 25345 | The uniform structure of t... |
| cnfldcusp 25346 | The field of complex numbe... |
| resscdrg 25347 | The real numbers are a sub... |
| cncdrg 25348 | The only complete subfield... |
| srabn 25349 | The subring algebra over a... |
| rlmbn 25350 | The ring module over a com... |
| ishl 25351 | The predicate "is a subcom... |
| hlbn 25352 | Every subcomplex Hilbert s... |
| hlcph 25353 | Every subcomplex Hilbert s... |
| hlphl 25354 | Every subcomplex Hilbert s... |
| hlcms 25355 | Every subcomplex Hilbert s... |
| hlprlem 25356 | Lemma for ~ hlpr . (Contr... |
| hlress 25357 | The scalar field of a subc... |
| hlpr 25358 | The scalar field of a subc... |
| ishl2 25359 | A Hilbert space is a compl... |
| cphssphl 25360 | A Banach subspace of a sub... |
| cmslssbn 25361 | A complete linear subspace... |
| cmscsscms 25362 | A closed subspace of a com... |
| bncssbn 25363 | A closed subspace of a Ban... |
| cssbn 25364 | A complete subspace of a n... |
| csschl 25365 | A complete subspace of a c... |
| cmslsschl 25366 | A complete linear subspace... |
| chlcsschl 25367 | A closed subspace of a sub... |
| retopn 25368 | The topology of the real n... |
| recms 25369 | The real numbers form a co... |
| reust 25370 | The Uniform structure of t... |
| recusp 25371 | The real numbers form a co... |
| rrxval 25376 | Value of the generalized E... |
| rrxbase 25377 | The base of the generalize... |
| rrxprds 25378 | Expand the definition of t... |
| rrxip 25379 | The inner product of the g... |
| rrxnm 25380 | The norm of the generalize... |
| rrxcph 25381 | Generalized Euclidean real... |
| rrxds 25382 | The distance over generali... |
| rrxvsca 25383 | The scalar product over ge... |
| rrxplusgvscavalb 25384 | The result of the addition... |
| rrxsca 25385 | The field of real numbers ... |
| rrx0 25386 | The zero ("origin") in a g... |
| rrx0el 25387 | The zero ("origin") in a g... |
| csbren 25388 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25389 | Triangle inequality in R^n... |
| rrxf 25390 | Euclidean vectors as funct... |
| rrxfsupp 25391 | Euclidean vectors are of f... |
| rrxsuppss 25392 | Support of Euclidean vecto... |
| rrxmvallem 25393 | Support of the function us... |
| rrxmval 25394 | The value of the Euclidean... |
| rrxmfval 25395 | The value of the Euclidean... |
| rrxmetlem 25396 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25397 | Euclidean space is a metri... |
| rrxdstprj1 25398 | The distance between two p... |
| rrxbasefi 25399 | The base of the generalize... |
| rrxdsfi 25400 | The distance over generali... |
| rrxmetfi 25401 | Euclidean space is a metri... |
| rrxdsfival 25402 | The value of the Euclidean... |
| ehlval 25403 | Value of the Euclidean spa... |
| ehlbase 25404 | The base of the Euclidean ... |
| ehl0base 25405 | The base of the Euclidean ... |
| ehl0 25406 | The Euclidean space of dim... |
| ehleudis 25407 | The Euclidean distance fun... |
| ehleudisval 25408 | The value of the Euclidean... |
| ehl1eudis 25409 | The Euclidean distance fun... |
| ehl1eudisval 25410 | The value of the Euclidean... |
| ehl2eudis 25411 | The Euclidean distance fun... |
| ehl2eudisval 25412 | The value of the Euclidean... |
| minveclem1 25413 | Lemma for ~ minvec . The ... |
| minveclem4c 25414 | Lemma for ~ minvec . The ... |
| minveclem2 25415 | Lemma for ~ minvec . Any ... |
| minveclem3a 25416 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25417 | Lemma for ~ minvec . The ... |
| minveclem3 25418 | Lemma for ~ minvec . The ... |
| minveclem4a 25419 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25420 | Lemma for ~ minvec . The ... |
| minveclem4 25421 | Lemma for ~ minvec . The ... |
| minveclem5 25422 | Lemma for ~ minvec . Disc... |
| minveclem6 25423 | Lemma for ~ minvec . Any ... |
| minveclem7 25424 | Lemma for ~ minvec . Sinc... |
| minvec 25425 | Minimizing vector theorem,... |
| pjthlem1 25426 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25427 | Lemma for ~ pjth . (Contr... |
| pjth 25428 | Projection Theorem: Any H... |
| pjth2 25429 | Projection Theorem with ab... |
| cldcss 25430 | Corollary of the Projectio... |
| cldcss2 25431 | Corollary of the Projectio... |
| hlhil 25432 | Corollary of the Projectio... |
| addcncf 25433 | The addition of two contin... |
| subcncf 25434 | The subtraction of two con... |
| mulcncf 25435 | The multiplication of two ... |
| divcncf 25436 | The quotient of two contin... |
| pmltpclem1 25437 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25438 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25439 | Any function on the reals ... |
| ivthlem1 25440 | Lemma for ~ ivth . The se... |
| ivthlem2 25441 | Lemma for ~ ivth . Show t... |
| ivthlem3 25442 | Lemma for ~ ivth , the int... |
| ivth 25443 | The intermediate value the... |
| ivth2 25444 | The intermediate value the... |
| ivthle 25445 | The intermediate value the... |
| ivthle2 25446 | The intermediate value the... |
| ivthicc 25447 | The interval between any t... |
| evthicc 25448 | Specialization of the Extr... |
| evthicc2 25449 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25450 | A continuous function on a... |
| ovolfcl 25455 | Closure for the interval e... |
| ovolfioo 25456 | Unpack the interval coveri... |
| ovolficc 25457 | Unpack the interval coveri... |
| ovolficcss 25458 | Any (closed) interval cove... |
| ovolfsval 25459 | The value of the interval ... |
| ovolfsf 25460 | Closure for the interval l... |
| ovolsf 25461 | Closure for the partial su... |
| ovolval 25462 | The value of the outer mea... |
| elovolmlem 25463 | Lemma for ~ elovolm and re... |
| elovolm 25464 | Elementhood in the set ` M... |
| elovolmr 25465 | Sufficient condition for e... |
| ovolmge0 25466 | The set ` M ` is composed ... |
| ovolcl 25467 | The volume of a set is an ... |
| ovollb 25468 | The outer volume is a lowe... |
| ovolgelb 25469 | The outer volume is the gr... |
| ovolge0 25470 | The volume of a set is alw... |
| ovolf 25471 | The domain and codomain of... |
| ovollecl 25472 | If an outer volume is boun... |
| ovolsslem 25473 | Lemma for ~ ovolss . (Con... |
| ovolss 25474 | The volume of a set is mon... |
| ovolsscl 25475 | If a set is contained in a... |
| ovolssnul 25476 | A subset of a nullset is n... |
| ovollb2lem 25477 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25478 | It is often more convenien... |
| ovolctb 25479 | The volume of a denumerabl... |
| ovolq 25480 | The rational numbers have ... |
| ovolctb2 25481 | The volume of a countable ... |
| ovol0 25482 | The empty set has 0 outer ... |
| ovolfi 25483 | A finite set has 0 outer L... |
| ovolsn 25484 | A singleton has 0 outer Le... |
| ovolunlem1a 25485 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25486 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25487 | Lemma for ~ ovolun . (Con... |
| ovolun 25488 | The Lebesgue outer measure... |
| ovolunnul 25489 | Adding a nullset does not ... |
| ovolfiniun 25490 | The Lebesgue outer measure... |
| ovoliunlem1 25491 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25492 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25493 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25494 | The Lebesgue outer measure... |
| ovoliun2 25495 | The Lebesgue outer measure... |
| ovoliunnul 25496 | A countable union of nulls... |
| shft2rab 25497 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25498 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25499 | Lemma for ~ ovolshft . (C... |
| ovolshft 25500 | The Lebesgue outer measure... |
| sca2rab 25501 | If ` B ` is a scale of ` A... |
| ovolscalem1 25502 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25503 | Lemma for ~ ovolshft . (C... |
| ovolsca 25504 | The Lebesgue outer measure... |
| ovolicc1 25505 | The measure of a closed in... |
| ovolicc2lem1 25506 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25507 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25508 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25509 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25510 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25511 | The measure of a closed in... |
| ovolicc 25512 | The measure of a closed in... |
| ovolicopnf 25513 | The measure of a right-unb... |
| ovolre 25514 | The measure of the real nu... |
| ismbl 25515 | The predicate " ` A ` is L... |
| ismbl2 25516 | From ~ ovolun , it suffice... |
| volres 25517 | A self-referencing abbrevi... |
| volf 25518 | The domain and codomain of... |
| mblvol 25519 | The volume of a measurable... |
| mblss 25520 | A measurable set is a subs... |
| mblsplit 25521 | The defining property of m... |
| volss 25522 | The Lebesgue measure is mo... |
| cmmbl 25523 | The complement of a measur... |
| nulmbl 25524 | A nullset is measurable. ... |
| nulmbl2 25525 | A set of outer measure zer... |
| unmbl 25526 | A union of measurable sets... |
| shftmbl 25527 | A shift of a measurable se... |
| 0mbl 25528 | The empty set is measurabl... |
| rembl 25529 | The set of all real number... |
| unidmvol 25530 | The union of the Lebesgue ... |
| inmbl 25531 | An intersection of measura... |
| difmbl 25532 | A difference of measurable... |
| finiunmbl 25533 | A finite union of measurab... |
| volun 25534 | The Lebesgue measure funct... |
| volinun 25535 | Addition of non-disjoint s... |
| volfiniun 25536 | The volume of a disjoint f... |
| iundisj 25537 | Rewrite a countable union ... |
| iundisj2 25538 | A disjoint union is disjoi... |
| voliunlem1 25539 | Lemma for ~ voliun . (Con... |
| voliunlem2 25540 | Lemma for ~ voliun . (Con... |
| voliunlem3 25541 | Lemma for ~ voliun . (Con... |
| iunmbl 25542 | The measurable sets are cl... |
| voliun 25543 | The Lebesgue measure funct... |
| volsuplem 25544 | Lemma for ~ volsup . (Con... |
| volsup 25545 | The volume of the limit of... |
| iunmbl2 25546 | The measurable sets are cl... |
| ioombl1lem1 25547 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25548 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25549 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25550 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25551 | An open right-unbounded in... |
| icombl1 25552 | A closed unbounded-above i... |
| icombl 25553 | A closed-below, open-above... |
| ioombl 25554 | An open real interval is m... |
| iccmbl 25555 | A closed real interval is ... |
| iccvolcl 25556 | A closed real interval has... |
| ovolioo 25557 | The measure of an open int... |
| volioo 25558 | The measure of an open int... |
| ioovolcl 25559 | An open real interval has ... |
| ovolfs2 25560 | Alternative expression for... |
| ioorcl2 25561 | An open interval with fini... |
| ioorf 25562 | Define a function from ope... |
| ioorval 25563 | Define a function from ope... |
| ioorinv2 25564 | The function ` F ` is an "... |
| ioorinv 25565 | The function ` F ` is an "... |
| ioorcl 25566 | The function ` F ` does no... |
| uniiccdif 25567 | A union of closed interval... |
| uniioovol 25568 | A disjoint union of open i... |
| uniiccvol 25569 | An almost-disjoint union o... |
| uniioombllem1 25570 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25571 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25572 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25573 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25574 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25575 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25576 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25577 | Lemma for ~ uniioombl . (... |
| uniioombl 25578 | A disjoint union of open i... |
| uniiccmbl 25579 | An almost-disjoint union o... |
| dyadf 25580 | The function ` F ` returns... |
| dyadval 25581 | Value of the dyadic ration... |
| dyadovol 25582 | Volume of a dyadic rationa... |
| dyadss 25583 | Two closed dyadic rational... |
| dyaddisjlem 25584 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25585 | Two closed dyadic rational... |
| dyadmaxlem 25586 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25587 | Any nonempty set of dyadic... |
| dyadmbllem 25588 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25589 | Any union of dyadic ration... |
| opnmbllem 25590 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25591 | All open sets are measurab... |
| opnmblALT 25592 | All open sets are measurab... |
| subopnmbl 25593 | Sets which are open in a m... |
| volsup2 25594 | The volume of ` A ` is the... |
| volcn 25595 | The function formed by res... |
| volivth 25596 | The Intermediate Value The... |
| vitalilem1 25597 | Lemma for ~ vitali . (Con... |
| vitalilem2 25598 | Lemma for ~ vitali . (Con... |
| vitalilem3 25599 | Lemma for ~ vitali . (Con... |
| vitalilem4 25600 | Lemma for ~ vitali . (Con... |
| vitalilem5 25601 | Lemma for ~ vitali . (Con... |
| vitali 25602 | If the reals can be well-o... |
| ismbf1 25613 | The predicate " ` F ` is a... |
| mbff 25614 | A measurable function is a... |
| mbfdm 25615 | The domain of a measurable... |
| mbfconstlem 25616 | Lemma for ~ mbfconst and r... |
| ismbf 25617 | The predicate " ` F ` is a... |
| ismbfcn 25618 | A complex function is meas... |
| mbfima 25619 | Definitional property of a... |
| mbfimaicc 25620 | The preimage of any closed... |
| mbfimasn 25621 | The preimage of a point un... |
| mbfconst 25622 | A constant function is mea... |
| mbf0 25623 | The empty function is meas... |
| mbfid 25624 | The identity function is m... |
| mbfmptcl 25625 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25626 | The domain of a measurable... |
| ismbfcn2 25627 | A complex function is meas... |
| ismbfd 25628 | Deduction to prove measura... |
| ismbf2d 25629 | Deduction to prove measura... |
| mbfeqalem1 25630 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25631 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25632 | If two functions are equal... |
| mbfres 25633 | The restriction of a measu... |
| mbfres2 25634 | Measurability of a piecewi... |
| mbfss 25635 | Change the domain of a mea... |
| mbfmulc2lem 25636 | Multiplication by a consta... |
| mbfmulc2re 25637 | Multiplication by a consta... |
| mbfmax 25638 | The maximum of two functio... |
| mbfneg 25639 | The negative of a measurab... |
| mbfpos 25640 | The positive part of a mea... |
| mbfposr 25641 | Converse to ~ mbfpos . (C... |
| mbfposb 25642 | A function is measurable i... |
| ismbf3d 25643 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25644 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25645 | The preimage of any open s... |
| mbfimaopn2 25646 | The preimage of any set op... |
| cncombf 25647 | The composition of a conti... |
| cnmbf 25648 | A continuous function is m... |
| mbfaddlem 25649 | The sum of two measurable ... |
| mbfadd 25650 | The sum of two measurable ... |
| mbfsub 25651 | The difference of two meas... |
| mbfmulc2 25652 | A complex constant times a... |
| mbfsup 25653 | The supremum of a sequence... |
| mbfinf 25654 | The infimum of a sequence ... |
| mbflimsup 25655 | The limit supremum of a se... |
| mbflimlem 25656 | The pointwise limit of a s... |
| mbflim 25657 | The pointwise limit of a s... |
| 0pval 25660 | The zero function evaluate... |
| 0plef 25661 | Two ways to say that the f... |
| 0pledm 25662 | Adjust the domain of the l... |
| isi1f 25663 | The predicate " ` F ` is a... |
| i1fmbf 25664 | Simple functions are measu... |
| i1ff 25665 | A simple function is a fun... |
| i1frn 25666 | A simple function has fini... |
| i1fima 25667 | Any preimage of a simple f... |
| i1fima2 25668 | Any preimage of a simple f... |
| i1fima2sn 25669 | Preimage of a singleton. ... |
| i1fd 25670 | A simplified set of assump... |
| i1f0rn 25671 | Any simple function takes ... |
| itg1val 25672 | The value of the integral ... |
| itg1val2 25673 | The value of the integral ... |
| itg1cl 25674 | Closure of the integral on... |
| itg1ge0 25675 | Closure of the integral on... |
| i1f0 25676 | The zero function is simpl... |
| itg10 25677 | The zero function has zero... |
| i1f1lem 25678 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25679 | Base case simple functions... |
| itg11 25680 | The integral of an indicat... |
| itg1addlem1 25681 | Decompose a preimage, whic... |
| i1faddlem 25682 | Decompose the preimage of ... |
| i1fmullem 25683 | Decompose the preimage of ... |
| i1fadd 25684 | The sum of two simple func... |
| i1fmul 25685 | The pointwise product of t... |
| itg1addlem2 25686 | Lemma for ~ itg1add . The... |
| itg1addlem3 25687 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25688 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 25689 | Lemma for ~ itg1add . (Co... |
| itg1add 25690 | The integral of a sum of s... |
| i1fmulclem 25691 | Decompose the preimage of ... |
| i1fmulc 25692 | A nonnegative constant tim... |
| itg1mulc 25693 | The integral of a constant... |
| i1fres 25694 | The "restriction" of a sim... |
| i1fpos 25695 | The positive part of a sim... |
| i1fposd 25696 | Deduction form of ~ i1fpos... |
| i1fsub 25697 | The difference of two simp... |
| itg1sub 25698 | The integral of a differen... |
| itg10a 25699 | The integral of a simple f... |
| itg1ge0a 25700 | The integral of an almost ... |
| itg1lea 25701 | Approximate version of ~ i... |
| itg1le 25702 | If one simple function dom... |
| itg1climres 25703 | Restricting the simple fun... |
| mbfi1fseqlem1 25704 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25705 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25706 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25707 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25708 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25709 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25710 | A characterization of meas... |
| mbfi1flimlem 25711 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25712 | Any real measurable functi... |
| mbfmullem2 25713 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25714 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25715 | The product of two measura... |
| itg2lcl 25716 | The set of lower sums is a... |
| itg2val 25717 | Value of the integral on n... |
| itg2l 25718 | Elementhood in the set ` L... |
| itg2lr 25719 | Sufficient condition for e... |
| xrge0f 25720 | A real function is a nonne... |
| itg2cl 25721 | The integral of a nonnegat... |
| itg2ub 25722 | The integral of a nonnegat... |
| itg2leub 25723 | Any upper bound on the int... |
| itg2ge0 25724 | The integral of a nonnegat... |
| itg2itg1 25725 | The integral of a nonnegat... |
| itg20 25726 | The integral of the zero f... |
| itg2lecl 25727 | If an ` S.2 ` integral is ... |
| itg2le 25728 | If one function dominates ... |
| itg2const 25729 | Integral of a constant fun... |
| itg2const2 25730 | When the base set of a con... |
| itg2seq 25731 | Definitional property of t... |
| itg2uba 25732 | Approximate version of ~ i... |
| itg2lea 25733 | Approximate version of ~ i... |
| itg2eqa 25734 | Approximate equality of in... |
| itg2mulclem 25735 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25736 | The integral of a nonnegat... |
| itg2splitlem 25737 | Lemma for ~ itg2split . (... |
| itg2split 25738 | The ` S.2 ` integral split... |
| itg2monolem1 25739 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25740 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25741 | Lemma for ~ itg2mono . (C... |
| itg2mono 25742 | The Monotone Convergence T... |
| itg2i1fseqle 25743 | Subject to the conditions ... |
| itg2i1fseq 25744 | Subject to the conditions ... |
| itg2i1fseq2 25745 | In an extension to the res... |
| itg2i1fseq3 25746 | Special case of ~ itg2i1fs... |
| itg2addlem 25747 | Lemma for ~ itg2add . (Co... |
| itg2add 25748 | The ` S.2 ` integral is li... |
| itg2gt0 25749 | If the function ` F ` is s... |
| itg2cnlem1 25750 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25751 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25752 | A sort of absolute continu... |
| ibllem 25753 | Conditioned equality theor... |
| isibl 25754 | The predicate " ` F ` is i... |
| isibl2 25755 | The predicate " ` F ` is i... |
| iblmbf 25756 | An integrable function is ... |
| iblitg 25757 | If a function is integrabl... |
| dfitg 25758 | Evaluate the class substit... |
| itgex 25759 | An integral is a set. (Co... |
| itgeq1f 25760 | Equality theorem for an in... |
| itgeq1fOLD 25761 | Obsolete version of ~ itge... |
| itgeq1 25762 | Equality theorem for an in... |
| nfitg1 25763 | Bound-variable hypothesis ... |
| nfitg 25764 | Bound-variable hypothesis ... |
| cbvitg 25765 | Change bound variable in a... |
| cbvitgv 25766 | Change bound variable in a... |
| itgeq2 25767 | Equality theorem for an in... |
| itgresr 25768 | The domain of an integral ... |
| itg0 25769 | The integral of anything o... |
| itgz 25770 | The integral of zero on an... |
| itgeq2dv 25771 | Equality theorem for an in... |
| itgmpt 25772 | Change bound variable in a... |
| itgcl 25773 | The integral of an integra... |
| itgvallem 25774 | Substitution lemma. (Cont... |
| itgvallem3 25775 | Lemma for ~ itgposval and ... |
| ibl0 25776 | The zero function is integ... |
| iblcnlem1 25777 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25778 | Expand out the universal q... |
| itgcnlem 25779 | Expand out the sum in ~ df... |
| iblrelem 25780 | Integrability of a real fu... |
| iblposlem 25781 | Lemma for ~ iblpos . (Con... |
| iblpos 25782 | Integrability of a nonnega... |
| iblre 25783 | Integrability of a real fu... |
| itgrevallem1 25784 | Lemma for ~ itgposval and ... |
| itgposval 25785 | The integral of a nonnegat... |
| itgreval 25786 | Decompose the integral of ... |
| itgrecl 25787 | Real closure of an integra... |
| iblcn 25788 | Integrability of a complex... |
| itgcnval 25789 | Decompose the integral of ... |
| itgre 25790 | Real part of an integral. ... |
| itgim 25791 | Imaginary part of an integ... |
| iblneg 25792 | The negative of an integra... |
| itgneg 25793 | Negation of an integral. ... |
| iblss 25794 | A subset of an integrable ... |
| iblss2 25795 | Change the domain of an in... |
| itgitg2 25796 | Transfer an integral using... |
| i1fibl 25797 | A simple function is integ... |
| itgitg1 25798 | Transfer an integral using... |
| itgle 25799 | Monotonicity of an integra... |
| itgge0 25800 | The integral of a positive... |
| itgss 25801 | Expand the set of an integ... |
| itgss2 25802 | Expand the set of an integ... |
| itgeqa 25803 | Approximate equality of in... |
| itgss3 25804 | Expand the set of an integ... |
| itgioo 25805 | Equality of integrals on o... |
| itgless 25806 | Expand the integral of a n... |
| iblconst 25807 | A constant function is int... |
| itgconst 25808 | Integral of a constant fun... |
| ibladdlem 25809 | Lemma for ~ ibladd . (Con... |
| ibladd 25810 | Add two integrals over the... |
| iblsub 25811 | Subtract two integrals ove... |
| itgaddlem1 25812 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25813 | Lemma for ~ itgadd . (Con... |
| itgadd 25814 | Add two integrals over the... |
| itgsub 25815 | Subtract two integrals ove... |
| itgfsum 25816 | Take a finite sum of integ... |
| iblabslem 25817 | Lemma for ~ iblabs . (Con... |
| iblabs 25818 | The absolute value of an i... |
| iblabsr 25819 | A measurable function is i... |
| iblmulc2 25820 | Multiply an integral by a ... |
| itgmulc2lem1 25821 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25822 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25823 | Multiply an integral by a ... |
| itgabs 25824 | The triangle inequality fo... |
| itgsplit 25825 | The ` S. ` integral splits... |
| itgspliticc 25826 | The ` S. ` integral splits... |
| itgsplitioo 25827 | The ` S. ` integral splits... |
| bddmulibl 25828 | A bounded function times a... |
| bddibl 25829 | A bounded function is inte... |
| cniccibl 25830 | A continuous function on a... |
| bddiblnc 25831 | Choice-free proof of ~ bdd... |
| cnicciblnc 25832 | Choice-free proof of ~ cni... |
| itggt0 25833 | The integral of a strictly... |
| itgcn 25834 | Transfer ~ itg2cn to the f... |
| ditgeq1 25837 | Equality theorem for the d... |
| ditgeq2 25838 | Equality theorem for the d... |
| ditgeq3 25839 | Equality theorem for the d... |
| ditgeq3dv 25840 | Equality theorem for the d... |
| ditgex 25841 | A directed integral is a s... |
| ditg0 25842 | Value of the directed inte... |
| cbvditg 25843 | Change bound variable in a... |
| cbvditgv 25844 | Change bound variable in a... |
| ditgpos 25845 | Value of the directed inte... |
| ditgneg 25846 | Value of the directed inte... |
| ditgcl 25847 | Closure of a directed inte... |
| ditgswap 25848 | Reverse a directed integra... |
| ditgsplitlem 25849 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25850 | This theorem is the raison... |
| reldv 25859 | The derivative function is... |
| limcvallem 25860 | Lemma for ~ ellimc . (Con... |
| limcfval 25861 | Value and set bounds on th... |
| ellimc 25862 | Value of the limit predica... |
| limcrcl 25863 | Reverse closure for the li... |
| limccl 25864 | Closure of the limit opera... |
| limcdif 25865 | It suffices to consider fu... |
| ellimc2 25866 | Write the definition of a ... |
| limcnlp 25867 | If ` B ` is not a limit po... |
| ellimc3 25868 | Write the epsilon-delta de... |
| limcflflem 25869 | Lemma for ~ limcflf . (Co... |
| limcflf 25870 | The limit operator can be ... |
| limcmo 25871 | If ` B ` is a limit point ... |
| limcmpt 25872 | Express the limit operator... |
| limcmpt2 25873 | Express the limit operator... |
| limcresi 25874 | Any limit of ` F ` is also... |
| limcres 25875 | If ` B ` is an interior po... |
| cnplimc 25876 | A function is continuous a... |
| cnlimc 25877 | ` F ` is a continuous func... |
| cnlimci 25878 | If ` F ` is a continuous f... |
| cnmptlimc 25879 | If ` F ` is a continuous f... |
| limccnp 25880 | If the limit of ` F ` at `... |
| limccnp2 25881 | The image of a convergent ... |
| limcco 25882 | Composition of two limits.... |
| limciun 25883 | A point is a limit of ` F ... |
| limcun 25884 | A point is a limit of ` F ... |
| dvlem 25885 | Closure for a difference q... |
| dvfval 25886 | Value and set bounds on th... |
| eldv 25887 | The differentiable predica... |
| dvcl 25888 | The derivative function ta... |
| dvbssntr 25889 | The set of differentiable ... |
| dvbss 25890 | The set of differentiable ... |
| dvbsss 25891 | The set of differentiable ... |
| perfdvf 25892 | The derivative is a functi... |
| recnprss 25893 | Both ` RR ` and ` CC ` are... |
| recnperf 25894 | Both ` RR ` and ` CC ` are... |
| dvfg 25895 | Explicitly write out the f... |
| dvf 25896 | The derivative is a functi... |
| dvfcn 25897 | The derivative is a functi... |
| dvreslem 25898 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25899 | Lemma for ~ dvres2 . (Con... |
| dvres 25900 | Restriction of a derivativ... |
| dvres2 25901 | Restriction of the base se... |
| dvres3 25902 | Restriction of a complex d... |
| dvres3a 25903 | Restriction of a complex d... |
| dvidlem 25904 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25905 | Derivative of a function r... |
| dvconst 25906 | Derivative of a constant f... |
| dvid 25907 | Derivative of the identity... |
| dvcnp 25908 | The difference quotient is... |
| dvcnp2 25909 | A function is continuous a... |
| dvcn 25910 | A differentiable function ... |
| dvnfval 25911 | Value of the iterated deri... |
| dvnff 25912 | The iterated derivative is... |
| dvn0 25913 | Zero times iterated deriva... |
| dvnp1 25914 | Successor iterated derivat... |
| dvn1 25915 | One times iterated derivat... |
| dvnf 25916 | The N-times derivative is ... |
| dvnbss 25917 | The set of N-times differe... |
| dvnadd 25918 | The ` N ` -th derivative o... |
| dvn2bss 25919 | An N-times differentiable ... |
| dvnres 25920 | Multiple derivative versio... |
| cpnfval 25921 | Condition for n-times cont... |
| fncpn 25922 | The ` C^n ` object is a fu... |
| elcpn 25923 | Condition for n-times cont... |
| cpnord 25924 | ` C^n ` conditions are ord... |
| cpncn 25925 | A ` C^n ` function is cont... |
| cpnres 25926 | The restriction of a ` C^n... |
| dvaddbr 25927 | The sum rule for derivativ... |
| dvmulbr 25928 | The product rule for deriv... |
| dvadd 25929 | The sum rule for derivativ... |
| dvmul 25930 | The product rule for deriv... |
| dvaddf 25931 | The sum rule for everywher... |
| dvmulf 25932 | The product rule for every... |
| dvcmul 25933 | The product rule when one ... |
| dvcmulf 25934 | The product rule when one ... |
| dvcobr 25935 | The chain rule for derivat... |
| dvco 25936 | The chain rule for derivat... |
| dvcof 25937 | The chain rule for everywh... |
| dvcjbr 25938 | The derivative of the conj... |
| dvcj 25939 | The derivative of the conj... |
| dvfre 25940 | The derivative of a real f... |
| dvnfre 25941 | The ` N ` -th derivative o... |
| dvexp 25942 | Derivative of a power func... |
| dvexp2 25943 | Derivative of an exponenti... |
| dvrec 25944 | Derivative of the reciproc... |
| dvmptres3 25945 | Function-builder for deriv... |
| dvmptid 25946 | Function-builder for deriv... |
| dvmptc 25947 | Function-builder for deriv... |
| dvmptcl 25948 | Closure lemma for ~ dvmptc... |
| dvmptadd 25949 | Function-builder for deriv... |
| dvmptmul 25950 | Function-builder for deriv... |
| dvmptres2 25951 | Function-builder for deriv... |
| dvmptres 25952 | Function-builder for deriv... |
| dvmptcmul 25953 | Function-builder for deriv... |
| dvmptdivc 25954 | Function-builder for deriv... |
| dvmptneg 25955 | Function-builder for deriv... |
| dvmptsub 25956 | Function-builder for deriv... |
| dvmptcj 25957 | Function-builder for deriv... |
| dvmptre 25958 | Function-builder for deriv... |
| dvmptim 25959 | Function-builder for deriv... |
| dvmptntr 25960 | Function-builder for deriv... |
| dvmptco 25961 | Function-builder for deriv... |
| dvrecg 25962 | Derivative of the reciproc... |
| dvmptdiv 25963 | Function-builder for deriv... |
| dvmptfsum 25964 | Function-builder for deriv... |
| dvcnvlem 25965 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25966 | A weak version of ~ dvcnvr... |
| dvexp3 25967 | Derivative of an exponenti... |
| dveflem 25968 | Derivative of the exponent... |
| dvef 25969 | Derivative of the exponent... |
| dvsincos 25970 | Derivative of the sine and... |
| dvsin 25971 | Derivative of the sine fun... |
| dvcos 25972 | Derivative of the cosine f... |
| dvferm1lem 25973 | Lemma for ~ dvferm . (Con... |
| dvferm1 25974 | One-sided version of ~ dvf... |
| dvferm2lem 25975 | Lemma for ~ dvferm . (Con... |
| dvferm2 25976 | One-sided version of ~ dvf... |
| dvferm 25977 | Fermat's theorem on statio... |
| rollelem 25978 | Lemma for ~ rolle . (Cont... |
| rolle 25979 | Rolle's theorem. If ` F `... |
| cmvth 25980 | Cauchy's Mean Value Theore... |
| mvth 25981 | The Mean Value Theorem. I... |
| dvlip 25982 | A function with derivative... |
| dvlipcn 25983 | A complex function with de... |
| dvlip2 25984 | Combine the results of ~ d... |
| c1liplem1 25985 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25986 | C^1 functions are Lipschit... |
| c1lip2 25987 | C^1 functions are Lipschit... |
| c1lip3 25988 | C^1 functions are Lipschit... |
| dveq0 25989 | If a continuous function h... |
| dv11cn 25990 | Two functions defined on a... |
| dvgt0lem1 25991 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25992 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25993 | A function on a closed int... |
| dvlt0 25994 | A function on a closed int... |
| dvge0 25995 | A function on a closed int... |
| dvle 25996 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25997 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25998 | Lemma for ~ dvivth . (Con... |
| dvivth 25999 | Darboux' theorem, or the i... |
| dvne0 26000 | A function on a closed int... |
| dvne0f1 26001 | A function on a closed int... |
| lhop1lem 26002 | Lemma for ~ lhop1 . (Cont... |
| lhop1 26003 | L'Hôpital's Rule for... |
| lhop2 26004 | L'Hôpital's Rule for... |
| lhop 26005 | L'Hôpital's Rule. I... |
| dvcnvrelem1 26006 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 26007 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 26008 | The derivative rule for in... |
| dvcvx 26009 | A real function with stric... |
| dvfsumle 26010 | Compare a finite sum to an... |
| dvfsumge 26011 | Compare a finite sum to an... |
| dvfsumabs 26012 | Compare a finite sum to an... |
| dvmptrecl 26013 | Real closure of a derivati... |
| dvfsumrlimf 26014 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 26015 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 26016 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 26017 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 26018 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 26019 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 26020 | Compare a finite sum to an... |
| dvfsumrlim2 26021 | Compare a finite sum to an... |
| dvfsumrlim3 26022 | Conjoin the statements of ... |
| dvfsum2 26023 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26024 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26025 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26026 | The Fundamental Theorem of... |
| ftc1lem3 26027 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26028 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26029 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26030 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26031 | The Fundamental Theorem of... |
| ftc1cn 26032 | Strengthen the assumptions... |
| ftc2 26033 | The Fundamental Theorem of... |
| ftc2ditglem 26034 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26035 | Directed integral analogue... |
| itgparts 26036 | Integration by parts. If ... |
| itgsubstlem 26037 | Lemma for ~ itgsubst . (C... |
| itgsubst 26038 | Integration by ` u ` -subs... |
| itgpowd 26039 | The integral of a monomial... |
| reldmmdeg 26044 | Multivariate degree is a b... |
| tdeglem1 26045 | Functionality of the total... |
| tdeglem3 26046 | Additivity of the total de... |
| tdeglem4 26047 | There is only one multi-in... |
| tdeglem2 26048 | Simplification of total de... |
| mdegfval 26049 | Value of the multivariate ... |
| mdegval 26050 | Value of the multivariate ... |
| mdegleb 26051 | Property of being of limit... |
| mdeglt 26052 | If there is an upper limit... |
| mdegldg 26053 | A nonzero polynomial has s... |
| mdegxrcl 26054 | Closure of polynomial degr... |
| mdegxrf 26055 | Functionality of polynomia... |
| mdegcl 26056 | Sharp closure for multivar... |
| mdeg0 26057 | Degree of the zero polynom... |
| mdegnn0cl 26058 | Degree of a nonzero polyno... |
| degltlem1 26059 | Theorem on arithmetic of e... |
| degltp1le 26060 | Theorem on arithmetic of e... |
| mdegaddle 26061 | The degree of a sum is at ... |
| mdegvscale 26062 | The degree of a scalar mul... |
| mdegvsca 26063 | The degree of a scalar mul... |
| mdegle0 26064 | A polynomial has nonpositi... |
| mdegmullem 26065 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26066 | The multivariate degree of... |
| deg1fval 26067 | Relate univariate polynomi... |
| deg1xrf 26068 | Functionality of univariat... |
| deg1xrcl 26069 | Closure of univariate poly... |
| deg1cl 26070 | Sharp closure of univariat... |
| mdegpropd 26071 | Property deduction for pol... |
| deg1fvi 26072 | Univariate polynomial degr... |
| deg1propd 26073 | Property deduction for pol... |
| deg1z 26074 | Degree of the zero univari... |
| deg1nn0cl 26075 | Degree of a nonzero univar... |
| deg1n0ima 26076 | Degree image of a set of p... |
| deg1nn0clb 26077 | A polynomial is nonzero if... |
| deg1lt0 26078 | A polynomial is zero iff i... |
| deg1ldg 26079 | A nonzero univariate polyn... |
| deg1ldgn 26080 | An index at which a polyno... |
| deg1ldgdomn 26081 | A nonzero univariate polyn... |
| deg1leb 26082 | Property of being of limit... |
| deg1val 26083 | Value of the univariate de... |
| deg1lt 26084 | If the degree of a univari... |
| deg1ge 26085 | Conversely, a nonzero coef... |
| coe1mul3 26086 | The coefficient vector of ... |
| coe1mul4 26087 | Value of the "leading" coe... |
| deg1addle 26088 | The degree of a sum is at ... |
| deg1addle2 26089 | If both factors have degre... |
| deg1add 26090 | Exact degree of a sum of t... |
| deg1vscale 26091 | The degree of a scalar tim... |
| deg1vsca 26092 | The degree of a scalar tim... |
| deg1invg 26093 | The degree of the negated ... |
| deg1suble 26094 | The degree of a difference... |
| deg1sub 26095 | Exact degree of a differen... |
| deg1mulle2 26096 | Produce a bound on the pro... |
| deg1sublt 26097 | Subtraction of two polynom... |
| deg1le0 26098 | A polynomial has nonpositi... |
| deg1sclle 26099 | A scalar polynomial has no... |
| deg1scl 26100 | A nonzero scalar polynomia... |
| deg1mul2 26101 | Degree of multiplication o... |
| deg1mul 26102 | Degree of multiplication o... |
| deg1mul3 26103 | Degree of multiplication o... |
| deg1mul3le 26104 | Degree of multiplication o... |
| deg1tmle 26105 | Limiting degree of a polyn... |
| deg1tm 26106 | Exact degree of a polynomi... |
| deg1pwle 26107 | Limiting degree of a varia... |
| deg1pw 26108 | Exact degree of a variable... |
| ply1nz 26109 | Univariate polynomials ove... |
| ply1nzb 26110 | Univariate polynomials are... |
| ply1domn 26111 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26112 | The ring of univariate pol... |
| ply1divmo 26123 | Uniqueness of a quotient i... |
| ply1divex 26124 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26125 | The division algorithm for... |
| ply1divalg2 26126 | Reverse the order of multi... |
| uc1pval 26127 | Value of the set of unitic... |
| isuc1p 26128 | Being a unitic polynomial.... |
| mon1pval 26129 | Value of the set of monic ... |
| ismon1p 26130 | Being a monic polynomial. ... |
| uc1pcl 26131 | Unitic polynomials are pol... |
| mon1pcl 26132 | Monic polynomials are poly... |
| uc1pn0 26133 | Unitic polynomials are not... |
| mon1pn0 26134 | Monic polynomials are not ... |
| uc1pdeg 26135 | Unitic polynomials have no... |
| uc1pldg 26136 | Unitic polynomials have un... |
| mon1pldg 26137 | Unitic polynomials have on... |
| mon1puc1p 26138 | Monic polynomials are unit... |
| uc1pmon1p 26139 | Make a unitic polynomial m... |
| deg1submon1p 26140 | The difference of two moni... |
| mon1pid 26141 | Monicity and degree of the... |
| q1pval 26142 | Value of the univariate po... |
| q1peqb 26143 | Characterizing property of... |
| q1pcl 26144 | Closure of the quotient by... |
| r1pval 26145 | Value of the polynomial re... |
| r1pcl 26146 | Closure of remainder follo... |
| r1pdeglt 26147 | The remainder has a degree... |
| r1pid 26148 | Express the original polyn... |
| r1pid2 26149 | Identity law for polynomia... |
| dvdsq1p 26150 | Divisibility in a polynomi... |
| dvdsr1p 26151 | Divisibility in a polynomi... |
| ply1remlem 26152 | A term of the form ` x - N... |
| ply1rem 26153 | The polynomial remainder t... |
| facth1 26154 | The factor theorem and its... |
| fta1glem1 26155 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26156 | Lemma for ~ fta1g . (Cont... |
| fta1g 26157 | The one-sided fundamental ... |
| fta1blem 26158 | Lemma for ~ fta1b . (Cont... |
| fta1b 26159 | The assumption that ` R ` ... |
| idomrootle 26160 | No element of an integral ... |
| drnguc1p 26161 | Over a division ring, all ... |
| ig1peu 26162 | There is a unique monic po... |
| ig1pval 26163 | Substitutions for the poly... |
| ig1pval2 26164 | Generator of the zero idea... |
| ig1pval3 26165 | Characterizing properties ... |
| ig1pcl 26166 | The monic generator of an ... |
| ig1pdvds 26167 | The monic generator of an ... |
| ig1prsp 26168 | Any ideal of polynomials o... |
| ply1lpir 26169 | The ring of polynomials ov... |
| ply1pid 26170 | The polynomials over a fie... |
| plyco0 26179 | Two ways to say that a fun... |
| plyval 26180 | Value of the polynomial se... |
| plybss 26181 | Reverse closure of the par... |
| elply 26182 | Definition of a polynomial... |
| elply2 26183 | The coefficient function c... |
| plyun0 26184 | The set of polynomials is ... |
| plyf 26185 | A polynomial is a function... |
| plyss 26186 | The polynomial set functio... |
| plyssc 26187 | Every polynomial ring is c... |
| elplyr 26188 | Sufficient condition for e... |
| elplyd 26189 | Sufficient condition for e... |
| ply1termlem 26190 | Lemma for ~ ply1term . (C... |
| ply1term 26191 | A one-term polynomial. (C... |
| plypow 26192 | A power is a polynomial. ... |
| plyconst 26193 | A constant function is a p... |
| ne0p 26194 | A test to show that a poly... |
| ply0 26195 | The zero function is a pol... |
| plyid 26196 | The identity function is a... |
| plyeq0lem 26197 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26198 | If a polynomial is zero at... |
| plypf1 26199 | Write the set of complex p... |
| plyaddlem1 26200 | Derive the coefficient fun... |
| plymullem1 26201 | Derive the coefficient fun... |
| plyaddlem 26202 | Lemma for ~ plyadd . (Con... |
| plymullem 26203 | Lemma for ~ plymul . (Con... |
| plyadd 26204 | The sum of two polynomials... |
| plymul 26205 | The product of two polynom... |
| plysub 26206 | The difference of two poly... |
| plyaddcl 26207 | The sum of two polynomials... |
| plymulcl 26208 | The product of two polynom... |
| plysubcl 26209 | The difference of two poly... |
| coeval 26210 | Value of the coefficient f... |
| coeeulem 26211 | Lemma for ~ coeeu . (Cont... |
| coeeu 26212 | Uniqueness of the coeffici... |
| coelem 26213 | Lemma for properties of th... |
| coeeq 26214 | If ` A ` satisfies the pro... |
| dgrval 26215 | Value of the degree functi... |
| dgrlem 26216 | Lemma for ~ dgrcl and simi... |
| coef 26217 | The domain and codomain of... |
| coef2 26218 | The domain and codomain of... |
| coef3 26219 | The domain and codomain of... |
| dgrcl 26220 | The degree of any polynomi... |
| dgrub 26221 | If the ` M ` -th coefficie... |
| dgrub2 26222 | All the coefficients above... |
| dgrlb 26223 | If all the coefficients ab... |
| coeidlem 26224 | Lemma for ~ coeid . (Cont... |
| coeid 26225 | Reconstruct a polynomial a... |
| coeid2 26226 | Reconstruct a polynomial a... |
| coeid3 26227 | Reconstruct a polynomial a... |
| plyco 26228 | The composition of two pol... |
| coeeq2 26229 | Compute the coefficient fu... |
| dgrle 26230 | Given an explicit expressi... |
| dgreq 26231 | If the highest term in a p... |
| 0dgr 26232 | A constant function has de... |
| 0dgrb 26233 | A function has degree zero... |
| dgrnznn 26234 | A nonzero polynomial with ... |
| coefv0 26235 | The result of evaluating a... |
| coeaddlem 26236 | Lemma for ~ coeadd and ~ d... |
| coemullem 26237 | Lemma for ~ coemul and ~ d... |
| coeadd 26238 | The coefficient function o... |
| coemul 26239 | A coefficient of a product... |
| coe11 26240 | The coefficient function i... |
| coemulhi 26241 | The leading coefficient of... |
| coemulc 26242 | The coefficient function i... |
| coe0 26243 | The coefficients of the ze... |
| coesub 26244 | The coefficient function o... |
| coe1termlem 26245 | The coefficient function o... |
| coe1term 26246 | The coefficient function o... |
| dgr1term 26247 | The degree of a monomial. ... |
| plycn 26248 | A polynomial is a continuo... |
| dgr0 26249 | The degree of the zero pol... |
| coeidp 26250 | The coefficients of the id... |
| dgrid 26251 | The degree of the identity... |
| dgreq0 26252 | The leading coefficient of... |
| dgrlt 26253 | Two ways to say that the d... |
| dgradd 26254 | The degree of a sum of pol... |
| dgradd2 26255 | The degree of a sum of pol... |
| dgrmul2 26256 | The degree of a product of... |
| dgrmul 26257 | The degree of a product of... |
| dgrmulc 26258 | Scalar multiplication by a... |
| dgrsub 26259 | The degree of a difference... |
| dgrcolem1 26260 | The degree of a compositio... |
| dgrcolem2 26261 | Lemma for ~ dgrco . (Cont... |
| dgrco 26262 | The degree of a compositio... |
| plycjlem 26263 | Lemma for ~ plycj and ~ co... |
| plycj 26264 | The double conjugation of ... |
| coecj 26265 | Double conjugation of a po... |
| plycjOLD 26266 | Obsolete version of ~ plyc... |
| coecjOLD 26267 | Obsolete version of ~ coec... |
| plyrecj 26268 | A polynomial with real coe... |
| plymul0or 26269 | Polynomial multiplication ... |
| ofmulrt 26270 | The set of roots of a prod... |
| plyreres 26271 | Real-coefficient polynomia... |
| dvply1 26272 | Derivative of a polynomial... |
| dvply2g 26273 | The derivative of a polyno... |
| dvply2 26274 | The derivative of a polyno... |
| dvnply2 26275 | Polynomials have polynomia... |
| dvnply 26276 | Polynomials have polynomia... |
| plycpn 26277 | Polynomials are smooth. (... |
| quotval 26280 | Value of the quotient func... |
| plydivlem1 26281 | Lemma for ~ plydivalg . (... |
| plydivlem2 26282 | Lemma for ~ plydivalg . (... |
| plydivlem3 26283 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26284 | Lemma for ~ plydivex . In... |
| plydivex 26285 | Lemma for ~ plydivalg . (... |
| plydiveu 26286 | Lemma for ~ plydivalg . (... |
| plydivalg 26287 | The division algorithm on ... |
| quotlem 26288 | Lemma for properties of th... |
| quotcl 26289 | The quotient of two polyno... |
| quotcl2 26290 | Closure of the quotient fu... |
| quotdgr 26291 | Remainder property of the ... |
| plyremlem 26292 | Closure of a linear factor... |
| plyrem 26293 | The polynomial remainder t... |
| facth 26294 | The factor theorem. If a ... |
| fta1lem 26295 | Lemma for ~ fta1 . (Contr... |
| fta1 26296 | The easy direction of the ... |
| quotcan 26297 | Exact division with a mult... |
| vieta1lem1 26298 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26299 | Lemma for ~ vieta1 : induc... |
| vieta1 26300 | The first-order Vieta's fo... |
| plyexmo 26301 | An infinite set of values ... |
| elaa 26304 | Elementhood in the set of ... |
| aacn 26305 | An algebraic number is a c... |
| aasscn 26306 | The algebraic numbers are ... |
| elqaalem1 26307 | Lemma for ~ elqaa . The f... |
| elqaalem2 26308 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26309 | Lemma for ~ elqaa . (Cont... |
| elqaa 26310 | The set of numbers generat... |
| qaa 26311 | Every rational number is a... |
| qssaa 26312 | The rational numbers are c... |
| iaa 26313 | The imaginary unit is alge... |
| aareccl 26314 | The reciprocal of an algeb... |
| aacjcl 26315 | The conjugate of an algebr... |
| aannenlem1 26316 | Lemma for ~ aannen . (Con... |
| aannenlem2 26317 | Lemma for ~ aannen . (Con... |
| aannenlem3 26318 | The algebraic numbers are ... |
| aannen 26319 | The algebraic numbers are ... |
| aalioulem1 26320 | Lemma for ~ aaliou . An i... |
| aalioulem2 26321 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26322 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26323 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26324 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26325 | Lemma for ~ aaliou . (Con... |
| aaliou 26326 | Liouville's theorem on dio... |
| geolim3 26327 | Geometric series convergen... |
| aaliou2 26328 | Liouville's approximation ... |
| aaliou2b 26329 | Liouville's approximation ... |
| aaliou3lem1 26330 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26331 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26332 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26333 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26334 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26335 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26336 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26337 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26338 | Example of a "Liouville nu... |
| aaliou3 26339 | Example of a "Liouville nu... |
| taylfvallem1 26344 | Lemma for ~ taylfval . (C... |
| taylfvallem 26345 | Lemma for ~ taylfval . (C... |
| taylfval 26346 | Define the Taylor polynomi... |
| eltayl 26347 | Value of the Taylor series... |
| taylf 26348 | The Taylor series defines ... |
| tayl0 26349 | The Taylor series is alway... |
| taylplem1 26350 | Lemma for ~ taylpfval and ... |
| taylplem2 26351 | Lemma for ~ taylpfval and ... |
| taylpfval 26352 | Define the Taylor polynomi... |
| taylpf 26353 | The Taylor polynomial is a... |
| taylpval 26354 | Value of the Taylor polyno... |
| taylply2 26355 | The Taylor polynomial is a... |
| taylply 26356 | The Taylor polynomial is a... |
| dvtaylp 26357 | The derivative of the Tayl... |
| dvntaylp 26358 | The ` M ` -th derivative o... |
| dvntaylp0 26359 | The first ` N ` derivative... |
| taylthlem1 26360 | Lemma for ~ taylth . This... |
| taylthlem2 26361 | Lemma for ~ taylth . (Con... |
| taylth 26362 | Taylor's theorem. The Tay... |
| ulmrel 26365 | The uniform limit relation... |
| ulmscl 26366 | Closure of the base set in... |
| ulmval 26367 | Express the predicate: Th... |
| ulmcl 26368 | Closure of a uniform limit... |
| ulmf 26369 | Closure of a uniform limit... |
| ulmpm 26370 | Closure of a uniform limit... |
| ulmf2 26371 | Closure of a uniform limit... |
| ulm2 26372 | Simplify ~ ulmval when ` F... |
| ulmi 26373 | The uniform limit property... |
| ulmclm 26374 | A uniform limit of functio... |
| ulmres 26375 | A sequence of functions co... |
| ulmshftlem 26376 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26377 | A sequence of functions co... |
| ulm0 26378 | Every function converges u... |
| ulmuni 26379 | A sequence of functions un... |
| ulmdm 26380 | Two ways to express that a... |
| ulmcaulem 26381 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26382 | A sequence of functions co... |
| ulmcau2 26383 | A sequence of functions co... |
| ulmss 26384 | A uniform limit of functio... |
| ulmbdd 26385 | A uniform limit of bounded... |
| ulmcn 26386 | A uniform limit of continu... |
| ulmdvlem1 26387 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26388 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26389 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26390 | If ` F ` is a sequence of ... |
| mtest 26391 | The Weierstrass M-test. I... |
| mtestbdd 26392 | Given the hypotheses of th... |
| mbfulm 26393 | A uniform limit of measura... |
| iblulm 26394 | A uniform limit of integra... |
| itgulm 26395 | A uniform limit of integra... |
| itgulm2 26396 | A uniform limit of integra... |
| pserval 26397 | Value of the function ` G ... |
| pserval2 26398 | Value of the function ` G ... |
| psergf 26399 | The sequence of terms in t... |
| radcnvlem1 26400 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26401 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26402 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26403 | Zero is always a convergen... |
| radcnvcl 26404 | The radius of convergence ... |
| radcnvlt1 26405 | If ` X ` is within the ope... |
| radcnvlt2 26406 | If ` X ` is within the ope... |
| radcnvle 26407 | If ` X ` is a convergent p... |
| dvradcnv 26408 | The radius of convergence ... |
| pserulm 26409 | If ` S ` is a region conta... |
| psercn2 26410 | Since by ~ pserulm the ser... |
| psercnlem2 26411 | Lemma for ~ psercn . (Con... |
| psercnlem1 26412 | Lemma for ~ psercn . (Con... |
| psercn 26413 | An infinite series converg... |
| pserdvlem1 26414 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26415 | Lemma for ~ pserdv . (Con... |
| pserdv 26416 | The derivative of a power ... |
| pserdv2 26417 | The derivative of a power ... |
| abelthlem1 26418 | Lemma for ~ abelth . (Con... |
| abelthlem2 26419 | Lemma for ~ abelth . The ... |
| abelthlem3 26420 | Lemma for ~ abelth . (Con... |
| abelthlem4 26421 | Lemma for ~ abelth . (Con... |
| abelthlem5 26422 | Lemma for ~ abelth . (Con... |
| abelthlem6 26423 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26424 | Lemma for ~ abelth . (Con... |
| abelthlem7 26425 | Lemma for ~ abelth . (Con... |
| abelthlem8 26426 | Lemma for ~ abelth . (Con... |
| abelthlem9 26427 | Lemma for ~ abelth . By a... |
| abelth 26428 | Abel's theorem. If the po... |
| abelth2 26429 | Abel's theorem, restricted... |
| efcn 26430 | The exponential function i... |
| sincn 26431 | Sine is continuous. (Cont... |
| coscn 26432 | Cosine is continuous. (Co... |
| reeff1olem 26433 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26434 | The real exponential funct... |
| reefiso 26435 | The exponential function o... |
| efcvx 26436 | The exponential function o... |
| reefgim 26437 | The exponential function i... |
| pilem1 26438 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26439 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26440 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26441 | ` _pi ` is between 2 and 4... |
| sinpi 26442 | The sine of ` _pi ` is 0. ... |
| pire 26443 | ` _pi ` is a real number. ... |
| picn 26444 | ` _pi ` is a complex numbe... |
| pipos 26445 | ` _pi ` is positive. (Con... |
| pine0 26446 | ` _pi ` is nonzero. (Cont... |
| pirp 26447 | ` _pi ` is a positive real... |
| negpicn 26448 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26449 | Lemma for ~ sinhalfpi and ... |
| halfpire 26450 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26451 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26452 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26453 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26454 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26455 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26456 | The cosine of ` -u _pi / 2... |
| efhalfpi 26457 | The exponential of ` _i _p... |
| cospi 26458 | The cosine of ` _pi ` is `... |
| efipi 26459 | The exponential of ` _i x.... |
| eulerid 26460 | Euler's identity. (Contri... |
| sin2pi 26461 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26462 | The cosine of ` 2 _pi ` is... |
| ef2pi 26463 | The exponential of ` 2 _pi... |
| ef2kpi 26464 | If ` K ` is an integer, th... |
| efper 26465 | The exponential function i... |
| sinperlem 26466 | Lemma for ~ sinper and ~ c... |
| sinper 26467 | The sine function is perio... |
| cosper 26468 | The cosine function is per... |
| sin2kpi 26469 | If ` K ` is an integer, th... |
| cos2kpi 26470 | If ` K ` is an integer, th... |
| sin2pim 26471 | Sine of a number subtracte... |
| cos2pim 26472 | Cosine of a number subtrac... |
| sinmpi 26473 | Sine of a number less ` _p... |
| cosmpi 26474 | Cosine of a number less ` ... |
| sinppi 26475 | Sine of a number plus ` _p... |
| cosppi 26476 | Cosine of a number plus ` ... |
| efimpi 26477 | The exponential function a... |
| sinhalfpip 26478 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26479 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26480 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26481 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26482 | Ptolemy's Theorem. This t... |
| sincosq1lem 26483 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26484 | The signs of the sine and ... |
| sincosq2sgn 26485 | The signs of the sine and ... |
| sincosq3sgn 26486 | The signs of the sine and ... |
| sincosq4sgn 26487 | The signs of the sine and ... |
| coseq00topi 26488 | Location of the zeroes of ... |
| coseq0negpitopi 26489 | Location of the zeroes of ... |
| tanrpcl 26490 | Positive real closure of t... |
| tangtx 26491 | The tangent function is gr... |
| tanabsge 26492 | The tangent function is gr... |
| sinq12gt0 26493 | The sine of a number stric... |
| sinq12ge0 26494 | The sine of a number betwe... |
| sinq34lt0t 26495 | The sine of a number stric... |
| cosq14gt0 26496 | The cosine of a number str... |
| cosq14ge0 26497 | The cosine of a number bet... |
| sincosq1eq 26498 | Complementarity of the sin... |
| sincos4thpi 26499 | The sine and cosine of ` _... |
| tan4thpi 26500 | The tangent of ` _pi / 4 `... |
| tan4thpiOLD 26501 | Obsolete version of ~ tan4... |
| sincos6thpi 26502 | The sine and cosine of ` _... |
| sincos3rdpi 26503 | The sine and cosine of ` _... |
| pigt3 26504 | ` _pi ` is greater than 3.... |
| pige3 26505 | ` _pi ` is greater than or... |
| pige3ALT 26506 | Alternate proof of ~ pige3... |
| abssinper 26507 | The absolute value of sine... |
| sinkpi 26508 | The sine of an integer mul... |
| coskpi 26509 | The absolute value of the ... |
| sineq0 26510 | A complex number whose sin... |
| coseq1 26511 | A complex number whose cos... |
| cos02pilt1 26512 | Cosine is less than one be... |
| cosq34lt1 26513 | Cosine is less than one in... |
| efeq1 26514 | A complex number whose exp... |
| cosne0 26515 | The cosine function has no... |
| cosordlem 26516 | Lemma for ~ cosord . (Con... |
| cosord 26517 | Cosine is decreasing over ... |
| cos0pilt1 26518 | Cosine is between minus on... |
| cos11 26519 | Cosine is one-to-one over ... |
| sinord 26520 | Sine is increasing over th... |
| recosf1o 26521 | The cosine function is a b... |
| resinf1o 26522 | The sine function is a bij... |
| tanord1 26523 | The tangent function is st... |
| tanord 26524 | The tangent function is st... |
| tanregt0 26525 | The real part of the tange... |
| negpitopissre 26526 | The interval ` ( -u _pi (,... |
| efgh 26527 | The exponential function o... |
| efif1olem1 26528 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26529 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26530 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26531 | The exponential function o... |
| efif1o 26532 | The exponential function o... |
| efifo 26533 | The exponential function o... |
| eff1olem 26534 | The exponential function m... |
| eff1o 26535 | The exponential function m... |
| efabl 26536 | The image of a subgroup of... |
| efsubm 26537 | The image of a subgroup of... |
| circgrp 26538 | The circle group ` T ` is ... |
| circsubm 26539 | The circle group ` T ` is ... |
| logrn 26544 | The range of the natural l... |
| ellogrn 26545 | Write out the property ` A... |
| dflog2 26546 | The natural logarithm func... |
| relogrn 26547 | The range of the natural l... |
| logrncn 26548 | The range of the natural l... |
| eff1o2 26549 | The exponential function r... |
| logf1o 26550 | The natural logarithm func... |
| dfrelog 26551 | The natural logarithm func... |
| relogf1o 26552 | The natural logarithm func... |
| logrncl 26553 | Closure of the natural log... |
| logcl 26554 | Closure of the natural log... |
| logimcl 26555 | Closure of the imaginary p... |
| logcld 26556 | The logarithm of a nonzero... |
| logimcld 26557 | The imaginary part of the ... |
| logimclad 26558 | The imaginary part of the ... |
| abslogimle 26559 | The imaginary part of the ... |
| logrnaddcl 26560 | The range of the natural l... |
| relogcl 26561 | Closure of the natural log... |
| eflog 26562 | Relationship between the n... |
| logeq0im1 26563 | If the logarithm of a numb... |
| logccne0 26564 | The logarithm isn't 0 if i... |
| logne0 26565 | Logarithm of a non-1 posit... |
| reeflog 26566 | Relationship between the n... |
| logef 26567 | Relationship between the n... |
| relogef 26568 | Relationship between the n... |
| logeftb 26569 | Relationship between the n... |
| relogeftb 26570 | Relationship between the n... |
| log1 26571 | The natural logarithm of `... |
| loge 26572 | The natural logarithm of `... |
| logi 26573 | The natural logarithm of `... |
| logneg 26574 | The natural logarithm of a... |
| logm1 26575 | The natural logarithm of n... |
| lognegb 26576 | If a number has imaginary ... |
| relogoprlem 26577 | Lemma for ~ relogmul and ~... |
| relogmul 26578 | The natural logarithm of t... |
| relogdiv 26579 | The natural logarithm of t... |
| explog 26580 | Exponentiation of a nonzer... |
| reexplog 26581 | Exponentiation of a positi... |
| relogexp 26582 | The natural logarithm of p... |
| relog 26583 | Real part of a logarithm. ... |
| relogiso 26584 | The natural logarithm func... |
| reloggim 26585 | The natural logarithm is a... |
| logltb 26586 | The natural logarithm func... |
| logfac 26587 | The logarithm of a factori... |
| eflogeq 26588 | Solve an equation involvin... |
| logleb 26589 | Natural logarithm preserve... |
| rplogcl 26590 | Closure of the logarithm f... |
| logge0 26591 | The logarithm of a number ... |
| logcj 26592 | The natural logarithm dist... |
| efiarg 26593 | The exponential of the "ar... |
| cosargd 26594 | The cosine of the argument... |
| cosarg0d 26595 | The cosine of the argument... |
| argregt0 26596 | Closure of the argument of... |
| argrege0 26597 | Closure of the argument of... |
| argimgt0 26598 | Closure of the argument of... |
| argimlt0 26599 | Closure of the argument of... |
| logimul 26600 | Multiplying a number by ` ... |
| logneg2 26601 | The logarithm of the negat... |
| logmul2 26602 | Generalization of ~ relogm... |
| logdiv2 26603 | Generalization of ~ relogd... |
| abslogle 26604 | Bound on the magnitude of ... |
| tanarg 26605 | The basic relation between... |
| logdivlti 26606 | The ` log x / x ` function... |
| logdivlt 26607 | The ` log x / x ` function... |
| logdivle 26608 | The ` log x / x ` function... |
| relogcld 26609 | Closure of the natural log... |
| reeflogd 26610 | Relationship between the n... |
| relogmuld 26611 | The natural logarithm of t... |
| relogdivd 26612 | The natural logarithm of t... |
| logled 26613 | Natural logarithm preserve... |
| relogefd 26614 | Relationship between the n... |
| rplogcld 26615 | Closure of the logarithm f... |
| logge0d 26616 | The logarithm of a number ... |
| logge0b 26617 | The logarithm of a number ... |
| loggt0b 26618 | The logarithm of a number ... |
| logle1b 26619 | The logarithm of a number ... |
| loglt1b 26620 | The logarithm of a number ... |
| divlogrlim 26621 | The inverse logarithm func... |
| logno1 26622 | The logarithm function is ... |
| dvrelog 26623 | The derivative of the real... |
| relogcn 26624 | The real logarithm functio... |
| ellogdm 26625 | Elementhood in the "contin... |
| logdmn0 26626 | A number in the continuous... |
| logdmnrp 26627 | A number in the continuous... |
| logdmss 26628 | The continuity domain of `... |
| logcnlem2 26629 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26630 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26631 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26632 | Lemma for ~ logcn . (Cont... |
| logcn 26633 | The logarithm function is ... |
| dvloglem 26634 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26635 | The "continuous domain" of... |
| logf1o2 26636 | The logarithm maps its con... |
| dvlog 26637 | The derivative of the comp... |
| dvlog2lem 26638 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26639 | The derivative of the comp... |
| advlog 26640 | The antiderivative of the ... |
| advlogexp 26641 | The antiderivative of a po... |
| efopnlem1 26642 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26643 | Lemma for ~ efopn . (Cont... |
| efopn 26644 | The exponential map is an ... |
| logtayllem 26645 | Lemma for ~ logtayl . (Co... |
| logtayl 26646 | The Taylor series for ` -u... |
| logtaylsum 26647 | The Taylor series for ` -u... |
| logtayl2 26648 | Power series expression fo... |
| logccv 26649 | The natural logarithm func... |
| cxpval 26650 | Value of the complex power... |
| cxpef 26651 | Value of the complex power... |
| 0cxp 26652 | Value of the complex power... |
| cxpexpz 26653 | Relate the complex power f... |
| cxpexp 26654 | Relate the complex power f... |
| logcxp 26655 | Logarithm of a complex pow... |
| cxp0 26656 | Value of the complex power... |
| cxp1 26657 | Value of the complex power... |
| 1cxp 26658 | Value of the complex power... |
| ecxp 26659 | Write the exponential func... |
| cxpcl 26660 | Closure of the complex pow... |
| recxpcl 26661 | Real closure of the comple... |
| rpcxpcl 26662 | Positive real closure of t... |
| cxpne0 26663 | Complex exponentiation is ... |
| cxpeq0 26664 | Complex exponentiation is ... |
| cxpadd 26665 | Sum of exponents law for c... |
| cxpp1 26666 | Value of a nonzero complex... |
| cxpneg 26667 | Value of a complex number ... |
| cxpsub 26668 | Exponent subtraction law f... |
| cxpge0 26669 | Nonnegative exponentiation... |
| mulcxplem 26670 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26671 | Complex exponentiation of ... |
| cxprec 26672 | Complex exponentiation of ... |
| divcxp 26673 | Complex exponentiation of ... |
| cxpmul 26674 | Product of exponents law f... |
| cxpmul2 26675 | Product of exponents law f... |
| cxproot 26676 | The complex power function... |
| cxpmul2z 26677 | Generalize ~ cxpmul2 to ne... |
| abscxp 26678 | Absolute value of a power,... |
| abscxp2 26679 | Absolute value of a power,... |
| cxplt 26680 | Ordering property for comp... |
| cxple 26681 | Ordering property for comp... |
| cxplea 26682 | Ordering property for comp... |
| cxple2 26683 | Ordering property for comp... |
| cxplt2 26684 | Ordering property for comp... |
| cxple2a 26685 | Ordering property for comp... |
| cxplt3 26686 | Ordering property for comp... |
| cxple3 26687 | Ordering property for comp... |
| cxpsqrtlem 26688 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26689 | The complex exponential fu... |
| logsqrt 26690 | Logarithm of a square root... |
| cxp0d 26691 | Value of the complex power... |
| cxp1d 26692 | Value of the complex power... |
| 1cxpd 26693 | Value of the complex power... |
| cxpcld 26694 | Closure of the complex pow... |
| cxpmul2d 26695 | Product of exponents law f... |
| 0cxpd 26696 | Value of the complex power... |
| cxpexpzd 26697 | Relate the complex power f... |
| cxpefd 26698 | Value of the complex power... |
| cxpne0d 26699 | Complex exponentiation is ... |
| cxpp1d 26700 | Value of a nonzero complex... |
| cxpnegd 26701 | Value of a complex number ... |
| cxpmul2zd 26702 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26703 | Sum of exponents law for c... |
| cxpsubd 26704 | Exponent subtraction law f... |
| cxpltd 26705 | Ordering property for comp... |
| cxpled 26706 | Ordering property for comp... |
| cxplead 26707 | Ordering property for comp... |
| divcxpd 26708 | Complex exponentiation of ... |
| recxpcld 26709 | Positive real closure of t... |
| cxpge0d 26710 | Nonnegative exponentiation... |
| cxple2ad 26711 | Ordering property for comp... |
| cxplt2d 26712 | Ordering property for comp... |
| cxple2d 26713 | Ordering property for comp... |
| mulcxpd 26714 | Complex exponentiation of ... |
| recxpf1lem 26715 | Complex exponentiation on ... |
| cxpsqrtth 26716 | Square root theorem over t... |
| 2irrexpq 26717 | There exist irrational num... |
| cxprecd 26718 | Complex exponentiation of ... |
| rpcxpcld 26719 | Positive real closure of t... |
| logcxpd 26720 | Logarithm of a complex pow... |
| cxplt3d 26721 | Ordering property for comp... |
| cxple3d 26722 | Ordering property for comp... |
| cxpmuld 26723 | Product of exponents law f... |
| cxpgt0d 26724 | A positive real raised to ... |
| cxpcom 26725 | Commutative law for real e... |
| dvcxp1 26726 | The derivative of a comple... |
| dvcxp2 26727 | The derivative of a comple... |
| dvsqrt 26728 | The derivative of the real... |
| dvcncxp1 26729 | Derivative of complex powe... |
| dvcnsqrt 26730 | Derivative of square root ... |
| cxpcn 26731 | Domain of continuity of th... |
| cxpcn2 26732 | Continuity of the complex ... |
| cxpcn3lem 26733 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26734 | Extend continuity of the c... |
| resqrtcn 26735 | Continuity of the real squ... |
| sqrtcn 26736 | Continuity of the square r... |
| cxpaddlelem 26737 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26738 | Ordering property for comp... |
| abscxpbnd 26739 | Bound on the absolute valu... |
| root1id 26740 | Property of an ` N ` -th r... |
| root1eq1 26741 | The only powers of an ` N ... |
| root1cj 26742 | Within the ` N ` -th roots... |
| cxpeq 26743 | Solve an equation involvin... |
| zrtelqelz 26744 | If the ` N ` -th root of a... |
| zrtdvds 26745 | A positive integer root di... |
| rtprmirr 26746 | The root of a prime number... |
| loglesqrt 26747 | An upper bound on the loga... |
| logreclem 26748 | Symmetry of the natural lo... |
| logrec 26749 | Logarithm of a reciprocal ... |
| logbval 26752 | Define the value of the ` ... |
| logbcl 26753 | General logarithm closure.... |
| logbid1 26754 | General logarithm is 1 whe... |
| logb1 26755 | The logarithm of ` 1 ` to ... |
| elogb 26756 | The general logarithm of a... |
| logbchbase 26757 | Change of base for logarit... |
| relogbval 26758 | Value of the general logar... |
| relogbcl 26759 | Closure of the general log... |
| relogbzcl 26760 | Closure of the general log... |
| relogbreexp 26761 | Power law for the general ... |
| relogbzexp 26762 | Power law for the general ... |
| relogbmul 26763 | The logarithm of the produ... |
| relogbmulexp 26764 | The logarithm of the produ... |
| relogbdiv 26765 | The logarithm of the quoti... |
| relogbexp 26766 | Identity law for general l... |
| nnlogbexp 26767 | Identity law for general l... |
| logbrec 26768 | Logarithm of a reciprocal ... |
| logbleb 26769 | The general logarithm func... |
| logblt 26770 | The general logarithm func... |
| relogbcxp 26771 | Identity law for the gener... |
| cxplogb 26772 | Identity law for the gener... |
| relogbcxpb 26773 | The logarithm is the inver... |
| logbmpt 26774 | The general logarithm to a... |
| logbf 26775 | The general logarithm to a... |
| logbfval 26776 | The general logarithm of a... |
| relogbf 26777 | The general logarithm to a... |
| logblog 26778 | The general logarithm to t... |
| logbgt0b 26779 | The logarithm of a positiv... |
| logbgcd1irr 26780 | The logarithm of an intege... |
| 2logb9irr 26781 | Example for ~ logbgcd1irr ... |
| logbprmirr 26782 | The logarithm of a prime t... |
| 2logb3irr 26783 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26784 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26785 | The square root of two to ... |
| 2irrexpqALT 26786 | Alternate proof of ~ 2irre... |
| angval 26787 | Define the angle function,... |
| angcan 26788 | Cancel a constant multipli... |
| angneg 26789 | Cancel a negative sign in ... |
| angvald 26790 | The (signed) angle between... |
| angcld 26791 | The (signed) angle between... |
| angrteqvd 26792 | Two vectors are at a right... |
| cosangneg2d 26793 | The cosine of the angle be... |
| angrtmuld 26794 | Perpendicularity of two ve... |
| ang180lem1 26795 | Lemma for ~ ang180 . Show... |
| ang180lem2 26796 | Lemma for ~ ang180 . Show... |
| ang180lem3 26797 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26798 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26799 | Lemma for ~ ang180 : Redu... |
| ang180 26800 | The sum of angles ` m A B ... |
| lawcoslem1 26801 | Lemma for ~ lawcos . Here... |
| lawcos 26802 | Law of cosines (also known... |
| pythag 26803 | Pythagorean theorem. Give... |
| isosctrlem1 26804 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26805 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26806 | Lemma for ~ isosctr . Cor... |
| isosctr 26807 | Isosceles triangle theorem... |
| ssscongptld 26808 | If two triangles have equa... |
| affineequiv 26809 | Equivalence between two wa... |
| affineequiv2 26810 | Equivalence between two wa... |
| affineequiv3 26811 | Equivalence between two wa... |
| affineequiv4 26812 | Equivalence between two wa... |
| affineequivne 26813 | Equivalence between two wa... |
| angpieqvdlem 26814 | Equivalence used in the pr... |
| angpieqvdlem2 26815 | Equivalence used in ~ angp... |
| angpined 26816 | If the angle at ABC is ` _... |
| angpieqvd 26817 | The angle ABC is ` _pi ` i... |
| chordthmlem 26818 | If ` M ` is the midpoint o... |
| chordthmlem2 26819 | If M is the midpoint of AB... |
| chordthmlem3 26820 | If M is the midpoint of AB... |
| chordthmlem4 26821 | If P is on the segment AB ... |
| chordthmlem5 26822 | If P is on the segment AB ... |
| chordthm 26823 | The intersecting chords th... |
| heron 26824 | Heron's formula gives the ... |
| quad2 26825 | The quadratic equation, wi... |
| quad 26826 | The quadratic equation. (... |
| 1cubrlem 26827 | The cube roots of unity. ... |
| 1cubr 26828 | The cube roots of unity. ... |
| dcubic1lem 26829 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26830 | Reverse direction of ~ dcu... |
| dcubic1 26831 | Forward direction of ~ dcu... |
| dcubic 26832 | Solutions to the depressed... |
| mcubic 26833 | Solutions to a monic cubic... |
| cubic2 26834 | The solution to the genera... |
| cubic 26835 | The cubic equation, which ... |
| binom4 26836 | Work out a quartic binomia... |
| dquartlem1 26837 | Lemma for ~ dquart . (Con... |
| dquartlem2 26838 | Lemma for ~ dquart . (Con... |
| dquart 26839 | Solve a depressed quartic ... |
| quart1cl 26840 | Closure lemmas for ~ quart... |
| quart1lem 26841 | Lemma for ~ quart1 . (Con... |
| quart1 26842 | Depress a quartic equation... |
| quartlem1 26843 | Lemma for ~ quart . (Cont... |
| quartlem2 26844 | Closure lemmas for ~ quart... |
| quartlem3 26845 | Closure lemmas for ~ quart... |
| quartlem4 26846 | Closure lemmas for ~ quart... |
| quart 26847 | The quartic equation, writ... |
| asinlem 26854 | The argument to the logari... |
| asinlem2 26855 | The argument to the logari... |
| asinlem3a 26856 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26857 | The argument to the logari... |
| asinf 26858 | Domain and codomain of the... |
| asincl 26859 | Closure for the arcsin fun... |
| acosf 26860 | Domain and codoamin of the... |
| acoscl 26861 | Closure for the arccos fun... |
| atandm 26862 | Since the property is a li... |
| atandm2 26863 | This form of ~ atandm is a... |
| atandm3 26864 | A compact form of ~ atandm... |
| atandm4 26865 | A compact form of ~ atandm... |
| atanf 26866 | Domain and codoamin of the... |
| atancl 26867 | Closure for the arctan fun... |
| asinval 26868 | Value of the arcsin functi... |
| acosval 26869 | Value of the arccos functi... |
| atanval 26870 | Value of the arctan functi... |
| atanre 26871 | A real number is in the do... |
| asinneg 26872 | The arcsine function is od... |
| acosneg 26873 | The negative symmetry rela... |
| efiasin 26874 | The exponential of the arc... |
| sinasin 26875 | The arcsine function is an... |
| cosacos 26876 | The arccosine function is ... |
| asinsinlem 26877 | Lemma for ~ asinsin . (Co... |
| asinsin 26878 | The arcsine function compo... |
| acoscos 26879 | The arccosine function is ... |
| asin1 26880 | The arcsine of ` 1 ` is ` ... |
| acos1 26881 | The arccosine of ` 1 ` is ... |
| reasinsin 26882 | The arcsine function compo... |
| asinsinb 26883 | Relationship between sine ... |
| acoscosb 26884 | Relationship between cosin... |
| asinbnd 26885 | The arcsine function has r... |
| acosbnd 26886 | The arccosine function has... |
| asinrebnd 26887 | Bounds on the arcsine func... |
| asinrecl 26888 | The arcsine function is re... |
| acosrecl 26889 | The arccosine function is ... |
| cosasin 26890 | The cosine of the arcsine ... |
| sinacos 26891 | The sine of the arccosine ... |
| atandmneg 26892 | The domain of the arctange... |
| atanneg 26893 | The arctangent function is... |
| atan0 26894 | The arctangent of zero is ... |
| atandmcj 26895 | The arctangent function di... |
| atancj 26896 | The arctangent function di... |
| atanrecl 26897 | The arctangent function is... |
| efiatan 26898 | Value of the exponential o... |
| atanlogaddlem 26899 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26900 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26901 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26902 | A variation on ~ atanlogad... |
| efiatan2 26903 | Value of the exponential o... |
| 2efiatan 26904 | Value of the exponential o... |
| tanatan 26905 | The arctangent function is... |
| atandmtan 26906 | The tangent function has r... |
| cosatan 26907 | The cosine of an arctangen... |
| cosatanne0 26908 | The arctangent function ha... |
| atantan 26909 | The arctangent function is... |
| atantanb 26910 | Relationship between tange... |
| atanbndlem 26911 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26912 | The arctangent function is... |
| atanord 26913 | The arctangent function is... |
| atan1 26914 | The arctangent of ` 1 ` is... |
| bndatandm 26915 | A point in the open unit d... |
| atans 26916 | The "domain of continuity"... |
| atans2 26917 | It suffices to show that `... |
| atansopn 26918 | The domain of continuity o... |
| atansssdm 26919 | The domain of continuity o... |
| ressatans 26920 | The real number line is a ... |
| dvatan 26921 | The derivative of the arct... |
| atancn 26922 | The arctangent is a contin... |
| atantayl 26923 | The Taylor series for ` ar... |
| atantayl2 26924 | The Taylor series for ` ar... |
| atantayl3 26925 | The Taylor series for ` ar... |
| leibpilem1 26926 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26927 | The Leibniz formula for ` ... |
| leibpi 26928 | The Leibniz formula for ` ... |
| leibpisum 26929 | The Leibniz formula for ` ... |
| log2cnv 26930 | Using the Taylor series fo... |
| log2tlbnd 26931 | Bound the error term in th... |
| log2ublem1 26932 | Lemma for ~ log2ub . The ... |
| log2ublem2 26933 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26934 | Lemma for ~ log2ub . In d... |
| log2ub 26935 | ` log 2 ` is less than ` 2... |
| log2le1 26936 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26937 | Lemma for ~ birthday . (C... |
| birthdaylem2 26938 | For general ` N ` and ` K ... |
| birthdaylem3 26939 | For general ` N ` and ` K ... |
| birthday 26940 | The Birthday Problem. The... |
| dmarea 26943 | The domain of the area fun... |
| areambl 26944 | The fibers of a measurable... |
| areass 26945 | A measurable region is a s... |
| dfarea 26946 | Rewrite ~ df-area self-ref... |
| areaf 26947 | Area measurement is a func... |
| areacl 26948 | The area of a measurable r... |
| areage0 26949 | The area of a measurable r... |
| areaval 26950 | The area of a measurable r... |
| rlimcnp 26951 | Relate a limit of a real-v... |
| rlimcnp2 26952 | Relate a limit of a real-v... |
| rlimcnp3 26953 | Relate a limit of a real-v... |
| xrlimcnp 26954 | Relate a limit of a real-v... |
| efrlim 26955 | The limit of the sequence ... |
| dfef2 26956 | The limit of the sequence ... |
| cxplim 26957 | A power to a negative expo... |
| sqrtlim 26958 | The inverse square root fu... |
| rlimcxp 26959 | Any power to a positive ex... |
| o1cxp 26960 | An eventually bounded func... |
| cxp2limlem 26961 | A linear factor grows slow... |
| cxp2lim 26962 | Any power grows slower tha... |
| cxploglim 26963 | The logarithm grows slower... |
| cxploglim2 26964 | Every power of the logarit... |
| divsqrtsumlem 26965 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26966 | The function ` F ` used in... |
| divsqrsum 26967 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26968 | A bound on the distance of... |
| divsqrtsumo1 26969 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26970 | Closure of a 0-1 linear co... |
| scvxcvx 26971 | A strictly convex function... |
| jensenlem1 26972 | Lemma for ~ jensen . (Con... |
| jensenlem2 26973 | Lemma for ~ jensen . (Con... |
| jensen 26974 | Jensen's inequality, a fin... |
| amgmlem 26975 | Lemma for ~ amgm . (Contr... |
| amgm 26976 | Inequality of arithmetic a... |
| logdifbnd 26979 | Bound on the difference of... |
| logdiflbnd 26980 | Lower bound on the differe... |
| emcllem1 26981 | Lemma for ~ emcl . The se... |
| emcllem2 26982 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26983 | Lemma for ~ emcl . The fu... |
| emcllem4 26984 | Lemma for ~ emcl . The di... |
| emcllem5 26985 | Lemma for ~ emcl . The pa... |
| emcllem6 26986 | Lemma for ~ emcl . By the... |
| emcllem7 26987 | Lemma for ~ emcl and ~ har... |
| emcl 26988 | Closure and bounds for the... |
| harmonicbnd 26989 | A bound on the harmonic se... |
| harmonicbnd2 26990 | A bound on the harmonic se... |
| emre 26991 | The Euler-Mascheroni const... |
| emgt0 26992 | The Euler-Mascheroni const... |
| harmonicbnd3 26993 | A bound on the harmonic se... |
| harmoniclbnd 26994 | A bound on the harmonic se... |
| harmonicubnd 26995 | A bound on the harmonic se... |
| harmonicbnd4 26996 | The asymptotic behavior of... |
| fsumharmonic 26997 | Bound a finite sum based o... |
| zetacvg 27000 | The zeta series is converg... |
| eldmgm 27007 | Elementhood in the set of ... |
| dmgmaddn0 27008 | If ` A ` is not a nonposit... |
| dmlogdmgm 27009 | If ` A ` is in the continu... |
| rpdmgm 27010 | A positive real number is ... |
| dmgmn0 27011 | If ` A ` is not a nonposit... |
| dmgmaddnn0 27012 | If ` A ` is not a nonposit... |
| dmgmdivn0 27013 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 27014 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 27015 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 27016 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 27017 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 27018 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 27019 | The series ` G ` is unifor... |
| lgamgulm 27020 | The series ` G ` is unifor... |
| lgamgulm2 27021 | Rewrite the limit of the s... |
| lgambdd 27022 | The log-Gamma function is ... |
| lgamucov 27023 | The ` U ` regions used in ... |
| lgamucov2 27024 | The ` U ` regions used in ... |
| lgamcvglem 27025 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27026 | The log-Gamma function is ... |
| lgamf 27027 | The log-Gamma function is ... |
| gamf 27028 | The Gamma function is a co... |
| gamcl 27029 | The exponential of the log... |
| eflgam 27030 | The exponential of the log... |
| gamne0 27031 | The Gamma function is neve... |
| igamval 27032 | Value of the inverse Gamma... |
| igamz 27033 | Value of the inverse Gamma... |
| igamgam 27034 | Value of the inverse Gamma... |
| igamlgam 27035 | Value of the inverse Gamma... |
| igamf 27036 | Closure of the inverse Gam... |
| igamcl 27037 | Closure of the inverse Gam... |
| gamigam 27038 | The Gamma function is the ... |
| lgamcvg 27039 | The series ` G ` converges... |
| lgamcvg2 27040 | The series ` G ` converges... |
| gamcvg 27041 | The pointwise exponential ... |
| lgamp1 27042 | The functional equation of... |
| gamp1 27043 | The functional equation of... |
| gamcvg2lem 27044 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27045 | An infinite product expres... |
| regamcl 27046 | The Gamma function is real... |
| relgamcl 27047 | The log-Gamma function is ... |
| rpgamcl 27048 | The log-Gamma function is ... |
| lgam1 27049 | The log-Gamma function at ... |
| gam1 27050 | The log-Gamma function at ... |
| facgam 27051 | The Gamma function general... |
| gamfac 27052 | The Gamma function general... |
| wilthlem1 27053 | The only elements that are... |
| wilthlem2 27054 | Lemma for ~ wilth : induct... |
| wilthlem3 27055 | Lemma for ~ wilth . Here ... |
| wilth 27056 | Wilson's theorem. A numbe... |
| wilthimp 27057 | The forward implication of... |
| ftalem1 27058 | Lemma for ~ fta : "growth... |
| ftalem2 27059 | Lemma for ~ fta . There e... |
| ftalem3 27060 | Lemma for ~ fta . There e... |
| ftalem4 27061 | Lemma for ~ fta : Closure... |
| ftalem5 27062 | Lemma for ~ fta : Main pr... |
| ftalem6 27063 | Lemma for ~ fta : Dischar... |
| ftalem7 27064 | Lemma for ~ fta . Shift t... |
| fta 27065 | The Fundamental Theorem of... |
| basellem1 27066 | Lemma for ~ basel . Closu... |
| basellem2 27067 | Lemma for ~ basel . Show ... |
| basellem3 27068 | Lemma for ~ basel . Using... |
| basellem4 27069 | Lemma for ~ basel . By ~ ... |
| basellem5 27070 | Lemma for ~ basel . Using... |
| basellem6 27071 | Lemma for ~ basel . The f... |
| basellem7 27072 | Lemma for ~ basel . The f... |
| basellem8 27073 | Lemma for ~ basel . The f... |
| basellem9 27074 | Lemma for ~ basel . Since... |
| basel 27075 | The sum of the inverse squ... |
| efnnfsumcl 27088 | Finite sum closure in the ... |
| ppisval 27089 | The set of primes less tha... |
| ppisval2 27090 | The set of primes less tha... |
| ppifi 27091 | The set of primes less tha... |
| prmdvdsfi 27092 | The set of prime divisors ... |
| chtf 27093 | Domain and codoamin of the... |
| chtcl 27094 | Real closure of the Chebys... |
| chtval 27095 | Value of the Chebyshev fun... |
| efchtcl 27096 | The Chebyshev function is ... |
| chtge0 27097 | The Chebyshev function is ... |
| vmaval 27098 | Value of the von Mangoldt ... |
| isppw 27099 | Two ways to say that ` A `... |
| isppw2 27100 | Two ways to say that ` A `... |
| vmappw 27101 | Value of the von Mangoldt ... |
| vmaprm 27102 | Value of the von Mangoldt ... |
| vmacl 27103 | Closure for the von Mangol... |
| vmaf 27104 | Functionality of the von M... |
| efvmacl 27105 | The von Mangoldt is closed... |
| vmage0 27106 | The von Mangoldt function ... |
| chpval 27107 | Value of the second Chebys... |
| chpf 27108 | Functionality of the secon... |
| chpcl 27109 | Closure for the second Che... |
| efchpcl 27110 | The second Chebyshev funct... |
| chpge0 27111 | The second Chebyshev funct... |
| ppival 27112 | Value of the prime-countin... |
| ppival2 27113 | Value of the prime-countin... |
| ppival2g 27114 | Value of the prime-countin... |
| ppif 27115 | Domain and codomain of the... |
| ppicl 27116 | Real closure of the prime-... |
| muval 27117 | The value of the Möbi... |
| muval1 27118 | The value of the Möbi... |
| muval2 27119 | The value of the Möbi... |
| isnsqf 27120 | Two ways to say that a num... |
| issqf 27121 | Two ways to say that a num... |
| sqfpc 27122 | The prime count of a squar... |
| dvdssqf 27123 | A divisor of a squarefree ... |
| sqf11 27124 | A squarefree number is com... |
| muf 27125 | The Möbius function i... |
| mucl 27126 | Closure of the Möbius... |
| sgmval 27127 | The value of the divisor f... |
| sgmval2 27128 | The value of the divisor f... |
| 0sgm 27129 | The value of the sum-of-di... |
| sgmf 27130 | The divisor function is a ... |
| sgmcl 27131 | Closure of the divisor fun... |
| sgmnncl 27132 | Closure of the divisor fun... |
| mule1 27133 | The Möbius function t... |
| chtfl 27134 | The Chebyshev function doe... |
| chpfl 27135 | The second Chebyshev funct... |
| ppiprm 27136 | The prime-counting functio... |
| ppinprm 27137 | The prime-counting functio... |
| chtprm 27138 | The Chebyshev function at ... |
| chtnprm 27139 | The Chebyshev function at ... |
| chpp1 27140 | The second Chebyshev funct... |
| chtwordi 27141 | The Chebyshev function is ... |
| chpwordi 27142 | The second Chebyshev funct... |
| chtdif 27143 | The difference of the Cheb... |
| efchtdvds 27144 | The exponentiated Chebyshe... |
| ppifl 27145 | The prime-counting functio... |
| ppip1le 27146 | The prime-counting functio... |
| ppiwordi 27147 | The prime-counting functio... |
| ppidif 27148 | The difference of the prim... |
| ppi1 27149 | The prime-counting functio... |
| cht1 27150 | The Chebyshev function at ... |
| vma1 27151 | The von Mangoldt function ... |
| chp1 27152 | The second Chebyshev funct... |
| ppi1i 27153 | Inference form of ~ ppiprm... |
| ppi2i 27154 | Inference form of ~ ppinpr... |
| ppi2 27155 | The prime-counting functio... |
| ppi3 27156 | The prime-counting functio... |
| cht2 27157 | The Chebyshev function at ... |
| cht3 27158 | The Chebyshev function at ... |
| ppinncl 27159 | Closure of the prime-count... |
| chtrpcl 27160 | Closure of the Chebyshev f... |
| ppieq0 27161 | The prime-counting functio... |
| ppiltx 27162 | The prime-counting functio... |
| prmorcht 27163 | Relate the primorial (prod... |
| mumullem1 27164 | Lemma for ~ mumul . A mul... |
| mumullem2 27165 | Lemma for ~ mumul . The p... |
| mumul 27166 | The Möbius function i... |
| sqff1o 27167 | There is a bijection from ... |
| fsumdvdsdiaglem 27168 | A "diagonal commutation" o... |
| fsumdvdsdiag 27169 | A "diagonal commutation" o... |
| fsumdvdscom 27170 | A double commutation of di... |
| dvdsppwf1o 27171 | A bijection between the di... |
| dvdsflf1o 27172 | A bijection from the numbe... |
| dvdsflsumcom 27173 | A sum commutation from ` s... |
| fsumfldivdiaglem 27174 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27175 | The right-hand side of ~ d... |
| musum 27176 | The sum of the Möbius... |
| musumsum 27177 | Evaluate a collapsing sum ... |
| muinv 27178 | The Möbius inversion ... |
| mpodvdsmulf1o 27179 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27180 | Product of two divisor sum... |
| dvdsmulf1o 27181 | If ` M ` and ` N ` are two... |
| sgmppw 27182 | The value of the divisor f... |
| 0sgmppw 27183 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27184 | The sum of divisors for a ... |
| 1sgm2ppw 27185 | The sum of the divisors of... |
| sgmmul 27186 | The divisor function for f... |
| ppiublem1 27187 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27188 | A prime greater than ` 3 `... |
| ppiub 27189 | An upper bound on the prim... |
| vmalelog 27190 | The von Mangoldt function ... |
| chtlepsi 27191 | The first Chebyshev functi... |
| chprpcl 27192 | Closure of the second Cheb... |
| chpeq0 27193 | The second Chebyshev funct... |
| chteq0 27194 | The first Chebyshev functi... |
| chtleppi 27195 | Upper bound on the ` theta... |
| chtublem 27196 | Lemma for ~ chtub . (Cont... |
| chtub 27197 | An upper bound on the Cheb... |
| fsumvma 27198 | Rewrite a sum over the von... |
| fsumvma2 27199 | Apply ~ fsumvma for the co... |
| pclogsum 27200 | The logarithmic analogue o... |
| vmasum 27201 | The sum of the von Mangold... |
| logfac2 27202 | Another expression for the... |
| chpval2 27203 | Express the second Chebysh... |
| chpchtsum 27204 | The second Chebyshev funct... |
| chpub 27205 | An upper bound on the seco... |
| logfacubnd 27206 | A simple upper bound on th... |
| logfaclbnd 27207 | A lower bound on the logar... |
| logfacbnd3 27208 | Show the stronger statemen... |
| logfacrlim 27209 | Combine the estimates ~ lo... |
| logexprlim 27210 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27211 | Write out ~ logfacrlim as ... |
| mersenne 27212 | A Mersenne prime is a prim... |
| perfect1 27213 | Euclid's contribution to t... |
| perfectlem1 27214 | Lemma for ~ perfect . (Co... |
| perfectlem2 27215 | Lemma for ~ perfect . (Co... |
| perfect 27216 | The Euclid-Euler theorem, ... |
| dchrval 27219 | Value of the group of Diri... |
| dchrbas 27220 | Base set of the group of D... |
| dchrelbas 27221 | A Dirichlet character is a... |
| dchrelbas2 27222 | A Dirichlet character is a... |
| dchrelbas3 27223 | A Dirichlet character is a... |
| dchrelbasd 27224 | A Dirichlet character is a... |
| dchrrcl 27225 | Reverse closure for a Diri... |
| dchrmhm 27226 | A Dirichlet character is a... |
| dchrf 27227 | A Dirichlet character is a... |
| dchrelbas4 27228 | A Dirichlet character is a... |
| dchrzrh1 27229 | Value of a Dirichlet chara... |
| dchrzrhcl 27230 | A Dirichlet character take... |
| dchrzrhmul 27231 | A Dirichlet character is c... |
| dchrplusg 27232 | Group operation on the gro... |
| dchrmul 27233 | Group operation on the gro... |
| dchrmulcl 27234 | Closure of the group opera... |
| dchrn0 27235 | A Dirichlet character is n... |
| dchr1cl 27236 | Closure of the principal D... |
| dchrmullid 27237 | Left identity for the prin... |
| dchrinvcl 27238 | Closure of the group inver... |
| dchrabl 27239 | The set of Dirichlet chara... |
| dchrfi 27240 | The group of Dirichlet cha... |
| dchrghm 27241 | A Dirichlet character rest... |
| dchr1 27242 | Value of the principal Dir... |
| dchreq 27243 | A Dirichlet character is d... |
| dchrresb 27244 | A Dirichlet character is d... |
| dchrabs 27245 | A Dirichlet character take... |
| dchrinv 27246 | The inverse of a Dirichlet... |
| dchrabs2 27247 | A Dirichlet character take... |
| dchr1re 27248 | The principal Dirichlet ch... |
| dchrptlem1 27249 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27250 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27251 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27252 | For any element other than... |
| dchrsum2 27253 | An orthogonality relation ... |
| dchrsum 27254 | An orthogonality relation ... |
| sumdchr2 27255 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27256 | There are exactly ` phi ( ... |
| sumdchr 27257 | An orthogonality relation ... |
| dchr2sum 27258 | An orthogonality relation ... |
| sum2dchr 27259 | An orthogonality relation ... |
| bcctr 27260 | Value of the central binom... |
| pcbcctr 27261 | Prime count of a central b... |
| bcmono 27262 | The binomial coefficient i... |
| bcmax 27263 | The binomial coefficient t... |
| bcp1ctr 27264 | Ratio of two central binom... |
| bclbnd 27265 | A bound on the binomial co... |
| efexple 27266 | Convert a bound on a power... |
| bpos1lem 27267 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27268 | Bertrand's postulate, chec... |
| bposlem1 27269 | An upper bound on the prim... |
| bposlem2 27270 | There are no odd primes in... |
| bposlem3 27271 | Lemma for ~ bpos . Since ... |
| bposlem4 27272 | Lemma for ~ bpos . (Contr... |
| bposlem5 27273 | Lemma for ~ bpos . Bound ... |
| bposlem6 27274 | Lemma for ~ bpos . By usi... |
| bposlem7 27275 | Lemma for ~ bpos . The fu... |
| bposlem8 27276 | Lemma for ~ bpos . Evalua... |
| bposlem9 27277 | Lemma for ~ bpos . Derive... |
| bpos 27278 | Bertrand's postulate: ther... |
| zabsle1 27281 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27282 | When ` a ` is coprime to t... |
| lgslem2 27283 | The set ` Z ` of all integ... |
| lgslem3 27284 | The set ` Z ` of all integ... |
| lgslem4 27285 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27286 | Value of the Legendre symb... |
| lgsfval 27287 | Value of the function ` F ... |
| lgsfcl2 27288 | The function ` F ` is clos... |
| lgscllem 27289 | The Legendre symbol is an ... |
| lgsfcl 27290 | Closure of the function ` ... |
| lgsfle1 27291 | The function ` F ` has mag... |
| lgsval2lem 27292 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27293 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27294 | The Legendre symbol is an ... |
| lgs0 27295 | The Legendre symbol when t... |
| lgscl 27296 | The Legendre symbol is an ... |
| lgsle1 27297 | The Legendre symbol has ab... |
| lgsval2 27298 | The Legendre symbol at a p... |
| lgs2 27299 | The Legendre symbol at ` 2... |
| lgsval3 27300 | The Legendre symbol at an ... |
| lgsvalmod 27301 | The Legendre symbol is equ... |
| lgsval4 27302 | Restate ~ lgsval for nonze... |
| lgsfcl3 27303 | Closure of the function ` ... |
| lgsval4a 27304 | Same as ~ lgsval4 for posi... |
| lgscl1 27305 | The value of the Legendre ... |
| lgsneg 27306 | The Legendre symbol is eit... |
| lgsneg1 27307 | The Legendre symbol for no... |
| lgsmod 27308 | The Legendre (Jacobi) symb... |
| lgsdilem 27309 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27310 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27311 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27312 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27313 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27314 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27315 | The Legendre symbol is com... |
| lgsdirprm 27316 | The Legendre symbol is com... |
| lgsdir 27317 | The Legendre symbol is com... |
| lgsdilem2 27318 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27319 | The Legendre symbol is com... |
| lgsne0 27320 | The Legendre symbol is non... |
| lgsabs1 27321 | The Legendre symbol is non... |
| lgssq 27322 | The Legendre symbol at a s... |
| lgssq2 27323 | The Legendre symbol at a s... |
| lgsprme0 27324 | The Legendre symbol at any... |
| 1lgs 27325 | The Legendre symbol at ` 1... |
| lgs1 27326 | The Legendre symbol at ` 1... |
| lgsmodeq 27327 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27328 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27329 | Variation on ~ lgsdir vali... |
| lgsdinn0 27330 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27331 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27332 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27333 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27334 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27335 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27336 | The Legendre symbol for od... |
| lgsqrmod 27337 | If the Legendre symbol of ... |
| lgsqrmodndvds 27338 | If the Legendre symbol of ... |
| lgsdchrval 27339 | The Legendre symbol functi... |
| lgsdchr 27340 | The Legendre symbol functi... |
| gausslemma2dlem0a 27341 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27342 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27343 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27344 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27345 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27346 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27347 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27348 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27349 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27350 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27351 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27352 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27353 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27354 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27355 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27356 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27357 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27358 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27359 | Gauss' Lemma (see also the... |
| lgseisenlem1 27360 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27361 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27362 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27363 | Lemma for ~ lgseisen . (C... |
| lgseisen 27364 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27365 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27366 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27367 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27368 | The Law of Quadratic Recip... |
| lgsquad2lem1 27369 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27370 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27371 | Extend ~ lgsquad to coprim... |
| lgsquad3 27372 | Extend ~ lgsquad2 to integ... |
| m1lgs 27373 | The first supplement to th... |
| 2lgslem1a1 27374 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27375 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27376 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27377 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27378 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27379 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27380 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27381 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27382 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27383 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27384 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27385 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27386 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27387 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27388 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27389 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27390 | The Legendre symbol for ` ... |
| 2lgslem4 27391 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27392 | The second supplement to t... |
| 2lgsoddprmlem1 27393 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27394 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27395 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27396 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27397 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27398 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27399 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27400 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27401 | The second supplement to t... |
| 2sqlem1 27402 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27403 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27404 | Fibonacci's identity (actu... |
| 2sqlem3 27405 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27406 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27407 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27408 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27409 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27410 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27411 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27412 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27413 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27414 | Lemma for ~ 2sq . (Contri... |
| 2sq 27415 | All primes of the form ` 4... |
| 2sqblem 27416 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27417 | The converse to ~ 2sq . (... |
| 2sq2 27418 | ` 2 ` is the sum of square... |
| 2sqn0 27419 | If the sum of two squares ... |
| 2sqcoprm 27420 | If the sum of two squares ... |
| 2sqmod 27421 | Given two decompositions o... |
| 2sqmo 27422 | There exists at most one d... |
| 2sqnn0 27423 | All primes of the form ` 4... |
| 2sqnn 27424 | All primes of the form ` 4... |
| addsq2reu 27425 | For each complex number ` ... |
| addsqn2reu 27426 | For each complex number ` ... |
| addsqrexnreu 27427 | For each complex number, t... |
| addsqnreup 27428 | There is no unique decompo... |
| addsq2nreurex 27429 | For each complex number ` ... |
| addsqn2reurex2 27430 | For each complex number ` ... |
| 2sqreulem1 27431 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27432 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27433 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27434 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27435 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27436 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27437 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27438 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27439 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27440 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27441 | There exists a unique deco... |
| 2sqreunn 27442 | There exists a unique deco... |
| 2sqreult 27443 | There exists a unique deco... |
| 2sqreultb 27444 | There exists a unique deco... |
| 2sqreunnlt 27445 | There exists a unique deco... |
| 2sqreunnltb 27446 | There exists a unique deco... |
| 2sqreuop 27447 | There exists a unique deco... |
| 2sqreuopnn 27448 | There exists a unique deco... |
| 2sqreuoplt 27449 | There exists a unique deco... |
| 2sqreuopltb 27450 | There exists a unique deco... |
| 2sqreuopnnlt 27451 | There exists a unique deco... |
| 2sqreuopnnltb 27452 | There exists a unique deco... |
| 2sqreuopb 27453 | There exists a unique deco... |
| chebbnd1lem1 27454 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27455 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27456 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27457 | The Chebyshev bound: The ... |
| chtppilimlem1 27458 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27459 | Lemma for ~ chtppilim . (... |
| chtppilim 27460 | The ` theta ` function is ... |
| chto1ub 27461 | The ` theta ` function is ... |
| chebbnd2 27462 | The Chebyshev bound, part ... |
| chto1lb 27463 | The ` theta ` function is ... |
| chpchtlim 27464 | The ` psi ` and ` theta ` ... |
| chpo1ub 27465 | The ` psi ` function is up... |
| chpo1ubb 27466 | The ` psi ` function is up... |
| vmadivsum 27467 | The sum of the von Mangold... |
| vmadivsumb 27468 | Give a total bound on the ... |
| rplogsumlem1 27469 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27470 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27471 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27472 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27473 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27474 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27475 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27476 | Lemma for ~ dchrisum . Le... |
| dchrisum 27477 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27478 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27479 | The sum of the Möbius... |
| dchrvmasumlem1 27480 | An alternative expression ... |
| dchrvmasum2lem 27481 | Give an expression for ` l... |
| dchrvmasum2if 27482 | Combine the results of ~ d... |
| dchrvmasumlem2 27483 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27484 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27485 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27486 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27487 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27488 | An asymptotic approximatio... |
| dchrvmaeq0 27489 | The set ` W ` is the colle... |
| dchrisum0fval 27490 | Value of the function ` F ... |
| dchrisum0fmul 27491 | The function ` F ` , the d... |
| dchrisum0ff 27492 | The function ` F ` is a re... |
| dchrisum0flblem1 27493 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27494 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27495 | The divisor sum of a real ... |
| dchrisum0fno1 27496 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27497 | A partial result along the... |
| dchrisum0re 27498 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27499 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27500 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27501 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27502 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27503 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27504 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27505 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27506 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27507 | The sum of the Möbius... |
| dchrvmasumlem 27508 | The sum of the Möbius... |
| dchrmusum 27509 | The sum of the Möbius... |
| dchrvmasum 27510 | The sum of the von Mangold... |
| rpvmasum 27511 | The sum of the von Mangold... |
| rplogsum 27512 | The sum of ` log p / p ` o... |
| dirith2 27513 | Dirichlet's theorem: there... |
| dirith 27514 | Dirichlet's theorem: there... |
| mudivsum 27515 | Asymptotic formula for ` s... |
| mulogsumlem 27516 | Lemma for ~ mulogsum . (C... |
| mulogsum 27517 | Asymptotic formula for ... |
| logdivsum 27518 | Asymptotic analysis of ... |
| mulog2sumlem1 27519 | Asymptotic formula for ... |
| mulog2sumlem2 27520 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27521 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27522 | Asymptotic formula for ... |
| vmalogdivsum2 27523 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27524 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27525 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27526 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27527 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27528 | The Möbius inverse of... |
| log2sumbnd 27529 | Bound on the difference be... |
| selberglem1 27530 | Lemma for ~ selberg . Est... |
| selberglem2 27531 | Lemma for ~ selberg . (Co... |
| selberglem3 27532 | Lemma for ~ selberg . Est... |
| selberg 27533 | Selberg's symmetry formula... |
| selbergb 27534 | Convert eventual boundedne... |
| selberg2lem 27535 | Lemma for ~ selberg2 . Eq... |
| selberg2 27536 | Selberg's symmetry formula... |
| selberg2b 27537 | Convert eventual boundedne... |
| chpdifbndlem1 27538 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27539 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27540 | A bound on the difference ... |
| logdivbnd 27541 | A bound on a sum of logs, ... |
| selberg3lem1 27542 | Introduce a log weighting ... |
| selberg3lem2 27543 | Lemma for ~ selberg3 . Eq... |
| selberg3 27544 | Introduce a log weighting ... |
| selberg4lem1 27545 | Lemma for ~ selberg4 . Eq... |
| selberg4 27546 | The Selberg symmetry formu... |
| pntrval 27547 | Define the residual of the... |
| pntrf 27548 | Functionality of the resid... |
| pntrmax 27549 | There is a bound on the re... |
| pntrsumo1 27550 | A bound on a sum over ` R ... |
| pntrsumbnd 27551 | A bound on a sum over ` R ... |
| pntrsumbnd2 27552 | A bound on a sum over ` R ... |
| selbergr 27553 | Selberg's symmetry formula... |
| selberg3r 27554 | Selberg's symmetry formula... |
| selberg4r 27555 | Selberg's symmetry formula... |
| selberg34r 27556 | The sum of ~ selberg3r and... |
| pntsval 27557 | Define the "Selberg functi... |
| pntsf 27558 | Functionality of the Selbe... |
| selbergs 27559 | Selberg's symmetry formula... |
| selbergsb 27560 | Selberg's symmetry formula... |
| pntsval2 27561 | The Selberg function can b... |
| pntrlog2bndlem1 27562 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27563 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27564 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27565 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27566 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27567 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27568 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27569 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27570 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27571 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27572 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27573 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27574 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27575 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27576 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27577 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27578 | Lemma for ~ pnt . Establi... |
| pntlemd 27579 | Lemma for ~ pnt . Closure... |
| pntlemc 27580 | Lemma for ~ pnt . Closure... |
| pntlema 27581 | Lemma for ~ pnt . Closure... |
| pntlemb 27582 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27583 | Lemma for ~ pnt . Closure... |
| pntlemh 27584 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27585 | Lemma for ~ pnt . The "na... |
| pntlemq 27586 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27587 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27588 | Lemma for ~ pnt . The ind... |
| pntlemi 27589 | Lemma for ~ pnt . Elimina... |
| pntlemf 27590 | Lemma for ~ pnt . Add up ... |
| pntlemk 27591 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27592 | Lemma for ~ pnt . Combine... |
| pntleme 27593 | Lemma for ~ pnt . Package... |
| pntlem3 27594 | Lemma for ~ pnt . Equatio... |
| pntlemp 27595 | Lemma for ~ pnt . Wrappin... |
| pntleml 27596 | Lemma for ~ pnt . Equatio... |
| pnt3 27597 | The Prime Number Theorem, ... |
| pnt2 27598 | The Prime Number Theorem, ... |
| pnt 27599 | The Prime Number Theorem: ... |
| abvcxp 27600 | Raising an absolute value ... |
| padicfval 27601 | Value of the p-adic absolu... |
| padicval 27602 | Value of the p-adic absolu... |
| ostth2lem1 27603 | Lemma for ~ ostth2 , altho... |
| qrngbas 27604 | The base set of the field ... |
| qdrng 27605 | The rationals form a divis... |
| qrng0 27606 | The zero element of the fi... |
| qrng1 27607 | The unity element of the f... |
| qrngneg 27608 | The additive inverse in th... |
| qrngdiv 27609 | The division operation in ... |
| qabvle 27610 | By using induction on ` N ... |
| qabvexp 27611 | Induct the product rule ~ ... |
| ostthlem1 27612 | Lemma for ~ ostth . If tw... |
| ostthlem2 27613 | Lemma for ~ ostth . Refin... |
| qabsabv 27614 | The regular absolute value... |
| padicabv 27615 | The p-adic absolute value ... |
| padicabvf 27616 | The p-adic absolute value ... |
| padicabvcxp 27617 | All positive powers of the... |
| ostth1 27618 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27619 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27620 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27621 | Lemma for ~ ostth2 . (Con... |
| ostth2 27622 | - Lemma for ~ ostth : regu... |
| ostth3 27623 | - Lemma for ~ ostth : p-ad... |
| ostth 27624 | Ostrowski's theorem, which... |
| elno 27631 | Membership in the surreals... |
| elnoOLD 27632 | Obsolete version of ~ elno... |
| ltsval 27633 | The value of the surreal l... |
| bdayval 27634 | The value of the birthday ... |
| nofun 27635 | A surreal is a function. ... |
| nodmon 27636 | The domain of a surreal is... |
| norn 27637 | The range of a surreal is ... |
| nofnbday 27638 | A surreal is a function ov... |
| nodmord 27639 | The domain of a surreal ha... |
| elno2 27640 | An alternative condition f... |
| elno3 27641 | Another condition for memb... |
| ltsval2 27642 | Alternate expression for s... |
| nofv 27643 | The function value of a su... |
| nosgnn0 27644 | ` (/) ` is not a surreal s... |
| nosgnn0i 27645 | If ` X ` is a surreal sign... |
| noreson 27646 | The restriction of a surre... |
| ltsintdifex 27647 |
If ` A |
| ltsres 27648 | If the restrictions of two... |
| noxp1o 27649 | The Cartesian product of a... |
| noseponlem 27650 | Lemma for ~ nosepon . Con... |
| nosepon 27651 | Given two unequal surreals... |
| noextend 27652 | Extending a surreal by one... |
| noextendseq 27653 | Extend a surreal by a sequ... |
| noextenddif 27654 | Calculate the place where ... |
| noextendlt 27655 | Extending a surreal with a... |
| noextendgt 27656 | Extending a surreal with a... |
| nolesgn2o 27657 | Given ` A ` less-than or e... |
| nolesgn2ores 27658 | Given ` A ` less-than or e... |
| nogesgn1o 27659 | Given ` A ` greater than o... |
| nogesgn1ores 27660 | Given ` A ` greater than o... |
| ltssolem1 27661 | Lemma for ~ ltsso . The "... |
| ltsso 27662 | Less-than totally orders t... |
| bdayfo 27663 | The birthday function maps... |
| fvnobday 27664 | The value of a surreal at ... |
| nosepnelem 27665 | Lemma for ~ nosepne . (Co... |
| nosepne 27666 | The value of two non-equal... |
| nosep1o 27667 | If the value of a surreal ... |
| nosep2o 27668 | If the value of a surreal ... |
| nosepdmlem 27669 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27670 | The first place two surrea... |
| nosepeq 27671 | The values of two surreals... |
| nosepssdm 27672 | Given two non-equal surrea... |
| nodenselem4 27673 | Lemma for ~ nodense . Sho... |
| nodenselem5 27674 | Lemma for ~ nodense . If ... |
| nodenselem6 27675 | The restriction of a surre... |
| nodenselem7 27676 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27677 | Lemma for ~ nodense . Giv... |
| nodense 27678 | Given two distinct surreal... |
| bdayimaon 27679 | Lemma for full-eta propert... |
| nolt02olem 27680 | Lemma for ~ nolt02o . If ... |
| nolt02o 27681 | Given ` A ` less-than ` B ... |
| nogt01o 27682 | Given ` A ` greater than `... |
| noresle 27683 | Restriction law for surrea... |
| nomaxmo 27684 | A class of surreals has at... |
| nominmo 27685 | A class of surreals has at... |
| nosupprefixmo 27686 | In any class of surreals, ... |
| noinfprefixmo 27687 | In any class of surreals, ... |
| nosupcbv 27688 | Lemma to change bound vari... |
| nosupno 27689 | The next several theorems ... |
| nosupdm 27690 | The domain of the surreal ... |
| nosupbday 27691 | Birthday bounding law for ... |
| nosupfv 27692 | The value of surreal supre... |
| nosupres 27693 | A restriction law for surr... |
| nosupbnd1lem1 27694 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27695 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27696 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27697 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27698 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27699 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27700 | Bounding law from below fo... |
| nosupbnd2lem1 27701 | Bounding law from above wh... |
| nosupbnd2 27702 | Bounding law from above fo... |
| noinfcbv 27703 | Change bound variables for... |
| noinfno 27704 | The next several theorems ... |
| noinfdm 27705 | Next, we calculate the dom... |
| noinfbday 27706 | Birthday bounding law for ... |
| noinffv 27707 | The value of surreal infim... |
| noinfres 27708 | The restriction of surreal... |
| noinfbnd1lem1 27709 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27710 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27711 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27712 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27713 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27714 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27715 | Bounding law from above fo... |
| noinfbnd2lem1 27716 | Bounding law from below wh... |
| noinfbnd2 27717 | Bounding law from below fo... |
| nosupinfsep 27718 | Given two sets of surreals... |
| noetasuplem1 27719 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27720 | Lemma for ~ noeta . The r... |
| noetasuplem3 27721 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27722 | Lemma for ~ noeta . When ... |
| noetainflem1 27723 | Lemma for ~ noeta . Estab... |
| noetainflem2 27724 | Lemma for ~ noeta . The r... |
| noetainflem3 27725 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27726 | Lemma for ~ noeta . If ` ... |
| noetalem1 27727 | Lemma for ~ noeta . Eithe... |
| noetalem2 27728 | Lemma for ~ noeta . The f... |
| noeta 27729 | The full-eta axiom for the... |
| ltsirr 27732 | Surreal less-than is irref... |
| ltstr 27733 | Surreal less-than is trans... |
| ltsasym 27734 | Surreal less-than is asymm... |
| ltslin 27735 | Surreal less-than obeys tr... |
| ltstrieq2 27736 | Trichotomy law for surreal... |
| ltstrine 27737 | Trichotomy law for surreal... |
| lenlts 27738 | Surreal less-than or equal... |
| ltnles 27739 | Surreal less-than in terms... |
| lesloe 27740 | Surreal less-than or equal... |
| lestri3 27741 | Trichotomy law for surreal... |
| lesnltd 27742 | Surreal less-than or equal... |
| ltsnled 27743 | Surreal less-than in terms... |
| lesloed 27744 | Surreal less-than or equal... |
| lestri3d 27745 | Trichotomy law for surreal... |
| ltlestr 27746 | Surreal transitive law. (... |
| leltstr 27747 | Surreal transitive law. (... |
| lestr 27748 | Surreal transitive law. (... |
| ltstrd 27749 | Surreal less-than is trans... |
| ltlestrd 27750 | Surreal less-than is trans... |
| leltstrd 27751 | Surreal less-than is trans... |
| lestrd 27752 | Surreal less-than or equal... |
| lesid 27753 | Surreal less-than or equal... |
| lestric 27754 | Surreal trichotomy law. (... |
| maxs1 27755 | A surreal is less than or ... |
| maxs2 27756 | A surreal is less than or ... |
| mins1 27757 | The minimum of two surreal... |
| mins2 27758 | The minimum of two surreal... |
| ltlesd 27759 | Surreal less-than implies ... |
| ltsne 27760 | Surreal less-than implies ... |
| ltlesnd 27761 | Surreal less-than in terms... |
| bdayfun 27762 | The birthday function is a... |
| bdayfn 27763 | The birthday function is a... |
| bdaydm 27764 | The birthday function's do... |
| bdayrn 27765 | The birthday function's ra... |
| bdayon 27766 | The value of the birthday ... |
| nobdaymin 27767 | Any non-empty class of sur... |
| nocvxminlem 27768 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27769 | Given a nonempty convex cl... |
| noprc 27770 | The surreal numbers are a ... |
| noeta2 27775 | A version of ~ noeta with ... |
| brslts 27776 | Binary relation form of th... |
| sltsex1 27777 | The first argument of surr... |
| sltsex2 27778 | The second argument of sur... |
| sltsss1 27779 | The first argument of surr... |
| sltsss2 27780 | The second argument of sur... |
| sltssep 27781 | The separation property of... |
| sltsd 27782 | Deduce surreal set less-th... |
| sltssnb 27783 | Surreal set less-than of t... |
| sltssn 27784 | Surreal set less-than of t... |
| sltssepc 27785 | Two elements of separated ... |
| sltssepcd 27786 | Two elements of separated ... |
| ssslts1 27787 | Relation between surreal s... |
| ssslts2 27788 | Relation between surreal s... |
| nulslts 27789 | The empty set is less-than... |
| nulsgts 27790 | The empty set is greater t... |
| nulsltsd 27791 | The empty set is less-than... |
| nulsgtsd 27792 | The empty set is greater t... |
| conway 27793 | Conway's Simplicity Theore... |
| cutsval 27794 | The value of the surreal c... |
| cutcuts 27795 | Cut properties of the surr... |
| cutscl 27796 | Closure law for surreal cu... |
| cutscld 27797 | Closure law for surreal cu... |
| cutbday 27798 | The birthday of the surrea... |
| eqcuts 27799 | Condition for equality to ... |
| eqcuts2 27800 | Condition for equality to ... |
| sltstr 27801 | Transitive law for surreal... |
| sltsun1 27802 | Union law for surreal set ... |
| sltsun2 27803 | Union law for surreal set ... |
| cutsun12 27804 | Union law for surreal cuts... |
| dmcuts 27805 | The domain of the surreal ... |
| cutsf 27806 | Functionality statement fo... |
| etaslts 27807 | A restatement of ~ noeta u... |
| etaslts2 27808 | A version of ~ etaslts wit... |
| cutbdaybnd 27809 | An upper bound on the birt... |
| cutbdaybnd2 27810 | An upper bound on the birt... |
| cutbdaybnd2lim 27811 | An upper bound on the birt... |
| cutbdaylt 27812 | If a surreal lies in a gap... |
| lesrec 27813 | A comparison law for surre... |
| lesrecd 27814 | A comparison law for surre... |
| ltsrec 27815 | A comparison law for surre... |
| ltsrecd 27816 | A comparison law for surre... |
| sltsdisj 27817 | If ` A ` preceeds ` B ` , ... |
| eqcuts3 27818 | A variant of the simplicit... |
| 0no 27823 | Surreal zero is a surreal.... |
| 1no 27824 | Surreal one is a surreal. ... |
| bday0 27825 | Calculate the birthday of ... |
| 0lt1s 27826 | Surreal zero is less than ... |
| bday0b 27827 | The only surreal with birt... |
| bday1 27828 | The birthday of surreal on... |
| cuteq0 27829 | Condition for a surreal cu... |
| cutneg 27830 | The simplest number greate... |
| cuteq1 27831 | Condition for a surreal cu... |
| gt0ne0s 27832 | A positive surreal is not ... |
| gt0ne0sd 27833 | A positive surreal is not ... |
| 1ne0s 27834 | Surreal zero does not equa... |
| rightge0 27835 | A surreal is non-negative ... |
| madeval 27846 | The value of the made by f... |
| madeval2 27847 | Alternative characterizati... |
| oldval 27848 | The value of the old optio... |
| newval 27849 | The value of the new optio... |
| madef 27850 | The made function is a fun... |
| oldf 27851 | The older function is a fu... |
| newf 27852 | The new function is a func... |
| old0 27853 | No surreal is older than `... |
| madessno 27854 | Made sets are surreals. (... |
| oldssno 27855 | Old sets are surreals. (C... |
| newssno 27856 | New sets are surreals. (C... |
| madeno 27857 | An element of a made set i... |
| oldno 27858 | An element of an old set i... |
| newno 27859 | An element of a new set is... |
| madenod 27860 | An element of a made set i... |
| oldnod 27861 | An element of an old set i... |
| newnod 27862 | An element of a new set is... |
| leftval 27863 | The value of the left opti... |
| rightval 27864 | The value of the right opt... |
| elleft 27865 | Membership in the left set... |
| elright 27866 | Membership in the right se... |
| leftlt 27867 | A member of a surreal's le... |
| rightgt 27868 | A member of a surreal's ri... |
| leftf 27869 | The functionality of the l... |
| rightf 27870 | The functionality of the r... |
| elmade 27871 | Membership in the made fun... |
| elmade2 27872 | Membership in the made fun... |
| elold 27873 | Membership in an old set. ... |
| sltsleft 27874 | A surreal is greater than ... |
| sltsright 27875 | A surreal is less than its... |
| lltr 27876 | The left options of a surr... |
| made0 27877 | The only surreal made on d... |
| new0 27878 | The only surreal new on da... |
| old1 27879 | The only surreal older tha... |
| madess 27880 | If ` A ` is less than or e... |
| oldssmade 27881 | The older-than set is a su... |
| oldmade 27882 | An element of an old set i... |
| oldmaded 27883 | An element of an old set i... |
| oldss 27884 | If ` A ` is less than or e... |
| leftssold 27885 | The left options are a sub... |
| rightssold 27886 | The right options are a su... |
| leftssno 27887 | The left set of a surreal ... |
| rightssno 27888 | The right set of a surreal... |
| leftold 27889 | An element of a left set i... |
| rightold 27890 | An element of a right set ... |
| leftno 27891 | An element of a left set i... |
| rightno 27892 | An element of a right set ... |
| leftoldd 27893 | An element of a left set i... |
| leftnod 27894 | An element of a left set i... |
| rightoldd 27895 | An element of a right set ... |
| rightnod 27896 | An element of a right set ... |
| madecut 27897 | Given a section that is a ... |
| madeun 27898 | The made set is the union ... |
| madeoldsuc 27899 | The made set is the old se... |
| oldsuc 27900 | The value of the old set a... |
| oldlim 27901 | The value of the old set a... |
| madebdayim 27902 | If a surreal is a member o... |
| oldbdayim 27903 | If ` X ` is in the old set... |
| oldirr 27904 | No surreal is a member of ... |
| leftirr 27905 | No surreal is a member of ... |
| rightirr 27906 | No surreal is a member of ... |
| left0s 27907 | The left set of ` 0s ` is ... |
| right0s 27908 | The right set of ` 0s ` is... |
| left1s 27909 | The left set of ` 1s ` is ... |
| right1s 27910 | The right set of ` 1s ` is... |
| lrold 27911 | The union of the left and ... |
| madebdaylemold 27912 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27913 | Lemma for ~ madebday . If... |
| madebday 27914 | A surreal is part of the s... |
| oldbday 27915 | A surreal is part of the s... |
| newbday 27916 | A surreal is an element of... |
| newbdayim 27917 | One direction of the bicon... |
| lrcut 27918 | A surreal is equal to the ... |
| cutsfo 27919 | The surreal cut function i... |
| ltsn0 27920 | If ` X ` is less than ` Y ... |
| lruneq 27921 | If two surreals share a bi... |
| ltslpss 27922 | If two surreals share a bi... |
| leslss 27923 | If two surreals ` A ` and ... |
| 0elold 27924 | Zero is in the old set of ... |
| 0elleft 27925 | Zero is in the left set of... |
| 0elright 27926 | Zero is in the right set o... |
| madefi 27927 | The made set of an ordinal... |
| oldfi 27928 | The old set of an ordinal ... |
| bdayiun 27929 | The birthday of a surreal ... |
| bdayle 27930 | A condition for bounding a... |
| sltsbday 27931 | Birthday comparison rule f... |
| cofslts 27932 | If every element of ` A ` ... |
| coinitslts 27933 | If ` B ` is coinitial with... |
| cofcut1 27934 | If ` C ` is cofinal with `... |
| cofcut1d 27935 | If ` C ` is cofinal with `... |
| cofcut2 27936 | If ` A ` and ` C ` are mut... |
| cofcut2d 27937 | If ` A ` and ` C ` are mut... |
| cofcutr 27938 | If ` X ` is the cut of ` A... |
| cofcutr1d 27939 | If ` X ` is the cut of ` A... |
| cofcutr2d 27940 | If ` X ` is the cut of ` A... |
| cofcutrtime 27941 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27942 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27943 | If ` X ` is a timely cut o... |
| cofss 27944 | Cofinality for a subset. ... |
| coiniss 27945 | Coinitiality for a subset.... |
| cutlt 27946 | Eliminating all elements b... |
| cutpos 27947 | Reduce the elements of a c... |
| cutmax 27948 | If ` A ` has a maximum, th... |
| cutmin 27949 | If ` B ` has a minimum, th... |
| cutminmax 27950 | If the left set of ` X ` h... |
| lrrecval 27953 | The next step in the devel... |
| lrrecval2 27954 | Next, we establish an alte... |
| lrrecpo 27955 | Now, we establish that ` R... |
| lrrecse 27956 | Next, we show that ` R ` i... |
| lrrecfr 27957 | Now we show that ` R ` is ... |
| lrrecpred 27958 | Finally, we calculate the ... |
| noinds 27959 | Induction principle for a ... |
| norecfn 27960 | Surreal recursion over one... |
| norecov 27961 | Calculate the value of the... |
| noxpordpo 27964 | To get through most of the... |
| noxpordfr 27965 | Next we establish the foun... |
| noxpordse 27966 | Next we establish the set-... |
| noxpordpred 27967 | Next we calculate the pred... |
| no2indlesm 27968 | Double induction on surrea... |
| no2inds 27969 | Double induction on surrea... |
| norec2fn 27970 | The double-recursion opera... |
| norec2ov 27971 | The value of the double-re... |
| no3inds 27972 | Triple induction over surr... |
| addsfn 27975 | Surreal addition is a func... |
| addsval 27976 | The value of surreal addit... |
| addsval2 27977 | The value of surreal addit... |
| addsrid 27978 | Surreal addition to zero i... |
| addsridd 27979 | Surreal addition to zero i... |
| addscom 27980 | Surreal addition commutes.... |
| addscomd 27981 | Surreal addition commutes.... |
| addslid 27982 | Surreal addition to zero i... |
| addsproplem1 27983 | Lemma for surreal addition... |
| addsproplem2 27984 | Lemma for surreal addition... |
| addsproplem3 27985 | Lemma for surreal addition... |
| addsproplem4 27986 | Lemma for surreal addition... |
| addsproplem5 27987 | Lemma for surreal addition... |
| addsproplem6 27988 | Lemma for surreal addition... |
| addsproplem7 27989 | Lemma for surreal addition... |
| addsprop 27990 | Inductively show that surr... |
| addcutslem 27991 | Lemma for ~ addcuts . Sho... |
| addcuts 27992 | Demonstrate the cut proper... |
| addcuts2 27993 | Show that the cut involved... |
| addscld 27994 | Surreal numbers are closed... |
| addscl 27995 | Surreal numbers are closed... |
| addsf 27996 | Function statement for sur... |
| addsfo 27997 | Surreal addition is onto. ... |
| peano2no 27998 | A theorem for surreals tha... |
| ltadds1im 27999 | Surreal less-than is prese... |
| ltadds2im 28000 | Surreal less-than is prese... |
| leadds1im 28001 | Surreal less-than or equal... |
| leadds2im 28002 | Surreal less-than or equal... |
| leadds1 28003 | Addition to both sides of ... |
| leadds2 28004 | Addition to both sides of ... |
| ltadds2 28005 | Addition to both sides of ... |
| ltadds1 28006 | Addition to both sides of ... |
| addscan2 28007 | Cancellation law for surre... |
| addscan1 28008 | Cancellation law for surre... |
| leadds1d 28009 | Addition to both sides of ... |
| leadds2d 28010 | Addition to both sides of ... |
| ltadds2d 28011 | Addition to both sides of ... |
| ltadds1d 28012 | Addition to both sides of ... |
| addscan2d 28013 | Cancellation law for surre... |
| addscan1d 28014 | Cancellation law for surre... |
| addsuniflem 28015 | Lemma for ~ addsunif . St... |
| addsunif 28016 | Uniformity theorem for sur... |
| addsasslem1 28017 | Lemma for addition associa... |
| addsasslem2 28018 | Lemma for addition associa... |
| addsass 28019 | Surreal addition is associ... |
| addsassd 28020 | Surreal addition is associ... |
| adds32d 28021 | Commutative/associative la... |
| adds12d 28022 | Commutative/associative la... |
| adds4d 28023 | Rearrangement of four term... |
| adds42d 28024 | Rearrangement of four term... |
| ltaddspos1d 28025 | Addition of a positive num... |
| ltaddspos2d 28026 | Addition of a positive num... |
| lt2addsd 28027 | Adding both sides of two s... |
| addsgt0d 28028 | The sum of two positive su... |
| ltsp1d 28029 | A surreal is less than its... |
| addsge01d 28030 | A surreal is less-than or ... |
| addbdaylem 28031 | Lemma for ~ addbday . (Co... |
| addbday 28032 | The birthday of the sum of... |
| negsfn 28037 | Surreal negation is a func... |
| subsfn 28038 | Surreal subtraction is a f... |
| negsval 28039 | The value of the surreal n... |
| neg0s 28040 | Negative surreal zero is s... |
| neg1s 28041 | An expression for negative... |
| negsproplem1 28042 | Lemma for surreal negation... |
| negsproplem2 28043 | Lemma for surreal negation... |
| negsproplem3 28044 | Lemma for surreal negation... |
| negsproplem4 28045 | Lemma for surreal negation... |
| negsproplem5 28046 | Lemma for surreal negation... |
| negsproplem6 28047 | Lemma for surreal negation... |
| negsproplem7 28048 | Lemma for surreal negation... |
| negsprop 28049 | Show closure and ordering ... |
| negscl 28050 | The surreals are closed un... |
| negscld 28051 | The surreals are closed un... |
| ltnegsim 28052 | The forward direction of t... |
| negcut 28053 | The cut properties of surr... |
| negcut2 28054 | The cut that defines surre... |
| negsid 28055 | Surreal addition of a numb... |
| negsidd 28056 | Surreal addition of a numb... |
| negsex 28057 | Every surreal has a negati... |
| negnegs 28058 | A surreal is equal to the ... |
| ltnegs 28059 | Negative of both sides of ... |
| lenegs 28060 | Negative of both sides of ... |
| ltnegsd 28061 | Negative of both sides of ... |
| lenegsd 28062 | Negative of both sides of ... |
| negs11 28063 | Surreal negation is one-to... |
| negsdi 28064 | Distribution of surreal ne... |
| lt0negs2d 28065 | Comparison of a surreal an... |
| negsf 28066 | Function statement for sur... |
| negsfo 28067 | Function statement for sur... |
| negsf1o 28068 | Surreal negation is a bije... |
| negsunif 28069 | Uniformity property for su... |
| negbdaylem 28070 | Lemma for ~ negbday . Bou... |
| negbday 28071 | Negation of a surreal numb... |
| negleft 28072 | The left set of the negati... |
| negright 28073 | The right set of the negat... |
| subsval 28074 | The value of surreal subtr... |
| subsvald 28075 | The value of surreal subtr... |
| subscl 28076 | Closure law for surreal su... |
| subscld 28077 | Closure law for surreal su... |
| subsf 28078 | Function statement for sur... |
| subsfo 28079 | Surreal subtraction is an ... |
| negsval2 28080 | Surreal negation in terms ... |
| negsval2d 28081 | Surreal negation in terms ... |
| subsid1 28082 | Identity law for subtracti... |
| subsid 28083 | Subtraction of a surreal f... |
| subadds 28084 | Relationship between addit... |
| subaddsd 28085 | Relationship between addit... |
| pncans 28086 | Cancellation law for surre... |
| pncan3s 28087 | Subtraction and addition o... |
| pncan2s 28088 | Cancellation law for surre... |
| npcans 28089 | Cancellation law for surre... |
| ltsubs1 28090 | Subtraction from both side... |
| ltsubs2 28091 | Subtraction from both side... |
| ltsubs1d 28092 | Subtraction from both side... |
| ltsubs2d 28093 | Subtraction from both side... |
| negsubsdi2d 28094 | Distribution of negative o... |
| addsubsassd 28095 | Associative-type law for s... |
| addsubsd 28096 | Law for surreal addition a... |
| ltsubsubsbd 28097 | Equivalence for the surrea... |
| ltsubsubs2bd 28098 | Equivalence for the surrea... |
| ltsubsubs3bd 28099 | Equivalence for the surrea... |
| lesubsubsbd 28100 | Equivalence for the surrea... |
| lesubsubs2bd 28101 | Equivalence for the surrea... |
| lesubsubs3bd 28102 | Equivalence for the surrea... |
| ltsubaddsd 28103 | Surreal less-than relation... |
| ltsubadds2d 28104 | Surreal less-than relation... |
| ltaddsubsd 28105 | Surreal less-than relation... |
| ltaddsubs2d 28106 | Surreal less-than relation... |
| lesubaddsd 28107 | Surreal less-than or equal... |
| subsubs4d 28108 | Law for double surreal sub... |
| subsubs2d 28109 | Law for double surreal sub... |
| lesubsd 28110 | Swap subtrahends in a surr... |
| nncansd 28111 | Cancellation law for surre... |
| posdifsd 28112 | Comparison of two surreals... |
| ltsubsposd 28113 | Subtraction of a positive ... |
| subsge0d 28114 | Non-negative subtraction. ... |
| addsubs4d 28115 | Rearrangement of four term... |
| ltsm1d 28116 | A surreal is greater than ... |
| subscan1d 28117 | Cancellation law for surre... |
| subscan2d 28118 | Cancellation law for surre... |
| subseq0d 28119 | The difference between two... |
| mulsfn 28122 | Surreal multiplication is ... |
| mulsval 28123 | The value of surreal multi... |
| mulsval2lem 28124 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28125 | The value of surreal multi... |
| muls01 28126 | Surreal multiplication by ... |
| mulsrid 28127 | Surreal one is a right ide... |
| mulsridd 28128 | Surreal one is a right ide... |
| mulsproplemcbv 28129 | Lemma for surreal multipli... |
| mulsproplem1 28130 | Lemma for surreal multipli... |
| mulsproplem2 28131 | Lemma for surreal multipli... |
| mulsproplem3 28132 | Lemma for surreal multipli... |
| mulsproplem4 28133 | Lemma for surreal multipli... |
| mulsproplem5 28134 | Lemma for surreal multipli... |
| mulsproplem6 28135 | Lemma for surreal multipli... |
| mulsproplem7 28136 | Lemma for surreal multipli... |
| mulsproplem8 28137 | Lemma for surreal multipli... |
| mulsproplem9 28138 | Lemma for surreal multipli... |
| mulsproplem10 28139 | Lemma for surreal multipli... |
| mulsproplem11 28140 | Lemma for surreal multipli... |
| mulsproplem12 28141 | Lemma for surreal multipli... |
| mulsproplem13 28142 | Lemma for surreal multipli... |
| mulsproplem14 28143 | Lemma for surreal multipli... |
| mulsprop 28144 | Surreals are closed under ... |
| mulcutlem 28145 | Lemma for ~ mulcut . Stat... |
| mulcut 28146 | Show the cut properties of... |
| mulcut2 28147 | Show that the cut involved... |
| mulscl 28148 | The surreals are closed un... |
| mulscld 28149 | The surreals are closed un... |
| ltmuls 28150 | An ordering relationship f... |
| ltmulsd 28151 | An ordering relationship f... |
| lemulsd 28152 | An ordering relationship f... |
| mulscom 28153 | Surreal multiplication com... |
| mulscomd 28154 | Surreal multiplication com... |
| muls02 28155 | Surreal multiplication by ... |
| mulslid 28156 | Surreal one is a left iden... |
| mulslidd 28157 | Surreal one is a left iden... |
| mulsgt0 28158 | The product of two positiv... |
| mulsgt0d 28159 | The product of two positiv... |
| mulsge0d 28160 | The product of two non-neg... |
| sltmuls1 28161 | One surreal set less-than ... |
| sltmuls2 28162 | One surreal set less-than ... |
| mulsuniflem 28163 | Lemma for ~ mulsunif . St... |
| mulsunif 28164 | Surreal multiplication has... |
| addsdilem1 28165 | Lemma for surreal distribu... |
| addsdilem2 28166 | Lemma for surreal distribu... |
| addsdilem3 28167 | Lemma for ~ addsdi . Show... |
| addsdilem4 28168 | Lemma for ~ addsdi . Show... |
| addsdi 28169 | Distributive law for surre... |
| addsdid 28170 | Distributive law for surre... |
| addsdird 28171 | Distributive law for surre... |
| subsdid 28172 | Distribution of surreal mu... |
| subsdird 28173 | Distribution of surreal mu... |
| mulnegs1d 28174 | Product with negative is n... |
| mulnegs2d 28175 | Product with negative is n... |
| mul2negsd 28176 | Surreal product of two neg... |
| mulsasslem1 28177 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28178 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28179 | Lemma for ~ mulsass . Dem... |
| mulsass 28180 | Associative law for surrea... |
| mulsassd 28181 | Associative law for surrea... |
| muls4d 28182 | Rearrangement of four surr... |
| mulsunif2lem 28183 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28184 | Alternate expression for s... |
| ltmuls2 28185 | Multiplication of both sid... |
| ltmuls2d 28186 | Multiplication of both sid... |
| ltmuls1d 28187 | Multiplication of both sid... |
| lemuls2d 28188 | Multiplication of both sid... |
| lemuls1d 28189 | Multiplication of both sid... |
| ltmulnegs1d 28190 | Multiplication of both sid... |
| ltmulnegs2d 28191 | Multiplication of both sid... |
| mulscan2dlem 28192 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28193 | Cancellation of surreal mu... |
| mulscan1d 28194 | Cancellation of surreal mu... |
| muls12d 28195 | Commutative/associative la... |
| lemuls1ad 28196 | Multiplication of both sid... |
| ltmuls12ad 28197 | Comparison of the product ... |
| divsmo 28198 | Uniqueness of surreal inve... |
| muls0ord 28199 | If a surreal product is ze... |
| mulsne0bd 28200 | The product of two nonzero... |
| divsval 28203 | The value of surreal divis... |
| norecdiv 28204 | If a surreal has a recipro... |
| noreceuw 28205 | If a surreal has a recipro... |
| recsne0 28206 | If a surreal has a recipro... |
| divmulsw 28207 | Relationship between surre... |
| divmulswd 28208 | Relationship between surre... |
| divsclw 28209 | Weak division closure law.... |
| divsclwd 28210 | Weak division closure law.... |
| divscan2wd 28211 | A weak cancellation law fo... |
| divscan1wd 28212 | A weak cancellation law fo... |
| ltdivmulswd 28213 | Surreal less-than relation... |
| ltdivmuls2wd 28214 | Surreal less-than relation... |
| ltmuldivswd 28215 | Surreal less-than relation... |
| ltmuldivs2wd 28216 | Surreal less-than relation... |
| divsasswd 28217 | An associative law for sur... |
| divs1 28218 | A surreal divided by one i... |
| divs1d 28219 | A surreal divided by one i... |
| precsexlemcbv 28220 | Lemma for surreal reciproc... |
| precsexlem1 28221 | Lemma for surreal reciproc... |
| precsexlem2 28222 | Lemma for surreal reciproc... |
| precsexlem3 28223 | Lemma for surreal reciproc... |
| precsexlem4 28224 | Lemma for surreal reciproc... |
| precsexlem5 28225 | Lemma for surreal reciproc... |
| precsexlem6 28226 | Lemma for surreal reciproc... |
| precsexlem7 28227 | Lemma for surreal reciproc... |
| precsexlem8 28228 | Lemma for surreal reciproc... |
| precsexlem9 28229 | Lemma for surreal reciproc... |
| precsexlem10 28230 | Lemma for surreal reciproc... |
| precsexlem11 28231 | Lemma for surreal reciproc... |
| precsex 28232 | Every positive surreal has... |
| recsex 28233 | A nonzero surreal has a re... |
| recsexd 28234 | A nonzero surreal has a re... |
| divmuls 28235 | Relationship between surre... |
| divmulsd 28236 | Relationship between surre... |
| divscl 28237 | Surreal division closure l... |
| divscld 28238 | Surreal division closure l... |
| divscan2d 28239 | A cancellation law for sur... |
| divscan1d 28240 | A cancellation law for sur... |
| ltdivmulsd 28241 | Surreal less-than relation... |
| ltdivmuls2d 28242 | Surreal less-than relation... |
| ltmuldivsd 28243 | Surreal less-than relation... |
| ltmuldivs2d 28244 | Surreal less-than relation... |
| divsassd 28245 | An associative law for sur... |
| divmuldivsd 28246 | Multiplication of two surr... |
| divdivs1d 28247 | Surreal division into a fr... |
| divsrecd 28248 | Relationship between surre... |
| divsdird 28249 | Distribution of surreal di... |
| divscan3d 28250 | A cancellation law for sur... |
| abssval 28253 | The value of surreal absol... |
| absscl 28254 | Closure law for surreal ab... |
| abssid 28255 | The absolute value of a no... |
| abs0s 28256 | The absolute value of surr... |
| abssnid 28257 | For a negative surreal, it... |
| absmuls 28258 | Surreal absolute value dis... |
| abssge0 28259 | The absolute value of a su... |
| abssor 28260 | The absolute value of a su... |
| absnegs 28261 | Surreal absolute value of ... |
| leabss 28262 | A surreal is less than or ... |
| abslts 28263 | Surreal absolute value and... |
| abssubs 28264 | Swapping order of surreal ... |
| elons 28267 | Membership in the class of... |
| onssno 28268 | The surreal ordinals are a... |
| onno 28269 | A surreal ordinal is a sur... |
| 0ons 28270 | Surreal zero is a surreal ... |
| 1ons 28271 | Surreal one is a surreal o... |
| elons2 28272 | A surreal is ordinal iff i... |
| elons2d 28273 | The cut of any set of surr... |
| onleft 28274 | The left set of a surreal ... |
| ltonold 28275 | The class of ordinals less... |
| ltonsex 28276 | The class of ordinals less... |
| oncutleft 28277 | A surreal ordinal is equal... |
| oncutlt 28278 | A surreal ordinal is the s... |
| bday11on 28279 | The birthday function is o... |
| onnolt 28280 | If a surreal ordinal is le... |
| onlts 28281 | Less-than is the same as b... |
| onles 28282 | Less-than or equal is the ... |
| onltsd 28283 | Less-than is the same as b... |
| onlesd 28284 | Less-than or equal is the ... |
| oniso 28285 | The birthday function rest... |
| onswe 28286 | Surreal less-than well-ord... |
| onsse 28287 | Surreal less-than is set-l... |
| onsis 28288 | Transfinite induction sche... |
| ons2ind 28289 | Double induction schema fo... |
| bdayons 28290 | The birthday of a surreal ... |
| onaddscl 28291 | The surreal ordinals are c... |
| onmulscl 28292 | The surreal ordinals are c... |
| addonbday 28293 | The birthday of the sum of... |
| peano2ons 28294 | The successor of a surreal... |
| onsbnd 28295 | The surreals of a given bi... |
| onsbnd2 28296 | The surreals of a given bi... |
| seqsex 28299 | Existence of the surreal s... |
| seqseq123d 28300 | Equality deduction for the... |
| nfseqs 28301 | Hypothesis builder for the... |
| seqsval 28302 | The value of the surreal s... |
| noseqex 28303 | The next several theorems ... |
| noseq0 28304 | The surreal ` A ` is a mem... |
| noseqp1 28305 | One plus an element of ` Z... |
| noseqind 28306 | Peano's inductive postulat... |
| noseqinds 28307 | Induction schema for surre... |
| noseqssno 28308 | A surreal sequence is a su... |
| noseqno 28309 | An element of a surreal se... |
| om2noseq0 28310 | The mapping ` G ` is a one... |
| om2noseqsuc 28311 | The value of ` G ` at a su... |
| om2noseqfo 28312 | Function statement for ` G... |
| om2noseqlt 28313 | Surreal less-than relation... |
| om2noseqlt2 28314 | The mapping ` G ` preserve... |
| om2noseqf1o 28315 | ` G ` is a bijection. (Co... |
| om2noseqiso 28316 | ` G ` is an isomorphism fr... |
| om2noseqoi 28317 | An alternative definition ... |
| om2noseqrdg 28318 | A helper lemma for the val... |
| noseqrdglem 28319 | A helper lemma for the val... |
| noseqrdgfn 28320 | The recursive definition g... |
| noseqrdg0 28321 | Initial value of a recursi... |
| noseqrdgsuc 28322 | Successor value of a recur... |
| seqsfn 28323 | The surreal sequence build... |
| seqs1 28324 | The value of the surreal s... |
| seqsp1 28325 | The value of the surreal s... |
| n0sexg 28330 | The set of all non-negativ... |
| n0sex 28331 | The set of all non-negativ... |
| nnsex 28332 | The set of all positive su... |
| peano5n0s 28333 | Peano's inductive postulat... |
| n0ssno 28334 | The non-negative surreal i... |
| nnssn0s 28335 | The positive surreal integ... |
| nnssno 28336 | The positive surreal integ... |
| n0no 28337 | A non-negative surreal int... |
| nnno 28338 | A positive surreal integer... |
| n0nod 28339 | A non-negative surreal int... |
| nnnod 28340 | A positive surreal integer... |
| nnn0s 28341 | A positive surreal integer... |
| nnn0sd 28342 | A positive surreal integer... |
| 0n0s 28343 | Peano postulate: ` 0s ` is... |
| peano2n0s 28344 | Peano postulate: the succe... |
| peano2n0sd 28345 | Peano postulate: the succe... |
| dfn0s2 28346 | Alternate definition of th... |
| n0sind 28347 | Principle of Mathematical ... |
| n0cut 28348 | A cut form for non-negativ... |
| n0cut2 28349 | A cut form for the success... |
| n0on 28350 | A surreal natural is a sur... |
| nnne0s 28351 | A surreal positive integer... |
| n0sge0 28352 | A non-negative integer is ... |
| nnsgt0 28353 | A positive integer is grea... |
| elnns 28354 | Membership in the positive... |
| elnns2 28355 | A positive surreal integer... |
| n0s0suc 28356 | A non-negative surreal int... |
| nnsge1 28357 | A positive surreal integer... |
| n0addscl 28358 | The non-negative surreal i... |
| n0mulscl 28359 | The non-negative surreal i... |
| nnaddscl 28360 | The positive surreal integ... |
| nnmulscl 28361 | The positive surreal integ... |
| 1n0s 28362 | Surreal one is a non-negat... |
| 1nns 28363 | Surreal one is a positive ... |
| peano2nns 28364 | Peano postulate for positi... |
| nnsrecgt0d 28365 | The reciprocal of a positi... |
| n0bday 28366 | A non-negative surreal int... |
| n0ssoldg 28367 | The non-negative surreal i... |
| n0ssold 28368 | The non-negative surreal i... |
| n0fincut 28369 | The simplest number greate... |
| onsfi 28370 | A surreal ordinal with a f... |
| eln0s2 28371 | A non-negative surreal int... |
| onltn0s 28372 | A surreal ordinal that is ... |
| n0cutlt 28373 | A non-negative surreal int... |
| seqn0sfn 28374 | The surreal sequence build... |
| eln0s 28375 | A non-negative surreal int... |
| n0s0m1 28376 | Every non-negative surreal... |
| n0subs 28377 | Subtraction of non-negativ... |
| n0subs2 28378 | Subtraction of non-negativ... |
| n0ltsp1le 28379 | Non-negative surreal order... |
| n0lesltp1 28380 | Non-negative surreal order... |
| n0lesm1lt 28381 | Non-negative surreal order... |
| n0lts1e0 28382 | A non-negative surreal int... |
| bdayn0p1 28383 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28384 | The birthday function rest... |
| n0p1nns 28385 | One plus a non-negative su... |
| dfnns2 28386 | Alternate definition of th... |
| nnsind 28387 | Principle of Mathematical ... |
| nn1m1nns 28388 | Every positive surreal int... |
| nnm1n0s 28389 | A positive surreal integer... |
| eucliddivs 28390 | Euclid's division lemma fo... |
| oldfib 28391 | The old set of an ordinal ... |
| zsex 28394 | The surreal integers form ... |
| zssno 28395 | The surreal integers are a... |
| zno 28396 | A surreal integer is a sur... |
| znod 28397 | A surreal integer is a sur... |
| elzs 28398 | Membership in the set of s... |
| nnzsubs 28399 | The difference of two surr... |
| nnzs 28400 | A positive surreal integer... |
| nnzsd 28401 | A positive surreal integer... |
| 0zs 28402 | Zero is a surreal integer.... |
| n0zs 28403 | A non-negative surreal int... |
| n0zsd 28404 | A non-negative surreal int... |
| 1zs 28405 | One is a surreal integer. ... |
| znegscl 28406 | The surreal integers are c... |
| znegscld 28407 | The surreal integers are c... |
| zaddscl 28408 | The surreal integers are c... |
| zaddscld 28409 | The surreal integers are c... |
| zsubscld 28410 | The surreal integers are c... |
| zmulscld 28411 | The surreal integers are c... |
| elzn0s 28412 | A surreal integer is a sur... |
| elzs2 28413 | A surreal integer is eithe... |
| eln0zs 28414 | Non-negative surreal integ... |
| elnnzs 28415 | Positive surreal integer p... |
| elznns 28416 | Surreal integer property e... |
| zn0subs 28417 | The non-negative differenc... |
| peano5uzs 28418 | Peano's inductive postulat... |
| uzsind 28419 | Induction on the upper sur... |
| zsbday 28420 | A surreal integer has a fi... |
| zcuts 28421 | A cut expression for surre... |
| zcuts0 28422 | Either the left or right s... |
| zsoring 28423 | The surreal integers form ... |
| 1p1e2s 28430 | One plus one is two. Surr... |
| no2times 28431 | Version of ~ 2times for su... |
| 2nns 28432 | Surreal two is a surreal n... |
| 2no 28433 | Surreal two is a surreal n... |
| 2ne0s 28434 | Surreal two is nonzero. (... |
| n0seo 28435 | A non-negative surreal int... |
| zseo 28436 | A surreal integer is eithe... |
| twocut 28437 | Two times the cut of zero ... |
| nohalf 28438 | An explicit expression for... |
| expsval 28439 | The value of surreal expon... |
| expnnsval 28440 | Value of surreal exponenti... |
| exps0 28441 | Surreal exponentiation to ... |
| exps1 28442 | Surreal exponentiation to ... |
| expsp1 28443 | Value of a surreal number ... |
| expscllem 28444 | Lemma for proving non-nega... |
| expscl 28445 | Closure law for surreal ex... |
| n0expscl 28446 | Closure law for non-negati... |
| nnexpscl 28447 | Closure law for positive s... |
| zexpscl 28448 | Closure law for surreal in... |
| expadds 28449 | Sum of exponents law for s... |
| expsne0 28450 | A non-negative surreal int... |
| expsgt0 28451 | A non-negative surreal int... |
| pw2recs 28452 | Any power of two has a mul... |
| pw2divscld 28453 | Division closure for power... |
| pw2divmulsd 28454 | Relationship between surre... |
| pw2divscan3d 28455 | Cancellation law for surre... |
| pw2divscan2d 28456 | A cancellation law for sur... |
| pw2divsassd 28457 | An associative law for div... |
| pw2divscan4d 28458 | Cancellation law for divis... |
| pw2gt0divsd 28459 | Division of a positive sur... |
| pw2ge0divsd 28460 | Divison of a non-negative ... |
| pw2divsrecd 28461 | Relationship between surre... |
| pw2divsdird 28462 | Distribution of surreal di... |
| pw2divsnegd 28463 | Move negative sign inside ... |
| pw2ltdivmulsd 28464 | Surreal less-than relation... |
| pw2ltmuldivs2d 28465 | Surreal less-than relation... |
| pw2ltsdiv1d 28466 | Surreal less-than relation... |
| avglts1d 28467 | Ordering property for aver... |
| avglts2d 28468 | Ordering property for aver... |
| pw2divs0d 28469 | Division into zero is zero... |
| pw2divsidd 28470 | Identity law for division ... |
| pw2ltdivmuls2d 28471 | Surreal less-than relation... |
| halfcut 28472 | Relate the cut of twice of... |
| addhalfcut 28473 | The cut of a surreal non-n... |
| pw2cut 28474 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28475 | Simplify ~ pw2cut in the c... |
| pw2cut2 28476 | Cut expression for powers ... |
| bdaypw2n0bndlem 28477 | Lemma for ~ bdaypw2n0bnd .... |
| bdaypw2n0bnd 28478 | Upper bound for the birthd... |
| bdaypw2bnd 28479 | Birthday bounding rule for... |
| bdayfinbndcbv 28480 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem1 28481 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem2 28482 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbnd 28483 | Given a non-negative integ... |
| z12bdaylem1 28484 | Lemma for ~ z12bday . Pro... |
| z12bdaylem2 28485 | Lemma for ~ z12bday . Sho... |
| elz12s 28486 | Membership in the dyadic f... |
| elz12si 28487 | Inference form of membersh... |
| z12sex 28488 | The class of dyadic fracti... |
| zz12s 28489 | A surreal integer is a dya... |
| z12no 28490 | A dyadic is a surreal. (C... |
| z12addscl 28491 | The dyadics are closed und... |
| z12negscl 28492 | The dyadics are closed und... |
| z12subscl 28493 | The dyadics are closed und... |
| z12shalf 28494 | Half of a dyadic is a dyad... |
| z12negsclb 28495 | A surreal is a dyadic frac... |
| z12zsodd 28496 | A dyadic fraction is eithe... |
| z12sge0 28497 | An expression for non-nega... |
| z12bdaylem 28498 | Lemma for ~ z12bday . Han... |
| z12bday 28499 | A dyadic fraction has a fi... |
| bdayfinlem 28500 | Lemma for ~ bdayfin . Han... |
| bdayfin 28501 | A surreal has a finite bir... |
| dfz12s2 28502 | The set of dyadic fraction... |
| elreno 28505 | Membership in the set of s... |
| reno 28506 | A surreal real is a surrea... |
| renod 28507 | A surreal real is a surrea... |
| recut 28508 | The cut involved in defini... |
| elreno2 28509 | Alternate characterization... |
| 0reno 28510 | Surreal zero is a surreal ... |
| 1reno 28511 | Surreal one is a surreal r... |
| renegscl 28512 | The surreal reals are clos... |
| readdscl 28513 | The surreal reals are clos... |
| remulscllem1 28514 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28515 | Lemma for ~ remulscl . Bo... |
| remulscl 28516 | The surreal reals are clos... |
| itvndx 28527 | Index value of the Interva... |
| lngndx 28528 | Index value of the "line" ... |
| itvid 28529 | Utility theorem: index-ind... |
| lngid 28530 | Utility theorem: index-ind... |
| slotsinbpsd 28531 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28532 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28533 | The slot for the line is n... |
| trkgstr 28534 | Functionality of a Tarski ... |
| trkgbas 28535 | The base set of a Tarski g... |
| trkgdist 28536 | The measure of a distance ... |
| trkgitv 28537 | The congruence relation in... |
| istrkgc 28544 | Property of being a Tarski... |
| istrkgb 28545 | Property of being a Tarski... |
| istrkgcb 28546 | Property of being a Tarski... |
| istrkge 28547 | Property of fulfilling Euc... |
| istrkgl 28548 | Building lines from the se... |
| istrkgld 28549 | Property of fulfilling the... |
| istrkg2ld 28550 | Property of fulfilling the... |
| istrkg3ld 28551 | Property of fulfilling the... |
| axtgcgrrflx 28552 | Axiom of reflexivity of co... |
| axtgcgrid 28553 | Axiom of identity of congr... |
| axtgsegcon 28554 | Axiom of segment construct... |
| axtg5seg 28555 | Five segments axiom, Axiom... |
| axtgbtwnid 28556 | Identity of Betweenness. ... |
| axtgpasch 28557 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28558 | Axiom of Continuity. Axio... |
| axtgcont 28559 | Axiom of Continuity. Axio... |
| axtglowdim2 28560 | Lower dimension axiom for ... |
| axtgupdim2 28561 | Upper dimension axiom for ... |
| axtgeucl 28562 | Euclid's Axiom. Axiom A10... |
| tgjustf 28563 | Given any function ` F ` ,... |
| tgjustr 28564 | Given any equivalence rela... |
| tgjustc1 28565 | A justification for using ... |
| tgjustc2 28566 | A justification for using ... |
| tgcgrcomimp 28567 | Congruence commutes on the... |
| tgcgrcomr 28568 | Congruence commutes on the... |
| tgcgrcoml 28569 | Congruence commutes on the... |
| tgcgrcomlr 28570 | Congruence commutes on bot... |
| tgcgreqb 28571 | Congruence and equality. ... |
| tgcgreq 28572 | Congruence and equality. ... |
| tgcgrneq 28573 | Congruence and equality. ... |
| tgcgrtriv 28574 | Degenerate segments are co... |
| tgcgrextend 28575 | Link congruence over a pai... |
| tgsegconeq 28576 | Two points that satisfy th... |
| tgbtwntriv2 28577 | Betweenness always holds f... |
| tgbtwncom 28578 | Betweenness commutes. The... |
| tgbtwncomb 28579 | Betweenness commutes, bico... |
| tgbtwnne 28580 | Betweenness and inequality... |
| tgbtwntriv1 28581 | Betweenness always holds f... |
| tgbtwnswapid 28582 | If you can swap the first ... |
| tgbtwnintr 28583 | Inner transitivity law for... |
| tgbtwnexch3 28584 | Exchange the first endpoin... |
| tgbtwnouttr2 28585 | Outer transitivity law for... |
| tgbtwnexch2 28586 | Exchange the outer point o... |
| tgbtwnouttr 28587 | Outer transitivity law for... |
| tgbtwnexch 28588 | Outer transitivity law for... |
| tgtrisegint 28589 | A line segment between two... |
| tglowdim1 28590 | Lower dimension axiom for ... |
| tglowdim1i 28591 | Lower dimension axiom for ... |
| tgldimor 28592 | Excluded-middle like state... |
| tgldim0eq 28593 | In dimension zero, any two... |
| tgldim0itv 28594 | In dimension zero, any two... |
| tgldim0cgr 28595 | In dimension zero, any two... |
| tgbtwndiff 28596 | There is always a ` c ` di... |
| tgdim01 28597 | In geometries of dimension... |
| tgifscgr 28598 | Inner five segment congrue... |
| tgcgrsub 28599 | Removing identical parts f... |
| iscgrg 28602 | The congruence property fo... |
| iscgrgd 28603 | The property for two seque... |
| iscgrglt 28604 | The property for two seque... |
| trgcgrg 28605 | The property for two trian... |
| trgcgr 28606 | Triangle congruence. (Con... |
| ercgrg 28607 | The shape congruence relat... |
| tgcgrxfr 28608 | A line segment can be divi... |
| cgr3id 28609 | Reflexivity law for three-... |
| cgr3simp1 28610 | Deduce segment congruence ... |
| cgr3simp2 28611 | Deduce segment congruence ... |
| cgr3simp3 28612 | Deduce segment congruence ... |
| cgr3swap12 28613 | Permutation law for three-... |
| cgr3swap23 28614 | Permutation law for three-... |
| cgr3swap13 28615 | Permutation law for three-... |
| cgr3rotr 28616 | Permutation law for three-... |
| cgr3rotl 28617 | Permutation law for three-... |
| trgcgrcom 28618 | Commutative law for three-... |
| cgr3tr 28619 | Transitivity law for three... |
| tgbtwnxfr 28620 | A condition for extending ... |
| tgcgr4 28621 | Two quadrilaterals to be c... |
| isismt 28624 | Property of being an isome... |
| ismot 28625 | Property of being an isome... |
| motcgr 28626 | Property of a motion: dist... |
| idmot 28627 | The identity is a motion. ... |
| motf1o 28628 | Motions are bijections. (... |
| motcl 28629 | Closure of motions. (Cont... |
| motco 28630 | The composition of two mot... |
| cnvmot 28631 | The converse of a motion i... |
| motplusg 28632 | The operation for motions ... |
| motgrp 28633 | The motions of a geometry ... |
| motcgrg 28634 | Property of a motion: dist... |
| motcgr3 28635 | Property of a motion: dist... |
| tglng 28636 | Lines of a Tarski Geometry... |
| tglnfn 28637 | Lines as functions. (Cont... |
| tglnunirn 28638 | Lines are sets of points. ... |
| tglnpt 28639 | Lines are sets of points. ... |
| tglngne 28640 | It takes two different poi... |
| tglngval 28641 | The line going through poi... |
| tglnssp 28642 | Lines are subset of the ge... |
| tgellng 28643 | Property of lying on the l... |
| tgcolg 28644 | We choose the notation ` (... |
| btwncolg1 28645 | Betweenness implies coline... |
| btwncolg2 28646 | Betweenness implies coline... |
| btwncolg3 28647 | Betweenness implies coline... |
| colcom 28648 | Swapping the points defini... |
| colrot1 28649 | Rotating the points defini... |
| colrot2 28650 | Rotating the points defini... |
| ncolcom 28651 | Swapping non-colinear poin... |
| ncolrot1 28652 | Rotating non-colinear poin... |
| ncolrot2 28653 | Rotating non-colinear poin... |
| tgdim01ln 28654 | In geometries of dimension... |
| ncoltgdim2 28655 | If there are three non-col... |
| lnxfr 28656 | Transfer law for colineari... |
| lnext 28657 | Extend a line with a missi... |
| tgfscgr 28658 | Congruence law for the gen... |
| lncgr 28659 | Congruence rule for lines.... |
| lnid 28660 | Identity law for points on... |
| tgidinside 28661 | Law for finding a point in... |
| tgbtwnconn1lem1 28662 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28663 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28664 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28665 | Connectivity law for betwe... |
| tgbtwnconn2 28666 | Another connectivity law f... |
| tgbtwnconn3 28667 | Inner connectivity law for... |
| tgbtwnconnln3 28668 | Derive colinearity from be... |
| tgbtwnconn22 28669 | Double connectivity law fo... |
| tgbtwnconnln1 28670 | Derive colinearity from be... |
| tgbtwnconnln2 28671 | Derive colinearity from be... |
| legval 28674 | Value of the less-than rel... |
| legov 28675 | Value of the less-than rel... |
| legov2 28676 | An equivalent definition o... |
| legid 28677 | Reflexivity of the less-th... |
| btwnleg 28678 | Betweenness implies less-t... |
| legtrd 28679 | Transitivity of the less-t... |
| legtri3 28680 | Equality from the less-tha... |
| legtrid 28681 | Trichotomy law for the les... |
| leg0 28682 | Degenerated (zero-length) ... |
| legeq 28683 | Deduce equality from "less... |
| legbtwn 28684 | Deduce betweenness from "l... |
| tgcgrsub2 28685 | Removing identical parts f... |
| ltgseg 28686 | The set ` E ` denotes the ... |
| ltgov 28687 | Strict "shorter than" geom... |
| legov3 28688 | An equivalent definition o... |
| legso 28689 | The "shorter than" relatio... |
| ishlg 28692 | Rays : Definition 6.1 of ... |
| hlcomb 28693 | The half-line relation com... |
| hlcomd 28694 | The half-line relation com... |
| hlne1 28695 | The half-line relation imp... |
| hlne2 28696 | The half-line relation imp... |
| hlln 28697 | The half-line relation imp... |
| hleqnid 28698 | The endpoint does not belo... |
| hlid 28699 | The half-line relation is ... |
| hltr 28700 | The half-line relation is ... |
| hlbtwn 28701 | Betweenness is a sufficien... |
| btwnhl1 28702 | Deduce half-line from betw... |
| btwnhl2 28703 | Deduce half-line from betw... |
| btwnhl 28704 | Swap betweenness for a hal... |
| lnhl 28705 | Either a point ` C ` on th... |
| hlcgrex 28706 | Construct a point on a hal... |
| hlcgreulem 28707 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28708 | The point constructed in ~... |
| btwnlng1 28709 | Betweenness implies coline... |
| btwnlng2 28710 | Betweenness implies coline... |
| btwnlng3 28711 | Betweenness implies coline... |
| lncom 28712 | Swapping the points defini... |
| lnrot1 28713 | Rotating the points defini... |
| lnrot2 28714 | Rotating the points defini... |
| ncolne1 28715 | Non-colinear points are di... |
| ncolne2 28716 | Non-colinear points are di... |
| tgisline 28717 | The property of being a pr... |
| tglnne 28718 | It takes two different poi... |
| tglndim0 28719 | There are no lines in dime... |
| tgelrnln 28720 | The property of being a pr... |
| tglineeltr 28721 | Transitivity law for lines... |
| tglineelsb2 28722 | If ` S ` lies on PQ , then... |
| tglinerflx1 28723 | Reflexivity law for line m... |
| tglinerflx2 28724 | Reflexivity law for line m... |
| tglinecom 28725 | Commutativity law for line... |
| tglinethru 28726 | If ` A ` is a line contain... |
| tghilberti1 28727 | There is a line through an... |
| tghilberti2 28728 | There is at most one line ... |
| tglinethrueu 28729 | There is a unique line goi... |
| tglnne0 28730 | A line ` A ` has at least ... |
| tglnpt2 28731 | Find a second point on a l... |
| tglineintmo 28732 | Two distinct lines interse... |
| tglineineq 28733 | Two distinct lines interse... |
| tglineneq 28734 | Given three non-colinear p... |
| tglineinteq 28735 | Two distinct lines interse... |
| ncolncol 28736 | Deduce non-colinearity fro... |
| coltr 28737 | A transitivity law for col... |
| coltr3 28738 | A transitivity law for col... |
| colline 28739 | Three points are colinear ... |
| tglowdim2l 28740 | Reformulation of the lower... |
| tglowdim2ln 28741 | There is always one point ... |
| mirreu3 28744 | Existential uniqueness of ... |
| mirval 28745 | Value of the point inversi... |
| mirfv 28746 | Value of the point inversi... |
| mircgr 28747 | Property of the image by t... |
| mirbtwn 28748 | Property of the image by t... |
| ismir 28749 | Property of the image by t... |
| mirf 28750 | Point inversion as functio... |
| mircl 28751 | Closure of the point inver... |
| mirmir 28752 | The point inversion functi... |
| mircom 28753 | Variation on ~ mirmir . (... |
| mirreu 28754 | Any point has a unique ant... |
| mireq 28755 | Equality deduction for poi... |
| mirinv 28756 | The only invariant point o... |
| mirne 28757 | Mirror of non-center point... |
| mircinv 28758 | The center point is invari... |
| mirf1o 28759 | The point inversion functi... |
| miriso 28760 | The point inversion functi... |
| mirbtwni 28761 | Point inversion preserves ... |
| mirbtwnb 28762 | Point inversion preserves ... |
| mircgrs 28763 | Point inversion preserves ... |
| mirmir2 28764 | Point inversion of a point... |
| mirmot 28765 | Point investion is a motio... |
| mirln 28766 | If two points are on the s... |
| mirln2 28767 | If a point and its mirror ... |
| mirconn 28768 | Point inversion of connect... |
| mirhl 28769 | If two points ` X ` and ` ... |
| mirbtwnhl 28770 | If the center of the point... |
| mirhl2 28771 | Deduce half-line relation ... |
| mircgrextend 28772 | Link congruence over a pai... |
| mirtrcgr 28773 | Point inversion of one poi... |
| mirauto 28774 | Point inversion preserves ... |
| miduniq 28775 | Uniqueness of the middle p... |
| miduniq1 28776 | Uniqueness of the middle p... |
| miduniq2 28777 | If two point inversions co... |
| colmid 28778 | Colinearity and equidistan... |
| symquadlem 28779 | Lemma of the symmetrical q... |
| krippenlem 28780 | Lemma for ~ krippen . We ... |
| krippen 28781 | Krippenlemma (German for c... |
| midexlem 28782 | Lemma for the existence of... |
| israg 28787 | Property for 3 points A, B... |
| ragcom 28788 | Commutative rule for right... |
| ragcol 28789 | The right angle property i... |
| ragmir 28790 | Right angle property is pr... |
| mirrag 28791 | Right angle is conserved b... |
| ragtrivb 28792 | Trivial right angle. Theo... |
| ragflat2 28793 | Deduce equality from two r... |
| ragflat 28794 | Deduce equality from two r... |
| ragtriva 28795 | Trivial right angle. Theo... |
| ragflat3 28796 | Right angle and colinearit... |
| ragcgr 28797 | Right angle and colinearit... |
| motrag 28798 | Right angles are preserved... |
| ragncol 28799 | Right angle implies non-co... |
| perpln1 28800 | Derive a line from perpend... |
| perpln2 28801 | Derive a line from perpend... |
| isperp 28802 | Property for 2 lines A, B ... |
| perpcom 28803 | The "perpendicular" relati... |
| perpneq 28804 | Two perpendicular lines ar... |
| isperp2 28805 | Property for 2 lines A, B,... |
| isperp2d 28806 | One direction of ~ isperp2... |
| ragperp 28807 | Deduce that two lines are ... |
| footexALT 28808 | Alternative version of ~ f... |
| footexlem1 28809 | Lemma for ~ footex . (Con... |
| footexlem2 28810 | Lemma for ~ footex . (Con... |
| footex 28811 | From a point ` C ` outside... |
| foot 28812 | From a point ` C ` outside... |
| footne 28813 | Uniqueness of the foot poi... |
| footeq 28814 | Uniqueness of the foot poi... |
| hlperpnel 28815 | A point on a half-line whi... |
| perprag 28816 | Deduce a right angle from ... |
| perpdragALT 28817 | Deduce a right angle from ... |
| perpdrag 28818 | Deduce a right angle from ... |
| colperp 28819 | Deduce a perpendicularity ... |
| colperpexlem1 28820 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28821 | Lemma for ~ colperpex . S... |
| colperpexlem3 28822 | Lemma for ~ colperpex . C... |
| colperpex 28823 | In dimension 2 and above, ... |
| mideulem2 28824 | Lemma for ~ opphllem , whi... |
| opphllem 28825 | Lemma 8.24 of [Schwabhause... |
| mideulem 28826 | Lemma for ~ mideu . We ca... |
| midex 28827 | Existence of the midpoint,... |
| mideu 28828 | Existence and uniqueness o... |
| islnopp 28829 | The property for two point... |
| islnoppd 28830 | Deduce that ` A ` and ` B ... |
| oppne1 28831 | Points lying on opposite s... |
| oppne2 28832 | Points lying on opposite s... |
| oppne3 28833 | Points lying on opposite s... |
| oppcom 28834 | Commutativity rule for "op... |
| opptgdim2 28835 | If two points opposite to ... |
| oppnid 28836 | The "opposite to a line" r... |
| opphllem1 28837 | Lemma for ~ opphl . (Cont... |
| opphllem2 28838 | Lemma for ~ opphl . Lemma... |
| opphllem3 28839 | Lemma for ~ opphl : We as... |
| opphllem4 28840 | Lemma for ~ opphl . (Cont... |
| opphllem5 28841 | Second part of Lemma 9.4 o... |
| opphllem6 28842 | First part of Lemma 9.4 of... |
| oppperpex 28843 | Restating ~ colperpex usin... |
| opphl 28844 | If two points ` A ` and ` ... |
| outpasch 28845 | Axiom of Pasch, outer form... |
| hlpasch 28846 | An application of the axio... |
| ishpg 28849 | Value of the half-plane re... |
| hpgbr 28850 | Half-planes : property for... |
| hpgne1 28851 | Points on the open half pl... |
| hpgne2 28852 | Points on the open half pl... |
| lnopp2hpgb 28853 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28854 | If two points lie on the o... |
| hpgerlem 28855 | Lemma for the proof that t... |
| hpgid 28856 | The half-plane relation is... |
| hpgcom 28857 | The half-plane relation co... |
| hpgtr 28858 | The half-plane relation is... |
| colopp 28859 | Opposite sides of a line f... |
| colhp 28860 | Half-plane relation for co... |
| hphl 28861 | If two points are on the s... |
| midf 28866 | Midpoint as a function. (... |
| midcl 28867 | Closure of the midpoint. ... |
| ismidb 28868 | Property of the midpoint. ... |
| midbtwn 28869 | Betweenness of midpoint. ... |
| midcgr 28870 | Congruence of midpoint. (... |
| midid 28871 | Midpoint of a null segment... |
| midcom 28872 | Commutativity rule for the... |
| mirmid 28873 | Point inversion preserves ... |
| lmieu 28874 | Uniqueness of the line mir... |
| lmif 28875 | Line mirror as a function.... |
| lmicl 28876 | Closure of the line mirror... |
| islmib 28877 | Property of the line mirro... |
| lmicom 28878 | The line mirroring functio... |
| lmilmi 28879 | Line mirroring is an invol... |
| lmireu 28880 | Any point has a unique ant... |
| lmieq 28881 | Equality deduction for lin... |
| lmiinv 28882 | The invariants of the line... |
| lmicinv 28883 | The mirroring line is an i... |
| lmimid 28884 | If we have a right angle, ... |
| lmif1o 28885 | The line mirroring functio... |
| lmiisolem 28886 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28887 | The line mirroring functio... |
| lmimot 28888 | Line mirroring is a motion... |
| hypcgrlem1 28889 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28890 | Lemma for ~ hypcgr , case ... |
| hypcgr 28891 | If the catheti of two righ... |
| lmiopp 28892 | Line mirroring produces po... |
| lnperpex 28893 | Existence of a perpendicul... |
| trgcopy 28894 | Triangle construction: a c... |
| trgcopyeulem 28895 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28896 | Triangle construction: a c... |
| iscgra 28899 | Property for two angles AB... |
| iscgra1 28900 | A special version of ~ isc... |
| iscgrad 28901 | Sufficient conditions for ... |
| cgrane1 28902 | Angles imply inequality. ... |
| cgrane2 28903 | Angles imply inequality. ... |
| cgrane3 28904 | Angles imply inequality. ... |
| cgrane4 28905 | Angles imply inequality. ... |
| cgrahl1 28906 | Angle congruence is indepe... |
| cgrahl2 28907 | Angle congruence is indepe... |
| cgracgr 28908 | First direction of proposi... |
| cgraid 28909 | Angle congruence is reflex... |
| cgraswap 28910 | Swap rays in a congruence ... |
| cgrcgra 28911 | Triangle congruence implie... |
| cgracom 28912 | Angle congruence commutes.... |
| cgratr 28913 | Angle congruence is transi... |
| flatcgra 28914 | Flat angles are congruent.... |
| cgraswaplr 28915 | Swap both side of angle co... |
| cgrabtwn 28916 | Angle congruence preserves... |
| cgrahl 28917 | Angle congruence preserves... |
| cgracol 28918 | Angle congruence preserves... |
| cgrancol 28919 | Angle congruence preserves... |
| dfcgra2 28920 | This is the full statement... |
| sacgr 28921 | Supplementary angles of co... |
| oacgr 28922 | Vertical angle theorem. V... |
| acopy 28923 | Angle construction. Theor... |
| acopyeu 28924 | Angle construction. Theor... |
| isinag 28928 | Property for point ` X ` t... |
| isinagd 28929 | Sufficient conditions for ... |
| inagflat 28930 | Any point lies in a flat a... |
| inagswap 28931 | Swap the order of the half... |
| inagne1 28932 | Deduce inequality from the... |
| inagne2 28933 | Deduce inequality from the... |
| inagne3 28934 | Deduce inequality from the... |
| inaghl 28935 | The "point lie in angle" r... |
| isleag 28937 | Geometrical "less than" pr... |
| isleagd 28938 | Sufficient condition for "... |
| leagne1 28939 | Deduce inequality from the... |
| leagne2 28940 | Deduce inequality from the... |
| leagne3 28941 | Deduce inequality from the... |
| leagne4 28942 | Deduce inequality from the... |
| cgrg3col4 28943 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28944 | First congruence theorem: ... |
| tgsas 28945 | First congruence theorem: ... |
| tgsas2 28946 | First congruence theorem: ... |
| tgsas3 28947 | First congruence theorem: ... |
| tgasa1 28948 | Second congruence theorem:... |
| tgasa 28949 | Second congruence theorem:... |
| tgsss1 28950 | Third congruence theorem: ... |
| tgsss2 28951 | Third congruence theorem: ... |
| tgsss3 28952 | Third congruence theorem: ... |
| dfcgrg2 28953 | Congruence for two triangl... |
| isoas 28954 | Congruence theorem for iso... |
| iseqlg 28957 | Property of a triangle bei... |
| iseqlgd 28958 | Condition for a triangle t... |
| f1otrgds 28959 | Convenient lemma for ~ f1o... |
| f1otrgitv 28960 | Convenient lemma for ~ f1o... |
| f1otrg 28961 | A bijection between bases ... |
| f1otrge 28962 | A bijection between bases ... |
| ttgval 28965 | Define a function to augme... |
| ttglem 28966 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 28967 | The base set of a subcompl... |
| ttgplusg 28968 | The addition operation of ... |
| ttgsub 28969 | The subtraction operation ... |
| ttgvsca 28970 | The scalar product of a su... |
| ttgds 28971 | The metric of a subcomplex... |
| ttgitvval 28972 | Betweenness for a subcompl... |
| ttgelitv 28973 | Betweenness for a subcompl... |
| ttgbtwnid 28974 | Any subcomplex module equi... |
| ttgcontlem1 28975 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28976 | Any metric space fulfills ... |
| cchhllem 28977 | Lemma for chlbas and chlvs... |
| elee 28984 | Membership in a Euclidean ... |
| mptelee 28985 | A condition for a mapping ... |
| mpteleeOLD 28986 | Obsolete version of ~ mpte... |
| eleenn 28987 | If ` A ` is in ` ( EE `` N... |
| eleei 28988 | The forward direction of ~... |
| eedimeq 28989 | A point belongs to at most... |
| brbtwn 28990 | The binary relation form o... |
| brcgr 28991 | The binary relation form o... |
| fveere 28992 | The function value of a po... |
| fveecn 28993 | The function value of a po... |
| eqeefv 28994 | Two points are equal iff t... |
| eqeelen 28995 | Two points are equal iff t... |
| brbtwn2 28996 | Alternate characterization... |
| colinearalglem1 28997 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28998 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28999 | Lemma for ~ colinearalg . ... |
| colinearalglem4 29000 | Lemma for ~ colinearalg . ... |
| colinearalg 29001 | An algebraic characterizat... |
| eleesub 29002 | Membership of a subtractio... |
| eleesubd 29003 | Membership of a subtractio... |
| axdimuniq 29004 | The unique dimension axiom... |
| axcgrrflx 29005 | ` A ` is as far from ` B `... |
| axcgrtr 29006 | Congruence is transitive. ... |
| axcgrid 29007 | If there is no distance be... |
| axsegconlem1 29008 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 29009 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 29010 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 29011 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 29012 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 29013 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 29014 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 29015 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 29016 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 29017 | Lemma for ~ axsegcon . Sh... |
| axsegcon 29018 | Any segment ` A B ` can be... |
| ax5seglem1 29019 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 29020 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 29021 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 29022 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 29023 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 29024 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 29025 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 29026 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 29027 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 29028 | Lemma for ~ ax5seg . Take... |
| ax5seg 29029 | The five segment axiom. T... |
| axbtwnid 29030 | Points are indivisible. T... |
| axpaschlem 29031 | Lemma for ~ axpasch . Set... |
| axpasch 29032 | The inner Pasch axiom. Ta... |
| axlowdimlem1 29033 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 29034 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 29035 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 29036 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 29037 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 29038 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 29039 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 29040 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 29041 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 29042 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 29043 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 29044 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 29045 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 29046 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 29047 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 29048 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 29049 | Lemma for ~ axlowdim . Es... |
| axlowdim1 29050 | The lower dimension axiom ... |
| axlowdim2 29051 | The lower two-dimensional ... |
| axlowdim 29052 | The general lower dimensio... |
| axeuclidlem 29053 | Lemma for ~ axeuclid . Ha... |
| axeuclid 29054 | Euclid's axiom. Take an a... |
| axcontlem1 29055 | Lemma for ~ axcont . Chan... |
| axcontlem2 29056 | Lemma for ~ axcont . The ... |
| axcontlem3 29057 | Lemma for ~ axcont . Give... |
| axcontlem4 29058 | Lemma for ~ axcont . Give... |
| axcontlem5 29059 | Lemma for ~ axcont . Comp... |
| axcontlem6 29060 | Lemma for ~ axcont . Stat... |
| axcontlem7 29061 | Lemma for ~ axcont . Give... |
| axcontlem8 29062 | Lemma for ~ axcont . A po... |
| axcontlem9 29063 | Lemma for ~ axcont . Give... |
| axcontlem10 29064 | Lemma for ~ axcont . Give... |
| axcontlem11 29065 | Lemma for ~ axcont . Elim... |
| axcontlem12 29066 | Lemma for ~ axcont . Elim... |
| axcont 29067 | The axiom of continuity. ... |
| eengv 29070 | The value of the Euclidean... |
| eengstr 29071 | The Euclidean geometry as ... |
| eengbas 29072 | The Base of the Euclidean ... |
| ebtwntg 29073 | The betweenness relation u... |
| ecgrtg 29074 | The congruence relation us... |
| elntg 29075 | The line definition in the... |
| elntg2 29076 | The line definition in the... |
| eengtrkg 29077 | The geometry structure for... |
| eengtrkge 29078 | The geometry structure for... |
| edgfid 29081 | Utility theorem: index-ind... |
| edgfndx 29082 | Index value of the ~ df-ed... |
| edgfndxnn 29083 | The index value of the edg... |
| edgfndxid 29084 | The value of the edge func... |
| basendxltedgfndx 29085 | The index value of the ` B... |
| basendxnedgfndx 29086 | The slots ` Base ` and ` .... |
| vtxval 29091 | The set of vertices of a g... |
| iedgval 29092 | The set of indexed edges o... |
| 1vgrex 29093 | A graph with at least one ... |
| opvtxval 29094 | The set of vertices of a g... |
| opvtxfv 29095 | The set of vertices of a g... |
| opvtxov 29096 | The set of vertices of a g... |
| opiedgval 29097 | The set of indexed edges o... |
| opiedgfv 29098 | The set of indexed edges o... |
| opiedgov 29099 | The set of indexed edges o... |
| opvtxfvi 29100 | The set of vertices of a g... |
| opiedgfvi 29101 | The set of indexed edges o... |
| funvtxdmge2val 29102 | The set of vertices of an ... |
| funiedgdmge2val 29103 | The set of indexed edges o... |
| funvtxdm2val 29104 | The set of vertices of an ... |
| funiedgdm2val 29105 | The set of indexed edges o... |
| funvtxval0 29106 | The set of vertices of an ... |
| basvtxval 29107 | The set of vertices of a g... |
| edgfiedgval 29108 | The set of indexed edges o... |
| funvtxval 29109 | The set of vertices of a g... |
| funiedgval 29110 | The set of indexed edges o... |
| structvtxvallem 29111 | Lemma for ~ structvtxval a... |
| structvtxval 29112 | The set of vertices of an ... |
| structiedg0val 29113 | The set of indexed edges o... |
| structgrssvtxlem 29114 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29115 | The set of vertices of a g... |
| structgrssiedg 29116 | The set of indexed edges o... |
| struct2grstr 29117 | A graph represented as an ... |
| struct2grvtx 29118 | The set of vertices of a g... |
| struct2griedg 29119 | The set of indexed edges o... |
| graop 29120 | Any representation of a gr... |
| grastruct 29121 | Any representation of a gr... |
| gropd 29122 | If any representation of a... |
| grstructd 29123 | If any representation of a... |
| gropeld 29124 | If any representation of a... |
| grstructeld 29125 | If any representation of a... |
| setsvtx 29126 | The vertices of a structur... |
| setsiedg 29127 | The (indexed) edges of a s... |
| snstrvtxval 29128 | The set of vertices of a g... |
| snstriedgval 29129 | The set of indexed edges o... |
| vtxval0 29130 | Degenerated case 1 for ver... |
| iedgval0 29131 | Degenerated case 1 for edg... |
| vtxvalsnop 29132 | Degenerated case 2 for ver... |
| iedgvalsnop 29133 | Degenerated case 2 for edg... |
| vtxval3sn 29134 | Degenerated case 3 for ver... |
| iedgval3sn 29135 | Degenerated case 3 for edg... |
| vtxvalprc 29136 | Degenerated case 4 for ver... |
| iedgvalprc 29137 | Degenerated case 4 for edg... |
| edgval 29140 | The edges of a graph. (Co... |
| iedgedg 29141 | An indexed edge is an edge... |
| edgopval 29142 | The edges of a graph repre... |
| edgov 29143 | The edges of a graph repre... |
| edgstruct 29144 | The edges of a graph repre... |
| edgiedgb 29145 | A set is an edge iff it is... |
| edg0iedg0 29146 | There is no edge in a grap... |
| isuhgr 29151 | The predicate "is an undir... |
| isushgr 29152 | The predicate "is an undir... |
| uhgrf 29153 | The edge function of an un... |
| ushgrf 29154 | The edge function of an un... |
| uhgrss 29155 | An edge is a subset of ver... |
| uhgreq12g 29156 | If two sets have the same ... |
| uhgrfun 29157 | The edge function of an un... |
| uhgrn0 29158 | An edge is a nonempty subs... |
| lpvtx 29159 | The endpoints of a loop (w... |
| ushgruhgr 29160 | An undirected simple hyper... |
| isuhgrop 29161 | The property of being an u... |
| uhgr0e 29162 | The empty graph, with vert... |
| uhgr0vb 29163 | The null graph, with no ve... |
| uhgr0 29164 | The null graph represented... |
| uhgrun 29165 | The union ` U ` of two (un... |
| uhgrunop 29166 | The union of two (undirect... |
| ushgrun 29167 | The union ` U ` of two (un... |
| ushgrunop 29168 | The union of two (undirect... |
| uhgrstrrepe 29169 | Replacing (or adding) the ... |
| incistruhgr 29170 | An _incidence structure_ `... |
| isupgr 29175 | The property of being an u... |
| wrdupgr 29176 | The property of being an u... |
| upgrf 29177 | The edge function of an un... |
| upgrfn 29178 | The edge function of an un... |
| upgrss 29179 | An edge is a subset of ver... |
| upgrn0 29180 | An edge is a nonempty subs... |
| upgrle 29181 | An edge of an undirected p... |
| upgrfi 29182 | An edge is a finite subset... |
| upgrex 29183 | An edge is an unordered pa... |
| upgrbi 29184 | Show that an unordered pai... |
| upgrop 29185 | A pseudograph represented ... |
| isumgr 29186 | The property of being an u... |
| isumgrs 29187 | The simplified property of... |
| wrdumgr 29188 | The property of being an u... |
| umgrf 29189 | The edge function of an un... |
| umgrfn 29190 | The edge function of an un... |
| umgredg2 29191 | An edge of a multigraph ha... |
| umgrbi 29192 | Show that an unordered pai... |
| upgruhgr 29193 | An undirected pseudograph ... |
| umgrupgr 29194 | An undirected multigraph i... |
| umgruhgr 29195 | An undirected multigraph i... |
| upgrle2 29196 | An edge of an undirected p... |
| umgrnloopv 29197 | In a multigraph, there is ... |
| umgredgprv 29198 | In a multigraph, an edge i... |
| umgrnloop 29199 | In a multigraph, there is ... |
| umgrnloop0 29200 | A multigraph has no loops.... |
| umgr0e 29201 | The empty graph, with vert... |
| upgr0e 29202 | The empty graph, with vert... |
| upgr1elem 29203 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29204 | A pseudograph with one edg... |
| upgr0eop 29205 | The empty graph, with vert... |
| upgr1eop 29206 | A pseudograph with one edg... |
| upgr0eopALT 29207 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29208 | Alternate proof of ~ upgr1... |
| upgrun 29209 | The union ` U ` of two pse... |
| upgrunop 29210 | The union of two pseudogra... |
| umgrun 29211 | The union ` U ` of two mul... |
| umgrunop 29212 | The union of two multigrap... |
| umgrislfupgrlem 29213 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29214 | A multigraph is a loop-fre... |
| lfgredgge2 29215 | An edge of a loop-free gra... |
| lfgrnloop 29216 | A loop-free graph has no l... |
| uhgredgiedgb 29217 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29218 | A hypergraph has no edges ... |
| uhgredgn0 29219 | An edge of a hypergraph is... |
| edguhgr 29220 | An edge of a hypergraph is... |
| uhgredgrnv 29221 | An edge of a hypergraph co... |
| uhgredgss 29222 | The set of edges of a hype... |
| upgredgss 29223 | The set of edges of a pseu... |
| umgredgss 29224 | The set of edges of a mult... |
| edgupgr 29225 | Properties of an edge of a... |
| edgumgr 29226 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29227 | In a hypergraph, a vertex ... |
| upgredg 29228 | For each edge in a pseudog... |
| umgredg 29229 | For each edge in a multigr... |
| upgrpredgv 29230 | An edge of a pseudograph a... |
| umgrpredgv 29231 | An edge of a multigraph al... |
| upgredg2vtx 29232 | For a vertex incident to a... |
| upgredgpr 29233 | If a proper pair (of verti... |
| edglnl 29234 | The edges incident with a ... |
| numedglnl 29235 | The number of edges incide... |
| umgredgne 29236 | An edge of a multigraph al... |
| umgrnloop2 29237 | A multigraph has no loops.... |
| umgredgnlp 29238 | An edge of a multigraph is... |
| isuspgr 29243 | The property of being a si... |
| isusgr 29244 | The property of being a si... |
| uspgrf 29245 | The edge function of a sim... |
| usgrf 29246 | The edge function of a sim... |
| isusgrs 29247 | The property of being a si... |
| usgrfs 29248 | The edge function of a sim... |
| usgrfun 29249 | The edge function of a sim... |
| usgredgss 29250 | The set of edges of a simp... |
| edgusgr 29251 | An edge of a simple graph ... |
| isuspgrop 29252 | The property of being an u... |
| isusgrop 29253 | The property of being an u... |
| usgrop 29254 | A simple graph represented... |
| isausgr 29255 | The property of an ordered... |
| ausgrusgrb 29256 | The equivalence of the def... |
| usgrausgri 29257 | A simple graph represented... |
| ausgrumgri 29258 | If an alternatively define... |
| ausgrusgri 29259 | The equivalence of the def... |
| usgrausgrb 29260 | The equivalence of the def... |
| usgredgop 29261 | An edge of a simple graph ... |
| usgrf1o 29262 | The edge function of a sim... |
| usgrf1 29263 | The edge function of a sim... |
| uspgrf1oedg 29264 | The edge function of a sim... |
| usgrss 29265 | An edge is a subset of ver... |
| uspgredgiedg 29266 | In a simple pseudograph, f... |
| uspgriedgedg 29267 | In a simple pseudograph, f... |
| uspgrushgr 29268 | A simple pseudograph is an... |
| uspgrupgr 29269 | A simple pseudograph is an... |
| uspgrupgrushgr 29270 | A graph is a simple pseudo... |
| usgruspgr 29271 | A simple graph is a simple... |
| usgrumgr 29272 | A simple graph is an undir... |
| usgrumgruspgr 29273 | A graph is a simple graph ... |
| usgruspgrb 29274 | A class is a simple graph ... |
| uspgruhgr 29275 | An undirected simple pseud... |
| usgrupgr 29276 | A simple graph is an undir... |
| usgruhgr 29277 | A simple graph is an undir... |
| usgrislfuspgr 29278 | A simple graph is a loop-f... |
| uspgrun 29279 | The union ` U ` of two sim... |
| uspgrunop 29280 | The union of two simple ps... |
| usgrun 29281 | The union ` U ` of two sim... |
| usgrunop 29282 | The union of two simple gr... |
| usgredg2 29283 | The value of the "edge fun... |
| usgredg2ALT 29284 | Alternate proof of ~ usgre... |
| usgredgprv 29285 | In a simple graph, an edge... |
| usgredgprvALT 29286 | Alternate proof of ~ usgre... |
| usgredgppr 29287 | An edge of a simple graph ... |
| usgrpredgv 29288 | An edge of a simple graph ... |
| edgssv2 29289 | An edge of a simple graph ... |
| usgredg 29290 | For each edge in a simple ... |
| usgrnloopv 29291 | In a simple graph, there i... |
| usgrnloopvALT 29292 | Alternate proof of ~ usgrn... |
| usgrnloop 29293 | In a simple graph, there i... |
| usgrnloopALT 29294 | Alternate proof of ~ usgrn... |
| usgrnloop0 29295 | A simple graph has no loop... |
| usgrnloop0ALT 29296 | Alternate proof of ~ usgrn... |
| usgredgne 29297 | An edge of a simple graph ... |
| usgrf1oedg 29298 | The edge function of a sim... |
| uhgr2edg 29299 | If a vertex is adjacent to... |
| umgr2edg 29300 | If a vertex is adjacent to... |
| usgr2edg 29301 | If a vertex is adjacent to... |
| umgr2edg1 29302 | If a vertex is adjacent to... |
| usgr2edg1 29303 | If a vertex is adjacent to... |
| umgrvad2edg 29304 | If a vertex is adjacent to... |
| umgr2edgneu 29305 | If a vertex is adjacent to... |
| usgrsizedg 29306 | In a simple graph, the siz... |
| usgredg3 29307 | The value of the "edge fun... |
| usgredg4 29308 | For a vertex incident to a... |
| usgredgreu 29309 | For a vertex incident to a... |
| usgredg2vtx 29310 | For a vertex incident to a... |
| uspgredg2vtxeu 29311 | For a vertex incident to a... |
| usgredg2vtxeu 29312 | For a vertex incident to a... |
| usgredg2vtxeuALT 29313 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29314 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29315 | In a simple pseudograph, t... |
| usgredg2vlem1 29316 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29317 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29318 | In a simple graph, the map... |
| usgriedgleord 29319 | Alternate version of ~ usg... |
| ushgredgedg 29320 | In a simple hypergraph the... |
| usgredgedg 29321 | In a simple graph there is... |
| ushgredgedgloop 29322 | In a simple hypergraph the... |
| uspgredgleord 29323 | In a simple pseudograph th... |
| usgredgleord 29324 | In a simple graph the numb... |
| usgredgleordALT 29325 | Alternate proof for ~ usgr... |
| usgrstrrepe 29326 | Replacing (or adding) the ... |
| usgr0e 29327 | The empty graph, with vert... |
| usgr0vb 29328 | The null graph, with no ve... |
| uhgr0v0e 29329 | The null graph, with no ve... |
| uhgr0vsize0 29330 | The size of a hypergraph w... |
| uhgr0edgfi 29331 | A graph of order 0 (i.e. w... |
| usgr0v 29332 | The null graph, with no ve... |
| uhgr0vusgr 29333 | The null graph, with no ve... |
| usgr0 29334 | The null graph represented... |
| uspgr1e 29335 | A simple pseudograph with ... |
| usgr1e 29336 | A simple graph with one ed... |
| usgr0eop 29337 | The empty graph, with vert... |
| uspgr1eop 29338 | A simple pseudograph with ... |
| uspgr1ewop 29339 | A simple pseudograph with ... |
| uspgr1v1eop 29340 | A simple pseudograph with ... |
| usgr1eop 29341 | A simple graph with (at le... |
| uspgr2v1e2w 29342 | A simple pseudograph with ... |
| usgr2v1e2w 29343 | A simple graph with two ve... |
| edg0usgr 29344 | A class without edges is a... |
| lfuhgr1v0e 29345 | A loop-free hypergraph wit... |
| usgr1vr 29346 | A simple graph with one ve... |
| usgr1v 29347 | A class with one (or no) v... |
| usgr1v0edg 29348 | A class with one (or no) v... |
| usgrexmpldifpr 29349 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29350 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29351 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29352 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29353 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29354 | ` G ` is a simple graph of... |
| griedg0prc 29355 | The class of empty graphs ... |
| griedg0ssusgr 29356 | The class of all simple gr... |
| usgrprc 29357 | The class of simple graphs... |
| relsubgr 29360 | The class of the subgraph ... |
| subgrv 29361 | If a class is a subgraph o... |
| issubgr 29362 | The property of a set to b... |
| issubgr2 29363 | The property of a set to b... |
| subgrprop 29364 | The properties of a subgra... |
| subgrprop2 29365 | The properties of a subgra... |
| uhgrissubgr 29366 | The property of a hypergra... |
| subgrprop3 29367 | The properties of a subgra... |
| egrsubgr 29368 | An empty graph consisting ... |
| 0grsubgr 29369 | The null graph (represente... |
| 0uhgrsubgr 29370 | The null graph (as hypergr... |
| uhgrsubgrself 29371 | A hypergraph is a subgraph... |
| subgrfun 29372 | The edge function of a sub... |
| subgruhgrfun 29373 | The edge function of a sub... |
| subgreldmiedg 29374 | An element of the domain o... |
| subgruhgredgd 29375 | An edge of a subgraph of a... |
| subumgredg2 29376 | An edge of a subgraph of a... |
| subuhgr 29377 | A subgraph of a hypergraph... |
| subupgr 29378 | A subgraph of a pseudograp... |
| subumgr 29379 | A subgraph of a multigraph... |
| subusgr 29380 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29381 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29382 | A spanning subgraph ` S ` ... |
| uhgrspan 29383 | A spanning subgraph ` S ` ... |
| upgrspan 29384 | A spanning subgraph ` S ` ... |
| umgrspan 29385 | A spanning subgraph ` S ` ... |
| usgrspan 29386 | A spanning subgraph ` S ` ... |
| uhgrspanop 29387 | A spanning subgraph of a h... |
| upgrspanop 29388 | A spanning subgraph of a p... |
| umgrspanop 29389 | A spanning subgraph of a m... |
| usgrspanop 29390 | A spanning subgraph of a s... |
| uhgrspan1lem1 29391 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29392 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29393 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29394 | The induced subgraph ` S `... |
| upgrreslem 29395 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29396 | Lemma for ~ umgrres and ~ ... |
| upgrres 29397 | A subgraph obtained by rem... |
| umgrres 29398 | A subgraph obtained by rem... |
| usgrres 29399 | A subgraph obtained by rem... |
| upgrres1lem1 29400 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29401 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29402 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29403 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29404 | A pseudograph obtained by ... |
| umgrres1 29405 | A multigraph obtained by r... |
| usgrres1 29406 | Restricting a simple graph... |
| isfusgr 29409 | The property of being a fi... |
| fusgrvtxfi 29410 | A finite simple graph has ... |
| isfusgrf1 29411 | The property of being a fi... |
| isfusgrcl 29412 | The property of being a fi... |
| fusgrusgr 29413 | A finite simple graph is a... |
| opfusgr 29414 | A finite simple graph repr... |
| usgredgffibi 29415 | The number of edges in a s... |
| fusgredgfi 29416 | In a finite simple graph t... |
| usgr1v0e 29417 | The size of a (finite) sim... |
| usgrfilem 29418 | In a finite simple graph, ... |
| fusgrfisbase 29419 | Induction base for ~ fusgr... |
| fusgrfisstep 29420 | Induction step in ~ fusgrf... |
| fusgrfis 29421 | A finite simple graph is o... |
| fusgrfupgrfs 29422 | A finite simple graph is a... |
| nbgrprc0 29425 | The set of neighbors is em... |
| nbgrcl 29426 | If a class ` X ` has at le... |
| nbgrval 29427 | The set of neighbors of a ... |
| dfnbgr2 29428 | Alternate definition of th... |
| dfnbgr3 29429 | Alternate definition of th... |
| nbgrnvtx0 29430 | If a class ` X ` is not a ... |
| nbgrel 29431 | Characterization of a neig... |
| nbgrisvtx 29432 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29433 | The neighbors of a vertex ... |
| nbuhgr 29434 | The set of neighbors of a ... |
| nbupgr 29435 | The set of neighbors of a ... |
| nbupgrel 29436 | A neighbor of a vertex in ... |
| nbumgrvtx 29437 | The set of neighbors of a ... |
| nbumgr 29438 | The set of neighbors of an... |
| nbusgrvtx 29439 | The set of neighbors of a ... |
| nbusgr 29440 | The set of neighbors of an... |
| nbgr2vtx1edg 29441 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29442 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29443 | If a hypergraph has two ve... |
| nbusgreledg 29444 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29445 | A vertex which is not endp... |
| nbgr0vtx 29446 | In a null graph (with no v... |
| nbgr0edglem 29447 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29448 | In an empty graph (with no... |
| nbgr1vtx 29449 | In a graph with one vertex... |
| nbgrnself 29450 | A vertex in a graph is not... |
| nbgrnself2 29451 | A class ` X ` is not a nei... |
| nbgrssovtx 29452 | The neighbors of a vertex ... |
| nbgrssvwo2 29453 | The neighbors of a vertex ... |
| nbgrsym 29454 | In a graph, the neighborho... |
| nbupgrres 29455 | The neighborhood of a vert... |
| usgrnbcnvfv 29456 | Applying the edge function... |
| nbusgredgeu 29457 | For each neighbor of a ver... |
| edgnbusgreu 29458 | For each edge incident to ... |
| nbusgredgeu0 29459 | For each neighbor of a ver... |
| nbusgrf1o0 29460 | The mapping of neighbors o... |
| nbusgrf1o1 29461 | The set of neighbors of a ... |
| nbusgrf1o 29462 | The set of neighbors of a ... |
| nbedgusgr 29463 | The number of neighbors of... |
| edgusgrnbfin 29464 | The number of neighbors of... |
| nbusgrfi 29465 | The class of neighbors of ... |
| nbfiusgrfi 29466 | The class of neighbors of ... |
| hashnbusgrnn0 29467 | The number of neighbors of... |
| nbfusgrlevtxm1 29468 | The number of neighbors of... |
| nbfusgrlevtxm2 29469 | If there is a vertex which... |
| nbusgrvtxm1 29470 | If the number of neighbors... |
| nb3grprlem1 29471 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29472 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29473 | The neighbors of a vertex ... |
| nb3grpr2 29474 | The neighbors of a vertex ... |
| nb3gr2nb 29475 | If the neighbors of two ve... |
| uvtxval 29478 | The set of all universal v... |
| uvtxel 29479 | A universal vertex, i.e. a... |
| uvtxisvtx 29480 | A universal vertex is a ve... |
| uvtxssvtx 29481 | The set of the universal v... |
| vtxnbuvtx 29482 | A universal vertex has all... |
| uvtxnbgrss 29483 | A universal vertex has all... |
| uvtxnbgrvtx 29484 | A universal vertex is neig... |
| uvtx0 29485 | There is no universal vert... |
| isuvtx 29486 | The set of all universal v... |
| uvtxel1 29487 | Characterization of a univ... |
| uvtx01vtx 29488 | If a graph/class has no ed... |
| uvtx2vtx1edg 29489 | If a graph has two vertice... |
| uvtx2vtx1edgb 29490 | If a hypergraph has two ve... |
| uvtxnbgr 29491 | A universal vertex has all... |
| uvtxnbgrb 29492 | A vertex is universal iff ... |
| uvtxusgr 29493 | The set of all universal v... |
| uvtxusgrel 29494 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29495 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29496 | If the number of neighbors... |
| uvtxnbvtxm1 29497 | A universal vertex has ` n... |
| nbupgruvtxres 29498 | The neighborhood of a univ... |
| uvtxupgrres 29499 | A universal vertex is univ... |
| cplgruvtxb 29504 | A graph ` G ` is complete ... |
| prcliscplgr 29505 | A proper class (representi... |
| iscplgr 29506 | The property of being a co... |
| iscplgrnb 29507 | A graph is complete iff al... |
| iscplgredg 29508 | A graph ` G ` is complete ... |
| iscusgr 29509 | The property of being a co... |
| cusgrusgr 29510 | A complete simple graph is... |
| cusgrcplgr 29511 | A complete simple graph is... |
| iscusgrvtx 29512 | A simple graph is complete... |
| cusgruvtxb 29513 | A simple graph is complete... |
| iscusgredg 29514 | A simple graph is complete... |
| cusgredg 29515 | In a complete simple graph... |
| cplgr0 29516 | The null graph (with no ve... |
| cusgr0 29517 | The null graph (with no ve... |
| cplgr0v 29518 | A null graph (with no vert... |
| cusgr0v 29519 | A graph with no vertices a... |
| cplgr1vlem 29520 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29521 | A graph with one vertex is... |
| cusgr1v 29522 | A graph with one vertex an... |
| cplgr2v 29523 | An undirected hypergraph w... |
| cplgr2vpr 29524 | An undirected hypergraph w... |
| nbcplgr 29525 | In a complete graph, each ... |
| cplgr3v 29526 | A pseudograph with three (... |
| cusgr3vnbpr 29527 | The neighbors of a vertex ... |
| cplgrop 29528 | A complete graph represent... |
| cusgrop 29529 | A complete simple graph re... |
| cusgrexilem1 29530 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29531 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29532 | An arbitrary set regarded ... |
| cusgrexilem2 29533 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29534 | An arbitrary set ` V ` reg... |
| cusgrexg 29535 | For each set there is a se... |
| structtousgr 29536 | Any (extensible) structure... |
| structtocusgr 29537 | Any (extensible) structure... |
| cffldtocusgr 29538 | The field of complex numbe... |
| cusgrres 29539 | Restricting a complete sim... |
| cusgrsizeindb0 29540 | Base case of the induction... |
| cusgrsizeindb1 29541 | Base case of the induction... |
| cusgrsizeindslem 29542 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29543 | Part 1 of induction step i... |
| cusgrsize2inds 29544 | Induction step in ~ cusgrs... |
| cusgrsize 29545 | The size of a finite compl... |
| cusgrfilem1 29546 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29547 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29548 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29549 | If the size of a complete ... |
| usgredgsscusgredg 29550 | A simple graph is a subgra... |
| usgrsscusgr 29551 | A simple graph is a subgra... |
| sizusglecusglem1 29552 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29553 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29554 | The size of a simple graph... |
| fusgrmaxsize 29555 | The maximum size of a fini... |
| vtxdgfval 29558 | The value of the vertex de... |
| vtxdgval 29559 | The degree of a vertex. (... |
| vtxdgfival 29560 | The degree of a vertex for... |
| vtxdgop 29561 | The vertex degree expresse... |
| vtxdgf 29562 | The vertex degree function... |
| vtxdgelxnn0 29563 | The degree of a vertex is ... |
| vtxdg0v 29564 | The degree of a vertex in ... |
| vtxdg0e 29565 | The degree of a vertex in ... |
| vtxdgfisnn0 29566 | The degree of a vertex in ... |
| vtxdgfisf 29567 | The vertex degree function... |
| vtxdeqd 29568 | Equality theorem for the v... |
| vtxduhgr0e 29569 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29570 | The degree of the vertex i... |
| vdumgr0 29571 | A vertex in a multigraph h... |
| vtxdun 29572 | The degree of a vertex in ... |
| vtxdfiun 29573 | The degree of a vertex in ... |
| vtxduhgrun 29574 | The degree of a vertex in ... |
| vtxduhgrfiun 29575 | The degree of a vertex in ... |
| vtxdlfgrval 29576 | The value of the vertex de... |
| vtxdumgrval 29577 | The value of the vertex de... |
| vtxdusgrval 29578 | The value of the vertex de... |
| vtxd0nedgb 29579 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29580 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29581 | The value of the vertex de... |
| vtxdusgrfvedg 29582 | The value of the vertex de... |
| vtxduhgr0nedg 29583 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29584 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29585 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29586 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29587 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29588 | The vertex degree function... |
| vtxdgfusgr 29589 | In a finite simple graph, ... |
| fusgrn0degnn0 29590 | In a nonempty, finite grap... |
| 1loopgruspgr 29591 | A graph with one edge whic... |
| 1loopgredg 29592 | The set of edges in a grap... |
| 1loopgrnb0 29593 | In a graph (simple pseudog... |
| 1loopgrvd2 29594 | The vertex degree of a one... |
| 1loopgrvd0 29595 | The vertex degree of a one... |
| 1hevtxdg0 29596 | The vertex degree of verte... |
| 1hevtxdg1 29597 | The vertex degree of verte... |
| 1hegrvtxdg1 29598 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29599 | The vertex degree of a gra... |
| 1egrvtxdg1 29600 | The vertex degree of a one... |
| 1egrvtxdg1r 29601 | The vertex degree of a one... |
| 1egrvtxdg0 29602 | The vertex degree of a one... |
| p1evtxdeqlem 29603 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29604 | If an edge ` E ` which doe... |
| p1evtxdp1 29605 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29606 | The set of vertices in a g... |
| uspgrloopvtxel 29607 | A vertex in a graph (simpl... |
| uspgrloopiedg 29608 | The set of edges in a grap... |
| uspgrloopedg 29609 | The set of edges in a grap... |
| uspgrloopnb0 29610 | In a graph (simple pseudog... |
| uspgrloopvd2 29611 | The vertex degree of a one... |
| umgr2v2evtx 29612 | The set of vertices in a m... |
| umgr2v2evtxel 29613 | A vertex in a multigraph w... |
| umgr2v2eiedg 29614 | The edge function in a mul... |
| umgr2v2eedg 29615 | The set of edges in a mult... |
| umgr2v2e 29616 | A multigraph with two edge... |
| umgr2v2enb1 29617 | In a multigraph with two e... |
| umgr2v2evd2 29618 | In a multigraph with two e... |
| hashnbusgrvd 29619 | In a simple graph, the num... |
| usgruvtxvdb 29620 | In a finite simple graph w... |
| vdiscusgrb 29621 | A finite simple graph with... |
| vdiscusgr 29622 | In a finite complete simpl... |
| vtxdusgradjvtx 29623 | The degree of a vertex in ... |
| usgrvd0nedg 29624 | If a vertex in a simple gr... |
| uhgrvd00 29625 | If every vertex in a hyper... |
| usgrvd00 29626 | If every vertex in a simpl... |
| vdegp1ai 29627 | The induction step for a v... |
| vdegp1bi 29628 | The induction step for a v... |
| vdegp1ci 29629 | The induction step for a v... |
| vtxdginducedm1lem1 29630 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29631 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29632 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29633 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29634 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29635 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29636 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29637 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29638 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29639 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29640 | Induction step of ~ finsum... |
| finsumvtxdg2size 29641 | The sum of the degrees of ... |
| fusgr1th 29642 | The sum of the degrees of ... |
| finsumvtxdgeven 29643 | The sum of the degrees of ... |
| vtxdgoddnumeven 29644 | The number of vertices of ... |
| fusgrvtxdgonume 29645 | The number of vertices of ... |
| isrgr 29650 | The property of a class be... |
| rgrprop 29651 | The properties of a k-regu... |
| isrusgr 29652 | The property of being a k-... |
| rusgrprop 29653 | The properties of a k-regu... |
| rusgrrgr 29654 | A k-regular simple graph i... |
| rusgrusgr 29655 | A k-regular simple graph i... |
| finrusgrfusgr 29656 | A finite regular simple gr... |
| isrusgr0 29657 | The property of being a k-... |
| rusgrprop0 29658 | The properties of a k-regu... |
| usgreqdrusgr 29659 | If all vertices in a simpl... |
| fusgrregdegfi 29660 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29661 | If all vertices in a nonem... |
| frusgrnn0 29662 | In a nonempty finite k-reg... |
| 0edg0rgr 29663 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29664 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29665 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29666 | A simple graph is 0-regula... |
| 0vtxrgr 29667 | A null graph (with no vert... |
| 0vtxrusgr 29668 | A graph with no vertices a... |
| 0uhgrrusgr 29669 | The null graph as hypergra... |
| 0grrusgr 29670 | The null graph represented... |
| 0grrgr 29671 | The null graph represented... |
| cusgrrusgr 29672 | A complete simple graph wi... |
| cusgrm1rusgr 29673 | A finite simple graph with... |
| rusgrpropnb 29674 | The properties of a k-regu... |
| rusgrpropedg 29675 | The properties of a k-regu... |
| rusgrpropadjvtx 29676 | The properties of a k-regu... |
| rusgrnumwrdl2 29677 | In a k-regular simple grap... |
| rusgr1vtxlem 29678 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29679 | If a k-regular simple grap... |
| rgrusgrprc 29680 | The class of 0-regular sim... |
| rusgrprc 29681 | The class of 0-regular sim... |
| rgrprc 29682 | The class of 0-regular gra... |
| rgrprcx 29683 | The class of 0-regular gra... |
| rgrx0ndm 29684 | 0 is not in the domain of ... |
| rgrx0nd 29685 | The potentially alternativ... |
| ewlksfval 29692 | The set of s-walks of edge... |
| isewlk 29693 | Conditions for a function ... |
| ewlkprop 29694 | Properties of an s-walk of... |
| ewlkinedg 29695 | The intersection (common v... |
| ewlkle 29696 | An s-walk of edges is also... |
| upgrewlkle2 29697 | In a pseudograph, there is... |
| wkslem1 29698 | Lemma 1 for walks to subst... |
| wkslem2 29699 | Lemma 2 for walks to subst... |
| wksfval 29700 | The set of walks (in an un... |
| iswlk 29701 | Properties of a pair of fu... |
| wlkprop 29702 | Properties of a walk. (Co... |
| wlkv 29703 | The classes involved in a ... |
| iswlkg 29704 | Generalization of ~ iswlk ... |
| wlkf 29705 | The mapping enumerating th... |
| wlkcl 29706 | A walk has length ` # ( F ... |
| wlkp 29707 | The mapping enumerating th... |
| wlkpwrd 29708 | The sequence of vertices o... |
| wlklenvp1 29709 | The number of vertices of ... |
| wksv 29710 | The class of walks is a se... |
| wlkn0 29711 | The sequence of vertices o... |
| wlklenvm1 29712 | The number of edges of a w... |
| ifpsnprss 29713 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29714 | Each pair of adjacent vert... |
| wlkvtxiedg 29715 | The vertices of a walk are... |
| relwlk 29716 | The set ` ( Walks `` G ) `... |
| wlkvv 29717 | If there is at least one w... |
| wlkop 29718 | A walk is an ordered pair.... |
| wlkcpr 29719 | A walk as class with two c... |
| wlk2f 29720 | If there is a walk ` W ` t... |
| wlkcomp 29721 | A walk expressed by proper... |
| wlkcompim 29722 | Implications for the prope... |
| wlkelwrd 29723 | The components of a walk a... |
| wlkeq 29724 | Conditions for two walks (... |
| edginwlk 29725 | The value of the edge func... |
| upgredginwlk 29726 | The value of the edge func... |
| iedginwlk 29727 | The value of the edge func... |
| wlkl1loop 29728 | A walk of length 1 from a ... |
| wlk1walk 29729 | A walk is a 1-walk "on the... |
| wlk1ewlk 29730 | A walk is an s-walk "on th... |
| upgriswlk 29731 | Properties of a pair of fu... |
| upgrwlkedg 29732 | The edges of a walk in a p... |
| upgrwlkcompim 29733 | Implications for the prope... |
| wlkvtxedg 29734 | The vertices of a walk are... |
| upgrwlkvtxedg 29735 | The pairs of connected ver... |
| uspgr2wlkeq 29736 | Conditions for two walks w... |
| uspgr2wlkeq2 29737 | Conditions for two walks w... |
| uspgr2wlkeqi 29738 | Conditions for two walks w... |
| umgrwlknloop 29739 | In a multigraph, each walk... |
| wlkv0 29740 | If there is a walk in the ... |
| g0wlk0 29741 | There is no walk in a null... |
| 0wlk0 29742 | There is no walk for the e... |
| wlk0prc 29743 | There is no walk in a null... |
| wlklenvclwlk 29744 | The number of vertices in ... |
| wlkson 29745 | The set of walks between t... |
| iswlkon 29746 | Properties of a pair of fu... |
| wlkonprop 29747 | Properties of a walk betwe... |
| wlkpvtx 29748 | A walk connects vertices. ... |
| wlkepvtx 29749 | The endpoints of a walk ar... |
| wlkoniswlk 29750 | A walk between two vertice... |
| wlkonwlk 29751 | A walk is a walk between i... |
| wlkonwlk1l 29752 | A walk is a walk from its ... |
| wlksoneq1eq2 29753 | Two walks with identical s... |
| wlkonl1iedg 29754 | If there is a walk between... |
| wlkon2n0 29755 | The length of a walk betwe... |
| 2wlklem 29756 | Lemma for theorems for wal... |
| upgr2wlk 29757 | Properties of a pair of fu... |
| wlkreslem 29758 | Lemma for ~ wlkres . (Con... |
| wlkres 29759 | The restriction ` <. H , Q... |
| redwlklem 29760 | Lemma for ~ redwlk . (Con... |
| redwlk 29761 | A walk ending at the last ... |
| wlkp1lem1 29762 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29763 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29764 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29765 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29766 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29767 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29768 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29769 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29770 | Append one path segment (e... |
| wlkdlem1 29771 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29772 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29773 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29774 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29775 | Two words representing a w... |
| lfgrwlkprop 29776 | Two adjacent vertices in a... |
| lfgriswlk 29777 | Conditions for a pair of f... |
| lfgrwlknloop 29778 | In a loop-free graph, each... |
| reltrls 29783 | The set ` ( Trails `` G ) ... |
| trlsfval 29784 | The set of trails (in an u... |
| istrl 29785 | Conditions for a pair of c... |
| trliswlk 29786 | A trail is a walk. (Contr... |
| trlf1 29787 | The enumeration ` F ` of a... |
| trlreslem 29788 | Lemma for ~ trlres . Form... |
| trlres 29789 | The restriction ` <. H , Q... |
| upgrtrls 29790 | The set of trails in a pse... |
| upgristrl 29791 | Properties of a pair of fu... |
| upgrf1istrl 29792 | Properties of a pair of a ... |
| wksonproplem 29793 | Lemma for theorems for pro... |
| trlsonfval 29794 | The set of trails between ... |
| istrlson 29795 | Properties of a pair of fu... |
| trlsonprop 29796 | Properties of a trail betw... |
| trlsonistrl 29797 | A trail between two vertic... |
| trlsonwlkon 29798 | A trail between two vertic... |
| trlontrl 29799 | A trail is a trail between... |
| relpths 29808 | The set ` ( Paths `` G ) `... |
| pthsfval 29809 | The set of paths (in an un... |
| spthsfval 29810 | The set of simple paths (i... |
| ispth 29811 | Conditions for a pair of c... |
| isspth 29812 | Conditions for a pair of c... |
| pthistrl 29813 | A path is a trail (in an u... |
| spthispth 29814 | A simple path is a path (i... |
| pthiswlk 29815 | A path is a walk (in an un... |
| spthiswlk 29816 | A simple path is a walk (i... |
| pthdivtx 29817 | The inner vertices of a pa... |
| pthdadjvtx 29818 | The adjacent vertices of a... |
| dfpth2 29819 | Alternate definition for a... |
| pthdifv 29820 | The vertices of a path are... |
| 2pthnloop 29821 | A path of length at least ... |
| upgr2pthnlp 29822 | A path of length at least ... |
| spthdifv 29823 | The vertices of a simple p... |
| spthdep 29824 | A simple path (at least of... |
| pthdepisspth 29825 | A path with different star... |
| upgrwlkdvdelem 29826 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29827 | In a pseudograph, all edge... |
| upgrspthswlk 29828 | The set of simple paths in... |
| upgrwlkdvspth 29829 | A walk consisting of diffe... |
| pthsonfval 29830 | The set of paths between t... |
| spthson 29831 | The set of simple paths be... |
| ispthson 29832 | Properties of a pair of fu... |
| isspthson 29833 | Properties of a pair of fu... |
| pthsonprop 29834 | Properties of a path betwe... |
| spthonprop 29835 | Properties of a simple pat... |
| pthonispth 29836 | A path between two vertice... |
| pthontrlon 29837 | A path between two vertice... |
| pthonpth 29838 | A path is a path between i... |
| isspthonpth 29839 | A pair of functions is a s... |
| spthonisspth 29840 | A simple path between to v... |
| spthonpthon 29841 | A simple path between two ... |
| spthonepeq 29842 | The endpoints of a simple ... |
| uhgrwkspthlem1 29843 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29844 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29845 | Any walk of length 1 betwe... |
| usgr2wlkneq 29846 | The vertices and edges are... |
| usgr2wlkspthlem1 29847 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29848 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29849 | In a simple graph, any wal... |
| usgr2trlncl 29850 | In a simple graph, any tra... |
| usgr2trlspth 29851 | In a simple graph, any tra... |
| usgr2pthspth 29852 | In a simple graph, any pat... |
| usgr2pthlem 29853 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29854 | In a simple graph, there i... |
| usgr2pth0 29855 | In a simply graph, there i... |
| pthdlem1 29856 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29857 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29858 | Lemma 2 for ~ pthd . (Con... |
| pthd 29859 | Two words representing a t... |
| clwlks 29862 | The set of closed walks (i... |
| isclwlk 29863 | A pair of functions repres... |
| clwlkiswlk 29864 | A closed walk is a walk (i... |
| clwlkwlk 29865 | Closed walks are walks (in... |
| clwlkswks 29866 | Closed walks are walks (in... |
| isclwlke 29867 | Properties of a pair of fu... |
| isclwlkupgr 29868 | Properties of a pair of fu... |
| clwlkcomp 29869 | A closed walk expressed by... |
| clwlkcompim 29870 | Implications for the prope... |
| upgrclwlkcompim 29871 | Implications for the prope... |
| clwlkcompbp 29872 | Basic properties of the co... |
| clwlkl1loop 29873 | A closed walk of length 1 ... |
| crcts 29878 | The set of circuits (in an... |
| cycls 29879 | The set of cycles (in an u... |
| iscrct 29880 | Sufficient and necessary c... |
| iscycl 29881 | Sufficient and necessary c... |
| crctprop 29882 | The properties of a circui... |
| cyclprop 29883 | The properties of a cycle:... |
| crctisclwlk 29884 | A circuit is a closed walk... |
| crctistrl 29885 | A circuit is a trail. (Co... |
| crctiswlk 29886 | A circuit is a walk. (Con... |
| cyclispth 29887 | A cycle is a path. (Contr... |
| cycliswlk 29888 | A cycle is a walk. (Contr... |
| cycliscrct 29889 | A cycle is a circuit. (Co... |
| cyclnumvtx 29890 | The number of vertices of ... |
| cyclnspth 29891 | A (non-trivial) cycle is n... |
| pthisspthorcycl 29892 | A path is either a simple ... |
| pthspthcyc 29893 | A pair ` <. F , P >. ` rep... |
| cyclispthon 29894 | A cycle is a path starting... |
| lfgrn1cycl 29895 | In a loop-free graph there... |
| usgr2trlncrct 29896 | In a simple graph, any tra... |
| umgrn1cycl 29897 | In a multigraph graph (wit... |
| uspgrn2crct 29898 | In a simple pseudograph th... |
| usgrn2cycl 29899 | In a simple graph there ar... |
| crctcshwlkn0lem1 29900 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29901 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29902 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29903 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29904 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29905 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29906 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29907 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29908 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29909 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29910 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29911 | Cyclically shifting the in... |
| crctcshwlk 29912 | Cyclically shifting the in... |
| crctcshtrl 29913 | Cyclically shifting the in... |
| crctcsh 29914 | Cyclically shifting the in... |
| wwlks 29925 | The set of walks (in an un... |
| iswwlks 29926 | A word over the set of ver... |
| wwlksn 29927 | The set of walks (in an un... |
| iswwlksn 29928 | A word over the set of ver... |
| wwlksnprcl 29929 | Derivation of the length o... |
| iswwlksnx 29930 | Properties of a word to re... |
| wwlkbp 29931 | Basic properties of a walk... |
| wwlknbp 29932 | Basic properties of a walk... |
| wwlknp 29933 | Properties of a set being ... |
| wwlknbp1 29934 | Other basic properties of ... |
| wwlknvtx 29935 | The symbols of a word ` W ... |
| wwlknllvtx 29936 | If a word ` W ` represents... |
| wwlknlsw 29937 | If a word represents a wal... |
| wspthsn 29938 | The set of simple paths of... |
| iswspthn 29939 | An element of the set of s... |
| wspthnp 29940 | Properties of a set being ... |
| wwlksnon 29941 | The set of walks of a fixe... |
| wspthsnon 29942 | The set of simple paths of... |
| iswwlksnon 29943 | The set of walks of a fixe... |
| wwlksnon0 29944 | Sufficient conditions for ... |
| wwlksonvtx 29945 | If a word ` W ` represents... |
| iswspthsnon 29946 | The set of simple paths of... |
| wwlknon 29947 | An element of the set of w... |
| wspthnon 29948 | An element of the set of s... |
| wspthnonp 29949 | Properties of a set being ... |
| wspthneq1eq2 29950 | Two simple paths with iden... |
| wwlksn0s 29951 | The set of all walks as wo... |
| wwlkssswrd 29952 | Walks (represented by word... |
| wwlksn0 29953 | A walk of length 0 is repr... |
| 0enwwlksnge1 29954 | In graphs without edges, t... |
| wwlkswwlksn 29955 | A walk of a fixed length a... |
| wwlkssswwlksn 29956 | The walks of a fixed lengt... |
| wlkiswwlks1 29957 | The sequence of vertices i... |
| wlklnwwlkln1 29958 | The sequence of vertices i... |
| wlkiswwlks2lem1 29959 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29960 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29961 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29962 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29963 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29964 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29965 | A walk as word corresponds... |
| wlkiswwlks 29966 | A walk as word corresponds... |
| wlkiswwlksupgr2 29967 | A walk as word corresponds... |
| wlkiswwlkupgr 29968 | A walk as word corresponds... |
| wlkswwlksf1o 29969 | The mapping of (ordinary) ... |
| wlkswwlksen 29970 | The set of walks as words ... |
| wwlksm1edg 29971 | Removing the trailing edge... |
| wlklnwwlkln2lem 29972 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29973 | A walk of length ` N ` as ... |
| wlklnwwlkn 29974 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29975 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29976 | A walk of length ` N ` as ... |
| wlknewwlksn 29977 | If a walk in a pseudograph... |
| wlknwwlksnbij 29978 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29979 | In a simple pseudograph, t... |
| wlknwwlksneqs 29980 | The set of walks of a fixe... |
| wwlkseq 29981 | Equality of two walks (as ... |
| wwlksnred 29982 | Reduction of a walk (as wo... |
| wwlksnext 29983 | Extension of a walk (as wo... |
| wwlksnextbi 29984 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29985 | For each walk (as word) of... |
| wwlksnredwwlkn0 29986 | For each walk (as word) of... |
| wwlksnextwrd 29987 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29988 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29989 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29990 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29991 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29992 | There is a bijection betwe... |
| wwlksnexthasheq 29993 | The number of the extensio... |
| disjxwwlksn 29994 | Sets of walks (as words) e... |
| wwlksnndef 29995 | Conditions for ` WWalksN `... |
| wwlksnfi 29996 | The number of walks repres... |
| wlksnfi 29997 | The number of walks of fix... |
| wlksnwwlknvbij 29998 | There is a bijection betwe... |
| wwlksnextproplem1 29999 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 30000 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 30001 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 30002 | Adding additional properti... |
| disjxwwlkn 30003 | Sets of walks (as words) e... |
| hashwwlksnext 30004 | Number of walks (as words)... |
| wwlksnwwlksnon 30005 | A walk of fixed length is ... |
| wspthsnwspthsnon 30006 | A simple path of fixed len... |
| wspthsnonn0vne 30007 | If the set of simple paths... |
| wspthsswwlkn 30008 | The set of simple paths of... |
| wspthnfi 30009 | In a finite graph, the set... |
| wwlksnonfi 30010 | In a finite graph, the set... |
| wspthsswwlknon 30011 | The set of simple paths of... |
| wspthnonfi 30012 | In a finite graph, the set... |
| wspniunwspnon 30013 | The set of nonempty simple... |
| wspn0 30014 | If there are no vertices, ... |
| 2wlkdlem1 30015 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 30016 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 30017 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 30018 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 30019 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 30020 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 30021 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 30022 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 30023 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 30024 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 30025 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 30026 | Construction of a walk fro... |
| 2wlkond 30027 | A walk of length 2 from on... |
| 2trld 30028 | Construction of a trail fr... |
| 2trlond 30029 | A trail of length 2 from o... |
| 2pthd 30030 | A path of length 2 from on... |
| 2spthd 30031 | A simple path of length 2 ... |
| 2pthond 30032 | A simple path of length 2 ... |
| 2pthon3v 30033 | For a vertex adjacent to t... |
| umgr2adedgwlklem 30034 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 30035 | In a multigraph, two adjac... |
| umgr2adedgwlkon 30036 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 30037 | Alternate proof for ~ umgr... |
| umgr2adedgspth 30038 | In a multigraph, two adjac... |
| umgr2wlk 30039 | In a multigraph, there is ... |
| umgr2wlkon 30040 | For each pair of adjacent ... |
| elwwlks2s3 30041 | A walk of length 2 as word... |
| midwwlks2s3 30042 | There is a vertex between ... |
| wwlks2onv 30043 | If a length 3 string repre... |
| elwwlks2ons3im 30044 | A walk as word of length 2... |
| elwwlks2ons3 30045 | For each walk of length 2 ... |
| s3wwlks2on 30046 | A length 3 string which re... |
| sps3wwlks2on 30047 | A length 3 string which re... |
| usgrwwlks2on 30048 | A walk of length 2 between... |
| umgrwwlks2on 30049 | A walk of length 2 between... |
| wwlks2onsym 30050 | There is a walk of length ... |
| elwwlks2on 30051 | A walk of length 2 between... |
| elwspths2on 30052 | A simple path of length 2 ... |
| elwspths2onw 30053 | A simple path of length 2 ... |
| wpthswwlks2on 30054 | For two different vertices... |
| 2wspdisj 30055 | All simple paths of length... |
| 2wspiundisj 30056 | All simple paths of length... |
| usgr2wspthons3 30057 | A simple path of length 2 ... |
| usgr2wspthon 30058 | A simple path of length 2 ... |
| elwwlks2 30059 | A walk of length 2 between... |
| elwspths2spth 30060 | A simple path of length 2 ... |
| rusgrnumwwlkl1 30061 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 30062 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 30063 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 30064 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 30065 | Induction base 1 for ~ rus... |
| rusgr0edg 30066 | Special case for graphs wi... |
| rusgrnumwwlks 30067 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30068 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30069 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30070 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30071 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30072 | In a ` K `-regular graph, ... |
| clwwlk 30075 | The set of closed walks (i... |
| isclwwlk 30076 | Properties of a word to re... |
| clwwlkbp 30077 | Basic properties of a clos... |
| clwwlkgt0 30078 | There is no empty closed w... |
| clwwlksswrd 30079 | Closed walks (represented ... |
| clwwlk1loop 30080 | A closed walk of length 1 ... |
| clwwlkccatlem 30081 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30082 | The concatenation of two w... |
| umgrclwwlkge2 30083 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30084 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30085 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30086 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30087 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30088 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30089 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30090 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30091 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30092 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30093 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30094 | A closed walk as word of l... |
| clwlkclwwlk2 30095 | A closed walk corresponds ... |
| clwlkclwwlkflem 30096 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30097 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30098 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30099 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30100 | ` F ` is a function from t... |
| clwlkclwwlkfo 30101 | ` F ` is a function from t... |
| clwlkclwwlkf1 30102 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30103 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30104 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30105 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30106 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30107 | Cyclically shifting a clos... |
| clwwisshclwwsn 30108 | Cyclically shifting a clos... |
| erclwwlkrel 30109 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30110 | Two classes are equivalent... |
| erclwwlkeqlen 30111 | If two classes are equival... |
| erclwwlkref 30112 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30113 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30114 | ` .~ ` is a transitive rel... |
| erclwwlk 30115 | ` .~ ` is an equivalence r... |
| clwwlkn 30118 | The set of closed walks of... |
| isclwwlkn 30119 | A word over the set of ver... |
| clwwlkn0 30120 | There is no closed walk of... |
| clwwlkneq0 30121 | Sufficient conditions for ... |
| clwwlkclwwlkn 30122 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30123 | The closed walks of a fixe... |
| clwwlknlen 30124 | The length of a word repre... |
| clwwlknnn 30125 | The length of a closed wal... |
| clwwlknwrd 30126 | A closed walk of a fixed l... |
| clwwlknbp 30127 | Basic properties of a clos... |
| isclwwlknx 30128 | Characterization of a word... |
| clwwlknp 30129 | Properties of a set being ... |
| clwwlknwwlksn 30130 | A word representing a clos... |
| clwwlknlbonbgr1 30131 | The last but one vertex in... |
| clwwlkinwwlk 30132 | If the initial vertex of a... |
| clwwlkn1 30133 | A closed walk of length 1 ... |
| loopclwwlkn1b 30134 | The singleton word consist... |
| clwwlkn1loopb 30135 | A word represents a closed... |
| clwwlkn2 30136 | A closed walk of length 2 ... |
| clwwlknfi 30137 | If there is only a finite ... |
| clwwlkel 30138 | Obtaining a closed walk (a... |
| clwwlkf 30139 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30140 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30141 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30142 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30143 | F is a 1-1 onto function, ... |
| clwwlken 30144 | The set of closed walks of... |
| clwwlknwwlkncl 30145 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30146 | A nonempty word over verti... |
| clwwlknwwlksnb 30147 | A word over vertices repre... |
| clwwlkext2edg 30148 | If a word concatenated wit... |
| wwlksext2clwwlk 30149 | If a word represents a wal... |
| wwlksubclwwlk 30150 | Any prefix of a word repre... |
| clwwnisshclwwsn 30151 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30152 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30153 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30154 | The set of cyclical shifts... |
| clwwlknccat 30155 | The concatenation of two w... |
| umgr2cwwk2dif 30156 | If a word represents a clo... |
| umgr2cwwkdifex 30157 | If a word represents a clo... |
| erclwwlknrel 30158 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30159 | Two classes are equivalent... |
| erclwwlkneqlen 30160 | If two classes are equival... |
| erclwwlknref 30161 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30162 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30163 | ` .~ ` is a transitive rel... |
| erclwwlkn 30164 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30165 | The quotient set of the se... |
| hashclwwlkn0 30166 | The number of closed walks... |
| eclclwwlkn1 30167 | An equivalence class accor... |
| eleclclwwlkn 30168 | A member of an equivalence... |
| hashecclwwlkn1 30169 | The size of every equivale... |
| umgrhashecclwwlk 30170 | The size of every equivale... |
| fusgrhashclwwlkn 30171 | The size of the set of clo... |
| clwwlkndivn 30172 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30173 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30174 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30175 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30176 | There is a one-to-one onto... |
| clwlkssizeeq 30177 | The size of the set of clo... |
| clwlksndivn 30178 | The size of the set of clo... |
| clwwlknonmpo 30181 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30182 | The set of closed walks on... |
| isclwwlknon 30183 | A word over the set of ver... |
| clwwlk0on0 30184 | There is no word over the ... |
| clwwlknon0 30185 | Sufficient conditions for ... |
| clwwlknonfin 30186 | In a finite graph ` G ` , ... |
| clwwlknonel 30187 | Characterization of a word... |
| clwwlknonccat 30188 | The concatenation of two w... |
| clwwlknon1 30189 | The set of closed walks on... |
| clwwlknon1loop 30190 | If there is a loop at vert... |
| clwwlknon1nloop 30191 | If there is no loop at ver... |
| clwwlknon1sn 30192 | The set of (closed) walks ... |
| clwwlknon1le1 30193 | There is at most one (clos... |
| clwwlknon2 30194 | The set of closed walks on... |
| clwwlknon2x 30195 | The set of closed walks on... |
| s2elclwwlknon2 30196 | Sufficient conditions of a... |
| clwwlknon2num 30197 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30198 | A word over vertices repre... |
| clwwlknonex2lem1 30199 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30200 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30201 | Extending a closed walk ` ... |
| clwwlknonex2e 30202 | Extending a closed walk ` ... |
| clwwlknondisj 30203 | The sets of closed walks o... |
| clwwlknun 30204 | The set of closed walks of... |
| clwwlkvbij 30205 | There is a bijection betwe... |
| 0ewlk 30206 | The empty set (empty seque... |
| 1ewlk 30207 | A sequence of 1 edge is an... |
| 0wlk 30208 | A pair of an empty set (of... |
| is0wlk 30209 | A pair of an empty set (of... |
| 0wlkonlem1 30210 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30211 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30212 | A walk of length 0 from a ... |
| 0wlkons1 30213 | A walk of length 0 from a ... |
| 0trl 30214 | A pair of an empty set (of... |
| is0trl 30215 | A pair of an empty set (of... |
| 0trlon 30216 | A trail of length 0 from a... |
| 0pth 30217 | A pair of an empty set (of... |
| 0spth 30218 | A pair of an empty set (of... |
| 0pthon 30219 | A path of length 0 from a ... |
| 0pthon1 30220 | A path of length 0 from a ... |
| 0pthonv 30221 | For each vertex there is a... |
| 0clwlk 30222 | A pair of an empty set (of... |
| 0clwlkv 30223 | Any vertex (more precisely... |
| 0clwlk0 30224 | There is no closed walk in... |
| 0crct 30225 | A pair of an empty set (of... |
| 0cycl 30226 | A pair of an empty set (of... |
| 1pthdlem1 30227 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30228 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30229 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30230 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30231 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30232 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30233 | In a graph with two vertic... |
| 1trld 30234 | In a graph with two vertic... |
| 1pthd 30235 | In a graph with two vertic... |
| 1pthond 30236 | In a graph with two vertic... |
| upgr1wlkdlem1 30237 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30238 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30239 | In a pseudograph with two ... |
| upgr1trld 30240 | In a pseudograph with two ... |
| upgr1pthd 30241 | In a pseudograph with two ... |
| upgr1pthond 30242 | In a pseudograph with two ... |
| lppthon 30243 | A loop (which is an edge a... |
| lp1cycl 30244 | A loop (which is an edge a... |
| 1pthon2v 30245 | For each pair of adjacent ... |
| 1pthon2ve 30246 | For each pair of adjacent ... |
| wlk2v2elem1 30247 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30248 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30249 | In a graph with two vertic... |
| ntrl2v2e 30250 | A walk which is not a trai... |
| 3wlkdlem1 30251 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30252 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30253 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30254 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30255 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30256 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30257 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30258 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30259 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30260 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30261 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30262 | Construction of a walk fro... |
| 3wlkond 30263 | A walk of length 3 from on... |
| 3trld 30264 | Construction of a trail fr... |
| 3trlond 30265 | A trail of length 3 from o... |
| 3pthd 30266 | A path of length 3 from on... |
| 3pthond 30267 | A path of length 3 from on... |
| 3spthd 30268 | A simple path of length 3 ... |
| 3spthond 30269 | A simple path of length 3 ... |
| 3cycld 30270 | Construction of a 3-cycle ... |
| 3cyclpd 30271 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30272 | If there is a cycle of len... |
| uhgr3cyclexlem 30273 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30274 | If there are three differe... |
| umgr3cyclex 30275 | If there are three (differ... |
| umgr3v3e3cycl 30276 | If and only if there is a ... |
| upgr4cycl4dv4e 30277 | If there is a cycle of len... |
| dfconngr1 30280 | Alternative definition of ... |
| isconngr 30281 | The property of being a co... |
| isconngr1 30282 | The property of being a co... |
| cusconngr 30283 | A complete hypergraph is c... |
| 0conngr 30284 | A graph without vertices i... |
| 0vconngr 30285 | A graph without vertices i... |
| 1conngr 30286 | A graph with (at most) one... |
| conngrv2edg 30287 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30288 | A vertex in a connected mu... |
| releupth 30291 | The set ` ( EulerPaths `` ... |
| eupths 30292 | The Eulerian paths on the ... |
| iseupth 30293 | The property " ` <. F , P ... |
| iseupthf1o 30294 | The property " ` <. F , P ... |
| eupthi 30295 | Properties of an Eulerian ... |
| eupthf1o 30296 | The ` F ` function in an E... |
| eupthfi 30297 | Any graph with an Eulerian... |
| eupthseg 30298 | The ` N ` -th edge in an e... |
| upgriseupth 30299 | The property " ` <. F , P ... |
| upgreupthi 30300 | Properties of an Eulerian ... |
| upgreupthseg 30301 | The ` N ` -th edge in an e... |
| eupthcl 30302 | An Eulerian path has lengt... |
| eupthistrl 30303 | An Eulerian path is a trai... |
| eupthiswlk 30304 | An Eulerian path is a walk... |
| eupthpf 30305 | The ` P ` function in an E... |
| eupth0 30306 | There is an Eulerian path ... |
| eupthres 30307 | The restriction ` <. H , Q... |
| eupthp1 30308 | Append one path segment to... |
| eupth2eucrct 30309 | Append one path segment to... |
| eupth2lem1 30310 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30311 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30312 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30313 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30314 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30315 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30316 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30317 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30318 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30319 | The effect on vertex degre... |
| eupth2lem3lem1 30320 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30321 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30322 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30323 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30324 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30325 | Formerly part of proof of ... |
| eupth2lem3lem7 30326 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30327 | Formerly part of proof of ... |
| eupth2lem3 30328 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30329 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30330 | Lemma for ~ eupth2 (induct... |
| eupth2 30331 | The only vertices of odd d... |
| eulerpathpr 30332 | A graph with an Eulerian p... |
| eulerpath 30333 | A pseudograph with an Eule... |
| eulercrct 30334 | A pseudograph with an Eule... |
| eucrctshift 30335 | Cyclically shifting the in... |
| eucrct2eupth1 30336 | Removing one edge ` ( I ``... |
| eucrct2eupth 30337 | Removing one edge ` ( I ``... |
| konigsbergvtx 30338 | The set of vertices of the... |
| konigsbergiedg 30339 | The indexed edges of the K... |
| konigsbergiedgw 30340 | The indexed edges of the K... |
| konigsbergssiedgwpr 30341 | Each subset of the indexed... |
| konigsbergssiedgw 30342 | Each subset of the indexed... |
| konigsbergumgr 30343 | The Königsberg graph ... |
| konigsberglem1 30344 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30345 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30346 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30347 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30348 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30349 | The Königsberg Bridge... |
| isfrgr 30352 | The property of being a fr... |
| frgrusgr 30353 | A friendship graph is a si... |
| frgr0v 30354 | Any null graph (set with n... |
| frgr0vb 30355 | Any null graph (without ve... |
| frgruhgr0v 30356 | Any null graph (without ve... |
| frgr0 30357 | The null graph (graph with... |
| frcond1 30358 | The friendship condition: ... |
| frcond2 30359 | The friendship condition: ... |
| frgreu 30360 | Variant of ~ frcond2 : An... |
| frcond3 30361 | The friendship condition, ... |
| frcond4 30362 | The friendship condition, ... |
| frgr1v 30363 | Any graph with (at most) o... |
| nfrgr2v 30364 | Any graph with two (differ... |
| frgr3vlem1 30365 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30366 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30367 | Any graph with three verti... |
| 1vwmgr 30368 | Every graph with one verte... |
| 3vfriswmgrlem 30369 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30370 | Every friendship graph wit... |
| 1to2vfriswmgr 30371 | Every friendship graph wit... |
| 1to3vfriswmgr 30372 | Every friendship graph wit... |
| 1to3vfriendship 30373 | The friendship theorem for... |
| 2pthfrgrrn 30374 | Between any two (different... |
| 2pthfrgrrn2 30375 | Between any two (different... |
| 2pthfrgr 30376 | Between any two (different... |
| 3cyclfrgrrn1 30377 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30378 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30379 | Every vertex in a friendsh... |
| 3cyclfrgr 30380 | Every vertex in a friendsh... |
| 4cycl2v2nb 30381 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30382 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30383 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30384 | A graph with a 4-cycle is ... |
| frgrnbnb 30385 | If two neighbors ` U ` and... |
| frgrconngr 30386 | A friendship graph is conn... |
| vdgn0frgrv2 30387 | A vertex in a friendship g... |
| vdgn1frgrv2 30388 | Any vertex in a friendship... |
| vdgn1frgrv3 30389 | Any vertex in a friendship... |
| vdgfrgrgt2 30390 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30391 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30392 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30393 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30394 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30395 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30396 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30397 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30398 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30399 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30400 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30401 | In a friendship graph, two... |
| frgrwopreglem4a 30402 | In a friendship graph any ... |
| frgrwopreglem5a 30403 | If a friendship graph has ... |
| frgrwopreglem1 30404 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30405 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30406 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30407 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30408 | According to statement 5 i... |
| frgrwopregbsn 30409 | According to statement 5 i... |
| frgrwopreg1 30410 | According to statement 5 i... |
| frgrwopreg2 30411 | According to statement 5 i... |
| frgrwopreglem5lem 30412 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30413 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30414 | Alternate direct proof of ... |
| frgrwopreg 30415 | In a friendship graph ther... |
| frgrregorufr0 30416 | In a friendship graph ther... |
| frgrregorufr 30417 | If there is a vertex havin... |
| frgrregorufrg 30418 | If there is a vertex havin... |
| frgr2wwlkeu 30419 | For two different vertices... |
| frgr2wwlkn0 30420 | In a friendship graph, the... |
| frgr2wwlk1 30421 | In a friendship graph, the... |
| frgr2wsp1 30422 | In a friendship graph, the... |
| frgr2wwlkeqm 30423 | If there is a (simple) pat... |
| frgrhash2wsp 30424 | The number of simple paths... |
| fusgreg2wsplem 30425 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30426 | The set of paths of length... |
| fusgreghash2wspv 30427 | According to statement 7 i... |
| fusgreg2wsp 30428 | In a finite simple graph, ... |
| 2wspmdisj 30429 | The sets of paths of lengt... |
| fusgreghash2wsp 30430 | In a finite k-regular grap... |
| frrusgrord0lem 30431 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30432 | If a nonempty finite frien... |
| frrusgrord 30433 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30434 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30435 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30436 | If the initial vertex of a... |
| clwwnonrepclwwnon 30437 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30438 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30439 | Value of operation ` C ` ,... |
| 2clwwlk2 30440 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30441 | Characterization of an ele... |
| 2clwwlk2clwwlk 30442 | An element of the value of... |
| numclwwlk1lem2foalem 30443 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30444 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30445 | Characterization of an ele... |
| numclwwlk1lem2foa 30446 | Going forth and back from ... |
| numclwwlk1lem2f 30447 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30448 | Value of the function ` T ... |
| numclwwlk1lem2f1 30449 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30450 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30451 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30452 | The set of double loops of... |
| numclwwlk1 30453 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30454 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30455 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30456 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30457 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30458 | The sets of the two repres... |
| wlkl0 30459 | There is exactly one walk ... |
| clwlknon2num 30460 | There are k walks of lengt... |
| numclwlk1lem1 30461 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30462 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30463 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30464 | Value of operation ` H ` ,... |
| numclwwlkovh 30465 | Value of operation ` H ` ,... |
| numclwwlkovq 30466 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30467 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30468 | In a friendship graph, for... |
| numclwlk2lem2f 30469 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30470 | Value of the function ` R ... |
| numclwlk2lem2f1o 30471 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30472 | In a friendship graph, the... |
| numclwwlk2 30473 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30474 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30475 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30476 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30477 | Statement 12 in [Huneke] p... |
| numclwwlk4 30478 | The total number of closed... |
| numclwwlk5lem 30479 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30480 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30481 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30482 | For a prime divisor ` P ` ... |
| numclwwlk7 30483 | Statement 14 in [Huneke] p... |
| numclwwlk8 30484 | The size of the set of clo... |
| frgrreggt1 30485 | If a finite nonempty frien... |
| frgrreg 30486 | If a finite nonempty frien... |
| frgrregord013 30487 | If a finite friendship gra... |
| frgrregord13 30488 | If a nonempty finite frien... |
| frgrogt3nreg 30489 | If a finite friendship gra... |
| friendshipgt3 30490 | The friendship theorem for... |
| friendship 30491 | The friendship theorem: I... |
| conventions 30492 |
H... |
| conventions-labels 30493 |
... |
| conventions-comments 30494 |
... |
| natded 30495 | Here are typical n... |
| ex-natded5.2 30496 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30497 | A more efficient proof of ... |
| ex-natded5.2i 30498 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30499 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30500 | A more efficient proof of ... |
| ex-natded5.3i 30501 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30502 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30503 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30504 | A more efficient proof of ... |
| ex-natded5.8 30505 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30506 | A more efficient proof of ... |
| ex-natded5.13 30507 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30508 | A more efficient proof of ... |
| ex-natded9.20 30509 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30510 | A more efficient proof of ... |
| ex-natded9.26 30511 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30512 | A more efficient proof of ... |
| ex-or 30513 | Example for ~ df-or . Exa... |
| ex-an 30514 | Example for ~ df-an . Exa... |
| ex-dif 30515 | Example for ~ df-dif . Ex... |
| ex-un 30516 | Example for ~ df-un . Exa... |
| ex-in 30517 | Example for ~ df-in . Exa... |
| ex-uni 30518 | Example for ~ df-uni . Ex... |
| ex-ss 30519 | Example for ~ df-ss . Exa... |
| ex-pss 30520 | Example for ~ df-pss . Ex... |
| ex-pw 30521 | Example for ~ df-pw . Exa... |
| ex-pr 30522 | Example for ~ df-pr . (Co... |
| ex-br 30523 | Example for ~ df-br . Exa... |
| ex-opab 30524 | Example for ~ df-opab . E... |
| ex-eprel 30525 | Example for ~ df-eprel . ... |
| ex-id 30526 | Example for ~ df-id . Exa... |
| ex-po 30527 | Example for ~ df-po . Exa... |
| ex-xp 30528 | Example for ~ df-xp . Exa... |
| ex-cnv 30529 | Example for ~ df-cnv . Ex... |
| ex-co 30530 | Example for ~ df-co . Exa... |
| ex-dm 30531 | Example for ~ df-dm . Exa... |
| ex-rn 30532 | Example for ~ df-rn . Exa... |
| ex-res 30533 | Example for ~ df-res . Ex... |
| ex-ima 30534 | Example for ~ df-ima . Ex... |
| ex-fv 30535 | Example for ~ df-fv . Exa... |
| ex-1st 30536 | Example for ~ df-1st . Ex... |
| ex-2nd 30537 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30538 | Example for ~ df-dec , 100... |
| ex-fl 30539 | Example for ~ df-fl . Exa... |
| ex-ceil 30540 | Example for ~ df-ceil . (... |
| ex-mod 30541 | Example for ~ df-mod . (C... |
| ex-exp 30542 | Example for ~ df-exp . (C... |
| ex-fac 30543 | Example for ~ df-fac . (C... |
| ex-bc 30544 | Example for ~ df-bc . (Co... |
| ex-hash 30545 | Example for ~ df-hash . (... |
| ex-sqrt 30546 | Example for ~ df-sqrt . (... |
| ex-abs 30547 | Example for ~ df-abs . (C... |
| ex-dvds 30548 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30549 | Example for ~ df-gcd . (C... |
| ex-lcm 30550 | Example for ~ df-lcm . (C... |
| ex-prmo 30551 | Example for ~ df-prmo : ` ... |
| aevdemo 30552 | Proof illustrating the com... |
| ex-ind-dvds 30553 | Example of a proof by indu... |
| ex-fpar 30554 | Formalized example provide... |
| avril1 30555 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30556 | The law of excluded middle... |
| helloworld 30557 | The classic "Hello world" ... |
| 1p1e2apr1 30558 | One plus one equals two. ... |
| eqid1 30559 | Law of identity (reflexivi... |
| 1div0apr 30560 | Division by zero is forbid... |
| topnfbey 30561 | Nothing seems to be imposs... |
| 9p10ne21 30562 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30563 | 9 + 10 equals 21. This as... |
| nrt2irr 30565 | The ` N ` -th root of 2 is... |
| nowisdomv 30566 | One's wisdom on matters of... |
| isplig 30569 | The predicate "is a planar... |
| ispligb 30570 | The predicate "is a planar... |
| tncp 30571 | In any planar incidence ge... |
| l2p 30572 | For any line in a planar i... |
| lpni 30573 | For any line in a planar i... |
| nsnlplig 30574 | There is no "one-point lin... |
| nsnlpligALT 30575 | Alternate version of ~ nsn... |
| n0lplig 30576 | There is no "empty line" i... |
| n0lpligALT 30577 | Alternate version of ~ n0l... |
| eulplig 30578 | Through two distinct point... |
| pliguhgr 30579 | Any planar incidence geome... |
| dummylink 30580 | Alias for ~ a1ii that may ... |
| id1 30581 | Alias for ~ idALT that may... |
| isgrpo 30590 | The predicate "is a group ... |
| isgrpoi 30591 | Properties that determine ... |
| grpofo 30592 | A group operation maps ont... |
| grpocl 30593 | Closure law for a group op... |
| grpolidinv 30594 | A group has a left identit... |
| grpon0 30595 | The base set of a group is... |
| grpoass 30596 | A group operation is assoc... |
| grpoidinvlem1 30597 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30598 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30599 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30600 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30601 | A group has a left and rig... |
| grpoideu 30602 | The left identity element ... |
| grporndm 30603 | A group's range in terms o... |
| 0ngrp 30604 | The empty set is not a gro... |
| gidval 30605 | The value of the identity ... |
| grpoidval 30606 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30607 | The identity element of a ... |
| grpoidinv2 30608 | A group's properties using... |
| grpolid 30609 | The identity element of a ... |
| grporid 30610 | The identity element of a ... |
| grporcan 30611 | Right cancellation law for... |
| grpoinveu 30612 | The left inverse element o... |
| grpoid 30613 | Two ways of saying that an... |
| grporn 30614 | The range of a group opera... |
| grpoinvfval 30615 | The inverse function of a ... |
| grpoinvval 30616 | The inverse of a group ele... |
| grpoinvcl 30617 | A group element's inverse ... |
| grpoinv 30618 | The properties of a group ... |
| grpolinv 30619 | The left inverse of a grou... |
| grporinv 30620 | The right inverse of a gro... |
| grpoinvid1 30621 | The inverse of a group ele... |
| grpoinvid2 30622 | The inverse of a group ele... |
| grpolcan 30623 | Left cancellation law for ... |
| grpo2inv 30624 | Double inverse law for gro... |
| grpoinvf 30625 | Mapping of the inverse fun... |
| grpoinvop 30626 | The inverse of the group o... |
| grpodivfval 30627 | Group division (or subtrac... |
| grpodivval 30628 | Group division (or subtrac... |
| grpodivinv 30629 | Group division by an inver... |
| grpoinvdiv 30630 | Inverse of a group divisio... |
| grpodivf 30631 | Mapping for group division... |
| grpodivcl 30632 | Closure of group division ... |
| grpodivdiv 30633 | Double group division. (C... |
| grpomuldivass 30634 | Associative-type law for m... |
| grpodivid 30635 | Division of a group member... |
| grponpcan 30636 | Cancellation law for group... |
| isablo 30639 | The predicate "is an Abeli... |
| ablogrpo 30640 | An Abelian group operation... |
| ablocom 30641 | An Abelian group operation... |
| ablo32 30642 | Commutative/associative la... |
| ablo4 30643 | Commutative/associative la... |
| isabloi 30644 | Properties that determine ... |
| ablomuldiv 30645 | Law for group multiplicati... |
| ablodivdiv 30646 | Law for double group divis... |
| ablodivdiv4 30647 | Law for double group divis... |
| ablodiv32 30648 | Swap the second and third ... |
| ablonncan 30649 | Cancellation law for group... |
| ablonnncan1 30650 | Cancellation law for group... |
| vcrel 30653 | The class of all complex v... |
| vciOLD 30654 | Obsolete version of ~ cvsi... |
| vcsm 30655 | Functionality of th scalar... |
| vccl 30656 | Closure of the scalar prod... |
| vcidOLD 30657 | Identity element for the s... |
| vcdi 30658 | Distributive law for the s... |
| vcdir 30659 | Distributive law for the s... |
| vcass 30660 | Associative law for the sc... |
| vc2OLD 30661 | A vector plus itself is tw... |
| vcablo 30662 | Vector addition is an Abel... |
| vcgrp 30663 | Vector addition is a group... |
| vclcan 30664 | Left cancellation law for ... |
| vczcl 30665 | The zero vector is a vecto... |
| vc0rid 30666 | The zero vector is a right... |
| vc0 30667 | Zero times a vector is the... |
| vcz 30668 | Anything times the zero ve... |
| vcm 30669 | Minus 1 times a vector is ... |
| isvclem 30670 | Lemma for ~ isvcOLD . (Co... |
| vcex 30671 | The components of a comple... |
| isvcOLD 30672 | The predicate "is a comple... |
| isvciOLD 30673 | Properties that determine ... |
| cnaddabloOLD 30674 | Obsolete version of ~ cnad... |
| cnidOLD 30675 | Obsolete version of ~ cnad... |
| cncvcOLD 30676 | Obsolete version of ~ cncv... |
| nvss 30686 | Structure of the class of ... |
| nvvcop 30687 | A normed complex vector sp... |
| nvrel 30695 | The class of all normed co... |
| vafval 30696 | Value of the function for ... |
| bafval 30697 | Value of the function for ... |
| smfval 30698 | Value of the function for ... |
| 0vfval 30699 | Value of the function for ... |
| nmcvfval 30700 | Value of the norm function... |
| nvop2 30701 | A normed complex vector sp... |
| nvvop 30702 | The vector space component... |
| isnvlem 30703 | Lemma for ~ isnv . (Contr... |
| nvex 30704 | The components of a normed... |
| isnv 30705 | The predicate "is a normed... |
| isnvi 30706 | Properties that determine ... |
| nvi 30707 | The properties of a normed... |
| nvvc 30708 | The vector space component... |
| nvablo 30709 | The vector addition operat... |
| nvgrp 30710 | The vector addition operat... |
| nvgf 30711 | Mapping for the vector add... |
| nvsf 30712 | Mapping for the scalar mul... |
| nvgcl 30713 | Closure law for the vector... |
| nvcom 30714 | The vector addition (group... |
| nvass 30715 | The vector addition (group... |
| nvadd32 30716 | Commutative/associative la... |
| nvrcan 30717 | Right cancellation law for... |
| nvadd4 30718 | Rearrangement of 4 terms i... |
| nvscl 30719 | Closure law for the scalar... |
| nvsid 30720 | Identity element for the s... |
| nvsass 30721 | Associative law for the sc... |
| nvscom 30722 | Commutative law for the sc... |
| nvdi 30723 | Distributive law for the s... |
| nvdir 30724 | Distributive law for the s... |
| nv2 30725 | A vector plus itself is tw... |
| vsfval 30726 | Value of the function for ... |
| nvzcl 30727 | Closure law for the zero v... |
| nv0rid 30728 | The zero vector is a right... |
| nv0lid 30729 | The zero vector is a left ... |
| nv0 30730 | Zero times a vector is the... |
| nvsz 30731 | Anything times the zero ve... |
| nvinv 30732 | Minus 1 times a vector is ... |
| nvinvfval 30733 | Function for the negative ... |
| nvm 30734 | Vector subtraction in term... |
| nvmval 30735 | Value of vector subtractio... |
| nvmval2 30736 | Value of vector subtractio... |
| nvmfval 30737 | Value of the function for ... |
| nvmf 30738 | Mapping for the vector sub... |
| nvmcl 30739 | Closure law for the vector... |
| nvnnncan1 30740 | Cancellation law for vecto... |
| nvmdi 30741 | Distributive law for scala... |
| nvnegneg 30742 | Double negative of a vecto... |
| nvmul0or 30743 | If a scalar product is zer... |
| nvrinv 30744 | A vector minus itself. (C... |
| nvlinv 30745 | Minus a vector plus itself... |
| nvpncan2 30746 | Cancellation law for vecto... |
| nvpncan 30747 | Cancellation law for vecto... |
| nvaddsub 30748 | Commutative/associative la... |
| nvnpcan 30749 | Cancellation law for a nor... |
| nvaddsub4 30750 | Rearrangement of 4 terms i... |
| nvmeq0 30751 | The difference between two... |
| nvmid 30752 | A vector minus itself is t... |
| nvf 30753 | Mapping for the norm funct... |
| nvcl 30754 | The norm of a normed compl... |
| nvcli 30755 | The norm of a normed compl... |
| nvs 30756 | Proportionality property o... |
| nvsge0 30757 | The norm of a scalar produ... |
| nvm1 30758 | The norm of the negative o... |
| nvdif 30759 | The norm of the difference... |
| nvpi 30760 | The norm of a vector plus ... |
| nvz0 30761 | The norm of a zero vector ... |
| nvz 30762 | The norm of a vector is ze... |
| nvtri 30763 | Triangle inequality for th... |
| nvmtri 30764 | Triangle inequality for th... |
| nvabs 30765 | Norm difference property o... |
| nvge0 30766 | The norm of a normed compl... |
| nvgt0 30767 | A nonzero norm is positive... |
| nv1 30768 | From any nonzero vector, c... |
| nvop 30769 | A complex inner product sp... |
| cnnv 30770 | The set of complex numbers... |
| cnnvg 30771 | The vector addition (group... |
| cnnvba 30772 | The base set of the normed... |
| cnnvs 30773 | The scalar product operati... |
| cnnvnm 30774 | The norm operation of the ... |
| cnnvm 30775 | The vector subtraction ope... |
| elimnv 30776 | Hypothesis elimination lem... |
| elimnvu 30777 | Hypothesis elimination lem... |
| imsval 30778 | Value of the induced metri... |
| imsdval 30779 | Value of the induced metri... |
| imsdval2 30780 | Value of the distance func... |
| nvnd 30781 | The norm of a normed compl... |
| imsdf 30782 | Mapping for the induced me... |
| imsmetlem 30783 | Lemma for ~ imsmet . (Con... |
| imsmet 30784 | The induced metric of a no... |
| imsxmet 30785 | The induced metric of a no... |
| cnims 30786 | The metric induced on the ... |
| vacn 30787 | Vector addition is jointly... |
| nmcvcn 30788 | The norm of a normed compl... |
| nmcnc 30789 | The norm of a normed compl... |
| smcnlem 30790 | Lemma for ~ smcn . (Contr... |
| smcn 30791 | Scalar multiplication is j... |
| vmcn 30792 | Vector subtraction is join... |
| dipfval 30795 | The inner product function... |
| ipval 30796 | Value of the inner product... |
| ipval2lem2 30797 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30798 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30799 | Lemma for ~ ipval3 . (Con... |
| ipval2 30800 | Expansion of the inner pro... |
| 4ipval2 30801 | Four times the inner produ... |
| ipval3 30802 | Expansion of the inner pro... |
| ipidsq 30803 | The inner product of a vec... |
| ipnm 30804 | Norm expressed in terms of... |
| dipcl 30805 | An inner product is a comp... |
| ipf 30806 | Mapping for the inner prod... |
| dipcj 30807 | The complex conjugate of a... |
| ipipcj 30808 | An inner product times its... |
| diporthcom 30809 | Orthogonality (meaning inn... |
| dip0r 30810 | Inner product with a zero ... |
| dip0l 30811 | Inner product with a zero ... |
| ipz 30812 | The inner product of a vec... |
| dipcn 30813 | Inner product is jointly c... |
| sspval 30816 | The set of all subspaces o... |
| isssp 30817 | The predicate "is a subspa... |
| sspid 30818 | A normed complex vector sp... |
| sspnv 30819 | A subspace is a normed com... |
| sspba 30820 | The base set of a subspace... |
| sspg 30821 | Vector addition on a subsp... |
| sspgval 30822 | Vector addition on a subsp... |
| ssps 30823 | Scalar multiplication on a... |
| sspsval 30824 | Scalar multiplication on a... |
| sspmlem 30825 | Lemma for ~ sspm and other... |
| sspmval 30826 | Vector addition on a subsp... |
| sspm 30827 | Vector subtraction on a su... |
| sspz 30828 | The zero vector of a subsp... |
| sspn 30829 | The norm on a subspace is ... |
| sspnval 30830 | The norm on a subspace in ... |
| sspimsval 30831 | The induced metric on a su... |
| sspims 30832 | The induced metric on a su... |
| lnoval 30845 | The set of linear operator... |
| islno 30846 | The predicate "is a linear... |
| lnolin 30847 | Basic linearity property o... |
| lnof 30848 | A linear operator is a map... |
| lno0 30849 | The value of a linear oper... |
| lnocoi 30850 | The composition of two lin... |
| lnoadd 30851 | Addition property of a lin... |
| lnosub 30852 | Subtraction property of a ... |
| lnomul 30853 | Scalar multiplication prop... |
| nvo00 30854 | Two ways to express a zero... |
| nmoofval 30855 | The operator norm function... |
| nmooval 30856 | The operator norm function... |
| nmosetre 30857 | The set in the supremum of... |
| nmosetn0 30858 | The set in the supremum of... |
| nmoxr 30859 | The norm of an operator is... |
| nmooge0 30860 | The norm of an operator is... |
| nmorepnf 30861 | The norm of an operator is... |
| nmoreltpnf 30862 | The norm of any operator i... |
| nmogtmnf 30863 | The norm of an operator is... |
| nmoolb 30864 | A lower bound for an opera... |
| nmoubi 30865 | An upper bound for an oper... |
| nmoub3i 30866 | An upper bound for an oper... |
| nmoub2i 30867 | An upper bound for an oper... |
| nmobndi 30868 | Two ways to express that a... |
| nmounbi 30869 | Two ways two express that ... |
| nmounbseqi 30870 | An unbounded operator dete... |
| nmounbseqiALT 30871 | Alternate shorter proof of... |
| nmobndseqi 30872 | A bounded sequence determi... |
| nmobndseqiALT 30873 | Alternate shorter proof of... |
| bloval 30874 | The class of bounded linea... |
| isblo 30875 | The predicate "is a bounde... |
| isblo2 30876 | The predicate "is a bounde... |
| bloln 30877 | A bounded operator is a li... |
| blof 30878 | A bounded operator is an o... |
| nmblore 30879 | The norm of a bounded oper... |
| 0ofval 30880 | The zero operator between ... |
| 0oval 30881 | Value of the zero operator... |
| 0oo 30882 | The zero operator is an op... |
| 0lno 30883 | The zero operator is linea... |
| nmoo0 30884 | The operator norm of the z... |
| 0blo 30885 | The zero operator is a bou... |
| nmlno0lem 30886 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30887 | The norm of a linear opera... |
| nmlno0 30888 | The norm of a linear opera... |
| nmlnoubi 30889 | An upper bound for the ope... |
| nmlnogt0 30890 | The norm of a nonzero line... |
| lnon0 30891 | The domain of a nonzero li... |
| nmblolbii 30892 | A lower bound for the norm... |
| nmblolbi 30893 | A lower bound for the norm... |
| isblo3i 30894 | The predicate "is a bounde... |
| blo3i 30895 | Properties that determine ... |
| blometi 30896 | Upper bound for the distan... |
| blocnilem 30897 | Lemma for ~ blocni and ~ l... |
| blocni 30898 | A linear operator is conti... |
| lnocni 30899 | If a linear operator is co... |
| blocn 30900 | A linear operator is conti... |
| blocn2 30901 | A bounded linear operator ... |
| ajfval 30902 | The adjoint function. (Co... |
| hmoval 30903 | The set of Hermitian (self... |
| ishmo 30904 | The predicate "is a hermit... |
| phnv 30907 | Every complex inner produc... |
| phrel 30908 | The class of all complex i... |
| phnvi 30909 | Every complex inner produc... |
| isphg 30910 | The predicate "is a comple... |
| phop 30911 | A complex inner product sp... |
| cncph 30912 | The set of complex numbers... |
| elimph 30913 | Hypothesis elimination lem... |
| elimphu 30914 | Hypothesis elimination lem... |
| isph 30915 | The predicate "is an inner... |
| phpar2 30916 | The parallelogram law for ... |
| phpar 30917 | The parallelogram law for ... |
| ip0i 30918 | A slight variant of Equati... |
| ip1ilem 30919 | Lemma for ~ ip1i . (Contr... |
| ip1i 30920 | Equation 6.47 of [Ponnusam... |
| ip2i 30921 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30922 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30923 | Distributive law for inner... |
| ipasslem1 30924 | Lemma for ~ ipassi . Show... |
| ipasslem2 30925 | Lemma for ~ ipassi . Show... |
| ipasslem3 30926 | Lemma for ~ ipassi . Show... |
| ipasslem4 30927 | Lemma for ~ ipassi . Show... |
| ipasslem5 30928 | Lemma for ~ ipassi . Show... |
| ipasslem7 30929 | Lemma for ~ ipassi . Show... |
| ipasslem8 30930 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30931 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30932 | Lemma for ~ ipassi . Show... |
| ipasslem11 30933 | Lemma for ~ ipassi . Show... |
| ipassi 30934 | Associative law for inner ... |
| dipdir 30935 | Distributive law for inner... |
| dipdi 30936 | Distributive law for inner... |
| ip2dii 30937 | Inner product of two sums.... |
| dipass 30938 | Associative law for inner ... |
| dipassr 30939 | "Associative" law for seco... |
| dipassr2 30940 | "Associative" law for inne... |
| dipsubdir 30941 | Distributive law for inner... |
| dipsubdi 30942 | Distributive law for inner... |
| pythi 30943 | The Pythagorean theorem fo... |
| siilem1 30944 | Lemma for ~ sii . (Contri... |
| siilem2 30945 | Lemma for ~ sii . (Contri... |
| siii 30946 | Inference from ~ sii . (C... |
| sii 30947 | Obsolete version of ~ ipca... |
| ipblnfi 30948 | A function ` F ` generated... |
| ip2eqi 30949 | Two vectors are equal iff ... |
| phoeqi 30950 | A condition implying that ... |
| ajmoi 30951 | Every operator has at most... |
| ajfuni 30952 | The adjoint function is a ... |
| ajfun 30953 | The adjoint function is a ... |
| ajval 30954 | Value of the adjoint funct... |
| iscbn 30957 | A complex Banach space is ... |
| cbncms 30958 | The induced metric on comp... |
| bnnv 30959 | Every complex Banach space... |
| bnrel 30960 | The class of all complex B... |
| bnsscmcl 30961 | A subspace of a Banach spa... |
| cnbn 30962 | The set of complex numbers... |
| ubthlem1 30963 | Lemma for ~ ubth . The fu... |
| ubthlem2 30964 | Lemma for ~ ubth . Given ... |
| ubthlem3 30965 | Lemma for ~ ubth . Prove ... |
| ubth 30966 | Uniform Boundedness Theore... |
| minvecolem1 30967 | Lemma for ~ minveco . The... |
| minvecolem2 30968 | Lemma for ~ minveco . Any... |
| minvecolem3 30969 | Lemma for ~ minveco . The... |
| minvecolem4a 30970 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30971 | Lemma for ~ minveco . The... |
| minvecolem4c 30972 | Lemma for ~ minveco . The... |
| minvecolem4 30973 | Lemma for ~ minveco . The... |
| minvecolem5 30974 | Lemma for ~ minveco . Dis... |
| minvecolem6 30975 | Lemma for ~ minveco . Any... |
| minvecolem7 30976 | Lemma for ~ minveco . Sin... |
| minveco 30977 | Minimizing vector theorem,... |
| ishlo 30980 | The predicate "is a comple... |
| hlobn 30981 | Every complex Hilbert spac... |
| hlph 30982 | Every complex Hilbert spac... |
| hlrel 30983 | The class of all complex H... |
| hlnv 30984 | Every complex Hilbert spac... |
| hlnvi 30985 | Every complex Hilbert spac... |
| hlvc 30986 | Every complex Hilbert spac... |
| hlcmet 30987 | The induced metric on a co... |
| hlmet 30988 | The induced metric on a co... |
| hlpar2 30989 | The parallelogram law sati... |
| hlpar 30990 | The parallelogram law sati... |
| hlex 30991 | The base set of a Hilbert ... |
| hladdf 30992 | Mapping for Hilbert space ... |
| hlcom 30993 | Hilbert space vector addit... |
| hlass 30994 | Hilbert space vector addit... |
| hl0cl 30995 | The Hilbert space zero vec... |
| hladdid 30996 | Hilbert space addition wit... |
| hlmulf 30997 | Mapping for Hilbert space ... |
| hlmulid 30998 | Hilbert space scalar multi... |
| hlmulass 30999 | Hilbert space scalar multi... |
| hldi 31000 | Hilbert space scalar multi... |
| hldir 31001 | Hilbert space scalar multi... |
| hlmul0 31002 | Hilbert space scalar multi... |
| hlipf 31003 | Mapping for Hilbert space ... |
| hlipcj 31004 | Conjugate law for Hilbert ... |
| hlipdir 31005 | Distributive law for Hilbe... |
| hlipass 31006 | Associative law for Hilber... |
| hlipgt0 31007 | The inner product of a Hil... |
| hlcompl 31008 | Completeness of a Hilbert ... |
| cnchl 31009 | The set of complex numbers... |
| htthlem 31010 | Lemma for ~ htth . The co... |
| htth 31011 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 31067 | The group (addition) opera... |
| h2hsm 31068 | The scalar product operati... |
| h2hnm 31069 | The norm function of Hilbe... |
| h2hvs 31070 | The vector subtraction ope... |
| h2hmetdval 31071 | Value of the distance func... |
| h2hcau 31072 | The Cauchy sequences of Hi... |
| h2hlm 31073 | The limit sequences of Hil... |
| axhilex-zf 31074 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31075 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31076 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31077 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31078 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31079 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31080 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31081 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31082 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31083 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31084 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31085 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31086 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31087 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31088 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31089 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31090 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31091 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31104 | The Hilbert space scalar p... |
| hvaddcl 31105 | Closure of vector addition... |
| hvmulcl 31106 | Closure of scalar multipli... |
| hvmulcli 31107 | Closure inference for scal... |
| hvsubf 31108 | Mapping domain and codomai... |
| hvsubval 31109 | Value of vector subtractio... |
| hvsubcl 31110 | Closure of vector subtract... |
| hvaddcli 31111 | Closure of vector addition... |
| hvcomi 31112 | Commutation of vector addi... |
| hvsubvali 31113 | Value of vector subtractio... |
| hvsubcli 31114 | Closure of vector subtract... |
| ifhvhv0 31115 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31116 | Addition with the zero vec... |
| hvmul0 31117 | Scalar multiplication with... |
| hvmul0or 31118 | If a scalar product is zer... |
| hvsubid 31119 | Subtraction of a vector fr... |
| hvnegid 31120 | Addition of negative of a ... |
| hv2neg 31121 | Two ways to express the ne... |
| hvaddlidi 31122 | Addition with the zero vec... |
| hvnegidi 31123 | Addition of negative of a ... |
| hv2negi 31124 | Two ways to express the ne... |
| hvm1neg 31125 | Convert minus one times a ... |
| hvaddsubval 31126 | Value of vector addition i... |
| hvadd32 31127 | Commutative/associative la... |
| hvadd12 31128 | Commutative/associative la... |
| hvadd4 31129 | Hilbert vector space addit... |
| hvsub4 31130 | Hilbert vector space addit... |
| hvaddsub12 31131 | Commutative/associative la... |
| hvpncan 31132 | Addition/subtraction cance... |
| hvpncan2 31133 | Addition/subtraction cance... |
| hvaddsubass 31134 | Associativity of sum and d... |
| hvpncan3 31135 | Subtraction and addition o... |
| hvmulcom 31136 | Scalar multiplication comm... |
| hvsubass 31137 | Hilbert vector space assoc... |
| hvsub32 31138 | Hilbert vector space commu... |
| hvmulassi 31139 | Scalar multiplication asso... |
| hvmulcomi 31140 | Scalar multiplication comm... |
| hvmul2negi 31141 | Double negative in scalar ... |
| hvsubdistr1 31142 | Scalar multiplication dist... |
| hvsubdistr2 31143 | Scalar multiplication dist... |
| hvdistr1i 31144 | Scalar multiplication dist... |
| hvsubdistr1i 31145 | Scalar multiplication dist... |
| hvassi 31146 | Hilbert vector space assoc... |
| hvadd32i 31147 | Hilbert vector space commu... |
| hvsubassi 31148 | Hilbert vector space assoc... |
| hvsub32i 31149 | Hilbert vector space commu... |
| hvadd12i 31150 | Hilbert vector space commu... |
| hvadd4i 31151 | Hilbert vector space addit... |
| hvsubsub4i 31152 | Hilbert vector space addit... |
| hvsubsub4 31153 | Hilbert vector space addit... |
| hv2times 31154 | Two times a vector. (Cont... |
| hvnegdii 31155 | Distribution of negative o... |
| hvsubeq0i 31156 | If the difference between ... |
| hvsubcan2i 31157 | Vector cancellation law. ... |
| hvaddcani 31158 | Cancellation law for vecto... |
| hvsubaddi 31159 | Relationship between vecto... |
| hvnegdi 31160 | Distribution of negative o... |
| hvsubeq0 31161 | If the difference between ... |
| hvaddeq0 31162 | If the sum of two vectors ... |
| hvaddcan 31163 | Cancellation law for vecto... |
| hvaddcan2 31164 | Cancellation law for vecto... |
| hvmulcan 31165 | Cancellation law for scala... |
| hvmulcan2 31166 | Cancellation law for scala... |
| hvsubcan 31167 | Cancellation law for vecto... |
| hvsubcan2 31168 | Cancellation law for vecto... |
| hvsub0 31169 | Subtraction of a zero vect... |
| hvsubadd 31170 | Relationship between vecto... |
| hvaddsub4 31171 | Hilbert vector space addit... |
| hicl 31173 | Closure of inner product. ... |
| hicli 31174 | Closure inference for inne... |
| his5 31179 | Associative law for inner ... |
| his52 31180 | Associative law for inner ... |
| his35 31181 | Move scalar multiplication... |
| his35i 31182 | Move scalar multiplication... |
| his7 31183 | Distributive law for inner... |
| hiassdi 31184 | Distributive/associative l... |
| his2sub 31185 | Distributive law for inner... |
| his2sub2 31186 | Distributive law for inner... |
| hire 31187 | A necessary and sufficient... |
| hiidrcl 31188 | Real closure of inner prod... |
| hi01 31189 | Inner product with the 0 v... |
| hi02 31190 | Inner product with the 0 v... |
| hiidge0 31191 | Inner product with self is... |
| his6 31192 | Zero inner product with se... |
| his1i 31193 | Conjugate law for inner pr... |
| abshicom 31194 | Commuted inner products ha... |
| hial0 31195 | A vector whose inner produ... |
| hial02 31196 | A vector whose inner produ... |
| hisubcomi 31197 | Two vector subtractions si... |
| hi2eq 31198 | Lemma used to prove equali... |
| hial2eq 31199 | Two vectors whose inner pr... |
| hial2eq2 31200 | Two vectors whose inner pr... |
| orthcom 31201 | Orthogonality commutes. (... |
| normlem0 31202 | Lemma used to derive prope... |
| normlem1 31203 | Lemma used to derive prope... |
| normlem2 31204 | Lemma used to derive prope... |
| normlem3 31205 | Lemma used to derive prope... |
| normlem4 31206 | Lemma used to derive prope... |
| normlem5 31207 | Lemma used to derive prope... |
| normlem6 31208 | Lemma used to derive prope... |
| normlem7 31209 | Lemma used to derive prope... |
| normlem8 31210 | Lemma used to derive prope... |
| normlem9 31211 | Lemma used to derive prope... |
| normlem7tALT 31212 | Lemma used to derive prope... |
| bcseqi 31213 | Equality case of Bunjakova... |
| normlem9at 31214 | Lemma used to derive prope... |
| dfhnorm2 31215 | Alternate definition of th... |
| normf 31216 | The norm function maps fro... |
| normval 31217 | The value of the norm of a... |
| normcl 31218 | Real closure of the norm o... |
| normge0 31219 | The norm of a vector is no... |
| normgt0 31220 | The norm of nonzero vector... |
| norm0 31221 | The norm of a zero vector.... |
| norm-i 31222 | Theorem 3.3(i) of [Beran] ... |
| normne0 31223 | A norm is nonzero iff its ... |
| normcli 31224 | Real closure of the norm o... |
| normsqi 31225 | The square of a norm. (Co... |
| norm-i-i 31226 | Theorem 3.3(i) of [Beran] ... |
| normsq 31227 | The square of a norm. (Co... |
| normsub0i 31228 | Two vectors are equal iff ... |
| normsub0 31229 | Two vectors are equal iff ... |
| norm-ii-i 31230 | Triangle inequality for no... |
| norm-ii 31231 | Triangle inequality for no... |
| norm-iii-i 31232 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31233 | Theorem 3.3(iii) of [Beran... |
| normsubi 31234 | Negative doesn't change th... |
| normpythi 31235 | Analogy to Pythagorean the... |
| normsub 31236 | Swapping order of subtract... |
| normneg 31237 | The norm of a vector equal... |
| normpyth 31238 | Analogy to Pythagorean the... |
| normpyc 31239 | Corollary to Pythagorean t... |
| norm3difi 31240 | Norm of differences around... |
| norm3adifii 31241 | Norm of differences around... |
| norm3lem 31242 | Lemma involving norm of di... |
| norm3dif 31243 | Norm of differences around... |
| norm3dif2 31244 | Norm of differences around... |
| norm3lemt 31245 | Lemma involving norm of di... |
| norm3adifi 31246 | Norm of differences around... |
| normpari 31247 | Parallelogram law for norm... |
| normpar 31248 | Parallelogram law for norm... |
| normpar2i 31249 | Corollary of parallelogram... |
| polid2i 31250 | Generalized polarization i... |
| polidi 31251 | Polarization identity. Re... |
| polid 31252 | Polarization identity. Re... |
| hilablo 31253 | Hilbert space vector addit... |
| hilid 31254 | The group identity element... |
| hilvc 31255 | Hilbert space is a complex... |
| hilnormi 31256 | Hilbert space norm in term... |
| hilhhi 31257 | Deduce the structure of Hi... |
| hhnv 31258 | Hilbert space is a normed ... |
| hhva 31259 | The group (addition) opera... |
| hhba 31260 | The base set of Hilbert sp... |
| hh0v 31261 | The zero vector of Hilbert... |
| hhsm 31262 | The scalar product operati... |
| hhvs 31263 | The vector subtraction ope... |
| hhnm 31264 | The norm function of Hilbe... |
| hhims 31265 | The induced metric of Hilb... |
| hhims2 31266 | Hilbert space distance met... |
| hhmet 31267 | The induced metric of Hilb... |
| hhxmet 31268 | The induced metric of Hilb... |
| hhmetdval 31269 | Value of the distance func... |
| hhip 31270 | The inner product operatio... |
| hhph 31271 | The Hilbert space of the H... |
| bcsiALT 31272 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31273 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31274 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31275 | Corollary of the Bunjakova... |
| bcs3 31276 | Corollary of the Bunjakova... |
| hcau 31277 | Member of the set of Cauch... |
| hcauseq 31278 | A Cauchy sequences on a Hi... |
| hcaucvg 31279 | A Cauchy sequence on a Hil... |
| seq1hcau 31280 | A sequence on a Hilbert sp... |
| hlimi 31281 | Express the predicate: Th... |
| hlimseqi 31282 | A sequence with a limit on... |
| hlimveci 31283 | Closure of the limit of a ... |
| hlimconvi 31284 | Convergence of a sequence ... |
| hlim2 31285 | The limit of a sequence on... |
| hlimadd 31286 | Limit of the sum of two se... |
| hilmet 31287 | The Hilbert space norm det... |
| hilxmet 31288 | The Hilbert space norm det... |
| hilmetdval 31289 | Value of the distance func... |
| hilims 31290 | Hilbert space distance met... |
| hhcau 31291 | The Cauchy sequences of Hi... |
| hhlm 31292 | The limit sequences of Hil... |
| hhcmpl 31293 | Lemma used for derivation ... |
| hilcompl 31294 | Lemma used for derivation ... |
| hhcms 31296 | The Hilbert space induced ... |
| hhhl 31297 | The Hilbert space structur... |
| hilcms 31298 | The Hilbert space norm det... |
| hilhl 31299 | The Hilbert space of the H... |
| issh 31301 | Subspace ` H ` of a Hilber... |
| issh2 31302 | Subspace ` H ` of a Hilber... |
| shss 31303 | A subspace is a subset of ... |
| shel 31304 | A member of a subspace of ... |
| shex 31305 | The set of subspaces of a ... |
| shssii 31306 | A closed subspace of a Hil... |
| sheli 31307 | A member of a subspace of ... |
| shelii 31308 | A member of a subspace of ... |
| sh0 31309 | The zero vector belongs to... |
| shaddcl 31310 | Closure of vector addition... |
| shmulcl 31311 | Closure of vector scalar m... |
| issh3 31312 | Subspace ` H ` of a Hilber... |
| shsubcl 31313 | Closure of vector subtract... |
| isch 31315 | Closed subspace ` H ` of a... |
| isch2 31316 | Closed subspace ` H ` of a... |
| chsh 31317 | A closed subspace is a sub... |
| chsssh 31318 | Closed subspaces are subsp... |
| chex 31319 | The set of closed subspace... |
| chshii 31320 | A closed subspace is a sub... |
| ch0 31321 | The zero vector belongs to... |
| chss 31322 | A closed subspace of a Hil... |
| chel 31323 | A member of a closed subsp... |
| chssii 31324 | A closed subspace of a Hil... |
| cheli 31325 | A member of a closed subsp... |
| chelii 31326 | A member of a closed subsp... |
| chlimi 31327 | The limit property of a cl... |
| hlim0 31328 | The zero sequence in Hilbe... |
| hlimcaui 31329 | If a sequence in Hilbert s... |
| hlimf 31330 | Function-like behavior of ... |
| hlimuni 31331 | A Hilbert space sequence c... |
| hlimreui 31332 | The limit of a Hilbert spa... |
| hlimeui 31333 | The limit of a Hilbert spa... |
| isch3 31334 | A Hilbert subspace is clos... |
| chcompl 31335 | Completeness of a closed s... |
| helch 31336 | The Hilbert lattice one (w... |
| ifchhv 31337 | Prove ` if ( A e. CH , A ,... |
| helsh 31338 | Hilbert space is a subspac... |
| shsspwh 31339 | Subspaces are subsets of H... |
| chsspwh 31340 | Closed subspaces are subse... |
| hsn0elch 31341 | The zero subspace belongs ... |
| norm1 31342 | From any nonzero Hilbert s... |
| norm1exi 31343 | A normalized vector exists... |
| norm1hex 31344 | A normalized vector can ex... |
| elch0 31347 | Membership in zero for clo... |
| h0elch 31348 | The zero subspace is a clo... |
| h0elsh 31349 | The zero subspace is a sub... |
| hhssva 31350 | The vector addition operat... |
| hhsssm 31351 | The scalar multiplication ... |
| hhssnm 31352 | The norm operation on a su... |
| issubgoilem 31353 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31354 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31355 | Abelian group property of ... |
| hhssablo 31356 | Abelian group property of ... |
| hhssnv 31357 | Normed complex vector spac... |
| hhssnvt 31358 | Normed complex vector spac... |
| hhsst 31359 | A member of ` SH ` is a su... |
| hhshsslem1 31360 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31361 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31362 | The predicate " ` H ` is a... |
| hhsssh2 31363 | The predicate " ` H ` is a... |
| hhssba 31364 | The base set of a subspace... |
| hhssvs 31365 | The vector subtraction ope... |
| hhssvsf 31366 | Mapping of the vector subt... |
| hhssims 31367 | Induced metric of a subspa... |
| hhssims2 31368 | Induced metric of a subspa... |
| hhssmet 31369 | Induced metric of a subspa... |
| hhssmetdval 31370 | Value of the distance func... |
| hhsscms 31371 | The induced metric of a cl... |
| hhssbnOLD 31372 | Obsolete version of ~ cssb... |
| ocval 31373 | Value of orthogonal comple... |
| ocel 31374 | Membership in orthogonal c... |
| shocel 31375 | Membership in orthogonal c... |
| ocsh 31376 | The orthogonal complement ... |
| shocsh 31377 | The orthogonal complement ... |
| ocss 31378 | An orthogonal complement i... |
| shocss 31379 | An orthogonal complement i... |
| occon 31380 | Contraposition law for ort... |
| occon2 31381 | Double contraposition for ... |
| occon2i 31382 | Double contraposition for ... |
| oc0 31383 | The zero vector belongs to... |
| ocorth 31384 | Members of a subset and it... |
| shocorth 31385 | Members of a subspace and ... |
| ococss 31386 | Inclusion in complement of... |
| shococss 31387 | Inclusion in complement of... |
| shorth 31388 | Members of orthogonal subs... |
| ocin 31389 | Intersection of a Hilbert ... |
| occon3 31390 | Hilbert lattice contraposi... |
| ocnel 31391 | A nonzero vector in the co... |
| chocvali 31392 | Value of the orthogonal co... |
| shuni 31393 | Two subspaces with trivial... |
| chocunii 31394 | Lemma for uniqueness part ... |
| pjhthmo 31395 | Projection Theorem, unique... |
| occllem 31396 | Lemma for ~ occl . (Contr... |
| occl 31397 | Closure of complement of H... |
| shoccl 31398 | Closure of complement of H... |
| choccl 31399 | Closure of complement of H... |
| choccli 31400 | Closure of ` CH ` orthocom... |
| shsval 31405 | Value of subspace sum of t... |
| shsss 31406 | The subspace sum is a subs... |
| shsel 31407 | Membership in the subspace... |
| shsel3 31408 | Membership in the subspace... |
| shseli 31409 | Membership in subspace sum... |
| shscli 31410 | Closure of subspace sum. ... |
| shscl 31411 | Closure of subspace sum. ... |
| shscom 31412 | Commutative law for subspa... |
| shsva 31413 | Vector sum belongs to subs... |
| shsel1 31414 | A subspace sum contains a ... |
| shsel2 31415 | A subspace sum contains a ... |
| shsvs 31416 | Vector subtraction belongs... |
| shsub1 31417 | Subspace sum is an upper b... |
| shsub2 31418 | Subspace sum is an upper b... |
| choc0 31419 | The orthocomplement of the... |
| choc1 31420 | The orthocomplement of the... |
| chocnul 31421 | Orthogonal complement of t... |
| shintcli 31422 | Closure of intersection of... |
| shintcl 31423 | The intersection of a none... |
| chintcli 31424 | The intersection of a none... |
| chintcl 31425 | The intersection (infimum)... |
| spanval 31426 | Value of the linear span o... |
| hsupval 31427 | Value of supremum of set o... |
| chsupval 31428 | The value of the supremum ... |
| spancl 31429 | The span of a subset of Hi... |
| elspancl 31430 | A member of a span is a ve... |
| shsupcl 31431 | Closure of the subspace su... |
| hsupcl 31432 | Closure of supremum of set... |
| chsupcl 31433 | Closure of supremum of sub... |
| hsupss 31434 | Subset relation for suprem... |
| chsupss 31435 | Subset relation for suprem... |
| hsupunss 31436 | The union of a set of Hilb... |
| chsupunss 31437 | The union of a set of clos... |
| spanss2 31438 | A subset of Hilbert space ... |
| shsupunss 31439 | The union of a set of subs... |
| spanid 31440 | A subspace of Hilbert spac... |
| spanss 31441 | Ordering relationship for ... |
| spanssoc 31442 | The span of a subset of Hi... |
| sshjval 31443 | Value of join for subsets ... |
| shjval 31444 | Value of join in ` SH ` . ... |
| chjval 31445 | Value of join in ` CH ` . ... |
| chjvali 31446 | Value of join in ` CH ` . ... |
| sshjval3 31447 | Value of join for subsets ... |
| sshjcl 31448 | Closure of join for subset... |
| shjcl 31449 | Closure of join in ` SH ` ... |
| chjcl 31450 | Closure of join in ` CH ` ... |
| shjcom 31451 | Commutative law for Hilber... |
| shless 31452 | Subset implies subset of s... |
| shlej1 31453 | Add disjunct to both sides... |
| shlej2 31454 | Add disjunct to both sides... |
| shincli 31455 | Closure of intersection of... |
| shscomi 31456 | Commutative law for subspa... |
| shsvai 31457 | Vector sum belongs to subs... |
| shsel1i 31458 | A subspace sum contains a ... |
| shsel2i 31459 | A subspace sum contains a ... |
| shsvsi 31460 | Vector subtraction belongs... |
| shunssi 31461 | Union is smaller than subs... |
| shunssji 31462 | Union is smaller than Hilb... |
| shsleji 31463 | Subspace sum is smaller th... |
| shjcomi 31464 | Commutative law for join i... |
| shsub1i 31465 | Subspace sum is an upper b... |
| shsub2i 31466 | Subspace sum is an upper b... |
| shub1i 31467 | Hilbert lattice join is an... |
| shjcli 31468 | Closure of ` CH ` join. (... |
| shjshcli 31469 | ` SH ` closure of join. (... |
| shlessi 31470 | Subset implies subset of s... |
| shlej1i 31471 | Add disjunct to both sides... |
| shlej2i 31472 | Add disjunct to both sides... |
| shslej 31473 | Subspace sum is smaller th... |
| shincl 31474 | Closure of intersection of... |
| shub1 31475 | Hilbert lattice join is an... |
| shub2 31476 | A subspace is a subset of ... |
| shsidmi 31477 | Idempotent law for Hilbert... |
| shslubi 31478 | The least upper bound law ... |
| shlesb1i 31479 | Hilbert lattice ordering i... |
| shsval2i 31480 | An alternate way to expres... |
| shsval3i 31481 | An alternate way to expres... |
| shmodsi 31482 | The modular law holds for ... |
| shmodi 31483 | The modular law is implied... |
| pjhthlem1 31484 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31485 | Lemma for ~ pjhth . (Cont... |
| pjhth 31486 | Projection Theorem: Any H... |
| pjhtheu 31487 | Projection Theorem: Any H... |
| pjhfval 31489 | The value of the projectio... |
| pjhval 31490 | Value of a projection. (C... |
| pjpreeq 31491 | Equality with a projection... |
| pjeq 31492 | Equality with a projection... |
| axpjcl 31493 | Closure of a projection in... |
| pjhcl 31494 | Closure of a projection in... |
| omlsilem 31495 | Lemma for orthomodular law... |
| omlsii 31496 | Subspace inference form of... |
| omlsi 31497 | Subspace form of orthomodu... |
| ococi 31498 | Complement of complement o... |
| ococ 31499 | Complement of complement o... |
| dfch2 31500 | Alternate definition of th... |
| ococin 31501 | The double complement is t... |
| hsupval2 31502 | Alternate definition of su... |
| chsupval2 31503 | The value of the supremum ... |
| sshjval2 31504 | Value of join in the set o... |
| chsupid 31505 | A subspace is the supremum... |
| chsupsn 31506 | Value of supremum of subse... |
| shlub 31507 | Hilbert lattice join is th... |
| shlubi 31508 | Hilbert lattice join is th... |
| pjhtheu2 31509 | Uniqueness of ` y ` for th... |
| pjcli 31510 | Closure of a projection in... |
| pjhcli 31511 | Closure of a projection in... |
| pjpjpre 31512 | Decomposition of a vector ... |
| axpjpj 31513 | Decomposition of a vector ... |
| pjclii 31514 | Closure of a projection in... |
| pjhclii 31515 | Closure of a projection in... |
| pjpj0i 31516 | Decomposition of a vector ... |
| pjpji 31517 | Decomposition of a vector ... |
| pjpjhth 31518 | Projection Theorem: Any H... |
| pjpjhthi 31519 | Projection Theorem: Any H... |
| pjop 31520 | Orthocomplement projection... |
| pjpo 31521 | Projection in terms of ort... |
| pjopi 31522 | Orthocomplement projection... |
| pjpoi 31523 | Projection in terms of ort... |
| pjoc1i 31524 | Projection of a vector in ... |
| pjchi 31525 | Projection of a vector in ... |
| pjoccl 31526 | The part of a vector that ... |
| pjoc1 31527 | Projection of a vector in ... |
| pjomli 31528 | Subspace form of orthomodu... |
| pjoml 31529 | Subspace form of orthomodu... |
| pjococi 31530 | Proof of orthocomplement t... |
| pjoc2i 31531 | Projection of a vector in ... |
| pjoc2 31532 | Projection of a vector in ... |
| sh0le 31533 | The zero subspace is the s... |
| ch0le 31534 | The zero subspace is the s... |
| shle0 31535 | No subspace is smaller tha... |
| chle0 31536 | No Hilbert lattice element... |
| chnlen0 31537 | A Hilbert lattice element ... |
| ch0pss 31538 | The zero subspace is a pro... |
| orthin 31539 | The intersection of orthog... |
| ssjo 31540 | The lattice join of a subs... |
| shne0i 31541 | A nonzero subspace has a n... |
| shs0i 31542 | Hilbert subspace sum with ... |
| shs00i 31543 | Two subspaces are zero iff... |
| ch0lei 31544 | The closed subspace zero i... |
| chle0i 31545 | No Hilbert closed subspace... |
| chne0i 31546 | A nonzero closed subspace ... |
| chocini 31547 | Intersection of a closed s... |
| chj0i 31548 | Join with lattice zero in ... |
| chm1i 31549 | Meet with lattice one in `... |
| chjcli 31550 | Closure of ` CH ` join. (... |
| chsleji 31551 | Subspace sum is smaller th... |
| chseli 31552 | Membership in subspace sum... |
| chincli 31553 | Closure of Hilbert lattice... |
| chsscon3i 31554 | Hilbert lattice contraposi... |
| chsscon1i 31555 | Hilbert lattice contraposi... |
| chsscon2i 31556 | Hilbert lattice contraposi... |
| chcon2i 31557 | Hilbert lattice contraposi... |
| chcon1i 31558 | Hilbert lattice contraposi... |
| chcon3i 31559 | Hilbert lattice contraposi... |
| chunssji 31560 | Union is smaller than ` CH... |
| chjcomi 31561 | Commutative law for join i... |
| chub1i 31562 | ` CH ` join is an upper bo... |
| chub2i 31563 | ` CH ` join is an upper bo... |
| chlubi 31564 | Hilbert lattice join is th... |
| chlubii 31565 | Hilbert lattice join is th... |
| chlej1i 31566 | Add join to both sides of ... |
| chlej2i 31567 | Add join to both sides of ... |
| chlej12i 31568 | Add join to both sides of ... |
| chlejb1i 31569 | Hilbert lattice ordering i... |
| chdmm1i 31570 | De Morgan's law for meet i... |
| chdmm2i 31571 | De Morgan's law for meet i... |
| chdmm3i 31572 | De Morgan's law for meet i... |
| chdmm4i 31573 | De Morgan's law for meet i... |
| chdmj1i 31574 | De Morgan's law for join i... |
| chdmj2i 31575 | De Morgan's law for join i... |
| chdmj3i 31576 | De Morgan's law for join i... |
| chdmj4i 31577 | De Morgan's law for join i... |
| chnlei 31578 | Equivalent expressions for... |
| chjassi 31579 | Associative law for Hilber... |
| chj00i 31580 | Two Hilbert lattice elemen... |
| chjoi 31581 | The join of a closed subsp... |
| chj1i 31582 | Join with Hilbert lattice ... |
| chm0i 31583 | Meet with Hilbert lattice ... |
| chm0 31584 | Meet with Hilbert lattice ... |
| shjshsi 31585 | Hilbert lattice join equal... |
| shjshseli 31586 | A closed subspace sum equa... |
| chne0 31587 | A nonzero closed subspace ... |
| chocin 31588 | Intersection of a closed s... |
| chssoc 31589 | A closed subspace less tha... |
| chj0 31590 | Join with Hilbert lattice ... |
| chslej 31591 | Subspace sum is smaller th... |
| chincl 31592 | Closure of Hilbert lattice... |
| chsscon3 31593 | Hilbert lattice contraposi... |
| chsscon1 31594 | Hilbert lattice contraposi... |
| chsscon2 31595 | Hilbert lattice contraposi... |
| chpsscon3 31596 | Hilbert lattice contraposi... |
| chpsscon1 31597 | Hilbert lattice contraposi... |
| chpsscon2 31598 | Hilbert lattice contraposi... |
| chjcom 31599 | Commutative law for Hilber... |
| chub1 31600 | Hilbert lattice join is gr... |
| chub2 31601 | Hilbert lattice join is gr... |
| chlub 31602 | Hilbert lattice join is th... |
| chlej1 31603 | Add join to both sides of ... |
| chlej2 31604 | Add join to both sides of ... |
| chlejb1 31605 | Hilbert lattice ordering i... |
| chlejb2 31606 | Hilbert lattice ordering i... |
| chnle 31607 | Equivalent expressions for... |
| chjo 31608 | The join of a closed subsp... |
| chabs1 31609 | Hilbert lattice absorption... |
| chabs2 31610 | Hilbert lattice absorption... |
| chabs1i 31611 | Hilbert lattice absorption... |
| chabs2i 31612 | Hilbert lattice absorption... |
| chjidm 31613 | Idempotent law for Hilbert... |
| chjidmi 31614 | Idempotent law for Hilbert... |
| chj12i 31615 | A rearrangement of Hilbert... |
| chj4i 31616 | Rearrangement of the join ... |
| chjjdiri 31617 | Hilbert lattice join distr... |
| chdmm1 31618 | De Morgan's law for meet i... |
| chdmm2 31619 | De Morgan's law for meet i... |
| chdmm3 31620 | De Morgan's law for meet i... |
| chdmm4 31621 | De Morgan's law for meet i... |
| chdmj1 31622 | De Morgan's law for join i... |
| chdmj2 31623 | De Morgan's law for join i... |
| chdmj3 31624 | De Morgan's law for join i... |
| chdmj4 31625 | De Morgan's law for join i... |
| chjass 31626 | Associative law for Hilber... |
| chj12 31627 | A rearrangement of Hilbert... |
| chj4 31628 | Rearrangement of the join ... |
| ledii 31629 | An ortholattice is distrib... |
| lediri 31630 | An ortholattice is distrib... |
| lejdii 31631 | An ortholattice is distrib... |
| lejdiri 31632 | An ortholattice is distrib... |
| ledi 31633 | An ortholattice is distrib... |
| spansn0 31634 | The span of the singleton ... |
| span0 31635 | The span of the empty set ... |
| elspani 31636 | Membership in the span of ... |
| spanuni 31637 | The span of a union is the... |
| spanun 31638 | The span of a union is the... |
| sshhococi 31639 | The join of two Hilbert sp... |
| hne0 31640 | Hilbert space has a nonzer... |
| chsup0 31641 | The supremum of the empty ... |
| h1deoi 31642 | Membership in orthocomplem... |
| h1dei 31643 | Membership in 1-dimensiona... |
| h1did 31644 | A generating vector belong... |
| h1dn0 31645 | A nonzero vector generates... |
| h1de2i 31646 | Membership in 1-dimensiona... |
| h1de2bi 31647 | Membership in 1-dimensiona... |
| h1de2ctlem 31648 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31649 | Membership in 1-dimensiona... |
| spansni 31650 | The span of a singleton in... |
| elspansni 31651 | Membership in the span of ... |
| spansn 31652 | The span of a singleton in... |
| spansnch 31653 | The span of a Hilbert spac... |
| spansnsh 31654 | The span of a Hilbert spac... |
| spansnchi 31655 | The span of a singleton in... |
| spansnid 31656 | A vector belongs to the sp... |
| spansnmul 31657 | A scalar product with a ve... |
| elspansncl 31658 | A member of a span of a si... |
| elspansn 31659 | Membership in the span of ... |
| elspansn2 31660 | Membership in the span of ... |
| spansncol 31661 | The singletons of collinea... |
| spansneleqi 31662 | Membership relation implie... |
| spansneleq 31663 | Membership relation that i... |
| spansnss 31664 | The span of the singleton ... |
| elspansn3 31665 | A member of the span of th... |
| elspansn4 31666 | A span membership conditio... |
| elspansn5 31667 | A vector belonging to both... |
| spansnss2 31668 | The span of the singleton ... |
| normcan 31669 | Cancellation-type law that... |
| pjspansn 31670 | A projection on the span o... |
| spansnpji 31671 | A subset of Hilbert space ... |
| spanunsni 31672 | The span of the union of a... |
| spanpr 31673 | The span of a pair of vect... |
| h1datomi 31674 | A 1-dimensional subspace i... |
| h1datom 31675 | A 1-dimensional subspace i... |
| cmbr 31677 | Binary relation expressing... |
| pjoml2i 31678 | Variation of orthomodular ... |
| pjoml3i 31679 | Variation of orthomodular ... |
| pjoml4i 31680 | Variation of orthomodular ... |
| pjoml5i 31681 | The orthomodular law. Rem... |
| pjoml6i 31682 | An equivalent of the ortho... |
| cmbri 31683 | Binary relation expressing... |
| cmcmlem 31684 | Commutation is symmetric. ... |
| cmcmi 31685 | Commutation is symmetric. ... |
| cmcm2i 31686 | Commutation with orthocomp... |
| cmcm3i 31687 | Commutation with orthocomp... |
| cmcm4i 31688 | Commutation with orthocomp... |
| cmbr2i 31689 | Alternate definition of th... |
| cmcmii 31690 | Commutation is symmetric. ... |
| cmcm2ii 31691 | Commutation with orthocomp... |
| cmcm3ii 31692 | Commutation with orthocomp... |
| cmbr3i 31693 | Alternate definition for t... |
| cmbr4i 31694 | Alternate definition for t... |
| lecmi 31695 | Comparable Hilbert lattice... |
| lecmii 31696 | Comparable Hilbert lattice... |
| cmj1i 31697 | A Hilbert lattice element ... |
| cmj2i 31698 | A Hilbert lattice element ... |
| cmm1i 31699 | A Hilbert lattice element ... |
| cmm2i 31700 | A Hilbert lattice element ... |
| cmbr3 31701 | Alternate definition for t... |
| cm0 31702 | The zero Hilbert lattice e... |
| cmidi 31703 | The commutes relation is r... |
| pjoml2 31704 | Variation of orthomodular ... |
| pjoml3 31705 | Variation of orthomodular ... |
| pjoml5 31706 | The orthomodular law. Rem... |
| cmcm 31707 | Commutation is symmetric. ... |
| cmcm3 31708 | Commutation with orthocomp... |
| cmcm2 31709 | Commutation with orthocomp... |
| lecm 31710 | Comparable Hilbert lattice... |
| fh1 31711 | Foulis-Holland Theorem. I... |
| fh2 31712 | Foulis-Holland Theorem. I... |
| cm2j 31713 | A lattice element that com... |
| fh1i 31714 | Foulis-Holland Theorem. I... |
| fh2i 31715 | Foulis-Holland Theorem. I... |
| fh3i 31716 | Variation of the Foulis-Ho... |
| fh4i 31717 | Variation of the Foulis-Ho... |
| cm2ji 31718 | A lattice element that com... |
| cm2mi 31719 | A lattice element that com... |
| qlax1i 31720 | One of the equations showi... |
| qlax2i 31721 | One of the equations showi... |
| qlax3i 31722 | One of the equations showi... |
| qlax4i 31723 | One of the equations showi... |
| qlax5i 31724 | One of the equations showi... |
| qlaxr1i 31725 | One of the conditions show... |
| qlaxr2i 31726 | One of the conditions show... |
| qlaxr4i 31727 | One of the conditions show... |
| qlaxr5i 31728 | One of the conditions show... |
| qlaxr3i 31729 | A variation of the orthomo... |
| chscllem1 31730 | Lemma for ~ chscl . (Cont... |
| chscllem2 31731 | Lemma for ~ chscl . (Cont... |
| chscllem3 31732 | Lemma for ~ chscl . (Cont... |
| chscllem4 31733 | Lemma for ~ chscl . (Cont... |
| chscl 31734 | The subspace sum of two cl... |
| osumi 31735 | If two closed subspaces of... |
| osumcori 31736 | Corollary of ~ osumi . (C... |
| osumcor2i 31737 | Corollary of ~ osumi , sho... |
| osum 31738 | If two closed subspaces of... |
| spansnji 31739 | The subspace sum of a clos... |
| spansnj 31740 | The subspace sum of a clos... |
| spansnscl 31741 | The subspace sum of a clos... |
| sumspansn 31742 | The sum of two vectors bel... |
| spansnm0i 31743 | The meet of different one-... |
| nonbooli 31744 | A Hilbert lattice with two... |
| spansncvi 31745 | Hilbert space has the cove... |
| spansncv 31746 | Hilbert space has the cove... |
| 5oalem1 31747 | Lemma for orthoarguesian l... |
| 5oalem2 31748 | Lemma for orthoarguesian l... |
| 5oalem3 31749 | Lemma for orthoarguesian l... |
| 5oalem4 31750 | Lemma for orthoarguesian l... |
| 5oalem5 31751 | Lemma for orthoarguesian l... |
| 5oalem6 31752 | Lemma for orthoarguesian l... |
| 5oalem7 31753 | Lemma for orthoarguesian l... |
| 5oai 31754 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31755 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31756 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31757 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31758 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31759 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31760 | Lemma for 3OA (weak) ortho... |
| 3oai 31761 | 3OA (weak) orthoarguesian ... |
| pjorthi 31762 | Projection components on o... |
| pjch1 31763 | Property of identity proje... |
| pjo 31764 | The orthogonal projection.... |
| pjcompi 31765 | Component of a projection.... |
| pjidmi 31766 | A projection is idempotent... |
| pjadjii 31767 | A projection is self-adjoi... |
| pjaddii 31768 | Projection of vector sum i... |
| pjinormii 31769 | The inner product of a pro... |
| pjmulii 31770 | Projection of (scalar) pro... |
| pjsubii 31771 | Projection of vector diffe... |
| pjsslem 31772 | Lemma for subset relations... |
| pjss2i 31773 | Subset relationship for pr... |
| pjssmii 31774 | Projection meet property. ... |
| pjssge0ii 31775 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31776 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31777 | The projection on a subspa... |
| pjadji 31778 | A projection is self-adjoi... |
| pjaddi 31779 | Projection of vector sum i... |
| pjinormi 31780 | The inner product of a pro... |
| pjsubi 31781 | Projection of vector diffe... |
| pjmuli 31782 | Projection of scalar produ... |
| pjige0i 31783 | The inner product of a pro... |
| pjige0 31784 | The inner product of a pro... |
| pjcjt2 31785 | The projection on a subspa... |
| pj0i 31786 | The projection of the zero... |
| pjch 31787 | Projection of a vector in ... |
| pjid 31788 | The projection of a vector... |
| pjvec 31789 | The set of vectors belongi... |
| pjocvec 31790 | The set of vectors belongi... |
| pjocini 31791 | Membership of projection i... |
| pjini 31792 | Membership of projection i... |
| pjjsi 31793 | A sufficient condition for... |
| pjfni 31794 | Functionality of a project... |
| pjrni 31795 | The range of a projection.... |
| pjfoi 31796 | A projection maps onto its... |
| pjfi 31797 | The mapping of a projectio... |
| pjvi 31798 | The value of a projection ... |
| pjhfo 31799 | A projection maps onto its... |
| pjrn 31800 | The range of a projection.... |
| pjhf 31801 | The mapping of a projectio... |
| pjfn 31802 | Functionality of a project... |
| pjsumi 31803 | The projection on a subspa... |
| pj11i 31804 | One-to-one correspondence ... |
| pjdsi 31805 | Vector decomposition into ... |
| pjds3i 31806 | Vector decomposition into ... |
| pj11 31807 | One-to-one correspondence ... |
| pjmfn 31808 | Functionality of the proje... |
| pjmf1 31809 | The projector function map... |
| pjoi0 31810 | The inner product of proje... |
| pjoi0i 31811 | The inner product of proje... |
| pjopythi 31812 | Pythagorean theorem for pr... |
| pjopyth 31813 | Pythagorean theorem for pr... |
| pjnormi 31814 | The norm of the projection... |
| pjpythi 31815 | Pythagorean theorem for pr... |
| pjneli 31816 | If a vector does not belon... |
| pjnorm 31817 | The norm of the projection... |
| pjpyth 31818 | Pythagorean theorem for pr... |
| pjnel 31819 | If a vector does not belon... |
| pjnorm2 31820 | A vector belongs to the su... |
| mayete3i 31821 | Mayet's equation E_3. Par... |
| mayetes3i 31822 | Mayet's equation E^*_3, de... |
| hosmval 31828 | Value of the sum of two Hi... |
| hommval 31829 | Value of the scalar produc... |
| hodmval 31830 | Value of the difference of... |
| hfsmval 31831 | Value of the sum of two Hi... |
| hfmmval 31832 | Value of the scalar produc... |
| hosval 31833 | Value of the sum of two Hi... |
| homval 31834 | Value of the scalar produc... |
| hodval 31835 | Value of the difference of... |
| hfsval 31836 | Value of the sum of two Hi... |
| hfmval 31837 | Value of the scalar produc... |
| hoscl 31838 | Closure of the sum of two ... |
| homcl 31839 | Closure of the scalar prod... |
| hodcl 31840 | Closure of the difference ... |
| ho0val 31843 | Value of the zero Hilbert ... |
| ho0f 31844 | Functionality of the zero ... |
| df0op2 31845 | Alternate definition of Hi... |
| dfiop2 31846 | Alternate definition of Hi... |
| hoif 31847 | Functionality of the Hilbe... |
| hoival 31848 | The value of the Hilbert s... |
| hoico1 31849 | Composition with the Hilbe... |
| hoico2 31850 | Composition with the Hilbe... |
| hoaddcl 31851 | The sum of Hilbert space o... |
| homulcl 31852 | The scalar product of a Hi... |
| hoeq 31853 | Equality of Hilbert space ... |
| hoeqi 31854 | Equality of Hilbert space ... |
| hoscli 31855 | Closure of Hilbert space o... |
| hodcli 31856 | Closure of Hilbert space o... |
| hocoi 31857 | Composition of Hilbert spa... |
| hococli 31858 | Closure of composition of ... |
| hocofi 31859 | Mapping of composition of ... |
| hocofni 31860 | Functionality of compositi... |
| hoaddcli 31861 | Mapping of sum of Hilbert ... |
| hosubcli 31862 | Mapping of difference of H... |
| hoaddfni 31863 | Functionality of sum of Hi... |
| hosubfni 31864 | Functionality of differenc... |
| hoaddcomi 31865 | Commutativity of sum of Hi... |
| hosubcl 31866 | Mapping of difference of H... |
| hoaddcom 31867 | Commutativity of sum of Hi... |
| hodsi 31868 | Relationship between Hilbe... |
| hoaddassi 31869 | Associativity of sum of Hi... |
| hoadd12i 31870 | Commutative/associative la... |
| hoadd32i 31871 | Commutative/associative la... |
| hocadddiri 31872 | Distributive law for Hilbe... |
| hocsubdiri 31873 | Distributive law for Hilbe... |
| ho2coi 31874 | Double composition of Hilb... |
| hoaddass 31875 | Associativity of sum of Hi... |
| hoadd32 31876 | Commutative/associative la... |
| hoadd4 31877 | Rearrangement of 4 terms i... |
| hocsubdir 31878 | Distributive law for Hilbe... |
| hoaddridi 31879 | Sum of a Hilbert space ope... |
| hodidi 31880 | Difference of a Hilbert sp... |
| ho0coi 31881 | Composition of the zero op... |
| hoid1i 31882 | Composition of Hilbert spa... |
| hoid1ri 31883 | Composition of Hilbert spa... |
| hoaddrid 31884 | Sum of a Hilbert space ope... |
| hodid 31885 | Difference of a Hilbert sp... |
| hon0 31886 | A Hilbert space operator i... |
| hodseqi 31887 | Subtraction and addition o... |
| ho0subi 31888 | Subtraction of Hilbert spa... |
| honegsubi 31889 | Relationship between Hilbe... |
| ho0sub 31890 | Subtraction of Hilbert spa... |
| hosubid1 31891 | The zero operator subtract... |
| honegsub 31892 | Relationship between Hilbe... |
| homullid 31893 | An operator equals its sca... |
| homco1 31894 | Associative law for scalar... |
| homulass 31895 | Scalar product associative... |
| hoadddi 31896 | Scalar product distributiv... |
| hoadddir 31897 | Scalar product reverse dis... |
| homul12 31898 | Swap first and second fact... |
| honegneg 31899 | Double negative of a Hilbe... |
| hosubneg 31900 | Relationship between opera... |
| hosubdi 31901 | Scalar product distributiv... |
| honegdi 31902 | Distribution of negative o... |
| honegsubdi 31903 | Distribution of negative o... |
| honegsubdi2 31904 | Distribution of negative o... |
| hosubsub2 31905 | Law for double subtraction... |
| hosub4 31906 | Rearrangement of 4 terms i... |
| hosubadd4 31907 | Rearrangement of 4 terms i... |
| hoaddsubass 31908 | Associative-type law for a... |
| hoaddsub 31909 | Law for operator addition ... |
| hosubsub 31910 | Law for double subtraction... |
| hosubsub4 31911 | Law for double subtraction... |
| ho2times 31912 | Two times a Hilbert space ... |
| hoaddsubassi 31913 | Associativity of sum and d... |
| hoaddsubi 31914 | Law for sum and difference... |
| hosd1i 31915 | Hilbert space operator sum... |
| hosd2i 31916 | Hilbert space operator sum... |
| hopncani 31917 | Hilbert space operator can... |
| honpcani 31918 | Hilbert space operator can... |
| hosubeq0i 31919 | If the difference between ... |
| honpncani 31920 | Hilbert space operator can... |
| ho01i 31921 | A condition implying that ... |
| ho02i 31922 | A condition implying that ... |
| hoeq1 31923 | A condition implying that ... |
| hoeq2 31924 | A condition implying that ... |
| adjmo 31925 | Every Hilbert space operat... |
| adjsym 31926 | Symmetry property of an ad... |
| eigrei 31927 | A necessary and sufficient... |
| eigre 31928 | A necessary and sufficient... |
| eigposi 31929 | A sufficient condition (fi... |
| eigorthi 31930 | A necessary and sufficient... |
| eigorth 31931 | A necessary and sufficient... |
| nmopval 31949 | Value of the norm of a Hil... |
| elcnop 31950 | Property defining a contin... |
| ellnop 31951 | Property defining a linear... |
| lnopf 31952 | A linear Hilbert space ope... |
| elbdop 31953 | Property defining a bounde... |
| bdopln 31954 | A bounded linear Hilbert s... |
| bdopf 31955 | A bounded linear Hilbert s... |
| nmopsetretALT 31956 | The set in the supremum of... |
| nmopsetretHIL 31957 | The set in the supremum of... |
| nmopsetn0 31958 | The set in the supremum of... |
| nmopxr 31959 | The norm of a Hilbert spac... |
| nmoprepnf 31960 | The norm of a Hilbert spac... |
| nmopgtmnf 31961 | The norm of a Hilbert spac... |
| nmopreltpnf 31962 | The norm of a Hilbert spac... |
| nmopre 31963 | The norm of a bounded oper... |
| elbdop2 31964 | Property defining a bounde... |
| elunop 31965 | Property defining a unitar... |
| elhmop 31966 | Property defining a Hermit... |
| hmopf 31967 | A Hermitian operator is a ... |
| hmopex 31968 | The class of Hermitian ope... |
| nmfnval 31969 | Value of the norm of a Hil... |
| nmfnsetre 31970 | The set in the supremum of... |
| nmfnsetn0 31971 | The set in the supremum of... |
| nmfnxr 31972 | The norm of any Hilbert sp... |
| nmfnrepnf 31973 | The norm of a Hilbert spac... |
| nlfnval 31974 | Value of the null space of... |
| elcnfn 31975 | Property defining a contin... |
| ellnfn 31976 | Property defining a linear... |
| lnfnf 31977 | A linear Hilbert space fun... |
| dfadj2 31978 | Alternate definition of th... |
| funadj 31979 | Functionality of the adjoi... |
| dmadjss 31980 | The domain of the adjoint ... |
| dmadjop 31981 | A member of the domain of ... |
| adjeu 31982 | Elementhood in the domain ... |
| adjval 31983 | Value of the adjoint funct... |
| adjval2 31984 | Value of the adjoint funct... |
| cnvadj 31985 | The adjoint function equal... |
| funcnvadj 31986 | The converse of the adjoin... |
| adj1o 31987 | The adjoint function maps ... |
| dmadjrn 31988 | The adjoint of an operator... |
| eigvecval 31989 | The set of eigenvectors of... |
| eigvalfval 31990 | The eigenvalues of eigenve... |
| specval 31991 | The value of the spectrum ... |
| speccl 31992 | The spectrum of an operato... |
| hhlnoi 31993 | The linear operators of Hi... |
| hhnmoi 31994 | The norm of an operator in... |
| hhbloi 31995 | A bounded linear operator ... |
| hh0oi 31996 | The zero operator in Hilbe... |
| hhcno 31997 | The continuous operators o... |
| hhcnf 31998 | The continuous functionals... |
| dmadjrnb 31999 | The adjoint of an operator... |
| nmoplb 32000 | A lower bound for an opera... |
| nmopub 32001 | An upper bound for an oper... |
| nmopub2tALT 32002 | An upper bound for an oper... |
| nmopub2tHIL 32003 | An upper bound for an oper... |
| nmopge0 32004 | The norm of any Hilbert sp... |
| nmopgt0 32005 | A linear Hilbert space ope... |
| cnopc 32006 | Basic continuity property ... |
| lnopl 32007 | Basic linearity property o... |
| unop 32008 | Basic inner product proper... |
| unopf1o 32009 | A unitary operator in Hilb... |
| unopnorm 32010 | A unitary operator is idem... |
| cnvunop 32011 | The inverse (converse) of ... |
| unopadj 32012 | The inverse (converse) of ... |
| unoplin 32013 | A unitary operator is line... |
| counop 32014 | The composition of two uni... |
| hmop 32015 | Basic inner product proper... |
| hmopre 32016 | The inner product of the v... |
| nmfnlb 32017 | A lower bound for a functi... |
| nmfnleub 32018 | An upper bound for the nor... |
| nmfnleub2 32019 | An upper bound for the nor... |
| nmfnge0 32020 | The norm of any Hilbert sp... |
| elnlfn 32021 | Membership in the null spa... |
| elnlfn2 32022 | Membership in the null spa... |
| cnfnc 32023 | Basic continuity property ... |
| lnfnl 32024 | Basic linearity property o... |
| adjcl 32025 | Closure of the adjoint of ... |
| adj1 32026 | Property of an adjoint Hil... |
| adj2 32027 | Property of an adjoint Hil... |
| adjeq 32028 | A property that determines... |
| adjadj 32029 | Double adjoint. Theorem 3... |
| adjvalval 32030 | Value of the value of the ... |
| unopadj2 32031 | The adjoint of a unitary o... |
| hmopadj 32032 | A Hermitian operator is se... |
| hmdmadj 32033 | Every Hermitian operator h... |
| hmopadj2 32034 | An operator is Hermitian i... |
| hmoplin 32035 | A Hermitian operator is li... |
| brafval 32036 | The bra of a vector, expre... |
| braval 32037 | A bra-ket juxtaposition, e... |
| braadd 32038 | Linearity property of bra ... |
| bramul 32039 | Linearity property of bra ... |
| brafn 32040 | The bra function is a func... |
| bralnfn 32041 | The Dirac bra function is ... |
| bracl 32042 | Closure of the bra functio... |
| bra0 32043 | The Dirac bra of the zero ... |
| brafnmul 32044 | Anti-linearity property of... |
| kbfval 32045 | The outer product of two v... |
| kbop 32046 | The outer product of two v... |
| kbval 32047 | The value of the operator ... |
| kbmul 32048 | Multiplication property of... |
| kbpj 32049 | If a vector ` A ` has norm... |
| eleigvec 32050 | Membership in the set of e... |
| eleigvec2 32051 | Membership in the set of e... |
| eleigveccl 32052 | Closure of an eigenvector ... |
| eigvalval 32053 | The eigenvalue of an eigen... |
| eigvalcl 32054 | An eigenvalue is a complex... |
| eigvec1 32055 | Property of an eigenvector... |
| eighmre 32056 | The eigenvalues of a Hermi... |
| eighmorth 32057 | Eigenvectors of a Hermitia... |
| nmopnegi 32058 | Value of the norm of the n... |
| lnop0 32059 | The value of a linear Hilb... |
| lnopmul 32060 | Multiplicative property of... |
| lnopli 32061 | Basic scalar product prope... |
| lnopfi 32062 | A linear Hilbert space ope... |
| lnop0i 32063 | The value of a linear Hilb... |
| lnopaddi 32064 | Additive property of a lin... |
| lnopmuli 32065 | Multiplicative property of... |
| lnopaddmuli 32066 | Sum/product property of a ... |
| lnopsubi 32067 | Subtraction property for a... |
| lnopsubmuli 32068 | Subtraction/product proper... |
| lnopmulsubi 32069 | Product/subtraction proper... |
| homco2 32070 | Move a scalar product out ... |
| idunop 32071 | The identity function (res... |
| 0cnop 32072 | The identically zero funct... |
| 0cnfn 32073 | The identically zero funct... |
| idcnop 32074 | The identity function (res... |
| idhmop 32075 | The Hilbert space identity... |
| 0hmop 32076 | The identically zero funct... |
| 0lnop 32077 | The identically zero funct... |
| 0lnfn 32078 | The identically zero funct... |
| nmop0 32079 | The norm of the zero opera... |
| nmfn0 32080 | The norm of the identicall... |
| hmopbdoptHIL 32081 | A Hermitian operator is a ... |
| hoddii 32082 | Distributive law for Hilbe... |
| hoddi 32083 | Distributive law for Hilbe... |
| nmop0h 32084 | The norm of any operator o... |
| idlnop 32085 | The identity function (res... |
| 0bdop 32086 | The identically zero opera... |
| adj0 32087 | Adjoint of the zero operat... |
| nmlnop0iALT 32088 | A linear operator with a z... |
| nmlnop0iHIL 32089 | A linear operator with a z... |
| nmlnopgt0i 32090 | A linear Hilbert space ope... |
| nmlnop0 32091 | A linear operator with a z... |
| nmlnopne0 32092 | A linear operator with a n... |
| lnopmi 32093 | The scalar product of a li... |
| lnophsi 32094 | The sum of two linear oper... |
| lnophdi 32095 | The difference of two line... |
| lnopcoi 32096 | The composition of two lin... |
| lnopco0i 32097 | The composition of a linea... |
| lnopeq0lem1 32098 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 32099 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 32100 | A condition implying that ... |
| lnopeqi 32101 | Two linear Hilbert space o... |
| lnopeq 32102 | Two linear Hilbert space o... |
| lnopunilem1 32103 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 32104 | Lemma for ~ lnopunii . (C... |
| lnopunii 32105 | If a linear operator (whos... |
| elunop2 32106 | An operator is unitary iff... |
| nmopun 32107 | Norm of a unitary Hilbert ... |
| unopbd 32108 | A unitary operator is a bo... |
| lnophmlem1 32109 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 32110 | Lemma for ~ lnophmi . (Co... |
| lnophmi 32111 | A linear operator is Hermi... |
| lnophm 32112 | A linear operator is Hermi... |
| hmops 32113 | The sum of two Hermitian o... |
| hmopm 32114 | The scalar product of a He... |
| hmopd 32115 | The difference of two Herm... |
| hmopco 32116 | The composition of two com... |
| nmbdoplbi 32117 | A lower bound for the norm... |
| nmbdoplb 32118 | A lower bound for the norm... |
| nmcexi 32119 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 32120 | The norm of a continuous l... |
| nmcoplbi 32121 | A lower bound for the norm... |
| nmcopex 32122 | The norm of a continuous l... |
| nmcoplb 32123 | A lower bound for the norm... |
| nmophmi 32124 | The norm of the scalar pro... |
| bdophmi 32125 | The scalar product of a bo... |
| lnconi 32126 | Lemma for ~ lnopconi and ~... |
| lnopconi 32127 | A condition equivalent to ... |
| lnopcon 32128 | A condition equivalent to ... |
| lnopcnbd 32129 | A linear operator is conti... |
| lncnopbd 32130 | A continuous linear operat... |
| lncnbd 32131 | A continuous linear operat... |
| lnopcnre 32132 | A linear operator is conti... |
| lnfnli 32133 | Basic property of a linear... |
| lnfnfi 32134 | A linear Hilbert space fun... |
| lnfn0i 32135 | The value of a linear Hilb... |
| lnfnaddi 32136 | Additive property of a lin... |
| lnfnmuli 32137 | Multiplicative property of... |
| lnfnaddmuli 32138 | Sum/product property of a ... |
| lnfnsubi 32139 | Subtraction property for a... |
| lnfn0 32140 | The value of a linear Hilb... |
| lnfnmul 32141 | Multiplicative property of... |
| nmbdfnlbi 32142 | A lower bound for the norm... |
| nmbdfnlb 32143 | A lower bound for the norm... |
| nmcfnexi 32144 | The norm of a continuous l... |
| nmcfnlbi 32145 | A lower bound for the norm... |
| nmcfnex 32146 | The norm of a continuous l... |
| nmcfnlb 32147 | A lower bound of the norm ... |
| lnfnconi 32148 | A condition equivalent to ... |
| lnfncon 32149 | A condition equivalent to ... |
| lnfncnbd 32150 | A linear functional is con... |
| imaelshi 32151 | The image of a subspace un... |
| rnelshi 32152 | The range of a linear oper... |
| nlelshi 32153 | The null space of a linear... |
| nlelchi 32154 | The null space of a contin... |
| riesz3i 32155 | A continuous linear functi... |
| riesz4i 32156 | A continuous linear functi... |
| riesz4 32157 | A continuous linear functi... |
| riesz1 32158 | Part 1 of the Riesz repres... |
| riesz2 32159 | Part 2 of the Riesz repres... |
| cnlnadjlem1 32160 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 32161 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 32162 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 32163 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32164 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32165 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32166 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32167 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32168 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32169 | Every continuous linear op... |
| cnlnadjeui 32170 | Every continuous linear op... |
| cnlnadjeu 32171 | Every continuous linear op... |
| cnlnadj 32172 | Every continuous linear op... |
| cnlnssadj 32173 | Every continuous linear Hi... |
| bdopssadj 32174 | Every bounded linear Hilbe... |
| bdopadj 32175 | Every bounded linear Hilbe... |
| adjbdln 32176 | The adjoint of a bounded l... |
| adjbdlnb 32177 | An operator is bounded and... |
| adjbd1o 32178 | The mapping of adjoints of... |
| adjlnop 32179 | The adjoint of an operator... |
| adjsslnop 32180 | Every operator with an adj... |
| nmopadjlei 32181 | Property of the norm of an... |
| nmopadjlem 32182 | Lemma for ~ nmopadji . (C... |
| nmopadji 32183 | Property of the norm of an... |
| adjeq0 32184 | An operator is zero iff it... |
| adjmul 32185 | The adjoint of the scalar ... |
| adjadd 32186 | The adjoint of the sum of ... |
| nmoptrii 32187 | Triangle inequality for th... |
| nmopcoi 32188 | Upper bound for the norm o... |
| bdophsi 32189 | The sum of two bounded lin... |
| bdophdi 32190 | The difference between two... |
| bdopcoi 32191 | The composition of two bou... |
| nmoptri2i 32192 | Triangle-type inequality f... |
| adjcoi 32193 | The adjoint of a compositi... |
| nmopcoadji 32194 | The norm of an operator co... |
| nmopcoadj2i 32195 | The norm of an operator co... |
| nmopcoadj0i 32196 | An operator composed with ... |
| unierri 32197 | If we approximate a chain ... |
| branmfn 32198 | The norm of the bra functi... |
| brabn 32199 | The bra of a vector is a b... |
| rnbra 32200 | The set of bras equals the... |
| bra11 32201 | The bra function maps vect... |
| bracnln 32202 | A bra is a continuous line... |
| cnvbraval 32203 | Value of the converse of t... |
| cnvbracl 32204 | Closure of the converse of... |
| cnvbrabra 32205 | The converse bra of the br... |
| bracnvbra 32206 | The bra of the converse br... |
| bracnlnval 32207 | The vector that a continuo... |
| cnvbramul 32208 | Multiplication property of... |
| kbass1 32209 | Dirac bra-ket associative ... |
| kbass2 32210 | Dirac bra-ket associative ... |
| kbass3 32211 | Dirac bra-ket associative ... |
| kbass4 32212 | Dirac bra-ket associative ... |
| kbass5 32213 | Dirac bra-ket associative ... |
| kbass6 32214 | Dirac bra-ket associative ... |
| leopg 32215 | Ordering relation for posi... |
| leop 32216 | Ordering relation for oper... |
| leop2 32217 | Ordering relation for oper... |
| leop3 32218 | Operator ordering in terms... |
| leoppos 32219 | Binary relation defining a... |
| leoprf2 32220 | The ordering relation for ... |
| leoprf 32221 | The ordering relation for ... |
| leopsq 32222 | The square of a Hermitian ... |
| 0leop 32223 | The zero operator is a pos... |
| idleop 32224 | The identity operator is a... |
| leopadd 32225 | The sum of two positive op... |
| leopmuli 32226 | The scalar product of a no... |
| leopmul 32227 | The scalar product of a po... |
| leopmul2i 32228 | Scalar product applied to ... |
| leoptri 32229 | The positive operator orde... |
| leoptr 32230 | The positive operator orde... |
| leopnmid 32231 | A bounded Hermitian operat... |
| nmopleid 32232 | A nonzero, bounded Hermiti... |
| opsqrlem1 32233 | Lemma for opsqri . (Contr... |
| opsqrlem2 32234 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32235 | Lemma for opsqri . (Contr... |
| opsqrlem4 32236 | Lemma for opsqri . (Contr... |
| opsqrlem5 32237 | Lemma for opsqri . (Contr... |
| opsqrlem6 32238 | Lemma for opsqri . (Contr... |
| pjhmopi 32239 | A projector is a Hermitian... |
| pjlnopi 32240 | A projector is a linear op... |
| pjnmopi 32241 | The operator norm of a pro... |
| pjbdlni 32242 | A projector is a bounded l... |
| pjhmop 32243 | A projection is a Hermitia... |
| hmopidmchi 32244 | An idempotent Hermitian op... |
| hmopidmpji 32245 | An idempotent Hermitian op... |
| hmopidmch 32246 | An idempotent Hermitian op... |
| hmopidmpj 32247 | An idempotent Hermitian op... |
| pjsdii 32248 | Distributive law for Hilbe... |
| pjddii 32249 | Distributive law for Hilbe... |
| pjsdi2i 32250 | Chained distributive law f... |
| pjcoi 32251 | Composition of projections... |
| pjcocli 32252 | Closure of composition of ... |
| pjcohcli 32253 | Closure of composition of ... |
| pjadjcoi 32254 | Adjoint of composition of ... |
| pjcofni 32255 | Functionality of compositi... |
| pjss1coi 32256 | Subset relationship for pr... |
| pjss2coi 32257 | Subset relationship for pr... |
| pjssmi 32258 | Projection meet property. ... |
| pjssge0i 32259 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32260 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32261 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32262 | Composition of projections... |
| pjscji 32263 | The projection of orthogon... |
| pjssumi 32264 | The projection on a subspa... |
| pjssposi 32265 | Projector ordering can be ... |
| pjordi 32266 | The definition of projecto... |
| pjssdif2i 32267 | The projection subspace of... |
| pjssdif1i 32268 | A necessary and sufficient... |
| pjimai 32269 | The image of a projection.... |
| pjidmcoi 32270 | A projection is idempotent... |
| pjoccoi 32271 | Composition of projections... |
| pjtoi 32272 | Subspace sum of projection... |
| pjoci 32273 | Projection of orthocomplem... |
| pjidmco 32274 | A projection operator is i... |
| dfpjop 32275 | Definition of projection o... |
| pjhmopidm 32276 | Two ways to express the se... |
| elpjidm 32277 | A projection operator is i... |
| elpjhmop 32278 | A projection operator is H... |
| 0leopj 32279 | A projector is a positive ... |
| pjadj2 32280 | A projector is self-adjoin... |
| pjadj3 32281 | A projector is self-adjoin... |
| elpjch 32282 | Reconstruction of the subs... |
| elpjrn 32283 | Reconstruction of the subs... |
| pjinvari 32284 | A closed subspace ` H ` wi... |
| pjin1i 32285 | Lemma for Theorem 1.22 of ... |
| pjin2i 32286 | Lemma for Theorem 1.22 of ... |
| pjin3i 32287 | Lemma for Theorem 1.22 of ... |
| pjclem1 32288 | Lemma for projection commu... |
| pjclem2 32289 | Lemma for projection commu... |
| pjclem3 32290 | Lemma for projection commu... |
| pjclem4a 32291 | Lemma for projection commu... |
| pjclem4 32292 | Lemma for projection commu... |
| pjci 32293 | Two subspaces commute iff ... |
| pjcmul1i 32294 | A necessary and sufficient... |
| pjcmul2i 32295 | The projection subspace of... |
| pjcohocli 32296 | Closure of composition of ... |
| pjadj2coi 32297 | Adjoint of double composit... |
| pj2cocli 32298 | Closure of double composit... |
| pj3lem1 32299 | Lemma for projection tripl... |
| pj3si 32300 | Stronger projection triple... |
| pj3i 32301 | Projection triplet theorem... |
| pj3cor1i 32302 | Projection triplet corolla... |
| pjs14i 32303 | Theorem S-14 of Watanabe, ... |
| isst 32306 | Property of a state. (Con... |
| ishst 32307 | Property of a complex Hilb... |
| sticl 32308 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32309 | Real closure of the value ... |
| hstcl 32310 | Closure of the value of a ... |
| hst1a 32311 | Unit value of a Hilbert-sp... |
| hstel2 32312 | Properties of a Hilbert-sp... |
| hstorth 32313 | Orthogonality property of ... |
| hstosum 32314 | Orthogonal sum property of... |
| hstoc 32315 | Sum of a Hilbert-space-val... |
| hstnmoc 32316 | Sum of norms of a Hilbert-... |
| stge0 32317 | The value of a state is no... |
| stle1 32318 | The value of a state is le... |
| hstle1 32319 | The norm of the value of a... |
| hst1h 32320 | The norm of a Hilbert-spac... |
| hst0h 32321 | The norm of a Hilbert-spac... |
| hstpyth 32322 | Pythagorean property of a ... |
| hstle 32323 | Ordering property of a Hil... |
| hstles 32324 | Ordering property of a Hil... |
| hstoh 32325 | A Hilbert-space-valued sta... |
| hst0 32326 | A Hilbert-space-valued sta... |
| sthil 32327 | The value of a state at th... |
| stj 32328 | The value of a state on a ... |
| sto1i 32329 | The state of a subspace pl... |
| sto2i 32330 | The state of the orthocomp... |
| stge1i 32331 | If a state is greater than... |
| stle0i 32332 | If a state is less than or... |
| stlei 32333 | Ordering law for states. ... |
| stlesi 32334 | Ordering law for states. ... |
| stji1i 32335 | Join of components of Sasa... |
| stm1i 32336 | State of component of unit... |
| stm1ri 32337 | State of component of unit... |
| stm1addi 32338 | Sum of states whose meet i... |
| staddi 32339 | If the sum of 2 states is ... |
| stm1add3i 32340 | Sum of states whose meet i... |
| stadd3i 32341 | If the sum of 3 states is ... |
| st0 32342 | The state of the zero subs... |
| strlem1 32343 | Lemma for strong state the... |
| strlem2 32344 | Lemma for strong state the... |
| strlem3a 32345 | Lemma for strong state the... |
| strlem3 32346 | Lemma for strong state the... |
| strlem4 32347 | Lemma for strong state the... |
| strlem5 32348 | Lemma for strong state the... |
| strlem6 32349 | Lemma for strong state the... |
| stri 32350 | Strong state theorem. The... |
| strb 32351 | Strong state theorem (bidi... |
| hstrlem2 32352 | Lemma for strong set of CH... |
| hstrlem3a 32353 | Lemma for strong set of CH... |
| hstrlem3 32354 | Lemma for strong set of CH... |
| hstrlem4 32355 | Lemma for strong set of CH... |
| hstrlem5 32356 | Lemma for strong set of CH... |
| hstrlem6 32357 | Lemma for strong set of CH... |
| hstri 32358 | Hilbert space admits a str... |
| hstrbi 32359 | Strong CH-state theorem (b... |
| largei 32360 | A Hilbert lattice admits a... |
| jplem1 32361 | Lemma for Jauch-Piron theo... |
| jplem2 32362 | Lemma for Jauch-Piron theo... |
| jpi 32363 | The function ` S ` , that ... |
| golem1 32364 | Lemma for Godowski's equat... |
| golem2 32365 | Lemma for Godowski's equat... |
| goeqi 32366 | Godowski's equation, shown... |
| stcltr1i 32367 | Property of a strong class... |
| stcltr2i 32368 | Property of a strong class... |
| stcltrlem1 32369 | Lemma for strong classical... |
| stcltrlem2 32370 | Lemma for strong classical... |
| stcltrthi 32371 | Theorem for classically st... |
| cvbr 32375 | Binary relation expressing... |
| cvbr2 32376 | Binary relation expressing... |
| cvcon3 32377 | Contraposition law for the... |
| cvpss 32378 | The covers relation implie... |
| cvnbtwn 32379 | The covers relation implie... |
| cvnbtwn2 32380 | The covers relation implie... |
| cvnbtwn3 32381 | The covers relation implie... |
| cvnbtwn4 32382 | The covers relation implie... |
| cvnsym 32383 | The covers relation is not... |
| cvnref 32384 | The covers relation is not... |
| cvntr 32385 | The covers relation is not... |
| spansncv2 32386 | Hilbert space has the cove... |
| mdbr 32387 | Binary relation expressing... |
| mdi 32388 | Consequence of the modular... |
| mdbr2 32389 | Binary relation expressing... |
| mdbr3 32390 | Binary relation expressing... |
| mdbr4 32391 | Binary relation expressing... |
| dmdbr 32392 | Binary relation expressing... |
| dmdmd 32393 | The dual modular pair prop... |
| mddmd 32394 | The modular pair property ... |
| dmdi 32395 | Consequence of the dual mo... |
| dmdbr2 32396 | Binary relation expressing... |
| dmdi2 32397 | Consequence of the dual mo... |
| dmdbr3 32398 | Binary relation expressing... |
| dmdbr4 32399 | Binary relation expressing... |
| dmdi4 32400 | Consequence of the dual mo... |
| dmdbr5 32401 | Binary relation expressing... |
| mddmd2 32402 | Relationship between modul... |
| mdsl0 32403 | A sublattice condition tha... |
| ssmd1 32404 | Ordering implies the modul... |
| ssmd2 32405 | Ordering implies the modul... |
| ssdmd1 32406 | Ordering implies the dual ... |
| ssdmd2 32407 | Ordering implies the dual ... |
| dmdsl3 32408 | Sublattice mapping for a d... |
| mdsl3 32409 | Sublattice mapping for a m... |
| mdslle1i 32410 | Order preservation of the ... |
| mdslle2i 32411 | Order preservation of the ... |
| mdslj1i 32412 | Join preservation of the o... |
| mdslj2i 32413 | Meet preservation of the r... |
| mdsl1i 32414 | If the modular pair proper... |
| mdsl2i 32415 | If the modular pair proper... |
| mdsl2bi 32416 | If the modular pair proper... |
| cvmdi 32417 | The covering property impl... |
| mdslmd1lem1 32418 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32419 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32420 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32421 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32422 | Preservation of the modula... |
| mdslmd2i 32423 | Preservation of the modula... |
| mdsldmd1i 32424 | Preservation of the dual m... |
| mdslmd3i 32425 | Modular pair conditions th... |
| mdslmd4i 32426 | Modular pair condition tha... |
| csmdsymi 32427 | Cross-symmetry implies M-s... |
| mdexchi 32428 | An exchange lemma for modu... |
| cvmd 32429 | The covering property impl... |
| cvdmd 32430 | The covering property impl... |
| ela 32432 | Atoms in a Hilbert lattice... |
| elat2 32433 | Expanded membership relati... |
| elatcv0 32434 | A Hilbert lattice element ... |
| atcv0 32435 | An atom covers the zero su... |
| atssch 32436 | Atoms are a subset of the ... |
| atelch 32437 | An atom is a Hilbert latti... |
| atne0 32438 | An atom is not the Hilbert... |
| atss 32439 | A lattice element smaller ... |
| atsseq 32440 | Two atoms in a subset rela... |
| atcveq0 32441 | A Hilbert lattice element ... |
| h1da 32442 | A 1-dimensional subspace i... |
| spansna 32443 | The span of the singleton ... |
| sh1dle 32444 | A 1-dimensional subspace i... |
| ch1dle 32445 | A 1-dimensional subspace i... |
| atom1d 32446 | The 1-dimensional subspace... |
| superpos 32447 | Superposition Principle. ... |
| chcv1 32448 | The Hilbert lattice has th... |
| chcv2 32449 | The Hilbert lattice has th... |
| chjatom 32450 | The join of a closed subsp... |
| shatomici 32451 | The lattice of Hilbert sub... |
| hatomici 32452 | The Hilbert lattice is ato... |
| hatomic 32453 | A Hilbert lattice is atomi... |
| shatomistici 32454 | The lattice of Hilbert sub... |
| hatomistici 32455 | ` CH ` is atomistic, i.e. ... |
| chpssati 32456 | Two Hilbert lattice elemen... |
| chrelati 32457 | The Hilbert lattice is rel... |
| chrelat2i 32458 | A consequence of relative ... |
| cvati 32459 | If a Hilbert lattice eleme... |
| cvbr4i 32460 | An alternate way to expres... |
| cvexchlem 32461 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32462 | The Hilbert lattice satisf... |
| chrelat2 32463 | A consequence of relative ... |
| chrelat3 32464 | A consequence of relative ... |
| chrelat3i 32465 | A consequence of the relat... |
| chrelat4i 32466 | A consequence of relative ... |
| cvexch 32467 | The Hilbert lattice satisf... |
| cvp 32468 | The Hilbert lattice satisf... |
| atnssm0 32469 | The meet of a Hilbert latt... |
| atnemeq0 32470 | The meet of distinct atoms... |
| atssma 32471 | The meet with an atom's su... |
| atcv0eq 32472 | Two atoms covering the zer... |
| atcv1 32473 | Two atoms covering the zer... |
| atexch 32474 | The Hilbert lattice satisf... |
| atomli 32475 | An assertion holding in at... |
| atoml2i 32476 | An assertion holding in at... |
| atordi 32477 | An ordering law for a Hilb... |
| atcvatlem 32478 | Lemma for ~ atcvati . (Co... |
| atcvati 32479 | A nonzero Hilbert lattice ... |
| atcvat2i 32480 | A Hilbert lattice element ... |
| atord 32481 | An ordering law for a Hilb... |
| atcvat2 32482 | A Hilbert lattice element ... |
| chirredlem1 32483 | Lemma for ~ chirredi . (C... |
| chirredlem2 32484 | Lemma for ~ chirredi . (C... |
| chirredlem3 32485 | Lemma for ~ chirredi . (C... |
| chirredlem4 32486 | Lemma for ~ chirredi . (C... |
| chirredi 32487 | The Hilbert lattice is irr... |
| chirred 32488 | The Hilbert lattice is irr... |
| atcvat3i 32489 | A condition implying that ... |
| atcvat4i 32490 | A condition implying exist... |
| atdmd 32491 | Two Hilbert lattice elemen... |
| atmd 32492 | Two Hilbert lattice elemen... |
| atmd2 32493 | Two Hilbert lattice elemen... |
| atabsi 32494 | Absorption of an incompara... |
| atabs2i 32495 | Absorption of an incompara... |
| mdsymlem1 32496 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32497 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32498 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32499 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32500 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32501 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32502 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32503 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32504 | M-symmetry of the Hilbert ... |
| mdsym 32505 | M-symmetry of the Hilbert ... |
| dmdsym 32506 | Dual M-symmetry of the Hil... |
| atdmd2 32507 | Two Hilbert lattice elemen... |
| sumdmdii 32508 | If the subspace sum of two... |
| cmmdi 32509 | Commuting subspaces form a... |
| cmdmdi 32510 | Commuting subspaces form a... |
| sumdmdlem 32511 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32512 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32513 | The subspace sum of two Hi... |
| dmdbr4ati 32514 | Dual modular pair property... |
| dmdbr5ati 32515 | Dual modular pair property... |
| dmdbr6ati 32516 | Dual modular pair property... |
| dmdbr7ati 32517 | Dual modular pair property... |
| mdoc1i 32518 | Orthocomplements form a mo... |
| mdoc2i 32519 | Orthocomplements form a mo... |
| dmdoc1i 32520 | Orthocomplements form a du... |
| dmdoc2i 32521 | Orthocomplements form a du... |
| mdcompli 32522 | A condition equivalent to ... |
| dmdcompli 32523 | A condition equivalent to ... |
| mddmdin0i 32524 | If dual modular implies mo... |
| cdjreui 32525 | A member of the sum of dis... |
| cdj1i 32526 | Two ways to express " ` A ... |
| cdj3lem1 32527 | A property of " ` A ` and ... |
| cdj3lem2 32528 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32529 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32530 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32531 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32532 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32533 | Lemma for ~ cdj3i . The s... |
| cdj3i 32534 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32535 | (_This theorem is a dummy ... |
| sa-abvi 32536 | A theorem about the univer... |
| xfree 32537 | A partial converse to ~ 19... |
| xfree2 32538 | A partial converse to ~ 19... |
| addltmulALT 32539 | A proof readability experi... |
| ad11antr 32540 | Deduction adding 11 conjun... |
| simp-12l 32541 | Simplification of a conjun... |
| simp-12r 32542 | Simplification of a conjun... |
| an52ds 32543 | Inference exchanging the l... |
| an62ds 32544 | Inference exchanging the l... |
| an72ds 32545 | Inference exchanging the l... |
| an82ds 32546 | Inference exchanging the l... |
| syl22anbrc 32547 | Syllogism inference. (Con... |
| bian1dOLD 32548 | Obsolete version of ~ bian... |
| orim12da 32549 | Deduce a disjunction from ... |
| or3di 32550 | Distributive law for disju... |
| or3dir 32551 | Distributive law for disju... |
| 3o1cs 32552 | Deduction eliminating disj... |
| 3o2cs 32553 | Deduction eliminating disj... |
| 3o3cs 32554 | Deduction eliminating disj... |
| 13an22anass 32555 | Associative law for four c... |
| sbc2iedf 32556 | Conversion of implicit sub... |
| rspc2daf 32557 | Double restricted speciali... |
| ralcom4f 32558 | Commutation of restricted ... |
| rexcom4f 32559 | Commutation of restricted ... |
| 19.9d2rf 32560 | A deduction version of one... |
| 19.9d2r 32561 | A deduction version of one... |
| r19.29ffa 32562 | A commonly used pattern ba... |
| n0limd 32563 | Deduction rule for nonempt... |
| reu6dv 32564 | A condition which implies ... |
| eqtrb 32565 | A transposition of equalit... |
| eqelbid 32566 | A variable elimination law... |
| opsbc2ie 32567 | Conversion of implicit sub... |
| opreu2reuALT 32568 | Correspondence between uni... |
| 2reucom 32571 | Double restricted existent... |
| 2reu2rex1 32572 | Double restricted existent... |
| 2reureurex 32573 | Double restricted existent... |
| 2reu2reu2 32574 | Double restricted existent... |
| opreu2reu1 32575 | Equivalent definition of t... |
| sq2reunnltb 32576 | There exists a unique deco... |
| addsqnot2reu 32577 | For each complex number ` ... |
| sbceqbidf 32578 | Equality theorem for class... |
| sbcies 32579 | A special version of class... |
| mo5f 32580 | Alternate definition of "a... |
| nmo 32581 | Negation of "at most one".... |
| reuxfrdf 32582 | Transfer existential uniqu... |
| rexunirn 32583 | Restricted existential qua... |
| rmoxfrd 32584 | Transfer "at most one" res... |
| rmoun 32585 | "At most one" restricted e... |
| rmounid 32586 | A case where an "at most o... |
| riotaeqbidva 32587 | Equivalent wff's yield equ... |
| dmrab 32588 | Domain of a restricted cla... |
| difrab2 32589 | Difference of two restrict... |
| elrabrd 32590 | Deduction version of ~ elr... |
| rabexgfGS 32591 | Separation Scheme in terms... |
| rabsnel 32592 | Truth implied by equality ... |
| rabsspr 32593 | Conditions for a restricte... |
| rabsstp 32594 | Conditions for a restricte... |
| 3unrab 32595 | Union of three restricted ... |
| foresf1o 32596 | From a surjective function... |
| rabfodom 32597 | Domination relation for re... |
| rabrexfi 32598 | Conditions for a class abs... |
| abrexdomjm 32599 | An indexed set is dominate... |
| abrexdom2jm 32600 | An indexed set is dominate... |
| abrexexd 32601 | Existence of a class abstr... |
| elabreximd 32602 | Class substitution in an i... |
| elabreximdv 32603 | Class substitution in an i... |
| abrexss 32604 | A necessary condition for ... |
| nelun 32605 | Negated membership for a u... |
| snsssng 32606 | If a singleton is a subset... |
| n0nsnel 32607 | If a class with one elemen... |
| inin 32608 | Intersection with an inter... |
| difininv 32609 | Condition for the intersec... |
| difeq 32610 | Rewriting an equation with... |
| eqdif 32611 | If both set differences of... |
| indifbi 32612 | Two ways to express equali... |
| diffib 32613 | Case where ~ diffi is a bi... |
| difxp1ss 32614 | Difference law for Cartesi... |
| difxp2ss 32615 | Difference law for Cartesi... |
| indifundif 32616 | A remarkable equation with... |
| elpwincl1 32617 | Closure of intersection wi... |
| elpwdifcl 32618 | Closure of class differenc... |
| elpwiuncl 32619 | Closure of indexed union w... |
| elpreq 32620 | Equality wihin a pair. (C... |
| prssad 32621 | If a pair is a subset of a... |
| prssbd 32622 | If a pair is a subset of a... |
| nelpr 32623 | A set ` A ` not in a pair ... |
| inpr0 32624 | Rewrite an empty intersect... |
| neldifpr1 32625 | The first element of a pai... |
| neldifpr2 32626 | The second element of a pa... |
| unidifsnel 32627 | The other element of a pai... |
| unidifsnne 32628 | The other element of a pai... |
| tpssg 32629 | An unordered triple of ele... |
| tpssd 32630 | Deduction version of tpssi... |
| tpssad 32631 | If an ordered triple is a ... |
| tpssbd 32632 | If an ordered triple is a ... |
| tpsscd 32633 | If an ordered triple is a ... |
| ifeqeqx 32634 | An equality theorem tailor... |
| elimifd 32635 | Elimination of a condition... |
| elim2if 32636 | Elimination of two conditi... |
| elim2ifim 32637 | Elimination of two conditi... |
| ifeq3da 32638 | Given an expression ` C ` ... |
| ifnetrue 32639 | Deduce truth from a condit... |
| ifnefals 32640 | Deduce falsehood from a co... |
| ifnebib 32641 | The converse of ~ ifbi hol... |
| ififcom 32642 | Commute two nested conditi... |
| uniinn0 32643 | Sufficient and necessary c... |
| uniin1 32644 | Union of intersection. Ge... |
| uniin2 32645 | Union of intersection. Ge... |
| difuncomp 32646 | Express a class difference... |
| elpwunicl 32647 | Closure of a set union wit... |
| cbviunf 32648 | Rule used to change the bo... |
| iuneq12daf 32649 | Equality deduction for ind... |
| iunin1f 32650 | Indexed union of intersect... |
| ssiun3 32651 | Subset equivalence for an ... |
| ssiun2sf 32652 | Subset relationship for an... |
| iuninc 32653 | The union of an increasing... |
| iundifdifd 32654 | The intersection of a set ... |
| iundifdif 32655 | The intersection of a set ... |
| iunrdx 32656 | Re-index an indexed union.... |
| iunpreima 32657 | Preimage of an indexed uni... |
| iunrnmptss 32658 | A subset relation for an i... |
| iunxunsn 32659 | Appending a set to an inde... |
| iunxunpr 32660 | Appending two sets to an i... |
| iunxpssiun1 32661 | Provide an upper bound for... |
| iinabrex 32662 | Rewriting an indexed inter... |
| disjnf 32663 | In case ` x ` is not free ... |
| cbvdisjf 32664 | Change bound variables in ... |
| disjss1f 32665 | A subset of a disjoint col... |
| disjeq1f 32666 | Equality theorem for disjo... |
| disjxun0 32667 | Simplify a disjoint union.... |
| disjdifprg 32668 | A trivial partition into a... |
| disjdifprg2 32669 | A trivial partition of a s... |
| disji2f 32670 | Property of a disjoint col... |
| disjif 32671 | Property of a disjoint col... |
| disjorf 32672 | Two ways to say that a col... |
| disjorsf 32673 | Two ways to say that a col... |
| disjif2 32674 | Property of a disjoint col... |
| disjabrex 32675 | Rewriting a disjoint colle... |
| disjabrexf 32676 | Rewriting a disjoint colle... |
| disjpreima 32677 | A preimage of a disjoint s... |
| disjrnmpt 32678 | Rewriting a disjoint colle... |
| disjin 32679 | If a collection is disjoin... |
| disjin2 32680 | If a collection is disjoin... |
| disjxpin 32681 | Derive a disjunction over ... |
| iundisjf 32682 | Rewrite a countable union ... |
| iundisj2f 32683 | A disjoint union is disjoi... |
| disjrdx 32684 | Re-index a disjunct collec... |
| disjex 32685 | Two ways to say that two c... |
| disjexc 32686 | A variant of ~ disjex , ap... |
| disjunsn 32687 | Append an element to a dis... |
| disjun0 32688 | Adding the empty element p... |
| disjiunel 32689 | A set of elements B of a d... |
| disjuniel 32690 | A set of elements B of a d... |
| xpdisjres 32691 | Restriction of a constant ... |
| opeldifid 32692 | Ordered pair elementhood o... |
| difres 32693 | Case when class difference... |
| imadifxp 32694 | Image of the difference wi... |
| relfi 32695 | A relation (set) is finite... |
| 0res 32696 | Restriction of the empty f... |
| fcoinver 32697 | Build an equivalence relat... |
| fcoinvbr 32698 | Binary relation for the eq... |
| breq1dd 32699 | Equality deduction for a b... |
| breq2dd 32700 | Equality deduction for a b... |
| brab2d 32701 | Expressing that two sets a... |
| brabgaf 32702 | The law of concretion for ... |
| brelg 32703 | Two things in a binary rel... |
| br8d 32704 | Substitution for an eight-... |
| fnfvor 32705 | Relation between two funct... |
| ofrco 32706 | Function relation between ... |
| opabdm 32707 | Domain of an ordered-pair ... |
| opabrn 32708 | Range of an ordered-pair c... |
| opabssi 32709 | Sufficient condition for a... |
| opabid2ss 32710 | One direction of ~ opabid2... |
| ssrelf 32711 | A subclass relationship de... |
| eqrelrd2 32712 | A version of ~ eqrelrdv2 w... |
| erbr3b 32713 | Biconditional for equivale... |
| iunsnima 32714 | Image of a singleton by an... |
| iunsnima2 32715 | Version of ~ iunsnima with... |
| fconst7v 32716 | An alternative way to expr... |
| constcof 32717 | Composition with a constan... |
| ac6sf2 32718 | Alternate version of ~ ac6... |
| ac6mapd 32719 | Axiom of choice equivalent... |
| fnresin 32720 | Restriction of a function ... |
| fresunsn 32721 | Recover the original funct... |
| f1o3d 32722 | Describe an implicit one-t... |
| eldmne0 32723 | A function of nonempty dom... |
| f1rnen 32724 | Equinumerosity of the rang... |
| f1oeq3dd 32725 | Equality deduction for one... |
| rinvf1o 32726 | Sufficient conditions for ... |
| fresf1o 32727 | Conditions for a restricti... |
| nfpconfp 32728 | The set of fixed points of... |
| fmptco1f1o 32729 | The action of composing (t... |
| cofmpt2 32730 | Express composition of a m... |
| f1mptrn 32731 | Express injection for a ma... |
| dfimafnf 32732 | Alternate definition of th... |
| funimass4f 32733 | Membership relation for th... |
| suppss2f 32734 | Show that the support of a... |
| ofrn 32735 | The range of the function ... |
| ofrn2 32736 | The range of the function ... |
| off2 32737 | The function operation pro... |
| ofresid 32738 | Applying an operation rest... |
| unipreima 32739 | Preimage of a class union.... |
| opfv 32740 | Value of a function produc... |
| xppreima 32741 | The preimage of a Cartesia... |
| 2ndimaxp 32742 | Image of a cartesian produ... |
| dmdju 32743 | Domain of a disjoint union... |
| djussxp2 32744 | Stronger version of ~ djus... |
| 2ndresdju 32745 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32746 | The ` 2nd ` function restr... |
| xppreima2 32747 | The preimage of a Cartesia... |
| abfmpunirn 32748 | Membership in a union of a... |
| rabfmpunirn 32749 | Membership in a union of a... |
| abfmpeld 32750 | Membership in an element o... |
| abfmpel 32751 | Membership in an element o... |
| fmptdF 32752 | Domain and codomain of the... |
| fmptcof2 32753 | Composition of two functio... |
| fcomptf 32754 | Express composition of two... |
| acunirnmpt 32755 | Axiom of choice for the un... |
| acunirnmpt2 32756 | Axiom of choice for the un... |
| acunirnmpt2f 32757 | Axiom of choice for the un... |
| aciunf1lem 32758 | Choice in an index union. ... |
| aciunf1 32759 | Choice in an index union. ... |
| ofoprabco 32760 | Function operation as a co... |
| ofpreima 32761 | Express the preimage of a ... |
| ofpreima2 32762 | Express the preimage of a ... |
| funcnv5mpt 32763 | Two ways to say that a fun... |
| funcnv4mpt 32764 | Two ways to say that a fun... |
| preimane 32765 | Different elements have di... |
| fnpreimac 32766 | Choose a set ` x ` contain... |
| fgreu 32767 | Exactly one point of a fun... |
| fcnvgreu 32768 | If the converse of a relat... |
| rnmposs 32769 | The range of an operation ... |
| mptssALT 32770 | Deduce subset relation of ... |
| dfcnv2 32771 | Alternative definition of ... |
| partfun2 32772 | Rewrite a function defined... |
| rnressnsn 32773 | The range of a restriction... |
| mpomptxf 32774 | Express a two-argument fun... |
| of0r 32775 | Function operation with th... |
| elmaprd 32776 | Deduction associated with ... |
| suppovss 32777 | A bound for the support of... |
| elsuppfnd 32778 | Deduce membership in the s... |
| fisuppov1 32779 | Formula building theorem f... |
| suppun2 32780 | The support of a union is ... |
| fdifsupp 32781 | Express the support of a f... |
| suppiniseg 32782 | Relation between the suppo... |
| fsuppinisegfi 32783 | The initial segment ` ( ``... |
| fressupp 32784 | The restriction of a funct... |
| fdifsuppconst 32785 | A function is a zero const... |
| ressupprn 32786 | The range of a function re... |
| supppreima 32787 | Express the support of a f... |
| fsupprnfi 32788 | Finite support implies fin... |
| mptiffisupp 32789 | Conditions for a mapping f... |
| cosnopne 32790 | Composition of two ordered... |
| cosnop 32791 | Composition of two ordered... |
| cnvprop 32792 | Converse of a pair of orde... |
| brprop 32793 | Binary relation for a pair... |
| mptprop 32794 | Rewrite pairs of ordered p... |
| coprprop 32795 | Composition of two pairs o... |
| fmptunsnop 32796 | Two ways to express a func... |
| gtiso 32797 | Two ways to write a strict... |
| isoun 32798 | Infer an isomorphism from ... |
| disjdsct 32799 | A disjoint collection is d... |
| df1stres 32800 | Definition for a restricti... |
| df2ndres 32801 | Definition for a restricti... |
| 1stpreimas 32802 | The preimage of a singleto... |
| 1stpreima 32803 | The preimage by ` 1st ` is... |
| 2ndpreima 32804 | The preimage by ` 2nd ` is... |
| curry2ima 32805 | The image of a curried fun... |
| preiman0 32806 | The preimage of a nonempty... |
| intimafv 32807 | The intersection of an ima... |
| snct 32808 | A singleton is countable. ... |
| prct 32809 | An unordered pair is count... |
| mpocti 32810 | An operation is countable ... |
| abrexct 32811 | An image set of a countabl... |
| mptctf 32812 | A countable mapping set is... |
| abrexctf 32813 | An image set of a countabl... |
| padct 32814 | Index a countable set with... |
| f1od2 32815 | Sufficient condition for a... |
| fcobij 32816 | Composing functions with a... |
| fcobijfs 32817 | Composing finitely support... |
| fcobijfs2 32818 | Composing finitely support... |
| suppss3 32819 | Deduce a function's suppor... |
| fsuppcurry1 32820 | Finite support of a currie... |
| fsuppcurry2 32821 | Finite support of a currie... |
| offinsupp1 32822 | Finite support for a funct... |
| ffs2 32823 | Rewrite a function's suppo... |
| ffsrn 32824 | The range of a finitely su... |
| cocnvf1o 32825 | Composing with the inverse... |
| resf1o 32826 | Restriction of functions t... |
| maprnin 32827 | Restricting the range of t... |
| fpwrelmapffslem 32828 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32829 | Define a canonical mapping... |
| fpwrelmapffs 32830 | Define a canonical mapping... |
| sgnval2 32831 | Value of the signum of a r... |
| creq0 32832 | The real representation of... |
| 1nei 32833 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32834 | An integer unit times itse... |
| nnmulge 32835 | Multiplying by a positive ... |
| submuladdd 32836 | The product of a differenc... |
| binom2subadd 32837 | The difference of the squa... |
| cjsubd 32838 | Complex conjugate distribu... |
| re0cj 32839 | The conjugate of a pure im... |
| receqid 32840 | Real numbers equal to thei... |
| pythagreim 32841 | A simplified version of th... |
| efiargd 32842 | The exponential of the "ar... |
| arginv 32843 | The argument of the invers... |
| argcj 32844 | The argument of the conjug... |
| quad3d 32845 | Variant of quadratic equat... |
| lt2addrd 32846 | If the right-hand side of ... |
| nn0mnfxrd 32847 | Nonnegative integers or mi... |
| xrlelttric 32848 | Trichotomy law for extende... |
| xaddeq0 32849 | Two extended reals which a... |
| rexmul2 32850 | If the result ` A ` of an ... |
| xrinfm 32851 | The extended real numbers ... |
| le2halvesd 32852 | A sum is less than the who... |
| xraddge02 32853 | A number is less than or e... |
| xrge0addge 32854 | A number is less than or e... |
| xlt2addrd 32855 | If the right-hand side of ... |
| xrge0infss 32856 | Any subset of nonnegative ... |
| xrge0infssd 32857 | Inequality deduction for i... |
| xrge0addcld 32858 | Nonnegative extended reals... |
| xrge0subcld 32859 | Condition for closure of n... |
| infxrge0lb 32860 | A member of a set of nonne... |
| infxrge0glb 32861 | The infimum of a set of no... |
| infxrge0gelb 32862 | The infimum of a set of no... |
| xrofsup 32863 | The supremum is preserved ... |
| supxrnemnf 32864 | The supremum of a nonempty... |
| xnn0gt0 32865 | Nonzero extended nonnegati... |
| xnn01gt 32866 | An extended nonnegative in... |
| nn0xmulclb 32867 | Finite multiplication in t... |
| xnn0nn0d 32868 | Conditions for an extended... |
| xnn0nnd 32869 | Conditions for an extended... |
| joiniooico 32870 | Disjoint joining an open i... |
| ubico 32871 | A right-open interval does... |
| xeqlelt 32872 | Equality in terms of 'less... |
| eliccelico 32873 | Relate elementhood to a cl... |
| elicoelioo 32874 | Relate elementhood to a cl... |
| iocinioc2 32875 | Intersection between two o... |
| xrdifh 32876 | Class difference of a half... |
| iocinif 32877 | Relate intersection of two... |
| difioo 32878 | The difference between two... |
| difico 32879 | The difference between two... |
| uzssico 32880 | Upper integer sets are a s... |
| fz2ssnn0 32881 | A finite set of sequential... |
| nndiffz1 32882 | Upper set of the positive ... |
| ssnnssfz 32883 | For any finite subset of `... |
| fzm1ne1 32884 | Elementhood of an integer ... |
| fzspl 32885 | Split the last element of ... |
| fzdif2 32886 | Split the last element of ... |
| fzodif2 32887 | Split the last element of ... |
| fzodif1 32888 | Set difference of two half... |
| fzsplit3 32889 | Split a finite interval of... |
| nn0diffz0 32890 | Upper set of the nonnegati... |
| bcm1n 32891 | The proportion of one bino... |
| iundisjfi 32892 | Rewrite a countable union ... |
| iundisj2fi 32893 | A disjoint union is disjoi... |
| iundisjcnt 32894 | Rewrite a countable union ... |
| iundisj2cnt 32895 | A countable disjoint union... |
| f1ocnt 32896 | Given a countable set ` A ... |
| fz1nnct 32897 | NN and integer ranges star... |
| fz1nntr 32898 | NN and integer ranges star... |
| fzo0opth 32899 | Equality for a half open i... |
| nn0difffzod 32900 | A nonnegative integer that... |
| suppssnn0 32901 | Show that the support of a... |
| hashunif 32902 | The cardinality of a disjo... |
| hashxpe 32903 | The size of the Cartesian ... |
| hashgt1 32904 | Restate "set contains at l... |
| hashpss 32905 | The size of a proper subse... |
| hashne0 32906 | Deduce that the size of a ... |
| hashimaf1 32907 | Taking the image of a set ... |
| elq2 32908 | Elementhood in the rationa... |
| znumd 32909 | Numerator of an integer. ... |
| zdend 32910 | Denominator of an integer.... |
| numdenneg 32911 | Numerator and denominator ... |
| divnumden2 32912 | Calculate the reduced form... |
| expgt0b 32913 | A real number ` A ` raised... |
| nn0split01 32914 | Split 0 and 1 from the non... |
| nn0disj01 32915 | The pair ` { 0 , 1 } ` doe... |
| nnindf 32916 | Principle of Mathematical ... |
| nn0min 32917 | Extracting the minimum pos... |
| subne0nn 32918 | A nonnegative difference i... |
| ltesubnnd 32919 | Subtracting an integer num... |
| fprodeq02 32920 | If one of the factors is z... |
| fprodex01 32921 | A product of factors equal... |
| prodpr 32922 | A product over a pair is t... |
| prodtp 32923 | A product over a triple is... |
| fsumub 32924 | An upper bound for a term ... |
| fsumiunle 32925 | Upper bound for a sum of n... |
| dfdec100 32926 | Split the hundreds from a ... |
| sgncl 32927 | Closure of the signum. (C... |
| sgnclre 32928 | Closure of the signum. (C... |
| sgnneg 32929 | Negation of the signum. (... |
| sgn3da 32930 | A conditional containing a... |
| sgnmul 32931 | Signum of a product. (Con... |
| sgnmulrp2 32932 | Multiplication by a positi... |
| sgnsub 32933 | Subtraction of a number of... |
| sgnnbi 32934 | Negative signum. (Contrib... |
| sgnpbi 32935 | Positive signum. (Contrib... |
| sgn0bi 32936 | Zero signum. (Contributed... |
| sgnsgn 32937 | Signum is idempotent. (Co... |
| sgnmulsgn 32938 | If two real numbers are of... |
| sgnmulsgp 32939 | If two real numbers are of... |
| nexple 32940 | A lower bound for an expon... |
| 2exple2exp 32941 | If a nonnegative integer `... |
| expevenpos 32942 | Even powers are positive. ... |
| oexpled 32943 | Odd power monomials are mo... |
| indsumin 32944 | Finite sum of a product wi... |
| prodindf 32945 | The product of indicators ... |
| indsn 32946 | The indicator function of ... |
| indf1o 32947 | The bijection between a po... |
| indpreima 32948 | A function with range ` { ... |
| indf1ofs 32949 | The bijection between fini... |
| indsupp 32950 | The support of the indicat... |
| indfsd 32951 | The indicator function of ... |
| indfsid 32952 | Conditions for a function ... |
| dp2eq1 32955 | Equality theorem for the d... |
| dp2eq2 32956 | Equality theorem for the d... |
| dp2eq1i 32957 | Equality theorem for the d... |
| dp2eq2i 32958 | Equality theorem for the d... |
| dp2eq12i 32959 | Equality theorem for the d... |
| dp20u 32960 | Add a zero in the tenths (... |
| dp20h 32961 | Add a zero in the unit pla... |
| dp2cl 32962 | Closure for the decimal fr... |
| dp2clq 32963 | Closure for a decimal frac... |
| rpdp2cl 32964 | Closure for a decimal frac... |
| rpdp2cl2 32965 | Closure for a decimal frac... |
| dp2lt10 32966 | Decimal fraction builds re... |
| dp2lt 32967 | Comparing two decimal frac... |
| dp2ltsuc 32968 | Comparing a decimal fracti... |
| dp2ltc 32969 | Comparing two decimal expa... |
| dpval 32972 | Define the value of the de... |
| dpcl 32973 | Prove that the closure of ... |
| dpfrac1 32974 | Prove a simple equivalence... |
| dpval2 32975 | Value of the decimal point... |
| dpval3 32976 | Value of the decimal point... |
| dpmul10 32977 | Multiply by 10 a decimal e... |
| decdiv10 32978 | Divide a decimal number by... |
| dpmul100 32979 | Multiply by 100 a decimal ... |
| dp3mul10 32980 | Multiply by 10 a decimal e... |
| dpmul1000 32981 | Multiply by 1000 a decimal... |
| dpval3rp 32982 | Value of the decimal point... |
| dp0u 32983 | Add a zero in the tenths p... |
| dp0h 32984 | Remove a zero in the units... |
| rpdpcl 32985 | Closure of the decimal poi... |
| dplt 32986 | Comparing two decimal expa... |
| dplti 32987 | Comparing a decimal expans... |
| dpgti 32988 | Comparing a decimal expans... |
| dpltc 32989 | Comparing two decimal inte... |
| dpexpp1 32990 | Add one zero to the mantis... |
| 0dp2dp 32991 | Multiply by 10 a decimal e... |
| dpadd2 32992 | Addition with one decimal,... |
| dpadd 32993 | Addition with one decimal.... |
| dpadd3 32994 | Addition with two decimals... |
| dpmul 32995 | Multiplication with one de... |
| dpmul4 32996 | An upper bound to multipli... |
| threehalves 32997 | Example theorem demonstrat... |
| 1mhdrd 32998 | Example theorem demonstrat... |
| xdivval 33001 | Value of division: the (un... |
| xrecex 33002 | Existence of reciprocal of... |
| xmulcand 33003 | Cancellation law for exten... |
| xreceu 33004 | Existential uniqueness of ... |
| xdivcld 33005 | Closure law for the extend... |
| xdivcl 33006 | Closure law for the extend... |
| xdivmul 33007 | Relationship between divis... |
| rexdiv 33008 | The extended real division... |
| xdivrec 33009 | Relationship between divis... |
| xdivid 33010 | A number divided by itself... |
| xdiv0 33011 | Division into zero is zero... |
| xdiv0rp 33012 | Division into zero is zero... |
| eliccioo 33013 | Membership in a closed int... |
| elxrge02 33014 | Elementhood in the set of ... |
| xdivpnfrp 33015 | Plus infinity divided by a... |
| rpxdivcld 33016 | Closure law for extended d... |
| xrpxdivcld 33017 | Closure law for extended d... |
| wrdres 33018 | Condition for the restrict... |
| wrdsplex 33019 | Existence of a split of a ... |
| wrdfsupp 33020 | A word has finite support.... |
| wrdpmcl 33021 | Closure of a word with per... |
| pfx1s2 33022 | The prefix of length 1 of ... |
| pfxrn2 33023 | The range of a prefix of a... |
| pfxrn3 33024 | Express the range of a pre... |
| pfxf1 33025 | Condition for a prefix to ... |
| s1f1 33026 | Conditions for a length 1 ... |
| s2rnOLD 33027 | Obsolete version of ~ s2rn... |
| s2f1 33028 | Conditions for a length 2 ... |
| s3rnOLD 33029 | Obsolete version of ~ s2rn... |
| s3f1 33030 | Conditions for a length 3 ... |
| s3clhash 33031 | Closure of the words of le... |
| ccatf1 33032 | Conditions for a concatena... |
| pfxlsw2ccat 33033 | Reconstruct a word from it... |
| ccatws1f1o 33034 | Conditions for the concate... |
| ccatws1f1olast 33035 | Two ways to reorder symbol... |
| wrdt2ind 33036 | Perform an induction over ... |
| swrdrn2 33037 | The range of a subword is ... |
| swrdrn3 33038 | Express the range of a sub... |
| swrdf1 33039 | Condition for a subword to... |
| swrdrndisj 33040 | Condition for the range of... |
| splfv3 33041 | Symbols to the right of a ... |
| 1cshid 33042 | Cyclically shifting a sing... |
| cshw1s2 33043 | Cyclically shifting a leng... |
| cshwrnid 33044 | Cyclically shifting a word... |
| cshf1o 33045 | Condition for the cyclic s... |
| ressplusf 33046 | The group operation functi... |
| ressnm 33047 | The norm in a restricted s... |
| abvpropd2 33048 | Weaker version of ~ abvpro... |
| ressprs 33049 | The restriction of a prose... |
| posrasymb 33050 | A poset ordering is asymme... |
| odutos 33051 | Being a toset is a self-du... |
| tlt2 33052 | In a Toset, two elements m... |
| tlt3 33053 | In a Toset, two elements m... |
| trleile 33054 | In a Toset, two elements m... |
| toslublem 33055 | Lemma for ~ toslub and ~ x... |
| toslub 33056 | In a toset, the lowest upp... |
| tosglblem 33057 | Lemma for ~ tosglb and ~ x... |
| tosglb 33058 | Same theorem as ~ toslub ,... |
| clatp0cl 33059 | The poset zero of a comple... |
| clatp1cl 33060 | The poset one of a complet... |
| mntoval 33065 | Operation value of the mon... |
| ismnt 33066 | Express the statement " ` ... |
| ismntd 33067 | Property of being a monoto... |
| mntf 33068 | A monotone function is a f... |
| mgcoval 33069 | Operation value of the mon... |
| mgcval 33070 | Monotone Galois connection... |
| mgcf1 33071 | The lower adjoint ` F ` of... |
| mgcf2 33072 | The upper adjoint ` G ` of... |
| mgccole1 33073 | An inequality for the kern... |
| mgccole2 33074 | Inequality for the closure... |
| mgcmnt1 33075 | The lower adjoint ` F ` of... |
| mgcmnt2 33076 | The upper adjoint ` G ` of... |
| mgcmntco 33077 | A Galois connection like s... |
| dfmgc2lem 33078 | Lemma for dfmgc2, backward... |
| dfmgc2 33079 | Alternate definition of th... |
| mgcmnt1d 33080 | Galois connection implies ... |
| mgcmnt2d 33081 | Galois connection implies ... |
| mgccnv 33082 | The inverse Galois connect... |
| pwrssmgc 33083 | Given a function ` F ` , e... |
| mgcf1olem1 33084 | Property of a Galois conne... |
| mgcf1olem2 33085 | Property of a Galois conne... |
| mgcf1o 33086 | Given a Galois connection,... |
| xrs0 33089 | The zero of the extended r... |
| xrslt 33090 | The "strictly less than" r... |
| xrsinvgval 33091 | The inversion operation in... |
| xrsmulgzz 33092 | The "multiple" function in... |
| xrstos 33093 | The extended real numbers ... |
| xrsclat 33094 | The extended real numbers ... |
| xrsp0 33095 | The poset 0 of the extende... |
| xrsp1 33096 | The poset 1 of the extende... |
| xrge00 33097 | The zero of the extended n... |
| xrge0mulgnn0 33098 | The group multiple functio... |
| xrge0addass 33099 | Associativity of extended ... |
| xrge0addgt0 33100 | The sum of nonnegative and... |
| xrge0adddir 33101 | Right-distributivity of ex... |
| xrge0adddi 33102 | Left-distributivity of ext... |
| xrge0npcan 33103 | Extended nonnegative real ... |
| fsumrp0cl 33104 | Closure of a finite sum of... |
| mndcld 33105 | Closure of the operation o... |
| mndassd 33106 | A monoid operation is asso... |
| mndlrinv 33107 | In a monoid, if an element... |
| mndlrinvb 33108 | In a monoid, if an element... |
| mndlactf1 33109 | If an element ` X ` of a m... |
| mndlactfo 33110 | An element ` X ` of a mono... |
| mndractf1 33111 | If an element ` X ` of a m... |
| mndractfo 33112 | An element ` X ` of a mono... |
| mndlactf1o 33113 | An element ` X ` of a mono... |
| mndractf1o 33114 | An element ` X ` of a mono... |
| cmn4d 33115 | Commutative/associative la... |
| cmn246135 33116 | Rearrange terms in a commu... |
| cmn145236 33117 | Rearrange terms in a commu... |
| submcld 33118 | Submonoids are closed unde... |
| abliso 33119 | The image of an Abelian gr... |
| lmhmghmd 33120 | A module homomorphism is a... |
| mhmimasplusg 33121 | Value of the operation of ... |
| lmhmimasvsca 33122 | Value of the scalar produc... |
| grpidcld 33123 | The identity element of a ... |
| grpinvinvd 33124 | Double inverse law for gro... |
| grpsubcld 33125 | Closure of group subtracti... |
| subgcld 33126 | A subgroup is closed under... |
| subgsubcld 33127 | A subgroup is closed under... |
| subgmulgcld 33128 | Closure of the group multi... |
| ressmulgnn0d 33129 | Values for the group multi... |
| ablcomd 33130 | An abelian group operation... |
| gsumsubg 33131 | The group sum in a subgrou... |
| gsumsra 33132 | The group sum in a subring... |
| gsummpt2co 33133 | Split a finite sum into a ... |
| gsummpt2d 33134 | Express a finite sum over ... |
| lmodvslmhm 33135 | Scalar multiplication in a... |
| gsumvsmul1 33136 | Pull a scalar multiplicati... |
| gsummptres 33137 | Extend a finite group sum ... |
| gsummptres2 33138 | Extend a finite group sum ... |
| gsummptfsres 33139 | Extend a finitely supporte... |
| gsummptf1od 33140 | Re-index a finite group su... |
| gsummptrev 33141 | Revert ordering in a group... |
| gsummptp1 33142 | Reindex a zero-based sum a... |
| gsummptfzsplitra 33143 | Split a group sum expresse... |
| gsummptfzsplitla 33144 | Split a group sum expresse... |
| gsummptfsf1o 33145 | Re-index a finite group su... |
| gsumfs2d 33146 | Express a finite sum over ... |
| gsumzresunsn 33147 | Append an element to a fin... |
| gsumpart 33148 | Express a group sum as a d... |
| gsumtp 33149 | Group sum of an unordered ... |
| gsumzrsum 33150 | Relate a group sum on ` ZZ... |
| gsummulgc2 33151 | A finite group sum multipl... |
| gsumhashmul 33152 | Express a group sum by gro... |
| gsummulsubdishift1 33153 | Distribute a subtraction o... |
| gsummulsubdishift2 33154 | Distribute a subtraction o... |
| gsummulsubdishift1s 33155 | Distribute a subtraction o... |
| gsummulsubdishift2s 33156 | Distribute a subtraction o... |
| suppgsumssiun 33157 | The support of a function ... |
| xrge0tsmsd 33158 | Any finite or infinite sum... |
| xrge0tsmsbi 33159 | Any limit of a finite or i... |
| xrge0tsmseq 33160 | Any limit of a finite or i... |
| gsumwun 33161 | In a commutative ring, a g... |
| gsumwrd2dccatlem 33162 | Lemma for ~ gsumwrd2dccat ... |
| gsumwrd2dccat 33163 | Rewrite a sum ranging over... |
| cntzun 33164 | The centralizer of a union... |
| cntzsnid 33165 | The centralizer of the ide... |
| cntrcrng 33166 | The center of a ring is a ... |
| symgfcoeu 33167 | Uniqueness property of per... |
| symgcom 33168 | Two permutations ` X ` and... |
| symgcom2 33169 | Two permutations ` X ` and... |
| symgcntz 33170 | All elements of a (finite)... |
| odpmco 33171 | The composition of two odd... |
| symgsubg 33172 | The value of the group sub... |
| pmtrprfv2 33173 | In a transposition of two ... |
| pmtrcnel 33174 | Composing a permutation ` ... |
| pmtrcnel2 33175 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33176 | Composing a permutation ` ... |
| fzo0pmtrlast 33177 | Reorder a half-open intege... |
| wrdpmtrlast 33178 | Reorder a word, so that th... |
| pmtridf1o 33179 | Transpositions of ` X ` an... |
| pmtridfv1 33180 | Value at X of the transpos... |
| pmtridfv2 33181 | Value at Y of the transpos... |
| psgnid 33182 | Permutation sign of the id... |
| psgndmfi 33183 | For a finite base set, the... |
| pmtrto1cl 33184 | Useful lemma for the follo... |
| psgnfzto1stlem 33185 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33186 | Value of our permutation `... |
| fzto1st1 33187 | Special case where the per... |
| fzto1st 33188 | The function moving one el... |
| fzto1stinvn 33189 | Value of the inverse of ou... |
| psgnfzto1st 33190 | The permutation sign for m... |
| tocycval 33193 | Value of the cycle builder... |
| tocycfv 33194 | Function value of a permut... |
| tocycfvres1 33195 | A cyclic permutation is a ... |
| tocycfvres2 33196 | A cyclic permutation is th... |
| cycpmfvlem 33197 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33198 | Value of a cycle function ... |
| cycpmfv2 33199 | Value of a cycle function ... |
| cycpmfv3 33200 | Values outside of the orbi... |
| cycpmcl 33201 | Cyclic permutations are pe... |
| tocycf 33202 | The permutation cycle buil... |
| tocyc01 33203 | Permutation cycles built f... |
| cycpm2tr 33204 | A cyclic permutation of 2 ... |
| cycpm2cl 33205 | Closure for the 2-cycles. ... |
| cyc2fv1 33206 | Function value of a 2-cycl... |
| cyc2fv2 33207 | Function value of a 2-cycl... |
| trsp2cyc 33208 | Exhibit the word a transpo... |
| cycpmco2f1 33209 | The word U used in ~ cycpm... |
| cycpmco2rn 33210 | The orbit of the compositi... |
| cycpmco2lem1 33211 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33212 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33213 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33214 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33215 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33216 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33217 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33218 | The composition of a cycli... |
| cyc2fvx 33219 | Function value of a 2-cycl... |
| cycpm3cl 33220 | Closure of the 3-cycles in... |
| cycpm3cl2 33221 | Closure of the 3-cycles in... |
| cyc3fv1 33222 | Function value of a 3-cycl... |
| cyc3fv2 33223 | Function value of a 3-cycl... |
| cyc3fv3 33224 | Function value of a 3-cycl... |
| cyc3co2 33225 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33226 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33227 | A formula for computing co... |
| cycpmrn 33228 | The range of the word used... |
| tocyccntz 33229 | All elements of a (finite)... |
| evpmval 33230 | Value of the set of even p... |
| cnmsgn0g 33231 | The neutral element of the... |
| evpmsubg 33232 | The alternating group is a... |
| evpmid 33233 | The identity is an even pe... |
| altgnsg 33234 | The alternating group ` ( ... |
| cyc3evpm 33235 | 3-Cycles are even permutat... |
| cyc3genpmlem 33236 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33237 | The alternating group ` A ... |
| cycpmgcl 33238 | Cyclic permutations are pe... |
| cycpmconjslem1 33239 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33240 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33241 | All cycles of the same len... |
| cyc3conja 33242 | All 3-cycles are conjugate... |
| sgnsv 33245 | The sign mapping. (Contri... |
| sgnsval 33246 | The sign value. (Contribu... |
| sgnsf 33247 | The sign function. (Contr... |
| fxpval 33250 | Value of the set of fixed ... |
| fxpss 33251 | The set of fixed points is... |
| fxpgaval 33252 | Value of the set of fixed ... |
| isfxp 33253 | Property of being a fixed ... |
| fxpgaeq 33254 | A fixed point ` X ` is inv... |
| conjga 33255 | Group conjugation induces ... |
| cntrval2 33256 | Express the center ` Z ` o... |
| fxpsubm 33257 | Provided the group action ... |
| fxpsubg 33258 | The fixed points of a grou... |
| fxpsubrg 33259 | The fixed points of a grou... |
| fxpsdrg 33260 | The fixed points of a grou... |
| inftmrel 33265 | The infinitesimal relation... |
| isinftm 33266 | Express ` x ` is infinites... |
| isarchi 33267 | Express the predicate " ` ... |
| pnfinf 33268 | Plus infinity is an infini... |
| xrnarchi 33269 | The completed real line is... |
| isarchi2 33270 | Alternative way to express... |
| submarchi 33271 | A submonoid is archimedean... |
| isarchi3 33272 | This is the usual definiti... |
| archirng 33273 | Property of Archimedean or... |
| archirngz 33274 | Property of Archimedean le... |
| archiexdiv 33275 | In an Archimedean group, g... |
| archiabllem1a 33276 | Lemma for ~ archiabl : In... |
| archiabllem1b 33277 | Lemma for ~ archiabl . (C... |
| archiabllem1 33278 | Archimedean ordered groups... |
| archiabllem2a 33279 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33280 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33281 | Lemma for ~ archiabl . (C... |
| archiabllem2 33282 | Archimedean ordered groups... |
| archiabl 33283 | Archimedean left- and righ... |
| isarchiofld 33284 | Axiom of Archimedes : a ch... |
| isslmd 33287 | The predicate "is a semimo... |
| slmdlema 33288 | Lemma for properties of a ... |
| lmodslmd 33289 | Left semimodules generaliz... |
| slmdcmn 33290 | A semimodule is a commutat... |
| slmdmnd 33291 | A semimodule is a monoid. ... |
| slmdsrg 33292 | The scalar component of a ... |
| slmdbn0 33293 | The base set of a semimodu... |
| slmdacl 33294 | Closure of ring addition f... |
| slmdmcl 33295 | Closure of ring multiplica... |
| slmdsn0 33296 | The set of scalars in a se... |
| slmdvacl 33297 | Closure of vector addition... |
| slmdass 33298 | Semiring left module vecto... |
| slmdvscl 33299 | Closure of scalar product ... |
| slmdvsdi 33300 | Distributive law for scala... |
| slmdvsdir 33301 | Distributive law for scala... |
| slmdvsass 33302 | Associative law for scalar... |
| slmd0cl 33303 | The ring zero in a semimod... |
| slmd1cl 33304 | The ring unity in a semiri... |
| slmdvs1 33305 | Scalar product with ring u... |
| slmd0vcl 33306 | The zero vector is a vecto... |
| slmd0vlid 33307 | Left identity law for the ... |
| slmd0vrid 33308 | Right identity law for the... |
| slmd0vs 33309 | Zero times a vector is the... |
| slmdvs0 33310 | Anything times the zero ve... |
| gsumvsca1 33311 | Scalar product of a finite... |
| gsumvsca2 33312 | Scalar product of a finite... |
| prmsimpcyc 33313 | A group of prime order is ... |
| ringrngd 33314 | A unital ring is a non-uni... |
| ringdi22 33315 | Expand the product of two ... |
| urpropd 33316 | Sufficient condition for r... |
| subrgmcld 33317 | A subring is closed under ... |
| ress1r 33318 | ` 1r ` is unaffected by re... |
| ringm1expp1 33319 | Ring exponentiation of min... |
| ringinvval 33320 | The ring inverse expressed... |
| dvrcan5 33321 | Cancellation law for commo... |
| subrgchr 33322 | If ` A ` is a subring of `... |
| rmfsupp2 33323 | A mapping of a multiplicat... |
| unitnz 33324 | In a nonzero ring, a unit ... |
| isunit2 33325 | Alternate definition of be... |
| isunit3 33326 | Alternate definition of be... |
| elrgspnlem1 33327 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem2 33328 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem3 33329 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem4 33330 | Lemma for ~ elrgspn . (Co... |
| elrgspn 33331 | Membership in the subring ... |
| elrgspnsubrunlem1 33332 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrunlem2 33333 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrun 33334 | Membership in the ring spa... |
| irrednzr 33335 | A ring with an irreducible... |
| 0ringsubrg 33336 | A subring of a zero ring i... |
| 0ringcring 33337 | The zero ring is commutati... |
| reldmrloc 33342 | Ring localization is a pro... |
| erlval 33343 | Value of the ring localiza... |
| rlocval 33344 | Expand the value of the ri... |
| erlcl1 33345 | Closure for the ring local... |
| erlcl2 33346 | Closure for the ring local... |
| erldi 33347 | Main property of the ring ... |
| erlbrd 33348 | Deduce the ring localizati... |
| erlbr2d 33349 | Deduce the ring localizati... |
| erler 33350 | The relation used to build... |
| elrlocbasi 33351 | Membership in the basis of... |
| rlocbas 33352 | The base set of a ring loc... |
| rlocaddval 33353 | Value of the addition in t... |
| rlocmulval 33354 | Value of the addition in t... |
| rloccring 33355 | The ring localization ` L ... |
| rloc0g 33356 | The zero of a ring localiz... |
| rloc1r 33357 | The multiplicative identit... |
| rlocf1 33358 | The embedding ` F ` of a r... |
| domnmuln0rd 33359 | In a domain, factors of a ... |
| domnprodn0 33360 | In a domain, a finite prod... |
| domnprodeq0 33361 | A product over a domain is... |
| domnpropd 33362 | If two structures have the... |
| idompropd 33363 | If two structures have the... |
| idomrcan 33364 | Right-cancellation law for... |
| domnlcanOLD 33365 | Obsolete version of ~ domn... |
| domnlcanbOLD 33366 | Obsolete version of ~ domn... |
| idomrcanOLD 33367 | Obsolete version of ~ idom... |
| 1rrg 33368 | The multiplicative identit... |
| rrgsubm 33369 | The left regular elements ... |
| subrdom 33370 | A subring of a domain is a... |
| subridom 33371 | A subring of an integral d... |
| subrfld 33372 | A subring of a field is an... |
| ricnzr1 33373 | A ring isomorphism maps a ... |
| ricdomn1 33374 | A ring isomorphism maps a ... |
| ricdomn 33375 | A ring is a domain if and ... |
| eufndx 33378 | Index value of the Euclide... |
| eufid 33379 | Utility theorem: index-ind... |
| ringinveu 33382 | If a ring unit element ` X... |
| isdrng4 33383 | A division ring is a ring ... |
| rndrhmcl 33384 | The image of a division ri... |
| qfld 33385 | The field of rational numb... |
| subsdrg 33386 | A subring of a sub-divisio... |
| sdrgdvcl 33387 | A sub-division-ring is clo... |
| sdrginvcl 33388 | A sub-division-ring is clo... |
| primefldchr 33389 | The characteristic of a pr... |
| fracval 33392 | Value of the field of frac... |
| fracbas 33393 | The base of the field of f... |
| fracerl 33394 | Rewrite the ring localizat... |
| fracf1 33395 | The embedding of a commuta... |
| fracfld 33396 | The field of fractions of ... |
| idomsubr 33397 | Every integral domain is i... |
| fldgenval 33400 | Value of the field generat... |
| fldgenssid 33401 | The field generated by a s... |
| fldgensdrg 33402 | A generated subfield is a ... |
| fldgenssv 33403 | A generated subfield is a ... |
| fldgenss 33404 | Generated subfields preser... |
| fldgenidfld 33405 | The subfield generated by ... |
| fldgenssp 33406 | The field generated by a s... |
| fldgenid 33407 | The subfield of a field ` ... |
| fldgenfld 33408 | A generated subfield is a ... |
| primefldgen1 33409 | The prime field of a divis... |
| 1fldgenq 33410 | The field of rational numb... |
| rhmdvd 33411 | A ring homomorphism preser... |
| kerunit 33412 | If a unit element lies in ... |
| reldmresv 33415 | The scalar restriction is ... |
| resvval 33416 | Value of structure restric... |
| resvid2 33417 | General behavior of trivia... |
| resvval2 33418 | Value of nontrivial struct... |
| resvsca 33419 | Base set of a structure re... |
| resvlem 33420 | Other elements of a scalar... |
| resvbas 33421 | ` Base ` is unaffected by ... |
| resvplusg 33422 | ` +g ` is unaffected by sc... |
| resvvsca 33423 | ` .s ` is unaffected by sc... |
| resvmulr 33424 | ` .r ` is unaffected by sc... |
| resv0g 33425 | ` 0g ` is unaffected by sc... |
| resv1r 33426 | ` 1r ` is unaffected by sc... |
| resvcmn 33427 | Scalar restriction preserv... |
| gzcrng 33428 | The gaussian integers form... |
| cnfldfld 33429 | The complex numbers form a... |
| reofld 33430 | The real numbers form an o... |
| nn0omnd 33431 | The nonnegative integers f... |
| gsumind 33432 | The group sum of an indica... |
| rearchi 33433 | The field of the real numb... |
| nn0archi 33434 | The monoid of the nonnegat... |
| xrge0slmod 33435 | The extended nonnegative r... |
| qusker 33436 | The kernel of a quotient m... |
| eqgvscpbl 33437 | The left coset equivalence... |
| qusvscpbl 33438 | The quotient map distribut... |
| qusvsval 33439 | Value of the scalar multip... |
| imaslmod 33440 | The image structure of a l... |
| imasmhm 33441 | Given a function ` F ` wit... |
| imasghm 33442 | Given a function ` F ` wit... |
| imasrhm 33443 | Given a function ` F ` wit... |
| imaslmhm 33444 | Given a function ` F ` wit... |
| quslmod 33445 | If ` G ` is a submodule in... |
| quslmhm 33446 | If ` G ` is a submodule of... |
| quslvec 33447 | If ` S ` is a vector subsp... |
| ecxpid 33448 | The equivalence class of a... |
| qsxpid 33449 | The quotient set of a cart... |
| qusxpid 33450 | The Group quotient equival... |
| qustriv 33451 | The quotient of a group ` ... |
| qustrivr 33452 | Converse of ~ qustriv . (... |
| znfermltl 33453 | Fermat's little theorem in... |
| islinds5 33454 | A set is linearly independ... |
| ellspds 33455 | Variation on ~ ellspd . (... |
| 0ellsp 33456 | Zero is in all spans. (Co... |
| 0nellinds 33457 | The group identity cannot ... |
| rspsnid 33458 | A principal ideal contains... |
| elrsp 33459 | Write the elements of a ri... |
| ellpi 33460 | Elementhood in a left prin... |
| lpirlidllpi 33461 | In a principal ideal ring,... |
| rspidlid 33462 | The ideal span of an ideal... |
| pidlnz 33463 | A principal ideal generate... |
| lbslsp 33464 | Any element of a left modu... |
| lindssn 33465 | Any singleton of a nonzero... |
| lindflbs 33466 | Conditions for an independ... |
| islbs5 33467 | An equivalent formulation ... |
| linds2eq 33468 | Deduce equality of element... |
| lindfpropd 33469 | Property deduction for lin... |
| lindspropd 33470 | Property deduction for lin... |
| dvdsruassoi 33471 | If two elements ` X ` and ... |
| dvdsruasso 33472 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33473 | A reformulation of ~ dvdsr... |
| dvdsrspss 33474 | In a ring, an element ` X ... |
| rspsnasso 33475 | Two elements ` X ` and ` Y... |
| unitprodclb 33476 | A finite product is a unit... |
| elgrplsmsn 33477 | Membership in a sumset wit... |
| lsmsnorb 33478 | The sumset of a group with... |
| lsmsnorb2 33479 | The sumset of a single ele... |
| elringlsm 33480 | Membership in a product of... |
| elringlsmd 33481 | Membership in a product of... |
| ringlsmss 33482 | Closure of the product of ... |
| ringlsmss1 33483 | The product of an ideal ` ... |
| ringlsmss2 33484 | The product with an ideal ... |
| lsmsnpridl 33485 | The product of the ring wi... |
| lsmsnidl 33486 | The product of the ring wi... |
| lsmidllsp 33487 | The sum of two ideals is t... |
| lsmidl 33488 | The sum of two ideals is a... |
| lsmssass 33489 | Group sum is associative, ... |
| grplsm0l 33490 | Sumset with the identity s... |
| grplsmid 33491 | The direct sum of an eleme... |
| quslsm 33492 | Express the image by the q... |
| qusbas2 33493 | Alternate definition of th... |
| qus0g 33494 | The identity element of a ... |
| qusima 33495 | The image of a subgroup by... |
| qusrn 33496 | The natural map from eleme... |
| nsgqus0 33497 | A normal subgroup ` N ` is... |
| nsgmgclem 33498 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33499 | There is a monotone Galois... |
| nsgqusf1olem1 33500 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33501 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33502 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33503 | The canonical projection h... |
| lmhmqusker 33504 | A surjective module homomo... |
| lmicqusker 33505 | The image ` H ` of a modul... |
| lidlmcld 33506 | An ideal is closed under l... |
| intlidl 33507 | The intersection of a none... |
| 0ringidl 33508 | The zero ideal is the only... |
| pidlnzb 33509 | A principal ideal is nonze... |
| lidlunitel 33510 | If an ideal ` I ` contains... |
| unitpidl1 33511 | The ideal ` I ` generated ... |
| rhmquskerlem 33512 | The mapping ` J ` induced ... |
| rhmqusker 33513 | A surjective ring homomorp... |
| ricqusker 33514 | The image ` H ` of a ring ... |
| elrspunidl 33515 | Elementhood in the span of... |
| elrspunsn 33516 | Membership to the span of ... |
| lidlincl 33517 | Ideals are closed under in... |
| idlinsubrg 33518 | The intersection between a... |
| rhmimaidl 33519 | The image of an ideal ` I ... |
| drngidl 33520 | A nonzero ring is a divisi... |
| drngidlhash 33521 | A ring is a division ring ... |
| prmidlval 33524 | The class of prime ideals ... |
| isprmidl 33525 | The predicate "is a prime ... |
| prmidlnr 33526 | A prime ideal is a proper ... |
| prmidl 33527 | The main property of a pri... |
| prmidl2 33528 | A condition that shows an ... |
| idlmulssprm 33529 | Let ` P ` be a prime ideal... |
| pridln1 33530 | A proper ideal cannot cont... |
| prmidlidl 33531 | A prime ideal is an ideal.... |
| prmidlssidl 33532 | Prime ideals as a subset o... |
| cringm4 33533 | Commutative/associative la... |
| isprmidlc 33534 | The predicate "is prime id... |
| prmidlc 33535 | Property of a prime ideal ... |
| 0ringprmidl 33536 | The trivial ring does not ... |
| prmidl0 33537 | The zero ideal of a commut... |
| rhmpreimaprmidl 33538 | The preimage of a prime id... |
| qsidomlem1 33539 | If the quotient ring of a ... |
| qsidomlem2 33540 | A quotient by a prime idea... |
| qsidom 33541 | An ideal ` I ` in the comm... |
| qsnzr 33542 | A quotient of a nonzero ri... |
| ssdifidllem 33543 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 33544 | Let ` R ` be a ring, and l... |
| ssdifidlprm 33545 | If the set ` S ` of ~ ssdi... |
| mxidlval 33548 | The set of maximal ideals ... |
| ismxidl 33549 | The predicate "is a maxima... |
| mxidlidl 33550 | A maximal ideal is an idea... |
| mxidlnr 33551 | A maximal ideal is proper.... |
| mxidlmax 33552 | A maximal ideal is a maxim... |
| mxidln1 33553 | One is not contained in an... |
| mxidlnzr 33554 | A ring with a maximal idea... |
| mxidlmaxv 33555 | An ideal ` I ` strictly co... |
| crngmxidl 33556 | In a commutative ring, max... |
| mxidlprm 33557 | Every maximal ideal is pri... |
| mxidlirredi 33558 | In an integral domain, the... |
| mxidlirred 33559 | In a principal ideal domai... |
| ssmxidllem 33560 | The set ` P ` used in the ... |
| ssmxidl 33561 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 33562 | In a nonzero ring, the zer... |
| drng0mxidl 33563 | In a division ring, the ze... |
| drngmxidl 33564 | The zero ideal is the only... |
| drngmxidlr 33565 | If a ring's only maximal i... |
| krull 33566 | Krull's theorem: Any nonz... |
| mxidlnzrb 33567 | A ring is nonzero if and o... |
| krullndrng 33568 | Krull's theorem for non-di... |
| opprabs 33569 | The opposite ring of the o... |
| oppreqg 33570 | Group coset equivalence re... |
| opprnsg 33571 | Normal subgroups of the op... |
| opprlidlabs 33572 | The ideals of the opposite... |
| oppr2idl 33573 | Two sided ideal of the opp... |
| opprmxidlabs 33574 | The maximal ideal of the o... |
| opprqusbas 33575 | The base of the quotient o... |
| opprqusplusg 33576 | The group operation of the... |
| opprqus0g 33577 | The group identity element... |
| opprqusmulr 33578 | The multiplication operati... |
| opprqus1r 33579 | The ring unity of the quot... |
| opprqusdrng 33580 | The quotient of the opposi... |
| qsdrngilem 33581 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33582 | A quotient by a maximal le... |
| qsdrnglem2 33583 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33584 | An ideal ` M ` is both lef... |
| qsfld 33585 | An ideal ` M ` in the comm... |
| mxidlprmALT 33586 | Every maximal ideal is pri... |
| drnglring 33587 | A division ring is a local... |
| dflring2 33588 | Alternate definition of a ... |
| dflringlem 33589 | Lemma for ~ dflring3 . If... |
| dflringlem2 33590 | Lemma for ~ dflring3 . In... |
| dflringlem3 33591 | Lemma for ~ dflring3 . In... |
| dflring3 33592 | Alternate definition of a ... |
| dflring4 33593 | Alternate definition of a ... |
| fldlring 33594 | A field is a local ring. ... |
| idlsrgstr 33597 | A constructed semiring of ... |
| idlsrgval 33598 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33599 | Base of the ideals of a ri... |
| idlsrgplusg 33600 | Additive operation of the ... |
| idlsrg0g 33601 | The zero ideal is the addi... |
| idlsrgmulr 33602 | Multiplicative operation o... |
| idlsrgtset 33603 | Topology component of the ... |
| idlsrgmulrval 33604 | Value of the ring multipli... |
| idlsrgmulrcl 33605 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33606 | In a commutative ring, the... |
| idlsrgmulrss2 33607 | The product of two ideals ... |
| idlsrgmulrssin 33608 | In a commutative ring, the... |
| idlsrgmnd 33609 | The ideals of a ring form ... |
| idlsrgcmnd 33610 | The ideals of a ring form ... |
| rprmval 33611 | The prime elements of a ri... |
| isrprm 33612 | Property for ` P ` to be a... |
| rprmcl 33613 | A ring prime is an element... |
| rprmdvds 33614 | If a ring prime ` Q ` divi... |
| rprmnz 33615 | A ring prime is nonzero. ... |
| rprmnunit 33616 | A ring prime is not a unit... |
| rsprprmprmidl 33617 | In a commutative ring, ide... |
| rsprprmprmidlb 33618 | An ideal generated by a si... |
| rprmndvdsr1 33619 | A ring prime element does ... |
| rprmasso 33620 | In an integral domain, the... |
| rprmasso2 33621 | In an integral domain, if ... |
| rprmasso3 33622 | In an integral domain, if ... |
| unitmulrprm 33623 | A ring unit multiplied by ... |
| rprmndvdsru 33624 | A ring prime element does ... |
| rprmirredlem 33625 | Lemma for ~ rprmirred . (... |
| rprmirred 33626 | In an integral domain, rin... |
| rprmirredb 33627 | In a principal ideal domai... |
| rprmdvdspow 33628 | If a prime element divides... |
| rprmdvdsprod 33629 | If a prime element ` Q ` d... |
| 1arithidomlem1 33630 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 33631 | Lemma for ~ 1arithidom : i... |
| 1arithidom 33632 | Uniqueness of prime factor... |
| isufd 33635 | The property of being a Un... |
| ufdprmidl 33636 | In a unique factorization ... |
| ufdidom 33637 | A nonzero unique factoriza... |
| pidufd 33638 | Every principal ideal doma... |
| 1arithufdlem1 33639 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 33640 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 33641 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 33642 | Lemma for ~ 1arithufd . N... |
| 1arithufd 33643 | Existence of a factorizati... |
| dfufd2lem 33644 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 33645 | Alternative definition of ... |
| zringidom 33646 | The ring of integers is an... |
| zringpid 33647 | The ring of integers is a ... |
| dfprm3 33648 | The (positive) prime eleme... |
| zringfrac 33649 | The field of fractions of ... |
| assaassd 33650 | Left-associative property ... |
| assaassrd 33651 | Right-associative property... |
| 0ringmon1p 33652 | There are no monic polynom... |
| fply1 33653 | Conditions for a function ... |
| ply1lvec 33654 | In a division ring, the un... |
| evls1fn 33655 | Functionality of the subri... |
| evls1dm 33656 | The domain of the subring ... |
| evls1fvf 33657 | The subring evaluation fun... |
| evl1fvf 33658 | The univariate polynomial ... |
| evl1fpws 33659 | Evaluation of a univariate... |
| ressply1evls1 33660 | Subring evaluation of a un... |
| ressdeg1 33661 | The degree of a univariate... |
| ressply10g 33662 | A restricted polynomial al... |
| ressply1mon1p 33663 | The monic polynomials of a... |
| ressply1invg 33664 | An element of a restricted... |
| ressply1sub 33665 | A restricted polynomial al... |
| ressasclcl 33666 | Closure of the univariate ... |
| evls1subd 33667 | Univariate polynomial eval... |
| deg1le0eq0 33668 | A polynomial with nonposit... |
| ply1asclunit 33669 | A nonzero scalar polynomia... |
| ply1unit 33670 | In a field ` F ` , a polyn... |
| evl1deg1 33671 | Evaluation of a univariate... |
| evl1deg2 33672 | Evaluation of a univariate... |
| evl1deg3 33673 | Evaluation of a univariate... |
| evls1monply1 33674 | Subring evaluation of a sc... |
| ply1dg1rt 33675 | Express the root ` - B / A... |
| ply1dg1rtn0 33676 | Polynomials of degree 1 ov... |
| ply1mulrtss 33677 | The roots of a factor ` F ... |
| deg1prod 33678 | Degree of a product of pol... |
| ply1dg3rt0irred 33679 | If a cubic polynomial over... |
| m1pmeq 33680 | If two monic polynomials `... |
| ply1fermltl 33681 | Fermat's little theorem fo... |
| coe1mon 33682 | Coefficient vector of a mo... |
| ply1moneq 33683 | Two monomials are equal if... |
| ply1coedeg 33684 | Decompose a univariate pol... |
| coe1zfv 33685 | The coefficients of the ze... |
| coe1vr1 33686 | Polynomial coefficient of ... |
| deg1vr 33687 | The degree of the variable... |
| vr1nz 33688 | A univariate polynomial va... |
| ply1degltel 33689 | Characterize elementhood i... |
| ply1degleel 33690 | Characterize elementhood i... |
| ply1degltlss 33691 | The space ` S ` of the uni... |
| gsummoncoe1fzo 33692 | A coefficient of the polyn... |
| gsummoncoe1fz 33693 | A coefficient of the polyn... |
| ply1gsumz 33694 | If a polynomial given as a... |
| deg1addlt 33695 | If both factors have degre... |
| ig1pnunit 33696 | The polynomial ideal gener... |
| ig1pmindeg 33697 | The polynomial ideal gener... |
| q1pdir 33698 | Distribution of univariate... |
| q1pvsca 33699 | Scalar multiplication prop... |
| r1pvsca 33700 | Scalar multiplication prop... |
| r1p0 33701 | Polynomial remainder opera... |
| r1pcyc 33702 | The polynomial remainder o... |
| r1padd1 33703 | Addition property of the p... |
| r1pid2OLD 33704 | Obsolete version of ~ r1pi... |
| r1plmhm 33705 | The univariate polynomial ... |
| r1pquslmic 33706 | The univariate polynomial ... |
| psrbasfsupp 33707 | Rewrite a finite support f... |
| psrnzr 33708 | The ring of power series o... |
| mplnzr 33709 | The multivariate polynomia... |
| 0mplrim 33710 | Build a ring isomorphism b... |
| 0mplric 33711 | Multivariate polynomials w... |
| mplasclco 33712 | Case where composing an al... |
| selvascl 33713 | The "variable selection" f... |
| selvply1rhmlema 33714 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlemb 33715 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem1 33716 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem2 33717 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem3 33718 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem4 33719 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem5 33720 | Lemma for ~ selvply1rhm . ... |
| selvply1rhm 33721 | Build a ring homomorphism ... |
| selvply1rhm0 33722 | The ring homomorphism ` H ... |
| mplidomlem 33723 | Lemma for ~ mplidom . (Co... |
| mplidom 33724 | The multivariate polynomia... |
| extvval 33727 | Value of the "variable ext... |
| extvfval 33728 | The "variable extension" f... |
| extvfv 33729 | The "variable extension" f... |
| extvfvv 33730 | The "variable extension" f... |
| extvfvvcl 33731 | Closure for the "variable ... |
| extvfvcl 33732 | Closure for the "variable ... |
| extvfvalf 33733 | The "variable extension" f... |
| mvrvalind 33734 | Value of the generating el... |
| mplmulmvr 33735 | Multiply a polynomial ` F ... |
| evlscaval 33736 | Polynomial evaluation for ... |
| evlvarval 33737 | Polynomial evaluation buil... |
| evlextv 33738 | Evaluating a variable-exte... |
| mplvrpmlem 33739 | Lemma for ~ mplvrpmga and ... |
| mplvrpmfgalem 33740 | Permuting variables in a m... |
| mplvrpmga 33741 | The action of permuting va... |
| mplvrpmmhm 33742 | The action of permuting va... |
| mplvrpmrhm 33743 | The action of permuting va... |
| psrgsum 33744 | Finite commutative sums of... |
| psrmon 33745 | A monomial is a power seri... |
| psrmonmul 33746 | The product of two power s... |
| psrmonmul2 33747 | The product of two power s... |
| psrmonprod 33748 | Finite product of bags of ... |
| mplgsum 33749 | Finite commutative sums of... |
| mplmonprod 33750 | Finite product of monomial... |
| splyval 33755 | The symmetric polynomials ... |
| splysubrg 33756 | The symmetric polynomials ... |
| issply 33757 | Conditions for being a sym... |
| esplyval 33758 | The elementary polynomials... |
| esplyfval 33759 | The ` K ` -th elementary p... |
| esplyfval0 33760 | The ` 0 ` -th elementary s... |
| esplyfval2 33761 | When ` K ` is out-of-bound... |
| esplylem 33762 | Lemma for ~ esplyfv and ot... |
| esplympl 33763 | Elementary symmetric polyn... |
| esplymhp 33764 | The ` K ` -th elementary s... |
| esplyfv1 33765 | Coefficient for the ` K ` ... |
| esplyfv 33766 | Coefficient for the ` K ` ... |
| esplysply 33767 | The ` K ` -th elementary s... |
| esplyfval3 33768 | Alternate expression for t... |
| esplyfval1 33769 | The first elementary symme... |
| esplyfvaln 33770 | The last elementary symmet... |
| esplyind 33771 | A recursive formula for th... |
| esplyindfv 33772 | A recursive formula for th... |
| esplyfvn 33773 | Express the last elementar... |
| vietadeg1 33774 | The degree of a product of... |
| vietalem 33775 | Lemma for ~ vieta : induct... |
| vieta 33776 | Vieta's Formulas: Coeffic... |
| sra1r 33777 | The unity element of a sub... |
| sradrng 33778 | Condition for a subring al... |
| sraidom 33779 | Condition for a subring al... |
| srasubrg 33780 | A subring of the original ... |
| sralvec 33781 | Given a sub division ring ... |
| srafldlvec 33782 | Given a subfield ` F ` of ... |
| resssra 33783 | The subring algebra of a r... |
| lsssra 33784 | A subring is a subspace of... |
| srapwov 33785 | The "power" operation on a... |
| drgext0g 33786 | The additive neutral eleme... |
| drgextvsca 33787 | The scalar multiplication ... |
| drgext0gsca 33788 | The additive neutral eleme... |
| drgextsubrg 33789 | The scalar field is a subr... |
| drgextlsp 33790 | The scalar field is a subs... |
| drgextgsum 33791 | Group sum in a division ri... |
| lvecdimfi 33792 | Finite version of ~ lvecdi... |
| exsslsb 33793 | Any finite generating set ... |
| lbslelsp 33794 | The size of a basis ` X ` ... |
| dimval 33797 | The dimension of a vector ... |
| dimvalfi 33798 | The dimension of a vector ... |
| dimcl 33799 | Closure of the vector spac... |
| lmimdim 33800 | Module isomorphisms preser... |
| lmicdim 33801 | Module isomorphisms preser... |
| lvecdim0i 33802 | A vector space of dimensio... |
| lvecdim0 33803 | A vector space of dimensio... |
| lssdimle 33804 | The dimension of a linear ... |
| dimpropd 33805 | If two structures have the... |
| rlmdim 33806 | The left vector space indu... |
| frlmdim 33807 | Dimension of a free left m... |
| tnglvec 33808 | Augmenting a structure wit... |
| tngdim 33809 | Dimension of a left vector... |
| rrxdim 33810 | Dimension of the generaliz... |
| matdim 33811 | Dimension of the space of ... |
| lbslsat 33812 | A nonzero vector ` X ` is ... |
| lsatdim 33813 | A line, spanned by a nonze... |
| drngdimgt0 33814 | The dimension of a vector ... |
| lmhmlvec2 33815 | A homomorphism of left vec... |
| kerlmhm 33816 | The kernel of a vector spa... |
| imlmhm 33817 | The image of a vector spac... |
| ply1degltdimlem 33818 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 33819 | The space ` S ` of the uni... |
| lindsunlem 33820 | Lemma for ~ lindsun . (Co... |
| lindsun 33821 | Condition for the union of... |
| lbsdiflsp0 33822 | The linear spans of two di... |
| dimkerim 33823 | Given a linear map ` F ` b... |
| qusdimsum 33824 | Let ` W ` be a vector spac... |
| fedgmullem1 33825 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33826 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33827 | The multiplicativity formu... |
| dimlssid 33828 | If the dimension of a line... |
| lvecendof1f1o 33829 | If an endomorphism ` U ` o... |
| lactlmhm 33830 | In an associative algebra ... |
| assalactf1o 33831 | In an associative algebra ... |
| assarrginv 33832 | If an element ` X ` of an ... |
| assafld 33833 | If an algebra ` A ` of fin... |
| relfldext 33840 | The field extension is a r... |
| brfldext 33841 | The field extension relati... |
| ccfldextrr 33842 | The field of the complex n... |
| fldextfld1 33843 | A field extension is only ... |
| fldextfld2 33844 | A field extension is only ... |
| fldextsubrg 33845 | Field extension implies a ... |
| sdrgfldext 33846 | A field ` E ` and any sub-... |
| fldextress 33847 | Field extension implies a ... |
| brfinext 33848 | The finite field extension... |
| extdgval 33849 | Value of the field extensi... |
| fldextsdrg 33850 | Deduce sub-division-ring f... |
| fldextsralvec 33851 | The subring algebra associ... |
| extdgcl 33852 | Closure of the field exten... |
| extdggt0 33853 | Degrees of field extension... |
| fldexttr 33854 | Field extension is a trans... |
| fldextid 33855 | The field extension relati... |
| extdgid 33856 | A trivial field extension ... |
| fldsdrgfldext 33857 | A sub-division-ring of a f... |
| fldsdrgfldext2 33858 | A sub-sub-division-ring of... |
| extdgmul 33859 | The multiplicativity formu... |
| finextfldext 33860 | A finite field extension i... |
| finexttrb 33861 | The extension ` E ` of ` K... |
| extdg1id 33862 | If the degree of the exten... |
| extdg1b 33863 | The degree of the extensio... |
| fldgenfldext 33864 | A subfield ` F ` extended ... |
| fldextchr 33865 | The characteristic of a su... |
| evls1fldgencl 33866 | Closure of the subring pol... |
| ccfldsrarelvec 33867 | The subring algebra of the... |
| ccfldextdgrr 33868 | The degree of the field ex... |
| fldextrspunlsplem 33869 | Lemma for ~ fldextrspunlsp... |
| fldextrspunlsp 33870 | Lemma for ~ fldextrspunfld... |
| fldextrspunlem1 33871 | Lemma for ~ fldextrspunfld... |
| fldextrspunfld 33872 | The ring generated by the ... |
| fldextrspunlem2 33873 | Part of the proof of Propo... |
| fldextrspundgle 33874 | Inequality involving the d... |
| fldextrspundglemul 33875 | Given two field extensions... |
| fldextrspundgdvdslem 33876 | Lemma for ~ fldextrspundgd... |
| fldextrspundgdvds 33877 | Given two finite extension... |
| fldext2rspun 33878 | Given two field extensions... |
| irngval 33881 | The elements of a field ` ... |
| elirng 33882 | Property for an element ` ... |
| irngss 33883 | All elements of a subring ... |
| irngssv 33884 | An integral element is an ... |
| 0ringirng 33885 | A zero ring ` R ` has no i... |
| irngnzply1lem 33886 | In the case of a field ` E... |
| irngnzply1 33887 | In the case of a field ` E... |
| extdgfialglem1 33888 | Lemma for ~ extdgfialg . ... |
| extdgfialglem2 33889 | Lemma for ~ extdgfialg . ... |
| extdgfialg 33890 | A finite field extension `... |
| bralgext 33893 | Express the fact that a fi... |
| finextalg 33894 | A finite field extension i... |
| ply1annidllem 33897 | Write the set ` Q ` of pol... |
| ply1annidl 33898 | The set ` Q ` of polynomia... |
| ply1annnr 33899 | The set ` Q ` of polynomia... |
| ply1annig1p 33900 | The ideal ` Q ` of polynom... |
| minplyval 33901 | Expand the value of the mi... |
| minplycl 33902 | The minimal polynomial is ... |
| ply1annprmidl 33903 | The set ` Q ` of polynomia... |
| minplymindeg 33904 | The minimal polynomial of ... |
| minplyann 33905 | The minimal polynomial for... |
| minplyirredlem 33906 | Lemma for ~ minplyirred . ... |
| minplyirred 33907 | A nonzero minimal polynomi... |
| irngnminplynz 33908 | Integral elements have non... |
| minplym1p 33909 | A minimal polynomial is mo... |
| minplynzm1p 33910 | If a minimal polynomial is... |
| minplyelirng 33911 | If the minimal polynomial ... |
| irredminply 33912 | An irreducible, monic, ann... |
| algextdeglem1 33913 | Lemma for ~ algextdeg . (... |
| algextdeglem2 33914 | Lemma for ~ algextdeg . B... |
| algextdeglem3 33915 | Lemma for ~ algextdeg . T... |
| algextdeglem4 33916 | Lemma for ~ algextdeg . B... |
| algextdeglem5 33917 | Lemma for ~ algextdeg . T... |
| algextdeglem6 33918 | Lemma for ~ algextdeg . B... |
| algextdeglem7 33919 | Lemma for ~ algextdeg . T... |
| algextdeglem8 33920 | Lemma for ~ algextdeg . T... |
| algextdeg 33921 | The degree of an algebraic... |
| rtelextdg2lem 33922 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 33923 | If an element ` X ` is a s... |
| fldext2chn 33924 | In a non-empty chain ` T `... |
| constrrtll 33927 | In the construction of con... |
| constrrtlc1 33928 | In the construction of con... |
| constrrtlc2 33929 | In the construction of con... |
| constrrtcclem 33930 | In the construction of con... |
| constrrtcc 33931 | In the construction of con... |
| isconstr 33932 | Property of being a constr... |
| constr0 33933 | The first step of the cons... |
| constrsuc 33934 | Membership in the successo... |
| constrlim 33935 | Limit step of the construc... |
| constrsscn 33936 | Closure of the constructib... |
| constrsslem 33937 | Lemma for ~ constrss . Th... |
| constr01 33938 | ` 0 ` and ` 1 ` are in all... |
| constrss 33939 | Constructed points are in ... |
| constrmon 33940 | The construction of constr... |
| constrconj 33941 | If a point ` X ` of the co... |
| constrfin 33942 | Each step of the construct... |
| constrelextdg2 33943 | If the ` N ` -th step ` ( ... |
| constrextdg2lem 33944 | Lemma for ~ constrextdg2 .... |
| constrextdg2 33945 | Any step ` ( C `` N ) ` of... |
| constrext2chnlem 33946 | Lemma for ~ constrext2chn ... |
| constrfiss 33947 | For any finite set ` A ` o... |
| constrllcllem 33948 | Constructible numbers are ... |
| constrlccllem 33949 | Constructible numbers are ... |
| constrcccllem 33950 | Constructible numbers are ... |
| constrcbvlem 33951 | Technical lemma for elimin... |
| constrllcl 33952 | Constructible numbers are ... |
| constrlccl 33953 | Constructible numbers are ... |
| constrcccl 33954 | Constructible numbers are ... |
| constrext2chn 33955 | If a constructible number ... |
| constrcn 33956 | Constructible numbers are ... |
| nn0constr 33957 | Nonnegative integers are c... |
| constraddcl 33958 | Constructive numbers are c... |
| constrnegcl 33959 | Constructible numbers are ... |
| zconstr 33960 | Integers are constructible... |
| constrdircl 33961 | Constructible numbers are ... |
| iconstr 33962 | The imaginary unit ` _i ` ... |
| constrremulcl 33963 | If two real numbers ` X ` ... |
| constrcjcl 33964 | Constructible numbers are ... |
| constrrecl 33965 | Constructible numbers are ... |
| constrimcl 33966 | Constructible numbers are ... |
| constrmulcl 33967 | Constructible numbers are ... |
| constrreinvcl 33968 | If a real number ` X ` is ... |
| constrinvcl 33969 | Constructible numbers are ... |
| constrcon 33970 | Contradiction of construct... |
| constrsdrg 33971 | Constructible numbers form... |
| constrfld 33972 | The constructible numbers ... |
| constrresqrtcl 33973 | If a positive real number ... |
| constrabscl 33974 | Constructible numbers are ... |
| constrsqrtcl 33975 | Constructible numbers are ... |
| 2sqr3minply 33976 | The polynomial ` ( ( X ^ 3... |
| 2sqr3nconstr 33977 | Doubling the cube is an im... |
| cos9thpiminplylem1 33978 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem2 33979 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem3 33980 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem4 33981 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem5 33982 | The constructed complex nu... |
| cos9thpiminplylem6 33983 | Evaluation of the polynomi... |
| cos9thpiminply 33984 | The polynomial ` ( ( X ^ 3... |
| cos9thpinconstrlem1 33985 | The complex number ` O ` ,... |
| cos9thpinconstrlem2 33986 | The complex number ` A ` i... |
| cos9thpinconstr 33987 | Trisecting an angle is an ... |
| trisecnconstr 33988 | Not all angles can be tris... |
| smatfval 33991 | Value of the submatrix. (... |
| smatrcl 33992 | Closure of the rectangular... |
| smatlem 33993 | Lemma for the next theorem... |
| smattl 33994 | Entries of a submatrix, to... |
| smattr 33995 | Entries of a submatrix, to... |
| smatbl 33996 | Entries of a submatrix, bo... |
| smatbr 33997 | Entries of a submatrix, bo... |
| smatcl 33998 | Closure of the square subm... |
| matmpo 33999 | Write a square matrix as a... |
| 1smat1 34000 | The submatrix of the ident... |
| submat1n 34001 | One case where the submatr... |
| submatres 34002 | Special case where the sub... |
| submateqlem1 34003 | Lemma for ~ submateq . (C... |
| submateqlem2 34004 | Lemma for ~ submateq . (C... |
| submateq 34005 | Sufficient condition for t... |
| submatminr1 34006 | If we take a submatrix by ... |
| lmatval 34009 | Value of the literal matri... |
| lmatfval 34010 | Entries of a literal matri... |
| lmatfvlem 34011 | Useful lemma to extract li... |
| lmatcl 34012 | Closure of the literal mat... |
| lmat22lem 34013 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 34014 | Entry of a 2x2 literal mat... |
| lmat22e12 34015 | Entry of a 2x2 literal mat... |
| lmat22e21 34016 | Entry of a 2x2 literal mat... |
| lmat22e22 34017 | Entry of a 2x2 literal mat... |
| lmat22det 34018 | The determinant of a liter... |
| mdetpmtr1 34019 | The determinant of a matri... |
| mdetpmtr2 34020 | The determinant of a matri... |
| mdetpmtr12 34021 | The determinant of a matri... |
| mdetlap1 34022 | A Laplace expansion of the... |
| madjusmdetlem1 34023 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 34024 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 34025 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 34026 | Lemma for ~ madjusmdet . ... |
| madjusmdet 34027 | Express the cofactor of th... |
| mdetlap 34028 | Laplace expansion of the d... |
| ist0cld 34029 | The predicate "is a T_0 sp... |
| txomap 34030 | Given two open maps ` F ` ... |
| qtopt1 34031 | If every equivalence class... |
| qtophaus 34032 | If an open map's graph in ... |
| circtopn 34033 | The topology of the unit c... |
| circcn 34034 | The function gluing the re... |
| reff 34035 | For any cover refinement, ... |
| locfinreflem 34036 | A locally finite refinemen... |
| locfinref 34037 | A locally finite refinemen... |
| iscref 34040 | The property that every op... |
| crefeq 34041 | Equality theorem for the "... |
| creftop 34042 | A space where every open c... |
| crefi 34043 | The property that every op... |
| crefdf 34044 | A formulation of ~ crefi e... |
| crefss 34045 | The "every open cover has ... |
| cmpcref 34046 | Equivalent definition of c... |
| cmpfiref 34047 | Every open cover of a Comp... |
| ldlfcntref 34050 | Every open cover of a Lind... |
| ispcmp 34053 | The predicate "is a paraco... |
| cmppcmp 34054 | Every compact space is par... |
| dispcmp 34055 | Every discrete space is pa... |
| pcmplfin 34056 | Given a paracompact topolo... |
| pcmplfinf 34057 | Given a paracompact topolo... |
| rspecval 34060 | Value of the spectrum of t... |
| rspecbas 34061 | The prime ideals form the ... |
| rspectset 34062 | Topology component of the ... |
| rspectopn 34063 | The topology component of ... |
| zarcls0 34064 | The closure of the identit... |
| zarcls1 34065 | The unit ideal ` B ` is th... |
| zarclsun 34066 | The union of two closed se... |
| zarclsiin 34067 | In a Zariski topology, the... |
| zarclsint 34068 | The intersection of a fami... |
| zarclssn 34069 | The closed points of Zaris... |
| zarcls 34070 | The open sets of the Zaris... |
| zartopn 34071 | The Zariski topology is a ... |
| zartop 34072 | The Zariski topology is a ... |
| zartopon 34073 | The points of the Zariski ... |
| zar0ring 34074 | The Zariski Topology of th... |
| zart0 34075 | The Zariski topology is T_... |
| zarmxt1 34076 | The Zariski topology restr... |
| zarcmplem 34077 | Lemma for ~ zarcmp . (Con... |
| zarcmp 34078 | The Zariski topology is co... |
| rspectps 34079 | The spectrum of a ring ` R... |
| rhmpreimacnlem 34080 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 34081 | The function mapping a pri... |
| metidval 34086 | Value of the metric identi... |
| metidss 34087 | As a relation, the metric ... |
| metidv 34088 | ` A ` and ` B ` identify b... |
| metideq 34089 | Basic property of the metr... |
| metider 34090 | The metric identification ... |
| pstmval 34091 | Value of the metric induce... |
| pstmfval 34092 | Function value of the metr... |
| pstmxmet 34093 | The metric induced by a ps... |
| hauseqcn 34094 | In a Hausdorff topology, t... |
| elunitge0 34095 | An element of the closed u... |
| unitssxrge0 34096 | The closed unit interval i... |
| unitdivcld 34097 | Necessary conditions for a... |
| iistmd 34098 | The closed unit interval f... |
| unicls 34099 | The union of the closed se... |
| tpr2tp 34100 | The usual topology on ` ( ... |
| tpr2uni 34101 | The usual topology on ` ( ... |
| xpinpreima 34102 | Rewrite the cartesian prod... |
| xpinpreima2 34103 | Rewrite the cartesian prod... |
| sqsscirc1 34104 | The complex square of side... |
| sqsscirc2 34105 | The complex square of side... |
| cnre2csqlem 34106 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 34107 | Image of a centered square... |
| tpr2rico 34108 | For any point of an open s... |
| cnvordtrestixx 34109 | The restriction of the 'gr... |
| prsdm 34110 | Domain of the relation of ... |
| prsrn 34111 | Range of the relation of a... |
| prsss 34112 | Relation of a subproset. ... |
| prsssdm 34113 | Domain of a subproset rela... |
| ordtprsval 34114 | Value of the order topolog... |
| ordtprsuni 34115 | Value of the order topolog... |
| ordtcnvNEW 34116 | The order dual generates t... |
| ordtrestNEW 34117 | The subspace topology of a... |
| ordtrest2NEWlem 34118 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 34119 | An interval-closed set ` A... |
| ordtconnlem1 34120 | Connectedness in the order... |
| ordtconn 34121 | Connectedness in the order... |
| mndpluscn 34122 | A mapping that is both a h... |
| mhmhmeotmd 34123 | Deduce a Topological Monoi... |
| rmulccn 34124 | Multiplication by a real c... |
| raddcn 34125 | Addition in the real numbe... |
| xrmulc1cn 34126 | The operation multiplying ... |
| fmcncfil 34127 | The image of a Cauchy filt... |
| xrge0hmph 34128 | The extended nonnegative r... |
| xrge0iifcnv 34129 | Define a bijection from ` ... |
| xrge0iifcv 34130 | The defined function's val... |
| xrge0iifiso 34131 | The defined bijection from... |
| xrge0iifhmeo 34132 | Expose a homeomorphism fro... |
| xrge0iifhom 34133 | The defined function from ... |
| xrge0iif1 34134 | Condition for the defined ... |
| xrge0iifmhm 34135 | The defined function from ... |
| xrge0pluscn 34136 | The addition operation of ... |
| xrge0mulc1cn 34137 | The operation multiplying ... |
| xrge0tps 34138 | The extended nonnegative r... |
| xrge0topn 34139 | The topology of the extend... |
| xrge0haus 34140 | The topology of the extend... |
| xrge0tmd 34141 | The extended nonnegative r... |
| xrge0tmdALT 34142 | Alternate proof of ~ xrge0... |
| lmlim 34143 | Relate a limit in a given ... |
| lmlimxrge0 34144 | Relate a limit in the nonn... |
| rge0scvg 34145 | Implication of convergence... |
| fsumcvg4 34146 | A serie with finite suppor... |
| pnfneige0 34147 | A neighborhood of ` +oo ` ... |
| lmxrge0 34148 | Express "sequence ` F ` co... |
| lmdvg 34149 | If a monotonic sequence of... |
| lmdvglim 34150 | If a monotonic real number... |
| pl1cn 34151 | A univariate polynomial is... |
| zringnm 34154 | The norm (function) for a ... |
| zzsnm 34155 | The norm of the ring of th... |
| zlm0 34156 | Zero of a ` ZZ ` -module. ... |
| zlm1 34157 | Unity element of a ` ZZ ` ... |
| zlmds 34158 | Distance in a ` ZZ ` -modu... |
| zlmtset 34159 | Topology in a ` ZZ ` -modu... |
| zlmnm 34160 | Norm of a ` ZZ ` -module (... |
| zhmnrg 34161 | The ` ZZ ` -module built f... |
| nmmulg 34162 | The norm of a group produc... |
| zrhnm 34163 | The norm of the image by `... |
| cnzh 34164 | The ` ZZ ` -module of ` CC... |
| rezh 34165 | The ` ZZ ` -module of ` RR... |
| qqhval 34168 | Value of the canonical hom... |
| zrhf1ker 34169 | The kernel of the homomorp... |
| zrhchr 34170 | The kernel of the homomorp... |
| zrhker 34171 | The kernel of the homomorp... |
| zrhunitpreima 34172 | The preimage by ` ZRHom ` ... |
| elzrhunit 34173 | Condition for the image by... |
| zrhneg 34174 | The canonical homomorphism... |
| zrhcntr 34175 | The canonical representati... |
| elzdif0 34176 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 34177 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 34178 | Value of the canonical hom... |
| qqhvval 34179 | Value of the canonical hom... |
| qqh0 34180 | The image of ` 0 ` by the ... |
| qqh1 34181 | The image of ` 1 ` by the ... |
| qqhf 34182 | ` QQHom ` as a function. ... |
| qqhvq 34183 | The image of a quotient by... |
| qqhghm 34184 | The ` QQHom ` homomorphism... |
| qqhrhm 34185 | The ` QQHom ` homomorphism... |
| qqhnm 34186 | The norm of the image by `... |
| qqhcn 34187 | The ` QQHom ` homomorphism... |
| qqhucn 34188 | The ` QQHom ` homomorphism... |
| rrhval 34192 | Value of the canonical hom... |
| rrhcn 34193 | If the topology of ` R ` i... |
| rrhf 34194 | If the topology of ` R ` i... |
| isrrext 34196 | Express the property " ` R... |
| rrextnrg 34197 | An extension of ` RR ` is ... |
| rrextdrg 34198 | An extension of ` RR ` is ... |
| rrextnlm 34199 | The norm of an extension o... |
| rrextchr 34200 | The ring characteristic of... |
| rrextcusp 34201 | An extension of ` RR ` is ... |
| rrexttps 34202 | An extension of ` RR ` is ... |
| rrexthaus 34203 | The topology of an extensi... |
| rrextust 34204 | The uniformity of an exten... |
| rerrext 34205 | The field of the real numb... |
| cnrrext 34206 | The field of the complex n... |
| qqtopn 34207 | The topology of the field ... |
| rrhfe 34208 | If ` R ` is an extension o... |
| rrhcne 34209 | If ` R ` is an extension o... |
| rrhqima 34210 | The ` RRHom ` homomorphism... |
| rrh0 34211 | The image of ` 0 ` by the ... |
| xrhval 34214 | The value of the embedding... |
| zrhre 34215 | The ` ZRHom ` homomorphism... |
| qqhre 34216 | The ` QQHom ` homomorphism... |
| rrhre 34217 | The ` RRHom ` homomorphism... |
| relmntop 34220 | Manifold is a relation. (... |
| ismntoplly 34221 | Property of being a manifo... |
| ismntop 34222 | Property of being a manifo... |
| esumex 34225 | An extended sum is a set b... |
| esumcl 34226 | Closure for extended sum i... |
| esumeq12dvaf 34227 | Equality deduction for ext... |
| esumeq12dva 34228 | Equality deduction for ext... |
| esumeq12d 34229 | Equality deduction for ext... |
| esumeq1 34230 | Equality theorem for an ex... |
| esumeq1d 34231 | Equality theorem for an ex... |
| esumeq2 34232 | Equality theorem for exten... |
| esumeq2d 34233 | Equality deduction for ext... |
| esumeq2dv 34234 | Equality deduction for ext... |
| esumeq2sdv 34235 | Equality deduction for ext... |
| nfesum1 34236 | Bound-variable hypothesis ... |
| nfesum2 34237 | Bound-variable hypothesis ... |
| cbvesum 34238 | Change bound variable in a... |
| cbvesumv 34239 | Change bound variable in a... |
| esumid 34240 | Identify the extended sum ... |
| esumgsum 34241 | A finite extended sum is t... |
| esumval 34242 | Develop the value of the e... |
| esumel 34243 | The extended sum is a limi... |
| esumnul 34244 | Extended sum over the empt... |
| esum0 34245 | Extended sum of zero. (Co... |
| esumf1o 34246 | Re-index an extended sum u... |
| esumc 34247 | Convert from the collectio... |
| esumrnmpt 34248 | Rewrite an extended sum in... |
| esumsplit 34249 | Split an extended sum into... |
| esummono 34250 | Extended sum is monotonic.... |
| esumpad 34251 | Extend an extended sum by ... |
| esumpad2 34252 | Remove zeroes from an exte... |
| esumadd 34253 | Addition of infinite sums.... |
| esumle 34254 | If all of the terms of an ... |
| gsumesum 34255 | Relate a group sum on ` ( ... |
| esumlub 34256 | The extended sum is the lo... |
| esumaddf 34257 | Addition of infinite sums.... |
| esumlef 34258 | If all of the terms of an ... |
| esumcst 34259 | The extended sum of a cons... |
| esumsnf 34260 | The extended sum of a sing... |
| esumsn 34261 | The extended sum of a sing... |
| esumpr 34262 | Extended sum over a pair. ... |
| esumpr2 34263 | Extended sum over a pair, ... |
| esumrnmpt2 34264 | Rewrite an extended sum in... |
| esumfzf 34265 | Formulating a partial exte... |
| esumfsup 34266 | Formulating an extended su... |
| esumfsupre 34267 | Formulating an extended su... |
| esumss 34268 | Change the index set to a ... |
| esumpinfval 34269 | The value of the extended ... |
| esumpfinvallem 34270 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34271 | The value of the extended ... |
| esumpfinvalf 34272 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34273 | The value of the extended ... |
| esumpcvgval 34274 | The value of the extended ... |
| esumpmono 34275 | The partial sums in an ext... |
| esumcocn 34276 | Lemma for ~ esummulc2 and ... |
| esummulc1 34277 | An extended sum multiplied... |
| esummulc2 34278 | An extended sum multiplied... |
| esumdivc 34279 | An extended sum divided by... |
| hashf2 34280 | Lemma for ~ hasheuni . (C... |
| hasheuni 34281 | The cardinality of a disjo... |
| esumcvg 34282 | The sequence of partial su... |
| esumcvg2 34283 | Simpler version of ~ esumc... |
| esumcvgsum 34284 | The value of the extended ... |
| esumsup 34285 | Express an extended sum as... |
| esumgect 34286 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34287 | All terms of a converging ... |
| esum2dlem 34288 | Lemma for ~ esum2d (finite... |
| esum2d 34289 | Write a double extended su... |
| esumiun 34290 | Sum over a nonnecessarily ... |
| ofceq 34293 | Equality theorem for funct... |
| ofcfval 34294 | Value of an operation appl... |
| ofcval 34295 | Evaluate a function/consta... |
| ofcfn 34296 | The function operation pro... |
| ofcfeqd2 34297 | Equality theorem for funct... |
| ofcfval3 34298 | General value of ` ( F oFC... |
| ofcf 34299 | The function/constant oper... |
| ofcfval2 34300 | The function operation exp... |
| ofcfval4 34301 | The function/constant oper... |
| ofcc 34302 | Left operation by a consta... |
| ofcof 34303 | Relate function operation ... |
| sigaex 34306 | Lemma for ~ issiga and ~ i... |
| sigaval 34307 | The set of sigma-algebra w... |
| issiga 34308 | An alternative definition ... |
| isrnsiga 34309 | The property of being a si... |
| 0elsiga 34310 | A sigma-algebra contains t... |
| baselsiga 34311 | A sigma-algebra contains i... |
| sigasspw 34312 | A sigma-algebra is a set o... |
| sigaclcu 34313 | A sigma-algebra is closed ... |
| sigaclcuni 34314 | A sigma-algebra is closed ... |
| sigaclfu 34315 | A sigma-algebra is closed ... |
| sigaclcu2 34316 | A sigma-algebra is closed ... |
| sigaclfu2 34317 | A sigma-algebra is closed ... |
| sigaclcu3 34318 | A sigma-algebra is closed ... |
| issgon 34319 | Property of being a sigma-... |
| sgon 34320 | A sigma-algebra is a sigma... |
| elsigass 34321 | An element of a sigma-alge... |
| elrnsiga 34322 | Dropping the base informat... |
| isrnsigau 34323 | The property of being a si... |
| unielsiga 34324 | A sigma-algebra contains i... |
| dmvlsiga 34325 | Lebesgue-measurable subset... |
| pwsiga 34326 | Any power set forms a sigm... |
| prsiga 34327 | The smallest possible sigm... |
| sigaclci 34328 | A sigma-algebra is closed ... |
| difelsiga 34329 | A sigma-algebra is closed ... |
| unelsiga 34330 | A sigma-algebra is closed ... |
| inelsiga 34331 | A sigma-algebra is closed ... |
| sigainb 34332 | Building a sigma-algebra f... |
| insiga 34333 | The intersection of a coll... |
| sigagenval 34336 | Value of the generated sig... |
| sigagensiga 34337 | A generated sigma-algebra ... |
| sgsiga 34338 | A generated sigma-algebra ... |
| unisg 34339 | The sigma-algebra generate... |
| dmsigagen 34340 | A sigma-algebra can be gen... |
| sssigagen 34341 | A set is a subset of the s... |
| sssigagen2 34342 | A subset of the generating... |
| elsigagen 34343 | Any element of a set is al... |
| elsigagen2 34344 | Any countable union of ele... |
| sigagenss 34345 | The generated sigma-algebr... |
| sigagenss2 34346 | Sufficient condition for i... |
| sigagenid 34347 | The sigma-algebra generate... |
| ispisys 34348 | The property of being a pi... |
| ispisys2 34349 | The property of being a pi... |
| inelpisys 34350 | Pi-systems are closed unde... |
| sigapisys 34351 | All sigma-algebras are pi-... |
| isldsys 34352 | The property of being a la... |
| pwldsys 34353 | The power set of the unive... |
| unelldsys 34354 | Lambda-systems are closed ... |
| sigaldsys 34355 | All sigma-algebras are lam... |
| ldsysgenld 34356 | The intersection of all la... |
| sigapildsyslem 34357 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34358 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34359 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34360 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34361 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34362 | The lambda system ` E ` ge... |
| dynkin 34363 | Dynkin's lambda-pi theorem... |
| isros 34364 | The property of being a ri... |
| rossspw 34365 | A ring of sets is a collec... |
| 0elros 34366 | A ring of sets contains th... |
| unelros 34367 | A ring of sets is closed u... |
| difelros 34368 | A ring of sets is closed u... |
| inelros 34369 | A ring of sets is closed u... |
| fiunelros 34370 | A ring of sets is closed u... |
| issros 34371 | The property of being a se... |
| srossspw 34372 | A semiring of sets is a co... |
| 0elsros 34373 | A semiring of sets contain... |
| inelsros 34374 | A semiring of sets is clos... |
| diffiunisros 34375 | In semiring of sets, compl... |
| rossros 34376 | Rings of sets are semiring... |
| brsiga 34379 | The Borel Algebra on real ... |
| brsigarn 34380 | The Borel Algebra is a sig... |
| brsigasspwrn 34381 | The Borel Algebra is a set... |
| unibrsiga 34382 | The union of the Borel Alg... |
| cldssbrsiga 34383 | A Borel Algebra contains a... |
| sxval 34386 | Value of the product sigma... |
| sxsiga 34387 | A product sigma-algebra is... |
| sxsigon 34388 | A product sigma-algebra is... |
| sxuni 34389 | The base set of a product ... |
| elsx 34390 | The cartesian product of t... |
| measbase 34393 | The base set of a measure ... |
| measval 34394 | The value of the ` measure... |
| ismeas 34395 | The property of being a me... |
| isrnmeas 34396 | The property of being a me... |
| dmmeas 34397 | The domain of a measure is... |
| measbasedom 34398 | The base set of a measure ... |
| measfrge0 34399 | A measure is a function ov... |
| measfn 34400 | A measure is a function on... |
| measvxrge0 34401 | The values of a measure ar... |
| measvnul 34402 | The measure of the empty s... |
| measge0 34403 | A measure is nonnegative. ... |
| measle0 34404 | If the measure of a given ... |
| measvun 34405 | The measure of a countable... |
| measxun2 34406 | The measure the union of t... |
| measun 34407 | The measure the union of t... |
| measvunilem 34408 | Lemma for ~ measvuni . (C... |
| measvunilem0 34409 | Lemma for ~ measvuni . (C... |
| measvuni 34410 | The measure of a countable... |
| measssd 34411 | A measure is monotone with... |
| measunl 34412 | A measure is sub-additive ... |
| measiuns 34413 | The measure of the union o... |
| measiun 34414 | A measure is sub-additive.... |
| meascnbl 34415 | A measure is continuous fr... |
| measinblem 34416 | Lemma for ~ measinb . (Co... |
| measinb 34417 | Building a measure restric... |
| measres 34418 | Building a measure restric... |
| measinb2 34419 | Building a measure restric... |
| measdivcst 34420 | Division of a measure by a... |
| measdivcstALTV 34421 | Alternate version of ~ mea... |
| cntmeas 34422 | The Counting measure is a ... |
| pwcntmeas 34423 | The counting measure is a ... |
| cntnevol 34424 | Counting and Lebesgue meas... |
| voliune 34425 | The Lebesgue measure funct... |
| volfiniune 34426 | The Lebesgue measure funct... |
| volmeas 34427 | The Lebesgue measure is a ... |
| ddeval1 34430 | Value of the delta measure... |
| ddeval0 34431 | Value of the delta measure... |
| ddemeas 34432 | The Dirac delta measure is... |
| relae 34436 | 'almost everywhere' is a r... |
| brae 34437 | 'almost everywhere' relati... |
| braew 34438 | 'almost everywhere' relati... |
| truae 34439 | A truth holds almost every... |
| aean 34440 | A conjunction holds almost... |
| faeval 34442 | Value of the 'almost every... |
| relfae 34443 | The 'almost everywhere' bu... |
| brfae 34444 | 'almost everywhere' relati... |
| ismbfm 34447 | The predicate " ` F ` is a... |
| elunirnmbfm 34448 | The property of being a me... |
| mbfmfun 34449 | A measurable function is a... |
| mbfmf 34450 | A measurable function as a... |
| mbfmcnvima 34451 | The preimage by a measurab... |
| isanmbfm 34452 | The predicate to be a meas... |
| mbfmbfmOLD 34453 | A measurable function to a... |
| mbfmbfm 34454 | A measurable function to a... |
| mbfmcst 34455 | A constant function is mea... |
| 1stmbfm 34456 | The first projection map i... |
| 2ndmbfm 34457 | The second projection map ... |
| imambfm 34458 | If the sigma-algebra in th... |
| cnmbfm 34459 | A continuous function is m... |
| mbfmco 34460 | The composition of two mea... |
| mbfmco2 34461 | The pair building of two m... |
| mbfmvolf 34462 | Measurable functions with ... |
| elmbfmvol2 34463 | Measurable functions with ... |
| mbfmcnt 34464 | All functions are measurab... |
| br2base 34465 | The base set for the gener... |
| dya2ub 34466 | An upper bound for a dyadi... |
| sxbrsigalem0 34467 | The closed half-spaces of ... |
| sxbrsigalem3 34468 | The sigma-algebra generate... |
| dya2iocival 34469 | The function ` I ` returns... |
| dya2iocress 34470 | Dyadic intervals are subse... |
| dya2iocbrsiga 34471 | Dyadic intervals are Borel... |
| dya2icobrsiga 34472 | Dyadic intervals are Borel... |
| dya2icoseg 34473 | For any point and any clos... |
| dya2icoseg2 34474 | For any point and any open... |
| dya2iocrfn 34475 | The function returning dya... |
| dya2iocct 34476 | The dyadic rectangle set i... |
| dya2iocnrect 34477 | For any point of an open r... |
| dya2iocnei 34478 | For any point of an open s... |
| dya2iocuni 34479 | Every open set of ` ( RR X... |
| dya2iocucvr 34480 | The dyadic rectangular set... |
| sxbrsigalem1 34481 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34482 | The sigma-algebra generate... |
| sxbrsigalem4 34483 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34484 | First direction for ~ sxbr... |
| sxbrsigalem6 34485 | First direction for ~ sxbr... |
| sxbrsiga 34486 | The product sigma-algebra ... |
| omsval 34489 | Value of the function mapp... |
| omsfval 34490 | Value of the outer measure... |
| omscl 34491 | A closure lemma for the co... |
| omsf 34492 | A constructed outer measur... |
| oms0 34493 | A constructed outer measur... |
| omsmon 34494 | A constructed outer measur... |
| omssubaddlem 34495 | For any small margin ` E `... |
| omssubadd 34496 | A constructed outer measur... |
| carsgval 34499 | Value of the Caratheodory ... |
| carsgcl 34500 | Closure of the Caratheodor... |
| elcarsg 34501 | Property of being a Carath... |
| baselcarsg 34502 | The universe set, ` O ` , ... |
| 0elcarsg 34503 | The empty set is Caratheod... |
| carsguni 34504 | The union of all Caratheod... |
| elcarsgss 34505 | Caratheodory measurable se... |
| difelcarsg 34506 | The Caratheodory measurabl... |
| inelcarsg 34507 | The Caratheodory measurabl... |
| unelcarsg 34508 | The Caratheodory-measurabl... |
| difelcarsg2 34509 | The Caratheodory-measurabl... |
| carsgmon 34510 | Utility lemma: Apply mono... |
| carsgsigalem 34511 | Lemma for the following th... |
| fiunelcarsg 34512 | The Caratheodory measurabl... |
| carsgclctunlem1 34513 | Lemma for ~ carsgclctun . ... |
| carsggect 34514 | The outer measure is count... |
| carsgclctunlem2 34515 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34516 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34517 | The Caratheodory measurabl... |
| carsgsiga 34518 | The Caratheodory measurabl... |
| omsmeas 34519 | The restriction of a const... |
| pmeasmono 34520 | This theorem's hypotheses ... |
| pmeasadd 34521 | A premeasure on a ring of ... |
| itgeq12dv 34522 | Equality theorem for an in... |
| sitgval 34528 | Value of the simple functi... |
| issibf 34529 | The predicate " ` F ` is a... |
| sibf0 34530 | The constant zero function... |
| sibfmbl 34531 | A simple function is measu... |
| sibff 34532 | A simple function is a fun... |
| sibfrn 34533 | A simple function has fini... |
| sibfima 34534 | Any preimage of a singleto... |
| sibfinima 34535 | The measure of the interse... |
| sibfof 34536 | Applying function operatio... |
| sitgfval 34537 | Value of the Bochner integ... |
| sitgclg 34538 | Closure of the Bochner int... |
| sitgclbn 34539 | Closure of the Bochner int... |
| sitgclcn 34540 | Closure of the Bochner int... |
| sitgclre 34541 | Closure of the Bochner int... |
| sitg0 34542 | The integral of the consta... |
| sitgf 34543 | The integral for simple fu... |
| sitgaddlemb 34544 | Lemma for * sitgadd . (Co... |
| sitmval 34545 | Value of the simple functi... |
| sitmfval 34546 | Value of the integral dist... |
| sitmcl 34547 | Closure of the integral di... |
| sitmf 34548 | The integral metric as a f... |
| oddpwdc 34550 | Lemma for ~ eulerpart . T... |
| oddpwdcv 34551 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 34552 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 34553 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 34554 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 34555 | Lemma for ~ eulerpart . (... |
| eulerpartlems 34556 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 34557 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 34558 | Lemma for ~ eulerpart . (... |
| eulerpartleme 34559 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 34560 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 34561 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 34562 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 34563 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 34564 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 34565 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 34566 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 34567 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 34568 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 34569 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 34570 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 34571 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 34572 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 34573 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 34574 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 34575 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 34576 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 34577 | Lemma for ~ eulerpart . (... |
| eulerpart 34578 | Euler's theorem on partiti... |
| subiwrd 34581 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 34582 | Length of a subword of an ... |
| iwrdsplit 34583 | Lemma for ~ sseqp1 . (Con... |
| sseqval 34584 | Value of the strong sequen... |
| sseqfv1 34585 | Value of the strong sequen... |
| sseqfn 34586 | A strong recursive sequenc... |
| sseqmw 34587 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 34588 | A strong recursive sequenc... |
| sseqfres 34589 | The first elements in the ... |
| sseqfv2 34590 | Value of the strong sequen... |
| sseqp1 34591 | Value of the strong sequen... |
| fiblem 34594 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 34595 | Value of the Fibonacci seq... |
| fib1 34596 | Value of the Fibonacci seq... |
| fibp1 34597 | Value of the Fibonacci seq... |
| fib2 34598 | Value of the Fibonacci seq... |
| fib3 34599 | Value of the Fibonacci seq... |
| fib4 34600 | Value of the Fibonacci seq... |
| fib5 34601 | Value of the Fibonacci seq... |
| fib6 34602 | Value of the Fibonacci seq... |
| elprob 34605 | The property of being a pr... |
| domprobmeas 34606 | A probability measure is a... |
| domprobsiga 34607 | The domain of a probabilit... |
| probtot 34608 | The probability of the uni... |
| prob01 34609 | A probability is an elemen... |
| probnul 34610 | The probability of the emp... |
| unveldomd 34611 | The universe is an element... |
| unveldom 34612 | The universe is an element... |
| nuleldmp 34613 | The empty set is an elemen... |
| probcun 34614 | The probability of the uni... |
| probun 34615 | The probability of the uni... |
| probdif 34616 | The probability of the dif... |
| probinc 34617 | A probability law is incre... |
| probdsb 34618 | The probability of the com... |
| probmeasd 34619 | A probability measure is a... |
| probvalrnd 34620 | The value of a probability... |
| probtotrnd 34621 | The probability of the uni... |
| totprobd 34622 | Law of total probability, ... |
| totprob 34623 | Law of total probability. ... |
| probfinmeasb 34624 | Build a probability measur... |
| probfinmeasbALTV 34625 | Alternate version of ~ pro... |
| probmeasb 34626 | Build a probability from a... |
| cndprobval 34629 | The value of the condition... |
| cndprobin 34630 | An identity linking condit... |
| cndprob01 34631 | The conditional probabilit... |
| cndprobtot 34632 | The conditional probabilit... |
| cndprobnul 34633 | The conditional probabilit... |
| cndprobprob 34634 | The conditional probabilit... |
| bayesth 34635 | Bayes Theorem. (Contribut... |
| rrvmbfm 34638 | A real-valued random varia... |
| isrrvv 34639 | Elementhood to the set of ... |
| rrvvf 34640 | A real-valued random varia... |
| rrvfn 34641 | A real-valued random varia... |
| rrvdm 34642 | The domain of a random var... |
| rrvrnss 34643 | The range of a random vari... |
| rrvf2 34644 | A real-valued random varia... |
| rrvdmss 34645 | The domain of a random var... |
| rrvfinvima 34646 | For a real-value random va... |
| 0rrv 34647 | The constant function equa... |
| rrvadd 34648 | The sum of two random vari... |
| rrvmulc 34649 | A random variable multipli... |
| rrvsum 34650 | An indexed sum of random v... |
| boolesineq 34651 | Boole's inequality (union ... |
| orvcval 34654 | Value of the preimage mapp... |
| orvcval2 34655 | Another way to express the... |
| elorvc 34656 | Elementhood of a preimage.... |
| orvcval4 34657 | The value of the preimage ... |
| orvcoel 34658 | If the relation produces o... |
| orvccel 34659 | If the relation produces c... |
| elorrvc 34660 | Elementhood of a preimage ... |
| orrvcval4 34661 | The value of the preimage ... |
| orrvcoel 34662 | If the relation produces o... |
| orrvccel 34663 | If the relation produces c... |
| orvcgteel 34664 | Preimage maps produced by ... |
| orvcelval 34665 | Preimage maps produced by ... |
| orvcelel 34666 | Preimage maps produced by ... |
| dstrvval 34667 | The value of the distribut... |
| dstrvprob 34668 | The distribution of a rand... |
| orvclteel 34669 | Preimage maps produced by ... |
| dstfrvel 34670 | Elementhood of preimage ma... |
| dstfrvunirn 34671 | The limit of all preimage ... |
| orvclteinc 34672 | Preimage maps produced by ... |
| dstfrvinc 34673 | A cumulative distribution ... |
| dstfrvclim1 34674 | The limit of the cumulativ... |
| coinfliplem 34675 | Division in the extended r... |
| coinflipprob 34676 | The ` P ` we defined for c... |
| coinflipspace 34677 | The space of our coin-flip... |
| coinflipuniv 34678 | The universe of our coin-f... |
| coinfliprv 34679 | The ` X ` we defined for c... |
| coinflippv 34680 | The probability of heads i... |
| coinflippvt 34681 | The probability of tails i... |
| ballotlemoex 34682 | ` O ` is a set. (Contribu... |
| ballotlem1 34683 | The size of the universe i... |
| ballotlemelo 34684 | Elementhood in ` O ` . (C... |
| ballotlem2 34685 | The probability that the f... |
| ballotlemfval 34686 | The value of ` F ` . (Con... |
| ballotlemfelz 34687 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 34688 | If the ` J ` th ballot is ... |
| ballotlemfc0 34689 | ` F ` takes value 0 betwee... |
| ballotlemfcc 34690 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 34691 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 34692 | ` ( F `` C ) ` always star... |
| ballotleme 34693 | Elements of ` E ` . (Cont... |
| ballotlemodife 34694 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 34695 | If the first pick is a vot... |
| ballotlem5 34696 | If A is not ahead througho... |
| ballotlemi 34697 | Value of ` I ` for a given... |
| ballotlemiex 34698 | Properties of ` ( I `` C )... |
| ballotlemi1 34699 | The first tie cannot be re... |
| ballotlemii 34700 | The first tie cannot be re... |
| ballotlemsup 34701 | The set of zeroes of ` F `... |
| ballotlemimin 34702 | ` ( I `` C ) ` is the firs... |
| ballotlemic 34703 | If the first vote is for B... |
| ballotlem1c 34704 | If the first vote is for A... |
| ballotlemsval 34705 | Value of ` S ` . (Contrib... |
| ballotlemsv 34706 | Value of ` S ` evaluated a... |
| ballotlemsgt1 34707 | ` S ` maps values less tha... |
| ballotlemsdom 34708 | Domain of ` S ` for a give... |
| ballotlemsel1i 34709 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 34710 | The defined ` S ` is a bij... |
| ballotlemsi 34711 | The image by ` S ` of the ... |
| ballotlemsima 34712 | The image by ` S ` of an i... |
| ballotlemieq 34713 | If two countings share the... |
| ballotlemrval 34714 | Value of ` R ` . (Contrib... |
| ballotlemscr 34715 | The image of ` ( R `` C ) ... |
| ballotlemrv 34716 | Value of ` R ` evaluated a... |
| ballotlemrv1 34717 | Value of ` R ` before the ... |
| ballotlemrv2 34718 | Value of ` R ` after the t... |
| ballotlemro 34719 | Range of ` R ` is included... |
| ballotlemgval 34720 | Expand the value of ` .^ `... |
| ballotlemgun 34721 | A property of the defined ... |
| ballotlemfg 34722 | Express the value of ` ( F... |
| ballotlemfrc 34723 | Express the value of ` ( F... |
| ballotlemfrci 34724 | Reverse counting preserves... |
| ballotlemfrceq 34725 | Value of ` F ` for a rever... |
| ballotlemfrcn0 34726 | Value of ` F ` for a rever... |
| ballotlemrc 34727 | Range of ` R ` . (Contrib... |
| ballotlemirc 34728 | Applying ` R ` does not ch... |
| ballotlemrinv0 34729 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 34730 | ` R ` is its own inverse :... |
| ballotlem1ri 34731 | When the vote on the first... |
| ballotlem7 34732 | ` R ` is a bijection betwe... |
| ballotlem8 34733 | There are as many counting... |
| ballotth 34734 | Bertrand's ballot problem ... |
| fzssfzo 34735 | Condition for an integer i... |
| gsumncl 34736 | Closure of a group sum in ... |
| gsumnunsn 34737 | Closure of a group sum in ... |
| ccatmulgnn0dir 34738 | Concatenation of words fol... |
| ofcccat 34739 | Letterwise operations on w... |
| ofcs1 34740 | Letterwise operations on a... |
| ofcs2 34741 | Letterwise operations on a... |
| plymul02 34742 | Product of a polynomial wi... |
| plymulx0 34743 | Coefficients of a polynomi... |
| plymulx 34744 | Coefficients of a polynomi... |
| plyrecld 34745 | Closure of a polynomial wi... |
| signsplypnf 34746 | The quotient of a polynomi... |
| signsply0 34747 | Lemma for the rule of sign... |
| signspval 34748 | The value of the skipping ... |
| signsw0glem 34749 | Neutral element property o... |
| signswbase 34750 | The base of ` W ` is the u... |
| signswplusg 34751 | The operation of ` W ` . ... |
| signsw0g 34752 | The neutral element of ` W... |
| signswmnd 34753 | ` W ` is a monoid structur... |
| signswrid 34754 | The zero-skipping operatio... |
| signswlid 34755 | The zero-skipping operatio... |
| signswn0 34756 | The zero-skipping operatio... |
| signswch 34757 | The zero-skipping operatio... |
| signslema 34758 | Computational part of ~~? ... |
| signstfv 34759 | Value of the zero-skipping... |
| signstfval 34760 | Value of the zero-skipping... |
| signstcl 34761 | Closure of the zero skippi... |
| signstf 34762 | The zero skipping sign wor... |
| signstlen 34763 | Length of the zero skippin... |
| signstf0 34764 | Sign of a single letter wo... |
| signstfvn 34765 | Zero-skipping sign in a wo... |
| signsvtn0 34766 | If the last letter is nonz... |
| signstfvp 34767 | Zero-skipping sign in a wo... |
| signstfvneq0 34768 | In case the first letter i... |
| signstfvcl 34769 | Closure of the zero skippi... |
| signstfvc 34770 | Zero-skipping sign in a wo... |
| signstres 34771 | Restriction of a zero skip... |
| signstfveq0a 34772 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 34773 | In case the last letter is... |
| signsvvfval 34774 | The value of ` V ` , which... |
| signsvvf 34775 | ` V ` is a function. (Con... |
| signsvf0 34776 | There is no change of sign... |
| signsvf1 34777 | In a single-letter word, w... |
| signsvfn 34778 | Number of changes in a wor... |
| signsvtp 34779 | Adding a letter of the sam... |
| signsvtn 34780 | Adding a letter of a diffe... |
| signsvfpn 34781 | Adding a letter of the sam... |
| signsvfnn 34782 | Adding a letter of a diffe... |
| signlem0 34783 | Adding a zero as the highe... |
| signshf 34784 | ` H ` , corresponding to t... |
| signshwrd 34785 | ` H ` , corresponding to t... |
| signshlen 34786 | Length of ` H ` , correspo... |
| signshnz 34787 | ` H ` is not the empty wor... |
| iblidicc 34788 | The identity function is i... |
| rpsqrtcn 34789 | Continuity of the real pos... |
| divsqrtid 34790 | A real number divided by i... |
| cxpcncf1 34791 | The power function on comp... |
| efmul2picn 34792 | Multiplying by ` ( _i x. (... |
| fct2relem 34793 | Lemma for ~ ftc2re . (Con... |
| ftc2re 34794 | The Fundamental Theorem of... |
| fdvposlt 34795 | Functions with a positive ... |
| fdvneggt 34796 | Functions with a negative ... |
| fdvposle 34797 | Functions with a nonnegati... |
| fdvnegge 34798 | Functions with a nonpositi... |
| prodfzo03 34799 | A product of three factors... |
| actfunsnf1o 34800 | The action ` F ` of extend... |
| actfunsnrndisj 34801 | The action ` F ` of extend... |
| itgexpif 34802 | The basis for the circle m... |
| fsum2dsub 34803 | Lemma for ~ breprexp - Re-... |
| reprval 34806 | Value of the representatio... |
| repr0 34807 | There is exactly one repre... |
| reprf 34808 | Members of the representat... |
| reprsum 34809 | Sums of values of the memb... |
| reprle 34810 | Upper bound to the terms i... |
| reprsuc 34811 | Express the representation... |
| reprfi 34812 | Bounded representations ar... |
| reprss 34813 | Representations with terms... |
| reprinrn 34814 | Representations with term ... |
| reprlt 34815 | There are no representatio... |
| hashreprin 34816 | Express a sum of represent... |
| reprgt 34817 | There are no representatio... |
| reprinfz1 34818 | For the representation of ... |
| reprfi2 34819 | Corollary of ~ reprinfz1 .... |
| reprfz1 34820 | Corollary of ~ reprinfz1 .... |
| hashrepr 34821 | Develop the number of repr... |
| reprpmtf1o 34822 | Transposing ` 0 ` and ` X ... |
| reprdifc 34823 | Express the representation... |
| chpvalz 34824 | Value of the second Chebys... |
| chtvalz 34825 | Value of the Chebyshev fun... |
| breprexplema 34826 | Lemma for ~ breprexp (indu... |
| breprexplemb 34827 | Lemma for ~ breprexp (clos... |
| breprexplemc 34828 | Lemma for ~ breprexp (indu... |
| breprexp 34829 | Express the ` S ` th power... |
| breprexpnat 34830 | Express the ` S ` th power... |
| vtsval 34833 | Value of the Vinogradov tr... |
| vtscl 34834 | Closure of the Vinogradov ... |
| vtsprod 34835 | Express the Vinogradov tri... |
| circlemeth 34836 | The Hardy, Littlewood and ... |
| circlemethnat 34837 | The Hardy, Littlewood and ... |
| circlevma 34838 | The Circle Method, where t... |
| circlemethhgt 34839 | The circle method, where t... |
| hgt750lemc 34843 | An upper bound to the summ... |
| hgt750lemd 34844 | An upper bound to the summ... |
| hgt749d 34845 | A deduction version of ~ a... |
| logdivsqrle 34846 | Conditions for ` ( ( log `... |
| hgt750lem 34847 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 34848 | Decimal multiplication gal... |
| hgt750lemf 34849 | Lemma for the statement 7.... |
| hgt750lemg 34850 | Lemma for the statement 7.... |
| oddprm2 34851 | Two ways to write the set ... |
| hgt750lemb 34852 | An upper bound on the cont... |
| hgt750lema 34853 | An upper bound on the cont... |
| hgt750leme 34854 | An upper bound on the cont... |
| tgoldbachgnn 34855 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 34856 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 34857 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 34858 | Odd integers greater than ... |
| tgoldbachgt 34859 | Odd integers greater than ... |
| istrkg2d 34862 | Property of fulfilling dim... |
| axtglowdim2ALTV 34863 | Alternate version of ~ axt... |
| axtgupdim2ALTV 34864 | Alternate version of ~ axt... |
| cgranbtwn 34865 | Null angle implies between... |
| btwnlng13 34866 | If ` Z ` is between ` X ` ... |
| morleylemrneab 34867 | Lemma for morley . (Contr... |
| afsval 34870 | Value of the AFS relation ... |
| brafs 34871 | Binary relation form of th... |
| tg5segofs 34872 | Rephrase ~ axtg5seg using ... |
| lpadval 34875 | Value of the ` leftpad ` f... |
| lpadlem1 34876 | Lemma for the ` leftpad ` ... |
| lpadlem3 34877 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 34878 | Length of a left-padded wo... |
| lpadlem2 34879 | Lemma for the ` leftpad ` ... |
| lpadlen2 34880 | Length of a left-padded wo... |
| lpadmax 34881 | Length of a left-padded wo... |
| lpadleft 34882 | The contents of prefix of ... |
| lpadright 34883 | The suffix of a left-padde... |
| bnj170 34896 | ` /\ ` -manipulation. (Co... |
| bnj240 34897 | ` /\ ` -manipulation. (Co... |
| bnj248 34898 | ` /\ ` -manipulation. (Co... |
| bnj250 34899 | ` /\ ` -manipulation. (Co... |
| bnj251 34900 | ` /\ ` -manipulation. (Co... |
| bnj252 34901 | ` /\ ` -manipulation. (Co... |
| bnj253 34902 | ` /\ ` -manipulation. (Co... |
| bnj255 34903 | ` /\ ` -manipulation. (Co... |
| bnj256 34904 | ` /\ ` -manipulation. (Co... |
| bnj257 34905 | ` /\ ` -manipulation. (Co... |
| bnj258 34906 | ` /\ ` -manipulation. (Co... |
| bnj268 34907 | ` /\ ` -manipulation. (Co... |
| bnj290 34908 | ` /\ ` -manipulation. (Co... |
| bnj291 34909 | ` /\ ` -manipulation. (Co... |
| bnj312 34910 | ` /\ ` -manipulation. (Co... |
| bnj334 34911 | ` /\ ` -manipulation. (Co... |
| bnj345 34912 | ` /\ ` -manipulation. (Co... |
| bnj422 34913 | ` /\ ` -manipulation. (Co... |
| bnj432 34914 | ` /\ ` -manipulation. (Co... |
| bnj446 34915 | ` /\ ` -manipulation. (Co... |
| bnj23 34916 | First-order logic and set ... |
| bnj31 34917 | First-order logic and set ... |
| bnj62 34918 | First-order logic and set ... |
| bnj89 34919 | First-order logic and set ... |
| bnj90 34920 | First-order logic and set ... |
| bnj101 34921 | First-order logic and set ... |
| bnj105 34922 | First-order logic and set ... |
| bnj115 34923 | First-order logic and set ... |
| bnj132 34924 | First-order logic and set ... |
| bnj133 34925 | First-order logic and set ... |
| bnj156 34926 | First-order logic and set ... |
| bnj158 34927 | First-order logic and set ... |
| bnj168 34928 | First-order logic and set ... |
| bnj206 34929 | First-order logic and set ... |
| bnj216 34930 | First-order logic and set ... |
| bnj219 34931 | First-order logic and set ... |
| bnj226 34932 | First-order logic and set ... |
| bnj228 34933 | First-order logic and set ... |
| bnj519 34934 | First-order logic and set ... |
| bnj524 34935 | First-order logic and set ... |
| bnj525 34936 | First-order logic and set ... |
| bnj534 34937 | First-order logic and set ... |
| bnj538 34938 | First-order logic and set ... |
| bnj529 34939 | First-order logic and set ... |
| bnj551 34940 | First-order logic and set ... |
| bnj563 34941 | First-order logic and set ... |
| bnj564 34942 | First-order logic and set ... |
| bnj593 34943 | First-order logic and set ... |
| bnj596 34944 | First-order logic and set ... |
| bnj610 34945 | Pass from equality ( ` x =... |
| bnj642 34946 | ` /\ ` -manipulation. (Co... |
| bnj643 34947 | ` /\ ` -manipulation. (Co... |
| bnj645 34948 | ` /\ ` -manipulation. (Co... |
| bnj658 34949 | ` /\ ` -manipulation. (Co... |
| bnj667 34950 | ` /\ ` -manipulation. (Co... |
| bnj705 34951 | ` /\ ` -manipulation. (Co... |
| bnj706 34952 | ` /\ ` -manipulation. (Co... |
| bnj707 34953 | ` /\ ` -manipulation. (Co... |
| bnj708 34954 | ` /\ ` -manipulation. (Co... |
| bnj721 34955 | ` /\ ` -manipulation. (Co... |
| bnj832 34956 | ` /\ ` -manipulation. (Co... |
| bnj835 34957 | ` /\ ` -manipulation. (Co... |
| bnj836 34958 | ` /\ ` -manipulation. (Co... |
| bnj837 34959 | ` /\ ` -manipulation. (Co... |
| bnj769 34960 | ` /\ ` -manipulation. (Co... |
| bnj770 34961 | ` /\ ` -manipulation. (Co... |
| bnj771 34962 | ` /\ ` -manipulation. (Co... |
| bnj887 34963 | ` /\ ` -manipulation. (Co... |
| bnj918 34964 | First-order logic and set ... |
| bnj919 34965 | First-order logic and set ... |
| bnj923 34966 | First-order logic and set ... |
| bnj927 34967 | First-order logic and set ... |
| bnj931 34968 | First-order logic and set ... |
| bnj937 34969 | First-order logic and set ... |
| bnj941 34970 | First-order logic and set ... |
| bnj945 34971 | Technical lemma for ~ bnj6... |
| bnj946 34972 | First-order logic and set ... |
| bnj951 34973 | ` /\ ` -manipulation. (Co... |
| bnj956 34974 | First-order logic and set ... |
| bnj976 34975 | First-order logic and set ... |
| bnj982 34976 | First-order logic and set ... |
| bnj1019 34977 | First-order logic and set ... |
| bnj1023 34978 | First-order logic and set ... |
| bnj1095 34979 | First-order logic and set ... |
| bnj1096 34980 | First-order logic and set ... |
| bnj1098 34981 | First-order logic and set ... |
| bnj1101 34982 | First-order logic and set ... |
| bnj1113 34983 | First-order logic and set ... |
| bnj1109 34984 | First-order logic and set ... |
| bnj1131 34985 | First-order logic and set ... |
| bnj1138 34986 | First-order logic and set ... |
| bnj1143 34987 | First-order logic and set ... |
| bnj1146 34988 | First-order logic and set ... |
| bnj1149 34989 | First-order logic and set ... |
| bnj1185 34990 | First-order logic and set ... |
| bnj1196 34991 | First-order logic and set ... |
| bnj1198 34992 | First-order logic and set ... |
| bnj1209 34993 | First-order logic and set ... |
| bnj1211 34994 | First-order logic and set ... |
| bnj1213 34995 | First-order logic and set ... |
| bnj1212 34996 | First-order logic and set ... |
| bnj1219 34997 | First-order logic and set ... |
| bnj1224 34998 | First-order logic and set ... |
| bnj1230 34999 | First-order logic and set ... |
| bnj1232 35000 | First-order logic and set ... |
| bnj1235 35001 | First-order logic and set ... |
| bnj1239 35002 | First-order logic and set ... |
| bnj1238 35003 | First-order logic and set ... |
| bnj1241 35004 | First-order logic and set ... |
| bnj1247 35005 | First-order logic and set ... |
| bnj1254 35006 | First-order logic and set ... |
| bnj1262 35007 | First-order logic and set ... |
| bnj1266 35008 | First-order logic and set ... |
| bnj1265 35009 | First-order logic and set ... |
| bnj1275 35010 | First-order logic and set ... |
| bnj1276 35011 | First-order logic and set ... |
| bnj1292 35012 | First-order logic and set ... |
| bnj1293 35013 | First-order logic and set ... |
| bnj1294 35014 | First-order logic and set ... |
| bnj1299 35015 | First-order logic and set ... |
| bnj1304 35016 | First-order logic and set ... |
| bnj1316 35017 | First-order logic and set ... |
| bnj1317 35018 | First-order logic and set ... |
| bnj1322 35019 | First-order logic and set ... |
| bnj1340 35020 | First-order logic and set ... |
| bnj1345 35021 | First-order logic and set ... |
| bnj1350 35022 | First-order logic and set ... |
| bnj1351 35023 | First-order logic and set ... |
| bnj1352 35024 | First-order logic and set ... |
| bnj1361 35025 | First-order logic and set ... |
| bnj1366 35026 | First-order logic and set ... |
| bnj1379 35027 | First-order logic and set ... |
| bnj1383 35028 | First-order logic and set ... |
| bnj1385 35029 | First-order logic and set ... |
| bnj1386 35030 | First-order logic and set ... |
| bnj1397 35031 | First-order logic and set ... |
| bnj1400 35032 | First-order logic and set ... |
| bnj1405 35033 | First-order logic and set ... |
| bnj1422 35034 | First-order logic and set ... |
| bnj1424 35035 | First-order logic and set ... |
| bnj1436 35036 | First-order logic and set ... |
| bnj1441 35037 | First-order logic and set ... |
| bnj1441g 35038 | First-order logic and set ... |
| bnj1454 35039 | First-order logic and set ... |
| bnj1459 35040 | First-order logic and set ... |
| bnj1464 35041 | Conversion of implicit sub... |
| bnj1465 35042 | First-order logic and set ... |
| bnj1468 35043 | Conversion of implicit sub... |
| bnj1476 35044 | First-order logic and set ... |
| bnj1502 35045 | First-order logic and set ... |
| bnj1503 35046 | First-order logic and set ... |
| bnj1517 35047 | First-order logic and set ... |
| bnj1521 35048 | First-order logic and set ... |
| bnj1533 35049 | First-order logic and set ... |
| bnj1534 35050 | First-order logic and set ... |
| bnj1536 35051 | First-order logic and set ... |
| bnj1538 35052 | First-order logic and set ... |
| bnj1541 35053 | First-order logic and set ... |
| bnj1542 35054 | First-order logic and set ... |
| bnj110 35055 | Well-founded induction res... |
| bnj157 35056 | Well-founded induction res... |
| bnj66 35057 | Technical lemma for ~ bnj6... |
| bnj91 35058 | First-order logic and set ... |
| bnj92 35059 | First-order logic and set ... |
| bnj93 35060 | Technical lemma for ~ bnj9... |
| bnj95 35061 | Technical lemma for ~ bnj1... |
| bnj96 35062 | Technical lemma for ~ bnj1... |
| bnj97 35063 | Technical lemma for ~ bnj1... |
| bnj98 35064 | Technical lemma for ~ bnj1... |
| bnj106 35065 | First-order logic and set ... |
| bnj118 35066 | First-order logic and set ... |
| bnj121 35067 | First-order logic and set ... |
| bnj124 35068 | Technical lemma for ~ bnj1... |
| bnj125 35069 | Technical lemma for ~ bnj1... |
| bnj126 35070 | Technical lemma for ~ bnj1... |
| bnj130 35071 | Technical lemma for ~ bnj1... |
| bnj149 35072 | Technical lemma for ~ bnj1... |
| bnj150 35073 | Technical lemma for ~ bnj1... |
| bnj151 35074 | Technical lemma for ~ bnj1... |
| bnj154 35075 | Technical lemma for ~ bnj1... |
| bnj155 35076 | Technical lemma for ~ bnj1... |
| bnj153 35077 | Technical lemma for ~ bnj8... |
| bnj207 35078 | Technical lemma for ~ bnj8... |
| bnj213 35079 | First-order logic and set ... |
| bnj222 35080 | Technical lemma for ~ bnj2... |
| bnj229 35081 | Technical lemma for ~ bnj5... |
| bnj517 35082 | Technical lemma for ~ bnj5... |
| bnj518 35083 | Technical lemma for ~ bnj8... |
| bnj523 35084 | Technical lemma for ~ bnj8... |
| bnj526 35085 | Technical lemma for ~ bnj8... |
| bnj528 35086 | Technical lemma for ~ bnj8... |
| bnj535 35087 | Technical lemma for ~ bnj8... |
| bnj539 35088 | Technical lemma for ~ bnj8... |
| bnj540 35089 | Technical lemma for ~ bnj8... |
| bnj543 35090 | Technical lemma for ~ bnj8... |
| bnj544 35091 | Technical lemma for ~ bnj8... |
| bnj545 35092 | Technical lemma for ~ bnj8... |
| bnj546 35093 | Technical lemma for ~ bnj8... |
| bnj548 35094 | Technical lemma for ~ bnj8... |
| bnj553 35095 | Technical lemma for ~ bnj8... |
| bnj554 35096 | Technical lemma for ~ bnj8... |
| bnj556 35097 | Technical lemma for ~ bnj8... |
| bnj557 35098 | Technical lemma for ~ bnj8... |
| bnj558 35099 | Technical lemma for ~ bnj8... |
| bnj561 35100 | Technical lemma for ~ bnj8... |
| bnj562 35101 | Technical lemma for ~ bnj8... |
| bnj570 35102 | Technical lemma for ~ bnj8... |
| bnj571 35103 | Technical lemma for ~ bnj8... |
| bnj605 35104 | Technical lemma. This lem... |
| bnj581 35105 | Technical lemma for ~ bnj5... |
| bnj589 35106 | Technical lemma for ~ bnj8... |
| bnj590 35107 | Technical lemma for ~ bnj8... |
| bnj591 35108 | Technical lemma for ~ bnj8... |
| bnj594 35109 | Technical lemma for ~ bnj8... |
| bnj580 35110 | Technical lemma for ~ bnj5... |
| bnj579 35111 | Technical lemma for ~ bnj8... |
| bnj602 35112 | Equality theorem for the `... |
| bnj607 35113 | Technical lemma for ~ bnj8... |
| bnj609 35114 | Technical lemma for ~ bnj8... |
| bnj611 35115 | Technical lemma for ~ bnj8... |
| bnj600 35116 | Technical lemma for ~ bnj8... |
| bnj601 35117 | Technical lemma for ~ bnj8... |
| bnj852 35118 | Technical lemma for ~ bnj6... |
| bnj864 35119 | Technical lemma for ~ bnj6... |
| bnj865 35120 | Technical lemma for ~ bnj6... |
| bnj873 35121 | Technical lemma for ~ bnj6... |
| bnj849 35122 | Technical lemma for ~ bnj6... |
| bnj882 35123 | Definition (using hypothes... |
| bnj18eq1 35124 | Equality theorem for trans... |
| bnj893 35125 | Property of ` _trCl ` . U... |
| bnj900 35126 | Technical lemma for ~ bnj6... |
| bnj906 35127 | Property of ` _trCl ` . (... |
| bnj908 35128 | Technical lemma for ~ bnj6... |
| bnj911 35129 | Technical lemma for ~ bnj6... |
| bnj916 35130 | Technical lemma for ~ bnj6... |
| bnj917 35131 | Technical lemma for ~ bnj6... |
| bnj934 35132 | Technical lemma for ~ bnj6... |
| bnj929 35133 | Technical lemma for ~ bnj6... |
| bnj938 35134 | Technical lemma for ~ bnj6... |
| bnj944 35135 | Technical lemma for ~ bnj6... |
| bnj953 35136 | Technical lemma for ~ bnj6... |
| bnj958 35137 | Technical lemma for ~ bnj6... |
| bnj1000 35138 | Technical lemma for ~ bnj8... |
| bnj965 35139 | Technical lemma for ~ bnj8... |
| bnj964 35140 | Technical lemma for ~ bnj6... |
| bnj966 35141 | Technical lemma for ~ bnj6... |
| bnj967 35142 | Technical lemma for ~ bnj6... |
| bnj969 35143 | Technical lemma for ~ bnj6... |
| bnj970 35144 | Technical lemma for ~ bnj6... |
| bnj910 35145 | Technical lemma for ~ bnj6... |
| bnj978 35146 | Technical lemma for ~ bnj6... |
| bnj981 35147 | Technical lemma for ~ bnj6... |
| bnj983 35148 | Technical lemma for ~ bnj6... |
| bnj984 35149 | Technical lemma for ~ bnj6... |
| bnj985v 35150 | Version of ~ bnj985 with a... |
| bnj985 35151 | Technical lemma for ~ bnj6... |
| bnj986 35152 | Technical lemma for ~ bnj6... |
| bnj996 35153 | Technical lemma for ~ bnj6... |
| bnj998 35154 | Technical lemma for ~ bnj6... |
| bnj999 35155 | Technical lemma for ~ bnj6... |
| bnj1001 35156 | Technical lemma for ~ bnj6... |
| bnj1006 35157 | Technical lemma for ~ bnj6... |
| bnj1014 35158 | Technical lemma for ~ bnj6... |
| bnj1015 35159 | Technical lemma for ~ bnj6... |
| bnj1018g 35160 | Version of ~ bnj1018 with ... |
| bnj1018 35161 | Technical lemma for ~ bnj6... |
| bnj1020 35162 | Technical lemma for ~ bnj6... |
| bnj1021 35163 | Technical lemma for ~ bnj6... |
| bnj907 35164 | Technical lemma for ~ bnj6... |
| bnj1029 35165 | Property of ` _trCl ` . (... |
| bnj1033 35166 | Technical lemma for ~ bnj6... |
| bnj1034 35167 | Technical lemma for ~ bnj6... |
| bnj1039 35168 | Technical lemma for ~ bnj6... |
| bnj1040 35169 | Technical lemma for ~ bnj6... |
| bnj1047 35170 | Technical lemma for ~ bnj6... |
| bnj1049 35171 | Technical lemma for ~ bnj6... |
| bnj1052 35172 | Technical lemma for ~ bnj6... |
| bnj1053 35173 | Technical lemma for ~ bnj6... |
| bnj1071 35174 | Technical lemma for ~ bnj6... |
| bnj1083 35175 | Technical lemma for ~ bnj6... |
| bnj1090 35176 | Technical lemma for ~ bnj6... |
| bnj1093 35177 | Technical lemma for ~ bnj6... |
| bnj1097 35178 | Technical lemma for ~ bnj6... |
| bnj1110 35179 | Technical lemma for ~ bnj6... |
| bnj1112 35180 | Technical lemma for ~ bnj6... |
| bnj1118 35181 | Technical lemma for ~ bnj6... |
| bnj1121 35182 | Technical lemma for ~ bnj6... |
| bnj1123 35183 | Technical lemma for ~ bnj6... |
| bnj1030 35184 | Technical lemma for ~ bnj6... |
| bnj1124 35185 | Property of ` _trCl ` . (... |
| bnj1133 35186 | Technical lemma for ~ bnj6... |
| bnj1128 35187 | Technical lemma for ~ bnj6... |
| bnj1127 35188 | Property of ` _trCl ` . (... |
| bnj1125 35189 | Property of ` _trCl ` . (... |
| bnj1145 35190 | Technical lemma for ~ bnj6... |
| bnj1147 35191 | Property of ` _trCl ` . (... |
| bnj1137 35192 | Property of ` _trCl ` . (... |
| bnj1148 35193 | Property of ` _pred ` . (... |
| bnj1136 35194 | Technical lemma for ~ bnj6... |
| bnj1152 35195 | Technical lemma for ~ bnj6... |
| bnj1154 35196 | Property of ` Fr ` . (Con... |
| bnj1171 35197 | Technical lemma for ~ bnj6... |
| bnj1172 35198 | Technical lemma for ~ bnj6... |
| bnj1173 35199 | Technical lemma for ~ bnj6... |
| bnj1174 35200 | Technical lemma for ~ bnj6... |
| bnj1175 35201 | Technical lemma for ~ bnj6... |
| bnj1176 35202 | Technical lemma for ~ bnj6... |
| bnj1177 35203 | Technical lemma for ~ bnj6... |
| bnj1186 35204 | Technical lemma for ~ bnj6... |
| bnj1190 35205 | Technical lemma for ~ bnj6... |
| bnj1189 35206 | Technical lemma for ~ bnj6... |
| bnj69 35207 | Existence of a minimal ele... |
| bnj1228 35208 | Existence of a minimal ele... |
| bnj1204 35209 | Well-founded induction. T... |
| bnj1234 35210 | Technical lemma for ~ bnj6... |
| bnj1245 35211 | Technical lemma for ~ bnj6... |
| bnj1256 35212 | Technical lemma for ~ bnj6... |
| bnj1259 35213 | Technical lemma for ~ bnj6... |
| bnj1253 35214 | Technical lemma for ~ bnj6... |
| bnj1279 35215 | Technical lemma for ~ bnj6... |
| bnj1286 35216 | Technical lemma for ~ bnj6... |
| bnj1280 35217 | Technical lemma for ~ bnj6... |
| bnj1296 35218 | Technical lemma for ~ bnj6... |
| bnj1309 35219 | Technical lemma for ~ bnj6... |
| bnj1307 35220 | Technical lemma for ~ bnj6... |
| bnj1311 35221 | Technical lemma for ~ bnj6... |
| bnj1318 35222 | Technical lemma for ~ bnj6... |
| bnj1326 35223 | Technical lemma for ~ bnj6... |
| bnj1321 35224 | Technical lemma for ~ bnj6... |
| bnj1364 35225 | Property of ` _FrSe ` . (... |
| bnj1371 35226 | Technical lemma for ~ bnj6... |
| bnj1373 35227 | Technical lemma for ~ bnj6... |
| bnj1374 35228 | Technical lemma for ~ bnj6... |
| bnj1384 35229 | Technical lemma for ~ bnj6... |
| bnj1388 35230 | Technical lemma for ~ bnj6... |
| bnj1398 35231 | Technical lemma for ~ bnj6... |
| bnj1413 35232 | Property of ` _trCl ` . (... |
| bnj1408 35233 | Technical lemma for ~ bnj1... |
| bnj1414 35234 | Property of ` _trCl ` . (... |
| bnj1415 35235 | Technical lemma for ~ bnj6... |
| bnj1416 35236 | Technical lemma for ~ bnj6... |
| bnj1418 35237 | Property of ` _pred ` . (... |
| bnj1417 35238 | Technical lemma for ~ bnj6... |
| bnj1421 35239 | Technical lemma for ~ bnj6... |
| bnj1444 35240 | Technical lemma for ~ bnj6... |
| bnj1445 35241 | Technical lemma for ~ bnj6... |
| bnj1446 35242 | Technical lemma for ~ bnj6... |
| bnj1447 35243 | Technical lemma for ~ bnj6... |
| bnj1448 35244 | Technical lemma for ~ bnj6... |
| bnj1449 35245 | Technical lemma for ~ bnj6... |
| bnj1442 35246 | Technical lemma for ~ bnj6... |
| bnj1450 35247 | Technical lemma for ~ bnj6... |
| bnj1423 35248 | Technical lemma for ~ bnj6... |
| bnj1452 35249 | Technical lemma for ~ bnj6... |
| bnj1466 35250 | Technical lemma for ~ bnj6... |
| bnj1467 35251 | Technical lemma for ~ bnj6... |
| bnj1463 35252 | Technical lemma for ~ bnj6... |
| bnj1489 35253 | Technical lemma for ~ bnj6... |
| bnj1491 35254 | Technical lemma for ~ bnj6... |
| bnj1312 35255 | Technical lemma for ~ bnj6... |
| bnj1493 35256 | Technical lemma for ~ bnj6... |
| bnj1497 35257 | Technical lemma for ~ bnj6... |
| bnj1498 35258 | Technical lemma for ~ bnj6... |
| bnj60 35259 | Well-founded recursion, pa... |
| bnj1514 35260 | Technical lemma for ~ bnj1... |
| bnj1518 35261 | Technical lemma for ~ bnj1... |
| bnj1519 35262 | Technical lemma for ~ bnj1... |
| bnj1520 35263 | Technical lemma for ~ bnj1... |
| bnj1501 35264 | Technical lemma for ~ bnj1... |
| bnj1500 35265 | Well-founded recursion, pa... |
| bnj1525 35266 | Technical lemma for ~ bnj1... |
| bnj1529 35267 | Technical lemma for ~ bnj1... |
| bnj1523 35268 | Technical lemma for ~ bnj1... |
| bnj1522 35269 | Well-founded recursion, pa... |
| nfan1c 35270 | Variant of ~ nfan and comm... |
| cbvex1v 35271 | Rule used to change bound ... |
| dvelimalcased 35272 | Eliminate a disjoint varia... |
| dvelimalcasei 35273 | Eliminate a disjoint varia... |
| dvelimexcased 35274 | Eliminate a disjoint varia... |
| dvelimexcasei 35275 | Eliminate a disjoint varia... |
| exdifsn 35276 | There exists an element in... |
| srcmpltd 35277 | If a statement is true for... |
| prsrcmpltd 35278 | If a statement is true for... |
| axnulALT2 35279 | Alternate proof of ~ axnul... |
| dff15 35280 | A one-to-one function in t... |
| f1resveqaeq 35281 | If a function restricted t... |
| f1resrcmplf1dlem 35282 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 35283 | If a function's restrictio... |
| funen1cnv 35284 | If a function is equinumer... |
| xoromon 35285 | ` _om ` is either an ordin... |
| fissorduni 35286 | The union (supremum) of a ... |
| fnrelpredd 35287 | A function that preserves ... |
| cardpred 35288 | The cardinality function p... |
| nummin 35289 | Every nonempty class of nu... |
| r11 35290 | Value of the cumulative hi... |
| r12 35291 | Value of the cumulative hi... |
| r1wf 35292 | Each stage in the cumulati... |
| elwf 35293 | An element of a well-found... |
| r1elcl 35294 | Each set of the cumulative... |
| rankval2b 35295 | Value of an alternate defi... |
| rankval4b 35296 | The rank of a set is the s... |
| rankfilimbi 35297 | If all elements in a finit... |
| rankfilimb 35298 | The rank of a finite well-... |
| r1filimi 35299 | If all elements in a finit... |
| r1filim 35300 | A finite set appears in th... |
| r1omfi 35301 | Hereditarily finite sets a... |
| r1omhf 35302 | A set is hereditarily fini... |
| r1ssel 35303 | A set is a subset of the v... |
| axnulALT3 35304 | Alternate proof of ~ axnul... |
| axprALT2 35305 | Alternate proof of ~ axpr ... |
| r1omfv 35306 | Value of the cumulative hi... |
| trssfir1om 35307 | If every element in a tran... |
| r1omhfb 35308 | The class of all hereditar... |
| prcinf 35309 | Any proper class is litera... |
| fineqvrep 35310 | If all sets are finite, th... |
| fineqvpow 35311 | If all sets are finite, th... |
| fineqvac 35312 | If all sets are finite, th... |
| fineqvacALT 35313 | Shorter proof of ~ fineqva... |
| fineqvomon 35314 | If all sets are finite, th... |
| fineqvomonb 35315 | All sets are finite iff al... |
| omprcomonb 35316 | The class of all finite or... |
| fineqvnttrclselem1 35317 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem2 35318 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem3 35319 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclse 35320 | A counterexample demonstra... |
| fineqvinfep 35321 | A counterexample demonstra... |
| axreg 35323 | Derivation of ~ ax-reg fro... |
| axregscl 35324 | A version of ~ ax-regs wit... |
| axregszf 35325 | Derivation of ~ zfregs usi... |
| setindregs 35326 | Set (epsilon) induction. ... |
| setinds2regs 35327 | Principle of set induction... |
| noinfepfnregs 35328 | There are no infinite desc... |
| noinfepregs 35329 | There are no infinite desc... |
| tz9.1regs 35330 | Every set has a transitive... |
| unir1regs 35331 | The cumulative hierarchy o... |
| trssfir1omregs 35332 | If every element in a tran... |
| r1omhfbregs 35333 | The class of all hereditar... |
| fineqvr1ombregs 35334 | All sets are finite iff al... |
| axregs 35335 | Derivation of ~ ax-regs fr... |
| axsepg2 35336 | A generalization of ~ ax-s... |
| axsepg3 35337 | A generalization of ~ ax-s... |
| axsepg3ALT 35338 | Alternate proof of ~ axsep... |
| axsepg4 35339 | A generalization of ~ ax-s... |
| axsepg5 35340 | A generalization of ~ ax-s... |
| axnulg 35341 | A generalization of ~ ax-n... |
| axpowg 35342 | A generalization of ~ ax-p... |
| axpowg2 35343 | A generalization of ~ ax-p... |
| axpowg3 35344 | A generalization of ~ ax-p... |
| gblacfnacd 35345 | If ` G ` is a global choic... |
| onvf1odlem1 35346 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem2 35347 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem3 35348 | Lemma for ~ onvf1od . The... |
| onvf1odlem4 35349 | Lemma for ~ onvf1od . If ... |
| onvf1od 35350 | If ` G ` is a global choic... |
| vonf1owev 35351 | If ` F ` is a bijection fr... |
| wevgblacfn 35352 | If ` R ` is a well-orderin... |
| zltp1ne 35353 | Integer ordering relation.... |
| nnltp1ne 35354 | Positive integer ordering ... |
| nn0ltp1ne 35355 | Nonnegative integer orderi... |
| 0nn0m1nnn0 35356 | A number is zero if and on... |
| f1resfz0f1d 35357 | If a function with a seque... |
| fisshasheq 35358 | A finite set is equal to i... |
| revpfxsfxrev 35359 | The reverse of a prefix of... |
| swrdrevpfx 35360 | A subword expressed in ter... |
| lfuhgr 35361 | A hypergraph is loop-free ... |
| lfuhgr2 35362 | A hypergraph is loop-free ... |
| lfuhgr3 35363 | A hypergraph is loop-free ... |
| cplgredgex 35364 | Any two (distinct) vertice... |
| cusgredgex 35365 | Any two (distinct) vertice... |
| cusgredgex2 35366 | Any two distinct vertices ... |
| pfxwlk 35367 | A prefix of a walk is a wa... |
| revwlk 35368 | The reverse of a walk is a... |
| revwlkb 35369 | Two words represent a walk... |
| swrdwlk 35370 | Two matching subwords of a... |
| pthhashvtx 35371 | A graph containing a path ... |
| spthcycl 35372 | A walk is a trivial path i... |
| usgrgt2cycl 35373 | A non-trivial cycle in a s... |
| usgrcyclgt2v 35374 | A simple graph with a non-... |
| subgrwlk 35375 | If a walk exists in a subg... |
| subgrtrl 35376 | If a trail exists in a sub... |
| subgrpth 35377 | If a path exists in a subg... |
| subgrcycl 35378 | If a cycle exists in a sub... |
| cusgr3cyclex 35379 | Every complete simple grap... |
| loop1cycl 35380 | A hypergraph has a cycle o... |
| 2cycld 35381 | Construction of a 2-cycle ... |
| 2cycl2d 35382 | Construction of a 2-cycle ... |
| umgr2cycllem 35383 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 35384 | A multigraph with two dist... |
| dfacycgr1 35387 | An alternate definition of... |
| isacycgr 35388 | The property of being an a... |
| isacycgr1 35389 | The property of being an a... |
| acycgrcycl 35390 | Any cycle in an acyclic gr... |
| acycgr0v 35391 | A null graph (with no vert... |
| acycgr1v 35392 | A multigraph with one vert... |
| acycgr2v 35393 | A simple graph with two ve... |
| prclisacycgr 35394 | A proper class (representi... |
| acycgrislfgr 35395 | An acyclic hypergraph is a... |
| upgracycumgr 35396 | An acyclic pseudograph is ... |
| umgracycusgr 35397 | An acyclic multigraph is a... |
| upgracycusgr 35398 | An acyclic pseudograph is ... |
| cusgracyclt3v 35399 | A complete simple graph is... |
| pthacycspth 35400 | A path in an acyclic graph... |
| acycgrsubgr 35401 | The subgraph of an acyclic... |
| quartfull 35408 | The quartic equation, writ... |
| deranglem 35409 | Lemma for derangements. (... |
| derangval 35410 | Define the derangement fun... |
| derangf 35411 | The derangement number is ... |
| derang0 35412 | The derangement number of ... |
| derangsn 35413 | The derangement number of ... |
| derangenlem 35414 | One half of ~ derangen . ... |
| derangen 35415 | The derangement number is ... |
| subfacval 35416 | The subfactorial is define... |
| derangen2 35417 | Write the derangement numb... |
| subfacf 35418 | The subfactorial is a func... |
| subfaclefac 35419 | The subfactorial is less t... |
| subfac0 35420 | The subfactorial at zero. ... |
| subfac1 35421 | The subfactorial at one. ... |
| subfacp1lem1 35422 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 35423 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 35424 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 35425 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 35426 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 35427 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 35428 | Lemma for ~ subfacp1 . By... |
| subfacp1 35429 | A two-term recurrence for ... |
| subfacval2 35430 | A closed-form expression f... |
| subfaclim 35431 | The subfactorial converges... |
| subfacval3 35432 | Another closed form expres... |
| derangfmla 35433 | The derangements formula, ... |
| erdszelem1 35434 | Lemma for ~ erdsze . (Con... |
| erdszelem2 35435 | Lemma for ~ erdsze . (Con... |
| erdszelem3 35436 | Lemma for ~ erdsze . (Con... |
| erdszelem4 35437 | Lemma for ~ erdsze . (Con... |
| erdszelem5 35438 | Lemma for ~ erdsze . (Con... |
| erdszelem6 35439 | Lemma for ~ erdsze . (Con... |
| erdszelem7 35440 | Lemma for ~ erdsze . (Con... |
| erdszelem8 35441 | Lemma for ~ erdsze . (Con... |
| erdszelem9 35442 | Lemma for ~ erdsze . (Con... |
| erdszelem10 35443 | Lemma for ~ erdsze . (Con... |
| erdszelem11 35444 | Lemma for ~ erdsze . (Con... |
| erdsze 35445 | The Erdős-Szekeres th... |
| erdsze2lem1 35446 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 35447 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 35448 | Generalize the statement o... |
| kur14lem1 35449 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 35450 | Lemma for ~ kur14 . Write... |
| kur14lem3 35451 | Lemma for ~ kur14 . A clo... |
| kur14lem4 35452 | Lemma for ~ kur14 . Compl... |
| kur14lem5 35453 | Lemma for ~ kur14 . Closu... |
| kur14lem6 35454 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 35455 | Lemma for ~ kur14 : main p... |
| kur14lem8 35456 | Lemma for ~ kur14 . Show ... |
| kur14lem9 35457 | Lemma for ~ kur14 . Since... |
| kur14lem10 35458 | Lemma for ~ kur14 . Disch... |
| kur14 35459 | Kuratowski's closure-compl... |
| ispconn 35466 | The property of being a pa... |
| pconncn 35467 | The property of being a pa... |
| pconntop 35468 | A simply connected space i... |
| issconn 35469 | The property of being a si... |
| sconnpconn 35470 | A simply connected space i... |
| sconntop 35471 | A simply connected space i... |
| sconnpht 35472 | A closed path in a simply ... |
| cnpconn 35473 | An image of a path-connect... |
| pconnconn 35474 | A path-connected space is ... |
| txpconn 35475 | The topological product of... |
| ptpconn 35476 | The topological product of... |
| indispconn 35477 | The indiscrete topology (o... |
| connpconn 35478 | A connected and locally pa... |
| qtoppconn 35479 | A quotient of a path-conne... |
| pconnpi1 35480 | All fundamental groups in ... |
| sconnpht2 35481 | Any two paths in a simply ... |
| sconnpi1 35482 | A path-connected topologic... |
| txsconnlem 35483 | Lemma for ~ txsconn . (Co... |
| txsconn 35484 | The topological product of... |
| cvxpconn 35485 | A convex subset of the com... |
| cvxsconn 35486 | A convex subset of the com... |
| blsconn 35487 | An open ball in the comple... |
| cnllysconn 35488 | The topology of the comple... |
| resconn 35489 | A subset of ` RR ` is simp... |
| ioosconn 35490 | An open interval is simply... |
| iccsconn 35491 | A closed interval is simpl... |
| retopsconn 35492 | The real numbers are simpl... |
| iccllysconn 35493 | A closed interval is local... |
| rellysconn 35494 | The real numbers are local... |
| iisconn 35495 | The unit interval is simpl... |
| iillysconn 35496 | The unit interval is local... |
| iinllyconn 35497 | The unit interval is local... |
| fncvm 35500 | Lemma for covering maps. ... |
| cvmscbv 35501 | Change bound variables in ... |
| iscvm 35502 | The property of being a co... |
| cvmtop1 35503 | Reverse closure for a cove... |
| cvmtop2 35504 | Reverse closure for a cove... |
| cvmcn 35505 | A covering map is a contin... |
| cvmcov 35506 | Property of a covering map... |
| cvmsrcl 35507 | Reverse closure for an eve... |
| cvmsi 35508 | One direction of ~ cvmsval... |
| cvmsval 35509 | Elementhood in the set ` S... |
| cvmsss 35510 | An even covering is a subs... |
| cvmsn0 35511 | An even covering is nonemp... |
| cvmsuni 35512 | An even covering of ` U ` ... |
| cvmsdisj 35513 | An even covering of ` U ` ... |
| cvmshmeo 35514 | Every element of an even c... |
| cvmsf1o 35515 | ` F ` , localized to an el... |
| cvmscld 35516 | The sets of an even coveri... |
| cvmsss2 35517 | An open subset of an evenl... |
| cvmcov2 35518 | The covering map property ... |
| cvmseu 35519 | Every element in ` U. T ` ... |
| cvmsiota 35520 | Identify the unique elemen... |
| cvmopnlem 35521 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 35522 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 35523 | A covering map is an open ... |
| cvmliftmolem1 35524 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 35525 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 35526 | A lift of a continuous fun... |
| cvmliftmo 35527 | A lift of a continuous fun... |
| cvmliftlem1 35528 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 35529 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 35530 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 35531 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 35532 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 35533 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 35534 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 35535 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 35536 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 35537 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 35538 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 35539 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 35540 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 35541 | Lemma for ~ cvmlift . Dis... |
| cvmlift 35542 | One of the important prope... |
| cvmfo 35543 | A covering map is an onto ... |
| cvmliftiota 35544 | Write out a function ` H `... |
| cvmlift2lem1 35545 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 35546 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 35547 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 35548 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 35549 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 35550 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 35551 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 35552 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 35553 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 35554 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 35555 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 35556 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 35557 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 35558 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 35559 | A two-dimensional version ... |
| cvmliftphtlem 35560 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 35561 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 35562 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 35563 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 35564 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 35565 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 35566 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 35567 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 35568 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 35569 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 35570 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 35571 | A general version of ~ cvm... |
| snmlff 35572 | The function ` F ` from ~ ... |
| snmlfval 35573 | The function ` F ` from ~ ... |
| snmlval 35574 | The property " ` A ` is si... |
| snmlflim 35575 | If ` A ` is simply normal,... |
| goel 35590 | A "Godel-set of membership... |
| goelel3xp 35591 | A "Godel-set of membership... |
| goeleq12bg 35592 | Two "Godel-set of membersh... |
| gonafv 35593 | The "Godel-set for the She... |
| goaleq12d 35594 | Equality of the "Godel-set... |
| gonanegoal 35595 | The Godel-set for the Shef... |
| satf 35596 | The satisfaction predicate... |
| satfsucom 35597 | The satisfaction predicate... |
| satfn 35598 | The satisfaction predicate... |
| satom 35599 | The satisfaction predicate... |
| satfvsucom 35600 | The satisfaction predicate... |
| satfv0 35601 | The value of the satisfact... |
| satfvsuclem1 35602 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 35603 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 35604 | The value of the satisfact... |
| satfv1lem 35605 | Lemma for ~ satfv1 . (Con... |
| satfv1 35606 | The value of the satisfact... |
| satfsschain 35607 | The binary relation of a s... |
| satfvsucsuc 35608 | The satisfaction predicate... |
| satfbrsuc 35609 | The binary relation of a s... |
| satfrel 35610 | The value of the satisfact... |
| satfdmlem 35611 | Lemma for ~ satfdm . (Con... |
| satfdm 35612 | The domain of the satisfac... |
| satfrnmapom 35613 | The range of the satisfact... |
| satfv0fun 35614 | The value of the satisfact... |
| satf0 35615 | The satisfaction predicate... |
| satf0sucom 35616 | The satisfaction predicate... |
| satf00 35617 | The value of the satisfact... |
| satf0suclem 35618 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 35619 | The value of the satisfact... |
| satf0op 35620 | An element of a value of t... |
| satf0n0 35621 | The value of the satisfact... |
| sat1el2xp 35622 | The first component of an ... |
| fmlafv 35623 | The valid Godel formulas o... |
| fmla 35624 | The set of all valid Godel... |
| fmla0 35625 | The valid Godel formulas o... |
| fmla0xp 35626 | The valid Godel formulas o... |
| fmlasuc0 35627 | The valid Godel formulas o... |
| fmlafvel 35628 | A class is a valid Godel f... |
| fmlasuc 35629 | The valid Godel formulas o... |
| fmla1 35630 | The valid Godel formulas o... |
| isfmlasuc 35631 | The characterization of a ... |
| fmlasssuc 35632 | The Godel formulas of heig... |
| fmlaomn0 35633 | The empty set is not a God... |
| fmlan0 35634 | The empty set is not a God... |
| gonan0 35635 | The "Godel-set of NAND" is... |
| goaln0 35636 | The "Godel-set of universa... |
| gonarlem 35637 | Lemma for ~ gonar (inducti... |
| gonar 35638 | If the "Godel-set of NAND"... |
| goalrlem 35639 | Lemma for ~ goalr (inducti... |
| goalr 35640 | If the "Godel-set of unive... |
| fmla0disjsuc 35641 | The set of valid Godel for... |
| fmlasucdisj 35642 | The valid Godel formulas o... |
| satfdmfmla 35643 | The domain of the satisfac... |
| satffunlem 35644 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 35645 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 35646 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 35647 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 35648 | The domain of an ordered p... |
| satffunlem2lem2 35649 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 35650 | Lemma 1 for ~ satffun : in... |
| satffunlem2 35651 | Lemma 2 for ~ satffun : in... |
| satffun 35652 | The value of the satisfact... |
| satff 35653 | The satisfaction predicate... |
| satfun 35654 | The satisfaction predicate... |
| satfvel 35655 | An element of the value of... |
| satfv0fvfmla0 35656 | The value of the satisfact... |
| satefv 35657 | The simplified satisfactio... |
| sate0 35658 | The simplified satisfactio... |
| satef 35659 | The simplified satisfactio... |
| sate0fv0 35660 | A simplified satisfaction ... |
| satefvfmla0 35661 | The simplified satisfactio... |
| sategoelfvb 35662 | Characterization of a valu... |
| sategoelfv 35663 | Condition of a valuation `... |
| ex-sategoelel 35664 | Example of a valuation of ... |
| ex-sategoel 35665 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 35666 | The value of the satisfact... |
| 2goelgoanfmla1 35667 | Two Godel-sets of membersh... |
| satefvfmla1 35668 | The simplified satisfactio... |
| ex-sategoelelomsuc 35669 | Example of a valuation of ... |
| ex-sategoelel12 35670 | Example of a valuation of ... |
| prv 35671 | The "proves" relation on a... |
| elnanelprv 35672 | The wff ` ( A e. B -/\ B e... |
| prv0 35673 | Every wff encoded as ` U `... |
| prv1n 35674 | No wff encoded as a Godel-... |
| mvtval 35743 | The set of variable typeco... |
| mrexval 35744 | The set of "raw expression... |
| mexval 35745 | The set of expressions, wh... |
| mexval2 35746 | The set of expressions, wh... |
| mdvval 35747 | The set of disjoint variab... |
| mvrsval 35748 | The set of variables in an... |
| mvrsfpw 35749 | The set of variables in an... |
| mrsubffval 35750 | The substitution of some v... |
| mrsubfval 35751 | The substitution of some v... |
| mrsubval 35752 | The substitution of some v... |
| mrsubcv 35753 | The value of a substituted... |
| mrsubvr 35754 | The value of a substituted... |
| mrsubff 35755 | A substitution is a functi... |
| mrsubrn 35756 | Although it is defined for... |
| mrsubff1 35757 | When restricted to complet... |
| mrsubff1o 35758 | When restricted to complet... |
| mrsub0 35759 | The value of the substitut... |
| mrsubf 35760 | A substitution is a functi... |
| mrsubccat 35761 | Substitution distributes o... |
| mrsubcn 35762 | A substitution does not ch... |
| elmrsubrn 35763 | Characterization of the su... |
| mrsubco 35764 | The composition of two sub... |
| mrsubvrs 35765 | The set of variables in a ... |
| msubffval 35766 | A substitution applied to ... |
| msubfval 35767 | A substitution applied to ... |
| msubval 35768 | A substitution applied to ... |
| msubrsub 35769 | A substitution applied to ... |
| msubty 35770 | The type of a substituted ... |
| elmsubrn 35771 | Characterization of substi... |
| msubrn 35772 | Although it is defined for... |
| msubff 35773 | A substitution is a functi... |
| msubco 35774 | The composition of two sub... |
| msubf 35775 | A substitution is a functi... |
| mvhfval 35776 | Value of the function mapp... |
| mvhval 35777 | Value of the function mapp... |
| mpstval 35778 | A pre-statement is an orde... |
| elmpst 35779 | Property of being a pre-st... |
| msrfval 35780 | Value of the reduct of a p... |
| msrval 35781 | Value of the reduct of a p... |
| mpstssv 35782 | A pre-statement is an orde... |
| mpst123 35783 | Decompose a pre-statement ... |
| mpstrcl 35784 | The elements of a pre-stat... |
| msrf 35785 | The reduct of a pre-statem... |
| msrrcl 35786 | If ` X ` and ` Y ` have th... |
| mstaval 35787 | Value of the set of statem... |
| msrid 35788 | The reduct of a statement ... |
| msrfo 35789 | The reduct of a pre-statem... |
| mstapst 35790 | A statement is a pre-state... |
| elmsta 35791 | Property of being a statem... |
| ismfs 35792 | A formal system is a tuple... |
| mfsdisj 35793 | The constants and variable... |
| mtyf2 35794 | The type function maps var... |
| mtyf 35795 | The type function maps var... |
| mvtss 35796 | The set of variable typeco... |
| maxsta 35797 | An axiom is a statement. ... |
| mvtinf 35798 | Each variable typecode has... |
| msubff1 35799 | When restricted to complet... |
| msubff1o 35800 | When restricted to complet... |
| mvhf 35801 | The function mapping varia... |
| mvhf1 35802 | The function mapping varia... |
| msubvrs 35803 | The set of variables in a ... |
| mclsrcl 35804 | Reverse closure for the cl... |
| mclsssvlem 35805 | Lemma for ~ mclsssv . (Co... |
| mclsval 35806 | The function mapping varia... |
| mclsssv 35807 | The closure of a set of ex... |
| ssmclslem 35808 | Lemma for ~ ssmcls . (Con... |
| vhmcls 35809 | All variable hypotheses ar... |
| ssmcls 35810 | The original expressions a... |
| ss2mcls 35811 | The closure is monotonic u... |
| mclsax 35812 | The closure is closed unde... |
| mclsind 35813 | Induction theorem for clos... |
| mppspstlem 35814 | Lemma for ~ mppspst . (Co... |
| mppsval 35815 | Definition of a provable p... |
| elmpps 35816 | Definition of a provable p... |
| mppspst 35817 | A provable pre-statement i... |
| mthmval 35818 | A theorem is a pre-stateme... |
| elmthm 35819 | A theorem is a pre-stateme... |
| mthmi 35820 | A statement whose reduct i... |
| mthmsta 35821 | A theorem is a pre-stateme... |
| mppsthm 35822 | A provable pre-statement i... |
| mthmblem 35823 | Lemma for ~ mthmb . (Cont... |
| mthmb 35824 | If two statements have the... |
| mthmpps 35825 | Given a theorem, there is ... |
| mclsppslem 35826 | The closure is closed unde... |
| mclspps 35827 | The closure is closed unde... |
| rexxfr3d 35881 | Transfer existential quant... |
| rexxfr3dALT 35882 | Longer proof of ~ rexxfr3d... |
| rspssbasd 35883 | The span of a set of ring ... |
| ellcsrspsn 35884 | Membership in a left coset... |
| ply1divalg3 35885 | Uniqueness of polynomial r... |
| r1peuqusdeg1 35886 | Uniqueness of polynomial r... |
| problem1 35908 | Practice problem 1. Clues... |
| problem2 35909 | Practice problem 2. Clues... |
| problem3 35910 | Practice problem 3. Clues... |
| problem4 35911 | Practice problem 4. Clues... |
| problem5 35912 | Practice problem 5. Clues... |
| quad3 35913 | Variant of quadratic equat... |
| climuzcnv 35914 | Utility lemma to convert b... |
| sinccvglem 35915 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 35916 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 35917 | The circumference of a cir... |
| elfzm12 35918 | Membership in a curtailed ... |
| nn0seqcvg 35919 | A strictly-decreasing nonn... |
| lediv2aALT 35920 | Division of both sides of ... |
| abs2sqlei 35921 | The absolute values of two... |
| abs2sqlti 35922 | The absolute values of two... |
| abs2sqle 35923 | The absolute values of two... |
| abs2sqlt 35924 | The absolute values of two... |
| abs2difi 35925 | Difference of absolute val... |
| abs2difabsi 35926 | Absolute value of differen... |
| 2thALT 35927 | Alternate proof of ~ 2th .... |
| orbi2iALT 35928 | Alternate proof of ~ orbi2... |
| pm3.48ALT 35929 | Alternate proof of ~ pm3.4... |
| 3jcadALT 35930 | Alternate proof of ~ 3jcad... |
| currybi 35931 | Biconditional version of C... |
| antnest 35932 | Suppose ` ph ` , ` ps ` ar... |
| antnestlaw3lem 35933 | Lemma for ~ antnestlaw3 . ... |
| antnestlaw1 35934 | A law of nested antecedent... |
| antnestlaw2 35935 | A law of nested antecedent... |
| antnestlaw3 35936 | A law of nested antecedent... |
| antnestALT 35937 | Alternative proof of ~ ant... |
| axextprim 35944 | ~ ax-ext without distinct ... |
| axrepprim 35945 | ~ ax-rep without distinct ... |
| axunprim 35946 | ~ ax-un without distinct v... |
| axpowprim 35947 | ~ ax-pow without distinct ... |
| axregprim 35948 | ~ ax-reg without distinct ... |
| axinfprim 35949 | ~ ax-inf without distinct ... |
| axacprim 35950 | ~ ax-ac without distinct v... |
| untelirr 35951 | We call a class "untanged"... |
| untuni 35952 | The union of a class is un... |
| untsucf 35953 | If a class is untangled, t... |
| unt0 35954 | The null set is untangled.... |
| untint 35955 | If there is an untangled e... |
| efrunt 35956 | If ` A ` is well-founded b... |
| untangtr 35957 | A transitive class is unta... |
| 3jaodd 35958 | Double deduction form of ~... |
| 3orit 35959 | Closed form of ~ 3ori . (... |
| biimpexp 35960 | A biconditional in the ant... |
| nepss 35961 | Two classes are unequal if... |
| 3ccased 35962 | Triple disjunction form of... |
| dfso3 35963 | Expansion of the definitio... |
| brtpid1 35964 | A binary relation involvin... |
| brtpid2 35965 | A binary relation involvin... |
| brtpid3 35966 | A binary relation involvin... |
| iota5f 35967 | A method for computing iot... |
| jath 35968 | Closed form of ~ ja . Pro... |
| xpab 35969 | Cartesian product of two c... |
| nnuni 35970 | The union of a finite ordi... |
| sqdivzi 35971 | Distribution of square ove... |
| supfz 35972 | The supremum of a finite s... |
| inffz 35973 | The infimum of a finite se... |
| fz0n 35974 | The sequence ` ( 0 ... ( N... |
| shftvalg 35975 | Value of a sequence shifte... |
| divcnvlin 35976 | Limit of the ratio of two ... |
| climlec3 35977 | Comparison of a constant t... |
| iexpire 35978 | ` _i ` raised to itself is... |
| bcneg1 35979 | The binomial coefficient o... |
| bcm1nt 35980 | The proportion of one bino... |
| bcprod 35981 | A product identity for bin... |
| bccolsum 35982 | A column-sum rule for bino... |
| iprodefisumlem 35983 | Lemma for ~ iprodefisum . ... |
| iprodefisum 35984 | Applying the exponential f... |
| iprodgam 35985 | An infinite product versio... |
| faclimlem1 35986 | Lemma for ~ faclim . Clos... |
| faclimlem2 35987 | Lemma for ~ faclim . Show... |
| faclimlem3 35988 | Lemma for ~ faclim . Alge... |
| faclim 35989 | An infinite product expres... |
| iprodfac 35990 | An infinite product expres... |
| faclim2 35991 | Another factorial limit du... |
| gcd32 35992 | Swap the second and third ... |
| gcdabsorb 35993 | Absorption law for gcd. (... |
| dftr6 35994 | A potential definition of ... |
| coep 35995 | Composition with the membe... |
| coepr 35996 | Composition with the conve... |
| dffr5 35997 | A quantifier-free definiti... |
| dfso2 35998 | Quantifier-free definition... |
| br8 35999 | Substitution for an eight-... |
| br6 36000 | Substitution for a six-pla... |
| br4 36001 | Substitution for a four-pl... |
| cnvco1 36002 | Another distributive law o... |
| cnvco2 36003 | Another distributive law o... |
| eldm3 36004 | Quantifier-free definition... |
| elrn3 36005 | Quantifier-free definition... |
| pocnv 36006 | The converse of a partial ... |
| socnv 36007 | The converse of a strict o... |
| elintfv 36008 | Membership in an intersect... |
| funpsstri 36009 | A condition for subset tri... |
| fundmpss 36010 | If a class ` F ` is a prop... |
| funsseq 36011 | Given two functions with e... |
| fununiq 36012 | The uniqueness condition o... |
| funbreq 36013 | An equality condition for ... |
| br1steq 36014 | Uniqueness condition for t... |
| br2ndeq 36015 | Uniqueness condition for t... |
| dfdm5 36016 | Definition of domain in te... |
| dfrn5 36017 | Definition of range in ter... |
| opelco3 36018 | Alternate way of saying th... |
| elima4 36019 | Quantifier-free expression... |
| fv1stcnv 36020 | The value of the converse ... |
| fv2ndcnv 36021 | The value of the converse ... |
| elpotr 36022 | A class of transitive sets... |
| dford5reg 36023 | Given ~ ax-reg , an ordina... |
| dfon2lem1 36024 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 36025 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 36026 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 36027 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 36028 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 36029 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 36030 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 36031 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 36032 | Lemma for ~ dfon2 . A cla... |
| dfon2 36033 | ` On ` consists of all set... |
| rdgprc0 36034 | The value of the recursive... |
| rdgprc 36035 | The value of the recursive... |
| dfrdg2 36036 | Alternate definition of th... |
| dfrdg3 36037 | Generalization of ~ dfrdg2... |
| axextdfeq 36038 | A version of ~ ax-ext for ... |
| ax8dfeq 36039 | A version of ~ ax-8 for us... |
| axextdist 36040 | ~ ax-ext with distinctors ... |
| axextbdist 36041 | ~ axextb with distinctors ... |
| 19.12b 36042 | Version of ~ 19.12vv with ... |
| exnel 36043 | There is always a set not ... |
| distel 36044 | Distinctors in terms of me... |
| axextndbi 36045 | ~ axextnd as a bicondition... |
| hbntg 36046 | A more general form of ~ h... |
| hbimtg 36047 | A more general and closed ... |
| hbaltg 36048 | A more general and closed ... |
| hbng 36049 | A more general form of ~ h... |
| hbimg 36050 | A more general form of ~ h... |
| wsuceq123 36055 | Equality theorem for well-... |
| wsuceq1 36056 | Equality theorem for well-... |
| wsuceq2 36057 | Equality theorem for well-... |
| wsuceq3 36058 | Equality theorem for well-... |
| nfwsuc 36059 | Bound-variable hypothesis ... |
| wlimeq12 36060 | Equality theorem for the l... |
| wlimeq1 36061 | Equality theorem for the l... |
| wlimeq2 36062 | Equality theorem for the l... |
| nfwlim 36063 | Bound-variable hypothesis ... |
| elwlim 36064 | Membership in the limit cl... |
| wzel 36065 | The zero of a well-founded... |
| wsuclem 36066 | Lemma for the supremum pro... |
| wsucex 36067 | Existence theorem for well... |
| wsuccl 36068 | If ` X ` is a set with an ... |
| wsuclb 36069 | A well-founded successor i... |
| wlimss 36070 | The class of limit points ... |
| txpss3v 36119 | A tail Cartesian product i... |
| txprel 36120 | A tail Cartesian product i... |
| brtxp 36121 | Characterize a ternary rel... |
| brtxp2 36122 | The binary relation over a... |
| dfpprod2 36123 | Expanded definition of par... |
| pprodcnveq 36124 | A converse law for paralle... |
| pprodss4v 36125 | The parallel product is a ... |
| brpprod 36126 | Characterize a quaternary ... |
| brpprod3a 36127 | Condition for parallel pro... |
| brpprod3b 36128 | Condition for parallel pro... |
| relsset 36129 | The subset class is a bina... |
| brsset 36130 | For sets, the ` SSet ` bin... |
| idsset 36131 | ` _I ` is equal to the int... |
| eltrans 36132 | Membership in the class of... |
| dfon3 36133 | A quantifier-free definiti... |
| dfon4 36134 | Another quantifier-free de... |
| brtxpsd 36135 | Expansion of a common form... |
| brtxpsd2 36136 | Another common abbreviatio... |
| brtxpsd3 36137 | A third common abbreviatio... |
| relbigcup 36138 | The ` Bigcup ` relationshi... |
| brbigcup 36139 | Binary relation over ` Big... |
| dfbigcup2 36140 | ` Bigcup ` using maps-to n... |
| fobigcup 36141 | ` Bigcup ` maps the univer... |
| fnbigcup 36142 | ` Bigcup ` is a function o... |
| fvbigcup 36143 | For sets, ` Bigcup ` yield... |
| elfix 36144 | Membership in the fixpoint... |
| elfix2 36145 | Alternative membership in ... |
| dffix2 36146 | The fixpoints of a class i... |
| fixssdm 36147 | The fixpoints of a class a... |
| fixssrn 36148 | The fixpoints of a class a... |
| fixcnv 36149 | The fixpoints of a class a... |
| fixun 36150 | The fixpoint operator dist... |
| ellimits 36151 | Membership in the class of... |
| limitssson 36152 | The class of all limit ord... |
| dfom5b 36153 | A quantifier-free definiti... |
| sscoid 36154 | A condition for subset and... |
| dffun10 36155 | Another potential definiti... |
| elfuns 36156 | Membership in the class of... |
| elfunsg 36157 | Closed form of ~ elfuns . ... |
| brsingle 36158 | The binary relation form o... |
| elsingles 36159 | Membership in the class of... |
| fnsingle 36160 | The singleton relationship... |
| fvsingle 36161 | The value of the singleton... |
| dfsingles2 36162 | Alternate definition of th... |
| snelsingles 36163 | A singleton is a member of... |
| dfiota3 36164 | A definition of iota using... |
| dffv5 36165 | Another quantifier-free de... |
| unisnif 36166 | Express union of singleton... |
| brimage 36167 | Binary relation form of th... |
| brimageg 36168 | Closed form of ~ brimage .... |
| funimage 36169 | ` Image A ` is a function.... |
| fnimage 36170 | ` Image R ` is a function ... |
| imageval 36171 | The image functor in maps-... |
| fvimage 36172 | Value of the image functor... |
| brcart 36173 | Binary relation form of th... |
| brdomain 36174 | Binary relation form of th... |
| brrange 36175 | Binary relation form of th... |
| brdomaing 36176 | Closed form of ~ brdomain ... |
| brrangeg 36177 | Closed form of ~ brrange .... |
| brimg 36178 | Binary relation form of th... |
| brapply 36179 | Binary relation form of th... |
| brcup 36180 | Binary relation form of th... |
| brcap 36181 | Binary relation form of th... |
| lemsuccf 36182 | Lemma for unfolding differ... |
| brsuccf 36183 | Binary relation form of th... |
| dfsuccf2 36184 | Alternate definition of Sc... |
| funpartlem 36185 | Lemma for ~ funpartfun . ... |
| funpartfun 36186 | The functional part of ` F... |
| funpartss 36187 | The functional part of ` F... |
| funpartfv 36188 | The function value of the ... |
| fullfunfnv 36189 | The full functional part o... |
| fullfunfv 36190 | The function value of the ... |
| brfullfun 36191 | A binary relation form con... |
| brrestrict 36192 | Binary relation form of th... |
| dfrecs2 36193 | A quantifier-free definiti... |
| dfrdg4 36194 | A quantifier-free definiti... |
| dfint3 36195 | Quantifier-free definition... |
| imagesset 36196 | The Image functor applied ... |
| brub 36197 | Binary relation form of th... |
| brlb 36198 | Binary relation form of th... |
| altopex 36203 | Alternative ordered pairs ... |
| altopthsn 36204 | Two alternate ordered pair... |
| altopeq12 36205 | Equality for alternate ord... |
| altopeq1 36206 | Equality for alternate ord... |
| altopeq2 36207 | Equality for alternate ord... |
| altopth1 36208 | Equality of the first memb... |
| altopth2 36209 | Equality of the second mem... |
| altopthg 36210 | Alternate ordered pair the... |
| altopthbg 36211 | Alternate ordered pair the... |
| altopth 36212 | The alternate ordered pair... |
| altopthb 36213 | Alternate ordered pair the... |
| altopthc 36214 | Alternate ordered pair the... |
| altopthd 36215 | Alternate ordered pair the... |
| altxpeq1 36216 | Equality for alternate Car... |
| altxpeq2 36217 | Equality for alternate Car... |
| elaltxp 36218 | Membership in alternate Ca... |
| altopelaltxp 36219 | Alternate ordered pair mem... |
| altxpsspw 36220 | An inclusion rule for alte... |
| altxpexg 36221 | The alternate Cartesian pr... |
| rankaltopb 36222 | Compute the rank of an alt... |
| nfaltop 36223 | Bound-variable hypothesis ... |
| sbcaltop 36224 | Distribution of class subs... |
| cgrrflx2d 36227 | Deduction form of ~ axcgrr... |
| cgrtr4d 36228 | Deduction form of ~ axcgrt... |
| cgrtr4and 36229 | Deduction form of ~ axcgrt... |
| cgrrflx 36230 | Reflexivity law for congru... |
| cgrrflxd 36231 | Deduction form of ~ cgrrfl... |
| cgrcomim 36232 | Congruence commutes on the... |
| cgrcom 36233 | Congruence commutes betwee... |
| cgrcomand 36234 | Deduction form of ~ cgrcom... |
| cgrtr 36235 | Transitivity law for congr... |
| cgrtrand 36236 | Deduction form of ~ cgrtr ... |
| cgrtr3 36237 | Transitivity law for congr... |
| cgrtr3and 36238 | Deduction form of ~ cgrtr3... |
| cgrcoml 36239 | Congruence commutes on the... |
| cgrcomr 36240 | Congruence commutes on the... |
| cgrcomlr 36241 | Congruence commutes on bot... |
| cgrcomland 36242 | Deduction form of ~ cgrcom... |
| cgrcomrand 36243 | Deduction form of ~ cgrcom... |
| cgrcomlrand 36244 | Deduction form of ~ cgrcom... |
| cgrtriv 36245 | Degenerate segments are co... |
| cgrid2 36246 | Identity law for congruenc... |
| cgrdegen 36247 | Two congruent segments are... |
| brofs 36248 | Binary relation form of th... |
| 5segofs 36249 | Rephrase ~ ax5seg using th... |
| ofscom 36250 | The outer five segment pre... |
| cgrextend 36251 | Link congruence over a pai... |
| cgrextendand 36252 | Deduction form of ~ cgrext... |
| segconeq 36253 | Two points that satisfy th... |
| segconeu 36254 | Existential uniqueness ver... |
| btwntriv2 36255 | Betweenness always holds f... |
| btwncomim 36256 | Betweenness commutes. Imp... |
| btwncom 36257 | Betweenness commutes. (Co... |
| btwncomand 36258 | Deduction form of ~ btwnco... |
| btwntriv1 36259 | Betweenness always holds f... |
| btwnswapid 36260 | If you can swap the first ... |
| btwnswapid2 36261 | If you can swap arguments ... |
| btwnintr 36262 | Inner transitivity law for... |
| btwnexch3 36263 | Exchange the first endpoin... |
| btwnexch3and 36264 | Deduction form of ~ btwnex... |
| btwnouttr2 36265 | Outer transitivity law for... |
| btwnexch2 36266 | Exchange the outer point o... |
| btwnouttr 36267 | Outer transitivity law for... |
| btwnexch 36268 | Outer transitivity law for... |
| btwnexchand 36269 | Deduction form of ~ btwnex... |
| btwndiff 36270 | There is always a ` c ` di... |
| trisegint 36271 | A line segment between two... |
| funtransport 36274 | The ` TransportTo ` relati... |
| fvtransport 36275 | Calculate the value of the... |
| transportcl 36276 | Closure law for segment tr... |
| transportprops 36277 | Calculate the defining pro... |
| brifs 36286 | Binary relation form of th... |
| ifscgr 36287 | Inner five segment congrue... |
| cgrsub 36288 | Removing identical parts f... |
| brcgr3 36289 | Binary relation form of th... |
| cgr3permute3 36290 | Permutation law for three-... |
| cgr3permute1 36291 | Permutation law for three-... |
| cgr3permute2 36292 | Permutation law for three-... |
| cgr3permute4 36293 | Permutation law for three-... |
| cgr3permute5 36294 | Permutation law for three-... |
| cgr3tr4 36295 | Transitivity law for three... |
| cgr3com 36296 | Commutativity law for thre... |
| cgr3rflx 36297 | Identity law for three-pla... |
| cgrxfr 36298 | A line segment can be divi... |
| btwnxfr 36299 | A condition for extending ... |
| colinrel 36300 | Colinearity is a relations... |
| brcolinear2 36301 | Alternate colinearity bina... |
| brcolinear 36302 | The binary relation form o... |
| colinearex 36303 | The colinear predicate exi... |
| colineardim1 36304 | If ` A ` is colinear with ... |
| colinearperm1 36305 | Permutation law for coline... |
| colinearperm3 36306 | Permutation law for coline... |
| colinearperm2 36307 | Permutation law for coline... |
| colinearperm4 36308 | Permutation law for coline... |
| colinearperm5 36309 | Permutation law for coline... |
| colineartriv1 36310 | Trivial case of colinearit... |
| colineartriv2 36311 | Trivial case of colinearit... |
| btwncolinear1 36312 | Betweenness implies coline... |
| btwncolinear2 36313 | Betweenness implies coline... |
| btwncolinear3 36314 | Betweenness implies coline... |
| btwncolinear4 36315 | Betweenness implies coline... |
| btwncolinear5 36316 | Betweenness implies coline... |
| btwncolinear6 36317 | Betweenness implies coline... |
| colinearxfr 36318 | Transfer law for colineari... |
| lineext 36319 | Extend a line with a missi... |
| brofs2 36320 | Change some conditions for... |
| brifs2 36321 | Change some conditions for... |
| brfs 36322 | Binary relation form of th... |
| fscgr 36323 | Congruence law for the gen... |
| linecgr 36324 | Congruence rule for lines.... |
| linecgrand 36325 | Deduction form of ~ linecg... |
| lineid 36326 | Identity law for points on... |
| idinside 36327 | Law for finding a point in... |
| endofsegid 36328 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 36329 | Deduction form of ~ endofs... |
| btwnconn1lem1 36330 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 36331 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 36332 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 36333 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 36334 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 36335 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 36336 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 36337 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 36338 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 36339 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 36340 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 36341 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 36342 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 36343 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 36344 | Connectitivy law for betwe... |
| btwnconn2 36345 | Another connectivity law f... |
| btwnconn3 36346 | Inner connectivity law for... |
| midofsegid 36347 | If two points fall in the ... |
| segcon2 36348 | Generalization of ~ axsegc... |
| brsegle 36351 | Binary relation form of th... |
| brsegle2 36352 | Alternate characterization... |
| seglecgr12im 36353 | Substitution law for segme... |
| seglecgr12 36354 | Substitution law for segme... |
| seglerflx 36355 | Segment comparison is refl... |
| seglemin 36356 | Any segment is at least as... |
| segletr 36357 | Segment less than is trans... |
| segleantisym 36358 | Antisymmetry law for segme... |
| seglelin 36359 | Linearity law for segment ... |
| btwnsegle 36360 | If ` B ` falls between ` A... |
| colinbtwnle 36361 | Given three colinear point... |
| broutsideof 36364 | Binary relation form of ` ... |
| broutsideof2 36365 | Alternate form of ` Outsid... |
| outsidene1 36366 | Outsideness implies inequa... |
| outsidene2 36367 | Outsideness implies inequa... |
| btwnoutside 36368 | A principle linking outsid... |
| broutsideof3 36369 | Characterization of outsid... |
| outsideofrflx 36370 | Reflexivity of outsideness... |
| outsideofcom 36371 | Commutativity law for outs... |
| outsideoftr 36372 | Transitivity law for outsi... |
| outsideofeq 36373 | Uniqueness law for ` Outsi... |
| outsideofeu 36374 | Given a nondegenerate ray,... |
| outsidele 36375 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 36376 | Outside of implies colinea... |
| funray 36383 | Show that the ` Ray ` rela... |
| fvray 36384 | Calculate the value of the... |
| funline 36385 | Show that the ` Line ` rel... |
| linedegen 36386 | When ` Line ` is applied w... |
| fvline 36387 | Calculate the value of the... |
| liness 36388 | A line is a subset of the ... |
| fvline2 36389 | Alternate definition of a ... |
| lineunray 36390 | A line is composed of a po... |
| lineelsb2 36391 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 36392 | Reflexivity law for line m... |
| linecom 36393 | Commutativity law for line... |
| linerflx2 36394 | Reflexivity law for line m... |
| ellines 36395 | Membership in the set of a... |
| linethru 36396 | If ` A ` is a line contain... |
| hilbert1.1 36397 | There is a line through an... |
| hilbert1.2 36398 | There is at most one line ... |
| linethrueu 36399 | There is a unique line goi... |
| lineintmo 36400 | Two distinct lines interse... |
| fwddifval 36405 | Calculate the value of the... |
| fwddifnval 36406 | The value of the forward d... |
| fwddifn0 36407 | The value of the n-iterate... |
| fwddifnp1 36408 | The value of the n-iterate... |
| rankung 36409 | The rank of the union of t... |
| ranksng 36410 | The rank of a singleton. ... |
| rankelg 36411 | The membership relation is... |
| rankpwg 36412 | The rank of a power set. ... |
| rank0 36413 | The rank of the empty set ... |
| rankeq1o 36414 | The only set with rank ` 1... |
| elhf 36417 | Membership in the heredita... |
| elhf2 36418 | Alternate form of membersh... |
| elhf2g 36419 | Hereditarily finiteness vi... |
| 0hf 36420 | The empty set is a heredit... |
| hfun 36421 | The union of two HF sets i... |
| hfsn 36422 | The singleton of an HF set... |
| hfadj 36423 | Adjoining one HF element t... |
| hfelhf 36424 | Any member of an HF set is... |
| hftr 36425 | The class of all hereditar... |
| hfext 36426 | Extensionality for HF sets... |
| hfuni 36427 | The union of an HF set is ... |
| hfpw 36428 | The power class of an HF s... |
| hfninf 36429 | ` _om ` is not hereditaril... |
| rmoeqi 36430 | Equality inference for res... |
| rmoeqbii 36431 | Equality inference for res... |
| reueqi 36432 | Equality inference for res... |
| reueqbii 36433 | Equality inference for res... |
| sbceqbii 36434 | Formula-building inference... |
| disjeq1i 36435 | Equality theorem for disjo... |
| disjeq12i 36436 | Equality theorem for disjo... |
| rabeqbii 36437 | Equality theorem for restr... |
| iuneq12i 36438 | Equality theorem for index... |
| iineq1i 36439 | Equality theorem for index... |
| iineq12i 36440 | Equality theorem for index... |
| riotaeqbii 36441 | Equivalent wff's and equal... |
| riotaeqi 36442 | Equal domains yield equal ... |
| ixpeq1i 36443 | Equality inference for inf... |
| ixpeq12i 36444 | Equality inference for inf... |
| sumeq2si 36445 | Equality inference for sum... |
| sumeq12si 36446 | Equality inference for sum... |
| prodeq2si 36447 | Equality inference for pro... |
| prodeq12si 36448 | Equality inference for pro... |
| itgeq12i 36449 | Equality inference for an ... |
| itgeq1i 36450 | Equality inference for an ... |
| itgeq2i 36451 | Equality inference for an ... |
| ditgeq123i 36452 | Equality inference for the... |
| ditgeq12i 36453 | Equality inference for the... |
| ditgeq3i 36454 | Equality inference for the... |
| rmoeqdv 36455 | Formula-building rule for ... |
| rmoeqbidv 36456 | Formula-building rule for ... |
| sbequbidv 36457 | Deduction substituting bot... |
| disjeq12dv 36458 | Equality theorem for disjo... |
| ixpeq12dv 36459 | Equality theorem for infin... |
| sumeq12sdv 36460 | Equality deduction for sum... |
| prodeq12sdv 36461 | Equality deduction for pro... |
| itgeq12sdv 36462 | Equality theorem for an in... |
| itgeq2sdv 36463 | Equality theorem for an in... |
| ditgeq123dv 36464 | Equality theorem for the d... |
| ditgeq12d 36465 | Equality theorem for the d... |
| ditgeq3sdv 36466 | Equality theorem for the d... |
| in-ax8 36467 | A proof of ~ ax-8 that doe... |
| ss-ax8 36468 | A proof of ~ ax-8 that doe... |
| cbvralvw2 36469 | Change bound variable and ... |
| cbvrexvw2 36470 | Change bound variable and ... |
| cbvrmovw2 36471 | Change bound variable and ... |
| cbvreuvw2 36472 | Change bound variable and ... |
| cbvsbcvw2 36473 | Change bound variable of a... |
| cbvcsbvw2 36474 | Change bound variable of a... |
| cbviunvw2 36475 | Change bound variable and ... |
| cbviinvw2 36476 | Change bound variable and ... |
| cbvmptvw2 36477 | Change bound variable and ... |
| cbvdisjvw2 36478 | Change bound variable and ... |
| cbvriotavw2 36479 | Change bound variable and ... |
| cbvoprab1vw 36480 | Change the first bound var... |
| cbvoprab2vw 36481 | Change the second bound va... |
| cbvoprab123vw 36482 | Change all bound variables... |
| cbvoprab23vw 36483 | Change the second and thir... |
| cbvoprab13vw 36484 | Change the first and third... |
| cbvmpovw2 36485 | Change bound variables and... |
| cbvmpo1vw2 36486 | Change domains and the fir... |
| cbvmpo2vw2 36487 | Change domains and the sec... |
| cbvixpvw2 36488 | Change bound variable and ... |
| cbvsumvw2 36489 | Change bound variable and ... |
| cbvprodvw2 36490 | Change bound variable and ... |
| cbvitgvw2 36491 | Change bound variable and ... |
| cbvditgvw2 36492 | Change bound variable and ... |
| cbvmodavw 36493 | Change bound variable in t... |
| cbveudavw 36494 | Change bound variable in t... |
| cbvrmodavw 36495 | Change bound variable in t... |
| cbvreudavw 36496 | Change bound variable in t... |
| cbvsbdavw 36497 | Change bound variable in p... |
| cbvsbdavw2 36498 | Change bound variable in p... |
| cbvabdavw 36499 | Change bound variable in c... |
| cbvsbcdavw 36500 | Change bound variable of a... |
| cbvsbcdavw2 36501 | Change bound variable of a... |
| cbvcsbdavw 36502 | Change bound variable of a... |
| cbvcsbdavw2 36503 | Change bound variable of a... |
| cbvrabdavw 36504 | Change bound variable in r... |
| cbviundavw 36505 | Change bound variable in i... |
| cbviindavw 36506 | Change bound variable in i... |
| cbvopab1davw 36507 | Change the first bound var... |
| cbvopab2davw 36508 | Change the second bound va... |
| cbvopabdavw 36509 | Change bound variables in ... |
| cbvmptdavw 36510 | Change bound variable in a... |
| cbvdisjdavw 36511 | Change bound variable in a... |
| cbviotadavw 36512 | Change bound variable in a... |
| cbvriotadavw 36513 | Change bound variable in a... |
| cbvoprab1davw 36514 | Change the first bound var... |
| cbvoprab2davw 36515 | Change the second bound va... |
| cbvoprab3davw 36516 | Change the third bound var... |
| cbvoprab123davw 36517 | Change all bound variables... |
| cbvoprab12davw 36518 | Change the first and secon... |
| cbvoprab23davw 36519 | Change the second and thir... |
| cbvoprab13davw 36520 | Change the first and third... |
| cbvixpdavw 36521 | Change bound variable in a... |
| cbvsumdavw 36522 | Change bound variable in a... |
| cbvproddavw 36523 | Change bound variable in a... |
| cbvitgdavw 36524 | Change bound variable in a... |
| cbvditgdavw 36525 | Change bound variable in a... |
| cbvrmodavw2 36526 | Change bound variable and ... |
| cbvreudavw2 36527 | Change bound variable and ... |
| cbvrabdavw2 36528 | Change bound variable and ... |
| cbviundavw2 36529 | Change bound variable and ... |
| cbviindavw2 36530 | Change bound variable and ... |
| cbvmptdavw2 36531 | Change bound variable and ... |
| cbvdisjdavw2 36532 | Change bound variable and ... |
| cbvriotadavw2 36533 | Change bound variable and ... |
| cbvmpodavw2 36534 | Change bound variable and ... |
| cbvmpo1davw2 36535 | Change first bound variabl... |
| cbvmpo2davw2 36536 | Change second bound variab... |
| cbvixpdavw2 36537 | Change bound variable and ... |
| cbvsumdavw2 36538 | Change bound variable and ... |
| cbvproddavw2 36539 | Change bound variable and ... |
| cbvitgdavw2 36540 | Change bound variable and ... |
| cbvditgdavw2 36541 | Change bound variable and ... |
| mpomulnzcnf 36542 | Multiplication maps nonzer... |
| a1i14 36543 | Add two antecedents to a w... |
| a1i24 36544 | Add two antecedents to a w... |
| exp5d 36545 | An exportation inference. ... |
| exp5g 36546 | An exportation inference. ... |
| exp5k 36547 | An exportation inference. ... |
| exp56 36548 | An exportation inference. ... |
| exp58 36549 | An exportation inference. ... |
| exp510 36550 | An exportation inference. ... |
| exp511 36551 | An exportation inference. ... |
| exp512 36552 | An exportation inference. ... |
| 3com12d 36553 | Commutation in consequent.... |
| imp5p 36554 | A triple importation infer... |
| imp5q 36555 | A triple importation infer... |
| ecase13d 36556 | Deduction for elimination ... |
| subtr 36557 | Transitivity of implicit s... |
| subtr2 36558 | Transitivity of implicit s... |
| trer 36559 | A relation intersected wit... |
| elicc3 36560 | An equivalent membership c... |
| finminlem 36561 | A useful lemma about finit... |
| gtinf 36562 | Any number greater than an... |
| opnrebl 36563 | A set is open in the stand... |
| opnrebl2 36564 | A set is open in the stand... |
| nn0prpwlem 36565 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 36566 | Two nonnegative integers a... |
| topbnd 36567 | Two equivalent expressions... |
| opnbnd 36568 | A set is open iff it is di... |
| cldbnd 36569 | A set is closed iff it con... |
| ntruni 36570 | A union of interiors is a ... |
| clsun 36571 | A pairwise union of closur... |
| clsint2 36572 | The closure of an intersec... |
| opnregcld 36573 | A set is regularly closed ... |
| cldregopn 36574 | A set if regularly open if... |
| neiin 36575 | Two neighborhoods intersec... |
| hmeoclda 36576 | Homeomorphisms preserve cl... |
| hmeocldb 36577 | Homeomorphisms preserve cl... |
| ivthALT 36578 | An alternate proof of the ... |
| fnerel 36581 | Fineness is a relation. (... |
| isfne 36582 | The predicate " ` B ` is f... |
| isfne4 36583 | The predicate " ` B ` is f... |
| isfne4b 36584 | A condition for a topology... |
| isfne2 36585 | The predicate " ` B ` is f... |
| isfne3 36586 | The predicate " ` B ` is f... |
| fnebas 36587 | A finer cover covers the s... |
| fnetg 36588 | A finer cover generates a ... |
| fnessex 36589 | If ` B ` is finer than ` A... |
| fneuni 36590 | If ` B ` is finer than ` A... |
| fneint 36591 | If a cover is finer than a... |
| fness 36592 | A cover is finer than its ... |
| fneref 36593 | Reflexivity of the finenes... |
| fnetr 36594 | Transitivity of the finene... |
| fneval 36595 | Two covers are finer than ... |
| fneer 36596 | Fineness intersected with ... |
| topfne 36597 | Fineness for covers corres... |
| topfneec 36598 | A cover is equivalent to a... |
| topfneec2 36599 | A topology is precisely id... |
| fnessref 36600 | A cover is finer iff it ha... |
| refssfne 36601 | A cover is a refinement if... |
| neibastop1 36602 | A collection of neighborho... |
| neibastop2lem 36603 | Lemma for ~ neibastop2 . ... |
| neibastop2 36604 | In the topology generated ... |
| neibastop3 36605 | The topology generated by ... |
| topmtcl 36606 | The meet of a collection o... |
| topmeet 36607 | Two equivalent formulation... |
| topjoin 36608 | Two equivalent formulation... |
| fnemeet1 36609 | The meet of a collection o... |
| fnemeet2 36610 | The meet of equivalence cl... |
| fnejoin1 36611 | Join of equivalence classe... |
| fnejoin2 36612 | Join of equivalence classe... |
| fgmin 36613 | Minimality property of a g... |
| neifg 36614 | The neighborhood filter of... |
| tailfval 36615 | The tail function for a di... |
| tailval 36616 | The tail of an element in ... |
| eltail 36617 | An element of a tail. (Co... |
| tailf 36618 | The tail function of a dir... |
| tailini 36619 | A tail contains its initia... |
| tailfb 36620 | The collection of tails of... |
| filnetlem1 36621 | Lemma for ~ filnet . Chan... |
| filnetlem2 36622 | Lemma for ~ filnet . The ... |
| filnetlem3 36623 | Lemma for ~ filnet . (Con... |
| filnetlem4 36624 | Lemma for ~ filnet . (Con... |
| filnet 36625 | A filter has the same conv... |
| tb-ax1 36626 | The first of three axioms ... |
| tb-ax2 36627 | The second of three axioms... |
| tb-ax3 36628 | The third of three axioms ... |
| tbsyl 36629 | The weak syllogism from Ta... |
| re1ax2lem 36630 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 36631 | ~ ax-2 rederived from the ... |
| naim1 36632 | Constructor theorem for ` ... |
| naim2 36633 | Constructor theorem for ` ... |
| naim1i 36634 | Constructor rule for ` -/\... |
| naim2i 36635 | Constructor rule for ` -/\... |
| naim12i 36636 | Constructor rule for ` -/\... |
| nabi1i 36637 | Constructor rule for ` -/\... |
| nabi2i 36638 | Constructor rule for ` -/\... |
| nabi12i 36639 | Constructor rule for ` -/\... |
| df3nandALT1 36642 | The double nand expressed ... |
| df3nandALT2 36643 | The double nand expressed ... |
| andnand1 36644 | Double and in terms of dou... |
| imnand2 36645 | An ` -> ` nand relation. ... |
| nalfal 36646 | Not all sets hold ` F. ` a... |
| nexntru 36647 | There does not exist a set... |
| nexfal 36648 | There does not exist a set... |
| neufal 36649 | There does not exist exact... |
| neutru 36650 | There does not exist exact... |
| nmotru 36651 | There does not exist at mo... |
| mofal 36652 | There exist at most one se... |
| nrmo 36653 | "At most one" restricted e... |
| meran1 36654 | A single axiom for proposi... |
| meran2 36655 | A single axiom for proposi... |
| meran3 36656 | A single axiom for proposi... |
| waj-ax 36657 | A single axiom for proposi... |
| lukshef-ax2 36658 | A single axiom for proposi... |
| arg-ax 36659 | A single axiom for proposi... |
| negsym1 36660 | In the paper "On Variable ... |
| imsym1 36661 | A symmetry with ` -> ` . ... |
| bisym1 36662 | A symmetry with ` <-> ` . ... |
| consym1 36663 | A symmetry with ` /\ ` . ... |
| dissym1 36664 | A symmetry with ` \/ ` . ... |
| nandsym1 36665 | A symmetry with ` -/\ ` . ... |
| unisym1 36666 | A symmetry with ` A. ` . ... |
| exisym1 36667 | A symmetry with ` E. ` . ... |
| unqsym1 36668 | A symmetry with ` E! ` . ... |
| amosym1 36669 | A symmetry with ` E* ` . ... |
| subsym1 36670 | A symmetry with ` [ x / y ... |
| ontopbas 36671 | An ordinal number is a top... |
| onsstopbas 36672 | The class of ordinal numbe... |
| onpsstopbas 36673 | The class of ordinal numbe... |
| ontgval 36674 | The topology generated fro... |
| ontgsucval 36675 | The topology generated fro... |
| onsuctop 36676 | A successor ordinal number... |
| onsuctopon 36677 | One of the topologies on a... |
| ordtoplem 36678 | Membership of the class of... |
| ordtop 36679 | An ordinal is a topology i... |
| onsucconni 36680 | A successor ordinal number... |
| onsucconn 36681 | A successor ordinal number... |
| ordtopconn 36682 | An ordinal topology is con... |
| onintopssconn 36683 | An ordinal topology is con... |
| onsuct0 36684 | A successor ordinal number... |
| ordtopt0 36685 | An ordinal topology is T_0... |
| onsucsuccmpi 36686 | The successor of a success... |
| onsucsuccmp 36687 | The successor of a success... |
| limsucncmpi 36688 | The successor of a limit o... |
| limsucncmp 36689 | The successor of a limit o... |
| ordcmp 36690 | An ordinal topology is com... |
| ssoninhaus 36691 | The ordinal topologies ` 1... |
| onint1 36692 | The ordinal T_1 spaces are... |
| oninhaus 36693 | The ordinal Hausdorff spac... |
| fveleq 36694 | Please add description her... |
| findfvcl 36695 | Please add description her... |
| findreccl 36696 | Please add description her... |
| findabrcl 36697 | Please add description her... |
| nnssi2 36698 | Convert a theorem for real... |
| nnssi3 36699 | Convert a theorem for real... |
| nndivsub 36700 | Please add description her... |
| nndivlub 36701 | A factor of a positive int... |
| ee7.2aOLD 36704 | Lemma for Euclid's Element... |
| weiunval 36705 | Value of the relation cons... |
| weiunlem 36706 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 36707 | Lemma for ~ weiunfr . (Co... |
| weiunpo 36708 | A partial ordering on an i... |
| weiunso 36709 | A strict ordering on an in... |
| weiunfr 36710 | A well-founded relation on... |
| weiunse 36711 | The relation constructed i... |
| weiunwe 36712 | A well-ordering on an inde... |
| numiunnum 36713 | An indexed union of sets i... |
| axtco 36714 | Axiom of Transitive Contai... |
| axtco1 36716 | Strong form of the Axiom o... |
| axtco2 36717 | Weak form of the Axiom of ... |
| axtco1from2 36718 | Strong form ~ axtco1 of th... |
| axtco1g 36719 | Strong form of the Axiom o... |
| axtco2g 36720 | Weak form of the Axiom of ... |
| axtcond 36721 | A version of the Axiom of ... |
| axuntco 36722 | Derivation of ~ ax-un from... |
| axnulregtco 36723 | Derivation of ~ ax-nul fro... |
| elALTtco 36724 | Derivation of ~ el from ~ ... |
| tz9.1ctco 36725 | Version of ~ tz9.1c derive... |
| tz9.1tco 36726 | Version of ~ tz9.1 derived... |
| tr0elw 36727 | Every nonempty transitive ... |
| tr0el 36728 | Every nonempty transitive ... |
| ttceq 36731 | Equality theorem for trans... |
| ttceqi 36732 | Equality inference for tra... |
| ttceqd 36733 | Equality deduction for tra... |
| nfttc 36734 | Bound-variable hypothesis ... |
| ttcid 36735 | The transitive closure con... |
| ttctr 36736 | The transitive closure of ... |
| ttctr2 36737 | The transitive closure of ... |
| ttctr3 36738 | The transitive closure of ... |
| ttcmin 36739 | The transitive closure of ... |
| ttcexrg 36740 | If the transitive closure ... |
| ttcss 36741 | A transitive closure conta... |
| ttcss2 36742 | The subclass relationship ... |
| ttcel 36743 | A transitive closure conta... |
| ttcel2 36744 | Elements turn into subclas... |
| ttctrid 36745 | The transitive closure of ... |
| ttcidm 36746 | The transitive closure ope... |
| ssttctr 36747 | Transitivity of ` A C_ TC+... |
| elttctr 36748 | Transitivity of ` A e. TC+... |
| dfttc2g 36749 | A shorter expression for t... |
| ttc0 36750 | The transitive closure of ... |
| ttc00 36751 | A class has an empty trans... |
| csbttc 36752 | Distribute proper substitu... |
| ttcuniun 36753 | Relationship between ` TC+... |
| ttciunun 36754 | Relationship between ` TC+... |
| ttcun 36755 | Distribute union of two cl... |
| ttcuni 36756 | Distribute union of a clas... |
| ttciun 36757 | Distribute indexed union t... |
| ttcpwss 36758 | The transitive closure of ... |
| ttcsnssg 36759 | The transitive closure is ... |
| ttcsnidg 36760 | The singleton transitive c... |
| ttcsnmin 36761 | The singleton transitive c... |
| ttcsng 36762 | Relationship between ` TC+... |
| ttcsnexg 36763 | If the transitive closure ... |
| ttcsnexbig 36764 | The transitive closure of ... |
| ttcsntrsucg 36765 | The singleton transitive c... |
| dfttc3gw 36766 | If the transitive closure ... |
| ttcwf 36767 | A set is well-founded iff ... |
| ttcwf2 36768 | If a transitive closure cl... |
| ttcwf3 36769 | The sets whose transitive ... |
| ttc0elw 36770 | If a transitive closure is... |
| dfttc4lem1 36771 | Lemma for ~ dfttc4 . (Con... |
| dfttc4lem2 36772 | Lemma for ~ dfttc4 . (Con... |
| dfttc4 36773 | An alternative expression ... |
| elttcirr 36774 | Irreflexivity of ` A e. TC... |
| ttcexg 36775 | The transitive closure of ... |
| ttcexbi 36776 | A class is a set iff its t... |
| dfttc3g 36777 | The transitive closure of ... |
| ttc0el 36778 | A transitive closure conta... |
| mh-setind 36779 | Principle of set induction... |
| mh-setindnd 36780 | A version of ~ mh-setind w... |
| regsfromregtco 36781 | Derivation of ~ ax-regs fr... |
| regsfromsetind 36782 | Derivation of ~ ax-regs fr... |
| regsfromunir1 36783 | Derivation of ~ ax-regs fr... |
| mh-inf3f1 36784 | A variant of ~ inf3 . If ... |
| mh-inf3sn 36785 | Version of ~ inf3 for the ... |
| mh-prprimbi 36786 | Shortest possible version ... |
| mh-unprimbi 36787 | Shortest possible version ... |
| mh-regprimbi 36788 | Shortest possible version ... |
| mh-infprim1bi 36789 | Shortest possible axiom of... |
| mh-infprim2bi 36790 | Shortest possible axiom of... |
| mh-infprim3bi 36791 | An axiom of infinity in pr... |
| dnival 36792 | Value of the "distance to ... |
| dnicld1 36793 | Closure theorem for the "d... |
| dnicld2 36794 | Closure theorem for the "d... |
| dnif 36795 | The "distance to nearest i... |
| dnizeq0 36796 | The distance to nearest in... |
| dnizphlfeqhlf 36797 | The distance to nearest in... |
| rddif2 36798 | Variant of ~ rddif . (Con... |
| dnibndlem1 36799 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 36800 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 36801 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 36802 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 36803 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 36804 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 36805 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 36806 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 36807 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 36808 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 36809 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 36810 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 36811 | Lemma for ~ dnibnd . (Con... |
| dnibnd 36812 | The "distance to nearest i... |
| dnicn 36813 | The "distance to nearest i... |
| knoppcnlem1 36814 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 36815 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 36816 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 36817 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 36818 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 36819 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 36820 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 36821 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 36822 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 36823 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 36824 | Lemma for ~ knoppcn . (Co... |
| knoppcn 36825 | The continuous nowhere dif... |
| knoppcld 36826 | Closure theorem for Knopp'... |
| unblimceq0lem 36827 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 36828 | If ` F ` is unbounded near... |
| unbdqndv1 36829 | If the difference quotient... |
| unbdqndv2lem1 36830 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 36831 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 36832 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 36833 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 36834 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 36835 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 36836 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 36837 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 36838 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 36839 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 36840 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 36841 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 36842 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 36843 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 36844 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 36845 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 36846 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 36847 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 36848 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 36849 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 36850 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 36851 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 36852 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 36853 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 36854 | Lemma for ~ knoppndv . (C... |
| knoppndv 36855 | The continuous nowhere dif... |
| knoppf 36856 | Knopp's function is a func... |
| knoppcn2 36857 | Variant of ~ knoppcn with ... |
| cnndvlem1 36858 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 36859 | Lemma for ~ cnndv . (Cont... |
| cnndv 36860 | There exists a continuous ... |
| bj-mp2c 36861 | A double _modus ponens_ in... |
| bj-mp2d 36862 | A double _modus ponens_ in... |
| bj-0 36863 | A syntactic theorem. See ... |
| bj-1 36864 | In this proof, the use of ... |
| bj-a1k 36865 | Weakening of ~ ax-1 . As ... |
| bj-poni 36866 | Inference associated with ... |
| bj-nnclav 36867 | When ` F. ` is substituted... |
| bj-nnclavi 36868 | Inference associated with ... |
| bj-nnclavc 36869 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 36870 | Inference associated with ... |
| bj-jarrii 36871 | Inference associated with ... |
| bj-imim21 36872 | The propositional function... |
| bj-imim21i 36873 | The propositional function... |
| bj-imim11 36874 | The propositional function... |
| bj-imim11i 36875 | The propositional function... |
| bj-peircestab 36876 | Over minimal implicational... |
| bj-stabpeirce 36877 | This minimal implicational... |
| bj-bisimpl 36878 | Implication from equivalen... |
| bj-bisimpr 36879 | Implication from equivalen... |
| bj-syl66ib 36880 | A mixed syllogism inferenc... |
| bj-orim2 36881 | Proof of ~ orim2 from the ... |
| bj-currypeirce 36882 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 36883 | Peirce's axiom ~ peirce im... |
| bj-animbi 36884 | Conjunction in terms of im... |
| bj-currypara 36885 | Curry's paradox. Note tha... |
| bj-con2com 36886 | A commuted form of the con... |
| bj-con2comi 36887 | Inference associated with ... |
| bj-nimn 36888 | If a formula is true, then... |
| bj-nimni 36889 | Inference associated with ... |
| bj-peircei 36890 | Inference associated with ... |
| bj-looinvi 36891 | Inference associated with ... |
| bj-looinvii 36892 | Inference associated with ... |
| bj-mt2bi 36893 | Version of ~ mt2 where the... |
| bj-fal 36894 | Shortening of ~ fal using ... |
| bj-ntrufal 36895 | The negation of a theorem ... |
| bj-dfnul2 36896 | Alternate definition of th... |
| bj-jaoi1 36897 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 36898 | Shortens ~ consensus (110>... |
| bj-dfbi4 36899 | Alternate definition of th... |
| bj-dfbi5 36900 | Alternate definition of th... |
| bj-dfbi6 36901 | Alternate definition of th... |
| bj-bijust0ALT 36902 | Alternate proof of ~ bijus... |
| bj-bijust00 36903 | A self-implication does no... |
| bj-consensus 36904 | Version of ~ consensus exp... |
| bj-consensusALT 36905 | Alternate proof of ~ bj-co... |
| bj-df-ifc 36906 | Candidate definition for t... |
| bj-dfif 36907 | Alternate definition of th... |
| bj-ififc 36908 | A biconditional connecting... |
| bj-imbi12 36909 | Uncurried (imported) form ... |
| bj-falor 36910 | Dual of ~ truan (which has... |
| bj-falor2 36911 | Dual of ~ truan . (Contri... |
| bj-bibibi 36912 | A property of the bicondit... |
| bj-imn3ani 36913 | Duplication of ~ bnj1224 .... |
| bj-andnotim 36914 | Two ways of expressing a c... |
| bj-bi3ant 36915 | This used to be in the mai... |
| bj-bisym 36916 | This used to be in the mai... |
| bj-bixor 36917 | Equivalence of two ternary... |
| bj-axdd2 36918 | This implication, proved u... |
| bj-axd2d 36919 | This implication, proved u... |
| bj-axtd 36920 | This implication, proved f... |
| bj-gl4 36921 | In a normal modal logic, t... |
| bj-axc4 36922 | Over minimal calculus, the... |
| prvlem1 36927 | An elementary property of ... |
| prvlem2 36928 | An elementary property of ... |
| bj-babygodel 36929 | See the section header com... |
| bj-babylob 36930 | See the section header com... |
| bj-godellob 36931 | Proof of Gödel's theo... |
| bj-exexalal 36932 | A lemma for changing bound... |
| bj-genr 36933 | Generalization rule on the... |
| bj-genl 36934 | Generalization rule on the... |
| bj-genan 36935 | Generalization rule on a c... |
| bj-mpgs 36936 | From a closed form theorem... |
| bj-almp 36937 | A quantified form of ~ ax-... |
| bj-sylggt 36938 | Stronger form of ~ sylgt ,... |
| bj-alrimg 36939 | The general form of the *a... |
| bj-sylgt2 36940 | Uncurried (imported) form ... |
| bj-nexdh 36941 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 36942 | Uncurried (imported) form ... |
| bj-alimii 36943 | Inference associated with ... |
| bj-ala1i 36944 | Add an antecedent in a uni... |
| bj-almpi 36945 | A quantified form of ~ mpi... |
| bj-almpig 36946 | A partially quantified for... |
| bj-alsyl 36947 | Syllogism under the univer... |
| bj-2alim 36948 | Closed form of ~ 2alimi . ... |
| bj-alimdh 36949 | General instance of ~ alim... |
| bj-alrimdh 36950 | Deduction form of Theorem ... |
| bj-alrimd 36951 | A slightly more general ~ ... |
| bj-exa1i 36952 | Add an antecedent in an ex... |
| bj-alanim 36953 | Closed form of ~ alanimi .... |
| bj-2albi 36954 | Closed form of ~ 2albii . ... |
| bj-notalbii 36955 | Equivalence of universal q... |
| bj-2exim 36956 | Closed form of ~ 2eximi . ... |
| bj-2exbi 36957 | Closed form of ~ 2exbii . ... |
| bj-3exbi 36958 | Closed form of ~ 3exbii . ... |
| bj-sylget 36959 | Dual statement of ~ sylgt ... |
| bj-sylget2 36960 | Uncurried (imported) form ... |
| bj-exlimg 36961 | The general form of the *e... |
| bj-sylge 36962 | Dual statement of ~ sylg (... |
| bj-exlimd 36963 | A slightly more general ~ ... |
| bj-nfimexal 36964 | A weak from of nonfreeness... |
| bj-exim 36965 | Theorem 19.22 of [Margaris... |
| bj-alexim 36966 | Closed form of ~ aleximi .... |
| bj-aleximiALT 36967 | Alternate proof of ~ alexi... |
| bj-hbxfrbi 36968 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 36969 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 36970 | Distribute quantifiers ove... |
| bj-exalimi 36971 | An inference for distribut... |
| bj-eximcom 36972 | A commuted form of ~ exim ... |
| bj-exalims 36973 | Distributing quantifiers o... |
| bj-exalimsi 36974 | An inference for distribut... |
| bj-axdd2ALT 36975 | Alternate proof of ~ bj-ax... |
| bj-ax12ig 36976 | A lemma used to prove a we... |
| bj-ax12i 36977 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 36978 | Closed form of ~ nfim and ... |
| bj-spimnfe 36979 | A universal specification ... |
| bj-spimenfa 36980 | An existential generalizat... |
| bj-spim 36981 | A lemma for universal spec... |
| bj-spime 36982 | A lemma for existential ge... |
| bj-cbvalimd0 36983 | A lemma for alpha-renaming... |
| bj-cbvalimdlem 36984 | A lemma for alpha-renaming... |
| bj-cbveximdlem 36985 | A lemma for alpha-renaming... |
| bj-cbvalimd 36986 | A lemma for alpha-renaming... |
| bj-cbveximd 36987 | A lemma for alpha-renaming... |
| bj-cbvalimdv 36988 | A lemma for alpha-renaming... |
| bj-cbveximdv 36989 | A lemma for alpha-renaming... |
| bj-spvw 36990 | Version of ~ spvw and ~ 19... |
| bj-spvew 36991 | Version of ~ 19.8v and ~ 1... |
| bj-alextruim 36992 | An equivalent expression f... |
| bj-exextruan 36993 | An equivalent expression f... |
| bj-cbvalvv 36994 | Universally quantifying ov... |
| bj-cbvexvv 36995 | Existentially quantifying ... |
| bj-cbvaw 36996 | Universally quantifying ov... |
| bj-cbvew 36997 | Existentially quantifying ... |
| bj-cbveaw 36998 | Universally quantifying ov... |
| bj-cbvaew 36999 | Exixtentially quantifying ... |
| bj-ax12wlem 37000 | A lemma used to prove a we... |
| bj-cbval 37001 | Changing a bound variable ... |
| bj-cbvex 37002 | Changing a bound variable ... |
| bj-df-sb 37005 | Proposed definition to rep... |
| bj-sbcex 37006 | Proof of ~ sbcex when taki... |
| bj-dfsbc 37007 | Proof of ~ df-sbc when tak... |
| bj-ssbeq 37008 | Substitution in an equalit... |
| bj-ssblem1 37009 | A lemma for the definiens ... |
| bj-ssblem2 37010 | An instance of ~ ax-11 pro... |
| bj-ax12v 37011 | A weaker form of ~ ax-12 a... |
| bj-ax12 37012 | Remove a DV condition from... |
| bj-ax12ssb 37013 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 37014 | Special case of ~ 19.41 pr... |
| bj-equsexval 37015 | Special case of ~ equsexv ... |
| bj-subst 37016 | Proof of ~ sbalex from cor... |
| bj-ssbid2 37017 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 37018 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 37019 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 37020 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 37021 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 37022 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 37023 | Proof of ~ ax6e (hence ~ a... |
| bj-spim0 37024 | A universal specialization... |
| bj-spimvwt 37025 | Closed form of ~ spimvw . ... |
| bj-spnfw 37026 | Theorem close to a closed ... |
| bj-cbvexiw 37027 | Change bound variable. Th... |
| bj-cbvexivw 37028 | Change bound variable. Th... |
| bj-modald 37029 | A short form of the axiom ... |
| bj-denot 37030 | A weakening of ~ ax-6 and ... |
| bj-eqs 37031 | A lemma for substitutions,... |
| bj-cbvexw 37032 | Change bound variable. Th... |
| bj-ax12w 37033 | The general statement that... |
| bj-ax89 37034 | A theorem which could be u... |
| bj-cleljusti 37035 | One direction of ~ cleljus... |
| bj-alcomexcom 37036 | Commutation of two existen... |
| bj-hbald 37037 | General statement that ~ h... |
| bj-hbalt 37038 | Closed form of (general in... |
| bj-hbal 37039 | More general instance of ~... |
| axc11n11 37040 | Proof of ~ axc11n from { ~... |
| axc11n11r 37041 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 37042 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 37043 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 37044 | Alternate proof of ~ bj-ax... |
| bj-sb 37045 | A weak variant of ~ sbid2 ... |
| bj-modalbe 37046 | The predicate-calculus ver... |
| bj-spst 37047 | Closed form of ~ sps . On... |
| bj-19.21bit 37048 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 37049 | Closed form of ~ 19.23bi .... |
| bj-nexrt 37050 | Closed form of ~ nexr . C... |
| bj-alrim 37051 | Closed form of ~ alrimi . ... |
| bj-alrim2 37052 | Uncurried (imported) form ... |
| bj-nfdt0 37053 | A theorem close to a close... |
| bj-nfdt 37054 | Closed form of ~ nf5d and ... |
| bj-nexdt 37055 | Closed form of ~ nexd . (... |
| bj-nexdvt 37056 | Closed form of ~ nexdv . ... |
| bj-alexbiex 37057 | Adding a second quantifier... |
| bj-exexbiex 37058 | Adding a second quantifier... |
| bj-alalbial 37059 | Adding a second quantifier... |
| bj-exalbial 37060 | Adding a second quantifier... |
| bj-19.9htbi 37061 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 37062 | Strengthening ~ hbnt by re... |
| bj-biexal1 37063 | A general FOL biconditiona... |
| bj-biexal2 37064 | When ` ph ` is substituted... |
| bj-biexal3 37065 | When ` ph ` is substituted... |
| bj-bialal 37066 | When ` ph ` is substituted... |
| bj-biexex 37067 | When ` ph ` is substituted... |
| bj-hbexd 37068 | A more general instance of... |
| bj-hbext 37069 | Closed form of ~ bj-hbex a... |
| bj-hbex 37070 | A more general instance of... |
| bj-nfalt 37071 | Closed form of ~ nfal . (... |
| bj-nfext 37072 | Closed form of ~ nfex . (... |
| bj-eeanvw 37073 | Version of ~ exdistrv with... |
| bj-modal4 37074 | First-order logic form of ... |
| bj-modal4e 37075 | First-order logic form of ... |
| bj-modalb 37076 | A short form of the axiom ... |
| bj-wnf1 37077 | When ` ph ` is substituted... |
| bj-wnf2 37078 | When ` ph ` is substituted... |
| bj-wnfanf 37079 | When ` ph ` is substituted... |
| bj-wnfenf 37080 | When ` ph ` is substituted... |
| bj-19.12 37081 | See ~ 19.12 . Could be la... |
| bj-substax12 37082 | Equivalent form of the axi... |
| bj-substw 37083 | Weak form of the LHS of ~ ... |
| bj-nnfa 37086 | Nonfreeness implies the eq... |
| bj-nnfad 37087 | Nonfreeness implies the eq... |
| bj-nnfai 37088 | Nonfreeness implies the eq... |
| bj-nnfe 37089 | Nonfreeness implies the eq... |
| bj-nnfed 37090 | Nonfreeness implies the eq... |
| bj-nnfei 37091 | Nonfreeness implies the eq... |
| bj-nnfea 37092 | Nonfreeness implies the eq... |
| bj-nnfead 37093 | Nonfreeness implies the eq... |
| bj-nnfeai 37094 | Nonfreeness implies the eq... |
| bj-alnnf 37095 | In deduction-style proofs,... |
| bj-alnnf2 37096 | If a proposition holds, th... |
| bj-dfnnf2 37097 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 37098 | New nonfreeness implies ol... |
| bj-nnfim1 37099 | A consequence of nonfreene... |
| bj-nnfim2 37100 | A consequence of nonfreene... |
| bj-nnftht 37101 | A variable is nonfree in a... |
| bj-nnfth 37102 | A variable is nonfree in a... |
| bj-nnf-alrim 37103 | Proof of the closed form o... |
| bj-stdpc5t 37104 | Alias of ~ bj-nnf-alrim fo... |
| bj-nnfbi 37105 | If two formulas are equiva... |
| bj-nnfbd0 37106 | If two formulas are equiva... |
| bj-nnfbii 37107 | If two formulas are equiva... |
| bj-nnfnt 37108 | A variable is nonfree in a... |
| bj-nnfnth 37109 | A variable is nonfree in t... |
| bj-nnfim 37110 | Nonfreeness in the anteced... |
| bj-nnfimd 37111 | Nonfreeness in the anteced... |
| bj-nnfan 37112 | Nonfreeness in both conjun... |
| bj-nnfand 37113 | Nonfreeness in both conjun... |
| bj-nnfor 37114 | Nonfreeness in both disjun... |
| bj-nnford 37115 | Nonfreeness in both disjun... |
| bj-nnfbit 37116 | Nonfreeness in both sides ... |
| bj-nnfbid 37117 | Nonfreeness in both sides ... |
| bj-nnf-exlim 37118 | Proof of the closed form o... |
| bj-19.21t 37119 | Statement ~ 19.21t proved ... |
| bj-19.23t 37120 | Statement ~ 19.23t proved ... |
| bj-19.36im 37121 | One direction of ~ 19.36 f... |
| bj-19.37im 37122 | One direction of ~ 19.37 f... |
| bj-19.42t 37123 | Closed form of ~ 19.42 fro... |
| bj-19.41t 37124 | Closed form of ~ 19.41 fro... |
| bj-pm11.53vw 37125 | Version of ~ pm11.53v with... |
| bj-nnfv 37126 | A non-occurring variable i... |
| bj-nnfbd 37127 | If two formulas are equiva... |
| bj-pm11.53a 37128 | A variant of ~ pm11.53v . ... |
| bj-equsvt 37129 | A variant of ~ equsv . (C... |
| bj-equsalvwd 37130 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 37131 | Variant of ~ equsexvw . (... |
| bj-nnf-spim 37132 | A universal specialization... |
| bj-nnf-spime 37133 | An existential generalizat... |
| bj-nnf-cbvaliv 37134 | The only DV conditions are... |
| bj-sbievwd 37135 | Variant of ~ sbievw . (Co... |
| bj-sbft 37136 | Version of ~ sbft using ` ... |
| bj-nnf-cbvali 37137 | Compared with ~ bj-nnf-cbv... |
| bj-nnf-cbval 37138 | Compared with ~ cbvalv1 , ... |
| bj-dfnnf3 37139 | Alternate definition of no... |
| bj-nfnnfTEMP 37140 | New nonfreeness is equival... |
| bj-wnfnf 37141 | When ` ph ` is substituted... |
| bj-nnfa1 37142 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 37143 | See ~ nfe1 . (Contributed... |
| bj-nnflemaa 37144 | One of four lemmas for non... |
| bj-nnflemee 37145 | One of four lemmas for non... |
| bj-nnflemae 37146 | One of four lemmas for non... |
| bj-nnflemea 37147 | One of four lemmas for non... |
| bj-nnfalt 37148 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 37149 | See ~ nfex and ~ bj-nfext ... |
| bj-pm11.53v 37150 | Version of ~ pm11.53v with... |
| bj-axc10 37151 | Alternate proof of ~ axc10... |
| bj-alequex 37152 | A fol lemma. See ~ aleque... |
| bj-spimt2 37153 | A step in the proof of ~ s... |
| bj-cbv3ta 37154 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 37155 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 37156 | A theorem close to a close... |
| bj-hbsb3 37157 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 37158 | A theorem close to a close... |
| bj-nfs1t2 37159 | A theorem close to a close... |
| bj-nfs1 37160 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 37161 | Version of ~ axc10 with a ... |
| bj-spimtv 37162 | Version of ~ spimt with a ... |
| bj-cbv3hv2 37163 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 37164 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 37165 | Version of ~ cbv2h with a ... |
| bj-cbv2v 37166 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 37167 | Version of ~ cbvald with a... |
| bj-cbvexdv 37168 | Version of ~ cbvexd with a... |
| bj-cbval2vv 37169 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 37170 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 37171 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 37172 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 37173 | Version of ~ cbvex4v with ... |
| bj-equsalhv 37174 | Version of ~ equsalh with ... |
| bj-axc11nv 37175 | Version of ~ axc11n with a... |
| bj-aecomsv 37176 | Version of ~ aecoms with a... |
| bj-axc11v 37177 | Version of ~ axc11 with a ... |
| bj-drnf2v 37178 | Version of ~ drnf2 with a ... |
| bj-equs45fv 37179 | Version of ~ equs45f with ... |
| bj-hbs1 37180 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 37181 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 37182 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 37183 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 37184 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 37185 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 37186 | Biconditional version of a... |
| bj-hbaeb 37187 | Biconditional version of ~... |
| bj-hbnaeb 37188 | Biconditional version of ~... |
| bj-dvv 37189 | A special instance of ~ bj... |
| bj-equsal1t 37190 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 37191 | Inference associated with ... |
| bj-equsal1 37192 | One direction of ~ equsal ... |
| bj-equsal2 37193 | One direction of ~ equsal ... |
| bj-equsal 37194 | Shorter proof of ~ equsal ... |
| stdpc5t 37195 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 37196 | More direct proof of ~ std... |
| 2stdpc5 37197 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 37198 | Proof of ~ 19.21t from ~ s... |
| exlimii 37199 | Inference associated with ... |
| ax11-pm 37200 | Proof of ~ ax-11 similar t... |
| ax6er 37201 | Commuted form of ~ ax6e . ... |
| exlimiieq1 37202 | Inferring a theorem when i... |
| exlimiieq2 37203 | Inferring a theorem when i... |
| ax11-pm2 37204 | Proof of ~ ax-11 from the ... |
| bj-sbsb 37205 | Biconditional showing two ... |
| bj-dfsb2 37206 | Alternate (dual) definitio... |
| bj-sbf3 37207 | Substitution has no effect... |
| bj-sbf4 37208 | Substitution has no effect... |
| bj-eu3f 37209 | Version of ~ eu3v where th... |
| bj-sblem1 37210 | Lemma for substitution. (... |
| bj-sblem2 37211 | Lemma for substitution. (... |
| bj-sblem 37212 | Lemma for substitution. (... |
| bj-sbievw1 37213 | Lemma for substitution. (... |
| bj-sbievw2 37214 | Lemma for substitution. (... |
| bj-sbievw 37215 | Lemma for substitution. C... |
| bj-sbievv 37216 | Version of ~ sbie with a s... |
| bj-moeub 37217 | Uniqueness is equivalent t... |
| bj-sbidmOLD 37218 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 37219 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 37220 | Curried (exported) form of... |
| bj-dvelimv 37221 | A version of ~ dvelim usin... |
| bj-nfeel2 37222 | Nonfreeness in a membershi... |
| bj-axc14nf 37223 | Proof of a version of ~ ax... |
| bj-axc14 37224 | Alternate proof of ~ axc14... |
| mobidvALT 37225 | Alternate proof of ~ mobid... |
| sbn1ALT 37226 | Alternate proof of ~ sbn1 ... |
| eliminable1 37227 | A theorem used to prove th... |
| eliminable2a 37228 | A theorem used to prove th... |
| eliminable2b 37229 | A theorem used to prove th... |
| eliminable2c 37230 | A theorem used to prove th... |
| eliminable3a 37231 | A theorem used to prove th... |
| eliminable3b 37232 | A theorem used to prove th... |
| eliminable-velab 37233 | A theorem used to prove th... |
| eliminable-veqab 37234 | A theorem used to prove th... |
| eliminable-abeqv 37235 | A theorem used to prove th... |
| eliminable-abeqab 37236 | A theorem used to prove th... |
| eliminable-abelv 37237 | A theorem used to prove th... |
| eliminable-abelab 37238 | A theorem used to prove th... |
| bj-denoteslem 37239 | Duplicate of ~ issettru an... |
| bj-denotesALTV 37240 | Moved to main as ~ iseqset... |
| bj-issettruALTV 37241 | Moved to main as ~ issettr... |
| bj-elabtru 37242 | This is as close as we can... |
| bj-issetwt 37243 | Closed form of ~ bj-issetw... |
| bj-issetw 37244 | The closest one can get to... |
| bj-issetiv 37245 | Version of ~ bj-isseti wit... |
| bj-isseti 37246 | Version of ~ isseti with a... |
| bj-ralvw 37247 | A weak version of ~ ralv n... |
| bj-rexvw 37248 | A weak version of ~ rexv n... |
| bj-rababw 37249 | A weak version of ~ rabab ... |
| bj-rexcom4bv 37250 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 37251 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 37252 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 37253 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 37254 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 37255 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 37256 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 37257 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 37258 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 37259 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 37260 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 37261 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 37262 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 37263 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 37264 | Remove from ~ spcimdv depe... |
| elelb 37265 | Equivalence between two co... |
| bj-pwvrelb 37266 | Characterization of the el... |
| bj-nfcsym 37267 | The nonfreeness quantifier... |
| bj-sbeqALT 37268 | Substitution in an equalit... |
| bj-sbeq 37269 | Distribute proper substitu... |
| bj-sbceqgALT 37270 | Distribute proper substitu... |
| bj-csbsnlem 37271 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 37272 | Substitution in a singleto... |
| bj-sbel1 37273 | Version of ~ sbcel1g when ... |
| bj-abv 37274 | The class of sets verifyin... |
| bj-abvALT 37275 | Alternate version of ~ bj-... |
| bj-ab0 37276 | The class of sets verifyin... |
| bj-abf 37277 | Shorter proof of ~ abf (wh... |
| bj-csbprc 37278 | More direct proof of ~ csb... |
| bj-exlimvmpi 37279 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 37280 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 37281 | Lemma for theorems of the ... |
| bj-exlimmpbir 37282 | Lemma for theorems of the ... |
| bj-vtoclf 37283 | Remove dependency on ~ ax-... |
| bj-vtocl 37284 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 37285 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 37286 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 37287 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 37288 | A version of ~ vtoclg with... |
| bj-rabeqbid 37289 | Version of ~ rabeqbidv wit... |
| bj-seex 37290 | Version of ~ seex with a d... |
| bj-nfcf 37291 | Version of ~ df-nfc with a... |
| bj-zfauscl 37292 | General version of ~ zfaus... |
| bj-elabd2ALT 37293 | Alternate proof of ~ elabd... |
| bj-unrab 37294 | Generalization of ~ unrab ... |
| bj-inrab 37295 | Generalization of ~ inrab ... |
| bj-inrab2 37296 | Shorter proof of ~ inrab .... |
| bj-inrab3 37297 | Generalization of ~ dfrab3... |
| bj-rabtr 37298 | Restricted class abstracti... |
| bj-rabtrALT 37299 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 37300 | Proof of ~ bj-rabtr found ... |
| bj-gabss 37303 | Inclusion of generalized c... |
| bj-gabssd 37304 | Inclusion of generalized c... |
| bj-gabeqd 37305 | Equality of generalized cl... |
| bj-gabeqis 37306 | Equality of generalized cl... |
| bj-elgab 37307 | Elements of a generalized ... |
| bj-gabima 37308 | Generalized class abstract... |
| bj-ru1 37311 | A version of Russell's par... |
| bj-ru 37312 | Remove dependency on ~ ax-... |
| currysetlem 37313 | Lemma for ~ currysetlem , ... |
| curryset 37314 | Curry's paradox in set the... |
| currysetlem1 37315 | Lemma for ~ currysetALT . ... |
| currysetlem2 37316 | Lemma for ~ currysetALT . ... |
| currysetlem3 37317 | Lemma for ~ currysetALT . ... |
| currysetALT 37318 | Alternate proof of ~ curry... |
| bj-n0i 37319 | Inference associated with ... |
| bj-disjsn01 37320 | Disjointness of the single... |
| bj-0nel1 37321 | The empty set does not bel... |
| bj-1nel0 37322 | ` 1o ` does not belong to ... |
| bj-xpimasn 37323 | The image of a singleton, ... |
| bj-xpima1sn 37324 | The image of a singleton b... |
| bj-xpima1snALT 37325 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 37326 | The image of a singleton b... |
| bj-xpnzex 37327 | If the first factor of a p... |
| bj-xpexg2 37328 | Curried (exported) form of... |
| bj-xpnzexb 37329 | If the first factor of a p... |
| bj-cleq 37330 | Substitution property for ... |
| bj-snsetex 37331 | The class of sets "whose s... |
| bj-clexab 37332 | Sethood of certain classes... |
| bj-sngleq 37335 | Substitution property for ... |
| bj-elsngl 37336 | Characterization of the el... |
| bj-snglc 37337 | Characterization of the el... |
| bj-snglss 37338 | The singletonization of a ... |
| bj-0nelsngl 37339 | The empty set is not a mem... |
| bj-snglinv 37340 | Inverse of singletonizatio... |
| bj-snglex 37341 | A class is a set if and on... |
| bj-tageq 37344 | Substitution property for ... |
| bj-eltag 37345 | Characterization of the el... |
| bj-0eltag 37346 | The empty set belongs to t... |
| bj-tagn0 37347 | The tagging of a class is ... |
| bj-tagss 37348 | The tagging of a class is ... |
| bj-snglsstag 37349 | The singletonization is in... |
| bj-sngltagi 37350 | The singletonization is in... |
| bj-sngltag 37351 | The singletonization and t... |
| bj-tagci 37352 | Characterization of the el... |
| bj-tagcg 37353 | Characterization of the el... |
| bj-taginv 37354 | Inverse of tagging. (Cont... |
| bj-tagex 37355 | A class is a set if and on... |
| bj-xtageq 37356 | The products of a given cl... |
| bj-xtagex 37357 | The product of a set and t... |
| bj-projeq 37360 | Substitution property for ... |
| bj-projeq2 37361 | Substitution property for ... |
| bj-projun 37362 | The class projection on a ... |
| bj-projex 37363 | Sethood of the class proje... |
| bj-projval 37364 | Value of the class project... |
| bj-1upleq 37367 | Substitution property for ... |
| bj-pr1eq 37370 | Substitution property for ... |
| bj-pr1un 37371 | The first projection prese... |
| bj-pr1val 37372 | Value of the first project... |
| bj-pr11val 37373 | Value of the first project... |
| bj-pr1ex 37374 | Sethood of the first proje... |
| bj-1uplth 37375 | The characteristic propert... |
| bj-1uplex 37376 | A monuple is a set if and ... |
| bj-1upln0 37377 | A monuple is nonempty. (C... |
| bj-2upleq 37380 | Substitution property for ... |
| bj-pr21val 37381 | Value of the first project... |
| bj-pr2eq 37384 | Substitution property for ... |
| bj-pr2un 37385 | The second projection pres... |
| bj-pr2val 37386 | Value of the second projec... |
| bj-pr22val 37387 | Value of the second projec... |
| bj-pr2ex 37388 | Sethood of the second proj... |
| bj-2uplth 37389 | The characteristic propert... |
| bj-2uplex 37390 | A couple is a set if and o... |
| bj-2upln0 37391 | A couple is nonempty. (Co... |
| bj-2upln1upl 37392 | A couple is never equal to... |
| bj-rcleqf 37393 | Relative version of ~ cleq... |
| bj-rcleq 37394 | Relative version of ~ dfcl... |
| bj-reabeq 37395 | Relative form of ~ eqabb .... |
| bj-disj2r 37396 | Relative version of ~ ssdi... |
| bj-sscon 37397 | Contraposition law for rel... |
| bj-abex 37398 | Two ways of stating that t... |
| bj-clex 37399 | Two ways of stating that a... |
| bj-axsn 37400 | Two ways of stating the ax... |
| bj-snexg 37402 | A singleton built on a set... |
| bj-snex 37403 | A singleton is a set. See... |
| bj-axbun 37404 | Two ways of stating the ax... |
| bj-unexg 37406 | Existence of binary unions... |
| bj-prexg 37407 | Existence of unordered pai... |
| bj-prex 37408 | Existence of unordered pai... |
| bj-axadj 37409 | Two ways of stating the ax... |
| bj-adjg1 37411 | Existence of the result of... |
| bj-snfromadj 37412 | Singleton from adjunction ... |
| bj-prfromadj 37413 | Unordered pair from adjunc... |
| bj-adjfrombun 37414 | Adjunction from singleton ... |
| eleq2w2ALT 37415 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 37416 | Alternate proof of ~ clel3... |
| bj-pw0ALT 37417 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 37418 | Quantitative version of ~ ... |
| bj-unirel 37419 | Quantitative version of ~ ... |
| bj-elpwg 37420 | If the intersection of two... |
| bj-velpwALT 37421 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 37422 | Alternate proof of ~ elpwg... |
| bj-vjust 37423 | Justification theorem for ... |
| bj-nul 37424 | Two formulations of the ax... |
| bj-nuliota 37425 | Definition of the empty se... |
| bj-nuliotaALT 37426 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 37427 | Alternate proof of ~ vtocl... |
| bj-elsn12g 37428 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 37429 | Biconditional version of ~... |
| bj-pwcfsdom 37430 | Remove hypothesis from ~ p... |
| bj-grur1 37431 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 37432 | The extension of a predica... |
| bj-dfid2ALT 37433 | Alternate version of ~ dfi... |
| bj-0nelopab 37434 | The empty set is never an ... |
| bj-brrelex12ALT 37435 | Two classes related by a b... |
| bj-epelg 37436 | The membership relation an... |
| bj-epelb 37437 | Two classes are related by... |
| bj-nsnid 37438 | A set does not contain the... |
| bj-rdg0gALT 37439 | Alternate proof of ~ rdg0g... |
| bj-axnul 37440 | Over the base theory ~ ax-... |
| bj-rep 37441 | Version of the axiom of re... |
| bj-axseprep 37442 | Axiom of separation (unive... |
| bj-axreprepsep 37443 | Strong axiom of replacemen... |
| bj-evaleq 37444 | Equality theorem for the `... |
| bj-evalfun 37445 | The evaluation at a class ... |
| bj-evalfn 37446 | The evaluation at a class ... |
| bj-evalf 37447 | The evaluation at a class ... |
| bj-evalval 37448 | Value of the evaluation at... |
| bj-evalid 37449 | The evaluation at a set of... |
| bj-ndxarg 37450 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 37451 | Closed general form of ~ s... |
| bj-rest00 37454 | An elementwise intersectio... |
| bj-restsn 37455 | An elementwise intersectio... |
| bj-restsnss 37456 | Special case of ~ bj-rests... |
| bj-restsnss2 37457 | Special case of ~ bj-rests... |
| bj-restsn0 37458 | An elementwise intersectio... |
| bj-restsn10 37459 | Special case of ~ bj-rests... |
| bj-restsnid 37460 | The elementwise intersecti... |
| bj-rest10 37461 | An elementwise intersectio... |
| bj-rest10b 37462 | Alternate version of ~ bj-... |
| bj-restn0 37463 | An elementwise intersectio... |
| bj-restn0b 37464 | Alternate version of ~ bj-... |
| bj-restpw 37465 | The elementwise intersecti... |
| bj-rest0 37466 | An elementwise intersectio... |
| bj-restb 37467 | An elementwise intersectio... |
| bj-restv 37468 | An elementwise intersectio... |
| bj-resta 37469 | An elementwise intersectio... |
| bj-restuni 37470 | The union of an elementwis... |
| bj-restuni2 37471 | The union of an elementwis... |
| bj-restreg 37472 | A reformulation of the axi... |
| bj-raldifsn 37473 | All elements in a set sati... |
| bj-0int 37474 | If ` A ` is a collection o... |
| bj-mooreset 37475 | A Moore collection is a se... |
| bj-ismoore 37478 | Characterization of Moore ... |
| bj-ismoored0 37479 | Necessary condition to be ... |
| bj-ismoored 37480 | Necessary condition to be ... |
| bj-ismoored2 37481 | Necessary condition to be ... |
| bj-ismooredr 37482 | Sufficient condition to be... |
| bj-ismooredr2 37483 | Sufficient condition to be... |
| bj-discrmoore 37484 | The powerclass ` ~P A ` is... |
| bj-0nmoore 37485 | The empty set is not a Moo... |
| bj-snmoore 37486 | A singleton is a Moore col... |
| bj-snmooreb 37487 | A singleton is a Moore col... |
| bj-prmoore 37488 | A pair formed of two neste... |
| bj-0nelmpt 37489 | The empty set is not an el... |
| bj-mptval 37490 | Value of a function given ... |
| bj-dfmpoa 37491 | An equivalent definition o... |
| bj-mpomptALT 37492 | Alternate proof of ~ mpomp... |
| setsstrset 37509 | Relation between ~ df-sets... |
| bj-nfald 37510 | Variant of ~ nfald . (Con... |
| bj-nfexd 37511 | Variant of ~ nfexd . (Con... |
| cgsex2gd 37512 | Implicit substitution infe... |
| copsex2gd 37513 | Implicit substitution infe... |
| copsex2d 37514 | Implicit substitution dedu... |
| copsex2b 37515 | Biconditional form of ~ co... |
| opelopabd 37516 | Membership of an ordered p... |
| opelopabb 37517 | Membership of an ordered p... |
| opelopabbv 37518 | Membership of an ordered p... |
| bj-opelrelex 37519 | The coordinates of an orde... |
| bj-opelresdm 37520 | If an ordered pair is in a... |
| bj-brresdm 37521 | If two classes are related... |
| brabd0 37522 | Expressing that two sets a... |
| brabd 37523 | Expressing that two sets a... |
| bj-brab2a1 37524 | "Unbounded" version of ~ b... |
| bj-opabssvv 37525 | A variant of ~ relopabiv (... |
| bj-funidres 37526 | The restricted identity re... |
| bj-opelidb 37527 | Characterization of the or... |
| bj-opelidb1 37528 | Characterization of the or... |
| bj-inexeqex 37529 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 37530 | If the intersection of two... |
| bj-opelid 37531 | Characterization of the or... |
| bj-ideqg 37532 | Characterization of the cl... |
| bj-ideqgALT 37533 | Alternate proof of ~ bj-id... |
| bj-ideqb 37534 | Characterization of classe... |
| bj-idres 37535 | Alternate expression for t... |
| bj-opelidres 37536 | Characterization of the or... |
| bj-idreseq 37537 | Sufficient condition for t... |
| bj-idreseqb 37538 | Characterization for two c... |
| bj-ideqg1 37539 | For sets, the identity rel... |
| bj-ideqg1ALT 37540 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 37541 | Characterization of the co... |
| bj-elid3 37542 | Characterization of the co... |
| bj-elid4 37543 | Characterization of the el... |
| bj-elid5 37544 | Characterization of the el... |
| bj-elid6 37545 | Characterization of the el... |
| bj-elid7 37546 | Characterization of the el... |
| bj-diagval 37549 | Value of the functionalize... |
| bj-diagval2 37550 | Value of the functionalize... |
| bj-eldiag 37551 | Characterization of the el... |
| bj-eldiag2 37552 | Characterization of the el... |
| bj-imdirvallem 37555 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 37556 | Value of the functionalize... |
| bj-imdirval2lem 37557 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 37558 | Value of the functionalize... |
| bj-imdirval3 37559 | Value of the functionalize... |
| bj-imdiridlem 37560 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 37561 | Functorial property of the... |
| bj-opelopabid 37562 | Membership in an ordered-p... |
| bj-opabco 37563 | Composition of ordered-pai... |
| bj-xpcossxp 37564 | The composition of two Car... |
| bj-imdirco 37565 | Functorial property of the... |
| bj-iminvval 37568 | Value of the functionalize... |
| bj-iminvval2 37569 | Value of the functionalize... |
| bj-iminvid 37570 | Functorial property of the... |
| bj-inftyexpitaufo 37577 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 37580 | An element of the circle a... |
| bj-inftyexpiinv 37583 | Utility theorem for the in... |
| bj-inftyexpiinj 37584 | Injectivity of the paramet... |
| bj-inftyexpidisj 37585 | An element of the circle a... |
| bj-ccinftydisj 37588 | The circle at infinity is ... |
| bj-elccinfty 37589 | A lemma for infinite exten... |
| bj-ccssccbar 37592 | Complex numbers are extend... |
| bj-ccinftyssccbar 37593 | Infinite extended complex ... |
| bj-pinftyccb 37596 | The class ` pinfty ` is an... |
| bj-pinftynrr 37597 | The extended complex numbe... |
| bj-minftyccb 37600 | The class ` minfty ` is an... |
| bj-minftynrr 37601 | The extended complex numbe... |
| bj-pinftynminfty 37602 | The extended complex numbe... |
| bj-rrhatsscchat 37611 | The real projective line i... |
| bj-imafv 37626 | If the direct image of a s... |
| bj-funun 37627 | Value of a function expres... |
| bj-fununsn1 37628 | Value of a function expres... |
| bj-fununsn2 37629 | Value of a function expres... |
| bj-fvsnun1 37630 | The value of a function wi... |
| bj-fvsnun2 37631 | The value of a function wi... |
| bj-fvmptunsn1 37632 | Value of a function expres... |
| bj-fvmptunsn2 37633 | Value of a function expres... |
| bj-iomnnom 37634 | The canonical bijection fr... |
| bj-smgrpssmgm 37643 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 37644 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 37645 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 37646 | Monoids are semigroups (el... |
| bj-cmnssmnd 37647 | Commutative monoids are mo... |
| bj-cmnssmndel 37648 | Commutative monoids are mo... |
| bj-grpssmnd 37649 | Groups are monoids. (Cont... |
| bj-grpssmndel 37650 | Groups are monoids (elemen... |
| bj-ablssgrp 37651 | Abelian groups are groups.... |
| bj-ablssgrpel 37652 | Abelian groups are groups ... |
| bj-ablsscmn 37653 | Abelian groups are commuta... |
| bj-ablsscmnel 37654 | Abelian groups are commuta... |
| bj-modssabl 37655 | (The additive groups of) m... |
| bj-vecssmod 37656 | Vector spaces are modules.... |
| bj-vecssmodel 37657 | Vector spaces are modules ... |
| bj-finsumval0 37660 | Value of a finite sum. (C... |
| bj-fvimacnv0 37661 | Variant of ~ fvimacnv wher... |
| bj-isvec 37662 | The predicate "is a vector... |
| bj-fldssdrng 37663 | Fields are division rings.... |
| bj-flddrng 37664 | Fields are division rings ... |
| bj-rrdrg 37665 | The field of real numbers ... |
| bj-isclm 37666 | The predicate "is a subcom... |
| bj-isrvec 37669 | The predicate "is a real v... |
| bj-rvecmod 37670 | Real vector spaces are mod... |
| bj-rvecssmod 37671 | Real vector spaces are mod... |
| bj-rvecrr 37672 | The field of scalars of a ... |
| bj-isrvecd 37673 | The predicate "is a real v... |
| bj-rvecvec 37674 | Real vector spaces are vec... |
| bj-isrvec2 37675 | The predicate "is a real v... |
| bj-rvecssvec 37676 | Real vector spaces are vec... |
| bj-rveccmod 37677 | Real vector spaces are sub... |
| bj-rvecsscmod 37678 | Real vector spaces are sub... |
| bj-rvecsscvec 37679 | Real vector spaces are sub... |
| bj-rveccvec 37680 | Real vector spaces are sub... |
| bj-rvecssabl 37681 | (The additive groups of) r... |
| bj-rvecabl 37682 | (The additive groups of) r... |
| bj-subcom 37683 | A consequence of commutati... |
| bj-lineqi 37684 | Solution of a (scalar) lin... |
| bj-bary1lem 37685 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 37686 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 37687 | Barycentric coordinates in... |
| bj-endval 37690 | Value of the monoid of end... |
| bj-endbase 37691 | Base set of the monoid of ... |
| bj-endcomp 37692 | Composition law of the mon... |
| bj-endmnd 37693 | The monoid of endomorphism... |
| taupilem3 37694 | Lemma for tau-related theo... |
| taupilemrplb 37695 | A set of positive reals ha... |
| taupilem1 37696 | Lemma for ~ taupi . A pos... |
| taupilem2 37697 | Lemma for ~ taupi . The s... |
| taupi 37698 | Relationship between ` _ta... |
| dfgcd3 37699 | Alternate definition of th... |
| irrdifflemf 37700 | Lemma for ~ irrdiff . The... |
| irrdiff 37701 | The irrationals are exactl... |
| qdiff 37702 | The rationals are exactly ... |
| qdiffALT 37703 | Alternate proof of ~ qdiff... |
| iccioo01 37704 | The closed unit interval i... |
| csbrecsg 37705 | Move class substitution in... |
| csbrdgg 37706 | Move class substitution in... |
| csboprabg 37707 | Move class substitution in... |
| csbmpo123 37708 | Move class substitution in... |
| con1bii2 37709 | A contraposition inference... |
| con2bii2 37710 | A contraposition inference... |
| vtoclefex 37711 | Implicit substitution of a... |
| rnmptsn 37712 | The range of a function ma... |
| f1omptsnlem 37713 | This is the core of the pr... |
| f1omptsn 37714 | A function mapping to sing... |
| mptsnunlem 37715 | This is the core of the pr... |
| mptsnun 37716 | A class ` B ` is equal to ... |
| dissneqlem 37717 | This is the core of the pr... |
| dissneq 37718 | Any topology that contains... |
| exlimim 37719 | Closed form of ~ exlimimd ... |
| exlimimd 37720 | Existential elimination ru... |
| exellim 37721 | Closed form of ~ exellimdd... |
| exellimddv 37722 | Eliminate an antecedent wh... |
| topdifinfindis 37723 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 37724 | This is the core of the pr... |
| topdifinffin 37725 | Part of Exercise 3 of [Mun... |
| topdifinf 37726 | Part of Exercise 3 of [Mun... |
| topdifinfeq 37727 | Two different ways of defi... |
| icorempo 37728 | Closed-below, open-above i... |
| icoreresf 37729 | Closed-below, open-above i... |
| icoreval 37730 | Value of the closed-below,... |
| icoreelrnab 37731 | Elementhood in the set of ... |
| isbasisrelowllem1 37732 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 37733 | Lemma for ~ isbasisrelowl ... |
| icoreclin 37734 | The set of closed-below, o... |
| isbasisrelowl 37735 | The set of all closed-belo... |
| icoreunrn 37736 | The union of all closed-be... |
| istoprelowl 37737 | The set of all closed-belo... |
| icoreelrn 37738 | A class abstraction which ... |
| iooelexlt 37739 | An element of an open inte... |
| relowlssretop 37740 | The lower limit topology o... |
| relowlpssretop 37741 | The lower limit topology o... |
| sucneqond 37742 | Inequality of an ordinal s... |
| sucneqoni 37743 | Inequality of an ordinal s... |
| onsucuni3 37744 | If an ordinal number has a... |
| 1oequni2o 37745 | The ordinal number ` 1o ` ... |
| rdgsucuni 37746 | If an ordinal number has a... |
| rdgeqoa 37747 | If a recursive function wi... |
| elxp8 37748 | Membership in a Cartesian ... |
| cbveud 37749 | Deduction used to change b... |
| cbvreud 37750 | Deduction used to change b... |
| difunieq 37751 | The difference of unions i... |
| inunissunidif 37752 | Theorem about subsets of t... |
| rdgellim 37753 | Elementhood in a recursive... |
| rdglimss 37754 | A recursive definition at ... |
| rdgssun 37755 | In a recursive definition ... |
| exrecfnlem 37756 | Lemma for ~ exrecfn . (Co... |
| exrecfn 37757 | Theorem about the existenc... |
| exrecfnpw 37758 | For any base set, a set wh... |
| finorwe 37759 | If the Axiom of Infinity i... |
| dffinxpf 37762 | This theorem is the same a... |
| finxpeq1 37763 | Equality theorem for Carte... |
| finxpeq2 37764 | Equality theorem for Carte... |
| csbfinxpg 37765 | Distribute proper substitu... |
| finxpreclem1 37766 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 37767 | Lemma for ` ^^ ` recursion... |
| finxp0 37768 | The value of Cartesian exp... |
| finxp1o 37769 | The value of Cartesian exp... |
| finxpreclem3 37770 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 37771 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 37772 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 37773 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 37774 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 37775 | The value of Cartesian exp... |
| finxp2o 37776 | The value of Cartesian exp... |
| finxp3o 37777 | The value of Cartesian exp... |
| finxpnom 37778 | Cartesian exponentiation w... |
| finxp00 37779 | Cartesian exponentiation o... |
| iunctb2 37780 | Using the axiom of countab... |
| domalom 37781 | A class which dominates ev... |
| isinf2 37782 | The converse of ~ isinf . ... |
| ctbssinf 37783 | Using the axiom of choice,... |
| ralssiun 37784 | The index set of an indexe... |
| nlpineqsn 37785 | For every point ` p ` of a... |
| nlpfvineqsn 37786 | Given a subset ` A ` of ` ... |
| fvineqsnf1 37787 | A theorem about functions ... |
| fvineqsneu 37788 | A theorem about functions ... |
| fvineqsneq 37789 | A theorem about functions ... |
| pibp16 37790 | Property P000016 of pi-bas... |
| pibp19 37791 | Property P000019 of pi-bas... |
| pibp21 37792 | Property P000021 of pi-bas... |
| pibt1 37793 | Theorem T000001 of pi-base... |
| pibt2 37794 | Theorem T000002 of pi-base... |
| wl-section-prop 37795 | Intuitionistic logic is no... |
| wl-section-boot 37799 | In this section, I provide... |
| wl-luk-imim1i 37800 | Inference adding common co... |
| wl-luk-syl 37801 | An inference version of th... |
| wl-luk-imtrid 37802 | A syllogism rule of infere... |
| wl-luk-pm2.18d 37803 | Deduction based on reducti... |
| wl-luk-con4i 37804 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 37805 | Inference rule. Copy of ~... |
| wl-luk-a1i 37806 | Inference rule. Copy of ~... |
| wl-luk-mpi 37807 | A nested _modus ponens_ in... |
| wl-luk-imim2i 37808 | Inference adding common an... |
| wl-luk-imtrdi 37809 | A syllogism rule of infere... |
| wl-luk-ax3 37810 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 37811 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 37812 | This theorem, called "Asse... |
| wl-luk-com12 37813 | Inference that swaps (comm... |
| wl-luk-pm2.21 37814 | From a wff and its negatio... |
| wl-luk-con1i 37815 | A contraposition inference... |
| wl-luk-ja 37816 | Inference joining the ante... |
| wl-luk-imim2 37817 | A closed form of syllogism... |
| wl-luk-a1d 37818 | Deduction introducing an e... |
| wl-luk-ax2 37819 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 37820 | Principle of identity. Th... |
| wl-luk-notnotr 37821 | Converse of double negatio... |
| wl-luk-pm2.04 37822 | Swap antecedents. Theorem... |
| wl-section-impchain 37823 | An implication like ` ( ps... |
| wl-impchain-mp-x 37824 | This series of theorems pr... |
| wl-impchain-mp-0 37825 | This theorem is the start ... |
| wl-impchain-mp-1 37826 | This theorem is in fact a ... |
| wl-impchain-mp-2 37827 | This theorem is in fact a ... |
| wl-impchain-com-1.x 37828 | It is often convenient to ... |
| wl-impchain-com-1.1 37829 | A degenerate form of antec... |
| wl-impchain-com-1.2 37830 | This theorem is in fact a ... |
| wl-impchain-com-1.3 37831 | This theorem is in fact a ... |
| wl-impchain-com-1.4 37832 | This theorem is in fact a ... |
| wl-impchain-com-n.m 37833 | This series of theorems al... |
| wl-impchain-com-2.3 37834 | This theorem is in fact a ... |
| wl-impchain-com-2.4 37835 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 37836 | This theorem is in fact a ... |
| wl-impchain-a1-x 37837 | If an implication chain is... |
| wl-impchain-a1-1 37838 | Inference rule, a copy of ... |
| wl-impchain-a1-2 37839 | Inference rule, a copy of ... |
| wl-impchain-a1-3 37840 | Inference rule, a copy of ... |
| wl-ifp-ncond1 37841 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 37842 | If one case of an ` if- ` ... |
| wl-ifpimpr 37843 | If one case of an ` if- ` ... |
| wl-ifp4impr 37844 | If one case of an ` if- ` ... |
| wl-df-3xor 37845 | Alternative definition of ... |
| wl-df3xor2 37846 | Alternative definition of ... |
| wl-df3xor3 37847 | Alternative form of ~ wl-d... |
| wl-3xortru 37848 | If the first input is true... |
| wl-3xorfal 37849 | If the first input is fals... |
| wl-3xorbi 37850 | Triple xor can be replaced... |
| wl-3xorbi2 37851 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 37852 | Equivalence theorem for tr... |
| wl-3xorbi123i 37853 | Equivalence theorem for tr... |
| wl-3xorrot 37854 | Rotation law for triple xo... |
| wl-3xorcoma 37855 | Commutative law for triple... |
| wl-3xorcomb 37856 | Commutative law for triple... |
| wl-3xornot1 37857 | Flipping the first input f... |
| wl-3xornot 37858 | Triple xor distributes ove... |
| wl-1xor 37859 | In the recursive scheme ... |
| wl-2xor 37860 | In the recursive scheme ... |
| wl-df-3mintru2 37861 | Alternative definition of ... |
| wl-df2-3mintru2 37862 | The adder carry in disjunc... |
| wl-df3-3mintru2 37863 | The adder carry in conjunc... |
| wl-df4-3mintru2 37864 | An alternative definition ... |
| wl-1mintru1 37865 | Using the recursion formul... |
| wl-1mintru2 37866 | Using the recursion formul... |
| wl-2mintru1 37867 | Using the recursion formul... |
| wl-2mintru2 37868 | Using the recursion formul... |
| wl-df3maxtru1 37869 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 37871 | A version of ~ ax-wl-13v w... |
| wl-cleq-0 37872 |
Disclaimer: |
| wl-cleq-1 37873 |
Disclaimer: |
| wl-cleq-2 37874 |
Disclaimer: |
| wl-cleq-3 37875 |
Disclaimer: |
| wl-cleq-4 37876 |
Disclaimer: |
| wl-cleq-5 37877 |
Disclaimer: |
| wl-cleq-6 37878 |
Disclaimer: |
| wl-df-clab 37881 | Disclaimer: The material ... |
| wl-isseteq 37882 | A class equal to a set var... |
| wl-ax12v2cl 37883 | The class version of ~ ax1... |
| wl-df.clab 37884 | Define class abstractions,... |
| wl-df.cleq 37885 | Define the equality connec... |
| wl-dfcleq.basic 37886 | This theorem is a conserva... |
| wl-dfcleq.just 37887 | The hypotheses added to th... |
| wl-df.clel 37888 | Define the membership conn... |
| wl-dfclel.basic 37889 | This theorem gives a conse... |
| wl-dfclel.just 37890 | Add a hypothesis to ~ wl-d... |
| wl-dfcleq 37891 | The defining characterizat... |
| wl-dfclel 37892 | The defining characterizat... |
| wl-mps 37893 | Replacing a nested consequ... |
| wl-syls1 37894 | Replacing a nested consequ... |
| wl-syls2 37895 | Replacing a nested anteced... |
| wl-embant 37896 | A true wff can always be a... |
| wl-orel12 37897 | In a conjunctive normal fo... |
| wl-cases2-dnf 37898 | A particular instance of ~... |
| wl-cbvmotv 37899 | Change bound variable. Us... |
| wl-moteq 37900 | Change bound variable. Us... |
| wl-motae 37901 | Change bound variable. Us... |
| wl-moae 37902 | Two ways to express "at mo... |
| wl-euae 37903 | Two ways to express "exact... |
| wl-nax6im 37904 | The following series of th... |
| wl-hbae1 37905 | This specialization of ~ h... |
| wl-naevhba1v 37906 | An instance of ~ hbn1w app... |
| wl-spae 37907 | Prove an instance of ~ sp ... |
| wl-speqv 37908 | Under the assumption ` -. ... |
| wl-19.8eqv 37909 | Under the assumption ` -. ... |
| wl-19.2reqv 37910 | Under the assumption ` -. ... |
| wl-nfalv 37911 | If ` x ` is not present in... |
| wl-nfimf1 37912 | An antecedent is irrelevan... |
| wl-nfae1 37913 | Unlike ~ nfae , this speci... |
| wl-nfnae1 37914 | Unlike ~ nfnae , this spec... |
| wl-aetr 37915 | A transitive law for varia... |
| wl-axc11r 37916 | Same as ~ axc11r , but usi... |
| wl-dral1d 37917 | A version of ~ dral1 with ... |
| wl-cbvalnaed 37918 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 37919 | A more general version of ... |
| wl-exeq 37920 | The semantics of ` E. x y ... |
| wl-aleq 37921 | The semantics of ` A. x y ... |
| wl-nfeqfb 37922 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 37923 | If ` y ` is not free in ` ... |
| wl-equsalvw 37924 | Version of ~ equsalv with ... |
| wl-equsald 37925 | Deduction version of ~ equ... |
| wl-equsaldv 37926 | Deduction version of ~ equ... |
| wl-equsal 37927 | A useful equivalence relat... |
| wl-equsal1t 37928 | The expression ` x = y ` i... |
| wl-equsalcom 37929 | This simple equivalence ea... |
| wl-equsal1i 37930 | The antecedent ` x = y ` i... |
| wl-sbid2ft 37931 | A more general version of ... |
| wl-cbvalsbi 37932 | Change bounded variables i... |
| wl-sbrimt 37933 | Substitution with a variab... |
| wl-sblimt 37934 | Substitution with a variab... |
| wl-sb9v 37935 | Commutation of quantificat... |
| wl-sb8ft 37936 | Substitution of variable i... |
| wl-sb8eft 37937 | Substitution of variable i... |
| wl-sb8t 37938 | Substitution of variable i... |
| wl-sb8et 37939 | Substitution of variable i... |
| wl-sbhbt 37940 | Closed form of ~ sbhb . C... |
| wl-sbnf1 37941 | Two ways expressing that `... |
| wl-equsb3 37942 | ~ equsb3 with a distinctor... |
| wl-equsb4 37943 | Substitution applied to an... |
| wl-2sb6d 37944 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 37945 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 37946 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 37947 | Version of ~ sbcom2 with a... |
| wl-sbalnae 37948 | A theorem used in eliminat... |
| wl-sbal1 37949 | A theorem used in eliminat... |
| wl-sbal2 37950 | Move quantifier in and out... |
| wl-2spsbbi 37951 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 37952 | This theorem provides a ba... |
| wl-lem-nexmo 37953 | This theorem provides a ba... |
| wl-lem-moexsb 37954 | The antecedent ` A. x ( ph... |
| wl-alanbii 37955 | This theorem extends ~ ala... |
| wl-mo2df 37956 | Version of ~ mof with a co... |
| wl-mo2tf 37957 | Closed form of ~ mof with ... |
| wl-eudf 37958 | Version of ~ eu6 with a co... |
| wl-eutf 37959 | Closed form of ~ eu6 with ... |
| wl-euequf 37960 | ~ euequ proved with a dist... |
| wl-mo2t 37961 | Closed form of ~ mof . (C... |
| wl-mo3t 37962 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 37963 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 37964 | Substitution of variable i... |
| wl-sb8eutv 37965 | Substitution of variable i... |
| wl-sb8mot 37966 | Substitution of variable i... |
| wl-sb8motv 37967 | Substitution of variable i... |
| wl-issetft 37968 | A closed form of ~ issetf ... |
| wl-axc11rc11 37969 | Proving ~ axc11r from ~ ax... |
| wl-clabv 37970 | Variant of ~ df-clab , whe... |
| wl-dfclab 37971 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 37972 | Using class abstraction in... |
| wl-clabt 37973 | Using class abstraction in... |
| wl-eujustlem1 37974 | Version of ~ cbvexvw with ... |
| rabiun 37975 | Abstraction restricted to ... |
| iundif1 37976 | Indexed union of class dif... |
| imadifss 37977 | The difference of images i... |
| cureq 37978 | Equality theorem for curry... |
| unceq 37979 | Equality theorem for uncur... |
| curf 37980 | Functional property of cur... |
| uncf 37981 | Functional property of unc... |
| curfv 37982 | Value of currying. (Contr... |
| uncov 37983 | Value of uncurrying. (Con... |
| curunc 37984 | Currying of uncurrying. (... |
| unccur 37985 | Uncurrying of currying. (... |
| phpreu 37986 | Theorem related to pigeonh... |
| finixpnum 37987 | A finite Cartesian product... |
| fin2solem 37988 | Lemma for ~ fin2so . (Con... |
| fin2so 37989 | Any totally ordered Tarski... |
| ltflcei 37990 | Theorem to move the floor ... |
| leceifl 37991 | Theorem to move the floor ... |
| sin2h 37992 | Half-angle rule for sine. ... |
| cos2h 37993 | Half-angle rule for cosine... |
| tan2h 37994 | Half-angle rule for tangen... |
| lindsadd 37995 | In a vector space, the uni... |
| lindsdom 37996 | A linearly independent set... |
| lindsenlbs 37997 | A maximal linearly indepen... |
| matunitlindflem1 37998 | One direction of ~ matunit... |
| matunitlindflem2 37999 | One direction of ~ matunit... |
| matunitlindf 38000 | A matrix over a field is i... |
| ptrest 38001 | Expressing a restriction o... |
| ptrecube 38002 | Any point in an open set o... |
| poimirlem1 38003 | Lemma for ~ poimir - the v... |
| poimirlem2 38004 | Lemma for ~ poimir - conse... |
| poimirlem3 38005 | Lemma for ~ poimir to add ... |
| poimirlem4 38006 | Lemma for ~ poimir connect... |
| poimirlem5 38007 | Lemma for ~ poimir to esta... |
| poimirlem6 38008 | Lemma for ~ poimir establi... |
| poimirlem7 38009 | Lemma for ~ poimir , simil... |
| poimirlem8 38010 | Lemma for ~ poimir , estab... |
| poimirlem9 38011 | Lemma for ~ poimir , estab... |
| poimirlem10 38012 | Lemma for ~ poimir establi... |
| poimirlem11 38013 | Lemma for ~ poimir connect... |
| poimirlem12 38014 | Lemma for ~ poimir connect... |
| poimirlem13 38015 | Lemma for ~ poimir - for a... |
| poimirlem14 38016 | Lemma for ~ poimir - for a... |
| poimirlem15 38017 | Lemma for ~ poimir , that ... |
| poimirlem16 38018 | Lemma for ~ poimir establi... |
| poimirlem17 38019 | Lemma for ~ poimir establi... |
| poimirlem18 38020 | Lemma for ~ poimir stating... |
| poimirlem19 38021 | Lemma for ~ poimir establi... |
| poimirlem20 38022 | Lemma for ~ poimir establi... |
| poimirlem21 38023 | Lemma for ~ poimir stating... |
| poimirlem22 38024 | Lemma for ~ poimir , that ... |
| poimirlem23 38025 | Lemma for ~ poimir , two w... |
| poimirlem24 38026 | Lemma for ~ poimir , two w... |
| poimirlem25 38027 | Lemma for ~ poimir stating... |
| poimirlem26 38028 | Lemma for ~ poimir showing... |
| poimirlem27 38029 | Lemma for ~ poimir showing... |
| poimirlem28 38030 | Lemma for ~ poimir , a var... |
| poimirlem29 38031 | Lemma for ~ poimir connect... |
| poimirlem30 38032 | Lemma for ~ poimir combini... |
| poimirlem31 38033 | Lemma for ~ poimir , assig... |
| poimirlem32 38034 | Lemma for ~ poimir , combi... |
| poimir 38035 | Poincare-Miranda theorem. ... |
| broucube 38036 | Brouwer - or as Kulpa call... |
| heicant 38037 | Heine-Cantor theorem: a co... |
| opnmbllem0 38038 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 38039 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 38040 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 38041 | The difference between two... |
| mblfinlem4 38042 | Backward direction of ~ is... |
| ismblfin 38043 | Measurability in terms of ... |
| ovoliunnfl 38044 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 38045 | Demonstration of ~ ovoliun... |
| voliunnfl 38046 | ~ voliun is incompatible w... |
| volsupnfl 38047 | ~ volsup is incompatible w... |
| mbfresfi 38048 | Measurability of a piecewi... |
| mbfposadd 38049 | If the sum of two measurab... |
| cnambfre 38050 | A real-valued, a.e. contin... |
| dvtanlem 38051 | Lemma for ~ dvtan - the do... |
| dvtan 38052 | Derivative of tangent. (C... |
| itg2addnclem 38053 | An alternate expression fo... |
| itg2addnclem2 38054 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 38055 | Lemma incomprehensible in ... |
| itg2addnc 38056 | Alternate proof of ~ itg2a... |
| itg2gt0cn 38057 | ~ itg2gt0 holds on functio... |
| ibladdnclem 38058 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 38059 | Choice-free analogue of ~ ... |
| itgaddnclem1 38060 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 38061 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 38062 | Choice-free analogue of ~ ... |
| iblsubnc 38063 | Choice-free analogue of ~ ... |
| itgsubnc 38064 | Choice-free analogue of ~ ... |
| iblabsnclem 38065 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 38066 | Choice-free analogue of ~ ... |
| iblmulc2nc 38067 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 38068 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 38069 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 38070 | Choice-free analogue of ~ ... |
| itgabsnc 38071 | Choice-free analogue of ~ ... |
| itggt0cn 38072 | ~ itggt0 holds for continu... |
| ftc1cnnclem 38073 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 38074 | Choice-free proof of ~ ftc... |
| ftc1anclem1 38075 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 38076 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 38077 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 38078 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 38079 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 38080 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 38081 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 38082 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 38083 | ~ ftc1a holds for function... |
| ftc2nc 38084 | Choice-free proof of ~ ftc... |
| asindmre 38085 | Real part of domain of dif... |
| dvasin 38086 | Derivative of arcsine. (C... |
| dvacos 38087 | Derivative of arccosine. ... |
| dvreasin 38088 | Real derivative of arcsine... |
| dvreacos 38089 | Real derivative of arccosi... |
| areacirclem1 38090 | Antiderivative of cross-se... |
| areacirclem2 38091 | Endpoint-inclusive continu... |
| areacirclem3 38092 | Integrability of cross-sec... |
| areacirclem4 38093 | Endpoint-inclusive continu... |
| areacirclem5 38094 | Finding the cross-section ... |
| areacirc 38095 | The area of a circle of ra... |
| unirep 38096 | Define a quantity whose de... |
| cover2 38097 | Two ways of expressing the... |
| cover2g 38098 | Two ways of expressing the... |
| brabg2 38099 | Relation by a binary relat... |
| opelopab3 38100 | Ordered pair membership in... |
| cocanfo 38101 | Cancellation of a surjecti... |
| brresi2 38102 | Restriction of a binary re... |
| fnopabeqd 38103 | Equality deduction for fun... |
| fvopabf4g 38104 | Function value of an opera... |
| fnopabco 38105 | Composition of a function ... |
| opropabco 38106 | Composition of an operator... |
| cocnv 38107 | Composition with a functio... |
| f1ocan1fv 38108 | Cancel a composition by a ... |
| f1ocan2fv 38109 | Cancel a composition by th... |
| inixp 38110 | Intersection of Cartesian ... |
| upixp 38111 | Universal property of the ... |
| abrexdom 38112 | An indexed set is dominate... |
| abrexdom2 38113 | An indexed set is dominate... |
| ac6gf 38114 | Axiom of Choice. (Contrib... |
| indexa 38115 | If for every element of an... |
| indexdom 38116 | If for every element of an... |
| frinfm 38117 | A subset of a well-founded... |
| welb 38118 | A nonempty subset of a wel... |
| supex2g 38119 | Existence of supremum. (C... |
| supclt 38120 | Closure of supremum. (Con... |
| supubt 38121 | Upper bound property of su... |
| filbcmb 38122 | Combine a finite set of lo... |
| fzmul 38123 | Membership of a product in... |
| sdclem2 38124 | Lemma for ~ sdc . (Contri... |
| sdclem1 38125 | Lemma for ~ sdc . (Contri... |
| sdc 38126 | Strong dependent choice. ... |
| fdc 38127 | Finite version of dependen... |
| fdc1 38128 | Variant of ~ fdc with no s... |
| seqpo 38129 | Two ways to say that a seq... |
| incsequz 38130 | An increasing sequence of ... |
| incsequz2 38131 | An increasing sequence of ... |
| nnubfi 38132 | A bounded above set of pos... |
| nninfnub 38133 | An infinite set of positiv... |
| subspopn 38134 | An open set is open in the... |
| neificl 38135 | Neighborhoods are closed u... |
| lpss2 38136 | Limit points of a subset a... |
| metf1o 38137 | Use a bijection with a met... |
| blssp 38138 | A ball in the subspace met... |
| mettrifi 38139 | Generalized triangle inequ... |
| lmclim2 38140 | A sequence in a metric spa... |
| geomcau 38141 | If the distance between co... |
| caures 38142 | The restriction of a Cauch... |
| caushft 38143 | A shifted Cauchy sequence ... |
| constcncf 38144 | A constant function is a c... |
| cnres2 38145 | The restriction of a conti... |
| cnresima 38146 | A continuous function is c... |
| cncfres 38147 | A continuous function on c... |
| istotbnd 38151 | The predicate "is a totall... |
| istotbnd2 38152 | The predicate "is a totall... |
| istotbnd3 38153 | A metric space is totally ... |
| totbndmet 38154 | The predicate "totally bou... |
| 0totbnd 38155 | The metric (there is only ... |
| sstotbnd2 38156 | Condition for a subset of ... |
| sstotbnd 38157 | Condition for a subset of ... |
| sstotbnd3 38158 | Use a net that is not nece... |
| totbndss 38159 | A subset of a totally boun... |
| equivtotbnd 38160 | If the metric ` M ` is "st... |
| isbnd 38162 | The predicate "is a bounde... |
| bndmet 38163 | A bounded metric space is ... |
| isbndx 38164 | A "bounded extended metric... |
| isbnd2 38165 | The predicate "is a bounde... |
| isbnd3 38166 | A metric space is bounded ... |
| isbnd3b 38167 | A metric space is bounded ... |
| bndss 38168 | A subset of a bounded metr... |
| blbnd 38169 | A ball is bounded. (Contr... |
| ssbnd 38170 | A subset of a metric space... |
| totbndbnd 38171 | A totally bounded metric s... |
| equivbnd 38172 | If the metric ` M ` is "st... |
| bnd2lem 38173 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 38174 | If balls are totally bound... |
| prdsbnd 38175 | The product metric over fi... |
| prdstotbnd 38176 | The product metric over fi... |
| prdsbnd2 38177 | If balls are totally bound... |
| cntotbnd 38178 | A subset of the complex nu... |
| cnpwstotbnd 38179 | A subset of ` A ^ I ` , wh... |
| ismtyval 38182 | The set of isometries betw... |
| isismty 38183 | The condition "is an isome... |
| ismtycnv 38184 | The inverse of an isometry... |
| ismtyima 38185 | The image of a ball under ... |
| ismtyhmeolem 38186 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 38187 | An isometry is a homeomorp... |
| ismtybndlem 38188 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 38189 | Isometries preserve bounde... |
| ismtyres 38190 | A restriction of an isomet... |
| heibor1lem 38191 | Lemma for ~ heibor1 . A c... |
| heibor1 38192 | One half of ~ heibor , tha... |
| heiborlem1 38193 | Lemma for ~ heibor . We w... |
| heiborlem2 38194 | Lemma for ~ heibor . Subs... |
| heiborlem3 38195 | Lemma for ~ heibor . Usin... |
| heiborlem4 38196 | Lemma for ~ heibor . Usin... |
| heiborlem5 38197 | Lemma for ~ heibor . The ... |
| heiborlem6 38198 | Lemma for ~ heibor . Sinc... |
| heiborlem7 38199 | Lemma for ~ heibor . Sinc... |
| heiborlem8 38200 | Lemma for ~ heibor . The ... |
| heiborlem9 38201 | Lemma for ~ heibor . Disc... |
| heiborlem10 38202 | Lemma for ~ heibor . The ... |
| heibor 38203 | Generalized Heine-Borel Th... |
| bfplem1 38204 | Lemma for ~ bfp . The seq... |
| bfplem2 38205 | Lemma for ~ bfp . Using t... |
| bfp 38206 | Banach fixed point theorem... |
| rrnval 38209 | The n-dimensional Euclidea... |
| rrnmval 38210 | The value of the Euclidean... |
| rrnmet 38211 | Euclidean space is a metri... |
| rrndstprj1 38212 | The distance between two p... |
| rrndstprj2 38213 | Bound on the distance betw... |
| rrncmslem 38214 | Lemma for ~ rrncms . (Con... |
| rrncms 38215 | Euclidean space is complet... |
| repwsmet 38216 | The supremum metric on ` R... |
| rrnequiv 38217 | The supremum metric on ` R... |
| rrntotbnd 38218 | A set in Euclidean space i... |
| rrnheibor 38219 | Heine-Borel theorem for Eu... |
| ismrer1 38220 | An isometry between ` RR `... |
| reheibor 38221 | Heine-Borel theorem for re... |
| iccbnd 38222 | A closed interval in ` RR ... |
| icccmpALT 38223 | A closed interval in ` RR ... |
| isass 38228 | The predicate "is an assoc... |
| isexid 38229 | The predicate ` G ` has a ... |
| ismgmOLD 38232 | Obsolete version of ~ ismg... |
| clmgmOLD 38233 | Obsolete version of ~ mgmc... |
| opidonOLD 38234 | Obsolete version of ~ mndp... |
| rngopidOLD 38235 | Obsolete version of ~ mndp... |
| opidon2OLD 38236 | Obsolete version of ~ mndp... |
| isexid2 38237 | If ` G e. ( Magma i^i ExId... |
| exidu1 38238 | Uniqueness of the left and... |
| idrval 38239 | The value of the identity ... |
| iorlid 38240 | A magma right and left ide... |
| cmpidelt 38241 | A magma right and left ide... |
| smgrpismgmOLD 38244 | Obsolete version of ~ sgrp... |
| issmgrpOLD 38245 | Obsolete version of ~ issg... |
| smgrpmgm 38246 | A semigroup is a magma. (... |
| smgrpassOLD 38247 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 38250 | Obsolete version of ~ mnds... |
| mndoisexid 38251 | A monoid has an identity e... |
| mndoismgmOLD 38252 | Obsolete version of ~ mndm... |
| mndomgmid 38253 | A monoid is a magma with a... |
| ismndo 38254 | The predicate "is a monoid... |
| ismndo1 38255 | The predicate "is a monoid... |
| ismndo2 38256 | The predicate "is a monoid... |
| grpomndo 38257 | A group is a monoid. (Con... |
| exidcl 38258 | Closure of the binary oper... |
| exidreslem 38259 | Lemma for ~ exidres and ~ ... |
| exidres 38260 | The restriction of a binar... |
| exidresid 38261 | The restriction of a binar... |
| ablo4pnp 38262 | A commutative/associative ... |
| grpoeqdivid 38263 | Two group elements are equ... |
| grposnOLD 38264 | The group operation for th... |
| elghomlem1OLD 38267 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 38268 | Obsolete as of 15-Mar-2020... |
| elghomOLD 38269 | Obsolete version of ~ isgh... |
| ghomlinOLD 38270 | Obsolete version of ~ ghml... |
| ghomidOLD 38271 | Obsolete version of ~ ghmi... |
| ghomf 38272 | Mapping property of a grou... |
| ghomco 38273 | The composition of two gro... |
| ghomdiv 38274 | Group homomorphisms preser... |
| grpokerinj 38275 | A group homomorphism is in... |
| relrngo 38278 | The class of all unital ri... |
| isrngo 38279 | The predicate "is a (unita... |
| isrngod 38280 | Conditions that determine ... |
| rngoi 38281 | The properties of a unital... |
| rngosm 38282 | Functionality of the multi... |
| rngocl 38283 | Closure of the multiplicat... |
| rngoid 38284 | The multiplication operati... |
| rngoideu 38285 | The unity element of a rin... |
| rngodi 38286 | Distributive law for the m... |
| rngodir 38287 | Distributive law for the m... |
| rngoass 38288 | Associative law for the mu... |
| rngo2 38289 | A ring element plus itself... |
| rngoablo 38290 | A ring's addition operatio... |
| rngoablo2 38291 | In a unital ring the addit... |
| rngogrpo 38292 | A ring's addition operatio... |
| rngone0 38293 | The base set of a ring is ... |
| rngogcl 38294 | Closure law for the additi... |
| rngocom 38295 | The addition operation of ... |
| rngoaass 38296 | The addition operation of ... |
| rngoa32 38297 | The addition operation of ... |
| rngoa4 38298 | Rearrangement of 4 terms i... |
| rngorcan 38299 | Right cancellation law for... |
| rngolcan 38300 | Left cancellation law for ... |
| rngo0cl 38301 | A ring has an additive ide... |
| rngo0rid 38302 | The additive identity of a... |
| rngo0lid 38303 | The additive identity of a... |
| rngolz 38304 | The zero of a unital ring ... |
| rngorz 38305 | The zero of a unital ring ... |
| rngosn3 38306 | Obsolete as of 25-Jan-2020... |
| rngosn4 38307 | Obsolete as of 25-Jan-2020... |
| rngosn6 38308 | Obsolete as of 25-Jan-2020... |
| rngonegcl 38309 | A ring is closed under neg... |
| rngoaddneg1 38310 | Adding the negative in a r... |
| rngoaddneg2 38311 | Adding the negative in a r... |
| rngosub 38312 | Subtraction in a ring, in ... |
| rngmgmbs4 38313 | The range of an internal o... |
| rngodm1dm2 38314 | In a unital ring the domai... |
| rngorn1 38315 | In a unital ring the range... |
| rngorn1eq 38316 | In a unital ring the range... |
| rngomndo 38317 | In a unital ring the multi... |
| rngoidmlem 38318 | The unity element of a rin... |
| rngolidm 38319 | The unity element of a rin... |
| rngoridm 38320 | The unity element of a rin... |
| rngo1cl 38321 | The unity element of a rin... |
| rngoueqz 38322 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 38323 | Negation in a ring is the ... |
| rngonegmn1r 38324 | Negation in a ring is the ... |
| rngoneglmul 38325 | Negation of a product in a... |
| rngonegrmul 38326 | Negation of a product in a... |
| rngosubdi 38327 | Ring multiplication distri... |
| rngosubdir 38328 | Ring multiplication distri... |
| zerdivemp1x 38329 | In a unital ring a left in... |
| isdivrngo 38332 | The predicate "is a divisi... |
| drngoi 38333 | The properties of a divisi... |
| gidsn 38334 | Obsolete as of 23-Jan-2020... |
| zrdivrng 38335 | The zero ring is not a div... |
| dvrunz 38336 | In a division ring the rin... |
| isgrpda 38337 | Properties that determine ... |
| isdrngo1 38338 | The predicate "is a divisi... |
| divrngcl 38339 | The product of two nonzero... |
| isdrngo2 38340 | A division ring is a ring ... |
| isdrngo3 38341 | A division ring is a ring ... |
| rngohomval 38346 | The set of ring homomorphi... |
| isrngohom 38347 | The predicate "is a ring h... |
| rngohomf 38348 | A ring homomorphism is a f... |
| rngohomcl 38349 | Closure law for a ring hom... |
| rngohom1 38350 | A ring homomorphism preser... |
| rngohomadd 38351 | Ring homomorphisms preserv... |
| rngohommul 38352 | Ring homomorphisms preserv... |
| rngogrphom 38353 | A ring homomorphism is a g... |
| rngohom0 38354 | A ring homomorphism preser... |
| rngohomsub 38355 | Ring homomorphisms preserv... |
| rngohomco 38356 | The composition of two rin... |
| rngokerinj 38357 | A ring homomorphism is inj... |
| rngoisoval 38359 | The set of ring isomorphis... |
| isrngoiso 38360 | The predicate "is a ring i... |
| rngoiso1o 38361 | A ring isomorphism is a bi... |
| rngoisohom 38362 | A ring isomorphism is a ri... |
| rngoisocnv 38363 | The inverse of a ring isom... |
| rngoisoco 38364 | The composition of two rin... |
| isriscg 38366 | The ring isomorphism relat... |
| isrisc 38367 | The ring isomorphism relat... |
| risc 38368 | The ring isomorphism relat... |
| risci 38369 | Determine that two rings a... |
| riscer 38370 | Ring isomorphism is an equ... |
| iscom2 38377 | A device to add commutativ... |
| iscrngo 38378 | The predicate "is a commut... |
| iscrngo2 38379 | The predicate "is a commut... |
| iscringd 38380 | Conditions that determine ... |
| flddivrng 38381 | A field is a division ring... |
| crngorngo 38382 | A commutative ring is a ri... |
| crngocom 38383 | The multiplication operati... |
| crngm23 38384 | Commutative/associative la... |
| crngm4 38385 | Commutative/associative la... |
| fldcrngo 38386 | A field is a commutative r... |
| isfld2 38387 | The predicate "is a field"... |
| crngohomfo 38388 | The image of a homomorphis... |
| idlval 38395 | The class of ideals of a r... |
| isidl 38396 | The predicate "is an ideal... |
| isidlc 38397 | The predicate "is an ideal... |
| idlss 38398 | An ideal of ` R ` is a sub... |
| idlcl 38399 | An element of an ideal is ... |
| idl0cl 38400 | An ideal contains ` 0 ` . ... |
| idladdcl 38401 | An ideal is closed under a... |
| idllmulcl 38402 | An ideal is closed under m... |
| idlrmulcl 38403 | An ideal is closed under m... |
| idlnegcl 38404 | An ideal is closed under n... |
| idlsubcl 38405 | An ideal is closed under s... |
| rngoidl 38406 | A ring ` R ` is an ` R ` i... |
| 0idl 38407 | The set containing only ` ... |
| 1idl 38408 | Two ways of expressing the... |
| 0rngo 38409 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 38410 | The only ideals in a divis... |
| intidl 38411 | The intersection of a none... |
| inidl 38412 | The intersection of two id... |
| unichnidl 38413 | The union of a nonempty ch... |
| keridl 38414 | The kernel of a ring homom... |
| pridlval 38415 | The class of prime ideals ... |
| ispridl 38416 | The predicate "is a prime ... |
| pridlidl 38417 | A prime ideal is an ideal.... |
| pridlnr 38418 | A prime ideal is a proper ... |
| pridl 38419 | The main property of a pri... |
| ispridl2 38420 | A condition that shows an ... |
| maxidlval 38421 | The set of maximal ideals ... |
| ismaxidl 38422 | The predicate "is a maxima... |
| maxidlidl 38423 | A maximal ideal is an idea... |
| maxidlnr 38424 | A maximal ideal is proper.... |
| maxidlmax 38425 | A maximal ideal is a maxim... |
| maxidln1 38426 | One is not contained in an... |
| maxidln0 38427 | A ring with a maximal idea... |
| isprrngo 38432 | The predicate "is a prime ... |
| prrngorngo 38433 | A prime ring is a ring. (... |
| smprngopr 38434 | A simple ring (one whose o... |
| divrngpr 38435 | A division ring is a prime... |
| isdmn 38436 | The predicate "is a domain... |
| isdmn2 38437 | The predicate "is a domain... |
| dmncrng 38438 | A domain is a commutative ... |
| dmnrngo 38439 | A domain is a ring. (Cont... |
| flddmn 38440 | A field is a domain. (Con... |
| igenval 38443 | The ideal generated by a s... |
| igenss 38444 | A set is a subset of the i... |
| igenidl 38445 | The ideal generated by a s... |
| igenmin 38446 | The ideal generated by a s... |
| igenidl2 38447 | The ideal generated by an ... |
| igenval2 38448 | The ideal generated by a s... |
| prnc 38449 | A principal ideal (an idea... |
| isfldidl 38450 | Determine if a ring is a f... |
| isfldidl2 38451 | Determine if a ring is a f... |
| ispridlc 38452 | The predicate "is a prime ... |
| pridlc 38453 | Property of a prime ideal ... |
| pridlc2 38454 | Property of a prime ideal ... |
| pridlc3 38455 | Property of a prime ideal ... |
| isdmn3 38456 | The predicate "is a domain... |
| dmnnzd 38457 | A domain has no zero-divis... |
| dmncan1 38458 | Cancellation law for domai... |
| dmncan2 38459 | Cancellation law for domai... |
| efald2 38460 | A proof by contradiction. ... |
| notbinot1 38461 | Simplification rule of neg... |
| bicontr 38462 | Biconditional of its own n... |
| impor 38463 | An equivalent formula for ... |
| orfa 38464 | The falsum ` F. ` can be r... |
| notbinot2 38465 | Commutation rule between n... |
| biimpor 38466 | A rewriting rule for bicon... |
| orfa1 38467 | Add a contradicting disjun... |
| orfa2 38468 | Remove a contradicting dis... |
| bifald 38469 | Infer the equivalence to a... |
| orsild 38470 | A lemma for not-or-not eli... |
| orsird 38471 | A lemma for not-or-not eli... |
| cnf1dd 38472 | A lemma for Conjunctive No... |
| cnf2dd 38473 | A lemma for Conjunctive No... |
| cnfn1dd 38474 | A lemma for Conjunctive No... |
| cnfn2dd 38475 | A lemma for Conjunctive No... |
| or32dd 38476 | A rearrangement of disjunc... |
| notornotel1 38477 | A lemma for not-or-not eli... |
| notornotel2 38478 | A lemma for not-or-not eli... |
| contrd 38479 | A proof by contradiction, ... |
| an12i 38480 | An inference from commutin... |
| exmid2 38481 | An excluded middle law. (... |
| selconj 38482 | An inference for selecting... |
| truconj 38483 | Add true as a conjunct. (... |
| orel 38484 | An inference for disjuncti... |
| negel 38485 | An inference for negation ... |
| botel 38486 | An inference for bottom el... |
| tradd 38487 | Add top ad a conjunct. (C... |
| gm-sbtru 38488 | Substitution does not chan... |
| sbfal 38489 | Substitution does not chan... |
| sbcani 38490 | Distribution of class subs... |
| sbcori 38491 | Distribution of class subs... |
| sbcimi 38492 | Distribution of class subs... |
| sbcni 38493 | Move class substitution in... |
| sbali 38494 | Discard class substitution... |
| sbexi 38495 | Discard class substitution... |
| sbcalf 38496 | Move universal quantifier ... |
| sbcexf 38497 | Move existential quantifie... |
| sbcalfi 38498 | Move universal quantifier ... |
| sbcexfi 38499 | Move existential quantifie... |
| spsbcdi 38500 | A lemma for eliminating a ... |
| alrimii 38501 | A lemma for introducing a ... |
| spesbcdi 38502 | A lemma for introducing an... |
| exlimddvf 38503 | A lemma for eliminating an... |
| exlimddvfi 38504 | A lemma for eliminating an... |
| sbceq1ddi 38505 | A lemma for eliminating in... |
| sbccom2lem 38506 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 38507 | Commutative law for double... |
| sbccom2f 38508 | Commutative law for double... |
| sbccom2fi 38509 | Commutative law for double... |
| csbcom2fi 38510 | Commutative law for double... |
| fald 38511 | Refutation of falsity, in ... |
| tsim1 38512 | A Tseitin axiom for logica... |
| tsim2 38513 | A Tseitin axiom for logica... |
| tsim3 38514 | A Tseitin axiom for logica... |
| tsbi1 38515 | A Tseitin axiom for logica... |
| tsbi2 38516 | A Tseitin axiom for logica... |
| tsbi3 38517 | A Tseitin axiom for logica... |
| tsbi4 38518 | A Tseitin axiom for logica... |
| tsxo1 38519 | A Tseitin axiom for logica... |
| tsxo2 38520 | A Tseitin axiom for logica... |
| tsxo3 38521 | A Tseitin axiom for logica... |
| tsxo4 38522 | A Tseitin axiom for logica... |
| tsan1 38523 | A Tseitin axiom for logica... |
| tsan2 38524 | A Tseitin axiom for logica... |
| tsan3 38525 | A Tseitin axiom for logica... |
| tsna1 38526 | A Tseitin axiom for logica... |
| tsna2 38527 | A Tseitin axiom for logica... |
| tsna3 38528 | A Tseitin axiom for logica... |
| tsor1 38529 | A Tseitin axiom for logica... |
| tsor2 38530 | A Tseitin axiom for logica... |
| tsor3 38531 | A Tseitin axiom for logica... |
| ts3an1 38532 | A Tseitin axiom for triple... |
| ts3an2 38533 | A Tseitin axiom for triple... |
| ts3an3 38534 | A Tseitin axiom for triple... |
| ts3or1 38535 | A Tseitin axiom for triple... |
| ts3or2 38536 | A Tseitin axiom for triple... |
| ts3or3 38537 | A Tseitin axiom for triple... |
| iuneq2f 38538 | Equality deduction for ind... |
| rabeq12f 38539 | Equality deduction for res... |
| csbeq12 38540 | Equality deduction for sub... |
| sbeqi 38541 | Equality deduction for sub... |
| ralbi12f 38542 | Equality deduction for res... |
| oprabbi 38543 | Equality deduction for cla... |
| mpobi123f 38544 | Equality deduction for map... |
| iuneq12f 38545 | Equality deduction for ind... |
| iineq12f 38546 | Equality deduction for ind... |
| opabbi 38547 | Equality deduction for cla... |
| mptbi12f 38548 | Equality deduction for map... |
| orcomdd 38549 | Commutativity of logic dis... |
| scottexf 38550 | A version of ~ scottex wit... |
| scott0f 38551 | A version of ~ scott0 with... |
| scottn0f 38552 | A version of ~ scott0f wit... |
| ac6s3f 38553 | Generalization of the Axio... |
| ac6s6 38554 | Generalization of the Axio... |
| ac6s6f 38555 | Generalization of the Axio... |
| el2v1 38611 | New way ( ~ elv , and the ... |
| el3v1 38612 | New way ( ~ elv , and the ... |
| el3v2 38613 | New way ( ~ elv , and the ... |
| el3v12 38614 | New way ( ~ elv , and the ... |
| el3v13 38615 | New way ( ~ elv , and the ... |
| el3v23 38616 | New way ( ~ elv , and the ... |
| anan 38617 | Multiple commutations in c... |
| triantru3 38618 | A wff is equivalent to its... |
| biorfd 38619 | A wff is equivalent to its... |
| eqbrtr 38620 | Substitution of equal clas... |
| eqbrb 38621 | Substitution of equal clas... |
| eqeltr 38622 | Substitution of equal clas... |
| eqelb 38623 | Substitution of equal clas... |
| eqeqan2d 38624 | Implication of introducing... |
| disjresin 38625 | The restriction to a disjo... |
| disjresdisj 38626 | The intersection of restri... |
| disjresdif 38627 | The difference between res... |
| disjresundif 38628 | Lemma for ~ ressucdifsn2 .... |
| inres2 38629 | Two ways of expressing the... |
| coideq 38630 | Equality theorem for compo... |
| nexmo1 38631 | If there is no case where ... |
| eqab2 38632 | Implication of a class abs... |
| r2alan 38633 | Double restricted universa... |
| ssrabi 38634 | Inference of restricted ab... |
| rabimbieq 38635 | Restricted equivalent wff'... |
| abeqin 38636 | Intersection with class ab... |
| abeqinbi 38637 | Intersection with class ab... |
| eqrabi 38638 | Class element of a restric... |
| rabeqel 38639 | Class element of a restric... |
| eqrelf 38640 | The equality connective be... |
| br1cnvinxp 38641 | Binary relation on the con... |
| releleccnv 38642 | Elementhood in a converse ... |
| releccnveq 38643 | Equality of converse ` R `... |
| xpv 38644 | Cartesian product of a cla... |
| vxp 38645 | Cartesian product of the u... |
| opelvvdif 38646 | Negated elementhood of ord... |
| vvdifopab 38647 | Ordered-pair class abstrac... |
| brvdif 38648 | Binary relation with unive... |
| brvdif2 38649 | Binary relation with unive... |
| brvvdif 38650 | Binary relation with the c... |
| brvbrvvdif 38651 | Binary relation with the c... |
| brcnvep 38652 | The converse of the binary... |
| elecALTV 38653 | Elementhood in the ` R ` -... |
| brcnvepres 38654 | Restricted converse epsilo... |
| brres2 38655 | Binary relation on a restr... |
| br1cnvres 38656 | Binary relation on the con... |
| elec1cnvres 38657 | Elementhood in the convers... |
| ec1cnvres 38658 | Converse restricted coset ... |
| eldmres 38659 | Elementhood in the domain ... |
| elrnres 38660 | Element of the range of a ... |
| eldmressnALTV 38661 | Element of the domain of a... |
| elrnressn 38662 | Element of the range of a ... |
| eldm4 38663 | Elementhood in a domain. ... |
| eldmres2 38664 | Elementhood in the domain ... |
| eldmres3 38665 | Elementhood in the domain ... |
| eceq1i 38666 | Equality theorem for ` C `... |
| ecres 38667 | Restricted coset of ` B ` ... |
| eccnvepres 38668 | Restricted converse epsilo... |
| eleccnvep 38669 | Elementhood in the convers... |
| eccnvep 38670 | The converse epsilon coset... |
| extep 38671 | Property of epsilon relati... |
| disjeccnvep 38672 | Property of the epsilon re... |
| eccnvepres2 38673 | The restricted converse ep... |
| eccnvepres3 38674 | Condition for a restricted... |
| eldmqsres 38675 | Elementhood in a restricte... |
| eldmqsres2 38676 | Elementhood in a restricte... |
| qsss1 38677 | Subclass theorem for quoti... |
| qseq1i 38678 | Equality theorem for quoti... |
| brinxprnres 38679 | Binary relation on a restr... |
| inxprnres 38680 | Restriction of a class as ... |
| dfres4 38681 | Alternate definition of th... |
| exan3 38682 | Equivalent expressions wit... |
| exanres 38683 | Equivalent expressions wit... |
| exanres3 38684 | Equivalent expressions wit... |
| exanres2 38685 | Equivalent expressions wit... |
| cnvepres 38686 | Restricted converse epsilo... |
| eqrel2 38687 | Equality of relations. (C... |
| rncnv 38688 | Range of converse is the d... |
| dfdm6 38689 | Alternate definition of do... |
| dfrn6 38690 | Alternate definition of ra... |
| rncnvepres 38691 | The range of the restricte... |
| dmecd 38692 | Equality of the coset of `... |
| dmec2d 38693 | Equality of the coset of `... |
| brid 38694 | Property of the identity b... |
| ideq2 38695 | For sets, the identity bin... |
| idresssidinxp 38696 | Condition for the identity... |
| idreseqidinxp 38697 | Condition for the identity... |
| extid 38698 | Property of identity relat... |
| inxpss 38699 | Two ways to say that an in... |
| idinxpss 38700 | Two ways to say that an in... |
| ref5 38701 | Two ways to say that an in... |
| inxpss3 38702 | Two ways to say that an in... |
| inxpss2 38703 | Two ways to say that inter... |
| inxpssidinxp 38704 | Two ways to say that inter... |
| idinxpssinxp 38705 | Two ways to say that inter... |
| idinxpssinxp2 38706 | Identity intersection with... |
| idinxpssinxp3 38707 | Identity intersection with... |
| idinxpssinxp4 38708 | Identity intersection with... |
| relcnveq3 38709 | Two ways of saying a relat... |
| relcnveq 38710 | Two ways of saying a relat... |
| relcnveq2 38711 | Two ways of saying a relat... |
| relcnveq4 38712 | Two ways of saying a relat... |
| qsresid 38713 | Simplification of a specia... |
| n0elqs 38714 | Two ways of expressing tha... |
| n0elqs2 38715 | Two ways of expressing tha... |
| rnresequniqs 38716 | The range of a restriction... |
| n0el2 38717 | Two ways of expressing tha... |
| cnvepresex 38718 | Sethood condition for the ... |
| cnvepima 38719 | The image of converse epsi... |
| inex3 38720 | Sufficient condition for t... |
| inxpex 38721 | Sufficient condition for a... |
| eqres 38722 | Converting a class constan... |
| brrabga 38723 | The law of concretion for ... |
| brcnvrabga 38724 | The law of concretion for ... |
| opideq 38725 | Equality conditions for or... |
| iss2 38726 | A subclass of the identity... |
| eldmcnv 38727 | Elementhood in a domain of... |
| dfrel5 38728 | Alternate definition of th... |
| dfrel6 38729 | Alternate definition of th... |
| cnvresrn 38730 | Converse restricted to ran... |
| relssinxpdmrn 38731 | Subset of restriction, spe... |
| cnvref4 38732 | Two ways to say that a rel... |
| cnvref5 38733 | Two ways to say that a rel... |
| ecin0 38734 | Two ways of saying that th... |
| ecinn0 38735 | Two ways of saying that th... |
| ineleq 38736 | Equivalence of restricted ... |
| inecmo 38737 | Equivalence of a double re... |
| inecmo2 38738 | Equivalence of a double re... |
| ineccnvmo 38739 | Equivalence of a double re... |
| alrmomorn 38740 | Equivalence of an "at most... |
| alrmomodm 38741 | Equivalence of an "at most... |
| ralmo 38742 | "At most one" can be restr... |
| ralrnmo 38743 | On the range, "at most one... |
| dmqsex 38744 | Sethood of the domain quot... |
| raldmqsmo 38745 | On the quotient carrier, "... |
| ralrmo3 38746 | Pull a restricted universa... |
| raldmqseu 38747 | Equivalence between "exact... |
| rsp3 38748 | From a restricted universa... |
| rsp3eq 38749 | From a restricted universa... |
| ineccnvmo2 38750 | Equivalence of a double un... |
| inecmo3 38751 | Equivalence of a double un... |
| moeu2 38752 | Uniqueness is equivalent t... |
| mopickr 38753 | "At most one" picks a vari... |
| moantr 38754 | Sufficient condition for t... |
| brabidgaw 38755 | The law of concretion for ... |
| brabidga 38756 | The law of concretion for ... |
| inxp2 38757 | Intersection with a Cartes... |
| opabf 38758 | A class abstraction of a c... |
| ec0 38759 | The empty-coset of a class... |
| brcnvin 38760 | Intersection with a conver... |
| ssdmral 38761 | Subclass of a domain. (Co... |
| xrnss3v 38763 | A range Cartesian product ... |
| xrnrel 38764 | A range Cartesian product ... |
| brxrn 38765 | Characterize a ternary rel... |
| brxrn2 38766 | A characterization of the ... |
| dfxrn2 38767 | Alternate definition of th... |
| brxrncnvep 38768 | The range product with con... |
| dmxrn 38769 | Domain of the range produc... |
| dmcnvep 38770 | Domain of converse epsilon... |
| dmxrncnvep 38771 | Domain of the range produc... |
| dmcnvepres 38772 | Domain of the restricted c... |
| dmuncnvepres 38773 | Domain of the union with t... |
| dmxrnuncnvepres 38774 | Domain of the combined rel... |
| ecun 38775 | The union coset of ` A ` .... |
| ecunres 38776 | The restricted union coset... |
| ecuncnvepres 38777 | The restricted union with ... |
| xrneq1 38778 | Equality theorem for the r... |
| xrneq1i 38779 | Equality theorem for the r... |
| xrneq1d 38780 | Equality theorem for the r... |
| xrneq2 38781 | Equality theorem for the r... |
| xrneq2i 38782 | Equality theorem for the r... |
| xrneq2d 38783 | Equality theorem for the r... |
| xrneq12 38784 | Equality theorem for the r... |
| xrneq12i 38785 | Equality theorem for the r... |
| xrneq12d 38786 | Equality theorem for the r... |
| elecxrn 38787 | Elementhood in the ` ( R |... |
| ecxrn 38788 | The ` ( R |X. S ) ` -coset... |
| relecxrn 38789 | The ` ( R |X. S ) ` -coset... |
| ecxrn2 38790 | The ` ( R |X. S ) ` -coset... |
| ecxrncnvep 38791 | The ` ( R |X. ``' _E ) ` -... |
| ecxrncnvep2 38792 | The ` ( R |X. ``' _E ) ` -... |
| disjressuc2 38793 | Double restricted quantifi... |
| disjecxrn 38794 | Two ways of saying that ` ... |
| disjecxrncnvep 38795 | Two ways of saying that co... |
| disjsuc2 38796 | Double restricted quantifi... |
| xrninxp 38797 | Intersection of a range Ca... |
| xrninxp2 38798 | Intersection of a range Ca... |
| xrninxpex 38799 | Sufficient condition for t... |
| inxpxrn 38800 | Two ways to express the in... |
| br1cnvxrn2 38801 | The converse of a binary r... |
| elec1cnvxrn2 38802 | Elementhood in the convers... |
| rnxrn 38803 | Range of the range Cartesi... |
| rnxrnres 38804 | Range of a range Cartesian... |
| rnxrncnvepres 38805 | Range of a range Cartesian... |
| rnxrnidres 38806 | Range of a range Cartesian... |
| xrnres 38807 | Two ways to express restri... |
| xrnres2 38808 | Two ways to express restri... |
| xrnres3 38809 | Two ways to express restri... |
| xrnres4 38810 | Two ways to express restri... |
| xrnresex 38811 | Sufficient condition for a... |
| xrnidresex 38812 | Sufficient condition for a... |
| xrncnvepresex 38813 | Sufficient condition for a... |
| dmxrncnvepres 38814 | Domain of the range produc... |
| dmxrncnvepres2 38815 | Domain of the range produc... |
| eldmxrncnvepres 38816 | Element of the domain of t... |
| eldmxrncnvepres2 38817 | Element of the domain of t... |
| eceldmqsxrncnvepres 38818 | An ` ( R |X. ( ``' _E |`` ... |
| eceldmqsxrncnvepres2 38819 | An ` ( R |X. ( ``' _E |`` ... |
| brin2 38820 | Binary relation on an inte... |
| brin3 38821 | Binary relation on an inte... |
| elrels2 38823 | The element of the relatio... |
| elrelsrel 38824 | The element of the relatio... |
| elrelsrelim 38825 | The element of the relatio... |
| elrels5 38826 | Equivalent expressions for... |
| elrels6 38827 | Equivalent expressions for... |
| dfqmap2 38829 | Alternate definition of th... |
| dfqmap3 38830 | Alternate definition of th... |
| ecqmap 38831 | ` QMap ` fibers are single... |
| ecqmap2 38832 | Fiber of ` QMap ` equals s... |
| qmapex 38833 | Quotient map exists if ` R... |
| relqmap 38834 | Quotient map is a relation... |
| dmqmap 38835 | ` QMap ` preserves the dom... |
| rnqmap 38836 | The range of the quotient ... |
| dfadjliftmap 38838 | Alternate (expanded) defin... |
| dfadjliftmap2 38839 | Alternate definition of th... |
| blockadjliftmap 38840 | A "two-stage" construction... |
| dfblockliftmap 38842 | Alternate definition of th... |
| dfblockliftmap2 38843 | Alternate definition of th... |
| dfsucmap3 38845 | Alternate definition of th... |
| dfsucmap2 38846 | Alternate definition of th... |
| dfsucmap4 38847 | Alternate definition of th... |
| brsucmap 38848 | Binary relation form of th... |
| relsucmap 38849 | The successor map is a rel... |
| dmsucmap 38850 | The domain of the successo... |
| dfsuccl2 38852 | Alternate definition of th... |
| mopre 38853 | There is at most one prede... |
| exeupre2 38854 | Whenever a predecessor exi... |
| dfsuccl3 38855 | Alternate definition of th... |
| dfsuccl4 38856 | Alternate definition that ... |
| dfpre 38858 | Alternate definition of th... |
| dfpre2 38859 | Alternate definition of th... |
| dfpre3 38860 | Alternate definition of th... |
| dfpred4 38861 | Alternate definition of th... |
| dfpre4 38862 | Alternate definition of th... |
| shiftstableeq2 38865 | Equality theorem for shift... |
| suceqsneq 38866 | One-to-one relationship be... |
| sucdifsn2 38867 | Absorption of union with a... |
| sucdifsn 38868 | The difference between the... |
| ressucdifsn2 38869 | The difference between res... |
| ressucdifsn 38870 | The difference between res... |
| sucmapsuc 38871 | A set is succeeded by its ... |
| sucmapleftuniq 38872 | Left uniqueness of the suc... |
| exeupre 38873 | Whenever a predecessor exi... |
| preex 38874 | The successor-predecessor ... |
| eupre2 38875 | Unique predecessor exists ... |
| eupre 38876 | Unique predecessor exists ... |
| presucmap 38877 | ` pre ` is really a predec... |
| preuniqval 38878 | Uniqueness/canonicity of `... |
| sucpre 38879 | ` suc ` is a right-inverse... |
| presuc 38880 | ` pre ` is a left-inverse ... |
| press 38881 | Predecessor is a subset of... |
| preel 38882 | Predecessor is a subset of... |
| dfcoss2 38885 | Alternate definition of th... |
| dfcoss3 38886 | Alternate definition of th... |
| dfcoss4 38887 | Alternate definition of th... |
| cosscnv 38888 | Class of cosets by the con... |
| coss1cnvres 38889 | Class of cosets by the con... |
| coss2cnvepres 38890 | Special case of ~ coss1cnv... |
| cossex 38891 | If ` A ` is a set then the... |
| cosscnvex 38892 | If ` A ` is a set then the... |
| 1cosscnvepresex 38893 | Sufficient condition for a... |
| 1cossxrncnvepresex 38894 | Sufficient condition for a... |
| relcoss 38895 | Cosets by ` R ` is a relat... |
| relcoels 38896 | Coelements on ` A ` is a r... |
| cossss 38897 | Subclass theorem for the c... |
| cosseq 38898 | Equality theorem for the c... |
| cosseqi 38899 | Equality theorem for the c... |
| cosseqd 38900 | Equality theorem for the c... |
| 1cossres 38901 | The class of cosets by a r... |
| dfcoels 38902 | Alternate definition of th... |
| brcoss 38903 | ` A ` and ` B ` are cosets... |
| brcoss2 38904 | Alternate form of the ` A ... |
| brcoss3 38905 | Alternate form of the ` A ... |
| brcosscnvcoss 38906 | For sets, the ` A ` and ` ... |
| brcoels 38907 | ` B ` and ` C ` are coelem... |
| cocossss 38908 | Two ways of saying that co... |
| cnvcosseq 38909 | The converse of cosets by ... |
| br2coss 38910 | Cosets by ` ,~ R ` binary ... |
| br1cossres 38911 | ` B ` and ` C ` are cosets... |
| br1cossres2 38912 | ` B ` and ` C ` are cosets... |
| brressn 38913 | Binary relation on a restr... |
| ressn2 38914 | A class ' R ' restricted t... |
| refressn 38915 | Any class ' R ' restricted... |
| antisymressn 38916 | Every class ' R ' restrict... |
| trressn 38917 | Any class ' R ' restricted... |
| relbrcoss 38918 | ` A ` and ` B ` are cosets... |
| br1cossinres 38919 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 38920 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 38921 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 38922 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 38923 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 38924 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 38925 | The domain of cosets is th... |
| dmcoss2 38926 | The domain of cosets is th... |
| rncossdmcoss 38927 | The range of cosets is the... |
| dm1cosscnvepres 38928 | The domain of cosets of th... |
| dmcoels 38929 | The domain of coelements i... |
| eldmcoss 38930 | Elementhood in the domain ... |
| eldmcoss2 38931 | Elementhood in the domain ... |
| eldm1cossres 38932 | Elementhood in the domain ... |
| eldm1cossres2 38933 | Elementhood in the domain ... |
| refrelcosslem 38934 | Lemma for the left side of... |
| refrelcoss3 38935 | The class of cosets by ` R... |
| refrelcoss2 38936 | The class of cosets by ` R... |
| symrelcoss3 38937 | The class of cosets by ` R... |
| symrelcoss2 38938 | The class of cosets by ` R... |
| cossssid 38939 | Equivalent expressions for... |
| cossssid2 38940 | Equivalent expressions for... |
| cossssid3 38941 | Equivalent expressions for... |
| cossssid4 38942 | Equivalent expressions for... |
| cossssid5 38943 | Equivalent expressions for... |
| brcosscnv 38944 | ` A ` and ` B ` are cosets... |
| brcosscnv2 38945 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 38946 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 38947 | Cosets by the converse ran... |
| cosscnvssid3 38948 | Equivalent expressions for... |
| cosscnvssid4 38949 | Equivalent expressions for... |
| cosscnvssid5 38950 | Equivalent expressions for... |
| coss0 38951 | Cosets by the empty set ar... |
| cossid 38952 | Cosets by the identity rel... |
| cosscnvid 38953 | Cosets by the converse ide... |
| trcoss 38954 | Sufficient condition for t... |
| eleccossin 38955 | Two ways of saying that th... |
| trcoss2 38956 | Equivalent expressions for... |
| cosselrels 38957 | Cosets of sets are element... |
| cnvelrels 38958 | The converse of a set is a... |
| cosscnvelrels 38959 | Cosets of converse sets ar... |
| dfssr2 38961 | Alternate definition of th... |
| relssr 38962 | The subset relation is a r... |
| brssr 38963 | The subset relation and su... |
| brssrid 38964 | Any set is a subset of its... |
| issetssr 38965 | Two ways of expressing set... |
| brssrres 38966 | Restricted subset binary r... |
| br1cnvssrres 38967 | Restricted converse subset... |
| brcnvssr 38968 | The converse of a subset r... |
| brcnvssrid 38969 | Any set is a converse subs... |
| br1cossxrncnvssrres 38970 | ` <. B , C >. ` and ` <. D... |
| extssr 38971 | Property of subset relatio... |
| dfrefrels2 38975 | Alternate definition of th... |
| dfrefrels3 38976 | Alternate definition of th... |
| dfrefrel2 38977 | Alternate definition of th... |
| dfrefrel3 38978 | Alternate definition of th... |
| dfrefrel5 38979 | Alternate definition of th... |
| elrefrels2 38980 | Element of the class of re... |
| elrefrels3 38981 | Element of the class of re... |
| elrefrelsrel 38982 | For sets, being an element... |
| refreleq 38983 | Equality theorem for refle... |
| refrelid 38984 | Identity relation is refle... |
| refrelcoss 38985 | The class of cosets by ` R... |
| refrelressn 38986 | Any class ' R ' restricted... |
| dfcnvrefrels2 38990 | Alternate definition of th... |
| dfcnvrefrels3 38991 | Alternate definition of th... |
| dfcnvrefrel2 38992 | Alternate definition of th... |
| dfcnvrefrel3 38993 | Alternate definition of th... |
| dfcnvrefrel4 38994 | Alternate definition of th... |
| dfcnvrefrel5 38995 | Alternate definition of th... |
| elcnvrefrels2 38996 | Element of the class of co... |
| elcnvrefrels3 38997 | Element of the class of co... |
| elcnvrefrelsrel 38998 | For sets, being an element... |
| cnvrefrelcoss2 38999 | Necessary and sufficient c... |
| cosselcnvrefrels2 39000 | Necessary and sufficient c... |
| cosselcnvrefrels3 39001 | Necessary and sufficient c... |
| cosselcnvrefrels4 39002 | Necessary and sufficient c... |
| cosselcnvrefrels5 39003 | Necessary and sufficient c... |
| dfsymrels2 39007 | Alternate definition of th... |
| dfsymrels3 39008 | Alternate definition of th... |
| elrelscnveq3 39009 | Two ways of saying a relat... |
| elrelscnveq 39010 | Two ways of saying a relat... |
| elrelscnveq2 39011 | Two ways of saying a relat... |
| elrelscnveq4 39012 | Two ways of saying a relat... |
| dfsymrels4 39013 | Alternate definition of th... |
| dfsymrels5 39014 | Alternate definition of th... |
| dfsymrel2 39015 | Alternate definition of th... |
| dfsymrel3 39016 | Alternate definition of th... |
| dfsymrel4 39017 | Alternate definition of th... |
| dfsymrel5 39018 | Alternate definition of th... |
| elsymrels2 39019 | Element of the class of sy... |
| elsymrels3 39020 | Element of the class of sy... |
| elsymrels4 39021 | Element of the class of sy... |
| elsymrels5 39022 | Element of the class of sy... |
| elsymrelsrel 39023 | For sets, being an element... |
| symreleq 39024 | Equality theorem for symme... |
| symrelim 39025 | Symmetric relation implies... |
| symrelcoss 39026 | The class of cosets by ` R... |
| idsymrel 39027 | The identity relation is s... |
| epnsymrel 39028 | The membership (epsilon) r... |
| symrefref2 39029 | Symmetry is a sufficient c... |
| symrefref3 39030 | Symmetry is a sufficient c... |
| refsymrels2 39031 | Elements of the class of r... |
| refsymrels3 39032 | Elements of the class of r... |
| refsymrel2 39033 | A relation which is reflex... |
| refsymrel3 39034 | A relation which is reflex... |
| elrefsymrels2 39035 | Elements of the class of r... |
| elrefsymrels3 39036 | Elements of the class of r... |
| elrefsymrelsrel 39037 | For sets, being an element... |
| dftrrels2 39041 | Alternate definition of th... |
| dftrrels3 39042 | Alternate definition of th... |
| dftrrel2 39043 | Alternate definition of th... |
| dftrrel3 39044 | Alternate definition of th... |
| eltrrels2 39045 | Element of the class of tr... |
| eltrrels3 39046 | Element of the class of tr... |
| eltrrelsrel 39047 | For sets, being an element... |
| trreleq 39048 | Equality theorem for the t... |
| trrelressn 39049 | Any class ' R ' restricted... |
| dfeqvrels2 39054 | Alternate definition of th... |
| dfeqvrels3 39055 | Alternate definition of th... |
| dfeqvrel2 39056 | Alternate definition of th... |
| dfeqvrel3 39057 | Alternate definition of th... |
| eleqvrels2 39058 | Element of the class of eq... |
| eleqvrels3 39059 | Element of the class of eq... |
| eleqvrelsrel 39060 | For sets, being an element... |
| elcoeleqvrels 39061 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 39062 | For sets, being an element... |
| eqvrelrel 39063 | An equivalence relation is... |
| eqvrelrefrel 39064 | An equivalence relation is... |
| eqvrelsymrel 39065 | An equivalence relation is... |
| eqvreltrrel 39066 | An equivalence relation is... |
| eqvrelim 39067 | Equivalence relation impli... |
| eqvreleq 39068 | Equality theorem for equiv... |
| eqvreleqi 39069 | Equality theorem for equiv... |
| eqvreleqd 39070 | Equality theorem for equiv... |
| eqvrelsym 39071 | An equivalence relation is... |
| eqvrelsymb 39072 | An equivalence relation is... |
| eqvreltr 39073 | An equivalence relation is... |
| eqvreltrd 39074 | A transitivity relation fo... |
| eqvreltr4d 39075 | A transitivity relation fo... |
| eqvrelref 39076 | An equivalence relation is... |
| eqvrelth 39077 | Basic property of equivale... |
| eqvrelcl 39078 | Elementhood in the field o... |
| eqvrelthi 39079 | Basic property of equivale... |
| eqvreldisj 39080 | Equivalence classes do not... |
| qsdisjALTV 39081 | Elements of a quotient set... |
| eqvrelqsel 39082 | If an element of a quotien... |
| eqvrelcoss 39083 | Two ways to express equiva... |
| eqvrelcoss3 39084 | Two ways to express equiva... |
| eqvrelcoss2 39085 | Two ways to express equiva... |
| eqvrelcoss4 39086 | Two ways to express equiva... |
| dfcoeleqvrels 39087 | Alternate definition of th... |
| dfcoeleqvrel 39088 | Alternate definition of th... |
| brredunds 39092 | Binary relation on the cla... |
| brredundsredund 39093 | For sets, binary relation ... |
| redundss3 39094 | Implication of redundancy ... |
| redundeq1 39095 | Equivalence of redundancy ... |
| redundpim3 39096 | Implication of redundancy ... |
| redundpbi1 39097 | Equivalence of redundancy ... |
| refrelsredund4 39098 | The naive version of the c... |
| refrelsredund2 39099 | The naive version of the c... |
| refrelsredund3 39100 | The naive version of the c... |
| refrelredund4 39101 | The naive version of the d... |
| refrelredund2 39102 | The naive version of the d... |
| refrelredund3 39103 | The naive version of the d... |
| dmqseq 39106 | Equality theorem for domai... |
| dmqseqi 39107 | Equality theorem for domai... |
| dmqseqd 39108 | Equality theorem for domai... |
| dmqseqeq1 39109 | Equality theorem for domai... |
| dmqseqeq1i 39110 | Equality theorem for domai... |
| dmqseqeq1d 39111 | Equality theorem for domai... |
| brdmqss 39112 | The domain quotient binary... |
| brdmqssqs 39113 | If ` A ` and ` R ` are set... |
| n0eldmqs 39114 | The empty set is not an el... |
| qseq 39115 | The quotient set equal to ... |
| n0eldmqseq 39116 | The empty set is not an el... |
| n0elim 39117 | Implication of that the em... |
| n0el3 39118 | Two ways of expressing tha... |
| cnvepresdmqss 39119 | The domain quotient binary... |
| cnvepresdmqs 39120 | The domain quotient predic... |
| unidmqs 39121 | The range of a relation is... |
| unidmqseq 39122 | The union of the domain qu... |
| dmqseqim 39123 | If the domain quotient of ... |
| dmqseqim2 39124 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 39125 | Elementhood in the domain ... |
| eldmqs1cossres 39126 | Elementhood in the domain ... |
| releldmqscoss 39127 | Elementhood in the domain ... |
| dmqscoelseq 39128 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 39129 | Two ways to express the eq... |
| brers 39134 | Binary equivalence relatio... |
| dferALTV2 39135 | Equivalence relation with ... |
| erALTVeq1 39136 | Equality theorem for equiv... |
| erALTVeq1i 39137 | Equality theorem for equiv... |
| erALTVeq1d 39138 | Equality theorem for equiv... |
| dfcomember 39139 | Alternate definition of th... |
| dfcomember2 39140 | Alternate definition of th... |
| dfcomember3 39141 | Alternate definition of th... |
| eqvreldmqs 39142 | Two ways to express comemb... |
| eqvreldmqs2 39143 | Two ways to express comemb... |
| brerser 39144 | Binary equivalence relatio... |
| erimeq2 39145 | Equivalence relation on it... |
| erimeq 39146 | Equivalence relation on it... |
| dffunsALTV 39150 | Alternate definition of th... |
| dffunsALTV2 39151 | Alternate definition of th... |
| dffunsALTV3 39152 | Alternate definition of th... |
| dffunsALTV4 39153 | Alternate definition of th... |
| dffunsALTV5 39154 | Alternate definition of th... |
| dffunALTV2 39155 | Alternate definition of th... |
| dffunALTV3 39156 | Alternate definition of th... |
| dffunALTV4 39157 | Alternate definition of th... |
| dffunALTV5 39158 | Alternate definition of th... |
| elfunsALTV 39159 | Elementhood in the class o... |
| elfunsALTV2 39160 | Elementhood in the class o... |
| elfunsALTV3 39161 | Elementhood in the class o... |
| elfunsALTV4 39162 | Elementhood in the class o... |
| elfunsALTV5 39163 | Elementhood in the class o... |
| elfunsALTVfunALTV 39164 | The element of the class o... |
| funALTVfun 39165 | Our definition of the func... |
| funALTVss 39166 | Subclass theorem for funct... |
| funALTVeq 39167 | Equality theorem for funct... |
| funALTVeqi 39168 | Equality inference for the... |
| funALTVeqd 39169 | Equality deduction for the... |
| dfdisjs 39175 | Alternate definition of th... |
| dfdisjs2 39176 | Alternate definition of th... |
| dfdisjs3 39177 | Alternate definition of th... |
| dfdisjs4 39178 | Alternate definition of th... |
| dfdisjs5 39179 | Alternate definition of th... |
| dfdisjALTV 39180 | Alternate definition of th... |
| dfdisjALTV2 39181 | Alternate definition of th... |
| dfdisjALTV3 39182 | Alternate definition of th... |
| dfdisjALTV4 39183 | Alternate definition of th... |
| dfdisjALTV5 39184 | Alternate definition of th... |
| dfdisjALTV5a 39185 | Alternate definition of th... |
| disjimeceqim 39186 | ` Disj ` implies coset-equ... |
| disjimeceqim2 39187 | ` Disj ` implies injectivi... |
| disjimeceqbi 39188 | ` Disj ` gives bicondition... |
| disjimeceqbi2 39189 | Injectivity of the block c... |
| disjimrmoeqec 39190 | Under ` Disj ` , every blo... |
| disjimdmqseq 39191 | Disjointness implies uniqu... |
| dfeldisj2 39192 | Alternate definition of th... |
| dfeldisj3 39193 | Alternate definition of th... |
| dfeldisj4 39194 | Alternate definition of th... |
| dfeldisj5 39195 | Alternate definition of th... |
| dfeldisj5a 39196 | Alternate definition of th... |
| eldisjim3 39197 | ` ElDisj ` elimination (tw... |
| eldisjdmqsim2 39198 | ElDisj of quotient implies... |
| eldisjdmqsim 39199 | Shared output implies equa... |
| suceldisj 39200 | Disjointness of successor ... |
| eldisjs 39201 | Elementhood in the class o... |
| eldisjs2 39202 | Elementhood in the class o... |
| eldisjs3 39203 | Elementhood in the class o... |
| eldisjs4 39204 | Elementhood in the class o... |
| eldisjs5 39205 | Elementhood in the class o... |
| eldisjsdisj 39206 | The element of the class o... |
| qmapeldisjs 39207 | When ` R ` is a set (e.g.,... |
| disjqmap2 39208 | Disjointness of ` QMap ` e... |
| disjqmap 39209 | Disjointness of ` QMap ` e... |
| eleldisjs 39210 | Elementhood in the disjoin... |
| eleldisjseldisj 39211 | The element of the disjoin... |
| disjrel 39212 | Disjoint relation is a rel... |
| disjss 39213 | Subclass theorem for disjo... |
| disjssi 39214 | Subclass theorem for disjo... |
| disjssd 39215 | Subclass theorem for disjo... |
| disjeq 39216 | Equality theorem for disjo... |
| disjeqi 39217 | Equality theorem for disjo... |
| disjeqd 39218 | Equality theorem for disjo... |
| disjdmqseqeq1 39219 | Lemma for the equality the... |
| eldisjss 39220 | Subclass theorem for disjo... |
| eldisjssi 39221 | Subclass theorem for disjo... |
| eldisjssd 39222 | Subclass theorem for disjo... |
| eldisjeq 39223 | Equality theorem for disjo... |
| eldisjeqi 39224 | Equality theorem for disjo... |
| eldisjeqd 39225 | Equality theorem for disjo... |
| disjres 39226 | Disjoint restriction. (Co... |
| eldisjn0elb 39227 | Two forms of disjoint elem... |
| disjxrn 39228 | Two ways of saying that a ... |
| disjxrnres5 39229 | Disjoint range Cartesian p... |
| disjorimxrn 39230 | Disjointness condition for... |
| disjimxrn 39231 | Disjointness condition for... |
| disjimres 39232 | Disjointness condition for... |
| disjimin 39233 | Disjointness condition for... |
| disjiminres 39234 | Disjointness condition for... |
| disjimxrnres 39235 | Disjointness condition for... |
| disjALTV0 39236 | The null class is disjoint... |
| disjALTVid 39237 | The class of identity rela... |
| disjALTVidres 39238 | The class of identity rela... |
| disjALTVinidres 39239 | The intersection with rest... |
| disjALTVxrnidres 39240 | The class of range Cartesi... |
| disjsuc 39241 | Disjoint range Cartesian p... |
| qmapeldisjsim 39242 | Injectivity of coset map f... |
| qmapeldisjsbi 39243 | Injectivity of coset map f... |
| rnqmapeleldisjsim 39244 | Element-disjointness of th... |
| dfantisymrel4 39246 | Alternate definition of th... |
| dfantisymrel5 39247 | Alternate definition of th... |
| antisymrelres 39248 | (Contributed by Peter Mazs... |
| antisymrelressn 39249 | (Contributed by Peter Mazs... |
| dfpart2 39254 | Alternate definition of th... |
| dfmembpart2 39255 | Alternate definition of th... |
| brparts 39256 | Binary partitions relation... |
| brparts2 39257 | Binary partitions relation... |
| brpartspart 39258 | Binary partition and the p... |
| parteq1 39259 | Equality theorem for parti... |
| parteq2 39260 | Equality theorem for parti... |
| parteq12 39261 | Equality theorem for parti... |
| parteq1i 39262 | Equality theorem for parti... |
| parteq1d 39263 | Equality theorem for parti... |
| partsuc2 39264 | Property of the partition.... |
| partsuc 39265 | Property of the partition.... |
| disjim 39266 | The "Divide et Aequivalere... |
| disjimi 39267 | Every disjoint relation ge... |
| detlem 39268 | If a relation is disjoint,... |
| eldisjim 39269 | If the elements of ` A ` a... |
| eldisjim2 39270 | Alternate form of ~ eldisj... |
| eqvrel0 39271 | The null class is an equiv... |
| det0 39272 | The cosets by the null cla... |
| eqvrelcoss0 39273 | The cosets by the null cla... |
| eqvrelid 39274 | The identity relation is a... |
| eqvrel1cossidres 39275 | The cosets by a restricted... |
| eqvrel1cossinidres 39276 | The cosets by an intersect... |
| eqvrel1cossxrnidres 39277 | The cosets by a range Cart... |
| detid 39278 | The cosets by the identity... |
| eqvrelcossid 39279 | The cosets by the identity... |
| detidres 39280 | The cosets by the restrict... |
| detinidres 39281 | The cosets by the intersec... |
| detxrnidres 39282 | The cosets by the range Ca... |
| disjlem14 39283 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 39284 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 39285 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 39286 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 39287 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 39288 | Lemma for ~ disjdmqseq via... |
| disjdmqs 39289 | If a relation is disjoint,... |
| disjdmqseq 39290 | If a relation is disjoint,... |
| eldisjn0el 39291 | Special case of ~ disjdmqs... |
| partim2 39292 | Disjoint relation on its n... |
| partim 39293 | Partition implies equivale... |
| partimeq 39294 | Partition implies that the... |
| eldisjlem19 39295 | Special case of ~ disjlem1... |
| membpartlem19 39296 | Together with ~ disjlem19 ... |
| petlem 39297 | If you can prove that the ... |
| petlemi 39298 | If you can prove disjointn... |
| pet02 39299 | Class ` A ` is a partition... |
| pet0 39300 | Class ` A ` is a partition... |
| petid2 39301 | Class ` A ` is a partition... |
| petid 39302 | A class is a partition by ... |
| petidres2 39303 | Class ` A ` is a partition... |
| petidres 39304 | A class is a partition by ... |
| petinidres2 39305 | Class ` A ` is a partition... |
| petinidres 39306 | A class is a partition by ... |
| petxrnidres2 39307 | Class ` A ` is a partition... |
| petxrnidres 39308 | A class is a partition by ... |
| eqvreldisj1 39309 | The elements of the quotie... |
| eqvreldisj2 39310 | The elements of the quotie... |
| eqvreldisj3 39311 | The elements of the quotie... |
| eqvreldisj4 39312 | Intersection with the conv... |
| eqvreldisj5 39313 | Range Cartesian product wi... |
| eqvrelqseqdisj2 39314 | Implication of ~ eqvreldis... |
| disjimeldisjdmqs 39315 | ` Disj ` implies element-d... |
| eldisjsim1 39316 | An element of the class of... |
| eldisjsim2 39317 | An element of the class of... |
| disjsssrels 39318 | The class of disjoint rela... |
| eldisjsim3 39319 | ` Disjs ` implies element-... |
| eldisjsim4 39320 | ` Disjs ` implies element-... |
| eldisjsim5 39321 | ` Disjs ` is closed under ... |
| eldisjs6 39322 | Elementhood in the class o... |
| eldisjs7 39323 | Elementhood in the class o... |
| dfdisjs6 39324 | Alternate definition of th... |
| dfdisjs7 39325 | Alternate definition of th... |
| fences3 39326 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 39327 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 39328 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 39329 | Lemma for the Partition-Eq... |
| mainer 39330 | The Main Theorem of Equiva... |
| partimcomember 39331 | Partition with general ` R... |
| mpet3 39332 | Member Partition-Equivalen... |
| cpet2 39333 | The conventional form of t... |
| cpet 39334 | The conventional form of M... |
| mpet 39335 | Member Partition-Equivalen... |
| mpet2 39336 | Member Partition-Equivalen... |
| mpets2 39337 | Member Partition-Equivalen... |
| mpets 39338 | Member Partition-Equivalen... |
| mainpart 39339 | Partition with general ` R... |
| fences 39340 | The Theorem of Fences by E... |
| fences2 39341 | The Theorem of Fences by E... |
| mainer2 39342 | The Main Theorem of Equiva... |
| mainerim 39343 | Every equivalence relation... |
| petincnvepres2 39344 | A partition-equivalence th... |
| petincnvepres 39345 | The shortest form of a par... |
| pet2 39346 | Partition-Equivalence Theo... |
| pet 39347 | Partition-Equivalence Theo... |
| pets 39348 | Partition-Equivalence Theo... |
| dmqsblocks 39349 | If the ~ pet span ` ( R |X... |
| dfpetparts2 39354 | Alternate definition of ` ... |
| dfpet2parts2 39355 | Grade stability applied to... |
| dfpeters2 39356 | Alternate definition of ` ... |
| typesafepets 39357 | Type-safe ~ pets scheme. ... |
| petseq 39358 | Generalized partition-equi... |
| pets2eq 39359 | Grade-stable generalized p... |
| prtlem60 39360 | Lemma for ~ prter3 . (Con... |
| bicomdd 39361 | Commute two sides of a bic... |
| jca2r 39362 | Inference conjoining the c... |
| jca3 39363 | Inference conjoining the c... |
| prtlem70 39364 | Lemma for ~ prter3 : a rea... |
| ibdr 39365 | Reverse of ~ ibd . (Contr... |
| prtlem100 39366 | Lemma for ~ prter3 . (Con... |
| prtlem5 39367 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 39368 | Lemma for ~ prter2 . (Con... |
| brabsb2 39369 | A closed form of ~ brabsb ... |
| eqbrrdv2 39370 | Other version of ~ eqbrrdi... |
| prtlem9 39371 | Lemma for ~ prter3 . (Con... |
| prtlem10 39372 | Lemma for ~ prter3 . (Con... |
| prtlem11 39373 | Lemma for ~ prter2 . (Con... |
| prtlem12 39374 | Lemma for ~ prtex and ~ pr... |
| prtlem13 39375 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 39376 | Lemma for ~ prtex , ~ prte... |
| prtlem400 39377 | Lemma for ~ prter2 and als... |
| erprt 39380 | The quotient set of an equ... |
| prtlem14 39381 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 39382 | Lemma for ~ prter1 and ~ p... |
| prtlem17 39383 | Lemma for ~ prter2 . (Con... |
| prtlem18 39384 | Lemma for ~ prter2 . (Con... |
| prtlem19 39385 | Lemma for ~ prter2 . (Con... |
| prter1 39386 | Every partition generates ... |
| prtex 39387 | The equivalence relation g... |
| prter2 39388 | The quotient set of the eq... |
| prter3 39389 | For every partition there ... |
| axc5 39400 | This theorem repeats ~ sp ... |
| ax4fromc4 39401 | Rederivation of Axiom ~ ax... |
| ax10fromc7 39402 | Rederivation of Axiom ~ ax... |
| ax6fromc10 39403 | Rederivation of Axiom ~ ax... |
| hba1-o 39404 | The setvar ` x ` is not fr... |
| axc4i-o 39405 | Inference version of ~ ax-... |
| equid1 39406 | Proof of ~ equid from our ... |
| equcomi1 39407 | Proof of ~ equcomi from ~ ... |
| aecom-o 39408 | Commutation law for identi... |
| aecoms-o 39409 | A commutation rule for ide... |
| hbae-o 39410 | All variables are effectiv... |
| dral1-o 39411 | Formula-building lemma for... |
| ax12fromc15 39412 | Rederivation of Axiom ~ ax... |
| ax13fromc9 39413 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 39414 | Axiom to quantify a variab... |
| sps-o 39415 | Generalization of antecede... |
| hbequid 39416 | Bound-variable hypothesis ... |
| nfequid-o 39417 | Bound-variable hypothesis ... |
| axc5c7 39418 | Proof of a single axiom th... |
| axc5c7toc5 39419 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 39420 | Rederivation of ~ ax-c7 fr... |
| axc711 39421 | Proof of a single axiom th... |
| nfa1-o 39422 | ` x ` is not free in ` A. ... |
| axc711toc7 39423 | Rederivation of ~ ax-c7 fr... |
| axc711to11 39424 | Rederivation of ~ ax-11 fr... |
| axc5c711 39425 | Proof of a single axiom th... |
| axc5c711toc5 39426 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 39427 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 39428 | Rederivation of ~ ax-11 fr... |
| equidqe 39429 | ~ equid with existential q... |
| axc5sp1 39430 | A special case of ~ ax-c5 ... |
| equidq 39431 | ~ equid with universal qua... |
| equid1ALT 39432 | Alternate proof of ~ equid... |
| axc11nfromc11 39433 | Rederivation of ~ ax-c11n ... |
| naecoms-o 39434 | A commutation rule for dis... |
| hbnae-o 39435 | All variables are effectiv... |
| dvelimf-o 39436 | Proof of ~ dvelimh that us... |
| dral2-o 39437 | Formula-building lemma for... |
| aev-o 39438 | A "distinctor elimination"... |
| ax5eq 39439 | Theorem to add distinct qu... |
| dveeq2-o 39440 | Quantifier introduction wh... |
| axc16g-o 39441 | A generalization of Axiom ... |
| dveeq1-o 39442 | Quantifier introduction wh... |
| dveeq1-o16 39443 | Version of ~ dveeq1 using ... |
| ax5el 39444 | Theorem to add distinct qu... |
| axc11n-16 39445 | This theorem shows that, g... |
| dveel2ALT 39446 | Alternate proof of ~ dveel... |
| ax12f 39447 | Basis step for constructin... |
| ax12eq 39448 | Basis step for constructin... |
| ax12el 39449 | Basis step for constructin... |
| ax12indn 39450 | Induction step for constru... |
| ax12indi 39451 | Induction step for constru... |
| ax12indalem 39452 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 39453 | Alternate proof of ~ ax12i... |
| ax12inda2 39454 | Induction step for constru... |
| ax12inda 39455 | Induction step for constru... |
| ax12v2-o 39456 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 39457 | Derive ~ ax-c15 from a hyp... |
| axc11-o 39458 | Show that ~ ax-c11 can be ... |
| fsumshftd 39459 | Index shift of a finite su... |
| riotaclbgBAD 39461 | Closure of restricted iota... |
| riotaclbBAD 39462 | Closure of restricted iota... |
| riotasvd 39463 | Deduction version of ~ rio... |
| riotasv2d 39464 | Value of description binde... |
| riotasv2s 39465 | The value of description b... |
| riotasv 39466 | Value of description binde... |
| riotasv3d 39467 | A property ` ch ` holding ... |
| elimhyps 39468 | A version of ~ elimhyp usi... |
| dedths 39469 | A version of weak deductio... |
| renegclALT 39470 | Closure law for negative o... |
| elimhyps2 39471 | Generalization of ~ elimhy... |
| dedths2 39472 | Generalization of ~ dedths... |
| nfcxfrdf 39473 | A utility lemma to transfe... |
| nfded 39474 | A deduction theorem that c... |
| nfded2 39475 | A deduction theorem that c... |
| nfunidALT2 39476 | Deduction version of ~ nfu... |
| nfunidALT 39477 | Deduction version of ~ nfu... |
| nfopdALT 39478 | Deduction version of bound... |
| cnaddcom 39479 | Recover the commutative la... |
| toycom 39480 | Show the commutative law f... |
| lshpset 39485 | The set of all hyperplanes... |
| islshp 39486 | The predicate "is a hyperp... |
| islshpsm 39487 | Hyperplane properties expr... |
| lshplss 39488 | A hyperplane is a subspace... |
| lshpne 39489 | A hyperplane is not equal ... |
| lshpnel 39490 | A hyperplane's generating ... |
| lshpnelb 39491 | The subspace sum of a hype... |
| lshpnel2N 39492 | Condition that determines ... |
| lshpne0 39493 | The member of the span in ... |
| lshpdisj 39494 | A hyperplane and the span ... |
| lshpcmp 39495 | If two hyperplanes are com... |
| lshpinN 39496 | The intersection of two di... |
| lsatset 39497 | The set of all 1-dim subsp... |
| islsat 39498 | The predicate "is a 1-dim ... |
| lsatlspsn2 39499 | The span of a nonzero sing... |
| lsatlspsn 39500 | The span of a nonzero sing... |
| islsati 39501 | A 1-dim subspace (atom) (o... |
| lsateln0 39502 | A 1-dim subspace (atom) (o... |
| lsatlss 39503 | The set of 1-dim subspaces... |
| lsatlssel 39504 | An atom is a subspace. (C... |
| lsatssv 39505 | An atom is a set of vector... |
| lsatn0 39506 | A 1-dim subspace (atom) of... |
| lsatspn0 39507 | The span of a vector is an... |
| lsator0sp 39508 | The span of a vector is ei... |
| lsatssn0 39509 | A subspace (or any class) ... |
| lsatcmp 39510 | If two atoms are comparabl... |
| lsatcmp2 39511 | If an atom is included in ... |
| lsatel 39512 | A nonzero vector in an ato... |
| lsatelbN 39513 | A nonzero vector in an ato... |
| lsat2el 39514 | Two atoms sharing a nonzer... |
| lsmsat 39515 | Convert comparison of atom... |
| lsatfixedN 39516 | Show equality with the spa... |
| lsmsatcv 39517 | Subspace sum has the cover... |
| lssatomic 39518 | The lattice of subspaces i... |
| lssats 39519 | The lattice of subspaces i... |
| lpssat 39520 | Two subspaces in a proper ... |
| lrelat 39521 | Subspaces are relatively a... |
| lssatle 39522 | The ordering of two subspa... |
| lssat 39523 | Two subspaces in a proper ... |
| islshpat 39524 | Hyperplane properties expr... |
| lcvfbr 39527 | The covers relation for a ... |
| lcvbr 39528 | The covers relation for a ... |
| lcvbr2 39529 | The covers relation for a ... |
| lcvbr3 39530 | The covers relation for a ... |
| lcvpss 39531 | The covers relation implie... |
| lcvnbtwn 39532 | The covers relation implie... |
| lcvntr 39533 | The covers relation is not... |
| lcvnbtwn2 39534 | The covers relation implie... |
| lcvnbtwn3 39535 | The covers relation implie... |
| lsmcv2 39536 | Subspace sum has the cover... |
| lcvat 39537 | If a subspace covers anoth... |
| lsatcv0 39538 | An atom covers the zero su... |
| lsatcveq0 39539 | A subspace covered by an a... |
| lsat0cv 39540 | A subspace is an atom iff ... |
| lcvexchlem1 39541 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 39542 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 39543 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 39544 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 39545 | Lemma for ~ lcvexch . (Co... |
| lcvexch 39546 | Subspaces satisfy the exch... |
| lcvp 39547 | Covering property of Defin... |
| lcv1 39548 | Covering property of a sub... |
| lcv2 39549 | Covering property of a sub... |
| lsatexch 39550 | The atom exchange property... |
| lsatnle 39551 | The meet of a subspace and... |
| lsatnem0 39552 | The meet of distinct atoms... |
| lsatexch1 39553 | The atom exch1ange propert... |
| lsatcv0eq 39554 | If the sum of two atoms co... |
| lsatcv1 39555 | Two atoms covering the zer... |
| lsatcvatlem 39556 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 39557 | A nonzero subspace less th... |
| lsatcvat2 39558 | A subspace covered by the ... |
| lsatcvat3 39559 | A condition implying that ... |
| islshpcv 39560 | Hyperplane properties expr... |
| l1cvpat 39561 | A subspace covered by the ... |
| l1cvat 39562 | Create an atom under an el... |
| lshpat 39563 | Create an atom under a hyp... |
| lflset 39566 | The set of linear function... |
| islfl 39567 | The predicate "is a linear... |
| lfli 39568 | Property of a linear funct... |
| islfld 39569 | Properties that determine ... |
| lflf 39570 | A linear functional is a f... |
| lflcl 39571 | A linear functional value ... |
| lfl0 39572 | A linear functional is zer... |
| lfladd 39573 | Property of a linear funct... |
| lflsub 39574 | Property of a linear funct... |
| lflmul 39575 | Property of a linear funct... |
| lfl0f 39576 | The zero function is a fun... |
| lfl1 39577 | A nonzero functional has a... |
| lfladdcl 39578 | Closure of addition of two... |
| lfladdcom 39579 | Commutativity of functiona... |
| lfladdass 39580 | Associativity of functiona... |
| lfladd0l 39581 | Functional addition with t... |
| lflnegcl 39582 | Closure of the negative of... |
| lflnegl 39583 | A functional plus its nega... |
| lflvscl 39584 | Closure of a scalar produc... |
| lflvsdi1 39585 | Distributive law for (righ... |
| lflvsdi2 39586 | Reverse distributive law f... |
| lflvsdi2a 39587 | Reverse distributive law f... |
| lflvsass 39588 | Associative law for (right... |
| lfl0sc 39589 | The (right vector space) s... |
| lflsc0N 39590 | The scalar product with th... |
| lfl1sc 39591 | The (right vector space) s... |
| lkrfval 39594 | The kernel of a functional... |
| lkrval 39595 | Value of the kernel of a f... |
| ellkr 39596 | Membership in the kernel o... |
| lkrval2 39597 | Value of the kernel of a f... |
| ellkr2 39598 | Membership in the kernel o... |
| lkrcl 39599 | A member of the kernel of ... |
| lkrf0 39600 | The value of a functional ... |
| lkr0f 39601 | The kernel of the zero fun... |
| lkrlss 39602 | The kernel of a linear fun... |
| lkrssv 39603 | The kernel of a linear fun... |
| lkrsc 39604 | The kernel of a nonzero sc... |
| lkrscss 39605 | The kernel of a scalar pro... |
| eqlkr 39606 | Two functionals with the s... |
| eqlkr2 39607 | Two functionals with the s... |
| eqlkr3 39608 | Two functionals with the s... |
| lkrlsp 39609 | The subspace sum of a kern... |
| lkrlsp2 39610 | The subspace sum of a kern... |
| lkrlsp3 39611 | The subspace sum of a kern... |
| lkrshp 39612 | The kernel of a nonzero fu... |
| lkrshp3 39613 | The kernels of nonzero fun... |
| lkrshpor 39614 | The kernel of a functional... |
| lkrshp4 39615 | A kernel is a hyperplane i... |
| lshpsmreu 39616 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 39617 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 39618 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 39619 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 39620 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 39621 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 39622 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 39623 | The set ` G ` defined by h... |
| lshpkr 39624 | The kernel of functional `... |
| lshpkrex 39625 | There exists a functional ... |
| lshpset2N 39626 | The set of all hyperplanes... |
| islshpkrN 39627 | The predicate "is a hyperp... |
| lfl1dim 39628 | Equivalent expressions for... |
| lfl1dim2N 39629 | Equivalent expressions for... |
| ldualset 39632 | Define the (left) dual of ... |
| ldualvbase 39633 | The vectors of a dual spac... |
| ldualelvbase 39634 | Utility theorem for conver... |
| ldualfvadd 39635 | Vector addition in the dua... |
| ldualvadd 39636 | Vector addition in the dua... |
| ldualvaddcl 39637 | The value of vector additi... |
| ldualvaddval 39638 | The value of the value of ... |
| ldualsca 39639 | The ring of scalars of the... |
| ldualsbase 39640 | Base set of scalar ring fo... |
| ldualsaddN 39641 | Scalar addition for the du... |
| ldualsmul 39642 | Scalar multiplication for ... |
| ldualfvs 39643 | Scalar product operation f... |
| ldualvs 39644 | Scalar product operation v... |
| ldualvsval 39645 | Value of scalar product op... |
| ldualvscl 39646 | The scalar product operati... |
| ldualvaddcom 39647 | Commutative law for vector... |
| ldualvsass 39648 | Associative law for scalar... |
| ldualvsass2 39649 | Associative law for scalar... |
| ldualvsdi1 39650 | Distributive law for scala... |
| ldualvsdi2 39651 | Reverse distributive law f... |
| ldualgrplem 39652 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 39653 | The dual of a vector space... |
| ldual0 39654 | The zero scalar of the dua... |
| ldual1 39655 | The unit scalar of the dua... |
| ldualneg 39656 | The negative of a scalar o... |
| ldual0v 39657 | The zero vector of the dua... |
| ldual0vcl 39658 | The dual zero vector is a ... |
| lduallmodlem 39659 | Lemma for ~ lduallmod . (... |
| lduallmod 39660 | The dual of a left module ... |
| lduallvec 39661 | The dual of a left vector ... |
| ldualvsub 39662 | The value of vector subtra... |
| ldualvsubcl 39663 | Closure of vector subtract... |
| ldualvsubval 39664 | The value of the value of ... |
| ldualssvscl 39665 | Closure of scalar product ... |
| ldualssvsubcl 39666 | Closure of vector subtract... |
| ldual0vs 39667 | Scalar zero times a functi... |
| lkr0f2 39668 | The kernel of the zero fun... |
| lduallkr3 39669 | The kernels of nonzero fun... |
| lkrpssN 39670 | Proper subset relation bet... |
| lkrin 39671 | Intersection of the kernel... |
| eqlkr4 39672 | Two functionals with the s... |
| ldual1dim 39673 | Equivalent expressions for... |
| ldualkrsc 39674 | The kernel of a nonzero sc... |
| lkrss 39675 | The kernel of a scalar pro... |
| lkrss2N 39676 | Two functionals with kerne... |
| lkreqN 39677 | Proportional functionals h... |
| lkrlspeqN 39678 | Condition for colinear fun... |
| isopos 39687 | The predicate "is an ortho... |
| opposet 39688 | Every orthoposet is a pose... |
| oposlem 39689 | Lemma for orthoposet prope... |
| op01dm 39690 | Conditions necessary for z... |
| op0cl 39691 | An orthoposet has a zero e... |
| op1cl 39692 | An orthoposet has a unity ... |
| op0le 39693 | Orthoposet zero is less th... |
| ople0 39694 | An element less than or eq... |
| opnlen0 39695 | An element not less than a... |
| lub0N 39696 | The least upper bound of t... |
| opltn0 39697 | A lattice element greater ... |
| ople1 39698 | Any element is less than t... |
| op1le 39699 | If the orthoposet unity is... |
| glb0N 39700 | The greatest lower bound o... |
| opoccl 39701 | Closure of orthocomplement... |
| opococ 39702 | Double negative law for or... |
| opcon3b 39703 | Contraposition law for ort... |
| opcon2b 39704 | Orthocomplement contraposi... |
| opcon1b 39705 | Orthocomplement contraposi... |
| oplecon3 39706 | Contraposition law for ort... |
| oplecon3b 39707 | Contraposition law for ort... |
| oplecon1b 39708 | Contraposition law for str... |
| opoc1 39709 | Orthocomplement of orthopo... |
| opoc0 39710 | Orthocomplement of orthopo... |
| opltcon3b 39711 | Contraposition law for str... |
| opltcon1b 39712 | Contraposition law for str... |
| opltcon2b 39713 | Contraposition law for str... |
| opexmid 39714 | Law of excluded middle for... |
| opnoncon 39715 | Law of contradiction for o... |
| riotaocN 39716 | The orthocomplement of the... |
| cmtfvalN 39717 | Value of commutes relation... |
| cmtvalN 39718 | Equivalence for commutes r... |
| isolat 39719 | The predicate "is an ortho... |
| ollat 39720 | An ortholattice is a latti... |
| olop 39721 | An ortholattice is an orth... |
| olposN 39722 | An ortholattice is a poset... |
| isolatiN 39723 | Properties that determine ... |
| oldmm1 39724 | De Morgan's law for meet i... |
| oldmm2 39725 | De Morgan's law for meet i... |
| oldmm3N 39726 | De Morgan's law for meet i... |
| oldmm4 39727 | De Morgan's law for meet i... |
| oldmj1 39728 | De Morgan's law for join i... |
| oldmj2 39729 | De Morgan's law for join i... |
| oldmj3 39730 | De Morgan's law for join i... |
| oldmj4 39731 | De Morgan's law for join i... |
| olj01 39732 | An ortholattice element jo... |
| olj02 39733 | An ortholattice element jo... |
| olm11 39734 | The meet of an ortholattic... |
| olm12 39735 | The meet of an ortholattic... |
| latmassOLD 39736 | Ortholattice meet is assoc... |
| latm12 39737 | A rearrangement of lattice... |
| latm32 39738 | A rearrangement of lattice... |
| latmrot 39739 | Rotate lattice meet of 3 c... |
| latm4 39740 | Rearrangement of lattice m... |
| latmmdiN 39741 | Lattice meet distributes o... |
| latmmdir 39742 | Lattice meet distributes o... |
| olm01 39743 | Meet with lattice zero is ... |
| olm02 39744 | Meet with lattice zero is ... |
| isoml 39745 | The predicate "is an ortho... |
| isomliN 39746 | Properties that determine ... |
| omlol 39747 | An orthomodular lattice is... |
| omlop 39748 | An orthomodular lattice is... |
| omllat 39749 | An orthomodular lattice is... |
| omllaw 39750 | The orthomodular law. (Co... |
| omllaw2N 39751 | Variation of orthomodular ... |
| omllaw3 39752 | Orthomodular law equivalen... |
| omllaw4 39753 | Orthomodular law equivalen... |
| omllaw5N 39754 | The orthomodular law. Rem... |
| cmtcomlemN 39755 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 39756 | Commutation is symmetric. ... |
| cmt2N 39757 | Commutation with orthocomp... |
| cmt3N 39758 | Commutation with orthocomp... |
| cmt4N 39759 | Commutation with orthocomp... |
| cmtbr2N 39760 | Alternate definition of th... |
| cmtbr3N 39761 | Alternate definition for t... |
| cmtbr4N 39762 | Alternate definition for t... |
| lecmtN 39763 | Ordered elements commute. ... |
| cmtidN 39764 | Any element commutes with ... |
| omlfh1N 39765 | Foulis-Holland Theorem, pa... |
| omlfh3N 39766 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 39767 | Analogue of modular law ~ ... |
| omlspjN 39768 | Contraction of a Sasaki pr... |
| cvrfval 39775 | Value of covers relation "... |
| cvrval 39776 | Binary relation expressing... |
| cvrlt 39777 | The covers relation implie... |
| cvrnbtwn 39778 | There is no element betwee... |
| ncvr1 39779 | No element covers the latt... |
| cvrletrN 39780 | Property of an element abo... |
| cvrval2 39781 | Binary relation expressing... |
| cvrnbtwn2 39782 | The covers relation implie... |
| cvrnbtwn3 39783 | The covers relation implie... |
| cvrcon3b 39784 | Contraposition law for the... |
| cvrle 39785 | The covers relation implie... |
| cvrnbtwn4 39786 | The covers relation implie... |
| cvrnle 39787 | The covers relation implie... |
| cvrne 39788 | The covers relation implie... |
| cvrnrefN 39789 | The covers relation is not... |
| cvrcmp 39790 | If two lattice elements th... |
| cvrcmp2 39791 | If two lattice elements co... |
| pats 39792 | The set of atoms in a pose... |
| isat 39793 | The predicate "is an atom"... |
| isat2 39794 | The predicate "is an atom"... |
| atcvr0 39795 | An atom covers zero. ( ~ ... |
| atbase 39796 | An atom is a member of the... |
| atssbase 39797 | The set of atoms is a subs... |
| 0ltat 39798 | An atom is greater than ze... |
| leatb 39799 | A poset element less than ... |
| leat 39800 | A poset element less than ... |
| leat2 39801 | A nonzero poset element le... |
| leat3 39802 | A poset element less than ... |
| meetat 39803 | The meet of any element wi... |
| meetat2 39804 | The meet of any element wi... |
| isatl 39806 | The predicate "is an atomi... |
| atllat 39807 | An atomic lattice is a lat... |
| atlpos 39808 | An atomic lattice is a pos... |
| atl0dm 39809 | Condition necessary for ze... |
| atl0cl 39810 | An atomic lattice has a ze... |
| atl0le 39811 | Orthoposet zero is less th... |
| atlle0 39812 | An element less than or eq... |
| atlltn0 39813 | A lattice element greater ... |
| isat3 39814 | The predicate "is an atom"... |
| atn0 39815 | An atom is not zero. ( ~ ... |
| atnle0 39816 | An atom is not less than o... |
| atlen0 39817 | A lattice element is nonze... |
| atcmp 39818 | If two atoms are comparabl... |
| atncmp 39819 | Frequently-used variation ... |
| atnlt 39820 | Two atoms cannot satisfy t... |
| atcvreq0 39821 | An element covered by an a... |
| atncvrN 39822 | Two atoms cannot satisfy t... |
| atlex 39823 | Every nonzero element of a... |
| atnle 39824 | Two ways of expressing "an... |
| atnem0 39825 | The meet of distinct atoms... |
| atlatmstc 39826 | An atomic, complete, ortho... |
| atlatle 39827 | The ordering of two Hilber... |
| atlrelat1 39828 | An atomistic lattice with ... |
| iscvlat 39830 | The predicate "is an atomi... |
| iscvlat2N 39831 | The predicate "is an atomi... |
| cvlatl 39832 | An atomic lattice with the... |
| cvllat 39833 | An atomic lattice with the... |
| cvlposN 39834 | An atomic lattice with the... |
| cvlexch1 39835 | An atomic covering lattice... |
| cvlexch2 39836 | An atomic covering lattice... |
| cvlexchb1 39837 | An atomic covering lattice... |
| cvlexchb2 39838 | An atomic covering lattice... |
| cvlexch3 39839 | An atomic covering lattice... |
| cvlexch4N 39840 | An atomic covering lattice... |
| cvlatexchb1 39841 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 39842 | A version of ~ cvlexchb2 f... |
| cvlatexch1 39843 | Atom exchange property. (... |
| cvlatexch2 39844 | Atom exchange property. (... |
| cvlatexch3 39845 | Atom exchange property. (... |
| cvlcvr1 39846 | The covering property. Pr... |
| cvlcvrp 39847 | A Hilbert lattice satisfie... |
| cvlatcvr1 39848 | An atom is covered by its ... |
| cvlatcvr2 39849 | An atom is covered by its ... |
| cvlsupr2 39850 | Two equivalent ways of exp... |
| cvlsupr3 39851 | Two equivalent ways of exp... |
| cvlsupr4 39852 | Consequence of superpositi... |
| cvlsupr5 39853 | Consequence of superpositi... |
| cvlsupr6 39854 | Consequence of superpositi... |
| cvlsupr7 39855 | Consequence of superpositi... |
| cvlsupr8 39856 | Consequence of superpositi... |
| ishlat1 39859 | The predicate "is a Hilber... |
| ishlat2 39860 | The predicate "is a Hilber... |
| ishlat3N 39861 | The predicate "is a Hilber... |
| ishlatiN 39862 | Properties that determine ... |
| hlomcmcv 39863 | A Hilbert lattice is ortho... |
| hloml 39864 | A Hilbert lattice is ortho... |
| hlclat 39865 | A Hilbert lattice is compl... |
| hlcvl 39866 | A Hilbert lattice is an at... |
| hlatl 39867 | A Hilbert lattice is atomi... |
| hlol 39868 | A Hilbert lattice is an or... |
| hlop 39869 | A Hilbert lattice is an or... |
| hllat 39870 | A Hilbert lattice is a lat... |
| hllatd 39871 | Deduction form of ~ hllat ... |
| hlomcmat 39872 | A Hilbert lattice is ortho... |
| hlpos 39873 | A Hilbert lattice is a pos... |
| hlatjcl 39874 | Closure of join operation.... |
| hlatjcom 39875 | Commutatitivity of join op... |
| hlatjidm 39876 | Idempotence of join operat... |
| hlatjass 39877 | Lattice join is associativ... |
| hlatj12 39878 | Swap 1st and 2nd members o... |
| hlatj32 39879 | Swap 2nd and 3rd members o... |
| hlatjrot 39880 | Rotate lattice join of 3 c... |
| hlatj4 39881 | Rearrangement of lattice j... |
| hlatlej1 39882 | A join's first argument is... |
| hlatlej2 39883 | A join's second argument i... |
| glbconN 39884 | De Morgan's law for GLB an... |
| glbconxN 39885 | De Morgan's law for GLB an... |
| atnlej1 39886 | If an atom is not less tha... |
| atnlej2 39887 | If an atom is not less tha... |
| hlsuprexch 39888 | A Hilbert lattice has the ... |
| hlexch1 39889 | A Hilbert lattice has the ... |
| hlexch2 39890 | A Hilbert lattice has the ... |
| hlexchb1 39891 | A Hilbert lattice has the ... |
| hlexchb2 39892 | A Hilbert lattice has the ... |
| hlsupr 39893 | A Hilbert lattice has the ... |
| hlsupr2 39894 | A Hilbert lattice has the ... |
| hlhgt4 39895 | A Hilbert lattice has a he... |
| hlhgt2 39896 | A Hilbert lattice has a he... |
| hl0lt1N 39897 | Lattice 0 is less than lat... |
| hlexch3 39898 | A Hilbert lattice has the ... |
| hlexch4N 39899 | A Hilbert lattice has the ... |
| hlatexchb1 39900 | A version of ~ hlexchb1 fo... |
| hlatexchb2 39901 | A version of ~ hlexchb2 fo... |
| hlatexch1 39902 | Atom exchange property. (... |
| hlatexch2 39903 | Atom exchange property. (... |
| hlatmstcOLDN 39904 | An atomic, complete, ortho... |
| hlatle 39905 | The ordering of two Hilber... |
| hlateq 39906 | The equality of two Hilber... |
| hlrelat1 39907 | An atomistic lattice with ... |
| hlrelat5N 39908 | An atomistic lattice with ... |
| hlrelat 39909 | A Hilbert lattice is relat... |
| hlrelat2 39910 | A consequence of relative ... |
| exatleN 39911 | A condition for an atom to... |
| hl2at 39912 | A Hilbert lattice has at l... |
| atex 39913 | At least one atom exists. ... |
| intnatN 39914 | If the intersection with a... |
| 2llnne2N 39915 | Condition implying that tw... |
| 2llnneN 39916 | Condition implying that tw... |
| cvr1 39917 | A Hilbert lattice has the ... |
| cvr2N 39918 | Less-than and covers equiv... |
| hlrelat3 39919 | The Hilbert lattice is rel... |
| cvrval3 39920 | Binary relation expressing... |
| cvrval4N 39921 | Binary relation expressing... |
| cvrval5 39922 | Binary relation expressing... |
| cvrp 39923 | A Hilbert lattice satisfie... |
| atcvr1 39924 | An atom is covered by its ... |
| atcvr2 39925 | An atom is covered by its ... |
| cvrexchlem 39926 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 39927 | A Hilbert lattice satisfie... |
| cvratlem 39928 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 39929 | A nonzero Hilbert lattice ... |
| ltltncvr 39930 | A chained strong ordering ... |
| ltcvrntr 39931 | Non-transitive condition f... |
| cvrntr 39932 | The covers relation is not... |
| atcvr0eq 39933 | The covers relation is not... |
| lnnat 39934 | A line (the join of two di... |
| atcvrj0 39935 | Two atoms covering the zer... |
| cvrat2 39936 | A Hilbert lattice element ... |
| atcvrneN 39937 | Inequality derived from at... |
| atcvrj1 39938 | Condition for an atom to b... |
| atcvrj2b 39939 | Condition for an atom to b... |
| atcvrj2 39940 | Condition for an atom to b... |
| atleneN 39941 | Inequality derived from at... |
| atltcvr 39942 | An equivalence of less-tha... |
| atle 39943 | Any nonzero element has an... |
| atlt 39944 | Two atoms are unequal iff ... |
| atlelt 39945 | Transfer less-than relatio... |
| 2atlt 39946 | Given an atom less than an... |
| atexchcvrN 39947 | Atom exchange property. V... |
| atexchltN 39948 | Atom exchange property. V... |
| cvrat3 39949 | A condition implying that ... |
| cvrat4 39950 | A condition implying exist... |
| cvrat42 39951 | Commuted version of ~ cvra... |
| 2atjm 39952 | The meet of a line (expres... |
| atbtwn 39953 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 39954 | There exists a 3rd atom ` ... |
| atbtwnex 39955 | Given atoms ` P ` in ` X `... |
| 3noncolr2 39956 | Two ways to express 3 non-... |
| 3noncolr1N 39957 | Two ways to express 3 non-... |
| hlatcon3 39958 | Atom exchange combined wit... |
| hlatcon2 39959 | Atom exchange combined wit... |
| 4noncolr3 39960 | A way to express 4 non-col... |
| 4noncolr2 39961 | A way to express 4 non-col... |
| 4noncolr1 39962 | A way to express 4 non-col... |
| athgt 39963 | A Hilbert lattice, whose h... |
| 3dim0 39964 | There exists a 3-dimension... |
| 3dimlem1 39965 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 39966 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 39967 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 39968 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 39969 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 39970 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 39971 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 39972 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 39973 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 39974 | Construct a 3-dimensional ... |
| 3dim2 39975 | Construct 2 new layers on ... |
| 3dim3 39976 | Construct a new layer on t... |
| 2dim 39977 | Generate a height-3 elemen... |
| 1dimN 39978 | An atom is covered by a he... |
| 1cvrco 39979 | The orthocomplement of an ... |
| 1cvratex 39980 | There exists an atom less ... |
| 1cvratlt 39981 | An atom less than or equal... |
| 1cvrjat 39982 | An element covered by the ... |
| 1cvrat 39983 | Create an atom under an el... |
| ps-1 39984 | The join of two atoms ` R ... |
| ps-2 39985 | Lattice analogue for the p... |
| 2atjlej 39986 | Two atoms are different if... |
| hlatexch3N 39987 | Rearrange join of atoms in... |
| hlatexch4 39988 | Exchange 2 atoms. (Contri... |
| ps-2b 39989 | Variation of projective ge... |
| 3atlem1 39990 | Lemma for ~ 3at . (Contri... |
| 3atlem2 39991 | Lemma for ~ 3at . (Contri... |
| 3atlem3 39992 | Lemma for ~ 3at . (Contri... |
| 3atlem4 39993 | Lemma for ~ 3at . (Contri... |
| 3atlem5 39994 | Lemma for ~ 3at . (Contri... |
| 3atlem6 39995 | Lemma for ~ 3at . (Contri... |
| 3atlem7 39996 | Lemma for ~ 3at . (Contri... |
| 3at 39997 | Any three non-colinear ato... |
| llnset 40012 | The set of lattice lines i... |
| islln 40013 | The predicate "is a lattic... |
| islln4 40014 | The predicate "is a lattic... |
| llni 40015 | Condition implying a latti... |
| llnbase 40016 | A lattice line is a lattic... |
| islln3 40017 | The predicate "is a lattic... |
| islln2 40018 | The predicate "is a lattic... |
| llni2 40019 | The join of two different ... |
| llnnleat 40020 | An atom cannot majorize a ... |
| llnneat 40021 | A lattice line is not an a... |
| 2atneat 40022 | The join of two distinct a... |
| llnn0 40023 | A lattice line is nonzero.... |
| islln2a 40024 | The predicate "is a lattic... |
| llnle 40025 | Any element greater than 0... |
| atcvrlln2 40026 | An atom under a line is co... |
| atcvrlln 40027 | An element covering an ato... |
| llnexatN 40028 | Given an atom on a line, t... |
| llncmp 40029 | If two lattice lines are c... |
| llnnlt 40030 | Two lattice lines cannot s... |
| 2llnmat 40031 | Two intersecting lines int... |
| 2at0mat0 40032 | Special case of ~ 2atmat0 ... |
| 2atmat0 40033 | The meet of two unequal li... |
| 2atm 40034 | An atom majorized by two d... |
| ps-2c 40035 | Variation of projective ge... |
| lplnset 40036 | The set of lattice planes ... |
| islpln 40037 | The predicate "is a lattic... |
| islpln4 40038 | The predicate "is a lattic... |
| lplni 40039 | Condition implying a latti... |
| islpln3 40040 | The predicate "is a lattic... |
| lplnbase 40041 | A lattice plane is a latti... |
| islpln5 40042 | The predicate "is a lattic... |
| islpln2 40043 | The predicate "is a lattic... |
| lplni2 40044 | The join of 3 different at... |
| lvolex3N 40045 | There is an atom outside o... |
| llnmlplnN 40046 | The intersection of a line... |
| lplnle 40047 | Any element greater than 0... |
| lplnnle2at 40048 | A lattice line (or atom) c... |
| lplnnleat 40049 | A lattice plane cannot maj... |
| lplnnlelln 40050 | A lattice plane is not les... |
| 2atnelpln 40051 | The join of two atoms is n... |
| lplnneat 40052 | No lattice plane is an ato... |
| lplnnelln 40053 | No lattice plane is a latt... |
| lplnn0N 40054 | A lattice plane is nonzero... |
| islpln2a 40055 | The predicate "is a lattic... |
| islpln2ah 40056 | The predicate "is a lattic... |
| lplnriaN 40057 | Property of a lattice plan... |
| lplnribN 40058 | Property of a lattice plan... |
| lplnric 40059 | Property of a lattice plan... |
| lplnri1 40060 | Property of a lattice plan... |
| lplnri2N 40061 | Property of a lattice plan... |
| lplnri3N 40062 | Property of a lattice plan... |
| lplnllnneN 40063 | Two lattice lines defined ... |
| llncvrlpln2 40064 | A lattice line under a lat... |
| llncvrlpln 40065 | An element covering a latt... |
| 2lplnmN 40066 | If the join of two lattice... |
| 2llnmj 40067 | The meet of two lattice li... |
| 2atmat 40068 | The meet of two intersecti... |
| lplncmp 40069 | If two lattice planes are ... |
| lplnexatN 40070 | Given a lattice line on a ... |
| lplnexllnN 40071 | Given an atom on a lattice... |
| lplnnlt 40072 | Two lattice planes cannot ... |
| 2llnjaN 40073 | The join of two different ... |
| 2llnjN 40074 | The join of two different ... |
| 2llnm2N 40075 | The meet of two different ... |
| 2llnm3N 40076 | Two lattice lines in a lat... |
| 2llnm4 40077 | Two lattice lines that maj... |
| 2llnmeqat 40078 | An atom equals the interse... |
| lvolset 40079 | The set of 3-dim lattice v... |
| islvol 40080 | The predicate "is a 3-dim ... |
| islvol4 40081 | The predicate "is a 3-dim ... |
| lvoli 40082 | Condition implying a 3-dim... |
| islvol3 40083 | The predicate "is a 3-dim ... |
| lvoli3 40084 | Condition implying a 3-dim... |
| lvolbase 40085 | A 3-dim lattice volume is ... |
| islvol5 40086 | The predicate "is a 3-dim ... |
| islvol2 40087 | The predicate "is a 3-dim ... |
| lvoli2 40088 | The join of 4 different at... |
| lvolnle3at 40089 | A lattice plane (or lattic... |
| lvolnleat 40090 | An atom cannot majorize a ... |
| lvolnlelln 40091 | A lattice line cannot majo... |
| lvolnlelpln 40092 | A lattice plane cannot maj... |
| 3atnelvolN 40093 | The join of 3 atoms is not... |
| 2atnelvolN 40094 | The join of two atoms is n... |
| lvolneatN 40095 | No lattice volume is an at... |
| lvolnelln 40096 | No lattice volume is a lat... |
| lvolnelpln 40097 | No lattice volume is a lat... |
| lvoln0N 40098 | A lattice volume is nonzer... |
| islvol2aN 40099 | The predicate "is a lattic... |
| 4atlem0a 40100 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 40101 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 40102 | Lemma for ~ 4at . (Contri... |
| 4atlem3 40103 | Lemma for ~ 4at . Break i... |
| 4atlem3a 40104 | Lemma for ~ 4at . Break i... |
| 4atlem3b 40105 | Lemma for ~ 4at . Break i... |
| 4atlem4a 40106 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 40107 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 40108 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 40109 | Lemma for ~ 4at . Frequen... |
| 4atlem9 40110 | Lemma for ~ 4at . Substit... |
| 4atlem10a 40111 | Lemma for ~ 4at . Substit... |
| 4atlem10b 40112 | Lemma for ~ 4at . Substit... |
| 4atlem10 40113 | Lemma for ~ 4at . Combine... |
| 4atlem11a 40114 | Lemma for ~ 4at . Substit... |
| 4atlem11b 40115 | Lemma for ~ 4at . Substit... |
| 4atlem11 40116 | Lemma for ~ 4at . Combine... |
| 4atlem12a 40117 | Lemma for ~ 4at . Substit... |
| 4atlem12b 40118 | Lemma for ~ 4at . Substit... |
| 4atlem12 40119 | Lemma for ~ 4at . Combine... |
| 4at 40120 | Four atoms determine a lat... |
| 4at2 40121 | Four atoms determine a lat... |
| lplncvrlvol2 40122 | A lattice line under a lat... |
| lplncvrlvol 40123 | An element covering a latt... |
| lvolcmp 40124 | If two lattice planes are ... |
| lvolnltN 40125 | Two lattice volumes cannot... |
| 2lplnja 40126 | The join of two different ... |
| 2lplnj 40127 | The join of two different ... |
| 2lplnm2N 40128 | The meet of two different ... |
| 2lplnmj 40129 | The meet of two lattice pl... |
| dalemkehl 40130 | Lemma for ~ dath . Freque... |
| dalemkelat 40131 | Lemma for ~ dath . Freque... |
| dalemkeop 40132 | Lemma for ~ dath . Freque... |
| dalempea 40133 | Lemma for ~ dath . Freque... |
| dalemqea 40134 | Lemma for ~ dath . Freque... |
| dalemrea 40135 | Lemma for ~ dath . Freque... |
| dalemsea 40136 | Lemma for ~ dath . Freque... |
| dalemtea 40137 | Lemma for ~ dath . Freque... |
| dalemuea 40138 | Lemma for ~ dath . Freque... |
| dalemyeo 40139 | Lemma for ~ dath . Freque... |
| dalemzeo 40140 | Lemma for ~ dath . Freque... |
| dalemclpjs 40141 | Lemma for ~ dath . Freque... |
| dalemclqjt 40142 | Lemma for ~ dath . Freque... |
| dalemclrju 40143 | Lemma for ~ dath . Freque... |
| dalem-clpjq 40144 | Lemma for ~ dath . Freque... |
| dalemceb 40145 | Lemma for ~ dath . Freque... |
| dalempeb 40146 | Lemma for ~ dath . Freque... |
| dalemqeb 40147 | Lemma for ~ dath . Freque... |
| dalemreb 40148 | Lemma for ~ dath . Freque... |
| dalemseb 40149 | Lemma for ~ dath . Freque... |
| dalemteb 40150 | Lemma for ~ dath . Freque... |
| dalemueb 40151 | Lemma for ~ dath . Freque... |
| dalempjqeb 40152 | Lemma for ~ dath . Freque... |
| dalemsjteb 40153 | Lemma for ~ dath . Freque... |
| dalemtjueb 40154 | Lemma for ~ dath . Freque... |
| dalemqrprot 40155 | Lemma for ~ dath . Freque... |
| dalemyeb 40156 | Lemma for ~ dath . Freque... |
| dalemcnes 40157 | Lemma for ~ dath . Freque... |
| dalempnes 40158 | Lemma for ~ dath . Freque... |
| dalemqnet 40159 | Lemma for ~ dath . Freque... |
| dalempjsen 40160 | Lemma for ~ dath . Freque... |
| dalemply 40161 | Lemma for ~ dath . Freque... |
| dalemsly 40162 | Lemma for ~ dath . Freque... |
| dalemswapyz 40163 | Lemma for ~ dath . Swap t... |
| dalemrot 40164 | Lemma for ~ dath . Rotate... |
| dalemrotyz 40165 | Lemma for ~ dath . Rotate... |
| dalem1 40166 | Lemma for ~ dath . Show t... |
| dalemcea 40167 | Lemma for ~ dath . Freque... |
| dalem2 40168 | Lemma for ~ dath . Show t... |
| dalemdea 40169 | Lemma for ~ dath . Freque... |
| dalemeea 40170 | Lemma for ~ dath . Freque... |
| dalem3 40171 | Lemma for ~ dalemdnee . (... |
| dalem4 40172 | Lemma for ~ dalemdnee . (... |
| dalemdnee 40173 | Lemma for ~ dath . Axis o... |
| dalem5 40174 | Lemma for ~ dath . Atom `... |
| dalem6 40175 | Lemma for ~ dath . Analog... |
| dalem7 40176 | Lemma for ~ dath . Analog... |
| dalem8 40177 | Lemma for ~ dath . Plane ... |
| dalem-cly 40178 | Lemma for ~ dalem9 . Cent... |
| dalem9 40179 | Lemma for ~ dath . Since ... |
| dalem10 40180 | Lemma for ~ dath . Atom `... |
| dalem11 40181 | Lemma for ~ dath . Analog... |
| dalem12 40182 | Lemma for ~ dath . Analog... |
| dalem13 40183 | Lemma for ~ dalem14 . (Co... |
| dalem14 40184 | Lemma for ~ dath . Planes... |
| dalem15 40185 | Lemma for ~ dath . The ax... |
| dalem16 40186 | Lemma for ~ dath . The at... |
| dalem17 40187 | Lemma for ~ dath . When p... |
| dalem18 40188 | Lemma for ~ dath . Show t... |
| dalem19 40189 | Lemma for ~ dath . Show t... |
| dalemccea 40190 | Lemma for ~ dath . Freque... |
| dalemddea 40191 | Lemma for ~ dath . Freque... |
| dalem-ccly 40192 | Lemma for ~ dath . Freque... |
| dalem-ddly 40193 | Lemma for ~ dath . Freque... |
| dalemccnedd 40194 | Lemma for ~ dath . Freque... |
| dalemclccjdd 40195 | Lemma for ~ dath . Freque... |
| dalemcceb 40196 | Lemma for ~ dath . Freque... |
| dalemswapyzps 40197 | Lemma for ~ dath . Swap t... |
| dalemrotps 40198 | Lemma for ~ dath . Rotate... |
| dalemcjden 40199 | Lemma for ~ dath . Show t... |
| dalem20 40200 | Lemma for ~ dath . Show t... |
| dalem21 40201 | Lemma for ~ dath . Show t... |
| dalem22 40202 | Lemma for ~ dath . Show t... |
| dalem23 40203 | Lemma for ~ dath . Show t... |
| dalem24 40204 | Lemma for ~ dath . Show t... |
| dalem25 40205 | Lemma for ~ dath . Show t... |
| dalem27 40206 | Lemma for ~ dath . Show t... |
| dalem28 40207 | Lemma for ~ dath . Lemma ... |
| dalem29 40208 | Lemma for ~ dath . Analog... |
| dalem30 40209 | Lemma for ~ dath . Analog... |
| dalem31N 40210 | Lemma for ~ dath . Analog... |
| dalem32 40211 | Lemma for ~ dath . Analog... |
| dalem33 40212 | Lemma for ~ dath . Analog... |
| dalem34 40213 | Lemma for ~ dath . Analog... |
| dalem35 40214 | Lemma for ~ dath . Analog... |
| dalem36 40215 | Lemma for ~ dath . Analog... |
| dalem37 40216 | Lemma for ~ dath . Analog... |
| dalem38 40217 | Lemma for ~ dath . Plane ... |
| dalem39 40218 | Lemma for ~ dath . Auxili... |
| dalem40 40219 | Lemma for ~ dath . Analog... |
| dalem41 40220 | Lemma for ~ dath . (Contr... |
| dalem42 40221 | Lemma for ~ dath . Auxili... |
| dalem43 40222 | Lemma for ~ dath . Planes... |
| dalem44 40223 | Lemma for ~ dath . Dummy ... |
| dalem45 40224 | Lemma for ~ dath . Dummy ... |
| dalem46 40225 | Lemma for ~ dath . Analog... |
| dalem47 40226 | Lemma for ~ dath . Analog... |
| dalem48 40227 | Lemma for ~ dath . Analog... |
| dalem49 40228 | Lemma for ~ dath . Analog... |
| dalem50 40229 | Lemma for ~ dath . Analog... |
| dalem51 40230 | Lemma for ~ dath . Constr... |
| dalem52 40231 | Lemma for ~ dath . Lines ... |
| dalem53 40232 | Lemma for ~ dath . The au... |
| dalem54 40233 | Lemma for ~ dath . Line `... |
| dalem55 40234 | Lemma for ~ dath . Lines ... |
| dalem56 40235 | Lemma for ~ dath . Analog... |
| dalem57 40236 | Lemma for ~ dath . Axis o... |
| dalem58 40237 | Lemma for ~ dath . Analog... |
| dalem59 40238 | Lemma for ~ dath . Analog... |
| dalem60 40239 | Lemma for ~ dath . ` B ` i... |
| dalem61 40240 | Lemma for ~ dath . Show t... |
| dalem62 40241 | Lemma for ~ dath . Elimin... |
| dalem63 40242 | Lemma for ~ dath . Combin... |
| dath 40243 | Desargues's theorem of pro... |
| dath2 40244 | Version of Desargues's the... |
| lineset 40245 | The set of lines in a Hilb... |
| isline 40246 | The predicate "is a line".... |
| islinei 40247 | Condition implying "is a l... |
| pointsetN 40248 | The set of points in a Hil... |
| ispointN 40249 | The predicate "is a point"... |
| atpointN 40250 | The singleton of an atom i... |
| psubspset 40251 | The set of projective subs... |
| ispsubsp 40252 | The predicate "is a projec... |
| ispsubsp2 40253 | The predicate "is a projec... |
| psubspi 40254 | Property of a projective s... |
| psubspi2N 40255 | Property of a projective s... |
| 0psubN 40256 | The empty set is a project... |
| snatpsubN 40257 | The singleton of an atom i... |
| pointpsubN 40258 | A point (singleton of an a... |
| linepsubN 40259 | A line is a projective sub... |
| atpsubN 40260 | The set of all atoms is a ... |
| psubssat 40261 | A projective subspace cons... |
| psubatN 40262 | A member of a projective s... |
| pmapfval 40263 | The projective map of a Hi... |
| pmapval 40264 | Value of the projective ma... |
| elpmap 40265 | Member of a projective map... |
| pmapssat 40266 | The projective map of a Hi... |
| pmapssbaN 40267 | A weakening of ~ pmapssat ... |
| pmaple 40268 | The projective map of a Hi... |
| pmap11 40269 | The projective map of a Hi... |
| pmapat 40270 | The projective map of an a... |
| elpmapat 40271 | Member of the projective m... |
| pmap0 40272 | Value of the projective ma... |
| pmapeq0 40273 | A projective map value is ... |
| pmap1N 40274 | Value of the projective ma... |
| pmapsub 40275 | The projective map of a Hi... |
| pmapglbx 40276 | The projective map of the ... |
| pmapglb 40277 | The projective map of the ... |
| pmapglb2N 40278 | The projective map of the ... |
| pmapglb2xN 40279 | The projective map of the ... |
| pmapmeet 40280 | The projective map of a me... |
| isline2 40281 | Definition of line in term... |
| linepmap 40282 | A line described with a pr... |
| isline3 40283 | Definition of line in term... |
| isline4N 40284 | Definition of line in term... |
| lneq2at 40285 | A line equals the join of ... |
| lnatexN 40286 | There is an atom in a line... |
| lnjatN 40287 | Given an atom in a line, t... |
| lncvrelatN 40288 | A lattice element covered ... |
| lncvrat 40289 | A line covers the atoms it... |
| lncmp 40290 | If two lines are comparabl... |
| 2lnat 40291 | Two intersecting lines int... |
| 2atm2atN 40292 | Two joins with a common at... |
| 2llnma1b 40293 | Generalization of ~ 2llnma... |
| 2llnma1 40294 | Two different intersecting... |
| 2llnma3r 40295 | Two different intersecting... |
| 2llnma2 40296 | Two different intersecting... |
| 2llnma2rN 40297 | Two different intersecting... |
| cdlema1N 40298 | A condition for required f... |
| cdlema2N 40299 | A condition for required f... |
| cdlemblem 40300 | Lemma for ~ cdlemb . (Con... |
| cdlemb 40301 | Given two atoms not less t... |
| paddfval 40304 | Projective subspace sum op... |
| paddval 40305 | Projective subspace sum op... |
| elpadd 40306 | Member of a projective sub... |
| elpaddn0 40307 | Member of projective subsp... |
| paddvaln0N 40308 | Projective subspace sum op... |
| elpaddri 40309 | Condition implying members... |
| elpaddatriN 40310 | Condition implying members... |
| elpaddat 40311 | Membership in a projective... |
| elpaddatiN 40312 | Consequence of membership ... |
| elpadd2at 40313 | Membership in a projective... |
| elpadd2at2 40314 | Membership in a projective... |
| paddunssN 40315 | Projective subspace sum in... |
| elpadd0 40316 | Member of projective subsp... |
| paddval0 40317 | Projective subspace sum wi... |
| padd01 40318 | Projective subspace sum wi... |
| padd02 40319 | Projective subspace sum wi... |
| paddcom 40320 | Projective subspace sum co... |
| paddssat 40321 | A projective subspace sum ... |
| sspadd1 40322 | A projective subspace sum ... |
| sspadd2 40323 | A projective subspace sum ... |
| paddss1 40324 | Subset law for projective ... |
| paddss2 40325 | Subset law for projective ... |
| paddss12 40326 | Subset law for projective ... |
| paddasslem1 40327 | Lemma for ~ paddass . (Co... |
| paddasslem2 40328 | Lemma for ~ paddass . (Co... |
| paddasslem3 40329 | Lemma for ~ paddass . Res... |
| paddasslem4 40330 | Lemma for ~ paddass . Com... |
| paddasslem5 40331 | Lemma for ~ paddass . Sho... |
| paddasslem6 40332 | Lemma for ~ paddass . (Co... |
| paddasslem7 40333 | Lemma for ~ paddass . Com... |
| paddasslem8 40334 | Lemma for ~ paddass . (Co... |
| paddasslem9 40335 | Lemma for ~ paddass . Com... |
| paddasslem10 40336 | Lemma for ~ paddass . Use... |
| paddasslem11 40337 | Lemma for ~ paddass . The... |
| paddasslem12 40338 | Lemma for ~ paddass . The... |
| paddasslem13 40339 | Lemma for ~ paddass . The... |
| paddasslem14 40340 | Lemma for ~ paddass . Rem... |
| paddasslem15 40341 | Lemma for ~ paddass . Use... |
| paddasslem16 40342 | Lemma for ~ paddass . Use... |
| paddasslem17 40343 | Lemma for ~ paddass . The... |
| paddasslem18 40344 | Lemma for ~ paddass . Com... |
| paddass 40345 | Projective subspace sum is... |
| padd12N 40346 | Commutative/associative la... |
| padd4N 40347 | Rearrangement of 4 terms i... |
| paddidm 40348 | Projective subspace sum is... |
| paddclN 40349 | The projective sum of two ... |
| paddssw1 40350 | Subset law for projective ... |
| paddssw2 40351 | Subset law for projective ... |
| paddss 40352 | Subset law for projective ... |
| pmodlem1 40353 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 40354 | Lemma for ~ pmod1i . (Con... |
| pmod1i 40355 | The modular law holds in a... |
| pmod2iN 40356 | Dual of the modular law. ... |
| pmodN 40357 | The modular law for projec... |
| pmodl42N 40358 | Lemma derived from modular... |
| pmapjoin 40359 | The projective map of the ... |
| pmapjat1 40360 | The projective map of the ... |
| pmapjat2 40361 | The projective map of the ... |
| pmapjlln1 40362 | The projective map of the ... |
| hlmod1i 40363 | A version of the modular l... |
| atmod1i1 40364 | Version of modular law ~ p... |
| atmod1i1m 40365 | Version of modular law ~ p... |
| atmod1i2 40366 | Version of modular law ~ p... |
| llnmod1i2 40367 | Version of modular law ~ p... |
| atmod2i1 40368 | Version of modular law ~ p... |
| atmod2i2 40369 | Version of modular law ~ p... |
| llnmod2i2 40370 | Version of modular law ~ p... |
| atmod3i1 40371 | Version of modular law tha... |
| atmod3i2 40372 | Version of modular law tha... |
| atmod4i1 40373 | Version of modular law tha... |
| atmod4i2 40374 | Version of modular law tha... |
| llnexchb2lem 40375 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 40376 | Line exchange property (co... |
| llnexch2N 40377 | Line exchange property (co... |
| dalawlem1 40378 | Lemma for ~ dalaw . Speci... |
| dalawlem2 40379 | Lemma for ~ dalaw . Utili... |
| dalawlem3 40380 | Lemma for ~ dalaw . First... |
| dalawlem4 40381 | Lemma for ~ dalaw . Secon... |
| dalawlem5 40382 | Lemma for ~ dalaw . Speci... |
| dalawlem6 40383 | Lemma for ~ dalaw . First... |
| dalawlem7 40384 | Lemma for ~ dalaw . Secon... |
| dalawlem8 40385 | Lemma for ~ dalaw . Speci... |
| dalawlem9 40386 | Lemma for ~ dalaw . Speci... |
| dalawlem10 40387 | Lemma for ~ dalaw . Combi... |
| dalawlem11 40388 | Lemma for ~ dalaw . First... |
| dalawlem12 40389 | Lemma for ~ dalaw . Secon... |
| dalawlem13 40390 | Lemma for ~ dalaw . Speci... |
| dalawlem14 40391 | Lemma for ~ dalaw . Combi... |
| dalawlem15 40392 | Lemma for ~ dalaw . Swap ... |
| dalaw 40393 | Desargues's law, derived f... |
| pclfvalN 40396 | The projective subspace cl... |
| pclvalN 40397 | Value of the projective su... |
| pclclN 40398 | Closure of the projective ... |
| elpclN 40399 | Membership in the projecti... |
| elpcliN 40400 | Implication of membership ... |
| pclssN 40401 | Ordering is preserved by s... |
| pclssidN 40402 | A set of atoms is included... |
| pclidN 40403 | The projective subspace cl... |
| pclbtwnN 40404 | A projective subspace sand... |
| pclunN 40405 | The projective subspace cl... |
| pclun2N 40406 | The projective subspace cl... |
| pclfinN 40407 | The projective subspace cl... |
| pclcmpatN 40408 | The set of projective subs... |
| polfvalN 40411 | The projective subspace po... |
| polvalN 40412 | Value of the projective su... |
| polval2N 40413 | Alternate expression for v... |
| polsubN 40414 | The polarity of a set of a... |
| polssatN 40415 | The polarity of a set of a... |
| pol0N 40416 | The polarity of the empty ... |
| pol1N 40417 | The polarity of the whole ... |
| 2pol0N 40418 | The closed subspace closur... |
| polpmapN 40419 | The polarity of a projecti... |
| 2polpmapN 40420 | Double polarity of a proje... |
| 2polvalN 40421 | Value of double polarity. ... |
| 2polssN 40422 | A set of atoms is a subset... |
| 3polN 40423 | Triple polarity cancels to... |
| polcon3N 40424 | Contraposition law for pol... |
| 2polcon4bN 40425 | Contraposition law for pol... |
| polcon2N 40426 | Contraposition law for pol... |
| polcon2bN 40427 | Contraposition law for pol... |
| pclss2polN 40428 | The projective subspace cl... |
| pcl0N 40429 | The projective subspace cl... |
| pcl0bN 40430 | The projective subspace cl... |
| pmaplubN 40431 | The LUB of a projective ma... |
| sspmaplubN 40432 | A set of atoms is a subset... |
| 2pmaplubN 40433 | Double projective map of a... |
| paddunN 40434 | The closure of the project... |
| poldmj1N 40435 | De Morgan's law for polari... |
| pmapj2N 40436 | The projective map of the ... |
| pmapocjN 40437 | The projective map of the ... |
| polatN 40438 | The polarity of the single... |
| 2polatN 40439 | Double polarity of the sin... |
| pnonsingN 40440 | The intersection of a set ... |
| psubclsetN 40443 | The set of closed projecti... |
| ispsubclN 40444 | The predicate "is a closed... |
| psubcliN 40445 | Property of a closed proje... |
| psubcli2N 40446 | Property of a closed proje... |
| psubclsubN 40447 | A closed projective subspa... |
| psubclssatN 40448 | A closed projective subspa... |
| pmapidclN 40449 | Projective map of the LUB ... |
| 0psubclN 40450 | The empty set is a closed ... |
| 1psubclN 40451 | The set of all atoms is a ... |
| atpsubclN 40452 | A point (singleton of an a... |
| pmapsubclN 40453 | A projective map value is ... |
| ispsubcl2N 40454 | Alternate predicate for "i... |
| psubclinN 40455 | The intersection of two cl... |
| paddatclN 40456 | The projective sum of a cl... |
| pclfinclN 40457 | The projective subspace cl... |
| linepsubclN 40458 | A line is a closed project... |
| polsubclN 40459 | A polarity is a closed pro... |
| poml4N 40460 | Orthomodular law for proje... |
| poml5N 40461 | Orthomodular law for proje... |
| poml6N 40462 | Orthomodular law for proje... |
| osumcllem1N 40463 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 40464 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 40465 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 40466 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 40467 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 40468 | Lemma for ~ osumclN . Use... |
| osumcllem7N 40469 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 40470 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 40471 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 40472 | Lemma for ~ osumclN . Con... |
| osumcllem11N 40473 | Lemma for ~ osumclN . (Co... |
| osumclN 40474 | Closure of orthogonal sum.... |
| pmapojoinN 40475 | For orthogonal elements, p... |
| pexmidN 40476 | Excluded middle law for cl... |
| pexmidlem1N 40477 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 40478 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 40479 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 40480 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 40481 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 40482 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 40483 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 40484 | Lemma for ~ pexmidN . The... |
| pexmidALTN 40485 | Excluded middle law for cl... |
| pl42lem1N 40486 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 40487 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 40488 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 40489 | Lemma for ~ pl42N . (Cont... |
| pl42N 40490 | Law holding in a Hilbert l... |
| watfvalN 40499 | The W atoms function. (Co... |
| watvalN 40500 | Value of the W atoms funct... |
| iswatN 40501 | The predicate "is a W atom... |
| lhpset 40502 | The set of co-atoms (latti... |
| islhp 40503 | The predicate "is a co-ato... |
| islhp2 40504 | The predicate "is a co-ato... |
| lhpbase 40505 | A co-atom is a member of t... |
| lhp1cvr 40506 | The lattice unity covers a... |
| lhplt 40507 | An atom under a co-atom is... |
| lhp2lt 40508 | The join of two atoms unde... |
| lhpexlt 40509 | There exists an atom less ... |
| lhp0lt 40510 | A co-atom is greater than ... |
| lhpn0 40511 | A co-atom is nonzero. TOD... |
| lhpexle 40512 | There exists an atom under... |
| lhpexnle 40513 | There exists an atom not u... |
| lhpexle1lem 40514 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 40515 | There exists an atom under... |
| lhpexle2lem 40516 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 40517 | There exists atom under a ... |
| lhpexle3lem 40518 | There exists atom under a ... |
| lhpexle3 40519 | There exists atom under a ... |
| lhpex2leN 40520 | There exist at least two d... |
| lhpoc 40521 | The orthocomplement of a c... |
| lhpoc2N 40522 | The orthocomplement of an ... |
| lhpocnle 40523 | The orthocomplement of a c... |
| lhpocat 40524 | The orthocomplement of a c... |
| lhpocnel 40525 | The orthocomplement of a c... |
| lhpocnel2 40526 | The orthocomplement of a c... |
| lhpjat1 40527 | The join of a co-atom (hyp... |
| lhpjat2 40528 | The join of a co-atom (hyp... |
| lhpj1 40529 | The join of a co-atom (hyp... |
| lhpmcvr 40530 | The meet of a lattice hype... |
| lhpmcvr2 40531 | Alternate way to express t... |
| lhpmcvr3 40532 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 40533 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 40534 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 40535 | Specialization of ~ lhpmcv... |
| lhpm0atN 40536 | If the meet of a lattice h... |
| lhpmat 40537 | An element covered by the ... |
| lhpmatb 40538 | An element covered by the ... |
| lhp2at0 40539 | Join and meet with differe... |
| lhp2atnle 40540 | Inequality for 2 different... |
| lhp2atne 40541 | Inequality for joins with ... |
| lhp2at0nle 40542 | Inequality for 2 different... |
| lhp2at0ne 40543 | Inequality for joins with ... |
| lhpelim 40544 | Eliminate an atom not unde... |
| lhpmod2i2 40545 | Modular law for hyperplane... |
| lhpmod6i1 40546 | Modular law for hyperplane... |
| lhprelat3N 40547 | The Hilbert lattice is rel... |
| cdlemb2 40548 | Given two atoms not under ... |
| lhple 40549 | Property of a lattice elem... |
| lhpat 40550 | Create an atom under a co-... |
| lhpat4N 40551 | Property of an atom under ... |
| lhpat2 40552 | Create an atom under a co-... |
| lhpat3 40553 | There is only one atom und... |
| 4atexlemk 40554 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 40555 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 40556 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 40557 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 40558 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 40559 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 40560 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 40561 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 40562 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 40563 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 40564 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 40565 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 40566 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 40567 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 40568 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 40569 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 40570 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 40571 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 40572 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 40573 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 40574 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 40575 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 40576 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 40577 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 40578 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 40579 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 40580 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 40581 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 40582 | Whenever there are at leas... |
| 4atex 40583 | Whenever there are at leas... |
| 4atex2 40584 | More general version of ~ ... |
| 4atex2-0aOLDN 40585 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 40586 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 40587 | Same as ~ 4atex2 except th... |
| 4atex3 40588 | More general version of ~ ... |
| lautset 40589 | The set of lattice automor... |
| islaut 40590 | The predicate "is a lattic... |
| lautle 40591 | Less-than or equal propert... |
| laut1o 40592 | A lattice automorphism is ... |
| laut11 40593 | One-to-one property of a l... |
| lautcl 40594 | A lattice automorphism val... |
| lautcnvclN 40595 | Reverse closure of a latti... |
| lautcnvle 40596 | Less-than or equal propert... |
| lautcnv 40597 | The converse of a lattice ... |
| lautlt 40598 | Less-than property of a la... |
| lautcvr 40599 | Covering property of a lat... |
| lautj 40600 | Meet property of a lattice... |
| lautm 40601 | Meet property of a lattice... |
| lauteq 40602 | A lattice automorphism arg... |
| idlaut 40603 | The identity function is a... |
| lautco 40604 | The composition of two lat... |
| pautsetN 40605 | The set of projective auto... |
| ispautN 40606 | The predicate "is a projec... |
| ldilfset 40615 | The mapping from fiducial ... |
| ldilset 40616 | The set of lattice dilatio... |
| isldil 40617 | The predicate "is a lattic... |
| ldillaut 40618 | A lattice dilation is an a... |
| ldil1o 40619 | A lattice dilation is a on... |
| ldilval 40620 | Value of a lattice dilatio... |
| idldil 40621 | The identity function is a... |
| ldilcnv 40622 | The converse of a lattice ... |
| ldilco 40623 | The composition of two lat... |
| ltrnfset 40624 | The set of all lattice tra... |
| ltrnset 40625 | The set of lattice transla... |
| isltrn 40626 | The predicate "is a lattic... |
| isltrn2N 40627 | The predicate "is a lattic... |
| ltrnu 40628 | Uniqueness property of a l... |
| ltrnldil 40629 | A lattice translation is a... |
| ltrnlaut 40630 | A lattice translation is a... |
| ltrn1o 40631 | A lattice translation is a... |
| ltrncl 40632 | Closure of a lattice trans... |
| ltrn11 40633 | One-to-one property of a l... |
| ltrncnvnid 40634 | If a translation is differ... |
| ltrncoidN 40635 | Two translations are equal... |
| ltrnle 40636 | Less-than or equal propert... |
| ltrncnvleN 40637 | Less-than or equal propert... |
| ltrnm 40638 | Lattice translation of a m... |
| ltrnj 40639 | Lattice translation of a m... |
| ltrncvr 40640 | Covering property of a lat... |
| ltrnval1 40641 | Value of a lattice transla... |
| ltrnid 40642 | A lattice translation is t... |
| ltrnnid 40643 | If a lattice translation i... |
| ltrnatb 40644 | The lattice translation of... |
| ltrncnvatb 40645 | The converse of the lattic... |
| ltrnel 40646 | The lattice translation of... |
| ltrnat 40647 | The lattice translation of... |
| ltrncnvat 40648 | The converse of the lattic... |
| ltrncnvel 40649 | The converse of the lattic... |
| ltrncoelN 40650 | Composition of lattice tra... |
| ltrncoat 40651 | Composition of lattice tra... |
| ltrncoval 40652 | Two ways to express value ... |
| ltrncnv 40653 | The converse of a lattice ... |
| ltrn11at 40654 | Frequently used one-to-one... |
| ltrneq2 40655 | The equality of two transl... |
| ltrneq 40656 | The equality of two transl... |
| idltrn 40657 | The identity function is a... |
| ltrnmw 40658 | Property of lattice transl... |
| dilfsetN 40659 | The mapping from fiducial ... |
| dilsetN 40660 | The set of dilations for a... |
| isdilN 40661 | The predicate "is a dilati... |
| trnfsetN 40662 | The mapping from fiducial ... |
| trnsetN 40663 | The set of translations fo... |
| istrnN 40664 | The predicate "is a transl... |
| trlfset 40667 | The set of all traces of l... |
| trlset 40668 | The set of traces of latti... |
| trlval 40669 | The value of the trace of ... |
| trlval2 40670 | The value of the trace of ... |
| trlcl 40671 | Closure of the trace of a ... |
| trlcnv 40672 | The trace of the converse ... |
| trljat1 40673 | The value of a translation... |
| trljat2 40674 | The value of a translation... |
| trljat3 40675 | The value of a translation... |
| trlat 40676 | If an atom differs from it... |
| trl0 40677 | If an atom not under the f... |
| trlator0 40678 | The trace of a lattice tra... |
| trlatn0 40679 | The trace of a lattice tra... |
| trlnidat 40680 | The trace of a lattice tra... |
| ltrnnidn 40681 | If a lattice translation i... |
| ltrnideq 40682 | Property of the identity l... |
| trlid0 40683 | The trace of the identity ... |
| trlnidatb 40684 | A lattice translation is n... |
| trlid0b 40685 | A lattice translation is t... |
| trlnid 40686 | Different translations wit... |
| ltrn2ateq 40687 | Property of the equality o... |
| ltrnateq 40688 | If any atom (under ` W ` )... |
| ltrnatneq 40689 | If any atom (under ` W ` )... |
| ltrnatlw 40690 | If the value of an atom eq... |
| trlle 40691 | The trace of a lattice tra... |
| trlne 40692 | The trace of a lattice tra... |
| trlnle 40693 | The atom not under the fid... |
| trlval3 40694 | The value of the trace of ... |
| trlval4 40695 | The value of the trace of ... |
| trlval5 40696 | The value of the trace of ... |
| arglem1N 40697 | Lemma for Desargues's law.... |
| cdlemc1 40698 | Part of proof of Lemma C i... |
| cdlemc2 40699 | Part of proof of Lemma C i... |
| cdlemc3 40700 | Part of proof of Lemma C i... |
| cdlemc4 40701 | Part of proof of Lemma C i... |
| cdlemc5 40702 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 40703 | Lemma for ~ cdlemc . (Con... |
| cdlemc 40704 | Lemma C in [Crawley] p. 11... |
| cdlemd1 40705 | Part of proof of Lemma D i... |
| cdlemd2 40706 | Part of proof of Lemma D i... |
| cdlemd3 40707 | Part of proof of Lemma D i... |
| cdlemd4 40708 | Part of proof of Lemma D i... |
| cdlemd5 40709 | Part of proof of Lemma D i... |
| cdlemd6 40710 | Part of proof of Lemma D i... |
| cdlemd7 40711 | Part of proof of Lemma D i... |
| cdlemd8 40712 | Part of proof of Lemma D i... |
| cdlemd9 40713 | Part of proof of Lemma D i... |
| cdlemd 40714 | If two translations agree ... |
| ltrneq3 40715 | Two translations agree at ... |
| cdleme00a 40716 | Part of proof of Lemma E i... |
| cdleme0aa 40717 | Part of proof of Lemma E i... |
| cdleme0a 40718 | Part of proof of Lemma E i... |
| cdleme0b 40719 | Part of proof of Lemma E i... |
| cdleme0c 40720 | Part of proof of Lemma E i... |
| cdleme0cp 40721 | Part of proof of Lemma E i... |
| cdleme0cq 40722 | Part of proof of Lemma E i... |
| cdleme0dN 40723 | Part of proof of Lemma E i... |
| cdleme0e 40724 | Part of proof of Lemma E i... |
| cdleme0fN 40725 | Part of proof of Lemma E i... |
| cdleme0gN 40726 | Part of proof of Lemma E i... |
| cdlemeulpq 40727 | Part of proof of Lemma E i... |
| cdleme01N 40728 | Part of proof of Lemma E i... |
| cdleme02N 40729 | Part of proof of Lemma E i... |
| cdleme0ex1N 40730 | Part of proof of Lemma E i... |
| cdleme0ex2N 40731 | Part of proof of Lemma E i... |
| cdleme0moN 40732 | Part of proof of Lemma E i... |
| cdleme1b 40733 | Part of proof of Lemma E i... |
| cdleme1 40734 | Part of proof of Lemma E i... |
| cdleme2 40735 | Part of proof of Lemma E i... |
| cdleme3b 40736 | Part of proof of Lemma E i... |
| cdleme3c 40737 | Part of proof of Lemma E i... |
| cdleme3d 40738 | Part of proof of Lemma E i... |
| cdleme3e 40739 | Part of proof of Lemma E i... |
| cdleme3fN 40740 | Part of proof of Lemma E i... |
| cdleme3g 40741 | Part of proof of Lemma E i... |
| cdleme3h 40742 | Part of proof of Lemma E i... |
| cdleme3fa 40743 | Part of proof of Lemma E i... |
| cdleme3 40744 | Part of proof of Lemma E i... |
| cdleme4 40745 | Part of proof of Lemma E i... |
| cdleme4a 40746 | Part of proof of Lemma E i... |
| cdleme5 40747 | Part of proof of Lemma E i... |
| cdleme6 40748 | Part of proof of Lemma E i... |
| cdleme7aa 40749 | Part of proof of Lemma E i... |
| cdleme7a 40750 | Part of proof of Lemma E i... |
| cdleme7b 40751 | Part of proof of Lemma E i... |
| cdleme7c 40752 | Part of proof of Lemma E i... |
| cdleme7d 40753 | Part of proof of Lemma E i... |
| cdleme7e 40754 | Part of proof of Lemma E i... |
| cdleme7ga 40755 | Part of proof of Lemma E i... |
| cdleme7 40756 | Part of proof of Lemma E i... |
| cdleme8 40757 | Part of proof of Lemma E i... |
| cdleme9a 40758 | Part of proof of Lemma E i... |
| cdleme9b 40759 | Utility lemma for Lemma E ... |
| cdleme9 40760 | Part of proof of Lemma E i... |
| cdleme10 40761 | Part of proof of Lemma E i... |
| cdleme8tN 40762 | Part of proof of Lemma E i... |
| cdleme9taN 40763 | Part of proof of Lemma E i... |
| cdleme9tN 40764 | Part of proof of Lemma E i... |
| cdleme10tN 40765 | Part of proof of Lemma E i... |
| cdleme16aN 40766 | Part of proof of Lemma E i... |
| cdleme11a 40767 | Part of proof of Lemma E i... |
| cdleme11c 40768 | Part of proof of Lemma E i... |
| cdleme11dN 40769 | Part of proof of Lemma E i... |
| cdleme11e 40770 | Part of proof of Lemma E i... |
| cdleme11fN 40771 | Part of proof of Lemma E i... |
| cdleme11g 40772 | Part of proof of Lemma E i... |
| cdleme11h 40773 | Part of proof of Lemma E i... |
| cdleme11j 40774 | Part of proof of Lemma E i... |
| cdleme11k 40775 | Part of proof of Lemma E i... |
| cdleme11l 40776 | Part of proof of Lemma E i... |
| cdleme11 40777 | Part of proof of Lemma E i... |
| cdleme12 40778 | Part of proof of Lemma E i... |
| cdleme13 40779 | Part of proof of Lemma E i... |
| cdleme14 40780 | Part of proof of Lemma E i... |
| cdleme15a 40781 | Part of proof of Lemma E i... |
| cdleme15b 40782 | Part of proof of Lemma E i... |
| cdleme15c 40783 | Part of proof of Lemma E i... |
| cdleme15d 40784 | Part of proof of Lemma E i... |
| cdleme15 40785 | Part of proof of Lemma E i... |
| cdleme16b 40786 | Part of proof of Lemma E i... |
| cdleme16c 40787 | Part of proof of Lemma E i... |
| cdleme16d 40788 | Part of proof of Lemma E i... |
| cdleme16e 40789 | Part of proof of Lemma E i... |
| cdleme16f 40790 | Part of proof of Lemma E i... |
| cdleme16g 40791 | Part of proof of Lemma E i... |
| cdleme16 40792 | Part of proof of Lemma E i... |
| cdleme17a 40793 | Part of proof of Lemma E i... |
| cdleme17b 40794 | Lemma leading to ~ cdleme1... |
| cdleme17c 40795 | Part of proof of Lemma E i... |
| cdleme17d1 40796 | Part of proof of Lemma E i... |
| cdleme0nex 40797 | Part of proof of Lemma E i... |
| cdleme18a 40798 | Part of proof of Lemma E i... |
| cdleme18b 40799 | Part of proof of Lemma E i... |
| cdleme18c 40800 | Part of proof of Lemma E i... |
| cdleme22gb 40801 | Utility lemma for Lemma E ... |
| cdleme18d 40802 | Part of proof of Lemma E i... |
| cdlemesner 40803 | Part of proof of Lemma E i... |
| cdlemedb 40804 | Part of proof of Lemma E i... |
| cdlemeda 40805 | Part of proof of Lemma E i... |
| cdlemednpq 40806 | Part of proof of Lemma E i... |
| cdlemednuN 40807 | Part of proof of Lemma E i... |
| cdleme20zN 40808 | Part of proof of Lemma E i... |
| cdleme20y 40809 | Part of proof of Lemma E i... |
| cdleme19a 40810 | Part of proof of Lemma E i... |
| cdleme19b 40811 | Part of proof of Lemma E i... |
| cdleme19c 40812 | Part of proof of Lemma E i... |
| cdleme19d 40813 | Part of proof of Lemma E i... |
| cdleme19e 40814 | Part of proof of Lemma E i... |
| cdleme19f 40815 | Part of proof of Lemma E i... |
| cdleme20aN 40816 | Part of proof of Lemma E i... |
| cdleme20bN 40817 | Part of proof of Lemma E i... |
| cdleme20c 40818 | Part of proof of Lemma E i... |
| cdleme20d 40819 | Part of proof of Lemma E i... |
| cdleme20e 40820 | Part of proof of Lemma E i... |
| cdleme20f 40821 | Part of proof of Lemma E i... |
| cdleme20g 40822 | Part of proof of Lemma E i... |
| cdleme20h 40823 | Part of proof of Lemma E i... |
| cdleme20i 40824 | Part of proof of Lemma E i... |
| cdleme20j 40825 | Part of proof of Lemma E i... |
| cdleme20k 40826 | Part of proof of Lemma E i... |
| cdleme20l1 40827 | Part of proof of Lemma E i... |
| cdleme20l2 40828 | Part of proof of Lemma E i... |
| cdleme20l 40829 | Part of proof of Lemma E i... |
| cdleme20m 40830 | Part of proof of Lemma E i... |
| cdleme20 40831 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 40832 | Part of proof of Lemma E i... |
| cdleme21b 40833 | Part of proof of Lemma E i... |
| cdleme21c 40834 | Part of proof of Lemma E i... |
| cdleme21at 40835 | Part of proof of Lemma E i... |
| cdleme21ct 40836 | Part of proof of Lemma E i... |
| cdleme21d 40837 | Part of proof of Lemma E i... |
| cdleme21e 40838 | Part of proof of Lemma E i... |
| cdleme21f 40839 | Part of proof of Lemma E i... |
| cdleme21g 40840 | Part of proof of Lemma E i... |
| cdleme21h 40841 | Part of proof of Lemma E i... |
| cdleme21i 40842 | Part of proof of Lemma E i... |
| cdleme21j 40843 | Combine ~ cdleme20 and ~ c... |
| cdleme21 40844 | Part of proof of Lemma E i... |
| cdleme21k 40845 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 40846 | Part of proof of Lemma E i... |
| cdleme22a 40847 | Part of proof of Lemma E i... |
| cdleme22b 40848 | Part of proof of Lemma E i... |
| cdleme22cN 40849 | Part of proof of Lemma E i... |
| cdleme22d 40850 | Part of proof of Lemma E i... |
| cdleme22e 40851 | Part of proof of Lemma E i... |
| cdleme22eALTN 40852 | Part of proof of Lemma E i... |
| cdleme22f 40853 | Part of proof of Lemma E i... |
| cdleme22f2 40854 | Part of proof of Lemma E i... |
| cdleme22g 40855 | Part of proof of Lemma E i... |
| cdleme23a 40856 | Part of proof of Lemma E i... |
| cdleme23b 40857 | Part of proof of Lemma E i... |
| cdleme23c 40858 | Part of proof of Lemma E i... |
| cdleme24 40859 | Quantified version of ~ cd... |
| cdleme25a 40860 | Lemma for ~ cdleme25b . (... |
| cdleme25b 40861 | Transform ~ cdleme24 . TO... |
| cdleme25c 40862 | Transform ~ cdleme25b . (... |
| cdleme25dN 40863 | Transform ~ cdleme25c . (... |
| cdleme25cl 40864 | Show closure of the unique... |
| cdleme25cv 40865 | Change bound variables in ... |
| cdleme26e 40866 | Part of proof of Lemma E i... |
| cdleme26ee 40867 | Part of proof of Lemma E i... |
| cdleme26eALTN 40868 | Part of proof of Lemma E i... |
| cdleme26fALTN 40869 | Part of proof of Lemma E i... |
| cdleme26f 40870 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 40871 | Part of proof of Lemma E i... |
| cdleme26f2 40872 | Part of proof of Lemma E i... |
| cdleme27cl 40873 | Part of proof of Lemma E i... |
| cdleme27a 40874 | Part of proof of Lemma E i... |
| cdleme27b 40875 | Lemma for ~ cdleme27N . (... |
| cdleme27N 40876 | Part of proof of Lemma E i... |
| cdleme28a 40877 | Lemma for ~ cdleme25b . T... |
| cdleme28b 40878 | Lemma for ~ cdleme25b . T... |
| cdleme28c 40879 | Part of proof of Lemma E i... |
| cdleme28 40880 | Quantified version of ~ cd... |
| cdleme29ex 40881 | Lemma for ~ cdleme29b . (... |
| cdleme29b 40882 | Transform ~ cdleme28 . (C... |
| cdleme29c 40883 | Transform ~ cdleme28b . (... |
| cdleme29cl 40884 | Show closure of the unique... |
| cdleme30a 40885 | Part of proof of Lemma E i... |
| cdleme31so 40886 | Part of proof of Lemma E i... |
| cdleme31sn 40887 | Part of proof of Lemma E i... |
| cdleme31sn1 40888 | Part of proof of Lemma E i... |
| cdleme31se 40889 | Part of proof of Lemma D i... |
| cdleme31se2 40890 | Part of proof of Lemma D i... |
| cdleme31sc 40891 | Part of proof of Lemma E i... |
| cdleme31sde 40892 | Part of proof of Lemma D i... |
| cdleme31snd 40893 | Part of proof of Lemma D i... |
| cdleme31sdnN 40894 | Part of proof of Lemma E i... |
| cdleme31sn1c 40895 | Part of proof of Lemma E i... |
| cdleme31sn2 40896 | Part of proof of Lemma E i... |
| cdleme31fv 40897 | Part of proof of Lemma E i... |
| cdleme31fv1 40898 | Part of proof of Lemma E i... |
| cdleme31fv1s 40899 | Part of proof of Lemma E i... |
| cdleme31fv2 40900 | Part of proof of Lemma E i... |
| cdleme31id 40901 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 40902 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 40903 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 40904 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 40905 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 40906 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 40907 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 40908 | Part of proof of Lemma E i... |
| cdlemefr29exN 40909 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 40910 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 40911 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 40912 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 40913 | TODO fix comment. (Contri... |
| cdlemefr29clN 40914 | Show closure of the unique... |
| cdleme43frv1snN 40915 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 40916 | Part of proof of Lemma E i... |
| cdlemefr32fva1 40917 | Part of proof of Lemma E i... |
| cdlemefr31fv1 40918 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 40919 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 40920 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 40921 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 40922 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 40923 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 40924 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 40925 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 40926 | Show closure of the unique... |
| cdleme43fsv1snlem 40927 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 40928 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 40929 | Part of proof of Lemma E i... |
| cdlemefs32fva1 40930 | Part of proof of Lemma E i... |
| cdlemefs31fv1 40931 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 40932 | Value of f(r) when r is an... |
| cdlemefs44 40933 | Value of f_s(r) when r is ... |
| cdlemefr45 40934 | Value of f(r) when r is an... |
| cdlemefr45e 40935 | Explicit expansion of ~ cd... |
| cdlemefs45 40936 | Value of f_s(r) when r is ... |
| cdlemefs45ee 40937 | Explicit expansion of ~ cd... |
| cdlemefs45eN 40938 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 40939 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 40940 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 40941 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 40942 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 40943 | Show closure of ` [_ R / s... |
| cdleme32fva 40944 | Part of proof of Lemma D i... |
| cdleme32fva1 40945 | Part of proof of Lemma D i... |
| cdleme32fvaw 40946 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 40947 | Part of proof of Lemma D i... |
| cdleme32a 40948 | Part of proof of Lemma D i... |
| cdleme32b 40949 | Part of proof of Lemma D i... |
| cdleme32c 40950 | Part of proof of Lemma D i... |
| cdleme32d 40951 | Part of proof of Lemma D i... |
| cdleme32e 40952 | Part of proof of Lemma D i... |
| cdleme32f 40953 | Part of proof of Lemma D i... |
| cdleme32le 40954 | Part of proof of Lemma D i... |
| cdleme35a 40955 | Part of proof of Lemma E i... |
| cdleme35fnpq 40956 | Part of proof of Lemma E i... |
| cdleme35b 40957 | Part of proof of Lemma E i... |
| cdleme35c 40958 | Part of proof of Lemma E i... |
| cdleme35d 40959 | Part of proof of Lemma E i... |
| cdleme35e 40960 | Part of proof of Lemma E i... |
| cdleme35f 40961 | Part of proof of Lemma E i... |
| cdleme35g 40962 | Part of proof of Lemma E i... |
| cdleme35h 40963 | Part of proof of Lemma E i... |
| cdleme35h2 40964 | Part of proof of Lemma E i... |
| cdleme35sn2aw 40965 | Part of proof of Lemma E i... |
| cdleme35sn3a 40966 | Part of proof of Lemma E i... |
| cdleme36a 40967 | Part of proof of Lemma E i... |
| cdleme36m 40968 | Part of proof of Lemma E i... |
| cdleme37m 40969 | Part of proof of Lemma E i... |
| cdleme38m 40970 | Part of proof of Lemma E i... |
| cdleme38n 40971 | Part of proof of Lemma E i... |
| cdleme39a 40972 | Part of proof of Lemma E i... |
| cdleme39n 40973 | Part of proof of Lemma E i... |
| cdleme40m 40974 | Part of proof of Lemma E i... |
| cdleme40n 40975 | Part of proof of Lemma E i... |
| cdleme40v 40976 | Part of proof of Lemma E i... |
| cdleme40w 40977 | Part of proof of Lemma E i... |
| cdleme42a 40978 | Part of proof of Lemma E i... |
| cdleme42c 40979 | Part of proof of Lemma E i... |
| cdleme42d 40980 | Part of proof of Lemma E i... |
| cdleme41sn3aw 40981 | Part of proof of Lemma E i... |
| cdleme41sn4aw 40982 | Part of proof of Lemma E i... |
| cdleme41snaw 40983 | Part of proof of Lemma E i... |
| cdleme41fva11 40984 | Part of proof of Lemma E i... |
| cdleme42b 40985 | Part of proof of Lemma E i... |
| cdleme42e 40986 | Part of proof of Lemma E i... |
| cdleme42f 40987 | Part of proof of Lemma E i... |
| cdleme42g 40988 | Part of proof of Lemma E i... |
| cdleme42h 40989 | Part of proof of Lemma E i... |
| cdleme42i 40990 | Part of proof of Lemma E i... |
| cdleme42k 40991 | Part of proof of Lemma E i... |
| cdleme42ke 40992 | Part of proof of Lemma E i... |
| cdleme42keg 40993 | Part of proof of Lemma E i... |
| cdleme42mN 40994 | Part of proof of Lemma E i... |
| cdleme42mgN 40995 | Part of proof of Lemma E i... |
| cdleme43aN 40996 | Part of proof of Lemma E i... |
| cdleme43bN 40997 | Lemma for Lemma E in [Craw... |
| cdleme43cN 40998 | Part of proof of Lemma E i... |
| cdleme43dN 40999 | Part of proof of Lemma E i... |
| cdleme46f2g2 41000 | Conversion for ` G ` to re... |
| cdleme46f2g1 41001 | Conversion for ` G ` to re... |
| cdleme17d2 41002 | Part of proof of Lemma E i... |
| cdleme17d3 41003 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 41004 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 41005 | Part of proof of Lemma E i... |
| cdleme48fv 41006 | Part of proof of Lemma D i... |
| cdleme48fvg 41007 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 41008 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 41009 | TODO: fix comment. TODO: ... |
| cdleme48b 41010 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 41011 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 41012 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 41013 | TODO: fix comment. (Contr... |
| cdleme4gfv 41014 | Part of proof of Lemma D i... |
| cdlemeg47b 41015 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 41016 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 41017 | Value of g_s(r) when r is ... |
| cdlemeg49le 41018 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 41019 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 41020 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 41021 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 41022 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 41023 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 41024 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 41025 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 41026 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 41027 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 41028 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 41029 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 41030 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 41031 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 41032 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 41033 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 41034 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 41035 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 41036 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 41037 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 41038 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 41039 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 41040 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 41041 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 41042 | TODO: fix comment. (Contr... |
| cdleme48gfv1 41043 | TODO: fix comment. (Contr... |
| cdleme48gfv 41044 | TODO: fix comment. (Contr... |
| cdleme48fgv 41045 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 41046 | Part of proof of Lemma D i... |
| cdleme50lebi 41047 | Part of proof of Lemma D i... |
| cdleme50eq 41048 | Part of proof of Lemma D i... |
| cdleme50f 41049 | Part of proof of Lemma D i... |
| cdleme50f1 41050 | Part of proof of Lemma D i... |
| cdleme50rnlem 41051 | Part of proof of Lemma D i... |
| cdleme50rn 41052 | Part of proof of Lemma D i... |
| cdleme50f1o 41053 | Part of proof of Lemma D i... |
| cdleme50laut 41054 | Part of proof of Lemma D i... |
| cdleme50ldil 41055 | Part of proof of Lemma D i... |
| cdleme50trn1 41056 | Part of proof that ` F ` i... |
| cdleme50trn2a 41057 | Part of proof that ` F ` i... |
| cdleme50trn2 41058 | Part of proof that ` F ` i... |
| cdleme50trn12 41059 | Part of proof that ` F ` i... |
| cdleme50trn3 41060 | Part of proof that ` F ` i... |
| cdleme50trn123 41061 | Part of proof that ` F ` i... |
| cdleme51finvfvN 41062 | Part of proof of Lemma E i... |
| cdleme51finvN 41063 | Part of proof of Lemma E i... |
| cdleme50ltrn 41064 | Part of proof of Lemma E i... |
| cdleme51finvtrN 41065 | Part of proof of Lemma E i... |
| cdleme50ex 41066 | Part of Lemma E in [Crawle... |
| cdleme 41067 | Lemma E in [Crawley] p. 11... |
| cdlemf1 41068 | Part of Lemma F in [Crawle... |
| cdlemf2 41069 | Part of Lemma F in [Crawle... |
| cdlemf 41070 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 41071 | ~ cdlemf with additional c... |
| cdlemftr3 41072 | Special case of ~ cdlemf s... |
| cdlemftr2 41073 | Special case of ~ cdlemf s... |
| cdlemftr1 41074 | Part of proof of Lemma G o... |
| cdlemftr0 41075 | Special case of ~ cdlemf s... |
| trlord 41076 | The ordering of two Hilber... |
| cdlemg1a 41077 | Shorter expression for ` G... |
| cdlemg1b2 41078 | This theorem can be used t... |
| cdlemg1idlemN 41079 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 41080 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 41081 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 41082 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 41083 | This theorem can be used t... |
| cdlemg1idN 41084 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 41085 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 41086 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 41087 | Version of ~ cdleme51finvt... |
| ltrniotaval 41088 | Value of the unique transl... |
| ltrniotacnvval 41089 | Converse value of the uniq... |
| ltrniotaidvalN 41090 | Value of the unique transl... |
| ltrniotavalbN 41091 | Value of the unique transl... |
| cdlemeiota 41092 | A translation is uniquely ... |
| cdlemg1ci2 41093 | Any function of the form o... |
| cdlemg1cN 41094 | Any translation belongs to... |
| cdlemg1cex 41095 | Any translation is one of ... |
| cdlemg2cN 41096 | Any translation belongs to... |
| cdlemg2dN 41097 | This theorem can be used t... |
| cdlemg2cex 41098 | Any translation is one of ... |
| cdlemg2ce 41099 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 41100 | Part of proof of Lemma E i... |
| cdlemg2fvlem 41101 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 41102 | ~ cdleme42keg with simpler... |
| cdlemg2idN 41103 | Version of ~ cdleme31id wi... |
| cdlemg3a 41104 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 41105 | TODO: Replace this with ~... |
| cdlemg2fv 41106 | Value of a translation in ... |
| cdlemg2fv2 41107 | Value of a translation in ... |
| cdlemg2k 41108 | ~ cdleme42keg with simpler... |
| cdlemg2kq 41109 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 41110 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 41111 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 41112 | TODO: Is there a simpler ... |
| cdlemb3 41113 | Given two atoms not under ... |
| cdlemg7fvbwN 41114 | Properties of a translatio... |
| cdlemg4a 41115 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 41116 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 41117 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 41118 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 41119 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 41120 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 41121 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 41122 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 41123 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 41124 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 41125 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 41126 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 41127 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 41128 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 41129 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 41130 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 41131 | Value of a translation com... |
| cdlemg7aN 41132 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 41133 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 41134 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 41135 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 41136 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 41137 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 41138 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 41139 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 41140 | The triples ` <. P , ( F `... |
| cdlemg9 41141 | The triples ` <. P , ( F `... |
| cdlemg10b 41142 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 41143 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 41144 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 41145 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 41146 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 41147 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 41148 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 41149 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 41150 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 41151 | The triples ` <. P , ( F `... |
| cdlemg12c 41152 | The triples ` <. P , ( F `... |
| cdlemg12d 41153 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 41154 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 41155 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 41156 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 41157 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 41158 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 41159 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 41160 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 41161 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 41162 | Eliminate the ` ( F `` P )... |
| cdlemg15 41163 | Eliminate the ` ( (... |
| cdlemg16 41164 | Part of proof of Lemma G o... |
| cdlemg16ALTN 41165 | This version of ~ cdlemg16... |
| cdlemg16z 41166 | Eliminate ` ( ( F `... |
| cdlemg16zz 41167 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 41168 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 41169 | Part of proof of Lemma G i... |
| cdlemg17dN 41170 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 41171 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 41172 | TODO: fix comment. (Contr... |
| cdlemg17f 41173 | TODO: fix comment. (Contr... |
| cdlemg17g 41174 | TODO: fix comment. (Contr... |
| cdlemg17h 41175 | TODO: fix comment. (Contr... |
| cdlemg17i 41176 | TODO: fix comment. (Contr... |
| cdlemg17ir 41177 | TODO: fix comment. (Contr... |
| cdlemg17j 41178 | TODO: fix comment. (Contr... |
| cdlemg17pq 41179 | Utility theorem for swappi... |
| cdlemg17bq 41180 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 41181 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 41182 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 41183 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 41184 | Part of Lemma G of [Crawle... |
| cdlemg18a 41185 | Show two lines are differe... |
| cdlemg18b 41186 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 41187 | Show two lines intersect a... |
| cdlemg18d 41188 | Show two lines intersect a... |
| cdlemg18 41189 | Show two lines intersect a... |
| cdlemg19a 41190 | Show two lines intersect a... |
| cdlemg19 41191 | Show two lines intersect a... |
| cdlemg20 41192 | Show two lines intersect a... |
| cdlemg21 41193 | Version of cdlemg19 with `... |
| cdlemg22 41194 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 41195 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 41196 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 41197 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 41198 | ~ cdlemg16zz restated for ... |
| cdlemg27a 41199 | For use with case when ` (... |
| cdlemg28a 41200 | Part of proof of Lemma G o... |
| cdlemg31b0N 41201 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 41202 | TODO: Fix comment. (Cont... |
| cdlemg27b 41203 | TODO: Fix comment. (Cont... |
| cdlemg31a 41204 | TODO: fix comment. (Contr... |
| cdlemg31b 41205 | TODO: fix comment. (Contr... |
| cdlemg31c 41206 | Show that when ` N ` is an... |
| cdlemg31d 41207 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 41208 | TODO: Fix comment. (Cont... |
| cdlemg33c0 41209 | TODO: Fix comment. (Cont... |
| cdlemg28b 41210 | Part of proof of Lemma G o... |
| cdlemg28 41211 | Part of proof of Lemma G o... |
| cdlemg29 41212 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 41213 | TODO: Fix comment. (Cont... |
| cdlemg33b 41214 | TODO: Fix comment. (Cont... |
| cdlemg33c 41215 | TODO: Fix comment. (Cont... |
| cdlemg33d 41216 | TODO: Fix comment. (Cont... |
| cdlemg33e 41217 | TODO: Fix comment. (Cont... |
| cdlemg33 41218 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 41219 | Use cdlemg33 to eliminate ... |
| cdlemg35 41220 | TODO: Fix comment. TODO:... |
| cdlemg36 41221 | Use cdlemg35 to eliminate ... |
| cdlemg38 41222 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 41223 | Eliminate ` =/= ` conditio... |
| cdlemg40 41224 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 41225 | Convert ~ cdlemg40 to func... |
| ltrnco 41226 | The composition of two tra... |
| trlcocnv 41227 | Swap the arguments of the ... |
| trlcoabs 41228 | Absorption into a composit... |
| trlcoabs2N 41229 | Absorption of the trace of... |
| trlcoat 41230 | The trace of a composition... |
| trlcocnvat 41231 | Commonly used special case... |
| trlconid 41232 | The composition of two dif... |
| trlcolem 41233 | Lemma for ~ trlco . (Cont... |
| trlco 41234 | The trace of a composition... |
| trlcone 41235 | If two translations have d... |
| cdlemg42 41236 | Part of proof of Lemma G o... |
| cdlemg43 41237 | Part of proof of Lemma G o... |
| cdlemg44a 41238 | Part of proof of Lemma G o... |
| cdlemg44b 41239 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 41240 | Part of proof of Lemma G o... |
| cdlemg47a 41241 | TODO: fix comment. TODO: ... |
| cdlemg46 41242 | Part of proof of Lemma G o... |
| cdlemg47 41243 | Part of proof of Lemma G o... |
| cdlemg48 41244 | Eliminate ` h ` from ~ cdl... |
| ltrncom 41245 | Composition is commutative... |
| ltrnco4 41246 | Rearrange a composition of... |
| trljco 41247 | Trace joined with trace of... |
| trljco2 41248 | Trace joined with trace of... |
| tgrpfset 41251 | The translation group maps... |
| tgrpset 41252 | The translation group for ... |
| tgrpbase 41253 | The base set of the transl... |
| tgrpopr 41254 | The group operation of the... |
| tgrpov 41255 | The group operation value ... |
| tgrpgrplem 41256 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 41257 | The translation group is a... |
| tgrpabl 41258 | The translation group is a... |
| tendofset 41265 | The set of all trace-prese... |
| tendoset 41266 | The set of trace-preservin... |
| istendo 41267 | The predicate "is a trace-... |
| tendotp 41268 | Trace-preserving property ... |
| istendod 41269 | Deduce the predicate "is a... |
| tendof 41270 | Functionality of a trace-p... |
| tendoeq1 41271 | Condition determining equa... |
| tendovalco 41272 | Value of composition of tr... |
| tendocoval 41273 | Value of composition of en... |
| tendocl 41274 | Closure of a trace-preserv... |
| tendoco2 41275 | Distribution of compositio... |
| tendoidcl 41276 | The identity is a trace-pr... |
| tendo1mul 41277 | Multiplicative identity mu... |
| tendo1mulr 41278 | Multiplicative identity mu... |
| tendococl 41279 | The composition of two tra... |
| tendoid 41280 | The identity value of a tr... |
| tendoeq2 41281 | Condition determining equa... |
| tendoplcbv 41282 | Define sum operation for t... |
| tendopl 41283 | Value of endomorphism sum ... |
| tendopl2 41284 | Value of result of endomor... |
| tendoplcl2 41285 | Value of result of endomor... |
| tendoplco2 41286 | Value of result of endomor... |
| tendopltp 41287 | Trace-preserving property ... |
| tendoplcl 41288 | Endomorphism sum is a trac... |
| tendoplcom 41289 | The endomorphism sum opera... |
| tendoplass 41290 | The endomorphism sum opera... |
| tendodi1 41291 | Endomorphism composition d... |
| tendodi2 41292 | Endomorphism composition d... |
| tendo0cbv 41293 | Define additive identity f... |
| tendo02 41294 | Value of additive identity... |
| tendo0co2 41295 | The additive identity trac... |
| tendo0tp 41296 | Trace-preserving property ... |
| tendo0cl 41297 | The additive identity is a... |
| tendo0pl 41298 | Property of the additive i... |
| tendo0plr 41299 | Property of the additive i... |
| tendoicbv 41300 | Define inverse function fo... |
| tendoi 41301 | Value of inverse endomorph... |
| tendoi2 41302 | Value of additive inverse ... |
| tendoicl 41303 | Closure of the additive in... |
| tendoipl 41304 | Property of the additive i... |
| tendoipl2 41305 | Property of the additive i... |
| erngfset 41306 | The division rings on trac... |
| erngset 41307 | The division ring on trace... |
| erngbase 41308 | The base set of the divisi... |
| erngfplus 41309 | Ring addition operation. ... |
| erngplus 41310 | Ring addition operation. ... |
| erngplus2 41311 | Ring addition operation. ... |
| erngfmul 41312 | Ring multiplication operat... |
| erngmul 41313 | Ring addition operation. ... |
| erngfset-rN 41314 | The division rings on trac... |
| erngset-rN 41315 | The division ring on trace... |
| erngbase-rN 41316 | The base set of the divisi... |
| erngfplus-rN 41317 | Ring addition operation. ... |
| erngplus-rN 41318 | Ring addition operation. ... |
| erngplus2-rN 41319 | Ring addition operation. ... |
| erngfmul-rN 41320 | Ring multiplication operat... |
| erngmul-rN 41321 | Ring addition operation. ... |
| cdlemh1 41322 | Part of proof of Lemma H o... |
| cdlemh2 41323 | Part of proof of Lemma H o... |
| cdlemh 41324 | Lemma H of [Crawley] p. 11... |
| cdlemi1 41325 | Part of proof of Lemma I o... |
| cdlemi2 41326 | Part of proof of Lemma I o... |
| cdlemi 41327 | Lemma I of [Crawley] p. 11... |
| cdlemj1 41328 | Part of proof of Lemma J o... |
| cdlemj2 41329 | Part of proof of Lemma J o... |
| cdlemj3 41330 | Part of proof of Lemma J o... |
| tendocan 41331 | Cancellation law: if the v... |
| tendoid0 41332 | A trace-preserving endomor... |
| tendo0mul 41333 | Additive identity multipli... |
| tendo0mulr 41334 | Additive identity multipli... |
| tendo1ne0 41335 | The identity (unity) is no... |
| tendoconid 41336 | The composition (product) ... |
| tendotr 41337 | The trace of the value of ... |
| cdlemk1 41338 | Part of proof of Lemma K o... |
| cdlemk2 41339 | Part of proof of Lemma K o... |
| cdlemk3 41340 | Part of proof of Lemma K o... |
| cdlemk4 41341 | Part of proof of Lemma K o... |
| cdlemk5a 41342 | Part of proof of Lemma K o... |
| cdlemk5 41343 | Part of proof of Lemma K o... |
| cdlemk6 41344 | Part of proof of Lemma K o... |
| cdlemk8 41345 | Part of proof of Lemma K o... |
| cdlemk9 41346 | Part of proof of Lemma K o... |
| cdlemk9bN 41347 | Part of proof of Lemma K o... |
| cdlemki 41348 | Part of proof of Lemma K o... |
| cdlemkvcl 41349 | Part of proof of Lemma K o... |
| cdlemk10 41350 | Part of proof of Lemma K o... |
| cdlemksv 41351 | Part of proof of Lemma K o... |
| cdlemksel 41352 | Part of proof of Lemma K o... |
| cdlemksat 41353 | Part of proof of Lemma K o... |
| cdlemksv2 41354 | Part of proof of Lemma K o... |
| cdlemk7 41355 | Part of proof of Lemma K o... |
| cdlemk11 41356 | Part of proof of Lemma K o... |
| cdlemk12 41357 | Part of proof of Lemma K o... |
| cdlemkoatnle 41358 | Utility lemma. (Contribut... |
| cdlemk13 41359 | Part of proof of Lemma K o... |
| cdlemkole 41360 | Utility lemma. (Contribut... |
| cdlemk14 41361 | Part of proof of Lemma K o... |
| cdlemk15 41362 | Part of proof of Lemma K o... |
| cdlemk16a 41363 | Part of proof of Lemma K o... |
| cdlemk16 41364 | Part of proof of Lemma K o... |
| cdlemk17 41365 | Part of proof of Lemma K o... |
| cdlemk1u 41366 | Part of proof of Lemma K o... |
| cdlemk5auN 41367 | Part of proof of Lemma K o... |
| cdlemk5u 41368 | Part of proof of Lemma K o... |
| cdlemk6u 41369 | Part of proof of Lemma K o... |
| cdlemkj 41370 | Part of proof of Lemma K o... |
| cdlemkuvN 41371 | Part of proof of Lemma K o... |
| cdlemkuel 41372 | Part of proof of Lemma K o... |
| cdlemkuat 41373 | Part of proof of Lemma K o... |
| cdlemkuv2 41374 | Part of proof of Lemma K o... |
| cdlemk18 41375 | Part of proof of Lemma K o... |
| cdlemk19 41376 | Part of proof of Lemma K o... |
| cdlemk7u 41377 | Part of proof of Lemma K o... |
| cdlemk11u 41378 | Part of proof of Lemma K o... |
| cdlemk12u 41379 | Part of proof of Lemma K o... |
| cdlemk21N 41380 | Part of proof of Lemma K o... |
| cdlemk20 41381 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 41382 | Utility lemma. (Contribut... |
| cdlemk13-2N 41383 | Part of proof of Lemma K o... |
| cdlemkole-2N 41384 | Utility lemma. (Contribut... |
| cdlemk14-2N 41385 | Part of proof of Lemma K o... |
| cdlemk15-2N 41386 | Part of proof of Lemma K o... |
| cdlemk16-2N 41387 | Part of proof of Lemma K o... |
| cdlemk17-2N 41388 | Part of proof of Lemma K o... |
| cdlemkj-2N 41389 | Part of proof of Lemma K o... |
| cdlemkuv-2N 41390 | Part of proof of Lemma K o... |
| cdlemkuel-2N 41391 | Part of proof of Lemma K o... |
| cdlemkuv2-2 41392 | Part of proof of Lemma K o... |
| cdlemk18-2N 41393 | Part of proof of Lemma K o... |
| cdlemk19-2N 41394 | Part of proof of Lemma K o... |
| cdlemk7u-2N 41395 | Part of proof of Lemma K o... |
| cdlemk11u-2N 41396 | Part of proof of Lemma K o... |
| cdlemk12u-2N 41397 | Part of proof of Lemma K o... |
| cdlemk21-2N 41398 | Part of proof of Lemma K o... |
| cdlemk20-2N 41399 | Part of proof of Lemma K o... |
| cdlemk22 41400 | Part of proof of Lemma K o... |
| cdlemk30 41401 | Part of proof of Lemma K o... |
| cdlemkuu 41402 | Convert between function a... |
| cdlemk31 41403 | Part of proof of Lemma K o... |
| cdlemk32 41404 | Part of proof of Lemma K o... |
| cdlemkuel-3 41405 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 41406 | Part of proof of Lemma K o... |
| cdlemk18-3N 41407 | Part of proof of Lemma K o... |
| cdlemk22-3 41408 | Part of proof of Lemma K o... |
| cdlemk23-3 41409 | Part of proof of Lemma K o... |
| cdlemk24-3 41410 | Part of proof of Lemma K o... |
| cdlemk25-3 41411 | Part of proof of Lemma K o... |
| cdlemk26b-3 41412 | Part of proof of Lemma K o... |
| cdlemk26-3 41413 | Part of proof of Lemma K o... |
| cdlemk27-3 41414 | Part of proof of Lemma K o... |
| cdlemk28-3 41415 | Part of proof of Lemma K o... |
| cdlemk33N 41416 | Part of proof of Lemma K o... |
| cdlemk34 41417 | Part of proof of Lemma K o... |
| cdlemk29-3 41418 | Part of proof of Lemma K o... |
| cdlemk35 41419 | Part of proof of Lemma K o... |
| cdlemk36 41420 | Part of proof of Lemma K o... |
| cdlemk37 41421 | Part of proof of Lemma K o... |
| cdlemk38 41422 | Part of proof of Lemma K o... |
| cdlemk39 41423 | Part of proof of Lemma K o... |
| cdlemk40 41424 | TODO: fix comment. (Contr... |
| cdlemk40t 41425 | TODO: fix comment. (Contr... |
| cdlemk40f 41426 | TODO: fix comment. (Contr... |
| cdlemk41 41427 | Part of proof of Lemma K o... |
| cdlemkfid1N 41428 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 41429 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 41430 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 41431 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 41432 | TODO: is this useful or sh... |
| cdlemky 41433 | Part of proof of Lemma K o... |
| cdlemkyu 41434 | Convert between function a... |
| cdlemkyuu 41435 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 41436 | Part of proof of Lemma K o... |
| cdlemk19ylem 41437 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 41438 | Part of proof of Lemma K o... |
| cdlemk19y 41439 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 41440 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 41441 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 41442 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 41443 | The value of the tau funct... |
| cdlemk35s 41444 | Substitution version of ~ ... |
| cdlemk35s-id 41445 | Substitution version of ~ ... |
| cdlemk39s 41446 | Substitution version of ~ ... |
| cdlemk39s-id 41447 | Substitution version of ~ ... |
| cdlemk42 41448 | Part of proof of Lemma K o... |
| cdlemk19xlem 41449 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 41450 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 41451 | Part of proof of Lemma K o... |
| cdlemk11tc 41452 | Part of proof of Lemma K o... |
| cdlemk11t 41453 | Part of proof of Lemma K o... |
| cdlemk45 41454 | Part of proof of Lemma K o... |
| cdlemk46 41455 | Part of proof of Lemma K o... |
| cdlemk47 41456 | Part of proof of Lemma K o... |
| cdlemk48 41457 | Part of proof of Lemma K o... |
| cdlemk49 41458 | Part of proof of Lemma K o... |
| cdlemk50 41459 | Part of proof of Lemma K o... |
| cdlemk51 41460 | Part of proof of Lemma K o... |
| cdlemk52 41461 | Part of proof of Lemma K o... |
| cdlemk53a 41462 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 41463 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 41464 | Part of proof of Lemma K o... |
| cdlemk54 41465 | Part of proof of Lemma K o... |
| cdlemk55a 41466 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 41467 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 41468 | Part of proof of Lemma K o... |
| cdlemkyyN 41469 | Part of proof of Lemma K o... |
| cdlemk43N 41470 | Part of proof of Lemma K o... |
| cdlemk35u 41471 | Substitution version of ~ ... |
| cdlemk55u1 41472 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 41473 | Part of proof of Lemma K o... |
| cdlemk39u1 41474 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 41475 | Part of proof of Lemma K o... |
| cdlemk19u1 41476 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 41477 | Part of Lemma K of [Crawle... |
| cdlemk56 41478 | Part of Lemma K of [Crawle... |
| cdlemk19w 41479 | Use a fixed element to eli... |
| cdlemk56w 41480 | Use a fixed element to eli... |
| cdlemk 41481 | Lemma K of [Crawley] p. 11... |
| tendoex 41482 | Generalization of Lemma K ... |
| cdleml1N 41483 | Part of proof of Lemma L o... |
| cdleml2N 41484 | Part of proof of Lemma L o... |
| cdleml3N 41485 | Part of proof of Lemma L o... |
| cdleml4N 41486 | Part of proof of Lemma L o... |
| cdleml5N 41487 | Part of proof of Lemma L o... |
| cdleml6 41488 | Part of proof of Lemma L o... |
| cdleml7 41489 | Part of proof of Lemma L o... |
| cdleml8 41490 | Part of proof of Lemma L o... |
| cdleml9 41491 | Part of proof of Lemma L o... |
| dva1dim 41492 | Two expressions for the 1-... |
| dvhb1dimN 41493 | Two expressions for the 1-... |
| erng1lem 41494 | Value of the endomorphism ... |
| erngdvlem1 41495 | Lemma for ~ eringring . (... |
| erngdvlem2N 41496 | Lemma for ~ eringring . (... |
| erngdvlem3 41497 | Lemma for ~ eringring . (... |
| erngdvlem4 41498 | Lemma for ~ erngdv . (Con... |
| eringring 41499 | An endomorphism ring is a ... |
| erngdv 41500 | An endomorphism ring is a ... |
| erng0g 41501 | The division ring zero of ... |
| erng1r 41502 | The division ring unity of... |
| erngdvlem1-rN 41503 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 41504 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 41505 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 41506 | Lemma for ~ erngdv . (Con... |
| erngring-rN 41507 | An endomorphism ring is a ... |
| erngdv-rN 41508 | An endomorphism ring is a ... |
| dvafset 41511 | The constructed partial ve... |
| dvaset 41512 | The constructed partial ve... |
| dvasca 41513 | The ring base set of the c... |
| dvabase 41514 | The ring base set of the c... |
| dvafplusg 41515 | Ring addition operation fo... |
| dvaplusg 41516 | Ring addition operation fo... |
| dvaplusgv 41517 | Ring addition operation fo... |
| dvafmulr 41518 | Ring multiplication operat... |
| dvamulr 41519 | Ring multiplication operat... |
| dvavbase 41520 | The vectors (vector base s... |
| dvafvadd 41521 | The vector sum operation f... |
| dvavadd 41522 | Ring addition operation fo... |
| dvafvsca 41523 | Ring addition operation fo... |
| dvavsca 41524 | Ring addition operation fo... |
| tendospcl 41525 | Closure of endomorphism sc... |
| tendospass 41526 | Associative law for endomo... |
| tendospdi1 41527 | Forward distributive law f... |
| tendocnv 41528 | Converse of a trace-preser... |
| tendospdi2 41529 | Reverse distributive law f... |
| tendospcanN 41530 | Cancellation law for trace... |
| dvaabl 41531 | The constructed partial ve... |
| dvalveclem 41532 | Lemma for ~ dvalvec . (Co... |
| dvalvec 41533 | The constructed partial ve... |
| dva0g 41534 | The zero vector of partial... |
| diaffval 41537 | The partial isomorphism A ... |
| diafval 41538 | The partial isomorphism A ... |
| diaval 41539 | The partial isomorphism A ... |
| diaelval 41540 | Member of the partial isom... |
| diafn 41541 | Functionality and domain o... |
| diadm 41542 | Domain of the partial isom... |
| diaeldm 41543 | Member of domain of the pa... |
| diadmclN 41544 | A member of domain of the ... |
| diadmleN 41545 | A member of domain of the ... |
| dian0 41546 | The value of the partial i... |
| dia0eldmN 41547 | The lattice zero belongs t... |
| dia1eldmN 41548 | The fiducial hyperplane (t... |
| diass 41549 | The value of the partial i... |
| diael 41550 | A member of the value of t... |
| diatrl 41551 | Trace of a member of the p... |
| diaelrnN 41552 | Any value of the partial i... |
| dialss 41553 | The value of partial isomo... |
| diaord 41554 | The partial isomorphism A ... |
| dia11N 41555 | The partial isomorphism A ... |
| diaf11N 41556 | The partial isomorphism A ... |
| diaclN 41557 | Closure of partial isomorp... |
| diacnvclN 41558 | Closure of partial isomorp... |
| dia0 41559 | The value of the partial i... |
| dia1N 41560 | The value of the partial i... |
| dia1elN 41561 | The largest subspace in th... |
| diaglbN 41562 | Partial isomorphism A of a... |
| diameetN 41563 | Partial isomorphism A of a... |
| diainN 41564 | Inverse partial isomorphis... |
| diaintclN 41565 | The intersection of partia... |
| diasslssN 41566 | The partial isomorphism A ... |
| diassdvaN 41567 | The partial isomorphism A ... |
| dia1dim 41568 | Two expressions for the 1-... |
| dia1dim2 41569 | Two expressions for a 1-di... |
| dia1dimid 41570 | A vector (translation) bel... |
| dia2dimlem1 41571 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 41572 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 41573 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 41574 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 41575 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 41576 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 41577 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 41578 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 41579 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 41580 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 41581 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 41582 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 41583 | Lemma for ~ dia2dim . Eli... |
| dia2dim 41584 | A two-dimensional subspace... |
| dvhfset 41587 | The constructed full vecto... |
| dvhset 41588 | The constructed full vecto... |
| dvhsca 41589 | The ring of scalars of the... |
| dvhbase 41590 | The ring base set of the c... |
| dvhfplusr 41591 | Ring addition operation fo... |
| dvhfmulr 41592 | Ring multiplication operat... |
| dvhmulr 41593 | Ring multiplication operat... |
| dvhvbase 41594 | The vectors (vector base s... |
| dvhelvbasei 41595 | Vector membership in the c... |
| dvhvaddcbv 41596 | Change bound variables to ... |
| dvhvaddval 41597 | The vector sum operation f... |
| dvhfvadd 41598 | The vector sum operation f... |
| dvhvadd 41599 | The vector sum operation f... |
| dvhopvadd 41600 | The vector sum operation f... |
| dvhopvadd2 41601 | The vector sum operation f... |
| dvhvaddcl 41602 | Closure of the vector sum ... |
| dvhvaddcomN 41603 | Commutativity of vector su... |
| dvhvaddass 41604 | Associativity of vector su... |
| dvhvscacbv 41605 | Change bound variables to ... |
| dvhvscaval 41606 | The scalar product operati... |
| dvhfvsca 41607 | Scalar product operation f... |
| dvhvsca 41608 | Scalar product operation f... |
| dvhopvsca 41609 | Scalar product operation f... |
| dvhvscacl 41610 | Closure of the scalar prod... |
| tendoinvcl 41611 | Closure of multiplicative ... |
| tendolinv 41612 | Left multiplicative invers... |
| tendorinv 41613 | Right multiplicative inver... |
| dvhgrp 41614 | The full vector space ` U ... |
| dvhlveclem 41615 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 41616 | The full vector space ` U ... |
| dvhlmod 41617 | The full vector space ` U ... |
| dvh0g 41618 | The zero vector of vector ... |
| dvheveccl 41619 | Properties of a unit vecto... |
| dvhopclN 41620 | Closure of a ` DVecH ` vec... |
| dvhopaddN 41621 | Sum of ` DVecH ` vectors e... |
| dvhopspN 41622 | Scalar product of ` DVecH ... |
| dvhopN 41623 | Decompose a ` DVecH ` vect... |
| dvhopellsm 41624 | Ordered pair membership in... |
| cdlemm10N 41625 | The image of the map ` G `... |
| docaffvalN 41628 | Subspace orthocomplement f... |
| docafvalN 41629 | Subspace orthocomplement f... |
| docavalN 41630 | Subspace orthocomplement f... |
| docaclN 41631 | Closure of subspace orthoc... |
| diaocN 41632 | Value of partial isomorphi... |
| doca2N 41633 | Double orthocomplement of ... |
| doca3N 41634 | Double orthocomplement of ... |
| dvadiaN 41635 | Any closed subspace is a m... |
| diarnN 41636 | Partial isomorphism A maps... |
| diaf1oN 41637 | The partial isomorphism A ... |
| djaffvalN 41640 | Subspace join for ` DVecA ... |
| djafvalN 41641 | Subspace join for ` DVecA ... |
| djavalN 41642 | Subspace join for ` DVecA ... |
| djaclN 41643 | Closure of subspace join f... |
| djajN 41644 | Transfer lattice join to `... |
| dibffval 41647 | The partial isomorphism B ... |
| dibfval 41648 | The partial isomorphism B ... |
| dibval 41649 | The partial isomorphism B ... |
| dibopelvalN 41650 | Member of the partial isom... |
| dibval2 41651 | Value of the partial isomo... |
| dibopelval2 41652 | Member of the partial isom... |
| dibval3N 41653 | Value of the partial isomo... |
| dibelval3 41654 | Member of the partial isom... |
| dibopelval3 41655 | Member of the partial isom... |
| dibelval1st 41656 | Membership in value of the... |
| dibelval1st1 41657 | Membership in value of the... |
| dibelval1st2N 41658 | Membership in value of the... |
| dibelval2nd 41659 | Membership in value of the... |
| dibn0 41660 | The value of the partial i... |
| dibfna 41661 | Functionality and domain o... |
| dibdiadm 41662 | Domain of the partial isom... |
| dibfnN 41663 | Functionality and domain o... |
| dibdmN 41664 | Domain of the partial isom... |
| dibeldmN 41665 | Member of domain of the pa... |
| dibord 41666 | The isomorphism B for a la... |
| dib11N 41667 | The isomorphism B for a la... |
| dibf11N 41668 | The partial isomorphism A ... |
| dibclN 41669 | Closure of partial isomorp... |
| dibvalrel 41670 | The value of partial isomo... |
| dib0 41671 | The value of partial isomo... |
| dib1dim 41672 | Two expressions for the 1-... |
| dibglbN 41673 | Partial isomorphism B of a... |
| dibintclN 41674 | The intersection of partia... |
| dib1dim2 41675 | Two expressions for a 1-di... |
| dibss 41676 | The partial isomorphism B ... |
| diblss 41677 | The value of partial isomo... |
| diblsmopel 41678 | Membership in subspace sum... |
| dicffval 41681 | The partial isomorphism C ... |
| dicfval 41682 | The partial isomorphism C ... |
| dicval 41683 | The partial isomorphism C ... |
| dicopelval 41684 | Membership in value of the... |
| dicelvalN 41685 | Membership in value of the... |
| dicval2 41686 | The partial isomorphism C ... |
| dicelval3 41687 | Member of the partial isom... |
| dicopelval2 41688 | Membership in value of the... |
| dicelval2N 41689 | Membership in value of the... |
| dicfnN 41690 | Functionality and domain o... |
| dicdmN 41691 | Domain of the partial isom... |
| dicvalrelN 41692 | The value of partial isomo... |
| dicssdvh 41693 | The partial isomorphism C ... |
| dicelval1sta 41694 | Membership in value of the... |
| dicelval1stN 41695 | Membership in value of the... |
| dicelval2nd 41696 | Membership in value of the... |
| dicvaddcl 41697 | Membership in value of the... |
| dicvscacl 41698 | Membership in value of the... |
| dicn0 41699 | The value of the partial i... |
| diclss 41700 | The value of partial isomo... |
| diclspsn 41701 | The value of isomorphism C... |
| cdlemn2 41702 | Part of proof of Lemma N o... |
| cdlemn2a 41703 | Part of proof of Lemma N o... |
| cdlemn3 41704 | Part of proof of Lemma N o... |
| cdlemn4 41705 | Part of proof of Lemma N o... |
| cdlemn4a 41706 | Part of proof of Lemma N o... |
| cdlemn5pre 41707 | Part of proof of Lemma N o... |
| cdlemn5 41708 | Part of proof of Lemma N o... |
| cdlemn6 41709 | Part of proof of Lemma N o... |
| cdlemn7 41710 | Part of proof of Lemma N o... |
| cdlemn8 41711 | Part of proof of Lemma N o... |
| cdlemn9 41712 | Part of proof of Lemma N o... |
| cdlemn10 41713 | Part of proof of Lemma N o... |
| cdlemn11a 41714 | Part of proof of Lemma N o... |
| cdlemn11b 41715 | Part of proof of Lemma N o... |
| cdlemn11c 41716 | Part of proof of Lemma N o... |
| cdlemn11pre 41717 | Part of proof of Lemma N o... |
| cdlemn11 41718 | Part of proof of Lemma N o... |
| cdlemn 41719 | Lemma N of [Crawley] p. 12... |
| dihordlem6 41720 | Part of proof of Lemma N o... |
| dihordlem7 41721 | Part of proof of Lemma N o... |
| dihordlem7b 41722 | Part of proof of Lemma N o... |
| dihjustlem 41723 | Part of proof after Lemma ... |
| dihjust 41724 | Part of proof after Lemma ... |
| dihord1 41725 | Part of proof after Lemma ... |
| dihord2a 41726 | Part of proof after Lemma ... |
| dihord2b 41727 | Part of proof after Lemma ... |
| dihord2cN 41728 | Part of proof after Lemma ... |
| dihord11b 41729 | Part of proof after Lemma ... |
| dihord10 41730 | Part of proof after Lemma ... |
| dihord11c 41731 | Part of proof after Lemma ... |
| dihord2pre 41732 | Part of proof after Lemma ... |
| dihord2pre2 41733 | Part of proof after Lemma ... |
| dihord2 41734 | Part of proof after Lemma ... |
| dihffval 41737 | The isomorphism H for a la... |
| dihfval 41738 | Isomorphism H for a lattic... |
| dihval 41739 | Value of isomorphism H for... |
| dihvalc 41740 | Value of isomorphism H for... |
| dihlsscpre 41741 | Closure of isomorphism H f... |
| dihvalcqpre 41742 | Value of isomorphism H for... |
| dihvalcq 41743 | Value of isomorphism H for... |
| dihvalb 41744 | Value of isomorphism H for... |
| dihopelvalbN 41745 | Ordered pair member of the... |
| dihvalcqat 41746 | Value of isomorphism H for... |
| dih1dimb 41747 | Two expressions for a 1-di... |
| dih1dimb2 41748 | Isomorphism H at an atom u... |
| dih1dimc 41749 | Isomorphism H at an atom n... |
| dib2dim 41750 | Extend ~ dia2dim to partia... |
| dih2dimb 41751 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 41752 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 41753 | Ordered pair member of the... |
| dihvalcq2 41754 | Value of isomorphism H for... |
| dihopelvalcpre 41755 | Member of value of isomorp... |
| dihopelvalc 41756 | Member of value of isomorp... |
| dihlss 41757 | The value of isomorphism H... |
| dihss 41758 | The value of isomorphism H... |
| dihssxp 41759 | An isomorphism H value is ... |
| dihopcl 41760 | Closure of an ordered pair... |
| xihopellsmN 41761 | Ordered pair membership in... |
| dihopellsm 41762 | Ordered pair membership in... |
| dihord6apre 41763 | Part of proof that isomorp... |
| dihord3 41764 | The isomorphism H for a la... |
| dihord4 41765 | The isomorphism H for a la... |
| dihord5b 41766 | Part of proof that isomorp... |
| dihord6b 41767 | Part of proof that isomorp... |
| dihord6a 41768 | Part of proof that isomorp... |
| dihord5apre 41769 | Part of proof that isomorp... |
| dihord5a 41770 | Part of proof that isomorp... |
| dihord 41771 | The isomorphism H is order... |
| dih11 41772 | The isomorphism H is one-t... |
| dihf11lem 41773 | Functionality of the isomo... |
| dihf11 41774 | The isomorphism H for a la... |
| dihfn 41775 | Functionality and domain o... |
| dihdm 41776 | Domain of isomorphism H. (... |
| dihcl 41777 | Closure of isomorphism H. ... |
| dihcnvcl 41778 | Closure of isomorphism H c... |
| dihcnvid1 41779 | The converse isomorphism o... |
| dihcnvid2 41780 | The isomorphism of a conve... |
| dihcnvord 41781 | Ordering property for conv... |
| dihcnv11 41782 | The converse of isomorphis... |
| dihsslss 41783 | The isomorphism H maps to ... |
| dihrnlss 41784 | The isomorphism H maps to ... |
| dihrnss 41785 | The isomorphism H maps to ... |
| dihvalrel 41786 | The value of isomorphism H... |
| dih0 41787 | The value of isomorphism H... |
| dih0bN 41788 | A lattice element is zero ... |
| dih0vbN 41789 | A vector is zero iff its s... |
| dih0cnv 41790 | The isomorphism H converse... |
| dih0rn 41791 | The zero subspace belongs ... |
| dih0sb 41792 | A subspace is zero iff the... |
| dih1 41793 | The value of isomorphism H... |
| dih1rn 41794 | The full vector space belo... |
| dih1cnv 41795 | The isomorphism H converse... |
| dihwN 41796 | Value of isomorphism H at ... |
| dihmeetlem1N 41797 | Isomorphism H of a conjunc... |
| dihglblem5apreN 41798 | A conjunction property of ... |
| dihglblem5aN 41799 | A conjunction property of ... |
| dihglblem2aN 41800 | Lemma for isomorphism H of... |
| dihglblem2N 41801 | The GLB of a set of lattic... |
| dihglblem3N 41802 | Isomorphism H of a lattice... |
| dihglblem3aN 41803 | Isomorphism H of a lattice... |
| dihglblem4 41804 | Isomorphism H of a lattice... |
| dihglblem5 41805 | Isomorphism H of a lattice... |
| dihmeetlem2N 41806 | Isomorphism H of a conjunc... |
| dihglbcpreN 41807 | Isomorphism H of a lattice... |
| dihglbcN 41808 | Isomorphism H of a lattice... |
| dihmeetcN 41809 | Isomorphism H of a lattice... |
| dihmeetbN 41810 | Isomorphism H of a lattice... |
| dihmeetbclemN 41811 | Lemma for isomorphism H of... |
| dihmeetlem3N 41812 | Lemma for isomorphism H of... |
| dihmeetlem4preN 41813 | Lemma for isomorphism H of... |
| dihmeetlem4N 41814 | Lemma for isomorphism H of... |
| dihmeetlem5 41815 | Part of proof that isomorp... |
| dihmeetlem6 41816 | Lemma for isomorphism H of... |
| dihmeetlem7N 41817 | Lemma for isomorphism H of... |
| dihjatc1 41818 | Lemma for isomorphism H of... |
| dihjatc2N 41819 | Isomorphism H of join with... |
| dihjatc3 41820 | Isomorphism H of join with... |
| dihmeetlem8N 41821 | Lemma for isomorphism H of... |
| dihmeetlem9N 41822 | Lemma for isomorphism H of... |
| dihmeetlem10N 41823 | Lemma for isomorphism H of... |
| dihmeetlem11N 41824 | Lemma for isomorphism H of... |
| dihmeetlem12N 41825 | Lemma for isomorphism H of... |
| dihmeetlem13N 41826 | Lemma for isomorphism H of... |
| dihmeetlem14N 41827 | Lemma for isomorphism H of... |
| dihmeetlem15N 41828 | Lemma for isomorphism H of... |
| dihmeetlem16N 41829 | Lemma for isomorphism H of... |
| dihmeetlem17N 41830 | Lemma for isomorphism H of... |
| dihmeetlem18N 41831 | Lemma for isomorphism H of... |
| dihmeetlem19N 41832 | Lemma for isomorphism H of... |
| dihmeetlem20N 41833 | Lemma for isomorphism H of... |
| dihmeetALTN 41834 | Isomorphism H of a lattice... |
| dih1dimatlem0 41835 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 41836 | Lemma for ~ dih1dimat . (... |
| dih1dimat 41837 | Any 1-dimensional subspace... |
| dihlsprn 41838 | The span of a vector belon... |
| dihlspsnssN 41839 | A subspace included in a 1... |
| dihlspsnat 41840 | The inverse isomorphism H ... |
| dihatlat 41841 | The isomorphism H of an at... |
| dihat 41842 | There exists at least one ... |
| dihpN 41843 | The value of isomorphism H... |
| dihlatat 41844 | The reverse isomorphism H ... |
| dihatexv 41845 | There is a nonzero vector ... |
| dihatexv2 41846 | There is a nonzero vector ... |
| dihglblem6 41847 | Isomorphism H of a lattice... |
| dihglb 41848 | Isomorphism H of a lattice... |
| dihglb2 41849 | Isomorphism H of a lattice... |
| dihmeet 41850 | Isomorphism H of a lattice... |
| dihintcl 41851 | The intersection of closed... |
| dihmeetcl 41852 | Closure of closed subspace... |
| dihmeet2 41853 | Reverse isomorphism H of a... |
| dochffval 41856 | Subspace orthocomplement f... |
| dochfval 41857 | Subspace orthocomplement f... |
| dochval 41858 | Subspace orthocomplement f... |
| dochval2 41859 | Subspace orthocomplement f... |
| dochcl 41860 | Closure of subspace orthoc... |
| dochlss 41861 | A subspace orthocomplement... |
| dochssv 41862 | A subspace orthocomplement... |
| dochfN 41863 | Domain and codomain of the... |
| dochvalr 41864 | Orthocomplement of a close... |
| doch0 41865 | Orthocomplement of the zer... |
| doch1 41866 | Orthocomplement of the uni... |
| dochoc0 41867 | The zero subspace is close... |
| dochoc1 41868 | The unit subspace (all vec... |
| dochvalr2 41869 | Orthocomplement of a close... |
| dochvalr3 41870 | Orthocomplement of a close... |
| doch2val2 41871 | Double orthocomplement for... |
| dochss 41872 | Subset law for orthocomple... |
| dochocss 41873 | Double negative law for or... |
| dochoc 41874 | Double negative law for or... |
| dochsscl 41875 | If a set of vectors is inc... |
| dochoccl 41876 | A set of vectors is closed... |
| dochord 41877 | Ordering law for orthocomp... |
| dochord2N 41878 | Ordering law for orthocomp... |
| dochord3 41879 | Ordering law for orthocomp... |
| doch11 41880 | Orthocomplement is one-to-... |
| dochsordN 41881 | Strict ordering law for or... |
| dochn0nv 41882 | An orthocomplement is nonz... |
| dihoml4c 41883 | Version of ~ dihoml4 with ... |
| dihoml4 41884 | Orthomodular law for const... |
| dochspss 41885 | The span of a set of vecto... |
| dochocsp 41886 | The span of an orthocomple... |
| dochspocN 41887 | The span of an orthocomple... |
| dochocsn 41888 | The double orthocomplement... |
| dochsncom 41889 | Swap vectors in an orthoco... |
| dochsat 41890 | The double orthocomplement... |
| dochshpncl 41891 | If a hyperplane is not clo... |
| dochlkr 41892 | Equivalent conditions for ... |
| dochkrshp 41893 | The closure of a kernel is... |
| dochkrshp2 41894 | Properties of the closure ... |
| dochkrshp3 41895 | Properties of the closure ... |
| dochkrshp4 41896 | Properties of the closure ... |
| dochdmj1 41897 | De Morgan-like law for sub... |
| dochnoncon 41898 | Law of noncontradiction. ... |
| dochnel2 41899 | A nonzero member of a subs... |
| dochnel 41900 | A nonzero vector doesn't b... |
| djhffval 41903 | Subspace join for ` DVecH ... |
| djhfval 41904 | Subspace join for ` DVecH ... |
| djhval 41905 | Subspace join for ` DVecH ... |
| djhval2 41906 | Value of subspace join for... |
| djhcl 41907 | Closure of subspace join f... |
| djhlj 41908 | Transfer lattice join to `... |
| djhljjN 41909 | Lattice join in terms of `... |
| djhjlj 41910 | ` DVecH ` vector space clo... |
| djhj 41911 | ` DVecH ` vector space clo... |
| djhcom 41912 | Subspace join commutes. (... |
| djhspss 41913 | Subspace span of union is ... |
| djhsumss 41914 | Subspace sum is a subset o... |
| dihsumssj 41915 | The subspace sum of two is... |
| djhunssN 41916 | Subspace union is a subset... |
| dochdmm1 41917 | De Morgan-like law for clo... |
| djhexmid 41918 | Excluded middle property o... |
| djh01 41919 | Closed subspace join with ... |
| djh02 41920 | Closed subspace join with ... |
| djhlsmcl 41921 | A closed subspace sum equa... |
| djhcvat42 41922 | A covering property. ( ~ ... |
| dihjatb 41923 | Isomorphism H of lattice j... |
| dihjatc 41924 | Isomorphism H of lattice j... |
| dihjatcclem1 41925 | Lemma for isomorphism H of... |
| dihjatcclem2 41926 | Lemma for isomorphism H of... |
| dihjatcclem3 41927 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 41928 | Lemma for isomorphism H of... |
| dihjatcc 41929 | Isomorphism H of lattice j... |
| dihjat 41930 | Isomorphism H of lattice j... |
| dihprrnlem1N 41931 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 41932 | Lemma for ~ dihprrn . (Co... |
| dihprrn 41933 | The span of a vector pair ... |
| djhlsmat 41934 | The sum of two subspace at... |
| dihjat1lem 41935 | Subspace sum of a closed s... |
| dihjat1 41936 | Subspace sum of a closed s... |
| dihsmsprn 41937 | Subspace sum of a closed s... |
| dihjat2 41938 | The subspace sum of a clos... |
| dihjat3 41939 | Isomorphism H of lattice j... |
| dihjat4 41940 | Transfer the subspace sum ... |
| dihjat6 41941 | Transfer the subspace sum ... |
| dihsmsnrn 41942 | The subspace sum of two si... |
| dihsmatrn 41943 | The subspace sum of a clos... |
| dihjat5N 41944 | Transfer lattice join with... |
| dvh4dimat 41945 | There is an atom that is o... |
| dvh3dimatN 41946 | There is an atom that is o... |
| dvh2dimatN 41947 | Given an atom, there exist... |
| dvh1dimat 41948 | There exists an atom. (Co... |
| dvh1dim 41949 | There exists a nonzero vec... |
| dvh4dimlem 41950 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 41951 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 41952 | There is a vector that is ... |
| dvh3dim 41953 | There is a vector that is ... |
| dvh4dimN 41954 | There is a vector that is ... |
| dvh3dim2 41955 | There is a vector that is ... |
| dvh3dim3N 41956 | There is a vector that is ... |
| dochsnnz 41957 | The orthocomplement of a s... |
| dochsatshp 41958 | The orthocomplement of a s... |
| dochsatshpb 41959 | The orthocomplement of a s... |
| dochsnshp 41960 | The orthocomplement of a n... |
| dochshpsat 41961 | A hyperplane is closed iff... |
| dochkrsat 41962 | The orthocomplement of a k... |
| dochkrsat2 41963 | The orthocomplement of a k... |
| dochsat0 41964 | The orthocomplement of a k... |
| dochkrsm 41965 | The subspace sum of a clos... |
| dochexmidat 41966 | Special case of excluded m... |
| dochexmidlem1 41967 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 41968 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 41969 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 41970 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 41971 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 41972 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 41973 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 41974 | Lemma for ~ dochexmid . T... |
| dochexmid 41975 | Excluded middle law for cl... |
| dochsnkrlem1 41976 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 41977 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 41978 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 41979 | A (closed) kernel expresse... |
| dochsnkr2 41980 | Kernel of the explicit fun... |
| dochsnkr2cl 41981 | The ` X ` determining func... |
| dochflcl 41982 | Closure of the explicit fu... |
| dochfl1 41983 | The value of the explicit ... |
| dochfln0 41984 | The value of a functional ... |
| dochkr1 41985 | A nonzero functional has a... |
| dochkr1OLDN 41986 | A nonzero functional has a... |
| lpolsetN 41989 | The set of polarities of a... |
| islpolN 41990 | The predicate "is a polari... |
| islpoldN 41991 | Properties that determine ... |
| lpolfN 41992 | Functionality of a polarit... |
| lpolvN 41993 | The polarity of the whole ... |
| lpolconN 41994 | Contraposition property of... |
| lpolsatN 41995 | The polarity of an atomic ... |
| lpolpolsatN 41996 | Property of a polarity. (... |
| dochpolN 41997 | The subspace orthocompleme... |
| lcfl1lem 41998 | Property of a functional w... |
| lcfl1 41999 | Property of a functional w... |
| lcfl2 42000 | Property of a functional w... |
| lcfl3 42001 | Property of a functional w... |
| lcfl4N 42002 | Property of a functional w... |
| lcfl5 42003 | Property of a functional w... |
| lcfl5a 42004 | Property of a functional w... |
| lcfl6lem 42005 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 42006 | Lemma for ~ lcfl7N . If t... |
| lcfl6 42007 | Property of a functional w... |
| lcfl7N 42008 | Property of a functional w... |
| lcfl8 42009 | Property of a functional w... |
| lcfl8a 42010 | Property of a functional w... |
| lcfl8b 42011 | Property of a nonzero func... |
| lcfl9a 42012 | Property implying that a f... |
| lclkrlem1 42013 | The set of functionals hav... |
| lclkrlem2a 42014 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 42015 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 42016 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 42017 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 42018 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 42019 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 42020 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 42021 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 42022 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 42023 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 42024 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 42025 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 42026 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 42027 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 42028 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 42029 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 42030 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 42031 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 42032 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 42033 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 42034 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 42035 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 42036 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 42037 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 42038 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 42039 | The set of functionals hav... |
| lclkr 42040 | The set of functionals wit... |
| lcfls1lem 42041 | Property of a functional w... |
| lcfls1N 42042 | Property of a functional w... |
| lcfls1c 42043 | Property of a functional w... |
| lclkrslem1 42044 | The set of functionals hav... |
| lclkrslem2 42045 | The set of functionals hav... |
| lclkrs 42046 | The set of functionals hav... |
| lclkrs2 42047 | The set of functionals wit... |
| lcfrvalsnN 42048 | Reconstruction from the du... |
| lcfrlem1 42049 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 42050 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 42051 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 42052 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 42053 | Lemma for ~ lcfr . The se... |
| lcfrlem6 42054 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 42055 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 42056 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 42057 | Lemma for ~ lcf1o . (This... |
| lcf1o 42058 | Define a function ` J ` th... |
| lcfrlem10 42059 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 42060 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 42061 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 42062 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 42063 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 42064 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 42065 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 42066 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 42067 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 42068 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 42069 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 42070 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 42071 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 42072 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 42073 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 42074 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 42075 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 42076 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 42077 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 42078 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 42079 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 42080 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 42081 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 42082 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 42083 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 42084 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 42085 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 42086 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 42087 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 42088 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 42089 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 42090 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 42091 | Lemma for ~ lcfr . Elimin... |
| lcfr 42092 | Reconstruction of a subspa... |
| lcdfval 42095 | Dual vector space of funct... |
| lcdval 42096 | Dual vector space of funct... |
| lcdval2 42097 | Dual vector space of funct... |
| lcdlvec 42098 | The dual vector space of f... |
| lcdlmod 42099 | The dual vector space of f... |
| lcdvbase 42100 | Vector base set of a dual ... |
| lcdvbasess 42101 | The vector base set of the... |
| lcdvbaselfl 42102 | A vector in the base set o... |
| lcdvbasecl 42103 | Closure of the value of a ... |
| lcdvadd 42104 | Vector addition for the cl... |
| lcdvaddval 42105 | The value of the value of ... |
| lcdsca 42106 | The ring of scalars of the... |
| lcdsbase 42107 | Base set of scalar ring fo... |
| lcdsadd 42108 | Scalar addition for the cl... |
| lcdsmul 42109 | Scalar multiplication for ... |
| lcdvs 42110 | Scalar product for the clo... |
| lcdvsval 42111 | Value of scalar product op... |
| lcdvscl 42112 | The scalar product operati... |
| lcdlssvscl 42113 | Closure of scalar product ... |
| lcdvsass 42114 | Associative law for scalar... |
| lcd0 42115 | The zero scalar of the clo... |
| lcd1 42116 | The unit scalar of the clo... |
| lcdneg 42117 | The unit scalar of the clo... |
| lcd0v 42118 | The zero functional in the... |
| lcd0v2 42119 | The zero functional in the... |
| lcd0vvalN 42120 | Value of the zero function... |
| lcd0vcl 42121 | Closure of the zero functi... |
| lcd0vs 42122 | A scalar zero times a func... |
| lcdvs0N 42123 | A scalar times the zero fu... |
| lcdvsub 42124 | The value of vector subtra... |
| lcdvsubval 42125 | The value of the value of ... |
| lcdlss 42126 | Subspaces of a dual vector... |
| lcdlss2N 42127 | Subspaces of a dual vector... |
| lcdlsp 42128 | Span in the set of functio... |
| lcdlkreqN 42129 | Colinear functionals have ... |
| lcdlkreq2N 42130 | Colinear functionals have ... |
| mapdffval 42133 | Projectivity from vector s... |
| mapdfval 42134 | Projectivity from vector s... |
| mapdval 42135 | Value of projectivity from... |
| mapdvalc 42136 | Value of projectivity from... |
| mapdval2N 42137 | Value of projectivity from... |
| mapdval3N 42138 | Value of projectivity from... |
| mapdval4N 42139 | Value of projectivity from... |
| mapdval5N 42140 | Value of projectivity from... |
| mapdordlem1a 42141 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 42142 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 42143 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 42144 | Lemma for ~ mapdord . Ord... |
| mapdord 42145 | Ordering property of the m... |
| mapd11 42146 | The map defined by ~ df-ma... |
| mapddlssN 42147 | The mapping of a subspace ... |
| mapdsn 42148 | Value of the map defined b... |
| mapdsn2 42149 | Value of the map defined b... |
| mapdsn3 42150 | Value of the map defined b... |
| mapd1dim2lem1N 42151 | Value of the map defined b... |
| mapdrvallem2 42152 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 42153 | Lemma for ~ mapdrval . (C... |
| mapdrval 42154 | Given a dual subspace ` R ... |
| mapd1o 42155 | The map defined by ~ df-ma... |
| mapdrn 42156 | Range of the map defined b... |
| mapdunirnN 42157 | Union of the range of the ... |
| mapdrn2 42158 | Range of the map defined b... |
| mapdcnvcl 42159 | Closure of the converse of... |
| mapdcl 42160 | Closure the value of the m... |
| mapdcnvid1N 42161 | Converse of the value of t... |
| mapdsord 42162 | Strong ordering property o... |
| mapdcl2 42163 | The mapping of a subspace ... |
| mapdcnvid2 42164 | Value of the converse of t... |
| mapdcnvordN 42165 | Ordering property of the c... |
| mapdcnv11N 42166 | The converse of the map de... |
| mapdcv 42167 | Covering property of the c... |
| mapdincl 42168 | Closure of dual subspace i... |
| mapdin 42169 | Subspace intersection is p... |
| mapdlsmcl 42170 | Closure of dual subspace s... |
| mapdlsm 42171 | Subspace sum is preserved ... |
| mapd0 42172 | Projectivity map of the ze... |
| mapdcnvatN 42173 | Atoms are preserved by the... |
| mapdat 42174 | Atoms are preserved by the... |
| mapdspex 42175 | The map of a span equals t... |
| mapdn0 42176 | Transfer nonzero property ... |
| mapdncol 42177 | Transfer non-colinearity f... |
| mapdindp 42178 | Transfer (part of) vector ... |
| mapdpglem1 42179 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 42180 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 42181 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 42182 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 42183 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 42184 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 42185 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 42186 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 42187 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 42188 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 42189 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 42190 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 42191 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 42192 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 42193 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 42194 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 42195 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 42196 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 42197 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 42198 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 42199 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 42200 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 42201 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 42202 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 42203 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 42204 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 42205 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 42206 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 42207 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 42208 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 42209 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 42210 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 42211 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 42212 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 42213 | Part 1 of proof of the fir... |
| baerlem3lem1 42214 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 42215 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 42216 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 42217 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 42218 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 42219 | Lemma for ~ baerlem5b . (... |
| baerlem3 42220 | An equality that holds whe... |
| baerlem5a 42221 | An equality that holds whe... |
| baerlem5b 42222 | An equality that holds whe... |
| baerlem5amN 42223 | An equality that holds whe... |
| baerlem5bmN 42224 | An equality that holds whe... |
| baerlem5abmN 42225 | An equality that holds whe... |
| mapdindp0 42226 | Vector independence lemma.... |
| mapdindp1 42227 | Vector independence lemma.... |
| mapdindp2 42228 | Vector independence lemma.... |
| mapdindp3 42229 | Vector independence lemma.... |
| mapdindp4 42230 | Vector independence lemma.... |
| mapdhval 42231 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 42232 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 42233 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 42234 | Lemmma for ~~? mapdh . (C... |
| mapdheq 42235 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 42236 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 42237 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 42238 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 42239 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 42240 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 42241 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 42242 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 42243 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 42244 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 42245 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 42246 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 42247 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 42248 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 42249 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 42250 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 42251 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 42252 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 42253 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 42254 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 42255 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 42256 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 42257 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 42258 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 42259 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 42260 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 42261 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 42262 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 42265 | Map from nonzero vectors t... |
| hvmapfval 42266 | Map from nonzero vectors t... |
| hvmapval 42267 | Value of map from nonzero ... |
| hvmapvalvalN 42268 | Value of value of map (i.e... |
| hvmapidN 42269 | The value of the vector to... |
| hvmap1o 42270 | The vector to functional m... |
| hvmapclN 42271 | Closure of the vector to f... |
| hvmap1o2 42272 | The vector to functional m... |
| hvmapcl2 42273 | Closure of the vector to f... |
| hvmaplfl 42274 | The vector to functional m... |
| hvmaplkr 42275 | Kernel of the vector to fu... |
| mapdhvmap 42276 | Relationship between ` map... |
| lspindp5 42277 | Obtain an independent vect... |
| hdmaplem1 42278 | Lemma to convert a frequen... |
| hdmaplem2N 42279 | Lemma to convert a frequen... |
| hdmaplem3 42280 | Lemma to convert a frequen... |
| hdmaplem4 42281 | Lemma to convert a frequen... |
| mapdh8a 42282 | Part of Part (8) in [Baer]... |
| mapdh8aa 42283 | Part of Part (8) in [Baer]... |
| mapdh8ab 42284 | Part of Part (8) in [Baer]... |
| mapdh8ac 42285 | Part of Part (8) in [Baer]... |
| mapdh8ad 42286 | Part of Part (8) in [Baer]... |
| mapdh8b 42287 | Part of Part (8) in [Baer]... |
| mapdh8c 42288 | Part of Part (8) in [Baer]... |
| mapdh8d0N 42289 | Part of Part (8) in [Baer]... |
| mapdh8d 42290 | Part of Part (8) in [Baer]... |
| mapdh8e 42291 | Part of Part (8) in [Baer]... |
| mapdh8g 42292 | Part of Part (8) in [Baer]... |
| mapdh8i 42293 | Part of Part (8) in [Baer]... |
| mapdh8j 42294 | Part of Part (8) in [Baer]... |
| mapdh8 42295 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 42296 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 42297 | Lemma for part (9) in [Bae... |
| hdmap1ffval 42302 | Preliminary map from vecto... |
| hdmap1fval 42303 | Preliminary map from vecto... |
| hdmap1vallem 42304 | Value of preliminary map f... |
| hdmap1val 42305 | Value of preliminary map f... |
| hdmap1val0 42306 | Value of preliminary map f... |
| hdmap1val2 42307 | Value of preliminary map f... |
| hdmap1eq 42308 | The defining equation for ... |
| hdmap1cbv 42309 | Frequently used lemma to c... |
| hdmap1valc 42310 | Connect the value of the p... |
| hdmap1cl 42311 | Convert closure theorem ~ ... |
| hdmap1eq2 42312 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 42313 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 42314 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 42315 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 42316 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 42317 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 42318 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 42319 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 42320 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 42321 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 42322 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 42323 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 42324 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 42325 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 42326 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 42327 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 42328 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 42329 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 42330 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 42331 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 42332 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 42333 | Map from vectors to functi... |
| hdmapfval 42334 | Map from vectors to functi... |
| hdmapval 42335 | Value of map from vectors ... |
| hdmapfnN 42336 | Functionality of map from ... |
| hdmapcl 42337 | Closure of map from vector... |
| hdmapval2lem 42338 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 42339 | Value of map from vectors ... |
| hdmapval0 42340 | Value of map from vectors ... |
| hdmapeveclem 42341 | Lemma for ~ hdmapevec . T... |
| hdmapevec 42342 | Value of map from vectors ... |
| hdmapevec2 42343 | The inner product of the r... |
| hdmapval3lemN 42344 | Value of map from vectors ... |
| hdmapval3N 42345 | Value of map from vectors ... |
| hdmap10lem 42346 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 42347 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 42348 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 42349 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 42350 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 42351 | Part of proof of part 12 i... |
| hdmapnzcl 42352 | Nonzero vector closure of ... |
| hdmapneg 42353 | Part of proof of part 12 i... |
| hdmapsub 42354 | Part of proof of part 12 i... |
| hdmap11 42355 | Part of proof of part 12 i... |
| hdmaprnlem1N 42356 | Part of proof of part 12 i... |
| hdmaprnlem3N 42357 | Part of proof of part 12 i... |
| hdmaprnlem3uN 42358 | Part of proof of part 12 i... |
| hdmaprnlem4tN 42359 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 42360 | Part of proof of part 12 i... |
| hdmaprnlem6N 42361 | Part of proof of part 12 i... |
| hdmaprnlem7N 42362 | Part of proof of part 12 i... |
| hdmaprnlem8N 42363 | Part of proof of part 12 i... |
| hdmaprnlem9N 42364 | Part of proof of part 12 i... |
| hdmaprnlem3eN 42365 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 42366 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 42367 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 42368 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 42369 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 42370 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 42371 | Part of proof of part 12 i... |
| hdmapf1oN 42372 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 42373 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 42374 | Prior to part 14 in [Baer]... |
| hdmap14lem1 42375 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 42376 | Prior to part 14 in [Baer]... |
| hdmap14lem3 42377 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 42378 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 42379 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 42380 | Case where ` F ` is zero. ... |
| hdmap14lem7 42381 | Combine cases of ` F ` . ... |
| hdmap14lem8 42382 | Part of proof of part 14 i... |
| hdmap14lem9 42383 | Part of proof of part 14 i... |
| hdmap14lem10 42384 | Part of proof of part 14 i... |
| hdmap14lem11 42385 | Part of proof of part 14 i... |
| hdmap14lem12 42386 | Lemma for proof of part 14... |
| hdmap14lem13 42387 | Lemma for proof of part 14... |
| hdmap14lem14 42388 | Part of proof of part 14 i... |
| hdmap14lem15 42389 | Part of proof of part 14 i... |
| hgmapffval 42392 | Map from the scalar divisi... |
| hgmapfval 42393 | Map from the scalar divisi... |
| hgmapval 42394 | Value of map from the scal... |
| hgmapfnN 42395 | Functionality of scalar si... |
| hgmapcl 42396 | Closure of scalar sigma ma... |
| hgmapdcl 42397 | Closure of the vector spac... |
| hgmapvs 42398 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 42399 | Value of the scalar sigma ... |
| hgmapval1 42400 | Value of the scalar sigma ... |
| hgmapadd 42401 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 42402 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 42403 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 42404 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 42405 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 42406 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 42407 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 42408 | Part of proof of part 16 i... |
| hgmap11 42409 | The scalar sigma map is on... |
| hgmapf1oN 42410 | The scalar sigma map is a ... |
| hgmapeq0 42411 | The scalar sigma map is ze... |
| hdmapipcl 42412 | The inner product (Hermiti... |
| hdmapln1 42413 | Linearity property that wi... |
| hdmaplna1 42414 | Additive property of first... |
| hdmaplns1 42415 | Subtraction property of fi... |
| hdmaplnm1 42416 | Multiplicative property of... |
| hdmaplna2 42417 | Additive property of secon... |
| hdmapglnm2 42418 | g-linear property of secon... |
| hdmapgln2 42419 | g-linear property that wil... |
| hdmaplkr 42420 | Kernel of the vector to du... |
| hdmapellkr 42421 | Membership in the kernel (... |
| hdmapip0 42422 | Zero property that will be... |
| hdmapip1 42423 | Construct a proportional v... |
| hdmapip0com 42424 | Commutation property of Ba... |
| hdmapinvlem1 42425 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 42426 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 42427 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 42428 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 42429 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 42430 | Involution property of sca... |
| hgmapvvlem2 42431 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 42432 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 42433 | Value of a double involuti... |
| hdmapglem7a 42434 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 42435 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 42436 | Lemma for ~ hdmapg . Line... |
| hdmapg 42437 | Apply the scalar sigma fun... |
| hdmapoc 42438 | Express our constructed or... |
| hlhilset 42441 | The final Hilbert space co... |
| hlhilsca 42442 | The scalar of the final co... |
| hlhilbase 42443 | The base set of the final ... |
| hlhilplus 42444 | The vector addition for th... |
| hlhilslem 42445 | Lemma for ~ hlhilsbase etc... |
| hlhilsbase 42446 | The scalar base set of the... |
| hlhilsplus 42447 | Scalar addition for the fi... |
| hlhilsmul 42448 | Scalar multiplication for ... |
| hlhilsbase2 42449 | The scalar base set of the... |
| hlhilsplus2 42450 | Scalar addition for the fi... |
| hlhilsmul2 42451 | Scalar multiplication for ... |
| hlhils0 42452 | The scalar ring zero for t... |
| hlhils1N 42453 | The scalar ring unity for ... |
| hlhilvsca 42454 | The scalar product for the... |
| hlhilip 42455 | Inner product operation fo... |
| hlhilipval 42456 | Value of inner product ope... |
| hlhilnvl 42457 | The involution operation o... |
| hlhillvec 42458 | The final constructed Hilb... |
| hlhildrng 42459 | The star division ring for... |
| hlhilsrnglem 42460 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 42461 | The star division ring for... |
| hlhil0 42462 | The zero vector for the fi... |
| hlhillsm 42463 | The vector sum operation f... |
| hlhilocv 42464 | The orthocomplement for th... |
| hlhillcs 42465 | The closed subspaces of th... |
| hlhilphllem 42466 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 42467 | Lemma for ~ hlhil . (Cont... |
| hlathil 42468 | Construction of a Hilbert ... |
| iscsrg 42471 | A commutative semiring is ... |
| rhmzrhval 42472 | Evaluation of integers acr... |
| zndvdchrrhm 42473 | Construction of a ring hom... |
| relogbcld 42474 | Closure of the general log... |
| relogbexpd 42475 | Identity law for general l... |
| relogbzexpd 42476 | Power law for the general ... |
| logblebd 42477 | The general logarithm is m... |
| uzindd 42478 | Induction on the upper int... |
| fzadd2d 42479 | Membership of a sum in a f... |
| fzne2d 42480 | Elementhood in a finite se... |
| eqfnfv2d2 42481 | Equality of functions is d... |
| fzsplitnd 42482 | Split a finite interval of... |
| fzsplitnr 42483 | Split a finite interval of... |
| addassnni 42484 | Associative law for additi... |
| addcomnni 42485 | Commutative law for additi... |
| mulassnni 42486 | Associative law for multip... |
| mulcomnni 42487 | Commutative law for multip... |
| gcdcomnni 42488 | Commutative law for gcd. ... |
| gcdnegnni 42489 | Negation invariance for gc... |
| neggcdnni 42490 | Negation invariance for gc... |
| bccl2d 42491 | Closure of the binomial co... |
| recbothd 42492 | Take reciprocal on both si... |
| gcdmultiplei 42493 | The GCD of a multiple of a... |
| gcdaddmzz2nni 42494 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 42495 | Adding a multiple of one o... |
| gcdnncli 42496 | Closure of the gcd operato... |
| muldvds1d 42497 | If a product divides an in... |
| muldvds2d 42498 | If a product divides an in... |
| nndivdvdsd 42499 | A positive integer divides... |
| nnproddivdvdsd 42500 | A product of natural numbe... |
| coprmdvds2d 42501 | If an integer is divisible... |
| imadomfi 42502 | An image of a function und... |
| 12gcd5e1 42503 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 42504 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 42505 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 42506 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 42507 | Calculate the least common... |
| 12lcm5e60 42508 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 42509 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 42510 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 42511 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 42512 | Useful equation to calcula... |
| lcm1un 42513 | Least common multiple of n... |
| lcm2un 42514 | Least common multiple of n... |
| lcm3un 42515 | Least common multiple of n... |
| lcm4un 42516 | Least common multiple of n... |
| lcm5un 42517 | Least common multiple of n... |
| lcm6un 42518 | Least common multiple of n... |
| lcm7un 42519 | Least common multiple of n... |
| lcm8un 42520 | Least common multiple of n... |
| 3factsumint1 42521 | Move constants out of inte... |
| 3factsumint2 42522 | Move constants out of inte... |
| 3factsumint3 42523 | Move constants out of inte... |
| 3factsumint4 42524 | Move constants out of inte... |
| 3factsumint 42525 | Helpful equation for lcm i... |
| resopunitintvd 42526 | Restrict continuous functi... |
| resclunitintvd 42527 | Restrict continuous functi... |
| resdvopclptsd 42528 | Restrict derivative on uni... |
| lcmineqlem1 42529 | Part of lcm inequality lem... |
| lcmineqlem2 42530 | Part of lcm inequality lem... |
| lcmineqlem3 42531 | Part of lcm inequality lem... |
| lcmineqlem4 42532 | Part of lcm inequality lem... |
| lcmineqlem5 42533 | Technical lemma for recipr... |
| lcmineqlem6 42534 | Part of lcm inequality lem... |
| lcmineqlem7 42535 | Derivative of 1-x for chai... |
| lcmineqlem8 42536 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 42537 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 42538 | Induction step of ~ lcmine... |
| lcmineqlem11 42539 | Induction step, continuati... |
| lcmineqlem12 42540 | Base case for induction. ... |
| lcmineqlem13 42541 | Induction proof for lcm in... |
| lcmineqlem14 42542 | Technical lemma for inequa... |
| lcmineqlem15 42543 | F times the least common m... |
| lcmineqlem16 42544 | Technical divisibility lem... |
| lcmineqlem17 42545 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 42546 | Technical lemma to shift f... |
| lcmineqlem19 42547 | Dividing implies inequalit... |
| lcmineqlem20 42548 | Inequality for lcm lemma. ... |
| lcmineqlem21 42549 | The lcm inequality lemma w... |
| lcmineqlem22 42550 | The lcm inequality lemma w... |
| lcmineqlem23 42551 | Penultimate step to the lc... |
| lcmineqlem 42552 | The least common multiple ... |
| 3exp7 42553 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 42554 | First inequality in inequa... |
| 3lexlogpow5ineq2 42555 | Second inequality in inequ... |
| 3lexlogpow5ineq4 42556 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 42557 | Combined inequality chain ... |
| 3lexlogpow2ineq1 42558 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 42559 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 42560 | Result for bound in AKS in... |
| intlewftc 42561 | Inequality inference by in... |
| aks4d1lem1 42562 | Technical lemma to reduce ... |
| aks4d1p1p1 42563 | Exponential law for finite... |
| dvrelog2 42564 | The derivative of the loga... |
| dvrelog3 42565 | The derivative of the loga... |
| dvrelog2b 42566 | Derivative of the binary l... |
| 0nonelalab 42567 | Technical lemma for open i... |
| dvrelogpow2b 42568 | Derivative of the power of... |
| aks4d1p1p3 42569 | Bound of a ceiling of the ... |
| aks4d1p1p2 42570 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 42571 | Technical step for inequal... |
| dvle2 42572 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 42573 | Inequality lift to differe... |
| aks4d1p1p7 42574 | Bound of intermediary of i... |
| aks4d1p1p5 42575 | Show inequality for existe... |
| aks4d1p1 42576 | Show inequality for existe... |
| aks4d1p2 42577 | Technical lemma for existe... |
| aks4d1p3 42578 | There exists a small enoug... |
| aks4d1p4 42579 | There exists a small enoug... |
| aks4d1p5 42580 | Show that ` N ` and ` R ` ... |
| aks4d1p6 42581 | The maximal prime power ex... |
| aks4d1p7d1 42582 | Technical step in AKS lemm... |
| aks4d1p7 42583 | Technical step in AKS lemm... |
| aks4d1p8d1 42584 | If a prime divides one num... |
| aks4d1p8d2 42585 | Any prime power dividing a... |
| aks4d1p8d3 42586 | The remainder of a divisio... |
| aks4d1p8 42587 | Show that ` N ` and ` R ` ... |
| aks4d1p9 42588 | Show that the order is bou... |
| aks4d1 42589 | Lemma 4.1 from ~ https://w... |
| fldhmf1 42590 | A field homomorphism is in... |
| isprimroot 42593 | The value of a primitive r... |
| isprimroot2 42594 | Alternative way of creatin... |
| mndmolinv 42595 | An element of a monoid tha... |
| linvh 42596 | If an element has a unique... |
| primrootsunit1 42597 | Primitive roots have left ... |
| primrootsunit 42598 | Primitive roots have left ... |
| primrootscoprmpow 42599 | Coprime powers of primitiv... |
| posbezout 42600 | Bezout's identity restrict... |
| primrootscoprf 42601 | Coprime powers of primitiv... |
| primrootscoprbij 42602 | A bijection between coprim... |
| primrootscoprbij2 42603 | A bijection between coprim... |
| remexz 42604 | Division with rest. (Cont... |
| primrootlekpowne0 42605 | There is no smaller power ... |
| primrootspoweq0 42606 | The power of a ` R ` -th p... |
| aks6d1c1p1 42607 | Definition of the introspe... |
| aks6d1c1p1rcl 42608 | Reverse closure of the int... |
| aks6d1c1p2 42609 | ` P ` and linear factors a... |
| aks6d1c1p3 42610 | In a field with a Frobeniu... |
| aks6d1c1p4 42611 | The product of polynomials... |
| aks6d1c1p5 42612 | The product of exponents i... |
| aks6d1c1p7 42613 | ` X ` is introspective to ... |
| aks6d1c1p6 42614 | If a polynomials ` F ` is ... |
| aks6d1c1p8 42615 | If a number ` E ` is intro... |
| aks6d1c1 42616 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 42617 | Polynomial evaluation buil... |
| aks6d1c2p1 42618 | In the AKS-theorem the sub... |
| aks6d1c2p2 42619 | Injective condition for co... |
| hashscontpowcl 42620 | Closure of E for ~ https:/... |
| hashscontpow1 42621 | Helper lemma for to prove ... |
| hashscontpow 42622 | If a set contains all ` N ... |
| aks6d1c3 42623 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 42624 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 42625 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 42626 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 42627 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 42628 | If the number of elements ... |
| hashnexinjle 42629 | If the number of elements ... |
| aks6d1c2 42630 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 42631 | Special case related to ~ ... |
| idomnnzpownz 42632 | A nonzero power in an inte... |
| idomnnzgmulnz 42633 | A finite product of nonzer... |
| ringexp0nn 42634 | Zero to the power of a pos... |
| aks6d1c5lem0 42635 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 42636 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 42637 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 42638 | Lemma for Claim 5, contrad... |
| aks6d1c5 42639 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 42640 | Degree multiplication is a... |
| deg1pow 42641 | Exact degree of a power of... |
| 5bc2eq10 42642 | The value of 5 choose 2. ... |
| facp2 42643 | The factorial of a success... |
| 2np3bcnp1 42644 | Part of induction step for... |
| 2ap1caineq 42645 | Inequality for Theorem 6.6... |
| sticksstones1 42646 | Different strictly monoton... |
| sticksstones2 42647 | The range function on stri... |
| sticksstones3 42648 | The range function on stri... |
| sticksstones4 42649 | Equinumerosity lemma for s... |
| sticksstones5 42650 | Count the number of strict... |
| sticksstones6 42651 | Function induces an order ... |
| sticksstones7 42652 | Closure property of sticks... |
| sticksstones8 42653 | Establish mapping between ... |
| sticksstones9 42654 | Establish mapping between ... |
| sticksstones10 42655 | Establish mapping between ... |
| sticksstones11 42656 | Establish bijective mappin... |
| sticksstones12a 42657 | Establish bijective mappin... |
| sticksstones12 42658 | Establish bijective mappin... |
| sticksstones13 42659 | Establish bijective mappin... |
| sticksstones14 42660 | Sticks and stones with def... |
| sticksstones15 42661 | Sticks and stones with alm... |
| sticksstones16 42662 | Sticks and stones with col... |
| sticksstones17 42663 | Extend sticks and stones t... |
| sticksstones18 42664 | Extend sticks and stones t... |
| sticksstones19 42665 | Extend sticks and stones t... |
| sticksstones20 42666 | Lift sticks and stones to ... |
| sticksstones21 42667 | Lift sticks and stones to ... |
| sticksstones22 42668 | Non-exhaustive sticks and ... |
| sticksstones23 42669 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 42670 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 42671 | Every primitive root is ro... |
| aks6d1c6lem3 42672 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 42673 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 42674 | Lemma to construct the map... |
| aks6d1c6isolem2 42675 | Lemma to construct the gro... |
| aks6d1c6isolem3 42676 | The preimage of a map send... |
| aks6d1c6lem5 42677 | Eliminate the size hypothe... |
| bcled 42678 | Inequality for binomial co... |
| bcle2d 42679 | Inequality for binomial co... |
| aks6d1c7lem1 42680 | The last set of inequaliti... |
| aks6d1c7lem2 42681 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 42682 | Remove lots of hypotheses ... |
| aks6d1c7lem4 42683 | In the AKS algorithm there... |
| aks6d1c7 42684 | ` N ` is a prime power if ... |
| rhmqusspan 42685 | Ring homomorphism out of a... |
| aks5lem1 42686 | Section 5 of ~ https://www... |
| aks5lem2 42687 | Lemma for section 5 ~ http... |
| ply1asclzrhval 42688 | Transfer results from alge... |
| aks5lem3a 42689 | Lemma for AKS section 5. ... |
| aks5lem4a 42690 | Lemma for AKS section 5, r... |
| aks5lem5a 42691 | Lemma for AKS, section 5, ... |
| aks5lem6 42692 | Connect results of section... |
| indstrd 42693 | Strong induction, deductio... |
| grpods 42694 | Relate sums of elements of... |
| unitscyglem1 42695 | Lemma for unitscyg . (Con... |
| unitscyglem2 42696 | Lemma for unitscyg . (Con... |
| unitscyglem3 42697 | Lemma for unitscyg . (Con... |
| unitscyglem4 42698 | Lemma for unitscyg . (Con... |
| unitscyglem5 42699 | Lemma for unitscyg . (Con... |
| aks5lem7 42700 | Lemma for aks5. We clean ... |
| aks5lem8 42701 | Lemma for aks5. Clean up ... |
| exfinfldd 42703 | For any prime ` P ` and an... |
| aks5 42704 | The AKS Primality test, gi... |
| jarrii 42705 | Inference associated with ... |
| intnanrt 42706 | Introduction of conjunct i... |
| ioin9i8 42707 | Miscellaneous inference cr... |
| jaodd 42708 | Double deduction form of ~... |
| syl3an12 42709 | A double syllogism inferen... |
| exbiii 42710 | Inference associated with ... |
| sbtd 42711 | A true statement is true u... |
| sbor2 42712 | One direction of ~ sbor , ... |
| sbalexi 42713 | Inference form of ~ sbalex... |
| nfalh 42714 | Version of ~ nfal with an ... |
| nfe2 42715 | An inner existential quant... |
| nfale2 42716 | An inner existential quant... |
| 19.9dev 42717 | ~ 19.9d in the case of an ... |
| 3rspcedvd 42718 | Triple application of ~ rs... |
| sn-axrep5v 42719 | A condensed form of ~ axre... |
| sn-axprlem3 42720 | ~ axprlem3 using only Tars... |
| sn-exelALT 42721 | Alternate proof of ~ exel ... |
| ssabdv 42722 | Deduction of abstraction s... |
| sn-iotalem 42723 | An unused lemma showing th... |
| sn-iotalemcor 42724 | Corollary of ~ sn-iotalem ... |
| abbi1sn 42725 | Originally part of ~ uniab... |
| brif2 42726 | Move a relation inside and... |
| brif12 42727 | Move a relation inside and... |
| pssexg 42728 | The proper subset of a set... |
| pssn0 42729 | A proper superset is nonem... |
| psspwb 42730 | Classes are proper subclas... |
| xppss12 42731 | Proper subset theorem for ... |
| elpwbi 42732 | Membership in a power set,... |
| imaopab 42733 | The image of a class of or... |
| eqresfnbd 42734 | Property of being the rest... |
| fmpocos 42735 | Composition of two functio... |
| ovmpogad 42736 | Value of an operation give... |
| ofun 42737 | A function operation of un... |
| dfqs3 42738 | Alternate definition of qu... |
| qseq12d 42739 | Equality theorem for quoti... |
| qsalrel 42740 | The quotient set is equal ... |
| supinf 42741 | The supremum is the infimu... |
| mapcod 42742 | Compose two mappings. (Co... |
| fisdomnn 42743 | A finite set is dominated ... |
| ltex 42744 | The less-than relation is ... |
| leex 42745 | The less-than-or-equal-to ... |
| subex 42746 | The subtraction operation ... |
| absex 42747 | The absolute value functio... |
| cjex 42748 | The conjugate function is ... |
| fzosumm1 42749 | Separate out the last term... |
| ccatcan2d 42750 | Cancellation law for conca... |
| c0exALT 42751 | Alternate proof of ~ c0ex ... |
| 0cnALT3 42752 | Alternate proof of ~ 0cn u... |
| elre0re 42753 | Specialized version of ~ 0... |
| lttrii 42754 | 'Less than' is transitive.... |
| remulcan2d 42755 | ~ mulcan2d for real number... |
| readdridaddlidd 42756 | Given some real number ` B... |
| 1p3e4 42757 | 1 + 3 = 4. (Contributed b... |
| 5ne0 42758 | The number 5 is nonzero. ... |
| 6ne0 42759 | The number 6 is nonzero. ... |
| 7ne0 42760 | The number 7 is nonzero. ... |
| 8ne0 42761 | The number 8 is nonzero. ... |
| 9ne0 42762 | The number 9 is nonzero. ... |
| sn-1ne2 42763 | A proof of ~ 1ne2 without ... |
| nnn1suc 42764 | A positive integer that is... |
| readdrcl2d 42765 | Reverse closure for additi... |
| mvrrsubd 42766 | Move a subtraction in the ... |
| laddrotrd 42767 | Rotate the variables right... |
| raddswap12d 42768 | Swap the first two variabl... |
| lsubrotld 42769 | Rotate the variables left ... |
| rsubrotld 42770 | Rotate the variables left ... |
| lsubswap23d 42771 | Swap the second and third ... |
| addsubeq4com 42772 | Relation between sums and ... |
| sqsumi 42773 | A sum squared. (Contribut... |
| negn0nposznnd 42774 | Lemma for ~ dffltz . (Con... |
| sqmid3api 42775 | Value of the square of the... |
| decaddcom 42776 | Commute ones place in addi... |
| sqn5i 42777 | The square of a number end... |
| sqn5ii 42778 | The square of a number end... |
| decpmulnc 42779 | Partial products algorithm... |
| decpmul 42780 | Partial products algorithm... |
| sqdeccom12 42781 | The square of a number in ... |
| sq3deccom12 42782 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 42783 | 4 times 5 equals 20. (Con... |
| 3rdpwhole 42784 | A third of a number plus t... |
| sq4 42785 | The square of 4 is 16. (C... |
| sq5 42786 | The square of 5 is 25. (C... |
| sq6 42787 | The square of 6 is 36. (C... |
| sq7 42788 | The square of 7 is 49. (C... |
| sq8 42789 | The square of 8 is 64. (C... |
| sq9 42790 | The square of 9 is 81. (C... |
| rpsscn 42791 | The positive reals are a s... |
| 4rp 42792 | 4 is a positive real. (Co... |
| 6rp 42793 | 6 is a positive real. (Co... |
| 7rp 42794 | 7 is a positive real. (Co... |
| 8rp 42795 | 8 is a positive real. (Co... |
| 9rp 42796 | 9 is a positive real. (Co... |
| 235t711 42797 | Calculate a product by lon... |
| ex-decpmul 42798 | Example usage of ~ decpmul... |
| eluzp1 42799 | Membership in a successor ... |
| sn-eluzp1l 42800 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 42801 | The sum of ` N ` constant ... |
| fz1sump1 42802 | Add one more term to a sum... |
| oddnumth 42803 | The Odd Number Theorem. T... |
| nicomachus 42804 | Nicomachus's Theorem. The... |
| sumcubes 42805 | The sum of the first ` N `... |
| ine1 42806 | ` _i ` is not 1. (Contrib... |
| 0tie0 42807 | 0 times ` _i ` equals 0. ... |
| it1ei 42808 | ` _i ` times 1 equals ` _i... |
| 1tiei 42809 | 1 times ` _i ` equals ` _i... |
| itrere 42810 | ` _i ` times a real is rea... |
| retire 42811 | A real times ` _i ` is rea... |
| iocioodisjd 42812 | Adjacent intervals where t... |
| rpabsid 42813 | A positive real is its own... |
| oexpreposd 42814 | Lemma for ~ dffltz . For ... |
| explt1d 42815 | A nonnegative real number ... |
| expeq1d 42816 | A nonnegative real number ... |
| expeqidd 42817 | A nonnegative real number ... |
| exp11d 42818 | ~ exp11nnd for nonzero int... |
| 0dvds0 42819 | 0 divides 0. (Contributed... |
| absdvdsabsb 42820 | Divisibility is invariant ... |
| gcdnn0id 42821 | The ` gcd ` of a nonnegati... |
| gcdle1d 42822 | The greatest common diviso... |
| gcdle2d 42823 | The greatest common diviso... |
| dvdsexpad 42824 | Deduction associated with ... |
| dvdsexpnn 42825 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 42826 | ~ dvdsexpnn generalized to... |
| dvdsexpb 42827 | ~ dvdssq generalized to po... |
| posqsqznn 42828 | When a positive rational s... |
| zdivgd 42829 | Two ways to express " ` N ... |
| efsubd 42830 | Difference of exponents la... |
| ef11d 42831 | General condition for the ... |
| logccne0d 42832 | The logarithm isn't 0 if i... |
| cxp112d 42833 | General condition for comp... |
| cxp111d 42834 | General condition for comp... |
| cxpi11d 42835 | ` _i ` to the powers of ` ... |
| logne0d 42836 | Deduction form of ~ logne0... |
| rxp112d 42837 | Real exponentiation is one... |
| log11d 42838 | The natural logarithm is o... |
| rplog11d 42839 | The natural logarithm is o... |
| rxp11d 42840 | Real exponentiation is one... |
| tanhalfpim 42841 | The tangent of ` _pi / 2 `... |
| sinpim 42842 | Sine of a number subtracte... |
| cospim 42843 | Cosine of a number subtrac... |
| tan3rdpi 42844 | The tangent of ` _pi / 3 `... |
| sin2t3rdpi 42845 | The sine of ` 2 x. ( _pi /... |
| cos2t3rdpi 42846 | The cosine of ` 2 x. ( _pi... |
| sin4t3rdpi 42847 | The sine of ` 4 x. ( _pi /... |
| cos4t3rdpi 42848 | The cosine of ` 4 x. ( _pi... |
| asin1half 42849 | The arcsine of ` 1 / 2 ` i... |
| acos1half 42850 | The arccosine of ` 1 / 2 `... |
| dvun 42851 | Condition for the union of... |
| redvmptabs 42852 | The derivative of the abso... |
| readvrec2 42853 | The antiderivative of 1/x ... |
| readvrec 42854 | For real numbers, the anti... |
| resuppsinopn 42855 | The support of sin ( ~ df-... |
| readvcot 42856 | Real antiderivative of cot... |
| resubval 42859 | Value of real subtraction,... |
| renegeulemv 42860 | Lemma for ~ renegeu and si... |
| renegeulem 42861 | Lemma for ~ renegeu and si... |
| renegeu 42862 | Existential uniqueness of ... |
| rernegcl 42863 | Closure law for negative r... |
| renegadd 42864 | Relationship between real ... |
| renegid 42865 | Addition of a real number ... |
| reneg0addlid 42866 | Negative zero is a left ad... |
| resubeulem1 42867 | Lemma for ~ resubeu . A v... |
| resubeulem2 42868 | Lemma for ~ resubeu . A v... |
| resubeu 42869 | Existential uniqueness of ... |
| rersubcl 42870 | Closure for real subtracti... |
| resubadd 42871 | Relation between real subt... |
| resubaddd 42872 | Relationship between subtr... |
| resubf 42873 | Real subtraction is an ope... |
| repncan2 42874 | Addition and subtraction o... |
| repncan3 42875 | Addition and subtraction o... |
| readdsub 42876 | Law for addition and subtr... |
| reladdrsub 42877 | Move LHS of a sum into RHS... |
| reltsub1 42878 | Subtraction from both side... |
| reltsubadd2 42879 | 'Less than' relationship b... |
| resubcan2 42880 | Cancellation law for real ... |
| resubsub4 42881 | Law for double subtraction... |
| rennncan2 42882 | Cancellation law for real ... |
| renpncan3 42883 | Cancellation law for real ... |
| repnpcan 42884 | Cancellation law for addit... |
| reppncan 42885 | Cancellation law for mixed... |
| resubidaddlidlem 42886 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 42887 | Any real number subtracted... |
| resubdi 42888 | Distribution of multiplica... |
| re1m1e0m0 42889 | Equality of two left-addit... |
| sn-00idlem1 42890 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 42891 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 42892 | Lemma for ~ sn-00id . (Co... |
| sn-00id 42893 | ~ 00id proven without ~ ax... |
| re0m0e0 42894 | Real number version of ~ 0... |
| readdlid 42895 | Real number version of ~ a... |
| sn-addlid 42896 | ~ addlid without ~ ax-mulc... |
| remul02 42897 | Real number version of ~ m... |
| sn-0ne2 42898 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 42899 | Real number version of ~ m... |
| sn-remul0ord 42900 | A product is zero iff one ... |
| resubid 42901 | Subtraction of a real numb... |
| readdrid 42902 | Real number version of ~ a... |
| resubid1 42903 | Real number version of ~ s... |
| renegneg 42904 | A real number is equal to ... |
| readdcan2 42905 | Commuted version of ~ read... |
| renegid2 42906 | Commuted version of ~ rene... |
| remulneg2d 42907 | Product with negative is n... |
| sn-it0e0 42908 | Proof of ~ it0e0 without ~... |
| sn-negex12 42909 | A combination of ~ cnegex ... |
| sn-negex 42910 | Proof of ~ cnegex without ... |
| sn-negex2 42911 | Proof of ~ cnegex2 without... |
| sn-addcand 42912 | ~ addcand without ~ ax-mul... |
| sn-addrid 42913 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 42914 | ~ addcan2d without ~ ax-mu... |
| reixi 42915 | ~ ixi without ~ ax-mulcom ... |
| rei4 42916 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 42917 | A number that sums to itse... |
| sn-mul01 42918 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 42919 | ~ negeu without ~ ax-mulco... |
| sn-subcl 42920 | ~ subcl without ~ ax-mulco... |
| sn-subf 42921 | ~ subf without ~ ax-mulcom... |
| resubeqsub 42922 | Equivalence between real s... |
| subresre 42923 | Subtraction restricted to ... |
| addinvcom 42924 | A number commutes with its... |
| remulinvcom 42925 | A left multiplicative inve... |
| remullid 42926 | Commuted version of ~ ax-1... |
| sn-1ticom 42927 | Lemma for ~ sn-mullid and ... |
| sn-mullid 42928 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 42929 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 42930 | The multiplicative inverse... |
| remulcand 42931 | Commuted version of ~ remu... |
| redivvald 42934 | Value of real division, wh... |
| rediveud 42935 | Existential uniqueness of ... |
| sn-redivcld 42936 | Closure law for real divis... |
| redivmuld 42937 | Relationship between divis... |
| redivmul2d 42938 | Relationship between divis... |
| redivcan2d 42939 | A cancellation law for div... |
| redivcan3d 42940 | A cancellation law for div... |
| rediveq0d 42941 | A ratio is zero iff the nu... |
| redivne0bd 42942 | The ratio of nonzero numbe... |
| rediveq1d 42943 | Equality in terms of unit ... |
| sn-rediv1d 42944 | A number divided by 1 is i... |
| sn-rediv0d 42945 | Division into zero is zero... |
| sn-redividd 42946 | A number divided by itself... |
| sn-rereccld 42947 | Closure law for reciprocal... |
| rerecne0d 42948 | The reciprocal of a nonzer... |
| rerecidd 42949 | Multiplication of a number... |
| rerecid2d 42950 | Multiplication of a number... |
| rerecrecd 42951 | A number is equal to the r... |
| redivrec2d 42952 | Relationship between divis... |
| rediv23d 42953 | A "commutative"/associativ... |
| redivdird 42954 | Distribution of division o... |
| rediv11d 42955 | One-to-one relationship fo... |
| sn-0tie0 42956 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 42957 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 42958 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 42959 | ~ ltaddneg without ~ ax-mu... |
| reposdif 42960 | Comparison of two numbers ... |
| relt0neg1 42961 | Comparison of a real and i... |
| relt0neg2 42962 | Comparison of a real and i... |
| sn-addlt0d 42963 | The sum of negative number... |
| sn-addgt0d 42964 | The sum of positive number... |
| sn-nnne0 42965 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 42966 | ~ elznn0nn restated using ... |
| nn0addcom 42967 | Addition is commutative fo... |
| zaddcomlem 42968 | Lemma for ~ zaddcom . (Co... |
| zaddcom 42969 | Addition is commutative fo... |
| renegmulnnass 42970 | Move multiplication by a n... |
| nn0mulcom 42971 | Multiplication is commutat... |
| zmulcomlem 42972 | Lemma for ~ zmulcom . (Co... |
| zmulcom 42973 | Multiplication is commutat... |
| mulgt0con1dlem 42974 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 42975 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 42976 | Lemma for ~ mulgt0b1d and ... |
| mulgt0b1d 42977 | Biconditional, deductive f... |
| sn-ltmul2d 42978 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 42979 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 42980 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 42981 | ~ ltp1 without ~ ax-mulcom... |
| sn-recgt0d 42982 | The reciprocal of a positi... |
| mulgt0b2d 42983 | Biconditional, deductive f... |
| sn-mulgt1d 42984 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 42985 | Negative one is a negative... |
| sn-reclt0d 42986 | The reciprocal of a negati... |
| mulltgt0d 42987 | Negative times positive is... |
| mullt0b1d 42988 | When the first term is neg... |
| mullt0b2d 42989 | When the second term is ne... |
| sn-mullt0d 42990 | The product of two negativ... |
| sn-msqgt0d 42991 | A nonzero square is positi... |
| sn-inelr 42992 | ~ inelr without ~ ax-mulco... |
| sn-itrere 42993 | ` _i ` times a real is rea... |
| sn-retire 42994 | Commuted version of ~ sn-i... |
| cnreeu 42995 | The reals in the expressio... |
| sn-sup2 42996 | ~ sup2 with exactly the sa... |
| sn-sup3d 42997 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 42998 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 42999 | ~ suprubd without ~ ax-mul... |
| sn-base0 43000 | Avoid axioms in ~ base0 by... |
| nelsubginvcld 43001 | The inverse of a non-subgr... |
| nelsubgcld 43002 | A non-subgroup-member plus... |
| nelsubgsubcld 43003 | A non-subgroup-member minu... |
| rnasclg 43004 | The set of injected scalar... |
| frlmfielbas 43005 | The vectors of a finite fr... |
| frlmfzwrd 43006 | A vector of a module with ... |
| frlmfzowrd 43007 | A vector of a module with ... |
| frlmfzolen 43008 | The dimension of a vector ... |
| frlmfzowrdb 43009 | The vectors of a module wi... |
| frlmfzoccat 43010 | The concatenation of two v... |
| frlmvscadiccat 43011 | Scalar multiplication dist... |
| grpasscan2d 43012 | An associative cancellatio... |
| grpcominv1 43013 | If two elements commute, t... |
| grpcominv2 43014 | If two elements commute, t... |
| finsubmsubg 43015 | A submonoid of a finite gr... |
| opprmndb 43016 | A class is a monoid if and... |
| opprgrpb 43017 | A class is a group if and ... |
| opprablb 43018 | A class is an Abelian grou... |
| imacrhmcl 43019 | The image of a commutative... |
| rimco 43020 | The composition of ring is... |
| rictr 43021 | Ring isomorphism is transi... |
| riccrng1 43022 | Ring isomorphism preserves... |
| riccrng 43023 | A ring is commutative if a... |
| domnexpgn0cl 43024 | In a domain, a (nonnegativ... |
| drnginvrn0d 43025 | A multiplicative inverse i... |
| drngmullcan 43026 | Cancellation of a nonzero ... |
| drngmulrcan 43027 | Cancellation of a nonzero ... |
| drnginvmuld 43028 | Inverse of a nonzero produ... |
| ricdrng1 43029 | A ring isomorphism maps a ... |
| ricdrng 43030 | A ring is a division ring ... |
| ricfld 43031 | A ring is a field if and o... |
| asclf1 43032 | Two ways of saying the sca... |
| abvexp 43033 | Move exponentiation in and... |
| fimgmcyclem 43034 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 43035 | Version of ~ odcl2 for fin... |
| fidomncyc 43036 | Version of ~ odcl2 for mul... |
| fiabv 43037 | In a finite domain (a fini... |
| lvecgrp 43038 | A vector space is a group.... |
| lvecring 43039 | The scalar component of a ... |
| frlm0vald 43040 | All coordinates of the zer... |
| frlmsnic 43041 | Given a free module with a... |
| uvccl 43042 | A unit vector is a vector.... |
| uvcn0 43043 | A unit vector is nonzero. ... |
| psrmnd 43044 | The ring of power series i... |
| mhmcopsr 43045 | The composition of a monoi... |
| mhmcoaddpsr 43046 | Show that the ring homomor... |
| rhmcomulpsr 43047 | Show that the ring homomor... |
| rhmpsr 43048 | Provide a ring homomorphis... |
| rhmpsr1 43049 | Provide a ring homomorphis... |
| evl0 43050 | The zero polynomial evalua... |
| evlsbagval 43051 | Polynomial evaluation buil... |
| evlvvvallem 43052 | Lemma for theorems using ~... |
| evlselvlem 43053 | Lemma for ~ evlselv . Use... |
| evlselv 43054 | Evaluating a selection of ... |
| fsuppind 43055 | Induction on functions ` F... |
| fsuppssindlem1 43056 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 43057 | Lemma for ~ fsuppssind . ... |
| fsuppssind 43058 | Induction on functions ` F... |
| mhpind 43059 | The homogeneous polynomial... |
| evlsmhpvvval 43060 | Give a formula for the eva... |
| mhphflem 43061 | Lemma for ~ mhphf . Add s... |
| mhphf 43062 | A homogeneous polynomial d... |
| mhphf2 43063 | A homogeneous polynomial d... |
| mhphf3 43064 | A homogeneous polynomial d... |
| mhphf4 43065 | A homogeneous polynomial d... |
| prjspval 43068 | Value of the projective sp... |
| prjsprel 43069 | Utility theorem regarding ... |
| prjspertr 43070 | The relation in ` PrjSp ` ... |
| prjsperref 43071 | The relation in ` PrjSp ` ... |
| prjspersym 43072 | The relation in ` PrjSp ` ... |
| prjsper 43073 | The relation used to defin... |
| prjspreln0 43074 | Two nonzero vectors are eq... |
| prjspvs 43075 | A nonzero multiple of a ve... |
| prjsprellsp 43076 | Two vectors are equivalent... |
| prjspeclsp 43077 | The vectors equivalent to ... |
| prjspval2 43078 | Alternate definition of pr... |
| prjspnval 43081 | Value of the n-dimensional... |
| prjspnerlem 43082 | A lemma showing that the e... |
| prjspnval2 43083 | Value of the n-dimensional... |
| prjspner 43084 | The relation used to defin... |
| prjspnvs 43085 | A nonzero multiple of a ve... |
| prjspnssbas 43086 | A projective point spans a... |
| prjspnn0 43087 | A projective point is none... |
| 0prjspnlem 43088 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 43089 | Any vector is equivalent t... |
| prjspner01 43090 | Any vector is equivalent t... |
| prjspner1 43091 | Two vectors whose zeroth c... |
| 0prjspnrel 43092 | In the zero-dimensional pr... |
| 0prjspn 43093 | A zero-dimensional project... |
| prjcrvfval 43096 | Value of the projective cu... |
| prjcrvval 43097 | Value of the projective cu... |
| prjcrv0 43098 | The "curve" (zero set) cor... |
| dffltz 43099 | Fermat's Last Theorem (FLT... |
| fltmul 43100 | A counterexample to FLT st... |
| fltdiv 43101 | A counterexample to FLT st... |
| flt0 43102 | A counterexample for FLT d... |
| fltdvdsabdvdsc 43103 | Any factor of both ` A ` a... |
| fltabcoprmex 43104 | A counterexample to FLT im... |
| fltaccoprm 43105 | A counterexample to FLT wi... |
| fltbccoprm 43106 | A counterexample to FLT wi... |
| fltabcoprm 43107 | A counterexample to FLT wi... |
| infdesc 43108 | Infinite descent. The hyp... |
| fltne 43109 | If a counterexample to FLT... |
| flt4lem 43110 | Raising a number to the fo... |
| flt4lem1 43111 | Satisfy the antecedent use... |
| flt4lem2 43112 | If ` A ` is even, ` B ` is... |
| flt4lem3 43113 | Equivalent to ~ pythagtrip... |
| flt4lem4 43114 | If the product of two copr... |
| flt4lem5 43115 | In the context of the lemm... |
| flt4lem5elem 43116 | Version of ~ fltaccoprm an... |
| flt4lem5a 43117 | Part 1 of Equation 1 of ... |
| flt4lem5b 43118 | Part 2 of Equation 1 of ... |
| flt4lem5c 43119 | Part 2 of Equation 2 of ... |
| flt4lem5d 43120 | Part 3 of Equation 2 of ... |
| flt4lem5e 43121 | Satisfy the hypotheses of ... |
| flt4lem5f 43122 | Final equation of ~... |
| flt4lem6 43123 | Remove shared factors in a... |
| flt4lem7 43124 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 43125 | Strengthening of Fermat's ... |
| fltltc 43126 | ` ( C ^ N ) ` is the large... |
| fltnltalem 43127 | Lemma for ~ fltnlta . A l... |
| fltnlta 43128 | In a Fermat counterexample... |
| iddii 43129 | Version of ~ a1ii with the... |
| bicomdALT 43130 | Alternate proof of ~ bicom... |
| alan 43131 | Alias for ~ 19.26 for easi... |
| exor 43132 | Alias for ~ 19.43 for easi... |
| rexor 43133 | Alias for ~ r19.43 for eas... |
| ruvALT 43134 | Alternate proof of ~ ruv w... |
| sn-wcdeq 43135 | Alternative to ~ wcdeq and... |
| sq45 43136 | 45 squared is 2025. (Cont... |
| sum9cubes 43137 | The sum of the first nine ... |
| sn-isghm 43138 | Longer proof of ~ isghm , ... |
| aprilfools2025 43139 | An abuse of notation. (Co... |
| nfa1w 43140 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 43141 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 43142 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 43143 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 43144 | Replace ~ ax-10 , ~ ax-11 ... |
| cu3addd 43145 | Cube of sum of three numbe... |
| negexpidd 43146 | The sum of a real number t... |
| rexlimdv3d 43147 | An extended version of ~ r... |
| 3cubeslem1 43148 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 43149 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 43150 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 43151 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 43152 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 43153 | Lemma for ~ 3cubes . This... |
| 3cubes 43154 | Every rational number is a... |
| rntrclfvOAI 43155 | The range of the transitiv... |
| moxfr 43156 | Transfer at-most-one betwe... |
| imaiinfv 43157 | Indexed intersection of an... |
| elrfi 43158 | Elementhood in a set of re... |
| elrfirn 43159 | Elementhood in a set of re... |
| elrfirn2 43160 | Elementhood in a set of re... |
| cmpfiiin 43161 | In a compact topology, a s... |
| ismrcd1 43162 | Any function from the subs... |
| ismrcd2 43163 | Second half of ~ ismrcd1 .... |
| istopclsd 43164 | A closure function which s... |
| ismrc 43165 | A function is a Moore clos... |
| isnacs 43168 | Expand definition of Noeth... |
| nacsfg 43169 | In a Noetherian-type closu... |
| isnacs2 43170 | Express Noetherian-type cl... |
| mrefg2 43171 | Slight variation on finite... |
| mrefg3 43172 | Slight variation on finite... |
| nacsacs 43173 | A closure system of Noethe... |
| isnacs3 43174 | A choice-free order equiva... |
| incssnn0 43175 | Transitivity induction of ... |
| nacsfix 43176 | An increasing sequence of ... |
| constmap 43177 | A constant (represented wi... |
| mapco2g 43178 | Renaming indices in a tupl... |
| mapco2 43179 | Post-composition (renaming... |
| mapfzcons 43180 | Extending a one-based mapp... |
| mapfzcons1 43181 | Recover prefix mapping fro... |
| mapfzcons1cl 43182 | A nonempty mapping has a p... |
| mapfzcons2 43183 | Recover added element from... |
| mptfcl 43184 | Interpret range of a maps-... |
| mzpclval 43189 | Substitution lemma for ` m... |
| elmzpcl 43190 | Double substitution lemma ... |
| mzpclall 43191 | The set of all functions w... |
| mzpcln0 43192 | Corollary of ~ mzpclall : ... |
| mzpcl1 43193 | Defining property 1 of a p... |
| mzpcl2 43194 | Defining property 2 of a p... |
| mzpcl34 43195 | Defining properties 3 and ... |
| mzpval 43196 | Value of the ` mzPoly ` fu... |
| dmmzp 43197 | ` mzPoly ` is defined for ... |
| mzpincl 43198 | Polynomial closedness is a... |
| mzpconst 43199 | Constant functions are pol... |
| mzpf 43200 | A polynomial function is a... |
| mzpproj 43201 | A projection function is p... |
| mzpadd 43202 | The pointwise sum of two p... |
| mzpmul 43203 | The pointwise product of t... |
| mzpconstmpt 43204 | A constant function expres... |
| mzpaddmpt 43205 | Sum of polynomial function... |
| mzpmulmpt 43206 | Product of polynomial func... |
| mzpsubmpt 43207 | The difference of two poly... |
| mzpnegmpt 43208 | Negation of a polynomial f... |
| mzpexpmpt 43209 | Raise a polynomial functio... |
| mzpindd 43210 | "Structural" induction to ... |
| mzpmfp 43211 | Relationship between multi... |
| mzpsubst 43212 | Substituting polynomials f... |
| mzprename 43213 | Simplified version of ~ mz... |
| mzpresrename 43214 | A polynomial is a polynomi... |
| mzpcompact2lem 43215 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 43216 | Polynomials are finitary o... |
| coeq0i 43217 | ~ coeq0 but without explic... |
| fzsplit1nn0 43218 | Split a finite 1-based set... |
| eldiophb 43221 | Initial expression of Diop... |
| eldioph 43222 | Condition for a set to be ... |
| diophrw 43223 | Renaming and adding unused... |
| eldioph2lem1 43224 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 43225 | Lemma for ~ eldioph2 . Co... |
| eldioph2 43226 | Construct a Diophantine se... |
| eldioph2b 43227 | While Diophantine sets wer... |
| eldiophelnn0 43228 | Remove antecedent on ` B `... |
| eldioph3b 43229 | Define Diophantine sets in... |
| eldioph3 43230 | Inference version of ~ eld... |
| ellz1 43231 | Membership in a lower set ... |
| lzunuz 43232 | The union of a lower set o... |
| fz1eqin 43233 | Express a one-based finite... |
| lzenom 43234 | Lower integers are countab... |
| elmapresaunres2 43235 | ~ fresaunres2 transposed t... |
| diophin 43236 | If two sets are Diophantin... |
| diophun 43237 | If two sets are Diophantin... |
| eldiophss 43238 | Diophantine sets are sets ... |
| diophrex 43239 | Projecting a Diophantine s... |
| eq0rabdioph 43240 | This is the first of a num... |
| eqrabdioph 43241 | Diophantine set builder fo... |
| 0dioph 43242 | The null set is Diophantin... |
| vdioph 43243 | The "universal" set (as la... |
| anrabdioph 43244 | Diophantine set builder fo... |
| orrabdioph 43245 | Diophantine set builder fo... |
| 3anrabdioph 43246 | Diophantine set builder fo... |
| 3orrabdioph 43247 | Diophantine set builder fo... |
| 2sbcrex 43248 | Exchange an existential qu... |
| sbc2rex 43249 | Exchange a substitution wi... |
| sbc4rex 43250 | Exchange a substitution wi... |
| sbcrot3 43251 | Rotate a sequence of three... |
| sbcrot5 43252 | Rotate a sequence of five ... |
| sbccomieg 43253 | Commute two explicit subst... |
| rexrabdioph 43254 | Diophantine set builder fo... |
| rexfrabdioph 43255 | Diophantine set builder fo... |
| 2rexfrabdioph 43256 | Diophantine set builder fo... |
| 3rexfrabdioph 43257 | Diophantine set builder fo... |
| 4rexfrabdioph 43258 | Diophantine set builder fo... |
| 6rexfrabdioph 43259 | Diophantine set builder fo... |
| 7rexfrabdioph 43260 | Diophantine set builder fo... |
| rabdiophlem1 43261 | Lemma for arithmetic dioph... |
| rabdiophlem2 43262 | Lemma for arithmetic dioph... |
| elnn0rabdioph 43263 | Diophantine set builder fo... |
| rexzrexnn0 43264 | Rewrite an existential qua... |
| lerabdioph 43265 | Diophantine set builder fo... |
| eluzrabdioph 43266 | Diophantine set builder fo... |
| elnnrabdioph 43267 | Diophantine set builder fo... |
| ltrabdioph 43268 | Diophantine set builder fo... |
| nerabdioph 43269 | Diophantine set builder fo... |
| dvdsrabdioph 43270 | Divisibility is a Diophant... |
| eldioph4b 43271 | Membership in ` Dioph ` ex... |
| eldioph4i 43272 | Forward-only version of ~ ... |
| diophren 43273 | Change variables in a Diop... |
| rabrenfdioph 43274 | Change variable numbers in... |
| rabren3dioph 43275 | Change variable numbers in... |
| fphpd 43276 | Pigeonhole principle expre... |
| fphpdo 43277 | Pigeonhole principle for s... |
| ctbnfien 43278 | An infinite subset of a co... |
| fiphp3d 43279 | Infinite pigeonhole princi... |
| rencldnfilem 43280 | Lemma for ~ rencldnfi . (... |
| rencldnfi 43281 | A set of real numbers whic... |
| irrapxlem1 43282 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 43283 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 43284 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 43285 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 43286 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 43287 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 43288 | Dirichlet's approximation ... |
| pellexlem1 43289 | Lemma for ~ pellex . Arit... |
| pellexlem2 43290 | Lemma for ~ pellex . Arit... |
| pellexlem3 43291 | Lemma for ~ pellex . To e... |
| pellexlem4 43292 | Lemma for ~ pellex . Invo... |
| pellexlem5 43293 | Lemma for ~ pellex . Invo... |
| pellexlem6 43294 | Lemma for ~ pellex . Doin... |
| pellex 43295 | Every Pell equation has a ... |
| pell1qrval 43306 | Value of the set of first-... |
| elpell1qr 43307 | Membership in a first-quad... |
| pell14qrval 43308 | Value of the set of positi... |
| elpell14qr 43309 | Membership in the set of p... |
| pell1234qrval 43310 | Value of the set of genera... |
| elpell1234qr 43311 | Membership in the set of g... |
| pell1234qrre 43312 | General Pell solutions are... |
| pell1234qrne0 43313 | No solution to a Pell equa... |
| pell1234qrreccl 43314 | General solutions of the P... |
| pell1234qrmulcl 43315 | General solutions of the P... |
| pell14qrss1234 43316 | A positive Pell solution i... |
| pell14qrre 43317 | A positive Pell solution i... |
| pell14qrne0 43318 | A positive Pell solution i... |
| pell14qrgt0 43319 | A positive Pell solution i... |
| pell14qrrp 43320 | A positive Pell solution i... |
| pell1234qrdich 43321 | A general Pell solution is... |
| elpell14qr2 43322 | A number is a positive Pel... |
| pell14qrmulcl 43323 | Positive Pell solutions ar... |
| pell14qrreccl 43324 | Positive Pell solutions ar... |
| pell14qrdivcl 43325 | Positive Pell solutions ar... |
| pell14qrexpclnn0 43326 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 43327 | Positive Pell solutions ar... |
| pell1qrss14 43328 | First-quadrant Pell soluti... |
| pell14qrdich 43329 | A positive Pell solution i... |
| pell1qrge1 43330 | A Pell solution in the fir... |
| pell1qr1 43331 | 1 is a Pell solution and i... |
| elpell1qr2 43332 | The first quadrant solutio... |
| pell1qrgaplem 43333 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 43334 | First-quadrant Pell soluti... |
| pell14qrgap 43335 | Positive Pell solutions ar... |
| pell14qrgapw 43336 | Positive Pell solutions ar... |
| pellqrexplicit 43337 | Condition for a calculated... |
| infmrgelbi 43338 | Any lower bound of a nonem... |
| pellqrex 43339 | There is a nontrivial solu... |
| pellfundval 43340 | Value of the fundamental s... |
| pellfundre 43341 | The fundamental solution o... |
| pellfundge 43342 | Lower bound on the fundame... |
| pellfundgt1 43343 | Weak lower bound on the Pe... |
| pellfundlb 43344 | A nontrivial first quadran... |
| pellfundglb 43345 | If a real is larger than t... |
| pellfundex 43346 | The fundamental solution a... |
| pellfund14gap 43347 | There are no solutions bet... |
| pellfundrp 43348 | The fundamental Pell solut... |
| pellfundne1 43349 | The fundamental Pell solut... |
| reglogcl 43350 | General logarithm is a rea... |
| reglogltb 43351 | General logarithm preserve... |
| reglogleb 43352 | General logarithm preserve... |
| reglogmul 43353 | Multiplication law for gen... |
| reglogexp 43354 | Power law for general log.... |
| reglogbas 43355 | General log of the base is... |
| reglog1 43356 | General log of 1 is 0. (C... |
| reglogexpbas 43357 | General log of a power of ... |
| pellfund14 43358 | Every positive Pell soluti... |
| pellfund14b 43359 | The positive Pell solution... |
| rmxfval 43364 | Value of the X sequence. ... |
| rmyfval 43365 | Value of the Y sequence. ... |
| rmspecsqrtnq 43366 | The discriminant used to d... |
| rmspecnonsq 43367 | The discriminant used to d... |
| qirropth 43368 | This lemma implements the ... |
| rmspecfund 43369 | The base of exponent used ... |
| rmxyelqirr 43370 | The solutions used to cons... |
| rmxypairf1o 43371 | The function used to extra... |
| rmxyelxp 43372 | Lemma for ~ frmx and ~ frm... |
| frmx 43373 | The X sequence is a nonneg... |
| frmy 43374 | The Y sequence is an integ... |
| rmxyval 43375 | Main definition of the X a... |
| rmspecpos 43376 | The discriminant used to d... |
| rmxycomplete 43377 | The X and Y sequences take... |
| rmxynorm 43378 | The X and Y sequences defi... |
| rmbaserp 43379 | The base of exponentiation... |
| rmxyneg 43380 | Negation law for X and Y s... |
| rmxyadd 43381 | Addition formula for X and... |
| rmxy1 43382 | Value of the X and Y seque... |
| rmxy0 43383 | Value of the X and Y seque... |
| rmxneg 43384 | Negation law (even functio... |
| rmx0 43385 | Value of X sequence at 0. ... |
| rmx1 43386 | Value of X sequence at 1. ... |
| rmxadd 43387 | Addition formula for X seq... |
| rmyneg 43388 | Negation formula for Y seq... |
| rmy0 43389 | Value of Y sequence at 0. ... |
| rmy1 43390 | Value of Y sequence at 1. ... |
| rmyadd 43391 | Addition formula for Y seq... |
| rmxp1 43392 | Special addition-of-1 form... |
| rmyp1 43393 | Special addition of 1 form... |
| rmxm1 43394 | Subtraction of 1 formula f... |
| rmym1 43395 | Subtraction of 1 formula f... |
| rmxluc 43396 | The X sequence is a Lucas ... |
| rmyluc 43397 | The Y sequence is a Lucas ... |
| rmyluc2 43398 | Lucas sequence property of... |
| rmxdbl 43399 | "Double-angle formula" for... |
| rmydbl 43400 | "Double-angle formula" for... |
| monotuz 43401 | A function defined on an u... |
| monotoddzzfi 43402 | A function which is odd an... |
| monotoddzz 43403 | A function (given implicit... |
| oddcomabszz 43404 | An odd function which take... |
| 2nn0ind 43405 | Induction on nonnegative i... |
| zindbi 43406 | Inductively transfer a pro... |
| rmxypos 43407 | For all nonnegative indice... |
| ltrmynn0 43408 | The Y-sequence is strictly... |
| ltrmxnn0 43409 | The X-sequence is strictly... |
| lermxnn0 43410 | The X-sequence is monotoni... |
| rmxnn 43411 | The X-sequence is defined ... |
| ltrmy 43412 | The Y-sequence is strictly... |
| rmyeq0 43413 | Y is zero only at zero. (... |
| rmyeq 43414 | Y is one-to-one. (Contrib... |
| lermy 43415 | Y is monotonic (non-strict... |
| rmynn 43416 | ` rmY ` is positive for po... |
| rmynn0 43417 | ` rmY ` is nonnegative for... |
| rmyabs 43418 | ` rmY ` commutes with ` ab... |
| jm2.24nn 43419 | X(n) is strictly greater t... |
| jm2.17a 43420 | First half of lemma 2.17 o... |
| jm2.17b 43421 | Weak form of the second ha... |
| jm2.17c 43422 | Second half of lemma 2.17 ... |
| jm2.24 43423 | Lemma 2.24 of [JonesMatija... |
| rmygeid 43424 | Y(n) increases faster than... |
| congtr 43425 | A wff of the form ` A || (... |
| congadd 43426 | If two pairs of numbers ar... |
| congmul 43427 | If two pairs of numbers ar... |
| congsym 43428 | Congruence mod ` A ` is a ... |
| congneg 43429 | If two integers are congru... |
| congsub 43430 | If two pairs of numbers ar... |
| congid 43431 | Every integer is congruent... |
| mzpcong 43432 | Polynomials commute with c... |
| congrep 43433 | Every integer is congruent... |
| congabseq 43434 | If two integers are congru... |
| acongid 43435 | A wff like that in this th... |
| acongsym 43436 | Symmetry of alternating co... |
| acongneg2 43437 | Negate right side of alter... |
| acongtr 43438 | Transitivity of alternatin... |
| acongeq12d 43439 | Substitution deduction for... |
| acongrep 43440 | Every integer is alternati... |
| fzmaxdif 43441 | Bound on the difference be... |
| fzneg 43442 | Reflection of a finite ran... |
| acongeq 43443 | Two numbers in the fundame... |
| dvdsacongtr 43444 | Alternating congruence pas... |
| coprmdvdsb 43445 | Multiplication by a coprim... |
| modabsdifz 43446 | Divisibility in terms of m... |
| dvdsabsmod0 43447 | Divisibility in terms of m... |
| jm2.18 43448 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 43449 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 43450 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 43451 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 43452 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 43453 | Lemma 2.19 of [JonesMatija... |
| jm2.21 43454 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 43455 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 43456 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 43457 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 43458 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 43459 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 43460 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 43461 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 43462 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 43463 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 43464 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 43465 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 43466 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 43467 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 43468 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 43469 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 43470 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 43471 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 43472 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 43473 | Lemma for ~ rmydioph . Us... |
| rmydioph 43474 | ~ jm2.27 restated in terms... |
| rmxdiophlem 43475 | X can be expressed in term... |
| rmxdioph 43476 | X is a Diophantine functio... |
| jm3.1lem1 43477 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 43478 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 43479 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 43480 | Diophantine expression for... |
| expdiophlem1 43481 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 43482 | Lemma for ~ expdioph . Ex... |
| expdioph 43483 | The exponential function i... |
| setindtr 43484 | Set induction for sets con... |
| setindtrs 43485 | Set induction scheme witho... |
| dford3lem1 43486 | Lemma for ~ dford3 . (Con... |
| dford3lem2 43487 | Lemma for ~ dford3 . (Con... |
| dford3 43488 | Ordinals are precisely the... |
| dford4 43489 | ~ dford3 expressed in prim... |
| wopprc 43490 | Unrelated: Wiener pairs t... |
| rpnnen3lem 43491 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 43492 | Dedekind cut injection of ... |
| axac10 43493 | Characterization of choice... |
| harinf 43494 | The Hartogs number of an i... |
| wdom2d2 43495 | Deduction for weak dominan... |
| ttac 43496 | Tarski's theorem about cho... |
| pw2f1ocnv 43497 | Define a bijection between... |
| pw2f1o2 43498 | Define a bijection between... |
| pw2f1o2val 43499 | Function value of the ~ pw... |
| pw2f1o2val2 43500 | Membership in a mapped set... |
| limsuc2 43501 | Limit ordinals in the sens... |
| wepwsolem 43502 | Transfer an ordering on ch... |
| wepwso 43503 | A well-ordering induces a ... |
| dnnumch1 43504 | Define an enumeration of a... |
| dnnumch2 43505 | Define an enumeration (wea... |
| dnnumch3lem 43506 | Value of the ordinal injec... |
| dnnumch3 43507 | Define an injection from a... |
| dnwech 43508 | Define a well-ordering fro... |
| fnwe2val 43509 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 43510 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 43511 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 43512 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 43513 | A well-ordering can be con... |
| aomclem1 43514 | Lemma for ~ dfac11 . This... |
| aomclem2 43515 | Lemma for ~ dfac11 . Succ... |
| aomclem3 43516 | Lemma for ~ dfac11 . Succ... |
| aomclem4 43517 | Lemma for ~ dfac11 . Limi... |
| aomclem5 43518 | Lemma for ~ dfac11 . Comb... |
| aomclem6 43519 | Lemma for ~ dfac11 . Tran... |
| aomclem7 43520 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 43521 | Lemma for ~ dfac11 . Perf... |
| dfac11 43522 | The right-hand side of thi... |
| kelac1 43523 | Kelley's choice, basic for... |
| kelac2lem 43524 | Lemma for ~ kelac2 and ~ d... |
| kelac2 43525 | Kelley's choice, most comm... |
| dfac21 43526 | Tychonoff's theorem is a c... |
| islmodfg 43529 | Property of a finitely gen... |
| islssfg 43530 | Property of a finitely gen... |
| islssfg2 43531 | Property of a finitely gen... |
| islssfgi 43532 | Finitely spanned subspaces... |
| fglmod 43533 | Finitely generated left mo... |
| lsmfgcl 43534 | The sum of two finitely ge... |
| islnm 43537 | Property of being a Noethe... |
| islnm2 43538 | Property of being a Noethe... |
| lnmlmod 43539 | A Noetherian left module i... |
| lnmlssfg 43540 | A submodule of Noetherian ... |
| lnmlsslnm 43541 | All submodules of a Noethe... |
| lnmfg 43542 | A Noetherian left module i... |
| kercvrlsm 43543 | The domain of a linear fun... |
| lmhmfgima 43544 | A homomorphism maps finite... |
| lnmepi 43545 | Epimorphic images of Noeth... |
| lmhmfgsplit 43546 | If the kernel and range of... |
| lmhmlnmsplit 43547 | If the kernel and range of... |
| lnmlmic 43548 | Noetherian is an invariant... |
| pwssplit4 43549 | Splitting for structure po... |
| filnm 43550 | Finite left modules are No... |
| pwslnmlem0 43551 | Zeroeth powers are Noether... |
| pwslnmlem1 43552 | First powers are Noetheria... |
| pwslnmlem2 43553 | A sum of powers is Noether... |
| pwslnm 43554 | Finite powers of Noetheria... |
| unxpwdom3 43555 | Weaker version of ~ unxpwd... |
| pwfi2f1o 43556 | The ~ pw2f1o bijection rel... |
| pwfi2en 43557 | Finitely supported indicat... |
| frlmpwfi 43558 | Formal linear combinations... |
| gicabl 43559 | Being Abelian is a group i... |
| imasgim 43560 | A relabeling of the elemen... |
| isnumbasgrplem1 43561 | A set which is equipollent... |
| harn0 43562 | The Hartogs number of a se... |
| numinfctb 43563 | A numerable infinite set c... |
| isnumbasgrplem2 43564 | If the (to be thought of a... |
| isnumbasgrplem3 43565 | Every nonempty numerable s... |
| isnumbasabl 43566 | A set is numerable iff it ... |
| isnumbasgrp 43567 | A set is numerable iff it ... |
| dfacbasgrp 43568 | A choice equivalent in abs... |
| islnr 43571 | Property of a left-Noether... |
| lnrring 43572 | Left-Noetherian rings are ... |
| lnrlnm 43573 | Left-Noetherian rings have... |
| islnr2 43574 | Property of being a left-N... |
| islnr3 43575 | Relate left-Noetherian rin... |
| lnr2i 43576 | Given an ideal in a left-N... |
| lpirlnr 43577 | Left principal ideal rings... |
| lnrfrlm 43578 | Finite-dimensional free mo... |
| lnrfg 43579 | Finitely-generated modules... |
| lnrfgtr 43580 | A submodule of a finitely ... |
| hbtlem1 43583 | Value of the leading coeff... |
| hbtlem2 43584 | Leading coefficient ideals... |
| hbtlem7 43585 | Functionality of leading c... |
| hbtlem4 43586 | The leading ideal function... |
| hbtlem3 43587 | The leading ideal function... |
| hbtlem5 43588 | The leading ideal function... |
| hbtlem6 43589 | There is a finite set of p... |
| hbt 43590 | The Hilbert Basis Theorem ... |
| dgrsub2 43595 | Subtracting two polynomial... |
| elmnc 43596 | Property of a monic polyno... |
| mncply 43597 | A monic polynomial is a po... |
| mnccoe 43598 | A monic polynomial has lea... |
| mncn0 43599 | A monic polynomial is not ... |
| dgraaval 43604 | Value of the degree functi... |
| dgraalem 43605 | Properties of the degree o... |
| dgraacl 43606 | Closure of the degree func... |
| dgraaf 43607 | Degree function on algebra... |
| dgraaub 43608 | Upper bound on degree of a... |
| dgraa0p 43609 | A rational polynomial of d... |
| mpaaeu 43610 | An algebraic number has ex... |
| mpaaval 43611 | Value of the minimal polyn... |
| mpaalem 43612 | Properties of the minimal ... |
| mpaacl 43613 | Minimal polynomial is a po... |
| mpaadgr 43614 | Minimal polynomial has deg... |
| mpaaroot 43615 | The minimal polynomial of ... |
| mpaamn 43616 | Minimal polynomial is moni... |
| itgoval 43621 | Value of the integral-over... |
| aaitgo 43622 | The standard algebraic num... |
| itgoss 43623 | An integral element is int... |
| itgocn 43624 | All integral elements are ... |
| cnsrexpcl 43625 | Exponentiation is closed i... |
| fsumcnsrcl 43626 | Finite sums are closed in ... |
| cnsrplycl 43627 | Polynomials are closed in ... |
| rgspnid 43628 | The span of a subring is i... |
| rngunsnply 43629 | Adjoining one element to a... |
| flcidc 43630 | Finite linear combinations... |
| algstr 43633 | Lemma to shorten proofs of... |
| algbase 43634 | The base set of a construc... |
| algaddg 43635 | The additive operation of ... |
| algmulr 43636 | The multiplicative operati... |
| algsca 43637 | The set of scalars of a co... |
| algvsca 43638 | The scalar product operati... |
| mendval 43639 | Value of the module endomo... |
| mendbas 43640 | Base set of the module end... |
| mendplusgfval 43641 | Addition in the module end... |
| mendplusg 43642 | A specific addition in the... |
| mendmulrfval 43643 | Multiplication in the modu... |
| mendmulr 43644 | A specific multiplication ... |
| mendsca 43645 | The module endomorphism al... |
| mendvscafval 43646 | Scalar multiplication in t... |
| mendvsca 43647 | A specific scalar multipli... |
| mendring 43648 | The module endomorphism al... |
| mendlmod 43649 | The module endomorphism al... |
| mendassa 43650 | The module endomorphism al... |
| idomodle 43651 | Limit on the number of ` N... |
| fiuneneq 43652 | Two finite sets of equal s... |
| idomsubgmo 43653 | The units of an integral d... |
| proot1mul 43654 | Any primitive ` N ` -th ro... |
| proot1hash 43655 | If an integral domain has ... |
| proot1ex 43656 | The complex field has prim... |
| mon1psubm 43659 | Monic polynomials are a mu... |
| deg1mhm 43660 | Homomorphic property of th... |
| cytpfn 43661 | Functionality of the cyclo... |
| cytpval 43662 | Substitutions for the Nth ... |
| fgraphopab 43663 | Express a function as a su... |
| fgraphxp 43664 | Express a function as a su... |
| hausgraph 43665 | The graph of a continuous ... |
| r1sssucd 43670 | Deductive form of ~ r1sssu... |
| iocunico 43671 | Split an open interval int... |
| iocinico 43672 | The intersection of two se... |
| iocmbl 43673 | An open-below, closed-abov... |
| cnioobibld 43674 | A bounded, continuous func... |
| arearect 43675 | The area of a rectangle wh... |
| areaquad 43676 | The area of a quadrilatera... |
| uniel 43677 | Two ways to say a union is... |
| unielss 43678 | Two ways to say the union ... |
| unielid 43679 | Two ways to say the union ... |
| ssunib 43680 | Two ways to say a class is... |
| rp-intrabeq 43681 | Equality theorem for supre... |
| rp-unirabeq 43682 | Equality theorem for infim... |
| onmaxnelsup 43683 | Two ways to say the maximu... |
| onsupneqmaxlim0 43684 | If the supremum of a class... |
| onsupcl2 43685 | The supremum of a set of o... |
| onuniintrab 43686 | The union of a set of ordi... |
| onintunirab 43687 | The intersection of a non-... |
| onsupnmax 43688 | If the union of a class of... |
| onsupuni 43689 | The supremum of a set of o... |
| onsupuni2 43690 | The supremum of a set of o... |
| onsupintrab 43691 | The supremum of a set of o... |
| onsupintrab2 43692 | The supremum of a set of o... |
| onsupcl3 43693 | The supremum of a set of o... |
| onsupex3 43694 | The supremum of a set of o... |
| onuniintrab2 43695 | The union of a set of ordi... |
| oninfint 43696 | The infimum of a non-empty... |
| oninfunirab 43697 | The infimum of a non-empty... |
| oninfcl2 43698 | The infimum of a non-empty... |
| onsupmaxb 43699 | The union of a class of or... |
| onexgt 43700 | For any ordinal, there is ... |
| onexomgt 43701 | For any ordinal, there is ... |
| omlimcl2 43702 | The product of a limit ord... |
| onexlimgt 43703 | For any ordinal, there is ... |
| onexoegt 43704 | For any ordinal, there is ... |
| oninfex2 43705 | The infimum of a non-empty... |
| onsupeqmax 43706 | Condition when the supremu... |
| onsupeqnmax 43707 | Condition when the supremu... |
| onsuplub 43708 | The supremum of a set of o... |
| onsupnub 43709 | An upper bound of a set of... |
| onfisupcl 43710 | Sufficient condition when ... |
| onelord 43711 | Every element of a ordinal... |
| onepsuc 43712 | Every ordinal is less than... |
| epsoon 43713 | The ordinals are strictly ... |
| epirron 43714 | The strict order on the or... |
| oneptr 43715 | The strict order on the or... |
| oneltr 43716 | The elementhood relation o... |
| oneptri 43717 | The strict, complete (line... |
| ordeldif 43718 | Membership in the differen... |
| ordeldifsucon 43719 | Membership in the differen... |
| ordeldif1o 43720 | Membership in the differen... |
| ordne0gt0 43721 | Ordinal zero is less than ... |
| ondif1i 43722 | Ordinal zero is less than ... |
| onsucelab 43723 | The successor of every ord... |
| dflim6 43724 | A limit ordinal is a nonze... |
| limnsuc 43725 | A limit ordinal is not an ... |
| onsucss 43726 | If one ordinal is less tha... |
| ordnexbtwnsuc 43727 | For any distinct pair of o... |
| orddif0suc 43728 | For any distinct pair of o... |
| onsucf1lem 43729 | For ordinals, the successo... |
| onsucf1olem 43730 | The successor operation is... |
| onsucrn 43731 | The successor operation is... |
| onsucf1o 43732 | The successor operation is... |
| dflim7 43733 | A limit ordinal is a nonze... |
| onov0suclim 43734 | Compactly express rules fo... |
| oa0suclim 43735 | Closed form expression of ... |
| om0suclim 43736 | Closed form expression of ... |
| oe0suclim 43737 | Closed form expression of ... |
| oaomoecl 43738 | The operations of addition... |
| onsupsucismax 43739 | If the union of a set of o... |
| onsssupeqcond 43740 | If for every element of a ... |
| limexissup 43741 | An ordinal which is a limi... |
| limiun 43742 | A limit ordinal is the uni... |
| limexissupab 43743 | An ordinal which is a limi... |
| om1om1r 43744 | Ordinal one is both a left... |
| oe0rif 43745 | Ordinal zero raised to any... |
| oasubex 43746 | While subtraction can't be... |
| nnamecl 43747 | Natural numbers are closed... |
| onsucwordi 43748 | The successor operation pr... |
| oalim2cl 43749 | The ordinal sum of any ord... |
| oaltublim 43750 | Given ` C ` is a limit ord... |
| oaordi3 43751 | Ordinal addition of the sa... |
| oaord3 43752 | When the same ordinal is a... |
| 1oaomeqom 43753 | Ordinal one plus omega is ... |
| oaabsb 43754 | The right addend absorbs t... |
| oaordnrex 43755 | When omega is added on the... |
| oaordnr 43756 | When the same ordinal is a... |
| omge1 43757 | Any nonzero ordinal produc... |
| omge2 43758 | Any nonzero ordinal produc... |
| omlim2 43759 | The nonzero product with a... |
| omord2lim 43760 | Given a limit ordinal, the... |
| omord2i 43761 | Ordinal multiplication of ... |
| omord2com 43762 | When the same nonzero ordi... |
| 2omomeqom 43763 | Ordinal two times omega is... |
| omnord1ex 43764 | When omega is multiplied o... |
| omnord1 43765 | When the same nonzero ordi... |
| oege1 43766 | Any nonzero ordinal power ... |
| oege2 43767 | Any power of an ordinal at... |
| rp-oelim2 43768 | The power of an ordinal at... |
| oeord2lim 43769 | Given a limit ordinal, the... |
| oeord2i 43770 | Ordinal exponentiation of ... |
| oeord2com 43771 | When the same base at leas... |
| nnoeomeqom 43772 | Any natural number at leas... |
| df3o2 43773 | Ordinal 3 is the unordered... |
| df3o3 43774 | Ordinal 3, fully expanded.... |
| oenord1ex 43775 | When ordinals two and thre... |
| oenord1 43776 | When two ordinals (both at... |
| oaomoencom 43777 | Ordinal addition, multipli... |
| oenassex 43778 | Ordinal two raised to two ... |
| oenass 43779 | Ordinal exponentiation is ... |
| cantnftermord 43780 | For terms of the form of a... |
| cantnfub 43781 | Given a finite number of t... |
| cantnfub2 43782 | Given a finite number of t... |
| bropabg 43783 | Equivalence for two classe... |
| cantnfresb 43784 | A Cantor normal form which... |
| cantnf2 43785 | For every ordinal, ` A ` ,... |
| oawordex2 43786 | If ` C ` is between ` A ` ... |
| nnawordexg 43787 | If an ordinal, ` B ` , is ... |
| succlg 43788 | Closure law for ordinal su... |
| dflim5 43789 | A limit ordinal is either ... |
| oacl2g 43790 | Closure law for ordinal ad... |
| onmcl 43791 | If an ordinal is less than... |
| omabs2 43792 | Ordinal multiplication by ... |
| omcl2 43793 | Closure law for ordinal mu... |
| omcl3g 43794 | Closure law for ordinal mu... |
| ordsssucb 43795 | An ordinal number is less ... |
| tfsconcatlem 43796 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 43797 | The concatenation of two t... |
| tfsconcatfn 43798 | The concatenation of two t... |
| tfsconcatfv1 43799 | An early value of the conc... |
| tfsconcatfv2 43800 | A latter value of the conc... |
| tfsconcatfv 43801 | The value of the concatena... |
| tfsconcatrn 43802 | The range of the concatena... |
| tfsconcatfo 43803 | The concatenation of two t... |
| tfsconcatb0 43804 | The concatentation with th... |
| tfsconcat0i 43805 | The concatentation with th... |
| tfsconcat0b 43806 | The concatentation with th... |
| tfsconcat00 43807 | The concatentation of two ... |
| tfsconcatrev 43808 | If the domain of a transfi... |
| tfsconcatrnss12 43809 | The range of the concatena... |
| tfsconcatrnss 43810 | The concatenation of trans... |
| tfsconcatrnsson 43811 | The concatenation of trans... |
| tfsnfin 43812 | A transfinite sequence is ... |
| rp-tfslim 43813 | The limit of a sequence of... |
| ofoafg 43814 | Addition operator for func... |
| ofoaf 43815 | Addition operator for func... |
| ofoafo 43816 | Addition operator for func... |
| ofoacl 43817 | Closure law for component ... |
| ofoaid1 43818 | Identity law for component... |
| ofoaid2 43819 | Identity law for component... |
| ofoaass 43820 | Component-wise addition of... |
| ofoacom 43821 | Component-wise addition of... |
| naddcnff 43822 | Addition operator for Cant... |
| naddcnffn 43823 | Addition operator for Cant... |
| naddcnffo 43824 | Addition of Cantor normal ... |
| naddcnfcl 43825 | Closure law for component-... |
| naddcnfcom 43826 | Component-wise ordinal add... |
| naddcnfid1 43827 | Identity law for component... |
| naddcnfid2 43828 | Identity law for component... |
| naddcnfass 43829 | Component-wise addition of... |
| onsucunifi 43830 | The successor to the union... |
| sucunisn 43831 | The successor to the union... |
| onsucunipr 43832 | The successor to the union... |
| onsucunitp 43833 | The successor to the union... |
| oaun3lem1 43834 | The class of all ordinal s... |
| oaun3lem2 43835 | The class of all ordinal s... |
| oaun3lem3 43836 | The class of all ordinal s... |
| oaun3lem4 43837 | The class of all ordinal s... |
| rp-abid 43838 | Two ways to express a clas... |
| oadif1lem 43839 | Express the set difference... |
| oadif1 43840 | Express the set difference... |
| oaun2 43841 | Ordinal addition as a unio... |
| oaun3 43842 | Ordinal addition as a unio... |
| naddov4 43843 | Alternate expression for n... |
| nadd2rabtr 43844 | The set of ordinals which ... |
| nadd2rabord 43845 | The set of ordinals which ... |
| nadd2rabex 43846 | The class of ordinals whic... |
| nadd2rabon 43847 | The set of ordinals which ... |
| nadd1rabtr 43848 | The set of ordinals which ... |
| nadd1rabord 43849 | The set of ordinals which ... |
| nadd1rabex 43850 | The class of ordinals whic... |
| nadd1rabon 43851 | The set of ordinals which ... |
| nadd1suc 43852 | Natural addition with 1 is... |
| naddass1 43853 | Natural addition of ordina... |
| naddgeoa 43854 | Natural addition results i... |
| naddonnn 43855 | Natural addition with a na... |
| naddwordnexlem0 43856 | When ` A ` is the sum of a... |
| naddwordnexlem1 43857 | When ` A ` is the sum of a... |
| naddwordnexlem2 43858 | When ` A ` is the sum of a... |
| naddwordnexlem3 43859 | When ` A ` is the sum of a... |
| oawordex3 43860 | When ` A ` is the sum of a... |
| naddwordnexlem4 43861 | When ` A ` is the sum of a... |
| ordsssucim 43862 | If an ordinal is less than... |
| insucid 43863 | The intersection of a clas... |
| oaltom 43864 | Multiplication eventually ... |
| oe2 43865 | Two ways to square an ordi... |
| omltoe 43866 | Exponentiation eventually ... |
| abeqabi 43867 | Generalized condition for ... |
| abpr 43868 | Condition for a class abst... |
| abtp 43869 | Condition for a class abst... |
| ralopabb 43870 | Restricted universal quant... |
| fpwfvss 43871 | Functions into a powerset ... |
| sdomne0 43872 | A class that strictly domi... |
| sdomne0d 43873 | A class that strictly domi... |
| safesnsupfiss 43874 | If ` B ` is a finite subse... |
| safesnsupfiub 43875 | If ` B ` is a finite subse... |
| safesnsupfidom1o 43876 | If ` B ` is a finite subse... |
| safesnsupfilb 43877 | If ` B ` is a finite subse... |
| isoeq145d 43878 | Equality deduction for iso... |
| resisoeq45d 43879 | Equality deduction for equ... |
| negslem1 43880 | An equivalence between ide... |
| nvocnvb 43881 | Equivalence to saying the ... |
| rp-brsslt 43882 | Binary relation form of a ... |
| nla0002 43883 | Extending a linear order t... |
| nla0003 43884 | Extending a linear order t... |
| nla0001 43885 | Extending a linear order t... |
| faosnf0.11b 43886 | ` B ` is called a non-limi... |
| dfno2 43887 | A surreal number, in the f... |
| onnoxpg 43888 | Every ordinal maps to a su... |
| onnobdayg 43889 | Every ordinal maps to a su... |
| bdaybndex 43890 | Bounds formed from the bir... |
| bdaybndbday 43891 | Bounds formed from the bir... |
| onnoxp 43892 | Every ordinal maps to a su... |
| onnoxpi 43893 | Every ordinal maps to a su... |
| 0fno 43894 | Ordinal zero maps to a sur... |
| 1fno 43895 | Ordinal one maps to a surr... |
| 2fno 43896 | Ordinal two maps to a surr... |
| 3fno 43897 | Ordinal three maps to a su... |
| 4fno 43898 | Ordinal four maps to a sur... |
| fnimafnex 43899 | The functional image of a ... |
| nlimsuc 43900 | A successor is not a limit... |
| nlim1NEW 43901 | 1 is not a limit ordinal. ... |
| nlim2NEW 43902 | 2 is not a limit ordinal. ... |
| nlim3 43903 | 3 is not a limit ordinal. ... |
| nlim4 43904 | 4 is not a limit ordinal. ... |
| oa1un 43905 | Given ` A e. On ` , let ` ... |
| oa1cl 43906 | ` A +o 1o ` is in ` On ` .... |
| 0finon 43907 | 0 is a finite ordinal. Se... |
| 1finon 43908 | 1 is a finite ordinal. Se... |
| 2finon 43909 | 2 is a finite ordinal. Se... |
| 3finon 43910 | 3 is a finite ordinal. Se... |
| 4finon 43911 | 4 is a finite ordinal. Se... |
| finona1cl 43912 | The finite ordinals are cl... |
| finonex 43913 | The finite ordinals are a ... |
| fzunt 43914 | Union of two adjacent fini... |
| fzuntd 43915 | Union of two adjacent fini... |
| fzunt1d 43916 | Union of two overlapping f... |
| fzuntgd 43917 | Union of two adjacent or o... |
| ifpan123g 43918 | Conjunction of conditional... |
| ifpan23 43919 | Conjunction of conditional... |
| ifpdfor2 43920 | Define or in terms of cond... |
| ifporcor 43921 | Corollary of commutation o... |
| ifpdfan2 43922 | Define and with conditiona... |
| ifpancor 43923 | Corollary of commutation o... |
| ifpdfor 43924 | Define or in terms of cond... |
| ifpdfan 43925 | Define and with conditiona... |
| ifpbi2 43926 | Equivalence theorem for co... |
| ifpbi3 43927 | Equivalence theorem for co... |
| ifpim1 43928 | Restate implication as con... |
| ifpnot 43929 | Restate negated wff as con... |
| ifpid2 43930 | Restate wff as conditional... |
| ifpim2 43931 | Restate implication as con... |
| ifpbi23 43932 | Equivalence theorem for co... |
| ifpbiidcor 43933 | Restatement of ~ biid . (... |
| ifpbicor 43934 | Corollary of commutation o... |
| ifpxorcor 43935 | Corollary of commutation o... |
| ifpbi1 43936 | Equivalence theorem for co... |
| ifpnot23 43937 | Negation of conditional lo... |
| ifpnotnotb 43938 | Factor conditional logic o... |
| ifpnorcor 43939 | Corollary of commutation o... |
| ifpnancor 43940 | Corollary of commutation o... |
| ifpnot23b 43941 | Negation of conditional lo... |
| ifpbiidcor2 43942 | Restatement of ~ biid . (... |
| ifpnot23c 43943 | Negation of conditional lo... |
| ifpnot23d 43944 | Negation of conditional lo... |
| ifpdfnan 43945 | Define nand as conditional... |
| ifpdfxor 43946 | Define xor as conditional ... |
| ifpbi12 43947 | Equivalence theorem for co... |
| ifpbi13 43948 | Equivalence theorem for co... |
| ifpbi123 43949 | Equivalence theorem for co... |
| ifpidg 43950 | Restate wff as conditional... |
| ifpid3g 43951 | Restate wff as conditional... |
| ifpid2g 43952 | Restate wff as conditional... |
| ifpid1g 43953 | Restate wff as conditional... |
| ifpim23g 43954 | Restate implication as con... |
| ifpim3 43955 | Restate implication as con... |
| ifpnim1 43956 | Restate negated implicatio... |
| ifpim4 43957 | Restate implication as con... |
| ifpnim2 43958 | Restate negated implicatio... |
| ifpim123g 43959 | Implication of conditional... |
| ifpim1g 43960 | Implication of conditional... |
| ifp1bi 43961 | Substitute the first eleme... |
| ifpbi1b 43962 | When the first variable is... |
| ifpimimb 43963 | Factor conditional logic o... |
| ifpororb 43964 | Factor conditional logic o... |
| ifpananb 43965 | Factor conditional logic o... |
| ifpnannanb 43966 | Factor conditional logic o... |
| ifpor123g 43967 | Disjunction of conditional... |
| ifpimim 43968 | Consequnce of implication.... |
| ifpbibib 43969 | Factor conditional logic o... |
| ifpxorxorb 43970 | Factor conditional logic o... |
| rp-fakeimass 43971 | A special case where impli... |
| rp-fakeanorass 43972 | A special case where a mix... |
| rp-fakeoranass 43973 | A special case where a mix... |
| rp-fakeinunass 43974 | A special case where a mix... |
| rp-fakeuninass 43975 | A special case where a mix... |
| rp-isfinite5 43976 | A set is said to be finite... |
| rp-isfinite6 43977 | A set is said to be finite... |
| intabssd 43978 | When for each element ` y ... |
| eu0 43979 | There is only one empty se... |
| epelon2 43980 | Over the ordinal numbers, ... |
| ontric3g 43981 | For all ` x , y e. On ` , ... |
| dfsucon 43982 | ` A ` is called a successo... |
| snen1g 43983 | A singleton is equinumerou... |
| snen1el 43984 | A singleton is equinumerou... |
| sn1dom 43985 | A singleton is dominated b... |
| pr2dom 43986 | An unordered pair is domin... |
| tr3dom 43987 | An unordered triple is dom... |
| ensucne0 43988 | A class equinumerous to a ... |
| ensucne0OLD 43989 | A class equinumerous to a ... |
| dfom6 43990 | Let ` _om ` be defined to ... |
| infordmin 43991 | ` _om ` is the smallest in... |
| iscard4 43992 | Two ways to express the pr... |
| minregex 43993 | Given any cardinal number ... |
| minregex2 43994 | Given any cardinal number ... |
| iscard5 43995 | Two ways to express the pr... |
| elrncard 43996 | Let us define a cardinal n... |
| harval3 43997 | ` ( har `` A ) ` is the le... |
| harval3on 43998 | For any ordinal number ` A... |
| omssrncard 43999 | All natural numbers are ca... |
| 0iscard 44000 | 0 is a cardinal number. (... |
| 1iscard 44001 | 1 is a cardinal number. (... |
| omiscard 44002 | ` _om ` is a cardinal numb... |
| sucomisnotcard 44003 | ` _om +o 1o ` is not a car... |
| nna1iscard 44004 | For any natural number, th... |
| har2o 44005 | The least cardinal greater... |
| en2pr 44006 | A class is equinumerous to... |
| pr2cv 44007 | If an unordered pair is eq... |
| pr2el1 44008 | If an unordered pair is eq... |
| pr2cv1 44009 | If an unordered pair is eq... |
| pr2el2 44010 | If an unordered pair is eq... |
| pr2cv2 44011 | If an unordered pair is eq... |
| pren2 44012 | An unordered pair is equin... |
| pr2eldif1 44013 | If an unordered pair is eq... |
| pr2eldif2 44014 | If an unordered pair is eq... |
| pren2d 44015 | A pair of two distinct set... |
| aleph1min 44016 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 44017 | ` aleph ` is a strictly or... |
| alephiso3 44018 | ` aleph ` is a strictly or... |
| pwelg 44019 | The powerclass is an eleme... |
| pwinfig 44020 | The powerclass of an infin... |
| pwinfi2 44021 | The powerclass of an infin... |
| pwinfi3 44022 | The powerclass of an infin... |
| pwinfi 44023 | The powerclass of an infin... |
| fipjust 44024 | A definition of the finite... |
| cllem0 44025 | The class of all sets with... |
| superficl 44026 | The class of all supersets... |
| superuncl 44027 | The class of all supersets... |
| ssficl 44028 | The class of all subsets o... |
| ssuncl 44029 | The class of all subsets o... |
| ssdifcl 44030 | The class of all subsets o... |
| sssymdifcl 44031 | The class of all subsets o... |
| fiinfi 44032 | If two classes have the fi... |
| rababg 44033 | Condition when restricted ... |
| elinintab 44034 | Two ways of saying a set i... |
| elmapintrab 44035 | Two ways to say a set is a... |
| elinintrab 44036 | Two ways of saying a set i... |
| inintabss 44037 | Upper bound on intersectio... |
| inintabd 44038 | Value of the intersection ... |
| xpinintabd 44039 | Value of the intersection ... |
| relintabex 44040 | If the intersection of a c... |
| elcnvcnvintab 44041 | Two ways of saying a set i... |
| relintab 44042 | Value of the intersection ... |
| nonrel 44043 | A non-relation is equal to... |
| elnonrel 44044 | Only an ordered pair where... |
| cnvssb 44045 | Subclass theorem for conve... |
| relnonrel 44046 | The non-relation part of a... |
| cnvnonrel 44047 | The converse of the non-re... |
| brnonrel 44048 | A non-relation cannot rela... |
| dmnonrel 44049 | The domain of the non-rela... |
| rnnonrel 44050 | The range of the non-relat... |
| resnonrel 44051 | A restriction of the non-r... |
| imanonrel 44052 | An image under the non-rel... |
| cononrel1 44053 | Composition with the non-r... |
| cononrel2 44054 | Composition with the non-r... |
| elmapintab 44055 | Two ways to say a set is a... |
| fvnonrel 44056 | The function value of any ... |
| elinlem 44057 | Two ways to say a set is a... |
| elcnvcnvlem 44058 | Two ways to say a set is a... |
| cnvcnvintabd 44059 | Value of the relationship ... |
| elcnvlem 44060 | Two ways to say a set is a... |
| elcnvintab 44061 | Two ways of saying a set i... |
| cnvintabd 44062 | Value of the converse of t... |
| undmrnresiss 44063 | Two ways of saying the ide... |
| reflexg 44064 | Two ways of saying a relat... |
| cnvssco 44065 | A condition weaker than re... |
| refimssco 44066 | Reflexive relations are su... |
| cleq2lem 44067 | Equality implies bijection... |
| cbvcllem 44068 | Change of bound variable i... |
| clublem 44069 | If a superset ` Y ` of ` X... |
| clss2lem 44070 | The closure of a property ... |
| dfid7 44071 | Definition of identity rel... |
| mptrcllem 44072 | Show two versions of a clo... |
| cotrintab 44073 | The intersection of a clas... |
| rclexi 44074 | The reflexive closure of a... |
| rtrclexlem 44075 | Existence of relation impl... |
| rtrclex 44076 | The reflexive-transitive c... |
| trclubgNEW 44077 | If a relation exists then ... |
| trclubNEW 44078 | If a relation exists then ... |
| trclexi 44079 | The transitive closure of ... |
| rtrclexi 44080 | The reflexive-transitive c... |
| clrellem 44081 | When the property ` ps ` h... |
| clcnvlem 44082 | When ` A ` , an upper boun... |
| cnvtrucl0 44083 | The converse of the trivia... |
| cnvrcl0 44084 | The converse of the reflex... |
| cnvtrcl0 44085 | The converse of the transi... |
| dmtrcl 44086 | The domain of the transiti... |
| rntrcl 44087 | The range of the transitiv... |
| dfrtrcl5 44088 | Definition of reflexive-tr... |
| trcleq2lemRP 44089 | Equality implies bijection... |
| sqrtcvallem1 44090 | Two ways of saying a compl... |
| reabsifneg 44091 | Alternate expression for t... |
| reabsifnpos 44092 | Alternate expression for t... |
| reabsifpos 44093 | Alternate expression for t... |
| reabsifnneg 44094 | Alternate expression for t... |
| reabssgn 44095 | Alternate expression for t... |
| sqrtcvallem2 44096 | Equivalent to saying that ... |
| sqrtcvallem3 44097 | Equivalent to saying that ... |
| sqrtcvallem4 44098 | Equivalent to saying that ... |
| sqrtcvallem5 44099 | Equivalent to saying that ... |
| sqrtcval 44100 | Explicit formula for the c... |
| sqrtcval2 44101 | Explicit formula for the c... |
| resqrtval 44102 | Real part of the complex s... |
| imsqrtval 44103 | Imaginary part of the comp... |
| resqrtvalex 44104 | Example for ~ resqrtval . ... |
| imsqrtvalex 44105 | Example for ~ imsqrtval . ... |
| al3im 44106 | Version of ~ ax-4 for a ne... |
| intima0 44107 | Two ways of expressing the... |
| elimaint 44108 | Element of image of inters... |
| cnviun 44109 | Converse of indexed union.... |
| imaiun1 44110 | The image of an indexed un... |
| coiun1 44111 | Composition with an indexe... |
| elintima 44112 | Element of intersection of... |
| intimass 44113 | The image under the inters... |
| intimass2 44114 | The image under the inters... |
| intimag 44115 | Requirement for the image ... |
| intimasn 44116 | Two ways to express the im... |
| intimasn2 44117 | Two ways to express the im... |
| ss2iundf 44118 | Subclass theorem for index... |
| ss2iundv 44119 | Subclass theorem for index... |
| cbviuneq12df 44120 | Rule used to change the bo... |
| cbviuneq12dv 44121 | Rule used to change the bo... |
| conrel1d 44122 | Deduction about compositio... |
| conrel2d 44123 | Deduction about compositio... |
| trrelind 44124 | The intersection of transi... |
| xpintrreld 44125 | The intersection of a tran... |
| restrreld 44126 | The restriction of a trans... |
| trrelsuperreldg 44127 | Concrete construction of a... |
| trficl 44128 | The class of all transitiv... |
| cnvtrrel 44129 | The converse of a transiti... |
| trrelsuperrel2dg 44130 | Concrete construction of a... |
| dfrcl2 44133 | Reflexive closure of a rel... |
| dfrcl3 44134 | Reflexive closure of a rel... |
| dfrcl4 44135 | Reflexive closure of a rel... |
| relexp2 44136 | A set operated on by the r... |
| relexpnul 44137 | If the domain and range of... |
| eliunov2 44138 | Membership in the indexed ... |
| eltrclrec 44139 | Membership in the indexed ... |
| elrtrclrec 44140 | Membership in the indexed ... |
| briunov2 44141 | Two classes related by the... |
| brmptiunrelexpd 44142 | If two elements are connec... |
| fvmptiunrelexplb0d 44143 | If the indexed union range... |
| fvmptiunrelexplb0da 44144 | If the indexed union range... |
| fvmptiunrelexplb1d 44145 | If the indexed union range... |
| brfvid 44146 | If two elements are connec... |
| brfvidRP 44147 | If two elements are connec... |
| fvilbd 44148 | A set is a subset of its i... |
| fvilbdRP 44149 | A set is a subset of its i... |
| brfvrcld 44150 | If two elements are connec... |
| brfvrcld2 44151 | If two elements are connec... |
| fvrcllb0d 44152 | A restriction of the ident... |
| fvrcllb0da 44153 | A restriction of the ident... |
| fvrcllb1d 44154 | A set is a subset of its i... |
| brtrclrec 44155 | Two classes related by the... |
| brrtrclrec 44156 | Two classes related by the... |
| briunov2uz 44157 | Two classes related by the... |
| eliunov2uz 44158 | Membership in the indexed ... |
| ov2ssiunov2 44159 | Any particular operator va... |
| relexp0eq 44160 | The zeroth power of relati... |
| iunrelexp0 44161 | Simplification of zeroth p... |
| relexpxpnnidm 44162 | Any positive power of a Ca... |
| relexpiidm 44163 | Any power of any restricti... |
| relexpss1d 44164 | The relational power of a ... |
| comptiunov2i 44165 | The composition two indexe... |
| corclrcl 44166 | The reflexive closure is i... |
| iunrelexpmin1 44167 | The indexed union of relat... |
| relexpmulnn 44168 | With exponents limited to ... |
| relexpmulg 44169 | With ordered exponents, th... |
| trclrelexplem 44170 | The union of relational po... |
| iunrelexpmin2 44171 | The indexed union of relat... |
| relexp01min 44172 | With exponents limited to ... |
| relexp1idm 44173 | Repeated raising a relatio... |
| relexp0idm 44174 | Repeated raising a relatio... |
| relexp0a 44175 | Absorption law for zeroth ... |
| relexpxpmin 44176 | The composition of powers ... |
| relexpaddss 44177 | The composition of two pow... |
| iunrelexpuztr 44178 | The indexed union of relat... |
| dftrcl3 44179 | Transitive closure of a re... |
| brfvtrcld 44180 | If two elements are connec... |
| fvtrcllb1d 44181 | A set is a subset of its i... |
| trclfvcom 44182 | The transitive closure of ... |
| cnvtrclfv 44183 | The converse of the transi... |
| cotrcltrcl 44184 | The transitive closure is ... |
| trclimalb2 44185 | Lower bound for image unde... |
| brtrclfv2 44186 | Two ways to indicate two e... |
| trclfvdecomr 44187 | The transitive closure of ... |
| trclfvdecoml 44188 | The transitive closure of ... |
| dmtrclfvRP 44189 | The domain of the transiti... |
| rntrclfvRP 44190 | The range of the transitiv... |
| rntrclfv 44191 | The range of the transitiv... |
| dfrtrcl3 44192 | Reflexive-transitive closu... |
| brfvrtrcld 44193 | If two elements are connec... |
| fvrtrcllb0d 44194 | A restriction of the ident... |
| fvrtrcllb0da 44195 | A restriction of the ident... |
| fvrtrcllb1d 44196 | A set is a subset of its i... |
| dfrtrcl4 44197 | Reflexive-transitive closu... |
| corcltrcl 44198 | The composition of the ref... |
| cortrcltrcl 44199 | Composition with the refle... |
| corclrtrcl 44200 | Composition with the refle... |
| cotrclrcl 44201 | The composition of the ref... |
| cortrclrcl 44202 | Composition with the refle... |
| cotrclrtrcl 44203 | Composition with the refle... |
| cortrclrtrcl 44204 | The reflexive-transitive c... |
| frege77d 44205 | If the images of both ` { ... |
| frege81d 44206 | If the image of ` U ` is a... |
| frege83d 44207 | If the image of the union ... |
| frege96d 44208 | If ` C ` follows ` A ` in ... |
| frege87d 44209 | If the images of both ` { ... |
| frege91d 44210 | If ` B ` follows ` A ` in ... |
| frege97d 44211 | If ` A ` contains all elem... |
| frege98d 44212 | If ` C ` follows ` A ` and... |
| frege102d 44213 | If either ` A ` and ` C ` ... |
| frege106d 44214 | If ` B ` follows ` A ` in ... |
| frege108d 44215 | If either ` A ` and ` C ` ... |
| frege109d 44216 | If ` A ` contains all elem... |
| frege114d 44217 | If either ` R ` relates ` ... |
| frege111d 44218 | If either ` A ` and ` C ` ... |
| frege122d 44219 | If ` F ` is a function, ` ... |
| frege124d 44220 | If ` F ` is a function, ` ... |
| frege126d 44221 | If ` F ` is a function, ` ... |
| frege129d 44222 | If ` F ` is a function and... |
| frege131d 44223 | If ` F ` is a function and... |
| frege133d 44224 | If ` F ` is a function and... |
| dfxor4 44225 | Express exclusive-or in te... |
| dfxor5 44226 | Express exclusive-or in te... |
| df3or2 44227 | Express triple-or in terms... |
| df3an2 44228 | Express triple-and in term... |
| nev 44229 | Express that not every set... |
| 0pssin 44230 | Express that an intersecti... |
| dfhe2 44233 | The property of relation `... |
| dfhe3 44234 | The property of relation `... |
| heeq12 44235 | Equality law for relations... |
| heeq1 44236 | Equality law for relations... |
| heeq2 44237 | Equality law for relations... |
| sbcheg 44238 | Distribute proper substitu... |
| hess 44239 | Subclass law for relations... |
| xphe 44240 | Any Cartesian product is h... |
| 0he 44241 | The empty relation is here... |
| 0heALT 44242 | The empty relation is here... |
| he0 44243 | Any relation is hereditary... |
| unhe1 44244 | The union of two relations... |
| snhesn 44245 | Any singleton is hereditar... |
| idhe 44246 | The identity relation is h... |
| psshepw 44247 | The relation between sets ... |
| sshepw 44248 | The relation between sets ... |
| rp-simp2-frege 44251 | Simplification of triple c... |
| rp-simp2 44252 | Simplification of triple c... |
| rp-frege3g 44253 | Add antecedent to ~ ax-fre... |
| frege3 44254 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 44255 | Double-use of ~ ax-frege2 ... |
| rp-frege24 44256 | Introducing an embedded an... |
| rp-frege4g 44257 | Deduction related to distr... |
| frege4 44258 | Special case of closed for... |
| frege5 44259 | A closed form of ~ syl . ... |
| rp-7frege 44260 | Distribute antecedent and ... |
| rp-4frege 44261 | Elimination of a nested an... |
| rp-6frege 44262 | Elimination of a nested an... |
| rp-8frege 44263 | Eliminate antecedent when ... |
| rp-frege25 44264 | Closed form for ~ a1dd . ... |
| frege6 44265 | A closed form of ~ imim2d ... |
| axfrege8 44266 | Swap antecedents. Identic... |
| frege7 44267 | A closed form of ~ syl6 . ... |
| frege26 44269 | Identical to ~ idd . Prop... |
| frege27 44270 | We cannot (at the same tim... |
| frege9 44271 | Closed form of ~ syl with ... |
| frege12 44272 | A closed form of ~ com23 .... |
| frege11 44273 | Elimination of a nested an... |
| frege24 44274 | Closed form for ~ a1d . D... |
| frege16 44275 | A closed form of ~ com34 .... |
| frege25 44276 | Closed form for ~ a1dd . ... |
| frege18 44277 | Closed form of a syllogism... |
| frege22 44278 | A closed form of ~ com45 .... |
| frege10 44279 | Result commuting anteceden... |
| frege17 44280 | A closed form of ~ com3l .... |
| frege13 44281 | A closed form of ~ com3r .... |
| frege14 44282 | Closed form of a deduction... |
| frege19 44283 | A closed form of ~ syl6 . ... |
| frege23 44284 | Syllogism followed by rota... |
| frege15 44285 | A closed form of ~ com4r .... |
| frege21 44286 | Replace antecedent in ante... |
| frege20 44287 | A closed form of ~ syl8 . ... |
| axfrege28 44288 | Contraposition. Identical... |
| frege29 44290 | Closed form of ~ con3d . ... |
| frege30 44291 | Commuted, closed form of ~... |
| axfrege31 44292 | Identical to ~ notnotr . ... |
| frege32 44294 | Deduce ~ con1 from ~ con3 ... |
| frege33 44295 | If ` ph ` or ` ps ` takes ... |
| frege34 44296 | If as a consequence of the... |
| frege35 44297 | Commuted, closed form of ~... |
| frege36 44298 | The case in which ` ps ` i... |
| frege37 44299 | If ` ch ` is a necessary c... |
| frege38 44300 | Identical to ~ pm2.21 . P... |
| frege39 44301 | Syllogism between ~ pm2.18... |
| frege40 44302 | Anything implies ~ pm2.18 ... |
| axfrege41 44303 | Identical to ~ notnot . A... |
| frege42 44305 | Not not ~ id . Propositio... |
| frege43 44306 | If there is a choice only ... |
| frege44 44307 | Similar to a commuted ~ pm... |
| frege45 44308 | Deduce ~ pm2.6 from ~ con1... |
| frege46 44309 | If ` ps ` holds when ` ph ... |
| frege47 44310 | Deduce consequence follows... |
| frege48 44311 | Closed form of syllogism w... |
| frege49 44312 | Closed form of deduction w... |
| frege50 44313 | Closed form of ~ jaoi . P... |
| frege51 44314 | Compare with ~ jaod . Pro... |
| axfrege52a 44315 | Justification for ~ ax-fre... |
| frege52aid 44317 | The case when the content ... |
| frege53aid 44318 | Specialization of ~ frege5... |
| frege53a 44319 | Lemma for ~ frege55a . Pr... |
| axfrege54a 44320 | Justification for ~ ax-fre... |
| frege54cor0a 44322 | Synonym for logical equiva... |
| frege54cor1a 44323 | Reflexive equality. (Cont... |
| frege55aid 44324 | Lemma for ~ frege57aid . ... |
| frege55lem1a 44325 | Necessary deduction regard... |
| frege55lem2a 44326 | Core proof of Proposition ... |
| frege55a 44327 | Proposition 55 of [Frege18... |
| frege55cor1a 44328 | Proposition 55 of [Frege18... |
| frege56aid 44329 | Lemma for ~ frege57aid . ... |
| frege56a 44330 | Proposition 56 of [Frege18... |
| frege57aid 44331 | This is the all important ... |
| frege57a 44332 | Analogue of ~ frege57aid .... |
| axfrege58a 44333 | Identical to ~ anifp . Ju... |
| frege58acor 44335 | Lemma for ~ frege59a . (C... |
| frege59a 44336 | A kind of Aristotelian inf... |
| frege60a 44337 | Swap antecedents of ~ ax-f... |
| frege61a 44338 | Lemma for ~ frege65a . Pr... |
| frege62a 44339 | A kind of Aristotelian inf... |
| frege63a 44340 | Proposition 63 of [Frege18... |
| frege64a 44341 | Lemma for ~ frege65a . Pr... |
| frege65a 44342 | A kind of Aristotelian inf... |
| frege66a 44343 | Swap antecedents of ~ freg... |
| frege67a 44344 | Lemma for ~ frege68a . Pr... |
| frege68a 44345 | Combination of applying a ... |
| axfrege52c 44346 | Justification for ~ ax-fre... |
| frege52b 44348 | The case when the content ... |
| frege53b 44349 | Lemma for frege102 (via ~ ... |
| axfrege54c 44350 | Reflexive equality of clas... |
| frege54b 44352 | Reflexive equality of sets... |
| frege54cor1b 44353 | Reflexive equality. (Cont... |
| frege55lem1b 44354 | Necessary deduction regard... |
| frege55lem2b 44355 | Lemma for ~ frege55b . Co... |
| frege55b 44356 | Lemma for ~ frege57b . Pr... |
| frege56b 44357 | Lemma for ~ frege57b . Pr... |
| frege57b 44358 | Analogue of ~ frege57aid .... |
| axfrege58b 44359 | If ` A. x ph ` is affirmed... |
| frege58bid 44361 | If ` A. x ph ` is affirmed... |
| frege58bcor 44362 | Lemma for ~ frege59b . (C... |
| frege59b 44363 | A kind of Aristotelian inf... |
| frege60b 44364 | Swap antecedents of ~ ax-f... |
| frege61b 44365 | Lemma for ~ frege65b . Pr... |
| frege62b 44366 | A kind of Aristotelian inf... |
| frege63b 44367 | Lemma for ~ frege91 . Pro... |
| frege64b 44368 | Lemma for ~ frege65b . Pr... |
| frege65b 44369 | A kind of Aristotelian inf... |
| frege66b 44370 | Swap antecedents of ~ freg... |
| frege67b 44371 | Lemma for ~ frege68b . Pr... |
| frege68b 44372 | Combination of applying a ... |
| frege53c 44373 | Proposition 53 of [Frege18... |
| frege54cor1c 44374 | Reflexive equality. (Cont... |
| frege55lem1c 44375 | Necessary deduction regard... |
| frege55lem2c 44376 | Core proof of Proposition ... |
| frege55c 44377 | Proposition 55 of [Frege18... |
| frege56c 44378 | Lemma for ~ frege57c . Pr... |
| frege57c 44379 | Swap order of implication ... |
| frege58c 44380 | Principle related to ~ sp ... |
| frege59c 44381 | A kind of Aristotelian inf... |
| frege60c 44382 | Swap antecedents of ~ freg... |
| frege61c 44383 | Lemma for ~ frege65c . Pr... |
| frege62c 44384 | A kind of Aristotelian inf... |
| frege63c 44385 | Analogue of ~ frege63b . ... |
| frege64c 44386 | Lemma for ~ frege65c . Pr... |
| frege65c 44387 | A kind of Aristotelian inf... |
| frege66c 44388 | Swap antecedents of ~ freg... |
| frege67c 44389 | Lemma for ~ frege68c . Pr... |
| frege68c 44390 | Combination of applying a ... |
| dffrege69 44391 | If from the proposition th... |
| frege70 44392 | Lemma for ~ frege72 . Pro... |
| frege71 44393 | Lemma for ~ frege72 . Pro... |
| frege72 44394 | If property ` A ` is hered... |
| frege73 44395 | Lemma for ~ frege87 . Pro... |
| frege74 44396 | If ` X ` has a property ` ... |
| frege75 44397 | If from the proposition th... |
| dffrege76 44398 | If from the two propositio... |
| frege77 44399 | If ` Y ` follows ` X ` in ... |
| frege78 44400 | Commuted form of ~ frege77... |
| frege79 44401 | Distributed form of ~ freg... |
| frege80 44402 | Add additional condition t... |
| frege81 44403 | If ` X ` has a property ` ... |
| frege82 44404 | Closed-form deduction base... |
| frege83 44405 | Apply commuted form of ~ f... |
| frege84 44406 | Commuted form of ~ frege81... |
| frege85 44407 | Commuted form of ~ frege77... |
| frege86 44408 | Conclusion about element o... |
| frege87 44409 | If ` Z ` is a result of an... |
| frege88 44410 | Commuted form of ~ frege87... |
| frege89 44411 | One direction of ~ dffrege... |
| frege90 44412 | Add antecedent to ~ frege8... |
| frege91 44413 | Every result of an applica... |
| frege92 44414 | Inference from ~ frege91 .... |
| frege93 44415 | Necessary condition for tw... |
| frege94 44416 | Looking one past a pair re... |
| frege95 44417 | Looking one past a pair re... |
| frege96 44418 | Every result of an applica... |
| frege97 44419 | The property of following ... |
| frege98 44420 | If ` Y ` follows ` X ` and... |
| dffrege99 44421 | If ` Z ` is identical with... |
| frege100 44422 | One direction of ~ dffrege... |
| frege101 44423 | Lemma for ~ frege102 . Pr... |
| frege102 44424 | If ` Z ` belongs to the ` ... |
| frege103 44425 | Proposition 103 of [Frege1... |
| frege104 44426 | Proposition 104 of [Frege1... |
| frege105 44427 | Proposition 105 of [Frege1... |
| frege106 44428 | Whatever follows ` X ` in ... |
| frege107 44429 | Proposition 107 of [Frege1... |
| frege108 44430 | If ` Y ` belongs to the ` ... |
| frege109 44431 | The property of belonging ... |
| frege110 44432 | Proposition 110 of [Frege1... |
| frege111 44433 | If ` Y ` belongs to the ` ... |
| frege112 44434 | Identity implies belonging... |
| frege113 44435 | Proposition 113 of [Frege1... |
| frege114 44436 | If ` X ` belongs to the ` ... |
| dffrege115 44437 | If from the circumstance t... |
| frege116 44438 | One direction of ~ dffrege... |
| frege117 44439 | Lemma for ~ frege118 . Pr... |
| frege118 44440 | Simplified application of ... |
| frege119 44441 | Lemma for ~ frege120 . Pr... |
| frege120 44442 | Simplified application of ... |
| frege121 44443 | Lemma for ~ frege122 . Pr... |
| frege122 44444 | If ` X ` is a result of an... |
| frege123 44445 | Lemma for ~ frege124 . Pr... |
| frege124 44446 | If ` X ` is a result of an... |
| frege125 44447 | Lemma for ~ frege126 . Pr... |
| frege126 44448 | If ` M ` follows ` Y ` in ... |
| frege127 44449 | Communte antecedents of ~ ... |
| frege128 44450 | Lemma for ~ frege129 . Pr... |
| frege129 44451 | If the procedure ` R ` is ... |
| frege130 44452 | Lemma for ~ frege131 . Pr... |
| frege131 44453 | If the procedure ` R ` is ... |
| frege132 44454 | Lemma for ~ frege133 . Pr... |
| frege133 44455 | If the procedure ` R ` is ... |
| enrelmap 44456 | The set of all possible re... |
| enrelmapr 44457 | The set of all possible re... |
| enmappw 44458 | The set of all mappings fr... |
| enmappwid 44459 | The set of all mappings fr... |
| rfovd 44460 | Value of the operator, ` (... |
| rfovfvd 44461 | Value of the operator, ` (... |
| rfovfvfvd 44462 | Value of the operator, ` (... |
| rfovcnvf1od 44463 | Properties of the operator... |
| rfovcnvd 44464 | Value of the converse of t... |
| rfovf1od 44465 | The value of the operator,... |
| rfovcnvfvd 44466 | Value of the converse of t... |
| fsovd 44467 | Value of the operator, ` (... |
| fsovrfovd 44468 | The operator which gives a... |
| fsovfvd 44469 | Value of the operator, ` (... |
| fsovfvfvd 44470 | Value of the operator, ` (... |
| fsovfd 44471 | The operator, ` ( A O B ) ... |
| fsovcnvlem 44472 | The ` O ` operator, which ... |
| fsovcnvd 44473 | The value of the converse ... |
| fsovcnvfvd 44474 | The value of the converse ... |
| fsovf1od 44475 | The value of ` ( A O B ) `... |
| dssmapfvd 44476 | Value of the duality opera... |
| dssmapfv2d 44477 | Value of the duality opera... |
| dssmapfv3d 44478 | Value of the duality opera... |
| dssmapnvod 44479 | For any base set ` B ` the... |
| dssmapf1od 44480 | For any base set ` B ` the... |
| dssmap2d 44481 | For any base set ` B ` the... |
| or3or 44482 | Decompose disjunction into... |
| andi3or 44483 | Distribute over triple dis... |
| uneqsn 44484 | If a union of classes is e... |
| brfvimex 44485 | If a binary relation holds... |
| brovmptimex 44486 | If a binary relation holds... |
| brovmptimex1 44487 | If a binary relation holds... |
| brovmptimex2 44488 | If a binary relation holds... |
| brcoffn 44489 | Conditions allowing the de... |
| brcofffn 44490 | Conditions allowing the de... |
| brco2f1o 44491 | Conditions allowing the de... |
| brco3f1o 44492 | Conditions allowing the de... |
| ntrclsbex 44493 | If (pseudo-)interior and (... |
| ntrclsrcomplex 44494 | The relative complement of... |
| neik0imk0p 44495 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 44496 | If an interior function is... |
| ntrkbimka 44497 | If the interiors of disjoi... |
| ntrk0kbimka 44498 | If the interiors of disjoi... |
| clsk3nimkb 44499 | If the base set is not emp... |
| clsk1indlem0 44500 | The ansatz closure functio... |
| clsk1indlem2 44501 | The ansatz closure functio... |
| clsk1indlem3 44502 | The ansatz closure functio... |
| clsk1indlem4 44503 | The ansatz closure functio... |
| clsk1indlem1 44504 | The ansatz closure functio... |
| clsk1independent 44505 | For generalized closure fu... |
| neik0pk1imk0 44506 | Kuratowski's K0' and K1 ax... |
| isotone1 44507 | Two different ways to say ... |
| isotone2 44508 | Two different ways to say ... |
| ntrk1k3eqk13 44509 | An interior function is bo... |
| ntrclsf1o 44510 | If (pseudo-)interior and (... |
| ntrclsnvobr 44511 | If (pseudo-)interior and (... |
| ntrclsiex 44512 | If (pseudo-)interior and (... |
| ntrclskex 44513 | If (pseudo-)interior and (... |
| ntrclsfv1 44514 | If (pseudo-)interior and (... |
| ntrclsfv2 44515 | If (pseudo-)interior and (... |
| ntrclselnel1 44516 | If (pseudo-)interior and (... |
| ntrclselnel2 44517 | If (pseudo-)interior and (... |
| ntrclsfv 44518 | The value of the interior ... |
| ntrclsfveq1 44519 | If interior and closure fu... |
| ntrclsfveq2 44520 | If interior and closure fu... |
| ntrclsfveq 44521 | If interior and closure fu... |
| ntrclsss 44522 | If interior and closure fu... |
| ntrclsneine0lem 44523 | If (pseudo-)interior and (... |
| ntrclsneine0 44524 | If (pseudo-)interior and (... |
| ntrclscls00 44525 | If (pseudo-)interior and (... |
| ntrclsiso 44526 | If (pseudo-)interior and (... |
| ntrclsk2 44527 | An interior function is co... |
| ntrclskb 44528 | The interiors of disjoint ... |
| ntrclsk3 44529 | The intersection of interi... |
| ntrclsk13 44530 | The interior of the inters... |
| ntrclsk4 44531 | Idempotence of the interio... |
| ntrneibex 44532 | If (pseudo-)interior and (... |
| ntrneircomplex 44533 | The relative complement of... |
| ntrneif1o 44534 | If (pseudo-)interior and (... |
| ntrneiiex 44535 | If (pseudo-)interior and (... |
| ntrneinex 44536 | If (pseudo-)interior and (... |
| ntrneicnv 44537 | If (pseudo-)interior and (... |
| ntrneifv1 44538 | If (pseudo-)interior and (... |
| ntrneifv2 44539 | If (pseudo-)interior and (... |
| ntrneiel 44540 | If (pseudo-)interior and (... |
| ntrneifv3 44541 | The value of the neighbors... |
| ntrneineine0lem 44542 | If (pseudo-)interior and (... |
| ntrneineine1lem 44543 | If (pseudo-)interior and (... |
| ntrneifv4 44544 | The value of the interior ... |
| ntrneiel2 44545 | Membership in iterated int... |
| ntrneineine0 44546 | If (pseudo-)interior and (... |
| ntrneineine1 44547 | If (pseudo-)interior and (... |
| ntrneicls00 44548 | If (pseudo-)interior and (... |
| ntrneicls11 44549 | If (pseudo-)interior and (... |
| ntrneiiso 44550 | If (pseudo-)interior and (... |
| ntrneik2 44551 | An interior function is co... |
| ntrneix2 44552 | An interior (closure) func... |
| ntrneikb 44553 | The interiors of disjoint ... |
| ntrneixb 44554 | The interiors (closures) o... |
| ntrneik3 44555 | The intersection of interi... |
| ntrneix3 44556 | The closure of the union o... |
| ntrneik13 44557 | The interior of the inters... |
| ntrneix13 44558 | The closure of the union o... |
| ntrneik4w 44559 | Idempotence of the interio... |
| ntrneik4 44560 | Idempotence of the interio... |
| clsneibex 44561 | If (pseudo-)closure and (p... |
| clsneircomplex 44562 | The relative complement of... |
| clsneif1o 44563 | If a (pseudo-)closure func... |
| clsneicnv 44564 | If a (pseudo-)closure func... |
| clsneikex 44565 | If closure and neighborhoo... |
| clsneinex 44566 | If closure and neighborhoo... |
| clsneiel1 44567 | If a (pseudo-)closure func... |
| clsneiel2 44568 | If a (pseudo-)closure func... |
| clsneifv3 44569 | Value of the neighborhoods... |
| clsneifv4 44570 | Value of the closure (inte... |
| neicvgbex 44571 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 44572 | The relative complement of... |
| neicvgf1o 44573 | If neighborhood and conver... |
| neicvgnvo 44574 | If neighborhood and conver... |
| neicvgnvor 44575 | If neighborhood and conver... |
| neicvgmex 44576 | If the neighborhoods and c... |
| neicvgnex 44577 | If the neighborhoods and c... |
| neicvgel1 44578 | A subset being an element ... |
| neicvgel2 44579 | The complement of a subset... |
| neicvgfv 44580 | The value of the neighborh... |
| ntrrn 44581 | The range of the interior ... |
| ntrf 44582 | The interior function of a... |
| ntrf2 44583 | The interior function is a... |
| ntrelmap 44584 | The interior function is a... |
| clsf2 44585 | The closure function is a ... |
| clselmap 44586 | The closure function is a ... |
| dssmapntrcls 44587 | The interior and closure o... |
| dssmapclsntr 44588 | The closure and interior o... |
| gneispa 44589 | Each point ` p ` of the ne... |
| gneispb 44590 | Given a neighborhood ` N `... |
| gneispace2 44591 | The predicate that ` F ` i... |
| gneispace3 44592 | The predicate that ` F ` i... |
| gneispace 44593 | The predicate that ` F ` i... |
| gneispacef 44594 | A generic neighborhood spa... |
| gneispacef2 44595 | A generic neighborhood spa... |
| gneispacefun 44596 | A generic neighborhood spa... |
| gneispacern 44597 | A generic neighborhood spa... |
| gneispacern2 44598 | A generic neighborhood spa... |
| gneispace0nelrn 44599 | A generic neighborhood spa... |
| gneispace0nelrn2 44600 | A generic neighborhood spa... |
| gneispace0nelrn3 44601 | A generic neighborhood spa... |
| gneispaceel 44602 | Every neighborhood of a po... |
| gneispaceel2 44603 | Every neighborhood of a po... |
| gneispacess 44604 | All supersets of a neighbo... |
| gneispacess2 44605 | All supersets of a neighbo... |
| k0004lem1 44606 | Application of ~ ssin to r... |
| k0004lem2 44607 | A mapping with a particula... |
| k0004lem3 44608 | When the value of a mappin... |
| k0004val 44609 | The topological simplex of... |
| k0004ss1 44610 | The topological simplex of... |
| k0004ss2 44611 | The topological simplex of... |
| k0004ss3 44612 | The topological simplex of... |
| k0004val0 44613 | The topological simplex of... |
| inductionexd 44614 | Simple induction example. ... |
| wwlemuld 44615 | Natural deduction form of ... |
| leeq1d 44616 | Specialization of ~ breq1d... |
| leeq2d 44617 | Specialization of ~ breq2d... |
| absmulrposd 44618 | Specialization of absmuld ... |
| imadisjld 44619 | Natural dduction form of o... |
| wnefimgd 44620 | The image of a mapping fro... |
| fco2d 44621 | Natural deduction form of ... |
| wfximgfd 44622 | The value of a function on... |
| extoimad 44623 | If |f(x)| <= C for all x t... |
| imo72b2lem0 44624 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 44625 | Natural deduction form of ... |
| imo72b2lem2 44626 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 44627 | Natural deduction form of ... |
| imo72b2lem1 44628 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 44629 | 'Less than or equal to' re... |
| lemuldiv4d 44630 | 'Less than or equal to' re... |
| imo72b2 44631 | IMO 1972 B2. (14th Intern... |
| int-addcomd 44632 | AdditionCommutativity gene... |
| int-addassocd 44633 | AdditionAssociativity gene... |
| int-addsimpd 44634 | AdditionSimplification gen... |
| int-mulcomd 44635 | MultiplicationCommutativit... |
| int-mulassocd 44636 | MultiplicationAssociativit... |
| int-mulsimpd 44637 | MultiplicationSimplificati... |
| int-leftdistd 44638 | AdditionMultiplicationLeft... |
| int-rightdistd 44639 | AdditionMultiplicationRigh... |
| int-sqdefd 44640 | SquareDefinition generator... |
| int-mul11d 44641 | First MultiplicationOne ge... |
| int-mul12d 44642 | Second MultiplicationOne g... |
| int-add01d 44643 | First AdditionZero generat... |
| int-add02d 44644 | Second AdditionZero genera... |
| int-sqgeq0d 44645 | SquareGEQZero generator ru... |
| int-eqprincd 44646 | PrincipleOfEquality genera... |
| int-eqtransd 44647 | EqualityTransitivity gener... |
| int-eqmvtd 44648 | EquMoveTerm generator rule... |
| int-eqineqd 44649 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 44650 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 44651 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 44652 | SecondPrincipleOfInequalit... |
| int-ineqtransd 44653 | InequalityTransitivity gen... |
| unitadd 44654 | Theorem used in conjunctio... |
| gsumws3 44655 | Valuation of a length 3 wo... |
| gsumws4 44656 | Valuation of a length 4 wo... |
| amgm2d 44657 | Arithmetic-geometric mean ... |
| amgm3d 44658 | Arithmetic-geometric mean ... |
| amgm4d 44659 | Arithmetic-geometric mean ... |
| spALT 44660 | ~ sp can be proven from th... |
| elnelneqd 44661 | Two classes are not equal ... |
| elnelneq2d 44662 | Two classes are not equal ... |
| rr-spce 44663 | Prove an existential. (Co... |
| rexlimdvaacbv 44664 | Unpack a restricted existe... |
| rexlimddvcbvw 44665 | Unpack a restricted existe... |
| rexlimddvcbv 44666 | Unpack a restricted existe... |
| rr-elrnmpt3d 44667 | Elementhood in an image se... |
| rr-phpd 44668 | Equivalent of ~ php withou... |
| tfindsd 44669 | Deduction associated with ... |
| mnringvald 44672 | Value of the monoid ring f... |
| mnringnmulrd 44673 | Components of a monoid rin... |
| mnringbased 44674 | The base set of a monoid r... |
| mnringbaserd 44675 | The base set of a monoid r... |
| mnringelbased 44676 | Membership in the base set... |
| mnringbasefd 44677 | Elements of a monoid ring ... |
| mnringbasefsuppd 44678 | Elements of a monoid ring ... |
| mnringaddgd 44679 | The additive operation of ... |
| mnring0gd 44680 | The additive identity of a... |
| mnring0g2d 44681 | The additive identity of a... |
| mnringmulrd 44682 | The ring product of a mono... |
| mnringscad 44683 | The scalar ring of a monoi... |
| mnringvscad 44684 | The scalar product of a mo... |
| mnringlmodd 44685 | Monoid rings are left modu... |
| mnringmulrvald 44686 | Value of multiplication in... |
| mnringmulrcld 44687 | Monoid rings are closed un... |
| gru0eld 44688 | A nonempty Grothendieck un... |
| grusucd 44689 | Grothendieck universes are... |
| r1rankcld 44690 | Any rank of the cumulative... |
| grur1cld 44691 | Grothendieck universes are... |
| grurankcld 44692 | Grothendieck universes are... |
| grurankrcld 44693 | If a Grothendieck universe... |
| scotteqd 44696 | Equality theorem for the S... |
| scotteq 44697 | Closed form of ~ scotteqd ... |
| nfscott 44698 | Bound-variable hypothesis ... |
| scottabf 44699 | Value of the Scott operati... |
| scottab 44700 | Value of the Scott operati... |
| scottabes 44701 | Value of the Scott operati... |
| scottss 44702 | Scott's trick produces a s... |
| elscottab 44703 | An element of the output o... |
| scottex2 44704 | ~ scottex expressed using ... |
| scotteld 44705 | The Scott operation sends ... |
| scottelrankd 44706 | Property of a Scott's tric... |
| scottrankd 44707 | Rank of a nonempty Scott's... |
| gruscottcld 44708 | If a Grothendieck universe... |
| dfcoll2 44711 | Alternate definition of th... |
| colleq12d 44712 | Equality theorem for the c... |
| colleq1 44713 | Equality theorem for the c... |
| colleq2 44714 | Equality theorem for the c... |
| nfcoll 44715 | Bound-variable hypothesis ... |
| collexd 44716 | The output of the collecti... |
| cpcolld 44717 | Property of the collection... |
| cpcoll2d 44718 | ~ cpcolld with an extra ex... |
| grucollcld 44719 | A Grothendieck universe co... |
| ismnu 44720 | The hypothesis of this the... |
| mnuop123d 44721 | Operations of a minimal un... |
| mnussd 44722 | Minimal universes are clos... |
| mnuss2d 44723 | ~ mnussd with arguments pr... |
| mnu0eld 44724 | A nonempty minimal univers... |
| mnuop23d 44725 | Second and third operation... |
| mnupwd 44726 | Minimal universes are clos... |
| mnusnd 44727 | Minimal universes are clos... |
| mnuprssd 44728 | A minimal universe contain... |
| mnuprss2d 44729 | Special case of ~ mnuprssd... |
| mnuop3d 44730 | Third operation of a minim... |
| mnuprdlem1 44731 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 44732 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 44733 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 44734 | Lemma for ~ mnuprd . Gene... |
| mnuprd 44735 | Minimal universes are clos... |
| mnuunid 44736 | Minimal universes are clos... |
| mnuund 44737 | Minimal universes are clos... |
| mnutrcld 44738 | Minimal universes contain ... |
| mnutrd 44739 | Minimal universes are tran... |
| mnurndlem1 44740 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 44741 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 44742 | Minimal universes contain ... |
| mnugrud 44743 | Minimal universes are Grot... |
| grumnudlem 44744 | Lemma for ~ grumnud . (Co... |
| grumnud 44745 | Grothendieck universes are... |
| grumnueq 44746 | The class of Grothendieck ... |
| expandan 44747 | Expand conjunction to prim... |
| expandexn 44748 | Expand an existential quan... |
| expandral 44749 | Expand a restricted univer... |
| expandrexn 44750 | Expand a restricted existe... |
| expandrex 44751 | Expand a restricted existe... |
| expanduniss 44752 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 44753 | Express the predicate on `... |
| rr-grothprimbi 44754 | Express "every set is cont... |
| inagrud 44755 | Inaccessible levels of the... |
| inaex 44756 | Assuming the Tarski-Grothe... |
| gruex 44757 | Assuming the Tarski-Grothe... |
| rr-groth 44758 | An equivalent of ~ ax-grot... |
| rr-grothprim 44759 | An equivalent of ~ ax-grot... |
| ismnushort 44760 | Express the predicate on `... |
| dfuniv2 44761 | Alternative definition of ... |
| rr-grothshortbi 44762 | Express "every set is cont... |
| rr-grothshort 44763 | A shorter equivalent of ~ ... |
| nanorxor 44764 | 'nand' is equivalent to th... |
| undisjrab 44765 | Union of two disjoint rest... |
| iso0 44766 | The empty set is an ` R , ... |
| ssrecnpr 44767 | ` RR ` is a subset of both... |
| seff 44768 | Let set ` S ` be the real ... |
| sblpnf 44769 | The infinity ball in the a... |
| prmunb2 44770 | The primes are unbounded. ... |
| dvgrat 44771 | Ratio test for divergence ... |
| cvgdvgrat 44772 | Ratio test for convergence... |
| radcnvrat 44773 | Let ` L ` be the limit, if... |
| reldvds 44774 | The divides relation is in... |
| nznngen 44775 | All positive integers in t... |
| nzss 44776 | The set of multiples of _m... |
| nzin 44777 | The intersection of the se... |
| nzprmdif 44778 | Subtract one prime's multi... |
| hashnzfz 44779 | Special case of ~ hashdvds... |
| hashnzfz2 44780 | Special case of ~ hashnzfz... |
| hashnzfzclim 44781 | As the upper bound ` K ` o... |
| caofcan 44782 | Transfer a cancellation la... |
| ofsubid 44783 | Function analogue of ~ sub... |
| ofmul12 44784 | Function analogue of ~ mul... |
| ofdivrec 44785 | Function analogue of ~ div... |
| ofdivcan4 44786 | Function analogue of ~ div... |
| ofdivdiv2 44787 | Function analogue of ~ div... |
| lhe4.4ex1a 44788 | Example of the Fundamental... |
| dvsconst 44789 | Derivative of a constant f... |
| dvsid 44790 | Derivative of the identity... |
| dvsef 44791 | Derivative of the exponent... |
| expgrowthi 44792 | Exponential growth and dec... |
| dvconstbi 44793 | The derivative of a functi... |
| expgrowth 44794 | Exponential growth and dec... |
| bccval 44797 | Value of the generalized b... |
| bcccl 44798 | Closure of the generalized... |
| bcc0 44799 | The generalized binomial c... |
| bccp1k 44800 | Generalized binomial coeff... |
| bccm1k 44801 | Generalized binomial coeff... |
| bccn0 44802 | Generalized binomial coeff... |
| bccn1 44803 | Generalized binomial coeff... |
| bccbc 44804 | The binomial coefficient a... |
| uzmptshftfval 44805 | When ` F ` is a maps-to fu... |
| dvradcnv2 44806 | The radius of convergence ... |
| binomcxplemwb 44807 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 44808 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 44809 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 44810 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 44811 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 44812 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 44813 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 44814 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 44815 | Lemma for ~ binomcxp . Wh... |
| binomcxp 44816 | Generalize the binomial th... |
| pm10.12 44817 | Theorem *10.12 in [Whitehe... |
| pm10.14 44818 | Theorem *10.14 in [Whitehe... |
| pm10.251 44819 | Theorem *10.251 in [Whiteh... |
| pm10.252 44820 | Theorem *10.252 in [Whiteh... |
| pm10.253 44821 | Theorem *10.253 in [Whiteh... |
| albitr 44822 | Theorem *10.301 in [Whiteh... |
| pm10.42 44823 | Theorem *10.42 in [Whitehe... |
| pm10.52 44824 | Theorem *10.52 in [Whitehe... |
| pm10.53 44825 | Theorem *10.53 in [Whitehe... |
| pm10.541 44826 | Theorem *10.541 in [Whiteh... |
| pm10.542 44827 | Theorem *10.542 in [Whiteh... |
| pm10.55 44828 | Theorem *10.55 in [Whitehe... |
| pm10.56 44829 | Theorem *10.56 in [Whitehe... |
| pm10.57 44830 | Theorem *10.57 in [Whitehe... |
| 2alanimi 44831 | Removes two universal quan... |
| 2al2imi 44832 | Removes two universal quan... |
| pm11.11 44833 | Theorem *11.11 in [Whitehe... |
| pm11.12 44834 | Theorem *11.12 in [Whitehe... |
| 19.21vv 44835 | Compare Theorem *11.3 in [... |
| 2alim 44836 | Theorem *11.32 in [Whitehe... |
| 2albi 44837 | Theorem *11.33 in [Whitehe... |
| 2exim 44838 | Theorem *11.34 in [Whitehe... |
| 2exbi 44839 | Theorem *11.341 in [Whiteh... |
| spsbce-2 44840 | Theorem *11.36 in [Whitehe... |
| 19.33-2 44841 | Theorem *11.421 in [Whiteh... |
| 19.36vv 44842 | Theorem *11.43 in [Whitehe... |
| 19.31vv 44843 | Theorem *11.44 in [Whitehe... |
| 19.37vv 44844 | Theorem *11.46 in [Whitehe... |
| 19.28vv 44845 | Theorem *11.47 in [Whitehe... |
| pm11.52 44846 | Theorem *11.52 in [Whitehe... |
| aaanv 44847 | Theorem *11.56 in [Whitehe... |
| pm11.57 44848 | Theorem *11.57 in [Whitehe... |
| pm11.58 44849 | Theorem *11.58 in [Whitehe... |
| pm11.59 44850 | Theorem *11.59 in [Whitehe... |
| pm11.6 44851 | Theorem *11.6 in [Whitehea... |
| pm11.61 44852 | Theorem *11.61 in [Whitehe... |
| pm11.62 44853 | Theorem *11.62 in [Whitehe... |
| pm11.63 44854 | Theorem *11.63 in [Whitehe... |
| pm11.7 44855 | Theorem *11.7 in [Whitehea... |
| pm11.71 44856 | Theorem *11.71 in [Whitehe... |
| sbeqal1 44857 | If ` x = y ` always implie... |
| sbeqal1i 44858 | Suppose you know ` x = y `... |
| sbeqal2i 44859 | If ` x = y ` implies ` x =... |
| axc5c4c711 44860 | Proof of a theorem that ca... |
| axc5c4c711toc5 44861 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 44862 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 44863 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 44864 | Rederivation of ~ ax-11 fr... |
| axc11next 44865 | This theorem shows that, g... |
| pm13.13a 44866 | One result of theorem *13.... |
| pm13.13b 44867 | Theorem *13.13 in [Whitehe... |
| pm13.14 44868 | Theorem *13.14 in [Whitehe... |
| pm13.192 44869 | Theorem *13.192 in [Whiteh... |
| pm13.193 44870 | Theorem *13.193 in [Whiteh... |
| pm13.194 44871 | Theorem *13.194 in [Whiteh... |
| pm13.195 44872 | Theorem *13.195 in [Whiteh... |
| pm13.196a 44873 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 44874 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 44875 | Theorem *13.22 in [Whitehe... |
| iotain 44876 | Equivalence between two di... |
| iotaexeu 44877 | The iota class exists. Th... |
| iotasbc 44878 | Definition *14.01 in [Whit... |
| iotasbc2 44879 | Theorem *14.111 in [Whiteh... |
| pm14.12 44880 | Theorem *14.12 in [Whitehe... |
| pm14.122a 44881 | Theorem *14.122 in [Whiteh... |
| pm14.122b 44882 | Theorem *14.122 in [Whiteh... |
| pm14.122c 44883 | Theorem *14.122 in [Whiteh... |
| pm14.123a 44884 | Theorem *14.123 in [Whiteh... |
| pm14.123b 44885 | Theorem *14.123 in [Whiteh... |
| pm14.123c 44886 | Theorem *14.123 in [Whiteh... |
| pm14.18 44887 | Theorem *14.18 in [Whitehe... |
| iotaequ 44888 | Theorem *14.2 in [Whitehea... |
| iotavalb 44889 | Theorem *14.202 in [Whiteh... |
| iotasbc5 44890 | Theorem *14.205 in [Whiteh... |
| pm14.24 44891 | Theorem *14.24 in [Whitehe... |
| iotavalsb 44892 | Theorem *14.242 in [Whiteh... |
| sbiota1 44893 | Theorem *14.25 in [Whitehe... |
| sbaniota 44894 | Theorem *14.26 in [Whitehe... |
| iotasbcq 44895 | Theorem *14.272 in [Whiteh... |
| elnev 44896 | Any set that contains one ... |
| rusbcALT 44897 | A version of Russell's par... |
| compeq 44898 | Equality between two ways ... |
| compne 44899 | The complement of ` A ` is... |
| compab 44900 | Two ways of saying "the co... |
| conss2 44901 | Contrapositive law for sub... |
| conss1 44902 | Contrapositive law for sub... |
| ralbidar 44903 | More general form of ~ ral... |
| rexbidar 44904 | More general form of ~ rex... |
| dropab1 44905 | Theorem to aid use of the ... |
| dropab2 44906 | Theorem to aid use of the ... |
| ipo0 44907 | If the identity relation p... |
| ifr0 44908 | A class that is founded by... |
| ordpss 44909 | ~ ordelpss with an anteced... |
| fvsb 44910 | Explicit substitution of a... |
| fveqsb 44911 | Implicit substitution of a... |
| xpexb 44912 | A Cartesian product exists... |
| trelpss 44913 | An element of a transitive... |
| addcomgi 44914 | Generalization of commutat... |
| addrval 44924 | Value of the operation of ... |
| subrval 44925 | Value of the operation of ... |
| mulvval 44926 | Value of the operation of ... |
| addrfv 44927 | Vector addition at a value... |
| subrfv 44928 | Vector subtraction at a va... |
| mulvfv 44929 | Scalar multiplication at a... |
| addrfn 44930 | Vector addition produces a... |
| subrfn 44931 | Vector subtraction produce... |
| mulvfn 44932 | Scalar multiplication prod... |
| addrcom 44933 | Vector addition is commuta... |
| idiALT 44937 | Placeholder for ~ idi . T... |
| exbir 44938 | Exportation implication al... |
| 3impexpbicom 44939 | Version of ~ 3impexp where... |
| 3impexpbicomi 44940 | Inference associated with ... |
| bi1imp 44941 | Importation inference simi... |
| bi2imp 44942 | Importation inference simi... |
| bi3impb 44943 | Similar to ~ 3impb with im... |
| bi3impa 44944 | Similar to ~ 3impa with im... |
| bi23impib 44945 | ~ 3impib with the inner im... |
| bi13impib 44946 | ~ 3impib with the outer im... |
| bi123impib 44947 | ~ 3impib with the implicat... |
| bi13impia 44948 | ~ 3impia with the outer im... |
| bi123impia 44949 | ~ 3impia with the implicat... |
| bi33imp12 44950 | ~ 3imp with innermost impl... |
| bi13imp23 44951 | ~ 3imp with outermost impl... |
| bi13imp2 44952 | Similar to ~ 3imp except t... |
| bi12imp3 44953 | Similar to ~ 3imp except a... |
| bi23imp1 44954 | Similar to ~ 3imp except a... |
| bi123imp0 44955 | Similar to ~ 3imp except a... |
| 4animp1 44956 | A single hypothesis unific... |
| 4an31 44957 | A rearrangement of conjunc... |
| 4an4132 44958 | A rearrangement of conjunc... |
| expcomdg 44959 | Biconditional form of ~ ex... |
| iidn3 44960 | ~ idn3 without virtual ded... |
| ee222 44961 | ~ e222 without virtual ded... |
| ee3bir 44962 | Right-biconditional form o... |
| ee13 44963 | ~ e13 without virtual dedu... |
| ee121 44964 | ~ e121 without virtual ded... |
| ee122 44965 | ~ e122 without virtual ded... |
| ee333 44966 | ~ e333 without virtual ded... |
| ee323 44967 | ~ e323 without virtual ded... |
| 3ornot23 44968 | If the second and third di... |
| orbi1r 44969 | ~ orbi1 with order of disj... |
| 3orbi123 44970 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 44971 | Closed form of ~ syl5 . D... |
| impexpd 44972 | The following User's Proof... |
| com3rgbi 44973 | The following User's Proof... |
| impexpdcom 44974 | The following User's Proof... |
| ee1111 44975 | Non-virtual deduction form... |
| pm2.43bgbi 44976 | Logical equivalence of a 2... |
| pm2.43cbi 44977 | Logical equivalence of a 3... |
| ee233 44978 | Non-virtual deduction form... |
| imbi13 44979 | Join three logical equival... |
| ee33 44980 | Non-virtual deduction form... |
| con5 44981 | Biconditional contrapositi... |
| con5i 44982 | Inference form of ~ con5 .... |
| exlimexi 44983 | Inference similar to Theor... |
| sb5ALT 44984 | Equivalence for substituti... |
| eexinst01 44985 | ~ exinst01 without virtual... |
| eexinst11 44986 | ~ exinst11 without virtual... |
| vk15.4j 44987 | Excercise 4j of Unit 15 of... |
| notnotrALT 44988 | Converse of double negatio... |
| con3ALT2 44989 | Contraposition. Alternate... |
| ssralv2 44990 | Quantification restricted ... |
| sbc3or 44991 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 44992 | Closed form of ~ alrimi wi... |
| rspsbc2 44993 | ~ rspsbc with two quantify... |
| sbcoreleleq 44994 | Substitution of a setvar v... |
| tratrb 44995 | If a class is transitive a... |
| ordelordALT 44996 | An element of an ordinal c... |
| sbcim2g 44997 | Distribution of class subs... |
| sbcbi 44998 | Implication form of ~ sbcb... |
| trsbc 44999 | Formula-building inference... |
| truniALT 45000 | The union of a class of tr... |
| onfrALTlem5 45001 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 45002 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 45003 | Lemma for ~ onfrALT . (Co... |
| ggen31 45004 | ~ gen31 without virtual de... |
| onfrALTlem2 45005 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 45006 | A theorem pertaining to th... |
| onfrALTlem1 45007 | Lemma for ~ onfrALT . (Co... |
| onfrALT 45008 | The membership relation is... |
| 19.41rg 45009 | Closed form of right-to-le... |
| opelopab4 45010 | Ordered pair membership in... |
| 2pm13.193 45011 | ~ pm13.193 for two variabl... |
| hbntal 45012 | A closed form of ~ hbn . ~... |
| hbimpg 45013 | A closed form of ~ hbim . ... |
| hbalg 45014 | Closed form of ~ hbal . D... |
| hbexg 45015 | Closed form of ~ nfex . D... |
| ax6e2eq 45016 | Alternate form of ~ ax6e f... |
| ax6e2nd 45017 | If at least two sets exist... |
| ax6e2ndeq 45018 | "At least two sets exist" ... |
| 2sb5nd 45019 | Equivalence for double sub... |
| 2uasbanh 45020 | Distribute the unabbreviat... |
| 2uasban 45021 | Distribute the unabbreviat... |
| e2ebind 45022 | Absorption of an existenti... |
| elpwgded 45023 | ~ elpwgdedVD in convention... |
| trelded 45024 | Deduction form of ~ trel .... |
| jaoded 45025 | Deduction form of ~ jao . ... |
| sbtT 45026 | A substitution into a theo... |
| not12an2impnot1 45027 | If a double conjunction is... |
| in1 45030 | Inference form of ~ df-vd1... |
| iin1 45031 | ~ in1 without virtual dedu... |
| dfvd1ir 45032 | Inference form of ~ df-vd1... |
| idn1 45033 | Virtual deduction identity... |
| dfvd1imp 45034 | Left-to-right part of defi... |
| dfvd1impr 45035 | Right-to-left part of defi... |
| dfvd2 45038 | Definition of a 2-hypothes... |
| dfvd2an 45041 | Definition of a 2-hypothes... |
| dfvd2ani 45042 | Inference form of ~ dfvd2a... |
| dfvd2anir 45043 | Right-to-left inference fo... |
| dfvd2i 45044 | Inference form of ~ dfvd2 ... |
| dfvd2ir 45045 | Right-to-left inference fo... |
| dfvd3 45050 | Definition of a 3-hypothes... |
| dfvd3i 45051 | Inference form of ~ dfvd3 ... |
| dfvd3ir 45052 | Right-to-left inference fo... |
| dfvd3an 45053 | Definition of a 3-hypothes... |
| dfvd3ani 45054 | Inference form of ~ dfvd3a... |
| dfvd3anir 45055 | Right-to-left inference fo... |
| vd01 45056 | A virtual hypothesis virtu... |
| vd02 45057 | Two virtual hypotheses vir... |
| vd03 45058 | A theorem is virtually inf... |
| vd12 45059 | A virtual deduction with 1... |
| vd13 45060 | A virtual deduction with 1... |
| vd23 45061 | A virtual deduction with 2... |
| dfvd2imp 45062 | The virtual deduction form... |
| dfvd2impr 45063 | A 2-antecedent nested impl... |
| in2 45064 | The virtual deduction intr... |
| int2 45065 | The virtual deduction intr... |
| iin2 45066 | ~ in2 without virtual dedu... |
| in2an 45067 | The virtual deduction intr... |
| in3 45068 | The virtual deduction intr... |
| iin3 45069 | ~ in3 without virtual dedu... |
| in3an 45070 | The virtual deduction intr... |
| int3 45071 | The virtual deduction intr... |
| idn2 45072 | Virtual deduction identity... |
| iden2 45073 | Virtual deduction identity... |
| idn3 45074 | Virtual deduction identity... |
| gen11 45075 | Virtual deduction generali... |
| gen11nv 45076 | Virtual deduction generali... |
| gen12 45077 | Virtual deduction generali... |
| gen21 45078 | Virtual deduction generali... |
| gen21nv 45079 | Virtual deduction form of ... |
| gen31 45080 | Virtual deduction generali... |
| gen22 45081 | Virtual deduction generali... |
| ggen22 45082 | ~ gen22 without virtual de... |
| exinst 45083 | Existential Instantiation.... |
| exinst01 45084 | Existential Instantiation.... |
| exinst11 45085 | Existential Instantiation.... |
| e1a 45086 | A Virtual deduction elimin... |
| el1 45087 | A Virtual deduction elimin... |
| e1bi 45088 | Biconditional form of ~ e1... |
| e1bir 45089 | Right biconditional form o... |
| e2 45090 | A virtual deduction elimin... |
| e2bi 45091 | Biconditional form of ~ e2... |
| e2bir 45092 | Right biconditional form o... |
| ee223 45093 | ~ e223 without virtual ded... |
| e223 45094 | A virtual deduction elimin... |
| e222 45095 | A virtual deduction elimin... |
| e220 45096 | A virtual deduction elimin... |
| ee220 45097 | ~ e220 without virtual ded... |
| e202 45098 | A virtual deduction elimin... |
| ee202 45099 | ~ e202 without virtual ded... |
| e022 45100 | A virtual deduction elimin... |
| ee022 45101 | ~ e022 without virtual ded... |
| e002 45102 | A virtual deduction elimin... |
| ee002 45103 | ~ e002 without virtual ded... |
| e020 45104 | A virtual deduction elimin... |
| ee020 45105 | ~ e020 without virtual ded... |
| e200 45106 | A virtual deduction elimin... |
| ee200 45107 | ~ e200 without virtual ded... |
| e221 45108 | A virtual deduction elimin... |
| ee221 45109 | ~ e221 without virtual ded... |
| e212 45110 | A virtual deduction elimin... |
| ee212 45111 | ~ e212 without virtual ded... |
| e122 45112 | A virtual deduction elimin... |
| e112 45113 | A virtual deduction elimin... |
| ee112 45114 | ~ e112 without virtual ded... |
| e121 45115 | A virtual deduction elimin... |
| e211 45116 | A virtual deduction elimin... |
| ee211 45117 | ~ e211 without virtual ded... |
| e210 45118 | A virtual deduction elimin... |
| ee210 45119 | ~ e210 without virtual ded... |
| e201 45120 | A virtual deduction elimin... |
| ee201 45121 | ~ e201 without virtual ded... |
| e120 45122 | A virtual deduction elimin... |
| ee120 45123 | Virtual deduction rule ~ e... |
| e021 45124 | A virtual deduction elimin... |
| ee021 45125 | ~ e021 without virtual ded... |
| e012 45126 | A virtual deduction elimin... |
| ee012 45127 | ~ e012 without virtual ded... |
| e102 45128 | A virtual deduction elimin... |
| ee102 45129 | ~ e102 without virtual ded... |
| e22 45130 | A virtual deduction elimin... |
| e22an 45131 | Conjunction form of ~ e22 ... |
| ee22an 45132 | ~ e22an without virtual de... |
| e111 45133 | A virtual deduction elimin... |
| e1111 45134 | A virtual deduction elimin... |
| e110 45135 | A virtual deduction elimin... |
| ee110 45136 | ~ e110 without virtual ded... |
| e101 45137 | A virtual deduction elimin... |
| ee101 45138 | ~ e101 without virtual ded... |
| e011 45139 | A virtual deduction elimin... |
| ee011 45140 | ~ e011 without virtual ded... |
| e100 45141 | A virtual deduction elimin... |
| ee100 45142 | ~ e100 without virtual ded... |
| e010 45143 | A virtual deduction elimin... |
| ee010 45144 | ~ e010 without virtual ded... |
| e001 45145 | A virtual deduction elimin... |
| ee001 45146 | ~ e001 without virtual ded... |
| e11 45147 | A virtual deduction elimin... |
| e11an 45148 | Conjunction form of ~ e11 ... |
| ee11an 45149 | ~ e11an without virtual de... |
| e01 45150 | A virtual deduction elimin... |
| e01an 45151 | Conjunction form of ~ e01 ... |
| ee01an 45152 | ~ e01an without virtual de... |
| e10 45153 | A virtual deduction elimin... |
| e10an 45154 | Conjunction form of ~ e10 ... |
| ee10an 45155 | ~ e10an without virtual de... |
| e02 45156 | A virtual deduction elimin... |
| e02an 45157 | Conjunction form of ~ e02 ... |
| ee02an 45158 | ~ e02an without virtual de... |
| eel021old 45159 | ~ el021old without virtual... |
| el021old 45160 | A virtual deduction elimin... |
| eel000cT 45161 | An elimination deduction. ... |
| eel0TT 45162 | An elimination deduction. ... |
| eelT00 45163 | An elimination deduction. ... |
| eelTTT 45164 | An elimination deduction. ... |
| eelT11 45165 | An elimination deduction. ... |
| eelT1 45166 | Syllogism inference combin... |
| eelT12 45167 | An elimination deduction. ... |
| eelTT1 45168 | An elimination deduction. ... |
| eelT01 45169 | An elimination deduction. ... |
| eel0T1 45170 | An elimination deduction. ... |
| eel12131 45171 | An elimination deduction. ... |
| eel2131 45172 | ~ syl2an with antecedents ... |
| eel3132 45173 | ~ syl2an with antecedents ... |
| eel0321old 45174 | ~ el0321old without virtua... |
| el0321old 45175 | A virtual deduction elimin... |
| eel2122old 45176 | ~ el2122old without virtua... |
| el2122old 45177 | A virtual deduction elimin... |
| eel0000 45178 | Elimination rule similar t... |
| eel00001 45179 | An elimination deduction. ... |
| eel00000 45180 | Elimination rule similar ~... |
| eel11111 45181 | Five-hypothesis eliminatio... |
| e12 45182 | A virtual deduction elimin... |
| e12an 45183 | Conjunction form of ~ e12 ... |
| el12 45184 | Virtual deduction form of ... |
| e20 45185 | A virtual deduction elimin... |
| e20an 45186 | Conjunction form of ~ e20 ... |
| ee20an 45187 | ~ e20an without virtual de... |
| e21 45188 | A virtual deduction elimin... |
| e21an 45189 | Conjunction form of ~ e21 ... |
| ee21an 45190 | ~ e21an without virtual de... |
| e333 45191 | A virtual deduction elimin... |
| e33 45192 | A virtual deduction elimin... |
| e33an 45193 | Conjunction form of ~ e33 ... |
| ee33an 45194 | ~ e33an without virtual de... |
| e3 45195 | Meta-connective form of ~ ... |
| e3bi 45196 | Biconditional form of ~ e3... |
| e3bir 45197 | Right biconditional form o... |
| e03 45198 | A virtual deduction elimin... |
| ee03 45199 | ~ e03 without virtual dedu... |
| e03an 45200 | Conjunction form of ~ e03 ... |
| ee03an 45201 | Conjunction form of ~ ee03... |
| e30 45202 | A virtual deduction elimin... |
| ee30 45203 | ~ e30 without virtual dedu... |
| e30an 45204 | A virtual deduction elimin... |
| ee30an 45205 | Conjunction form of ~ ee30... |
| e13 45206 | A virtual deduction elimin... |
| e13an 45207 | A virtual deduction elimin... |
| ee13an 45208 | ~ e13an without virtual de... |
| e31 45209 | A virtual deduction elimin... |
| ee31 45210 | ~ e31 without virtual dedu... |
| e31an 45211 | A virtual deduction elimin... |
| ee31an 45212 | ~ e31an without virtual de... |
| e23 45213 | A virtual deduction elimin... |
| e23an 45214 | A virtual deduction elimin... |
| ee23an 45215 | ~ e23an without virtual de... |
| e32 45216 | A virtual deduction elimin... |
| ee32 45217 | ~ e32 without virtual dedu... |
| e32an 45218 | A virtual deduction elimin... |
| ee32an 45219 | ~ e33an without virtual de... |
| e123 45220 | A virtual deduction elimin... |
| ee123 45221 | ~ e123 without virtual ded... |
| el123 45222 | A virtual deduction elimin... |
| e233 45223 | A virtual deduction elimin... |
| e323 45224 | A virtual deduction elimin... |
| e000 45225 | A virtual deduction elimin... |
| e00 45226 | Elimination rule identical... |
| e00an 45227 | Elimination rule identical... |
| eel00cT 45228 | An elimination deduction. ... |
| eelTT 45229 | An elimination deduction. ... |
| e0a 45230 | Elimination rule identical... |
| eelT 45231 | An elimination deduction. ... |
| eel0cT 45232 | An elimination deduction. ... |
| eelT0 45233 | An elimination deduction. ... |
| e0bi 45234 | Elimination rule identical... |
| e0bir 45235 | Elimination rule identical... |
| uun0.1 45236 | Convention notation form o... |
| un0.1 45237 | ` T. ` is the constant tru... |
| uunT1 45238 | A deduction unionizing a n... |
| uunT1p1 45239 | A deduction unionizing a n... |
| uunT21 45240 | A deduction unionizing a n... |
| uun121 45241 | A deduction unionizing a n... |
| uun121p1 45242 | A deduction unionizing a n... |
| uun132 45243 | A deduction unionizing a n... |
| uun132p1 45244 | A deduction unionizing a n... |
| anabss7p1 45245 | A deduction unionizing a n... |
| un10 45246 | A unionizing deduction. (... |
| un01 45247 | A unionizing deduction. (... |
| un2122 45248 | A deduction unionizing a n... |
| uun2131 45249 | A deduction unionizing a n... |
| uun2131p1 45250 | A deduction unionizing a n... |
| uunTT1 45251 | A deduction unionizing a n... |
| uunTT1p1 45252 | A deduction unionizing a n... |
| uunTT1p2 45253 | A deduction unionizing a n... |
| uunT11 45254 | A deduction unionizing a n... |
| uunT11p1 45255 | A deduction unionizing a n... |
| uunT11p2 45256 | A deduction unionizing a n... |
| uunT12 45257 | A deduction unionizing a n... |
| uunT12p1 45258 | A deduction unionizing a n... |
| uunT12p2 45259 | A deduction unionizing a n... |
| uunT12p3 45260 | A deduction unionizing a n... |
| uunT12p4 45261 | A deduction unionizing a n... |
| uunT12p5 45262 | A deduction unionizing a n... |
| uun111 45263 | A deduction unionizing a n... |
| 3anidm12p1 45264 | A deduction unionizing a n... |
| 3anidm12p2 45265 | A deduction unionizing a n... |
| uun123 45266 | A deduction unionizing a n... |
| uun123p1 45267 | A deduction unionizing a n... |
| uun123p2 45268 | A deduction unionizing a n... |
| uun123p3 45269 | A deduction unionizing a n... |
| uun123p4 45270 | A deduction unionizing a n... |
| uun2221 45271 | A deduction unionizing a n... |
| uun2221p1 45272 | A deduction unionizing a n... |
| uun2221p2 45273 | A deduction unionizing a n... |
| 3impdirp1 45274 | A deduction unionizing a n... |
| 3impcombi 45275 | A 1-hypothesis proposition... |
| trsspwALT 45276 | Virtual deduction proof of... |
| trsspwALT2 45277 | Virtual deduction proof of... |
| trsspwALT3 45278 | Short predicate calculus p... |
| sspwtr 45279 | Virtual deduction proof of... |
| sspwtrALT 45280 | Virtual deduction proof of... |
| sspwtrALT2 45281 | Short predicate calculus p... |
| pwtrVD 45282 | Virtual deduction proof of... |
| pwtrrVD 45283 | Virtual deduction proof of... |
| suctrALT 45284 | The successor of a transit... |
| snssiALTVD 45285 | Virtual deduction proof of... |
| snssiALT 45286 | If a class is an element o... |
| snsslVD 45287 | Virtual deduction proof of... |
| snssl 45288 | If a singleton is a subcla... |
| snelpwrVD 45289 | Virtual deduction proof of... |
| unipwrVD 45290 | Virtual deduction proof of... |
| unipwr 45291 | A class is a subclass of t... |
| sstrALT2VD 45292 | Virtual deduction proof of... |
| sstrALT2 45293 | Virtual deduction proof of... |
| suctrALT2VD 45294 | Virtual deduction proof of... |
| suctrALT2 45295 | Virtual deduction proof of... |
| elex2VD 45296 | Virtual deduction proof of... |
| elex22VD 45297 | Virtual deduction proof of... |
| eqsbc2VD 45298 | Virtual deduction proof of... |
| zfregs2VD 45299 | Virtual deduction proof of... |
| tpid3gVD 45300 | Virtual deduction proof of... |
| en3lplem1VD 45301 | Virtual deduction proof of... |
| en3lplem2VD 45302 | Virtual deduction proof of... |
| en3lpVD 45303 | Virtual deduction proof of... |
| simplbi2VD 45304 | Virtual deduction proof of... |
| 3ornot23VD 45305 | Virtual deduction proof of... |
| orbi1rVD 45306 | Virtual deduction proof of... |
| bitr3VD 45307 | Virtual deduction proof of... |
| 3orbi123VD 45308 | Virtual deduction proof of... |
| sbc3orgVD 45309 | Virtual deduction proof of... |
| 19.21a3con13vVD 45310 | Virtual deduction proof of... |
| exbirVD 45311 | Virtual deduction proof of... |
| exbiriVD 45312 | Virtual deduction proof of... |
| rspsbc2VD 45313 | Virtual deduction proof of... |
| 3impexpVD 45314 | Virtual deduction proof of... |
| 3impexpbicomVD 45315 | Virtual deduction proof of... |
| 3impexpbicomiVD 45316 | Virtual deduction proof of... |
| sbcoreleleqVD 45317 | Virtual deduction proof of... |
| hbra2VD 45318 | Virtual deduction proof of... |
| tratrbVD 45319 | Virtual deduction proof of... |
| al2imVD 45320 | Virtual deduction proof of... |
| syl5impVD 45321 | Virtual deduction proof of... |
| idiVD 45322 | Virtual deduction proof of... |
| ancomstVD 45323 | Closed form of ~ ancoms . ... |
| ssralv2VD 45324 | Quantification restricted ... |
| ordelordALTVD 45325 | An element of an ordinal c... |
| equncomVD 45326 | If a class equals the unio... |
| equncomiVD 45327 | Inference form of ~ equnco... |
| sucidALTVD 45328 | A set belongs to its succe... |
| sucidALT 45329 | A set belongs to its succe... |
| sucidVD 45330 | A set belongs to its succe... |
| imbi12VD 45331 | Implication form of ~ imbi... |
| imbi13VD 45332 | Join three logical equival... |
| sbcim2gVD 45333 | Distribution of class subs... |
| sbcbiVD 45334 | Implication form of ~ sbcb... |
| trsbcVD 45335 | Formula-building inference... |
| truniALTVD 45336 | The union of a class of tr... |
| ee33VD 45337 | Non-virtual deduction form... |
| trintALTVD 45338 | The intersection of a clas... |
| trintALT 45339 | The intersection of a clas... |
| undif3VD 45340 | The first equality of Exer... |
| sbcssgVD 45341 | Virtual deduction proof of... |
| csbingVD 45342 | Virtual deduction proof of... |
| onfrALTlem5VD 45343 | Virtual deduction proof of... |
| onfrALTlem4VD 45344 | Virtual deduction proof of... |
| onfrALTlem3VD 45345 | Virtual deduction proof of... |
| simplbi2comtVD 45346 | Virtual deduction proof of... |
| onfrALTlem2VD 45347 | Virtual deduction proof of... |
| onfrALTlem1VD 45348 | Virtual deduction proof of... |
| onfrALTVD 45349 | Virtual deduction proof of... |
| csbeq2gVD 45350 | Virtual deduction proof of... |
| csbsngVD 45351 | Virtual deduction proof of... |
| csbxpgVD 45352 | Virtual deduction proof of... |
| csbresgVD 45353 | Virtual deduction proof of... |
| csbrngVD 45354 | Virtual deduction proof of... |
| csbima12gALTVD 45355 | Virtual deduction proof of... |
| csbunigVD 45356 | Virtual deduction proof of... |
| csbfv12gALTVD 45357 | Virtual deduction proof of... |
| con5VD 45358 | Virtual deduction proof of... |
| relopabVD 45359 | Virtual deduction proof of... |
| 19.41rgVD 45360 | Virtual deduction proof of... |
| 2pm13.193VD 45361 | Virtual deduction proof of... |
| hbimpgVD 45362 | Virtual deduction proof of... |
| hbalgVD 45363 | Virtual deduction proof of... |
| hbexgVD 45364 | Virtual deduction proof of... |
| ax6e2eqVD 45365 | The following User's Proof... |
| ax6e2ndVD 45366 | The following User's Proof... |
| ax6e2ndeqVD 45367 | The following User's Proof... |
| 2sb5ndVD 45368 | The following User's Proof... |
| 2uasbanhVD 45369 | The following User's Proof... |
| e2ebindVD 45370 | The following User's Proof... |
| sb5ALTVD 45371 | The following User's Proof... |
| vk15.4jVD 45372 | The following User's Proof... |
| notnotrALTVD 45373 | The following User's Proof... |
| con3ALTVD 45374 | The following User's Proof... |
| elpwgdedVD 45375 | Membership in a power clas... |
| sspwimp 45376 | If a class is a subclass o... |
| sspwimpVD 45377 | The following User's Proof... |
| sspwimpcf 45378 | If a class is a subclass o... |
| sspwimpcfVD 45379 | The following User's Proof... |
| suctrALTcf 45380 | The successor of a transit... |
| suctrALTcfVD 45381 | The following User's Proof... |
| suctrALT3 45382 | The successor of a transit... |
| sspwimpALT 45383 | If a class is a subclass o... |
| unisnALT 45384 | A set equals the union of ... |
| notnotrALT2 45385 | Converse of double negatio... |
| sspwimpALT2 45386 | If a class is a subclass o... |
| e2ebindALT 45387 | Absorption of an existenti... |
| ax6e2ndALT 45388 | If at least two sets exist... |
| ax6e2ndeqALT 45389 | "At least two sets exist" ... |
| 2sb5ndALT 45390 | Equivalence for double sub... |
| chordthmALT 45391 | The intersecting chords th... |
| isosctrlem1ALT 45392 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 45393 | The indexed union of conne... |
| iunconnALT 45394 | The indexed union of conne... |
| sineq0ALT 45395 | A complex number whose sin... |
| rspesbcd 45396 | Restricted quantifier vers... |
| rext0 45397 | Nonempty existential quant... |
| dfbi1ALTa 45398 | Version of ~ dfbi1ALT usin... |
| simprimi 45399 | Inference associated with ... |
| dfbi1ALTb 45400 | Further shorten ~ dfbi1ALT... |
| relpeq1 45403 | Equality theorem for relat... |
| relpeq2 45404 | Equality theorem for relat... |
| relpeq3 45405 | Equality theorem for relat... |
| relpeq4 45406 | Equality theorem for relat... |
| relpeq5 45407 | Equality theorem for relat... |
| nfrelp 45408 | Bound-variable hypothesis ... |
| relpf 45409 | A relation-preserving func... |
| relprel 45410 | A relation-preserving func... |
| relpmin 45411 | A preimage of a minimal el... |
| relpfrlem 45412 | Lemma for ~ relpfr . Prov... |
| relpfr 45413 | If the image of a set unde... |
| orbitex 45414 | Orbits exist. Given a set... |
| orbitinit 45415 | A set is contained in its ... |
| orbitcl 45416 | The orbit under a function... |
| orbitclmpt 45417 | Version of ~ orbitcl using... |
| trwf 45418 | The class of well-founded ... |
| rankrelp 45419 | The rank function preserve... |
| wffr 45420 | The class of well-founded ... |
| trfr 45421 | A transitive class well-fo... |
| tcfr 45422 | A set is well-founded if a... |
| xpwf 45423 | The Cartesian product of t... |
| dmwf 45424 | The domain of a well-found... |
| rnwf 45425 | The range of a well-founde... |
| relwf 45426 | A relation is a well-found... |
| ralabso 45427 | Simplification of restrict... |
| rexabso 45428 | Simplification of restrict... |
| ralabsod 45429 | Deduction form of ~ ralabs... |
| rexabsod 45430 | Deduction form of ~ rexabs... |
| ralabsobidv 45431 | Formula-building lemma for... |
| rexabsobidv 45432 | Formula-building lemma for... |
| ssabso 45433 | The notion " ` x ` is a su... |
| disjabso 45434 | Disjointness is absolute f... |
| n0abso 45435 | Nonemptiness is absolute f... |
| traxext 45436 | A transitive class models ... |
| modelaxreplem1 45437 | Lemma for ~ modelaxrep . ... |
| modelaxreplem2 45438 | Lemma for ~ modelaxrep . ... |
| modelaxreplem3 45439 | Lemma for ~ modelaxrep . ... |
| modelaxrep 45440 | Conditions which guarantee... |
| ssclaxsep 45441 | A class that is closed und... |
| 0elaxnul 45442 | A class that contains the ... |
| pwclaxpow 45443 | Suppose ` M ` is a transit... |
| prclaxpr 45444 | A class that is closed und... |
| uniclaxun 45445 | A class that is closed und... |
| sswfaxreg 45446 | A subclass of the class of... |
| omssaxinf2 45447 | A class that contains all ... |
| omelaxinf2 45448 | A transitive class that co... |
| dfac5prim 45449 | ~ dfac5 expanded into prim... |
| ac8prim 45450 | ~ ac8 expanded into primit... |
| modelac8prim 45451 | If ` M ` is a transitive c... |
| wfaxext 45452 | The class of well-founded ... |
| wfaxrep 45453 | The class of well-founded ... |
| wfaxsep 45454 | The class of well-founded ... |
| wfaxnul 45455 | The class of well-founded ... |
| wfaxpow 45456 | The class of well-founded ... |
| wfaxpr 45457 | The class of well-founded ... |
| wfaxun 45458 | The class of well-founded ... |
| wfaxreg 45459 | The class of well-founded ... |
| wfaxinf2 45460 | The class of well-founded ... |
| wfac8prim 45461 | The class of well-founded ... |
| brpermmodel 45462 | The membership relation in... |
| brpermmodelcnv 45463 | Ordinary membership expres... |
| permaxext 45464 | The Axiom of Extensionalit... |
| permaxrep 45465 | The Axiom of Replacement ~... |
| permaxsep 45466 | The Axiom of Separation ~ ... |
| permaxnul 45467 | The Null Set Axiom ~ ax-nu... |
| permaxpow 45468 | The Axiom of Power Sets ~ ... |
| permaxpr 45469 | The Axiom of Pairing ~ ax-... |
| permaxun 45470 | The Axiom of Union ~ ax-un... |
| permaxinf2lem 45471 | Lemma for ~ permaxinf2 . ... |
| permaxinf2 45472 | The Axiom of Infinity ~ ax... |
| permac8prim 45473 | The Axiom of Choice ~ ac8p... |
| nregmodelf1o 45474 | Define a permutation ` F `... |
| nregmodellem 45475 | Lemma for ~ nregmodel . (... |
| nregmodel 45476 | The Axiom of Regularity ~ ... |
| nregmodelaxext 45477 | The Axiom of Extensionalit... |
| evth2f 45478 | A version of ~ evth2 using... |
| elunif 45479 | A version of ~ eluni using... |
| rzalf 45480 | A version of ~ rzal using ... |
| fvelrnbf 45481 | A version of ~ fvelrnb usi... |
| rfcnpre1 45482 | If F is a continuous funct... |
| ubelsupr 45483 | If U belongs to A and U is... |
| fsumcnf 45484 | A finite sum of functions ... |
| mulltgt0 45485 | The product of a negative ... |
| rspcegf 45486 | A version of ~ rspcev usin... |
| rabexgf 45487 | A version of ~ rabexg usin... |
| fcnre 45488 | A function continuous with... |
| sumsnd 45489 | A sum of a singleton is th... |
| evthf 45490 | A version of ~ evth using ... |
| cnfex 45491 | The class of continuous fu... |
| fnchoice 45492 | For a finite set, a choice... |
| refsumcn 45493 | A finite sum of continuous... |
| rfcnpre2 45494 | If ` F ` is a continuous f... |
| cncmpmax 45495 | When the hypothesis for th... |
| rfcnpre3 45496 | If F is a continuous funct... |
| rfcnpre4 45497 | If F is a continuous funct... |
| sumpair 45498 | Sum of two distinct comple... |
| rfcnnnub 45499 | Given a real continuous fu... |
| refsum2cnlem1 45500 | This is the core Lemma for... |
| refsum2cn 45501 | The sum of two continuus r... |
| adantlllr 45502 | Deduction adding a conjunc... |
| 3adantlr3 45503 | Deduction adding a conjunc... |
| 3adantll2 45504 | Deduction adding a conjunc... |
| 3adantll3 45505 | Deduction adding a conjunc... |
| ssnel 45506 | If not element of a set, t... |
| sncldre 45507 | A singleton is closed w.r.... |
| n0p 45508 | A polynomial with a nonzer... |
| pm2.65ni 45509 | Inference rule for proof b... |
| iuneq2df 45510 | Equality deduction for ind... |
| nnfoctb 45511 | There exists a mapping fro... |
| elpwinss 45512 | An element of the powerset... |
| unidmex 45513 | If ` F ` is a set, then ` ... |
| ndisj2 45514 | A non-disjointness conditi... |
| zenom 45515 | The set of integer numbers... |
| uzwo4 45516 | Well-ordering principle: a... |
| unisn0 45517 | The union of the singleton... |
| ssin0 45518 | If two classes are disjoin... |
| inabs3 45519 | Absorption law for interse... |
| pwpwuni 45520 | Relationship between power... |
| disjiun2 45521 | In a disjoint collection, ... |
| 0pwfi 45522 | The empty set is in any po... |
| ssinss2d 45523 | Intersection preserves sub... |
| zct 45524 | The set of integer numbers... |
| pwfin0 45525 | A finite set always belong... |
| uzct 45526 | An upper integer set is co... |
| iunxsnf 45527 | A singleton index picks ou... |
| fiiuncl 45528 | If a set is closed under t... |
| iunp1 45529 | The addition of the next s... |
| fiunicl 45530 | If a set is closed under t... |
| ixpeq2d 45531 | Equality theorem for infin... |
| disjxp1 45532 | The sets of a cartesian pr... |
| disjsnxp 45533 | The sets in the cartesian ... |
| eliind 45534 | Membership in indexed inte... |
| rspcef 45535 | Restricted existential spe... |
| ixpssmapc 45536 | An infinite Cartesian prod... |
| elintd 45537 | Membership in class inters... |
| ssdf 45538 | A sufficient condition for... |
| brneqtrd 45539 | Substitution of equal clas... |
| ssnct 45540 | A set containing an uncoun... |
| ssuniint 45541 | Sufficient condition for b... |
| elintdv 45542 | Membership in class inters... |
| ssd 45543 | A sufficient condition for... |
| ralimralim 45544 | Introducing any antecedent... |
| snelmap 45545 | Membership of the element ... |
| xrnmnfpnf 45546 | An extended real that is n... |
| iuneq1i 45547 | Equality theorem for index... |
| ssinc 45548 | Inclusion relation for a m... |
| ssdec 45549 | Inclusion relation for a m... |
| elixpconstg 45550 | Membership in an infinite ... |
| iineq1d 45551 | Equality theorem for index... |
| metpsmet 45552 | A metric is a pseudometric... |
| ixpssixp 45553 | Subclass theorem for infin... |
| ballss3 45554 | A sufficient condition for... |
| iunincfi 45555 | Given a sequence of increa... |
| nsstr 45556 | If it's not a subclass, it... |
| rexanuz3 45557 | Combine two different uppe... |
| cbvmpo2 45558 | Rule to change the second ... |
| cbvmpo1 45559 | Rule to change the first b... |
| eliuniin 45560 | Indexed union of indexed i... |
| ssabf 45561 | Subclass of a class abstra... |
| pssnssi 45562 | A proper subclass does not... |
| rabidim2 45563 | Membership in a restricted... |
| eluni2f 45564 | Membership in class union.... |
| eliin2f 45565 | Membership in indexed inte... |
| nssd 45566 | Negation of subclass relat... |
| iineq12dv 45567 | Equality deduction for ind... |
| supxrcld 45568 | The supremum of an arbitra... |
| elrestd 45569 | A sufficient condition for... |
| eliuniincex 45570 | Counterexample to show tha... |
| eliincex 45571 | Counterexample to show tha... |
| eliinid 45572 | Membership in an indexed i... |
| abssf 45573 | Class abstraction in a sub... |
| supxrubd 45574 | A member of a set of exten... |
| ssrabf 45575 | Subclass of a restricted c... |
| ssrabdf 45576 | Subclass of a restricted c... |
| eliin2 45577 | Membership in indexed inte... |
| ssrab2f 45578 | Subclass relation for a re... |
| restuni3 45579 | The underlying set of a su... |
| rabssf 45580 | Restricted class abstracti... |
| eliuniin2 45581 | Indexed union of indexed i... |
| restuni4 45582 | The underlying set of a su... |
| restuni6 45583 | The underlying set of a su... |
| restuni5 45584 | The underlying set of a su... |
| unirestss 45585 | The union of an elementwis... |
| iniin1 45586 | Indexed intersection of in... |
| iniin2 45587 | Indexed intersection of in... |
| cbvrabv2 45588 | A more general version of ... |
| cbvrabv2w 45589 | A more general version of ... |
| iinssiin 45590 | Subset implication for an ... |
| eliind2 45591 | Membership in indexed inte... |
| iinssd 45592 | Subset implication for an ... |
| rabbida2 45593 | Equivalent wff's yield equ... |
| iinexd 45594 | The existence of an indexe... |
| rabexf 45595 | Separation Scheme in terms... |
| rabbida3 45596 | Equivalent wff's yield equ... |
| r19.36vf 45597 | Restricted quantifier vers... |
| raleqd 45598 | Equality deduction for res... |
| iinssf 45599 | Subset implication for an ... |
| iinssdf 45600 | Subset implication for an ... |
| resabs2i 45601 | Absorption law for restric... |
| ssdf2 45602 | A sufficient condition for... |
| rabssd 45603 | Restricted class abstracti... |
| rexnegd 45604 | Minus a real number. (Con... |
| rexlimd3 45605 | * Inference from Theorem 1... |
| nel1nelini 45606 | Membership in an intersect... |
| nel2nelini 45607 | Membership in an intersect... |
| eliunid 45608 | Membership in indexed unio... |
| reximdd 45609 | Deduction from Theorem 19.... |
| inopnd 45610 | The intersection of two op... |
| ss2rabdf 45611 | Deduction of restricted ab... |
| restopn3 45612 | If ` A ` is open, then ` A... |
| restopnssd 45613 | A topology restricted to a... |
| restsubel 45614 | A subset belongs in the sp... |
| toprestsubel 45615 | A subset is open in the to... |
| rabidd 45616 | An "identity" law of concr... |
| iunssdf 45617 | Subset theorem for an inde... |
| iinss2d 45618 | Subset implication for an ... |
| r19.3rzf 45619 | Restricted quantification ... |
| r19.28zf 45620 | Restricted quantifier vers... |
| iindif2f 45621 | Indexed intersection of cl... |
| ralfal 45622 | Two ways of expressing emp... |
| archd 45623 | Archimedean property of re... |
| nimnbi 45624 | If an implication is false... |
| nimnbi2 45625 | If an implication is false... |
| notbicom 45626 | Commutative law for the ne... |
| rexeqif 45627 | Equality inference for res... |
| rspced 45628 | Restricted existential spe... |
| fnresdmss 45629 | A function does not change... |
| fmptsnxp 45630 | Maps-to notation and Carte... |
| fvmpt2bd 45631 | Value of a function given ... |
| rnmptfi 45632 | The range of a function wi... |
| fresin2 45633 | Restriction of a function ... |
| ffi 45634 | A function with finite dom... |
| suprnmpt 45635 | An explicit bound for the ... |
| rnffi 45636 | The range of a function wi... |
| mptelpm 45637 | A function in maps-to nota... |
| rnmptpr 45638 | Range of a function define... |
| resmpti 45639 | Restriction of the mapping... |
| founiiun 45640 | Union expressed as an inde... |
| rnresun 45641 | Distribution law for range... |
| elrnmptf 45642 | The range of a function in... |
| rnmptssrn 45643 | Inclusion relation for two... |
| disjf1 45644 | A 1 to 1 mapping built fro... |
| rnsnf 45645 | The range of a function wh... |
| wessf1ornlem 45646 | Given a function ` F ` on ... |
| wessf1orn 45647 | Given a function ` F ` on ... |
| nelrnres 45648 | If ` A ` is not in the ran... |
| disjrnmpt2 45649 | Disjointness of the range ... |
| elrnmpt1sf 45650 | Elementhood in an image se... |
| founiiun0 45651 | Union expressed as an inde... |
| disjf1o 45652 | A bijection built from dis... |
| disjinfi 45653 | Only a finite number of di... |
| fvovco 45654 | Value of the composition o... |
| ssnnf1octb 45655 | There exists a bijection b... |
| nnf1oxpnn 45656 | There is a bijection betwe... |
| projf1o 45657 | A biijection from a set to... |
| fvmap 45658 | Function value for a membe... |
| fvixp2 45659 | Projection of a factor of ... |
| choicefi 45660 | For a finite set, a choice... |
| mpct 45661 | The exponentiation of a co... |
| cnmetcoval 45662 | Value of the distance func... |
| fcomptss 45663 | Express composition of two... |
| elmapsnd 45664 | Membership in a set expone... |
| mapss2 45665 | Subset inheritance for set... |
| difmap 45666 | Difference of two sets exp... |
| unirnmap 45667 | Given a subset of a set ex... |
| inmap 45668 | Intersection of two sets e... |
| fcoss 45669 | Composition of two mapping... |
| fsneqrn 45670 | Equality condition for two... |
| difmapsn 45671 | Difference of two sets exp... |
| mapssbi 45672 | Subset inheritance for set... |
| unirnmapsn 45673 | Equality theorem for a sub... |
| iunmapss 45674 | The indexed union of set e... |
| ssmapsn 45675 | A subset ` C ` of a set ex... |
| iunmapsn 45676 | The indexed union of set e... |
| absfico 45677 | Mapping domain and codomai... |
| icof 45678 | The set of left-closed rig... |
| elpmrn 45679 | The range of a partial fun... |
| imaexi 45680 | The image of a set is a se... |
| axccdom 45681 | Relax the constraint on ax... |
| dmmptdff 45682 | The domain of the mapping ... |
| dmmptdf 45683 | The domain of the mapping ... |
| elpmi2 45684 | The domain of a partial fu... |
| dmrelrnrel 45685 | A relation preserving func... |
| elrnmpoid 45686 | Membership in the range of... |
| axccd 45687 | An alternative version of ... |
| axccd2 45688 | An alternative version of ... |
| feqresmptf 45689 | Express a restricted funct... |
| dmmptssf 45690 | The domain of a mapping is... |
| dmmptdf2 45691 | The domain of the mapping ... |
| dmuz 45692 | Domain of the upper intege... |
| fmptd2f 45693 | Domain and codomain of the... |
| mpteq1df 45694 | An equality theorem for th... |
| mptexf 45695 | If the domain of a functio... |
| fvmpt4 45696 | Value of a function given ... |
| fmptf 45697 | Functionality of the mappi... |
| resimass 45698 | The image of a restriction... |
| mptssid 45699 | The mapping operation expr... |
| mptfnd 45700 | The maps-to notation defin... |
| rnmptlb 45701 | Boundness below of the ran... |
| rnmptbddlem 45702 | Boundness of the range of ... |
| rnmptbdd 45703 | Boundness of the range of ... |
| funimaeq 45704 | Membership relation for th... |
| rnmptssf 45705 | The range of a function gi... |
| rnmptbd2lem 45706 | Boundness below of the ran... |
| rnmptbd2 45707 | Boundness below of the ran... |
| infnsuprnmpt 45708 | The indexed infimum of rea... |
| suprclrnmpt 45709 | Closure of the indexed sup... |
| suprubrnmpt2 45710 | A member of a nonempty ind... |
| suprubrnmpt 45711 | A member of a nonempty ind... |
| rnmptssdf 45712 | The range of a function gi... |
| rnmptbdlem 45713 | Boundness above of the ran... |
| rnmptbd 45714 | Boundness above of the ran... |
| rnmptss2 45715 | The range of a function gi... |
| elmptima 45716 | The image of a function in... |
| ralrnmpt3 45717 | A restricted quantifier ov... |
| rnmptssbi 45718 | The range of a function gi... |
| imass2d 45719 | Subset theorem for image. ... |
| imassmpt 45720 | Membership relation for th... |
| fpmd 45721 | A total function is a part... |
| fconst7 45722 | An alternative way to expr... |
| fnmptif 45723 | Functionality and domain o... |
| dmmptif 45724 | Domain of the mapping oper... |
| mpteq2dfa 45725 | Slightly more general equa... |
| dmmpt1 45726 | The domain of the mapping ... |
| fmptff 45727 | Functionality of the mappi... |
| fvmptelcdmf 45728 | The value of a function at... |
| fmptdff 45729 | A version of ~ fmptd using... |
| fvmpt2df 45730 | Deduction version of ~ fvm... |
| rn1st 45731 | The range of a function wi... |
| rnmptssff 45732 | The range of a function gi... |
| rnmptssdff 45733 | The range of a function gi... |
| fvmpt4d 45734 | Value of a function given ... |
| sub2times 45735 | Subtracting from a number,... |
| nnxrd 45736 | A natural number is an ext... |
| nnxr 45737 | A natural number is an ext... |
| abssubrp 45738 | The distance of two distin... |
| elfzfzo 45739 | Relationship between membe... |
| oddfl 45740 | Odd number representation ... |
| abscosbd 45741 | Bound for the absolute val... |
| mul13d 45742 | Commutative/associative la... |
| negpilt0 45743 | Negative ` _pi ` is negati... |
| dstregt0 45744 | A complex number ` A ` tha... |
| subadd4b 45745 | Rearrangement of 4 terms i... |
| xrlttri5d 45746 | Not equal and not larger i... |
| zltlesub 45747 | If an integer ` N ` is les... |
| divlt0gt0d 45748 | The ratio of a negative nu... |
| subsub23d 45749 | Swap subtrahend and result... |
| 2timesgt 45750 | Double of a positive real ... |
| reopn 45751 | The reals are open with re... |
| sub31 45752 | Swap the first and third t... |
| nnne1ge2 45753 | A positive integer which i... |
| lefldiveq 45754 | A closed enough, smaller r... |
| negsubdi3d 45755 | Distribution of negative o... |
| ltdiv2dd 45756 | Division of a positive num... |
| abssinbd 45757 | Bound for the absolute val... |
| halffl 45758 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 45759 | Ordering relation for a st... |
| hashssle 45760 | The size of a subset of a ... |
| lttri5d 45761 | Not equal and not larger i... |
| fzisoeu 45762 | A finite ordered set has a... |
| lt3addmuld 45763 | If three real numbers are ... |
| absnpncan2d 45764 | Triangular inequality, com... |
| fperiodmullem 45765 | A function with period ` T... |
| fperiodmul 45766 | A function with period T i... |
| upbdrech 45767 | Choice of an upper bound f... |
| lt4addmuld 45768 | If four real numbers are l... |
| absnpncan3d 45769 | Triangular inequality, com... |
| upbdrech2 45770 | Choice of an upper bound f... |
| ssfiunibd 45771 | A finite union of bounded ... |
| fzdifsuc2 45772 | Remove a successor from th... |
| fzsscn 45773 | A finite sequence of integ... |
| divcan8d 45774 | A cancellation law for div... |
| dmmcand 45775 | Cancellation law for divis... |
| fzssre 45776 | A finite sequence of integ... |
| bccld 45777 | A binomial coefficient, in... |
| fzssnn0 45778 | A finite set of sequential... |
| xreqle 45779 | Equality implies 'less tha... |
| xaddlidd 45780 | ` 0 ` is a left identity f... |
| xadd0ge 45781 | A number is less than or e... |
| xrleneltd 45782 | 'Less than or equal to' an... |
| xaddcomd 45783 | The extended real addition... |
| supxrre3 45784 | The supremum of a nonempty... |
| uzfissfz 45785 | For any finite subset of t... |
| xleadd2d 45786 | Addition of extended reals... |
| suprltrp 45787 | The supremum of a nonempty... |
| xleadd1d 45788 | Addition of extended reals... |
| xreqled 45789 | Equality implies 'less tha... |
| xrgepnfd 45790 | An extended real greater t... |
| xrge0nemnfd 45791 | A nonnegative extended rea... |
| supxrgere 45792 | If a real number can be ap... |
| iuneqfzuzlem 45793 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 45794 | If two unions indexed by u... |
| xle2addd 45795 | Adding both side of two in... |
| supxrgelem 45796 | If an extended real number... |
| supxrge 45797 | If an extended real number... |
| suplesup 45798 | If any element of ` A ` ca... |
| infxrglb 45799 | The infimum of a set of ex... |
| xadd0ge2 45800 | A number is less than or e... |
| nepnfltpnf 45801 | An extended real that is n... |
| ltadd12dd 45802 | Addition to both sides of ... |
| nemnftgtmnft 45803 | An extended real that is n... |
| xrgtso 45804 | 'Greater than' is a strict... |
| rpex 45805 | The positive reals form a ... |
| xrge0ge0 45806 | A nonnegative extended rea... |
| xrssre 45807 | A subset of extended reals... |
| ssuzfz 45808 | A finite subset of the upp... |
| absfun 45809 | The absolute value is a fu... |
| infrpge 45810 | The infimum of a nonempty,... |
| xrlexaddrp 45811 | If an extended real number... |
| supsubc 45812 | The supremum function dist... |
| xralrple2 45813 | Show that ` A ` is less th... |
| nnuzdisj 45814 | The first ` N ` elements o... |
| ltdivgt1 45815 | Divsion by a number greate... |
| xrltned 45816 | 'Less than' implies not eq... |
| nnsplit 45817 | Express the set of positiv... |
| divdiv3d 45818 | Division into a fraction. ... |
| abslt2sqd 45819 | Comparison of the square o... |
| qenom 45820 | The set of rational number... |
| qct 45821 | The set of rational number... |
| lenlteq 45822 | 'less than or equal to' bu... |
| xrred 45823 | An extended real that is n... |
| rr2sscn2 45824 | The cartesian square of ` ... |
| infxr 45825 | The infimum of a set of ex... |
| infxrunb2 45826 | The infimum of an unbounde... |
| infxrbnd2 45827 | The infimum of a bounded-b... |
| infleinflem1 45828 | Lemma for ~ infleinf , cas... |
| infleinflem2 45829 | Lemma for ~ infleinf , whe... |
| infleinf 45830 | If any element of ` B ` ca... |
| xralrple4 45831 | Show that ` A ` is less th... |
| xralrple3 45832 | Show that ` A ` is less th... |
| eluzelzd 45833 | A member of an upper set o... |
| suplesup2 45834 | If any element of ` A ` is... |
| recnnltrp 45835 | ` N ` is a natural number ... |
| nnn0 45836 | The set of positive intege... |
| fzct 45837 | A finite set of sequential... |
| rpgtrecnn 45838 | Any positive real number i... |
| fzossuz 45839 | A half-open integer interv... |
| infxrrefi 45840 | The real and extended real... |
| xrralrecnnle 45841 | Show that ` A ` is less th... |
| fzoct 45842 | A finite set of sequential... |
| frexr 45843 | A function taking real val... |
| nnrecrp 45844 | The reciprocal of a positi... |
| reclt0d 45845 | The reciprocal of a negati... |
| lt0neg1dd 45846 | If a number is negative, i... |
| infxrcld 45847 | The infimum of an arbitrar... |
| xrralrecnnge 45848 | Show that ` A ` is less th... |
| reclt0 45849 | The reciprocal of a negati... |
| ltmulneg 45850 | Multiplying by a negative ... |
| allbutfi 45851 | For all but finitely many.... |
| ltdiv23neg 45852 | Swap denominator with othe... |
| xreqnltd 45853 | A consequence of trichotom... |
| mnfnre2 45854 | Minus infinity is not a re... |
| zssxr 45855 | The integers are a subset ... |
| fisupclrnmpt 45856 | A nonempty finite indexed ... |
| supxrunb3 45857 | The supremum of an unbound... |
| fimaxre4 45858 | A nonempty finite set of r... |
| ren0 45859 | The set of reals is nonemp... |
| eluzelz2 45860 | A member of an upper set o... |
| resabs2d 45861 | Absorption law for restric... |
| uzid2 45862 | Membership of the least me... |
| supxrleubrnmpt 45863 | The supremum of a nonempty... |
| uzssre2 45864 | An upper set of integers i... |
| uzssd 45865 | Subset relationship for tw... |
| eluzd 45866 | Membership in an upper set... |
| infxrlbrnmpt2 45867 | A member of a nonempty ind... |
| xrre4 45868 | An extended real is real i... |
| uz0 45869 | The upper integers functio... |
| eluzelz2d 45870 | A member of an upper set o... |
| infleinf2 45871 | If any element in ` B ` is... |
| unb2ltle 45872 | "Unbounded below" expresse... |
| uzidd2 45873 | Membership of the least me... |
| uzssd2 45874 | Subset relationship for tw... |
| rexabslelem 45875 | An indexed set of absolute... |
| rexabsle 45876 | An indexed set of absolute... |
| allbutfiinf 45877 | Given a "for all but finit... |
| supxrrernmpt 45878 | The real and extended real... |
| suprleubrnmpt 45879 | The supremum of a nonempty... |
| infrnmptle 45880 | An indexed infimum of exte... |
| infxrunb3 45881 | The infimum of an unbounde... |
| uzn0d 45882 | The upper integers are all... |
| uzssd3 45883 | Subset relationship for tw... |
| rexabsle2 45884 | An indexed set of absolute... |
| infxrunb3rnmpt 45885 | The infimum of an unbounde... |
| supxrre3rnmpt 45886 | The indexed supremum of a ... |
| uzublem 45887 | A set of reals, indexed by... |
| uzub 45888 | A set of reals, indexed by... |
| ssrexr 45889 | A subset of the reals is a... |
| supxrmnf2 45890 | Removing minus infinity fr... |
| supxrcli 45891 | The supremum of an arbitra... |
| uzid3 45892 | Membership of the least me... |
| infxrlesupxr 45893 | The supremum of a nonempty... |
| xnegeqd 45894 | Equality of two extended n... |
| xnegrecl 45895 | The extended real negative... |
| xnegnegi 45896 | Extended real version of ~... |
| xnegeqi 45897 | Equality of two extended n... |
| nfxnegd 45898 | Deduction version of ~ nfx... |
| xnegnegd 45899 | Extended real version of ~... |
| uzred 45900 | An upper integer is a real... |
| xnegcli 45901 | Closure of extended real n... |
| supminfrnmpt 45902 | The indexed supremum of a ... |
| infxrpnf 45903 | Adding plus infinity to a ... |
| infxrrnmptcl 45904 | The infimum of an arbitrar... |
| leneg2d 45905 | Negative of one side of 'l... |
| supxrltinfxr 45906 | The supremum of the empty ... |
| max1d 45907 | A number is less than or e... |
| supxrleubrnmptf 45908 | The supremum of a nonempty... |
| nleltd 45909 | 'Not less than or equal to... |
| zxrd 45910 | An integer is an extended ... |
| infxrgelbrnmpt 45911 | The infimum of an indexed ... |
| rphalfltd 45912 | Half of a positive real is... |
| uzssz2 45913 | An upper set of integers i... |
| leneg3d 45914 | Negative of one side of 'l... |
| max2d 45915 | A number is less than or e... |
| uzn0bi 45916 | The upper integers functio... |
| xnegrecl2 45917 | If the extended real negat... |
| nfxneg 45918 | Bound-variable hypothesis ... |
| uzxrd 45919 | An upper integer is an ext... |
| infxrpnf2 45920 | Removing plus infinity fro... |
| supminfxr 45921 | The extended real suprema ... |
| infrpgernmpt 45922 | The infimum of a nonempty,... |
| xnegre 45923 | An extended real is real i... |
| xnegrecl2d 45924 | If the extended real negat... |
| uzxr 45925 | An upper integer is an ext... |
| supminfxr2 45926 | The extended real suprema ... |
| xnegred 45927 | An extended real is real i... |
| supminfxrrnmpt 45928 | The indexed supremum of a ... |
| min1d 45929 | The minimum of two numbers... |
| min2d 45930 | The minimum of two numbers... |
| xrnpnfmnf 45931 | An extended real that is n... |
| uzsscn 45932 | An upper set of integers i... |
| absimnre 45933 | The absolute value of the ... |
| uzsscn2 45934 | An upper set of integers i... |
| xrtgcntopre 45935 | The standard topologies on... |
| absimlere 45936 | The absolute value of the ... |
| rpssxr 45937 | The positive reals are a s... |
| monoordxrv 45938 | Ordering relation for a mo... |
| monoordxr 45939 | Ordering relation for a mo... |
| monoord2xrv 45940 | Ordering relation for a mo... |
| monoord2xr 45941 | Ordering relation for a mo... |
| xrpnf 45942 | An extended real is plus i... |
| xlenegcon1 45943 | Extended real version of ~... |
| xlenegcon2 45944 | Extended real version of ~... |
| pimxrneun 45945 | The preimage of a set of e... |
| caucvgbf 45946 | A function is convergent i... |
| cvgcau 45947 | A convergent function is C... |
| cvgcaule 45948 | A convergent function is C... |
| rexanuz2nf 45949 | A simple counterexample re... |
| gtnelioc 45950 | A real number larger than ... |
| ioossioc 45951 | An open interval is a subs... |
| ioondisj2 45952 | A condition for two open i... |
| ioondisj1 45953 | A condition for two open i... |
| ioogtlb 45954 | An element of a closed int... |
| evthiccabs 45955 | Extreme Value Theorem on y... |
| ltnelicc 45956 | A real number smaller than... |
| eliood 45957 | Membership in an open real... |
| iooabslt 45958 | An upper bound for the dis... |
| gtnelicc 45959 | A real number greater than... |
| iooinlbub 45960 | An open interval has empty... |
| iocgtlb 45961 | An element of a left-open ... |
| iocleub 45962 | An element of a left-open ... |
| eliccd 45963 | Membership in a closed rea... |
| eliccre 45964 | A member of a closed inter... |
| eliooshift 45965 | Element of an open interva... |
| eliocd 45966 | Membership in a left-open ... |
| icoltub 45967 | An element of a left-close... |
| eliocre 45968 | A member of a left-open ri... |
| iooltub 45969 | An element of an open inte... |
| ioontr 45970 | The interior of an interva... |
| snunioo1 45971 | The closure of one end of ... |
| lbioc 45972 | A left-open right-closed i... |
| ioomidp 45973 | The midpoint is an element... |
| iccdifioo 45974 | If the open inverval is re... |
| iccdifprioo 45975 | An open interval is the cl... |
| ioossioobi 45976 | Biconditional form of ~ io... |
| iccshift 45977 | A closed interval shifted ... |
| iccsuble 45978 | An upper bound to the dist... |
| iocopn 45979 | A left-open right-closed i... |
| eliccelioc 45980 | Membership in a closed int... |
| iooshift 45981 | An open interval shifted b... |
| iccintsng 45982 | Intersection of two adiace... |
| icoiccdif 45983 | Left-closed right-open int... |
| icoopn 45984 | A left-closed right-open i... |
| icoub 45985 | A left-closed, right-open ... |
| eliccxrd 45986 | Membership in a closed rea... |
| pnfel0pnf 45987 | ` +oo ` is a nonnegative e... |
| eliccnelico 45988 | An element of a closed int... |
| eliccelicod 45989 | A member of a closed inter... |
| ge0xrre 45990 | A nonnegative extended rea... |
| ge0lere 45991 | A nonnegative extended Rea... |
| elicores 45992 | Membership in a left-close... |
| inficc 45993 | The infimum of a nonempty ... |
| qinioo 45994 | The rational numbers are d... |
| lenelioc 45995 | A real number smaller than... |
| ioonct 45996 | A nonempty open interval i... |
| xrgtnelicc 45997 | A real number greater than... |
| iccdificc 45998 | The difference of two clos... |
| iocnct 45999 | A nonempty left-open, righ... |
| iccnct 46000 | A closed interval, with mo... |
| iooiinicc 46001 | A closed interval expresse... |
| iccgelbd 46002 | An element of a closed int... |
| iooltubd 46003 | An element of an open inte... |
| icoltubd 46004 | An element of a left-close... |
| qelioo 46005 | The rational numbers are d... |
| tgqioo2 46006 | Every open set of reals is... |
| iccleubd 46007 | An element of a closed int... |
| elioored 46008 | A member of an open interv... |
| ioogtlbd 46009 | An element of a closed int... |
| ioofun 46010 | ` (,) ` is a function. (C... |
| icomnfinre 46011 | A left-closed, right-open,... |
| sqrlearg 46012 | The square compared with i... |
| ressiocsup 46013 | If the supremum belongs to... |
| ressioosup 46014 | If the supremum does not b... |
| iooiinioc 46015 | A left-open, right-closed ... |
| ressiooinf 46016 | If the infimum does not be... |
| iocleubd 46017 | An element of a left-open ... |
| uzinico 46018 | An upper interval of integ... |
| preimaiocmnf 46019 | Preimage of a right-closed... |
| uzinico2 46020 | An upper interval of integ... |
| uzinico3 46021 | An upper interval of integ... |
| dmico 46022 | The domain of the closed-b... |
| ndmico 46023 | The closed-below, open-abo... |
| uzubioo 46024 | The upper integers are unb... |
| uzubico 46025 | The upper integers are unb... |
| uzubioo2 46026 | The upper integers are unb... |
| uzubico2 46027 | The upper integers are unb... |
| iocgtlbd 46028 | An element of a left-open ... |
| xrtgioo2 46029 | The topology on the extend... |
| fsummulc1f 46030 | Closure of a finite sum of... |
| fsumnncl 46031 | Closure of a nonempty, fin... |
| fsumge0cl 46032 | The finite sum of nonnegat... |
| fsumf1of 46033 | Re-index a finite sum usin... |
| fsumiunss 46034 | Sum over a disjoint indexe... |
| fsumreclf 46035 | Closure of a finite sum of... |
| fsumlessf 46036 | A shorter sum of nonnegati... |
| fsumsupp0 46037 | Finite sum of function val... |
| fsumsermpt 46038 | A finite sum expressed in ... |
| fmul01 46039 | Multiplying a finite numbe... |
| fmulcl 46040 | If ' Y ' is closed under t... |
| fmuldfeqlem1 46041 | induction step for the pro... |
| fmuldfeq 46042 | X and Z are two equivalent... |
| fmul01lt1lem1 46043 | Given a finite multiplicat... |
| fmul01lt1lem2 46044 | Given a finite multiplicat... |
| fmul01lt1 46045 | Given a finite multiplicat... |
| cncfmptss 46046 | A continuous complex funct... |
| rrpsscn 46047 | The positive reals are a s... |
| mulc1cncfg 46048 | A version of ~ mulc1cncf u... |
| infrglb 46049 | The infimum of a nonempty ... |
| expcnfg 46050 | If ` F ` is a complex cont... |
| prodeq2ad 46051 | Equality deduction for pro... |
| fprodsplit1 46052 | Separate out a term in a f... |
| fprodexp 46053 | Positive integer exponenti... |
| fprodabs2 46054 | The absolute value of a fi... |
| fprod0 46055 | A finite product with a ze... |
| mccllem 46056 | * Induction step for ~ mcc... |
| mccl 46057 | A multinomial coefficient,... |
| fprodcnlem 46058 | A finite product of functi... |
| fprodcn 46059 | A finite product of functi... |
| clim1fr1 46060 | A class of sequences of fr... |
| isumneg 46061 | Negation of a converging s... |
| climrec 46062 | Limit of the reciprocal of... |
| climmulf 46063 | A version of ~ climmul usi... |
| climexp 46064 | The limit of natural power... |
| climinf 46065 | A bounded monotonic noninc... |
| climsuselem1 46066 | The subsequence index ` I ... |
| climsuse 46067 | A subsequence ` G ` of a c... |
| climrecf 46068 | A version of ~ climrec usi... |
| climneg 46069 | Complex limit of the negat... |
| climinff 46070 | A version of ~ climinf usi... |
| climdivf 46071 | Limit of the ratio of two ... |
| climreeq 46072 | If ` F ` is a real functio... |
| ellimciota 46073 | An explicit value for the ... |
| climaddf 46074 | A version of ~ climadd usi... |
| mullimc 46075 | Limit of the product of tw... |
| ellimcabssub0 46076 | An equivalent condition fo... |
| limcdm0 46077 | If a function has empty do... |
| islptre 46078 | An equivalence condition f... |
| limccog 46079 | Limit of the composition o... |
| limciccioolb 46080 | The limit of a function at... |
| climf 46081 | Express the predicate: Th... |
| mullimcf 46082 | Limit of the multiplicatio... |
| constlimc 46083 | Limit of constant function... |
| rexlim2d 46084 | Inference removing two res... |
| idlimc 46085 | Limit of the identity func... |
| divcnvg 46086 | The sequence of reciprocal... |
| limcperiod 46087 | If ` F ` is a periodic fun... |
| limcrecl 46088 | If ` F ` is a real-valued ... |
| sumnnodd 46089 | A series indexed by ` NN `... |
| lptioo2 46090 | The upper bound of an open... |
| lptioo1 46091 | The lower bound of an open... |
| limcmptdm 46092 | The domain of a maps-to fu... |
| clim2f 46093 | Express the predicate: Th... |
| limcicciooub 46094 | The limit of a function at... |
| ltmod 46095 | A sufficient condition for... |
| islpcn 46096 | A characterization for a l... |
| lptre2pt 46097 | If a set in the real line ... |
| limsupre 46098 | If a sequence is bounded, ... |
| limcresiooub 46099 | The left limit doesn't cha... |
| limcresioolb 46100 | The right limit doesn't ch... |
| limcleqr 46101 | If the left and the right ... |
| lptioo2cn 46102 | The upper bound of an open... |
| lptioo1cn 46103 | The lower bound of an open... |
| neglimc 46104 | Limit of the negative func... |
| addlimc 46105 | Sum of two limits. (Contr... |
| 0ellimcdiv 46106 | If the numerator converges... |
| clim2cf 46107 | Express the predicate ` F ... |
| limclner 46108 | For a limit point, both fr... |
| sublimc 46109 | Subtraction of two limits.... |
| reclimc 46110 | Limit of the reciprocal of... |
| clim0cf 46111 | Express the predicate ` F ... |
| limclr 46112 | For a limit point, both fr... |
| divlimc 46113 | Limit of the quotient of t... |
| expfac 46114 | Factorial grows faster tha... |
| climconstmpt 46115 | A constant sequence conver... |
| climresmpt 46116 | A function restricted to u... |
| climsubmpt 46117 | Limit of the difference of... |
| climsubc2mpt 46118 | Limit of the difference of... |
| climsubc1mpt 46119 | Limit of the difference of... |
| fnlimfv 46120 | The value of the limit fun... |
| climreclf 46121 | The limit of a convergent ... |
| climeldmeq 46122 | Two functions that are eve... |
| climf2 46123 | Express the predicate: Th... |
| fnlimcnv 46124 | The sequence of function v... |
| climeldmeqmpt 46125 | Two functions that are eve... |
| climfveq 46126 | Two functions that are eve... |
| clim2f2 46127 | Express the predicate: Th... |
| climfveqmpt 46128 | Two functions that are eve... |
| climd 46129 | Express the predicate: Th... |
| clim2d 46130 | The limit of complex numbe... |
| fnlimfvre 46131 | The limit function of real... |
| allbutfifvre 46132 | Given a sequence of real-v... |
| climleltrp 46133 | The limit of complex numbe... |
| fnlimfvre2 46134 | The limit function of real... |
| fnlimf 46135 | The limit function of real... |
| fnlimabslt 46136 | A sequence of function val... |
| climfveqf 46137 | Two functions that are eve... |
| climmptf 46138 | Exhibit a function ` G ` w... |
| climfveqmpt3 46139 | Two functions that are eve... |
| climeldmeqf 46140 | Two functions that are eve... |
| climreclmpt 46141 | The limit of B convergent ... |
| limsupref 46142 | If a sequence is bounded, ... |
| limsupbnd1f 46143 | If a sequence is eventuall... |
| climbddf 46144 | A converging sequence of c... |
| climeqf 46145 | Two functions that are eve... |
| climeldmeqmpt3 46146 | Two functions that are eve... |
| limsupcld 46147 | Closure of the superior li... |
| climfv 46148 | The limit of a convergent ... |
| limsupval3 46149 | The superior limit of an i... |
| climfveqmpt2 46150 | Two functions that are eve... |
| limsup0 46151 | The superior limit of the ... |
| climeldmeqmpt2 46152 | Two functions that are eve... |
| limsupresre 46153 | The supremum limit of a fu... |
| climeqmpt 46154 | Two functions that are eve... |
| climfvd 46155 | The limit of a convergent ... |
| limsuplesup 46156 | An upper bound for the sup... |
| limsupresico 46157 | The superior limit doesn't... |
| limsuppnfdlem 46158 | If the restriction of a fu... |
| limsuppnfd 46159 | If the restriction of a fu... |
| limsupresuz 46160 | If the real part of the do... |
| limsupub 46161 | If the limsup is not ` +oo... |
| limsupres 46162 | The superior limit of a re... |
| climinf2lem 46163 | A convergent, nonincreasin... |
| climinf2 46164 | A convergent, nonincreasin... |
| limsupvaluz 46165 | The superior limit, when t... |
| limsupresuz2 46166 | If the domain of a functio... |
| limsuppnflem 46167 | If the restriction of a fu... |
| limsuppnf 46168 | If the restriction of a fu... |
| limsupubuzlem 46169 | If the limsup is not ` +oo... |
| limsupubuz 46170 | For a real-valued function... |
| climinf2mpt 46171 | A bounded below, monotonic... |
| climinfmpt 46172 | A bounded below, monotonic... |
| climinf3 46173 | A convergent, nonincreasin... |
| limsupvaluzmpt 46174 | The superior limit, when t... |
| limsupequzmpt2 46175 | Two functions that are eve... |
| limsupubuzmpt 46176 | If the limsup is not ` +oo... |
| limsupmnflem 46177 | The superior limit of a fu... |
| limsupmnf 46178 | The superior limit of a fu... |
| limsupequzlem 46179 | Two functions that are eve... |
| limsupequz 46180 | Two functions that are eve... |
| limsupre2lem 46181 | Given a function on the ex... |
| limsupre2 46182 | Given a function on the ex... |
| limsupmnfuzlem 46183 | The superior limit of a fu... |
| limsupmnfuz 46184 | The superior limit of a fu... |
| limsupequzmptlem 46185 | Two functions that are eve... |
| limsupequzmpt 46186 | Two functions that are eve... |
| limsupre2mpt 46187 | Given a function on the ex... |
| limsupequzmptf 46188 | Two functions that are eve... |
| limsupre3lem 46189 | Given a function on the ex... |
| limsupre3 46190 | Given a function on the ex... |
| limsupre3mpt 46191 | Given a function on the ex... |
| limsupre3uzlem 46192 | Given a function on the ex... |
| limsupre3uz 46193 | Given a function on the ex... |
| limsupreuz 46194 | Given a function on the re... |
| limsupvaluz2 46195 | The superior limit, when t... |
| limsupreuzmpt 46196 | Given a function on the re... |
| supcnvlimsup 46197 | If a function on a set of ... |
| supcnvlimsupmpt 46198 | If a function on a set of ... |
| 0cnv 46199 | If ` (/) ` is a complex nu... |
| climuzlem 46200 | Express the predicate: Th... |
| climuz 46201 | Express the predicate: Th... |
| lmbr3v 46202 | Express the binary relatio... |
| climisp 46203 | If a sequence converges to... |
| lmbr3 46204 | Express the binary relatio... |
| climrescn 46205 | A sequence converging w.r.... |
| climxrrelem 46206 | If a sequence ranging over... |
| climxrre 46207 | If a sequence ranging over... |
| limsuplt2 46210 | The defining property of t... |
| liminfgord 46211 | Ordering property of the i... |
| limsupvald 46212 | The superior limit of a se... |
| limsupresicompt 46213 | The superior limit doesn't... |
| limsupcli 46214 | Closure of the superior li... |
| liminfgf 46215 | Closure of the inferior li... |
| liminfval 46216 | The inferior limit of a se... |
| climlimsup 46217 | A sequence of real numbers... |
| limsupge 46218 | The defining property of t... |
| liminfgval 46219 | Value of the inferior limi... |
| liminfcl 46220 | Closure of the inferior li... |
| liminfvald 46221 | The inferior limit of a se... |
| liminfval5 46222 | The inferior limit of an i... |
| limsupresxr 46223 | The superior limit of a fu... |
| liminfresxr 46224 | The inferior limit of a fu... |
| liminfval2 46225 | The superior limit, relati... |
| climlimsupcex 46226 | Counterexample for ~ climl... |
| liminfcld 46227 | Closure of the inferior li... |
| liminfresico 46228 | The inferior limit doesn't... |
| limsup10exlem 46229 | The range of the given fun... |
| limsup10ex 46230 | The superior limit of a fu... |
| liminf10ex 46231 | The inferior limit of a fu... |
| liminflelimsuplem 46232 | The superior limit is grea... |
| liminflelimsup 46233 | The superior limit is grea... |
| limsupgtlem 46234 | For any positive real, the... |
| limsupgt 46235 | Given a sequence of real n... |
| liminfresre 46236 | The inferior limit of a fu... |
| liminfresicompt 46237 | The inferior limit doesn't... |
| liminfltlimsupex 46238 | An example where the ` lim... |
| liminfgelimsup 46239 | The inferior limit is grea... |
| liminfvalxr 46240 | Alternate definition of ` ... |
| liminfresuz 46241 | If the real part of the do... |
| liminflelimsupuz 46242 | The superior limit is grea... |
| liminfvalxrmpt 46243 | Alternate definition of ` ... |
| liminfresuz2 46244 | If the domain of a functio... |
| liminfgelimsupuz 46245 | The inferior limit is grea... |
| liminfval4 46246 | Alternate definition of ` ... |
| liminfval3 46247 | Alternate definition of ` ... |
| liminfequzmpt2 46248 | Two functions that are eve... |
| liminfvaluz 46249 | Alternate definition of ` ... |
| liminf0 46250 | The inferior limit of the ... |
| limsupval4 46251 | Alternate definition of ` ... |
| liminfvaluz2 46252 | Alternate definition of ` ... |
| liminfvaluz3 46253 | Alternate definition of ` ... |
| liminflelimsupcex 46254 | A counterexample for ~ lim... |
| limsupvaluz3 46255 | Alternate definition of ` ... |
| liminfvaluz4 46256 | Alternate definition of ` ... |
| limsupvaluz4 46257 | Alternate definition of ` ... |
| climliminflimsupd 46258 | If a sequence of real numb... |
| liminfreuzlem 46259 | Given a function on the re... |
| liminfreuz 46260 | Given a function on the re... |
| liminfltlem 46261 | Given a sequence of real n... |
| liminflt 46262 | Given a sequence of real n... |
| climliminf 46263 | A sequence of real numbers... |
| liminflimsupclim 46264 | A sequence of real numbers... |
| climliminflimsup 46265 | A sequence of real numbers... |
| climliminflimsup2 46266 | A sequence of real numbers... |
| climliminflimsup3 46267 | A sequence of real numbers... |
| climliminflimsup4 46268 | A sequence of real numbers... |
| limsupub2 46269 | A extended real valued fun... |
| limsupubuz2 46270 | A sequence with values in ... |
| xlimpnfxnegmnf 46271 | A sequence converges to ` ... |
| liminflbuz2 46272 | A sequence with values in ... |
| liminfpnfuz 46273 | The inferior limit of a fu... |
| liminflimsupxrre 46274 | A sequence with values in ... |
| xlimrel 46277 | The limit on extended real... |
| xlimres 46278 | A function converges iff i... |
| xlimcl 46279 | The limit of a sequence of... |
| rexlimddv2 46280 | Restricted existential eli... |
| xlimclim 46281 | Given a sequence of reals,... |
| xlimconst 46282 | A constant sequence conver... |
| climxlim 46283 | A converging sequence in t... |
| xlimbr 46284 | Express the binary relatio... |
| fuzxrpmcn 46285 | A function mapping from an... |
| cnrefiisplem 46286 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 46287 | A non-real, complex number... |
| xlimxrre 46288 | If a sequence ranging over... |
| xlimmnfvlem1 46289 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 46290 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 46291 | A function converges to mi... |
| xlimconst2 46292 | A sequence that eventually... |
| xlimpnfvlem1 46293 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 46294 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 46295 | A function converges to pl... |
| xlimclim2lem 46296 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 46297 | Given a sequence of extend... |
| xlimmnf 46298 | A function converges to mi... |
| xlimpnf 46299 | A function converges to pl... |
| xlimmnfmpt 46300 | A function converges to pl... |
| xlimpnfmpt 46301 | A function converges to pl... |
| climxlim2lem 46302 | In this lemma for ~ climxl... |
| climxlim2 46303 | A sequence of extended rea... |
| dfxlim2v 46304 | An alternative definition ... |
| dfxlim2 46305 | An alternative definition ... |
| climresd 46306 | A function restricted to u... |
| climresdm 46307 | A real function converges ... |
| dmclimxlim 46308 | A real valued sequence tha... |
| xlimmnflimsup2 46309 | A sequence of extended rea... |
| xlimuni 46310 | An infinite sequence conve... |
| xlimclimdm 46311 | A sequence of extended rea... |
| xlimfun 46312 | The convergence relation o... |
| xlimmnflimsup 46313 | If a sequence of extended ... |
| xlimdm 46314 | Two ways to express that a... |
| xlimpnfxnegmnf2 46315 | A sequence converges to ` ... |
| xlimresdm 46316 | A function converges in th... |
| xlimpnfliminf 46317 | If a sequence of extended ... |
| xlimpnfliminf2 46318 | A sequence of extended rea... |
| xlimliminflimsup 46319 | A sequence of extended rea... |
| xlimlimsupleliminf 46320 | A sequence of extended rea... |
| coseq0 46321 | A complex number whose cos... |
| sinmulcos 46322 | Multiplication formula for... |
| coskpi2 46323 | The cosine of an integer m... |
| cosnegpi 46324 | The cosine of negative ` _... |
| sinaover2ne0 46325 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 46326 | The cosine of an integer m... |
| mulcncff 46327 | The multiplication of two ... |
| cncfmptssg 46328 | A continuous complex funct... |
| constcncfg 46329 | A constant function is a c... |
| idcncfg 46330 | The identity function is a... |
| cncfshift 46331 | A periodic continuous func... |
| resincncf 46332 | ` sin ` restricted to real... |
| addccncf2 46333 | Adding a constant is a con... |
| 0cnf 46334 | The empty set is a continu... |
| fsumcncf 46335 | The finite sum of continuo... |
| cncfperiod 46336 | A periodic continuous func... |
| subcncff 46337 | The subtraction of two con... |
| negcncfg 46338 | The opposite of a continuo... |
| cnfdmsn 46339 | A function with a singleto... |
| cncfcompt 46340 | Composition of continuous ... |
| addcncff 46341 | The sum of two continuous ... |
| ioccncflimc 46342 | Limit at the upper bound o... |
| cncfuni 46343 | A complex function on a su... |
| icccncfext 46344 | A continuous function on a... |
| cncficcgt0 46345 | A the absolute value of a ... |
| icocncflimc 46346 | Limit at the lower bound, ... |
| cncfdmsn 46347 | A complex function with a ... |
| divcncff 46348 | The quotient of two contin... |
| cncfshiftioo 46349 | A periodic continuous func... |
| cncfiooicclem1 46350 | A continuous function ` F ... |
| cncfiooicc 46351 | A continuous function ` F ... |
| cncfiooiccre 46352 | A continuous function ` F ... |
| cncfioobdlem 46353 | ` G ` actually extends ` F... |
| cncfioobd 46354 | A continuous function ` F ... |
| jumpncnp 46355 | Jump discontinuity or disc... |
| cxpcncf2 46356 | The complex power function... |
| fprodcncf 46357 | The finite product of cont... |
| add1cncf 46358 | Addition to a constant is ... |
| add2cncf 46359 | Addition to a constant is ... |
| sub1cncfd 46360 | Subtracting a constant is ... |
| sub2cncfd 46361 | Subtraction from a constan... |
| fprodsub2cncf 46362 | ` F ` is continuous. (Con... |
| fprodadd2cncf 46363 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 46364 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 46365 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 46366 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 46367 | The sequence ` S ` of fini... |
| dvsinexp 46368 | The derivative of sin^N . ... |
| dvcosre 46369 | The real derivative of the... |
| dvsinax 46370 | Derivative exercise: the d... |
| dvsubf 46371 | The subtraction rule for e... |
| dvmptconst 46372 | Function-builder for deriv... |
| dvcnre 46373 | From complex differentiati... |
| dvmptidg 46374 | Function-builder for deriv... |
| dvresntr 46375 | Function-builder for deriv... |
| fperdvper 46376 | The derivative of a period... |
| dvasinbx 46377 | Derivative exercise: the d... |
| dvresioo 46378 | Restriction of a derivativ... |
| dvdivf 46379 | The quotient rule for ever... |
| dvdivbd 46380 | A sufficient condition for... |
| dvsubcncf 46381 | A sufficient condition for... |
| dvmulcncf 46382 | A sufficient condition for... |
| dvcosax 46383 | Derivative exercise: the d... |
| dvdivcncf 46384 | A sufficient condition for... |
| dvbdfbdioolem1 46385 | Given a function with boun... |
| dvbdfbdioolem2 46386 | A function on an open inte... |
| dvbdfbdioo 46387 | A function on an open inte... |
| ioodvbdlimc1lem1 46388 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 46389 | Limit at the lower bound o... |
| ioodvbdlimc1 46390 | A real function with bound... |
| ioodvbdlimc2lem 46391 | Limit at the upper bound o... |
| ioodvbdlimc2 46392 | A real function with bound... |
| dvdmsscn 46393 | ` X ` is a subset of ` CC ... |
| dvmptmulf 46394 | Function-builder for deriv... |
| dvnmptdivc 46395 | Function-builder for itera... |
| dvdsn1add 46396 | If ` K ` divides ` N ` but... |
| dvxpaek 46397 | Derivative of the polynomi... |
| dvnmptconst 46398 | The ` N ` -th derivative o... |
| dvnxpaek 46399 | The ` n ` -th derivative o... |
| dvnmul 46400 | Function-builder for the `... |
| dvmptfprodlem 46401 | Induction step for ~ dvmpt... |
| dvmptfprod 46402 | Function-builder for deriv... |
| dvnprodlem1 46403 | ` D ` is bijective. (Cont... |
| dvnprodlem2 46404 | Induction step for ~ dvnpr... |
| dvnprodlem3 46405 | The multinomial formula fo... |
| dvnprod 46406 | The multinomial formula fo... |
| itgsin0pilem1 46407 | Calculation of the integra... |
| ibliccsinexp 46408 | sin^n on a closed interval... |
| itgsin0pi 46409 | Calculation of the integra... |
| iblioosinexp 46410 | sin^n on an open integral ... |
| itgsinexplem1 46411 | Integration by parts is ap... |
| itgsinexp 46412 | A recursive formula for th... |
| iblconstmpt 46413 | A constant function is int... |
| itgeq1d 46414 | Equality theorem for an in... |
| mbfres2cn 46415 | Measurability of a piecewi... |
| vol0 46416 | The measure of the empty s... |
| ditgeqiooicc 46417 | A function ` F ` on an ope... |
| volge0 46418 | The volume of a set is alw... |
| cnbdibl 46419 | A continuous bounded funct... |
| snmbl 46420 | A singleton is measurable.... |
| ditgeq3d 46421 | Equality theorem for the d... |
| iblempty 46422 | The empty function is inte... |
| iblsplit 46423 | The union of two integrabl... |
| volsn 46424 | A singleton has 0 Lebesgue... |
| itgvol0 46425 | If the domani is negligibl... |
| itgcoscmulx 46426 | Exercise: the integral of ... |
| iblsplitf 46427 | A version of ~ iblsplit us... |
| ibliooicc 46428 | If a function is integrabl... |
| volioc 46429 | The measure of a left-open... |
| iblspltprt 46430 | If a function is integrabl... |
| itgsincmulx 46431 | Exercise: the integral of ... |
| itgsubsticclem 46432 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 46433 | Integration by u-substitut... |
| itgioocnicc 46434 | The integral of a piecewis... |
| iblcncfioo 46435 | A continuous function ` F ... |
| itgspltprt 46436 | The ` S. ` integral splits... |
| itgiccshift 46437 | The integral of a function... |
| itgperiod 46438 | The integral of a periodic... |
| itgsbtaddcnst 46439 | Integral substitution, add... |
| volico 46440 | The measure of left-closed... |
| sublevolico 46441 | The Lebesgue measure of a ... |
| dmvolss 46442 | Lebesgue measurable sets a... |
| ismbl3 46443 | The predicate " ` A ` is L... |
| volioof 46444 | The function that assigns ... |
| ovolsplit 46445 | The Lebesgue outer measure... |
| fvvolioof 46446 | The function value of the ... |
| volioore 46447 | The measure of an open int... |
| fvvolicof 46448 | The function value of the ... |
| voliooico 46449 | An open interval and a lef... |
| ismbl4 46450 | The predicate " ` A ` is L... |
| volioofmpt 46451 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 46452 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 46453 | The Lebesgue measure of op... |
| volicofmpt 46454 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 46455 | The Lebesgue measure of a ... |
| voliccico 46456 | A closed interval and a le... |
| mbfdmssre 46457 | The domain of a measurable... |
| stoweidlem1 46458 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 46459 | lemma for ~ stoweid : here... |
| stoweidlem3 46460 | Lemma for ~ stoweid : if `... |
| stoweidlem4 46461 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 46462 | There exists a δ as ... |
| stoweidlem6 46463 | Lemma for ~ stoweid : two ... |
| stoweidlem7 46464 | This lemma is used to prov... |
| stoweidlem8 46465 | Lemma for ~ stoweid : two ... |
| stoweidlem9 46466 | Lemma for ~ stoweid : here... |
| stoweidlem10 46467 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 46468 | This lemma is used to prov... |
| stoweidlem12 46469 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 46470 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 46471 | There exists a ` k ` as in... |
| stoweidlem15 46472 | This lemma is used to prov... |
| stoweidlem16 46473 | Lemma for ~ stoweid . The... |
| stoweidlem17 46474 | This lemma proves that the... |
| stoweidlem18 46475 | This theorem proves Lemma ... |
| stoweidlem19 46476 | If a set of real functions... |
| stoweidlem20 46477 | If a set A of real functio... |
| stoweidlem21 46478 | Once the Stone Weierstrass... |
| stoweidlem22 46479 | If a set of real functions... |
| stoweidlem23 46480 | This lemma is used to prov... |
| stoweidlem24 46481 | This lemma proves that for... |
| stoweidlem25 46482 | This lemma proves that for... |
| stoweidlem26 46483 | This lemma is used to prov... |
| stoweidlem27 46484 | This lemma is used to prov... |
| stoweidlem28 46485 | There exists a δ as ... |
| stoweidlem29 46486 | When the hypothesis for th... |
| stoweidlem30 46487 | This lemma is used to prov... |
| stoweidlem31 46488 | This lemma is used to prov... |
| stoweidlem32 46489 | If a set A of real functio... |
| stoweidlem33 46490 | If a set of real functions... |
| stoweidlem34 46491 | This lemma proves that for... |
| stoweidlem35 46492 | This lemma is used to prov... |
| stoweidlem36 46493 | This lemma is used to prov... |
| stoweidlem37 46494 | This lemma is used to prov... |
| stoweidlem38 46495 | This lemma is used to prov... |
| stoweidlem39 46496 | This lemma is used to prov... |
| stoweidlem40 46497 | This lemma proves that q_n... |
| stoweidlem41 46498 | This lemma is used to prov... |
| stoweidlem42 46499 | This lemma is used to prov... |
| stoweidlem43 46500 | This lemma is used to prov... |
| stoweidlem44 46501 | This lemma is used to prov... |
| stoweidlem45 46502 | This lemma proves that, gi... |
| stoweidlem46 46503 | This lemma proves that set... |
| stoweidlem47 46504 | Subtracting a constant fro... |
| stoweidlem48 46505 | This lemma is used to prov... |
| stoweidlem49 46506 | There exists a function q_... |
| stoweidlem50 46507 | This lemma proves that set... |
| stoweidlem51 46508 | There exists a function x ... |
| stoweidlem52 46509 | There exists a neighborhoo... |
| stoweidlem53 46510 | This lemma is used to prov... |
| stoweidlem54 46511 | There exists a function ` ... |
| stoweidlem55 46512 | This lemma proves the exis... |
| stoweidlem56 46513 | This theorem proves Lemma ... |
| stoweidlem57 46514 | There exists a function x ... |
| stoweidlem58 46515 | This theorem proves Lemma ... |
| stoweidlem59 46516 | This lemma proves that the... |
| stoweidlem60 46517 | This lemma proves that the... |
| stoweidlem61 46518 | This lemma proves that the... |
| stoweidlem62 46519 | This theorem proves the St... |
| stoweid 46520 | This theorem proves the St... |
| stowei 46521 | This theorem proves the St... |
| wallispilem1 46522 | ` I ` is monotone: increas... |
| wallispilem2 46523 | A first set of properties ... |
| wallispilem3 46524 | I maps to real values. (C... |
| wallispilem4 46525 | ` F ` maps to explicit exp... |
| wallispilem5 46526 | The sequence ` H ` converg... |
| wallispi 46527 | Wallis' formula for π :... |
| wallispi2lem1 46528 | An intermediate step betwe... |
| wallispi2lem2 46529 | Two expressions are proven... |
| wallispi2 46530 | An alternative version of ... |
| stirlinglem1 46531 | A simple limit of fraction... |
| stirlinglem2 46532 | ` A ` maps to positive rea... |
| stirlinglem3 46533 | Long but simple algebraic ... |
| stirlinglem4 46534 | Algebraic manipulation of ... |
| stirlinglem5 46535 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 46536 | A series that converges to... |
| stirlinglem7 46537 | Algebraic manipulation of ... |
| stirlinglem8 46538 | If ` A ` converges to ` C ... |
| stirlinglem9 46539 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 46540 | A bound for any B(N)-B(N +... |
| stirlinglem11 46541 | ` B ` is decreasing. (Con... |
| stirlinglem12 46542 | The sequence ` B ` is boun... |
| stirlinglem13 46543 | ` B ` is decreasing and ha... |
| stirlinglem14 46544 | The sequence ` A ` converg... |
| stirlinglem15 46545 | The Stirling's formula is ... |
| stirling 46546 | Stirling's approximation f... |
| stirlingr 46547 | Stirling's approximation f... |
| dirkerval 46548 | The N_th Dirichlet kernel.... |
| dirker2re 46549 | The Dirichlet kernel value... |
| dirkerdenne0 46550 | The Dirichlet kernel denom... |
| dirkerval2 46551 | The N_th Dirichlet kernel ... |
| dirkerre 46552 | The Dirichlet kernel at an... |
| dirkerper 46553 | the Dirichlet kernel has p... |
| dirkerf 46554 | For any natural number ` N... |
| dirkertrigeqlem1 46555 | Sum of an even number of a... |
| dirkertrigeqlem2 46556 | Trigonometric equality lem... |
| dirkertrigeqlem3 46557 | Trigonometric equality lem... |
| dirkertrigeq 46558 | Trigonometric equality for... |
| dirkeritg 46559 | The definite integral of t... |
| dirkercncflem1 46560 | If ` Y ` is a multiple of ... |
| dirkercncflem2 46561 | Lemma used to prove that t... |
| dirkercncflem3 46562 | The Dirichlet kernel is co... |
| dirkercncflem4 46563 | The Dirichlet kernel is co... |
| dirkercncf 46564 | For any natural number ` N... |
| fourierdlem1 46565 | A partition interval is a ... |
| fourierdlem2 46566 | Membership in a partition.... |
| fourierdlem3 46567 | Membership in a partition.... |
| fourierdlem4 46568 | ` E ` is a function that m... |
| fourierdlem5 46569 | ` S ` is a function. (Con... |
| fourierdlem6 46570 | ` X ` is in the periodic p... |
| fourierdlem7 46571 | The difference between the... |
| fourierdlem8 46572 | A partition interval is a ... |
| fourierdlem9 46573 | ` H ` is a complex functio... |
| fourierdlem10 46574 | Condition on the bounds of... |
| fourierdlem11 46575 | If there is a partition, t... |
| fourierdlem12 46576 | A point of a partition is ... |
| fourierdlem13 46577 | Value of ` V ` in terms of... |
| fourierdlem14 46578 | Given the partition ` V ` ... |
| fourierdlem15 46579 | The range of the partition... |
| fourierdlem16 46580 | The coefficients of the fo... |
| fourierdlem17 46581 | The defined ` L ` is actua... |
| fourierdlem18 46582 | The function ` S ` is cont... |
| fourierdlem19 46583 | If two elements of ` D ` h... |
| fourierdlem20 46584 | Every interval in the part... |
| fourierdlem21 46585 | The coefficients of the fo... |
| fourierdlem22 46586 | The coefficients of the fo... |
| fourierdlem23 46587 | If ` F ` is continuous and... |
| fourierdlem24 46588 | A sufficient condition for... |
| fourierdlem25 46589 | If ` C ` is not in the ran... |
| fourierdlem26 46590 | Periodic image of a point ... |
| fourierdlem27 46591 | A partition open interval ... |
| fourierdlem28 46592 | Derivative of ` ( F `` ( X... |
| fourierdlem29 46593 | Explicit function value fo... |
| fourierdlem30 46594 | Sum of three small pieces ... |
| fourierdlem31 46595 | If ` A ` is finite and for... |
| fourierdlem32 46596 | Limit of a continuous func... |
| fourierdlem33 46597 | Limit of a continuous func... |
| fourierdlem34 46598 | A partition is one to one.... |
| fourierdlem35 46599 | There is a single point in... |
| fourierdlem36 46600 | ` F ` is an isomorphism. ... |
| fourierdlem37 46601 | ` I ` is a function that m... |
| fourierdlem38 46602 | The function ` F ` is cont... |
| fourierdlem39 46603 | Integration by parts of ... |
| fourierdlem40 46604 | ` H ` is a continuous func... |
| fourierdlem41 46605 | Lemma used to prove that e... |
| fourierdlem42 46606 | The set of points in a mov... |
| fourierdlem43 46607 | ` K ` is a real function. ... |
| fourierdlem44 46608 | A condition for having ` (... |
| fourierdlem46 46609 | The function ` F ` has a l... |
| fourierdlem47 46610 | For ` r ` large enough, th... |
| fourierdlem48 46611 | The given periodic functio... |
| fourierdlem49 46612 | The given periodic functio... |
| fourierdlem50 46613 | Continuity of ` O ` and it... |
| fourierdlem51 46614 | ` X ` is in the periodic p... |
| fourierdlem52 46615 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 46616 | The limit of ` F ( s ) ` a... |
| fourierdlem54 46617 | Given a partition ` Q ` an... |
| fourierdlem55 46618 | ` U ` is a real function. ... |
| fourierdlem56 46619 | Derivative of the ` K ` fu... |
| fourierdlem57 46620 | The derivative of ` O ` . ... |
| fourierdlem58 46621 | The derivative of ` K ` is... |
| fourierdlem59 46622 | The derivative of ` H ` is... |
| fourierdlem60 46623 | Given a differentiable fun... |
| fourierdlem61 46624 | Given a differentiable fun... |
| fourierdlem62 46625 | The function ` K ` is cont... |
| fourierdlem63 46626 | The upper bound of interva... |
| fourierdlem64 46627 | The partition ` V ` is fin... |
| fourierdlem65 46628 | The distance of two adjace... |
| fourierdlem66 46629 | Value of the ` G ` functio... |
| fourierdlem67 46630 | ` G ` is a function. (Con... |
| fourierdlem68 46631 | The derivative of ` O ` is... |
| fourierdlem69 46632 | A piecewise continuous fun... |
| fourierdlem70 46633 | A piecewise continuous fun... |
| fourierdlem71 46634 | A periodic piecewise conti... |
| fourierdlem72 46635 | The derivative of ` O ` is... |
| fourierdlem73 46636 | A version of the Riemann L... |
| fourierdlem74 46637 | Given a piecewise smooth f... |
| fourierdlem75 46638 | Given a piecewise smooth f... |
| fourierdlem76 46639 | Continuity of ` O ` and it... |
| fourierdlem77 46640 | If ` H ` is bounded, then ... |
| fourierdlem78 46641 | ` G ` is continuous when r... |
| fourierdlem79 46642 | ` E ` projects every inter... |
| fourierdlem80 46643 | The derivative of ` O ` is... |
| fourierdlem81 46644 | The integral of a piecewis... |
| fourierdlem82 46645 | Integral by substitution, ... |
| fourierdlem83 46646 | The fourier partial sum fo... |
| fourierdlem84 46647 | If ` F ` is piecewise cont... |
| fourierdlem85 46648 | Limit of the function ` G ... |
| fourierdlem86 46649 | Continuity of ` O ` and it... |
| fourierdlem87 46650 | The integral of ` G ` goes... |
| fourierdlem88 46651 | Given a piecewise continuo... |
| fourierdlem89 46652 | Given a piecewise continuo... |
| fourierdlem90 46653 | Given a piecewise continuo... |
| fourierdlem91 46654 | Given a piecewise continuo... |
| fourierdlem92 46655 | The integral of a piecewis... |
| fourierdlem93 46656 | Integral by substitution (... |
| fourierdlem94 46657 | For a piecewise smooth fun... |
| fourierdlem95 46658 | Algebraic manipulation of ... |
| fourierdlem96 46659 | limit for ` F ` at the low... |
| fourierdlem97 46660 | ` F ` is continuous on the... |
| fourierdlem98 46661 | ` F ` is continuous on the... |
| fourierdlem99 46662 | limit for ` F ` at the upp... |
| fourierdlem100 46663 | A piecewise continuous fun... |
| fourierdlem101 46664 | Integral by substitution f... |
| fourierdlem102 46665 | For a piecewise smooth fun... |
| fourierdlem103 46666 | The half lower part of the... |
| fourierdlem104 46667 | The half upper part of the... |
| fourierdlem105 46668 | A piecewise continuous fun... |
| fourierdlem106 46669 | For a piecewise smooth fun... |
| fourierdlem107 46670 | The integral of a piecewis... |
| fourierdlem108 46671 | The integral of a piecewis... |
| fourierdlem109 46672 | The integral of a piecewis... |
| fourierdlem110 46673 | The integral of a piecewis... |
| fourierdlem111 46674 | The fourier partial sum fo... |
| fourierdlem112 46675 | Here abbreviations (local ... |
| fourierdlem113 46676 | Fourier series convergence... |
| fourierdlem114 46677 | Fourier series convergence... |
| fourierdlem115 46678 | Fourier serier convergence... |
| fourierd 46679 | Fourier series convergence... |
| fourierclimd 46680 | Fourier series convergence... |
| fourierclim 46681 | Fourier series convergence... |
| fourier 46682 | Fourier series convergence... |
| fouriercnp 46683 | If ` F ` is continuous at ... |
| fourier2 46684 | Fourier series convergence... |
| sqwvfoura 46685 | Fourier coefficients for t... |
| sqwvfourb 46686 | Fourier series ` B ` coeff... |
| fourierswlem 46687 | The Fourier series for the... |
| fouriersw 46688 | Fourier series convergence... |
| fouriercn 46689 | If the derivative of ` F `... |
| elaa2lem 46690 | Elementhood in the set of ... |
| elaa2 46691 | Elementhood in the set of ... |
| etransclem1 46692 | ` H ` is a function. (Con... |
| etransclem2 46693 | Derivative of ` G ` . (Co... |
| etransclem3 46694 | The given ` if ` term is a... |
| etransclem4 46695 | ` F ` expressed as a finit... |
| etransclem5 46696 | A change of bound variable... |
| etransclem6 46697 | A change of bound variable... |
| etransclem7 46698 | The given product is an in... |
| etransclem8 46699 | ` F ` is a function. (Con... |
| etransclem9 46700 | If ` K ` divides ` N ` but... |
| etransclem10 46701 | The given ` if ` term is a... |
| etransclem11 46702 | A change of bound variable... |
| etransclem12 46703 | ` C ` applied to ` N ` . ... |
| etransclem13 46704 | ` F ` applied to ` Y ` . ... |
| etransclem14 46705 | Value of the term ` T ` , ... |
| etransclem15 46706 | Value of the term ` T ` , ... |
| etransclem16 46707 | Every element in the range... |
| etransclem17 46708 | The ` N ` -th derivative o... |
| etransclem18 46709 | The given function is inte... |
| etransclem19 46710 | The ` N ` -th derivative o... |
| etransclem20 46711 | ` H ` is smooth. (Contrib... |
| etransclem21 46712 | The ` N ` -th derivative o... |
| etransclem22 46713 | The ` N ` -th derivative o... |
| etransclem23 46714 | This is the claim proof in... |
| etransclem24 46715 | ` P ` divides the I -th de... |
| etransclem25 46716 | ` P ` factorial divides th... |
| etransclem26 46717 | Every term in the sum of t... |
| etransclem27 46718 | The ` N ` -th derivative o... |
| etransclem28 46719 | ` ( P - 1 ) ` factorial di... |
| etransclem29 46720 | The ` N ` -th derivative o... |
| etransclem30 46721 | The ` N ` -th derivative o... |
| etransclem31 46722 | The ` N ` -th derivative o... |
| etransclem32 46723 | This is the proof for the ... |
| etransclem33 46724 | ` F ` is smooth. (Contrib... |
| etransclem34 46725 | The ` N ` -th derivative o... |
| etransclem35 46726 | ` P ` does not divide the ... |
| etransclem36 46727 | The ` N ` -th derivative o... |
| etransclem37 46728 | ` ( P - 1 ) ` factorial di... |
| etransclem38 46729 | ` P ` divides the I -th de... |
| etransclem39 46730 | ` G ` is a function. (Con... |
| etransclem40 46731 | The ` N ` -th derivative o... |
| etransclem41 46732 | ` P ` does not divide the ... |
| etransclem42 46733 | The ` N ` -th derivative o... |
| etransclem43 46734 | ` G ` is a continuous func... |
| etransclem44 46735 | The given finite sum is no... |
| etransclem45 46736 | ` K ` is an integer. (Con... |
| etransclem46 46737 | This is the proof for equa... |
| etransclem47 46738 | ` _e ` is transcendental. ... |
| etransclem48 46739 | ` _e ` is transcendental. ... |
| etransc 46740 | ` _e ` is transcendental. ... |
| rrxtopn 46741 | The topology of the genera... |
| rrxngp 46742 | Generalized Euclidean real... |
| rrxtps 46743 | Generalized Euclidean real... |
| rrxtopnfi 46744 | The topology of the n-dime... |
| rrxtopon 46745 | The topology on generalize... |
| rrxtop 46746 | The topology on generalize... |
| rrndistlt 46747 | Given two points in the sp... |
| rrxtoponfi 46748 | The topology on n-dimensio... |
| rrxunitopnfi 46749 | The base set of the standa... |
| rrxtopn0 46750 | The topology of the zero-d... |
| qndenserrnbllem 46751 | n-dimensional rational num... |
| qndenserrnbl 46752 | n-dimensional rational num... |
| rrxtopn0b 46753 | The topology of the zero-d... |
| qndenserrnopnlem 46754 | n-dimensional rational num... |
| qndenserrnopn 46755 | n-dimensional rational num... |
| qndenserrn 46756 | n-dimensional rational num... |
| rrxsnicc 46757 | A multidimensional singlet... |
| rrnprjdstle 46758 | The distance between two p... |
| rrndsmet 46759 | ` D ` is a metric for the ... |
| rrndsxmet 46760 | ` D ` is an extended metri... |
| ioorrnopnlem 46761 | The a point in an indexed ... |
| ioorrnopn 46762 | The indexed product of ope... |
| ioorrnopnxrlem 46763 | Given a point ` F ` that b... |
| ioorrnopnxr 46764 | The indexed product of ope... |
| issal 46771 | Express the predicate " ` ... |
| pwsal 46772 | The power set of a given s... |
| salunicl 46773 | SAlg sigma-algebra is clos... |
| saluncl 46774 | The union of two sets in a... |
| prsal 46775 | The pair of the empty set ... |
| saldifcl 46776 | The complement of an eleme... |
| 0sal 46777 | The empty set belongs to e... |
| salgenval 46778 | The sigma-algebra generate... |
| saliunclf 46779 | SAlg sigma-algebra is clos... |
| saliuncl 46780 | SAlg sigma-algebra is clos... |
| salincl 46781 | The intersection of two se... |
| saluni 46782 | A set is an element of any... |
| saliinclf 46783 | SAlg sigma-algebra is clos... |
| saliincl 46784 | SAlg sigma-algebra is clos... |
| saldifcl2 46785 | The difference of two elem... |
| intsaluni 46786 | The union of an arbitrary ... |
| intsal 46787 | The arbitrary intersection... |
| salgenn0 46788 | The set used in the defini... |
| salgencl 46789 | ` SalGen ` actually genera... |
| issald 46790 | Sufficient condition to pr... |
| salexct 46791 | An example of nontrivial s... |
| sssalgen 46792 | A set is a subset of the s... |
| salgenss 46793 | The sigma-algebra generate... |
| salgenuni 46794 | The base set of the sigma-... |
| issalgend 46795 | One side of ~ dfsalgen2 . ... |
| salexct2 46796 | An example of a subset tha... |
| unisalgen 46797 | The union of a set belongs... |
| dfsalgen2 46798 | Alternate characterization... |
| salexct3 46799 | An example of a sigma-alge... |
| salgencntex 46800 | This counterexample shows ... |
| salgensscntex 46801 | This counterexample shows ... |
| issalnnd 46802 | Sufficient condition to pr... |
| dmvolsal 46803 | Lebesgue measurable sets f... |
| saldifcld 46804 | The complement of an eleme... |
| saluncld 46805 | The union of two sets in a... |
| salgencld 46806 | ` SalGen ` actually genera... |
| 0sald 46807 | The empty set belongs to e... |
| iooborel 46808 | An open interval is a Bore... |
| salincld 46809 | The intersection of two se... |
| salunid 46810 | A set is an element of any... |
| unisalgen2 46811 | The union of a set belongs... |
| bor1sal 46812 | The Borel sigma-algebra on... |
| iocborel 46813 | A left-open, right-closed ... |
| subsaliuncllem 46814 | A subspace sigma-algebra i... |
| subsaliuncl 46815 | A subspace sigma-algebra i... |
| subsalsal 46816 | A subspace sigma-algebra i... |
| subsaluni 46817 | A set belongs to the subsp... |
| salrestss 46818 | A sigma-algebra restricted... |
| sge0rnre 46821 | When ` sum^ ` is applied t... |
| fge0icoicc 46822 | If ` F ` maps to nonnegati... |
| sge0val 46823 | The value of the sum of no... |
| fge0npnf 46824 | If ` F ` maps to nonnegati... |
| sge0rnn0 46825 | The range used in the defi... |
| sge0vald 46826 | The value of the sum of no... |
| fge0iccico 46827 | A range of nonnegative ext... |
| gsumge0cl 46828 | Closure of group sum, for ... |
| sge0reval 46829 | Value of the sum of nonneg... |
| sge0pnfval 46830 | If a term in the sum of no... |
| fge0iccre 46831 | A range of nonnegative ext... |
| sge0z 46832 | Any nonnegative extended s... |
| sge00 46833 | The sum of nonnegative ext... |
| fsumlesge0 46834 | Every finite subsum of non... |
| sge0revalmpt 46835 | Value of the sum of nonneg... |
| sge0sn 46836 | A sum of a nonnegative ext... |
| sge0tsms 46837 | ` sum^ ` applied to a nonn... |
| sge0cl 46838 | The arbitrary sum of nonne... |
| sge0f1o 46839 | Re-index a nonnegative ext... |
| sge0snmpt 46840 | A sum of a nonnegative ext... |
| sge0ge0 46841 | The sum of nonnegative ext... |
| sge0xrcl 46842 | The arbitrary sum of nonne... |
| sge0repnf 46843 | The of nonnegative extende... |
| sge0fsum 46844 | The arbitrary sum of a fin... |
| sge0rern 46845 | If the sum of nonnegative ... |
| sge0supre 46846 | If the arbitrary sum of no... |
| sge0fsummpt 46847 | The arbitrary sum of a fin... |
| sge0sup 46848 | The arbitrary sum of nonne... |
| sge0less 46849 | A shorter sum of nonnegati... |
| sge0rnbnd 46850 | The range used in the defi... |
| sge0pr 46851 | Sum of a pair of nonnegati... |
| sge0gerp 46852 | The arbitrary sum of nonne... |
| sge0pnffigt 46853 | If the sum of nonnegative ... |
| sge0ssre 46854 | If a sum of nonnegative ex... |
| sge0lefi 46855 | A sum of nonnegative exten... |
| sge0lessmpt 46856 | A shorter sum of nonnegati... |
| sge0ltfirp 46857 | If the sum of nonnegative ... |
| sge0prle 46858 | The sum of a pair of nonne... |
| sge0gerpmpt 46859 | The arbitrary sum of nonne... |
| sge0resrnlem 46860 | The sum of nonnegative ext... |
| sge0resrn 46861 | The sum of nonnegative ext... |
| sge0ssrempt 46862 | If a sum of nonnegative ex... |
| sge0resplit 46863 | ` sum^ ` splits into two p... |
| sge0le 46864 | If all of the terms of sum... |
| sge0ltfirpmpt 46865 | If the extended sum of non... |
| sge0split 46866 | Split a sum of nonnegative... |
| sge0lempt 46867 | If all of the terms of sum... |
| sge0splitmpt 46868 | Split a sum of nonnegative... |
| sge0ss 46869 | Change the index set to a ... |
| sge0iunmptlemfi 46870 | Sum of nonnegative extende... |
| sge0p1 46871 | The addition of the next t... |
| sge0iunmptlemre 46872 | Sum of nonnegative extende... |
| sge0fodjrnlem 46873 | Re-index a nonnegative ext... |
| sge0fodjrn 46874 | Re-index a nonnegative ext... |
| sge0iunmpt 46875 | Sum of nonnegative extende... |
| sge0iun 46876 | Sum of nonnegative extende... |
| sge0nemnf 46877 | The generalized sum of non... |
| sge0rpcpnf 46878 | The sum of an infinite num... |
| sge0rernmpt 46879 | If the sum of nonnegative ... |
| sge0lefimpt 46880 | A sum of nonnegative exten... |
| nn0ssge0 46881 | Nonnegative integers are n... |
| sge0clmpt 46882 | The generalized sum of non... |
| sge0ltfirpmpt2 46883 | If the extended sum of non... |
| sge0isum 46884 | If a series of nonnegative... |
| sge0xrclmpt 46885 | The generalized sum of non... |
| sge0xp 46886 | Combine two generalized su... |
| sge0isummpt 46887 | If a series of nonnegative... |
| sge0ad2en 46888 | The value of the infinite ... |
| sge0isummpt2 46889 | If a series of nonnegative... |
| sge0xaddlem1 46890 | The extended addition of t... |
| sge0xaddlem2 46891 | The extended addition of t... |
| sge0xadd 46892 | The extended addition of t... |
| sge0fsummptf 46893 | The generalized sum of a f... |
| sge0snmptf 46894 | A sum of a nonnegative ext... |
| sge0ge0mpt 46895 | The sum of nonnegative ext... |
| sge0repnfmpt 46896 | The of nonnegative extende... |
| sge0pnffigtmpt 46897 | If the generalized sum of ... |
| sge0splitsn 46898 | Separate out a term in a g... |
| sge0pnffsumgt 46899 | If the sum of nonnegative ... |
| sge0gtfsumgt 46900 | If the generalized sum of ... |
| sge0uzfsumgt 46901 | If a real number is smalle... |
| sge0pnfmpt 46902 | If a term in the sum of no... |
| sge0seq 46903 | A series of nonnegative re... |
| sge0reuz 46904 | Value of the generalized s... |
| sge0reuzb 46905 | Value of the generalized s... |
| ismea 46908 | Express the predicate " ` ... |
| dmmeasal 46909 | The domain of a measure is... |
| meaf 46910 | A measure is a function th... |
| mea0 46911 | The measure of the empty s... |
| nnfoctbdjlem 46912 | There exists a mapping fro... |
| nnfoctbdj 46913 | There exists a mapping fro... |
| meadjuni 46914 | The measure of the disjoin... |
| meacl 46915 | The measure of a set is a ... |
| iundjiunlem 46916 | The sets in the sequence `... |
| iundjiun 46917 | Given a sequence ` E ` of ... |
| meaxrcl 46918 | The measure of a set is an... |
| meadjun 46919 | The measure of the union o... |
| meassle 46920 | The measure of a set is gr... |
| meaunle 46921 | The measure of the union o... |
| meadjiunlem 46922 | The sum of nonnegative ext... |
| meadjiun 46923 | The measure of the disjoin... |
| ismeannd 46924 | Sufficient condition to pr... |
| meaiunlelem 46925 | The measure of the union o... |
| meaiunle 46926 | The measure of the union o... |
| psmeasurelem 46927 | ` M ` applied to a disjoin... |
| psmeasure 46928 | Point supported measure, R... |
| voliunsge0lem 46929 | The Lebesgue measure funct... |
| voliunsge0 46930 | The Lebesgue measure funct... |
| volmea 46931 | The Lebesgue measure on th... |
| meage0 46932 | If the measure of a measur... |
| meadjunre 46933 | The measure of the union o... |
| meassre 46934 | If the measure of a measur... |
| meale0eq0 46935 | A measure that is less tha... |
| meadif 46936 | The measure of the differe... |
| meaiuninclem 46937 | Measures are continuous fr... |
| meaiuninc 46938 | Measures are continuous fr... |
| meaiuninc2 46939 | Measures are continuous fr... |
| meaiunincf 46940 | Measures are continuous fr... |
| meaiuninc3v 46941 | Measures are continuous fr... |
| meaiuninc3 46942 | Measures are continuous fr... |
| meaiininclem 46943 | Measures are continuous fr... |
| meaiininc 46944 | Measures are continuous fr... |
| meaiininc2 46945 | Measures are continuous fr... |
| caragenval 46950 | The sigma-algebra generate... |
| isome 46951 | Express the predicate " ` ... |
| caragenel 46952 | Membership in the Caratheo... |
| omef 46953 | An outer measure is a func... |
| ome0 46954 | The outer measure of the e... |
| omessle 46955 | The outer measure of a set... |
| omedm 46956 | The domain of an outer mea... |
| caragensplit 46957 | If ` E ` is in the set gen... |
| caragenelss 46958 | An element of the Caratheo... |
| carageneld 46959 | Membership in the Caratheo... |
| omecl 46960 | The outer measure of a set... |
| caragenss 46961 | The sigma-algebra generate... |
| omeunile 46962 | The outer measure of the u... |
| caragen0 46963 | The empty set belongs to a... |
| omexrcl 46964 | The outer measure of a set... |
| caragenunidm 46965 | The base set of an outer m... |
| caragensspw 46966 | The sigma-algebra generate... |
| omessre 46967 | If the outer measure of a ... |
| caragenuni 46968 | The base set of the sigma-... |
| caragenuncllem 46969 | The Caratheodory's constru... |
| caragenuncl 46970 | The Caratheodory's constru... |
| caragendifcl 46971 | The Caratheodory's constru... |
| caragenfiiuncl 46972 | The Caratheodory's constru... |
| omeunle 46973 | The outer measure of the u... |
| omeiunle 46974 | The outer measure of the i... |
| omelesplit 46975 | The outer measure of a set... |
| omeiunltfirp 46976 | If the outer measure of a ... |
| omeiunlempt 46977 | The outer measure of the i... |
| carageniuncllem1 46978 | The outer measure of ` A i... |
| carageniuncllem2 46979 | The Caratheodory's constru... |
| carageniuncl 46980 | The Caratheodory's constru... |
| caragenunicl 46981 | The Caratheodory's constru... |
| caragensal 46982 | Caratheodory's method gene... |
| caratheodorylem1 46983 | Lemma used to prove that C... |
| caratheodorylem2 46984 | Caratheodory's constructio... |
| caratheodory 46985 | Caratheodory's constructio... |
| 0ome 46986 | The map that assigns 0 to ... |
| isomenndlem 46987 | ` O ` is sub-additive w.r.... |
| isomennd 46988 | Sufficient condition to pr... |
| caragenel2d 46989 | Membership in the Caratheo... |
| omege0 46990 | If the outer measure of a ... |
| omess0 46991 | If the outer measure of a ... |
| caragencmpl 46992 | A measure built with the C... |
| vonval 46997 | Value of the Lebesgue meas... |
| ovnval 46998 | Value of the Lebesgue oute... |
| elhoi 46999 | Membership in a multidimen... |
| icoresmbl 47000 | A closed-below, open-above... |
| hoissre 47001 | The projection of a half-o... |
| ovnval2 47002 | Value of the Lebesgue oute... |
| volicorecl 47003 | The Lebesgue measure of a ... |
| hoiprodcl 47004 | The pre-measure of half-op... |
| hoicvr 47005 | ` I ` is a countable set o... |
| hoissrrn 47006 | A half-open interval is a ... |
| ovn0val 47007 | The Lebesgue outer measure... |
| ovnn0val 47008 | The value of a (multidimen... |
| ovnval2b 47009 | Value of the Lebesgue oute... |
| volicorescl 47010 | The Lebesgue measure of a ... |
| ovnprodcl 47011 | The product used in the de... |
| hoiprodcl2 47012 | The pre-measure of half-op... |
| hoicvrrex 47013 | Any subset of the multidim... |
| ovnsupge0 47014 | The set used in the defini... |
| ovnlecvr 47015 | Given a subset of multidim... |
| ovnpnfelsup 47016 | ` +oo ` is an element of t... |
| ovnsslelem 47017 | The (multidimensional, non... |
| ovnssle 47018 | The (multidimensional) Leb... |
| ovnlerp 47019 | The Lebesgue outer measure... |
| ovnf 47020 | The Lebesgue outer measure... |
| ovncvrrp 47021 | The Lebesgue outer measure... |
| ovn0lem 47022 | For any finite dimension, ... |
| ovn0 47023 | For any finite dimension, ... |
| ovncl 47024 | The Lebesgue outer measure... |
| ovn02 47025 | For the zero-dimensional s... |
| ovnxrcl 47026 | The Lebesgue outer measure... |
| ovnsubaddlem1 47027 | The Lebesgue outer measure... |
| ovnsubaddlem2 47028 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 47029 | ` ( voln* `` X ) ` is suba... |
| ovnome 47030 | ` ( voln* `` X ) ` is an o... |
| vonmea 47031 | ` ( voln `` X ) ` is a mea... |
| volicon0 47032 | The measure of a nonempty ... |
| hsphoif 47033 | ` H ` is a function (that ... |
| hoidmvval 47034 | The dimensional volume of ... |
| hoissrrn2 47035 | A half-open interval is a ... |
| hsphoival 47036 | ` H ` is a function (that ... |
| hoiprodcl3 47037 | The pre-measure of half-op... |
| volicore 47038 | The Lebesgue measure of a ... |
| hoidmvcl 47039 | The dimensional volume of ... |
| hoidmv0val 47040 | The dimensional volume of ... |
| hoidmvn0val 47041 | The dimensional volume of ... |
| hsphoidmvle2 47042 | The dimensional volume of ... |
| hsphoidmvle 47043 | The dimensional volume of ... |
| hoidmvval0 47044 | The dimensional volume of ... |
| hoiprodp1 47045 | The dimensional volume of ... |
| sge0hsphoire 47046 | If the generalized sum of ... |
| hoidmvval0b 47047 | The dimensional volume of ... |
| hoidmv1lelem1 47048 | The supremum of ` U ` belo... |
| hoidmv1lelem2 47049 | This is the contradiction ... |
| hoidmv1lelem3 47050 | The dimensional volume of ... |
| hoidmv1le 47051 | The dimensional volume of ... |
| hoidmvlelem1 47052 | The supremum of ` U ` belo... |
| hoidmvlelem2 47053 | This is the contradiction ... |
| hoidmvlelem3 47054 | This is the contradiction ... |
| hoidmvlelem4 47055 | The dimensional volume of ... |
| hoidmvlelem5 47056 | The dimensional volume of ... |
| hoidmvle 47057 | The dimensional volume of ... |
| ovnhoilem1 47058 | The Lebesgue outer measure... |
| ovnhoilem2 47059 | The Lebesgue outer measure... |
| ovnhoi 47060 | The Lebesgue outer measure... |
| dmovn 47061 | The domain of the Lebesgue... |
| hoicoto2 47062 | The half-open interval exp... |
| dmvon 47063 | Lebesgue measurable n-dime... |
| hoi2toco 47064 | The half-open interval exp... |
| hoidifhspval 47065 | ` D ` is a function that r... |
| hspval 47066 | The value of the half-spac... |
| ovnlecvr2 47067 | Given a subset of multidim... |
| ovncvr2 47068 | ` B ` and ` T ` are the le... |
| dmovnsal 47069 | The domain of the Lebesgue... |
| unidmovn 47070 | Base set of the n-dimensio... |
| rrnmbl 47071 | The set of n-dimensional R... |
| hoidifhspval2 47072 | ` D ` is a function that r... |
| hspdifhsp 47073 | A n-dimensional half-open ... |
| unidmvon 47074 | Base set of the n-dimensio... |
| hoidifhspf 47075 | ` D ` is a function that r... |
| hoidifhspval3 47076 | ` D ` is a function that r... |
| hoidifhspdmvle 47077 | The dimensional volume of ... |
| voncmpl 47078 | The Lebesgue measure is co... |
| hoiqssbllem1 47079 | The center of the n-dimens... |
| hoiqssbllem2 47080 | The center of the n-dimens... |
| hoiqssbllem3 47081 | A n-dimensional ball conta... |
| hoiqssbl 47082 | A n-dimensional ball conta... |
| hspmbllem1 47083 | Any half-space of the n-di... |
| hspmbllem2 47084 | Any half-space of the n-di... |
| hspmbllem3 47085 | Any half-space of the n-di... |
| hspmbl 47086 | Any half-space of the n-di... |
| hoimbllem 47087 | Any n-dimensional half-ope... |
| hoimbl 47088 | Any n-dimensional half-ope... |
| opnvonmbllem1 47089 | The half-open interval exp... |
| opnvonmbllem2 47090 | An open subset of the n-di... |
| opnvonmbl 47091 | An open subset of the n-di... |
| opnssborel 47092 | Open sets of a generalized... |
| borelmbl 47093 | All Borel subsets of the n... |
| volicorege0 47094 | The Lebesgue measure of a ... |
| isvonmbl 47095 | The predicate " ` A ` is m... |
| mblvon 47096 | The n-dimensional Lebesgue... |
| vonmblss 47097 | n-dimensional Lebesgue mea... |
| volico2 47098 | The measure of left-closed... |
| vonmblss2 47099 | n-dimensional Lebesgue mea... |
| ovolval2lem 47100 | The value of the Lebesgue ... |
| ovolval2 47101 | The value of the Lebesgue ... |
| ovnsubadd2lem 47102 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 47103 | ` ( voln* `` X ) ` is suba... |
| ovolval3 47104 | The value of the Lebesgue ... |
| ovnsplit 47105 | The n-dimensional Lebesgue... |
| ovolval4lem1 47106 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 47107 | The value of the Lebesgue ... |
| ovolval4 47108 | The value of the Lebesgue ... |
| ovolval5lem1 47109 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 47110 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 47111 | The value of the Lebesgue ... |
| ovolval5 47112 | The value of the Lebesgue ... |
| ovnovollem1 47113 | if ` F ` is a cover of ` B... |
| ovnovollem2 47114 | if ` I ` is a cover of ` (... |
| ovnovollem3 47115 | The 1-dimensional Lebesgue... |
| ovnovol 47116 | The 1-dimensional Lebesgue... |
| vonvolmbllem 47117 | If a subset ` B ` of real ... |
| vonvolmbl 47118 | A subset of Real numbers i... |
| vonvol 47119 | The 1-dimensional Lebesgue... |
| vonvolmbl2 47120 | A subset ` X ` of the spac... |
| vonvol2 47121 | The 1-dimensional Lebesgue... |
| hoimbl2 47122 | Any n-dimensional half-ope... |
| voncl 47123 | The Lebesgue measure of a ... |
| vonhoi 47124 | The Lebesgue outer measure... |
| vonxrcl 47125 | The Lebesgue measure of a ... |
| ioosshoi 47126 | A n-dimensional open inter... |
| vonn0hoi 47127 | The Lebesgue outer measure... |
| von0val 47128 | The Lebesgue measure (for ... |
| vonhoire 47129 | The Lebesgue measure of a ... |
| iinhoiicclem 47130 | A n-dimensional closed int... |
| iinhoiicc 47131 | A n-dimensional closed int... |
| iunhoiioolem 47132 | A n-dimensional open inter... |
| iunhoiioo 47133 | A n-dimensional open inter... |
| ioovonmbl 47134 | Any n-dimensional open int... |
| iccvonmbllem 47135 | Any n-dimensional closed i... |
| iccvonmbl 47136 | Any n-dimensional closed i... |
| vonioolem1 47137 | The sequence of the measur... |
| vonioolem2 47138 | The n-dimensional Lebesgue... |
| vonioo 47139 | The n-dimensional Lebesgue... |
| vonicclem1 47140 | The sequence of the measur... |
| vonicclem2 47141 | The n-dimensional Lebesgue... |
| vonicc 47142 | The n-dimensional Lebesgue... |
| snvonmbl 47143 | A n-dimensional singleton ... |
| vonn0ioo 47144 | The n-dimensional Lebesgue... |
| vonn0icc 47145 | The n-dimensional Lebesgue... |
| ctvonmbl 47146 | Any n-dimensional countabl... |
| vonn0ioo2 47147 | The n-dimensional Lebesgue... |
| vonsn 47148 | The n-dimensional Lebesgue... |
| vonn0icc2 47149 | The n-dimensional Lebesgue... |
| vonct 47150 | The n-dimensional Lebesgue... |
| vitali2 47151 | There are non-measurable s... |
| pimltmnf2f 47154 | Given a real-valued functi... |
| pimltmnf2 47155 | Given a real-valued functi... |
| preimagelt 47156 | The preimage of a right-op... |
| preimalegt 47157 | The preimage of a left-ope... |
| pimconstlt0 47158 | Given a constant function,... |
| pimconstlt1 47159 | Given a constant function,... |
| pimltpnff 47160 | Given a real-valued functi... |
| pimltpnf 47161 | Given a real-valued functi... |
| pimgtpnf2f 47162 | Given a real-valued functi... |
| pimgtpnf2 47163 | Given a real-valued functi... |
| salpreimagelt 47164 | If all the preimages of le... |
| pimrecltpos 47165 | The preimage of an unbound... |
| salpreimalegt 47166 | If all the preimages of ri... |
| pimiooltgt 47167 | The preimage of an open in... |
| preimaicomnf 47168 | Preimage of an open interv... |
| pimltpnf2f 47169 | Given a real-valued functi... |
| pimltpnf2 47170 | Given a real-valued functi... |
| pimgtmnf2 47171 | Given a real-valued functi... |
| pimdecfgtioc 47172 | Given a nonincreasing func... |
| pimincfltioc 47173 | Given a nondecreasing func... |
| pimdecfgtioo 47174 | Given a nondecreasing func... |
| pimincfltioo 47175 | Given a nondecreasing func... |
| preimaioomnf 47176 | Preimage of an open interv... |
| preimageiingt 47177 | A preimage of a left-close... |
| preimaleiinlt 47178 | A preimage of a left-open,... |
| pimgtmnff 47179 | Given a real-valued functi... |
| pimgtmnf 47180 | Given a real-valued functi... |
| pimrecltneg 47181 | The preimage of an unbound... |
| salpreimagtge 47182 | If all the preimages of le... |
| salpreimaltle 47183 | If all the preimages of ri... |
| issmflem 47184 | The predicate " ` F ` is a... |
| issmf 47185 | The predicate " ` F ` is a... |
| salpreimalelt 47186 | If all the preimages of ri... |
| salpreimagtlt 47187 | If all the preimages of le... |
| smfpreimalt 47188 | Given a function measurabl... |
| smff 47189 | A function measurable w.r.... |
| smfdmss 47190 | The domain of a function m... |
| issmff 47191 | The predicate " ` F ` is a... |
| issmfd 47192 | A sufficient condition for... |
| smfpreimaltf 47193 | Given a function measurabl... |
| issmfdf 47194 | A sufficient condition for... |
| sssmf 47195 | The restriction of a sigma... |
| mbfresmf 47196 | A real-valued measurable f... |
| cnfsmf 47197 | A continuous function is m... |
| incsmflem 47198 | A nondecreasing function i... |
| incsmf 47199 | A real-valued, nondecreasi... |
| smfsssmf 47200 | If a function is measurabl... |
| issmflelem 47201 | The predicate " ` F ` is a... |
| issmfle 47202 | The predicate " ` F ` is a... |
| smfpimltmpt 47203 | Given a function measurabl... |
| smfpimltxr 47204 | Given a function measurabl... |
| issmfdmpt 47205 | A sufficient condition for... |
| smfconst 47206 | Given a sigma-algebra over... |
| sssmfmpt 47207 | The restriction of a sigma... |
| cnfrrnsmf 47208 | A function, continuous fro... |
| smfid 47209 | The identity function is B... |
| bormflebmf 47210 | A Borel measurable functio... |
| smfpreimale 47211 | Given a function measurabl... |
| issmfgtlem 47212 | The predicate " ` F ` is a... |
| issmfgt 47213 | The predicate " ` F ` is a... |
| issmfled 47214 | A sufficient condition for... |
| smfpimltxrmptf 47215 | Given a function measurabl... |
| smfpimltxrmpt 47216 | Given a function measurabl... |
| smfmbfcex 47217 | A constant function, with ... |
| issmfgtd 47218 | A sufficient condition for... |
| smfpreimagt 47219 | Given a function measurabl... |
| smfaddlem1 47220 | Given the sum of two funct... |
| smfaddlem2 47221 | The sum of two sigma-measu... |
| smfadd 47222 | The sum of two sigma-measu... |
| decsmflem 47223 | A nonincreasing function i... |
| decsmf 47224 | A real-valued, nonincreasi... |
| smfpreimagtf 47225 | Given a function measurabl... |
| issmfgelem 47226 | The predicate " ` F ` is a... |
| issmfge 47227 | The predicate " ` F ` is a... |
| smflimlem1 47228 | Lemma for the proof that t... |
| smflimlem2 47229 | Lemma for the proof that t... |
| smflimlem3 47230 | The limit of sigma-measura... |
| smflimlem4 47231 | Lemma for the proof that t... |
| smflimlem5 47232 | Lemma for the proof that t... |
| smflimlem6 47233 | Lemma for the proof that t... |
| smflim 47234 | The limit of sigma-measura... |
| nsssmfmbflem 47235 | The sigma-measurable funct... |
| nsssmfmbf 47236 | The sigma-measurable funct... |
| smfpimgtxr 47237 | Given a function measurabl... |
| smfpimgtmpt 47238 | Given a function measurabl... |
| smfpreimage 47239 | Given a function measurabl... |
| mbfpsssmf 47240 | Real-valued measurable fun... |
| smfpimgtxrmptf 47241 | Given a function measurabl... |
| smfpimgtxrmpt 47242 | Given a function measurabl... |
| smfpimioompt 47243 | Given a function measurabl... |
| smfpimioo 47244 | Given a function measurabl... |
| smfresal 47245 | Given a sigma-measurable f... |
| smfrec 47246 | The reciprocal of a sigma-... |
| smfres 47247 | The restriction of sigma-m... |
| smfmullem1 47248 | The multiplication of two ... |
| smfmullem2 47249 | The multiplication of two ... |
| smfmullem3 47250 | The multiplication of two ... |
| smfmullem4 47251 | The multiplication of two ... |
| smfmul 47252 | The multiplication of two ... |
| smfmulc1 47253 | A sigma-measurable functio... |
| smfdiv 47254 | The fraction of two sigma-... |
| smfpimbor1lem1 47255 | Every open set belongs to ... |
| smfpimbor1lem2 47256 | Given a sigma-measurable f... |
| smfpimbor1 47257 | Given a sigma-measurable f... |
| smf2id 47258 | Twice the identity functio... |
| smfco 47259 | The composition of a Borel... |
| smfneg 47260 | The negative of a sigma-me... |
| smffmptf 47261 | A function measurable w.r.... |
| smffmpt 47262 | A function measurable w.r.... |
| smflim2 47263 | The limit of a sequence of... |
| smfpimcclem 47264 | Lemma for ~ smfpimcc given... |
| smfpimcc 47265 | Given a countable set of s... |
| issmfle2d 47266 | A sufficient condition for... |
| smflimmpt 47267 | The limit of a sequence of... |
| smfsuplem1 47268 | The supremum of a countabl... |
| smfsuplem2 47269 | The supremum of a countabl... |
| smfsuplem3 47270 | The supremum of a countabl... |
| smfsup 47271 | The supremum of a countabl... |
| smfsupmpt 47272 | The supremum of a countabl... |
| smfsupxr 47273 | The supremum of a countabl... |
| smfinflem 47274 | The infimum of a countable... |
| smfinf 47275 | The infimum of a countable... |
| smfinfmpt 47276 | The infimum of a countable... |
| smflimsuplem1 47277 | If ` H ` converges, the ` ... |
| smflimsuplem2 47278 | The superior limit of a se... |
| smflimsuplem3 47279 | The limit of the ` ( H `` ... |
| smflimsuplem4 47280 | If ` H ` converges, the ` ... |
| smflimsuplem5 47281 | ` H ` converges to the sup... |
| smflimsuplem6 47282 | The superior limit of a se... |
| smflimsuplem7 47283 | The superior limit of a se... |
| smflimsuplem8 47284 | The superior limit of a se... |
| smflimsup 47285 | The superior limit of a se... |
| smflimsupmpt 47286 | The superior limit of a se... |
| smfliminflem 47287 | The inferior limit of a co... |
| smfliminf 47288 | The inferior limit of a co... |
| smfliminfmpt 47289 | The inferior limit of a co... |
| adddmmbl 47290 | If two functions have doma... |
| adddmmbl2 47291 | If two functions have doma... |
| muldmmbl 47292 | If two functions have doma... |
| muldmmbl2 47293 | If two functions have doma... |
| smfdmmblpimne 47294 | If a measurable function w... |
| smfdivdmmbl 47295 | If a functions and a sigma... |
| smfpimne 47296 | Given a function measurabl... |
| smfpimne2 47297 | Given a function measurabl... |
| smfdivdmmbl2 47298 | If a functions and a sigma... |
| fsupdm 47299 | The domain of the sup func... |
| fsupdm2 47300 | The domain of the sup func... |
| smfsupdmmbllem 47301 | If a countable set of sigm... |
| smfsupdmmbl 47302 | If a countable set of sigm... |
| finfdm 47303 | The domain of the inf func... |
| finfdm2 47304 | The domain of the inf func... |
| smfinfdmmbllem 47305 | If a countable set of sigm... |
| smfinfdmmbl 47306 | If a countable set of sigm... |
| sigarval 47307 | Define the signed area by ... |
| sigarim 47308 | Signed area takes value in... |
| sigarac 47309 | Signed area is anticommuta... |
| sigaraf 47310 | Signed area is additive by... |
| sigarmf 47311 | Signed area is additive (w... |
| sigaras 47312 | Signed area is additive by... |
| sigarms 47313 | Signed area is additive (w... |
| sigarls 47314 | Signed area is linear by t... |
| sigarid 47315 | Signed area of a flat para... |
| sigarexp 47316 | Expand the signed area for... |
| sigarperm 47317 | Signed area ` ( A - C ) G ... |
| sigardiv 47318 | If signed area between vec... |
| sigarimcd 47319 | Signed area takes value in... |
| sigariz 47320 | If signed area is zero, th... |
| sigarcol 47321 | Given three points ` A ` ,... |
| sharhght 47322 | Let ` A B C ` be a triangl... |
| sigaradd 47323 | Subtracting (double) area ... |
| cevathlem1 47324 | Ceva's theorem first lemma... |
| cevathlem2 47325 | Ceva's theorem second lemm... |
| cevath 47326 | Ceva's theorem. Let ` A B... |
| simpcntrab 47327 | The center of a simple gro... |
| et-ltneverrefl 47328 | Less-than class is never r... |
| et-equeucl 47329 | Alternative proof that equ... |
| et-sqrtnegnre 47330 | The square root of a negat... |
| quantgodel 47331 | There can be no formula as... |
| quantgodelALT 47332 | There can be no formula as... |
| ormklocald 47333 | If elements of a certain s... |
| ormkglobd 47334 | If all adjacent elements o... |
| natlocalincr 47335 | Global monotonicity on hal... |
| natglobalincr 47336 | Local monotonicity on half... |
| chnsubseqword 47337 | A subsequence of a chain i... |
| chnsubseqwl 47338 | A subsequence of a chain h... |
| chnsubseq 47339 | An order-preserving subseq... |
| chnsuslle 47340 | Length of a subsequence is... |
| chnerlem1 47341 | In a chain constructed on ... |
| chnerlem2 47342 | Lemma for ~ chner where th... |
| chnerlem3 47343 | Lemma for ~ chner - tricho... |
| chner 47344 | Any two elements are equiv... |
| nthrucw 47345 | Some number sets form a ch... |
| evenwodadd 47346 | If an integer is multiplie... |
| squeezedltsq 47347 | If a real value is squeeze... |
| sin3t 47348 | Triple-angle formula for s... |
| cos3t 47349 | Triple-angle formula for c... |
| sin5tlem1 47350 | Lemma 1 for quintupled ang... |
| sin5tlem2 47351 | Lemma 2 for quintupled ang... |
| sin5tlem3 47352 | Lemma 3 for quintupled ang... |
| sin5tlem4 47353 | Lemma 4 for quintupled ang... |
| sin5tlem5 47354 | Lemma 5 for quintupled ang... |
| sin5t 47355 | Five-times-angle formula f... |
| cos5t 47356 | Five-times-angle formula f... |
| cos5teq 47357 | Five-times-angle formula f... |
| goldrarr 47358 | The golden ratio is a real... |
| goldrasin 47359 | Alternative trigonometric ... |
| goldrapos 47360 | Golden ratio is positive. ... |
| goldrarp 47361 | The golden ratio is a posi... |
| goldracos5teq 47362 | Lemma 1 for determining th... |
| goldratmolem2 47363 | Lemma 2 for determining th... |
| lambert0 47364 | A value of Lambert W (prod... |
| lamberte 47365 | A value of Lambert W (prod... |
| cjnpoly 47366 | Complex conjugation operat... |
| tannpoly 47367 | The tangent function is no... |
| sinnpoly 47368 | Sine function is not a pol... |
| hirstL-ax3 47369 | The third axiom of a syste... |
| ax3h 47370 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 47371 | A closed form showing (a i... |
| aibandbiaiaiffb 47372 | A closed form showing (a i... |
| notatnand 47373 | Do not use. Use intnanr i... |
| aistia 47374 | Given a is equivalent to `... |
| aisfina 47375 | Given a is equivalent to `... |
| bothtbothsame 47376 | Given both a, b are equiva... |
| bothfbothsame 47377 | Given both a, b are equiva... |
| aiffbbtat 47378 | Given a is equivalent to b... |
| aisbbisfaisf 47379 | Given a is equivalent to b... |
| axorbtnotaiffb 47380 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 47381 | Given a is equivalent to (... |
| axorbciffatcxorb 47382 | Given a is equivalent to (... |
| aibnbna 47383 | Given a implies b, (not b)... |
| aibnbaif 47384 | Given a implies b, not b, ... |
| aiffbtbat 47385 | Given a is equivalent to b... |
| astbstanbst 47386 | Given a is equivalent to T... |
| aistbistaandb 47387 | Given a is equivalent to T... |
| aisbnaxb 47388 | Given a is equivalent to b... |
| atbiffatnnb 47389 | If a implies b, then a imp... |
| bisaiaisb 47390 | Application of bicom1 with... |
| atbiffatnnbalt 47391 | If a implies b, then a imp... |
| abnotbtaxb 47392 | Assuming a, not b, there e... |
| abnotataxb 47393 | Assuming not a, b, there e... |
| conimpf 47394 | Assuming a, not b, and a i... |
| conimpfalt 47395 | Assuming a, not b, and a i... |
| aistbisfiaxb 47396 | Given a is equivalent to T... |
| aisfbistiaxb 47397 | Given a is equivalent to F... |
| aifftbifffaibif 47398 | Given a is equivalent to T... |
| aifftbifffaibifff 47399 | Given a is equivalent to T... |
| atnaiana 47400 | Given a, it is not the cas... |
| ainaiaandna 47401 | Given a, a implies it is n... |
| abcdta 47402 | Given (((a and b) and c) a... |
| abcdtb 47403 | Given (((a and b) and c) a... |
| abcdtc 47404 | Given (((a and b) and c) a... |
| abcdtd 47405 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 47406 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 47407 | Operands in a biconditiona... |
| nabctnabc 47408 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 47409 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 47410 | From one negated implicati... |
| twonotinotbothi 47411 | From these two negated imp... |
| clifte 47412 | show d is the same as an i... |
| cliftet 47413 | show d is the same as an i... |
| clifteta 47414 | show d is the same as an i... |
| cliftetb 47415 | show d is the same as an i... |
| confun 47416 | Given the hypotheses there... |
| confun2 47417 | Confun simplified to two p... |
| confun3 47418 | Confun's more complex form... |
| confun4 47419 | An attempt at derivative. ... |
| confun5 47420 | An attempt at derivative. ... |
| plcofph 47421 | Given, a,b and a "definiti... |
| pldofph 47422 | Given, a,b c, d, "definiti... |
| plvcofph 47423 | Given, a,b,d, and "definit... |
| plvcofphax 47424 | Given, a,b,d, and "definit... |
| plvofpos 47425 | rh is derivable because ON... |
| mdandyv0 47426 | Given the equivalences set... |
| mdandyv1 47427 | Given the equivalences set... |
| mdandyv2 47428 | Given the equivalences set... |
| mdandyv3 47429 | Given the equivalences set... |
| mdandyv4 47430 | Given the equivalences set... |
| mdandyv5 47431 | Given the equivalences set... |
| mdandyv6 47432 | Given the equivalences set... |
| mdandyv7 47433 | Given the equivalences set... |
| mdandyv8 47434 | Given the equivalences set... |
| mdandyv9 47435 | Given the equivalences set... |
| mdandyv10 47436 | Given the equivalences set... |
| mdandyv11 47437 | Given the equivalences set... |
| mdandyv12 47438 | Given the equivalences set... |
| mdandyv13 47439 | Given the equivalences set... |
| mdandyv14 47440 | Given the equivalences set... |
| mdandyv15 47441 | Given the equivalences set... |
| mdandyvr0 47442 | Given the equivalences set... |
| mdandyvr1 47443 | Given the equivalences set... |
| mdandyvr2 47444 | Given the equivalences set... |
| mdandyvr3 47445 | Given the equivalences set... |
| mdandyvr4 47446 | Given the equivalences set... |
| mdandyvr5 47447 | Given the equivalences set... |
| mdandyvr6 47448 | Given the equivalences set... |
| mdandyvr7 47449 | Given the equivalences set... |
| mdandyvr8 47450 | Given the equivalences set... |
| mdandyvr9 47451 | Given the equivalences set... |
| mdandyvr10 47452 | Given the equivalences set... |
| mdandyvr11 47453 | Given the equivalences set... |
| mdandyvr12 47454 | Given the equivalences set... |
| mdandyvr13 47455 | Given the equivalences set... |
| mdandyvr14 47456 | Given the equivalences set... |
| mdandyvr15 47457 | Given the equivalences set... |
| mdandyvrx0 47458 | Given the exclusivities se... |
| mdandyvrx1 47459 | Given the exclusivities se... |
| mdandyvrx2 47460 | Given the exclusivities se... |
| mdandyvrx3 47461 | Given the exclusivities se... |
| mdandyvrx4 47462 | Given the exclusivities se... |
| mdandyvrx5 47463 | Given the exclusivities se... |
| mdandyvrx6 47464 | Given the exclusivities se... |
| mdandyvrx7 47465 | Given the exclusivities se... |
| mdandyvrx8 47466 | Given the exclusivities se... |
| mdandyvrx9 47467 | Given the exclusivities se... |
| mdandyvrx10 47468 | Given the exclusivities se... |
| mdandyvrx11 47469 | Given the exclusivities se... |
| mdandyvrx12 47470 | Given the exclusivities se... |
| mdandyvrx13 47471 | Given the exclusivities se... |
| mdandyvrx14 47472 | Given the exclusivities se... |
| mdandyvrx15 47473 | Given the exclusivities se... |
| H15NH16TH15IH16 47474 | Given 15 hypotheses and a ... |
| dandysum2p2e4 47475 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 47476 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 47477 | Replacement of a nested an... |
| adh-minim 47478 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 47479 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 47480 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 47481 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 47482 | Fourth lemma for the deriv... |
| adh-minim-ax1 47483 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 47484 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 47485 | Sixth lemma for the deriva... |
| adh-minim-ax2c 47486 | Derivation of a commuted f... |
| adh-minim-ax2 47487 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 47488 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 47489 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 47490 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 47491 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 47492 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 47493 | Third lemma for the deriva... |
| adh-minimp-sylsimp 47494 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 47495 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 47496 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 47497 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 47498 | Fourth lemma for the deriv... |
| adh-minimp-ax2 47499 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 47500 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 47501 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 47502 | If a class with one elemen... |
| eusnsn 47503 | There is a unique element ... |
| absnsb 47504 | If the class abstraction `... |
| euabsneu 47505 | Another way to express exi... |
| elprneb 47506 | An element of a proper uno... |
| oppr 47507 | Equality for ordered pairs... |
| opprb 47508 | Equality for unordered pai... |
| or2expropbilem1 47509 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 47510 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 47511 | If two classes are strictl... |
| eubrv 47512 | If there is a unique set w... |
| eubrdm 47513 | If there is a unique set w... |
| eldmressn 47514 | Element of the domain of a... |
| iota0def 47515 | Example for a defined iota... |
| iota0ndef 47516 | Example for an undefined i... |
| fveqvfvv 47517 | If a function's value at a... |
| fnresfnco 47518 | Composition of two functio... |
| funcoressn 47519 | A composition restricted t... |
| funressnfv 47520 | A restriction to a singlet... |
| funressndmfvrn 47521 | The value of a function ` ... |
| funressnvmo 47522 | A function restricted to a... |
| funressnmo 47523 | A function restricted to a... |
| funressneu 47524 | There is exactly one value... |
| fresfo 47525 | Conditions for a restricti... |
| fsetsniunop 47526 | The class of all functions... |
| fsetabsnop 47527 | The class of all functions... |
| fsetsnf 47528 | The mapping of an element ... |
| fsetsnf1 47529 | The mapping of an element ... |
| fsetsnfo 47530 | The mapping of an element ... |
| fsetsnf1o 47531 | The mapping of an element ... |
| fsetsnprcnex 47532 | The class of all functions... |
| cfsetssfset 47533 | The class of constant func... |
| cfsetsnfsetfv 47534 | The function value of the ... |
| cfsetsnfsetf 47535 | The mapping of the class o... |
| cfsetsnfsetf1 47536 | The mapping of the class o... |
| cfsetsnfsetfo 47537 | The mapping of the class o... |
| cfsetsnfsetf1o 47538 | The mapping of the class o... |
| fsetprcnexALT 47539 | First version of proof for... |
| fcoreslem1 47540 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 47541 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 47542 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 47543 | Lemma 4 for ~ fcores . (C... |
| fcores 47544 | Every composite function `... |
| fcoresf1lem 47545 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 47546 | If a composition is inject... |
| fcoresf1b 47547 | A composition is injective... |
| fcoresfo 47548 | If a composition is surjec... |
| fcoresfob 47549 | A composition is surjectiv... |
| fcoresf1ob 47550 | A composition is bijective... |
| f1cof1blem 47551 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 47552 | The composition of three b... |
| 3f1oss2 47553 | The composition of three b... |
| f1cof1b 47554 | If the range of ` F ` equa... |
| funfocofob 47555 | If the domain of a functio... |
| fnfocofob 47556 | If the domain of a functio... |
| focofob 47557 | If the domain of a functio... |
| f1ocof1ob 47558 | If the range of ` F ` equa... |
| f1ocof1ob2 47559 | If the range of ` F ` equa... |
| aiotajust 47561 | Soundness justification th... |
| dfaiota2 47563 | Alternate definition of th... |
| reuabaiotaiota 47564 | The iota and the alternate... |
| reuaiotaiota 47565 | The iota and the alternate... |
| aiotaexb 47566 | The alternate iota over a ... |
| aiotavb 47567 | The alternate iota over a ... |
| aiotaint 47568 | This is to ~ df-aiota what... |
| dfaiota3 47569 | Alternate definition of ` ... |
| iotan0aiotaex 47570 | If the iota over a wff ` p... |
| aiotaexaiotaiota 47571 | The alternate iota over a ... |
| aiotaval 47572 | Theorem 8.19 in [Quine] p.... |
| aiota0def 47573 | Example for a defined alte... |
| aiota0ndef 47574 | Example for an undefined a... |
| r19.32 47575 | Theorem 19.32 of [Margaris... |
| rexsb 47576 | An equivalent expression f... |
| rexrsb 47577 | An equivalent expression f... |
| 2rexsb 47578 | An equivalent expression f... |
| 2rexrsb 47579 | An equivalent expression f... |
| cbvral2 47580 | Change bound variables of ... |
| cbvrex2 47581 | Change bound variables of ... |
| ralndv1 47582 | Example for a theorem abou... |
| ralndv2 47583 | Second example for a theor... |
| reuf1odnf 47584 | There is exactly one eleme... |
| reuf1od 47585 | There is exactly one eleme... |
| euoreqb 47586 | There is a set which is eq... |
| 2reu3 47587 | Double restricted existent... |
| 2reu7 47588 | Two equivalent expressions... |
| 2reu8 47589 | Two equivalent expressions... |
| 2reu8i 47590 | Implication of a double re... |
| 2reuimp0 47591 | Implication of a double re... |
| 2reuimp 47592 | Implication of a double re... |
| ralbinrald 47599 | Elemination of a restricte... |
| nvelim 47600 | If a class is the universa... |
| alneu 47601 | If a statement holds for a... |
| eu2ndop1stv 47602 | If there is a unique secon... |
| dfateq12d 47603 | Equality deduction for "de... |
| nfdfat 47604 | Bound-variable hypothesis ... |
| dfdfat2 47605 | Alternate definition of th... |
| fundmdfat 47606 | A function is defined at a... |
| dfatprc 47607 | A function is not defined ... |
| dfatelrn 47608 | The value of a function ` ... |
| dfafv2 47609 | Alternative definition of ... |
| afveq12d 47610 | Equality deduction for fun... |
| afveq1 47611 | Equality theorem for funct... |
| afveq2 47612 | Equality theorem for funct... |
| nfafv 47613 | Bound-variable hypothesis ... |
| csbafv12g 47614 | Move class substitution in... |
| afvfundmfveq 47615 | If a class is a function r... |
| afvnfundmuv 47616 | If a set is not in the dom... |
| ndmafv 47617 | The value of a class outsi... |
| afvvdm 47618 | If the function value of a... |
| nfunsnafv 47619 | If the restriction of a cl... |
| afvvfunressn 47620 | If the function value of a... |
| afvprc 47621 | A function's value at a pr... |
| afvvv 47622 | If a function's value at a... |
| afvpcfv0 47623 | If the value of the altern... |
| afvnufveq 47624 | The value of the alternati... |
| afvvfveq 47625 | The value of the alternati... |
| afv0fv0 47626 | If the value of the altern... |
| afvfvn0fveq 47627 | If the function's value at... |
| afv0nbfvbi 47628 | The function's value at an... |
| afvfv0bi 47629 | The function's value at an... |
| afveu 47630 | The value of a function at... |
| fnbrafvb 47631 | Equivalence of function va... |
| fnopafvb 47632 | Equivalence of function va... |
| funbrafvb 47633 | Equivalence of function va... |
| funopafvb 47634 | Equivalence of function va... |
| funbrafv 47635 | The second argument of a b... |
| funbrafv2b 47636 | Function value in terms of... |
| dfafn5a 47637 | Representation of a functi... |
| dfafn5b 47638 | Representation of a functi... |
| fnrnafv 47639 | The range of a function ex... |
| afvelrnb 47640 | A member of a function's r... |
| afvelrnb0 47641 | A member of a function's r... |
| dfaimafn 47642 | Alternate definition of th... |
| dfaimafn2 47643 | Alternate definition of th... |
| afvelima 47644 | Function value in an image... |
| afvelrn 47645 | A function's value belongs... |
| fnafvelrn 47646 | A function's value belongs... |
| fafvelcdm 47647 | A function's value belongs... |
| ffnafv 47648 | A function maps to a class... |
| afvres 47649 | The value of a restricted ... |
| tz6.12-afv 47650 | Function value. Theorem 6... |
| tz6.12-1-afv 47651 | Function value (Theorem 6.... |
| dmfcoafv 47652 | Domains of a function comp... |
| afvco2 47653 | Value of a function compos... |
| rlimdmafv 47654 | Two ways to express that a... |
| aoveq123d 47655 | Equality deduction for ope... |
| nfaov 47656 | Bound-variable hypothesis ... |
| csbaovg 47657 | Move class substitution in... |
| aovfundmoveq 47658 | If a class is a function r... |
| aovnfundmuv 47659 | If an ordered pair is not ... |
| ndmaov 47660 | The value of an operation ... |
| ndmaovg 47661 | The value of an operation ... |
| aovvdm 47662 | If the operation value of ... |
| nfunsnaov 47663 | If the restriction of a cl... |
| aovvfunressn 47664 | If the operation value of ... |
| aovprc 47665 | The value of an operation ... |
| aovrcl 47666 | Reverse closure for an ope... |
| aovpcov0 47667 | If the alternative value o... |
| aovnuoveq 47668 | The alternative value of t... |
| aovvoveq 47669 | The alternative value of t... |
| aov0ov0 47670 | If the alternative value o... |
| aovovn0oveq 47671 | If the operation's value a... |
| aov0nbovbi 47672 | The operation's value on a... |
| aovov0bi 47673 | The operation's value on a... |
| rspceaov 47674 | A frequently used special ... |
| fnotaovb 47675 | Equivalence of operation v... |
| ffnaov 47676 | An operation maps to a cla... |
| faovcl 47677 | Closure law for an operati... |
| aovmpt4g 47678 | Value of a function given ... |
| aoprssdm 47679 | Domain of closure of an op... |
| ndmaovcl 47680 | The "closure" of an operat... |
| ndmaovrcl 47681 | Reverse closure law, in co... |
| ndmaovcom 47682 | Any operation is commutati... |
| ndmaovass 47683 | Any operation is associati... |
| ndmaovdistr 47684 | Any operation is distribut... |
| dfatafv2iota 47687 | If a function is defined a... |
| ndfatafv2 47688 | The alternate function val... |
| ndfatafv2undef 47689 | The alternate function val... |
| dfatafv2ex 47690 | The alternate function val... |
| afv2ex 47691 | The alternate function val... |
| afv2eq12d 47692 | Equality deduction for fun... |
| afv2eq1 47693 | Equality theorem for funct... |
| afv2eq2 47694 | Equality theorem for funct... |
| nfafv2 47695 | Bound-variable hypothesis ... |
| csbafv212g 47696 | Move class substitution in... |
| fexafv2ex 47697 | The alternate function val... |
| ndfatafv2nrn 47698 | The alternate function val... |
| ndmafv2nrn 47699 | The value of a class outsi... |
| funressndmafv2rn 47700 | The alternate function val... |
| afv2ndefb 47701 | Two ways to say that an al... |
| nfunsnafv2 47702 | If the restriction of a cl... |
| afv2prc 47703 | A function's value at a pr... |
| dfatafv2rnb 47704 | The alternate function val... |
| afv2orxorb 47705 | If a set is in the range o... |
| dmafv2rnb 47706 | The alternate function val... |
| fundmafv2rnb 47707 | The alternate function val... |
| afv2elrn 47708 | An alternate function valu... |
| afv20defat 47709 | If the alternate function ... |
| fnafv2elrn 47710 | An alternate function valu... |
| fafv2elcdm 47711 | An alternate function valu... |
| fafv2elrnb 47712 | An alternate function valu... |
| fcdmvafv2v 47713 | If the codomain of a funct... |
| tz6.12-2-afv2 47714 | Function value when ` F ` ... |
| afv2eu 47715 | The value of a function at... |
| afv2res 47716 | The value of a restricted ... |
| tz6.12-afv2 47717 | Function value (Theorem 6.... |
| tz6.12-1-afv2 47718 | Function value (Theorem 6.... |
| tz6.12c-afv2 47719 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 47720 | Corollary of Theorem 6.12(... |
| funressnbrafv2 47721 | The second argument of a b... |
| dfatbrafv2b 47722 | Equivalence of function va... |
| dfatopafv2b 47723 | Equivalence of function va... |
| funbrafv2 47724 | The second argument of a b... |
| fnbrafv2b 47725 | Equivalence of function va... |
| fnopafv2b 47726 | Equivalence of function va... |
| funbrafv22b 47727 | Equivalence of function va... |
| funopafv2b 47728 | Equivalence of function va... |
| dfatsnafv2 47729 | Singleton of function valu... |
| dfafv23 47730 | A definition of function v... |
| dfatdmfcoafv2 47731 | Domain of a function compo... |
| dfatcolem 47732 | Lemma for ~ dfatco . (Con... |
| dfatco 47733 | The predicate "defined at"... |
| afv2co2 47734 | Value of a function compos... |
| rlimdmafv2 47735 | Two ways to express that a... |
| dfafv22 47736 | Alternate definition of ` ... |
| afv2ndeffv0 47737 | If the alternate function ... |
| dfatafv2eqfv 47738 | If a function is defined a... |
| afv2rnfveq 47739 | If the alternate function ... |
| afv20fv0 47740 | If the alternate function ... |
| afv2fvn0fveq 47741 | If the function's value at... |
| afv2fv0 47742 | If the function's value at... |
| afv2fv0b 47743 | The function's value at an... |
| afv2fv0xorb 47744 | If a set is in the range o... |
| an4com24 47745 | Rearrangement of 4 conjunc... |
| 3an4ancom24 47746 | Commutative law for a conj... |
| 4an21 47747 | Rearrangement of 4 conjunc... |
| dfnelbr2 47750 | Alternate definition of th... |
| nelbr 47751 | The binary relation of a s... |
| nelbrim 47752 | If a set is related to ano... |
| nelbrnel 47753 | A set is related to anothe... |
| nelbrnelim 47754 | If a set is related to ano... |
| ralralimp 47755 | Selecting one of two alter... |
| otiunsndisjX 47756 | The union of singletons co... |
| fvifeq 47757 | Equality of function value... |
| rnfdmpr 47758 | The range of a one-to-one ... |
| imarnf1pr 47759 | The image of the range of ... |
| funop1 47760 | A function is an ordered p... |
| fun2dmnopgexmpl 47761 | A function with a domain c... |
| opabresex0d 47762 | A collection of ordered pa... |
| opabbrfex0d 47763 | A collection of ordered pa... |
| opabresexd 47764 | A collection of ordered pa... |
| opabbrfexd 47765 | A collection of ordered pa... |
| f1oresf1orab 47766 | Build a bijection by restr... |
| f1oresf1o 47767 | Build a bijection by restr... |
| f1oresf1o2 47768 | Build a bijection by restr... |
| fvmptrab 47769 | Value of a function mappin... |
| fvmptrabdm 47770 | Value of a function mappin... |
| cnambpcma 47771 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 47772 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 47773 | The sum of two complex num... |
| leaddsuble 47774 | Addition and subtraction o... |
| 2leaddle2 47775 | If two real numbers are le... |
| ltnltne 47776 | Variant of trichotomy law ... |
| p1lep2 47777 | A real number increasd by ... |
| ltsubsubaddltsub 47778 | If the result of subtracti... |
| zm1nn 47779 | An integer minus 1 is posi... |
| readdcnnred 47780 | The sum of a real number a... |
| resubcnnred 47781 | The difference of a real n... |
| recnmulnred 47782 | The product of a real numb... |
| cndivrenred 47783 | The quotient of an imagina... |
| sqrtnegnre 47784 | The square root of a negat... |
| nn0resubcl 47785 | Closure law for subtractio... |
| zgeltp1eq 47786 | If an integer is between a... |
| 1t10e1p1e11 47787 | 11 is 1 times 10 to the po... |
| deccarry 47788 | Add 1 to a 2 digit number ... |
| eluzge0nn0 47789 | If an integer is greater t... |
| nltle2tri 47790 | Negated extended trichotom... |
| ssfz12 47791 | Subset relationship for fi... |
| elfz2z 47792 | Membership of an integer i... |
| 2elfz3nn0 47793 | If there are two elements ... |
| fz0addcom 47794 | The addition of two member... |
| 2elfz2melfz 47795 | If the sum of two integers... |
| fz0addge0 47796 | The sum of two integers in... |
| elfzlble 47797 | Membership of an integer i... |
| elfzelfzlble 47798 | Membership of an element o... |
| elfz2nn 47799 | A member of a finite set o... |
| fzopred 47800 | Join a predecessor to the ... |
| fzopredsuc 47801 | Join a predecessor and a s... |
| 1fzopredsuc 47802 | Join 0 and a successor to ... |
| el1fzopredsuc 47803 | An element of an open inte... |
| subsubelfzo0 47804 | Subtracting a difference f... |
| 2ffzoeq 47805 | Two functions over a half-... |
| elfzo2nn 47806 | A member of a half-open ra... |
| nnmul2 47807 | If one factor of a product... |
| nnmul2b 47808 | A factor of a product of i... |
| 2ltceilhalf 47809 | The ceiling of half of an ... |
| ceilhalfgt1 47810 | The ceiling of half of an ... |
| ceilhalfelfzo1 47811 | A positive integer less th... |
| gpgedgvtx1lem 47812 | Lemma for ~ gpgedgvtx1 . ... |
| 2tceilhalfelfzo1 47813 | Two times a positive integ... |
| ceilbi 47814 | A condition equivalent to ... |
| ceilhalf1 47815 | The ceiling of one half is... |
| rehalfge1 47816 | Half of a real number grea... |
| ceilhalfnn 47817 | The ceiling of half of a p... |
| 1elfzo1ceilhalf1 47818 | 1 is in the half-open inte... |
| nnge2recfl0 47819 | The floor of the reciproca... |
| flmrecm1 47820 | The floor of an integer mi... |
| fldivmod 47821 | Expressing the floor of a ... |
| ceildivmod 47822 | Expressing the ceiling of ... |
| ceil5half3 47823 | The ceiling of half of 5 i... |
| submodaddmod 47824 | Subtraction and addition m... |
| difltmodne 47825 | Two nonnegative integers a... |
| zplusmodne 47826 | A nonnegative integer is n... |
| addmodne 47827 | The sum of a nonnegative i... |
| plusmod5ne 47828 | A nonnegative integer is n... |
| zp1modne 47829 | An integer is not itself p... |
| p1modne 47830 | A nonnegative integer is n... |
| m1modne 47831 | A nonnegative integer is n... |
| minusmod5ne 47832 | A nonnegative integer is n... |
| submodlt 47833 | The difference of an eleme... |
| submodneaddmod 47834 | An integer minus ` B ` is ... |
| m1modnep2mod 47835 | A nonnegative integer minu... |
| minusmodnep2tmod 47836 | A nonnegative integer minu... |
| m1mod0mod1 47837 | An integer decreased by 1 ... |
| elmod2 47838 | An integer modulo 2 is eit... |
| mod0mul 47839 | If an integer is 0 modulo ... |
| modn0mul 47840 | If an integer is not 0 mod... |
| m1modmmod 47841 | An integer decreased by 1 ... |
| difmodm1lt 47842 | The difference between an ... |
| 8mod5e3 47843 | 8 modulo 5 is 3. (Contrib... |
| modmkpkne 47844 | If an integer minus a cons... |
| modmknepk 47845 | A nonnegative integer less... |
| modlt0b 47846 | An integer with an absolut... |
| mod2addne 47847 | The sums of a nonnegative ... |
| modm1nep1 47848 | A nonnegative integer less... |
| modm2nep1 47849 | A nonnegative integer less... |
| modp2nep1 47850 | A nonnegative integer less... |
| modm1nep2 47851 | A nonnegative integer less... |
| modm1nem2 47852 | A nonnegative integer less... |
| modm1p1ne 47853 | If an integer minus one eq... |
| smonoord 47854 | Ordering relation for a st... |
| 2timesltsq 47855 | Two times an integer great... |
| 2timesltsqm1 47856 | Two times an integer great... |
| fsummsndifre 47857 | A finite sum with one of i... |
| fsumsplitsndif 47858 | Separate out a term in a f... |
| fsummmodsndifre 47859 | A finite sum of summands m... |
| fsummmodsnunz 47860 | A finite sum of summands m... |
| nndivides2 47861 | Definition of the divides ... |
| facnn0dvdsfac 47862 | The factorial of a nonnega... |
| muldvdsfacgt 47863 | The product of two differe... |
| muldvdsfacm1 47864 | The product of two differe... |
| setsidel 47865 | The injected slot is an el... |
| setsnidel 47866 | The injected slot is an el... |
| setsv 47867 | The value of the structure... |
| preimafvsnel 47868 | The preimage of a function... |
| preimafvn0 47869 | The preimage of a function... |
| uniimafveqt 47870 | The union of the image of ... |
| uniimaprimaeqfv 47871 | The union of the image of ... |
| setpreimafvex 47872 | The class ` P ` of all pre... |
| elsetpreimafvb 47873 | The characterization of an... |
| elsetpreimafv 47874 | An element of the class ` ... |
| elsetpreimafvssdm 47875 | An element of the class ` ... |
| fvelsetpreimafv 47876 | There is an element in a p... |
| preimafvelsetpreimafv 47877 | The preimage of a function... |
| preimafvsspwdm 47878 | The class ` P ` of all pre... |
| 0nelsetpreimafv 47879 | The empty set is not an el... |
| elsetpreimafvbi 47880 | An element of the preimage... |
| elsetpreimafveqfv 47881 | The elements of the preima... |
| eqfvelsetpreimafv 47882 | If an element of the domai... |
| elsetpreimafvrab 47883 | An element of the preimage... |
| imaelsetpreimafv 47884 | The image of an element of... |
| uniimaelsetpreimafv 47885 | The union of the image of ... |
| elsetpreimafveq 47886 | If two preimages of functi... |
| fundcmpsurinjlem1 47887 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 47888 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 47889 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 47890 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 47891 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 47892 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 47893 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 47894 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 47895 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 47896 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 47897 | Every function ` F : A -->... |
| fundcmpsurinj 47898 | Every function ` F : A -->... |
| fundcmpsurbijinj 47899 | Every function ` F : A -->... |
| fundcmpsurinjimaid 47900 | Every function ` F : A -->... |
| fundcmpsurinjALT 47901 | Alternate proof of ~ fundc... |
| iccpval 47904 | Partition consisting of a ... |
| iccpart 47905 | A special partition. Corr... |
| iccpartimp 47906 | Implications for a class b... |
| iccpartres 47907 | The restriction of a parti... |
| iccpartxr 47908 | If there is a partition, t... |
| iccpartgtprec 47909 | If there is a partition, t... |
| iccpartipre 47910 | If there is a partition, t... |
| iccpartiltu 47911 | If there is a partition, t... |
| iccpartigtl 47912 | If there is a partition, t... |
| iccpartlt 47913 | If there is a partition, t... |
| iccpartltu 47914 | If there is a partition, t... |
| iccpartgtl 47915 | If there is a partition, t... |
| iccpartgt 47916 | If there is a partition, t... |
| iccpartleu 47917 | If there is a partition, t... |
| iccpartgel 47918 | If there is a partition, t... |
| iccpartrn 47919 | If there is a partition, t... |
| iccpartf 47920 | The range of the partition... |
| iccpartel 47921 | If there is a partition, t... |
| iccelpart 47922 | An element of any partitio... |
| iccpartiun 47923 | A half-open interval of ex... |
| icceuelpartlem 47924 | Lemma for ~ icceuelpart . ... |
| icceuelpart 47925 | An element of a partitione... |
| iccpartdisj 47926 | The segments of a partitio... |
| iccpartnel 47927 | A point of a partition is ... |
| fargshiftfv 47928 | If a class is a function, ... |
| fargshiftf 47929 | If a class is a function, ... |
| fargshiftf1 47930 | If a function is 1-1, then... |
| fargshiftfo 47931 | If a function is onto, the... |
| fargshiftfva 47932 | The values of a shifted fu... |
| lswn0 47933 | The last symbol of a nonem... |
| nfich1 47936 | The first interchangeable ... |
| nfich2 47937 | The second interchangeable... |
| ichv 47938 | Setvar variables are inter... |
| ichf 47939 | Setvar variables are inter... |
| ichid 47940 | A setvar variable is alway... |
| icht 47941 | A theorem is interchangeab... |
| ichbidv 47942 | Formula building rule for ... |
| ichcircshi 47943 | The setvar variables are i... |
| ichan 47944 | If two setvar variables ar... |
| ichn 47945 | Negation does not affect i... |
| ichim 47946 | Formula building rule for ... |
| dfich2 47947 | Alternate definition of th... |
| ichcom 47948 | The interchangeability of ... |
| ichbi12i 47949 | Equivalence for interchang... |
| icheqid 47950 | In an equality for the sam... |
| icheq 47951 | In an equality of setvar v... |
| ichnfimlem 47952 | Lemma for ~ ichnfim : A s... |
| ichnfim 47953 | If in an interchangeabilit... |
| ichnfb 47954 | If ` x ` and ` y ` are int... |
| ichal 47955 | Move a universal quantifie... |
| ich2al 47956 | Two setvar variables are a... |
| ich2ex 47957 | Two setvar variables are a... |
| ichexmpl1 47958 | Example for interchangeabl... |
| ichexmpl2 47959 | Example for interchangeabl... |
| ich2exprop 47960 | If the setvar variables ar... |
| ichnreuop 47961 | If the setvar variables ar... |
| ichreuopeq 47962 | If the setvar variables ar... |
| sprid 47963 | Two identical representati... |
| elsprel 47964 | An unordered pair is an el... |
| spr0nelg 47965 | The empty set is not an el... |
| sprval 47968 | The set of all unordered p... |
| sprvalpw 47969 | The set of all unordered p... |
| sprssspr 47970 | The set of all unordered p... |
| spr0el 47971 | The empty set is not an un... |
| sprvalpwn0 47972 | The set of all unordered p... |
| sprel 47973 | An element of the set of a... |
| prssspr 47974 | An element of a subset of ... |
| prelspr 47975 | An unordered pair of eleme... |
| prsprel 47976 | The elements of a pair fro... |
| prsssprel 47977 | The elements of a pair fro... |
| sprvalpwle2 47978 | The set of all unordered p... |
| sprsymrelfvlem 47979 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 47980 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 47981 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 47982 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 47983 | The value of the function ... |
| sprsymrelf 47984 | The mapping ` F ` is a fun... |
| sprsymrelf1 47985 | The mapping ` F ` is a one... |
| sprsymrelfo 47986 | The mapping ` F ` is a fun... |
| sprsymrelf1o 47987 | The mapping ` F ` is a bij... |
| sprbisymrel 47988 | There is a bijection betwe... |
| sprsymrelen 47989 | The class ` P ` of subsets... |
| prpair 47990 | Characterization of a prop... |
| prproropf1olem0 47991 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 47992 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 47993 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 47994 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 47995 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 47996 | There is a bijection betwe... |
| prproropen 47997 | The set of proper pairs an... |
| prproropreud 47998 | There is exactly one order... |
| pairreueq 47999 | Two equivalent representat... |
| paireqne 48000 | Two sets are not equal iff... |
| prprval 48003 | The set of all proper unor... |
| prprvalpw 48004 | The set of all proper unor... |
| prprelb 48005 | An element of the set of a... |
| prprelprb 48006 | A set is an element of the... |
| prprspr2 48007 | The set of all proper unor... |
| prprsprreu 48008 | There is a unique proper u... |
| prprreueq 48009 | There is a unique proper u... |
| sbcpr 48010 | The proper substitution of... |
| reupr 48011 | There is a unique unordere... |
| reuprpr 48012 | There is a unique proper u... |
| poprelb 48013 | Equality for unordered pai... |
| 2exopprim 48014 | The existence of an ordere... |
| reuopreuprim 48015 | There is a unique unordere... |
| nprmmul1 48016 | Special factorization of a... |
| nprmmul2 48017 | Special factorization of a... |
| nprmmul3 48018 | Special factorization of a... |
| fmtno 48021 | The ` N ` th Fermat number... |
| fmtnoge3 48022 | Each Fermat number is grea... |
| fmtnonn 48023 | Each Fermat number is a po... |
| fmtnom1nn 48024 | A Fermat number minus one ... |
| fmtnoodd 48025 | Each Fermat number is odd.... |
| fmtnorn 48026 | A Fermat number is a funct... |
| fmtnof1 48027 | The enumeration of the Fer... |
| fmtnoinf 48028 | The set of Fermat numbers ... |
| fmtnorec1 48029 | The first recurrence relat... |
| sqrtpwpw2p 48030 | The floor of the square ro... |
| fmtnosqrt 48031 | The floor of the square ro... |
| fmtno0 48032 | The ` 0 ` th Fermat number... |
| fmtno1 48033 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 48034 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 48035 | The second recurrence rela... |
| fmtnodvds 48036 | Any Fermat number divides ... |
| goldbachthlem1 48037 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 48038 | Lemma 2 for ~ goldbachth .... |
| goldbachth 48039 | Goldbach's theorem: Two d... |
| fmtnorec3 48040 | The third recurrence relat... |
| fmtnorec4 48041 | The fourth recurrence rela... |
| fmtno2 48042 | The ` 2 ` nd Fermat number... |
| fmtno3 48043 | The ` 3 ` rd Fermat number... |
| fmtno4 48044 | The ` 4 ` th Fermat number... |
| fmtno5lem1 48045 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 48046 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 48047 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 48048 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 48049 | The ` 5 ` th Fermat number... |
| fmtno0prm 48050 | The ` 0 ` th Fermat number... |
| fmtno1prm 48051 | The ` 1 ` st Fermat number... |
| fmtno2prm 48052 | The ` 2 ` nd Fermat number... |
| 257prm 48053 | 257 is a prime number (the... |
| fmtno3prm 48054 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 48055 | Any power of two is coprim... |
| fmtnoprmfac1lem 48056 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 48057 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 48058 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 48059 | Divisor of Fermat number (... |
| fmtnofac2lem 48060 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 48061 | Divisor of Fermat number (... |
| fmtnofac1 48062 | Divisor of Fermat number (... |
| fmtno4sqrt 48063 | The floor of the square ro... |
| fmtno4prmfac 48064 | If P was a (prime) factor ... |
| fmtno4prmfac193 48065 | If P was a (prime) factor ... |
| fmtno4nprmfac193 48066 | 193 is not a (prime) facto... |
| fmtno4prm 48067 | The ` 4 `-th Fermat number... |
| 65537prm 48068 | 65537 is a prime number (t... |
| fmtnofz04prm 48069 | The first five Fermat numb... |
| fmtnole4prm 48070 | The first five Fermat numb... |
| fmtno5faclem1 48071 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 48072 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 48073 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 48074 | The factorization of the `... |
| fmtno5nprm 48075 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 48076 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 48077 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 48078 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 48079 | The mapping of a Fermat nu... |
| prminf2 48080 | The set of prime numbers i... |
| 2pwp1prm 48081 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 48082 | Every prime number of the ... |
| m2prm 48083 | The second Mersenne number... |
| m3prm 48084 | The third Mersenne number ... |
| flsqrt 48085 | A condition equivalent to ... |
| flsqrt5 48086 | The floor of the square ro... |
| 3ndvds4 48087 | 3 does not divide 4. (Con... |
| 139prmALT 48088 | 139 is a prime number. In... |
| 31prm 48089 | 31 is a prime number. In ... |
| m5prm 48090 | The fifth Mersenne number ... |
| 127prm 48091 | 127 is a prime number. (C... |
| m7prm 48092 | The seventh Mersenne numbe... |
| m11nprm 48093 | The eleventh Mersenne numb... |
| mod42tp1mod8 48094 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 48095 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 48096 | If ` P ` is a Sophie Germa... |
| lighneallem1 48097 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 48098 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 48099 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 48100 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 48101 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 48102 | Lemma 3 for ~ lighneal . ... |
| lighneal 48103 | If a power of a prime ` P ... |
| modexp2m1d 48104 | The square of an integer w... |
| proththdlem 48105 | Lemma for ~ proththd . (C... |
| proththd 48106 | Proth's theorem (1878). I... |
| 5tcu2e40 48107 | 5 times the cube of 2 is 4... |
| 3exp4mod41 48108 | 3 to the fourth power is -... |
| 41prothprmlem1 48109 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 48110 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 48111 | 41 is a _Proth prime_. (C... |
| nprmdvdsfacm1lem1 48112 | Lemma 1 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem2 48113 | Lemma 2 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem3 48114 | Lemma 3 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem4 48115 | Lemma 4 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1 48116 | A non-prime integer greate... |
| ppivalnnprm 48117 | Value of a term of the pri... |
| ppivalnnnprmge6 48118 | Value of a term of the pri... |
| ppivalnn4 48119 | Value of the term of the p... |
| ppivalnnnprm 48120 | Value of a term of the pri... |
| indprm 48121 | An indicator function for ... |
| indprmfz 48122 | An indicator function for ... |
| ppi1sum 48123 | Value of the prime-countin... |
| ppivalnn 48124 | Value of the prime-countin... |
| quad1 48125 | A condition for a quadrati... |
| requad01 48126 | A condition for a quadrati... |
| requad1 48127 | A condition for a quadrati... |
| requad2 48128 | A condition for a quadrati... |
| iseven 48133 | The predicate "is an even ... |
| isodd 48134 | The predicate "is an odd n... |
| evenz 48135 | An even number is an integ... |
| oddz 48136 | An odd number is an intege... |
| evendiv2z 48137 | The result of dividing an ... |
| oddp1div2z 48138 | The result of dividing an ... |
| oddm1div2z 48139 | The result of dividing an ... |
| isodd2 48140 | The predicate "is an odd n... |
| dfodd2 48141 | Alternate definition for o... |
| dfodd6 48142 | Alternate definition for o... |
| dfeven4 48143 | Alternate definition for e... |
| evenm1odd 48144 | The predecessor of an even... |
| evenp1odd 48145 | The successor of an even n... |
| oddp1eveni 48146 | The successor of an odd nu... |
| oddm1eveni 48147 | The predecessor of an odd ... |
| evennodd 48148 | An even number is not an o... |
| oddneven 48149 | An odd number is not an ev... |
| enege 48150 | The negative of an even nu... |
| onego 48151 | The negative of an odd num... |
| m1expevenALTV 48152 | Exponentiation of -1 by an... |
| m1expoddALTV 48153 | Exponentiation of -1 by an... |
| dfeven2 48154 | Alternate definition for e... |
| dfodd3 48155 | Alternate definition for o... |
| iseven2 48156 | The predicate "is an even ... |
| isodd3 48157 | The predicate "is an odd n... |
| 2dvdseven 48158 | 2 divides an even number. ... |
| m2even 48159 | A multiple of 2 is an even... |
| 2ndvdsodd 48160 | 2 does not divide an odd n... |
| 2dvdsoddp1 48161 | 2 divides an odd number in... |
| 2dvdsoddm1 48162 | 2 divides an odd number de... |
| dfeven3 48163 | Alternate definition for e... |
| dfodd4 48164 | Alternate definition for o... |
| dfodd5 48165 | Alternate definition for o... |
| zefldiv2ALTV 48166 | The floor of an even numbe... |
| zofldiv2ALTV 48167 | The floor of an odd number... |
| oddflALTV 48168 | Odd number representation ... |
| iseven5 48169 | The predicate "is an even ... |
| isodd7 48170 | The predicate "is an odd n... |
| dfeven5 48171 | Alternate definition for e... |
| dfodd7 48172 | Alternate definition for o... |
| gcd2odd1 48173 | The greatest common diviso... |
| zneoALTV 48174 | No even integer equals an ... |
| zeoALTV 48175 | An integer is even or odd.... |
| zeo2ALTV 48176 | An integer is even or odd ... |
| nneoALTV 48177 | A positive integer is even... |
| nneoiALTV 48178 | A positive integer is even... |
| odd2np1ALTV 48179 | An integer is odd iff it i... |
| oddm1evenALTV 48180 | An integer is odd iff its ... |
| oddp1evenALTV 48181 | An integer is odd iff its ... |
| oexpnegALTV 48182 | The exponential of the neg... |
| oexpnegnz 48183 | The exponential of the neg... |
| bits0ALTV 48184 | Value of the zeroth bit. ... |
| bits0eALTV 48185 | The zeroth bit of an even ... |
| bits0oALTV 48186 | The zeroth bit of an odd n... |
| divgcdoddALTV 48187 | Either ` A / ( A gcd B ) `... |
| opoeALTV 48188 | The sum of two odds is eve... |
| opeoALTV 48189 | The sum of an odd and an e... |
| omoeALTV 48190 | The difference of two odds... |
| omeoALTV 48191 | The difference of an odd a... |
| oddprmALTV 48192 | A prime not equal to ` 2 `... |
| 0evenALTV 48193 | 0 is an even number. (Con... |
| 0noddALTV 48194 | 0 is not an odd number. (... |
| 1oddALTV 48195 | 1 is an odd number. (Cont... |
| 1nevenALTV 48196 | 1 is not an even number. ... |
| 2evenALTV 48197 | 2 is an even number. (Con... |
| 2noddALTV 48198 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 48199 | An odd nonnegative integer... |
| nnoALTV 48200 | An alternate characterizat... |
| nn0oALTV 48201 | An alternate characterizat... |
| nn0e 48202 | An alternate characterizat... |
| nneven 48203 | An alternate characterizat... |
| nn0onn0exALTV 48204 | For each odd nonnegative i... |
| nn0enn0exALTV 48205 | For each even nonnegative ... |
| nnennexALTV 48206 | For each even positive int... |
| nnpw2evenALTV 48207 | 2 to the power of a positi... |
| epoo 48208 | The sum of an even and an ... |
| emoo 48209 | The difference of an even ... |
| epee 48210 | The sum of two even number... |
| emee 48211 | The difference of two even... |
| evensumeven 48212 | If a summand is even, the ... |
| 3odd 48213 | 3 is an odd number. (Cont... |
| 4even 48214 | 4 is an even number. (Con... |
| 5odd 48215 | 5 is an odd number. (Cont... |
| 6even 48216 | 6 is an even number. (Con... |
| 7odd 48217 | 7 is an odd number. (Cont... |
| 8even 48218 | 8 is an even number. (Con... |
| evenprm2 48219 | A prime number is even iff... |
| oddprmne2 48220 | Every prime number not bei... |
| oddprmuzge3 48221 | A prime number which is od... |
| evenltle 48222 | If an even number is great... |
| odd2prm2 48223 | If an odd number is the su... |
| even3prm2 48224 | If an even number is the s... |
| mogoldbblem 48225 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 48226 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 48227 | Lemma for ~ perfectALTV . ... |
| perfectALTV 48228 | The Euclid-Euler theorem, ... |
| fppr 48231 | The set of Fermat pseudopr... |
| fpprmod 48232 | The set of Fermat pseudopr... |
| fpprel 48233 | A Fermat pseudoprime to th... |
| fpprbasnn 48234 | The base of a Fermat pseud... |
| fpprnn 48235 | A Fermat pseudoprime to th... |
| fppr2odd 48236 | A Fermat pseudoprime to th... |
| 11t31e341 48237 | 341 is the product of 11 a... |
| 2exp340mod341 48238 | Eight to the eighth power ... |
| 341fppr2 48239 | 341 is the (smallest) _Pou... |
| 4fppr1 48240 | 4 is the (smallest) Fermat... |
| 8exp8mod9 48241 | Eight to the eighth power ... |
| 9fppr8 48242 | 9 is the (smallest) Fermat... |
| dfwppr 48243 | Alternate definition of a ... |
| fpprwppr 48244 | A Fermat pseudoprime to th... |
| fpprwpprb 48245 | An integer ` X ` which is ... |
| fpprel2 48246 | An alternate definition fo... |
| nfermltl8rev 48247 | Fermat's little theorem wi... |
| nfermltl2rev 48248 | Fermat's little theorem wi... |
| nfermltlrev 48249 | Fermat's little theorem re... |
| isgbe 48256 | The predicate "is an even ... |
| isgbow 48257 | The predicate "is a weak o... |
| isgbo 48258 | The predicate "is an odd G... |
| gbeeven 48259 | An even Goldbach number is... |
| gbowodd 48260 | A weak odd Goldbach number... |
| gbogbow 48261 | A (strong) odd Goldbach nu... |
| gboodd 48262 | An odd Goldbach number is ... |
| gbepos 48263 | Any even Goldbach number i... |
| gbowpos 48264 | Any weak odd Goldbach numb... |
| gbopos 48265 | Any odd Goldbach number is... |
| gbegt5 48266 | Any even Goldbach number i... |
| gbowgt5 48267 | Any weak odd Goldbach numb... |
| gbowge7 48268 | Any weak odd Goldbach numb... |
| gboge9 48269 | Any odd Goldbach number is... |
| gbege6 48270 | Any even Goldbach number i... |
| gbpart6 48271 | The Goldbach partition of ... |
| gbpart7 48272 | The (weak) Goldbach partit... |
| gbpart8 48273 | The Goldbach partition of ... |
| gbpart9 48274 | The (strong) Goldbach part... |
| gbpart11 48275 | The (strong) Goldbach part... |
| 6gbe 48276 | 6 is an even Goldbach numb... |
| 7gbow 48277 | 7 is a weak odd Goldbach n... |
| 8gbe 48278 | 8 is an even Goldbach numb... |
| 9gbo 48279 | 9 is an odd Goldbach numbe... |
| 11gbo 48280 | 11 is an odd Goldbach numb... |
| stgoldbwt 48281 | If the strong ternary Gold... |
| sbgoldbwt 48282 | If the strong binary Goldb... |
| sbgoldbst 48283 | If the strong binary Goldb... |
| sbgoldbaltlem1 48284 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 48285 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 48286 | An alternate (related to t... |
| sbgoldbb 48287 | If the strong binary Goldb... |
| sgoldbeven3prm 48288 | If the binary Goldbach con... |
| sbgoldbm 48289 | If the strong binary Goldb... |
| mogoldbb 48290 | If the modern version of t... |
| sbgoldbmb 48291 | The strong binary Goldbach... |
| sbgoldbo 48292 | If the strong binary Goldb... |
| nnsum3primes4 48293 | 4 is the sum of at most 3 ... |
| nnsum4primes4 48294 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 48295 | Every prime is "the sum of... |
| nnsum4primesprm 48296 | Every prime is "the sum of... |
| nnsum3primesgbe 48297 | Any even Goldbach number i... |
| nnsum4primesgbe 48298 | Any even Goldbach number i... |
| nnsum3primesle9 48299 | Every integer greater than... |
| nnsum4primesle9 48300 | Every integer greater than... |
| nnsum4primesodd 48301 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 48302 | If the (strong) ternary Go... |
| evengpop3 48303 | If the (weak) ternary Gold... |
| evengpoap3 48304 | If the (strong) ternary Go... |
| nnsum4primeseven 48305 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 48306 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 48307 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 48308 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 48309 | If the binary Goldbach con... |
| bgoldbtbndlem1 48310 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 48311 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 48312 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 48313 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 48314 | If the binary Goldbach con... |
| tgoldbachgtALTV 48317 | Variant of Thierry Arnoux'... |
| bgoldbachlt 48318 | The binary Goldbach conjec... |
| tgblthelfgott 48320 | The ternary Goldbach conje... |
| tgoldbachlt 48321 | The ternary Goldbach conje... |
| tgoldbach 48322 | The ternary Goldbach conje... |
| clnbgrprc0 48325 | The closed neighborhood is... |
| clnbgrcl 48326 | If a class ` X ` has at le... |
| clnbgrval 48327 | The closed neighborhood of... |
| dfclnbgr2 48328 | Alternate definition of th... |
| dfclnbgr4 48329 | Alternate definition of th... |
| elclnbgrelnbgr 48330 | An element of the closed n... |
| dfclnbgr3 48331 | Alternate definition of th... |
| clnbgrnvtx0 48332 | If a class ` X ` is not a ... |
| clnbgrel 48333 | Characterization of a memb... |
| clnbgrvtxel 48334 | Every vertex ` K ` is a me... |
| clnbgrisvtx 48335 | Every member ` N ` of the ... |
| clnbgrssvtx 48336 | The closed neighborhood of... |
| clnbgrn0 48337 | The closed neighborhood of... |
| clnbupgr 48338 | The closed neighborhood of... |
| clnbupgrel 48339 | A member of the closed nei... |
| clnbupgreli 48340 | A member of the closed nei... |
| clnbgr0vtx 48341 | In a null graph (with no v... |
| clnbgr0edg 48342 | In an empty graph (with no... |
| clnbgrsym 48343 | In a graph, the closed nei... |
| predgclnbgrel 48344 | If a (not necessarily prop... |
| clnbgredg 48345 | A vertex connected by an e... |
| clnbgrssedg 48346 | The vertices connected by ... |
| edgusgrclnbfin 48347 | The size of the closed nei... |
| clnbusgrfi 48348 | The closed neighborhood of... |
| clnbfiusgrfi 48349 | The closed neighborhood of... |
| clnbgrlevtx 48350 | The size of the closed nei... |
| dfsclnbgr2 48351 | Alternate definition of th... |
| sclnbgrel 48352 | Characterization of a memb... |
| sclnbgrelself 48353 | A vertex ` N ` is a member... |
| sclnbgrisvtx 48354 | Every member ` X ` of the ... |
| dfclnbgr5 48355 | Alternate definition of th... |
| dfnbgr5 48356 | Alternate definition of th... |
| dfnbgrss 48357 | Subset chain for different... |
| dfvopnbgr2 48358 | Alternate definition of th... |
| vopnbgrel 48359 | Characterization of a memb... |
| vopnbgrelself 48360 | A vertex ` N ` is a member... |
| dfclnbgr6 48361 | Alternate definition of th... |
| dfnbgr6 48362 | Alternate definition of th... |
| dfsclnbgr6 48363 | Alternate definition of a ... |
| dfnbgrss2 48364 | Subset chain for different... |
| isisubgr 48367 | The subgraph induced by a ... |
| isubgriedg 48368 | The edges of an induced su... |
| isubgrvtxuhgr 48369 | The subgraph induced by th... |
| isubgredgss 48370 | The edges of an induced su... |
| isubgredg 48371 | An edge of an induced subg... |
| isubgrvtx 48372 | The vertices of an induced... |
| isubgruhgr 48373 | An induced subgraph of a h... |
| isubgrsubgr 48374 | An induced subgraph of a h... |
| isubgrupgr 48375 | An induced subgraph of a p... |
| isubgrumgr 48376 | An induced subgraph of a m... |
| isubgrusgr 48377 | An induced subgraph of a s... |
| isubgr0uhgr 48378 | The subgraph induced by an... |
| grimfn 48384 | The graph isomorphism func... |
| grimdmrel 48385 | The domain of the graph is... |
| isgrim 48387 | An isomorphism of graphs i... |
| grimprop 48388 | Properties of an isomorphi... |
| grimf1o 48389 | An isomorphism of graphs i... |
| grimidvtxedg 48390 | The identity relation rest... |
| grimid 48391 | The identity relation rest... |
| grimuhgr 48392 | If there is a graph isomor... |
| grimcnv 48393 | The converse of a graph is... |
| grimco 48394 | The composition of graph i... |
| uhgrimedgi 48395 | An isomorphism between gra... |
| uhgrimedg 48396 | An isomorphism between gra... |
| uhgrimprop 48397 | An isomorphism between hyp... |
| isuspgrim0lem 48398 | An isomorphism of simple p... |
| isuspgrim0 48399 | An isomorphism of simple p... |
| isuspgrimlem 48400 | Lemma for ~ isuspgrim . (... |
| isuspgrim 48401 | A class is an isomorphism ... |
| upgrimwlklem1 48402 | Lemma 1 for ~ upgrimwlk an... |
| upgrimwlklem2 48403 | Lemma 2 for ~ upgrimwlk . ... |
| upgrimwlklem3 48404 | Lemma 3 for ~ upgrimwlk . ... |
| upgrimwlklem4 48405 | Lemma 4 for ~ upgrimwlk . ... |
| upgrimwlklem5 48406 | Lemma 5 for ~ upgrimwlk . ... |
| upgrimwlk 48407 | Graph isomorphisms between... |
| upgrimwlklen 48408 | Graph isomorphisms between... |
| upgrimtrlslem1 48409 | Lemma 1 for ~ upgrimtrls .... |
| upgrimtrlslem2 48410 | Lemma 2 for ~ upgrimtrls .... |
| upgrimtrls 48411 | Graph isomorphisms between... |
| upgrimpthslem1 48412 | Lemma 1 for ~ upgrimpths .... |
| upgrimpthslem2 48413 | Lemma 2 for ~ upgrimpths .... |
| upgrimpths 48414 | Graph isomorphisms between... |
| upgrimspths 48415 | Graph isomorphisms between... |
| upgrimcycls 48416 | Graph isomorphisms between... |
| brgric 48417 | The relation "is isomorphi... |
| brgrici 48418 | Prove that two graphs are ... |
| gricrcl 48419 | Reverse closure of the "is... |
| dfgric2 48420 | Alternate, explicit defini... |
| gricbri 48421 | Implications of two graphs... |
| gricushgr 48422 | The "is isomorphic to" rel... |
| gricuspgr 48423 | The "is isomorphic to" rel... |
| gricrel 48424 | The "is isomorphic to" rel... |
| gricref 48425 | Graph isomorphism is refle... |
| gricsym 48426 | Graph isomorphism is symme... |
| gricsymb 48427 | Graph isomorphism is symme... |
| grictr 48428 | Graph isomorphism is trans... |
| gricer 48429 | Isomorphism is an equivale... |
| gricen 48430 | Isomorphic graphs have equ... |
| opstrgric 48431 | A graph represented as an ... |
| ushggricedg 48432 | A simple hypergraph (with ... |
| cycldlenngric 48433 | Two simple pseudographs ar... |
| isubgrgrim 48434 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 48435 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 48436 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 48437 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 48438 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 48439 | Graph isomorphisms between... |
| grimedg 48440 | For two isomorphic graphs,... |
| grimedgi 48441 | Graph isomorphisms map edg... |
| grtriproplem 48444 | Lemma for ~ grtriprop . (... |
| grtri 48445 | The triangles in a graph. ... |
| grtriprop 48446 | The properties of a triang... |
| grtrif1o 48447 | Any bijection onto a trian... |
| isgrtri 48448 | A triangle in a graph. (C... |
| grtrissvtx 48449 | A triangle is a subset of ... |
| grtriclwlk3 48450 | A triangle induces a close... |
| cycl3grtrilem 48451 | Lemma for ~ cycl3grtri . ... |
| cycl3grtri 48452 | The vertices of a cycle of... |
| grtrimap 48453 | Conditions for mapping tri... |
| grimgrtri 48454 | Graph isomorphisms map tri... |
| usgrgrtrirex 48455 | Conditions for a simple gr... |
| stgrfv 48458 | The star graph S_N. (Contr... |
| stgrvtx 48459 | The vertices of the star g... |
| stgriedg 48460 | The indexed edges of the s... |
| stgredg 48461 | The edges of the star grap... |
| stgredgel 48462 | An edge of the star graph ... |
| stgredgiun 48463 | The edges of the star grap... |
| stgrusgra 48464 | The star graph S_N is a si... |
| stgr0 48465 | The star graph S_0 consist... |
| stgr1 48466 | The star graph S_1 consist... |
| stgrvtx0 48467 | The center ("internal node... |
| stgrorder 48468 | The order of a star graph ... |
| stgrnbgr0 48469 | All vertices of a star gra... |
| stgrclnbgr0 48470 | All vertices of a star gra... |
| isubgr3stgrlem1 48471 | Lemma 1 for ~ isubgr3stgr ... |
| isubgr3stgrlem2 48472 | Lemma 2 for ~ isubgr3stgr ... |
| isubgr3stgrlem3 48473 | Lemma 3 for ~ isubgr3stgr ... |
| isubgr3stgrlem4 48474 | Lemma 4 for ~ isubgr3stgr ... |
| isubgr3stgrlem5 48475 | Lemma 5 for ~ isubgr3stgr ... |
| isubgr3stgrlem6 48476 | Lemma 6 for ~ isubgr3stgr ... |
| isubgr3stgrlem7 48477 | Lemma 7 for ~ isubgr3stgr ... |
| isubgr3stgrlem8 48478 | Lemma 8 for ~ isubgr3stgr ... |
| isubgr3stgrlem9 48479 | Lemma 9 for ~ isubgr3stgr ... |
| isubgr3stgr 48480 | If a vertex of a simple gr... |
| grlimfn 48484 | The graph local isomorphis... |
| grlimdmrel 48485 | The domain of the graph lo... |
| isgrlim 48487 | A local isomorphism of gra... |
| isgrlim2 48488 | A local isomorphism of gra... |
| grlimprop 48489 | Properties of a local isom... |
| grlimf1o 48490 | A local isomorphism of gra... |
| grlimprop2 48491 | Properties of a local isom... |
| uhgrimgrlim 48492 | An isomorphism of hypergra... |
| uspgrlimlem1 48493 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 48494 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 48495 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 48496 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 48497 | A local isomorphism of sim... |
| usgrlimprop 48498 | Properties of a local isom... |
| clnbgrvtxedg 48499 | An edge ` E ` containing a... |
| grlimedgclnbgr 48500 | For two locally isomorphic... |
| grlimprclnbgr 48501 | For two locally isomorphic... |
| grlimprclnbgredg 48502 | For two locally isomorphic... |
| grlimpredg 48503 | For two locally isomorphic... |
| grlimprclnbgrvtx 48504 | For two locally isomorphic... |
| grlimgredgex 48505 | Local isomorphisms between... |
| grlimgrtrilem1 48506 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 48507 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 48508 | If one of two locally isom... |
| brgrlic 48509 | The relation "is locally i... |
| brgrilci 48510 | Prove that two graphs are ... |
| grlicrel 48511 | The "is locally isomorphic... |
| grlicrcl 48512 | Reverse closure of the "is... |
| dfgrlic2 48513 | Alternate, explicit defini... |
| grilcbri 48514 | Implications of two graphs... |
| dfgrlic3 48515 | Alternate, explicit defini... |
| grilcbri2 48516 | Implications of two graphs... |
| grlicref 48517 | Graph local isomorphism is... |
| grlicsym 48518 | Graph local isomorphism is... |
| grlicsymb 48519 | Graph local isomorphism is... |
| grlictr 48520 | Graph local isomorphism is... |
| grlicer 48521 | Local isomorphism is an eq... |
| grlicen 48522 | Locally isomorphic graphs ... |
| gricgrlic 48523 | Isomorphic hypergraphs are... |
| clnbgr3stgrgrlim 48524 | If all (closed) neighborho... |
| clnbgr3stgrgrlic 48525 | If all (closed) neighborho... |
| usgrexmpl1lem 48526 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 48527 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 48528 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 48529 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 48530 | ` G ` contains a triangle ... |
| usgrexmpl2lem 48531 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 48532 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 48533 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 48534 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 48535 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 48536 | The neighborhood of the fi... |
| usgrexmpl2nb1 48537 | The neighborhood of the se... |
| usgrexmpl2nb2 48538 | The neighborhood of the th... |
| usgrexmpl2nb3 48539 | The neighborhood of the fo... |
| usgrexmpl2nb4 48540 | The neighborhood of the fi... |
| usgrexmpl2nb5 48541 | The neighborhood of the si... |
| usgrexmpl2trifr 48542 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 48543 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 48544 | The graphs ` H ` and ` G `... |
| gpgov 48547 | The generalized Petersen g... |
| gpgvtx 48548 | The vertices of the genera... |
| gpgiedg 48549 | The indexed edges of the g... |
| gpgedg 48550 | The edges of the generaliz... |
| gpgiedgdmellem 48551 | Lemma for ~ gpgiedgdmel an... |
| gpgvtxel 48552 | A vertex in a generalized ... |
| gpgvtxel2 48553 | The second component of a ... |
| gpgiedgdmel 48554 | An index of edges of the g... |
| gpgedgel 48555 | An edge in a generalized P... |
| gpgprismgriedgdmel 48556 | An index of edges of the g... |
| gpgprismgriedgdmss 48557 | A subset of the index of e... |
| gpgvtx0 48558 | The outside vertices in a ... |
| gpgvtx1 48559 | The inside vertices in a g... |
| opgpgvtx 48560 | A vertex in a generalized ... |
| gpgusgralem 48561 | Lemma for ~ gpgusgra . (C... |
| gpgusgra 48562 | The generalized Petersen g... |
| gpgprismgrusgra 48563 | The generalized Petersen g... |
| gpgorder 48564 | The order of the generaliz... |
| gpg5order 48565 | The order of a generalized... |
| gpgedgvtx0 48566 | The edges starting at an o... |
| gpgedgvtx1 48567 | The edges starting at an i... |
| gpgvtxedg0 48568 | The edges starting at an o... |
| gpgvtxedg1 48569 | The edges starting at an i... |
| gpgedgiov 48570 | The edges of the generaliz... |
| gpgedg2ov 48571 | The edges of the generaliz... |
| gpgedg2iv 48572 | The edges of the generaliz... |
| gpg5nbgrvtx03starlem1 48573 | Lemma 1 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem2 48574 | Lemma 2 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem3 48575 | Lemma 3 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx13starlem1 48576 | Lemma 1 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem2 48577 | Lemma 2 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem3 48578 | Lemma 3 for ~ gpg5nbgr3sta... |
| gpgnbgrvtx0 48579 | The (open) neighborhood of... |
| gpgnbgrvtx1 48580 | The (open) neighborhood of... |
| gpg3nbgrvtx0 48581 | In a generalized Petersen ... |
| gpg3nbgrvtx0ALT 48582 | In a generalized Petersen ... |
| gpg3nbgrvtx1 48583 | In a generalized Petersen ... |
| gpgcubic 48584 | Every generalized Petersen... |
| gpg5nbgrvtx03star 48585 | In a generalized Petersen ... |
| gpg5nbgr3star 48586 | In a generalized Petersen ... |
| gpgvtxdg3 48587 | Every vertex in a generali... |
| gpg3kgrtriexlem1 48588 | Lemma 1 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem2 48589 | Lemma 2 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem3 48590 | Lemma 3 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem4 48591 | Lemma 4 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem5 48592 | Lemma 5 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem6 48593 | Lemma 6 for ~ gpg3kgrtriex... |
| gpg3kgrtriex 48594 | All generalized Petersen g... |
| gpg5gricstgr3 48595 | Each closed neighborhood i... |
| pglem 48596 | Lemma for theorems about P... |
| pgjsgr 48597 | A Petersen graph is a simp... |
| gpg5grlim 48598 | A local isomorphism betwee... |
| gpg5grlic 48599 | The two generalized Peters... |
| gpgprismgr4cycllem1 48600 | Lemma 1 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem2 48601 | Lemma 2 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem3 48602 | Lemma 3 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem4 48603 | Lemma 4 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem5 48604 | Lemma 5 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem6 48605 | Lemma 6 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem7 48606 | Lemma 7 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem8 48607 | Lemma 8 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem9 48608 | Lemma 9 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem10 48609 | Lemma 10 for ~ gpgprismgr4... |
| gpgprismgr4cycllem11 48610 | Lemma 11 for ~ gpgprismgr4... |
| gpgprismgr4cycl0 48611 | The generalized Petersen g... |
| gpgprismgr4cyclex 48612 | The generalized Petersen g... |
| pgnioedg1 48613 | An inside and an outside v... |
| pgnioedg2 48614 | An inside and an outside v... |
| pgnioedg3 48615 | An inside and an outside v... |
| pgnioedg4 48616 | An inside and an outside v... |
| pgnioedg5 48617 | An inside and an outside v... |
| pgnbgreunbgrlem1 48618 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem1 48619 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem2 48620 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem3 48621 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2 48622 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem3 48623 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem4 48624 | Lemma 4 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem1 48625 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem2 48626 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem3 48627 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5 48628 | Lemma 5 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem6 48629 | Lemma 6 for ~ pgnbgreunbgr... |
| pgnbgreunbgr 48630 | In a Petersen graph, two d... |
| pgn4cyclex 48631 | A cycle in a Petersen grap... |
| pg4cyclnex 48632 | In the Petersen graph G(5,... |
| gpg5ngric 48633 | The two generalized Peters... |
| lgricngricex 48634 | There are two different lo... |
| gpg5edgnedg 48635 | Two consecutive (according... |
| grlimedgnedg 48636 | In general, the image of a... |
| 1hegrlfgr 48637 | A graph ` G ` with one hyp... |
| upwlksfval 48640 | The set of simple walks (i... |
| isupwlk 48641 | Properties of a pair of fu... |
| isupwlkg 48642 | Generalization of ~ isupwl... |
| upwlkbprop 48643 | Basic properties of a simp... |
| upwlkwlk 48644 | A simple walk is a walk. ... |
| upgrwlkupwlk 48645 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 48646 | In a pseudograph, the defi... |
| upgrisupwlkALT 48647 | Alternate proof of ~ upgri... |
| upgredgssspr 48648 | The set of edges of a pseu... |
| uspgropssxp 48649 | The set ` G ` of "simple p... |
| uspgrsprfv 48650 | The value of the function ... |
| uspgrsprf 48651 | The mapping ` F ` is a fun... |
| uspgrsprf1 48652 | The mapping ` F ` is a one... |
| uspgrsprfo 48653 | The mapping ` F ` is a fun... |
| uspgrsprf1o 48654 | The mapping ` F ` is a bij... |
| uspgrex 48655 | The class ` G ` of all "si... |
| uspgrbispr 48656 | There is a bijection betwe... |
| uspgrspren 48657 | The set ` G ` of the "simp... |
| uspgrymrelen 48658 | The set ` G ` of the "simp... |
| uspgrbisymrel 48659 | There is a bijection betwe... |
| uspgrbisymrelALT 48660 | Alternate proof of ~ uspgr... |
| ovn0dmfun 48661 | If a class operation value... |
| xpsnopab 48662 | A Cartesian product with a... |
| xpiun 48663 | A Cartesian product expres... |
| ovn0ssdmfun 48664 | If a class' operation valu... |
| fnxpdmdm 48665 | The domain of the domain o... |
| cnfldsrngbas 48666 | The base set of a subring ... |
| cnfldsrngadd 48667 | The group addition operati... |
| cnfldsrngmul 48668 | The ring multiplication op... |
| plusfreseq 48669 | If the empty set is not co... |
| mgmplusfreseq 48670 | If the empty set is not co... |
| 0mgm 48671 | A set with an empty base s... |
| opmpoismgm 48672 | A structure with a group a... |
| copissgrp 48673 | A structure with a constan... |
| copisnmnd 48674 | A structure with a constan... |
| 0nodd 48675 | 0 is not an odd integer. ... |
| 1odd 48676 | 1 is an odd integer. (Con... |
| 2nodd 48677 | 2 is not an odd integer. ... |
| oddibas 48678 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 48679 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 48680 | The structure of all odd i... |
| nnsgrpmgm 48681 | The structure of positive ... |
| nnsgrp 48682 | The structure of positive ... |
| nnsgrpnmnd 48683 | The structure of positive ... |
| nn0mnd 48684 | The set of nonnegative int... |
| gsumsplit2f 48685 | Split a group sum into two... |
| gsumdifsndf 48686 | Extract a summand from a f... |
| gsumfsupp 48687 | A group sum of a family ca... |
| iscllaw 48694 | The predicate "is a closed... |
| iscomlaw 48695 | The predicate "is a commut... |
| clcllaw 48696 | Closure of a closed operat... |
| isasslaw 48697 | The predicate "is an assoc... |
| asslawass 48698 | Associativity of an associ... |
| mgmplusgiopALT 48699 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 48700 | Slot 2 (group operation) o... |
| intopval 48707 | The internal (binary) oper... |
| intop 48708 | An internal (binary) opera... |
| clintopval 48709 | The closed (internal binar... |
| assintopval 48710 | The associative (closed in... |
| assintopmap 48711 | The associative (closed in... |
| isclintop 48712 | The predicate "is a closed... |
| clintop 48713 | A closed (internal binary)... |
| assintop 48714 | An associative (closed int... |
| isassintop 48715 | The predicate "is an assoc... |
| clintopcllaw 48716 | The closure law holds for ... |
| assintopcllaw 48717 | The closure low holds for ... |
| assintopasslaw 48718 | The associative low holds ... |
| assintopass 48719 | An associative (closed int... |
| ismgmALT 48728 | The predicate "is a magma"... |
| iscmgmALT 48729 | The predicate "is a commut... |
| issgrpALT 48730 | The predicate "is a semigr... |
| iscsgrpALT 48731 | The predicate "is a commut... |
| mgm2mgm 48732 | Equivalence of the two def... |
| sgrp2sgrp 48733 | Equivalence of the two def... |
| lmod0rng 48734 | If the scalar ring of a mo... |
| nzrneg1ne0 48735 | The additive inverse of th... |
| lidldomn1 48736 | If a (left) ideal (which i... |
| lidlabl 48737 | A (left) ideal of a ring i... |
| lidlrng 48738 | A (left) ideal of a ring i... |
| zlidlring 48739 | The zero (left) ideal of a... |
| uzlidlring 48740 | Only the zero (left) ideal... |
| lidldomnnring 48741 | A (left) ideal of a domain... |
| 0even 48742 | 0 is an even integer. (Co... |
| 1neven 48743 | 1 is not an even integer. ... |
| 2even 48744 | 2 is an even integer. (Co... |
| 2zlidl 48745 | The even integers are a (l... |
| 2zrng 48746 | The ring of integers restr... |
| 2zrngbas 48747 | The base set of R is the s... |
| 2zrngadd 48748 | The group addition operati... |
| 2zrng0 48749 | The additive identity of R... |
| 2zrngamgm 48750 | R is an (additive) magma. ... |
| 2zrngasgrp 48751 | R is an (additive) semigro... |
| 2zrngamnd 48752 | R is an (additive) monoid.... |
| 2zrngacmnd 48753 | R is a commutative (additi... |
| 2zrngagrp 48754 | R is an (additive) group. ... |
| 2zrngaabl 48755 | R is an (additive) abelian... |
| 2zrngmul 48756 | The ring multiplication op... |
| 2zrngmmgm 48757 | R is a (multiplicative) ma... |
| 2zrngmsgrp 48758 | R is a (multiplicative) se... |
| 2zrngALT 48759 | The ring of integers restr... |
| 2zrngnmlid 48760 | R has no multiplicative (l... |
| 2zrngnmrid 48761 | R has no multiplicative (r... |
| 2zrngnmlid2 48762 | R has no multiplicative (l... |
| 2zrngnring 48763 | R is not a unital ring. (... |
| cznrnglem 48764 | Lemma for ~ cznrng : The ... |
| cznabel 48765 | The ring constructed from ... |
| cznrng 48766 | The ring constructed from ... |
| cznnring 48767 | The ring constructed from ... |
| rngcvalALTV 48770 | Value of the category of n... |
| rngcbasALTV 48771 | Set of objects of the cate... |
| rngchomfvalALTV 48772 | Set of arrows of the categ... |
| rngchomALTV 48773 | Set of arrows of the categ... |
| elrngchomALTV 48774 | A morphism of non-unital r... |
| rngccofvalALTV 48775 | Composition in the categor... |
| rngccoALTV 48776 | Composition in the categor... |
| rngccatidALTV 48777 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 48778 | The category of non-unital... |
| rngcidALTV 48779 | The identity arrow in the ... |
| rngcsectALTV 48780 | A section in the category ... |
| rngcinvALTV 48781 | An inverse in the category... |
| rngcisoALTV 48782 | An isomorphism in the cate... |
| rngchomffvalALTV 48783 | The value of the functiona... |
| rngchomrnghmresALTV 48784 | The value of the functiona... |
| rngcrescrhmALTV 48785 | The category of non-unital... |
| rhmsubcALTVlem1 48786 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 48787 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 48788 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 48789 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 48790 | According to ~ df-subc , t... |
| rhmsubcALTVcat 48791 | The restriction of the cat... |
| ringcvalALTV 48794 | Value of the category of r... |
| funcringcsetcALTV2lem1 48795 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 48796 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 48797 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 48798 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 48799 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 48800 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 48801 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 48802 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 48803 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 48804 | The "natural forgetful fun... |
| ringcbasALTV 48805 | Set of objects of the cate... |
| ringchomfvalALTV 48806 | Set of arrows of the categ... |
| ringchomALTV 48807 | Set of arrows of the categ... |
| elringchomALTV 48808 | A morphism of rings is a f... |
| ringccofvalALTV 48809 | Composition in the categor... |
| ringccoALTV 48810 | Composition in the categor... |
| ringccatidALTV 48811 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 48812 | The category of rings is a... |
| ringcidALTV 48813 | The identity arrow in the ... |
| ringcsectALTV 48814 | A section in the category ... |
| ringcinvALTV 48815 | An inverse in the category... |
| ringcisoALTV 48816 | An isomorphism in the cate... |
| ringcbasbasALTV 48817 | An element of the base set... |
| funcringcsetclem1ALTV 48818 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 48819 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 48820 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 48821 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 48822 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 48823 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 48824 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 48825 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 48826 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 48827 | The "natural forgetful fun... |
| srhmsubcALTVlem1 48828 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 48829 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 48830 | According to ~ df-subc , t... |
| sringcatALTV 48831 | The restriction of the cat... |
| crhmsubcALTV 48832 | According to ~ df-subc , t... |
| cringcatALTV 48833 | The restriction of the cat... |
| drhmsubcALTV 48834 | According to ~ df-subc , t... |
| drngcatALTV 48835 | The restriction of the cat... |
| fldcatALTV 48836 | The restriction of the cat... |
| fldcALTV 48837 | The restriction of the cat... |
| fldhmsubcALTV 48838 | According to ~ df-subc , t... |
| eliunxp2 48839 | Membership in a union of C... |
| mpomptx2 48840 | Express a two-argument fun... |
| cbvmpox2 48841 | Rule to change the bound v... |
| dmmpossx2 48842 | The domain of a mapping is... |
| mpoexxg2 48843 | Existence of an operation ... |
| ovmpordxf 48844 | Value of an operation give... |
| ovmpordx 48845 | Value of an operation give... |
| ovmpox2 48846 | The value of an operation ... |
| fdmdifeqresdif 48847 | The restriction of a condi... |
| ofaddmndmap 48848 | The function operation app... |
| mapsnop 48849 | A singleton of an ordered ... |
| fprmappr 48850 | A function with a domain o... |
| mapprop 48851 | An unordered pair containi... |
| ztprmneprm 48852 | A prime is not an integer ... |
| 2t6m3t4e0 48853 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 48854 | For any finite subset of `... |
| nn0sumltlt 48855 | If the sum of two nonnegat... |
| bcpascm1 48856 | Pascal's rule for the bino... |
| altgsumbc 48857 | The sum of binomial coeffi... |
| altgsumbcALT 48858 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 48859 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 48860 | An element of the (base se... |
| zlmodzxz0 48861 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 48862 | The scalar multiplication ... |
| zlmodzxzadd 48863 | The addition of the ` ZZ `... |
| zlmodzxzsubm 48864 | The subtraction of the ` Z... |
| zlmodzxzsub 48865 | The subtraction of the ` Z... |
| mgpsumunsn 48866 | Extract a summand/factor f... |
| mgpsumz 48867 | If the group sum for the m... |
| mgpsumn 48868 | If the group sum for the m... |
| exple2lt6 48869 | A nonnegative integer to t... |
| pgrple2abl 48870 | Every symmetric group on a... |
| pgrpgt2nabl 48871 | Every symmetric group on a... |
| invginvrid 48872 | Identity for a multiplicat... |
| rmsupp0 48873 | The support of a mapping o... |
| domnmsuppn0 48874 | The support of a mapping o... |
| rmsuppss 48875 | The support of a mapping o... |
| scmsuppss 48876 | The support of a mapping o... |
| rmsuppfi 48877 | The support of a mapping o... |
| rmfsupp 48878 | A mapping of a multiplicat... |
| scmsuppfi 48879 | The support of a mapping o... |
| scmfsupp 48880 | A mapping of a scalar mult... |
| suppmptcfin 48881 | The support of a mapping w... |
| mptcfsupp 48882 | A mapping with value 0 exc... |
| fsuppmptdmf 48883 | A mapping with a finite do... |
| lmodvsmdi 48884 | Multiple distributive law ... |
| gsumlsscl 48885 | Closure of a group sum in ... |
| assaascl0 48886 | The scalar 0 embedded into... |
| assaascl1 48887 | The scalar 1 embedded into... |
| ply1vr1smo 48888 | The variable in a polynomi... |
| ply1sclrmsm 48889 | The ring multiplication of... |
| coe1sclmulval 48890 | The value of the coefficie... |
| ply1mulgsumlem1 48891 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 48892 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 48893 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 48894 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 48895 | The product of two polynom... |
| evl1at0 48896 | Polynomial evaluation for ... |
| evl1at1 48897 | Polynomial evaluation for ... |
| linply1 48898 | A term of the form ` x - C... |
| lineval 48899 | A term of the form ` x - C... |
| linevalexample 48900 | The polynomial ` x - 3 ` o... |
| dmatALTval 48905 | The algebra of ` N ` x ` N... |
| dmatALTbas 48906 | The base set of the algebr... |
| dmatALTbasel 48907 | An element of the base set... |
| dmatbas 48908 | The set of all ` N ` x ` N... |
| lincop 48913 | A linear combination as op... |
| lincval 48914 | The value of a linear comb... |
| dflinc2 48915 | Alternative definition of ... |
| lcoop 48916 | A linear combination as op... |
| lcoval 48917 | The value of a linear comb... |
| lincfsuppcl 48918 | A linear combination of ve... |
| linccl 48919 | A linear combination of ve... |
| lincval0 48920 | The value of an empty line... |
| lincvalsng 48921 | The linear combination ove... |
| lincvalsn 48922 | The linear combination ove... |
| lincvalpr 48923 | The linear combination ove... |
| lincval1 48924 | The linear combination ove... |
| lcosn0 48925 | Properties of a linear com... |
| lincvalsc0 48926 | The linear combination whe... |
| lcoc0 48927 | Properties of a linear com... |
| linc0scn0 48928 | If a set contains the zero... |
| lincdifsn 48929 | A vector is a linear combi... |
| linc1 48930 | A vector is a linear combi... |
| lincellss 48931 | A linear combination of a ... |
| lco0 48932 | The set of empty linear co... |
| lcoel0 48933 | The zero vector is always ... |
| lincsum 48934 | The sum of two linear comb... |
| lincscm 48935 | A linear combinations mult... |
| lincsumcl 48936 | The sum of two linear comb... |
| lincscmcl 48937 | The multiplication of a li... |
| lincsumscmcl 48938 | The sum of a linear combin... |
| lincolss 48939 | According to the statement... |
| ellcoellss 48940 | Every linear combination o... |
| lcoss 48941 | A set of vectors of a modu... |
| lspsslco 48942 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 48943 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 48944 | Equivalence of a _span_ of... |
| rellininds 48948 | The class defining the rel... |
| linindsv 48950 | The classes of the module ... |
| islininds 48951 | The property of being a li... |
| linindsi 48952 | The implications of being ... |
| linindslinci 48953 | The implications of being ... |
| islinindfis 48954 | The property of being a li... |
| islinindfiss 48955 | The property of being a li... |
| linindscl 48956 | A linearly independent set... |
| lindepsnlininds 48957 | A linearly dependent subse... |
| islindeps 48958 | The property of being a li... |
| lincext1 48959 | Property 1 of an extension... |
| lincext2 48960 | Property 2 of an extension... |
| lincext3 48961 | Property 3 of an extension... |
| lindslinindsimp1 48962 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 48963 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 48964 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 48965 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 48966 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 48967 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 48968 | Implication 2 for ~ lindsl... |
| lindslininds 48969 | Equivalence of definitions... |
| linds0 48970 | The empty set is always a ... |
| el0ldep 48971 | A set containing the zero ... |
| el0ldepsnzr 48972 | A set containing the zero ... |
| lindsrng01 48973 | Any subset of a module is ... |
| lindszr 48974 | Any subset of a module ove... |
| snlindsntorlem 48975 | Lemma for ~ snlindsntor . ... |
| snlindsntor 48976 | A singleton is linearly in... |
| ldepsprlem 48977 | Lemma for ~ ldepspr . (Co... |
| ldepspr 48978 | If a vector is a scalar mu... |
| lincresunit3lem3 48979 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 48980 | Lemma 1 for properties of ... |
| lincresunitlem2 48981 | Lemma for properties of a ... |
| lincresunit1 48982 | Property 1 of a specially ... |
| lincresunit2 48983 | Property 2 of a specially ... |
| lincresunit3lem1 48984 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 48985 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 48986 | Property 3 of a specially ... |
| lincreslvec3 48987 | Property 3 of a specially ... |
| islindeps2 48988 | Conditions for being a lin... |
| islininds2 48989 | Implication of being a lin... |
| isldepslvec2 48990 | Alternative definition of ... |
| lindssnlvec 48991 | A singleton not containing... |
| lmod1lem1 48992 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 48993 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 48994 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 48995 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 48996 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 48997 | The (smallest) structure r... |
| lmod1zr 48998 | The (smallest) structure r... |
| lmod1zrnlvec 48999 | There is a (left) module (... |
| lmodn0 49000 | Left modules exist. (Cont... |
| zlmodzxzequa 49001 | Example of an equation wit... |
| zlmodzxznm 49002 | Example of a linearly depe... |
| zlmodzxzldeplem 49003 | A and B are not equal. (C... |
| zlmodzxzequap 49004 | Example of an equation wit... |
| zlmodzxzldeplem1 49005 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 49006 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 49007 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 49008 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 49009 | { A , B } is a linearly de... |
| ldepsnlinclem1 49010 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 49011 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 49012 | The class of all (left) ve... |
| ldepsnlinc 49013 | The reverse implication of... |
| ldepslinc 49014 | For (left) vector spaces, ... |
| suppdm 49015 | If the range of a function... |
| eluz2cnn0n1 49016 | An integer greater than 1 ... |
| divge1b 49017 | The ratio of a real number... |
| divgt1b 49018 | The ratio of a real number... |
| ltsubaddb 49019 | Equivalence for the "less ... |
| ltsubsubb 49020 | Equivalence for the "less ... |
| ltsubadd2b 49021 | Equivalence for the "less ... |
| divsub1dir 49022 | Distribution of division o... |
| expnegico01 49023 | An integer greater than 1 ... |
| elfzolborelfzop1 49024 | An element of a half-open ... |
| pw2m1lepw2m1 49025 | 2 to the power of a positi... |
| zgtp1leeq 49026 | If an integer is between a... |
| flsubz 49027 | An integer can be moved in... |
| nn0onn0ex 49028 | For each odd nonnegative i... |
| nn0enn0ex 49029 | For each even nonnegative ... |
| nnennex 49030 | For each even positive int... |
| nneop 49031 | A positive integer is even... |
| nneom 49032 | A positive integer is even... |
| nn0eo 49033 | A nonnegative integer is e... |
| nnpw2even 49034 | 2 to the power of a positi... |
| zefldiv2 49035 | The floor of an even integ... |
| zofldiv2 49036 | The floor of an odd intege... |
| nn0ofldiv2 49037 | The floor of an odd nonneg... |
| flnn0div2ge 49038 | The floor of a positive in... |
| flnn0ohalf 49039 | The floor of the half of a... |
| logcxp0 49040 | Logarithm of a complex pow... |
| regt1loggt0 49041 | The natural logarithm for ... |
| fdivval 49044 | The quotient of two functi... |
| fdivmpt 49045 | The quotient of two functi... |
| fdivmptf 49046 | The quotient of two functi... |
| refdivmptf 49047 | The quotient of two functi... |
| fdivpm 49048 | The quotient of two functi... |
| refdivpm 49049 | The quotient of two functi... |
| fdivmptfv 49050 | The function value of a qu... |
| refdivmptfv 49051 | The function value of a qu... |
| bigoval 49054 | Set of functions of order ... |
| elbigofrcl 49055 | Reverse closure of the "bi... |
| elbigo 49056 | Properties of a function o... |
| elbigo2 49057 | Properties of a function o... |
| elbigo2r 49058 | Sufficient condition for a... |
| elbigof 49059 | A function of order G(x) i... |
| elbigodm 49060 | The domain of a function o... |
| elbigoimp 49061 | The defining property of a... |
| elbigolo1 49062 | A function (into the posit... |
| rege1logbrege0 49063 | The general logarithm, wit... |
| rege1logbzge0 49064 | The general logarithm, wit... |
| fllogbd 49065 | A real number is between t... |
| relogbmulbexp 49066 | The logarithm of the produ... |
| relogbdivb 49067 | The logarithm of the quoti... |
| logbge0b 49068 | The logarithm of a number ... |
| logblt1b 49069 | The logarithm of a number ... |
| fldivexpfllog2 49070 | The floor of a positive re... |
| nnlog2ge0lt1 49071 | A positive integer is 1 if... |
| logbpw2m1 49072 | The floor of the binary lo... |
| fllog2 49073 | The floor of the binary lo... |
| blenval 49076 | The binary length of an in... |
| blen0 49077 | The binary length of 0. (... |
| blenn0 49078 | The binary length of a "nu... |
| blenre 49079 | The binary length of a pos... |
| blennn 49080 | The binary length of a pos... |
| blennnelnn 49081 | The binary length of a pos... |
| blennn0elnn 49082 | The binary length of a non... |
| blenpw2 49083 | The binary length of a pow... |
| blenpw2m1 49084 | The binary length of a pow... |
| nnpw2blen 49085 | A positive integer is betw... |
| nnpw2blenfzo 49086 | A positive integer is betw... |
| nnpw2blenfzo2 49087 | A positive integer is eith... |
| nnpw2pmod 49088 | Every positive integer can... |
| blen1 49089 | The binary length of 1. (... |
| blen2 49090 | The binary length of 2. (... |
| nnpw2p 49091 | Every positive integer can... |
| nnpw2pb 49092 | A number is a positive int... |
| blen1b 49093 | The binary length of a non... |
| blennnt2 49094 | The binary length of a pos... |
| nnolog2flm1 49095 | The floor of the binary lo... |
| blennn0em1 49096 | The binary length of the h... |
| blennngt2o2 49097 | The binary length of an od... |
| blengt1fldiv2p1 49098 | The binary length of an in... |
| blennn0e2 49099 | The binary length of an ev... |
| digfval 49102 | Operation to obtain the ` ... |
| digval 49103 | The ` K ` th digit of a no... |
| digvalnn0 49104 | The ` K ` th digit of a no... |
| nn0digval 49105 | The ` K ` th digit of a no... |
| dignn0fr 49106 | The digits of the fraction... |
| dignn0ldlem 49107 | Lemma for ~ dignnld . (Co... |
| dignnld 49108 | The leading digits of a po... |
| dig2nn0ld 49109 | The leading digits of a po... |
| dig2nn1st 49110 | The first (relevant) digit... |
| dig0 49111 | All digits of 0 are 0. (C... |
| digexp 49112 | The ` K ` th digit of a po... |
| dig1 49113 | All but one digits of 1 ar... |
| 0dig1 49114 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 49115 | The integers 0 and 1 corre... |
| dig2nn0 49116 | A digit of a nonnegative i... |
| 0dig2nn0e 49117 | The last bit of an even in... |
| 0dig2nn0o 49118 | The last bit of an odd int... |
| dig2bits 49119 | The ` K ` th digit of a no... |
| dignn0flhalflem1 49120 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 49121 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 49122 | The digits of the half of ... |
| dignn0flhalf 49123 | The digits of the rounded ... |
| nn0sumshdiglemA 49124 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 49125 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 49126 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 49127 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 49128 | A nonnegative integer can ... |
| nn0mulfsum 49129 | Trivial algorithm to calcu... |
| nn0mullong 49130 | Standard algorithm (also k... |
| naryfval 49133 | The set of the n-ary (endo... |
| naryfvalixp 49134 | The set of the n-ary (endo... |
| naryfvalel 49135 | An n-ary (endo)function on... |
| naryrcl 49136 | Reverse closure for n-ary ... |
| naryfvalelfv 49137 | The value of an n-ary (end... |
| naryfvalelwrdf 49138 | An n-ary (endo)function on... |
| 0aryfvalel 49139 | A nullary (endo)function o... |
| 0aryfvalelfv 49140 | The value of a nullary (en... |
| 1aryfvalel 49141 | A unary (endo)function on ... |
| fv1arycl 49142 | Closure of a unary (endo)f... |
| 1arympt1 49143 | A unary (endo)function in ... |
| 1arympt1fv 49144 | The value of a unary (endo... |
| 1arymaptfv 49145 | The value of the mapping o... |
| 1arymaptf 49146 | The mapping of unary (endo... |
| 1arymaptf1 49147 | The mapping of unary (endo... |
| 1arymaptfo 49148 | The mapping of unary (endo... |
| 1arymaptf1o 49149 | The mapping of unary (endo... |
| 1aryenef 49150 | The set of unary (endo)fun... |
| 1aryenefmnd 49151 | The set of unary (endo)fun... |
| 2aryfvalel 49152 | A binary (endo)function on... |
| fv2arycl 49153 | Closure of a binary (endo)... |
| 2arympt 49154 | A binary (endo)function in... |
| 2arymptfv 49155 | The value of a binary (end... |
| 2arymaptfv 49156 | The value of the mapping o... |
| 2arymaptf 49157 | The mapping of binary (end... |
| 2arymaptf1 49158 | The mapping of binary (end... |
| 2arymaptfo 49159 | The mapping of binary (end... |
| 2arymaptf1o 49160 | The mapping of binary (end... |
| 2aryenef 49161 | The set of binary (endo)fu... |
| itcoval 49166 | The value of the function ... |
| itcoval0 49167 | A function iterated zero t... |
| itcoval1 49168 | A function iterated once. ... |
| itcoval2 49169 | A function iterated twice.... |
| itcoval3 49170 | A function iterated three ... |
| itcoval0mpt 49171 | A mapping iterated zero ti... |
| itcovalsuc 49172 | The value of the function ... |
| itcovalsucov 49173 | The value of the function ... |
| itcovalendof 49174 | The n-th iterate of an end... |
| itcovalpclem1 49175 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 49176 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 49177 | The value of the function ... |
| itcovalt2lem2lem1 49178 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 49179 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 49180 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 49181 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 49182 | The value of the function ... |
| ackvalsuc1mpt 49183 | The Ackermann function at ... |
| ackvalsuc1 49184 | The Ackermann function at ... |
| ackval0 49185 | The Ackermann function at ... |
| ackval1 49186 | The Ackermann function at ... |
| ackval2 49187 | The Ackermann function at ... |
| ackval3 49188 | The Ackermann function at ... |
| ackendofnn0 49189 | The Ackermann function at ... |
| ackfnnn0 49190 | The Ackermann function at ... |
| ackval0val 49191 | The Ackermann function at ... |
| ackvalsuc0val 49192 | The Ackermann function at ... |
| ackvalsucsucval 49193 | The Ackermann function at ... |
| ackval0012 49194 | The Ackermann function at ... |
| ackval1012 49195 | The Ackermann function at ... |
| ackval2012 49196 | The Ackermann function at ... |
| ackval3012 49197 | The Ackermann function at ... |
| ackval40 49198 | The Ackermann function at ... |
| ackval41a 49199 | The Ackermann function at ... |
| ackval41 49200 | The Ackermann function at ... |
| ackval42 49201 | The Ackermann function at ... |
| ackval42a 49202 | The Ackermann function at ... |
| ackval50 49203 | The Ackermann function at ... |
| fv1prop 49204 | The function value of unor... |
| fv2prop 49205 | The function value of unor... |
| submuladdmuld 49206 | Transformation of a sum of... |
| affinecomb1 49207 | Combination of two real af... |
| affinecomb2 49208 | Combination of two real af... |
| affineid 49209 | Identity of an affine comb... |
| 1subrec1sub 49210 | Subtract the reciprocal of... |
| resum2sqcl 49211 | The sum of two squares of ... |
| resum2sqgt0 49212 | The sum of the square of a... |
| resum2sqrp 49213 | The sum of the square of a... |
| resum2sqorgt0 49214 | The sum of the square of t... |
| reorelicc 49215 | Membership in and outside ... |
| rrx2pxel 49216 | The x-coordinate of a poin... |
| rrx2pyel 49217 | The y-coordinate of a poin... |
| prelrrx2 49218 | An unordered pair of order... |
| prelrrx2b 49219 | An unordered pair of order... |
| rrx2pnecoorneor 49220 | If two different points ` ... |
| rrx2pnedifcoorneor 49221 | If two different points ` ... |
| rrx2pnedifcoorneorr 49222 | If two different points ` ... |
| rrx2xpref1o 49223 | There is a bijection betwe... |
| rrx2xpreen 49224 | The set of points in the t... |
| rrx2plord 49225 | The lexicographical orderi... |
| rrx2plord1 49226 | The lexicographical orderi... |
| rrx2plord2 49227 | The lexicographical orderi... |
| rrx2plordisom 49228 | The set of points in the t... |
| rrx2plordso 49229 | The lexicographical orderi... |
| ehl2eudisval0 49230 | The Euclidean distance of ... |
| ehl2eudis0lt 49231 | An upper bound of the Eucl... |
| lines 49236 | The lines passing through ... |
| line 49237 | The line passing through t... |
| rrxlines 49238 | Definition of lines passin... |
| rrxline 49239 | The line passing through t... |
| rrxlinesc 49240 | Definition of lines passin... |
| rrxlinec 49241 | The line passing through t... |
| eenglngeehlnmlem1 49242 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 49243 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 49244 | The line definition in the... |
| rrx2line 49245 | The line passing through t... |
| rrx2vlinest 49246 | The vertical line passing ... |
| rrx2linest 49247 | The line passing through t... |
| rrx2linesl 49248 | The line passing through t... |
| rrx2linest2 49249 | The line passing through t... |
| elrrx2linest2 49250 | The line passing through t... |
| spheres 49251 | The spheres for given cent... |
| sphere 49252 | A sphere with center ` X `... |
| rrxsphere 49253 | The sphere with center ` M... |
| 2sphere 49254 | The sphere with center ` M... |
| 2sphere0 49255 | The sphere around the orig... |
| line2ylem 49256 | Lemma for ~ line2y . This... |
| line2 49257 | Example for a line ` G ` p... |
| line2xlem 49258 | Lemma for ~ line2x . This... |
| line2x 49259 | Example for a horizontal l... |
| line2y 49260 | Example for a vertical lin... |
| itsclc0lem1 49261 | Lemma for theorems about i... |
| itsclc0lem2 49262 | Lemma for theorems about i... |
| itsclc0lem3 49263 | Lemma for theorems about i... |
| itscnhlc0yqe 49264 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 49265 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 49266 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 49267 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 49268 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 49269 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 49270 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 49271 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 49272 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 49273 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 49274 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 49275 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 49276 | The intersection points of... |
| itsclc0b 49277 | The intersection points of... |
| itsclinecirc0 49278 | The intersection points of... |
| itsclinecirc0b 49279 | The intersection points of... |
| itsclinecirc0in 49280 | The intersection points of... |
| itsclquadb 49281 | Quadratic equation for the... |
| itsclquadeu 49282 | Quadratic equation for the... |
| 2itscplem1 49283 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 49284 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 49285 | Lemma D for ~ 2itscp . (C... |
| 2itscp 49286 | A condition for a quadrati... |
| itscnhlinecirc02plem1 49287 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 49288 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 49289 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 49290 | Intersection of a nonhoriz... |
| inlinecirc02plem 49291 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 49292 | Intersection of a line wit... |
| inlinecirc02preu 49293 | Intersection of a line wit... |
| pm4.71da 49294 | Deduction converting a bic... |
| logic1 49295 | Distribution of implicatio... |
| logic1a 49296 | Variant of ~ logic1 . (Co... |
| logic2 49297 | Variant of ~ logic1 . (Co... |
| pm5.32dav 49298 | Distribution of implicatio... |
| pm5.32dra 49299 | Reverse distribution of im... |
| exp12bd 49300 | The import-export theorem ... |
| mpbiran3d 49301 | Equivalence with a conjunc... |
| mpbiran4d 49302 | Equivalence with a conjunc... |
| dtrucor3 49303 | An example of how ~ ax-5 w... |
| ralbidb 49304 | Formula-building rule for ... |
| ralbidc 49305 | Formula-building rule for ... |
| r19.41dv 49306 | A complex deduction form o... |
| rmotru 49307 | Two ways of expressing "at... |
| reutru 49308 | Two ways of expressing "ex... |
| reutruALT 49309 | Alternate proof of ~ reutr... |
| reueqbidva 49310 | Formula-building rule for ... |
| reuxfr1dd 49311 | Transfer existential uniqu... |
| ssdisjd 49312 | Subset preserves disjointn... |
| ssdisjdr 49313 | Subset preserves disjointn... |
| disjdifb 49314 | Relative complement is ant... |
| predisj 49315 | Preimages of disjoint sets... |
| vsn 49316 | The singleton of the unive... |
| mosn 49317 | "At most one" element in a... |
| mo0 49318 | "At most one" element in a... |
| mosssn 49319 | "At most one" element in a... |
| mo0sn 49320 | Two ways of expressing "at... |
| mosssn2 49321 | Two ways of expressing "at... |
| unilbss 49322 | Superclass of the greatest... |
| iuneq0 49323 | An indexed union is empty ... |
| iineq0 49324 | An indexed intersection is... |
| iunlub 49325 | The indexed union is the t... |
| iinglb 49326 | The indexed intersection i... |
| iuneqconst2 49327 | Indexed union of identical... |
| iineqconst2 49328 | Indexed intersection of id... |
| inpw 49329 | Two ways of expressing a c... |
| opth1neg 49330 | Two ordered pairs are not ... |
| opth2neg 49331 | Two ordered pairs are not ... |
| brab2dd 49332 | Expressing that two sets a... |
| brab2ddw 49333 | Expressing that two sets a... |
| brab2ddw2 49334 | Expressing that two sets a... |
| iinxp 49335 | Indexed intersection of Ca... |
| intxp 49336 | Intersection of Cartesian ... |
| coxp 49337 | Composition with a Cartesi... |
| cosn 49338 | Composition with an ordere... |
| cosni 49339 | Composition with an ordere... |
| inisegn0a 49340 | The inverse image of a sin... |
| dmrnxp 49341 | A Cartesian product is the... |
| mof0 49342 | There is at most one funct... |
| mof02 49343 | A variant of ~ mof0 . (Co... |
| mof0ALT 49344 | Alternate proof of ~ mof0 ... |
| eufsnlem 49345 | There is exactly one funct... |
| eufsn 49346 | There is exactly one funct... |
| eufsn2 49347 | There is exactly one funct... |
| mofsn 49348 | There is at most one funct... |
| mofsn2 49349 | There is at most one funct... |
| mofsssn 49350 | There is at most one funct... |
| mofmo 49351 | There is at most one funct... |
| mofeu 49352 | The uniqueness of a functi... |
| elfvne0 49353 | If a function value has a ... |
| fdomne0 49354 | A function with non-empty ... |
| f1sn2g 49355 | A function that maps a sin... |
| f102g 49356 | A function that maps the e... |
| f1mo 49357 | A function that maps a set... |
| f002 49358 | A function with an empty c... |
| map0cor 49359 | A function exists iff an e... |
| ffvbr 49360 | Relation with function val... |
| xpco2 49361 | Composition of a Cartesian... |
| ovsng 49362 | The operation value of a s... |
| ovsng2 49363 | The operation value of a s... |
| ovsn 49364 | The operation value of a s... |
| ovsn2 49365 | The operation value of a s... |
| fvconstr 49366 | Two ways of expressing ` A... |
| fvconstrn0 49367 | Two ways of expressing ` A... |
| fvconstr2 49368 | Two ways of expressing ` A... |
| ovmpt4d 49369 | Deduction version of ~ ovm... |
| eqfnovd 49370 | Deduction for equality of ... |
| fonex 49371 | The domain of a surjection... |
| eloprab1st2nd 49372 | Reconstruction of a nested... |
| fmpodg 49373 | Domain and codomain of the... |
| fmpod 49374 | Domain and codomain of the... |
| resinsnlem 49375 | Lemma for ~ resinsnALT . ... |
| resinsn 49376 | Restriction to the interse... |
| resinsnALT 49377 | Restriction to the interse... |
| dftpos5 49378 | Alternate definition of ` ... |
| dftpos6 49379 | Alternate definition of ` ... |
| dmtposss 49380 | The domain of ` tpos F ` i... |
| tposres0 49381 | The transposition of a set... |
| tposresg 49382 | The transposition restrict... |
| tposrescnv 49383 | The transposition restrict... |
| tposres2 49384 | The transposition restrict... |
| tposres3 49385 | The transposition restrict... |
| tposres 49386 | The transposition restrict... |
| tposresxp 49387 | The transposition restrict... |
| tposf1o 49388 | Condition of a bijective t... |
| tposid 49389 | Swap an ordered pair. (Co... |
| tposidres 49390 | Swap an ordered pair. (Co... |
| tposidf1o 49391 | The swap function, or the ... |
| tposideq 49392 | Two ways of expressing the... |
| tposideq2 49393 | Two ways of expressing the... |
| ixpv 49394 | Infinite Cartesian product... |
| fvconst0ci 49395 | A constant function's valu... |
| fvconstdomi 49396 | A constant function's valu... |
| f1omo 49397 | There is at most one eleme... |
| f1omoOLD 49398 | Obsolete version of ~ f1om... |
| f1omoALT 49399 | There is at most one eleme... |
| iccin 49400 | Intersection of two closed... |
| iccdisj2 49401 | If the upper bound of one ... |
| iccdisj 49402 | If the upper bound of one ... |
| slotresfo 49403 | The condition of a structu... |
| mreuniss 49404 | The union of a collection ... |
| clduni 49405 | The union of closed sets i... |
| opncldeqv 49406 | Conditions on open sets ar... |
| opndisj 49407 | Two ways of saying that tw... |
| clddisj 49408 | Two ways of saying that tw... |
| neircl 49409 | Reverse closure of the nei... |
| opnneilem 49410 | Lemma factoring out common... |
| opnneir 49411 | If something is true for a... |
| opnneirv 49412 | A variant of ~ opnneir wit... |
| opnneilv 49413 | The converse of ~ opnneir ... |
| opnneil 49414 | A variant of ~ opnneilv . ... |
| opnneieqv 49415 | The equivalence between ne... |
| opnneieqvv 49416 | The equivalence between ne... |
| restcls2lem 49417 | A closed set in a subspace... |
| restcls2 49418 | A closed set in a subspace... |
| restclsseplem 49419 | Lemma for ~ restclssep . ... |
| restclssep 49420 | Two disjoint closed sets i... |
| cnneiima 49421 | Given a continuous functio... |
| iooii 49422 | Open intervals are open se... |
| icccldii 49423 | Closed intervals are close... |
| i0oii 49424 | ` ( 0 [,) A ) ` is open in... |
| io1ii 49425 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 49426 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 49427 | Open neighborhood and neig... |
| sepnsepo 49428 | Open neighborhood and neig... |
| sepdisj 49429 | Separated sets are disjoin... |
| seposep 49430 | If two sets are separated ... |
| sepcsepo 49431 | If two sets are separated ... |
| sepfsepc 49432 | If two sets are separated ... |
| seppsepf 49433 | If two sets are precisely ... |
| seppcld 49434 | If two sets are precisely ... |
| isnrm4 49435 | A topological space is nor... |
| dfnrm2 49436 | A topological space is nor... |
| dfnrm3 49437 | A topological space is nor... |
| iscnrm3lem1 49438 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 49439 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem4 49440 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 49441 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 49442 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 49443 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 49444 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 49445 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 49446 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 49447 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 49448 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 49449 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 49450 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 49451 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 49452 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 49453 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 49454 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 49455 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 49456 | A completely normal topolo... |
| iscnrm3v 49457 | A topology is completely n... |
| iscnrm4 49458 | A completely normal topolo... |
| isprsd 49459 | Property of being a preord... |
| lubeldm2 49460 | Member of the domain of th... |
| glbeldm2 49461 | Member of the domain of th... |
| lubeldm2d 49462 | Member of the domain of th... |
| glbeldm2d 49463 | Member of the domain of th... |
| lubsscl 49464 | If a subset of ` S ` conta... |
| glbsscl 49465 | If a subset of ` S ` conta... |
| lubprlem 49466 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 49467 | The set of two comparable ... |
| lubpr 49468 | The LUB of the set of two ... |
| glbprlem 49469 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 49470 | The set of two comparable ... |
| glbpr 49471 | The GLB of the set of two ... |
| joindm2 49472 | The join of any two elemen... |
| joindm3 49473 | The join of any two elemen... |
| meetdm2 49474 | The meet of any two elemen... |
| meetdm3 49475 | The meet of any two elemen... |
| posjidm 49476 | Poset join is idempotent. ... |
| posmidm 49477 | Poset meet is idempotent. ... |
| resiposbas 49478 | Construct a poset ( ~ resi... |
| resipos 49479 | A set equipped with an ord... |
| exbaspos 49480 | There exists a poset for a... |
| exbasprs 49481 | There exists a preordered ... |
| basresposfo 49482 | The base function restrict... |
| basresprsfo 49483 | The base function restrict... |
| posnex 49484 | The class of posets is a p... |
| prsnex 49485 | The class of preordered se... |
| toslat 49486 | A toset is a lattice. (Co... |
| isclatd 49487 | The predicate "is a comple... |
| intubeu 49488 | Existential uniqueness of ... |
| unilbeu 49489 | Existential uniqueness of ... |
| ipolublem 49490 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 49491 | The domain of the LUB of t... |
| ipolub 49492 | The LUB of the inclusion p... |
| ipoglblem 49493 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 49494 | The domain of the GLB of t... |
| ipoglb 49495 | The GLB of the inclusion p... |
| ipolub0 49496 | The LUB of the empty set i... |
| ipolub00 49497 | The LUB of the empty set i... |
| ipoglb0 49498 | The GLB of the empty set i... |
| mrelatlubALT 49499 | Least upper bounds in a Mo... |
| mrelatglbALT 49500 | Greatest lower bounds in a... |
| mreclat 49501 | A Moore space is a complet... |
| topclat 49502 | A topology is a complete l... |
| toplatglb0 49503 | The empty intersection in ... |
| toplatlub 49504 | Least upper bounds in a to... |
| toplatglb 49505 | Greatest lower bounds in a... |
| toplatjoin 49506 | Joins in a topology are re... |
| toplatmeet 49507 | Meets in a topology are re... |
| topdlat 49508 | A topology is a distributi... |
| elmgpcntrd 49509 | The center of a ring. (Co... |
| asclelbasALT 49510 | Alternate proof of ~ ascle... |
| asclcntr 49511 | The algebra scalar lifting... |
| asclcom 49512 | Scalars are commutative af... |
| homf0 49513 | The base is empty iff the ... |
| catprslem 49514 | Lemma for ~ catprs . (Con... |
| catprs 49515 | A preorder can be extracte... |
| catprs2 49516 | A category equipped with t... |
| catprsc 49517 | A construction of the preo... |
| catprsc2 49518 | An alternate construction ... |
| endmndlem 49519 | A diagonal hom-set in a ca... |
| oppccatb 49520 | An opposite category is a ... |
| oppcmndclem 49521 | Lemma for ~ oppcmndc . Ev... |
| oppcendc 49522 | The opposite category of a... |
| oppcmndc 49523 | The opposite category of a... |
| idmon 49524 | An identity arrow, or an i... |
| idepi 49525 | An identity arrow, or an i... |
| sectrcl 49526 | Reverse closure for sectio... |
| sectrcl2 49527 | Reverse closure for sectio... |
| invrcl 49528 | Reverse closure for invers... |
| invrcl2 49529 | Reverse closure for invers... |
| isinv2 49530 | The property " ` F ` is an... |
| isisod 49531 | The predicate "is an isomo... |
| upeu2lem 49532 | Lemma for ~ upeu2 . There... |
| sectfn 49533 | The function value of the ... |
| invfn 49534 | The function value of the ... |
| isofnALT 49535 | The function value of the ... |
| isofval2 49536 | Function value of the func... |
| isorcl 49537 | Reverse closure for isomor... |
| isorcl2 49538 | Reverse closure for isomor... |
| isoval2 49539 | The isomorphisms are the d... |
| sectpropdlem 49540 | Lemma for ~ sectpropd . (... |
| sectpropd 49541 | Two structures with the sa... |
| invpropdlem 49542 | Lemma for ~ invpropd . (C... |
| invpropd 49543 | Two structures with the sa... |
| isopropdlem 49544 | Lemma for ~ isopropd . (C... |
| isopropd 49545 | Two structures with the sa... |
| cicfn 49546 | ` ~=c ` is a function on `... |
| cicrcl2 49547 | Isomorphism implies the st... |
| oppccic 49548 | Isomorphic objects are iso... |
| relcic 49549 | The set of isomorphic obje... |
| cicerALT 49550 | Isomorphism is an equivale... |
| cic1st2nd 49551 | Reconstruction of a pair o... |
| cic1st2ndbr 49552 | Rewrite the predicate of i... |
| cicpropdlem 49553 | Lemma for ~ cicpropd . (C... |
| cicpropd 49554 | Two structures with the sa... |
| oppccicb 49555 | Isomorphic objects are iso... |
| oppcciceq 49556 | The opposite category has ... |
| dmdm 49557 | The double domain of a fun... |
| iinfssclem1 49558 | Lemma for ~ iinfssc . (Co... |
| iinfssclem2 49559 | Lemma for ~ iinfssc . (Co... |
| iinfssclem3 49560 | Lemma for ~ iinfssc . (Co... |
| iinfssc 49561 | Indexed intersection of su... |
| iinfsubc 49562 | Indexed intersection of su... |
| iinfprg 49563 | Indexed intersection of fu... |
| infsubc 49564 | The intersection of two su... |
| infsubc2 49565 | The intersection of two su... |
| infsubc2d 49566 | The intersection of two su... |
| discsubclem 49567 | Lemma for ~ discsubc . (C... |
| discsubc 49568 | A discrete category, whose... |
| iinfconstbaslem 49569 | Lemma for ~ iinfconstbas .... |
| iinfconstbas 49570 | The discrete category is t... |
| nelsubclem 49571 | Lemma for ~ nelsubc . (Co... |
| nelsubc 49572 | An empty "hom-set" for non... |
| nelsubc2 49573 | An empty "hom-set" for non... |
| nelsubc3lem 49574 | Lemma for ~ nelsubc3 . (C... |
| nelsubc3 49575 | Remark 4.2(2) of [Adamek] ... |
| ssccatid 49576 | A category ` C ` restricte... |
| resccatlem 49577 | Lemma for ~ resccat . (Co... |
| resccat 49578 | A class ` C ` restricted b... |
| reldmfunc 49579 | The domain of ` Func ` is ... |
| func1st2nd 49580 | Rewrite the functor predic... |
| func1st 49581 | Extract the first member o... |
| func2nd 49582 | Extract the second member ... |
| funcrcl2 49583 | Reverse closure for a func... |
| funcrcl3 49584 | Reverse closure for a func... |
| funcf2lem 49585 | A utility theorem for prov... |
| funcf2lem2 49586 | A utility theorem for prov... |
| 0funcglem 49587 | Lemma for ~ 0funcg . (Con... |
| 0funcg2 49588 | The functor from the empty... |
| 0funcg 49589 | The functor from the empty... |
| 0funclem 49590 | Lemma for ~ 0funcALT . (C... |
| 0func 49591 | The functor from the empty... |
| 0funcALT 49592 | Alternate proof of ~ 0func... |
| func0g 49593 | The source category of a f... |
| func0g2 49594 | The source category of a f... |
| initc 49595 | Sets with empty base are t... |
| cofu1st2nd 49596 | Rewrite the functor compos... |
| rescofuf 49597 | The restriction of functor... |
| cofu1a 49598 | Value of the object part o... |
| cofu2a 49599 | Value of the morphism part... |
| cofucla 49600 | The composition of two fun... |
| funchomf 49601 | Source categories of a fun... |
| idfurcl 49602 | Reverse closure for an ide... |
| idfu1stf1o 49603 | The identity functor/inclu... |
| idfu1stalem 49604 | Lemma for ~ idfu1sta . (C... |
| idfu1sta 49605 | Value of the object part o... |
| idfu1a 49606 | Value of the object part o... |
| idfu2nda 49607 | Value of the morphism part... |
| imasubclem1 49608 | Lemma for ~ imasubc . (Co... |
| imasubclem2 49609 | Lemma for ~ imasubc . (Co... |
| imasubclem3 49610 | Lemma for ~ imasubc . (Co... |
| imaf1homlem 49611 | Lemma for ~ imaf1hom and o... |
| imaf1hom 49612 | The hom-set of an image of... |
| imaidfu2lem 49613 | Lemma for ~ imaidfu2 . (C... |
| imaidfu 49614 | The image of the identity ... |
| imaidfu2 49615 | The image of the identity ... |
| cofid1a 49616 | Express the object part of... |
| cofid2a 49617 | Express the morphism part ... |
| cofid1 49618 | Express the object part of... |
| cofid2 49619 | Express the morphism part ... |
| cofidvala 49620 | The property " ` F ` is a ... |
| cofidf2a 49621 | If " ` F ` is a section of... |
| cofidf1a 49622 | If " ` F ` is a section of... |
| cofidval 49623 | The property " ` <. F , G ... |
| cofidf2 49624 | If " ` F ` is a section of... |
| cofidf1 49625 | If " ` <. F , G >. ` is a ... |
| oppffn 49628 | ` oppFunc ` is a function ... |
| reldmoppf 49629 | The domain of ` oppFunc ` ... |
| oppfvalg 49630 | Value of the opposite func... |
| oppfrcllem 49631 | Lemma for ~ oppfrcl . (Co... |
| oppfrcl 49632 | If an opposite functor of ... |
| oppfrcl2 49633 | If an opposite functor of ... |
| oppfrcl3 49634 | If an opposite functor of ... |
| oppf1st2nd 49635 | Rewrite the opposite funct... |
| 2oppf 49636 | The double opposite functo... |
| eloppf 49637 | The pre-image of a non-emp... |
| eloppf2 49638 | Both components of a pre-i... |
| oppfvallem 49639 | Lemma for ~ oppfval . (Co... |
| oppfval 49640 | Value of the opposite func... |
| oppfval2 49641 | Value of the opposite func... |
| oppfval3 49642 | Value of the opposite func... |
| oppf1 49643 | Value of the object part o... |
| oppf2 49644 | Value of the morphism part... |
| oppfoppc 49645 | The opposite functor is a ... |
| oppfoppc2 49646 | The opposite functor is a ... |
| funcoppc2 49647 | A functor on opposite cate... |
| funcoppc4 49648 | A functor on opposite cate... |
| funcoppc5 49649 | A functor on opposite cate... |
| 2oppffunc 49650 | The opposite functor of an... |
| funcoppc3 49651 | A functor on opposite cate... |
| oppff1 49652 | The operation generating o... |
| oppff1o 49653 | The operation generating o... |
| cofuoppf 49654 | Composition of opposite fu... |
| imasubc 49655 | An image of a full functor... |
| imasubc2 49656 | An image of a full functor... |
| imassc 49657 | An image of a functor sati... |
| imaid 49658 | An image of a functor pres... |
| imaf1co 49659 | An image of a functor whos... |
| imasubc3 49660 | An image of a functor inje... |
| fthcomf 49661 | Source categories of a fai... |
| idfth 49662 | The inclusion functor is a... |
| idemb 49663 | The inclusion functor is a... |
| idsubc 49664 | The source category of an ... |
| idfullsubc 49665 | The source category of an ... |
| cofidfth 49666 | If " ` F ` is a section of... |
| fulloppf 49667 | The opposite functor of a ... |
| fthoppf 49668 | The opposite functor of a ... |
| ffthoppf 49669 | The opposite functor of a ... |
| upciclem1 49670 | Lemma for ~ upcic , ~ upeu... |
| upciclem2 49671 | Lemma for ~ upciclem3 and ... |
| upciclem3 49672 | Lemma for ~ upciclem4 . (... |
| upciclem4 49673 | Lemma for ~ upcic and ~ up... |
| upcic 49674 | A universal property defin... |
| upeu 49675 | A universal property defin... |
| upeu2 49676 | Generate new universal mor... |
| reldmup 49679 | The domain of ` UP ` is a ... |
| upfval 49680 | Function value of the clas... |
| upfval2 49681 | Function value of the clas... |
| upfval3 49682 | Function value of the clas... |
| isuplem 49683 | Lemma for ~ isup and other... |
| isup 49684 | The predicate "is a univer... |
| uppropd 49685 | If two categories have the... |
| reldmup2 49686 | The domain of ` ( D UP E )... |
| relup 49687 | The set of universal pairs... |
| uprcl 49688 | Reverse closure for the cl... |
| up1st2nd 49689 | Rewrite the universal prop... |
| up1st2ndr 49690 | Combine separated parts in... |
| up1st2ndb 49691 | Combine/separate parts in ... |
| up1st2nd2 49692 | Rewrite the universal prop... |
| uprcl2 49693 | Reverse closure for the cl... |
| uprcl3 49694 | Reverse closure for the cl... |
| uprcl4 49695 | Reverse closure for the cl... |
| uprcl5 49696 | Reverse closure for the cl... |
| uobrcl 49697 | Reverse closure for univer... |
| isup2 49698 | The universal property of ... |
| upeu3 49699 | The universal pair ` <. X ... |
| upeu4 49700 | Generate a new universal m... |
| uptposlem 49701 | Lemma for ~ uptpos . (Con... |
| uptpos 49702 | Rewrite the predicate of u... |
| oppcuprcl4 49703 | Reverse closure for the cl... |
| oppcuprcl3 49704 | Reverse closure for the cl... |
| oppcuprcl5 49705 | Reverse closure for the cl... |
| oppcuprcl2 49706 | Reverse closure for the cl... |
| uprcl2a 49707 | Reverse closure for the cl... |
| oppfuprcl 49708 | Reverse closure for the cl... |
| oppfuprcl2 49709 | Reverse closure for the cl... |
| oppcup3lem 49710 | Lemma for ~ oppcup3 . (Co... |
| oppcup 49711 | The universal pair ` <. X ... |
| oppcup2 49712 | The universal property for... |
| oppcup3 49713 | The universal property for... |
| uptrlem1 49714 | Lemma for ~ uptr . (Contr... |
| uptrlem2 49715 | Lemma for ~ uptr . (Contr... |
| uptrlem3 49716 | Lemma for ~ uptr . (Contr... |
| uptr 49717 | Universal property and ful... |
| uptri 49718 | Universal property and ful... |
| uptra 49719 | Universal property and ful... |
| uptrar 49720 | Universal property and ful... |
| uptrai 49721 | Universal property and ful... |
| uobffth 49722 | A fully faithful functor g... |
| uobeqw 49723 | If a full functor (in fact... |
| uobeq 49724 | If a full functor (in fact... |
| uptr2 49725 | Universal property and ful... |
| uptr2a 49726 | Universal property and ful... |
| isnatd 49727 | Property of being a natura... |
| natrcl2 49728 | Reverse closure for a natu... |
| natrcl3 49729 | Reverse closure for a natu... |
| catbas 49730 | The base of the category s... |
| cathomfval 49731 | The hom-sets of the catego... |
| catcofval 49732 | Composition of the categor... |
| natoppf 49733 | A natural transformation i... |
| natoppf2 49734 | A natural transformation i... |
| natoppfb 49735 | A natural transformation i... |
| initoo2 49736 | An initial object is an ob... |
| termoo2 49737 | A terminal object is an ob... |
| zeroo2 49738 | A zero object is an object... |
| oppcinito 49739 | Initial objects are termin... |
| oppctermo 49740 | Terminal objects are initi... |
| oppczeroo 49741 | Zero objects are zero in t... |
| termoeu2 49742 | Terminal objects are essen... |
| initopropdlemlem 49743 | Lemma for ~ initopropdlem ... |
| initopropdlem 49744 | Lemma for ~ initopropd . ... |
| termopropdlem 49745 | Lemma for ~ termopropd . ... |
| zeroopropdlem 49746 | Lemma for ~ zeroopropd . ... |
| initopropd 49747 | Two structures with the sa... |
| termopropd 49748 | Two structures with the sa... |
| zeroopropd 49749 | Two structures with the sa... |
| reldmxpc 49750 | The binary product of cate... |
| reldmxpcALT 49751 | Alternate proof of ~ reldm... |
| elxpcbasex1 49752 | A non-empty base set of th... |
| elxpcbasex1ALT 49753 | Alternate proof of ~ elxpc... |
| elxpcbasex2 49754 | A non-empty base set of th... |
| elxpcbasex2ALT 49755 | Alternate proof of ~ elxpc... |
| xpcfucbas 49756 | The base set of the produc... |
| xpcfuchomfval 49757 | Set of morphisms of the bi... |
| xpcfuchom 49758 | Set of morphisms of the bi... |
| xpcfuchom2 49759 | Value of the set of morphi... |
| xpcfucco2 49760 | Value of composition in th... |
| xpcfuccocl 49761 | The composition of two nat... |
| xpcfucco3 49762 | Value of composition in th... |
| dfswapf2 49765 | Alternate definition of ` ... |
| swapfval 49766 | Value of the swap functor.... |
| swapfelvv 49767 | A swap functor is an order... |
| swapf2fvala 49768 | The morphism part of the s... |
| swapf2fval 49769 | The morphism part of the s... |
| swapf1vala 49770 | The object part of the swa... |
| swapf1val 49771 | The object part of the swa... |
| swapf2fn 49772 | The morphism part of the s... |
| swapf1a 49773 | The object part of the swa... |
| swapf2vala 49774 | The morphism part of the s... |
| swapf2a 49775 | The morphism part of the s... |
| swapf1 49776 | The object part of the swa... |
| swapf2val 49777 | The morphism part of the s... |
| swapf2 49778 | The morphism part of the s... |
| swapf1f1o 49779 | The object part of the swa... |
| swapf2f1o 49780 | The morphism part of the s... |
| swapf2f1oa 49781 | The morphism part of the s... |
| swapf2f1oaALT 49782 | Alternate proof of ~ swapf... |
| swapfid 49783 | Each identity morphism in ... |
| swapfida 49784 | Each identity morphism in ... |
| swapfcoa 49785 | Composition in the source ... |
| swapffunc 49786 | The swap functor is a func... |
| swapfffth 49787 | The swap functor is a full... |
| swapffunca 49788 | The swap functor is a func... |
| swapfiso 49789 | The swap functor is an iso... |
| swapciso 49790 | The product category is ca... |
| oppc1stflem 49791 | A utility theorem for prov... |
| oppc1stf 49792 | The opposite functor of th... |
| oppc2ndf 49793 | The opposite functor of th... |
| 1stfpropd 49794 | If two categories have the... |
| 2ndfpropd 49795 | If two categories have the... |
| diagpropd 49796 | If two categories have the... |
| cofuswapfcl 49797 | The bifunctor pre-composed... |
| cofuswapf1 49798 | The object part of a bifun... |
| cofuswapf2 49799 | The morphism part of a bif... |
| tposcurf1cl 49800 | The partially evaluated tr... |
| tposcurf11 49801 | Value of the double evalua... |
| tposcurf12 49802 | The partially evaluated tr... |
| tposcurf1 49803 | Value of the object part o... |
| tposcurf2 49804 | Value of the transposed cu... |
| tposcurf2val 49805 | Value of a component of th... |
| tposcurf2cl 49806 | The transposed curry funct... |
| tposcurfcl 49807 | The transposed curry funct... |
| diag1 49808 | The constant functor of ` ... |
| diag1a 49809 | The constant functor of ` ... |
| diag1f1lem 49810 | The object part of the dia... |
| diag1f1 49811 | The object part of the dia... |
| diag2f1lem 49812 | Lemma for ~ diag2f1 . The... |
| diag2f1 49813 | If ` B ` is non-empty, the... |
| fucofulem1 49814 | Lemma for proving functor ... |
| fucofulem2 49815 | Lemma for proving functor ... |
| fuco2el 49816 | Equivalence of product fun... |
| fuco2eld 49817 | Equivalence of product fun... |
| fuco2eld2 49818 | Equivalence of product fun... |
| fuco2eld3 49819 | Equivalence of product fun... |
| fucofvalg 49822 | Value of the function givi... |
| fucofval 49823 | Value of the function givi... |
| fucoelvv 49824 | A functor composition bifu... |
| fuco1 49825 | The object part of the fun... |
| fucof1 49826 | The object part of the fun... |
| fuco2 49827 | The morphism part of the f... |
| fucofn2 49828 | The morphism part of the f... |
| fucofvalne 49829 | Value of the function givi... |
| fuco11 49830 | The object part of the fun... |
| fuco11cl 49831 | The object part of the fun... |
| fuco11a 49832 | The object part of the fun... |
| fuco112 49833 | The object part of the fun... |
| fuco111 49834 | The object part of the fun... |
| fuco111x 49835 | The object part of the fun... |
| fuco112x 49836 | The object part of the fun... |
| fuco112xa 49837 | The object part of the fun... |
| fuco11id 49838 | The identity morphism of t... |
| fuco11idx 49839 | The identity morphism of t... |
| fuco21 49840 | The morphism part of the f... |
| fuco11b 49841 | The object part of the fun... |
| fuco11bALT 49842 | Alternate proof of ~ fuco1... |
| fuco22 49843 | The morphism part of the f... |
| fucofn22 49844 | The morphism part of the f... |
| fuco23 49845 | The morphism part of the f... |
| fuco22natlem1 49846 | Lemma for ~ fuco22nat . T... |
| fuco22natlem2 49847 | Lemma for ~ fuco22nat . T... |
| fuco22natlem3 49848 | Combine ~ fuco22natlem2 wi... |
| fuco22natlem 49849 | The composed natural trans... |
| fuco22nat 49850 | The composed natural trans... |
| fucof21 49851 | The morphism part of the f... |
| fucoid 49852 | Each identity morphism in ... |
| fucoid2 49853 | Each identity morphism in ... |
| fuco22a 49854 | The morphism part of the f... |
| fuco23alem 49855 | The naturality property ( ... |
| fuco23a 49856 | The morphism part of the f... |
| fucocolem1 49857 | Lemma for ~ fucoco . Asso... |
| fucocolem2 49858 | Lemma for ~ fucoco . The ... |
| fucocolem3 49859 | Lemma for ~ fucoco . The ... |
| fucocolem4 49860 | Lemma for ~ fucoco . The ... |
| fucoco 49861 | Composition in the source ... |
| fucoco2 49862 | Composition in the source ... |
| fucofunc 49863 | The functor composition bi... |
| fucofunca 49864 | The functor composition bi... |
| fucolid 49865 | Post-compose a natural tra... |
| fucorid 49866 | Pre-composing a natural tr... |
| fucorid2 49867 | Pre-composing a natural tr... |
| postcofval 49868 | Value of the post-composit... |
| postcofcl 49869 | The post-composition funct... |
| precofvallem 49870 | Lemma for ~ precofval to e... |
| precofval 49871 | Value of the pre-compositi... |
| precofvalALT 49872 | Alternate proof of ~ preco... |
| precofval2 49873 | Value of the pre-compositi... |
| precofcl 49874 | The pre-composition functo... |
| precofval3 49875 | Value of the pre-compositi... |
| precoffunc 49876 | The pre-composition functo... |
| reldmprcof 49879 | The domain of ` -o.F ` is ... |
| prcofvalg 49880 | Value of the pre-compositi... |
| prcofvala 49881 | Value of the pre-compositi... |
| prcofval 49882 | Value of the pre-compositi... |
| prcofpropd 49883 | If the categories have the... |
| prcofelvv 49884 | The pre-composition functo... |
| reldmprcof1 49885 | The domain of the object p... |
| reldmprcof2 49886 | The domain of the morphism... |
| prcoftposcurfuco 49887 | The pre-composition functo... |
| prcoftposcurfucoa 49888 | The pre-composition functo... |
| prcoffunc 49889 | The pre-composition functo... |
| prcoffunca 49890 | The pre-composition functo... |
| prcoffunca2 49891 | The pre-composition functo... |
| prcof1 49892 | The object part of the pre... |
| prcof2a 49893 | The morphism part of the p... |
| prcof2 49894 | The morphism part of the p... |
| prcof21a 49895 | The morphism part of the p... |
| prcof22a 49896 | The morphism part of the p... |
| prcofdiag1 49897 | A constant functor pre-com... |
| prcofdiag 49898 | A diagonal functor post-co... |
| catcrcl 49899 | Reverse closure for the ca... |
| catcrcl2 49900 | Reverse closure for the ca... |
| elcatchom 49901 | A morphism of the category... |
| catcsect 49902 | The property " ` F ` is a ... |
| catcinv 49903 | The property " ` F ` is an... |
| catcisoi 49904 | A functor is an isomorphis... |
| uobeq2 49905 | If a full functor (in fact... |
| uobeq3 49906 | An isomorphism between cat... |
| opf11 49907 | The object part of the op ... |
| opf12 49908 | The object part of the op ... |
| opf2fval 49909 | The morphism part of the o... |
| opf2 49910 | The morphism part of the o... |
| fucoppclem 49911 | Lemma for ~ fucoppc . (Co... |
| fucoppcid 49912 | The opposite category of f... |
| fucoppcco 49913 | The opposite category of f... |
| fucoppc 49914 | The isomorphism from the o... |
| fucoppcffth 49915 | A fully faithful functor f... |
| fucoppcfunc 49916 | A functor from the opposit... |
| fucoppccic 49917 | The opposite category of f... |
| oppfdiag1 49918 | A constant functor for opp... |
| oppfdiag1a 49919 | A constant functor for opp... |
| oppfdiag 49920 | A diagonal functor for opp... |
| isthinc 49923 | The predicate "is a thin c... |
| isthinc2 49924 | A thin category is a categ... |
| isthinc3 49925 | A thin category is a categ... |
| thincc 49926 | A thin category is a categ... |
| thinccd 49927 | A thin category is a categ... |
| thincssc 49928 | A thin category is a categ... |
| isthincd2lem1 49929 | Lemma for ~ isthincd2 and ... |
| thincmo2 49930 | Morphisms in the same hom-... |
| thinchom 49931 | A non-empty hom-set of a t... |
| thincmo 49932 | There is at most one morph... |
| thincmoALT 49933 | Alternate proof of ~ thinc... |
| thincmod 49934 | At most one morphism in ea... |
| thincn0eu 49935 | In a thin category, a hom-... |
| thincid 49936 | In a thin category, a morp... |
| thincmon 49937 | In a thin category, all mo... |
| thincepi 49938 | In a thin category, all mo... |
| isthincd2lem2 49939 | Lemma for ~ isthincd2 . (... |
| isthincd 49940 | The predicate "is a thin c... |
| isthincd2 49941 | The predicate " ` C ` is a... |
| oppcthin 49942 | The opposite category of a... |
| oppcthinco 49943 | If the opposite category o... |
| oppcthinendc 49944 | The opposite category of a... |
| oppcthinendcALT 49945 | Alternate proof of ~ oppct... |
| thincpropd 49946 | Two structures with the sa... |
| subthinc 49947 | A subcategory of a thin ca... |
| functhinclem1 49948 | Lemma for ~ functhinc . G... |
| functhinclem2 49949 | Lemma for ~ functhinc . (... |
| functhinclem3 49950 | Lemma for ~ functhinc . T... |
| functhinclem4 49951 | Lemma for ~ functhinc . O... |
| functhinc 49952 | A functor to a thin catego... |
| functhincfun 49953 | A functor to a thin catego... |
| fullthinc 49954 | A functor to a thin catego... |
| fullthinc2 49955 | A full functor to a thin c... |
| thincfth 49956 | A functor from a thin cate... |
| thincciso 49957 | Two thin categories are is... |
| thinccisod 49958 | Two thin categories are is... |
| thincciso2 49959 | Categories isomorphic to a... |
| thincciso3 49960 | Categories isomorphic to a... |
| thincciso4 49961 | Two isomorphic categories ... |
| 0thincg 49962 | Any structure with an empt... |
| 0thinc 49963 | The empty category (see ~ ... |
| indcthing 49964 | An indiscrete category, i.... |
| discthing 49965 | A discrete category, i.e.,... |
| indthinc 49966 | An indiscrete category in ... |
| indthincALT 49967 | An alternate proof of ~ in... |
| prsthinc 49968 | Preordered sets as categor... |
| setcthin 49969 | A category of sets all of ... |
| setc2othin 49970 | The category ` ( SetCat ``... |
| thincsect 49971 | In a thin category, one mo... |
| thincsect2 49972 | In a thin category, ` F ` ... |
| thincinv 49973 | In a thin category, ` F ` ... |
| thinciso 49974 | In a thin category, ` F : ... |
| thinccic 49975 | In a thin category, two ob... |
| istermc 49978 | The predicate "is a termin... |
| istermc2 49979 | The predicate "is a termin... |
| istermc3 49980 | The predicate "is a termin... |
| termcthin 49981 | A terminal category is a t... |
| termcthind 49982 | A terminal category is a t... |
| termccd 49983 | A terminal category is a c... |
| termcbas 49984 | The base of a terminal cat... |
| termco 49985 | The object of a terminal c... |
| termcbas2 49986 | The base of a terminal cat... |
| termcbasmo 49987 | Two objects in a terminal ... |
| termchomn0 49988 | All hom-sets of a terminal... |
| termchommo 49989 | All morphisms of a termina... |
| termcid 49990 | The morphism of a terminal... |
| termcid2 49991 | The morphism of a terminal... |
| termchom 49992 | The hom-set of a terminal ... |
| termchom2 49993 | The hom-set of a terminal ... |
| setcsnterm 49994 | The category of one set, e... |
| setc1oterm 49995 | The category ` ( SetCat ``... |
| setc1obas 49996 | The base of the trivial ca... |
| setc1ohomfval 49997 | Set of morphisms of the tr... |
| setc1ocofval 49998 | Composition in the trivial... |
| setc1oid 49999 | The identity morphism of t... |
| funcsetc1ocl 50000 | The functor to the trivial... |
| funcsetc1o 50001 | Value of the functor to th... |
| isinito2lem 50002 | The predicate "is an initi... |
| isinito2 50003 | The predicate "is an initi... |
| isinito3 50004 | The predicate "is an initi... |
| dfinito4 50005 | An alternate definition of... |
| dftermo4 50006 | An alternate definition of... |
| termcpropd 50007 | Two structures with the sa... |
| oppctermhom 50008 | The opposite category of a... |
| oppctermco 50009 | The opposite category of a... |
| oppcterm 50010 | The opposite category of a... |
| functermclem 50011 | Lemma for ~ functermc . (... |
| functermc 50012 | Functor to a terminal cate... |
| functermc2 50013 | Functor to a terminal cate... |
| functermceu 50014 | There exists a unique func... |
| fulltermc 50015 | A functor to a terminal ca... |
| fulltermc2 50016 | Given a full functor to a ... |
| termcterm 50017 | A terminal category is a t... |
| termcterm2 50018 | A terminal object of the c... |
| termcterm3 50019 | In the category of small c... |
| termcciso 50020 | A category is isomorphic t... |
| termccisoeu 50021 | The isomorphism between te... |
| termc2 50022 | If there exists a unique f... |
| termc 50023 | Alternate definition of ` ... |
| dftermc2 50024 | Alternate definition of ` ... |
| eufunclem 50025 | If there exists a unique f... |
| eufunc 50026 | If there exists a unique f... |
| idfudiag1lem 50027 | Lemma for ~ idfudiag1bas a... |
| idfudiag1bas 50028 | If the identity functor of... |
| idfudiag1 50029 | If the identity functor of... |
| euendfunc 50030 | If there exists a unique e... |
| euendfunc2 50031 | If there exists a unique e... |
| termcarweu 50032 | There exists a unique disj... |
| arweuthinc 50033 | If a structure has a uniqu... |
| arweutermc 50034 | If a structure has a uniqu... |
| dftermc3 50035 | Alternate definition of ` ... |
| termcfuncval 50036 | The value of a functor fro... |
| diag1f1olem 50037 | To any functor from a term... |
| diag1f1o 50038 | The object part of the dia... |
| termcnatval 50039 | Value of natural transform... |
| diag2f1olem 50040 | Lemma for ~ diag2f1o . (C... |
| diag2f1o 50041 | If ` D ` is terminal, the ... |
| diagffth 50042 | The diagonal functor is a ... |
| diagciso 50043 | The diagonal functor is an... |
| diagcic 50044 | Any category ` C ` is isom... |
| funcsn 50045 | The category of one functo... |
| fucterm 50046 | The category of functors t... |
| 0fucterm 50047 | The category of functors f... |
| termfucterm 50048 | All functors between two t... |
| cofuterm 50049 | Post-compose with a functo... |
| uobeqterm 50050 | Universal objects and term... |
| isinito4 50051 | The predicate "is an initi... |
| isinito4a 50052 | The predicate "is an initi... |
| prstcval 50055 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 50056 | Lemma for ~ prstcnid and ~... |
| prstcnid 50057 | Components other than ` Ho... |
| prstcbas 50058 | The base set is unchanged.... |
| prstcleval 50059 | Value of the less-than-or-... |
| prstcle 50060 | Value of the less-than-or-... |
| prstcocval 50061 | Orthocomplementation is un... |
| prstcoc 50062 | Orthocomplementation is un... |
| prstchomval 50063 | Hom-sets of the constructe... |
| prstcprs 50064 | The category is a preorder... |
| prstcthin 50065 | The preordered set is equi... |
| prstchom 50066 | Hom-sets of the constructe... |
| prstchom2 50067 | Hom-sets of the constructe... |
| prstchom2ALT 50068 | Hom-sets of the constructe... |
| oduoppcbas 50069 | The dual of a preordered s... |
| oduoppcciso 50070 | The dual of a preordered s... |
| postcpos 50071 | The converted category is ... |
| postcposALT 50072 | Alternate proof of ~ postc... |
| postc 50073 | The converted category is ... |
| discsntermlem 50074 | A singlegon is an element ... |
| basrestermcfolem 50075 | An element of the class of... |
| discbas 50076 | A discrete category (a cat... |
| discthin 50077 | A discrete category (a cat... |
| discsnterm 50078 | A discrete category (a cat... |
| basrestermcfo 50079 | The base function restrict... |
| termcnex 50080 | The class of all terminal ... |
| mndtcval 50083 | Value of the category buil... |
| mndtcbasval 50084 | The base set of the catego... |
| mndtcbas 50085 | The category built from a ... |
| mndtcob 50086 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 50087 | Two objects in a category ... |
| mndtchom 50088 | The only hom-set of the ca... |
| mndtcco 50089 | The composition of the cat... |
| mndtcco2 50090 | The composition of the cat... |
| mndtccatid 50091 | Lemma for ~ mndtccat and ~... |
| mndtccat 50092 | The function value is a ca... |
| mndtcid 50093 | The identity morphism, or ... |
| oppgoppchom 50094 | The converted opposite mon... |
| oppgoppcco 50095 | The converted opposite mon... |
| oppgoppcid 50096 | The converted opposite mon... |
| grptcmon 50097 | All morphisms in a categor... |
| grptcepi 50098 | All morphisms in a categor... |
| 2arwcatlem1 50099 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem2 50100 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem3 50101 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem4 50102 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem5 50103 | Lemma for ~ 2arwcat . (Co... |
| 2arwcat 50104 | The condition for a struct... |
| incat 50105 | Constructing a category wi... |
| setc1onsubc 50106 | Construct a category with ... |
| cnelsubclem 50107 | Lemma for ~ cnelsubc . (C... |
| cnelsubc 50108 | Remark 4.2(2) of [Adamek] ... |
| lanfn 50113 | ` Lan ` is a function on `... |
| ranfn 50114 | ` Ran ` is a function on `... |
| reldmlan 50115 | The domain of ` Lan ` is a... |
| reldmran 50116 | The domain of ` Ran ` is a... |
| lanfval 50117 | Value of the function gene... |
| ranfval 50118 | Value of the function gene... |
| lanpropd 50119 | If the categories have the... |
| ranpropd 50120 | If the categories have the... |
| reldmlan2 50121 | The domain of ` ( P Lan E ... |
| reldmran2 50122 | The domain of ` ( P Ran E ... |
| lanval 50123 | Value of the set of left K... |
| ranval 50124 | Value of the set of right ... |
| lanrcl 50125 | Reverse closure for left K... |
| ranrcl 50126 | Reverse closure for right ... |
| rellan 50127 | The set of left Kan extens... |
| relran 50128 | The set of right Kan exten... |
| islan 50129 | A left Kan extension is a ... |
| islan2 50130 | A left Kan extension is a ... |
| lanval2 50131 | The set of left Kan extens... |
| isran 50132 | A right Kan extension is a... |
| isran2 50133 | A right Kan extension is a... |
| ranval2 50134 | The set of right Kan exten... |
| ranval3 50135 | The set of right Kan exten... |
| lanrcl2 50136 | Reverse closure for left K... |
| lanrcl3 50137 | Reverse closure for left K... |
| lanrcl4 50138 | The first component of a l... |
| lanrcl5 50139 | The second component of a ... |
| ranrcl2 50140 | Reverse closure for right ... |
| ranrcl3 50141 | Reverse closure for right ... |
| ranrcl4lem 50142 | Lemma for ~ ranrcl4 and ~ ... |
| ranrcl4 50143 | The first component of a r... |
| ranrcl5 50144 | The second component of a ... |
| lanup 50145 | The universal property of ... |
| ranup 50146 | The universal property of ... |
| reldmlmd 50151 | The domain of ` Limit ` is... |
| reldmcmd 50152 | The domain of ` Colimit ` ... |
| lmdfval 50153 | Function value of ` Limit ... |
| cmdfval 50154 | Function value of ` Colimi... |
| lmdrcl 50155 | Reverse closure for a limi... |
| cmdrcl 50156 | Reverse closure for a coli... |
| reldmlmd2 50157 | The domain of ` ( C Limit ... |
| reldmcmd2 50158 | The domain of ` ( C Colimi... |
| lmdfval2 50159 | The set of limits of a dia... |
| cmdfval2 50160 | The set of colimits of a d... |
| lmdpropd 50161 | If the categories have the... |
| cmdpropd 50162 | If the categories have the... |
| rellmd 50163 | The set of limits of a dia... |
| relcmd 50164 | The set of colimits of a d... |
| concl 50165 | A natural transformation f... |
| coccl 50166 | A natural transformation t... |
| concom 50167 | A cone to a diagram commut... |
| coccom 50168 | A co-cone to a diagram com... |
| islmd 50169 | The universal property of ... |
| iscmd 50170 | The universal property of ... |
| lmddu 50171 | The duality of limits and ... |
| cmddu 50172 | The duality of limits and ... |
| initocmd 50173 | Initial objects are the ob... |
| termolmd 50174 | Terminal objects are the o... |
| lmdran 50175 | To each limit of a diagram... |
| cmdlan 50176 | To each colimit of a diagr... |
| nfintd 50177 | Bound-variable hypothesis ... |
| nfiund 50178 | Bound-variable hypothesis ... |
| nfiundg 50179 | Bound-variable hypothesis ... |
| iunord 50180 | The indexed union of a col... |
| iunordi 50181 | The indexed union of a col... |
| spd 50182 | Specialization deduction, ... |
| spcdvw 50183 | A version of ~ spcdv where... |
| tfis2d 50184 | Transfinite Induction Sche... |
| bnd2d 50185 | Deduction form of ~ bnd2 .... |
| dffun3f 50186 | Alternate definition of fu... |
| setrecseq 50189 | Equality theorem for set r... |
| nfsetrecs 50190 | Bound-variable hypothesis ... |
| setrec1lem1 50191 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 50192 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 50193 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 50194 | Lemma for ~ setrec1 . If ... |
| setrec1 50195 | This is the first of two f... |
| setrec2fun 50196 | This is the second of two ... |
| setrec2lem1 50197 | Lemma for ~ setrec2 . The... |
| setrec2lem2 50198 | Lemma for ~ setrec2 . The... |
| setrec2 50199 | This is the second of two ... |
| setrec2v 50200 | Version of ~ setrec2 with ... |
| setrec2mpt 50201 | Version of ~ setrec2 where... |
| setis 50202 | Version of ~ setrec2 expre... |
| elsetrecslem 50203 | Lemma for ~ elsetrecs . A... |
| elsetrecs 50204 | A set ` A ` is an element ... |
| setrecsss 50205 | The ` setrecs ` operator r... |
| setrecsres 50206 | A recursively generated cl... |
| vsetrec 50207 | Construct ` _V ` using set... |
| 0setrec 50208 | If a function sends the em... |
| onsetreclem1 50209 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 50210 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 50211 | Lemma for ~ onsetrec . (C... |
| onsetrec 50212 | Construct ` On ` using set... |
| elpglem1 50215 | Lemma for ~ elpg . (Contr... |
| elpglem2 50216 | Lemma for ~ elpg . (Contr... |
| elpglem3 50217 | Lemma for ~ elpg . (Contr... |
| elpg 50218 | Membership in the class of... |
| pgindlem 50219 | Lemma for ~ pgind . (Cont... |
| pgindnf 50220 | Version of ~ pgind with ex... |
| pgind 50221 | Induction on partizan game... |
| sbidd 50222 | An identity theorem for su... |
| sbidd-misc 50223 | An identity theorem for su... |
| gte-lte 50228 | Simple relationship betwee... |
| gt-lt 50229 | Simple relationship betwee... |
| gte-lteh 50230 | Relationship between ` <_ ... |
| gt-lth 50231 | Relationship between ` < `... |
| ex-gt 50232 | Simple example of ` > ` , ... |
| ex-gte 50233 | Simple example of ` >_ ` ,... |
| sinhval-named 50240 | Value of the named sinh fu... |
| coshval-named 50241 | Value of the named cosh fu... |
| tanhval-named 50242 | Value of the named tanh fu... |
| sinh-conventional 50243 | Conventional definition of... |
| sinhpcosh 50244 | Prove that ` ( sinh `` A )... |
| secval 50251 | Value of the secant functi... |
| cscval 50252 | Value of the cosecant func... |
| cotval 50253 | Value of the cotangent fun... |
| seccl 50254 | The closure of the secant ... |
| csccl 50255 | The closure of the cosecan... |
| cotcl 50256 | The closure of the cotange... |
| reseccl 50257 | The closure of the secant ... |
| recsccl 50258 | The closure of the cosecan... |
| recotcl 50259 | The closure of the cotange... |
| recsec 50260 | The reciprocal of secant i... |
| reccsc 50261 | The reciprocal of cosecant... |
| reccot 50262 | The reciprocal of cotangen... |
| rectan 50263 | The reciprocal of tangent ... |
| sec0 50264 | The value of the secant fu... |
| onetansqsecsq 50265 | Prove the tangent squared ... |
| cotsqcscsq 50266 | Prove the tangent squared ... |
| ifnmfalse 50267 | If A is not a member of B,... |
| logb2aval 50268 | Define the value of the ` ... |
| mvlraddi 50275 | Move the right term in a s... |
| assraddsubi 50276 | Associate RHS addition-sub... |
| joinlmuladdmuli 50277 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 50278 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 50279 | Join AB-CB into (A-C) on L... |
| mvlrmuld 50280 | Move the right term in a p... |
| mvlrmuli 50281 | Move the right term in a p... |
| i2linesi 50282 | Solve for the intersection... |
| i2linesd 50283 | Solve for the intersection... |
| alimp-surprise 50284 | Demonstrate that when usin... |
| alimp-no-surprise 50285 | There is no "surprise" in ... |
| empty-surprise 50286 | Demonstrate that when usin... |
| empty-surprise2 50287 | "Prove" that false is true... |
| eximp-surprise 50288 | Show what implication insi... |
| eximp-surprise2 50289 | Show that "there exists" w... |
| alsconv 50294 | There is an equivalence be... |
| alsi1d 50295 | Deduction rule: Given "al... |
| alsi2d 50296 | Deduction rule: Given "al... |
| alsc1d 50297 | Deduction rule: Given "al... |
| alsc2d 50298 | Deduction rule: Given "al... |
| alscn0d 50299 | Deduction rule: Given "al... |
| alsi-no-surprise 50300 | Demonstrate that there is ... |
| 5m4e1 50301 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 50302 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 50303 | Resolution rule. This is ... |
| testable 50304 | In classical logic all wff... |
| aacllem 50305 | Lemma for other theorems a... |
| amgmwlem 50306 | Weighted version of ~ amgm... |
| amgmlemALT 50307 | Alternate proof of ~ amgml... |
| amgmw2d 50308 | Weighted arithmetic-geomet... |
| young2d 50309 | Young's inequality for ` n... |
| Copyright terms: Public domain | W3C validator |