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.... |
| impt 178 | Importation theorem ~ pm3.... |
| pm2.61d 179 | Deduction eliminating an a... |
| pm2.61d1 180 | Inference eliminating an a... |
| pm2.61d2 181 | Inference eliminating an a... |
| pm2.61i 182 | Inference eliminating an a... |
| pm2.61ii 183 | Inference eliminating two ... |
| pm2.61nii 184 | Inference eliminating two ... |
| pm2.61iii 185 | Inference eliminating thre... |
| ja 186 | Inference joining the ante... |
| jad 187 | Deduction form of ~ ja . ... |
| pm2.01 188 | Weak Clavius law. If a fo... |
| pm2.01i 189 | Inference associated with ... |
| pm2.01d 190 | Deduction based on reducti... |
| pm2.6 191 | Theorem *2.6 of [Whitehead... |
| pm2.61 192 | Theorem *2.61 of [Whitehea... |
| pm2.65 193 | Theorem *2.65 of [Whitehea... |
| pm2.65i 194 | Inference for proof by con... |
| pm2.21dd 195 | A contradiction implies an... |
| pm2.65d 196 | Deduction for proof by con... |
| mto 197 | The rule of modus tollens.... |
| mtod 198 | Modus tollens deduction. ... |
| mtoi 199 | Modus tollens inference. ... |
| mt2 200 | A rule similar to modus to... |
| mt3 201 | A rule similar to modus to... |
| peirce 202 | Peirce's axiom. A non-int... |
| looinv 203 | The Inversion Axiom of the... |
| bijust0 204 | A self-implication (see ~ ... |
| bijust 205 | Theorem used to justify th... |
| impbi 208 | Property of the biconditio... |
| impbii 209 | Infer an equivalence from ... |
| impbidd 210 | Deduce an equivalence from... |
| impbid21d 211 | Deduce an equivalence from... |
| impbid 212 | Deduce an equivalence from... |
| dfbi1 213 | Relate the biconditional c... |
| dfbi1ALT 214 | Alternate proof of ~ dfbi1... |
| biimp 215 | Property of the biconditio... |
| biimpi 216 | Infer an implication from ... |
| sylbi 217 | A mixed syllogism inferenc... |
| sylib 218 | A mixed syllogism inferenc... |
| sylbb 219 | A mixed syllogism inferenc... |
| biimpr 220 | Property of the biconditio... |
| bicom1 221 | Commutative law for the bi... |
| bicom 222 | Commutative law for the bi... |
| bicomd 223 | Commute two sides of a bic... |
| bicomi 224 | Inference from commutative... |
| impbid1 225 | Infer an equivalence from ... |
| impbid2 226 | Infer an equivalence from ... |
| impcon4bid 227 | A variation on ~ impbid wi... |
| biimpri 228 | Infer a converse implicati... |
| biimpd 229 | Deduce an implication from... |
| mpbi 230 | An inference from a bicond... |
| mpbir 231 | An inference from a bicond... |
| mpbid 232 | A deduction from a bicondi... |
| mpbii 233 | An inference from a nested... |
| sylibr 234 | A mixed syllogism inferenc... |
| sylbir 235 | A mixed syllogism inferenc... |
| sylbbr 236 | A mixed syllogism inferenc... |
| sylbb1 237 | A mixed syllogism inferenc... |
| sylbb2 238 | A mixed syllogism inferenc... |
| sylibd 239 | A syllogism deduction. (C... |
| sylbid 240 | A syllogism deduction. (C... |
| mpbidi 241 | A deduction from a bicondi... |
| biimtrid 242 | A mixed syllogism inferenc... |
| biimtrrid 243 | A mixed syllogism inferenc... |
| imbitrid 244 | A mixed syllogism inferenc... |
| syl5ibcom 245 | A mixed syllogism inferenc... |
| imbitrrid 246 | A mixed syllogism inferenc... |
| syl5ibrcom 247 | A mixed syllogism inferenc... |
| biimprd 248 | Deduce a converse implicat... |
| biimpcd 249 | Deduce a commuted implicat... |
| biimprcd 250 | Deduce a converse commuted... |
| imbitrdi 251 | A mixed syllogism inferenc... |
| imbitrrdi 252 | A mixed syllogism inferenc... |
| biimtrdi 253 | A mixed syllogism inferenc... |
| biimtrrdi 254 | A mixed syllogism inferenc... |
| syl7bi 255 | A mixed syllogism inferenc... |
| syl8ib 256 | A syllogism rule of infere... |
| mpbird 257 | A deduction from a bicondi... |
| mpbiri 258 | An inference from a nested... |
| sylibrd 259 | A syllogism deduction. (C... |
| sylbird 260 | A syllogism deduction. (C... |
| biid 261 | Principle of identity for ... |
| biidd 262 | Principle of identity with... |
| pm5.1im 263 | Two propositions are equiv... |
| 2th 264 | Two truths are equivalent.... |
| 2thd 265 | Two truths are equivalent.... |
| monothetic 266 | Two self-implications (see... |
| ibi 267 | Inference that converts a ... |
| ibir 268 | Inference that converts a ... |
| ibd 269 | Deduction that converts a ... |
| pm5.74 270 | Distribution of implicatio... |
| pm5.74i 271 | Distribution of implicatio... |
| pm5.74ri 272 | Distribution of implicatio... |
| pm5.74d 273 | Distribution of implicatio... |
| pm5.74rd 274 | Distribution of implicatio... |
| bitri 275 | An inference from transiti... |
| bitr2i 276 | An inference from transiti... |
| bitr3i 277 | An inference from transiti... |
| bitr4i 278 | An inference from transiti... |
| bitrd 279 | Deduction form of ~ bitri ... |
| bitr2d 280 | Deduction form of ~ bitr2i... |
| bitr3d 281 | Deduction form of ~ bitr3i... |
| bitr4d 282 | Deduction form of ~ bitr4i... |
| bitrid 283 | A syllogism inference from... |
| bitr2id 284 | A syllogism inference from... |
| bitr3id 285 | A syllogism inference from... |
| bitr3di 286 | A syllogism inference from... |
| bitrdi 287 | A syllogism inference from... |
| bitr2di 288 | A syllogism inference from... |
| bitr4di 289 | A syllogism inference from... |
| bitr4id 290 | A syllogism inference from... |
| 3imtr3i 291 | A mixed syllogism inferenc... |
| 3imtr4i 292 | A mixed syllogism inferenc... |
| 3imtr3d 293 | More general version of ~ ... |
| 3imtr4d 294 | More general version of ~ ... |
| 3imtr3g 295 | More general version of ~ ... |
| 3imtr4g 296 | More general version of ~ ... |
| 3bitri 297 | A chained inference from t... |
| 3bitrri 298 | A chained inference from t... |
| 3bitr2i 299 | A chained inference from t... |
| 3bitr2ri 300 | A chained inference from t... |
| 3bitr3i 301 | A chained inference from t... |
| 3bitr3ri 302 | A chained inference from t... |
| 3bitr4i 303 | A chained inference from t... |
| 3bitr4ri 304 | A chained inference from t... |
| 3bitrd 305 | Deduction from transitivit... |
| 3bitrrd 306 | Deduction from transitivit... |
| 3bitr2d 307 | Deduction from transitivit... |
| 3bitr2rd 308 | Deduction from transitivit... |
| 3bitr3d 309 | Deduction from transitivit... |
| 3bitr3rd 310 | Deduction from transitivit... |
| 3bitr4d 311 | Deduction from transitivit... |
| 3bitr4rd 312 | Deduction from transitivit... |
| 3bitr3g 313 | More general version of ~ ... |
| 3bitr4g 314 | More general version of ~ ... |
| notnotb 315 | Double negation. Theorem ... |
| con34b 316 | A biconditional form of co... |
| con4bid 317 | A contraposition deduction... |
| notbid 318 | Deduction negating both si... |
| notbi 319 | Contraposition. Theorem *... |
| notbii 320 | Negate both sides of a log... |
| con4bii 321 | A contraposition inference... |
| mtbi 322 | An inference from a bicond... |
| mtbir 323 | An inference from a bicond... |
| mtbid 324 | A deduction from a bicondi... |
| mtbird 325 | A deduction from a bicondi... |
| mtbii 326 | An inference from a bicond... |
| mtbiri 327 | An inference from a bicond... |
| sylnib 328 | A mixed syllogism inferenc... |
| sylnibr 329 | A mixed syllogism inferenc... |
| sylnbi 330 | A mixed syllogism inferenc... |
| sylnbir 331 | A mixed syllogism inferenc... |
| xchnxbi 332 | Replacement of a subexpres... |
| xchnxbir 333 | Replacement of a subexpres... |
| xchbinx 334 | Replacement of a subexpres... |
| xchbinxr 335 | Replacement of a subexpres... |
| imbi2i 336 | Introduce an antecedent to... |
| bibi2i 337 | Inference adding a bicondi... |
| bibi1i 338 | Inference adding a bicondi... |
| bibi12i 339 | The equivalence of two equ... |
| imbi2d 340 | Deduction adding an antece... |
| imbi1d 341 | Deduction adding a consequ... |
| bibi2d 342 | Deduction adding a bicondi... |
| bibi1d 343 | Deduction adding a bicondi... |
| imbi12d 344 | Deduction joining two equi... |
| bibi12d 345 | Deduction joining two equi... |
| imbi12 346 | Closed form of ~ imbi12i .... |
| imbi1 347 | Theorem *4.84 of [Whitehea... |
| imbi2 348 | Theorem *4.85 of [Whitehea... |
| imbi1i 349 | Introduce a consequent to ... |
| imbi12i 350 | Join two logical equivalen... |
| bibi1 351 | Theorem *4.86 of [Whitehea... |
| bitr3 352 | Closed nested implication ... |
| con2bi 353 | Contraposition. Theorem *... |
| con2bid 354 | A contraposition deduction... |
| con1bid 355 | A contraposition deduction... |
| con1bii 356 | A contraposition inference... |
| con2bii 357 | A contraposition inference... |
| con1b 358 | Contraposition. Bidirecti... |
| con2b 359 | Contraposition. Bidirecti... |
| biimt 360 | A wff is equivalent to its... |
| pm5.5 361 | Theorem *5.5 of [Whitehead... |
| a1bi 362 | Inference introducing a th... |
| mt2bi 363 | A false consequent falsifi... |
| mtt 364 | Modus-tollens-like theorem... |
| imnot 365 | If a proposition is false,... |
| pm5.501 366 | Theorem *5.501 of [Whitehe... |
| ibib 367 | Implication in terms of im... |
| ibibr 368 | Implication in terms of im... |
| tbt 369 | A wff is equivalent to its... |
| nbn2 370 | The negation of a wff is e... |
| bibif 371 | Transfer negation via an e... |
| nbn 372 | The negation of a wff is e... |
| nbn3 373 | Transfer falsehood via equ... |
| pm5.21im 374 | Two propositions are equiv... |
| 2false 375 | Two falsehoods are equival... |
| 2falsed 376 | Two falsehoods are equival... |
| pm5.21ni 377 | Two propositions implying ... |
| pm5.21nii 378 | Eliminate an antecedent im... |
| pm5.21ndd 379 | Eliminate an antecedent im... |
| bija 380 | Combine antecedents into a... |
| pm5.18 381 | Theorem *5.18 of [Whitehea... |
| xor3 382 | Two ways to express "exclu... |
| nbbn 383 | Move negation outside of b... |
| biass 384 | Associative law for the bi... |
| biluk 385 | Lukasiewicz's shortest axi... |
| pm5.19 386 | Theorem *5.19 of [Whitehea... |
| bi2.04 387 | Logical equivalence of com... |
| pm5.4 388 | Antecedent absorption impl... |
| imdi 389 | Distributive law for impli... |
| pm5.41 390 | Theorem *5.41 of [Whitehea... |
| imbibi 391 | The antecedent of one side... |
| pm4.8 392 | Theorem *4.8 of [Whitehead... |
| pm4.81 393 | A formula is equivalent to... |
| imim21b 394 | Simplify an implication be... |
| pm4.63 397 | Theorem *4.63 of [Whitehea... |
| pm4.67 398 | Theorem *4.67 of [Whitehea... |
| imnan 399 | Express an implication in ... |
| imnani 400 | Infer an implication from ... |
| iman 401 | Implication in terms of co... |
| pm3.24 402 | Law of noncontradiction. ... |
| annim 403 | Express a conjunction in t... |
| pm4.61 404 | Theorem *4.61 of [Whitehea... |
| pm4.65 405 | Theorem *4.65 of [Whitehea... |
| imp 406 | Importation inference. (C... |
| impcom 407 | Importation inference with... |
| con3dimp 408 | Variant of ~ con3d with im... |
| mpnanrd 409 | Eliminate the right side o... |
| impd 410 | Importation deduction. (C... |
| impcomd 411 | Importation deduction with... |
| ex 412 | Exportation inference. (T... |
| expcom 413 | Exportation inference with... |
| expdcom 414 | Commuted form of ~ expd . ... |
| expd 415 | Exportation deduction. (C... |
| expcomd 416 | Deduction form of ~ expcom... |
| imp31 417 | An importation inference. ... |
| imp32 418 | An importation inference. ... |
| exp31 419 | An exportation inference. ... |
| exp32 420 | An exportation inference. ... |
| imp4b 421 | An importation inference. ... |
| imp4a 422 | An importation inference. ... |
| imp4c 423 | An importation inference. ... |
| imp4d 424 | An importation inference. ... |
| imp41 425 | An importation inference. ... |
| imp42 426 | An importation inference. ... |
| imp43 427 | An importation inference. ... |
| imp44 428 | An importation inference. ... |
| imp45 429 | An importation inference. ... |
| exp4b 430 | An exportation inference. ... |
| exp4a 431 | An exportation inference. ... |
| exp4c 432 | An exportation inference. ... |
| exp4d 433 | An exportation inference. ... |
| exp41 434 | An exportation inference. ... |
| exp42 435 | An exportation inference. ... |
| exp43 436 | An exportation inference. ... |
| exp44 437 | An exportation inference. ... |
| exp45 438 | An exportation inference. ... |
| imp5d 439 | An importation inference. ... |
| imp5a 440 | An importation inference. ... |
| imp5g 441 | An importation inference. ... |
| imp55 442 | An importation inference. ... |
| imp511 443 | An importation inference. ... |
| exp5c 444 | An exportation inference. ... |
| exp5j 445 | An exportation inference. ... |
| exp5l 446 | An exportation inference. ... |
| exp53 447 | An exportation inference. ... |
| pm3.3 448 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 449 | Theorem *3.31 (Imp) of [Wh... |
| impexp 450 | Import-export theorem. Pa... |
| impancom 451 | Mixed importation/commutat... |
| expdimp 452 | A deduction version of exp... |
| expimpd 453 | Exportation followed by a ... |
| impr 454 | Import a wff into a right ... |
| impl 455 | Export a wff from a left c... |
| expr 456 | Export a wff from a right ... |
| expl 457 | Export a wff from a left c... |
| ancoms 458 | Inference commuting conjun... |
| pm3.22 459 | Theorem *3.22 of [Whitehea... |
| ancom 460 | Commutative law for conjun... |
| ancomd 461 | Commutation of conjuncts i... |
| biancomi 462 | Commuting conjunction in a... |
| biancomd 463 | Commuting conjunction in a... |
| ancomst 464 | Closed form of ~ ancoms . ... |
| ancomsd 465 | Deduction commuting conjun... |
| anasss 466 | Associative law for conjun... |
| anassrs 467 | Associative law for conjun... |
| anass 468 | Associative law for conjun... |
| pm3.2 469 | Join antecedents with conj... |
| pm3.2i 470 | Infer conjunction of premi... |
| pm3.21 471 | Join antecedents with conj... |
| pm3.43i 472 | Nested conjunction of ante... |
| pm3.43 473 | Theorem *3.43 (Comp) of [W... |
| dfbi2 474 | A theorem similar to the s... |
| dfbi 475 | Definition ~ df-bi rewritt... |
| biimpa 476 | Importation inference from... |
| biimpar 477 | Importation inference from... |
| biimpac 478 | Importation inference from... |
| biimparc 479 | Importation inference from... |
| adantr 480 | Inference adding a conjunc... |
| adantl 481 | Inference adding a conjunc... |
| simpl 482 | Elimination of a conjunct.... |
| simpli 483 | Inference eliminating a co... |
| simpr 484 | Elimination of a conjunct.... |
| simpri 485 | Inference eliminating a co... |
| intnan 486 | Introduction of conjunct i... |
| intnanr 487 | Introduction of conjunct i... |
| intnand 488 | Introduction of conjunct i... |
| intnanrd 489 | Introduction of conjunct i... |
| adantld 490 | Deduction adding a conjunc... |
| adantrd 491 | Deduction adding a conjunc... |
| pm3.41 492 | Theorem *3.41 of [Whitehea... |
| pm3.42 493 | Theorem *3.42 of [Whitehea... |
| simpld 494 | Deduction eliminating a co... |
| simprd 495 | Deduction eliminating a co... |
| simprbi 496 | Deduction eliminating a co... |
| simplbi 497 | Deduction eliminating a co... |
| simprbda 498 | Deduction eliminating a co... |
| simplbda 499 | Deduction eliminating a co... |
| simplbi2 500 | Deduction eliminating a co... |
| simplbi2comt 501 | Closed form of ~ simplbi2c... |
| simplbi2com 502 | A deduction eliminating a ... |
| simpl2im 503 | Implication from an elimin... |
| simplbiim 504 | Implication from an elimin... |
| impel 505 | An inference for implicati... |
| mpan9 506 | Modus ponens conjoining di... |
| sylan9 507 | Nested syllogism inference... |
| sylan9r 508 | Nested syllogism inference... |
| sylan9bb 509 | Nested syllogism inference... |
| sylan9bbr 510 | Nested syllogism inference... |
| jca 511 | Deduce conjunction of the ... |
| jcad 512 | Deduction conjoining the c... |
| jca2 513 | Inference conjoining the c... |
| jca31 514 | Join three consequents. (... |
| jca32 515 | Join three consequents. (... |
| jcai 516 | Deduction replacing implic... |
| jcab 517 | Distributive law for impli... |
| pm4.76 518 | Theorem *4.76 of [Whitehea... |
| jctil 519 | Inference conjoining a the... |
| jctir 520 | Inference conjoining a the... |
| jccir 521 | Inference conjoining a con... |
| jccil 522 | Inference conjoining a con... |
| jctl 523 | Inference conjoining a the... |
| jctr 524 | Inference conjoining a the... |
| jctild 525 | Deduction conjoining a the... |
| jctird 526 | Deduction conjoining a the... |
| iba 527 | Introduction of antecedent... |
| ibar 528 | Introduction of antecedent... |
| biantru 529 | A wff is equivalent to its... |
| biantrur 530 | A wff is equivalent to its... |
| biantrud 531 | A wff is equivalent to its... |
| biantrurd 532 | A wff is equivalent to its... |
| bianfi 533 | A wff conjoined with false... |
| bianfd 534 | A wff conjoined with false... |
| baib 535 | Move conjunction outside o... |
| baibr 536 | Move conjunction outside o... |
| rbaibr 537 | Move conjunction outside o... |
| rbaib 538 | Move conjunction outside o... |
| baibd 539 | Move conjunction outside o... |
| rbaibd 540 | Move conjunction outside o... |
| bianabs 541 | Absorb a hypothesis into t... |
| pm5.44 542 | Theorem *5.44 of [Whitehea... |
| pm5.42 543 | Theorem *5.42 of [Whitehea... |
| ancl 544 | Conjoin antecedent to left... |
| anclb 545 | Conjoin antecedent to left... |
| ancr 546 | Conjoin antecedent to righ... |
| ancrb 547 | Conjoin antecedent to righ... |
| ancli 548 | Deduction conjoining antec... |
| ancri 549 | Deduction conjoining antec... |
| ancld 550 | Deduction conjoining antec... |
| ancrd 551 | Deduction conjoining antec... |
| impac 552 | Importation with conjuncti... |
| anc2l 553 | Conjoin antecedent to left... |
| anc2r 554 | Conjoin antecedent to righ... |
| anc2li 555 | Deduction conjoining antec... |
| anc2ri 556 | Deduction conjoining antec... |
| pm4.71 557 | Implication in terms of bi... |
| pm4.71r 558 | Implication in terms of bi... |
| pm4.71i 559 | Inference converting an im... |
| pm4.71ri 560 | Inference converting an im... |
| pm4.71d 561 | Deduction converting an im... |
| pm4.71rd 562 | Deduction converting an im... |
| pm4.24 563 | Theorem *4.24 of [Whitehea... |
| anidm 564 | Idempotent law for conjunc... |
| anidmdbi 565 | Conjunction idempotence wi... |
| anidms 566 | Inference from idempotent ... |
| imdistan 567 | Distribution of implicatio... |
| imdistani 568 | Distribution of implicatio... |
| imdistanri 569 | Distribution of implicatio... |
| imdistand 570 | Distribution of implicatio... |
| imdistanda 571 | Distribution of implicatio... |
| pm5.3 572 | Theorem *5.3 of [Whitehead... |
| pm5.32 573 | Distribution of implicatio... |
| pm5.32i 574 | Distribution of implicatio... |
| pm5.32ri 575 | Distribution of implicatio... |
| bianim 576 | Exchanging conjunction in ... |
| pm5.32d 577 | Distribution of implicatio... |
| pm5.32rd 578 | Distribution of implicatio... |
| pm5.32da 579 | Distribution of implicatio... |
| sylan 580 | A syllogism inference. (C... |
| sylanb 581 | A syllogism inference. (C... |
| sylanbr 582 | A syllogism inference. (C... |
| sylanbrc 583 | Syllogism inference. (Con... |
| syl2anc 584 | Syllogism inference combin... |
| syl2anc2 585 | Double syllogism inference... |
| sylancl 586 | Syllogism inference combin... |
| sylancr 587 | Syllogism inference combin... |
| sylancom 588 | Syllogism inference with c... |
| sylanblc 589 | Syllogism inference combin... |
| sylanblrc 590 | Syllogism inference combin... |
| syldan 591 | A syllogism deduction with... |
| sylbida 592 | A syllogism deduction. (C... |
| sylan2 593 | A syllogism inference. (C... |
| sylan2b 594 | A syllogism inference. (C... |
| sylan2br 595 | A syllogism inference. (C... |
| syl2an 596 | A double syllogism inferen... |
| syl2anr 597 | A double syllogism inferen... |
| syl2anb 598 | A double syllogism inferen... |
| syl2anbr 599 | A double syllogism inferen... |
| sylancb 600 | A syllogism inference comb... |
| sylancbr 601 | A syllogism inference comb... |
| syldanl 602 | A syllogism deduction with... |
| syland 603 | A syllogism deduction. (C... |
| sylani 604 | A syllogism inference. (C... |
| sylan2d 605 | A syllogism deduction. (C... |
| sylan2i 606 | A syllogism inference. (C... |
| syl2ani 607 | A syllogism inference. (C... |
| syl2and 608 | A syllogism deduction. (C... |
| anim12d 609 | Conjoin antecedents and co... |
| anim12d1 610 | Variant of ~ anim12d where... |
| anim1d 611 | Add a conjunct to right of... |
| anim2d 612 | Add a conjunct to left of ... |
| anim12i 613 | Conjoin antecedents and co... |
| anim12ci 614 | Variant of ~ anim12i with ... |
| anim1i 615 | Introduce conjunct to both... |
| anim1ci 616 | Introduce conjunct to both... |
| anim2i 617 | Introduce conjunct to both... |
| anim12ii 618 | Conjoin antecedents and co... |
| anim12dan 619 | Conjoin antecedents and co... |
| im2anan9 620 | Deduction joining nested i... |
| im2anan9r 621 | Deduction joining nested i... |
| pm3.45 622 | Theorem *3.45 (Fact) of [W... |
| anbi2i 623 | Introduce a left conjunct ... |
| anbi1i 624 | Introduce a right conjunct... |
| anbi2ci 625 | Variant of ~ anbi2i with c... |
| anbi1ci 626 | Variant of ~ anbi1i with c... |
| bianbi 627 | Exchanging conjunction in ... |
| anbi12i 628 | Conjoin both sides of two ... |
| anbi12ci 629 | Variant of ~ anbi12i with ... |
| anbi2d 630 | Deduction adding a left co... |
| anbi1d 631 | Deduction adding a right c... |
| anbi12d 632 | Deduction joining two equi... |
| anbi1 633 | Introduce a right conjunct... |
| anbi2 634 | Introduce a left conjunct ... |
| anbi1cd 635 | Introduce a proposition as... |
| an2anr 636 | Double commutation in conj... |
| pm4.38 637 | Theorem *4.38 of [Whitehea... |
| bi2anan9 638 | Deduction joining two equi... |
| bi2anan9r 639 | Deduction joining two equi... |
| bi2bian9 640 | Deduction joining two bico... |
| anbiim 641 | Adding biconditional when ... |
| bianass 642 | An inference to merge two ... |
| bianassc 643 | An inference to merge two ... |
| an21 644 | Swap two conjuncts. (Cont... |
| an12 645 | Swap two conjuncts. Note ... |
| an32 646 | A rearrangement of conjunc... |
| an13 647 | A rearrangement of conjunc... |
| an31 648 | A rearrangement of conjunc... |
| an12s 649 | Swap two conjuncts in ante... |
| ancom2s 650 | Inference commuting a nest... |
| an13s 651 | Swap two conjuncts in ante... |
| an32s 652 | Swap two conjuncts in ante... |
| ancom1s 653 | Inference commuting a nest... |
| an31s 654 | Swap two conjuncts in ante... |
| anass1rs 655 | Commutative-associative la... |
| an4 656 | Rearrangement of 4 conjunc... |
| an42 657 | Rearrangement of 4 conjunc... |
| an43 658 | Rearrangement of 4 conjunc... |
| an3 659 | A rearrangement of conjunc... |
| an4s 660 | Inference rearranging 4 co... |
| an42s 661 | Inference rearranging 4 co... |
| anabs1 662 | Absorption into embedded c... |
| anabs5 663 | Absorption into embedded c... |
| anabs7 664 | Absorption into embedded c... |
| anabsan 665 | Absorption of antecedent w... |
| anabss1 666 | Absorption of antecedent i... |
| anabss4 667 | Absorption of antecedent i... |
| anabss5 668 | Absorption of antecedent i... |
| anabsi5 669 | Absorption of antecedent i... |
| anabsi6 670 | Absorption of antecedent i... |
| anabsi7 671 | Absorption of antecedent i... |
| anabsi8 672 | Absorption of antecedent i... |
| anabss7 673 | Absorption of antecedent i... |
| anabsan2 674 | Absorption of antecedent w... |
| anabss3 675 | Absorption of antecedent i... |
| anandi 676 | Distribution of conjunctio... |
| anandir 677 | Distribution of conjunctio... |
| anandis 678 | Inference that undistribut... |
| anandirs 679 | Inference that undistribut... |
| sylanl1 680 | A syllogism inference. (C... |
| sylanl2 681 | A syllogism inference. (C... |
| sylanr1 682 | A syllogism inference. (C... |
| sylanr2 683 | A syllogism inference. (C... |
| syl6an 684 | A syllogism deduction comb... |
| syl2an2r 685 | ~ syl2anr with antecedents... |
| syl2an2 686 | ~ syl2an with antecedents ... |
| mpdan 687 | An inference based on modu... |
| mpancom 688 | An inference based on modu... |
| mpidan 689 | A deduction which "stacks"... |
| mpan 690 | An inference based on modu... |
| mpan2 691 | An inference based on modu... |
| mp2an 692 | An inference based on modu... |
| mp4an 693 | An inference based on modu... |
| mpan2d 694 | A deduction based on modus... |
| mpand 695 | A deduction based on modus... |
| mpani 696 | An inference based on modu... |
| mpan2i 697 | An inference based on modu... |
| mp2ani 698 | An inference based on modu... |
| mp2and 699 | A deduction based on modus... |
| mpanl1 700 | An inference based on modu... |
| mpanl2 701 | An inference based on modu... |
| mpanl12 702 | An inference based on modu... |
| mpanr1 703 | An inference based on modu... |
| mpanr2 704 | An inference based on modu... |
| mpanr12 705 | An inference based on modu... |
| mpanlr1 706 | An inference based on modu... |
| mpbirand 707 | Detach truth from conjunct... |
| mpbiran2d 708 | Detach truth from conjunct... |
| mpbiran 709 | Detach truth from conjunct... |
| mpbiran2 710 | Detach truth from conjunct... |
| mpbir2an 711 | Detach a conjunction of tr... |
| mpbi2and 712 | Detach a conjunction of tr... |
| mpbir2and 713 | Detach a conjunction of tr... |
| adantll 714 | Deduction adding a conjunc... |
| adantlr 715 | Deduction adding a conjunc... |
| adantrl 716 | Deduction adding a conjunc... |
| adantrr 717 | Deduction adding a conjunc... |
| adantlll 718 | Deduction adding a conjunc... |
| adantllr 719 | Deduction adding a conjunc... |
| adantlrl 720 | Deduction adding a conjunc... |
| adantlrr 721 | Deduction adding a conjunc... |
| adantrll 722 | Deduction adding a conjunc... |
| adantrlr 723 | Deduction adding a conjunc... |
| adantrrl 724 | Deduction adding a conjunc... |
| adantrrr 725 | Deduction adding a conjunc... |
| ad2antrr 726 | Deduction adding two conju... |
| ad2antlr 727 | Deduction adding two conju... |
| ad2antrl 728 | Deduction adding two conju... |
| ad2antll 729 | Deduction adding conjuncts... |
| ad3antrrr 730 | Deduction adding three con... |
| ad3antlr 731 | Deduction adding three con... |
| ad4antr 732 | Deduction adding 4 conjunc... |
| ad4antlr 733 | Deduction adding 4 conjunc... |
| ad5antr 734 | Deduction adding 5 conjunc... |
| ad5antlr 735 | Deduction adding 5 conjunc... |
| ad6antr 736 | Deduction adding 6 conjunc... |
| ad6antlr 737 | Deduction adding 6 conjunc... |
| ad7antr 738 | Deduction adding 7 conjunc... |
| ad7antlr 739 | Deduction adding 7 conjunc... |
| ad8antr 740 | Deduction adding 8 conjunc... |
| ad8antlr 741 | Deduction adding 8 conjunc... |
| ad9antr 742 | Deduction adding 9 conjunc... |
| ad9antlr 743 | Deduction adding 9 conjunc... |
| ad10antr 744 | Deduction adding 10 conjun... |
| ad10antlr 745 | Deduction adding 10 conjun... |
| ad2ant2l 746 | Deduction adding two conju... |
| ad2ant2r 747 | Deduction adding two conju... |
| ad2ant2lr 748 | Deduction adding two conju... |
| ad2ant2rl 749 | Deduction adding two conju... |
| adantl3r 750 | Deduction adding 1 conjunc... |
| ad4ant13 751 | Deduction adding conjuncts... |
| ad4ant14 752 | Deduction adding conjuncts... |
| ad4ant23 753 | Deduction adding conjuncts... |
| ad4ant24 754 | Deduction adding conjuncts... |
| adantl4r 755 | Deduction adding 1 conjunc... |
| ad5ant13 756 | Deduction adding conjuncts... |
| ad5ant14 757 | Deduction adding conjuncts... |
| ad5ant15 758 | Deduction adding conjuncts... |
| ad5ant23 759 | Deduction adding conjuncts... |
| ad5ant24 760 | Deduction adding conjuncts... |
| ad5ant25 761 | Deduction adding conjuncts... |
| adantl5r 762 | Deduction adding 1 conjunc... |
| adantl6r 763 | Deduction adding 1 conjunc... |
| pm3.33 764 | Theorem *3.33 (Syll) of [W... |
| pm3.34 765 | Theorem *3.34 (Syll) of [W... |
| simpll 766 | Simplification of a conjun... |
| simplld 767 | Deduction form of ~ simpll... |
| simplr 768 | Simplification of a conjun... |
| simplrd 769 | Deduction eliminating a do... |
| simprl 770 | Simplification of a conjun... |
| simprld 771 | Deduction eliminating a do... |
| simprr 772 | Simplification of a conjun... |
| simprrd 773 | Deduction form of ~ simprr... |
| simplll 774 | Simplification of a conjun... |
| simpllr 775 | Simplification of a conjun... |
| simplrl 776 | Simplification of a conjun... |
| simplrr 777 | Simplification of a conjun... |
| simprll 778 | Simplification of a conjun... |
| simprlr 779 | Simplification of a conjun... |
| simprrl 780 | Simplification of a conjun... |
| simprrr 781 | Simplification of a conjun... |
| simp-4l 782 | Simplification of a conjun... |
| simp-4r 783 | Simplification of a conjun... |
| simp-5l 784 | Simplification of a conjun... |
| simp-5r 785 | Simplification of a conjun... |
| simp-6l 786 | Simplification of a conjun... |
| simp-6r 787 | Simplification of a conjun... |
| simp-7l 788 | Simplification of a conjun... |
| simp-7r 789 | Simplification of a conjun... |
| simp-8l 790 | Simplification of a conjun... |
| simp-8r 791 | Simplification of a conjun... |
| simp-9l 792 | Simplification of a conjun... |
| simp-9r 793 | Simplification of a conjun... |
| simp-10l 794 | Simplification of a conjun... |
| simp-10r 795 | Simplification of a conjun... |
| simp-11l 796 | Simplification of a conjun... |
| simp-11r 797 | Simplification of a conjun... |
| pm2.01da 798 | Deduction based on reducti... |
| pm2.18da 799 | Deduction based on reducti... |
| impbida 800 | Deduce an equivalence from... |
| pm5.21nd 801 | Eliminate an antecedent im... |
| pm3.35 802 | Conjunctive detachment. T... |
| pm5.74da 803 | Distribution of implicatio... |
| bitr 804 | Theorem *4.22 of [Whitehea... |
| biantr 805 | A transitive law of equiva... |
| pm4.14 806 | Theorem *4.14 of [Whitehea... |
| pm3.37 807 | Theorem *3.37 (Transp) of ... |
| anim12 808 | Conjoin antecedents and co... |
| pm3.4 809 | Conjunction implies implic... |
| exbiri 810 | Inference form of ~ exbir ... |
| pm2.61ian 811 | Elimination of an antecede... |
| pm2.61dan 812 | Elimination of an antecede... |
| pm2.61ddan 813 | Elimination of two anteced... |
| pm2.61dda 814 | Elimination of two anteced... |
| mtand 815 | A modus tollens deduction.... |
| pm2.65da 816 | Deduction for proof by con... |
| condan 817 | Proof by contradiction. (... |
| biadan 818 | An implication is equivale... |
| biadani 819 | Inference associated with ... |
| biadaniALT 820 | Alternate proof of ~ biada... |
| biadanii 821 | Inference associated with ... |
| biadanid 822 | Deduction associated with ... |
| pm5.1 823 | Two propositions are equiv... |
| pm5.21 824 | Two propositions are equiv... |
| pm5.35 825 | Theorem *5.35 of [Whitehea... |
| abai 826 | Introduce one conjunct as ... |
| pm4.45im 827 | Conjunction with implicati... |
| impimprbi 828 | An implication and its rev... |
| nan 829 | Theorem to move a conjunct... |
| pm5.31 830 | Theorem *5.31 of [Whitehea... |
| pm5.31r 831 | Variant of ~ pm5.31 . (Co... |
| pm4.15 832 | Theorem *4.15 of [Whitehea... |
| pm5.36 833 | Theorem *5.36 of [Whitehea... |
| annotanannot 834 | A conjunction with a negat... |
| pm5.33 835 | Theorem *5.33 of [Whitehea... |
| syl12anc 836 | Syllogism combined with co... |
| syl21anc 837 | Syllogism combined with co... |
| syl22anc 838 | Syllogism combined with co... |
| bibiad 839 | Eliminate an hypothesis ` ... |
| syl1111anc 840 | Four-hypothesis eliminatio... |
| syldbl2 841 | Stacked hypotheseis implie... |
| mpsyl4anc 842 | An elimination deduction. ... |
| pm4.87 843 | Theorem *4.87 of [Whitehea... |
| bimsc1 844 | Removal of conjunct from o... |
| a2and 845 | Deduction distributing a c... |
| animpimp2impd 846 | Deduction deriving nested ... |
| pm4.64 849 | Theorem *4.64 of [Whitehea... |
| pm4.66 850 | Theorem *4.66 of [Whitehea... |
| pm2.53 851 | Theorem *2.53 of [Whitehea... |
| pm2.54 852 | Theorem *2.54 of [Whitehea... |
| imor 853 | Implication in terms of di... |
| imori 854 | Infer disjunction from imp... |
| imorri 855 | Infer implication from dis... |
| pm4.62 856 | Theorem *4.62 of [Whitehea... |
| jaoi 857 | Inference disjoining the a... |
| jao1i 858 | Add a disjunct in the ante... |
| jaod 859 | Deduction disjoining the a... |
| mpjaod 860 | Eliminate a disjunction in... |
| ori 861 | Infer implication from dis... |
| orri 862 | Infer disjunction from imp... |
| orrd 863 | Deduce disjunction from im... |
| ord 864 | Deduce implication from di... |
| orci 865 | Deduction introducing a di... |
| olci 866 | Deduction introducing a di... |
| orc 867 | Introduction of a disjunct... |
| olc 868 | Introduction of a disjunct... |
| pm1.4 869 | Axiom *1.4 of [WhiteheadRu... |
| orcom 870 | Commutative law for disjun... |
| orcomd 871 | Commutation of disjuncts i... |
| orcoms 872 | Commutation of disjuncts i... |
| orcd 873 | Deduction introducing a di... |
| olcd 874 | Deduction introducing a di... |
| orcs 875 | Deduction eliminating disj... |
| olcs 876 | Deduction eliminating disj... |
| olcnd 877 | A lemma for Conjunctive No... |
| orcnd 878 | A lemma for Conjunctive No... |
| mtord 879 | A modus tollens deduction ... |
| pm3.2ni 880 | Infer negated disjunction ... |
| pm2.45 881 | Theorem *2.45 of [Whitehea... |
| pm2.46 882 | Theorem *2.46 of [Whitehea... |
| pm2.47 883 | Theorem *2.47 of [Whitehea... |
| pm2.48 884 | Theorem *2.48 of [Whitehea... |
| pm2.49 885 | Theorem *2.49 of [Whitehea... |
| norbi 886 | If neither of two proposit... |
| nbior 887 | If two propositions are no... |
| orel1 888 | Elimination of disjunction... |
| pm2.25 889 | Theorem *2.25 of [Whitehea... |
| orel2 890 | Elimination of disjunction... |
| pm2.67-2 891 | Slight generalization of T... |
| pm2.67 892 | Theorem *2.67 of [Whitehea... |
| curryax 893 | A non-intuitionistic posit... |
| exmid 894 | Law of excluded middle, al... |
| exmidd 895 | Law of excluded middle in ... |
| pm2.1 896 | Theorem *2.1 of [Whitehead... |
| pm2.13 897 | Theorem *2.13 of [Whitehea... |
| pm2.621 898 | Theorem *2.621 of [Whitehe... |
| pm2.62 899 | Theorem *2.62 of [Whitehea... |
| pm2.68 900 | Theorem *2.68 of [Whitehea... |
| dfor2 901 | Logical 'or' expressed in ... |
| pm2.07 902 | Theorem *2.07 of [Whitehea... |
| pm1.2 903 | Axiom *1.2 of [WhiteheadRu... |
| oridm 904 | Idempotent law for disjunc... |
| pm4.25 905 | Theorem *4.25 of [Whitehea... |
| pm2.4 906 | Theorem *2.4 of [Whitehead... |
| pm2.41 907 | Theorem *2.41 of [Whitehea... |
| orim12i 908 | Disjoin antecedents and co... |
| orim1i 909 | Introduce disjunct to both... |
| orim2i 910 | Introduce disjunct to both... |
| orim12dALT 911 | Alternate proof of ~ orim1... |
| orbi2i 912 | Inference adding a left di... |
| orbi1i 913 | Inference adding a right d... |
| orbi12i 914 | Infer the disjunction of t... |
| orbi2d 915 | Deduction adding a left di... |
| orbi1d 916 | Deduction adding a right d... |
| orbi1 917 | Theorem *4.37 of [Whitehea... |
| orbi12d 918 | Deduction joining two equi... |
| pm1.5 919 | Axiom *1.5 (Assoc) of [Whi... |
| or12 920 | Swap two disjuncts. (Cont... |
| orass 921 | Associative law for disjun... |
| pm2.31 922 | Theorem *2.31 of [Whitehea... |
| pm2.32 923 | Theorem *2.32 of [Whitehea... |
| pm2.3 924 | Theorem *2.3 of [Whitehead... |
| or32 925 | A rearrangement of disjunc... |
| or4 926 | Rearrangement of 4 disjunc... |
| or42 927 | Rearrangement of 4 disjunc... |
| orordi 928 | Distribution of disjunctio... |
| orordir 929 | Distribution of disjunctio... |
| orimdi 930 | Disjunction distributes ov... |
| pm2.76 931 | Theorem *2.76 of [Whitehea... |
| pm2.85 932 | Theorem *2.85 of [Whitehea... |
| pm2.75 933 | Theorem *2.75 of [Whitehea... |
| pm4.78 934 | Implication distributes ov... |
| biort 935 | A disjunction with a true ... |
| biorf 936 | A wff is equivalent to its... |
| biortn 937 | A wff is equivalent to its... |
| biorfi 938 | The dual of ~ biorf is not... |
| biorfri 939 | A wff is equivalent to its... |
| biorfriOLD 940 | Obsolete version of ~ bior... |
| pm2.26 941 | Theorem *2.26 of [Whitehea... |
| pm2.63 942 | Theorem *2.63 of [Whitehea... |
| pm2.64 943 | Theorem *2.64 of [Whitehea... |
| pm2.42 944 | Theorem *2.42 of [Whitehea... |
| pm5.11g 945 | A general instance of Theo... |
| pm5.11 946 | Theorem *5.11 of [Whitehea... |
| pm5.12 947 | Theorem *5.12 of [Whitehea... |
| pm5.14 948 | Theorem *5.14 of [Whitehea... |
| pm5.13 949 | Theorem *5.13 of [Whitehea... |
| pm5.55 950 | Theorem *5.55 of [Whitehea... |
| pm4.72 951 | Implication in terms of bi... |
| imimorb 952 | Simplify an implication be... |
| oibabs 953 | Absorption of disjunction ... |
| orbidi 954 | Disjunction distributes ov... |
| pm5.7 955 | Disjunction distributes ov... |
| jaao 956 | Inference conjoining and d... |
| jaoa 957 | Inference disjoining and c... |
| jaoian 958 | Inference disjoining the a... |
| jaodan 959 | Deduction disjoining the a... |
| mpjaodan 960 | Eliminate a disjunction in... |
| pm3.44 961 | Theorem *3.44 of [Whitehea... |
| jao 962 | Disjunction of antecedents... |
| jaob 963 | Disjunction of antecedents... |
| pm4.77 964 | Theorem *4.77 of [Whitehea... |
| pm3.48 965 | Theorem *3.48 of [Whitehea... |
| orim12d 966 | Disjoin antecedents and co... |
| orim1d 967 | Disjoin antecedents and co... |
| orim2d 968 | Disjoin antecedents and co... |
| orim2 969 | Axiom *1.6 (Sum) of [White... |
| pm2.38 970 | Theorem *2.38 of [Whitehea... |
| pm2.36 971 | Theorem *2.36 of [Whitehea... |
| pm2.37 972 | Theorem *2.37 of [Whitehea... |
| pm2.81 973 | Theorem *2.81 of [Whitehea... |
| pm2.8 974 | Theorem *2.8 of [Whitehead... |
| pm2.73 975 | Theorem *2.73 of [Whitehea... |
| pm2.74 976 | Theorem *2.74 of [Whitehea... |
| pm2.82 977 | Theorem *2.82 of [Whitehea... |
| pm4.39 978 | Theorem *4.39 of [Whitehea... |
| animorl 979 | Conjunction implies disjun... |
| animorr 980 | Conjunction implies disjun... |
| animorlr 981 | Conjunction implies disjun... |
| animorrl 982 | Conjunction implies disjun... |
| ianor 983 | Negated conjunction in ter... |
| anor 984 | Conjunction in terms of di... |
| ioran 985 | Negated disjunction in ter... |
| pm4.52 986 | Theorem *4.52 of [Whitehea... |
| pm4.53 987 | Theorem *4.53 of [Whitehea... |
| pm4.54 988 | Theorem *4.54 of [Whitehea... |
| pm4.55 989 | Theorem *4.55 of [Whitehea... |
| pm4.56 990 | Theorem *4.56 of [Whitehea... |
| oran 991 | Disjunction in terms of co... |
| pm4.57 992 | Theorem *4.57 of [Whitehea... |
| pm3.1 993 | Theorem *3.1 of [Whitehead... |
| pm3.11 994 | Theorem *3.11 of [Whitehea... |
| pm3.12 995 | Theorem *3.12 of [Whitehea... |
| pm3.13 996 | Theorem *3.13 of [Whitehea... |
| pm3.14 997 | Theorem *3.14 of [Whitehea... |
| pm4.44 998 | Theorem *4.44 of [Whitehea... |
| pm4.45 999 | Theorem *4.45 of [Whitehea... |
| orabs 1000 | Absorption of redundant in... |
| oranabs 1001 | Absorb a disjunct into a c... |
| pm5.61 1002 | Theorem *5.61 of [Whitehea... |
| pm5.6 1003 | Conjunction in antecedent ... |
| orcanai 1004 | Change disjunction in cons... |
| pm4.79 1005 | Theorem *4.79 of [Whitehea... |
| pm5.53 1006 | Theorem *5.53 of [Whitehea... |
| ordi 1007 | Distributive law for disju... |
| ordir 1008 | Distributive law for disju... |
| andi 1009 | Distributive law for conju... |
| andir 1010 | Distributive law for conju... |
| orddi 1011 | Double distributive law fo... |
| anddi 1012 | Double distributive law fo... |
| pm5.17 1013 | Theorem *5.17 of [Whitehea... |
| pm5.15 1014 | Theorem *5.15 of [Whitehea... |
| pm5.16 1015 | Theorem *5.16 of [Whitehea... |
| xor 1016 | Two ways to express exclus... |
| nbi2 1017 | Two ways to express "exclu... |
| xordi 1018 | Conjunction distributes ov... |
| pm5.54 1019 | Theorem *5.54 of [Whitehea... |
| pm5.62 1020 | Theorem *5.62 of [Whitehea... |
| pm5.63 1021 | Theorem *5.63 of [Whitehea... |
| niabn 1022 | Miscellaneous inference re... |
| ninba 1023 | Miscellaneous inference re... |
| pm4.43 1024 | Theorem *4.43 of [Whitehea... |
| pm4.82 1025 | Theorem *4.82 of [Whitehea... |
| pm4.83 1026 | Theorem *4.83 of [Whitehea... |
| pclem6 1027 | Negation inferred from emb... |
| bigolden 1028 | Dijkstra-Scholten's Golden... |
| pm5.71 1029 | Theorem *5.71 of [Whitehea... |
| pm5.75 1030 | Theorem *5.75 of [Whitehea... |
| ecase2d 1031 | Deduction for elimination ... |
| ecase3 1032 | Inference for elimination ... |
| ecase 1033 | Inference for elimination ... |
| ecase3d 1034 | Deduction for elimination ... |
| ecased 1035 | Deduction for elimination ... |
| ecase3ad 1036 | Deduction for elimination ... |
| ccase 1037 | Inference for combining ca... |
| ccased 1038 | Deduction for combining ca... |
| ccase2 1039 | Inference for combining ca... |
| 4cases 1040 | Inference eliminating two ... |
| 4casesdan 1041 | Deduction eliminating two ... |
| cases 1042 | Case disjunction according... |
| dedlem0a 1043 | Lemma for an alternate ver... |
| dedlem0b 1044 | Lemma for an alternate ver... |
| dedlema 1045 | Lemma for weak deduction t... |
| dedlemb 1046 | Lemma for weak deduction t... |
| cases2 1047 | Case disjunction according... |
| cases2ALT 1048 | Alternate proof of ~ cases... |
| dfbi3 1049 | An alternate definition of... |
| pm5.24 1050 | Theorem *5.24 of [Whitehea... |
| 4exmid 1051 | The disjunction of the fou... |
| consensus 1052 | The consensus theorem. Th... |
| pm4.42 1053 | Theorem *4.42 of [Whitehea... |
| prlem1 1054 | A specialized lemma for se... |
| prlem2 1055 | A specialized lemma for se... |
| oplem1 1056 | A specialized lemma for se... |
| dn1 1057 | A single axiom for Boolean... |
| bianir 1058 | A closed form of ~ mpbir ,... |
| jaoi2 1059 | Inference removing a negat... |
| jaoi3 1060 | Inference separating a dis... |
| ornld 1061 | Selecting one statement fr... |
| dfifp2 1064 | Alternate definition of th... |
| dfifp3 1065 | Alternate definition of th... |
| dfifp4 1066 | Alternate definition of th... |
| dfifp5 1067 | Alternate definition of th... |
| dfifp6 1068 | Alternate definition of th... |
| dfifp7 1069 | Alternate definition of th... |
| ifpdfbi 1070 | Define the biconditional a... |
| anifp 1071 | The conditional operator i... |
| ifpor 1072 | The conditional operator i... |
| ifpn 1073 | Conditional operator for t... |
| ifptru 1074 | Value of the conditional o... |
| ifpfal 1075 | Value of the conditional o... |
| ifpid 1076 | Value of the conditional o... |
| casesifp 1077 | Version of ~ cases express... |
| ifpbi123d 1078 | Equivalence deduction for ... |
| ifpbi23d 1079 | Equivalence deduction for ... |
| ifpimpda 1080 | Separation of the values o... |
| 1fpid3 1081 | The value of the condition... |
| elimh 1082 | Hypothesis builder for the... |
| dedt 1083 | The weak deduction theorem... |
| con3ALT 1084 | Proof of ~ con3 from its a... |
| 3orass 1089 | Associative law for triple... |
| 3orel1 1090 | Partial elimination of a t... |
| 3orrot 1091 | Rotation law for triple di... |
| 3orcoma 1092 | Commutation law for triple... |
| 3orcomb 1093 | Commutation law for triple... |
| 3anass 1094 | Associative law for triple... |
| 3anan12 1095 | Convert triple conjunction... |
| 3anan32 1096 | Convert triple conjunction... |
| 3ancoma 1097 | Commutation law for triple... |
| 3ancomb 1098 | Commutation law for triple... |
| 3anrot 1099 | Rotation law for triple co... |
| 3anrev 1100 | Reversal law for triple co... |
| anandi3 1101 | Distribution of triple con... |
| anandi3r 1102 | Distribution of triple con... |
| 3anidm 1103 | Idempotent law for conjunc... |
| 3an4anass 1104 | Associative law for four c... |
| 3ioran 1105 | Negated triple disjunction... |
| 3ianor 1106 | Negated triple conjunction... |
| 3anor 1107 | Triple conjunction express... |
| 3oran 1108 | Triple disjunction in term... |
| 3impa 1109 | Importation from double to... |
| 3imp 1110 | Importation inference. (C... |
| 3imp31 1111 | The importation inference ... |
| 3imp231 1112 | Importation inference. (C... |
| 3imp21 1113 | The importation inference ... |
| 3impb 1114 | Importation from double to... |
| bi23imp13 1115 | ~ 3imp with middle implica... |
| 3impib 1116 | Importation to triple conj... |
| 3impia 1117 | Importation to triple conj... |
| 3expa 1118 | Exportation from triple to... |
| 3exp 1119 | Exportation inference. (C... |
| 3expb 1120 | Exportation from triple to... |
| 3expia 1121 | Exportation from triple co... |
| 3expib 1122 | Exportation from triple co... |
| 3com12 1123 | Commutation in antecedent.... |
| 3com13 1124 | Commutation in antecedent.... |
| 3comr 1125 | Commutation in antecedent.... |
| 3com23 1126 | Commutation in antecedent.... |
| 3coml 1127 | Commutation in antecedent.... |
| 3jca 1128 | Join consequents with conj... |
| 3jcad 1129 | Deduction conjoining the c... |
| 3adant1 1130 | Deduction adding a conjunc... |
| 3adant2 1131 | Deduction adding a conjunc... |
| 3adant3 1132 | Deduction adding a conjunc... |
| 3ad2ant1 1133 | Deduction adding conjuncts... |
| 3ad2ant2 1134 | Deduction adding conjuncts... |
| 3ad2ant3 1135 | Deduction adding conjuncts... |
| simp1 1136 | Simplification of triple c... |
| simp2 1137 | Simplification of triple c... |
| simp3 1138 | Simplification of triple c... |
| simp1i 1139 | Infer a conjunct from a tr... |
| simp2i 1140 | Infer a conjunct from a tr... |
| simp3i 1141 | Infer a conjunct from a tr... |
| simp1d 1142 | Deduce a conjunct from a t... |
| simp2d 1143 | Deduce a conjunct from a t... |
| simp3d 1144 | Deduce a conjunct from a t... |
| simp1bi 1145 | Deduce a conjunct from a t... |
| simp2bi 1146 | Deduce a conjunct from a t... |
| simp3bi 1147 | Deduce a conjunct from a t... |
| 3simpa 1148 | Simplification of triple c... |
| 3simpb 1149 | Simplification of triple c... |
| 3simpc 1150 | Simplification of triple c... |
| 3anim123i 1151 | Join antecedents and conse... |
| 3anim1i 1152 | Add two conjuncts to antec... |
| 3anim2i 1153 | Add two conjuncts to antec... |
| 3anim3i 1154 | Add two conjuncts to antec... |
| 3anbi123i 1155 | Join 3 biconditionals with... |
| 3orbi123i 1156 | Join 3 biconditionals with... |
| 3anbi1i 1157 | Inference adding two conju... |
| 3anbi2i 1158 | Inference adding two conju... |
| 3anbi3i 1159 | Inference adding two conju... |
| syl3an 1160 | A triple syllogism inferen... |
| syl3anb 1161 | A triple syllogism inferen... |
| syl3anbr 1162 | A triple syllogism inferen... |
| syl3an1 1163 | A syllogism inference. (C... |
| syl3an2 1164 | A syllogism inference. (C... |
| syl3an3 1165 | A syllogism inference. (C... |
| syl3an132 1166 | ~ syl2an with antecedents ... |
| 3adantl1 1167 | Deduction adding a conjunc... |
| 3adantl2 1168 | Deduction adding a conjunc... |
| 3adantl3 1169 | Deduction adding a conjunc... |
| 3adantr1 1170 | Deduction adding a conjunc... |
| 3adantr2 1171 | Deduction adding a conjunc... |
| 3adantr3 1172 | Deduction adding a conjunc... |
| ad4ant123 1173 | Deduction adding conjuncts... |
| ad4ant124 1174 | Deduction adding conjuncts... |
| ad4ant134 1175 | Deduction adding conjuncts... |
| ad4ant234 1176 | Deduction adding conjuncts... |
| 3adant1l 1177 | Deduction adding a conjunc... |
| 3adant1r 1178 | Deduction adding a conjunc... |
| 3adant2l 1179 | Deduction adding a conjunc... |
| 3adant2r 1180 | Deduction adding a conjunc... |
| 3adant3l 1181 | Deduction adding a conjunc... |
| 3adant3r 1182 | Deduction adding a conjunc... |
| 3adant3r1 1183 | Deduction adding a conjunc... |
| 3adant3r2 1184 | Deduction adding a conjunc... |
| 3adant3r3 1185 | Deduction adding a conjunc... |
| 3ad2antl1 1186 | Deduction adding conjuncts... |
| 3ad2antl2 1187 | Deduction adding conjuncts... |
| 3ad2antl3 1188 | Deduction adding conjuncts... |
| 3ad2antr1 1189 | Deduction adding conjuncts... |
| 3ad2antr2 1190 | Deduction adding conjuncts... |
| 3ad2antr3 1191 | Deduction adding conjuncts... |
| simpl1 1192 | Simplification of conjunct... |
| simpl2 1193 | Simplification of conjunct... |
| simpl3 1194 | Simplification of conjunct... |
| simpr1 1195 | Simplification of conjunct... |
| simpr2 1196 | Simplification of conjunct... |
| simpr3 1197 | Simplification of conjunct... |
| simp1l 1198 | Simplification of triple c... |
| simp1r 1199 | Simplification of triple c... |
| simp2l 1200 | Simplification of triple c... |
| simp2r 1201 | Simplification of triple c... |
| simp3l 1202 | Simplification of triple c... |
| simp3r 1203 | Simplification of triple c... |
| simp11 1204 | Simplification of doubly t... |
| simp12 1205 | Simplification of doubly t... |
| simp13 1206 | Simplification of doubly t... |
| simp21 1207 | Simplification of doubly t... |
| simp22 1208 | Simplification of doubly t... |
| simp23 1209 | Simplification of doubly t... |
| simp31 1210 | Simplification of doubly t... |
| simp32 1211 | Simplification of doubly t... |
| simp33 1212 | Simplification of doubly t... |
| simpll1 1213 | Simplification of conjunct... |
| simpll2 1214 | Simplification of conjunct... |
| simpll3 1215 | Simplification of conjunct... |
| simplr1 1216 | Simplification of conjunct... |
| simplr2 1217 | Simplification of conjunct... |
| simplr3 1218 | Simplification of conjunct... |
| simprl1 1219 | Simplification of conjunct... |
| simprl2 1220 | Simplification of conjunct... |
| simprl3 1221 | Simplification of conjunct... |
| simprr1 1222 | Simplification of conjunct... |
| simprr2 1223 | Simplification of conjunct... |
| simprr3 1224 | Simplification of conjunct... |
| simpl1l 1225 | Simplification of conjunct... |
| simpl1r 1226 | Simplification of conjunct... |
| simpl2l 1227 | Simplification of conjunct... |
| simpl2r 1228 | Simplification of conjunct... |
| simpl3l 1229 | Simplification of conjunct... |
| simpl3r 1230 | Simplification of conjunct... |
| simpr1l 1231 | Simplification of conjunct... |
| simpr1r 1232 | Simplification of conjunct... |
| simpr2l 1233 | Simplification of conjunct... |
| simpr2r 1234 | Simplification of conjunct... |
| simpr3l 1235 | Simplification of conjunct... |
| simpr3r 1236 | Simplification of conjunct... |
| simp1ll 1237 | Simplification of conjunct... |
| simp1lr 1238 | Simplification of conjunct... |
| simp1rl 1239 | Simplification of conjunct... |
| simp1rr 1240 | Simplification of conjunct... |
| simp2ll 1241 | Simplification of conjunct... |
| simp2lr 1242 | Simplification of conjunct... |
| simp2rl 1243 | Simplification of conjunct... |
| simp2rr 1244 | Simplification of conjunct... |
| simp3ll 1245 | Simplification of conjunct... |
| simp3lr 1246 | Simplification of conjunct... |
| simp3rl 1247 | Simplification of conjunct... |
| simp3rr 1248 | Simplification of conjunct... |
| simpl11 1249 | Simplification of conjunct... |
| simpl12 1250 | Simplification of conjunct... |
| simpl13 1251 | Simplification of conjunct... |
| simpl21 1252 | Simplification of conjunct... |
| simpl22 1253 | Simplification of conjunct... |
| simpl23 1254 | Simplification of conjunct... |
| simpl31 1255 | Simplification of conjunct... |
| simpl32 1256 | Simplification of conjunct... |
| simpl33 1257 | Simplification of conjunct... |
| simpr11 1258 | Simplification of conjunct... |
| simpr12 1259 | Simplification of conjunct... |
| simpr13 1260 | Simplification of conjunct... |
| simpr21 1261 | Simplification of conjunct... |
| simpr22 1262 | Simplification of conjunct... |
| simpr23 1263 | Simplification of conjunct... |
| simpr31 1264 | Simplification of conjunct... |
| simpr32 1265 | Simplification of conjunct... |
| simpr33 1266 | Simplification of conjunct... |
| simp1l1 1267 | Simplification of conjunct... |
| simp1l2 1268 | Simplification of conjunct... |
| simp1l3 1269 | Simplification of conjunct... |
| simp1r1 1270 | Simplification of conjunct... |
| simp1r2 1271 | Simplification of conjunct... |
| simp1r3 1272 | Simplification of conjunct... |
| simp2l1 1273 | Simplification of conjunct... |
| simp2l2 1274 | Simplification of conjunct... |
| simp2l3 1275 | Simplification of conjunct... |
| simp2r1 1276 | Simplification of conjunct... |
| simp2r2 1277 | Simplification of conjunct... |
| simp2r3 1278 | Simplification of conjunct... |
| simp3l1 1279 | Simplification of conjunct... |
| simp3l2 1280 | Simplification of conjunct... |
| simp3l3 1281 | Simplification of conjunct... |
| simp3r1 1282 | Simplification of conjunct... |
| simp3r2 1283 | Simplification of conjunct... |
| simp3r3 1284 | Simplification of conjunct... |
| simp11l 1285 | Simplification of conjunct... |
| simp11r 1286 | Simplification of conjunct... |
| simp12l 1287 | Simplification of conjunct... |
| simp12r 1288 | Simplification of conjunct... |
| simp13l 1289 | Simplification of conjunct... |
| simp13r 1290 | Simplification of conjunct... |
| simp21l 1291 | Simplification of conjunct... |
| simp21r 1292 | Simplification of conjunct... |
| simp22l 1293 | Simplification of conjunct... |
| simp22r 1294 | Simplification of conjunct... |
| simp23l 1295 | Simplification of conjunct... |
| simp23r 1296 | Simplification of conjunct... |
| simp31l 1297 | Simplification of conjunct... |
| simp31r 1298 | Simplification of conjunct... |
| simp32l 1299 | Simplification of conjunct... |
| simp32r 1300 | Simplification of conjunct... |
| simp33l 1301 | Simplification of conjunct... |
| simp33r 1302 | Simplification of conjunct... |
| simp111 1303 | Simplification of conjunct... |
| simp112 1304 | Simplification of conjunct... |
| simp113 1305 | Simplification of conjunct... |
| simp121 1306 | Simplification of conjunct... |
| simp122 1307 | Simplification of conjunct... |
| simp123 1308 | Simplification of conjunct... |
| simp131 1309 | Simplification of conjunct... |
| simp132 1310 | Simplification of conjunct... |
| simp133 1311 | Simplification of conjunct... |
| simp211 1312 | Simplification of conjunct... |
| simp212 1313 | Simplification of conjunct... |
| simp213 1314 | Simplification of conjunct... |
| simp221 1315 | Simplification of conjunct... |
| simp222 1316 | Simplification of conjunct... |
| simp223 1317 | Simplification of conjunct... |
| simp231 1318 | Simplification of conjunct... |
| simp232 1319 | Simplification of conjunct... |
| simp233 1320 | Simplification of conjunct... |
| simp311 1321 | Simplification of conjunct... |
| simp312 1322 | Simplification of conjunct... |
| simp313 1323 | Simplification of conjunct... |
| simp321 1324 | Simplification of conjunct... |
| simp322 1325 | Simplification of conjunct... |
| simp323 1326 | Simplification of conjunct... |
| simp331 1327 | Simplification of conjunct... |
| simp332 1328 | Simplification of conjunct... |
| simp333 1329 | Simplification of conjunct... |
| 3anibar 1330 | Remove a hypothesis from t... |
| 3mix1 1331 | Introduction in triple dis... |
| 3mix2 1332 | Introduction in triple dis... |
| 3mix3 1333 | Introduction in triple dis... |
| 3mix1i 1334 | Introduction in triple dis... |
| 3mix2i 1335 | Introduction in triple dis... |
| 3mix3i 1336 | Introduction in triple dis... |
| 3mix1d 1337 | Deduction introducing trip... |
| 3mix2d 1338 | Deduction introducing trip... |
| 3mix3d 1339 | Deduction introducing trip... |
| 3pm3.2i 1340 | Infer conjunction of premi... |
| pm3.2an3 1341 | Version of ~ pm3.2 for a t... |
| mpbir3an 1342 | Detach a conjunction of tr... |
| mpbir3and 1343 | Detach a conjunction of tr... |
| syl3anbrc 1344 | Syllogism inference. (Con... |
| syl21anbrc 1345 | Syllogism inference. (Con... |
| 3imp3i2an 1346 | An elimination deduction. ... |
| ex3 1347 | Apply ~ ex to a hypothesis... |
| 3imp1 1348 | Importation to left triple... |
| 3impd 1349 | Importation deduction for ... |
| 3imp2 1350 | Importation to right tripl... |
| 3impdi 1351 | Importation inference (und... |
| 3impdir 1352 | Importation inference (und... |
| 3exp1 1353 | Exportation from left trip... |
| 3expd 1354 | Exportation deduction for ... |
| 3exp2 1355 | Exportation from right tri... |
| exp5o 1356 | A triple exportation infer... |
| exp516 1357 | A triple exportation infer... |
| exp520 1358 | A triple exportation infer... |
| 3impexp 1359 | Version of ~ impexp for a ... |
| 3an1rs 1360 | Swap conjuncts. (Contribu... |
| 3anassrs 1361 | Associative law for conjun... |
| 4anpull2 1362 | An equivalence of two four... |
| ad5ant245 1363 | Deduction adding conjuncts... |
| ad5ant234 1364 | Deduction adding conjuncts... |
| ad5ant235 1365 | Deduction adding conjuncts... |
| ad5ant123 1366 | Deduction adding conjuncts... |
| ad5ant124 1367 | Deduction adding conjuncts... |
| ad5ant125 1368 | Deduction adding conjuncts... |
| ad5ant134 1369 | Deduction adding conjuncts... |
| ad5ant135 1370 | Deduction adding conjuncts... |
| ad5ant145 1371 | Deduction adding conjuncts... |
| ad5ant2345 1372 | Deduction adding conjuncts... |
| syl3anc 1373 | Syllogism combined with co... |
| syl13anc 1374 | Syllogism combined with co... |
| syl31anc 1375 | Syllogism combined with co... |
| syl112anc 1376 | Syllogism combined with co... |
| syl121anc 1377 | Syllogism combined with co... |
| syl211anc 1378 | Syllogism combined with co... |
| syl23anc 1379 | Syllogism combined with co... |
| syl32anc 1380 | Syllogism combined with co... |
| syl122anc 1381 | Syllogism combined with co... |
| syl212anc 1382 | Syllogism combined with co... |
| syl221anc 1383 | Syllogism combined with co... |
| syl113anc 1384 | Syllogism combined with co... |
| syl131anc 1385 | Syllogism combined with co... |
| syl311anc 1386 | Syllogism combined with co... |
| syl33anc 1387 | Syllogism combined with co... |
| syl222anc 1388 | Syllogism combined with co... |
| syl123anc 1389 | Syllogism combined with co... |
| syl132anc 1390 | Syllogism combined with co... |
| syl213anc 1391 | Syllogism combined with co... |
| syl231anc 1392 | Syllogism combined with co... |
| syl312anc 1393 | Syllogism combined with co... |
| syl321anc 1394 | Syllogism combined with co... |
| syl133anc 1395 | Syllogism combined with co... |
| syl313anc 1396 | Syllogism combined with co... |
| syl331anc 1397 | Syllogism combined with co... |
| syl223anc 1398 | Syllogism combined with co... |
| syl232anc 1399 | Syllogism combined with co... |
| syl322anc 1400 | Syllogism combined with co... |
| syl233anc 1401 | Syllogism combined with co... |
| syl323anc 1402 | Syllogism combined with co... |
| syl332anc 1403 | Syllogism combined with co... |
| syl333anc 1404 | A syllogism inference comb... |
| syl3an1b 1405 | A syllogism inference. (C... |
| syl3an2b 1406 | A syllogism inference. (C... |
| syl3an3b 1407 | A syllogism inference. (C... |
| syl3an1br 1408 | A syllogism inference. (C... |
| syl3an2br 1409 | A syllogism inference. (C... |
| syl3an3br 1410 | A syllogism inference. (C... |
| syld3an3 1411 | A syllogism inference. (C... |
| syld3an1 1412 | A syllogism inference. (C... |
| syld3an2 1413 | A syllogism inference. (C... |
| syl3anl1 1414 | A syllogism inference. (C... |
| syl3anl2 1415 | A syllogism inference. (C... |
| syl3anl3 1416 | A syllogism inference. (C... |
| syl3anl 1417 | A triple syllogism inferen... |
| syl3anr1 1418 | A syllogism inference. (C... |
| syl3anr2 1419 | A syllogism inference. (C... |
| syl3anr3 1420 | A syllogism inference. (C... |
| 3anidm12 1421 | Inference from idempotent ... |
| 3anidm13 1422 | Inference from idempotent ... |
| 3anidm23 1423 | Inference from idempotent ... |
| syl2an3an 1424 | ~ syl3an with antecedents ... |
| syl2an23an 1425 | Deduction related to ~ syl... |
| 3ori 1426 | Infer implication from tri... |
| 3jao 1427 | Disjunction of three antec... |
| 3jaob 1428 | Disjunction of three antec... |
| 3jaobOLD 1429 | Obsolete version of ~ 3jao... |
| 3jaoi 1430 | Disjunction of three antec... |
| 3jaod 1431 | Disjunction of three antec... |
| 3jaoian 1432 | Disjunction of three antec... |
| 3jaodan 1433 | Disjunction of three antec... |
| mpjao3dan 1434 | Eliminate a three-way disj... |
| 3jaao 1435 | Inference conjoining and d... |
| syl3an9b 1436 | Nested syllogism inference... |
| 3orbi123d 1437 | Deduction joining 3 equiva... |
| 3anbi123d 1438 | Deduction joining 3 equiva... |
| 3anbi12d 1439 | Deduction conjoining and a... |
| 3anbi13d 1440 | Deduction conjoining and a... |
| 3anbi23d 1441 | Deduction conjoining and a... |
| 3anbi1d 1442 | Deduction adding conjuncts... |
| 3anbi2d 1443 | Deduction adding conjuncts... |
| 3anbi3d 1444 | Deduction adding conjuncts... |
| 3anim123d 1445 | Deduction joining 3 implic... |
| 3orim123d 1446 | Deduction joining 3 implic... |
| an6 1447 | Rearrangement of 6 conjunc... |
| 3an6 1448 | Analogue of ~ an4 for trip... |
| 3or6 1449 | Analogue of ~ or4 for trip... |
| mp3an1 1450 | An inference based on modu... |
| mp3an2 1451 | An inference based on modu... |
| mp3an3 1452 | An inference based on modu... |
| mp3an12 1453 | An inference based on modu... |
| mp3an13 1454 | An inference based on modu... |
| mp3an23 1455 | An inference based on modu... |
| mp3an1i 1456 | An inference based on modu... |
| mp3anl1 1457 | An inference based on modu... |
| mp3anl2 1458 | An inference based on modu... |
| mp3anl3 1459 | An inference based on modu... |
| mp3anr1 1460 | An inference based on modu... |
| mp3anr2 1461 | An inference based on modu... |
| mp3anr3 1462 | An inference based on modu... |
| mp3an 1463 | An inference based on modu... |
| mpd3an3 1464 | An inference based on modu... |
| mpd3an23 1465 | An inference based on modu... |
| mp3and 1466 | A deduction based on modus... |
| mp3an12i 1467 | ~ mp3an with antecedents i... |
| mp3an2i 1468 | ~ mp3an with antecedents i... |
| mp3an3an 1469 | ~ mp3an with antecedents i... |
| mp3an2ani 1470 | An elimination deduction. ... |
| biimp3a 1471 | Infer implication from a l... |
| biimp3ar 1472 | Infer implication from a l... |
| 3anandis 1473 | Inference that undistribut... |
| 3anandirs 1474 | Inference that undistribut... |
| ecase23d 1475 | Deduction for elimination ... |
| 3ecase 1476 | Inference for elimination ... |
| 3bior1fd 1477 | A disjunction is equivalen... |
| 3bior1fand 1478 | A disjunction is equivalen... |
| 3bior2fd 1479 | A wff is equivalent to its... |
| 3biant1d 1480 | A conjunction is equivalen... |
| intn3an1d 1481 | Introduction of a triple c... |
| intn3an2d 1482 | Introduction of a triple c... |
| intn3an3d 1483 | Introduction of a triple c... |
| an3andi 1484 | Distribution of conjunctio... |
| an33rean 1485 | Rearrange a 9-fold conjunc... |
| 3orel2 1486 | Partial elimination of a t... |
| 3orel2OLD 1487 | Obsolete version of ~ 3ore... |
| 3orel3 1488 | Partial elimination of a t... |
| 3orel13 1489 | Elimination of two disjunc... |
| 3pm3.2ni 1490 | Triple negated disjunction... |
| an42ds 1491 | Inference exchanging the l... |
| nanan 1494 | Conjunction in terms of al... |
| dfnan2 1495 | Alternative denial in term... |
| nanor 1496 | Alternative denial in term... |
| nancom 1497 | Alternative denial is comm... |
| nannan 1498 | Nested alternative denials... |
| nanim 1499 | Implication in terms of al... |
| nannot 1500 | Negation in terms of alter... |
| nanbi 1501 | Biconditional in terms of ... |
| nanbi1 1502 | Introduce a right anti-con... |
| nanbi2 1503 | Introduce a left anti-conj... |
| nanbi12 1504 | Join two logical equivalen... |
| nanbi1i 1505 | Introduce a right anti-con... |
| nanbi2i 1506 | Introduce a left anti-conj... |
| nanbi12i 1507 | Join two logical equivalen... |
| nanbi1d 1508 | Introduce a right anti-con... |
| nanbi2d 1509 | Introduce a left anti-conj... |
| nanbi12d 1510 | Join two logical equivalen... |
| nanass 1511 | A characterization of when... |
| xnor 1514 | Two ways to write XNOR (ex... |
| xorcom 1515 | The connector ` \/_ ` is c... |
| xorass 1516 | The connector ` \/_ ` is a... |
| excxor 1517 | This tautology shows that ... |
| xor2 1518 | Two ways to express "exclu... |
| xoror 1519 | Exclusive disjunction impl... |
| xornan 1520 | Exclusive disjunction impl... |
| xornan2 1521 | XOR implies NAND (written ... |
| xorneg2 1522 | The connector ` \/_ ` is n... |
| xorneg1 1523 | The connector ` \/_ ` is n... |
| xorneg 1524 | The connector ` \/_ ` is u... |
| xorbi12i 1525 | Equality property for excl... |
| xorbi12d 1526 | Equality property for excl... |
| anxordi 1527 | Conjunction distributes ov... |
| xorexmid 1528 | Exclusive-or variant of th... |
| norcom 1531 | The connector ` -\/ ` is c... |
| nornot 1532 | ` -. ` is expressible via ... |
| noran 1533 | ` /\ ` is expressible via ... |
| noror 1534 | ` \/ ` is expressible via ... |
| norasslem1 1535 | This lemma shows the equiv... |
| norasslem2 1536 | This lemma specializes ~ b... |
| norasslem3 1537 | This lemma specializes ~ b... |
| norass 1538 | A characterization of when... |
| trujust 1543 | Soundness justification th... |
| tru 1545 | The truth value ` T. ` is ... |
| dftru2 1546 | An alternate definition of... |
| trut 1547 | A proposition is equivalen... |
| mptru 1548 | Eliminate ` T. ` as an ant... |
| tbtru 1549 | A proposition is equivalen... |
| bitru 1550 | A theorem is equivalent to... |
| trud 1551 | Anything implies ` T. ` . ... |
| truan 1552 | True can be removed from a... |
| fal 1555 | The truth value ` F. ` is ... |
| nbfal 1556 | The negation of a proposit... |
| bifal 1557 | A contradiction is equival... |
| falim 1558 | The truth value ` F. ` imp... |
| falimd 1559 | The truth value ` F. ` imp... |
| dfnot 1560 | Given falsum ` F. ` , we c... |
| inegd 1561 | Negation introduction rule... |
| efald 1562 | Deduction based on reducti... |
| pm2.21fal 1563 | If a wff and its negation ... |
| truimtru 1564 | A ` -> ` identity. (Contr... |
| truimfal 1565 | A ` -> ` identity. (Contr... |
| falimtru 1566 | A ` -> ` identity. (Contr... |
| falimfal 1567 | A ` -> ` identity. (Contr... |
| nottru 1568 | A ` -. ` identity. (Contr... |
| notfal 1569 | A ` -. ` identity. (Contr... |
| trubitru 1570 | A ` <-> ` identity. (Cont... |
| falbitru 1571 | A ` <-> ` identity. (Cont... |
| trubifal 1572 | A ` <-> ` identity. (Cont... |
| falbifal 1573 | A ` <-> ` identity. (Cont... |
| truantru 1574 | A ` /\ ` identity. (Contr... |
| truanfal 1575 | A ` /\ ` identity. (Contr... |
| falantru 1576 | A ` /\ ` identity. (Contr... |
| falanfal 1577 | A ` /\ ` identity. (Contr... |
| truortru 1578 | A ` \/ ` identity. (Contr... |
| truorfal 1579 | A ` \/ ` identity. (Contr... |
| falortru 1580 | A ` \/ ` identity. (Contr... |
| falorfal 1581 | A ` \/ ` identity. (Contr... |
| trunantru 1582 | A ` -/\ ` identity. (Cont... |
| trunanfal 1583 | A ` -/\ ` identity. (Cont... |
| falnantru 1584 | A ` -/\ ` identity. (Cont... |
| falnanfal 1585 | A ` -/\ ` identity. (Cont... |
| truxortru 1586 | A ` \/_ ` identity. (Cont... |
| truxorfal 1587 | A ` \/_ ` identity. (Cont... |
| falxortru 1588 | A ` \/_ ` identity. (Cont... |
| falxorfal 1589 | A ` \/_ ` identity. (Cont... |
| trunortru 1590 | A ` -\/ ` identity. (Cont... |
| trunorfal 1591 | A ` -\/ ` identity. (Cont... |
| falnortru 1592 | A ` -\/ ` identity. (Cont... |
| falnorfal 1593 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1596 | Equality theorem for the a... |
| hadbi123i 1597 | Equality theorem for the a... |
| hadass 1598 | Associative law for the ad... |
| hadbi 1599 | The adder sum is the same ... |
| hadcoma 1600 | Commutative law for the ad... |
| hadcomb 1601 | Commutative law for the ad... |
| hadrot 1602 | Rotation law for the adder... |
| hadnot 1603 | The adder sum distributes ... |
| had1 1604 | If the first input is true... |
| had0 1605 | If the first input is fals... |
| hadifp 1606 | The value of the adder sum... |
| cador 1609 | The adder carry in disjunc... |
| cadan 1610 | The adder carry in conjunc... |
| cadbi123d 1611 | Equality theorem for the a... |
| cadbi123i 1612 | Equality theorem for the a... |
| cadcoma 1613 | Commutative law for the ad... |
| cadcomb 1614 | Commutative law for the ad... |
| cadrot 1615 | Rotation law for the adder... |
| cadnot 1616 | The adder carry distribute... |
| cad11 1617 | If (at least) two inputs a... |
| cad1 1618 | If one input is true, then... |
| cad0 1619 | If one input is false, the... |
| cadifp 1620 | The value of the carry is,... |
| cadtru 1621 | The adder carry is true as... |
| minimp 1622 | A single axiom for minimal... |
| minimp-syllsimp 1623 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1624 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1625 | Derivation of a commuted f... |
| minimp-ax2 1626 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1627 | Derivation of ~ pm2.43 (al... |
| impsingle 1628 | The shortest single axiom ... |
| impsingle-step4 1629 | Derivation of impsingle-st... |
| impsingle-step8 1630 | Derivation of impsingle-st... |
| impsingle-ax1 1631 | Derivation of impsingle-ax... |
| impsingle-step15 1632 | Derivation of impsingle-st... |
| impsingle-step18 1633 | Derivation of impsingle-st... |
| impsingle-step19 1634 | Derivation of impsingle-st... |
| impsingle-step20 1635 | Derivation of impsingle-st... |
| impsingle-step21 1636 | Derivation of impsingle-st... |
| impsingle-step22 1637 | Derivation of impsingle-st... |
| impsingle-step25 1638 | Derivation of impsingle-st... |
| impsingle-imim1 1639 | Derivation of impsingle-im... |
| impsingle-peirce 1640 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1641 | Derivation of ~ ax-2 from ... |
| meredith 1642 | Carew Meredith's sole axio... |
| merlem1 1643 | Step 3 of Meredith's proof... |
| merlem2 1644 | Step 4 of Meredith's proof... |
| merlem3 1645 | Step 7 of Meredith's proof... |
| merlem4 1646 | Step 8 of Meredith's proof... |
| merlem5 1647 | Step 11 of Meredith's proo... |
| merlem6 1648 | Step 12 of Meredith's proo... |
| merlem7 1649 | Between steps 14 and 15 of... |
| merlem8 1650 | Step 15 of Meredith's proo... |
| merlem9 1651 | Step 18 of Meredith's proo... |
| merlem10 1652 | Step 19 of Meredith's proo... |
| merlem11 1653 | Step 20 of Meredith's proo... |
| merlem12 1654 | Step 28 of Meredith's proo... |
| merlem13 1655 | Step 35 of Meredith's proo... |
| luk-1 1656 | 1 of 3 axioms for proposit... |
| luk-2 1657 | 2 of 3 axioms for proposit... |
| luk-3 1658 | 3 of 3 axioms for proposit... |
| luklem1 1659 | Used to rederive standard ... |
| luklem2 1660 | Used to rederive standard ... |
| luklem3 1661 | Used to rederive standard ... |
| luklem4 1662 | Used to rederive standard ... |
| luklem5 1663 | Used to rederive standard ... |
| luklem6 1664 | Used to rederive standard ... |
| luklem7 1665 | Used to rederive standard ... |
| luklem8 1666 | Used to rederive standard ... |
| ax1 1667 | Standard propositional axi... |
| ax2 1668 | Standard propositional axi... |
| ax3 1669 | Standard propositional axi... |
| nic-dfim 1670 | This theorem "defines" imp... |
| nic-dfneg 1671 | This theorem "defines" neg... |
| nic-mp 1672 | Derive Nicod's rule of mod... |
| nic-mpALT 1673 | A direct proof of ~ nic-mp... |
| nic-ax 1674 | Nicod's axiom derived from... |
| nic-axALT 1675 | A direct proof of ~ nic-ax... |
| nic-imp 1676 | Inference for ~ nic-mp usi... |
| nic-idlem1 1677 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1678 | Lemma for ~ nic-id . Infe... |
| nic-id 1679 | Theorem ~ id expressed wit... |
| nic-swap 1680 | The connector ` -/\ ` is s... |
| nic-isw1 1681 | Inference version of ~ nic... |
| nic-isw2 1682 | Inference for swapping nes... |
| nic-iimp1 1683 | Inference version of ~ nic... |
| nic-iimp2 1684 | Inference version of ~ nic... |
| nic-idel 1685 | Inference to remove the tr... |
| nic-ich 1686 | Chained inference. (Contr... |
| nic-idbl 1687 | Double the terms. Since d... |
| nic-bijust 1688 | Biconditional justificatio... |
| nic-bi1 1689 | Inference to extract one s... |
| nic-bi2 1690 | Inference to extract the o... |
| nic-stdmp 1691 | Derive the standard modus ... |
| nic-luk1 1692 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1693 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1694 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1695 | This alternative axiom for... |
| lukshefth1 1696 | Lemma for ~ renicax . (Co... |
| lukshefth2 1697 | Lemma for ~ renicax . (Co... |
| renicax 1698 | A rederivation of ~ nic-ax... |
| tbw-bijust 1699 | Justification for ~ tbw-ne... |
| tbw-negdf 1700 | The definition of negation... |
| tbw-ax1 1701 | The first of four axioms i... |
| tbw-ax2 1702 | The second of four axioms ... |
| tbw-ax3 1703 | The third of four axioms i... |
| tbw-ax4 1704 | The fourth of four axioms ... |
| tbwsyl 1705 | Used to rederive the Lukas... |
| tbwlem1 1706 | Used to rederive the Lukas... |
| tbwlem2 1707 | Used to rederive the Lukas... |
| tbwlem3 1708 | Used to rederive the Lukas... |
| tbwlem4 1709 | Used to rederive the Lukas... |
| tbwlem5 1710 | Used to rederive the Lukas... |
| re1luk1 1711 | ~ luk-1 derived from the T... |
| re1luk2 1712 | ~ luk-2 derived from the T... |
| re1luk3 1713 | ~ luk-3 derived from the T... |
| merco1 1714 | A single axiom for proposi... |
| merco1lem1 1715 | Used to rederive the Tarsk... |
| retbwax4 1716 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1717 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1718 | Used to rederive the Tarsk... |
| merco1lem3 1719 | Used to rederive the Tarsk... |
| merco1lem4 1720 | Used to rederive the Tarsk... |
| merco1lem5 1721 | Used to rederive the Tarsk... |
| merco1lem6 1722 | Used to rederive the Tarsk... |
| merco1lem7 1723 | Used to rederive the Tarsk... |
| retbwax3 1724 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1725 | Used to rederive the Tarsk... |
| merco1lem9 1726 | Used to rederive the Tarsk... |
| merco1lem10 1727 | Used to rederive the Tarsk... |
| merco1lem11 1728 | Used to rederive the Tarsk... |
| merco1lem12 1729 | Used to rederive the Tarsk... |
| merco1lem13 1730 | Used to rederive the Tarsk... |
| merco1lem14 1731 | Used to rederive the Tarsk... |
| merco1lem15 1732 | Used to rederive the Tarsk... |
| merco1lem16 1733 | Used to rederive the Tarsk... |
| merco1lem17 1734 | Used to rederive the Tarsk... |
| merco1lem18 1735 | Used to rederive the Tarsk... |
| retbwax1 1736 | ~ tbw-ax1 rederived from ~... |
| merco2 1737 | A single axiom for proposi... |
| mercolem1 1738 | Used to rederive the Tarsk... |
| mercolem2 1739 | Used to rederive the Tarsk... |
| mercolem3 1740 | Used to rederive the Tarsk... |
| mercolem4 1741 | Used to rederive the Tarsk... |
| mercolem5 1742 | Used to rederive the Tarsk... |
| mercolem6 1743 | Used to rederive the Tarsk... |
| mercolem7 1744 | Used to rederive the Tarsk... |
| mercolem8 1745 | Used to rederive the Tarsk... |
| re1tbw1 1746 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1747 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1748 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1749 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1750 | Justification for ~ rb-imd... |
| rb-imdf 1751 | The definition of implicat... |
| anmp 1752 | Modus ponens for ` { \/ , ... |
| rb-ax1 1753 | The first of four axioms i... |
| rb-ax2 1754 | The second of four axioms ... |
| rb-ax3 1755 | The third of four axioms i... |
| rb-ax4 1756 | The fourth of four axioms ... |
| rbsyl 1757 | Used to rederive the Lukas... |
| rblem1 1758 | Used to rederive the Lukas... |
| rblem2 1759 | Used to rederive the Lukas... |
| rblem3 1760 | Used to rederive the Lukas... |
| rblem4 1761 | Used to rederive the Lukas... |
| rblem5 1762 | Used to rederive the Lukas... |
| rblem6 1763 | Used to rederive the Lukas... |
| rblem7 1764 | Used to rederive the Lukas... |
| re1axmp 1765 | ~ ax-mp derived from Russe... |
| re2luk1 1766 | ~ luk-1 derived from Russe... |
| re2luk2 1767 | ~ luk-2 derived from Russe... |
| re2luk3 1768 | ~ luk-3 derived from Russe... |
| mptnan 1769 | Modus ponendo tollens 1, o... |
| mptxor 1770 | Modus ponendo tollens 2, o... |
| mtpor 1771 | Modus tollendo ponens (inc... |
| mtpxor 1772 | Modus tollendo ponens (ori... |
| stoic1a 1773 | Stoic logic Thema 1 (part ... |
| stoic1b 1774 | Stoic logic Thema 1 (part ... |
| stoic2a 1775 | Stoic logic Thema 2 versio... |
| stoic2b 1776 | Stoic logic Thema 2 versio... |
| stoic3 1777 | Stoic logic Thema 3. Stat... |
| stoic4a 1778 | Stoic logic Thema 4 versio... |
| stoic4b 1779 | Stoic logic Thema 4 versio... |
| alnex 1782 | Universal quantification o... |
| eximal 1783 | An equivalence between an ... |
| nf2 1786 | Alternate definition of no... |
| nf3 1787 | Alternate definition of no... |
| nf4 1788 | Alternate definition of no... |
| nfi 1789 | Deduce that ` x ` is not f... |
| nfri 1790 | Consequence of the definit... |
| nfd 1791 | Deduce that ` x ` is not f... |
| nfrd 1792 | Consequence of the definit... |
| nftht 1793 | Closed form of ~ nfth . (... |
| nfntht 1794 | Closed form of ~ nfnth . ... |
| nfntht2 1795 | Closed form of ~ nfnth . ... |
| gen2 1797 | Generalization applied twi... |
| mpg 1798 | Modus ponens combined with... |
| mpgbi 1799 | Modus ponens on biconditio... |
| mpgbir 1800 | Modus ponens on biconditio... |
| nex 1801 | Generalization rule for ne... |
| nfth 1802 | No variable is (effectivel... |
| nfnth 1803 | No variable is (effectivel... |
| hbth 1804 | No variable is (effectivel... |
| nftru 1805 | The true constant has no f... |
| nffal 1806 | The false constant has no ... |
| sptruw 1807 | Version of ~ sp when ` ph ... |
| altru 1808 | For all sets, ` T. ` is tr... |
| alfal 1809 | For all sets, ` -. F. ` is... |
| alim 1811 | Restatement of Axiom ~ ax-... |
| alimi 1812 | Inference quantifying both... |
| 2alimi 1813 | Inference doubly quantifyi... |
| ala1 1814 | Add an antecedent in a uni... |
| al2im 1815 | Closed form of ~ al2imi . ... |
| al2imi 1816 | Inference quantifying ante... |
| alanimi 1817 | Variant of ~ al2imi with c... |
| alimdh 1818 | Deduction form of Theorem ... |
| albi 1819 | Theorem 19.15 of [Margaris... |
| albii 1820 | Inference adding universal... |
| 2albii 1821 | Inference adding two unive... |
| 3albii 1822 | Inference adding three uni... |
| sylgt 1823 | Closed form of ~ sylg . (... |
| sylg 1824 | A syllogism combined with ... |
| alrimih 1825 | Inference form of Theorem ... |
| hbxfrbi 1826 | A utility lemma to transfe... |
| alex 1827 | Universal quantifier in te... |
| exnal 1828 | Existential quantification... |
| 2nalexn 1829 | Part of theorem *11.5 in [... |
| 2exnaln 1830 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1831 | Theorem *11.25 in [Whitehe... |
| alimex 1832 | An equivalence between an ... |
| aleximi 1833 | A variant of ~ al2imi : in... |
| alexbii 1834 | Biconditional form of ~ al... |
| exim 1835 | Theorem 19.22 of [Margaris... |
| eximi 1836 | Inference adding existenti... |
| 2eximi 1837 | Inference adding two exist... |
| eximii 1838 | Inference associated with ... |
| exa1 1839 | Add an antecedent in an ex... |
| 19.38 1840 | Theorem 19.38 of [Margaris... |
| 19.38a 1841 | Under a nonfreeness hypoth... |
| 19.38b 1842 | Under a nonfreeness hypoth... |
| imnang 1843 | Quantified implication in ... |
| alinexa 1844 | A transformation of quanti... |
| exnalimn 1845 | Existential quantification... |
| alexn 1846 | A relationship between two... |
| 2exnexn 1847 | Theorem *11.51 in [Whitehe... |
| exbi 1848 | Theorem 19.18 of [Margaris... |
| exbii 1849 | Inference adding existenti... |
| 2exbii 1850 | Inference adding two exist... |
| 3exbii 1851 | Inference adding three exi... |
| nfbiit 1852 | Equivalence theorem for th... |
| nfbii 1853 | Equality theorem for the n... |
| nfxfr 1854 | A utility lemma to transfe... |
| nfxfrd 1855 | A utility lemma to transfe... |
| nfnbi 1856 | A variable is nonfree in a... |
| nfnt 1857 | If a variable is nonfree i... |
| nfn 1858 | Inference associated with ... |
| nfnd 1859 | Deduction associated with ... |
| exanali 1860 | A transformation of quanti... |
| 2exanali 1861 | Theorem *11.521 in [Whiteh... |
| exancom 1862 | Commutation of conjunction... |
| exan 1863 | Place a conjunct in the sc... |
| alrimdh 1864 | Deduction form of Theorem ... |
| eximdh 1865 | Deduction from Theorem 19.... |
| nexdh 1866 | Deduction for generalizati... |
| albidh 1867 | Formula-building rule for ... |
| exbidh 1868 | Formula-building rule for ... |
| exsimpl 1869 | Simplification of an exist... |
| exsimpr 1870 | Simplification of an exist... |
| 19.26 1871 | Theorem 19.26 of [Margaris... |
| 19.26-2 1872 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1873 | Theorem ~ 19.26 with tripl... |
| 19.29 1874 | Theorem 19.29 of [Margaris... |
| 19.29r 1875 | Variation of ~ 19.29 . (C... |
| 19.29r2 1876 | Variation of ~ 19.29r with... |
| 19.29x 1877 | Variation of ~ 19.29 with ... |
| 19.35 1878 | Theorem 19.35 of [Margaris... |
| 19.35i 1879 | Inference associated with ... |
| 19.35ri 1880 | Inference associated with ... |
| 19.25 1881 | Theorem 19.25 of [Margaris... |
| 19.30 1882 | Theorem 19.30 of [Margaris... |
| 19.43 1883 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1884 | Obsolete proof of ~ 19.43 ... |
| 19.33 1885 | Theorem 19.33 of [Margaris... |
| 19.33b 1886 | The antecedent provides a ... |
| 19.40 1887 | Theorem 19.40 of [Margaris... |
| 19.40-2 1888 | Theorem *11.42 in [Whitehe... |
| 19.40b 1889 | The antecedent provides a ... |
| albiim 1890 | Split a biconditional and ... |
| 2albiim 1891 | Split a biconditional and ... |
| exintrbi 1892 | Add/remove a conjunct in t... |
| exintr 1893 | Introduce a conjunct in th... |
| alsyl 1894 | Universally quantified and... |
| nfimd 1895 | If in a context ` x ` is n... |
| nfimt 1896 | Closed form of ~ nfim and ... |
| nfim 1897 | If ` x ` is not free in ` ... |
| nfand 1898 | If in a context ` x ` is n... |
| nf3and 1899 | Deduction form of bound-va... |
| nfan 1900 | If ` x ` is not free in ` ... |
| nfnan 1901 | If ` x ` is not free in ` ... |
| nf3an 1902 | If ` x ` is not free in ` ... |
| nfbid 1903 | If in a context ` x ` is n... |
| nfbi 1904 | If ` x ` is not free in ` ... |
| nfor 1905 | If ` x ` is not free in ` ... |
| nf3or 1906 | If ` x ` is not free in ` ... |
| empty 1907 | Two characterizations of t... |
| emptyex 1908 | On the empty domain, any e... |
| emptyal 1909 | On the empty domain, any u... |
| emptynf 1910 | On the empty domain, any v... |
| ax5d 1912 | Version of ~ ax-5 with ant... |
| ax5e 1913 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1914 | If a formula holds for som... |
| nfv 1915 | If ` x ` is not present in... |
| nfvd 1916 | ~ nfv with antecedent. Us... |
| alimdv 1917 | Deduction form of Theorem ... |
| eximdv 1918 | Deduction form of Theorem ... |
| 2alimdv 1919 | Deduction form of Theorem ... |
| 2eximdv 1920 | Deduction form of Theorem ... |
| albidv 1921 | Formula-building rule for ... |
| exbidv 1922 | Formula-building rule for ... |
| nfbidv 1923 | An equality theorem for no... |
| 2albidv 1924 | Formula-building rule for ... |
| 2exbidv 1925 | Formula-building rule for ... |
| 3exbidv 1926 | Formula-building rule for ... |
| 4exbidv 1927 | Formula-building rule for ... |
| alrimiv 1928 | Inference form of Theorem ... |
| alrimivv 1929 | Inference form of Theorem ... |
| alrimdv 1930 | Deduction form of Theorem ... |
| exlimiv 1931 | Inference form of Theorem ... |
| exlimiiv 1932 | Inference (Rule C) associa... |
| exlimivv 1933 | Inference form of Theorem ... |
| exlimdv 1934 | Deduction form of Theorem ... |
| exlimdvv 1935 | Deduction form of Theorem ... |
| exlimddv 1936 | Existential elimination ru... |
| nexdv 1937 | Deduction for generalizati... |
| 2ax5 1938 | Quantification of two vari... |
| stdpc5v 1939 | Version of ~ stdpc5 with a... |
| 19.21v 1940 | Version of ~ 19.21 with a ... |
| 19.32v 1941 | Version of ~ 19.32 with a ... |
| 19.31v 1942 | Version of ~ 19.31 with a ... |
| 19.23v 1943 | Version of ~ 19.23 with a ... |
| 19.23vv 1944 | Theorem ~ 19.23v extended ... |
| pm11.53v 1945 | Version of ~ pm11.53 with ... |
| 19.36imv 1946 | One direction of ~ 19.36v ... |
| 19.36iv 1947 | Inference associated with ... |
| 19.37imv 1948 | One direction of ~ 19.37v ... |
| 19.37iv 1949 | Inference associated with ... |
| 19.41v 1950 | Version of ~ 19.41 with a ... |
| 19.41vv 1951 | Version of ~ 19.41 with tw... |
| 19.41vvv 1952 | Version of ~ 19.41 with th... |
| 19.41vvvv 1953 | Version of ~ 19.41 with fo... |
| 19.42v 1954 | Version of ~ 19.42 with a ... |
| exdistr 1955 | Distribution of existentia... |
| exdistrv 1956 | Distribute a pair of exist... |
| 4exdistrv 1957 | Distribute two pairs of ex... |
| 19.42vv 1958 | Version of ~ 19.42 with tw... |
| exdistr2 1959 | Distribution of existentia... |
| 19.42vvv 1960 | Version of ~ 19.42 with th... |
| 3exdistr 1961 | Distribution of existentia... |
| 4exdistr 1962 | Distribution of existentia... |
| weq 1963 | Extend wff definition to i... |
| speimfw 1964 | Specialization, with addit... |
| speimfwALT 1965 | Alternate proof of ~ speim... |
| spimfw 1966 | Specialization, with addit... |
| ax12i 1967 | Inference that has ~ ax-12... |
| ax6v 1969 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1970 | At least one individual ex... |
| spimw 1971 | Specialization. Lemma 8 o... |
| spimew 1972 | Existential introduction, ... |
| speiv 1973 | Inference from existential... |
| speivw 1974 | Version of ~ spei with a d... |
| exgen 1975 | Rule of existential genera... |
| extru 1976 | There exists a variable su... |
| 19.2 1977 | Theorem 19.2 of [Margaris]... |
| 19.2d 1978 | Deduction associated with ... |
| 19.8w 1979 | Weak version of ~ 19.8a an... |
| spnfw 1980 | Weak version of ~ sp . Us... |
| spfalw 1981 | Version of ~ sp when ` ph ... |
| spvw 1982 | Version of ~ sp when ` x `... |
| 19.3v 1983 | Version of ~ 19.3 with a d... |
| 19.8v 1984 | Version of ~ 19.8a with a ... |
| 19.9v 1985 | Version of ~ 19.9 with a d... |
| spimevw 1986 | Existential introduction, ... |
| spimvw 1987 | A weak form of specializat... |
| spsv 1988 | Generalization of antecede... |
| spvv 1989 | Specialization, using impl... |
| chvarvv 1990 | Implicit substitution of `... |
| 19.39 1991 | Theorem 19.39 of [Margaris... |
| 19.24 1992 | Theorem 19.24 of [Margaris... |
| 19.34 1993 | Theorem 19.34 of [Margaris... |
| 19.36v 1994 | Version of ~ 19.36 with a ... |
| 19.12vvv 1995 | Version of ~ 19.12vv with ... |
| 19.27v 1996 | Version of ~ 19.27 with a ... |
| 19.28v 1997 | Version of ~ 19.28 with a ... |
| 19.37v 1998 | Version of ~ 19.37 with a ... |
| 19.44v 1999 | Version of ~ 19.44 with a ... |
| 19.45v 2000 | Version of ~ 19.45 with a ... |
| equs4v 2001 | Version of ~ equs4 with a ... |
| alequexv 2002 | Version of ~ equs4v with i... |
| exsbim 2003 | One direction of the equiv... |
| equsv 2004 | If a formula does not cont... |
| equsalvw 2005 | Version of ~ equsalv with ... |
| equsexvw 2006 | Version of ~ equsexv with ... |
| cbvaliw 2007 | Change bound variable. Us... |
| cbvalivw 2008 | Change bound variable. Us... |
| ax7v 2010 | Weakened version of ~ ax-7... |
| ax7v1 2011 | First of two weakened vers... |
| ax7v2 2012 | Second of two weakened ver... |
| equid 2013 | Identity law for equality.... |
| nfequid 2014 | Bound-variable hypothesis ... |
| equcomiv 2015 | Weaker form of ~ equcomi w... |
| ax6evr 2016 | A commuted form of ~ ax6ev... |
| ax7 2017 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2018 | Commutative law for equali... |
| equcom 2019 | Commutative law for equali... |
| equcomd 2020 | Deduction form of ~ equcom... |
| equcoms 2021 | An inference commuting equ... |
| equtr 2022 | A transitive law for equal... |
| equtrr 2023 | A transitive law for equal... |
| equeuclr 2024 | Commuted version of ~ eque... |
| equeucl 2025 | Equality is a left-Euclide... |
| equequ1 2026 | An equivalence law for equ... |
| equequ2 2027 | An equivalence law for equ... |
| equtr2 2028 | Equality is a left-Euclide... |
| stdpc6 2029 | One of the two equality ax... |
| equvinv 2030 | A variable introduction la... |
| equvinva 2031 | A modified version of the ... |
| equvelv 2032 | A biconditional form of ~ ... |
| ax13b 2033 | An equivalence between two... |
| spfw 2034 | Weak version of ~ sp . Us... |
| spw 2035 | Weak version of the specia... |
| cbvalw 2036 | Change bound variable. Us... |
| cbvalvw 2037 | Change bound variable. Us... |
| cbvexvw 2038 | Change bound variable. Us... |
| cbvaldvaw 2039 | Rule used to change the bo... |
| cbvexdvaw 2040 | Rule used to change the bo... |
| cbval2vw 2041 | Rule used to change bound ... |
| cbvex2vw 2042 | Rule used to change bound ... |
| cbvex4vw 2043 | Rule used to change bound ... |
| alcomimw 2044 | Weak version of ~ ax-11 . ... |
| excomimw 2045 | Weak version of ~ excomim ... |
| alcomw 2046 | Weak version of ~ alcom an... |
| excomw 2047 | Weak version of ~ excom an... |
| hbn1fw 2048 | Weak version of ~ ax-10 fr... |
| hbn1w 2049 | Weak version of ~ hbn1 . ... |
| hba1w 2050 | Weak version of ~ hba1 . ... |
| hbe1w 2051 | Weak version of ~ hbe1 . ... |
| hbalw 2052 | Weak version of ~ hbal . ... |
| 19.8aw 2053 | If a formula is true, then... |
| exexw 2054 | Existential quantification... |
| spaev 2055 | A special instance of ~ sp... |
| cbvaev 2056 | Change bound variable in a... |
| aevlem0 2057 | Lemma for ~ aevlem . Inst... |
| aevlem 2058 | Lemma for ~ aev and ~ axc1... |
| aeveq 2059 | The antecedent ` A. x x = ... |
| aev 2060 | A "distinctor elimination"... |
| aev2 2061 | A version of ~ aev with tw... |
| hbaev 2062 | All variables are effectiv... |
| naev 2063 | If some set variables can ... |
| naev2 2064 | Generalization of ~ hbnaev... |
| hbnaev 2065 | Any variable is free in ` ... |
| sbjust 2066 | Justification theorem for ... |
| dfsb 2069 | Simplify definition ~ df-s... |
| sbtlem 2070 | In the case of ~ sbt , the... |
| sbt 2071 | A substitution into a theo... |
| sbtru 2072 | The result of substituting... |
| stdpc4 2073 | The specialization axiom o... |
| sbtALT 2074 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2075 | A double specialization us... |
| sbi1 2076 | Distribute substitution ov... |
| spsbim 2077 | Distribute substitution ov... |
| spsbbi 2078 | Biconditional property for... |
| sbimi 2079 | Distribute substitution ov... |
| sb2imi 2080 | Distribute substitution ov... |
| sbbii 2081 | Infer substitution into bo... |
| 2sbbii 2082 | Infer double substitution ... |
| sbimdv 2083 | Deduction substituting bot... |
| sbbidv 2084 | Deduction substituting bot... |
| sban 2085 | Conjunction inside and out... |
| sb3an 2086 | Threefold conjunction insi... |
| spsbe 2087 | Existential generalization... |
| sbequ 2088 | Equality property for subs... |
| sbequi 2089 | An equality theorem for su... |
| sb6 2090 | Alternate definition of su... |
| 2sb6 2091 | Equivalence for double sub... |
| sb1v 2092 | One direction of ~ sb5 , p... |
| sbv 2093 | Substitution for a variabl... |
| sbcom4 2094 | Commutativity law for subs... |
| pm11.07 2095 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2096 | Substitution in an implica... |
| sbbiiev 2097 | An equivalence of substitu... |
| sbievw 2098 | Conversion of implicit sub... |
| sbievwOLD 2099 | Obsolete version of ~ sbie... |
| sbiedvw 2100 | Conversion of implicit sub... |
| 2sbievw 2101 | Conversion of double impli... |
| sbcom3vv 2102 | Substituting ` y ` for ` x... |
| sbievw2 2103 | ~ sbievw applied twice, av... |
| sbco2vv 2104 | A composition law for subs... |
| cbvsbv 2105 | Change the bound variable ... |
| sbco4lem 2106 | Lemma for ~ sbco4 . It re... |
| sbco4 2107 | Two ways of exchanging two... |
| equsb3 2108 | Substitution in an equalit... |
| equsb3r 2109 | Substitution applied to th... |
| equsb1v 2110 | Substitution applied to an... |
| nsb 2111 | Any substitution in an alw... |
| sbn1 2112 | One direction of ~ sbn , u... |
| wel 2114 | Extend wff definition to i... |
| ax8v 2116 | Weakened version of ~ ax-8... |
| ax8v1 2117 | First of two weakened vers... |
| ax8v2 2118 | Second of two weakened ver... |
| ax8 2119 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2120 | An identity law for the no... |
| elsb1 2121 | Substitution for the first... |
| cleljust 2122 | When the class variables i... |
| ax9v 2124 | Weakened version of ~ ax-9... |
| ax9v1 2125 | First of two weakened vers... |
| ax9v2 2126 | Second of two weakened ver... |
| ax9 2127 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2128 | An identity law for the no... |
| elequ2g 2129 | A form of ~ elequ2 with a ... |
| elsb2 2130 | Substitution for the secon... |
| elequ12 2131 | An identity law for the no... |
| ru0 2132 | The FOL statement used in ... |
| ax6dgen 2133 | Tarski's system uses the w... |
| ax10w 2134 | Weak version of ~ ax-10 fr... |
| ax11w 2135 | Weak version of ~ ax-11 fr... |
| ax11dgen 2136 | Degenerate instance of ~ a... |
| ax12wlem 2137 | Lemma for weak version of ... |
| ax12w 2138 | Weak version of ~ ax-12 fr... |
| ax12dgen 2139 | Degenerate instance of ~ a... |
| ax12wdemo 2140 | Example of an application ... |
| ax13w 2141 | Weak version (principal in... |
| ax13dgen1 2142 | Degenerate instance of ~ a... |
| ax13dgen2 2143 | Degenerate instance of ~ a... |
| ax13dgen3 2144 | Degenerate instance of ~ a... |
| ax13dgen4 2145 | Degenerate instance of ~ a... |
| hbn1 2147 | Alias for ~ ax-10 to be us... |
| hbe1 2148 | The setvar ` x ` is not fr... |
| hbe1a 2149 | Dual statement of ~ hbe1 .... |
| nf5-1 2150 | One direction of ~ nf5 can... |
| nf5i 2151 | Deduce that ` x ` is not f... |
| nf5dh 2152 | Deduce that ` x ` is not f... |
| nf5dv 2153 | Apply the definition of no... |
| nfnaew 2154 | All variables are effectiv... |
| nfe1 2155 | The setvar ` x ` is not fr... |
| nfa1 2156 | The setvar ` x ` is not fr... |
| nfna1 2157 | A convenience theorem part... |
| nfia1 2158 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2159 | The setvar ` x ` is not fr... |
| modal5 2160 | The analogue in our predic... |
| nfs1v 2161 | The setvar ` x ` is not fr... |
| alcoms 2163 | Swap quantifiers in an ant... |
| alcom 2164 | Theorem 19.5 of [Margaris]... |
| alrot3 2165 | Theorem *11.21 in [Whitehe... |
| alrot4 2166 | Rotate four universal quan... |
| excom 2167 | Theorem 19.11 of [Margaris... |
| excomim 2168 | One direction of Theorem 1... |
| excom13 2169 | Swap 1st and 3rd existenti... |
| exrot3 2170 | Rotate existential quantif... |
| exrot4 2171 | Rotate existential quantif... |
| hbal 2172 | If ` x ` is not free in ` ... |
| hbald 2173 | Deduction form of bound-va... |
| sbal 2174 | Move universal quantifier ... |
| sbalv 2175 | Quantify with new variable... |
| hbsbw 2176 | If ` z ` is not free in ` ... |
| hbsbwOLD 2177 | Obsolete version of ~ hbsb... |
| sbcom2 2178 | Commutativity law for subs... |
| sbco4lemOLD 2179 | Obsolete version of ~ sbco... |
| sbco4OLD 2180 | Obsolete version of ~ sbco... |
| nfa2 2181 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2183 | This is essentially Axiom ... |
| ax12v2 2184 | It is possible to remove a... |
| ax12ev2 2185 | Version of ~ ax12v2 rewrit... |
| 19.8a 2186 | If a wff is true, it is tr... |
| 19.8ad 2187 | If a wff is true, it is tr... |
| sp 2188 | Specialization. A univers... |
| spi 2189 | Inference rule of universa... |
| sps 2190 | Generalization of antecede... |
| 2sp 2191 | A double specialization (s... |
| spsd 2192 | Deduction generalizing ant... |
| 19.2g 2193 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2194 | Inference form of ~ 19.21 ... |
| 19.21bbi 2195 | Inference removing two uni... |
| 19.23bi 2196 | Inference form of Theorem ... |
| nexr 2197 | Inference associated with ... |
| qexmid 2198 | Quantified excluded middle... |
| nf5r 2199 | Consequence of the definit... |
| nf5ri 2200 | Consequence of the definit... |
| nf5rd 2201 | Consequence of the definit... |
| spimedv 2202 | Deduction version of ~ spi... |
| spimefv 2203 | Version of ~ spime with a ... |
| nfim1 2204 | A closed form of ~ nfim . ... |
| nfan1 2205 | A closed form of ~ nfan . ... |
| 19.3t 2206 | Closed form of ~ 19.3 and ... |
| 19.3 2207 | A wff may be quantified wi... |
| 19.9d 2208 | A deduction version of one... |
| 19.9t 2209 | Closed form of ~ 19.9 and ... |
| 19.9 2210 | A wff may be existentially... |
| 19.21t 2211 | Closed form of Theorem 19.... |
| 19.21 2212 | Theorem 19.21 of [Margaris... |
| stdpc5 2213 | An axiom scheme of standar... |
| 19.21-2 2214 | Version of ~ 19.21 with tw... |
| 19.23t 2215 | Closed form of Theorem 19.... |
| 19.23 2216 | Theorem 19.23 of [Margaris... |
| alimd 2217 | Deduction form of Theorem ... |
| alrimi 2218 | Inference form of Theorem ... |
| alrimdd 2219 | Deduction form of Theorem ... |
| alrimd 2220 | Deduction form of Theorem ... |
| eximd 2221 | Deduction form of Theorem ... |
| exlimi 2222 | Inference associated with ... |
| exlimd 2223 | Deduction form of Theorem ... |
| exlimimdd 2224 | Existential elimination ru... |
| exlimdd 2225 | Existential elimination ru... |
| nexd 2226 | Deduction for generalizati... |
| albid 2227 | Formula-building rule for ... |
| exbid 2228 | Formula-building rule for ... |
| nfbidf 2229 | An equality theorem for ef... |
| 19.16 2230 | Theorem 19.16 of [Margaris... |
| 19.17 2231 | Theorem 19.17 of [Margaris... |
| 19.27 2232 | Theorem 19.27 of [Margaris... |
| 19.28 2233 | Theorem 19.28 of [Margaris... |
| 19.19 2234 | Theorem 19.19 of [Margaris... |
| 19.36 2235 | Theorem 19.36 of [Margaris... |
| 19.36i 2236 | Inference associated with ... |
| 19.37 2237 | Theorem 19.37 of [Margaris... |
| 19.32 2238 | Theorem 19.32 of [Margaris... |
| 19.31 2239 | Theorem 19.31 of [Margaris... |
| 19.41 2240 | Theorem 19.41 of [Margaris... |
| 19.42 2241 | Theorem 19.42 of [Margaris... |
| 19.44 2242 | Theorem 19.44 of [Margaris... |
| 19.45 2243 | Theorem 19.45 of [Margaris... |
| spimfv 2244 | Specialization, using impl... |
| chvarfv 2245 | Implicit substitution of `... |
| cbv3v2 2246 | Version of ~ cbv3 with two... |
| sbalex 2247 | Equivalence of two ways to... |
| sbalexOLD 2248 | Obsolete version of ~ sbal... |
| sb4av 2249 | Version of ~ sb4a with a d... |
| sbimd 2250 | Deduction substituting bot... |
| sbbid 2251 | Deduction substituting bot... |
| 2sbbid 2252 | Deduction doubly substitut... |
| sbequ1 2253 | An equality theorem for su... |
| sbequ2 2254 | An equality theorem for su... |
| stdpc7 2255 | One of the two equality ax... |
| sbequ12 2256 | An equality theorem for su... |
| sbequ12r 2257 | An equality theorem for su... |
| sbelx 2258 | Elimination of substitutio... |
| sbequ12a 2259 | An equality theorem for su... |
| sbid 2260 | An identity theorem for su... |
| sbcov 2261 | A composition law for subs... |
| sbcovOLD 2262 | Obsolete version of ~ sbco... |
| sb6a 2263 | Equivalence for substituti... |
| sbid2vw 2264 | Reverting substitution yie... |
| axc16g 2265 | Generalization of ~ axc16 ... |
| axc16 2266 | Proof of older axiom ~ ax-... |
| axc16gb 2267 | Biconditional strengthenin... |
| axc16nf 2268 | If ~ dtru is false, then t... |
| axc11v 2269 | Version of ~ axc11 with a ... |
| axc11rv 2270 | Version of ~ axc11r with a... |
| drsb2 2271 | Formula-building lemma for... |
| equsalv 2272 | An equivalence related to ... |
| equsexv 2273 | An equivalence related to ... |
| sbft 2274 | Substitution has no effect... |
| sbf 2275 | Substitution for a variabl... |
| sbf2 2276 | Substitution has no effect... |
| sbh 2277 | Substitution for a variabl... |
| hbs1 2278 | The setvar ` x ` is not fr... |
| nfs1f 2279 | If ` x ` is not free in ` ... |
| sb5 2280 | Alternate definition of su... |
| equs5av 2281 | A property related to subs... |
| 2sb5 2282 | Equivalence for double sub... |
| dfsb7 2283 | An alternate definition of... |
| sbn 2284 | Negation inside and outsid... |
| sbex 2285 | Move existential quantifie... |
| nf5 2286 | Alternate definition of ~ ... |
| nf6 2287 | An alternate definition of... |
| nf5d 2288 | Deduce that ` x ` is not f... |
| nf5di 2289 | Since the converse holds b... |
| 19.9h 2290 | A wff may be existentially... |
| 19.21h 2291 | Theorem 19.21 of [Margaris... |
| 19.23h 2292 | Theorem 19.23 of [Margaris... |
| exlimih 2293 | Inference associated with ... |
| exlimdh 2294 | Deduction form of Theorem ... |
| equsalhw 2295 | Version of ~ equsalh with ... |
| equsexhv 2296 | An equivalence related to ... |
| hba1 2297 | The setvar ` x ` is not fr... |
| hbnt 2298 | Closed theorem version of ... |
| hbn 2299 | If ` x ` is not free in ` ... |
| hbnd 2300 | Deduction form of bound-va... |
| hbim1 2301 | A closed form of ~ hbim . ... |
| hbimd 2302 | Deduction form of bound-va... |
| hbim 2303 | If ` x ` is not free in ` ... |
| hban 2304 | If ` x ` is not free in ` ... |
| hb3an 2305 | If ` x ` is not free in ` ... |
| sbi2 2306 | Introduction of implicatio... |
| sbim 2307 | Implication inside and out... |
| sbrim 2308 | Substitution in an implica... |
| sblim 2309 | Substitution in an implica... |
| sbor 2310 | Disjunction inside and out... |
| sbbi 2311 | Equivalence inside and out... |
| sblbis 2312 | Introduce left bicondition... |
| sbrbis 2313 | Introduce right biconditio... |
| sbrbif 2314 | Introduce right biconditio... |
| sbnf 2315 | Move nonfree predicate in ... |
| sbnfOLD 2316 | Obsolete version of ~ sbnf... |
| sbiev 2317 | Conversion of implicit sub... |
| sbievOLD 2318 | Obsolete version of ~ sbie... |
| sbiedw 2319 | Conversion of implicit sub... |
| axc7 2320 | Show that the original axi... |
| axc7e 2321 | Abbreviated version of ~ a... |
| modal-b 2322 | The analogue in our predic... |
| 19.9ht 2323 | A closed version of ~ 19.9... |
| axc4 2324 | Show that the original axi... |
| axc4i 2325 | Inference version of ~ axc... |
| nfal 2326 | If ` x ` is not free in ` ... |
| nfex 2327 | If ` x ` is not free in ` ... |
| hbex 2328 | If ` x ` is not free in ` ... |
| nfnf 2329 | If ` x ` is not free in ` ... |
| 19.12 2330 | Theorem 19.12 of [Margaris... |
| nfald 2331 | Deduction form of ~ nfal .... |
| nfexd 2332 | If ` x ` is not free in ` ... |
| nfsbv 2333 | If ` z ` is not free in ` ... |
| sbco2v 2334 | A composition law for subs... |
| aaan 2335 | Distribute universal quant... |
| eeor 2336 | Distribute existential qua... |
| cbv3v 2337 | Rule used to change bound ... |
| cbv1v 2338 | Rule used to change bound ... |
| cbv2w 2339 | Rule used to change bound ... |
| cbvaldw 2340 | Deduction used to change b... |
| cbvexdw 2341 | Deduction used to change b... |
| cbv3hv 2342 | Rule used to change bound ... |
| cbvalv1 2343 | Rule used to change bound ... |
| cbvexv1 2344 | Rule used to change bound ... |
| cbval2v 2345 | Rule used to change bound ... |
| cbvex2v 2346 | Rule used to change bound ... |
| dvelimhw 2347 | Proof of ~ dvelimh without... |
| pm11.53 2348 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2349 | Special case of ~ 19.12 wh... |
| eean 2350 | Distribute existential qua... |
| eeanv 2351 | Distribute a pair of exist... |
| eeeanv 2352 | Distribute three existenti... |
| ee4anv 2353 | Distribute two pairs of ex... |
| ee4anvOLD 2354 | Obsolete version of ~ ee4a... |
| sb8v 2355 | Substitution of variable i... |
| sb8f 2356 | Substitution of variable i... |
| sb8ef 2357 | Substitution of variable i... |
| 2sb8ef 2358 | An equivalent expression f... |
| sb6rfv 2359 | Reversed substitution. Ve... |
| sbnf2 2360 | Two ways of expressing " `... |
| exsb 2361 | An equivalent expression f... |
| 2exsb 2362 | An equivalent expression f... |
| sbbib 2363 | Reversal of substitution. ... |
| sbbibvv 2364 | Reversal of substitution. ... |
| cbvsbvf 2365 | Change the bound variable ... |
| cleljustALT 2366 | Alternate proof of ~ clelj... |
| cleljustALT2 2367 | Alternate proof of ~ clelj... |
| equs5aALT 2368 | Alternate proof of ~ equs5... |
| equs5eALT 2369 | Alternate proof of ~ equs5... |
| axc11r 2370 | Same as ~ axc11 but with r... |
| dral1v 2371 | Formula-building lemma for... |
| drex1v 2372 | Formula-building lemma for... |
| drnf1v 2373 | Formula-building lemma for... |
| ax13v 2375 | A weaker version of ~ ax-1... |
| ax13lem1 2376 | A version of ~ ax13v with ... |
| ax13 2377 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2378 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2379 | An equation between setvar... |
| dveeq2 2380 | Quantifier introduction wh... |
| nfeqf1 2381 | An equation between setvar... |
| dveeq1 2382 | Quantifier introduction wh... |
| nfeqf 2383 | A variable is effectively ... |
| axc9 2384 | Derive set.mm's original ~... |
| ax6e 2385 | At least one individual ex... |
| ax6 2386 | Theorem showing that ~ ax-... |
| axc10 2387 | Show that the original axi... |
| spimt 2388 | Closed theorem form of ~ s... |
| spim 2389 | Specialization, using impl... |
| spimed 2390 | Deduction version of ~ spi... |
| spime 2391 | Existential introduction, ... |
| spimv 2392 | A version of ~ spim with a... |
| spimvALT 2393 | Alternate proof of ~ spimv... |
| spimev 2394 | Distinct-variable version ... |
| spv 2395 | Specialization, using impl... |
| spei 2396 | Inference from existential... |
| chvar 2397 | Implicit substitution of `... |
| chvarv 2398 | Implicit substitution of `... |
| cbv3 2399 | Rule used to change bound ... |
| cbval 2400 | Rule used to change bound ... |
| cbvex 2401 | Rule used to change bound ... |
| cbvalv 2402 | Rule used to change bound ... |
| cbvexv 2403 | Rule used to change bound ... |
| cbv1 2404 | Rule used to change bound ... |
| cbv2 2405 | Rule used to change bound ... |
| cbv3h 2406 | Rule used to change bound ... |
| cbv1h 2407 | Rule used to change bound ... |
| cbv2h 2408 | Rule used to change bound ... |
| cbvald 2409 | Deduction used to change b... |
| cbvexd 2410 | Deduction used to change b... |
| cbvaldva 2411 | Rule used to change the bo... |
| cbvexdva 2412 | Rule used to change the bo... |
| cbval2 2413 | Rule used to change bound ... |
| cbvex2 2414 | Rule used to change bound ... |
| cbval2vv 2415 | Rule used to change bound ... |
| cbvex2vv 2416 | Rule used to change bound ... |
| cbvex4v 2417 | Rule used to change bound ... |
| equs4 2418 | Lemma used in proofs of im... |
| equsal 2419 | An equivalence related to ... |
| equsex 2420 | An equivalence related to ... |
| equsexALT 2421 | Alternate proof of ~ equse... |
| equsalh 2422 | An equivalence related to ... |
| equsexh 2423 | An equivalence related to ... |
| axc15 2424 | Derivation of set.mm's ori... |
| ax12 2425 | Rederivation of Axiom ~ ax... |
| ax12b 2426 | A bidirectional version of... |
| ax13ALT 2427 | Alternate proof of ~ ax13 ... |
| axc11n 2428 | Derive set.mm's original ~... |
| aecom 2429 | Commutation law for identi... |
| aecoms 2430 | A commutation rule for ide... |
| naecoms 2431 | A commutation rule for dis... |
| axc11 2432 | Show that ~ ax-c11 can be ... |
| hbae 2433 | All variables are effectiv... |
| hbnae 2434 | All variables are effectiv... |
| nfae 2435 | All variables are effectiv... |
| nfnae 2436 | All variables are effectiv... |
| hbnaes 2437 | Rule that applies ~ hbnae ... |
| axc16i 2438 | Inference with ~ axc16 as ... |
| axc16nfALT 2439 | Alternate proof of ~ axc16... |
| dral2 2440 | Formula-building lemma for... |
| dral1 2441 | Formula-building lemma for... |
| dral1ALT 2442 | Alternate proof of ~ dral1... |
| drex1 2443 | Formula-building lemma for... |
| drex2 2444 | Formula-building lemma for... |
| drnf1 2445 | Formula-building lemma for... |
| drnf2 2446 | Formula-building lemma for... |
| nfald2 2447 | Variation on ~ nfald which... |
| nfexd2 2448 | Variation on ~ nfexd which... |
| exdistrf 2449 | Distribution of existentia... |
| dvelimf 2450 | Version of ~ dvelimv witho... |
| dvelimdf 2451 | Deduction form of ~ dvelim... |
| dvelimh 2452 | Version of ~ dvelim withou... |
| dvelim 2453 | This theorem can be used t... |
| dvelimv 2454 | Similar to ~ dvelim with f... |
| dvelimnf 2455 | Version of ~ dvelim using ... |
| dveeq2ALT 2456 | Alternate proof of ~ dveeq... |
| equvini 2457 | A variable introduction la... |
| equvel 2458 | A variable elimination law... |
| equs5a 2459 | A property related to subs... |
| equs5e 2460 | A property related to subs... |
| equs45f 2461 | Two ways of expressing sub... |
| equs5 2462 | Lemma used in proofs of su... |
| dveel1 2463 | Quantifier introduction wh... |
| dveel2 2464 | Quantifier introduction wh... |
| axc14 2465 | Axiom ~ ax-c14 is redundan... |
| sb6x 2466 | Equivalence involving subs... |
| sbequ5 2467 | Substitution does not chan... |
| sbequ6 2468 | Substitution does not chan... |
| sb5rf 2469 | Reversed substitution. Us... |
| sb6rf 2470 | Reversed substitution. Fo... |
| ax12vALT 2471 | Alternate proof of ~ ax12v... |
| 2ax6elem 2472 | We can always find values ... |
| 2ax6e 2473 | We can always find values ... |
| 2sb5rf 2474 | Reversed double substituti... |
| 2sb6rf 2475 | Reversed double substituti... |
| sbel2x 2476 | Elimination of double subs... |
| sb4b 2477 | Simplified definition of s... |
| sb3b 2478 | Simplified definition of s... |
| sb3 2479 | One direction of a simplif... |
| sb1 2480 | One direction of a simplif... |
| sb2 2481 | One direction of a simplif... |
| sb4a 2482 | A version of one implicati... |
| dfsb1 2483 | Alternate definition of su... |
| hbsb2 2484 | Bound-variable hypothesis ... |
| nfsb2 2485 | Bound-variable hypothesis ... |
| hbsb2a 2486 | Special case of a bound-va... |
| sb4e 2487 | One direction of a simplif... |
| hbsb2e 2488 | Special case of a bound-va... |
| hbsb3 2489 | If ` y ` is not free in ` ... |
| nfs1 2490 | If ` y ` is not free in ` ... |
| axc16ALT 2491 | Alternate proof of ~ axc16... |
| axc16gALT 2492 | Alternate proof of ~ axc16... |
| equsb1 2493 | Substitution applied to an... |
| equsb2 2494 | Substitution applied to an... |
| dfsb2 2495 | An alternate definition of... |
| dfsb3 2496 | An alternate definition of... |
| drsb1 2497 | Formula-building lemma for... |
| sb2ae 2498 | In the case of two success... |
| sb6f 2499 | Equivalence for substituti... |
| sb5f 2500 | Equivalence for substituti... |
| nfsb4t 2501 | A variable not free in a p... |
| nfsb4 2502 | A variable not free in a p... |
| sbequ8 2503 | Elimination of equality fr... |
| sbie 2504 | Conversion of implicit sub... |
| sbied 2505 | Conversion of implicit sub... |
| sbiedv 2506 | Conversion of implicit sub... |
| 2sbiev 2507 | Conversion of double impli... |
| sbcom3 2508 | Substituting ` y ` for ` x... |
| sbco 2509 | A composition law for subs... |
| sbid2 2510 | An identity law for substi... |
| sbid2v 2511 | An identity law for substi... |
| sbidm 2512 | An idempotent law for subs... |
| sbco2 2513 | A composition law for subs... |
| sbco2d 2514 | A composition law for subs... |
| sbco3 2515 | A composition law for subs... |
| sbcom 2516 | A commutativity law for su... |
| sbtrt 2517 | Partially closed form of ~... |
| sbtr 2518 | A partial converse to ~ sb... |
| sb8 2519 | Substitution of variable i... |
| sb8e 2520 | Substitution of variable i... |
| sb9 2521 | Commutation of quantificat... |
| sb9i 2522 | Commutation of quantificat... |
| sbhb 2523 | Two ways of expressing " `... |
| nfsbd 2524 | Deduction version of ~ nfs... |
| nfsb 2525 | If ` z ` is not free in ` ... |
| hbsb 2526 | If ` z ` is not free in ` ... |
| sb7f 2527 | This version of ~ dfsb7 do... |
| sb7h 2528 | This version of ~ dfsb7 do... |
| sb10f 2529 | Hao Wang's identity axiom ... |
| sbal1 2530 | Check out ~ sbal for a ver... |
| sbal2 2531 | Move quantifier in and out... |
| 2sb8e 2532 | An equivalent expression f... |
| dfmoeu 2533 | An elementary proof of ~ m... |
| dfeumo 2534 | An elementary proof showin... |
| mojust 2536 | Soundness justification th... |
| dfmo 2538 | Simplify definition ~ df-m... |
| nexmo 2539 | Nonexistence implies uniqu... |
| exmo 2540 | Any proposition holds for ... |
| moabs 2541 | Absorption of existence co... |
| moim 2542 | The at-most-one quantifier... |
| moimi 2543 | The at-most-one quantifier... |
| moimdv 2544 | The at-most-one quantifier... |
| mobi 2545 | Equivalence theorem for th... |
| mobii 2546 | Formula-building rule for ... |
| mobidv 2547 | Formula-building rule for ... |
| mobid 2548 | Formula-building rule for ... |
| moa1 2549 | If an implication holds fo... |
| moan 2550 | "At most one" is still the... |
| moani 2551 | "At most one" is still tru... |
| moor 2552 | "At most one" is still the... |
| mooran1 2553 | "At most one" imports disj... |
| mooran2 2554 | "At most one" exports disj... |
| nfmo1 2555 | Bound-variable hypothesis ... |
| nfmod2 2556 | Bound-variable hypothesis ... |
| nfmodv 2557 | Bound-variable hypothesis ... |
| nfmov 2558 | Bound-variable hypothesis ... |
| nfmod 2559 | Bound-variable hypothesis ... |
| nfmo 2560 | Bound-variable hypothesis ... |
| mof 2561 | Version of ~ df-mo with di... |
| mo3 2562 | Alternate definition of th... |
| mo 2563 | Equivalent definitions of ... |
| mo4 2564 | At-most-one quantifier exp... |
| mo4f 2565 | At-most-one quantifier exp... |
| eu3v 2568 | An alternate way to expres... |
| eujust 2569 | Soundness justification th... |
| eujustALT 2570 | Alternate proof of ~ eujus... |
| eu6lem 2571 | Lemma of ~ eu6im . A diss... |
| eu6 2572 | Alternate definition of th... |
| eu6im 2573 | One direction of ~ eu6 nee... |
| euf 2574 | Version of ~ eu6 with disj... |
| euex 2575 | Existential uniqueness imp... |
| eumo 2576 | Existential uniqueness imp... |
| eumoi 2577 | Uniqueness inferred from e... |
| exmoeub 2578 | Existence implies that uni... |
| exmoeu 2579 | Existence is equivalent to... |
| moeuex 2580 | Uniqueness implies that ex... |
| moeu 2581 | Uniqueness is equivalent t... |
| eubi 2582 | Equivalence theorem for th... |
| eubii 2583 | Introduce unique existenti... |
| eubidv 2584 | Formula-building rule for ... |
| eubid 2585 | Formula-building rule for ... |
| nfeu1 2586 | Bound-variable hypothesis ... |
| nfeu1ALT 2587 | Alternate proof of ~ nfeu1... |
| nfeud2 2588 | Bound-variable hypothesis ... |
| nfeudw 2589 | Bound-variable hypothesis ... |
| nfeud 2590 | Bound-variable hypothesis ... |
| nfeuw 2591 | Bound-variable hypothesis ... |
| nfeu 2592 | Bound-variable hypothesis ... |
| dfeu 2593 | Rederive ~ df-eu from the ... |
| dfmo2 2594 | Rederive ~ df-mo from the ... |
| euequ 2595 | There exists a unique set ... |
| sb8eulem 2596 | Lemma. Factor out the com... |
| sb8euv 2597 | Variable substitution in u... |
| sb8eu 2598 | Variable substitution in u... |
| sb8mo 2599 | Variable substitution for ... |
| cbvmovw 2600 | Change bound variable. Us... |
| cbvmow 2601 | Rule used to change bound ... |
| cbvmo 2602 | Rule used to change bound ... |
| cbveuvw 2603 | Change bound variable. Us... |
| cbveuw 2604 | Version of ~ cbveu with a ... |
| cbveu 2605 | Rule used to change bound ... |
| cbveuALT 2606 | Alternative proof of ~ cbv... |
| eu2 2607 | An alternate way of defini... |
| eu1 2608 | An alternate way to expres... |
| euor 2609 | Introduce a disjunct into ... |
| euorv 2610 | Introduce a disjunct into ... |
| euor2 2611 | Introduce or eliminate a d... |
| sbmo 2612 | Substitution into an at-mo... |
| eu4 2613 | Uniqueness using implicit ... |
| euimmo 2614 | Existential uniqueness imp... |
| euim 2615 | Add unique existential qua... |
| moanimlem 2616 | Factor out the common proo... |
| moanimv 2617 | Introduction of a conjunct... |
| moanim 2618 | Introduction of a conjunct... |
| euan 2619 | Introduction of a conjunct... |
| moanmo 2620 | Nested at-most-one quantif... |
| moaneu 2621 | Nested at-most-one and uni... |
| euanv 2622 | Introduction of a conjunct... |
| mopick 2623 | "At most one" picks a vari... |
| moexexlem 2624 | Factor out the proof skele... |
| 2moexv 2625 | Double quantification with... |
| moexexvw 2626 | "At most one" double quant... |
| 2moswapv 2627 | A condition allowing to sw... |
| 2euswapv 2628 | A condition allowing to sw... |
| 2euexv 2629 | Double quantification with... |
| 2exeuv 2630 | Double existential uniquen... |
| eupick 2631 | Existential uniqueness "pi... |
| eupicka 2632 | Version of ~ eupick with c... |
| eupickb 2633 | Existential uniqueness "pi... |
| eupickbi 2634 | Theorem *14.26 in [Whitehe... |
| mopick2 2635 | "At most one" can show the... |
| moexex 2636 | "At most one" double quant... |
| moexexv 2637 | "At most one" double quant... |
| 2moex 2638 | Double quantification with... |
| 2euex 2639 | Double quantification with... |
| 2eumo 2640 | Nested unique existential ... |
| 2eu2ex 2641 | Double existential uniquen... |
| 2moswap 2642 | A condition allowing to sw... |
| 2euswap 2643 | A condition allowing to sw... |
| 2exeu 2644 | Double existential uniquen... |
| 2mo2 2645 | Two ways of expressing "th... |
| 2mo 2646 | Two ways of expressing "th... |
| 2mos 2647 | Double "there exists at mo... |
| 2mosOLD 2648 | Obsolete version of ~ 2mos... |
| 2eu1 2649 | Double existential uniquen... |
| 2eu1v 2650 | Double existential uniquen... |
| 2eu2 2651 | Double existential uniquen... |
| 2eu3 2652 | Double existential uniquen... |
| 2eu4 2653 | This theorem provides us w... |
| 2eu5 2654 | An alternate definition of... |
| 2eu6 2655 | Two equivalent expressions... |
| 2eu7 2656 | Two equivalent expressions... |
| 2eu8 2657 | Two equivalent expressions... |
| euae 2658 | Two ways to express "exact... |
| exists1 2659 | Two ways to express "exact... |
| exists2 2660 | A condition implying that ... |
| barbara 2661 | "Barbara", one of the fund... |
| celarent 2662 | "Celarent", one of the syl... |
| darii 2663 | "Darii", one of the syllog... |
| dariiALT 2664 | Alternate proof of ~ darii... |
| ferio 2665 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2666 | Lemma for ~ barbari and th... |
| barbari 2667 | "Barbari", one of the syll... |
| barbariALT 2668 | Alternate proof of ~ barba... |
| celaront 2669 | "Celaront", one of the syl... |
| cesare 2670 | "Cesare", one of the syllo... |
| camestres 2671 | "Camestres", one of the sy... |
| festino 2672 | "Festino", one of the syll... |
| festinoALT 2673 | Alternate proof of ~ festi... |
| baroco 2674 | "Baroco", one of the syllo... |
| barocoALT 2675 | Alternate proof of ~ festi... |
| cesaro 2676 | "Cesaro", one of the syllo... |
| camestros 2677 | "Camestros", one of the sy... |
| datisi 2678 | "Datisi", one of the syllo... |
| disamis 2679 | "Disamis", one of the syll... |
| ferison 2680 | "Ferison", one of the syll... |
| bocardo 2681 | "Bocardo", one of the syll... |
| darapti 2682 | "Darapti", one of the syll... |
| daraptiALT 2683 | Alternate proof of ~ darap... |
| felapton 2684 | "Felapton", one of the syl... |
| calemes 2685 | "Calemes", one of the syll... |
| dimatis 2686 | "Dimatis", one of the syll... |
| fresison 2687 | "Fresison", one of the syl... |
| calemos 2688 | "Calemos", one of the syll... |
| fesapo 2689 | "Fesapo", one of the syllo... |
| bamalip 2690 | "Bamalip", one of the syll... |
| axia1 2691 | Left 'and' elimination (in... |
| axia2 2692 | Right 'and' elimination (i... |
| axia3 2693 | 'And' introduction (intuit... |
| axin1 2694 | 'Not' introduction (intuit... |
| axin2 2695 | 'Not' elimination (intuiti... |
| axio 2696 | Definition of 'or' (intuit... |
| axi4 2697 | Specialization (intuitioni... |
| axi5r 2698 | Converse of ~ axc4 (intuit... |
| axial 2699 | The setvar ` x ` is not fr... |
| axie1 2700 | The setvar ` x ` is not fr... |
| axie2 2701 | A key property of existent... |
| axi9 2702 | Axiom of existence (intuit... |
| axi10 2703 | Axiom of Quantifier Substi... |
| axi12 2704 | Axiom of Quantifier Introd... |
| axbnd 2705 | Axiom of Bundling (intuiti... |
| axexte 2707 | The axiom of extensionalit... |
| axextg 2708 | A generalization of the ax... |
| axextb 2709 | A bidirectional version of... |
| axextmo 2710 | There exists at most one s... |
| nulmo 2711 | There exists at most one e... |
| eleq1ab 2714 | Extension (in the sense of... |
| cleljustab 2715 | Extension of ~ cleljust fr... |
| abid 2716 | Simplification of class ab... |
| vexwt 2717 | A standard theorem of pred... |
| vexw 2718 | If ` ph ` is a theorem, th... |
| vextru 2719 | Every setvar is a member o... |
| nfsab1 2720 | Bound-variable hypothesis ... |
| hbab1 2721 | Bound-variable hypothesis ... |
| hbab 2722 | Bound-variable hypothesis ... |
| hbabg 2723 | Bound-variable hypothesis ... |
| nfsab 2724 | Bound-variable hypothesis ... |
| nfsabg 2725 | Bound-variable hypothesis ... |
| dfcleq 2727 | The defining characterizat... |
| cvjust 2728 | Every set is a class. Pro... |
| ax9ALT 2729 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2730 | A weaker version of ~ eleq... |
| eqriv 2731 | Infer equality of classes ... |
| eqrdv 2732 | Deduce equality of classes... |
| eqrdav 2733 | Deduce equality of classes... |
| eqid 2734 | Law of identity (reflexivi... |
| eqidd 2735 | Class identity law with an... |
| eqeq1d 2736 | Deduction from equality to... |
| eqeq1dALT 2737 | Alternate proof of ~ eqeq1... |
| eqeq1 2738 | Equality implies equivalen... |
| eqeq1i 2739 | Inference from equality to... |
| eqcomd 2740 | Deduction from commutative... |
| eqcom 2741 | Commutative law for class ... |
| eqcoms 2742 | Inference applying commuta... |
| eqcomi 2743 | Inference from commutative... |
| neqcomd 2744 | Commute an inequality. (C... |
| eqeq2d 2745 | Deduction from equality to... |
| eqeq2 2746 | Equality implies equivalen... |
| eqeq2i 2747 | Inference from equality to... |
| eqeqan12d 2748 | A useful inference for sub... |
| eqeqan12rd 2749 | A useful inference for sub... |
| eqeq12d 2750 | A useful inference for sub... |
| eqeq12 2751 | Equality relationship amon... |
| eqeq12i 2752 | A useful inference for sub... |
| eqeqan12dALT 2753 | Alternate proof of ~ eqeqa... |
| eqtr 2754 | Transitive law for class e... |
| eqtr2 2755 | A transitive law for class... |
| eqtr3 2756 | A transitive law for class... |
| eqtri 2757 | An equality transitivity i... |
| eqtr2i 2758 | An equality transitivity i... |
| eqtr3i 2759 | An equality transitivity i... |
| eqtr4i 2760 | An equality transitivity i... |
| 3eqtri 2761 | An inference from three ch... |
| 3eqtrri 2762 | An inference from three ch... |
| 3eqtr2i 2763 | An inference from three ch... |
| 3eqtr2ri 2764 | An inference from three ch... |
| 3eqtr3i 2765 | An inference from three ch... |
| 3eqtr3ri 2766 | An inference from three ch... |
| 3eqtr4i 2767 | An inference from three ch... |
| 3eqtr4ri 2768 | An inference from three ch... |
| eqtrd 2769 | An equality transitivity d... |
| eqtr2d 2770 | An equality transitivity d... |
| eqtr3d 2771 | An equality transitivity e... |
| eqtr4d 2772 | An equality transitivity e... |
| 3eqtrd 2773 | A deduction from three cha... |
| 3eqtrrd 2774 | A deduction from three cha... |
| 3eqtr2d 2775 | A deduction from three cha... |
| 3eqtr2rd 2776 | A deduction from three cha... |
| 3eqtr3d 2777 | A deduction from three cha... |
| 3eqtr3rd 2778 | A deduction from three cha... |
| 3eqtr4d 2779 | A deduction from three cha... |
| 3eqtr4rd 2780 | A deduction from three cha... |
| eqtrid 2781 | An equality transitivity d... |
| eqtr2id 2782 | An equality transitivity d... |
| eqtr3id 2783 | An equality transitivity d... |
| eqtr3di 2784 | An equality transitivity d... |
| eqtrdi 2785 | An equality transitivity d... |
| eqtr2di 2786 | An equality transitivity d... |
| eqtr4di 2787 | An equality transitivity d... |
| eqtr4id 2788 | An equality transitivity d... |
| sylan9eq 2789 | An equality transitivity d... |
| sylan9req 2790 | An equality transitivity d... |
| sylan9eqr 2791 | An equality transitivity d... |
| 3eqtr3g 2792 | A chained equality inferen... |
| 3eqtr3a 2793 | A chained equality inferen... |
| 3eqtr4g 2794 | A chained equality inferen... |
| 3eqtr4a 2795 | A chained equality inferen... |
| eq2tri 2796 | A compound transitive infe... |
| iseqsetvlem 2797 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2798 | Alternate proof of ~ iseqs... |
| abbi 2799 | Equivalent formulas yield ... |
| abbidv 2800 | Equivalent wff's yield equ... |
| abbii 2801 | Equivalent wff's yield equ... |
| abbid 2802 | Equivalent wff's yield equ... |
| abbib 2803 | Equal class abstractions r... |
| cbvabv 2804 | Rule used to change bound ... |
| cbvabw 2805 | Rule used to change bound ... |
| cbvab 2806 | Rule used to change bound ... |
| eqabbw 2807 | Version of ~ eqabb using i... |
| eqabcbw 2808 | Version of ~ eqabcb using ... |
| dfclel 2810 | Characterization of the el... |
| elex2 2811 | If a class contains anothe... |
| issettru 2812 | Weak version of ~ isset . ... |
| iseqsetv-clel 2813 | Alternate proof of ~ iseqs... |
| issetlem 2814 | Lemma for ~ elisset and ~ ... |
| elissetv 2815 | An element of a class exis... |
| elisset 2816 | An element of a class exis... |
| eleq1w 2817 | Weaker version of ~ eleq1 ... |
| eleq2w 2818 | Weaker version of ~ eleq2 ... |
| eleq1d 2819 | Deduction from equality to... |
| eleq2d 2820 | Deduction from equality to... |
| eleq2dALT 2821 | Alternate proof of ~ eleq2... |
| eleq1 2822 | Equality implies equivalen... |
| eleq2 2823 | Equality implies equivalen... |
| eleq12 2824 | Equality implies equivalen... |
| eleq1i 2825 | Inference from equality to... |
| eleq2i 2826 | Inference from equality to... |
| eleq12i 2827 | Inference from equality to... |
| eleq12d 2828 | Deduction from equality to... |
| eleq1a 2829 | A transitive-type law rela... |
| eqeltri 2830 | Substitution of equal clas... |
| eqeltrri 2831 | Substitution of equal clas... |
| eleqtri 2832 | Substitution of equal clas... |
| eleqtrri 2833 | Substitution of equal clas... |
| eqeltrd 2834 | Substitution of equal clas... |
| eqeltrrd 2835 | Deduction that substitutes... |
| eleqtrd 2836 | Deduction that substitutes... |
| eleqtrrd 2837 | Deduction that substitutes... |
| eqeltrid 2838 | A membership and equality ... |
| eqeltrrid 2839 | A membership and equality ... |
| eleqtrid 2840 | A membership and equality ... |
| eleqtrrid 2841 | A membership and equality ... |
| eqeltrdi 2842 | A membership and equality ... |
| eqeltrrdi 2843 | A membership and equality ... |
| eleqtrdi 2844 | A membership and equality ... |
| eleqtrrdi 2845 | A membership and equality ... |
| 3eltr3i 2846 | Substitution of equal clas... |
| 3eltr4i 2847 | Substitution of equal clas... |
| 3eltr3d 2848 | Substitution of equal clas... |
| 3eltr4d 2849 | Substitution of equal clas... |
| 3eltr3g 2850 | Substitution of equal clas... |
| 3eltr4g 2851 | Substitution of equal clas... |
| eleq2s 2852 | Substitution of equal clas... |
| eqneltri 2853 | If a class is not an eleme... |
| eqneltrd 2854 | If a class is not an eleme... |
| eqneltrrd 2855 | If a class is not an eleme... |
| neleqtrd 2856 | If a class is not an eleme... |
| neleqtrrd 2857 | If a class is not an eleme... |
| nelneq 2858 | A way of showing two class... |
| nelneq2 2859 | A way of showing two class... |
| eqsb1 2860 | Substitution for the left-... |
| clelsb1 2861 | Substitution for the first... |
| clelsb2 2862 | Substitution for the secon... |
| cleqh 2863 | Establish equality between... |
| hbxfreq 2864 | A utility lemma to transfe... |
| hblem 2865 | Change the free variable o... |
| hblemg 2866 | Change the free variable o... |
| eqabdv 2867 | Deduction from a wff to a ... |
| eqabcdv 2868 | Deduction from a wff to a ... |
| eqabi 2869 | Equality of a class variab... |
| abid1 2870 | Every class is equal to a ... |
| abid2 2871 | A simplification of class ... |
| eqab 2872 | One direction of ~ eqabb i... |
| eqabb 2873 | Equality of a class variab... |
| eqabcb 2874 | Equality of a class variab... |
| eqabrd 2875 | Equality of a class variab... |
| eqabri 2876 | Equality of a class variab... |
| eqabcri 2877 | Equality of a class variab... |
| clelab 2878 | Membership of a class vari... |
| clabel 2879 | Membership of a class abst... |
| sbab 2880 | The right-hand side of the... |
| nfcjust 2882 | Justification theorem for ... |
| nfci 2884 | Deduce that a class ` A ` ... |
| nfcii 2885 | Deduce that a class ` A ` ... |
| nfcr 2886 | Consequence of the not-fre... |
| nfcrALT 2887 | Alternate version of ~ nfc... |
| nfcri 2888 | Consequence of the not-fre... |
| nfcd 2889 | Deduce that a class ` A ` ... |
| nfcrd 2890 | Consequence of the not-fre... |
| nfcrii 2891 | Consequence of the not-fre... |
| nfceqdf 2892 | An equality theorem for ef... |
| nfceqi 2893 | Equality theorem for class... |
| nfcxfr 2894 | A utility lemma to transfe... |
| nfcxfrd 2895 | A utility lemma to transfe... |
| nfcv 2896 | If ` x ` is disjoint from ... |
| nfcvd 2897 | If ` x ` is disjoint from ... |
| nfab1 2898 | Bound-variable hypothesis ... |
| nfnfc1 2899 | The setvar ` x ` is bound ... |
| clelsb1fw 2900 | Substitution for the first... |
| clelsb1f 2901 | Substitution for the first... |
| nfab 2902 | Bound-variable hypothesis ... |
| nfabg 2903 | Bound-variable hypothesis ... |
| nfaba1 2904 | Bound-variable hypothesis ... |
| nfaba1OLD 2905 | Obsolete version of ~ nfab... |
| nfaba1g 2906 | Bound-variable hypothesis ... |
| nfeqd 2907 | Hypothesis builder for equ... |
| nfeld 2908 | Hypothesis builder for ele... |
| nfnfc 2909 | Hypothesis builder for ` F... |
| nfeq 2910 | Hypothesis builder for equ... |
| nfel 2911 | Hypothesis builder for ele... |
| nfeq1 2912 | Hypothesis builder for equ... |
| nfel1 2913 | Hypothesis builder for ele... |
| nfeq2 2914 | Hypothesis builder for equ... |
| nfel2 2915 | Hypothesis builder for ele... |
| drnfc1 2916 | Formula-building lemma for... |
| drnfc2 2917 | Formula-building lemma for... |
| nfabdw 2918 | Bound-variable hypothesis ... |
| nfabd 2919 | Bound-variable hypothesis ... |
| nfabd2 2920 | Bound-variable hypothesis ... |
| dvelimdc 2921 | Deduction form of ~ dvelim... |
| dvelimc 2922 | Version of ~ dvelim for cl... |
| nfcvf 2923 | If ` x ` and ` y ` are dis... |
| nfcvf2 2924 | If ` x ` and ` y ` are dis... |
| cleqf 2925 | Establish equality between... |
| eqabf 2926 | Equality of a class variab... |
| abid2f 2927 | A simplification of class ... |
| abid2fOLD 2928 | Obsolete version of ~ abid... |
| sbabel 2929 | Theorem to move a substitu... |
| neii 2932 | Inference associated with ... |
| neir 2933 | Inference associated with ... |
| nne 2934 | Negation of inequality. (... |
| neneqd 2935 | Deduction eliminating ineq... |
| neneq 2936 | From inequality to non-equ... |
| neqned 2937 | If it is not the case that... |
| neqne 2938 | From non-equality to inequ... |
| neirr 2939 | No class is unequal to its... |
| exmidne 2940 | Excluded middle with equal... |
| eqneqall 2941 | A contradiction concerning... |
| nonconne 2942 | Law of noncontradiction wi... |
| necon3ad 2943 | Contrapositive law deducti... |
| necon3bd 2944 | Contrapositive law deducti... |
| necon2ad 2945 | Contrapositive inference f... |
| necon2bd 2946 | Contrapositive inference f... |
| necon1ad 2947 | Contrapositive deduction f... |
| necon1bd 2948 | Contrapositive deduction f... |
| necon4ad 2949 | Contrapositive inference f... |
| necon4bd 2950 | Contrapositive inference f... |
| necon3d 2951 | Contrapositive law deducti... |
| necon1d 2952 | Contrapositive law deducti... |
| necon2d 2953 | Contrapositive inference f... |
| necon4d 2954 | Contrapositive inference f... |
| necon3ai 2955 | Contrapositive inference f... |
| necon3bi 2956 | Contrapositive inference f... |
| necon1ai 2957 | Contrapositive inference f... |
| necon1bi 2958 | Contrapositive inference f... |
| necon2ai 2959 | Contrapositive inference f... |
| necon2bi 2960 | Contrapositive inference f... |
| necon4ai 2961 | Contrapositive inference f... |
| necon3i 2962 | Contrapositive inference f... |
| necon1i 2963 | Contrapositive inference f... |
| necon2i 2964 | Contrapositive inference f... |
| necon4i 2965 | Contrapositive inference f... |
| necon3abid 2966 | Deduction from equality to... |
| necon3bbid 2967 | Deduction from equality to... |
| necon1abid 2968 | Contrapositive deduction f... |
| necon1bbid 2969 | Contrapositive inference f... |
| necon4abid 2970 | Contrapositive law deducti... |
| necon4bbid 2971 | Contrapositive law deducti... |
| necon2abid 2972 | Contrapositive deduction f... |
| necon2bbid 2973 | Contrapositive deduction f... |
| necon3bid 2974 | Deduction from equality to... |
| necon4bid 2975 | Contrapositive law deducti... |
| necon3abii 2976 | Deduction from equality to... |
| necon3bbii 2977 | Deduction from equality to... |
| necon1abii 2978 | Contrapositive inference f... |
| necon1bbii 2979 | Contrapositive inference f... |
| necon2abii 2980 | Contrapositive inference f... |
| necon2bbii 2981 | Contrapositive inference f... |
| necon3bii 2982 | Inference from equality to... |
| necom 2983 | Commutation of inequality.... |
| necomi 2984 | Inference from commutative... |
| necomd 2985 | Deduction from commutative... |
| nesym 2986 | Characterization of inequa... |
| nesymi 2987 | Inference associated with ... |
| nesymir 2988 | Inference associated with ... |
| neeq1d 2989 | Deduction for inequality. ... |
| neeq2d 2990 | Deduction for inequality. ... |
| neeq12d 2991 | Deduction for inequality. ... |
| neeq1 2992 | Equality theorem for inequ... |
| neeq2 2993 | Equality theorem for inequ... |
| neeq1i 2994 | Inference for inequality. ... |
| neeq2i 2995 | Inference for inequality. ... |
| neeq12i 2996 | Inference for inequality. ... |
| eqnetrd 2997 | Substitution of equal clas... |
| eqnetrrd 2998 | Substitution of equal clas... |
| neeqtrd 2999 | Substitution of equal clas... |
| eqnetri 3000 | Substitution of equal clas... |
| eqnetrri 3001 | Substitution of equal clas... |
| neeqtri 3002 | Substitution of equal clas... |
| neeqtrri 3003 | Substitution of equal clas... |
| neeqtrrd 3004 | Substitution of equal clas... |
| eqnetrrid 3005 | A chained equality inferen... |
| 3netr3d 3006 | Substitution of equality i... |
| 3netr4d 3007 | Substitution of equality i... |
| 3netr3g 3008 | Substitution of equality i... |
| 3netr4g 3009 | Substitution of equality i... |
| nebi 3010 | Contraposition law for ine... |
| pm13.18 3011 | Theorem *13.18 in [Whitehe... |
| pm13.181 3012 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3013 | Inference eliminating an i... |
| pm2.21ddne 3014 | A contradiction implies an... |
| pm2.61ne 3015 | Deduction eliminating an i... |
| pm2.61dne 3016 | Deduction eliminating an i... |
| pm2.61dane 3017 | Deduction eliminating an i... |
| pm2.61da2ne 3018 | Deduction eliminating two ... |
| pm2.61da3ne 3019 | Deduction eliminating thre... |
| pm2.61iine 3020 | Equality version of ~ pm2.... |
| mteqand 3021 | A modus tollens deduction ... |
| neor 3022 | Logical OR with an equalit... |
| neanior 3023 | A De Morgan's law for ineq... |
| ne3anior 3024 | A De Morgan's law for ineq... |
| neorian 3025 | A De Morgan's law for ineq... |
| nemtbir 3026 | An inference from an inequ... |
| nelne1 3027 | Two classes are different ... |
| nelne2 3028 | Two classes are different ... |
| nelelne 3029 | Two classes are different ... |
| neneor 3030 | If two classes are differe... |
| nfne 3031 | Bound-variable hypothesis ... |
| nfned 3032 | Bound-variable hypothesis ... |
| nabbib 3033 | Not equivalent wff's corre... |
| neli 3036 | Inference associated with ... |
| nelir 3037 | Inference associated with ... |
| nelcon3d 3038 | Contrapositive law deducti... |
| neleq12d 3039 | Equality theorem for negat... |
| neleq1 3040 | Equality theorem for negat... |
| neleq2 3041 | Equality theorem for negat... |
| nfnel 3042 | Bound-variable hypothesis ... |
| nfneld 3043 | Bound-variable hypothesis ... |
| nnel 3044 | Negation of negated member... |
| elnelne1 3045 | Two classes are different ... |
| elnelne2 3046 | Two classes are different ... |
| pm2.24nel 3047 | A contradiction concerning... |
| pm2.61danel 3048 | Deduction eliminating an e... |
| rgen 3051 | Generalization rule for re... |
| ralel 3052 | All elements of a class ar... |
| rgenw 3053 | Generalization rule for re... |
| rgen2w 3054 | Generalization rule for re... |
| mprg 3055 | Modus ponens combined with... |
| mprgbir 3056 | Modus ponens on biconditio... |
| raln 3057 | Restricted universally qua... |
| ralnex 3060 | Relationship between restr... |
| dfrex2 3061 | Relationship between restr... |
| nrex 3062 | Inference adding restricte... |
| alral 3063 | Universal quantification i... |
| rexex 3064 | Restricted existence impli... |
| rextru 3065 | Two ways of expressing tha... |
| ralimi2 3066 | Inference quantifying both... |
| reximi2 3067 | Inference quantifying both... |
| ralimia 3068 | Inference quantifying both... |
| reximia 3069 | Inference quantifying both... |
| ralimiaa 3070 | Inference quantifying both... |
| ralimi 3071 | Inference quantifying both... |
| reximi 3072 | Inference quantifying both... |
| ral2imi 3073 | Inference quantifying ante... |
| ralim 3074 | Distribution of restricted... |
| rexim 3075 | Theorem 19.22 of [Margaris... |
| ralbii2 3076 | Inference adding different... |
| rexbii2 3077 | Inference adding different... |
| ralbiia 3078 | Inference adding restricte... |
| rexbiia 3079 | Inference adding restricte... |
| ralbii 3080 | Inference adding restricte... |
| rexbii 3081 | Inference adding restricte... |
| ralanid 3082 | Cancellation law for restr... |
| rexanid 3083 | Cancellation law for restr... |
| ralcom3 3084 | A commutation law for rest... |
| dfral2 3085 | Relationship between restr... |
| rexnal 3086 | Relationship between restr... |
| ralinexa 3087 | A transformation of restri... |
| rexanali 3088 | A transformation of restri... |
| ralbi 3089 | Distribute a restricted un... |
| rexbi 3090 | Distribute restricted quan... |
| ralrexbid 3091 | Formula-building rule for ... |
| r19.35 3092 | Restricted quantifier vers... |
| r19.26m 3093 | Version of ~ 19.26 and ~ r... |
| r19.26 3094 | Restricted quantifier vers... |
| r19.26-3 3095 | Version of ~ r19.26 with t... |
| ralbiim 3096 | Split a biconditional and ... |
| r19.29 3097 | Restricted quantifier vers... |
| r19.29r 3098 | Restricted quantifier vers... |
| r19.29imd 3099 | Theorem 19.29 of [Margaris... |
| r19.40 3100 | Restricted quantifier vers... |
| r19.30 3101 | Restricted quantifier vers... |
| r19.43 3102 | Restricted quantifier vers... |
| 3r19.43 3103 | Restricted quantifier vers... |
| 2ralimi 3104 | Inference quantifying both... |
| 3ralimi 3105 | Inference quantifying both... |
| 4ralimi 3106 | Inference quantifying both... |
| 5ralimi 3107 | Inference quantifying both... |
| 6ralimi 3108 | Inference quantifying both... |
| 2ralbii 3109 | Inference adding two restr... |
| 2rexbii 3110 | Inference adding two restr... |
| 3ralbii 3111 | Inference adding three res... |
| 4ralbii 3112 | Inference adding four rest... |
| 2ralbiim 3113 | Split a biconditional and ... |
| ralnex2 3114 | Relationship between two r... |
| ralnex3 3115 | Relationship between three... |
| rexnal2 3116 | Relationship between two r... |
| rexnal3 3117 | Relationship between three... |
| nrexralim 3118 | Negation of a complex pred... |
| r19.26-2 3119 | Restricted quantifier vers... |
| 2r19.29 3120 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3121 | Theorem 19.29 of [Margaris... |
| r2allem 3122 | Lemma factoring out common... |
| r2exlem 3123 | Lemma factoring out common... |
| hbralrimi 3124 | Inference from Theorem 19.... |
| ralrimiv 3125 | Inference from Theorem 19.... |
| ralrimiva 3126 | Inference from Theorem 19.... |
| rexlimiva 3127 | Inference from Theorem 19.... |
| rexlimiv 3128 | Inference from Theorem 19.... |
| nrexdv 3129 | Deduction adding restricte... |
| ralrimivw 3130 | Inference from Theorem 19.... |
| rexlimivw 3131 | Weaker version of ~ rexlim... |
| ralrimdv 3132 | Inference from Theorem 19.... |
| rexlimdv 3133 | Inference from Theorem 19.... |
| ralrimdva 3134 | Inference from Theorem 19.... |
| rexlimdva 3135 | Inference from Theorem 19.... |
| rexlimdvaa 3136 | Inference from Theorem 19.... |
| rexlimdva2 3137 | Inference from Theorem 19.... |
| r19.29an 3138 | A commonly used pattern in... |
| rexlimdv3a 3139 | Inference from Theorem 19.... |
| rexlimdvw 3140 | Inference from Theorem 19.... |
| rexlimddv 3141 | Restricted existential eli... |
| r19.29a 3142 | A commonly used pattern in... |
| ralimdv2 3143 | Inference quantifying both... |
| reximdv2 3144 | Deduction quantifying both... |
| reximdvai 3145 | Deduction quantifying both... |
| ralimdva 3146 | Deduction quantifying both... |
| reximdva 3147 | Deduction quantifying both... |
| ralimdv 3148 | Deduction quantifying both... |
| reximdv 3149 | Deduction from Theorem 19.... |
| reximddv 3150 | Deduction from Theorem 19.... |
| reximddv3 3151 | Deduction from Theorem 19.... |
| reximssdv 3152 | Derivation of a restricted... |
| ralbidv2 3153 | Formula-building rule for ... |
| rexbidv2 3154 | Formula-building rule for ... |
| ralbidva 3155 | Formula-building rule for ... |
| rexbidva 3156 | Formula-building rule for ... |
| ralbidv 3157 | Formula-building rule for ... |
| rexbidv 3158 | Formula-building rule for ... |
| r19.21v 3159 | Restricted quantifier vers... |
| r19.37v 3160 | Restricted quantifier vers... |
| r19.23v 3161 | Restricted quantifier vers... |
| r19.36v 3162 | Restricted quantifier vers... |
| r19.27v 3163 | Restricted quantitifer ver... |
| r19.41v 3164 | Restricted quantifier vers... |
| r19.28v 3165 | Restricted quantifier vers... |
| r19.42v 3166 | Restricted quantifier vers... |
| r19.32v 3167 | Restricted quantifier vers... |
| r19.45v 3168 | Restricted quantifier vers... |
| r19.44v 3169 | One direction of a restric... |
| r2al 3170 | Double restricted universa... |
| r2ex 3171 | Double restricted existent... |
| r3al 3172 | Triple restricted universa... |
| r3ex 3173 | Triple existential quantif... |
| rgen2 3174 | Generalization rule for re... |
| ralrimivv 3175 | Inference from Theorem 19.... |
| rexlimivv 3176 | Inference from Theorem 19.... |
| ralrimivva 3177 | Inference from Theorem 19.... |
| ralrimdvv 3178 | Inference from Theorem 19.... |
| rgen3 3179 | Generalization rule for re... |
| ralrimivvva 3180 | Inference from Theorem 19.... |
| ralimdvva 3181 | Deduction doubly quantifyi... |
| reximdvva 3182 | Deduction doubly quantifyi... |
| ralimdvv 3183 | Deduction doubly quantifyi... |
| ralimdvvOLD 3184 | Obsolete version of ~ rali... |
| ralimd4v 3185 | Deduction quadrupally quan... |
| ralimd4vOLD 3186 | Obsolete version of ~ rali... |
| ralimd6v 3187 | Deduction sextupally quant... |
| ralimd6vOLD 3188 | Obsolete version of ~ rali... |
| ralrimdvva 3189 | Inference from Theorem 19.... |
| rexlimdvv 3190 | Inference from Theorem 19.... |
| rexlimdvva 3191 | Inference from Theorem 19.... |
| rexlimdvvva 3192 | Inference from Theorem 19.... |
| reximddv2 3193 | Double deduction from Theo... |
| r19.29vva 3194 | A commonly used pattern ba... |
| 2rexbiia 3195 | Inference adding two restr... |
| 2ralbidva 3196 | Formula-building rule for ... |
| 2rexbidva 3197 | Formula-building rule for ... |
| 2ralbidv 3198 | Formula-building rule for ... |
| 2rexbidv 3199 | Formula-building rule for ... |
| rexralbidv 3200 | Formula-building rule for ... |
| 3ralbidv 3201 | Formula-building rule for ... |
| 4ralbidv 3202 | Formula-building rule for ... |
| 6ralbidv 3203 | Formula-building rule for ... |
| r19.41vv 3204 | Version of ~ r19.41v with ... |
| reeanlem 3205 | Lemma factoring out common... |
| reeanv 3206 | Rearrange restricted exist... |
| 3reeanv 3207 | Rearrange three restricted... |
| 2ralor 3208 | Distribute restricted univ... |
| risset 3209 | Two ways to say " ` A ` be... |
| nelb 3210 | A definition of ` -. A e. ... |
| rspw 3211 | Restricted specialization.... |
| cbvralvw 3212 | Change the bound variable ... |
| cbvrexvw 3213 | Change the bound variable ... |
| cbvraldva 3214 | Rule used to change the bo... |
| cbvrexdva 3215 | Rule used to change the bo... |
| cbvral2vw 3216 | Change bound variables of ... |
| cbvrex2vw 3217 | Change bound variables of ... |
| cbvral3vw 3218 | Change bound variables of ... |
| cbvral4vw 3219 | Change bound variables of ... |
| cbvral6vw 3220 | Change bound variables of ... |
| cbvral8vw 3221 | Change bound variables of ... |
| rsp 3222 | Restricted specialization.... |
| rspa 3223 | Restricted specialization.... |
| rspe 3224 | Restricted specialization.... |
| rspec 3225 | Specialization rule for re... |
| r19.21bi 3226 | Inference from Theorem 19.... |
| r19.21be 3227 | Inference from Theorem 19.... |
| r19.21t 3228 | Restricted quantifier vers... |
| r19.21 3229 | Restricted quantifier vers... |
| r19.23t 3230 | Closed theorem form of ~ r... |
| r19.23 3231 | Restricted quantifier vers... |
| ralrimi 3232 | Inference from Theorem 19.... |
| ralrimia 3233 | Inference from Theorem 19.... |
| rexlimi 3234 | Restricted quantifier vers... |
| ralimdaa 3235 | Deduction quantifying both... |
| reximdai 3236 | Deduction from Theorem 19.... |
| r19.37 3237 | Restricted quantifier vers... |
| r19.41 3238 | Restricted quantifier vers... |
| ralrimd 3239 | Inference from Theorem 19.... |
| rexlimd2 3240 | Version of ~ rexlimd with ... |
| rexlimd 3241 | Deduction form of ~ rexlim... |
| r19.29af2 3242 | A commonly used pattern ba... |
| r19.29af 3243 | A commonly used pattern ba... |
| reximd2a 3244 | Deduction quantifying both... |
| ralbida 3245 | Formula-building rule for ... |
| rexbida 3246 | Formula-building rule for ... |
| ralbid 3247 | Formula-building rule for ... |
| rexbid 3248 | Formula-building rule for ... |
| rexbidvALT 3249 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3250 | Alternate proof of ~ rexbi... |
| rsp2 3251 | Restricted specialization,... |
| rsp2e 3252 | Restricted specialization.... |
| rspec2 3253 | Specialization rule for re... |
| rspec3 3254 | Specialization rule for re... |
| r2alf 3255 | Double restricted universa... |
| r2exf 3256 | Double restricted existent... |
| 2ralbida 3257 | Formula-building rule for ... |
| nfra1 3258 | The setvar ` x ` is not fr... |
| nfre1 3259 | The setvar ` x ` is not fr... |
| ralcom4 3260 | Commutation of restricted ... |
| rexcom4 3261 | Commutation of restricted ... |
| ralcom 3262 | Commutation of restricted ... |
| rexcom 3263 | Commutation of restricted ... |
| rexcom4a 3264 | Specialized existential co... |
| ralrot3 3265 | Rotate three restricted un... |
| ralcom13 3266 | Swap first and third restr... |
| rexcom13 3267 | Swap first and third restr... |
| rexrot4 3268 | Rotate four restricted exi... |
| 2ex2rexrot 3269 | Rotate two existential qua... |
| nfra2w 3270 | Similar to Lemma 24 of [Mo... |
| hbra1 3271 | The setvar ` x ` is not fr... |
| ralcomf 3272 | Commutation of restricted ... |
| rexcomf 3273 | Commutation of restricted ... |
| cbvralfw 3274 | Rule used to change bound ... |
| cbvrexfw 3275 | Rule used to change bound ... |
| cbvralw 3276 | Rule used to change bound ... |
| cbvrexw 3277 | Rule used to change bound ... |
| hbral 3278 | Bound-variable hypothesis ... |
| nfraldw 3279 | Deduction version of ~ nfr... |
| nfrexdw 3280 | Deduction version of ~ nfr... |
| nfralw 3281 | Bound-variable hypothesis ... |
| nfrexw 3282 | Bound-variable hypothesis ... |
| r19.12 3283 | Restricted quantifier vers... |
| reean 3284 | Rearrange restricted exist... |
| cbvralsvw 3285 | Change bound variable by u... |
| cbvrexsvw 3286 | Change bound variable by u... |
| cbvralsvwOLD 3287 | Obsolete version of ~ cbvr... |
| cbvralsvwOLDOLD 3288 | Obsolete version of ~ cbvr... |
| cbvrexsvwOLD 3289 | Obsolete version of ~ cbvr... |
| rexeq 3290 | Equality theorem for restr... |
| raleq 3291 | Equality theorem for restr... |
| raleqi 3292 | Equality inference for res... |
| rexeqi 3293 | Equality inference for res... |
| raleqdv 3294 | Equality deduction for res... |
| rexeqdv 3295 | Equality deduction for res... |
| raleqtrdv 3296 | Substitution of equal clas... |
| rexeqtrdv 3297 | Substitution of equal clas... |
| raleqtrrdv 3298 | Substitution of equal clas... |
| rexeqtrrdv 3299 | Substitution of equal clas... |
| raleqbidva 3300 | Equality deduction for res... |
| rexeqbidva 3301 | Equality deduction for res... |
| raleqbidvv 3302 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3303 | Obsolete version of ~ rale... |
| rexeqbidvv 3304 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3305 | Obsolete version of ~ rexe... |
| raleqbi1dv 3306 | Equality deduction for res... |
| rexeqbi1dv 3307 | Equality deduction for res... |
| raleqOLD 3308 | Obsolete version of ~ rale... |
| rexeqOLD 3309 | Obsolete version of ~ rale... |
| raleleq 3310 | All elements of a class ar... |
| raleleqOLD 3311 | Obsolete version of ~ rale... |
| raleqbii 3312 | Equality deduction for res... |
| rexeqbii 3313 | Equality deduction for res... |
| raleqbidv 3314 | Equality deduction for res... |
| rexeqbidv 3315 | Equality deduction for res... |
| cbvraldva2 3316 | Rule used to change the bo... |
| cbvrexdva2 3317 | Rule used to change the bo... |
| cbvraldvaOLD 3318 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3319 | Obsolete version of ~ cbvr... |
| sbralie 3320 | Implicit to explicit subst... |
| sbralieALT 3321 | Alternative shorter proof ... |
| sbralieOLD 3322 | Obsolete version of ~ sbra... |
| raleqf 3323 | Equality theorem for restr... |
| rexeqf 3324 | Equality theorem for restr... |
| rexeqfOLD 3325 | Obsolete version of ~ rexe... |
| raleqbid 3326 | Equality deduction for res... |
| rexeqbid 3327 | Equality deduction for res... |
| cbvralf 3328 | Rule used to change bound ... |
| cbvrexf 3329 | Rule used to change bound ... |
| cbvral 3330 | Rule used to change bound ... |
| cbvrex 3331 | Rule used to change bound ... |
| cbvralv 3332 | Change the bound variable ... |
| cbvrexv 3333 | Change the bound variable ... |
| cbvralsv 3334 | Change bound variable by u... |
| cbvrexsv 3335 | Change bound variable by u... |
| cbvral2v 3336 | Change bound variables of ... |
| cbvrex2v 3337 | Change bound variables of ... |
| cbvral3v 3338 | Change bound variables of ... |
| rgen2a 3339 | Generalization rule for re... |
| nfrald 3340 | Deduction version of ~ nfr... |
| nfrexd 3341 | Deduction version of ~ nfr... |
| nfral 3342 | Bound-variable hypothesis ... |
| nfrex 3343 | Bound-variable hypothesis ... |
| nfra2 3344 | Similar to Lemma 24 of [Mo... |
| ralcom2 3345 | Commutation of restricted ... |
| reu5 3350 | Restricted uniqueness in t... |
| reurmo 3351 | Restricted existential uni... |
| reurex 3352 | Restricted unique existenc... |
| mormo 3353 | Unrestricted "at most one"... |
| rmobiia 3354 | Formula-building rule for ... |
| reubiia 3355 | Formula-building rule for ... |
| rmobii 3356 | Formula-building rule for ... |
| reubii 3357 | Formula-building rule for ... |
| rmoanid 3358 | Cancellation law for restr... |
| reuanid 3359 | Cancellation law for restr... |
| 2reu2rex 3360 | Double restricted existent... |
| rmobidva 3361 | Formula-building rule for ... |
| reubidva 3362 | Formula-building rule for ... |
| rmobidv 3363 | Formula-building rule for ... |
| reubidv 3364 | Formula-building rule for ... |
| reueubd 3365 | Restricted existential uni... |
| rmo5 3366 | Restricted "at most one" i... |
| nrexrmo 3367 | Nonexistence implies restr... |
| moel 3368 | "At most one" element in a... |
| cbvrmovw 3369 | Change the bound variable ... |
| cbvreuvw 3370 | Change the bound variable ... |
| rmobida 3371 | Formula-building rule for ... |
| reubida 3372 | Formula-building rule for ... |
| cbvrmow 3373 | Change the bound variable ... |
| cbvreuw 3374 | Change the bound variable ... |
| nfrmo1 3375 | The setvar ` x ` is not fr... |
| nfreu1 3376 | The setvar ` x ` is not fr... |
| nfrmow 3377 | Bound-variable hypothesis ... |
| nfreuw 3378 | Bound-variable hypothesis ... |
| rmoeq1 3379 | Equality theorem for restr... |
| reueq1 3380 | Equality theorem for restr... |
| rmoeq1OLD 3381 | Obsolete version of ~ rmoe... |
| reueq1OLD 3382 | Obsolete version of ~ reue... |
| rmoeqd 3383 | Equality deduction for res... |
| reueqd 3384 | Equality deduction for res... |
| reueqdv 3385 | Formula-building rule for ... |
| reueqbidv 3386 | Formula-building rule for ... |
| rmoeq1f 3387 | Equality theorem for restr... |
| reueq1f 3388 | Equality theorem for restr... |
| cbvreu 3389 | Change the bound variable ... |
| cbvrmo 3390 | Change the bound variable ... |
| cbvrmov 3391 | Change the bound variable ... |
| cbvreuv 3392 | Change the bound variable ... |
| nfrmod 3393 | Deduction version of ~ nfr... |
| nfreud 3394 | Deduction version of ~ nfr... |
| nfrmo 3395 | Bound-variable hypothesis ... |
| nfreu 3396 | Bound-variable hypothesis ... |
| rabbidva2 3399 | Equivalent wff's yield equ... |
| rabbia2 3400 | Equivalent wff's yield equ... |
| rabbiia 3401 | Equivalent formulas yield ... |
| rabbii 3402 | Equivalent wff's correspon... |
| rabbidva 3403 | Equivalent wff's yield equ... |
| rabbidv 3404 | Equivalent wff's yield equ... |
| rabbieq 3405 | Equivalent wff's correspon... |
| rabswap 3406 | Swap with a membership rel... |
| cbvrabv 3407 | Rule to change the bound v... |
| rabeqcda 3408 | When ` ps ` is always true... |
| rabeqc 3409 | A restricted class abstrac... |
| rabeqi 3410 | Equality theorem for restr... |
| rabeq 3411 | Equality theorem for restr... |
| rabeqdv 3412 | Equality of restricted cla... |
| rabeqbidva 3413 | Equality of restricted cla... |
| rabeqbidvaOLD 3414 | Obsolete version of ~ rabe... |
| rabeqbidv 3415 | Equality of restricted cla... |
| rabrabi 3416 | Abstract builder restricte... |
| nfrab1 3417 | The abstraction variable i... |
| rabid 3418 | An "identity" law of concr... |
| rabidim1 3419 | Membership in a restricted... |
| reqabi 3420 | Inference from equality of... |
| rabrab 3421 | Abstract builder restricte... |
| rabbida4 3422 | Version of ~ rabbidva2 wit... |
| rabbida 3423 | Equivalent wff's yield equ... |
| rabbid 3424 | Version of ~ rabbidv with ... |
| rabeqd 3425 | Deduction form of ~ rabeq ... |
| rabeqbida 3426 | Version of ~ rabeqbidva wi... |
| rabbi 3427 | Equivalent wff's correspon... |
| rabid2f 3428 | An "identity" law for rest... |
| rabid2im 3429 | One direction of ~ rabid2 ... |
| rabid2 3430 | An "identity" law for rest... |
| rabeqf 3431 | Equality theorem for restr... |
| cbvrabw 3432 | Rule to change the bound v... |
| cbvrabwOLD 3433 | Obsolete version of ~ cbvr... |
| nfrabw 3434 | A variable not free in a w... |
| rabbidaOLD 3435 | Obsolete version of ~ rabb... |
| nfrab 3436 | A variable not free in a w... |
| cbvrab 3437 | Rule to change the bound v... |
| vjust 3439 | Justification theorem for ... |
| dfv2 3441 | Alternate definition of th... |
| vex 3442 | All setvar variables are s... |
| elv 3443 | If a proposition is implie... |
| elvd 3444 | If a proposition is implie... |
| el2v 3445 | If a proposition is implie... |
| el3v 3446 | If a proposition is implie... |
| el3v3 3447 | If a proposition is implie... |
| eqv 3448 | The universe contains ever... |
| eqvf 3449 | The universe contains ever... |
| abv 3450 | The class of sets verifyin... |
| abvALT 3451 | Alternate proof of ~ abv ,... |
| isset 3452 | Two ways to express that "... |
| cbvexeqsetf 3453 | The expression ` E. x x = ... |
| issetft 3454 | Closed theorem form of ~ i... |
| issetf 3455 | A version of ~ isset that ... |
| isseti 3456 | A way to say " ` A ` is a ... |
| issetri 3457 | A way to say " ` A ` is a ... |
| eqvisset 3458 | A class equal to a variabl... |
| elex 3459 | If a class is a member of ... |
| elexOLD 3460 | Obsolete version of ~ elex... |
| elexi 3461 | If a class is a member of ... |
| elexd 3462 | If a class is a member of ... |
| elex22 3463 | If two classes each contai... |
| prcnel 3464 | A proper class doesn't bel... |
| ralv 3465 | A universal quantifier res... |
| rexv 3466 | An existential quantifier ... |
| reuv 3467 | A unique existential quant... |
| rmov 3468 | An at-most-one quantifier ... |
| rabab 3469 | A class abstraction restri... |
| rexcom4b 3470 | Specialized existential co... |
| ceqsal1t 3471 | One direction of ~ ceqsalt... |
| ceqsalt 3472 | Closed theorem version of ... |
| ceqsralt 3473 | Restricted quantifier vers... |
| ceqsalg 3474 | A representation of explic... |
| ceqsalgALT 3475 | Alternate proof of ~ ceqsa... |
| ceqsal 3476 | A representation of explic... |
| ceqsalALT 3477 | A representation of explic... |
| ceqsalv 3478 | A representation of explic... |
| ceqsralv 3479 | Restricted quantifier vers... |
| gencl 3480 | Implicit substitution for ... |
| 2gencl 3481 | Implicit substitution for ... |
| 3gencl 3482 | Implicit substitution for ... |
| cgsexg 3483 | Implicit substitution infe... |
| cgsex2g 3484 | Implicit substitution infe... |
| cgsex4g 3485 | An implicit substitution i... |
| cgsex4gOLD 3486 | Obsolete version of ~ cgse... |
| ceqsex 3487 | Elimination of an existent... |
| ceqsexv 3488 | Elimination of an existent... |
| ceqsexv2d 3489 | Elimination of an existent... |
| ceqsexv2dOLD 3490 | Obsolete version of ~ ceqs... |
| ceqsex2 3491 | Elimination of two existen... |
| ceqsex2v 3492 | Elimination of two existen... |
| ceqsex3v 3493 | Elimination of three exist... |
| ceqsex4v 3494 | Elimination of four existe... |
| ceqsex6v 3495 | Elimination of six existen... |
| ceqsex8v 3496 | Elimination of eight exist... |
| gencbvex 3497 | Change of bound variable u... |
| gencbvex2 3498 | Restatement of ~ gencbvex ... |
| gencbval 3499 | Change of bound variable u... |
| sbhypf 3500 | Introduce an explicit subs... |
| spcimgft 3501 | Closed theorem form of ~ s... |
| spcimgfi1 3502 | A closed version of ~ spci... |
| spcimgfi1OLD 3503 | Obsolete version of ~ spci... |
| spcgft 3504 | A closed version of ~ spcg... |
| spcimgf 3505 | Rule of specialization, us... |
| spcimegf 3506 | Existential specialization... |
| vtoclgft 3507 | Closed theorem form of ~ v... |
| vtocleg 3508 | Implicit substitution of a... |
| vtoclg 3509 | Implicit substitution of a... |
| vtocle 3510 | Implicit substitution of a... |
| vtocleOLD 3511 | Obsolete version of ~ vtoc... |
| vtoclbg 3512 | Implicit substitution of a... |
| vtocl 3513 | Implicit substitution of a... |
| vtoclOLD 3514 | Obsolete version of ~ vtoc... |
| vtocldf 3515 | Implicit substitution of a... |
| vtocld 3516 | Implicit substitution of a... |
| vtocl2d 3517 | Implicit substitution of t... |
| vtoclef 3518 | Implicit substitution of a... |
| vtoclf 3519 | Implicit substitution of a... |
| vtocl2 3520 | Implicit substitution of c... |
| vtocl3 3521 | Implicit substitution of c... |
| vtoclb 3522 | Implicit substitution of a... |
| vtoclgf 3523 | Implicit substitution of a... |
| vtoclg1f 3524 | Version of ~ vtoclgf with ... |
| vtocl2gf 3525 | Implicit substitution of a... |
| vtocl3gf 3526 | Implicit substitution of a... |
| vtocl2g 3527 | Implicit substitution of 2... |
| vtocl3g 3528 | Implicit substitution of a... |
| vtoclgaf 3529 | Implicit substitution of a... |
| vtoclga 3530 | Implicit substitution of a... |
| vtocl2ga 3531 | Implicit substitution of 2... |
| vtocl2gaf 3532 | Implicit substitution of 2... |
| vtocl2gafOLD 3533 | Obsolete version of ~ vtoc... |
| vtocl3gaf 3534 | Implicit substitution of 3... |
| vtocl3gafOLD 3535 | Obsolete version of ~ vtoc... |
| vtocl3ga 3536 | Implicit substitution of 3... |
| vtocl3gaOLD 3537 | Obsolete version of ~ vtoc... |
| vtocl4g 3538 | Implicit substitution of 4... |
| vtocl4ga 3539 | Implicit substitution of 4... |
| vtocl4gaOLD 3540 | Obsolete version of ~ vtoc... |
| vtoclegft 3541 | Implicit substitution of a... |
| vtoclri 3542 | Implicit substitution of a... |
| spcgf 3543 | Rule of specialization, us... |
| spcegf 3544 | Existential specialization... |
| spcimdv 3545 | Restricted specialization,... |
| spcdv 3546 | Rule of specialization, us... |
| spcimedv 3547 | Restricted existential spe... |
| spcgv 3548 | Rule of specialization, us... |
| spcegv 3549 | Existential specialization... |
| spcedv 3550 | Existential specialization... |
| spc2egv 3551 | Existential specialization... |
| spc2gv 3552 | Specialization with two qu... |
| spc2ed 3553 | Existential specialization... |
| spc2d 3554 | Specialization with 2 quan... |
| spc3egv 3555 | Existential specialization... |
| spc3gv 3556 | Specialization with three ... |
| spcv 3557 | Rule of specialization, us... |
| spcev 3558 | Existential specialization... |
| spc2ev 3559 | Existential specialization... |
| rspct 3560 | A closed version of ~ rspc... |
| rspcdf 3561 | Restricted specialization,... |
| rspc 3562 | Restricted specialization,... |
| rspce 3563 | Restricted existential spe... |
| rspcimdv 3564 | Restricted specialization,... |
| rspcimedv 3565 | Restricted existential spe... |
| rspcdv 3566 | Restricted specialization,... |
| rspcedv 3567 | Restricted existential spe... |
| rspcebdv 3568 | Restricted existential spe... |
| rspcdv2 3569 | Restricted specialization,... |
| rspcv 3570 | Restricted specialization,... |
| rspccv 3571 | Restricted specialization,... |
| rspcva 3572 | Restricted specialization,... |
| rspccva 3573 | Restricted specialization,... |
| rspcev 3574 | Restricted existential spe... |
| rspcdva 3575 | Restricted specialization,... |
| rspcedvd 3576 | Restricted existential spe... |
| rspcedvdw 3577 | Version of ~ rspcedvd wher... |
| rspceb2dv 3578 | Restricted existential spe... |
| rspcime 3579 | Prove a restricted existen... |
| rspceaimv 3580 | Restricted existential spe... |
| rspcedeq1vd 3581 | Restricted existential spe... |
| rspcedeq2vd 3582 | Restricted existential spe... |
| rspc2 3583 | Restricted specialization ... |
| rspc2gv 3584 | Restricted specialization ... |
| rspc2v 3585 | 2-variable restricted spec... |
| rspc2va 3586 | 2-variable restricted spec... |
| rspc2ev 3587 | 2-variable restricted exis... |
| 2rspcedvdw 3588 | Double application of ~ rs... |
| rspc2dv 3589 | 2-variable restricted spec... |
| rspc3v 3590 | 3-variable restricted spec... |
| rspc3ev 3591 | 3-variable restricted exis... |
| 3rspcedvdw 3592 | Triple application of ~ rs... |
| rspc3dv 3593 | 3-variable restricted spec... |
| rspc4v 3594 | 4-variable restricted spec... |
| rspc6v 3595 | 6-variable restricted spec... |
| rspc8v 3596 | 8-variable restricted spec... |
| rspceeqv 3597 | Restricted existential spe... |
| ralxpxfr2d 3598 | Transfer a universal quant... |
| rexraleqim 3599 | Statement following from e... |
| eqvincg 3600 | A variable introduction la... |
| eqvinc 3601 | A variable introduction la... |
| eqvincf 3602 | A variable introduction la... |
| alexeqg 3603 | Two ways to express substi... |
| ceqex 3604 | Equality implies equivalen... |
| ceqsexg 3605 | A representation of explic... |
| ceqsexgv 3606 | Elimination of an existent... |
| ceqsrexv 3607 | Elimination of a restricte... |
| ceqsrexbv 3608 | Elimination of a restricte... |
| ceqsralbv 3609 | Elimination of a restricte... |
| ceqsrex2v 3610 | Elimination of a restricte... |
| clel2g 3611 | Alternate definition of me... |
| clel2 3612 | Alternate definition of me... |
| clel3g 3613 | Alternate definition of me... |
| clel3 3614 | Alternate definition of me... |
| clel4g 3615 | Alternate definition of me... |
| clel4 3616 | Alternate definition of me... |
| clel5 3617 | Alternate definition of cl... |
| pm13.183 3618 | Compare theorem *13.183 in... |
| rr19.3v 3619 | Restricted quantifier vers... |
| rr19.28v 3620 | Restricted quantifier vers... |
| elab6g 3621 | Membership in a class abst... |
| elabd2 3622 | Membership in a class abst... |
| elabd3 3623 | Membership in a class abst... |
| elabgt 3624 | Membership in a class abst... |
| elabgtOLD 3625 | Obsolete version of ~ elab... |
| elabgtOLDOLD 3626 | Obsolete version of ~ elab... |
| elabgf 3627 | Membership in a class abst... |
| elabf 3628 | Membership in a class abst... |
| elabg 3629 | Membership in a class abst... |
| elabgw 3630 | Membership in a class abst... |
| elab2gw 3631 | Membership in a class abst... |
| elab 3632 | Membership in a class abst... |
| elab2g 3633 | Membership in a class abst... |
| elabd 3634 | Explicit demonstration the... |
| elab2 3635 | Membership in a class abst... |
| elab4g 3636 | Membership in a class abst... |
| elab3gf 3637 | Membership in a class abst... |
| elab3g 3638 | Membership in a class abst... |
| elab3 3639 | Membership in a class abst... |
| elrabi 3640 | Implication for the member... |
| elrabf 3641 | Membership in a restricted... |
| rabtru 3642 | Abstract builder using the... |
| elrab3t 3643 | Membership in a restricted... |
| elrab 3644 | Membership in a restricted... |
| elrab3 3645 | Membership in a restricted... |
| elrabd 3646 | Membership in a restricted... |
| elrab2 3647 | Membership in a restricted... |
| elrab2w 3648 | Membership in a restricted... |
| ralab 3649 | Universal quantification o... |
| ralrab 3650 | Universal quantification o... |
| rexab 3651 | Existential quantification... |
| rexrab 3652 | Existential quantification... |
| ralab2 3653 | Universal quantification o... |
| ralrab2 3654 | Universal quantification o... |
| rexab2 3655 | Existential quantification... |
| rexrab2 3656 | Existential quantification... |
| reurab 3657 | Restricted existential uni... |
| abidnf 3658 | Identity used to create cl... |
| dedhb 3659 | A deduction theorem for co... |
| class2seteq 3660 | Writing a set as a class a... |
| nelrdva 3661 | Deduce negative membership... |
| eqeu 3662 | A condition which implies ... |
| moeq 3663 | There exists at most one s... |
| eueq 3664 | A class is a set if and on... |
| eueqi 3665 | There exists a unique set ... |
| eueq2 3666 | Equality has existential u... |
| eueq3 3667 | Equality has existential u... |
| moeq3 3668 | "At most one" property of ... |
| mosub 3669 | "At most one" remains true... |
| mo2icl 3670 | Theorem for inferring "at ... |
| mob2 3671 | Consequence of "at most on... |
| moi2 3672 | Consequence of "at most on... |
| mob 3673 | Equality implied by "at mo... |
| moi 3674 | Equality implied by "at mo... |
| morex 3675 | Derive membership from uni... |
| euxfr2w 3676 | Transfer existential uniqu... |
| euxfrw 3677 | Transfer existential uniqu... |
| euxfr2 3678 | Transfer existential uniqu... |
| euxfr 3679 | Transfer existential uniqu... |
| euind 3680 | Existential uniqueness via... |
| reu2 3681 | A way to express restricte... |
| reu6 3682 | A way to express restricte... |
| reu3 3683 | A way to express restricte... |
| reu6i 3684 | A condition which implies ... |
| eqreu 3685 | A condition which implies ... |
| rmo4 3686 | Restricted "at most one" u... |
| reu4 3687 | Restricted uniqueness usin... |
| reu7 3688 | Restricted uniqueness usin... |
| reu8 3689 | Restricted uniqueness usin... |
| rmo3f 3690 | Restricted "at most one" u... |
| rmo4f 3691 | Restricted "at most one" u... |
| reu2eqd 3692 | Deduce equality from restr... |
| reueq 3693 | Equality has existential u... |
| rmoeq 3694 | Equality's restricted exis... |
| rmoan 3695 | Restricted "at most one" s... |
| rmoim 3696 | Restricted "at most one" i... |
| rmoimia 3697 | Restricted "at most one" i... |
| rmoimi 3698 | Restricted "at most one" i... |
| rmoimi2 3699 | Restricted "at most one" i... |
| 2reu5a 3700 | Double restricted existent... |
| reuimrmo 3701 | Restricted uniqueness impl... |
| 2reuswap 3702 | A condition allowing swap ... |
| 2reuswap2 3703 | A condition allowing swap ... |
| reuxfrd 3704 | Transfer existential uniqu... |
| reuxfr 3705 | Transfer existential uniqu... |
| reuxfr1d 3706 | Transfer existential uniqu... |
| reuxfr1ds 3707 | Transfer existential uniqu... |
| reuxfr1 3708 | Transfer existential uniqu... |
| reuind 3709 | Existential uniqueness via... |
| 2rmorex 3710 | Double restricted quantifi... |
| 2reu5lem1 3711 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3712 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3713 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3714 | Double restricted existent... |
| 2reurmo 3715 | Double restricted quantifi... |
| 2reurex 3716 | Double restricted quantifi... |
| 2rmoswap 3717 | A condition allowing to sw... |
| 2rexreu 3718 | Double restricted existent... |
| cdeqi 3721 | Deduce conditional equalit... |
| cdeqri 3722 | Property of conditional eq... |
| cdeqth 3723 | Deduce conditional equalit... |
| cdeqnot 3724 | Distribute conditional equ... |
| cdeqal 3725 | Distribute conditional equ... |
| cdeqab 3726 | Distribute conditional equ... |
| cdeqal1 3727 | Distribute conditional equ... |
| cdeqab1 3728 | Distribute conditional equ... |
| cdeqim 3729 | Distribute conditional equ... |
| cdeqcv 3730 | Conditional equality for s... |
| cdeqeq 3731 | Distribute conditional equ... |
| cdeqel 3732 | Distribute conditional equ... |
| nfcdeq 3733 | If we have a conditional e... |
| nfccdeq 3734 | Variation of ~ nfcdeq for ... |
| rru 3735 | Relative version of Russel... |
| ru 3736 | Russell's Paradox. Propos... |
| ruOLD 3737 | Obsolete version of ~ ru a... |
| dfsbcq 3740 | Proper substitution of a c... |
| dfsbcq2 3741 | This theorem, which is sim... |
| sbsbc 3742 | Show that ~ df-sb and ~ df... |
| sbceq1d 3743 | Equality theorem for class... |
| sbceq1dd 3744 | Equality theorem for class... |
| sbceqbid 3745 | Equality theorem for class... |
| sbc8g 3746 | This is the closest we can... |
| sbc2or 3747 | The disjunction of two equ... |
| sbcex 3748 | By our definition of prope... |
| sbceq1a 3749 | Equality theorem for class... |
| sbceq2a 3750 | Equality theorem for class... |
| spsbc 3751 | Specialization: if a formu... |
| spsbcd 3752 | Specialization: if a formu... |
| sbcth 3753 | A substitution into a theo... |
| sbcthdv 3754 | Deduction version of ~ sbc... |
| sbcid 3755 | An identity theorem for su... |
| nfsbc1d 3756 | Deduction version of ~ nfs... |
| nfsbc1 3757 | Bound-variable hypothesis ... |
| nfsbc1v 3758 | Bound-variable hypothesis ... |
| nfsbcdw 3759 | Deduction version of ~ nfs... |
| nfsbcw 3760 | Bound-variable hypothesis ... |
| sbccow 3761 | A composition law for clas... |
| nfsbcd 3762 | Deduction version of ~ nfs... |
| nfsbc 3763 | Bound-variable hypothesis ... |
| sbcco 3764 | A composition law for clas... |
| sbcco2 3765 | A composition law for clas... |
| sbc5 3766 | An equivalence for class s... |
| sbc5ALT 3767 | Alternate proof of ~ sbc5 ... |
| sbc6g 3768 | An equivalence for class s... |
| sbc6 3769 | An equivalence for class s... |
| sbc7 3770 | An equivalence for class s... |
| cbvsbcw 3771 | Change bound variables in ... |
| cbvsbcvw 3772 | Change the bound variable ... |
| cbvsbc 3773 | Change bound variables in ... |
| cbvsbcv 3774 | Change the bound variable ... |
| sbciegft 3775 | Conversion of implicit sub... |
| sbciegftOLD 3776 | Obsolete version of ~ sbci... |
| sbciegf 3777 | Conversion of implicit sub... |
| sbcieg 3778 | Conversion of implicit sub... |
| sbcie2g 3779 | Conversion of implicit sub... |
| sbcie 3780 | Conversion of implicit sub... |
| sbciedf 3781 | Conversion of implicit sub... |
| sbcied 3782 | Conversion of implicit sub... |
| sbcied2 3783 | Conversion of implicit sub... |
| elrabsf 3784 | Membership in a restricted... |
| eqsbc1 3785 | Substitution for the left-... |
| sbcng 3786 | Move negation in and out o... |
| sbcimg 3787 | Distribution of class subs... |
| sbcan 3788 | Distribution of class subs... |
| sbcor 3789 | Distribution of class subs... |
| sbcbig 3790 | Distribution of class subs... |
| sbcn1 3791 | Move negation in and out o... |
| sbcim1 3792 | Distribution of class subs... |
| sbcbid 3793 | Formula-building deduction... |
| sbcbidv 3794 | Formula-building deduction... |
| sbcbii 3795 | Formula-building inference... |
| sbcbi1 3796 | Distribution of class subs... |
| sbcbi2 3797 | Substituting into equivale... |
| sbcal 3798 | Move universal quantifier ... |
| sbcex2 3799 | Move existential quantifie... |
| sbceqal 3800 | Class version of one impli... |
| sbeqalb 3801 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3802 | Substitution for the right... |
| sbc3an 3803 | Distribution of class subs... |
| sbcel1v 3804 | Class substitution into a ... |
| sbcel2gv 3805 | Class substitution into a ... |
| sbcel21v 3806 | Class substitution into a ... |
| sbcimdv 3807 | Substitution analogue of T... |
| sbctt 3808 | Substitution for a variabl... |
| sbcgf 3809 | Substitution for a variabl... |
| sbc19.21g 3810 | Substitution for a variabl... |
| sbcg 3811 | Substitution for a variabl... |
| sbcgfi 3812 | Substitution for a variabl... |
| sbc2iegf 3813 | Conversion of implicit sub... |
| sbc2ie 3814 | Conversion of implicit sub... |
| sbc2iedv 3815 | Conversion of implicit sub... |
| sbc3ie 3816 | Conversion of implicit sub... |
| sbccomlem 3817 | Lemma for ~ sbccom . (Con... |
| sbccomlemOLD 3818 | Obsolete version of ~ sbcc... |
| sbccom 3819 | Commutative law for double... |
| sbcralt 3820 | Interchange class substitu... |
| sbcrext 3821 | Interchange class substitu... |
| sbcralg 3822 | Interchange class substitu... |
| sbcrex 3823 | Interchange class substitu... |
| sbcreu 3824 | Interchange class substitu... |
| reu8nf 3825 | Restricted uniqueness usin... |
| sbcabel 3826 | Interchange class substitu... |
| rspsbc 3827 | Restricted quantifier vers... |
| rspsbca 3828 | Restricted quantifier vers... |
| rspesbca 3829 | Existence form of ~ rspsbc... |
| spesbc 3830 | Existence form of ~ spsbc ... |
| spesbcd 3831 | form of ~ spsbc . (Contri... |
| sbcth2 3832 | A substitution into a theo... |
| ra4v 3833 | Version of ~ ra4 with a di... |
| ra4 3834 | Restricted quantifier vers... |
| rmo2 3835 | Alternate definition of re... |
| rmo2i 3836 | Condition implying restric... |
| rmo3 3837 | Restricted "at most one" u... |
| rmob 3838 | Consequence of "at most on... |
| rmoi 3839 | Consequence of "at most on... |
| rmob2 3840 | Consequence of "restricted... |
| rmoi2 3841 | Consequence of "restricted... |
| rmoanim 3842 | Introduction of a conjunct... |
| rmoanimALT 3843 | Alternate proof of ~ rmoan... |
| reuan 3844 | Introduction of a conjunct... |
| 2reu1 3845 | Double restricted existent... |
| 2reu2 3846 | Double restricted existent... |
| csb2 3849 | Alternate expression for t... |
| csbeq1 3850 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3851 | Equality deduction for pro... |
| csbeq2 3852 | Substituting into equivale... |
| csbeq2d 3853 | Formula-building deduction... |
| csbeq2dv 3854 | Formula-building deduction... |
| csbeq2i 3855 | Formula-building inference... |
| csbeq12dv 3856 | Formula-building inference... |
| cbvcsbw 3857 | Change bound variables in ... |
| cbvcsb 3858 | Change bound variables in ... |
| cbvcsbv 3859 | Change the bound variable ... |
| csbid 3860 | Analogue of ~ sbid for pro... |
| csbeq1a 3861 | Equality theorem for prope... |
| csbcow 3862 | Composition law for chaine... |
| csbco 3863 | Composition law for chaine... |
| csbtt 3864 | Substitution doesn't affec... |
| csbconstgf 3865 | Substitution doesn't affec... |
| csbconstg 3866 | Substitution doesn't affec... |
| csbgfi 3867 | Substitution for a variabl... |
| csbconstgi 3868 | The proper substitution of... |
| nfcsb1d 3869 | Bound-variable hypothesis ... |
| nfcsb1 3870 | Bound-variable hypothesis ... |
| nfcsb1v 3871 | Bound-variable hypothesis ... |
| nfcsbd 3872 | Deduction version of ~ nfc... |
| nfcsbw 3873 | Bound-variable hypothesis ... |
| nfcsb 3874 | Bound-variable hypothesis ... |
| csbhypf 3875 | Introduce an explicit subs... |
| csbiebt 3876 | Conversion of implicit sub... |
| csbiedf 3877 | Conversion of implicit sub... |
| csbieb 3878 | Bidirectional conversion b... |
| csbiebg 3879 | Bidirectional conversion b... |
| csbiegf 3880 | Conversion of implicit sub... |
| csbief 3881 | Conversion of implicit sub... |
| csbie 3882 | Conversion of implicit sub... |
| csbied 3883 | Conversion of implicit sub... |
| csbied2 3884 | Conversion of implicit sub... |
| csbie2t 3885 | Conversion of implicit sub... |
| csbie2 3886 | Conversion of implicit sub... |
| csbie2g 3887 | Conversion of implicit sub... |
| cbvrabcsfw 3888 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3889 | A more general version of ... |
| cbvrexcsf 3890 | A more general version of ... |
| cbvreucsf 3891 | A more general version of ... |
| cbvrabcsf 3892 | A more general version of ... |
| cbvralv2 3893 | Rule used to change the bo... |
| cbvrexv2 3894 | Rule used to change the bo... |
| rspc2vd 3895 | Deduction version of 2-var... |
| difjust 3901 | Soundness justification th... |
| unjust 3903 | Soundness justification th... |
| injust 3905 | Soundness justification th... |
| dfin5 3907 | Alternate definition for t... |
| dfdif2 3908 | Alternate definition of cl... |
| eldif 3909 | Expansion of membership in... |
| eldifd 3910 | If a class is in one class... |
| eldifad 3911 | If a class is in the diffe... |
| eldifbd 3912 | If a class is in the diffe... |
| elneeldif 3913 | The elements of a set diff... |
| velcomp 3914 | Characterization of setvar... |
| elin 3915 | Expansion of membership in... |
| dfss2 3917 | Alternate definition of th... |
| dfss 3918 | Variant of subclass defini... |
| dfss3 3920 | Alternate definition of su... |
| dfss6 3921 | Alternate definition of su... |
| dfssf 3922 | Equivalence for subclass r... |
| dfss3f 3923 | Equivalence for subclass r... |
| nfss 3924 | If ` x ` is not free in ` ... |
| ssel 3925 | Membership relationships f... |
| ssel2 3926 | Membership relationships f... |
| sseli 3927 | Membership implication fro... |
| sselii 3928 | Membership inference from ... |
| sselid 3929 | Membership inference from ... |
| sseld 3930 | Membership deduction from ... |
| sselda 3931 | Membership deduction from ... |
| sseldd 3932 | Membership inference from ... |
| ssneld 3933 | If a class is not in anoth... |
| ssneldd 3934 | If an element is not in a ... |
| ssriv 3935 | Inference based on subclas... |
| ssrd 3936 | Deduction based on subclas... |
| ssrdv 3937 | Deduction based on subclas... |
| sstr2 3938 | Transitivity of subclass r... |
| sstr2OLD 3939 | Obsolete version of ~ sstr... |
| sstr 3940 | Transitivity of subclass r... |
| sstri 3941 | Subclass transitivity infe... |
| sstrd 3942 | Subclass transitivity dedu... |
| sstrid 3943 | Subclass transitivity dedu... |
| sstrdi 3944 | Subclass transitivity dedu... |
| sylan9ss 3945 | A subclass transitivity de... |
| sylan9ssr 3946 | A subclass transitivity de... |
| eqss 3947 | The subclass relationship ... |
| eqssi 3948 | Infer equality from two su... |
| eqssd 3949 | Equality deduction from tw... |
| sssseq 3950 | If a class is a subclass o... |
| eqrd 3951 | Deduce equality of classes... |
| eqri 3952 | Infer equality of classes ... |
| eqelssd 3953 | Equality deduction from su... |
| ssid 3954 | Any class is a subclass of... |
| ssidd 3955 | Weakening of ~ ssid . (Co... |
| ssv 3956 | Any class is a subclass of... |
| sseq1 3957 | Equality theorem for subcl... |
| sseq2 3958 | Equality theorem for the s... |
| sseq12 3959 | Equality theorem for the s... |
| sseq1i 3960 | An equality inference for ... |
| sseq2i 3961 | An equality inference for ... |
| sseq12i 3962 | An equality inference for ... |
| sseq1d 3963 | An equality deduction for ... |
| sseq2d 3964 | An equality deduction for ... |
| sseq12d 3965 | An equality deduction for ... |
| eqsstrd 3966 | Substitution of equality i... |
| eqsstrrd 3967 | Substitution of equality i... |
| sseqtrd 3968 | Substitution of equality i... |
| sseqtrrd 3969 | Substitution of equality i... |
| eqsstrid 3970 | A chained subclass and equ... |
| eqsstrrid 3971 | A chained subclass and equ... |
| sseqtrdi 3972 | A chained subclass and equ... |
| sseqtrrdi 3973 | A chained subclass and equ... |
| sseqtrid 3974 | Subclass transitivity dedu... |
| sseqtrrid 3975 | Subclass transitivity dedu... |
| eqsstrdi 3976 | A chained subclass and equ... |
| eqsstrrdi 3977 | A chained subclass and equ... |
| eqsstri 3978 | Substitution of equality i... |
| eqsstrri 3979 | Substitution of equality i... |
| sseqtri 3980 | Substitution of equality i... |
| sseqtrri 3981 | Substitution of equality i... |
| 3sstr3i 3982 | Substitution of equality i... |
| 3sstr4i 3983 | Substitution of equality i... |
| 3sstr3g 3984 | Substitution of equality i... |
| 3sstr4g 3985 | Substitution of equality i... |
| 3sstr3d 3986 | Substitution of equality i... |
| 3sstr4d 3987 | Substitution of equality i... |
| eqimssd 3988 | Equality implies inclusion... |
| eqimsscd 3989 | Equality implies inclusion... |
| eqimss 3990 | Equality implies inclusion... |
| eqimss2 3991 | Equality implies inclusion... |
| eqimssi 3992 | Infer subclass relationshi... |
| eqimss2i 3993 | Infer subclass relationshi... |
| nssne1 3994 | Two classes are different ... |
| nssne2 3995 | Two classes are different ... |
| nss 3996 | Negation of subclass relat... |
| nelss 3997 | Demonstrate by witnesses t... |
| ssrexf 3998 | Restricted existential qua... |
| ssrmof 3999 | "At most one" existential ... |
| ssralv 4000 | Quantification restricted ... |
| ssrexv 4001 | Existential quantification... |
| ss2ralv 4002 | Two quantifications restri... |
| ss2rexv 4003 | Two existential quantifica... |
| ssralvOLD 4004 | Obsolete version of ~ ssra... |
| ssrexvOLD 4005 | Obsolete version of ~ ssre... |
| ralss 4006 | Restricted universal quant... |
| rexss 4007 | Restricted existential qua... |
| ralssOLD 4008 | Obsolete version of ~ rals... |
| rexssOLD 4009 | Obsolete version of ~ rexs... |
| ss2abim 4010 | Class abstractions in a su... |
| ss2ab 4011 | Class abstractions in a su... |
| abss 4012 | Class abstraction in a sub... |
| ssab 4013 | Subclass of a class abstra... |
| ssabral 4014 | The relation for a subclas... |
| ss2abdv 4015 | Deduction of abstraction s... |
| ss2abi 4016 | Inference of abstraction s... |
| abssdv 4017 | Deduction of abstraction s... |
| abssi 4018 | Inference of abstraction s... |
| ss2rab 4019 | Restricted abstraction cla... |
| rabss 4020 | Restricted class abstracti... |
| ssrab 4021 | Subclass of a restricted c... |
| ss2rabd 4022 | Subclass of a restricted c... |
| ssrabdv 4023 | Subclass of a restricted c... |
| rabssdv 4024 | Subclass of a restricted c... |
| ss2rabdv 4025 | Deduction of restricted ab... |
| ss2rabi 4026 | Inference of restricted ab... |
| rabss2 4027 | Subclass law for restricte... |
| rabss2OLD 4028 | Obsolete version of ~ rabs... |
| ssab2 4029 | Subclass relation for the ... |
| ssrab2 4030 | Subclass relation for a re... |
| rabss3d 4031 | Subclass law for restricte... |
| ssrab3 4032 | Subclass relation for a re... |
| rabssrabd 4033 | Subclass of a restricted c... |
| ssrabeq 4034 | If the restricting class o... |
| rabssab 4035 | A restricted class is a su... |
| eqrrabd 4036 | Deduce equality with a res... |
| uniiunlem 4037 | A subset relationship usef... |
| dfpss2 4038 | Alternate definition of pr... |
| dfpss3 4039 | Alternate definition of pr... |
| psseq1 4040 | Equality theorem for prope... |
| psseq2 4041 | Equality theorem for prope... |
| psseq1i 4042 | An equality inference for ... |
| psseq2i 4043 | An equality inference for ... |
| psseq12i 4044 | An equality inference for ... |
| psseq1d 4045 | An equality deduction for ... |
| psseq2d 4046 | An equality deduction for ... |
| psseq12d 4047 | An equality deduction for ... |
| pssss 4048 | A proper subclass is a sub... |
| pssne 4049 | Two classes in a proper su... |
| pssssd 4050 | Deduce subclass from prope... |
| pssned 4051 | Proper subclasses are uneq... |
| sspss 4052 | Subclass in terms of prope... |
| pssirr 4053 | Proper subclass is irrefle... |
| pssn2lp 4054 | Proper subclass has no 2-c... |
| sspsstri 4055 | Two ways of stating tricho... |
| ssnpss 4056 | Partial trichotomy law for... |
| psstr 4057 | Transitive law for proper ... |
| sspsstr 4058 | Transitive law for subclas... |
| psssstr 4059 | Transitive law for subclas... |
| psstrd 4060 | Proper subclass inclusion ... |
| sspsstrd 4061 | Transitivity involving sub... |
| psssstrd 4062 | Transitivity involving sub... |
| npss 4063 | A class is not a proper su... |
| ssnelpss 4064 | A subclass missing a membe... |
| ssnelpssd 4065 | Subclass inclusion with on... |
| ssexnelpss 4066 | If there is an element of ... |
| dfdif3 4067 | Alternate definition of cl... |
| dfdif3OLD 4068 | Obsolete version of ~ dfdi... |
| difeq1 4069 | Equality theorem for class... |
| difeq2 4070 | Equality theorem for class... |
| difeq12 4071 | Equality theorem for class... |
| difeq1i 4072 | Inference adding differenc... |
| difeq2i 4073 | Inference adding differenc... |
| difeq12i 4074 | Equality inference for cla... |
| difeq1d 4075 | Deduction adding differenc... |
| difeq2d 4076 | Deduction adding differenc... |
| difeq12d 4077 | Equality deduction for cla... |
| difeqri 4078 | Inference from membership ... |
| nfdif 4079 | Bound-variable hypothesis ... |
| nfdifOLD 4080 | Obsolete version of ~ nfdi... |
| eldifi 4081 | Implication of membership ... |
| eldifn 4082 | Implication of membership ... |
| elndif 4083 | A set does not belong to a... |
| neldif 4084 | Implication of membership ... |
| difdif 4085 | Double class difference. ... |
| difss 4086 | Subclass relationship for ... |
| difssd 4087 | A difference of two classe... |
| difss2 4088 | If a class is contained in... |
| difss2d 4089 | If a class is contained in... |
| ssdifss 4090 | Preservation of a subclass... |
| ddif 4091 | Double complement under un... |
| ssconb 4092 | Contraposition law for sub... |
| sscon 4093 | Contraposition law for sub... |
| ssdif 4094 | Difference law for subsets... |
| ssdifd 4095 | If ` A ` is contained in `... |
| sscond 4096 | If ` A ` is contained in `... |
| ssdifssd 4097 | If ` A ` is contained in `... |
| ssdif2d 4098 | If ` A ` is contained in `... |
| raldifb 4099 | Restricted universal quant... |
| rexdifi 4100 | Restricted existential qua... |
| complss 4101 | Complementation reverses i... |
| compleq 4102 | Two classes are equal if a... |
| elun 4103 | Expansion of membership in... |
| elunnel1 4104 | A member of a union that i... |
| elunnel2 4105 | A member of a union that i... |
| uneqri 4106 | Inference from membership ... |
| unidm 4107 | Idempotent law for union o... |
| uncom 4108 | Commutative law for union ... |
| equncom 4109 | If a class equals the unio... |
| equncomi 4110 | Inference form of ~ equnco... |
| uneq1 4111 | Equality theorem for the u... |
| uneq2 4112 | Equality theorem for the u... |
| uneq12 4113 | Equality theorem for the u... |
| uneq1i 4114 | Inference adding union to ... |
| uneq2i 4115 | Inference adding union to ... |
| uneq12i 4116 | Equality inference for the... |
| uneq1d 4117 | Deduction adding union to ... |
| uneq2d 4118 | Deduction adding union to ... |
| uneq12d 4119 | Equality deduction for the... |
| nfun 4120 | Bound-variable hypothesis ... |
| nfunOLD 4121 | Obsolete version of ~ nfun... |
| unass 4122 | Associative law for union ... |
| un12 4123 | A rearrangement of union. ... |
| un23 4124 | A rearrangement of union. ... |
| un4 4125 | A rearrangement of the uni... |
| unundi 4126 | Union distributes over its... |
| unundir 4127 | Union distributes over its... |
| ssun1 4128 | Subclass relationship for ... |
| ssun2 4129 | Subclass relationship for ... |
| ssun3 4130 | Subclass law for union of ... |
| ssun4 4131 | Subclass law for union of ... |
| elun1 4132 | Membership law for union o... |
| elun2 4133 | Membership law for union o... |
| elunant 4134 | A statement is true for ev... |
| unss1 4135 | Subclass law for union of ... |
| ssequn1 4136 | A relationship between sub... |
| unss2 4137 | Subclass law for union of ... |
| unss12 4138 | Subclass law for union of ... |
| ssequn2 4139 | A relationship between sub... |
| unss 4140 | The union of two subclasse... |
| unssi 4141 | An inference showing the u... |
| unssd 4142 | A deduction showing the un... |
| unssad 4143 | If ` ( A u. B ) ` is conta... |
| unssbd 4144 | If ` ( A u. B ) ` is conta... |
| ssun 4145 | A condition that implies i... |
| rexun 4146 | Restricted existential qua... |
| ralunb 4147 | Restricted quantification ... |
| ralun 4148 | Restricted quantification ... |
| elini 4149 | Membership in an intersect... |
| elind 4150 | Deduce membership in an in... |
| elinel1 4151 | Membership in an intersect... |
| elinel2 4152 | Membership in an intersect... |
| elin2 4153 | Membership in a class defi... |
| elin1d 4154 | Elementhood in the first s... |
| elin2d 4155 | Elementhood in the first s... |
| elin3 4156 | Membership in a class defi... |
| nel1nelin 4157 | Membership in an intersect... |
| nel2nelin 4158 | Membership in an intersect... |
| incom 4159 | Commutative law for inters... |
| ineqcom 4160 | Two ways of expressing tha... |
| ineqcomi 4161 | Two ways of expressing tha... |
| ineqri 4162 | Inference from membership ... |
| ineq1 4163 | Equality theorem for inter... |
| ineq2 4164 | Equality theorem for inter... |
| ineq12 4165 | Equality theorem for inter... |
| ineq1i 4166 | Equality inference for int... |
| ineq2i 4167 | Equality inference for int... |
| ineq12i 4168 | Equality inference for int... |
| ineq1d 4169 | Equality deduction for int... |
| ineq2d 4170 | Equality deduction for int... |
| ineq12d 4171 | Equality deduction for int... |
| ineqan12d 4172 | Equality deduction for int... |
| sseqin2 4173 | A relationship between sub... |
| nfin 4174 | Bound-variable hypothesis ... |
| nfinOLD 4175 | Obsolete version of ~ nfin... |
| rabbi2dva 4176 | Deduction from a wff to a ... |
| inidm 4177 | Idempotent law for interse... |
| inass 4178 | Associative law for inters... |
| in12 4179 | A rearrangement of interse... |
| in32 4180 | A rearrangement of interse... |
| in13 4181 | A rearrangement of interse... |
| in31 4182 | A rearrangement of interse... |
| inrot 4183 | Rotate the intersection of... |
| in4 4184 | Rearrangement of intersect... |
| inindi 4185 | Intersection distributes o... |
| inindir 4186 | Intersection distributes o... |
| inss1 4187 | The intersection of two cl... |
| inss2 4188 | The intersection of two cl... |
| ssin 4189 | Subclass of intersection. ... |
| ssini 4190 | An inference showing that ... |
| ssind 4191 | A deduction showing that a... |
| ssrin 4192 | Add right intersection to ... |
| sslin 4193 | Add left intersection to s... |
| ssrind 4194 | Add right intersection to ... |
| ss2in 4195 | Intersection of subclasses... |
| ssinss1 4196 | Intersection preserves sub... |
| ssinss1d 4197 | Intersection preserves sub... |
| inss 4198 | Inclusion of an intersecti... |
| ralin 4199 | Restricted universal quant... |
| rexin 4200 | Restricted existential qua... |
| dfss7 4201 | Alternate definition of su... |
| symdifcom 4204 | Symmetric difference commu... |
| symdifeq1 4205 | Equality theorem for symme... |
| symdifeq2 4206 | Equality theorem for symme... |
| nfsymdif 4207 | Hypothesis builder for sym... |
| elsymdif 4208 | Membership in a symmetric ... |
| dfsymdif4 4209 | Alternate definition of th... |
| elsymdifxor 4210 | Membership in a symmetric ... |
| dfsymdif2 4211 | Alternate definition of th... |
| symdifass 4212 | Symmetric difference is as... |
| difsssymdif 4213 | The symmetric difference c... |
| difsymssdifssd 4214 | If the symmetric differenc... |
| unabs 4215 | Absorption law for union. ... |
| inabs 4216 | Absorption law for interse... |
| nssinpss 4217 | Negation of subclass expre... |
| nsspssun 4218 | Negation of subclass expre... |
| dfss4 4219 | Subclass defined in terms ... |
| dfun2 4220 | An alternate definition of... |
| dfin2 4221 | An alternate definition of... |
| difin 4222 | Difference with intersecti... |
| ssdifim 4223 | Implication of a class dif... |
| ssdifsym 4224 | Symmetric class difference... |
| dfss5 4225 | Alternate definition of su... |
| dfun3 4226 | Union defined in terms of ... |
| dfin3 4227 | Intersection defined in te... |
| dfin4 4228 | Alternate definition of th... |
| invdif 4229 | Intersection with universa... |
| indif 4230 | Intersection with class di... |
| indif2 4231 | Bring an intersection in a... |
| indif1 4232 | Bring an intersection in a... |
| indifcom 4233 | Commutation law for inters... |
| indi 4234 | Distributive law for inter... |
| undi 4235 | Distributive law for union... |
| indir 4236 | Distributive law for inter... |
| undir 4237 | Distributive law for union... |
| unineq 4238 | Infer equality from equali... |
| uneqin 4239 | Equality of union and inte... |
| difundi 4240 | Distributive law for class... |
| difundir 4241 | Distributive law for class... |
| difindi 4242 | Distributive law for class... |
| difindir 4243 | Distributive law for class... |
| indifdi 4244 | Distribute intersection ov... |
| indifdir 4245 | Distribute intersection ov... |
| difdif2 4246 | Class difference by a clas... |
| undm 4247 | De Morgan's law for union.... |
| indm 4248 | De Morgan's law for inters... |
| difun1 4249 | A relationship involving d... |
| undif3 4250 | An equality involving clas... |
| difin2 4251 | Represent a class differen... |
| dif32 4252 | Swap second and third argu... |
| difabs 4253 | Absorption-like law for cl... |
| sscon34b 4254 | Relative complementation r... |
| rcompleq 4255 | Two subclasses are equal i... |
| dfsymdif3 4256 | Alternate definition of th... |
| unabw 4257 | Union of two class abstrac... |
| unab 4258 | Union of two class abstrac... |
| inab 4259 | Intersection of two class ... |
| difab 4260 | Difference of two class ab... |
| abanssl 4261 | A class abstraction with a... |
| abanssr 4262 | A class abstraction with a... |
| notabw 4263 | A class abstraction define... |
| notab 4264 | A class abstraction define... |
| unrab 4265 | Union of two restricted cl... |
| inrab 4266 | Intersection of two restri... |
| inrab2 4267 | Intersection with a restri... |
| difrab 4268 | Difference of two restrict... |
| dfrab3 4269 | Alternate definition of re... |
| dfrab2 4270 | Alternate definition of re... |
| rabdif 4271 | Move difference in and out... |
| notrab 4272 | Complementation of restric... |
| dfrab3ss 4273 | Restricted class abstracti... |
| rabun2 4274 | Abstraction restricted to ... |
| reuun2 4275 | Transfer uniqueness to a s... |
| reuss2 4276 | Transfer uniqueness to a s... |
| reuss 4277 | Transfer uniqueness to a s... |
| reuun1 4278 | Transfer uniqueness to a s... |
| reupick 4279 | Restricted uniqueness "pic... |
| reupick3 4280 | Restricted uniqueness "pic... |
| reupick2 4281 | Restricted uniqueness "pic... |
| euelss 4282 | Transfer uniqueness of an ... |
| dfnul4 4285 | Alternate definition of th... |
| dfnul2 4286 | Alternate definition of th... |
| dfnul3 4287 | Alternate definition of th... |
| noel 4288 | The empty set has no eleme... |
| nel02 4289 | The empty set has no eleme... |
| n0i 4290 | If a class has elements, t... |
| ne0i 4291 | If a class has elements, t... |
| ne0d 4292 | Deduction form of ~ ne0i .... |
| n0ii 4293 | If a class has elements, t... |
| ne0ii 4294 | If a class has elements, t... |
| vn0 4295 | The universal class is not... |
| vn0ALT 4296 | Alternate proof of ~ vn0 .... |
| eq0f 4297 | A class is equal to the em... |
| neq0f 4298 | A class is not empty if an... |
| n0f 4299 | A class is nonempty if and... |
| eq0 4300 | A class is equal to the em... |
| eq0ALT 4301 | Alternate proof of ~ eq0 .... |
| neq0 4302 | A class is not empty if an... |
| n0 4303 | A class is nonempty if and... |
| nel0 4304 | From the general negation ... |
| reximdva0 4305 | Restricted existence deduc... |
| rspn0 4306 | Specialization for restric... |
| n0rex 4307 | There is an element in a n... |
| ssn0rex 4308 | There is an element in a c... |
| n0moeu 4309 | A case of equivalence of "... |
| rex0 4310 | Vacuous restricted existen... |
| reu0 4311 | Vacuous restricted uniquen... |
| rmo0 4312 | Vacuous restricted at-most... |
| 0el 4313 | Membership of the empty se... |
| n0el 4314 | Negated membership of the ... |
| eqeuel 4315 | A condition which implies ... |
| ssdif0 4316 | Subclass expressed in term... |
| difn0 4317 | If the difference of two s... |
| pssdifn0 4318 | A proper subclass has a no... |
| pssdif 4319 | A proper subclass has a no... |
| ndisj 4320 | Express that an intersecti... |
| inn0f 4321 | A nonempty intersection. ... |
| inn0 4322 | A nonempty intersection. ... |
| difin0ss 4323 | Difference, intersection, ... |
| inssdif0 4324 | Intersection, subclass, an... |
| inindif 4325 | The intersection and class... |
| difid 4326 | The difference between a c... |
| difidALT 4327 | Alternate proof of ~ difid... |
| dif0 4328 | The difference between a c... |
| ab0w 4329 | The class of sets verifyin... |
| ab0 4330 | The class of sets verifyin... |
| ab0ALT 4331 | Alternate proof of ~ ab0 ,... |
| dfnf5 4332 | Characterization of nonfre... |
| ab0orv 4333 | The class abstraction defi... |
| ab0orvALT 4334 | Alternate proof of ~ ab0or... |
| abn0 4335 | Nonempty class abstraction... |
| rab0 4336 | Any restricted class abstr... |
| rabeq0w 4337 | Condition for a restricted... |
| rabeq0 4338 | Condition for a restricted... |
| rabn0 4339 | Nonempty restricted class ... |
| rabxm 4340 | Law of excluded middle, in... |
| rabnc 4341 | Law of noncontradiction, i... |
| elneldisj 4342 | The set of elements ` s ` ... |
| elnelun 4343 | The union of the set of el... |
| un0 4344 | The union of a class with ... |
| in0 4345 | The intersection of a clas... |
| 0un 4346 | The union of the empty set... |
| 0in 4347 | The intersection of the em... |
| inv1 4348 | The intersection of a clas... |
| unv 4349 | The union of a class with ... |
| 0ss 4350 | The null set is a subset o... |
| ss0b 4351 | Any subset of the empty se... |
| ss0 4352 | Any subset of the empty se... |
| sseq0 4353 | A subclass of an empty cla... |
| ssn0 4354 | A class with a nonempty su... |
| 0dif 4355 | The difference between the... |
| abf 4356 | A class abstraction determ... |
| eq0rdv 4357 | Deduction for equality to ... |
| eq0rdvALT 4358 | Alternate proof of ~ eq0rd... |
| csbprc 4359 | The proper substitution of... |
| csb0 4360 | The proper substitution of... |
| sbcel12 4361 | Distribute proper substitu... |
| sbceqg 4362 | Distribute proper substitu... |
| sbceqi 4363 | Distribution of class subs... |
| sbcnel12g 4364 | Distribute proper substitu... |
| sbcne12 4365 | Distribute proper substitu... |
| sbcel1g 4366 | Move proper substitution i... |
| sbceq1g 4367 | Move proper substitution t... |
| sbcel2 4368 | Move proper substitution i... |
| sbceq2g 4369 | Move proper substitution t... |
| csbcom 4370 | Commutative law for double... |
| sbcnestgfw 4371 | Nest the composition of tw... |
| csbnestgfw 4372 | Nest the composition of tw... |
| sbcnestgw 4373 | Nest the composition of tw... |
| csbnestgw 4374 | Nest the composition of tw... |
| sbcco3gw 4375 | Composition of two substit... |
| sbcnestgf 4376 | Nest the composition of tw... |
| csbnestgf 4377 | Nest the composition of tw... |
| sbcnestg 4378 | Nest the composition of tw... |
| csbnestg 4379 | Nest the composition of tw... |
| sbcco3g 4380 | Composition of two substit... |
| csbco3g 4381 | Composition of two class s... |
| csbnest1g 4382 | Nest the composition of tw... |
| csbidm 4383 | Idempotent law for class s... |
| csbvarg 4384 | The proper substitution of... |
| csbvargi 4385 | The proper substitution of... |
| sbccsb 4386 | Substitution into a wff ex... |
| sbccsb2 4387 | Substitution into a wff ex... |
| rspcsbela 4388 | Special case related to ~ ... |
| sbnfc2 4389 | Two ways of expressing " `... |
| csbab 4390 | Move substitution into a c... |
| csbun 4391 | Distribution of class subs... |
| csbin 4392 | Distribute proper substitu... |
| csbie2df 4393 | Conversion of implicit sub... |
| 2nreu 4394 | If there are two different... |
| un00 4395 | Two classes are empty iff ... |
| vss 4396 | Only the universal class h... |
| 0pss 4397 | The null set is a proper s... |
| npss0 4398 | No set is a proper subset ... |
| pssv 4399 | Any non-universal class is... |
| disj 4400 | Two ways of saying that tw... |
| disjr 4401 | Two ways of saying that tw... |
| disj1 4402 | Two ways of saying that tw... |
| reldisj 4403 | Two ways of saying that tw... |
| disj3 4404 | Two ways of saying that tw... |
| disjne 4405 | Members of disjoint sets a... |
| disjeq0 4406 | Two disjoint sets are equa... |
| disjel 4407 | A set can't belong to both... |
| disj2 4408 | Two ways of saying that tw... |
| disj4 4409 | Two ways of saying that tw... |
| ssdisj 4410 | Intersection with a subcla... |
| disjpss 4411 | A class is a proper subset... |
| undisj1 4412 | The union of disjoint clas... |
| undisj2 4413 | The union of disjoint clas... |
| ssindif0 4414 | Subclass expressed in term... |
| inelcm 4415 | The intersection of classe... |
| minel 4416 | A minimum element of a cla... |
| undif4 4417 | Distribute union over diff... |
| disjssun 4418 | Subset relation for disjoi... |
| vdif0 4419 | Universal class equality i... |
| difrab0eq 4420 | If the difference between ... |
| pssnel 4421 | A proper subclass has a me... |
| disjdif 4422 | A class and its relative c... |
| disjdifr 4423 | A class and its relative c... |
| difin0 4424 | The difference of a class ... |
| unvdif 4425 | The union of a class and i... |
| undif1 4426 | Absorption of difference b... |
| undif2 4427 | Absorption of difference b... |
| undifabs 4428 | Absorption of difference b... |
| inundif 4429 | The intersection and class... |
| disjdif2 4430 | The difference of a class ... |
| difun2 4431 | Absorption of union by dif... |
| undif 4432 | Union of complementary par... |
| undifr 4433 | Union of complementary par... |
| undifrOLD 4434 | Obsolete version of ~ undi... |
| undif5 4435 | An equality involving clas... |
| ssdifin0 4436 | A subset of a difference d... |
| ssdifeq0 4437 | A class is a subclass of i... |
| ssundif 4438 | A condition equivalent to ... |
| difcom 4439 | Swap the arguments of a cl... |
| pssdifcom1 4440 | Two ways to express overla... |
| pssdifcom2 4441 | Two ways to express non-co... |
| difdifdir 4442 | Distributive law for class... |
| uneqdifeq 4443 | Two ways to say that ` A `... |
| raldifeq 4444 | Equality theorem for restr... |
| rzal 4445 | Vacuous quantification is ... |
| rzalALT 4446 | Alternate proof of ~ rzal ... |
| rexn0 4447 | Restricted existential qua... |
| ralf0 4448 | The quantification of a fa... |
| ral0 4449 | Vacuous universal quantifi... |
| r19.2z 4450 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4451 | A response to the notion t... |
| r19.3rz 4452 | Restricted quantification ... |
| r19.28z 4453 | Restricted quantifier vers... |
| r19.3rzv 4454 | Restricted quantification ... |
| r19.3rzvOLD 4455 | Obsolete version of ~ r19.... |
| r19.9rzv 4456 | Restricted quantification ... |
| r19.28zv 4457 | Restricted quantifier vers... |
| r19.37zv 4458 | Restricted quantifier vers... |
| r19.45zv 4459 | Restricted version of Theo... |
| r19.44zv 4460 | Restricted version of Theo... |
| r19.27z 4461 | Restricted quantifier vers... |
| r19.27zv 4462 | Restricted quantifier vers... |
| r19.36zv 4463 | Restricted quantifier vers... |
| ralnralall 4464 | A contradiction concerning... |
| falseral0 4465 | A false statement can only... |
| falseral0OLD 4466 | Obsolete version of ~ fals... |
| ralidmw 4467 | Idempotent law for restric... |
| ralidm 4468 | Idempotent law for restric... |
| raaan 4469 | Rearrange restricted quant... |
| raaanv 4470 | Rearrange restricted quant... |
| sbss 4471 | Set substitution into the ... |
| sbcssg 4472 | Distribute proper substitu... |
| raaan2 4473 | Rearrange restricted quant... |
| 2reu4lem 4474 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4475 | Definition of double restr... |
| csbdif 4476 | Distribution of class subs... |
| dfif2 4479 | An alternate definition of... |
| dfif6 4480 | An alternate definition of... |
| ifeq1 4481 | Equality theorem for condi... |
| ifeq2 4482 | Equality theorem for condi... |
| iftrue 4483 | Value of the conditional o... |
| iftruei 4484 | Inference associated with ... |
| iftrued 4485 | Value of the conditional o... |
| iffalse 4486 | Value of the conditional o... |
| iffalsei 4487 | Inference associated with ... |
| iffalsed 4488 | Value of the conditional o... |
| ifnefalse 4489 | When values are unequal, b... |
| iftrueb 4490 | When the branches are not ... |
| ifsb 4491 | Distribute a function over... |
| dfif3 4492 | Alternate definition of th... |
| dfif4 4493 | Alternate definition of th... |
| dfif5 4494 | Alternate definition of th... |
| ifssun 4495 | A conditional class is inc... |
| ifeq12 4496 | Equality theorem for condi... |
| ifeq1d 4497 | Equality deduction for con... |
| ifeq2d 4498 | Equality deduction for con... |
| ifeq12d 4499 | Equality deduction for con... |
| ifbi 4500 | Equivalence theorem for co... |
| ifbid 4501 | Equivalence deduction for ... |
| ifbieq1d 4502 | Equivalence/equality deduc... |
| ifbieq2i 4503 | Equivalence/equality infer... |
| ifbieq2d 4504 | Equivalence/equality deduc... |
| ifbieq12i 4505 | Equivalence deduction for ... |
| ifbieq12d 4506 | Equivalence deduction for ... |
| nfifd 4507 | Deduction form of ~ nfif .... |
| nfif 4508 | Bound-variable hypothesis ... |
| ifeq1da 4509 | Conditional equality. (Co... |
| ifeq2da 4510 | Conditional equality. (Co... |
| ifeq12da 4511 | Equivalence deduction for ... |
| ifbieq12d2 4512 | Equivalence deduction for ... |
| ifclda 4513 | Conditional closure. (Con... |
| ifeqda 4514 | Separation of the values o... |
| elimif 4515 | Elimination of a condition... |
| ifbothda 4516 | A wff ` th ` containing a ... |
| ifboth 4517 | A wff ` th ` containing a ... |
| ifid 4518 | Identical true and false a... |
| eqif 4519 | Expansion of an equality w... |
| ifval 4520 | Another expression of the ... |
| elif 4521 | Membership in a conditiona... |
| ifel 4522 | Membership of a conditiona... |
| ifcl 4523 | Membership (closure) of a ... |
| ifcld 4524 | Membership (closure) of a ... |
| ifcli 4525 | Inference associated with ... |
| ifexd 4526 | Existence of the condition... |
| ifexg 4527 | Existence of the condition... |
| ifex 4528 | Existence of the condition... |
| ifeqor 4529 | The possible values of a c... |
| ifnot 4530 | Negating the first argumen... |
| ifan 4531 | Rewrite a conjunction in a... |
| ifor 4532 | Rewrite a disjunction in a... |
| 2if2 4533 | Resolve two nested conditi... |
| ifcomnan 4534 | Commute the conditions in ... |
| csbif 4535 | Distribute proper substitu... |
| dedth 4536 | Weak deduction theorem tha... |
| dedth2h 4537 | Weak deduction theorem eli... |
| dedth3h 4538 | Weak deduction theorem eli... |
| dedth4h 4539 | Weak deduction theorem eli... |
| dedth2v 4540 | Weak deduction theorem for... |
| dedth3v 4541 | Weak deduction theorem for... |
| dedth4v 4542 | Weak deduction theorem for... |
| elimhyp 4543 | Eliminate a hypothesis con... |
| elimhyp2v 4544 | Eliminate a hypothesis con... |
| elimhyp3v 4545 | Eliminate a hypothesis con... |
| elimhyp4v 4546 | Eliminate a hypothesis con... |
| elimel 4547 | Eliminate a membership hyp... |
| elimdhyp 4548 | Version of ~ elimhyp where... |
| keephyp 4549 | Transform a hypothesis ` p... |
| keephyp2v 4550 | Keep a hypothesis containi... |
| keephyp3v 4551 | Keep a hypothesis containi... |
| pwjust 4553 | Soundness justification th... |
| elpwg 4555 | Membership in a power clas... |
| elpw 4556 | Membership in a power clas... |
| velpw 4557 | Setvar variable membership... |
| elpwd 4558 | Membership in a power clas... |
| elpwi 4559 | Subset relation implied by... |
| elpwb 4560 | Characterization of the el... |
| elpwid 4561 | An element of a power clas... |
| elelpwi 4562 | If ` A ` belongs to a part... |
| sspw 4563 | The powerclass preserves i... |
| sspwi 4564 | The powerclass preserves i... |
| sspwd 4565 | The powerclass preserves i... |
| pweq 4566 | Equality theorem for power... |
| pweqALT 4567 | Alternate proof of ~ pweq ... |
| pweqi 4568 | Equality inference for pow... |
| pweqd 4569 | Equality deduction for pow... |
| pwunss 4570 | The power class of the uni... |
| nfpw 4571 | Bound-variable hypothesis ... |
| pwidg 4572 | A set is an element of its... |
| pwidb 4573 | A class is an element of i... |
| pwid 4574 | A set is a member of its p... |
| pwss 4575 | Subclass relationship for ... |
| pwundif 4576 | Break up the power class o... |
| snjust 4577 | Soundness justification th... |
| sneq 4588 | Equality theorem for singl... |
| sneqi 4589 | Equality inference for sin... |
| sneqd 4590 | Equality deduction for sin... |
| dfsn2 4591 | Alternate definition of si... |
| elsng 4592 | There is exactly one eleme... |
| elsn 4593 | There is exactly one eleme... |
| velsn 4594 | There is only one element ... |
| elsni 4595 | There is at most one eleme... |
| elsnd 4596 | There is at most one eleme... |
| rabsneq 4597 | Equality of class abstract... |
| absn 4598 | Condition for a class abst... |
| dfpr2 4599 | Alternate definition of a ... |
| dfsn2ALT 4600 | Alternate definition of si... |
| elprg 4601 | A member of a pair of clas... |
| elpri 4602 | If a class is an element o... |
| elpr 4603 | A member of a pair of clas... |
| elpr2g 4604 | A member of a pair of sets... |
| elpr2 4605 | A member of a pair of sets... |
| elprn1 4606 | A member of an unordered p... |
| elprn2 4607 | A member of an unordered p... |
| nelpr2 4608 | If a class is not an eleme... |
| nelpr1 4609 | If a class is not an eleme... |
| nelpri 4610 | If an element doesn't matc... |
| prneli 4611 | If an element doesn't matc... |
| nelprd 4612 | If an element doesn't matc... |
| eldifpr 4613 | Membership in a set with t... |
| rexdifpr 4614 | Restricted existential qua... |
| snidg 4615 | A set is a member of its s... |
| snidb 4616 | A class is a set iff it is... |
| snid 4617 | A set is a member of its s... |
| vsnid 4618 | A setvar variable is a mem... |
| elsn2g 4619 | There is exactly one eleme... |
| elsn2 4620 | There is exactly one eleme... |
| nelsn 4621 | If a class is not equal to... |
| rabeqsn 4622 | Conditions for a restricte... |
| rabsssn 4623 | Conditions for a restricte... |
| rabeqsnd 4624 | Conditions for a restricte... |
| ralsnsg 4625 | Substitution expressed in ... |
| rexsns 4626 | Restricted existential qua... |
| rexsngf 4627 | Restricted existential qua... |
| ralsngf 4628 | Restricted universal quant... |
| reusngf 4629 | Restricted existential uni... |
| ralsng 4630 | Substitution expressed in ... |
| rexsng 4631 | Restricted existential qua... |
| reusng 4632 | Restricted existential uni... |
| 2ralsng 4633 | Substitution expressed in ... |
| rexreusng 4634 | Restricted existential uni... |
| exsnrex 4635 | There is a set being the e... |
| ralsn 4636 | Convert a universal quanti... |
| rexsn 4637 | Convert an existential qua... |
| elunsn 4638 | Elementhood in a union wit... |
| elpwunsn 4639 | Membership in an extension... |
| eqoreldif 4640 | An element of a set is eit... |
| eltpg 4641 | Members of an unordered tr... |
| eldiftp 4642 | Membership in a set with t... |
| eltpi 4643 | A member of an unordered t... |
| eltp 4644 | A member of an unordered t... |
| el7g 4645 | Members of a set with seve... |
| dftp2 4646 | Alternate definition of un... |
| nfpr 4647 | Bound-variable hypothesis ... |
| ifpr 4648 | Membership of a conditiona... |
| ralprgf 4649 | Convert a restricted unive... |
| rexprgf 4650 | Convert a restricted exist... |
| ralprg 4651 | Convert a restricted unive... |
| rexprg 4652 | Convert a restricted exist... |
| raltpg 4653 | Convert a restricted unive... |
| rextpg 4654 | Convert a restricted exist... |
| ralpr 4655 | Convert a restricted unive... |
| rexpr 4656 | Convert a restricted exist... |
| reuprg0 4657 | Convert a restricted exist... |
| reuprg 4658 | Convert a restricted exist... |
| reurexprg 4659 | Convert a restricted exist... |
| raltp 4660 | Convert a universal quanti... |
| rextp 4661 | Convert an existential qua... |
| nfsn 4662 | Bound-variable hypothesis ... |
| csbsng 4663 | Distribute proper substitu... |
| csbprg 4664 | Distribute proper substitu... |
| elinsn 4665 | If the intersection of two... |
| disjsn 4666 | Intersection with the sing... |
| disjsn2 4667 | Two distinct singletons ar... |
| disjpr2 4668 | Two completely distinct un... |
| disjprsn 4669 | The disjoint intersection ... |
| disjtpsn 4670 | The disjoint intersection ... |
| disjtp2 4671 | Two completely distinct un... |
| snprc 4672 | The singleton of a proper ... |
| snnzb 4673 | A singleton is nonempty if... |
| rmosn 4674 | A restricted at-most-one q... |
| r19.12sn 4675 | Special case of ~ r19.12 w... |
| rabsn 4676 | Condition where a restrict... |
| rabsnifsb 4677 | A restricted class abstrac... |
| rabsnif 4678 | A restricted class abstrac... |
| rabrsn 4679 | A restricted class abstrac... |
| euabsn2 4680 | Another way to express exi... |
| euabsn 4681 | Another way to express exi... |
| reusn 4682 | A way to express restricte... |
| absneu 4683 | Restricted existential uni... |
| rabsneu 4684 | Restricted existential uni... |
| eusn 4685 | Two ways to express " ` A ... |
| rabsnt 4686 | Truth implied by equality ... |
| prcom 4687 | Commutative law for unorde... |
| preq1 4688 | Equality theorem for unord... |
| preq2 4689 | Equality theorem for unord... |
| preq12 4690 | Equality theorem for unord... |
| preq1i 4691 | Equality inference for uno... |
| preq2i 4692 | Equality inference for uno... |
| preq12i 4693 | Equality inference for uno... |
| preq1d 4694 | Equality deduction for uno... |
| preq2d 4695 | Equality deduction for uno... |
| preq12d 4696 | Equality deduction for uno... |
| tpeq1 4697 | Equality theorem for unord... |
| tpeq2 4698 | Equality theorem for unord... |
| tpeq3 4699 | Equality theorem for unord... |
| tpeq1d 4700 | Equality theorem for unord... |
| tpeq2d 4701 | Equality theorem for unord... |
| tpeq3d 4702 | Equality theorem for unord... |
| tpeq123d 4703 | Equality theorem for unord... |
| tprot 4704 | Rotation of the elements o... |
| tpcoma 4705 | Swap 1st and 2nd members o... |
| tpcomb 4706 | Swap 2nd and 3rd members o... |
| tpass 4707 | Split off the first elemen... |
| qdass 4708 | Two ways to write an unord... |
| qdassr 4709 | Two ways to write an unord... |
| tpidm12 4710 | Unordered triple ` { A , A... |
| tpidm13 4711 | Unordered triple ` { A , B... |
| tpidm23 4712 | Unordered triple ` { A , B... |
| tpidm 4713 | Unordered triple ` { A , A... |
| tppreq3 4714 | An unordered triple is an ... |
| prid1g 4715 | An unordered pair contains... |
| prid2g 4716 | An unordered pair contains... |
| prid1 4717 | An unordered pair contains... |
| prid2 4718 | An unordered pair contains... |
| ifpprsnss 4719 | An unordered pair is a sin... |
| prprc1 4720 | A proper class vanishes in... |
| prprc2 4721 | A proper class vanishes in... |
| prprc 4722 | An unordered pair containi... |
| tpid1 4723 | One of the three elements ... |
| tpid1g 4724 | Closed theorem form of ~ t... |
| tpid2 4725 | One of the three elements ... |
| tpid2g 4726 | Closed theorem form of ~ t... |
| tpid3g 4727 | Closed theorem form of ~ t... |
| tpid3 4728 | One of the three elements ... |
| snnzg 4729 | The singleton of a set is ... |
| snn0d 4730 | The singleton of a set is ... |
| snnz 4731 | The singleton of a set is ... |
| prnz 4732 | A pair containing a set is... |
| prnzg 4733 | A pair containing a set is... |
| tpnz 4734 | An unordered triple contai... |
| tpnzd 4735 | An unordered triple contai... |
| raltpd 4736 | Convert a universal quanti... |
| snssb 4737 | Characterization of the in... |
| snssg 4738 | The singleton formed on a ... |
| snss 4739 | The singleton of an elemen... |
| eldifsn 4740 | Membership in a set with a... |
| eldifsnd 4741 | Membership in a set with a... |
| ssdifsn 4742 | Subset of a set with an el... |
| elpwdifsn 4743 | A subset of a set is an el... |
| eldifsni 4744 | Membership in a set with a... |
| eldifsnneq 4745 | An element of a difference... |
| neldifsn 4746 | The class ` A ` is not in ... |
| neldifsnd 4747 | The class ` A ` is not in ... |
| rexdifsn 4748 | Restricted existential qua... |
| raldifsni 4749 | Rearrangement of a propert... |
| raldifsnb 4750 | Restricted universal quant... |
| eldifvsn 4751 | A set is an element of the... |
| difsn 4752 | An element not in a set ca... |
| difprsnss 4753 | Removal of a singleton fro... |
| difprsn1 4754 | Removal of a singleton fro... |
| difprsn2 4755 | Removal of a singleton fro... |
| diftpsn3 4756 | Removal of a singleton fro... |
| difpr 4757 | Removing two elements as p... |
| tpprceq3 4758 | An unordered triple is an ... |
| tppreqb 4759 | An unordered triple is an ... |
| difsnb 4760 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4761 | ` ( B \ { A } ) ` is a pro... |
| snssi 4762 | The singleton of an elemen... |
| snssd 4763 | The singleton of an elemen... |
| difsnid 4764 | If we remove a single elem... |
| eldifeldifsn 4765 | An element of a difference... |
| pw0 4766 | Compute the power set of t... |
| pwpw0 4767 | Compute the power set of t... |
| snsspr1 4768 | A singleton is a subset of... |
| snsspr2 4769 | A singleton is a subset of... |
| snsstp1 4770 | A singleton is a subset of... |
| snsstp2 4771 | A singleton is a subset of... |
| snsstp3 4772 | A singleton is a subset of... |
| prssg 4773 | A pair of elements of a cl... |
| prss 4774 | A pair of elements of a cl... |
| prssi 4775 | A pair of elements of a cl... |
| prssd 4776 | Deduction version of ~ prs... |
| prsspwg 4777 | An unordered pair belongs ... |
| ssprss 4778 | A pair as subset of a pair... |
| ssprsseq 4779 | A proper pair is a subset ... |
| sssn 4780 | The subsets of a singleton... |
| ssunsn2 4781 | The property of being sand... |
| ssunsn 4782 | Possible values for a set ... |
| eqsn 4783 | Two ways to express that a... |
| eqsnd 4784 | Deduce that a set is a sin... |
| eqsndOLD 4785 | Obsolete version of ~ eqsn... |
| issn 4786 | A sufficient condition for... |
| n0snor2el 4787 | A nonempty set is either a... |
| ssunpr 4788 | Possible values for a set ... |
| sspr 4789 | The subsets of a pair. (C... |
| sstp 4790 | The subsets of an unordere... |
| tpss 4791 | An unordered triple of ele... |
| tpssi 4792 | An unordered triple of ele... |
| sneqrg 4793 | Closed form of ~ sneqr . ... |
| sneqr 4794 | If the singletons of two s... |
| snsssn 4795 | If a singleton is a subset... |
| mosneq 4796 | There exists at most one s... |
| sneqbg 4797 | Two singletons of sets are... |
| snsspw 4798 | The singleton of a class i... |
| prsspw 4799 | An unordered pair belongs ... |
| preq1b 4800 | Biconditional equality lem... |
| preq2b 4801 | Biconditional equality lem... |
| preqr1 4802 | Reverse equality lemma for... |
| preqr2 4803 | Reverse equality lemma for... |
| preq12b 4804 | Equality relationship for ... |
| opthpr 4805 | An unordered pair has the ... |
| preqr1g 4806 | Reverse equality lemma for... |
| preq12bg 4807 | Closed form of ~ preq12b .... |
| prneimg 4808 | Two pairs are not equal if... |
| prneimg2 4809 | Two pairs are not equal if... |
| prnebg 4810 | A (proper) pair is not equ... |
| pr1eqbg 4811 | A (proper) pair is equal t... |
| pr1nebg 4812 | A (proper) pair is not equ... |
| preqsnd 4813 | Equivalence for a pair equ... |
| prnesn 4814 | A proper unordered pair is... |
| prneprprc 4815 | A proper unordered pair is... |
| preqsn 4816 | Equivalence for a pair equ... |
| preq12nebg 4817 | Equality relationship for ... |
| prel12g 4818 | Equality of two unordered ... |
| opthprneg 4819 | An unordered pair has the ... |
| elpreqprlem 4820 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4821 | Equality and membership ru... |
| elpreqprb 4822 | A set is an element of an ... |
| elpr2elpr 4823 | For an element ` A ` of an... |
| dfopif 4824 | Rewrite ~ df-op using ` if... |
| dfopg 4825 | Value of the ordered pair ... |
| dfop 4826 | Value of an ordered pair w... |
| opeq1 4827 | Equality theorem for order... |
| opeq2 4828 | Equality theorem for order... |
| opeq12 4829 | Equality theorem for order... |
| opeq1i 4830 | Equality inference for ord... |
| opeq2i 4831 | Equality inference for ord... |
| opeq12i 4832 | Equality inference for ord... |
| opeq1d 4833 | Equality deduction for ord... |
| opeq2d 4834 | Equality deduction for ord... |
| opeq12d 4835 | Equality deduction for ord... |
| oteq1 4836 | Equality theorem for order... |
| oteq2 4837 | Equality theorem for order... |
| oteq3 4838 | Equality theorem for order... |
| oteq1d 4839 | Equality deduction for ord... |
| oteq2d 4840 | Equality deduction for ord... |
| oteq3d 4841 | Equality deduction for ord... |
| oteq123d 4842 | Equality deduction for ord... |
| nfop 4843 | Bound-variable hypothesis ... |
| nfopd 4844 | Deduction version of bound... |
| csbopg 4845 | Distribution of class subs... |
| opidg 4846 | The ordered pair ` <. A , ... |
| opid 4847 | The ordered pair ` <. A , ... |
| ralunsn 4848 | Restricted quantification ... |
| 2ralunsn 4849 | Double restricted quantifi... |
| opprc 4850 | Expansion of an ordered pa... |
| opprc1 4851 | Expansion of an ordered pa... |
| opprc2 4852 | Expansion of an ordered pa... |
| oprcl 4853 | If an ordered pair has an ... |
| pwsn 4854 | The power set of a singlet... |
| pwpr 4855 | The power set of an unorde... |
| pwtp 4856 | The power set of an unorde... |
| pwpwpw0 4857 | Compute the power set of t... |
| pwv 4858 | The power class of the uni... |
| prproe 4859 | For an element of a proper... |
| 3elpr2eq 4860 | If there are three element... |
| dfuni2 4863 | Alternate definition of cl... |
| eluni 4864 | Membership in class union.... |
| eluni2 4865 | Membership in class union.... |
| elunii 4866 | Membership in class union.... |
| nfunid 4867 | Deduction version of ~ nfu... |
| nfuni 4868 | Bound-variable hypothesis ... |
| uniss 4869 | Subclass relationship for ... |
| unissi 4870 | Subclass relationship for ... |
| unissd 4871 | Subclass relationship for ... |
| unieq 4872 | Equality theorem for class... |
| unieqi 4873 | Inference of equality of t... |
| unieqd 4874 | Deduction of equality of t... |
| eluniab 4875 | Membership in union of a c... |
| elunirab 4876 | Membership in union of a c... |
| uniprg 4877 | The union of a pair is the... |
| unipr 4878 | The union of a pair is the... |
| unisng 4879 | A set equals the union of ... |
| unisn 4880 | A set equals the union of ... |
| unisnv 4881 | A set equals the union of ... |
| unisn3 4882 | Union of a singleton in th... |
| dfnfc2 4883 | An alternative statement o... |
| uniun 4884 | The class union of the uni... |
| uniin 4885 | The class union of the int... |
| ssuni 4886 | Subclass relationship for ... |
| uni0b 4887 | The union of a set is empt... |
| uni0c 4888 | The union of a set is empt... |
| uni0 4889 | The union of the empty set... |
| uni0OLD 4890 | Obsolete version of ~ uni0... |
| csbuni 4891 | Distribute proper substitu... |
| elssuni 4892 | An element of a class is a... |
| unissel 4893 | Condition turning a subcla... |
| unissb 4894 | Relationship involving mem... |
| uniss2 4895 | A subclass condition on th... |
| unidif 4896 | If the difference ` A \ B ... |
| ssunieq 4897 | Relationship implying unio... |
| unimax 4898 | Any member of a class is t... |
| pwuni 4899 | A class is a subclass of t... |
| dfint2 4902 | Alternate definition of cl... |
| inteq 4903 | Equality law for intersect... |
| inteqi 4904 | Equality inference for cla... |
| inteqd 4905 | Equality deduction for cla... |
| elint 4906 | Membership in class inters... |
| elint2 4907 | Membership in class inters... |
| elintg 4908 | Membership in class inters... |
| elinti 4909 | Membership in class inters... |
| nfint 4910 | Bound-variable hypothesis ... |
| elintabg 4911 | Two ways of saying a set i... |
| elintab 4912 | Membership in the intersec... |
| elintrab 4913 | Membership in the intersec... |
| elintrabg 4914 | Membership in the intersec... |
| int0 4915 | The intersection of the em... |
| intss1 4916 | An element of a class incl... |
| ssint 4917 | Subclass of a class inters... |
| ssintab 4918 | Subclass of the intersecti... |
| ssintub 4919 | Subclass of the least uppe... |
| ssmin 4920 | Subclass of the minimum va... |
| intmin 4921 | Any member of a class is t... |
| intss 4922 | Intersection of subclasses... |
| intssuni 4923 | The intersection of a none... |
| ssintrab 4924 | Subclass of the intersecti... |
| unissint 4925 | If the union of a class is... |
| intssuni2 4926 | Subclass relationship for ... |
| intminss 4927 | Under subset ordering, the... |
| intmin2 4928 | Any set is the smallest of... |
| intmin3 4929 | Under subset ordering, the... |
| intmin4 4930 | Elimination of a conjunct ... |
| intab 4931 | The intersection of a spec... |
| int0el 4932 | The intersection of a clas... |
| intun 4933 | The class intersection of ... |
| intprg 4934 | The intersection of a pair... |
| intpr 4935 | The intersection of a pair... |
| intsng 4936 | Intersection of a singleto... |
| intsn 4937 | The intersection of a sing... |
| uniintsn 4938 | Two ways to express " ` A ... |
| uniintab 4939 | The union and the intersec... |
| intunsn 4940 | Theorem joining a singleto... |
| rint0 4941 | Relative intersection of a... |
| elrint 4942 | Membership in a restricted... |
| elrint2 4943 | Membership in a restricted... |
| eliun 4948 | Membership in indexed unio... |
| eliin 4949 | Membership in indexed inte... |
| eliuni 4950 | Membership in an indexed u... |
| eliund 4951 | Membership in indexed unio... |
| iuncom 4952 | Commutation of indexed uni... |
| iuncom4 4953 | Commutation of union with ... |
| iunconst 4954 | Indexed union of a constan... |
| iinconst 4955 | Indexed intersection of a ... |
| iuneqconst 4956 | Indexed union of identical... |
| iuniin 4957 | Law combining indexed unio... |
| iinssiun 4958 | An indexed intersection is... |
| iunss1 4959 | Subclass theorem for index... |
| iinss1 4960 | Subclass theorem for index... |
| iuneq1 4961 | Equality theorem for index... |
| iineq1 4962 | Equality theorem for index... |
| ss2iun 4963 | Subclass theorem for index... |
| iuneq2 4964 | Equality theorem for index... |
| iineq2 4965 | Equality theorem for index... |
| iuneq2i 4966 | Equality inference for ind... |
| iineq2i 4967 | Equality inference for ind... |
| iineq2d 4968 | Equality deduction for ind... |
| iuneq2dv 4969 | Equality deduction for ind... |
| iineq2dv 4970 | Equality deduction for ind... |
| iuneq12df 4971 | Equality deduction for ind... |
| iuneq1d 4972 | Equality theorem for index... |
| iuneq12dOLD 4973 | Obsolete version of ~ iune... |
| iuneq12d 4974 | Equality deduction for ind... |
| iuneq2d 4975 | Equality deduction for ind... |
| nfiun 4976 | Bound-variable hypothesis ... |
| nfiin 4977 | Bound-variable hypothesis ... |
| nfiung 4978 | Bound-variable hypothesis ... |
| nfiing 4979 | Bound-variable hypothesis ... |
| nfiu1 4980 | Bound-variable hypothesis ... |
| nfiu1OLD 4981 | Obsolete version of ~ nfiu... |
| nfii1 4982 | Bound-variable hypothesis ... |
| dfiun2g 4983 | Alternate definition of in... |
| dfiin2g 4984 | Alternate definition of in... |
| dfiun2 4985 | Alternate definition of in... |
| dfiin2 4986 | Alternate definition of in... |
| dfiunv2 4987 | Define double indexed unio... |
| cbviun 4988 | Rule used to change the bo... |
| cbviin 4989 | Change bound variables in ... |
| cbviung 4990 | Rule used to change the bo... |
| cbviing 4991 | Change bound variables in ... |
| cbviunv 4992 | Rule used to change the bo... |
| cbviinv 4993 | Change bound variables in ... |
| cbviunvg 4994 | Rule used to change the bo... |
| cbviinvg 4995 | Change bound variables in ... |
| iunssf 4996 | Subset theorem for an inde... |
| iunssfOLD 4997 | Obsolete version of ~ iuns... |
| iunss 4998 | Subset theorem for an inde... |
| iunssOLD 4999 | Obsolete version of ~ iuns... |
| ssiun 5000 | Subset implication for an ... |
| ssiun2 5001 | Identity law for subset of... |
| ssiun2s 5002 | Subset relationship for an... |
| iunss2 5003 | A subclass condition on th... |
| iunssd 5004 | Subset theorem for an inde... |
| iunab 5005 | The indexed union of a cla... |
| iunrab 5006 | The indexed union of a res... |
| iunxdif2 5007 | Indexed union with a class... |
| ssiinf 5008 | Subset theorem for an inde... |
| ssiin 5009 | Subset theorem for an inde... |
| iinss 5010 | Subset implication for an ... |
| iinss2 5011 | An indexed intersection is... |
| uniiun 5012 | Class union in terms of in... |
| intiin 5013 | Class intersection in term... |
| iunid 5014 | An indexed union of single... |
| iun0 5015 | An indexed union of the em... |
| 0iun 5016 | An empty indexed union is ... |
| 0iin 5017 | An empty indexed intersect... |
| viin 5018 | Indexed intersection with ... |
| iunsn 5019 | Indexed union of a singlet... |
| iunn0 5020 | There is a nonempty class ... |
| iinab 5021 | Indexed intersection of a ... |
| iinrab 5022 | Indexed intersection of a ... |
| iinrab2 5023 | Indexed intersection of a ... |
| iunin2 5024 | Indexed union of intersect... |
| iunin1 5025 | Indexed union of intersect... |
| iinun2 5026 | Indexed intersection of un... |
| iundif2 5027 | Indexed union of class dif... |
| iindif1 5028 | Indexed intersection of cl... |
| 2iunin 5029 | Rearrange indexed unions o... |
| iindif2 5030 | Indexed intersection of cl... |
| iinin2 5031 | Indexed intersection of in... |
| iinin1 5032 | Indexed intersection of in... |
| iinvdif 5033 | The indexed intersection o... |
| elriin 5034 | Elementhood in a relative ... |
| riin0 5035 | Relative intersection of a... |
| riinn0 5036 | Relative intersection of a... |
| riinrab 5037 | Relative intersection of a... |
| symdif0 5038 | Symmetric difference with ... |
| symdifv 5039 | The symmetric difference w... |
| symdifid 5040 | The symmetric difference o... |
| iinxsng 5041 | A singleton index picks ou... |
| iinxprg 5042 | Indexed intersection with ... |
| iunxsng 5043 | A singleton index picks ou... |
| iunxsn 5044 | A singleton index picks ou... |
| iunxsngf 5045 | A singleton index picks ou... |
| iunun 5046 | Separate a union in an ind... |
| iunxun 5047 | Separate a union in the in... |
| iunxdif3 5048 | An indexed union where som... |
| iunxprg 5049 | A pair index picks out two... |
| iunxiun 5050 | Separate an indexed union ... |
| iinuni 5051 | A relationship involving u... |
| iununi 5052 | A relationship involving u... |
| sspwuni 5053 | Subclass relationship for ... |
| pwssb 5054 | Two ways to express a coll... |
| elpwpw 5055 | Characterization of the el... |
| pwpwab 5056 | The double power class wri... |
| pwpwssunieq 5057 | The class of sets whose un... |
| elpwuni 5058 | Relationship for power cla... |
| iinpw 5059 | The power class of an inte... |
| iunpwss 5060 | Inclusion of an indexed un... |
| intss2 5061 | A nonempty intersection of... |
| rintn0 5062 | Relative intersection of a... |
| dfdisj2 5065 | Alternate definition for d... |
| disjss2 5066 | If each element of a colle... |
| disjeq2 5067 | Equality theorem for disjo... |
| disjeq2dv 5068 | Equality deduction for dis... |
| disjss1 5069 | A subset of a disjoint col... |
| disjeq1 5070 | Equality theorem for disjo... |
| disjeq1d 5071 | Equality theorem for disjo... |
| disjeq12d 5072 | Equality theorem for disjo... |
| cbvdisj 5073 | Change bound variables in ... |
| cbvdisjv 5074 | Change bound variables in ... |
| nfdisjw 5075 | Bound-variable hypothesis ... |
| nfdisj 5076 | Bound-variable hypothesis ... |
| nfdisj1 5077 | Bound-variable hypothesis ... |
| disjor 5078 | Two ways to say that a col... |
| disjors 5079 | Two ways to say that a col... |
| disji2 5080 | Property of a disjoint col... |
| disji 5081 | Property of a disjoint col... |
| invdisj 5082 | If there is a function ` C... |
| invdisjrab 5083 | The restricted class abstr... |
| disjiun 5084 | A disjoint collection yiel... |
| disjord 5085 | Conditions for a collectio... |
| disjiunb 5086 | Two ways to say that a col... |
| disjiund 5087 | Conditions for a collectio... |
| sndisj 5088 | Any collection of singleto... |
| 0disj 5089 | Any collection of empty se... |
| disjxsn 5090 | A singleton collection is ... |
| disjx0 5091 | An empty collection is dis... |
| disjprg 5092 | A pair collection is disjo... |
| disjxiun 5093 | An indexed union of a disj... |
| disjxun 5094 | The union of two disjoint ... |
| disjss3 5095 | Expand a disjoint collecti... |
| breq 5098 | Equality theorem for binar... |
| breq1 5099 | Equality theorem for a bin... |
| breq2 5100 | Equality theorem for a bin... |
| breq12 5101 | Equality theorem for a bin... |
| breqi 5102 | Equality inference for bin... |
| breq1i 5103 | Equality inference for a b... |
| breq2i 5104 | Equality inference for a b... |
| breq12i 5105 | Equality inference for a b... |
| breq1d 5106 | Equality deduction for a b... |
| breqd 5107 | Equality deduction for a b... |
| breq2d 5108 | Equality deduction for a b... |
| breq12d 5109 | Equality deduction for a b... |
| breq123d 5110 | Equality deduction for a b... |
| breqdi 5111 | Equality deduction for a b... |
| breqan12d 5112 | Equality deduction for a b... |
| breqan12rd 5113 | Equality deduction for a b... |
| eqnbrtrd 5114 | Substitution of equal clas... |
| nbrne1 5115 | Two classes are different ... |
| nbrne2 5116 | Two classes are different ... |
| eqbrtri 5117 | Substitution of equal clas... |
| eqbrtrd 5118 | Substitution of equal clas... |
| eqbrtrri 5119 | Substitution of equal clas... |
| eqbrtrrd 5120 | Substitution of equal clas... |
| breqtri 5121 | Substitution of equal clas... |
| breqtrd 5122 | Substitution of equal clas... |
| breqtrri 5123 | Substitution of equal clas... |
| breqtrrd 5124 | Substitution of equal clas... |
| 3brtr3i 5125 | Substitution of equality i... |
| 3brtr4i 5126 | Substitution of equality i... |
| 3brtr3d 5127 | Substitution of equality i... |
| 3brtr4d 5128 | Substitution of equality i... |
| 3brtr3g 5129 | Substitution of equality i... |
| 3brtr4g 5130 | Substitution of equality i... |
| eqbrtrid 5131 | A chained equality inferen... |
| eqbrtrrid 5132 | A chained equality inferen... |
| breqtrid 5133 | A chained equality inferen... |
| breqtrrid 5134 | A chained equality inferen... |
| eqbrtrdi 5135 | A chained equality inferen... |
| eqbrtrrdi 5136 | A chained equality inferen... |
| breqtrdi 5137 | A chained equality inferen... |
| breqtrrdi 5138 | A chained equality inferen... |
| ssbrd 5139 | Deduction from a subclass ... |
| ssbr 5140 | Implication from a subclas... |
| ssbri 5141 | Inference from a subclass ... |
| nfbrd 5142 | Deduction version of bound... |
| nfbr 5143 | Bound-variable hypothesis ... |
| brab1 5144 | Relationship between a bin... |
| br0 5145 | The empty binary relation ... |
| brne0 5146 | If two sets are in a binar... |
| brun 5147 | The union of two binary re... |
| brin 5148 | The intersection of two re... |
| brdif 5149 | The difference of two bina... |
| sbcbr123 5150 | Move substitution in and o... |
| sbcbr 5151 | Move substitution in and o... |
| sbcbr12g 5152 | Move substitution in and o... |
| sbcbr1g 5153 | Move substitution in and o... |
| sbcbr2g 5154 | Move substitution in and o... |
| brsymdif 5155 | Characterization of the sy... |
| brralrspcev 5156 | Restricted existential spe... |
| brimralrspcev 5157 | Restricted existential spe... |
| opabss 5160 | The collection of ordered ... |
| opabbid 5161 | Equivalent wff's yield equ... |
| opabbidv 5162 | Equivalent wff's yield equ... |
| opabbii 5163 | Equivalent wff's yield equ... |
| nfopabd 5164 | Bound-variable hypothesis ... |
| nfopab 5165 | Bound-variable hypothesis ... |
| nfopab1 5166 | The first abstraction vari... |
| nfopab2 5167 | The second abstraction var... |
| cbvopab 5168 | Rule used to change bound ... |
| cbvopabv 5169 | Rule used to change bound ... |
| cbvopab1 5170 | Change first bound variabl... |
| cbvopab1g 5171 | Change first bound variabl... |
| cbvopab2 5172 | Change second bound variab... |
| cbvopab1s 5173 | Change first bound variabl... |
| cbvopab1v 5174 | Rule used to change the fi... |
| cbvopab2v 5175 | Rule used to change the se... |
| unopab 5176 | Union of two ordered pair ... |
| mpteq12da 5179 | An equality inference for ... |
| mpteq12df 5180 | An equality inference for ... |
| mpteq12f 5181 | An equality theorem for th... |
| mpteq12dva 5182 | An equality inference for ... |
| mpteq12dv 5183 | An equality inference for ... |
| mpteq12 5184 | An equality theorem for th... |
| mpteq1 5185 | An equality theorem for th... |
| mpteq1d 5186 | An equality theorem for th... |
| mpteq1i 5187 | An equality theorem for th... |
| mpteq2da 5188 | Slightly more general equa... |
| mpteq2dva 5189 | Slightly more general equa... |
| mpteq2dv 5190 | An equality inference for ... |
| mpteq2ia 5191 | An equality inference for ... |
| mpteq2i 5192 | An equality inference for ... |
| mpteq12i 5193 | An equality inference for ... |
| nfmpt 5194 | Bound-variable hypothesis ... |
| nfmpt1 5195 | Bound-variable hypothesis ... |
| cbvmptf 5196 | Rule to change the bound v... |
| cbvmptfg 5197 | Rule to change the bound v... |
| cbvmpt 5198 | Rule to change the bound v... |
| cbvmptg 5199 | Rule to change the bound v... |
| cbvmptv 5200 | Rule to change the bound v... |
| cbvmptvg 5201 | Rule to change the bound v... |
| mptv 5202 | Function with universal do... |
| dftr2 5205 | An alternate way of defini... |
| dftr2c 5206 | Variant of ~ dftr2 with co... |
| dftr5 5207 | An alternate way of defini... |
| dftr3 5208 | An alternate way of defini... |
| dftr4 5209 | An alternate way of defini... |
| treq 5210 | Equality theorem for the t... |
| trel 5211 | In a transitive class, the... |
| trel3 5212 | In a transitive class, the... |
| trss 5213 | An element of a transitive... |
| trin 5214 | The intersection of transi... |
| tr0 5215 | The empty set is transitiv... |
| trv 5216 | The universe is transitive... |
| triun 5217 | An indexed union of a clas... |
| truni 5218 | The union of a class of tr... |
| triin 5219 | An indexed intersection of... |
| trint 5220 | The intersection of a clas... |
| trintss 5221 | Any nonempty transitive cl... |
| axrep1 5223 | The version of the Axiom o... |
| axreplem 5224 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5225 | Axiom of Replacement expre... |
| axrep3 5226 | Axiom of Replacement sligh... |
| axrep4v 5227 | Version of ~ axrep4 with a... |
| axrep4 5228 | A more traditional version... |
| axrep4OLD 5229 | Obsolete version of ~ axre... |
| axrep5 5230 | Axiom of Replacement (simi... |
| axrep6 5231 | A condensed form of ~ ax-r... |
| axrep6OLD 5232 | Obsolete version of ~ axre... |
| axrep6g 5233 | ~ axrep6 in class notation... |
| zfrepclf 5234 | An inference based on the ... |
| zfrep3cl 5235 | An inference based on the ... |
| zfrep4 5236 | A version of Replacement u... |
| axsepgfromrep 5237 | A more general version ~ a... |
| axsep 5238 | Axiom scheme of separation... |
| axsepg 5240 | A more general version of ... |
| zfauscl 5241 | Separation Scheme (Aussond... |
| sepexlem 5242 | Lemma for ~ sepex . Use ~... |
| sepex 5243 | Convert implication to equ... |
| sepexi 5244 | Convert implication to equ... |
| bm1.3iiOLD 5245 | Obsolete version of ~ sepe... |
| ax6vsep 5246 | Derive ~ ax6v (a weakened ... |
| axnulALT 5247 | Alternate proof of ~ axnul... |
| axnul 5248 | The Null Set Axiom of ZF s... |
| 0ex 5250 | The Null Set Axiom of ZF s... |
| al0ssb 5251 | The empty set is the uniqu... |
| sseliALT 5252 | Alternate proof of ~ sseli... |
| csbexg 5253 | The existence of proper su... |
| csbex 5254 | The existence of proper su... |
| unisn2 5255 | A version of ~ unisn witho... |
| nalset 5256 | No set contains all sets. ... |
| vnex 5257 | The universal class does n... |
| vprc 5258 | The universal class is not... |
| nvel 5259 | The universal class does n... |
| inex1 5260 | Separation Scheme (Aussond... |
| inex2 5261 | Separation Scheme (Aussond... |
| inex1g 5262 | Closed-form, generalized S... |
| inex2g 5263 | Sufficient condition for a... |
| ssex 5264 | The subset of a set is als... |
| ssexi 5265 | The subset of a set is als... |
| ssexg 5266 | The subset of a set is als... |
| ssexd 5267 | A subclass of a set is a s... |
| abexd 5268 | Conditions for a class abs... |
| abex 5269 | Conditions for a class abs... |
| prcssprc 5270 | The superclass of a proper... |
| sselpwd 5271 | Elementhood to a power set... |
| difexg 5272 | Existence of a difference.... |
| difexi 5273 | Existence of a difference,... |
| difexd 5274 | Existence of a difference.... |
| zfausab 5275 | Separation Scheme (Aussond... |
| elpw2g 5276 | Membership in a power clas... |
| elpw2 5277 | Membership in a power clas... |
| elpwi2 5278 | Membership in a power clas... |
| rabelpw 5279 | A restricted class abstrac... |
| rabexg 5280 | Separation Scheme in terms... |
| rabexgOLD 5281 | Obsolete version of ~ rabe... |
| rabex 5282 | Separation Scheme in terms... |
| rabexd 5283 | Separation Scheme in terms... |
| rabex2 5284 | Separation Scheme in terms... |
| rab2ex 5285 | A class abstraction based ... |
| elssabg 5286 | Membership in a class abst... |
| intex 5287 | The intersection of a none... |
| intnex 5288 | If a class intersection is... |
| intexab 5289 | The intersection of a none... |
| intexrab 5290 | The intersection of a none... |
| iinexg 5291 | The existence of a class i... |
| intabs 5292 | Absorption of a redundant ... |
| inuni 5293 | The intersection of a unio... |
| axpweq 5294 | Two equivalent ways to exp... |
| pwnss 5295 | The power set of a set is ... |
| pwne 5296 | No set equals its power se... |
| difelpw 5297 | A difference is an element... |
| class2set 5298 | The class of elements of `... |
| 0elpw 5299 | Every power class contains... |
| pwne0 5300 | A power class is never emp... |
| 0nep0 5301 | The empty set and its powe... |
| 0inp0 5302 | Something cannot be equal ... |
| unidif0 5303 | The removal of the empty s... |
| eqsnuniex 5304 | If a class is equal to the... |
| iin0 5305 | An indexed intersection of... |
| notzfaus 5306 | In the Separation Scheme ~... |
| intv 5307 | The intersection of the un... |
| zfpow 5309 | Axiom of Power Sets expres... |
| axpow2 5310 | A variant of the Axiom of ... |
| axpow3 5311 | A variant of the Axiom of ... |
| elALT2 5312 | Alternate proof of ~ el us... |
| dtruALT2 5313 | Alternate proof of ~ dtru ... |
| dtrucor 5314 | Corollary of ~ dtru . Thi... |
| dtrucor2 5315 | The theorem form of the de... |
| dvdemo1 5316 | Demonstration of a theorem... |
| dvdemo2 5317 | Demonstration of a theorem... |
| nfnid 5318 | A setvar variable is not f... |
| nfcvb 5319 | The "distinctor" expressio... |
| vpwex 5320 | Power set axiom: the power... |
| pwexg 5321 | Power set axiom expressed ... |
| pwexd 5322 | Deduction version of the p... |
| pwex 5323 | Power set axiom expressed ... |
| pwel 5324 | Quantitative version of ~ ... |
| abssexg 5325 | Existence of a class of su... |
| snexALT 5326 | Alternate proof of ~ snex ... |
| p0ex 5327 | The power set of the empty... |
| p0exALT 5328 | Alternate proof of ~ p0ex ... |
| pp0ex 5329 | The power set of the power... |
| ord3ex 5330 | The ordinal number 3 is a ... |
| dtruALT 5331 | Alternate proof of ~ dtru ... |
| axc16b 5332 | This theorem shows that Ax... |
| eunex 5333 | Existential uniqueness imp... |
| eusv1 5334 | Two ways to express single... |
| eusvnf 5335 | Even if ` x ` is free in `... |
| eusvnfb 5336 | Two ways to say that ` A (... |
| eusv2i 5337 | Two ways to express single... |
| eusv2nf 5338 | Two ways to express single... |
| eusv2 5339 | Two ways to express single... |
| reusv1 5340 | Two ways to express single... |
| reusv2lem1 5341 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5342 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5343 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5344 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5345 | Lemma for ~ reusv2 . (Con... |
| reusv2 5346 | Two ways to express single... |
| reusv3i 5347 | Two ways of expressing exi... |
| reusv3 5348 | Two ways to express single... |
| eusv4 5349 | Two ways to express single... |
| alxfr 5350 | Transfer universal quantif... |
| ralxfrd 5351 | Transfer universal quantif... |
| rexxfrd 5352 | Transfer existential quant... |
| ralxfr2d 5353 | Transfer universal quantif... |
| rexxfr2d 5354 | Transfer existential quant... |
| ralxfrd2 5355 | Transfer universal quantif... |
| rexxfrd2 5356 | Transfer existence from a ... |
| ralxfr 5357 | Transfer universal quantif... |
| ralxfrALT 5358 | Alternate proof of ~ ralxf... |
| rexxfr 5359 | Transfer existence from a ... |
| rabxfrd 5360 | Membership in a restricted... |
| rabxfr 5361 | Membership in a restricted... |
| reuhypd 5362 | A theorem useful for elimi... |
| reuhyp 5363 | A theorem useful for elimi... |
| zfpair 5364 | The Axiom of Pairing of Ze... |
| axprALT 5365 | Alternate proof of ~ axpr ... |
| axprlem1 5366 | Lemma for ~ axpr . There ... |
| axprlem2 5367 | Lemma for ~ axpr . There ... |
| axprlem3 5368 | Lemma for ~ axpr . Elimin... |
| axprlem4 5369 | Lemma for ~ axpr . If an ... |
| axpr 5370 | Unabbreviated version of t... |
| axprlem3OLD 5371 | Obsolete version of ~ axpr... |
| axprlem4OLD 5372 | Obsolete version of ~ axpr... |
| axprlem5OLD 5373 | Obsolete version of ~ axpr... |
| axprOLD 5374 | Obsolete version of ~ axpr... |
| zfpair2 5376 | Derive the abbreviated ver... |
| vsnex 5377 | A singleton built on a set... |
| snexg 5378 | A singleton built on a set... |
| snex 5379 | A singleton is a set. The... |
| prex 5380 | The Axiom of Pairing using... |
| exel 5381 | There exist two sets, one ... |
| exexneq 5382 | There exist two different ... |
| exneq 5383 | Given any set (the " ` y `... |
| dtru 5384 | Given any set (the " ` y `... |
| el 5385 | Any set is an element of s... |
| sels 5386 | If a class is a set, then ... |
| selsALT 5387 | Alternate proof of ~ sels ... |
| elALT 5388 | Alternate proof of ~ el , ... |
| snelpwg 5389 | A singleton of a set is a ... |
| snelpwi 5390 | If a set is a member of a ... |
| snelpw 5391 | A singleton of a set is a ... |
| prelpw 5392 | An unordered pair of two s... |
| prelpwi 5393 | If two sets are members of... |
| rext 5394 | A theorem similar to exten... |
| sspwb 5395 | The powerclass constructio... |
| unipw 5396 | A class equals the union o... |
| univ 5397 | The union of the universe ... |
| pwtr 5398 | A class is transitive iff ... |
| ssextss 5399 | An extensionality-like pri... |
| ssext 5400 | An extensionality-like pri... |
| nssss 5401 | Negation of subclass relat... |
| pweqb 5402 | Classes are equal if and o... |
| intidg 5403 | The intersection of all se... |
| moabex 5404 | "At most one" existence im... |
| moabexOLD 5405 | Obsolete version of ~ moab... |
| rmorabex 5406 | Restricted "at most one" e... |
| euabex 5407 | The abstraction of a wff w... |
| nnullss 5408 | A nonempty class (even if ... |
| exss 5409 | Restricted existence in a ... |
| opex 5410 | An ordered pair of classes... |
| otex 5411 | An ordered triple of class... |
| elopg 5412 | Characterization of the el... |
| elop 5413 | Characterization of the el... |
| opi1 5414 | One of the two elements in... |
| opi2 5415 | One of the two elements of... |
| opeluu 5416 | Each member of an ordered ... |
| op1stb 5417 | Extract the first member o... |
| brv 5418 | Two classes are always in ... |
| opnz 5419 | An ordered pair is nonempt... |
| opnzi 5420 | An ordered pair is nonempt... |
| opth1 5421 | Equality of the first memb... |
| opth 5422 | The ordered pair theorem. ... |
| opthg 5423 | Ordered pair theorem. ` C ... |
| opth1g 5424 | Equality of the first memb... |
| opthg2 5425 | Ordered pair theorem. (Co... |
| opth2 5426 | Ordered pair theorem. (Co... |
| opthneg 5427 | Two ordered pairs are not ... |
| opthne 5428 | Two ordered pairs are not ... |
| otth2 5429 | Ordered triple theorem, wi... |
| otth 5430 | Ordered triple theorem. (... |
| otthg 5431 | Ordered triple theorem, cl... |
| otthne 5432 | Contrapositive of the orde... |
| eqvinop 5433 | A variable introduction la... |
| sbcop1 5434 | The proper substitution of... |
| sbcop 5435 | The proper substitution of... |
| copsexgw 5436 | Version of ~ copsexg with ... |
| copsexg 5437 | Substitution of class ` A ... |
| copsex2t 5438 | Closed theorem form of ~ c... |
| copsex2g 5439 | Implicit substitution infe... |
| copsex2dv 5440 | Implicit substitution dedu... |
| copsex4g 5441 | An implicit substitution i... |
| 0nelop 5442 | A property of ordered pair... |
| opwo0id 5443 | An ordered pair is equal t... |
| opeqex 5444 | Equivalence of existence i... |
| oteqex2 5445 | Equivalence of existence i... |
| oteqex 5446 | Equivalence of existence i... |
| opcom 5447 | An ordered pair commutes i... |
| moop2 5448 | "At most one" property of ... |
| opeqsng 5449 | Equivalence for an ordered... |
| opeqsn 5450 | Equivalence for an ordered... |
| opeqpr 5451 | Equivalence for an ordered... |
| snopeqop 5452 | Equivalence for an ordered... |
| propeqop 5453 | Equivalence for an ordered... |
| propssopi 5454 | If a pair of ordered pairs... |
| snopeqopsnid 5455 | Equivalence for an ordered... |
| mosubopt 5456 | "At most one" remains true... |
| mosubop 5457 | "At most one" remains true... |
| euop2 5458 | Transfer existential uniqu... |
| euotd 5459 | Prove existential uniquene... |
| opthwiener 5460 | Justification theorem for ... |
| uniop 5461 | The union of an ordered pa... |
| uniopel 5462 | Ordered pair membership is... |
| opthhausdorff 5463 | Justification theorem for ... |
| opthhausdorff0 5464 | Justification theorem for ... |
| otsndisj 5465 | The singletons consisting ... |
| otiunsndisj 5466 | The union of singletons co... |
| iunopeqop 5467 | Implication of an ordered ... |
| brsnop 5468 | Binary relation for an ord... |
| brtp 5469 | A necessary and sufficient... |
| opabidw 5470 | The law of concretion. Sp... |
| opabid 5471 | The law of concretion. Sp... |
| elopabw 5472 | Membership in a class abst... |
| elopab 5473 | Membership in a class abst... |
| rexopabb 5474 | Restricted existential qua... |
| vopelopabsb 5475 | The law of concretion in t... |
| opelopabsb 5476 | The law of concretion in t... |
| brabsb 5477 | The law of concretion in t... |
| opelopabt 5478 | Closed theorem form of ~ o... |
| opelopabga 5479 | The law of concretion. Th... |
| brabga 5480 | The law of concretion for ... |
| opelopab2a 5481 | Ordered pair membership in... |
| opelopaba 5482 | The law of concretion. Th... |
| braba 5483 | The law of concretion for ... |
| opelopabg 5484 | The law of concretion. Th... |
| brabg 5485 | The law of concretion for ... |
| opelopabgf 5486 | The law of concretion. Th... |
| opelopab2 5487 | Ordered pair membership in... |
| opelopab 5488 | The law of concretion. Th... |
| brab 5489 | The law of concretion for ... |
| opelopabaf 5490 | The law of concretion. Th... |
| opelopabf 5491 | The law of concretion. Th... |
| ssopab2 5492 | Equivalence of ordered pai... |
| ssopab2bw 5493 | Equivalence of ordered pai... |
| eqopab2bw 5494 | Equivalence of ordered pai... |
| ssopab2b 5495 | Equivalence of ordered pai... |
| ssopab2i 5496 | Inference of ordered pair ... |
| ssopab2dv 5497 | Inference of ordered pair ... |
| eqopab2b 5498 | Equivalence of ordered pai... |
| opabn0 5499 | Nonempty ordered pair clas... |
| opab0 5500 | Empty ordered pair class a... |
| csbopab 5501 | Move substitution into a c... |
| csbopabgALT 5502 | Move substitution into a c... |
| csbmpt12 5503 | Move substitution into a m... |
| csbmpt2 5504 | Move substitution into the... |
| iunopab 5505 | Move indexed union inside ... |
| elopabr 5506 | Membership in an ordered-p... |
| elopabran 5507 | Membership in an ordered-p... |
| rbropapd 5508 | Properties of a pair in an... |
| rbropap 5509 | Properties of a pair in a ... |
| 2rbropap 5510 | Properties of a pair in a ... |
| 0nelopab 5511 | The empty set is never an ... |
| brabv 5512 | If two classes are in a re... |
| pwin 5513 | The power class of the int... |
| pwssun 5514 | The power class of the uni... |
| pwun 5515 | The power class of the uni... |
| dfid4 5518 | The identity function expr... |
| dfid2 5519 | Alternate definition of th... |
| dfid3 5520 | A stronger version of ~ df... |
| epelg 5523 | The membership relation an... |
| epeli 5524 | The membership relation an... |
| epel 5525 | The membership relation an... |
| 0sn0ep 5526 | An example for the members... |
| epn0 5527 | The membership relation is... |
| poss 5532 | Subset theorem for the par... |
| poeq1 5533 | Equality theorem for parti... |
| poeq2 5534 | Equality theorem for parti... |
| poeq12d 5535 | Equality deduction for par... |
| nfpo 5536 | Bound-variable hypothesis ... |
| nfso 5537 | Bound-variable hypothesis ... |
| pocl 5538 | Characteristic properties ... |
| ispod 5539 | Sufficient conditions for ... |
| swopolem 5540 | Perform the substitutions ... |
| swopo 5541 | A strict weak order is a p... |
| poirr 5542 | A partial order is irrefle... |
| potr 5543 | A partial order is a trans... |
| po2nr 5544 | A partial order has no 2-c... |
| po3nr 5545 | A partial order has no 3-c... |
| po2ne 5546 | Two sets related by a part... |
| po0 5547 | Any relation is a partial ... |
| pofun 5548 | The inverse image of a par... |
| sopo 5549 | A strict linear order is a... |
| soss 5550 | Subset theorem for the str... |
| soeq1 5551 | Equality theorem for the s... |
| soeq2 5552 | Equality theorem for the s... |
| soeq12d 5553 | Equality deduction for tot... |
| sonr 5554 | A strict order relation is... |
| sotr 5555 | A strict order relation is... |
| sotrd 5556 | Transitivity law for stric... |
| solin 5557 | A strict order relation is... |
| so2nr 5558 | A strict order relation ha... |
| so3nr 5559 | A strict order relation ha... |
| sotric 5560 | A strict order relation sa... |
| sotrieq 5561 | Trichotomy law for strict ... |
| sotrieq2 5562 | Trichotomy law for strict ... |
| soasym 5563 | Asymmetry law for strict o... |
| sotr2 5564 | A transitivity relation. ... |
| issod 5565 | An irreflexive, transitive... |
| issoi 5566 | An irreflexive, transitive... |
| isso2i 5567 | Deduce strict ordering fro... |
| so0 5568 | Any relation is a strict o... |
| somo 5569 | A totally ordered set has ... |
| sotrine 5570 | Trichotomy law for strict ... |
| sotr3 5571 | Transitivity law for stric... |
| dffr6 5578 | Alternate definition of ~ ... |
| frd 5579 | A nonempty subset of an ` ... |
| fri 5580 | A nonempty subset of an ` ... |
| seex 5581 | The ` R ` -preimage of an ... |
| exse 5582 | Any relation on a set is s... |
| dffr2 5583 | Alternate definition of we... |
| dffr2ALT 5584 | Alternate proof of ~ dffr2... |
| frc 5585 | Property of well-founded r... |
| frss 5586 | Subset theorem for the wel... |
| sess1 5587 | Subset theorem for the set... |
| sess2 5588 | Subset theorem for the set... |
| freq1 5589 | Equality theorem for the w... |
| freq2 5590 | Equality theorem for the w... |
| freq12d 5591 | Equality deduction for wel... |
| seeq1 5592 | Equality theorem for the s... |
| seeq2 5593 | Equality theorem for the s... |
| seeq12d 5594 | Equality deduction for the... |
| nffr 5595 | Bound-variable hypothesis ... |
| nfse 5596 | Bound-variable hypothesis ... |
| nfwe 5597 | Bound-variable hypothesis ... |
| frirr 5598 | A well-founded relation is... |
| fr2nr 5599 | A well-founded relation ha... |
| fr0 5600 | Any relation is well-found... |
| frminex 5601 | If an element of a well-fo... |
| efrirr 5602 | A well-founded class does ... |
| efrn2lp 5603 | A well-founded class conta... |
| epse 5604 | The membership relation is... |
| tz7.2 5605 | Similar to Theorem 7.2 of ... |
| dfepfr 5606 | An alternate way of saying... |
| epfrc 5607 | A subset of a well-founded... |
| wess 5608 | Subset theorem for the wel... |
| weeq1 5609 | Equality theorem for the w... |
| weeq2 5610 | Equality theorem for the w... |
| weeq12d 5611 | Equality deduction for wel... |
| wefr 5612 | A well-ordering is well-fo... |
| weso 5613 | A well-ordering is a stric... |
| wecmpep 5614 | The elements of a class we... |
| wetrep 5615 | On a class well-ordered by... |
| wefrc 5616 | A nonempty subclass of a c... |
| we0 5617 | Any relation is a well-ord... |
| wereu 5618 | A nonempty subset of an ` ... |
| wereu2 5619 | A nonempty subclass of an ... |
| xpeq1 5636 | Equality theorem for Carte... |
| xpss12 5637 | Subset theorem for Cartesi... |
| xpss 5638 | A Cartesian product is inc... |
| inxpssres 5639 | Intersection with a Cartes... |
| relxp 5640 | A Cartesian product is a r... |
| xpss1 5641 | Subset relation for Cartes... |
| xpss2 5642 | Subset relation for Cartes... |
| xpeq2 5643 | Equality theorem for Carte... |
| elxpi 5644 | Membership in a Cartesian ... |
| elxp 5645 | Membership in a Cartesian ... |
| elxp2 5646 | Membership in a Cartesian ... |
| xpeq12 5647 | Equality theorem for Carte... |
| xpeq1i 5648 | Equality inference for Car... |
| xpeq2i 5649 | Equality inference for Car... |
| xpeq12i 5650 | Equality inference for Car... |
| xpeq1d 5651 | Equality deduction for Car... |
| xpeq2d 5652 | Equality deduction for Car... |
| xpeq12d 5653 | Equality deduction for Car... |
| sqxpeqd 5654 | Equality deduction for a C... |
| nfxp 5655 | Bound-variable hypothesis ... |
| 0nelxp 5656 | The empty set is not a mem... |
| 0nelelxp 5657 | A member of a Cartesian pr... |
| opelxp 5658 | Ordered pair membership in... |
| opelxpi 5659 | Ordered pair membership in... |
| opelxpii 5660 | Ordered pair membership in... |
| opelxpd 5661 | Ordered pair membership in... |
| opelvv 5662 | Ordered pair membership in... |
| opelvvg 5663 | Ordered pair membership in... |
| opelxp1 5664 | The first member of an ord... |
| opelxp2 5665 | The second member of an or... |
| otelxp 5666 | Ordered triple membership ... |
| otelxp1 5667 | The first member of an ord... |
| otel3xp 5668 | An ordered triple is an el... |
| opabssxpd 5669 | An ordered-pair class abst... |
| rabxp 5670 | Class abstraction restrict... |
| brxp 5671 | Binary relation on a Carte... |
| pwvrel 5672 | A set is a binary relation... |
| pwvabrel 5673 | The powerclass of the cart... |
| brrelex12 5674 | Two classes related by a b... |
| brrelex1 5675 | If two classes are related... |
| brrelex2 5676 | If two classes are related... |
| brrelex12i 5677 | Two classes that are relat... |
| brrelex1i 5678 | The first argument of a bi... |
| brrelex2i 5679 | The second argument of a b... |
| nprrel12 5680 | Proper classes are not rel... |
| nprrel 5681 | No proper class is related... |
| 0nelrel0 5682 | A binary relation does not... |
| 0nelrel 5683 | A binary relation does not... |
| fconstmpt 5684 | Representation of a consta... |
| vtoclr 5685 | Variable to class conversi... |
| opthprc 5686 | Justification theorem for ... |
| brel 5687 | Two things in a binary rel... |
| elxp3 5688 | Membership in a Cartesian ... |
| opeliunxp 5689 | Membership in a union of C... |
| opeliun2xp 5690 | Membership of an ordered p... |
| xpundi 5691 | Distributive law for Carte... |
| xpundir 5692 | Distributive law for Carte... |
| xpiundi 5693 | Distributive law for Carte... |
| xpiundir 5694 | Distributive law for Carte... |
| iunxpconst 5695 | Membership in a union of C... |
| xpun 5696 | The Cartesian product of t... |
| elvv 5697 | Membership in universal cl... |
| elvvv 5698 | Membership in universal cl... |
| elvvuni 5699 | An ordered pair contains i... |
| brinxp2 5700 | Intersection of binary rel... |
| brinxp 5701 | Intersection of binary rel... |
| opelinxp 5702 | Ordered pair element in an... |
| poinxp 5703 | Intersection of partial or... |
| soinxp 5704 | Intersection of total orde... |
| frinxp 5705 | Intersection of well-found... |
| seinxp 5706 | Intersection of set-like r... |
| weinxp 5707 | Intersection of well-order... |
| posn 5708 | Partial ordering of a sing... |
| sosn 5709 | Strict ordering on a singl... |
| frsn 5710 | Founded relation on a sing... |
| wesn 5711 | Well-ordering of a singlet... |
| elopaelxp 5712 | Membership in an ordered-p... |
| bropaex12 5713 | Two classes related by an ... |
| opabssxp 5714 | An abstraction relation is... |
| brab2a 5715 | The law of concretion for ... |
| optocl 5716 | Implicit substitution of c... |
| optoclOLD 5717 | Obsolete version of ~ opto... |
| 2optocl 5718 | Implicit substitution of c... |
| 3optocl 5719 | Implicit substitution of c... |
| opbrop 5720 | Ordered pair membership in... |
| 0xp 5721 | The Cartesian product with... |
| xp0 5722 | The Cartesian product with... |
| csbxp 5723 | Distribute proper substitu... |
| releq 5724 | Equality theorem for the r... |
| releqi 5725 | Equality inference for the... |
| releqd 5726 | Equality deduction for the... |
| nfrel 5727 | Bound-variable hypothesis ... |
| sbcrel 5728 | Distribute proper substitu... |
| relss 5729 | Subclass theorem for relat... |
| ssrel 5730 | A subclass relationship de... |
| eqrel 5731 | Extensionality principle f... |
| ssrel2 5732 | A subclass relationship de... |
| ssrel3 5733 | Subclass relation in anoth... |
| relssi 5734 | Inference from subclass pr... |
| relssdv 5735 | Deduction from subclass pr... |
| eqrelriv 5736 | Inference from extensional... |
| eqrelriiv 5737 | Inference from extensional... |
| eqbrriv 5738 | Inference from extensional... |
| eqrelrdv 5739 | Deduce equality of relatio... |
| eqbrrdv 5740 | Deduction from extensional... |
| eqbrrdiv 5741 | Deduction from extensional... |
| eqrelrdv2 5742 | A version of ~ eqrelrdv . ... |
| ssrelrel 5743 | A subclass relationship de... |
| eqrelrel 5744 | Extensionality principle f... |
| elrel 5745 | A member of a relation is ... |
| rel0 5746 | The empty set is a relatio... |
| nrelv 5747 | The universal class is not... |
| relsng 5748 | A singleton is a relation ... |
| relsnb 5749 | An at-most-singleton is a ... |
| relsnopg 5750 | A singleton of an ordered ... |
| relsn 5751 | A singleton is a relation ... |
| relsnop 5752 | A singleton of an ordered ... |
| copsex2gb 5753 | Implicit substitution infe... |
| copsex2ga 5754 | Implicit substitution infe... |
| elopaba 5755 | Membership in an ordered-p... |
| xpsspw 5756 | A Cartesian product is inc... |
| unixpss 5757 | The double class union of ... |
| relun 5758 | The union of two relations... |
| relin1 5759 | The intersection with a re... |
| relin2 5760 | The intersection with a re... |
| relinxp 5761 | Intersection with a Cartes... |
| reldif 5762 | A difference cutting down ... |
| reliun 5763 | An indexed union is a rela... |
| reliin 5764 | An indexed intersection is... |
| reluni 5765 | The union of a class is a ... |
| relint 5766 | The intersection of a clas... |
| relopabiv 5767 | A class of ordered pairs i... |
| relopabv 5768 | A class of ordered pairs i... |
| relopabi 5769 | A class of ordered pairs i... |
| relopabiALT 5770 | Alternate proof of ~ relop... |
| relopab 5771 | A class of ordered pairs i... |
| mptrel 5772 | The maps-to notation alway... |
| reli 5773 | The identity relation is a... |
| rele 5774 | The membership relation is... |
| opabid2 5775 | A relation expressed as an... |
| inopab 5776 | Intersection of two ordere... |
| difopab 5777 | Difference of two ordered-... |
| inxp 5778 | Intersection of two Cartes... |
| inxpOLD 5779 | Obsolete version of ~ inxp... |
| xpindi 5780 | Distributive law for Carte... |
| xpindir 5781 | Distributive law for Carte... |
| xpiindi 5782 | Distributive law for Carte... |
| xpriindi 5783 | Distributive law for Carte... |
| eliunxp 5784 | Membership in a union of C... |
| opeliunxp2 5785 | Membership in a union of C... |
| raliunxp 5786 | Write a double restricted ... |
| rexiunxp 5787 | Write a double restricted ... |
| ralxp 5788 | Universal quantification r... |
| rexxp 5789 | Existential quantification... |
| exopxfr 5790 | Transfer ordered-pair exis... |
| exopxfr2 5791 | Transfer ordered-pair exis... |
| djussxp 5792 | Disjoint union is a subset... |
| ralxpf 5793 | Version of ~ ralxp with bo... |
| rexxpf 5794 | Version of ~ rexxp with bo... |
| iunxpf 5795 | Indexed union on a Cartesi... |
| opabbi2dv 5796 | Deduce equality of a relat... |
| relop 5797 | A necessary and sufficient... |
| ideqg 5798 | For sets, the identity rel... |
| ideq 5799 | For sets, the identity rel... |
| ididg 5800 | A set is identical to itse... |
| issetid 5801 | Two ways of expressing set... |
| coss1 5802 | Subclass theorem for compo... |
| coss2 5803 | Subclass theorem for compo... |
| coeq1 5804 | Equality theorem for compo... |
| coeq2 5805 | Equality theorem for compo... |
| coeq1i 5806 | Equality inference for com... |
| coeq2i 5807 | Equality inference for com... |
| coeq1d 5808 | Equality deduction for com... |
| coeq2d 5809 | Equality deduction for com... |
| coeq12i 5810 | Equality inference for com... |
| coeq12d 5811 | Equality deduction for com... |
| nfco 5812 | Bound-variable hypothesis ... |
| brcog 5813 | Ordered pair membership in... |
| opelco2g 5814 | Ordered pair membership in... |
| brcogw 5815 | Ordered pair membership in... |
| eqbrrdva 5816 | Deduction from extensional... |
| brco 5817 | Binary relation on a compo... |
| opelco 5818 | Ordered pair membership in... |
| cnvss 5819 | Subset theorem for convers... |
| cnveq 5820 | Equality theorem for conve... |
| cnveqi 5821 | Equality inference for con... |
| cnveqd 5822 | Equality deduction for con... |
| elcnv 5823 | Membership in a converse r... |
| elcnv2 5824 | Membership in a converse r... |
| nfcnv 5825 | Bound-variable hypothesis ... |
| brcnvg 5826 | The converse of a binary r... |
| opelcnvg 5827 | Ordered-pair membership in... |
| opelcnv 5828 | Ordered-pair membership in... |
| brcnv 5829 | The converse of a binary r... |
| csbcnv 5830 | Move class substitution in... |
| csbcnvgALT 5831 | Move class substitution in... |
| cnvco 5832 | Distributive law of conver... |
| cnvuni 5833 | The converse of a class un... |
| dfdm3 5834 | Alternate definition of do... |
| dfrn2 5835 | Alternate definition of ra... |
| dfrn3 5836 | Alternate definition of ra... |
| elrn2g 5837 | Membership in a range. (C... |
| elrng 5838 | Membership in a range. (C... |
| elrn2 5839 | Membership in a range. (C... |
| elrn 5840 | Membership in a range. (C... |
| ssrelrn 5841 | If a relation is a subset ... |
| dfdm4 5842 | Alternate definition of do... |
| dfdmf 5843 | Definition of domain, usin... |
| csbdm 5844 | Distribute proper substitu... |
| eldmg 5845 | Domain membership. Theore... |
| eldm2g 5846 | Domain membership. Theore... |
| eldm 5847 | Membership in a domain. T... |
| eldm2 5848 | Membership in a domain. T... |
| dmss 5849 | Subset theorem for domain.... |
| dmeq 5850 | Equality theorem for domai... |
| dmeqi 5851 | Equality inference for dom... |
| dmeqd 5852 | Equality deduction for dom... |
| opeldmd 5853 | Membership of first of an ... |
| opeldm 5854 | Membership of first of an ... |
| breldm 5855 | Membership of first of a b... |
| breldmg 5856 | Membership of first of a b... |
| dmun 5857 | The domain of a union is t... |
| dmin 5858 | The domain of an intersect... |
| breldmd 5859 | Membership of first of a b... |
| dmiun 5860 | The domain of an indexed u... |
| dmuni 5861 | The domain of a union. Pa... |
| dmopab 5862 | The domain of a class of o... |
| dmopabelb 5863 | A set is an element of the... |
| dmopab2rex 5864 | The domain of an ordered p... |
| dmopabss 5865 | Upper bound for the domain... |
| dmopab3 5866 | The domain of a restricted... |
| dm0 5867 | The domain of the empty se... |
| dmi 5868 | The domain of the identity... |
| dmv 5869 | The domain of the universe... |
| dmep 5870 | The domain of the membersh... |
| dm0rn0 5871 | An empty domain is equival... |
| dm0rn0OLD 5872 | Obsolete version of ~ dm0r... |
| rn0 5873 | The range of the empty set... |
| rnep 5874 | The range of the membershi... |
| reldm0 5875 | A relation is empty iff it... |
| dmxp 5876 | The domain of a Cartesian ... |
| dmxpid 5877 | The domain of a Cartesian ... |
| dmxpin 5878 | The domain of the intersec... |
| xpid11 5879 | The Cartesian square is a ... |
| dmcnvcnv 5880 | The domain of the double c... |
| rncnvcnv 5881 | The range of the double co... |
| elreldm 5882 | The first member of an ord... |
| rneq 5883 | Equality theorem for range... |
| rneqi 5884 | Equality inference for ran... |
| rneqd 5885 | Equality deduction for ran... |
| rnss 5886 | Subset theorem for range. ... |
| rnssi 5887 | Subclass inference for ran... |
| brelrng 5888 | The second argument of a b... |
| brelrn 5889 | The second argument of a b... |
| opelrn 5890 | Membership of second membe... |
| releldm 5891 | The first argument of a bi... |
| relelrn 5892 | The second argument of a b... |
| releldmb 5893 | Membership in a domain. (... |
| relelrnb 5894 | Membership in a range. (C... |
| releldmi 5895 | The first argument of a bi... |
| relelrni 5896 | The second argument of a b... |
| dfrnf 5897 | Definition of range, using... |
| nfdm 5898 | Bound-variable hypothesis ... |
| nfrn 5899 | Bound-variable hypothesis ... |
| dmiin 5900 | Domain of an intersection.... |
| rnopab 5901 | The range of a class of or... |
| rnopabss 5902 | Upper bound for the range ... |
| rnopab3 5903 | The range of a restricted ... |
| rnmpt 5904 | The range of a function in... |
| elrnmpt 5905 | The range of a function in... |
| elrnmpt1s 5906 | Elementhood in an image se... |
| elrnmpt1 5907 | Elementhood in an image se... |
| elrnmptg 5908 | Membership in the range of... |
| elrnmpti 5909 | Membership in the range of... |
| elrnmptd 5910 | The range of a function in... |
| elrnmpt1d 5911 | Elementhood in an image se... |
| elrnmptdv 5912 | Elementhood in the range o... |
| elrnmpt2d 5913 | Elementhood in the range o... |
| nelrnmpt 5914 | Non-membership in the rang... |
| dfiun3g 5915 | Alternate definition of in... |
| dfiin3g 5916 | Alternate definition of in... |
| dfiun3 5917 | Alternate definition of in... |
| dfiin3 5918 | Alternate definition of in... |
| riinint 5919 | Express a relative indexed... |
| relrn0 5920 | A relation is empty iff it... |
| dmrnssfld 5921 | The domain and range of a ... |
| dmcoss 5922 | Domain of a composition. ... |
| dmcossOLD 5923 | Obsolete version of ~ dmco... |
| rncoss 5924 | Range of a composition. (... |
| dmcosseq 5925 | Domain of a composition. ... |
| dmcosseqOLD 5926 | Obsolete version of ~ dmco... |
| dmcosseqOLDOLD 5927 | Obsolete version of ~ dmco... |
| dmcoeq 5928 | Domain of a composition. ... |
| rncoeq 5929 | Range of a composition. (... |
| reseq1 5930 | Equality theorem for restr... |
| reseq2 5931 | Equality theorem for restr... |
| reseq1i 5932 | Equality inference for res... |
| reseq2i 5933 | Equality inference for res... |
| reseq12i 5934 | Equality inference for res... |
| reseq1d 5935 | Equality deduction for res... |
| reseq2d 5936 | Equality deduction for res... |
| reseq12d 5937 | Equality deduction for res... |
| nfres 5938 | Bound-variable hypothesis ... |
| csbres 5939 | Distribute proper substitu... |
| res0 5940 | A restriction to the empty... |
| dfres3 5941 | Alternate definition of re... |
| opelres 5942 | Ordered pair elementhood i... |
| brres 5943 | Binary relation on a restr... |
| opelresi 5944 | Ordered pair membership in... |
| brresi 5945 | Binary relation on a restr... |
| opres 5946 | Ordered pair membership in... |
| resieq 5947 | A restricted identity rela... |
| opelidres 5948 | ` <. A , A >. ` belongs to... |
| resres 5949 | The restriction of a restr... |
| resundi 5950 | Distributive law for restr... |
| resundir 5951 | Distributive law for restr... |
| resindi 5952 | Class restriction distribu... |
| resindir 5953 | Class restriction distribu... |
| inres 5954 | Move intersection into cla... |
| resdifcom 5955 | Commutative law for restri... |
| resiun1 5956 | Distribution of restrictio... |
| resiun2 5957 | Distribution of restrictio... |
| resss 5958 | A class includes its restr... |
| rescom 5959 | Commutative law for restri... |
| ssres 5960 | Subclass theorem for restr... |
| ssres2 5961 | Subclass theorem for restr... |
| relres 5962 | A restriction is a relatio... |
| resabs1 5963 | Absorption law for restric... |
| resabs1i 5964 | Absorption law for restric... |
| resabs1d 5965 | Absorption law for restric... |
| resabs2 5966 | Absorption law for restric... |
| residm 5967 | Idempotent law for restric... |
| dmresss 5968 | The domain of a restrictio... |
| dmres 5969 | The domain of a restrictio... |
| ssdmres 5970 | A domain restricted to a s... |
| dmresexg 5971 | The domain of a restrictio... |
| resima 5972 | A restriction to an image.... |
| resima2 5973 | Image under a restricted c... |
| rnresss 5974 | The range of a restriction... |
| xpssres 5975 | Restriction of a constant ... |
| elinxp 5976 | Membership in an intersect... |
| elres 5977 | Membership in a restrictio... |
| elsnres 5978 | Membership in restriction ... |
| relssres 5979 | Simplification law for res... |
| dmressnsn 5980 | The domain of a restrictio... |
| eldmressnsn 5981 | The element of the domain ... |
| eldmeldmressn 5982 | An element of the domain (... |
| resdm 5983 | A relation restricted to i... |
| resexg 5984 | The restriction of a set i... |
| resexd 5985 | The restriction of a set i... |
| resex 5986 | The restriction of a set i... |
| resindm 5987 | When restricting a relatio... |
| resdmdfsn 5988 | Restricting a relation to ... |
| reldisjun 5989 | Split a relation into two ... |
| relresdm1 5990 | Restriction of a disjoint ... |
| resopab 5991 | Restriction of a class abs... |
| iss 5992 | A subclass of the identity... |
| resopab2 5993 | Restriction of a class abs... |
| resmpt 5994 | Restriction of the mapping... |
| resmpt3 5995 | Unconditional restriction ... |
| resmptf 5996 | Restriction of the mapping... |
| resmptd 5997 | Restriction of the mapping... |
| dfres2 5998 | Alternate definition of th... |
| mptss 5999 | Sufficient condition for i... |
| elimampt 6000 | Membership in the image of... |
| elidinxp 6001 | Characterization of the el... |
| elidinxpid 6002 | Characterization of the el... |
| elrid 6003 | Characterization of the el... |
| idinxpres 6004 | The intersection of the id... |
| idinxpresid 6005 | The intersection of the id... |
| idssxp 6006 | A diagonal set as a subset... |
| opabresid 6007 | The restricted identity re... |
| mptresid 6008 | The restricted identity re... |
| dmresi 6009 | The domain of a restricted... |
| restidsing 6010 | Restriction of the identit... |
| iresn0n0 6011 | The identity function rest... |
| imaeq1 6012 | Equality theorem for image... |
| imaeq2 6013 | Equality theorem for image... |
| imaeq1i 6014 | Equality theorem for image... |
| imaeq2i 6015 | Equality theorem for image... |
| imaeq1d 6016 | Equality theorem for image... |
| imaeq2d 6017 | Equality theorem for image... |
| imaeq12d 6018 | Equality theorem for image... |
| dfima2 6019 | Alternate definition of im... |
| dfima3 6020 | Alternate definition of im... |
| elimag 6021 | Membership in an image. T... |
| elima 6022 | Membership in an image. T... |
| elima2 6023 | Membership in an image. T... |
| elima3 6024 | Membership in an image. T... |
| nfima 6025 | Bound-variable hypothesis ... |
| nfimad 6026 | Deduction version of bound... |
| imadmrn 6027 | The image of the domain of... |
| imassrn 6028 | The image of a class is a ... |
| mptima 6029 | Image of a function in map... |
| mptimass 6030 | Image of a function in map... |
| imai 6031 | Image under the identity r... |
| rnresi 6032 | The range of the restricte... |
| resiima 6033 | The image of a restriction... |
| ima0 6034 | Image of the empty set. T... |
| 0ima 6035 | Image under the empty rela... |
| csbima12 6036 | Move class substitution in... |
| imadisj 6037 | A class whose image under ... |
| imadisjlnd 6038 | Deduction form of one nega... |
| cnvimass 6039 | A preimage under any class... |
| cnvimarndm 6040 | The preimage of the range ... |
| imasng 6041 | The image of a singleton. ... |
| relimasn 6042 | The image of a singleton. ... |
| elrelimasn 6043 | Elementhood in the image o... |
| elimasng1 6044 | Membership in an image of ... |
| elimasn1 6045 | Membership in an image of ... |
| elimasng 6046 | Membership in an image of ... |
| elimasn 6047 | Membership in an image of ... |
| elimasni 6048 | Membership in an image of ... |
| args 6049 | Two ways to express the cl... |
| elinisegg 6050 | Membership in the inverse ... |
| eliniseg 6051 | Membership in the inverse ... |
| epin 6052 | Any set is equal to its pr... |
| epini 6053 | Any set is equal to its pr... |
| iniseg 6054 | An idiom that signifies an... |
| inisegn0 6055 | Nonemptiness of an initial... |
| dffr3 6056 | Alternate definition of we... |
| dfse2 6057 | Alternate definition of se... |
| imass1 6058 | Subset theorem for image. ... |
| imass2 6059 | Subset theorem for image. ... |
| ndmima 6060 | The image of a singleton o... |
| relcnv 6061 | A converse is a relation. ... |
| relbrcnvg 6062 | When ` R ` is a relation, ... |
| eliniseg2 6063 | Eliminate the class existe... |
| relbrcnv 6064 | When ` R ` is a relation, ... |
| relco 6065 | A composition is a relatio... |
| cotrg 6066 | Two ways of saying that th... |
| cotr 6067 | Two ways of saying a relat... |
| idrefALT 6068 | Alternate proof of ~ idref... |
| cnvsym 6069 | Two ways of saying a relat... |
| intasym 6070 | Two ways of saying a relat... |
| asymref 6071 | Two ways of saying a relat... |
| asymref2 6072 | Two ways of saying a relat... |
| intirr 6073 | Two ways of saying a relat... |
| brcodir 6074 | Two ways of saying that tw... |
| codir 6075 | Two ways of saying a relat... |
| qfto 6076 | A quantifier-free way of e... |
| xpidtr 6077 | A Cartesian square is a tr... |
| trin2 6078 | The intersection of two tr... |
| poirr2 6079 | A partial order is irrefle... |
| trinxp 6080 | The relation induced by a ... |
| soirri 6081 | A strict order relation is... |
| sotri 6082 | A strict order relation is... |
| son2lpi 6083 | A strict order relation ha... |
| sotri2 6084 | A transitivity relation. ... |
| sotri3 6085 | A transitivity relation. ... |
| poleloe 6086 | Express "less than or equa... |
| poltletr 6087 | Transitive law for general... |
| somin1 6088 | Property of a minimum in a... |
| somincom 6089 | Commutativity of minimum i... |
| somin2 6090 | Property of a minimum in a... |
| soltmin 6091 | Being less than a minimum,... |
| cnvopab 6092 | The converse of a class ab... |
| cnvopabOLD 6093 | Obsolete version of ~ cnvo... |
| mptcnv 6094 | The converse of a mapping ... |
| cnv0 6095 | The converse of the empty ... |
| cnv0OLD 6096 | Obsolete version of ~ cnv0... |
| cnvi 6097 | The converse of the identi... |
| cnvun 6098 | The converse of a union is... |
| cnvdif 6099 | Distributive law for conve... |
| cnvin 6100 | Distributive law for conve... |
| rnun 6101 | Distributive law for range... |
| rnin 6102 | The range of an intersecti... |
| rniun 6103 | The range of an indexed un... |
| rnuni 6104 | The range of a union. Par... |
| imaundi 6105 | Distributive law for image... |
| imaundir 6106 | The image of a union. (Co... |
| imadifssran 6107 | Condition for the range of... |
| cnvimassrndm 6108 | The preimage of a superset... |
| dminss 6109 | An upper bound for interse... |
| imainss 6110 | An upper bound for interse... |
| inimass 6111 | The image of an intersecti... |
| inimasn 6112 | The intersection of the im... |
| cnvxp 6113 | The converse of a Cartesia... |
| xp0OLD 6114 | Obsolete version of ~ xp0 ... |
| xpnz 6115 | The Cartesian product of n... |
| xpeq0 6116 | At least one member of an ... |
| xpdisj1 6117 | Cartesian products with di... |
| xpdisj2 6118 | Cartesian products with di... |
| xpsndisj 6119 | Cartesian products with tw... |
| difxp 6120 | Difference of Cartesian pr... |
| difxp1 6121 | Difference law for Cartesi... |
| difxp2 6122 | Difference law for Cartesi... |
| djudisj 6123 | Disjoint unions with disjo... |
| xpdifid 6124 | The set of distinct couple... |
| resdisj 6125 | A double restriction to di... |
| rnxp 6126 | The range of a Cartesian p... |
| dmxpss 6127 | The domain of a Cartesian ... |
| rnxpss 6128 | The range of a Cartesian p... |
| rnxpid 6129 | The range of a Cartesian s... |
| ssxpb 6130 | A Cartesian product subcla... |
| xp11 6131 | The Cartesian product of n... |
| xpcan 6132 | Cancellation law for Carte... |
| xpcan2 6133 | Cancellation law for Carte... |
| ssrnres 6134 | Two ways to express surjec... |
| rninxp 6135 | Two ways to express surjec... |
| dminxp 6136 | Two ways to express totali... |
| imainrect 6137 | Image by a restricted and ... |
| xpima 6138 | Direct image by a Cartesia... |
| xpima1 6139 | Direct image by a Cartesia... |
| xpima2 6140 | Direct image by a Cartesia... |
| xpimasn 6141 | Direct image of a singleto... |
| sossfld 6142 | The base set of a strict o... |
| sofld 6143 | The base set of a nonempty... |
| cnvcnv3 6144 | The set of all ordered pai... |
| dfrel2 6145 | Alternate definition of re... |
| dfrel4v 6146 | A relation can be expresse... |
| dfrel4 6147 | A relation can be expresse... |
| cnvcnv 6148 | The double converse of a c... |
| cnvcnv2 6149 | The double converse of a c... |
| cnvcnvss 6150 | The double converse of a c... |
| cnvrescnv 6151 | Two ways to express the co... |
| cnveqb 6152 | Equality theorem for conve... |
| cnveq0 6153 | A relation empty iff its c... |
| dfrel3 6154 | Alternate definition of re... |
| elid 6155 | Characterization of the el... |
| dmresv 6156 | The domain of a universal ... |
| rnresv 6157 | The range of a universal r... |
| dfrn4 6158 | Range defined in terms of ... |
| csbrn 6159 | Distribute proper substitu... |
| rescnvcnv 6160 | The restriction of the dou... |
| cnvcnvres 6161 | The double converse of the... |
| imacnvcnv 6162 | The image of the double co... |
| dmsnn0 6163 | The domain of a singleton ... |
| rnsnn0 6164 | The range of a singleton i... |
| dmsn0 6165 | The domain of the singleto... |
| cnvsn0 6166 | The converse of the single... |
| dmsn0el 6167 | The domain of a singleton ... |
| relsn2 6168 | A singleton is a relation ... |
| dmsnopg 6169 | The domain of a singleton ... |
| dmsnopss 6170 | The domain of a singleton ... |
| dmpropg 6171 | The domain of an unordered... |
| dmsnop 6172 | The domain of a singleton ... |
| dmprop 6173 | The domain of an unordered... |
| dmtpop 6174 | The domain of an unordered... |
| cnvcnvsn 6175 | Double converse of a singl... |
| dmsnsnsn 6176 | The domain of the singleto... |
| rnsnopg 6177 | The range of a singleton o... |
| rnpropg 6178 | The range of a pair of ord... |
| cnvsng 6179 | Converse of a singleton of... |
| rnsnop 6180 | The range of a singleton o... |
| op1sta 6181 | Extract the first member o... |
| cnvsn 6182 | Converse of a singleton of... |
| op2ndb 6183 | Extract the second member ... |
| op2nda 6184 | Extract the second member ... |
| opswap 6185 | Swap the members of an ord... |
| cnvresima 6186 | An image under the convers... |
| resdm2 6187 | A class restricted to its ... |
| resdmres 6188 | Restriction to the domain ... |
| resresdm 6189 | A restriction by an arbitr... |
| imadmres 6190 | The image of the domain of... |
| resdmss 6191 | Subset relationship for th... |
| resdifdi 6192 | Distributive law for restr... |
| resdifdir 6193 | Distributive law for restr... |
| mptpreima 6194 | The preimage of a function... |
| mptiniseg 6195 | Converse singleton image o... |
| dmmpt 6196 | The domain of the mapping ... |
| dmmptss 6197 | The domain of a mapping is... |
| dmmptg 6198 | The domain of the mapping ... |
| rnmpt0f 6199 | The range of a function in... |
| rnmptn0 6200 | The range of a function in... |
| dfco2 6201 | Alternate definition of a ... |
| dfco2a 6202 | Generalization of ~ dfco2 ... |
| coundi 6203 | Class composition distribu... |
| coundir 6204 | Class composition distribu... |
| cores 6205 | Restricted first member of... |
| resco 6206 | Associative law for the re... |
| imaco 6207 | Image of the composition o... |
| rnco 6208 | The range of the compositi... |
| rncoOLD 6209 | Obsolete version of ~ rnco... |
| rnco2 6210 | The range of the compositi... |
| dmco 6211 | The domain of a compositio... |
| coeq0 6212 | A composition of two relat... |
| coiun 6213 | Composition with an indexe... |
| cocnvcnv1 6214 | A composition is not affec... |
| cocnvcnv2 6215 | A composition is not affec... |
| cores2 6216 | Absorption of a reverse (p... |
| co02 6217 | Composition with the empty... |
| co01 6218 | Composition with the empty... |
| coi1 6219 | Composition with the ident... |
| coi2 6220 | Composition with the ident... |
| coires1 6221 | Composition with a restric... |
| coass 6222 | Associative law for class ... |
| relcnvtrg 6223 | General form of ~ relcnvtr... |
| relcnvtr 6224 | A relation is transitive i... |
| relssdmrn 6225 | A relation is included in ... |
| resssxp 6226 | If the ` R ` -image of a c... |
| cnvssrndm 6227 | The converse is a subset o... |
| cossxp 6228 | Composition as a subset of... |
| relrelss 6229 | Two ways to describe the s... |
| unielrel 6230 | The membership relation fo... |
| relfld 6231 | The double union of a rela... |
| relresfld 6232 | Restriction of a relation ... |
| relcoi2 6233 | Composition with the ident... |
| relcoi1 6234 | Composition with the ident... |
| unidmrn 6235 | The double union of the co... |
| relcnvfld 6236 | if ` R ` is a relation, it... |
| dfdm2 6237 | Alternate definition of do... |
| unixp 6238 | The double class union of ... |
| unixp0 6239 | A Cartesian product is emp... |
| unixpid 6240 | Field of a Cartesian squar... |
| ressn 6241 | Restriction of a class to ... |
| cnviin 6242 | The converse of an interse... |
| cnvpo 6243 | The converse of a partial ... |
| cnvso 6244 | The converse of a strict o... |
| xpco 6245 | Composition of two Cartesi... |
| xpcoid 6246 | Composition of two Cartesi... |
| elsnxp 6247 | Membership in a Cartesian ... |
| reu3op 6248 | There is a unique ordered ... |
| reuop 6249 | There is a unique ordered ... |
| opreu2reurex 6250 | There is a unique ordered ... |
| opreu2reu 6251 | If there is a unique order... |
| dfpo2 6252 | Quantifier-free definition... |
| csbcog 6253 | Distribute proper substitu... |
| snres0 6254 | Condition for restriction ... |
| imaindm 6255 | The image is unaffected by... |
| predeq123 6258 | Equality theorem for the p... |
| predeq1 6259 | Equality theorem for the p... |
| predeq2 6260 | Equality theorem for the p... |
| predeq3 6261 | Equality theorem for the p... |
| nfpred 6262 | Bound-variable hypothesis ... |
| csbpredg 6263 | Move class substitution in... |
| predpredss 6264 | If ` A ` is a subset of ` ... |
| predss 6265 | The predecessor class of `... |
| sspred 6266 | Another subset/predecessor... |
| dfpred2 6267 | An alternate definition of... |
| dfpred3 6268 | An alternate definition of... |
| dfpred3g 6269 | An alternate definition of... |
| elpredgg 6270 | Membership in a predecesso... |
| elpredg 6271 | Membership in a predecesso... |
| elpredimg 6272 | Membership in a predecesso... |
| elpredim 6273 | Membership in a predecesso... |
| elpred 6274 | Membership in a predecesso... |
| predexg 6275 | The predecessor class exis... |
| dffr4 6276 | Alternate definition of we... |
| predel 6277 | Membership in the predeces... |
| predtrss 6278 | If ` R ` is transitive ove... |
| predpo 6279 | Property of the predecesso... |
| predso 6280 | Property of the predecesso... |
| setlikespec 6281 | If ` R ` is set-like in ` ... |
| predidm 6282 | Idempotent law for the pre... |
| predin 6283 | Intersection law for prede... |
| predun 6284 | Union law for predecessor ... |
| preddif 6285 | Difference law for predece... |
| predep 6286 | The predecessor under the ... |
| trpred 6287 | The class of predecessors ... |
| preddowncl 6288 | A property of classes that... |
| predpoirr 6289 | Given a partial ordering, ... |
| predfrirr 6290 | Given a well-founded relat... |
| pred0 6291 | The predecessor class over... |
| dfse3 6292 | Alternate definition of se... |
| predrelss 6293 | Subset carries from relati... |
| predprc 6294 | The predecessor of a prope... |
| predres 6295 | Predecessor class is unaff... |
| frpomin 6296 | Every nonempty (possibly p... |
| frpomin2 6297 | Every nonempty (possibly p... |
| frpoind 6298 | The principle of well-foun... |
| frpoinsg 6299 | Well-Founded Induction Sch... |
| frpoins2fg 6300 | Well-Founded Induction sch... |
| frpoins2g 6301 | Well-Founded Induction sch... |
| frpoins3g 6302 | Well-Founded Induction sch... |
| tz6.26 6303 | All nonempty subclasses of... |
| tz6.26i 6304 | All nonempty subclasses of... |
| wfi 6305 | The Principle of Well-Orde... |
| wfii 6306 | The Principle of Well-Orde... |
| wfisg 6307 | Well-Ordered Induction Sch... |
| wfis 6308 | Well-Ordered Induction Sch... |
| wfis2fg 6309 | Well-Ordered Induction Sch... |
| wfis2f 6310 | Well-Ordered Induction sch... |
| wfis2g 6311 | Well-Ordered Induction Sch... |
| wfis2 6312 | Well-Ordered Induction sch... |
| wfis3 6313 | Well-Ordered Induction sch... |
| ordeq 6322 | Equality theorem for the o... |
| elong 6323 | An ordinal number is an or... |
| elon 6324 | An ordinal number is an or... |
| eloni 6325 | An ordinal number has the ... |
| elon2 6326 | An ordinal number is an or... |
| limeq 6327 | Equality theorem for the l... |
| ordwe 6328 | Membership well-orders eve... |
| ordtr 6329 | An ordinal class is transi... |
| ordfr 6330 | Membership is well-founded... |
| ordelss 6331 | An element of an ordinal c... |
| trssord 6332 | A transitive subclass of a... |
| ordirr 6333 | No ordinal class is a memb... |
| nordeq 6334 | A member of an ordinal cla... |
| ordn2lp 6335 | An ordinal class cannot be... |
| tz7.5 6336 | A nonempty subclass of an ... |
| ordelord 6337 | An element of an ordinal c... |
| tron 6338 | The class of all ordinal n... |
| ordelon 6339 | An element of an ordinal c... |
| onelon 6340 | An element of an ordinal n... |
| tz7.7 6341 | A transitive class belongs... |
| ordelssne 6342 | For ordinal classes, membe... |
| ordelpss 6343 | For ordinal classes, membe... |
| ordsseleq 6344 | For ordinal classes, inclu... |
| ordin 6345 | The intersection of two or... |
| onin 6346 | The intersection of two or... |
| ordtri3or 6347 | A trichotomy law for ordin... |
| ordtri1 6348 | A trichotomy law for ordin... |
| ontri1 6349 | A trichotomy law for ordin... |
| ordtri2 6350 | A trichotomy law for ordin... |
| ordtri3 6351 | A trichotomy law for ordin... |
| ordtri4 6352 | A trichotomy law for ordin... |
| orddisj 6353 | An ordinal class and its s... |
| onfr 6354 | The ordinal class is well-... |
| onelpss 6355 | Relationship between membe... |
| onsseleq 6356 | Relationship between subse... |
| onelss 6357 | An element of an ordinal n... |
| oneltri 6358 | The elementhood relation o... |
| ordtr1 6359 | Transitive law for ordinal... |
| ordtr2 6360 | Transitive law for ordinal... |
| ordtr3 6361 | Transitive law for ordinal... |
| ontr1 6362 | Transitive law for ordinal... |
| ontr2 6363 | Transitive law for ordinal... |
| onelssex 6364 | Ordinal less than is equiv... |
| ordunidif 6365 | The union of an ordinal st... |
| ordintdif 6366 | If ` B ` is smaller than `... |
| onintss 6367 | If a property is true for ... |
| oneqmini 6368 | A way to show that an ordi... |
| ord0 6369 | The empty set is an ordina... |
| 0elon 6370 | The empty set is an ordina... |
| ord0eln0 6371 | A nonempty ordinal contain... |
| on0eln0 6372 | An ordinal number contains... |
| dflim2 6373 | An alternate definition of... |
| inton 6374 | The intersection of the cl... |
| nlim0 6375 | The empty set is not a lim... |
| limord 6376 | A limit ordinal is ordinal... |
| limuni 6377 | A limit ordinal is its own... |
| limuni2 6378 | The union of a limit ordin... |
| 0ellim 6379 | A limit ordinal contains t... |
| limelon 6380 | A limit ordinal class that... |
| onn0 6381 | The class of all ordinal n... |
| suceqd 6382 | Deduction associated with ... |
| suceq 6383 | Equality of successors. (... |
| elsuci 6384 | Membership in a successor.... |
| elsucg 6385 | Membership in a successor.... |
| elsuc2g 6386 | Variant of membership in a... |
| elsuc 6387 | Membership in a successor.... |
| elsuc2 6388 | Membership in a successor.... |
| nfsuc 6389 | Bound-variable hypothesis ... |
| elelsuc 6390 | Membership in a successor.... |
| sucel 6391 | Membership of a successor ... |
| suc0 6392 | The successor of the empty... |
| sucprc 6393 | A proper class is its own ... |
| unisucs 6394 | The union of the successor... |
| unisucg 6395 | A transitive class is equa... |
| unisuc 6396 | A transitive class is equa... |
| sssucid 6397 | A class is included in its... |
| sucidg 6398 | Part of Proposition 7.23 o... |
| sucid 6399 | A set belongs to its succe... |
| nsuceq0 6400 | No successor is empty. (C... |
| eqelsuc 6401 | A set belongs to the succe... |
| iunsuc 6402 | Inductive definition for t... |
| suctr 6403 | The successor of a transit... |
| trsuc 6404 | A set whose successor belo... |
| trsucss 6405 | A member of the successor ... |
| ordsssuc 6406 | An ordinal is a subset of ... |
| onsssuc 6407 | A subset of an ordinal num... |
| ordsssuc2 6408 | An ordinal subset of an or... |
| onmindif 6409 | When its successor is subt... |
| ordnbtwn 6410 | There is no set between an... |
| onnbtwn 6411 | There is no set between an... |
| sucssel 6412 | A set whose successor is a... |
| orddif 6413 | Ordinal derived from its s... |
| orduniss 6414 | An ordinal class includes ... |
| ordtri2or 6415 | A trichotomy law for ordin... |
| ordtri2or2 6416 | A trichotomy law for ordin... |
| ordtri2or3 6417 | A consequence of total ord... |
| ordelinel 6418 | The intersection of two or... |
| ordssun 6419 | Property of a subclass of ... |
| ordequn 6420 | The maximum (i.e. union) o... |
| ordun 6421 | The maximum (i.e., union) ... |
| onunel 6422 | The union of two ordinals ... |
| ordunisssuc 6423 | A subclass relationship fo... |
| suc11 6424 | The successor operation be... |
| onun2 6425 | The union of two ordinals ... |
| ontr 6426 | An ordinal number is a tra... |
| onunisuc 6427 | An ordinal number is equal... |
| onordi 6428 | An ordinal number is an or... |
| onirri 6429 | An ordinal number is not a... |
| oneli 6430 | A member of an ordinal num... |
| onelssi 6431 | A member of an ordinal num... |
| onssneli 6432 | An ordering law for ordina... |
| onssnel2i 6433 | An ordering law for ordina... |
| onelini 6434 | An element of an ordinal n... |
| oneluni 6435 | An ordinal number equals i... |
| onunisuci 6436 | An ordinal number is equal... |
| onsseli 6437 | Subset is equivalent to me... |
| onun2i 6438 | The union of two ordinal n... |
| unizlim 6439 | An ordinal equal to its ow... |
| on0eqel 6440 | An ordinal number either e... |
| snsn0non 6441 | The singleton of the singl... |
| onxpdisj 6442 | Ordinal numbers and ordere... |
| onnev 6443 | The class of ordinal numbe... |
| iotajust 6445 | Soundness justification th... |
| dfiota2 6447 | Alternate definition for d... |
| nfiota1 6448 | Bound-variable hypothesis ... |
| nfiotadw 6449 | Deduction version of ~ nfi... |
| nfiotaw 6450 | Bound-variable hypothesis ... |
| nfiotad 6451 | Deduction version of ~ nfi... |
| nfiota 6452 | Bound-variable hypothesis ... |
| cbviotaw 6453 | Change bound variables in ... |
| cbviotavw 6454 | Change bound variables in ... |
| cbviota 6455 | Change bound variables in ... |
| cbviotav 6456 | Change bound variables in ... |
| sb8iota 6457 | Variable substitution in d... |
| iotaeq 6458 | Equality theorem for descr... |
| iotabi 6459 | Equivalence theorem for de... |
| uniabio 6460 | Part of Theorem 8.17 in [Q... |
| iotaval2 6461 | Version of ~ iotaval using... |
| iotauni2 6462 | Version of ~ iotauni using... |
| iotanul2 6463 | Version of ~ iotanul using... |
| iotaval 6464 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6465 | The ` iota ` class is a su... |
| iotaex 6466 | Theorem 8.23 in [Quine] p.... |
| iotauni 6467 | Equivalence between two di... |
| iotaint 6468 | Equivalence between two di... |
| iota1 6469 | Property of iota. (Contri... |
| iotanul 6470 | Theorem 8.22 in [Quine] p.... |
| iota4 6471 | Theorem *14.22 in [Whitehe... |
| iota4an 6472 | Theorem *14.23 in [Whitehe... |
| iota5 6473 | A method for computing iot... |
| iotabidv 6474 | Formula-building deduction... |
| iotabii 6475 | Formula-building deduction... |
| iotacl 6476 | Membership law for descrip... |
| iota2df 6477 | A condition that allows to... |
| iota2d 6478 | A condition that allows to... |
| iota2 6479 | The unique element such th... |
| iotan0 6480 | Representation of "the uni... |
| sniota 6481 | A class abstraction with a... |
| dfiota4 6482 | The ` iota ` operation usi... |
| csbiota 6483 | Class substitution within ... |
| dffun2 6500 | Alternate definition of a ... |
| dffun6 6501 | Alternate definition of a ... |
| dffun3 6502 | Alternate definition of fu... |
| dffun4 6503 | Alternate definition of a ... |
| dffun5 6504 | Alternate definition of fu... |
| dffun6f 6505 | Definition of function, us... |
| funmo 6506 | A function has at most one... |
| funrel 6507 | A function is a relation. ... |
| 0nelfun 6508 | A function does not contai... |
| funss 6509 | Subclass theorem for funct... |
| funeq 6510 | Equality theorem for funct... |
| funeqi 6511 | Equality inference for the... |
| funeqd 6512 | Equality deduction for the... |
| nffun 6513 | Bound-variable hypothesis ... |
| sbcfung 6514 | Distribute proper substitu... |
| funeu 6515 | There is exactly one value... |
| funeu2 6516 | There is exactly one value... |
| dffun7 6517 | Alternate definition of a ... |
| dffun8 6518 | Alternate definition of a ... |
| dffun9 6519 | Alternate definition of a ... |
| funfn 6520 | A class is a function if a... |
| funfnd 6521 | A function is a function o... |
| funi 6522 | The identity relation is a... |
| nfunv 6523 | The universal class is not... |
| funopg 6524 | A Kuratowski ordered pair ... |
| funopab 6525 | A class of ordered pairs i... |
| funopabeq 6526 | A class of ordered pairs o... |
| funopab4 6527 | A class of ordered pairs o... |
| funmpt 6528 | A function in maps-to nota... |
| funmpt2 6529 | Functionality of a class g... |
| funco 6530 | The composition of two fun... |
| funresfunco 6531 | Composition of two functio... |
| funres 6532 | A restriction of a functio... |
| funresd 6533 | A restriction of a functio... |
| funssres 6534 | The restriction of a funct... |
| fun2ssres 6535 | Equality of restrictions o... |
| funun 6536 | The union of functions wit... |
| fununmo 6537 | If the union of classes is... |
| fununfun 6538 | If the union of classes is... |
| fundif 6539 | A function with removed el... |
| funcnvsn 6540 | The converse singleton of ... |
| funsng 6541 | A singleton of an ordered ... |
| fnsng 6542 | Functionality and domain o... |
| funsn 6543 | A singleton of an ordered ... |
| funprg 6544 | A set of two pairs is a fu... |
| funtpg 6545 | A set of three pairs is a ... |
| funpr 6546 | A function with a domain o... |
| funtp 6547 | A function with a domain o... |
| fnsn 6548 | Functionality and domain o... |
| fnprg 6549 | Function with a domain of ... |
| fntpg 6550 | Function with a domain of ... |
| fntp 6551 | A function with a domain o... |
| funcnvpr 6552 | The converse pair of order... |
| funcnvtp 6553 | The converse triple of ord... |
| funcnvqp 6554 | The converse quadruple of ... |
| fun0 6555 | The empty set is a functio... |
| funcnv0 6556 | The converse of the empty ... |
| funcnvcnv 6557 | The double converse of a f... |
| funcnv2 6558 | A simpler equivalence for ... |
| funcnv 6559 | The converse of a class is... |
| funcnv3 6560 | A condition showing a clas... |
| fun2cnv 6561 | The double converse of a c... |
| svrelfun 6562 | A single-valued relation i... |
| fncnv 6563 | Single-rootedness (see ~ f... |
| fun11 6564 | Two ways of stating that `... |
| fununi 6565 | The union of a chain (with... |
| funin 6566 | The intersection with a fu... |
| funres11 6567 | The restriction of a one-t... |
| funcnvres 6568 | The converse of a restrict... |
| cnvresid 6569 | Converse of a restricted i... |
| funcnvres2 6570 | The converse of a restrict... |
| funimacnv 6571 | The image of the preimage ... |
| funimass1 6572 | A kind of contraposition l... |
| funimass2 6573 | A kind of contraposition l... |
| imadif 6574 | The image of a difference ... |
| imain 6575 | The image of an intersecti... |
| f1imadifssran 6576 | Condition for the range of... |
| funimaexg 6577 | Axiom of Replacement using... |
| funimaex 6578 | The image of a set under a... |
| isarep1 6579 | Part of a study of the Axi... |
| isarep2 6580 | Part of a study of the Axi... |
| fneq1 6581 | Equality theorem for funct... |
| fneq2 6582 | Equality theorem for funct... |
| fneq1d 6583 | Equality deduction for fun... |
| fneq2d 6584 | Equality deduction for fun... |
| fneq12d 6585 | Equality deduction for fun... |
| fneq12 6586 | Equality theorem for funct... |
| fneq1i 6587 | Equality inference for fun... |
| fneq2i 6588 | Equality inference for fun... |
| nffn 6589 | Bound-variable hypothesis ... |
| fnfun 6590 | A function with domain is ... |
| fnfund 6591 | A function with domain is ... |
| fnrel 6592 | A function with domain is ... |
| fndm 6593 | The domain of a function. ... |
| fndmi 6594 | The domain of a function. ... |
| fndmd 6595 | The domain of a function. ... |
| funfni 6596 | Inference to convert a fun... |
| fndmu 6597 | A function has a unique do... |
| fnbr 6598 | The first argument of bina... |
| fnop 6599 | The first argument of an o... |
| fneu 6600 | There is exactly one value... |
| fneu2 6601 | There is exactly one value... |
| fnunres1 6602 | Restriction of a disjoint ... |
| fnunres2 6603 | Restriction of a disjoint ... |
| fnun 6604 | The union of two functions... |
| fnund 6605 | The union of two functions... |
| fnunop 6606 | Extension of a function wi... |
| fncofn 6607 | Composition of a function ... |
| fnco 6608 | Composition of two functio... |
| fnresdm 6609 | A function does not change... |
| fnresdisj 6610 | A function restricted to a... |
| 2elresin 6611 | Membership in two function... |
| fnssresb 6612 | Restriction of a function ... |
| fnssres 6613 | Restriction of a function ... |
| fnssresd 6614 | Restriction of a function ... |
| fnresin1 6615 | Restriction of a function'... |
| fnresin2 6616 | Restriction of a function'... |
| fnres 6617 | An equivalence for functio... |
| idfn 6618 | The identity relation is a... |
| fnresi 6619 | The restricted identity re... |
| fnima 6620 | The image of a function's ... |
| fn0 6621 | A function with empty doma... |
| fnimadisj 6622 | A class that is disjoint w... |
| fnimaeq0 6623 | Images under a function ne... |
| dfmpt3 6624 | Alternate definition for t... |
| mptfnf 6625 | The maps-to notation defin... |
| fnmptf 6626 | The maps-to notation defin... |
| fnopabg 6627 | Functionality and domain o... |
| fnopab 6628 | Functionality and domain o... |
| mptfng 6629 | The maps-to notation defin... |
| fnmpt 6630 | The maps-to notation defin... |
| fnmptd 6631 | The maps-to notation defin... |
| mpt0 6632 | A mapping operation with e... |
| fnmpti 6633 | Functionality and domain o... |
| dmmpti 6634 | Domain of the mapping oper... |
| dmmptd 6635 | The domain of the mapping ... |
| mptun 6636 | Union of mappings which ar... |
| partfun 6637 | Rewrite a function defined... |
| feq1 6638 | Equality theorem for funct... |
| feq2 6639 | Equality theorem for funct... |
| feq3 6640 | Equality theorem for funct... |
| feq23 6641 | Equality theorem for funct... |
| feq1d 6642 | Equality deduction for fun... |
| feq1dd 6643 | Equality deduction for fun... |
| feq2d 6644 | Equality deduction for fun... |
| feq3d 6645 | Equality deduction for fun... |
| feq2dd 6646 | Equality deduction for fun... |
| feq3dd 6647 | Equality deduction for fun... |
| feq12d 6648 | Equality deduction for fun... |
| feq123d 6649 | Equality deduction for fun... |
| feq123 6650 | Equality theorem for funct... |
| feq1i 6651 | Equality inference for fun... |
| feq2i 6652 | Equality inference for fun... |
| feq12i 6653 | Equality inference for fun... |
| feq23i 6654 | Equality inference for fun... |
| feq23d 6655 | Equality deduction for fun... |
| nff 6656 | Bound-variable hypothesis ... |
| sbcfng 6657 | Distribute proper substitu... |
| sbcfg 6658 | Distribute proper substitu... |
| elimf 6659 | Eliminate a mapping hypoth... |
| ffn 6660 | A mapping is a function wi... |
| ffnd 6661 | A mapping is a function wi... |
| dffn2 6662 | Any function is a mapping ... |
| ffun 6663 | A mapping is a function. ... |
| ffund 6664 | A mapping is a function, d... |
| frel 6665 | A mapping is a relation. ... |
| freld 6666 | A mapping is a relation. ... |
| frn 6667 | The range of a mapping. (... |
| frnd 6668 | Deduction form of ~ frn . ... |
| fdm 6669 | The domain of a mapping. ... |
| fdmd 6670 | Deduction form of ~ fdm . ... |
| fdmi 6671 | Inference associated with ... |
| dffn3 6672 | A function maps to its ran... |
| ffrn 6673 | A function maps to its ran... |
| ffrnb 6674 | Characterization of a func... |
| ffrnbd 6675 | A function maps to its ran... |
| fss 6676 | Expanding the codomain of ... |
| fssd 6677 | Expanding the codomain of ... |
| fssdmd 6678 | Expressing that a class is... |
| fssdm 6679 | Expressing that a class is... |
| fimass 6680 | The image of a class under... |
| fimassd 6681 | The image of a class is a ... |
| fimacnv 6682 | The preimage of the codoma... |
| fcof 6683 | Composition of a function ... |
| fco 6684 | Composition of two functio... |
| fcod 6685 | Composition of two mapping... |
| fco2 6686 | Functionality of a composi... |
| fssxp 6687 | A mapping is a class of or... |
| funssxp 6688 | Two ways of specifying a p... |
| ffdm 6689 | A mapping is a partial fun... |
| ffdmd 6690 | The domain of a function. ... |
| fdmrn 6691 | A different way to write `... |
| funcofd 6692 | Composition of two functio... |
| opelf 6693 | The members of an ordered ... |
| fun 6694 | The union of two functions... |
| fun2 6695 | The union of two functions... |
| fun2d 6696 | The union of functions wit... |
| fnfco 6697 | Composition of two functio... |
| fssres 6698 | Restriction of a function ... |
| fssresd 6699 | Restriction of a function ... |
| fssres2 6700 | Restriction of a restricte... |
| fresin 6701 | An identity for the mappin... |
| resasplit 6702 | If two functions agree on ... |
| fresaun 6703 | The union of two functions... |
| fresaunres2 6704 | From the union of two func... |
| fresaunres1 6705 | From the union of two func... |
| fcoi1 6706 | Composition of a mapping a... |
| fcoi2 6707 | Composition of restricted ... |
| feu 6708 | There is exactly one value... |
| fcnvres 6709 | The converse of a restrict... |
| fimacnvdisj 6710 | The preimage of a class di... |
| fint 6711 | Function into an intersect... |
| fin 6712 | Mapping into an intersecti... |
| f0 6713 | The empty function. (Cont... |
| f00 6714 | A class is a function with... |
| f0bi 6715 | A function with empty doma... |
| f0dom0 6716 | A function is empty iff it... |
| f0rn0 6717 | If there is no element in ... |
| fconst 6718 | A Cartesian product with a... |
| fconstg 6719 | A Cartesian product with a... |
| fnconstg 6720 | A Cartesian product with a... |
| fconst6g 6721 | Constant function with loo... |
| fconst6 6722 | A constant function as a m... |
| f1eq1 6723 | Equality theorem for one-t... |
| f1eq2 6724 | Equality theorem for one-t... |
| f1eq3 6725 | Equality theorem for one-t... |
| nff1 6726 | Bound-variable hypothesis ... |
| dff12 6727 | Alternate definition of a ... |
| f1f 6728 | A one-to-one mapping is a ... |
| f1fn 6729 | A one-to-one mapping is a ... |
| f1fun 6730 | A one-to-one mapping is a ... |
| f1rel 6731 | A one-to-one onto mapping ... |
| f1dm 6732 | The domain of a one-to-one... |
| f1ss 6733 | A function that is one-to-... |
| f1ssr 6734 | A function that is one-to-... |
| f1ssres 6735 | A function that is one-to-... |
| f1resf1 6736 | The restriction of an inje... |
| f1cnvcnv 6737 | Two ways to express that a... |
| f1cof1 6738 | Composition of two one-to-... |
| f1co 6739 | Composition of one-to-one ... |
| foeq1 6740 | Equality theorem for onto ... |
| foeq2 6741 | Equality theorem for onto ... |
| foeq3 6742 | Equality theorem for onto ... |
| nffo 6743 | Bound-variable hypothesis ... |
| fof 6744 | An onto mapping is a mappi... |
| fofun 6745 | An onto mapping is a funct... |
| fofn 6746 | An onto mapping is a funct... |
| forn 6747 | The codomain of an onto fu... |
| dffo2 6748 | Alternate definition of an... |
| foima 6749 | The image of the domain of... |
| dffn4 6750 | A function maps onto its r... |
| funforn 6751 | A function maps its domain... |
| fodmrnu 6752 | An onto function has uniqu... |
| fimadmfo 6753 | A function is a function o... |
| fores 6754 | Restriction of an onto fun... |
| fimadmfoALT 6755 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6756 | The preimage of the codoma... |
| focofo 6757 | Composition of onto functi... |
| foco 6758 | Composition of onto functi... |
| foconst 6759 | A nonzero constant functio... |
| f1oeq1 6760 | Equality theorem for one-t... |
| f1oeq2 6761 | Equality theorem for one-t... |
| f1oeq3 6762 | Equality theorem for one-t... |
| f1oeq23 6763 | Equality theorem for one-t... |
| f1eq123d 6764 | Equality deduction for one... |
| foeq123d 6765 | Equality deduction for ont... |
| f1oeq123d 6766 | Equality deduction for one... |
| f1oeq1d 6767 | Equality deduction for one... |
| f1oeq2d 6768 | Equality deduction for one... |
| f1oeq3d 6769 | Equality deduction for one... |
| nff1o 6770 | Bound-variable hypothesis ... |
| f1of1 6771 | A one-to-one onto mapping ... |
| f1of 6772 | A one-to-one onto mapping ... |
| f1ofn 6773 | A one-to-one onto mapping ... |
| f1ofun 6774 | A one-to-one onto mapping ... |
| f1orel 6775 | A one-to-one onto mapping ... |
| f1odm 6776 | The domain of a one-to-one... |
| dff1o2 6777 | Alternate definition of on... |
| dff1o3 6778 | Alternate definition of on... |
| f1ofo 6779 | A one-to-one onto function... |
| dff1o4 6780 | Alternate definition of on... |
| dff1o5 6781 | Alternate definition of on... |
| f1orn 6782 | A one-to-one function maps... |
| f1f1orn 6783 | A one-to-one function maps... |
| f1ocnv 6784 | The converse of a one-to-o... |
| f1ocnvb 6785 | A relation is a one-to-one... |
| f1ores 6786 | The restriction of a one-t... |
| f1orescnv 6787 | The converse of a one-to-o... |
| f1imacnv 6788 | Preimage of an image. (Co... |
| foimacnv 6789 | A reverse version of ~ f1i... |
| foun 6790 | The union of two onto func... |
| f1oun 6791 | The union of two one-to-on... |
| f1un 6792 | The union of two one-to-on... |
| resdif 6793 | The restriction of a one-t... |
| resin 6794 | The restriction of a one-t... |
| f1oco 6795 | Composition of one-to-one ... |
| f1cnv 6796 | The converse of an injecti... |
| funcocnv2 6797 | Composition with the conve... |
| fococnv2 6798 | The composition of an onto... |
| f1ococnv2 6799 | The composition of a one-t... |
| f1cocnv2 6800 | Composition of an injectiv... |
| f1ococnv1 6801 | The composition of a one-t... |
| f1cocnv1 6802 | Composition of an injectiv... |
| funcoeqres 6803 | Express a constraint on a ... |
| f1ssf1 6804 | A subset of an injective f... |
| f10 6805 | The empty set maps one-to-... |
| f10d 6806 | The empty set maps one-to-... |
| f1o00 6807 | One-to-one onto mapping of... |
| fo00 6808 | Onto mapping of the empty ... |
| f1o0 6809 | One-to-one onto mapping of... |
| f1oi 6810 | A restriction of the ident... |
| f1oiOLD 6811 | Obsolete version of ~ f1oi... |
| f1ovi 6812 | The identity relation is a... |
| f1osn 6813 | A singleton of an ordered ... |
| f1osng 6814 | A singleton of an ordered ... |
| f1sng 6815 | A singleton of an ordered ... |
| fsnd 6816 | A singleton of an ordered ... |
| f1oprswap 6817 | A two-element swap is a bi... |
| f1oprg 6818 | An unordered pair of order... |
| tz6.12-2 6819 | Function value when ` F ` ... |
| tz6.12-2OLD 6820 | Obsolete version of ~ tz6.... |
| fveu 6821 | The value of a function at... |
| brprcneu 6822 | If ` A ` is a proper class... |
| brprcneuALT 6823 | Alternate proof of ~ brprc... |
| fvprc 6824 | A function's value at a pr... |
| fvprcALT 6825 | Alternate proof of ~ fvprc... |
| rnfvprc 6826 | The range of a function va... |
| fv2 6827 | Alternate definition of fu... |
| dffv3 6828 | A definition of function v... |
| dffv4 6829 | The previous definition of... |
| elfv 6830 | Membership in a function v... |
| fveq1 6831 | Equality theorem for funct... |
| fveq2 6832 | Equality theorem for funct... |
| fveq1i 6833 | Equality inference for fun... |
| fveq1d 6834 | Equality deduction for fun... |
| fveq2i 6835 | Equality inference for fun... |
| fveq2d 6836 | Equality deduction for fun... |
| 2fveq3 6837 | Equality theorem for neste... |
| fveq12i 6838 | Equality deduction for fun... |
| fveq12d 6839 | Equality deduction for fun... |
| fveqeq2d 6840 | Equality deduction for fun... |
| fveqeq2 6841 | Equality deduction for fun... |
| nffv 6842 | Bound-variable hypothesis ... |
| nffvmpt1 6843 | Bound-variable hypothesis ... |
| nffvd 6844 | Deduction version of bound... |
| fvex 6845 | The value of a class exist... |
| fvexi 6846 | The value of a class exist... |
| fvexd 6847 | The value of a class exist... |
| fvif 6848 | Move a conditional outside... |
| iffv 6849 | Move a conditional outside... |
| fv3 6850 | Alternate definition of th... |
| fvres 6851 | The value of a restricted ... |
| fvresd 6852 | The value of a restricted ... |
| funssfv 6853 | The value of a member of t... |
| tz6.12c 6854 | Corollary of Theorem 6.12(... |
| tz6.12-1 6855 | Function value. Theorem 6... |
| tz6.12 6856 | Function value. Theorem 6... |
| tz6.12f 6857 | Function value, using boun... |
| tz6.12i 6858 | Corollary of Theorem 6.12(... |
| fvbr0 6859 | Two possibilities for the ... |
| fvrn0 6860 | A function value is a memb... |
| fvn0fvelrn 6861 | If the value of a function... |
| elfvunirn 6862 | A function value is a subs... |
| fvssunirn 6863 | The result of a function v... |
| ndmfv 6864 | The value of a class outsi... |
| ndmfvrcl 6865 | Reverse closure law for fu... |
| elfvdm 6866 | If a function value has a ... |
| elfvex 6867 | If a function value has a ... |
| elfvexd 6868 | If a function value has a ... |
| eliman0 6869 | A nonempty function value ... |
| nfvres 6870 | The value of a non-member ... |
| nfunsn 6871 | If the restriction of a cl... |
| fvfundmfvn0 6872 | If the "value of a class" ... |
| 0fv 6873 | Function value of the empt... |
| fv2prc 6874 | A function value of a func... |
| elfv2ex 6875 | If a function value of a f... |
| fveqres 6876 | Equal values imply equal v... |
| csbfv12 6877 | Move class substitution in... |
| csbfv2g 6878 | Move class substitution in... |
| csbfv 6879 | Substitution for a functio... |
| funbrfv 6880 | The second argument of a b... |
| funopfv 6881 | The second element in an o... |
| fnbrfvb 6882 | Equivalence of function va... |
| fnopfvb 6883 | Equivalence of function va... |
| fvelima2 6884 | Function value in an image... |
| funbrfvb 6885 | Equivalence of function va... |
| funopfvb 6886 | Equivalence of function va... |
| fnbrfvb2 6887 | Version of ~ fnbrfvb for f... |
| fdmeu 6888 | There is exactly one codom... |
| funbrfv2b 6889 | Function value in terms of... |
| dffn5 6890 | Representation of a functi... |
| fnrnfv 6891 | The range of a function ex... |
| fvelrnb 6892 | A member of a function's r... |
| foelcdmi 6893 | A member of a surjective f... |
| dfimafn 6894 | Alternate definition of th... |
| dfimafn2 6895 | Alternate definition of th... |
| funimass4 6896 | Membership relation for th... |
| fvelima 6897 | Function value in an image... |
| funimassd 6898 | Sufficient condition for t... |
| fvelimad 6899 | Function value in an image... |
| feqmptd 6900 | Deduction form of ~ dffn5 ... |
| feqresmpt 6901 | Express a restricted funct... |
| feqmptdf 6902 | Deduction form of ~ dffn5f... |
| dffn5f 6903 | Representation of a functi... |
| fvelimab 6904 | Function value in an image... |
| fvelimabd 6905 | Deduction form of ~ fvelim... |
| fimarab 6906 | Expressing the image of a ... |
| unima 6907 | Image of a union. (Contri... |
| fvi 6908 | The value of the identity ... |
| fviss 6909 | The value of the identity ... |
| fniinfv 6910 | The indexed intersection o... |
| fnsnfv 6911 | Singleton of function valu... |
| opabiotafun 6912 | Define a function whose va... |
| opabiotadm 6913 | Define a function whose va... |
| opabiota 6914 | Define a function whose va... |
| fnimapr 6915 | The image of a pair under ... |
| fnimatpd 6916 | The image of an unordered ... |
| ssimaex 6917 | The existence of a subimag... |
| ssimaexg 6918 | The existence of a subimag... |
| funfv 6919 | A simplified expression fo... |
| funfv2 6920 | The value of a function. ... |
| funfv2f 6921 | The value of a function. ... |
| fvun 6922 | Value of the union of two ... |
| fvun1 6923 | The value of a union when ... |
| fvun2 6924 | The value of a union when ... |
| fvun1d 6925 | The value of a union when ... |
| fvun2d 6926 | The value of a union when ... |
| dffv2 6927 | Alternate definition of fu... |
| dmfco 6928 | Domains of a function comp... |
| fvco2 6929 | Value of a function compos... |
| fvco 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 ... |
| fvtresfn 6941 | Functionality of a tuple-r... |
| fvmpts 6942 | Value of a function given ... |
| fvmpt3 6943 | Value of a function given ... |
| fvmpt3i 6944 | Value of a function given ... |
| fvmptdf 6945 | Deduction version of ~ fvm... |
| fvmptd 6946 | Deduction version of ~ fvm... |
| fvmptd2 6947 | Deduction version of ~ fvm... |
| mptrcl 6948 | Reverse closure for a mapp... |
| fvmpt2i 6949 | Value of a function given ... |
| fvmpt2 6950 | Value of a function given ... |
| fvmptss 6951 | If all the values of the m... |
| fvmpt2d 6952 | Deduction version of ~ fvm... |
| fvmptex 6953 | Express a function ` F ` w... |
| fvmptd3f 6954 | Alternate deduction versio... |
| fvmptd2f 6955 | Alternate deduction versio... |
| fvmptdv 6956 | Alternate deduction versio... |
| fvmptdv2 6957 | Alternate deduction versio... |
| mpteqb 6958 | Bidirectional equality the... |
| fvmptt 6959 | Closed theorem form of ~ f... |
| fvmptf 6960 | Value of a function given ... |
| fvmptnf 6961 | The value of a function gi... |
| fvmptd3 6962 | Deduction version of ~ fvm... |
| fvmptd4 6963 | Deduction version of ~ fvm... |
| fvmptn 6964 | This somewhat non-intuitiv... |
| fvmptss2 6965 | A mapping always evaluates... |
| elfvmptrab1w 6966 | Implications for the value... |
| elfvmptrab1 6967 | Implications for the value... |
| elfvmptrab 6968 | Implications for the value... |
| fvopab4ndm 6969 | Value of a function given ... |
| fvmptndm 6970 | Value of a function given ... |
| fvmptrabfv 6971 | Value of a function mappin... |
| fvopab5 6972 | The value of a function th... |
| fvopab6 6973 | Value of a function given ... |
| eqfnfv 6974 | Equality of functions is d... |
| eqfnfv2 6975 | Equality of functions is d... |
| eqfnfv3 6976 | Derive equality of functio... |
| eqfnfvd 6977 | Deduction for equality of ... |
| eqfnfv2f 6978 | Equality of functions is d... |
| eqfunfv 6979 | Equality of functions is d... |
| eqfnun 6980 | Two functions on ` A u. B ... |
| fvreseq0 6981 | Equality of restricted fun... |
| fvreseq1 6982 | Equality of a function res... |
| fvreseq 6983 | Equality of restricted fun... |
| fnmptfvd 6984 | A function with a given do... |
| fndmdif 6985 | Two ways to express the lo... |
| fndmdifcom 6986 | The difference set between... |
| fndmdifeq0 6987 | The difference set of two ... |
| fndmin 6988 | Two ways to express the lo... |
| fneqeql 6989 | Two functions are equal if... |
| fneqeql2 6990 | Two functions are equal if... |
| fnreseql 6991 | Two functions are equal on... |
| chfnrn 6992 | The range of a choice func... |
| funfvop 6993 | Ordered pair with function... |
| funfvbrb 6994 | Two ways to say that ` A `... |
| fvimacnvi 6995 | A member of a preimage is ... |
| fvimacnv 6996 | The argument of a function... |
| funimass3 6997 | A kind of contraposition l... |
| funimass5 6998 | A subclass of a preimage i... |
| funconstss 6999 | Two ways of specifying tha... |
| fvimacnvALT 7000 | Alternate proof of ~ fvima... |
| elpreima 7001 | Membership in the preimage... |
| elpreimad 7002 | Membership in the preimage... |
| fniniseg 7003 | Membership in the preimage... |
| fncnvima2 7004 | Inverse images under funct... |
| fniniseg2 7005 | Inverse point images under... |
| unpreima 7006 | Preimage of a union. (Con... |
| inpreima 7007 | Preimage of an intersectio... |
| difpreima 7008 | Preimage of a difference. ... |
| respreima 7009 | The preimage of a restrict... |
| cnvimainrn 7010 | The preimage of the inters... |
| sspreima 7011 | The preimage of a subset i... |
| iinpreima 7012 | Preimage of an intersectio... |
| intpreima 7013 | Preimage of an intersectio... |
| fimacnvinrn 7014 | Taking the converse image ... |
| fimacnvinrn2 7015 | Taking the converse image ... |
| rescnvimafod 7016 | The restriction of a funct... |
| fvn0ssdmfun 7017 | If a class' function value... |
| fnopfv 7018 | Ordered pair with function... |
| fvelrn 7019 | A function's value belongs... |
| nelrnfvne 7020 | A function value cannot be... |
| fveqdmss 7021 | If the empty set is not co... |
| fveqressseq 7022 | If the empty set is not co... |
| fnfvelrn 7023 | A function's value belongs... |
| ffvelcdm 7024 | A function's value belongs... |
| fnfvelrnd 7025 | A function's value belongs... |
| ffvelcdmi 7026 | A function's value belongs... |
| ffvelcdmda 7027 | A function's value belongs... |
| ffvelcdmd 7028 | A function's value belongs... |
| feldmfvelcdm 7029 | A class is an element of t... |
| rexrn 7030 | Restricted existential qua... |
| ralrn 7031 | Restricted universal quant... |
| elrnrexdm 7032 | For any element in the ran... |
| elrnrexdmb 7033 | For any element in the ran... |
| eldmrexrn 7034 | For any element in the dom... |
| eldmrexrnb 7035 | For any element in the dom... |
| fvcofneq 7036 | The values of two function... |
| ralrnmptw 7037 | A restricted quantifier ov... |
| rexrnmptw 7038 | A restricted quantifier ov... |
| ralrnmpt 7039 | A restricted quantifier ov... |
| rexrnmpt 7040 | A restricted quantifier ov... |
| f0cli 7041 | Unconditional closure of a... |
| dff2 7042 | Alternate definition of a ... |
| dff3 7043 | Alternate definition of a ... |
| dff4 7044 | Alternate definition of a ... |
| dffo3 7045 | An onto mapping expressed ... |
| dffo4 7046 | Alternate definition of an... |
| dffo5 7047 | Alternate definition of an... |
| exfo 7048 | A relation equivalent to t... |
| dffo3f 7049 | An onto mapping expressed ... |
| foelrn 7050 | Property of a surjective f... |
| foelrnf 7051 | Property of a surjective f... |
| foco2 7052 | If a composition of two fu... |
| fmpt 7053 | Functionality of the mappi... |
| f1ompt 7054 | Express bijection for a ma... |
| fmpti 7055 | Functionality of the mappi... |
| fvmptelcdm 7056 | The value of a function at... |
| fmptd 7057 | Domain and codomain of the... |
| fmpttd 7058 | Version of ~ fmptd with in... |
| fmpt3d 7059 | Domain and codomain of the... |
| fmptdf 7060 | A version of ~ fmptd using... |
| fompt 7061 | Express being onto for a m... |
| ffnfv 7062 | A function maps to a class... |
| ffnfvf 7063 | A function maps to a class... |
| fnfvrnss 7064 | An upper bound for range d... |
| fcdmssb 7065 | A function is a function i... |
| rnmptss 7066 | The range of an operation ... |
| fmpt2d 7067 | Domain and codomain of the... |
| ffvresb 7068 | A necessary and sufficient... |
| fssrescdmd 7069 | Restriction of a function ... |
| f1oresrab 7070 | Build a bijection between ... |
| f1ossf1o 7071 | Restricting a bijection, w... |
| fmptco 7072 | Composition of two functio... |
| fmptcof 7073 | Version of ~ fmptco where ... |
| fmptcos 7074 | Composition of two functio... |
| cofmpt 7075 | Express composition of a m... |
| fcompt 7076 | Express composition of two... |
| fcoconst 7077 | Composition with a constan... |
| fsn 7078 | A function maps a singleto... |
| fsn2 7079 | A function that maps a sin... |
| fsng 7080 | A function maps a singleto... |
| fsn2g 7081 | A function that maps a sin... |
| xpsng 7082 | The Cartesian product of t... |
| xpprsng 7083 | The Cartesian product of a... |
| xpsn 7084 | The Cartesian product of t... |
| f1o2sn 7085 | A singleton consisting in ... |
| residpr 7086 | Restriction of the identit... |
| dfmpt 7087 | Alternate definition for t... |
| fnasrn 7088 | A function expressed as th... |
| idref 7089 | Two ways to state that a r... |
| funiun 7090 | A function is a union of s... |
| funopsn 7091 | If a function is an ordere... |
| funop 7092 | An ordered pair is a funct... |
| funopdmsn 7093 | The domain of a function w... |
| funsndifnop 7094 | A singleton of an ordered ... |
| funsneqopb 7095 | A singleton of an ordered ... |
| ressnop0 7096 | If ` A ` is not in ` C ` ,... |
| fpr 7097 | A function with a domain o... |
| fprg 7098 | A function with a domain o... |
| ftpg 7099 | A function with a domain o... |
| ftp 7100 | A function with a domain o... |
| fnressn 7101 | A function restricted to a... |
| funressn 7102 | A function restricted to a... |
| fressnfv 7103 | The value of a function re... |
| fvrnressn 7104 | If the value of a function... |
| fvressn 7105 | The value of a function re... |
| fvconst 7106 | The value of a constant fu... |
| fnsnr 7107 | If a class belongs to a fu... |
| fnsnbg 7108 | A function's domain is a s... |
| fnsnb 7109 | A function whose domain is... |
| fnsnbOLD 7110 | Obsolete version of ~ fnsn... |
| fmptsn 7111 | Express a singleton functi... |
| fmptsng 7112 | Express a singleton functi... |
| fmptsnd 7113 | Express a singleton functi... |
| fmptap 7114 | Append an additional value... |
| fmptapd 7115 | Append an additional value... |
| fmptpr 7116 | Express a pair function in... |
| fvresi 7117 | The value of a restricted ... |
| fninfp 7118 | Express the class of fixed... |
| fnelfp 7119 | Property of a fixed point ... |
| fndifnfp 7120 | Express the class of non-f... |
| fnelnfp 7121 | Property of a non-fixed po... |
| fnnfpeq0 7122 | A function is the identity... |
| fvunsn 7123 | Remove an ordered pair not... |
| fvsng 7124 | The value of a singleton o... |
| fvsn 7125 | The value of a singleton o... |
| fvsnun1 7126 | The value of a function wi... |
| fvsnun2 7127 | The value of a function wi... |
| fnsnsplit 7128 | Split a function into a si... |
| fsnunf 7129 | Adjoining a point to a fun... |
| fsnunf2 7130 | Adjoining a point to a pun... |
| fsnunfv 7131 | Recover the added point fr... |
| fsnunres 7132 | Recover the original funct... |
| funresdfunsn 7133 | Restricting a function to ... |
| fvpr1g 7134 | The value of a function wi... |
| fvpr2g 7135 | The value of a function wi... |
| fvpr1 7136 | The value of a function wi... |
| fvpr2 7137 | The value of a function wi... |
| fprb 7138 | A condition for functionho... |
| fvtp1 7139 | The first value of a funct... |
| fvtp2 7140 | The second value of a func... |
| fvtp3 7141 | The third value of a funct... |
| fvtp1g 7142 | The value of a function wi... |
| fvtp2g 7143 | The value of a function wi... |
| fvtp3g 7144 | The value of a function wi... |
| tpres 7145 | An unordered triple of ord... |
| fvconst2g 7146 | The value of a constant fu... |
| fconst2g 7147 | A constant function expres... |
| fvconst2 7148 | The value of a constant fu... |
| fconst2 7149 | A constant function expres... |
| fconst5 7150 | Two ways to express that a... |
| rnmptc 7151 | Range of a constant functi... |
| fnprb 7152 | A function whose domain ha... |
| fntpb 7153 | A function whose domain ha... |
| fnpr2g 7154 | A function whose domain ha... |
| fpr2g 7155 | A function that maps a pai... |
| fconstfv 7156 | A constant function expres... |
| fconst3 7157 | Two ways to express a cons... |
| fconst4 7158 | Two ways to express a cons... |
| resfunexg 7159 | The restriction of a funct... |
| resiexd 7160 | The restriction of the ide... |
| fnex 7161 | If the domain of a functio... |
| fnexd 7162 | If the domain of a functio... |
| funex 7163 | If the domain of a functio... |
| opabex 7164 | Existence of a function ex... |
| mptexg 7165 | If the domain of a functio... |
| mptexgf 7166 | If the domain of a functio... |
| mptex 7167 | If the domain of a functio... |
| mptexd 7168 | If the domain of a functio... |
| mptrabex 7169 | If the domain of a functio... |
| fex 7170 | If the domain of a mapping... |
| fexd 7171 | If the domain of a mapping... |
| mptfvmpt 7172 | A function in maps-to nota... |
| eufnfv 7173 | A function is uniquely det... |
| funfvima 7174 | A function's value in a pr... |
| funfvima2 7175 | A function's value in an i... |
| funfvima2d 7176 | A function's value in a pr... |
| fnfvima 7177 | The function value of an o... |
| fnfvimad 7178 | A function's value belongs... |
| resfvresima 7179 | The value of the function ... |
| funfvima3 7180 | A class including a functi... |
| ralima 7181 | Universal quantification u... |
| rexima 7182 | Existential quantification... |
| reximaOLD 7183 | Obsolete version of ~ rexi... |
| ralimaOLD 7184 | Obsolete version of ~ rali... |
| fvclss 7185 | Upper bound for the class ... |
| elabrex 7186 | Elementhood in an image se... |
| elabrexg 7187 | Elementhood in an image se... |
| abrexco 7188 | Composition of two image m... |
| imaiun 7189 | The image of an indexed un... |
| imauni 7190 | The image of a union is th... |
| fniunfv 7191 | The indexed union of a fun... |
| funiunfv 7192 | The indexed union of a fun... |
| funiunfvf 7193 | The indexed union of a fun... |
| eluniima 7194 | Membership in the union of... |
| elunirn 7195 | Membership in the union of... |
| elunirnALT 7196 | Alternate proof of ~ eluni... |
| fnunirn 7197 | Membership in a union of s... |
| dff13 7198 | A one-to-one function in t... |
| dff13f 7199 | A one-to-one function in t... |
| f1veqaeq 7200 | If the values of a one-to-... |
| f1cofveqaeq 7201 | If the values of a composi... |
| f1cofveqaeqALT 7202 | Alternate proof of ~ f1cof... |
| dff14i 7203 | A one-to-one function maps... |
| 2f1fvneq 7204 | If two one-to-one function... |
| f1mpt 7205 | Express injection for a ma... |
| f1fveq 7206 | Equality of function value... |
| f1elima 7207 | Membership in the image of... |
| f1imass 7208 | Taking images under a one-... |
| f1imaeq 7209 | Taking images under a one-... |
| f1imapss 7210 | Taking images under a one-... |
| fpropnf1 7211 | A function, given by an un... |
| f1dom3fv3dif 7212 | The function values for a ... |
| f1dom3el3dif 7213 | The codomain of a 1-1 func... |
| dff14a 7214 | A one-to-one function in t... |
| dff14b 7215 | A one-to-one function in t... |
| f1ounsn 7216 | Extension of a bijection b... |
| f12dfv 7217 | A one-to-one function with... |
| f13dfv 7218 | A one-to-one function with... |
| dff1o6 7219 | A one-to-one onto function... |
| f1ocnvfv1 7220 | The converse value of the ... |
| f1ocnvfv2 7221 | The value of the converse ... |
| f1ocnvfv 7222 | Relationship between the v... |
| f1ocnvfvb 7223 | Relationship between the v... |
| nvof1o 7224 | An involution is a bijecti... |
| nvocnv 7225 | The converse of an involut... |
| f1cdmsn 7226 | If a one-to-one function w... |
| fsnex 7227 | Relate a function with a s... |
| f1prex 7228 | Relate a one-to-one functi... |
| f1ocnvdm 7229 | The value of the converse ... |
| f1ocnvfvrneq 7230 | If the values of a one-to-... |
| fcof1 7231 | An application is injectiv... |
| fcofo 7232 | An application is surjecti... |
| cbvfo 7233 | Change bound variable betw... |
| cbvexfo 7234 | Change bound variable betw... |
| cocan1 7235 | An injection is left-cance... |
| cocan2 7236 | A surjection is right-canc... |
| fcof1oinvd 7237 | Show that a function is th... |
| fcof1od 7238 | A function is bijective if... |
| 2fcoidinvd 7239 | Show that a function is th... |
| fcof1o 7240 | Show that two functions ar... |
| 2fvcoidd 7241 | Show that the composition ... |
| 2fvidf1od 7242 | A function is bijective if... |
| 2fvidinvd 7243 | Show that two functions ar... |
| foeqcnvco 7244 | Condition for function equ... |
| f1eqcocnv 7245 | Condition for function equ... |
| fveqf1o 7246 | Given a bijection ` F ` , ... |
| f1ocoima 7247 | The composition of two bij... |
| nf1const 7248 | A constant function from a... |
| nf1oconst 7249 | A constant function from a... |
| f1ofvswap 7250 | Swapping two values in a b... |
| fvf1pr 7251 | Values of a one-to-one fun... |
| fliftrel 7252 | ` F ` , a function lift, i... |
| fliftel 7253 | Elementhood in the relatio... |
| fliftel1 7254 | Elementhood in the relatio... |
| fliftcnv 7255 | Converse of the relation `... |
| fliftfun 7256 | The function ` F ` is the ... |
| fliftfund 7257 | The function ` F ` is the ... |
| fliftfuns 7258 | The function ` F ` is the ... |
| fliftf 7259 | The domain and range of th... |
| fliftval 7260 | The value of the function ... |
| isoeq1 7261 | Equality theorem for isomo... |
| isoeq2 7262 | Equality theorem for isomo... |
| isoeq3 7263 | Equality theorem for isomo... |
| isoeq4 7264 | Equality theorem for isomo... |
| isoeq5 7265 | Equality theorem for isomo... |
| nfiso 7266 | Bound-variable hypothesis ... |
| isof1o 7267 | An isomorphism is a one-to... |
| isof1oidb 7268 | A function is a bijection ... |
| isof1oopb 7269 | A function is a bijection ... |
| isorel 7270 | An isomorphism connects bi... |
| soisores 7271 | Express the condition of i... |
| soisoi 7272 | Infer isomorphism from one... |
| isoid 7273 | Identity law for isomorphi... |
| isocnv 7274 | Converse law for isomorphi... |
| isocnv2 7275 | Converse law for isomorphi... |
| isocnv3 7276 | Complementation law for is... |
| isores2 7277 | An isomorphism from one we... |
| isores1 7278 | An isomorphism from one we... |
| isores3 7279 | Induced isomorphism on a s... |
| isotr 7280 | Composition (transitive) l... |
| isomin 7281 | Isomorphisms preserve mini... |
| isoini 7282 | Isomorphisms preserve init... |
| isoini2 7283 | Isomorphisms are isomorphi... |
| isofrlem 7284 | Lemma for ~ isofr . (Cont... |
| isoselem 7285 | Lemma for ~ isose . (Cont... |
| isofr 7286 | An isomorphism preserves w... |
| isose 7287 | An isomorphism preserves s... |
| isofr2 7288 | A weak form of ~ isofr tha... |
| isopolem 7289 | Lemma for ~ isopo . (Cont... |
| isopo 7290 | An isomorphism preserves t... |
| isosolem 7291 | Lemma for ~ isoso . (Cont... |
| isoso 7292 | An isomorphism preserves t... |
| isowe 7293 | An isomorphism preserves t... |
| isowe2 7294 | A weak form of ~ isowe tha... |
| f1oiso 7295 | Any one-to-one onto functi... |
| f1oiso2 7296 | Any one-to-one onto functi... |
| f1owe 7297 | Well-ordering of isomorphi... |
| weniso 7298 | A set-like well-ordering h... |
| weisoeq 7299 | Thus, there is at most one... |
| weisoeq2 7300 | Thus, there is at most one... |
| knatar 7301 | The Knaster-Tarski theorem... |
| fvresval 7302 | The value of a restricted ... |
| funeldmb 7303 | If ` (/) ` is not part of ... |
| eqfunresadj 7304 | Law for adjoining an eleme... |
| eqfunressuc 7305 | Law for equality of restri... |
| fnssintima 7306 | Condition for subset of an... |
| imaeqsexvOLD 7307 | Obsolete version of ~ rexi... |
| imaeqsalvOLD 7308 | Obsolete version of ~ rali... |
| fnimasnd 7309 | The image of a function by... |
| canth 7310 | No set ` A ` is equinumero... |
| ncanth 7311 | Cantor's theorem fails for... |
| riotaeqdv 7314 | Formula-building deduction... |
| riotabidv 7315 | Formula-building deduction... |
| riotaeqbidv 7316 | Equality deduction for res... |
| riotaex 7317 | Restricted iota is a set. ... |
| riotav 7318 | An iota restricted to the ... |
| riotauni 7319 | Restricted iota in terms o... |
| nfriota1 7320 | The abstraction variable i... |
| nfriotadw 7321 | Deduction version of ~ nfr... |
| cbvriotaw 7322 | Change bound variable in a... |
| cbvriotavw 7323 | Change bound variable in a... |
| nfriotad 7324 | Deduction version of ~ nfr... |
| nfriota 7325 | A variable not free in a w... |
| cbvriota 7326 | Change bound variable in a... |
| cbvriotav 7327 | Change bound variable in a... |
| csbriota 7328 | Interchange class substitu... |
| riotacl2 7329 | Membership law for "the un... |
| riotacl 7330 | Closure of restricted iota... |
| riotasbc 7331 | Substitution law for descr... |
| riotabidva 7332 | Equivalent wff's yield equ... |
| riotabiia 7333 | Equivalent wff's yield equ... |
| riota1 7334 | Property of restricted iot... |
| riota1a 7335 | Property of iota. (Contri... |
| riota2df 7336 | A deduction version of ~ r... |
| riota2f 7337 | This theorem shows a condi... |
| riota2 7338 | This theorem shows a condi... |
| riotaeqimp 7339 | If two restricted iota des... |
| riotaprop 7340 | Properties of a restricted... |
| riota5f 7341 | A method for computing res... |
| riota5 7342 | A method for computing res... |
| riotass2 7343 | Restriction of a unique el... |
| riotass 7344 | Restriction of a unique el... |
| moriotass 7345 | Restriction of a unique el... |
| snriota 7346 | A restricted class abstrac... |
| riotaxfrd 7347 | Change the variable ` x ` ... |
| eusvobj2 7348 | Specify the same property ... |
| eusvobj1 7349 | Specify the same object in... |
| f1ofveu 7350 | There is one domain elemen... |
| f1ocnvfv3 7351 | Value of the converse of a... |
| riotaund 7352 | Restricted iota equals the... |
| riotassuni 7353 | The restricted iota class ... |
| riotaclb 7354 | Bidirectional closure of r... |
| riotarab 7355 | Restricted iota of a restr... |
| oveq 7362 | Equality theorem for opera... |
| oveq1 7363 | Equality theorem for opera... |
| oveq2 7364 | Equality theorem for opera... |
| oveq12 7365 | Equality theorem for opera... |
| oveq1i 7366 | Equality inference for ope... |
| oveq2i 7367 | Equality inference for ope... |
| oveq12i 7368 | Equality inference for ope... |
| oveqi 7369 | Equality inference for ope... |
| oveq123i 7370 | Equality inference for ope... |
| oveq1d 7371 | Equality deduction for ope... |
| oveq2d 7372 | Equality deduction for ope... |
| oveqd 7373 | Equality deduction for ope... |
| oveq12d 7374 | Equality deduction for ope... |
| oveqan12d 7375 | Equality deduction for ope... |
| oveqan12rd 7376 | Equality deduction for ope... |
| oveq123d 7377 | Equality deduction for ope... |
| fvoveq1d 7378 | Equality deduction for nes... |
| fvoveq1 7379 | Equality theorem for neste... |
| ovanraleqv 7380 | Equality theorem for a con... |
| imbrov2fvoveq 7381 | Equality theorem for neste... |
| ovrspc2v 7382 | If an operation value is a... |
| oveqrspc2v 7383 | Restricted specialization ... |
| oveqdr 7384 | Equality of two operations... |
| nfovd 7385 | Deduction version of bound... |
| nfov 7386 | Bound-variable hypothesis ... |
| oprabidw 7387 | The law of concretion. Sp... |
| oprabid 7388 | The law of concretion. Sp... |
| ovex 7389 | The result of an operation... |
| ovexi 7390 | The result of an operation... |
| ovexd 7391 | The result of an operation... |
| ovssunirn 7392 | The result of an operation... |
| 0ov 7393 | Operation value of the emp... |
| ovprc 7394 | The value of an operation ... |
| ovprc1 7395 | The value of an operation ... |
| ovprc2 7396 | The value of an operation ... |
| ovrcl 7397 | Reverse closure for an ope... |
| elfvov1 7398 | Utility theorem: reverse c... |
| elfvov2 7399 | Utility theorem: reverse c... |
| csbov123 7400 | Move class substitution in... |
| csbov 7401 | Move class substitution in... |
| csbov12g 7402 | Move class substitution in... |
| csbov1g 7403 | Move class substitution in... |
| csbov2g 7404 | Move class substitution in... |
| rspceov 7405 | A frequently used special ... |
| elovimad 7406 | Elementhood of the image s... |
| fnbrovb 7407 | Value of a binary operatio... |
| fnotovb 7408 | Equivalence of operation v... |
| opabbrex 7409 | A collection of ordered pa... |
| opabresex2 7410 | Restrictions of a collecti... |
| fvmptopab 7411 | The function value of a ma... |
| f1opr 7412 | Condition for an operation... |
| brfvopab 7413 | The classes involved in a ... |
| dfoprab2 7414 | Class abstraction for oper... |
| reloprab 7415 | An operation class abstrac... |
| oprabv 7416 | If a pair and a class are ... |
| nfoprab1 7417 | The abstraction variables ... |
| nfoprab2 7418 | The abstraction variables ... |
| nfoprab3 7419 | The abstraction variables ... |
| nfoprab 7420 | Bound-variable hypothesis ... |
| oprabbid 7421 | Equivalent wff's yield equ... |
| oprabbidv 7422 | Equivalent wff's yield equ... |
| oprabbii 7423 | Equivalent wff's yield equ... |
| ssoprab2 7424 | Equivalence of ordered pai... |
| ssoprab2b 7425 | Equivalence of ordered pai... |
| eqoprab2bw 7426 | Equivalence of ordered pai... |
| eqoprab2b 7427 | Equivalence of ordered pai... |
| mpoeq123 7428 | An equality theorem for th... |
| mpoeq12 7429 | An equality theorem for th... |
| mpoeq123dva 7430 | An equality deduction for ... |
| mpoeq123dv 7431 | An equality deduction for ... |
| mpoeq123i 7432 | An equality inference for ... |
| mpoeq3dva 7433 | Slightly more general equa... |
| mpoeq3ia 7434 | An equality inference for ... |
| mpoeq3dv 7435 | An equality deduction for ... |
| nfmpo1 7436 | Bound-variable hypothesis ... |
| nfmpo2 7437 | Bound-variable hypothesis ... |
| nfmpo 7438 | Bound-variable hypothesis ... |
| 0mpo0 7439 | A mapping operation with e... |
| mpo0v 7440 | A mapping operation with e... |
| mpo0 7441 | A mapping operation with e... |
| oprab4 7442 | Two ways to state the doma... |
| cbvoprab1 7443 | Rule used to change first ... |
| cbvoprab2 7444 | Change the second bound va... |
| cbvoprab12 7445 | Rule used to change first ... |
| cbvoprab12v 7446 | Rule used to change first ... |
| cbvoprab3 7447 | Rule used to change the th... |
| cbvoprab3v 7448 | Rule used to change the th... |
| cbvmpox 7449 | Rule to change the bound v... |
| cbvmpo 7450 | Rule to change the bound v... |
| cbvmpov 7451 | Rule to change the bound v... |
| elimdelov 7452 | Eliminate a hypothesis whi... |
| brif1 7453 | Move a relation inside and... |
| ovif 7454 | Move a conditional outside... |
| ovif2 7455 | Move a conditional outside... |
| ovif12 7456 | Move a conditional outside... |
| ifov 7457 | Move a conditional outside... |
| ifmpt2v 7458 | Move a conditional inside ... |
| dmoprab 7459 | The domain of an operation... |
| dmoprabss 7460 | The domain of an operation... |
| rnoprab 7461 | The range of an operation ... |
| rnoprab2 7462 | The range of a restricted ... |
| reldmoprab 7463 | The domain of an operation... |
| oprabss 7464 | Structure of an operation ... |
| eloprabga 7465 | The law of concretion for ... |
| eloprabg 7466 | The law of concretion for ... |
| ssoprab2i 7467 | Inference of operation cla... |
| mpov 7468 | Operation with universal d... |
| mpomptx 7469 | Express a two-argument fun... |
| mpompt 7470 | Express a two-argument fun... |
| mpodifsnif 7471 | A mapping with two argumen... |
| mposnif 7472 | A mapping with two argumen... |
| fconstmpo 7473 | Representation of a consta... |
| resoprab 7474 | Restriction of an operatio... |
| resoprab2 7475 | Restriction of an operator... |
| resmpo 7476 | Restriction of the mapping... |
| funoprabg 7477 | "At most one" is a suffici... |
| funoprab 7478 | "At most one" is a suffici... |
| fnoprabg 7479 | Functionality and domain o... |
| mpofun 7480 | The maps-to notation for a... |
| fnoprab 7481 | Functionality and domain o... |
| ffnov 7482 | An operation maps to a cla... |
| fovcld 7483 | Closure law for an operati... |
| fovcl 7484 | Closure law for an operati... |
| eqfnov 7485 | Equality of two operations... |
| eqfnov2 7486 | Two operators with the sam... |
| fnov 7487 | Representation of a functi... |
| mpo2eqb 7488 | Bidirectional equality the... |
| rnmpo 7489 | The range of an operation ... |
| reldmmpo 7490 | The domain of an operation... |
| elrnmpog 7491 | Membership in the range of... |
| elrnmpo 7492 | Membership in the range of... |
| elimampo 7493 | Membership in the image of... |
| elrnmpores 7494 | Membership in the range of... |
| ralrnmpo 7495 | A restricted quantifier ov... |
| rexrnmpo 7496 | A restricted quantifier ov... |
| ovid 7497 | The value of an operation ... |
| ovidig 7498 | The value of an operation ... |
| ovidi 7499 | The value of an operation ... |
| ov 7500 | The value of an operation ... |
| ovigg 7501 | The value of an operation ... |
| ovig 7502 | The value of an operation ... |
| ovmpt4g 7503 | Value of a function given ... |
| ovmpos 7504 | Value of a function given ... |
| ov2gf 7505 | The value of an operation ... |
| ovmpodxf 7506 | Value of an operation give... |
| ovmpodx 7507 | Value of an operation give... |
| ovmpod 7508 | Value of an operation give... |
| ovmpox 7509 | The value of an operation ... |
| ovmpoga 7510 | Value of an operation give... |
| ovmpoa 7511 | Value of an operation give... |
| ovmpodf 7512 | Alternate deduction versio... |
| ovmpodv 7513 | Alternate deduction versio... |
| ovmpodv2 7514 | Alternate deduction versio... |
| ovmpog 7515 | Value of an operation give... |
| ovmpo 7516 | Value of an operation give... |
| ovmpot 7517 | The value of an operation ... |
| fvmpopr2d 7518 | Value of an operation give... |
| ov3 7519 | The value of an operation ... |
| ov6g 7520 | The value of an operation ... |
| ovg 7521 | The value of an operation ... |
| ovres 7522 | The value of a restricted ... |
| ovresd 7523 | Lemma for converting metri... |
| oprres 7524 | The restriction of an oper... |
| oprssov 7525 | The value of a member of t... |
| fovcdm 7526 | An operation's value belon... |
| fovcdmda 7527 | An operation's value belon... |
| fovcdmd 7528 | An operation's value belon... |
| fnrnov 7529 | The range of an operation ... |
| foov 7530 | An onto mapping of an oper... |
| fnovrn 7531 | An operation's value belon... |
| ovelrn 7532 | A member of an operation's... |
| funimassov 7533 | Membership relation for th... |
| ovelimab 7534 | Operation value in an imag... |
| ovima0 7535 | An operation value is a me... |
| ovconst2 7536 | The value of a constant op... |
| oprssdm 7537 | Domain of closure of an op... |
| nssdmovg 7538 | The value of an operation ... |
| ndmovg 7539 | The value of an operation ... |
| ndmov 7540 | The value of an operation ... |
| ndmovcl 7541 | The closure of an operatio... |
| ndmovrcl 7542 | Reverse closure law, when ... |
| ndmovcom 7543 | Any operation is commutati... |
| ndmovass 7544 | Any operation is associati... |
| ndmovdistr 7545 | Any operation is distribut... |
| ndmovord 7546 | Elimination of redundant a... |
| ndmovordi 7547 | Elimination of redundant a... |
| caovclg 7548 | Convert an operation closu... |
| caovcld 7549 | Convert an operation closu... |
| caovcl 7550 | Convert an operation closu... |
| caovcomg 7551 | Convert an operation commu... |
| caovcomd 7552 | Convert an operation commu... |
| caovcom 7553 | Convert an operation commu... |
| caovassg 7554 | Convert an operation assoc... |
| caovassd 7555 | Convert an operation assoc... |
| caovass 7556 | Convert an operation assoc... |
| caovcang 7557 | Convert an operation cance... |
| caovcand 7558 | Convert an operation cance... |
| caovcanrd 7559 | Commute the arguments of a... |
| caovcan 7560 | Convert an operation cance... |
| caovordig 7561 | Convert an operation order... |
| caovordid 7562 | Convert an operation order... |
| caovordg 7563 | Convert an operation order... |
| caovordd 7564 | Convert an operation order... |
| caovord2d 7565 | Operation ordering law wit... |
| caovord3d 7566 | Ordering law. (Contribute... |
| caovord 7567 | Convert an operation order... |
| caovord2 7568 | Operation ordering law wit... |
| caovord3 7569 | Ordering law. (Contribute... |
| caovdig 7570 | Convert an operation distr... |
| caovdid 7571 | Convert an operation distr... |
| caovdir2d 7572 | Convert an operation distr... |
| caovdirg 7573 | Convert an operation rever... |
| caovdird 7574 | Convert an operation distr... |
| caovdi 7575 | Convert an operation distr... |
| caov32d 7576 | Rearrange arguments in a c... |
| caov12d 7577 | Rearrange arguments in a c... |
| caov31d 7578 | Rearrange arguments in a c... |
| caov13d 7579 | Rearrange arguments in a c... |
| caov4d 7580 | Rearrange arguments in a c... |
| caov411d 7581 | Rearrange arguments in a c... |
| caov42d 7582 | Rearrange arguments in a c... |
| caov32 7583 | Rearrange arguments in a c... |
| caov12 7584 | Rearrange arguments in a c... |
| caov31 7585 | Rearrange arguments in a c... |
| caov13 7586 | Rearrange arguments in a c... |
| caov4 7587 | Rearrange arguments in a c... |
| caov411 7588 | Rearrange arguments in a c... |
| caov42 7589 | Rearrange arguments in a c... |
| caovdir 7590 | Reverse distributive law. ... |
| caovdilem 7591 | Lemma used by real number ... |
| caovlem2 7592 | Lemma used in real number ... |
| caovmo 7593 | Uniqueness of inverse elem... |
| imaeqexov 7594 | Substitute an operation va... |
| imaeqalov 7595 | Substitute an operation va... |
| mpondm0 7596 | The value of an operation ... |
| elmpocl 7597 | If a two-parameter class i... |
| elmpocl1 7598 | If a two-parameter class i... |
| elmpocl2 7599 | If a two-parameter class i... |
| elovmpod 7600 | Utility lemma for two-para... |
| elovmpo 7601 | Utility lemma for two-para... |
| elovmporab 7602 | Implications for the value... |
| elovmporab1w 7603 | Implications for the value... |
| elovmporab1 7604 | Implications for the value... |
| 2mpo0 7605 | If the operation value of ... |
| relmptopab 7606 | Any function to sets of or... |
| f1ocnvd 7607 | Describe an implicit one-t... |
| f1od 7608 | Describe an implicit one-t... |
| f1ocnv2d 7609 | Describe an implicit one-t... |
| f1o2d 7610 | Describe an implicit one-t... |
| f1opw2 7611 | A one-to-one mapping induc... |
| f1opw 7612 | A one-to-one mapping induc... |
| elovmpt3imp 7613 | If the value of a function... |
| ovmpt3rab1 7614 | The value of an operation ... |
| ovmpt3rabdm 7615 | If the value of a function... |
| elovmpt3rab1 7616 | Implications for the value... |
| elovmpt3rab 7617 | Implications for the value... |
| ofeqd 7622 | Equality theorem for funct... |
| ofeq 7623 | Equality theorem for funct... |
| ofreq 7624 | Equality theorem for funct... |
| ofexg 7625 | A function operation restr... |
| nfof 7626 | Hypothesis builder for fun... |
| nfofr 7627 | Hypothesis builder for fun... |
| ofrfvalg 7628 | Value of a relation applie... |
| offval 7629 | Value of an operation appl... |
| ofrfval 7630 | Value of a relation applie... |
| ofval 7631 | Evaluate a function operat... |
| ofrval 7632 | Exhibit a function relatio... |
| offn 7633 | The function operation pro... |
| offun 7634 | The function operation pro... |
| offval2f 7635 | The function operation exp... |
| ofmresval 7636 | Value of a restriction of ... |
| fnfvof 7637 | Function value of a pointw... |
| off 7638 | The function operation pro... |
| ofres 7639 | Restrict the operands of a... |
| offval2 7640 | The function operation exp... |
| ofrfval2 7641 | The function relation acti... |
| offvalfv 7642 | The function operation exp... |
| ofmpteq 7643 | Value of a pointwise opera... |
| coof 7644 | The composition of a _homo... |
| ofco 7645 | The composition of a funct... |
| offveq 7646 | Convert an identity of the... |
| offveqb 7647 | Equivalent expressions for... |
| ofc1 7648 | Left operation by a consta... |
| ofc2 7649 | Right operation by a const... |
| ofc12 7650 | Function operation on two ... |
| caofref 7651 | Transfer a reflexive law t... |
| caofinvl 7652 | Transfer a left inverse la... |
| caofid0l 7653 | Transfer a left identity l... |
| caofid0r 7654 | Transfer a right identity ... |
| caofid1 7655 | Transfer a right absorptio... |
| caofid2 7656 | Transfer a right absorptio... |
| caofcom 7657 | Transfer a commutative law... |
| caofidlcan 7658 | Transfer a cancellation/id... |
| caofrss 7659 | Transfer a relation subset... |
| caofass 7660 | Transfer an associative la... |
| caoftrn 7661 | Transfer a transitivity la... |
| caofdi 7662 | Transfer a distributive la... |
| caofdir 7663 | Transfer a reverse distrib... |
| caonncan 7664 | Transfer ~ nncan -shaped l... |
| relrpss 7667 | The proper subset relation... |
| brrpssg 7668 | The proper subset relation... |
| brrpss 7669 | The proper subset relation... |
| porpss 7670 | Every class is partially o... |
| sorpss 7671 | Express strict ordering un... |
| sorpssi 7672 | Property of a chain of set... |
| sorpssun 7673 | A chain of sets is closed ... |
| sorpssin 7674 | A chain of sets is closed ... |
| sorpssuni 7675 | In a chain of sets, a maxi... |
| sorpssint 7676 | In a chain of sets, a mini... |
| sorpsscmpl 7677 | The componentwise compleme... |
| zfun 7679 | Axiom of Union expressed w... |
| axun2 7680 | A variant of the Axiom of ... |
| uniex2 7681 | The Axiom of Union using t... |
| vuniex 7682 | The union of a setvar is a... |
| uniexg 7683 | The ZF Axiom of Union in c... |
| uniex 7684 | The Axiom of Union in clas... |
| uniexd 7685 | Deduction version of the Z... |
| unexg 7686 | The union of two sets is a... |
| unex 7687 | The union of two sets is a... |
| unexOLD 7688 | Obsolete version of ~ unex... |
| tpex 7689 | An unordered triple of cla... |
| unexb 7690 | Existence of union is equi... |
| unexbOLD 7691 | Obsolete version of ~ unex... |
| unexgOLD 7692 | Obsolete version of ~ unex... |
| xpexg 7693 | The Cartesian product of t... |
| xpexd 7694 | The Cartesian product of t... |
| 3xpexg 7695 | The Cartesian product of t... |
| xpex 7696 | The Cartesian product of t... |
| unexd 7697 | The union of two sets is a... |
| sqxpexg 7698 | The Cartesian square of a ... |
| abnexg 7699 | Sufficient condition for a... |
| abnex 7700 | Sufficient condition for a... |
| snnex 7701 | The class of all singleton... |
| pwnex 7702 | The class of all power set... |
| difex2 7703 | If the subtrahend of a cla... |
| difsnexi 7704 | If the difference of a cla... |
| uniuni 7705 | Expression for double unio... |
| uniexr 7706 | Converse of the Axiom of U... |
| uniexb 7707 | The Axiom of Union and its... |
| pwexr 7708 | Converse of the Axiom of P... |
| pwexb 7709 | The Axiom of Power Sets an... |
| elpwpwel 7710 | A class belongs to a doubl... |
| eldifpw 7711 | Membership in a power clas... |
| elpwun 7712 | Membership in the power cl... |
| pwuncl 7713 | Power classes are closed u... |
| iunpw 7714 | An indexed union of a powe... |
| fr3nr 7715 | A well-founded relation ha... |
| epne3 7716 | A well-founded class conta... |
| dfwe2 7717 | Alternate definition of we... |
| epweon 7718 | The membership relation we... |
| epweonALT 7719 | Alternate proof of ~ epweo... |
| ordon 7720 | The class of all ordinal n... |
| onprc 7721 | No set contains all ordina... |
| ssorduni 7722 | The union of a class of or... |
| ssonuni 7723 | The union of a set of ordi... |
| ssonunii 7724 | The union of a set of ordi... |
| ordeleqon 7725 | A way to express the ordin... |
| ordsson 7726 | Any ordinal class is a sub... |
| dford5 7727 | A class is ordinal iff it ... |
| onss 7728 | An ordinal number is a sub... |
| predon 7729 | The predecessor of an ordi... |
| ssonprc 7730 | Two ways of saying a class... |
| onuni 7731 | The union of an ordinal nu... |
| orduni 7732 | The union of an ordinal cl... |
| onint 7733 | The intersection (infimum)... |
| onint0 7734 | The intersection of a clas... |
| onssmin 7735 | A nonempty class of ordina... |
| onminesb 7736 | If a property is true for ... |
| onminsb 7737 | If a property is true for ... |
| oninton 7738 | The intersection of a none... |
| onintrab 7739 | The intersection of a clas... |
| onintrab2 7740 | An existence condition equ... |
| onnmin 7741 | No member of a set of ordi... |
| onnminsb 7742 | An ordinal number smaller ... |
| oneqmin 7743 | A way to show that an ordi... |
| uniordint 7744 | The union of a set of ordi... |
| onminex 7745 | If a wff is true for an or... |
| sucon 7746 | The class of all ordinal n... |
| sucexb 7747 | A successor exists iff its... |
| sucexg 7748 | The successor of a set is ... |
| sucex 7749 | The successor of a set is ... |
| onmindif2 7750 | The minimum of a class of ... |
| ordsuci 7751 | The successor of an ordina... |
| sucexeloni 7752 | If the successor of an ord... |
| onsuc 7753 | The successor of an ordina... |
| ordsuc 7754 | A class is ordinal if and ... |
| ordpwsuc 7755 | The collection of ordinals... |
| onpwsuc 7756 | The collection of ordinal ... |
| onsucb 7757 | A class is an ordinal numb... |
| ordsucss 7758 | The successor of an elemen... |
| onpsssuc 7759 | An ordinal number is a pro... |
| ordelsuc 7760 | A set belongs to an ordina... |
| onsucmin 7761 | The successor of an ordina... |
| ordsucelsuc 7762 | Membership is inherited by... |
| ordsucsssuc 7763 | The subclass relationship ... |
| ordsucuniel 7764 | Given an element ` A ` of ... |
| ordsucun 7765 | The successor of the maxim... |
| ordunpr 7766 | The maximum of two ordinal... |
| ordunel 7767 | The maximum of two ordinal... |
| onsucuni 7768 | A class of ordinal numbers... |
| ordsucuni 7769 | An ordinal class is a subc... |
| orduniorsuc 7770 | An ordinal class is either... |
| unon 7771 | The class of all ordinal n... |
| ordunisuc 7772 | An ordinal class is equal ... |
| orduniss2 7773 | The union of the ordinal s... |
| onsucuni2 7774 | A successor ordinal is the... |
| 0elsuc 7775 | The successor of an ordina... |
| limon 7776 | The class of ordinal numbe... |
| onuniorsuc 7777 | An ordinal number is eithe... |
| onssi 7778 | An ordinal number is a sub... |
| onsuci 7779 | The successor of an ordina... |
| onuninsuci 7780 | An ordinal is equal to its... |
| onsucssi 7781 | A set belongs to an ordina... |
| nlimsucg 7782 | A successor is not a limit... |
| orduninsuc 7783 | An ordinal class is equal ... |
| ordunisuc2 7784 | An ordinal equal to its un... |
| ordzsl 7785 | An ordinal is zero, a succ... |
| onzsl 7786 | An ordinal number is zero,... |
| dflim3 7787 | An alternate definition of... |
| dflim4 7788 | An alternate definition of... |
| limsuc 7789 | The successor of a member ... |
| limsssuc 7790 | A class includes a limit o... |
| nlimon 7791 | Two ways to express the cl... |
| limuni3 7792 | The union of a nonempty cl... |
| tfi 7793 | The Principle of Transfini... |
| tfisg 7794 | A closed form of ~ tfis . ... |
| tfis 7795 | Transfinite Induction Sche... |
| tfis2f 7796 | Transfinite Induction Sche... |
| tfis2 7797 | Transfinite Induction Sche... |
| tfis3 7798 | Transfinite Induction Sche... |
| tfisi 7799 | A transfinite induction sc... |
| tfinds 7800 | Principle of Transfinite I... |
| tfindsg 7801 | Transfinite Induction (inf... |
| tfindsg2 7802 | Transfinite Induction (inf... |
| tfindes 7803 | Transfinite Induction with... |
| tfinds2 7804 | Transfinite Induction (inf... |
| tfinds3 7805 | Principle of Transfinite I... |
| dfom2 7808 | An alternate definition of... |
| elom 7809 | Membership in omega. The ... |
| omsson 7810 | Omega is a subset of ` On ... |
| limomss 7811 | The class of natural numbe... |
| nnon 7812 | A natural number is an ord... |
| nnoni 7813 | A natural number is an ord... |
| nnord 7814 | A natural number is ordina... |
| trom 7815 | The class of finite ordina... |
| ordom 7816 | The class of finite ordina... |
| elnn 7817 | A member of a natural numb... |
| omon 7818 | The class of natural numbe... |
| omelon2 7819 | Omega is an ordinal number... |
| nnlim 7820 | A natural number is not a ... |
| omssnlim 7821 | The class of natural numbe... |
| limom 7822 | Omega is a limit ordinal. ... |
| peano2b 7823 | A class belongs to omega i... |
| nnsuc 7824 | A nonzero natural number i... |
| omsucne 7825 | A natural number is not th... |
| ssnlim 7826 | An ordinal subclass of non... |
| omsinds 7827 | Strong (or "total") induct... |
| omun 7828 | The union of two finite or... |
| peano1 7829 | Zero is a natural number. ... |
| peano2 7830 | The successor of any natur... |
| peano3 7831 | The successor of any natur... |
| peano4 7832 | Two natural numbers are eq... |
| peano5 7833 | The induction postulate: a... |
| nn0suc 7834 | A natural number is either... |
| find 7835 | The Principle of Finite In... |
| finds 7836 | Principle of Finite Induct... |
| findsg 7837 | Principle of Finite Induct... |
| finds2 7838 | Principle of Finite Induct... |
| finds1 7839 | Principle of Finite Induct... |
| findes 7840 | Finite induction with expl... |
| dmexg 7841 | The domain of a set is a s... |
| rnexg 7842 | The range of a set is a se... |
| dmexd 7843 | The domain of a set is a s... |
| fndmexd 7844 | If a function is a set, it... |
| dmfex 7845 | If a mapping is a set, its... |
| fndmexb 7846 | The domain of a function i... |
| fdmexb 7847 | The domain of a function i... |
| dmfexALT 7848 | Alternate proof of ~ dmfex... |
| dmex 7849 | The domain of a set is a s... |
| rnex 7850 | The range of a set is a se... |
| iprc 7851 | The identity function is a... |
| resiexg 7852 | The existence of a restric... |
| imaexg 7853 | The image of a set is a se... |
| imaex 7854 | The image of a set is a se... |
| rnexd 7855 | The range of a set is a se... |
| imaexd 7856 | The image of a set is a se... |
| exse2 7857 | Any set relation is set-li... |
| xpexr 7858 | If a Cartesian product is ... |
| xpexr2 7859 | If a nonempty Cartesian pr... |
| xpexcnv 7860 | A condition where the conv... |
| soex 7861 | If the relation in a stric... |
| elxp4 7862 | Membership in a Cartesian ... |
| elxp5 7863 | Membership in a Cartesian ... |
| cnvexg 7864 | The converse of a set is a... |
| cnvex 7865 | The converse of a set is a... |
| relcnvexb 7866 | A relation is a set iff it... |
| f1oexrnex 7867 | If the range of a 1-1 onto... |
| f1oexbi 7868 | There is a one-to-one onto... |
| coexg 7869 | The composition of two set... |
| coex 7870 | The composition of two set... |
| coexd 7871 | The composition of two set... |
| funcnvuni 7872 | The union of a chain (with... |
| fun11uni 7873 | The union of a chain (with... |
| resf1extb 7874 | Extension of an injection ... |
| resf1ext2b 7875 | Extension of an injection ... |
| fex2 7876 | A function with bounded do... |
| fabexd 7877 | Existence of a set of func... |
| fabexg 7878 | Existence of a set of func... |
| fabexgOLD 7879 | Obsolete version of ~ fabe... |
| fabex 7880 | Existence of a set of func... |
| mapex 7881 | The class of all functions... |
| f1oabexg 7882 | The class of all 1-1-onto ... |
| f1oabexgOLD 7883 | Obsolete version of ~ f1oa... |
| fiunlem 7884 | Lemma for ~ fiun and ~ f1i... |
| fiun 7885 | The union of a chain (with... |
| f1iun 7886 | The union of a chain (with... |
| fviunfun 7887 | The function value of an i... |
| ffoss 7888 | Relationship between a map... |
| f11o 7889 | Relationship between one-t... |
| resfunexgALT 7890 | Alternate proof of ~ resfu... |
| cofunexg 7891 | Existence of a composition... |
| cofunex2g 7892 | Existence of a composition... |
| fnexALT 7893 | Alternate proof of ~ fnex ... |
| funexw 7894 | Weak version of ~ funex th... |
| mptexw 7895 | Weak version of ~ mptex th... |
| funrnex 7896 | If the domain of a functio... |
| zfrep6 7897 | A version of the Axiom of ... |
| focdmex 7898 | If the domain of an onto f... |
| f1dmex 7899 | If the codomain of a one-t... |
| f1ovv 7900 | The codomain/range of a 1-... |
| fvclex 7901 | Existence of the class of ... |
| fvresex 7902 | Existence of the class of ... |
| abrexexg 7903 | Existence of a class abstr... |
| abrexex 7904 | Existence of a class abstr... |
| iunexg 7905 | The existence of an indexe... |
| abrexex2g 7906 | Existence of an existentia... |
| opabex3d 7907 | Existence of an ordered pa... |
| opabex3rd 7908 | Existence of an ordered pa... |
| opabex3 7909 | Existence of an ordered pa... |
| iunex 7910 | The existence of an indexe... |
| abrexex2 7911 | Existence of an existentia... |
| abexssex 7912 | Existence of a class abstr... |
| abexex 7913 | A condition where a class ... |
| f1oweALT 7914 | Alternate proof of ~ f1owe... |
| wemoiso 7915 | Thus, there is at most one... |
| wemoiso2 7916 | Thus, there is at most one... |
| oprabexd 7917 | Existence of an operator a... |
| oprabex 7918 | Existence of an operation ... |
| oprabex3 7919 | Existence of an operation ... |
| oprabrexex2 7920 | Existence of an existentia... |
| ab2rexex 7921 | Existence of a class abstr... |
| ab2rexex2 7922 | Existence of an existentia... |
| xpexgALT 7923 | Alternate proof of ~ xpexg... |
| offval3 7924 | General value of ` ( F oF ... |
| offres 7925 | Pointwise combination comm... |
| ofmres 7926 | Equivalent expressions for... |
| ofmresex 7927 | Existence of a restriction... |
| mptcnfimad 7928 | The converse of a mapping ... |
| 1stval 7933 | The value of the function ... |
| 2ndval 7934 | The value of the function ... |
| 1stnpr 7935 | Value of the first-member ... |
| 2ndnpr 7936 | Value of the second-member... |
| 1st0 7937 | The value of the first-mem... |
| 2nd0 7938 | The value of the second-me... |
| op1st 7939 | Extract the first member o... |
| op2nd 7940 | Extract the second member ... |
| op1std 7941 | Extract the first member o... |
| op2ndd 7942 | Extract the second member ... |
| op1stg 7943 | Extract the first member o... |
| op2ndg 7944 | Extract the second member ... |
| ot1stg 7945 | Extract the first member o... |
| ot2ndg 7946 | Extract the second member ... |
| ot3rdg 7947 | Extract the third member o... |
| 1stval2 7948 | Alternate value of the fun... |
| 2ndval2 7949 | Alternate value of the fun... |
| oteqimp 7950 | The components of an order... |
| fo1st 7951 | The ` 1st ` function maps ... |
| fo2nd 7952 | The ` 2nd ` function maps ... |
| br1steqg 7953 | Uniqueness condition for t... |
| br2ndeqg 7954 | Uniqueness condition for t... |
| f1stres 7955 | Mapping of a restriction o... |
| f2ndres 7956 | Mapping of a restriction o... |
| fo1stres 7957 | Onto mapping of a restrict... |
| fo2ndres 7958 | Onto mapping of a restrict... |
| 1st2val 7959 | Value of an alternate defi... |
| 2nd2val 7960 | Value of an alternate defi... |
| 1stcof 7961 | Composition of the first m... |
| 2ndcof 7962 | Composition of the second ... |
| xp1st 7963 | Location of the first elem... |
| xp2nd 7964 | Location of the second ele... |
| elxp6 7965 | Membership in a Cartesian ... |
| elxp7 7966 | Membership in a Cartesian ... |
| eqopi 7967 | Equality with an ordered p... |
| xp2 7968 | Representation of Cartesia... |
| unielxp 7969 | The membership relation fo... |
| 1st2nd2 7970 | Reconstruction of a member... |
| 1st2ndb 7971 | Reconstruction of an order... |
| xpopth 7972 | An ordered pair theorem fo... |
| eqop 7973 | Two ways to express equali... |
| eqop2 7974 | Two ways to express equali... |
| op1steq 7975 | Two ways of expressing tha... |
| opreuopreu 7976 | There is a unique ordered ... |
| el2xptp 7977 | A member of a nested Carte... |
| el2xptp0 7978 | A member of a nested Carte... |
| el2xpss 7979 | Version of ~ elrel for tri... |
| 2nd1st 7980 | Swap the members of an ord... |
| 1st2nd 7981 | Reconstruction of a member... |
| 1stdm 7982 | The first ordered pair com... |
| 2ndrn 7983 | The second ordered pair co... |
| 1st2ndbr 7984 | Express an element of a re... |
| releldm2 7985 | Two ways of expressing mem... |
| reldm 7986 | An expression for the doma... |
| releldmdifi 7987 | One way of expressing memb... |
| funfv1st2nd 7988 | The function value for the... |
| funelss 7989 | If the first component of ... |
| funeldmdif 7990 | Two ways of expressing mem... |
| sbcopeq1a 7991 | Equality theorem for subst... |
| csbopeq1a 7992 | Equality theorem for subst... |
| sbcoteq1a 7993 | Equality theorem for subst... |
| dfopab2 7994 | A way to define an ordered... |
| dfoprab3s 7995 | A way to define an operati... |
| dfoprab3 7996 | Operation class abstractio... |
| dfoprab4 7997 | Operation class abstractio... |
| dfoprab4f 7998 | Operation class abstractio... |
| opabex2 7999 | Condition for an operation... |
| opabn1stprc 8000 | An ordered-pair class abst... |
| opiota 8001 | The property of a uniquely... |
| cnvoprab 8002 | The converse of a class ab... |
| dfxp3 8003 | Define the Cartesian produ... |
| elopabi 8004 | A consequence of membershi... |
| eloprabi 8005 | A consequence of membershi... |
| mpomptsx 8006 | Express a two-argument fun... |
| mpompts 8007 | Express a two-argument fun... |
| dmmpossx 8008 | The domain of a mapping is... |
| fmpox 8009 | Functionality, domain and ... |
| fmpo 8010 | Functionality, domain and ... |
| fnmpo 8011 | Functionality and domain o... |
| fnmpoi 8012 | Functionality and domain o... |
| dmmpo 8013 | Domain of a class given by... |
| ovmpoelrn 8014 | An operation's value belon... |
| dmmpoga 8015 | Domain of an operation giv... |
| dmmpog 8016 | Domain of an operation giv... |
| mpoexxg 8017 | Existence of an operation ... |
| mpoexg 8018 | Existence of an operation ... |
| mpoexga 8019 | If the domain of an operat... |
| mpoexw 8020 | Weak version of ~ mpoex th... |
| mpoex 8021 | If the domain of an operat... |
| mptmpoopabbrd 8022 | The operation value of a f... |
| mptmpoopabbrdOLD 8023 | Obsolete version of ~ mptm... |
| mptmpoopabovd 8024 | The operation value of a f... |
| el2mpocsbcl 8025 | If the operation value of ... |
| el2mpocl 8026 | If the operation value of ... |
| fnmpoovd 8027 | A function with a Cartesia... |
| offval22 8028 | The function operation exp... |
| brovpreldm 8029 | If a binary relation holds... |
| bropopvvv 8030 | If a binary relation holds... |
| bropfvvvvlem 8031 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8032 | If a binary relation holds... |
| ovmptss 8033 | If all the values of the m... |
| relmpoopab 8034 | Any function to sets of or... |
| fmpoco 8035 | Composition of two functio... |
| oprabco 8036 | Composition of a function ... |
| oprab2co 8037 | Composition of operator ab... |
| df1st2 8038 | An alternate possible defi... |
| df2nd2 8039 | An alternate possible defi... |
| 1stconst 8040 | The mapping of a restricti... |
| 2ndconst 8041 | The mapping of a restricti... |
| dfmpo 8042 | Alternate definition for t... |
| mposn 8043 | An operation (in maps-to n... |
| curry1 8044 | Composition with ` ``' ( 2... |
| curry1val 8045 | The value of a curried fun... |
| curry1f 8046 | Functionality of a curried... |
| curry2 8047 | Composition with ` ``' ( 1... |
| curry2f 8048 | Functionality of a curried... |
| curry2val 8049 | The value of a curried fun... |
| cnvf1olem 8050 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8051 | Describe a function that m... |
| fparlem1 8052 | Lemma for ~ fpar . (Contr... |
| fparlem2 8053 | Lemma for ~ fpar . (Contr... |
| fparlem3 8054 | Lemma for ~ fpar . (Contr... |
| fparlem4 8055 | Lemma for ~ fpar . (Contr... |
| fpar 8056 | Merge two functions in par... |
| fsplit 8057 | A function that can be use... |
| fsplitfpar 8058 | Merge two functions with a... |
| offsplitfpar 8059 | Express the function opera... |
| f2ndf 8060 | The ` 2nd ` (second compon... |
| fo2ndf 8061 | The ` 2nd ` (second compon... |
| f1o2ndf1 8062 | The ` 2nd ` (second compon... |
| opco1 8063 | Value of an operation prec... |
| opco2 8064 | Value of an operation prec... |
| opco1i 8065 | Inference form of ~ opco1 ... |
| frxp 8066 | A lexicographical ordering... |
| xporderlem 8067 | Lemma for lexicographical ... |
| poxp 8068 | A lexicographical ordering... |
| soxp 8069 | A lexicographical ordering... |
| wexp 8070 | A lexicographical ordering... |
| fnwelem 8071 | Lemma for ~ fnwe . (Contr... |
| fnwe 8072 | A variant on lexicographic... |
| fnse 8073 | Condition for the well-ord... |
| fvproj 8074 | Value of a function on ord... |
| fimaproj 8075 | Image of a cartesian produ... |
| ralxpes 8076 | A version of ~ ralxp with ... |
| ralxp3f 8077 | Restricted for all over a ... |
| ralxp3 8078 | Restricted for all over a ... |
| ralxp3es 8079 | Restricted for-all over a ... |
| frpoins3xpg 8080 | Special case of founded pa... |
| frpoins3xp3g 8081 | Special case of founded pa... |
| xpord2lem 8082 | Lemma for Cartesian produc... |
| poxp2 8083 | Another way of partially o... |
| frxp2 8084 | Another way of giving a we... |
| xpord2pred 8085 | Calculate the predecessor ... |
| sexp2 8086 | Condition for the relation... |
| xpord2indlem 8087 | Induction over the Cartesi... |
| xpord2ind 8088 | Induction over the Cartesi... |
| xpord3lem 8089 | Lemma for triple ordering.... |
| poxp3 8090 | Triple Cartesian product p... |
| frxp3 8091 | Give well-foundedness over... |
| xpord3pred 8092 | Calculate the predecsessor... |
| sexp3 8093 | Show that the triple order... |
| xpord3inddlem 8094 | Induction over the triple ... |
| xpord3indd 8095 | Induction over the triple ... |
| xpord3ind 8096 | Induction over the triple ... |
| orderseqlem 8097 | Lemma for ~ poseq and ~ so... |
| poseq 8098 | A partial ordering of ordi... |
| soseq 8099 | A linear ordering of ordin... |
| suppval 8102 | The value of the operation... |
| supp0prc 8103 | The support of a class is ... |
| suppvalbr 8104 | The value of the operation... |
| supp0 8105 | The support of the empty s... |
| suppval1 8106 | The value of the operation... |
| suppvalfng 8107 | The value of the operation... |
| suppvalfn 8108 | The value of the operation... |
| elsuppfng 8109 | An element of the support ... |
| elsuppfn 8110 | An element of the support ... |
| fvdifsupp 8111 | Function value is zero out... |
| cnvimadfsn 8112 | The support of functions "... |
| suppimacnvss 8113 | The support of functions "... |
| suppimacnv 8114 | Support sets of functions ... |
| fsuppeq 8115 | Two ways of writing the su... |
| fsuppeqg 8116 | Version of ~ fsuppeq avoid... |
| suppssdm 8117 | The support of a function ... |
| suppsnop 8118 | The support of a singleton... |
| snopsuppss 8119 | The support of a singleton... |
| fvn0elsupp 8120 | If the function value for ... |
| fvn0elsuppb 8121 | The function value for a g... |
| rexsupp 8122 | Existential quantification... |
| ressuppss 8123 | The support of the restric... |
| suppun 8124 | The support of a class/fun... |
| ressuppssdif 8125 | The support of the restric... |
| mptsuppdifd 8126 | The support of a function ... |
| mptsuppd 8127 | The support of a function ... |
| extmptsuppeq 8128 | The support of an extended... |
| suppfnss 8129 | The support of a function ... |
| funsssuppss 8130 | The support of a function ... |
| fnsuppres 8131 | Two ways to express restri... |
| fnsuppeq0 8132 | The support of a function ... |
| fczsupp0 8133 | The support of a constant ... |
| suppss 8134 | Show that the support of a... |
| suppssr 8135 | A function is zero outside... |
| suppssrg 8136 | A function is zero outside... |
| suppssov1 8137 | Formula building theorem f... |
| suppssov2 8138 | Formula building theorem f... |
| suppssof1 8139 | Formula building theorem f... |
| suppss2 8140 | Show that the support of a... |
| suppsssn 8141 | Show that the support of a... |
| suppssfv 8142 | Formula building theorem f... |
| suppofssd 8143 | Condition for the support ... |
| suppofss1d 8144 | Condition for the support ... |
| suppofss2d 8145 | Condition for the support ... |
| suppco 8146 | The support of the composi... |
| suppcoss 8147 | The support of the composi... |
| supp0cosupp0 8148 | The support of the composi... |
| imacosupp 8149 | The image of the support o... |
| opeliunxp2f 8150 | Membership in a union of C... |
| mpoxeldm 8151 | If there is an element of ... |
| mpoxneldm 8152 | If the first argument of a... |
| mpoxopn0yelv 8153 | If there is an element of ... |
| mpoxopynvov0g 8154 | If the second argument of ... |
| mpoxopxnop0 8155 | If the first argument of a... |
| mpoxopx0ov0 8156 | If the first argument of a... |
| mpoxopxprcov0 8157 | If the components of the f... |
| mpoxopynvov0 8158 | If the second argument of ... |
| mpoxopoveq 8159 | Value of an operation give... |
| mpoxopovel 8160 | Element of the value of an... |
| mpoxopoveqd 8161 | Value of an operation give... |
| brovex 8162 | A binary relation of the v... |
| brovmpoex 8163 | A binary relation of the v... |
| sprmpod 8164 | The extension of a binary ... |
| tposss 8167 | Subset theorem for transpo... |
| tposeq 8168 | Equality theorem for trans... |
| tposeqd 8169 | Equality theorem for trans... |
| tposssxp 8170 | The transposition is a sub... |
| reltpos 8171 | The transposition is a rel... |
| brtpos2 8172 | Value of the transposition... |
| brtpos0 8173 | The behavior of ` tpos ` w... |
| reldmtpos 8174 | Necessary and sufficient c... |
| brtpos 8175 | The transposition swaps ar... |
| ottpos 8176 | The transposition swaps th... |
| relbrtpos 8177 | The transposition swaps ar... |
| dmtpos 8178 | The domain of ` tpos F ` w... |
| rntpos 8179 | The range of ` tpos F ` wh... |
| tposexg 8180 | The transposition of a set... |
| ovtpos 8181 | The transposition swaps th... |
| tposfun 8182 | The transposition of a fun... |
| dftpos2 8183 | Alternate definition of ` ... |
| dftpos3 8184 | Alternate definition of ` ... |
| dftpos4 8185 | Alternate definition of ` ... |
| tpostpos 8186 | Value of the double transp... |
| tpostpos2 8187 | Value of the double transp... |
| tposfn2 8188 | The domain of a transposit... |
| tposfo2 8189 | Condition for a surjective... |
| tposf2 8190 | The domain and codomain of... |
| tposf12 8191 | Condition for an injective... |
| tposf1o2 8192 | Condition of a bijective t... |
| tposfo 8193 | The domain and codomain/ra... |
| tposf 8194 | The domain and codomain of... |
| tposfn 8195 | Functionality of a transpo... |
| tpos0 8196 | Transposition of the empty... |
| tposco 8197 | Transposition of a composi... |
| tpossym 8198 | Two ways to say a function... |
| tposeqi 8199 | Equality theorem for trans... |
| tposex 8200 | A transposition is a set. ... |
| nftpos 8201 | Hypothesis builder for tra... |
| tposoprab 8202 | Transposition of a class o... |
| tposmpo 8203 | Transposition of a two-arg... |
| tposconst 8204 | The transposition of a con... |
| mpocurryd 8209 | The currying of an operati... |
| mpocurryvald 8210 | The value of a curried ope... |
| fvmpocurryd 8211 | The value of the value of ... |
| pwuninel2 8214 | Proof of ~ pwuninel under ... |
| pwuninel 8215 | The powerclass of the unio... |
| undefval 8216 | Value of the undefined val... |
| undefnel2 8217 | The undefined value genera... |
| undefnel 8218 | The undefined value genera... |
| undefne0 8219 | The undefined value genera... |
| frecseq123 8222 | Equality theorem for the w... |
| nffrecs 8223 | Bound-variable hypothesis ... |
| csbfrecsg 8224 | Move class substitution in... |
| fpr3g 8225 | Functions defined by well-... |
| frrlem1 8226 | Lemma for well-founded rec... |
| frrlem2 8227 | Lemma for well-founded rec... |
| frrlem3 8228 | Lemma for well-founded rec... |
| frrlem4 8229 | Lemma for well-founded rec... |
| frrlem5 8230 | Lemma for well-founded rec... |
| frrlem6 8231 | Lemma for well-founded rec... |
| frrlem7 8232 | Lemma for well-founded rec... |
| frrlem8 8233 | Lemma for well-founded rec... |
| frrlem9 8234 | Lemma for well-founded rec... |
| frrlem10 8235 | Lemma for well-founded rec... |
| frrlem11 8236 | Lemma for well-founded rec... |
| frrlem12 8237 | Lemma for well-founded rec... |
| frrlem13 8238 | Lemma for well-founded rec... |
| frrlem14 8239 | Lemma for well-founded rec... |
| fprlem1 8240 | Lemma for well-founded rec... |
| fprlem2 8241 | Lemma for well-founded rec... |
| fpr2a 8242 | Weak version of ~ fpr2 whi... |
| fpr1 8243 | Law of well-founded recurs... |
| fpr2 8244 | Law of well-founded recurs... |
| fpr3 8245 | Law of well-founded recurs... |
| frrrel 8246 | Show without using the axi... |
| frrdmss 8247 | Show without using the axi... |
| frrdmcl 8248 | Show without using the axi... |
| fprfung 8249 | A "function" defined by we... |
| fprresex 8250 | The restriction of a funct... |
| wrecseq123 8253 | General equality theorem f... |
| nfwrecs 8254 | Bound-variable hypothesis ... |
| wrecseq1 8255 | Equality theorem for the w... |
| wrecseq2 8256 | Equality theorem for the w... |
| wrecseq3 8257 | Equality theorem for the w... |
| csbwrecsg 8258 | Move class substitution in... |
| wfr3g 8259 | Functions defined by well-... |
| wfrrel 8260 | The well-ordered recursion... |
| wfrdmss 8261 | The domain of the well-ord... |
| wfrdmcl 8262 | The predecessor class of a... |
| wfrfun 8263 | The "function" generated b... |
| wfrresex 8264 | Show without using the axi... |
| wfr2a 8265 | A weak version of ~ wfr2 w... |
| wfr1 8266 | The Principle of Well-Orde... |
| wfr2 8267 | The Principle of Well-Orde... |
| wfr3 8268 | The principle of Well-Orde... |
| iunon 8269 | The indexed union of a set... |
| iinon 8270 | The nonempty indexed inter... |
| onfununi 8271 | A property of functions on... |
| onovuni 8272 | A variant of ~ onfununi fo... |
| onoviun 8273 | A variant of ~ onovuni wit... |
| onnseq 8274 | There are no length ` _om ... |
| dfsmo2 8277 | Alternate definition of a ... |
| issmo 8278 | Conditions for which ` A `... |
| issmo2 8279 | Alternate definition of a ... |
| smoeq 8280 | Equality theorem for stric... |
| smodm 8281 | The domain of a strictly m... |
| smores 8282 | A strictly monotone functi... |
| smores3 8283 | A strictly monotone functi... |
| smores2 8284 | A strictly monotone ordina... |
| smodm2 8285 | The domain of a strictly m... |
| smofvon2 8286 | The function values of a s... |
| iordsmo 8287 | The identity relation rest... |
| smo0 8288 | The null set is a strictly... |
| smofvon 8289 | If ` B ` is a strictly mon... |
| smoel 8290 | If ` x ` is less than ` y ... |
| smoiun 8291 | The value of a strictly mo... |
| smoiso 8292 | If ` F ` is an isomorphism... |
| smoel2 8293 | A strictly monotone ordina... |
| smo11 8294 | A strictly monotone ordina... |
| smoord 8295 | A strictly monotone ordina... |
| smoword 8296 | A strictly monotone ordina... |
| smogt 8297 | A strictly monotone ordina... |
| smocdmdom 8298 | The codomain of a strictly... |
| smoiso2 8299 | The strictly monotone ordi... |
| dfrecs3 8302 | The old definition of tran... |
| recseq 8303 | Equality theorem for ` rec... |
| nfrecs 8304 | Bound-variable hypothesis ... |
| tfrlem1 8305 | A technical lemma for tran... |
| tfrlem3a 8306 | Lemma for transfinite recu... |
| tfrlem3 8307 | Lemma for transfinite recu... |
| tfrlem4 8308 | Lemma for transfinite recu... |
| tfrlem5 8309 | Lemma for transfinite recu... |
| recsfval 8310 | Lemma for transfinite recu... |
| tfrlem6 8311 | Lemma for transfinite recu... |
| tfrlem7 8312 | Lemma for transfinite recu... |
| tfrlem8 8313 | Lemma for transfinite recu... |
| tfrlem9 8314 | Lemma for transfinite recu... |
| tfrlem9a 8315 | Lemma for transfinite recu... |
| tfrlem10 8316 | Lemma for transfinite recu... |
| tfrlem11 8317 | Lemma for transfinite recu... |
| tfrlem12 8318 | Lemma for transfinite recu... |
| tfrlem13 8319 | Lemma for transfinite recu... |
| tfrlem14 8320 | Lemma for transfinite recu... |
| tfrlem15 8321 | Lemma for transfinite recu... |
| tfrlem16 8322 | Lemma for finite recursion... |
| tfr1a 8323 | A weak version of ~ tfr1 w... |
| tfr2a 8324 | A weak version of ~ tfr2 w... |
| tfr2b 8325 | Without assuming ~ ax-rep ... |
| tfr1 8326 | Principle of Transfinite R... |
| tfr2 8327 | Principle of Transfinite R... |
| tfr3 8328 | Principle of Transfinite R... |
| tfr1ALT 8329 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8330 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8331 | Alternate proof of ~ tfr3 ... |
| recsfnon 8332 | Strong transfinite recursi... |
| recsval 8333 | Strong transfinite recursi... |
| tz7.44lem1 8334 | The ordered pair abstracti... |
| tz7.44-1 8335 | The value of ` F ` at ` (/... |
| tz7.44-2 8336 | The value of ` F ` at a su... |
| tz7.44-3 8337 | The value of ` F ` at a li... |
| rdgeq1 8340 | Equality theorem for the r... |
| rdgeq2 8341 | Equality theorem for the r... |
| rdgeq12 8342 | Equality theorem for the r... |
| nfrdg 8343 | Bound-variable hypothesis ... |
| rdglem1 8344 | Lemma used with the recurs... |
| rdgfun 8345 | The recursive definition g... |
| rdgdmlim 8346 | The domain of the recursiv... |
| rdgfnon 8347 | The recursive definition g... |
| rdgvalg 8348 | Value of the recursive def... |
| rdgval 8349 | Value of the recursive def... |
| rdg0 8350 | The initial value of the r... |
| rdgseg 8351 | The initial segments of th... |
| rdgsucg 8352 | The value of the recursive... |
| rdgsuc 8353 | The value of the recursive... |
| rdglimg 8354 | The value of the recursive... |
| rdglim 8355 | The value of the recursive... |
| rdg0g 8356 | The initial value of the r... |
| rdgsucmptf 8357 | The value of the recursive... |
| rdgsucmptnf 8358 | The value of the recursive... |
| rdgsucmpt2 8359 | This version of ~ rdgsucmp... |
| rdgsucmpt 8360 | The value of the recursive... |
| rdglim2 8361 | The value of the recursive... |
| rdglim2a 8362 | The value of the recursive... |
| rdg0n 8363 | If ` A ` is a proper class... |
| frfnom 8364 | The function generated by ... |
| fr0g 8365 | The initial value resultin... |
| frsuc 8366 | The successor value result... |
| frsucmpt 8367 | The successor value result... |
| frsucmptn 8368 | The value of the finite re... |
| frsucmpt2 8369 | The successor value result... |
| tz7.48lem 8370 | A way of showing an ordina... |
| tz7.48-2 8371 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8372 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8373 | Proposition 7.48(3) of [Ta... |
| tz7.49 8374 | Proposition 7.49 of [Takeu... |
| tz7.49c 8375 | Corollary of Proposition 7... |
| seqomlem0 8378 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8379 | Lemma for ` seqom ` . The... |
| seqomlem2 8380 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8381 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8382 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8383 | Equality theorem for ` seq... |
| fnseqom 8384 | An index-aware recursive d... |
| seqom0g 8385 | Value of an index-aware re... |
| seqomsuc 8386 | Value of an index-aware re... |
| omsucelsucb 8387 | Membership is inherited by... |
| df1o2 8402 | Expanded value of the ordi... |
| df2o3 8403 | Expanded value of the ordi... |
| df2o2 8404 | Expanded value of the ordi... |
| 1oex 8405 | Ordinal 1 is a set. (Cont... |
| 2oex 8406 | ` 2o ` is a set. (Contrib... |
| 1on 8407 | Ordinal 1 is an ordinal nu... |
| 2on 8408 | Ordinal 2 is an ordinal nu... |
| 2on0 8409 | Ordinal two is not zero. ... |
| ord3 8410 | Ordinal 3 is an ordinal cl... |
| 3on 8411 | Ordinal 3 is an ordinal nu... |
| 4on 8412 | Ordinal 4 is an ordinal nu... |
| 1n0 8413 | Ordinal one is not equal t... |
| nlim1 8414 | 1 is not a limit ordinal. ... |
| nlim2 8415 | 2 is not a limit ordinal. ... |
| xp01disj 8416 | Cartesian products with th... |
| xp01disjl 8417 | Cartesian products with th... |
| ordgt0ge1 8418 | Two ways to express that a... |
| ordge1n0 8419 | An ordinal greater than or... |
| el1o 8420 | Membership in ordinal one.... |
| ord1eln01 8421 | An ordinal that is not 0 o... |
| ord2eln012 8422 | An ordinal that is not 0, ... |
| 1ellim 8423 | A limit ordinal contains 1... |
| 2ellim 8424 | A limit ordinal contains 2... |
| dif1o 8425 | Two ways to say that ` A `... |
| ondif1 8426 | Two ways to say that ` A `... |
| ondif2 8427 | Two ways to say that ` A `... |
| 2oconcl 8428 | Closure of the pair swappi... |
| 0lt1o 8429 | Ordinal zero is less than ... |
| dif20el 8430 | An ordinal greater than on... |
| 0we1 8431 | The empty set is a well-or... |
| brwitnlem 8432 | Lemma for relations which ... |
| fnoa 8433 | Functionality and domain o... |
| fnom 8434 | Functionality and domain o... |
| fnoe 8435 | Functionality and domain o... |
| oav 8436 | Value of ordinal addition.... |
| omv 8437 | Value of ordinal multiplic... |
| oe0lem 8438 | A helper lemma for ~ oe0 a... |
| oev 8439 | Value of ordinal exponenti... |
| oevn0 8440 | Value of ordinal exponenti... |
| oa0 8441 | Addition with zero. Propo... |
| om0 8442 | Ordinal multiplication wit... |
| oe0m 8443 | Value of zero raised to an... |
| om0x 8444 | Ordinal multiplication wit... |
| oe0m0 8445 | Ordinal exponentiation wit... |
| oe0m1 8446 | Ordinal exponentiation wit... |
| oe0 8447 | Ordinal exponentiation wit... |
| oev2 8448 | Alternate value of ordinal... |
| oasuc 8449 | Addition with successor. ... |
| oesuclem 8450 | Lemma for ~ oesuc . (Cont... |
| omsuc 8451 | Multiplication with succes... |
| oesuc 8452 | Ordinal exponentiation wit... |
| onasuc 8453 | Addition with successor. ... |
| onmsuc 8454 | Multiplication with succes... |
| onesuc 8455 | Exponentiation with a succ... |
| oa1suc 8456 | Addition with 1 is same as... |
| oalim 8457 | Ordinal addition with a li... |
| omlim 8458 | Ordinal multiplication wit... |
| oelim 8459 | Ordinal exponentiation wit... |
| oacl 8460 | Closure law for ordinal ad... |
| omcl 8461 | Closure law for ordinal mu... |
| oecl 8462 | Closure law for ordinal ex... |
| oa0r 8463 | Ordinal addition with zero... |
| om0r 8464 | Ordinal multiplication wit... |
| o1p1e2 8465 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8466 | 2 + 2 = 4 for ordinal numb... |
| om1 8467 | Ordinal multiplication wit... |
| om1r 8468 | Ordinal multiplication wit... |
| oe1 8469 | Ordinal exponentiation wit... |
| oe1m 8470 | Ordinal exponentiation wit... |
| oaordi 8471 | Ordering property of ordin... |
| oaord 8472 | Ordering property of ordin... |
| oacan 8473 | Left cancellation law for ... |
| oaword 8474 | Weak ordering property of ... |
| oawordri 8475 | Weak ordering property of ... |
| oaord1 8476 | An ordinal is less than it... |
| oaword1 8477 | An ordinal is less than or... |
| oaword2 8478 | An ordinal is less than or... |
| oawordeulem 8479 | Lemma for ~ oawordex . (C... |
| oawordeu 8480 | Existence theorem for weak... |
| oawordexr 8481 | Existence theorem for weak... |
| oawordex 8482 | Existence theorem for weak... |
| oaordex 8483 | Existence theorem for orde... |
| oa00 8484 | An ordinal sum is zero iff... |
| oalimcl 8485 | The ordinal sum with a lim... |
| oaass 8486 | Ordinal addition is associ... |
| oarec 8487 | Recursive definition of or... |
| oaf1o 8488 | Left addition by a constan... |
| oacomf1olem 8489 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8490 | Define a bijection from ` ... |
| omordi 8491 | Ordering property of ordin... |
| omord2 8492 | Ordering property of ordin... |
| omord 8493 | Ordering property of ordin... |
| omcan 8494 | Left cancellation law for ... |
| omword 8495 | Weak ordering property of ... |
| omwordi 8496 | Weak ordering property of ... |
| omwordri 8497 | Weak ordering property of ... |
| omword1 8498 | An ordinal is less than or... |
| omword2 8499 | An ordinal is less than or... |
| om00 8500 | The product of two ordinal... |
| om00el 8501 | The product of two nonzero... |
| omordlim 8502 | Ordering involving the pro... |
| omlimcl 8503 | The product of any nonzero... |
| odi 8504 | Distributive law for ordin... |
| omass 8505 | Multiplication of ordinal ... |
| oneo 8506 | If an ordinal number is ev... |
| omeulem1 8507 | Lemma for ~ omeu : existen... |
| omeulem2 8508 | Lemma for ~ omeu : uniquen... |
| omopth2 8509 | An ordered pair-like theor... |
| omeu 8510 | The division algorithm for... |
| om2 8511 | Two ways to double an ordi... |
| oen0 8512 | Ordinal exponentiation wit... |
| oeordi 8513 | Ordering law for ordinal e... |
| oeord 8514 | Ordering property of ordin... |
| oecan 8515 | Left cancellation law for ... |
| oeword 8516 | Weak ordering property of ... |
| oewordi 8517 | Weak ordering property of ... |
| oewordri 8518 | Weak ordering property of ... |
| oeworde 8519 | Ordinal exponentiation com... |
| oeordsuc 8520 | Ordering property of ordin... |
| oelim2 8521 | Ordinal exponentiation wit... |
| oeoalem 8522 | Lemma for ~ oeoa . (Contr... |
| oeoa 8523 | Sum of exponents law for o... |
| oeoelem 8524 | Lemma for ~ oeoe . (Contr... |
| oeoe 8525 | Product of exponents law f... |
| oelimcl 8526 | The ordinal exponential wi... |
| oeeulem 8527 | Lemma for ~ oeeu . (Contr... |
| oeeui 8528 | The division algorithm for... |
| oeeu 8529 | The division algorithm for... |
| nna0 8530 | Addition with zero. Theor... |
| nnm0 8531 | Multiplication with zero. ... |
| nnasuc 8532 | Addition with successor. ... |
| nnmsuc 8533 | Multiplication with succes... |
| nnesuc 8534 | Exponentiation with a succ... |
| nna0r 8535 | Addition to zero. Remark ... |
| nnm0r 8536 | Multiplication with zero. ... |
| nnacl 8537 | Closure of addition of nat... |
| nnmcl 8538 | Closure of multiplication ... |
| nnecl 8539 | Closure of exponentiation ... |
| nnacli 8540 | ` _om ` is closed under ad... |
| nnmcli 8541 | ` _om ` is closed under mu... |
| nnarcl 8542 | Reverse closure law for ad... |
| nnacom 8543 | Addition of natural number... |
| nnaordi 8544 | Ordering property of addit... |
| nnaord 8545 | Ordering property of addit... |
| nnaordr 8546 | Ordering property of addit... |
| nnawordi 8547 | Adding to both sides of an... |
| nnaass 8548 | Addition of natural number... |
| nndi 8549 | Distributive law for natur... |
| nnmass 8550 | Multiplication of natural ... |
| nnmsucr 8551 | Multiplication with succes... |
| nnmcom 8552 | Multiplication of natural ... |
| nnaword 8553 | Weak ordering property of ... |
| nnacan 8554 | Cancellation law for addit... |
| nnaword1 8555 | Weak ordering property of ... |
| nnaword2 8556 | Weak ordering property of ... |
| nnmordi 8557 | Ordering property of multi... |
| nnmord 8558 | Ordering property of multi... |
| nnmword 8559 | Weak ordering property of ... |
| nnmcan 8560 | Cancellation law for multi... |
| nnmwordi 8561 | Weak ordering property of ... |
| nnmwordri 8562 | Weak ordering property of ... |
| nnawordex 8563 | Equivalence for weak order... |
| nnaordex 8564 | Equivalence for ordering. ... |
| nnaordex2 8565 | Equivalence for ordering. ... |
| 1onn 8566 | The ordinal 1 is a natural... |
| 1onnALT 8567 | Shorter proof of ~ 1onn us... |
| 2onn 8568 | The ordinal 2 is a natural... |
| 2onnALT 8569 | Shorter proof of ~ 2onn us... |
| 3onn 8570 | The ordinal 3 is a natural... |
| 4onn 8571 | The ordinal 4 is a natural... |
| 1one2o 8572 | Ordinal one is not ordinal... |
| oaabslem 8573 | Lemma for ~ oaabs . (Cont... |
| oaabs 8574 | Ordinal addition absorbs a... |
| oaabs2 8575 | The absorption law ~ oaabs... |
| omabslem 8576 | Lemma for ~ omabs . (Cont... |
| omabs 8577 | Ordinal multiplication is ... |
| nnm1 8578 | Multiply an element of ` _... |
| nnm2 8579 | Multiply an element of ` _... |
| nn2m 8580 | Multiply an element of ` _... |
| nnneo 8581 | If a natural number is eve... |
| nneob 8582 | A natural number is even i... |
| omsmolem 8583 | Lemma for ~ omsmo . (Cont... |
| omsmo 8584 | A strictly monotonic ordin... |
| omopthlem1 8585 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8586 | Lemma for ~ omopthi . (Co... |
| omopthi 8587 | An ordered pair theorem fo... |
| omopth 8588 | An ordered pair theorem fo... |
| nnasmo 8589 | There is at most one left ... |
| eldifsucnn 8590 | Condition for membership i... |
| on2recsfn 8593 | Show that double recursion... |
| on2recsov 8594 | Calculate the value of the... |
| on2ind 8595 | Double induction over ordi... |
| on3ind 8596 | Triple induction over ordi... |
| coflton 8597 | Cofinality theorem for ord... |
| cofon1 8598 | Cofinality theorem for ord... |
| cofon2 8599 | Cofinality theorem for ord... |
| cofonr 8600 | Inverse cofinality law for... |
| naddfn 8601 | Natural addition is a func... |
| naddcllem 8602 | Lemma for ordinal addition... |
| naddcl 8603 | Closure law for natural ad... |
| naddov 8604 | The value of natural addit... |
| naddov2 8605 | Alternate expression for n... |
| naddov3 8606 | Alternate expression for n... |
| naddf 8607 | Function statement for nat... |
| naddcom 8608 | Natural addition commutes.... |
| naddrid 8609 | Ordinal zero is the additi... |
| naddlid 8610 | Ordinal zero is the additi... |
| naddssim 8611 | Ordinal less-than-or-equal... |
| naddelim 8612 | Ordinal less-than is prese... |
| naddel1 8613 | Ordinal less-than is not a... |
| naddel2 8614 | Ordinal less-than is not a... |
| naddss1 8615 | Ordinal less-than-or-equal... |
| naddss2 8616 | Ordinal less-than-or-equal... |
| naddword1 8617 | Weak-ordering principle fo... |
| naddword2 8618 | Weak-ordering principle fo... |
| naddunif 8619 | Uniformity theorem for nat... |
| naddasslem1 8620 | Lemma for ~ naddass . Exp... |
| naddasslem2 8621 | Lemma for ~ naddass . Exp... |
| naddass 8622 | Natural ordinal addition i... |
| nadd32 8623 | Commutative/associative la... |
| nadd4 8624 | Rearragement of terms in a... |
| nadd42 8625 | Rearragement of terms in a... |
| naddel12 8626 | Natural addition to both s... |
| naddsuc2 8627 | Natural addition with succ... |
| naddoa 8628 | Natural addition of a natu... |
| omnaddcl 8629 | The naturals are closed un... |
| dfer2 8634 | Alternate definition of eq... |
| dfec2 8636 | Alternate definition of ` ... |
| ecexg 8637 | An equivalence class modul... |
| ecexr 8638 | A nonempty equivalence cla... |
| ereq1 8640 | Equality theorem for equiv... |
| ereq2 8641 | Equality theorem for equiv... |
| errel 8642 | An equivalence relation is... |
| erdm 8643 | The domain of an equivalen... |
| ercl 8644 | Elementhood in the field o... |
| ersym 8645 | An equivalence relation is... |
| ercl2 8646 | Elementhood in the field o... |
| ersymb 8647 | An equivalence relation is... |
| ertr 8648 | An equivalence relation is... |
| ertrd 8649 | A transitivity relation fo... |
| ertr2d 8650 | A transitivity relation fo... |
| ertr3d 8651 | A transitivity relation fo... |
| ertr4d 8652 | A transitivity relation fo... |
| erref 8653 | An equivalence relation is... |
| ercnv 8654 | The converse of an equival... |
| errn 8655 | The range and domain of an... |
| erssxp 8656 | An equivalence relation is... |
| erex 8657 | An equivalence relation is... |
| erexb 8658 | An equivalence relation is... |
| iserd 8659 | A reflexive, symmetric, tr... |
| iseri 8660 | A reflexive, symmetric, tr... |
| iseriALT 8661 | Alternate proof of ~ iseri... |
| brinxper 8662 | Conditions for a reflexive... |
| brdifun 8663 | Evaluate the incomparabili... |
| swoer 8664 | Incomparability under a st... |
| swoord1 8665 | The incomparability equiva... |
| swoord2 8666 | The incomparability equiva... |
| swoso 8667 | If the incomparability rel... |
| eqerlem 8668 | Lemma for ~ eqer . (Contr... |
| eqer 8669 | Equivalence relation invol... |
| ider 8670 | The identity relation is a... |
| 0er 8671 | The empty set is an equiva... |
| eceq1 8672 | Equality theorem for equiv... |
| eceq1d 8673 | Equality theorem for equiv... |
| eceq2 8674 | Equality theorem for equiv... |
| eceq2i 8675 | Equality theorem for the `... |
| eceq2d 8676 | Equality theorem for the `... |
| elecg 8677 | Membership in an equivalen... |
| ecref 8678 | All elements are in their ... |
| elec 8679 | Membership in an equivalen... |
| relelec 8680 | Membership in an equivalen... |
| elecres 8681 | Elementhood in the restric... |
| elecreseq 8682 | The restricted coset of ` ... |
| elecex 8683 | Condition for a coset to b... |
| ecss 8684 | An equivalence class is a ... |
| ecdmn0 8685 | A representative of a none... |
| ereldm 8686 | Equality of equivalence cl... |
| erth 8687 | Basic property of equivale... |
| erth2 8688 | Basic property of equivale... |
| erthi 8689 | Basic property of equivale... |
| erdisj 8690 | Equivalence classes do not... |
| ecidsn 8691 | An equivalence class modul... |
| qseq1 8692 | Equality theorem for quoti... |
| qseq2 8693 | Equality theorem for quoti... |
| qseq2i 8694 | Equality theorem for quoti... |
| qseq1d 8695 | Equality theorem for quoti... |
| qseq2d 8696 | Equality theorem for quoti... |
| qseq12 8697 | Equality theorem for quoti... |
| 0qs 8698 | Quotient set with the empt... |
| elqsg 8699 | Closed form of ~ elqs . (... |
| elqs 8700 | Membership in a quotient s... |
| elqsi 8701 | Membership in a quotient s... |
| elqsecl 8702 | Membership in a quotient s... |
| ecelqs 8703 | Membership of an equivalen... |
| ecelqsw 8704 | Membership of an equivalen... |
| ecelqsi 8705 | Membership of an equivalen... |
| ecopqsi 8706 | "Closure" law for equivale... |
| qsexg 8707 | A quotient set exists. (C... |
| qsex 8708 | A quotient set exists. (C... |
| uniqs 8709 | The union of a quotient se... |
| uniqsw 8710 | The union of a quotient se... |
| qsss 8711 | A quotient set is a set of... |
| uniqs2 8712 | The union of a quotient se... |
| snec 8713 | The singleton of an equiva... |
| ecqs 8714 | Equivalence class in terms... |
| ecid 8715 | A set is equal to its cose... |
| qsid 8716 | A set is equal to its quot... |
| ectocld 8717 | Implicit substitution of c... |
| ectocl 8718 | Implicit substitution of c... |
| elqsn0 8719 | A quotient set does not co... |
| ecelqsdm 8720 | Membership of an equivalen... |
| ecelqsdmb 8721 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8722 | ` R ` -coset in its domain... |
| xpider 8723 | A Cartesian square is an e... |
| iiner 8724 | The intersection of a none... |
| riiner 8725 | The relative intersection ... |
| erinxp 8726 | A restricted equivalence r... |
| ecinxp 8727 | Restrict the relation in a... |
| qsinxp 8728 | Restrict the equivalence r... |
| qsdisj 8729 | Members of a quotient set ... |
| qsdisj2 8730 | A quotient set is a disjoi... |
| qsel 8731 | If an element of a quotien... |
| uniinqs 8732 | Class union distributes ov... |
| qliftlem 8733 | Lemma for theorems about a... |
| qliftrel 8734 | ` F ` , a function lift, i... |
| qliftel 8735 | Elementhood in the relatio... |
| qliftel1 8736 | Elementhood in the relatio... |
| qliftfun 8737 | The function ` F ` is the ... |
| qliftfund 8738 | The function ` F ` is the ... |
| qliftfuns 8739 | The function ` F ` is the ... |
| qliftf 8740 | The domain and codomain of... |
| qliftval 8741 | The value of the function ... |
| ecoptocl 8742 | Implicit substitution of c... |
| 2ecoptocl 8743 | Implicit substitution of c... |
| 3ecoptocl 8744 | Implicit substitution of c... |
| brecop 8745 | Binary relation on a quoti... |
| brecop2 8746 | Binary relation on a quoti... |
| eroveu 8747 | Lemma for ~ erov and ~ ero... |
| erovlem 8748 | Lemma for ~ erov and ~ ero... |
| erov 8749 | The value of an operation ... |
| eroprf 8750 | Functionality of an operat... |
| erov2 8751 | The value of an operation ... |
| eroprf2 8752 | Functionality of an operat... |
| ecopoveq 8753 | This is the first of sever... |
| ecopovsym 8754 | Assuming the operation ` F... |
| ecopovtrn 8755 | Assuming that operation ` ... |
| ecopover 8756 | Assuming that operation ` ... |
| eceqoveq 8757 | Equality of equivalence re... |
| ecovcom 8758 | Lemma used to transfer a c... |
| ecovass 8759 | Lemma used to transfer an ... |
| ecovdi 8760 | Lemma used to transfer a d... |
| mapprc 8765 | When ` A ` is a proper cla... |
| pmex 8766 | The class of all partial f... |
| mapexOLD 8767 | Obsolete version of ~ mape... |
| fnmap 8768 | Set exponentiation has a u... |
| fnpm 8769 | Partial function exponenti... |
| reldmmap 8770 | Set exponentiation is a we... |
| mapvalg 8771 | The value of set exponenti... |
| pmvalg 8772 | The value of the partial m... |
| mapval 8773 | The value of set exponenti... |
| elmapg 8774 | Membership relation for se... |
| elmapd 8775 | Deduction form of ~ elmapg... |
| elmapdd 8776 | Deduction associated with ... |
| mapdm0 8777 | The empty set is the only ... |
| elpmg 8778 | The predicate "is a partia... |
| elpm2g 8779 | The predicate "is a partia... |
| elpm2r 8780 | Sufficient condition for b... |
| elpmi 8781 | A partial function is a fu... |
| pmfun 8782 | A partial function is a fu... |
| elmapex 8783 | Eliminate antecedent for m... |
| elmapi 8784 | A mapping is a function, f... |
| mapfset 8785 | If ` B ` is a set, the val... |
| mapssfset 8786 | The value of the set expon... |
| mapfoss 8787 | The value of the set expon... |
| fsetsspwxp 8788 | The class of all functions... |
| fset0 8789 | The set of functions from ... |
| fsetdmprc0 8790 | The set of functions with ... |
| fsetex 8791 | The set of functions betwe... |
| f1setex 8792 | The set of injections betw... |
| fosetex 8793 | The set of surjections bet... |
| f1osetex 8794 | The set of bijections betw... |
| fsetfcdm 8795 | The class of functions wit... |
| fsetfocdm 8796 | The class of functions wit... |
| fsetprcnex 8797 | The class of all functions... |
| fsetcdmex 8798 | The class of all functions... |
| fsetexb 8799 | The class of all functions... |
| elmapfn 8800 | A mapping is a function wi... |
| elmapfun 8801 | A mapping is always a func... |
| elmapssres 8802 | A restricted mapping is a ... |
| elmapssresd 8803 | A restricted mapping is a ... |
| fpmg 8804 | A total function is a part... |
| pmss12g 8805 | Subset relation for the se... |
| pmresg 8806 | Elementhood of a restricte... |
| elmap 8807 | Membership relation for se... |
| mapval2 8808 | Alternate expression for t... |
| elpm 8809 | The predicate "is a partia... |
| elpm2 8810 | The predicate "is a partia... |
| fpm 8811 | A total function is a part... |
| mapsspm 8812 | Set exponentiation is a su... |
| pmsspw 8813 | Partial maps are a subset ... |
| mapsspw 8814 | Set exponentiation is a su... |
| mapfvd 8815 | The value of a function th... |
| elmapresaun 8816 | ~ fresaun transposed to ma... |
| fvmptmap 8817 | Special case of ~ fvmpt fo... |
| map0e 8818 | Set exponentiation with an... |
| map0b 8819 | Set exponentiation with an... |
| map0g 8820 | Set exponentiation is empt... |
| 0map0sn0 8821 | The set of mappings of the... |
| mapsnd 8822 | The value of set exponenti... |
| map0 8823 | Set exponentiation is empt... |
| mapsn 8824 | The value of set exponenti... |
| mapss 8825 | Subset inheritance for set... |
| fdiagfn 8826 | Functionality of the diago... |
| fvdiagfn 8827 | Functionality of the diago... |
| mapsnconst 8828 | Every singleton map is a c... |
| mapsncnv 8829 | Expression for the inverse... |
| mapsnf1o2 8830 | Explicit bijection between... |
| mapsnf1o3 8831 | Explicit bijection in the ... |
| ralxpmap 8832 | Quantification over functi... |
| dfixp 8835 | Eliminate the expression `... |
| ixpsnval 8836 | The value of an infinite C... |
| elixp2 8837 | Membership in an infinite ... |
| fvixp 8838 | Projection of a factor of ... |
| ixpfn 8839 | A nuple is a function. (C... |
| elixp 8840 | Membership in an infinite ... |
| elixpconst 8841 | Membership in an infinite ... |
| ixpconstg 8842 | Infinite Cartesian product... |
| ixpconst 8843 | Infinite Cartesian product... |
| ixpeq1 8844 | Equality theorem for infin... |
| ixpeq1d 8845 | Equality theorem for infin... |
| ss2ixp 8846 | Subclass theorem for infin... |
| ixpeq2 8847 | Equality theorem for infin... |
| ixpeq2dva 8848 | Equality theorem for infin... |
| ixpeq2dv 8849 | Equality theorem for infin... |
| cbvixp 8850 | Change bound variable in a... |
| cbvixpv 8851 | Change bound variable in a... |
| nfixpw 8852 | Bound-variable hypothesis ... |
| nfixp 8853 | Bound-variable hypothesis ... |
| nfixp1 8854 | The index variable in an i... |
| ixpprc 8855 | A cartesian product of pro... |
| ixpf 8856 | A member of an infinite Ca... |
| uniixp 8857 | The union of an infinite C... |
| ixpexg 8858 | The existence of an infini... |
| ixpin 8859 | The intersection of two in... |
| ixpiin 8860 | The indexed intersection o... |
| ixpint 8861 | The intersection of a coll... |
| ixp0x 8862 | An infinite Cartesian prod... |
| ixpssmap2g 8863 | An infinite Cartesian prod... |
| ixpssmapg 8864 | An infinite Cartesian prod... |
| 0elixp 8865 | Membership of the empty se... |
| ixpn0 8866 | The infinite Cartesian pro... |
| ixp0 8867 | The infinite Cartesian pro... |
| ixpssmap 8868 | An infinite Cartesian prod... |
| resixp 8869 | Restriction of an element ... |
| undifixp 8870 | Union of two projections o... |
| mptelixpg 8871 | Condition for an explicit ... |
| resixpfo 8872 | Restriction of elements of... |
| elixpsn 8873 | Membership in a class of s... |
| ixpsnf1o 8874 | A bijection between a clas... |
| mapsnf1o 8875 | A bijection between a set ... |
| boxriin 8876 | A rectangular subset of a ... |
| boxcutc 8877 | The relative complement of... |
| relen 8886 | Equinumerosity is a relati... |
| reldom 8887 | Dominance is a relation. ... |
| relsdom 8888 | Strict dominance is a rela... |
| encv 8889 | If two classes are equinum... |
| breng 8890 | Equinumerosity relation. ... |
| bren 8891 | Equinumerosity relation. ... |
| brdom2g 8892 | Dominance relation. This ... |
| brdomg 8893 | Dominance relation. (Cont... |
| brdomi 8894 | Dominance relation. (Cont... |
| brdom 8895 | Dominance relation. (Cont... |
| domen 8896 | Dominance in terms of equi... |
| domeng 8897 | Dominance in terms of equi... |
| ctex 8898 | A countable set is a set. ... |
| f1oen4g 8899 | The domain and range of a ... |
| f1dom4g 8900 | The domain of a one-to-one... |
| f1oen3g 8901 | The domain and range of a ... |
| f1dom3g 8902 | The domain of a one-to-one... |
| f1oen2g 8903 | The domain and range of a ... |
| f1dom2g 8904 | The domain of a one-to-one... |
| f1oeng 8905 | The domain and range of a ... |
| f1domg 8906 | The domain of a one-to-one... |
| f1oen 8907 | The domain and range of a ... |
| f1dom 8908 | The domain of a one-to-one... |
| brsdom 8909 | Strict dominance relation,... |
| isfi 8910 | Express " ` A ` is finite"... |
| enssdom 8911 | Equinumerosity implies dom... |
| enssdomOLD 8912 | Obsolete version of ~ enss... |
| dfdom2 8913 | Alternate definition of do... |
| endom 8914 | Equinumerosity implies dom... |
| sdomdom 8915 | Strict dominance implies d... |
| sdomnen 8916 | Strict dominance implies n... |
| brdom2 8917 | Dominance in terms of stri... |
| bren2 8918 | Equinumerosity expressed i... |
| enrefg 8919 | Equinumerosity is reflexiv... |
| enref 8920 | Equinumerosity is reflexiv... |
| eqeng 8921 | Equality implies equinumer... |
| domrefg 8922 | Dominance is reflexive. (... |
| en2d 8923 | Equinumerosity inference f... |
| en3d 8924 | Equinumerosity inference f... |
| en2i 8925 | Equinumerosity inference f... |
| en3i 8926 | Equinumerosity inference f... |
| dom2lem 8927 | A mapping (first hypothesi... |
| dom2d 8928 | A mapping (first hypothesi... |
| dom3d 8929 | A mapping (first hypothesi... |
| dom2 8930 | A mapping (first hypothesi... |
| dom3 8931 | A mapping (first hypothesi... |
| idssen 8932 | Equality implies equinumer... |
| domssl 8933 | If ` A ` is a subset of ` ... |
| domssr 8934 | If ` C ` is a superset of ... |
| ssdomg 8935 | A set dominates its subset... |
| ener 8936 | Equinumerosity is an equiv... |
| ensymb 8937 | Symmetry of equinumerosity... |
| ensym 8938 | Symmetry of equinumerosity... |
| ensymi 8939 | Symmetry of equinumerosity... |
| ensymd 8940 | Symmetry of equinumerosity... |
| entr 8941 | Transitivity of equinumero... |
| domtr 8942 | Transitivity of dominance ... |
| entri 8943 | A chained equinumerosity i... |
| entr2i 8944 | A chained equinumerosity i... |
| entr3i 8945 | A chained equinumerosity i... |
| entr4i 8946 | A chained equinumerosity i... |
| endomtr 8947 | Transitivity of equinumero... |
| domentr 8948 | Transitivity of dominance ... |
| f1imaeng 8949 | If a function is one-to-on... |
| f1imaen2g 8950 | If a function is one-to-on... |
| f1imaen3g 8951 | If a set function is one-t... |
| f1imaen 8952 | If a function is one-to-on... |
| en0 8953 | The empty set is equinumer... |
| en0ALT 8954 | Shorter proof of ~ en0 , d... |
| en0r 8955 | The empty set is equinumer... |
| ensn1 8956 | A singleton is equinumerou... |
| ensn1g 8957 | A singleton is equinumerou... |
| enpr1g 8958 | ` { A , A } ` has only one... |
| en1 8959 | A set is equinumerous to o... |
| en1b 8960 | A set is equinumerous to o... |
| reuen1 8961 | Two ways to express "exact... |
| euen1 8962 | Two ways to express "exact... |
| euen1b 8963 | Two ways to express " ` A ... |
| en1uniel 8964 | A singleton contains its s... |
| 2dom 8965 | A set that dominates ordin... |
| fundmen 8966 | A function is equinumerous... |
| fundmeng 8967 | A function is equinumerous... |
| cnven 8968 | A relational set is equinu... |
| cnvct 8969 | If a set is countable, so ... |
| fndmeng 8970 | A function is equinumerate... |
| mapsnend 8971 | Set exponentiation to a si... |
| mapsnen 8972 | Set exponentiation to a si... |
| snmapen 8973 | Set exponentiation: a sing... |
| snmapen1 8974 | Set exponentiation: a sing... |
| map1 8975 | Set exponentiation: ordina... |
| en2sn 8976 | Two singletons are equinum... |
| 0fi 8977 | The empty set is finite. ... |
| snfi 8978 | A singleton is finite. (C... |
| fiprc 8979 | The class of finite sets i... |
| unen 8980 | Equinumerosity of union of... |
| enrefnn 8981 | Equinumerosity is reflexiv... |
| en2prd 8982 | Two proper unordered pairs... |
| enpr2d 8983 | A pair with distinct eleme... |
| ssct 8984 | Any subset of a countable ... |
| difsnen 8985 | All decrements of a set ar... |
| domdifsn 8986 | Dominance over a set with ... |
| xpsnen 8987 | A set is equinumerous to i... |
| xpsneng 8988 | A set is equinumerous to i... |
| xp1en 8989 | One times a cardinal numbe... |
| endisj 8990 | Any two sets are equinumer... |
| undom 8991 | Dominance law for union. ... |
| xpcomf1o 8992 | The canonical bijection fr... |
| xpcomco 8993 | Composition with the bijec... |
| xpcomen 8994 | Commutative law for equinu... |
| xpcomeng 8995 | Commutative law for equinu... |
| xpsnen2g 8996 | A set is equinumerous to i... |
| xpassen 8997 | Associative law for equinu... |
| xpdom2 8998 | Dominance law for Cartesia... |
| xpdom2g 8999 | Dominance law for Cartesia... |
| xpdom1g 9000 | Dominance law for Cartesia... |
| xpdom3 9001 | A set is dominated by its ... |
| xpdom1 9002 | Dominance law for Cartesia... |
| domunsncan 9003 | A singleton cancellation l... |
| omxpenlem 9004 | Lemma for ~ omxpen . (Con... |
| omxpen 9005 | The cardinal and ordinal p... |
| omf1o 9006 | Construct an explicit bije... |
| pw2f1olem 9007 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9008 | The power set of a set is ... |
| pw2eng 9009 | The power set of a set is ... |
| pw2en 9010 | The power set of a set is ... |
| fopwdom 9011 | Covering implies injection... |
| enfixsn 9012 | Given two equipollent sets... |
| sbthlem1 9013 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9014 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9015 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9016 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9017 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9018 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9019 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9020 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9021 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9022 | Lemma for ~ sbth . (Contr... |
| sbth 9023 | Schroeder-Bernstein Theore... |
| sbthb 9024 | Schroeder-Bernstein Theore... |
| sbthcl 9025 | Schroeder-Bernstein Theore... |
| dfsdom2 9026 | Alternate definition of st... |
| brsdom2 9027 | Alternate definition of st... |
| sdomnsym 9028 | Strict dominance is asymme... |
| domnsym 9029 | Theorem 22(i) of [Suppes] ... |
| 0domg 9030 | Any set dominates the empt... |
| dom0 9031 | A set dominated by the emp... |
| 0sdomg 9032 | A set strictly dominates t... |
| 0dom 9033 | Any set dominates the empt... |
| 0sdom 9034 | A set strictly dominates t... |
| sdom0 9035 | The empty set does not str... |
| sdomdomtr 9036 | Transitivity of strict dom... |
| sdomentr 9037 | Transitivity of strict dom... |
| domsdomtr 9038 | Transitivity of dominance ... |
| ensdomtr 9039 | Transitivity of equinumero... |
| sdomirr 9040 | Strict dominance is irrefl... |
| sdomtr 9041 | Strict dominance is transi... |
| sdomn2lp 9042 | Strict dominance has no 2-... |
| enen1 9043 | Equality-like theorem for ... |
| enen2 9044 | Equality-like theorem for ... |
| domen1 9045 | Equality-like theorem for ... |
| domen2 9046 | Equality-like theorem for ... |
| sdomen1 9047 | Equality-like theorem for ... |
| sdomen2 9048 | Equality-like theorem for ... |
| domtriord 9049 | Dominance is trichotomous ... |
| sdomel 9050 | For ordinals, strict domin... |
| sdomdif 9051 | The difference of a set fr... |
| onsdominel 9052 | An ordinal with more eleme... |
| domunsn 9053 | Dominance over a set with ... |
| fodomr 9054 | There exists a mapping fro... |
| pwdom 9055 | Injection of sets implies ... |
| canth2 9056 | Cantor's Theorem. No set ... |
| canth2g 9057 | Cantor's theorem with the ... |
| 2pwuninel 9058 | The power set of the power... |
| 2pwne 9059 | No set equals the power se... |
| disjen 9060 | A stronger form of ~ pwuni... |
| disjenex 9061 | Existence version of ~ dis... |
| domss2 9062 | A corollary of ~ disjenex ... |
| domssex2 9063 | A corollary of ~ disjenex ... |
| domssex 9064 | Weakening of ~ domssex2 to... |
| xpf1o 9065 | Construct a bijection on a... |
| xpen 9066 | Equinumerosity law for Car... |
| mapen 9067 | Two set exponentiations ar... |
| mapdom1 9068 | Order-preserving property ... |
| mapxpen 9069 | Equinumerosity law for dou... |
| xpmapenlem 9070 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9071 | Equinumerosity law for set... |
| mapunen 9072 | Equinumerosity law for set... |
| map2xp 9073 | A cardinal power with expo... |
| mapdom2 9074 | Order-preserving property ... |
| mapdom3 9075 | Set exponentiation dominat... |
| pwen 9076 | If two sets are equinumero... |
| ssenen 9077 | Equinumerosity of equinume... |
| limenpsi 9078 | A limit ordinal is equinum... |
| limensuci 9079 | A limit ordinal is equinum... |
| limensuc 9080 | A limit ordinal is equinum... |
| infensuc 9081 | Any infinite ordinal is eq... |
| dif1enlem 9082 | Lemma for ~ rexdif1en and ... |
| rexdif1en 9083 | If a set is equinumerous t... |
| dif1en 9084 | If a set ` A ` is equinume... |
| dif1ennn 9085 | If a set ` A ` is equinume... |
| findcard 9086 | Schema for induction on th... |
| findcard2 9087 | Schema for induction on th... |
| findcard2s 9088 | Variation of ~ findcard2 r... |
| findcard2d 9089 | Deduction version of ~ fin... |
| nnfi 9090 | Natural numbers are finite... |
| pssnn 9091 | A proper subset of a natur... |
| ssnnfi 9092 | A subset of a natural numb... |
| unfi 9093 | The union of two finite se... |
| unfid 9094 | The union of two finite se... |
| ssfi 9095 | A subset of a finite set i... |
| ssfiALT 9096 | Shorter proof of ~ ssfi us... |
| diffi 9097 | If ` A ` is finite, ` ( A ... |
| cnvfi 9098 | If a set is finite, its co... |
| pwssfi 9099 | Every element of the power... |
| fnfi 9100 | A version of ~ fnex for fi... |
| f1oenfi 9101 | If the domain of a one-to-... |
| f1oenfirn 9102 | If the range of a one-to-o... |
| f1domfi 9103 | If the codomain of a one-t... |
| f1domfi2 9104 | If the domain of a one-to-... |
| enreffi 9105 | Equinumerosity is reflexiv... |
| ensymfib 9106 | Symmetry of equinumerosity... |
| entrfil 9107 | Transitivity of equinumero... |
| enfii 9108 | A set equinumerous to a fi... |
| enfi 9109 | Equinumerous sets have the... |
| enfiALT 9110 | Shorter proof of ~ enfi us... |
| domfi 9111 | A set dominated by a finit... |
| entrfi 9112 | Transitivity of equinumero... |
| entrfir 9113 | Transitivity of equinumero... |
| domtrfil 9114 | Transitivity of dominance ... |
| domtrfi 9115 | Transitivity of dominance ... |
| domtrfir 9116 | Transitivity of dominance ... |
| f1imaenfi 9117 | If a function is one-to-on... |
| ssdomfi 9118 | A finite set dominates its... |
| ssdomfi2 9119 | A set dominates its finite... |
| sbthfilem 9120 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9121 | Schroeder-Bernstein Theore... |
| domnsymfi 9122 | If a set dominates a finit... |
| sdomdomtrfi 9123 | Transitivity of strict dom... |
| domsdomtrfi 9124 | Transitivity of dominance ... |
| sucdom2 9125 | Strict dominance of a set ... |
| phplem1 9126 | Lemma for Pigeonhole Princ... |
| phplem2 9127 | Lemma for Pigeonhole Princ... |
| nneneq 9128 | Two equinumerous natural n... |
| php 9129 | Pigeonhole Principle. A n... |
| php2 9130 | Corollary of Pigeonhole Pr... |
| php3 9131 | Corollary of Pigeonhole Pr... |
| php4 9132 | Corollary of the Pigeonhol... |
| php5 9133 | Corollary of the Pigeonhol... |
| phpeqd 9134 | Corollary of the Pigeonhol... |
| nndomog 9135 | Cardinal ordering agrees w... |
| onomeneq 9136 | An ordinal number equinume... |
| onfin 9137 | An ordinal number is finit... |
| ordfin 9138 | A generalization of ~ onfi... |
| onfin2 9139 | A set is a natural number ... |
| nndomo 9140 | Cardinal ordering agrees w... |
| nnsdomo 9141 | Cardinal ordering agrees w... |
| sucdom 9142 | Strict dominance of a set ... |
| snnen2o 9143 | A singleton ` { A } ` is n... |
| 0sdom1dom 9144 | Strict dominance over 0 is... |
| 0sdom1domALT 9145 | Alternate proof of ~ 0sdom... |
| 1sdom2 9146 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9147 | Alternate proof of ~ 1sdom... |
| sdom1 9148 | A set has less than one me... |
| modom 9149 | Two ways to express "at mo... |
| modom2 9150 | Two ways to express "at mo... |
| rex2dom 9151 | A set that has at least 2 ... |
| 1sdom2dom 9152 | Strict dominance over 1 is... |
| 1sdom 9153 | A set that strictly domina... |
| unxpdomlem1 9154 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9155 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9156 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9157 | Cartesian product dominate... |
| unxpdom2 9158 | Corollary of ~ unxpdom . ... |
| sucxpdom 9159 | Cartesian product dominate... |
| pssinf 9160 | A set equinumerous to a pr... |
| fisseneq 9161 | A finite set is equal to i... |
| ominf 9162 | The set of natural numbers... |
| isinf 9163 | Any set that is not finite... |
| fineqvlem 9164 | Lemma for ~ fineqv . (Con... |
| fineqv 9165 | If the Axiom of Infinity i... |
| xpfir 9166 | The components of a nonemp... |
| ssfid 9167 | A subset of a finite set i... |
| infi 9168 | The intersection of two se... |
| rabfi 9169 | A restricted class built f... |
| finresfin 9170 | The restriction of a finit... |
| f1finf1o 9171 | Any injection from one fin... |
| nfielex 9172 | If a class is not finite, ... |
| en1eqsn 9173 | A set with one element is ... |
| en1eqsnbi 9174 | A set containing an elemen... |
| dif1ennnALT 9175 | Alternate proof of ~ dif1e... |
| enp1ilem 9176 | Lemma for uses of ~ enp1i ... |
| enp1i 9177 | Proof induction for ~ en2 ... |
| en2 9178 | A set equinumerous to ordi... |
| en3 9179 | A set equinumerous to ordi... |
| en4 9180 | A set equinumerous to ordi... |
| findcard3 9181 | Schema for strong inductio... |
| ac6sfi 9182 | A version of ~ ac6s for fi... |
| frfi 9183 | A partial order is well-fo... |
| fimax2g 9184 | A finite set has a maximum... |
| fimaxg 9185 | A finite set has a maximum... |
| fisupg 9186 | Lemma showing existence an... |
| wofi 9187 | A total order on a finite ... |
| ordunifi 9188 | The maximum of a finite co... |
| nnunifi 9189 | The union (supremum) of a ... |
| unblem1 9190 | Lemma for ~ unbnn . After... |
| unblem2 9191 | Lemma for ~ unbnn . The v... |
| unblem3 9192 | Lemma for ~ unbnn . The v... |
| unblem4 9193 | Lemma for ~ unbnn . The f... |
| unbnn 9194 | Any unbounded subset of na... |
| unbnn2 9195 | Version of ~ unbnn that do... |
| isfinite2 9196 | Any set strictly dominated... |
| nnsdomg 9197 | Omega strictly dominates a... |
| isfiniteg 9198 | A set is finite iff it is ... |
| infsdomnn 9199 | An infinite set strictly d... |
| infn0 9200 | An infinite set is not emp... |
| infn0ALT 9201 | Shorter proof of ~ infn0 u... |
| fin2inf 9202 | This (useless) theorem, wh... |
| unfilem1 9203 | Lemma for proving that the... |
| unfilem2 9204 | Lemma for proving that the... |
| unfilem3 9205 | Lemma for proving that the... |
| unfir 9206 | If a union is finite, the ... |
| unfib 9207 | A union is finite if and o... |
| unfi2 9208 | The union of two finite se... |
| difinf 9209 | An infinite set ` A ` minu... |
| fodomfi 9210 | An onto function implies d... |
| fofi 9211 | If an onto function has a ... |
| f1fi 9212 | If a 1-to-1 function has a... |
| imafi 9213 | Images of finite sets are ... |
| imafiOLD 9214 | Obsolete version of ~ imaf... |
| pwfir 9215 | If the power set of a set ... |
| pwfilem 9216 | Lemma for ~ pwfi . (Contr... |
| pwfi 9217 | The power set of a finite ... |
| xpfi 9218 | The Cartesian product of t... |
| 3xpfi 9219 | The Cartesian product of t... |
| domunfican 9220 | A finite set union cancell... |
| infcntss 9221 | Every infinite set has a d... |
| prfi 9222 | An unordered pair is finit... |
| prfiALT 9223 | Shorter proof of ~ prfi us... |
| tpfi 9224 | An unordered triple is fin... |
| fiint 9225 | Equivalent ways of stating... |
| fodomfir 9226 | There exists a mapping fro... |
| fodomfib 9227 | Equivalence of an onto map... |
| fodomfiOLD 9228 | Obsolete version of ~ fodo... |
| fodomfibOLD 9229 | Obsolete version of ~ fodo... |
| fofinf1o 9230 | Any surjection from one fi... |
| rneqdmfinf1o 9231 | Any function from a finite... |
| fidomdm 9232 | Any finite set dominates i... |
| dmfi 9233 | The domain of a finite set... |
| fundmfibi 9234 | A function is finite if an... |
| resfnfinfin 9235 | The restriction of a funct... |
| residfi 9236 | A restricted identity func... |
| cnvfiALT 9237 | Shorter proof of ~ cnvfi u... |
| rnfi 9238 | The range of a finite set ... |
| f1dmvrnfibi 9239 | A one-to-one function whos... |
| f1vrnfibi 9240 | A one-to-one function whic... |
| iunfi 9241 | The finite union of finite... |
| unifi 9242 | The finite union of finite... |
| unifi2 9243 | The finite union of finite... |
| infssuni 9244 | If an infinite set ` A ` i... |
| unirnffid 9245 | The union of the range of ... |
| mapfi 9246 | Set exponentiation of fini... |
| ixpfi 9247 | A Cartesian product of fin... |
| ixpfi2 9248 | A Cartesian product of fin... |
| mptfi 9249 | A finite mapping set is fi... |
| abrexfi 9250 | An image set from a finite... |
| cnvimamptfin 9251 | A preimage of a mapping wi... |
| elfpw 9252 | Membership in a class of f... |
| unifpw 9253 | A set is the union of its ... |
| f1opwfi 9254 | A one-to-one mapping induc... |
| fissuni 9255 | A finite subset of a union... |
| fipreima 9256 | Given a finite subset ` A ... |
| finsschain 9257 | A finite subset of the uni... |
| indexfi 9258 | If for every element of a ... |
| imafi2 9259 | The image by a finite set ... |
| unifi3 9260 | If a union is finite, then... |
| tfsnfin2 9261 | A transfinite sequence is ... |
| relfsupp 9264 | The property of a function... |
| relprcnfsupp 9265 | A proper class is never fi... |
| isfsupp 9266 | The property of a class to... |
| isfsuppd 9267 | Deduction form of ~ isfsup... |
| funisfsupp 9268 | The property of a function... |
| fsuppimp 9269 | Implications of a class be... |
| fsuppimpd 9270 | A finitely supported funct... |
| fsuppfund 9271 | A finitely supported funct... |
| fisuppfi 9272 | A function on a finite set... |
| fidmfisupp 9273 | A function with a finite d... |
| finnzfsuppd 9274 | If a function is zero outs... |
| fdmfisuppfi 9275 | The support of a function ... |
| fdmfifsupp 9276 | A function with a finite d... |
| fsuppmptdm 9277 | A mapping with a finite do... |
| fndmfisuppfi 9278 | The support of a function ... |
| fndmfifsupp 9279 | A function with a finite d... |
| suppeqfsuppbi 9280 | If two functions have the ... |
| suppssfifsupp 9281 | If the support of a functi... |
| fsuppsssupp 9282 | If the support of a functi... |
| fsuppsssuppgd 9283 | If the support of a functi... |
| fsuppss 9284 | A subset of a finitely sup... |
| fsuppssov1 9285 | Formula building theorem f... |
| fsuppxpfi 9286 | The cartesian product of t... |
| fczfsuppd 9287 | A constant function with v... |
| fsuppun 9288 | The union of two finitely ... |
| fsuppunfi 9289 | The union of the support o... |
| fsuppunbi 9290 | If the union of two classe... |
| 0fsupp 9291 | The empty set is a finitel... |
| snopfsupp 9292 | A singleton containing an ... |
| funsnfsupp 9293 | Finite support for a funct... |
| fsuppres 9294 | The restriction of a finit... |
| fmptssfisupp 9295 | The restriction of a mappi... |
| ressuppfi 9296 | If the support of the rest... |
| resfsupp 9297 | If the restriction of a fu... |
| resfifsupp 9298 | The restriction of a funct... |
| ffsuppbi 9299 | Two ways of saying that a ... |
| fsuppmptif 9300 | A function mapping an argu... |
| sniffsupp 9301 | A function mapping all but... |
| fsuppcolem 9302 | Lemma for ~ fsuppco . For... |
| fsuppco 9303 | The composition of a 1-1 f... |
| fsuppco2 9304 | The composition of a funct... |
| fsuppcor 9305 | The composition of a funct... |
| mapfienlem1 9306 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9307 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9308 | Lemma 3 for ~ mapfien . (... |
| mapfien 9309 | A bijection of the base se... |
| mapfien2 9310 | Equinumerousity relation f... |
| fival 9313 | The set of all the finite ... |
| elfi 9314 | Specific properties of an ... |
| elfi2 9315 | The empty intersection nee... |
| elfir 9316 | Sufficient condition for a... |
| intrnfi 9317 | Sufficient condition for t... |
| iinfi 9318 | An indexed intersection of... |
| inelfi 9319 | The intersection of two se... |
| ssfii 9320 | Any element of a set ` A `... |
| fi0 9321 | The set of finite intersec... |
| fieq0 9322 | A set is empty iff the cla... |
| fiin 9323 | The elements of ` ( fi `` ... |
| dffi2 9324 | The set of finite intersec... |
| fiss 9325 | Subset relationship for fu... |
| inficl 9326 | A set which is closed unde... |
| fipwuni 9327 | The set of finite intersec... |
| fisn 9328 | A singleton is closed unde... |
| fiuni 9329 | The union of the finite in... |
| fipwss 9330 | If a set is a family of su... |
| elfiun 9331 | A finite intersection of e... |
| dffi3 9332 | The set of finite intersec... |
| fifo 9333 | Describe a surjection from... |
| marypha1lem 9334 | Core induction for Philip ... |
| marypha1 9335 | (Philip) Hall's marriage t... |
| marypha2lem1 9336 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9337 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9338 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9339 | Lemma for ~ marypha2 . Pr... |
| marypha2 9340 | Version of ~ marypha1 usin... |
| dfsup2 9345 | Quantifier-free definition... |
| supeq1 9346 | Equality theorem for supre... |
| supeq1d 9347 | Equality deduction for sup... |
| supeq1i 9348 | Equality inference for sup... |
| supeq2 9349 | Equality theorem for supre... |
| supeq3 9350 | Equality theorem for supre... |
| supeq123d 9351 | Equality deduction for sup... |
| nfsup 9352 | Hypothesis builder for sup... |
| supmo 9353 | Any class ` B ` has at mos... |
| supexd 9354 | A supremum is a set. (Con... |
| supeu 9355 | A supremum is unique. Sim... |
| supval2 9356 | Alternate expression for t... |
| eqsup 9357 | Sufficient condition for a... |
| eqsupd 9358 | Sufficient condition for a... |
| supcl 9359 | A supremum belongs to its ... |
| supub 9360 | A supremum is an upper bou... |
| suplub 9361 | A supremum is the least up... |
| suplub2 9362 | Bidirectional form of ~ su... |
| supnub 9363 | An upper bound is not less... |
| supssd 9364 | Inequality deduction for s... |
| supex 9365 | A supremum is a set. (Con... |
| sup00 9366 | The supremum under an empt... |
| sup0riota 9367 | The supremum of an empty s... |
| sup0 9368 | The supremum of an empty s... |
| supmax 9369 | The greatest element of a ... |
| fisup2g 9370 | A finite set satisfies the... |
| fisupcl 9371 | A nonempty finite set cont... |
| supgtoreq 9372 | The supremum of a finite s... |
| suppr 9373 | The supremum of a pair. (... |
| supsn 9374 | The supremum of a singleto... |
| supisolem 9375 | Lemma for ~ supiso . (Con... |
| supisoex 9376 | Lemma for ~ supiso . (Con... |
| supiso 9377 | Image of a supremum under ... |
| infeq1 9378 | Equality theorem for infim... |
| infeq1d 9379 | Equality deduction for inf... |
| infeq1i 9380 | Equality inference for inf... |
| infeq2 9381 | Equality theorem for infim... |
| infeq3 9382 | Equality theorem for infim... |
| infeq123d 9383 | Equality deduction for inf... |
| nfinf 9384 | Hypothesis builder for inf... |
| infexd 9385 | An infimum is a set. (Con... |
| eqinf 9386 | Sufficient condition for a... |
| eqinfd 9387 | Sufficient condition for a... |
| infval 9388 | Alternate expression for t... |
| infcllem 9389 | Lemma for ~ infcl , ~ infl... |
| infcl 9390 | An infimum belongs to its ... |
| inflb 9391 | An infimum is a lower boun... |
| infglb 9392 | An infimum is the greatest... |
| infglbb 9393 | Bidirectional form of ~ in... |
| infnlb 9394 | A lower bound is not great... |
| infssd 9395 | Inequality deduction for i... |
| infex 9396 | An infimum is a set. (Con... |
| infmin 9397 | The smallest element of a ... |
| infmo 9398 | Any class ` B ` has at mos... |
| infeu 9399 | An infimum is unique. (Co... |
| fimin2g 9400 | A finite set has a minimum... |
| fiming 9401 | A finite set has a minimum... |
| fiinfg 9402 | Lemma showing existence an... |
| fiinf2g 9403 | A finite set satisfies the... |
| fiinfcl 9404 | A nonempty finite set cont... |
| infltoreq 9405 | The infimum of a finite se... |
| infpr 9406 | The infimum of a pair. (C... |
| infsupprpr 9407 | The infimum of a proper pa... |
| infsn 9408 | The infimum of a singleton... |
| inf00 9409 | The infimum regarding an e... |
| infempty 9410 | The infimum of an empty se... |
| infiso 9411 | Image of an infimum under ... |
| dfoi 9414 | Rewrite ~ df-oi with abbre... |
| oieq1 9415 | Equality theorem for ordin... |
| oieq2 9416 | Equality theorem for ordin... |
| nfoi 9417 | Hypothesis builder for ord... |
| ordiso2 9418 | Generalize ~ ordiso to pro... |
| ordiso 9419 | Order-isomorphic ordinal n... |
| ordtypecbv 9420 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9421 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9422 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9423 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9424 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9425 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9426 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9427 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9428 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9429 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9430 | Lemma for ~ ordtype . Usi... |
| oi0 9431 | Definition of the ordinal ... |
| oicl 9432 | The order type of the well... |
| oif 9433 | The order isomorphism of t... |
| oiiso2 9434 | The order isomorphism of t... |
| ordtype 9435 | For any set-like well-orde... |
| oiiniseg 9436 | ` ran F ` is an initial se... |
| ordtype2 9437 | For any set-like well-orde... |
| oiexg 9438 | The order isomorphism on a... |
| oion 9439 | The order type of the well... |
| oiiso 9440 | The order isomorphism of t... |
| oien 9441 | The order type of a well-o... |
| oieu 9442 | Uniqueness of the unique o... |
| oismo 9443 | When ` A ` is a subclass o... |
| oiid 9444 | The order type of an ordin... |
| hartogslem1 9445 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9446 | Lemma for ~ hartogs . (Co... |
| hartogs 9447 | The class of ordinals domi... |
| wofib 9448 | The only sets which are we... |
| wemaplem1 9449 | Value of the lexicographic... |
| wemaplem2 9450 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9451 | Lemma for ~ wemapso . Tra... |
| wemappo 9452 | Construct lexicographic or... |
| wemapsolem 9453 | Lemma for ~ wemapso . (Co... |
| wemapso 9454 | Construct lexicographic or... |
| wemapso2lem 9455 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9456 | An alternative to having a... |
| card2on 9457 | The alternate definition o... |
| card2inf 9458 | The alternate definition o... |
| harf 9461 | Functionality of the Harto... |
| harcl 9462 | Values of the Hartogs func... |
| harval 9463 | Function value of the Hart... |
| elharval 9464 | The Hartogs number of a se... |
| harndom 9465 | The Hartogs number of a se... |
| harword 9466 | Weak ordering property of ... |
| relwdom 9469 | Weak dominance is a relati... |
| brwdom 9470 | Property of weak dominance... |
| brwdomi 9471 | Property of weak dominance... |
| brwdomn0 9472 | Weak dominance over nonemp... |
| 0wdom 9473 | Any set weakly dominates t... |
| fowdom 9474 | An onto function implies w... |
| wdomref 9475 | Reflexivity of weak domina... |
| brwdom2 9476 | Alternate characterization... |
| domwdom 9477 | Weak dominance is implied ... |
| wdomtr 9478 | Transitivity of weak domin... |
| wdomen1 9479 | Equality-like theorem for ... |
| wdomen2 9480 | Equality-like theorem for ... |
| wdompwdom 9481 | Weak dominance strengthens... |
| canthwdom 9482 | Cantor's Theorem, stated u... |
| wdom2d 9483 | Deduce weak dominance from... |
| wdomd 9484 | Deduce weak dominance from... |
| brwdom3 9485 | Condition for weak dominan... |
| brwdom3i 9486 | Weak dominance implies exi... |
| unwdomg 9487 | Weak dominance of a (disjo... |
| xpwdomg 9488 | Weak dominance of a Cartes... |
| wdomima2g 9489 | A set is weakly dominant o... |
| wdomimag 9490 | A set is weakly dominant o... |
| unxpwdom2 9491 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9492 | If a Cartesian product is ... |
| ixpiunwdom 9493 | Describe an onto function ... |
| harwdom 9494 | The value of the Hartogs f... |
| axreg2 9496 | Axiom of Regularity expres... |
| zfregcl 9497 | The Axiom of Regularity wi... |
| zfregclOLD 9498 | Obsolete version of ~ zfre... |
| zfreg 9499 | The Axiom of Regularity us... |
| elirrv 9500 | The membership relation is... |
| elirrvOLD 9501 | Obsolete version of ~ elir... |
| elirr 9502 | No class is a member of it... |
| elneq 9503 | A class is not equal to an... |
| nelaneq 9504 | A class is not an element ... |
| nelaneqOLD 9505 | Obsolete version of ~ nela... |
| epinid0 9506 | The membership relation an... |
| sucprcreg 9507 | A class is equal to its su... |
| ruv 9508 | The Russell class is equal... |
| ruALT 9509 | Alternate proof of ~ ru , ... |
| disjcsn 9510 | A class is disjoint from i... |
| zfregfr 9511 | The membership relation is... |
| elirrvALT 9512 | Alternate proof of ~ elirr... |
| en2lp 9513 | No class has 2-cycle membe... |
| elnanel 9514 | Two classes are not elemen... |
| cnvepnep 9515 | The membership (epsilon) r... |
| epnsym 9516 | The membership (epsilon) r... |
| elnotel 9517 | A class cannot be an eleme... |
| elnel 9518 | A class cannot be an eleme... |
| en3lplem1 9519 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9520 | Lemma for ~ en3lp . (Cont... |
| en3lp 9521 | No class has 3-cycle membe... |
| preleqg 9522 | Equality of two unordered ... |
| preleq 9523 | Equality of two unordered ... |
| preleqALT 9524 | Alternate proof of ~ prele... |
| opthreg 9525 | Theorem for alternate repr... |
| suc11reg 9526 | The successor operation be... |
| dford2 9527 | Assuming ~ ax-reg , an ord... |
| inf0 9528 | Existence of ` _om ` impli... |
| inf1 9529 | Variation of Axiom of Infi... |
| inf2 9530 | Variation of Axiom of Infi... |
| inf3lema 9531 | Lemma for our Axiom of Inf... |
| inf3lemb 9532 | Lemma for our Axiom of Inf... |
| inf3lemc 9533 | Lemma for our Axiom of Inf... |
| inf3lemd 9534 | Lemma for our Axiom of Inf... |
| inf3lem1 9535 | Lemma for our Axiom of Inf... |
| inf3lem2 9536 | Lemma for our Axiom of Inf... |
| inf3lem3 9537 | Lemma for our Axiom of Inf... |
| inf3lem4 9538 | Lemma for our Axiom of Inf... |
| inf3lem5 9539 | Lemma for our Axiom of Inf... |
| inf3lem6 9540 | Lemma for our Axiom of Inf... |
| inf3lem7 9541 | Lemma for our Axiom of Inf... |
| inf3 9542 | Our Axiom of Infinity ~ ax... |
| infeq5i 9543 | Half of ~ infeq5 . (Contr... |
| infeq5 9544 | The statement "there exist... |
| zfinf 9546 | Axiom of Infinity expresse... |
| axinf2 9547 | A standard version of Axio... |
| zfinf2 9549 | A standard version of the ... |
| omex 9550 | The existence of omega (th... |
| axinf 9551 | The first version of the A... |
| inf5 9552 | The statement "there exist... |
| omelon 9553 | Omega is an ordinal number... |
| dfom3 9554 | The class of natural numbe... |
| elom3 9555 | A simplification of ~ elom... |
| dfom4 9556 | A simplification of ~ df-o... |
| dfom5 9557 | ` _om ` is the smallest li... |
| oancom 9558 | Ordinal addition is not co... |
| isfinite 9559 | A set is finite iff it is ... |
| fict 9560 | A finite set is countable ... |
| nnsdom 9561 | A natural number is strict... |
| omenps 9562 | Omega is equinumerous to a... |
| omensuc 9563 | The set of natural numbers... |
| infdifsn 9564 | Removing a singleton from ... |
| infdiffi 9565 | Removing a finite set from... |
| unbnn3 9566 | Any unbounded subset of na... |
| noinfep 9567 | Using the Axiom of Regular... |
| cantnffval 9570 | The value of the Cantor no... |
| cantnfdm 9571 | The domain of the Cantor n... |
| cantnfvalf 9572 | Lemma for ~ cantnf . The ... |
| cantnfs 9573 | Elementhood in the set of ... |
| cantnfcl 9574 | Basic properties of the or... |
| cantnfval 9575 | The value of the Cantor no... |
| cantnfval2 9576 | Alternate expression for t... |
| cantnfsuc 9577 | The value of the recursive... |
| cantnfle 9578 | A lower bound on the ` CNF... |
| cantnflt 9579 | An upper bound on the part... |
| cantnflt2 9580 | An upper bound on the ` CN... |
| cantnff 9581 | The ` CNF ` function is a ... |
| cantnf0 9582 | The value of the zero func... |
| cantnfrescl 9583 | A function is finitely sup... |
| cantnfres 9584 | The ` CNF ` function respe... |
| cantnfp1lem1 9585 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9586 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9587 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9588 | If ` F ` is created by add... |
| oemapso 9589 | The relation ` T ` is a st... |
| oemapval 9590 | Value of the relation ` T ... |
| oemapvali 9591 | If ` F < G ` , then there ... |
| cantnflem1a 9592 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9593 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9594 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9595 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9596 | Lemma for ~ cantnf . This... |
| cantnflem2 9597 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9598 | Lemma for ~ cantnf . Here... |
| cantnflem4 9599 | Lemma for ~ cantnf . Comp... |
| cantnf 9600 | The Cantor Normal Form the... |
| oemapwe 9601 | The lexicographic order on... |
| cantnffval2 9602 | An alternate definition of... |
| cantnff1o 9603 | Simplify the isomorphism o... |
| wemapwe 9604 | Construct lexicographic or... |
| oef1o 9605 | A bijection of the base se... |
| cnfcomlem 9606 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9607 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9608 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9609 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9610 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9611 | Any infinite ordinal ` B `... |
| cnfcom3clem 9612 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9613 | Wrap the construction of ~... |
| ttrcleq 9616 | Equality theorem for trans... |
| nfttrcld 9617 | Bound variable hypothesis ... |
| nfttrcl 9618 | Bound variable hypothesis ... |
| relttrcl 9619 | The transitive closure of ... |
| brttrcl 9620 | Characterization of elemen... |
| brttrcl2 9621 | Characterization of elemen... |
| ssttrcl 9622 | If ` R ` is a relation, th... |
| ttrcltr 9623 | The transitive closure of ... |
| ttrclresv 9624 | The transitive closure of ... |
| ttrclco 9625 | Composition law for the tr... |
| cottrcl 9626 | Composition law for the tr... |
| ttrclss 9627 | If ` R ` is a subclass of ... |
| dmttrcl 9628 | The domain of a transitive... |
| rnttrcl 9629 | The range of a transitive ... |
| ttrclexg 9630 | If ` R ` is a set, then so... |
| dfttrcl2 9631 | When ` R ` is a set and a ... |
| ttrclselem1 9632 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9633 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9634 | If ` R ` is set-like over ... |
| trcl 9635 | For any set ` A ` , show t... |
| tz9.1 9636 | Every set has a transitive... |
| tz9.1c 9637 | Alternate expression for t... |
| epfrs 9638 | The strong form of the Axi... |
| zfregs 9639 | The strong form of the Axi... |
| zfregs2 9640 | Alternate strong form of t... |
| tcvalg 9643 | Value of the transitive cl... |
| tcid 9644 | Defining property of the t... |
| tctr 9645 | Defining property of the t... |
| tcmin 9646 | Defining property of the t... |
| tc2 9647 | A variant of the definitio... |
| tcsni 9648 | The transitive closure of ... |
| tcss 9649 | The transitive closure fun... |
| tcel 9650 | The transitive closure fun... |
| tcidm 9651 | The transitive closure fun... |
| tc0 9652 | The transitive closure of ... |
| tc00 9653 | The transitive closure is ... |
| setind 9654 | Set (epsilon) induction. ... |
| setind2 9655 | Set (epsilon) induction, s... |
| setinds 9656 | Principle of set induction... |
| setinds2f 9657 | ` _E ` induction schema, u... |
| setinds2 9658 | ` _E ` induction schema, u... |
| frmin 9659 | Every (possibly proper) su... |
| frind 9660 | A subclass of a well-found... |
| frinsg 9661 | Well-Founded Induction Sch... |
| frins 9662 | Well-Founded Induction Sch... |
| frins2f 9663 | Well-Founded Induction sch... |
| frins2 9664 | Well-Founded Induction sch... |
| frins3 9665 | Well-Founded Induction sch... |
| frr3g 9666 | Functions defined by well-... |
| frrlem15 9667 | Lemma for general well-fou... |
| frrlem16 9668 | Lemma for general well-fou... |
| frr1 9669 | Law of general well-founde... |
| frr2 9670 | Law of general well-founde... |
| frr3 9671 | Law of general well-founde... |
| r1funlim 9676 | The cumulative hierarchy o... |
| r1fnon 9677 | The cumulative hierarchy o... |
| r10 9678 | Value of the cumulative hi... |
| r1sucg 9679 | Value of the cumulative hi... |
| r1suc 9680 | Value of the cumulative hi... |
| r1limg 9681 | Value of the cumulative hi... |
| r1lim 9682 | Value of the cumulative hi... |
| r1fin 9683 | The first ` _om ` levels o... |
| r1sdom 9684 | Each stage in the cumulati... |
| r111 9685 | The cumulative hierarchy i... |
| r1tr 9686 | The cumulative hierarchy o... |
| r1tr2 9687 | The union of a cumulative ... |
| r1ordg 9688 | Ordering relation for the ... |
| r1ord3g 9689 | Ordering relation for the ... |
| r1ord 9690 | Ordering relation for the ... |
| r1ord2 9691 | Ordering relation for the ... |
| r1ord3 9692 | Ordering relation for the ... |
| r1sssuc 9693 | The value of the cumulativ... |
| r1pwss 9694 | Each set of the cumulative... |
| r1sscl 9695 | Each set of the cumulative... |
| r1val1 9696 | The value of the cumulativ... |
| tz9.12lem1 9697 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9698 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9699 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9700 | A set is well-founded if a... |
| tz9.13 9701 | Every set is well-founded,... |
| tz9.13g 9702 | Every set is well-founded,... |
| rankwflemb 9703 | Two ways of saying a set i... |
| rankf 9704 | The domain and codomain of... |
| rankon 9705 | The rank of a set is an or... |
| r1elwf 9706 | Any member of the cumulati... |
| rankvalb 9707 | Value of the rank function... |
| rankr1ai 9708 | One direction of ~ rankr1a... |
| rankvaln 9709 | Value of the rank function... |
| rankidb 9710 | Identity law for the rank ... |
| rankdmr1 9711 | A rank is a member of the ... |
| rankr1ag 9712 | A version of ~ rankr1a tha... |
| rankr1bg 9713 | A relationship between ran... |
| r1rankidb 9714 | Any set is a subset of the... |
| r1elssi 9715 | The range of the ` R1 ` fu... |
| r1elss 9716 | The range of the ` R1 ` fu... |
| pwwf 9717 | A power set is well-founde... |
| sswf 9718 | A subset of a well-founded... |
| snwf 9719 | A singleton is well-founde... |
| unwf 9720 | A binary union is well-fou... |
| prwf 9721 | An unordered pair is well-... |
| opwf 9722 | An ordered pair is well-fo... |
| unir1 9723 | The cumulative hierarchy o... |
| jech9.3 9724 | Every set belongs to some ... |
| rankwflem 9725 | Every set is well-founded,... |
| rankval 9726 | Value of the rank function... |
| rankvalg 9727 | Value of the rank function... |
| rankval2 9728 | Value of an alternate defi... |
| uniwf 9729 | A union is well-founded if... |
| rankr1clem 9730 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9731 | A relationship between the... |
| rankidn 9732 | A relationship between the... |
| rankpwi 9733 | The rank of a power set. ... |
| rankelb 9734 | The membership relation is... |
| wfelirr 9735 | A well-founded set is not ... |
| rankval3b 9736 | The value of the rank func... |
| ranksnb 9737 | The rank of a singleton. ... |
| rankonidlem 9738 | Lemma for ~ rankonid . (C... |
| rankonid 9739 | The rank of an ordinal num... |
| onwf 9740 | The ordinals are all well-... |
| onssr1 9741 | Initial segments of the or... |
| rankr1g 9742 | A relationship between the... |
| rankid 9743 | Identity law for the rank ... |
| rankr1 9744 | A relationship between the... |
| ssrankr1 9745 | A relationship between an ... |
| rankr1a 9746 | A relationship between ran... |
| r1val2 9747 | The value of the cumulativ... |
| r1val3 9748 | The value of the cumulativ... |
| rankel 9749 | The membership relation is... |
| rankval3 9750 | The value of the rank func... |
| bndrank 9751 | Any class whose elements h... |
| unbndrank 9752 | The elements of a proper c... |
| rankpw 9753 | The rank of a power set. ... |
| ranklim 9754 | The rank of a set belongs ... |
| r1pw 9755 | A stronger property of ` R... |
| r1pwALT 9756 | Alternate shorter proof of... |
| r1pwcl 9757 | The cumulative hierarchy o... |
| rankssb 9758 | The subset relation is inh... |
| rankss 9759 | The subset relation is inh... |
| rankunb 9760 | The rank of the union of t... |
| rankprb 9761 | The rank of an unordered p... |
| rankopb 9762 | The rank of an ordered pai... |
| rankuni2b 9763 | The value of the rank func... |
| ranksn 9764 | The rank of a singleton. ... |
| rankuni2 9765 | The rank of a union. Part... |
| rankun 9766 | The rank of the union of t... |
| rankpr 9767 | The rank of an unordered p... |
| rankop 9768 | The rank of an ordered pai... |
| r1rankid 9769 | Any set is a subset of the... |
| rankeq0b 9770 | A set is empty iff its ran... |
| rankeq0 9771 | A set is empty iff its ran... |
| rankr1id 9772 | The rank of the hierarchy ... |
| rankuni 9773 | The rank of a union. Part... |
| rankr1b 9774 | A relationship between ran... |
| ranksuc 9775 | The rank of a successor. ... |
| rankuniss 9776 | Upper bound of the rank of... |
| rankval4 9777 | The rank of a set is the s... |
| rankbnd 9778 | The rank of a set is bound... |
| rankbnd2 9779 | The rank of a set is bound... |
| rankc1 9780 | A relationship that can be... |
| rankc2 9781 | A relationship that can be... |
| rankelun 9782 | Rank membership is inherit... |
| rankelpr 9783 | Rank membership is inherit... |
| rankelop 9784 | Rank membership is inherit... |
| rankxpl 9785 | A lower bound on the rank ... |
| rankxpu 9786 | An upper bound on the rank... |
| rankfu 9787 | An upper bound on the rank... |
| rankmapu 9788 | An upper bound on the rank... |
| rankxplim 9789 | The rank of a Cartesian pr... |
| rankxplim2 9790 | If the rank of a Cartesian... |
| rankxplim3 9791 | The rank of a Cartesian pr... |
| rankxpsuc 9792 | The rank of a Cartesian pr... |
| tcwf 9793 | The transitive closure fun... |
| tcrank 9794 | This theorem expresses two... |
| scottex 9795 | Scott's trick collects all... |
| scott0 9796 | Scott's trick collects all... |
| scottexs 9797 | Theorem scheme version of ... |
| scott0s 9798 | Theorem scheme version of ... |
| cplem1 9799 | Lemma for the Collection P... |
| cplem2 9800 | Lemma for the Collection P... |
| cp 9801 | Collection Principle. Thi... |
| bnd 9802 | A very strong generalizati... |
| bnd2 9803 | A variant of the Boundedne... |
| kardex 9804 | The collection of all sets... |
| karden 9805 | If we allow the Axiom of R... |
| htalem 9806 | Lemma for defining an emul... |
| hta 9807 | A ZFC emulation of Hilbert... |
| djueq12 9814 | Equality theorem for disjo... |
| djueq1 9815 | Equality theorem for disjo... |
| djueq2 9816 | Equality theorem for disjo... |
| nfdju 9817 | Bound-variable hypothesis ... |
| djuex 9818 | The disjoint union of sets... |
| djuexb 9819 | The disjoint union of two ... |
| djulcl 9820 | Left closure of disjoint u... |
| djurcl 9821 | Right closure of disjoint ... |
| djulf1o 9822 | The left injection functio... |
| djurf1o 9823 | The right injection functi... |
| inlresf 9824 | The left injection restric... |
| inlresf1 9825 | The left injection restric... |
| inrresf 9826 | The right injection restri... |
| inrresf1 9827 | The right injection restri... |
| djuin 9828 | The images of any classes ... |
| djur 9829 | A member of a disjoint uni... |
| djuss 9830 | A disjoint union is a subc... |
| djuunxp 9831 | The union of a disjoint un... |
| djuexALT 9832 | Alternate proof of ~ djuex... |
| eldju1st 9833 | The first component of an ... |
| eldju2ndl 9834 | The second component of an... |
| eldju2ndr 9835 | The second component of an... |
| djuun 9836 | The disjoint union of two ... |
| 1stinl 9837 | The first component of the... |
| 2ndinl 9838 | The second component of th... |
| 1stinr 9839 | The first component of the... |
| 2ndinr 9840 | The second component of th... |
| updjudhf 9841 | The mapping of an element ... |
| updjudhcoinlf 9842 | The composition of the map... |
| updjudhcoinrg 9843 | The composition of the map... |
| updjud 9844 | Universal property of the ... |
| cardf2 9853 | The cardinality function i... |
| cardon 9854 | The cardinal number of a s... |
| isnum2 9855 | A way to express well-orde... |
| isnumi 9856 | A set equinumerous to an o... |
| ennum 9857 | Equinumerous sets are equi... |
| finnum 9858 | Every finite set is numera... |
| onenon 9859 | Every ordinal number is nu... |
| tskwe 9860 | A Tarski set is well-order... |
| xpnum 9861 | The cartesian product of n... |
| cardval3 9862 | An alternate definition of... |
| cardid2 9863 | Any numerable set is equin... |
| isnum3 9864 | A set is numerable iff it ... |
| oncardval 9865 | The value of the cardinal ... |
| oncardid 9866 | Any ordinal number is equi... |
| cardonle 9867 | The cardinal of an ordinal... |
| card0 9868 | The cardinality of the emp... |
| cardidm 9869 | The cardinality function i... |
| oncard 9870 | A set is a cardinal number... |
| ficardom 9871 | The cardinal number of a f... |
| ficardid 9872 | A finite set is equinumero... |
| cardnn 9873 | The cardinality of a natur... |
| cardnueq0 9874 | The empty set is the only ... |
| cardne 9875 | No member of a cardinal nu... |
| carden2a 9876 | If two sets have equal non... |
| carden2b 9877 | If two sets are equinumero... |
| card1 9878 | A set has cardinality one ... |
| cardsn 9879 | A singleton has cardinalit... |
| carddomi2 9880 | Two sets have the dominanc... |
| sdomsdomcardi 9881 | A set strictly dominates i... |
| cardlim 9882 | An infinite cardinal is a ... |
| cardsdomelir 9883 | A cardinal strictly domina... |
| cardsdomel 9884 | A cardinal strictly domina... |
| iscard 9885 | Two ways to express the pr... |
| iscard2 9886 | Two ways to express the pr... |
| carddom2 9887 | Two numerable sets have th... |
| harcard 9888 | The class of ordinal numbe... |
| cardprclem 9889 | Lemma for ~ cardprc . (Co... |
| cardprc 9890 | The class of all cardinal ... |
| carduni 9891 | The union of a set of card... |
| cardiun 9892 | The indexed union of a set... |
| cardennn 9893 | If ` A ` is equinumerous t... |
| cardsucinf 9894 | The cardinality of the suc... |
| cardsucnn 9895 | The cardinality of the suc... |
| cardom 9896 | The set of natural numbers... |
| carden2 9897 | Two numerable sets are equ... |
| cardsdom2 9898 | A numerable set is strictl... |
| domtri2 9899 | Trichotomy of dominance fo... |
| nnsdomel 9900 | Strict dominance and eleme... |
| cardval2 9901 | An alternate version of th... |
| isinffi 9902 | An infinite set contains s... |
| fidomtri 9903 | Trichotomy of dominance wi... |
| fidomtri2 9904 | Trichotomy of dominance wi... |
| harsdom 9905 | The Hartogs number of a we... |
| onsdom 9906 | Any well-orderable set is ... |
| harval2 9907 | An alternate expression fo... |
| harsucnn 9908 | The next cardinal after a ... |
| cardmin2 9909 | The smallest ordinal that ... |
| pm54.43lem 9910 | In Theorem *54.43 of [Whit... |
| pm54.43 9911 | Theorem *54.43 of [Whitehe... |
| enpr2 9912 | An unordered pair with dis... |
| pr2ne 9913 | If an unordered pair has t... |
| prdom2 9914 | An unordered pair has at m... |
| en2eqpr 9915 | Building a set with two el... |
| en2eleq 9916 | Express a set of pair card... |
| en2other2 9917 | Taking the other element t... |
| dif1card 9918 | The cardinality of a nonem... |
| leweon 9919 | Lexicographical order is a... |
| r0weon 9920 | A set-like well-ordering o... |
| infxpenlem 9921 | Lemma for ~ infxpen . (Co... |
| infxpen 9922 | Every infinite ordinal is ... |
| xpomen 9923 | The Cartesian product of o... |
| xpct 9924 | The cartesian product of t... |
| infxpidm2 9925 | Every infinite well-ordera... |
| infxpenc 9926 | A canonical version of ~ i... |
| infxpenc2lem1 9927 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9928 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9929 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9930 | Existence form of ~ infxpe... |
| iunmapdisj 9931 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9932 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9933 | Lemma for ~ fseqen . (Con... |
| fseqdom 9934 | One half of ~ fseqen . (C... |
| fseqen 9935 | A set that is equinumerous... |
| infpwfidom 9936 | The collection of finite s... |
| dfac8alem 9937 | Lemma for ~ dfac8a . If t... |
| dfac8a 9938 | Numeration theorem: every ... |
| dfac8b 9939 | The well-ordering theorem:... |
| dfac8clem 9940 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9941 | If the union of a set is w... |
| ac10ct 9942 | A proof of the well-orderi... |
| ween 9943 | A set is numerable iff it ... |
| ac5num 9944 | A version of ~ ac5b with t... |
| ondomen 9945 | If a set is dominated by a... |
| numdom 9946 | A set dominated by a numer... |
| ssnum 9947 | A subset of a numerable se... |
| onssnum 9948 | All subsets of the ordinal... |
| indcardi 9949 | Indirect strong induction ... |
| acnrcl 9950 | Reverse closure for the ch... |
| acneq 9951 | Equality theorem for the c... |
| isacn 9952 | The property of being a ch... |
| acni 9953 | The property of being a ch... |
| acni2 9954 | The property of being a ch... |
| acni3 9955 | The property of being a ch... |
| acnlem 9956 | Construct a mapping satisf... |
| numacn 9957 | A well-orderable set has c... |
| finacn 9958 | Every set has finite choic... |
| acndom 9959 | A set with long choice seq... |
| acnnum 9960 | A set ` X ` which has choi... |
| acnen 9961 | The class of choice sets o... |
| acndom2 9962 | A set smaller than one wit... |
| acnen2 9963 | The class of sets with cho... |
| fodomacn 9964 | A version of ~ fodom that ... |
| fodomnum 9965 | A version of ~ fodom that ... |
| fonum 9966 | A surjection maps numerabl... |
| numwdom 9967 | A surjection maps numerabl... |
| fodomfi2 9968 | Onto functions define domi... |
| wdomfil 9969 | Weak dominance agrees with... |
| infpwfien 9970 | Any infinite well-orderabl... |
| inffien 9971 | The set of finite intersec... |
| wdomnumr 9972 | Weak dominance agrees with... |
| alephfnon 9973 | The aleph function is a fu... |
| aleph0 9974 | The first infinite cardina... |
| alephlim 9975 | Value of the aleph functio... |
| alephsuc 9976 | Value of the aleph functio... |
| alephon 9977 | An aleph is an ordinal num... |
| alephcard 9978 | Every aleph is a cardinal ... |
| alephnbtwn 9979 | No cardinal can be sandwic... |
| alephnbtwn2 9980 | No set has equinumerosity ... |
| alephordilem1 9981 | Lemma for ~ alephordi . (... |
| alephordi 9982 | Strict ordering property o... |
| alephord 9983 | Ordering property of the a... |
| alephord2 9984 | Ordering property of the a... |
| alephord2i 9985 | Ordering property of the a... |
| alephord3 9986 | Ordering property of the a... |
| alephsucdom 9987 | A set dominated by an alep... |
| alephsuc2 9988 | An alternate representatio... |
| alephdom 9989 | Relationship between inclu... |
| alephgeom 9990 | Every aleph is greater tha... |
| alephislim 9991 | Every aleph is a limit ord... |
| aleph11 9992 | The aleph function is one-... |
| alephf1 9993 | The aleph function is a on... |
| alephsdom 9994 | If an ordinal is smaller t... |
| alephdom2 9995 | A dominated initial ordina... |
| alephle 9996 | The argument of the aleph ... |
| cardaleph 9997 | Given any transfinite card... |
| cardalephex 9998 | Every transfinite cardinal... |
| infenaleph 9999 | An infinite numerable set ... |
| isinfcard 10000 | Two ways to express the pr... |
| iscard3 10001 | Two ways to express the pr... |
| cardnum 10002 | Two ways to express the cl... |
| alephinit 10003 | An infinite initial ordina... |
| carduniima 10004 | The union of the image of ... |
| cardinfima 10005 | If a mapping to cardinals ... |
| alephiso 10006 | Aleph is an order isomorph... |
| alephprc 10007 | The class of all transfini... |
| alephsson 10008 | The class of transfinite c... |
| unialeph 10009 | The union of the class of ... |
| alephsmo 10010 | The aleph function is stri... |
| alephf1ALT 10011 | Alternate proof of ~ aleph... |
| alephfplem1 10012 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10013 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10014 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10015 | Lemma for ~ alephfp . (Co... |
| alephfp 10016 | The aleph function has a f... |
| alephfp2 10017 | The aleph function has at ... |
| alephval3 10018 | An alternate way to expres... |
| alephsucpw2 10019 | The power set of an aleph ... |
| mappwen 10020 | Power rule for cardinal ar... |
| finnisoeu 10021 | A finite totally ordered s... |
| iunfictbso 10022 | Countability of a countabl... |
| aceq1 10025 | Equivalence of two version... |
| aceq0 10026 | Equivalence of two version... |
| aceq2 10027 | Equivalence of two version... |
| aceq3lem 10028 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10029 | Equivalence of two version... |
| dfac4 10030 | Equivalence of two version... |
| dfac5lem1 10031 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10032 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10033 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10034 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10035 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4OLD 10036 | Obsolete version of ~ dfac... |
| dfac5 10037 | Equivalence of two version... |
| dfac2a 10038 | Our Axiom of Choice (in th... |
| dfac2b 10039 | Axiom of Choice (first for... |
| dfac2 10040 | Axiom of Choice (first for... |
| dfac7 10041 | Equivalence of the Axiom o... |
| dfac0 10042 | Equivalence of two version... |
| dfac1 10043 | Equivalence of two version... |
| dfac8 10044 | A proof of the equivalency... |
| dfac9 10045 | Equivalence of the axiom o... |
| dfac10 10046 | Axiom of Choice equivalent... |
| dfac10c 10047 | Axiom of Choice equivalent... |
| dfac10b 10048 | Axiom of Choice equivalent... |
| acacni 10049 | A choice equivalent: every... |
| dfacacn 10050 | A choice equivalent: every... |
| dfac13 10051 | The axiom of choice holds ... |
| dfac12lem1 10052 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10053 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10054 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10055 | The axiom of choice holds ... |
| dfac12k 10056 | Equivalence of ~ dfac12 an... |
| dfac12a 10057 | The axiom of choice holds ... |
| dfac12 10058 | The axiom of choice holds ... |
| kmlem1 10059 | Lemma for 5-quantifier AC ... |
| kmlem2 10060 | Lemma for 5-quantifier AC ... |
| kmlem3 10061 | Lemma for 5-quantifier AC ... |
| kmlem4 10062 | Lemma for 5-quantifier AC ... |
| kmlem5 10063 | Lemma for 5-quantifier AC ... |
| kmlem6 10064 | Lemma for 5-quantifier AC ... |
| kmlem7 10065 | Lemma for 5-quantifier AC ... |
| kmlem8 10066 | Lemma for 5-quantifier AC ... |
| kmlem9 10067 | Lemma for 5-quantifier AC ... |
| kmlem10 10068 | Lemma for 5-quantifier AC ... |
| kmlem11 10069 | Lemma for 5-quantifier AC ... |
| kmlem12 10070 | Lemma for 5-quantifier AC ... |
| kmlem13 10071 | Lemma for 5-quantifier AC ... |
| kmlem14 10072 | Lemma for 5-quantifier AC ... |
| kmlem15 10073 | Lemma for 5-quantifier AC ... |
| kmlem16 10074 | Lemma for 5-quantifier AC ... |
| dfackm 10075 | Equivalence of the Axiom o... |
| undjudom 10076 | Cardinal addition dominate... |
| endjudisj 10077 | Equinumerosity of a disjoi... |
| djuen 10078 | Disjoint unions of equinum... |
| djuenun 10079 | Disjoint union is equinume... |
| dju1en 10080 | Cardinal addition with car... |
| dju1dif 10081 | Adding and subtracting one... |
| dju1p1e2 10082 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10083 | Alternate proof of ~ dju1p... |
| dju0en 10084 | Cardinal addition with car... |
| xp2dju 10085 | Two times a cardinal numbe... |
| djucomen 10086 | Commutative law for cardin... |
| djuassen 10087 | Associative law for cardin... |
| xpdjuen 10088 | Cardinal multiplication di... |
| mapdjuen 10089 | Sum of exponents law for c... |
| pwdjuen 10090 | Sum of exponents law for c... |
| djudom1 10091 | Ordering law for cardinal ... |
| djudom2 10092 | Ordering law for cardinal ... |
| djudoml 10093 | A set is dominated by its ... |
| djuxpdom 10094 | Cartesian product dominate... |
| djufi 10095 | The disjoint union of two ... |
| cdainflem 10096 | Any partition of omega int... |
| djuinf 10097 | A set is infinite iff the ... |
| infdju1 10098 | An infinite set is equinum... |
| pwdju1 10099 | The sum of a powerset with... |
| pwdjuidm 10100 | If the natural numbers inj... |
| djulepw 10101 | If ` A ` is idempotent und... |
| onadju 10102 | The cardinal and ordinal s... |
| cardadju 10103 | The cardinal sum is equinu... |
| djunum 10104 | The disjoint union of two ... |
| unnum 10105 | The union of two numerable... |
| nnadju 10106 | The cardinal and ordinal s... |
| nnadjuALT 10107 | Shorter proof of ~ nnadju ... |
| ficardadju 10108 | The disjoint union of fini... |
| ficardun 10109 | The cardinality of the uni... |
| ficardun2 10110 | The cardinality of the uni... |
| pwsdompw 10111 | Lemma for ~ domtriom . Th... |
| unctb 10112 | The union of two countable... |
| infdjuabs 10113 | Absorption law for additio... |
| infunabs 10114 | An infinite set is equinum... |
| infdju 10115 | The sum of two cardinal nu... |
| infdif 10116 | The cardinality of an infi... |
| infdif2 10117 | Cardinality ordering for a... |
| infxpdom 10118 | Dominance law for multipli... |
| infxpabs 10119 | Absorption law for multipl... |
| infunsdom1 10120 | The union of two sets that... |
| infunsdom 10121 | The union of two sets that... |
| infxp 10122 | Absorption law for multipl... |
| pwdjudom 10123 | A property of dominance ov... |
| infpss 10124 | Every infinite set has an ... |
| infmap2 10125 | An exponentiation law for ... |
| ackbij2lem1 10126 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10127 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10128 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10129 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10130 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10131 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10132 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10133 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10134 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10135 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10136 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10137 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10138 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10139 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10140 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10141 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10142 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10143 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10144 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10145 | The Ackermann bijection, p... |
| ackbij1b 10146 | The Ackermann bijection, p... |
| ackbij2lem2 10147 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10148 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10149 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10150 | The Ackermann bijection, p... |
| r1om 10151 | The set of hereditarily fi... |
| fictb 10152 | A set is countable iff its... |
| cflem 10153 | A lemma used to simplify c... |
| cflemOLD 10154 | Obsolete version of ~ cfle... |
| cfval 10155 | Value of the cofinality fu... |
| cff 10156 | Cofinality is a function o... |
| cfub 10157 | An upper bound on cofinali... |
| cflm 10158 | Value of the cofinality fu... |
| cf0 10159 | Value of the cofinality fu... |
| cardcf 10160 | Cofinality is a cardinal n... |
| cflecard 10161 | Cofinality is bounded by t... |
| cfle 10162 | Cofinality is bounded by i... |
| cfon 10163 | The cofinality of any set ... |
| cfeq0 10164 | Only the ordinal zero has ... |
| cfsuc 10165 | Value of the cofinality fu... |
| cff1 10166 | There is always a map from... |
| cfflb 10167 | If there is a cofinal map ... |
| cfval2 10168 | Another expression for the... |
| coflim 10169 | A simpler expression for t... |
| cflim3 10170 | Another expression for the... |
| cflim2 10171 | The cofinality function is... |
| cfom 10172 | Value of the cofinality fu... |
| cfss 10173 | There is a cofinal subset ... |
| cfslb 10174 | Any cofinal subset of ` A ... |
| cfslbn 10175 | Any subset of ` A ` smalle... |
| cfslb2n 10176 | Any small collection of sm... |
| cofsmo 10177 | Any cofinal map implies th... |
| cfsmolem 10178 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10179 | The map in ~ cff1 can be a... |
| cfcoflem 10180 | Lemma for ~ cfcof , showin... |
| coftr 10181 | If there is a cofinal map ... |
| cfcof 10182 | If there is a cofinal map ... |
| cfidm 10183 | The cofinality function is... |
| alephsing 10184 | The cofinality of a limit ... |
| sornom 10185 | The range of a single-step... |
| isfin1a 10200 | Definition of a Ia-finite ... |
| fin1ai 10201 | Property of a Ia-finite se... |
| isfin2 10202 | Definition of a II-finite ... |
| fin2i 10203 | Property of a II-finite se... |
| isfin3 10204 | Definition of a III-finite... |
| isfin4 10205 | Definition of a IV-finite ... |
| fin4i 10206 | Infer that a set is IV-inf... |
| isfin5 10207 | Definition of a V-finite s... |
| isfin6 10208 | Definition of a VI-finite ... |
| isfin7 10209 | Definition of a VII-finite... |
| sdom2en01 10210 | A set with less than two e... |
| infpssrlem1 10211 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10212 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10213 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10214 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10215 | Lemma for ~ infpssr . (Co... |
| infpssr 10216 | Dedekind infinity implies ... |
| fin4en1 10217 | Dedekind finite is a cardi... |
| ssfin4 10218 | Dedekind finite sets have ... |
| domfin4 10219 | A set dominated by a Dedek... |
| ominf4 10220 | ` _om ` is Dedekind infini... |
| infpssALT 10221 | Alternate proof of ~ infps... |
| isfin4-2 10222 | Alternate definition of IV... |
| isfin4p1 10223 | Alternate definition of IV... |
| fin23lem7 10224 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10225 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10226 | A II-finite set contains m... |
| isfin2-2 10227 | ` Fin2 ` expressed in term... |
| ssfin2 10228 | A subset of a II-finite se... |
| enfin2i 10229 | II-finiteness is a cardina... |
| fin23lem24 10230 | Lemma for ~ fin23 . In a ... |
| fincssdom 10231 | In a chain of finite sets,... |
| fin23lem25 10232 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10233 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10234 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10235 | Lemma for ~ fin23 but coul... |
| fin23lem27 10236 | The mapping constructed in... |
| isfin3ds 10237 | Property of a III-finite s... |
| ssfin3ds 10238 | A subset of a III-finite s... |
| fin23lem12 10239 | The beginning of the proof... |
| fin23lem13 10240 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10241 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10242 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10243 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10244 | Lemma for ~ fin23 . The f... |
| fin23lem20 10245 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10246 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10247 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10248 | Lemma for ~ fin23 . The r... |
| fin23lem29 10249 | Lemma for ~ fin23 . The r... |
| fin23lem30 10250 | Lemma for ~ fin23 . The r... |
| fin23lem31 10251 | Lemma for ~ fin23 . The r... |
| fin23lem32 10252 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10253 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10254 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10255 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10256 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10257 | Lemma for ~ fin23 . The c... |
| fin23lem39 10258 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10259 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10260 | Lemma for ~ fin23 . A set... |
| isf32lem1 10261 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10262 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10263 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10264 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10265 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10266 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10267 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10268 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10269 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10270 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10271 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10272 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10273 | One half of ~ isfin3-2 . ... |
| isf33lem 10274 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10275 | Weakly Dedekind-infinite s... |
| isfin3-3 10276 | Weakly Dedekind-infinite s... |
| fin33i 10277 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10278 | Lemma for ~ compsscnv . (... |
| compsscnv 10279 | Complementation on a power... |
| isf34lem1 10280 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10281 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10282 | Complementation is an anti... |
| isf34lem3 10283 | Lemma for ~ isfin3-4 . (C... |
| compss 10284 | Express image under of the... |
| isf34lem4 10285 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10286 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10287 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10288 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10289 | Inference from ~ isfin3-4 ... |
| isfin3-4 10290 | Weakly Dedekind-infinite s... |
| fin11a 10291 | Every I-finite set is Ia-f... |
| enfin1ai 10292 | Ia-finiteness is a cardina... |
| isfin1-2 10293 | A set is finite in the usu... |
| isfin1-3 10294 | A set is I-finite iff ever... |
| isfin1-4 10295 | A set is I-finite iff ever... |
| dffin1-5 10296 | Compact quantifier-free ve... |
| fin23 10297 | Every II-finite set (every... |
| fin34 10298 | Every III-finite set is IV... |
| isfin5-2 10299 | Alternate definition of V-... |
| fin45 10300 | Every IV-finite set is V-f... |
| fin56 10301 | Every V-finite set is VI-f... |
| fin17 10302 | Every I-finite set is VII-... |
| fin67 10303 | Every VI-finite set is VII... |
| isfin7-2 10304 | A set is VII-finite iff it... |
| fin71num 10305 | A well-orderable set is VI... |
| dffin7-2 10306 | Class form of ~ isfin7-2 .... |
| dfacfin7 10307 | Axiom of Choice equivalent... |
| fin1a2lem1 10308 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10309 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10310 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10311 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10312 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10313 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10314 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10315 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10316 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10317 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10318 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10319 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10320 | Lemma for ~ fin1a2 . (Con... |
| fin12 10321 | Weak theorem which skips I... |
| fin1a2s 10322 | An II-infinite set can hav... |
| fin1a2 10323 | Every Ia-finite set is II-... |
| itunifval 10324 | Function value of iterated... |
| itunifn 10325 | Functionality of the itera... |
| ituni0 10326 | A zero-fold iterated union... |
| itunisuc 10327 | Successor iterated union. ... |
| itunitc1 10328 | Each union iterate is a me... |
| itunitc 10329 | The union of all union ite... |
| ituniiun 10330 | Unwrap an iterated union f... |
| hsmexlem7 10331 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10332 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10333 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10334 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10335 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10336 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10337 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10338 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10339 | Lemma for ~ hsmex . (Cont... |
| hsmex 10340 | The collection of heredita... |
| hsmex2 10341 | The set of hereditary size... |
| hsmex3 10342 | The set of hereditary size... |
| axcc2lem 10344 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10345 | A possibly more useful ver... |
| axcc3 10346 | A possibly more useful ver... |
| axcc4 10347 | A version of ~ axcc3 that ... |
| acncc 10348 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10349 | Relax the constraint on ~ ... |
| domtriomlem 10350 | Lemma for ~ domtriom . (C... |
| domtriom 10351 | Trichotomy of equinumerosi... |
| fin41 10352 | Under countable choice, th... |
| dominf 10353 | A nonempty set that is a s... |
| dcomex 10355 | The Axiom of Dependent Cho... |
| axdc2lem 10356 | Lemma for ~ axdc2 . We co... |
| axdc2 10357 | An apparent strengthening ... |
| axdc3lem 10358 | The class ` S ` of finite ... |
| axdc3lem2 10359 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10360 | Simple substitution lemma ... |
| axdc3lem4 10361 | Lemma for ~ axdc3 . We ha... |
| axdc3 10362 | Dependent Choice. Axiom D... |
| axdc4lem 10363 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10364 | A more general version of ... |
| axcclem 10365 | Lemma for ~ axcc . (Contr... |
| axcc 10366 | Although CC can be proven ... |
| zfac 10368 | Axiom of Choice expressed ... |
| ac2 10369 | Axiom of Choice equivalent... |
| ac3 10370 | Axiom of Choice using abbr... |
| axac3 10372 | This theorem asserts that ... |
| ackm 10373 | A remarkable equivalent to... |
| axac2 10374 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10375 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10376 | Apply a choice equivalent.... |
| cardeqv 10377 | All sets are well-orderabl... |
| numth3 10378 | All sets are well-orderabl... |
| numth2 10379 | Numeration theorem: any se... |
| numth 10380 | Numeration theorem: every ... |
| ac7 10381 | An Axiom of Choice equival... |
| ac7g 10382 | An Axiom of Choice equival... |
| ac4 10383 | Equivalent of Axiom of Cho... |
| ac4c 10384 | Equivalent of Axiom of Cho... |
| ac5 10385 | An Axiom of Choice equival... |
| ac5b 10386 | Equivalent of Axiom of Cho... |
| ac6num 10387 | A version of ~ ac6 which t... |
| ac6 10388 | Equivalent of Axiom of Cho... |
| ac6c4 10389 | Equivalent of Axiom of Cho... |
| ac6c5 10390 | Equivalent of Axiom of Cho... |
| ac9 10391 | An Axiom of Choice equival... |
| ac6s 10392 | Equivalent of Axiom of Cho... |
| ac6n 10393 | Equivalent of Axiom of Cho... |
| ac6s2 10394 | Generalization of the Axio... |
| ac6s3 10395 | Generalization of the Axio... |
| ac6sg 10396 | ~ ac6s with sethood as ant... |
| ac6sf 10397 | Version of ~ ac6 with boun... |
| ac6s4 10398 | Generalization of the Axio... |
| ac6s5 10399 | Generalization of the Axio... |
| ac8 10400 | An Axiom of Choice equival... |
| ac9s 10401 | An Axiom of Choice equival... |
| numthcor 10402 | Any set is strictly domina... |
| weth 10403 | Well-ordering theorem: any... |
| zorn2lem1 10404 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10405 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10406 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10407 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10408 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10409 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10410 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10411 | Zorn's Lemma of [Monk1] p.... |
| zorng 10412 | Zorn's Lemma. If the unio... |
| zornn0g 10413 | Variant of Zorn's lemma ~ ... |
| zorn2 10414 | Zorn's Lemma of [Monk1] p.... |
| zorn 10415 | Zorn's Lemma. If the unio... |
| zornn0 10416 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10417 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10418 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10419 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10420 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10421 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10422 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10423 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10424 | The Teichmüller-Tukey... |
| ttukeyg 10425 | The Teichmüller-Tukey... |
| ttukey 10426 | The Teichmüller-Tukey... |
| axdclem 10427 | Lemma for ~ axdc . (Contr... |
| axdclem2 10428 | Lemma for ~ axdc . Using ... |
| axdc 10429 | This theorem derives ~ ax-... |
| fodomg 10430 | An onto function implies d... |
| fodom 10431 | An onto function implies d... |
| dmct 10432 | The domain of a countable ... |
| rnct 10433 | The range of a countable s... |
| fodomb 10434 | Equivalence of an onto map... |
| wdomac 10435 | When assuming AC, weak and... |
| brdom3 10436 | Equivalence to a dominance... |
| brdom5 10437 | An equivalence to a domina... |
| brdom4 10438 | An equivalence to a domina... |
| brdom7disj 10439 | An equivalence to a domina... |
| brdom6disj 10440 | An equivalence to a domina... |
| fin71ac 10441 | Once we allow AC, the "str... |
| imadomg 10442 | An image of a function und... |
| fimact 10443 | The image by a function of... |
| fnrndomg 10444 | The range of a function is... |
| fnct 10445 | If the domain of a functio... |
| mptct 10446 | A countable mapping set is... |
| iunfo 10447 | Existence of an onto funct... |
| iundom2g 10448 | An upper bound for the car... |
| iundomg 10449 | An upper bound for the car... |
| iundom 10450 | An upper bound for the car... |
| unidom 10451 | An upper bound for the car... |
| uniimadom 10452 | An upper bound for the car... |
| uniimadomf 10453 | An upper bound for the car... |
| cardval 10454 | The value of the cardinal ... |
| cardid 10455 | Any set is equinumerous to... |
| cardidg 10456 | Any set is equinumerous to... |
| cardidd 10457 | Any set is equinumerous to... |
| cardf 10458 | The cardinality function i... |
| carden 10459 | Two sets are equinumerous ... |
| cardeq0 10460 | Only the empty set has car... |
| unsnen 10461 | Equinumerosity of a set wi... |
| carddom 10462 | Two sets have the dominanc... |
| cardsdom 10463 | Two sets have the strict d... |
| domtri 10464 | Trichotomy law for dominan... |
| entric 10465 | Trichotomy of equinumerosi... |
| entri2 10466 | Trichotomy of dominance an... |
| entri3 10467 | Trichotomy of dominance. ... |
| sdomsdomcard 10468 | A set strictly dominates i... |
| canth3 10469 | Cantor's theorem in terms ... |
| infxpidm 10470 | Every infinite class is eq... |
| ondomon 10471 | The class of ordinals domi... |
| cardmin 10472 | The smallest ordinal that ... |
| ficard 10473 | A set is finite iff its ca... |
| infinfg 10474 | Equivalence between two in... |
| infinf 10475 | Equivalence between two in... |
| unirnfdomd 10476 | The union of the range of ... |
| konigthlem 10477 | Lemma for ~ konigth . (Co... |
| konigth 10478 | Konig's Theorem. If ` m (... |
| alephsucpw 10479 | The power set of an aleph ... |
| aleph1 10480 | The set exponentiation of ... |
| alephval2 10481 | An alternate way to expres... |
| dominfac 10482 | A nonempty set that is a s... |
| iunctb 10483 | The countable union of cou... |
| unictb 10484 | The countable union of cou... |
| infmap 10485 | An exponentiation law for ... |
| alephadd 10486 | The sum of two alephs is t... |
| alephmul 10487 | The product of two alephs ... |
| alephexp1 10488 | An exponentiation law for ... |
| alephsuc3 10489 | An alternate representatio... |
| alephexp2 10490 | An expression equinumerous... |
| alephreg 10491 | A successor aleph is regul... |
| pwcfsdom 10492 | A corollary of Konig's The... |
| cfpwsdom 10493 | A corollary of Konig's The... |
| alephom 10494 | From ~ canth2 , we know th... |
| smobeth 10495 | The beth function is stric... |
| nd1 10496 | A lemma for proving condit... |
| nd2 10497 | A lemma for proving condit... |
| nd3 10498 | A lemma for proving condit... |
| nd4 10499 | A lemma for proving condit... |
| axextnd 10500 | A version of the Axiom of ... |
| axrepndlem1 10501 | Lemma for the Axiom of Rep... |
| axrepndlem2 10502 | Lemma for the Axiom of Rep... |
| axrepnd 10503 | A version of the Axiom of ... |
| axunndlem1 10504 | Lemma for the Axiom of Uni... |
| axunnd 10505 | A version of the Axiom of ... |
| axpowndlem1 10506 | Lemma for the Axiom of Pow... |
| axpowndlem2 10507 | Lemma for the Axiom of Pow... |
| axpowndlem3 10508 | Lemma for the Axiom of Pow... |
| axpowndlem4 10509 | Lemma for the Axiom of Pow... |
| axpownd 10510 | A version of the Axiom of ... |
| axregndlem1 10511 | Lemma for the Axiom of Reg... |
| axregndlem2 10512 | Lemma for the Axiom of Reg... |
| axregnd 10513 | A version of the Axiom of ... |
| axinfndlem1 10514 | Lemma for the Axiom of Inf... |
| axinfnd 10515 | A version of the Axiom of ... |
| axacndlem1 10516 | Lemma for the Axiom of Cho... |
| axacndlem2 10517 | Lemma for the Axiom of Cho... |
| axacndlem3 10518 | Lemma for the Axiom of Cho... |
| axacndlem4 10519 | Lemma for the Axiom of Cho... |
| axacndlem5 10520 | Lemma for the Axiom of Cho... |
| axacnd 10521 | A version of the Axiom of ... |
| zfcndext 10522 | Axiom of Extensionality ~ ... |
| zfcndrep 10523 | Axiom of Replacement ~ ax-... |
| zfcndun 10524 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10525 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10526 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10527 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10528 | Axiom of Choice ~ ax-ac , ... |
| elgch 10531 | Elementhood in the collect... |
| fingch 10532 | A finite set is a GCH-set.... |
| gchi 10533 | The only GCH-sets which ha... |
| gchen1 10534 | If ` A <_ B < ~P A ` , and... |
| gchen2 10535 | If ` A < B <_ ~P A ` , and... |
| gchor 10536 | If ` A <_ B <_ ~P A ` , an... |
| engch 10537 | The property of being a GC... |
| gchdomtri 10538 | Under certain conditions, ... |
| fpwwe2cbv 10539 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10540 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10541 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10542 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10543 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10544 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10545 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10546 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10547 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10548 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10549 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10550 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10551 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10552 | Given any function ` F ` f... |
| fpwwecbv 10553 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10554 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10555 | Given any function ` F ` f... |
| canth4 10556 | An "effective" form of Can... |
| canthnumlem 10557 | Lemma for ~ canthnum . (C... |
| canthnum 10558 | The set of well-orderable ... |
| canthwelem 10559 | Lemma for ~ canthwe . (Co... |
| canthwe 10560 | The set of well-orders of ... |
| canthp1lem1 10561 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10562 | Lemma for ~ canthp1 . (Co... |
| canthp1 10563 | A slightly stronger form o... |
| finngch 10564 | The exclusion of finite se... |
| gchdju1 10565 | An infinite GCH-set is ide... |
| gchinf 10566 | An infinite GCH-set is Ded... |
| pwfseqlem1 10567 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10568 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10569 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10570 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10571 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10572 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10573 | The powerset of a Dedekind... |
| pwxpndom2 10574 | The powerset of a Dedekind... |
| pwxpndom 10575 | The powerset of a Dedekind... |
| pwdjundom 10576 | The powerset of a Dedekind... |
| gchdjuidm 10577 | An infinite GCH-set is ide... |
| gchxpidm 10578 | An infinite GCH-set is ide... |
| gchpwdom 10579 | A relationship between dom... |
| gchaleph 10580 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10581 | If ` ( aleph `` A ) ` and ... |
| hargch 10582 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10583 | If ` ( aleph `` suc A ) ` ... |
| gch2 10584 | It is sufficient to requir... |
| gch3 10585 | An equivalent formulation ... |
| gch-kn 10586 | The equivalence of two ver... |
| gchaclem 10587 | Lemma for ~ gchac (obsolet... |
| gchhar 10588 | A "local" form of ~ gchac ... |
| gchacg 10589 | A "local" form of ~ gchac ... |
| gchac 10590 | The Generalized Continuum ... |
| elwina 10595 | Conditions of weak inacces... |
| elina 10596 | Conditions of strong inacc... |
| winaon 10597 | A weakly inaccessible card... |
| inawinalem 10598 | Lemma for ~ inawina . (Co... |
| inawina 10599 | Every strongly inaccessibl... |
| omina 10600 | ` _om ` is a strongly inac... |
| winacard 10601 | A weakly inaccessible card... |
| winainflem 10602 | A weakly inaccessible card... |
| winainf 10603 | A weakly inaccessible card... |
| winalim 10604 | A weakly inaccessible card... |
| winalim2 10605 | A nontrivial weakly inacce... |
| winafp 10606 | A nontrivial weakly inacce... |
| winafpi 10607 | This theorem, which states... |
| gchina 10608 | Assuming the GCH, weakly a... |
| iswun 10613 | Properties of a weak unive... |
| wuntr 10614 | A weak universe is transit... |
| wununi 10615 | A weak universe is closed ... |
| wunpw 10616 | A weak universe is closed ... |
| wunelss 10617 | The elements of a weak uni... |
| wunpr 10618 | A weak universe is closed ... |
| wunun 10619 | A weak universe is closed ... |
| wuntp 10620 | A weak universe is closed ... |
| wunss 10621 | A weak universe is closed ... |
| wunin 10622 | A weak universe is closed ... |
| wundif 10623 | A weak universe is closed ... |
| wunint 10624 | A weak universe is closed ... |
| wunsn 10625 | A weak universe is closed ... |
| wunsuc 10626 | A weak universe is closed ... |
| wun0 10627 | A weak universe contains t... |
| wunr1om 10628 | A weak universe is infinit... |
| wunom 10629 | A weak universe contains a... |
| wunfi 10630 | A weak universe contains a... |
| wunop 10631 | A weak universe is closed ... |
| wunot 10632 | A weak universe is closed ... |
| wunxp 10633 | A weak universe is closed ... |
| wunpm 10634 | A weak universe is closed ... |
| wunmap 10635 | A weak universe is closed ... |
| wunf 10636 | A weak universe is closed ... |
| wundm 10637 | A weak universe is closed ... |
| wunrn 10638 | A weak universe is closed ... |
| wuncnv 10639 | A weak universe is closed ... |
| wunres 10640 | A weak universe is closed ... |
| wunfv 10641 | A weak universe is closed ... |
| wunco 10642 | A weak universe is closed ... |
| wuntpos 10643 | A weak universe is closed ... |
| intwun 10644 | The intersection of a coll... |
| r1limwun 10645 | Each limit stage in the cu... |
| r1wunlim 10646 | The weak universes in the ... |
| wunex2 10647 | Construct a weak universe ... |
| wunex 10648 | Construct a weak universe ... |
| uniwun 10649 | Every set is contained in ... |
| wunex3 10650 | Construct a weak universe ... |
| wuncval 10651 | Value of the weak universe... |
| wuncid 10652 | The weak universe closure ... |
| wunccl 10653 | The weak universe closure ... |
| wuncss 10654 | The weak universe closure ... |
| wuncidm 10655 | The weak universe closure ... |
| wuncval2 10656 | Our earlier expression for... |
| eltskg 10659 | Properties of a Tarski cla... |
| eltsk2g 10660 | Properties of a Tarski cla... |
| tskpwss 10661 | First axiom of a Tarski cl... |
| tskpw 10662 | Second axiom of a Tarski c... |
| tsken 10663 | Third axiom of a Tarski cl... |
| 0tsk 10664 | The empty set is a (transi... |
| tsksdom 10665 | An element of a Tarski cla... |
| tskssel 10666 | A part of a Tarski class s... |
| tskss 10667 | The subsets of an element ... |
| tskin 10668 | The intersection of two el... |
| tsksn 10669 | A singleton of an element ... |
| tsktrss 10670 | A transitive element of a ... |
| tsksuc 10671 | If an element of a Tarski ... |
| tsk0 10672 | A nonempty Tarski class co... |
| tsk1 10673 | One is an element of a non... |
| tsk2 10674 | Two is an element of a non... |
| 2domtsk 10675 | If a Tarski class is not e... |
| tskr1om 10676 | A nonempty Tarski class is... |
| tskr1om2 10677 | A nonempty Tarski class co... |
| tskinf 10678 | A nonempty Tarski class is... |
| tskpr 10679 | If ` A ` and ` B ` are mem... |
| tskop 10680 | If ` A ` and ` B ` are mem... |
| tskxpss 10681 | A Cartesian product of two... |
| tskwe2 10682 | A Tarski class is well-ord... |
| inttsk 10683 | The intersection of a coll... |
| inar1 10684 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10685 | Alternate proof of ~ r1om ... |
| rankcf 10686 | Any set must be at least a... |
| inatsk 10687 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10688 | The set of hereditarily fi... |
| tskord 10689 | A Tarski class contains al... |
| tskcard 10690 | An even more direct relati... |
| r1tskina 10691 | There is a direct relation... |
| tskuni 10692 | The union of an element of... |
| tskwun 10693 | A nonempty transitive Tars... |
| tskint 10694 | The intersection of an ele... |
| tskun 10695 | The union of two elements ... |
| tskxp 10696 | The Cartesian product of t... |
| tskmap 10697 | Set exponentiation is an e... |
| tskurn 10698 | A transitive Tarski class ... |
| elgrug 10701 | Properties of a Grothendie... |
| grutr 10702 | A Grothendieck universe is... |
| gruelss 10703 | A Grothendieck universe is... |
| grupw 10704 | A Grothendieck universe co... |
| gruss 10705 | Any subset of an element o... |
| grupr 10706 | A Grothendieck universe co... |
| gruurn 10707 | A Grothendieck universe co... |
| gruiun 10708 | If ` B ( x ) ` is a family... |
| gruuni 10709 | A Grothendieck universe co... |
| grurn 10710 | A Grothendieck universe co... |
| gruima 10711 | A Grothendieck universe co... |
| gruel 10712 | Any element of an element ... |
| grusn 10713 | A Grothendieck universe co... |
| gruop 10714 | A Grothendieck universe co... |
| gruun 10715 | A Grothendieck universe co... |
| gruxp 10716 | A Grothendieck universe co... |
| grumap 10717 | A Grothendieck universe co... |
| gruixp 10718 | A Grothendieck universe co... |
| gruiin 10719 | A Grothendieck universe co... |
| gruf 10720 | A Grothendieck universe co... |
| gruen 10721 | A Grothendieck universe co... |
| gruwun 10722 | A nonempty Grothendieck un... |
| intgru 10723 | The intersection of a fami... |
| ingru 10724 | The intersection of a univ... |
| wfgru 10725 | The wellfounded part of a ... |
| grudomon 10726 | Each ordinal that is compa... |
| gruina 10727 | If a Grothendieck universe... |
| grur1a 10728 | A characterization of Grot... |
| grur1 10729 | A characterization of Grot... |
| grutsk1 10730 | Grothendieck universes are... |
| grutsk 10731 | Grothendieck universes are... |
| axgroth5 10733 | The Tarski-Grothendieck ax... |
| axgroth2 10734 | Alternate version of the T... |
| grothpw 10735 | Derive the Axiom of Power ... |
| grothpwex 10736 | Derive the Axiom of Power ... |
| axgroth6 10737 | The Tarski-Grothendieck ax... |
| grothomex 10738 | The Tarski-Grothendieck Ax... |
| grothac 10739 | The Tarski-Grothendieck Ax... |
| axgroth3 10740 | Alternate version of the T... |
| axgroth4 10741 | Alternate version of the T... |
| grothprimlem 10742 | Lemma for ~ grothprim . E... |
| grothprim 10743 | The Tarski-Grothendieck Ax... |
| grothtsk 10744 | The Tarski-Grothendieck Ax... |
| inaprc 10745 | An equivalent to the Tarsk... |
| tskmval 10748 | Value of our tarski map. ... |
| tskmid 10749 | The set ` A ` is an elemen... |
| tskmcl 10750 | A Tarski class that contai... |
| sstskm 10751 | Being a part of ` ( tarski... |
| eltskm 10752 | Belonging to ` ( tarskiMap... |
| elni 10785 | Membership in the class of... |
| elni2 10786 | Membership in the class of... |
| pinn 10787 | A positive integer is a na... |
| pion 10788 | A positive integer is an o... |
| piord 10789 | A positive integer is ordi... |
| niex 10790 | The class of positive inte... |
| 0npi 10791 | The empty set is not a pos... |
| 1pi 10792 | Ordinal 'one' is a positiv... |
| addpiord 10793 | Positive integer addition ... |
| mulpiord 10794 | Positive integer multiplic... |
| mulidpi 10795 | 1 is an identity element f... |
| ltpiord 10796 | Positive integer 'less tha... |
| ltsopi 10797 | Positive integer 'less tha... |
| ltrelpi 10798 | Positive integer 'less tha... |
| dmaddpi 10799 | Domain of addition on posi... |
| dmmulpi 10800 | Domain of multiplication o... |
| addclpi 10801 | Closure of addition of pos... |
| mulclpi 10802 | Closure of multiplication ... |
| addcompi 10803 | Addition of positive integ... |
| addasspi 10804 | Addition of positive integ... |
| mulcompi 10805 | Multiplication of positive... |
| mulasspi 10806 | Multiplication of positive... |
| distrpi 10807 | Multiplication of positive... |
| addcanpi 10808 | Addition cancellation law ... |
| mulcanpi 10809 | Multiplication cancellatio... |
| addnidpi 10810 | There is no identity eleme... |
| ltexpi 10811 | Ordering on positive integ... |
| ltapi 10812 | Ordering property of addit... |
| ltmpi 10813 | Ordering property of multi... |
| 1lt2pi 10814 | One is less than two (one ... |
| nlt1pi 10815 | No positive integer is les... |
| indpi 10816 | Principle of Finite Induct... |
| enqbreq 10828 | Equivalence relation for p... |
| enqbreq2 10829 | Equivalence relation for p... |
| enqer 10830 | The equivalence relation f... |
| enqex 10831 | The equivalence relation f... |
| nqex 10832 | The class of positive frac... |
| 0nnq 10833 | The empty set is not a pos... |
| elpqn 10834 | Each positive fraction is ... |
| ltrelnq 10835 | Positive fraction 'less th... |
| pinq 10836 | The representatives of pos... |
| 1nq 10837 | The positive fraction 'one... |
| nqereu 10838 | There is a unique element ... |
| nqerf 10839 | Corollary of ~ nqereu : th... |
| nqercl 10840 | Corollary of ~ nqereu : cl... |
| nqerrel 10841 | Any member of ` ( N. X. N.... |
| nqerid 10842 | Corollary of ~ nqereu : th... |
| enqeq 10843 | Corollary of ~ nqereu : if... |
| nqereq 10844 | The function ` /Q ` acts a... |
| addpipq2 10845 | Addition of positive fract... |
| addpipq 10846 | Addition of positive fract... |
| addpqnq 10847 | Addition of positive fract... |
| mulpipq2 10848 | Multiplication of positive... |
| mulpipq 10849 | Multiplication of positive... |
| mulpqnq 10850 | Multiplication of positive... |
| ordpipq 10851 | Ordering of positive fract... |
| ordpinq 10852 | Ordering of positive fract... |
| addpqf 10853 | Closure of addition on pos... |
| addclnq 10854 | Closure of addition on pos... |
| mulpqf 10855 | Closure of multiplication ... |
| mulclnq 10856 | Closure of multiplication ... |
| addnqf 10857 | Domain of addition on posi... |
| mulnqf 10858 | Domain of multiplication o... |
| addcompq 10859 | Addition of positive fract... |
| addcomnq 10860 | Addition of positive fract... |
| mulcompq 10861 | Multiplication of positive... |
| mulcomnq 10862 | Multiplication of positive... |
| adderpqlem 10863 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10864 | Lemma for ~ mulerpq . (Co... |
| adderpq 10865 | Addition is compatible wit... |
| mulerpq 10866 | Multiplication is compatib... |
| addassnq 10867 | Addition of positive fract... |
| mulassnq 10868 | Multiplication of positive... |
| mulcanenq 10869 | Lemma for distributive law... |
| distrnq 10870 | Multiplication of positive... |
| 1nqenq 10871 | The equivalence class of r... |
| mulidnq 10872 | Multiplication identity el... |
| recmulnq 10873 | Relationship between recip... |
| recidnq 10874 | A positive fraction times ... |
| recclnq 10875 | Closure law for positive f... |
| recrecnq 10876 | Reciprocal of reciprocal o... |
| dmrecnq 10877 | Domain of reciprocal on po... |
| ltsonq 10878 | 'Less than' is a strict or... |
| lterpq 10879 | Compatibility of ordering ... |
| ltanq 10880 | Ordering property of addit... |
| ltmnq 10881 | Ordering property of multi... |
| 1lt2nq 10882 | One is less than two (one ... |
| ltaddnq 10883 | The sum of two fractions i... |
| ltexnq 10884 | Ordering on positive fract... |
| halfnq 10885 | One-half of any positive f... |
| nsmallnq 10886 | The is no smallest positiv... |
| ltbtwnnq 10887 | There exists a number betw... |
| ltrnq 10888 | Ordering property of recip... |
| archnq 10889 | For any fraction, there is... |
| npex 10895 | The class of positive real... |
| elnp 10896 | Membership in positive rea... |
| elnpi 10897 | Membership in positive rea... |
| prn0 10898 | A positive real is not emp... |
| prpssnq 10899 | A positive real is a subse... |
| elprnq 10900 | A positive real is a set o... |
| 0npr 10901 | The empty set is not a pos... |
| prcdnq 10902 | A positive real is closed ... |
| prub 10903 | A positive fraction not in... |
| prnmax 10904 | A positive real has no lar... |
| npomex 10905 | A simplifying observation,... |
| prnmadd 10906 | A positive real has no lar... |
| ltrelpr 10907 | Positive real 'less than' ... |
| genpv 10908 | Value of general operation... |
| genpelv 10909 | Membership in value of gen... |
| genpprecl 10910 | Pre-closure law for genera... |
| genpdm 10911 | Domain of general operatio... |
| genpn0 10912 | The result of an operation... |
| genpss 10913 | The result of an operation... |
| genpnnp 10914 | The result of an operation... |
| genpcd 10915 | Downward closure of an ope... |
| genpnmax 10916 | An operation on positive r... |
| genpcl 10917 | Closure of an operation on... |
| genpass 10918 | Associativity of an operat... |
| plpv 10919 | Value of addition on posit... |
| mpv 10920 | Value of multiplication on... |
| dmplp 10921 | Domain of addition on posi... |
| dmmp 10922 | Domain of multiplication o... |
| nqpr 10923 | The canonical embedding of... |
| 1pr 10924 | The positive real number '... |
| addclprlem1 10925 | Lemma to prove downward cl... |
| addclprlem2 10926 | Lemma to prove downward cl... |
| addclpr 10927 | Closure of addition on pos... |
| mulclprlem 10928 | Lemma to prove downward cl... |
| mulclpr 10929 | Closure of multiplication ... |
| addcompr 10930 | Addition of positive reals... |
| addasspr 10931 | Addition of positive reals... |
| mulcompr 10932 | Multiplication of positive... |
| mulasspr 10933 | Multiplication of positive... |
| distrlem1pr 10934 | Lemma for distributive law... |
| distrlem4pr 10935 | Lemma for distributive law... |
| distrlem5pr 10936 | Lemma for distributive law... |
| distrpr 10937 | Multiplication of positive... |
| 1idpr 10938 | 1 is an identity element f... |
| ltprord 10939 | Positive real 'less than' ... |
| psslinpr 10940 | Proper subset is a linear ... |
| ltsopr 10941 | Positive real 'less than' ... |
| prlem934 10942 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10943 | The sum of two positive re... |
| ltaddpr2 10944 | The sum of two positive re... |
| ltexprlem1 10945 | Lemma for Proposition 9-3.... |
| ltexprlem2 10946 | Lemma for Proposition 9-3.... |
| ltexprlem3 10947 | Lemma for Proposition 9-3.... |
| ltexprlem4 10948 | Lemma for Proposition 9-3.... |
| ltexprlem5 10949 | Lemma for Proposition 9-3.... |
| ltexprlem6 10950 | Lemma for Proposition 9-3.... |
| ltexprlem7 10951 | Lemma for Proposition 9-3.... |
| ltexpri 10952 | Proposition 9-3.5(iv) of [... |
| ltaprlem 10953 | Lemma for Proposition 9-3.... |
| ltapr 10954 | Ordering property of addit... |
| addcanpr 10955 | Addition cancellation law ... |
| prlem936 10956 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 10957 | Lemma for Proposition 9-3.... |
| reclem3pr 10958 | Lemma for Proposition 9-3.... |
| reclem4pr 10959 | Lemma for Proposition 9-3.... |
| recexpr 10960 | The reciprocal of a positi... |
| suplem1pr 10961 | The union of a nonempty, b... |
| suplem2pr 10962 | The union of a set of posi... |
| supexpr 10963 | The union of a nonempty, b... |
| enrer 10972 | The equivalence relation f... |
| nrex1 10973 | The class of signed reals ... |
| enrbreq 10974 | Equivalence relation for s... |
| enreceq 10975 | Equivalence class equality... |
| enrex 10976 | The equivalence relation f... |
| ltrelsr 10977 | Signed real 'less than' is... |
| addcmpblnr 10978 | Lemma showing compatibilit... |
| mulcmpblnrlem 10979 | Lemma used in lemma showin... |
| mulcmpblnr 10980 | Lemma showing compatibilit... |
| prsrlem1 10981 | Decomposing signed reals i... |
| addsrmo 10982 | There is at most one resul... |
| mulsrmo 10983 | There is at most one resul... |
| addsrpr 10984 | Addition of signed reals i... |
| mulsrpr 10985 | Multiplication of signed r... |
| ltsrpr 10986 | Ordering of signed reals i... |
| gt0srpr 10987 | Greater than zero in terms... |
| 0nsr 10988 | The empty set is not a sig... |
| 0r 10989 | The constant ` 0R ` is a s... |
| 1sr 10990 | The constant ` 1R ` is a s... |
| m1r 10991 | The constant ` -1R ` is a ... |
| addclsr 10992 | Closure of addition on sig... |
| mulclsr 10993 | Closure of multiplication ... |
| dmaddsr 10994 | Domain of addition on sign... |
| dmmulsr 10995 | Domain of multiplication o... |
| addcomsr 10996 | Addition of signed reals i... |
| addasssr 10997 | Addition of signed reals i... |
| mulcomsr 10998 | Multiplication of signed r... |
| mulasssr 10999 | Multiplication of signed r... |
| distrsr 11000 | Multiplication of signed r... |
| m1p1sr 11001 | Minus one plus one is zero... |
| m1m1sr 11002 | Minus one times minus one ... |
| ltsosr 11003 | Signed real 'less than' is... |
| 0lt1sr 11004 | 0 is less than 1 for signe... |
| 1ne0sr 11005 | 1 and 0 are distinct for s... |
| 0idsr 11006 | The signed real number 0 i... |
| 1idsr 11007 | 1 is an identity element f... |
| 00sr 11008 | A signed real times 0 is 0... |
| ltasr 11009 | Ordering property of addit... |
| pn0sr 11010 | A signed real plus its neg... |
| negexsr 11011 | Existence of negative sign... |
| recexsrlem 11012 | The reciprocal of a positi... |
| addgt0sr 11013 | The sum of two positive si... |
| mulgt0sr 11014 | The product of two positiv... |
| sqgt0sr 11015 | The square of a nonzero si... |
| recexsr 11016 | The reciprocal of a nonzer... |
| mappsrpr 11017 | Mapping from positive sign... |
| ltpsrpr 11018 | Mapping of order from posi... |
| map2psrpr 11019 | Equivalence for positive s... |
| supsrlem 11020 | Lemma for supremum theorem... |
| supsr 11021 | A nonempty, bounded set of... |
| opelcn 11038 | Ordered pair membership in... |
| opelreal 11039 | Ordered pair membership in... |
| elreal 11040 | Membership in class of rea... |
| elreal2 11041 | Ordered pair membership in... |
| 0ncn 11042 | The empty set is not a com... |
| ltrelre 11043 | 'Less than' is a relation ... |
| addcnsr 11044 | Addition of complex number... |
| mulcnsr 11045 | Multiplication of complex ... |
| eqresr 11046 | Equality of real numbers i... |
| addresr 11047 | Addition of real numbers i... |
| mulresr 11048 | Multiplication of real num... |
| ltresr 11049 | Ordering of real subset of... |
| ltresr2 11050 | Ordering of real subset of... |
| dfcnqs 11051 | Technical trick to permit ... |
| addcnsrec 11052 | Technical trick to permit ... |
| mulcnsrec 11053 | Technical trick to permit ... |
| axaddf 11054 | Addition is an operation o... |
| axmulf 11055 | Multiplication is an opera... |
| axcnex 11056 | The complex numbers form a... |
| axresscn 11057 | The real numbers are a sub... |
| ax1cn 11058 | 1 is a complex number. Ax... |
| axicn 11059 | ` _i ` is a complex number... |
| axaddcl 11060 | Closure law for addition o... |
| axaddrcl 11061 | Closure law for addition i... |
| axmulcl 11062 | Closure law for multiplica... |
| axmulrcl 11063 | Closure law for multiplica... |
| axmulcom 11064 | Multiplication of complex ... |
| axaddass 11065 | Addition of complex number... |
| axmulass 11066 | Multiplication of complex ... |
| axdistr 11067 | Distributive law for compl... |
| axi2m1 11068 | i-squared equals -1 (expre... |
| ax1ne0 11069 | 1 and 0 are distinct. Axi... |
| ax1rid 11070 | ` 1 ` is an identity eleme... |
| axrnegex 11071 | Existence of negative of r... |
| axrrecex 11072 | Existence of reciprocal of... |
| axcnre 11073 | A complex number can be ex... |
| axpre-lttri 11074 | Ordering on reals satisfie... |
| axpre-lttrn 11075 | Ordering on reals is trans... |
| axpre-ltadd 11076 | Ordering property of addit... |
| axpre-mulgt0 11077 | The product of two positiv... |
| axpre-sup 11078 | A nonempty, bounded-above ... |
| wuncn 11079 | A weak universe containing... |
| cnex 11105 | Alias for ~ ax-cnex . See... |
| addcl 11106 | Alias for ~ ax-addcl , for... |
| readdcl 11107 | Alias for ~ ax-addrcl , fo... |
| mulcl 11108 | Alias for ~ ax-mulcl , for... |
| remulcl 11109 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11110 | Alias for ~ ax-mulcom , fo... |
| addass 11111 | Alias for ~ ax-addass , fo... |
| mulass 11112 | Alias for ~ ax-mulass , fo... |
| adddi 11113 | Alias for ~ ax-distr , for... |
| recn 11114 | A real number is a complex... |
| reex 11115 | The real numbers form a se... |
| reelprrecn 11116 | Reals are a subset of the ... |
| cnelprrecn 11117 | Complex numbers are a subs... |
| mpoaddf 11118 | Addition is an operation o... |
| mpomulf 11119 | Multiplication is an opera... |
| elimne0 11120 | Hypothesis for weak deduct... |
| adddir 11121 | Distributive law for compl... |
| 0cn 11122 | Zero is a complex number. ... |
| 0cnd 11123 | Zero is a complex number, ... |
| c0ex 11124 | Zero is a set. (Contribut... |
| 1cnd 11125 | One is a complex number, d... |
| 1ex 11126 | One is a set. (Contribute... |
| cnre 11127 | Alias for ~ ax-cnre , for ... |
| mulrid 11128 | The number 1 is an identit... |
| mullid 11129 | Identity law for multiplic... |
| 1re 11130 | The number 1 is real. Thi... |
| 1red 11131 | The number 1 is real, dedu... |
| 0re 11132 | The number 0 is real. Rem... |
| 0red 11133 | The number 0 is real, dedu... |
| mulridi 11134 | Identity law for multiplic... |
| mullidi 11135 | Identity law for multiplic... |
| addcli 11136 | Closure law for addition. ... |
| mulcli 11137 | Closure law for multiplica... |
| mulcomi 11138 | Commutative law for multip... |
| mulcomli 11139 | Commutative law for multip... |
| addassi 11140 | Associative law for additi... |
| mulassi 11141 | Associative law for multip... |
| adddii 11142 | Distributive law (left-dis... |
| adddiri 11143 | Distributive law (right-di... |
| recni 11144 | A real number is a complex... |
| readdcli 11145 | Closure law for addition o... |
| remulcli 11146 | Closure law for multiplica... |
| mulridd 11147 | Identity law for multiplic... |
| mullidd 11148 | Identity law for multiplic... |
| addcld 11149 | Closure law for addition. ... |
| mulcld 11150 | Closure law for multiplica... |
| mulcomd 11151 | Commutative law for multip... |
| addassd 11152 | Associative law for additi... |
| mulassd 11153 | Associative law for multip... |
| adddid 11154 | Distributive law (left-dis... |
| adddird 11155 | Distributive law (right-di... |
| adddirp1d 11156 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11157 | Join AB+CB into (A+C) on L... |
| recnd 11158 | Deduction from real number... |
| readdcld 11159 | Closure law for addition o... |
| remulcld 11160 | Closure law for multiplica... |
| pnfnre 11171 | Plus infinity is not a rea... |
| pnfnre2 11172 | Plus infinity is not a rea... |
| mnfnre 11173 | Minus infinity is not a re... |
| ressxr 11174 | The standard reals are a s... |
| rexpssxrxp 11175 | The Cartesian product of s... |
| rexr 11176 | A standard real is an exte... |
| 0xr 11177 | Zero is an extended real. ... |
| renepnf 11178 | No (finite) real equals pl... |
| renemnf 11179 | No real equals minus infin... |
| rexrd 11180 | A standard real is an exte... |
| renepnfd 11181 | No (finite) real equals pl... |
| renemnfd 11182 | No real equals minus infin... |
| pnfex 11183 | Plus infinity exists. (Co... |
| pnfxr 11184 | Plus infinity belongs to t... |
| pnfnemnf 11185 | Plus and minus infinity ar... |
| mnfnepnf 11186 | Minus and plus infinity ar... |
| mnfxr 11187 | Minus infinity belongs to ... |
| rexri 11188 | A standard real is an exte... |
| 1xr 11189 | ` 1 ` is an extended real ... |
| renfdisj 11190 | The reals and the infiniti... |
| ltrelxr 11191 | "Less than" is a relation ... |
| ltrel 11192 | "Less than" is a relation.... |
| lerelxr 11193 | "Less than or equal to" is... |
| lerel 11194 | "Less than or equal to" is... |
| xrlenlt 11195 | "Less than or equal to" ex... |
| xrlenltd 11196 | "Less than or equal to" ex... |
| xrltnle 11197 | "Less than" expressed in t... |
| xrltnled 11198 | 'Less than' in terms of 'l... |
| xrnltled 11199 | "Not less than" implies "l... |
| ssxr 11200 | The three (non-exclusive) ... |
| ltxrlt 11201 | The standard less-than ` <... |
| axlttri 11202 | Ordering on reals satisfie... |
| axlttrn 11203 | Ordering on reals is trans... |
| axltadd 11204 | Ordering property of addit... |
| axmulgt0 11205 | The product of two positiv... |
| axsup 11206 | A nonempty, bounded-above ... |
| lttr 11207 | Alias for ~ axlttrn , for ... |
| mulgt0 11208 | The product of two positiv... |
| lenlt 11209 | 'Less than or equal to' ex... |
| ltnle 11210 | 'Less than' expressed in t... |
| ltso 11211 | 'Less than' is a strict or... |
| gtso 11212 | 'Greater than' is a strict... |
| lttri2 11213 | Consequence of trichotomy.... |
| lttri3 11214 | Trichotomy law for 'less t... |
| lttri4 11215 | Trichotomy law for 'less t... |
| letri3 11216 | Trichotomy law. (Contribu... |
| leloe 11217 | 'Less than or equal to' ex... |
| eqlelt 11218 | Equality in terms of 'less... |
| ltle 11219 | 'Less than' implies 'less ... |
| leltne 11220 | 'Less than or equal to' im... |
| lelttr 11221 | Transitive law. (Contribu... |
| leltletr 11222 | Transitive law, weaker for... |
| ltletr 11223 | Transitive law. (Contribu... |
| ltleletr 11224 | Transitive law, weaker for... |
| letr 11225 | Transitive law. (Contribu... |
| ltnr 11226 | 'Less than' is irreflexive... |
| leid 11227 | 'Less than or equal to' is... |
| ltne 11228 | 'Less than' implies not eq... |
| ltnsym 11229 | 'Less than' is not symmetr... |
| ltnsym2 11230 | 'Less than' is antisymmetr... |
| letric 11231 | Trichotomy law. (Contribu... |
| ltlen 11232 | 'Less than' expressed in t... |
| eqle 11233 | Equality implies 'less tha... |
| eqled 11234 | Equality implies 'less tha... |
| ltadd2 11235 | Addition to both sides of ... |
| ne0gt0 11236 | A nonzero nonnegative numb... |
| lecasei 11237 | Ordering elimination by ca... |
| lelttric 11238 | Trichotomy law. (Contribu... |
| ltlecasei 11239 | Ordering elimination by ca... |
| ltnri 11240 | 'Less than' is irreflexive... |
| eqlei 11241 | Equality implies 'less tha... |
| eqlei2 11242 | Equality implies 'less tha... |
| gtneii 11243 | 'Less than' implies not eq... |
| ltneii 11244 | 'Greater than' implies not... |
| lttri2i 11245 | Consequence of trichotomy.... |
| lttri3i 11246 | Consequence of trichotomy.... |
| letri3i 11247 | Consequence of trichotomy.... |
| leloei 11248 | 'Less than or equal to' in... |
| ltleni 11249 | 'Less than' expressed in t... |
| ltnsymi 11250 | 'Less than' is not symmetr... |
| lenlti 11251 | 'Less than or equal to' in... |
| ltnlei 11252 | 'Less than' in terms of 'l... |
| ltlei 11253 | 'Less than' implies 'less ... |
| ltleii 11254 | 'Less than' implies 'less ... |
| ltnei 11255 | 'Less than' implies not eq... |
| letrii 11256 | Trichotomy law for 'less t... |
| lttri 11257 | 'Less than' is transitive.... |
| lelttri 11258 | 'Less than or equal to', '... |
| ltletri 11259 | 'Less than', 'less than or... |
| letri 11260 | 'Less than or equal to' is... |
| le2tri3i 11261 | Extended trichotomy law fo... |
| ltadd2i 11262 | Addition to both sides of ... |
| mulgt0i 11263 | The product of two positiv... |
| mulgt0ii 11264 | The product of two positiv... |
| ltnrd 11265 | 'Less than' is irreflexive... |
| gtned 11266 | 'Less than' implies not eq... |
| ltned 11267 | 'Greater than' implies not... |
| ne0gt0d 11268 | A nonzero nonnegative numb... |
| lttrid 11269 | Ordering on reals satisfie... |
| lttri2d 11270 | Consequence of trichotomy.... |
| lttri3d 11271 | Consequence of trichotomy.... |
| lttri4d 11272 | Trichotomy law for 'less t... |
| letri3d 11273 | Consequence of trichotomy.... |
| leloed 11274 | 'Less than or equal to' in... |
| eqleltd 11275 | Equality in terms of 'less... |
| ltlend 11276 | 'Less than' expressed in t... |
| lenltd 11277 | 'Less than or equal to' in... |
| ltnled 11278 | 'Less than' in terms of 'l... |
| ltled 11279 | 'Less than' implies 'less ... |
| ltnsymd 11280 | 'Less than' implies 'less ... |
| nltled 11281 | 'Not less than ' implies '... |
| lensymd 11282 | 'Less than or equal to' im... |
| letrid 11283 | Trichotomy law for 'less t... |
| leltned 11284 | 'Less than or equal to' im... |
| leneltd 11285 | 'Less than or equal to' an... |
| mulgt0d 11286 | The product of two positiv... |
| ltadd2d 11287 | Addition to both sides of ... |
| letrd 11288 | Transitive law deduction f... |
| lelttrd 11289 | Transitive law deduction f... |
| ltadd2dd 11290 | Addition to both sides of ... |
| ltletrd 11291 | Transitive law deduction f... |
| lttrd 11292 | Transitive law deduction f... |
| lelttrdi 11293 | If a number is less than a... |
| dedekind 11294 | The Dedekind cut theorem. ... |
| dedekindle 11295 | The Dedekind cut theorem, ... |
| mul12 11296 | Commutative/associative la... |
| mul32 11297 | Commutative/associative la... |
| mul31 11298 | Commutative/associative la... |
| mul4 11299 | Rearrangement of 4 factors... |
| mul4r 11300 | Rearrangement of 4 factors... |
| muladd11 11301 | A simple product of sums e... |
| 1p1times 11302 | Two times a number. (Cont... |
| peano2cn 11303 | A theorem for complex numb... |
| peano2re 11304 | A theorem for reals analog... |
| readdcan 11305 | Cancellation law for addit... |
| 00id 11306 | ` 0 ` is its own additive ... |
| mul02lem1 11307 | Lemma for ~ mul02 . If an... |
| mul02lem2 11308 | Lemma for ~ mul02 . Zero ... |
| mul02 11309 | Multiplication by ` 0 ` . ... |
| mul01 11310 | Multiplication by ` 0 ` . ... |
| addrid 11311 | ` 0 ` is an additive ident... |
| cnegex 11312 | Existence of the negative ... |
| cnegex2 11313 | Existence of a left invers... |
| addlid 11314 | ` 0 ` is a left identity f... |
| addcan 11315 | Cancellation law for addit... |
| addcan2 11316 | Cancellation law for addit... |
| addcom 11317 | Addition commutes. This u... |
| addridi 11318 | ` 0 ` is an additive ident... |
| addlidi 11319 | ` 0 ` is a left identity f... |
| mul02i 11320 | Multiplication by 0. Theo... |
| mul01i 11321 | Multiplication by ` 0 ` . ... |
| addcomi 11322 | Addition commutes. Based ... |
| addcomli 11323 | Addition commutes. (Contr... |
| addcani 11324 | Cancellation law for addit... |
| addcan2i 11325 | Cancellation law for addit... |
| mul12i 11326 | Commutative/associative la... |
| mul32i 11327 | Commutative/associative la... |
| mul4i 11328 | Rearrangement of 4 factors... |
| mul02d 11329 | Multiplication by 0. Theo... |
| mul01d 11330 | Multiplication by ` 0 ` . ... |
| addridd 11331 | ` 0 ` is an additive ident... |
| addlidd 11332 | ` 0 ` is a left identity f... |
| addcomd 11333 | Addition commutes. Based ... |
| addcand 11334 | Cancellation law for addit... |
| addcan2d 11335 | Cancellation law for addit... |
| addcanad 11336 | Cancelling a term on the l... |
| addcan2ad 11337 | Cancelling a term on the r... |
| addneintrd 11338 | Introducing a term on the ... |
| addneintr2d 11339 | Introducing a term on the ... |
| mul12d 11340 | Commutative/associative la... |
| mul32d 11341 | Commutative/associative la... |
| mul31d 11342 | Commutative/associative la... |
| mul4d 11343 | Rearrangement of 4 factors... |
| muladd11r 11344 | A simple product of sums e... |
| comraddd 11345 | Commute RHS addition, in d... |
| comraddi 11346 | Commute RHS addition. See... |
| ltaddneg 11347 | Adding a negative number t... |
| ltaddnegr 11348 | Adding a negative number t... |
| add12 11349 | Commutative/associative la... |
| add32 11350 | Commutative/associative la... |
| add32r 11351 | Commutative/associative la... |
| add4 11352 | Rearrangement of 4 terms i... |
| add42 11353 | Rearrangement of 4 terms i... |
| add12i 11354 | Commutative/associative la... |
| add32i 11355 | Commutative/associative la... |
| add4i 11356 | Rearrangement of 4 terms i... |
| add42i 11357 | Rearrangement of 4 terms i... |
| add12d 11358 | Commutative/associative la... |
| add32d 11359 | Commutative/associative la... |
| add4d 11360 | Rearrangement of 4 terms i... |
| add42d 11361 | Rearrangement of 4 terms i... |
| 0cnALT 11366 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11367 | Alternate proof of ~ 0cnAL... |
| negeu 11368 | Existential uniqueness of ... |
| subval 11369 | Value of subtraction, whic... |
| negeq 11370 | Equality theorem for negat... |
| negeqi 11371 | Equality inference for neg... |
| negeqd 11372 | Equality deduction for neg... |
| nfnegd 11373 | Deduction version of ~ nfn... |
| nfneg 11374 | Bound-variable hypothesis ... |
| csbnegg 11375 | Move class substitution in... |
| negex 11376 | A negative is a set. (Con... |
| subcl 11377 | Closure law for subtractio... |
| negcl 11378 | Closure law for negative. ... |
| negicn 11379 | ` -u _i ` is a complex num... |
| subf 11380 | Subtraction is an operatio... |
| subadd 11381 | Relationship between subtr... |
| subadd2 11382 | Relationship between subtr... |
| subsub23 11383 | Swap subtrahend and result... |
| pncan 11384 | Cancellation law for subtr... |
| pncan2 11385 | Cancellation law for subtr... |
| pncan3 11386 | Subtraction and addition o... |
| npcan 11387 | Cancellation law for subtr... |
| addsubass 11388 | Associative-type law for a... |
| addsub 11389 | Law for addition and subtr... |
| subadd23 11390 | Commutative/associative la... |
| addsub12 11391 | Commutative/associative la... |
| 2addsub 11392 | Law for subtraction and ad... |
| addsubeq4 11393 | Relation between sums and ... |
| pncan3oi 11394 | Subtraction and addition o... |
| mvrraddi 11395 | Move the right term in a s... |
| mvrladdi 11396 | Move the left term in a su... |
| mvlladdi 11397 | Move the left term in a su... |
| subid 11398 | Subtraction of a number fr... |
| subid1 11399 | Identity law for subtracti... |
| npncan 11400 | Cancellation law for subtr... |
| nppcan 11401 | Cancellation law for subtr... |
| nnpcan 11402 | Cancellation law for subtr... |
| nppcan3 11403 | Cancellation law for subtr... |
| subcan2 11404 | Cancellation law for subtr... |
| subeq0 11405 | If the difference between ... |
| npncan2 11406 | Cancellation law for subtr... |
| subsub2 11407 | Law for double subtraction... |
| nncan 11408 | Cancellation law for subtr... |
| subsub 11409 | Law for double subtraction... |
| nppcan2 11410 | Cancellation law for subtr... |
| subsub3 11411 | Law for double subtraction... |
| subsub4 11412 | Law for double subtraction... |
| sub32 11413 | Swap the second and third ... |
| nnncan 11414 | Cancellation law for subtr... |
| nnncan1 11415 | Cancellation law for subtr... |
| nnncan2 11416 | Cancellation law for subtr... |
| npncan3 11417 | Cancellation law for subtr... |
| pnpcan 11418 | Cancellation law for mixed... |
| pnpcan2 11419 | Cancellation law for mixed... |
| pnncan 11420 | Cancellation law for mixed... |
| ppncan 11421 | Cancellation law for mixed... |
| addsub4 11422 | Rearrangement of 4 terms i... |
| subadd4 11423 | Rearrangement of 4 terms i... |
| sub4 11424 | Rearrangement of 4 terms i... |
| neg0 11425 | Minus 0 equals 0. (Contri... |
| negid 11426 | Addition of a number and i... |
| negsub 11427 | Relationship between subtr... |
| subneg 11428 | Relationship between subtr... |
| negneg 11429 | A number is equal to the n... |
| neg11 11430 | Negative is one-to-one. (... |
| negcon1 11431 | Negative contraposition la... |
| negcon2 11432 | Negative contraposition la... |
| negeq0 11433 | A number is zero iff its n... |
| subcan 11434 | Cancellation law for subtr... |
| negsubdi 11435 | Distribution of negative o... |
| negdi 11436 | Distribution of negative o... |
| negdi2 11437 | Distribution of negative o... |
| negsubdi2 11438 | Distribution of negative o... |
| neg2sub 11439 | Relationship between subtr... |
| renegcli 11440 | Closure law for negative o... |
| resubcli 11441 | Closure law for subtractio... |
| renegcl 11442 | Closure law for negative o... |
| resubcl 11443 | Closure law for subtractio... |
| negreb 11444 | The negative of a real is ... |
| peano2cnm 11445 | "Reverse" second Peano pos... |
| peano2rem 11446 | "Reverse" second Peano pos... |
| negcli 11447 | Closure law for negative. ... |
| negidi 11448 | Addition of a number and i... |
| negnegi 11449 | A number is equal to the n... |
| subidi 11450 | Subtraction of a number fr... |
| subid1i 11451 | Identity law for subtracti... |
| negne0bi 11452 | A number is nonzero iff it... |
| negrebi 11453 | The negative of a real is ... |
| negne0i 11454 | The negative of a nonzero ... |
| subcli 11455 | Closure law for subtractio... |
| pncan3i 11456 | Subtraction and addition o... |
| negsubi 11457 | Relationship between subtr... |
| subnegi 11458 | Relationship between subtr... |
| subeq0i 11459 | If the difference between ... |
| neg11i 11460 | Negative is one-to-one. (... |
| negcon1i 11461 | Negative contraposition la... |
| negcon2i 11462 | Negative contraposition la... |
| negdii 11463 | Distribution of negative o... |
| negsubdii 11464 | Distribution of negative o... |
| negsubdi2i 11465 | Distribution of negative o... |
| subaddi 11466 | Relationship between subtr... |
| subadd2i 11467 | Relationship between subtr... |
| subaddrii 11468 | Relationship between subtr... |
| subsub23i 11469 | Swap subtrahend and result... |
| addsubassi 11470 | Associative-type law for s... |
| addsubi 11471 | Law for subtraction and ad... |
| subcani 11472 | Cancellation law for subtr... |
| subcan2i 11473 | Cancellation law for subtr... |
| pnncani 11474 | Cancellation law for mixed... |
| addsub4i 11475 | Rearrangement of 4 terms i... |
| 0reALT 11476 | Alternate proof of ~ 0re .... |
| negcld 11477 | Closure law for negative. ... |
| subidd 11478 | Subtraction of a number fr... |
| subid1d 11479 | Identity law for subtracti... |
| negidd 11480 | Addition of a number and i... |
| negnegd 11481 | A number is equal to the n... |
| negeq0d 11482 | A number is zero iff its n... |
| negne0bd 11483 | A number is nonzero iff it... |
| negcon1d 11484 | Contraposition law for una... |
| negcon1ad 11485 | Contraposition law for una... |
| neg11ad 11486 | The negatives of two compl... |
| negned 11487 | If two complex numbers are... |
| negne0d 11488 | The negative of a nonzero ... |
| negrebd 11489 | The negative of a real is ... |
| subcld 11490 | Closure law for subtractio... |
| pncand 11491 | Cancellation law for subtr... |
| pncan2d 11492 | Cancellation law for subtr... |
| pncan3d 11493 | Subtraction and addition o... |
| npcand 11494 | Cancellation law for subtr... |
| nncand 11495 | Cancellation law for subtr... |
| negsubd 11496 | Relationship between subtr... |
| subnegd 11497 | Relationship between subtr... |
| subeq0d 11498 | If the difference between ... |
| subne0d 11499 | Two unequal numbers have n... |
| subeq0ad 11500 | The difference of two comp... |
| subne0ad 11501 | If the difference of two c... |
| neg11d 11502 | If the difference between ... |
| negdid 11503 | Distribution of negative o... |
| negdi2d 11504 | Distribution of negative o... |
| negsubdid 11505 | Distribution of negative o... |
| negsubdi2d 11506 | Distribution of negative o... |
| neg2subd 11507 | Relationship between subtr... |
| subaddd 11508 | Relationship between subtr... |
| subadd2d 11509 | Relationship between subtr... |
| addsubassd 11510 | Associative-type law for s... |
| addsubd 11511 | Law for subtraction and ad... |
| subadd23d 11512 | Commutative/associative la... |
| addsub12d 11513 | Commutative/associative la... |
| npncand 11514 | Cancellation law for subtr... |
| nppcand 11515 | Cancellation law for subtr... |
| nppcan2d 11516 | Cancellation law for subtr... |
| nppcan3d 11517 | Cancellation law for subtr... |
| subsubd 11518 | Law for double subtraction... |
| subsub2d 11519 | Law for double subtraction... |
| subsub3d 11520 | Law for double subtraction... |
| subsub4d 11521 | Law for double subtraction... |
| sub32d 11522 | Swap the second and third ... |
| nnncand 11523 | Cancellation law for subtr... |
| nnncan1d 11524 | Cancellation law for subtr... |
| nnncan2d 11525 | Cancellation law for subtr... |
| npncan3d 11526 | Cancellation law for subtr... |
| pnpcand 11527 | Cancellation law for mixed... |
| pnpcan2d 11528 | Cancellation law for mixed... |
| pnncand 11529 | Cancellation law for mixed... |
| ppncand 11530 | Cancellation law for mixed... |
| subcand 11531 | Cancellation law for subtr... |
| subcan2d 11532 | Cancellation law for subtr... |
| subcanad 11533 | Cancellation law for subtr... |
| subneintrd 11534 | Introducing subtraction on... |
| subcan2ad 11535 | Cancellation law for subtr... |
| subneintr2d 11536 | Introducing subtraction on... |
| addsub4d 11537 | Rearrangement of 4 terms i... |
| subadd4d 11538 | Rearrangement of 4 terms i... |
| sub4d 11539 | Rearrangement of 4 terms i... |
| 2addsubd 11540 | Law for subtraction and ad... |
| addsubeq4d 11541 | Relation between sums and ... |
| subsubadd23 11542 | Swap the second and the th... |
| addsubsub23 11543 | Swap the second and the th... |
| subeqxfrd 11544 | Transfer two terms of a su... |
| mvlraddd 11545 | Move the right term in a s... |
| mvlladdd 11546 | Move the left term in a su... |
| mvrraddd 11547 | Move the right term in a s... |
| mvrladdd 11548 | Move the left term in a su... |
| assraddsubd 11549 | Associate RHS addition-sub... |
| subaddeqd 11550 | Transfer two terms of a su... |
| addlsub 11551 | Left-subtraction: Subtrac... |
| addrsub 11552 | Right-subtraction: Subtra... |
| subexsub 11553 | A subtraction law: Exchan... |
| addid0 11554 | If adding a number to a an... |
| addn0nid 11555 | Adding a nonzero number to... |
| pnpncand 11556 | Addition/subtraction cance... |
| subeqrev 11557 | Reverse the order of subtr... |
| addeq0 11558 | Two complex numbers add up... |
| pncan1 11559 | Cancellation law for addit... |
| npcan1 11560 | Cancellation law for subtr... |
| subeq0bd 11561 | If two complex numbers are... |
| renegcld 11562 | Closure law for negative o... |
| resubcld 11563 | Closure law for subtractio... |
| negn0 11564 | The image under negation o... |
| negf1o 11565 | Negation is an isomorphism... |
| kcnktkm1cn 11566 | k times k minus 1 is a com... |
| muladd 11567 | Product of two sums. (Con... |
| subdi 11568 | Distribution of multiplica... |
| subdir 11569 | Distribution of multiplica... |
| ine0 11570 | The imaginary unit ` _i ` ... |
| mulneg1 11571 | Product with negative is n... |
| mulneg2 11572 | The product with a negativ... |
| mulneg12 11573 | Swap the negative sign in ... |
| mul2neg 11574 | Product of two negatives. ... |
| submul2 11575 | Convert a subtraction to a... |
| mulm1 11576 | Product with minus one is ... |
| addneg1mul 11577 | Addition with product with... |
| mulsub 11578 | Product of two differences... |
| mulsub2 11579 | Swap the order of subtract... |
| mulm1i 11580 | Product with minus one is ... |
| mulneg1i 11581 | Product with negative is n... |
| mulneg2i 11582 | Product with negative is n... |
| mul2negi 11583 | Product of two negatives. ... |
| subdii 11584 | Distribution of multiplica... |
| subdiri 11585 | Distribution of multiplica... |
| muladdi 11586 | Product of two sums. (Con... |
| mulm1d 11587 | Product with minus one is ... |
| mulneg1d 11588 | Product with negative is n... |
| mulneg2d 11589 | Product with negative is n... |
| mul2negd 11590 | Product of two negatives. ... |
| subdid 11591 | Distribution of multiplica... |
| subdird 11592 | Distribution of multiplica... |
| muladdd 11593 | Product of two sums. (Con... |
| mulsubd 11594 | Product of two differences... |
| muls1d 11595 | Multiplication by one minu... |
| mulsubfacd 11596 | Multiplication followed by... |
| addmulsub 11597 | The product of a sum and a... |
| subaddmulsub 11598 | The difference with a prod... |
| mulsubaddmulsub 11599 | A special difference of a ... |
| gt0ne0 11600 | Positive implies nonzero. ... |
| lt0ne0 11601 | A number which is less tha... |
| ltadd1 11602 | Addition to both sides of ... |
| leadd1 11603 | Addition to both sides of ... |
| leadd2 11604 | Addition to both sides of ... |
| ltsubadd 11605 | 'Less than' relationship b... |
| ltsubadd2 11606 | 'Less than' relationship b... |
| lesubadd 11607 | 'Less than or equal to' re... |
| lesubadd2 11608 | 'Less than or equal to' re... |
| ltaddsub 11609 | 'Less than' relationship b... |
| ltaddsub2 11610 | 'Less than' relationship b... |
| leaddsub 11611 | 'Less than or equal to' re... |
| leaddsub2 11612 | 'Less than or equal to' re... |
| suble 11613 | Swap subtrahends in an ine... |
| lesub 11614 | Swap subtrahends in an ine... |
| ltsub23 11615 | 'Less than' relationship b... |
| ltsub13 11616 | 'Less than' relationship b... |
| le2add 11617 | Adding both sides of two '... |
| ltleadd 11618 | Adding both sides of two o... |
| leltadd 11619 | Adding both sides of two o... |
| lt2add 11620 | Adding both sides of two '... |
| addgt0 11621 | The sum of 2 positive numb... |
| addgegt0 11622 | The sum of nonnegative and... |
| addgtge0 11623 | The sum of nonnegative and... |
| addge0 11624 | The sum of 2 nonnegative n... |
| ltaddpos 11625 | Adding a positive number t... |
| ltaddpos2 11626 | Adding a positive number t... |
| ltsubpos 11627 | Subtracting a positive num... |
| posdif 11628 | Comparison of two numbers ... |
| lesub1 11629 | Subtraction from both side... |
| lesub2 11630 | Subtraction of both sides ... |
| ltsub1 11631 | Subtraction from both side... |
| ltsub2 11632 | Subtraction of both sides ... |
| lt2sub 11633 | Subtracting both sides of ... |
| le2sub 11634 | Subtracting both sides of ... |
| ltneg 11635 | Negative of both sides of ... |
| ltnegcon1 11636 | Contraposition of negative... |
| ltnegcon2 11637 | Contraposition of negative... |
| leneg 11638 | Negative of both sides of ... |
| lenegcon1 11639 | Contraposition of negative... |
| lenegcon2 11640 | Contraposition of negative... |
| lt0neg1 11641 | Comparison of a number and... |
| lt0neg2 11642 | Comparison of a number and... |
| le0neg1 11643 | Comparison of a number and... |
| le0neg2 11644 | Comparison of a number and... |
| addge01 11645 | A number is less than or e... |
| addge02 11646 | A number is less than or e... |
| add20 11647 | Two nonnegative numbers ar... |
| subge0 11648 | Nonnegative subtraction. ... |
| suble0 11649 | Nonpositive subtraction. ... |
| leaddle0 11650 | The sum of a real number a... |
| subge02 11651 | Nonnegative subtraction. ... |
| lesub0 11652 | Lemma to show a nonnegativ... |
| mulge0 11653 | The product of two nonnega... |
| mullt0 11654 | The product of two negativ... |
| msqgt0 11655 | A nonzero square is positi... |
| msqge0 11656 | A square is nonnegative. ... |
| 0lt1 11657 | 0 is less than 1. Theorem... |
| 0le1 11658 | 0 is less than or equal to... |
| relin01 11659 | An interval law for less t... |
| ltordlem 11660 | Lemma for ~ ltord1 . (Con... |
| ltord1 11661 | Infer an ordering relation... |
| leord1 11662 | Infer an ordering relation... |
| eqord1 11663 | A strictly increasing real... |
| ltord2 11664 | Infer an ordering relation... |
| leord2 11665 | Infer an ordering relation... |
| eqord2 11666 | A strictly decreasing real... |
| wloglei 11667 | Form of ~ wlogle where bot... |
| wlogle 11668 | If the predicate ` ch ( x ... |
| leidi 11669 | 'Less than or equal to' is... |
| gt0ne0i 11670 | Positive means nonzero (us... |
| gt0ne0ii 11671 | Positive implies nonzero. ... |
| msqgt0i 11672 | A nonzero square is positi... |
| msqge0i 11673 | A square is nonnegative. ... |
| addgt0i 11674 | Addition of 2 positive num... |
| addge0i 11675 | Addition of 2 nonnegative ... |
| addgegt0i 11676 | Addition of nonnegative an... |
| addgt0ii 11677 | Addition of 2 positive num... |
| add20i 11678 | Two nonnegative numbers ar... |
| ltnegi 11679 | Negative of both sides of ... |
| lenegi 11680 | Negative of both sides of ... |
| ltnegcon2i 11681 | Contraposition of negative... |
| mulge0i 11682 | The product of two nonnega... |
| lesub0i 11683 | Lemma to show a nonnegativ... |
| ltaddposi 11684 | Adding a positive number t... |
| posdifi 11685 | Comparison of two numbers ... |
| ltnegcon1i 11686 | Contraposition of negative... |
| lenegcon1i 11687 | Contraposition of negative... |
| subge0i 11688 | Nonnegative subtraction. ... |
| ltadd1i 11689 | Addition to both sides of ... |
| leadd1i 11690 | Addition to both sides of ... |
| leadd2i 11691 | Addition to both sides of ... |
| ltsubaddi 11692 | 'Less than' relationship b... |
| lesubaddi 11693 | 'Less than or equal to' re... |
| ltsubadd2i 11694 | 'Less than' relationship b... |
| lesubadd2i 11695 | 'Less than or equal to' re... |
| ltaddsubi 11696 | 'Less than' relationship b... |
| lt2addi 11697 | Adding both side of two in... |
| le2addi 11698 | Adding both side of two in... |
| gt0ne0d 11699 | Positive implies nonzero. ... |
| lt0ne0d 11700 | Something less than zero i... |
| leidd 11701 | 'Less than or equal to' is... |
| msqgt0d 11702 | A nonzero square is positi... |
| msqge0d 11703 | A square is nonnegative. ... |
| lt0neg1d 11704 | Comparison of a number and... |
| lt0neg2d 11705 | Comparison of a number and... |
| le0neg1d 11706 | Comparison of a number and... |
| le0neg2d 11707 | Comparison of a number and... |
| addgegt0d 11708 | Addition of nonnegative an... |
| addgtge0d 11709 | Addition of positive and n... |
| addgt0d 11710 | Addition of 2 positive num... |
| addge0d 11711 | Addition of 2 nonnegative ... |
| mulge0d 11712 | The product of two nonnega... |
| ltnegd 11713 | Negative of both sides of ... |
| lenegd 11714 | Negative of both sides of ... |
| ltnegcon1d 11715 | Contraposition of negative... |
| ltnegcon2d 11716 | Contraposition of negative... |
| lenegcon1d 11717 | Contraposition of negative... |
| lenegcon2d 11718 | Contraposition of negative... |
| ltaddposd 11719 | Adding a positive number t... |
| ltaddpos2d 11720 | Adding a positive number t... |
| ltsubposd 11721 | Subtracting a positive num... |
| posdifd 11722 | Comparison of two numbers ... |
| addge01d 11723 | A number is less than or e... |
| addge02d 11724 | A number is less than or e... |
| subge0d 11725 | Nonnegative subtraction. ... |
| suble0d 11726 | Nonpositive subtraction. ... |
| subge02d 11727 | Nonnegative subtraction. ... |
| ltadd1d 11728 | Addition to both sides of ... |
| leadd1d 11729 | Addition to both sides of ... |
| leadd2d 11730 | Addition to both sides of ... |
| ltsubaddd 11731 | 'Less than' relationship b... |
| lesubaddd 11732 | 'Less than or equal to' re... |
| ltsubadd2d 11733 | 'Less than' relationship b... |
| lesubadd2d 11734 | 'Less than or equal to' re... |
| ltaddsubd 11735 | 'Less than' relationship b... |
| ltaddsub2d 11736 | 'Less than' relationship b... |
| leaddsub2d 11737 | 'Less than or equal to' re... |
| subled 11738 | Swap subtrahends in an ine... |
| lesubd 11739 | Swap subtrahends in an ine... |
| ltsub23d 11740 | 'Less than' relationship b... |
| ltsub13d 11741 | 'Less than' relationship b... |
| lesub1d 11742 | Subtraction from both side... |
| lesub2d 11743 | Subtraction of both sides ... |
| ltsub1d 11744 | Subtraction from both side... |
| ltsub2d 11745 | Subtraction of both sides ... |
| ltadd1dd 11746 | Addition to both sides of ... |
| ltsub1dd 11747 | Subtraction from both side... |
| ltsub2dd 11748 | Subtraction of both sides ... |
| leadd1dd 11749 | Addition to both sides of ... |
| leadd2dd 11750 | Addition to both sides of ... |
| lesub1dd 11751 | Subtraction from both side... |
| lesub2dd 11752 | Subtraction of both sides ... |
| lesub3d 11753 | The result of subtracting ... |
| le2addd 11754 | Adding both side of two in... |
| le2subd 11755 | Subtracting both sides of ... |
| ltleaddd 11756 | Adding both sides of two o... |
| leltaddd 11757 | Adding both sides of two o... |
| lt2addd 11758 | Adding both side of two in... |
| lt2subd 11759 | Subtracting both sides of ... |
| possumd 11760 | Condition for a positive s... |
| sublt0d 11761 | When a subtraction gives a... |
| ltaddsublt 11762 | Addition and subtraction o... |
| 1le1 11763 | One is less than or equal ... |
| ixi 11764 | ` _i ` times itself is min... |
| recextlem1 11765 | Lemma for ~ recex . (Cont... |
| recextlem2 11766 | Lemma for ~ recex . (Cont... |
| recex 11767 | Existence of reciprocal of... |
| mulcand 11768 | Cancellation law for multi... |
| mulcan2d 11769 | Cancellation law for multi... |
| mulcanad 11770 | Cancellation of a nonzero ... |
| mulcan2ad 11771 | Cancellation of a nonzero ... |
| mulcan 11772 | Cancellation law for multi... |
| mulcan2 11773 | Cancellation law for multi... |
| mulcani 11774 | Cancellation law for multi... |
| mul0or 11775 | If a product is zero, one ... |
| mulne0b 11776 | The product of two nonzero... |
| mulne0 11777 | The product of two nonzero... |
| mulne0i 11778 | The product of two nonzero... |
| muleqadd 11779 | Property of numbers whose ... |
| receu 11780 | Existential uniqueness of ... |
| mulnzcnf 11781 | Multiplication maps nonzer... |
| mul0ori 11782 | If a product is zero, one ... |
| mul0ord 11783 | If a product is zero, one ... |
| msq0i 11784 | A number is zero iff its s... |
| msq0d 11785 | A number is zero iff its s... |
| mulne0bd 11786 | The product of two nonzero... |
| mulne0d 11787 | The product of two nonzero... |
| mulcan1g 11788 | A generalized form of the ... |
| mulcan2g 11789 | A generalized form of the ... |
| mulne0bad 11790 | A factor of a nonzero comp... |
| mulne0bbd 11791 | A factor of a nonzero comp... |
| 1div0 11794 | You can't divide by zero, ... |
| 1div0OLD 11795 | Obsolete version of ~ 1div... |
| divval 11796 | Value of division: if ` A ... |
| divmul 11797 | Relationship between divis... |
| divmul2 11798 | Relationship between divis... |
| divmul3 11799 | Relationship between divis... |
| divcl 11800 | Closure law for division. ... |
| reccl 11801 | Closure law for reciprocal... |
| divcan2 11802 | A cancellation law for div... |
| divcan1 11803 | A cancellation law for div... |
| diveq0 11804 | A ratio is zero iff the nu... |
| divne0b 11805 | The ratio of nonzero numbe... |
| divne0 11806 | The ratio of nonzero numbe... |
| recne0 11807 | The reciprocal of a nonzer... |
| recid 11808 | Multiplication of a number... |
| recid2 11809 | Multiplication of a number... |
| divrec 11810 | Relationship between divis... |
| divrec2 11811 | Relationship between divis... |
| divass 11812 | An associative law for div... |
| div23 11813 | A commutative/associative ... |
| div32 11814 | A commutative/associative ... |
| div13 11815 | A commutative/associative ... |
| div12 11816 | A commutative/associative ... |
| divmulass 11817 | An associative law for div... |
| divmulasscom 11818 | An associative/commutative... |
| divdir 11819 | Distribution of division o... |
| divcan3 11820 | A cancellation law for div... |
| divcan4 11821 | A cancellation law for div... |
| div11 11822 | One-to-one relationship fo... |
| div11OLD 11823 | Obsolete version of ~ div1... |
| diveq1 11824 | Equality in terms of unit ... |
| divid 11825 | A number divided by itself... |
| dividOLD 11826 | Obsolete version of ~ divi... |
| div0 11827 | Division into zero is zero... |
| div0OLD 11828 | Obsolete version of ~ div0... |
| div1 11829 | A number divided by 1 is i... |
| 1div1e1 11830 | 1 divided by 1 is 1. (Con... |
| divneg 11831 | Move negative sign inside ... |
| muldivdir 11832 | Distribution of division o... |
| divsubdir 11833 | Distribution of division o... |
| subdivcomb1 11834 | Bring a term in a subtract... |
| subdivcomb2 11835 | Bring a term in a subtract... |
| recrec 11836 | A number is equal to the r... |
| rec11 11837 | Reciprocal is one-to-one. ... |
| rec11r 11838 | Mutual reciprocals. (Cont... |
| divmuldiv 11839 | Multiplication of two rati... |
| divdivdiv 11840 | Division of two ratios. T... |
| divcan5 11841 | Cancellation of common fac... |
| divmul13 11842 | Swap the denominators in t... |
| divmul24 11843 | Swap the numerators in the... |
| divmuleq 11844 | Cross-multiply in an equal... |
| recdiv 11845 | The reciprocal of a ratio.... |
| divcan6 11846 | Cancellation of inverted f... |
| divdiv32 11847 | Swap denominators in a div... |
| divcan7 11848 | Cancel equal divisors in a... |
| dmdcan 11849 | Cancellation law for divis... |
| divdiv1 11850 | Division into a fraction. ... |
| divdiv2 11851 | Division by a fraction. (... |
| recdiv2 11852 | Division into a reciprocal... |
| ddcan 11853 | Cancellation in a double d... |
| divadddiv 11854 | Addition of two ratios. T... |
| divsubdiv 11855 | Subtraction of two ratios.... |
| conjmul 11856 | Two numbers whose reciproc... |
| rereccl 11857 | Closure law for reciprocal... |
| redivcl 11858 | Closure law for division o... |
| eqneg 11859 | A number equal to its nega... |
| eqnegd 11860 | A complex number equals it... |
| eqnegad 11861 | If a complex number equals... |
| div2neg 11862 | Quotient of two negatives.... |
| divneg2 11863 | Move negative sign inside ... |
| recclzi 11864 | Closure law for reciprocal... |
| recne0zi 11865 | The reciprocal of a nonzer... |
| recidzi 11866 | Multiplication of a number... |
| div1i 11867 | A number divided by 1 is i... |
| eqnegi 11868 | A number equal to its nega... |
| reccli 11869 | Closure law for reciprocal... |
| recidi 11870 | Multiplication of a number... |
| recreci 11871 | A number is equal to the r... |
| dividi 11872 | A number divided by itself... |
| div0i 11873 | Division into zero is zero... |
| divclzi 11874 | Closure law for division. ... |
| divcan1zi 11875 | A cancellation law for div... |
| divcan2zi 11876 | A cancellation law for div... |
| divreczi 11877 | Relationship between divis... |
| divcan3zi 11878 | A cancellation law for div... |
| divcan4zi 11879 | A cancellation law for div... |
| rec11i 11880 | Reciprocal is one-to-one. ... |
| divcli 11881 | Closure law for division. ... |
| divcan2i 11882 | A cancellation law for div... |
| divcan1i 11883 | A cancellation law for div... |
| divreci 11884 | Relationship between divis... |
| divcan3i 11885 | A cancellation law for div... |
| divcan4i 11886 | A cancellation law for div... |
| divne0i 11887 | The ratio of nonzero numbe... |
| rec11ii 11888 | Reciprocal is one-to-one. ... |
| divasszi 11889 | An associative law for div... |
| divmulzi 11890 | Relationship between divis... |
| divdirzi 11891 | Distribution of division o... |
| divdiv23zi 11892 | Swap denominators in a div... |
| divmuli 11893 | Relationship between divis... |
| divdiv32i 11894 | Swap denominators in a div... |
| divassi 11895 | An associative law for div... |
| divdiri 11896 | Distribution of division o... |
| div23i 11897 | A commutative/associative ... |
| div11i 11898 | One-to-one relationship fo... |
| divmuldivi 11899 | Multiplication of two rati... |
| divmul13i 11900 | Swap denominators of two r... |
| divadddivi 11901 | Addition of two ratios. T... |
| divdivdivi 11902 | Division of two ratios. T... |
| rerecclzi 11903 | Closure law for reciprocal... |
| rereccli 11904 | Closure law for reciprocal... |
| redivclzi 11905 | Closure law for division o... |
| redivcli 11906 | Closure law for division o... |
| div1d 11907 | A number divided by 1 is i... |
| reccld 11908 | Closure law for reciprocal... |
| recne0d 11909 | The reciprocal of a nonzer... |
| recidd 11910 | Multiplication of a number... |
| recid2d 11911 | Multiplication of a number... |
| recrecd 11912 | A number is equal to the r... |
| dividd 11913 | A number divided by itself... |
| div0d 11914 | Division into zero is zero... |
| divcld 11915 | Closure law for division. ... |
| divcan1d 11916 | A cancellation law for div... |
| divcan2d 11917 | A cancellation law for div... |
| divrecd 11918 | Relationship between divis... |
| divrec2d 11919 | Relationship between divis... |
| divcan3d 11920 | A cancellation law for div... |
| divcan4d 11921 | A cancellation law for div... |
| diveq0d 11922 | A ratio is zero iff the nu... |
| diveq1d 11923 | Equality in terms of unit ... |
| diveq1ad 11924 | The quotient of two comple... |
| diveq0ad 11925 | A fraction of complex numb... |
| divne1d 11926 | If two complex numbers are... |
| divne0bd 11927 | A ratio is zero iff the nu... |
| divnegd 11928 | Move negative sign inside ... |
| divneg2d 11929 | Move negative sign inside ... |
| div2negd 11930 | Quotient of two negatives.... |
| divne0d 11931 | The ratio of nonzero numbe... |
| recdivd 11932 | The reciprocal of a ratio.... |
| recdiv2d 11933 | Division into a reciprocal... |
| divcan6d 11934 | Cancellation of inverted f... |
| ddcand 11935 | Cancellation in a double d... |
| rec11d 11936 | Reciprocal is one-to-one. ... |
| divmuld 11937 | Relationship between divis... |
| div32d 11938 | A commutative/associative ... |
| div13d 11939 | A commutative/associative ... |
| divdiv32d 11940 | Swap denominators in a div... |
| divcan5d 11941 | Cancellation of common fac... |
| divcan5rd 11942 | Cancellation of common fac... |
| divcan7d 11943 | Cancel equal divisors in a... |
| dmdcand 11944 | Cancellation law for divis... |
| dmdcan2d 11945 | Cancellation law for divis... |
| divdiv1d 11946 | Division into a fraction. ... |
| divdiv2d 11947 | Division by a fraction. (... |
| divmul2d 11948 | Relationship between divis... |
| divmul3d 11949 | Relationship between divis... |
| divassd 11950 | An associative law for div... |
| div12d 11951 | A commutative/associative ... |
| div23d 11952 | A commutative/associative ... |
| divdird 11953 | Distribution of division o... |
| divsubdird 11954 | Distribution of division o... |
| div11d 11955 | One-to-one relationship fo... |
| divmuldivd 11956 | Multiplication of two rati... |
| divmul13d 11957 | Swap denominators of two r... |
| divmul24d 11958 | Swap the numerators in the... |
| divadddivd 11959 | Addition of two ratios. T... |
| divsubdivd 11960 | Subtraction of two ratios.... |
| divmuleqd 11961 | Cross-multiply in an equal... |
| divdivdivd 11962 | Division of two ratios. T... |
| diveq1bd 11963 | If two complex numbers are... |
| div2sub 11964 | Swap the order of subtract... |
| div2subd 11965 | Swap subtrahend and minuen... |
| rereccld 11966 | Closure law for reciprocal... |
| redivcld 11967 | Closure law for division o... |
| subrecd 11968 | Subtraction of reciprocals... |
| subrec 11969 | Subtraction of reciprocals... |
| subreci 11970 | Subtraction of reciprocals... |
| mvllmuld 11971 | Move the left term in a pr... |
| mvllmuli 11972 | Move the left term in a pr... |
| ldiv 11973 | Left-division. (Contribut... |
| rdiv 11974 | Right-division. (Contribu... |
| mdiv 11975 | A division law. (Contribu... |
| lineq 11976 | Solution of a (scalar) lin... |
| elimgt0 11977 | Hypothesis for weak deduct... |
| elimge0 11978 | Hypothesis for weak deduct... |
| ltp1 11979 | A number is less than itse... |
| lep1 11980 | A number is less than or e... |
| ltm1 11981 | A number minus 1 is less t... |
| lem1 11982 | A number minus 1 is less t... |
| letrp1 11983 | A transitive property of '... |
| p1le 11984 | A transitive property of p... |
| recgt0 11985 | The reciprocal of a positi... |
| prodgt0 11986 | Infer that a multiplicand ... |
| prodgt02 11987 | Infer that a multiplier is... |
| ltmul1a 11988 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 11989 | Multiplication of both sid... |
| ltmul2 11990 | Multiplication of both sid... |
| lemul1 11991 | Multiplication of both sid... |
| lemul2 11992 | Multiplication of both sid... |
| lemul1a 11993 | Multiplication of both sid... |
| lemul2a 11994 | Multiplication of both sid... |
| ltmul12a 11995 | Comparison of product of t... |
| lemul12b 11996 | Comparison of product of t... |
| lemul12a 11997 | Comparison of product of t... |
| mulgt1OLD 11998 | Obsolete version of ~ mulg... |
| ltmulgt11 11999 | Multiplication by a number... |
| ltmulgt12 12000 | Multiplication by a number... |
| mulgt1 12001 | The product of two numbers... |
| lemulge11 12002 | Multiplication by a number... |
| lemulge12 12003 | Multiplication by a number... |
| ltdiv1 12004 | Division of both sides of ... |
| lediv1 12005 | Division of both sides of ... |
| gt0div 12006 | Division of a positive num... |
| ge0div 12007 | Division of a nonnegative ... |
| divgt0 12008 | The ratio of two positive ... |
| divge0 12009 | The ratio of nonnegative a... |
| mulge0b 12010 | A condition for multiplica... |
| mulle0b 12011 | A condition for multiplica... |
| mulsuble0b 12012 | A condition for multiplica... |
| ltmuldiv 12013 | 'Less than' relationship b... |
| ltmuldiv2 12014 | 'Less than' relationship b... |
| ltdivmul 12015 | 'Less than' relationship b... |
| ledivmul 12016 | 'Less than or equal to' re... |
| ltdivmul2 12017 | 'Less than' relationship b... |
| lt2mul2div 12018 | 'Less than' relationship b... |
| ledivmul2 12019 | 'Less than or equal to' re... |
| lemuldiv 12020 | 'Less than or equal' relat... |
| lemuldiv2 12021 | 'Less than or equal' relat... |
| ltrec 12022 | The reciprocal of both sid... |
| lerec 12023 | The reciprocal of both sid... |
| lt2msq1 12024 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12025 | Two nonnegative numbers co... |
| ltdiv2 12026 | Division of a positive num... |
| ltrec1 12027 | Reciprocal swap in a 'less... |
| lerec2 12028 | Reciprocal swap in a 'less... |
| ledivdiv 12029 | Invert ratios of positive ... |
| lediv2 12030 | Division of a positive num... |
| ltdiv23 12031 | Swap denominator with othe... |
| lediv23 12032 | Swap denominator with othe... |
| lediv12a 12033 | Comparison of ratio of two... |
| lediv2a 12034 | Division of both sides of ... |
| reclt1 12035 | The reciprocal of a positi... |
| recgt1 12036 | The reciprocal of a positi... |
| recgt1i 12037 | The reciprocal of a number... |
| recp1lt1 12038 | Construct a number less th... |
| recreclt 12039 | Given a positive number ` ... |
| le2msq 12040 | The square function on non... |
| msq11 12041 | The square of a nonnegativ... |
| ledivp1 12042 | "Less than or equal to" an... |
| squeeze0 12043 | If a nonnegative number is... |
| ltp1i 12044 | A number is less than itse... |
| recgt0i 12045 | The reciprocal of a positi... |
| recgt0ii 12046 | The reciprocal of a positi... |
| prodgt0i 12047 | Infer that a multiplicand ... |
| divgt0i 12048 | The ratio of two positive ... |
| divge0i 12049 | The ratio of nonnegative a... |
| ltreci 12050 | The reciprocal of both sid... |
| lereci 12051 | The reciprocal of both sid... |
| lt2msqi 12052 | The square function on non... |
| le2msqi 12053 | The square function on non... |
| msq11i 12054 | The square of a nonnegativ... |
| divgt0i2i 12055 | The ratio of two positive ... |
| ltrecii 12056 | The reciprocal of both sid... |
| divgt0ii 12057 | The ratio of two positive ... |
| ltmul1i 12058 | Multiplication of both sid... |
| ltdiv1i 12059 | Division of both sides of ... |
| ltmuldivi 12060 | 'Less than' relationship b... |
| ltmul2i 12061 | Multiplication of both sid... |
| lemul1i 12062 | Multiplication of both sid... |
| lemul2i 12063 | Multiplication of both sid... |
| ltdiv23i 12064 | Swap denominator with othe... |
| ledivp1i 12065 | "Less than or equal to" an... |
| ltdivp1i 12066 | Less-than and division rel... |
| ltdiv23ii 12067 | Swap denominator with othe... |
| ltmul1ii 12068 | Multiplication of both sid... |
| ltdiv1ii 12069 | Division of both sides of ... |
| ltp1d 12070 | A number is less than itse... |
| lep1d 12071 | A number is less than or e... |
| ltm1d 12072 | A number minus 1 is less t... |
| lem1d 12073 | A number minus 1 is less t... |
| recgt0d 12074 | The reciprocal of a positi... |
| divgt0d 12075 | The ratio of two positive ... |
| mulgt1d 12076 | The product of two numbers... |
| lemulge11d 12077 | Multiplication by a number... |
| lemulge12d 12078 | Multiplication by a number... |
| lemul1ad 12079 | Multiplication of both sid... |
| lemul2ad 12080 | Multiplication of both sid... |
| ltmul12ad 12081 | Comparison of product of t... |
| lemul12ad 12082 | Comparison of product of t... |
| lemul12bd 12083 | Comparison of product of t... |
| fimaxre 12084 | A finite set of real numbe... |
| fimaxre2 12085 | A nonempty finite set of r... |
| fimaxre3 12086 | A nonempty finite set of r... |
| fiminre 12087 | A nonempty finite set of r... |
| fiminre2 12088 | A nonempty finite set of r... |
| negfi 12089 | The negation of a finite s... |
| lbreu 12090 | If a set of reals contains... |
| lbcl 12091 | If a set of reals contains... |
| lble 12092 | If a set of reals contains... |
| lbinf 12093 | If a set of reals contains... |
| lbinfcl 12094 | If a set of reals contains... |
| lbinfle 12095 | If a set of reals contains... |
| sup2 12096 | A nonempty, bounded-above ... |
| sup3 12097 | A version of the completen... |
| infm3lem 12098 | Lemma for ~ infm3 . (Cont... |
| infm3 12099 | The completeness axiom for... |
| suprcl 12100 | Closure of supremum of a n... |
| suprub 12101 | A member of a nonempty bou... |
| suprubd 12102 | Natural deduction form of ... |
| suprcld 12103 | Natural deduction form of ... |
| suprlub 12104 | The supremum of a nonempty... |
| suprnub 12105 | An upper bound is not less... |
| suprleub 12106 | The supremum of a nonempty... |
| supaddc 12107 | The supremum function dist... |
| supadd 12108 | The supremum function dist... |
| supmul1 12109 | The supremum function dist... |
| supmullem1 12110 | Lemma for ~ supmul . (Con... |
| supmullem2 12111 | Lemma for ~ supmul . (Con... |
| supmul 12112 | The supremum function dist... |
| sup3ii 12113 | A version of the completen... |
| suprclii 12114 | Closure of supremum of a n... |
| suprubii 12115 | A member of a nonempty bou... |
| suprlubii 12116 | The supremum of a nonempty... |
| suprnubii 12117 | An upper bound is not less... |
| suprleubii 12118 | The supremum of a nonempty... |
| riotaneg 12119 | The negative of the unique... |
| negiso 12120 | Negation is an order anti-... |
| dfinfre 12121 | The infimum of a set of re... |
| infrecl 12122 | Closure of infimum of a no... |
| infrenegsup 12123 | The infimum of a set of re... |
| infregelb 12124 | Any lower bound of a nonem... |
| infrelb 12125 | If a nonempty set of real ... |
| infrefilb 12126 | The infimum of a finite se... |
| supfirege 12127 | The supremum of a finite s... |
| neg1cn 12128 | -1 is a complex number. (... |
| neg1rr 12129 | -1 is a real number. (Con... |
| neg1ne0 12130 | -1 is nonzero. (Contribut... |
| neg1lt0 12131 | -1 is less than 0. (Contr... |
| negneg1e1 12132 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12133 | The imaginary unit ` _i ` ... |
| rimul 12134 | A real number times the im... |
| cru 12135 | The representation of comp... |
| crne0 12136 | The real representation of... |
| creur 12137 | The real part of a complex... |
| creui 12138 | The imaginary part of a co... |
| cju 12139 | The complex conjugate of a... |
| ofsubeq0 12140 | Function analogue of ~ sub... |
| ofnegsub 12141 | Function analogue of ~ neg... |
| ofsubge0 12142 | Function analogue of ~ sub... |
| nnexALT 12145 | Alternate proof of ~ nnex ... |
| peano5nni 12146 | Peano's inductive postulat... |
| nnssre 12147 | The positive integers are ... |
| nnsscn 12148 | The positive integers are ... |
| nnex 12149 | The set of positive intege... |
| nnre 12150 | A positive integer is a re... |
| nncn 12151 | A positive integer is a co... |
| nnrei 12152 | A positive integer is a re... |
| nncni 12153 | A positive integer is a co... |
| 1nn 12154 | Peano postulate: 1 is a po... |
| peano2nn 12155 | Peano postulate: a success... |
| dfnn2 12156 | Alternate definition of th... |
| dfnn3 12157 | Alternate definition of th... |
| nnred 12158 | A positive integer is a re... |
| nncnd 12159 | A positive integer is a co... |
| peano2nnd 12160 | Peano postulate: a success... |
| nnind 12161 | Principle of Mathematical ... |
| nnindALT 12162 | Principle of Mathematical ... |
| nnindd 12163 | Principle of Mathematical ... |
| nn1m1nn 12164 | Every positive integer is ... |
| nn1suc 12165 | If a statement holds for 1... |
| nnaddcl 12166 | Closure of addition of pos... |
| nnmulcl 12167 | Closure of multiplication ... |
| nnmulcli 12168 | Closure of multiplication ... |
| nnmtmip 12169 | "Minus times minus is plus... |
| nn2ge 12170 | There exists a positive in... |
| nnge1 12171 | A positive integer is one ... |
| nngt1ne1 12172 | A positive integer is grea... |
| nnle1eq1 12173 | A positive integer is less... |
| nngt0 12174 | A positive integer is posi... |
| nnnlt1 12175 | A positive integer is not ... |
| nnnle0 12176 | A positive integer is not ... |
| nnne0 12177 | A positive integer is nonz... |
| nnneneg 12178 | No positive integer is equ... |
| 0nnn 12179 | Zero is not a positive int... |
| 0nnnALT 12180 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12181 | Alternate version of ~ nnn... |
| nngt0i 12182 | A positive integer is posi... |
| nnne0i 12183 | A positive integer is nonz... |
| nndivre 12184 | The quotient of a real and... |
| nnrecre 12185 | The reciprocal of a positi... |
| nnrecgt0 12186 | The reciprocal of a positi... |
| nnsub 12187 | Subtraction of positive in... |
| nnsubi 12188 | Subtraction of positive in... |
| nndiv 12189 | Two ways to express " ` A ... |
| nndivtr 12190 | Transitive property of div... |
| nnge1d 12191 | A positive integer is one ... |
| nngt0d 12192 | A positive integer is posi... |
| nnne0d 12193 | A positive integer is nonz... |
| nnrecred 12194 | The reciprocal of a positi... |
| nnaddcld 12195 | Closure of addition of pos... |
| nnmulcld 12196 | Closure of multiplication ... |
| nndivred 12197 | A positive integer is one ... |
| 0ne1 12214 | Zero is different from one... |
| 1m1e0 12215 | One minus one equals zero.... |
| 2nn 12216 | 2 is a positive integer. ... |
| 2re 12217 | The number 2 is real. (Co... |
| 2cn 12218 | The number 2 is a complex ... |
| 2cnALT 12219 | Alternate proof of ~ 2cn .... |
| 2ex 12220 | The number 2 is a set. (C... |
| 2cnd 12221 | The number 2 is a complex ... |
| 3nn 12222 | 3 is a positive integer. ... |
| 3re 12223 | The number 3 is real. (Co... |
| 3cn 12224 | The number 3 is a complex ... |
| 3ex 12225 | The number 3 is a set. (C... |
| 4nn 12226 | 4 is a positive integer. ... |
| 4re 12227 | The number 4 is real. (Co... |
| 4cn 12228 | The number 4 is a complex ... |
| 5nn 12229 | 5 is a positive integer. ... |
| 5re 12230 | The number 5 is real. (Co... |
| 5cn 12231 | The number 5 is a complex ... |
| 6nn 12232 | 6 is a positive integer. ... |
| 6re 12233 | The number 6 is real. (Co... |
| 6cn 12234 | The number 6 is a complex ... |
| 7nn 12235 | 7 is a positive integer. ... |
| 7re 12236 | The number 7 is real. (Co... |
| 7cn 12237 | The number 7 is a complex ... |
| 8nn 12238 | 8 is a positive integer. ... |
| 8re 12239 | The number 8 is real. (Co... |
| 8cn 12240 | The number 8 is a complex ... |
| 9nn 12241 | 9 is a positive integer. ... |
| 9re 12242 | The number 9 is real. (Co... |
| 9cn 12243 | The number 9 is a complex ... |
| 0le0 12244 | Zero is nonnegative. (Con... |
| 0le2 12245 | The number 0 is less than ... |
| 2pos 12246 | The number 2 is positive. ... |
| 2ne0 12247 | The number 2 is nonzero. ... |
| 3pos 12248 | The number 3 is positive. ... |
| 3ne0 12249 | The number 3 is nonzero. ... |
| 4pos 12250 | The number 4 is positive. ... |
| 4ne0 12251 | The number 4 is nonzero. ... |
| 5pos 12252 | The number 5 is positive. ... |
| 6pos 12253 | The number 6 is positive. ... |
| 7pos 12254 | The number 7 is positive. ... |
| 8pos 12255 | The number 8 is positive. ... |
| 9pos 12256 | The number 9 is positive. ... |
| 1pneg1e0 12257 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12258 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12259 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12260 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12261 | Function value at ` N + 1 ... |
| 1p0e1 12262 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12263 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12264 | 2 - 1 = 1. The result is ... |
| 1e2m1 12265 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12266 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12267 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12268 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12269 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12270 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12271 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12272 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12273 | Two plus two equals four. ... |
| 2times 12274 | Two times a number. (Cont... |
| times2 12275 | A number times 2. (Contri... |
| 2timesi 12276 | Two times a number. (Cont... |
| times2i 12277 | A number times 2. (Contri... |
| 2txmxeqx 12278 | Two times a complex number... |
| 2div2e1 12279 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12280 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12281 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12282 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12283 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12284 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12285 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12286 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12287 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12288 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12289 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12290 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12291 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12292 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12293 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12294 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12295 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12296 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12297 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12298 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12299 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12300 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12301 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12302 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12303 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12304 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12305 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12306 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12307 | 2 times 0 equals 0. (Cont... |
| 4div2e2 12308 | One half of four is two. ... |
| 1lt2 12309 | 1 is less than 2. (Contri... |
| 2lt3 12310 | 2 is less than 3. (Contri... |
| 1lt3 12311 | 1 is less than 3. (Contri... |
| 3lt4 12312 | 3 is less than 4. (Contri... |
| 2lt4 12313 | 2 is less than 4. (Contri... |
| 1lt4 12314 | 1 is less than 4. (Contri... |
| 4lt5 12315 | 4 is less than 5. (Contri... |
| 3lt5 12316 | 3 is less than 5. (Contri... |
| 2lt5 12317 | 2 is less than 5. (Contri... |
| 1lt5 12318 | 1 is less than 5. (Contri... |
| 5lt6 12319 | 5 is less than 6. (Contri... |
| 4lt6 12320 | 4 is less than 6. (Contri... |
| 3lt6 12321 | 3 is less than 6. (Contri... |
| 2lt6 12322 | 2 is less than 6. (Contri... |
| 1lt6 12323 | 1 is less than 6. (Contri... |
| 6lt7 12324 | 6 is less than 7. (Contri... |
| 5lt7 12325 | 5 is less than 7. (Contri... |
| 4lt7 12326 | 4 is less than 7. (Contri... |
| 3lt7 12327 | 3 is less than 7. (Contri... |
| 2lt7 12328 | 2 is less than 7. (Contri... |
| 1lt7 12329 | 1 is less than 7. (Contri... |
| 7lt8 12330 | 7 is less than 8. (Contri... |
| 6lt8 12331 | 6 is less than 8. (Contri... |
| 5lt8 12332 | 5 is less than 8. (Contri... |
| 4lt8 12333 | 4 is less than 8. (Contri... |
| 3lt8 12334 | 3 is less than 8. (Contri... |
| 2lt8 12335 | 2 is less than 8. (Contri... |
| 1lt8 12336 | 1 is less than 8. (Contri... |
| 8lt9 12337 | 8 is less than 9. (Contri... |
| 7lt9 12338 | 7 is less than 9. (Contri... |
| 6lt9 12339 | 6 is less than 9. (Contri... |
| 5lt9 12340 | 5 is less than 9. (Contri... |
| 4lt9 12341 | 4 is less than 9. (Contri... |
| 3lt9 12342 | 3 is less than 9. (Contri... |
| 2lt9 12343 | 2 is less than 9. (Contri... |
| 1lt9 12344 | 1 is less than 9. (Contri... |
| 0ne2 12345 | 0 is not equal to 2. (Con... |
| 1ne2 12346 | 1 is not equal to 2. (Con... |
| 1le2 12347 | 1 is less than or equal to... |
| 2cnne0 12348 | 2 is a nonzero complex num... |
| 2rene0 12349 | 2 is a nonzero real number... |
| 1le3 12350 | 1 is less than or equal to... |
| neg1mulneg1e1 12351 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12352 | One-half is real. (Contri... |
| halfcn 12353 | One-half is a complex numb... |
| halfgt0 12354 | One-half is greater than z... |
| halfge0 12355 | One-half is not negative. ... |
| halflt1 12356 | One-half is less than one.... |
| 2halves 12357 | Two halves make a whole. ... |
| 1mhlfehlf 12358 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12359 | An eighth of four thirds i... |
| halfthird 12360 | Half minus a third. (Cont... |
| halfpm6th 12361 | One half plus or minus one... |
| it0e0 12362 | i times 0 equals 0. (Cont... |
| 2mulicn 12363 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12364 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12365 | Closure of half of a numbe... |
| rehalfcl 12366 | Real closure of half. (Co... |
| half0 12367 | Half of a number is zero i... |
| halfpos2 12368 | A number is positive iff i... |
| halfpos 12369 | A positive number is great... |
| halfnneg2 12370 | A number is nonnegative if... |
| halfaddsubcl 12371 | Closure of half-sum and ha... |
| halfaddsub 12372 | Sum and difference of half... |
| subhalfhalf 12373 | Subtracting the half of a ... |
| lt2halves 12374 | A sum is less than the who... |
| addltmul 12375 | Sum is less than product f... |
| nominpos 12376 | There is no smallest posit... |
| avglt1 12377 | Ordering property for aver... |
| avglt2 12378 | Ordering property for aver... |
| avgle1 12379 | Ordering property for aver... |
| avgle2 12380 | Ordering property for aver... |
| avgle 12381 | The average of two numbers... |
| 2timesd 12382 | Two times a number. (Cont... |
| times2d 12383 | A number times 2. (Contri... |
| halfcld 12384 | Closure of half of a numbe... |
| 2halvesd 12385 | Two halves make a whole. ... |
| rehalfcld 12386 | Real closure of half. (Co... |
| lt2halvesd 12387 | A sum is less than the who... |
| rehalfcli 12388 | Half a real number is real... |
| lt2addmuld 12389 | If two real numbers are le... |
| add1p1 12390 | Adding two times 1 to a nu... |
| sub1m1 12391 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12392 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12393 | A complex number increased... |
| div4p1lem1div2 12394 | An integer greater than 5,... |
| nnunb 12395 | The set of positive intege... |
| arch 12396 | Archimedean property of re... |
| nnrecl 12397 | There exists a positive in... |
| bndndx 12398 | A bounded real sequence ` ... |
| elnn0 12401 | Nonnegative integers expre... |
| nnssnn0 12402 | Positive naturals are a su... |
| nn0ssre 12403 | Nonnegative integers are a... |
| nn0sscn 12404 | Nonnegative integers are a... |
| nn0ex 12405 | The set of nonnegative int... |
| nnnn0 12406 | A positive integer is a no... |
| nnnn0i 12407 | A positive integer is a no... |
| nn0re 12408 | A nonnegative integer is a... |
| nn0cn 12409 | A nonnegative integer is a... |
| nn0rei 12410 | A nonnegative integer is a... |
| nn0cni 12411 | A nonnegative integer is a... |
| dfn2 12412 | The set of positive intege... |
| elnnne0 12413 | The positive integer prope... |
| 0nn0 12414 | 0 is a nonnegative integer... |
| 1nn0 12415 | 1 is a nonnegative integer... |
| 2nn0 12416 | 2 is a nonnegative integer... |
| 3nn0 12417 | 3 is a nonnegative integer... |
| 4nn0 12418 | 4 is a nonnegative integer... |
| 5nn0 12419 | 5 is a nonnegative integer... |
| 6nn0 12420 | 6 is a nonnegative integer... |
| 7nn0 12421 | 7 is a nonnegative integer... |
| 8nn0 12422 | 8 is a nonnegative integer... |
| 9nn0 12423 | 9 is a nonnegative integer... |
| nn0ge0 12424 | A nonnegative integer is g... |
| nn0nlt0 12425 | A nonnegative integer is n... |
| nn0ge0i 12426 | Nonnegative integers are n... |
| nn0le0eq0 12427 | A nonnegative integer is l... |
| nn0p1gt0 12428 | A nonnegative integer incr... |
| nnnn0addcl 12429 | A positive integer plus a ... |
| nn0nnaddcl 12430 | A nonnegative integer plus... |
| 0mnnnnn0 12431 | The result of subtracting ... |
| un0addcl 12432 | If ` S ` is closed under a... |
| un0mulcl 12433 | If ` S ` is closed under m... |
| nn0addcl 12434 | Closure of addition of non... |
| nn0mulcl 12435 | Closure of multiplication ... |
| nn0addcli 12436 | Closure of addition of non... |
| nn0mulcli 12437 | Closure of multiplication ... |
| nn0p1nn 12438 | A nonnegative integer plus... |
| peano2nn0 12439 | Second Peano postulate for... |
| nnm1nn0 12440 | A positive integer minus 1... |
| elnn0nn 12441 | The nonnegative integer pr... |
| elnnnn0 12442 | The positive integer prope... |
| elnnnn0b 12443 | The positive integer prope... |
| elnnnn0c 12444 | The positive integer prope... |
| nn0addge1 12445 | A number is less than or e... |
| nn0addge2 12446 | A number is less than or e... |
| nn0addge1i 12447 | A number is less than or e... |
| nn0addge2i 12448 | A number is less than or e... |
| nn0sub 12449 | Subtraction of nonnegative... |
| ltsubnn0 12450 | Subtracting a nonnegative ... |
| nn0negleid 12451 | A nonnegative integer is g... |
| difgtsumgt 12452 | If the difference of a rea... |
| nn0le2x 12453 | A nonnegative integer is l... |
| nn0le2xi 12454 | A nonnegative integer is l... |
| nn0lele2xi 12455 | 'Less than or equal to' im... |
| fcdmnn0supp 12456 | Two ways to write the supp... |
| fcdmnn0fsupp 12457 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12458 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12459 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12460 | A positive integer is a no... |
| nn0red 12461 | A nonnegative integer is a... |
| nn0cnd 12462 | A nonnegative integer is a... |
| nn0ge0d 12463 | A nonnegative integer is g... |
| nn0addcld 12464 | Closure of addition of non... |
| nn0mulcld 12465 | Closure of multiplication ... |
| nn0readdcl 12466 | Closure law for addition o... |
| nn0n0n1ge2 12467 | A nonnegative integer whic... |
| nn0n0n1ge2b 12468 | A nonnegative integer is n... |
| nn0ge2m1nn 12469 | If a nonnegative integer i... |
| nn0ge2m1nn0 12470 | If a nonnegative integer i... |
| nn0nndivcl 12471 | Closure law for dividing o... |
| elxnn0 12474 | An extended nonnegative in... |
| nn0ssxnn0 12475 | The standard nonnegative i... |
| nn0xnn0 12476 | A standard nonnegative int... |
| xnn0xr 12477 | An extended nonnegative in... |
| 0xnn0 12478 | Zero is an extended nonneg... |
| pnf0xnn0 12479 | Positive infinity is an ex... |
| nn0nepnf 12480 | No standard nonnegative in... |
| nn0xnn0d 12481 | A standard nonnegative int... |
| nn0nepnfd 12482 | No standard nonnegative in... |
| xnn0nemnf 12483 | No extended nonnegative in... |
| xnn0xrnemnf 12484 | The extended nonnegative i... |
| xnn0nnn0pnf 12485 | An extended nonnegative in... |
| elz 12488 | Membership in the set of i... |
| nnnegz 12489 | The negative of a positive... |
| zre 12490 | An integer is a real. (Co... |
| zcn 12491 | An integer is a complex nu... |
| zrei 12492 | An integer is a real numbe... |
| zssre 12493 | The integers are a subset ... |
| zsscn 12494 | The integers are a subset ... |
| zex 12495 | The set of integers exists... |
| elnnz 12496 | Positive integer property ... |
| 0z 12497 | Zero is an integer. (Cont... |
| 0zd 12498 | Zero is an integer, deduct... |
| elnn0z 12499 | Nonnegative integer proper... |
| elznn0nn 12500 | Integer property expressed... |
| elznn0 12501 | Integer property expressed... |
| elznn 12502 | Integer property expressed... |
| zle0orge1 12503 | There is no integer in the... |
| elz2 12504 | Membership in the set of i... |
| dfz2 12505 | Alternative definition of ... |
| zexALT 12506 | Alternate proof of ~ zex .... |
| nnz 12507 | A positive integer is an i... |
| nnssz 12508 | Positive integers are a su... |
| nn0ssz 12509 | Nonnegative integers are a... |
| nn0z 12510 | A nonnegative integer is a... |
| nn0zd 12511 | A nonnegative integer is a... |
| nnzd 12512 | A positive integer is an i... |
| nnzi 12513 | A positive integer is an i... |
| nn0zi 12514 | A nonnegative integer is a... |
| elnnz1 12515 | Positive integer property ... |
| znnnlt1 12516 | An integer is not a positi... |
| nnzrab 12517 | Positive integers expresse... |
| nn0zrab 12518 | Nonnegative integers expre... |
| 1z 12519 | One is an integer. (Contr... |
| 1zzd 12520 | One is an integer, deducti... |
| 2z 12521 | 2 is an integer. (Contrib... |
| 3z 12522 | 3 is an integer. (Contrib... |
| 4z 12523 | 4 is an integer. (Contrib... |
| znegcl 12524 | Closure law for negative i... |
| neg1z 12525 | -1 is an integer. (Contri... |
| znegclb 12526 | A complex number is an int... |
| nn0negz 12527 | The negative of a nonnegat... |
| nn0negzi 12528 | The negative of a nonnegat... |
| zaddcl 12529 | Closure of addition of int... |
| peano2z 12530 | Second Peano postulate gen... |
| zsubcl 12531 | Closure of subtraction of ... |
| peano2zm 12532 | "Reverse" second Peano pos... |
| zletr 12533 | Transitive law of ordering... |
| zrevaddcl 12534 | Reverse closure law for ad... |
| znnsub 12535 | The positive difference of... |
| znn0sub 12536 | The nonnegative difference... |
| nzadd 12537 | The sum of a real number n... |
| zmulcl 12538 | Closure of multiplication ... |
| zltp1le 12539 | Integer ordering relation.... |
| zleltp1 12540 | Integer ordering relation.... |
| zlem1lt 12541 | Integer ordering relation.... |
| zltlem1 12542 | Integer ordering relation.... |
| zltlem1d 12543 | Integer ordering relation,... |
| zgt0ge1 12544 | An integer greater than ` ... |
| nnleltp1 12545 | Positive integer ordering ... |
| nnltp1le 12546 | Positive integer ordering ... |
| nnaddm1cl 12547 | Closure of addition of pos... |
| nn0ltp1le 12548 | Nonnegative integer orderi... |
| nn0leltp1 12549 | Nonnegative integer orderi... |
| nn0ltlem1 12550 | Nonnegative integer orderi... |
| nn0sub2 12551 | Subtraction of nonnegative... |
| nn0lt10b 12552 | A nonnegative integer less... |
| nn0lt2 12553 | A nonnegative integer less... |
| nn0le2is012 12554 | A nonnegative integer whic... |
| nn0lem1lt 12555 | Nonnegative integer orderi... |
| nnlem1lt 12556 | Positive integer ordering ... |
| nnltlem1 12557 | Positive integer ordering ... |
| nnm1ge0 12558 | A positive integer decreas... |
| nn0ge0div 12559 | Division of a nonnegative ... |
| zdiv 12560 | Two ways to express " ` M ... |
| zdivadd 12561 | Property of divisibility: ... |
| zdivmul 12562 | Property of divisibility: ... |
| zextle 12563 | An extensionality-like pro... |
| zextlt 12564 | An extensionality-like pro... |
| recnz 12565 | The reciprocal of a number... |
| btwnnz 12566 | A number between an intege... |
| gtndiv 12567 | A larger number does not d... |
| halfnz 12568 | One-half is not an integer... |
| 3halfnz 12569 | Three halves is not an int... |
| suprzcl 12570 | The supremum of a bounded-... |
| prime 12571 | Two ways to express " ` A ... |
| msqznn 12572 | The square of a nonzero in... |
| zneo 12573 | No even integer equals an ... |
| nneo 12574 | A positive integer is even... |
| nneoi 12575 | A positive integer is even... |
| zeo 12576 | An integer is even or odd.... |
| zeo2 12577 | An integer is even or odd ... |
| peano2uz2 12578 | Second Peano postulate for... |
| peano5uzi 12579 | Peano's inductive postulat... |
| peano5uzti 12580 | Peano's inductive postulat... |
| dfuzi 12581 | An expression for the uppe... |
| uzind 12582 | Induction on the upper int... |
| uzind2 12583 | Induction on the upper int... |
| uzind3 12584 | Induction on the upper int... |
| nn0ind 12585 | Principle of Mathematical ... |
| nn0indALT 12586 | Principle of Mathematical ... |
| nn0indd 12587 | Principle of Mathematical ... |
| fzind 12588 | Induction on the integers ... |
| fnn0ind 12589 | Induction on the integers ... |
| nn0ind-raph 12590 | Principle of Mathematical ... |
| zindd 12591 | Principle of Mathematical ... |
| fzindd 12592 | Induction on the integers ... |
| btwnz 12593 | Any real number can be san... |
| zred 12594 | An integer is a real numbe... |
| zcnd 12595 | An integer is a complex nu... |
| znegcld 12596 | Closure law for negative i... |
| peano2zd 12597 | Deduction from second Pean... |
| zaddcld 12598 | Closure of addition of int... |
| zsubcld 12599 | Closure of subtraction of ... |
| zmulcld 12600 | Closure of multiplication ... |
| znnn0nn 12601 | The negative of a negative... |
| zadd2cl 12602 | Increasing an integer by 2... |
| zriotaneg 12603 | The negative of the unique... |
| suprfinzcl 12604 | The supremum of a nonempty... |
| 9p1e10 12607 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12608 | Version of the definition ... |
| decex 12609 | A decimal number is a set.... |
| deceq1 12610 | Equality theorem for the d... |
| deceq2 12611 | Equality theorem for the d... |
| deceq1i 12612 | Equality theorem for the d... |
| deceq2i 12613 | Equality theorem for the d... |
| deceq12i 12614 | Equality theorem for the d... |
| numnncl 12615 | Closure for a numeral (wit... |
| num0u 12616 | Add a zero in the units pl... |
| num0h 12617 | Add a zero in the higher p... |
| numcl 12618 | Closure for a decimal inte... |
| numsuc 12619 | The successor of a decimal... |
| deccl 12620 | Closure for a numeral. (C... |
| 10nn 12621 | 10 is a positive integer. ... |
| 10pos 12622 | The number 10 is positive.... |
| 10nn0 12623 | 10 is a nonnegative intege... |
| 10re 12624 | The number 10 is real. (C... |
| decnncl 12625 | Closure for a numeral. (C... |
| dec0u 12626 | Add a zero in the units pl... |
| dec0h 12627 | Add a zero in the higher p... |
| numnncl2 12628 | Closure for a decimal inte... |
| decnncl2 12629 | Closure for a decimal inte... |
| numlt 12630 | Comparing two decimal inte... |
| numltc 12631 | Comparing two decimal inte... |
| le9lt10 12632 | A "decimal digit" (i.e. a ... |
| declt 12633 | Comparing two decimal inte... |
| decltc 12634 | Comparing two decimal inte... |
| declth 12635 | Comparing two decimal inte... |
| decsuc 12636 | The successor of a decimal... |
| 3declth 12637 | Comparing two decimal inte... |
| 3decltc 12638 | Comparing two decimal inte... |
| decle 12639 | Comparing two decimal inte... |
| decleh 12640 | Comparing two decimal inte... |
| declei 12641 | Comparing a digit to a dec... |
| numlti 12642 | Comparing a digit to a dec... |
| declti 12643 | Comparing a digit to a dec... |
| decltdi 12644 | Comparing a digit to a dec... |
| numsucc 12645 | The successor of a decimal... |
| decsucc 12646 | The successor of a decimal... |
| 1e0p1 12647 | The successor of zero. (C... |
| dec10p 12648 | Ten plus an integer. (Con... |
| numma 12649 | Perform a multiply-add of ... |
| nummac 12650 | Perform a multiply-add of ... |
| numma2c 12651 | Perform a multiply-add of ... |
| numadd 12652 | Add two decimal integers `... |
| numaddc 12653 | Add two decimal integers `... |
| nummul1c 12654 | The product of a decimal i... |
| nummul2c 12655 | The product of a decimal i... |
| decma 12656 | Perform a multiply-add of ... |
| decmac 12657 | Perform a multiply-add of ... |
| decma2c 12658 | Perform a multiply-add of ... |
| decadd 12659 | Add two numerals ` M ` and... |
| decaddc 12660 | Add two numerals ` M ` and... |
| decaddc2 12661 | Add two numerals ` M ` and... |
| decrmanc 12662 | Perform a multiply-add of ... |
| decrmac 12663 | Perform a multiply-add of ... |
| decaddm10 12664 | The sum of two multiples o... |
| decaddi 12665 | Add two numerals ` M ` and... |
| decaddci 12666 | Add two numerals ` M ` and... |
| decaddci2 12667 | Add two numerals ` M ` and... |
| decsubi 12668 | Difference between a numer... |
| decmul1 12669 | The product of a numeral w... |
| decmul1c 12670 | The product of a numeral w... |
| decmul2c 12671 | The product of a numeral w... |
| decmulnc 12672 | The product of a numeral w... |
| 11multnc 12673 | The product of 11 (as nume... |
| decmul10add 12674 | A multiplication of a numb... |
| 6p5lem 12675 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12676 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12677 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12678 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12679 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12680 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12681 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12682 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12683 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12684 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12685 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12686 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12687 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12688 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12689 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12690 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12691 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12692 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12693 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12694 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12695 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12696 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12697 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12698 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12699 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12700 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12701 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12702 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12703 | 4 times 3 equals 12. (Con... |
| 4t4e16 12704 | 4 times 4 equals 16. (Con... |
| 5t2e10 12705 | 5 times 2 equals 10. (Con... |
| 5t3e15 12706 | 5 times 3 equals 15. (Con... |
| 5t4e20 12707 | 5 times 4 equals 20. (Con... |
| 5t5e25 12708 | 5 times 5 equals 25. (Con... |
| 6t2e12 12709 | 6 times 2 equals 12. (Con... |
| 6t3e18 12710 | 6 times 3 equals 18. (Con... |
| 6t4e24 12711 | 6 times 4 equals 24. (Con... |
| 6t5e30 12712 | 6 times 5 equals 30. (Con... |
| 6t6e36 12713 | 6 times 6 equals 36. (Con... |
| 7t2e14 12714 | 7 times 2 equals 14. (Con... |
| 7t3e21 12715 | 7 times 3 equals 21. (Con... |
| 7t4e28 12716 | 7 times 4 equals 28. (Con... |
| 7t5e35 12717 | 7 times 5 equals 35. (Con... |
| 7t6e42 12718 | 7 times 6 equals 42. (Con... |
| 7t7e49 12719 | 7 times 7 equals 49. (Con... |
| 8t2e16 12720 | 8 times 2 equals 16. (Con... |
| 8t3e24 12721 | 8 times 3 equals 24. (Con... |
| 8t4e32 12722 | 8 times 4 equals 32. (Con... |
| 8t5e40 12723 | 8 times 5 equals 40. (Con... |
| 8t6e48 12724 | 8 times 6 equals 48. (Con... |
| 8t7e56 12725 | 8 times 7 equals 56. (Con... |
| 8t8e64 12726 | 8 times 8 equals 64. (Con... |
| 9t2e18 12727 | 9 times 2 equals 18. (Con... |
| 9t3e27 12728 | 9 times 3 equals 27. (Con... |
| 9t4e36 12729 | 9 times 4 equals 36. (Con... |
| 9t5e45 12730 | 9 times 5 equals 45. (Con... |
| 9t6e54 12731 | 9 times 6 equals 54. (Con... |
| 9t7e63 12732 | 9 times 7 equals 63. (Con... |
| 9t8e72 12733 | 9 times 8 equals 72. (Con... |
| 9t9e81 12734 | 9 times 9 equals 81. (Con... |
| 9t11e99 12735 | 9 times 11 equals 99. (Co... |
| 9lt10 12736 | 9 is less than 10. (Contr... |
| 8lt10 12737 | 8 is less than 10. (Contr... |
| 7lt10 12738 | 7 is less than 10. (Contr... |
| 6lt10 12739 | 6 is less than 10. (Contr... |
| 5lt10 12740 | 5 is less than 10. (Contr... |
| 4lt10 12741 | 4 is less than 10. (Contr... |
| 3lt10 12742 | 3 is less than 10. (Contr... |
| 2lt10 12743 | 2 is less than 10. (Contr... |
| 1lt10 12744 | 1 is less than 10. (Contr... |
| decbin0 12745 | Decompose base 4 into base... |
| decbin2 12746 | Decompose base 4 into base... |
| decbin3 12747 | Decompose base 4 into base... |
| 5recm6rec 12748 | One fifth minus one sixth.... |
| uzval 12751 | The value of the upper int... |
| uzf 12752 | The domain and codomain of... |
| eluz1 12753 | Membership in the upper se... |
| eluzel2 12754 | Implication of membership ... |
| eluz2 12755 | Membership in an upper set... |
| eluzmn 12756 | Membership in an earlier u... |
| eluz1i 12757 | Membership in an upper set... |
| eluzuzle 12758 | An integer in an upper set... |
| eluzelz 12759 | A member of an upper set o... |
| eluzelre 12760 | A member of an upper set o... |
| eluzelcn 12761 | A member of an upper set o... |
| eluzle 12762 | Implication of membership ... |
| eluz 12763 | Membership in an upper set... |
| uzid 12764 | Membership of the least me... |
| uzidd 12765 | Membership of the least me... |
| uzn0 12766 | The upper integers are all... |
| uztrn 12767 | Transitive law for sets of... |
| uztrn2 12768 | Transitive law for sets of... |
| uzneg 12769 | Contraposition law for upp... |
| uzssz 12770 | An upper set of integers i... |
| uzssre 12771 | An upper set of integers i... |
| uzss 12772 | Subset relationship for tw... |
| uztric 12773 | Totality of the ordering r... |
| uz11 12774 | The upper integers functio... |
| eluzp1m1 12775 | Membership in the next upp... |
| eluzp1l 12776 | Strict ordering implied by... |
| eluzp1p1 12777 | Membership in the next upp... |
| eluzadd 12778 | Membership in a later uppe... |
| eluzsub 12779 | Membership in an earlier u... |
| eluzaddi 12780 | Membership in a later uppe... |
| eluzsubi 12781 | Membership in an earlier u... |
| subeluzsub 12782 | Membership of a difference... |
| uzm1 12783 | Choices for an element of ... |
| uznn0sub 12784 | The nonnegative difference... |
| uzin 12785 | Intersection of two upper ... |
| uzp1 12786 | Choices for an element of ... |
| nn0uz 12787 | Nonnegative integers expre... |
| nnuz 12788 | Positive integers expresse... |
| elnnuz 12789 | A positive integer express... |
| elnn0uz 12790 | A nonnegative integer expr... |
| 1eluzge0 12791 | 1 is an integer greater th... |
| 2eluzge0 12792 | 2 is an integer greater th... |
| 2eluzge1 12793 | 2 is an integer greater th... |
| 5eluz3 12794 | 5 is an integer greater th... |
| uzuzle23 12795 | An integer greater than or... |
| uzuzle24 12796 | An integer greater than or... |
| uzuzle34 12797 | An integer greater than or... |
| uzuzle35 12798 | An integer greater than or... |
| eluz2nn 12799 | An integer greater than or... |
| eluz3nn 12800 | An integer greater than or... |
| eluz4nn 12801 | An integer greater than or... |
| eluz5nn 12802 | An integer greater than or... |
| eluzge2nn0 12803 | If an integer is greater t... |
| eluz2n0 12804 | An integer greater than or... |
| uz3m2nn 12805 | An integer greater than or... |
| uznnssnn 12806 | The upper integers startin... |
| raluz 12807 | Restricted universal quant... |
| raluz2 12808 | Restricted universal quant... |
| rexuz 12809 | Restricted existential qua... |
| rexuz2 12810 | Restricted existential qua... |
| 2rexuz 12811 | Double existential quantif... |
| peano2uz 12812 | Second Peano postulate for... |
| peano2uzs 12813 | Second Peano postulate for... |
| peano2uzr 12814 | Reversed second Peano axio... |
| uzaddcl 12815 | Addition closure law for a... |
| nn0pzuz 12816 | The sum of a nonnegative i... |
| uzind4 12817 | Induction on the upper set... |
| uzind4ALT 12818 | Induction on the upper set... |
| uzind4s 12819 | Induction on the upper set... |
| uzind4s2 12820 | Induction on the upper set... |
| uzind4i 12821 | Induction on the upper int... |
| uzwo 12822 | Well-ordering principle: a... |
| uzwo2 12823 | Well-ordering principle: a... |
| nnwo 12824 | Well-ordering principle: a... |
| nnwof 12825 | Well-ordering principle: a... |
| nnwos 12826 | Well-ordering principle: a... |
| indstr 12827 | Strong Mathematical Induct... |
| eluznn0 12828 | Membership in a nonnegativ... |
| eluznn 12829 | Membership in a positive u... |
| eluz2b1 12830 | Two ways to say "an intege... |
| eluz2gt1 12831 | An integer greater than or... |
| eluz2b2 12832 | Two ways to say "an intege... |
| eluz2b3 12833 | Two ways to say "an intege... |
| uz2m1nn 12834 | One less than an integer g... |
| 1nuz2 12835 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12836 | A positive integer is eith... |
| uz2mulcl 12837 | Closure of multiplication ... |
| indstr2 12838 | Strong Mathematical Induct... |
| uzinfi 12839 | Extract the lower bound of... |
| nninf 12840 | The infimum of the set of ... |
| nn0inf 12841 | The infimum of the set of ... |
| infssuzle 12842 | The infimum of a subset of... |
| infssuzcl 12843 | The infimum of a subset of... |
| ublbneg 12844 | The image under negation o... |
| eqreznegel 12845 | Two ways to express the im... |
| supminf 12846 | The supremum of a bounded-... |
| lbzbi 12847 | If a set of reals is bound... |
| zsupss 12848 | Any nonempty bounded subse... |
| suprzcl2 12849 | The supremum of a bounded-... |
| suprzub 12850 | The supremum of a bounded-... |
| uzsupss 12851 | Any bounded subset of an u... |
| nn01to3 12852 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12853 | Alternate proof of ~ nn0ge... |
| uzwo3 12854 | Well-ordering principle: a... |
| zmin 12855 | There is a unique smallest... |
| zmax 12856 | There is a unique largest ... |
| zbtwnre 12857 | There is a unique integer ... |
| rebtwnz 12858 | There is a unique greatest... |
| elq 12861 | Membership in the set of r... |
| qmulz 12862 | If ` A ` is rational, then... |
| znq 12863 | The ratio of an integer an... |
| qre 12864 | A rational number is a rea... |
| zq 12865 | An integer is a rational n... |
| qred 12866 | A rational number is a rea... |
| zssq 12867 | The integers are a subset ... |
| nn0ssq 12868 | The nonnegative integers a... |
| nnssq 12869 | The positive integers are ... |
| qssre 12870 | The rationals are a subset... |
| qsscn 12871 | The rationals are a subset... |
| qex 12872 | The set of rational number... |
| nnq 12873 | A positive integer is rati... |
| qcn 12874 | A rational number is a com... |
| qexALT 12875 | Alternate proof of ~ qex .... |
| qaddcl 12876 | Closure of addition of rat... |
| qnegcl 12877 | Closure law for the negati... |
| qmulcl 12878 | Closure of multiplication ... |
| qsubcl 12879 | Closure of subtraction of ... |
| qreccl 12880 | Closure of reciprocal of r... |
| qdivcl 12881 | Closure of division of rat... |
| qrevaddcl 12882 | Reverse closure law for ad... |
| nnrecq 12883 | The reciprocal of a positi... |
| irradd 12884 | The sum of an irrational n... |
| irrmul 12885 | The product of an irration... |
| elpq 12886 | A positive rational is the... |
| elpqb 12887 | A class is a positive rati... |
| rpnnen1lem2 12888 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12889 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12890 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12891 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12892 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12893 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12894 | One half of ~ rpnnen , whe... |
| reexALT 12895 | Alternate proof of ~ reex ... |
| cnref1o 12896 | There is a natural one-to-... |
| cnexALT 12897 | The set of complex numbers... |
| xrex 12898 | The set of extended reals ... |
| mpoaddex 12899 | The addition operation is ... |
| addex 12900 | The addition operation is ... |
| mpomulex 12901 | The multiplication operati... |
| mulex 12902 | The multiplication operati... |
| elrp 12905 | Membership in the set of p... |
| elrpii 12906 | Membership in the set of p... |
| 1rp 12907 | 1 is a positive real. (Co... |
| 2rp 12908 | 2 is a positive real. (Co... |
| 3rp 12909 | 3 is a positive real. (Co... |
| 5rp 12910 | 5 is a positive real. (Co... |
| rpssre 12911 | The positive reals are a s... |
| rpre 12912 | A positive real is a real.... |
| rpxr 12913 | A positive real is an exte... |
| rpcn 12914 | A positive real is a compl... |
| nnrp 12915 | A positive integer is a po... |
| rpgt0 12916 | A positive real is greater... |
| rpge0 12917 | A positive real is greater... |
| rpregt0 12918 | A positive real is a posit... |
| rprege0 12919 | A positive real is a nonne... |
| rpne0 12920 | A positive real is nonzero... |
| rprene0 12921 | A positive real is a nonze... |
| rpcnne0 12922 | A positive real is a nonze... |
| neglt 12923 | The negative of a positive... |
| rpcndif0 12924 | A positive real number is ... |
| ralrp 12925 | Quantification over positi... |
| rexrp 12926 | Quantification over positi... |
| rpaddcl 12927 | Closure law for addition o... |
| rpmulcl 12928 | Closure law for multiplica... |
| rpmtmip 12929 | "Minus times minus is plus... |
| rpdivcl 12930 | Closure law for division o... |
| rpreccl 12931 | Closure law for reciprocat... |
| rphalfcl 12932 | Closure law for half of a ... |
| rpgecl 12933 | A number greater than or e... |
| rphalflt 12934 | Half of a positive real is... |
| rerpdivcl 12935 | Closure law for division o... |
| ge0p1rp 12936 | A nonnegative number plus ... |
| rpneg 12937 | Either a nonzero real or i... |
| negelrp 12938 | Elementhood of a negation ... |
| negelrpd 12939 | The negation of a negative... |
| 0nrp 12940 | Zero is not a positive rea... |
| ltsubrp 12941 | Subtracting a positive rea... |
| ltaddrp 12942 | Adding a positive number t... |
| difrp 12943 | Two ways to say one number... |
| elrpd 12944 | Membership in the set of p... |
| nnrpd 12945 | A positive integer is a po... |
| zgt1rpn0n1 12946 | An integer greater than 1 ... |
| rpred 12947 | A positive real is a real.... |
| rpxrd 12948 | A positive real is an exte... |
| rpcnd 12949 | A positive real is a compl... |
| rpgt0d 12950 | A positive real is greater... |
| rpge0d 12951 | A positive real is greater... |
| rpne0d 12952 | A positive real is nonzero... |
| rpregt0d 12953 | A positive real is real an... |
| rprege0d 12954 | A positive real is real an... |
| rprene0d 12955 | A positive real is a nonze... |
| rpcnne0d 12956 | A positive real is a nonze... |
| rpreccld 12957 | Closure law for reciprocat... |
| rprecred 12958 | Closure law for reciprocat... |
| rphalfcld 12959 | Closure law for half of a ... |
| reclt1d 12960 | The reciprocal of a positi... |
| recgt1d 12961 | The reciprocal of a positi... |
| rpaddcld 12962 | Closure law for addition o... |
| rpmulcld 12963 | Closure law for multiplica... |
| rpdivcld 12964 | Closure law for division o... |
| ltrecd 12965 | The reciprocal of both sid... |
| lerecd 12966 | The reciprocal of both sid... |
| ltrec1d 12967 | Reciprocal swap in a 'less... |
| lerec2d 12968 | Reciprocal swap in a 'less... |
| lediv2ad 12969 | Division of both sides of ... |
| ltdiv2d 12970 | Division of a positive num... |
| lediv2d 12971 | Division of a positive num... |
| ledivdivd 12972 | Invert ratios of positive ... |
| divge1 12973 | The ratio of a number over... |
| divlt1lt 12974 | A real number divided by a... |
| divle1le 12975 | A real number divided by a... |
| ledivge1le 12976 | If a number is less than o... |
| ge0p1rpd 12977 | A nonnegative number plus ... |
| rerpdivcld 12978 | Closure law for division o... |
| ltsubrpd 12979 | Subtracting a positive rea... |
| ltaddrpd 12980 | Adding a positive number t... |
| ltaddrp2d 12981 | Adding a positive number t... |
| ltmulgt11d 12982 | Multiplication by a number... |
| ltmulgt12d 12983 | Multiplication by a number... |
| gt0divd 12984 | Division of a positive num... |
| ge0divd 12985 | Division of a nonnegative ... |
| rpgecld 12986 | A number greater than or e... |
| divge0d 12987 | The ratio of nonnegative a... |
| ltmul1d 12988 | The ratio of nonnegative a... |
| ltmul2d 12989 | Multiplication of both sid... |
| lemul1d 12990 | Multiplication of both sid... |
| lemul2d 12991 | Multiplication of both sid... |
| ltdiv1d 12992 | Division of both sides of ... |
| lediv1d 12993 | Division of both sides of ... |
| ltmuldivd 12994 | 'Less than' relationship b... |
| ltmuldiv2d 12995 | 'Less than' relationship b... |
| lemuldivd 12996 | 'Less than or equal to' re... |
| lemuldiv2d 12997 | 'Less than or equal to' re... |
| ltdivmuld 12998 | 'Less than' relationship b... |
| ltdivmul2d 12999 | 'Less than' relationship b... |
| ledivmuld 13000 | 'Less than or equal to' re... |
| ledivmul2d 13001 | 'Less than or equal to' re... |
| ltmul1dd 13002 | The ratio of nonnegative a... |
| ltmul2dd 13003 | Multiplication of both sid... |
| ltdiv1dd 13004 | Division of both sides of ... |
| lediv1dd 13005 | Division of both sides of ... |
| lediv12ad 13006 | Comparison of ratio of two... |
| mul2lt0rlt0 13007 | If the result of a multipl... |
| mul2lt0rgt0 13008 | If the result of a multipl... |
| mul2lt0llt0 13009 | If the result of a multipl... |
| mul2lt0lgt0 13010 | If the result of a multipl... |
| mul2lt0bi 13011 | If the result of a multipl... |
| prodge0rd 13012 | Infer that a multiplicand ... |
| prodge0ld 13013 | Infer that a multiplier is... |
| ltdiv23d 13014 | Swap denominator with othe... |
| lediv23d 13015 | Swap denominator with othe... |
| lt2mul2divd 13016 | The ratio of nonnegative a... |
| nnledivrp 13017 | Division of a positive int... |
| nn0ledivnn 13018 | Division of a nonnegative ... |
| addlelt 13019 | If the sum of a real numbe... |
| ge2halflem1 13020 | Half of an integer greater... |
| ltxr 13027 | The 'less than' binary rel... |
| elxr 13028 | Membership in the set of e... |
| xrnemnf 13029 | An extended real other tha... |
| xrnepnf 13030 | An extended real other tha... |
| xrltnr 13031 | The extended real 'less th... |
| ltpnf 13032 | Any (finite) real is less ... |
| ltpnfd 13033 | Any (finite) real is less ... |
| 0ltpnf 13034 | Zero is less than plus inf... |
| mnflt 13035 | Minus infinity is less tha... |
| mnfltd 13036 | Minus infinity is less tha... |
| mnflt0 13037 | Minus infinity is less tha... |
| mnfltpnf 13038 | Minus infinity is less tha... |
| mnfltxr 13039 | Minus infinity is less tha... |
| pnfnlt 13040 | No extended real is greate... |
| nltmnf 13041 | No extended real is less t... |
| pnfge 13042 | Plus infinity is an upper ... |
| pnfged 13043 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13044 | An extended nonnegative in... |
| 0lepnf 13045 | 0 less than or equal to po... |
| xnn0ge0 13046 | An extended nonnegative in... |
| mnfle 13047 | Minus infinity is less tha... |
| mnfled 13048 | Minus infinity is less tha... |
| xrltnsym 13049 | Ordering on the extended r... |
| xrltnsym2 13050 | 'Less than' is antisymmetr... |
| xrlttri 13051 | Ordering on the extended r... |
| xrlttr 13052 | Ordering on the extended r... |
| xrltso 13053 | 'Less than' is a strict or... |
| xrlttri2 13054 | Trichotomy law for 'less t... |
| xrlttri3 13055 | Trichotomy law for 'less t... |
| xrleloe 13056 | 'Less than or equal' expre... |
| xrleltne 13057 | 'Less than or equal to' im... |
| xrltlen 13058 | 'Less than' expressed in t... |
| dfle2 13059 | Alternative definition of ... |
| dflt2 13060 | Alternative definition of ... |
| xrltle 13061 | 'Less than' implies 'less ... |
| xrltled 13062 | 'Less than' implies 'less ... |
| xrleid 13063 | 'Less than or equal to' is... |
| xrleidd 13064 | 'Less than or equal to' is... |
| xrletri 13065 | Trichotomy law for extende... |
| xrletri3 13066 | Trichotomy law for extende... |
| xrletrid 13067 | Trichotomy law for extende... |
| xrlelttr 13068 | Transitive law for orderin... |
| xrltletr 13069 | Transitive law for orderin... |
| xrletr 13070 | Transitive law for orderin... |
| xrlttrd 13071 | Transitive law for orderin... |
| xrlelttrd 13072 | Transitive law for orderin... |
| xrltletrd 13073 | Transitive law for orderin... |
| xrletrd 13074 | Transitive law for orderin... |
| xrltne 13075 | 'Less than' implies not eq... |
| xrgtned 13076 | 'Greater than' implies not... |
| nltpnft 13077 | An extended real is not le... |
| xgepnf 13078 | An extended real which is ... |
| ngtmnft 13079 | An extended real is not gr... |
| xlemnf 13080 | An extended real which is ... |
| xrrebnd 13081 | An extended real is real i... |
| xrre 13082 | A way of proving that an e... |
| xrre2 13083 | An extended real between t... |
| xrre3 13084 | A way of proving that an e... |
| ge0gtmnf 13085 | A nonnegative extended rea... |
| ge0nemnf 13086 | A nonnegative extended rea... |
| xrrege0 13087 | A nonnegative extended rea... |
| xrmax1 13088 | An extended real is less t... |
| xrmax2 13089 | An extended real is less t... |
| xrmin1 13090 | The minimum of two extende... |
| xrmin2 13091 | The minimum of two extende... |
| xrmaxeq 13092 | The maximum of two extende... |
| xrmineq 13093 | The minimum of two extende... |
| xrmaxlt 13094 | Two ways of saying the max... |
| xrltmin 13095 | Two ways of saying an exte... |
| xrmaxle 13096 | Two ways of saying the max... |
| xrlemin 13097 | Two ways of saying a numbe... |
| max1 13098 | A number is less than or e... |
| max1ALT 13099 | A number is less than or e... |
| max2 13100 | A number is less than or e... |
| 2resupmax 13101 | The supremum of two real n... |
| min1 13102 | The minimum of two numbers... |
| min2 13103 | The minimum of two numbers... |
| maxle 13104 | Two ways of saying the max... |
| lemin 13105 | Two ways of saying a numbe... |
| maxlt 13106 | Two ways of saying the max... |
| ltmin 13107 | Two ways of saying a numbe... |
| lemaxle 13108 | A real number which is les... |
| max0sub 13109 | Decompose a real number in... |
| ifle 13110 | An if statement transforms... |
| z2ge 13111 | There exists an integer gr... |
| qbtwnre 13112 | The rational numbers are d... |
| qbtwnxr 13113 | The rational numbers are d... |
| qsqueeze 13114 | If a nonnegative real is l... |
| qextltlem 13115 | Lemma for ~ qextlt and qex... |
| qextlt 13116 | An extensionality-like pro... |
| qextle 13117 | An extensionality-like pro... |
| xralrple 13118 | Show that ` A ` is less th... |
| alrple 13119 | Show that ` A ` is less th... |
| xnegeq 13120 | Equality of two extended n... |
| xnegex 13121 | A negative extended real e... |
| xnegpnf 13122 | Minus ` +oo ` . Remark of... |
| xnegmnf 13123 | Minus ` -oo ` . Remark of... |
| rexneg 13124 | Minus a real number. Rema... |
| xneg0 13125 | The negative of zero. (Co... |
| xnegcl 13126 | Closure of extended real n... |
| xnegneg 13127 | Extended real version of ~... |
| xneg11 13128 | Extended real version of ~... |
| xltnegi 13129 | Forward direction of ~ xlt... |
| xltneg 13130 | Extended real version of ~... |
| xleneg 13131 | Extended real version of ~... |
| xlt0neg1 13132 | Extended real version of ~... |
| xlt0neg2 13133 | Extended real version of ~... |
| xle0neg1 13134 | Extended real version of ~... |
| xle0neg2 13135 | Extended real version of ~... |
| xaddval 13136 | Value of the extended real... |
| xaddf 13137 | The extended real addition... |
| xmulval 13138 | Value of the extended real... |
| xaddpnf1 13139 | Addition of positive infin... |
| xaddpnf2 13140 | Addition of positive infin... |
| xaddmnf1 13141 | Addition of negative infin... |
| xaddmnf2 13142 | Addition of negative infin... |
| pnfaddmnf 13143 | Addition of positive and n... |
| mnfaddpnf 13144 | Addition of negative and p... |
| rexadd 13145 | The extended real addition... |
| rexsub 13146 | Extended real subtraction ... |
| rexaddd 13147 | The extended real addition... |
| xnn0xaddcl 13148 | The extended nonnegative i... |
| xaddnemnf 13149 | Closure of extended real a... |
| xaddnepnf 13150 | Closure of extended real a... |
| xnegid 13151 | Extended real version of ~... |
| xaddcl 13152 | The extended real addition... |
| xaddcom 13153 | The extended real addition... |
| xaddrid 13154 | Extended real version of ~... |
| xaddlid 13155 | Extended real version of ~... |
| xaddridd 13156 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13157 | Extended nonnegative integ... |
| xnn0lenn0nn0 13158 | An extended nonnegative in... |
| xnn0le2is012 13159 | An extended nonnegative in... |
| xnn0xadd0 13160 | The sum of two extended no... |
| xnegdi 13161 | Extended real version of ~... |
| xaddass 13162 | Associativity of extended ... |
| xaddass2 13163 | Associativity of extended ... |
| xpncan 13164 | Extended real version of ~... |
| xnpcan 13165 | Extended real version of ~... |
| xleadd1a 13166 | Extended real version of ~... |
| xleadd2a 13167 | Commuted form of ~ xleadd1... |
| xleadd1 13168 | Weakened version of ~ xlea... |
| xltadd1 13169 | Extended real version of ~... |
| xltadd2 13170 | Extended real version of ~... |
| xaddge0 13171 | The sum of nonnegative ext... |
| xle2add 13172 | Extended real version of ~... |
| xlt2add 13173 | Extended real version of ~... |
| xsubge0 13174 | Extended real version of ~... |
| xposdif 13175 | Extended real version of ~... |
| xlesubadd 13176 | Under certain conditions, ... |
| xmullem 13177 | Lemma for ~ rexmul . (Con... |
| xmullem2 13178 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13179 | Extended real multiplicati... |
| xmul01 13180 | Extended real version of ~... |
| xmul02 13181 | Extended real version of ~... |
| xmulneg1 13182 | Extended real version of ~... |
| xmulneg2 13183 | Extended real version of ~... |
| rexmul 13184 | The extended real multipli... |
| xmulf 13185 | The extended real multipli... |
| xmulcl 13186 | Closure of extended real m... |
| xmulpnf1 13187 | Multiplication by plus inf... |
| xmulpnf2 13188 | Multiplication by plus inf... |
| xmulmnf1 13189 | Multiplication by minus in... |
| xmulmnf2 13190 | Multiplication by minus in... |
| xmulpnf1n 13191 | Multiplication by plus inf... |
| xmulrid 13192 | Extended real version of ~... |
| xmullid 13193 | Extended real version of ~... |
| xmulm1 13194 | Extended real version of ~... |
| xmulasslem2 13195 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13196 | Extended real version of ~... |
| xmulge0 13197 | Extended real version of ~... |
| xmulasslem 13198 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13199 | Lemma for ~ xmulass . (Co... |
| xmulass 13200 | Associativity of the exten... |
| xlemul1a 13201 | Extended real version of ~... |
| xlemul2a 13202 | Extended real version of ~... |
| xlemul1 13203 | Extended real version of ~... |
| xlemul2 13204 | Extended real version of ~... |
| xltmul1 13205 | Extended real version of ~... |
| xltmul2 13206 | Extended real version of ~... |
| xadddilem 13207 | Lemma for ~ xadddi . (Con... |
| xadddi 13208 | Distributive property for ... |
| xadddir 13209 | Commuted version of ~ xadd... |
| xadddi2 13210 | The assumption that the mu... |
| xadddi2r 13211 | Commuted version of ~ xadd... |
| x2times 13212 | Extended real version of ~... |
| xnegcld 13213 | Closure of extended real n... |
| xaddcld 13214 | The extended real addition... |
| xmulcld 13215 | Closure of extended real m... |
| xadd4d 13216 | Rearrangement of 4 terms i... |
| xnn0add4d 13217 | Rearrangement of 4 terms i... |
| xrsupexmnf 13218 | Adding minus infinity to a... |
| xrinfmexpnf 13219 | Adding plus infinity to a ... |
| xrsupsslem 13220 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13221 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13222 | Any subset of extended rea... |
| xrinfmss 13223 | Any subset of extended rea... |
| xrinfmss2 13224 | Any subset of extended rea... |
| xrub 13225 | By quantifying only over r... |
| supxr 13226 | The supremum of a set of e... |
| supxr2 13227 | The supremum of a set of e... |
| supxrcl 13228 | The supremum of an arbitra... |
| supxrun 13229 | The supremum of the union ... |
| supxrmnf 13230 | Adding minus infinity to a... |
| supxrpnf 13231 | The supremum of a set of e... |
| supxrunb1 13232 | The supremum of an unbound... |
| supxrunb2 13233 | The supremum of an unbound... |
| supxrbnd1 13234 | The supremum of a bounded-... |
| supxrbnd2 13235 | The supremum of a bounded-... |
| xrsup0 13236 | The supremum of an empty s... |
| supxrub 13237 | A member of a set of exten... |
| supxrlub 13238 | The supremum of a set of e... |
| supxrleub 13239 | The supremum of a set of e... |
| supxrre 13240 | The real and extended real... |
| supxrbnd 13241 | The supremum of a bounded-... |
| supxrgtmnf 13242 | The supremum of a nonempty... |
| supxrre1 13243 | The supremum of a nonempty... |
| supxrre2 13244 | The supremum of a nonempty... |
| supxrss 13245 | Smaller sets of extended r... |
| xrsupssd 13246 | Inequality deduction for s... |
| infxrcl 13247 | The infimum of an arbitrar... |
| infxrlb 13248 | A member of a set of exten... |
| infxrgelb 13249 | The infimum of a set of ex... |
| infxrre 13250 | The real and extended real... |
| infxrmnf 13251 | The infinimum of a set of ... |
| xrinf0 13252 | The infimum of the empty s... |
| infxrss 13253 | Larger sets of extended re... |
| reltre 13254 | For all real numbers there... |
| rpltrp 13255 | For all positive real numb... |
| reltxrnmnf 13256 | For all extended real numb... |
| infmremnf 13257 | The infimum of the reals i... |
| infmrp1 13258 | The infimum of the positiv... |
| ixxval 13267 | Value of the interval func... |
| elixx1 13268 | Membership in an interval ... |
| ixxf 13269 | The set of intervals of ex... |
| ixxex 13270 | The set of intervals of ex... |
| ixxssxr 13271 | The set of intervals of ex... |
| elixx3g 13272 | Membership in a set of ope... |
| ixxssixx 13273 | An interval is a subset of... |
| ixxdisj 13274 | Split an interval into dis... |
| ixxun 13275 | Split an interval into two... |
| ixxin 13276 | Intersection of two interv... |
| ixxss1 13277 | Subset relationship for in... |
| ixxss2 13278 | Subset relationship for in... |
| ixxss12 13279 | Subset relationship for in... |
| ixxub 13280 | Extract the upper bound of... |
| ixxlb 13281 | Extract the lower bound of... |
| iooex 13282 | The set of open intervals ... |
| iooval 13283 | Value of the open interval... |
| ioo0 13284 | An empty open interval of ... |
| ioon0 13285 | An open interval of extend... |
| ndmioo 13286 | The open interval function... |
| iooid 13287 | An open interval with iden... |
| elioo3g 13288 | Membership in a set of ope... |
| elioore 13289 | A member of an open interv... |
| lbioo 13290 | An open interval does not ... |
| ubioo 13291 | An open interval does not ... |
| iooval2 13292 | Value of the open interval... |
| iooin 13293 | Intersection of two open i... |
| iooss1 13294 | Subset relationship for op... |
| iooss2 13295 | Subset relationship for op... |
| iocval 13296 | Value of the open-below, c... |
| icoval 13297 | Value of the closed-below,... |
| iccval 13298 | Value of the closed interv... |
| elioo1 13299 | Membership in an open inte... |
| elioo2 13300 | Membership in an open inte... |
| elioc1 13301 | Membership in an open-belo... |
| elico1 13302 | Membership in a closed-bel... |
| elicc1 13303 | Membership in a closed int... |
| iccid 13304 | A closed interval with ide... |
| ico0 13305 | An empty open interval of ... |
| ioc0 13306 | An empty open interval of ... |
| icc0 13307 | An empty closed interval o... |
| dfrp2 13308 | Alternate definition of th... |
| elicod 13309 | Membership in a left-close... |
| icogelb 13310 | An element of a left-close... |
| icogelbd 13311 | An element of a left-close... |
| elicore 13312 | A member of a left-closed ... |
| ubioc1 13313 | The upper bound belongs to... |
| lbico1 13314 | The lower bound belongs to... |
| iccleub 13315 | An element of a closed int... |
| iccgelb 13316 | An element of a closed int... |
| elioo5 13317 | Membership in an open inte... |
| eliooxr 13318 | A nonempty open interval s... |
| eliooord 13319 | Ordering implied by a memb... |
| elioo4g 13320 | Membership in an open inte... |
| ioossre 13321 | An open interval is a set ... |
| ioosscn 13322 | An open interval is a set ... |
| elioc2 13323 | Membership in an open-belo... |
| elico2 13324 | Membership in a closed-bel... |
| elicc2 13325 | Membership in a closed rea... |
| elicc2i 13326 | Inference for membership i... |
| elicc4 13327 | Membership in a closed rea... |
| iccss 13328 | Condition for a closed int... |
| iccssioo 13329 | Condition for a closed int... |
| icossico 13330 | Condition for a closed-bel... |
| iccss2 13331 | Condition for a closed int... |
| iccssico 13332 | Condition for a closed int... |
| iccssioo2 13333 | Condition for a closed int... |
| iccssico2 13334 | Condition for a closed int... |
| icossico2d 13335 | Condition for a closed-bel... |
| ioomax 13336 | The open interval from min... |
| iccmax 13337 | The closed interval from m... |
| ioopos 13338 | The set of positive reals ... |
| ioorp 13339 | The set of positive reals ... |
| iooshf 13340 | Shift the arguments of the... |
| iocssre 13341 | A closed-above interval wi... |
| icossre 13342 | A closed-below interval wi... |
| iccssre 13343 | A closed real interval is ... |
| iccssxr 13344 | A closed interval is a set... |
| iocssxr 13345 | An open-below, closed-abov... |
| icossxr 13346 | A closed-below, open-above... |
| ioossicc 13347 | An open interval is a subs... |
| iccssred 13348 | A closed real interval is ... |
| eliccxr 13349 | A member of a closed inter... |
| icossicc 13350 | A closed-below, open-above... |
| iocssicc 13351 | A closed-above, open-below... |
| ioossico 13352 | An open interval is a subs... |
| iocssioo 13353 | Condition for a closed int... |
| icossioo 13354 | Condition for a closed int... |
| ioossioo 13355 | Condition for an open inte... |
| iccsupr 13356 | A nonempty subset of a clo... |
| elioopnf 13357 | Membership in an unbounded... |
| elioomnf 13358 | Membership in an unbounded... |
| elicopnf 13359 | Membership in a closed unb... |
| repos 13360 | Two ways of saying that a ... |
| ioof 13361 | The set of open intervals ... |
| iccf 13362 | The set of closed interval... |
| unirnioo 13363 | The union of the range of ... |
| dfioo2 13364 | Alternate definition of th... |
| ioorebas 13365 | Open intervals are element... |
| xrge0neqmnf 13366 | A nonnegative extended rea... |
| xrge0nre 13367 | An extended real which is ... |
| elrege0 13368 | The predicate "is a nonneg... |
| nn0rp0 13369 | A nonnegative integer is a... |
| rge0ssre 13370 | Nonnegative real numbers a... |
| elxrge0 13371 | Elementhood in the set of ... |
| 0e0icopnf 13372 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13373 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13374 | The nonnegative reals are ... |
| ge0mulcl 13375 | The nonnegative reals are ... |
| ge0xaddcl 13376 | The nonnegative reals are ... |
| ge0xmulcl 13377 | The nonnegative extended r... |
| lbicc2 13378 | The lower bound of a close... |
| ubicc2 13379 | The upper bound of a close... |
| elicc01 13380 | Membership in the closed r... |
| elunitrn 13381 | The closed unit interval i... |
| elunitcn 13382 | The closed unit interval i... |
| 0elunit 13383 | Zero is an element of the ... |
| 1elunit 13384 | One is an element of the c... |
| iooneg 13385 | Membership in a negated op... |
| iccneg 13386 | Membership in a negated cl... |
| icoshft 13387 | A shifted real is a member... |
| icoshftf1o 13388 | Shifting a closed-below, o... |
| icoun 13389 | The union of two adjacent ... |
| icodisj 13390 | Adjacent left-closed right... |
| ioounsn 13391 | The union of an open inter... |
| snunioo 13392 | The closure of one end of ... |
| snunico 13393 | The closure of the open en... |
| snunioc 13394 | The closure of the open en... |
| prunioo 13395 | The closure of an open rea... |
| ioodisj 13396 | If the upper bound of one ... |
| ioojoin 13397 | Join two open intervals to... |
| difreicc 13398 | The class difference of ` ... |
| iccsplit 13399 | Split a closed interval in... |
| iccshftr 13400 | Membership in a shifted in... |
| iccshftri 13401 | Membership in a shifted in... |
| iccshftl 13402 | Membership in a shifted in... |
| iccshftli 13403 | Membership in a shifted in... |
| iccdil 13404 | Membership in a dilated in... |
| iccdili 13405 | Membership in a dilated in... |
| icccntr 13406 | Membership in a contracted... |
| icccntri 13407 | Membership in a contracted... |
| divelunit 13408 | A condition for a ratio to... |
| lincmb01cmp 13409 | A linear combination of tw... |
| iccf1o 13410 | Describe a bijection from ... |
| iccen 13411 | Any nontrivial closed inte... |
| xov1plusxeqvd 13412 | A complex number ` X ` is ... |
| unitssre 13413 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13414 | The closed unit interval i... |
| supicc 13415 | Supremum of a bounded set ... |
| supiccub 13416 | The supremum of a bounded ... |
| supicclub 13417 | The supremum of a bounded ... |
| supicclub2 13418 | The supremum of a bounded ... |
| zltaddlt1le 13419 | The sum of an integer and ... |
| xnn0xrge0 13420 | An extended nonnegative in... |
| fzval 13423 | The value of a finite set ... |
| fzval2 13424 | An alternative way of expr... |
| fzf 13425 | Establish the domain and c... |
| elfz1 13426 | Membership in a finite set... |
| elfz 13427 | Membership in a finite set... |
| elfz2 13428 | Membership in a finite set... |
| elfzd 13429 | Membership in a finite set... |
| elfz5 13430 | Membership in a finite set... |
| elfz4 13431 | Membership in a finite set... |
| elfzuzb 13432 | Membership in a finite set... |
| eluzfz 13433 | Membership in a finite set... |
| elfzuz 13434 | A member of a finite set o... |
| elfzuz3 13435 | Membership in a finite set... |
| elfzel2 13436 | Membership in a finite set... |
| elfzel1 13437 | Membership in a finite set... |
| elfzelz 13438 | A member of a finite set o... |
| elfzelzd 13439 | A member of a finite set o... |
| fzssz 13440 | A finite sequence of integ... |
| elfzle1 13441 | A member of a finite set o... |
| elfzle2 13442 | A member of a finite set o... |
| elfzuz2 13443 | Implication of membership ... |
| elfzle3 13444 | Membership in a finite set... |
| eluzfz1 13445 | Membership in a finite set... |
| eluzfz2 13446 | Membership in a finite set... |
| eluzfz2b 13447 | Membership in a finite set... |
| elfz3 13448 | Membership in a finite set... |
| elfz1eq 13449 | Membership in a finite set... |
| elfzubelfz 13450 | If there is a member in a ... |
| peano2fzr 13451 | A Peano-postulate-like the... |
| fzn0 13452 | Properties of a finite int... |
| fz0 13453 | A finite set of sequential... |
| fzn 13454 | A finite set of sequential... |
| fzen 13455 | A shifted finite set of se... |
| fz1n 13456 | A 1-based finite set of se... |
| 0nelfz1 13457 | 0 is not an element of a f... |
| 0fz1 13458 | Two ways to say a finite 1... |
| fz10 13459 | There are no integers betw... |
| uzsubsubfz 13460 | Membership of an integer g... |
| uzsubsubfz1 13461 | Membership of an integer g... |
| ige3m2fz 13462 | Membership of an integer g... |
| fzsplit2 13463 | Split a finite interval of... |
| fzsplit 13464 | Split a finite interval of... |
| fzdisj 13465 | Condition for two finite i... |
| fz01en 13466 | 0-based and 1-based finite... |
| elfznn 13467 | A member of a finite set o... |
| elfz1end 13468 | A nonempty finite range of... |
| fz1ssnn 13469 | A finite set of positive i... |
| fznn0sub 13470 | Subtraction closure for a ... |
| fzmmmeqm 13471 | Subtracting the difference... |
| fzaddel 13472 | Membership of a sum in a f... |
| fzadd2 13473 | Membership of a sum in a f... |
| fzsubel 13474 | Membership of a difference... |
| fzopth 13475 | A finite set of sequential... |
| fzass4 13476 | Two ways to express a nond... |
| fzss1 13477 | Subset relationship for fi... |
| fzss2 13478 | Subset relationship for fi... |
| fzssuz 13479 | A finite set of sequential... |
| fzsn 13480 | A finite interval of integ... |
| fzssp1 13481 | Subset relationship for fi... |
| fzssnn 13482 | Finite sets of sequential ... |
| ssfzunsnext 13483 | A subset of a finite seque... |
| ssfzunsn 13484 | A subset of a finite seque... |
| fzsuc 13485 | Join a successor to the en... |
| fzpred 13486 | Join a predecessor to the ... |
| fzpreddisj 13487 | A finite set of sequential... |
| elfzp1 13488 | Append an element to a fin... |
| fzp1ss 13489 | Subset relationship for fi... |
| fzelp1 13490 | Membership in a set of seq... |
| fzp1elp1 13491 | Add one to an element of a... |
| fznatpl1 13492 | Shift membership in a fini... |
| fzpr 13493 | A finite interval of integ... |
| fztp 13494 | A finite interval of integ... |
| fz12pr 13495 | An integer range between 1... |
| fzsuc2 13496 | Join a successor to the en... |
| fzp1disj 13497 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13498 | Remove a successor from th... |
| fzprval 13499 | Two ways of defining the f... |
| fztpval 13500 | Two ways of defining the f... |
| fzrev 13501 | Reversal of start and end ... |
| fzrev2 13502 | Reversal of start and end ... |
| fzrev2i 13503 | Reversal of start and end ... |
| fzrev3 13504 | The "complement" of a memb... |
| fzrev3i 13505 | The "complement" of a memb... |
| fznn 13506 | Finite set of sequential i... |
| elfz1b 13507 | Membership in a 1-based fi... |
| elfz1uz 13508 | Membership in a 1-based fi... |
| elfzm11 13509 | Membership in a finite set... |
| uzsplit 13510 | Express an upper integer s... |
| uzdisj 13511 | The first ` N ` elements o... |
| fseq1p1m1 13512 | Add/remove an item to/from... |
| fseq1m1p1 13513 | Add/remove an item to/from... |
| fz1sbc 13514 | Quantification over a one-... |
| elfzp1b 13515 | An integer is a member of ... |
| elfzm1b 13516 | An integer is a member of ... |
| elfzp12 13517 | Options for membership in ... |
| fzne1 13518 | Elementhood in a finite se... |
| fzdif1 13519 | Split the first element of... |
| fz0dif1 13520 | Split the first element of... |
| fzm1 13521 | Choices for an element of ... |
| fzneuz 13522 | No finite set of sequentia... |
| fznuz 13523 | Disjointness of the upper ... |
| uznfz 13524 | Disjointness of the upper ... |
| fzp1nel 13525 | One plus the upper bound o... |
| fzrevral 13526 | Reversal of scanning order... |
| fzrevral2 13527 | Reversal of scanning order... |
| fzrevral3 13528 | Reversal of scanning order... |
| fzshftral 13529 | Shift the scanning order i... |
| ige2m1fz1 13530 | Membership of an integer g... |
| ige2m1fz 13531 | Membership in a 0-based fi... |
| elfz2nn0 13532 | Membership in a finite set... |
| fznn0 13533 | Characterization of a fini... |
| elfznn0 13534 | A member of a finite set o... |
| elfz3nn0 13535 | The upper bound of a nonem... |
| fz0ssnn0 13536 | Finite sets of sequential ... |
| fz1ssfz0 13537 | Subset relationship for fi... |
| 0elfz 13538 | 0 is an element of a finit... |
| nn0fz0 13539 | A nonnegative integer is a... |
| elfz0add 13540 | An element of a finite set... |
| fz0sn 13541 | An integer range from 0 to... |
| fz0tp 13542 | An integer range from 0 to... |
| fz0to3un2pr 13543 | An integer range from 0 to... |
| fz0to4untppr 13544 | An integer range from 0 to... |
| fz0to5un2tp 13545 | An integer range from 0 to... |
| elfz0ubfz0 13546 | An element of a finite set... |
| elfz0fzfz0 13547 | A member of a finite set o... |
| fz0fzelfz0 13548 | If a member of a finite se... |
| fznn0sub2 13549 | Subtraction closure for a ... |
| uzsubfz0 13550 | Membership of an integer g... |
| fz0fzdiffz0 13551 | The difference of an integ... |
| elfzmlbm 13552 | Subtracting the lower boun... |
| elfzmlbp 13553 | Subtracting the lower boun... |
| fzctr 13554 | Lemma for theorems about t... |
| difelfzle 13555 | The difference of two inte... |
| difelfznle 13556 | The difference of two inte... |
| nn0split 13557 | Express the set of nonnega... |
| nn0disj 13558 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13559 | A finite set of sequential... |
| fvffz0 13560 | The function value of a fu... |
| 1fv 13561 | A function on a singleton.... |
| 4fvwrd4 13562 | The first four function va... |
| 2ffzeq 13563 | Two functions over 0-based... |
| preduz 13564 | The value of the predecess... |
| prednn 13565 | The value of the predecess... |
| prednn0 13566 | The value of the predecess... |
| predfz 13567 | Calculate the predecessor ... |
| fzof 13570 | Functionality of the half-... |
| elfzoel1 13571 | Reverse closure for half-o... |
| elfzoel2 13572 | Reverse closure for half-o... |
| elfzoelz 13573 | Reverse closure for half-o... |
| fzoval 13574 | Value of the half-open int... |
| elfzo 13575 | Membership in a half-open ... |
| elfzo2 13576 | Membership in a half-open ... |
| elfzouz 13577 | Membership in a half-open ... |
| nelfzo 13578 | An integer not being a mem... |
| fzolb 13579 | The left endpoint of a hal... |
| fzolb2 13580 | The left endpoint of a hal... |
| elfzole1 13581 | A member in a half-open in... |
| elfzolt2 13582 | A member in a half-open in... |
| elfzolt3 13583 | Membership in a half-open ... |
| elfzolt2b 13584 | A member in a half-open in... |
| elfzolt3b 13585 | Membership in a half-open ... |
| elfzop1le2 13586 | A member in a half-open in... |
| fzonel 13587 | A half-open range does not... |
| elfzouz2 13588 | The upper bound of a half-... |
| elfzofz 13589 | A half-open range is conta... |
| elfzo3 13590 | Express membership in a ha... |
| fzon0 13591 | A half-open integer interv... |
| fzossfz 13592 | A half-open range is conta... |
| fzossz 13593 | A half-open integer interv... |
| fzon 13594 | A half-open set of sequent... |
| fzo0n 13595 | A half-open range of nonne... |
| fzonlt0 13596 | A half-open integer range ... |
| fzo0 13597 | Half-open sets with equal ... |
| fzonnsub 13598 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13599 | If ` M < N ` then ` N - M ... |
| fzoss1 13600 | Subset relationship for ha... |
| fzoss2 13601 | Subset relationship for ha... |
| fzossrbm1 13602 | Subset of a half-open rang... |
| fzo0ss1 13603 | Subset relationship for ha... |
| fzossnn0 13604 | A half-open integer range ... |
| fzospliti 13605 | One direction of splitting... |
| fzosplit 13606 | Split a half-open integer ... |
| fzodisj 13607 | Abutting half-open integer... |
| fzouzsplit 13608 | Split an upper integer set... |
| fzouzdisj 13609 | A half-open integer range ... |
| fzoun 13610 | A half-open integer range ... |
| fzodisjsn 13611 | A half-open integer range ... |
| prinfzo0 13612 | The intersection of a half... |
| lbfzo0 13613 | An integer is strictly gre... |
| elfzo0 13614 | Membership in a half-open ... |
| elfzo0z 13615 | Membership in a half-open ... |
| nn0p1elfzo 13616 | A nonnegative integer incr... |
| elfzo0le 13617 | A member in a half-open ra... |
| elfzolem1 13618 | A member in a half-open in... |
| elfzo0subge1 13619 | The difference of the uppe... |
| elfzo0suble 13620 | The difference of the uppe... |
| elfzonn0 13621 | A member of a half-open ra... |
| fzonmapblen 13622 | The result of subtracting ... |
| fzofzim 13623 | If a nonnegative integer i... |
| fz1fzo0m1 13624 | Translation of one between... |
| fzossnn 13625 | Half-open integer ranges s... |
| elfzo1 13626 | Membership in a half-open ... |
| fzo1lb 13627 | 1 is the left endpoint of ... |
| 1elfzo1 13628 | 1 is in a half-open range ... |
| fzo1fzo0n0 13629 | An integer between 1 and a... |
| fzo0n0 13630 | A half-open integer range ... |
| fzoaddel 13631 | Translate membership in a ... |
| fzo0addel 13632 | Translate membership in a ... |
| fzo0addelr 13633 | Translate membership in a ... |
| fzoaddel2 13634 | Translate membership in a ... |
| elfzoextl 13635 | Membership of an integer i... |
| elfzoext 13636 | Membership of an integer i... |
| elincfzoext 13637 | Membership of an increased... |
| fzosubel 13638 | Translate membership in a ... |
| fzosubel2 13639 | Membership in a translated... |
| fzosubel3 13640 | Membership in a translated... |
| eluzgtdifelfzo 13641 | Membership of the differen... |
| ige2m2fzo 13642 | Membership of an integer g... |
| fzocatel 13643 | Translate membership in a ... |
| ubmelfzo 13644 | If an integer in a 1-based... |
| elfzodifsumelfzo 13645 | If an integer is in a half... |
| elfzom1elp1fzo 13646 | Membership of an integer i... |
| elfzom1elfzo 13647 | Membership in a half-open ... |
| fzval3 13648 | Expressing a closed intege... |
| fz0add1fz1 13649 | Translate membership in a ... |
| fzosn 13650 | Expressing a singleton as ... |
| elfzomin 13651 | Membership of an integer i... |
| zpnn0elfzo 13652 | Membership of an integer i... |
| zpnn0elfzo1 13653 | Membership of an integer i... |
| fzosplitsnm1 13654 | Removing a singleton from ... |
| elfzonlteqm1 13655 | If an element of a half-op... |
| fzonn0p1 13656 | A nonnegative integer is a... |
| fzossfzop1 13657 | A half-open range of nonne... |
| fzonn0p1p1 13658 | If a nonnegative integer i... |
| elfzom1p1elfzo 13659 | Increasing an element of a... |
| fzo0ssnn0 13660 | Half-open integer ranges s... |
| fzo01 13661 | Expressing the singleton o... |
| fzo12sn 13662 | A 1-based half-open intege... |
| fzo13pr 13663 | A 1-based half-open intege... |
| fzo0to2pr 13664 | A half-open integer range ... |
| fz01pr 13665 | An integer range between 0... |
| fzo0to3tp 13666 | A half-open integer range ... |
| fzo0to42pr 13667 | A half-open integer range ... |
| fzo1to4tp 13668 | A half-open integer range ... |
| fzo0sn0fzo1 13669 | A half-open range of nonne... |
| elfzo0l 13670 | A member of a half-open ra... |
| fzoend 13671 | The endpoint of a half-ope... |
| fzo0end 13672 | The endpoint of a zero-bas... |
| ssfzo12 13673 | Subset relationship for ha... |
| ssfzoulel 13674 | If a half-open integer ran... |
| ssfzo12bi 13675 | Subset relationship for ha... |
| fzoopth 13676 | A half-open integer range ... |
| ubmelm1fzo 13677 | The result of subtracting ... |
| fzofzp1 13678 | If a point is in a half-op... |
| fzofzp1b 13679 | If a point is in a half-op... |
| elfzom1b 13680 | An integer is a member of ... |
| elfzom1elp1fzo1 13681 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13682 | Membership of a positive i... |
| elfzonelfzo 13683 | If an element of a half-op... |
| elfzodif0 13684 | If an integer ` M ` is in ... |
| fzonfzoufzol 13685 | If an element of a half-op... |
| elfzomelpfzo 13686 | An integer increased by an... |
| elfznelfzo 13687 | A value in a finite set of... |
| elfznelfzob 13688 | A value in a finite set of... |
| peano2fzor 13689 | A Peano-postulate-like the... |
| fzosplitsn 13690 | Extending a half-open rang... |
| fzosplitpr 13691 | Extending a half-open inte... |
| fzosplitprm1 13692 | Extending a half-open inte... |
| fzosplitsni 13693 | Membership in a half-open ... |
| fzisfzounsn 13694 | A finite interval of integ... |
| elfzr 13695 | A member of a finite inter... |
| elfzlmr 13696 | A member of a finite inter... |
| elfz0lmr 13697 | A member of a finite inter... |
| fzone1 13698 | Elementhood in a half-open... |
| fzom1ne1 13699 | Elementhood in a half-open... |
| fzostep1 13700 | Two possibilities for a nu... |
| fzoshftral 13701 | Shift the scanning order i... |
| fzind2 13702 | Induction on the integers ... |
| fvinim0ffz 13703 | The function values for th... |
| injresinjlem 13704 | Lemma for ~ injresinj . (... |
| injresinj 13705 | A function whose restricti... |
| subfzo0 13706 | The difference between two... |
| fvf1tp 13707 | Values of a one-to-one fun... |
| flval 13712 | Value of the floor (greate... |
| flcl 13713 | The floor (greatest intege... |
| reflcl 13714 | The floor (greatest intege... |
| fllelt 13715 | A basic property of the fl... |
| flcld 13716 | The floor (greatest intege... |
| flle 13717 | A basic property of the fl... |
| flltp1 13718 | A basic property of the fl... |
| fllep1 13719 | A basic property of the fl... |
| fraclt1 13720 | The fractional part of a r... |
| fracle1 13721 | The fractional part of a r... |
| fracge0 13722 | The fractional part of a r... |
| flge 13723 | The floor function value i... |
| fllt 13724 | The floor function value i... |
| flflp1 13725 | Move floor function betwee... |
| flid 13726 | An integer is its own floo... |
| flidm 13727 | The floor function is idem... |
| flidz 13728 | A real number equals its f... |
| flltnz 13729 | The floor of a non-integer... |
| flwordi 13730 | Ordering relation for the ... |
| flword2 13731 | Ordering relation for the ... |
| flval2 13732 | An alternate way to define... |
| flval3 13733 | An alternate way to define... |
| flbi 13734 | A condition equivalent to ... |
| flbi2 13735 | A condition equivalent to ... |
| adddivflid 13736 | The floor of a sum of an i... |
| ico01fl0 13737 | The floor of a real number... |
| flge0nn0 13738 | The floor of a number grea... |
| flge1nn 13739 | The floor of a number grea... |
| fldivnn0 13740 | The floor function of a di... |
| refldivcl 13741 | The floor function of a di... |
| divfl0 13742 | The floor of a fraction is... |
| fladdz 13743 | An integer can be moved in... |
| flzadd 13744 | An integer can be moved in... |
| flmulnn0 13745 | Move a nonnegative integer... |
| btwnzge0 13746 | A real bounded between an ... |
| 2tnp1ge0ge0 13747 | Two times an integer plus ... |
| flhalf 13748 | Ordering relation for the ... |
| fldivle 13749 | The floor function of a di... |
| fldivnn0le 13750 | The floor function of a di... |
| flltdivnn0lt 13751 | The floor function of a di... |
| ltdifltdiv 13752 | If the dividend of a divis... |
| fldiv4p1lem1div2 13753 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13754 | The floor of an integer gr... |
| fldiv4lem1div2 13755 | The floor of a positive in... |
| ceilval 13756 | The value of the ceiling f... |
| dfceil2 13757 | Alternative definition of ... |
| ceilval2 13758 | The value of the ceiling f... |
| ceicl 13759 | The ceiling function retur... |
| ceilcl 13760 | Closure of the ceiling fun... |
| ceilcld 13761 | Closure of the ceiling fun... |
| ceige 13762 | The ceiling of a real numb... |
| ceilge 13763 | The ceiling of a real numb... |
| ceilged 13764 | The ceiling of a real numb... |
| ceim1l 13765 | One less than the ceiling ... |
| ceilm1lt 13766 | One less than the ceiling ... |
| ceile 13767 | The ceiling of a real numb... |
| ceille 13768 | The ceiling of a real numb... |
| ceilid 13769 | An integer is its own ceil... |
| ceilidz 13770 | A real number equals its c... |
| flleceil 13771 | The floor of a real number... |
| fleqceilz 13772 | A real number is an intege... |
| quoremz 13773 | Quotient and remainder of ... |
| quoremnn0 13774 | Quotient and remainder of ... |
| quoremnn0ALT 13775 | Alternate proof of ~ quore... |
| intfrac2 13776 | Decompose a real into inte... |
| intfracq 13777 | Decompose a rational numbe... |
| fldiv 13778 | Cancellation of the embedd... |
| fldiv2 13779 | Cancellation of an embedde... |
| fznnfl 13780 | Finite set of sequential i... |
| uzsup 13781 | An upper set of integers i... |
| ioopnfsup 13782 | An upper set of reals is u... |
| icopnfsup 13783 | An upper set of reals is u... |
| rpsup 13784 | The positive reals are unb... |
| resup 13785 | The real numbers are unbou... |
| xrsup 13786 | The extended real numbers ... |
| modval 13789 | The value of the modulo op... |
| modvalr 13790 | The value of the modulo op... |
| modcl 13791 | Closure law for the modulo... |
| flpmodeq 13792 | Partition of a division in... |
| modcld 13793 | Closure law for the modulo... |
| mod0 13794 | ` A mod B ` is zero iff ` ... |
| mulmod0 13795 | The product of an integer ... |
| negmod0 13796 | ` A ` is divisible by ` B ... |
| modge0 13797 | The modulo operation is no... |
| modlt 13798 | The modulo operation is le... |
| modelico 13799 | Modular reduction produces... |
| moddiffl 13800 | Value of the modulo operat... |
| moddifz 13801 | The modulo operation diffe... |
| modfrac 13802 | The fractional part of a n... |
| flmod 13803 | The floor function express... |
| intfrac 13804 | Break a number into its in... |
| zmod10 13805 | An integer modulo 1 is 0. ... |
| zmod1congr 13806 | Two arbitrary integers are... |
| modmulnn 13807 | Move a positive integer in... |
| modvalp1 13808 | The value of the modulo op... |
| zmodcl 13809 | Closure law for the modulo... |
| zmodcld 13810 | Closure law for the modulo... |
| zmodfz 13811 | An integer mod ` B ` lies ... |
| zmodfzo 13812 | An integer mod ` B ` lies ... |
| zmodfzp1 13813 | An integer mod ` B ` lies ... |
| modid 13814 | Identity law for modulo. ... |
| modid0 13815 | A positive real number mod... |
| modid2 13816 | Identity law for modulo. ... |
| zmodid2 13817 | Identity law for modulo re... |
| zmodidfzo 13818 | Identity law for modulo re... |
| zmodidfzoimp 13819 | Identity law for modulo re... |
| 0mod 13820 | Special case: 0 modulo a p... |
| 1mod 13821 | Special case: 1 modulo a r... |
| modabs 13822 | Absorption law for modulo.... |
| modabs2 13823 | Absorption law for modulo.... |
| modcyc 13824 | The modulo operation is pe... |
| modcyc2 13825 | The modulo operation is pe... |
| modadd1 13826 | Addition property of the m... |
| modaddb 13827 | Addition property of the m... |
| modaddid 13828 | The sums of two nonnegativ... |
| modaddabs 13829 | Absorption law for modulo.... |
| modaddmod 13830 | The sum of a real number m... |
| muladdmodid 13831 | The sum of a positive real... |
| mulp1mod1 13832 | The product of an integer ... |
| muladdmod 13833 | A real number is the sum o... |
| modmuladd 13834 | Decomposition of an intege... |
| modmuladdim 13835 | Implication of a decomposi... |
| modmuladdnn0 13836 | Implication of a decomposi... |
| negmod 13837 | The negation of a number m... |
| m1modnnsub1 13838 | Minus one modulo a positiv... |
| m1modge3gt1 13839 | Minus one modulo an intege... |
| addmodid 13840 | The sum of a positive inte... |
| addmodidr 13841 | The sum of a positive inte... |
| modadd2mod 13842 | The sum of a real number m... |
| modm1p1mod0 13843 | If a real number modulo a ... |
| modltm1p1mod 13844 | If a real number modulo a ... |
| modmul1 13845 | Multiplication property of... |
| modmul12d 13846 | Multiplication property of... |
| modnegd 13847 | Negation property of the m... |
| modadd12d 13848 | Additive property of the m... |
| modsub12d 13849 | Subtraction property of th... |
| modsubmod 13850 | The difference of a real n... |
| modsubmodmod 13851 | The difference of a real n... |
| 2txmodxeq0 13852 | Two times a positive real ... |
| 2submod 13853 | If a real number is betwee... |
| modifeq2int 13854 | If a nonnegative integer i... |
| modaddmodup 13855 | The sum of an integer modu... |
| modaddmodlo 13856 | The sum of an integer modu... |
| modmulmod 13857 | The product of a real numb... |
| modmulmodr 13858 | The product of an integer ... |
| modaddmulmod 13859 | The sum of a real number a... |
| moddi 13860 | Distribute multiplication ... |
| modsubdir 13861 | Distribute the modulo oper... |
| modeqmodmin 13862 | A real number equals the d... |
| modirr 13863 | A number modulo an irratio... |
| modfzo0difsn 13864 | For a number within a half... |
| modsumfzodifsn 13865 | The sum of a number within... |
| modlteq 13866 | Two nonnegative integers l... |
| addmodlteq 13867 | Two nonnegative integers l... |
| om2uz0i 13868 | The mapping ` G ` is a one... |
| om2uzsuci 13869 | The value of ` G ` (see ~ ... |
| om2uzuzi 13870 | The value ` G ` (see ~ om2... |
| om2uzlti 13871 | Less-than relation for ` G... |
| om2uzlt2i 13872 | The mapping ` G ` (see ~ o... |
| om2uzrani 13873 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13874 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13875 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13876 | An alternative definition ... |
| om2uzrdg 13877 | A helper lemma for the val... |
| uzrdglem 13878 | A helper lemma for the val... |
| uzrdgfni 13879 | The recursive definition g... |
| uzrdg0i 13880 | Initial value of a recursi... |
| uzrdgsuci 13881 | Successor value of a recur... |
| ltweuz 13882 | ` < ` is a well-founded re... |
| ltwenn 13883 | Less than well-orders the ... |
| ltwefz 13884 | Less than well-orders a se... |
| uzenom 13885 | An upper integer set is de... |
| uzinf 13886 | An upper integer set is in... |
| nnnfi 13887 | The set of positive intege... |
| uzrdgxfr 13888 | Transfer the value of the ... |
| fzennn 13889 | The cardinality of a finit... |
| fzen2 13890 | The cardinality of a finit... |
| cardfz 13891 | The cardinality of a finit... |
| hashgf1o 13892 | ` G ` maps ` _om ` one-to-... |
| fzfi 13893 | A finite interval of integ... |
| fzfid 13894 | Commonly used special case... |
| fzofi 13895 | Half-open integer sets are... |
| fsequb 13896 | The values of a finite rea... |
| fsequb2 13897 | The values of a finite rea... |
| fseqsupcl 13898 | The values of a finite rea... |
| fseqsupubi 13899 | The values of a finite rea... |
| nn0ennn 13900 | The nonnegative integers a... |
| nnenom 13901 | The set of positive intege... |
| nnct 13902 | ` NN ` is countable. (Con... |
| uzindi 13903 | Indirect strong induction ... |
| axdc4uzlem 13904 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13905 | A version of ~ axdc4 that ... |
| ssnn0fi 13906 | A subset of the nonnegativ... |
| rabssnn0fi 13907 | A subset of the nonnegativ... |
| uzsinds 13908 | Strong (or "total") induct... |
| nnsinds 13909 | Strong (or "total") induct... |
| nn0sinds 13910 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13911 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13912 | If all functions of a fini... |
| fsuppmapnn0fiubex 13913 | If all functions of a fini... |
| fsuppmapnn0fiub0 13914 | If all functions of a fini... |
| suppssfz 13915 | Condition for a function o... |
| fsuppmapnn0ub 13916 | If a function over the non... |
| fsuppmapnn0fz 13917 | If a function over the non... |
| mptnn0fsupp 13918 | A mapping from the nonnega... |
| mptnn0fsuppd 13919 | A mapping from the nonnega... |
| mptnn0fsuppr 13920 | A finitely supported mappi... |
| f13idfv 13921 | A one-to-one function with... |
| seqex 13924 | Existence of the sequence ... |
| seqeq1 13925 | Equality theorem for the s... |
| seqeq2 13926 | Equality theorem for the s... |
| seqeq3 13927 | Equality theorem for the s... |
| seqeq1d 13928 | Equality deduction for the... |
| seqeq2d 13929 | Equality deduction for the... |
| seqeq3d 13930 | Equality deduction for the... |
| seqeq123d 13931 | Equality deduction for the... |
| nfseq 13932 | Hypothesis builder for the... |
| seqval 13933 | Value of the sequence buil... |
| seqfn 13934 | The sequence builder funct... |
| seq1 13935 | Value of the sequence buil... |
| seq1i 13936 | Value of the sequence buil... |
| seqp1 13937 | Value of the sequence buil... |
| seqexw 13938 | Weak version of ~ seqex th... |
| seqp1d 13939 | Value of the sequence buil... |
| seqm1 13940 | Value of the sequence buil... |
| seqcl2 13941 | Closure properties of the ... |
| seqf2 13942 | Range of the recursive seq... |
| seqcl 13943 | Closure properties of the ... |
| seqf 13944 | Range of the recursive seq... |
| seqfveq2 13945 | Equality of sequences. (C... |
| seqfeq2 13946 | Equality of sequences. (C... |
| seqfveq 13947 | Equality of sequences. (C... |
| seqfeq 13948 | Equality of sequences. (C... |
| seqshft2 13949 | Shifting the index set of ... |
| seqres 13950 | Restricting its characteri... |
| serf 13951 | An infinite series of comp... |
| serfre 13952 | An infinite series of real... |
| monoord 13953 | Ordering relation for a mo... |
| monoord2 13954 | Ordering relation for a mo... |
| sermono 13955 | The partial sums in an inf... |
| seqsplit 13956 | Split a sequence into two ... |
| seq1p 13957 | Removing the first term fr... |
| seqcaopr3 13958 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 13959 | The sum of two infinite se... |
| seqcaopr 13960 | The sum of two infinite se... |
| seqf1olem2a 13961 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 13962 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 13963 | Lemma for ~ seqf1o . (Con... |
| seqf1o 13964 | Rearrange a sum via an arb... |
| seradd 13965 | The sum of two infinite se... |
| sersub 13966 | The difference of two infi... |
| seqid3 13967 | A sequence that consists e... |
| seqid 13968 | Discarding the first few t... |
| seqid2 13969 | The last few partial sums ... |
| seqhomo 13970 | Apply a homomorphism to a ... |
| seqz 13971 | If the operation ` .+ ` ha... |
| seqfeq4 13972 | Equality of series under d... |
| seqfeq3 13973 | Equality of series under d... |
| seqdistr 13974 | The distributive property ... |
| ser0 13975 | The value of the partial s... |
| ser0f 13976 | A zero-valued infinite ser... |
| serge0 13977 | A finite sum of nonnegativ... |
| serle 13978 | Comparison of partial sums... |
| ser1const 13979 | Value of the partial serie... |
| seqof 13980 | Distribute function operat... |
| seqof2 13981 | Distribute function operat... |
| expval 13984 | Value of exponentiation to... |
| expnnval 13985 | Value of exponentiation to... |
| exp0 13986 | Value of a complex number ... |
| 0exp0e1 13987 | The zeroth power of zero e... |
| exp1 13988 | Value of a complex number ... |
| expp1 13989 | Value of a complex number ... |
| expneg 13990 | Value of a complex number ... |
| expneg2 13991 | Value of a complex number ... |
| expn1 13992 | A complex number raised to... |
| expcllem 13993 | Lemma for proving nonnegat... |
| expcl2lem 13994 | Lemma for proving integer ... |
| nnexpcl 13995 | Closure of exponentiation ... |
| nn0expcl 13996 | Closure of exponentiation ... |
| zexpcl 13997 | Closure of exponentiation ... |
| qexpcl 13998 | Closure of exponentiation ... |
| reexpcl 13999 | Closure of exponentiation ... |
| expcl 14000 | Closure law for nonnegativ... |
| rpexpcl 14001 | Closure law for integer ex... |
| qexpclz 14002 | Closure of integer exponen... |
| reexpclz 14003 | Closure of integer exponen... |
| expclzlem 14004 | Lemma for ~ expclz . (Con... |
| expclz 14005 | Closure law for integer ex... |
| m1expcl2 14006 | Closure of integer exponen... |
| m1expcl 14007 | Closure of exponentiation ... |
| zexpcld 14008 | Closure of exponentiation ... |
| nn0expcli 14009 | Closure of exponentiation ... |
| nn0sqcl 14010 | The square of a nonnegativ... |
| expm1t 14011 | Exponentiation in terms of... |
| 1exp 14012 | Value of 1 raised to an in... |
| expeq0 14013 | A positive integer power i... |
| expne0 14014 | A positive integer power i... |
| expne0i 14015 | An integer power is nonzer... |
| expgt0 14016 | A positive real raised to ... |
| expnegz 14017 | Value of a nonzero complex... |
| 0exp 14018 | Value of zero raised to a ... |
| expge0 14019 | A nonnegative real raised ... |
| expge1 14020 | A real greater than or equ... |
| expgt1 14021 | A real greater than 1 rais... |
| mulexp 14022 | Nonnegative integer expone... |
| mulexpz 14023 | Integer exponentiation of ... |
| exprec 14024 | Integer exponentiation of ... |
| expadd 14025 | Sum of exponents law for n... |
| expaddzlem 14026 | Lemma for ~ expaddz . (Co... |
| expaddz 14027 | Sum of exponents law for i... |
| expmul 14028 | Product of exponents law f... |
| expmulz 14029 | Product of exponents law f... |
| m1expeven 14030 | Exponentiation of negative... |
| expsub 14031 | Exponent subtraction law f... |
| expp1z 14032 | Value of a nonzero complex... |
| expm1 14033 | Value of a nonzero complex... |
| expdiv 14034 | Nonnegative integer expone... |
| sqval 14035 | Value of the square of a c... |
| sqneg 14036 | The square of the negative... |
| sqnegd 14037 | The square of the negative... |
| sqsubswap 14038 | Swap the order of subtract... |
| sqcl 14039 | Closure of square. (Contr... |
| sqmul 14040 | Distribution of squaring o... |
| sqeq0 14041 | A complex number is zero i... |
| sqdiv 14042 | Distribution of squaring o... |
| sqdivid 14043 | The square of a nonzero co... |
| sqne0 14044 | A complex number is nonzer... |
| resqcl 14045 | Closure of squaring in rea... |
| resqcld 14046 | Closure of squaring in rea... |
| sqgt0 14047 | The square of a nonzero re... |
| sqn0rp 14048 | The square of a nonzero re... |
| nnsqcl 14049 | The positive naturals are ... |
| zsqcl 14050 | Integers are closed under ... |
| qsqcl 14051 | The square of a rational i... |
| sq11 14052 | The square function is one... |
| nn0sq11 14053 | The square function is one... |
| lt2sq 14054 | The square function is inc... |
| le2sq 14055 | The square function is non... |
| le2sq2 14056 | The square function is non... |
| sqge0 14057 | The square of a real is no... |
| sqge0d 14058 | The square of a real is no... |
| zsqcl2 14059 | The square of an integer i... |
| 0expd 14060 | Value of zero raised to a ... |
| exp0d 14061 | Value of a complex number ... |
| exp1d 14062 | Value of a complex number ... |
| expeq0d 14063 | If a positive integer powe... |
| sqvald 14064 | Value of square. Inferenc... |
| sqcld 14065 | Closure of square. (Contr... |
| sqeq0d 14066 | A number is zero iff its s... |
| expcld 14067 | Closure law for nonnegativ... |
| expp1d 14068 | Value of a complex number ... |
| expaddd 14069 | Sum of exponents law for n... |
| expmuld 14070 | Product of exponents law f... |
| sqrecd 14071 | Square of reciprocal is re... |
| expclzd 14072 | Closure law for integer ex... |
| expne0d 14073 | A nonnegative integer powe... |
| expnegd 14074 | Value of a nonzero complex... |
| exprecd 14075 | An integer power of a reci... |
| expp1zd 14076 | Value of a nonzero complex... |
| expm1d 14077 | Value of a nonzero complex... |
| expsubd 14078 | Exponent subtraction law f... |
| sqmuld 14079 | Distribution of squaring o... |
| sqdivd 14080 | Distribution of squaring o... |
| expdivd 14081 | Nonnegative integer expone... |
| mulexpd 14082 | Nonnegative integer expone... |
| znsqcld 14083 | The square of a nonzero in... |
| reexpcld 14084 | Closure of exponentiation ... |
| expge0d 14085 | A nonnegative real raised ... |
| expge1d 14086 | A real greater than or equ... |
| ltexp2a 14087 | Exponent ordering relation... |
| expmordi 14088 | Base ordering relationship... |
| rpexpmord 14089 | Base ordering relationship... |
| expcan 14090 | Cancellation law for integ... |
| ltexp2 14091 | Strict ordering law for ex... |
| leexp2 14092 | Ordering law for exponenti... |
| leexp2a 14093 | Weak ordering relationship... |
| ltexp2r 14094 | The integer powers of a fi... |
| leexp2r 14095 | Weak ordering relationship... |
| leexp1a 14096 | Weak base ordering relatio... |
| leexp1ad 14097 | Weak base ordering relatio... |
| exple1 14098 | A real between 0 and 1 inc... |
| expubnd 14099 | An upper bound on ` A ^ N ... |
| sumsqeq0 14100 | The sum of two squres of r... |
| sqvali 14101 | Value of square. Inferenc... |
| sqcli 14102 | Closure of square. (Contr... |
| sqeq0i 14103 | A complex number is zero i... |
| sqrecii 14104 | The square of a reciprocal... |
| sqmuli 14105 | Distribution of squaring o... |
| sqdivi 14106 | Distribution of squaring o... |
| resqcli 14107 | Closure of square in reals... |
| sqgt0i 14108 | The square of a nonzero re... |
| sqge0i 14109 | The square of a real is no... |
| lt2sqi 14110 | The square function on non... |
| le2sqi 14111 | The square function on non... |
| sq11i 14112 | The square function is one... |
| sq0 14113 | The square of 0 is 0. (Co... |
| sq0i 14114 | If a number is zero, then ... |
| sq0id 14115 | If a number is zero, then ... |
| sq1 14116 | The square of 1 is 1. (Co... |
| neg1sqe1 14117 | The square of ` -u 1 ` is ... |
| sq2 14118 | The square of 2 is 4. (Co... |
| sq3 14119 | The square of 3 is 9. (Co... |
| sq4e2t8 14120 | The square of 4 is 2 times... |
| cu2 14121 | The cube of 2 is 8. (Cont... |
| irec 14122 | The reciprocal of ` _i ` .... |
| i2 14123 | ` _i ` squared. (Contribu... |
| i3 14124 | ` _i ` cubed. (Contribute... |
| i4 14125 | ` _i ` to the fourth power... |
| nnlesq 14126 | A positive integer is less... |
| zzlesq 14127 | An integer is less than or... |
| iexpcyc 14128 | Taking ` _i ` to the ` K `... |
| expnass 14129 | A counterexample showing t... |
| sqlecan 14130 | Cancel one factor of a squ... |
| subsq 14131 | Factor the difference of t... |
| subsq2 14132 | Express the difference of ... |
| binom2i 14133 | The square of a binomial. ... |
| subsqi 14134 | Factor the difference of t... |
| sqeqori 14135 | The squares of two complex... |
| subsq0i 14136 | The two solutions to the d... |
| sqeqor 14137 | The squares of two complex... |
| binom2 14138 | The square of a binomial. ... |
| binom2d 14139 | Deduction form of ~ binom2... |
| binom21 14140 | Special case of ~ binom2 w... |
| binom2sub 14141 | Expand the square of a sub... |
| binom2sub1 14142 | Special case of ~ binom2su... |
| binom2subi 14143 | Expand the square of a sub... |
| mulbinom2 14144 | The square of a binomial w... |
| binom3 14145 | The cube of a binomial. (... |
| sq01 14146 | If a complex number equals... |
| zesq 14147 | An integer is even iff its... |
| nnesq 14148 | A positive integer is even... |
| crreczi 14149 | Reciprocal of a complex nu... |
| bernneq 14150 | Bernoulli's inequality, du... |
| bernneq2 14151 | Variation of Bernoulli's i... |
| bernneq3 14152 | A corollary of ~ bernneq .... |
| expnbnd 14153 | Exponentiation with a base... |
| expnlbnd 14154 | The reciprocal of exponent... |
| expnlbnd2 14155 | The reciprocal of exponent... |
| expmulnbnd 14156 | Exponentiation with a base... |
| digit2 14157 | Two ways to express the ` ... |
| digit1 14158 | Two ways to express the ` ... |
| modexp 14159 | Exponentiation property of... |
| discr1 14160 | A nonnegative quadratic fo... |
| discr 14161 | If a quadratic polynomial ... |
| expnngt1 14162 | If an integer power with a... |
| expnngt1b 14163 | An integer power with an i... |
| sqoddm1div8 14164 | A squared odd number minus... |
| nnsqcld 14165 | The naturals are closed un... |
| nnexpcld 14166 | Closure of exponentiation ... |
| nn0expcld 14167 | Closure of exponentiation ... |
| rpexpcld 14168 | Closure law for exponentia... |
| ltexp2rd 14169 | The power of a positive nu... |
| reexpclzd 14170 | Closure of exponentiation ... |
| sqgt0d 14171 | The square of a nonzero re... |
| ltexp2d 14172 | Ordering relationship for ... |
| leexp2d 14173 | Ordering law for exponenti... |
| expcand 14174 | Ordering relationship for ... |
| leexp2ad 14175 | Ordering relationship for ... |
| leexp2rd 14176 | Ordering relationship for ... |
| lt2sqd 14177 | The square function on non... |
| le2sqd 14178 | The square function on non... |
| sq11d 14179 | The square function is one... |
| ltexp1d 14180 | Elevating to a positive po... |
| ltexp1dd 14181 | Raising both sides of 'les... |
| exp11nnd 14182 | The function elevating non... |
| mulsubdivbinom2 14183 | The square of a binomial w... |
| muldivbinom2 14184 | The square of a binomial w... |
| sq10 14185 | The square of 10 is 100. ... |
| sq10e99m1 14186 | The square of 10 is 99 plu... |
| 3dec 14187 | A "decimal constructor" wh... |
| nn0le2msqi 14188 | The square function on non... |
| nn0opthlem1 14189 | A rather pretty lemma for ... |
| nn0opthlem2 14190 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14191 | An ordered pair theorem fo... |
| nn0opth2i 14192 | An ordered pair theorem fo... |
| nn0opth2 14193 | An ordered pair theorem fo... |
| facnn 14196 | Value of the factorial fun... |
| fac0 14197 | The factorial of 0. (Cont... |
| fac1 14198 | The factorial of 1. (Cont... |
| facp1 14199 | The factorial of a success... |
| fac2 14200 | The factorial of 2. (Cont... |
| fac3 14201 | The factorial of 3. (Cont... |
| fac4 14202 | The factorial of 4. (Cont... |
| facnn2 14203 | Value of the factorial fun... |
| faccl 14204 | Closure of the factorial f... |
| faccld 14205 | Closure of the factorial f... |
| facmapnn 14206 | The factorial function res... |
| facne0 14207 | The factorial function is ... |
| facdiv 14208 | A positive integer divides... |
| facndiv 14209 | No positive integer (great... |
| facwordi 14210 | Ordering property of facto... |
| faclbnd 14211 | A lower bound for the fact... |
| faclbnd2 14212 | A lower bound for the fact... |
| faclbnd3 14213 | A lower bound for the fact... |
| faclbnd4lem1 14214 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14215 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14216 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14217 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14218 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14219 | The factorial function gro... |
| faclbnd6 14220 | Geometric lower bound for ... |
| facubnd 14221 | An upper bound for the fac... |
| facavg 14222 | The product of two factori... |
| bcval 14225 | Value of the binomial coef... |
| bcval2 14226 | Value of the binomial coef... |
| bcval3 14227 | Value of the binomial coef... |
| bcval4 14228 | Value of the binomial coef... |
| bcrpcl 14229 | Closure of the binomial co... |
| bccmpl 14230 | "Complementing" its second... |
| bcn0 14231 | ` N ` choose 0 is 1. Rema... |
| bc0k 14232 | The binomial coefficient "... |
| bcnn 14233 | ` N ` choose ` N ` is 1. ... |
| bcn1 14234 | Binomial coefficient: ` N ... |
| bcnp1n 14235 | Binomial coefficient: ` N ... |
| bcm1k 14236 | The proportion of one bino... |
| bcp1n 14237 | The proportion of one bino... |
| bcp1nk 14238 | The proportion of one bino... |
| bcval5 14239 | Write out the top and bott... |
| bcn2 14240 | Binomial coefficient: ` N ... |
| bcp1m1 14241 | Compute the binomial coeff... |
| bcpasc 14242 | Pascal's rule for the bino... |
| bccl 14243 | A binomial coefficient, in... |
| bccl2 14244 | A binomial coefficient, in... |
| bcn2m1 14245 | Compute the binomial coeff... |
| bcn2p1 14246 | Compute the binomial coeff... |
| permnn 14247 | The number of permutations... |
| bcnm1 14248 | The binomial coefficient o... |
| 4bc3eq4 14249 | The value of four choose t... |
| 4bc2eq6 14250 | The value of four choose t... |
| hashkf 14253 | The finite part of the siz... |
| hashgval 14254 | The value of the ` # ` fun... |
| hashginv 14255 | The converse of ` G ` maps... |
| hashinf 14256 | The value of the ` # ` fun... |
| hashbnd 14257 | If ` A ` has size bounded ... |
| hashfxnn0 14258 | The size function is a fun... |
| hashf 14259 | The size function maps all... |
| hashxnn0 14260 | The value of the hash func... |
| hashresfn 14261 | Restriction of the domain ... |
| dmhashres 14262 | Restriction of the domain ... |
| hashnn0pnf 14263 | The value of the hash func... |
| hashnnn0genn0 14264 | If the size of a set is no... |
| hashnemnf 14265 | The size of a set is never... |
| hashv01gt1 14266 | The size of a set is eithe... |
| hashfz1 14267 | The set ` ( 1 ... N ) ` ha... |
| hashen 14268 | Two finite sets have the s... |
| hasheni 14269 | Equinumerous sets have the... |
| hasheqf1o 14270 | The size of two finite set... |
| fiinfnf1o 14271 | There is no bijection betw... |
| hasheqf1oi 14272 | The size of two sets is eq... |
| hashf1rn 14273 | The size of a finite set w... |
| hasheqf1od 14274 | The size of two sets is eq... |
| fz1eqb 14275 | Two possibly-empty 1-based... |
| hashcard 14276 | The size function of the c... |
| hashcl 14277 | Closure of the ` # ` funct... |
| hashxrcl 14278 | Extended real closure of t... |
| hashclb 14279 | Reverse closure of the ` #... |
| nfile 14280 | The size of any infinite s... |
| hashvnfin 14281 | A set of finite size is a ... |
| hashnfinnn0 14282 | The size of an infinite se... |
| isfinite4 14283 | A finite set is equinumero... |
| hasheq0 14284 | Two ways of saying a set i... |
| hashneq0 14285 | Two ways of saying a set i... |
| hashgt0n0 14286 | If the size of a set is gr... |
| hashnncl 14287 | Positive natural closure o... |
| hash0 14288 | The empty set has size zer... |
| hashelne0d 14289 | A set with an element has ... |
| hashsng 14290 | The size of a singleton. ... |
| hashen1 14291 | A set has size 1 if and on... |
| hash1elsn 14292 | A set of size 1 with a kno... |
| hashrabrsn 14293 | The size of a restricted c... |
| hashrabsn01 14294 | The size of a restricted c... |
| hashrabsn1 14295 | If the size of a restricte... |
| hashfn 14296 | A function is equinumerous... |
| fseq1hash 14297 | The value of the size func... |
| hashgadd 14298 | ` G ` maps ordinal additio... |
| hashgval2 14299 | A short expression for the... |
| hashdom 14300 | Dominance relation for the... |
| hashdomi 14301 | Non-strict order relation ... |
| hashsdom 14302 | Strict dominance relation ... |
| hashun 14303 | The size of the union of d... |
| hashun2 14304 | The size of the union of f... |
| hashun3 14305 | The size of the union of f... |
| hashinfxadd 14306 | The extended real addition... |
| hashunx 14307 | The size of the union of d... |
| hashge0 14308 | The cardinality of a set i... |
| hashgt0 14309 | The cardinality of a nonem... |
| hashge1 14310 | The cardinality of a nonem... |
| 1elfz0hash 14311 | 1 is an element of the fin... |
| hashnn0n0nn 14312 | If a nonnegative integer i... |
| hashunsng 14313 | The size of the union of a... |
| hashunsngx 14314 | The size of the union of a... |
| hashunsnggt 14315 | The size of a set is great... |
| hashprg 14316 | The size of an unordered p... |
| elprchashprn2 14317 | If one element of an unord... |
| hashprb 14318 | The size of an unordered p... |
| hashprdifel 14319 | The elements of an unorder... |
| prhash2ex 14320 | There is (at least) one se... |
| hashle00 14321 | If the size of a set is le... |
| hashgt0elex 14322 | If the size of a set is gr... |
| hashgt0elexb 14323 | The size of a set is great... |
| hashp1i 14324 | Size of a finite ordinal. ... |
| hash1 14325 | Size of a finite ordinal. ... |
| hash2 14326 | Size of a finite ordinal. ... |
| hash3 14327 | Size of a finite ordinal. ... |
| hash4 14328 | Size of a finite ordinal. ... |
| pr0hash2ex 14329 | There is (at least) one se... |
| hashss 14330 | The size of a subset is le... |
| prsshashgt1 14331 | The size of a superset of ... |
| hashin 14332 | The size of the intersecti... |
| hashssdif 14333 | The size of the difference... |
| hashdif 14334 | The size of the difference... |
| hashdifsn 14335 | The size of the difference... |
| hashdifpr 14336 | The size of the difference... |
| hashsn01 14337 | The size of a singleton is... |
| hashsnle1 14338 | The size of a singleton is... |
| hashsnlei 14339 | Get an upper bound on a co... |
| hash1snb 14340 | The size of a set is 1 if ... |
| euhash1 14341 | The size of a set is 1 in ... |
| hash1n0 14342 | If the size of a set is 1 ... |
| hashgt12el 14343 | In a set with more than on... |
| hashgt12el2 14344 | In a set with more than on... |
| hashgt23el 14345 | A set with more than two e... |
| hashunlei 14346 | Get an upper bound on a co... |
| hashsslei 14347 | Get an upper bound on a co... |
| hashfz 14348 | Value of the numeric cardi... |
| fzsdom2 14349 | Condition for finite range... |
| hashfzo 14350 | Cardinality of a half-open... |
| hashfzo0 14351 | Cardinality of a half-open... |
| hashfzp1 14352 | Value of the numeric cardi... |
| hashfz0 14353 | Value of the numeric cardi... |
| hashxplem 14354 | Lemma for ~ hashxp . (Con... |
| hashxp 14355 | The size of the Cartesian ... |
| hashmap 14356 | The size of the set expone... |
| hashpw 14357 | The size of the power set ... |
| hashfun 14358 | A finite set is a function... |
| hashres 14359 | The number of elements of ... |
| hashreshashfun 14360 | The number of elements of ... |
| hashimarn 14361 | The size of the image of a... |
| hashimarni 14362 | If the size of the image o... |
| hashfundm 14363 | The size of a set function... |
| hashf1dmrn 14364 | The size of the domain of ... |
| hashf1dmcdm 14365 | The size of the domain of ... |
| resunimafz0 14366 | TODO-AV: Revise using ` F... |
| fnfz0hash 14367 | The size of a function on ... |
| ffz0hash 14368 | The size of a function on ... |
| fnfz0hashnn0 14369 | The size of a function on ... |
| ffzo0hash 14370 | The size of a function on ... |
| fnfzo0hash 14371 | The size of a function on ... |
| fnfzo0hashnn0 14372 | The value of the size func... |
| hashbclem 14373 | Lemma for ~ hashbc : induc... |
| hashbc 14374 | The binomial coefficient c... |
| hashfacen 14375 | The number of bijections b... |
| hashf1lem1 14376 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14377 | Lemma for ~ hashf1 . (Con... |
| hashf1 14378 | The permutation number ` |... |
| hashfac 14379 | A factorial counts the num... |
| leiso 14380 | Two ways to write a strict... |
| leisorel 14381 | Version of ~ isorel for st... |
| fz1isolem 14382 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14383 | Any finite ordered set has... |
| ishashinf 14384 | Any set that is not finite... |
| seqcoll 14385 | The function ` F ` contain... |
| seqcoll2 14386 | The function ` F ` contain... |
| phphashd 14387 | Corollary of the Pigeonhol... |
| phphashrd 14388 | Corollary of the Pigeonhol... |
| hashprlei 14389 | An unordered pair has at m... |
| hash2pr 14390 | A set of size two is an un... |
| hash2prde 14391 | A set of size two is an un... |
| hash2exprb 14392 | A set of size two is an un... |
| hash2prb 14393 | A set of size two is a pro... |
| prprrab 14394 | The set of proper pairs of... |
| nehash2 14395 | The cardinality of a set w... |
| hash2prd 14396 | A set of size two is an un... |
| hash2pwpr 14397 | If the size of a subset of... |
| hashle2pr 14398 | A nonempty set of size les... |
| hashle2prv 14399 | A nonempty subset of a pow... |
| pr2pwpr 14400 | The set of subsets of a pa... |
| hashge2el2dif 14401 | A set with size at least 2... |
| hashge2el2difr 14402 | A set with at least 2 diff... |
| hashge2el2difb 14403 | A set has size at least 2 ... |
| hashdmpropge2 14404 | The size of the domain of ... |
| hashtplei 14405 | An unordered triple has at... |
| hashtpg 14406 | The size of an unordered t... |
| hash7g 14407 | The size of an unordered s... |
| hashge3el3dif 14408 | A set with size at least 3... |
| elss2prb 14409 | An element of the set of s... |
| hash2sspr 14410 | A subset of size two is an... |
| exprelprel 14411 | If there is an element of ... |
| hash3tr 14412 | A set of size three is an ... |
| hash1to3 14413 | If the size of a set is be... |
| hash3tpde 14414 | A set of size three is an ... |
| hash3tpexb 14415 | A set of size three is an ... |
| hash3tpb 14416 | A set of size three is a p... |
| tpf1ofv0 14417 | The value of a one-to-one ... |
| tpf1ofv1 14418 | The value of a one-to-one ... |
| tpf1ofv2 14419 | The value of a one-to-one ... |
| tpf 14420 | A function into a (proper)... |
| tpfo 14421 | A function onto a (proper)... |
| tpf1o 14422 | A bijection onto a (proper... |
| fundmge2nop0 14423 | A function with a domain c... |
| fundmge2nop 14424 | A function with a domain c... |
| fun2dmnop0 14425 | A function with a domain c... |
| fun2dmnop 14426 | A function with a domain c... |
| hashdifsnp1 14427 | If the size of a set is a ... |
| fi1uzind 14428 | Properties of an ordered p... |
| brfi1uzind 14429 | Properties of a binary rel... |
| brfi1ind 14430 | Properties of a binary rel... |
| brfi1indALT 14431 | Alternate proof of ~ brfi1... |
| opfi1uzind 14432 | Properties of an ordered p... |
| opfi1ind 14433 | Properties of an ordered p... |
| iswrd 14436 | Property of being a word o... |
| wrdval 14437 | Value of the set of words ... |
| iswrdi 14438 | A zero-based sequence is a... |
| wrdf 14439 | A word is a zero-based seq... |
| wrdfd 14440 | A word is a zero-based seq... |
| iswrdb 14441 | A word over an alphabet is... |
| wrddm 14442 | The indices of a word (i.e... |
| sswrd 14443 | The set of words respects ... |
| snopiswrd 14444 | A singleton of an ordered ... |
| wrdexg 14445 | The set of words over a se... |
| wrdexb 14446 | The set of words over a se... |
| wrdexi 14447 | The set of words over a se... |
| wrdsymbcl 14448 | A symbol within a word ove... |
| wrdfn 14449 | A word is a function with ... |
| wrdv 14450 | A word over an alphabet is... |
| wrdlndm 14451 | The length of a word is no... |
| iswrdsymb 14452 | An arbitrary word is a wor... |
| wrdfin 14453 | A word is a finite set. (... |
| lencl 14454 | The length of a word is a ... |
| lennncl 14455 | The length of a nonempty w... |
| wrdffz 14456 | A word is a function from ... |
| wrdeq 14457 | Equality theorem for the s... |
| wrdeqi 14458 | Equality theorem for the s... |
| iswrddm0 14459 | A function with empty doma... |
| wrd0 14460 | The empty set is a word (t... |
| 0wrd0 14461 | The empty word is the only... |
| ffz0iswrd 14462 | A sequence with zero-based... |
| wrdsymb 14463 | A word is a word over the ... |
| nfwrd 14464 | Hypothesis builder for ` W... |
| csbwrdg 14465 | Class substitution for the... |
| wrdnval 14466 | Words of a fixed length ar... |
| wrdmap 14467 | Words as a mapping. (Cont... |
| hashwrdn 14468 | If there is only a finite ... |
| wrdnfi 14469 | If there is only a finite ... |
| wrdsymb0 14470 | A symbol at a position "ou... |
| wrdlenge1n0 14471 | A word with length at leas... |
| len0nnbi 14472 | The length of a word is a ... |
| wrdlenge2n0 14473 | A word with length at leas... |
| wrdsymb1 14474 | The first symbol of a none... |
| wrdlen1 14475 | A word of length 1 starts ... |
| fstwrdne 14476 | The first symbol of a none... |
| fstwrdne0 14477 | The first symbol of a none... |
| eqwrd 14478 | Two words are equal iff th... |
| elovmpowrd 14479 | Implications for the value... |
| elovmptnn0wrd 14480 | Implications for the value... |
| wrdred1 14481 | A word truncated by a symb... |
| wrdred1hash 14482 | The length of a word trunc... |
| lsw 14485 | Extract the last symbol of... |
| lsw0 14486 | The last symbol of an empt... |
| lsw0g 14487 | The last symbol of an empt... |
| lsw1 14488 | The last symbol of a word ... |
| lswcl 14489 | Closure of the last symbol... |
| lswlgt0cl 14490 | The last symbol of a nonem... |
| ccatfn 14493 | The concatenation operator... |
| ccatfval 14494 | Value of the concatenation... |
| ccatcl 14495 | The concatenation of two w... |
| ccatlen 14496 | The length of a concatenat... |
| ccat0 14497 | The concatenation of two w... |
| ccatval1 14498 | Value of a symbol in the l... |
| ccatval2 14499 | Value of a symbol in the r... |
| ccatval3 14500 | Value of a symbol in the r... |
| elfzelfzccat 14501 | An element of a finite set... |
| ccatvalfn 14502 | The concatenation of two w... |
| ccatdmss 14503 | The domain of a concatenat... |
| ccatsymb 14504 | The symbol at a given posi... |
| ccatfv0 14505 | The first symbol of a conc... |
| ccatval1lsw 14506 | The last symbol of the lef... |
| ccatval21sw 14507 | The first symbol of the ri... |
| ccatlid 14508 | Concatenation of a word by... |
| ccatrid 14509 | Concatenation of a word by... |
| ccatass 14510 | Associative law for concat... |
| ccatrn 14511 | The range of a concatenate... |
| ccatidid 14512 | Concatenation of the empty... |
| lswccatn0lsw 14513 | The last symbol of a word ... |
| lswccat0lsw 14514 | The last symbol of a word ... |
| ccatalpha 14515 | A concatenation of two arb... |
| ccatrcl1 14516 | Reverse closure of a conca... |
| ids1 14519 | Identity function protecti... |
| s1val 14520 | Value of a singleton word.... |
| s1rn 14521 | The range of a singleton w... |
| s1eq 14522 | Equality theorem for a sin... |
| s1eqd 14523 | Equality theorem for a sin... |
| s1cl 14524 | A singleton word is a word... |
| s1cld 14525 | A singleton word is a word... |
| s1prc 14526 | Value of a singleton word ... |
| s1cli 14527 | A singleton word is a word... |
| s1len 14528 | Length of a singleton word... |
| s1nz 14529 | A singleton word is not th... |
| s1dm 14530 | The domain of a singleton ... |
| s1dmALT 14531 | Alternate version of ~ s1d... |
| s1fv 14532 | Sole symbol of a singleton... |
| lsws1 14533 | The last symbol of a singl... |
| eqs1 14534 | A word of length 1 is a si... |
| wrdl1exs1 14535 | A word of length 1 is a si... |
| wrdl1s1 14536 | A word of length 1 is a si... |
| s111 14537 | The singleton word functio... |
| ccatws1cl 14538 | The concatenation of a wor... |
| ccatws1clv 14539 | The concatenation of a wor... |
| ccat2s1cl 14540 | The concatenation of two s... |
| ccats1alpha 14541 | A concatenation of a word ... |
| ccatws1len 14542 | The length of the concaten... |
| ccatws1lenp1b 14543 | The length of a word is ` ... |
| wrdlenccats1lenm1 14544 | The length of a word is th... |
| ccat2s1len 14545 | The length of the concaten... |
| ccatw2s1cl 14546 | The concatenation of a wor... |
| ccatw2s1len 14547 | The length of the concaten... |
| ccats1val1 14548 | Value of a symbol in the l... |
| ccats1val2 14549 | Value of the symbol concat... |
| ccat1st1st 14550 | The first symbol of a word... |
| ccat2s1p1 14551 | Extract the first of two c... |
| ccat2s1p2 14552 | Extract the second of two ... |
| ccatw2s1ass 14553 | Associative law for a conc... |
| ccatws1n0 14554 | The concatenation of a wor... |
| ccatws1ls 14555 | The last symbol of the con... |
| lswccats1 14556 | The last symbol of a word ... |
| lswccats1fst 14557 | The last symbol of a nonem... |
| ccatw2s1p1 14558 | Extract the symbol of the ... |
| ccatw2s1p2 14559 | Extract the second of two ... |
| ccat2s1fvw 14560 | Extract a symbol of a word... |
| ccat2s1fst 14561 | The first symbol of the co... |
| swrdnznd 14564 | The value of a subword ope... |
| swrdval 14565 | Value of a subword. (Cont... |
| swrd00 14566 | A zero length substring. ... |
| swrdcl 14567 | Closure of the subword ext... |
| swrdval2 14568 | Value of the subword extra... |
| swrdlen 14569 | Length of an extracted sub... |
| swrdfv 14570 | A symbol in an extracted s... |
| swrdfv0 14571 | The first symbol in an ext... |
| swrdf 14572 | A subword of a word is a f... |
| swrdvalfn 14573 | Value of the subword extra... |
| swrdrn 14574 | The range of a subword of ... |
| swrdlend 14575 | The value of the subword e... |
| swrdnd 14576 | The value of the subword e... |
| swrdnd2 14577 | Value of the subword extra... |
| swrdnnn0nd 14578 | The value of a subword ope... |
| swrdnd0 14579 | The value of a subword ope... |
| swrd0 14580 | A subword of an empty set ... |
| swrdrlen 14581 | Length of a right-anchored... |
| swrdlen2 14582 | Length of an extracted sub... |
| swrdfv2 14583 | A symbol in an extracted s... |
| swrdwrdsymb 14584 | A subword is a word over t... |
| swrdsb0eq 14585 | Two subwords with the same... |
| swrdsbslen 14586 | Two subwords with the same... |
| swrdspsleq 14587 | Two words have a common su... |
| swrds1 14588 | Extract a single symbol fr... |
| swrdlsw 14589 | Extract the last single sy... |
| ccatswrd 14590 | Joining two adjacent subwo... |
| swrdccat2 14591 | Recover the right half of ... |
| pfxnndmnd 14594 | The value of a prefix oper... |
| pfxval 14595 | Value of a prefix operatio... |
| pfx00 14596 | The zero length prefix is ... |
| pfx0 14597 | A prefix of an empty set i... |
| pfxval0 14598 | Value of a prefix operatio... |
| pfxcl 14599 | Closure of the prefix extr... |
| pfxmpt 14600 | Value of the prefix extrac... |
| pfxres 14601 | Value of the prefix extrac... |
| pfxf 14602 | A prefix of a word is a fu... |
| pfxfn 14603 | Value of the prefix extrac... |
| pfxfv 14604 | A symbol in a prefix of a ... |
| pfxlen 14605 | Length of a prefix. (Cont... |
| pfxid 14606 | A word is a prefix of itse... |
| pfxrn 14607 | The range of a prefix of a... |
| pfxn0 14608 | A prefix consisting of at ... |
| pfxnd 14609 | The value of a prefix oper... |
| pfxnd0 14610 | The value of a prefix oper... |
| pfxwrdsymb 14611 | A prefix of a word is a wo... |
| addlenpfx 14612 | The sum of the lengths of ... |
| pfxfv0 14613 | The first symbol of a pref... |
| pfxtrcfv 14614 | A symbol in a word truncat... |
| pfxtrcfv0 14615 | The first symbol in a word... |
| pfxfvlsw 14616 | The last symbol in a nonem... |
| pfxeq 14617 | The prefixes of two words ... |
| pfxtrcfvl 14618 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14619 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14620 | Two (nonempty) words are e... |
| disjwrdpfx 14621 | Sets of words are disjoint... |
| ccatpfx 14622 | Concatenating a prefix wit... |
| pfxccat1 14623 | Recover the left half of a... |
| pfx1 14624 | The prefix of length one o... |
| swrdswrdlem 14625 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14626 | A subword of a subword is ... |
| pfxswrd 14627 | A prefix of a subword is a... |
| swrdpfx 14628 | A subword of a prefix is a... |
| pfxpfx 14629 | A prefix of a prefix is a ... |
| pfxpfxid 14630 | A prefix of a prefix with ... |
| pfxcctswrd 14631 | The concatenation of the p... |
| lenpfxcctswrd 14632 | The length of the concaten... |
| lenrevpfxcctswrd 14633 | The length of the concaten... |
| pfxlswccat 14634 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14635 | The last symbol of a word ... |
| ccats1pfxeqrex 14636 | There exists a symbol such... |
| ccatopth 14637 | An ~ opth -like theorem fo... |
| ccatopth2 14638 | An ~ opth -like theorem fo... |
| ccatlcan 14639 | Concatenation of words is ... |
| ccatrcan 14640 | Concatenation of words is ... |
| wrdeqs1cat 14641 | Decompose a nonempty word ... |
| cats1un 14642 | Express a word with an ext... |
| wrdind 14643 | Perform induction over the... |
| wrd2ind 14644 | Perform induction over the... |
| swrdccatfn 14645 | The subword of a concatena... |
| swrdccatin1 14646 | The subword of a concatena... |
| pfxccatin12lem4 14647 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14648 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14649 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14650 | The subword of a concatena... |
| pfxccatin12lem2c 14651 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14652 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14653 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14654 | The subword of a concatena... |
| pfxccat3 14655 | The subword of a concatena... |
| swrdccat 14656 | The subword of a concatena... |
| pfxccatpfx1 14657 | A prefix of a concatenatio... |
| pfxccatpfx2 14658 | A prefix of a concatenatio... |
| pfxccat3a 14659 | A prefix of a concatenatio... |
| swrdccat3blem 14660 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14661 | A suffix of a concatenatio... |
| pfxccatid 14662 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14663 | A word is a prefix of a wo... |
| swrdccatin1d 14664 | The subword of a concatena... |
| swrdccatin2d 14665 | The subword of a concatena... |
| pfxccatin12d 14666 | The subword of a concatena... |
| reuccatpfxs1lem 14667 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14668 | There is a unique word hav... |
| reuccatpfxs1v 14669 | There is a unique word hav... |
| splval 14672 | Value of the substring rep... |
| splcl 14673 | Closure of the substring r... |
| splid 14674 | Splicing a subword for the... |
| spllen 14675 | The length of a splice. (... |
| splfv1 14676 | Symbols to the left of a s... |
| splfv2a 14677 | Symbols within the replace... |
| splval2 14678 | Value of a splice, assumin... |
| revval 14681 | Value of the word reversin... |
| revcl 14682 | The reverse of a word is a... |
| revlen 14683 | The reverse of a word has ... |
| revfv 14684 | Reverse of a word at a poi... |
| rev0 14685 | The empty word is its own ... |
| revs1 14686 | Singleton words are their ... |
| revccat 14687 | Antiautomorphic property o... |
| revrev 14688 | Reversal is an involution ... |
| reps 14691 | Construct a function mappi... |
| repsundef 14692 | A function mapping a half-... |
| repsconst 14693 | Construct a function mappi... |
| repsf 14694 | The constructed function m... |
| repswsymb 14695 | The symbols of a "repeated... |
| repsw 14696 | A function mapping a half-... |
| repswlen 14697 | The length of a "repeated ... |
| repsw0 14698 | The "repeated symbol word"... |
| repsdf2 14699 | Alternative definition of ... |
| repswsymball 14700 | All the symbols of a "repe... |
| repswsymballbi 14701 | A word is a "repeated symb... |
| repswfsts 14702 | The first symbol of a none... |
| repswlsw 14703 | The last symbol of a nonem... |
| repsw1 14704 | The "repeated symbol word"... |
| repswswrd 14705 | A subword of a "repeated s... |
| repswpfx 14706 | A prefix of a repeated sym... |
| repswccat 14707 | The concatenation of two "... |
| repswrevw 14708 | The reverse of a "repeated... |
| cshfn 14711 | Perform a cyclical shift f... |
| cshword 14712 | Perform a cyclical shift f... |
| cshnz 14713 | A cyclical shift is the em... |
| 0csh0 14714 | Cyclically shifting an emp... |
| cshw0 14715 | A word cyclically shifted ... |
| cshwmodn 14716 | Cyclically shifting a word... |
| cshwsublen 14717 | Cyclically shifting a word... |
| cshwn 14718 | A word cyclically shifted ... |
| cshwcl 14719 | A cyclically shifted word ... |
| cshwlen 14720 | The length of a cyclically... |
| cshwf 14721 | A cyclically shifted word ... |
| cshwfn 14722 | A cyclically shifted word ... |
| cshwrn 14723 | The range of a cyclically ... |
| cshwidxmod 14724 | The symbol at a given inde... |
| cshwidxmodr 14725 | The symbol at a given inde... |
| cshwidx0mod 14726 | The symbol at index 0 of a... |
| cshwidx0 14727 | The symbol at index 0 of a... |
| cshwidxm1 14728 | The symbol at index ((n-N)... |
| cshwidxm 14729 | The symbol at index (n-N) ... |
| cshwidxn 14730 | The symbol at index (n-1) ... |
| cshf1 14731 | Cyclically shifting a word... |
| cshinj 14732 | If a word is injectiv (reg... |
| repswcshw 14733 | A cyclically shifted "repe... |
| 2cshw 14734 | Cyclically shifting a word... |
| 2cshwid 14735 | Cyclically shifting a word... |
| lswcshw 14736 | The last symbol of a word ... |
| 2cshwcom 14737 | Cyclically shifting a word... |
| cshwleneq 14738 | If the results of cyclical... |
| 3cshw 14739 | Cyclically shifting a word... |
| cshweqdif2 14740 | If cyclically shifting two... |
| cshweqdifid 14741 | If cyclically shifting a w... |
| cshweqrep 14742 | If cyclically shifting a w... |
| cshw1 14743 | If cyclically shifting a w... |
| cshw1repsw 14744 | If cyclically shifting a w... |
| cshwsexa 14745 | The class of (different!) ... |
| 2cshwcshw 14746 | If a word is a cyclically ... |
| scshwfzeqfzo 14747 | For a nonempty word the se... |
| cshwcshid 14748 | A cyclically shifted word ... |
| cshwcsh2id 14749 | A cyclically shifted word ... |
| cshimadifsn 14750 | The image of a cyclically ... |
| cshimadifsn0 14751 | The image of a cyclically ... |
| wrdco 14752 | Mapping a word by a functi... |
| lenco 14753 | Length of a mapped word is... |
| s1co 14754 | Mapping of a singleton wor... |
| revco 14755 | Mapping of words (i.e., a ... |
| ccatco 14756 | Mapping of words commutes ... |
| cshco 14757 | Mapping of words commutes ... |
| swrdco 14758 | Mapping of words commutes ... |
| pfxco 14759 | Mapping of words commutes ... |
| lswco 14760 | Mapping of (nonempty) word... |
| repsco 14761 | Mapping of words commutes ... |
| cats1cld 14776 | Closure of concatenation w... |
| cats1co 14777 | Closure of concatenation w... |
| cats1cli 14778 | Closure of concatenation w... |
| cats1fvn 14779 | The last symbol of a conca... |
| cats1fv 14780 | A symbol other than the la... |
| cats1len 14781 | The length of concatenatio... |
| cats1cat 14782 | Closure of concatenation w... |
| cats2cat 14783 | Closure of concatenation o... |
| s2eqd 14784 | Equality theorem for a dou... |
| s3eqd 14785 | Equality theorem for a len... |
| s4eqd 14786 | Equality theorem for a len... |
| s5eqd 14787 | Equality theorem for a len... |
| s6eqd 14788 | Equality theorem for a len... |
| s7eqd 14789 | Equality theorem for a len... |
| s8eqd 14790 | Equality theorem for a len... |
| s3eq2 14791 | Equality theorem for a len... |
| s2cld 14792 | A doubleton word is a word... |
| s3cld 14793 | A length 3 string is a wor... |
| s4cld 14794 | A length 4 string is a wor... |
| s5cld 14795 | A length 5 string is a wor... |
| s6cld 14796 | A length 6 string is a wor... |
| s7cld 14797 | A length 7 string is a wor... |
| s8cld 14798 | A length 8 string is a wor... |
| s2cl 14799 | A doubleton word is a word... |
| s3cl 14800 | A length 3 string is a wor... |
| s2cli 14801 | A doubleton word is a word... |
| s3cli 14802 | A length 3 string is a wor... |
| s4cli 14803 | A length 4 string is a wor... |
| s5cli 14804 | A length 5 string is a wor... |
| s6cli 14805 | A length 6 string is a wor... |
| s7cli 14806 | A length 7 string is a wor... |
| s8cli 14807 | A length 8 string is a wor... |
| s2fv0 14808 | Extract the first symbol f... |
| s2fv1 14809 | Extract the second symbol ... |
| s2len 14810 | The length of a doubleton ... |
| s2dm 14811 | The domain of a doubleton ... |
| s3fv0 14812 | Extract the first symbol f... |
| s3fv1 14813 | Extract the second symbol ... |
| s3fv2 14814 | Extract the third symbol f... |
| s3len 14815 | The length of a length 3 s... |
| s4fv0 14816 | Extract the first symbol f... |
| s4fv1 14817 | Extract the second symbol ... |
| s4fv2 14818 | Extract the third symbol f... |
| s4fv3 14819 | Extract the fourth symbol ... |
| s4len 14820 | The length of a length 4 s... |
| s5len 14821 | The length of a length 5 s... |
| s6len 14822 | The length of a length 6 s... |
| s7len 14823 | The length of a length 7 s... |
| s8len 14824 | The length of a length 8 s... |
| lsws2 14825 | The last symbol of a doubl... |
| lsws3 14826 | The last symbol of a 3 let... |
| lsws4 14827 | The last symbol of a 4 let... |
| s2prop 14828 | A length 2 word is an unor... |
| s2dmALT 14829 | Alternate version of ~ s2d... |
| s3tpop 14830 | A length 3 word is an unor... |
| s4prop 14831 | A length 4 word is a union... |
| s3fn 14832 | A length 3 word is a funct... |
| funcnvs1 14833 | The converse of a singleto... |
| funcnvs2 14834 | The converse of a length 2... |
| funcnvs3 14835 | The converse of a length 3... |
| funcnvs4 14836 | The converse of a length 4... |
| s2f1o 14837 | A length 2 word with mutua... |
| f1oun2prg 14838 | A union of unordered pairs... |
| s4f1o 14839 | A length 4 word with mutua... |
| s4dom 14840 | The domain of a length 4 w... |
| s2co 14841 | Mapping a doubleton word b... |
| s3co 14842 | Mapping a length 3 string ... |
| s0s1 14843 | Concatenation of fixed len... |
| s1s2 14844 | Concatenation of fixed len... |
| s1s3 14845 | Concatenation of fixed len... |
| s1s4 14846 | Concatenation of fixed len... |
| s1s5 14847 | Concatenation of fixed len... |
| s1s6 14848 | Concatenation of fixed len... |
| s1s7 14849 | Concatenation of fixed len... |
| s2s2 14850 | Concatenation of fixed len... |
| s4s2 14851 | Concatenation of fixed len... |
| s4s3 14852 | Concatenation of fixed len... |
| s4s4 14853 | Concatenation of fixed len... |
| s3s4 14854 | Concatenation of fixed len... |
| s2s5 14855 | Concatenation of fixed len... |
| s5s2 14856 | Concatenation of fixed len... |
| s2eq2s1eq 14857 | Two length 2 words are equ... |
| s2eq2seq 14858 | Two length 2 words are equ... |
| s3eqs2s1eq 14859 | Two length 3 words are equ... |
| s3eq3seq 14860 | Two length 3 words are equ... |
| swrds2 14861 | Extract two adjacent symbo... |
| swrds2m 14862 | Extract two adjacent symbo... |
| wrdlen2i 14863 | Implications of a word of ... |
| wrd2pr2op 14864 | A word of length two repre... |
| wrdlen2 14865 | A word of length two. (Co... |
| wrdlen2s2 14866 | A word of length two as do... |
| wrdl2exs2 14867 | A word of length two is a ... |
| pfx2 14868 | A prefix of length two. (... |
| wrd3tpop 14869 | A word of length three rep... |
| wrdlen3s3 14870 | A word of length three as ... |
| repsw2 14871 | The "repeated symbol word"... |
| repsw3 14872 | The "repeated symbol word"... |
| swrd2lsw 14873 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14874 | Two words of length at lea... |
| ccatw2s1ccatws2 14875 | The concatenation of a wor... |
| ccat2s1fvwALT 14876 | Alternate proof of ~ ccat2... |
| wwlktovf 14877 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14878 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14879 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14880 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14881 | There is a bijection betwe... |
| eqwrds3 14882 | A word is equal with a len... |
| wrdl3s3 14883 | A word of length 3 is a le... |
| s2rn 14884 | Range of a length 2 string... |
| s3rn 14885 | Range of a length 3 string... |
| s7rn 14886 | Range of a length 7 string... |
| s7f1o 14887 | A length 7 word with mutua... |
| s3sndisj 14888 | The singletons consisting ... |
| s3iunsndisj 14889 | The union of singletons co... |
| ofccat 14890 | Letterwise operations on w... |
| ofs1 14891 | Letterwise operations on a... |
| ofs2 14892 | Letterwise operations on a... |
| coss12d 14893 | Subset deduction for compo... |
| trrelssd 14894 | The composition of subclas... |
| xpcogend 14895 | The most interesting case ... |
| xpcoidgend 14896 | If two classes are not dis... |
| cotr2g 14897 | Two ways of saying that th... |
| cotr2 14898 | Two ways of saying a relat... |
| cotr3 14899 | Two ways of saying a relat... |
| coemptyd 14900 | Deduction about compositio... |
| xptrrel 14901 | The cross product is alway... |
| 0trrel 14902 | The empty class is a trans... |
| cleq1lem 14903 | Equality implies bijection... |
| cleq1 14904 | Equality of relations impl... |
| clsslem 14905 | The closure of a subclass ... |
| trcleq1 14910 | Equality of relations impl... |
| trclsslem 14911 | The transitive closure (as... |
| trcleq2lem 14912 | Equality implies bijection... |
| cvbtrcl 14913 | Change of bound variable i... |
| trcleq12lem 14914 | Equality implies bijection... |
| trclexlem 14915 | Existence of relation impl... |
| trclublem 14916 | If a relation exists then ... |
| trclubi 14917 | The Cartesian product of t... |
| trclubgi 14918 | The union with the Cartesi... |
| trclub 14919 | The Cartesian product of t... |
| trclubg 14920 | The union with the Cartesi... |
| trclfv 14921 | The transitive closure of ... |
| brintclab 14922 | Two ways to express a bina... |
| brtrclfv 14923 | Two ways of expressing the... |
| brcnvtrclfv 14924 | Two ways of expressing the... |
| brtrclfvcnv 14925 | Two ways of expressing the... |
| brcnvtrclfvcnv 14926 | Two ways of expressing the... |
| trclfvss 14927 | The transitive closure (as... |
| trclfvub 14928 | The transitive closure of ... |
| trclfvlb 14929 | The transitive closure of ... |
| trclfvcotr 14930 | The transitive closure of ... |
| trclfvlb2 14931 | The transitive closure of ... |
| trclfvlb3 14932 | The transitive closure of ... |
| cotrtrclfv 14933 | The transitive closure of ... |
| trclidm 14934 | The transitive closure of ... |
| trclun 14935 | Transitive closure of a un... |
| trclfvg 14936 | The value of the transitiv... |
| trclfvcotrg 14937 | The value of the transitiv... |
| reltrclfv 14938 | The transitive closure of ... |
| dmtrclfv 14939 | The domain of the transiti... |
| reldmrelexp 14942 | The domain of the repeated... |
| relexp0g 14943 | A relation composed zero t... |
| relexp0 14944 | A relation composed zero t... |
| relexp0d 14945 | A relation composed zero t... |
| relexpsucnnr 14946 | A reduction for relation e... |
| relexp1g 14947 | A relation composed once i... |
| dfid5 14948 | Identity relation is equal... |
| dfid6 14949 | Identity relation expresse... |
| relexp1d 14950 | A relation composed once i... |
| relexpsucnnl 14951 | A reduction for relation e... |
| relexpsucl 14952 | A reduction for relation e... |
| relexpsucr 14953 | A reduction for relation e... |
| relexpsucrd 14954 | A reduction for relation e... |
| relexpsucld 14955 | A reduction for relation e... |
| relexpcnv 14956 | Commutation of converse an... |
| relexpcnvd 14957 | Commutation of converse an... |
| relexp0rel 14958 | The exponentiation of a cl... |
| relexprelg 14959 | The exponentiation of a cl... |
| relexprel 14960 | The exponentiation of a re... |
| relexpreld 14961 | The exponentiation of a re... |
| relexpnndm 14962 | The domain of an exponenti... |
| relexpdmg 14963 | The domain of an exponenti... |
| relexpdm 14964 | The domain of an exponenti... |
| relexpdmd 14965 | The domain of an exponenti... |
| relexpnnrn 14966 | The range of an exponentia... |
| relexprng 14967 | The range of an exponentia... |
| relexprn 14968 | The range of an exponentia... |
| relexprnd 14969 | The range of an exponentia... |
| relexpfld 14970 | The field of an exponentia... |
| relexpfldd 14971 | The field of an exponentia... |
| relexpaddnn 14972 | Relation composition becom... |
| relexpuzrel 14973 | The exponentiation of a cl... |
| relexpaddg 14974 | Relation composition becom... |
| relexpaddd 14975 | Relation composition becom... |
| rtrclreclem1 14978 | The reflexive, transitive ... |
| dfrtrclrec2 14979 | If two elements are connec... |
| rtrclreclem2 14980 | The reflexive, transitive ... |
| rtrclreclem3 14981 | The reflexive, transitive ... |
| rtrclreclem4 14982 | The reflexive, transitive ... |
| dfrtrcl2 14983 | The two definitions ` t* `... |
| relexpindlem 14984 | Principle of transitive in... |
| relexpind 14985 | Principle of transitive in... |
| rtrclind 14986 | Principle of transitive in... |
| shftlem 14989 | Two ways to write a shifte... |
| shftuz 14990 | A shift of the upper integ... |
| shftfval 14991 | The value of the sequence ... |
| shftdm 14992 | Domain of a relation shift... |
| shftfib 14993 | Value of a fiber of the re... |
| shftfn 14994 | Functionality and domain o... |
| shftval 14995 | Value of a sequence shifte... |
| shftval2 14996 | Value of a sequence shifte... |
| shftval3 14997 | Value of a sequence shifte... |
| shftval4 14998 | Value of a sequence shifte... |
| shftval5 14999 | Value of a shifted sequenc... |
| shftf 15000 | Functionality of a shifted... |
| 2shfti 15001 | Composite shift operations... |
| shftidt2 15002 | Identity law for the shift... |
| shftidt 15003 | Identity law for the shift... |
| shftcan1 15004 | Cancellation law for the s... |
| shftcan2 15005 | Cancellation law for the s... |
| seqshft 15006 | Shifting the index set of ... |
| sgnval 15009 | Value of the signum functi... |
| sgn0 15010 | The signum of 0 is 0. (Co... |
| sgnp 15011 | The signum of a positive e... |
| sgnrrp 15012 | The signum of a positive r... |
| sgn1 15013 | The signum of 1 is 1. (Co... |
| sgnpnf 15014 | The signum of ` +oo ` is 1... |
| sgnn 15015 | The signum of a negative e... |
| sgnmnf 15016 | The signum of ` -oo ` is -... |
| cjval 15023 | The value of the conjugate... |
| cjth 15024 | The defining property of t... |
| cjf 15025 | Domain and codomain of the... |
| cjcl 15026 | The conjugate of a complex... |
| reval 15027 | The value of the real part... |
| imval 15028 | The value of the imaginary... |
| imre 15029 | The imaginary part of a co... |
| reim 15030 | The real part of a complex... |
| recl 15031 | The real part of a complex... |
| imcl 15032 | The imaginary part of a co... |
| ref 15033 | Domain and codomain of the... |
| imf 15034 | Domain and codomain of the... |
| crre 15035 | The real part of a complex... |
| crim 15036 | The real part of a complex... |
| replim 15037 | Reconstruct a complex numb... |
| remim 15038 | Value of the conjugate of ... |
| reim0 15039 | The imaginary part of a re... |
| reim0b 15040 | A number is real iff its i... |
| rereb 15041 | A number is real iff it eq... |
| mulre 15042 | A product with a nonzero r... |
| rere 15043 | A real number equals its r... |
| cjreb 15044 | A number is real iff it eq... |
| recj 15045 | Real part of a complex con... |
| reneg 15046 | Real part of negative. (C... |
| readd 15047 | Real part distributes over... |
| resub 15048 | Real part distributes over... |
| remullem 15049 | Lemma for ~ remul , ~ immu... |
| remul 15050 | Real part of a product. (... |
| remul2 15051 | Real part of a product. (... |
| rediv 15052 | Real part of a division. ... |
| imcj 15053 | Imaginary part of a comple... |
| imneg 15054 | The imaginary part of a ne... |
| imadd 15055 | Imaginary part distributes... |
| imsub 15056 | Imaginary part distributes... |
| immul 15057 | Imaginary part of a produc... |
| immul2 15058 | Imaginary part of a produc... |
| imdiv 15059 | Imaginary part of a divisi... |
| cjre 15060 | A real number equals its c... |
| cjcj 15061 | The conjugate of the conju... |
| cjadd 15062 | Complex conjugate distribu... |
| cjmul 15063 | Complex conjugate distribu... |
| ipcnval 15064 | Standard inner product on ... |
| cjmulrcl 15065 | A complex number times its... |
| cjmulval 15066 | A complex number times its... |
| cjmulge0 15067 | A complex number times its... |
| cjneg 15068 | Complex conjugate of negat... |
| addcj 15069 | A number plus its conjugat... |
| cjsub 15070 | Complex conjugate distribu... |
| cjexp 15071 | Complex conjugate of posit... |
| imval2 15072 | The imaginary part of a nu... |
| re0 15073 | The real part of zero. (C... |
| im0 15074 | The imaginary part of zero... |
| re1 15075 | The real part of one. (Co... |
| im1 15076 | The imaginary part of one.... |
| rei 15077 | The real part of ` _i ` . ... |
| imi 15078 | The imaginary part of ` _i... |
| cj0 15079 | The conjugate of zero. (C... |
| cji 15080 | The complex conjugate of t... |
| cjreim 15081 | The conjugate of a represe... |
| cjreim2 15082 | The conjugate of the repre... |
| cj11 15083 | Complex conjugate is a one... |
| cjne0 15084 | A number is nonzero iff it... |
| cjdiv 15085 | Complex conjugate distribu... |
| cnrecnv 15086 | The inverse to the canonic... |
| sqeqd 15087 | A deduction for showing tw... |
| recli 15088 | The real part of a complex... |
| imcli 15089 | The imaginary part of a co... |
| cjcli 15090 | Closure law for complex co... |
| replimi 15091 | Construct a complex number... |
| cjcji 15092 | The conjugate of the conju... |
| reim0bi 15093 | A number is real iff its i... |
| rerebi 15094 | A real number equals its r... |
| cjrebi 15095 | A number is real iff it eq... |
| recji 15096 | Real part of a complex con... |
| imcji 15097 | Imaginary part of a comple... |
| cjmulrcli 15098 | A complex number times its... |
| cjmulvali 15099 | A complex number times its... |
| cjmulge0i 15100 | A complex number times its... |
| renegi 15101 | Real part of negative. (C... |
| imnegi 15102 | Imaginary part of negative... |
| cjnegi 15103 | Complex conjugate of negat... |
| addcji 15104 | A number plus its conjugat... |
| readdi 15105 | Real part distributes over... |
| imaddi 15106 | Imaginary part distributes... |
| remuli 15107 | Real part of a product. (... |
| immuli 15108 | Imaginary part of a produc... |
| cjaddi 15109 | Complex conjugate distribu... |
| cjmuli 15110 | Complex conjugate distribu... |
| ipcni 15111 | Standard inner product on ... |
| cjdivi 15112 | Complex conjugate distribu... |
| crrei 15113 | The real part of a complex... |
| crimi 15114 | The imaginary part of a co... |
| recld 15115 | The real part of a complex... |
| imcld 15116 | The imaginary part of a co... |
| cjcld 15117 | Closure law for complex co... |
| replimd 15118 | Construct a complex number... |
| remimd 15119 | Value of the conjugate of ... |
| cjcjd 15120 | The conjugate of the conju... |
| reim0bd 15121 | A number is real iff its i... |
| rerebd 15122 | A real number equals its r... |
| cjrebd 15123 | A number is real iff it eq... |
| cjne0d 15124 | A number is nonzero iff it... |
| recjd 15125 | Real part of a complex con... |
| imcjd 15126 | Imaginary part of a comple... |
| cjmulrcld 15127 | A complex number times its... |
| cjmulvald 15128 | A complex number times its... |
| cjmulge0d 15129 | A complex number times its... |
| renegd 15130 | Real part of negative. (C... |
| imnegd 15131 | Imaginary part of negative... |
| cjnegd 15132 | Complex conjugate of negat... |
| addcjd 15133 | A number plus its conjugat... |
| cjexpd 15134 | Complex conjugate of posit... |
| readdd 15135 | Real part distributes over... |
| imaddd 15136 | Imaginary part distributes... |
| resubd 15137 | Real part distributes over... |
| imsubd 15138 | Imaginary part distributes... |
| remuld 15139 | Real part of a product. (... |
| immuld 15140 | Imaginary part of a produc... |
| cjaddd 15141 | Complex conjugate distribu... |
| cjmuld 15142 | Complex conjugate distribu... |
| ipcnd 15143 | Standard inner product on ... |
| cjdivd 15144 | Complex conjugate distribu... |
| rered 15145 | A real number equals its r... |
| reim0d 15146 | The imaginary part of a re... |
| cjred 15147 | A real number equals its c... |
| remul2d 15148 | Real part of a product. (... |
| immul2d 15149 | Imaginary part of a produc... |
| redivd 15150 | Real part of a division. ... |
| imdivd 15151 | Imaginary part of a divisi... |
| crred 15152 | The real part of a complex... |
| crimd 15153 | The imaginary part of a co... |
| sqrtval 15158 | Value of square root funct... |
| absval 15159 | The absolute value (modulu... |
| rennim 15160 | A real number does not lie... |
| cnpart 15161 | The specification of restr... |
| sqrt0 15162 | The square root of zero is... |
| 01sqrexlem1 15163 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15164 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15165 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15166 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15167 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15168 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15169 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15170 | Existence of a square root... |
| resqrex 15171 | Existence of a square root... |
| sqrmo 15172 | Uniqueness for the square ... |
| resqreu 15173 | Existence and uniqueness f... |
| resqrtcl 15174 | Closure of the square root... |
| resqrtthlem 15175 | Lemma for ~ resqrtth . (C... |
| resqrtth 15176 | Square root theorem over t... |
| remsqsqrt 15177 | Square of square root. (C... |
| sqrtge0 15178 | The square root function i... |
| sqrtgt0 15179 | The square root function i... |
| sqrtmul 15180 | Square root distributes ov... |
| sqrtle 15181 | Square root is monotonic. ... |
| sqrtlt 15182 | Square root is strictly mo... |
| sqrt11 15183 | The square root function i... |
| sqrt00 15184 | A square root is zero iff ... |
| rpsqrtcl 15185 | The square root of a posit... |
| sqrtdiv 15186 | Square root distributes ov... |
| sqrtneglem 15187 | The square root of a negat... |
| sqrtneg 15188 | The square root of a negat... |
| sqrtsq2 15189 | Relationship between squar... |
| sqrtsq 15190 | Square root of square. (C... |
| sqrtmsq 15191 | Square root of square. (C... |
| sqrt1 15192 | The square root of 1 is 1.... |
| sqrt4 15193 | The square root of 4 is 2.... |
| sqrt9 15194 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15195 | The square root of 2 is bo... |
| sqrtm1 15196 | The imaginary unit is the ... |
| nn0sqeq1 15197 | A natural number with squa... |
| absneg 15198 | Absolute value of the nega... |
| abscl 15199 | Real closure of absolute v... |
| abscj 15200 | The absolute value of a nu... |
| absvalsq 15201 | Square of value of absolut... |
| absvalsq2 15202 | Square of value of absolut... |
| sqabsadd 15203 | Square of absolute value o... |
| sqabssub 15204 | Square of absolute value o... |
| absval2 15205 | Value of absolute value fu... |
| abs0 15206 | The absolute value of 0. ... |
| absi 15207 | The absolute value of the ... |
| absge0 15208 | Absolute value is nonnegat... |
| absrpcl 15209 | The absolute value of a no... |
| abs00 15210 | The absolute value of a nu... |
| abs00ad 15211 | A complex number is zero i... |
| abs00bd 15212 | If a complex number is zer... |
| absreimsq 15213 | Square of the absolute val... |
| absreim 15214 | Absolute value of a number... |
| absmul 15215 | Absolute value distributes... |
| absdiv 15216 | Absolute value distributes... |
| absid 15217 | A nonnegative number is it... |
| abs1 15218 | The absolute value of one ... |
| absnid 15219 | For a negative number, its... |
| leabs 15220 | A real number is less than... |
| absor 15221 | The absolute value of a re... |
| absre 15222 | Absolute value of a real n... |
| absresq 15223 | Square of the absolute val... |
| absmod0 15224 | ` A ` is divisible by ` B ... |
| absexp 15225 | Absolute value of positive... |
| absexpz 15226 | Absolute value of integer ... |
| abssq 15227 | Square can be moved in and... |
| sqabs 15228 | The squares of two reals a... |
| absrele 15229 | The absolute value of a co... |
| absimle 15230 | The absolute value of a co... |
| max0add 15231 | The sum of the positive an... |
| absz 15232 | A real number is an intege... |
| nn0abscl 15233 | The absolute value of an i... |
| zabscl 15234 | The absolute value of an i... |
| zabs0b 15235 | An integer has an absolute... |
| abslt 15236 | Absolute value and 'less t... |
| absle 15237 | Absolute value and 'less t... |
| abssubne0 15238 | If the absolute value of a... |
| absdiflt 15239 | The absolute value of a di... |
| absdifle 15240 | The absolute value of a di... |
| elicc4abs 15241 | Membership in a symmetric ... |
| lenegsq 15242 | Comparison to a nonnegativ... |
| releabs 15243 | The real part of a number ... |
| recval 15244 | Reciprocal expressed with ... |
| absidm 15245 | The absolute value functio... |
| absgt0 15246 | The absolute value of a no... |
| nnabscl 15247 | The absolute value of a no... |
| abssub 15248 | Swapping order of subtract... |
| abssubge0 15249 | Absolute value of a nonneg... |
| abssuble0 15250 | Absolute value of a nonpos... |
| absmax 15251 | The maximum of two numbers... |
| abstri 15252 | Triangle inequality for ab... |
| abs3dif 15253 | Absolute value of differen... |
| abs2dif 15254 | Difference of absolute val... |
| abs2dif2 15255 | Difference of absolute val... |
| abs2difabs 15256 | Absolute value of differen... |
| abs1m 15257 | For any complex number, th... |
| recan 15258 | Cancellation law involving... |
| absf 15259 | Mapping domain and codomai... |
| abs3lem 15260 | Lemma involving absolute v... |
| abslem2 15261 | Lemma involving absolute v... |
| rddif 15262 | The difference between a r... |
| absrdbnd 15263 | Bound on the absolute valu... |
| fzomaxdiflem 15264 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15265 | A bound on the separation ... |
| uzin2 15266 | The upper integers are clo... |
| rexanuz 15267 | Combine two different uppe... |
| rexanre 15268 | Combine two different uppe... |
| rexfiuz 15269 | Combine finitely many diff... |
| rexuz3 15270 | Restrict the base of the u... |
| rexanuz2 15271 | Combine two different uppe... |
| r19.29uz 15272 | A version of ~ 19.29 for u... |
| r19.2uz 15273 | A version of ~ r19.2z for ... |
| rexuzre 15274 | Convert an upper real quan... |
| rexico 15275 | Restrict the base of an up... |
| cau3lem 15276 | Lemma for ~ cau3 . (Contr... |
| cau3 15277 | Convert between three-quan... |
| cau4 15278 | Change the base of a Cauch... |
| caubnd2 15279 | A Cauchy sequence of compl... |
| caubnd 15280 | A Cauchy sequence of compl... |
| sqreulem 15281 | Lemma for ~ sqreu : write ... |
| sqreu 15282 | Existence and uniqueness f... |
| sqrtcl 15283 | Closure of the square root... |
| sqrtthlem 15284 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15285 | Mapping domain and codomai... |
| sqrtth 15286 | Square root theorem over t... |
| sqrtrege0 15287 | The square root function m... |
| eqsqrtor 15288 | Solve an equation containi... |
| eqsqrtd 15289 | A deduction for showing th... |
| eqsqrt2d 15290 | A deduction for showing th... |
| amgm2 15291 | Arithmetic-geometric mean ... |
| sqrtthi 15292 | Square root theorem. Theo... |
| sqrtcli 15293 | The square root of a nonne... |
| sqrtgt0i 15294 | The square root of a posit... |
| sqrtmsqi 15295 | Square root of square. (C... |
| sqrtsqi 15296 | Square root of square. (C... |
| sqsqrti 15297 | Square of square root. (C... |
| sqrtge0i 15298 | The square root of a nonne... |
| absidi 15299 | A nonnegative number is it... |
| absnidi 15300 | A negative number is the n... |
| leabsi 15301 | A real number is less than... |
| absori 15302 | The absolute value of a re... |
| absrei 15303 | Absolute value of a real n... |
| sqrtpclii 15304 | The square root of a posit... |
| sqrtgt0ii 15305 | The square root of a posit... |
| sqrt11i 15306 | The square root function i... |
| sqrtmuli 15307 | Square root distributes ov... |
| sqrtmulii 15308 | Square root distributes ov... |
| sqrtmsq2i 15309 | Relationship between squar... |
| sqrtlei 15310 | Square root is monotonic. ... |
| sqrtlti 15311 | Square root is strictly mo... |
| abslti 15312 | Absolute value and 'less t... |
| abslei 15313 | Absolute value and 'less t... |
| cnsqrt00 15314 | A square root of a complex... |
| absvalsqi 15315 | Square of value of absolut... |
| absvalsq2i 15316 | Square of value of absolut... |
| abscli 15317 | Real closure of absolute v... |
| absge0i 15318 | Absolute value is nonnegat... |
| absval2i 15319 | Value of absolute value fu... |
| abs00i 15320 | The absolute value of a nu... |
| absgt0i 15321 | The absolute value of a no... |
| absnegi 15322 | Absolute value of negative... |
| abscji 15323 | The absolute value of a nu... |
| releabsi 15324 | The real part of a number ... |
| abssubi 15325 | Swapping order of subtract... |
| absmuli 15326 | Absolute value distributes... |
| sqabsaddi 15327 | Square of absolute value o... |
| sqabssubi 15328 | Square of absolute value o... |
| absdivzi 15329 | Absolute value distributes... |
| abstrii 15330 | Triangle inequality for ab... |
| abs3difi 15331 | Absolute value of differen... |
| abs3lemi 15332 | Lemma involving absolute v... |
| rpsqrtcld 15333 | The square root of a posit... |
| sqrtgt0d 15334 | The square root of a posit... |
| absnidd 15335 | A negative number is the n... |
| leabsd 15336 | A real number is less than... |
| absord 15337 | The absolute value of a re... |
| absred 15338 | Absolute value of a real n... |
| resqrtcld 15339 | The square root of a nonne... |
| sqrtmsqd 15340 | Square root of square. (C... |
| sqrtsqd 15341 | Square root of square. (C... |
| sqrtge0d 15342 | The square root of a nonne... |
| sqrtnegd 15343 | The square root of a negat... |
| absidd 15344 | A nonnegative number is it... |
| sqrtdivd 15345 | Square root distributes ov... |
| sqrtmuld 15346 | Square root distributes ov... |
| sqrtsq2d 15347 | Relationship between squar... |
| sqrtled 15348 | Square root is monotonic. ... |
| sqrtltd 15349 | Square root is strictly mo... |
| sqr11d 15350 | The square root function i... |
| nn0absid 15351 | A nonnegative integer is i... |
| nn0absidi 15352 | A nonnegative integer is i... |
| absltd 15353 | Absolute value and 'less t... |
| absled 15354 | Absolute value and 'less t... |
| abssubge0d 15355 | Absolute value of a nonneg... |
| abssuble0d 15356 | Absolute value of a nonpos... |
| absdifltd 15357 | The absolute value of a di... |
| absdifled 15358 | The absolute value of a di... |
| icodiamlt 15359 | Two elements in a half-ope... |
| abscld 15360 | Real closure of absolute v... |
| sqrtcld 15361 | Closure of the square root... |
| sqrtrege0d 15362 | The real part of the squar... |
| sqsqrtd 15363 | Square root theorem. Theo... |
| msqsqrtd 15364 | Square root theorem. Theo... |
| sqr00d 15365 | A square root is zero iff ... |
| absvalsqd 15366 | Square of value of absolut... |
| absvalsq2d 15367 | Square of value of absolut... |
| absge0d 15368 | Absolute value is nonnegat... |
| absval2d 15369 | Value of absolute value fu... |
| abs00d 15370 | The absolute value of a nu... |
| absne0d 15371 | The absolute value of a nu... |
| absrpcld 15372 | The absolute value of a no... |
| absnegd 15373 | Absolute value of negative... |
| abscjd 15374 | The absolute value of a nu... |
| releabsd 15375 | The real part of a number ... |
| absexpd 15376 | Absolute value of positive... |
| abssubd 15377 | Swapping order of subtract... |
| absmuld 15378 | Absolute value distributes... |
| absdivd 15379 | Absolute value distributes... |
| abstrid 15380 | Triangle inequality for ab... |
| abs2difd 15381 | Difference of absolute val... |
| abs2dif2d 15382 | Difference of absolute val... |
| abs2difabsd 15383 | Absolute value of differen... |
| abs3difd 15384 | Absolute value of differen... |
| abs3lemd 15385 | Lemma involving absolute v... |
| reusq0 15386 | A complex number is the sq... |
| bhmafibid1cn 15387 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15388 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15389 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15390 | The Brahmagupta-Fibonacci ... |
| limsupgord 15393 | Ordering property of the s... |
| limsupcl 15394 | Closure of the superior li... |
| limsupval 15395 | The superior limit of an i... |
| limsupgf 15396 | Closure of the superior li... |
| limsupgval 15397 | Value of the superior limi... |
| limsupgle 15398 | The defining property of t... |
| limsuple 15399 | The defining property of t... |
| limsuplt 15400 | The defining property of t... |
| limsupval2 15401 | The superior limit, relati... |
| limsupgre 15402 | If a sequence of real numb... |
| limsupbnd1 15403 | If a sequence is eventuall... |
| limsupbnd2 15404 | If a sequence is eventuall... |
| climrel 15413 | The limit relation is a re... |
| rlimrel 15414 | The limit relation is a re... |
| clim 15415 | Express the predicate: Th... |
| rlim 15416 | Express the predicate: Th... |
| rlim2 15417 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15418 | Use strictly less-than in ... |
| rlim3 15419 | Restrict the range of the ... |
| climcl 15420 | Closure of the limit of a ... |
| rlimpm 15421 | Closure of a function with... |
| rlimf 15422 | Closure of a function with... |
| rlimss 15423 | Domain closure of a functi... |
| rlimcl 15424 | Closure of the limit of a ... |
| clim2 15425 | Express the predicate: Th... |
| clim2c 15426 | Express the predicate ` F ... |
| clim0 15427 | Express the predicate ` F ... |
| clim0c 15428 | Express the predicate ` F ... |
| rlim0 15429 | Express the predicate ` B ... |
| rlim0lt 15430 | Use strictly less-than in ... |
| climi 15431 | Convergence of a sequence ... |
| climi2 15432 | Convergence of a sequence ... |
| climi0 15433 | Convergence of a sequence ... |
| rlimi 15434 | Convergence at infinity of... |
| rlimi2 15435 | Convergence at infinity of... |
| ello1 15436 | Elementhood in the set of ... |
| ello12 15437 | Elementhood in the set of ... |
| ello12r 15438 | Sufficient condition for e... |
| lo1f 15439 | An eventually upper bounde... |
| lo1dm 15440 | An eventually upper bounde... |
| lo1bdd 15441 | The defining property of a... |
| ello1mpt 15442 | Elementhood in the set of ... |
| ello1mpt2 15443 | Elementhood in the set of ... |
| ello1d 15444 | Sufficient condition for e... |
| lo1bdd2 15445 | If an eventually bounded f... |
| lo1bddrp 15446 | Refine ~ o1bdd2 to give a ... |
| elo1 15447 | Elementhood in the set of ... |
| elo12 15448 | Elementhood in the set of ... |
| elo12r 15449 | Sufficient condition for e... |
| o1f 15450 | An eventually bounded func... |
| o1dm 15451 | An eventually bounded func... |
| o1bdd 15452 | The defining property of a... |
| lo1o1 15453 | A function is eventually b... |
| lo1o12 15454 | A function is eventually b... |
| elo1mpt 15455 | Elementhood in the set of ... |
| elo1mpt2 15456 | Elementhood in the set of ... |
| elo1d 15457 | Sufficient condition for e... |
| o1lo1 15458 | A real function is eventua... |
| o1lo12 15459 | A lower bounded real funct... |
| o1lo1d 15460 | A real eventually bounded ... |
| icco1 15461 | Derive eventual boundednes... |
| o1bdd2 15462 | If an eventually bounded f... |
| o1bddrp 15463 | Refine ~ o1bdd2 to give a ... |
| climconst 15464 | An (eventually) constant s... |
| rlimconst 15465 | A constant sequence conver... |
| rlimclim1 15466 | Forward direction of ~ rli... |
| rlimclim 15467 | A sequence on an upper int... |
| climrlim2 15468 | Produce a real limit from ... |
| climconst2 15469 | A constant sequence conver... |
| climz 15470 | The zero sequence converge... |
| rlimuni 15471 | A real function whose doma... |
| rlimdm 15472 | Two ways to express that a... |
| climuni 15473 | An infinite sequence of co... |
| fclim 15474 | The limit relation is func... |
| climdm 15475 | Two ways to express that a... |
| climeu 15476 | An infinite sequence of co... |
| climreu 15477 | An infinite sequence of co... |
| climmo 15478 | An infinite sequence of co... |
| rlimres 15479 | The restriction of a funct... |
| lo1res 15480 | The restriction of an even... |
| o1res 15481 | The restriction of an even... |
| rlimres2 15482 | The restriction of a funct... |
| lo1res2 15483 | The restriction of a funct... |
| o1res2 15484 | The restriction of a funct... |
| lo1resb 15485 | The restriction of a funct... |
| rlimresb 15486 | The restriction of a funct... |
| o1resb 15487 | The restriction of a funct... |
| climeq 15488 | Two functions that are eve... |
| lo1eq 15489 | Two functions that are eve... |
| rlimeq 15490 | Two functions that are eve... |
| o1eq 15491 | Two functions that are eve... |
| climmpt 15492 | Exhibit a function ` G ` w... |
| 2clim 15493 | If two sequences converge ... |
| climmpt2 15494 | Relate an integer limit on... |
| climshftlem 15495 | A shifted function converg... |
| climres 15496 | A function restricted to u... |
| climshft 15497 | A shifted function converg... |
| serclim0 15498 | The zero series converges ... |
| rlimcld2 15499 | If ` D ` is a closed set i... |
| rlimrege0 15500 | The limit of a sequence of... |
| rlimrecl 15501 | The limit of a real sequen... |
| rlimge0 15502 | The limit of a sequence of... |
| climshft2 15503 | A shifted function converg... |
| climrecl 15504 | The limit of a convergent ... |
| climge0 15505 | A nonnegative sequence con... |
| climabs0 15506 | Convergence to zero of the... |
| o1co 15507 | Sufficient condition for t... |
| o1compt 15508 | Sufficient condition for t... |
| rlimcn1 15509 | Image of a limit under a c... |
| rlimcn1b 15510 | Image of a limit under a c... |
| rlimcn3 15511 | Image of a limit under a c... |
| rlimcn2 15512 | Image of a limit under a c... |
| climcn1 15513 | Image of a limit under a c... |
| climcn2 15514 | Image of a limit under a c... |
| addcn2 15515 | Complex number addition is... |
| subcn2 15516 | Complex number subtraction... |
| mulcn2 15517 | Complex number multiplicat... |
| reccn2 15518 | The reciprocal function is... |
| cn1lem 15519 | A sufficient condition for... |
| abscn2 15520 | The absolute value functio... |
| cjcn2 15521 | The complex conjugate func... |
| recn2 15522 | The real part function is ... |
| imcn2 15523 | The imaginary part functio... |
| climcn1lem 15524 | The limit of a continuous ... |
| climabs 15525 | Limit of the absolute valu... |
| climcj 15526 | Limit of the complex conju... |
| climre 15527 | Limit of the real part of ... |
| climim 15528 | Limit of the imaginary par... |
| rlimmptrcl 15529 | Reverse closure for a real... |
| rlimabs 15530 | Limit of the absolute valu... |
| rlimcj 15531 | Limit of the complex conju... |
| rlimre 15532 | Limit of the real part of ... |
| rlimim 15533 | Limit of the imaginary par... |
| o1of2 15534 | Show that a binary operati... |
| o1add 15535 | The sum of two eventually ... |
| o1mul 15536 | The product of two eventua... |
| o1sub 15537 | The difference of two even... |
| rlimo1 15538 | Any function with a finite... |
| rlimdmo1 15539 | A convergent function is e... |
| o1rlimmul 15540 | The product of an eventual... |
| o1const 15541 | A constant function is eve... |
| lo1const 15542 | A constant function is eve... |
| lo1mptrcl 15543 | Reverse closure for an eve... |
| o1mptrcl 15544 | Reverse closure for an eve... |
| o1add2 15545 | The sum of two eventually ... |
| o1mul2 15546 | The product of two eventua... |
| o1sub2 15547 | The product of two eventua... |
| lo1add 15548 | The sum of two eventually ... |
| lo1mul 15549 | The product of an eventual... |
| lo1mul2 15550 | The product of an eventual... |
| o1dif 15551 | If the difference of two f... |
| lo1sub 15552 | The difference of an event... |
| climadd 15553 | Limit of the sum of two co... |
| climmul 15554 | Limit of the product of tw... |
| climsub 15555 | Limit of the difference of... |
| climaddc1 15556 | Limit of a constant ` C ` ... |
| climaddc2 15557 | Limit of a constant ` C ` ... |
| climmulc2 15558 | Limit of a sequence multip... |
| climsubc1 15559 | Limit of a constant ` C ` ... |
| climsubc2 15560 | Limit of a constant ` C ` ... |
| climle 15561 | Comparison of the limits o... |
| climsqz 15562 | Convergence of a sequence ... |
| climsqz2 15563 | Convergence of a sequence ... |
| rlimadd 15564 | Limit of the sum of two co... |
| rlimsub 15565 | Limit of the difference of... |
| rlimmul 15566 | Limit of the product of tw... |
| rlimdiv 15567 | Limit of the quotient of t... |
| rlimneg 15568 | Limit of the negative of a... |
| rlimle 15569 | Comparison of the limits o... |
| rlimsqzlem 15570 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15571 | Convergence of a sequence ... |
| rlimsqz2 15572 | Convergence of a sequence ... |
| lo1le 15573 | Transfer eventual upper bo... |
| o1le 15574 | Transfer eventual boundedn... |
| rlimno1 15575 | A function whose inverse c... |
| clim2ser 15576 | The limit of an infinite s... |
| clim2ser2 15577 | The limit of an infinite s... |
| iserex 15578 | An infinite series converg... |
| isermulc2 15579 | Multiplication of an infin... |
| climlec2 15580 | Comparison of a constant t... |
| iserle 15581 | Comparison of the limits o... |
| iserge0 15582 | The limit of an infinite s... |
| climub 15583 | The limit of a monotonic s... |
| climserle 15584 | The partial sums of a conv... |
| isershft 15585 | Index shift of the limit o... |
| isercolllem1 15586 | Lemma for ~ isercoll . (C... |
| isercolllem2 15587 | Lemma for ~ isercoll . (C... |
| isercolllem3 15588 | Lemma for ~ isercoll . (C... |
| isercoll 15589 | Rearrange an infinite seri... |
| isercoll2 15590 | Generalize ~ isercoll so t... |
| climsup 15591 | A bounded monotonic sequen... |
| climcau 15592 | A converging sequence of c... |
| climbdd 15593 | A converging sequence of c... |
| caucvgrlem 15594 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15595 | A Cauchy sequence of real ... |
| caucvgrlem2 15596 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15597 | A Cauchy sequence of compl... |
| caurcvg 15598 | A Cauchy sequence of real ... |
| caurcvg2 15599 | A Cauchy sequence of real ... |
| caucvg 15600 | A Cauchy sequence of compl... |
| caucvgb 15601 | A function is convergent i... |
| serf0 15602 | If an infinite series conv... |
| iseraltlem1 15603 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15604 | Lemma for ~ iseralt . The... |
| iseraltlem3 15605 | Lemma for ~ iseralt . Fro... |
| iseralt 15606 | The alternating series tes... |
| sumex 15609 | A sum is a set. (Contribu... |
| sumeq1 15610 | Equality theorem for a sum... |
| nfsum1 15611 | Bound-variable hypothesis ... |
| nfsum 15612 | Bound-variable hypothesis ... |
| sumeq2w 15613 | Equality theorem for sum, ... |
| sumeq2ii 15614 | Equality theorem for sum, ... |
| sumeq2 15615 | Equality theorem for sum. ... |
| cbvsum 15616 | Change bound variable in a... |
| cbvsumv 15617 | Change bound variable in a... |
| sumeq1i 15618 | Equality inference for sum... |
| sumeq2i 15619 | Equality inference for sum... |
| sumeq12i 15620 | Equality inference for sum... |
| sumeq1d 15621 | Equality deduction for sum... |
| sumeq2d 15622 | Equality deduction for sum... |
| sumeq2dv 15623 | Equality deduction for sum... |
| sumeq2sdv 15624 | Equality deduction for sum... |
| sumeq2sdvOLD 15625 | Obsolete version of ~ sume... |
| 2sumeq2dv 15626 | Equality deduction for dou... |
| sumeq12dv 15627 | Equality deduction for sum... |
| sumeq12rdv 15628 | Equality deduction for sum... |
| sum2id 15629 | The second class argument ... |
| sumfc 15630 | A lemma to facilitate conv... |
| fz1f1o 15631 | A lemma for working with f... |
| sumrblem 15632 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15633 | The sequence of partial su... |
| sumrb 15634 | Rebase the starting point ... |
| summolem3 15635 | Lemma for ~ summo . (Cont... |
| summolem2a 15636 | Lemma for ~ summo . (Cont... |
| summolem2 15637 | Lemma for ~ summo . (Cont... |
| summo 15638 | A sum has at most one limi... |
| zsum 15639 | Series sum with index set ... |
| isum 15640 | Series sum with an upper i... |
| fsum 15641 | The value of a sum over a ... |
| sum0 15642 | Any sum over the empty set... |
| sumz 15643 | Any sum of zero over a sum... |
| fsumf1o 15644 | Re-index a finite sum usin... |
| sumss 15645 | Change the index set to a ... |
| fsumss 15646 | Change the index set to a ... |
| sumss2 15647 | Change the index set of a ... |
| fsumcvg2 15648 | The sequence of partial su... |
| fsumsers 15649 | Special case of series sum... |
| fsumcvg3 15650 | A finite sum is convergent... |
| fsumser 15651 | A finite sum expressed in ... |
| fsumcl2lem 15652 | - Lemma for finite sum clo... |
| fsumcllem 15653 | - Lemma for finite sum clo... |
| fsumcl 15654 | Closure of a finite sum of... |
| fsumrecl 15655 | Closure of a finite sum of... |
| fsumzcl 15656 | Closure of a finite sum of... |
| fsumnn0cl 15657 | Closure of a finite sum of... |
| fsumrpcl 15658 | Closure of a finite sum of... |
| fsumclf 15659 | Closure of a finite sum of... |
| fsumzcl2 15660 | A finite sum with integer ... |
| fsumadd 15661 | The sum of two finite sums... |
| fsumsplit 15662 | Split a sum into two parts... |
| fsumsplitf 15663 | Split a sum into two parts... |
| sumsnf 15664 | A sum of a singleton is th... |
| fsumsplitsn 15665 | Separate out a term in a f... |
| fsumsplit1 15666 | Separate out a term in a f... |
| sumsn 15667 | A sum of a singleton is th... |
| fsum1 15668 | The finite sum of ` A ( k ... |
| sumpr 15669 | A sum over a pair is the s... |
| sumtp 15670 | A sum over a triple is the... |
| sumsns 15671 | A sum of a singleton is th... |
| fsumm1 15672 | Separate out the last term... |
| fzosump1 15673 | Separate out the last term... |
| fsum1p 15674 | Separate out the first ter... |
| fsummsnunz 15675 | A finite sum all of whose ... |
| fsumsplitsnun 15676 | Separate out a term in a f... |
| fsump1 15677 | The addition of the next t... |
| isumclim 15678 | An infinite sum equals the... |
| isumclim2 15679 | A converging series conver... |
| isumclim3 15680 | The sequence of partial fi... |
| sumnul 15681 | The sum of a non-convergen... |
| isumcl 15682 | The sum of a converging in... |
| isummulc2 15683 | An infinite sum multiplied... |
| isummulc1 15684 | An infinite sum multiplied... |
| isumdivc 15685 | An infinite sum divided by... |
| isumrecl 15686 | The sum of a converging in... |
| isumge0 15687 | An infinite sum of nonnega... |
| isumadd 15688 | Addition of infinite sums.... |
| sumsplit 15689 | Split a sum into two parts... |
| fsump1i 15690 | Optimized version of ~ fsu... |
| fsum2dlem 15691 | Lemma for ~ fsum2d - induc... |
| fsum2d 15692 | Write a double sum as a su... |
| fsumxp 15693 | Combine two sums into a si... |
| fsumcnv 15694 | Transform a region of summ... |
| fsumcom2 15695 | Interchange order of summa... |
| fsumcom 15696 | Interchange order of summa... |
| fsum0diaglem 15697 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15698 | Two ways to express "the s... |
| mptfzshft 15699 | 1-1 onto function in maps-... |
| fsumrev 15700 | Reversal of a finite sum. ... |
| fsumshft 15701 | Index shift of a finite su... |
| fsumshftm 15702 | Negative index shift of a ... |
| fsumrev2 15703 | Reversal of a finite sum. ... |
| fsum0diag2 15704 | Two ways to express "the s... |
| fsummulc2 15705 | A finite sum multiplied by... |
| fsummulc1 15706 | A finite sum multiplied by... |
| fsumdivc 15707 | A finite sum divided by a ... |
| fsumneg 15708 | Negation of a finite sum. ... |
| fsumsub 15709 | Split a finite sum over a ... |
| fsum2mul 15710 | Separate the nested sum of... |
| fsumconst 15711 | The sum of constant terms ... |
| fsumdifsnconst 15712 | The sum of constant terms ... |
| modfsummodslem1 15713 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15714 | Induction step for ~ modfs... |
| modfsummod 15715 | A finite sum modulo a posi... |
| fsumge0 15716 | If all of the terms of a f... |
| fsumless 15717 | A shorter sum of nonnegati... |
| fsumge1 15718 | A sum of nonnegative numbe... |
| fsum00 15719 | A sum of nonnegative numbe... |
| fsumle 15720 | If all of the terms of fin... |
| fsumlt 15721 | If every term in one finit... |
| fsumabs 15722 | Generalized triangle inequ... |
| telfsumo 15723 | Sum of a telescoping serie... |
| telfsumo2 15724 | Sum of a telescoping serie... |
| telfsum 15725 | Sum of a telescoping serie... |
| telfsum2 15726 | Sum of a telescoping serie... |
| fsumparts 15727 | Summation by parts. (Cont... |
| fsumrelem 15728 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15729 | The real part of a sum. (... |
| fsumim 15730 | The imaginary part of a su... |
| fsumcj 15731 | The complex conjugate of a... |
| fsumrlim 15732 | Limit of a finite sum of c... |
| fsumo1 15733 | The finite sum of eventual... |
| o1fsum 15734 | If ` A ( k ) ` is O(1), th... |
| seqabs 15735 | Generalized triangle inequ... |
| iserabs 15736 | Generalized triangle inequ... |
| cvgcmp 15737 | A comparison test for conv... |
| cvgcmpub 15738 | An upper bound for the lim... |
| cvgcmpce 15739 | A comparison test for conv... |
| abscvgcvg 15740 | An absolutely convergent s... |
| climfsum 15741 | Limit of a finite sum of c... |
| fsumiun 15742 | Sum over a disjoint indexe... |
| hashiun 15743 | The cardinality of a disjo... |
| hash2iun 15744 | The cardinality of a neste... |
| hash2iun1dif1 15745 | The cardinality of a neste... |
| hashrabrex 15746 | The number of elements in ... |
| hashuni 15747 | The cardinality of a disjo... |
| qshash 15748 | The cardinality of a set w... |
| ackbijnn 15749 | Translate the Ackermann bi... |
| binomlem 15750 | Lemma for ~ binom (binomia... |
| binom 15751 | The binomial theorem: ` ( ... |
| binom1p 15752 | Special case of the binomi... |
| binom11 15753 | Special case of the binomi... |
| binom1dif 15754 | A summation for the differ... |
| bcxmaslem1 15755 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15756 | Parallel summation (Christ... |
| incexclem 15757 | Lemma for ~ incexc . (Con... |
| incexc 15758 | The inclusion/exclusion pr... |
| incexc2 15759 | The inclusion/exclusion pr... |
| isumshft 15760 | Index shift of an infinite... |
| isumsplit 15761 | Split off the first ` N ` ... |
| isum1p 15762 | The infinite sum of a conv... |
| isumnn0nn 15763 | Sum from 0 to infinity in ... |
| isumrpcl 15764 | The infinite sum of positi... |
| isumle 15765 | Comparison of two infinite... |
| isumless 15766 | A finite sum of nonnegativ... |
| isumsup2 15767 | An infinite sum of nonnega... |
| isumsup 15768 | An infinite sum of nonnega... |
| isumltss 15769 | A partial sum of a series ... |
| climcndslem1 15770 | Lemma for ~ climcnds : bou... |
| climcndslem2 15771 | Lemma for ~ climcnds : bou... |
| climcnds 15772 | The Cauchy condensation te... |
| divrcnv 15773 | The sequence of reciprocal... |
| divcnv 15774 | The sequence of reciprocal... |
| flo1 15775 | The floor function satisfi... |
| divcnvshft 15776 | Limit of a ratio function.... |
| supcvg 15777 | Extract a sequence ` f ` i... |
| infcvgaux1i 15778 | Auxiliary theorem for appl... |
| infcvgaux2i 15779 | Auxiliary theorem for appl... |
| harmonic 15780 | The harmonic series ` H ` ... |
| arisum 15781 | Arithmetic series sum of t... |
| arisum2 15782 | Arithmetic series sum of t... |
| trireciplem 15783 | Lemma for ~ trirecip . Sh... |
| trirecip 15784 | The sum of the reciprocals... |
| expcnv 15785 | A sequence of powers of a ... |
| explecnv 15786 | A sequence of terms conver... |
| geoserg 15787 | The value of the finite ge... |
| geoser 15788 | The value of the finite ge... |
| pwdif 15789 | The difference of two numb... |
| pwm1geoser 15790 | The n-th power of a number... |
| geolim 15791 | The partial sums in the in... |
| geolim2 15792 | The partial sums in the ge... |
| georeclim 15793 | The limit of a geometric s... |
| geo2sum 15794 | The value of the finite ge... |
| geo2sum2 15795 | The value of the finite ge... |
| geo2lim 15796 | The value of the infinite ... |
| geomulcvg 15797 | The geometric series conve... |
| geoisum 15798 | The infinite sum of ` 1 + ... |
| geoisumr 15799 | The infinite sum of recipr... |
| geoisum1 15800 | The infinite sum of ` A ^ ... |
| geoisum1c 15801 | The infinite sum of ` A x.... |
| 0.999... 15802 | The recurring decimal 0.99... |
| geoihalfsum 15803 | Prove that the infinite ge... |
| cvgrat 15804 | Ratio test for convergence... |
| mertenslem1 15805 | Lemma for ~ mertens . (Co... |
| mertenslem2 15806 | Lemma for ~ mertens . (Co... |
| mertens 15807 | Mertens' theorem. If ` A ... |
| prodf 15808 | An infinite product of com... |
| clim2prod 15809 | The limit of an infinite p... |
| clim2div 15810 | The limit of an infinite p... |
| prodfmul 15811 | The product of two infinit... |
| prodf1 15812 | The value of the partial p... |
| prodf1f 15813 | A one-valued infinite prod... |
| prodfclim1 15814 | The constant one product c... |
| prodfn0 15815 | No term of a nonzero infin... |
| prodfrec 15816 | The reciprocal of an infin... |
| prodfdiv 15817 | The quotient of two infini... |
| ntrivcvg 15818 | A non-trivially converging... |
| ntrivcvgn0 15819 | A product that converges t... |
| ntrivcvgfvn0 15820 | Any value of a product seq... |
| ntrivcvgtail 15821 | A tail of a non-trivially ... |
| ntrivcvgmullem 15822 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15823 | The product of two non-tri... |
| prodex 15826 | A product is a set. (Cont... |
| prodeq1f 15827 | Equality theorem for a pro... |
| prodeq1 15828 | Equality theorem for a pro... |
| nfcprod1 15829 | Bound-variable hypothesis ... |
| nfcprod 15830 | Bound-variable hypothesis ... |
| prodeq2w 15831 | Equality theorem for produ... |
| prodeq2ii 15832 | Equality theorem for produ... |
| prodeq2 15833 | Equality theorem for produ... |
| cbvprod 15834 | Change bound variable in a... |
| cbvprodv 15835 | Change bound variable in a... |
| cbvprodi 15836 | Change bound variable in a... |
| prodeq1i 15837 | Equality inference for pro... |
| prodeq1iOLD 15838 | Obsolete version of ~ prod... |
| prodeq2i 15839 | Equality inference for pro... |
| prodeq12i 15840 | Equality inference for pro... |
| prodeq1d 15841 | Equality deduction for pro... |
| prodeq2d 15842 | Equality deduction for pro... |
| prodeq2dv 15843 | Equality deduction for pro... |
| prodeq2sdv 15844 | Equality deduction for pro... |
| prodeq2sdvOLD 15845 | Obsolete version of ~ prod... |
| 2cprodeq2dv 15846 | Equality deduction for dou... |
| prodeq12dv 15847 | Equality deduction for pro... |
| prodeq12rdv 15848 | Equality deduction for pro... |
| prod2id 15849 | The second class argument ... |
| prodrblem 15850 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15851 | The sequence of partial pr... |
| prodrblem2 15852 | Lemma for ~ prodrb . (Con... |
| prodrb 15853 | Rebase the starting point ... |
| prodmolem3 15854 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15855 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15856 | Lemma for ~ prodmo . (Con... |
| prodmo 15857 | A product has at most one ... |
| zprod 15858 | Series product with index ... |
| iprod 15859 | Series product with an upp... |
| zprodn0 15860 | Nonzero series product wit... |
| iprodn0 15861 | Nonzero series product wit... |
| fprod 15862 | The value of a product ove... |
| fprodntriv 15863 | A non-triviality lemma for... |
| prod0 15864 | A product over the empty s... |
| prod1 15865 | Any product of one over a ... |
| prodfc 15866 | A lemma to facilitate conv... |
| fprodf1o 15867 | Re-index a finite product ... |
| prodss 15868 | Change the index set to a ... |
| fprodss 15869 | Change the index set to a ... |
| fprodser 15870 | A finite product expressed... |
| fprodcl2lem 15871 | Finite product closure lem... |
| fprodcllem 15872 | Finite product closure lem... |
| fprodcl 15873 | Closure of a finite produc... |
| fprodrecl 15874 | Closure of a finite produc... |
| fprodzcl 15875 | Closure of a finite produc... |
| fprodnncl 15876 | Closure of a finite produc... |
| fprodrpcl 15877 | Closure of a finite produc... |
| fprodnn0cl 15878 | Closure of a finite produc... |
| fprodcllemf 15879 | Finite product closure lem... |
| fprodreclf 15880 | Closure of a finite produc... |
| fprodmul 15881 | The product of two finite ... |
| fproddiv 15882 | The quotient of two finite... |
| prodsn 15883 | A product of a singleton i... |
| fprod1 15884 | A finite product of only o... |
| prodsnf 15885 | A product of a singleton i... |
| climprod1 15886 | The limit of a product ove... |
| fprodsplit 15887 | Split a finite product int... |
| fprodm1 15888 | Separate out the last term... |
| fprod1p 15889 | Separate out the first ter... |
| fprodp1 15890 | Multiply in the last term ... |
| fprodm1s 15891 | Separate out the last term... |
| fprodp1s 15892 | Multiply in the last term ... |
| prodsns 15893 | A product of the singleton... |
| fprodfac 15894 | Factorial using product no... |
| fprodabs 15895 | The absolute value of a fi... |
| fprodeq0 15896 | Any finite product contain... |
| fprodshft 15897 | Shift the index of a finit... |
| fprodrev 15898 | Reversal of a finite produ... |
| fprodconst 15899 | The product of constant te... |
| fprodn0 15900 | A finite product of nonzer... |
| fprod2dlem 15901 | Lemma for ~ fprod2d - indu... |
| fprod2d 15902 | Write a double product as ... |
| fprodxp 15903 | Combine two products into ... |
| fprodcnv 15904 | Transform a product region... |
| fprodcom2 15905 | Interchange order of multi... |
| fprodcom 15906 | Interchange product order.... |
| fprod0diag 15907 | Two ways to express "the p... |
| fproddivf 15908 | The quotient of two finite... |
| fprodsplitf 15909 | Split a finite product int... |
| fprodsplitsn 15910 | Separate out a term in a f... |
| fprodsplit1f 15911 | Separate out a term in a f... |
| fprodn0f 15912 | A finite product of nonzer... |
| fprodclf 15913 | Closure of a finite produc... |
| fprodge0 15914 | If all the terms of a fini... |
| fprodeq0g 15915 | Any finite product contain... |
| fprodge1 15916 | If all of the terms of a f... |
| fprodle 15917 | If all the terms of two fi... |
| fprodmodd 15918 | If all factors of two fini... |
| iprodclim 15919 | An infinite product equals... |
| iprodclim2 15920 | A converging product conve... |
| iprodclim3 15921 | The sequence of partial fi... |
| iprodcl 15922 | The product of a non-trivi... |
| iprodrecl 15923 | The product of a non-trivi... |
| iprodmul 15924 | Multiplication of infinite... |
| risefacval 15929 | The value of the rising fa... |
| fallfacval 15930 | The value of the falling f... |
| risefacval2 15931 | One-based value of rising ... |
| fallfacval2 15932 | One-based value of falling... |
| fallfacval3 15933 | A product representation o... |
| risefaccllem 15934 | Lemma for rising factorial... |
| fallfaccllem 15935 | Lemma for falling factoria... |
| risefaccl 15936 | Closure law for rising fac... |
| fallfaccl 15937 | Closure law for falling fa... |
| rerisefaccl 15938 | Closure law for rising fac... |
| refallfaccl 15939 | Closure law for falling fa... |
| nnrisefaccl 15940 | Closure law for rising fac... |
| zrisefaccl 15941 | Closure law for rising fac... |
| zfallfaccl 15942 | Closure law for falling fa... |
| nn0risefaccl 15943 | Closure law for rising fac... |
| rprisefaccl 15944 | Closure law for rising fac... |
| risefallfac 15945 | A relationship between ris... |
| fallrisefac 15946 | A relationship between fal... |
| risefall0lem 15947 | Lemma for ~ risefac0 and ~... |
| risefac0 15948 | The value of the rising fa... |
| fallfac0 15949 | The value of the falling f... |
| risefacp1 15950 | The value of the rising fa... |
| fallfacp1 15951 | The value of the falling f... |
| risefacp1d 15952 | The value of the rising fa... |
| fallfacp1d 15953 | The value of the falling f... |
| risefac1 15954 | The value of rising factor... |
| fallfac1 15955 | The value of falling facto... |
| risefacfac 15956 | Relate rising factorial to... |
| fallfacfwd 15957 | The forward difference of ... |
| 0fallfac 15958 | The value of the zero fall... |
| 0risefac 15959 | The value of the zero risi... |
| binomfallfaclem1 15960 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 15961 | Lemma for ~ binomfallfac .... |
| binomfallfac 15962 | A version of the binomial ... |
| binomrisefac 15963 | A version of the binomial ... |
| fallfacval4 15964 | Represent the falling fact... |
| bcfallfac 15965 | Binomial coefficient in te... |
| fallfacfac 15966 | Relate falling factorial t... |
| bpolylem 15969 | Lemma for ~ bpolyval . (C... |
| bpolyval 15970 | The value of the Bernoulli... |
| bpoly0 15971 | The value of the Bernoulli... |
| bpoly1 15972 | The value of the Bernoulli... |
| bpolycl 15973 | Closure law for Bernoulli ... |
| bpolysum 15974 | A sum for Bernoulli polyno... |
| bpolydiflem 15975 | Lemma for ~ bpolydif . (C... |
| bpolydif 15976 | Calculate the difference b... |
| fsumkthpow 15977 | A closed-form expression f... |
| bpoly2 15978 | The Bernoulli polynomials ... |
| bpoly3 15979 | The Bernoulli polynomials ... |
| bpoly4 15980 | The Bernoulli polynomials ... |
| fsumcube 15981 | Express the sum of cubes i... |
| eftcl 15994 | Closure of a term in the s... |
| reeftcl 15995 | The terms of the series ex... |
| eftabs 15996 | The absolute value of a te... |
| eftval 15997 | The value of a term in the... |
| efcllem 15998 | Lemma for ~ efcl . The se... |
| ef0lem 15999 | The series defining the ex... |
| efval 16000 | Value of the exponential f... |
| esum 16001 | Value of Euler's constant ... |
| eff 16002 | Domain and codomain of the... |
| efcl 16003 | Closure law for the expone... |
| efcld 16004 | Closure law for the expone... |
| efval2 16005 | Value of the exponential f... |
| efcvg 16006 | The series that defines th... |
| efcvgfsum 16007 | Exponential function conve... |
| reefcl 16008 | The exponential function i... |
| reefcld 16009 | The exponential function i... |
| ere 16010 | Euler's constant ` _e ` = ... |
| ege2le3 16011 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16012 | Value of the exponential f... |
| efcj 16013 | The exponential of a compl... |
| efaddlem 16014 | Lemma for ~ efadd (exponen... |
| efadd 16015 | Sum of exponents law for e... |
| fprodefsum 16016 | Move the exponential funct... |
| efcan 16017 | Cancellation law for expon... |
| efne0d 16018 | The exponential of a compl... |
| efne0 16019 | The exponential of a compl... |
| efne0OLD 16020 | Obsolete version of ~ efne... |
| efneg 16021 | The exponential of the opp... |
| eff2 16022 | The exponential function m... |
| efsub 16023 | Difference of exponents la... |
| efexp 16024 | The exponential of an inte... |
| efzval 16025 | Value of the exponential f... |
| efgt0 16026 | The exponential of a real ... |
| rpefcl 16027 | The exponential of a real ... |
| rpefcld 16028 | The exponential of a real ... |
| eftlcvg 16029 | The tail series of the exp... |
| eftlcl 16030 | Closure of the sum of an i... |
| reeftlcl 16031 | Closure of the sum of an i... |
| eftlub 16032 | An upper bound on the abso... |
| efsep 16033 | Separate out the next term... |
| effsumlt 16034 | The partial sums of the se... |
| eft0val 16035 | The value of the first ter... |
| ef4p 16036 | Separate out the first fou... |
| efgt1p2 16037 | The exponential of a posit... |
| efgt1p 16038 | The exponential of a posit... |
| efgt1 16039 | The exponential of a posit... |
| eflt 16040 | The exponential function o... |
| efle 16041 | The exponential function o... |
| reef11 16042 | The exponential function o... |
| reeff1 16043 | The exponential function m... |
| eflegeo 16044 | The exponential function o... |
| sinval 16045 | Value of the sine function... |
| cosval 16046 | Value of the cosine functi... |
| sinf 16047 | Domain and codomain of the... |
| cosf 16048 | Domain and codomain of the... |
| sincl 16049 | Closure of the sine functi... |
| coscl 16050 | Closure of the cosine func... |
| tanval 16051 | Value of the tangent funct... |
| tancl 16052 | The closure of the tangent... |
| sincld 16053 | Closure of the sine functi... |
| coscld 16054 | Closure of the cosine func... |
| tancld 16055 | Closure of the tangent fun... |
| tanval2 16056 | Express the tangent functi... |
| tanval3 16057 | Express the tangent functi... |
| resinval 16058 | The sine of a real number ... |
| recosval 16059 | The cosine of a real numbe... |
| efi4p 16060 | Separate out the first fou... |
| resin4p 16061 | Separate out the first fou... |
| recos4p 16062 | Separate out the first fou... |
| resincl 16063 | The sine of a real number ... |
| recoscl 16064 | The cosine of a real numbe... |
| retancl 16065 | The closure of the tangent... |
| resincld 16066 | Closure of the sine functi... |
| recoscld 16067 | Closure of the cosine func... |
| retancld 16068 | Closure of the tangent fun... |
| sinneg 16069 | The sine of a negative is ... |
| cosneg 16070 | The cosines of a number an... |
| tanneg 16071 | The tangent of a negative ... |
| sin0 16072 | Value of the sine function... |
| cos0 16073 | Value of the cosine functi... |
| tan0 16074 | The value of the tangent f... |
| efival 16075 | The exponential function i... |
| efmival 16076 | The exponential function i... |
| sinhval 16077 | Value of the hyperbolic si... |
| coshval 16078 | Value of the hyperbolic co... |
| resinhcl 16079 | The hyperbolic sine of a r... |
| rpcoshcl 16080 | The hyperbolic cosine of a... |
| recoshcl 16081 | The hyperbolic cosine of a... |
| retanhcl 16082 | The hyperbolic tangent of ... |
| tanhlt1 16083 | The hyperbolic tangent of ... |
| tanhbnd 16084 | The hyperbolic tangent of ... |
| efeul 16085 | Eulerian representation of... |
| efieq 16086 | The exponentials of two im... |
| sinadd 16087 | Addition formula for sine.... |
| cosadd 16088 | Addition formula for cosin... |
| tanaddlem 16089 | A useful intermediate step... |
| tanadd 16090 | Addition formula for tange... |
| sinsub 16091 | Sine of difference. (Cont... |
| cossub 16092 | Cosine of difference. (Co... |
| addsin 16093 | Sum of sines. (Contribute... |
| subsin 16094 | Difference of sines. (Con... |
| sinmul 16095 | Product of sines can be re... |
| cosmul 16096 | Product of cosines can be ... |
| addcos 16097 | Sum of cosines. (Contribu... |
| subcos 16098 | Difference of cosines. (C... |
| sincossq 16099 | Sine squared plus cosine s... |
| sin2t 16100 | Double-angle formula for s... |
| cos2t 16101 | Double-angle formula for c... |
| cos2tsin 16102 | Double-angle formula for c... |
| sinbnd 16103 | The sine of a real number ... |
| cosbnd 16104 | The cosine of a real numbe... |
| sinbnd2 16105 | The sine of a real number ... |
| cosbnd2 16106 | The cosine of a real numbe... |
| ef01bndlem 16107 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16108 | Bounds on the sine of a po... |
| cos01bnd 16109 | Bounds on the cosine of a ... |
| cos1bnd 16110 | Bounds on the cosine of 1.... |
| cos2bnd 16111 | Bounds on the cosine of 2.... |
| sinltx 16112 | The sine of a positive rea... |
| sin01gt0 16113 | The sine of a positive rea... |
| cos01gt0 16114 | The cosine of a positive r... |
| sin02gt0 16115 | The sine of a positive rea... |
| sincos1sgn 16116 | The signs of the sine and ... |
| sincos2sgn 16117 | The signs of the sine and ... |
| sin4lt0 16118 | The sine of 4 is negative.... |
| absefi 16119 | The absolute value of the ... |
| absef 16120 | The absolute value of the ... |
| absefib 16121 | A complex number is real i... |
| efieq1re 16122 | A number whose imaginary e... |
| demoivre 16123 | De Moivre's Formula. Proo... |
| demoivreALT 16124 | Alternate proof of ~ demoi... |
| eirrlem 16127 | Lemma for ~ eirr . (Contr... |
| eirr 16128 | ` _e ` is irrational. (Co... |
| egt2lt3 16129 | Euler's constant ` _e ` = ... |
| epos 16130 | Euler's constant ` _e ` is... |
| epr 16131 | Euler's constant ` _e ` is... |
| ene0 16132 | ` _e ` is not 0. (Contrib... |
| ene1 16133 | ` _e ` is not 1. (Contrib... |
| xpnnen 16134 | The Cartesian product of t... |
| znnen 16135 | The set of integers and th... |
| qnnen 16136 | The rational numbers are c... |
| rpnnen2lem1 16137 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16138 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16139 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16140 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16141 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16142 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16143 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16144 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16145 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16146 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16147 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16148 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16149 | The other half of ~ rpnnen... |
| rpnnen 16150 | The cardinality of the con... |
| rexpen 16151 | The real numbers are equin... |
| cpnnen 16152 | The complex numbers are eq... |
| rucALT 16153 | Alternate proof of ~ ruc .... |
| ruclem1 16154 | Lemma for ~ ruc (the reals... |
| ruclem2 16155 | Lemma for ~ ruc . Orderin... |
| ruclem3 16156 | Lemma for ~ ruc . The con... |
| ruclem4 16157 | Lemma for ~ ruc . Initial... |
| ruclem6 16158 | Lemma for ~ ruc . Domain ... |
| ruclem7 16159 | Lemma for ~ ruc . Success... |
| ruclem8 16160 | Lemma for ~ ruc . The int... |
| ruclem9 16161 | Lemma for ~ ruc . The fir... |
| ruclem10 16162 | Lemma for ~ ruc . Every f... |
| ruclem11 16163 | Lemma for ~ ruc . Closure... |
| ruclem12 16164 | Lemma for ~ ruc . The sup... |
| ruclem13 16165 | Lemma for ~ ruc . There i... |
| ruc 16166 | The set of positive intege... |
| resdomq 16167 | The set of rationals is st... |
| aleph1re 16168 | There are at least aleph-o... |
| aleph1irr 16169 | There are at least aleph-o... |
| cnso 16170 | The complex numbers can be... |
| sqrt2irrlem 16171 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16172 | The square root of 2 is ir... |
| sqrt2re 16173 | The square root of 2 exist... |
| sqrt2irr0 16174 | The square root of 2 is an... |
| nthruc 16175 | The sequence ` NN ` , ` ZZ... |
| nthruz 16176 | The sequence ` NN ` , ` NN... |
| divides 16179 | Define the divides relatio... |
| dvdsval2 16180 | One nonzero integer divide... |
| dvdsval3 16181 | One nonzero integer divide... |
| dvdszrcl 16182 | Reverse closure for the di... |
| dvdsmod0 16183 | If a positive integer divi... |
| p1modz1 16184 | If a number greater than 1... |
| dvdsmodexp 16185 | If a positive integer divi... |
| nndivdvds 16186 | Strong form of ~ dvdsval2 ... |
| nndivides 16187 | Definition of the divides ... |
| moddvds 16188 | Two ways to say ` A == B `... |
| modm1div 16189 | An integer greater than on... |
| addmulmodb 16190 | An integer plus a product ... |
| dvds0lem 16191 | A lemma to assist theorems... |
| dvds1lem 16192 | A lemma to assist theorems... |
| dvds2lem 16193 | A lemma to assist theorems... |
| iddvds 16194 | An integer divides itself.... |
| 1dvds 16195 | 1 divides any integer. Th... |
| dvds0 16196 | Any integer divides 0. Th... |
| negdvdsb 16197 | An integer divides another... |
| dvdsnegb 16198 | An integer divides another... |
| absdvdsb 16199 | An integer divides another... |
| dvdsabsb 16200 | An integer divides another... |
| 0dvds 16201 | Only 0 is divisible by 0. ... |
| dvdsmul1 16202 | An integer divides a multi... |
| dvdsmul2 16203 | An integer divides a multi... |
| iddvdsexp 16204 | An integer divides a posit... |
| muldvds1 16205 | If a product divides an in... |
| muldvds2 16206 | If a product divides an in... |
| dvdscmul 16207 | Multiplication by a consta... |
| dvdsmulc 16208 | Multiplication by a consta... |
| dvdscmulr 16209 | Cancellation law for the d... |
| dvdsmulcr 16210 | Cancellation law for the d... |
| summodnegmod 16211 | The sum of two integers mo... |
| difmod0 16212 | The difference of two inte... |
| modmulconst 16213 | Constant multiplication in... |
| dvds2ln 16214 | If an integer divides each... |
| dvds2add 16215 | If an integer divides each... |
| dvds2sub 16216 | If an integer divides each... |
| dvds2addd 16217 | Deduction form of ~ dvds2a... |
| dvds2subd 16218 | Deduction form of ~ dvds2s... |
| dvdstr 16219 | The divides relation is tr... |
| dvdstrd 16220 | The divides relation is tr... |
| dvdsmultr1 16221 | If an integer divides anot... |
| dvdsmultr1d 16222 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16223 | If an integer divides anot... |
| dvdsmultr2d 16224 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16225 | If an integer divides eith... |
| dvdssub2 16226 | If an integer divides a di... |
| dvdsadd 16227 | An integer divides another... |
| dvdsaddr 16228 | An integer divides another... |
| dvdssub 16229 | An integer divides another... |
| dvdssubr 16230 | An integer divides another... |
| dvdsadd2b 16231 | Adding a multiple of the b... |
| dvdsaddre2b 16232 | Adding a multiple of the b... |
| fsumdvds 16233 | If every term in a sum is ... |
| dvdslelem 16234 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16235 | The divisors of a positive... |
| dvdsleabs 16236 | The divisors of a nonzero ... |
| dvdsleabs2 16237 | Transfer divisibility to a... |
| dvdsabseq 16238 | If two integers divide eac... |
| dvdseq 16239 | If two nonnegative integer... |
| divconjdvds 16240 | If a nonzero integer ` M `... |
| dvdsdivcl 16241 | The complement of a diviso... |
| dvdsflip 16242 | An involution of the divis... |
| dvdsssfz1 16243 | The set of divisors of a n... |
| dvds1 16244 | The only nonnegative integ... |
| alzdvds 16245 | Only 0 is divisible by all... |
| dvdsext 16246 | Poset extensionality for d... |
| fzm1ndvds 16247 | No number between ` 1 ` an... |
| fzo0dvdseq 16248 | Zero is the only one of th... |
| fzocongeq 16249 | Two different elements of ... |
| addmodlteqALT 16250 | Two nonnegative integers l... |
| dvdsfac 16251 | A positive integer divides... |
| dvdsexp2im 16252 | If an integer divides anot... |
| dvdsexp 16253 | A power divides a power wi... |
| dvdsmod 16254 | Any number ` K ` whose mod... |
| mulmoddvds 16255 | If an integer is divisible... |
| 3dvds 16256 | A rule for divisibility by... |
| 3dvdsdec 16257 | A decimal number is divisi... |
| 3dvds2dec 16258 | A decimal number is divisi... |
| fprodfvdvdsd 16259 | A finite product of intege... |
| fproddvdsd 16260 | A finite product of intege... |
| evenelz 16261 | An even number is an integ... |
| zeo3 16262 | An integer is even or odd.... |
| zeo4 16263 | An integer is even or odd ... |
| zeneo 16264 | No even integer equals an ... |
| odd2np1lem 16265 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16266 | An integer is odd iff it i... |
| even2n 16267 | An integer is even iff it ... |
| oddm1even 16268 | An integer is odd iff its ... |
| oddp1even 16269 | An integer is odd iff its ... |
| oexpneg 16270 | The exponential of the neg... |
| mod2eq0even 16271 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16272 | An integer is 1 modulo 2 i... |
| oddnn02np1 16273 | A nonnegative integer is o... |
| oddge22np1 16274 | An integer greater than on... |
| evennn02n 16275 | A nonnegative integer is e... |
| evennn2n 16276 | A positive integer is even... |
| 2tp1odd 16277 | A number which is twice an... |
| mulsucdiv2z 16278 | An integer multiplied with... |
| sqoddm1div8z 16279 | A squared odd number minus... |
| 2teven 16280 | A number which is twice an... |
| zeo5 16281 | An integer is either even ... |
| evend2 16282 | An integer is even iff its... |
| oddp1d2 16283 | An integer is odd iff its ... |
| zob 16284 | Alternate characterization... |
| oddm1d2 16285 | An integer is odd iff its ... |
| ltoddhalfle 16286 | An integer is less than ha... |
| halfleoddlt 16287 | An integer is greater than... |
| opoe 16288 | The sum of two odds is eve... |
| omoe 16289 | The difference of two odds... |
| opeo 16290 | The sum of an odd and an e... |
| omeo 16291 | The difference of an odd a... |
| z0even 16292 | 2 divides 0. That means 0... |
| n2dvds1 16293 | 2 does not divide 1. That... |
| n2dvdsm1 16294 | 2 does not divide -1. Tha... |
| z2even 16295 | 2 divides 2. That means 2... |
| n2dvds3 16296 | 2 does not divide 3. That... |
| z4even 16297 | 2 divides 4. That means 4... |
| 4dvdseven 16298 | An integer which is divisi... |
| m1expe 16299 | Exponentiation of -1 by an... |
| m1expo 16300 | Exponentiation of -1 by an... |
| m1exp1 16301 | Exponentiation of negative... |
| nn0enne 16302 | A positive integer is an e... |
| nn0ehalf 16303 | The half of an even nonneg... |
| nnehalf 16304 | The half of an even positi... |
| nn0onn 16305 | An odd nonnegative integer... |
| nn0o1gt2 16306 | An odd nonnegative integer... |
| nno 16307 | An alternate characterizat... |
| nn0o 16308 | An alternate characterizat... |
| nn0ob 16309 | Alternate characterization... |
| nn0oddm1d2 16310 | A positive integer is odd ... |
| nnoddm1d2 16311 | A positive integer is odd ... |
| sumeven 16312 | If every term in a sum is ... |
| sumodd 16313 | If every term in a sum is ... |
| evensumodd 16314 | If every term in a sum wit... |
| oddsumodd 16315 | If every term in a sum wit... |
| pwp1fsum 16316 | The n-th power of a number... |
| oddpwp1fsum 16317 | An odd power of a number i... |
| divalglem0 16318 | Lemma for ~ divalg . (Con... |
| divalglem1 16319 | Lemma for ~ divalg . (Con... |
| divalglem2 16320 | Lemma for ~ divalg . (Con... |
| divalglem4 16321 | Lemma for ~ divalg . (Con... |
| divalglem5 16322 | Lemma for ~ divalg . (Con... |
| divalglem6 16323 | Lemma for ~ divalg . (Con... |
| divalglem7 16324 | Lemma for ~ divalg . (Con... |
| divalglem8 16325 | Lemma for ~ divalg . (Con... |
| divalglem9 16326 | Lemma for ~ divalg . (Con... |
| divalglem10 16327 | Lemma for ~ divalg . (Con... |
| divalg 16328 | The division algorithm (th... |
| divalgb 16329 | Express the division algor... |
| divalg2 16330 | The division algorithm (th... |
| divalgmod 16331 | The result of the ` mod ` ... |
| divalgmodcl 16332 | The result of the ` mod ` ... |
| modremain 16333 | The result of the modulo o... |
| ndvdssub 16334 | Corollary of the division ... |
| ndvdsadd 16335 | Corollary of the division ... |
| ndvdsp1 16336 | Special case of ~ ndvdsadd... |
| ndvdsi 16337 | A quick test for non-divis... |
| 5ndvds3 16338 | 5 does not divide 3. (Con... |
| 5ndvds6 16339 | 5 does not divide 6. (Con... |
| flodddiv4 16340 | The floor of an odd intege... |
| fldivndvdslt 16341 | The floor of an integer di... |
| flodddiv4lt 16342 | The floor of an odd number... |
| flodddiv4t2lthalf 16343 | The floor of an odd number... |
| bitsfval 16348 | Expand the definition of t... |
| bitsval 16349 | Expand the definition of t... |
| bitsval2 16350 | Expand the definition of t... |
| bitsss 16351 | The set of bits of an inte... |
| bitsf 16352 | The ` bits ` function is a... |
| bits0 16353 | Value of the zeroth bit. ... |
| bits0e 16354 | The zeroth bit of an even ... |
| bits0o 16355 | The zeroth bit of an odd n... |
| bitsp1 16356 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16357 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16358 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16359 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16360 | The bits of a number are a... |
| bitsmod 16361 | Truncating the bit sequenc... |
| bitsfi 16362 | Every number is associated... |
| bitscmp 16363 | The bit complement of ` N ... |
| 0bits 16364 | The bits of zero. (Contri... |
| m1bits 16365 | The bits of negative one. ... |
| bitsinv1lem 16366 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16367 | There is an explicit inver... |
| bitsinv2 16368 | There is an explicit inver... |
| bitsf1ocnv 16369 | The ` bits ` function rest... |
| bitsf1o 16370 | The ` bits ` function rest... |
| bitsf1 16371 | The ` bits ` function is a... |
| 2ebits 16372 | The bits of a power of two... |
| bitsinv 16373 | The inverse of the ` bits ... |
| bitsinvp1 16374 | Recursive definition of th... |
| sadadd2lem2 16375 | The core of the proof of ~... |
| sadfval 16377 | Define the addition of two... |
| sadcf 16378 | The carry sequence is a se... |
| sadc0 16379 | The initial element of the... |
| sadcp1 16380 | The carry sequence (which ... |
| sadval 16381 | The full adder sequence is... |
| sadcaddlem 16382 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16383 | Non-recursive definition o... |
| sadadd2lem 16384 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16385 | Sum of initial segments of... |
| sadadd3 16386 | Sum of initial segments of... |
| sadcl 16387 | The sum of two sequences i... |
| sadcom 16388 | The adder sequence functio... |
| saddisjlem 16389 | Lemma for ~ sadadd . (Con... |
| saddisj 16390 | The sum of disjoint sequen... |
| sadaddlem 16391 | Lemma for ~ sadadd . (Con... |
| sadadd 16392 | For sequences that corresp... |
| sadid1 16393 | The adder sequence functio... |
| sadid2 16394 | The adder sequence functio... |
| sadasslem 16395 | Lemma for ~ sadass . (Con... |
| sadass 16396 | Sequence addition is assoc... |
| sadeq 16397 | Any element of a sequence ... |
| bitsres 16398 | Restrict the bits of a num... |
| bitsuz 16399 | The bits of a number are a... |
| bitsshft 16400 | Shifting a bit sequence to... |
| smufval 16402 | The multiplication of two ... |
| smupf 16403 | The sequence of partial su... |
| smup0 16404 | The initial element of the... |
| smupp1 16405 | The initial element of the... |
| smuval 16406 | Define the addition of two... |
| smuval2 16407 | The partial sum sequence s... |
| smupvallem 16408 | If ` A ` only has elements... |
| smucl 16409 | The product of two sequenc... |
| smu01lem 16410 | Lemma for ~ smu01 and ~ sm... |
| smu01 16411 | Multiplication of a sequen... |
| smu02 16412 | Multiplication of a sequen... |
| smupval 16413 | Rewrite the elements of th... |
| smup1 16414 | Rewrite ~ smupp1 using onl... |
| smueqlem 16415 | Any element of a sequence ... |
| smueq 16416 | Any element of a sequence ... |
| smumullem 16417 | Lemma for ~ smumul . (Con... |
| smumul 16418 | For sequences that corresp... |
| gcdval 16421 | The value of the ` gcd ` o... |
| gcd0val 16422 | The value, by convention, ... |
| gcdn0val 16423 | The value of the ` gcd ` o... |
| gcdcllem1 16424 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16425 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16426 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16427 | Closure of the ` gcd ` ope... |
| gcddvds 16428 | The gcd of two integers di... |
| dvdslegcd 16429 | An integer which divides b... |
| nndvdslegcd 16430 | A positive integer which d... |
| gcdcl 16431 | Closure of the ` gcd ` ope... |
| gcdnncl 16432 | Closure of the ` gcd ` ope... |
| gcdcld 16433 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16434 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16435 | An integer is the product ... |
| divgcdz 16436 | An integer divided by the ... |
| gcdf 16437 | Domain and codomain of the... |
| gcdcom 16438 | The ` gcd ` operator is co... |
| gcdcomd 16439 | The ` gcd ` operator is co... |
| divgcdnn 16440 | A positive integer divided... |
| divgcdnnr 16441 | A positive integer divided... |
| gcdeq0 16442 | The gcd of two integers is... |
| gcdn0gt0 16443 | The gcd of two integers is... |
| gcd0id 16444 | The gcd of 0 and an intege... |
| gcdid0 16445 | The gcd of an integer and ... |
| nn0gcdid0 16446 | The gcd of a nonnegative i... |
| gcdneg 16447 | Negating one operand of th... |
| neggcd 16448 | Negating one operand of th... |
| gcdaddmlem 16449 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16450 | Adding a multiple of one o... |
| gcdadd 16451 | The GCD of two numbers is ... |
| gcdid 16452 | The gcd of a number and it... |
| gcd1 16453 | The gcd of a number with 1... |
| gcdabs1 16454 | ` gcd ` of the absolute va... |
| gcdabs2 16455 | ` gcd ` of the absolute va... |
| gcdabs 16456 | The gcd of two integers is... |
| modgcd 16457 | The gcd remains unchanged ... |
| 1gcd 16458 | The GCD of one and an inte... |
| gcdmultipled 16459 | The greatest common diviso... |
| gcdmultiplez 16460 | The GCD of a multiple of a... |
| gcdmultiple 16461 | The GCD of a multiple of a... |
| dvdsgcdidd 16462 | The greatest common diviso... |
| 6gcd4e2 16463 | The greatest common diviso... |
| bezoutlem1 16464 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16465 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16466 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16467 | Lemma for ~ bezout . (Con... |
| bezout 16468 | Bézout's identity: ... |
| dvdsgcd 16469 | An integer which divides e... |
| dvdsgcdb 16470 | Biconditional form of ~ dv... |
| dfgcd2 16471 | Alternate definition of th... |
| gcdass 16472 | Associative law for ` gcd ... |
| mulgcd 16473 | Distribute multiplication ... |
| absmulgcd 16474 | Distribute absolute value ... |
| mulgcdr 16475 | Reverse distribution law f... |
| gcddiv 16476 | Division law for GCD. (Con... |
| gcdzeq 16477 | A positive integer ` A ` i... |
| gcdeq 16478 | ` A ` is equal to its gcd ... |
| dvdssqim 16479 | Unidirectional form of ~ d... |
| dvdsexpim 16480 | If two numbers are divisib... |
| dvdsmulgcd 16481 | A divisibility equivalent ... |
| rpmulgcd 16482 | If ` K ` and ` M ` are rel... |
| rplpwr 16483 | If ` A ` and ` B ` are rel... |
| rprpwr 16484 | If ` A ` and ` B ` are rel... |
| rppwr 16485 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16486 | If ` A ` and ` B ` are rel... |
| sqgcd 16487 | Square distributes over gc... |
| expgcd 16488 | Exponentiation distributes... |
| nn0expgcd 16489 | Exponentiation distributes... |
| zexpgcd 16490 | Exponentiation distributes... |
| dvdssqlem 16491 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16492 | Two numbers are divisible ... |
| bezoutr 16493 | Partial converse to ~ bezo... |
| bezoutr1 16494 | Converse of ~ bezout for w... |
| nn0seqcvgd 16495 | A strictly-decreasing nonn... |
| seq1st 16496 | A sequence whose iteration... |
| algr0 16497 | The value of the algorithm... |
| algrf 16498 | An algorithm is a step fun... |
| algrp1 16499 | The value of the algorithm... |
| alginv 16500 | If ` I ` is an invariant o... |
| algcvg 16501 | One way to prove that an a... |
| algcvgblem 16502 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16503 | Two ways of expressing tha... |
| algcvga 16504 | The countdown function ` C... |
| algfx 16505 | If ` F ` reaches a fixed p... |
| eucalgval2 16506 | The value of the step func... |
| eucalgval 16507 | Euclid's Algorithm ~ eucal... |
| eucalgf 16508 | Domain and codomain of the... |
| eucalginv 16509 | The invariant of the step ... |
| eucalglt 16510 | The second member of the s... |
| eucalgcvga 16511 | Once Euclid's Algorithm ha... |
| eucalg 16512 | Euclid's Algorithm compute... |
| lcmval 16517 | Value of the ` lcm ` opera... |
| lcmcom 16518 | The ` lcm ` operator is co... |
| lcm0val 16519 | The value, by convention, ... |
| lcmn0val 16520 | The value of the ` lcm ` o... |
| lcmcllem 16521 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16522 | Closure of the ` lcm ` ope... |
| dvdslcm 16523 | The lcm of two integers is... |
| lcmledvds 16524 | A positive integer which b... |
| lcmeq0 16525 | The lcm of two integers is... |
| lcmcl 16526 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16527 | The greatest common diviso... |
| lcmneg 16528 | Negating one operand of th... |
| neglcm 16529 | Negating one operand of th... |
| lcmabs 16530 | The lcm of two integers is... |
| lcmgcdlem 16531 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16532 | The product of two numbers... |
| lcmdvds 16533 | The lcm of two integers di... |
| lcmid 16534 | The lcm of an integer and ... |
| lcm1 16535 | The lcm of an integer and ... |
| lcmgcdnn 16536 | The product of two positiv... |
| lcmgcdeq 16537 | Two integers' absolute val... |
| lcmdvdsb 16538 | Biconditional form of ~ lc... |
| lcmass 16539 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16540 | The least common multiple ... |
| 6lcm4e12 16541 | The least common multiple ... |
| absproddvds 16542 | The absolute value of the ... |
| absprodnn 16543 | The absolute value of the ... |
| fissn0dvds 16544 | For each finite subset of ... |
| fissn0dvdsn0 16545 | For each finite subset of ... |
| lcmfval 16546 | Value of the ` _lcm ` func... |
| lcmf0val 16547 | The value, by convention, ... |
| lcmfn0val 16548 | The value of the ` _lcm ` ... |
| lcmfnnval 16549 | The value of the ` _lcm ` ... |
| lcmfcllem 16550 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16551 | Closure of the ` _lcm ` fu... |
| lcmfpr 16552 | The value of the ` _lcm ` ... |
| lcmfcl 16553 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16554 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16555 | The least common multiple ... |
| dvdslcmf 16556 | The least common multiple ... |
| lcmfledvds 16557 | A positive integer which i... |
| lcmf 16558 | Characterization of the le... |
| lcmf0 16559 | The least common multiple ... |
| lcmfsn 16560 | The least common multiple ... |
| lcmftp 16561 | The least common multiple ... |
| lcmfunsnlem1 16562 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16563 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16564 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16565 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16566 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16567 | The least common multiple ... |
| lcmfdvdsb 16568 | Biconditional form of ~ lc... |
| lcmfunsn 16569 | The ` _lcm ` function for ... |
| lcmfun 16570 | The ` _lcm ` function for ... |
| lcmfass 16571 | Associative law for the ` ... |
| lcmf2a3a4e12 16572 | The least common multiple ... |
| lcmflefac 16573 | The least common multiple ... |
| coprmgcdb 16574 | Two positive integers are ... |
| ncoprmgcdne1b 16575 | Two positive integers are ... |
| ncoprmgcdgt1b 16576 | Two positive integers are ... |
| coprmdvds1 16577 | If two positive integers a... |
| coprmdvds 16578 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16579 | If an integer is divisible... |
| mulgcddvds 16580 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16581 | If ` M ` is relatively pri... |
| qredeq 16582 | Two equal reduced fraction... |
| qredeu 16583 | Every rational number has ... |
| rpmul 16584 | If ` K ` is relatively pri... |
| rpdvds 16585 | If ` K ` is relatively pri... |
| coprmprod 16586 | The product of the element... |
| coprmproddvdslem 16587 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16588 | If a positive integer is d... |
| congr 16589 | Definition of congruence b... |
| divgcdcoprm0 16590 | Integers divided by gcd ar... |
| divgcdcoprmex 16591 | Integers divided by gcd ar... |
| cncongr1 16592 | One direction of the bicon... |
| cncongr2 16593 | The other direction of the... |
| cncongr 16594 | Cancellability of Congruen... |
| cncongrcoprm 16595 | Corollary 1 of Cancellabil... |
| isprm 16598 | The predicate "is a prime ... |
| prmnn 16599 | A prime number is a positi... |
| prmz 16600 | A prime number is an integ... |
| prmssnn 16601 | The prime numbers are a su... |
| prmex 16602 | The set of prime numbers e... |
| 0nprm 16603 | 0 is not a prime number. ... |
| 1nprm 16604 | 1 is not a prime number. ... |
| 1idssfct 16605 | The positive divisors of a... |
| isprm2lem 16606 | Lemma for ~ isprm2 . (Con... |
| isprm2 16607 | The predicate "is a prime ... |
| isprm3 16608 | The predicate "is a prime ... |
| isprm4 16609 | The predicate "is a prime ... |
| prmind2 16610 | A variation on ~ prmind as... |
| prmind 16611 | Perform induction over the... |
| dvdsprime 16612 | If ` M ` divides a prime, ... |
| nprm 16613 | A product of two integers ... |
| nprmi 16614 | An inference for composite... |
| dvdsnprmd 16615 | If a number is divisible b... |
| prm2orodd 16616 | A prime number is either 2... |
| 2prm 16617 | 2 is a prime number. (Con... |
| 2mulprm 16618 | A multiple of two is prime... |
| 3prm 16619 | 3 is a prime number. (Con... |
| 4nprm 16620 | 4 is not a prime number. ... |
| prmuz2 16621 | A prime number is an integ... |
| prmgt1 16622 | A prime number is an integ... |
| prmm2nn0 16623 | Subtracting 2 from a prime... |
| oddprmgt2 16624 | An odd prime is greater th... |
| oddprmge3 16625 | An odd prime is greater th... |
| ge2nprmge4 16626 | A composite integer greate... |
| sqnprm 16627 | A square is never prime. ... |
| dvdsprm 16628 | An integer greater than or... |
| exprmfct 16629 | Every integer greater than... |
| prmdvdsfz 16630 | Each integer greater than ... |
| nprmdvds1 16631 | No prime number divides 1.... |
| isprm5 16632 | One need only check prime ... |
| isprm7 16633 | One need only check prime ... |
| maxprmfct 16634 | The set of prime factors o... |
| divgcdodd 16635 | Either ` A / ( A gcd B ) `... |
| coprm 16636 | A prime number either divi... |
| prmrp 16637 | Unequal prime numbers are ... |
| euclemma 16638 | Euclid's lemma. A prime n... |
| isprm6 16639 | A number is prime iff it s... |
| prmdvdsexp 16640 | A prime divides a positive... |
| prmdvdsexpb 16641 | A prime divides a positive... |
| prmdvdsexpr 16642 | If a prime divides a nonne... |
| prmdvdssq 16643 | Condition for a prime divi... |
| prmexpb 16644 | Two positive prime powers ... |
| prmfac1 16645 | The factorial of a number ... |
| dvdszzq 16646 | Divisibility for an intege... |
| rpexp 16647 | If two numbers ` A ` and `... |
| rpexp1i 16648 | Relative primality passes ... |
| rpexp12i 16649 | Relative primality passes ... |
| prmndvdsfaclt 16650 | A prime number does not di... |
| prmdvdsbc 16651 | Condition for a prime numb... |
| prmdvdsncoprmbd 16652 | Two positive integers are ... |
| ncoprmlnprm 16653 | If two positive integers a... |
| cncongrprm 16654 | Corollary 2 of Cancellabil... |
| isevengcd2 16655 | The predicate "is an even ... |
| isoddgcd1 16656 | The predicate "is an odd n... |
| 3lcm2e6 16657 | The least common multiple ... |
| qnumval 16662 | Value of the canonical num... |
| qdenval 16663 | Value of the canonical den... |
| qnumdencl 16664 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16665 | The canonical numerator of... |
| qdencl 16666 | The canonical denominator ... |
| fnum 16667 | Canonical numerator define... |
| fden 16668 | Canonical denominator defi... |
| qnumdenbi 16669 | Two numbers are the canoni... |
| qnumdencoprm 16670 | The canonical representati... |
| qeqnumdivden 16671 | Recover a rational number ... |
| qmuldeneqnum 16672 | Multiplying a rational by ... |
| divnumden 16673 | Calculate the reduced form... |
| divdenle 16674 | Reducing a quotient never ... |
| qnumgt0 16675 | A rational is positive iff... |
| qgt0numnn 16676 | A rational is positive iff... |
| nn0gcdsq 16677 | Squaring commutes with GCD... |
| zgcdsq 16678 | ~ nn0gcdsq extended to int... |
| numdensq 16679 | Squaring a rational square... |
| numsq 16680 | Square commutes with canon... |
| densq 16681 | Square commutes with canon... |
| qden1elz 16682 | A rational is an integer i... |
| zsqrtelqelz 16683 | If an integer has a ration... |
| nonsq 16684 | Any integer strictly betwe... |
| numdenexp 16685 | Elevating a rational numbe... |
| numexp 16686 | Elevating to a nonnegative... |
| denexp 16687 | Elevating to a nonnegative... |
| phival 16692 | Value of the Euler ` phi `... |
| phicl2 16693 | Bounds and closure for the... |
| phicl 16694 | Closure for the value of t... |
| phibndlem 16695 | Lemma for ~ phibnd . (Con... |
| phibnd 16696 | A slightly tighter bound o... |
| phicld 16697 | Closure for the value of t... |
| phi1 16698 | Value of the Euler ` phi `... |
| dfphi2 16699 | Alternate definition of th... |
| hashdvds 16700 | The number of numbers in a... |
| phiprmpw 16701 | Value of the Euler ` phi `... |
| phiprm 16702 | Value of the Euler ` phi `... |
| crth 16703 | The Chinese Remainder Theo... |
| phimullem 16704 | Lemma for ~ phimul . (Con... |
| phimul 16705 | The Euler ` phi ` function... |
| eulerthlem1 16706 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16707 | Lemma for ~ eulerth . (Co... |
| eulerth 16708 | Euler's theorem, a general... |
| fermltl 16709 | Fermat's little theorem. ... |
| prmdiv 16710 | Show an explicit expressio... |
| prmdiveq 16711 | The modular inverse of ` A... |
| prmdivdiv 16712 | The (modular) inverse of t... |
| hashgcdlem 16713 | A correspondence between e... |
| dvdsfi 16714 | A natural number has finit... |
| hashgcdeq 16715 | Number of initial positive... |
| phisum 16716 | The divisor sum identity o... |
| odzval 16717 | Value of the order functio... |
| odzcllem 16718 | - Lemma for ~ odzcl , show... |
| odzcl 16719 | The order of a group eleme... |
| odzid 16720 | Any element raised to the ... |
| odzdvds 16721 | The only powers of ` A ` t... |
| odzphi 16722 | The order of any group ele... |
| modprm1div 16723 | A prime number divides an ... |
| m1dvdsndvds 16724 | If an integer minus 1 is d... |
| modprminv 16725 | Show an explicit expressio... |
| modprminveq 16726 | The modular inverse of ` A... |
| vfermltl 16727 | Variant of Fermat's little... |
| vfermltlALT 16728 | Alternate proof of ~ vferm... |
| powm2modprm 16729 | If an integer minus 1 is d... |
| reumodprminv 16730 | For any prime number and f... |
| modprm0 16731 | For two positive integers ... |
| nnnn0modprm0 16732 | For a positive integer and... |
| modprmn0modprm0 16733 | For an integer not being 0... |
| coprimeprodsq 16734 | If three numbers are copri... |
| coprimeprodsq2 16735 | If three numbers are copri... |
| oddprm 16736 | A prime not equal to ` 2 `... |
| nnoddn2prm 16737 | A prime not equal to ` 2 `... |
| oddn2prm 16738 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16739 | A number is a prime number... |
| prm23lt5 16740 | A prime less than 5 is eit... |
| prm23ge5 16741 | A prime is either 2 or 3 o... |
| pythagtriplem1 16742 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16743 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16744 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16745 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16746 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16747 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16748 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16749 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16750 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16751 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16752 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16753 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16754 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16755 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16756 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16757 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16758 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16759 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16760 | Parameterize the Pythagore... |
| iserodd 16761 | Collect the odd terms in a... |
| pclem 16764 | - Lemma for the prime powe... |
| pcprecl 16765 | Closure of the prime power... |
| pcprendvds 16766 | Non-divisibility property ... |
| pcprendvds2 16767 | Non-divisibility property ... |
| pcpre1 16768 | Value of the prime power p... |
| pcpremul 16769 | Multiplicative property of... |
| pcval 16770 | The value of the prime pow... |
| pceulem 16771 | Lemma for ~ pceu . (Contr... |
| pceu 16772 | Uniqueness for the prime p... |
| pczpre 16773 | Connect the prime count pr... |
| pczcl 16774 | Closure of the prime power... |
| pccl 16775 | Closure of the prime power... |
| pccld 16776 | Closure of the prime power... |
| pcmul 16777 | Multiplication property of... |
| pcdiv 16778 | Division property of the p... |
| pcqmul 16779 | Multiplication property of... |
| pc0 16780 | The value of the prime pow... |
| pc1 16781 | Value of the prime count f... |
| pcqcl 16782 | Closure of the general pri... |
| pcqdiv 16783 | Division property of the p... |
| pcrec 16784 | Prime power of a reciproca... |
| pcexp 16785 | Prime power of an exponent... |
| pcxnn0cl 16786 | Extended nonnegative integ... |
| pcxcl 16787 | Extended real closure of t... |
| pcge0 16788 | The prime count of an inte... |
| pczdvds 16789 | Defining property of the p... |
| pcdvds 16790 | Defining property of the p... |
| pczndvds 16791 | Defining property of the p... |
| pcndvds 16792 | Defining property of the p... |
| pczndvds2 16793 | The remainder after dividi... |
| pcndvds2 16794 | The remainder after dividi... |
| pcdvdsb 16795 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16796 | There are a positive numbe... |
| pceq0 16797 | There are zero powers of a... |
| pcidlem 16798 | The prime count of a prime... |
| pcid 16799 | The prime count of a prime... |
| pcneg 16800 | The prime count of a negat... |
| pcabs 16801 | The prime count of an abso... |
| pcdvdstr 16802 | The prime count increases ... |
| pcgcd1 16803 | The prime count of a GCD i... |
| pcgcd 16804 | The prime count of a GCD i... |
| pc2dvds 16805 | A characterization of divi... |
| pc11 16806 | The prime count function, ... |
| pcz 16807 | The prime count function c... |
| pcprmpw2 16808 | Self-referential expressio... |
| pcprmpw 16809 | Self-referential expressio... |
| dvdsprmpweq 16810 | If a positive integer divi... |
| dvdsprmpweqnn 16811 | If an integer greater than... |
| dvdsprmpweqle 16812 | If a positive integer divi... |
| difsqpwdvds 16813 | If the difference of two s... |
| pcaddlem 16814 | Lemma for ~ pcadd . The o... |
| pcadd 16815 | An inequality for the prim... |
| pcadd2 16816 | The inequality of ~ pcadd ... |
| pcmptcl 16817 | Closure for the prime powe... |
| pcmpt 16818 | Construct a function with ... |
| pcmpt2 16819 | Dividing two prime count m... |
| pcmptdvds 16820 | The partial products of th... |
| pcprod 16821 | The product of the primes ... |
| sumhash 16822 | The sum of 1 over a set is... |
| fldivp1 16823 | The difference between the... |
| pcfaclem 16824 | Lemma for ~ pcfac . (Cont... |
| pcfac 16825 | Calculate the prime count ... |
| pcbc 16826 | Calculate the prime count ... |
| qexpz 16827 | If a power of a rational n... |
| expnprm 16828 | A second or higher power o... |
| oddprmdvds 16829 | Every positive integer whi... |
| prmpwdvds 16830 | A relation involving divis... |
| pockthlem 16831 | Lemma for ~ pockthg . (Co... |
| pockthg 16832 | The generalized Pocklingto... |
| pockthi 16833 | Pocklington's theorem, whi... |
| unbenlem 16834 | Lemma for ~ unben . (Cont... |
| unben 16835 | An unbounded set of positi... |
| infpnlem1 16836 | Lemma for ~ infpn . The s... |
| infpnlem2 16837 | Lemma for ~ infpn . For a... |
| infpn 16838 | There exist infinitely man... |
| infpn2 16839 | There exist infinitely man... |
| prmunb 16840 | The primes are unbounded. ... |
| prminf 16841 | There are an infinite numb... |
| prmreclem1 16842 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16843 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16844 | Lemma for ~ prmrec . The ... |
| prmreclem4 16845 | Lemma for ~ prmrec . Show... |
| prmreclem5 16846 | Lemma for ~ prmrec . Here... |
| prmreclem6 16847 | Lemma for ~ prmrec . If t... |
| prmrec 16848 | The sum of the reciprocals... |
| 1arithlem1 16849 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16850 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16851 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16852 | Lemma for ~ 1arith . (Con... |
| 1arith 16853 | Fundamental theorem of ari... |
| 1arith2 16854 | Fundamental theorem of ari... |
| elgz 16857 | Elementhood in the gaussia... |
| gzcn 16858 | A gaussian integer is a co... |
| zgz 16859 | An integer is a gaussian i... |
| igz 16860 | ` _i ` is a gaussian integ... |
| gznegcl 16861 | The gaussian integers are ... |
| gzcjcl 16862 | The gaussian integers are ... |
| gzaddcl 16863 | The gaussian integers are ... |
| gzmulcl 16864 | The gaussian integers are ... |
| gzreim 16865 | Construct a gaussian integ... |
| gzsubcl 16866 | The gaussian integers are ... |
| gzabssqcl 16867 | The squared norm of a gaus... |
| 4sqlem5 16868 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16869 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16870 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16871 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16872 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16873 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16874 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16875 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16876 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16877 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16878 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16879 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16880 | Euler's four-square identi... |
| 4sqlem11 16881 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16882 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16883 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16884 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16885 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16886 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16887 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16888 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16889 | Lemma for ~ 4sq . The pro... |
| 4sq 16890 | Lagrange's four-square the... |
| vdwapfval 16897 | Define the arithmetic prog... |
| vdwapf 16898 | The arithmetic progression... |
| vdwapval 16899 | Value of the arithmetic pr... |
| vdwapun 16900 | Remove the first element o... |
| vdwapid1 16901 | The first element of an ar... |
| vdwap0 16902 | Value of a length-1 arithm... |
| vdwap1 16903 | Value of a length-1 arithm... |
| vdwmc 16904 | The predicate " The ` <. R... |
| vdwmc2 16905 | Expand out the definition ... |
| vdwpc 16906 | The predicate " The colori... |
| vdwlem1 16907 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16908 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16909 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16910 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16911 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16912 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16913 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16914 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16915 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16916 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16917 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16918 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16919 | Lemma for ~ vdw . Main in... |
| vdw 16920 | Van der Waerden's theorem.... |
| vdwnnlem1 16921 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16922 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16923 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16924 | Van der Waerden's theorem,... |
| ramtlecl 16926 | The set ` T ` of numbers w... |
| hashbcval 16928 | Value of the "binomial set... |
| hashbccl 16929 | The binomial set is a fini... |
| hashbcss 16930 | Subset relation for the bi... |
| hashbc0 16931 | The set of subsets of size... |
| hashbc2 16932 | The size of the binomial s... |
| 0hashbc 16933 | There are no subsets of th... |
| ramval 16934 | The value of the Ramsey nu... |
| ramcl2lem 16935 | Lemma for extended real cl... |
| ramtcl 16936 | The Ramsey number has the ... |
| ramtcl2 16937 | The Ramsey number is an in... |
| ramtub 16938 | The Ramsey number is a low... |
| ramub 16939 | The Ramsey number is a low... |
| ramub2 16940 | It is sufficient to check ... |
| rami 16941 | The defining property of a... |
| ramcl2 16942 | The Ramsey number is eithe... |
| ramxrcl 16943 | The Ramsey number is an ex... |
| ramubcl 16944 | If the Ramsey number is up... |
| ramlb 16945 | Establish a lower bound on... |
| 0ram 16946 | The Ramsey number when ` M... |
| 0ram2 16947 | The Ramsey number when ` M... |
| ram0 16948 | The Ramsey number when ` R... |
| 0ramcl 16949 | Lemma for ~ ramcl : Exist... |
| ramz2 16950 | The Ramsey number when ` F... |
| ramz 16951 | The Ramsey number when ` F... |
| ramub1lem1 16952 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16953 | Lemma for ~ ramub1 . (Con... |
| ramub1 16954 | Inductive step for Ramsey'... |
| ramcl 16955 | Ramsey's theorem: the Rams... |
| ramsey 16956 | Ramsey's theorem with the ... |
| prmoval 16959 | Value of the primorial fun... |
| prmocl 16960 | Closure of the primorial f... |
| prmone0 16961 | The primorial function is ... |
| prmo0 16962 | The primorial of 0. (Cont... |
| prmo1 16963 | The primorial of 1. (Cont... |
| prmop1 16964 | The primorial of a success... |
| prmonn2 16965 | Value of the primorial fun... |
| prmo2 16966 | The primorial of 2. (Cont... |
| prmo3 16967 | The primorial of 3. (Cont... |
| prmdvdsprmo 16968 | The primorial of a number ... |
| prmdvdsprmop 16969 | The primorial of a number ... |
| fvprmselelfz 16970 | The value of the prime sel... |
| fvprmselgcd1 16971 | The greatest common diviso... |
| prmolefac 16972 | The primorial of a positiv... |
| prmodvdslcmf 16973 | The primorial of a nonnega... |
| prmolelcmf 16974 | The primorial of a positiv... |
| prmgaplem1 16975 | Lemma for ~ prmgap : The ... |
| prmgaplem2 16976 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 16977 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 16978 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 16979 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 16980 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 16981 | Lemma for ~ prmgap : for e... |
| prmgaplem6 16982 | Lemma for ~ prmgap : for e... |
| prmgaplem7 16983 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 16984 | Lemma for ~ prmgap . (Con... |
| prmgap 16985 | The prime gap theorem: for... |
| prmgaplcm 16986 | Alternate proof of ~ prmga... |
| prmgapprmolem 16987 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 16988 | Alternate proof of ~ prmga... |
| dec2dvds 16989 | Divisibility by two is obv... |
| dec5dvds 16990 | Divisibility by five is ob... |
| dec5dvds2 16991 | Divisibility by five is ob... |
| dec5nprm 16992 | A decimal number greater t... |
| dec2nprm 16993 | A decimal number greater t... |
| modxai 16994 | Add exponents in a power m... |
| mod2xi 16995 | Double exponents in a powe... |
| modxp1i 16996 | Add one to an exponent in ... |
| mod2xnegi 16997 | Version of ~ mod2xi with a... |
| modsubi 16998 | Subtract from within a mod... |
| gcdi 16999 | Calculate a GCD via Euclid... |
| gcdmodi 17000 | Calculate a GCD via Euclid... |
| numexp0 17001 | Calculate an integer power... |
| numexp1 17002 | Calculate an integer power... |
| numexpp1 17003 | Calculate an integer power... |
| numexp2x 17004 | Double an integer power. ... |
| decsplit0b 17005 | Split a decimal number int... |
| decsplit0 17006 | Split a decimal number int... |
| decsplit1 17007 | Split a decimal number int... |
| decsplit 17008 | Split a decimal number int... |
| karatsuba 17009 | The Karatsuba multiplicati... |
| 2exp4 17010 | Two to the fourth power is... |
| 2exp5 17011 | Two to the fifth power is ... |
| 2exp6 17012 | Two to the sixth power is ... |
| 2exp7 17013 | Two to the seventh power i... |
| 2exp8 17014 | Two to the eighth power is... |
| 2exp11 17015 | Two to the eleventh power ... |
| 2exp16 17016 | Two to the sixteenth power... |
| 3exp3 17017 | Three to the third power i... |
| 2expltfac 17018 | The factorial grows faster... |
| cshwsidrepsw 17019 | If cyclically shifting a w... |
| cshwsidrepswmod0 17020 | If cyclically shifting a w... |
| cshwshashlem1 17021 | If cyclically shifting a w... |
| cshwshashlem2 17022 | If cyclically shifting a w... |
| cshwshashlem3 17023 | If cyclically shifting a w... |
| cshwsdisj 17024 | The singletons resulting b... |
| cshwsiun 17025 | The set of (different!) wo... |
| cshwsex 17026 | The class of (different!) ... |
| cshws0 17027 | The size of the set of (di... |
| cshwrepswhash1 17028 | The size of the set of (di... |
| cshwshashnsame 17029 | If a word (not consisting ... |
| cshwshash 17030 | If a word has a length bei... |
| prmlem0 17031 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17032 | A quick proof skeleton to ... |
| prmlem1 17033 | A quick proof skeleton to ... |
| 5prm 17034 | 5 is a prime number. (Con... |
| 6nprm 17035 | 6 is not a prime number. ... |
| 7prm 17036 | 7 is a prime number. (Con... |
| 8nprm 17037 | 8 is not a prime number. ... |
| 9nprm 17038 | 9 is not a prime number. ... |
| 10nprm 17039 | 10 is not a prime number. ... |
| 11prm 17040 | 11 is a prime number. (Co... |
| 13prm 17041 | 13 is a prime number. (Co... |
| 17prm 17042 | 17 is a prime number. (Co... |
| 19prm 17043 | 19 is a prime number. (Co... |
| 23prm 17044 | 23 is a prime number. (Co... |
| prmlem2 17045 | Our last proving session g... |
| 37prm 17046 | 37 is a prime number. (Co... |
| 43prm 17047 | 43 is a prime number. (Co... |
| 83prm 17048 | 83 is a prime number. (Co... |
| 139prm 17049 | 139 is a prime number. (C... |
| 163prm 17050 | 163 is a prime number. (C... |
| 317prm 17051 | 317 is a prime number. (C... |
| 631prm 17052 | 631 is a prime number. (C... |
| prmo4 17053 | The primorial of 4. (Cont... |
| prmo5 17054 | The primorial of 5. (Cont... |
| prmo6 17055 | The primorial of 6. (Cont... |
| 1259lem1 17056 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17057 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17058 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17059 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17060 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17061 | 1259 is a prime number. (... |
| 2503lem1 17062 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17063 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17064 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17065 | 2503 is a prime number. (... |
| 4001lem1 17066 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17067 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17068 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17069 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17070 | 4001 is a prime number. (... |
| brstruct 17073 | The structure relation is ... |
| isstruct2 17074 | The property of being a st... |
| structex 17075 | A structure is a set. (Co... |
| structn0fun 17076 | A structure without the em... |
| isstruct 17077 | The property of being a st... |
| structcnvcnv 17078 | Two ways to express the re... |
| structfung 17079 | The converse of the conver... |
| structfun 17080 | Convert between two kinds ... |
| structfn 17081 | Convert between two kinds ... |
| strleun 17082 | Combine two structures int... |
| strle1 17083 | Make a structure from a si... |
| strle2 17084 | Make a structure from a pa... |
| strle3 17085 | Make a structure from a tr... |
| sbcie2s 17086 | A special version of class... |
| sbcie3s 17087 | A special version of class... |
| reldmsets 17090 | The structure override ope... |
| setsvalg 17091 | Value of the structure rep... |
| setsval 17092 | Value of the structure rep... |
| fvsetsid 17093 | The value of the structure... |
| fsets 17094 | The structure replacement ... |
| setsdm 17095 | The domain of a structure ... |
| setsfun 17096 | A structure with replaceme... |
| setsfun0 17097 | A structure with replaceme... |
| setsn0fun 17098 | The value of the structure... |
| setsstruct2 17099 | An extensible structure wi... |
| setsexstruct2 17100 | An extensible structure wi... |
| setsstruct 17101 | An extensible structure wi... |
| wunsets 17102 | Closure of structure repla... |
| setsres 17103 | The structure replacement ... |
| setsabs 17104 | Replacing the same compone... |
| setscom 17105 | Different components can b... |
| sloteq 17108 | Equality theorem for the `... |
| slotfn 17109 | A slot is a function on se... |
| strfvnd 17110 | Deduction version of ~ str... |
| strfvn 17111 | Value of a structure compo... |
| strfvss 17112 | A structure component extr... |
| wunstr 17113 | Closure of a structure ind... |
| str0 17114 | All components of the empt... |
| strfvi 17115 | Structure slot extractors ... |
| fveqprc 17116 | Lemma for showing the equa... |
| oveqprc 17117 | Lemma for showing the equa... |
| wunndx 17120 | Closure of the index extra... |
| ndxarg 17121 | Get the numeric argument f... |
| ndxid 17122 | A structure component extr... |
| strndxid 17123 | The value of a structure c... |
| setsidvald 17124 | Value of the structure rep... |
| strfvd 17125 | Deduction version of ~ str... |
| strfv2d 17126 | Deduction version of ~ str... |
| strfv2 17127 | A variation on ~ strfv to ... |
| strfv 17128 | Extract a structure compon... |
| strfv3 17129 | Variant on ~ strfv for lar... |
| strssd 17130 | Deduction version of ~ str... |
| strss 17131 | Propagate component extrac... |
| setsid 17132 | Value of the structure rep... |
| setsnid 17133 | Value of the structure rep... |
| baseval 17136 | Value of the base set extr... |
| baseid 17137 | Utility theorem: index-ind... |
| basfn 17138 | The base set extractor is ... |
| base0 17139 | The base set of the empty ... |
| elbasfv 17140 | Utility theorem: reverse c... |
| elbasov 17141 | Utility theorem: reverse c... |
| strov2rcl 17142 | Partial reverse closure fo... |
| basendx 17143 | Index value of the base se... |
| basendxnn 17144 | The index value of the bas... |
| basndxelwund 17145 | The index of the base set ... |
| basprssdmsets 17146 | The pair of the base index... |
| opelstrbas 17147 | The base set of a structur... |
| 1strstr 17148 | A constructed one-slot str... |
| 1strbas 17149 | The base set of a construc... |
| 1strwunbndx 17150 | A constructed one-slot str... |
| 1strwun 17151 | A constructed one-slot str... |
| 2strstr 17152 | A constructed two-slot str... |
| 2strbas 17153 | The base set of a construc... |
| 2strop 17154 | The other slot of a constr... |
| reldmress 17157 | The structure restriction ... |
| ressval 17158 | Value of structure restric... |
| ressid2 17159 | General behavior of trivia... |
| ressval2 17160 | Value of nontrivial struct... |
| ressbas 17161 | Base set of a structure re... |
| ressbasssg 17162 | The base set of a restrict... |
| ressbas2 17163 | Base set of a structure re... |
| ressbasss 17164 | The base set of a restrict... |
| ressbasssOLD 17165 | Obsolete version of ~ ress... |
| ressbasss2 17166 | The base set of a restrict... |
| resseqnbas 17167 | The components of an exten... |
| ress0 17168 | All restrictions of the nu... |
| ressid 17169 | Behavior of trivial restri... |
| ressinbas 17170 | Restriction only cares abo... |
| ressval3d 17171 | Value of structure restric... |
| ressress 17172 | Restriction composition la... |
| ressabs 17173 | Restriction absorption law... |
| wunress 17174 | Closure of structure restr... |
| plusgndx 17201 | Index value of the ~ df-pl... |
| plusgid 17202 | Utility theorem: index-ind... |
| plusgndxnn 17203 | The index of the slot for ... |
| basendxltplusgndx 17204 | The index of the slot for ... |
| basendxnplusgndx 17205 | The slot for the base set ... |
| grpstr 17206 | A constructed group is a s... |
| grpbase 17207 | The base set of a construc... |
| grpplusg 17208 | The operation of a constru... |
| ressplusg 17209 | ` +g ` is unaffected by re... |
| grpbasex 17210 | The base of an explicitly ... |
| grpplusgx 17211 | The operation of an explic... |
| mulrndx 17212 | Index value of the ~ df-mu... |
| mulridx 17213 | Utility theorem: index-ind... |
| basendxnmulrndx 17214 | The slot for the base set ... |
| plusgndxnmulrndx 17215 | The slot for the group (ad... |
| rngstr 17216 | A constructed ring is a st... |
| rngbase 17217 | The base set of a construc... |
| rngplusg 17218 | The additive operation of ... |
| rngmulr 17219 | The multiplicative operati... |
| starvndx 17220 | Index value of the ~ df-st... |
| starvid 17221 | Utility theorem: index-ind... |
| starvndxnbasendx 17222 | The slot for the involutio... |
| starvndxnplusgndx 17223 | The slot for the involutio... |
| starvndxnmulrndx 17224 | The slot for the involutio... |
| ressmulr 17225 | ` .r ` is unaffected by re... |
| ressstarv 17226 | ` *r ` is unaffected by re... |
| srngstr 17227 | A constructed star ring is... |
| srngbase 17228 | The base set of a construc... |
| srngplusg 17229 | The addition operation of ... |
| srngmulr 17230 | The multiplication operati... |
| srnginvl 17231 | The involution function of... |
| scandx 17232 | Index value of the ~ df-sc... |
| scaid 17233 | Utility theorem: index-ind... |
| scandxnbasendx 17234 | The slot for the scalar is... |
| scandxnplusgndx 17235 | The slot for the scalar fi... |
| scandxnmulrndx 17236 | The slot for the scalar fi... |
| vscandx 17237 | Index value of the ~ df-vs... |
| vscaid 17238 | Utility theorem: index-ind... |
| vscandxnbasendx 17239 | The slot for the scalar pr... |
| vscandxnplusgndx 17240 | The slot for the scalar pr... |
| vscandxnmulrndx 17241 | The slot for the scalar pr... |
| vscandxnscandx 17242 | The slot for the scalar pr... |
| lmodstr 17243 | A constructed left module ... |
| lmodbase 17244 | The base set of a construc... |
| lmodplusg 17245 | The additive operation of ... |
| lmodsca 17246 | The set of scalars of a co... |
| lmodvsca 17247 | The scalar product operati... |
| ipndx 17248 | Index value of the ~ df-ip... |
| ipid 17249 | Utility theorem: index-ind... |
| ipndxnbasendx 17250 | The slot for the inner pro... |
| ipndxnplusgndx 17251 | The slot for the inner pro... |
| ipndxnmulrndx 17252 | The slot for the inner pro... |
| slotsdifipndx 17253 | The slot for the scalar is... |
| ipsstr 17254 | Lemma to shorten proofs of... |
| ipsbase 17255 | The base set of a construc... |
| ipsaddg 17256 | The additive operation of ... |
| ipsmulr 17257 | The multiplicative operati... |
| ipssca 17258 | The set of scalars of a co... |
| ipsvsca 17259 | The scalar product operati... |
| ipsip 17260 | The multiplicative operati... |
| resssca 17261 | ` Scalar ` is unaffected b... |
| ressvsca 17262 | ` .s ` is unaffected by re... |
| ressip 17263 | The inner product is unaff... |
| phlstr 17264 | A constructed pre-Hilbert ... |
| phlbase 17265 | The base set of a construc... |
| phlplusg 17266 | The additive operation of ... |
| phlsca 17267 | The ring of scalars of a c... |
| phlvsca 17268 | The scalar product operati... |
| phlip 17269 | The inner product (Hermiti... |
| tsetndx 17270 | Index value of the ~ df-ts... |
| tsetid 17271 | Utility theorem: index-ind... |
| tsetndxnn 17272 | The index of the slot for ... |
| basendxlttsetndx 17273 | The index of the slot for ... |
| tsetndxnbasendx 17274 | The slot for the topology ... |
| tsetndxnplusgndx 17275 | The slot for the topology ... |
| tsetndxnmulrndx 17276 | The slot for the topology ... |
| tsetndxnstarvndx 17277 | The slot for the topology ... |
| slotstnscsi 17278 | The slots ` Scalar ` , ` .... |
| topgrpstr 17279 | A constructed topological ... |
| topgrpbas 17280 | The base set of a construc... |
| topgrpplusg 17281 | The additive operation of ... |
| topgrptset 17282 | The topology of a construc... |
| resstset 17283 | ` TopSet ` is unaffected b... |
| plendx 17284 | Index value of the ~ df-pl... |
| pleid 17285 | Utility theorem: self-refe... |
| plendxnn 17286 | The index value of the ord... |
| basendxltplendx 17287 | The index value of the ` B... |
| plendxnbasendx 17288 | The slot for the order is ... |
| plendxnplusgndx 17289 | The slot for the "less tha... |
| plendxnmulrndx 17290 | The slot for the "less tha... |
| plendxnscandx 17291 | The slot for the "less tha... |
| plendxnvscandx 17292 | The slot for the "less tha... |
| slotsdifplendx 17293 | The index of the slot for ... |
| otpsstr 17294 | Functionality of a topolog... |
| otpsbas 17295 | The base set of a topologi... |
| otpstset 17296 | The open sets of a topolog... |
| otpsle 17297 | The order of a topological... |
| ressle 17298 | ` le ` is unaffected by re... |
| ocndx 17299 | Index value of the ~ df-oc... |
| ocid 17300 | Utility theorem: index-ind... |
| basendxnocndx 17301 | The slot for the orthocomp... |
| plendxnocndx 17302 | The slot for the orthocomp... |
| dsndx 17303 | Index value of the ~ df-ds... |
| dsid 17304 | Utility theorem: index-ind... |
| dsndxnn 17305 | The index of the slot for ... |
| basendxltdsndx 17306 | The index of the slot for ... |
| dsndxnbasendx 17307 | The slot for the distance ... |
| dsndxnplusgndx 17308 | The slot for the distance ... |
| dsndxnmulrndx 17309 | The slot for the distance ... |
| slotsdnscsi 17310 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17311 | The slot for the distance ... |
| slotsdifdsndx 17312 | The index of the slot for ... |
| unifndx 17313 | Index value of the ~ df-un... |
| unifid 17314 | Utility theorem: index-ind... |
| unifndxnn 17315 | The index of the slot for ... |
| basendxltunifndx 17316 | The index of the slot for ... |
| unifndxnbasendx 17317 | The slot for the uniform s... |
| unifndxntsetndx 17318 | The slot for the uniform s... |
| slotsdifunifndx 17319 | The index of the slot for ... |
| ressunif 17320 | ` UnifSet ` is unaffected ... |
| odrngstr 17321 | Functionality of an ordere... |
| odrngbas 17322 | The base set of an ordered... |
| odrngplusg 17323 | The addition operation of ... |
| odrngmulr 17324 | The multiplication operati... |
| odrngtset 17325 | The open sets of an ordere... |
| odrngle 17326 | The order of an ordered me... |
| odrngds 17327 | The metric of an ordered m... |
| ressds 17328 | ` dist ` is unaffected by ... |
| homndx 17329 | Index value of the ~ df-ho... |
| homid 17330 | Utility theorem: index-ind... |
| ccondx 17331 | Index value of the ~ df-cc... |
| ccoid 17332 | Utility theorem: index-ind... |
| slotsbhcdif 17333 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17334 | The index of the slot for ... |
| slotsdifocndx 17335 | The index of the slot for ... |
| resshom 17336 | ` Hom ` is unaffected by r... |
| ressco 17337 | ` comp ` is unaffected by ... |
| restfn 17342 | The subspace topology oper... |
| topnfn 17343 | The topology extractor fun... |
| restval 17344 | The subspace topology indu... |
| elrest 17345 | The predicate "is an open ... |
| elrestr 17346 | Sufficient condition for b... |
| 0rest 17347 | Value of the structure res... |
| restid2 17348 | The subspace topology over... |
| restsspw 17349 | The subspace topology is a... |
| firest 17350 | The finite intersections o... |
| restid 17351 | The subspace topology of t... |
| topnval 17352 | Value of the topology extr... |
| topnid 17353 | Value of the topology extr... |
| topnpropd 17354 | The topology extractor fun... |
| reldmprds 17366 | The structure product is a... |
| prdsbasex 17368 | Lemma for structure produc... |
| imasvalstr 17369 | An image structure value i... |
| prdsvalstr 17370 | Structure product value is... |
| prdsbaslem 17371 | Lemma for ~ prdsbas and si... |
| prdsvallem 17372 | Lemma for ~ prdsval . (Co... |
| prdsval 17373 | Value of the structure pro... |
| prdssca 17374 | Scalar ring of a structure... |
| prdsbas 17375 | Base set of a structure pr... |
| prdsplusg 17376 | Addition in a structure pr... |
| prdsmulr 17377 | Multiplication in a struct... |
| prdsvsca 17378 | Scalar multiplication in a... |
| prdsip 17379 | Inner product in a structu... |
| prdsle 17380 | Structure product weak ord... |
| prdsless 17381 | Closure of the order relat... |
| prdsds 17382 | Structure product distance... |
| prdsdsfn 17383 | Structure product distance... |
| prdstset 17384 | Structure product topology... |
| prdshom 17385 | Structure product hom-sets... |
| prdsco 17386 | Structure product composit... |
| prdsbas2 17387 | The base set of a structur... |
| prdsbasmpt 17388 | A constructed tuple is a p... |
| prdsbasfn 17389 | Points in the structure pr... |
| prdsbasprj 17390 | Each point in a structure ... |
| prdsplusgval 17391 | Value of a componentwise s... |
| prdsplusgfval 17392 | Value of a structure produ... |
| prdsmulrval 17393 | Value of a componentwise r... |
| prdsmulrfval 17394 | Value of a structure produ... |
| prdsleval 17395 | Value of the product order... |
| prdsdsval 17396 | Value of the metric in a s... |
| prdsvscaval 17397 | Scalar multiplication in a... |
| prdsvscafval 17398 | Scalar multiplication of a... |
| prdsbas3 17399 | The base set of an indexed... |
| prdsbasmpt2 17400 | A constructed tuple is a p... |
| prdsbascl 17401 | An element of the base has... |
| prdsdsval2 17402 | Value of the metric in a s... |
| prdsdsval3 17403 | Value of the metric in a s... |
| pwsval 17404 | Value of a structure power... |
| pwsbas 17405 | Base set of a structure po... |
| pwselbasb 17406 | Membership in the base set... |
| pwselbas 17407 | An element of a structure ... |
| pwselbasr 17408 | The reverse direction of ~... |
| pwsplusgval 17409 | Value of addition in a str... |
| pwsmulrval 17410 | Value of multiplication in... |
| pwsle 17411 | Ordering in a structure po... |
| pwsleval 17412 | Ordering in a structure po... |
| pwsvscafval 17413 | Scalar multiplication in a... |
| pwsvscaval 17414 | Scalar multiplication of a... |
| pwssca 17415 | The ring of scalars of a s... |
| pwsdiagel 17416 | Membership of diagonal ele... |
| pwssnf1o 17417 | Triviality of singleton po... |
| imasval 17430 | Value of an image structur... |
| imasbas 17431 | The base set of an image s... |
| imasds 17432 | The distance function of a... |
| imasdsfn 17433 | The distance function is a... |
| imasdsval 17434 | The distance function of a... |
| imasdsval2 17435 | The distance function of a... |
| imasplusg 17436 | The group operation in an ... |
| imasmulr 17437 | The ring multiplication in... |
| imassca 17438 | The scalar field of an ima... |
| imasvsca 17439 | The scalar multiplication ... |
| imasip 17440 | The inner product of an im... |
| imastset 17441 | The topology of an image s... |
| imasle 17442 | The ordering of an image s... |
| f1ocpbllem 17443 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17444 | An injection is compatible... |
| f1ovscpbl 17445 | An injection is compatible... |
| f1olecpbl 17446 | An injection is compatible... |
| imasaddfnlem 17447 | The image structure operat... |
| imasaddvallem 17448 | The operation of an image ... |
| imasaddflem 17449 | The image set operations a... |
| imasaddfn 17450 | The image structure's grou... |
| imasaddval 17451 | The value of an image stru... |
| imasaddf 17452 | The image structure's grou... |
| imasmulfn 17453 | The image structure's ring... |
| imasmulval 17454 | The value of an image stru... |
| imasmulf 17455 | The image structure's ring... |
| imasvscafn 17456 | The image structure's scal... |
| imasvscaval 17457 | The value of an image stru... |
| imasvscaf 17458 | The image structure's scal... |
| imasless 17459 | The order relation defined... |
| imasleval 17460 | The value of the image str... |
| qusval 17461 | Value of a quotient struct... |
| quslem 17462 | The function in ~ qusval i... |
| qusin 17463 | Restrict the equivalence r... |
| qusbas 17464 | Base set of a quotient str... |
| quss 17465 | The scalar field of a quot... |
| divsfval 17466 | Value of the function in ~... |
| ercpbllem 17467 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17468 | Translate the function com... |
| erlecpbl 17469 | Translate the relation com... |
| qusaddvallem 17470 | Value of an operation defi... |
| qusaddflem 17471 | The operation of a quotien... |
| qusaddval 17472 | The addition in a quotient... |
| qusaddf 17473 | The addition in a quotient... |
| qusmulval 17474 | The multiplication in a qu... |
| qusmulf 17475 | The multiplication in a qu... |
| fnpr2o 17476 | Function with a domain of ... |
| fnpr2ob 17477 | Biconditional version of ~... |
| fvpr0o 17478 | The value of a function wi... |
| fvpr1o 17479 | The value of a function wi... |
| fvprif 17480 | The value of the pair func... |
| xpsfrnel 17481 | Elementhood in the target ... |
| xpsfeq 17482 | A function on ` 2o ` is de... |
| xpsfrnel2 17483 | Elementhood in the target ... |
| xpscf 17484 | Equivalent condition for t... |
| xpsfval 17485 | The value of the function ... |
| xpsff1o 17486 | The function appearing in ... |
| xpsfrn 17487 | A short expression for the... |
| xpsff1o2 17488 | The function appearing in ... |
| xpsval 17489 | Value of the binary struct... |
| xpsrnbas 17490 | The indexed structure prod... |
| xpsbas 17491 | The base set of the binary... |
| xpsaddlem 17492 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17493 | Value of the addition oper... |
| xpsmul 17494 | Value of the multiplicatio... |
| xpssca 17495 | Value of the scalar field ... |
| xpsvsca 17496 | Value of the scalar multip... |
| xpsless 17497 | Closure of the ordering in... |
| xpsle 17498 | Value of the ordering in a... |
| ismre 17507 | Property of being a Moore ... |
| fnmre 17508 | The Moore collection gener... |
| mresspw 17509 | A Moore collection is a su... |
| mress 17510 | A Moore-closed subset is a... |
| mre1cl 17511 | In any Moore collection th... |
| mreintcl 17512 | A nonempty collection of c... |
| mreiincl 17513 | A nonempty indexed interse... |
| mrerintcl 17514 | The relative intersection ... |
| mreriincl 17515 | The relative intersection ... |
| mreincl 17516 | Two closed sets have a clo... |
| mreuni 17517 | Since the entire base set ... |
| mreunirn 17518 | Two ways to express the no... |
| ismred 17519 | Properties that determine ... |
| ismred2 17520 | Properties that determine ... |
| mremre 17521 | The Moore collections of s... |
| submre 17522 | The subcollection of a clo... |
| xrsle 17523 | The ordering of the extend... |
| xrge0le 17524 | The "less than or equal to... |
| xrsbas 17525 | The base set of the extend... |
| xrge0base 17526 | The base of the extended n... |
| mrcflem 17527 | The domain and codomain of... |
| fnmrc 17528 | Moore-closure is a well-be... |
| mrcfval 17529 | Value of the function expr... |
| mrcf 17530 | The Moore closure is a fun... |
| mrcval 17531 | Evaluation of the Moore cl... |
| mrccl 17532 | The Moore closure of a set... |
| mrcsncl 17533 | The Moore closure of a sin... |
| mrcid 17534 | The closure of a closed se... |
| mrcssv 17535 | The closure of a set is a ... |
| mrcidb 17536 | A set is closed iff it is ... |
| mrcss 17537 | Closure preserves subset o... |
| mrcssid 17538 | The closure of a set is a ... |
| mrcidb2 17539 | A set is closed iff it con... |
| mrcidm 17540 | The closure operation is i... |
| mrcsscl 17541 | The closure is the minimal... |
| mrcuni 17542 | Idempotence of closure und... |
| mrcun 17543 | Idempotence of closure und... |
| mrcssvd 17544 | The Moore closure of a set... |
| mrcssd 17545 | Moore closure preserves su... |
| mrcssidd 17546 | A set is contained in its ... |
| mrcidmd 17547 | Moore closure is idempoten... |
| mressmrcd 17548 | In a Moore system, if a se... |
| submrc 17549 | In a closure system which ... |
| mrieqvlemd 17550 | In a Moore system, if ` Y ... |
| mrisval 17551 | Value of the set of indepe... |
| ismri 17552 | Criterion for a set to be ... |
| ismri2 17553 | Criterion for a subset of ... |
| ismri2d 17554 | Criterion for a subset of ... |
| ismri2dd 17555 | Definition of independence... |
| mriss 17556 | An independent set of a Mo... |
| mrissd 17557 | An independent set of a Mo... |
| ismri2dad 17558 | Consequence of a set in a ... |
| mrieqvd 17559 | In a Moore system, a set i... |
| mrieqv2d 17560 | In a Moore system, a set i... |
| mrissmrcd 17561 | In a Moore system, if an i... |
| mrissmrid 17562 | In a Moore system, subsets... |
| mreexd 17563 | In a Moore system, the clo... |
| mreexmrid 17564 | In a Moore system whose cl... |
| mreexexlemd 17565 | This lemma is used to gene... |
| mreexexlem2d 17566 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17567 | Base case of the induction... |
| mreexexlem4d 17568 | Induction step of the indu... |
| mreexexd 17569 | Exchange-type theorem. In... |
| mreexdomd 17570 | In a Moore system whose cl... |
| mreexfidimd 17571 | In a Moore system whose cl... |
| isacs 17572 | A set is an algebraic clos... |
| acsmre 17573 | Algebraic closure systems ... |
| isacs2 17574 | In the definition of an al... |
| acsfiel 17575 | A set is closed in an alge... |
| acsfiel2 17576 | A set is closed in an alge... |
| acsmred 17577 | An algebraic closure syste... |
| isacs1i 17578 | A closure system determine... |
| mreacs 17579 | Algebraicity is a composab... |
| acsfn 17580 | Algebraicity of a conditio... |
| acsfn0 17581 | Algebraicity of a point cl... |
| acsfn1 17582 | Algebraicity of a one-argu... |
| acsfn1c 17583 | Algebraicity of a one-argu... |
| acsfn2 17584 | Algebraicity of a two-argu... |
| iscat 17593 | The predicate "is a catego... |
| iscatd 17594 | Properties that determine ... |
| catidex 17595 | Each object in a category ... |
| catideu 17596 | Each object in a category ... |
| cidfval 17597 | Each object in a category ... |
| cidval 17598 | Each object in a category ... |
| cidffn 17599 | The identity arrow constru... |
| cidfn 17600 | The identity arrow operato... |
| catidd 17601 | Deduce the identity arrow ... |
| iscatd2 17602 | Version of ~ iscatd with a... |
| catidcl 17603 | Each object in a category ... |
| catlid 17604 | Left identity property of ... |
| catrid 17605 | Right identity property of... |
| catcocl 17606 | Closure of a composition a... |
| catass 17607 | Associativity of compositi... |
| catcone0 17608 | Composition of non-empty h... |
| 0catg 17609 | Any structure with an empt... |
| 0cat 17610 | The empty set is a categor... |
| homffval 17611 | Value of the functionalize... |
| fnhomeqhomf 17612 | If the Hom-set operation i... |
| homfval 17613 | Value of the functionalize... |
| homffn 17614 | The functionalized Hom-set... |
| homfeq 17615 | Condition for two categori... |
| homfeqd 17616 | If two structures have the... |
| homfeqbas 17617 | Deduce equality of base se... |
| homfeqval 17618 | Value of the functionalize... |
| comfffval 17619 | Value of the functionalize... |
| comffval 17620 | Value of the functionalize... |
| comfval 17621 | Value of the functionalize... |
| comfffval2 17622 | Value of the functionalize... |
| comffval2 17623 | Value of the functionalize... |
| comfval2 17624 | Value of the functionalize... |
| comfffn 17625 | The functionalized composi... |
| comffn 17626 | The functionalized composi... |
| comfeq 17627 | Condition for two categori... |
| comfeqd 17628 | Condition for two categori... |
| comfeqval 17629 | Equality of two compositio... |
| catpropd 17630 | Two structures with the sa... |
| cidpropd 17631 | Two structures with the sa... |
| oppcval 17634 | Value of the opposite cate... |
| oppchomfval 17635 | Hom-sets of the opposite c... |
| oppchom 17636 | Hom-sets of the opposite c... |
| oppccofval 17637 | Composition in the opposit... |
| oppcco 17638 | Composition in the opposit... |
| oppcbas 17639 | Base set of an opposite ca... |
| oppccatid 17640 | Lemma for ~ oppccat . (Co... |
| oppchomf 17641 | Hom-sets of the opposite c... |
| oppcid 17642 | Identity function of an op... |
| oppccat 17643 | An opposite category is a ... |
| 2oppcbas 17644 | The double opposite catego... |
| 2oppchomf 17645 | The double opposite catego... |
| 2oppccomf 17646 | The double opposite catego... |
| oppchomfpropd 17647 | If two categories have the... |
| oppccomfpropd 17648 | If two categories have the... |
| oppccatf 17649 | ` oppCat ` restricted to `... |
| monfval 17654 | Definition of a monomorphi... |
| ismon 17655 | Definition of a monomorphi... |
| ismon2 17656 | Write out the monomorphism... |
| monhom 17657 | A monomorphism is a morphi... |
| moni 17658 | Property of a monomorphism... |
| monpropd 17659 | If two categories have the... |
| oppcmon 17660 | A monomorphism in the oppo... |
| oppcepi 17661 | An epimorphism in the oppo... |
| isepi 17662 | Definition of an epimorphi... |
| isepi2 17663 | Write out the epimorphism ... |
| epihom 17664 | An epimorphism is a morphi... |
| epii 17665 | Property of an epimorphism... |
| sectffval 17672 | Value of the section opera... |
| sectfval 17673 | Value of the section relat... |
| sectss 17674 | The section relation is a ... |
| issect 17675 | The property " ` F ` is a ... |
| issect2 17676 | Property of being a sectio... |
| sectcan 17677 | If ` G ` is a section of `... |
| sectco 17678 | Composition of two section... |
| isofval 17679 | Function value of the func... |
| invffval 17680 | Value of the inverse relat... |
| invfval 17681 | Value of the inverse relat... |
| isinv 17682 | Value of the inverse relat... |
| invss 17683 | The inverse relation is a ... |
| invsym 17684 | The inverse relation is sy... |
| invsym2 17685 | The inverse relation is sy... |
| invfun 17686 | The inverse relation is a ... |
| isoval 17687 | The isomorphisms are the d... |
| inviso1 17688 | If ` G ` is an inverse to ... |
| inviso2 17689 | If ` G ` is an inverse to ... |
| invf 17690 | The inverse relation is a ... |
| invf1o 17691 | The inverse relation is a ... |
| invinv 17692 | The inverse of the inverse... |
| invco 17693 | The composition of two iso... |
| dfiso2 17694 | Alternate definition of an... |
| dfiso3 17695 | Alternate definition of an... |
| inveq 17696 | If there are two inverses ... |
| isofn 17697 | The function value of the ... |
| isohom 17698 | An isomorphism is a homomo... |
| isoco 17699 | The composition of two iso... |
| oppcsect 17700 | A section in the opposite ... |
| oppcsect2 17701 | A section in the opposite ... |
| oppcinv 17702 | An inverse in the opposite... |
| oppciso 17703 | An isomorphism in the oppo... |
| sectmon 17704 | If ` F ` is a section of `... |
| monsect 17705 | If ` F ` is a monomorphism... |
| sectepi 17706 | If ` F ` is a section of `... |
| episect 17707 | If ` F ` is an epimorphism... |
| sectid 17708 | The identity is a section ... |
| invid 17709 | The inverse of the identit... |
| idiso 17710 | The identity is an isomorp... |
| idinv 17711 | The inverse of the identit... |
| invisoinvl 17712 | The inverse of an isomorph... |
| invisoinvr 17713 | The inverse of an isomorph... |
| invcoisoid 17714 | The inverse of an isomorph... |
| isocoinvid 17715 | The inverse of an isomorph... |
| rcaninv 17716 | Right cancellation of an i... |
| cicfval 17719 | The set of isomorphic obje... |
| brcic 17720 | The relation "is isomorphi... |
| cic 17721 | Objects ` X ` and ` Y ` in... |
| brcici 17722 | Prove that two objects are... |
| cicref 17723 | Isomorphism is reflexive. ... |
| ciclcl 17724 | Isomorphism implies the le... |
| cicrcl 17725 | Isomorphism implies the ri... |
| cicsym 17726 | Isomorphism is symmetric. ... |
| cictr 17727 | Isomorphism is transitive.... |
| cicer 17728 | Isomorphism is an equivale... |
| sscrel 17735 | The subcategory subset rel... |
| brssc 17736 | The subcategory subset rel... |
| sscpwex 17737 | An analogue of ~ pwex for ... |
| subcrcl 17738 | Reverse closure for the su... |
| sscfn1 17739 | The subcategory subset rel... |
| sscfn2 17740 | The subcategory subset rel... |
| ssclem 17741 | Lemma for ~ ssc1 and simil... |
| isssc 17742 | Value of the subcategory s... |
| ssc1 17743 | Infer subset relation on o... |
| ssc2 17744 | Infer subset relation on m... |
| sscres 17745 | Any function restricted to... |
| sscid 17746 | The subcategory subset rel... |
| ssctr 17747 | The subcategory subset rel... |
| ssceq 17748 | The subcategory subset rel... |
| rescval 17749 | Value of the category rest... |
| rescval2 17750 | Value of the category rest... |
| rescbas 17751 | Base set of the category r... |
| reschom 17752 | Hom-sets of the category r... |
| reschomf 17753 | Hom-sets of the category r... |
| rescco 17754 | Composition in the categor... |
| rescabs 17755 | Restriction absorption law... |
| rescabs2 17756 | Restriction absorption law... |
| issubc 17757 | Elementhood in the set of ... |
| issubc2 17758 | Elementhood in the set of ... |
| 0ssc 17759 | For any category ` C ` , t... |
| 0subcat 17760 | For any category ` C ` , t... |
| catsubcat 17761 | For any category ` C ` , `... |
| subcssc 17762 | An element in the set of s... |
| subcfn 17763 | An element in the set of s... |
| subcss1 17764 | The objects of a subcatego... |
| subcss2 17765 | The morphisms of a subcate... |
| subcidcl 17766 | The identity of the origin... |
| subccocl 17767 | A subcategory is closed un... |
| subccatid 17768 | A subcategory is a categor... |
| subcid 17769 | The identity in a subcateg... |
| subccat 17770 | A subcategory is a categor... |
| issubc3 17771 | Alternate definition of a ... |
| fullsubc 17772 | The full subcategory gener... |
| fullresc 17773 | The category formed by str... |
| resscat 17774 | A category restricted to a... |
| subsubc 17775 | A subcategory of a subcate... |
| relfunc 17784 | The set of functors is a r... |
| funcrcl 17785 | Reverse closure for a func... |
| isfunc 17786 | Value of the set of functo... |
| isfuncd 17787 | Deduce that an operation i... |
| funcf1 17788 | The object part of a funct... |
| funcixp 17789 | The morphism part of a fun... |
| funcf2 17790 | The morphism part of a fun... |
| funcfn2 17791 | The morphism part of a fun... |
| funcid 17792 | A functor maps each identi... |
| funcco 17793 | A functor maps composition... |
| funcsect 17794 | The image of a section und... |
| funcinv 17795 | The image of an inverse un... |
| funciso 17796 | The image of an isomorphis... |
| funcoppc 17797 | A functor on categories yi... |
| idfuval 17798 | Value of the identity func... |
| idfu2nd 17799 | Value of the morphism part... |
| idfu2 17800 | Value of the morphism part... |
| idfu1st 17801 | Value of the object part o... |
| idfu1 17802 | Value of the object part o... |
| idfucl 17803 | The identity functor is a ... |
| cofuval 17804 | Value of the composition o... |
| cofu1st 17805 | Value of the object part o... |
| cofu1 17806 | Value of the object part o... |
| cofu2nd 17807 | Value of the morphism part... |
| cofu2 17808 | Value of the morphism part... |
| cofuval2 17809 | Value of the composition o... |
| cofucl 17810 | The composition of two fun... |
| cofuass 17811 | Functor composition is ass... |
| cofulid 17812 | The identity functor is a ... |
| cofurid 17813 | The identity functor is a ... |
| resfval 17814 | Value of the functor restr... |
| resfval2 17815 | Value of the functor restr... |
| resf1st 17816 | Value of the functor restr... |
| resf2nd 17817 | Value of the functor restr... |
| funcres 17818 | A functor restricted to a ... |
| funcres2b 17819 | Condition for a functor to... |
| funcres2 17820 | A functor into a restricte... |
| idfusubc0 17821 | The identity functor for a... |
| idfusubc 17822 | The identity functor for a... |
| wunfunc 17823 | A weak universe is closed ... |
| funcpropd 17824 | If two categories have the... |
| funcres2c 17825 | Condition for a functor to... |
| fullfunc 17830 | A full functor is a functo... |
| fthfunc 17831 | A faithful functor is a fu... |
| relfull 17832 | The set of full functors i... |
| relfth 17833 | The set of faithful functo... |
| isfull 17834 | Value of the set of full f... |
| isfull2 17835 | Equivalent condition for a... |
| fullfo 17836 | The morphism map of a full... |
| fulli 17837 | The morphism map of a full... |
| isfth 17838 | Value of the set of faithf... |
| isfth2 17839 | Equivalent condition for a... |
| isffth2 17840 | A fully faithful functor i... |
| fthf1 17841 | The morphism map of a fait... |
| fthi 17842 | The morphism map of a fait... |
| ffthf1o 17843 | The morphism map of a full... |
| fullpropd 17844 | If two categories have the... |
| fthpropd 17845 | If two categories have the... |
| fulloppc 17846 | The opposite functor of a ... |
| fthoppc 17847 | The opposite functor of a ... |
| ffthoppc 17848 | The opposite functor of a ... |
| fthsect 17849 | A faithful functor reflect... |
| fthinv 17850 | A faithful functor reflect... |
| fthmon 17851 | A faithful functor reflect... |
| fthepi 17852 | A faithful functor reflect... |
| ffthiso 17853 | A fully faithful functor r... |
| fthres2b 17854 | Condition for a faithful f... |
| fthres2c 17855 | Condition for a faithful f... |
| fthres2 17856 | A faithful functor into a ... |
| idffth 17857 | The identity functor is a ... |
| cofull 17858 | The composition of two ful... |
| cofth 17859 | The composition of two fai... |
| coffth 17860 | The composition of two ful... |
| rescfth 17861 | The inclusion functor from... |
| ressffth 17862 | The inclusion functor from... |
| fullres2c 17863 | Condition for a full funct... |
| ffthres2c 17864 | Condition for a fully fait... |
| inclfusubc 17865 | The "inclusion functor" fr... |
| fnfuc 17870 | The ` FuncCat ` operation ... |
| natfval 17871 | Value of the function givi... |
| isnat 17872 | Property of being a natura... |
| isnat2 17873 | Property of being a natura... |
| natffn 17874 | The natural transformation... |
| natrcl 17875 | Reverse closure for a natu... |
| nat1st2nd 17876 | Rewrite the natural transf... |
| natixp 17877 | A natural transformation i... |
| natcl 17878 | A component of a natural t... |
| natfn 17879 | A natural transformation i... |
| nati 17880 | Naturality property of a n... |
| wunnat 17881 | A weak universe is closed ... |
| catstr 17882 | A category structure is a ... |
| fucval 17883 | Value of the functor categ... |
| fuccofval 17884 | Value of the functor categ... |
| fucbas 17885 | The objects of the functor... |
| fuchom 17886 | The morphisms in the funct... |
| fucco 17887 | Value of the composition o... |
| fuccoval 17888 | Value of the functor categ... |
| fuccocl 17889 | The composition of two nat... |
| fucidcl 17890 | The identity natural trans... |
| fuclid 17891 | Left identity of natural t... |
| fucrid 17892 | Right identity of natural ... |
| fucass 17893 | Associativity of natural t... |
| fuccatid 17894 | The functor category is a ... |
| fuccat 17895 | The functor category is a ... |
| fucid 17896 | The identity morphism in t... |
| fucsect 17897 | Two natural transformation... |
| fucinv 17898 | Two natural transformation... |
| invfuc 17899 | If ` V ( x ) ` is an inver... |
| fuciso 17900 | A natural transformation i... |
| natpropd 17901 | If two categories have the... |
| fucpropd 17902 | If two categories have the... |
| initofn 17909 | ` InitO ` is a function on... |
| termofn 17910 | ` TermO ` is a function on... |
| zeroofn 17911 | ` ZeroO ` is a function on... |
| initorcl 17912 | Reverse closure for an ini... |
| termorcl 17913 | Reverse closure for a term... |
| zeroorcl 17914 | Reverse closure for a zero... |
| initoval 17915 | The value of the initial o... |
| termoval 17916 | The value of the terminal ... |
| zerooval 17917 | The value of the zero obje... |
| isinito 17918 | The predicate "is an initi... |
| istermo 17919 | The predicate "is a termin... |
| iszeroo 17920 | The predicate "is a zero o... |
| isinitoi 17921 | Implication of a class bei... |
| istermoi 17922 | Implication of a class bei... |
| initoid 17923 | For an initial object, the... |
| termoid 17924 | For a terminal object, the... |
| dfinito2 17925 | An initial object is a ter... |
| dftermo2 17926 | A terminal object is an in... |
| dfinito3 17927 | An alternate definition of... |
| dftermo3 17928 | An alternate definition of... |
| initoo 17929 | An initial object is an ob... |
| termoo 17930 | A terminal object is an ob... |
| iszeroi 17931 | Implication of a class bei... |
| 2initoinv 17932 | Morphisms between two init... |
| initoeu1 17933 | Initial objects are essent... |
| initoeu1w 17934 | Initial objects are essent... |
| initoeu2lem0 17935 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17936 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17937 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17938 | Initial objects are essent... |
| 2termoinv 17939 | Morphisms between two term... |
| termoeu1 17940 | Terminal objects are essen... |
| termoeu1w 17941 | Terminal objects are essen... |
| homarcl 17950 | Reverse closure for an arr... |
| homafval 17951 | Value of the disjointified... |
| homaf 17952 | Functionality of the disjo... |
| homaval 17953 | Value of the disjointified... |
| elhoma 17954 | Value of the disjointified... |
| elhomai 17955 | Produce an arrow from a mo... |
| elhomai2 17956 | Produce an arrow from a mo... |
| homarcl2 17957 | Reverse closure for the do... |
| homarel 17958 | An arrow is an ordered pai... |
| homa1 17959 | The first component of an ... |
| homahom2 17960 | The second component of an... |
| homahom 17961 | The second component of an... |
| homadm 17962 | The domain of an arrow wit... |
| homacd 17963 | The codomain of an arrow w... |
| homadmcd 17964 | Decompose an arrow into do... |
| arwval 17965 | The set of arrows is the u... |
| arwrcl 17966 | The first component of an ... |
| arwhoma 17967 | An arrow is contained in t... |
| homarw 17968 | A hom-set is a subset of t... |
| arwdm 17969 | The domain of an arrow is ... |
| arwcd 17970 | The codomain of an arrow i... |
| dmaf 17971 | The domain function is a f... |
| cdaf 17972 | The codomain function is a... |
| arwhom 17973 | The second component of an... |
| arwdmcd 17974 | Decompose an arrow into do... |
| idafval 17979 | Value of the identity arro... |
| idaval 17980 | Value of the identity arro... |
| ida2 17981 | Morphism part of the ident... |
| idahom 17982 | Domain and codomain of the... |
| idadm 17983 | Domain of the identity arr... |
| idacd 17984 | Codomain of the identity a... |
| idaf 17985 | The identity arrow functio... |
| coafval 17986 | The value of the compositi... |
| eldmcoa 17987 | A pair ` <. G , F >. ` is ... |
| dmcoass 17988 | The domain of composition ... |
| homdmcoa 17989 | If ` F : X --> Y ` and ` G... |
| coaval 17990 | Value of composition for c... |
| coa2 17991 | The morphism part of arrow... |
| coahom 17992 | The composition of two com... |
| coapm 17993 | Composition of arrows is a... |
| arwlid 17994 | Left identity of a categor... |
| arwrid 17995 | Right identity of a catego... |
| arwass 17996 | Associativity of compositi... |
| setcval 17999 | Value of the category of s... |
| setcbas 18000 | Set of objects of the cate... |
| setchomfval 18001 | Set of arrows of the categ... |
| setchom 18002 | Set of arrows of the categ... |
| elsetchom 18003 | A morphism of sets is a fu... |
| setccofval 18004 | Composition in the categor... |
| setcco 18005 | Composition in the categor... |
| setccatid 18006 | Lemma for ~ setccat . (Co... |
| setccat 18007 | The category of sets is a ... |
| setcid 18008 | The identity arrow in the ... |
| setcmon 18009 | A monomorphism of sets is ... |
| setcepi 18010 | An epimorphism of sets is ... |
| setcsect 18011 | A section in the category ... |
| setcinv 18012 | An inverse in the category... |
| setciso 18013 | An isomorphism in the cate... |
| resssetc 18014 | The restriction of the cat... |
| funcsetcres2 18015 | A functor into a smaller c... |
| setc2obas 18016 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18017 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18018 | The category of sets in a ... |
| cat1 18019 | The definition of category... |
| catcval 18022 | Value of the category of c... |
| catcbas 18023 | Set of objects of the cate... |
| catchomfval 18024 | Set of arrows of the categ... |
| catchom 18025 | Set of arrows of the categ... |
| catccofval 18026 | Composition in the categor... |
| catcco 18027 | Composition in the categor... |
| catccatid 18028 | Lemma for ~ catccat . (Co... |
| catcid 18029 | The identity arrow in the ... |
| catccat 18030 | The category of categories... |
| resscatc 18031 | The restriction of the cat... |
| catcisolem 18032 | Lemma for ~ catciso . (Co... |
| catciso 18033 | A functor is an isomorphis... |
| catcbascl 18034 | An element of the base set... |
| catcslotelcl 18035 | A slot entry of an element... |
| catcbaselcl 18036 | The base set of an element... |
| catchomcl 18037 | The Hom-set of an element ... |
| catcccocl 18038 | The composition operation ... |
| catcoppccl 18039 | The category of categories... |
| catcfuccl 18040 | The category of categories... |
| fncnvimaeqv 18041 | The inverse images of the ... |
| bascnvimaeqv 18042 | The inverse image of the u... |
| estrcval 18045 | Value of the category of e... |
| estrcbas 18046 | Set of objects of the cate... |
| estrchomfval 18047 | Set of morphisms ("arrows"... |
| estrchom 18048 | The morphisms between exte... |
| elestrchom 18049 | A morphism between extensi... |
| estrccofval 18050 | Composition in the categor... |
| estrcco 18051 | Composition in the categor... |
| estrcbasbas 18052 | An element of the base set... |
| estrccatid 18053 | Lemma for ~ estrccat . (C... |
| estrccat 18054 | The category of extensible... |
| estrcid 18055 | The identity arrow in the ... |
| estrchomfn 18056 | The Hom-set operation in t... |
| estrchomfeqhom 18057 | The functionalized Hom-set... |
| estrreslem1 18058 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18059 | Lemma 2 for ~ estrres . (... |
| estrres 18060 | Any restriction of a categ... |
| funcestrcsetclem1 18061 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18062 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18063 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18064 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18065 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18066 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18067 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18068 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18069 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18070 | The "natural forgetful fun... |
| fthestrcsetc 18071 | The "natural forgetful fun... |
| fullestrcsetc 18072 | The "natural forgetful fun... |
| equivestrcsetc 18073 | The "natural forgetful fun... |
| setc1strwun 18074 | A constructed one-slot str... |
| funcsetcestrclem1 18075 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18076 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18077 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18078 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18079 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18080 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18081 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18082 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18083 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18084 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18085 | The "embedding functor" fr... |
| fthsetcestrc 18086 | The "embedding functor" fr... |
| fullsetcestrc 18087 | The "embedding functor" fr... |
| embedsetcestrc 18088 | The "embedding functor" fr... |
| fnxpc 18097 | The binary product of cate... |
| xpcval 18098 | Value of the binary produc... |
| xpcbas 18099 | Set of objects of the bina... |
| xpchomfval 18100 | Set of morphisms of the bi... |
| xpchom 18101 | Set of morphisms of the bi... |
| relxpchom 18102 | A hom-set in the binary pr... |
| xpccofval 18103 | Value of composition in th... |
| xpcco 18104 | Value of composition in th... |
| xpcco1st 18105 | Value of composition in th... |
| xpcco2nd 18106 | Value of composition in th... |
| xpchom2 18107 | Value of the set of morphi... |
| xpcco2 18108 | Value of composition in th... |
| xpccatid 18109 | The product of two categor... |
| xpcid 18110 | The identity morphism in t... |
| xpccat 18111 | The product of two categor... |
| 1stfval 18112 | Value of the first project... |
| 1stf1 18113 | Value of the first project... |
| 1stf2 18114 | Value of the first project... |
| 2ndfval 18115 | Value of the first project... |
| 2ndf1 18116 | Value of the first project... |
| 2ndf2 18117 | Value of the first project... |
| 1stfcl 18118 | The first projection funct... |
| 2ndfcl 18119 | The second projection func... |
| prfval 18120 | Value of the pairing funct... |
| prf1 18121 | Value of the pairing funct... |
| prf2fval 18122 | Value of the pairing funct... |
| prf2 18123 | Value of the pairing funct... |
| prfcl 18124 | The pairing of functors ` ... |
| prf1st 18125 | Cancellation of pairing wi... |
| prf2nd 18126 | Cancellation of pairing wi... |
| 1st2ndprf 18127 | Break a functor into a pro... |
| catcxpccl 18128 | The category of categories... |
| xpcpropd 18129 | If two categories have the... |
| evlfval 18138 | Value of the evaluation fu... |
| evlf2 18139 | Value of the evaluation fu... |
| evlf2val 18140 | Value of the evaluation na... |
| evlf1 18141 | Value of the evaluation fu... |
| evlfcllem 18142 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18143 | The evaluation functor is ... |
| curfval 18144 | Value of the curry functor... |
| curf1fval 18145 | Value of the object part o... |
| curf1 18146 | Value of the object part o... |
| curf11 18147 | Value of the double evalua... |
| curf12 18148 | The partially evaluated cu... |
| curf1cl 18149 | The partially evaluated cu... |
| curf2 18150 | Value of the curry functor... |
| curf2val 18151 | Value of a component of th... |
| curf2cl 18152 | The curry functor at a mor... |
| curfcl 18153 | The curry functor of a fun... |
| curfpropd 18154 | If two categories have the... |
| uncfval 18155 | Value of the uncurry funct... |
| uncfcl 18156 | The uncurry operation take... |
| uncf1 18157 | Value of the uncurry funct... |
| uncf2 18158 | Value of the uncurry funct... |
| curfuncf 18159 | Cancellation of curry with... |
| uncfcurf 18160 | Cancellation of uncurry wi... |
| diagval 18161 | Define the diagonal functo... |
| diagcl 18162 | The diagonal functor is a ... |
| diag1cl 18163 | The constant functor of ` ... |
| diag11 18164 | Value of the constant func... |
| diag12 18165 | Value of the constant func... |
| diag2 18166 | Value of the diagonal func... |
| diag2cl 18167 | The diagonal functor at a ... |
| curf2ndf 18168 | As shown in ~ diagval , th... |
| hofval 18173 | Value of the Hom functor, ... |
| hof1fval 18174 | The object part of the Hom... |
| hof1 18175 | The object part of the Hom... |
| hof2fval 18176 | The morphism part of the H... |
| hof2val 18177 | The morphism part of the H... |
| hof2 18178 | The morphism part of the H... |
| hofcllem 18179 | Lemma for ~ hofcl . (Cont... |
| hofcl 18180 | Closure of the Hom functor... |
| oppchofcl 18181 | Closure of the opposite Ho... |
| yonval 18182 | Value of the Yoneda embedd... |
| yoncl 18183 | The Yoneda embedding is a ... |
| yon1cl 18184 | The Yoneda embedding at an... |
| yon11 18185 | Value of the Yoneda embedd... |
| yon12 18186 | Value of the Yoneda embedd... |
| yon2 18187 | Value of the Yoneda embedd... |
| hofpropd 18188 | If two categories have the... |
| yonpropd 18189 | If two categories have the... |
| oppcyon 18190 | Value of the opposite Yone... |
| oyoncl 18191 | The opposite Yoneda embedd... |
| oyon1cl 18192 | The opposite Yoneda embedd... |
| yonedalem1 18193 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18194 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18195 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18196 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18197 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18198 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18199 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18200 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18201 | Lemma for ~ yoneda . (Con... |
| yonedainv 18202 | The Yoneda Lemma with expl... |
| yonffthlem 18203 | Lemma for ~ yonffth . (Co... |
| yoneda 18204 | The Yoneda Lemma. There i... |
| yonffth 18205 | The Yoneda Lemma. The Yon... |
| yoniso 18206 | If the codomain is recover... |
| oduval 18209 | Value of an order dual str... |
| oduleval 18210 | Value of the less-equal re... |
| oduleg 18211 | Truth of the less-equal re... |
| odubas 18212 | Base set of an order dual ... |
| isprs 18217 | Property of being a preord... |
| prslem 18218 | Lemma for ~ prsref and ~ p... |
| prsref 18219 | "Less than or equal to" is... |
| prstr 18220 | "Less than or equal to" is... |
| oduprs 18221 | Being a proset is a self-d... |
| isdrs 18222 | Property of being a direct... |
| drsdir 18223 | Direction of a directed se... |
| drsprs 18224 | A directed set is a proset... |
| drsbn0 18225 | The base of a directed set... |
| drsdirfi 18226 | Any _finite_ number of ele... |
| isdrs2 18227 | Directed sets may be defin... |
| ispos 18235 | The predicate "is a poset"... |
| ispos2 18236 | A poset is an antisymmetri... |
| posprs 18237 | A poset is a proset. (Con... |
| posi 18238 | Lemma for poset properties... |
| posref 18239 | A poset ordering is reflex... |
| posasymb 18240 | A poset ordering is asymme... |
| postr 18241 | A poset ordering is transi... |
| 0pos 18242 | Technical lemma to simplif... |
| isposd 18243 | Properties that determine ... |
| isposi 18244 | Properties that determine ... |
| isposix 18245 | Properties that determine ... |
| pospropd 18246 | Posethood is determined on... |
| odupos 18247 | Being a poset is a self-du... |
| oduposb 18248 | Being a poset is a self-du... |
| pltfval 18250 | Value of the less-than rel... |
| pltval 18251 | Less-than relation. ( ~ d... |
| pltle 18252 | "Less than" implies "less ... |
| pltne 18253 | The "less than" relation i... |
| pltirr 18254 | The "less than" relation i... |
| pleval2i 18255 | One direction of ~ pleval2... |
| pleval2 18256 | "Less than or equal to" in... |
| pltnle 18257 | "Less than" implies not co... |
| pltval3 18258 | Alternate expression for t... |
| pltnlt 18259 | The less-than relation imp... |
| pltn2lp 18260 | The less-than relation has... |
| plttr 18261 | The less-than relation is ... |
| pltletr 18262 | Transitive law for chained... |
| plelttr 18263 | Transitive law for chained... |
| pospo 18264 | Write a poset structure in... |
| lubfval 18269 | Value of the least upper b... |
| lubdm 18270 | Domain of the least upper ... |
| lubfun 18271 | The LUB is a function. (C... |
| lubeldm 18272 | Member of the domain of th... |
| lubelss 18273 | A member of the domain of ... |
| lubeu 18274 | Unique existence proper of... |
| lubval 18275 | Value of the least upper b... |
| lubcl 18276 | The least upper bound func... |
| lubprop 18277 | Properties of greatest low... |
| luble 18278 | The greatest lower bound i... |
| lublecllem 18279 | Lemma for ~ lublecl and ~ ... |
| lublecl 18280 | The set of all elements le... |
| lubid 18281 | The LUB of elements less t... |
| glbfval 18282 | Value of the greatest lowe... |
| glbdm 18283 | Domain of the greatest low... |
| glbfun 18284 | The GLB is a function. (C... |
| glbeldm 18285 | Member of the domain of th... |
| glbelss 18286 | A member of the domain of ... |
| glbeu 18287 | Unique existence proper of... |
| glbval 18288 | Value of the greatest lowe... |
| glbcl 18289 | The least upper bound func... |
| glbprop 18290 | Properties of greatest low... |
| glble 18291 | The greatest lower bound i... |
| joinfval 18292 | Value of join function for... |
| joinfval2 18293 | Value of join function for... |
| joindm 18294 | Domain of join function fo... |
| joindef 18295 | Two ways to say that a joi... |
| joinval 18296 | Join value. Since both si... |
| joincl 18297 | Closure of join of element... |
| joindmss 18298 | Subset property of domain ... |
| joinval2lem 18299 | Lemma for ~ joinval2 and ~... |
| joinval2 18300 | Value of join for a poset ... |
| joineu 18301 | Uniqueness of join of elem... |
| joinlem 18302 | Lemma for join properties.... |
| lejoin1 18303 | A join's first argument is... |
| lejoin2 18304 | A join's second argument i... |
| joinle 18305 | A join is less than or equ... |
| meetfval 18306 | Value of meet function for... |
| meetfval2 18307 | Value of meet function for... |
| meetdm 18308 | Domain of meet function fo... |
| meetdef 18309 | Two ways to say that a mee... |
| meetval 18310 | Meet value. Since both si... |
| meetcl 18311 | Closure of meet of element... |
| meetdmss 18312 | Subset property of domain ... |
| meetval2lem 18313 | Lemma for ~ meetval2 and ~... |
| meetval2 18314 | Value of meet for a poset ... |
| meeteu 18315 | Uniqueness of meet of elem... |
| meetlem 18316 | Lemma for meet properties.... |
| lemeet1 18317 | A meet's first argument is... |
| lemeet2 18318 | A meet's second argument i... |
| meetle 18319 | A meet is less than or equ... |
| joincomALT 18320 | The join of a poset is com... |
| joincom 18321 | The join of a poset is com... |
| meetcomALT 18322 | The meet of a poset is com... |
| meetcom 18323 | The meet of a poset is com... |
| join0 18324 | Lemma for ~ odumeet . (Co... |
| meet0 18325 | Lemma for ~ odujoin . (Co... |
| odulub 18326 | Least upper bounds in a du... |
| odujoin 18327 | Joins in a dual order are ... |
| oduglb 18328 | Greatest lower bounds in a... |
| odumeet 18329 | Meets in a dual order are ... |
| poslubmo 18330 | Least upper bounds in a po... |
| posglbmo 18331 | Greatest lower bounds in a... |
| poslubd 18332 | Properties which determine... |
| poslubdg 18333 | Properties which determine... |
| posglbdg 18334 | Properties which determine... |
| istos 18337 | The predicate "is a toset"... |
| tosso 18338 | Write the totally ordered ... |
| tospos 18339 | A Toset is a Poset. (Cont... |
| tleile 18340 | In a Toset, any two elemen... |
| tltnle 18341 | In a Toset, "less than" is... |
| p0val 18346 | Value of poset zero. (Con... |
| p1val 18347 | Value of poset zero. (Con... |
| p0le 18348 | Any element is less than o... |
| ple1 18349 | Any element is less than o... |
| resspos 18350 | The restriction of a Poset... |
| resstos 18351 | The restriction of a Toset... |
| islat 18354 | The predicate "is a lattic... |
| odulatb 18355 | Being a lattice is self-du... |
| odulat 18356 | Being a lattice is self-du... |
| latcl2 18357 | The join and meet of any t... |
| latlem 18358 | Lemma for lattice properti... |
| latpos 18359 | A lattice is a poset. (Co... |
| latjcl 18360 | Closure of join operation ... |
| latmcl 18361 | Closure of meet operation ... |
| latref 18362 | A lattice ordering is refl... |
| latasymb 18363 | A lattice ordering is asym... |
| latasym 18364 | A lattice ordering is asym... |
| lattr 18365 | A lattice ordering is tran... |
| latasymd 18366 | Deduce equality from latti... |
| lattrd 18367 | A lattice ordering is tran... |
| latjcom 18368 | The join of a lattice comm... |
| latlej1 18369 | A join's first argument is... |
| latlej2 18370 | A join's second argument i... |
| latjle12 18371 | A join is less than or equ... |
| latleeqj1 18372 | "Less than or equal to" in... |
| latleeqj2 18373 | "Less than or equal to" in... |
| latjlej1 18374 | Add join to both sides of ... |
| latjlej2 18375 | Add join to both sides of ... |
| latjlej12 18376 | Add join to both sides of ... |
| latnlej 18377 | An idiom to express that a... |
| latnlej1l 18378 | An idiom to express that a... |
| latnlej1r 18379 | An idiom to express that a... |
| latnlej2 18380 | An idiom to express that a... |
| latnlej2l 18381 | An idiom to express that a... |
| latnlej2r 18382 | An idiom to express that a... |
| latjidm 18383 | Lattice join is idempotent... |
| latmcom 18384 | The join of a lattice comm... |
| latmle1 18385 | A meet is less than or equ... |
| latmle2 18386 | A meet is less than or equ... |
| latlem12 18387 | An element is less than or... |
| latleeqm1 18388 | "Less than or equal to" in... |
| latleeqm2 18389 | "Less than or equal to" in... |
| latmlem1 18390 | Add meet to both sides of ... |
| latmlem2 18391 | Add meet to both sides of ... |
| latmlem12 18392 | Add join to both sides of ... |
| latnlemlt 18393 | Negation of "less than or ... |
| latnle 18394 | Equivalent expressions for... |
| latmidm 18395 | Lattice meet is idempotent... |
| latabs1 18396 | Lattice absorption law. F... |
| latabs2 18397 | Lattice absorption law. F... |
| latledi 18398 | An ortholattice is distrib... |
| latmlej11 18399 | Ordering of a meet and joi... |
| latmlej12 18400 | Ordering of a meet and joi... |
| latmlej21 18401 | Ordering of a meet and joi... |
| latmlej22 18402 | Ordering of a meet and joi... |
| lubsn 18403 | The least upper bound of a... |
| latjass 18404 | Lattice join is associativ... |
| latj12 18405 | Swap 1st and 2nd members o... |
| latj32 18406 | Swap 2nd and 3rd members o... |
| latj13 18407 | Swap 1st and 3rd members o... |
| latj31 18408 | Swap 2nd and 3rd members o... |
| latjrot 18409 | Rotate lattice join of 3 c... |
| latj4 18410 | Rearrangement of lattice j... |
| latj4rot 18411 | Rotate lattice join of 4 c... |
| latjjdi 18412 | Lattice join distributes o... |
| latjjdir 18413 | Lattice join distributes o... |
| mod1ile 18414 | The weak direction of the ... |
| mod2ile 18415 | The weak direction of the ... |
| latmass 18416 | Lattice meet is associativ... |
| latdisdlem 18417 | Lemma for ~ latdisd . (Co... |
| latdisd 18418 | In a lattice, joins distri... |
| isclat 18421 | The predicate "is a comple... |
| clatpos 18422 | A complete lattice is a po... |
| clatlem 18423 | Lemma for properties of a ... |
| clatlubcl 18424 | Any subset of the base set... |
| clatlubcl2 18425 | Any subset of the base set... |
| clatglbcl 18426 | Any subset of the base set... |
| clatglbcl2 18427 | Any subset of the base set... |
| oduclatb 18428 | Being a complete lattice i... |
| clatl 18429 | A complete lattice is a la... |
| isglbd 18430 | Properties that determine ... |
| lublem 18431 | Lemma for the least upper ... |
| lubub 18432 | The LUB of a complete latt... |
| lubl 18433 | The LUB of a complete latt... |
| lubss 18434 | Subset law for least upper... |
| lubel 18435 | An element of a set is les... |
| lubun 18436 | The LUB of a union. (Cont... |
| clatglb 18437 | Properties of greatest low... |
| clatglble 18438 | The greatest lower bound i... |
| clatleglb 18439 | Two ways of expressing "le... |
| clatglbss 18440 | Subset law for greatest lo... |
| isdlat 18443 | Property of being a distri... |
| dlatmjdi 18444 | In a distributive lattice,... |
| dlatl 18445 | A distributive lattice is ... |
| odudlatb 18446 | The dual of a distributive... |
| dlatjmdi 18447 | In a distributive lattice,... |
| ipostr 18450 | The structure of ~ df-ipo ... |
| ipoval 18451 | Value of the inclusion pos... |
| ipobas 18452 | Base set of the inclusion ... |
| ipolerval 18453 | Relation of the inclusion ... |
| ipotset 18454 | Topology of the inclusion ... |
| ipole 18455 | Weak order condition of th... |
| ipolt 18456 | Strict order condition of ... |
| ipopos 18457 | The inclusion poset on a f... |
| isipodrs 18458 | Condition for a family of ... |
| ipodrscl 18459 | Direction by inclusion as ... |
| ipodrsfi 18460 | Finite upper bound propert... |
| fpwipodrs 18461 | The finite subsets of any ... |
| ipodrsima 18462 | The monotone image of a di... |
| isacs3lem 18463 | An algebraic closure syste... |
| acsdrsel 18464 | An algebraic closure syste... |
| isacs4lem 18465 | In a closure system in whi... |
| isacs5lem 18466 | If closure commutes with d... |
| acsdrscl 18467 | In an algebraic closure sy... |
| acsficl 18468 | A closure in an algebraic ... |
| isacs5 18469 | A closure system is algebr... |
| isacs4 18470 | A closure system is algebr... |
| isacs3 18471 | A closure system is algebr... |
| acsficld 18472 | In an algebraic closure sy... |
| acsficl2d 18473 | In an algebraic closure sy... |
| acsfiindd 18474 | In an algebraic closure sy... |
| acsmapd 18475 | In an algebraic closure sy... |
| acsmap2d 18476 | In an algebraic closure sy... |
| acsinfd 18477 | In an algebraic closure sy... |
| acsdomd 18478 | In an algebraic closure sy... |
| acsinfdimd 18479 | In an algebraic closure sy... |
| acsexdimd 18480 | In an algebraic closure sy... |
| mrelatglb 18481 | Greatest lower bounds in a... |
| mrelatglb0 18482 | The empty intersection in ... |
| mrelatlub 18483 | Least upper bounds in a Mo... |
| mreclatBAD 18484 | A Moore space is a complet... |
| isps 18489 | The predicate "is a poset"... |
| psrel 18490 | A poset is a relation. (C... |
| psref2 18491 | A poset is antisymmetric a... |
| pstr2 18492 | A poset is transitive. (C... |
| pslem 18493 | Lemma for ~ psref and othe... |
| psdmrn 18494 | The domain and range of a ... |
| psref 18495 | A poset is reflexive. (Co... |
| psrn 18496 | The range of a poset equal... |
| psasym 18497 | A poset is antisymmetric. ... |
| pstr 18498 | A poset is transitive. (C... |
| cnvps 18499 | The converse of a poset is... |
| cnvpsb 18500 | The converse of a poset is... |
| psss 18501 | Any subset of a partially ... |
| psssdm2 18502 | Field of a subposet. (Con... |
| psssdm 18503 | Field of a subposet. (Con... |
| istsr 18504 | The predicate is a toset. ... |
| istsr2 18505 | The predicate is a toset. ... |
| tsrlin 18506 | A toset is a linear order.... |
| tsrlemax 18507 | Two ways of saying a numbe... |
| tsrps 18508 | A toset is a poset. (Cont... |
| cnvtsr 18509 | The converse of a toset is... |
| tsrss 18510 | Any subset of a totally or... |
| ledm 18511 | The domain of ` <_ ` is ` ... |
| lern 18512 | The range of ` <_ ` is ` R... |
| lefld 18513 | The field of the 'less or ... |
| letsr 18514 | The "less than or equal to... |
| isdir 18519 | A condition for a relation... |
| reldir 18520 | A direction is a relation.... |
| dirdm 18521 | A direction's domain is eq... |
| dirref 18522 | A direction is reflexive. ... |
| dirtr 18523 | A direction is transitive.... |
| dirge 18524 | For any two elements of a ... |
| tsrdir 18525 | A totally ordered set is a... |
| ischn 18528 | Property of being a chain.... |
| chnwrd 18529 | A chain is an ordered sequ... |
| chnltm1 18530 | Basic property of a chain.... |
| pfxchn 18531 | A prefix of a chain is sti... |
| nfchnd 18532 | Bound-variable hypothesis ... |
| chneq1 18533 | Equality theorem for chain... |
| chneq2 18534 | Equality theorem for chain... |
| chneq12 18535 | Equality theorem for chain... |
| chnrss 18536 | Chains under a relation ar... |
| chndss 18537 | Chains with an alphabet ar... |
| chnrdss 18538 | Subset theorem for chains.... |
| chnexg 18539 | Chains with a set given fo... |
| nulchn 18540 | Empty set is an increasing... |
| s1chn 18541 | A singleton word is always... |
| chnind 18542 | Induction over a chain. S... |
| chnub 18543 | In a chain, the last eleme... |
| chnlt 18544 | Compare any two elements i... |
| chnso 18545 | A chain induces a total or... |
| chnccats1 18546 | Extend a chain with a sing... |
| chnccat 18547 | Concatenate two chains. (... |
| chnrev 18548 | Reverse of a chain is chai... |
| chnflenfi 18549 | There is a finite number o... |
| chnf 18550 | A chain is a zero-based fi... |
| chnpof1 18551 | A chain under relation whi... |
| chnpoadomd 18552 | A chain under relation whi... |
| chnpolleha 18553 | A chain under relation whi... |
| chnpolfz 18554 | Provided that chain's rela... |
| chnfi 18555 | There is a finite number o... |
| chninf 18556 | There is an infinite numbe... |
| chnfibg 18557 | Given a partial order, the... |
| ex-chn1 18558 | Example: a doubleton of tw... |
| ex-chn2 18559 | Example: sequence <" ZZ NN... |
| ismgm 18564 | The predicate "is a magma"... |
| ismgmn0 18565 | The predicate "is a magma"... |
| mgmcl 18566 | Closure of the operation o... |
| isnmgm 18567 | A condition for a structur... |
| mgmsscl 18568 | If the base set of a magma... |
| plusffval 18569 | The group addition operati... |
| plusfval 18570 | The group addition operati... |
| plusfeq 18571 | If the addition operation ... |
| plusffn 18572 | The group addition operati... |
| mgmplusf 18573 | The group addition functio... |
| mgmpropd 18574 | If two structures have the... |
| ismgmd 18575 | Deduce a magma from its pr... |
| issstrmgm 18576 | Characterize a substructur... |
| intopsn 18577 | The internal operation for... |
| mgmb1mgm1 18578 | The only magma with a base... |
| mgm0 18579 | Any set with an empty base... |
| mgm0b 18580 | The structure with an empt... |
| mgm1 18581 | The structure with one ele... |
| opifismgm 18582 | A structure with a group a... |
| mgmidmo 18583 | A two-sided identity eleme... |
| grpidval 18584 | The value of the identity ... |
| grpidpropd 18585 | If two structures have the... |
| fn0g 18586 | The group zero extractor i... |
| 0g0 18587 | The identity element funct... |
| ismgmid 18588 | The identity element of a ... |
| mgmidcl 18589 | The identity element of a ... |
| mgmlrid 18590 | The identity element of a ... |
| ismgmid2 18591 | Show that a given element ... |
| lidrideqd 18592 | If there is a left and rig... |
| lidrididd 18593 | If there is a left and rig... |
| grpidd 18594 | Deduce the identity elemen... |
| mgmidsssn0 18595 | Property of the set of ide... |
| grpinvalem 18596 | Lemma for ~ grpinva . (Co... |
| grpinva 18597 | Deduce right inverse from ... |
| grprida 18598 | Deduce right identity from... |
| gsumvalx 18599 | Expand out the substitutio... |
| gsumval 18600 | Expand out the substitutio... |
| gsumpropd 18601 | The group sum depends only... |
| gsumpropd2lem 18602 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18603 | A stronger version of ~ gs... |
| gsummgmpropd 18604 | A stronger version of ~ gs... |
| gsumress 18605 | The group sum in a substru... |
| gsumval1 18606 | Value of the group sum ope... |
| gsum0 18607 | Value of the empty group s... |
| gsumval2a 18608 | Value of the group sum ope... |
| gsumval2 18609 | Value of the group sum ope... |
| gsumsplit1r 18610 | Splitting off the rightmos... |
| gsumprval 18611 | Value of the group sum ope... |
| gsumpr12val 18612 | Value of the group sum ope... |
| mgmhmrcl 18617 | Reverse closure of a magma... |
| submgmrcl 18618 | Reverse closure for submag... |
| ismgmhm 18619 | Property of a magma homomo... |
| mgmhmf 18620 | A magma homomorphism is a ... |
| mgmhmpropd 18621 | Magma homomorphism depends... |
| mgmhmlin 18622 | A magma homomorphism prese... |
| mgmhmf1o 18623 | A magma homomorphism is bi... |
| idmgmhm 18624 | The identity homomorphism ... |
| issubmgm 18625 | Expand definition of a sub... |
| issubmgm2 18626 | Submagmas are subsets that... |
| rabsubmgmd 18627 | Deduction for proving that... |
| submgmss 18628 | Submagmas are subsets of t... |
| submgmid 18629 | Every magma is trivially a... |
| submgmcl 18630 | Submagmas are closed under... |
| submgmmgm 18631 | Submagmas are themselves m... |
| submgmbas 18632 | The base set of a submagma... |
| subsubmgm 18633 | A submagma of a submagma i... |
| resmgmhm 18634 | Restriction of a magma hom... |
| resmgmhm2 18635 | One direction of ~ resmgmh... |
| resmgmhm2b 18636 | Restriction of the codomai... |
| mgmhmco 18637 | The composition of magma h... |
| mgmhmima 18638 | The homomorphic image of a... |
| mgmhmeql 18639 | The equalizer of two magma... |
| submgmacs 18640 | Submagmas are an algebraic... |
| issgrp 18643 | The predicate "is a semigr... |
| issgrpv 18644 | The predicate "is a semigr... |
| issgrpn0 18645 | The predicate "is a semigr... |
| isnsgrp 18646 | A condition for a structur... |
| sgrpmgm 18647 | A semigroup is a magma. (... |
| sgrpass 18648 | A semigroup operation is a... |
| sgrpcl 18649 | Closure of the operation o... |
| sgrp0 18650 | Any set with an empty base... |
| sgrp0b 18651 | The structure with an empt... |
| sgrp1 18652 | The structure with one ele... |
| issgrpd 18653 | Deduce a semigroup from it... |
| sgrppropd 18654 | If two structures are sets... |
| prdsplusgsgrpcl 18655 | Structure product pointwis... |
| prdssgrpd 18656 | The product of a family of... |
| ismnddef 18659 | The predicate "is a monoid... |
| ismnd 18660 | The predicate "is a monoid... |
| isnmnd 18661 | A condition for a structur... |
| sgrpidmnd 18662 | A semigroup with an identi... |
| mndsgrp 18663 | A monoid is a semigroup. ... |
| mndmgm 18664 | A monoid is a magma. (Con... |
| mndcl 18665 | Closure of the operation o... |
| mndass 18666 | A monoid operation is asso... |
| mndid 18667 | A monoid has a two-sided i... |
| mndideu 18668 | The two-sided identity ele... |
| mnd32g 18669 | Commutative/associative la... |
| mnd12g 18670 | Commutative/associative la... |
| mnd4g 18671 | Commutative/associative la... |
| mndidcl 18672 | The identity element of a ... |
| mndbn0 18673 | The base set of a monoid i... |
| hashfinmndnn 18674 | A finite monoid has positi... |
| mndplusf 18675 | The group addition operati... |
| mndlrid 18676 | A monoid's identity elemen... |
| mndlid 18677 | The identity element of a ... |
| mndrid 18678 | The identity element of a ... |
| ismndd 18679 | Deduce a monoid from its p... |
| mndpfo 18680 | The addition operation of ... |
| mndfo 18681 | The addition operation of ... |
| mndpropd 18682 | If two structures have the... |
| mndprop 18683 | If two structures have the... |
| issubmnd 18684 | Characterize a submonoid b... |
| ress0g 18685 | ` 0g ` is unaffected by re... |
| submnd0 18686 | The zero of a submonoid is... |
| mndinvmod 18687 | Uniqueness of an inverse e... |
| mndpsuppss 18688 | The support of a mapping o... |
| mndpsuppfi 18689 | The support of a mapping o... |
| mndpfsupp 18690 | A mapping of a scalar mult... |
| prdsplusgcl 18691 | Structure product pointwis... |
| prdsidlem 18692 | Characterization of identi... |
| prdsmndd 18693 | The product of a family of... |
| prds0g 18694 | The identity in a product ... |
| pwsmnd 18695 | The structure power of a m... |
| pws0g 18696 | The identity in a structur... |
| imasmnd2 18697 | The image structure of a m... |
| imasmnd 18698 | The image structure of a m... |
| imasmndf1 18699 | The image of a monoid unde... |
| xpsmnd 18700 | The binary product of mono... |
| xpsmnd0 18701 | The identity element of a ... |
| mnd1 18702 | The (smallest) structure r... |
| mnd1id 18703 | The singleton element of a... |
| ismhm 18708 | Property of a monoid homom... |
| ismhmd 18709 | Deduction version of ~ ism... |
| mhmrcl1 18710 | Reverse closure of a monoi... |
| mhmrcl2 18711 | Reverse closure of a monoi... |
| mhmf 18712 | A monoid homomorphism is a... |
| ismhm0 18713 | Property of a monoid homom... |
| mhmismgmhm 18714 | Each monoid homomorphism i... |
| mhmpropd 18715 | Monoid homomorphism depend... |
| mhmlin 18716 | A monoid homomorphism comm... |
| mhm0 18717 | A monoid homomorphism pres... |
| idmhm 18718 | The identity homomorphism ... |
| mhmf1o 18719 | A monoid homomorphism is b... |
| mndvcl 18720 | Tuple-wise additive closur... |
| mndvass 18721 | Tuple-wise associativity i... |
| mndvlid 18722 | Tuple-wise left identity i... |
| mndvrid 18723 | Tuple-wise right identity ... |
| mhmvlin 18724 | Tuple extension of monoid ... |
| submrcl 18725 | Reverse closure for submon... |
| issubm 18726 | Expand definition of a sub... |
| issubm2 18727 | Submonoids are subsets tha... |
| issubmndb 18728 | The submonoid predicate. ... |
| issubmd 18729 | Deduction for proving a su... |
| mndissubm 18730 | If the base set of a monoi... |
| resmndismnd 18731 | If the base set of a monoi... |
| submss 18732 | Submonoids are subsets of ... |
| submid 18733 | Every monoid is trivially ... |
| subm0cl 18734 | Submonoids contain zero. ... |
| submcl 18735 | Submonoids are closed unde... |
| submmnd 18736 | Submonoids are themselves ... |
| submbas 18737 | The base set of a submonoi... |
| subm0 18738 | Submonoids have the same i... |
| subsubm 18739 | A submonoid of a submonoid... |
| 0subm 18740 | The zero submonoid of an a... |
| insubm 18741 | The intersection of two su... |
| 0mhm 18742 | The constant zero linear f... |
| resmhm 18743 | Restriction of a monoid ho... |
| resmhm2 18744 | One direction of ~ resmhm2... |
| resmhm2b 18745 | Restriction of the codomai... |
| mhmco 18746 | The composition of monoid ... |
| mhmimalem 18747 | Lemma for ~ mhmima and sim... |
| mhmima 18748 | The homomorphic image of a... |
| mhmeql 18749 | The equalizer of two monoi... |
| submacs 18750 | Submonoids are an algebrai... |
| mndind 18751 | Induction in a monoid. In... |
| prdspjmhm 18752 | A projection from a produc... |
| pwspjmhm 18753 | A projection from a struct... |
| pwsdiagmhm 18754 | Diagonal monoid homomorphi... |
| pwsco1mhm 18755 | Right composition with a f... |
| pwsco2mhm 18756 | Left composition with a mo... |
| gsumvallem2 18757 | Lemma for properties of th... |
| gsumsubm 18758 | Evaluate a group sum in a ... |
| gsumz 18759 | Value of a group sum over ... |
| gsumwsubmcl 18760 | Closure of the composite i... |
| gsumws1 18761 | A singleton composite reco... |
| gsumwcl 18762 | Closure of the composite o... |
| gsumsgrpccat 18763 | Homomorphic property of no... |
| gsumccat 18764 | Homomorphic property of co... |
| gsumws2 18765 | Valuation of a pair in a m... |
| gsumccatsn 18766 | Homomorphic property of co... |
| gsumspl 18767 | The primary purpose of the... |
| gsumwmhm 18768 | Behavior of homomorphisms ... |
| gsumwspan 18769 | The submonoid generated by... |
| frmdval 18774 | Value of the free monoid c... |
| frmdbas 18775 | The base set of a free mon... |
| frmdelbas 18776 | An element of the base set... |
| frmdplusg 18777 | The monoid operation of a ... |
| frmdadd 18778 | Value of the monoid operat... |
| vrmdfval 18779 | The canonical injection fr... |
| vrmdval 18780 | The value of the generatin... |
| vrmdf 18781 | The mapping from the index... |
| frmdmnd 18782 | A free monoid is a monoid.... |
| frmd0 18783 | The identity of the free m... |
| frmdsssubm 18784 | The set of words taking va... |
| frmdgsum 18785 | Any word in a free monoid ... |
| frmdss2 18786 | A subset of generators is ... |
| frmdup1 18787 | Any assignment of the gene... |
| frmdup2 18788 | The evaluation map has the... |
| frmdup3lem 18789 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18790 | Universal property of the ... |
| efmnd 18793 | The monoid of endofunction... |
| efmndbas 18794 | The base set of the monoid... |
| efmndbasabf 18795 | The base set of the monoid... |
| elefmndbas 18796 | Two ways of saying a funct... |
| elefmndbas2 18797 | Two ways of saying a funct... |
| efmndbasf 18798 | Elements in the monoid of ... |
| efmndhash 18799 | The monoid of endofunction... |
| efmndbasfi 18800 | The monoid of endofunction... |
| efmndfv 18801 | The function value of an e... |
| efmndtset 18802 | The topology of the monoid... |
| efmndplusg 18803 | The group operation of a m... |
| efmndov 18804 | The value of the group ope... |
| efmndcl 18805 | The group operation of the... |
| efmndtopn 18806 | The topology of the monoid... |
| symggrplem 18807 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18808 | The monoid of endofunction... |
| efmndsgrp 18809 | The monoid of endofunction... |
| ielefmnd 18810 | The identity function rest... |
| efmndid 18811 | The identity function rest... |
| efmndmnd 18812 | The monoid of endofunction... |
| efmnd0nmnd 18813 | Even the monoid of endofun... |
| efmndbas0 18814 | The base set of the monoid... |
| efmnd1hash 18815 | The monoid of endofunction... |
| efmnd1bas 18816 | The monoid of endofunction... |
| efmnd2hash 18817 | The monoid of endofunction... |
| submefmnd 18818 | If the base set of a monoi... |
| sursubmefmnd 18819 | The set of surjective endo... |
| injsubmefmnd 18820 | The set of injective endof... |
| idressubmefmnd 18821 | The singleton containing o... |
| idresefmnd 18822 | The structure with the sin... |
| smndex1ibas 18823 | The modulo function ` I ` ... |
| smndex1iidm 18824 | The modulo function ` I ` ... |
| smndex1gbas 18825 | The constant functions ` (... |
| smndex1gid 18826 | The composition of a const... |
| smndex1igid 18827 | The composition of the mod... |
| smndex1basss 18828 | The modulo function ` I ` ... |
| smndex1bas 18829 | The base set of the monoid... |
| smndex1mgm 18830 | The monoid of endofunction... |
| smndex1sgrp 18831 | The monoid of endofunction... |
| smndex1mndlem 18832 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18833 | The monoid of endofunction... |
| smndex1id 18834 | The modulo function ` I ` ... |
| smndex1n0mnd 18835 | The identity of the monoid... |
| nsmndex1 18836 | The base set ` B ` of the ... |
| smndex2dbas 18837 | The doubling function ` D ... |
| smndex2dnrinv 18838 | The doubling function ` D ... |
| smndex2hbas 18839 | The halving functions ` H ... |
| smndex2dlinvh 18840 | The halving functions ` H ... |
| mgm2nsgrplem1 18841 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18842 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18843 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18844 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18845 | A small magma (with two el... |
| sgrp2nmndlem1 18846 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18847 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18848 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18849 | A small semigroup (with tw... |
| sgrp2rid2ex 18850 | A small semigroup (with tw... |
| sgrp2nmndlem4 18851 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18852 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18853 | A small semigroup (with tw... |
| mgmnsgrpex 18854 | There is a magma which is ... |
| sgrpnmndex 18855 | There is a semigroup which... |
| sgrpssmgm 18856 | The class of all semigroup... |
| mndsssgrp 18857 | The class of all monoids i... |
| pwmndgplus 18858 | The operation of the monoi... |
| pwmndid 18859 | The identity of the monoid... |
| pwmnd 18860 | The power set of a class `... |
| isgrp 18867 | The predicate "is a group"... |
| grpmnd 18868 | A group is a monoid. (Con... |
| grpcl 18869 | Closure of the operation o... |
| grpass 18870 | A group operation is assoc... |
| grpinvex 18871 | Every member of a group ha... |
| grpideu 18872 | The two-sided identity ele... |
| grpassd 18873 | A group operation is assoc... |
| grpmndd 18874 | A group is a monoid. (Con... |
| grpcld 18875 | Closure of the operation o... |
| grpplusf 18876 | The group addition operati... |
| grpplusfo 18877 | The group addition operati... |
| resgrpplusfrn 18878 | The underlying set of a gr... |
| grppropd 18879 | If two structures have the... |
| grpprop 18880 | If two structures have the... |
| grppropstr 18881 | Generalize a specific 2-el... |
| grpss 18882 | Show that a structure exte... |
| isgrpd2e 18883 | Deduce a group from its pr... |
| isgrpd2 18884 | Deduce a group from its pr... |
| isgrpde 18885 | Deduce a group from its pr... |
| isgrpd 18886 | Deduce a group from its pr... |
| isgrpi 18887 | Properties that determine ... |
| grpsgrp 18888 | A group is a semigroup. (... |
| grpmgmd 18889 | A group is a magma, deduct... |
| dfgrp2 18890 | Alternate definition of a ... |
| dfgrp2e 18891 | Alternate definition of a ... |
| isgrpix 18892 | Properties that determine ... |
| grpidcl 18893 | The identity element of a ... |
| grpbn0 18894 | The base set of a group is... |
| grplid 18895 | The identity element of a ... |
| grprid 18896 | The identity element of a ... |
| grplidd 18897 | The identity element of a ... |
| grpridd 18898 | The identity element of a ... |
| grpn0 18899 | A group is not empty. (Co... |
| hashfingrpnn 18900 | A finite group has positiv... |
| grprcan 18901 | Right cancellation law for... |
| grpinveu 18902 | The left inverse element o... |
| grpid 18903 | Two ways of saying that an... |
| isgrpid2 18904 | Properties showing that an... |
| grpidd2 18905 | Deduce the identity elemen... |
| grpinvfval 18906 | The inverse function of a ... |
| grpinvfvalALT 18907 | Shorter proof of ~ grpinvf... |
| grpinvval 18908 | The inverse of a group ele... |
| grpinvfn 18909 | Functionality of the group... |
| grpinvfvi 18910 | The group inverse function... |
| grpsubfval 18911 | Group subtraction (divisio... |
| grpsubfvalALT 18912 | Shorter proof of ~ grpsubf... |
| grpsubval 18913 | Group subtraction (divisio... |
| grpinvf 18914 | The group inversion operat... |
| grpinvcl 18915 | A group element's inverse ... |
| grpinvcld 18916 | A group element's inverse ... |
| grplinv 18917 | The left inverse of a grou... |
| grprinv 18918 | The right inverse of a gro... |
| grpinvid1 18919 | The inverse of a group ele... |
| grpinvid2 18920 | The inverse of a group ele... |
| isgrpinv 18921 | Properties showing that a ... |
| grplinvd 18922 | The left inverse of a grou... |
| grprinvd 18923 | The right inverse of a gro... |
| grplrinv 18924 | In a group, every member h... |
| grpidinv2 18925 | A group's properties using... |
| grpidinv 18926 | A group has a left and rig... |
| grpinvid 18927 | The inverse of the identit... |
| grplcan 18928 | Left cancellation law for ... |
| grpasscan1 18929 | An associative cancellatio... |
| grpasscan2 18930 | An associative cancellatio... |
| grpidrcan 18931 | If right adding an element... |
| grpidlcan 18932 | If left adding an element ... |
| grpinvinv 18933 | Double inverse law for gro... |
| grpinvcnv 18934 | The group inverse is its o... |
| grpinv11 18935 | The group inverse is one-t... |
| grpinv11OLD 18936 | Obsolete version of ~ grpi... |
| grpinvf1o 18937 | The group inverse is a one... |
| grpinvnz 18938 | The inverse of a nonzero g... |
| grpinvnzcl 18939 | The inverse of a nonzero g... |
| grpsubinv 18940 | Subtraction of an inverse.... |
| grplmulf1o 18941 | Left multiplication by a g... |
| grpraddf1o 18942 | Right addition by a group ... |
| grpinvpropd 18943 | If two structures have the... |
| grpidssd 18944 | If the base set of a group... |
| grpinvssd 18945 | If the base set of a group... |
| grpinvadd 18946 | The inverse of the group o... |
| grpsubf 18947 | Functionality of group sub... |
| grpsubcl 18948 | Closure of group subtracti... |
| grpsubrcan 18949 | Right cancellation law for... |
| grpinvsub 18950 | Inverse of a group subtrac... |
| grpinvval2 18951 | A ~ df-neg -like equation ... |
| grpsubid 18952 | Subtraction of a group ele... |
| grpsubid1 18953 | Subtraction of the identit... |
| grpsubeq0 18954 | If the difference between ... |
| grpsubadd0sub 18955 | Subtraction expressed as a... |
| grpsubadd 18956 | Relationship between group... |
| grpsubsub 18957 | Double group subtraction. ... |
| grpaddsubass 18958 | Associative-type law for g... |
| grppncan 18959 | Cancellation law for subtr... |
| grpnpcan 18960 | Cancellation law for subtr... |
| grpsubsub4 18961 | Double group subtraction (... |
| grppnpcan2 18962 | Cancellation law for mixed... |
| grpnpncan 18963 | Cancellation law for group... |
| grpnpncan0 18964 | Cancellation law for group... |
| grpnnncan2 18965 | Cancellation law for group... |
| dfgrp3lem 18966 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18967 | Alternate definition of a ... |
| dfgrp3e 18968 | Alternate definition of a ... |
| grplactfval 18969 | The left group action of e... |
| grplactval 18970 | The value of the left grou... |
| grplactcnv 18971 | The left group action of e... |
| grplactf1o 18972 | The left group action of e... |
| grpsubpropd 18973 | Weak property deduction fo... |
| grpsubpropd2 18974 | Strong property deduction ... |
| grp1 18975 | The (smallest) structure r... |
| grp1inv 18976 | The inverse function of th... |
| prdsinvlem 18977 | Characterization of invers... |
| prdsgrpd 18978 | The product of a family of... |
| prdsinvgd 18979 | Negation in a product of g... |
| pwsgrp 18980 | A structure power of a gro... |
| pwsinvg 18981 | Negation in a group power.... |
| pwssub 18982 | Subtraction in a group pow... |
| imasgrp2 18983 | The image structure of a g... |
| imasgrp 18984 | The image structure of a g... |
| imasgrpf1 18985 | The image of a group under... |
| qusgrp2 18986 | Prove that a quotient stru... |
| xpsgrp 18987 | The binary product of grou... |
| xpsinv 18988 | Value of the negation oper... |
| xpsgrpsub 18989 | Value of the subtraction o... |
| mhmlem 18990 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18991 | A surjective monoid morphi... |
| mhmmnd 18992 | The image of a monoid ` G ... |
| mhmfmhm 18993 | The function fulfilling th... |
| ghmgrp 18994 | The image of a group ` G `... |
| mulgfval 18997 | Group multiple (exponentia... |
| mulgfvalALT 18998 | Shorter proof of ~ mulgfva... |
| mulgval 18999 | Value of the group multipl... |
| mulgfn 19000 | Functionality of the group... |
| mulgfvi 19001 | The group multiple operati... |
| mulg0 19002 | Group multiple (exponentia... |
| mulgnn 19003 | Group multiple (exponentia... |
| ressmulgnn 19004 | Values for the group multi... |
| ressmulgnn0 19005 | Values for the group multi... |
| ressmulgnnd 19006 | Values for the group multi... |
| mulgnngsum 19007 | Group multiple (exponentia... |
| mulgnn0gsum 19008 | Group multiple (exponentia... |
| mulg1 19009 | Group multiple (exponentia... |
| mulgnnp1 19010 | Group multiple (exponentia... |
| mulg2 19011 | Group multiple (exponentia... |
| mulgnegnn 19012 | Group multiple (exponentia... |
| mulgnn0p1 19013 | Group multiple (exponentia... |
| mulgnnsubcl 19014 | Closure of the group multi... |
| mulgnn0subcl 19015 | Closure of the group multi... |
| mulgsubcl 19016 | Closure of the group multi... |
| mulgnncl 19017 | Closure of the group multi... |
| mulgnn0cl 19018 | Closure of the group multi... |
| mulgcl 19019 | Closure of the group multi... |
| mulgneg 19020 | Group multiple (exponentia... |
| mulgnegneg 19021 | The inverse of a negative ... |
| mulgm1 19022 | Group multiple (exponentia... |
| mulgnn0cld 19023 | Closure of the group multi... |
| mulgcld 19024 | Deduction associated with ... |
| mulgaddcomlem 19025 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19026 | The group multiple operato... |
| mulginvcom 19027 | The group multiple operato... |
| mulginvinv 19028 | The group multiple operato... |
| mulgnn0z 19029 | A group multiple of the id... |
| mulgz 19030 | A group multiple of the id... |
| mulgnndir 19031 | Sum of group multiples, fo... |
| mulgnn0dir 19032 | Sum of group multiples, ge... |
| mulgdirlem 19033 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19034 | Sum of group multiples, ge... |
| mulgp1 19035 | Group multiple (exponentia... |
| mulgneg2 19036 | Group multiple (exponentia... |
| mulgnnass 19037 | Product of group multiples... |
| mulgnn0ass 19038 | Product of group multiples... |
| mulgass 19039 | Product of group multiples... |
| mulgassr 19040 | Reversed product of group ... |
| mulgmodid 19041 | Casting out multiples of t... |
| mulgsubdir 19042 | Distribution of group mult... |
| mhmmulg 19043 | A homomorphism of monoids ... |
| mulgpropd 19044 | Two structures with the sa... |
| submmulgcl 19045 | Closure of the group multi... |
| submmulg 19046 | A group multiple is the sa... |
| pwsmulg 19047 | Value of a group multiple ... |
| issubg 19054 | The subgroup predicate. (... |
| subgss 19055 | A subgroup is a subset. (... |
| subgid 19056 | A group is a subgroup of i... |
| subggrp 19057 | A subgroup is a group. (C... |
| subgbas 19058 | The base of the restricted... |
| subgrcl 19059 | Reverse closure for the su... |
| subg0 19060 | A subgroup of a group must... |
| subginv 19061 | The inverse of an element ... |
| subg0cl 19062 | The group identity is an e... |
| subginvcl 19063 | The inverse of an element ... |
| subgcl 19064 | A subgroup is closed under... |
| subgsubcl 19065 | A subgroup is closed under... |
| subgsub 19066 | The subtraction of element... |
| subgmulgcl 19067 | Closure of the group multi... |
| subgmulg 19068 | A group multiple is the sa... |
| issubg2 19069 | Characterize the subgroups... |
| issubgrpd2 19070 | Prove a subgroup by closur... |
| issubgrpd 19071 | Prove a subgroup by closur... |
| issubg3 19072 | A subgroup is a symmetric ... |
| issubg4 19073 | A subgroup is a nonempty s... |
| grpissubg 19074 | If the base set of a group... |
| resgrpisgrp 19075 | If the base set of a group... |
| subgsubm 19076 | A subgroup is a submonoid.... |
| subsubg 19077 | A subgroup of a subgroup i... |
| subgint 19078 | The intersection of a none... |
| 0subg 19079 | The zero subgroup of an ar... |
| trivsubgd 19080 | The only subgroup of a tri... |
| trivsubgsnd 19081 | The only subgroup of a tri... |
| isnsg 19082 | Property of being a normal... |
| isnsg2 19083 | Weaken the condition of ~ ... |
| nsgbi 19084 | Defining property of a nor... |
| nsgsubg 19085 | A normal subgroup is a sub... |
| nsgconj 19086 | The conjugation of an elem... |
| isnsg3 19087 | A subgroup is normal iff t... |
| subgacs 19088 | Subgroups are an algebraic... |
| nsgacs 19089 | Normal subgroups form an a... |
| elnmz 19090 | Elementhood in the normali... |
| nmzbi 19091 | Defining property of the n... |
| nmzsubg 19092 | The normalizer N_G(S) of a... |
| ssnmz 19093 | A subgroup is a subset of ... |
| isnsg4 19094 | A subgroup is normal iff i... |
| nmznsg 19095 | Any subgroup is a normal s... |
| 0nsg 19096 | The zero subgroup is norma... |
| nsgid 19097 | The whole group is a norma... |
| 0idnsgd 19098 | The whole group and the ze... |
| trivnsgd 19099 | The only normal subgroup o... |
| triv1nsgd 19100 | A trivial group has exactl... |
| 1nsgtrivd 19101 | A group with exactly one n... |
| releqg 19102 | The left coset equivalence... |
| eqgfval 19103 | Value of the subgroup left... |
| eqgval 19104 | Value of the subgroup left... |
| eqger 19105 | The subgroup coset equival... |
| eqglact 19106 | A left coset can be expres... |
| eqgid 19107 | The left coset containing ... |
| eqgen 19108 | Each coset is equipotent t... |
| eqgcpbl 19109 | The subgroup coset equival... |
| eqg0el 19110 | Equivalence class of a quo... |
| quselbas 19111 | Membership in the base set... |
| quseccl0 19112 | Closure of the quotient ma... |
| qusgrp 19113 | If ` Y ` is a normal subgr... |
| quseccl 19114 | Closure of the quotient ma... |
| qusadd 19115 | Value of the group operati... |
| qus0 19116 | Value of the group identit... |
| qusinv 19117 | Value of the group inverse... |
| qussub 19118 | Value of the group subtrac... |
| ecqusaddd 19119 | Addition of equivalence cl... |
| ecqusaddcl 19120 | Closure of the addition in... |
| lagsubg2 19121 | Lagrange's theorem for fin... |
| lagsubg 19122 | Lagrange's theorem for Gro... |
| eqg0subg 19123 | The coset equivalence rela... |
| eqg0subgecsn 19124 | The equivalence classes mo... |
| qus0subgbas 19125 | The base set of a quotient... |
| qus0subgadd 19126 | The addition in a quotient... |
| cycsubmel 19127 | Characterization of an ele... |
| cycsubmcl 19128 | The set of nonnegative int... |
| cycsubm 19129 | The set of nonnegative int... |
| cyccom 19130 | Condition for an operation... |
| cycsubmcom 19131 | The operation of a monoid ... |
| cycsubggend 19132 | The cyclic subgroup genera... |
| cycsubgcl 19133 | The set of integer powers ... |
| cycsubgss 19134 | The cyclic subgroup genera... |
| cycsubg 19135 | The cyclic group generated... |
| cycsubgcld 19136 | The cyclic subgroup genera... |
| cycsubg2 19137 | The subgroup generated by ... |
| cycsubg2cl 19138 | Any multiple of an element... |
| reldmghm 19141 | Lemma for group homomorphi... |
| isghm 19142 | Property of being a homomo... |
| isghmOLD 19143 | Obsolete version of ~ isgh... |
| isghm3 19144 | Property of a group homomo... |
| ghmgrp1 19145 | A group homomorphism is on... |
| ghmgrp2 19146 | A group homomorphism is on... |
| ghmf 19147 | A group homomorphism is a ... |
| ghmlin 19148 | A homomorphism of groups i... |
| ghmid 19149 | A homomorphism of groups p... |
| ghminv 19150 | A homomorphism of groups p... |
| ghmsub 19151 | Linearity of subtraction t... |
| isghmd 19152 | Deduction for a group homo... |
| ghmmhm 19153 | A group homomorphism is a ... |
| ghmmhmb 19154 | Group homomorphisms and mo... |
| ghmmulg 19155 | A group homomorphism prese... |
| ghmrn 19156 | The range of a homomorphis... |
| 0ghm 19157 | The constant zero linear f... |
| idghm 19158 | The identity homomorphism ... |
| resghm 19159 | Restriction of a homomorph... |
| resghm2 19160 | One direction of ~ resghm2... |
| resghm2b 19161 | Restriction of the codomai... |
| ghmghmrn 19162 | A group homomorphism from ... |
| ghmco 19163 | The composition of group h... |
| ghmima 19164 | The image of a subgroup un... |
| ghmpreima 19165 | The inverse image of a sub... |
| ghmeql 19166 | The equalizer of two group... |
| ghmnsgima 19167 | The image of a normal subg... |
| ghmnsgpreima 19168 | The inverse image of a nor... |
| ghmker 19169 | The kernel of a homomorphi... |
| ghmeqker 19170 | Two source points map to t... |
| pwsdiagghm 19171 | Diagonal homomorphism into... |
| f1ghm0to0 19172 | If a group homomorphism ` ... |
| ghmf1 19173 | Two ways of saying a group... |
| kerf1ghm 19174 | A group homomorphism ` F `... |
| ghmf1o 19175 | A bijective group homomorp... |
| conjghm 19176 | Conjugation is an automorp... |
| conjsubg 19177 | A conjugated subgroup is a... |
| conjsubgen 19178 | A conjugated subgroup is e... |
| conjnmz 19179 | A subgroup is unchanged un... |
| conjnmzb 19180 | Alternative condition for ... |
| conjnsg 19181 | A normal subgroup is uncha... |
| qusghm 19182 | If ` Y ` is a normal subgr... |
| ghmpropd 19183 | Group homomorphism depends... |
| gimfn 19188 | The group isomorphism func... |
| isgim 19189 | An isomorphism of groups i... |
| gimf1o 19190 | An isomorphism of groups i... |
| gimghm 19191 | An isomorphism of groups i... |
| isgim2 19192 | A group isomorphism is a h... |
| subggim 19193 | Behavior of subgroups unde... |
| gimcnv 19194 | The converse of a group is... |
| gimco 19195 | The composition of group i... |
| gim0to0 19196 | A group isomorphism maps t... |
| brgic 19197 | The relation "is isomorphi... |
| brgici 19198 | Prove isomorphic by an exp... |
| gicref 19199 | Isomorphism is reflexive. ... |
| giclcl 19200 | Isomorphism implies the le... |
| gicrcl 19201 | Isomorphism implies the ri... |
| gicsym 19202 | Isomorphism is symmetric. ... |
| gictr 19203 | Isomorphism is transitive.... |
| gicer 19204 | Isomorphism is an equivale... |
| gicen 19205 | Isomorphic groups have equ... |
| gicsubgen 19206 | A less trivial example of ... |
| ghmqusnsglem1 19207 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19208 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19209 | The mapping ` H ` induced ... |
| ghmquskerlem1 19210 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19211 | In the case of theorem ~ g... |
| ghmquskerlem2 19212 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19213 | The mapping ` H ` induced ... |
| ghmqusker 19214 | A surjective group homomor... |
| gicqusker 19215 | The image ` H ` of a group... |
| isga 19218 | The predicate "is a (left)... |
| gagrp 19219 | The left argument of a gro... |
| gaset 19220 | The right argument of a gr... |
| gagrpid 19221 | The identity of the group ... |
| gaf 19222 | The mapping of the group a... |
| gafo 19223 | A group action is onto its... |
| gaass 19224 | An "associative" property ... |
| ga0 19225 | The action of a group on t... |
| gaid 19226 | The trivial action of a gr... |
| subgga 19227 | A subgroup acts on its par... |
| gass 19228 | A subset of a group action... |
| gasubg 19229 | The restriction of a group... |
| gaid2 19230 | A group operation is a lef... |
| galcan 19231 | The action of a particular... |
| gacan 19232 | Group inverses cancel in a... |
| gapm 19233 | The action of a particular... |
| gaorb 19234 | The orbit equivalence rela... |
| gaorber 19235 | The orbit equivalence rela... |
| gastacl 19236 | The stabilizer subgroup in... |
| gastacos 19237 | Write the coset relation f... |
| orbstafun 19238 | Existence and uniqueness f... |
| orbstaval 19239 | Value of the function at a... |
| orbsta 19240 | The Orbit-Stabilizer theor... |
| orbsta2 19241 | Relation between the size ... |
| cntrval 19246 | Substitute definition of t... |
| cntzfval 19247 | First level substitution f... |
| cntzval 19248 | Definition substitution fo... |
| elcntz 19249 | Elementhood in the central... |
| cntzel 19250 | Membership in a centralize... |
| cntzsnval 19251 | Special substitution for t... |
| elcntzsn 19252 | Value of the centralizer o... |
| sscntz 19253 | A centralizer expression f... |
| cntzrcl 19254 | Reverse closure for elemen... |
| cntzssv 19255 | The centralizer is uncondi... |
| cntzi 19256 | Membership in a centralize... |
| elcntr 19257 | Elementhood in the center ... |
| cntrss 19258 | The center is a subset of ... |
| cntri 19259 | Defining property of the c... |
| resscntz 19260 | Centralizer in a substruct... |
| cntzsgrpcl 19261 | Centralizers are closed un... |
| cntz2ss 19262 | Centralizers reverse the s... |
| cntzrec 19263 | Reciprocity relationship f... |
| cntziinsn 19264 | Express any centralizer as... |
| cntzsubm 19265 | Centralizers in a monoid a... |
| cntzsubg 19266 | Centralizers in a group ar... |
| cntzidss 19267 | If the elements of ` S ` c... |
| cntzmhm 19268 | Centralizers in a monoid a... |
| cntzmhm2 19269 | Centralizers in a monoid a... |
| cntrsubgnsg 19270 | A central subgroup is norm... |
| cntrnsg 19271 | The center of a group is a... |
| oppgval 19274 | Value of the opposite grou... |
| oppgplusfval 19275 | Value of the addition oper... |
| oppgplus 19276 | Value of the addition oper... |
| setsplusg 19277 | The other components of an... |
| oppgbas 19278 | Base set of an opposite gr... |
| oppgtset 19279 | Topology of an opposite gr... |
| oppgtopn 19280 | Topology of an opposite gr... |
| oppgmnd 19281 | The opposite of a monoid i... |
| oppgmndb 19282 | Bidirectional form of ~ op... |
| oppgid 19283 | Zero in a monoid is a symm... |
| oppggrp 19284 | The opposite of a group is... |
| oppggrpb 19285 | Bidirectional form of ~ op... |
| oppginv 19286 | Inverses in a group are a ... |
| invoppggim 19287 | The inverse is an antiauto... |
| oppggic 19288 | Every group is (naturally)... |
| oppgsubm 19289 | Being a submonoid is a sym... |
| oppgsubg 19290 | Being a subgroup is a symm... |
| oppgcntz 19291 | A centralizer in a group i... |
| oppgcntr 19292 | The center of a group is t... |
| gsumwrev 19293 | A sum in an opposite monoi... |
| oppgle 19294 | less-than relation of an o... |
| oppglt 19295 | less-than relation of an o... |
| symgval 19298 | The value of the symmetric... |
| symgbas 19299 | The base set of the symmet... |
| elsymgbas2 19300 | Two ways of saying a funct... |
| elsymgbas 19301 | Two ways of saying a funct... |
| symgbasf1o 19302 | Elements in the symmetric ... |
| symgbasf 19303 | A permutation (element of ... |
| symgbasmap 19304 | A permutation (element of ... |
| symghash 19305 | The symmetric group on ` n... |
| symgbasfi 19306 | The symmetric group on a f... |
| symgfv 19307 | The function value of a pe... |
| symgfvne 19308 | The function values of a p... |
| symgressbas 19309 | The symmetric group on ` A... |
| symgplusg 19310 | The group operation of a s... |
| symgov 19311 | The value of the group ope... |
| symgcl 19312 | The group operation of the... |
| idresperm 19313 | The identity function rest... |
| symgmov1 19314 | For a permutation of a set... |
| symgmov2 19315 | For a permutation of a set... |
| symgbas0 19316 | The base set of the symmet... |
| symg1hash 19317 | The symmetric group on a s... |
| symg1bas 19318 | The symmetric group on a s... |
| symg2hash 19319 | The symmetric group on a (... |
| symg2bas 19320 | The symmetric group on a p... |
| 0symgefmndeq 19321 | The symmetric group on the... |
| snsymgefmndeq 19322 | The symmetric group on a s... |
| symgpssefmnd 19323 | For a set ` A ` with more ... |
| symgvalstruct 19324 | The value of the symmetric... |
| symgsubmefmnd 19325 | The symmetric group on a s... |
| symgtset 19326 | The topology of the symmet... |
| symggrp 19327 | The symmetric group on a s... |
| symgid 19328 | The group identity element... |
| symginv 19329 | The group inverse in the s... |
| symgsubmefmndALT 19330 | The symmetric group on a s... |
| galactghm 19331 | The currying of a group ac... |
| lactghmga 19332 | The converse of ~ galactgh... |
| symgtopn 19333 | The topology of the symmet... |
| symgga 19334 | The symmetric group induce... |
| pgrpsubgsymgbi 19335 | Every permutation group is... |
| pgrpsubgsymg 19336 | Every permutation group is... |
| idressubgsymg 19337 | The singleton containing o... |
| idrespermg 19338 | The structure with the sin... |
| cayleylem1 19339 | Lemma for ~ cayley . (Con... |
| cayleylem2 19340 | Lemma for ~ cayley . (Con... |
| cayley 19341 | Cayley's Theorem (construc... |
| cayleyth 19342 | Cayley's Theorem (existenc... |
| symgfix2 19343 | If a permutation does not ... |
| symgextf 19344 | The extension of a permuta... |
| symgextfv 19345 | The function value of the ... |
| symgextfve 19346 | The function value of the ... |
| symgextf1lem 19347 | Lemma for ~ symgextf1 . (... |
| symgextf1 19348 | The extension of a permuta... |
| symgextfo 19349 | The extension of a permuta... |
| symgextf1o 19350 | The extension of a permuta... |
| symgextsymg 19351 | The extension of a permuta... |
| symgextres 19352 | The restriction of the ext... |
| gsumccatsymgsn 19353 | Homomorphic property of co... |
| gsmsymgrfixlem1 19354 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19355 | The composition of permuta... |
| fvcosymgeq 19356 | The values of two composit... |
| gsmsymgreqlem1 19357 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19358 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19359 | Two combination of permuta... |
| symgfixelq 19360 | A permutation of a set fix... |
| symgfixels 19361 | The restriction of a permu... |
| symgfixelsi 19362 | The restriction of a permu... |
| symgfixf 19363 | The mapping of a permutati... |
| symgfixf1 19364 | The mapping of a permutati... |
| symgfixfolem1 19365 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19366 | The mapping of a permutati... |
| symgfixf1o 19367 | The mapping of a permutati... |
| f1omvdmvd 19370 | A permutation of any class... |
| f1omvdcnv 19371 | A permutation and its inve... |
| mvdco 19372 | Composing two permutations... |
| f1omvdconj 19373 | Conjugation of a permutati... |
| f1otrspeq 19374 | A transposition is charact... |
| f1omvdco2 19375 | If exactly one of two perm... |
| f1omvdco3 19376 | If a point is moved by exa... |
| pmtrfval 19377 | The function generating tr... |
| pmtrval 19378 | A generated transposition,... |
| pmtrfv 19379 | General value of mapping a... |
| pmtrprfv 19380 | In a transposition of two ... |
| pmtrprfv3 19381 | In a transposition of two ... |
| pmtrf 19382 | Functionality of a transpo... |
| pmtrmvd 19383 | A transposition moves prec... |
| pmtrrn 19384 | Transposing two points giv... |
| pmtrfrn 19385 | A transposition (as a kind... |
| pmtrffv 19386 | Mapping of a point under a... |
| pmtrrn2 19387 | For any transposition ther... |
| pmtrfinv 19388 | A transposition function i... |
| pmtrfmvdn0 19389 | A transposition moves at l... |
| pmtrff1o 19390 | A transposition function i... |
| pmtrfcnv 19391 | A transposition function i... |
| pmtrfb 19392 | An intrinsic characterizat... |
| pmtrfconj 19393 | Any conjugate of a transpo... |
| symgsssg 19394 | The symmetric group has su... |
| symgfisg 19395 | The symmetric group has a ... |
| symgtrf 19396 | Transpositions are element... |
| symggen 19397 | The span of the transposit... |
| symggen2 19398 | A finite permutation group... |
| symgtrinv 19399 | To invert a permutation re... |
| pmtr3ncomlem1 19400 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19401 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19402 | Transpositions over sets w... |
| pmtrdifellem1 19403 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19404 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19405 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19406 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19407 | A transposition of element... |
| pmtrdifwrdellem1 19408 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19409 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19410 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19411 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19412 | A sequence of transpositio... |
| pmtrdifwrdel2 19413 | A sequence of transpositio... |
| pmtrprfval 19414 | The transpositions on a pa... |
| pmtrprfvalrn 19415 | The range of the transposi... |
| psgnunilem1 19420 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19421 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19422 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19423 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19424 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19425 | Addition and subtraction o... |
| psgnuni 19426 | If the same permutation ca... |
| psgnfval 19427 | Function definition of the... |
| psgnfn 19428 | Functionality and domain o... |
| psgndmsubg 19429 | The finitary permutations ... |
| psgneldm 19430 | Property of being a finita... |
| psgneldm2 19431 | The finitary permutations ... |
| psgneldm2i 19432 | A sequence of transpositio... |
| psgneu 19433 | A finitary permutation has... |
| psgnval 19434 | Value of the permutation s... |
| psgnvali 19435 | A finitary permutation has... |
| psgnvalii 19436 | Any representation of a pe... |
| psgnpmtr 19437 | All transpositions are odd... |
| psgn0fv0 19438 | The permutation sign funct... |
| sygbasnfpfi 19439 | The class of non-fixed poi... |
| psgnfvalfi 19440 | Function definition of the... |
| psgnvalfi 19441 | Value of the permutation s... |
| psgnran 19442 | The range of the permutati... |
| gsmtrcl 19443 | The group sum of transposi... |
| psgnfitr 19444 | A permutation of a finite ... |
| psgnfieu 19445 | A permutation of a finite ... |
| pmtrsn 19446 | The value of the transposi... |
| psgnsn 19447 | The permutation sign funct... |
| psgnprfval 19448 | The permutation sign funct... |
| psgnprfval1 19449 | The permutation sign of th... |
| psgnprfval2 19450 | The permutation sign of th... |
| odfval 19459 | Value of the order functio... |
| odfvalALT 19460 | Shorter proof of ~ odfval ... |
| odval 19461 | Second substitution for th... |
| odlem1 19462 | The group element order is... |
| odcl 19463 | The order of a group eleme... |
| odf 19464 | Functionality of the group... |
| odid 19465 | Any element to the power o... |
| odlem2 19466 | Any positive annihilator o... |
| odmodnn0 19467 | Reduce the argument of a g... |
| mndodconglem 19468 | Lemma for ~ mndodcong . (... |
| mndodcong 19469 | If two multipliers are con... |
| mndodcongi 19470 | If two multipliers are con... |
| oddvdsnn0 19471 | The only multiples of ` A ... |
| odnncl 19472 | If a nonzero multiple of a... |
| odmod 19473 | Reduce the argument of a g... |
| oddvds 19474 | The only multiples of ` A ... |
| oddvdsi 19475 | Any group element is annih... |
| odcong 19476 | If two multipliers are con... |
| odeq 19477 | The ~ oddvds property uniq... |
| odval2 19478 | A non-conditional definiti... |
| odcld 19479 | The order of a group eleme... |
| odm1inv 19480 | The (order-1)th multiple o... |
| odmulgid 19481 | A relationship between the... |
| odmulg2 19482 | The order of a multiple di... |
| odmulg 19483 | Relationship between the o... |
| odmulgeq 19484 | A multiple of a point of f... |
| odbezout 19485 | If ` N ` is coprime to the... |
| od1 19486 | The order of the group ide... |
| odeq1 19487 | The group identity is the ... |
| odinv 19488 | The order of the inverse o... |
| odf1 19489 | The multiples of an elemen... |
| odinf 19490 | The multiples of an elemen... |
| dfod2 19491 | An alternative definition ... |
| odcl2 19492 | The order of an element of... |
| oddvds2 19493 | The order of an element of... |
| finodsubmsubg 19494 | A submonoid whose elements... |
| 0subgALT 19495 | A shorter proof of ~ 0subg... |
| submod 19496 | The order of an element is... |
| subgod 19497 | The order of an element is... |
| odsubdvds 19498 | The order of an element of... |
| odf1o1 19499 | An element with zero order... |
| odf1o2 19500 | An element with nonzero or... |
| odhash 19501 | An element of zero order g... |
| odhash2 19502 | If an element has nonzero ... |
| odhash3 19503 | An element which generates... |
| odngen 19504 | A cyclic subgroup of size ... |
| gexval 19505 | Value of the exponent of a... |
| gexlem1 19506 | The group element order is... |
| gexcl 19507 | The exponent of a group is... |
| gexid 19508 | Any element to the power o... |
| gexlem2 19509 | Any positive annihilator o... |
| gexdvdsi 19510 | Any group element is annih... |
| gexdvds 19511 | The only ` N ` that annihi... |
| gexdvds2 19512 | An integer divides the gro... |
| gexod 19513 | Any group element is annih... |
| gexcl3 19514 | If the order of every grou... |
| gexnnod 19515 | Every group element has fi... |
| gexcl2 19516 | The exponent of a finite g... |
| gexdvds3 19517 | The exponent of a finite g... |
| gex1 19518 | A group or monoid has expo... |
| ispgp 19519 | A group is a ` P ` -group ... |
| pgpprm 19520 | Reverse closure for the fi... |
| pgpgrp 19521 | Reverse closure for the se... |
| pgpfi1 19522 | A finite group with order ... |
| pgp0 19523 | The identity subgroup is a... |
| subgpgp 19524 | A subgroup of a p-group is... |
| sylow1lem1 19525 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19526 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19527 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19528 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19529 | Lemma for ~ sylow1 . Usin... |
| sylow1 19530 | Sylow's first theorem. If... |
| odcau 19531 | Cauchy's theorem for the o... |
| pgpfi 19532 | The converse to ~ pgpfi1 .... |
| pgpfi2 19533 | Alternate version of ~ pgp... |
| pgphash 19534 | The order of a p-group. (... |
| isslw 19535 | The property of being a Sy... |
| slwprm 19536 | Reverse closure for the fi... |
| slwsubg 19537 | A Sylow ` P ` -subgroup is... |
| slwispgp 19538 | Defining property of a Syl... |
| slwpss 19539 | A proper superset of a Syl... |
| slwpgp 19540 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19541 | Every ` P ` -subgroup is c... |
| slwn0 19542 | Every finite group contain... |
| subgslw 19543 | A Sylow subgroup that is c... |
| sylow2alem1 19544 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19545 | Lemma for ~ sylow2a . All... |
| sylow2a 19546 | A named lemma of Sylow's s... |
| sylow2blem1 19547 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19548 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19549 | Sylow's second theorem. P... |
| sylow2b 19550 | Sylow's second theorem. A... |
| slwhash 19551 | A sylow subgroup has cardi... |
| fislw 19552 | The sylow subgroups of a f... |
| sylow2 19553 | Sylow's second theorem. S... |
| sylow3lem1 19554 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19555 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19556 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19557 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19558 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19559 | Lemma for ~ sylow3 , secon... |
| sylow3 19560 | Sylow's third theorem. Th... |
| lsmfval 19565 | The subgroup sum function ... |
| lsmvalx 19566 | Subspace sum value (for a ... |
| lsmelvalx 19567 | Subspace sum membership (f... |
| lsmelvalix 19568 | Subspace sum membership (f... |
| oppglsm 19569 | The subspace sum operation... |
| lsmssv 19570 | Subgroup sum is a subset o... |
| lsmless1x 19571 | Subset implies subgroup su... |
| lsmless2x 19572 | Subset implies subgroup su... |
| lsmub1x 19573 | Subgroup sum is an upper b... |
| lsmub2x 19574 | Subgroup sum is an upper b... |
| lsmval 19575 | Subgroup sum value (for a ... |
| lsmelval 19576 | Subgroup sum membership (f... |
| lsmelvali 19577 | Subgroup sum membership (f... |
| lsmelvalm 19578 | Subgroup sum membership an... |
| lsmelvalmi 19579 | Membership of vector subtr... |
| lsmsubm 19580 | The sum of two commuting s... |
| lsmsubg 19581 | The sum of two commuting s... |
| lsmcom2 19582 | Subgroup sum commutes. (C... |
| smndlsmidm 19583 | The direct product is idem... |
| lsmub1 19584 | Subgroup sum is an upper b... |
| lsmub2 19585 | Subgroup sum is an upper b... |
| lsmunss 19586 | Union of subgroups is a su... |
| lsmless1 19587 | Subset implies subgroup su... |
| lsmless2 19588 | Subset implies subgroup su... |
| lsmless12 19589 | Subset implies subgroup su... |
| lsmidm 19590 | Subgroup sum is idempotent... |
| lsmlub 19591 | The least upper bound prop... |
| lsmss1 19592 | Subgroup sum with a subset... |
| lsmss1b 19593 | Subgroup sum with a subset... |
| lsmss2 19594 | Subgroup sum with a subset... |
| lsmss2b 19595 | Subgroup sum with a subset... |
| lsmass 19596 | Subgroup sum is associativ... |
| mndlsmidm 19597 | Subgroup sum is idempotent... |
| lsm01 19598 | Subgroup sum with the zero... |
| lsm02 19599 | Subgroup sum with the zero... |
| subglsm 19600 | The subgroup sum evaluated... |
| lssnle 19601 | Equivalent expressions for... |
| lsmmod 19602 | The modular law holds for ... |
| lsmmod2 19603 | Modular law dual for subgr... |
| lsmpropd 19604 | If two structures have the... |
| cntzrecd 19605 | Commute the "subgroups com... |
| lsmcntz 19606 | The "subgroups commute" pr... |
| lsmcntzr 19607 | The "subgroups commute" pr... |
| lsmdisj 19608 | Disjointness from a subgro... |
| lsmdisj2 19609 | Association of the disjoin... |
| lsmdisj3 19610 | Association of the disjoin... |
| lsmdisjr 19611 | Disjointness from a subgro... |
| lsmdisj2r 19612 | Association of the disjoin... |
| lsmdisj3r 19613 | Association of the disjoin... |
| lsmdisj2a 19614 | Association of the disjoin... |
| lsmdisj2b 19615 | Association of the disjoin... |
| lsmdisj3a 19616 | Association of the disjoin... |
| lsmdisj3b 19617 | Association of the disjoin... |
| subgdisj1 19618 | Vectors belonging to disjo... |
| subgdisj2 19619 | Vectors belonging to disjo... |
| subgdisjb 19620 | Vectors belonging to disjo... |
| pj1fval 19621 | The left projection functi... |
| pj1val 19622 | The left projection functi... |
| pj1eu 19623 | Uniqueness of a left proje... |
| pj1f 19624 | The left projection functi... |
| pj2f 19625 | The right projection funct... |
| pj1id 19626 | Any element of a direct su... |
| pj1eq 19627 | Any element of a direct su... |
| pj1lid 19628 | The left projection functi... |
| pj1rid 19629 | The left projection functi... |
| pj1ghm 19630 | The left projection functi... |
| pj1ghm2 19631 | The left projection functi... |
| lsmhash 19632 | The order of the direct pr... |
| efgmval 19639 | Value of the formal invers... |
| efgmf 19640 | The formal inverse operati... |
| efgmnvl 19641 | The inversion function on ... |
| efgrcl 19642 | Lemma for ~ efgval . (Con... |
| efglem 19643 | Lemma for ~ efgval . (Con... |
| efgval 19644 | Value of the free group co... |
| efger 19645 | Value of the free group co... |
| efgi 19646 | Value of the free group co... |
| efgi0 19647 | Value of the free group co... |
| efgi1 19648 | Value of the free group co... |
| efgtf 19649 | Value of the free group co... |
| efgtval 19650 | Value of the extension fun... |
| efgval2 19651 | Value of the free group co... |
| efgi2 19652 | Value of the free group co... |
| efgtlen 19653 | Value of the free group co... |
| efginvrel2 19654 | The inverse of the reverse... |
| efginvrel1 19655 | The inverse of the reverse... |
| efgsf 19656 | Value of the auxiliary fun... |
| efgsdm 19657 | Elementhood in the domain ... |
| efgsval 19658 | Value of the auxiliary fun... |
| efgsdmi 19659 | Property of the last link ... |
| efgsval2 19660 | Value of the auxiliary fun... |
| efgsrel 19661 | The start and end of any e... |
| efgs1 19662 | A singleton of an irreduci... |
| efgs1b 19663 | Every extension sequence e... |
| efgsp1 19664 | If ` F ` is an extension s... |
| efgsres 19665 | An initial segment of an e... |
| efgsfo 19666 | For any word, there is a s... |
| efgredlema 19667 | The reduced word that form... |
| efgredlemf 19668 | Lemma for ~ efgredleme . ... |
| efgredlemg 19669 | Lemma for ~ efgred . (Con... |
| efgredleme 19670 | Lemma for ~ efgred . (Con... |
| efgredlemd 19671 | The reduced word that form... |
| efgredlemc 19672 | The reduced word that form... |
| efgredlemb 19673 | The reduced word that form... |
| efgredlem 19674 | The reduced word that form... |
| efgred 19675 | The reduced word that form... |
| efgrelexlema 19676 | If two words ` A , B ` are... |
| efgrelexlemb 19677 | If two words ` A , B ` are... |
| efgrelex 19678 | If two words ` A , B ` are... |
| efgredeu 19679 | There is a unique reduced ... |
| efgred2 19680 | Two extension sequences ha... |
| efgcpbllema 19681 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19682 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19683 | Two extension sequences ha... |
| efgcpbl2 19684 | Two extension sequences ha... |
| frgpval 19685 | Value of the free group co... |
| frgpcpbl 19686 | Compatibility of the group... |
| frgp0 19687 | The free group is a group.... |
| frgpeccl 19688 | Closure of the quotient ma... |
| frgpgrp 19689 | The free group is a group.... |
| frgpadd 19690 | Addition in the free group... |
| frgpinv 19691 | The inverse of an element ... |
| frgpmhm 19692 | The "natural map" from wor... |
| vrgpfval 19693 | The canonical injection fr... |
| vrgpval 19694 | The value of the generatin... |
| vrgpf 19695 | The mapping from the index... |
| vrgpinv 19696 | The inverse of a generatin... |
| frgpuptf 19697 | Any assignment of the gene... |
| frgpuptinv 19698 | Any assignment of the gene... |
| frgpuplem 19699 | Any assignment of the gene... |
| frgpupf 19700 | Any assignment of the gene... |
| frgpupval 19701 | Any assignment of the gene... |
| frgpup1 19702 | Any assignment of the gene... |
| frgpup2 19703 | The evaluation map has the... |
| frgpup3lem 19704 | The evaluation map has the... |
| frgpup3 19705 | Universal property of the ... |
| 0frgp 19706 | The free group on zero gen... |
| isabl 19711 | The predicate "is an Abeli... |
| ablgrp 19712 | An Abelian group is a grou... |
| ablgrpd 19713 | An Abelian group is a grou... |
| ablcmn 19714 | An Abelian group is a comm... |
| ablcmnd 19715 | An Abelian group is a comm... |
| iscmn 19716 | The predicate "is a commut... |
| isabl2 19717 | The predicate "is an Abeli... |
| cmnpropd 19718 | If two structures have the... |
| ablpropd 19719 | If two structures have the... |
| ablprop 19720 | If two structures have the... |
| iscmnd 19721 | Properties that determine ... |
| isabld 19722 | Properties that determine ... |
| isabli 19723 | Properties that determine ... |
| cmnmnd 19724 | A commutative monoid is a ... |
| cmncom 19725 | A commutative monoid is co... |
| ablcom 19726 | An Abelian group operation... |
| cmn32 19727 | Commutative/associative la... |
| cmn4 19728 | Commutative/associative la... |
| cmn12 19729 | Commutative/associative la... |
| abl32 19730 | Commutative/associative la... |
| cmnmndd 19731 | A commutative monoid is a ... |
| cmnbascntr 19732 | The base set of a commutat... |
| rinvmod 19733 | Uniqueness of a right inve... |
| ablinvadd 19734 | The inverse of an Abelian ... |
| ablsub2inv 19735 | Abelian group subtraction ... |
| ablsubadd 19736 | Relationship between Abeli... |
| ablsub4 19737 | Commutative/associative su... |
| abladdsub4 19738 | Abelian group addition/sub... |
| abladdsub 19739 | Associative-type law for g... |
| ablsubadd23 19740 | Commutative/associative la... |
| ablsubaddsub 19741 | Double subtraction and add... |
| ablpncan2 19742 | Cancellation law for subtr... |
| ablpncan3 19743 | A cancellation law for Abe... |
| ablsubsub 19744 | Law for double subtraction... |
| ablsubsub4 19745 | Law for double subtraction... |
| ablpnpcan 19746 | Cancellation law for mixed... |
| ablnncan 19747 | Cancellation law for group... |
| ablsub32 19748 | Swap the second and third ... |
| ablnnncan 19749 | Cancellation law for group... |
| ablnnncan1 19750 | Cancellation law for group... |
| ablsubsub23 19751 | Swap subtrahend and result... |
| mulgnn0di 19752 | Group multiple of a sum, f... |
| mulgdi 19753 | Group multiple of a sum. ... |
| mulgmhm 19754 | The map from ` x ` to ` n ... |
| mulgghm 19755 | The map from ` x ` to ` n ... |
| mulgsubdi 19756 | Group multiple of a differ... |
| ghmfghm 19757 | The function fulfilling th... |
| ghmcmn 19758 | The image of a commutative... |
| ghmabl 19759 | The image of an abelian gr... |
| invghm 19760 | The inversion map is a gro... |
| eqgabl 19761 | Value of the subgroup cose... |
| qusecsub 19762 | Two subgroup cosets are eq... |
| subgabl 19763 | A subgroup of an abelian g... |
| subcmn 19764 | A submonoid of a commutati... |
| submcmn 19765 | A submonoid of a commutati... |
| submcmn2 19766 | A submonoid is commutative... |
| cntzcmn 19767 | The centralizer of any sub... |
| cntzcmnss 19768 | Any subset in a commutativ... |
| cntrcmnd 19769 | The center of a monoid is ... |
| cntrabl 19770 | The center of a group is a... |
| cntzspan 19771 | If the generators commute,... |
| cntzcmnf 19772 | Discharge the centralizer ... |
| ghmplusg 19773 | The pointwise sum of two l... |
| ablnsg 19774 | Every subgroup of an abeli... |
| odadd1 19775 | The order of a product in ... |
| odadd2 19776 | The order of a product in ... |
| odadd 19777 | The order of a product is ... |
| gex2abl 19778 | A group with exponent 2 (o... |
| gexexlem 19779 | Lemma for ~ gexex . (Cont... |
| gexex 19780 | In an abelian group with f... |
| torsubg 19781 | The set of all elements of... |
| oddvdssubg 19782 | The set of all elements wh... |
| lsmcomx 19783 | Subgroup sum commutes (ext... |
| ablcntzd 19784 | All subgroups in an abelia... |
| lsmcom 19785 | Subgroup sum commutes. (C... |
| lsmsubg2 19786 | The sum of two subgroups i... |
| lsm4 19787 | Commutative/associative la... |
| prdscmnd 19788 | The product of a family of... |
| prdsabld 19789 | The product of a family of... |
| pwscmn 19790 | The structure power on a c... |
| pwsabl 19791 | The structure power on an ... |
| qusabl 19792 | If ` Y ` is a subgroup of ... |
| abl1 19793 | The (smallest) structure r... |
| abln0 19794 | Abelian groups (and theref... |
| cnaddablx 19795 | The complex numbers are an... |
| cnaddabl 19796 | The complex numbers are an... |
| cnaddid 19797 | The group identity element... |
| cnaddinv 19798 | Value of the group inverse... |
| zaddablx 19799 | The integers are an Abelia... |
| frgpnabllem1 19800 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19801 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19802 | The free group on two or m... |
| imasabl 19803 | The image structure of an ... |
| iscyg 19806 | Definition of a cyclic gro... |
| iscyggen 19807 | The property of being a cy... |
| iscyggen2 19808 | The property of being a cy... |
| iscyg2 19809 | A cyclic group is a group ... |
| cyggeninv 19810 | The inverse of a cyclic ge... |
| cyggenod 19811 | An element is the generato... |
| cyggenod2 19812 | In an infinite cyclic grou... |
| iscyg3 19813 | Definition of a cyclic gro... |
| iscygd 19814 | Definition of a cyclic gro... |
| iscygodd 19815 | Show that a group with an ... |
| cycsubmcmn 19816 | The set of nonnegative int... |
| cyggrp 19817 | A cyclic group is a group.... |
| cygabl 19818 | A cyclic group is abelian.... |
| cygctb 19819 | A cyclic group is countabl... |
| 0cyg 19820 | The trivial group is cycli... |
| prmcyg 19821 | A group with prime order i... |
| lt6abl 19822 | A group with fewer than ` ... |
| ghmcyg 19823 | The image of a cyclic grou... |
| cyggex2 19824 | The exponent of a cyclic g... |
| cyggex 19825 | The exponent of a finite c... |
| cyggexb 19826 | A finite abelian group is ... |
| giccyg 19827 | Cyclicity is a group prope... |
| cycsubgcyg 19828 | The cyclic subgroup genera... |
| cycsubgcyg2 19829 | The cyclic subgroup genera... |
| gsumval3a 19830 | Value of the group sum ope... |
| gsumval3eu 19831 | The group sum as defined i... |
| gsumval3lem1 19832 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19833 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19834 | Value of the group sum ope... |
| gsumcllem 19835 | Lemma for ~ gsumcl and rel... |
| gsumzres 19836 | Extend a finite group sum ... |
| gsumzcl2 19837 | Closure of a finite group ... |
| gsumzcl 19838 | Closure of a finite group ... |
| gsumzf1o 19839 | Re-index a finite group su... |
| gsumres 19840 | Extend a finite group sum ... |
| gsumcl2 19841 | Closure of a finite group ... |
| gsumcl 19842 | Closure of a finite group ... |
| gsumf1o 19843 | Re-index a finite group su... |
| gsumreidx 19844 | Re-index a finite group su... |
| gsumzsubmcl 19845 | Closure of a group sum in ... |
| gsumsubmcl 19846 | Closure of a group sum in ... |
| gsumsubgcl 19847 | Closure of a group sum in ... |
| gsumzaddlem 19848 | The sum of two group sums.... |
| gsumzadd 19849 | The sum of two group sums.... |
| gsumadd 19850 | The sum of two group sums.... |
| gsummptfsadd 19851 | The sum of two group sums ... |
| gsummptfidmadd 19852 | The sum of two group sums ... |
| gsummptfidmadd2 19853 | The sum of two group sums ... |
| gsumzsplit 19854 | Split a group sum into two... |
| gsumsplit 19855 | Split a group sum into two... |
| gsumsplit2 19856 | Split a group sum into two... |
| gsummptfidmsplit 19857 | Split a group sum expresse... |
| gsummptfidmsplitres 19858 | Split a group sum expresse... |
| gsummptfzsplit 19859 | Split a group sum expresse... |
| gsummptfzsplitl 19860 | Split a group sum expresse... |
| gsumconst 19861 | Sum of a constant series. ... |
| gsumconstf 19862 | Sum of a constant series. ... |
| gsummptshft 19863 | Index shift of a finite gr... |
| gsumzmhm 19864 | Apply a group homomorphism... |
| gsummhm 19865 | Apply a group homomorphism... |
| gsummhm2 19866 | Apply a group homomorphism... |
| gsummptmhm 19867 | Apply a group homomorphism... |
| gsummulglem 19868 | Lemma for ~ gsummulg and ~... |
| gsummulg 19869 | Nonnegative multiple of a ... |
| gsummulgz 19870 | Integer multiple of a grou... |
| gsumzoppg 19871 | The opposite of a group su... |
| gsumzinv 19872 | Inverse of a group sum. (... |
| gsuminv 19873 | Inverse of a group sum. (... |
| gsummptfidminv 19874 | Inverse of a group sum exp... |
| gsumsub 19875 | The difference of two grou... |
| gsummptfssub 19876 | The difference of two grou... |
| gsummptfidmsub 19877 | The difference of two grou... |
| gsumsnfd 19878 | Group sum of a singleton, ... |
| gsumsnd 19879 | Group sum of a singleton, ... |
| gsumsnf 19880 | Group sum of a singleton, ... |
| gsumsn 19881 | Group sum of a singleton. ... |
| gsumpr 19882 | Group sum of a pair. (Con... |
| gsumzunsnd 19883 | Append an element to a fin... |
| gsumunsnfd 19884 | Append an element to a fin... |
| gsumunsnd 19885 | Append an element to a fin... |
| gsumunsnf 19886 | Append an element to a fin... |
| gsumunsn 19887 | Append an element to a fin... |
| gsumdifsnd 19888 | Extract a summand from a f... |
| gsumpt 19889 | Sum of a family that is no... |
| gsummptf1o 19890 | Re-index a finite group su... |
| gsummptun 19891 | Group sum of a disjoint un... |
| gsummpt1n0 19892 | If only one summand in a f... |
| gsummptif1n0 19893 | If only one summand in a f... |
| gsummptcl 19894 | Closure of a finite group ... |
| gsummptfif1o 19895 | Re-index a finite group su... |
| gsummptfzcl 19896 | Closure of a finite group ... |
| gsum2dlem1 19897 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19898 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19899 | Write a sum over a two-dim... |
| gsum2d2lem 19900 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19901 | Write a group sum over a t... |
| gsumcom2 19902 | Two-dimensional commutatio... |
| gsumxp 19903 | Write a group sum over a c... |
| gsumcom 19904 | Commute the arguments of a... |
| gsumcom3 19905 | A commutative law for fini... |
| gsumcom3fi 19906 | A commutative law for fini... |
| gsumxp2 19907 | Write a group sum over a c... |
| prdsgsum 19908 | Finite commutative sums in... |
| pwsgsum 19909 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19910 | Replacing a finitely suppo... |
| nn0gsumfz 19911 | Replacing a finitely suppo... |
| nn0gsumfz0 19912 | Replacing a finitely suppo... |
| gsummptnn0fz 19913 | A final group sum over a f... |
| gsummptnn0fzfv 19914 | A final group sum over a f... |
| telgsumfzslem 19915 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19916 | Telescoping group sum rang... |
| telgsumfz 19917 | Telescoping group sum rang... |
| telgsumfz0s 19918 | Telescoping finite group s... |
| telgsumfz0 19919 | Telescoping finite group s... |
| telgsums 19920 | Telescoping finitely suppo... |
| telgsum 19921 | Telescoping finitely suppo... |
| reldmdprd 19926 | The domain of the internal... |
| dmdprd 19927 | The domain of definition o... |
| dmdprdd 19928 | Show that a given family i... |
| dprddomprc 19929 | A family of subgroups inde... |
| dprddomcld 19930 | If a family of subgroups i... |
| dprdval0prc 19931 | The internal direct produc... |
| dprdval 19932 | The value of the internal ... |
| eldprd 19933 | A class ` A ` is an intern... |
| dprdgrp 19934 | Reverse closure for the in... |
| dprdf 19935 | The function ` S ` is a fa... |
| dprdf2 19936 | The function ` S ` is a fa... |
| dprdcntz 19937 | The function ` S ` is a fa... |
| dprddisj 19938 | The function ` S ` is a fa... |
| dprdw 19939 | The property of being a fi... |
| dprdwd 19940 | A mapping being a finitely... |
| dprdff 19941 | A finitely supported funct... |
| dprdfcl 19942 | A finitely supported funct... |
| dprdffsupp 19943 | A finitely supported funct... |
| dprdfcntz 19944 | A function on the elements... |
| dprdssv 19945 | The internal direct produc... |
| dprdfid 19946 | A function mapping all but... |
| eldprdi 19947 | The domain of definition o... |
| dprdfinv 19948 | Take the inverse of a grou... |
| dprdfadd 19949 | Take the sum of group sums... |
| dprdfsub 19950 | Take the difference of gro... |
| dprdfeq0 19951 | The zero function is the o... |
| dprdf11 19952 | Two group sums over a dire... |
| dprdsubg 19953 | The internal direct produc... |
| dprdub 19954 | Each factor is a subset of... |
| dprdlub 19955 | The direct product is smal... |
| dprdspan 19956 | The direct product is the ... |
| dprdres 19957 | Restriction of a direct pr... |
| dprdss 19958 | Create a direct product by... |
| dprdz 19959 | A family consisting entire... |
| dprd0 19960 | The empty family is an int... |
| dprdf1o 19961 | Rearrange the index set of... |
| dprdf1 19962 | Rearrange the index set of... |
| subgdmdprd 19963 | A direct product in a subg... |
| subgdprd 19964 | A direct product in a subg... |
| dprdsn 19965 | A singleton family is an i... |
| dmdprdsplitlem 19966 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19967 | The function ` S ` is a fa... |
| dprddisj2 19968 | The function ` S ` is a fa... |
| dprd2dlem2 19969 | The direct product of a co... |
| dprd2dlem1 19970 | The direct product of a co... |
| dprd2da 19971 | The direct product of a co... |
| dprd2db 19972 | The direct product of a co... |
| dprd2d2 19973 | The direct product of a co... |
| dmdprdsplit2lem 19974 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19975 | The direct product splits ... |
| dmdprdsplit 19976 | The direct product splits ... |
| dprdsplit 19977 | The direct product is the ... |
| dmdprdpr 19978 | A singleton family is an i... |
| dprdpr 19979 | A singleton family is an i... |
| dpjlem 19980 | Lemma for theorems about d... |
| dpjcntz 19981 | The two subgroups that app... |
| dpjdisj 19982 | The two subgroups that app... |
| dpjlsm 19983 | The two subgroups that app... |
| dpjfval 19984 | Value of the direct produc... |
| dpjval 19985 | Value of the direct produc... |
| dpjf 19986 | The ` X ` -th index projec... |
| dpjidcl 19987 | The key property of projec... |
| dpjeq 19988 | Decompose a group sum into... |
| dpjid 19989 | The key property of projec... |
| dpjlid 19990 | The ` X ` -th index projec... |
| dpjrid 19991 | The ` Y ` -th index projec... |
| dpjghm 19992 | The direct product is the ... |
| dpjghm2 19993 | The direct product is the ... |
| ablfacrplem 19994 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 19995 | A finite abelian group who... |
| ablfacrp2 19996 | The factors ` K , L ` of ~... |
| ablfac1lem 19997 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 19998 | The factors of ~ ablfac1b ... |
| ablfac1b 19999 | Any abelian group is the d... |
| ablfac1c 20000 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20001 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20002 | The factorization of ~ abl... |
| pgpfac1lem1 20003 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20004 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20005 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20006 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20007 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20008 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20009 | Factorization of a finite ... |
| pgpfaclem1 20010 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20011 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20012 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20013 | Full factorization of a fi... |
| ablfaclem1 20014 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20015 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20016 | Lemma for ~ ablfac . (Con... |
| ablfac 20017 | The Fundamental Theorem of... |
| ablfac2 20018 | Choose generators for each... |
| issimpg 20021 | The predicate "is a simple... |
| issimpgd 20022 | Deduce a simple group from... |
| simpggrp 20023 | A simple group is a group.... |
| simpggrpd 20024 | A simple group is a group.... |
| simpg2nsg 20025 | A simple group has two nor... |
| trivnsimpgd 20026 | Trivial groups are not sim... |
| simpgntrivd 20027 | Simple groups are nontrivi... |
| simpgnideld 20028 | A simple group contains a ... |
| simpgnsgd 20029 | The only normal subgroups ... |
| simpgnsgeqd 20030 | A normal subgroup of a sim... |
| 2nsgsimpgd 20031 | If any normal subgroup of ... |
| simpgnsgbid 20032 | A nontrivial group is simp... |
| ablsimpnosubgd 20033 | A subgroup of an abelian s... |
| ablsimpg1gend 20034 | An abelian simple group is... |
| ablsimpgcygd 20035 | An abelian simple group is... |
| ablsimpgfindlem1 20036 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20037 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20038 | Relationship between the o... |
| ablsimpgfind 20039 | An abelian simple group is... |
| fincygsubgd 20040 | The subgroup referenced in... |
| fincygsubgodd 20041 | Calculate the order of a s... |
| fincygsubgodexd 20042 | A finite cyclic group has ... |
| prmgrpsimpgd 20043 | A group of prime order is ... |
| ablsimpgprmd 20044 | An abelian simple group ha... |
| ablsimpgd 20045 | An abelian group is simple... |
| isomnd 20050 | A (left) ordered monoid is... |
| isogrp 20051 | A (left-)ordered group is ... |
| ogrpgrp 20052 | A left-ordered group is a ... |
| omndmnd 20053 | A left-ordered monoid is a... |
| omndtos 20054 | A left-ordered monoid is a... |
| omndadd 20055 | In an ordered monoid, the ... |
| omndaddr 20056 | In a right ordered monoid,... |
| omndadd2d 20057 | In a commutative left orde... |
| omndadd2rd 20058 | In a left- and right- orde... |
| submomnd 20059 | A submonoid of an ordered ... |
| omndmul2 20060 | In an ordered monoid, the ... |
| omndmul3 20061 | In an ordered monoid, the ... |
| omndmul 20062 | In a commutative ordered m... |
| ogrpinv0le 20063 | In an ordered group, the o... |
| ogrpsub 20064 | In an ordered group, the o... |
| ogrpaddlt 20065 | In an ordered group, stric... |
| ogrpaddltbi 20066 | In a right ordered group, ... |
| ogrpaddltrd 20067 | In a right ordered group, ... |
| ogrpaddltrbid 20068 | In a right ordered group, ... |
| ogrpsublt 20069 | In an ordered group, stric... |
| ogrpinv0lt 20070 | In an ordered group, the o... |
| ogrpinvlt 20071 | In an ordered group, the o... |
| gsumle 20072 | A finite sum in an ordered... |
| fnmgp 20075 | The multiplicative group o... |
| mgpval 20076 | Value of the multiplicatio... |
| mgpplusg 20077 | Value of the group operati... |
| mgpbas 20078 | Base set of the multiplica... |
| mgpsca 20079 | The multiplication monoid ... |
| mgptset 20080 | Topology component of the ... |
| mgptopn 20081 | Topology of the multiplica... |
| mgpds 20082 | Distance function of the m... |
| mgpress 20083 | Subgroup commutes with the... |
| prdsmgp 20084 | The multiplicative monoid ... |
| isrng 20087 | The predicate "is a non-un... |
| rngabl 20088 | A non-unital ring is an (a... |
| rngmgp 20089 | A non-unital ring is a sem... |
| rngmgpf 20090 | Restricted functionality o... |
| rnggrp 20091 | A non-unital ring is a (ad... |
| rngass 20092 | Associative law for the mu... |
| rngdi 20093 | Distributive law for the m... |
| rngdir 20094 | Distributive law for the m... |
| rngacl 20095 | Closure of the addition op... |
| rng0cl 20096 | The zero element of a non-... |
| rngcl 20097 | Closure of the multiplicat... |
| rnglz 20098 | The zero of a non-unital r... |
| rngrz 20099 | The zero of a non-unital r... |
| rngmneg1 20100 | Negation of a product in a... |
| rngmneg2 20101 | Negation of a product in a... |
| rngm2neg 20102 | Double negation of a produ... |
| rngansg 20103 | Every additive subgroup of... |
| rngsubdi 20104 | Ring multiplication distri... |
| rngsubdir 20105 | Ring multiplication distri... |
| isrngd 20106 | Properties that determine ... |
| rngpropd 20107 | If two structures have the... |
| prdsmulrngcl 20108 | Closure of the multiplicat... |
| prdsrngd 20109 | A product of non-unital ri... |
| imasrng 20110 | The image structure of a n... |
| imasrngf1 20111 | The image of a non-unital ... |
| xpsrngd 20112 | A product of two non-unita... |
| qusrng 20113 | The quotient structure of ... |
| ringidval 20116 | The value of the unity ele... |
| dfur2 20117 | The multiplicative identit... |
| ringurd 20118 | Deduce the unity element o... |
| issrg 20121 | The predicate "is a semiri... |
| srgcmn 20122 | A semiring is a commutativ... |
| srgmnd 20123 | A semiring is a monoid. (... |
| srgmgp 20124 | A semiring is a monoid und... |
| srgdilem 20125 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20126 | Closure of the multiplicat... |
| srgass 20127 | Associative law for the mu... |
| srgideu 20128 | The unity element of a sem... |
| srgfcl 20129 | Functionality of the multi... |
| srgdi 20130 | Distributive law for the m... |
| srgdir 20131 | Distributive law for the m... |
| srgidcl 20132 | The unity element of a sem... |
| srg0cl 20133 | The zero element of a semi... |
| srgidmlem 20134 | Lemma for ~ srglidm and ~ ... |
| srglidm 20135 | The unity element of a sem... |
| srgridm 20136 | The unity element of a sem... |
| issrgid 20137 | Properties showing that an... |
| srgacl 20138 | Closure of the addition op... |
| srgcom 20139 | Commutativity of the addit... |
| srgrz 20140 | The zero of a semiring is ... |
| srglz 20141 | The zero of a semiring is ... |
| srgisid 20142 | In a semiring, the only le... |
| o2timesd 20143 | An element of a ring-like ... |
| rglcom4d 20144 | Restricted commutativity o... |
| srgo2times 20145 | A semiring element plus it... |
| srgcom4lem 20146 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20147 | Restricted commutativity o... |
| srg1zr 20148 | The only semiring with a b... |
| srgen1zr 20149 | The only semiring with one... |
| srgmulgass 20150 | An associative property be... |
| srgpcomp 20151 | If two elements of a semir... |
| srgpcompp 20152 | If two elements of a semir... |
| srgpcomppsc 20153 | If two elements of a semir... |
| srglmhm 20154 | Left-multiplication in a s... |
| srgrmhm 20155 | Right-multiplication in a ... |
| srgsummulcr 20156 | A finite semiring sum mult... |
| sgsummulcl 20157 | A finite semiring sum mult... |
| srg1expzeq1 20158 | The exponentiation (by a n... |
| srgbinomlem1 20159 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20160 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20161 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20162 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20163 | Lemma for ~ srgbinom . In... |
| srgbinom 20164 | The binomial theorem for c... |
| csrgbinom 20165 | The binomial theorem for c... |
| isring 20170 | The predicate "is a (unita... |
| ringgrp 20171 | A ring is a group. (Contr... |
| ringmgp 20172 | A ring is a monoid under m... |
| iscrng 20173 | A commutative ring is a ri... |
| crngmgp 20174 | A commutative ring's multi... |
| ringgrpd 20175 | A ring is a group. (Contr... |
| ringmnd 20176 | A ring is a monoid under a... |
| ringmgm 20177 | A ring is a magma. (Contr... |
| crngring 20178 | A commutative ring is a ri... |
| crngringd 20179 | A commutative ring is a ri... |
| crnggrpd 20180 | A commutative ring is a gr... |
| mgpf 20181 | Restricted functionality o... |
| ringdilem 20182 | Properties of a unital rin... |
| ringcl 20183 | Closure of the multiplicat... |
| crngcom 20184 | A commutative ring's multi... |
| iscrng2 20185 | A commutative ring is a ri... |
| ringass 20186 | Associative law for multip... |
| ringideu 20187 | The unity element of a rin... |
| crngcomd 20188 | Multiplication is commutat... |
| crngbascntr 20189 | The base set of a commutat... |
| ringassd 20190 | Associative law for multip... |
| crng12d 20191 | Commutative/associative la... |
| crng32d 20192 | Commutative/associative la... |
| ringcld 20193 | Closure of the multiplicat... |
| ringdi 20194 | Distributive law for the m... |
| ringdir 20195 | Distributive law for the m... |
| ringdid 20196 | Distributive law for the m... |
| ringdird 20197 | Distributive law for the m... |
| ringidcl 20198 | The unity element of a rin... |
| ringidcld 20199 | The unity element of a rin... |
| ring0cl 20200 | The zero element of a ring... |
| ringidmlem 20201 | Lemma for ~ ringlidm and ~... |
| ringlidm 20202 | The unity element of a rin... |
| ringridm 20203 | The unity element of a rin... |
| isringid 20204 | Properties showing that an... |
| ringlidmd 20205 | The unity element of a rin... |
| ringridmd 20206 | The unity element of a rin... |
| ringid 20207 | The multiplication operati... |
| ringo2times 20208 | A ring element plus itself... |
| ringadd2 20209 | A ring element plus itself... |
| ringidss 20210 | A subset of the multiplica... |
| ringacl 20211 | Closure of the addition op... |
| ringcomlem 20212 | Lemma for ~ ringcom . Thi... |
| ringcom 20213 | Commutativity of the addit... |
| ringabl 20214 | A ring is an Abelian group... |
| ringcmn 20215 | A ring is a commutative mo... |
| ringabld 20216 | A ring is an Abelian group... |
| ringcmnd 20217 | A ring is a commutative mo... |
| ringrng 20218 | A unital ring is a non-uni... |
| ringssrng 20219 | The unital rings are non-u... |
| isringrng 20220 | The predicate "is a unital... |
| ringpropd 20221 | If two structures have the... |
| crngpropd 20222 | If two structures have the... |
| ringprop 20223 | If two structures have the... |
| isringd 20224 | Properties that determine ... |
| iscrngd 20225 | Properties that determine ... |
| ringlz 20226 | The zero of a unital ring ... |
| ringrz 20227 | The zero of a unital ring ... |
| ringlzd 20228 | The zero of a unital ring ... |
| ringrzd 20229 | The zero of a unital ring ... |
| ringsrg 20230 | Any ring is also a semirin... |
| ring1eq0 20231 | If one and zero are equal,... |
| ring1ne0 20232 | If a ring has at least two... |
| ringinvnz1ne0 20233 | In a unital ring, a left i... |
| ringinvnzdiv 20234 | In a unital ring, a left i... |
| ringnegl 20235 | Negation in a ring is the ... |
| ringnegr 20236 | Negation in a ring is the ... |
| ringmneg1 20237 | Negation of a product in a... |
| ringmneg2 20238 | Negation of a product in a... |
| ringm2neg 20239 | Double negation of a produ... |
| ringsubdi 20240 | Ring multiplication distri... |
| ringsubdir 20241 | Ring multiplication distri... |
| mulgass2 20242 | An associative property be... |
| ring1 20243 | The (smallest) structure r... |
| ringn0 20244 | Rings exist. (Contributed... |
| ringlghm 20245 | Left-multiplication in a r... |
| ringrghm 20246 | Right-multiplication in a ... |
| gsummulc1OLD 20247 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20248 | Obsolete version of ~ gsum... |
| gsummulc1 20249 | A finite ring sum multipli... |
| gsummulc2 20250 | A finite ring sum multipli... |
| gsummgp0 20251 | If one factor in a finite ... |
| gsumdixp 20252 | Distribute a binary produc... |
| prdsmulrcl 20253 | A structure product of rin... |
| prdsringd 20254 | A product of rings is a ri... |
| prdscrngd 20255 | A product of commutative r... |
| prds1 20256 | Value of the ring unity in... |
| pwsring 20257 | A structure power of a rin... |
| pws1 20258 | Value of the ring unity in... |
| pwscrng 20259 | A structure power of a com... |
| pwsmgp 20260 | The multiplicative group o... |
| pwspjmhmmgpd 20261 | The projection given by ~ ... |
| pwsexpg 20262 | Value of a group exponenti... |
| pwsgprod 20263 | Finite products in a power... |
| imasring 20264 | The image structure of a r... |
| imasringf1 20265 | The image of a ring under ... |
| xpsringd 20266 | A product of two rings is ... |
| xpsring1d 20267 | The multiplicative identit... |
| qusring2 20268 | The quotient structure of ... |
| crngbinom 20269 | The binomial theorem for c... |
| opprval 20272 | Value of the opposite ring... |
| opprmulfval 20273 | Value of the multiplicatio... |
| opprmul 20274 | Value of the multiplicatio... |
| crngoppr 20275 | In a commutative ring, the... |
| opprlem 20276 | Lemma for ~ opprbas and ~ ... |
| opprbas 20277 | Base set of an opposite ri... |
| oppradd 20278 | Addition operation of an o... |
| opprrng 20279 | An opposite non-unital rin... |
| opprrngb 20280 | A class is a non-unital ri... |
| opprring 20281 | An opposite ring is a ring... |
| opprringb 20282 | Bidirectional form of ~ op... |
| oppr0 20283 | Additive identity of an op... |
| oppr1 20284 | Multiplicative identity of... |
| opprneg 20285 | The negative function in a... |
| opprsubg 20286 | Being a subgroup is a symm... |
| mulgass3 20287 | An associative property be... |
| reldvdsr 20294 | The divides relation is a ... |
| dvdsrval 20295 | Value of the divides relat... |
| dvdsr 20296 | Value of the divides relat... |
| dvdsr2 20297 | Value of the divides relat... |
| dvdsrmul 20298 | A left-multiple of ` X ` i... |
| dvdsrcl 20299 | Closure of a dividing elem... |
| dvdsrcl2 20300 | Closure of a dividing elem... |
| dvdsrid 20301 | An element in a (unital) r... |
| dvdsrtr 20302 | Divisibility is transitive... |
| dvdsrmul1 20303 | The divisibility relation ... |
| dvdsrneg 20304 | An element divides its neg... |
| dvdsr01 20305 | In a ring, zero is divisib... |
| dvdsr02 20306 | Only zero is divisible by ... |
| isunit 20307 | Property of being a unit o... |
| 1unit 20308 | The multiplicative identit... |
| unitcl 20309 | A unit is an element of th... |
| unitss 20310 | The set of units is contai... |
| opprunit 20311 | Being a unit is a symmetri... |
| crngunit 20312 | Property of being a unit i... |
| dvdsunit 20313 | A divisor of a unit is a u... |
| unitmulcl 20314 | The product of units is a ... |
| unitmulclb 20315 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20316 | The base set of the group ... |
| unitgrp 20317 | The group of units is a gr... |
| unitabl 20318 | The group of units of a co... |
| unitgrpid 20319 | The identity of the group ... |
| unitsubm 20320 | The group of units is a su... |
| invrfval 20323 | Multiplicative inverse fun... |
| unitinvcl 20324 | The inverse of a unit exis... |
| unitinvinv 20325 | The inverse of the inverse... |
| ringinvcl 20326 | The inverse of a unit is a... |
| unitlinv 20327 | A unit times its inverse i... |
| unitrinv 20328 | A unit times its inverse i... |
| 1rinv 20329 | The inverse of the ring un... |
| 0unit 20330 | The additive identity is a... |
| unitnegcl 20331 | The negative of a unit is ... |
| ringunitnzdiv 20332 | In a unitary ring, a unit ... |
| ring1nzdiv 20333 | In a unitary ring, the rin... |
| dvrfval 20336 | Division operation in a ri... |
| dvrval 20337 | Division operation in a ri... |
| dvrcl 20338 | Closure of division operat... |
| unitdvcl 20339 | The units are closed under... |
| dvrid 20340 | A ring element divided by ... |
| dvr1 20341 | A ring element divided by ... |
| dvrass 20342 | An associative law for div... |
| dvrcan1 20343 | A cancellation law for div... |
| dvrcan3 20344 | A cancellation law for div... |
| dvreq1 20345 | Equality in terms of ratio... |
| dvrdir 20346 | Distributive law for the d... |
| rdivmuldivd 20347 | Multiplication of two rati... |
| ringinvdv 20348 | Write the inverse function... |
| rngidpropd 20349 | The ring unity depends onl... |
| dvdsrpropd 20350 | The divisibility relation ... |
| unitpropd 20351 | The set of units depends o... |
| invrpropd 20352 | The ring inverse function ... |
| isirred 20353 | An irreducible element of ... |
| isnirred 20354 | The property of being a no... |
| isirred2 20355 | Expand out the class diffe... |
| opprirred 20356 | Irreducibility is symmetri... |
| irredn0 20357 | The additive identity is n... |
| irredcl 20358 | An irreducible element is ... |
| irrednu 20359 | An irreducible element is ... |
| irredn1 20360 | The multiplicative identit... |
| irredrmul 20361 | The product of an irreduci... |
| irredlmul 20362 | The product of a unit and ... |
| irredmul 20363 | If product of two elements... |
| irredneg 20364 | The negative of an irreduc... |
| irrednegb 20365 | An element is irreducible ... |
| rnghmrcl 20372 | Reverse closure of a non-u... |
| rnghmfn 20373 | The mapping of two non-uni... |
| rnghmval 20374 | The set of the non-unital ... |
| isrnghm 20375 | A function is a non-unital... |
| isrnghmmul 20376 | A function is a non-unital... |
| rnghmmgmhm 20377 | A non-unital ring homomorp... |
| rnghmval2 20378 | The non-unital ring homomo... |
| isrngim 20379 | An isomorphism of non-unit... |
| rngimrcl 20380 | Reverse closure for an iso... |
| rnghmghm 20381 | A non-unital ring homomorp... |
| rnghmf 20382 | A ring homomorphism is a f... |
| rnghmmul 20383 | A homomorphism of non-unit... |
| isrnghm2d 20384 | Demonstration of non-unita... |
| isrnghmd 20385 | Demonstration of non-unita... |
| rnghmf1o 20386 | A non-unital ring homomorp... |
| isrngim2 20387 | An isomorphism of non-unit... |
| rngimf1o 20388 | An isomorphism of non-unit... |
| rngimrnghm 20389 | An isomorphism of non-unit... |
| rngimcnv 20390 | The converse of an isomorp... |
| rnghmco 20391 | The composition of non-uni... |
| idrnghm 20392 | The identity homomorphism ... |
| c0mgm 20393 | The constant mapping to ze... |
| c0mhm 20394 | The constant mapping to ze... |
| c0ghm 20395 | The constant mapping to ze... |
| c0snmgmhm 20396 | The constant mapping to ze... |
| c0snmhm 20397 | The constant mapping to ze... |
| c0snghm 20398 | The constant mapping to ze... |
| rngisomfv1 20399 | If there is a non-unital r... |
| rngisom1 20400 | If there is a non-unital r... |
| rngisomring 20401 | If there is a non-unital r... |
| rngisomring1 20402 | If there is a non-unital r... |
| dfrhm2 20408 | The property of a ring hom... |
| rhmrcl1 20410 | Reverse closure of a ring ... |
| rhmrcl2 20411 | Reverse closure of a ring ... |
| isrhm 20412 | A function is a ring homom... |
| rhmmhm 20413 | A ring homomorphism is a h... |
| rhmisrnghm 20414 | Each unital ring homomorph... |
| rimrcl 20415 | Reverse closure for an iso... |
| isrim0 20416 | A ring isomorphism is a ho... |
| rhmghm 20417 | A ring homomorphism is an ... |
| rhmf 20418 | A ring homomorphism is a f... |
| rhmmul 20419 | A homomorphism of rings pr... |
| isrhm2d 20420 | Demonstration of ring homo... |
| isrhmd 20421 | Demonstration of ring homo... |
| rhm1 20422 | Ring homomorphisms are req... |
| idrhm 20423 | The identity homomorphism ... |
| rhmf1o 20424 | A ring homomorphism is bij... |
| isrim 20425 | An isomorphism of rings is... |
| rimf1o 20426 | An isomorphism of rings is... |
| rimrhm 20427 | A ring isomorphism is a ho... |
| rimgim 20428 | An isomorphism of rings is... |
| rimisrngim 20429 | Each unital ring isomorphi... |
| rhmfn 20430 | The mapping of two rings t... |
| rhmval 20431 | The ring homomorphisms bet... |
| rhmco 20432 | The composition of ring ho... |
| pwsco1rhm 20433 | Right composition with a f... |
| pwsco2rhm 20434 | Left composition with a ri... |
| brric 20435 | The relation "is isomorphi... |
| brrici 20436 | Prove isomorphic by an exp... |
| brric2 20437 | The relation "is isomorphi... |
| ricgic 20438 | If two rings are (ring) is... |
| rhmdvdsr 20439 | A ring homomorphism preser... |
| rhmopp 20440 | A ring homomorphism is als... |
| elrhmunit 20441 | Ring homomorphisms preserv... |
| rhmunitinv 20442 | Ring homomorphisms preserv... |
| isnzr 20445 | Property of a nonzero ring... |
| nzrnz 20446 | One and zero are different... |
| nzrring 20447 | A nonzero ring is a ring. ... |
| nzrringOLD 20448 | Obsolete version of ~ nzrr... |
| isnzr2 20449 | Equivalent characterizatio... |
| isnzr2hash 20450 | Equivalent characterizatio... |
| nzrpropd 20451 | If two structures have the... |
| opprnzrb 20452 | The opposite of a nonzero ... |
| opprnzr 20453 | The opposite of a nonzero ... |
| ringelnzr 20454 | A ring is nonzero if it ha... |
| nzrunit 20455 | A unit is nonzero in any n... |
| 0ringnnzr 20456 | A ring is a zero ring iff ... |
| 0ring 20457 | If a ring has only one ele... |
| 0ringdif 20458 | A zero ring is a ring whic... |
| 0ringbas 20459 | The base set of a zero rin... |
| 0ring01eq 20460 | In a ring with only one el... |
| 01eq0ring 20461 | If the zero and the identi... |
| 01eq0ringOLD 20462 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20463 | In a unital ring the zero ... |
| 0ring1eq0 20464 | In a zero ring, a ring whi... |
| c0rhm 20465 | The constant mapping to ze... |
| c0rnghm 20466 | The constant mapping to ze... |
| zrrnghm 20467 | The constant mapping to ze... |
| nrhmzr 20468 | There is no ring homomorph... |
| islring 20471 | The predicate "is a local ... |
| lringnzr 20472 | A local ring is a nonzero ... |
| lringring 20473 | A local ring is a ring. (... |
| lringnz 20474 | A local ring is a nonzero ... |
| lringuplu 20475 | If the sum of two elements... |
| issubrng 20478 | The subring of non-unital ... |
| subrngss 20479 | A subring is a subset. (C... |
| subrngid 20480 | Every non-unital ring is a... |
| subrngrng 20481 | A subring is a non-unital ... |
| subrngrcl 20482 | Reverse closure for a subr... |
| subrngsubg 20483 | A subring is a subgroup. ... |
| subrngringnsg 20484 | A subring is a normal subg... |
| subrngbas 20485 | Base set of a subring stru... |
| subrng0 20486 | A subring always has the s... |
| subrngacl 20487 | A subring is closed under ... |
| subrngmcl 20488 | A subring is closed under ... |
| issubrng2 20489 | Characterize the subrings ... |
| opprsubrng 20490 | Being a subring is a symme... |
| subrngint 20491 | The intersection of a none... |
| subrngin 20492 | The intersection of two su... |
| subrngmre 20493 | The subrings of a non-unit... |
| subsubrng 20494 | A subring of a subring is ... |
| subsubrng2 20495 | The set of subrings of a s... |
| rhmimasubrnglem 20496 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20497 | The homomorphic image of a... |
| cntzsubrng 20498 | Centralizers in a non-unit... |
| subrngpropd 20499 | If two structures have the... |
| issubrg 20502 | The subring predicate. (C... |
| subrgss 20503 | A subring is a subset. (C... |
| subrgid 20504 | Every ring is a subring of... |
| subrgring 20505 | A subring is a ring. (Con... |
| subrgcrng 20506 | A subring of a commutative... |
| subrgrcl 20507 | Reverse closure for a subr... |
| subrgsubg 20508 | A subring is a subgroup. ... |
| subrgsubrng 20509 | A subring of a unital ring... |
| subrg0 20510 | A subring always has the s... |
| subrg1cl 20511 | A subring contains the mul... |
| subrgbas 20512 | Base set of a subring stru... |
| subrg1 20513 | A subring always has the s... |
| subrgacl 20514 | A subring is closed under ... |
| subrgmcl 20515 | A subring is closed under ... |
| subrgsubm 20516 | A subring is a submonoid o... |
| subrgdvds 20517 | If an element divides anot... |
| subrguss 20518 | A unit of a subring is a u... |
| subrginv 20519 | A subring always has the s... |
| subrgdv 20520 | A subring always has the s... |
| subrgunit 20521 | An element of a ring is a ... |
| subrgugrp 20522 | The units of a subring for... |
| issubrg2 20523 | Characterize the subrings ... |
| opprsubrg 20524 | Being a subring is a symme... |
| subrgnzr 20525 | A subring of a nonzero rin... |
| subrgint 20526 | The intersection of a none... |
| subrgin 20527 | The intersection of two su... |
| subrgmre 20528 | The subrings of a ring are... |
| subsubrg 20529 | A subring of a subring is ... |
| subsubrg2 20530 | The set of subrings of a s... |
| issubrg3 20531 | A subring is an additive s... |
| resrhm 20532 | Restriction of a ring homo... |
| resrhm2b 20533 | Restriction of the codomai... |
| rhmeql 20534 | The equalizer of two ring ... |
| rhmima 20535 | The homomorphic image of a... |
| rnrhmsubrg 20536 | The range of a ring homomo... |
| cntzsubr 20537 | Centralizers in a ring are... |
| pwsdiagrhm 20538 | Diagonal homomorphism into... |
| subrgpropd 20539 | If two structures have the... |
| rhmpropd 20540 | Ring homomorphism depends ... |
| rgspnval 20543 | Value of the ring-span of ... |
| rgspncl 20544 | The ring-span of a set is ... |
| rgspnssid 20545 | The ring-span of a set con... |
| rgspnmin 20546 | The ring-span is contained... |
| rngcval 20549 | Value of the category of n... |
| rnghmresfn 20550 | The class of non-unital ri... |
| rnghmresel 20551 | An element of the non-unit... |
| rngcbas 20552 | Set of objects of the cate... |
| rngchomfval 20553 | Set of arrows of the categ... |
| rngchom 20554 | Set of arrows of the categ... |
| elrngchom 20555 | A morphism of non-unital r... |
| rngchomfeqhom 20556 | The functionalized Hom-set... |
| rngccofval 20557 | Composition in the categor... |
| rngcco 20558 | Composition in the categor... |
| dfrngc2 20559 | Alternate definition of th... |
| rnghmsscmap2 20560 | The non-unital ring homomo... |
| rnghmsscmap 20561 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20562 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20563 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20564 | The non-unital ring homomo... |
| rngccat 20565 | The category of non-unital... |
| rngcid 20566 | The identity arrow in the ... |
| rngcsect 20567 | A section in the category ... |
| rngcinv 20568 | An inverse in the category... |
| rngciso 20569 | An isomorphism in the cate... |
| rngcifuestrc 20570 | The "inclusion functor" fr... |
| funcrngcsetc 20571 | The "natural forgetful fun... |
| funcrngcsetcALT 20572 | Alternate proof of ~ funcr... |
| zrinitorngc 20573 | The zero ring is an initia... |
| zrtermorngc 20574 | The zero ring is a termina... |
| zrzeroorngc 20575 | The zero ring is a zero ob... |
| ringcval 20578 | Value of the category of u... |
| rhmresfn 20579 | The class of unital ring h... |
| rhmresel 20580 | An element of the unital r... |
| ringcbas 20581 | Set of objects of the cate... |
| ringchomfval 20582 | Set of arrows of the categ... |
| ringchom 20583 | Set of arrows of the categ... |
| elringchom 20584 | A morphism of unital rings... |
| ringchomfeqhom 20585 | The functionalized Hom-set... |
| ringccofval 20586 | Composition in the categor... |
| ringcco 20587 | Composition in the categor... |
| dfringc2 20588 | Alternate definition of th... |
| rhmsscmap2 20589 | The unital ring homomorphi... |
| rhmsscmap 20590 | The unital ring homomorphi... |
| rhmsubcsetclem1 20591 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20592 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20593 | The unital ring homomorphi... |
| ringccat 20594 | The category of unital rin... |
| ringcid 20595 | The identity arrow in the ... |
| rhmsscrnghm 20596 | The unital ring homomorphi... |
| rhmsubcrngclem1 20597 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20598 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20599 | The unital ring homomorphi... |
| rngcresringcat 20600 | The restriction of the cat... |
| ringcsect 20601 | A section in the category ... |
| ringcinv 20602 | An inverse in the category... |
| ringciso 20603 | An isomorphism in the cate... |
| ringcbasbas 20604 | An element of the base set... |
| funcringcsetc 20605 | The "natural forgetful fun... |
| zrtermoringc 20606 | The zero ring is a termina... |
| zrninitoringc 20607 | The zero ring is not an in... |
| srhmsubclem1 20608 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20609 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20610 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20611 | According to ~ df-subc , t... |
| sringcat 20612 | The restriction of the cat... |
| crhmsubc 20613 | According to ~ df-subc , t... |
| cringcat 20614 | The restriction of the cat... |
| rngcrescrhm 20615 | The category of non-unital... |
| rhmsubclem1 20616 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20617 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20618 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20619 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20620 | According to ~ df-subc , t... |
| rhmsubccat 20621 | The restriction of the cat... |
| rrgval 20628 | Value of the set or left-r... |
| isrrg 20629 | Membership in the set of l... |
| rrgeq0i 20630 | Property of a left-regular... |
| rrgeq0 20631 | Left-multiplication by a l... |
| rrgsupp 20632 | Left multiplication by a l... |
| rrgss 20633 | Left-regular elements are ... |
| unitrrg 20634 | Units are regular elements... |
| rrgnz 20635 | In a nonzero ring, the zer... |
| isdomn 20636 | Expand definition of a dom... |
| domnnzr 20637 | A domain is a nonzero ring... |
| domnring 20638 | A domain is a ring. (Cont... |
| domneq0 20639 | In a domain, a product is ... |
| domnmuln0 20640 | In a domain, a product of ... |
| isdomn5 20641 | The equivalence between th... |
| isdomn2 20642 | A ring is a domain iff all... |
| isdomn2OLD 20643 | Obsolete version of ~ isdo... |
| domnrrg 20644 | In a domain, a nonzero ele... |
| isdomn6 20645 | A ring is a domain iff the... |
| isdomn3 20646 | Nonzero elements form a mu... |
| isdomn4 20647 | A ring is a domain iff it ... |
| opprdomnb 20648 | A class is a domain if and... |
| opprdomn 20649 | The opposite of a domain i... |
| isdomn4r 20650 | A ring is a domain iff it ... |
| domnlcanb 20651 | Left-cancellation law for ... |
| domnlcan 20652 | Left-cancellation law for ... |
| domnrcanb 20653 | Right-cancellation law for... |
| domnrcan 20654 | Right-cancellation law for... |
| domneq0r 20655 | Right multiplication by a ... |
| isidom 20656 | An integral domain is a co... |
| idomdomd 20657 | An integral domain is a do... |
| idomcringd 20658 | An integral domain is a co... |
| idomringd 20659 | An integral domain is a ri... |
| isdrng 20664 | The predicate "is a divisi... |
| drngunit 20665 | Elementhood in the set of ... |
| drngui 20666 | The set of units of a divi... |
| drngring 20667 | A division ring is a ring.... |
| drngringd 20668 | A division ring is a ring.... |
| drnggrpd 20669 | A division ring is a group... |
| drnggrp 20670 | A division ring is a group... |
| isfld 20671 | A field is a commutative d... |
| flddrngd 20672 | A field is a division ring... |
| fldcrngd 20673 | A field is a commutative r... |
| isdrng2 20674 | A division ring can equiva... |
| drngprop 20675 | If two structures have the... |
| drngmgp 20676 | A division ring contains a... |
| drngid 20677 | A division ring's unity is... |
| drngunz 20678 | A division ring's unity is... |
| drngnzr 20679 | A division ring is a nonze... |
| drngdomn 20680 | A division ring is a domai... |
| drngmcl 20681 | The product of two nonzero... |
| drngmclOLD 20682 | Obsolete version of ~ drng... |
| drngid2 20683 | Properties showing that an... |
| drnginvrcl 20684 | Closure of the multiplicat... |
| drnginvrn0 20685 | The multiplicative inverse... |
| drnginvrcld 20686 | Closure of the multiplicat... |
| drnginvrl 20687 | Property of the multiplica... |
| drnginvrr 20688 | Property of the multiplica... |
| drnginvrld 20689 | Property of the multiplica... |
| drnginvrrd 20690 | Property of the multiplica... |
| drngmul0or 20691 | A product is zero iff one ... |
| drngmul0orOLD 20692 | Obsolete version of ~ drng... |
| drngmulne0 20693 | A product is nonzero iff b... |
| drngmuleq0 20694 | An element is zero iff its... |
| opprdrng 20695 | The opposite of a division... |
| isdrngd 20696 | Properties that characteri... |
| isdrngrd 20697 | Properties that characteri... |
| isdrngdOLD 20698 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20699 | Obsolete version of ~ isdr... |
| drngpropd 20700 | If two structures have the... |
| fldpropd 20701 | If two structures have the... |
| fldidom 20702 | A field is an integral dom... |
| fidomndrnglem 20703 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20704 | A finite domain is a divis... |
| fiidomfld 20705 | A finite integral domain i... |
| rng1nnzr 20706 | The (smallest) structure r... |
| ring1zr 20707 | The only (unital) ring wit... |
| rngen1zr 20708 | The only (unital) ring wit... |
| ringen1zr 20709 | The only unital ring with ... |
| rng1nfld 20710 | The zero ring is not a fie... |
| issubdrg 20711 | Characterize the subfields... |
| drhmsubc 20712 | According to ~ df-subc , t... |
| drngcat 20713 | The restriction of the cat... |
| fldcat 20714 | The restriction of the cat... |
| fldc 20715 | The restriction of the cat... |
| fldhmsubc 20716 | According to ~ df-subc , t... |
| issdrg 20719 | Property of a division sub... |
| sdrgrcl 20720 | Reverse closure for a sub-... |
| sdrgdrng 20721 | A sub-division-ring is a d... |
| sdrgsubrg 20722 | A sub-division-ring is a s... |
| sdrgid 20723 | Every division ring is a d... |
| sdrgss 20724 | A division subring is a su... |
| sdrgbas 20725 | Base set of a sub-division... |
| issdrg2 20726 | Property of a division sub... |
| sdrgunit 20727 | A unit of a sub-division-r... |
| imadrhmcl 20728 | The image of a (nontrivial... |
| fldsdrgfld 20729 | A sub-division-ring of a f... |
| acsfn1p 20730 | Construction of a closure ... |
| subrgacs 20731 | Closure property of subrin... |
| sdrgacs 20732 | Closure property of divisi... |
| cntzsdrg 20733 | Centralizers in division r... |
| subdrgint 20734 | The intersection of a none... |
| sdrgint 20735 | The intersection of a none... |
| primefld 20736 | The smallest sub division ... |
| primefld0cl 20737 | The prime field contains t... |
| primefld1cl 20738 | The prime field contains t... |
| abvfval 20741 | Value of the set of absolu... |
| isabv 20742 | Elementhood in the set of ... |
| isabvd 20743 | Properties that determine ... |
| abvrcl 20744 | Reverse closure for the ab... |
| abvfge0 20745 | An absolute value is a fun... |
| abvf 20746 | An absolute value is a fun... |
| abvcl 20747 | An absolute value is a fun... |
| abvge0 20748 | The absolute value of a nu... |
| abveq0 20749 | The value of an absolute v... |
| abvne0 20750 | The absolute value of a no... |
| abvgt0 20751 | The absolute value of a no... |
| abvmul 20752 | An absolute value distribu... |
| abvtri 20753 | An absolute value satisfie... |
| abv0 20754 | The absolute value of zero... |
| abv1z 20755 | The absolute value of one ... |
| abv1 20756 | The absolute value of one ... |
| abvneg 20757 | The absolute value of a ne... |
| abvsubtri 20758 | An absolute value satisfie... |
| abvrec 20759 | The absolute value distrib... |
| abvdiv 20760 | The absolute value distrib... |
| abvdom 20761 | Any ring with an absolute ... |
| abvres 20762 | The restriction of an abso... |
| abvtrivd 20763 | The trivial absolute value... |
| abvtrivg 20764 | The trivial absolute value... |
| abvtriv 20765 | The trivial absolute value... |
| abvpropd 20766 | If two structures have the... |
| abvn0b 20767 | Another characterization o... |
| staffval 20772 | The functionalization of t... |
| stafval 20773 | The functionalization of t... |
| staffn 20774 | The functionalization is e... |
| issrng 20775 | The predicate "is a star r... |
| srngrhm 20776 | The involution function in... |
| srngring 20777 | A star ring is a ring. (C... |
| srngcnv 20778 | The involution function in... |
| srngf1o 20779 | The involution function in... |
| srngcl 20780 | The involution function in... |
| srngnvl 20781 | The involution function in... |
| srngadd 20782 | The involution function in... |
| srngmul 20783 | The involution function in... |
| srng1 20784 | The conjugate of the ring ... |
| srng0 20785 | The conjugate of the ring ... |
| issrngd 20786 | Properties that determine ... |
| idsrngd 20787 | A commutative ring is a st... |
| isorng 20792 | An ordered ring is a ring ... |
| orngring 20793 | An ordered ring is a ring.... |
| orngogrp 20794 | An ordered ring is an orde... |
| isofld 20795 | An ordered field is a fiel... |
| orngmul 20796 | In an ordered ring, the or... |
| orngsqr 20797 | In an ordered ring, all sq... |
| ornglmulle 20798 | In an ordered ring, multip... |
| orngrmulle 20799 | In an ordered ring, multip... |
| ornglmullt 20800 | In an ordered ring, multip... |
| orngrmullt 20801 | In an ordered ring, multip... |
| orngmullt 20802 | In an ordered ring, the st... |
| ofldfld 20803 | An ordered field is a fiel... |
| ofldtos 20804 | An ordered field is a tota... |
| orng0le1 20805 | In an ordered ring, the ri... |
| ofldlt1 20806 | In an ordered field, the r... |
| suborng 20807 | Every subring of an ordere... |
| subofld 20808 | Every subfield of an order... |
| islmod 20813 | The predicate "is a left m... |
| lmodlema 20814 | Lemma for properties of a ... |
| islmodd 20815 | Properties that determine ... |
| lmodgrp 20816 | A left module is a group. ... |
| lmodring 20817 | The scalar component of a ... |
| lmodfgrp 20818 | The scalar component of a ... |
| lmodgrpd 20819 | A left module is a group. ... |
| lmodbn0 20820 | The base set of a left mod... |
| lmodacl 20821 | Closure of ring addition f... |
| lmodmcl 20822 | Closure of ring multiplica... |
| lmodsn0 20823 | The set of scalars in a le... |
| lmodvacl 20824 | Closure of vector addition... |
| lmodass 20825 | Left module vector sum is ... |
| lmodlcan 20826 | Left cancellation law for ... |
| lmodvscl 20827 | Closure of scalar product ... |
| lmodvscld 20828 | Closure of scalar product ... |
| scaffval 20829 | The scalar multiplication ... |
| scafval 20830 | The scalar multiplication ... |
| scafeq 20831 | If the scalar multiplicati... |
| scaffn 20832 | The scalar multiplication ... |
| lmodscaf 20833 | The scalar multiplication ... |
| lmodvsdi 20834 | Distributive law for scala... |
| lmodvsdir 20835 | Distributive law for scala... |
| lmodvsass 20836 | Associative law for scalar... |
| lmod0cl 20837 | The ring zero in a left mo... |
| lmod1cl 20838 | The ring unity in a left m... |
| lmodvs1 20839 | Scalar product with the ri... |
| lmod0vcl 20840 | The zero vector is a vecto... |
| lmod0vlid 20841 | Left identity law for the ... |
| lmod0vrid 20842 | Right identity law for the... |
| lmod0vid 20843 | Identity equivalent to the... |
| lmod0vs 20844 | Zero times a vector is the... |
| lmodvs0 20845 | Anything times the zero ve... |
| lmodvsmmulgdi 20846 | Distributive law for a gro... |
| lmodfopnelem1 20847 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20848 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20849 | The (functionalized) opera... |
| lcomf 20850 | A linear-combination sum i... |
| lcomfsupp 20851 | A linear-combination sum i... |
| lmodvnegcl 20852 | Closure of vector negative... |
| lmodvnegid 20853 | Addition of a vector with ... |
| lmodvneg1 20854 | Minus 1 times a vector is ... |
| lmodvsneg 20855 | Multiplication of a vector... |
| lmodvsubcl 20856 | Closure of vector subtract... |
| lmodcom 20857 | Left module vector sum is ... |
| lmodabl 20858 | A left module is an abelia... |
| lmodcmn 20859 | A left module is a commuta... |
| lmodnegadd 20860 | Distribute negation throug... |
| lmod4 20861 | Commutative/associative la... |
| lmodvsubadd 20862 | Relationship between vecto... |
| lmodvaddsub4 20863 | Vector addition/subtractio... |
| lmodvpncan 20864 | Addition/subtraction cance... |
| lmodvnpcan 20865 | Cancellation law for vecto... |
| lmodvsubval2 20866 | Value of vector subtractio... |
| lmodsubvs 20867 | Subtraction of a scalar pr... |
| lmodsubdi 20868 | Scalar multiplication dist... |
| lmodsubdir 20869 | Scalar multiplication dist... |
| lmodsubeq0 20870 | If the difference between ... |
| lmodsubid 20871 | Subtraction of a vector fr... |
| lmodvsghm 20872 | Scalar multiplication of t... |
| lmodprop2d 20873 | If two structures have the... |
| lmodpropd 20874 | If two structures have the... |
| gsumvsmul 20875 | Pull a scalar multiplicati... |
| mptscmfsupp0 20876 | A mapping to a scalar prod... |
| mptscmfsuppd 20877 | A function mapping to a sc... |
| rmodislmodlem 20878 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20879 | The right module ` R ` ind... |
| lssset 20882 | The set of all (not necess... |
| islss 20883 | The predicate "is a subspa... |
| islssd 20884 | Properties that determine ... |
| lssss 20885 | A subspace is a set of vec... |
| lssel 20886 | A subspace member is a vec... |
| lss1 20887 | The set of vectors in a le... |
| lssuni 20888 | The union of all subspaces... |
| lssn0 20889 | A subspace is not empty. ... |
| 00lss 20890 | The empty structure has no... |
| lsscl 20891 | Closure property of a subs... |
| lssvacl 20892 | Closure of vector addition... |
| lssvsubcl 20893 | Closure of vector subtract... |
| lssvancl1 20894 | Non-closure: if one vector... |
| lssvancl2 20895 | Non-closure: if one vector... |
| lss0cl 20896 | The zero vector belongs to... |
| lsssn0 20897 | The singleton of the zero ... |
| lss0ss 20898 | The zero subspace is inclu... |
| lssle0 20899 | No subspace is smaller tha... |
| lssne0 20900 | A nonzero subspace has a n... |
| lssvneln0 20901 | A vector ` X ` which doesn... |
| lssneln0 20902 | A vector ` X ` which doesn... |
| lssssr 20903 | Conclude subspace ordering... |
| lssvscl 20904 | Closure of scalar product ... |
| lssvnegcl 20905 | Closure of negative vector... |
| lsssubg 20906 | All subspaces are subgroup... |
| lsssssubg 20907 | All subspaces are subgroup... |
| islss3 20908 | A linear subspace of a mod... |
| lsslmod 20909 | A submodule is a module. ... |
| lsslss 20910 | The subspaces of a subspac... |
| islss4 20911 | A linear subspace is a sub... |
| lss1d 20912 | One-dimensional subspace (... |
| lssintcl 20913 | The intersection of a none... |
| lssincl 20914 | The intersection of two su... |
| lssmre 20915 | The subspaces of a module ... |
| lssacs 20916 | Submodules are an algebrai... |
| prdsvscacl 20917 | Pointwise scalar multiplic... |
| prdslmodd 20918 | The product of a family of... |
| pwslmod 20919 | A structure power of a lef... |
| lspfval 20922 | The span function for a le... |
| lspf 20923 | The span function on a lef... |
| lspval 20924 | The span of a set of vecto... |
| lspcl 20925 | The span of a set of vecto... |
| lspsncl 20926 | The span of a singleton is... |
| lspprcl 20927 | The span of a pair is a su... |
| lsptpcl 20928 | The span of an unordered t... |
| lspsnsubg 20929 | The span of a singleton is... |
| 00lsp 20930 | ~ fvco4i lemma for linear ... |
| lspid 20931 | The span of a subspace is ... |
| lspssv 20932 | A span is a set of vectors... |
| lspss 20933 | Span preserves subset orde... |
| lspssid 20934 | A set of vectors is a subs... |
| lspidm 20935 | The span of a set of vecto... |
| lspun 20936 | The span of union is the s... |
| lspssp 20937 | If a set of vectors is a s... |
| mrclsp 20938 | Moore closure generalizes ... |
| lspsnss 20939 | The span of the singleton ... |
| ellspsn3 20940 | A member of the span of th... |
| lspprss 20941 | The span of a pair of vect... |
| lspsnid 20942 | A vector belongs to the sp... |
| ellspsn6 20943 | Relationship between a vec... |
| ellspsn5b 20944 | Relationship between a vec... |
| ellspsn5 20945 | Relationship between a vec... |
| lspprid1 20946 | A member of a pair of vect... |
| lspprid2 20947 | A member of a pair of vect... |
| lspprvacl 20948 | The sum of two vectors bel... |
| lssats2 20949 | A way to express atomistic... |
| ellspsni 20950 | A scalar product with a ve... |
| lspsn 20951 | Span of the singleton of a... |
| ellspsn 20952 | Member of span of the sing... |
| lspsnvsi 20953 | Span of a scalar product o... |
| lspsnss2 20954 | Comparable spans of single... |
| lspsnneg 20955 | Negation does not change t... |
| lspsnsub 20956 | Swapping subtraction order... |
| lspsn0 20957 | Span of the singleton of t... |
| lsp0 20958 | Span of the empty set. (C... |
| lspuni0 20959 | Union of the span of the e... |
| lspun0 20960 | The span of a union with t... |
| lspsneq0 20961 | Span of the singleton is t... |
| lspsneq0b 20962 | Equal singleton spans impl... |
| lmodindp1 20963 | Two independent (non-colin... |
| lsslsp 20964 | Spans in submodules corres... |
| lsslspOLD 20965 | Obsolete version of ~ lssl... |
| lss0v 20966 | The zero vector in a submo... |
| lsspropd 20967 | If two structures have the... |
| lsppropd 20968 | If two structures have the... |
| reldmlmhm 20975 | Lemma for module homomorph... |
| lmimfn 20976 | Lemma for module isomorphi... |
| islmhm 20977 | Property of being a homomo... |
| islmhm3 20978 | Property of a module homom... |
| lmhmlem 20979 | Non-quantified consequence... |
| lmhmsca 20980 | A homomorphism of left mod... |
| lmghm 20981 | A homomorphism of left mod... |
| lmhmlmod2 20982 | A homomorphism of left mod... |
| lmhmlmod1 20983 | A homomorphism of left mod... |
| lmhmf 20984 | A homomorphism of left mod... |
| lmhmlin 20985 | A homomorphism of left mod... |
| lmodvsinv 20986 | Multiplication of a vector... |
| lmodvsinv2 20987 | Multiplying a negated vect... |
| islmhm2 20988 | A one-equation proof of li... |
| islmhmd 20989 | Deduction for a module hom... |
| 0lmhm 20990 | The constant zero linear f... |
| idlmhm 20991 | The identity function on a... |
| invlmhm 20992 | The negative function on a... |
| lmhmco 20993 | The composition of two mod... |
| lmhmplusg 20994 | The pointwise sum of two l... |
| lmhmvsca 20995 | The pointwise scalar produ... |
| lmhmf1o 20996 | A bijective module homomor... |
| lmhmima 20997 | The image of a subspace un... |
| lmhmpreima 20998 | The inverse image of a sub... |
| lmhmlsp 20999 | Homomorphisms preserve spa... |
| lmhmrnlss 21000 | The range of a homomorphis... |
| lmhmkerlss 21001 | The kernel of a homomorphi... |
| reslmhm 21002 | Restriction of a homomorph... |
| reslmhm2 21003 | Expansion of the codomain ... |
| reslmhm2b 21004 | Expansion of the codomain ... |
| lmhmeql 21005 | The equalizer of two modul... |
| lspextmo 21006 | A linear function is compl... |
| pwsdiaglmhm 21007 | Diagonal homomorphism into... |
| pwssplit0 21008 | Splitting for structure po... |
| pwssplit1 21009 | Splitting for structure po... |
| pwssplit2 21010 | Splitting for structure po... |
| pwssplit3 21011 | Splitting for structure po... |
| islmim 21012 | An isomorphism of left mod... |
| lmimf1o 21013 | An isomorphism of left mod... |
| lmimlmhm 21014 | An isomorphism of modules ... |
| lmimgim 21015 | An isomorphism of modules ... |
| islmim2 21016 | An isomorphism of left mod... |
| lmimcnv 21017 | The converse of a bijectiv... |
| brlmic 21018 | The relation "is isomorphi... |
| brlmici 21019 | Prove isomorphic by an exp... |
| lmiclcl 21020 | Isomorphism implies the le... |
| lmicrcl 21021 | Isomorphism implies the ri... |
| lmicsym 21022 | Module isomorphism is symm... |
| lmhmpropd 21023 | Module homomorphism depend... |
| islbs 21026 | The predicate " ` B ` is a... |
| lbsss 21027 | A basis is a set of vector... |
| lbsel 21028 | An element of a basis is a... |
| lbssp 21029 | The span of a basis is the... |
| lbsind 21030 | A basis is linearly indepe... |
| lbsind2 21031 | A basis is linearly indepe... |
| lbspss 21032 | No proper subset of a basi... |
| lsmcl 21033 | The sum of two subspaces i... |
| lsmspsn 21034 | Member of subspace sum of ... |
| lsmelval2 21035 | Subspace sum membership in... |
| lsmsp 21036 | Subspace sum in terms of s... |
| lsmsp2 21037 | Subspace sum of spans of s... |
| lsmssspx 21038 | Subspace sum (in its exten... |
| lsmpr 21039 | The span of a pair of vect... |
| lsppreli 21040 | A vector expressed as a su... |
| lsmelpr 21041 | Two ways to say that a vec... |
| lsppr0 21042 | The span of a vector paire... |
| lsppr 21043 | Span of a pair of vectors.... |
| lspprel 21044 | Member of the span of a pa... |
| lspprabs 21045 | Absorption of vector sum i... |
| lspvadd 21046 | The span of a vector sum i... |
| lspsntri 21047 | Triangle-type inequality f... |
| lspsntrim 21048 | Triangle-type inequality f... |
| lbspropd 21049 | If two structures have the... |
| pj1lmhm 21050 | The left projection functi... |
| pj1lmhm2 21051 | The left projection functi... |
| islvec 21054 | The predicate "is a left v... |
| lvecdrng 21055 | The set of scalars of a le... |
| lveclmod 21056 | A left vector space is a l... |
| lveclmodd 21057 | A vector space is a left m... |
| lvecgrpd 21058 | A vector space is a group.... |
| lsslvec 21059 | A vector subspace is a vec... |
| lmhmlvec 21060 | The property for modules t... |
| lvecvs0or 21061 | If a scalar product is zer... |
| lvecvsn0 21062 | A scalar product is nonzer... |
| lssvs0or 21063 | If a scalar product belong... |
| lvecvscan 21064 | Cancellation law for scala... |
| lvecvscan2 21065 | Cancellation law for scala... |
| lvecinv 21066 | Invert coefficient of scal... |
| lspsnvs 21067 | A nonzero scalar product d... |
| lspsneleq 21068 | Membership relation that i... |
| lspsncmp 21069 | Comparable spans of nonzer... |
| lspsnne1 21070 | Two ways to express that v... |
| lspsnne2 21071 | Two ways to express that v... |
| lspsnnecom 21072 | Swap two vectors with diff... |
| lspabs2 21073 | Absorption law for span of... |
| lspabs3 21074 | Absorption law for span of... |
| lspsneq 21075 | Equal spans of singletons ... |
| lspsneu 21076 | Nonzero vectors with equal... |
| ellspsn4 21077 | A member of the span of th... |
| lspdisj 21078 | The span of a vector not i... |
| lspdisjb 21079 | A nonzero vector is not in... |
| lspdisj2 21080 | Unequal spans are disjoint... |
| lspfixed 21081 | Show membership in the spa... |
| lspexch 21082 | Exchange property for span... |
| lspexchn1 21083 | Exchange property for span... |
| lspexchn2 21084 | Exchange property for span... |
| lspindpi 21085 | Partial independence prope... |
| lspindp1 21086 | Alternate way to say 3 vec... |
| lspindp2l 21087 | Alternate way to say 3 vec... |
| lspindp2 21088 | Alternate way to say 3 vec... |
| lspindp3 21089 | Independence of 2 vectors ... |
| lspindp4 21090 | (Partial) independence of ... |
| lvecindp 21091 | Compute the ` X ` coeffici... |
| lvecindp2 21092 | Sums of independent vector... |
| lspsnsubn0 21093 | Unequal singleton spans im... |
| lsmcv 21094 | Subspace sum has the cover... |
| lspsolvlem 21095 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21096 | If ` X ` is in the span of... |
| lssacsex 21097 | In a vector space, subspac... |
| lspsnat 21098 | There is no subspace stric... |
| lspsncv0 21099 | The span of a singleton co... |
| lsppratlem1 21100 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21101 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21102 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21103 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21104 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21105 | Lemma for ~ lspprat . Neg... |
| lspprat 21106 | A proper subspace of the s... |
| islbs2 21107 | An equivalent formulation ... |
| islbs3 21108 | An equivalent formulation ... |
| lbsacsbs 21109 | Being a basis in a vector ... |
| lvecdim 21110 | The dimension theorem for ... |
| lbsextlem1 21111 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21112 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21113 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21114 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21115 | For any linearly independe... |
| lbsext 21116 | For any linearly independe... |
| lbsexg 21117 | Every vector space has a b... |
| lbsex 21118 | Every vector space has a b... |
| lvecprop2d 21119 | If two structures have the... |
| lvecpropd 21120 | If two structures have the... |
| sraval 21125 | Lemma for ~ srabase throug... |
| sralem 21126 | Lemma for ~ srabase and si... |
| srabase 21127 | Base set of a subring alge... |
| sraaddg 21128 | Additive operation of a su... |
| sramulr 21129 | Multiplicative operation o... |
| srasca 21130 | The set of scalars of a su... |
| sravsca 21131 | The scalar product operati... |
| sraip 21132 | The inner product operatio... |
| sratset 21133 | Topology component of a su... |
| sratopn 21134 | Topology component of a su... |
| srads 21135 | Distance function of a sub... |
| sraring 21136 | Condition for a subring al... |
| sralmod 21137 | The subring algebra is a l... |
| sralmod0 21138 | The subring module inherit... |
| issubrgd 21139 | Prove a subring by closure... |
| rlmfn 21140 | ` ringLMod ` is a function... |
| rlmval 21141 | Value of the ring module. ... |
| rlmval2 21142 | Value of the ring module e... |
| rlmbas 21143 | Base set of the ring modul... |
| rlmplusg 21144 | Vector addition in the rin... |
| rlm0 21145 | Zero vector in the ring mo... |
| rlmsub 21146 | Subtraction in the ring mo... |
| rlmmulr 21147 | Ring multiplication in the... |
| rlmsca 21148 | Scalars in the ring module... |
| rlmsca2 21149 | Scalars in the ring module... |
| rlmvsca 21150 | Scalar multiplication in t... |
| rlmtopn 21151 | Topology component of the ... |
| rlmds 21152 | Metric component of the ri... |
| rlmlmod 21153 | The ring module is a modul... |
| rlmlvec 21154 | The ring module over a div... |
| rlmlsm 21155 | Subgroup sum of the ring m... |
| rlmvneg 21156 | Vector negation in the rin... |
| rlmscaf 21157 | Functionalized scalar mult... |
| ixpsnbasval 21158 | The value of an infinite C... |
| lidlval 21163 | Value of the set of ring i... |
| rspval 21164 | Value of the ring span fun... |
| lidlss 21165 | An ideal is a subset of th... |
| lidlssbas 21166 | The base set of the restri... |
| lidlbas 21167 | A (left) ideal of a ring i... |
| islidl 21168 | Predicate of being a (left... |
| rnglidlmcl 21169 | A (left) ideal containing ... |
| rngridlmcl 21170 | A right ideal (which is a ... |
| dflidl2rng 21171 | Alternate (the usual textb... |
| isridlrng 21172 | A right ideal is a left id... |
| lidl0cl 21173 | An ideal contains 0. (Con... |
| lidlacl 21174 | An ideal is closed under a... |
| lidlnegcl 21175 | An ideal contains negative... |
| lidlsubg 21176 | An ideal is a subgroup of ... |
| lidlsubcl 21177 | An ideal is closed under s... |
| lidlmcl 21178 | An ideal is closed under l... |
| lidl1el 21179 | An ideal contains 1 iff it... |
| dflidl2 21180 | Alternate (the usual textb... |
| lidl0ALT 21181 | Alternate proof for ~ lidl... |
| rnglidl0 21182 | Every non-unital ring cont... |
| lidl0 21183 | Every ring contains a zero... |
| lidl1ALT 21184 | Alternate proof for ~ lidl... |
| rnglidl1 21185 | The base set of every non-... |
| lidl1 21186 | Every ring contains a unit... |
| lidlacs 21187 | The ideal system is an alg... |
| rspcl 21188 | The span of a set of ring ... |
| rspssid 21189 | The span of a set of ring ... |
| rsp1 21190 | The span of the identity e... |
| rsp0 21191 | The span of the zero eleme... |
| rspssp 21192 | The ideal span of a set of... |
| elrspsn 21193 | Membership in a principal ... |
| mrcrsp 21194 | Moore closure generalizes ... |
| lidlnz 21195 | A nonzero ideal contains a... |
| drngnidl 21196 | A division ring has only t... |
| lidlrsppropd 21197 | The left ideals and ring s... |
| rnglidlmmgm 21198 | The multiplicative group o... |
| rnglidlmsgrp 21199 | The multiplicative group o... |
| rnglidlrng 21200 | A (left) ideal of a non-un... |
| lidlnsg 21201 | An ideal is a normal subgr... |
| 2idlval 21204 | Definition of a two-sided ... |
| isridl 21205 | A right ideal is a left id... |
| 2idlelb 21206 | Membership in a two-sided ... |
| 2idllidld 21207 | A two-sided ideal is a lef... |
| 2idlridld 21208 | A two-sided ideal is a rig... |
| df2idl2rng 21209 | Alternate (the usual textb... |
| df2idl2 21210 | Alternate (the usual textb... |
| ridl0 21211 | Every ring contains a zero... |
| ridl1 21212 | Every ring contains a unit... |
| 2idl0 21213 | Every ring contains a zero... |
| 2idl1 21214 | Every ring contains a unit... |
| 2idlss 21215 | A two-sided ideal is a sub... |
| 2idlbas 21216 | The base set of a two-side... |
| 2idlelbas 21217 | The base set of a two-side... |
| rng2idlsubrng 21218 | A two-sided ideal of a non... |
| rng2idlnsg 21219 | A two-sided ideal of a non... |
| rng2idl0 21220 | The zero (additive identit... |
| rng2idlsubgsubrng 21221 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21222 | A two-sided ideal of a non... |
| rng2idlsubg0 21223 | The zero (additive identit... |
| 2idlcpblrng 21224 | The coset equivalence rela... |
| 2idlcpbl 21225 | The coset equivalence rela... |
| qus2idrng 21226 | The quotient of a non-unit... |
| qus1 21227 | The multiplicative identit... |
| qusring 21228 | If ` S ` is a two-sided id... |
| qusrhm 21229 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21230 | The preimage of an ideal b... |
| kerlidl 21231 | The kernel of a ring homom... |
| qusmul2idl 21232 | Value of the ring operatio... |
| crngridl 21233 | In a commutative ring, the... |
| crng2idl 21234 | In a commutative ring, a t... |
| qusmulrng 21235 | Value of the multiplicatio... |
| quscrng 21236 | The quotient of a commutat... |
| qusmulcrng 21237 | Value of the ring operatio... |
| rhmqusnsg 21238 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21239 | The ring unity of a two-si... |
| rngqiprngghmlem1 21240 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21241 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21242 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21243 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21244 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21245 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21246 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21247 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21248 | The base set of the produc... |
| rngqiprng 21249 | The product of the quotien... |
| rngqiprngimf 21250 | ` F ` is a function from (... |
| rngqiprngimfv 21251 | The value of the function ... |
| rngqiprngghm 21252 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21253 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21254 | ` F ` is a function from (... |
| rngqiprnglin 21255 | ` F ` is linear with respe... |
| rngqiprngho 21256 | ` F ` is a homomorphism of... |
| rngqiprngim 21257 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21258 | The unity of a two-sided i... |
| rngringbdlem1 21259 | In a unital ring, the quot... |
| rngringbdlem2 21260 | A non-unital ring is unita... |
| rngringbd 21261 | A non-unital ring is unita... |
| ring2idlqus 21262 | For every unital ring ther... |
| ring2idlqusb 21263 | A non-unital ring is unita... |
| rngqiprngfulem1 21264 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21265 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21266 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21267 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21268 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21269 | The ring unity of the prod... |
| rngqiprngfu 21270 | The function value of ` F ... |
| rngqiprngu 21271 | If a non-unital ring has a... |
| ring2idlqus1 21272 | If a non-unital ring has a... |
| lpival 21277 | Value of the set of princi... |
| islpidl 21278 | Property of being a princi... |
| lpi0 21279 | The zero ideal is always p... |
| lpi1 21280 | The unit ideal is always p... |
| islpir 21281 | Principal ideal rings are ... |
| lpiss 21282 | Principal ideals are a sub... |
| islpir2 21283 | Principal ideal rings are ... |
| lpirring 21284 | Principal ideal rings are ... |
| drnglpir 21285 | Division rings are princip... |
| rspsn 21286 | Membership in principal id... |
| lidldvgen 21287 | An element generates an id... |
| lpigen 21288 | An ideal is principal iff ... |
| cnfldstr 21309 | The field of complex numbe... |
| cnfldex 21310 | The field of complex numbe... |
| cnfldbas 21311 | The base set of the field ... |
| mpocnfldadd 21312 | The addition operation of ... |
| cnfldadd 21313 | The addition operation of ... |
| mpocnfldmul 21314 | The multiplication operati... |
| cnfldmul 21315 | The multiplication operati... |
| cnfldcj 21316 | The conjugation operation ... |
| cnfldtset 21317 | The topology component of ... |
| cnfldle 21318 | The ordering of the field ... |
| cnfldds 21319 | The metric of the field of... |
| cnfldunif 21320 | The uniform structure comp... |
| cnfldfun 21321 | The field of complex numbe... |
| cnfldfunALT 21322 | The field of complex numbe... |
| dfcnfldOLD 21323 | Obsolete version of ~ df-c... |
| cnfldstrOLD 21324 | Obsolete version of ~ cnfl... |
| cnfldexOLD 21325 | Obsolete version of ~ cnfl... |
| cnfldbasOLD 21326 | Obsolete version of ~ cnfl... |
| cnfldaddOLD 21327 | Obsolete version of ~ cnfl... |
| cnfldmulOLD 21328 | Obsolete version of ~ cnfl... |
| cnfldcjOLD 21329 | Obsolete version of ~ cnfl... |
| cnfldtsetOLD 21330 | Obsolete version of ~ cnfl... |
| cnfldleOLD 21331 | Obsolete version of ~ cnfl... |
| cnflddsOLD 21332 | Obsolete version of ~ cnfl... |
| cnfldunifOLD 21333 | Obsolete version of ~ cnfl... |
| cnfldfunOLD 21334 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLD 21335 | Obsolete version of ~ cnfl... |
| xrsstr 21336 | The extended real structur... |
| xrsex 21337 | The extended real structur... |
| xrsadd 21338 | The addition operation of ... |
| xrsmul 21339 | The multiplication operati... |
| xrstset 21340 | The topology component of ... |
| cncrng 21341 | The complex numbers form a... |
| cncrngOLD 21342 | Obsolete version of ~ cncr... |
| cnring 21343 | The complex numbers form a... |
| xrsmcmn 21344 | The "multiplicative group"... |
| cnfld0 21345 | Zero is the zero element o... |
| cnfld1 21346 | One is the unity element o... |
| cnfld1OLD 21347 | Obsolete version of ~ cnfl... |
| cnfldneg 21348 | The additive inverse in th... |
| cnfldplusf 21349 | The functionalized additio... |
| cnfldsub 21350 | The subtraction operator i... |
| cndrng 21351 | The complex numbers form a... |
| cndrngOLD 21352 | Obsolete version of ~ cndr... |
| cnflddiv 21353 | The division operation in ... |
| cnflddivOLD 21354 | Obsolete version of ~ cnfl... |
| cnfldinv 21355 | The multiplicative inverse... |
| cnfldmulg 21356 | The group multiple functio... |
| cnfldexp 21357 | The exponentiation operato... |
| cnsrng 21358 | The complex numbers form a... |
| xrsmgm 21359 | The "additive group" of th... |
| xrsnsgrp 21360 | The "additive group" of th... |
| xrsmgmdifsgrp 21361 | The "additive group" of th... |
| xrsds 21362 | The metric of the extended... |
| xrsdsval 21363 | The metric of the extended... |
| xrsdsreval 21364 | The metric of the extended... |
| xrsdsreclblem 21365 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21366 | The metric of the extended... |
| cnsubmlem 21367 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21368 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21369 | Lemma for ~ resubdrg and f... |
| cnsubrglemOLD 21370 | Obsolete version of ~ cnsu... |
| cnsubdrglem 21371 | Lemma for ~ resubdrg and f... |
| qsubdrg 21372 | The rational numbers form ... |
| zsubrg 21373 | The integers form a subrin... |
| gzsubrg 21374 | The gaussian integers form... |
| nn0subm 21375 | The nonnegative integers f... |
| rege0subm 21376 | The nonnegative reals form... |
| absabv 21377 | The regular absolute value... |
| zsssubrg 21378 | The integers are a subset ... |
| qsssubdrg 21379 | The rational numbers are a... |
| cnsubrg 21380 | There are no subrings of t... |
| cnmgpabl 21381 | The unit group of the comp... |
| cnmgpid 21382 | The group identity element... |
| cnmsubglem 21383 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21384 | The positive reals form a ... |
| gzrngunitlem 21385 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21386 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21387 | Relate a group sum on ` CC... |
| regsumfsum 21388 | Relate a group sum on ` ( ... |
| expmhm 21389 | Exponentiation is a monoid... |
| nn0srg 21390 | The nonnegative integers f... |
| rge0srg 21391 | The nonnegative real numbe... |
| xrge0plusg 21392 | The additive law of the ex... |
| xrs1mnd 21393 | The extended real numbers,... |
| xrs10 21394 | The zero of the extended r... |
| xrs1cmn 21395 | The extended real numbers ... |
| xrge0subm 21396 | The nonnegative extended r... |
| xrge0cmn 21397 | The nonnegative extended r... |
| xrge0omnd 21398 | The nonnegative extended r... |
| zringcrng 21401 | The ring of integers is a ... |
| zringring 21402 | The ring of integers is a ... |
| zringrng 21403 | The ring of integers is a ... |
| zringabl 21404 | The ring of integers is an... |
| zringgrp 21405 | The ring of integers is an... |
| zringbas 21406 | The integers are the base ... |
| zringplusg 21407 | The addition operation of ... |
| zringsub 21408 | The subtraction of element... |
| zringmulg 21409 | The multiplication (group ... |
| zringmulr 21410 | The multiplication operati... |
| zring0 21411 | The zero element of the ri... |
| zring1 21412 | The unity element of the r... |
| zringnzr 21413 | The ring of integers is a ... |
| dvdsrzring 21414 | Ring divisibility in the r... |
| zringlpirlem1 21415 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21416 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21417 | Lemma for ~ zringlpir . A... |
| zringinvg 21418 | The additive inverse of an... |
| zringunit 21419 | The units of ` ZZ ` are th... |
| zringlpir 21420 | The integers are a princip... |
| zringndrg 21421 | The integers are not a div... |
| zringcyg 21422 | The integers are a cyclic ... |
| zringsubgval 21423 | Subtraction in the ring of... |
| zringmpg 21424 | The multiplicative group o... |
| prmirredlem 21425 | A positive integer is irre... |
| dfprm2 21426 | The positive irreducible e... |
| prmirred 21427 | The irreducible elements o... |
| expghm 21428 | Exponentiation is a group ... |
| mulgghm2 21429 | The powers of a group elem... |
| mulgrhm 21430 | The powers of the element ... |
| mulgrhm2 21431 | The powers of the element ... |
| irinitoringc 21432 | The ring of integers is an... |
| nzerooringczr 21433 | There is no zero object in... |
| pzriprnglem1 21434 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21435 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21436 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21437 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21438 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21439 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21440 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21441 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21442 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21443 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21444 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21445 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21446 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21447 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21448 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21449 | The ring unity of the ring... |
| pzriprng 21450 | The non-unital ring ` ( ZZ... |
| pzriprng1 21451 | The ring unity of the ring... |
| zrhval 21460 | Define the unique homomorp... |
| zrhval2 21461 | Alternate value of the ` Z... |
| zrhmulg 21462 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21463 | The ` ZRHom ` homomorphism... |
| zrhrhm 21464 | The ` ZRHom ` homomorphism... |
| zrh1 21465 | Interpretation of 1 in a r... |
| zrh0 21466 | Interpretation of 0 in a r... |
| zrhpropd 21467 | The ` ZZ ` ring homomorphi... |
| zlmval 21468 | Augment an abelian group w... |
| zlmlem 21469 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21470 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21471 | Group operation of a ` ZZ ... |
| zlmmulr 21472 | Ring operation of a ` ZZ `... |
| zlmsca 21473 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21474 | Scalar multiplication oper... |
| zlmlmod 21475 | The ` ZZ ` -module operati... |
| chrval 21476 | Definition substitution of... |
| chrcl 21477 | Closure of the characteris... |
| chrid 21478 | The canonical ` ZZ ` ring ... |
| chrdvds 21479 | The ` ZZ ` ring homomorphi... |
| chrcong 21480 | If two integers are congru... |
| dvdschrmulg 21481 | In a ring, any multiple of... |
| fermltlchr 21482 | A generalization of Fermat... |
| chrnzr 21483 | Nonzero rings are precisel... |
| chrrhm 21484 | The characteristic restric... |
| domnchr 21485 | The characteristic of a do... |
| znlidl 21486 | The set ` n ZZ ` is an ide... |
| zncrng2 21487 | Making a commutative ring ... |
| znval 21488 | The value of the ` Z/nZ ` ... |
| znle 21489 | The value of the ` Z/nZ ` ... |
| znval2 21490 | Self-referential expressio... |
| znbaslem 21491 | Lemma for ~ znbas . (Cont... |
| znbas2 21492 | The base set of ` Z/nZ ` i... |
| znadd 21493 | The additive structure of ... |
| znmul 21494 | The multiplicative structu... |
| znzrh 21495 | The ` ZZ ` ring homomorphi... |
| znbas 21496 | The base set of ` Z/nZ ` s... |
| zncrng 21497 | ` Z/nZ ` is a commutative ... |
| znzrh2 21498 | The ` ZZ ` ring homomorphi... |
| znzrhval 21499 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21500 | The ` ZZ ` ring homomorphi... |
| zncyg 21501 | The group ` ZZ / n ZZ ` is... |
| zndvds 21502 | Express equality of equiva... |
| zndvds0 21503 | Special case of ~ zndvds w... |
| znf1o 21504 | The function ` F ` enumera... |
| zzngim 21505 | The ` ZZ ` ring homomorphi... |
| znle2 21506 | The ordering of the ` Z/nZ... |
| znleval 21507 | The ordering of the ` Z/nZ... |
| znleval2 21508 | The ordering of the ` Z/nZ... |
| zntoslem 21509 | Lemma for ~ zntos . (Cont... |
| zntos 21510 | The ` Z/nZ ` structure is ... |
| znhash 21511 | The ` Z/nZ ` structure has... |
| znfi 21512 | The ` Z/nZ ` structure is ... |
| znfld 21513 | The ` Z/nZ ` structure is ... |
| znidomb 21514 | The ` Z/nZ ` structure is ... |
| znchr 21515 | Cyclic rings are defined b... |
| znunit 21516 | The units of ` Z/nZ ` are ... |
| znunithash 21517 | The size of the unit group... |
| znrrg 21518 | The regular elements of ` ... |
| cygznlem1 21519 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21520 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21521 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21522 | A cyclic group with ` n ` ... |
| cygzn 21523 | A cyclic group with ` n ` ... |
| cygth 21524 | The "fundamental theorem o... |
| cyggic 21525 | Cyclic groups are isomorph... |
| frgpcyg 21526 | A free group is cyclic iff... |
| freshmansdream 21527 | For a prime number ` P ` ,... |
| frobrhm 21528 | In a commutative ring with... |
| ofldchr 21529 | The characteristic of an o... |
| cnmsgnsubg 21530 | The signs form a multiplic... |
| cnmsgnbas 21531 | The base set of the sign s... |
| cnmsgngrp 21532 | The group of signs under m... |
| psgnghm 21533 | The sign is a homomorphism... |
| psgnghm2 21534 | The sign is a homomorphism... |
| psgninv 21535 | The sign of a permutation ... |
| psgnco 21536 | Multiplicativity of the pe... |
| zrhpsgnmhm 21537 | Embedding of permutation s... |
| zrhpsgninv 21538 | The embedded sign of a per... |
| evpmss 21539 | Even permutations are perm... |
| psgnevpmb 21540 | A class is an even permuta... |
| psgnodpm 21541 | A permutation which is odd... |
| psgnevpm 21542 | A permutation which is eve... |
| psgnodpmr 21543 | If a permutation has sign ... |
| zrhpsgnevpm 21544 | The sign of an even permut... |
| zrhpsgnodpm 21545 | The sign of an odd permuta... |
| cofipsgn 21546 | Composition of any class `... |
| zrhpsgnelbas 21547 | Embedding of permutation s... |
| zrhcopsgnelbas 21548 | Embedding of permutation s... |
| evpmodpmf1o 21549 | The function for performin... |
| pmtrodpm 21550 | A transposition is an odd ... |
| psgnfix1 21551 | A permutation of a finite ... |
| psgnfix2 21552 | A permutation of a finite ... |
| psgndiflemB 21553 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21554 | Lemma 2 for ~ psgndif . (... |
| psgndif 21555 | Embedding of permutation s... |
| copsgndif 21556 | Embedding of permutation s... |
| rebase 21559 | The base of the field of r... |
| remulg 21560 | The multiplication (group ... |
| resubdrg 21561 | The real numbers form a di... |
| resubgval 21562 | Subtraction in the field o... |
| replusg 21563 | The addition operation of ... |
| remulr 21564 | The multiplication operati... |
| re0g 21565 | The zero element of the fi... |
| re1r 21566 | The unity element of the f... |
| rele2 21567 | The ordering relation of t... |
| relt 21568 | The ordering relation of t... |
| reds 21569 | The distance of the field ... |
| redvr 21570 | The division operation of ... |
| retos 21571 | The real numbers are a tot... |
| refld 21572 | The real numbers form a fi... |
| refldcj 21573 | The conjugation operation ... |
| resrng 21574 | The real numbers form a st... |
| regsumsupp 21575 | The group sum over the rea... |
| rzgrp 21576 | The quotient group ` RR / ... |
| isphl 21581 | The predicate "is a genera... |
| phllvec 21582 | A pre-Hilbert space is a l... |
| phllmod 21583 | A pre-Hilbert space is a l... |
| phlsrng 21584 | The scalar ring of a pre-H... |
| phllmhm 21585 | The inner product of a pre... |
| ipcl 21586 | Closure of the inner produ... |
| ipcj 21587 | Conjugate of an inner prod... |
| iporthcom 21588 | Orthogonality (meaning inn... |
| ip0l 21589 | Inner product with a zero ... |
| ip0r 21590 | Inner product with a zero ... |
| ipeq0 21591 | The inner product of a vec... |
| ipdir 21592 | Distributive law for inner... |
| ipdi 21593 | Distributive law for inner... |
| ip2di 21594 | Distributive law for inner... |
| ipsubdir 21595 | Distributive law for inner... |
| ipsubdi 21596 | Distributive law for inner... |
| ip2subdi 21597 | Distributive law for inner... |
| ipass 21598 | Associative law for inner ... |
| ipassr 21599 | "Associative" law for seco... |
| ipassr2 21600 | "Associative" law for inne... |
| ipffval 21601 | The inner product operatio... |
| ipfval 21602 | The inner product operatio... |
| ipfeq 21603 | If the inner product opera... |
| ipffn 21604 | The inner product operatio... |
| phlipf 21605 | The inner product operatio... |
| ip2eq 21606 | Two vectors are equal iff ... |
| isphld 21607 | Properties that determine ... |
| phlpropd 21608 | If two structures have the... |
| ssipeq 21609 | The inner product on a sub... |
| phssipval 21610 | The inner product on a sub... |
| phssip 21611 | The inner product (as a fu... |
| phlssphl 21612 | A subspace of an inner pro... |
| ocvfval 21619 | The orthocomplement operat... |
| ocvval 21620 | Value of the orthocompleme... |
| elocv 21621 | Elementhood in the orthoco... |
| ocvi 21622 | Property of a member of th... |
| ocvss 21623 | The orthocomplement of a s... |
| ocvocv 21624 | A set is contained in its ... |
| ocvlss 21625 | The orthocomplement of a s... |
| ocv2ss 21626 | Orthocomplements reverse s... |
| ocvin 21627 | An orthocomplement has tri... |
| ocvsscon 21628 | Two ways to say that ` S `... |
| ocvlsp 21629 | The orthocomplement of a l... |
| ocv0 21630 | The orthocomplement of the... |
| ocvz 21631 | The orthocomplement of the... |
| ocv1 21632 | The orthocomplement of the... |
| unocv 21633 | The orthocomplement of a u... |
| iunocv 21634 | The orthocomplement of an ... |
| cssval 21635 | The set of closed subspace... |
| iscss 21636 | The predicate "is a closed... |
| cssi 21637 | Property of a closed subsp... |
| cssss 21638 | A closed subspace is a sub... |
| iscss2 21639 | It is sufficient to prove ... |
| ocvcss 21640 | The orthocomplement of any... |
| cssincl 21641 | The zero subspace is a clo... |
| css0 21642 | The zero subspace is a clo... |
| css1 21643 | The whole space is a close... |
| csslss 21644 | A closed subspace of a pre... |
| lsmcss 21645 | A subset of a pre-Hilbert ... |
| cssmre 21646 | The closed subspaces of a ... |
| mrccss 21647 | The Moore closure correspo... |
| thlval 21648 | Value of the Hilbert latti... |
| thlbas 21649 | Base set of the Hilbert la... |
| thlle 21650 | Ordering on the Hilbert la... |
| thlleval 21651 | Ordering on the Hilbert la... |
| thloc 21652 | Orthocomplement on the Hil... |
| pjfval 21659 | The value of the projectio... |
| pjdm 21660 | A subspace is in the domai... |
| pjpm 21661 | The projection map is a pa... |
| pjfval2 21662 | Value of the projection ma... |
| pjval 21663 | Value of the projection ma... |
| pjdm2 21664 | A subspace is in the domai... |
| pjff 21665 | A projection is a linear o... |
| pjf 21666 | A projection is a function... |
| pjf2 21667 | A projection is a function... |
| pjfo 21668 | A projection is a surjecti... |
| pjcss 21669 | A projection subspace is a... |
| ocvpj 21670 | The orthocomplement of a p... |
| ishil 21671 | The predicate "is a Hilber... |
| ishil2 21672 | The predicate "is a Hilber... |
| isobs 21673 | The predicate "is an ortho... |
| obsip 21674 | The inner product of two e... |
| obsipid 21675 | A basis element has length... |
| obsrcl 21676 | Reverse closure for an ort... |
| obsss 21677 | An orthonormal basis is a ... |
| obsne0 21678 | A basis element is nonzero... |
| obsocv 21679 | An orthonormal basis has t... |
| obs2ocv 21680 | The double orthocomplement... |
| obselocv 21681 | A basis element is in the ... |
| obs2ss 21682 | A basis has no proper subs... |
| obslbs 21683 | An orthogonal basis is a l... |
| reldmdsmm 21686 | The direct sum is a well-b... |
| dsmmval 21687 | Value of the module direct... |
| dsmmbase 21688 | Base set of the module dir... |
| dsmmval2 21689 | Self-referential definitio... |
| dsmmbas2 21690 | Base set of the direct sum... |
| dsmmfi 21691 | For finite products, the d... |
| dsmmelbas 21692 | Membership in the finitely... |
| dsmm0cl 21693 | The all-zero vector is con... |
| dsmmacl 21694 | The finite hull is closed ... |
| prdsinvgd2 21695 | Negation of a single coord... |
| dsmmsubg 21696 | The finite hull of a produ... |
| dsmmlss 21697 | The finite hull of a produ... |
| dsmmlmod 21698 | The direct sum of a family... |
| frlmval 21701 | Value of the "free module"... |
| frlmlmod 21702 | The free module is a modul... |
| frlmpws 21703 | The free module as a restr... |
| frlmlss 21704 | The base set of the free m... |
| frlmpwsfi 21705 | The finite free module is ... |
| frlmsca 21706 | The ring of scalars of a f... |
| frlm0 21707 | Zero in a free module (rin... |
| frlmbas 21708 | Base set of the free modul... |
| frlmelbas 21709 | Membership in the base set... |
| frlmrcl 21710 | If a free module is inhabi... |
| frlmbasfsupp 21711 | Elements of the free modul... |
| frlmbasmap 21712 | Elements of the free modul... |
| frlmbasf 21713 | Elements of the free modul... |
| frlmlvec 21714 | The free module over a div... |
| frlmfibas 21715 | The base set of the finite... |
| elfrlmbasn0 21716 | If the dimension of a free... |
| frlmplusgval 21717 | Addition in a free module.... |
| frlmsubgval 21718 | Subtraction in a free modu... |
| frlmvscafval 21719 | Scalar multiplication in a... |
| frlmvplusgvalc 21720 | Coordinates of a sum with ... |
| frlmvscaval 21721 | Coordinates of a scalar mu... |
| frlmplusgvalb 21722 | Addition in a free module ... |
| frlmvscavalb 21723 | Scalar multiplication in a... |
| frlmvplusgscavalb 21724 | Addition combined with sca... |
| frlmgsum 21725 | Finite commutative sums in... |
| frlmsplit2 21726 | Restriction is homomorphic... |
| frlmsslss 21727 | A subset of a free module ... |
| frlmsslss2 21728 | A subset of a free module ... |
| frlmbas3 21729 | An element of the base set... |
| mpofrlmd 21730 | Elements of the free modul... |
| frlmip 21731 | The inner product of a fre... |
| frlmipval 21732 | The inner product of a fre... |
| frlmphllem 21733 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21734 | Conditions for a free modu... |
| uvcfval 21737 | Value of the unit-vector g... |
| uvcval 21738 | Value of a single unit vec... |
| uvcvval 21739 | Value of a unit vector coo... |
| uvcvvcl 21740 | A coordinate of a unit vec... |
| uvcvvcl2 21741 | A unit vector coordinate i... |
| uvcvv1 21742 | The unit vector is one at ... |
| uvcvv0 21743 | The unit vector is zero at... |
| uvcff 21744 | Domain and codomain of the... |
| uvcf1 21745 | In a nonzero ring, each un... |
| uvcresum 21746 | Any element of a free modu... |
| frlmssuvc1 21747 | A scalar multiple of a uni... |
| frlmssuvc2 21748 | A nonzero scalar multiple ... |
| frlmsslsp 21749 | A subset of a free module ... |
| frlmlbs 21750 | The unit vectors comprise ... |
| frlmup1 21751 | Any assignment of unit vec... |
| frlmup2 21752 | The evaluation map has the... |
| frlmup3 21753 | The range of such an evalu... |
| frlmup4 21754 | Universal property of the ... |
| ellspd 21755 | The elements of the span o... |
| elfilspd 21756 | Simplified version of ~ el... |
| rellindf 21761 | The independent-family pre... |
| islinds 21762 | Property of an independent... |
| linds1 21763 | An independent set of vect... |
| linds2 21764 | An independent set of vect... |
| islindf 21765 | Property of an independent... |
| islinds2 21766 | Expanded property of an in... |
| islindf2 21767 | Property of an independent... |
| lindff 21768 | Functional property of a l... |
| lindfind 21769 | A linearly independent fam... |
| lindsind 21770 | A linearly independent set... |
| lindfind2 21771 | In a linearly independent ... |
| lindsind2 21772 | In a linearly independent ... |
| lindff1 21773 | A linearly independent fam... |
| lindfrn 21774 | The range of an independen... |
| f1lindf 21775 | Rearranging and deleting e... |
| lindfres 21776 | Any restriction of an inde... |
| lindsss 21777 | Any subset of an independe... |
| f1linds 21778 | A family constructed from ... |
| islindf3 21779 | In a nonzero ring, indepen... |
| lindfmm 21780 | Linear independence of a f... |
| lindsmm 21781 | Linear independence of a s... |
| lindsmm2 21782 | The monomorphic image of a... |
| lsslindf 21783 | Linear independence is unc... |
| lsslinds 21784 | Linear independence is unc... |
| islbs4 21785 | A basis is an independent ... |
| lbslinds 21786 | A basis is independent. (... |
| islinds3 21787 | A subset is linearly indep... |
| islinds4 21788 | A set is independent in a ... |
| lmimlbs 21789 | The isomorphic image of a ... |
| lmiclbs 21790 | Having a basis is an isomo... |
| islindf4 21791 | A family is independent if... |
| islindf5 21792 | A family is independent if... |
| indlcim 21793 | An independent, spanning f... |
| lbslcic 21794 | A module with a basis is i... |
| lmisfree 21795 | A module has a basis iff i... |
| lvecisfrlm 21796 | Every vector space is isom... |
| lmimco 21797 | The composition of two iso... |
| lmictra 21798 | Module isomorphism is tran... |
| uvcf1o 21799 | In a nonzero ring, the map... |
| uvcendim 21800 | In a nonzero ring, the num... |
| frlmisfrlm 21801 | A free module is isomorphi... |
| frlmiscvec 21802 | Every free module is isomo... |
| isassa 21809 | The properties of an assoc... |
| assalem 21810 | The properties of an assoc... |
| assaass 21811 | Left-associative property ... |
| assaassr 21812 | Right-associative property... |
| assalmod 21813 | An associative algebra is ... |
| assaring 21814 | An associative algebra is ... |
| assasca 21815 | The scalars of an associat... |
| assa2ass 21816 | Left- and right-associativ... |
| assa2ass2 21817 | Left- and right-associativ... |
| isassad 21818 | Sufficient condition for b... |
| issubassa3 21819 | A subring that is also a s... |
| issubassa 21820 | The subalgebras of an asso... |
| sraassab 21821 | A subring algebra is an as... |
| sraassa 21822 | The subring algebra over a... |
| sraassaOLD 21823 | Obsolete version of ~ sraa... |
| rlmassa 21824 | The ring module over a com... |
| assapropd 21825 | If two structures have the... |
| aspval 21826 | Value of the algebraic clo... |
| asplss 21827 | The algebraic span of a se... |
| aspid 21828 | The algebraic span of a su... |
| aspsubrg 21829 | The algebraic span of a se... |
| aspss 21830 | Span preserves subset orde... |
| aspssid 21831 | A set of vectors is a subs... |
| asclfval 21832 | Function value of the alge... |
| asclval 21833 | Value of a mapped algebra ... |
| asclfn 21834 | Unconditional functionalit... |
| asclf 21835 | The algebra scalar lifting... |
| asclghm 21836 | The algebra scalar lifting... |
| asclelbas 21837 | Lifted scalars are in the ... |
| ascl0 21838 | The scalar 0 embedded into... |
| ascl1 21839 | The scalar 1 embedded into... |
| asclmul1 21840 | Left multiplication by a l... |
| asclmul2 21841 | Right multiplication by a ... |
| ascldimul 21842 | The algebra scalar lifting... |
| asclinvg 21843 | The group inverse (negatio... |
| asclrhm 21844 | The algebra scalar lifting... |
| rnascl 21845 | The set of lifted scalars ... |
| issubassa2 21846 | A subring of a unital alge... |
| rnasclsubrg 21847 | The scalar multiples of th... |
| rnasclmulcl 21848 | (Vector) multiplication is... |
| rnasclassa 21849 | The scalar multiples of th... |
| ressascl 21850 | The lifting of scalars is ... |
| asclpropd 21851 | If two structures have the... |
| aspval2 21852 | The algebraic closure is t... |
| assamulgscmlem1 21853 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21854 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21855 | Exponentiation of a scalar... |
| asclmulg 21856 | Apply group multiplication... |
| zlmassa 21857 | The ` ZZ ` -module operati... |
| reldmpsr 21868 | The multivariate power ser... |
| psrval 21869 | Value of the multivariate ... |
| psrvalstr 21870 | The multivariate power ser... |
| psrbag 21871 | Elementhood in the set of ... |
| psrbagf 21872 | A finite bag is a function... |
| psrbagfsupp 21873 | Finite bags have finite su... |
| snifpsrbag 21874 | A bag containing one eleme... |
| fczpsrbag 21875 | The constant function equa... |
| psrbaglesupp 21876 | The support of a dominated... |
| psrbaglecl 21877 | The set of finite bags is ... |
| psrbagaddcl 21878 | The sum of two finite bags... |
| psrbagcon 21879 | The analogue of the statem... |
| psrbaglefi 21880 | There are finitely many ba... |
| psrbagconcl 21881 | The complement of a bag is... |
| psrbagleadd1 21882 | The analogue of " ` X <_ F... |
| psrbagconf1o 21883 | Bag complementation is a b... |
| gsumbagdiaglem 21884 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21885 | Two-dimensional commutatio... |
| psrass1lem 21886 | A group sum commutation us... |
| psrbas 21887 | The base set of the multiv... |
| psrelbas 21888 | An element of the set of p... |
| psrelbasfun 21889 | An element of the set of p... |
| psrplusg 21890 | The addition operation of ... |
| psradd 21891 | The addition operation of ... |
| psraddcl 21892 | Closure of the power serie... |
| psraddclOLD 21893 | Obsolete version of ~ psra... |
| rhmpsrlem1 21894 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21895 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21896 | The multiplication operati... |
| psrmulfval 21897 | The multiplication operati... |
| psrmulval 21898 | The multiplication operati... |
| psrmulcllem 21899 | Closure of the power serie... |
| psrmulcl 21900 | Closure of the power serie... |
| psrsca 21901 | The scalar field of the mu... |
| psrvscafval 21902 | The scalar multiplication ... |
| psrvsca 21903 | The scalar multiplication ... |
| psrvscaval 21904 | The scalar multiplication ... |
| psrvscacl 21905 | Closure of the power serie... |
| psr0cl 21906 | The zero element of the ri... |
| psr0lid 21907 | The zero element of the ri... |
| psrnegcl 21908 | The negative function in t... |
| psrlinv 21909 | The negative function in t... |
| psrgrp 21910 | The ring of power series i... |
| psr0 21911 | The zero element of the ri... |
| psrneg 21912 | The negative function of t... |
| psrlmod 21913 | The ring of power series i... |
| psr1cl 21914 | The identity element of th... |
| psrlidm 21915 | The identity element of th... |
| psrridm 21916 | The identity element of th... |
| psrass1 21917 | Associative identity for t... |
| psrdi 21918 | Distributive law for the r... |
| psrdir 21919 | Distributive law for the r... |
| psrass23l 21920 | Associative identity for t... |
| psrcom 21921 | Commutative law for the ri... |
| psrass23 21922 | Associative identities for... |
| psrring 21923 | The ring of power series i... |
| psr1 21924 | The identity element of th... |
| psrcrng 21925 | The ring of power series i... |
| psrassa 21926 | The ring of power series i... |
| resspsrbas 21927 | A restricted power series ... |
| resspsradd 21928 | A restricted power series ... |
| resspsrmul 21929 | A restricted power series ... |
| resspsrvsca 21930 | A restricted power series ... |
| subrgpsr 21931 | A subring of the base ring... |
| psrascl 21932 | Value of the scalar inject... |
| psrasclcl 21933 | A scalar is lifted into a ... |
| mvrfval 21934 | Value of the generating el... |
| mvrval 21935 | Value of the generating el... |
| mvrval2 21936 | Value of the generating el... |
| mvrid 21937 | The ` X i ` -th coefficien... |
| mvrf 21938 | The power series variable ... |
| mvrf1 21939 | The power series variable ... |
| mvrcl2 21940 | A power series variable is... |
| reldmmpl 21941 | The multivariate polynomia... |
| mplval 21942 | Value of the set of multiv... |
| mplbas 21943 | Base set of the set of mul... |
| mplelbas 21944 | Property of being a polyno... |
| mvrcl 21945 | A power series variable is... |
| mvrf2 21946 | The power series/polynomia... |
| mplrcl 21947 | Reverse closure for the po... |
| mplelsfi 21948 | A polynomial treated as a ... |
| mplval2 21949 | Self-referential expressio... |
| mplbasss 21950 | The set of polynomials is ... |
| mplelf 21951 | A polynomial is defined as... |
| mplsubglem 21952 | If ` A ` is an ideal of se... |
| mpllsslem 21953 | If ` A ` is an ideal of su... |
| mplsubglem2 21954 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21955 | The set of polynomials is ... |
| mpllss 21956 | The set of polynomials is ... |
| mplsubrglem 21957 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21958 | The set of polynomials is ... |
| mpl0 21959 | The zero polynomial. (Con... |
| mplplusg 21960 | Value of addition in a pol... |
| mplmulr 21961 | Value of multiplication in... |
| mpladd 21962 | The addition operation on ... |
| mplneg 21963 | The negative function on m... |
| mplmul 21964 | The multiplication operati... |
| mpl1 21965 | The identity element of th... |
| mplsca 21966 | The scalar field of a mult... |
| mplvsca2 21967 | The scalar multiplication ... |
| mplvsca 21968 | The scalar multiplication ... |
| mplvscaval 21969 | The scalar multiplication ... |
| mplgrp 21970 | The polynomial ring is a g... |
| mpllmod 21971 | The polynomial ring is a l... |
| mplring 21972 | The polynomial ring is a r... |
| mpllvec 21973 | The polynomial ring is a v... |
| mplcrng 21974 | The polynomial ring is a c... |
| mplassa 21975 | The polynomial ring is an ... |
| mplringd 21976 | The polynomial ring is a r... |
| mpllmodd 21977 | The polynomial ring is a l... |
| mplascl0 21978 | The zero scalar as a polyn... |
| mplascl1 21979 | The one scalar as a polyno... |
| ressmplbas2 21980 | The base set of a restrict... |
| ressmplbas 21981 | A restricted polynomial al... |
| ressmpladd 21982 | A restricted polynomial al... |
| ressmplmul 21983 | A restricted polynomial al... |
| ressmplvsca 21984 | A restricted power series ... |
| subrgmpl 21985 | A subring of the base ring... |
| subrgmvr 21986 | The variables in a subring... |
| subrgmvrf 21987 | The variables in a polynom... |
| mplmon 21988 | A monomial is a polynomial... |
| mplmonmul 21989 | The product of two monomia... |
| mplcoe1 21990 | Decompose a polynomial int... |
| mplcoe3 21991 | Decompose a monomial in on... |
| mplcoe5lem 21992 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21993 | Decompose a monomial into ... |
| mplcoe2 21994 | Decompose a monomial into ... |
| mplbas2 21995 | An alternative expression ... |
| ltbval 21996 | Value of the well-order on... |
| ltbwe 21997 | The finite bag order is a ... |
| reldmopsr 21998 | Lemma for ordered power se... |
| opsrval 21999 | The value of the "ordered ... |
| opsrle 22000 | An alternative expression ... |
| opsrval2 22001 | Self-referential expressio... |
| opsrbaslem 22002 | Get a component of the ord... |
| opsrbas 22003 | The base set of the ordere... |
| opsrplusg 22004 | The addition operation of ... |
| opsrmulr 22005 | The multiplication operati... |
| opsrvsca 22006 | The scalar product operati... |
| opsrsca 22007 | The scalar ring of the ord... |
| opsrtoslem1 22008 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22009 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22010 | The ordered power series s... |
| opsrso 22011 | The ordered power series s... |
| opsrcrng 22012 | The ring of ordered power ... |
| opsrassa 22013 | The ring of ordered power ... |
| mplmon2 22014 | Express a scaled monomial.... |
| psrbag0 22015 | The empty bag is a bag. (... |
| psrbagsn 22016 | A singleton bag is a bag. ... |
| mplascl 22017 | Value of the scalar inject... |
| mplasclf 22018 | The scalar injection is a ... |
| subrgascl 22019 | The scalar injection funct... |
| subrgasclcl 22020 | The scalars in a polynomia... |
| mplmon2cl 22021 | A scaled monomial is a pol... |
| mplmon2mul 22022 | Product of scaled monomial... |
| mplind 22023 | Prove a property of polyno... |
| mplcoe4 22024 | Decompose a polynomial int... |
| evlslem4 22029 | The support of a tensor pr... |
| psrbagev1 22030 | A bag of multipliers provi... |
| psrbagev2 22031 | Closure of a sum using a b... |
| evlslem2 22032 | A linear function on the p... |
| evlslem3 22033 | Lemma for ~ evlseu . Poly... |
| evlslem6 22034 | Lemma for ~ evlseu . Fini... |
| evlslem1 22035 | Lemma for ~ evlseu , give ... |
| evlseu 22036 | For a given interpretation... |
| reldmevls 22037 | Well-behaved binary operat... |
| mpfrcl 22038 | Reverse closure for the se... |
| evlsval 22039 | Value of the polynomial ev... |
| evlsval2 22040 | Characterizing properties ... |
| evlsrhm 22041 | Polynomial evaluation is a... |
| evlsval3 22042 | Give a formula for the pol... |
| evlsvval 22043 | Give a formula for the eva... |
| evlsvvvallem 22044 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 22045 | Lemma for theorems using ~... |
| evlsvvval 22046 | Give a formula for the eva... |
| evlssca 22047 | Polynomial evaluation maps... |
| evlsvar 22048 | Polynomial evaluation maps... |
| evlsgsumadd 22049 | Polynomial evaluation maps... |
| evlsgsummul 22050 | Polynomial evaluation maps... |
| evlspw 22051 | Polynomial evaluation for ... |
| evlsvarpw 22052 | Polynomial evaluation for ... |
| evlval 22053 | Value of the simple/same r... |
| evlrhm 22054 | The simple evaluation map ... |
| evlcl 22055 | A polynomial over the ring... |
| evladdval 22056 | Polynomial evaluation buil... |
| evlmulval 22057 | Polynomial evaluation buil... |
| evlsscasrng 22058 | The evaluation of a scalar... |
| evlsca 22059 | Simple polynomial evaluati... |
| evlsvarsrng 22060 | The evaluation of the vari... |
| evlvar 22061 | Simple polynomial evaluati... |
| mpfconst 22062 | Constants are multivariate... |
| mpfproj 22063 | Projections are multivaria... |
| mpfsubrg 22064 | Polynomial functions are a... |
| mpff 22065 | Polynomial functions are f... |
| mpfaddcl 22066 | The sum of multivariate po... |
| mpfmulcl 22067 | The product of multivariat... |
| mpfind 22068 | Prove a property of polyno... |
| selvffval 22074 | Value of the "variable sel... |
| selvfval 22075 | Value of the "variable sel... |
| selvval 22076 | Value of the "variable sel... |
| reldmmhp 22078 | The domain of the homogene... |
| mhpfval 22079 | Value of the "homogeneous ... |
| mhpval 22080 | Value of the "homogeneous ... |
| ismhp 22081 | Property of being a homoge... |
| ismhp2 22082 | Deduce a homogeneous polyn... |
| ismhp3 22083 | A polynomial is homogeneou... |
| mhprcl 22084 | Reverse closure for homoge... |
| mhpmpl 22085 | A homogeneous polynomial i... |
| mhpdeg 22086 | All nonzero terms of a hom... |
| mhp0cl 22087 | The zero polynomial is hom... |
| mhpsclcl 22088 | A scalar (or constant) pol... |
| mhpvarcl 22089 | A power series variable is... |
| mhpmulcl 22090 | A product of homogeneous p... |
| mhppwdeg 22091 | Degree of a homogeneous po... |
| mhpaddcl 22092 | Homogeneous polynomials ar... |
| mhpinvcl 22093 | Homogeneous polynomials ar... |
| mhpsubg 22094 | Homogeneous polynomials fo... |
| mhpvscacl 22095 | Homogeneous polynomials ar... |
| mhplss 22096 | Homogeneous polynomials fo... |
| psdffval 22098 | Value of the power series ... |
| psdfval 22099 | Give a map between power s... |
| psdval 22100 | Evaluate the partial deriv... |
| psdcoef 22101 | Coefficient of a term of t... |
| psdcl 22102 | The derivative of a power ... |
| psdmplcl 22103 | The derivative of a polyno... |
| psdadd 22104 | The derivative of a sum is... |
| psdvsca 22105 | The derivative of a scaled... |
| psdmullem 22106 | Lemma for ~ psdmul . Tran... |
| psdmul 22107 | Product rule for power ser... |
| psd1 22108 | The derivative of one is z... |
| psdascl 22109 | The derivative of a consta... |
| psdmvr 22110 | The partial derivative of ... |
| psdpw 22111 | Power rule for partial der... |
| psr1baslem 22123 | The set of finite bags on ... |
| psr1val 22124 | Value of the ring of univa... |
| psr1crng 22125 | The ring of univariate pow... |
| psr1assa 22126 | The ring of univariate pow... |
| psr1tos 22127 | The ordered power series s... |
| psr1bas2 22128 | The base set of the ring o... |
| psr1bas 22129 | The base set of the ring o... |
| vr1val 22130 | The value of the generator... |
| vr1cl2 22131 | The variable ` X ` is a me... |
| ply1val 22132 | The value of the set of un... |
| ply1bas 22133 | The value of the base set ... |
| ply1basOLD 22134 | Obsolete version of ~ ply1... |
| ply1lss 22135 | Univariate polynomials for... |
| ply1subrg 22136 | Univariate polynomials for... |
| ply1crng 22137 | The ring of univariate pol... |
| ply1assa 22138 | The ring of univariate pol... |
| psr1bascl 22139 | A univariate power series ... |
| psr1basf 22140 | Univariate power series ba... |
| ply1basf 22141 | Univariate polynomial base... |
| ply1bascl 22142 | A univariate polynomial is... |
| ply1bascl2 22143 | A univariate polynomial is... |
| coe1fval 22144 | Value of the univariate po... |
| coe1fv 22145 | Value of an evaluated coef... |
| fvcoe1 22146 | Value of a multivariate co... |
| coe1fval3 22147 | Univariate power series co... |
| coe1f2 22148 | Functionality of univariat... |
| coe1fval2 22149 | Univariate polynomial coef... |
| coe1f 22150 | Functionality of univariat... |
| coe1fvalcl 22151 | A coefficient of a univari... |
| coe1sfi 22152 | Finite support of univaria... |
| coe1fsupp 22153 | The coefficient vector of ... |
| mptcoe1fsupp 22154 | A mapping involving coeffi... |
| coe1ae0 22155 | The coefficient vector of ... |
| vr1cl 22156 | The generator of a univari... |
| opsr0 22157 | Zero in the ordered power ... |
| opsr1 22158 | One in the ordered power s... |
| psr1plusg 22159 | Value of addition in a uni... |
| psr1vsca 22160 | Value of scalar multiplica... |
| psr1mulr 22161 | Value of multiplication in... |
| ply1plusg 22162 | Value of addition in a uni... |
| ply1vsca 22163 | Value of scalar multiplica... |
| ply1mulr 22164 | Value of multiplication in... |
| ply1ass23l 22165 | Associative identity with ... |
| ressply1bas2 22166 | The base set of a restrict... |
| ressply1bas 22167 | A restricted polynomial al... |
| ressply1add 22168 | A restricted polynomial al... |
| ressply1mul 22169 | A restricted polynomial al... |
| ressply1vsca 22170 | A restricted power series ... |
| subrgply1 22171 | A subring of the base ring... |
| gsumply1subr 22172 | Evaluate a group sum in a ... |
| psrbaspropd 22173 | Property deduction for pow... |
| psrplusgpropd 22174 | Property deduction for pow... |
| mplbaspropd 22175 | Property deduction for pol... |
| psropprmul 22176 | Reversing multiplication i... |
| ply1opprmul 22177 | Reversing multiplication i... |
| 00ply1bas 22178 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22179 | Protection compatibility o... |
| ply1plusgfvi 22180 | Protection compatibility o... |
| ply1baspropd 22181 | Property deduction for uni... |
| ply1plusgpropd 22182 | Property deduction for uni... |
| opsrring 22183 | Ordered power series form ... |
| opsrlmod 22184 | Ordered power series form ... |
| psr1ring 22185 | Univariate power series fo... |
| ply1ring 22186 | Univariate polynomials for... |
| psr1lmod 22187 | Univariate power series fo... |
| psr1sca 22188 | Scalars of a univariate po... |
| psr1sca2 22189 | Scalars of a univariate po... |
| ply1lmod 22190 | Univariate polynomials for... |
| ply1sca 22191 | Scalars of a univariate po... |
| ply1sca2 22192 | Scalars of a univariate po... |
| ply1ascl0 22193 | The zero scalar as a polyn... |
| ply1ascl1 22194 | The multiplicative identit... |
| ply1mpl0 22195 | The univariate polynomial ... |
| ply10s0 22196 | Zero times a univariate po... |
| ply1mpl1 22197 | The univariate polynomial ... |
| ply1ascl 22198 | The univariate polynomial ... |
| subrg1ascl 22199 | The scalar injection funct... |
| subrg1asclcl 22200 | The scalars in a polynomia... |
| subrgvr1 22201 | The variables in a subring... |
| subrgvr1cl 22202 | The variables in a polynom... |
| coe1z 22203 | The coefficient vector of ... |
| coe1add 22204 | The coefficient vector of ... |
| coe1addfv 22205 | A particular coefficient o... |
| coe1subfv 22206 | A particular coefficient o... |
| coe1mul2lem1 22207 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22208 | An equivalence for ~ coe1m... |
| coe1mul2 22209 | The coefficient vector of ... |
| coe1mul 22210 | The coefficient vector of ... |
| ply1moncl 22211 | Closure of the expression ... |
| ply1tmcl 22212 | Closure of the expression ... |
| coe1tm 22213 | Coefficient vector of a po... |
| coe1tmfv1 22214 | Nonzero coefficient of a p... |
| coe1tmfv2 22215 | Zero coefficient of a poly... |
| coe1tmmul2 22216 | Coefficient vector of a po... |
| coe1tmmul 22217 | Coefficient vector of a po... |
| coe1tmmul2fv 22218 | Function value of a right-... |
| coe1pwmul 22219 | Coefficient vector of a po... |
| coe1pwmulfv 22220 | Function value of a right-... |
| ply1scltm 22221 | A scalar is a term with ze... |
| coe1sclmul 22222 | Coefficient vector of a po... |
| coe1sclmulfv 22223 | A single coefficient of a ... |
| coe1sclmul2 22224 | Coefficient vector of a po... |
| ply1sclf 22225 | A scalar polynomial is a p... |
| ply1sclcl 22226 | The value of the algebra s... |
| coe1scl 22227 | Coefficient vector of a sc... |
| ply1sclid 22228 | Recover the base scalar fr... |
| ply1sclf1 22229 | The polynomial scalar func... |
| ply1scl0 22230 | The zero scalar is zero. ... |
| ply1scl0OLD 22231 | Obsolete version of ~ ply1... |
| ply1scln0 22232 | Nonzero scalars create non... |
| ply1scl1 22233 | The one scalar is the unit... |
| ply1scl1OLD 22234 | Obsolete version of ~ ply1... |
| coe1id 22235 | Coefficient vector of the ... |
| ply1idvr1 22236 | The identity of a polynomi... |
| ply1idvr1OLD 22237 | Obsolete version of ~ ply1... |
| cply1mul 22238 | The product of two constan... |
| ply1coefsupp 22239 | The decomposition of a uni... |
| ply1coe 22240 | Decompose a univariate pol... |
| eqcoe1ply1eq 22241 | Two polynomials over the s... |
| ply1coe1eq 22242 | Two polynomials over the s... |
| cply1coe0 22243 | All but the first coeffici... |
| cply1coe0bi 22244 | A polynomial is constant (... |
| coe1fzgsumdlem 22245 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22246 | Value of an evaluated coef... |
| ply1scleq 22247 | Equality of a constant pol... |
| ply1chr 22248 | The characteristic of a po... |
| gsumsmonply1 22249 | A finite group sum of scal... |
| gsummoncoe1 22250 | A coefficient of the polyn... |
| gsumply1eq 22251 | Two univariate polynomials... |
| lply1binom 22252 | The binomial theorem for l... |
| lply1binomsc 22253 | The binomial theorem for l... |
| ply1fermltlchr 22254 | Fermat's little theorem fo... |
| reldmevls1 22259 | Well-behaved binary operat... |
| ply1frcl 22260 | Reverse closure for the se... |
| evls1fval 22261 | Value of the univariate po... |
| evls1val 22262 | Value of the univariate po... |
| evls1rhmlem 22263 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22264 | Polynomial evaluation is a... |
| evls1sca 22265 | Univariate polynomial eval... |
| evls1gsumadd 22266 | Univariate polynomial eval... |
| evls1gsummul 22267 | Univariate polynomial eval... |
| evls1pw 22268 | Univariate polynomial eval... |
| evls1varpw 22269 | Univariate polynomial eval... |
| evl1fval 22270 | Value of the simple/same r... |
| evl1val 22271 | Value of the simple/same r... |
| evl1fval1lem 22272 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22273 | Value of the simple/same r... |
| evl1rhm 22274 | Polynomial evaluation is a... |
| fveval1fvcl 22275 | The function value of the ... |
| evl1sca 22276 | Polynomial evaluation maps... |
| evl1scad 22277 | Polynomial evaluation buil... |
| evl1var 22278 | Polynomial evaluation maps... |
| evl1vard 22279 | Polynomial evaluation buil... |
| evls1var 22280 | Univariate polynomial eval... |
| evls1scasrng 22281 | The evaluation of a scalar... |
| evls1varsrng 22282 | The evaluation of the vari... |
| evl1addd 22283 | Polynomial evaluation buil... |
| evl1subd 22284 | Polynomial evaluation buil... |
| evl1muld 22285 | Polynomial evaluation buil... |
| evl1vsd 22286 | Polynomial evaluation buil... |
| evl1expd 22287 | Polynomial evaluation buil... |
| pf1const 22288 | Constants are polynomial f... |
| pf1id 22289 | The identity is a polynomi... |
| pf1subrg 22290 | Polynomial functions are a... |
| pf1rcl 22291 | Reverse closure for the se... |
| pf1f 22292 | Polynomial functions are f... |
| mpfpf1 22293 | Convert a multivariate pol... |
| pf1mpf 22294 | Convert a univariate polyn... |
| pf1addcl 22295 | The sum of multivariate po... |
| pf1mulcl 22296 | The product of multivariat... |
| pf1ind 22297 | Prove a property of polyno... |
| evl1gsumdlem 22298 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22299 | Polynomial evaluation buil... |
| evl1gsumadd 22300 | Univariate polynomial eval... |
| evl1gsumaddval 22301 | Value of a univariate poly... |
| evl1gsummul 22302 | Univariate polynomial eval... |
| evl1varpw 22303 | Univariate polynomial eval... |
| evl1varpwval 22304 | Value of a univariate poly... |
| evl1scvarpw 22305 | Univariate polynomial eval... |
| evl1scvarpwval 22306 | Value of a univariate poly... |
| evl1gsummon 22307 | Value of a univariate poly... |
| evls1scafv 22308 | Value of the univariate po... |
| evls1expd 22309 | Univariate polynomial eval... |
| evls1varpwval 22310 | Univariate polynomial eval... |
| evls1fpws 22311 | Evaluation of a univariate... |
| ressply1evl 22312 | Evaluation of a univariate... |
| evls1addd 22313 | Univariate polynomial eval... |
| evls1muld 22314 | Univariate polynomial eval... |
| evls1vsca 22315 | Univariate polynomial eval... |
| asclply1subcl 22316 | Closure of the algebra sca... |
| evls1fvcl 22317 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22318 | The function ` F ` mapping... |
| evls1maplmhm 22319 | The function ` F ` mapping... |
| evls1maprnss 22320 | The function ` F ` mapping... |
| evl1maprhm 22321 | The function ` F ` mapping... |
| mhmcompl 22322 | The composition of a monoi... |
| mhmcoaddmpl 22323 | Show that the ring homomor... |
| rhmcomulmpl 22324 | Show that the ring homomor... |
| rhmmpl 22325 | Provide a ring homomorphis... |
| ply1vscl 22326 | Closure of scalar multipli... |
| mhmcoply1 22327 | The composition of a monoi... |
| rhmply1 22328 | Provide a ring homomorphis... |
| rhmply1vr1 22329 | A ring homomorphism betwee... |
| rhmply1vsca 22330 | Apply a ring homomorphism ... |
| rhmply1mon 22331 | Apply a ring homomorphism ... |
| mamufval 22334 | Functional value of the ma... |
| mamuval 22335 | Multiplication of two matr... |
| mamufv 22336 | A cell in the multiplicati... |
| mamudm 22337 | The domain of the matrix m... |
| mamufacex 22338 | Every solution of the equa... |
| mamures 22339 | Rows in a matrix product a... |
| grpvlinv 22340 | Tuple-wise left inverse in... |
| grpvrinv 22341 | Tuple-wise right inverse i... |
| ringvcl 22342 | Tuple-wise multiplication ... |
| mamucl 22343 | Operation closure of matri... |
| mamuass 22344 | Matrix multiplication is a... |
| mamudi 22345 | Matrix multiplication dist... |
| mamudir 22346 | Matrix multiplication dist... |
| mamuvs1 22347 | Matrix multiplication dist... |
| mamuvs2 22348 | Matrix multiplication dist... |
| matbas0pc 22351 | There is no matrix with a ... |
| matbas0 22352 | There is no matrix for a n... |
| matval 22353 | Value of the matrix algebr... |
| matrcl 22354 | Reverse closure for the ma... |
| matbas 22355 | The matrix ring has the sa... |
| matplusg 22356 | The matrix ring has the sa... |
| matsca 22357 | The matrix ring has the sa... |
| matvsca 22358 | The matrix ring has the sa... |
| mat0 22359 | The matrix ring has the sa... |
| matinvg 22360 | The matrix ring has the sa... |
| mat0op 22361 | Value of a zero matrix as ... |
| matsca2 22362 | The scalars of the matrix ... |
| matbas2 22363 | The base set of the matrix... |
| matbas2i 22364 | A matrix is a function. (... |
| matbas2d 22365 | The base set of the matrix... |
| eqmat 22366 | Two square matrices of the... |
| matecl 22367 | Each entry (according to W... |
| matecld 22368 | Each entry (according to W... |
| matplusg2 22369 | Addition in the matrix rin... |
| matvsca2 22370 | Scalar multiplication in t... |
| matlmod 22371 | The matrix ring is a linea... |
| matgrp 22372 | The matrix ring is a group... |
| matvscl 22373 | Closure of the scalar mult... |
| matsubg 22374 | The matrix ring has the sa... |
| matplusgcell 22375 | Addition in the matrix rin... |
| matsubgcell 22376 | Subtraction in the matrix ... |
| matinvgcell 22377 | Additive inversion in the ... |
| matvscacell 22378 | Scalar multiplication in t... |
| matgsum 22379 | Finite commutative sums in... |
| matmulr 22380 | Multiplication in the matr... |
| mamumat1cl 22381 | The identity matrix (as op... |
| mat1comp 22382 | The components of the iden... |
| mamulid 22383 | The identity matrix (as op... |
| mamurid 22384 | The identity matrix (as op... |
| matring 22385 | Existence of the matrix ri... |
| matassa 22386 | Existence of the matrix al... |
| matmulcell 22387 | Multiplication in the matr... |
| mpomatmul 22388 | Multiplication of two N x ... |
| mat1 22389 | Value of an identity matri... |
| mat1ov 22390 | Entries of an identity mat... |
| mat1bas 22391 | The identity matrix is a m... |
| matsc 22392 | The identity matrix multip... |
| ofco2 22393 | Distribution law for the f... |
| oftpos 22394 | The transposition of the v... |
| mattposcl 22395 | The transpose of a square ... |
| mattpostpos 22396 | The transpose of the trans... |
| mattposvs 22397 | The transposition of a mat... |
| mattpos1 22398 | The transposition of the i... |
| tposmap 22399 | The transposition of an I ... |
| mamutpos 22400 | Behavior of transposes in ... |
| mattposm 22401 | Multiplying two transposed... |
| matgsumcl 22402 | Closure of a group sum ove... |
| madetsumid 22403 | The identity summand in th... |
| matepmcl 22404 | Each entry of a matrix wit... |
| matepm2cl 22405 | Each entry of a matrix wit... |
| madetsmelbas 22406 | A summand of the determina... |
| madetsmelbas2 22407 | A summand of the determina... |
| mat0dimbas0 22408 | The empty set is the one a... |
| mat0dim0 22409 | The zero of the algebra of... |
| mat0dimid 22410 | The identity of the algebr... |
| mat0dimscm 22411 | The scalar multiplication ... |
| mat0dimcrng 22412 | The algebra of matrices wi... |
| mat1dimelbas 22413 | A matrix with dimension 1 ... |
| mat1dimbas 22414 | A matrix with dimension 1 ... |
| mat1dim0 22415 | The zero of the algebra of... |
| mat1dimid 22416 | The identity of the algebr... |
| mat1dimscm 22417 | The scalar multiplication ... |
| mat1dimmul 22418 | The ring multiplication in... |
| mat1dimcrng 22419 | The algebra of matrices wi... |
| mat1f1o 22420 | There is a 1-1 function fr... |
| mat1rhmval 22421 | The value of the ring homo... |
| mat1rhmelval 22422 | The value of the ring homo... |
| mat1rhmcl 22423 | The value of the ring homo... |
| mat1f 22424 | There is a function from a... |
| mat1ghm 22425 | There is a group homomorph... |
| mat1mhm 22426 | There is a monoid homomorp... |
| mat1rhm 22427 | There is a ring homomorphi... |
| mat1rngiso 22428 | There is a ring isomorphis... |
| mat1ric 22429 | A ring is isomorphic to th... |
| dmatval 22434 | The set of ` N ` x ` N ` d... |
| dmatel 22435 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22436 | An ` N ` x ` N ` diagonal ... |
| dmatid 22437 | The identity matrix is a d... |
| dmatelnd 22438 | An extradiagonal entry of ... |
| dmatmul 22439 | The product of two diagona... |
| dmatsubcl 22440 | The difference of two diag... |
| dmatsgrp 22441 | The set of diagonal matric... |
| dmatmulcl 22442 | The product of two diagona... |
| dmatsrng 22443 | The set of diagonal matric... |
| dmatcrng 22444 | The subring of diagonal ma... |
| dmatscmcl 22445 | The multiplication of a di... |
| scmatval 22446 | The set of ` N ` x ` N ` s... |
| scmatel 22447 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22448 | A scalar matrix can be exp... |
| scmatscmide 22449 | An entry of a scalar matri... |
| scmatscmiddistr 22450 | Distributive law for scala... |
| scmatmat 22451 | An ` N ` x ` N ` scalar ma... |
| scmate 22452 | An entry of an ` N ` x ` N... |
| scmatmats 22453 | The set of an ` N ` x ` N ... |
| scmateALT 22454 | Alternate proof of ~ scmat... |
| scmatscm 22455 | The multiplication of a ma... |
| scmatid 22456 | The identity matrix is a s... |
| scmatdmat 22457 | A scalar matrix is a diago... |
| scmataddcl 22458 | The sum of two scalar matr... |
| scmatsubcl 22459 | The difference of two scal... |
| scmatmulcl 22460 | The product of two scalar ... |
| scmatsgrp 22461 | The set of scalar matrices... |
| scmatsrng 22462 | The set of scalar matrices... |
| scmatcrng 22463 | The subring of scalar matr... |
| scmatsgrp1 22464 | The set of scalar matrices... |
| scmatsrng1 22465 | The set of scalar matrices... |
| smatvscl 22466 | Closure of the scalar mult... |
| scmatlss 22467 | The set of scalar matrices... |
| scmatstrbas 22468 | The set of scalar matrices... |
| scmatrhmval 22469 | The value of the ring homo... |
| scmatrhmcl 22470 | The value of the ring homo... |
| scmatf 22471 | There is a function from a... |
| scmatfo 22472 | There is a function from a... |
| scmatf1 22473 | There is a 1-1 function fr... |
| scmatf1o 22474 | There is a bijection betwe... |
| scmatghm 22475 | There is a group homomorph... |
| scmatmhm 22476 | There is a monoid homomorp... |
| scmatrhm 22477 | There is a ring homomorphi... |
| scmatrngiso 22478 | There is a ring isomorphis... |
| scmatric 22479 | A ring is isomorphic to ev... |
| mat0scmat 22480 | The empty matrix over a ri... |
| mat1scmat 22481 | A 1-dimensional matrix ove... |
| mvmulfval 22484 | Functional value of the ma... |
| mvmulval 22485 | Multiplication of a vector... |
| mvmulfv 22486 | A cell/element in the vect... |
| mavmulval 22487 | Multiplication of a vector... |
| mavmulfv 22488 | A cell/element in the vect... |
| mavmulcl 22489 | Multiplication of an NxN m... |
| 1mavmul 22490 | Multiplication of the iden... |
| mavmulass 22491 | Associativity of the multi... |
| mavmuldm 22492 | The domain of the matrix v... |
| mavmulsolcl 22493 | Every solution of the equa... |
| mavmul0 22494 | Multiplication of a 0-dime... |
| mavmul0g 22495 | The result of the 0-dimens... |
| mvmumamul1 22496 | The multiplication of an M... |
| mavmumamul1 22497 | The multiplication of an N... |
| marrepfval 22502 | First substitution for the... |
| marrepval0 22503 | Second substitution for th... |
| marrepval 22504 | Third substitution for the... |
| marrepeval 22505 | An entry of a matrix with ... |
| marrepcl 22506 | Closure of the row replace... |
| marepvfval 22507 | First substitution for the... |
| marepvval0 22508 | Second substitution for th... |
| marepvval 22509 | Third substitution for the... |
| marepveval 22510 | An entry of a matrix with ... |
| marepvcl 22511 | Closure of the column repl... |
| ma1repvcl 22512 | Closure of the column repl... |
| ma1repveval 22513 | An entry of an identity ma... |
| mulmarep1el 22514 | Element by element multipl... |
| mulmarep1gsum1 22515 | The sum of element by elem... |
| mulmarep1gsum2 22516 | The sum of element by elem... |
| 1marepvmarrepid 22517 | Replacing the ith row by 0... |
| submabas 22520 | Any subset of the index se... |
| submafval 22521 | First substitution for a s... |
| submaval0 22522 | Second substitution for a ... |
| submaval 22523 | Third substitution for a s... |
| submaeval 22524 | An entry of a submatrix of... |
| 1marepvsma1 22525 | The submatrix of the ident... |
| mdetfval 22528 | First substitution for the... |
| mdetleib 22529 | Full substitution of our d... |
| mdetleib2 22530 | Leibniz' formula can also ... |
| nfimdetndef 22531 | The determinant is not def... |
| mdetfval1 22532 | First substitution of an a... |
| mdetleib1 22533 | Full substitution of an al... |
| mdet0pr 22534 | The determinant function f... |
| mdet0f1o 22535 | The determinant function f... |
| mdet0fv0 22536 | The determinant of the emp... |
| mdetf 22537 | Functionality of the deter... |
| mdetcl 22538 | The determinant evaluates ... |
| m1detdiag 22539 | The determinant of a 1-dim... |
| mdetdiaglem 22540 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22541 | The determinant of a diago... |
| mdetdiagid 22542 | The determinant of a diago... |
| mdet1 22543 | The determinant of the ide... |
| mdetrlin 22544 | The determinant function i... |
| mdetrsca 22545 | The determinant function i... |
| mdetrsca2 22546 | The determinant function i... |
| mdetr0 22547 | The determinant of a matri... |
| mdet0 22548 | The determinant of the zer... |
| mdetrlin2 22549 | The determinant function i... |
| mdetralt 22550 | The determinant function i... |
| mdetralt2 22551 | The determinant function i... |
| mdetero 22552 | The determinant function i... |
| mdettpos 22553 | Determinant is invariant u... |
| mdetunilem1 22554 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22555 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22556 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22557 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22558 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22559 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22560 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22561 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22562 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22563 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22564 | According to the definitio... |
| mdetmul 22565 | Multiplicativity of the de... |
| m2detleiblem1 22566 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22567 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22568 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22569 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22570 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22571 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22572 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22573 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22578 | Lemma for ~ maducoeval2 . ... |
| madufval 22579 | First substitution for the... |
| maduval 22580 | Second substitution for th... |
| maducoeval 22581 | An entry of the adjunct (c... |
| maducoeval2 22582 | An entry of the adjunct (c... |
| maduf 22583 | Creating the adjunct of ma... |
| madutpos 22584 | The adjuct of a transposed... |
| madugsum 22585 | The determinant of a matri... |
| madurid 22586 | Multiplying a matrix with ... |
| madulid 22587 | Multiplying the adjunct of... |
| minmar1fval 22588 | First substitution for the... |
| minmar1val0 22589 | Second substitution for th... |
| minmar1val 22590 | Third substitution for the... |
| minmar1eval 22591 | An entry of a matrix for a... |
| minmar1marrep 22592 | The minor matrix is a spec... |
| minmar1cl 22593 | Closure of the row replace... |
| maducoevalmin1 22594 | The coefficients of an adj... |
| symgmatr01lem 22595 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22596 | Applying a permutation tha... |
| gsummatr01lem1 22597 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22598 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22599 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22600 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22601 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22602 | Replacing a row of a squar... |
| smadiadetlem0 22603 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22604 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22605 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22606 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22607 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22608 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22609 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22610 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22611 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22612 | The determinant of a subma... |
| smadiadetglem1 22613 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22614 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22615 | The determinant of a squar... |
| smadiadetg0 22616 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22617 | The determinant of a squar... |
| invrvald 22618 | If a matrix multiplied wit... |
| matinv 22619 | The inverse of a matrix is... |
| matunit 22620 | A matrix is a unit in the ... |
| slesolvec 22621 | Every solution of a system... |
| slesolinv 22622 | The solution of a system o... |
| slesolinvbi 22623 | The solution of a system o... |
| slesolex 22624 | Every system of linear equ... |
| cramerimplem1 22625 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22626 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22627 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22628 | One direction of Cramer's ... |
| cramerlem1 22629 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22630 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22631 | Lemma 3 for ~ cramer . (C... |
| cramer0 22632 | Special case of Cramer's r... |
| cramer 22633 | Cramer's rule. According ... |
| pmatring 22634 | The set of polynomial matr... |
| pmatlmod 22635 | The set of polynomial matr... |
| pmatassa 22636 | The set of polynomial matr... |
| pmat0op 22637 | The zero polynomial matrix... |
| pmat1op 22638 | The identity polynomial ma... |
| pmat1ovd 22639 | Entries of the identity po... |
| pmat0opsc 22640 | The zero polynomial matrix... |
| pmat1opsc 22641 | The identity polynomial ma... |
| pmat1ovscd 22642 | Entries of the identity po... |
| pmatcoe1fsupp 22643 | For a polynomial matrix th... |
| 1pmatscmul 22644 | The scalar product of the ... |
| cpmat 22651 | Value of the constructor o... |
| cpmatpmat 22652 | A constant polynomial matr... |
| cpmatel 22653 | Property of a constant pol... |
| cpmatelimp 22654 | Implication of a set being... |
| cpmatel2 22655 | Another property of a cons... |
| cpmatelimp2 22656 | Another implication of a s... |
| 1elcpmat 22657 | The identity of the ring o... |
| cpmatacl 22658 | The set of all constant po... |
| cpmatinvcl 22659 | The set of all constant po... |
| cpmatmcllem 22660 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22661 | The set of all constant po... |
| cpmatsubgpmat 22662 | The set of all constant po... |
| cpmatsrgpmat 22663 | The set of all constant po... |
| 0elcpmat 22664 | The zero of the ring of al... |
| mat2pmatfval 22665 | Value of the matrix transf... |
| mat2pmatval 22666 | The result of a matrix tra... |
| mat2pmatvalel 22667 | A (matrix) element of the ... |
| mat2pmatbas 22668 | The result of a matrix tra... |
| mat2pmatbas0 22669 | The result of a matrix tra... |
| mat2pmatf 22670 | The matrix transformation ... |
| mat2pmatf1 22671 | The matrix transformation ... |
| mat2pmatghm 22672 | The transformation of matr... |
| mat2pmatmul 22673 | The transformation of matr... |
| mat2pmat1 22674 | The transformation of the ... |
| mat2pmatmhm 22675 | The transformation of matr... |
| mat2pmatrhm 22676 | The transformation of matr... |
| mat2pmatlin 22677 | The transformation of matr... |
| 0mat2pmat 22678 | The transformed zero matri... |
| idmatidpmat 22679 | The transformed identity m... |
| d0mat2pmat 22680 | The transformed empty set ... |
| d1mat2pmat 22681 | The transformation of a ma... |
| mat2pmatscmxcl 22682 | A transformed matrix multi... |
| m2cpm 22683 | The result of a matrix tra... |
| m2cpmf 22684 | The matrix transformation ... |
| m2cpmf1 22685 | The matrix transformation ... |
| m2cpmghm 22686 | The transformation of matr... |
| m2cpmmhm 22687 | The transformation of matr... |
| m2cpmrhm 22688 | The transformation of matr... |
| m2pmfzmap 22689 | The transformed values of ... |
| m2pmfzgsumcl 22690 | Closure of the sum of scal... |
| cpm2mfval 22691 | Value of the inverse matri... |
| cpm2mval 22692 | The result of an inverse m... |
| cpm2mvalel 22693 | A (matrix) element of the ... |
| cpm2mf 22694 | The inverse matrix transfo... |
| m2cpminvid 22695 | The inverse transformation... |
| m2cpminvid2lem 22696 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22697 | The transformation applied... |
| m2cpmfo 22698 | The matrix transformation ... |
| m2cpmf1o 22699 | The matrix transformation ... |
| m2cpmrngiso 22700 | The transformation of matr... |
| matcpmric 22701 | The ring of matrices over ... |
| m2cpminv 22702 | The inverse matrix transfo... |
| m2cpminv0 22703 | The inverse matrix transfo... |
| decpmatval0 22706 | The matrix consisting of t... |
| decpmatval 22707 | The matrix consisting of t... |
| decpmate 22708 | An entry of the matrix con... |
| decpmatcl 22709 | Closure of the decompositi... |
| decpmataa0 22710 | The matrix consisting of t... |
| decpmatfsupp 22711 | The mapping to the matrice... |
| decpmatid 22712 | The matrix consisting of t... |
| decpmatmullem 22713 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22714 | The matrix consisting of t... |
| decpmatmulsumfsupp 22715 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22716 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22717 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22718 | Write a polynomial matrix ... |
| pmatcollpw2lem 22719 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22720 | Write a polynomial matrix ... |
| monmatcollpw 22721 | The matrix consisting of t... |
| pmatcollpwlem 22722 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22723 | Write a polynomial matrix ... |
| pmatcollpwfi 22724 | Write a polynomial matrix ... |
| pmatcollpw3lem 22725 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22726 | Write a polynomial matrix ... |
| pmatcollpw3fi 22727 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22728 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22729 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22730 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22731 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22732 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22733 | Write a scalar matrix over... |
| pm2mpf1lem 22736 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22737 | Value of the transformatio... |
| pm2mpfval 22738 | A polynomial matrix transf... |
| pm2mpcl 22739 | The transformation of poly... |
| pm2mpf 22740 | The transformation of poly... |
| pm2mpf1 22741 | The transformation of poly... |
| pm2mpcoe1 22742 | A coefficient of the polyn... |
| idpm2idmp 22743 | The transformation of the ... |
| mptcoe1matfsupp 22744 | The mapping extracting the... |
| mply1topmatcllem 22745 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22746 | A polynomial over matrices... |
| mply1topmatcl 22747 | A polynomial over matrices... |
| mp2pm2mplem1 22748 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22749 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22750 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22751 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22752 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22753 | A polynomial over matrices... |
| pm2mpghmlem2 22754 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22755 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22756 | The transformation of poly... |
| pm2mpf1o 22757 | The transformation of poly... |
| pm2mpghm 22758 | The transformation of poly... |
| pm2mpgrpiso 22759 | The transformation of poly... |
| pm2mpmhmlem1 22760 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22761 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22762 | The transformation of poly... |
| pm2mprhm 22763 | The transformation of poly... |
| pm2mprngiso 22764 | The transformation of poly... |
| pmmpric 22765 | The ring of polynomial mat... |
| monmat2matmon 22766 | The transformation of a po... |
| pm2mp 22767 | The transformation of a su... |
| chmatcl 22770 | Closure of the characteris... |
| chmatval 22771 | The entries of the charact... |
| chpmatfval 22772 | Value of the characteristi... |
| chpmatval 22773 | The characteristic polynom... |
| chpmatply1 22774 | The characteristic polynom... |
| chpmatval2 22775 | The characteristic polynom... |
| chpmat0d 22776 | The characteristic polynom... |
| chpmat1dlem 22777 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22778 | The characteristic polynom... |
| chpdmatlem0 22779 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22780 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22781 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22782 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22783 | The characteristic polynom... |
| chpscmat 22784 | The characteristic polynom... |
| chpscmat0 22785 | The characteristic polynom... |
| chpscmatgsumbin 22786 | The characteristic polynom... |
| chpscmatgsummon 22787 | The characteristic polynom... |
| chp0mat 22788 | The characteristic polynom... |
| chpidmat 22789 | The characteristic polynom... |
| chmaidscmat 22790 | The characteristic polynom... |
| fvmptnn04if 22791 | The function values of a m... |
| fvmptnn04ifa 22792 | The function value of a ma... |
| fvmptnn04ifb 22793 | The function value of a ma... |
| fvmptnn04ifc 22794 | The function value of a ma... |
| fvmptnn04ifd 22795 | The function value of a ma... |
| chfacfisf 22796 | The "characteristic factor... |
| chfacfisfcpmat 22797 | The "characteristic factor... |
| chfacffsupp 22798 | The "characteristic factor... |
| chfacfscmulcl 22799 | Closure of a scaled value ... |
| chfacfscmul0 22800 | A scaled value of the "cha... |
| chfacfscmulfsupp 22801 | A mapping of scaled values... |
| chfacfscmulgsum 22802 | Breaking up a sum of value... |
| chfacfpmmulcl 22803 | Closure of the value of th... |
| chfacfpmmul0 22804 | The value of the "characte... |
| chfacfpmmulfsupp 22805 | A mapping of values of the... |
| chfacfpmmulgsum 22806 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22807 | Breaking up a sum of value... |
| cayhamlem1 22808 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22809 | The right-hand fundamental... |
| cpmidgsum 22810 | Representation of the iden... |
| cpmidgsumm2pm 22811 | Representation of the iden... |
| cpmidpmatlem1 22812 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22813 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22814 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22815 | Representation of the iden... |
| cpmadugsumlemB 22816 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22817 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22818 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22819 | The product of the charact... |
| cpmadugsum 22820 | The product of the charact... |
| cpmidgsum2 22821 | Representation of the iden... |
| cpmidg2sum 22822 | Equality of two sums repre... |
| cpmadumatpolylem1 22823 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22824 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22825 | The product of the charact... |
| cayhamlem2 22826 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22827 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22828 | The coefficients of the ch... |
| cayhamlem3 22829 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22830 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22831 | The Cayley-Hamilton theore... |
| cayleyhamilton 22832 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22833 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22834 | The Cayley-Hamilton theore... |
| istopg 22837 | Express the predicate " ` ... |
| istop2g 22838 | Express the predicate " ` ... |
| uniopn 22839 | The union of a subset of a... |
| iunopn 22840 | The indexed union of a sub... |
| inopn 22841 | The intersection of two op... |
| fitop 22842 | A topology is closed under... |
| fiinopn 22843 | The intersection of a none... |
| iinopn 22844 | The intersection of a none... |
| unopn 22845 | The union of two open sets... |
| 0opn 22846 | The empty set is an open s... |
| 0ntop 22847 | The empty set is not a top... |
| topopn 22848 | The underlying set of a to... |
| eltopss 22849 | A member of a topology is ... |
| riinopn 22850 | A finite indexed relative ... |
| rintopn 22851 | A finite relative intersec... |
| istopon 22854 | Property of being a topolo... |
| topontop 22855 | A topology on a given base... |
| toponuni 22856 | The base set of a topology... |
| topontopi 22857 | A topology on a given base... |
| toponunii 22858 | The base set of a topology... |
| toptopon 22859 | Alternative definition of ... |
| toptopon2 22860 | A topology is the same thi... |
| topontopon 22861 | A topology on a set is a t... |
| funtopon 22862 | The class ` TopOn ` is a f... |
| toponrestid 22863 | Given a topology on a set,... |
| toponsspwpw 22864 | The set of topologies on a... |
| dmtopon 22865 | The domain of ` TopOn ` is... |
| fntopon 22866 | The class ` TopOn ` is a f... |
| toprntopon 22867 | A topology is the same thi... |
| toponmax 22868 | The base set of a topology... |
| toponss 22869 | A member of a topology is ... |
| toponcom 22870 | If ` K ` is a topology on ... |
| toponcomb 22871 | Biconditional form of ~ to... |
| topgele 22872 | The topologies over the sa... |
| topsn 22873 | The only topology on a sin... |
| istps 22876 | Express the predicate "is ... |
| istps2 22877 | Express the predicate "is ... |
| tpsuni 22878 | The base set of a topologi... |
| tpstop 22879 | The topology extractor on ... |
| tpspropd 22880 | A topological space depend... |
| tpsprop2d 22881 | A topological space depend... |
| topontopn 22882 | Express the predicate "is ... |
| tsettps 22883 | If the topology component ... |
| istpsi 22884 | Properties that determine ... |
| eltpsg 22885 | Properties that determine ... |
| eltpsi 22886 | Properties that determine ... |
| isbasisg 22889 | Express the predicate "the... |
| isbasis2g 22890 | Express the predicate "the... |
| isbasis3g 22891 | Express the predicate "the... |
| basis1 22892 | Property of a basis. (Con... |
| basis2 22893 | Property of a basis. (Con... |
| fiinbas 22894 | If a set is closed under f... |
| basdif0 22895 | A basis is not affected by... |
| baspartn 22896 | A disjoint system of sets ... |
| tgval 22897 | The topology generated by ... |
| tgval2 22898 | Definition of a topology g... |
| eltg 22899 | Membership in a topology g... |
| eltg2 22900 | Membership in a topology g... |
| eltg2b 22901 | Membership in a topology g... |
| eltg4i 22902 | An open set in a topology ... |
| eltg3i 22903 | The union of a set of basi... |
| eltg3 22904 | Membership in a topology g... |
| tgval3 22905 | Alternate expression for t... |
| tg1 22906 | Property of a member of a ... |
| tg2 22907 | Property of a member of a ... |
| bastg 22908 | A member of a basis is a s... |
| unitg 22909 | The topology generated by ... |
| tgss 22910 | Subset relation for genera... |
| tgcl 22911 | Show that a basis generate... |
| tgclb 22912 | The property ~ tgcl can be... |
| tgtopon 22913 | A basis generates a topolo... |
| topbas 22914 | A topology is its own basi... |
| tgtop 22915 | A topology is its own basi... |
| eltop 22916 | Membership in a topology, ... |
| eltop2 22917 | Membership in a topology. ... |
| eltop3 22918 | Membership in a topology. ... |
| fibas 22919 | A collection of finite int... |
| tgdom 22920 | A space has no more open s... |
| tgiun 22921 | The indexed union of a set... |
| tgidm 22922 | The topology generator fun... |
| bastop 22923 | Two ways to express that a... |
| tgtop11 22924 | The topology generation fu... |
| 0top 22925 | The singleton of the empty... |
| en1top 22926 | ` { (/) } ` is the only to... |
| en2top 22927 | If a topology has two elem... |
| tgss3 22928 | A criterion for determinin... |
| tgss2 22929 | A criterion for determinin... |
| basgen 22930 | Given a topology ` J ` , s... |
| basgen2 22931 | Given a topology ` J ` , s... |
| 2basgen 22932 | Conditions that determine ... |
| tgfiss 22933 | If a subbase is included i... |
| tgdif0 22934 | A generated topology is no... |
| bastop1 22935 | A subset of a topology is ... |
| bastop2 22936 | A version of ~ bastop1 tha... |
| distop 22937 | The discrete topology on a... |
| topnex 22938 | The class of all topologie... |
| distopon 22939 | The discrete topology on a... |
| sn0topon 22940 | The singleton of the empty... |
| sn0top 22941 | The singleton of the empty... |
| indislem 22942 | A lemma to eliminate some ... |
| indistopon 22943 | The indiscrete topology on... |
| indistop 22944 | The indiscrete topology on... |
| indisuni 22945 | The base set of the indisc... |
| fctop 22946 | The finite complement topo... |
| fctop2 22947 | The finite complement topo... |
| cctop 22948 | The countable complement t... |
| ppttop 22949 | The particular point topol... |
| pptbas 22950 | The particular point topol... |
| epttop 22951 | The excluded point topolog... |
| indistpsx 22952 | The indiscrete topology on... |
| indistps 22953 | The indiscrete topology on... |
| indistps2 22954 | The indiscrete topology on... |
| indistpsALT 22955 | The indiscrete topology on... |
| indistps2ALT 22956 | The indiscrete topology on... |
| distps 22957 | The discrete topology on a... |
| fncld 22964 | The closed-set generator i... |
| cldval 22965 | The set of closed sets of ... |
| ntrfval 22966 | The interior function on t... |
| clsfval 22967 | The closure function on th... |
| cldrcl 22968 | Reverse closure of the clo... |
| iscld 22969 | The predicate "the class `... |
| iscld2 22970 | A subset of the underlying... |
| cldss 22971 | A closed set is a subset o... |
| cldss2 22972 | The set of closed sets is ... |
| cldopn 22973 | The complement of a closed... |
| isopn2 22974 | A subset of the underlying... |
| opncld 22975 | The complement of an open ... |
| difopn 22976 | The difference of a closed... |
| topcld 22977 | The underlying set of a to... |
| ntrval 22978 | The interior of a subset o... |
| clsval 22979 | The closure of a subset of... |
| 0cld 22980 | The empty set is closed. ... |
| iincld 22981 | The indexed intersection o... |
| intcld 22982 | The intersection of a set ... |
| uncld 22983 | The union of two closed se... |
| cldcls 22984 | A closed subset equals its... |
| incld 22985 | The intersection of two cl... |
| riincld 22986 | An indexed relative inters... |
| iuncld 22987 | A finite indexed union of ... |
| unicld 22988 | A finite union of closed s... |
| clscld 22989 | The closure of a subset of... |
| clsf 22990 | The closure function is a ... |
| ntropn 22991 | The interior of a subset o... |
| clsval2 22992 | Express closure in terms o... |
| ntrval2 22993 | Interior expressed in term... |
| ntrdif 22994 | An interior of a complemen... |
| clsdif 22995 | A closure of a complement ... |
| clsss 22996 | Subset relationship for cl... |
| ntrss 22997 | Subset relationship for in... |
| sscls 22998 | A subset of a topology's u... |
| ntrss2 22999 | A subset includes its inte... |
| ssntr 23000 | An open subset of a set is... |
| clsss3 23001 | The closure of a subset of... |
| ntrss3 23002 | The interior of a subset o... |
| ntrin 23003 | A pairwise intersection of... |
| cmclsopn 23004 | The complement of a closur... |
| cmntrcld 23005 | The complement of an inter... |
| iscld3 23006 | A subset is closed iff it ... |
| iscld4 23007 | A subset is closed iff it ... |
| isopn3 23008 | A subset is open iff it eq... |
| clsidm 23009 | The closure operation is i... |
| ntridm 23010 | The interior operation is ... |
| clstop 23011 | The closure of a topology'... |
| ntrtop 23012 | The interior of a topology... |
| 0ntr 23013 | A subset with an empty int... |
| clsss2 23014 | If a subset is included in... |
| elcls 23015 | Membership in a closure. ... |
| elcls2 23016 | Membership in a closure. ... |
| clsndisj 23017 | Any open set containing a ... |
| ntrcls0 23018 | A subset whose closure has... |
| ntreq0 23019 | Two ways to say that a sub... |
| cldmre 23020 | The closed sets of a topol... |
| mrccls 23021 | Moore closure generalizes ... |
| cls0 23022 | The closure of the empty s... |
| ntr0 23023 | The interior of the empty ... |
| isopn3i 23024 | An open subset equals its ... |
| elcls3 23025 | Membership in a closure in... |
| opncldf1 23026 | A bijection useful for con... |
| opncldf2 23027 | The values of the open-clo... |
| opncldf3 23028 | The values of the converse... |
| isclo 23029 | A set ` A ` is clopen iff ... |
| isclo2 23030 | A set ` A ` is clopen iff ... |
| discld 23031 | The open sets of a discret... |
| sn0cld 23032 | The closed sets of the top... |
| indiscld 23033 | The closed sets of an indi... |
| mretopd 23034 | A Moore collection which i... |
| toponmre 23035 | The topologies over a give... |
| cldmreon 23036 | The closed sets of a topol... |
| iscldtop 23037 | A family is the closed set... |
| mreclatdemoBAD 23038 | The closed subspaces of a ... |
| neifval 23041 | Value of the neighborhood ... |
| neif 23042 | The neighborhood function ... |
| neiss2 23043 | A set with a neighborhood ... |
| neival 23044 | Value of the set of neighb... |
| isnei 23045 | The predicate "the class `... |
| neiint 23046 | An intuitive definition of... |
| isneip 23047 | The predicate "the class `... |
| neii1 23048 | A neighborhood is included... |
| neisspw 23049 | The neighborhoods of any s... |
| neii2 23050 | Property of a neighborhood... |
| neiss 23051 | Any neighborhood of a set ... |
| ssnei 23052 | A set is included in any o... |
| elnei 23053 | A point belongs to any of ... |
| 0nnei 23054 | The empty set is not a nei... |
| neips 23055 | A neighborhood of a set is... |
| opnneissb 23056 | An open set is a neighborh... |
| opnssneib 23057 | Any superset of an open se... |
| ssnei2 23058 | Any subset ` M ` of ` X ` ... |
| neindisj 23059 | Any neighborhood of an ele... |
| opnneiss 23060 | An open set is a neighborh... |
| opnneip 23061 | An open set is a neighborh... |
| opnnei 23062 | A set is open iff it is a ... |
| tpnei 23063 | The underlying set of a to... |
| neiuni 23064 | The union of the neighborh... |
| neindisj2 23065 | A point ` P ` belongs to t... |
| topssnei 23066 | A finer topology has more ... |
| innei 23067 | The intersection of two ne... |
| opnneiid 23068 | Only an open set is a neig... |
| neissex 23069 | For any neighborhood ` N `... |
| 0nei 23070 | The empty set is a neighbo... |
| neipeltop 23071 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23072 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23073 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23074 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23075 | If, to each element ` P ` ... |
| lpfval 23080 | The limit point function o... |
| lpval 23081 | The set of limit points of... |
| islp 23082 | The predicate "the class `... |
| lpsscls 23083 | The limit points of a subs... |
| lpss 23084 | The limit points of a subs... |
| lpdifsn 23085 | ` P ` is a limit point of ... |
| lpss3 23086 | Subset relationship for li... |
| islp2 23087 | The predicate " ` P ` is a... |
| islp3 23088 | The predicate " ` P ` is a... |
| maxlp 23089 | A point is a limit point o... |
| clslp 23090 | The closure of a subset of... |
| islpi 23091 | A point belonging to a set... |
| cldlp 23092 | A subset of a topological ... |
| isperf 23093 | Definition of a perfect sp... |
| isperf2 23094 | Definition of a perfect sp... |
| isperf3 23095 | A perfect space is a topol... |
| perflp 23096 | The limit points of a perf... |
| perfi 23097 | Property of a perfect spac... |
| perftop 23098 | A perfect space is a topol... |
| restrcl 23099 | Reverse closure for the su... |
| restbas 23100 | A subspace topology basis ... |
| tgrest 23101 | A subspace can be generate... |
| resttop 23102 | A subspace topology is a t... |
| resttopon 23103 | A subspace topology is a t... |
| restuni 23104 | The underlying set of a su... |
| stoig 23105 | The topological space buil... |
| restco 23106 | Composition of subspaces. ... |
| restabs 23107 | Equivalence of being a sub... |
| restin 23108 | When the subspace region i... |
| restuni2 23109 | The underlying set of a su... |
| resttopon2 23110 | The underlying set of a su... |
| rest0 23111 | The subspace topology indu... |
| restsn 23112 | The only subspace topology... |
| restsn2 23113 | The subspace topology indu... |
| restcld 23114 | A closed set of a subspace... |
| restcldi 23115 | A closed set is closed in ... |
| restcldr 23116 | A set which is closed in t... |
| restopnb 23117 | If ` B ` is an open subset... |
| ssrest 23118 | If ` K ` is a finer topolo... |
| restopn2 23119 | If ` A ` is open, then ` B... |
| restdis 23120 | A subspace of a discrete t... |
| restfpw 23121 | The restriction of the set... |
| neitr 23122 | The neighborhood of a trac... |
| restcls 23123 | A closure in a subspace to... |
| restntr 23124 | An interior in a subspace ... |
| restlp 23125 | The limit points of a subs... |
| restperf 23126 | Perfection of a subspace. ... |
| perfopn 23127 | An open subset of a perfec... |
| resstopn 23128 | The topology of a restrict... |
| resstps 23129 | A restricted topological s... |
| ordtbaslem 23130 | Lemma for ~ ordtbas . In ... |
| ordtval 23131 | Value of the order topolog... |
| ordtuni 23132 | Value of the order topolog... |
| ordtbas2 23133 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23134 | In a total order, the fini... |
| ordttopon 23135 | Value of the order topolog... |
| ordtopn1 23136 | An upward ray ` ( P , +oo ... |
| ordtopn2 23137 | A downward ray ` ( -oo , P... |
| ordtopn3 23138 | An open interval ` ( A , B... |
| ordtcld1 23139 | A downward ray ` ( -oo , P... |
| ordtcld2 23140 | An upward ray ` [ P , +oo ... |
| ordtcld3 23141 | A closed interval ` [ A , ... |
| ordttop 23142 | The order topology is a to... |
| ordtcnv 23143 | The order dual generates t... |
| ordtrest 23144 | The subspace topology of a... |
| ordtrest2lem 23145 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23146 | An interval-closed set ` A... |
| letopon 23147 | The topology of the extend... |
| letop 23148 | The topology of the extend... |
| letopuni 23149 | The topology of the extend... |
| xrstopn 23150 | The topology component of ... |
| xrstps 23151 | The extended real number s... |
| leordtvallem1 23152 | Lemma for ~ leordtval . (... |
| leordtvallem2 23153 | Lemma for ~ leordtval . (... |
| leordtval2 23154 | The topology of the extend... |
| leordtval 23155 | The topology of the extend... |
| iccordt 23156 | A closed interval is close... |
| iocpnfordt 23157 | An unbounded above open in... |
| icomnfordt 23158 | An unbounded above open in... |
| iooordt 23159 | An open interval is open i... |
| reordt 23160 | The real numbers are an op... |
| lecldbas 23161 | The set of closed interval... |
| pnfnei 23162 | A neighborhood of ` +oo ` ... |
| mnfnei 23163 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23164 | The restriction of the les... |
| ordtresticc 23165 | The restriction of the les... |
| lmrel 23172 | The topological space conv... |
| lmrcl 23173 | Reverse closure for the co... |
| lmfval 23174 | The relation "sequence ` f... |
| cnfval 23175 | The set of all continuous ... |
| cnpfval 23176 | The function mapping the p... |
| iscn 23177 | The predicate "the class `... |
| cnpval 23178 | The set of all functions f... |
| iscnp 23179 | The predicate "the class `... |
| iscn2 23180 | The predicate "the class `... |
| iscnp2 23181 | The predicate "the class `... |
| cntop1 23182 | Reverse closure for a cont... |
| cntop2 23183 | Reverse closure for a cont... |
| cnptop1 23184 | Reverse closure for a func... |
| cnptop2 23185 | Reverse closure for a func... |
| iscnp3 23186 | The predicate "the class `... |
| cnprcl 23187 | Reverse closure for a func... |
| cnf 23188 | A continuous function is a... |
| cnpf 23189 | A continuous function at p... |
| cnpcl 23190 | The value of a continuous ... |
| cnf2 23191 | A continuous function is a... |
| cnpf2 23192 | A continuous function at p... |
| cnprcl2 23193 | Reverse closure for a func... |
| tgcn 23194 | The continuity predicate w... |
| tgcnp 23195 | The "continuous at a point... |
| subbascn 23196 | The continuity predicate w... |
| ssidcn 23197 | The identity function is a... |
| cnpimaex 23198 | Property of a function con... |
| idcn 23199 | A restricted identity func... |
| lmbr 23200 | Express the binary relatio... |
| lmbr2 23201 | Express the binary relatio... |
| lmbrf 23202 | Express the binary relatio... |
| lmconst 23203 | A constant sequence conver... |
| lmcvg 23204 | Convergence property of a ... |
| iscnp4 23205 | The predicate "the class `... |
| cnpnei 23206 | A condition for continuity... |
| cnima 23207 | An open subset of the codo... |
| cnco 23208 | The composition of two con... |
| cnpco 23209 | The composition of a funct... |
| cnclima 23210 | A closed subset of the cod... |
| iscncl 23211 | A characterization of a co... |
| cncls2i 23212 | Property of the preimage o... |
| cnntri 23213 | Property of the preimage o... |
| cnclsi 23214 | Property of the image of a... |
| cncls2 23215 | Continuity in terms of clo... |
| cncls 23216 | Continuity in terms of clo... |
| cnntr 23217 | Continuity in terms of int... |
| cnss1 23218 | If the topology ` K ` is f... |
| cnss2 23219 | If the topology ` K ` is f... |
| cncnpi 23220 | A continuous function is c... |
| cnsscnp 23221 | The set of continuous func... |
| cncnp 23222 | A continuous function is c... |
| cncnp2 23223 | A continuous function is c... |
| cnnei 23224 | Continuity in terms of nei... |
| cnconst2 23225 | A constant function is con... |
| cnconst 23226 | A constant function is con... |
| cnrest 23227 | Continuity of a restrictio... |
| cnrest2 23228 | Equivalence of continuity ... |
| cnrest2r 23229 | Equivalence of continuity ... |
| cnpresti 23230 | One direction of ~ cnprest... |
| cnprest 23231 | Equivalence of continuity ... |
| cnprest2 23232 | Equivalence of point-conti... |
| cndis 23233 | Every function is continuo... |
| cnindis 23234 | Every function is continuo... |
| cnpdis 23235 | If ` A ` is an isolated po... |
| paste 23236 | Pasting lemma. If ` A ` a... |
| lmfpm 23237 | If ` F ` converges, then `... |
| lmfss 23238 | Inclusion of a function ha... |
| lmcl 23239 | Closure of a limit. (Cont... |
| lmss 23240 | Limit on a subspace. (Con... |
| sslm 23241 | A finer topology has fewer... |
| lmres 23242 | A function converges iff i... |
| lmff 23243 | If ` F ` converges, there ... |
| lmcls 23244 | Any convergent sequence of... |
| lmcld 23245 | Any convergent sequence of... |
| lmcnp 23246 | The image of a convergent ... |
| lmcn 23247 | The image of a convergent ... |
| ist0 23262 | The predicate "is a T_0 sp... |
| ist1 23263 | The predicate "is a T_1 sp... |
| ishaus 23264 | The predicate "is a Hausdo... |
| iscnrm 23265 | The property of being comp... |
| t0sep 23266 | Any two topologically indi... |
| t0dist 23267 | Any two distinct points in... |
| t1sncld 23268 | In a T_1 space, singletons... |
| t1ficld 23269 | In a T_1 space, finite set... |
| hausnei 23270 | Neighborhood property of a... |
| t0top 23271 | A T_0 space is a topologic... |
| t1top 23272 | A T_1 space is a topologic... |
| haustop 23273 | A Hausdorff space is a top... |
| isreg 23274 | The predicate "is a regula... |
| regtop 23275 | A regular space is a topol... |
| regsep 23276 | In a regular space, every ... |
| isnrm 23277 | The predicate "is a normal... |
| nrmtop 23278 | A normal space is a topolo... |
| cnrmtop 23279 | A completely normal space ... |
| iscnrm2 23280 | The property of being comp... |
| ispnrm 23281 | The property of being perf... |
| pnrmnrm 23282 | A perfectly normal space i... |
| pnrmtop 23283 | A perfectly normal space i... |
| pnrmcld 23284 | A closed set in a perfectl... |
| pnrmopn 23285 | An open set in a perfectly... |
| ist0-2 23286 | The predicate "is a T_0 sp... |
| ist0-3 23287 | The predicate "is a T_0 sp... |
| cnt0 23288 | The preimage of a T_0 topo... |
| ist1-2 23289 | An alternate characterizat... |
| t1t0 23290 | A T_1 space is a T_0 space... |
| ist1-3 23291 | A space is T_1 iff every p... |
| cnt1 23292 | The preimage of a T_1 topo... |
| ishaus2 23293 | Express the predicate " ` ... |
| haust1 23294 | A Hausdorff space is a T_1... |
| hausnei2 23295 | The Hausdorff condition st... |
| cnhaus 23296 | The preimage of a Hausdorf... |
| nrmsep3 23297 | In a normal space, given a... |
| nrmsep2 23298 | In a normal space, any two... |
| nrmsep 23299 | In a normal space, disjoin... |
| isnrm2 23300 | An alternate characterizat... |
| isnrm3 23301 | A topological space is nor... |
| cnrmi 23302 | A subspace of a completely... |
| cnrmnrm 23303 | A completely normal space ... |
| restcnrm 23304 | A subspace of a completely... |
| resthauslem 23305 | Lemma for ~ resthaus and s... |
| lpcls 23306 | The limit points of the cl... |
| perfcls 23307 | A subset of a perfect spac... |
| restt0 23308 | A subspace of a T_0 topolo... |
| restt1 23309 | A subspace of a T_1 topolo... |
| resthaus 23310 | A subspace of a Hausdorff ... |
| t1sep2 23311 | Any two points in a T_1 sp... |
| t1sep 23312 | Any two distinct points in... |
| sncld 23313 | A singleton is closed in a... |
| sshauslem 23314 | Lemma for ~ sshaus and sim... |
| sst0 23315 | A topology finer than a T_... |
| sst1 23316 | A topology finer than a T_... |
| sshaus 23317 | A topology finer than a Ha... |
| regsep2 23318 | In a regular space, a clos... |
| isreg2 23319 | A topological space is reg... |
| dnsconst 23320 | If a continuous mapping to... |
| ordtt1 23321 | The order topology is T_1 ... |
| lmmo 23322 | A sequence in a Hausdorff ... |
| lmfun 23323 | The convergence relation i... |
| dishaus 23324 | A discrete topology is Hau... |
| ordthauslem 23325 | Lemma for ~ ordthaus . (C... |
| ordthaus 23326 | The order topology of a to... |
| xrhaus 23327 | The topology of the extend... |
| iscmp 23330 | The predicate "is a compac... |
| cmpcov 23331 | An open cover of a compact... |
| cmpcov2 23332 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23333 | Combine ~ cmpcov with ~ ac... |
| cncmp 23334 | Compactness is respected b... |
| fincmp 23335 | A finite topology is compa... |
| 0cmp 23336 | The singleton of the empty... |
| cmptop 23337 | A compact topology is a to... |
| rncmp 23338 | The image of a compact set... |
| imacmp 23339 | The image of a compact set... |
| discmp 23340 | A discrete topology is com... |
| cmpsublem 23341 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23342 | Two equivalent ways of des... |
| tgcmp 23343 | A topology generated by a ... |
| cmpcld 23344 | A closed subset of a compa... |
| uncmp 23345 | The union of two compact s... |
| fiuncmp 23346 | A finite union of compact ... |
| sscmp 23347 | A subset of a compact topo... |
| hauscmplem 23348 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23349 | A compact subspace of a T2... |
| cmpfi 23350 | If a topology is compact a... |
| cmpfii 23351 | In a compact topology, a s... |
| bwth 23352 | The glorious Bolzano-Weier... |
| isconn 23355 | The predicate ` J ` is a c... |
| isconn2 23356 | The predicate ` J ` is a c... |
| connclo 23357 | The only nonempty clopen s... |
| conndisj 23358 | If a topology is connected... |
| conntop 23359 | A connected topology is a ... |
| indisconn 23360 | The indiscrete topology (o... |
| dfconn2 23361 | An alternate definition of... |
| connsuba 23362 | Connectedness for a subspa... |
| connsub 23363 | Two equivalent ways of say... |
| cnconn 23364 | Connectedness is respected... |
| nconnsubb 23365 | Disconnectedness for a sub... |
| connsubclo 23366 | If a clopen set meets a co... |
| connima 23367 | The image of a connected s... |
| conncn 23368 | A continuous function from... |
| iunconnlem 23369 | Lemma for ~ iunconn . (Co... |
| iunconn 23370 | The indexed union of conne... |
| unconn 23371 | The union of two connected... |
| clsconn 23372 | The closure of a connected... |
| conncompid 23373 | The connected component co... |
| conncompconn 23374 | The connected component co... |
| conncompss 23375 | The connected component co... |
| conncompcld 23376 | The connected component co... |
| conncompclo 23377 | The connected component co... |
| t1connperf 23378 | A connected T_1 space is p... |
| is1stc 23383 | The predicate "is a first-... |
| is1stc2 23384 | An equivalent way of sayin... |
| 1stctop 23385 | A first-countable topology... |
| 1stcclb 23386 | A property of points in a ... |
| 1stcfb 23387 | For any point ` A ` in a f... |
| is2ndc 23388 | The property of being seco... |
| 2ndctop 23389 | A second-countable topolog... |
| 2ndci 23390 | A countable basis generate... |
| 2ndcsb 23391 | Having a countable subbase... |
| 2ndcredom 23392 | A second-countable space h... |
| 2ndc1stc 23393 | A second-countable space i... |
| 1stcrestlem 23394 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23395 | A subspace of a first-coun... |
| 2ndcrest 23396 | A subspace of a second-cou... |
| 2ndcctbss 23397 | If a topology is second-co... |
| 2ndcdisj 23398 | Any disjoint family of ope... |
| 2ndcdisj2 23399 | Any disjoint collection of... |
| 2ndcomap 23400 | A surjective continuous op... |
| 2ndcsep 23401 | A second-countable topolog... |
| dis2ndc 23402 | A discrete space is second... |
| 1stcelcls 23403 | A point belongs to the clo... |
| 1stccnp 23404 | A mapping is continuous at... |
| 1stccn 23405 | A mapping ` X --> Y ` , wh... |
| islly 23410 | The property of being a lo... |
| isnlly 23411 | The property of being an n... |
| llyeq 23412 | Equality theorem for the `... |
| nllyeq 23413 | Equality theorem for the `... |
| llytop 23414 | A locally ` A ` space is a... |
| nllytop 23415 | A locally ` A ` space is a... |
| llyi 23416 | The property of a locally ... |
| nllyi 23417 | The property of an n-local... |
| nlly2i 23418 | Eliminate the neighborhood... |
| llynlly 23419 | A locally ` A ` space is n... |
| llyssnlly 23420 | A locally ` A ` space is n... |
| llyss 23421 | The "locally" predicate re... |
| nllyss 23422 | The "n-locally" predicate ... |
| subislly 23423 | The property of a subspace... |
| restnlly 23424 | If the property ` A ` pass... |
| restlly 23425 | If the property ` A ` pass... |
| islly2 23426 | An alternative expression ... |
| llyrest 23427 | An open subspace of a loca... |
| nllyrest 23428 | An open subspace of an n-l... |
| loclly 23429 | If ` A ` is a local proper... |
| llyidm 23430 | Idempotence of the "locall... |
| nllyidm 23431 | Idempotence of the "n-loca... |
| toplly 23432 | A topology is locally a to... |
| topnlly 23433 | A topology is n-locally a ... |
| hauslly 23434 | A Hausdorff space is local... |
| hausnlly 23435 | A Hausdorff space is n-loc... |
| hausllycmp 23436 | A compact Hausdorff space ... |
| cldllycmp 23437 | A closed subspace of a loc... |
| lly1stc 23438 | First-countability is a lo... |
| dislly 23439 | The discrete space ` ~P X ... |
| disllycmp 23440 | A discrete space is locall... |
| dis1stc 23441 | A discrete space is first-... |
| hausmapdom 23442 | If ` X ` is a first-counta... |
| hauspwdom 23443 | Simplify the cardinal ` A ... |
| refrel 23450 | Refinement is a relation. ... |
| isref 23451 | The property of being a re... |
| refbas 23452 | A refinement covers the sa... |
| refssex 23453 | Every set in a refinement ... |
| ssref 23454 | A subcover is a refinement... |
| refref 23455 | Reflexivity of refinement.... |
| reftr 23456 | Refinement is transitive. ... |
| refun0 23457 | Adding the empty set prese... |
| isptfin 23458 | The statement "is a point-... |
| islocfin 23459 | The statement "is a locall... |
| finptfin 23460 | A finite cover is a point-... |
| ptfinfin 23461 | A point covered by a point... |
| finlocfin 23462 | A finite cover of a topolo... |
| locfintop 23463 | A locally finite cover cov... |
| locfinbas 23464 | A locally finite cover mus... |
| locfinnei 23465 | A point covered by a local... |
| lfinpfin 23466 | A locally finite cover is ... |
| lfinun 23467 | Adding a finite set preser... |
| locfincmp 23468 | For a compact space, the l... |
| unisngl 23469 | Taking the union of the se... |
| dissnref 23470 | The set of singletons is a... |
| dissnlocfin 23471 | The set of singletons is l... |
| locfindis 23472 | The locally finite covers ... |
| locfincf 23473 | A locally finite cover in ... |
| comppfsc 23474 | A space where every open c... |
| kgenval 23477 | Value of the compact gener... |
| elkgen 23478 | Value of the compact gener... |
| kgeni 23479 | Property of the open sets ... |
| kgentopon 23480 | The compact generator gene... |
| kgenuni 23481 | The base set of the compac... |
| kgenftop 23482 | The compact generator gene... |
| kgenf 23483 | The compact generator is a... |
| kgentop 23484 | A compactly generated spac... |
| kgenss 23485 | The compact generator gene... |
| kgenhaus 23486 | The compact generator gene... |
| kgencmp 23487 | The compact generator topo... |
| kgencmp2 23488 | The compact generator topo... |
| kgenidm 23489 | The compact generator is i... |
| iskgen2 23490 | A space is compactly gener... |
| iskgen3 23491 | Derive the usual definitio... |
| llycmpkgen2 23492 | A locally compact space is... |
| cmpkgen 23493 | A compact space is compact... |
| llycmpkgen 23494 | A locally compact space is... |
| 1stckgenlem 23495 | The one-point compactifica... |
| 1stckgen 23496 | A first-countable space is... |
| kgen2ss 23497 | The compact generator pres... |
| kgencn 23498 | A function from a compactl... |
| kgencn2 23499 | A function ` F : J --> K `... |
| kgencn3 23500 | The set of continuous func... |
| kgen2cn 23501 | A continuous function is a... |
| txval 23506 | Value of the binary topolo... |
| txuni2 23507 | The underlying set of the ... |
| txbasex 23508 | The basis for the product ... |
| txbas 23509 | The set of Cartesian produ... |
| eltx 23510 | A set in a product is open... |
| txtop 23511 | The product of two topolog... |
| ptval 23512 | The value of the product t... |
| ptpjpre1 23513 | The preimage of a projecti... |
| elpt 23514 | Elementhood in the bases o... |
| elptr 23515 | A basic open set in the pr... |
| elptr2 23516 | A basic open set in the pr... |
| ptbasid 23517 | The base set of the produc... |
| ptuni2 23518 | The base set for the produ... |
| ptbasin 23519 | The basis for a product to... |
| ptbasin2 23520 | The basis for a product to... |
| ptbas 23521 | The basis for a product to... |
| ptpjpre2 23522 | The basis for a product to... |
| ptbasfi 23523 | The basis for the product ... |
| pttop 23524 | The product topology is a ... |
| ptopn 23525 | A basic open set in the pr... |
| ptopn2 23526 | A sub-basic open set in th... |
| xkotf 23527 | Functionality of function ... |
| xkobval 23528 | Alternative expression for... |
| xkoval 23529 | Value of the compact-open ... |
| xkotop 23530 | The compact-open topology ... |
| xkoopn 23531 | A basic open set of the co... |
| txtopi 23532 | The product of two topolog... |
| txtopon 23533 | The underlying set of the ... |
| txuni 23534 | The underlying set of the ... |
| txunii 23535 | The underlying set of the ... |
| ptuni 23536 | The base set for the produ... |
| ptunimpt 23537 | Base set of a product topo... |
| pttopon 23538 | The base set for the produ... |
| pttoponconst 23539 | The base set for a product... |
| ptuniconst 23540 | The base set for a product... |
| xkouni 23541 | The base set of the compac... |
| xkotopon 23542 | The base set of the compac... |
| ptval2 23543 | The value of the product t... |
| txopn 23544 | The product of two open se... |
| txcld 23545 | The product of two closed ... |
| txcls 23546 | Closure of a rectangle in ... |
| txss12 23547 | Subset property of the top... |
| txbasval 23548 | It is sufficient to consid... |
| neitx 23549 | The Cartesian product of t... |
| txcnpi 23550 | Continuity of a two-argume... |
| tx1cn 23551 | Continuity of the first pr... |
| tx2cn 23552 | Continuity of the second p... |
| ptpjcn 23553 | Continuity of a projection... |
| ptpjopn 23554 | The projection map is an o... |
| ptcld 23555 | A closed box in the produc... |
| ptcldmpt 23556 | A closed box in the produc... |
| ptclsg 23557 | The closure of a box in th... |
| ptcls 23558 | The closure of a box in th... |
| dfac14lem 23559 | Lemma for ~ dfac14 . By e... |
| dfac14 23560 | Theorem ~ ptcls is an equi... |
| xkoccn 23561 | The "constant function" fu... |
| txcnp 23562 | If two functions are conti... |
| ptcnplem 23563 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23564 | If every projection of a f... |
| upxp 23565 | Universal property of the ... |
| txcnmpt 23566 | A map into the product of ... |
| uptx 23567 | Universal property of the ... |
| txcn 23568 | A map into the product of ... |
| ptcn 23569 | If every projection of a f... |
| prdstopn 23570 | Topology of a structure pr... |
| prdstps 23571 | A structure product of top... |
| pwstps 23572 | A structure power of a top... |
| txrest 23573 | The subspace of a topologi... |
| txdis 23574 | The topological product of... |
| txindislem 23575 | Lemma for ~ txindis . (Co... |
| txindis 23576 | The topological product of... |
| txdis1cn 23577 | A function is jointly cont... |
| txlly 23578 | If the property ` A ` is p... |
| txnlly 23579 | If the property ` A ` is p... |
| pthaus 23580 | The product of a collectio... |
| ptrescn 23581 | Restriction is a continuou... |
| txtube 23582 | The "tube lemma". If ` X ... |
| txcmplem1 23583 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23584 | Lemma for ~ txcmp . (Cont... |
| txcmp 23585 | The topological product of... |
| txcmpb 23586 | The topological product of... |
| hausdiag 23587 | A topology is Hausdorff if... |
| hauseqlcld 23588 | In a Hausdorff topology, t... |
| txhaus 23589 | The topological product of... |
| txlm 23590 | Two sequences converge iff... |
| lmcn2 23591 | The image of a convergent ... |
| tx1stc 23592 | The topological product of... |
| tx2ndc 23593 | The topological product of... |
| txkgen 23594 | The topological product of... |
| xkohaus 23595 | If the codomain space is H... |
| xkoptsub 23596 | The compact-open topology ... |
| xkopt 23597 | The compact-open topology ... |
| xkopjcn 23598 | Continuity of a projection... |
| xkoco1cn 23599 | If ` F ` is a continuous f... |
| xkoco2cn 23600 | If ` F ` is a continuous f... |
| xkococnlem 23601 | Continuity of the composit... |
| xkococn 23602 | Continuity of the composit... |
| cnmptid 23603 | The identity function is c... |
| cnmptc 23604 | A constant function is con... |
| cnmpt11 23605 | The composition of continu... |
| cnmpt11f 23606 | The composition of continu... |
| cnmpt1t 23607 | The composition of continu... |
| cnmpt12f 23608 | The composition of continu... |
| cnmpt12 23609 | The composition of continu... |
| cnmpt1st 23610 | The projection onto the fi... |
| cnmpt2nd 23611 | The projection onto the se... |
| cnmpt2c 23612 | A constant function is con... |
| cnmpt21 23613 | The composition of continu... |
| cnmpt21f 23614 | The composition of continu... |
| cnmpt2t 23615 | The composition of continu... |
| cnmpt22 23616 | The composition of continu... |
| cnmpt22f 23617 | The composition of continu... |
| cnmpt1res 23618 | The restriction of a conti... |
| cnmpt2res 23619 | The restriction of a conti... |
| cnmptcom 23620 | The argument converse of a... |
| cnmptkc 23621 | The curried first projecti... |
| cnmptkp 23622 | The evaluation of the inne... |
| cnmptk1 23623 | The composition of a curri... |
| cnmpt1k 23624 | The composition of a one-a... |
| cnmptkk 23625 | The composition of two cur... |
| xkofvcn 23626 | Joint continuity of the fu... |
| cnmptk1p 23627 | The evaluation of a currie... |
| cnmptk2 23628 | The uncurrying of a currie... |
| xkoinjcn 23629 | Continuity of "injection",... |
| cnmpt2k 23630 | The currying of a two-argu... |
| txconn 23631 | The topological product of... |
| imasnopn 23632 | If a relation graph is ope... |
| imasncld 23633 | If a relation graph is clo... |
| imasncls 23634 | If a relation graph is clo... |
| qtopval 23637 | Value of the quotient topo... |
| qtopval2 23638 | Value of the quotient topo... |
| elqtop 23639 | Value of the quotient topo... |
| qtopres 23640 | The quotient topology is u... |
| qtoptop2 23641 | The quotient topology is a... |
| qtoptop 23642 | The quotient topology is a... |
| elqtop2 23643 | Value of the quotient topo... |
| qtopuni 23644 | The base set of the quotie... |
| elqtop3 23645 | Value of the quotient topo... |
| qtoptopon 23646 | The base set of the quotie... |
| qtopid 23647 | A quotient map is a contin... |
| idqtop 23648 | The quotient topology indu... |
| qtopcmplem 23649 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23650 | A quotient of a compact sp... |
| qtopconn 23651 | A quotient of a connected ... |
| qtopkgen 23652 | A quotient of a compactly ... |
| basqtop 23653 | An injection maps bases to... |
| tgqtop 23654 | An injection maps generate... |
| qtopcld 23655 | The property of being a cl... |
| qtopcn 23656 | Universal property of a qu... |
| qtopss 23657 | A surjective continuous fu... |
| qtopeu 23658 | Universal property of the ... |
| qtoprest 23659 | If ` A ` is a saturated op... |
| qtopomap 23660 | If ` F ` is a surjective c... |
| qtopcmap 23661 | If ` F ` is a surjective c... |
| imastopn 23662 | The topology of an image s... |
| imastps 23663 | The image of a topological... |
| qustps 23664 | A quotient structure is a ... |
| kqfval 23665 | Value of the function appe... |
| kqfeq 23666 | Two points in the Kolmogor... |
| kqffn 23667 | The topological indistingu... |
| kqval 23668 | Value of the quotient topo... |
| kqtopon 23669 | The Kolmogorov quotient is... |
| kqid 23670 | The topological indistingu... |
| ist0-4 23671 | The topological indistingu... |
| kqfvima 23672 | When the image set is open... |
| kqsat 23673 | Any open set is saturated ... |
| kqdisj 23674 | A version of ~ imain for t... |
| kqcldsat 23675 | Any closed set is saturate... |
| kqopn 23676 | The topological indistingu... |
| kqcld 23677 | The topological indistingu... |
| kqt0lem 23678 | Lemma for ~ kqt0 . (Contr... |
| isr0 23679 | The property " ` J ` is an... |
| r0cld 23680 | The analogue of the T_1 ax... |
| regr1lem 23681 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23682 | A Kolmogorov quotient of a... |
| kqreglem1 23683 | A Kolmogorov quotient of a... |
| kqreglem2 23684 | If the Kolmogorov quotient... |
| kqnrmlem1 23685 | A Kolmogorov quotient of a... |
| kqnrmlem2 23686 | If the Kolmogorov quotient... |
| kqtop 23687 | The Kolmogorov quotient is... |
| kqt0 23688 | The Kolmogorov quotient is... |
| kqf 23689 | The Kolmogorov quotient is... |
| r0sep 23690 | The separation property of... |
| nrmr0reg 23691 | A normal R_0 space is also... |
| regr1 23692 | A regular space is R_1, wh... |
| kqreg 23693 | The Kolmogorov quotient of... |
| kqnrm 23694 | The Kolmogorov quotient of... |
| hmeofn 23699 | The set of homeomorphisms ... |
| hmeofval 23700 | The set of all the homeomo... |
| ishmeo 23701 | The predicate F is a homeo... |
| hmeocn 23702 | A homeomorphism is continu... |
| hmeocnvcn 23703 | The converse of a homeomor... |
| hmeocnv 23704 | The converse of a homeomor... |
| hmeof1o2 23705 | A homeomorphism is a 1-1-o... |
| hmeof1o 23706 | A homeomorphism is a 1-1-o... |
| hmeoima 23707 | The image of an open set b... |
| hmeoopn 23708 | Homeomorphisms preserve op... |
| hmeocld 23709 | Homeomorphisms preserve cl... |
| hmeocls 23710 | Homeomorphisms preserve cl... |
| hmeontr 23711 | Homeomorphisms preserve in... |
| hmeoimaf1o 23712 | The function mapping open ... |
| hmeores 23713 | The restriction of a homeo... |
| hmeoco 23714 | The composite of two homeo... |
| idhmeo 23715 | The identity function is a... |
| hmeocnvb 23716 | The converse of a homeomor... |
| hmeoqtop 23717 | A homeomorphism is a quoti... |
| hmph 23718 | Express the predicate ` J ... |
| hmphi 23719 | If there is a homeomorphis... |
| hmphtop 23720 | Reverse closure for the ho... |
| hmphtop1 23721 | The relation "being homeom... |
| hmphtop2 23722 | The relation "being homeom... |
| hmphref 23723 | "Is homeomorphic to" is re... |
| hmphsym 23724 | "Is homeomorphic to" is sy... |
| hmphtr 23725 | "Is homeomorphic to" is tr... |
| hmpher 23726 | "Is homeomorphic to" is an... |
| hmphen 23727 | Homeomorphisms preserve th... |
| hmphsymb 23728 | "Is homeomorphic to" is sy... |
| haushmphlem 23729 | Lemma for ~ haushmph and s... |
| cmphmph 23730 | Compactness is a topologic... |
| connhmph 23731 | Connectedness is a topolog... |
| t0hmph 23732 | T_0 is a topological prope... |
| t1hmph 23733 | T_1 is a topological prope... |
| haushmph 23734 | Hausdorff-ness is a topolo... |
| reghmph 23735 | Regularity is a topologica... |
| nrmhmph 23736 | Normality is a topological... |
| hmph0 23737 | A topology homeomorphic to... |
| hmphdis 23738 | Homeomorphisms preserve to... |
| hmphindis 23739 | Homeomorphisms preserve to... |
| indishmph 23740 | Equinumerous sets equipped... |
| hmphen2 23741 | Homeomorphisms preserve th... |
| cmphaushmeo 23742 | A continuous bijection fro... |
| ordthmeolem 23743 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23744 | An order isomorphism is a ... |
| txhmeo 23745 | Lift a pair of homeomorphi... |
| txswaphmeolem 23746 | Show inverse for the "swap... |
| txswaphmeo 23747 | There is a homeomorphism f... |
| pt1hmeo 23748 | The canonical homeomorphis... |
| ptuncnv 23749 | Exhibit the converse funct... |
| ptunhmeo 23750 | Define a homeomorphism fro... |
| xpstopnlem1 23751 | The function ` F ` used in... |
| xpstps 23752 | A binary product of topolo... |
| xpstopnlem2 23753 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23754 | The topology on a binary p... |
| ptcmpfi 23755 | A topological product of f... |
| xkocnv 23756 | The inverse of the "curryi... |
| xkohmeo 23757 | The Exponential Law for to... |
| qtopf1 23758 | If a quotient map is injec... |
| qtophmeo 23759 | If two functions on a base... |
| t0kq 23760 | A topological space is T_0... |
| kqhmph 23761 | A topological space is T_0... |
| ist1-5lem 23762 | Lemma for ~ ist1-5 and sim... |
| t1r0 23763 | A T_1 space is R_0. That ... |
| ist1-5 23764 | A topological space is T_1... |
| ishaus3 23765 | A topological space is Hau... |
| nrmreg 23766 | A normal T_1 space is regu... |
| reghaus 23767 | A regular T_0 space is Hau... |
| nrmhaus 23768 | A T_1 normal space is Haus... |
| elmptrab 23769 | Membership in a one-parame... |
| elmptrab2 23770 | Membership in a one-parame... |
| isfbas 23771 | The predicate " ` F ` is a... |
| fbasne0 23772 | There are no empty filter ... |
| 0nelfb 23773 | No filter base contains th... |
| fbsspw 23774 | A filter base on a set is ... |
| fbelss 23775 | An element of the filter b... |
| fbdmn0 23776 | The domain of a filter bas... |
| isfbas2 23777 | The predicate " ` F ` is a... |
| fbasssin 23778 | A filter base contains sub... |
| fbssfi 23779 | A filter base contains sub... |
| fbssint 23780 | A filter base contains sub... |
| fbncp 23781 | A filter base does not con... |
| fbun 23782 | A necessary and sufficient... |
| fbfinnfr 23783 | No filter base containing ... |
| opnfbas 23784 | The collection of open sup... |
| trfbas2 23785 | Conditions for the trace o... |
| trfbas 23786 | Conditions for the trace o... |
| isfil 23789 | The predicate "is a filter... |
| filfbas 23790 | A filter is a filter base.... |
| 0nelfil 23791 | The empty set doesn't belo... |
| fileln0 23792 | An element of a filter is ... |
| filsspw 23793 | A filter is a subset of th... |
| filelss 23794 | An element of a filter is ... |
| filss 23795 | A filter is closed under t... |
| filin 23796 | A filter is closed under t... |
| filtop 23797 | The underlying set belongs... |
| isfil2 23798 | Derive the standard axioms... |
| isfildlem 23799 | Lemma for ~ isfild . (Con... |
| isfild 23800 | Sufficient condition for a... |
| filfi 23801 | A filter is closed under t... |
| filinn0 23802 | The intersection of two el... |
| filintn0 23803 | A filter has the finite in... |
| filn0 23804 | The empty set is not a fil... |
| infil 23805 | The intersection of two fi... |
| snfil 23806 | A singleton is a filter. ... |
| fbasweak 23807 | A filter base on any set i... |
| snfbas 23808 | Condition for a singleton ... |
| fsubbas 23809 | A condition for a set to g... |
| fbasfip 23810 | A filter base has the fini... |
| fbunfip 23811 | A helpful lemma for showin... |
| fgval 23812 | The filter generating clas... |
| elfg 23813 | A condition for elements o... |
| ssfg 23814 | A filter base is a subset ... |
| fgss 23815 | A bigger base generates a ... |
| fgss2 23816 | A condition for a filter t... |
| fgfil 23817 | A filter generates itself.... |
| elfilss 23818 | An element belongs to a fi... |
| filfinnfr 23819 | No filter containing a fin... |
| fgcl 23820 | A generated filter is a fi... |
| fgabs 23821 | Absorption law for filter ... |
| neifil 23822 | The neighborhoods of a non... |
| filunibas 23823 | Recover the base set from ... |
| filunirn 23824 | Two ways to express a filt... |
| filconn 23825 | A filter gives rise to a c... |
| fbasrn 23826 | Given a filter on a domain... |
| filuni 23827 | The union of a nonempty se... |
| trfil1 23828 | Conditions for the trace o... |
| trfil2 23829 | Conditions for the trace o... |
| trfil3 23830 | Conditions for the trace o... |
| trfilss 23831 | If ` A ` is a member of th... |
| fgtr 23832 | If ` A ` is a member of th... |
| trfg 23833 | The trace operation and th... |
| trnei 23834 | The trace, over a set ` A ... |
| cfinfil 23835 | Relative complements of th... |
| csdfil 23836 | The set of all elements wh... |
| supfil 23837 | The supersets of a nonempt... |
| zfbas 23838 | The set of upper sets of i... |
| uzrest 23839 | The restriction of the set... |
| uzfbas 23840 | The set of upper sets of i... |
| isufil 23845 | The property of being an u... |
| ufilfil 23846 | An ultrafilter is a filter... |
| ufilss 23847 | For any subset of the base... |
| ufilb 23848 | The complement is in an ul... |
| ufilmax 23849 | Any filter finer than an u... |
| isufil2 23850 | The maximal property of an... |
| ufprim 23851 | An ultrafilter is a prime ... |
| trufil 23852 | Conditions for the trace o... |
| filssufilg 23853 | A filter is contained in s... |
| filssufil 23854 | A filter is contained in s... |
| isufl 23855 | Define the (strong) ultraf... |
| ufli 23856 | Property of a set that sat... |
| numufl 23857 | Consequence of ~ filssufil... |
| fiufl 23858 | A finite set satisfies the... |
| acufl 23859 | The axiom of choice implie... |
| ssufl 23860 | If ` Y ` is a subset of ` ... |
| ufileu 23861 | If the ultrafilter contain... |
| filufint 23862 | A filter is equal to the i... |
| uffix 23863 | Lemma for ~ fixufil and ~ ... |
| fixufil 23864 | The condition describing a... |
| uffixfr 23865 | An ultrafilter is either f... |
| uffix2 23866 | A classification of fixed ... |
| uffixsn 23867 | The singleton of the gener... |
| ufildom1 23868 | An ultrafilter is generate... |
| uffinfix 23869 | An ultrafilter containing ... |
| cfinufil 23870 | An ultrafilter is free iff... |
| ufinffr 23871 | An infinite subset is cont... |
| ufilen 23872 | Any infinite set has an ul... |
| ufildr 23873 | An ultrafilter gives rise ... |
| fin1aufil 23874 | There are no definable fre... |
| fmval 23885 | Introduce a function that ... |
| fmfil 23886 | A mapping filter is a filt... |
| fmf 23887 | Pushing-forward via a func... |
| fmss 23888 | A finer filter produces a ... |
| elfm 23889 | An element of a mapping fi... |
| elfm2 23890 | An element of a mapping fi... |
| fmfg 23891 | The image filter of a filt... |
| elfm3 23892 | An alternate formulation o... |
| imaelfm 23893 | An image of a filter eleme... |
| rnelfmlem 23894 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23895 | A condition for a filter t... |
| fmfnfmlem1 23896 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23897 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23898 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23899 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23900 | A filter finer than an ima... |
| fmufil 23901 | An image filter of an ultr... |
| fmid 23902 | The filter map applied to ... |
| fmco 23903 | Composition of image filte... |
| ufldom 23904 | The ultrafilter lemma prop... |
| flimval 23905 | The set of limit points of... |
| elflim2 23906 | The predicate "is a limit ... |
| flimtop 23907 | Reverse closure for the li... |
| flimneiss 23908 | A filter contains the neig... |
| flimnei 23909 | A filter contains all of t... |
| flimelbas 23910 | A limit point of a filter ... |
| flimfil 23911 | Reverse closure for the li... |
| flimtopon 23912 | Reverse closure for the li... |
| elflim 23913 | The predicate "is a limit ... |
| flimss2 23914 | A limit point of a filter ... |
| flimss1 23915 | A limit point of a filter ... |
| neiflim 23916 | A point is a limit point o... |
| flimopn 23917 | The condition for being a ... |
| fbflim 23918 | A condition for a filter t... |
| fbflim2 23919 | A condition for a filter b... |
| flimclsi 23920 | The convergent points of a... |
| hausflimlem 23921 | If ` A ` and ` B ` are bot... |
| hausflimi 23922 | One direction of ~ hausfli... |
| hausflim 23923 | A condition for a topology... |
| flimcf 23924 | Fineness is properly chara... |
| flimrest 23925 | The set of limit points in... |
| flimclslem 23926 | Lemma for ~ flimcls . (Co... |
| flimcls 23927 | Closure in terms of filter... |
| flimsncls 23928 | If ` A ` is a limit point ... |
| hauspwpwf1 23929 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23930 | If ` X ` is a Hausdorff sp... |
| flffval 23931 | Given a topology and a fil... |
| flfval 23932 | Given a function from a fi... |
| flfnei 23933 | The property of being a li... |
| flfneii 23934 | A neighborhood of a limit ... |
| isflf 23935 | The property of being a li... |
| flfelbas 23936 | A limit point of a functio... |
| flffbas 23937 | Limit points of a function... |
| flftg 23938 | Limit points of a function... |
| hausflf 23939 | If a function has its valu... |
| hausflf2 23940 | If a convergent function h... |
| cnpflfi 23941 | Forward direction of ~ cnp... |
| cnpflf2 23942 | ` F ` is continuous at poi... |
| cnpflf 23943 | Continuity of a function a... |
| cnflf 23944 | A function is continuous i... |
| cnflf2 23945 | A function is continuous i... |
| flfcnp 23946 | A continuous function pres... |
| lmflf 23947 | The topological limit rela... |
| txflf 23948 | Two sequences converge in ... |
| flfcnp2 23949 | The image of a convergent ... |
| fclsval 23950 | The set of all cluster poi... |
| isfcls 23951 | A cluster point of a filte... |
| fclsfil 23952 | Reverse closure for the cl... |
| fclstop 23953 | Reverse closure for the cl... |
| fclstopon 23954 | Reverse closure for the cl... |
| isfcls2 23955 | A cluster point of a filte... |
| fclsopn 23956 | Write the cluster point co... |
| fclsopni 23957 | An open neighborhood of a ... |
| fclselbas 23958 | A cluster point is in the ... |
| fclsneii 23959 | A neighborhood of a cluste... |
| fclssscls 23960 | The set of cluster points ... |
| fclsnei 23961 | Cluster points in terms of... |
| supnfcls 23962 | The filter of supersets of... |
| fclsbas 23963 | Cluster points in terms of... |
| fclsss1 23964 | A finer topology has fewer... |
| fclsss2 23965 | A finer filter has fewer c... |
| fclsrest 23966 | The set of cluster points ... |
| fclscf 23967 | Characterization of finene... |
| flimfcls 23968 | A limit point is a cluster... |
| fclsfnflim 23969 | A filter clusters at a poi... |
| flimfnfcls 23970 | A filter converges to a po... |
| fclscmpi 23971 | Forward direction of ~ fcl... |
| fclscmp 23972 | A space is compact iff eve... |
| uffclsflim 23973 | The cluster points of an u... |
| ufilcmp 23974 | A space is compact iff eve... |
| fcfval 23975 | The set of cluster points ... |
| isfcf 23976 | The property of being a cl... |
| fcfnei 23977 | The property of being a cl... |
| fcfelbas 23978 | A cluster point of a funct... |
| fcfneii 23979 | A neighborhood of a cluste... |
| flfssfcf 23980 | A limit point of a functio... |
| uffcfflf 23981 | If the domain filter is an... |
| cnpfcfi 23982 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23983 | A function ` F ` is contin... |
| cnfcf 23984 | Continuity of a function i... |
| flfcntr 23985 | A continuous function's va... |
| alexsublem 23986 | Lemma for ~ alexsub . (Co... |
| alexsub 23987 | The Alexander Subbase Theo... |
| alexsubb 23988 | Biconditional form of the ... |
| alexsubALTlem1 23989 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23990 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23991 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23992 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23993 | The Alexander Subbase Theo... |
| ptcmplem1 23994 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23995 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23996 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23997 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23998 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 23999 | Tychonoff's theorem: The ... |
| ptcmp 24000 | Tychonoff's theorem: The ... |
| cnextval 24003 | The function applying cont... |
| cnextfval 24004 | The continuous extension o... |
| cnextrel 24005 | In the general case, a con... |
| cnextfun 24006 | If the target space is Hau... |
| cnextfvval 24007 | The value of the continuou... |
| cnextf 24008 | Extension by continuity. ... |
| cnextcn 24009 | Extension by continuity. ... |
| cnextfres1 24010 | ` F ` and its extension by... |
| cnextfres 24011 | ` F ` and its extension by... |
| istmd 24016 | The predicate "is a topolo... |
| tmdmnd 24017 | A topological monoid is a ... |
| tmdtps 24018 | A topological monoid is a ... |
| istgp 24019 | The predicate "is a topolo... |
| tgpgrp 24020 | A topological group is a g... |
| tgptmd 24021 | A topological group is a t... |
| tgptps 24022 | A topological group is a t... |
| tmdtopon 24023 | The topology of a topologi... |
| tgptopon 24024 | The topology of a topologi... |
| tmdcn 24025 | In a topological monoid, t... |
| tgpcn 24026 | In a topological group, th... |
| tgpinv 24027 | In a topological group, th... |
| grpinvhmeo 24028 | The inverse function in a ... |
| cnmpt1plusg 24029 | Continuity of the group su... |
| cnmpt2plusg 24030 | Continuity of the group su... |
| tmdcn2 24031 | Write out the definition o... |
| tgpsubcn 24032 | In a topological group, th... |
| istgp2 24033 | A group with a topology is... |
| tmdmulg 24034 | In a topological monoid, t... |
| tgpmulg 24035 | In a topological group, th... |
| tgpmulg2 24036 | In a topological monoid, t... |
| tmdgsum 24037 | In a topological monoid, t... |
| tmdgsum2 24038 | For any neighborhood ` U `... |
| oppgtmd 24039 | The opposite of a topologi... |
| oppgtgp 24040 | The opposite of a topologi... |
| distgp 24041 | Any group equipped with th... |
| indistgp 24042 | Any group equipped with th... |
| efmndtmd 24043 | The monoid of endofunction... |
| tmdlactcn 24044 | The left group action of e... |
| tgplacthmeo 24045 | The left group action of e... |
| submtmd 24046 | A submonoid of a topologic... |
| subgtgp 24047 | A subgroup of a topologica... |
| symgtgp 24048 | The symmetric group is a t... |
| subgntr 24049 | A subgroup of a topologica... |
| opnsubg 24050 | An open subgroup of a topo... |
| clssubg 24051 | The closure of a subgroup ... |
| clsnsg 24052 | The closure of a normal su... |
| cldsubg 24053 | A subgroup of finite index... |
| tgpconncompeqg 24054 | The connected component co... |
| tgpconncomp 24055 | The identity component, th... |
| tgpconncompss 24056 | The identity component is ... |
| ghmcnp 24057 | A group homomorphism on to... |
| snclseqg 24058 | The coset of the closure o... |
| tgphaus 24059 | A topological group is Hau... |
| tgpt1 24060 | Hausdorff and T1 are equiv... |
| tgpt0 24061 | Hausdorff and T0 are equiv... |
| qustgpopn 24062 | A quotient map in a topolo... |
| qustgplem 24063 | Lemma for ~ qustgp . (Con... |
| qustgp 24064 | The quotient of a topologi... |
| qustgphaus 24065 | The quotient of a topologi... |
| prdstmdd 24066 | The product of a family of... |
| prdstgpd 24067 | The product of a family of... |
| tsmsfbas 24070 | The collection of all sets... |
| tsmslem1 24071 | The finite partial sums of... |
| tsmsval2 24072 | Definition of the topologi... |
| tsmsval 24073 | Definition of the topologi... |
| tsmspropd 24074 | The group sum depends only... |
| eltsms 24075 | The property of being a su... |
| tsmsi 24076 | The property of being a su... |
| tsmscl 24077 | A sum in a topological gro... |
| haustsms 24078 | In a Hausdorff topological... |
| haustsms2 24079 | In a Hausdorff topological... |
| tsmscls 24080 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24081 | The convergent points of a... |
| tsmsid 24082 | If a sum is finite, the us... |
| haustsmsid 24083 | In a Hausdorff topological... |
| tsms0 24084 | The sum of zero is zero. ... |
| tsmssubm 24085 | Evaluate an infinite group... |
| tsmsres 24086 | Extend an infinite group s... |
| tsmsf1o 24087 | Re-index an infinite group... |
| tsmsmhm 24088 | Apply a continuous group h... |
| tsmsadd 24089 | The sum of two infinite gr... |
| tsmsinv 24090 | Inverse of an infinite gro... |
| tsmssub 24091 | The difference of two infi... |
| tgptsmscls 24092 | A sum in a topological gro... |
| tgptsmscld 24093 | The set of limit points to... |
| tsmssplit 24094 | Split a topological group ... |
| tsmsxplem1 24095 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24096 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24097 | Write a sum over a two-dim... |
| istrg 24106 | Express the predicate " ` ... |
| trgtmd 24107 | The multiplicative monoid ... |
| istdrg 24108 | Express the predicate " ` ... |
| tdrgunit 24109 | The unit group of a topolo... |
| trgtgp 24110 | A topological ring is a to... |
| trgtmd2 24111 | A topological ring is a to... |
| trgtps 24112 | A topological ring is a to... |
| trgring 24113 | A topological ring is a ri... |
| trggrp 24114 | A topological ring is a gr... |
| tdrgtrg 24115 | A topological division rin... |
| tdrgdrng 24116 | A topological division rin... |
| tdrgring 24117 | A topological division rin... |
| tdrgtmd 24118 | A topological division rin... |
| tdrgtps 24119 | A topological division rin... |
| istdrg2 24120 | A topological-ring divisio... |
| mulrcn 24121 | The functionalization of t... |
| invrcn2 24122 | The multiplicative inverse... |
| invrcn 24123 | The multiplicative inverse... |
| cnmpt1mulr 24124 | Continuity of ring multipl... |
| cnmpt2mulr 24125 | Continuity of ring multipl... |
| dvrcn 24126 | The division function is c... |
| istlm 24127 | The predicate " ` W ` is a... |
| vscacn 24128 | The scalar multiplication ... |
| tlmtmd 24129 | A topological module is a ... |
| tlmtps 24130 | A topological module is a ... |
| tlmlmod 24131 | A topological module is a ... |
| tlmtrg 24132 | The scalar ring of a topol... |
| tlmscatps 24133 | The scalar ring of a topol... |
| istvc 24134 | A topological vector space... |
| tvctdrg 24135 | The scalar field of a topo... |
| cnmpt1vsca 24136 | Continuity of scalar multi... |
| cnmpt2vsca 24137 | Continuity of scalar multi... |
| tlmtgp 24138 | A topological vector space... |
| tvctlm 24139 | A topological vector space... |
| tvclmod 24140 | A topological vector space... |
| tvclvec 24141 | A topological vector space... |
| ustfn 24144 | The defined uniform struct... |
| ustval 24145 | The class of all uniform s... |
| isust 24146 | The predicate " ` U ` is a... |
| ustssxp 24147 | Entourages are subsets of ... |
| ustssel 24148 | A uniform structure is upw... |
| ustbasel 24149 | The full set is always an ... |
| ustincl 24150 | A uniform structure is clo... |
| ustdiag 24151 | The diagonal set is includ... |
| ustinvel 24152 | If ` V ` is an entourage, ... |
| ustexhalf 24153 | For each entourage ` V ` t... |
| ustrel 24154 | The elements of uniform st... |
| ustfilxp 24155 | A uniform structure on a n... |
| ustne0 24156 | A uniform structure cannot... |
| ustssco 24157 | In an uniform structure, a... |
| ustexsym 24158 | In an uniform structure, f... |
| ustex2sym 24159 | In an uniform structure, f... |
| ustex3sym 24160 | In an uniform structure, f... |
| ustref 24161 | Any element of the base se... |
| ust0 24162 | The unique uniform structu... |
| ustn0 24163 | The empty set is not an un... |
| ustund 24164 | If two intersecting sets `... |
| ustelimasn 24165 | Any point ` A ` is near en... |
| ustneism 24166 | For a point ` A ` in ` X `... |
| ustbas2 24167 | Second direction for ~ ust... |
| ustuni 24168 | The set union of a uniform... |
| ustbas 24169 | Recover the base of an uni... |
| ustimasn 24170 | Lemma for ~ ustuqtop . (C... |
| trust 24171 | The trace of a uniform str... |
| utopval 24174 | The topology induced by a ... |
| elutop 24175 | Open sets in the topology ... |
| utoptop 24176 | The topology induced by a ... |
| utopbas 24177 | The base of the topology i... |
| utoptopon 24178 | Topology induced by a unif... |
| restutop 24179 | Restriction of a topology ... |
| restutopopn 24180 | The restriction of the top... |
| ustuqtoplem 24181 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24182 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24183 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24184 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24185 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24186 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24187 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24188 | For a given uniform struct... |
| utopsnneiplem 24189 | The neighborhoods of a poi... |
| utopsnneip 24190 | The neighborhoods of a poi... |
| utopsnnei 24191 | Images of singletons by en... |
| utop2nei 24192 | For any symmetrical entour... |
| utop3cls 24193 | Relation between a topolog... |
| utopreg 24194 | All Hausdorff uniform spac... |
| ussval 24201 | The uniform structure on u... |
| ussid 24202 | In case the base of the ` ... |
| isusp 24203 | The predicate ` W ` is a u... |
| ressuss 24204 | Value of the uniform struc... |
| ressust 24205 | The uniform structure of a... |
| ressusp 24206 | The restriction of a unifo... |
| tusval 24207 | The value of the uniform s... |
| tuslem 24208 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24209 | The base set of a construc... |
| tusunif 24210 | The uniform structure of a... |
| tususs 24211 | The uniform structure of a... |
| tustopn 24212 | The topology induced by a ... |
| tususp 24213 | A constructed uniform spac... |
| tustps 24214 | A constructed uniform spac... |
| uspreg 24215 | If a uniform space is Haus... |
| ucnval 24218 | The set of all uniformly c... |
| isucn 24219 | The predicate " ` F ` is a... |
| isucn2 24220 | The predicate " ` F ` is a... |
| ucnimalem 24221 | Reformulate the ` G ` func... |
| ucnima 24222 | An equivalent statement of... |
| ucnprima 24223 | The preimage by a uniforml... |
| iducn 24224 | The identity is uniformly ... |
| cstucnd 24225 | A constant function is uni... |
| ucncn 24226 | Uniform continuity implies... |
| iscfilu 24229 | The predicate " ` F ` is a... |
| cfilufbas 24230 | A Cauchy filter base is a ... |
| cfiluexsm 24231 | For a Cauchy filter base a... |
| fmucndlem 24232 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24233 | The image of a Cauchy filt... |
| cfilufg 24234 | The filter generated by a ... |
| trcfilu 24235 | Condition for the trace of... |
| cfiluweak 24236 | A Cauchy filter base is al... |
| neipcfilu 24237 | In an uniform space, a nei... |
| iscusp 24240 | The predicate " ` W ` is a... |
| cuspusp 24241 | A complete uniform space i... |
| cuspcvg 24242 | In a complete uniform spac... |
| iscusp2 24243 | The predicate " ` W ` is a... |
| cnextucn 24244 | Extension by continuity. ... |
| ucnextcn 24245 | Extension by continuity. ... |
| ispsmet 24246 | Express the predicate " ` ... |
| psmetdmdm 24247 | Recover the base set from ... |
| psmetf 24248 | The distance function of a... |
| psmetcl 24249 | Closure of the distance fu... |
| psmet0 24250 | The distance function of a... |
| psmettri2 24251 | Triangle inequality for th... |
| psmetsym 24252 | The distance function of a... |
| psmettri 24253 | Triangle inequality for th... |
| psmetge0 24254 | The distance function of a... |
| psmetxrge0 24255 | The distance function of a... |
| psmetres2 24256 | Restriction of a pseudomet... |
| psmetlecl 24257 | Real closure of an extende... |
| distspace 24258 | A set ` X ` together with ... |
| ismet 24265 | Express the predicate " ` ... |
| isxmet 24266 | Express the predicate " ` ... |
| ismeti 24267 | Properties that determine ... |
| isxmetd 24268 | Properties that determine ... |
| isxmet2d 24269 | It is safe to only require... |
| metflem 24270 | Lemma for ~ metf and other... |
| xmetf 24271 | Mapping of the distance fu... |
| metf 24272 | Mapping of the distance fu... |
| xmetcl 24273 | Closure of the distance fu... |
| metcl 24274 | Closure of the distance fu... |
| ismet2 24275 | An extended metric is a me... |
| metxmet 24276 | A metric is an extended me... |
| xmetdmdm 24277 | Recover the base set from ... |
| metdmdm 24278 | Recover the base set from ... |
| xmetunirn 24279 | Two ways to express an ext... |
| xmeteq0 24280 | The value of an extended m... |
| meteq0 24281 | The value of a metric is z... |
| xmettri2 24282 | Triangle inequality for th... |
| mettri2 24283 | Triangle inequality for th... |
| xmet0 24284 | The distance function of a... |
| met0 24285 | The distance function of a... |
| xmetge0 24286 | The distance function of a... |
| metge0 24287 | The distance function of a... |
| xmetlecl 24288 | Real closure of an extende... |
| xmetsym 24289 | The distance function of a... |
| xmetpsmet 24290 | An extended metric is a ps... |
| xmettpos 24291 | The distance function of a... |
| metsym 24292 | The distance function of a... |
| xmettri 24293 | Triangle inequality for th... |
| mettri 24294 | Triangle inequality for th... |
| xmettri3 24295 | Triangle inequality for th... |
| mettri3 24296 | Triangle inequality for th... |
| xmetrtri 24297 | One half of the reverse tr... |
| xmetrtri2 24298 | The reverse triangle inequ... |
| metrtri 24299 | Reverse triangle inequalit... |
| xmetgt0 24300 | The distance function of a... |
| metgt0 24301 | The distance function of a... |
| metn0 24302 | A metric space is nonempty... |
| xmetres2 24303 | Restriction of an extended... |
| metreslem 24304 | Lemma for ~ metres . (Con... |
| metres2 24305 | Lemma for ~ metres . (Con... |
| xmetres 24306 | A restriction of an extend... |
| metres 24307 | A restriction of a metric ... |
| 0met 24308 | The empty metric. (Contri... |
| prdsdsf 24309 | The product metric is a fu... |
| prdsxmetlem 24310 | The product metric is an e... |
| prdsxmet 24311 | The product metric is an e... |
| prdsmet 24312 | The product metric is a me... |
| ressprdsds 24313 | Restriction of a product m... |
| resspwsds 24314 | Restriction of a power met... |
| imasdsf1olem 24315 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24316 | The distance function is t... |
| imasf1oxmet 24317 | The image of an extended m... |
| imasf1omet 24318 | The image of a metric is a... |
| xpsdsfn 24319 | Closure of the metric in a... |
| xpsdsfn2 24320 | Closure of the metric in a... |
| xpsxmetlem 24321 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24322 | A product metric of extend... |
| xpsdsval 24323 | Value of the metric in a b... |
| xpsmet 24324 | The direct product of two ... |
| blfvalps 24325 | The value of the ball func... |
| blfval 24326 | The value of the ball func... |
| blvalps 24327 | The ball around a point ` ... |
| blval 24328 | The ball around a point ` ... |
| elblps 24329 | Membership in a ball. (Co... |
| elbl 24330 | Membership in a ball. (Co... |
| elbl2ps 24331 | Membership in a ball. (Co... |
| elbl2 24332 | Membership in a ball. (Co... |
| elbl3ps 24333 | Membership in a ball, with... |
| elbl3 24334 | Membership in a ball, with... |
| blcomps 24335 | Commute the arguments to t... |
| blcom 24336 | Commute the arguments to t... |
| xblpnfps 24337 | The infinity ball in an ex... |
| xblpnf 24338 | The infinity ball in an ex... |
| blpnf 24339 | The infinity ball in a sta... |
| bldisj 24340 | Two balls are disjoint if ... |
| blgt0 24341 | A nonempty ball implies th... |
| bl2in 24342 | Two balls are disjoint if ... |
| xblss2ps 24343 | One ball is contained in a... |
| xblss2 24344 | One ball is contained in a... |
| blss2ps 24345 | One ball is contained in a... |
| blss2 24346 | One ball is contained in a... |
| blhalf 24347 | A ball of radius ` R / 2 `... |
| blfps 24348 | Mapping of a ball. (Contr... |
| blf 24349 | Mapping of a ball. (Contr... |
| blrnps 24350 | Membership in the range of... |
| blrn 24351 | Membership in the range of... |
| xblcntrps 24352 | A ball contains its center... |
| xblcntr 24353 | A ball contains its center... |
| blcntrps 24354 | A ball contains its center... |
| blcntr 24355 | A ball contains its center... |
| xbln0 24356 | A ball is nonempty iff the... |
| bln0 24357 | A ball is not empty. (Con... |
| blelrnps 24358 | A ball belongs to the set ... |
| blelrn 24359 | A ball belongs to the set ... |
| blssm 24360 | A ball is a subset of the ... |
| unirnblps 24361 | The union of the set of ba... |
| unirnbl 24362 | The union of the set of ba... |
| blin 24363 | The intersection of two ba... |
| ssblps 24364 | The size of a ball increas... |
| ssbl 24365 | The size of a ball increas... |
| blssps 24366 | Any point ` P ` in a ball ... |
| blss 24367 | Any point ` P ` in a ball ... |
| blssexps 24368 | Two ways to express the ex... |
| blssex 24369 | Two ways to express the ex... |
| ssblex 24370 | A nested ball exists whose... |
| blin2 24371 | Given any two balls and a ... |
| blbas 24372 | The balls of a metric spac... |
| blres 24373 | A ball in a restricted met... |
| xmeterval 24374 | Value of the "finitely sep... |
| xmeter 24375 | The "finitely separated" r... |
| xmetec 24376 | The equivalence classes un... |
| blssec 24377 | A ball centered at ` P ` i... |
| blpnfctr 24378 | The infinity ball in an ex... |
| xmetresbl 24379 | An extended metric restric... |
| mopnval 24380 | An open set is a subset of... |
| mopntopon 24381 | The set of open sets of a ... |
| mopntop 24382 | The set of open sets of a ... |
| mopnuni 24383 | The union of all open sets... |
| elmopn 24384 | The defining property of a... |
| mopnfss 24385 | The family of open sets of... |
| mopnm 24386 | The base set of a metric s... |
| elmopn2 24387 | A defining property of an ... |
| mopnss 24388 | An open set of a metric sp... |
| isxms 24389 | Express the predicate " ` ... |
| isxms2 24390 | Express the predicate " ` ... |
| isms 24391 | Express the predicate " ` ... |
| isms2 24392 | Express the predicate " ` ... |
| xmstopn 24393 | The topology component of ... |
| mstopn 24394 | The topology component of ... |
| xmstps 24395 | An extended metric space i... |
| msxms 24396 | A metric space is an exten... |
| mstps 24397 | A metric space is a topolo... |
| xmsxmet 24398 | The distance function, sui... |
| msmet 24399 | The distance function, sui... |
| msf 24400 | The distance function of a... |
| xmsxmet2 24401 | The distance function, sui... |
| msmet2 24402 | The distance function, sui... |
| mscl 24403 | Closure of the distance fu... |
| xmscl 24404 | Closure of the distance fu... |
| xmsge0 24405 | The distance function in a... |
| xmseq0 24406 | The distance between two p... |
| xmssym 24407 | The distance function in a... |
| xmstri2 24408 | Triangle inequality for th... |
| mstri2 24409 | Triangle inequality for th... |
| xmstri 24410 | Triangle inequality for th... |
| mstri 24411 | Triangle inequality for th... |
| xmstri3 24412 | Triangle inequality for th... |
| mstri3 24413 | Triangle inequality for th... |
| msrtri 24414 | Reverse triangle inequalit... |
| xmspropd 24415 | Property deduction for an ... |
| mspropd 24416 | Property deduction for a m... |
| setsmsbas 24417 | The base set of a construc... |
| setsmsds 24418 | The distance function of a... |
| setsmstset 24419 | The topology of a construc... |
| setsmstopn 24420 | The topology of a construc... |
| setsxms 24421 | The constructed metric spa... |
| setsms 24422 | The constructed metric spa... |
| tmsval 24423 | For any metric there is an... |
| tmslem 24424 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24425 | The base set of a construc... |
| tmsds 24426 | The metric of a constructe... |
| tmstopn 24427 | The topology of a construc... |
| tmsxms 24428 | The constructed metric spa... |
| tmsms 24429 | The constructed metric spa... |
| imasf1obl 24430 | The image of a metric spac... |
| imasf1oxms 24431 | The image of a metric spac... |
| imasf1oms 24432 | The image of a metric spac... |
| prdsbl 24433 | A ball in the product metr... |
| mopni 24434 | An open set of a metric sp... |
| mopni2 24435 | An open set of a metric sp... |
| mopni3 24436 | An open set of a metric sp... |
| blssopn 24437 | The balls of a metric spac... |
| unimopn 24438 | The union of a collection ... |
| mopnin 24439 | The intersection of two op... |
| mopn0 24440 | The empty set is an open s... |
| rnblopn 24441 | A ball of a metric space i... |
| blopn 24442 | A ball of a metric space i... |
| neibl 24443 | The neighborhoods around a... |
| blnei 24444 | A ball around a point is a... |
| lpbl 24445 | Every ball around a limit ... |
| blsscls2 24446 | A smaller closed ball is c... |
| blcld 24447 | A "closed ball" in a metri... |
| blcls 24448 | The closure of an open bal... |
| blsscls 24449 | If two concentric balls ha... |
| metss 24450 | Two ways of saying that me... |
| metequiv 24451 | Two ways of saying that tw... |
| metequiv2 24452 | If there is a sequence of ... |
| metss2lem 24453 | Lemma for ~ metss2 . (Con... |
| metss2 24454 | If the metric ` D ` is "st... |
| comet 24455 | The composition of an exte... |
| stdbdmetval 24456 | Value of the standard boun... |
| stdbdxmet 24457 | The standard bounded metri... |
| stdbdmet 24458 | The standard bounded metri... |
| stdbdbl 24459 | The standard bounded metri... |
| stdbdmopn 24460 | The standard bounded metri... |
| mopnex 24461 | The topology generated by ... |
| methaus 24462 | The topology generated by ... |
| met1stc 24463 | The topology generated by ... |
| met2ndci 24464 | A separable metric space (... |
| met2ndc 24465 | A metric space is second-c... |
| metrest 24466 | Two alternate formulations... |
| ressxms 24467 | The restriction of a metri... |
| ressms 24468 | The restriction of a metri... |
| prdsmslem1 24469 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24470 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24471 | Lemma for ~ prdsxms . The... |
| prdsxms 24472 | The indexed product struct... |
| prdsms 24473 | The indexed product struct... |
| pwsxms 24474 | A power of an extended met... |
| pwsms 24475 | A power of a metric space ... |
| xpsxms 24476 | A binary product of metric... |
| xpsms 24477 | A binary product of metric... |
| tmsxps 24478 | Express the product of two... |
| tmsxpsmopn 24479 | Express the product of two... |
| tmsxpsval 24480 | Value of the product of tw... |
| tmsxpsval2 24481 | Value of the product of tw... |
| metcnp3 24482 | Two ways to express that `... |
| metcnp 24483 | Two ways to say a mapping ... |
| metcnp2 24484 | Two ways to say a mapping ... |
| metcn 24485 | Two ways to say a mapping ... |
| metcnpi 24486 | Epsilon-delta property of ... |
| metcnpi2 24487 | Epsilon-delta property of ... |
| metcnpi3 24488 | Epsilon-delta property of ... |
| txmetcnp 24489 | Continuity of a binary ope... |
| txmetcn 24490 | Continuity of a binary ope... |
| metuval 24491 | Value of the uniform struc... |
| metustel 24492 | Define a filter base ` F `... |
| metustss 24493 | Range of the elements of t... |
| metustrel 24494 | Elements of the filter bas... |
| metustto 24495 | Any two elements of the fi... |
| metustid 24496 | The identity diagonal is i... |
| metustsym 24497 | Elements of the filter bas... |
| metustexhalf 24498 | For any element ` A ` of t... |
| metustfbas 24499 | The filter base generated ... |
| metust 24500 | The uniform structure gene... |
| cfilucfil 24501 | Given a metric ` D ` and a... |
| metuust 24502 | The uniform structure gene... |
| cfilucfil2 24503 | Given a metric ` D ` and a... |
| blval2 24504 | The ball around a point ` ... |
| elbl4 24505 | Membership in a ball, alte... |
| metuel 24506 | Elementhood in the uniform... |
| metuel2 24507 | Elementhood in the uniform... |
| metustbl 24508 | The "section" image of an ... |
| psmetutop 24509 | The topology induced by a ... |
| xmetutop 24510 | The topology induced by a ... |
| xmsusp 24511 | If the uniform set of a me... |
| restmetu 24512 | The uniform structure gene... |
| metucn 24513 | Uniform continuity in metr... |
| dscmet 24514 | The discrete metric on any... |
| dscopn 24515 | The discrete metric genera... |
| nrmmetd 24516 | Show that a group norm gen... |
| abvmet 24517 | An absolute value ` F ` ge... |
| nmfval 24530 | The value of the norm func... |
| nmval 24531 | The value of the norm as t... |
| nmfval0 24532 | The value of the norm func... |
| nmfval2 24533 | The value of the norm func... |
| nmval2 24534 | The value of the norm on a... |
| nmf2 24535 | The norm on a metric group... |
| nmpropd 24536 | Weak property deduction fo... |
| nmpropd2 24537 | Strong property deduction ... |
| isngp 24538 | The property of being a no... |
| isngp2 24539 | The property of being a no... |
| isngp3 24540 | The property of being a no... |
| ngpgrp 24541 | A normed group is a group.... |
| ngpms 24542 | A normed group is a metric... |
| ngpxms 24543 | A normed group is an exten... |
| ngptps 24544 | A normed group is a topolo... |
| ngpmet 24545 | The (induced) metric of a ... |
| ngpds 24546 | Value of the distance func... |
| ngpdsr 24547 | Value of the distance func... |
| ngpds2 24548 | Write the distance between... |
| ngpds2r 24549 | Write the distance between... |
| ngpds3 24550 | Write the distance between... |
| ngpds3r 24551 | Write the distance between... |
| ngprcan 24552 | Cancel right addition insi... |
| ngplcan 24553 | Cancel left addition insid... |
| isngp4 24554 | Express the property of be... |
| ngpinvds 24555 | Two elements are the same ... |
| ngpsubcan 24556 | Cancel right subtraction i... |
| nmf 24557 | The norm on a normed group... |
| nmcl 24558 | The norm of a normed group... |
| nmge0 24559 | The norm of a normed group... |
| nmeq0 24560 | The identity is the only e... |
| nmne0 24561 | The norm of a nonzero elem... |
| nmrpcl 24562 | The norm of a nonzero elem... |
| nminv 24563 | The norm of a negated elem... |
| nmmtri 24564 | The triangle inequality fo... |
| nmsub 24565 | The norm of the difference... |
| nmrtri 24566 | Reverse triangle inequalit... |
| nm2dif 24567 | Inequality for the differe... |
| nmtri 24568 | The triangle inequality fo... |
| nmtri2 24569 | Triangle inequality for th... |
| ngpi 24570 | The properties of a normed... |
| nm0 24571 | Norm of the identity eleme... |
| nmgt0 24572 | The norm of a nonzero elem... |
| sgrim 24573 | The induced metric on a su... |
| sgrimval 24574 | The induced metric on a su... |
| subgnm 24575 | The norm in a subgroup. (... |
| subgnm2 24576 | A substructure assigns the... |
| subgngp 24577 | A normed group restricted ... |
| ngptgp 24578 | A normed abelian group is ... |
| ngppropd 24579 | Property deduction for a n... |
| reldmtng 24580 | The function ` toNrmGrp ` ... |
| tngval 24581 | Value of the function whic... |
| tnglem 24582 | Lemma for ~ tngbas and sim... |
| tngbas 24583 | The base set of a structur... |
| tngplusg 24584 | The group addition of a st... |
| tng0 24585 | The group identity of a st... |
| tngmulr 24586 | The ring multiplication of... |
| tngsca 24587 | The scalar ring of a struc... |
| tngvsca 24588 | The scalar multiplication ... |
| tngip 24589 | The inner product operatio... |
| tngds 24590 | The metric function of a s... |
| tngtset 24591 | The topology generated by ... |
| tngtopn 24592 | The topology generated by ... |
| tngnm 24593 | The topology generated by ... |
| tngngp2 24594 | A norm turns a group into ... |
| tngngpd 24595 | Derive the axioms for a no... |
| tngngp 24596 | Derive the axioms for a no... |
| tnggrpr 24597 | If a structure equipped wi... |
| tngngp3 24598 | Alternate definition of a ... |
| nrmtngdist 24599 | The augmentation of a norm... |
| nrmtngnrm 24600 | The augmentation of a norm... |
| tngngpim 24601 | The induced metric of a no... |
| isnrg 24602 | A normed ring is a ring wi... |
| nrgabv 24603 | The norm of a normed ring ... |
| nrgngp 24604 | A normed ring is a normed ... |
| nrgring 24605 | A normed ring is a ring. ... |
| nmmul 24606 | The norm of a product in a... |
| nrgdsdi 24607 | Distribute a distance calc... |
| nrgdsdir 24608 | Distribute a distance calc... |
| nm1 24609 | The norm of one in a nonze... |
| unitnmn0 24610 | The norm of a unit is nonz... |
| nminvr 24611 | The norm of an inverse in ... |
| nmdvr 24612 | The norm of a division in ... |
| nrgdomn 24613 | A nonzero normed ring is a... |
| nrgtgp 24614 | A normed ring is a topolog... |
| subrgnrg 24615 | A normed ring restricted t... |
| tngnrg 24616 | Given any absolute value o... |
| isnlm 24617 | A normed (left) module is ... |
| nmvs 24618 | Defining property of a nor... |
| nlmngp 24619 | A normed module is a norme... |
| nlmlmod 24620 | A normed module is a left ... |
| nlmnrg 24621 | The scalar component of a ... |
| nlmngp2 24622 | The scalar component of a ... |
| nlmdsdi 24623 | Distribute a distance calc... |
| nlmdsdir 24624 | Distribute a distance calc... |
| nlmmul0or 24625 | If a scalar product is zer... |
| sranlm 24626 | The subring algebra over a... |
| nlmvscnlem2 24627 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24628 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24629 | The scalar multiplication ... |
| rlmnlm 24630 | The ring module over a nor... |
| rlmnm 24631 | The norm function in the r... |
| nrgtrg 24632 | A normed ring is a topolog... |
| nrginvrcnlem 24633 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24634 | The ring inverse function ... |
| nrgtdrg 24635 | A normed division ring is ... |
| nlmtlm 24636 | A normed module is a topol... |
| isnvc 24637 | A normed vector space is j... |
| nvcnlm 24638 | A normed vector space is a... |
| nvclvec 24639 | A normed vector space is a... |
| nvclmod 24640 | A normed vector space is a... |
| isnvc2 24641 | A normed vector space is j... |
| nvctvc 24642 | A normed vector space is a... |
| lssnlm 24643 | A subspace of a normed mod... |
| lssnvc 24644 | A subspace of a normed vec... |
| rlmnvc 24645 | The ring module over a nor... |
| ngpocelbl 24646 | Membership of an off-cente... |
| nmoffn 24653 | The function producing ope... |
| reldmnghm 24654 | Lemma for normed group hom... |
| reldmnmhm 24655 | Lemma for module homomorph... |
| nmofval 24656 | Value of the operator norm... |
| nmoval 24657 | Value of the operator norm... |
| nmogelb 24658 | Property of the operator n... |
| nmolb 24659 | Any upper bound on the val... |
| nmolb2d 24660 | Any upper bound on the val... |
| nmof 24661 | The operator norm is a fun... |
| nmocl 24662 | The operator norm of an op... |
| nmoge0 24663 | The operator norm of an op... |
| nghmfval 24664 | A normed group homomorphis... |
| isnghm 24665 | A normed group homomorphis... |
| isnghm2 24666 | A normed group homomorphis... |
| isnghm3 24667 | A normed group homomorphis... |
| bddnghm 24668 | A bounded group homomorphi... |
| nghmcl 24669 | A normed group homomorphis... |
| nmoi 24670 | The operator norm achieves... |
| nmoix 24671 | The operator norm is a bou... |
| nmoi2 24672 | The operator norm is a bou... |
| nmoleub 24673 | The operator norm, defined... |
| nghmrcl1 24674 | Reverse closure for a norm... |
| nghmrcl2 24675 | Reverse closure for a norm... |
| nghmghm 24676 | A normed group homomorphis... |
| nmo0 24677 | The operator norm of the z... |
| nmoeq0 24678 | The operator norm is zero ... |
| nmoco 24679 | An upper bound on the oper... |
| nghmco 24680 | The composition of normed ... |
| nmotri 24681 | Triangle inequality for th... |
| nghmplusg 24682 | The sum of two bounded lin... |
| 0nghm 24683 | The zero operator is a nor... |
| nmoid 24684 | The operator norm of the i... |
| idnghm 24685 | The identity operator is a... |
| nmods 24686 | Upper bound for the distan... |
| nghmcn 24687 | A normed group homomorphis... |
| isnmhm 24688 | A normed module homomorphi... |
| nmhmrcl1 24689 | Reverse closure for a norm... |
| nmhmrcl2 24690 | Reverse closure for a norm... |
| nmhmlmhm 24691 | A normed module homomorphi... |
| nmhmnghm 24692 | A normed module homomorphi... |
| nmhmghm 24693 | A normed module homomorphi... |
| isnmhm2 24694 | A normed module homomorphi... |
| nmhmcl 24695 | A normed module homomorphi... |
| idnmhm 24696 | The identity operator is a... |
| 0nmhm 24697 | The zero operator is a bou... |
| nmhmco 24698 | The composition of bounded... |
| nmhmplusg 24699 | The sum of two bounded lin... |
| qtopbaslem 24700 | The set of open intervals ... |
| qtopbas 24701 | The set of open intervals ... |
| retopbas 24702 | A basis for the standard t... |
| retop 24703 | The standard topology on t... |
| uniretop 24704 | The underlying set of the ... |
| retopon 24705 | The standard topology on t... |
| retps 24706 | The standard topological s... |
| iooretop 24707 | Open intervals are open se... |
| icccld 24708 | Closed intervals are close... |
| icopnfcld 24709 | Right-unbounded closed int... |
| iocmnfcld 24710 | Left-unbounded closed inte... |
| qdensere 24711 | ` QQ ` is dense in the sta... |
| cnmetdval 24712 | Value of the distance func... |
| cnmet 24713 | The absolute value metric ... |
| cnxmet 24714 | The absolute value metric ... |
| cnbl0 24715 | Two ways to write the open... |
| cnblcld 24716 | Two ways to write the clos... |
| cnfldms 24717 | The complex number field i... |
| cnfldxms 24718 | The complex number field i... |
| cnfldtps 24719 | The complex number field i... |
| cnfldnm 24720 | The norm of the field of c... |
| cnngp 24721 | The complex numbers form a... |
| cnnrg 24722 | The complex numbers form a... |
| cnfldtopn 24723 | The topology of the comple... |
| cnfldtopon 24724 | The topology of the comple... |
| cnfldtop 24725 | The topology of the comple... |
| cnfldhaus 24726 | The topology of the comple... |
| unicntop 24727 | The underlying set of the ... |
| cnopn 24728 | The set of complex numbers... |
| cnn0opn 24729 | The set of nonzero complex... |
| zringnrg 24730 | The ring of integers is a ... |
| remetdval 24731 | Value of the distance func... |
| remet 24732 | The absolute value metric ... |
| rexmet 24733 | The absolute value metric ... |
| bl2ioo 24734 | A ball in terms of an open... |
| ioo2bl 24735 | An open interval of reals ... |
| ioo2blex 24736 | An open interval of reals ... |
| blssioo 24737 | The balls of the standard ... |
| tgioo 24738 | The topology generated by ... |
| qdensere2 24739 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24740 | An open ball in the comple... |
| rehaus 24741 | The standard topology on t... |
| tgqioo 24742 | The topology generated by ... |
| re2ndc 24743 | The standard topology on t... |
| resubmet 24744 | The subspace topology indu... |
| tgioo2 24745 | The standard topology on t... |
| rerest 24746 | The subspace topology indu... |
| tgioo4 24747 | The standard topology on t... |
| tgioo3 24748 | The standard topology on t... |
| xrtgioo 24749 | The topology on the extend... |
| xrrest 24750 | The subspace topology indu... |
| xrrest2 24751 | The subspace topology indu... |
| xrsxmet 24752 | The metric on the extended... |
| xrsdsre 24753 | The metric on the extended... |
| xrsblre 24754 | Any ball of the metric of ... |
| xrsmopn 24755 | The metric on the extended... |
| zcld 24756 | The integers are a closed ... |
| recld2 24757 | The real numbers are a clo... |
| zcld2 24758 | The integers are a closed ... |
| zdis 24759 | The integers are a discret... |
| sszcld 24760 | Every subset of the intege... |
| reperflem 24761 | A subset of the real numbe... |
| reperf 24762 | The real numbers are a per... |
| cnperf 24763 | The complex numbers are a ... |
| iccntr 24764 | The interior of a closed i... |
| icccmplem1 24765 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24766 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24767 | Lemma for ~ icccmp . (Con... |
| icccmp 24768 | A closed interval in ` RR ... |
| reconnlem1 24769 | Lemma for ~ reconn . Conn... |
| reconnlem2 24770 | Lemma for ~ reconn . (Con... |
| reconn 24771 | A subset of the reals is c... |
| retopconn 24772 | Corollary of ~ reconn . T... |
| iccconn 24773 | A closed interval is conne... |
| opnreen 24774 | Every nonempty open set is... |
| rectbntr0 24775 | A countable subset of the ... |
| xrge0gsumle 24776 | A finite sum in the nonneg... |
| xrge0tsms 24777 | Any finite or infinite sum... |
| xrge0tsms2 24778 | Any finite or infinite sum... |
| metdcnlem 24779 | The metric function of a m... |
| xmetdcn2 24780 | The metric function of an ... |
| xmetdcn 24781 | The metric function of an ... |
| metdcn2 24782 | The metric function of a m... |
| metdcn 24783 | The metric function of a m... |
| msdcn 24784 | The metric function of a m... |
| cnmpt1ds 24785 | Continuity of the metric f... |
| cnmpt2ds 24786 | Continuity of the metric f... |
| nmcn 24787 | The norm of a normed group... |
| ngnmcncn 24788 | The norm of a normed group... |
| abscn 24789 | The absolute value functio... |
| metdsval 24790 | Value of the "distance to ... |
| metdsf 24791 | The distance from a point ... |
| metdsge 24792 | The distance from the poin... |
| metds0 24793 | If a point is in a set, it... |
| metdstri 24794 | A generalization of the tr... |
| metdsle 24795 | The distance from a point ... |
| metdsre 24796 | The distance from a point ... |
| metdseq0 24797 | The distance from a point ... |
| metdscnlem 24798 | Lemma for ~ metdscn . (Co... |
| metdscn 24799 | The function ` F ` which g... |
| metdscn2 24800 | The function ` F ` which g... |
| metnrmlem1a 24801 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24802 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24803 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24804 | Lemma for ~ metnrm . (Con... |
| metnrm 24805 | A metric space is normal. ... |
| metreg 24806 | A metric space is regular.... |
| addcnlem 24807 | Lemma for ~ addcn , ~ subc... |
| addcn 24808 | Complex number addition is... |
| subcn 24809 | Complex number subtraction... |
| mulcn 24810 | Complex number multiplicat... |
| divcnOLD 24811 | Obsolete version of ~ divc... |
| mpomulcn 24812 | Complex number multiplicat... |
| divcn 24813 | Complex number division is... |
| cnfldtgp 24814 | The complex numbers form a... |
| fsumcn 24815 | A finite sum of functions ... |
| fsum2cn 24816 | Version of ~ fsumcn for tw... |
| expcn 24817 | The power function on comp... |
| divccn 24818 | Division by a nonzero cons... |
| expcnOLD 24819 | Obsolete version of ~ expc... |
| divccnOLD 24820 | Obsolete version of ~ divc... |
| sqcn 24821 | The square function on com... |
| iitopon 24826 | The unit interval is a top... |
| iitop 24827 | The unit interval is a top... |
| iiuni 24828 | The base set of the unit i... |
| dfii2 24829 | Alternate definition of th... |
| dfii3 24830 | Alternate definition of th... |
| dfii4 24831 | Alternate definition of th... |
| dfii5 24832 | The unit interval expresse... |
| iicmp 24833 | The unit interval is compa... |
| iiconn 24834 | The unit interval is conne... |
| cncfval 24835 | The value of the continuou... |
| elcncf 24836 | Membership in the set of c... |
| elcncf2 24837 | Version of ~ elcncf with a... |
| cncfrss 24838 | Reverse closure of the con... |
| cncfrss2 24839 | Reverse closure of the con... |
| cncff 24840 | A continuous complex funct... |
| cncfi 24841 | Defining property of a con... |
| elcncf1di 24842 | Membership in the set of c... |
| elcncf1ii 24843 | Membership in the set of c... |
| rescncf 24844 | A continuous complex funct... |
| cncfcdm 24845 | Change the codomain of a c... |
| cncfss 24846 | The set of continuous func... |
| climcncf 24847 | Image of a limit under a c... |
| abscncf 24848 | Absolute value is continuo... |
| recncf 24849 | Real part is continuous. ... |
| imcncf 24850 | Imaginary part is continuo... |
| cjcncf 24851 | Complex conjugate is conti... |
| mulc1cncf 24852 | Multiplication by a consta... |
| divccncf 24853 | Division by a constant is ... |
| cncfco 24854 | The composition of two con... |
| cncfcompt2 24855 | Composition of continuous ... |
| cncfmet 24856 | Relate complex function co... |
| cncfcn 24857 | Relate complex function co... |
| cncfcn1 24858 | Relate complex function co... |
| cncfmptc 24859 | A constant function is a c... |
| cncfmptid 24860 | The identity function is a... |
| cncfmpt1f 24861 | Composition of continuous ... |
| cncfmpt2f 24862 | Composition of continuous ... |
| cncfmpt2ss 24863 | Composition of continuous ... |
| addccncf 24864 | Adding a constant is a con... |
| idcncf 24865 | The identity function is a... |
| sub1cncf 24866 | Subtracting a constant is ... |
| sub2cncf 24867 | Subtraction from a constan... |
| cdivcncf 24868 | Division with a constant n... |
| negcncf 24869 | The negative function is c... |
| negcncfOLD 24870 | Obsolete version of ~ negc... |
| negfcncf 24871 | The negative of a continuo... |
| abscncfALT 24872 | Absolute value is continuo... |
| cncfcnvcn 24873 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24874 | The power function on comp... |
| cnmptre 24875 | Lemma for ~ iirevcn and re... |
| cnmpopc 24876 | Piecewise definition of a ... |
| iirev 24877 | Reverse the unit interval.... |
| iirevcn 24878 | The reversion function is ... |
| iihalf1 24879 | Map the first half of ` II... |
| iihalf1cn 24880 | The first half function is... |
| iihalf1cnOLD 24881 | Obsolete version of ~ iiha... |
| iihalf2 24882 | Map the second half of ` I... |
| iihalf2cn 24883 | The second half function i... |
| iihalf2cnOLD 24884 | Obsolete version of ~ iiha... |
| elii1 24885 | Divide the unit interval i... |
| elii2 24886 | Divide the unit interval i... |
| iimulcl 24887 | The unit interval is close... |
| iimulcn 24888 | Multiplication is a contin... |
| iimulcnOLD 24889 | Obsolete version of ~ iimu... |
| icoopnst 24890 | A half-open interval start... |
| iocopnst 24891 | A half-open interval endin... |
| icchmeo 24892 | The natural bijection from... |
| icchmeoOLD 24893 | Obsolete version of ~ icch... |
| icopnfcnv 24894 | Define a bijection from ` ... |
| icopnfhmeo 24895 | The defined bijection from... |
| iccpnfcnv 24896 | Define a bijection from ` ... |
| iccpnfhmeo 24897 | The defined bijection from... |
| xrhmeo 24898 | The bijection from ` [ -u ... |
| xrhmph 24899 | The extended reals are hom... |
| xrcmp 24900 | The topology of the extend... |
| xrconn 24901 | The topology of the extend... |
| icccvx 24902 | A linear combination of tw... |
| oprpiece1res1 24903 | Restriction to the first p... |
| oprpiece1res2 24904 | Restriction to the second ... |
| cnrehmeo 24905 | The canonical bijection fr... |
| cnrehmeoOLD 24906 | Obsolete version of ~ cnre... |
| cnheiborlem 24907 | Lemma for ~ cnheibor . (C... |
| cnheibor 24908 | Heine-Borel theorem for co... |
| cnllycmp 24909 | The topology on the comple... |
| rellycmp 24910 | The topology on the reals ... |
| bndth 24911 | The Boundedness Theorem. ... |
| evth 24912 | The Extreme Value Theorem.... |
| evth2 24913 | The Extreme Value Theorem,... |
| lebnumlem1 24914 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24915 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24916 | Lemma for ~ lebnum . By t... |
| lebnum 24917 | The Lebesgue number lemma,... |
| xlebnum 24918 | Generalize ~ lebnum to ext... |
| lebnumii 24919 | Specialize the Lebesgue nu... |
| ishtpy 24925 | Membership in the class of... |
| htpycn 24926 | A homotopy is a continuous... |
| htpyi 24927 | A homotopy evaluated at it... |
| ishtpyd 24928 | Deduction for membership i... |
| htpycom 24929 | Given a homotopy from ` F ... |
| htpyid 24930 | A homotopy from a function... |
| htpyco1 24931 | Compose a homotopy with a ... |
| htpyco2 24932 | Compose a homotopy with a ... |
| htpycc 24933 | Concatenate two homotopies... |
| isphtpy 24934 | Membership in the class of... |
| phtpyhtpy 24935 | A path homotopy is a homot... |
| phtpycn 24936 | A path homotopy is a conti... |
| phtpyi 24937 | Membership in the class of... |
| phtpy01 24938 | Two path-homotopic paths h... |
| isphtpyd 24939 | Deduction for membership i... |
| isphtpy2d 24940 | Deduction for membership i... |
| phtpycom 24941 | Given a homotopy from ` F ... |
| phtpyid 24942 | A homotopy from a path to ... |
| phtpyco2 24943 | Compose a path homotopy wi... |
| phtpycc 24944 | Concatenate two path homot... |
| phtpcrel 24946 | The path homotopy relation... |
| isphtpc 24947 | The relation "is path homo... |
| phtpcer 24948 | Path homotopy is an equiva... |
| phtpc01 24949 | Path homotopic paths have ... |
| reparphti 24950 | Lemma for ~ reparpht . (C... |
| reparphtiOLD 24951 | Obsolete version of ~ repa... |
| reparpht 24952 | Reparametrization lemma. ... |
| phtpcco2 24953 | Compose a path homotopy wi... |
| pcofval 24964 | The value of the path conc... |
| pcoval 24965 | The concatenation of two p... |
| pcovalg 24966 | Evaluate the concatenation... |
| pcoval1 24967 | Evaluate the concatenation... |
| pco0 24968 | The starting point of a pa... |
| pco1 24969 | The ending point of a path... |
| pcoval2 24970 | Evaluate the concatenation... |
| pcocn 24971 | The concatenation of two p... |
| copco 24972 | The composition of a conca... |
| pcohtpylem 24973 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24974 | Homotopy invariance of pat... |
| pcoptcl 24975 | A constant function is a p... |
| pcopt 24976 | Concatenation with a point... |
| pcopt2 24977 | Concatenation with a point... |
| pcoass 24978 | Order of concatenation doe... |
| pcorevcl 24979 | Closure for a reversed pat... |
| pcorevlem 24980 | Lemma for ~ pcorev . Prov... |
| pcorev 24981 | Concatenation with the rev... |
| pcorev2 24982 | Concatenation with the rev... |
| pcophtb 24983 | The path homotopy equivale... |
| om1val 24984 | The definition of the loop... |
| om1bas 24985 | The base set of the loop s... |
| om1elbas 24986 | Elementhood in the base se... |
| om1addcl 24987 | Closure of the group opera... |
| om1plusg 24988 | The group operation (which... |
| om1tset 24989 | The topology of the loop s... |
| om1opn 24990 | The topology of the loop s... |
| pi1val 24991 | The definition of the fund... |
| pi1bas 24992 | The base set of the fundam... |
| pi1blem 24993 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24994 | Another way to write the l... |
| pi1bas2 24995 | The base set of the fundam... |
| pi1eluni 24996 | Elementhood in the base se... |
| pi1bas3 24997 | The base set of the fundam... |
| pi1cpbl 24998 | The group operation, loop ... |
| elpi1 24999 | The elements of the fundam... |
| elpi1i 25000 | The elements of the fundam... |
| pi1addf 25001 | The group operation of ` p... |
| pi1addval 25002 | The concatenation of two p... |
| pi1grplem 25003 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25004 | The fundamental group is a... |
| pi1id 25005 | The identity element of th... |
| pi1inv 25006 | An inverse in the fundamen... |
| pi1xfrf 25007 | Functionality of the loop ... |
| pi1xfrval 25008 | The value of the loop tran... |
| pi1xfr 25009 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25010 | Given a path ` F ` between... |
| pi1xfrcnv 25011 | Given a path ` F ` between... |
| pi1xfrgim 25012 | The mapping ` G ` between ... |
| pi1cof 25013 | Functionality of the loop ... |
| pi1coval 25014 | The value of the loop tran... |
| pi1coghm 25015 | The mapping ` G ` between ... |
| isclm 25018 | A subcomplex module is a l... |
| clmsca 25019 | The ring of scalars ` F ` ... |
| clmsubrg 25020 | The base set of the ring o... |
| clmlmod 25021 | A subcomplex module is a l... |
| clmgrp 25022 | A subcomplex module is an ... |
| clmabl 25023 | A subcomplex module is an ... |
| clmring 25024 | The scalar ring of a subco... |
| clmfgrp 25025 | The scalar ring of a subco... |
| clm0 25026 | The zero of the scalar rin... |
| clm1 25027 | The identity of the scalar... |
| clmadd 25028 | The addition of the scalar... |
| clmmul 25029 | The multiplication of the ... |
| clmcj 25030 | The conjugation of the sca... |
| isclmi 25031 | Reverse direction of ~ isc... |
| clmzss 25032 | The scalar ring of a subco... |
| clmsscn 25033 | The scalar ring of a subco... |
| clmsub 25034 | Subtraction in the scalar ... |
| clmneg 25035 | Negation in the scalar rin... |
| clmneg1 25036 | Minus one is in the scalar... |
| clmabs 25037 | Norm in the scalar ring of... |
| clmacl 25038 | Closure of ring addition f... |
| clmmcl 25039 | Closure of ring multiplica... |
| clmsubcl 25040 | Closure of ring subtractio... |
| lmhmclm 25041 | The domain of a linear ope... |
| clmvscl 25042 | Closure of scalar product ... |
| clmvsass 25043 | Associative law for scalar... |
| clmvscom 25044 | Commutative law for the sc... |
| clmvsdir 25045 | Distributive law for scala... |
| clmvsdi 25046 | Distributive law for scala... |
| clmvs1 25047 | Scalar product with ring u... |
| clmvs2 25048 | A vector plus itself is tw... |
| clm0vs 25049 | Zero times a vector is the... |
| clmopfne 25050 | The (functionalized) opera... |
| isclmp 25051 | The predicate "is a subcom... |
| isclmi0 25052 | Properties that determine ... |
| clmvneg1 25053 | Minus 1 times a vector is ... |
| clmvsneg 25054 | Multiplication of a vector... |
| clmmulg 25055 | The group multiple functio... |
| clmsubdir 25056 | Scalar multiplication dist... |
| clmpm1dir 25057 | Subtractive distributive l... |
| clmnegneg 25058 | Double negative of a vecto... |
| clmnegsubdi2 25059 | Distribution of negative o... |
| clmsub4 25060 | Rearrangement of 4 terms i... |
| clmvsrinv 25061 | A vector minus itself. (C... |
| clmvslinv 25062 | Minus a vector plus itself... |
| clmvsubval 25063 | Value of vector subtractio... |
| clmvsubval2 25064 | Value of vector subtractio... |
| clmvz 25065 | Two ways to express the ne... |
| zlmclm 25066 | The ` ZZ ` -module operati... |
| clmzlmvsca 25067 | The scalar product of a su... |
| nmoleub2lem 25068 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25069 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25070 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25071 | The operator norm is the s... |
| nmoleub2b 25072 | The operator norm is the s... |
| nmoleub3 25073 | The operator norm is the s... |
| nmhmcn 25074 | A linear operator over a n... |
| cmodscexp 25075 | The powers of ` _i ` belon... |
| cmodscmulexp 25076 | The scalar product of a ve... |
| cvslvec 25079 | A subcomplex vector space ... |
| cvsclm 25080 | A subcomplex vector space ... |
| iscvs 25081 | A subcomplex vector space ... |
| iscvsp 25082 | The predicate "is a subcom... |
| iscvsi 25083 | Properties that determine ... |
| cvsi 25084 | The properties of a subcom... |
| cvsunit 25085 | Unit group of the scalar r... |
| cvsdiv 25086 | Division of the scalar rin... |
| cvsdivcl 25087 | The scalar field of a subc... |
| cvsmuleqdivd 25088 | An equality involving rati... |
| cvsdiveqd 25089 | An equality involving rati... |
| cnlmodlem1 25090 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25091 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25092 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25093 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25094 | The set of complex numbers... |
| cnstrcvs 25095 | The set of complex numbers... |
| cnrbas 25096 | The set of complex numbers... |
| cnrlmod 25097 | The complex left module of... |
| cnrlvec 25098 | The complex left module of... |
| cncvs 25099 | The complex left module of... |
| recvs 25100 | The field of the real numb... |
| qcvs 25101 | The field of rational numb... |
| zclmncvs 25102 | The ring of integers as le... |
| isncvsngp 25103 | A normed subcomplex vector... |
| isncvsngpd 25104 | Properties that determine ... |
| ncvsi 25105 | The properties of a normed... |
| ncvsprp 25106 | Proportionality property o... |
| ncvsge0 25107 | The norm of a scalar produ... |
| ncvsm1 25108 | The norm of the opposite o... |
| ncvsdif 25109 | The norm of the difference... |
| ncvspi 25110 | The norm of a vector plus ... |
| ncvs1 25111 | From any nonzero vector of... |
| cnrnvc 25112 | The module of complex numb... |
| cnncvs 25113 | The module of complex numb... |
| cnnm 25114 | The norm of the normed sub... |
| ncvspds 25115 | Value of the distance func... |
| cnindmet 25116 | The metric induced on the ... |
| cnncvsaddassdemo 25117 | Derive the associative law... |
| cnncvsmulassdemo 25118 | Derive the associative law... |
| cnncvsabsnegdemo 25119 | Derive the absolute value ... |
| iscph 25124 | A subcomplex pre-Hilbert s... |
| cphphl 25125 | A subcomplex pre-Hilbert s... |
| cphnlm 25126 | A subcomplex pre-Hilbert s... |
| cphngp 25127 | A subcomplex pre-Hilbert s... |
| cphlmod 25128 | A subcomplex pre-Hilbert s... |
| cphlvec 25129 | A subcomplex pre-Hilbert s... |
| cphnvc 25130 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25131 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25132 | Lemma for ~ cphreccl . (C... |
| cphsca 25133 | A subcomplex pre-Hilbert s... |
| cphsubrg 25134 | The scalar field of a subc... |
| cphreccl 25135 | The scalar field of a subc... |
| cphdivcl 25136 | The scalar field of a subc... |
| cphcjcl 25137 | The scalar field of a subc... |
| cphsqrtcl 25138 | The scalar field of a subc... |
| cphabscl 25139 | The scalar field of a subc... |
| cphsqrtcl2 25140 | The scalar field of a subc... |
| cphsqrtcl3 25141 | If the scalar field of a s... |
| cphqss 25142 | The scalar field of a subc... |
| cphclm 25143 | A subcomplex pre-Hilbert s... |
| cphnmvs 25144 | Norm of a scalar product. ... |
| cphipcl 25145 | An inner product is a memb... |
| cphnmfval 25146 | The value of the norm in a... |
| cphnm 25147 | The square of the norm is ... |
| nmsq 25148 | The square of the norm is ... |
| cphnmf 25149 | The norm of a vector is a ... |
| cphnmcl 25150 | The norm of a vector is a ... |
| reipcl 25151 | An inner product of an ele... |
| ipge0 25152 | The inner product in a sub... |
| cphipcj 25153 | Conjugate of an inner prod... |
| cphipipcj 25154 | An inner product times its... |
| cphorthcom 25155 | Orthogonality (meaning inn... |
| cphip0l 25156 | Inner product with a zero ... |
| cphip0r 25157 | Inner product with a zero ... |
| cphipeq0 25158 | The inner product of a vec... |
| cphdir 25159 | Distributive law for inner... |
| cphdi 25160 | Distributive law for inner... |
| cph2di 25161 | Distributive law for inner... |
| cphsubdir 25162 | Distributive law for inner... |
| cphsubdi 25163 | Distributive law for inner... |
| cph2subdi 25164 | Distributive law for inner... |
| cphass 25165 | Associative law for inner ... |
| cphassr 25166 | "Associative" law for seco... |
| cph2ass 25167 | Move scalar multiplication... |
| cphassi 25168 | Associative law for the fi... |
| cphassir 25169 | "Associative" law for the ... |
| cphpyth 25170 | The pythagorean theorem fo... |
| tcphex 25171 | Lemma for ~ tcphbas and si... |
| tcphval 25172 | Define a function to augme... |
| tcphbas 25173 | The base set of a subcompl... |
| tchplusg 25174 | The addition operation of ... |
| tcphsub 25175 | The subtraction operation ... |
| tcphmulr 25176 | The ring operation of a su... |
| tcphsca 25177 | The scalar field of a subc... |
| tcphvsca 25178 | The scalar multiplication ... |
| tcphip 25179 | The inner product of a sub... |
| tcphtopn 25180 | The topology of a subcompl... |
| tcphphl 25181 | Augmentation of a subcompl... |
| tchnmfval 25182 | The norm of a subcomplex p... |
| tcphnmval 25183 | The norm of a subcomplex p... |
| cphtcphnm 25184 | The norm of a norm-augment... |
| tcphds 25185 | The distance of a pre-Hilb... |
| phclm 25186 | A pre-Hilbert space whose ... |
| tcphcphlem3 25187 | Lemma for ~ tcphcph : real... |
| ipcau2 25188 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25189 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25190 | Lemma for ~ tcphcph : homo... |
| tcphcph 25191 | The standard definition of... |
| ipcau 25192 | The Cauchy-Schwarz inequal... |
| nmparlem 25193 | Lemma for ~ nmpar . (Cont... |
| nmpar 25194 | A subcomplex pre-Hilbert s... |
| cphipval2 25195 | Value of the inner product... |
| 4cphipval2 25196 | Four times the inner produ... |
| cphipval 25197 | Value of the inner product... |
| ipcnlem2 25198 | The inner product operatio... |
| ipcnlem1 25199 | The inner product operatio... |
| ipcn 25200 | The inner product operatio... |
| cnmpt1ip 25201 | Continuity of inner produc... |
| cnmpt2ip 25202 | Continuity of inner produc... |
| csscld 25203 | A "closed subspace" in a s... |
| clsocv 25204 | The orthogonal complement ... |
| cphsscph 25205 | A subspace of a subcomplex... |
| lmmbr 25212 | Express the binary relatio... |
| lmmbr2 25213 | Express the binary relatio... |
| lmmbr3 25214 | Express the binary relatio... |
| lmmcvg 25215 | Convergence property of a ... |
| lmmbrf 25216 | Express the binary relatio... |
| lmnn 25217 | A condition that implies c... |
| cfilfval 25218 | The set of Cauchy filters ... |
| iscfil 25219 | The property of being a Ca... |
| iscfil2 25220 | The property of being a Ca... |
| cfilfil 25221 | A Cauchy filter is a filte... |
| cfili 25222 | Property of a Cauchy filte... |
| cfil3i 25223 | A Cauchy filter contains b... |
| cfilss 25224 | A filter finer than a Cauc... |
| fgcfil 25225 | The Cauchy filter conditio... |
| fmcfil 25226 | The Cauchy filter conditio... |
| iscfil3 25227 | A filter is Cauchy iff it ... |
| cfilfcls 25228 | Similar to ultrafilters ( ... |
| caufval 25229 | The set of Cauchy sequence... |
| iscau 25230 | Express the property " ` F... |
| iscau2 25231 | Express the property " ` F... |
| iscau3 25232 | Express the Cauchy sequenc... |
| iscau4 25233 | Express the property " ` F... |
| iscauf 25234 | Express the property " ` F... |
| caun0 25235 | A metric with a Cauchy seq... |
| caufpm 25236 | Inclusion of a Cauchy sequ... |
| caucfil 25237 | A Cauchy sequence predicat... |
| iscmet 25238 | The property " ` D ` is a ... |
| cmetcvg 25239 | The convergence of a Cauch... |
| cmetmet 25240 | A complete metric space is... |
| cmetmeti 25241 | A complete metric space is... |
| cmetcaulem 25242 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25243 | The convergence of a Cauch... |
| iscmet3lem3 25244 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25245 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25246 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25247 | The property " ` D ` is a ... |
| iscmet2 25248 | A metric ` D ` is complete... |
| cfilresi 25249 | A Cauchy filter on a metri... |
| cfilres 25250 | Cauchy filter on a metric ... |
| caussi 25251 | Cauchy sequence on a metri... |
| causs 25252 | Cauchy sequence on a metri... |
| equivcfil 25253 | If the metric ` D ` is "st... |
| equivcau 25254 | If the metric ` D ` is "st... |
| lmle 25255 | If the distance from each ... |
| nglmle 25256 | If the norm of each member... |
| lmclim 25257 | Relate a limit on the metr... |
| lmclimf 25258 | Relate a limit on the metr... |
| metelcls 25259 | A point belongs to the clo... |
| metcld 25260 | A subset of a metric space... |
| metcld2 25261 | A subset of a metric space... |
| caubl 25262 | Sufficient condition to en... |
| caublcls 25263 | The convergent point of a ... |
| metcnp4 25264 | Two ways to say a mapping ... |
| metcn4 25265 | Two ways to say a mapping ... |
| iscmet3i 25266 | Properties that determine ... |
| lmcau 25267 | Every convergent sequence ... |
| flimcfil 25268 | Every convergent filter in... |
| metsscmetcld 25269 | A complete subspace of a m... |
| cmetss 25270 | A subspace of a complete m... |
| equivcmet 25271 | If two metrics are strongl... |
| relcmpcmet 25272 | If ` D ` is a metric space... |
| cmpcmet 25273 | A compact metric space is ... |
| cfilucfil3 25274 | Given a metric ` D ` and a... |
| cfilucfil4 25275 | Given a metric ` D ` and a... |
| cncmet 25276 | The set of complex numbers... |
| recmet 25277 | The real numbers are a com... |
| bcthlem1 25278 | Lemma for ~ bcth . Substi... |
| bcthlem2 25279 | Lemma for ~ bcth . The ba... |
| bcthlem3 25280 | Lemma for ~ bcth . The li... |
| bcthlem4 25281 | Lemma for ~ bcth . Given ... |
| bcthlem5 25282 | Lemma for ~ bcth . The pr... |
| bcth 25283 | Baire's Category Theorem. ... |
| bcth2 25284 | Baire's Category Theorem, ... |
| bcth3 25285 | Baire's Category Theorem, ... |
| isbn 25292 | A Banach space is a normed... |
| bnsca 25293 | The scalar field of a Bana... |
| bnnvc 25294 | A Banach space is a normed... |
| bnnlm 25295 | A Banach space is a normed... |
| bnngp 25296 | A Banach space is a normed... |
| bnlmod 25297 | A Banach space is a left m... |
| bncms 25298 | A Banach space is a comple... |
| iscms 25299 | A complete metric space is... |
| cmscmet 25300 | The induced metric on a co... |
| bncmet 25301 | The induced metric on Bana... |
| cmsms 25302 | A complete metric space is... |
| cmspropd 25303 | Property deduction for a c... |
| cmssmscld 25304 | The restriction of a metri... |
| cmsss 25305 | The restriction of a compl... |
| lssbn 25306 | A subspace of a Banach spa... |
| cmetcusp1 25307 | If the uniform set of a co... |
| cmetcusp 25308 | The uniform space generate... |
| cncms 25309 | The field of complex numbe... |
| cnflduss 25310 | The uniform structure of t... |
| cnfldcusp 25311 | The field of complex numbe... |
| resscdrg 25312 | The real numbers are a sub... |
| cncdrg 25313 | The only complete subfield... |
| srabn 25314 | The subring algebra over a... |
| rlmbn 25315 | The ring module over a com... |
| ishl 25316 | The predicate "is a subcom... |
| hlbn 25317 | Every subcomplex Hilbert s... |
| hlcph 25318 | Every subcomplex Hilbert s... |
| hlphl 25319 | Every subcomplex Hilbert s... |
| hlcms 25320 | Every subcomplex Hilbert s... |
| hlprlem 25321 | Lemma for ~ hlpr . (Contr... |
| hlress 25322 | The scalar field of a subc... |
| hlpr 25323 | The scalar field of a subc... |
| ishl2 25324 | A Hilbert space is a compl... |
| cphssphl 25325 | A Banach subspace of a sub... |
| cmslssbn 25326 | A complete linear subspace... |
| cmscsscms 25327 | A closed subspace of a com... |
| bncssbn 25328 | A closed subspace of a Ban... |
| cssbn 25329 | A complete subspace of a n... |
| csschl 25330 | A complete subspace of a c... |
| cmslsschl 25331 | A complete linear subspace... |
| chlcsschl 25332 | A closed subspace of a sub... |
| retopn 25333 | The topology of the real n... |
| recms 25334 | The real numbers form a co... |
| reust 25335 | The Uniform structure of t... |
| recusp 25336 | The real numbers form a co... |
| rrxval 25341 | Value of the generalized E... |
| rrxbase 25342 | The base of the generalize... |
| rrxprds 25343 | Expand the definition of t... |
| rrxip 25344 | The inner product of the g... |
| rrxnm 25345 | The norm of the generalize... |
| rrxcph 25346 | Generalized Euclidean real... |
| rrxds 25347 | The distance over generali... |
| rrxvsca 25348 | The scalar product over ge... |
| rrxplusgvscavalb 25349 | The result of the addition... |
| rrxsca 25350 | The field of real numbers ... |
| rrx0 25351 | The zero ("origin") in a g... |
| rrx0el 25352 | The zero ("origin") in a g... |
| csbren 25353 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25354 | Triangle inequality in R^n... |
| rrxf 25355 | Euclidean vectors as funct... |
| rrxfsupp 25356 | Euclidean vectors are of f... |
| rrxsuppss 25357 | Support of Euclidean vecto... |
| rrxmvallem 25358 | Support of the function us... |
| rrxmval 25359 | The value of the Euclidean... |
| rrxmfval 25360 | The value of the Euclidean... |
| rrxmetlem 25361 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25362 | Euclidean space is a metri... |
| rrxdstprj1 25363 | The distance between two p... |
| rrxbasefi 25364 | The base of the generalize... |
| rrxdsfi 25365 | The distance over generali... |
| rrxmetfi 25366 | Euclidean space is a metri... |
| rrxdsfival 25367 | The value of the Euclidean... |
| ehlval 25368 | Value of the Euclidean spa... |
| ehlbase 25369 | The base of the Euclidean ... |
| ehl0base 25370 | The base of the Euclidean ... |
| ehl0 25371 | The Euclidean space of dim... |
| ehleudis 25372 | The Euclidean distance fun... |
| ehleudisval 25373 | The value of the Euclidean... |
| ehl1eudis 25374 | The Euclidean distance fun... |
| ehl1eudisval 25375 | The value of the Euclidean... |
| ehl2eudis 25376 | The Euclidean distance fun... |
| ehl2eudisval 25377 | The value of the Euclidean... |
| minveclem1 25378 | Lemma for ~ minvec . The ... |
| minveclem4c 25379 | Lemma for ~ minvec . The ... |
| minveclem2 25380 | Lemma for ~ minvec . Any ... |
| minveclem3a 25381 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25382 | Lemma for ~ minvec . The ... |
| minveclem3 25383 | Lemma for ~ minvec . The ... |
| minveclem4a 25384 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25385 | Lemma for ~ minvec . The ... |
| minveclem4 25386 | Lemma for ~ minvec . The ... |
| minveclem5 25387 | Lemma for ~ minvec . Disc... |
| minveclem6 25388 | Lemma for ~ minvec . Any ... |
| minveclem7 25389 | Lemma for ~ minvec . Sinc... |
| minvec 25390 | Minimizing vector theorem,... |
| pjthlem1 25391 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25392 | Lemma for ~ pjth . (Contr... |
| pjth 25393 | Projection Theorem: Any H... |
| pjth2 25394 | Projection Theorem with ab... |
| cldcss 25395 | Corollary of the Projectio... |
| cldcss2 25396 | Corollary of the Projectio... |
| hlhil 25397 | Corollary of the Projectio... |
| addcncf 25398 | The addition of two contin... |
| subcncf 25399 | The subtraction of two con... |
| mulcncf 25400 | The multiplication of two ... |
| mulcncfOLD 25401 | Obsolete version of ~ mulc... |
| divcncf 25402 | The quotient of two contin... |
| pmltpclem1 25403 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25404 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25405 | Any function on the reals ... |
| ivthlem1 25406 | Lemma for ~ ivth . The se... |
| ivthlem2 25407 | Lemma for ~ ivth . Show t... |
| ivthlem3 25408 | Lemma for ~ ivth , the int... |
| ivth 25409 | The intermediate value the... |
| ivth2 25410 | The intermediate value the... |
| ivthle 25411 | The intermediate value the... |
| ivthle2 25412 | The intermediate value the... |
| ivthicc 25413 | The interval between any t... |
| evthicc 25414 | Specialization of the Extr... |
| evthicc2 25415 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25416 | A continuous function on a... |
| ovolfcl 25421 | Closure for the interval e... |
| ovolfioo 25422 | Unpack the interval coveri... |
| ovolficc 25423 | Unpack the interval coveri... |
| ovolficcss 25424 | Any (closed) interval cove... |
| ovolfsval 25425 | The value of the interval ... |
| ovolfsf 25426 | Closure for the interval l... |
| ovolsf 25427 | Closure for the partial su... |
| ovolval 25428 | The value of the outer mea... |
| elovolmlem 25429 | Lemma for ~ elovolm and re... |
| elovolm 25430 | Elementhood in the set ` M... |
| elovolmr 25431 | Sufficient condition for e... |
| ovolmge0 25432 | The set ` M ` is composed ... |
| ovolcl 25433 | The volume of a set is an ... |
| ovollb 25434 | The outer volume is a lowe... |
| ovolgelb 25435 | The outer volume is the gr... |
| ovolge0 25436 | The volume of a set is alw... |
| ovolf 25437 | The domain and codomain of... |
| ovollecl 25438 | If an outer volume is boun... |
| ovolsslem 25439 | Lemma for ~ ovolss . (Con... |
| ovolss 25440 | The volume of a set is mon... |
| ovolsscl 25441 | If a set is contained in a... |
| ovolssnul 25442 | A subset of a nullset is n... |
| ovollb2lem 25443 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25444 | It is often more convenien... |
| ovolctb 25445 | The volume of a denumerabl... |
| ovolq 25446 | The rational numbers have ... |
| ovolctb2 25447 | The volume of a countable ... |
| ovol0 25448 | The empty set has 0 outer ... |
| ovolfi 25449 | A finite set has 0 outer L... |
| ovolsn 25450 | A singleton has 0 outer Le... |
| ovolunlem1a 25451 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25452 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25453 | Lemma for ~ ovolun . (Con... |
| ovolun 25454 | The Lebesgue outer measure... |
| ovolunnul 25455 | Adding a nullset does not ... |
| ovolfiniun 25456 | The Lebesgue outer measure... |
| ovoliunlem1 25457 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25458 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25459 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25460 | The Lebesgue outer measure... |
| ovoliun2 25461 | The Lebesgue outer measure... |
| ovoliunnul 25462 | A countable union of nulls... |
| shft2rab 25463 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25464 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25465 | Lemma for ~ ovolshft . (C... |
| ovolshft 25466 | The Lebesgue outer measure... |
| sca2rab 25467 | If ` B ` is a scale of ` A... |
| ovolscalem1 25468 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25469 | Lemma for ~ ovolshft . (C... |
| ovolsca 25470 | The Lebesgue outer measure... |
| ovolicc1 25471 | The measure of a closed in... |
| ovolicc2lem1 25472 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25473 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25474 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25475 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25476 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25477 | The measure of a closed in... |
| ovolicc 25478 | The measure of a closed in... |
| ovolicopnf 25479 | The measure of a right-unb... |
| ovolre 25480 | The measure of the real nu... |
| ismbl 25481 | The predicate " ` A ` is L... |
| ismbl2 25482 | From ~ ovolun , it suffice... |
| volres 25483 | A self-referencing abbrevi... |
| volf 25484 | The domain and codomain of... |
| mblvol 25485 | The volume of a measurable... |
| mblss 25486 | A measurable set is a subs... |
| mblsplit 25487 | The defining property of m... |
| volss 25488 | The Lebesgue measure is mo... |
| cmmbl 25489 | The complement of a measur... |
| nulmbl 25490 | A nullset is measurable. ... |
| nulmbl2 25491 | A set of outer measure zer... |
| unmbl 25492 | A union of measurable sets... |
| shftmbl 25493 | A shift of a measurable se... |
| 0mbl 25494 | The empty set is measurabl... |
| rembl 25495 | The set of all real number... |
| unidmvol 25496 | The union of the Lebesgue ... |
| inmbl 25497 | An intersection of measura... |
| difmbl 25498 | A difference of measurable... |
| finiunmbl 25499 | A finite union of measurab... |
| volun 25500 | The Lebesgue measure funct... |
| volinun 25501 | Addition of non-disjoint s... |
| volfiniun 25502 | The volume of a disjoint f... |
| iundisj 25503 | Rewrite a countable union ... |
| iundisj2 25504 | A disjoint union is disjoi... |
| voliunlem1 25505 | Lemma for ~ voliun . (Con... |
| voliunlem2 25506 | Lemma for ~ voliun . (Con... |
| voliunlem3 25507 | Lemma for ~ voliun . (Con... |
| iunmbl 25508 | The measurable sets are cl... |
| voliun 25509 | The Lebesgue measure funct... |
| volsuplem 25510 | Lemma for ~ volsup . (Con... |
| volsup 25511 | The volume of the limit of... |
| iunmbl2 25512 | The measurable sets are cl... |
| ioombl1lem1 25513 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25514 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25515 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25516 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25517 | An open right-unbounded in... |
| icombl1 25518 | A closed unbounded-above i... |
| icombl 25519 | A closed-below, open-above... |
| ioombl 25520 | An open real interval is m... |
| iccmbl 25521 | A closed real interval is ... |
| iccvolcl 25522 | A closed real interval has... |
| ovolioo 25523 | The measure of an open int... |
| volioo 25524 | The measure of an open int... |
| ioovolcl 25525 | An open real interval has ... |
| ovolfs2 25526 | Alternative expression for... |
| ioorcl2 25527 | An open interval with fini... |
| ioorf 25528 | Define a function from ope... |
| ioorval 25529 | Define a function from ope... |
| ioorinv2 25530 | The function ` F ` is an "... |
| ioorinv 25531 | The function ` F ` is an "... |
| ioorcl 25532 | The function ` F ` does no... |
| uniiccdif 25533 | A union of closed interval... |
| uniioovol 25534 | A disjoint union of open i... |
| uniiccvol 25535 | An almost-disjoint union o... |
| uniioombllem1 25536 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25537 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25538 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25539 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25540 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25541 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25542 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25543 | Lemma for ~ uniioombl . (... |
| uniioombl 25544 | A disjoint union of open i... |
| uniiccmbl 25545 | An almost-disjoint union o... |
| dyadf 25546 | The function ` F ` returns... |
| dyadval 25547 | Value of the dyadic ration... |
| dyadovol 25548 | Volume of a dyadic rationa... |
| dyadss 25549 | Two closed dyadic rational... |
| dyaddisjlem 25550 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25551 | Two closed dyadic rational... |
| dyadmaxlem 25552 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25553 | Any nonempty set of dyadic... |
| dyadmbllem 25554 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25555 | Any union of dyadic ration... |
| opnmbllem 25556 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25557 | All open sets are measurab... |
| opnmblALT 25558 | All open sets are measurab... |
| subopnmbl 25559 | Sets which are open in a m... |
| volsup2 25560 | The volume of ` A ` is the... |
| volcn 25561 | The function formed by res... |
| volivth 25562 | The Intermediate Value The... |
| vitalilem1 25563 | Lemma for ~ vitali . (Con... |
| vitalilem2 25564 | Lemma for ~ vitali . (Con... |
| vitalilem3 25565 | Lemma for ~ vitali . (Con... |
| vitalilem4 25566 | Lemma for ~ vitali . (Con... |
| vitalilem5 25567 | Lemma for ~ vitali . (Con... |
| vitali 25568 | If the reals can be well-o... |
| ismbf1 25579 | The predicate " ` F ` is a... |
| mbff 25580 | A measurable function is a... |
| mbfdm 25581 | The domain of a measurable... |
| mbfconstlem 25582 | Lemma for ~ mbfconst and r... |
| ismbf 25583 | The predicate " ` F ` is a... |
| ismbfcn 25584 | A complex function is meas... |
| mbfima 25585 | Definitional property of a... |
| mbfimaicc 25586 | The preimage of any closed... |
| mbfimasn 25587 | The preimage of a point un... |
| mbfconst 25588 | A constant function is mea... |
| mbf0 25589 | The empty function is meas... |
| mbfid 25590 | The identity function is m... |
| mbfmptcl 25591 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25592 | The domain of a measurable... |
| ismbfcn2 25593 | A complex function is meas... |
| ismbfd 25594 | Deduction to prove measura... |
| ismbf2d 25595 | Deduction to prove measura... |
| mbfeqalem1 25596 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25597 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25598 | If two functions are equal... |
| mbfres 25599 | The restriction of a measu... |
| mbfres2 25600 | Measurability of a piecewi... |
| mbfss 25601 | Change the domain of a mea... |
| mbfmulc2lem 25602 | Multiplication by a consta... |
| mbfmulc2re 25603 | Multiplication by a consta... |
| mbfmax 25604 | The maximum of two functio... |
| mbfneg 25605 | The negative of a measurab... |
| mbfpos 25606 | The positive part of a mea... |
| mbfposr 25607 | Converse to ~ mbfpos . (C... |
| mbfposb 25608 | A function is measurable i... |
| ismbf3d 25609 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25610 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25611 | The preimage of any open s... |
| mbfimaopn2 25612 | The preimage of any set op... |
| cncombf 25613 | The composition of a conti... |
| cnmbf 25614 | A continuous function is m... |
| mbfaddlem 25615 | The sum of two measurable ... |
| mbfadd 25616 | The sum of two measurable ... |
| mbfsub 25617 | The difference of two meas... |
| mbfmulc2 25618 | A complex constant times a... |
| mbfsup 25619 | The supremum of a sequence... |
| mbfinf 25620 | The infimum of a sequence ... |
| mbflimsup 25621 | The limit supremum of a se... |
| mbflimlem 25622 | The pointwise limit of a s... |
| mbflim 25623 | The pointwise limit of a s... |
| 0pval 25626 | The zero function evaluate... |
| 0plef 25627 | Two ways to say that the f... |
| 0pledm 25628 | Adjust the domain of the l... |
| isi1f 25629 | The predicate " ` F ` is a... |
| i1fmbf 25630 | Simple functions are measu... |
| i1ff 25631 | A simple function is a fun... |
| i1frn 25632 | A simple function has fini... |
| i1fima 25633 | Any preimage of a simple f... |
| i1fima2 25634 | Any preimage of a simple f... |
| i1fima2sn 25635 | Preimage of a singleton. ... |
| i1fd 25636 | A simplified set of assump... |
| i1f0rn 25637 | Any simple function takes ... |
| itg1val 25638 | The value of the integral ... |
| itg1val2 25639 | The value of the integral ... |
| itg1cl 25640 | Closure of the integral on... |
| itg1ge0 25641 | Closure of the integral on... |
| i1f0 25642 | The zero function is simpl... |
| itg10 25643 | The zero function has zero... |
| i1f1lem 25644 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25645 | Base case simple functions... |
| itg11 25646 | The integral of an indicat... |
| itg1addlem1 25647 | Decompose a preimage, whic... |
| i1faddlem 25648 | Decompose the preimage of ... |
| i1fmullem 25649 | Decompose the preimage of ... |
| i1fadd 25650 | The sum of two simple func... |
| i1fmul 25651 | The pointwise product of t... |
| itg1addlem2 25652 | Lemma for ~ itg1add . The... |
| itg1addlem3 25653 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25654 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 25655 | Lemma for ~ itg1add . (Co... |
| itg1add 25656 | The integral of a sum of s... |
| i1fmulclem 25657 | Decompose the preimage of ... |
| i1fmulc 25658 | A nonnegative constant tim... |
| itg1mulc 25659 | The integral of a constant... |
| i1fres 25660 | The "restriction" of a sim... |
| i1fpos 25661 | The positive part of a sim... |
| i1fposd 25662 | Deduction form of ~ i1fpos... |
| i1fsub 25663 | The difference of two simp... |
| itg1sub 25664 | The integral of a differen... |
| itg10a 25665 | The integral of a simple f... |
| itg1ge0a 25666 | The integral of an almost ... |
| itg1lea 25667 | Approximate version of ~ i... |
| itg1le 25668 | If one simple function dom... |
| itg1climres 25669 | Restricting the simple fun... |
| mbfi1fseqlem1 25670 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25671 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25672 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25673 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25674 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25675 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25676 | A characterization of meas... |
| mbfi1flimlem 25677 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25678 | Any real measurable functi... |
| mbfmullem2 25679 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25680 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25681 | The product of two measura... |
| itg2lcl 25682 | The set of lower sums is a... |
| itg2val 25683 | Value of the integral on n... |
| itg2l 25684 | Elementhood in the set ` L... |
| itg2lr 25685 | Sufficient condition for e... |
| xrge0f 25686 | A real function is a nonne... |
| itg2cl 25687 | The integral of a nonnegat... |
| itg2ub 25688 | The integral of a nonnegat... |
| itg2leub 25689 | Any upper bound on the int... |
| itg2ge0 25690 | The integral of a nonnegat... |
| itg2itg1 25691 | The integral of a nonnegat... |
| itg20 25692 | The integral of the zero f... |
| itg2lecl 25693 | If an ` S.2 ` integral is ... |
| itg2le 25694 | If one function dominates ... |
| itg2const 25695 | Integral of a constant fun... |
| itg2const2 25696 | When the base set of a con... |
| itg2seq 25697 | Definitional property of t... |
| itg2uba 25698 | Approximate version of ~ i... |
| itg2lea 25699 | Approximate version of ~ i... |
| itg2eqa 25700 | Approximate equality of in... |
| itg2mulclem 25701 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25702 | The integral of a nonnegat... |
| itg2splitlem 25703 | Lemma for ~ itg2split . (... |
| itg2split 25704 | The ` S.2 ` integral split... |
| itg2monolem1 25705 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25706 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25707 | Lemma for ~ itg2mono . (C... |
| itg2mono 25708 | The Monotone Convergence T... |
| itg2i1fseqle 25709 | Subject to the conditions ... |
| itg2i1fseq 25710 | Subject to the conditions ... |
| itg2i1fseq2 25711 | In an extension to the res... |
| itg2i1fseq3 25712 | Special case of ~ itg2i1fs... |
| itg2addlem 25713 | Lemma for ~ itg2add . (Co... |
| itg2add 25714 | The ` S.2 ` integral is li... |
| itg2gt0 25715 | If the function ` F ` is s... |
| itg2cnlem1 25716 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25717 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25718 | A sort of absolute continu... |
| ibllem 25719 | Conditioned equality theor... |
| isibl 25720 | The predicate " ` F ` is i... |
| isibl2 25721 | The predicate " ` F ` is i... |
| iblmbf 25722 | An integrable function is ... |
| iblitg 25723 | If a function is integrabl... |
| dfitg 25724 | Evaluate the class substit... |
| itgex 25725 | An integral is a set. (Co... |
| itgeq1f 25726 | Equality theorem for an in... |
| itgeq1fOLD 25727 | Obsolete version of ~ itge... |
| itgeq1 25728 | Equality theorem for an in... |
| nfitg1 25729 | Bound-variable hypothesis ... |
| nfitg 25730 | Bound-variable hypothesis ... |
| cbvitg 25731 | Change bound variable in a... |
| cbvitgv 25732 | Change bound variable in a... |
| itgeq2 25733 | Equality theorem for an in... |
| itgresr 25734 | The domain of an integral ... |
| itg0 25735 | The integral of anything o... |
| itgz 25736 | The integral of zero on an... |
| itgeq2dv 25737 | Equality theorem for an in... |
| itgmpt 25738 | Change bound variable in a... |
| itgcl 25739 | The integral of an integra... |
| itgvallem 25740 | Substitution lemma. (Cont... |
| itgvallem3 25741 | Lemma for ~ itgposval and ... |
| ibl0 25742 | The zero function is integ... |
| iblcnlem1 25743 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25744 | Expand out the universal q... |
| itgcnlem 25745 | Expand out the sum in ~ df... |
| iblrelem 25746 | Integrability of a real fu... |
| iblposlem 25747 | Lemma for ~ iblpos . (Con... |
| iblpos 25748 | Integrability of a nonnega... |
| iblre 25749 | Integrability of a real fu... |
| itgrevallem1 25750 | Lemma for ~ itgposval and ... |
| itgposval 25751 | The integral of a nonnegat... |
| itgreval 25752 | Decompose the integral of ... |
| itgrecl 25753 | Real closure of an integra... |
| iblcn 25754 | Integrability of a complex... |
| itgcnval 25755 | Decompose the integral of ... |
| itgre 25756 | Real part of an integral. ... |
| itgim 25757 | Imaginary part of an integ... |
| iblneg 25758 | The negative of an integra... |
| itgneg 25759 | Negation of an integral. ... |
| iblss 25760 | A subset of an integrable ... |
| iblss2 25761 | Change the domain of an in... |
| itgitg2 25762 | Transfer an integral using... |
| i1fibl 25763 | A simple function is integ... |
| itgitg1 25764 | Transfer an integral using... |
| itgle 25765 | Monotonicity of an integra... |
| itgge0 25766 | The integral of a positive... |
| itgss 25767 | Expand the set of an integ... |
| itgss2 25768 | Expand the set of an integ... |
| itgeqa 25769 | Approximate equality of in... |
| itgss3 25770 | Expand the set of an integ... |
| itgioo 25771 | Equality of integrals on o... |
| itgless 25772 | Expand the integral of a n... |
| iblconst 25773 | A constant function is int... |
| itgconst 25774 | Integral of a constant fun... |
| ibladdlem 25775 | Lemma for ~ ibladd . (Con... |
| ibladd 25776 | Add two integrals over the... |
| iblsub 25777 | Subtract two integrals ove... |
| itgaddlem1 25778 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25779 | Lemma for ~ itgadd . (Con... |
| itgadd 25780 | Add two integrals over the... |
| itgsub 25781 | Subtract two integrals ove... |
| itgfsum 25782 | Take a finite sum of integ... |
| iblabslem 25783 | Lemma for ~ iblabs . (Con... |
| iblabs 25784 | The absolute value of an i... |
| iblabsr 25785 | A measurable function is i... |
| iblmulc2 25786 | Multiply an integral by a ... |
| itgmulc2lem1 25787 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25788 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25789 | Multiply an integral by a ... |
| itgabs 25790 | The triangle inequality fo... |
| itgsplit 25791 | The ` S. ` integral splits... |
| itgspliticc 25792 | The ` S. ` integral splits... |
| itgsplitioo 25793 | The ` S. ` integral splits... |
| bddmulibl 25794 | A bounded function times a... |
| bddibl 25795 | A bounded function is inte... |
| cniccibl 25796 | A continuous function on a... |
| bddiblnc 25797 | Choice-free proof of ~ bdd... |
| cnicciblnc 25798 | Choice-free proof of ~ cni... |
| itggt0 25799 | The integral of a strictly... |
| itgcn 25800 | Transfer ~ itg2cn to the f... |
| ditgeq1 25803 | Equality theorem for the d... |
| ditgeq2 25804 | Equality theorem for the d... |
| ditgeq3 25805 | Equality theorem for the d... |
| ditgeq3dv 25806 | Equality theorem for the d... |
| ditgex 25807 | A directed integral is a s... |
| ditg0 25808 | Value of the directed inte... |
| cbvditg 25809 | Change bound variable in a... |
| cbvditgv 25810 | Change bound variable in a... |
| ditgpos 25811 | Value of the directed inte... |
| ditgneg 25812 | Value of the directed inte... |
| ditgcl 25813 | Closure of a directed inte... |
| ditgswap 25814 | Reverse a directed integra... |
| ditgsplitlem 25815 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25816 | This theorem is the raison... |
| reldv 25825 | The derivative function is... |
| limcvallem 25826 | Lemma for ~ ellimc . (Con... |
| limcfval 25827 | Value and set bounds on th... |
| ellimc 25828 | Value of the limit predica... |
| limcrcl 25829 | Reverse closure for the li... |
| limccl 25830 | Closure of the limit opera... |
| limcdif 25831 | It suffices to consider fu... |
| ellimc2 25832 | Write the definition of a ... |
| limcnlp 25833 | If ` B ` is not a limit po... |
| ellimc3 25834 | Write the epsilon-delta de... |
| limcflflem 25835 | Lemma for ~ limcflf . (Co... |
| limcflf 25836 | The limit operator can be ... |
| limcmo 25837 | If ` B ` is a limit point ... |
| limcmpt 25838 | Express the limit operator... |
| limcmpt2 25839 | Express the limit operator... |
| limcresi 25840 | Any limit of ` F ` is also... |
| limcres 25841 | If ` B ` is an interior po... |
| cnplimc 25842 | A function is continuous a... |
| cnlimc 25843 | ` F ` is a continuous func... |
| cnlimci 25844 | If ` F ` is a continuous f... |
| cnmptlimc 25845 | If ` F ` is a continuous f... |
| limccnp 25846 | If the limit of ` F ` at `... |
| limccnp2 25847 | The image of a convergent ... |
| limcco 25848 | Composition of two limits.... |
| limciun 25849 | A point is a limit of ` F ... |
| limcun 25850 | A point is a limit of ` F ... |
| dvlem 25851 | Closure for a difference q... |
| dvfval 25852 | Value and set bounds on th... |
| eldv 25853 | The differentiable predica... |
| dvcl 25854 | The derivative function ta... |
| dvbssntr 25855 | The set of differentiable ... |
| dvbss 25856 | The set of differentiable ... |
| dvbsss 25857 | The set of differentiable ... |
| perfdvf 25858 | The derivative is a functi... |
| recnprss 25859 | Both ` RR ` and ` CC ` are... |
| recnperf 25860 | Both ` RR ` and ` CC ` are... |
| dvfg 25861 | Explicitly write out the f... |
| dvf 25862 | The derivative is a functi... |
| dvfcn 25863 | The derivative is a functi... |
| dvreslem 25864 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25865 | Lemma for ~ dvres2 . (Con... |
| dvres 25866 | Restriction of a derivativ... |
| dvres2 25867 | Restriction of the base se... |
| dvres3 25868 | Restriction of a complex d... |
| dvres3a 25869 | Restriction of a complex d... |
| dvidlem 25870 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25871 | Derivative of a function r... |
| dvconst 25872 | Derivative of a constant f... |
| dvid 25873 | Derivative of the identity... |
| dvcnp 25874 | The difference quotient is... |
| dvcnp2 25875 | A function is continuous a... |
| dvcnp2OLD 25876 | Obsolete version of ~ dvcn... |
| dvcn 25877 | A differentiable function ... |
| dvnfval 25878 | Value of the iterated deri... |
| dvnff 25879 | The iterated derivative is... |
| dvn0 25880 | Zero times iterated deriva... |
| dvnp1 25881 | Successor iterated derivat... |
| dvn1 25882 | One times iterated derivat... |
| dvnf 25883 | The N-times derivative is ... |
| dvnbss 25884 | The set of N-times differe... |
| dvnadd 25885 | The ` N ` -th derivative o... |
| dvn2bss 25886 | An N-times differentiable ... |
| dvnres 25887 | Multiple derivative versio... |
| cpnfval 25888 | Condition for n-times cont... |
| fncpn 25889 | The ` C^n ` object is a fu... |
| elcpn 25890 | Condition for n-times cont... |
| cpnord 25891 | ` C^n ` conditions are ord... |
| cpncn 25892 | A ` C^n ` function is cont... |
| cpnres 25893 | The restriction of a ` C^n... |
| dvaddbr 25894 | The sum rule for derivativ... |
| dvmulbr 25895 | The product rule for deriv... |
| dvmulbrOLD 25896 | Obsolete version of ~ dvmu... |
| dvadd 25897 | The sum rule for derivativ... |
| dvmul 25898 | The product rule for deriv... |
| dvaddf 25899 | The sum rule for everywher... |
| dvmulf 25900 | The product rule for every... |
| dvcmul 25901 | The product rule when one ... |
| dvcmulf 25902 | The product rule when one ... |
| dvcobr 25903 | The chain rule for derivat... |
| dvcobrOLD 25904 | Obsolete version of ~ dvco... |
| dvco 25905 | The chain rule for derivat... |
| dvcof 25906 | The chain rule for everywh... |
| dvcjbr 25907 | The derivative of the conj... |
| dvcj 25908 | The derivative of the conj... |
| dvfre 25909 | The derivative of a real f... |
| dvnfre 25910 | The ` N ` -th derivative o... |
| dvexp 25911 | Derivative of a power func... |
| dvexp2 25912 | Derivative of an exponenti... |
| dvrec 25913 | Derivative of the reciproc... |
| dvmptres3 25914 | Function-builder for deriv... |
| dvmptid 25915 | Function-builder for deriv... |
| dvmptc 25916 | Function-builder for deriv... |
| dvmptcl 25917 | Closure lemma for ~ dvmptc... |
| dvmptadd 25918 | Function-builder for deriv... |
| dvmptmul 25919 | Function-builder for deriv... |
| dvmptres2 25920 | Function-builder for deriv... |
| dvmptres 25921 | Function-builder for deriv... |
| dvmptcmul 25922 | Function-builder for deriv... |
| dvmptdivc 25923 | Function-builder for deriv... |
| dvmptneg 25924 | Function-builder for deriv... |
| dvmptsub 25925 | Function-builder for deriv... |
| dvmptcj 25926 | Function-builder for deriv... |
| dvmptre 25927 | Function-builder for deriv... |
| dvmptim 25928 | Function-builder for deriv... |
| dvmptntr 25929 | Function-builder for deriv... |
| dvmptco 25930 | Function-builder for deriv... |
| dvrecg 25931 | Derivative of the reciproc... |
| dvmptdiv 25932 | Function-builder for deriv... |
| dvmptfsum 25933 | Function-builder for deriv... |
| dvcnvlem 25934 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25935 | A weak version of ~ dvcnvr... |
| dvexp3 25936 | Derivative of an exponenti... |
| dveflem 25937 | Derivative of the exponent... |
| dvef 25938 | Derivative of the exponent... |
| dvsincos 25939 | Derivative of the sine and... |
| dvsin 25940 | Derivative of the sine fun... |
| dvcos 25941 | Derivative of the cosine f... |
| dvferm1lem 25942 | Lemma for ~ dvferm . (Con... |
| dvferm1 25943 | One-sided version of ~ dvf... |
| dvferm2lem 25944 | Lemma for ~ dvferm . (Con... |
| dvferm2 25945 | One-sided version of ~ dvf... |
| dvferm 25946 | Fermat's theorem on statio... |
| rollelem 25947 | Lemma for ~ rolle . (Cont... |
| rolle 25948 | Rolle's theorem. If ` F `... |
| cmvth 25949 | Cauchy's Mean Value Theore... |
| cmvthOLD 25950 | Obsolete version of ~ cmvt... |
| mvth 25951 | The Mean Value Theorem. I... |
| dvlip 25952 | A function with derivative... |
| dvlipcn 25953 | A complex function with de... |
| dvlip2 25954 | Combine the results of ~ d... |
| c1liplem1 25955 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25956 | C^1 functions are Lipschit... |
| c1lip2 25957 | C^1 functions are Lipschit... |
| c1lip3 25958 | C^1 functions are Lipschit... |
| dveq0 25959 | If a continuous function h... |
| dv11cn 25960 | Two functions defined on a... |
| dvgt0lem1 25961 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25962 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25963 | A function on a closed int... |
| dvlt0 25964 | A function on a closed int... |
| dvge0 25965 | A function on a closed int... |
| dvle 25966 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25967 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25968 | Lemma for ~ dvivth . (Con... |
| dvivth 25969 | Darboux' theorem, or the i... |
| dvne0 25970 | A function on a closed int... |
| dvne0f1 25971 | A function on a closed int... |
| lhop1lem 25972 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25973 | L'Hôpital's Rule for... |
| lhop2 25974 | L'Hôpital's Rule for... |
| lhop 25975 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25976 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25977 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25978 | The derivative rule for in... |
| dvcvx 25979 | A real function with stric... |
| dvfsumle 25980 | Compare a finite sum to an... |
| dvfsumleOLD 25981 | Obsolete version of ~ dvfs... |
| dvfsumge 25982 | Compare a finite sum to an... |
| dvfsumabs 25983 | Compare a finite sum to an... |
| dvmptrecl 25984 | Real closure of a derivati... |
| dvfsumrlimf 25985 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25986 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25987 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2OLD 25988 | Obsolete version of ~ dvfs... |
| dvfsumlem3 25989 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25990 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25991 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25992 | Compare a finite sum to an... |
| dvfsumrlim2 25993 | Compare a finite sum to an... |
| dvfsumrlim3 25994 | Conjoin the statements of ... |
| dvfsum2 25995 | The reverse of ~ dvfsumrli... |
| ftc1lem1 25996 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 25997 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 25998 | The Fundamental Theorem of... |
| ftc1lem3 25999 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26000 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26001 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26002 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26003 | The Fundamental Theorem of... |
| ftc1cn 26004 | Strengthen the assumptions... |
| ftc2 26005 | The Fundamental Theorem of... |
| ftc2ditglem 26006 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26007 | Directed integral analogue... |
| itgparts 26008 | Integration by parts. If ... |
| itgsubstlem 26009 | Lemma for ~ itgsubst . (C... |
| itgsubst 26010 | Integration by ` u ` -subs... |
| itgpowd 26011 | The integral of a monomial... |
| reldmmdeg 26016 | Multivariate degree is a b... |
| tdeglem1 26017 | Functionality of the total... |
| tdeglem3 26018 | Additivity of the total de... |
| tdeglem4 26019 | There is only one multi-in... |
| tdeglem2 26020 | Simplification of total de... |
| mdegfval 26021 | Value of the multivariate ... |
| mdegval 26022 | Value of the multivariate ... |
| mdegleb 26023 | Property of being of limit... |
| mdeglt 26024 | If there is an upper limit... |
| mdegldg 26025 | A nonzero polynomial has s... |
| mdegxrcl 26026 | Closure of polynomial degr... |
| mdegxrf 26027 | Functionality of polynomia... |
| mdegcl 26028 | Sharp closure for multivar... |
| mdeg0 26029 | Degree of the zero polynom... |
| mdegnn0cl 26030 | Degree of a nonzero polyno... |
| degltlem1 26031 | Theorem on arithmetic of e... |
| degltp1le 26032 | Theorem on arithmetic of e... |
| mdegaddle 26033 | The degree of a sum is at ... |
| mdegvscale 26034 | The degree of a scalar mul... |
| mdegvsca 26035 | The degree of a scalar mul... |
| mdegle0 26036 | A polynomial has nonpositi... |
| mdegmullem 26037 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26038 | The multivariate degree of... |
| deg1fval 26039 | Relate univariate polynomi... |
| deg1xrf 26040 | Functionality of univariat... |
| deg1xrcl 26041 | Closure of univariate poly... |
| deg1cl 26042 | Sharp closure of univariat... |
| mdegpropd 26043 | Property deduction for pol... |
| deg1fvi 26044 | Univariate polynomial degr... |
| deg1propd 26045 | Property deduction for pol... |
| deg1z 26046 | Degree of the zero univari... |
| deg1nn0cl 26047 | Degree of a nonzero univar... |
| deg1n0ima 26048 | Degree image of a set of p... |
| deg1nn0clb 26049 | A polynomial is nonzero if... |
| deg1lt0 26050 | A polynomial is zero iff i... |
| deg1ldg 26051 | A nonzero univariate polyn... |
| deg1ldgn 26052 | An index at which a polyno... |
| deg1ldgdomn 26053 | A nonzero univariate polyn... |
| deg1leb 26054 | Property of being of limit... |
| deg1val 26055 | Value of the univariate de... |
| deg1lt 26056 | If the degree of a univari... |
| deg1ge 26057 | Conversely, a nonzero coef... |
| coe1mul3 26058 | The coefficient vector of ... |
| coe1mul4 26059 | Value of the "leading" coe... |
| deg1addle 26060 | The degree of a sum is at ... |
| deg1addle2 26061 | If both factors have degre... |
| deg1add 26062 | Exact degree of a sum of t... |
| deg1vscale 26063 | The degree of a scalar tim... |
| deg1vsca 26064 | The degree of a scalar tim... |
| deg1invg 26065 | The degree of the negated ... |
| deg1suble 26066 | The degree of a difference... |
| deg1sub 26067 | Exact degree of a differen... |
| deg1mulle2 26068 | Produce a bound on the pro... |
| deg1sublt 26069 | Subtraction of two polynom... |
| deg1le0 26070 | A polynomial has nonpositi... |
| deg1sclle 26071 | A scalar polynomial has no... |
| deg1scl 26072 | A nonzero scalar polynomia... |
| deg1mul2 26073 | Degree of multiplication o... |
| deg1mul 26074 | Degree of multiplication o... |
| deg1mul3 26075 | Degree of multiplication o... |
| deg1mul3le 26076 | Degree of multiplication o... |
| deg1tmle 26077 | Limiting degree of a polyn... |
| deg1tm 26078 | Exact degree of a polynomi... |
| deg1pwle 26079 | Limiting degree of a varia... |
| deg1pw 26080 | Exact degree of a variable... |
| ply1nz 26081 | Univariate polynomials ove... |
| ply1nzb 26082 | Univariate polynomials are... |
| ply1domn 26083 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26084 | The ring of univariate pol... |
| ply1divmo 26095 | Uniqueness of a quotient i... |
| ply1divex 26096 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26097 | The division algorithm for... |
| ply1divalg2 26098 | Reverse the order of multi... |
| uc1pval 26099 | Value of the set of unitic... |
| isuc1p 26100 | Being a unitic polynomial.... |
| mon1pval 26101 | Value of the set of monic ... |
| ismon1p 26102 | Being a monic polynomial. ... |
| uc1pcl 26103 | Unitic polynomials are pol... |
| mon1pcl 26104 | Monic polynomials are poly... |
| uc1pn0 26105 | Unitic polynomials are not... |
| mon1pn0 26106 | Monic polynomials are not ... |
| uc1pdeg 26107 | Unitic polynomials have no... |
| uc1pldg 26108 | Unitic polynomials have un... |
| mon1pldg 26109 | Unitic polynomials have on... |
| mon1puc1p 26110 | Monic polynomials are unit... |
| uc1pmon1p 26111 | Make a unitic polynomial m... |
| deg1submon1p 26112 | The difference of two moni... |
| mon1pid 26113 | Monicity and degree of the... |
| q1pval 26114 | Value of the univariate po... |
| q1peqb 26115 | Characterizing property of... |
| q1pcl 26116 | Closure of the quotient by... |
| r1pval 26117 | Value of the polynomial re... |
| r1pcl 26118 | Closure of remainder follo... |
| r1pdeglt 26119 | The remainder has a degree... |
| r1pid 26120 | Express the original polyn... |
| r1pid2 26121 | Identity law for polynomia... |
| dvdsq1p 26122 | Divisibility in a polynomi... |
| dvdsr1p 26123 | Divisibility in a polynomi... |
| ply1remlem 26124 | A term of the form ` x - N... |
| ply1rem 26125 | The polynomial remainder t... |
| facth1 26126 | The factor theorem and its... |
| fta1glem1 26127 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26128 | Lemma for ~ fta1g . (Cont... |
| fta1g 26129 | The one-sided fundamental ... |
| fta1blem 26130 | Lemma for ~ fta1b . (Cont... |
| fta1b 26131 | The assumption that ` R ` ... |
| idomrootle 26132 | No element of an integral ... |
| drnguc1p 26133 | Over a division ring, all ... |
| ig1peu 26134 | There is a unique monic po... |
| ig1pval 26135 | Substitutions for the poly... |
| ig1pval2 26136 | Generator of the zero idea... |
| ig1pval3 26137 | Characterizing properties ... |
| ig1pcl 26138 | The monic generator of an ... |
| ig1pdvds 26139 | The monic generator of an ... |
| ig1prsp 26140 | Any ideal of polynomials o... |
| ply1lpir 26141 | The ring of polynomials ov... |
| ply1pid 26142 | The polynomials over a fie... |
| plyco0 26151 | Two ways to say that a fun... |
| plyval 26152 | Value of the polynomial se... |
| plybss 26153 | Reverse closure of the par... |
| elply 26154 | Definition of a polynomial... |
| elply2 26155 | The coefficient function c... |
| plyun0 26156 | The set of polynomials is ... |
| plyf 26157 | A polynomial is a function... |
| plyss 26158 | The polynomial set functio... |
| plyssc 26159 | Every polynomial ring is c... |
| elplyr 26160 | Sufficient condition for e... |
| elplyd 26161 | Sufficient condition for e... |
| ply1termlem 26162 | Lemma for ~ ply1term . (C... |
| ply1term 26163 | A one-term polynomial. (C... |
| plypow 26164 | A power is a polynomial. ... |
| plyconst 26165 | A constant function is a p... |
| ne0p 26166 | A test to show that a poly... |
| ply0 26167 | The zero function is a pol... |
| plyid 26168 | The identity function is a... |
| plyeq0lem 26169 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26170 | If a polynomial is zero at... |
| plypf1 26171 | Write the set of complex p... |
| plyaddlem1 26172 | Derive the coefficient fun... |
| plymullem1 26173 | Derive the coefficient fun... |
| plyaddlem 26174 | Lemma for ~ plyadd . (Con... |
| plymullem 26175 | Lemma for ~ plymul . (Con... |
| plyadd 26176 | The sum of two polynomials... |
| plymul 26177 | The product of two polynom... |
| plysub 26178 | The difference of two poly... |
| plyaddcl 26179 | The sum of two polynomials... |
| plymulcl 26180 | The product of two polynom... |
| plysubcl 26181 | The difference of two poly... |
| coeval 26182 | Value of the coefficient f... |
| coeeulem 26183 | Lemma for ~ coeeu . (Cont... |
| coeeu 26184 | Uniqueness of the coeffici... |
| coelem 26185 | Lemma for properties of th... |
| coeeq 26186 | If ` A ` satisfies the pro... |
| dgrval 26187 | Value of the degree functi... |
| dgrlem 26188 | Lemma for ~ dgrcl and simi... |
| coef 26189 | The domain and codomain of... |
| coef2 26190 | The domain and codomain of... |
| coef3 26191 | The domain and codomain of... |
| dgrcl 26192 | The degree of any polynomi... |
| dgrub 26193 | If the ` M ` -th coefficie... |
| dgrub2 26194 | All the coefficients above... |
| dgrlb 26195 | If all the coefficients ab... |
| coeidlem 26196 | Lemma for ~ coeid . (Cont... |
| coeid 26197 | Reconstruct a polynomial a... |
| coeid2 26198 | Reconstruct a polynomial a... |
| coeid3 26199 | Reconstruct a polynomial a... |
| plyco 26200 | The composition of two pol... |
| coeeq2 26201 | Compute the coefficient fu... |
| dgrle 26202 | Given an explicit expressi... |
| dgreq 26203 | If the highest term in a p... |
| 0dgr 26204 | A constant function has de... |
| 0dgrb 26205 | A function has degree zero... |
| dgrnznn 26206 | A nonzero polynomial with ... |
| coefv0 26207 | The result of evaluating a... |
| coeaddlem 26208 | Lemma for ~ coeadd and ~ d... |
| coemullem 26209 | Lemma for ~ coemul and ~ d... |
| coeadd 26210 | The coefficient function o... |
| coemul 26211 | A coefficient of a product... |
| coe11 26212 | The coefficient function i... |
| coemulhi 26213 | The leading coefficient of... |
| coemulc 26214 | The coefficient function i... |
| coe0 26215 | The coefficients of the ze... |
| coesub 26216 | The coefficient function o... |
| coe1termlem 26217 | The coefficient function o... |
| coe1term 26218 | The coefficient function o... |
| dgr1term 26219 | The degree of a monomial. ... |
| plycn 26220 | A polynomial is a continuo... |
| plycnOLD 26221 | Obsolete version of ~ plyc... |
| dgr0 26222 | The degree of the zero pol... |
| coeidp 26223 | The coefficients of the id... |
| dgrid 26224 | The degree of the identity... |
| dgreq0 26225 | The leading coefficient of... |
| dgrlt 26226 | Two ways to say that the d... |
| dgradd 26227 | The degree of a sum of pol... |
| dgradd2 26228 | The degree of a sum of pol... |
| dgrmul2 26229 | The degree of a product of... |
| dgrmul 26230 | The degree of a product of... |
| dgrmulc 26231 | Scalar multiplication by a... |
| dgrsub 26232 | The degree of a difference... |
| dgrcolem1 26233 | The degree of a compositio... |
| dgrcolem2 26234 | Lemma for ~ dgrco . (Cont... |
| dgrco 26235 | The degree of a compositio... |
| plycjlem 26236 | Lemma for ~ plycj and ~ co... |
| plycj 26237 | The double conjugation of ... |
| coecj 26238 | Double conjugation of a po... |
| plycjOLD 26239 | Obsolete version of ~ plyc... |
| coecjOLD 26240 | Obsolete version of ~ coec... |
| plyrecj 26241 | A polynomial with real coe... |
| plymul0or 26242 | Polynomial multiplication ... |
| ofmulrt 26243 | The set of roots of a prod... |
| plyreres 26244 | Real-coefficient polynomia... |
| dvply1 26245 | Derivative of a polynomial... |
| dvply2g 26246 | The derivative of a polyno... |
| dvply2gOLD 26247 | Obsolete version of ~ dvpl... |
| dvply2 26248 | The derivative of a polyno... |
| dvnply2 26249 | Polynomials have polynomia... |
| dvnply 26250 | Polynomials have polynomia... |
| plycpn 26251 | Polynomials are smooth. (... |
| quotval 26254 | Value of the quotient func... |
| plydivlem1 26255 | Lemma for ~ plydivalg . (... |
| plydivlem2 26256 | Lemma for ~ plydivalg . (... |
| plydivlem3 26257 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26258 | Lemma for ~ plydivex . In... |
| plydivex 26259 | Lemma for ~ plydivalg . (... |
| plydiveu 26260 | Lemma for ~ plydivalg . (... |
| plydivalg 26261 | The division algorithm on ... |
| quotlem 26262 | Lemma for properties of th... |
| quotcl 26263 | The quotient of two polyno... |
| quotcl2 26264 | Closure of the quotient fu... |
| quotdgr 26265 | Remainder property of the ... |
| plyremlem 26266 | Closure of a linear factor... |
| plyrem 26267 | The polynomial remainder t... |
| facth 26268 | The factor theorem. If a ... |
| fta1lem 26269 | Lemma for ~ fta1 . (Contr... |
| fta1 26270 | The easy direction of the ... |
| quotcan 26271 | Exact division with a mult... |
| vieta1lem1 26272 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26273 | Lemma for ~ vieta1 : induc... |
| vieta1 26274 | The first-order Vieta's fo... |
| plyexmo 26275 | An infinite set of values ... |
| elaa 26278 | Elementhood in the set of ... |
| aacn 26279 | An algebraic number is a c... |
| aasscn 26280 | The algebraic numbers are ... |
| elqaalem1 26281 | Lemma for ~ elqaa . The f... |
| elqaalem2 26282 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26283 | Lemma for ~ elqaa . (Cont... |
| elqaa 26284 | The set of numbers generat... |
| qaa 26285 | Every rational number is a... |
| qssaa 26286 | The rational numbers are c... |
| iaa 26287 | The imaginary unit is alge... |
| aareccl 26288 | The reciprocal of an algeb... |
| aacjcl 26289 | The conjugate of an algebr... |
| aannenlem1 26290 | Lemma for ~ aannen . (Con... |
| aannenlem2 26291 | Lemma for ~ aannen . (Con... |
| aannenlem3 26292 | The algebraic numbers are ... |
| aannen 26293 | The algebraic numbers are ... |
| aalioulem1 26294 | Lemma for ~ aaliou . An i... |
| aalioulem2 26295 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26296 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26297 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26298 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26299 | Lemma for ~ aaliou . (Con... |
| aaliou 26300 | Liouville's theorem on dio... |
| geolim3 26301 | Geometric series convergen... |
| aaliou2 26302 | Liouville's approximation ... |
| aaliou2b 26303 | Liouville's approximation ... |
| aaliou3lem1 26304 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26305 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26306 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26307 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26308 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26309 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26310 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26311 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26312 | Example of a "Liouville nu... |
| aaliou3 26313 | Example of a "Liouville nu... |
| taylfvallem1 26318 | Lemma for ~ taylfval . (C... |
| taylfvallem 26319 | Lemma for ~ taylfval . (C... |
| taylfval 26320 | Define the Taylor polynomi... |
| eltayl 26321 | Value of the Taylor series... |
| taylf 26322 | The Taylor series defines ... |
| tayl0 26323 | The Taylor series is alway... |
| taylplem1 26324 | Lemma for ~ taylpfval and ... |
| taylplem2 26325 | Lemma for ~ taylpfval and ... |
| taylpfval 26326 | Define the Taylor polynomi... |
| taylpf 26327 | The Taylor polynomial is a... |
| taylpval 26328 | Value of the Taylor polyno... |
| taylply2 26329 | The Taylor polynomial is a... |
| taylply2OLD 26330 | Obsolete version of ~ tayl... |
| taylply 26331 | The Taylor polynomial is a... |
| dvtaylp 26332 | The derivative of the Tayl... |
| dvntaylp 26333 | The ` M ` -th derivative o... |
| dvntaylp0 26334 | The first ` N ` derivative... |
| taylthlem1 26335 | Lemma for ~ taylth . This... |
| taylthlem2 26336 | Lemma for ~ taylth . (Con... |
| taylthlem2OLD 26337 | Obsolete version of ~ tayl... |
| taylth 26338 | Taylor's theorem. The Tay... |
| ulmrel 26341 | The uniform limit relation... |
| ulmscl 26342 | Closure of the base set in... |
| ulmval 26343 | Express the predicate: Th... |
| ulmcl 26344 | Closure of a uniform limit... |
| ulmf 26345 | Closure of a uniform limit... |
| ulmpm 26346 | Closure of a uniform limit... |
| ulmf2 26347 | Closure of a uniform limit... |
| ulm2 26348 | Simplify ~ ulmval when ` F... |
| ulmi 26349 | The uniform limit property... |
| ulmclm 26350 | A uniform limit of functio... |
| ulmres 26351 | A sequence of functions co... |
| ulmshftlem 26352 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26353 | A sequence of functions co... |
| ulm0 26354 | Every function converges u... |
| ulmuni 26355 | A sequence of functions un... |
| ulmdm 26356 | Two ways to express that a... |
| ulmcaulem 26357 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26358 | A sequence of functions co... |
| ulmcau2 26359 | A sequence of functions co... |
| ulmss 26360 | A uniform limit of functio... |
| ulmbdd 26361 | A uniform limit of bounded... |
| ulmcn 26362 | A uniform limit of continu... |
| ulmdvlem1 26363 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26364 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26365 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26366 | If ` F ` is a sequence of ... |
| mtest 26367 | The Weierstrass M-test. I... |
| mtestbdd 26368 | Given the hypotheses of th... |
| mbfulm 26369 | A uniform limit of measura... |
| iblulm 26370 | A uniform limit of integra... |
| itgulm 26371 | A uniform limit of integra... |
| itgulm2 26372 | A uniform limit of integra... |
| pserval 26373 | Value of the function ` G ... |
| pserval2 26374 | Value of the function ` G ... |
| psergf 26375 | The sequence of terms in t... |
| radcnvlem1 26376 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26377 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26378 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26379 | Zero is always a convergen... |
| radcnvcl 26380 | The radius of convergence ... |
| radcnvlt1 26381 | If ` X ` is within the ope... |
| radcnvlt2 26382 | If ` X ` is within the ope... |
| radcnvle 26383 | If ` X ` is a convergent p... |
| dvradcnv 26384 | The radius of convergence ... |
| pserulm 26385 | If ` S ` is a region conta... |
| psercn2 26386 | Since by ~ pserulm the ser... |
| psercn2OLD 26387 | Obsolete version of ~ pser... |
| psercnlem2 26388 | Lemma for ~ psercn . (Con... |
| psercnlem1 26389 | Lemma for ~ psercn . (Con... |
| psercn 26390 | An infinite series converg... |
| pserdvlem1 26391 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26392 | Lemma for ~ pserdv . (Con... |
| pserdv 26393 | The derivative of a power ... |
| pserdv2 26394 | The derivative of a power ... |
| abelthlem1 26395 | Lemma for ~ abelth . (Con... |
| abelthlem2 26396 | Lemma for ~ abelth . The ... |
| abelthlem3 26397 | Lemma for ~ abelth . (Con... |
| abelthlem4 26398 | Lemma for ~ abelth . (Con... |
| abelthlem5 26399 | Lemma for ~ abelth . (Con... |
| abelthlem6 26400 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26401 | Lemma for ~ abelth . (Con... |
| abelthlem7 26402 | Lemma for ~ abelth . (Con... |
| abelthlem8 26403 | Lemma for ~ abelth . (Con... |
| abelthlem9 26404 | Lemma for ~ abelth . By a... |
| abelth 26405 | Abel's theorem. If the po... |
| abelth2 26406 | Abel's theorem, restricted... |
| efcn 26407 | The exponential function i... |
| sincn 26408 | Sine is continuous. (Cont... |
| coscn 26409 | Cosine is continuous. (Co... |
| reeff1olem 26410 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26411 | The real exponential funct... |
| reefiso 26412 | The exponential function o... |
| efcvx 26413 | The exponential function o... |
| reefgim 26414 | The exponential function i... |
| pilem1 26415 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26416 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26417 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26418 | ` _pi ` is between 2 and 4... |
| sinpi 26419 | The sine of ` _pi ` is 0. ... |
| pire 26420 | ` _pi ` is a real number. ... |
| picn 26421 | ` _pi ` is a complex numbe... |
| pipos 26422 | ` _pi ` is positive. (Con... |
| pine0 26423 | ` _pi ` is nonzero. (Cont... |
| pirp 26424 | ` _pi ` is a positive real... |
| negpicn 26425 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26426 | Lemma for ~ sinhalfpi and ... |
| halfpire 26427 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26428 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26429 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26430 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26431 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26432 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26433 | The cosine of ` -u _pi / 2... |
| efhalfpi 26434 | The exponential of ` _i _p... |
| cospi 26435 | The cosine of ` _pi ` is `... |
| efipi 26436 | The exponential of ` _i x.... |
| eulerid 26437 | Euler's identity. (Contri... |
| sin2pi 26438 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26439 | The cosine of ` 2 _pi ` is... |
| ef2pi 26440 | The exponential of ` 2 _pi... |
| ef2kpi 26441 | If ` K ` is an integer, th... |
| efper 26442 | The exponential function i... |
| sinperlem 26443 | Lemma for ~ sinper and ~ c... |
| sinper 26444 | The sine function is perio... |
| cosper 26445 | The cosine function is per... |
| sin2kpi 26446 | If ` K ` is an integer, th... |
| cos2kpi 26447 | If ` K ` is an integer, th... |
| sin2pim 26448 | Sine of a number subtracte... |
| cos2pim 26449 | Cosine of a number subtrac... |
| sinmpi 26450 | Sine of a number less ` _p... |
| cosmpi 26451 | Cosine of a number less ` ... |
| sinppi 26452 | Sine of a number plus ` _p... |
| cosppi 26453 | Cosine of a number plus ` ... |
| efimpi 26454 | The exponential function a... |
| sinhalfpip 26455 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26456 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26457 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26458 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26459 | Ptolemy's Theorem. This t... |
| sincosq1lem 26460 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26461 | The signs of the sine and ... |
| sincosq2sgn 26462 | The signs of the sine and ... |
| sincosq3sgn 26463 | The signs of the sine and ... |
| sincosq4sgn 26464 | The signs of the sine and ... |
| coseq00topi 26465 | Location of the zeroes of ... |
| coseq0negpitopi 26466 | Location of the zeroes of ... |
| tanrpcl 26467 | Positive real closure of t... |
| tangtx 26468 | The tangent function is gr... |
| tanabsge 26469 | The tangent function is gr... |
| sinq12gt0 26470 | The sine of a number stric... |
| sinq12ge0 26471 | The sine of a number betwe... |
| sinq34lt0t 26472 | The sine of a number stric... |
| cosq14gt0 26473 | The cosine of a number str... |
| cosq14ge0 26474 | The cosine of a number bet... |
| sincosq1eq 26475 | Complementarity of the sin... |
| sincos4thpi 26476 | The sine and cosine of ` _... |
| tan4thpi 26477 | The tangent of ` _pi / 4 `... |
| tan4thpiOLD 26478 | Obsolete version of ~ tan4... |
| sincos6thpi 26479 | The sine and cosine of ` _... |
| sincos3rdpi 26480 | The sine and cosine of ` _... |
| pigt3 26481 | ` _pi ` is greater than 3.... |
| pige3 26482 | ` _pi ` is greater than or... |
| pige3ALT 26483 | Alternate proof of ~ pige3... |
| abssinper 26484 | The absolute value of sine... |
| sinkpi 26485 | The sine of an integer mul... |
| coskpi 26486 | The absolute value of the ... |
| sineq0 26487 | A complex number whose sin... |
| coseq1 26488 | A complex number whose cos... |
| cos02pilt1 26489 | Cosine is less than one be... |
| cosq34lt1 26490 | Cosine is less than one in... |
| efeq1 26491 | A complex number whose exp... |
| cosne0 26492 | The cosine function has no... |
| cosordlem 26493 | Lemma for ~ cosord . (Con... |
| cosord 26494 | Cosine is decreasing over ... |
| cos0pilt1 26495 | Cosine is between minus on... |
| cos11 26496 | Cosine is one-to-one over ... |
| sinord 26497 | Sine is increasing over th... |
| recosf1o 26498 | The cosine function is a b... |
| resinf1o 26499 | The sine function is a bij... |
| tanord1 26500 | The tangent function is st... |
| tanord 26501 | The tangent function is st... |
| tanregt0 26502 | The real part of the tange... |
| negpitopissre 26503 | The interval ` ( -u _pi (,... |
| efgh 26504 | The exponential function o... |
| efif1olem1 26505 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26506 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26507 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26508 | The exponential function o... |
| efif1o 26509 | The exponential function o... |
| efifo 26510 | The exponential function o... |
| eff1olem 26511 | The exponential function m... |
| eff1o 26512 | The exponential function m... |
| efabl 26513 | The image of a subgroup of... |
| efsubm 26514 | The image of a subgroup of... |
| circgrp 26515 | The circle group ` T ` is ... |
| circsubm 26516 | The circle group ` T ` is ... |
| logrn 26521 | The range of the natural l... |
| ellogrn 26522 | Write out the property ` A... |
| dflog2 26523 | The natural logarithm func... |
| relogrn 26524 | The range of the natural l... |
| logrncn 26525 | The range of the natural l... |
| eff1o2 26526 | The exponential function r... |
| logf1o 26527 | The natural logarithm func... |
| dfrelog 26528 | The natural logarithm func... |
| relogf1o 26529 | The natural logarithm func... |
| logrncl 26530 | Closure of the natural log... |
| logcl 26531 | Closure of the natural log... |
| logimcl 26532 | Closure of the imaginary p... |
| logcld 26533 | The logarithm of a nonzero... |
| logimcld 26534 | The imaginary part of the ... |
| logimclad 26535 | The imaginary part of the ... |
| abslogimle 26536 | The imaginary part of the ... |
| logrnaddcl 26537 | The range of the natural l... |
| relogcl 26538 | Closure of the natural log... |
| eflog 26539 | Relationship between the n... |
| logeq0im1 26540 | If the logarithm of a numb... |
| logccne0 26541 | The logarithm isn't 0 if i... |
| logne0 26542 | Logarithm of a non-1 posit... |
| reeflog 26543 | Relationship between the n... |
| logef 26544 | Relationship between the n... |
| relogef 26545 | Relationship between the n... |
| logeftb 26546 | Relationship between the n... |
| relogeftb 26547 | Relationship between the n... |
| log1 26548 | The natural logarithm of `... |
| loge 26549 | The natural logarithm of `... |
| logi 26550 | The natural logarithm of `... |
| logneg 26551 | The natural logarithm of a... |
| logm1 26552 | The natural logarithm of n... |
| lognegb 26553 | If a number has imaginary ... |
| relogoprlem 26554 | Lemma for ~ relogmul and ~... |
| relogmul 26555 | The natural logarithm of t... |
| relogdiv 26556 | The natural logarithm of t... |
| explog 26557 | Exponentiation of a nonzer... |
| reexplog 26558 | Exponentiation of a positi... |
| relogexp 26559 | The natural logarithm of p... |
| relog 26560 | Real part of a logarithm. ... |
| relogiso 26561 | The natural logarithm func... |
| reloggim 26562 | The natural logarithm is a... |
| logltb 26563 | The natural logarithm func... |
| logfac 26564 | The logarithm of a factori... |
| eflogeq 26565 | Solve an equation involvin... |
| logleb 26566 | Natural logarithm preserve... |
| rplogcl 26567 | Closure of the logarithm f... |
| logge0 26568 | The logarithm of a number ... |
| logcj 26569 | The natural logarithm dist... |
| efiarg 26570 | The exponential of the "ar... |
| cosargd 26571 | The cosine of the argument... |
| cosarg0d 26572 | The cosine of the argument... |
| argregt0 26573 | Closure of the argument of... |
| argrege0 26574 | Closure of the argument of... |
| argimgt0 26575 | Closure of the argument of... |
| argimlt0 26576 | Closure of the argument of... |
| logimul 26577 | Multiplying a number by ` ... |
| logneg2 26578 | The logarithm of the negat... |
| logmul2 26579 | Generalization of ~ relogm... |
| logdiv2 26580 | Generalization of ~ relogd... |
| abslogle 26581 | Bound on the magnitude of ... |
| tanarg 26582 | The basic relation between... |
| logdivlti 26583 | The ` log x / x ` function... |
| logdivlt 26584 | The ` log x / x ` function... |
| logdivle 26585 | The ` log x / x ` function... |
| relogcld 26586 | Closure of the natural log... |
| reeflogd 26587 | Relationship between the n... |
| relogmuld 26588 | The natural logarithm of t... |
| relogdivd 26589 | The natural logarithm of t... |
| logled 26590 | Natural logarithm preserve... |
| relogefd 26591 | Relationship between the n... |
| rplogcld 26592 | Closure of the logarithm f... |
| logge0d 26593 | The logarithm of a number ... |
| logge0b 26594 | The logarithm of a number ... |
| loggt0b 26595 | The logarithm of a number ... |
| logle1b 26596 | The logarithm of a number ... |
| loglt1b 26597 | The logarithm of a number ... |
| divlogrlim 26598 | The inverse logarithm func... |
| logno1 26599 | The logarithm function is ... |
| dvrelog 26600 | The derivative of the real... |
| relogcn 26601 | The real logarithm functio... |
| ellogdm 26602 | Elementhood in the "contin... |
| logdmn0 26603 | A number in the continuous... |
| logdmnrp 26604 | A number in the continuous... |
| logdmss 26605 | The continuity domain of `... |
| logcnlem2 26606 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26607 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26608 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26609 | Lemma for ~ logcn . (Cont... |
| logcn 26610 | The logarithm function is ... |
| dvloglem 26611 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26612 | The "continuous domain" of... |
| logf1o2 26613 | The logarithm maps its con... |
| dvlog 26614 | The derivative of the comp... |
| dvlog2lem 26615 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26616 | The derivative of the comp... |
| advlog 26617 | The antiderivative of the ... |
| advlogexp 26618 | The antiderivative of a po... |
| efopnlem1 26619 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26620 | Lemma for ~ efopn . (Cont... |
| efopn 26621 | The exponential map is an ... |
| logtayllem 26622 | Lemma for ~ logtayl . (Co... |
| logtayl 26623 | The Taylor series for ` -u... |
| logtaylsum 26624 | The Taylor series for ` -u... |
| logtayl2 26625 | Power series expression fo... |
| logccv 26626 | The natural logarithm func... |
| cxpval 26627 | Value of the complex power... |
| cxpef 26628 | Value of the complex power... |
| 0cxp 26629 | Value of the complex power... |
| cxpexpz 26630 | Relate the complex power f... |
| cxpexp 26631 | Relate the complex power f... |
| logcxp 26632 | Logarithm of a complex pow... |
| cxp0 26633 | Value of the complex power... |
| cxp1 26634 | Value of the complex power... |
| 1cxp 26635 | Value of the complex power... |
| ecxp 26636 | Write the exponential func... |
| cxpcl 26637 | Closure of the complex pow... |
| recxpcl 26638 | Real closure of the comple... |
| rpcxpcl 26639 | Positive real closure of t... |
| cxpne0 26640 | Complex exponentiation is ... |
| cxpeq0 26641 | Complex exponentiation is ... |
| cxpadd 26642 | Sum of exponents law for c... |
| cxpp1 26643 | Value of a nonzero complex... |
| cxpneg 26644 | Value of a complex number ... |
| cxpsub 26645 | Exponent subtraction law f... |
| cxpge0 26646 | Nonnegative exponentiation... |
| mulcxplem 26647 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26648 | Complex exponentiation of ... |
| cxprec 26649 | Complex exponentiation of ... |
| divcxp 26650 | Complex exponentiation of ... |
| cxpmul 26651 | Product of exponents law f... |
| cxpmul2 26652 | Product of exponents law f... |
| cxproot 26653 | The complex power function... |
| cxpmul2z 26654 | Generalize ~ cxpmul2 to ne... |
| abscxp 26655 | Absolute value of a power,... |
| abscxp2 26656 | Absolute value of a power,... |
| cxplt 26657 | Ordering property for comp... |
| cxple 26658 | Ordering property for comp... |
| cxplea 26659 | Ordering property for comp... |
| cxple2 26660 | Ordering property for comp... |
| cxplt2 26661 | Ordering property for comp... |
| cxple2a 26662 | Ordering property for comp... |
| cxplt3 26663 | Ordering property for comp... |
| cxple3 26664 | Ordering property for comp... |
| cxpsqrtlem 26665 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26666 | The complex exponential fu... |
| logsqrt 26667 | Logarithm of a square root... |
| cxp0d 26668 | Value of the complex power... |
| cxp1d 26669 | Value of the complex power... |
| 1cxpd 26670 | Value of the complex power... |
| cxpcld 26671 | Closure of the complex pow... |
| cxpmul2d 26672 | Product of exponents law f... |
| 0cxpd 26673 | Value of the complex power... |
| cxpexpzd 26674 | Relate the complex power f... |
| cxpefd 26675 | Value of the complex power... |
| cxpne0d 26676 | Complex exponentiation is ... |
| cxpp1d 26677 | Value of a nonzero complex... |
| cxpnegd 26678 | Value of a complex number ... |
| cxpmul2zd 26679 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26680 | Sum of exponents law for c... |
| cxpsubd 26681 | Exponent subtraction law f... |
| cxpltd 26682 | Ordering property for comp... |
| cxpled 26683 | Ordering property for comp... |
| cxplead 26684 | Ordering property for comp... |
| divcxpd 26685 | Complex exponentiation of ... |
| recxpcld 26686 | Positive real closure of t... |
| cxpge0d 26687 | Nonnegative exponentiation... |
| cxple2ad 26688 | Ordering property for comp... |
| cxplt2d 26689 | Ordering property for comp... |
| cxple2d 26690 | Ordering property for comp... |
| mulcxpd 26691 | Complex exponentiation of ... |
| recxpf1lem 26692 | Complex exponentiation on ... |
| cxpsqrtth 26693 | Square root theorem over t... |
| 2irrexpq 26694 | There exist irrational num... |
| cxprecd 26695 | Complex exponentiation of ... |
| rpcxpcld 26696 | Positive real closure of t... |
| logcxpd 26697 | Logarithm of a complex pow... |
| cxplt3d 26698 | Ordering property for comp... |
| cxple3d 26699 | Ordering property for comp... |
| cxpmuld 26700 | Product of exponents law f... |
| cxpgt0d 26701 | A positive real raised to ... |
| cxpcom 26702 | Commutative law for real e... |
| dvcxp1 26703 | The derivative of a comple... |
| dvcxp2 26704 | The derivative of a comple... |
| dvsqrt 26705 | The derivative of the real... |
| dvcncxp1 26706 | Derivative of complex powe... |
| dvcnsqrt 26707 | Derivative of square root ... |
| cxpcn 26708 | Domain of continuity of th... |
| cxpcnOLD 26709 | Obsolete version of ~ cxpc... |
| cxpcn2 26710 | Continuity of the complex ... |
| cxpcn3lem 26711 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26712 | Extend continuity of the c... |
| resqrtcn 26713 | Continuity of the real squ... |
| sqrtcn 26714 | Continuity of the square r... |
| cxpaddlelem 26715 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26716 | Ordering property for comp... |
| abscxpbnd 26717 | Bound on the absolute valu... |
| root1id 26718 | Property of an ` N ` -th r... |
| root1eq1 26719 | The only powers of an ` N ... |
| root1cj 26720 | Within the ` N ` -th roots... |
| cxpeq 26721 | Solve an equation involvin... |
| zrtelqelz 26722 | If the ` N ` -th root of a... |
| zrtdvds 26723 | A positive integer root di... |
| rtprmirr 26724 | The root of a prime number... |
| loglesqrt 26725 | An upper bound on the loga... |
| logreclem 26726 | Symmetry of the natural lo... |
| logrec 26727 | Logarithm of a reciprocal ... |
| logbval 26730 | Define the value of the ` ... |
| logbcl 26731 | General logarithm closure.... |
| logbid1 26732 | General logarithm is 1 whe... |
| logb1 26733 | The logarithm of ` 1 ` to ... |
| elogb 26734 | The general logarithm of a... |
| logbchbase 26735 | Change of base for logarit... |
| relogbval 26736 | Value of the general logar... |
| relogbcl 26737 | Closure of the general log... |
| relogbzcl 26738 | Closure of the general log... |
| relogbreexp 26739 | Power law for the general ... |
| relogbzexp 26740 | Power law for the general ... |
| relogbmul 26741 | The logarithm of the produ... |
| relogbmulexp 26742 | The logarithm of the produ... |
| relogbdiv 26743 | The logarithm of the quoti... |
| relogbexp 26744 | Identity law for general l... |
| nnlogbexp 26745 | Identity law for general l... |
| logbrec 26746 | Logarithm of a reciprocal ... |
| logbleb 26747 | The general logarithm func... |
| logblt 26748 | The general logarithm func... |
| relogbcxp 26749 | Identity law for the gener... |
| cxplogb 26750 | Identity law for the gener... |
| relogbcxpb 26751 | The logarithm is the inver... |
| logbmpt 26752 | The general logarithm to a... |
| logbf 26753 | The general logarithm to a... |
| logbfval 26754 | The general logarithm of a... |
| relogbf 26755 | The general logarithm to a... |
| logblog 26756 | The general logarithm to t... |
| logbgt0b 26757 | The logarithm of a positiv... |
| logbgcd1irr 26758 | The logarithm of an intege... |
| 2logb9irr 26759 | Example for ~ logbgcd1irr ... |
| logbprmirr 26760 | The logarithm of a prime t... |
| 2logb3irr 26761 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26762 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26763 | The square root of two to ... |
| 2irrexpqALT 26764 | Alternate proof of ~ 2irre... |
| angval 26765 | Define the angle function,... |
| angcan 26766 | Cancel a constant multipli... |
| angneg 26767 | Cancel a negative sign in ... |
| angvald 26768 | The (signed) angle between... |
| angcld 26769 | The (signed) angle between... |
| angrteqvd 26770 | Two vectors are at a right... |
| cosangneg2d 26771 | The cosine of the angle be... |
| angrtmuld 26772 | Perpendicularity of two ve... |
| ang180lem1 26773 | Lemma for ~ ang180 . Show... |
| ang180lem2 26774 | Lemma for ~ ang180 . Show... |
| ang180lem3 26775 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26776 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26777 | Lemma for ~ ang180 : Redu... |
| ang180 26778 | The sum of angles ` m A B ... |
| lawcoslem1 26779 | Lemma for ~ lawcos . Here... |
| lawcos 26780 | Law of cosines (also known... |
| pythag 26781 | Pythagorean theorem. Give... |
| isosctrlem1 26782 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26783 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26784 | Lemma for ~ isosctr . Cor... |
| isosctr 26785 | Isosceles triangle theorem... |
| ssscongptld 26786 | If two triangles have equa... |
| affineequiv 26787 | Equivalence between two wa... |
| affineequiv2 26788 | Equivalence between two wa... |
| affineequiv3 26789 | Equivalence between two wa... |
| affineequiv4 26790 | Equivalence between two wa... |
| affineequivne 26791 | Equivalence between two wa... |
| angpieqvdlem 26792 | Equivalence used in the pr... |
| angpieqvdlem2 26793 | Equivalence used in ~ angp... |
| angpined 26794 | If the angle at ABC is ` _... |
| angpieqvd 26795 | The angle ABC is ` _pi ` i... |
| chordthmlem 26796 | If ` M ` is the midpoint o... |
| chordthmlem2 26797 | If M is the midpoint of AB... |
| chordthmlem3 26798 | If M is the midpoint of AB... |
| chordthmlem4 26799 | If P is on the segment AB ... |
| chordthmlem5 26800 | If P is on the segment AB ... |
| chordthm 26801 | The intersecting chords th... |
| heron 26802 | Heron's formula gives the ... |
| quad2 26803 | The quadratic equation, wi... |
| quad 26804 | The quadratic equation. (... |
| 1cubrlem 26805 | The cube roots of unity. ... |
| 1cubr 26806 | The cube roots of unity. ... |
| dcubic1lem 26807 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26808 | Reverse direction of ~ dcu... |
| dcubic1 26809 | Forward direction of ~ dcu... |
| dcubic 26810 | Solutions to the depressed... |
| mcubic 26811 | Solutions to a monic cubic... |
| cubic2 26812 | The solution to the genera... |
| cubic 26813 | The cubic equation, which ... |
| binom4 26814 | Work out a quartic binomia... |
| dquartlem1 26815 | Lemma for ~ dquart . (Con... |
| dquartlem2 26816 | Lemma for ~ dquart . (Con... |
| dquart 26817 | Solve a depressed quartic ... |
| quart1cl 26818 | Closure lemmas for ~ quart... |
| quart1lem 26819 | Lemma for ~ quart1 . (Con... |
| quart1 26820 | Depress a quartic equation... |
| quartlem1 26821 | Lemma for ~ quart . (Cont... |
| quartlem2 26822 | Closure lemmas for ~ quart... |
| quartlem3 26823 | Closure lemmas for ~ quart... |
| quartlem4 26824 | Closure lemmas for ~ quart... |
| quart 26825 | The quartic equation, writ... |
| asinlem 26832 | The argument to the logari... |
| asinlem2 26833 | The argument to the logari... |
| asinlem3a 26834 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26835 | The argument to the logari... |
| asinf 26836 | Domain and codomain of the... |
| asincl 26837 | Closure for the arcsin fun... |
| acosf 26838 | Domain and codoamin of the... |
| acoscl 26839 | Closure for the arccos fun... |
| atandm 26840 | Since the property is a li... |
| atandm2 26841 | This form of ~ atandm is a... |
| atandm3 26842 | A compact form of ~ atandm... |
| atandm4 26843 | A compact form of ~ atandm... |
| atanf 26844 | Domain and codoamin of the... |
| atancl 26845 | Closure for the arctan fun... |
| asinval 26846 | Value of the arcsin functi... |
| acosval 26847 | Value of the arccos functi... |
| atanval 26848 | Value of the arctan functi... |
| atanre 26849 | A real number is in the do... |
| asinneg 26850 | The arcsine function is od... |
| acosneg 26851 | The negative symmetry rela... |
| efiasin 26852 | The exponential of the arc... |
| sinasin 26853 | The arcsine function is an... |
| cosacos 26854 | The arccosine function is ... |
| asinsinlem 26855 | Lemma for ~ asinsin . (Co... |
| asinsin 26856 | The arcsine function compo... |
| acoscos 26857 | The arccosine function is ... |
| asin1 26858 | The arcsine of ` 1 ` is ` ... |
| acos1 26859 | The arccosine of ` 1 ` is ... |
| reasinsin 26860 | The arcsine function compo... |
| asinsinb 26861 | Relationship between sine ... |
| acoscosb 26862 | Relationship between cosin... |
| asinbnd 26863 | The arcsine function has r... |
| acosbnd 26864 | The arccosine function has... |
| asinrebnd 26865 | Bounds on the arcsine func... |
| asinrecl 26866 | The arcsine function is re... |
| acosrecl 26867 | The arccosine function is ... |
| cosasin 26868 | The cosine of the arcsine ... |
| sinacos 26869 | The sine of the arccosine ... |
| atandmneg 26870 | The domain of the arctange... |
| atanneg 26871 | The arctangent function is... |
| atan0 26872 | The arctangent of zero is ... |
| atandmcj 26873 | The arctangent function di... |
| atancj 26874 | The arctangent function di... |
| atanrecl 26875 | The arctangent function is... |
| efiatan 26876 | Value of the exponential o... |
| atanlogaddlem 26877 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26878 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26879 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26880 | A variation on ~ atanlogad... |
| efiatan2 26881 | Value of the exponential o... |
| 2efiatan 26882 | Value of the exponential o... |
| tanatan 26883 | The arctangent function is... |
| atandmtan 26884 | The tangent function has r... |
| cosatan 26885 | The cosine of an arctangen... |
| cosatanne0 26886 | The arctangent function ha... |
| atantan 26887 | The arctangent function is... |
| atantanb 26888 | Relationship between tange... |
| atanbndlem 26889 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26890 | The arctangent function is... |
| atanord 26891 | The arctangent function is... |
| atan1 26892 | The arctangent of ` 1 ` is... |
| bndatandm 26893 | A point in the open unit d... |
| atans 26894 | The "domain of continuity"... |
| atans2 26895 | It suffices to show that `... |
| atansopn 26896 | The domain of continuity o... |
| atansssdm 26897 | The domain of continuity o... |
| ressatans 26898 | The real number line is a ... |
| dvatan 26899 | The derivative of the arct... |
| atancn 26900 | The arctangent is a contin... |
| atantayl 26901 | The Taylor series for ` ar... |
| atantayl2 26902 | The Taylor series for ` ar... |
| atantayl3 26903 | The Taylor series for ` ar... |
| leibpilem1 26904 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26905 | The Leibniz formula for ` ... |
| leibpi 26906 | The Leibniz formula for ` ... |
| leibpisum 26907 | The Leibniz formula for ` ... |
| log2cnv 26908 | Using the Taylor series fo... |
| log2tlbnd 26909 | Bound the error term in th... |
| log2ublem1 26910 | Lemma for ~ log2ub . The ... |
| log2ublem2 26911 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26912 | Lemma for ~ log2ub . In d... |
| log2ub 26913 | ` log 2 ` is less than ` 2... |
| log2le1 26914 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26915 | Lemma for ~ birthday . (C... |
| birthdaylem2 26916 | For general ` N ` and ` K ... |
| birthdaylem3 26917 | For general ` N ` and ` K ... |
| birthday 26918 | The Birthday Problem. The... |
| dmarea 26921 | The domain of the area fun... |
| areambl 26922 | The fibers of a measurable... |
| areass 26923 | A measurable region is a s... |
| dfarea 26924 | Rewrite ~ df-area self-ref... |
| areaf 26925 | Area measurement is a func... |
| areacl 26926 | The area of a measurable r... |
| areage0 26927 | The area of a measurable r... |
| areaval 26928 | The area of a measurable r... |
| rlimcnp 26929 | Relate a limit of a real-v... |
| rlimcnp2 26930 | Relate a limit of a real-v... |
| rlimcnp3 26931 | Relate a limit of a real-v... |
| xrlimcnp 26932 | Relate a limit of a real-v... |
| efrlim 26933 | The limit of the sequence ... |
| efrlimOLD 26934 | Obsolete version of ~ efrl... |
| dfef2 26935 | The limit of the sequence ... |
| cxplim 26936 | A power to a negative expo... |
| sqrtlim 26937 | The inverse square root fu... |
| rlimcxp 26938 | Any power to a positive ex... |
| o1cxp 26939 | An eventually bounded func... |
| cxp2limlem 26940 | A linear factor grows slow... |
| cxp2lim 26941 | Any power grows slower tha... |
| cxploglim 26942 | The logarithm grows slower... |
| cxploglim2 26943 | Every power of the logarit... |
| divsqrtsumlem 26944 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26945 | The function ` F ` used in... |
| divsqrsum 26946 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26947 | A bound on the distance of... |
| divsqrtsumo1 26948 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26949 | Closure of a 0-1 linear co... |
| scvxcvx 26950 | A strictly convex function... |
| jensenlem1 26951 | Lemma for ~ jensen . (Con... |
| jensenlem2 26952 | Lemma for ~ jensen . (Con... |
| jensen 26953 | Jensen's inequality, a fin... |
| amgmlem 26954 | Lemma for ~ amgm . (Contr... |
| amgm 26955 | Inequality of arithmetic a... |
| logdifbnd 26958 | Bound on the difference of... |
| logdiflbnd 26959 | Lower bound on the differe... |
| emcllem1 26960 | Lemma for ~ emcl . The se... |
| emcllem2 26961 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26962 | Lemma for ~ emcl . The fu... |
| emcllem4 26963 | Lemma for ~ emcl . The di... |
| emcllem5 26964 | Lemma for ~ emcl . The pa... |
| emcllem6 26965 | Lemma for ~ emcl . By the... |
| emcllem7 26966 | Lemma for ~ emcl and ~ har... |
| emcl 26967 | Closure and bounds for the... |
| harmonicbnd 26968 | A bound on the harmonic se... |
| harmonicbnd2 26969 | A bound on the harmonic se... |
| emre 26970 | The Euler-Mascheroni const... |
| emgt0 26971 | The Euler-Mascheroni const... |
| harmonicbnd3 26972 | A bound on the harmonic se... |
| harmoniclbnd 26973 | A bound on the harmonic se... |
| harmonicubnd 26974 | A bound on the harmonic se... |
| harmonicbnd4 26975 | The asymptotic behavior of... |
| fsumharmonic 26976 | Bound a finite sum based o... |
| zetacvg 26979 | The zeta series is converg... |
| eldmgm 26986 | Elementhood in the set of ... |
| dmgmaddn0 26987 | If ` A ` is not a nonposit... |
| dmlogdmgm 26988 | If ` A ` is in the continu... |
| rpdmgm 26989 | A positive real number is ... |
| dmgmn0 26990 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26991 | If ` A ` is not a nonposit... |
| dmgmdivn0 26992 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26993 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26994 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26995 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26996 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26997 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26998 | The series ` G ` is unifor... |
| lgamgulm 26999 | The series ` G ` is unifor... |
| lgamgulm2 27000 | Rewrite the limit of the s... |
| lgambdd 27001 | The log-Gamma function is ... |
| lgamucov 27002 | The ` U ` regions used in ... |
| lgamucov2 27003 | The ` U ` regions used in ... |
| lgamcvglem 27004 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27005 | The log-Gamma function is ... |
| lgamf 27006 | The log-Gamma function is ... |
| gamf 27007 | The Gamma function is a co... |
| gamcl 27008 | The exponential of the log... |
| eflgam 27009 | The exponential of the log... |
| gamne0 27010 | The Gamma function is neve... |
| igamval 27011 | Value of the inverse Gamma... |
| igamz 27012 | Value of the inverse Gamma... |
| igamgam 27013 | Value of the inverse Gamma... |
| igamlgam 27014 | Value of the inverse Gamma... |
| igamf 27015 | Closure of the inverse Gam... |
| igamcl 27016 | Closure of the inverse Gam... |
| gamigam 27017 | The Gamma function is the ... |
| lgamcvg 27018 | The series ` G ` converges... |
| lgamcvg2 27019 | The series ` G ` converges... |
| gamcvg 27020 | The pointwise exponential ... |
| lgamp1 27021 | The functional equation of... |
| gamp1 27022 | The functional equation of... |
| gamcvg2lem 27023 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27024 | An infinite product expres... |
| regamcl 27025 | The Gamma function is real... |
| relgamcl 27026 | The log-Gamma function is ... |
| rpgamcl 27027 | The log-Gamma function is ... |
| lgam1 27028 | The log-Gamma function at ... |
| gam1 27029 | The log-Gamma function at ... |
| facgam 27030 | The Gamma function general... |
| gamfac 27031 | The Gamma function general... |
| wilthlem1 27032 | The only elements that are... |
| wilthlem2 27033 | Lemma for ~ wilth : induct... |
| wilthlem3 27034 | Lemma for ~ wilth . Here ... |
| wilth 27035 | Wilson's theorem. A numbe... |
| wilthimp 27036 | The forward implication of... |
| ftalem1 27037 | Lemma for ~ fta : "growth... |
| ftalem2 27038 | Lemma for ~ fta . There e... |
| ftalem3 27039 | Lemma for ~ fta . There e... |
| ftalem4 27040 | Lemma for ~ fta : Closure... |
| ftalem5 27041 | Lemma for ~ fta : Main pr... |
| ftalem6 27042 | Lemma for ~ fta : Dischar... |
| ftalem7 27043 | Lemma for ~ fta . Shift t... |
| fta 27044 | The Fundamental Theorem of... |
| basellem1 27045 | Lemma for ~ basel . Closu... |
| basellem2 27046 | Lemma for ~ basel . Show ... |
| basellem3 27047 | Lemma for ~ basel . Using... |
| basellem4 27048 | Lemma for ~ basel . By ~ ... |
| basellem5 27049 | Lemma for ~ basel . Using... |
| basellem6 27050 | Lemma for ~ basel . The f... |
| basellem7 27051 | Lemma for ~ basel . The f... |
| basellem8 27052 | Lemma for ~ basel . The f... |
| basellem9 27053 | Lemma for ~ basel . Since... |
| basel 27054 | The sum of the inverse squ... |
| efnnfsumcl 27067 | Finite sum closure in the ... |
| ppisval 27068 | The set of primes less tha... |
| ppisval2 27069 | The set of primes less tha... |
| ppifi 27070 | The set of primes less tha... |
| prmdvdsfi 27071 | The set of prime divisors ... |
| chtf 27072 | Domain and codoamin of the... |
| chtcl 27073 | Real closure of the Chebys... |
| chtval 27074 | Value of the Chebyshev fun... |
| efchtcl 27075 | The Chebyshev function is ... |
| chtge0 27076 | The Chebyshev function is ... |
| vmaval 27077 | Value of the von Mangoldt ... |
| isppw 27078 | Two ways to say that ` A `... |
| isppw2 27079 | Two ways to say that ` A `... |
| vmappw 27080 | Value of the von Mangoldt ... |
| vmaprm 27081 | Value of the von Mangoldt ... |
| vmacl 27082 | Closure for the von Mangol... |
| vmaf 27083 | Functionality of the von M... |
| efvmacl 27084 | The von Mangoldt is closed... |
| vmage0 27085 | The von Mangoldt function ... |
| chpval 27086 | Value of the second Chebys... |
| chpf 27087 | Functionality of the secon... |
| chpcl 27088 | Closure for the second Che... |
| efchpcl 27089 | The second Chebyshev funct... |
| chpge0 27090 | The second Chebyshev funct... |
| ppival 27091 | Value of the prime-countin... |
| ppival2 27092 | Value of the prime-countin... |
| ppival2g 27093 | Value of the prime-countin... |
| ppif 27094 | Domain and codomain of the... |
| ppicl 27095 | Real closure of the prime-... |
| muval 27096 | The value of the Möbi... |
| muval1 27097 | The value of the Möbi... |
| muval2 27098 | The value of the Möbi... |
| isnsqf 27099 | Two ways to say that a num... |
| issqf 27100 | Two ways to say that a num... |
| sqfpc 27101 | The prime count of a squar... |
| dvdssqf 27102 | A divisor of a squarefree ... |
| sqf11 27103 | A squarefree number is com... |
| muf 27104 | The Möbius function i... |
| mucl 27105 | Closure of the Möbius... |
| sgmval 27106 | The value of the divisor f... |
| sgmval2 27107 | The value of the divisor f... |
| 0sgm 27108 | The value of the sum-of-di... |
| sgmf 27109 | The divisor function is a ... |
| sgmcl 27110 | Closure of the divisor fun... |
| sgmnncl 27111 | Closure of the divisor fun... |
| mule1 27112 | The Möbius function t... |
| chtfl 27113 | The Chebyshev function doe... |
| chpfl 27114 | The second Chebyshev funct... |
| ppiprm 27115 | The prime-counting functio... |
| ppinprm 27116 | The prime-counting functio... |
| chtprm 27117 | The Chebyshev function at ... |
| chtnprm 27118 | The Chebyshev function at ... |
| chpp1 27119 | The second Chebyshev funct... |
| chtwordi 27120 | The Chebyshev function is ... |
| chpwordi 27121 | The second Chebyshev funct... |
| chtdif 27122 | The difference of the Cheb... |
| efchtdvds 27123 | The exponentiated Chebyshe... |
| ppifl 27124 | The prime-counting functio... |
| ppip1le 27125 | The prime-counting functio... |
| ppiwordi 27126 | The prime-counting functio... |
| ppidif 27127 | The difference of the prim... |
| ppi1 27128 | The prime-counting functio... |
| cht1 27129 | The Chebyshev function at ... |
| vma1 27130 | The von Mangoldt function ... |
| chp1 27131 | The second Chebyshev funct... |
| ppi1i 27132 | Inference form of ~ ppiprm... |
| ppi2i 27133 | Inference form of ~ ppinpr... |
| ppi2 27134 | The prime-counting functio... |
| ppi3 27135 | The prime-counting functio... |
| cht2 27136 | The Chebyshev function at ... |
| cht3 27137 | The Chebyshev function at ... |
| ppinncl 27138 | Closure of the prime-count... |
| chtrpcl 27139 | Closure of the Chebyshev f... |
| ppieq0 27140 | The prime-counting functio... |
| ppiltx 27141 | The prime-counting functio... |
| prmorcht 27142 | Relate the primorial (prod... |
| mumullem1 27143 | Lemma for ~ mumul . A mul... |
| mumullem2 27144 | Lemma for ~ mumul . The p... |
| mumul 27145 | The Möbius function i... |
| sqff1o 27146 | There is a bijection from ... |
| fsumdvdsdiaglem 27147 | A "diagonal commutation" o... |
| fsumdvdsdiag 27148 | A "diagonal commutation" o... |
| fsumdvdscom 27149 | A double commutation of di... |
| dvdsppwf1o 27150 | A bijection between the di... |
| dvdsflf1o 27151 | A bijection from the numbe... |
| dvdsflsumcom 27152 | A sum commutation from ` s... |
| fsumfldivdiaglem 27153 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27154 | The right-hand side of ~ d... |
| musum 27155 | The sum of the Möbius... |
| musumsum 27156 | Evaluate a collapsing sum ... |
| muinv 27157 | The Möbius inversion ... |
| mpodvdsmulf1o 27158 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27159 | Product of two divisor sum... |
| dvdsmulf1o 27160 | If ` M ` and ` N ` are two... |
| fsumdvdsmulOLD 27161 | Obsolete version of ~ fsum... |
| sgmppw 27162 | The value of the divisor f... |
| 0sgmppw 27163 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27164 | The sum of divisors for a ... |
| 1sgm2ppw 27165 | The sum of the divisors of... |
| sgmmul 27166 | The divisor function for f... |
| ppiublem1 27167 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27168 | A prime greater than ` 3 `... |
| ppiub 27169 | An upper bound on the prim... |
| vmalelog 27170 | The von Mangoldt function ... |
| chtlepsi 27171 | The first Chebyshev functi... |
| chprpcl 27172 | Closure of the second Cheb... |
| chpeq0 27173 | The second Chebyshev funct... |
| chteq0 27174 | The first Chebyshev functi... |
| chtleppi 27175 | Upper bound on the ` theta... |
| chtublem 27176 | Lemma for ~ chtub . (Cont... |
| chtub 27177 | An upper bound on the Cheb... |
| fsumvma 27178 | Rewrite a sum over the von... |
| fsumvma2 27179 | Apply ~ fsumvma for the co... |
| pclogsum 27180 | The logarithmic analogue o... |
| vmasum 27181 | The sum of the von Mangold... |
| logfac2 27182 | Another expression for the... |
| chpval2 27183 | Express the second Chebysh... |
| chpchtsum 27184 | The second Chebyshev funct... |
| chpub 27185 | An upper bound on the seco... |
| logfacubnd 27186 | A simple upper bound on th... |
| logfaclbnd 27187 | A lower bound on the logar... |
| logfacbnd3 27188 | Show the stronger statemen... |
| logfacrlim 27189 | Combine the estimates ~ lo... |
| logexprlim 27190 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27191 | Write out ~ logfacrlim as ... |
| mersenne 27192 | A Mersenne prime is a prim... |
| perfect1 27193 | Euclid's contribution to t... |
| perfectlem1 27194 | Lemma for ~ perfect . (Co... |
| perfectlem2 27195 | Lemma for ~ perfect . (Co... |
| perfect 27196 | The Euclid-Euler theorem, ... |
| dchrval 27199 | Value of the group of Diri... |
| dchrbas 27200 | Base set of the group of D... |
| dchrelbas 27201 | A Dirichlet character is a... |
| dchrelbas2 27202 | A Dirichlet character is a... |
| dchrelbas3 27203 | A Dirichlet character is a... |
| dchrelbasd 27204 | A Dirichlet character is a... |
| dchrrcl 27205 | Reverse closure for a Diri... |
| dchrmhm 27206 | A Dirichlet character is a... |
| dchrf 27207 | A Dirichlet character is a... |
| dchrelbas4 27208 | A Dirichlet character is a... |
| dchrzrh1 27209 | Value of a Dirichlet chara... |
| dchrzrhcl 27210 | A Dirichlet character take... |
| dchrzrhmul 27211 | A Dirichlet character is c... |
| dchrplusg 27212 | Group operation on the gro... |
| dchrmul 27213 | Group operation on the gro... |
| dchrmulcl 27214 | Closure of the group opera... |
| dchrn0 27215 | A Dirichlet character is n... |
| dchr1cl 27216 | Closure of the principal D... |
| dchrmullid 27217 | Left identity for the prin... |
| dchrinvcl 27218 | Closure of the group inver... |
| dchrabl 27219 | The set of Dirichlet chara... |
| dchrfi 27220 | The group of Dirichlet cha... |
| dchrghm 27221 | A Dirichlet character rest... |
| dchr1 27222 | Value of the principal Dir... |
| dchreq 27223 | A Dirichlet character is d... |
| dchrresb 27224 | A Dirichlet character is d... |
| dchrabs 27225 | A Dirichlet character take... |
| dchrinv 27226 | The inverse of a Dirichlet... |
| dchrabs2 27227 | A Dirichlet character take... |
| dchr1re 27228 | The principal Dirichlet ch... |
| dchrptlem1 27229 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27230 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27231 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27232 | For any element other than... |
| dchrsum2 27233 | An orthogonality relation ... |
| dchrsum 27234 | An orthogonality relation ... |
| sumdchr2 27235 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27236 | There are exactly ` phi ( ... |
| sumdchr 27237 | An orthogonality relation ... |
| dchr2sum 27238 | An orthogonality relation ... |
| sum2dchr 27239 | An orthogonality relation ... |
| bcctr 27240 | Value of the central binom... |
| pcbcctr 27241 | Prime count of a central b... |
| bcmono 27242 | The binomial coefficient i... |
| bcmax 27243 | The binomial coefficient t... |
| bcp1ctr 27244 | Ratio of two central binom... |
| bclbnd 27245 | A bound on the binomial co... |
| efexple 27246 | Convert a bound on a power... |
| bpos1lem 27247 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27248 | Bertrand's postulate, chec... |
| bposlem1 27249 | An upper bound on the prim... |
| bposlem2 27250 | There are no odd primes in... |
| bposlem3 27251 | Lemma for ~ bpos . Since ... |
| bposlem4 27252 | Lemma for ~ bpos . (Contr... |
| bposlem5 27253 | Lemma for ~ bpos . Bound ... |
| bposlem6 27254 | Lemma for ~ bpos . By usi... |
| bposlem7 27255 | Lemma for ~ bpos . The fu... |
| bposlem8 27256 | Lemma for ~ bpos . Evalua... |
| bposlem9 27257 | Lemma for ~ bpos . Derive... |
| bpos 27258 | Bertrand's postulate: ther... |
| zabsle1 27261 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27262 | When ` a ` is coprime to t... |
| lgslem2 27263 | The set ` Z ` of all integ... |
| lgslem3 27264 | The set ` Z ` of all integ... |
| lgslem4 27265 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27266 | Value of the Legendre symb... |
| lgsfval 27267 | Value of the function ` F ... |
| lgsfcl2 27268 | The function ` F ` is clos... |
| lgscllem 27269 | The Legendre symbol is an ... |
| lgsfcl 27270 | Closure of the function ` ... |
| lgsfle1 27271 | The function ` F ` has mag... |
| lgsval2lem 27272 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27273 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27274 | The Legendre symbol is an ... |
| lgs0 27275 | The Legendre symbol when t... |
| lgscl 27276 | The Legendre symbol is an ... |
| lgsle1 27277 | The Legendre symbol has ab... |
| lgsval2 27278 | The Legendre symbol at a p... |
| lgs2 27279 | The Legendre symbol at ` 2... |
| lgsval3 27280 | The Legendre symbol at an ... |
| lgsvalmod 27281 | The Legendre symbol is equ... |
| lgsval4 27282 | Restate ~ lgsval for nonze... |
| lgsfcl3 27283 | Closure of the function ` ... |
| lgsval4a 27284 | Same as ~ lgsval4 for posi... |
| lgscl1 27285 | The value of the Legendre ... |
| lgsneg 27286 | The Legendre symbol is eit... |
| lgsneg1 27287 | The Legendre symbol for no... |
| lgsmod 27288 | The Legendre (Jacobi) symb... |
| lgsdilem 27289 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27290 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27291 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27292 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27293 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27294 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27295 | The Legendre symbol is com... |
| lgsdirprm 27296 | The Legendre symbol is com... |
| lgsdir 27297 | The Legendre symbol is com... |
| lgsdilem2 27298 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27299 | The Legendre symbol is com... |
| lgsne0 27300 | The Legendre symbol is non... |
| lgsabs1 27301 | The Legendre symbol is non... |
| lgssq 27302 | The Legendre symbol at a s... |
| lgssq2 27303 | The Legendre symbol at a s... |
| lgsprme0 27304 | The Legendre symbol at any... |
| 1lgs 27305 | The Legendre symbol at ` 1... |
| lgs1 27306 | The Legendre symbol at ` 1... |
| lgsmodeq 27307 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27308 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27309 | Variation on ~ lgsdir vali... |
| lgsdinn0 27310 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27311 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27312 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27313 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27314 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27315 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27316 | The Legendre symbol for od... |
| lgsqrmod 27317 | If the Legendre symbol of ... |
| lgsqrmodndvds 27318 | If the Legendre symbol of ... |
| lgsdchrval 27319 | The Legendre symbol functi... |
| lgsdchr 27320 | The Legendre symbol functi... |
| gausslemma2dlem0a 27321 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27322 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27323 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27324 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27325 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27326 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27327 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27328 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27329 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27330 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27331 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27332 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27333 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27334 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27335 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27336 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27337 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27338 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27339 | Gauss' Lemma (see also the... |
| lgseisenlem1 27340 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27341 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27342 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27343 | Lemma for ~ lgseisen . (C... |
| lgseisen 27344 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27345 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27346 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27347 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27348 | The Law of Quadratic Recip... |
| lgsquad2lem1 27349 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27350 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27351 | Extend ~ lgsquad to coprim... |
| lgsquad3 27352 | Extend ~ lgsquad2 to integ... |
| m1lgs 27353 | The first supplement to th... |
| 2lgslem1a1 27354 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27355 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27356 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27357 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27358 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27359 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27360 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27361 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27362 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27363 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27364 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27365 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27366 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27367 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27368 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27369 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27370 | The Legendre symbol for ` ... |
| 2lgslem4 27371 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27372 | The second supplement to t... |
| 2lgsoddprmlem1 27373 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27374 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27375 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27376 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27377 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27378 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27379 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27380 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27381 | The second supplement to t... |
| 2sqlem1 27382 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27383 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27384 | Fibonacci's identity (actu... |
| 2sqlem3 27385 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27386 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27387 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27388 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27389 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27390 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27391 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27392 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27393 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27394 | Lemma for ~ 2sq . (Contri... |
| 2sq 27395 | All primes of the form ` 4... |
| 2sqblem 27396 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27397 | The converse to ~ 2sq . (... |
| 2sq2 27398 | ` 2 ` is the sum of square... |
| 2sqn0 27399 | If the sum of two squares ... |
| 2sqcoprm 27400 | If the sum of two squares ... |
| 2sqmod 27401 | Given two decompositions o... |
| 2sqmo 27402 | There exists at most one d... |
| 2sqnn0 27403 | All primes of the form ` 4... |
| 2sqnn 27404 | All primes of the form ` 4... |
| addsq2reu 27405 | For each complex number ` ... |
| addsqn2reu 27406 | For each complex number ` ... |
| addsqrexnreu 27407 | For each complex number, t... |
| addsqnreup 27408 | There is no unique decompo... |
| addsq2nreurex 27409 | For each complex number ` ... |
| addsqn2reurex2 27410 | For each complex number ` ... |
| 2sqreulem1 27411 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27412 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27413 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27414 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27415 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27416 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27417 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27418 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27419 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27420 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27421 | There exists a unique deco... |
| 2sqreunn 27422 | There exists a unique deco... |
| 2sqreult 27423 | There exists a unique deco... |
| 2sqreultb 27424 | There exists a unique deco... |
| 2sqreunnlt 27425 | There exists a unique deco... |
| 2sqreunnltb 27426 | There exists a unique deco... |
| 2sqreuop 27427 | There exists a unique deco... |
| 2sqreuopnn 27428 | There exists a unique deco... |
| 2sqreuoplt 27429 | There exists a unique deco... |
| 2sqreuopltb 27430 | There exists a unique deco... |
| 2sqreuopnnlt 27431 | There exists a unique deco... |
| 2sqreuopnnltb 27432 | There exists a unique deco... |
| 2sqreuopb 27433 | There exists a unique deco... |
| chebbnd1lem1 27434 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27435 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27436 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27437 | The Chebyshev bound: The ... |
| chtppilimlem1 27438 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27439 | Lemma for ~ chtppilim . (... |
| chtppilim 27440 | The ` theta ` function is ... |
| chto1ub 27441 | The ` theta ` function is ... |
| chebbnd2 27442 | The Chebyshev bound, part ... |
| chto1lb 27443 | The ` theta ` function is ... |
| chpchtlim 27444 | The ` psi ` and ` theta ` ... |
| chpo1ub 27445 | The ` psi ` function is up... |
| chpo1ubb 27446 | The ` psi ` function is up... |
| vmadivsum 27447 | The sum of the von Mangold... |
| vmadivsumb 27448 | Give a total bound on the ... |
| rplogsumlem1 27449 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27450 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27451 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27452 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27453 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27454 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27455 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27456 | Lemma for ~ dchrisum . Le... |
| dchrisum 27457 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27458 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27459 | The sum of the Möbius... |
| dchrvmasumlem1 27460 | An alternative expression ... |
| dchrvmasum2lem 27461 | Give an expression for ` l... |
| dchrvmasum2if 27462 | Combine the results of ~ d... |
| dchrvmasumlem2 27463 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27464 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27465 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27466 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27467 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27468 | An asymptotic approximatio... |
| dchrvmaeq0 27469 | The set ` W ` is the colle... |
| dchrisum0fval 27470 | Value of the function ` F ... |
| dchrisum0fmul 27471 | The function ` F ` , the d... |
| dchrisum0ff 27472 | The function ` F ` is a re... |
| dchrisum0flblem1 27473 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27474 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27475 | The divisor sum of a real ... |
| dchrisum0fno1 27476 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27477 | A partial result along the... |
| dchrisum0re 27478 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27479 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27480 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27481 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27482 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27483 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27484 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27485 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27486 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27487 | The sum of the Möbius... |
| dchrvmasumlem 27488 | The sum of the Möbius... |
| dchrmusum 27489 | The sum of the Möbius... |
| dchrvmasum 27490 | The sum of the von Mangold... |
| rpvmasum 27491 | The sum of the von Mangold... |
| rplogsum 27492 | The sum of ` log p / p ` o... |
| dirith2 27493 | Dirichlet's theorem: there... |
| dirith 27494 | Dirichlet's theorem: there... |
| mudivsum 27495 | Asymptotic formula for ` s... |
| mulogsumlem 27496 | Lemma for ~ mulogsum . (C... |
| mulogsum 27497 | Asymptotic formula for ... |
| logdivsum 27498 | Asymptotic analysis of ... |
| mulog2sumlem1 27499 | Asymptotic formula for ... |
| mulog2sumlem2 27500 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27501 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27502 | Asymptotic formula for ... |
| vmalogdivsum2 27503 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27504 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27505 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27506 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27507 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27508 | The Möbius inverse of... |
| log2sumbnd 27509 | Bound on the difference be... |
| selberglem1 27510 | Lemma for ~ selberg . Est... |
| selberglem2 27511 | Lemma for ~ selberg . (Co... |
| selberglem3 27512 | Lemma for ~ selberg . Est... |
| selberg 27513 | Selberg's symmetry formula... |
| selbergb 27514 | Convert eventual boundedne... |
| selberg2lem 27515 | Lemma for ~ selberg2 . Eq... |
| selberg2 27516 | Selberg's symmetry formula... |
| selberg2b 27517 | Convert eventual boundedne... |
| chpdifbndlem1 27518 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27519 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27520 | A bound on the difference ... |
| logdivbnd 27521 | A bound on a sum of logs, ... |
| selberg3lem1 27522 | Introduce a log weighting ... |
| selberg3lem2 27523 | Lemma for ~ selberg3 . Eq... |
| selberg3 27524 | Introduce a log weighting ... |
| selberg4lem1 27525 | Lemma for ~ selberg4 . Eq... |
| selberg4 27526 | The Selberg symmetry formu... |
| pntrval 27527 | Define the residual of the... |
| pntrf 27528 | Functionality of the resid... |
| pntrmax 27529 | There is a bound on the re... |
| pntrsumo1 27530 | A bound on a sum over ` R ... |
| pntrsumbnd 27531 | A bound on a sum over ` R ... |
| pntrsumbnd2 27532 | A bound on a sum over ` R ... |
| selbergr 27533 | Selberg's symmetry formula... |
| selberg3r 27534 | Selberg's symmetry formula... |
| selberg4r 27535 | Selberg's symmetry formula... |
| selberg34r 27536 | The sum of ~ selberg3r and... |
| pntsval 27537 | Define the "Selberg functi... |
| pntsf 27538 | Functionality of the Selbe... |
| selbergs 27539 | Selberg's symmetry formula... |
| selbergsb 27540 | Selberg's symmetry formula... |
| pntsval2 27541 | The Selberg function can b... |
| pntrlog2bndlem1 27542 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27543 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27544 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27545 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27546 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27547 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27548 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27549 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27550 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27551 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27552 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27553 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27554 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27555 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27556 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27557 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27558 | Lemma for ~ pnt . Establi... |
| pntlemd 27559 | Lemma for ~ pnt . Closure... |
| pntlemc 27560 | Lemma for ~ pnt . Closure... |
| pntlema 27561 | Lemma for ~ pnt . Closure... |
| pntlemb 27562 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27563 | Lemma for ~ pnt . Closure... |
| pntlemh 27564 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27565 | Lemma for ~ pnt . The "na... |
| pntlemq 27566 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27567 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27568 | Lemma for ~ pnt . The ind... |
| pntlemi 27569 | Lemma for ~ pnt . Elimina... |
| pntlemf 27570 | Lemma for ~ pnt . Add up ... |
| pntlemk 27571 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27572 | Lemma for ~ pnt . Combine... |
| pntleme 27573 | Lemma for ~ pnt . Package... |
| pntlem3 27574 | Lemma for ~ pnt . Equatio... |
| pntlemp 27575 | Lemma for ~ pnt . Wrappin... |
| pntleml 27576 | Lemma for ~ pnt . Equatio... |
| pnt3 27577 | The Prime Number Theorem, ... |
| pnt2 27578 | The Prime Number Theorem, ... |
| pnt 27579 | The Prime Number Theorem: ... |
| abvcxp 27580 | Raising an absolute value ... |
| padicfval 27581 | Value of the p-adic absolu... |
| padicval 27582 | Value of the p-adic absolu... |
| ostth2lem1 27583 | Lemma for ~ ostth2 , altho... |
| qrngbas 27584 | The base set of the field ... |
| qdrng 27585 | The rationals form a divis... |
| qrng0 27586 | The zero element of the fi... |
| qrng1 27587 | The unity element of the f... |
| qrngneg 27588 | The additive inverse in th... |
| qrngdiv 27589 | The division operation in ... |
| qabvle 27590 | By using induction on ` N ... |
| qabvexp 27591 | Induct the product rule ~ ... |
| ostthlem1 27592 | Lemma for ~ ostth . If tw... |
| ostthlem2 27593 | Lemma for ~ ostth . Refin... |
| qabsabv 27594 | The regular absolute value... |
| padicabv 27595 | The p-adic absolute value ... |
| padicabvf 27596 | The p-adic absolute value ... |
| padicabvcxp 27597 | All positive powers of the... |
| ostth1 27598 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27599 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27600 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27601 | Lemma for ~ ostth2 . (Con... |
| ostth2 27602 | - Lemma for ~ ostth : regu... |
| ostth3 27603 | - Lemma for ~ ostth : p-ad... |
| ostth 27604 | Ostrowski's theorem, which... |
| elno 27611 | Membership in the surreals... |
| elnoOLD 27612 | Obsolete version of ~ elno... |
| sltval 27613 | The value of the surreal l... |
| bdayval 27614 | The value of the birthday ... |
| nofun 27615 | A surreal is a function. ... |
| nodmon 27616 | The domain of a surreal is... |
| norn 27617 | The range of a surreal is ... |
| nofnbday 27618 | A surreal is a function ov... |
| nodmord 27619 | The domain of a surreal ha... |
| elno2 27620 | An alternative condition f... |
| elno3 27621 | Another condition for memb... |
| sltval2 27622 | Alternate expression for s... |
| nofv 27623 | The function value of a su... |
| nosgnn0 27624 | ` (/) ` is not a surreal s... |
| nosgnn0i 27625 | If ` X ` is a surreal sign... |
| noreson 27626 | The restriction of a surre... |
| sltintdifex 27627 |
If ` A |
| sltres 27628 | If the restrictions of two... |
| noxp1o 27629 | The Cartesian product of a... |
| noseponlem 27630 | Lemma for ~ nosepon . Con... |
| nosepon 27631 | Given two unequal surreals... |
| noextend 27632 | Extending a surreal by one... |
| noextendseq 27633 | Extend a surreal by a sequ... |
| noextenddif 27634 | Calculate the place where ... |
| noextendlt 27635 | Extending a surreal with a... |
| noextendgt 27636 | Extending a surreal with a... |
| nolesgn2o 27637 | Given ` A ` less-than or e... |
| nolesgn2ores 27638 | Given ` A ` less-than or e... |
| nogesgn1o 27639 | Given ` A ` greater than o... |
| nogesgn1ores 27640 | Given ` A ` greater than o... |
| sltsolem1 27641 | Lemma for ~ sltso . The "... |
| sltso 27642 | Less-than totally orders t... |
| bdayfo 27643 | The birthday function maps... |
| fvnobday 27644 | The value of a surreal at ... |
| nosepnelem 27645 | Lemma for ~ nosepne . (Co... |
| nosepne 27646 | The value of two non-equal... |
| nosep1o 27647 | If the value of a surreal ... |
| nosep2o 27648 | If the value of a surreal ... |
| nosepdmlem 27649 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27650 | The first place two surrea... |
| nosepeq 27651 | The values of two surreals... |
| nosepssdm 27652 | Given two non-equal surrea... |
| nodenselem4 27653 | Lemma for ~ nodense . Sho... |
| nodenselem5 27654 | Lemma for ~ nodense . If ... |
| nodenselem6 27655 | The restriction of a surre... |
| nodenselem7 27656 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27657 | Lemma for ~ nodense . Giv... |
| nodense 27658 | Given two distinct surreal... |
| bdayimaon 27659 | Lemma for full-eta propert... |
| nolt02olem 27660 | Lemma for ~ nolt02o . If ... |
| nolt02o 27661 | Given ` A ` less-than ` B ... |
| nogt01o 27662 | Given ` A ` greater than `... |
| noresle 27663 | Restriction law for surrea... |
| nomaxmo 27664 | A class of surreals has at... |
| nominmo 27665 | A class of surreals has at... |
| nosupprefixmo 27666 | In any class of surreals, ... |
| noinfprefixmo 27667 | In any class of surreals, ... |
| nosupcbv 27668 | Lemma to change bound vari... |
| nosupno 27669 | The next several theorems ... |
| nosupdm 27670 | The domain of the surreal ... |
| nosupbday 27671 | Birthday bounding law for ... |
| nosupfv 27672 | The value of surreal supre... |
| nosupres 27673 | A restriction law for surr... |
| nosupbnd1lem1 27674 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27675 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27676 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27677 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27678 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27679 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27680 | Bounding law from below fo... |
| nosupbnd2lem1 27681 | Bounding law from above wh... |
| nosupbnd2 27682 | Bounding law from above fo... |
| noinfcbv 27683 | Change bound variables for... |
| noinfno 27684 | The next several theorems ... |
| noinfdm 27685 | Next, we calculate the dom... |
| noinfbday 27686 | Birthday bounding law for ... |
| noinffv 27687 | The value of surreal infim... |
| noinfres 27688 | The restriction of surreal... |
| noinfbnd1lem1 27689 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27690 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27691 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27692 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27693 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27694 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27695 | Bounding law from above fo... |
| noinfbnd2lem1 27696 | Bounding law from below wh... |
| noinfbnd2 27697 | Bounding law from below fo... |
| nosupinfsep 27698 | Given two sets of surreals... |
| noetasuplem1 27699 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27700 | Lemma for ~ noeta . The r... |
| noetasuplem3 27701 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27702 | Lemma for ~ noeta . When ... |
| noetainflem1 27703 | Lemma for ~ noeta . Estab... |
| noetainflem2 27704 | Lemma for ~ noeta . The r... |
| noetainflem3 27705 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27706 | Lemma for ~ noeta . If ` ... |
| noetalem1 27707 | Lemma for ~ noeta . Eithe... |
| noetalem2 27708 | Lemma for ~ noeta . The f... |
| noeta 27709 | The full-eta axiom for the... |
| sltirr 27712 | Surreal less-than is irref... |
| slttr 27713 | Surreal less-than is trans... |
| sltasym 27714 | Surreal less-than is asymm... |
| sltlin 27715 | Surreal less-than obeys tr... |
| slttrieq2 27716 | Trichotomy law for surreal... |
| slttrine 27717 | Trichotomy law for surreal... |
| slenlt 27718 | Surreal less-than or equal... |
| sltnle 27719 | Surreal less-than in terms... |
| sleloe 27720 | Surreal less-than or equal... |
| sletri3 27721 | Trichotomy law for surreal... |
| sltletr 27722 | Surreal transitive law. (... |
| slelttr 27723 | Surreal transitive law. (... |
| sletr 27724 | Surreal transitive law. (... |
| slttrd 27725 | Surreal less-than is trans... |
| sltletrd 27726 | Surreal less-than is trans... |
| slelttrd 27727 | Surreal less-than is trans... |
| sletrd 27728 | Surreal less-than or equal... |
| slerflex 27729 | Surreal less-than or equal... |
| sletric 27730 | Surreal trichotomy law. (... |
| maxs1 27731 | A surreal is less than or ... |
| maxs2 27732 | A surreal is less than or ... |
| mins1 27733 | The minimum of two surreal... |
| mins2 27734 | The minimum of two surreal... |
| sltled 27735 | Surreal less-than implies ... |
| sltne 27736 | Surreal less-than implies ... |
| sltlend 27737 | Surreal less-than in terms... |
| bdayfun 27738 | The birthday function is a... |
| bdayfn 27739 | The birthday function is a... |
| bdaydm 27740 | The birthday function's do... |
| bdayrn 27741 | The birthday function's ra... |
| bdayelon 27742 | The value of the birthday ... |
| nobdaymin 27743 | Any non-empty class of sur... |
| nocvxminlem 27744 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27745 | Given a nonempty convex cl... |
| noprc 27746 | The surreal numbers are a ... |
| noeta2 27751 | A version of ~ noeta with ... |
| brsslt 27752 | Binary relation form of th... |
| ssltex1 27753 | The first argument of surr... |
| ssltex2 27754 | The second argument of sur... |
| ssltss1 27755 | The first argument of surr... |
| ssltss2 27756 | The second argument of sur... |
| ssltsep 27757 | The separation property of... |
| ssltd 27758 | Deduce surreal set less-th... |
| ssltsnb 27759 | Surreal set less-than of t... |
| ssltsn 27760 | Surreal set less-than of t... |
| ssltsepc 27761 | Two elements of separated ... |
| ssltsepcd 27762 | Two elements of separated ... |
| sssslt1 27763 | Relation between surreal s... |
| sssslt2 27764 | Relation between surreal s... |
| nulsslt 27765 | The empty set is less-than... |
| nulssgt 27766 | The empty set is greater t... |
| conway 27767 | Conway's Simplicity Theore... |
| scutval 27768 | The value of the surreal c... |
| scutcut 27769 | Cut properties of the surr... |
| scutcl 27770 | Closure law for surreal cu... |
| scutcld 27771 | Closure law for surreal cu... |
| scutbday 27772 | The birthday of the surrea... |
| eqscut 27773 | Condition for equality to ... |
| eqscut2 27774 | Condition for equality to ... |
| sslttr 27775 | Transitive law for surreal... |
| ssltun1 27776 | Union law for surreal set ... |
| ssltun2 27777 | Union law for surreal set ... |
| scutun12 27778 | Union law for surreal cuts... |
| dmscut 27779 | The domain of the surreal ... |
| scutf 27780 | Functionality statement fo... |
| etasslt 27781 | A restatement of ~ noeta u... |
| etasslt2 27782 | A version of ~ etasslt wit... |
| scutbdaybnd 27783 | An upper bound on the birt... |
| scutbdaybnd2 27784 | An upper bound on the birt... |
| scutbdaybnd2lim 27785 | An upper bound on the birt... |
| scutbdaylt 27786 | If a surreal lies in a gap... |
| slerec 27787 | A comparison law for surre... |
| slerecd 27788 | A comparison law for surre... |
| sltrec 27789 | A comparison law for surre... |
| sltrecd 27790 | A comparison law for surre... |
| ssltdisj 27791 | If ` A ` preceeds ` B ` , ... |
| eqscut3 27792 | A variant of the simplicit... |
| 0sno 27797 | Surreal zero is a surreal.... |
| 1sno 27798 | Surreal one is a surreal. ... |
| bday0s 27799 | Calculate the birthday of ... |
| 0slt1s 27800 | Surreal zero is less than ... |
| bday0b 27801 | The only surreal with birt... |
| bday1s 27802 | The birthday of surreal on... |
| cuteq0 27803 | Condition for a surreal cu... |
| cutneg 27804 | The simplest number greate... |
| cuteq1 27805 | Condition for a surreal cu... |
| sgt0ne0 27806 | A positive surreal is not ... |
| sgt0ne0d 27807 | A positive surreal is not ... |
| 1sne0s 27808 | Surreal zero does not equa... |
| rightpos 27809 | A surreal is non-negative ... |
| madeval 27820 | The value of the made by f... |
| madeval2 27821 | Alternative characterizati... |
| oldval 27822 | The value of the old optio... |
| newval 27823 | The value of the new optio... |
| madef 27824 | The made function is a fun... |
| oldf 27825 | The older function is a fu... |
| newf 27826 | The new function is a func... |
| old0 27827 | No surreal is older than `... |
| madessno 27828 | Made sets are surreals. (... |
| oldssno 27829 | Old sets are surreals. (C... |
| newssno 27830 | New sets are surreals. (C... |
| leftval 27831 | The value of the left opti... |
| rightval 27832 | The value of the right opt... |
| elleft 27833 | Membership in the left set... |
| elright 27834 | Membership in the right se... |
| leftlt 27835 | A member of a surreal's le... |
| rightgt 27836 | A member of a surreal's ri... |
| leftf 27837 | The functionality of the l... |
| rightf 27838 | The functionality of the r... |
| elmade 27839 | Membership in the made fun... |
| elmade2 27840 | Membership in the made fun... |
| elold 27841 | Membership in an old set. ... |
| ssltleft 27842 | A surreal is greater than ... |
| ssltright 27843 | A surreal is less than its... |
| lltropt 27844 | The left options of a surr... |
| made0 27845 | The only surreal made on d... |
| new0 27846 | The only surreal new on da... |
| old1 27847 | The only surreal older tha... |
| madess 27848 | If ` A ` is less than or e... |
| oldssmade 27849 | The older-than set is a su... |
| oldss 27850 | If ` A ` is less than or e... |
| leftssold 27851 | The left options are a sub... |
| rightssold 27852 | The right options are a su... |
| leftssno 27853 | The left set of a surreal ... |
| rightssno 27854 | The right set of a surreal... |
| madecut 27855 | Given a section that is a ... |
| madeun 27856 | The made set is the union ... |
| madeoldsuc 27857 | The made set is the old se... |
| oldsuc 27858 | The value of the old set a... |
| oldlim 27859 | The value of the old set a... |
| madebdayim 27860 | If a surreal is a member o... |
| oldbdayim 27861 | If ` X ` is in the old set... |
| oldirr 27862 | No surreal is a member of ... |
| leftirr 27863 | No surreal is a member of ... |
| rightirr 27864 | No surreal is a member of ... |
| left0s 27865 | The left set of ` 0s ` is ... |
| right0s 27866 | The right set of ` 0s ` is... |
| left1s 27867 | The left set of ` 1s ` is ... |
| right1s 27868 | The right set of ` 1s ` is... |
| lrold 27869 | The union of the left and ... |
| madebdaylemold 27870 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27871 | Lemma for ~ madebday . If... |
| madebday 27872 | A surreal is part of the s... |
| oldbday 27873 | A surreal is part of the s... |
| newbday 27874 | A surreal is an element of... |
| newbdayim 27875 | One direction of the bicon... |
| lrcut 27876 | A surreal is equal to the ... |
| scutfo 27877 | The surreal cut function i... |
| sltn0 27878 | If ` X ` is less than ` Y ... |
| lruneq 27879 | If two surreals share a bi... |
| sltlpss 27880 | If two surreals share a bi... |
| slelss 27881 | If two surreals ` A ` and ... |
| 0elold 27882 | Zero is in the old set of ... |
| 0elleft 27883 | Zero is in the left set of... |
| 0elright 27884 | Zero is in the right set o... |
| madefi 27885 | The made set of an ordinal... |
| oldfi 27886 | The old set of an ordinal ... |
| bdayiun 27887 | The birthday of a surreal ... |
| bdayle 27888 | A condition for bounding a... |
| cofsslt 27889 | If every element of ` A ` ... |
| coinitsslt 27890 | If ` B ` is coinitial with... |
| cofcut1 27891 | If ` C ` is cofinal with `... |
| cofcut1d 27892 | If ` C ` is cofinal with `... |
| cofcut2 27893 | If ` A ` and ` C ` are mut... |
| cofcut2d 27894 | If ` A ` and ` C ` are mut... |
| cofcutr 27895 | If ` X ` is the cut of ` A... |
| cofcutr1d 27896 | If ` X ` is the cut of ` A... |
| cofcutr2d 27897 | If ` X ` is the cut of ` A... |
| cofcutrtime 27898 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27899 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27900 | If ` X ` is a timely cut o... |
| cofss 27901 | Cofinality for a subset. ... |
| coiniss 27902 | Coinitiality for a subset.... |
| cutlt 27903 | Eliminating all elements b... |
| cutpos 27904 | Reduce the elements of a c... |
| cutmax 27905 | If ` A ` has a maximum, th... |
| cutmin 27906 | If ` B ` has a minimum, th... |
| lrrecval 27909 | The next step in the devel... |
| lrrecval2 27910 | Next, we establish an alte... |
| lrrecpo 27911 | Now, we establish that ` R... |
| lrrecse 27912 | Next, we show that ` R ` i... |
| lrrecfr 27913 | Now we show that ` R ` is ... |
| lrrecpred 27914 | Finally, we calculate the ... |
| noinds 27915 | Induction principle for a ... |
| norecfn 27916 | Surreal recursion over one... |
| norecov 27917 | Calculate the value of the... |
| noxpordpo 27920 | To get through most of the... |
| noxpordfr 27921 | Next we establish the foun... |
| noxpordse 27922 | Next we establish the set-... |
| noxpordpred 27923 | Next we calculate the pred... |
| no2indslem 27924 | Double induction on surrea... |
| no2inds 27925 | Double induction on surrea... |
| norec2fn 27926 | The double-recursion opera... |
| norec2ov 27927 | The value of the double-re... |
| no3inds 27928 | Triple induction over surr... |
| addsfn 27931 | Surreal addition is a func... |
| addsval 27932 | The value of surreal addit... |
| addsval2 27933 | The value of surreal addit... |
| addsrid 27934 | Surreal addition to zero i... |
| addsridd 27935 | Surreal addition to zero i... |
| addscom 27936 | Surreal addition commutes.... |
| addscomd 27937 | Surreal addition commutes.... |
| addslid 27938 | Surreal addition to zero i... |
| addsproplem1 27939 | Lemma for surreal addition... |
| addsproplem2 27940 | Lemma for surreal addition... |
| addsproplem3 27941 | Lemma for surreal addition... |
| addsproplem4 27942 | Lemma for surreal addition... |
| addsproplem5 27943 | Lemma for surreal addition... |
| addsproplem6 27944 | Lemma for surreal addition... |
| addsproplem7 27945 | Lemma for surreal addition... |
| addsprop 27946 | Inductively show that surr... |
| addscutlem 27947 | Lemma for ~ addscut . Sho... |
| addscut 27948 | Demonstrate the cut proper... |
| addscut2 27949 | Show that the cut involved... |
| addscld 27950 | Surreal numbers are closed... |
| addscl 27951 | Surreal numbers are closed... |
| addsf 27952 | Function statement for sur... |
| addsfo 27953 | Surreal addition is onto. ... |
| peano2no 27954 | A theorem for surreals tha... |
| sltadd1im 27955 | Surreal less-than is prese... |
| sltadd2im 27956 | Surreal less-than is prese... |
| sleadd1im 27957 | Surreal less-than or equal... |
| sleadd2im 27958 | Surreal less-than or equal... |
| sleadd1 27959 | Addition to both sides of ... |
| sleadd2 27960 | Addition to both sides of ... |
| sltadd2 27961 | Addition to both sides of ... |
| sltadd1 27962 | Addition to both sides of ... |
| addscan2 27963 | Cancellation law for surre... |
| addscan1 27964 | Cancellation law for surre... |
| sleadd1d 27965 | Addition to both sides of ... |
| sleadd2d 27966 | Addition to both sides of ... |
| sltadd2d 27967 | Addition to both sides of ... |
| sltadd1d 27968 | Addition to both sides of ... |
| addscan2d 27969 | Cancellation law for surre... |
| addscan1d 27970 | Cancellation law for surre... |
| addsuniflem 27971 | Lemma for ~ addsunif . St... |
| addsunif 27972 | Uniformity theorem for sur... |
| addsasslem1 27973 | Lemma for addition associa... |
| addsasslem2 27974 | Lemma for addition associa... |
| addsass 27975 | Surreal addition is associ... |
| addsassd 27976 | Surreal addition is associ... |
| adds32d 27977 | Commutative/associative la... |
| adds12d 27978 | Commutative/associative la... |
| adds4d 27979 | Rearrangement of four term... |
| adds42d 27980 | Rearrangement of four term... |
| sltaddpos1d 27981 | Addition of a positive num... |
| sltaddpos2d 27982 | Addition of a positive num... |
| slt2addd 27983 | Adding both sides of two s... |
| addsgt0d 27984 | The sum of two positive su... |
| sltp1d 27985 | A surreal is less than its... |
| addsbdaylem 27986 | Lemma for ~ addsbday . (C... |
| addsbday 27987 | The birthday of the sum of... |
| negsfn 27992 | Surreal negation is a func... |
| subsfn 27993 | Surreal subtraction is a f... |
| negsval 27994 | The value of the surreal n... |
| negs0s 27995 | Negative surreal zero is s... |
| negs1s 27996 | An expression for negative... |
| negsproplem1 27997 | Lemma for surreal negation... |
| negsproplem2 27998 | Lemma for surreal negation... |
| negsproplem3 27999 | Lemma for surreal negation... |
| negsproplem4 28000 | Lemma for surreal negation... |
| negsproplem5 28001 | Lemma for surreal negation... |
| negsproplem6 28002 | Lemma for surreal negation... |
| negsproplem7 28003 | Lemma for surreal negation... |
| negsprop 28004 | Show closure and ordering ... |
| negscl 28005 | The surreals are closed un... |
| negscld 28006 | The surreals are closed un... |
| sltnegim 28007 | The forward direction of t... |
| negscut 28008 | The cut properties of surr... |
| negscut2 28009 | The cut that defines surre... |
| negsid 28010 | Surreal addition of a numb... |
| negsidd 28011 | Surreal addition of a numb... |
| negsex 28012 | Every surreal has a negati... |
| negnegs 28013 | A surreal is equal to the ... |
| sltneg 28014 | Negative of both sides of ... |
| sleneg 28015 | Negative of both sides of ... |
| sltnegd 28016 | Negative of both sides of ... |
| slenegd 28017 | Negative of both sides of ... |
| negs11 28018 | Surreal negation is one-to... |
| negsdi 28019 | Distribution of surreal ne... |
| slt0neg2d 28020 | Comparison of a surreal an... |
| negsf 28021 | Function statement for sur... |
| negsfo 28022 | Function statement for sur... |
| negsf1o 28023 | Surreal negation is a bije... |
| negsunif 28024 | Uniformity property for su... |
| negsbdaylem 28025 | Lemma for ~ negsbday . Bo... |
| negsbday 28026 | Negation of a surreal numb... |
| negsleft 28027 | The left set of the negati... |
| negsright 28028 | The right set of the negat... |
| subsval 28029 | The value of surreal subtr... |
| subsvald 28030 | The value of surreal subtr... |
| subscl 28031 | Closure law for surreal su... |
| subscld 28032 | Closure law for surreal su... |
| subsf 28033 | Function statement for sur... |
| subsfo 28034 | Surreal subtraction is an ... |
| negsval2 28035 | Surreal negation in terms ... |
| negsval2d 28036 | Surreal negation in terms ... |
| subsid1 28037 | Identity law for subtracti... |
| subsid 28038 | Subtraction of a surreal f... |
| subadds 28039 | Relationship between addit... |
| subaddsd 28040 | Relationship between addit... |
| pncans 28041 | Cancellation law for surre... |
| pncan3s 28042 | Subtraction and addition o... |
| pncan2s 28043 | Cancellation law for surre... |
| npcans 28044 | Cancellation law for surre... |
| sltsub1 28045 | Subtraction from both side... |
| sltsub2 28046 | Subtraction from both side... |
| sltsub1d 28047 | Subtraction from both side... |
| sltsub2d 28048 | Subtraction from both side... |
| negsubsdi2d 28049 | Distribution of negative o... |
| addsubsassd 28050 | Associative-type law for s... |
| addsubsd 28051 | Law for surreal addition a... |
| sltsubsubbd 28052 | Equivalence for the surrea... |
| sltsubsub2bd 28053 | Equivalence for the surrea... |
| sltsubsub3bd 28054 | Equivalence for the surrea... |
| slesubsubbd 28055 | Equivalence for the surrea... |
| slesubsub2bd 28056 | Equivalence for the surrea... |
| slesubsub3bd 28057 | Equivalence for the surrea... |
| sltsubaddd 28058 | Surreal less-than relation... |
| sltsubadd2d 28059 | Surreal less-than relation... |
| sltaddsubd 28060 | Surreal less-than relation... |
| sltaddsub2d 28061 | Surreal less-than relation... |
| slesubaddd 28062 | Surreal less-than or equal... |
| subsubs4d 28063 | Law for double surreal sub... |
| subsubs2d 28064 | Law for double surreal sub... |
| slesubd 28065 | Swap subtrahends in a surr... |
| nncansd 28066 | Cancellation law for surre... |
| posdifsd 28067 | Comparison of two surreals... |
| sltsubposd 28068 | Subtraction of a positive ... |
| subsge0d 28069 | Non-negative subtraction. ... |
| addsubs4d 28070 | Rearrangement of four term... |
| sltm1d 28071 | A surreal is greater than ... |
| subscan1d 28072 | Cancellation law for surre... |
| subscan2d 28073 | Cancellation law for surre... |
| subseq0d 28074 | The difference between two... |
| mulsfn 28077 | Surreal multiplication is ... |
| mulsval 28078 | The value of surreal multi... |
| mulsval2lem 28079 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28080 | The value of surreal multi... |
| muls01 28081 | Surreal multiplication by ... |
| mulsrid 28082 | Surreal one is a right ide... |
| mulsridd 28083 | Surreal one is a right ide... |
| mulsproplemcbv 28084 | Lemma for surreal multipli... |
| mulsproplem1 28085 | Lemma for surreal multipli... |
| mulsproplem2 28086 | Lemma for surreal multipli... |
| mulsproplem3 28087 | Lemma for surreal multipli... |
| mulsproplem4 28088 | Lemma for surreal multipli... |
| mulsproplem5 28089 | Lemma for surreal multipli... |
| mulsproplem6 28090 | Lemma for surreal multipli... |
| mulsproplem7 28091 | Lemma for surreal multipli... |
| mulsproplem8 28092 | Lemma for surreal multipli... |
| mulsproplem9 28093 | Lemma for surreal multipli... |
| mulsproplem10 28094 | Lemma for surreal multipli... |
| mulsproplem11 28095 | Lemma for surreal multipli... |
| mulsproplem12 28096 | Lemma for surreal multipli... |
| mulsproplem13 28097 | Lemma for surreal multipli... |
| mulsproplem14 28098 | Lemma for surreal multipli... |
| mulsprop 28099 | Surreals are closed under ... |
| mulscutlem 28100 | Lemma for ~ mulscut . Sta... |
| mulscut 28101 | Show the cut properties of... |
| mulscut2 28102 | Show that the cut involved... |
| mulscl 28103 | The surreals are closed un... |
| mulscld 28104 | The surreals are closed un... |
| sltmul 28105 | An ordering relationship f... |
| sltmuld 28106 | An ordering relationship f... |
| slemuld 28107 | An ordering relationship f... |
| mulscom 28108 | Surreal multiplication com... |
| mulscomd 28109 | Surreal multiplication com... |
| muls02 28110 | Surreal multiplication by ... |
| mulslid 28111 | Surreal one is a left iden... |
| mulslidd 28112 | Surreal one is a left iden... |
| mulsgt0 28113 | The product of two positiv... |
| mulsgt0d 28114 | The product of two positiv... |
| mulsge0d 28115 | The product of two non-neg... |
| ssltmul1 28116 | One surreal set less-than ... |
| ssltmul2 28117 | One surreal set less-than ... |
| mulsuniflem 28118 | Lemma for ~ mulsunif . St... |
| mulsunif 28119 | Surreal multiplication has... |
| addsdilem1 28120 | Lemma for surreal distribu... |
| addsdilem2 28121 | Lemma for surreal distribu... |
| addsdilem3 28122 | Lemma for ~ addsdi . Show... |
| addsdilem4 28123 | Lemma for ~ addsdi . Show... |
| addsdi 28124 | Distributive law for surre... |
| addsdid 28125 | Distributive law for surre... |
| addsdird 28126 | Distributive law for surre... |
| subsdid 28127 | Distribution of surreal mu... |
| subsdird 28128 | Distribution of surreal mu... |
| mulnegs1d 28129 | Product with negative is n... |
| mulnegs2d 28130 | Product with negative is n... |
| mul2negsd 28131 | Surreal product of two neg... |
| mulsasslem1 28132 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28133 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28134 | Lemma for ~ mulsass . Dem... |
| mulsass 28135 | Associative law for surrea... |
| mulsassd 28136 | Associative law for surrea... |
| muls4d 28137 | Rearrangement of four surr... |
| mulsunif2lem 28138 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28139 | Alternate expression for s... |
| sltmul2 28140 | Multiplication of both sid... |
| sltmul2d 28141 | Multiplication of both sid... |
| sltmul1d 28142 | Multiplication of both sid... |
| slemul2d 28143 | Multiplication of both sid... |
| slemul1d 28144 | Multiplication of both sid... |
| sltmulneg1d 28145 | Multiplication of both sid... |
| sltmulneg2d 28146 | Multiplication of both sid... |
| mulscan2dlem 28147 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28148 | Cancellation of surreal mu... |
| mulscan1d 28149 | Cancellation of surreal mu... |
| muls12d 28150 | Commutative/associative la... |
| slemul1ad 28151 | Multiplication of both sid... |
| sltmul12ad 28152 | Comparison of the product ... |
| divsmo 28153 | Uniqueness of surreal inve... |
| muls0ord 28154 | If a surreal product is ze... |
| mulsne0bd 28155 | The product of two non-zer... |
| divsval 28158 | The value of surreal divis... |
| norecdiv 28159 | If a surreal has a recipro... |
| noreceuw 28160 | If a surreal has a recipro... |
| recsne0 28161 | If a surreal has a recipro... |
| divsmulw 28162 | Relationship between surre... |
| divsmulwd 28163 | Relationship between surre... |
| divsclw 28164 | Weak division closure law.... |
| divsclwd 28165 | Weak division closure law.... |
| divscan2wd 28166 | A weak cancellation law fo... |
| divscan1wd 28167 | A weak cancellation law fo... |
| sltdivmulwd 28168 | Surreal less-than relation... |
| sltdivmul2wd 28169 | Surreal less-than relation... |
| sltmuldivwd 28170 | Surreal less-than relation... |
| sltmuldiv2wd 28171 | Surreal less-than relation... |
| divsasswd 28172 | An associative law for sur... |
| divs1 28173 | A surreal divided by one i... |
| precsexlemcbv 28174 | Lemma for surreal reciproc... |
| precsexlem1 28175 | Lemma for surreal reciproc... |
| precsexlem2 28176 | Lemma for surreal reciproc... |
| precsexlem3 28177 | Lemma for surreal reciproc... |
| precsexlem4 28178 | Lemma for surreal reciproc... |
| precsexlem5 28179 | Lemma for surreal reciproc... |
| precsexlem6 28180 | Lemma for surreal reciproc... |
| precsexlem7 28181 | Lemma for surreal reciproc... |
| precsexlem8 28182 | Lemma for surreal reciproc... |
| precsexlem9 28183 | Lemma for surreal reciproc... |
| precsexlem10 28184 | Lemma for surreal reciproc... |
| precsexlem11 28185 | Lemma for surreal reciproc... |
| precsex 28186 | Every positive surreal has... |
| recsex 28187 | A non-zero surreal has a r... |
| recsexd 28188 | A non-zero surreal has a r... |
| divsmul 28189 | Relationship between surre... |
| divsmuld 28190 | Relationship between surre... |
| divscl 28191 | Surreal division closure l... |
| divscld 28192 | Surreal division closure l... |
| divscan2d 28193 | A cancellation law for sur... |
| divscan1d 28194 | A cancellation law for sur... |
| sltdivmuld 28195 | Surreal less-than relation... |
| sltdivmul2d 28196 | Surreal less-than relation... |
| sltmuldivd 28197 | Surreal less-than relation... |
| sltmuldiv2d 28198 | Surreal less-than relation... |
| divsassd 28199 | An associative law for sur... |
| divmuldivsd 28200 | Multiplication of two surr... |
| divdivs1d 28201 | Surreal division into a fr... |
| divsrecd 28202 | Relationship between surre... |
| divsdird 28203 | Distribution of surreal di... |
| divscan3d 28204 | A cancellation law for sur... |
| abssval 28207 | The value of surreal absol... |
| absscl 28208 | Closure law for surreal ab... |
| abssid 28209 | The absolute value of a no... |
| abs0s 28210 | The absolute value of surr... |
| abssnid 28211 | For a negative surreal, it... |
| absmuls 28212 | Surreal absolute value dis... |
| abssge0 28213 | The absolute value of a su... |
| abssor 28214 | The absolute value of a su... |
| abssneg 28215 | Surreal absolute value of ... |
| sleabs 28216 | A surreal is less than or ... |
| absslt 28217 | Surreal absolute value and... |
| absssub 28218 | Swapping order of surreal ... |
| elons 28221 | Membership in the class of... |
| onssno 28222 | The surreal ordinals are a... |
| onsno 28223 | A surreal ordinal is a sur... |
| 0ons 28224 | Surreal zero is a surreal ... |
| 1ons 28225 | Surreal one is a surreal o... |
| elons2 28226 | A surreal is ordinal iff i... |
| elons2d 28227 | The cut of any set of surr... |
| onsleft 28228 | The left set of a surreal ... |
| sltonold 28229 | The class of ordinals less... |
| sltonex 28230 | The class of ordinals less... |
| onscutleft 28231 | A surreal ordinal is equal... |
| onscutlt 28232 | A surreal ordinal is the s... |
| bday11on 28233 | The birthday function is o... |
| onnolt 28234 | If a surreal ordinal is le... |
| onslt 28235 | Less-than is the same as b... |
| onsiso 28236 | The birthday function rest... |
| onswe 28237 | Surreal less-than well-ord... |
| onsse 28238 | Surreal less-than is set-l... |
| onsis 28239 | Transfinite induction sche... |
| bdayon 28240 | The birthday of a surreal ... |
| onaddscl 28241 | The surreal ordinals are c... |
| onmulscl 28242 | The surreal ordinals are c... |
| peano2ons 28243 | The successor of a surreal... |
| seqsex 28246 | Existence of the surreal s... |
| seqseq123d 28247 | Equality deduction for the... |
| nfseqs 28248 | Hypothesis builder for the... |
| seqsval 28249 | The value of the surreal s... |
| noseqex 28250 | The next several theorems ... |
| noseq0 28251 | The surreal ` A ` is a mem... |
| noseqp1 28252 | One plus an element of ` Z... |
| noseqind 28253 | Peano's inductive postulat... |
| noseqinds 28254 | Induction schema for surre... |
| noseqssno 28255 | A surreal sequence is a su... |
| noseqno 28256 | An element of a surreal se... |
| om2noseq0 28257 | The mapping ` G ` is a one... |
| om2noseqsuc 28258 | The value of ` G ` at a su... |
| om2noseqfo 28259 | Function statement for ` G... |
| om2noseqlt 28260 | Surreal less-than relation... |
| om2noseqlt2 28261 | The mapping ` G ` preserve... |
| om2noseqf1o 28262 | ` G ` is a bijection. (Co... |
| om2noseqiso 28263 | ` G ` is an isomorphism fr... |
| om2noseqoi 28264 | An alternative definition ... |
| om2noseqrdg 28265 | A helper lemma for the val... |
| noseqrdglem 28266 | A helper lemma for the val... |
| noseqrdgfn 28267 | The recursive definition g... |
| noseqrdg0 28268 | Initial value of a recursi... |
| noseqrdgsuc 28269 | Successor value of a recur... |
| seqsfn 28270 | The surreal sequence build... |
| seqs1 28271 | The value of the surreal s... |
| seqsp1 28272 | The value of the surreal s... |
| n0sexg 28277 | The set of all non-negativ... |
| n0sex 28278 | The set of all non-negativ... |
| nnsex 28279 | The set of all positive su... |
| peano5n0s 28280 | Peano's inductive postulat... |
| n0ssno 28281 | The non-negative surreal i... |
| nnssn0s 28282 | The positive surreal integ... |
| nnssno 28283 | The positive surreal integ... |
| n0sno 28284 | A non-negative surreal int... |
| nnsno 28285 | A positive surreal integer... |
| n0snod 28286 | A non-negative surreal int... |
| nnsnod 28287 | A positive surreal integer... |
| nnn0s 28288 | A positive surreal integer... |
| nnn0sd 28289 | A positive surreal integer... |
| 0n0s 28290 | Peano postulate: ` 0s ` is... |
| peano2n0s 28291 | Peano postulate: the succe... |
| dfn0s2 28292 | Alternate definition of th... |
| n0sind 28293 | Principle of Mathematical ... |
| n0scut 28294 | A cut form for non-negativ... |
| n0scut2 28295 | A cut form for the success... |
| n0ons 28296 | A surreal natural is a sur... |
| nnne0s 28297 | A surreal positive integer... |
| n0sge0 28298 | A non-negative integer is ... |
| nnsgt0 28299 | A positive integer is grea... |
| elnns 28300 | Membership in the positive... |
| elnns2 28301 | A positive surreal integer... |
| n0s0suc 28302 | A non-negative surreal int... |
| nnsge1 28303 | A positive surreal integer... |
| n0addscl 28304 | The non-negative surreal i... |
| n0mulscl 28305 | The non-negative surreal i... |
| nnaddscl 28306 | The positive surreal integ... |
| nnmulscl 28307 | The positive surreal integ... |
| 1n0s 28308 | Surreal one is a non-negat... |
| 1nns 28309 | Surreal one is a positive ... |
| peano2nns 28310 | Peano postulate for positi... |
| nnsrecgt0d 28311 | The reciprocal of a positi... |
| n0sbday 28312 | A non-negative surreal int... |
| n0ssoldg 28313 | The non-negative surreal i... |
| n0ssold 28314 | The non-negative surreal i... |
| n0sfincut 28315 | The simplest number greate... |
| onsfi 28316 | A surreal ordinal with a f... |
| onltn0s 28317 | A surreal ordinal that is ... |
| n0cutlt 28318 | A non-negative surreal int... |
| seqn0sfn 28319 | The surreal sequence build... |
| eln0s 28320 | A non-negative surreal int... |
| n0s0m1 28321 | Every non-negative surreal... |
| n0subs 28322 | Subtraction of non-negativ... |
| n0subs2 28323 | Subtraction of non-negativ... |
| n0sltp1le 28324 | Non-negative surreal order... |
| n0sleltp1 28325 | Non-negative surreal order... |
| n0slem1lt 28326 | Non-negative surreal order... |
| bdayn0p1 28327 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28328 | The birthday function rest... |
| n0p1nns 28329 | One plus a non-negative su... |
| dfnns2 28330 | Alternate definition of th... |
| nnsind 28331 | Principle of Mathematical ... |
| nn1m1nns 28332 | Every positive surreal int... |
| nnm1n0s 28333 | A positive surreal integer... |
| eucliddivs 28334 | Euclid's division lemma fo... |
| oldfib 28335 | The old set of an ordinal ... |
| zsex 28338 | The surreal integers form ... |
| zssno 28339 | The surreal integers are a... |
| zno 28340 | A surreal integer is a sur... |
| znod 28341 | A surreal integer is a sur... |
| elzs 28342 | Membership in the set of s... |
| nnzsubs 28343 | The difference of two surr... |
| nnzs 28344 | A positive surreal integer... |
| nnzsd 28345 | A positive surreal integer... |
| 0zs 28346 | Zero is a surreal integer.... |
| n0zs 28347 | A non-negative surreal int... |
| n0zsd 28348 | A non-negative surreal int... |
| 1zs 28349 | One is a surreal integer. ... |
| znegscl 28350 | The surreal integers are c... |
| znegscld 28351 | The surreal integers are c... |
| zaddscl 28352 | The surreal integers are c... |
| zaddscld 28353 | The surreal integers are c... |
| zsubscld 28354 | The surreal integers are c... |
| zmulscld 28355 | The surreal integers are c... |
| elzn0s 28356 | A surreal integer is a sur... |
| elzs2 28357 | A surreal integer is eithe... |
| eln0zs 28358 | Non-negative surreal integ... |
| elnnzs 28359 | Positive surreal integer p... |
| elznns 28360 | Surreal integer property e... |
| zn0subs 28361 | The non-negative differenc... |
| peano5uzs 28362 | Peano's inductive postulat... |
| uzsind 28363 | Induction on the upper sur... |
| zsbday 28364 | A surreal integer has a fi... |
| zscut 28365 | A cut expression for surre... |
| zscut0 28366 | Either the left or right s... |
| zsoring 28367 | The surreal integers form ... |
| 1p1e2s 28374 | One plus one is two. Surr... |
| no2times 28375 | Version of ~ 2times for su... |
| 2nns 28376 | Surreal two is a surreal n... |
| 2sno 28377 | Surreal two is a surreal n... |
| 2ne0s 28378 | Surreal two is non-zero. ... |
| n0seo 28379 | A non-negative surreal int... |
| zseo 28380 | A surreal integer is eithe... |
| twocut 28381 | Two times the cut of zero ... |
| nohalf 28382 | An explicit expression for... |
| expsval 28383 | The value of surreal expon... |
| expsnnval 28384 | Value of surreal exponenti... |
| exps0 28385 | Surreal exponentiation to ... |
| exps1 28386 | Surreal exponentiation to ... |
| expsp1 28387 | Value of a surreal number ... |
| expscllem 28388 | Lemma for proving non-nega... |
| expscl 28389 | Closure law for surreal ex... |
| n0expscl 28390 | Closure law for non-negati... |
| nnexpscl 28391 | Closure law for positive s... |
| zexpscl 28392 | Closure law for surreal in... |
| expadds 28393 | Sum of exponents law for s... |
| expsne0 28394 | A non-negative surreal int... |
| expsgt0 28395 | A non-negative surreal int... |
| pw2recs 28396 | Any power of two has a mul... |
| pw2divscld 28397 | Division closure for power... |
| pw2divsmuld 28398 | Relationship between surre... |
| pw2divscan3d 28399 | Cancellation law for surre... |
| pw2divscan2d 28400 | A cancellation law for sur... |
| pw2divsassd 28401 | An associative law for div... |
| pw2divscan4d 28402 | Cancellation law for divis... |
| pw2gt0divsd 28403 | Division of a positive sur... |
| pw2ge0divsd 28404 | Divison of a non-negative ... |
| pw2divsrecd 28405 | Relationship between surre... |
| pw2divsdird 28406 | Distribution of surreal di... |
| pw2divsnegd 28407 | Move negative sign inside ... |
| pw2sltdivmuld 28408 | Surreal less-than relation... |
| pw2sltmuldiv2d 28409 | Surreal less-than relation... |
| pw2sltdiv1d 28410 | Surreal less-than relation... |
| avgslt1d 28411 | Ordering property for aver... |
| avgslt2d 28412 | Ordering property for aver... |
| pw2divs0d 28413 | Division into zero is zero... |
| pw2divsidd 28414 | Identity law for division ... |
| halfcut 28415 | Relate the cut of twice of... |
| addhalfcut 28416 | The cut of a surreal non-n... |
| pw2cut 28417 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28418 | Simplify ~ pw2cut in the c... |
| pw2cut2 28419 | Cut expression for powers ... |
| bdaypw2n0s 28420 | Upper bound for the birthd... |
| elzs12 28421 | Membership in the dyadic f... |
| elzs12i 28422 | Inference form of membersh... |
| zs12ex 28423 | The class of dyadic fracti... |
| zzs12 28424 | A surreal integer is a dya... |
| zs12no 28425 | A dyadic is a surreal. (C... |
| zs12addscl 28426 | The dyadics are closed und... |
| zs12negscl 28427 | The dyadics are closed und... |
| zs12subscl 28428 | The dyadics are closed und... |
| zs12half 28429 | Half of a dyadic is a dyad... |
| zs12negsclb 28430 | A surreal is a dyadic frac... |
| zs12zodd 28431 | A dyadic fraction is eithe... |
| zs12ge0 28432 | An expression for non-nega... |
| zs12bday 28433 | A dyadic fraction has a fi... |
| elreno 28436 | Membership in the set of s... |
| reno 28437 | A surreal real is a surrea... |
| renod 28438 | A surreal real is a surrea... |
| recut 28439 | The cut involved in defini... |
| elreno2 28440 | Alternate characterization... |
| 0reno 28441 | Surreal zero is a surreal ... |
| 1reno 28442 | Surreal one is a surreal r... |
| renegscl 28443 | The surreal reals are clos... |
| readdscl 28444 | The surreal reals are clos... |
| remulscllem1 28445 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28446 | Lemma for ~ remulscl . Bo... |
| remulscl 28447 | The surreal reals are clos... |
| itvndx 28458 | Index value of the Interva... |
| lngndx 28459 | Index value of the "line" ... |
| itvid 28460 | Utility theorem: index-ind... |
| lngid 28461 | Utility theorem: index-ind... |
| slotsinbpsd 28462 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28463 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28464 | The slot for the line is n... |
| trkgstr 28465 | Functionality of a Tarski ... |
| trkgbas 28466 | The base set of a Tarski g... |
| trkgdist 28467 | The measure of a distance ... |
| trkgitv 28468 | The congruence relation in... |
| istrkgc 28475 | Property of being a Tarski... |
| istrkgb 28476 | Property of being a Tarski... |
| istrkgcb 28477 | Property of being a Tarski... |
| istrkge 28478 | Property of fulfilling Euc... |
| istrkgl 28479 | Building lines from the se... |
| istrkgld 28480 | Property of fulfilling the... |
| istrkg2ld 28481 | Property of fulfilling the... |
| istrkg3ld 28482 | Property of fulfilling the... |
| axtgcgrrflx 28483 | Axiom of reflexivity of co... |
| axtgcgrid 28484 | Axiom of identity of congr... |
| axtgsegcon 28485 | Axiom of segment construct... |
| axtg5seg 28486 | Five segments axiom, Axiom... |
| axtgbtwnid 28487 | Identity of Betweenness. ... |
| axtgpasch 28488 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28489 | Axiom of Continuity. Axio... |
| axtgcont 28490 | Axiom of Continuity. Axio... |
| axtglowdim2 28491 | Lower dimension axiom for ... |
| axtgupdim2 28492 | Upper dimension axiom for ... |
| axtgeucl 28493 | Euclid's Axiom. Axiom A10... |
| tgjustf 28494 | Given any function ` F ` ,... |
| tgjustr 28495 | Given any equivalence rela... |
| tgjustc1 28496 | A justification for using ... |
| tgjustc2 28497 | A justification for using ... |
| tgcgrcomimp 28498 | Congruence commutes on the... |
| tgcgrcomr 28499 | Congruence commutes on the... |
| tgcgrcoml 28500 | Congruence commutes on the... |
| tgcgrcomlr 28501 | Congruence commutes on bot... |
| tgcgreqb 28502 | Congruence and equality. ... |
| tgcgreq 28503 | Congruence and equality. ... |
| tgcgrneq 28504 | Congruence and equality. ... |
| tgcgrtriv 28505 | Degenerate segments are co... |
| tgcgrextend 28506 | Link congruence over a pai... |
| tgsegconeq 28507 | Two points that satisfy th... |
| tgbtwntriv2 28508 | Betweenness always holds f... |
| tgbtwncom 28509 | Betweenness commutes. The... |
| tgbtwncomb 28510 | Betweenness commutes, bico... |
| tgbtwnne 28511 | Betweenness and inequality... |
| tgbtwntriv1 28512 | Betweenness always holds f... |
| tgbtwnswapid 28513 | If you can swap the first ... |
| tgbtwnintr 28514 | Inner transitivity law for... |
| tgbtwnexch3 28515 | Exchange the first endpoin... |
| tgbtwnouttr2 28516 | Outer transitivity law for... |
| tgbtwnexch2 28517 | Exchange the outer point o... |
| tgbtwnouttr 28518 | Outer transitivity law for... |
| tgbtwnexch 28519 | Outer transitivity law for... |
| tgtrisegint 28520 | A line segment between two... |
| tglowdim1 28521 | Lower dimension axiom for ... |
| tglowdim1i 28522 | Lower dimension axiom for ... |
| tgldimor 28523 | Excluded-middle like state... |
| tgldim0eq 28524 | In dimension zero, any two... |
| tgldim0itv 28525 | In dimension zero, any two... |
| tgldim0cgr 28526 | In dimension zero, any two... |
| tgbtwndiff 28527 | There is always a ` c ` di... |
| tgdim01 28528 | In geometries of dimension... |
| tgifscgr 28529 | Inner five segment congrue... |
| tgcgrsub 28530 | Removing identical parts f... |
| iscgrg 28533 | The congruence property fo... |
| iscgrgd 28534 | The property for two seque... |
| iscgrglt 28535 | The property for two seque... |
| trgcgrg 28536 | The property for two trian... |
| trgcgr 28537 | Triangle congruence. (Con... |
| ercgrg 28538 | The shape congruence relat... |
| tgcgrxfr 28539 | A line segment can be divi... |
| cgr3id 28540 | Reflexivity law for three-... |
| cgr3simp1 28541 | Deduce segment congruence ... |
| cgr3simp2 28542 | Deduce segment congruence ... |
| cgr3simp3 28543 | Deduce segment congruence ... |
| cgr3swap12 28544 | Permutation law for three-... |
| cgr3swap23 28545 | Permutation law for three-... |
| cgr3swap13 28546 | Permutation law for three-... |
| cgr3rotr 28547 | Permutation law for three-... |
| cgr3rotl 28548 | Permutation law for three-... |
| trgcgrcom 28549 | Commutative law for three-... |
| cgr3tr 28550 | Transitivity law for three... |
| tgbtwnxfr 28551 | A condition for extending ... |
| tgcgr4 28552 | Two quadrilaterals to be c... |
| isismt 28555 | Property of being an isome... |
| ismot 28556 | Property of being an isome... |
| motcgr 28557 | Property of a motion: dist... |
| idmot 28558 | The identity is a motion. ... |
| motf1o 28559 | Motions are bijections. (... |
| motcl 28560 | Closure of motions. (Cont... |
| motco 28561 | The composition of two mot... |
| cnvmot 28562 | The converse of a motion i... |
| motplusg 28563 | The operation for motions ... |
| motgrp 28564 | The motions of a geometry ... |
| motcgrg 28565 | Property of a motion: dist... |
| motcgr3 28566 | Property of a motion: dist... |
| tglng 28567 | Lines of a Tarski Geometry... |
| tglnfn 28568 | Lines as functions. (Cont... |
| tglnunirn 28569 | Lines are sets of points. ... |
| tglnpt 28570 | Lines are sets of points. ... |
| tglngne 28571 | It takes two different poi... |
| tglngval 28572 | The line going through poi... |
| tglnssp 28573 | Lines are subset of the ge... |
| tgellng 28574 | Property of lying on the l... |
| tgcolg 28575 | We choose the notation ` (... |
| btwncolg1 28576 | Betweenness implies coline... |
| btwncolg2 28577 | Betweenness implies coline... |
| btwncolg3 28578 | Betweenness implies coline... |
| colcom 28579 | Swapping the points defini... |
| colrot1 28580 | Rotating the points defini... |
| colrot2 28581 | Rotating the points defini... |
| ncolcom 28582 | Swapping non-colinear poin... |
| ncolrot1 28583 | Rotating non-colinear poin... |
| ncolrot2 28584 | Rotating non-colinear poin... |
| tgdim01ln 28585 | In geometries of dimension... |
| ncoltgdim2 28586 | If there are three non-col... |
| lnxfr 28587 | Transfer law for colineari... |
| lnext 28588 | Extend a line with a missi... |
| tgfscgr 28589 | Congruence law for the gen... |
| lncgr 28590 | Congruence rule for lines.... |
| lnid 28591 | Identity law for points on... |
| tgidinside 28592 | Law for finding a point in... |
| tgbtwnconn1lem1 28593 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28594 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28595 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28596 | Connectivity law for betwe... |
| tgbtwnconn2 28597 | Another connectivity law f... |
| tgbtwnconn3 28598 | Inner connectivity law for... |
| tgbtwnconnln3 28599 | Derive colinearity from be... |
| tgbtwnconn22 28600 | Double connectivity law fo... |
| tgbtwnconnln1 28601 | Derive colinearity from be... |
| tgbtwnconnln2 28602 | Derive colinearity from be... |
| legval 28605 | Value of the less-than rel... |
| legov 28606 | Value of the less-than rel... |
| legov2 28607 | An equivalent definition o... |
| legid 28608 | Reflexivity of the less-th... |
| btwnleg 28609 | Betweenness implies less-t... |
| legtrd 28610 | Transitivity of the less-t... |
| legtri3 28611 | Equality from the less-tha... |
| legtrid 28612 | Trichotomy law for the les... |
| leg0 28613 | Degenerated (zero-length) ... |
| legeq 28614 | Deduce equality from "less... |
| legbtwn 28615 | Deduce betweenness from "l... |
| tgcgrsub2 28616 | Removing identical parts f... |
| ltgseg 28617 | The set ` E ` denotes the ... |
| ltgov 28618 | Strict "shorter than" geom... |
| legov3 28619 | An equivalent definition o... |
| legso 28620 | The "shorter than" relatio... |
| ishlg 28623 | Rays : Definition 6.1 of ... |
| hlcomb 28624 | The half-line relation com... |
| hlcomd 28625 | The half-line relation com... |
| hlne1 28626 | The half-line relation imp... |
| hlne2 28627 | The half-line relation imp... |
| hlln 28628 | The half-line relation imp... |
| hleqnid 28629 | The endpoint does not belo... |
| hlid 28630 | The half-line relation is ... |
| hltr 28631 | The half-line relation is ... |
| hlbtwn 28632 | Betweenness is a sufficien... |
| btwnhl1 28633 | Deduce half-line from betw... |
| btwnhl2 28634 | Deduce half-line from betw... |
| btwnhl 28635 | Swap betweenness for a hal... |
| lnhl 28636 | Either a point ` C ` on th... |
| hlcgrex 28637 | Construct a point on a hal... |
| hlcgreulem 28638 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28639 | The point constructed in ~... |
| btwnlng1 28640 | Betweenness implies coline... |
| btwnlng2 28641 | Betweenness implies coline... |
| btwnlng3 28642 | Betweenness implies coline... |
| lncom 28643 | Swapping the points defini... |
| lnrot1 28644 | Rotating the points defini... |
| lnrot2 28645 | Rotating the points defini... |
| ncolne1 28646 | Non-colinear points are di... |
| ncolne2 28647 | Non-colinear points are di... |
| tgisline 28648 | The property of being a pr... |
| tglnne 28649 | It takes two different poi... |
| tglndim0 28650 | There are no lines in dime... |
| tgelrnln 28651 | The property of being a pr... |
| tglineeltr 28652 | Transitivity law for lines... |
| tglineelsb2 28653 | If ` S ` lies on PQ , then... |
| tglinerflx1 28654 | Reflexivity law for line m... |
| tglinerflx2 28655 | Reflexivity law for line m... |
| tglinecom 28656 | Commutativity law for line... |
| tglinethru 28657 | If ` A ` is a line contain... |
| tghilberti1 28658 | There is a line through an... |
| tghilberti2 28659 | There is at most one line ... |
| tglinethrueu 28660 | There is a unique line goi... |
| tglnne0 28661 | A line ` A ` has at least ... |
| tglnpt2 28662 | Find a second point on a l... |
| tglineintmo 28663 | Two distinct lines interse... |
| tglineineq 28664 | Two distinct lines interse... |
| tglineneq 28665 | Given three non-colinear p... |
| tglineinteq 28666 | Two distinct lines interse... |
| ncolncol 28667 | Deduce non-colinearity fro... |
| coltr 28668 | A transitivity law for col... |
| coltr3 28669 | A transitivity law for col... |
| colline 28670 | Three points are colinear ... |
| tglowdim2l 28671 | Reformulation of the lower... |
| tglowdim2ln 28672 | There is always one point ... |
| mirreu3 28675 | Existential uniqueness of ... |
| mirval 28676 | Value of the point inversi... |
| mirfv 28677 | Value of the point inversi... |
| mircgr 28678 | Property of the image by t... |
| mirbtwn 28679 | Property of the image by t... |
| ismir 28680 | Property of the image by t... |
| mirf 28681 | Point inversion as functio... |
| mircl 28682 | Closure of the point inver... |
| mirmir 28683 | The point inversion functi... |
| mircom 28684 | Variation on ~ mirmir . (... |
| mirreu 28685 | Any point has a unique ant... |
| mireq 28686 | Equality deduction for poi... |
| mirinv 28687 | The only invariant point o... |
| mirne 28688 | Mirror of non-center point... |
| mircinv 28689 | The center point is invari... |
| mirf1o 28690 | The point inversion functi... |
| miriso 28691 | The point inversion functi... |
| mirbtwni 28692 | Point inversion preserves ... |
| mirbtwnb 28693 | Point inversion preserves ... |
| mircgrs 28694 | Point inversion preserves ... |
| mirmir2 28695 | Point inversion of a point... |
| mirmot 28696 | Point investion is a motio... |
| mirln 28697 | If two points are on the s... |
| mirln2 28698 | If a point and its mirror ... |
| mirconn 28699 | Point inversion of connect... |
| mirhl 28700 | If two points ` X ` and ` ... |
| mirbtwnhl 28701 | If the center of the point... |
| mirhl2 28702 | Deduce half-line relation ... |
| mircgrextend 28703 | Link congruence over a pai... |
| mirtrcgr 28704 | Point inversion of one poi... |
| mirauto 28705 | Point inversion preserves ... |
| miduniq 28706 | Uniqueness of the middle p... |
| miduniq1 28707 | Uniqueness of the middle p... |
| miduniq2 28708 | If two point inversions co... |
| colmid 28709 | Colinearity and equidistan... |
| symquadlem 28710 | Lemma of the symetrial qua... |
| krippenlem 28711 | Lemma for ~ krippen . We ... |
| krippen 28712 | Krippenlemma (German for c... |
| midexlem 28713 | Lemma for the existence of... |
| israg 28718 | Property for 3 points A, B... |
| ragcom 28719 | Commutative rule for right... |
| ragcol 28720 | The right angle property i... |
| ragmir 28721 | Right angle property is pr... |
| mirrag 28722 | Right angle is conserved b... |
| ragtrivb 28723 | Trivial right angle. Theo... |
| ragflat2 28724 | Deduce equality from two r... |
| ragflat 28725 | Deduce equality from two r... |
| ragtriva 28726 | Trivial right angle. Theo... |
| ragflat3 28727 | Right angle and colinearit... |
| ragcgr 28728 | Right angle and colinearit... |
| motrag 28729 | Right angles are preserved... |
| ragncol 28730 | Right angle implies non-co... |
| perpln1 28731 | Derive a line from perpend... |
| perpln2 28732 | Derive a line from perpend... |
| isperp 28733 | Property for 2 lines A, B ... |
| perpcom 28734 | The "perpendicular" relati... |
| perpneq 28735 | Two perpendicular lines ar... |
| isperp2 28736 | Property for 2 lines A, B,... |
| isperp2d 28737 | One direction of ~ isperp2... |
| ragperp 28738 | Deduce that two lines are ... |
| footexALT 28739 | Alternative version of ~ f... |
| footexlem1 28740 | Lemma for ~ footex . (Con... |
| footexlem2 28741 | Lemma for ~ footex . (Con... |
| footex 28742 | From a point ` C ` outside... |
| foot 28743 | From a point ` C ` outside... |
| footne 28744 | Uniqueness of the foot poi... |
| footeq 28745 | Uniqueness of the foot poi... |
| hlperpnel 28746 | A point on a half-line whi... |
| perprag 28747 | Deduce a right angle from ... |
| perpdragALT 28748 | Deduce a right angle from ... |
| perpdrag 28749 | Deduce a right angle from ... |
| colperp 28750 | Deduce a perpendicularity ... |
| colperpexlem1 28751 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28752 | Lemma for ~ colperpex . S... |
| colperpexlem3 28753 | Lemma for ~ colperpex . C... |
| colperpex 28754 | In dimension 2 and above, ... |
| mideulem2 28755 | Lemma for ~ opphllem , whi... |
| opphllem 28756 | Lemma 8.24 of [Schwabhause... |
| mideulem 28757 | Lemma for ~ mideu . We ca... |
| midex 28758 | Existence of the midpoint,... |
| mideu 28759 | Existence and uniqueness o... |
| islnopp 28760 | The property for two point... |
| islnoppd 28761 | Deduce that ` A ` and ` B ... |
| oppne1 28762 | Points lying on opposite s... |
| oppne2 28763 | Points lying on opposite s... |
| oppne3 28764 | Points lying on opposite s... |
| oppcom 28765 | Commutativity rule for "op... |
| opptgdim2 28766 | If two points opposite to ... |
| oppnid 28767 | The "opposite to a line" r... |
| opphllem1 28768 | Lemma for ~ opphl . (Cont... |
| opphllem2 28769 | Lemma for ~ opphl . Lemma... |
| opphllem3 28770 | Lemma for ~ opphl : We as... |
| opphllem4 28771 | Lemma for ~ opphl . (Cont... |
| opphllem5 28772 | Second part of Lemma 9.4 o... |
| opphllem6 28773 | First part of Lemma 9.4 of... |
| oppperpex 28774 | Restating ~ colperpex usin... |
| opphl 28775 | If two points ` A ` and ` ... |
| outpasch 28776 | Axiom of Pasch, outer form... |
| hlpasch 28777 | An application of the axio... |
| ishpg 28780 | Value of the half-plane re... |
| hpgbr 28781 | Half-planes : property for... |
| hpgne1 28782 | Points on the open half pl... |
| hpgne2 28783 | Points on the open half pl... |
| lnopp2hpgb 28784 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28785 | If two points lie on the o... |
| hpgerlem 28786 | Lemma for the proof that t... |
| hpgid 28787 | The half-plane relation is... |
| hpgcom 28788 | The half-plane relation co... |
| hpgtr 28789 | The half-plane relation is... |
| colopp 28790 | Opposite sides of a line f... |
| colhp 28791 | Half-plane relation for co... |
| hphl 28792 | If two points are on the s... |
| midf 28797 | Midpoint as a function. (... |
| midcl 28798 | Closure of the midpoint. ... |
| ismidb 28799 | Property of the midpoint. ... |
| midbtwn 28800 | Betweenness of midpoint. ... |
| midcgr 28801 | Congruence of midpoint. (... |
| midid 28802 | Midpoint of a null segment... |
| midcom 28803 | Commutativity rule for the... |
| mirmid 28804 | Point inversion preserves ... |
| lmieu 28805 | Uniqueness of the line mir... |
| lmif 28806 | Line mirror as a function.... |
| lmicl 28807 | Closure of the line mirror... |
| islmib 28808 | Property of the line mirro... |
| lmicom 28809 | The line mirroring functio... |
| lmilmi 28810 | Line mirroring is an invol... |
| lmireu 28811 | Any point has a unique ant... |
| lmieq 28812 | Equality deduction for lin... |
| lmiinv 28813 | The invariants of the line... |
| lmicinv 28814 | The mirroring line is an i... |
| lmimid 28815 | If we have a right angle, ... |
| lmif1o 28816 | The line mirroring functio... |
| lmiisolem 28817 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28818 | The line mirroring functio... |
| lmimot 28819 | Line mirroring is a motion... |
| hypcgrlem1 28820 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28821 | Lemma for ~ hypcgr , case ... |
| hypcgr 28822 | If the catheti of two righ... |
| lmiopp 28823 | Line mirroring produces po... |
| lnperpex 28824 | Existence of a perpendicul... |
| trgcopy 28825 | Triangle construction: a c... |
| trgcopyeulem 28826 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28827 | Triangle construction: a c... |
| iscgra 28830 | Property for two angles AB... |
| iscgra1 28831 | A special version of ~ isc... |
| iscgrad 28832 | Sufficient conditions for ... |
| cgrane1 28833 | Angles imply inequality. ... |
| cgrane2 28834 | Angles imply inequality. ... |
| cgrane3 28835 | Angles imply inequality. ... |
| cgrane4 28836 | Angles imply inequality. ... |
| cgrahl1 28837 | Angle congruence is indepe... |
| cgrahl2 28838 | Angle congruence is indepe... |
| cgracgr 28839 | First direction of proposi... |
| cgraid 28840 | Angle congruence is reflex... |
| cgraswap 28841 | Swap rays in a congruence ... |
| cgrcgra 28842 | Triangle congruence implie... |
| cgracom 28843 | Angle congruence commutes.... |
| cgratr 28844 | Angle congruence is transi... |
| flatcgra 28845 | Flat angles are congruent.... |
| cgraswaplr 28846 | Swap both side of angle co... |
| cgrabtwn 28847 | Angle congruence preserves... |
| cgrahl 28848 | Angle congruence preserves... |
| cgracol 28849 | Angle congruence preserves... |
| cgrancol 28850 | Angle congruence preserves... |
| dfcgra2 28851 | This is the full statement... |
| sacgr 28852 | Supplementary angles of co... |
| oacgr 28853 | Vertical angle theorem. V... |
| acopy 28854 | Angle construction. Theor... |
| acopyeu 28855 | Angle construction. Theor... |
| isinag 28859 | Property for point ` X ` t... |
| isinagd 28860 | Sufficient conditions for ... |
| inagflat 28861 | Any point lies in a flat a... |
| inagswap 28862 | Swap the order of the half... |
| inagne1 28863 | Deduce inequality from the... |
| inagne2 28864 | Deduce inequality from the... |
| inagne3 28865 | Deduce inequality from the... |
| inaghl 28866 | The "point lie in angle" r... |
| isleag 28868 | Geometrical "less than" pr... |
| isleagd 28869 | Sufficient condition for "... |
| leagne1 28870 | Deduce inequality from the... |
| leagne2 28871 | Deduce inequality from the... |
| leagne3 28872 | Deduce inequality from the... |
| leagne4 28873 | Deduce inequality from the... |
| cgrg3col4 28874 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28875 | First congruence theorem: ... |
| tgsas 28876 | First congruence theorem: ... |
| tgsas2 28877 | First congruence theorem: ... |
| tgsas3 28878 | First congruence theorem: ... |
| tgasa1 28879 | Second congruence theorem:... |
| tgasa 28880 | Second congruence theorem:... |
| tgsss1 28881 | Third congruence theorem: ... |
| tgsss2 28882 | Third congruence theorem: ... |
| tgsss3 28883 | Third congruence theorem: ... |
| dfcgrg2 28884 | Congruence for two triangl... |
| isoas 28885 | Congruence theorem for iso... |
| iseqlg 28888 | Property of a triangle bei... |
| iseqlgd 28889 | Condition for a triangle t... |
| f1otrgds 28890 | Convenient lemma for ~ f1o... |
| f1otrgitv 28891 | Convenient lemma for ~ f1o... |
| f1otrg 28892 | A bijection between bases ... |
| f1otrge 28893 | A bijection between bases ... |
| ttgval 28896 | Define a function to augme... |
| ttglem 28897 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 28898 | The base set of a subcompl... |
| ttgplusg 28899 | The addition operation of ... |
| ttgsub 28900 | The subtraction operation ... |
| ttgvsca 28901 | The scalar product of a su... |
| ttgds 28902 | The metric of a subcomplex... |
| ttgitvval 28903 | Betweenness for a subcompl... |
| ttgelitv 28904 | Betweenness for a subcompl... |
| ttgbtwnid 28905 | Any subcomplex module equi... |
| ttgcontlem1 28906 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28907 | Any metric space fulfills ... |
| cchhllem 28908 | Lemma for chlbas and chlvs... |
| elee 28915 | Membership in a Euclidean ... |
| mptelee 28916 | A condition for a mapping ... |
| mpteleeOLD 28917 | Obsolete version of ~ mpte... |
| eleenn 28918 | If ` A ` is in ` ( EE `` N... |
| eleei 28919 | The forward direction of ~... |
| eedimeq 28920 | A point belongs to at most... |
| brbtwn 28921 | The binary relation form o... |
| brcgr 28922 | The binary relation form o... |
| fveere 28923 | The function value of a po... |
| fveecn 28924 | The function value of a po... |
| eqeefv 28925 | Two points are equal iff t... |
| eqeelen 28926 | Two points are equal iff t... |
| brbtwn2 28927 | Alternate characterization... |
| colinearalglem1 28928 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28929 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28930 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28931 | Lemma for ~ colinearalg . ... |
| colinearalg 28932 | An algebraic characterizat... |
| eleesub 28933 | Membership of a subtractio... |
| eleesubd 28934 | Membership of a subtractio... |
| axdimuniq 28935 | The unique dimension axiom... |
| axcgrrflx 28936 | ` A ` is as far from ` B `... |
| axcgrtr 28937 | Congruence is transitive. ... |
| axcgrid 28938 | If there is no distance be... |
| axsegconlem1 28939 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28940 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28941 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28942 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28943 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28944 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28945 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28946 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28947 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28948 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28949 | Any segment ` A B ` can be... |
| ax5seglem1 28950 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28951 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28952 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28953 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28954 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28955 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28956 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28957 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28958 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28959 | Lemma for ~ ax5seg . Take... |
| ax5seg 28960 | The five segment axiom. T... |
| axbtwnid 28961 | Points are indivisible. T... |
| axpaschlem 28962 | Lemma for ~ axpasch . Set... |
| axpasch 28963 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28964 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 28965 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 28966 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 28967 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 28968 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 28969 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 28970 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 28971 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 28972 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 28973 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 28974 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 28975 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 28976 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 28977 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 28978 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 28979 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 28980 | Lemma for ~ axlowdim . Es... |
| axlowdim1 28981 | The lower dimension axiom ... |
| axlowdim2 28982 | The lower two-dimensional ... |
| axlowdim 28983 | The general lower dimensio... |
| axeuclidlem 28984 | Lemma for ~ axeuclid . Ha... |
| axeuclid 28985 | Euclid's axiom. Take an a... |
| axcontlem1 28986 | Lemma for ~ axcont . Chan... |
| axcontlem2 28987 | Lemma for ~ axcont . The ... |
| axcontlem3 28988 | Lemma for ~ axcont . Give... |
| axcontlem4 28989 | Lemma for ~ axcont . Give... |
| axcontlem5 28990 | Lemma for ~ axcont . Comp... |
| axcontlem6 28991 | Lemma for ~ axcont . Stat... |
| axcontlem7 28992 | Lemma for ~ axcont . Give... |
| axcontlem8 28993 | Lemma for ~ axcont . A po... |
| axcontlem9 28994 | Lemma for ~ axcont . Give... |
| axcontlem10 28995 | Lemma for ~ axcont . Give... |
| axcontlem11 28996 | Lemma for ~ axcont . Elim... |
| axcontlem12 28997 | Lemma for ~ axcont . Elim... |
| axcont 28998 | The axiom of continuity. ... |
| eengv 29001 | The value of the Euclidean... |
| eengstr 29002 | The Euclidean geometry as ... |
| eengbas 29003 | The Base of the Euclidean ... |
| ebtwntg 29004 | The betweenness relation u... |
| ecgrtg 29005 | The congruence relation us... |
| elntg 29006 | The line definition in the... |
| elntg2 29007 | The line definition in the... |
| eengtrkg 29008 | The geometry structure for... |
| eengtrkge 29009 | The geometry structure for... |
| edgfid 29012 | Utility theorem: index-ind... |
| edgfndx 29013 | Index value of the ~ df-ed... |
| edgfndxnn 29014 | The index value of the edg... |
| edgfndxid 29015 | The value of the edge func... |
| basendxltedgfndx 29016 | The index value of the ` B... |
| basendxnedgfndx 29017 | The slots ` Base ` and ` .... |
| vtxval 29022 | The set of vertices of a g... |
| iedgval 29023 | The set of indexed edges o... |
| 1vgrex 29024 | A graph with at least one ... |
| opvtxval 29025 | The set of vertices of a g... |
| opvtxfv 29026 | The set of vertices of a g... |
| opvtxov 29027 | The set of vertices of a g... |
| opiedgval 29028 | The set of indexed edges o... |
| opiedgfv 29029 | The set of indexed edges o... |
| opiedgov 29030 | The set of indexed edges o... |
| opvtxfvi 29031 | The set of vertices of a g... |
| opiedgfvi 29032 | The set of indexed edges o... |
| funvtxdmge2val 29033 | The set of vertices of an ... |
| funiedgdmge2val 29034 | The set of indexed edges o... |
| funvtxdm2val 29035 | The set of vertices of an ... |
| funiedgdm2val 29036 | The set of indexed edges o... |
| funvtxval0 29037 | The set of vertices of an ... |
| basvtxval 29038 | The set of vertices of a g... |
| edgfiedgval 29039 | The set of indexed edges o... |
| funvtxval 29040 | The set of vertices of a g... |
| funiedgval 29041 | The set of indexed edges o... |
| structvtxvallem 29042 | Lemma for ~ structvtxval a... |
| structvtxval 29043 | The set of vertices of an ... |
| structiedg0val 29044 | The set of indexed edges o... |
| structgrssvtxlem 29045 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29046 | The set of vertices of a g... |
| structgrssiedg 29047 | The set of indexed edges o... |
| struct2grstr 29048 | A graph represented as an ... |
| struct2grvtx 29049 | The set of vertices of a g... |
| struct2griedg 29050 | The set of indexed edges o... |
| graop 29051 | Any representation of a gr... |
| grastruct 29052 | Any representation of a gr... |
| gropd 29053 | If any representation of a... |
| grstructd 29054 | If any representation of a... |
| gropeld 29055 | If any representation of a... |
| grstructeld 29056 | If any representation of a... |
| setsvtx 29057 | The vertices of a structur... |
| setsiedg 29058 | The (indexed) edges of a s... |
| snstrvtxval 29059 | The set of vertices of a g... |
| snstriedgval 29060 | The set of indexed edges o... |
| vtxval0 29061 | Degenerated case 1 for ver... |
| iedgval0 29062 | Degenerated case 1 for edg... |
| vtxvalsnop 29063 | Degenerated case 2 for ver... |
| iedgvalsnop 29064 | Degenerated case 2 for edg... |
| vtxval3sn 29065 | Degenerated case 3 for ver... |
| iedgval3sn 29066 | Degenerated case 3 for edg... |
| vtxvalprc 29067 | Degenerated case 4 for ver... |
| iedgvalprc 29068 | Degenerated case 4 for edg... |
| edgval 29071 | The edges of a graph. (Co... |
| iedgedg 29072 | An indexed edge is an edge... |
| edgopval 29073 | The edges of a graph repre... |
| edgov 29074 | The edges of a graph repre... |
| edgstruct 29075 | The edges of a graph repre... |
| edgiedgb 29076 | A set is an edge iff it is... |
| edg0iedg0 29077 | There is no edge in a grap... |
| isuhgr 29082 | The predicate "is an undir... |
| isushgr 29083 | The predicate "is an undir... |
| uhgrf 29084 | The edge function of an un... |
| ushgrf 29085 | The edge function of an un... |
| uhgrss 29086 | An edge is a subset of ver... |
| uhgreq12g 29087 | If two sets have the same ... |
| uhgrfun 29088 | The edge function of an un... |
| uhgrn0 29089 | An edge is a nonempty subs... |
| lpvtx 29090 | The endpoints of a loop (w... |
| ushgruhgr 29091 | An undirected simple hyper... |
| isuhgrop 29092 | The property of being an u... |
| uhgr0e 29093 | The empty graph, with vert... |
| uhgr0vb 29094 | The null graph, with no ve... |
| uhgr0 29095 | The null graph represented... |
| uhgrun 29096 | The union ` U ` of two (un... |
| uhgrunop 29097 | The union of two (undirect... |
| ushgrun 29098 | The union ` U ` of two (un... |
| ushgrunop 29099 | The union of two (undirect... |
| uhgrstrrepe 29100 | Replacing (or adding) the ... |
| incistruhgr 29101 | An _incidence structure_ `... |
| isupgr 29106 | The property of being an u... |
| wrdupgr 29107 | The property of being an u... |
| upgrf 29108 | The edge function of an un... |
| upgrfn 29109 | The edge function of an un... |
| upgrss 29110 | An edge is a subset of ver... |
| upgrn0 29111 | An edge is a nonempty subs... |
| upgrle 29112 | An edge of an undirected p... |
| upgrfi 29113 | An edge is a finite subset... |
| upgrex 29114 | An edge is an unordered pa... |
| upgrbi 29115 | Show that an unordered pai... |
| upgrop 29116 | A pseudograph represented ... |
| isumgr 29117 | The property of being an u... |
| isumgrs 29118 | The simplified property of... |
| wrdumgr 29119 | The property of being an u... |
| umgrf 29120 | The edge function of an un... |
| umgrfn 29121 | The edge function of an un... |
| umgredg2 29122 | An edge of a multigraph ha... |
| umgrbi 29123 | Show that an unordered pai... |
| upgruhgr 29124 | An undirected pseudograph ... |
| umgrupgr 29125 | An undirected multigraph i... |
| umgruhgr 29126 | An undirected multigraph i... |
| upgrle2 29127 | An edge of an undirected p... |
| umgrnloopv 29128 | In a multigraph, there is ... |
| umgredgprv 29129 | In a multigraph, an edge i... |
| umgrnloop 29130 | In a multigraph, there is ... |
| umgrnloop0 29131 | A multigraph has no loops.... |
| umgr0e 29132 | The empty graph, with vert... |
| upgr0e 29133 | The empty graph, with vert... |
| upgr1elem 29134 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29135 | A pseudograph with one edg... |
| upgr0eop 29136 | The empty graph, with vert... |
| upgr1eop 29137 | A pseudograph with one edg... |
| upgr0eopALT 29138 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29139 | Alternate proof of ~ upgr1... |
| upgrun 29140 | The union ` U ` of two pse... |
| upgrunop 29141 | The union of two pseudogra... |
| umgrun 29142 | The union ` U ` of two mul... |
| umgrunop 29143 | The union of two multigrap... |
| umgrislfupgrlem 29144 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29145 | A multigraph is a loop-fre... |
| lfgredgge2 29146 | An edge of a loop-free gra... |
| lfgrnloop 29147 | A loop-free graph has no l... |
| uhgredgiedgb 29148 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29149 | A hypergraph has no edges ... |
| uhgredgn0 29150 | An edge of a hypergraph is... |
| edguhgr 29151 | An edge of a hypergraph is... |
| uhgredgrnv 29152 | An edge of a hypergraph co... |
| uhgredgss 29153 | The set of edges of a hype... |
| upgredgss 29154 | The set of edges of a pseu... |
| umgredgss 29155 | The set of edges of a mult... |
| edgupgr 29156 | Properties of an edge of a... |
| edgumgr 29157 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29158 | In a hypergraph, a vertex ... |
| upgredg 29159 | For each edge in a pseudog... |
| umgredg 29160 | For each edge in a multigr... |
| upgrpredgv 29161 | An edge of a pseudograph a... |
| umgrpredgv 29162 | An edge of a multigraph al... |
| upgredg2vtx 29163 | For a vertex incident to a... |
| upgredgpr 29164 | If a proper pair (of verti... |
| edglnl 29165 | The edges incident with a ... |
| numedglnl 29166 | The number of edges incide... |
| umgredgne 29167 | An edge of a multigraph al... |
| umgrnloop2 29168 | A multigraph has no loops.... |
| umgredgnlp 29169 | An edge of a multigraph is... |
| isuspgr 29174 | The property of being a si... |
| isusgr 29175 | The property of being a si... |
| uspgrf 29176 | The edge function of a sim... |
| usgrf 29177 | The edge function of a sim... |
| isusgrs 29178 | The property of being a si... |
| usgrfs 29179 | The edge function of a sim... |
| usgrfun 29180 | The edge function of a sim... |
| usgredgss 29181 | The set of edges of a simp... |
| edgusgr 29182 | An edge of a simple graph ... |
| isuspgrop 29183 | The property of being an u... |
| isusgrop 29184 | The property of being an u... |
| usgrop 29185 | A simple graph represented... |
| isausgr 29186 | The property of an ordered... |
| ausgrusgrb 29187 | The equivalence of the def... |
| usgrausgri 29188 | A simple graph represented... |
| ausgrumgri 29189 | If an alternatively define... |
| ausgrusgri 29190 | The equivalence of the def... |
| usgrausgrb 29191 | The equivalence of the def... |
| usgredgop 29192 | An edge of a simple graph ... |
| usgrf1o 29193 | The edge function of a sim... |
| usgrf1 29194 | The edge function of a sim... |
| uspgrf1oedg 29195 | The edge function of a sim... |
| usgrss 29196 | An edge is a subset of ver... |
| uspgredgiedg 29197 | In a simple pseudograph, f... |
| uspgriedgedg 29198 | In a simple pseudograph, f... |
| uspgrushgr 29199 | A simple pseudograph is an... |
| uspgrupgr 29200 | A simple pseudograph is an... |
| uspgrupgrushgr 29201 | A graph is a simple pseudo... |
| usgruspgr 29202 | A simple graph is a simple... |
| usgrumgr 29203 | A simple graph is an undir... |
| usgrumgruspgr 29204 | A graph is a simple graph ... |
| usgruspgrb 29205 | A class is a simple graph ... |
| uspgruhgr 29206 | An undirected simple pseud... |
| usgrupgr 29207 | A simple graph is an undir... |
| usgruhgr 29208 | A simple graph is an undir... |
| usgrislfuspgr 29209 | A simple graph is a loop-f... |
| uspgrun 29210 | The union ` U ` of two sim... |
| uspgrunop 29211 | The union of two simple ps... |
| usgrun 29212 | The union ` U ` of two sim... |
| usgrunop 29213 | The union of two simple gr... |
| usgredg2 29214 | The value of the "edge fun... |
| usgredg2ALT 29215 | Alternate proof of ~ usgre... |
| usgredgprv 29216 | In a simple graph, an edge... |
| usgredgprvALT 29217 | Alternate proof of ~ usgre... |
| usgredgppr 29218 | An edge of a simple graph ... |
| usgrpredgv 29219 | An edge of a simple graph ... |
| edgssv2 29220 | An edge of a simple graph ... |
| usgredg 29221 | For each edge in a simple ... |
| usgrnloopv 29222 | In a simple graph, there i... |
| usgrnloopvALT 29223 | Alternate proof of ~ usgrn... |
| usgrnloop 29224 | In a simple graph, there i... |
| usgrnloopALT 29225 | Alternate proof of ~ usgrn... |
| usgrnloop0 29226 | A simple graph has no loop... |
| usgrnloop0ALT 29227 | Alternate proof of ~ usgrn... |
| usgredgne 29228 | An edge of a simple graph ... |
| usgrf1oedg 29229 | The edge function of a sim... |
| uhgr2edg 29230 | If a vertex is adjacent to... |
| umgr2edg 29231 | If a vertex is adjacent to... |
| usgr2edg 29232 | If a vertex is adjacent to... |
| umgr2edg1 29233 | If a vertex is adjacent to... |
| usgr2edg1 29234 | If a vertex is adjacent to... |
| umgrvad2edg 29235 | If a vertex is adjacent to... |
| umgr2edgneu 29236 | If a vertex is adjacent to... |
| usgrsizedg 29237 | In a simple graph, the siz... |
| usgredg3 29238 | The value of the "edge fun... |
| usgredg4 29239 | For a vertex incident to a... |
| usgredgreu 29240 | For a vertex incident to a... |
| usgredg2vtx 29241 | For a vertex incident to a... |
| uspgredg2vtxeu 29242 | For a vertex incident to a... |
| usgredg2vtxeu 29243 | For a vertex incident to a... |
| usgredg2vtxeuALT 29244 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29245 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29246 | In a simple pseudograph, t... |
| usgredg2vlem1 29247 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29248 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29249 | In a simple graph, the map... |
| usgriedgleord 29250 | Alternate version of ~ usg... |
| ushgredgedg 29251 | In a simple hypergraph the... |
| usgredgedg 29252 | In a simple graph there is... |
| ushgredgedgloop 29253 | In a simple hypergraph the... |
| uspgredgleord 29254 | In a simple pseudograph th... |
| usgredgleord 29255 | In a simple graph the numb... |
| usgredgleordALT 29256 | Alternate proof for ~ usgr... |
| usgrstrrepe 29257 | Replacing (or adding) the ... |
| usgr0e 29258 | The empty graph, with vert... |
| usgr0vb 29259 | The null graph, with no ve... |
| uhgr0v0e 29260 | The null graph, with no ve... |
| uhgr0vsize0 29261 | The size of a hypergraph w... |
| uhgr0edgfi 29262 | A graph of order 0 (i.e. w... |
| usgr0v 29263 | The null graph, with no ve... |
| uhgr0vusgr 29264 | The null graph, with no ve... |
| usgr0 29265 | The null graph represented... |
| uspgr1e 29266 | A simple pseudograph with ... |
| usgr1e 29267 | A simple graph with one ed... |
| usgr0eop 29268 | The empty graph, with vert... |
| uspgr1eop 29269 | A simple pseudograph with ... |
| uspgr1ewop 29270 | A simple pseudograph with ... |
| uspgr1v1eop 29271 | A simple pseudograph with ... |
| usgr1eop 29272 | A simple graph with (at le... |
| uspgr2v1e2w 29273 | A simple pseudograph with ... |
| usgr2v1e2w 29274 | A simple graph with two ve... |
| edg0usgr 29275 | A class without edges is a... |
| lfuhgr1v0e 29276 | A loop-free hypergraph wit... |
| usgr1vr 29277 | A simple graph with one ve... |
| usgr1v 29278 | A class with one (or no) v... |
| usgr1v0edg 29279 | A class with one (or no) v... |
| usgrexmpldifpr 29280 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29281 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29282 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29283 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29284 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29285 | ` G ` is a simple graph of... |
| griedg0prc 29286 | The class of empty graphs ... |
| griedg0ssusgr 29287 | The class of all simple gr... |
| usgrprc 29288 | The class of simple graphs... |
| relsubgr 29291 | The class of the subgraph ... |
| subgrv 29292 | If a class is a subgraph o... |
| issubgr 29293 | The property of a set to b... |
| issubgr2 29294 | The property of a set to b... |
| subgrprop 29295 | The properties of a subgra... |
| subgrprop2 29296 | The properties of a subgra... |
| uhgrissubgr 29297 | The property of a hypergra... |
| subgrprop3 29298 | The properties of a subgra... |
| egrsubgr 29299 | An empty graph consisting ... |
| 0grsubgr 29300 | The null graph (represente... |
| 0uhgrsubgr 29301 | The null graph (as hypergr... |
| uhgrsubgrself 29302 | A hypergraph is a subgraph... |
| subgrfun 29303 | The edge function of a sub... |
| subgruhgrfun 29304 | The edge function of a sub... |
| subgreldmiedg 29305 | An element of the domain o... |
| subgruhgredgd 29306 | An edge of a subgraph of a... |
| subumgredg2 29307 | An edge of a subgraph of a... |
| subuhgr 29308 | A subgraph of a hypergraph... |
| subupgr 29309 | A subgraph of a pseudograp... |
| subumgr 29310 | A subgraph of a multigraph... |
| subusgr 29311 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29312 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29313 | A spanning subgraph ` S ` ... |
| uhgrspan 29314 | A spanning subgraph ` S ` ... |
| upgrspan 29315 | A spanning subgraph ` S ` ... |
| umgrspan 29316 | A spanning subgraph ` S ` ... |
| usgrspan 29317 | A spanning subgraph ` S ` ... |
| uhgrspanop 29318 | A spanning subgraph of a h... |
| upgrspanop 29319 | A spanning subgraph of a p... |
| umgrspanop 29320 | A spanning subgraph of a m... |
| usgrspanop 29321 | A spanning subgraph of a s... |
| uhgrspan1lem1 29322 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29323 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29324 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29325 | The induced subgraph ` S `... |
| upgrreslem 29326 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29327 | Lemma for ~ umgrres and ~ ... |
| upgrres 29328 | A subgraph obtained by rem... |
| umgrres 29329 | A subgraph obtained by rem... |
| usgrres 29330 | A subgraph obtained by rem... |
| upgrres1lem1 29331 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29332 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29333 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29334 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29335 | A pseudograph obtained by ... |
| umgrres1 29336 | A multigraph obtained by r... |
| usgrres1 29337 | Restricting a simple graph... |
| isfusgr 29340 | The property of being a fi... |
| fusgrvtxfi 29341 | A finite simple graph has ... |
| isfusgrf1 29342 | The property of being a fi... |
| isfusgrcl 29343 | The property of being a fi... |
| fusgrusgr 29344 | A finite simple graph is a... |
| opfusgr 29345 | A finite simple graph repr... |
| usgredgffibi 29346 | The number of edges in a s... |
| fusgredgfi 29347 | In a finite simple graph t... |
| usgr1v0e 29348 | The size of a (finite) sim... |
| usgrfilem 29349 | In a finite simple graph, ... |
| fusgrfisbase 29350 | Induction base for ~ fusgr... |
| fusgrfisstep 29351 | Induction step in ~ fusgrf... |
| fusgrfis 29352 | A finite simple graph is o... |
| fusgrfupgrfs 29353 | A finite simple graph is a... |
| nbgrprc0 29356 | The set of neighbors is em... |
| nbgrcl 29357 | If a class ` X ` has at le... |
| nbgrval 29358 | The set of neighbors of a ... |
| dfnbgr2 29359 | Alternate definition of th... |
| dfnbgr3 29360 | Alternate definition of th... |
| nbgrnvtx0 29361 | If a class ` X ` is not a ... |
| nbgrel 29362 | Characterization of a neig... |
| nbgrisvtx 29363 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29364 | The neighbors of a vertex ... |
| nbuhgr 29365 | The set of neighbors of a ... |
| nbupgr 29366 | The set of neighbors of a ... |
| nbupgrel 29367 | A neighbor of a vertex in ... |
| nbumgrvtx 29368 | The set of neighbors of a ... |
| nbumgr 29369 | The set of neighbors of an... |
| nbusgrvtx 29370 | The set of neighbors of a ... |
| nbusgr 29371 | The set of neighbors of an... |
| nbgr2vtx1edg 29372 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29373 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29374 | If a hypergraph has two ve... |
| nbusgreledg 29375 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29376 | A vertex which is not endp... |
| nbgr0vtx 29377 | In a null graph (with no v... |
| nbgr0edglem 29378 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29379 | In an empty graph (with no... |
| nbgr1vtx 29380 | In a graph with one vertex... |
| nbgrnself 29381 | A vertex in a graph is not... |
| nbgrnself2 29382 | A class ` X ` is not a nei... |
| nbgrssovtx 29383 | The neighbors of a vertex ... |
| nbgrssvwo2 29384 | The neighbors of a vertex ... |
| nbgrsym 29385 | In a graph, the neighborho... |
| nbupgrres 29386 | The neighborhood of a vert... |
| usgrnbcnvfv 29387 | Applying the edge function... |
| nbusgredgeu 29388 | For each neighbor of a ver... |
| edgnbusgreu 29389 | For each edge incident to ... |
| nbusgredgeu0 29390 | For each neighbor of a ver... |
| nbusgrf1o0 29391 | The mapping of neighbors o... |
| nbusgrf1o1 29392 | The set of neighbors of a ... |
| nbusgrf1o 29393 | The set of neighbors of a ... |
| nbedgusgr 29394 | The number of neighbors of... |
| edgusgrnbfin 29395 | The number of neighbors of... |
| nbusgrfi 29396 | The class of neighbors of ... |
| nbfiusgrfi 29397 | The class of neighbors of ... |
| hashnbusgrnn0 29398 | The number of neighbors of... |
| nbfusgrlevtxm1 29399 | The number of neighbors of... |
| nbfusgrlevtxm2 29400 | If there is a vertex which... |
| nbusgrvtxm1 29401 | If the number of neighbors... |
| nb3grprlem1 29402 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29403 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29404 | The neighbors of a vertex ... |
| nb3grpr2 29405 | The neighbors of a vertex ... |
| nb3gr2nb 29406 | If the neighbors of two ve... |
| uvtxval 29409 | The set of all universal v... |
| uvtxel 29410 | A universal vertex, i.e. a... |
| uvtxisvtx 29411 | A universal vertex is a ve... |
| uvtxssvtx 29412 | The set of the universal v... |
| vtxnbuvtx 29413 | A universal vertex has all... |
| uvtxnbgrss 29414 | A universal vertex has all... |
| uvtxnbgrvtx 29415 | A universal vertex is neig... |
| uvtx0 29416 | There is no universal vert... |
| isuvtx 29417 | The set of all universal v... |
| uvtxel1 29418 | Characterization of a univ... |
| uvtx01vtx 29419 | If a graph/class has no ed... |
| uvtx2vtx1edg 29420 | If a graph has two vertice... |
| uvtx2vtx1edgb 29421 | If a hypergraph has two ve... |
| uvtxnbgr 29422 | A universal vertex has all... |
| uvtxnbgrb 29423 | A vertex is universal iff ... |
| uvtxusgr 29424 | The set of all universal v... |
| uvtxusgrel 29425 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29426 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29427 | If the number of neighbors... |
| uvtxnbvtxm1 29428 | A universal vertex has ` n... |
| nbupgruvtxres 29429 | The neighborhood of a univ... |
| uvtxupgrres 29430 | A universal vertex is univ... |
| cplgruvtxb 29435 | A graph ` G ` is complete ... |
| prcliscplgr 29436 | A proper class (representi... |
| iscplgr 29437 | The property of being a co... |
| iscplgrnb 29438 | A graph is complete iff al... |
| iscplgredg 29439 | A graph ` G ` is complete ... |
| iscusgr 29440 | The property of being a co... |
| cusgrusgr 29441 | A complete simple graph is... |
| cusgrcplgr 29442 | A complete simple graph is... |
| iscusgrvtx 29443 | A simple graph is complete... |
| cusgruvtxb 29444 | A simple graph is complete... |
| iscusgredg 29445 | A simple graph is complete... |
| cusgredg 29446 | In a complete simple graph... |
| cplgr0 29447 | The null graph (with no ve... |
| cusgr0 29448 | The null graph (with no ve... |
| cplgr0v 29449 | A null graph (with no vert... |
| cusgr0v 29450 | A graph with no vertices a... |
| cplgr1vlem 29451 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29452 | A graph with one vertex is... |
| cusgr1v 29453 | A graph with one vertex an... |
| cplgr2v 29454 | An undirected hypergraph w... |
| cplgr2vpr 29455 | An undirected hypergraph w... |
| nbcplgr 29456 | In a complete graph, each ... |
| cplgr3v 29457 | A pseudograph with three (... |
| cusgr3vnbpr 29458 | The neighbors of a vertex ... |
| cplgrop 29459 | A complete graph represent... |
| cusgrop 29460 | A complete simple graph re... |
| cusgrexilem1 29461 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29462 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29463 | An arbitrary set regarded ... |
| cusgrexilem2 29464 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29465 | An arbitrary set ` V ` reg... |
| cusgrexg 29466 | For each set there is a se... |
| structtousgr 29467 | Any (extensible) structure... |
| structtocusgr 29468 | Any (extensible) structure... |
| cffldtocusgr 29469 | The field of complex numbe... |
| cffldtocusgrOLD 29470 | Obsolete version of ~ cffl... |
| cusgrres 29471 | Restricting a complete sim... |
| cusgrsizeindb0 29472 | Base case of the induction... |
| cusgrsizeindb1 29473 | Base case of the induction... |
| cusgrsizeindslem 29474 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29475 | Part 1 of induction step i... |
| cusgrsize2inds 29476 | Induction step in ~ cusgrs... |
| cusgrsize 29477 | The size of a finite compl... |
| cusgrfilem1 29478 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29479 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29480 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29481 | If the size of a complete ... |
| usgredgsscusgredg 29482 | A simple graph is a subgra... |
| usgrsscusgr 29483 | A simple graph is a subgra... |
| sizusglecusglem1 29484 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29485 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29486 | The size of a simple graph... |
| fusgrmaxsize 29487 | The maximum size of a fini... |
| vtxdgfval 29490 | The value of the vertex de... |
| vtxdgval 29491 | The degree of a vertex. (... |
| vtxdgfival 29492 | The degree of a vertex for... |
| vtxdgop 29493 | The vertex degree expresse... |
| vtxdgf 29494 | The vertex degree function... |
| vtxdgelxnn0 29495 | The degree of a vertex is ... |
| vtxdg0v 29496 | The degree of a vertex in ... |
| vtxdg0e 29497 | The degree of a vertex in ... |
| vtxdgfisnn0 29498 | The degree of a vertex in ... |
| vtxdgfisf 29499 | The vertex degree function... |
| vtxdeqd 29500 | Equality theorem for the v... |
| vtxduhgr0e 29501 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29502 | The degree of the vertex i... |
| vdumgr0 29503 | A vertex in a multigraph h... |
| vtxdun 29504 | The degree of a vertex in ... |
| vtxdfiun 29505 | The degree of a vertex in ... |
| vtxduhgrun 29506 | The degree of a vertex in ... |
| vtxduhgrfiun 29507 | The degree of a vertex in ... |
| vtxdlfgrval 29508 | The value of the vertex de... |
| vtxdumgrval 29509 | The value of the vertex de... |
| vtxdusgrval 29510 | The value of the vertex de... |
| vtxd0nedgb 29511 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29512 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29513 | The value of the vertex de... |
| vtxdusgrfvedg 29514 | The value of the vertex de... |
| vtxduhgr0nedg 29515 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29516 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29517 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29518 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29519 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29520 | The vertex degree function... |
| vtxdgfusgr 29521 | In a finite simple graph, ... |
| fusgrn0degnn0 29522 | In a nonempty, finite grap... |
| 1loopgruspgr 29523 | A graph with one edge whic... |
| 1loopgredg 29524 | The set of edges in a grap... |
| 1loopgrnb0 29525 | In a graph (simple pseudog... |
| 1loopgrvd2 29526 | The vertex degree of a one... |
| 1loopgrvd0 29527 | The vertex degree of a one... |
| 1hevtxdg0 29528 | The vertex degree of verte... |
| 1hevtxdg1 29529 | The vertex degree of verte... |
| 1hegrvtxdg1 29530 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29531 | The vertex degree of a gra... |
| 1egrvtxdg1 29532 | The vertex degree of a one... |
| 1egrvtxdg1r 29533 | The vertex degree of a one... |
| 1egrvtxdg0 29534 | The vertex degree of a one... |
| p1evtxdeqlem 29535 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29536 | If an edge ` E ` which doe... |
| p1evtxdp1 29537 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29538 | The set of vertices in a g... |
| uspgrloopvtxel 29539 | A vertex in a graph (simpl... |
| uspgrloopiedg 29540 | The set of edges in a grap... |
| uspgrloopedg 29541 | The set of edges in a grap... |
| uspgrloopnb0 29542 | In a graph (simple pseudog... |
| uspgrloopvd2 29543 | The vertex degree of a one... |
| umgr2v2evtx 29544 | The set of vertices in a m... |
| umgr2v2evtxel 29545 | A vertex in a multigraph w... |
| umgr2v2eiedg 29546 | The edge function in a mul... |
| umgr2v2eedg 29547 | The set of edges in a mult... |
| umgr2v2e 29548 | A multigraph with two edge... |
| umgr2v2enb1 29549 | In a multigraph with two e... |
| umgr2v2evd2 29550 | In a multigraph with two e... |
| hashnbusgrvd 29551 | In a simple graph, the num... |
| usgruvtxvdb 29552 | In a finite simple graph w... |
| vdiscusgrb 29553 | A finite simple graph with... |
| vdiscusgr 29554 | In a finite complete simpl... |
| vtxdusgradjvtx 29555 | The degree of a vertex in ... |
| usgrvd0nedg 29556 | If a vertex in a simple gr... |
| uhgrvd00 29557 | If every vertex in a hyper... |
| usgrvd00 29558 | If every vertex in a simpl... |
| vdegp1ai 29559 | The induction step for a v... |
| vdegp1bi 29560 | The induction step for a v... |
| vdegp1ci 29561 | The induction step for a v... |
| vtxdginducedm1lem1 29562 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29563 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29564 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29565 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29566 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29567 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29568 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29569 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29570 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29571 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29572 | Induction step of ~ finsum... |
| finsumvtxdg2size 29573 | The sum of the degrees of ... |
| fusgr1th 29574 | The sum of the degrees of ... |
| finsumvtxdgeven 29575 | The sum of the degrees of ... |
| vtxdgoddnumeven 29576 | The number of vertices of ... |
| fusgrvtxdgonume 29577 | The number of vertices of ... |
| isrgr 29582 | The property of a class be... |
| rgrprop 29583 | The properties of a k-regu... |
| isrusgr 29584 | The property of being a k-... |
| rusgrprop 29585 | The properties of a k-regu... |
| rusgrrgr 29586 | A k-regular simple graph i... |
| rusgrusgr 29587 | A k-regular simple graph i... |
| finrusgrfusgr 29588 | A finite regular simple gr... |
| isrusgr0 29589 | The property of being a k-... |
| rusgrprop0 29590 | The properties of a k-regu... |
| usgreqdrusgr 29591 | If all vertices in a simpl... |
| fusgrregdegfi 29592 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29593 | If all vertices in a nonem... |
| frusgrnn0 29594 | In a nonempty finite k-reg... |
| 0edg0rgr 29595 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29596 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29597 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29598 | A simple graph is 0-regula... |
| 0vtxrgr 29599 | A null graph (with no vert... |
| 0vtxrusgr 29600 | A graph with no vertices a... |
| 0uhgrrusgr 29601 | The null graph as hypergra... |
| 0grrusgr 29602 | The null graph represented... |
| 0grrgr 29603 | The null graph represented... |
| cusgrrusgr 29604 | A complete simple graph wi... |
| cusgrm1rusgr 29605 | A finite simple graph with... |
| rusgrpropnb 29606 | The properties of a k-regu... |
| rusgrpropedg 29607 | The properties of a k-regu... |
| rusgrpropadjvtx 29608 | The properties of a k-regu... |
| rusgrnumwrdl2 29609 | In a k-regular simple grap... |
| rusgr1vtxlem 29610 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29611 | If a k-regular simple grap... |
| rgrusgrprc 29612 | The class of 0-regular sim... |
| rusgrprc 29613 | The class of 0-regular sim... |
| rgrprc 29614 | The class of 0-regular gra... |
| rgrprcx 29615 | The class of 0-regular gra... |
| rgrx0ndm 29616 | 0 is not in the domain of ... |
| rgrx0nd 29617 | The potentially alternativ... |
| ewlksfval 29624 | The set of s-walks of edge... |
| isewlk 29625 | Conditions for a function ... |
| ewlkprop 29626 | Properties of an s-walk of... |
| ewlkinedg 29627 | The intersection (common v... |
| ewlkle 29628 | An s-walk of edges is also... |
| upgrewlkle2 29629 | In a pseudograph, there is... |
| wkslem1 29630 | Lemma 1 for walks to subst... |
| wkslem2 29631 | Lemma 2 for walks to subst... |
| wksfval 29632 | The set of walks (in an un... |
| iswlk 29633 | Properties of a pair of fu... |
| wlkprop 29634 | Properties of a walk. (Co... |
| wlkv 29635 | The classes involved in a ... |
| iswlkg 29636 | Generalization of ~ iswlk ... |
| wlkf 29637 | The mapping enumerating th... |
| wlkcl 29638 | A walk has length ` # ( F ... |
| wlkp 29639 | The mapping enumerating th... |
| wlkpwrd 29640 | The sequence of vertices o... |
| wlklenvp1 29641 | The number of vertices of ... |
| wksv 29642 | The class of walks is a se... |
| wlkn0 29643 | The sequence of vertices o... |
| wlklenvm1 29644 | The number of edges of a w... |
| ifpsnprss 29645 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29646 | Each pair of adjacent vert... |
| wlkvtxiedg 29647 | The vertices of a walk are... |
| relwlk 29648 | The set ` ( Walks `` G ) `... |
| wlkvv 29649 | If there is at least one w... |
| wlkop 29650 | A walk is an ordered pair.... |
| wlkcpr 29651 | A walk as class with two c... |
| wlk2f 29652 | If there is a walk ` W ` t... |
| wlkcomp 29653 | A walk expressed by proper... |
| wlkcompim 29654 | Implications for the prope... |
| wlkelwrd 29655 | The components of a walk a... |
| wlkeq 29656 | Conditions for two walks (... |
| edginwlk 29657 | The value of the edge func... |
| upgredginwlk 29658 | The value of the edge func... |
| iedginwlk 29659 | The value of the edge func... |
| wlkl1loop 29660 | A walk of length 1 from a ... |
| wlk1walk 29661 | A walk is a 1-walk "on the... |
| wlk1ewlk 29662 | A walk is an s-walk "on th... |
| upgriswlk 29663 | Properties of a pair of fu... |
| upgrwlkedg 29664 | The edges of a walk in a p... |
| upgrwlkcompim 29665 | Implications for the prope... |
| wlkvtxedg 29666 | The vertices of a walk are... |
| upgrwlkvtxedg 29667 | The pairs of connected ver... |
| uspgr2wlkeq 29668 | Conditions for two walks w... |
| uspgr2wlkeq2 29669 | Conditions for two walks w... |
| uspgr2wlkeqi 29670 | Conditions for two walks w... |
| umgrwlknloop 29671 | In a multigraph, each walk... |
| wlkv0 29672 | If there is a walk in the ... |
| g0wlk0 29673 | There is no walk in a null... |
| 0wlk0 29674 | There is no walk for the e... |
| wlk0prc 29675 | There is no walk in a null... |
| wlklenvclwlk 29676 | The number of vertices in ... |
| wlkson 29677 | The set of walks between t... |
| iswlkon 29678 | Properties of a pair of fu... |
| wlkonprop 29679 | Properties of a walk betwe... |
| wlkpvtx 29680 | A walk connects vertices. ... |
| wlkepvtx 29681 | The endpoints of a walk ar... |
| wlkoniswlk 29682 | A walk between two vertice... |
| wlkonwlk 29683 | A walk is a walk between i... |
| wlkonwlk1l 29684 | A walk is a walk from its ... |
| wlksoneq1eq2 29685 | Two walks with identical s... |
| wlkonl1iedg 29686 | If there is a walk between... |
| wlkon2n0 29687 | The length of a walk betwe... |
| 2wlklem 29688 | Lemma for theorems for wal... |
| upgr2wlk 29689 | Properties of a pair of fu... |
| wlkreslem 29690 | Lemma for ~ wlkres . (Con... |
| wlkres 29691 | The restriction ` <. H , Q... |
| redwlklem 29692 | Lemma for ~ redwlk . (Con... |
| redwlk 29693 | A walk ending at the last ... |
| wlkp1lem1 29694 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29695 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29696 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29697 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29698 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29699 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29700 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29701 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29702 | Append one path segment (e... |
| wlkdlem1 29703 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29704 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29705 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29706 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29707 | Two words representing a w... |
| lfgrwlkprop 29708 | Two adjacent vertices in a... |
| lfgriswlk 29709 | Conditions for a pair of f... |
| lfgrwlknloop 29710 | In a loop-free graph, each... |
| reltrls 29715 | The set ` ( Trails `` G ) ... |
| trlsfval 29716 | The set of trails (in an u... |
| istrl 29717 | Conditions for a pair of c... |
| trliswlk 29718 | A trail is a walk. (Contr... |
| trlf1 29719 | The enumeration ` F ` of a... |
| trlreslem 29720 | Lemma for ~ trlres . Form... |
| trlres 29721 | The restriction ` <. H , Q... |
| upgrtrls 29722 | The set of trails in a pse... |
| upgristrl 29723 | Properties of a pair of fu... |
| upgrf1istrl 29724 | Properties of a pair of a ... |
| wksonproplem 29725 | Lemma for theorems for pro... |
| trlsonfval 29726 | The set of trails between ... |
| istrlson 29727 | Properties of a pair of fu... |
| trlsonprop 29728 | Properties of a trail betw... |
| trlsonistrl 29729 | A trail between two vertic... |
| trlsonwlkon 29730 | A trail between two vertic... |
| trlontrl 29731 | A trail is a trail between... |
| relpths 29740 | The set ` ( Paths `` G ) `... |
| pthsfval 29741 | The set of paths (in an un... |
| spthsfval 29742 | The set of simple paths (i... |
| ispth 29743 | Conditions for a pair of c... |
| isspth 29744 | Conditions for a pair of c... |
| pthistrl 29745 | A path is a trail (in an u... |
| spthispth 29746 | A simple path is a path (i... |
| pthiswlk 29747 | A path is a walk (in an un... |
| spthiswlk 29748 | A simple path is a walk (i... |
| pthdivtx 29749 | The inner vertices of a pa... |
| pthdadjvtx 29750 | The adjacent vertices of a... |
| dfpth2 29751 | Alternate definition for a... |
| pthdifv 29752 | The vertices of a path are... |
| 2pthnloop 29753 | A path of length at least ... |
| upgr2pthnlp 29754 | A path of length at least ... |
| spthdifv 29755 | The vertices of a simple p... |
| spthdep 29756 | A simple path (at least of... |
| pthdepisspth 29757 | A path with different star... |
| upgrwlkdvdelem 29758 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29759 | In a pseudograph, all edge... |
| upgrspthswlk 29760 | The set of simple paths in... |
| upgrwlkdvspth 29761 | A walk consisting of diffe... |
| pthsonfval 29762 | The set of paths between t... |
| spthson 29763 | The set of simple paths be... |
| ispthson 29764 | Properties of a pair of fu... |
| isspthson 29765 | Properties of a pair of fu... |
| pthsonprop 29766 | Properties of a path betwe... |
| spthonprop 29767 | Properties of a simple pat... |
| pthonispth 29768 | A path between two vertice... |
| pthontrlon 29769 | A path between two vertice... |
| pthonpth 29770 | A path is a path between i... |
| isspthonpth 29771 | A pair of functions is a s... |
| spthonisspth 29772 | A simple path between to v... |
| spthonpthon 29773 | A simple path between two ... |
| spthonepeq 29774 | The endpoints of a simple ... |
| uhgrwkspthlem1 29775 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29776 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29777 | Any walk of length 1 betwe... |
| usgr2wlkneq 29778 | The vertices and edges are... |
| usgr2wlkspthlem1 29779 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29780 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29781 | In a simple graph, any wal... |
| usgr2trlncl 29782 | In a simple graph, any tra... |
| usgr2trlspth 29783 | In a simple graph, any tra... |
| usgr2pthspth 29784 | In a simple graph, any pat... |
| usgr2pthlem 29785 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29786 | In a simple graph, there i... |
| usgr2pth0 29787 | In a simply graph, there i... |
| pthdlem1 29788 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29789 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29790 | Lemma 2 for ~ pthd . (Con... |
| pthd 29791 | Two words representing a t... |
| clwlks 29794 | The set of closed walks (i... |
| isclwlk 29795 | A pair of functions repres... |
| clwlkiswlk 29796 | A closed walk is a walk (i... |
| clwlkwlk 29797 | Closed walks are walks (in... |
| clwlkswks 29798 | Closed walks are walks (in... |
| isclwlke 29799 | Properties of a pair of fu... |
| isclwlkupgr 29800 | Properties of a pair of fu... |
| clwlkcomp 29801 | A closed walk expressed by... |
| clwlkcompim 29802 | Implications for the prope... |
| upgrclwlkcompim 29803 | Implications for the prope... |
| clwlkcompbp 29804 | Basic properties of the co... |
| clwlkl1loop 29805 | A closed walk of length 1 ... |
| crcts 29810 | The set of circuits (in an... |
| cycls 29811 | The set of cycles (in an u... |
| iscrct 29812 | Sufficient and necessary c... |
| iscycl 29813 | Sufficient and necessary c... |
| crctprop 29814 | The properties of a circui... |
| cyclprop 29815 | The properties of a cycle:... |
| crctisclwlk 29816 | A circuit is a closed walk... |
| crctistrl 29817 | A circuit is a trail. (Co... |
| crctiswlk 29818 | A circuit is a walk. (Con... |
| cyclispth 29819 | A cycle is a path. (Contr... |
| cycliswlk 29820 | A cycle is a walk. (Contr... |
| cycliscrct 29821 | A cycle is a circuit. (Co... |
| cyclnumvtx 29822 | The number of vertices of ... |
| cyclnspth 29823 | A (non-trivial) cycle is n... |
| pthisspthorcycl 29824 | A path is either a simple ... |
| pthspthcyc 29825 | A pair ` <. F , P >. ` rep... |
| cyclispthon 29826 | A cycle is a path starting... |
| lfgrn1cycl 29827 | In a loop-free graph there... |
| usgr2trlncrct 29828 | In a simple graph, any tra... |
| umgrn1cycl 29829 | In a multigraph graph (wit... |
| uspgrn2crct 29830 | In a simple pseudograph th... |
| usgrn2cycl 29831 | In a simple graph there ar... |
| crctcshwlkn0lem1 29832 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29833 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29834 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29835 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29836 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29837 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29838 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29839 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29840 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29841 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29842 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29843 | Cyclically shifting the in... |
| crctcshwlk 29844 | Cyclically shifting the in... |
| crctcshtrl 29845 | Cyclically shifting the in... |
| crctcsh 29846 | Cyclically shifting the in... |
| wwlks 29857 | The set of walks (in an un... |
| iswwlks 29858 | A word over the set of ver... |
| wwlksn 29859 | The set of walks (in an un... |
| iswwlksn 29860 | A word over the set of ver... |
| wwlksnprcl 29861 | Derivation of the length o... |
| iswwlksnx 29862 | Properties of a word to re... |
| wwlkbp 29863 | Basic properties of a walk... |
| wwlknbp 29864 | Basic properties of a walk... |
| wwlknp 29865 | Properties of a set being ... |
| wwlknbp1 29866 | Other basic properties of ... |
| wwlknvtx 29867 | The symbols of a word ` W ... |
| wwlknllvtx 29868 | If a word ` W ` represents... |
| wwlknlsw 29869 | If a word represents a wal... |
| wspthsn 29870 | The set of simple paths of... |
| iswspthn 29871 | An element of the set of s... |
| wspthnp 29872 | Properties of a set being ... |
| wwlksnon 29873 | The set of walks of a fixe... |
| wspthsnon 29874 | The set of simple paths of... |
| iswwlksnon 29875 | The set of walks of a fixe... |
| wwlksnon0 29876 | Sufficient conditions for ... |
| wwlksonvtx 29877 | If a word ` W ` represents... |
| iswspthsnon 29878 | The set of simple paths of... |
| wwlknon 29879 | An element of the set of w... |
| wspthnon 29880 | An element of the set of s... |
| wspthnonp 29881 | Properties of a set being ... |
| wspthneq1eq2 29882 | Two simple paths with iden... |
| wwlksn0s 29883 | The set of all walks as wo... |
| wwlkssswrd 29884 | Walks (represented by word... |
| wwlksn0 29885 | A walk of length 0 is repr... |
| 0enwwlksnge1 29886 | In graphs without edges, t... |
| wwlkswwlksn 29887 | A walk of a fixed length a... |
| wwlkssswwlksn 29888 | The walks of a fixed lengt... |
| wlkiswwlks1 29889 | The sequence of vertices i... |
| wlklnwwlkln1 29890 | The sequence of vertices i... |
| wlkiswwlks2lem1 29891 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29892 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29893 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29894 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29895 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29896 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29897 | A walk as word corresponds... |
| wlkiswwlks 29898 | A walk as word corresponds... |
| wlkiswwlksupgr2 29899 | A walk as word corresponds... |
| wlkiswwlkupgr 29900 | A walk as word corresponds... |
| wlkswwlksf1o 29901 | The mapping of (ordinary) ... |
| wlkswwlksen 29902 | The set of walks as words ... |
| wwlksm1edg 29903 | Removing the trailing edge... |
| wlklnwwlkln2lem 29904 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29905 | A walk of length ` N ` as ... |
| wlklnwwlkn 29906 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29907 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29908 | A walk of length ` N ` as ... |
| wlknewwlksn 29909 | If a walk in a pseudograph... |
| wlknwwlksnbij 29910 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29911 | In a simple pseudograph, t... |
| wlknwwlksneqs 29912 | The set of walks of a fixe... |
| wwlkseq 29913 | Equality of two walks (as ... |
| wwlksnred 29914 | Reduction of a walk (as wo... |
| wwlksnext 29915 | Extension of a walk (as wo... |
| wwlksnextbi 29916 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29917 | For each walk (as word) of... |
| wwlksnredwwlkn0 29918 | For each walk (as word) of... |
| wwlksnextwrd 29919 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29920 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29921 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29922 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29923 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29924 | There is a bijection betwe... |
| wwlksnexthasheq 29925 | The number of the extensio... |
| disjxwwlksn 29926 | Sets of walks (as words) e... |
| wwlksnndef 29927 | Conditions for ` WWalksN `... |
| wwlksnfi 29928 | The number of walks repres... |
| wlksnfi 29929 | The number of walks of fix... |
| wlksnwwlknvbij 29930 | There is a bijection betwe... |
| wwlksnextproplem1 29931 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29932 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29933 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29934 | Adding additional properti... |
| disjxwwlkn 29935 | Sets of walks (as words) e... |
| hashwwlksnext 29936 | Number of walks (as words)... |
| wwlksnwwlksnon 29937 | A walk of fixed length is ... |
| wspthsnwspthsnon 29938 | A simple path of fixed len... |
| wspthsnonn0vne 29939 | If the set of simple paths... |
| wspthsswwlkn 29940 | The set of simple paths of... |
| wspthnfi 29941 | In a finite graph, the set... |
| wwlksnonfi 29942 | In a finite graph, the set... |
| wspthsswwlknon 29943 | The set of simple paths of... |
| wspthnonfi 29944 | In a finite graph, the set... |
| wspniunwspnon 29945 | The set of nonempty simple... |
| wspn0 29946 | If there are no vertices, ... |
| 2wlkdlem1 29947 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29948 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29949 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29950 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29951 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29952 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29953 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29954 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29955 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29956 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29957 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29958 | Construction of a walk fro... |
| 2wlkond 29959 | A walk of length 2 from on... |
| 2trld 29960 | Construction of a trail fr... |
| 2trlond 29961 | A trail of length 2 from o... |
| 2pthd 29962 | A path of length 2 from on... |
| 2spthd 29963 | A simple path of length 2 ... |
| 2pthond 29964 | A simple path of length 2 ... |
| 2pthon3v 29965 | For a vertex adjacent to t... |
| umgr2adedgwlklem 29966 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 29967 | In a multigraph, two adjac... |
| umgr2adedgwlkon 29968 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 29969 | Alternate proof for ~ umgr... |
| umgr2adedgspth 29970 | In a multigraph, two adjac... |
| umgr2wlk 29971 | In a multigraph, there is ... |
| umgr2wlkon 29972 | For each pair of adjacent ... |
| elwwlks2s3 29973 | A walk of length 2 as word... |
| midwwlks2s3 29974 | There is a vertex between ... |
| wwlks2onv 29975 | If a length 3 string repre... |
| elwwlks2ons3im 29976 | A walk as word of length 2... |
| elwwlks2ons3 29977 | For each walk of length 2 ... |
| s3wwlks2on 29978 | A length 3 string which re... |
| sps3wwlks2on 29979 | A length 3 string which re... |
| usgrwwlks2on 29980 | A walk of length 2 between... |
| umgrwwlks2on 29981 | A walk of length 2 between... |
| wwlks2onsym 29982 | There is a walk of length ... |
| elwwlks2on 29983 | A walk of length 2 between... |
| elwspths2on 29984 | A simple path of length 2 ... |
| elwspths2onw 29985 | A simple path of length 2 ... |
| wpthswwlks2on 29986 | For two different vertices... |
| 2wspdisj 29987 | All simple paths of length... |
| 2wspiundisj 29988 | All simple paths of length... |
| usgr2wspthons3 29989 | A simple path of length 2 ... |
| usgr2wspthon 29990 | A simple path of length 2 ... |
| elwwlks2 29991 | A walk of length 2 between... |
| elwspths2spth 29992 | A simple path of length 2 ... |
| rusgrnumwwlkl1 29993 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 29994 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 29995 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 29996 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 29997 | Induction base 1 for ~ rus... |
| rusgr0edg 29998 | Special case for graphs wi... |
| rusgrnumwwlks 29999 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30000 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30001 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30002 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30003 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30004 | In a ` K `-regular graph, ... |
| clwwlk 30007 | The set of closed walks (i... |
| isclwwlk 30008 | Properties of a word to re... |
| clwwlkbp 30009 | Basic properties of a clos... |
| clwwlkgt0 30010 | There is no empty closed w... |
| clwwlksswrd 30011 | Closed walks (represented ... |
| clwwlk1loop 30012 | A closed walk of length 1 ... |
| clwwlkccatlem 30013 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30014 | The concatenation of two w... |
| umgrclwwlkge2 30015 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30016 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30017 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30018 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30019 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30020 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30021 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30022 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30023 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30024 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30025 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30026 | A closed walk as word of l... |
| clwlkclwwlk2 30027 | A closed walk corresponds ... |
| clwlkclwwlkflem 30028 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30029 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30030 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30031 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30032 | ` F ` is a function from t... |
| clwlkclwwlkfo 30033 | ` F ` is a function from t... |
| clwlkclwwlkf1 30034 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30035 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30036 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30037 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30038 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30039 | Cyclically shifting a clos... |
| clwwisshclwwsn 30040 | Cyclically shifting a clos... |
| erclwwlkrel 30041 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30042 | Two classes are equivalent... |
| erclwwlkeqlen 30043 | If two classes are equival... |
| erclwwlkref 30044 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30045 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30046 | ` .~ ` is a transitive rel... |
| erclwwlk 30047 | ` .~ ` is an equivalence r... |
| clwwlkn 30050 | The set of closed walks of... |
| isclwwlkn 30051 | A word over the set of ver... |
| clwwlkn0 30052 | There is no closed walk of... |
| clwwlkneq0 30053 | Sufficient conditions for ... |
| clwwlkclwwlkn 30054 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30055 | The closed walks of a fixe... |
| clwwlknlen 30056 | The length of a word repre... |
| clwwlknnn 30057 | The length of a closed wal... |
| clwwlknwrd 30058 | A closed walk of a fixed l... |
| clwwlknbp 30059 | Basic properties of a clos... |
| isclwwlknx 30060 | Characterization of a word... |
| clwwlknp 30061 | Properties of a set being ... |
| clwwlknwwlksn 30062 | A word representing a clos... |
| clwwlknlbonbgr1 30063 | The last but one vertex in... |
| clwwlkinwwlk 30064 | If the initial vertex of a... |
| clwwlkn1 30065 | A closed walk of length 1 ... |
| loopclwwlkn1b 30066 | The singleton word consist... |
| clwwlkn1loopb 30067 | A word represents a closed... |
| clwwlkn2 30068 | A closed walk of length 2 ... |
| clwwlknfi 30069 | If there is only a finite ... |
| clwwlkel 30070 | Obtaining a closed walk (a... |
| clwwlkf 30071 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30072 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30073 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30074 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30075 | F is a 1-1 onto function, ... |
| clwwlken 30076 | The set of closed walks of... |
| clwwlknwwlkncl 30077 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30078 | A nonempty word over verti... |
| clwwlknwwlksnb 30079 | A word over vertices repre... |
| clwwlkext2edg 30080 | If a word concatenated wit... |
| wwlksext2clwwlk 30081 | If a word represents a wal... |
| wwlksubclwwlk 30082 | Any prefix of a word repre... |
| clwwnisshclwwsn 30083 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30084 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30085 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30086 | The set of cyclical shifts... |
| clwwlknccat 30087 | The concatenation of two w... |
| umgr2cwwk2dif 30088 | If a word represents a clo... |
| umgr2cwwkdifex 30089 | If a word represents a clo... |
| erclwwlknrel 30090 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30091 | Two classes are equivalent... |
| erclwwlkneqlen 30092 | If two classes are equival... |
| erclwwlknref 30093 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30094 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30095 | ` .~ ` is a transitive rel... |
| erclwwlkn 30096 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30097 | The quotient set of the se... |
| hashclwwlkn0 30098 | The number of closed walks... |
| eclclwwlkn1 30099 | An equivalence class accor... |
| eleclclwwlkn 30100 | A member of an equivalence... |
| hashecclwwlkn1 30101 | The size of every equivale... |
| umgrhashecclwwlk 30102 | The size of every equivale... |
| fusgrhashclwwlkn 30103 | The size of the set of clo... |
| clwwlkndivn 30104 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30105 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30106 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30107 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30108 | There is a one-to-one onto... |
| clwlkssizeeq 30109 | The size of the set of clo... |
| clwlksndivn 30110 | The size of the set of clo... |
| clwwlknonmpo 30113 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30114 | The set of closed walks on... |
| isclwwlknon 30115 | A word over the set of ver... |
| clwwlk0on0 30116 | There is no word over the ... |
| clwwlknon0 30117 | Sufficient conditions for ... |
| clwwlknonfin 30118 | In a finite graph ` G ` , ... |
| clwwlknonel 30119 | Characterization of a word... |
| clwwlknonccat 30120 | The concatenation of two w... |
| clwwlknon1 30121 | The set of closed walks on... |
| clwwlknon1loop 30122 | If there is a loop at vert... |
| clwwlknon1nloop 30123 | If there is no loop at ver... |
| clwwlknon1sn 30124 | The set of (closed) walks ... |
| clwwlknon1le1 30125 | There is at most one (clos... |
| clwwlknon2 30126 | The set of closed walks on... |
| clwwlknon2x 30127 | The set of closed walks on... |
| s2elclwwlknon2 30128 | Sufficient conditions of a... |
| clwwlknon2num 30129 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30130 | A word over vertices repre... |
| clwwlknonex2lem1 30131 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30132 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30133 | Extending a closed walk ` ... |
| clwwlknonex2e 30134 | Extending a closed walk ` ... |
| clwwlknondisj 30135 | The sets of closed walks o... |
| clwwlknun 30136 | The set of closed walks of... |
| clwwlkvbij 30137 | There is a bijection betwe... |
| 0ewlk 30138 | The empty set (empty seque... |
| 1ewlk 30139 | A sequence of 1 edge is an... |
| 0wlk 30140 | A pair of an empty set (of... |
| is0wlk 30141 | A pair of an empty set (of... |
| 0wlkonlem1 30142 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30143 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30144 | A walk of length 0 from a ... |
| 0wlkons1 30145 | A walk of length 0 from a ... |
| 0trl 30146 | A pair of an empty set (of... |
| is0trl 30147 | A pair of an empty set (of... |
| 0trlon 30148 | A trail of length 0 from a... |
| 0pth 30149 | A pair of an empty set (of... |
| 0spth 30150 | A pair of an empty set (of... |
| 0pthon 30151 | A path of length 0 from a ... |
| 0pthon1 30152 | A path of length 0 from a ... |
| 0pthonv 30153 | For each vertex there is a... |
| 0clwlk 30154 | A pair of an empty set (of... |
| 0clwlkv 30155 | Any vertex (more precisely... |
| 0clwlk0 30156 | There is no closed walk in... |
| 0crct 30157 | A pair of an empty set (of... |
| 0cycl 30158 | A pair of an empty set (of... |
| 1pthdlem1 30159 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30160 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30161 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30162 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30163 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30164 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30165 | In a graph with two vertic... |
| 1trld 30166 | In a graph with two vertic... |
| 1pthd 30167 | In a graph with two vertic... |
| 1pthond 30168 | In a graph with two vertic... |
| upgr1wlkdlem1 30169 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30170 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30171 | In a pseudograph with two ... |
| upgr1trld 30172 | In a pseudograph with two ... |
| upgr1pthd 30173 | In a pseudograph with two ... |
| upgr1pthond 30174 | In a pseudograph with two ... |
| lppthon 30175 | A loop (which is an edge a... |
| lp1cycl 30176 | A loop (which is an edge a... |
| 1pthon2v 30177 | For each pair of adjacent ... |
| 1pthon2ve 30178 | For each pair of adjacent ... |
| wlk2v2elem1 30179 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30180 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30181 | In a graph with two vertic... |
| ntrl2v2e 30182 | A walk which is not a trai... |
| 3wlkdlem1 30183 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30184 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30185 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30186 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30187 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30188 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30189 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30190 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30191 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30192 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30193 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30194 | Construction of a walk fro... |
| 3wlkond 30195 | A walk of length 3 from on... |
| 3trld 30196 | Construction of a trail fr... |
| 3trlond 30197 | A trail of length 3 from o... |
| 3pthd 30198 | A path of length 3 from on... |
| 3pthond 30199 | A path of length 3 from on... |
| 3spthd 30200 | A simple path of length 3 ... |
| 3spthond 30201 | A simple path of length 3 ... |
| 3cycld 30202 | Construction of a 3-cycle ... |
| 3cyclpd 30203 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30204 | If there is a cycle of len... |
| uhgr3cyclexlem 30205 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30206 | If there are three differe... |
| umgr3cyclex 30207 | If there are three (differ... |
| umgr3v3e3cycl 30208 | If and only if there is a ... |
| upgr4cycl4dv4e 30209 | If there is a cycle of len... |
| dfconngr1 30212 | Alternative definition of ... |
| isconngr 30213 | The property of being a co... |
| isconngr1 30214 | The property of being a co... |
| cusconngr 30215 | A complete hypergraph is c... |
| 0conngr 30216 | A graph without vertices i... |
| 0vconngr 30217 | A graph without vertices i... |
| 1conngr 30218 | A graph with (at most) one... |
| conngrv2edg 30219 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30220 | A vertex in a connected mu... |
| releupth 30223 | The set ` ( EulerPaths `` ... |
| eupths 30224 | The Eulerian paths on the ... |
| iseupth 30225 | The property " ` <. F , P ... |
| iseupthf1o 30226 | The property " ` <. F , P ... |
| eupthi 30227 | Properties of an Eulerian ... |
| eupthf1o 30228 | The ` F ` function in an E... |
| eupthfi 30229 | Any graph with an Eulerian... |
| eupthseg 30230 | The ` N ` -th edge in an e... |
| upgriseupth 30231 | The property " ` <. F , P ... |
| upgreupthi 30232 | Properties of an Eulerian ... |
| upgreupthseg 30233 | The ` N ` -th edge in an e... |
| eupthcl 30234 | An Eulerian path has lengt... |
| eupthistrl 30235 | An Eulerian path is a trai... |
| eupthiswlk 30236 | An Eulerian path is a walk... |
| eupthpf 30237 | The ` P ` function in an E... |
| eupth0 30238 | There is an Eulerian path ... |
| eupthres 30239 | The restriction ` <. H , Q... |
| eupthp1 30240 | Append one path segment to... |
| eupth2eucrct 30241 | Append one path segment to... |
| eupth2lem1 30242 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30243 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30244 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30245 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30246 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30247 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30248 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30249 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30250 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30251 | Formerly part of proof of ... |
| eupth2lem3lem1 30252 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30253 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30254 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30255 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30256 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30257 | Formerly part of proof of ... |
| eupth2lem3lem7 30258 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30259 | Formerly part of proof of ... |
| eupth2lem3 30260 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30261 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30262 | Lemma for ~ eupth2 (induct... |
| eupth2 30263 | The only vertices of odd d... |
| eulerpathpr 30264 | A graph with an Eulerian p... |
| eulerpath 30265 | A pseudograph with an Eule... |
| eulercrct 30266 | A pseudograph with an Eule... |
| eucrctshift 30267 | Cyclically shifting the in... |
| eucrct2eupth1 30268 | Removing one edge ` ( I ``... |
| eucrct2eupth 30269 | Removing one edge ` ( I ``... |
| konigsbergvtx 30270 | The set of vertices of the... |
| konigsbergiedg 30271 | The indexed edges of the K... |
| konigsbergiedgw 30272 | The indexed edges of the K... |
| konigsbergssiedgwpr 30273 | Each subset of the indexed... |
| konigsbergssiedgw 30274 | Each subset of the indexed... |
| konigsbergumgr 30275 | The Königsberg graph ... |
| konigsberglem1 30276 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30277 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30278 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30279 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30280 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30281 | The Königsberg Bridge... |
| isfrgr 30284 | The property of being a fr... |
| frgrusgr 30285 | A friendship graph is a si... |
| frgr0v 30286 | Any null graph (set with n... |
| frgr0vb 30287 | Any null graph (without ve... |
| frgruhgr0v 30288 | Any null graph (without ve... |
| frgr0 30289 | The null graph (graph with... |
| frcond1 30290 | The friendship condition: ... |
| frcond2 30291 | The friendship condition: ... |
| frgreu 30292 | Variant of ~ frcond2 : An... |
| frcond3 30293 | The friendship condition, ... |
| frcond4 30294 | The friendship condition, ... |
| frgr1v 30295 | Any graph with (at most) o... |
| nfrgr2v 30296 | Any graph with two (differ... |
| frgr3vlem1 30297 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30298 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30299 | Any graph with three verti... |
| 1vwmgr 30300 | Every graph with one verte... |
| 3vfriswmgrlem 30301 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30302 | Every friendship graph wit... |
| 1to2vfriswmgr 30303 | Every friendship graph wit... |
| 1to3vfriswmgr 30304 | Every friendship graph wit... |
| 1to3vfriendship 30305 | The friendship theorem for... |
| 2pthfrgrrn 30306 | Between any two (different... |
| 2pthfrgrrn2 30307 | Between any two (different... |
| 2pthfrgr 30308 | Between any two (different... |
| 3cyclfrgrrn1 30309 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30310 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30311 | Every vertex in a friendsh... |
| 3cyclfrgr 30312 | Every vertex in a friendsh... |
| 4cycl2v2nb 30313 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30314 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30315 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30316 | A graph with a 4-cycle is ... |
| frgrnbnb 30317 | If two neighbors ` U ` and... |
| frgrconngr 30318 | A friendship graph is conn... |
| vdgn0frgrv2 30319 | A vertex in a friendship g... |
| vdgn1frgrv2 30320 | Any vertex in a friendship... |
| vdgn1frgrv3 30321 | Any vertex in a friendship... |
| vdgfrgrgt2 30322 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30323 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30324 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30325 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30326 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30327 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30328 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30329 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30330 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30331 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30332 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30333 | In a friendship graph, two... |
| frgrwopreglem4a 30334 | In a friendship graph any ... |
| frgrwopreglem5a 30335 | If a friendship graph has ... |
| frgrwopreglem1 30336 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30337 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30338 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30339 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30340 | According to statement 5 i... |
| frgrwopregbsn 30341 | According to statement 5 i... |
| frgrwopreg1 30342 | According to statement 5 i... |
| frgrwopreg2 30343 | According to statement 5 i... |
| frgrwopreglem5lem 30344 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30345 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30346 | Alternate direct proof of ... |
| frgrwopreg 30347 | In a friendship graph ther... |
| frgrregorufr0 30348 | In a friendship graph ther... |
| frgrregorufr 30349 | If there is a vertex havin... |
| frgrregorufrg 30350 | If there is a vertex havin... |
| frgr2wwlkeu 30351 | For two different vertices... |
| frgr2wwlkn0 30352 | In a friendship graph, the... |
| frgr2wwlk1 30353 | In a friendship graph, the... |
| frgr2wsp1 30354 | In a friendship graph, the... |
| frgr2wwlkeqm 30355 | If there is a (simple) pat... |
| frgrhash2wsp 30356 | The number of simple paths... |
| fusgreg2wsplem 30357 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30358 | The set of paths of length... |
| fusgreghash2wspv 30359 | According to statement 7 i... |
| fusgreg2wsp 30360 | In a finite simple graph, ... |
| 2wspmdisj 30361 | The sets of paths of lengt... |
| fusgreghash2wsp 30362 | In a finite k-regular grap... |
| frrusgrord0lem 30363 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30364 | If a nonempty finite frien... |
| frrusgrord 30365 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30366 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30367 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30368 | If the initial vertex of a... |
| clwwnonrepclwwnon 30369 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30370 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30371 | Value of operation ` C ` ,... |
| 2clwwlk2 30372 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30373 | Characterization of an ele... |
| 2clwwlk2clwwlk 30374 | An element of the value of... |
| numclwwlk1lem2foalem 30375 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30376 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30377 | Characterization of an ele... |
| numclwwlk1lem2foa 30378 | Going forth and back from ... |
| numclwwlk1lem2f 30379 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30380 | Value of the function ` T ... |
| numclwwlk1lem2f1 30381 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30382 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30383 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30384 | The set of double loops of... |
| numclwwlk1 30385 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30386 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30387 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30388 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30389 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30390 | The sets of the two repres... |
| wlkl0 30391 | There is exactly one walk ... |
| clwlknon2num 30392 | There are k walks of lengt... |
| numclwlk1lem1 30393 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30394 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30395 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30396 | Value of operation ` H ` ,... |
| numclwwlkovh 30397 | Value of operation ` H ` ,... |
| numclwwlkovq 30398 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30399 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30400 | In a friendship graph, for... |
| numclwlk2lem2f 30401 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30402 | Value of the function ` R ... |
| numclwlk2lem2f1o 30403 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30404 | In a friendship graph, the... |
| numclwwlk2 30405 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30406 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30407 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30408 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30409 | Statement 12 in [Huneke] p... |
| numclwwlk4 30410 | The total number of closed... |
| numclwwlk5lem 30411 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30412 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30413 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30414 | For a prime divisor ` P ` ... |
| numclwwlk7 30415 | Statement 14 in [Huneke] p... |
| numclwwlk8 30416 | The size of the set of clo... |
| frgrreggt1 30417 | If a finite nonempty frien... |
| frgrreg 30418 | If a finite nonempty frien... |
| frgrregord013 30419 | If a finite friendship gra... |
| frgrregord13 30420 | If a nonempty finite frien... |
| frgrogt3nreg 30421 | If a finite friendship gra... |
| friendshipgt3 30422 | The friendship theorem for... |
| friendship 30423 | The friendship theorem: I... |
| conventions 30424 |
H... |
| conventions-labels 30425 |
... |
| conventions-comments 30426 |
... |
| natded 30427 | Here are typical n... |
| ex-natded5.2 30428 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30429 | A more efficient proof of ... |
| ex-natded5.2i 30430 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30431 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30432 | A more efficient proof of ... |
| ex-natded5.3i 30433 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30434 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30435 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30436 | A more efficient proof of ... |
| ex-natded5.8 30437 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30438 | A more efficient proof of ... |
| ex-natded5.13 30439 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30440 | A more efficient proof of ... |
| ex-natded9.20 30441 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30442 | A more efficient proof of ... |
| ex-natded9.26 30443 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30444 | A more efficient proof of ... |
| ex-or 30445 | Example for ~ df-or . Exa... |
| ex-an 30446 | Example for ~ df-an . Exa... |
| ex-dif 30447 | Example for ~ df-dif . Ex... |
| ex-un 30448 | Example for ~ df-un . Exa... |
| ex-in 30449 | Example for ~ df-in . Exa... |
| ex-uni 30450 | Example for ~ df-uni . Ex... |
| ex-ss 30451 | Example for ~ df-ss . Exa... |
| ex-pss 30452 | Example for ~ df-pss . Ex... |
| ex-pw 30453 | Example for ~ df-pw . Exa... |
| ex-pr 30454 | Example for ~ df-pr . (Co... |
| ex-br 30455 | Example for ~ df-br . Exa... |
| ex-opab 30456 | Example for ~ df-opab . E... |
| ex-eprel 30457 | Example for ~ df-eprel . ... |
| ex-id 30458 | Example for ~ df-id . Exa... |
| ex-po 30459 | Example for ~ df-po . Exa... |
| ex-xp 30460 | Example for ~ df-xp . Exa... |
| ex-cnv 30461 | Example for ~ df-cnv . Ex... |
| ex-co 30462 | Example for ~ df-co . Exa... |
| ex-dm 30463 | Example for ~ df-dm . Exa... |
| ex-rn 30464 | Example for ~ df-rn . Exa... |
| ex-res 30465 | Example for ~ df-res . Ex... |
| ex-ima 30466 | Example for ~ df-ima . Ex... |
| ex-fv 30467 | Example for ~ df-fv . Exa... |
| ex-1st 30468 | Example for ~ df-1st . Ex... |
| ex-2nd 30469 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30470 | Example for ~ df-dec , 100... |
| ex-fl 30471 | Example for ~ df-fl . Exa... |
| ex-ceil 30472 | Example for ~ df-ceil . (... |
| ex-mod 30473 | Example for ~ df-mod . (C... |
| ex-exp 30474 | Example for ~ df-exp . (C... |
| ex-fac 30475 | Example for ~ df-fac . (C... |
| ex-bc 30476 | Example for ~ df-bc . (Co... |
| ex-hash 30477 | Example for ~ df-hash . (... |
| ex-sqrt 30478 | Example for ~ df-sqrt . (... |
| ex-abs 30479 | Example for ~ df-abs . (C... |
| ex-dvds 30480 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30481 | Example for ~ df-gcd . (C... |
| ex-lcm 30482 | Example for ~ df-lcm . (C... |
| ex-prmo 30483 | Example for ~ df-prmo : ` ... |
| aevdemo 30484 | Proof illustrating the com... |
| ex-ind-dvds 30485 | Example of a proof by indu... |
| ex-fpar 30486 | Formalized example provide... |
| avril1 30487 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30488 | The law of excluded middle... |
| helloworld 30489 | The classic "Hello world" ... |
| 1p1e2apr1 30490 | One plus one equals two. ... |
| eqid1 30491 | Law of identity (reflexivi... |
| 1div0apr 30492 | Division by zero is forbid... |
| topnfbey 30493 | Nothing seems to be imposs... |
| 9p10ne21 30494 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30495 | 9 + 10 equals 21. This as... |
| nrt2irr 30497 | The ` N ` -th root of 2 is... |
| isplig 30500 | The predicate "is a planar... |
| ispligb 30501 | The predicate "is a planar... |
| tncp 30502 | In any planar incidence ge... |
| l2p 30503 | For any line in a planar i... |
| lpni 30504 | For any line in a planar i... |
| nsnlplig 30505 | There is no "one-point lin... |
| nsnlpligALT 30506 | Alternate version of ~ nsn... |
| n0lplig 30507 | There is no "empty line" i... |
| n0lpligALT 30508 | Alternate version of ~ n0l... |
| eulplig 30509 | Through two distinct point... |
| pliguhgr 30510 | Any planar incidence geome... |
| dummylink 30511 | Alias for ~ a1ii that may ... |
| id1 30512 | Alias for ~ idALT that may... |
| isgrpo 30521 | The predicate "is a group ... |
| isgrpoi 30522 | Properties that determine ... |
| grpofo 30523 | A group operation maps ont... |
| grpocl 30524 | Closure law for a group op... |
| grpolidinv 30525 | A group has a left identit... |
| grpon0 30526 | The base set of a group is... |
| grpoass 30527 | A group operation is assoc... |
| grpoidinvlem1 30528 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30529 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30530 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30531 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30532 | A group has a left and rig... |
| grpoideu 30533 | The left identity element ... |
| grporndm 30534 | A group's range in terms o... |
| 0ngrp 30535 | The empty set is not a gro... |
| gidval 30536 | The value of the identity ... |
| grpoidval 30537 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30538 | The identity element of a ... |
| grpoidinv2 30539 | A group's properties using... |
| grpolid 30540 | The identity element of a ... |
| grporid 30541 | The identity element of a ... |
| grporcan 30542 | Right cancellation law for... |
| grpoinveu 30543 | The left inverse element o... |
| grpoid 30544 | Two ways of saying that an... |
| grporn 30545 | The range of a group opera... |
| grpoinvfval 30546 | The inverse function of a ... |
| grpoinvval 30547 | The inverse of a group ele... |
| grpoinvcl 30548 | A group element's inverse ... |
| grpoinv 30549 | The properties of a group ... |
| grpolinv 30550 | The left inverse of a grou... |
| grporinv 30551 | The right inverse of a gro... |
| grpoinvid1 30552 | The inverse of a group ele... |
| grpoinvid2 30553 | The inverse of a group ele... |
| grpolcan 30554 | Left cancellation law for ... |
| grpo2inv 30555 | Double inverse law for gro... |
| grpoinvf 30556 | Mapping of the inverse fun... |
| grpoinvop 30557 | The inverse of the group o... |
| grpodivfval 30558 | Group division (or subtrac... |
| grpodivval 30559 | Group division (or subtrac... |
| grpodivinv 30560 | Group division by an inver... |
| grpoinvdiv 30561 | Inverse of a group divisio... |
| grpodivf 30562 | Mapping for group division... |
| grpodivcl 30563 | Closure of group division ... |
| grpodivdiv 30564 | Double group division. (C... |
| grpomuldivass 30565 | Associative-type law for m... |
| grpodivid 30566 | Division of a group member... |
| grponpcan 30567 | Cancellation law for group... |
| isablo 30570 | The predicate "is an Abeli... |
| ablogrpo 30571 | An Abelian group operation... |
| ablocom 30572 | An Abelian group operation... |
| ablo32 30573 | Commutative/associative la... |
| ablo4 30574 | Commutative/associative la... |
| isabloi 30575 | Properties that determine ... |
| ablomuldiv 30576 | Law for group multiplicati... |
| ablodivdiv 30577 | Law for double group divis... |
| ablodivdiv4 30578 | Law for double group divis... |
| ablodiv32 30579 | Swap the second and third ... |
| ablonncan 30580 | Cancellation law for group... |
| ablonnncan1 30581 | Cancellation law for group... |
| vcrel 30584 | The class of all complex v... |
| vciOLD 30585 | Obsolete version of ~ cvsi... |
| vcsm 30586 | Functionality of th scalar... |
| vccl 30587 | Closure of the scalar prod... |
| vcidOLD 30588 | Identity element for the s... |
| vcdi 30589 | Distributive law for the s... |
| vcdir 30590 | Distributive law for the s... |
| vcass 30591 | Associative law for the sc... |
| vc2OLD 30592 | A vector plus itself is tw... |
| vcablo 30593 | Vector addition is an Abel... |
| vcgrp 30594 | Vector addition is a group... |
| vclcan 30595 | Left cancellation law for ... |
| vczcl 30596 | The zero vector is a vecto... |
| vc0rid 30597 | The zero vector is a right... |
| vc0 30598 | Zero times a vector is the... |
| vcz 30599 | Anything times the zero ve... |
| vcm 30600 | Minus 1 times a vector is ... |
| isvclem 30601 | Lemma for ~ isvcOLD . (Co... |
| vcex 30602 | The components of a comple... |
| isvcOLD 30603 | The predicate "is a comple... |
| isvciOLD 30604 | Properties that determine ... |
| cnaddabloOLD 30605 | Obsolete version of ~ cnad... |
| cnidOLD 30606 | Obsolete version of ~ cnad... |
| cncvcOLD 30607 | Obsolete version of ~ cncv... |
| nvss 30617 | Structure of the class of ... |
| nvvcop 30618 | A normed complex vector sp... |
| nvrel 30626 | The class of all normed co... |
| vafval 30627 | Value of the function for ... |
| bafval 30628 | Value of the function for ... |
| smfval 30629 | Value of the function for ... |
| 0vfval 30630 | Value of the function for ... |
| nmcvfval 30631 | Value of the norm function... |
| nvop2 30632 | A normed complex vector sp... |
| nvvop 30633 | The vector space component... |
| isnvlem 30634 | Lemma for ~ isnv . (Contr... |
| nvex 30635 | The components of a normed... |
| isnv 30636 | The predicate "is a normed... |
| isnvi 30637 | Properties that determine ... |
| nvi 30638 | The properties of a normed... |
| nvvc 30639 | The vector space component... |
| nvablo 30640 | The vector addition operat... |
| nvgrp 30641 | The vector addition operat... |
| nvgf 30642 | Mapping for the vector add... |
| nvsf 30643 | Mapping for the scalar mul... |
| nvgcl 30644 | Closure law for the vector... |
| nvcom 30645 | The vector addition (group... |
| nvass 30646 | The vector addition (group... |
| nvadd32 30647 | Commutative/associative la... |
| nvrcan 30648 | Right cancellation law for... |
| nvadd4 30649 | Rearrangement of 4 terms i... |
| nvscl 30650 | Closure law for the scalar... |
| nvsid 30651 | Identity element for the s... |
| nvsass 30652 | Associative law for the sc... |
| nvscom 30653 | Commutative law for the sc... |
| nvdi 30654 | Distributive law for the s... |
| nvdir 30655 | Distributive law for the s... |
| nv2 30656 | A vector plus itself is tw... |
| vsfval 30657 | Value of the function for ... |
| nvzcl 30658 | Closure law for the zero v... |
| nv0rid 30659 | The zero vector is a right... |
| nv0lid 30660 | The zero vector is a left ... |
| nv0 30661 | Zero times a vector is the... |
| nvsz 30662 | Anything times the zero ve... |
| nvinv 30663 | Minus 1 times a vector is ... |
| nvinvfval 30664 | Function for the negative ... |
| nvm 30665 | Vector subtraction in term... |
| nvmval 30666 | Value of vector subtractio... |
| nvmval2 30667 | Value of vector subtractio... |
| nvmfval 30668 | Value of the function for ... |
| nvmf 30669 | Mapping for the vector sub... |
| nvmcl 30670 | Closure law for the vector... |
| nvnnncan1 30671 | Cancellation law for vecto... |
| nvmdi 30672 | Distributive law for scala... |
| nvnegneg 30673 | Double negative of a vecto... |
| nvmul0or 30674 | If a scalar product is zer... |
| nvrinv 30675 | A vector minus itself. (C... |
| nvlinv 30676 | Minus a vector plus itself... |
| nvpncan2 30677 | Cancellation law for vecto... |
| nvpncan 30678 | Cancellation law for vecto... |
| nvaddsub 30679 | Commutative/associative la... |
| nvnpcan 30680 | Cancellation law for a nor... |
| nvaddsub4 30681 | Rearrangement of 4 terms i... |
| nvmeq0 30682 | The difference between two... |
| nvmid 30683 | A vector minus itself is t... |
| nvf 30684 | Mapping for the norm funct... |
| nvcl 30685 | The norm of a normed compl... |
| nvcli 30686 | The norm of a normed compl... |
| nvs 30687 | Proportionality property o... |
| nvsge0 30688 | The norm of a scalar produ... |
| nvm1 30689 | The norm of the negative o... |
| nvdif 30690 | The norm of the difference... |
| nvpi 30691 | The norm of a vector plus ... |
| nvz0 30692 | The norm of a zero vector ... |
| nvz 30693 | The norm of a vector is ze... |
| nvtri 30694 | Triangle inequality for th... |
| nvmtri 30695 | Triangle inequality for th... |
| nvabs 30696 | Norm difference property o... |
| nvge0 30697 | The norm of a normed compl... |
| nvgt0 30698 | A nonzero norm is positive... |
| nv1 30699 | From any nonzero vector, c... |
| nvop 30700 | A complex inner product sp... |
| cnnv 30701 | The set of complex numbers... |
| cnnvg 30702 | The vector addition (group... |
| cnnvba 30703 | The base set of the normed... |
| cnnvs 30704 | The scalar product operati... |
| cnnvnm 30705 | The norm operation of the ... |
| cnnvm 30706 | The vector subtraction ope... |
| elimnv 30707 | Hypothesis elimination lem... |
| elimnvu 30708 | Hypothesis elimination lem... |
| imsval 30709 | Value of the induced metri... |
| imsdval 30710 | Value of the induced metri... |
| imsdval2 30711 | Value of the distance func... |
| nvnd 30712 | The norm of a normed compl... |
| imsdf 30713 | Mapping for the induced me... |
| imsmetlem 30714 | Lemma for ~ imsmet . (Con... |
| imsmet 30715 | The induced metric of a no... |
| imsxmet 30716 | The induced metric of a no... |
| cnims 30717 | The metric induced on the ... |
| vacn 30718 | Vector addition is jointly... |
| nmcvcn 30719 | The norm of a normed compl... |
| nmcnc 30720 | The norm of a normed compl... |
| smcnlem 30721 | Lemma for ~ smcn . (Contr... |
| smcn 30722 | Scalar multiplication is j... |
| vmcn 30723 | Vector subtraction is join... |
| dipfval 30726 | The inner product function... |
| ipval 30727 | Value of the inner product... |
| ipval2lem2 30728 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30729 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30730 | Lemma for ~ ipval3 . (Con... |
| ipval2 30731 | Expansion of the inner pro... |
| 4ipval2 30732 | Four times the inner produ... |
| ipval3 30733 | Expansion of the inner pro... |
| ipidsq 30734 | The inner product of a vec... |
| ipnm 30735 | Norm expressed in terms of... |
| dipcl 30736 | An inner product is a comp... |
| ipf 30737 | Mapping for the inner prod... |
| dipcj 30738 | The complex conjugate of a... |
| ipipcj 30739 | An inner product times its... |
| diporthcom 30740 | Orthogonality (meaning inn... |
| dip0r 30741 | Inner product with a zero ... |
| dip0l 30742 | Inner product with a zero ... |
| ipz 30743 | The inner product of a vec... |
| dipcn 30744 | Inner product is jointly c... |
| sspval 30747 | The set of all subspaces o... |
| isssp 30748 | The predicate "is a subspa... |
| sspid 30749 | A normed complex vector sp... |
| sspnv 30750 | A subspace is a normed com... |
| sspba 30751 | The base set of a subspace... |
| sspg 30752 | Vector addition on a subsp... |
| sspgval 30753 | Vector addition on a subsp... |
| ssps 30754 | Scalar multiplication on a... |
| sspsval 30755 | Scalar multiplication on a... |
| sspmlem 30756 | Lemma for ~ sspm and other... |
| sspmval 30757 | Vector addition on a subsp... |
| sspm 30758 | Vector subtraction on a su... |
| sspz 30759 | The zero vector of a subsp... |
| sspn 30760 | The norm on a subspace is ... |
| sspnval 30761 | The norm on a subspace in ... |
| sspimsval 30762 | The induced metric on a su... |
| sspims 30763 | The induced metric on a su... |
| lnoval 30776 | The set of linear operator... |
| islno 30777 | The predicate "is a linear... |
| lnolin 30778 | Basic linearity property o... |
| lnof 30779 | A linear operator is a map... |
| lno0 30780 | The value of a linear oper... |
| lnocoi 30781 | The composition of two lin... |
| lnoadd 30782 | Addition property of a lin... |
| lnosub 30783 | Subtraction property of a ... |
| lnomul 30784 | Scalar multiplication prop... |
| nvo00 30785 | Two ways to express a zero... |
| nmoofval 30786 | The operator norm function... |
| nmooval 30787 | The operator norm function... |
| nmosetre 30788 | The set in the supremum of... |
| nmosetn0 30789 | The set in the supremum of... |
| nmoxr 30790 | The norm of an operator is... |
| nmooge0 30791 | The norm of an operator is... |
| nmorepnf 30792 | The norm of an operator is... |
| nmoreltpnf 30793 | The norm of any operator i... |
| nmogtmnf 30794 | The norm of an operator is... |
| nmoolb 30795 | A lower bound for an opera... |
| nmoubi 30796 | An upper bound for an oper... |
| nmoub3i 30797 | An upper bound for an oper... |
| nmoub2i 30798 | An upper bound for an oper... |
| nmobndi 30799 | Two ways to express that a... |
| nmounbi 30800 | Two ways two express that ... |
| nmounbseqi 30801 | An unbounded operator dete... |
| nmounbseqiALT 30802 | Alternate shorter proof of... |
| nmobndseqi 30803 | A bounded sequence determi... |
| nmobndseqiALT 30804 | Alternate shorter proof of... |
| bloval 30805 | The class of bounded linea... |
| isblo 30806 | The predicate "is a bounde... |
| isblo2 30807 | The predicate "is a bounde... |
| bloln 30808 | A bounded operator is a li... |
| blof 30809 | A bounded operator is an o... |
| nmblore 30810 | The norm of a bounded oper... |
| 0ofval 30811 | The zero operator between ... |
| 0oval 30812 | Value of the zero operator... |
| 0oo 30813 | The zero operator is an op... |
| 0lno 30814 | The zero operator is linea... |
| nmoo0 30815 | The operator norm of the z... |
| 0blo 30816 | The zero operator is a bou... |
| nmlno0lem 30817 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30818 | The norm of a linear opera... |
| nmlno0 30819 | The norm of a linear opera... |
| nmlnoubi 30820 | An upper bound for the ope... |
| nmlnogt0 30821 | The norm of a nonzero line... |
| lnon0 30822 | The domain of a nonzero li... |
| nmblolbii 30823 | A lower bound for the norm... |
| nmblolbi 30824 | A lower bound for the norm... |
| isblo3i 30825 | The predicate "is a bounde... |
| blo3i 30826 | Properties that determine ... |
| blometi 30827 | Upper bound for the distan... |
| blocnilem 30828 | Lemma for ~ blocni and ~ l... |
| blocni 30829 | A linear operator is conti... |
| lnocni 30830 | If a linear operator is co... |
| blocn 30831 | A linear operator is conti... |
| blocn2 30832 | A bounded linear operator ... |
| ajfval 30833 | The adjoint function. (Co... |
| hmoval 30834 | The set of Hermitian (self... |
| ishmo 30835 | The predicate "is a hermit... |
| phnv 30838 | Every complex inner produc... |
| phrel 30839 | The class of all complex i... |
| phnvi 30840 | Every complex inner produc... |
| isphg 30841 | The predicate "is a comple... |
| phop 30842 | A complex inner product sp... |
| cncph 30843 | The set of complex numbers... |
| elimph 30844 | Hypothesis elimination lem... |
| elimphu 30845 | Hypothesis elimination lem... |
| isph 30846 | The predicate "is an inner... |
| phpar2 30847 | The parallelogram law for ... |
| phpar 30848 | The parallelogram law for ... |
| ip0i 30849 | A slight variant of Equati... |
| ip1ilem 30850 | Lemma for ~ ip1i . (Contr... |
| ip1i 30851 | Equation 6.47 of [Ponnusam... |
| ip2i 30852 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30853 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30854 | Distributive law for inner... |
| ipasslem1 30855 | Lemma for ~ ipassi . Show... |
| ipasslem2 30856 | Lemma for ~ ipassi . Show... |
| ipasslem3 30857 | Lemma for ~ ipassi . Show... |
| ipasslem4 30858 | Lemma for ~ ipassi . Show... |
| ipasslem5 30859 | Lemma for ~ ipassi . Show... |
| ipasslem7 30860 | Lemma for ~ ipassi . Show... |
| ipasslem8 30861 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30862 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30863 | Lemma for ~ ipassi . Show... |
| ipasslem11 30864 | Lemma for ~ ipassi . Show... |
| ipassi 30865 | Associative law for inner ... |
| dipdir 30866 | Distributive law for inner... |
| dipdi 30867 | Distributive law for inner... |
| ip2dii 30868 | Inner product of two sums.... |
| dipass 30869 | Associative law for inner ... |
| dipassr 30870 | "Associative" law for seco... |
| dipassr2 30871 | "Associative" law for inne... |
| dipsubdir 30872 | Distributive law for inner... |
| dipsubdi 30873 | Distributive law for inner... |
| pythi 30874 | The Pythagorean theorem fo... |
| siilem1 30875 | Lemma for ~ sii . (Contri... |
| siilem2 30876 | Lemma for ~ sii . (Contri... |
| siii 30877 | Inference from ~ sii . (C... |
| sii 30878 | Obsolete version of ~ ipca... |
| ipblnfi 30879 | A function ` F ` generated... |
| ip2eqi 30880 | Two vectors are equal iff ... |
| phoeqi 30881 | A condition implying that ... |
| ajmoi 30882 | Every operator has at most... |
| ajfuni 30883 | The adjoint function is a ... |
| ajfun 30884 | The adjoint function is a ... |
| ajval 30885 | Value of the adjoint funct... |
| iscbn 30888 | A complex Banach space is ... |
| cbncms 30889 | The induced metric on comp... |
| bnnv 30890 | Every complex Banach space... |
| bnrel 30891 | The class of all complex B... |
| bnsscmcl 30892 | A subspace of a Banach spa... |
| cnbn 30893 | The set of complex numbers... |
| ubthlem1 30894 | Lemma for ~ ubth . The fu... |
| ubthlem2 30895 | Lemma for ~ ubth . Given ... |
| ubthlem3 30896 | Lemma for ~ ubth . Prove ... |
| ubth 30897 | Uniform Boundedness Theore... |
| minvecolem1 30898 | Lemma for ~ minveco . The... |
| minvecolem2 30899 | Lemma for ~ minveco . Any... |
| minvecolem3 30900 | Lemma for ~ minveco . The... |
| minvecolem4a 30901 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30902 | Lemma for ~ minveco . The... |
| minvecolem4c 30903 | Lemma for ~ minveco . The... |
| minvecolem4 30904 | Lemma for ~ minveco . The... |
| minvecolem5 30905 | Lemma for ~ minveco . Dis... |
| minvecolem6 30906 | Lemma for ~ minveco . Any... |
| minvecolem7 30907 | Lemma for ~ minveco . Sin... |
| minveco 30908 | Minimizing vector theorem,... |
| ishlo 30911 | The predicate "is a comple... |
| hlobn 30912 | Every complex Hilbert spac... |
| hlph 30913 | Every complex Hilbert spac... |
| hlrel 30914 | The class of all complex H... |
| hlnv 30915 | Every complex Hilbert spac... |
| hlnvi 30916 | Every complex Hilbert spac... |
| hlvc 30917 | Every complex Hilbert spac... |
| hlcmet 30918 | The induced metric on a co... |
| hlmet 30919 | The induced metric on a co... |
| hlpar2 30920 | The parallelogram law sati... |
| hlpar 30921 | The parallelogram law sati... |
| hlex 30922 | The base set of a Hilbert ... |
| hladdf 30923 | Mapping for Hilbert space ... |
| hlcom 30924 | Hilbert space vector addit... |
| hlass 30925 | Hilbert space vector addit... |
| hl0cl 30926 | The Hilbert space zero vec... |
| hladdid 30927 | Hilbert space addition wit... |
| hlmulf 30928 | Mapping for Hilbert space ... |
| hlmulid 30929 | Hilbert space scalar multi... |
| hlmulass 30930 | Hilbert space scalar multi... |
| hldi 30931 | Hilbert space scalar multi... |
| hldir 30932 | Hilbert space scalar multi... |
| hlmul0 30933 | Hilbert space scalar multi... |
| hlipf 30934 | Mapping for Hilbert space ... |
| hlipcj 30935 | Conjugate law for Hilbert ... |
| hlipdir 30936 | Distributive law for Hilbe... |
| hlipass 30937 | Associative law for Hilber... |
| hlipgt0 30938 | The inner product of a Hil... |
| hlcompl 30939 | Completeness of a Hilbert ... |
| cnchl 30940 | The set of complex numbers... |
| htthlem 30941 | Lemma for ~ htth . The co... |
| htth 30942 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 30998 | The group (addition) opera... |
| h2hsm 30999 | The scalar product operati... |
| h2hnm 31000 | The norm function of Hilbe... |
| h2hvs 31001 | The vector subtraction ope... |
| h2hmetdval 31002 | Value of the distance func... |
| h2hcau 31003 | The Cauchy sequences of Hi... |
| h2hlm 31004 | The limit sequences of Hil... |
| axhilex-zf 31005 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31006 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31007 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31008 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31009 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31010 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31011 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31012 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31013 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31014 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31015 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31016 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31017 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31018 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31019 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31020 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31021 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31022 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31035 | The Hilbert space scalar p... |
| hvaddcl 31036 | Closure of vector addition... |
| hvmulcl 31037 | Closure of scalar multipli... |
| hvmulcli 31038 | Closure inference for scal... |
| hvsubf 31039 | Mapping domain and codomai... |
| hvsubval 31040 | Value of vector subtractio... |
| hvsubcl 31041 | Closure of vector subtract... |
| hvaddcli 31042 | Closure of vector addition... |
| hvcomi 31043 | Commutation of vector addi... |
| hvsubvali 31044 | Value of vector subtractio... |
| hvsubcli 31045 | Closure of vector subtract... |
| ifhvhv0 31046 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31047 | Addition with the zero vec... |
| hvmul0 31048 | Scalar multiplication with... |
| hvmul0or 31049 | If a scalar product is zer... |
| hvsubid 31050 | Subtraction of a vector fr... |
| hvnegid 31051 | Addition of negative of a ... |
| hv2neg 31052 | Two ways to express the ne... |
| hvaddlidi 31053 | Addition with the zero vec... |
| hvnegidi 31054 | Addition of negative of a ... |
| hv2negi 31055 | Two ways to express the ne... |
| hvm1neg 31056 | Convert minus one times a ... |
| hvaddsubval 31057 | Value of vector addition i... |
| hvadd32 31058 | Commutative/associative la... |
| hvadd12 31059 | Commutative/associative la... |
| hvadd4 31060 | Hilbert vector space addit... |
| hvsub4 31061 | Hilbert vector space addit... |
| hvaddsub12 31062 | Commutative/associative la... |
| hvpncan 31063 | Addition/subtraction cance... |
| hvpncan2 31064 | Addition/subtraction cance... |
| hvaddsubass 31065 | Associativity of sum and d... |
| hvpncan3 31066 | Subtraction and addition o... |
| hvmulcom 31067 | Scalar multiplication comm... |
| hvsubass 31068 | Hilbert vector space assoc... |
| hvsub32 31069 | Hilbert vector space commu... |
| hvmulassi 31070 | Scalar multiplication asso... |
| hvmulcomi 31071 | Scalar multiplication comm... |
| hvmul2negi 31072 | Double negative in scalar ... |
| hvsubdistr1 31073 | Scalar multiplication dist... |
| hvsubdistr2 31074 | Scalar multiplication dist... |
| hvdistr1i 31075 | Scalar multiplication dist... |
| hvsubdistr1i 31076 | Scalar multiplication dist... |
| hvassi 31077 | Hilbert vector space assoc... |
| hvadd32i 31078 | Hilbert vector space commu... |
| hvsubassi 31079 | Hilbert vector space assoc... |
| hvsub32i 31080 | Hilbert vector space commu... |
| hvadd12i 31081 | Hilbert vector space commu... |
| hvadd4i 31082 | Hilbert vector space addit... |
| hvsubsub4i 31083 | Hilbert vector space addit... |
| hvsubsub4 31084 | Hilbert vector space addit... |
| hv2times 31085 | Two times a vector. (Cont... |
| hvnegdii 31086 | Distribution of negative o... |
| hvsubeq0i 31087 | If the difference between ... |
| hvsubcan2i 31088 | Vector cancellation law. ... |
| hvaddcani 31089 | Cancellation law for vecto... |
| hvsubaddi 31090 | Relationship between vecto... |
| hvnegdi 31091 | Distribution of negative o... |
| hvsubeq0 31092 | If the difference between ... |
| hvaddeq0 31093 | If the sum of two vectors ... |
| hvaddcan 31094 | Cancellation law for vecto... |
| hvaddcan2 31095 | Cancellation law for vecto... |
| hvmulcan 31096 | Cancellation law for scala... |
| hvmulcan2 31097 | Cancellation law for scala... |
| hvsubcan 31098 | Cancellation law for vecto... |
| hvsubcan2 31099 | Cancellation law for vecto... |
| hvsub0 31100 | Subtraction of a zero vect... |
| hvsubadd 31101 | Relationship between vecto... |
| hvaddsub4 31102 | Hilbert vector space addit... |
| hicl 31104 | Closure of inner product. ... |
| hicli 31105 | Closure inference for inne... |
| his5 31110 | Associative law for inner ... |
| his52 31111 | Associative law for inner ... |
| his35 31112 | Move scalar multiplication... |
| his35i 31113 | Move scalar multiplication... |
| his7 31114 | Distributive law for inner... |
| hiassdi 31115 | Distributive/associative l... |
| his2sub 31116 | Distributive law for inner... |
| his2sub2 31117 | Distributive law for inner... |
| hire 31118 | A necessary and sufficient... |
| hiidrcl 31119 | Real closure of inner prod... |
| hi01 31120 | Inner product with the 0 v... |
| hi02 31121 | Inner product with the 0 v... |
| hiidge0 31122 | Inner product with self is... |
| his6 31123 | Zero inner product with se... |
| his1i 31124 | Conjugate law for inner pr... |
| abshicom 31125 | Commuted inner products ha... |
| hial0 31126 | A vector whose inner produ... |
| hial02 31127 | A vector whose inner produ... |
| hisubcomi 31128 | Two vector subtractions si... |
| hi2eq 31129 | Lemma used to prove equali... |
| hial2eq 31130 | Two vectors whose inner pr... |
| hial2eq2 31131 | Two vectors whose inner pr... |
| orthcom 31132 | Orthogonality commutes. (... |
| normlem0 31133 | Lemma used to derive prope... |
| normlem1 31134 | Lemma used to derive prope... |
| normlem2 31135 | Lemma used to derive prope... |
| normlem3 31136 | Lemma used to derive prope... |
| normlem4 31137 | Lemma used to derive prope... |
| normlem5 31138 | Lemma used to derive prope... |
| normlem6 31139 | Lemma used to derive prope... |
| normlem7 31140 | Lemma used to derive prope... |
| normlem8 31141 | Lemma used to derive prope... |
| normlem9 31142 | Lemma used to derive prope... |
| normlem7tALT 31143 | Lemma used to derive prope... |
| bcseqi 31144 | Equality case of Bunjakova... |
| normlem9at 31145 | Lemma used to derive prope... |
| dfhnorm2 31146 | Alternate definition of th... |
| normf 31147 | The norm function maps fro... |
| normval 31148 | The value of the norm of a... |
| normcl 31149 | Real closure of the norm o... |
| normge0 31150 | The norm of a vector is no... |
| normgt0 31151 | The norm of nonzero vector... |
| norm0 31152 | The norm of a zero vector.... |
| norm-i 31153 | Theorem 3.3(i) of [Beran] ... |
| normne0 31154 | A norm is nonzero iff its ... |
| normcli 31155 | Real closure of the norm o... |
| normsqi 31156 | The square of a norm. (Co... |
| norm-i-i 31157 | Theorem 3.3(i) of [Beran] ... |
| normsq 31158 | The square of a norm. (Co... |
| normsub0i 31159 | Two vectors are equal iff ... |
| normsub0 31160 | Two vectors are equal iff ... |
| norm-ii-i 31161 | Triangle inequality for no... |
| norm-ii 31162 | Triangle inequality for no... |
| norm-iii-i 31163 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31164 | Theorem 3.3(iii) of [Beran... |
| normsubi 31165 | Negative doesn't change th... |
| normpythi 31166 | Analogy to Pythagorean the... |
| normsub 31167 | Swapping order of subtract... |
| normneg 31168 | The norm of a vector equal... |
| normpyth 31169 | Analogy to Pythagorean the... |
| normpyc 31170 | Corollary to Pythagorean t... |
| norm3difi 31171 | Norm of differences around... |
| norm3adifii 31172 | Norm of differences around... |
| norm3lem 31173 | Lemma involving norm of di... |
| norm3dif 31174 | Norm of differences around... |
| norm3dif2 31175 | Norm of differences around... |
| norm3lemt 31176 | Lemma involving norm of di... |
| norm3adifi 31177 | Norm of differences around... |
| normpari 31178 | Parallelogram law for norm... |
| normpar 31179 | Parallelogram law for norm... |
| normpar2i 31180 | Corollary of parallelogram... |
| polid2i 31181 | Generalized polarization i... |
| polidi 31182 | Polarization identity. Re... |
| polid 31183 | Polarization identity. Re... |
| hilablo 31184 | Hilbert space vector addit... |
| hilid 31185 | The group identity element... |
| hilvc 31186 | Hilbert space is a complex... |
| hilnormi 31187 | Hilbert space norm in term... |
| hilhhi 31188 | Deduce the structure of Hi... |
| hhnv 31189 | Hilbert space is a normed ... |
| hhva 31190 | The group (addition) opera... |
| hhba 31191 | The base set of Hilbert sp... |
| hh0v 31192 | The zero vector of Hilbert... |
| hhsm 31193 | The scalar product operati... |
| hhvs 31194 | The vector subtraction ope... |
| hhnm 31195 | The norm function of Hilbe... |
| hhims 31196 | The induced metric of Hilb... |
| hhims2 31197 | Hilbert space distance met... |
| hhmet 31198 | The induced metric of Hilb... |
| hhxmet 31199 | The induced metric of Hilb... |
| hhmetdval 31200 | Value of the distance func... |
| hhip 31201 | The inner product operatio... |
| hhph 31202 | The Hilbert space of the H... |
| bcsiALT 31203 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31204 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31205 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31206 | Corollary of the Bunjakova... |
| bcs3 31207 | Corollary of the Bunjakova... |
| hcau 31208 | Member of the set of Cauch... |
| hcauseq 31209 | A Cauchy sequences on a Hi... |
| hcaucvg 31210 | A Cauchy sequence on a Hil... |
| seq1hcau 31211 | A sequence on a Hilbert sp... |
| hlimi 31212 | Express the predicate: Th... |
| hlimseqi 31213 | A sequence with a limit on... |
| hlimveci 31214 | Closure of the limit of a ... |
| hlimconvi 31215 | Convergence of a sequence ... |
| hlim2 31216 | The limit of a sequence on... |
| hlimadd 31217 | Limit of the sum of two se... |
| hilmet 31218 | The Hilbert space norm det... |
| hilxmet 31219 | The Hilbert space norm det... |
| hilmetdval 31220 | Value of the distance func... |
| hilims 31221 | Hilbert space distance met... |
| hhcau 31222 | The Cauchy sequences of Hi... |
| hhlm 31223 | The limit sequences of Hil... |
| hhcmpl 31224 | Lemma used for derivation ... |
| hilcompl 31225 | Lemma used for derivation ... |
| hhcms 31227 | The Hilbert space induced ... |
| hhhl 31228 | The Hilbert space structur... |
| hilcms 31229 | The Hilbert space norm det... |
| hilhl 31230 | The Hilbert space of the H... |
| issh 31232 | Subspace ` H ` of a Hilber... |
| issh2 31233 | Subspace ` H ` of a Hilber... |
| shss 31234 | A subspace is a subset of ... |
| shel 31235 | A member of a subspace of ... |
| shex 31236 | The set of subspaces of a ... |
| shssii 31237 | A closed subspace of a Hil... |
| sheli 31238 | A member of a subspace of ... |
| shelii 31239 | A member of a subspace of ... |
| sh0 31240 | The zero vector belongs to... |
| shaddcl 31241 | Closure of vector addition... |
| shmulcl 31242 | Closure of vector scalar m... |
| issh3 31243 | Subspace ` H ` of a Hilber... |
| shsubcl 31244 | Closure of vector subtract... |
| isch 31246 | Closed subspace ` H ` of a... |
| isch2 31247 | Closed subspace ` H ` of a... |
| chsh 31248 | A closed subspace is a sub... |
| chsssh 31249 | Closed subspaces are subsp... |
| chex 31250 | The set of closed subspace... |
| chshii 31251 | A closed subspace is a sub... |
| ch0 31252 | The zero vector belongs to... |
| chss 31253 | A closed subspace of a Hil... |
| chel 31254 | A member of a closed subsp... |
| chssii 31255 | A closed subspace of a Hil... |
| cheli 31256 | A member of a closed subsp... |
| chelii 31257 | A member of a closed subsp... |
| chlimi 31258 | The limit property of a cl... |
| hlim0 31259 | The zero sequence in Hilbe... |
| hlimcaui 31260 | If a sequence in Hilbert s... |
| hlimf 31261 | Function-like behavior of ... |
| hlimuni 31262 | A Hilbert space sequence c... |
| hlimreui 31263 | The limit of a Hilbert spa... |
| hlimeui 31264 | The limit of a Hilbert spa... |
| isch3 31265 | A Hilbert subspace is clos... |
| chcompl 31266 | Completeness of a closed s... |
| helch 31267 | The Hilbert lattice one (w... |
| ifchhv 31268 | Prove ` if ( A e. CH , A ,... |
| helsh 31269 | Hilbert space is a subspac... |
| shsspwh 31270 | Subspaces are subsets of H... |
| chsspwh 31271 | Closed subspaces are subse... |
| hsn0elch 31272 | The zero subspace belongs ... |
| norm1 31273 | From any nonzero Hilbert s... |
| norm1exi 31274 | A normalized vector exists... |
| norm1hex 31275 | A normalized vector can ex... |
| elch0 31278 | Membership in zero for clo... |
| h0elch 31279 | The zero subspace is a clo... |
| h0elsh 31280 | The zero subspace is a sub... |
| hhssva 31281 | The vector addition operat... |
| hhsssm 31282 | The scalar multiplication ... |
| hhssnm 31283 | The norm operation on a su... |
| issubgoilem 31284 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31285 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31286 | Abelian group property of ... |
| hhssablo 31287 | Abelian group property of ... |
| hhssnv 31288 | Normed complex vector spac... |
| hhssnvt 31289 | Normed complex vector spac... |
| hhsst 31290 | A member of ` SH ` is a su... |
| hhshsslem1 31291 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31292 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31293 | The predicate " ` H ` is a... |
| hhsssh2 31294 | The predicate " ` H ` is a... |
| hhssba 31295 | The base set of a subspace... |
| hhssvs 31296 | The vector subtraction ope... |
| hhssvsf 31297 | Mapping of the vector subt... |
| hhssims 31298 | Induced metric of a subspa... |
| hhssims2 31299 | Induced metric of a subspa... |
| hhssmet 31300 | Induced metric of a subspa... |
| hhssmetdval 31301 | Value of the distance func... |
| hhsscms 31302 | The induced metric of a cl... |
| hhssbnOLD 31303 | Obsolete version of ~ cssb... |
| ocval 31304 | Value of orthogonal comple... |
| ocel 31305 | Membership in orthogonal c... |
| shocel 31306 | Membership in orthogonal c... |
| ocsh 31307 | The orthogonal complement ... |
| shocsh 31308 | The orthogonal complement ... |
| ocss 31309 | An orthogonal complement i... |
| shocss 31310 | An orthogonal complement i... |
| occon 31311 | Contraposition law for ort... |
| occon2 31312 | Double contraposition for ... |
| occon2i 31313 | Double contraposition for ... |
| oc0 31314 | The zero vector belongs to... |
| ocorth 31315 | Members of a subset and it... |
| shocorth 31316 | Members of a subspace and ... |
| ococss 31317 | Inclusion in complement of... |
| shococss 31318 | Inclusion in complement of... |
| shorth 31319 | Members of orthogonal subs... |
| ocin 31320 | Intersection of a Hilbert ... |
| occon3 31321 | Hilbert lattice contraposi... |
| ocnel 31322 | A nonzero vector in the co... |
| chocvali 31323 | Value of the orthogonal co... |
| shuni 31324 | Two subspaces with trivial... |
| chocunii 31325 | Lemma for uniqueness part ... |
| pjhthmo 31326 | Projection Theorem, unique... |
| occllem 31327 | Lemma for ~ occl . (Contr... |
| occl 31328 | Closure of complement of H... |
| shoccl 31329 | Closure of complement of H... |
| choccl 31330 | Closure of complement of H... |
| choccli 31331 | Closure of ` CH ` orthocom... |
| shsval 31336 | Value of subspace sum of t... |
| shsss 31337 | The subspace sum is a subs... |
| shsel 31338 | Membership in the subspace... |
| shsel3 31339 | Membership in the subspace... |
| shseli 31340 | Membership in subspace sum... |
| shscli 31341 | Closure of subspace sum. ... |
| shscl 31342 | Closure of subspace sum. ... |
| shscom 31343 | Commutative law for subspa... |
| shsva 31344 | Vector sum belongs to subs... |
| shsel1 31345 | A subspace sum contains a ... |
| shsel2 31346 | A subspace sum contains a ... |
| shsvs 31347 | Vector subtraction belongs... |
| shsub1 31348 | Subspace sum is an upper b... |
| shsub2 31349 | Subspace sum is an upper b... |
| choc0 31350 | The orthocomplement of the... |
| choc1 31351 | The orthocomplement of the... |
| chocnul 31352 | Orthogonal complement of t... |
| shintcli 31353 | Closure of intersection of... |
| shintcl 31354 | The intersection of a none... |
| chintcli 31355 | The intersection of a none... |
| chintcl 31356 | The intersection (infimum)... |
| spanval 31357 | Value of the linear span o... |
| hsupval 31358 | Value of supremum of set o... |
| chsupval 31359 | The value of the supremum ... |
| spancl 31360 | The span of a subset of Hi... |
| elspancl 31361 | A member of a span is a ve... |
| shsupcl 31362 | Closure of the subspace su... |
| hsupcl 31363 | Closure of supremum of set... |
| chsupcl 31364 | Closure of supremum of sub... |
| hsupss 31365 | Subset relation for suprem... |
| chsupss 31366 | Subset relation for suprem... |
| hsupunss 31367 | The union of a set of Hilb... |
| chsupunss 31368 | The union of a set of clos... |
| spanss2 31369 | A subset of Hilbert space ... |
| shsupunss 31370 | The union of a set of subs... |
| spanid 31371 | A subspace of Hilbert spac... |
| spanss 31372 | Ordering relationship for ... |
| spanssoc 31373 | The span of a subset of Hi... |
| sshjval 31374 | Value of join for subsets ... |
| shjval 31375 | Value of join in ` SH ` . ... |
| chjval 31376 | Value of join in ` CH ` . ... |
| chjvali 31377 | Value of join in ` CH ` . ... |
| sshjval3 31378 | Value of join for subsets ... |
| sshjcl 31379 | Closure of join for subset... |
| shjcl 31380 | Closure of join in ` SH ` ... |
| chjcl 31381 | Closure of join in ` CH ` ... |
| shjcom 31382 | Commutative law for Hilber... |
| shless 31383 | Subset implies subset of s... |
| shlej1 31384 | Add disjunct to both sides... |
| shlej2 31385 | Add disjunct to both sides... |
| shincli 31386 | Closure of intersection of... |
| shscomi 31387 | Commutative law for subspa... |
| shsvai 31388 | Vector sum belongs to subs... |
| shsel1i 31389 | A subspace sum contains a ... |
| shsel2i 31390 | A subspace sum contains a ... |
| shsvsi 31391 | Vector subtraction belongs... |
| shunssi 31392 | Union is smaller than subs... |
| shunssji 31393 | Union is smaller than Hilb... |
| shsleji 31394 | Subspace sum is smaller th... |
| shjcomi 31395 | Commutative law for join i... |
| shsub1i 31396 | Subspace sum is an upper b... |
| shsub2i 31397 | Subspace sum is an upper b... |
| shub1i 31398 | Hilbert lattice join is an... |
| shjcli 31399 | Closure of ` CH ` join. (... |
| shjshcli 31400 | ` SH ` closure of join. (... |
| shlessi 31401 | Subset implies subset of s... |
| shlej1i 31402 | Add disjunct to both sides... |
| shlej2i 31403 | Add disjunct to both sides... |
| shslej 31404 | Subspace sum is smaller th... |
| shincl 31405 | Closure of intersection of... |
| shub1 31406 | Hilbert lattice join is an... |
| shub2 31407 | A subspace is a subset of ... |
| shsidmi 31408 | Idempotent law for Hilbert... |
| shslubi 31409 | The least upper bound law ... |
| shlesb1i 31410 | Hilbert lattice ordering i... |
| shsval2i 31411 | An alternate way to expres... |
| shsval3i 31412 | An alternate way to expres... |
| shmodsi 31413 | The modular law holds for ... |
| shmodi 31414 | The modular law is implied... |
| pjhthlem1 31415 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31416 | Lemma for ~ pjhth . (Cont... |
| pjhth 31417 | Projection Theorem: Any H... |
| pjhtheu 31418 | Projection Theorem: Any H... |
| pjhfval 31420 | The value of the projectio... |
| pjhval 31421 | Value of a projection. (C... |
| pjpreeq 31422 | Equality with a projection... |
| pjeq 31423 | Equality with a projection... |
| axpjcl 31424 | Closure of a projection in... |
| pjhcl 31425 | Closure of a projection in... |
| omlsilem 31426 | Lemma for orthomodular law... |
| omlsii 31427 | Subspace inference form of... |
| omlsi 31428 | Subspace form of orthomodu... |
| ococi 31429 | Complement of complement o... |
| ococ 31430 | Complement of complement o... |
| dfch2 31431 | Alternate definition of th... |
| ococin 31432 | The double complement is t... |
| hsupval2 31433 | Alternate definition of su... |
| chsupval2 31434 | The value of the supremum ... |
| sshjval2 31435 | Value of join in the set o... |
| chsupid 31436 | A subspace is the supremum... |
| chsupsn 31437 | Value of supremum of subse... |
| shlub 31438 | Hilbert lattice join is th... |
| shlubi 31439 | Hilbert lattice join is th... |
| pjhtheu2 31440 | Uniqueness of ` y ` for th... |
| pjcli 31441 | Closure of a projection in... |
| pjhcli 31442 | Closure of a projection in... |
| pjpjpre 31443 | Decomposition of a vector ... |
| axpjpj 31444 | Decomposition of a vector ... |
| pjclii 31445 | Closure of a projection in... |
| pjhclii 31446 | Closure of a projection in... |
| pjpj0i 31447 | Decomposition of a vector ... |
| pjpji 31448 | Decomposition of a vector ... |
| pjpjhth 31449 | Projection Theorem: Any H... |
| pjpjhthi 31450 | Projection Theorem: Any H... |
| pjop 31451 | Orthocomplement projection... |
| pjpo 31452 | Projection in terms of ort... |
| pjopi 31453 | Orthocomplement projection... |
| pjpoi 31454 | Projection in terms of ort... |
| pjoc1i 31455 | Projection of a vector in ... |
| pjchi 31456 | Projection of a vector in ... |
| pjoccl 31457 | The part of a vector that ... |
| pjoc1 31458 | Projection of a vector in ... |
| pjomli 31459 | Subspace form of orthomodu... |
| pjoml 31460 | Subspace form of orthomodu... |
| pjococi 31461 | Proof of orthocomplement t... |
| pjoc2i 31462 | Projection of a vector in ... |
| pjoc2 31463 | Projection of a vector in ... |
| sh0le 31464 | The zero subspace is the s... |
| ch0le 31465 | The zero subspace is the s... |
| shle0 31466 | No subspace is smaller tha... |
| chle0 31467 | No Hilbert lattice element... |
| chnlen0 31468 | A Hilbert lattice element ... |
| ch0pss 31469 | The zero subspace is a pro... |
| orthin 31470 | The intersection of orthog... |
| ssjo 31471 | The lattice join of a subs... |
| shne0i 31472 | A nonzero subspace has a n... |
| shs0i 31473 | Hilbert subspace sum with ... |
| shs00i 31474 | Two subspaces are zero iff... |
| ch0lei 31475 | The closed subspace zero i... |
| chle0i 31476 | No Hilbert closed subspace... |
| chne0i 31477 | A nonzero closed subspace ... |
| chocini 31478 | Intersection of a closed s... |
| chj0i 31479 | Join with lattice zero in ... |
| chm1i 31480 | Meet with lattice one in `... |
| chjcli 31481 | Closure of ` CH ` join. (... |
| chsleji 31482 | Subspace sum is smaller th... |
| chseli 31483 | Membership in subspace sum... |
| chincli 31484 | Closure of Hilbert lattice... |
| chsscon3i 31485 | Hilbert lattice contraposi... |
| chsscon1i 31486 | Hilbert lattice contraposi... |
| chsscon2i 31487 | Hilbert lattice contraposi... |
| chcon2i 31488 | Hilbert lattice contraposi... |
| chcon1i 31489 | Hilbert lattice contraposi... |
| chcon3i 31490 | Hilbert lattice contraposi... |
| chunssji 31491 | Union is smaller than ` CH... |
| chjcomi 31492 | Commutative law for join i... |
| chub1i 31493 | ` CH ` join is an upper bo... |
| chub2i 31494 | ` CH ` join is an upper bo... |
| chlubi 31495 | Hilbert lattice join is th... |
| chlubii 31496 | Hilbert lattice join is th... |
| chlej1i 31497 | Add join to both sides of ... |
| chlej2i 31498 | Add join to both sides of ... |
| chlej12i 31499 | Add join to both sides of ... |
| chlejb1i 31500 | Hilbert lattice ordering i... |
| chdmm1i 31501 | De Morgan's law for meet i... |
| chdmm2i 31502 | De Morgan's law for meet i... |
| chdmm3i 31503 | De Morgan's law for meet i... |
| chdmm4i 31504 | De Morgan's law for meet i... |
| chdmj1i 31505 | De Morgan's law for join i... |
| chdmj2i 31506 | De Morgan's law for join i... |
| chdmj3i 31507 | De Morgan's law for join i... |
| chdmj4i 31508 | De Morgan's law for join i... |
| chnlei 31509 | Equivalent expressions for... |
| chjassi 31510 | Associative law for Hilber... |
| chj00i 31511 | Two Hilbert lattice elemen... |
| chjoi 31512 | The join of a closed subsp... |
| chj1i 31513 | Join with Hilbert lattice ... |
| chm0i 31514 | Meet with Hilbert lattice ... |
| chm0 31515 | Meet with Hilbert lattice ... |
| shjshsi 31516 | Hilbert lattice join equal... |
| shjshseli 31517 | A closed subspace sum equa... |
| chne0 31518 | A nonzero closed subspace ... |
| chocin 31519 | Intersection of a closed s... |
| chssoc 31520 | A closed subspace less tha... |
| chj0 31521 | Join with Hilbert lattice ... |
| chslej 31522 | Subspace sum is smaller th... |
| chincl 31523 | Closure of Hilbert lattice... |
| chsscon3 31524 | Hilbert lattice contraposi... |
| chsscon1 31525 | Hilbert lattice contraposi... |
| chsscon2 31526 | Hilbert lattice contraposi... |
| chpsscon3 31527 | Hilbert lattice contraposi... |
| chpsscon1 31528 | Hilbert lattice contraposi... |
| chpsscon2 31529 | Hilbert lattice contraposi... |
| chjcom 31530 | Commutative law for Hilber... |
| chub1 31531 | Hilbert lattice join is gr... |
| chub2 31532 | Hilbert lattice join is gr... |
| chlub 31533 | Hilbert lattice join is th... |
| chlej1 31534 | Add join to both sides of ... |
| chlej2 31535 | Add join to both sides of ... |
| chlejb1 31536 | Hilbert lattice ordering i... |
| chlejb2 31537 | Hilbert lattice ordering i... |
| chnle 31538 | Equivalent expressions for... |
| chjo 31539 | The join of a closed subsp... |
| chabs1 31540 | Hilbert lattice absorption... |
| chabs2 31541 | Hilbert lattice absorption... |
| chabs1i 31542 | Hilbert lattice absorption... |
| chabs2i 31543 | Hilbert lattice absorption... |
| chjidm 31544 | Idempotent law for Hilbert... |
| chjidmi 31545 | Idempotent law for Hilbert... |
| chj12i 31546 | A rearrangement of Hilbert... |
| chj4i 31547 | Rearrangement of the join ... |
| chjjdiri 31548 | Hilbert lattice join distr... |
| chdmm1 31549 | De Morgan's law for meet i... |
| chdmm2 31550 | De Morgan's law for meet i... |
| chdmm3 31551 | De Morgan's law for meet i... |
| chdmm4 31552 | De Morgan's law for meet i... |
| chdmj1 31553 | De Morgan's law for join i... |
| chdmj2 31554 | De Morgan's law for join i... |
| chdmj3 31555 | De Morgan's law for join i... |
| chdmj4 31556 | De Morgan's law for join i... |
| chjass 31557 | Associative law for Hilber... |
| chj12 31558 | A rearrangement of Hilbert... |
| chj4 31559 | Rearrangement of the join ... |
| ledii 31560 | An ortholattice is distrib... |
| lediri 31561 | An ortholattice is distrib... |
| lejdii 31562 | An ortholattice is distrib... |
| lejdiri 31563 | An ortholattice is distrib... |
| ledi 31564 | An ortholattice is distrib... |
| spansn0 31565 | The span of the singleton ... |
| span0 31566 | The span of the empty set ... |
| elspani 31567 | Membership in the span of ... |
| spanuni 31568 | The span of a union is the... |
| spanun 31569 | The span of a union is the... |
| sshhococi 31570 | The join of two Hilbert sp... |
| hne0 31571 | Hilbert space has a nonzer... |
| chsup0 31572 | The supremum of the empty ... |
| h1deoi 31573 | Membership in orthocomplem... |
| h1dei 31574 | Membership in 1-dimensiona... |
| h1did 31575 | A generating vector belong... |
| h1dn0 31576 | A nonzero vector generates... |
| h1de2i 31577 | Membership in 1-dimensiona... |
| h1de2bi 31578 | Membership in 1-dimensiona... |
| h1de2ctlem 31579 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31580 | Membership in 1-dimensiona... |
| spansni 31581 | The span of a singleton in... |
| elspansni 31582 | Membership in the span of ... |
| spansn 31583 | The span of a singleton in... |
| spansnch 31584 | The span of a Hilbert spac... |
| spansnsh 31585 | The span of a Hilbert spac... |
| spansnchi 31586 | The span of a singleton in... |
| spansnid 31587 | A vector belongs to the sp... |
| spansnmul 31588 | A scalar product with a ve... |
| elspansncl 31589 | A member of a span of a si... |
| elspansn 31590 | Membership in the span of ... |
| elspansn2 31591 | Membership in the span of ... |
| spansncol 31592 | The singletons of collinea... |
| spansneleqi 31593 | Membership relation implie... |
| spansneleq 31594 | Membership relation that i... |
| spansnss 31595 | The span of the singleton ... |
| elspansn3 31596 | A member of the span of th... |
| elspansn4 31597 | A span membership conditio... |
| elspansn5 31598 | A vector belonging to both... |
| spansnss2 31599 | The span of the singleton ... |
| normcan 31600 | Cancellation-type law that... |
| pjspansn 31601 | A projection on the span o... |
| spansnpji 31602 | A subset of Hilbert space ... |
| spanunsni 31603 | The span of the union of a... |
| spanpr 31604 | The span of a pair of vect... |
| h1datomi 31605 | A 1-dimensional subspace i... |
| h1datom 31606 | A 1-dimensional subspace i... |
| cmbr 31608 | Binary relation expressing... |
| pjoml2i 31609 | Variation of orthomodular ... |
| pjoml3i 31610 | Variation of orthomodular ... |
| pjoml4i 31611 | Variation of orthomodular ... |
| pjoml5i 31612 | The orthomodular law. Rem... |
| pjoml6i 31613 | An equivalent of the ortho... |
| cmbri 31614 | Binary relation expressing... |
| cmcmlem 31615 | Commutation is symmetric. ... |
| cmcmi 31616 | Commutation is symmetric. ... |
| cmcm2i 31617 | Commutation with orthocomp... |
| cmcm3i 31618 | Commutation with orthocomp... |
| cmcm4i 31619 | Commutation with orthocomp... |
| cmbr2i 31620 | Alternate definition of th... |
| cmcmii 31621 | Commutation is symmetric. ... |
| cmcm2ii 31622 | Commutation with orthocomp... |
| cmcm3ii 31623 | Commutation with orthocomp... |
| cmbr3i 31624 | Alternate definition for t... |
| cmbr4i 31625 | Alternate definition for t... |
| lecmi 31626 | Comparable Hilbert lattice... |
| lecmii 31627 | Comparable Hilbert lattice... |
| cmj1i 31628 | A Hilbert lattice element ... |
| cmj2i 31629 | A Hilbert lattice element ... |
| cmm1i 31630 | A Hilbert lattice element ... |
| cmm2i 31631 | A Hilbert lattice element ... |
| cmbr3 31632 | Alternate definition for t... |
| cm0 31633 | The zero Hilbert lattice e... |
| cmidi 31634 | The commutes relation is r... |
| pjoml2 31635 | Variation of orthomodular ... |
| pjoml3 31636 | Variation of orthomodular ... |
| pjoml5 31637 | The orthomodular law. Rem... |
| cmcm 31638 | Commutation is symmetric. ... |
| cmcm3 31639 | Commutation with orthocomp... |
| cmcm2 31640 | Commutation with orthocomp... |
| lecm 31641 | Comparable Hilbert lattice... |
| fh1 31642 | Foulis-Holland Theorem. I... |
| fh2 31643 | Foulis-Holland Theorem. I... |
| cm2j 31644 | A lattice element that com... |
| fh1i 31645 | Foulis-Holland Theorem. I... |
| fh2i 31646 | Foulis-Holland Theorem. I... |
| fh3i 31647 | Variation of the Foulis-Ho... |
| fh4i 31648 | Variation of the Foulis-Ho... |
| cm2ji 31649 | A lattice element that com... |
| cm2mi 31650 | A lattice element that com... |
| qlax1i 31651 | One of the equations showi... |
| qlax2i 31652 | One of the equations showi... |
| qlax3i 31653 | One of the equations showi... |
| qlax4i 31654 | One of the equations showi... |
| qlax5i 31655 | One of the equations showi... |
| qlaxr1i 31656 | One of the conditions show... |
| qlaxr2i 31657 | One of the conditions show... |
| qlaxr4i 31658 | One of the conditions show... |
| qlaxr5i 31659 | One of the conditions show... |
| qlaxr3i 31660 | A variation of the orthomo... |
| chscllem1 31661 | Lemma for ~ chscl . (Cont... |
| chscllem2 31662 | Lemma for ~ chscl . (Cont... |
| chscllem3 31663 | Lemma for ~ chscl . (Cont... |
| chscllem4 31664 | Lemma for ~ chscl . (Cont... |
| chscl 31665 | The subspace sum of two cl... |
| osumi 31666 | If two closed subspaces of... |
| osumcori 31667 | Corollary of ~ osumi . (C... |
| osumcor2i 31668 | Corollary of ~ osumi , sho... |
| osum 31669 | If two closed subspaces of... |
| spansnji 31670 | The subspace sum of a clos... |
| spansnj 31671 | The subspace sum of a clos... |
| spansnscl 31672 | The subspace sum of a clos... |
| sumspansn 31673 | The sum of two vectors bel... |
| spansnm0i 31674 | The meet of different one-... |
| nonbooli 31675 | A Hilbert lattice with two... |
| spansncvi 31676 | Hilbert space has the cove... |
| spansncv 31677 | Hilbert space has the cove... |
| 5oalem1 31678 | Lemma for orthoarguesian l... |
| 5oalem2 31679 | Lemma for orthoarguesian l... |
| 5oalem3 31680 | Lemma for orthoarguesian l... |
| 5oalem4 31681 | Lemma for orthoarguesian l... |
| 5oalem5 31682 | Lemma for orthoarguesian l... |
| 5oalem6 31683 | Lemma for orthoarguesian l... |
| 5oalem7 31684 | Lemma for orthoarguesian l... |
| 5oai 31685 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31686 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31687 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31688 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31689 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31690 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31691 | Lemma for 3OA (weak) ortho... |
| 3oai 31692 | 3OA (weak) orthoarguesian ... |
| pjorthi 31693 | Projection components on o... |
| pjch1 31694 | Property of identity proje... |
| pjo 31695 | The orthogonal projection.... |
| pjcompi 31696 | Component of a projection.... |
| pjidmi 31697 | A projection is idempotent... |
| pjadjii 31698 | A projection is self-adjoi... |
| pjaddii 31699 | Projection of vector sum i... |
| pjinormii 31700 | The inner product of a pro... |
| pjmulii 31701 | Projection of (scalar) pro... |
| pjsubii 31702 | Projection of vector diffe... |
| pjsslem 31703 | Lemma for subset relations... |
| pjss2i 31704 | Subset relationship for pr... |
| pjssmii 31705 | Projection meet property. ... |
| pjssge0ii 31706 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31707 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31708 | The projection on a subspa... |
| pjadji 31709 | A projection is self-adjoi... |
| pjaddi 31710 | Projection of vector sum i... |
| pjinormi 31711 | The inner product of a pro... |
| pjsubi 31712 | Projection of vector diffe... |
| pjmuli 31713 | Projection of scalar produ... |
| pjige0i 31714 | The inner product of a pro... |
| pjige0 31715 | The inner product of a pro... |
| pjcjt2 31716 | The projection on a subspa... |
| pj0i 31717 | The projection of the zero... |
| pjch 31718 | Projection of a vector in ... |
| pjid 31719 | The projection of a vector... |
| pjvec 31720 | The set of vectors belongi... |
| pjocvec 31721 | The set of vectors belongi... |
| pjocini 31722 | Membership of projection i... |
| pjini 31723 | Membership of projection i... |
| pjjsi 31724 | A sufficient condition for... |
| pjfni 31725 | Functionality of a project... |
| pjrni 31726 | The range of a projection.... |
| pjfoi 31727 | A projection maps onto its... |
| pjfi 31728 | The mapping of a projectio... |
| pjvi 31729 | The value of a projection ... |
| pjhfo 31730 | A projection maps onto its... |
| pjrn 31731 | The range of a projection.... |
| pjhf 31732 | The mapping of a projectio... |
| pjfn 31733 | Functionality of a project... |
| pjsumi 31734 | The projection on a subspa... |
| pj11i 31735 | One-to-one correspondence ... |
| pjdsi 31736 | Vector decomposition into ... |
| pjds3i 31737 | Vector decomposition into ... |
| pj11 31738 | One-to-one correspondence ... |
| pjmfn 31739 | Functionality of the proje... |
| pjmf1 31740 | The projector function map... |
| pjoi0 31741 | The inner product of proje... |
| pjoi0i 31742 | The inner product of proje... |
| pjopythi 31743 | Pythagorean theorem for pr... |
| pjopyth 31744 | Pythagorean theorem for pr... |
| pjnormi 31745 | The norm of the projection... |
| pjpythi 31746 | Pythagorean theorem for pr... |
| pjneli 31747 | If a vector does not belon... |
| pjnorm 31748 | The norm of the projection... |
| pjpyth 31749 | Pythagorean theorem for pr... |
| pjnel 31750 | If a vector does not belon... |
| pjnorm2 31751 | A vector belongs to the su... |
| mayete3i 31752 | Mayet's equation E_3. Par... |
| mayetes3i 31753 | Mayet's equation E^*_3, de... |
| hosmval 31759 | Value of the sum of two Hi... |
| hommval 31760 | Value of the scalar produc... |
| hodmval 31761 | Value of the difference of... |
| hfsmval 31762 | Value of the sum of two Hi... |
| hfmmval 31763 | Value of the scalar produc... |
| hosval 31764 | Value of the sum of two Hi... |
| homval 31765 | Value of the scalar produc... |
| hodval 31766 | Value of the difference of... |
| hfsval 31767 | Value of the sum of two Hi... |
| hfmval 31768 | Value of the scalar produc... |
| hoscl 31769 | Closure of the sum of two ... |
| homcl 31770 | Closure of the scalar prod... |
| hodcl 31771 | Closure of the difference ... |
| ho0val 31774 | Value of the zero Hilbert ... |
| ho0f 31775 | Functionality of the zero ... |
| df0op2 31776 | Alternate definition of Hi... |
| dfiop2 31777 | Alternate definition of Hi... |
| hoif 31778 | Functionality of the Hilbe... |
| hoival 31779 | The value of the Hilbert s... |
| hoico1 31780 | Composition with the Hilbe... |
| hoico2 31781 | Composition with the Hilbe... |
| hoaddcl 31782 | The sum of Hilbert space o... |
| homulcl 31783 | The scalar product of a Hi... |
| hoeq 31784 | Equality of Hilbert space ... |
| hoeqi 31785 | Equality of Hilbert space ... |
| hoscli 31786 | Closure of Hilbert space o... |
| hodcli 31787 | Closure of Hilbert space o... |
| hocoi 31788 | Composition of Hilbert spa... |
| hococli 31789 | Closure of composition of ... |
| hocofi 31790 | Mapping of composition of ... |
| hocofni 31791 | Functionality of compositi... |
| hoaddcli 31792 | Mapping of sum of Hilbert ... |
| hosubcli 31793 | Mapping of difference of H... |
| hoaddfni 31794 | Functionality of sum of Hi... |
| hosubfni 31795 | Functionality of differenc... |
| hoaddcomi 31796 | Commutativity of sum of Hi... |
| hosubcl 31797 | Mapping of difference of H... |
| hoaddcom 31798 | Commutativity of sum of Hi... |
| hodsi 31799 | Relationship between Hilbe... |
| hoaddassi 31800 | Associativity of sum of Hi... |
| hoadd12i 31801 | Commutative/associative la... |
| hoadd32i 31802 | Commutative/associative la... |
| hocadddiri 31803 | Distributive law for Hilbe... |
| hocsubdiri 31804 | Distributive law for Hilbe... |
| ho2coi 31805 | Double composition of Hilb... |
| hoaddass 31806 | Associativity of sum of Hi... |
| hoadd32 31807 | Commutative/associative la... |
| hoadd4 31808 | Rearrangement of 4 terms i... |
| hocsubdir 31809 | Distributive law for Hilbe... |
| hoaddridi 31810 | Sum of a Hilbert space ope... |
| hodidi 31811 | Difference of a Hilbert sp... |
| ho0coi 31812 | Composition of the zero op... |
| hoid1i 31813 | Composition of Hilbert spa... |
| hoid1ri 31814 | Composition of Hilbert spa... |
| hoaddrid 31815 | Sum of a Hilbert space ope... |
| hodid 31816 | Difference of a Hilbert sp... |
| hon0 31817 | A Hilbert space operator i... |
| hodseqi 31818 | Subtraction and addition o... |
| ho0subi 31819 | Subtraction of Hilbert spa... |
| honegsubi 31820 | Relationship between Hilbe... |
| ho0sub 31821 | Subtraction of Hilbert spa... |
| hosubid1 31822 | The zero operator subtract... |
| honegsub 31823 | Relationship between Hilbe... |
| homullid 31824 | An operator equals its sca... |
| homco1 31825 | Associative law for scalar... |
| homulass 31826 | Scalar product associative... |
| hoadddi 31827 | Scalar product distributiv... |
| hoadddir 31828 | Scalar product reverse dis... |
| homul12 31829 | Swap first and second fact... |
| honegneg 31830 | Double negative of a Hilbe... |
| hosubneg 31831 | Relationship between opera... |
| hosubdi 31832 | Scalar product distributiv... |
| honegdi 31833 | Distribution of negative o... |
| honegsubdi 31834 | Distribution of negative o... |
| honegsubdi2 31835 | Distribution of negative o... |
| hosubsub2 31836 | Law for double subtraction... |
| hosub4 31837 | Rearrangement of 4 terms i... |
| hosubadd4 31838 | Rearrangement of 4 terms i... |
| hoaddsubass 31839 | Associative-type law for a... |
| hoaddsub 31840 | Law for operator addition ... |
| hosubsub 31841 | Law for double subtraction... |
| hosubsub4 31842 | Law for double subtraction... |
| ho2times 31843 | Two times a Hilbert space ... |
| hoaddsubassi 31844 | Associativity of sum and d... |
| hoaddsubi 31845 | Law for sum and difference... |
| hosd1i 31846 | Hilbert space operator sum... |
| hosd2i 31847 | Hilbert space operator sum... |
| hopncani 31848 | Hilbert space operator can... |
| honpcani 31849 | Hilbert space operator can... |
| hosubeq0i 31850 | If the difference between ... |
| honpncani 31851 | Hilbert space operator can... |
| ho01i 31852 | A condition implying that ... |
| ho02i 31853 | A condition implying that ... |
| hoeq1 31854 | A condition implying that ... |
| hoeq2 31855 | A condition implying that ... |
| adjmo 31856 | Every Hilbert space operat... |
| adjsym 31857 | Symmetry property of an ad... |
| eigrei 31858 | A necessary and sufficient... |
| eigre 31859 | A necessary and sufficient... |
| eigposi 31860 | A sufficient condition (fi... |
| eigorthi 31861 | A necessary and sufficient... |
| eigorth 31862 | A necessary and sufficient... |
| nmopval 31880 | Value of the norm of a Hil... |
| elcnop 31881 | Property defining a contin... |
| ellnop 31882 | Property defining a linear... |
| lnopf 31883 | A linear Hilbert space ope... |
| elbdop 31884 | Property defining a bounde... |
| bdopln 31885 | A bounded linear Hilbert s... |
| bdopf 31886 | A bounded linear Hilbert s... |
| nmopsetretALT 31887 | The set in the supremum of... |
| nmopsetretHIL 31888 | The set in the supremum of... |
| nmopsetn0 31889 | The set in the supremum of... |
| nmopxr 31890 | The norm of a Hilbert spac... |
| nmoprepnf 31891 | The norm of a Hilbert spac... |
| nmopgtmnf 31892 | The norm of a Hilbert spac... |
| nmopreltpnf 31893 | The norm of a Hilbert spac... |
| nmopre 31894 | The norm of a bounded oper... |
| elbdop2 31895 | Property defining a bounde... |
| elunop 31896 | Property defining a unitar... |
| elhmop 31897 | Property defining a Hermit... |
| hmopf 31898 | A Hermitian operator is a ... |
| hmopex 31899 | The class of Hermitian ope... |
| nmfnval 31900 | Value of the norm of a Hil... |
| nmfnsetre 31901 | The set in the supremum of... |
| nmfnsetn0 31902 | The set in the supremum of... |
| nmfnxr 31903 | The norm of any Hilbert sp... |
| nmfnrepnf 31904 | The norm of a Hilbert spac... |
| nlfnval 31905 | Value of the null space of... |
| elcnfn 31906 | Property defining a contin... |
| ellnfn 31907 | Property defining a linear... |
| lnfnf 31908 | A linear Hilbert space fun... |
| dfadj2 31909 | Alternate definition of th... |
| funadj 31910 | Functionality of the adjoi... |
| dmadjss 31911 | The domain of the adjoint ... |
| dmadjop 31912 | A member of the domain of ... |
| adjeu 31913 | Elementhood in the domain ... |
| adjval 31914 | Value of the adjoint funct... |
| adjval2 31915 | Value of the adjoint funct... |
| cnvadj 31916 | The adjoint function equal... |
| funcnvadj 31917 | The converse of the adjoin... |
| adj1o 31918 | The adjoint function maps ... |
| dmadjrn 31919 | The adjoint of an operator... |
| eigvecval 31920 | The set of eigenvectors of... |
| eigvalfval 31921 | The eigenvalues of eigenve... |
| specval 31922 | The value of the spectrum ... |
| speccl 31923 | The spectrum of an operato... |
| hhlnoi 31924 | The linear operators of Hi... |
| hhnmoi 31925 | The norm of an operator in... |
| hhbloi 31926 | A bounded linear operator ... |
| hh0oi 31927 | The zero operator in Hilbe... |
| hhcno 31928 | The continuous operators o... |
| hhcnf 31929 | The continuous functionals... |
| dmadjrnb 31930 | The adjoint of an operator... |
| nmoplb 31931 | A lower bound for an opera... |
| nmopub 31932 | An upper bound for an oper... |
| nmopub2tALT 31933 | An upper bound for an oper... |
| nmopub2tHIL 31934 | An upper bound for an oper... |
| nmopge0 31935 | The norm of any Hilbert sp... |
| nmopgt0 31936 | A linear Hilbert space ope... |
| cnopc 31937 | Basic continuity property ... |
| lnopl 31938 | Basic linearity property o... |
| unop 31939 | Basic inner product proper... |
| unopf1o 31940 | A unitary operator in Hilb... |
| unopnorm 31941 | A unitary operator is idem... |
| cnvunop 31942 | The inverse (converse) of ... |
| unopadj 31943 | The inverse (converse) of ... |
| unoplin 31944 | A unitary operator is line... |
| counop 31945 | The composition of two uni... |
| hmop 31946 | Basic inner product proper... |
| hmopre 31947 | The inner product of the v... |
| nmfnlb 31948 | A lower bound for a functi... |
| nmfnleub 31949 | An upper bound for the nor... |
| nmfnleub2 31950 | An upper bound for the nor... |
| nmfnge0 31951 | The norm of any Hilbert sp... |
| elnlfn 31952 | Membership in the null spa... |
| elnlfn2 31953 | Membership in the null spa... |
| cnfnc 31954 | Basic continuity property ... |
| lnfnl 31955 | Basic linearity property o... |
| adjcl 31956 | Closure of the adjoint of ... |
| adj1 31957 | Property of an adjoint Hil... |
| adj2 31958 | Property of an adjoint Hil... |
| adjeq 31959 | A property that determines... |
| adjadj 31960 | Double adjoint. Theorem 3... |
| adjvalval 31961 | Value of the value of the ... |
| unopadj2 31962 | The adjoint of a unitary o... |
| hmopadj 31963 | A Hermitian operator is se... |
| hmdmadj 31964 | Every Hermitian operator h... |
| hmopadj2 31965 | An operator is Hermitian i... |
| hmoplin 31966 | A Hermitian operator is li... |
| brafval 31967 | The bra of a vector, expre... |
| braval 31968 | A bra-ket juxtaposition, e... |
| braadd 31969 | Linearity property of bra ... |
| bramul 31970 | Linearity property of bra ... |
| brafn 31971 | The bra function is a func... |
| bralnfn 31972 | The Dirac bra function is ... |
| bracl 31973 | Closure of the bra functio... |
| bra0 31974 | The Dirac bra of the zero ... |
| brafnmul 31975 | Anti-linearity property of... |
| kbfval 31976 | The outer product of two v... |
| kbop 31977 | The outer product of two v... |
| kbval 31978 | The value of the operator ... |
| kbmul 31979 | Multiplication property of... |
| kbpj 31980 | If a vector ` A ` has norm... |
| eleigvec 31981 | Membership in the set of e... |
| eleigvec2 31982 | Membership in the set of e... |
| eleigveccl 31983 | Closure of an eigenvector ... |
| eigvalval 31984 | The eigenvalue of an eigen... |
| eigvalcl 31985 | An eigenvalue is a complex... |
| eigvec1 31986 | Property of an eigenvector... |
| eighmre 31987 | The eigenvalues of a Hermi... |
| eighmorth 31988 | Eigenvectors of a Hermitia... |
| nmopnegi 31989 | Value of the norm of the n... |
| lnop0 31990 | The value of a linear Hilb... |
| lnopmul 31991 | Multiplicative property of... |
| lnopli 31992 | Basic scalar product prope... |
| lnopfi 31993 | A linear Hilbert space ope... |
| lnop0i 31994 | The value of a linear Hilb... |
| lnopaddi 31995 | Additive property of a lin... |
| lnopmuli 31996 | Multiplicative property of... |
| lnopaddmuli 31997 | Sum/product property of a ... |
| lnopsubi 31998 | Subtraction property for a... |
| lnopsubmuli 31999 | Subtraction/product proper... |
| lnopmulsubi 32000 | Product/subtraction proper... |
| homco2 32001 | Move a scalar product out ... |
| idunop 32002 | The identity function (res... |
| 0cnop 32003 | The identically zero funct... |
| 0cnfn 32004 | The identically zero funct... |
| idcnop 32005 | The identity function (res... |
| idhmop 32006 | The Hilbert space identity... |
| 0hmop 32007 | The identically zero funct... |
| 0lnop 32008 | The identically zero funct... |
| 0lnfn 32009 | The identically zero funct... |
| nmop0 32010 | The norm of the zero opera... |
| nmfn0 32011 | The norm of the identicall... |
| hmopbdoptHIL 32012 | A Hermitian operator is a ... |
| hoddii 32013 | Distributive law for Hilbe... |
| hoddi 32014 | Distributive law for Hilbe... |
| nmop0h 32015 | The norm of any operator o... |
| idlnop 32016 | The identity function (res... |
| 0bdop 32017 | The identically zero opera... |
| adj0 32018 | Adjoint of the zero operat... |
| nmlnop0iALT 32019 | A linear operator with a z... |
| nmlnop0iHIL 32020 | A linear operator with a z... |
| nmlnopgt0i 32021 | A linear Hilbert space ope... |
| nmlnop0 32022 | A linear operator with a z... |
| nmlnopne0 32023 | A linear operator with a n... |
| lnopmi 32024 | The scalar product of a li... |
| lnophsi 32025 | The sum of two linear oper... |
| lnophdi 32026 | The difference of two line... |
| lnopcoi 32027 | The composition of two lin... |
| lnopco0i 32028 | The composition of a linea... |
| lnopeq0lem1 32029 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 32030 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 32031 | A condition implying that ... |
| lnopeqi 32032 | Two linear Hilbert space o... |
| lnopeq 32033 | Two linear Hilbert space o... |
| lnopunilem1 32034 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 32035 | Lemma for ~ lnopunii . (C... |
| lnopunii 32036 | If a linear operator (whos... |
| elunop2 32037 | An operator is unitary iff... |
| nmopun 32038 | Norm of a unitary Hilbert ... |
| unopbd 32039 | A unitary operator is a bo... |
| lnophmlem1 32040 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 32041 | Lemma for ~ lnophmi . (Co... |
| lnophmi 32042 | A linear operator is Hermi... |
| lnophm 32043 | A linear operator is Hermi... |
| hmops 32044 | The sum of two Hermitian o... |
| hmopm 32045 | The scalar product of a He... |
| hmopd 32046 | The difference of two Herm... |
| hmopco 32047 | The composition of two com... |
| nmbdoplbi 32048 | A lower bound for the norm... |
| nmbdoplb 32049 | A lower bound for the norm... |
| nmcexi 32050 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 32051 | The norm of a continuous l... |
| nmcoplbi 32052 | A lower bound for the norm... |
| nmcopex 32053 | The norm of a continuous l... |
| nmcoplb 32054 | A lower bound for the norm... |
| nmophmi 32055 | The norm of the scalar pro... |
| bdophmi 32056 | The scalar product of a bo... |
| lnconi 32057 | Lemma for ~ lnopconi and ~... |
| lnopconi 32058 | A condition equivalent to ... |
| lnopcon 32059 | A condition equivalent to ... |
| lnopcnbd 32060 | A linear operator is conti... |
| lncnopbd 32061 | A continuous linear operat... |
| lncnbd 32062 | A continuous linear operat... |
| lnopcnre 32063 | A linear operator is conti... |
| lnfnli 32064 | Basic property of a linear... |
| lnfnfi 32065 | A linear Hilbert space fun... |
| lnfn0i 32066 | The value of a linear Hilb... |
| lnfnaddi 32067 | Additive property of a lin... |
| lnfnmuli 32068 | Multiplicative property of... |
| lnfnaddmuli 32069 | Sum/product property of a ... |
| lnfnsubi 32070 | Subtraction property for a... |
| lnfn0 32071 | The value of a linear Hilb... |
| lnfnmul 32072 | Multiplicative property of... |
| nmbdfnlbi 32073 | A lower bound for the norm... |
| nmbdfnlb 32074 | A lower bound for the norm... |
| nmcfnexi 32075 | The norm of a continuous l... |
| nmcfnlbi 32076 | A lower bound for the norm... |
| nmcfnex 32077 | The norm of a continuous l... |
| nmcfnlb 32078 | A lower bound of the norm ... |
| lnfnconi 32079 | A condition equivalent to ... |
| lnfncon 32080 | A condition equivalent to ... |
| lnfncnbd 32081 | A linear functional is con... |
| imaelshi 32082 | The image of a subspace un... |
| rnelshi 32083 | The range of a linear oper... |
| nlelshi 32084 | The null space of a linear... |
| nlelchi 32085 | The null space of a contin... |
| riesz3i 32086 | A continuous linear functi... |
| riesz4i 32087 | A continuous linear functi... |
| riesz4 32088 | A continuous linear functi... |
| riesz1 32089 | Part 1 of the Riesz repres... |
| riesz2 32090 | Part 2 of the Riesz repres... |
| cnlnadjlem1 32091 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 32092 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 32093 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 32094 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32095 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32096 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32097 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32098 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32099 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32100 | Every continuous linear op... |
| cnlnadjeui 32101 | Every continuous linear op... |
| cnlnadjeu 32102 | Every continuous linear op... |
| cnlnadj 32103 | Every continuous linear op... |
| cnlnssadj 32104 | Every continuous linear Hi... |
| bdopssadj 32105 | Every bounded linear Hilbe... |
| bdopadj 32106 | Every bounded linear Hilbe... |
| adjbdln 32107 | The adjoint of a bounded l... |
| adjbdlnb 32108 | An operator is bounded and... |
| adjbd1o 32109 | The mapping of adjoints of... |
| adjlnop 32110 | The adjoint of an operator... |
| adjsslnop 32111 | Every operator with an adj... |
| nmopadjlei 32112 | Property of the norm of an... |
| nmopadjlem 32113 | Lemma for ~ nmopadji . (C... |
| nmopadji 32114 | Property of the norm of an... |
| adjeq0 32115 | An operator is zero iff it... |
| adjmul 32116 | The adjoint of the scalar ... |
| adjadd 32117 | The adjoint of the sum of ... |
| nmoptrii 32118 | Triangle inequality for th... |
| nmopcoi 32119 | Upper bound for the norm o... |
| bdophsi 32120 | The sum of two bounded lin... |
| bdophdi 32121 | The difference between two... |
| bdopcoi 32122 | The composition of two bou... |
| nmoptri2i 32123 | Triangle-type inequality f... |
| adjcoi 32124 | The adjoint of a compositi... |
| nmopcoadji 32125 | The norm of an operator co... |
| nmopcoadj2i 32126 | The norm of an operator co... |
| nmopcoadj0i 32127 | An operator composed with ... |
| unierri 32128 | If we approximate a chain ... |
| branmfn 32129 | The norm of the bra functi... |
| brabn 32130 | The bra of a vector is a b... |
| rnbra 32131 | The set of bras equals the... |
| bra11 32132 | The bra function maps vect... |
| bracnln 32133 | A bra is a continuous line... |
| cnvbraval 32134 | Value of the converse of t... |
| cnvbracl 32135 | Closure of the converse of... |
| cnvbrabra 32136 | The converse bra of the br... |
| bracnvbra 32137 | The bra of the converse br... |
| bracnlnval 32138 | The vector that a continuo... |
| cnvbramul 32139 | Multiplication property of... |
| kbass1 32140 | Dirac bra-ket associative ... |
| kbass2 32141 | Dirac bra-ket associative ... |
| kbass3 32142 | Dirac bra-ket associative ... |
| kbass4 32143 | Dirac bra-ket associative ... |
| kbass5 32144 | Dirac bra-ket associative ... |
| kbass6 32145 | Dirac bra-ket associative ... |
| leopg 32146 | Ordering relation for posi... |
| leop 32147 | Ordering relation for oper... |
| leop2 32148 | Ordering relation for oper... |
| leop3 32149 | Operator ordering in terms... |
| leoppos 32150 | Binary relation defining a... |
| leoprf2 32151 | The ordering relation for ... |
| leoprf 32152 | The ordering relation for ... |
| leopsq 32153 | The square of a Hermitian ... |
| 0leop 32154 | The zero operator is a pos... |
| idleop 32155 | The identity operator is a... |
| leopadd 32156 | The sum of two positive op... |
| leopmuli 32157 | The scalar product of a no... |
| leopmul 32158 | The scalar product of a po... |
| leopmul2i 32159 | Scalar product applied to ... |
| leoptri 32160 | The positive operator orde... |
| leoptr 32161 | The positive operator orde... |
| leopnmid 32162 | A bounded Hermitian operat... |
| nmopleid 32163 | A nonzero, bounded Hermiti... |
| opsqrlem1 32164 | Lemma for opsqri . (Contr... |
| opsqrlem2 32165 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32166 | Lemma for opsqri . (Contr... |
| opsqrlem4 32167 | Lemma for opsqri . (Contr... |
| opsqrlem5 32168 | Lemma for opsqri . (Contr... |
| opsqrlem6 32169 | Lemma for opsqri . (Contr... |
| pjhmopi 32170 | A projector is a Hermitian... |
| pjlnopi 32171 | A projector is a linear op... |
| pjnmopi 32172 | The operator norm of a pro... |
| pjbdlni 32173 | A projector is a bounded l... |
| pjhmop 32174 | A projection is a Hermitia... |
| hmopidmchi 32175 | An idempotent Hermitian op... |
| hmopidmpji 32176 | An idempotent Hermitian op... |
| hmopidmch 32177 | An idempotent Hermitian op... |
| hmopidmpj 32178 | An idempotent Hermitian op... |
| pjsdii 32179 | Distributive law for Hilbe... |
| pjddii 32180 | Distributive law for Hilbe... |
| pjsdi2i 32181 | Chained distributive law f... |
| pjcoi 32182 | Composition of projections... |
| pjcocli 32183 | Closure of composition of ... |
| pjcohcli 32184 | Closure of composition of ... |
| pjadjcoi 32185 | Adjoint of composition of ... |
| pjcofni 32186 | Functionality of compositi... |
| pjss1coi 32187 | Subset relationship for pr... |
| pjss2coi 32188 | Subset relationship for pr... |
| pjssmi 32189 | Projection meet property. ... |
| pjssge0i 32190 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32191 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32192 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32193 | Composition of projections... |
| pjscji 32194 | The projection of orthogon... |
| pjssumi 32195 | The projection on a subspa... |
| pjssposi 32196 | Projector ordering can be ... |
| pjordi 32197 | The definition of projecto... |
| pjssdif2i 32198 | The projection subspace of... |
| pjssdif1i 32199 | A necessary and sufficient... |
| pjimai 32200 | The image of a projection.... |
| pjidmcoi 32201 | A projection is idempotent... |
| pjoccoi 32202 | Composition of projections... |
| pjtoi 32203 | Subspace sum of projection... |
| pjoci 32204 | Projection of orthocomplem... |
| pjidmco 32205 | A projection operator is i... |
| dfpjop 32206 | Definition of projection o... |
| pjhmopidm 32207 | Two ways to express the se... |
| elpjidm 32208 | A projection operator is i... |
| elpjhmop 32209 | A projection operator is H... |
| 0leopj 32210 | A projector is a positive ... |
| pjadj2 32211 | A projector is self-adjoin... |
| pjadj3 32212 | A projector is self-adjoin... |
| elpjch 32213 | Reconstruction of the subs... |
| elpjrn 32214 | Reconstruction of the subs... |
| pjinvari 32215 | A closed subspace ` H ` wi... |
| pjin1i 32216 | Lemma for Theorem 1.22 of ... |
| pjin2i 32217 | Lemma for Theorem 1.22 of ... |
| pjin3i 32218 | Lemma for Theorem 1.22 of ... |
| pjclem1 32219 | Lemma for projection commu... |
| pjclem2 32220 | Lemma for projection commu... |
| pjclem3 32221 | Lemma for projection commu... |
| pjclem4a 32222 | Lemma for projection commu... |
| pjclem4 32223 | Lemma for projection commu... |
| pjci 32224 | Two subspaces commute iff ... |
| pjcmul1i 32225 | A necessary and sufficient... |
| pjcmul2i 32226 | The projection subspace of... |
| pjcohocli 32227 | Closure of composition of ... |
| pjadj2coi 32228 | Adjoint of double composit... |
| pj2cocli 32229 | Closure of double composit... |
| pj3lem1 32230 | Lemma for projection tripl... |
| pj3si 32231 | Stronger projection triple... |
| pj3i 32232 | Projection triplet theorem... |
| pj3cor1i 32233 | Projection triplet corolla... |
| pjs14i 32234 | Theorem S-14 of Watanabe, ... |
| isst 32237 | Property of a state. (Con... |
| ishst 32238 | Property of a complex Hilb... |
| sticl 32239 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32240 | Real closure of the value ... |
| hstcl 32241 | Closure of the value of a ... |
| hst1a 32242 | Unit value of a Hilbert-sp... |
| hstel2 32243 | Properties of a Hilbert-sp... |
| hstorth 32244 | Orthogonality property of ... |
| hstosum 32245 | Orthogonal sum property of... |
| hstoc 32246 | Sum of a Hilbert-space-val... |
| hstnmoc 32247 | Sum of norms of a Hilbert-... |
| stge0 32248 | The value of a state is no... |
| stle1 32249 | The value of a state is le... |
| hstle1 32250 | The norm of the value of a... |
| hst1h 32251 | The norm of a Hilbert-spac... |
| hst0h 32252 | The norm of a Hilbert-spac... |
| hstpyth 32253 | Pythagorean property of a ... |
| hstle 32254 | Ordering property of a Hil... |
| hstles 32255 | Ordering property of a Hil... |
| hstoh 32256 | A Hilbert-space-valued sta... |
| hst0 32257 | A Hilbert-space-valued sta... |
| sthil 32258 | The value of a state at th... |
| stj 32259 | The value of a state on a ... |
| sto1i 32260 | The state of a subspace pl... |
| sto2i 32261 | The state of the orthocomp... |
| stge1i 32262 | If a state is greater than... |
| stle0i 32263 | If a state is less than or... |
| stlei 32264 | Ordering law for states. ... |
| stlesi 32265 | Ordering law for states. ... |
| stji1i 32266 | Join of components of Sasa... |
| stm1i 32267 | State of component of unit... |
| stm1ri 32268 | State of component of unit... |
| stm1addi 32269 | Sum of states whose meet i... |
| staddi 32270 | If the sum of 2 states is ... |
| stm1add3i 32271 | Sum of states whose meet i... |
| stadd3i 32272 | If the sum of 3 states is ... |
| st0 32273 | The state of the zero subs... |
| strlem1 32274 | Lemma for strong state the... |
| strlem2 32275 | Lemma for strong state the... |
| strlem3a 32276 | Lemma for strong state the... |
| strlem3 32277 | Lemma for strong state the... |
| strlem4 32278 | Lemma for strong state the... |
| strlem5 32279 | Lemma for strong state the... |
| strlem6 32280 | Lemma for strong state the... |
| stri 32281 | Strong state theorem. The... |
| strb 32282 | Strong state theorem (bidi... |
| hstrlem2 32283 | Lemma for strong set of CH... |
| hstrlem3a 32284 | Lemma for strong set of CH... |
| hstrlem3 32285 | Lemma for strong set of CH... |
| hstrlem4 32286 | Lemma for strong set of CH... |
| hstrlem5 32287 | Lemma for strong set of CH... |
| hstrlem6 32288 | Lemma for strong set of CH... |
| hstri 32289 | Hilbert space admits a str... |
| hstrbi 32290 | Strong CH-state theorem (b... |
| largei 32291 | A Hilbert lattice admits a... |
| jplem1 32292 | Lemma for Jauch-Piron theo... |
| jplem2 32293 | Lemma for Jauch-Piron theo... |
| jpi 32294 | The function ` S ` , that ... |
| golem1 32295 | Lemma for Godowski's equat... |
| golem2 32296 | Lemma for Godowski's equat... |
| goeqi 32297 | Godowski's equation, shown... |
| stcltr1i 32298 | Property of a strong class... |
| stcltr2i 32299 | Property of a strong class... |
| stcltrlem1 32300 | Lemma for strong classical... |
| stcltrlem2 32301 | Lemma for strong classical... |
| stcltrthi 32302 | Theorem for classically st... |
| cvbr 32306 | Binary relation expressing... |
| cvbr2 32307 | Binary relation expressing... |
| cvcon3 32308 | Contraposition law for the... |
| cvpss 32309 | The covers relation implie... |
| cvnbtwn 32310 | The covers relation implie... |
| cvnbtwn2 32311 | The covers relation implie... |
| cvnbtwn3 32312 | The covers relation implie... |
| cvnbtwn4 32313 | The covers relation implie... |
| cvnsym 32314 | The covers relation is not... |
| cvnref 32315 | The covers relation is not... |
| cvntr 32316 | The covers relation is not... |
| spansncv2 32317 | Hilbert space has the cove... |
| mdbr 32318 | Binary relation expressing... |
| mdi 32319 | Consequence of the modular... |
| mdbr2 32320 | Binary relation expressing... |
| mdbr3 32321 | Binary relation expressing... |
| mdbr4 32322 | Binary relation expressing... |
| dmdbr 32323 | Binary relation expressing... |
| dmdmd 32324 | The dual modular pair prop... |
| mddmd 32325 | The modular pair property ... |
| dmdi 32326 | Consequence of the dual mo... |
| dmdbr2 32327 | Binary relation expressing... |
| dmdi2 32328 | Consequence of the dual mo... |
| dmdbr3 32329 | Binary relation expressing... |
| dmdbr4 32330 | Binary relation expressing... |
| dmdi4 32331 | Consequence of the dual mo... |
| dmdbr5 32332 | Binary relation expressing... |
| mddmd2 32333 | Relationship between modul... |
| mdsl0 32334 | A sublattice condition tha... |
| ssmd1 32335 | Ordering implies the modul... |
| ssmd2 32336 | Ordering implies the modul... |
| ssdmd1 32337 | Ordering implies the dual ... |
| ssdmd2 32338 | Ordering implies the dual ... |
| dmdsl3 32339 | Sublattice mapping for a d... |
| mdsl3 32340 | Sublattice mapping for a m... |
| mdslle1i 32341 | Order preservation of the ... |
| mdslle2i 32342 | Order preservation of the ... |
| mdslj1i 32343 | Join preservation of the o... |
| mdslj2i 32344 | Meet preservation of the r... |
| mdsl1i 32345 | If the modular pair proper... |
| mdsl2i 32346 | If the modular pair proper... |
| mdsl2bi 32347 | If the modular pair proper... |
| cvmdi 32348 | The covering property impl... |
| mdslmd1lem1 32349 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32350 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32351 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32352 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32353 | Preservation of the modula... |
| mdslmd2i 32354 | Preservation of the modula... |
| mdsldmd1i 32355 | Preservation of the dual m... |
| mdslmd3i 32356 | Modular pair conditions th... |
| mdslmd4i 32357 | Modular pair condition tha... |
| csmdsymi 32358 | Cross-symmetry implies M-s... |
| mdexchi 32359 | An exchange lemma for modu... |
| cvmd 32360 | The covering property impl... |
| cvdmd 32361 | The covering property impl... |
| ela 32363 | Atoms in a Hilbert lattice... |
| elat2 32364 | Expanded membership relati... |
| elatcv0 32365 | A Hilbert lattice element ... |
| atcv0 32366 | An atom covers the zero su... |
| atssch 32367 | Atoms are a subset of the ... |
| atelch 32368 | An atom is a Hilbert latti... |
| atne0 32369 | An atom is not the Hilbert... |
| atss 32370 | A lattice element smaller ... |
| atsseq 32371 | Two atoms in a subset rela... |
| atcveq0 32372 | A Hilbert lattice element ... |
| h1da 32373 | A 1-dimensional subspace i... |
| spansna 32374 | The span of the singleton ... |
| sh1dle 32375 | A 1-dimensional subspace i... |
| ch1dle 32376 | A 1-dimensional subspace i... |
| atom1d 32377 | The 1-dimensional subspace... |
| superpos 32378 | Superposition Principle. ... |
| chcv1 32379 | The Hilbert lattice has th... |
| chcv2 32380 | The Hilbert lattice has th... |
| chjatom 32381 | The join of a closed subsp... |
| shatomici 32382 | The lattice of Hilbert sub... |
| hatomici 32383 | The Hilbert lattice is ato... |
| hatomic 32384 | A Hilbert lattice is atomi... |
| shatomistici 32385 | The lattice of Hilbert sub... |
| hatomistici 32386 | ` CH ` is atomistic, i.e. ... |
| chpssati 32387 | Two Hilbert lattice elemen... |
| chrelati 32388 | The Hilbert lattice is rel... |
| chrelat2i 32389 | A consequence of relative ... |
| cvati 32390 | If a Hilbert lattice eleme... |
| cvbr4i 32391 | An alternate way to expres... |
| cvexchlem 32392 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32393 | The Hilbert lattice satisf... |
| chrelat2 32394 | A consequence of relative ... |
| chrelat3 32395 | A consequence of relative ... |
| chrelat3i 32396 | A consequence of the relat... |
| chrelat4i 32397 | A consequence of relative ... |
| cvexch 32398 | The Hilbert lattice satisf... |
| cvp 32399 | The Hilbert lattice satisf... |
| atnssm0 32400 | The meet of a Hilbert latt... |
| atnemeq0 32401 | The meet of distinct atoms... |
| atssma 32402 | The meet with an atom's su... |
| atcv0eq 32403 | Two atoms covering the zer... |
| atcv1 32404 | Two atoms covering the zer... |
| atexch 32405 | The Hilbert lattice satisf... |
| atomli 32406 | An assertion holding in at... |
| atoml2i 32407 | An assertion holding in at... |
| atordi 32408 | An ordering law for a Hilb... |
| atcvatlem 32409 | Lemma for ~ atcvati . (Co... |
| atcvati 32410 | A nonzero Hilbert lattice ... |
| atcvat2i 32411 | A Hilbert lattice element ... |
| atord 32412 | An ordering law for a Hilb... |
| atcvat2 32413 | A Hilbert lattice element ... |
| chirredlem1 32414 | Lemma for ~ chirredi . (C... |
| chirredlem2 32415 | Lemma for ~ chirredi . (C... |
| chirredlem3 32416 | Lemma for ~ chirredi . (C... |
| chirredlem4 32417 | Lemma for ~ chirredi . (C... |
| chirredi 32418 | The Hilbert lattice is irr... |
| chirred 32419 | The Hilbert lattice is irr... |
| atcvat3i 32420 | A condition implying that ... |
| atcvat4i 32421 | A condition implying exist... |
| atdmd 32422 | Two Hilbert lattice elemen... |
| atmd 32423 | Two Hilbert lattice elemen... |
| atmd2 32424 | Two Hilbert lattice elemen... |
| atabsi 32425 | Absorption of an incompara... |
| atabs2i 32426 | Absorption of an incompara... |
| mdsymlem1 32427 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32428 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32429 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32430 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32431 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32432 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32433 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32434 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32435 | M-symmetry of the Hilbert ... |
| mdsym 32436 | M-symmetry of the Hilbert ... |
| dmdsym 32437 | Dual M-symmetry of the Hil... |
| atdmd2 32438 | Two Hilbert lattice elemen... |
| sumdmdii 32439 | If the subspace sum of two... |
| cmmdi 32440 | Commuting subspaces form a... |
| cmdmdi 32441 | Commuting subspaces form a... |
| sumdmdlem 32442 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32443 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32444 | The subspace sum of two Hi... |
| dmdbr4ati 32445 | Dual modular pair property... |
| dmdbr5ati 32446 | Dual modular pair property... |
| dmdbr6ati 32447 | Dual modular pair property... |
| dmdbr7ati 32448 | Dual modular pair property... |
| mdoc1i 32449 | Orthocomplements form a mo... |
| mdoc2i 32450 | Orthocomplements form a mo... |
| dmdoc1i 32451 | Orthocomplements form a du... |
| dmdoc2i 32452 | Orthocomplements form a du... |
| mdcompli 32453 | A condition equivalent to ... |
| dmdcompli 32454 | A condition equivalent to ... |
| mddmdin0i 32455 | If dual modular implies mo... |
| cdjreui 32456 | A member of the sum of dis... |
| cdj1i 32457 | Two ways to express " ` A ... |
| cdj3lem1 32458 | A property of " ` A ` and ... |
| cdj3lem2 32459 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32460 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32461 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32462 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32463 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32464 | Lemma for ~ cdj3i . The s... |
| cdj3i 32465 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32466 | (_This theorem is a dummy ... |
| sa-abvi 32467 | A theorem about the univer... |
| xfree 32468 | A partial converse to ~ 19... |
| xfree2 32469 | A partial converse to ~ 19... |
| addltmulALT 32470 | A proof readability experi... |
| ad11antr 32471 | Deduction adding 11 conjun... |
| simp-12l 32472 | Simplification of a conjun... |
| simp-12r 32473 | Simplification of a conjun... |
| an52ds 32474 | Inference exchanging the l... |
| an62ds 32475 | Inference exchanging the l... |
| an72ds 32476 | Inference exchanging the l... |
| an82ds 32477 | Inference exchanging the l... |
| syl22anbrc 32478 | Syllogism inference. (Con... |
| bian1d 32479 | Adding a superfluous conju... |
| bian1dOLD 32480 | Obsolete version of ~ bian... |
| orim12da 32481 | Deduce a disjunction from ... |
| or3di 32482 | Distributive law for disju... |
| or3dir 32483 | Distributive law for disju... |
| 3o1cs 32484 | Deduction eliminating disj... |
| 3o2cs 32485 | Deduction eliminating disj... |
| 3o3cs 32486 | Deduction eliminating disj... |
| 13an22anass 32487 | Associative law for four c... |
| sbc2iedf 32488 | Conversion of implicit sub... |
| rspc2daf 32489 | Double restricted speciali... |
| ralcom4f 32490 | Commutation of restricted ... |
| rexcom4f 32491 | Commutation of restricted ... |
| 19.9d2rf 32492 | A deduction version of one... |
| 19.9d2r 32493 | A deduction version of one... |
| r19.29ffa 32494 | A commonly used pattern ba... |
| n0limd 32495 | Deduction rule for nonempt... |
| reu6dv 32496 | A condition which implies ... |
| eqtrb 32497 | A transposition of equalit... |
| eqelbid 32498 | A variable elimination law... |
| opsbc2ie 32499 | Conversion of implicit sub... |
| opreu2reuALT 32500 | Correspondence between uni... |
| 2reucom 32503 | Double restricted existent... |
| 2reu2rex1 32504 | Double restricted existent... |
| 2reureurex 32505 | Double restricted existent... |
| 2reu2reu2 32506 | Double restricted existent... |
| opreu2reu1 32507 | Equivalent definition of t... |
| sq2reunnltb 32508 | There exists a unique deco... |
| addsqnot2reu 32509 | For each complex number ` ... |
| sbceqbidf 32510 | Equality theorem for class... |
| sbcies 32511 | A special version of class... |
| mo5f 32512 | Alternate definition of "a... |
| nmo 32513 | Negation of "at most one".... |
| reuxfrdf 32514 | Transfer existential uniqu... |
| rexunirn 32515 | Restricted existential qua... |
| rmoxfrd 32516 | Transfer "at most one" res... |
| rmoun 32517 | "At most one" restricted e... |
| rmounid 32518 | A case where an "at most o... |
| riotaeqbidva 32519 | Equivalent wff's yield equ... |
| dmrab 32520 | Domain of a restricted cla... |
| difrab2 32521 | Difference of two restrict... |
| elrabrd 32522 | Deduction version of ~ elr... |
| rabexgfGS 32523 | Separation Scheme in terms... |
| rabsnel 32524 | Truth implied by equality ... |
| rabsspr 32525 | Conditions for a restricte... |
| rabsstp 32526 | Conditions for a restricte... |
| 3unrab 32527 | Union of three restricted ... |
| foresf1o 32528 | From a surjective function... |
| rabfodom 32529 | Domination relation for re... |
| rabrexfi 32530 | Conditions for a class abs... |
| abrexdomjm 32531 | An indexed set is dominate... |
| abrexdom2jm 32532 | An indexed set is dominate... |
| abrexexd 32533 | Existence of a class abstr... |
| elabreximd 32534 | Class substitution in an i... |
| elabreximdv 32535 | Class substitution in an i... |
| abrexss 32536 | A necessary condition for ... |
| nelun 32537 | Negated membership for a u... |
| snsssng 32538 | If a singleton is a subset... |
| n0nsnel 32539 | If a class with one elemen... |
| inin 32540 | Intersection with an inter... |
| difininv 32541 | Condition for the intersec... |
| difeq 32542 | Rewriting an equation with... |
| eqdif 32543 | If both set differences of... |
| indifbi 32544 | Two ways to express equali... |
| diffib 32545 | Case where ~ diffi is a bi... |
| difxp1ss 32546 | Difference law for Cartesi... |
| difxp2ss 32547 | Difference law for Cartesi... |
| indifundif 32548 | A remarkable equation with... |
| elpwincl1 32549 | Closure of intersection wi... |
| elpwdifcl 32550 | Closure of class differenc... |
| elpwiuncl 32551 | Closure of indexed union w... |
| elpreq 32552 | Equality wihin a pair. (C... |
| prssad 32553 | If a pair is a subset of a... |
| prssbd 32554 | If a pair is a subset of a... |
| nelpr 32555 | A set ` A ` not in a pair ... |
| inpr0 32556 | Rewrite an empty intersect... |
| neldifpr1 32557 | The first element of a pai... |
| neldifpr2 32558 | The second element of a pa... |
| unidifsnel 32559 | The other element of a pai... |
| unidifsnne 32560 | The other element of a pai... |
| tpssg 32561 | An unordered triple of ele... |
| tpssd 32562 | Deduction version of tpssi... |
| tpssad 32563 | If an ordered triple is a ... |
| tpssbd 32564 | If an ordered triple is a ... |
| tpsscd 32565 | If an ordered triple is a ... |
| ifeqeqx 32566 | An equality theorem tailor... |
| elimifd 32567 | Elimination of a condition... |
| elim2if 32568 | Elimination of two conditi... |
| elim2ifim 32569 | Elimination of two conditi... |
| ifeq3da 32570 | Given an expression ` C ` ... |
| ifnetrue 32571 | Deduce truth from a condit... |
| ifnefals 32572 | Deduce falsehood from a co... |
| ifnebib 32573 | The converse of ~ ifbi hol... |
| uniinn0 32574 | Sufficient and necessary c... |
| uniin1 32575 | Union of intersection. Ge... |
| uniin2 32576 | Union of intersection. Ge... |
| difuncomp 32577 | Express a class difference... |
| elpwunicl 32578 | Closure of a set union wit... |
| cbviunf 32579 | Rule used to change the bo... |
| iuneq12daf 32580 | Equality deduction for ind... |
| iunin1f 32581 | Indexed union of intersect... |
| ssiun3 32582 | Subset equivalence for an ... |
| ssiun2sf 32583 | Subset relationship for an... |
| iuninc 32584 | The union of an increasing... |
| iundifdifd 32585 | The intersection of a set ... |
| iundifdif 32586 | The intersection of a set ... |
| iunrdx 32587 | Re-index an indexed union.... |
| iunpreima 32588 | Preimage of an indexed uni... |
| iunrnmptss 32589 | A subset relation for an i... |
| iunxunsn 32590 | Appending a set to an inde... |
| iunxunpr 32591 | Appending two sets to an i... |
| iunxpssiun1 32592 | Provide an upper bound for... |
| iinabrex 32593 | Rewriting an indexed inter... |
| disjnf 32594 | In case ` x ` is not free ... |
| cbvdisjf 32595 | Change bound variables in ... |
| disjss1f 32596 | A subset of a disjoint col... |
| disjeq1f 32597 | Equality theorem for disjo... |
| disjxun0 32598 | Simplify a disjoint union.... |
| disjdifprg 32599 | A trivial partition into a... |
| disjdifprg2 32600 | A trivial partition of a s... |
| disji2f 32601 | Property of a disjoint col... |
| disjif 32602 | Property of a disjoint col... |
| disjorf 32603 | Two ways to say that a col... |
| disjorsf 32604 | Two ways to say that a col... |
| disjif2 32605 | Property of a disjoint col... |
| disjabrex 32606 | Rewriting a disjoint colle... |
| disjabrexf 32607 | Rewriting a disjoint colle... |
| disjpreima 32608 | A preimage of a disjoint s... |
| disjrnmpt 32609 | Rewriting a disjoint colle... |
| disjin 32610 | If a collection is disjoin... |
| disjin2 32611 | If a collection is disjoin... |
| disjxpin 32612 | Derive a disjunction over ... |
| iundisjf 32613 | Rewrite a countable union ... |
| iundisj2f 32614 | A disjoint union is disjoi... |
| disjrdx 32615 | Re-index a disjunct collec... |
| disjex 32616 | Two ways to say that two c... |
| disjexc 32617 | A variant of ~ disjex , ap... |
| disjunsn 32618 | Append an element to a dis... |
| disjun0 32619 | Adding the empty element p... |
| disjiunel 32620 | A set of elements B of a d... |
| disjuniel 32621 | A set of elements B of a d... |
| xpdisjres 32622 | Restriction of a constant ... |
| opeldifid 32623 | Ordered pair elementhood o... |
| difres 32624 | Case when class difference... |
| imadifxp 32625 | Image of the difference wi... |
| relfi 32626 | A relation (set) is finite... |
| 0res 32627 | Restriction of the empty f... |
| fcoinver 32628 | Build an equivalence relat... |
| fcoinvbr 32629 | Binary relation for the eq... |
| breq1dd 32630 | Equality deduction for a b... |
| breq2dd 32631 | Equality deduction for a b... |
| brab2d 32632 | Expressing that two sets a... |
| brabgaf 32633 | The law of concretion for ... |
| brelg 32634 | Two things in a binary rel... |
| br8d 32635 | Substitution for an eight-... |
| fnfvor 32636 | Relation between two funct... |
| ofrco 32637 | Function relation between ... |
| opabdm 32638 | Domain of an ordered-pair ... |
| opabrn 32639 | Range of an ordered-pair c... |
| opabssi 32640 | Sufficient condition for a... |
| opabid2ss 32641 | One direction of ~ opabid2... |
| ssrelf 32642 | A subclass relationship de... |
| eqrelrd2 32643 | A version of ~ eqrelrdv2 w... |
| erbr3b 32644 | Biconditional for equivale... |
| iunsnima 32645 | Image of a singleton by an... |
| iunsnima2 32646 | Version of ~ iunsnima with... |
| fconst7v 32647 | An alternative way to expr... |
| constcof 32648 | Composition with a constan... |
| ac6sf2 32649 | Alternate version of ~ ac6... |
| ac6mapd 32650 | Axiom of choice equivalent... |
| fnresin 32651 | Restriction of a function ... |
| fresunsn 32652 | Recover the original funct... |
| f1o3d 32653 | Describe an implicit one-t... |
| eldmne0 32654 | A function of nonempty dom... |
| f1rnen 32655 | Equinumerosity of the rang... |
| f1oeq3dd 32656 | Equality deduction for one... |
| rinvf1o 32657 | Sufficient conditions for ... |
| fresf1o 32658 | Conditions for a restricti... |
| nfpconfp 32659 | The set of fixed points of... |
| fmptco1f1o 32660 | The action of composing (t... |
| cofmpt2 32661 | Express composition of a m... |
| f1mptrn 32662 | Express injection for a ma... |
| dfimafnf 32663 | Alternate definition of th... |
| funimass4f 32664 | Membership relation for th... |
| suppss2f 32665 | Show that the support of a... |
| ofrn 32666 | The range of the function ... |
| ofrn2 32667 | The range of the function ... |
| off2 32668 | The function operation pro... |
| ofresid 32669 | Applying an operation rest... |
| unipreima 32670 | Preimage of a class union.... |
| opfv 32671 | Value of a function produc... |
| xppreima 32672 | The preimage of a Cartesia... |
| 2ndimaxp 32673 | Image of a cartesian produ... |
| dmdju 32674 | Domain of a disjoint union... |
| djussxp2 32675 | Stronger version of ~ djus... |
| 2ndresdju 32676 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32677 | The ` 2nd ` function restr... |
| xppreima2 32678 | The preimage of a Cartesia... |
| abfmpunirn 32679 | Membership in a union of a... |
| rabfmpunirn 32680 | Membership in a union of a... |
| abfmpeld 32681 | Membership in an element o... |
| abfmpel 32682 | Membership in an element o... |
| fmptdF 32683 | Domain and codomain of the... |
| fmptcof2 32684 | Composition of two functio... |
| fcomptf 32685 | Express composition of two... |
| acunirnmpt 32686 | Axiom of choice for the un... |
| acunirnmpt2 32687 | Axiom of choice for the un... |
| acunirnmpt2f 32688 | Axiom of choice for the un... |
| aciunf1lem 32689 | Choice in an index union. ... |
| aciunf1 32690 | Choice in an index union. ... |
| ofoprabco 32691 | Function operation as a co... |
| ofpreima 32692 | Express the preimage of a ... |
| ofpreima2 32693 | Express the preimage of a ... |
| funcnvmpt 32694 | Condition for a function i... |
| funcnv5mpt 32695 | Two ways to say that a fun... |
| funcnv4mpt 32696 | Two ways to say that a fun... |
| preimane 32697 | Different elements have di... |
| fnpreimac 32698 | Choose a set ` x ` contain... |
| fgreu 32699 | Exactly one point of a fun... |
| fcnvgreu 32700 | If the converse of a relat... |
| rnmposs 32701 | The range of an operation ... |
| mptssALT 32702 | Deduce subset relation of ... |
| dfcnv2 32703 | Alternative definition of ... |
| partfun2 32704 | Rewrite a function defined... |
| rnressnsn 32705 | The range of a restriction... |
| mpomptxf 32706 | Express a two-argument fun... |
| of0r 32707 | Function operation with th... |
| elmaprd 32708 | Deduction associated with ... |
| suppovss 32709 | A bound for the support of... |
| elsuppfnd 32710 | Deduce membership in the s... |
| fisuppov1 32711 | Formula building theorem f... |
| suppun2 32712 | The support of a union is ... |
| fdifsupp 32713 | Express the support of a f... |
| suppiniseg 32714 | Relation between the suppo... |
| fsuppinisegfi 32715 | The initial segment ` ( ``... |
| fressupp 32716 | The restriction of a funct... |
| fdifsuppconst 32717 | A function is a zero const... |
| ressupprn 32718 | The range of a function re... |
| supppreima 32719 | Express the support of a f... |
| fsupprnfi 32720 | Finite support implies fin... |
| mptiffisupp 32721 | Conditions for a mapping f... |
| cosnopne 32722 | Composition of two ordered... |
| cosnop 32723 | Composition of two ordered... |
| cnvprop 32724 | Converse of a pair of orde... |
| brprop 32725 | Binary relation for a pair... |
| mptprop 32726 | Rewrite pairs of ordered p... |
| coprprop 32727 | Composition of two pairs o... |
| fmptunsnop 32728 | Two ways to express a func... |
| gtiso 32729 | Two ways to write a strict... |
| isoun 32730 | Infer an isomorphism from ... |
| disjdsct 32731 | A disjoint collection is d... |
| df1stres 32732 | Definition for a restricti... |
| df2ndres 32733 | Definition for a restricti... |
| 1stpreimas 32734 | The preimage of a singleto... |
| 1stpreima 32735 | The preimage by ` 1st ` is... |
| 2ndpreima 32736 | The preimage by ` 2nd ` is... |
| curry2ima 32737 | The image of a curried fun... |
| preiman0 32738 | The preimage of a nonempty... |
| intimafv 32739 | The intersection of an ima... |
| snct 32740 | A singleton is countable. ... |
| prct 32741 | An unordered pair is count... |
| mpocti 32742 | An operation is countable ... |
| abrexct 32743 | An image set of a countabl... |
| mptctf 32744 | A countable mapping set is... |
| abrexctf 32745 | An image set of a countabl... |
| padct 32746 | Index a countable set with... |
| f1od2 32747 | Sufficient condition for a... |
| fcobij 32748 | Composing functions with a... |
| fcobijfs 32749 | Composing finitely support... |
| fcobijfs2 32750 | Composing finitely support... |
| suppss3 32751 | Deduce a function's suppor... |
| fsuppcurry1 32752 | Finite support of a currie... |
| fsuppcurry2 32753 | Finite support of a currie... |
| offinsupp1 32754 | Finite support for a funct... |
| ffs2 32755 | Rewrite a function's suppo... |
| ffsrn 32756 | The range of a finitely su... |
| cocnvf1o 32757 | Composing with the inverse... |
| resf1o 32758 | Restriction of functions t... |
| maprnin 32759 | Restricting the range of t... |
| fpwrelmapffslem 32760 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32761 | Define a canonical mapping... |
| fpwrelmapffs 32762 | Define a canonical mapping... |
| sgnval2 32763 | Value of the signum of a r... |
| creq0 32764 | The real representation of... |
| 1nei 32765 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32766 | An integer unit times itse... |
| nnmulge 32767 | Multiplying by a positive ... |
| submuladdd 32768 | The product of a differenc... |
| muldivdid 32769 | Distribution of division o... |
| binom2subadd 32770 | The difference of the squa... |
| cjsubd 32771 | Complex conjugate distribu... |
| re0cj 32772 | The conjugate of a pure im... |
| receqid 32773 | Real numbers equal to thei... |
| pythagreim 32774 | A simplified version of th... |
| efiargd 32775 | The exponential of the "ar... |
| arginv 32776 | The argument of the invers... |
| argcj 32777 | The argument of the conjug... |
| quad3d 32778 | Variant of quadratic equat... |
| lt2addrd 32779 | If the right-hand side of ... |
| nn0mnfxrd 32780 | Nonnegative integers or mi... |
| xrlelttric 32781 | Trichotomy law for extende... |
| xaddeq0 32782 | Two extended reals which a... |
| rexmul2 32783 | If the result ` A ` of an ... |
| xrinfm 32784 | The extended real numbers ... |
| le2halvesd 32785 | A sum is less than the who... |
| xraddge02 32786 | A number is less than or e... |
| xrge0addge 32787 | A number is less than or e... |
| xlt2addrd 32788 | If the right-hand side of ... |
| xrge0infss 32789 | Any subset of nonnegative ... |
| xrge0infssd 32790 | Inequality deduction for i... |
| xrge0addcld 32791 | Nonnegative extended reals... |
| xrge0subcld 32792 | Condition for closure of n... |
| infxrge0lb 32793 | A member of a set of nonne... |
| infxrge0glb 32794 | The infimum of a set of no... |
| infxrge0gelb 32795 | The infimum of a set of no... |
| xrofsup 32796 | The supremum is preserved ... |
| supxrnemnf 32797 | The supremum of a nonempty... |
| xnn0gt0 32798 | Nonzero extended nonnegati... |
| xnn01gt 32799 | An extended nonnegative in... |
| nn0xmulclb 32800 | Finite multiplication in t... |
| xnn0nn0d 32801 | Conditions for an extended... |
| xnn0nnd 32802 | Conditions for an extended... |
| joiniooico 32803 | Disjoint joining an open i... |
| ubico 32804 | A right-open interval does... |
| xeqlelt 32805 | Equality in terms of 'less... |
| eliccelico 32806 | Relate elementhood to a cl... |
| elicoelioo 32807 | Relate elementhood to a cl... |
| iocinioc2 32808 | Intersection between two o... |
| xrdifh 32809 | Class difference of a half... |
| iocinif 32810 | Relate intersection of two... |
| difioo 32811 | The difference between two... |
| difico 32812 | The difference between two... |
| uzssico 32813 | Upper integer sets are a s... |
| fz2ssnn0 32814 | A finite set of sequential... |
| nndiffz1 32815 | Upper set of the positive ... |
| ssnnssfz 32816 | For any finite subset of `... |
| fzm1ne1 32817 | Elementhood of an integer ... |
| fzspl 32818 | Split the last element of ... |
| fzdif2 32819 | Split the last element of ... |
| fzodif2 32820 | Split the last element of ... |
| fzodif1 32821 | Set difference of two half... |
| fzsplit3 32822 | Split a finite interval of... |
| nn0diffz0 32823 | Upper set of the nonnegati... |
| bcm1n 32824 | The proportion of one bino... |
| iundisjfi 32825 | Rewrite a countable union ... |
| iundisj2fi 32826 | A disjoint union is disjoi... |
| iundisjcnt 32827 | Rewrite a countable union ... |
| iundisj2cnt 32828 | A countable disjoint union... |
| f1ocnt 32829 | Given a countable set ` A ... |
| fz1nnct 32830 | NN and integer ranges star... |
| fz1nntr 32831 | NN and integer ranges star... |
| fzo0opth 32832 | Equality for a half open i... |
| nn0difffzod 32833 | A nonnegative integer that... |
| suppssnn0 32834 | Show that the support of a... |
| hashunif 32835 | The cardinality of a disjo... |
| hashxpe 32836 | The size of the Cartesian ... |
| hashgt1 32837 | Restate "set contains at l... |
| hashpss 32838 | The size of a proper subse... |
| hashne0 32839 | Deduce that the size of a ... |
| hashimaf1 32840 | Taking the image of a set ... |
| elq2 32841 | Elementhood in the rationa... |
| znumd 32842 | Numerator of an integer. ... |
| zdend 32843 | Denominator of an integer.... |
| numdenneg 32844 | Numerator and denominator ... |
| divnumden2 32845 | Calculate the reduced form... |
| expgt0b 32846 | A real number ` A ` raised... |
| nn0split01 32847 | Split 0 and 1 from the non... |
| nn0disj01 32848 | The pair ` { 0 , 1 } ` doe... |
| nnindf 32849 | Principle of Mathematical ... |
| nn0min 32850 | Extracting the minimum pos... |
| subne0nn 32851 | A nonnegative difference i... |
| ltesubnnd 32852 | Subtracting an integer num... |
| fprodeq02 32853 | If one of the factors is z... |
| pr01ssre 32854 | The range of the indicator... |
| fprodex01 32855 | A product of factors equal... |
| prodpr 32856 | A product over a pair is t... |
| prodtp 32857 | A product over a triple is... |
| fsumub 32858 | An upper bound for a term ... |
| fsumiunle 32859 | Upper bound for a sum of n... |
| dfdec100 32860 | Split the hundreds from a ... |
| sgncl 32861 | Closure of the signum. (C... |
| sgnclre 32862 | Closure of the signum. (C... |
| sgnneg 32863 | Negation of the signum. (... |
| sgn3da 32864 | A conditional containing a... |
| sgnmul 32865 | Signum of a product. (Con... |
| sgnmulrp2 32866 | Multiplication by a positi... |
| sgnsub 32867 | Subtraction of a number of... |
| sgnnbi 32868 | Negative signum. (Contrib... |
| sgnpbi 32869 | Positive signum. (Contrib... |
| sgn0bi 32870 | Zero signum. (Contributed... |
| sgnsgn 32871 | Signum is idempotent. (Co... |
| sgnmulsgn 32872 | If two real numbers are of... |
| sgnmulsgp 32873 | If two real numbers are of... |
| nexple 32874 | A lower bound for an expon... |
| 2exple2exp 32875 | If a nonnegative integer `... |
| expevenpos 32876 | Even powers are positive. ... |
| oexpled 32877 | Odd power monomials are mo... |
| indv 32880 | Value of the indicator fun... |
| indval 32881 | Value of the indicator fun... |
| indval2 32882 | Alternate value of the ind... |
| indf 32883 | An indicator function as a... |
| indfval 32884 | Value of the indicator fun... |
| ind1 32885 | Value of the indicator fun... |
| ind0 32886 | Value of the indicator fun... |
| ind1a 32887 | Value of the indicator fun... |
| indconst0 32888 | Indicator of the empty set... |
| indconst1 32889 | Indicator of the whole set... |
| indpi1 32890 | Preimage of the singleton ... |
| indsum 32891 | Finite sum of a product wi... |
| indsumin 32892 | Finite sum of a product wi... |
| prodindf 32893 | The product of indicators ... |
| indsn 32894 | The indicator function of ... |
| indf1o 32895 | The bijection between a po... |
| indpreima 32896 | A function with range ` { ... |
| indf1ofs 32897 | The bijection between fini... |
| indsupp 32898 | The support of the indicat... |
| indfsd 32899 | The indicator function of ... |
| indfsid 32900 | Conditions for a function ... |
| dp2eq1 32903 | Equality theorem for the d... |
| dp2eq2 32904 | Equality theorem for the d... |
| dp2eq1i 32905 | Equality theorem for the d... |
| dp2eq2i 32906 | Equality theorem for the d... |
| dp2eq12i 32907 | Equality theorem for the d... |
| dp20u 32908 | Add a zero in the tenths (... |
| dp20h 32909 | Add a zero in the unit pla... |
| dp2cl 32910 | Closure for the decimal fr... |
| dp2clq 32911 | Closure for a decimal frac... |
| rpdp2cl 32912 | Closure for a decimal frac... |
| rpdp2cl2 32913 | Closure for a decimal frac... |
| dp2lt10 32914 | Decimal fraction builds re... |
| dp2lt 32915 | Comparing two decimal frac... |
| dp2ltsuc 32916 | Comparing a decimal fracti... |
| dp2ltc 32917 | Comparing two decimal expa... |
| dpval 32920 | Define the value of the de... |
| dpcl 32921 | Prove that the closure of ... |
| dpfrac1 32922 | Prove a simple equivalence... |
| dpval2 32923 | Value of the decimal point... |
| dpval3 32924 | Value of the decimal point... |
| dpmul10 32925 | Multiply by 10 a decimal e... |
| decdiv10 32926 | Divide a decimal number by... |
| dpmul100 32927 | Multiply by 100 a decimal ... |
| dp3mul10 32928 | Multiply by 10 a decimal e... |
| dpmul1000 32929 | Multiply by 1000 a decimal... |
| dpval3rp 32930 | Value of the decimal point... |
| dp0u 32931 | Add a zero in the tenths p... |
| dp0h 32932 | Remove a zero in the units... |
| rpdpcl 32933 | Closure of the decimal poi... |
| dplt 32934 | Comparing two decimal expa... |
| dplti 32935 | Comparing a decimal expans... |
| dpgti 32936 | Comparing a decimal expans... |
| dpltc 32937 | Comparing two decimal inte... |
| dpexpp1 32938 | Add one zero to the mantis... |
| 0dp2dp 32939 | Multiply by 10 a decimal e... |
| dpadd2 32940 | Addition with one decimal,... |
| dpadd 32941 | Addition with one decimal.... |
| dpadd3 32942 | Addition with two decimals... |
| dpmul 32943 | Multiplication with one de... |
| dpmul4 32944 | An upper bound to multipli... |
| threehalves 32945 | Example theorem demonstrat... |
| 1mhdrd 32946 | Example theorem demonstrat... |
| xdivval 32949 | Value of division: the (un... |
| xrecex 32950 | Existence of reciprocal of... |
| xmulcand 32951 | Cancellation law for exten... |
| xreceu 32952 | Existential uniqueness of ... |
| xdivcld 32953 | Closure law for the extend... |
| xdivcl 32954 | Closure law for the extend... |
| xdivmul 32955 | Relationship between divis... |
| rexdiv 32956 | The extended real division... |
| xdivrec 32957 | Relationship between divis... |
| xdivid 32958 | A number divided by itself... |
| xdiv0 32959 | Division into zero is zero... |
| xdiv0rp 32960 | Division into zero is zero... |
| eliccioo 32961 | Membership in a closed int... |
| elxrge02 32962 | Elementhood in the set of ... |
| xdivpnfrp 32963 | Plus infinity divided by a... |
| rpxdivcld 32964 | Closure law for extended d... |
| xrpxdivcld 32965 | Closure law for extended d... |
| wrdres 32966 | Condition for the restrict... |
| wrdsplex 32967 | Existence of a split of a ... |
| wrdfsupp 32968 | A word has finite support.... |
| wrdpmcl 32969 | Closure of a word with per... |
| pfx1s2 32970 | The prefix of length 1 of ... |
| pfxrn2 32971 | The range of a prefix of a... |
| pfxrn3 32972 | Express the range of a pre... |
| pfxf1 32973 | Condition for a prefix to ... |
| s1f1 32974 | Conditions for a length 1 ... |
| s2rnOLD 32975 | Obsolete version of ~ s2rn... |
| s2f1 32976 | Conditions for a length 2 ... |
| s3rnOLD 32977 | Obsolete version of ~ s2rn... |
| s3f1 32978 | Conditions for a length 3 ... |
| s3clhash 32979 | Closure of the words of le... |
| ccatf1 32980 | Conditions for a concatena... |
| pfxlsw2ccat 32981 | Reconstruct a word from it... |
| ccatws1f1o 32982 | Conditions for the concate... |
| ccatws1f1olast 32983 | Two ways to reorder symbol... |
| wrdt2ind 32984 | Perform an induction over ... |
| swrdrn2 32985 | The range of a subword is ... |
| swrdrn3 32986 | Express the range of a sub... |
| swrdf1 32987 | Condition for a subword to... |
| swrdrndisj 32988 | Condition for the range of... |
| splfv3 32989 | Symbols to the right of a ... |
| 1cshid 32990 | Cyclically shifting a sing... |
| cshw1s2 32991 | Cyclically shifting a leng... |
| cshwrnid 32992 | Cyclically shifting a word... |
| cshf1o 32993 | Condition for the cyclic s... |
| ressplusf 32994 | The group operation functi... |
| ressnm 32995 | The norm in a restricted s... |
| abvpropd2 32996 | Weaker version of ~ abvpro... |
| ressprs 32997 | The restriction of a prose... |
| posrasymb 32998 | A poset ordering is asymet... |
| odutos 32999 | Being a toset is a self-du... |
| tlt2 33000 | In a Toset, two elements m... |
| tlt3 33001 | In a Toset, two elements m... |
| trleile 33002 | In a Toset, two elements m... |
| toslublem 33003 | Lemma for ~ toslub and ~ x... |
| toslub 33004 | In a toset, the lowest upp... |
| tosglblem 33005 | Lemma for ~ tosglb and ~ x... |
| tosglb 33006 | Same theorem as ~ toslub ,... |
| clatp0cl 33007 | The poset zero of a comple... |
| clatp1cl 33008 | The poset one of a complet... |
| mntoval 33013 | Operation value of the mon... |
| ismnt 33014 | Express the statement " ` ... |
| ismntd 33015 | Property of being a monoto... |
| mntf 33016 | A monotone function is a f... |
| mgcoval 33017 | Operation value of the mon... |
| mgcval 33018 | Monotone Galois connection... |
| mgcf1 33019 | The lower adjoint ` F ` of... |
| mgcf2 33020 | The upper adjoint ` G ` of... |
| mgccole1 33021 | An inequality for the kern... |
| mgccole2 33022 | Inequality for the closure... |
| mgcmnt1 33023 | The lower adjoint ` F ` of... |
| mgcmnt2 33024 | The upper adjoint ` G ` of... |
| mgcmntco 33025 | A Galois connection like s... |
| dfmgc2lem 33026 | Lemma for dfmgc2, backward... |
| dfmgc2 33027 | Alternate definition of th... |
| mgcmnt1d 33028 | Galois connection implies ... |
| mgcmnt2d 33029 | Galois connection implies ... |
| mgccnv 33030 | The inverse Galois connect... |
| pwrssmgc 33031 | Given a function ` F ` , e... |
| mgcf1olem1 33032 | Property of a Galois conne... |
| mgcf1olem2 33033 | Property of a Galois conne... |
| mgcf1o 33034 | Given a Galois connection,... |
| xrs0 33037 | The zero of the extended r... |
| xrslt 33038 | The "strictly less than" r... |
| xrsinvgval 33039 | The inversion operation in... |
| xrsmulgzz 33040 | The "multiple" function in... |
| xrstos 33041 | The extended real numbers ... |
| xrsclat 33042 | The extended real numbers ... |
| xrsp0 33043 | The poset 0 of the extende... |
| xrsp1 33044 | The poset 1 of the extende... |
| xrge00 33045 | The zero of the extended n... |
| xrge0mulgnn0 33046 | The group multiple functio... |
| xrge0addass 33047 | Associativity of extended ... |
| xrge0addgt0 33048 | The sum of nonnegative and... |
| xrge0adddir 33049 | Right-distributivity of ex... |
| xrge0adddi 33050 | Left-distributivity of ext... |
| xrge0npcan 33051 | Extended nonnegative real ... |
| fsumrp0cl 33052 | Closure of a finite sum of... |
| mndcld 33053 | Closure of the operation o... |
| mndassd 33054 | A monoid operation is asso... |
| mndlrinv 33055 | In a monoid, if an element... |
| mndlrinvb 33056 | In a monoid, if an element... |
| mndlactf1 33057 | If an element ` X ` of a m... |
| mndlactfo 33058 | An element ` X ` of a mono... |
| mndractf1 33059 | If an element ` X ` of a m... |
| mndractfo 33060 | An element ` X ` of a mono... |
| mndlactf1o 33061 | An element ` X ` of a mono... |
| mndractf1o 33062 | An element ` X ` of a mono... |
| cmn4d 33063 | Commutative/associative la... |
| cmn246135 33064 | Rearrange terms in a commu... |
| cmn145236 33065 | Rearrange terms in a commu... |
| submcld 33066 | Submonoids are closed unde... |
| abliso 33067 | The image of an Abelian gr... |
| lmhmghmd 33068 | A module homomorphism is a... |
| mhmimasplusg 33069 | Value of the operation of ... |
| lmhmimasvsca 33070 | Value of the scalar produc... |
| grpinvinvd 33071 | Double inverse law for gro... |
| grpsubcld 33072 | Closure of group subtracti... |
| subgcld 33073 | A subgroup is closed under... |
| subgsubcld 33074 | A subgroup is closed under... |
| subgmulgcld 33075 | Closure of the group multi... |
| ressmulgnn0d 33076 | Values for the group multi... |
| ablcomd 33077 | An abelian group operation... |
| gsumsubg 33078 | The group sum in a subgrou... |
| gsumsra 33079 | The group sum in a subring... |
| gsummpt2co 33080 | Split a finite sum into a ... |
| gsummpt2d 33081 | Express a finite sum over ... |
| lmodvslmhm 33082 | Scalar multiplication in a... |
| gsumvsmul1 33083 | Pull a scalar multiplicati... |
| gsummptres 33084 | Extend a finite group sum ... |
| gsummptres2 33085 | Extend a finite group sum ... |
| gsummptfsres 33086 | Extend a finitely supporte... |
| gsummptf1od 33087 | Re-index a finite group su... |
| gsummptrev 33088 | Revert ordering in a group... |
| gsummptp1 33089 | Reindex a zero-based sum a... |
| gsummptfzsplitra 33090 | Split a group sum expresse... |
| gsummptfzsplitla 33091 | Split a group sum expresse... |
| gsummptfsf1o 33092 | Re-index a finite group su... |
| gsumfs2d 33093 | Express a finite sum over ... |
| gsumzresunsn 33094 | Append an element to a fin... |
| gsumpart 33095 | Express a group sum as a d... |
| gsumtp 33096 | Group sum of an unordered ... |
| gsumzrsum 33097 | Relate a group sum on ` ZZ... |
| gsummulgc2 33098 | A finite group sum multipl... |
| gsumhashmul 33099 | Express a group sum by gro... |
| gsummulsubdishift1 33100 | Distribute a subtraction o... |
| gsummulsubdishift2 33101 | Distribute a subtraction o... |
| gsummulsubdishift1s 33102 | Distribute a subtraction o... |
| gsummulsubdishift2s 33103 | Distribute a subtraction o... |
| xrge0tsmsd 33104 | Any finite or infinite sum... |
| xrge0tsmsbi 33105 | Any limit of a finite or i... |
| xrge0tsmseq 33106 | Any limit of a finite or i... |
| gsumwun 33107 | In a commutative ring, a g... |
| gsumwrd2dccatlem 33108 | Lemma for ~ gsumwrd2dccat ... |
| gsumwrd2dccat 33109 | Rewrite a sum ranging over... |
| cntzun 33110 | The centralizer of a union... |
| cntzsnid 33111 | The centralizer of the ide... |
| cntrcrng 33112 | The center of a ring is a ... |
| symgfcoeu 33113 | Uniqueness property of per... |
| symgcom 33114 | Two permutations ` X ` and... |
| symgcom2 33115 | Two permutations ` X ` and... |
| symgcntz 33116 | All elements of a (finite)... |
| odpmco 33117 | The composition of two odd... |
| symgsubg 33118 | The value of the group sub... |
| pmtrprfv2 33119 | In a transposition of two ... |
| pmtrcnel 33120 | Composing a permutation ` ... |
| pmtrcnel2 33121 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33122 | Composing a permutation ` ... |
| fzo0pmtrlast 33123 | Reorder a half-open intege... |
| wrdpmtrlast 33124 | Reorder a word, so that th... |
| pmtridf1o 33125 | Transpositions of ` X ` an... |
| pmtridfv1 33126 | Value at X of the transpos... |
| pmtridfv2 33127 | Value at Y of the transpos... |
| psgnid 33128 | Permutation sign of the id... |
| psgndmfi 33129 | For a finite base set, the... |
| pmtrto1cl 33130 | Useful lemma for the follo... |
| psgnfzto1stlem 33131 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33132 | Value of our permutation `... |
| fzto1st1 33133 | Special case where the per... |
| fzto1st 33134 | The function moving one el... |
| fzto1stinvn 33135 | Value of the inverse of ou... |
| psgnfzto1st 33136 | The permutation sign for m... |
| tocycval 33139 | Value of the cycle builder... |
| tocycfv 33140 | Function value of a permut... |
| tocycfvres1 33141 | A cyclic permutation is a ... |
| tocycfvres2 33142 | A cyclic permutation is th... |
| cycpmfvlem 33143 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33144 | Value of a cycle function ... |
| cycpmfv2 33145 | Value of a cycle function ... |
| cycpmfv3 33146 | Values outside of the orbi... |
| cycpmcl 33147 | Cyclic permutations are pe... |
| tocycf 33148 | The permutation cycle buil... |
| tocyc01 33149 | Permutation cycles built f... |
| cycpm2tr 33150 | A cyclic permutation of 2 ... |
| cycpm2cl 33151 | Closure for the 2-cycles. ... |
| cyc2fv1 33152 | Function value of a 2-cycl... |
| cyc2fv2 33153 | Function value of a 2-cycl... |
| trsp2cyc 33154 | Exhibit the word a transpo... |
| cycpmco2f1 33155 | The word U used in ~ cycpm... |
| cycpmco2rn 33156 | The orbit of the compositi... |
| cycpmco2lem1 33157 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33158 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33159 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33160 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33161 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33162 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33163 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33164 | The composition of a cycli... |
| cyc2fvx 33165 | Function value of a 2-cycl... |
| cycpm3cl 33166 | Closure of the 3-cycles in... |
| cycpm3cl2 33167 | Closure of the 3-cycles in... |
| cyc3fv1 33168 | Function value of a 3-cycl... |
| cyc3fv2 33169 | Function value of a 3-cycl... |
| cyc3fv3 33170 | Function value of a 3-cycl... |
| cyc3co2 33171 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33172 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33173 | A formula for computing co... |
| cycpmrn 33174 | The range of the word used... |
| tocyccntz 33175 | All elements of a (finite)... |
| evpmval 33176 | Value of the set of even p... |
| cnmsgn0g 33177 | The neutral element of the... |
| evpmsubg 33178 | The alternating group is a... |
| evpmid 33179 | The identity is an even pe... |
| altgnsg 33180 | The alternating group ` ( ... |
| cyc3evpm 33181 | 3-Cycles are even permutat... |
| cyc3genpmlem 33182 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33183 | The alternating group ` A ... |
| cycpmgcl 33184 | Cyclic permutations are pe... |
| cycpmconjslem1 33185 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33186 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33187 | All cycles of the same len... |
| cyc3conja 33188 | All 3-cycles are conjugate... |
| sgnsv 33191 | The sign mapping. (Contri... |
| sgnsval 33192 | The sign value. (Contribu... |
| sgnsf 33193 | The sign function. (Contr... |
| fxpval 33196 | Value of the set of fixed ... |
| fxpss 33197 | The set of fixed points is... |
| fxpgaval 33198 | Value of the set of fixed ... |
| isfxp 33199 | Property of being a fixed ... |
| fxpgaeq 33200 | A fixed point ` X ` is inv... |
| conjga 33201 | Group conjugation induces ... |
| cntrval2 33202 | Express the center ` Z ` o... |
| fxpsubm 33203 | Provided the group action ... |
| fxpsubg 33204 | The fixed points of a grou... |
| fxpsubrg 33205 | The fixed points of a grou... |
| fxpsdrg 33206 | The fixed points of a grou... |
| inftmrel 33211 | The infinitesimal relation... |
| isinftm 33212 | Express ` x ` is infinites... |
| isarchi 33213 | Express the predicate " ` ... |
| pnfinf 33214 | Plus infinity is an infini... |
| xrnarchi 33215 | The completed real line is... |
| isarchi2 33216 | Alternative way to express... |
| submarchi 33217 | A submonoid is archimedean... |
| isarchi3 33218 | This is the usual definiti... |
| archirng 33219 | Property of Archimedean or... |
| archirngz 33220 | Property of Archimedean le... |
| archiexdiv 33221 | In an Archimedean group, g... |
| archiabllem1a 33222 | Lemma for ~ archiabl : In... |
| archiabllem1b 33223 | Lemma for ~ archiabl . (C... |
| archiabllem1 33224 | Archimedean ordered groups... |
| archiabllem2a 33225 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33226 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33227 | Lemma for ~ archiabl . (C... |
| archiabllem2 33228 | Archimedean ordered groups... |
| archiabl 33229 | Archimedean left- and righ... |
| isarchiofld 33230 | Axiom of Archimedes : a ch... |
| isslmd 33233 | The predicate "is a semimo... |
| slmdlema 33234 | Lemma for properties of a ... |
| lmodslmd 33235 | Left semimodules generaliz... |
| slmdcmn 33236 | A semimodule is a commutat... |
| slmdmnd 33237 | A semimodule is a monoid. ... |
| slmdsrg 33238 | The scalar component of a ... |
| slmdbn0 33239 | The base set of a semimodu... |
| slmdacl 33240 | Closure of ring addition f... |
| slmdmcl 33241 | Closure of ring multiplica... |
| slmdsn0 33242 | The set of scalars in a se... |
| slmdvacl 33243 | Closure of vector addition... |
| slmdass 33244 | Semiring left module vecto... |
| slmdvscl 33245 | Closure of scalar product ... |
| slmdvsdi 33246 | Distributive law for scala... |
| slmdvsdir 33247 | Distributive law for scala... |
| slmdvsass 33248 | Associative law for scalar... |
| slmd0cl 33249 | The ring zero in a semimod... |
| slmd1cl 33250 | The ring unity in a semiri... |
| slmdvs1 33251 | Scalar product with ring u... |
| slmd0vcl 33252 | The zero vector is a vecto... |
| slmd0vlid 33253 | Left identity law for the ... |
| slmd0vrid 33254 | Right identity law for the... |
| slmd0vs 33255 | Zero times a vector is the... |
| slmdvs0 33256 | Anything times the zero ve... |
| gsumvsca1 33257 | Scalar product of a finite... |
| gsumvsca2 33258 | Scalar product of a finite... |
| prmsimpcyc 33259 | A group of prime order is ... |
| ringrngd 33260 | A unital ring is a non-uni... |
| ringdi22 33261 | Expand the product of two ... |
| urpropd 33262 | Sufficient condition for r... |
| subrgmcld 33263 | A subring is closed under ... |
| ress1r 33264 | ` 1r ` is unaffected by re... |
| ringm1expp1 33265 | Ring exponentiation of min... |
| ringinvval 33266 | The ring inverse expressed... |
| dvrcan5 33267 | Cancellation law for commo... |
| subrgchr 33268 | If ` A ` is a subring of `... |
| rmfsupp2 33269 | A mapping of a multiplicat... |
| unitnz 33270 | In a nonzero ring, a unit ... |
| isunit2 33271 | Alternate definition of be... |
| isunit3 33272 | Alternate definition of be... |
| elrgspnlem1 33273 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem2 33274 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem3 33275 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem4 33276 | Lemma for ~ elrgspn . (Co... |
| elrgspn 33277 | Membership in the subring ... |
| elrgspnsubrunlem1 33278 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrunlem2 33279 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrun 33280 | Membership in the ring spa... |
| irrednzr 33281 | A ring with an irreducible... |
| 0ringsubrg 33282 | A subring of a zero ring i... |
| 0ringcring 33283 | The zero ring is commutati... |
| reldmrloc 33288 | Ring localization is a pro... |
| erlval 33289 | Value of the ring localiza... |
| rlocval 33290 | Expand the value of the ri... |
| erlcl1 33291 | Closure for the ring local... |
| erlcl2 33292 | Closure for the ring local... |
| erldi 33293 | Main property of the ring ... |
| erlbrd 33294 | Deduce the ring localizati... |
| erlbr2d 33295 | Deduce the ring localizati... |
| erler 33296 | The relation used to build... |
| elrlocbasi 33297 | Membership in the basis of... |
| rlocbas 33298 | The base set of a ring loc... |
| rlocaddval 33299 | Value of the addition in t... |
| rlocmulval 33300 | Value of the addition in t... |
| rloccring 33301 | The ring localization ` L ... |
| rloc0g 33302 | The zero of a ring localiz... |
| rloc1r 33303 | The multiplicative identit... |
| rlocf1 33304 | The embedding ` F ` of a r... |
| domnmuln0rd 33305 | In a domain, factors of a ... |
| domnprodn0 33306 | In a domain, a finite prod... |
| domnprodeq0 33307 | A product over a domain is... |
| domnpropd 33308 | If two structures have the... |
| idompropd 33309 | If two structures have the... |
| idomrcan 33310 | Right-cancellation law for... |
| domnlcanOLD 33311 | Obsolete version of ~ domn... |
| domnlcanbOLD 33312 | Obsolete version of ~ domn... |
| idomrcanOLD 33313 | Obsolete version of ~ idom... |
| 1rrg 33314 | The multiplicative identit... |
| rrgsubm 33315 | The left regular elements ... |
| subrdom 33316 | A subring of a domain is a... |
| subridom 33317 | A subring of an integral d... |
| subrfld 33318 | A subring of a field is an... |
| eufndx 33321 | Index value of the Euclide... |
| eufid 33322 | Utility theorem: index-ind... |
| ringinveu 33325 | If a ring unit element ` X... |
| isdrng4 33326 | A division ring is a ring ... |
| rndrhmcl 33327 | The image of a division ri... |
| qfld 33328 | The field of rational numb... |
| subsdrg 33329 | A subring of a sub-divisio... |
| sdrgdvcl 33330 | A sub-division-ring is clo... |
| sdrginvcl 33331 | A sub-division-ring is clo... |
| primefldchr 33332 | The characteristic of a pr... |
| fracval 33335 | Value of the field of frac... |
| fracbas 33336 | The base of the field of f... |
| fracerl 33337 | Rewrite the ring localizat... |
| fracf1 33338 | The embedding of a commuta... |
| fracfld 33339 | The field of fractions of ... |
| idomsubr 33340 | Every integral domain is i... |
| fldgenval 33343 | Value of the field generat... |
| fldgenssid 33344 | The field generated by a s... |
| fldgensdrg 33345 | A generated subfield is a ... |
| fldgenssv 33346 | A generated subfield is a ... |
| fldgenss 33347 | Generated subfields preser... |
| fldgenidfld 33348 | The subfield generated by ... |
| fldgenssp 33349 | The field generated by a s... |
| fldgenid 33350 | The subfield of a field ` ... |
| fldgenfld 33351 | A generated subfield is a ... |
| primefldgen1 33352 | The prime field of a divis... |
| 1fldgenq 33353 | The field of rational numb... |
| rhmdvd 33354 | A ring homomorphism preser... |
| kerunit 33355 | If a unit element lies in ... |
| reldmresv 33358 | The scalar restriction is ... |
| resvval 33359 | Value of structure restric... |
| resvid2 33360 | General behavior of trivia... |
| resvval2 33361 | Value of nontrivial struct... |
| resvsca 33362 | Base set of a structure re... |
| resvlem 33363 | Other elements of a scalar... |
| resvbas 33364 | ` Base ` is unaffected by ... |
| resvplusg 33365 | ` +g ` is unaffected by sc... |
| resvvsca 33366 | ` .s ` is unaffected by sc... |
| resvmulr 33367 | ` .r ` is unaffected by sc... |
| resv0g 33368 | ` 0g ` is unaffected by sc... |
| resv1r 33369 | ` 1r ` is unaffected by sc... |
| resvcmn 33370 | Scalar restriction preserv... |
| gzcrng 33371 | The gaussian integers form... |
| cnfldfld 33372 | The complex numbers form a... |
| reofld 33373 | The real numbers form an o... |
| nn0omnd 33374 | The nonnegative integers f... |
| gsumind 33375 | The group sum of an indica... |
| rearchi 33376 | The field of the real numb... |
| nn0archi 33377 | The monoid of the nonnegat... |
| xrge0slmod 33378 | The extended nonnegative r... |
| qusker 33379 | The kernel of a quotient m... |
| eqgvscpbl 33380 | The left coset equivalence... |
| qusvscpbl 33381 | The quotient map distribut... |
| qusvsval 33382 | Value of the scalar multip... |
| imaslmod 33383 | The image structure of a l... |
| imasmhm 33384 | Given a function ` F ` wit... |
| imasghm 33385 | Given a function ` F ` wit... |
| imasrhm 33386 | Given a function ` F ` wit... |
| imaslmhm 33387 | Given a function ` F ` wit... |
| quslmod 33388 | If ` G ` is a submodule in... |
| quslmhm 33389 | If ` G ` is a submodule of... |
| quslvec 33390 | If ` S ` is a vector subsp... |
| ecxpid 33391 | The equivalence class of a... |
| qsxpid 33392 | The quotient set of a cart... |
| qusxpid 33393 | The Group quotient equival... |
| qustriv 33394 | The quotient of a group ` ... |
| qustrivr 33395 | Converse of ~ qustriv . (... |
| znfermltl 33396 | Fermat's little theorem in... |
| islinds5 33397 | A set is linearly independ... |
| ellspds 33398 | Variation on ~ ellspd . (... |
| 0ellsp 33399 | Zero is in all spans. (Co... |
| 0nellinds 33400 | The group identity cannot ... |
| rspsnid 33401 | A principal ideal contains... |
| elrsp 33402 | Write the elements of a ri... |
| ellpi 33403 | Elementhood in a left prin... |
| lpirlidllpi 33404 | In a principal ideal ring,... |
| rspidlid 33405 | The ideal span of an ideal... |
| pidlnz 33406 | A principal ideal generate... |
| lbslsp 33407 | Any element of a left modu... |
| lindssn 33408 | Any singleton of a nonzero... |
| lindflbs 33409 | Conditions for an independ... |
| islbs5 33410 | An equivalent formulation ... |
| linds2eq 33411 | Deduce equality of element... |
| lindfpropd 33412 | Property deduction for lin... |
| lindspropd 33413 | Property deduction for lin... |
| dvdsruassoi 33414 | If two elements ` X ` and ... |
| dvdsruasso 33415 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33416 | A reformulation of ~ dvdsr... |
| dvdsrspss 33417 | In a ring, an element ` X ... |
| rspsnasso 33418 | Two elements ` X ` and ` Y... |
| unitprodclb 33419 | A finite product is a unit... |
| elgrplsmsn 33420 | Membership in a sumset wit... |
| lsmsnorb 33421 | The sumset of a group with... |
| lsmsnorb2 33422 | The sumset of a single ele... |
| elringlsm 33423 | Membership in a product of... |
| elringlsmd 33424 | Membership in a product of... |
| ringlsmss 33425 | Closure of the product of ... |
| ringlsmss1 33426 | The product of an ideal ` ... |
| ringlsmss2 33427 | The product with an ideal ... |
| lsmsnpridl 33428 | The product of the ring wi... |
| lsmsnidl 33429 | The product of the ring wi... |
| lsmidllsp 33430 | The sum of two ideals is t... |
| lsmidl 33431 | The sum of two ideals is a... |
| lsmssass 33432 | Group sum is associative, ... |
| grplsm0l 33433 | Sumset with the identity s... |
| grplsmid 33434 | The direct sum of an eleme... |
| quslsm 33435 | Express the image by the q... |
| qusbas2 33436 | Alternate definition of th... |
| qus0g 33437 | The identity element of a ... |
| qusima 33438 | The image of a subgroup by... |
| qusrn 33439 | The natural map from eleme... |
| nsgqus0 33440 | A normal subgroup ` N ` is... |
| nsgmgclem 33441 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33442 | There is a monotone Galois... |
| nsgqusf1olem1 33443 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33444 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33445 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33446 | The canonical projection h... |
| lmhmqusker 33447 | A surjective module homomo... |
| lmicqusker 33448 | The image ` H ` of a modul... |
| lidlmcld 33449 | An ideal is closed under l... |
| intlidl 33450 | The intersection of a none... |
| 0ringidl 33451 | The zero ideal is the only... |
| pidlnzb 33452 | A principal ideal is nonze... |
| lidlunitel 33453 | If an ideal ` I ` contains... |
| unitpidl1 33454 | The ideal ` I ` generated ... |
| rhmquskerlem 33455 | The mapping ` J ` induced ... |
| rhmqusker 33456 | A surjective ring homomorp... |
| ricqusker 33457 | The image ` H ` of a ring ... |
| elrspunidl 33458 | Elementhood in the span of... |
| elrspunsn 33459 | Membership to the span of ... |
| lidlincl 33460 | Ideals are closed under in... |
| idlinsubrg 33461 | The intersection between a... |
| rhmimaidl 33462 | The image of an ideal ` I ... |
| drngidl 33463 | A nonzero ring is a divisi... |
| drngidlhash 33464 | A ring is a division ring ... |
| prmidlval 33467 | The class of prime ideals ... |
| isprmidl 33468 | The predicate "is a prime ... |
| prmidlnr 33469 | A prime ideal is a proper ... |
| prmidl 33470 | The main property of a pri... |
| prmidl2 33471 | A condition that shows an ... |
| idlmulssprm 33472 | Let ` P ` be a prime ideal... |
| pridln1 33473 | A proper ideal cannot cont... |
| prmidlidl 33474 | A prime ideal is an ideal.... |
| prmidlssidl 33475 | Prime ideals as a subset o... |
| cringm4 33476 | Commutative/associative la... |
| isprmidlc 33477 | The predicate "is prime id... |
| prmidlc 33478 | Property of a prime ideal ... |
| 0ringprmidl 33479 | The trivial ring does not ... |
| prmidl0 33480 | The zero ideal of a commut... |
| rhmpreimaprmidl 33481 | The preimage of a prime id... |
| qsidomlem1 33482 | If the quotient ring of a ... |
| qsidomlem2 33483 | A quotient by a prime idea... |
| qsidom 33484 | An ideal ` I ` in the comm... |
| qsnzr 33485 | A quotient of a non-zero r... |
| ssdifidllem 33486 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 33487 | Let ` R ` be a ring, and l... |
| ssdifidlprm 33488 | If the set ` S ` of ~ ssdi... |
| mxidlval 33491 | The set of maximal ideals ... |
| ismxidl 33492 | The predicate "is a maxima... |
| mxidlidl 33493 | A maximal ideal is an idea... |
| mxidlnr 33494 | A maximal ideal is proper.... |
| mxidlmax 33495 | A maximal ideal is a maxim... |
| mxidln1 33496 | One is not contained in an... |
| mxidlnzr 33497 | A ring with a maximal idea... |
| mxidlmaxv 33498 | An ideal ` I ` strictly co... |
| crngmxidl 33499 | In a commutative ring, max... |
| mxidlprm 33500 | Every maximal ideal is pri... |
| mxidlirredi 33501 | In an integral domain, the... |
| mxidlirred 33502 | In a principal ideal domai... |
| ssmxidllem 33503 | The set ` P ` used in the ... |
| ssmxidl 33504 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 33505 | In a nonzero ring, the zer... |
| drng0mxidl 33506 | In a division ring, the ze... |
| drngmxidl 33507 | The zero ideal is the only... |
| drngmxidlr 33508 | If a ring's only maximal i... |
| krull 33509 | Krull's theorem: Any nonz... |
| mxidlnzrb 33510 | A ring is nonzero if and o... |
| krullndrng 33511 | Krull's theorem for non-di... |
| opprabs 33512 | The opposite ring of the o... |
| oppreqg 33513 | Group coset equivalence re... |
| opprnsg 33514 | Normal subgroups of the op... |
| opprlidlabs 33515 | The ideals of the opposite... |
| oppr2idl 33516 | Two sided ideal of the opp... |
| opprmxidlabs 33517 | The maximal ideal of the o... |
| opprqusbas 33518 | The base of the quotient o... |
| opprqusplusg 33519 | The group operation of the... |
| opprqus0g 33520 | The group identity element... |
| opprqusmulr 33521 | The multiplication operati... |
| opprqus1r 33522 | The ring unity of the quot... |
| opprqusdrng 33523 | The quotient of the opposi... |
| qsdrngilem 33524 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33525 | A quotient by a maximal le... |
| qsdrnglem2 33526 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33527 | An ideal ` M ` is both lef... |
| qsfld 33528 | An ideal ` M ` in the comm... |
| mxidlprmALT 33529 | Every maximal ideal is pri... |
| idlsrgstr 33532 | A constructed semiring of ... |
| idlsrgval 33533 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33534 | Base of the ideals of a ri... |
| idlsrgplusg 33535 | Additive operation of the ... |
| idlsrg0g 33536 | The zero ideal is the addi... |
| idlsrgmulr 33537 | Multiplicative operation o... |
| idlsrgtset 33538 | Topology component of the ... |
| idlsrgmulrval 33539 | Value of the ring multipli... |
| idlsrgmulrcl 33540 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33541 | In a commutative ring, the... |
| idlsrgmulrss2 33542 | The product of two ideals ... |
| idlsrgmulrssin 33543 | In a commutative ring, the... |
| idlsrgmnd 33544 | The ideals of a ring form ... |
| idlsrgcmnd 33545 | The ideals of a ring form ... |
| rprmval 33546 | The prime elements of a ri... |
| isrprm 33547 | Property for ` P ` to be a... |
| rprmcl 33548 | A ring prime is an element... |
| rprmdvds 33549 | If a ring prime ` Q ` divi... |
| rprmnz 33550 | A ring prime is nonzero. ... |
| rprmnunit 33551 | A ring prime is not a unit... |
| rsprprmprmidl 33552 | In a commutative ring, ide... |
| rsprprmprmidlb 33553 | In an integral domain, an ... |
| rprmndvdsr1 33554 | A ring prime element does ... |
| rprmasso 33555 | In an integral domain, the... |
| rprmasso2 33556 | In an integral domain, if ... |
| rprmasso3 33557 | In an integral domain, if ... |
| unitmulrprm 33558 | A ring unit multiplied by ... |
| rprmndvdsru 33559 | A ring prime element does ... |
| rprmirredlem 33560 | Lemma for ~ rprmirred . (... |
| rprmirred 33561 | In an integral domain, rin... |
| rprmirredb 33562 | In a principal ideal domai... |
| rprmdvdspow 33563 | If a prime element divides... |
| rprmdvdsprod 33564 | If a prime element ` Q ` d... |
| 1arithidomlem1 33565 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 33566 | Lemma for ~ 1arithidom : i... |
| 1arithidom 33567 | Uniqueness of prime factor... |
| isufd 33570 | The property of being a Un... |
| ufdprmidl 33571 | In a unique factorization ... |
| ufdidom 33572 | A nonzero unique factoriza... |
| pidufd 33573 | Every principal ideal doma... |
| 1arithufdlem1 33574 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 33575 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 33576 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 33577 | Lemma for ~ 1arithufd . N... |
| 1arithufd 33578 | Existence of a factorizati... |
| dfufd2lem 33579 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 33580 | Alternative definition of ... |
| zringidom 33581 | The ring of integers is an... |
| zringpid 33582 | The ring of integers is a ... |
| dfprm3 33583 | The (positive) prime eleme... |
| zringfrac 33584 | The field of fractions of ... |
| assaassd 33585 | Left-associative property ... |
| assaassrd 33586 | Right-associative property... |
| 0ringmon1p 33587 | There are no monic polynom... |
| fply1 33588 | Conditions for a function ... |
| ply1lvec 33589 | In a division ring, the un... |
| evls1fn 33590 | Functionality of the subri... |
| evls1dm 33591 | The domain of the subring ... |
| evls1fvf 33592 | The subring evaluation fun... |
| evl1fvf 33593 | The univariate polynomial ... |
| evl1fpws 33594 | Evaluation of a univariate... |
| ressply1evls1 33595 | Subring evaluation of a un... |
| ressdeg1 33596 | The degree of a univariate... |
| ressply10g 33597 | A restricted polynomial al... |
| ressply1mon1p 33598 | The monic polynomials of a... |
| ressply1invg 33599 | An element of a restricted... |
| ressply1sub 33600 | A restricted polynomial al... |
| ressasclcl 33601 | Closure of the univariate ... |
| evls1subd 33602 | Univariate polynomial eval... |
| deg1le0eq0 33603 | A polynomial with nonposit... |
| ply1asclunit 33604 | A non-zero scalar polynomi... |
| ply1unit 33605 | In a field ` F ` , a polyn... |
| evl1deg1 33606 | Evaluation of a univariate... |
| evl1deg2 33607 | Evaluation of a univariate... |
| evl1deg3 33608 | Evaluation of a univariate... |
| evls1monply1 33609 | Subring evaluation of a sc... |
| ply1dg1rt 33610 | Express the root ` - B / A... |
| ply1dg1rtn0 33611 | Polynomials of degree 1 ov... |
| ply1mulrtss 33612 | The roots of a factor ` F ... |
| deg1prod 33613 | Degree of a product of pol... |
| ply1dg3rt0irred 33614 | If a cubic polynomial over... |
| m1pmeq 33615 | If two monic polynomials `... |
| ply1fermltl 33616 | Fermat's little theorem fo... |
| coe1mon 33617 | Coefficient vector of a mo... |
| ply1moneq 33618 | Two monomials are equal if... |
| ply1coedeg 33619 | Decompose a univariate pol... |
| coe1zfv 33620 | The coefficients of the ze... |
| coe1vr1 33621 | Polynomial coefficient of ... |
| deg1vr 33622 | The degree of the variable... |
| vr1nz 33623 | A univariate polynomial va... |
| ply1degltel 33624 | Characterize elementhood i... |
| ply1degleel 33625 | Characterize elementhood i... |
| ply1degltlss 33626 | The space ` S ` of the uni... |
| gsummoncoe1fzo 33627 | A coefficient of the polyn... |
| gsummoncoe1fz 33628 | A coefficient of the polyn... |
| ply1gsumz 33629 | If a polynomial given as a... |
| deg1addlt 33630 | If both factors have degre... |
| ig1pnunit 33631 | The polynomial ideal gener... |
| ig1pmindeg 33632 | The polynomial ideal gener... |
| q1pdir 33633 | Distribution of univariate... |
| q1pvsca 33634 | Scalar multiplication prop... |
| r1pvsca 33635 | Scalar multiplication prop... |
| r1p0 33636 | Polynomial remainder opera... |
| r1pcyc 33637 | The polynomial remainder o... |
| r1padd1 33638 | Addition property of the p... |
| r1pid2OLD 33639 | Obsolete version of ~ r1pi... |
| r1plmhm 33640 | The univariate polynomial ... |
| r1pquslmic 33641 | The univariate polynomial ... |
| psrbasfsupp 33642 | Rewrite a finite support f... |
| extvval 33645 | Value of the "variable ext... |
| extvfval 33646 | The "variable extension" f... |
| extvfv 33647 | The "variable extension" f... |
| extvfvv 33648 | The "variable extension" f... |
| extvfvvcl 33649 | Closure for the "variable ... |
| extvfvcl 33650 | Closure for the "variable ... |
| extvfvalf 33651 | The "variable extension" f... |
| mvrvalind 33652 | Value of the generating el... |
| mplmulmvr 33653 | Multiply a polynomial ` F ... |
| evlscaval 33654 | Polynomial evaluation for ... |
| evlvarval 33655 | Polynomial evaluation buil... |
| evlextv 33656 | Evaluating a variable-exte... |
| mplvrpmlem 33657 | Lemma for ~ mplvrpmga and ... |
| mplvrpmfgalem 33658 | Permuting variables in a m... |
| mplvrpmga 33659 | The action of permuting va... |
| mplvrpmmhm 33660 | The action of permuting va... |
| mplvrpmrhm 33661 | The action of permuting va... |
| splyval 33666 | The symmetric polynomials ... |
| splysubrg 33667 | The symmetric polynomials ... |
| issply 33668 | Conditions for being a sym... |
| esplyval 33669 | The elementary polynomials... |
| esplyfval 33670 | The ` K ` -th elementary p... |
| esplyfval0 33671 | The ` 0 ` -th elementary s... |
| esplyfval2 33672 | When ` K ` is out-of-bound... |
| esplylem 33673 | Lemma for ~ esplyfv and ot... |
| esplympl 33674 | Elementary symmetric polyn... |
| esplymhp 33675 | The ` K ` -th elementary s... |
| esplyfv1 33676 | Coefficient for the ` K ` ... |
| esplyfv 33677 | Coefficient for the ` K ` ... |
| esplysply 33678 | The ` K ` -th elementary s... |
| esplyfval3 33679 | Alternate expression for t... |
| esplyind 33680 | A recursive formula for th... |
| esplyindfv 33681 | A recursive formula for th... |
| esplyfvn 33682 | Express the last elementar... |
| vietadeg1 33683 | The degree of a product of... |
| vietalem 33684 | Lemma for ~ vieta : induct... |
| vieta 33685 | Vieta's Formulas: Coeffic... |
| sra1r 33686 | The unity element of a sub... |
| sradrng 33687 | Condition for a subring al... |
| sraidom 33688 | Condition for a subring al... |
| srasubrg 33689 | A subring of the original ... |
| sralvec 33690 | Given a sub division ring ... |
| srafldlvec 33691 | Given a subfield ` F ` of ... |
| resssra 33692 | The subring algebra of a r... |
| lsssra 33693 | A subring is a subspace of... |
| srapwov 33694 | The "power" operation on a... |
| drgext0g 33695 | The additive neutral eleme... |
| drgextvsca 33696 | The scalar multiplication ... |
| drgext0gsca 33697 | The additive neutral eleme... |
| drgextsubrg 33698 | The scalar field is a subr... |
| drgextlsp 33699 | The scalar field is a subs... |
| drgextgsum 33700 | Group sum in a division ri... |
| lvecdimfi 33701 | Finite version of ~ lvecdi... |
| exsslsb 33702 | Any finite generating set ... |
| lbslelsp 33703 | The size of a basis ` X ` ... |
| dimval 33706 | The dimension of a vector ... |
| dimvalfi 33707 | The dimension of a vector ... |
| dimcl 33708 | Closure of the vector spac... |
| lmimdim 33709 | Module isomorphisms preser... |
| lmicdim 33710 | Module isomorphisms preser... |
| lvecdim0i 33711 | A vector space of dimensio... |
| lvecdim0 33712 | A vector space of dimensio... |
| lssdimle 33713 | The dimension of a linear ... |
| dimpropd 33714 | If two structures have the... |
| rlmdim 33715 | The left vector space indu... |
| rgmoddimOLD 33716 | Obsolete version of ~ rlmd... |
| frlmdim 33717 | Dimension of a free left m... |
| tnglvec 33718 | Augmenting a structure wit... |
| tngdim 33719 | Dimension of a left vector... |
| rrxdim 33720 | Dimension of the generaliz... |
| matdim 33721 | Dimension of the space of ... |
| lbslsat 33722 | A nonzero vector ` X ` is ... |
| lsatdim 33723 | A line, spanned by a nonze... |
| drngdimgt0 33724 | The dimension of a vector ... |
| lmhmlvec2 33725 | A homomorphism of left vec... |
| kerlmhm 33726 | The kernel of a vector spa... |
| imlmhm 33727 | The image of a vector spac... |
| ply1degltdimlem 33728 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 33729 | The space ` S ` of the uni... |
| lindsunlem 33730 | Lemma for ~ lindsun . (Co... |
| lindsun 33731 | Condition for the union of... |
| lbsdiflsp0 33732 | The linear spans of two di... |
| dimkerim 33733 | Given a linear map ` F ` b... |
| qusdimsum 33734 | Let ` W ` be a vector spac... |
| fedgmullem1 33735 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33736 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33737 | The multiplicativity formu... |
| dimlssid 33738 | If the dimension of a line... |
| lvecendof1f1o 33739 | If an endomorphism ` U ` o... |
| lactlmhm 33740 | In an associative algebra ... |
| assalactf1o 33741 | In an associative algebra ... |
| assarrginv 33742 | If an element ` X ` of an ... |
| assafld 33743 | If an algebra ` A ` of fin... |
| relfldext 33750 | The field extension is a r... |
| brfldext 33751 | The field extension relati... |
| ccfldextrr 33752 | The field of the complex n... |
| fldextfld1 33753 | A field extension is only ... |
| fldextfld2 33754 | A field extension is only ... |
| fldextsubrg 33755 | Field extension implies a ... |
| sdrgfldext 33756 | A field ` E ` and any sub-... |
| fldextress 33757 | Field extension implies a ... |
| brfinext 33758 | The finite field extension... |
| extdgval 33759 | Value of the field extensi... |
| fldextsdrg 33760 | Deduce sub-division-ring f... |
| fldextsralvec 33761 | The subring algebra associ... |
| extdgcl 33762 | Closure of the field exten... |
| extdggt0 33763 | Degrees of field extension... |
| fldexttr 33764 | Field extension is a trans... |
| fldextid 33765 | The field extension relati... |
| extdgid 33766 | A trivial field extension ... |
| fldsdrgfldext 33767 | A sub-division-ring of a f... |
| fldsdrgfldext2 33768 | A sub-sub-division-ring of... |
| extdgmul 33769 | The multiplicativity formu... |
| finextfldext 33770 | A finite field extension i... |
| finexttrb 33771 | The extension ` E ` of ` K... |
| extdg1id 33772 | If the degree of the exten... |
| extdg1b 33773 | The degree of the extensio... |
| fldgenfldext 33774 | A subfield ` F ` extended ... |
| fldextchr 33775 | The characteristic of a su... |
| evls1fldgencl 33776 | Closure of the subring pol... |
| ccfldsrarelvec 33777 | The subring algebra of the... |
| ccfldextdgrr 33778 | The degree of the field ex... |
| fldextrspunlsplem 33779 | Lemma for ~ fldextrspunlsp... |
| fldextrspunlsp 33780 | Lemma for ~ fldextrspunfld... |
| fldextrspunlem1 33781 | Lemma for ~ fldextrspunfld... |
| fldextrspunfld 33782 | The ring generated by the ... |
| fldextrspunlem2 33783 | Part of the proof of Propo... |
| fldextrspundgle 33784 | Inequality involving the d... |
| fldextrspundglemul 33785 | Given two field extensions... |
| fldextrspundgdvdslem 33786 | Lemma for ~ fldextrspundgd... |
| fldextrspundgdvds 33787 | Given two finite extension... |
| fldext2rspun 33788 | Given two field extensions... |
| irngval 33791 | The elements of a field ` ... |
| elirng 33792 | Property for an element ` ... |
| irngss 33793 | All elements of a subring ... |
| irngssv 33794 | An integral element is an ... |
| 0ringirng 33795 | A zero ring ` R ` has no i... |
| irngnzply1lem 33796 | In the case of a field ` E... |
| irngnzply1 33797 | In the case of a field ` E... |
| extdgfialglem1 33798 | Lemma for ~ extdgfialg . ... |
| extdgfialglem2 33799 | Lemma for ~ extdgfialg . ... |
| extdgfialg 33800 | A finite field extension `... |
| bralgext 33803 | Express the fact that a fi... |
| finextalg 33804 | A finite field extension i... |
| ply1annidllem 33807 | Write the set ` Q ` of pol... |
| ply1annidl 33808 | The set ` Q ` of polynomia... |
| ply1annnr 33809 | The set ` Q ` of polynomia... |
| ply1annig1p 33810 | The ideal ` Q ` of polynom... |
| minplyval 33811 | Expand the value of the mi... |
| minplycl 33812 | The minimal polynomial is ... |
| ply1annprmidl 33813 | The set ` Q ` of polynomia... |
| minplymindeg 33814 | The minimal polynomial of ... |
| minplyann 33815 | The minimal polynomial for... |
| minplyirredlem 33816 | Lemma for ~ minplyirred . ... |
| minplyirred 33817 | A nonzero minimal polynomi... |
| irngnminplynz 33818 | Integral elements have non... |
| minplym1p 33819 | A minimal polynomial is mo... |
| minplynzm1p 33820 | If a minimal polynomial is... |
| minplyelirng 33821 | If the minimial polynomial... |
| irredminply 33822 | An irreducible, monic, ann... |
| algextdeglem1 33823 | Lemma for ~ algextdeg . (... |
| algextdeglem2 33824 | Lemma for ~ algextdeg . B... |
| algextdeglem3 33825 | Lemma for ~ algextdeg . T... |
| algextdeglem4 33826 | Lemma for ~ algextdeg . B... |
| algextdeglem5 33827 | Lemma for ~ algextdeg . T... |
| algextdeglem6 33828 | Lemma for ~ algextdeg . B... |
| algextdeglem7 33829 | Lemma for ~ algextdeg . T... |
| algextdeglem8 33830 | Lemma for ~ algextdeg . T... |
| algextdeg 33831 | The degree of an algebraic... |
| rtelextdg2lem 33832 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 33833 | If an element ` X ` is a s... |
| fldext2chn 33834 | In a non-empty chain ` T `... |
| constrrtll 33837 | In the construction of con... |
| constrrtlc1 33838 | In the construction of con... |
| constrrtlc2 33839 | In the construction of con... |
| constrrtcclem 33840 | In the construction of con... |
| constrrtcc 33841 | In the construction of con... |
| isconstr 33842 | Property of being a constr... |
| constr0 33843 | The first step of the cons... |
| constrsuc 33844 | Membership in the successo... |
| constrlim 33845 | Limit step of the construc... |
| constrsscn 33846 | Closure of the constructib... |
| constrsslem 33847 | Lemma for ~ constrss . Th... |
| constr01 33848 | ` 0 ` and ` 1 ` are in all... |
| constrss 33849 | Constructed points are in ... |
| constrmon 33850 | The construction of constr... |
| constrconj 33851 | If a point ` X ` of the co... |
| constrfin 33852 | Each step of the construct... |
| constrelextdg2 33853 | If the ` N ` -th step ` ( ... |
| constrextdg2lem 33854 | Lemma for ~ constrextdg2 (... |
| constrextdg2 33855 | Any step ` ( C `` N ) ` of... |
| constrext2chnlem 33856 | Lemma for ~ constrext2chn ... |
| constrfiss 33857 | For any finite set ` A ` o... |
| constrllcllem 33858 | Constructible numbers are ... |
| constrlccllem 33859 | Constructible numbers are ... |
| constrcccllem 33860 | Constructible numbers are ... |
| constrcbvlem 33861 | Technical lemma for elimin... |
| constrllcl 33862 | Constructible numbers are ... |
| constrlccl 33863 | Constructible numbers are ... |
| constrcccl 33864 | Constructible numbers are ... |
| constrext2chn 33865 | If a constructible number ... |
| constrcn 33866 | Constructible numbers are ... |
| nn0constr 33867 | Nonnegative integers are c... |
| constraddcl 33868 | Constructive numbers are c... |
| constrnegcl 33869 | Constructible numbers are ... |
| zconstr 33870 | Integers are constructible... |
| constrdircl 33871 | Constructible numbers are ... |
| iconstr 33872 | The imaginary unit ` _i ` ... |
| constrremulcl 33873 | If two real numbers ` X ` ... |
| constrcjcl 33874 | Constructible numbers are ... |
| constrrecl 33875 | Constructible numbers are ... |
| constrimcl 33876 | Constructible numbers are ... |
| constrmulcl 33877 | Constructible numbers are ... |
| constrreinvcl 33878 | If a real number ` X ` is ... |
| constrinvcl 33879 | Constructible numbers are ... |
| constrcon 33880 | Contradiction of construct... |
| constrsdrg 33881 | Constructible numbers form... |
| constrfld 33882 | The constructible numbers ... |
| constrresqrtcl 33883 | If a positive real number ... |
| constrabscl 33884 | Constructible numbers are ... |
| constrsqrtcl 33885 | Constructible numbers are ... |
| 2sqr3minply 33886 | The polynomial ` ( ( X ^ 3... |
| 2sqr3nconstr 33887 | Doubling the cube is an im... |
| cos9thpiminplylem1 33888 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem2 33889 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem3 33890 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem4 33891 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem5 33892 | The constructed complex nu... |
| cos9thpiminplylem6 33893 | Evaluation of the polynomi... |
| cos9thpiminply 33894 | The polynomial ` ( ( X ^ 3... |
| cos9thpinconstrlem1 33895 | The complex number ` O ` ,... |
| cos9thpinconstrlem2 33896 | The complex number ` A ` i... |
| cos9thpinconstr 33897 | Trisecting an angle is an ... |
| trisecnconstr 33898 | Not all angles can be tris... |
| smatfval 33901 | Value of the submatrix. (... |
| smatrcl 33902 | Closure of the rectangular... |
| smatlem 33903 | Lemma for the next theorem... |
| smattl 33904 | Entries of a submatrix, to... |
| smattr 33905 | Entries of a submatrix, to... |
| smatbl 33906 | Entries of a submatrix, bo... |
| smatbr 33907 | Entries of a submatrix, bo... |
| smatcl 33908 | Closure of the square subm... |
| matmpo 33909 | Write a square matrix as a... |
| 1smat1 33910 | The submatrix of the ident... |
| submat1n 33911 | One case where the submatr... |
| submatres 33912 | Special case where the sub... |
| submateqlem1 33913 | Lemma for ~ submateq . (C... |
| submateqlem2 33914 | Lemma for ~ submateq . (C... |
| submateq 33915 | Sufficient condition for t... |
| submatminr1 33916 | If we take a submatrix by ... |
| lmatval 33919 | Value of the literal matri... |
| lmatfval 33920 | Entries of a literal matri... |
| lmatfvlem 33921 | Useful lemma to extract li... |
| lmatcl 33922 | Closure of the literal mat... |
| lmat22lem 33923 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 33924 | Entry of a 2x2 literal mat... |
| lmat22e12 33925 | Entry of a 2x2 literal mat... |
| lmat22e21 33926 | Entry of a 2x2 literal mat... |
| lmat22e22 33927 | Entry of a 2x2 literal mat... |
| lmat22det 33928 | The determinant of a liter... |
| mdetpmtr1 33929 | The determinant of a matri... |
| mdetpmtr2 33930 | The determinant of a matri... |
| mdetpmtr12 33931 | The determinant of a matri... |
| mdetlap1 33932 | A Laplace expansion of the... |
| madjusmdetlem1 33933 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 33934 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 33935 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 33936 | Lemma for ~ madjusmdet . ... |
| madjusmdet 33937 | Express the cofactor of th... |
| mdetlap 33938 | Laplace expansion of the d... |
| ist0cld 33939 | The predicate "is a T_0 sp... |
| txomap 33940 | Given two open maps ` F ` ... |
| qtopt1 33941 | If every equivalence class... |
| qtophaus 33942 | If an open map's graph in ... |
| circtopn 33943 | The topology of the unit c... |
| circcn 33944 | The function gluing the re... |
| reff 33945 | For any cover refinement, ... |
| locfinreflem 33946 | A locally finite refinemen... |
| locfinref 33947 | A locally finite refinemen... |
| iscref 33950 | The property that every op... |
| crefeq 33951 | Equality theorem for the "... |
| creftop 33952 | A space where every open c... |
| crefi 33953 | The property that every op... |
| crefdf 33954 | A formulation of ~ crefi e... |
| crefss 33955 | The "every open cover has ... |
| cmpcref 33956 | Equivalent definition of c... |
| cmpfiref 33957 | Every open cover of a Comp... |
| ldlfcntref 33960 | Every open cover of a Lind... |
| ispcmp 33963 | The predicate "is a paraco... |
| cmppcmp 33964 | Every compact space is par... |
| dispcmp 33965 | Every discrete space is pa... |
| pcmplfin 33966 | Given a paracompact topolo... |
| pcmplfinf 33967 | Given a paracompact topolo... |
| rspecval 33970 | Value of the spectrum of t... |
| rspecbas 33971 | The prime ideals form the ... |
| rspectset 33972 | Topology component of the ... |
| rspectopn 33973 | The topology component of ... |
| zarcls0 33974 | The closure of the identit... |
| zarcls1 33975 | The unit ideal ` B ` is th... |
| zarclsun 33976 | The union of two closed se... |
| zarclsiin 33977 | In a Zariski topology, the... |
| zarclsint 33978 | The intersection of a fami... |
| zarclssn 33979 | The closed points of Zaris... |
| zarcls 33980 | The open sets of the Zaris... |
| zartopn 33981 | The Zariski topology is a ... |
| zartop 33982 | The Zariski topology is a ... |
| zartopon 33983 | The points of the Zariski ... |
| zar0ring 33984 | The Zariski Topology of th... |
| zart0 33985 | The Zariski topology is T_... |
| zarmxt1 33986 | The Zariski topology restr... |
| zarcmplem 33987 | Lemma for ~ zarcmp . (Con... |
| zarcmp 33988 | The Zariski topology is co... |
| rspectps 33989 | The spectrum of a ring ` R... |
| rhmpreimacnlem 33990 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 33991 | The function mapping a pri... |
| metidval 33996 | Value of the metric identi... |
| metidss 33997 | As a relation, the metric ... |
| metidv 33998 | ` A ` and ` B ` identify b... |
| metideq 33999 | Basic property of the metr... |
| metider 34000 | The metric identification ... |
| pstmval 34001 | Value of the metric induce... |
| pstmfval 34002 | Function value of the metr... |
| pstmxmet 34003 | The metric induced by a ps... |
| hauseqcn 34004 | In a Hausdorff topology, t... |
| elunitge0 34005 | An element of the closed u... |
| unitssxrge0 34006 | The closed unit interval i... |
| unitdivcld 34007 | Necessary conditions for a... |
| iistmd 34008 | The closed unit interval f... |
| unicls 34009 | The union of the closed se... |
| tpr2tp 34010 | The usual topology on ` ( ... |
| tpr2uni 34011 | The usual topology on ` ( ... |
| xpinpreima 34012 | Rewrite the cartesian prod... |
| xpinpreima2 34013 | Rewrite the cartesian prod... |
| sqsscirc1 34014 | The complex square of side... |
| sqsscirc2 34015 | The complex square of side... |
| cnre2csqlem 34016 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 34017 | Image of a centered square... |
| tpr2rico 34018 | For any point of an open s... |
| cnvordtrestixx 34019 | The restriction of the 'gr... |
| prsdm 34020 | Domain of the relation of ... |
| prsrn 34021 | Range of the relation of a... |
| prsss 34022 | Relation of a subproset. ... |
| prsssdm 34023 | Domain of a subproset rela... |
| ordtprsval 34024 | Value of the order topolog... |
| ordtprsuni 34025 | Value of the order topolog... |
| ordtcnvNEW 34026 | The order dual generates t... |
| ordtrestNEW 34027 | The subspace topology of a... |
| ordtrest2NEWlem 34028 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 34029 | An interval-closed set ` A... |
| ordtconnlem1 34030 | Connectedness in the order... |
| ordtconn 34031 | Connectedness in the order... |
| mndpluscn 34032 | A mapping that is both a h... |
| mhmhmeotmd 34033 | Deduce a Topological Monoi... |
| rmulccn 34034 | Multiplication by a real c... |
| raddcn 34035 | Addition in the real numbe... |
| xrmulc1cn 34036 | The operation multiplying ... |
| fmcncfil 34037 | The image of a Cauchy filt... |
| xrge0hmph 34038 | The extended nonnegative r... |
| xrge0iifcnv 34039 | Define a bijection from ` ... |
| xrge0iifcv 34040 | The defined function's val... |
| xrge0iifiso 34041 | The defined bijection from... |
| xrge0iifhmeo 34042 | Expose a homeomorphism fro... |
| xrge0iifhom 34043 | The defined function from ... |
| xrge0iif1 34044 | Condition for the defined ... |
| xrge0iifmhm 34045 | The defined function from ... |
| xrge0pluscn 34046 | The addition operation of ... |
| xrge0mulc1cn 34047 | The operation multiplying ... |
| xrge0tps 34048 | The extended nonnegative r... |
| xrge0topn 34049 | The topology of the extend... |
| xrge0haus 34050 | The topology of the extend... |
| xrge0tmd 34051 | The extended nonnegative r... |
| xrge0tmdALT 34052 | Alternate proof of ~ xrge0... |
| lmlim 34053 | Relate a limit in a given ... |
| lmlimxrge0 34054 | Relate a limit in the nonn... |
| rge0scvg 34055 | Implication of convergence... |
| fsumcvg4 34056 | A serie with finite suppor... |
| pnfneige0 34057 | A neighborhood of ` +oo ` ... |
| lmxrge0 34058 | Express "sequence ` F ` co... |
| lmdvg 34059 | If a monotonic sequence of... |
| lmdvglim 34060 | If a monotonic real number... |
| pl1cn 34061 | A univariate polynomial is... |
| zringnm 34064 | The norm (function) for a ... |
| zzsnm 34065 | The norm of the ring of th... |
| zlm0 34066 | Zero of a ` ZZ ` -module. ... |
| zlm1 34067 | Unity element of a ` ZZ ` ... |
| zlmds 34068 | Distance in a ` ZZ ` -modu... |
| zlmtset 34069 | Topology in a ` ZZ ` -modu... |
| zlmnm 34070 | Norm of a ` ZZ ` -module (... |
| zhmnrg 34071 | The ` ZZ ` -module built f... |
| nmmulg 34072 | The norm of a group produc... |
| zrhnm 34073 | The norm of the image by `... |
| cnzh 34074 | The ` ZZ ` -module of ` CC... |
| rezh 34075 | The ` ZZ ` -module of ` RR... |
| qqhval 34078 | Value of the canonical hom... |
| zrhf1ker 34079 | The kernel of the homomorp... |
| zrhchr 34080 | The kernel of the homomorp... |
| zrhker 34081 | The kernel of the homomorp... |
| zrhunitpreima 34082 | The preimage by ` ZRHom ` ... |
| elzrhunit 34083 | Condition for the image by... |
| zrhneg 34084 | The canonical homomorphism... |
| zrhcntr 34085 | The canonical representati... |
| elzdif0 34086 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 34087 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 34088 | Value of the canonical hom... |
| qqhvval 34089 | Value of the canonical hom... |
| qqh0 34090 | The image of ` 0 ` by the ... |
| qqh1 34091 | The image of ` 1 ` by the ... |
| qqhf 34092 | ` QQHom ` as a function. ... |
| qqhvq 34093 | The image of a quotient by... |
| qqhghm 34094 | The ` QQHom ` homomorphism... |
| qqhrhm 34095 | The ` QQHom ` homomorphism... |
| qqhnm 34096 | The norm of the image by `... |
| qqhcn 34097 | The ` QQHom ` homomorphism... |
| qqhucn 34098 | The ` QQHom ` homomorphism... |
| rrhval 34102 | Value of the canonical hom... |
| rrhcn 34103 | If the topology of ` R ` i... |
| rrhf 34104 | If the topology of ` R ` i... |
| isrrext 34106 | Express the property " ` R... |
| rrextnrg 34107 | An extension of ` RR ` is ... |
| rrextdrg 34108 | An extension of ` RR ` is ... |
| rrextnlm 34109 | The norm of an extension o... |
| rrextchr 34110 | The ring characteristic of... |
| rrextcusp 34111 | An extension of ` RR ` is ... |
| rrexttps 34112 | An extension of ` RR ` is ... |
| rrexthaus 34113 | The topology of an extensi... |
| rrextust 34114 | The uniformity of an exten... |
| rerrext 34115 | The field of the real numb... |
| cnrrext 34116 | The field of the complex n... |
| qqtopn 34117 | The topology of the field ... |
| rrhfe 34118 | If ` R ` is an extension o... |
| rrhcne 34119 | If ` R ` is an extension o... |
| rrhqima 34120 | The ` RRHom ` homomorphism... |
| rrh0 34121 | The image of ` 0 ` by the ... |
| xrhval 34124 | The value of the embedding... |
| zrhre 34125 | The ` ZRHom ` homomorphism... |
| qqhre 34126 | The ` QQHom ` homomorphism... |
| rrhre 34127 | The ` RRHom ` homomorphism... |
| relmntop 34130 | Manifold is a relation. (... |
| ismntoplly 34131 | Property of being a manifo... |
| ismntop 34132 | Property of being a manifo... |
| esumex 34135 | An extended sum is a set b... |
| esumcl 34136 | Closure for extended sum i... |
| esumeq12dvaf 34137 | Equality deduction for ext... |
| esumeq12dva 34138 | Equality deduction for ext... |
| esumeq12d 34139 | Equality deduction for ext... |
| esumeq1 34140 | Equality theorem for an ex... |
| esumeq1d 34141 | Equality theorem for an ex... |
| esumeq2 34142 | Equality theorem for exten... |
| esumeq2d 34143 | Equality deduction for ext... |
| esumeq2dv 34144 | Equality deduction for ext... |
| esumeq2sdv 34145 | Equality deduction for ext... |
| nfesum1 34146 | Bound-variable hypothesis ... |
| nfesum2 34147 | Bound-variable hypothesis ... |
| cbvesum 34148 | Change bound variable in a... |
| cbvesumv 34149 | Change bound variable in a... |
| esumid 34150 | Identify the extended sum ... |
| esumgsum 34151 | A finite extended sum is t... |
| esumval 34152 | Develop the value of the e... |
| esumel 34153 | The extended sum is a limi... |
| esumnul 34154 | Extended sum over the empt... |
| esum0 34155 | Extended sum of zero. (Co... |
| esumf1o 34156 | Re-index an extended sum u... |
| esumc 34157 | Convert from the collectio... |
| esumrnmpt 34158 | Rewrite an extended sum in... |
| esumsplit 34159 | Split an extended sum into... |
| esummono 34160 | Extended sum is monotonic.... |
| esumpad 34161 | Extend an extended sum by ... |
| esumpad2 34162 | Remove zeroes from an exte... |
| esumadd 34163 | Addition of infinite sums.... |
| esumle 34164 | If all of the terms of an ... |
| gsumesum 34165 | Relate a group sum on ` ( ... |
| esumlub 34166 | The extended sum is the lo... |
| esumaddf 34167 | Addition of infinite sums.... |
| esumlef 34168 | If all of the terms of an ... |
| esumcst 34169 | The extended sum of a cons... |
| esumsnf 34170 | The extended sum of a sing... |
| esumsn 34171 | The extended sum of a sing... |
| esumpr 34172 | Extended sum over a pair. ... |
| esumpr2 34173 | Extended sum over a pair, ... |
| esumrnmpt2 34174 | Rewrite an extended sum in... |
| esumfzf 34175 | Formulating a partial exte... |
| esumfsup 34176 | Formulating an extended su... |
| esumfsupre 34177 | Formulating an extended su... |
| esumss 34178 | Change the index set to a ... |
| esumpinfval 34179 | The value of the extended ... |
| esumpfinvallem 34180 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34181 | The value of the extended ... |
| esumpfinvalf 34182 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34183 | The value of the extended ... |
| esumpcvgval 34184 | The value of the extended ... |
| esumpmono 34185 | The partial sums in an ext... |
| esumcocn 34186 | Lemma for ~ esummulc2 and ... |
| esummulc1 34187 | An extended sum multiplied... |
| esummulc2 34188 | An extended sum multiplied... |
| esumdivc 34189 | An extended sum divided by... |
| hashf2 34190 | Lemma for ~ hasheuni . (C... |
| hasheuni 34191 | The cardinality of a disjo... |
| esumcvg 34192 | The sequence of partial su... |
| esumcvg2 34193 | Simpler version of ~ esumc... |
| esumcvgsum 34194 | The value of the extended ... |
| esumsup 34195 | Express an extended sum as... |
| esumgect 34196 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34197 | All terms of a converging ... |
| esum2dlem 34198 | Lemma for ~ esum2d (finite... |
| esum2d 34199 | Write a double extended su... |
| esumiun 34200 | Sum over a nonnecessarily ... |
| ofceq 34203 | Equality theorem for funct... |
| ofcfval 34204 | Value of an operation appl... |
| ofcval 34205 | Evaluate a function/consta... |
| ofcfn 34206 | The function operation pro... |
| ofcfeqd2 34207 | Equality theorem for funct... |
| ofcfval3 34208 | General value of ` ( F oFC... |
| ofcf 34209 | The function/constant oper... |
| ofcfval2 34210 | The function operation exp... |
| ofcfval4 34211 | The function/constant oper... |
| ofcc 34212 | Left operation by a consta... |
| ofcof 34213 | Relate function operation ... |
| sigaex 34216 | Lemma for ~ issiga and ~ i... |
| sigaval 34217 | The set of sigma-algebra w... |
| issiga 34218 | An alternative definition ... |
| isrnsiga 34219 | The property of being a si... |
| 0elsiga 34220 | A sigma-algebra contains t... |
| baselsiga 34221 | A sigma-algebra contains i... |
| sigasspw 34222 | A sigma-algebra is a set o... |
| sigaclcu 34223 | A sigma-algebra is closed ... |
| sigaclcuni 34224 | A sigma-algebra is closed ... |
| sigaclfu 34225 | A sigma-algebra is closed ... |
| sigaclcu2 34226 | A sigma-algebra is closed ... |
| sigaclfu2 34227 | A sigma-algebra is closed ... |
| sigaclcu3 34228 | A sigma-algebra is closed ... |
| issgon 34229 | Property of being a sigma-... |
| sgon 34230 | A sigma-algebra is a sigma... |
| elsigass 34231 | An element of a sigma-alge... |
| elrnsiga 34232 | Dropping the base informat... |
| isrnsigau 34233 | The property of being a si... |
| unielsiga 34234 | A sigma-algebra contains i... |
| dmvlsiga 34235 | Lebesgue-measurable subset... |
| pwsiga 34236 | Any power set forms a sigm... |
| prsiga 34237 | The smallest possible sigm... |
| sigaclci 34238 | A sigma-algebra is closed ... |
| difelsiga 34239 | A sigma-algebra is closed ... |
| unelsiga 34240 | A sigma-algebra is closed ... |
| inelsiga 34241 | A sigma-algebra is closed ... |
| sigainb 34242 | Building a sigma-algebra f... |
| insiga 34243 | The intersection of a coll... |
| sigagenval 34246 | Value of the generated sig... |
| sigagensiga 34247 | A generated sigma-algebra ... |
| sgsiga 34248 | A generated sigma-algebra ... |
| unisg 34249 | The sigma-algebra generate... |
| dmsigagen 34250 | A sigma-algebra can be gen... |
| sssigagen 34251 | A set is a subset of the s... |
| sssigagen2 34252 | A subset of the generating... |
| elsigagen 34253 | Any element of a set is al... |
| elsigagen2 34254 | Any countable union of ele... |
| sigagenss 34255 | The generated sigma-algebr... |
| sigagenss2 34256 | Sufficient condition for i... |
| sigagenid 34257 | The sigma-algebra generate... |
| ispisys 34258 | The property of being a pi... |
| ispisys2 34259 | The property of being a pi... |
| inelpisys 34260 | Pi-systems are closed unde... |
| sigapisys 34261 | All sigma-algebras are pi-... |
| isldsys 34262 | The property of being a la... |
| pwldsys 34263 | The power set of the unive... |
| unelldsys 34264 | Lambda-systems are closed ... |
| sigaldsys 34265 | All sigma-algebras are lam... |
| ldsysgenld 34266 | The intersection of all la... |
| sigapildsyslem 34267 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34268 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34269 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34270 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34271 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34272 | The lambda system ` E ` ge... |
| dynkin 34273 | Dynkin's lambda-pi theorem... |
| isros 34274 | The property of being a ri... |
| rossspw 34275 | A ring of sets is a collec... |
| 0elros 34276 | A ring of sets contains th... |
| unelros 34277 | A ring of sets is closed u... |
| difelros 34278 | A ring of sets is closed u... |
| inelros 34279 | A ring of sets is closed u... |
| fiunelros 34280 | A ring of sets is closed u... |
| issros 34281 | The property of being a se... |
| srossspw 34282 | A semiring of sets is a co... |
| 0elsros 34283 | A semiring of sets contain... |
| inelsros 34284 | A semiring of sets is clos... |
| diffiunisros 34285 | In semiring of sets, compl... |
| rossros 34286 | Rings of sets are semiring... |
| brsiga 34289 | The Borel Algebra on real ... |
| brsigarn 34290 | The Borel Algebra is a sig... |
| brsigasspwrn 34291 | The Borel Algebra is a set... |
| unibrsiga 34292 | The union of the Borel Alg... |
| cldssbrsiga 34293 | A Borel Algebra contains a... |
| sxval 34296 | Value of the product sigma... |
| sxsiga 34297 | A product sigma-algebra is... |
| sxsigon 34298 | A product sigma-algebra is... |
| sxuni 34299 | The base set of a product ... |
| elsx 34300 | The cartesian product of t... |
| measbase 34303 | The base set of a measure ... |
| measval 34304 | The value of the ` measure... |
| ismeas 34305 | The property of being a me... |
| isrnmeas 34306 | The property of being a me... |
| dmmeas 34307 | The domain of a measure is... |
| measbasedom 34308 | The base set of a measure ... |
| measfrge0 34309 | A measure is a function ov... |
| measfn 34310 | A measure is a function on... |
| measvxrge0 34311 | The values of a measure ar... |
| measvnul 34312 | The measure of the empty s... |
| measge0 34313 | A measure is nonnegative. ... |
| measle0 34314 | If the measure of a given ... |
| measvun 34315 | The measure of a countable... |
| measxun2 34316 | The measure the union of t... |
| measun 34317 | The measure the union of t... |
| measvunilem 34318 | Lemma for ~ measvuni . (C... |
| measvunilem0 34319 | Lemma for ~ measvuni . (C... |
| measvuni 34320 | The measure of a countable... |
| measssd 34321 | A measure is monotone with... |
| measunl 34322 | A measure is sub-additive ... |
| measiuns 34323 | The measure of the union o... |
| measiun 34324 | A measure is sub-additive.... |
| meascnbl 34325 | A measure is continuous fr... |
| measinblem 34326 | Lemma for ~ measinb . (Co... |
| measinb 34327 | Building a measure restric... |
| measres 34328 | Building a measure restric... |
| measinb2 34329 | Building a measure restric... |
| measdivcst 34330 | Division of a measure by a... |
| measdivcstALTV 34331 | Alternate version of ~ mea... |
| cntmeas 34332 | The Counting measure is a ... |
| pwcntmeas 34333 | The counting measure is a ... |
| cntnevol 34334 | Counting and Lebesgue meas... |
| voliune 34335 | The Lebesgue measure funct... |
| volfiniune 34336 | The Lebesgue measure funct... |
| volmeas 34337 | The Lebesgue measure is a ... |
| ddeval1 34340 | Value of the delta measure... |
| ddeval0 34341 | Value of the delta measure... |
| ddemeas 34342 | The Dirac delta measure is... |
| relae 34346 | 'almost everywhere' is a r... |
| brae 34347 | 'almost everywhere' relati... |
| braew 34348 | 'almost everywhere' relati... |
| truae 34349 | A truth holds almost every... |
| aean 34350 | A conjunction holds almost... |
| faeval 34352 | Value of the 'almost every... |
| relfae 34353 | The 'almost everywhere' bu... |
| brfae 34354 | 'almost everywhere' relati... |
| ismbfm 34357 | The predicate " ` F ` is a... |
| elunirnmbfm 34358 | The property of being a me... |
| mbfmfun 34359 | A measurable function is a... |
| mbfmf 34360 | A measurable function as a... |
| mbfmcnvima 34361 | The preimage by a measurab... |
| isanmbfm 34362 | The predicate to be a meas... |
| mbfmbfmOLD 34363 | A measurable function to a... |
| mbfmbfm 34364 | A measurable function to a... |
| mbfmcst 34365 | A constant function is mea... |
| 1stmbfm 34366 | The first projection map i... |
| 2ndmbfm 34367 | The second projection map ... |
| imambfm 34368 | If the sigma-algebra in th... |
| cnmbfm 34369 | A continuous function is m... |
| mbfmco 34370 | The composition of two mea... |
| mbfmco2 34371 | The pair building of two m... |
| mbfmvolf 34372 | Measurable functions with ... |
| elmbfmvol2 34373 | Measurable functions with ... |
| mbfmcnt 34374 | All functions are measurab... |
| br2base 34375 | The base set for the gener... |
| dya2ub 34376 | An upper bound for a dyadi... |
| sxbrsigalem0 34377 | The closed half-spaces of ... |
| sxbrsigalem3 34378 | The sigma-algebra generate... |
| dya2iocival 34379 | The function ` I ` returns... |
| dya2iocress 34380 | Dyadic intervals are subse... |
| dya2iocbrsiga 34381 | Dyadic intervals are Borel... |
| dya2icobrsiga 34382 | Dyadic intervals are Borel... |
| dya2icoseg 34383 | For any point and any clos... |
| dya2icoseg2 34384 | For any point and any open... |
| dya2iocrfn 34385 | The function returning dya... |
| dya2iocct 34386 | The dyadic rectangle set i... |
| dya2iocnrect 34387 | For any point of an open r... |
| dya2iocnei 34388 | For any point of an open s... |
| dya2iocuni 34389 | Every open set of ` ( RR X... |
| dya2iocucvr 34390 | The dyadic rectangular set... |
| sxbrsigalem1 34391 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34392 | The sigma-algebra generate... |
| sxbrsigalem4 34393 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34394 | First direction for ~ sxbr... |
| sxbrsigalem6 34395 | First direction for ~ sxbr... |
| sxbrsiga 34396 | The product sigma-algebra ... |
| omsval 34399 | Value of the function mapp... |
| omsfval 34400 | Value of the outer measure... |
| omscl 34401 | A closure lemma for the co... |
| omsf 34402 | A constructed outer measur... |
| oms0 34403 | A constructed outer measur... |
| omsmon 34404 | A constructed outer measur... |
| omssubaddlem 34405 | For any small margin ` E `... |
| omssubadd 34406 | A constructed outer measur... |
| carsgval 34409 | Value of the Caratheodory ... |
| carsgcl 34410 | Closure of the Caratheodor... |
| elcarsg 34411 | Property of being a Carath... |
| baselcarsg 34412 | The universe set, ` O ` , ... |
| 0elcarsg 34413 | The empty set is Caratheod... |
| carsguni 34414 | The union of all Caratheod... |
| elcarsgss 34415 | Caratheodory measurable se... |
| difelcarsg 34416 | The Caratheodory measurabl... |
| inelcarsg 34417 | The Caratheodory measurabl... |
| unelcarsg 34418 | The Caratheodory-measurabl... |
| difelcarsg2 34419 | The Caratheodory-measurabl... |
| carsgmon 34420 | Utility lemma: Apply mono... |
| carsgsigalem 34421 | Lemma for the following th... |
| fiunelcarsg 34422 | The Caratheodory measurabl... |
| carsgclctunlem1 34423 | Lemma for ~ carsgclctun . ... |
| carsggect 34424 | The outer measure is count... |
| carsgclctunlem2 34425 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34426 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34427 | The Caratheodory measurabl... |
| carsgsiga 34428 | The Caratheodory measurabl... |
| omsmeas 34429 | The restriction of a const... |
| pmeasmono 34430 | This theorem's hypotheses ... |
| pmeasadd 34431 | A premeasure on a ring of ... |
| itgeq12dv 34432 | Equality theorem for an in... |
| sitgval 34438 | Value of the simple functi... |
| issibf 34439 | The predicate " ` F ` is a... |
| sibf0 34440 | The constant zero function... |
| sibfmbl 34441 | A simple function is measu... |
| sibff 34442 | A simple function is a fun... |
| sibfrn 34443 | A simple function has fini... |
| sibfima 34444 | Any preimage of a singleto... |
| sibfinima 34445 | The measure of the interse... |
| sibfof 34446 | Applying function operatio... |
| sitgfval 34447 | Value of the Bochner integ... |
| sitgclg 34448 | Closure of the Bochner int... |
| sitgclbn 34449 | Closure of the Bochner int... |
| sitgclcn 34450 | Closure of the Bochner int... |
| sitgclre 34451 | Closure of the Bochner int... |
| sitg0 34452 | The integral of the consta... |
| sitgf 34453 | The integral for simple fu... |
| sitgaddlemb 34454 | Lemma for * sitgadd . (Co... |
| sitmval 34455 | Value of the simple functi... |
| sitmfval 34456 | Value of the integral dist... |
| sitmcl 34457 | Closure of the integral di... |
| sitmf 34458 | The integral metric as a f... |
| oddpwdc 34460 | Lemma for ~ eulerpart . T... |
| oddpwdcv 34461 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 34462 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 34463 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 34464 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 34465 | Lemma for ~ eulerpart . (... |
| eulerpartlems 34466 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 34467 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 34468 | Lemma for ~ eulerpart . (... |
| eulerpartleme 34469 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 34470 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 34471 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 34472 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 34473 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 34474 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 34475 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 34476 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 34477 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 34478 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 34479 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 34480 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 34481 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 34482 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 34483 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 34484 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 34485 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 34486 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 34487 | Lemma for ~ eulerpart . (... |
| eulerpart 34488 | Euler's theorem on partiti... |
| subiwrd 34491 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 34492 | Length of a subword of an ... |
| iwrdsplit 34493 | Lemma for ~ sseqp1 . (Con... |
| sseqval 34494 | Value of the strong sequen... |
| sseqfv1 34495 | Value of the strong sequen... |
| sseqfn 34496 | A strong recursive sequenc... |
| sseqmw 34497 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 34498 | A strong recursive sequenc... |
| sseqfres 34499 | The first elements in the ... |
| sseqfv2 34500 | Value of the strong sequen... |
| sseqp1 34501 | Value of the strong sequen... |
| fiblem 34504 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 34505 | Value of the Fibonacci seq... |
| fib1 34506 | Value of the Fibonacci seq... |
| fibp1 34507 | Value of the Fibonacci seq... |
| fib2 34508 | Value of the Fibonacci seq... |
| fib3 34509 | Value of the Fibonacci seq... |
| fib4 34510 | Value of the Fibonacci seq... |
| fib5 34511 | Value of the Fibonacci seq... |
| fib6 34512 | Value of the Fibonacci seq... |
| elprob 34515 | The property of being a pr... |
| domprobmeas 34516 | A probability measure is a... |
| domprobsiga 34517 | The domain of a probabilit... |
| probtot 34518 | The probability of the uni... |
| prob01 34519 | A probability is an elemen... |
| probnul 34520 | The probability of the emp... |
| unveldomd 34521 | The universe is an element... |
| unveldom 34522 | The universe is an element... |
| nuleldmp 34523 | The empty set is an elemen... |
| probcun 34524 | The probability of the uni... |
| probun 34525 | The probability of the uni... |
| probdif 34526 | The probability of the dif... |
| probinc 34527 | A probability law is incre... |
| probdsb 34528 | The probability of the com... |
| probmeasd 34529 | A probability measure is a... |
| probvalrnd 34530 | The value of a probability... |
| probtotrnd 34531 | The probability of the uni... |
| totprobd 34532 | Law of total probability, ... |
| totprob 34533 | Law of total probability. ... |
| probfinmeasb 34534 | Build a probability measur... |
| probfinmeasbALTV 34535 | Alternate version of ~ pro... |
| probmeasb 34536 | Build a probability from a... |
| cndprobval 34539 | The value of the condition... |
| cndprobin 34540 | An identity linking condit... |
| cndprob01 34541 | The conditional probabilit... |
| cndprobtot 34542 | The conditional probabilit... |
| cndprobnul 34543 | The conditional probabilit... |
| cndprobprob 34544 | The conditional probabilit... |
| bayesth 34545 | Bayes Theorem. (Contribut... |
| rrvmbfm 34548 | A real-valued random varia... |
| isrrvv 34549 | Elementhood to the set of ... |
| rrvvf 34550 | A real-valued random varia... |
| rrvfn 34551 | A real-valued random varia... |
| rrvdm 34552 | The domain of a random var... |
| rrvrnss 34553 | The range of a random vari... |
| rrvf2 34554 | A real-valued random varia... |
| rrvdmss 34555 | The domain of a random var... |
| rrvfinvima 34556 | For a real-value random va... |
| 0rrv 34557 | The constant function equa... |
| rrvadd 34558 | The sum of two random vari... |
| rrvmulc 34559 | A random variable multipli... |
| rrvsum 34560 | An indexed sum of random v... |
| boolesineq 34561 | Boole's inequality (union ... |
| orvcval 34564 | Value of the preimage mapp... |
| orvcval2 34565 | Another way to express the... |
| elorvc 34566 | Elementhood of a preimage.... |
| orvcval4 34567 | The value of the preimage ... |
| orvcoel 34568 | If the relation produces o... |
| orvccel 34569 | If the relation produces c... |
| elorrvc 34570 | Elementhood of a preimage ... |
| orrvcval4 34571 | The value of the preimage ... |
| orrvcoel 34572 | If the relation produces o... |
| orrvccel 34573 | If the relation produces c... |
| orvcgteel 34574 | Preimage maps produced by ... |
| orvcelval 34575 | Preimage maps produced by ... |
| orvcelel 34576 | Preimage maps produced by ... |
| dstrvval 34577 | The value of the distribut... |
| dstrvprob 34578 | The distribution of a rand... |
| orvclteel 34579 | Preimage maps produced by ... |
| dstfrvel 34580 | Elementhood of preimage ma... |
| dstfrvunirn 34581 | The limit of all preimage ... |
| orvclteinc 34582 | Preimage maps produced by ... |
| dstfrvinc 34583 | A cumulative distribution ... |
| dstfrvclim1 34584 | The limit of the cumulativ... |
| coinfliplem 34585 | Division in the extended r... |
| coinflipprob 34586 | The ` P ` we defined for c... |
| coinflipspace 34587 | The space of our coin-flip... |
| coinflipuniv 34588 | The universe of our coin-f... |
| coinfliprv 34589 | The ` X ` we defined for c... |
| coinflippv 34590 | The probability of heads i... |
| coinflippvt 34591 | The probability of tails i... |
| ballotlemoex 34592 | ` O ` is a set. (Contribu... |
| ballotlem1 34593 | The size of the universe i... |
| ballotlemelo 34594 | Elementhood in ` O ` . (C... |
| ballotlem2 34595 | The probability that the f... |
| ballotlemfval 34596 | The value of ` F ` . (Con... |
| ballotlemfelz 34597 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 34598 | If the ` J ` th ballot is ... |
| ballotlemfc0 34599 | ` F ` takes value 0 betwee... |
| ballotlemfcc 34600 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 34601 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 34602 | ` ( F `` C ) ` always star... |
| ballotleme 34603 | Elements of ` E ` . (Cont... |
| ballotlemodife 34604 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 34605 | If the first pick is a vot... |
| ballotlem5 34606 | If A is not ahead througho... |
| ballotlemi 34607 | Value of ` I ` for a given... |
| ballotlemiex 34608 | Properties of ` ( I `` C )... |
| ballotlemi1 34609 | The first tie cannot be re... |
| ballotlemii 34610 | The first tie cannot be re... |
| ballotlemsup 34611 | The set of zeroes of ` F `... |
| ballotlemimin 34612 | ` ( I `` C ) ` is the firs... |
| ballotlemic 34613 | If the first vote is for B... |
| ballotlem1c 34614 | If the first vote is for A... |
| ballotlemsval 34615 | Value of ` S ` . (Contrib... |
| ballotlemsv 34616 | Value of ` S ` evaluated a... |
| ballotlemsgt1 34617 | ` S ` maps values less tha... |
| ballotlemsdom 34618 | Domain of ` S ` for a give... |
| ballotlemsel1i 34619 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 34620 | The defined ` S ` is a bij... |
| ballotlemsi 34621 | The image by ` S ` of the ... |
| ballotlemsima 34622 | The image by ` S ` of an i... |
| ballotlemieq 34623 | If two countings share the... |
| ballotlemrval 34624 | Value of ` R ` . (Contrib... |
| ballotlemscr 34625 | The image of ` ( R `` C ) ... |
| ballotlemrv 34626 | Value of ` R ` evaluated a... |
| ballotlemrv1 34627 | Value of ` R ` before the ... |
| ballotlemrv2 34628 | Value of ` R ` after the t... |
| ballotlemro 34629 | Range of ` R ` is included... |
| ballotlemgval 34630 | Expand the value of ` .^ `... |
| ballotlemgun 34631 | A property of the defined ... |
| ballotlemfg 34632 | Express the value of ` ( F... |
| ballotlemfrc 34633 | Express the value of ` ( F... |
| ballotlemfrci 34634 | Reverse counting preserves... |
| ballotlemfrceq 34635 | Value of ` F ` for a rever... |
| ballotlemfrcn0 34636 | Value of ` F ` for a rever... |
| ballotlemrc 34637 | Range of ` R ` . (Contrib... |
| ballotlemirc 34638 | Applying ` R ` does not ch... |
| ballotlemrinv0 34639 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 34640 | ` R ` is its own inverse :... |
| ballotlem1ri 34641 | When the vote on the first... |
| ballotlem7 34642 | ` R ` is a bijection betwe... |
| ballotlem8 34643 | There are as many counting... |
| ballotth 34644 | Bertrand's ballot problem ... |
| fzssfzo 34645 | Condition for an integer i... |
| gsumncl 34646 | Closure of a group sum in ... |
| gsumnunsn 34647 | Closure of a group sum in ... |
| ccatmulgnn0dir 34648 | Concatenation of words fol... |
| ofcccat 34649 | Letterwise operations on w... |
| ofcs1 34650 | Letterwise operations on a... |
| ofcs2 34651 | Letterwise operations on a... |
| plymul02 34652 | Product of a polynomial wi... |
| plymulx0 34653 | Coefficients of a polynomi... |
| plymulx 34654 | Coefficients of a polynomi... |
| plyrecld 34655 | Closure of a polynomial wi... |
| signsplypnf 34656 | The quotient of a polynomi... |
| signsply0 34657 | Lemma for the rule of sign... |
| signspval 34658 | The value of the skipping ... |
| signsw0glem 34659 | Neutral element property o... |
| signswbase 34660 | The base of ` W ` is the u... |
| signswplusg 34661 | The operation of ` W ` . ... |
| signsw0g 34662 | The neutral element of ` W... |
| signswmnd 34663 | ` W ` is a monoid structur... |
| signswrid 34664 | The zero-skipping operatio... |
| signswlid 34665 | The zero-skipping operatio... |
| signswn0 34666 | The zero-skipping operatio... |
| signswch 34667 | The zero-skipping operatio... |
| signslema 34668 | Computational part of ~~? ... |
| signstfv 34669 | Value of the zero-skipping... |
| signstfval 34670 | Value of the zero-skipping... |
| signstcl 34671 | Closure of the zero skippi... |
| signstf 34672 | The zero skipping sign wor... |
| signstlen 34673 | Length of the zero skippin... |
| signstf0 34674 | Sign of a single letter wo... |
| signstfvn 34675 | Zero-skipping sign in a wo... |
| signsvtn0 34676 | If the last letter is nonz... |
| signstfvp 34677 | Zero-skipping sign in a wo... |
| signstfvneq0 34678 | In case the first letter i... |
| signstfvcl 34679 | Closure of the zero skippi... |
| signstfvc 34680 | Zero-skipping sign in a wo... |
| signstres 34681 | Restriction of a zero skip... |
| signstfveq0a 34682 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 34683 | In case the last letter is... |
| signsvvfval 34684 | The value of ` V ` , which... |
| signsvvf 34685 | ` V ` is a function. (Con... |
| signsvf0 34686 | There is no change of sign... |
| signsvf1 34687 | In a single-letter word, w... |
| signsvfn 34688 | Number of changes in a wor... |
| signsvtp 34689 | Adding a letter of the sam... |
| signsvtn 34690 | Adding a letter of a diffe... |
| signsvfpn 34691 | Adding a letter of the sam... |
| signsvfnn 34692 | Adding a letter of a diffe... |
| signlem0 34693 | Adding a zero as the highe... |
| signshf 34694 | ` H ` , corresponding to t... |
| signshwrd 34695 | ` H ` , corresponding to t... |
| signshlen 34696 | Length of ` H ` , correspo... |
| signshnz 34697 | ` H ` is not the empty wor... |
| iblidicc 34698 | The identity function is i... |
| rpsqrtcn 34699 | Continuity of the real pos... |
| divsqrtid 34700 | A real number divided by i... |
| cxpcncf1 34701 | The power function on comp... |
| efmul2picn 34702 | Multiplying by ` ( _i x. (... |
| fct2relem 34703 | Lemma for ~ ftc2re . (Con... |
| ftc2re 34704 | The Fundamental Theorem of... |
| fdvposlt 34705 | Functions with a positive ... |
| fdvneggt 34706 | Functions with a negative ... |
| fdvposle 34707 | Functions with a nonnegati... |
| fdvnegge 34708 | Functions with a nonpositi... |
| prodfzo03 34709 | A product of three factors... |
| actfunsnf1o 34710 | The action ` F ` of extend... |
| actfunsnrndisj 34711 | The action ` F ` of extend... |
| itgexpif 34712 | The basis for the circle m... |
| fsum2dsub 34713 | Lemma for ~ breprexp - Re-... |
| reprval 34716 | Value of the representatio... |
| repr0 34717 | There is exactly one repre... |
| reprf 34718 | Members of the representat... |
| reprsum 34719 | Sums of values of the memb... |
| reprle 34720 | Upper bound to the terms i... |
| reprsuc 34721 | Express the representation... |
| reprfi 34722 | Bounded representations ar... |
| reprss 34723 | Representations with terms... |
| reprinrn 34724 | Representations with term ... |
| reprlt 34725 | There are no representatio... |
| hashreprin 34726 | Express a sum of represent... |
| reprgt 34727 | There are no representatio... |
| reprinfz1 34728 | For the representation of ... |
| reprfi2 34729 | Corollary of ~ reprinfz1 .... |
| reprfz1 34730 | Corollary of ~ reprinfz1 .... |
| hashrepr 34731 | Develop the number of repr... |
| reprpmtf1o 34732 | Transposing ` 0 ` and ` X ... |
| reprdifc 34733 | Express the representation... |
| chpvalz 34734 | Value of the second Chebys... |
| chtvalz 34735 | Value of the Chebyshev fun... |
| breprexplema 34736 | Lemma for ~ breprexp (indu... |
| breprexplemb 34737 | Lemma for ~ breprexp (clos... |
| breprexplemc 34738 | Lemma for ~ breprexp (indu... |
| breprexp 34739 | Express the ` S ` th power... |
| breprexpnat 34740 | Express the ` S ` th power... |
| vtsval 34743 | Value of the Vinogradov tr... |
| vtscl 34744 | Closure of the Vinogradov ... |
| vtsprod 34745 | Express the Vinogradov tri... |
| circlemeth 34746 | The Hardy, Littlewood and ... |
| circlemethnat 34747 | The Hardy, Littlewood and ... |
| circlevma 34748 | The Circle Method, where t... |
| circlemethhgt 34749 | The circle method, where t... |
| hgt750lemc 34753 | An upper bound to the summ... |
| hgt750lemd 34754 | An upper bound to the summ... |
| hgt749d 34755 | A deduction version of ~ a... |
| logdivsqrle 34756 | Conditions for ` ( ( log `... |
| hgt750lem 34757 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 34758 | Decimal multiplication gal... |
| hgt750lemf 34759 | Lemma for the statement 7.... |
| hgt750lemg 34760 | Lemma for the statement 7.... |
| oddprm2 34761 | Two ways to write the set ... |
| hgt750lemb 34762 | An upper bound on the cont... |
| hgt750lema 34763 | An upper bound on the cont... |
| hgt750leme 34764 | An upper bound on the cont... |
| tgoldbachgnn 34765 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 34766 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 34767 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 34768 | Odd integers greater than ... |
| tgoldbachgt 34769 | Odd integers greater than ... |
| istrkg2d 34772 | Property of fulfilling dim... |
| axtglowdim2ALTV 34773 | Alternate version of ~ axt... |
| axtgupdim2ALTV 34774 | Alternate version of ~ axt... |
| afsval 34777 | Value of the AFS relation ... |
| brafs 34778 | Binary relation form of th... |
| tg5segofs 34779 | Rephrase ~ axtg5seg using ... |
| lpadval 34782 | Value of the ` leftpad ` f... |
| lpadlem1 34783 | Lemma for the ` leftpad ` ... |
| lpadlem3 34784 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 34785 | Length of a left-padded wo... |
| lpadlem2 34786 | Lemma for the ` leftpad ` ... |
| lpadlen2 34787 | Length of a left-padded wo... |
| lpadmax 34788 | Length of a left-padded wo... |
| lpadleft 34789 | The contents of prefix of ... |
| lpadright 34790 | The suffix of a left-padde... |
| bnj170 34803 | ` /\ ` -manipulation. (Co... |
| bnj240 34804 | ` /\ ` -manipulation. (Co... |
| bnj248 34805 | ` /\ ` -manipulation. (Co... |
| bnj250 34806 | ` /\ ` -manipulation. (Co... |
| bnj251 34807 | ` /\ ` -manipulation. (Co... |
| bnj252 34808 | ` /\ ` -manipulation. (Co... |
| bnj253 34809 | ` /\ ` -manipulation. (Co... |
| bnj255 34810 | ` /\ ` -manipulation. (Co... |
| bnj256 34811 | ` /\ ` -manipulation. (Co... |
| bnj257 34812 | ` /\ ` -manipulation. (Co... |
| bnj258 34813 | ` /\ ` -manipulation. (Co... |
| bnj268 34814 | ` /\ ` -manipulation. (Co... |
| bnj290 34815 | ` /\ ` -manipulation. (Co... |
| bnj291 34816 | ` /\ ` -manipulation. (Co... |
| bnj312 34817 | ` /\ ` -manipulation. (Co... |
| bnj334 34818 | ` /\ ` -manipulation. (Co... |
| bnj345 34819 | ` /\ ` -manipulation. (Co... |
| bnj422 34820 | ` /\ ` -manipulation. (Co... |
| bnj432 34821 | ` /\ ` -manipulation. (Co... |
| bnj446 34822 | ` /\ ` -manipulation. (Co... |
| bnj23 34823 | First-order logic and set ... |
| bnj31 34824 | First-order logic and set ... |
| bnj62 34825 | First-order logic and set ... |
| bnj89 34826 | First-order logic and set ... |
| bnj90 34827 | First-order logic and set ... |
| bnj101 34828 | First-order logic and set ... |
| bnj105 34829 | First-order logic and set ... |
| bnj115 34830 | First-order logic and set ... |
| bnj132 34831 | First-order logic and set ... |
| bnj133 34832 | First-order logic and set ... |
| bnj156 34833 | First-order logic and set ... |
| bnj158 34834 | First-order logic and set ... |
| bnj168 34835 | First-order logic and set ... |
| bnj206 34836 | First-order logic and set ... |
| bnj216 34837 | First-order logic and set ... |
| bnj219 34838 | First-order logic and set ... |
| bnj226 34839 | First-order logic and set ... |
| bnj228 34840 | First-order logic and set ... |
| bnj519 34841 | First-order logic and set ... |
| bnj524 34842 | First-order logic and set ... |
| bnj525 34843 | First-order logic and set ... |
| bnj534 34844 | First-order logic and set ... |
| bnj538 34845 | First-order logic and set ... |
| bnj529 34846 | First-order logic and set ... |
| bnj551 34847 | First-order logic and set ... |
| bnj563 34848 | First-order logic and set ... |
| bnj564 34849 | First-order logic and set ... |
| bnj593 34850 | First-order logic and set ... |
| bnj596 34851 | First-order logic and set ... |
| bnj610 34852 | Pass from equality ( ` x =... |
| bnj642 34853 | ` /\ ` -manipulation. (Co... |
| bnj643 34854 | ` /\ ` -manipulation. (Co... |
| bnj645 34855 | ` /\ ` -manipulation. (Co... |
| bnj658 34856 | ` /\ ` -manipulation. (Co... |
| bnj667 34857 | ` /\ ` -manipulation. (Co... |
| bnj705 34858 | ` /\ ` -manipulation. (Co... |
| bnj706 34859 | ` /\ ` -manipulation. (Co... |
| bnj707 34860 | ` /\ ` -manipulation. (Co... |
| bnj708 34861 | ` /\ ` -manipulation. (Co... |
| bnj721 34862 | ` /\ ` -manipulation. (Co... |
| bnj832 34863 | ` /\ ` -manipulation. (Co... |
| bnj835 34864 | ` /\ ` -manipulation. (Co... |
| bnj836 34865 | ` /\ ` -manipulation. (Co... |
| bnj837 34866 | ` /\ ` -manipulation. (Co... |
| bnj769 34867 | ` /\ ` -manipulation. (Co... |
| bnj770 34868 | ` /\ ` -manipulation. (Co... |
| bnj771 34869 | ` /\ ` -manipulation. (Co... |
| bnj887 34870 | ` /\ ` -manipulation. (Co... |
| bnj918 34871 | First-order logic and set ... |
| bnj919 34872 | First-order logic and set ... |
| bnj923 34873 | First-order logic and set ... |
| bnj927 34874 | First-order logic and set ... |
| bnj931 34875 | First-order logic and set ... |
| bnj937 34876 | First-order logic and set ... |
| bnj941 34877 | First-order logic and set ... |
| bnj945 34878 | Technical lemma for ~ bnj6... |
| bnj946 34879 | First-order logic and set ... |
| bnj951 34880 | ` /\ ` -manipulation. (Co... |
| bnj956 34881 | First-order logic and set ... |
| bnj976 34882 | First-order logic and set ... |
| bnj982 34883 | First-order logic and set ... |
| bnj1019 34884 | First-order logic and set ... |
| bnj1023 34885 | First-order logic and set ... |
| bnj1095 34886 | First-order logic and set ... |
| bnj1096 34887 | First-order logic and set ... |
| bnj1098 34888 | First-order logic and set ... |
| bnj1101 34889 | First-order logic and set ... |
| bnj1113 34890 | First-order logic and set ... |
| bnj1109 34891 | First-order logic and set ... |
| bnj1131 34892 | First-order logic and set ... |
| bnj1138 34893 | First-order logic and set ... |
| bnj1142 34894 | First-order logic and set ... |
| bnj1143 34895 | First-order logic and set ... |
| bnj1146 34896 | First-order logic and set ... |
| bnj1149 34897 | First-order logic and set ... |
| bnj1185 34898 | First-order logic and set ... |
| bnj1196 34899 | First-order logic and set ... |
| bnj1198 34900 | First-order logic and set ... |
| bnj1209 34901 | First-order logic and set ... |
| bnj1211 34902 | First-order logic and set ... |
| bnj1213 34903 | First-order logic and set ... |
| bnj1212 34904 | First-order logic and set ... |
| bnj1219 34905 | First-order logic and set ... |
| bnj1224 34906 | First-order logic and set ... |
| bnj1230 34907 | First-order logic and set ... |
| bnj1232 34908 | First-order logic and set ... |
| bnj1235 34909 | First-order logic and set ... |
| bnj1239 34910 | First-order logic and set ... |
| bnj1238 34911 | First-order logic and set ... |
| bnj1241 34912 | First-order logic and set ... |
| bnj1247 34913 | First-order logic and set ... |
| bnj1254 34914 | First-order logic and set ... |
| bnj1262 34915 | First-order logic and set ... |
| bnj1266 34916 | First-order logic and set ... |
| bnj1265 34917 | First-order logic and set ... |
| bnj1275 34918 | First-order logic and set ... |
| bnj1276 34919 | First-order logic and set ... |
| bnj1292 34920 | First-order logic and set ... |
| bnj1293 34921 | First-order logic and set ... |
| bnj1294 34922 | First-order logic and set ... |
| bnj1299 34923 | First-order logic and set ... |
| bnj1304 34924 | First-order logic and set ... |
| bnj1316 34925 | First-order logic and set ... |
| bnj1317 34926 | First-order logic and set ... |
| bnj1322 34927 | First-order logic and set ... |
| bnj1340 34928 | First-order logic and set ... |
| bnj1345 34929 | First-order logic and set ... |
| bnj1350 34930 | First-order logic and set ... |
| bnj1351 34931 | First-order logic and set ... |
| bnj1352 34932 | First-order logic and set ... |
| bnj1361 34933 | First-order logic and set ... |
| bnj1366 34934 | First-order logic and set ... |
| bnj1379 34935 | First-order logic and set ... |
| bnj1383 34936 | First-order logic and set ... |
| bnj1385 34937 | First-order logic and set ... |
| bnj1386 34938 | First-order logic and set ... |
| bnj1397 34939 | First-order logic and set ... |
| bnj1400 34940 | First-order logic and set ... |
| bnj1405 34941 | First-order logic and set ... |
| bnj1422 34942 | First-order logic and set ... |
| bnj1424 34943 | First-order logic and set ... |
| bnj1436 34944 | First-order logic and set ... |
| bnj1441 34945 | First-order logic and set ... |
| bnj1441g 34946 | First-order logic and set ... |
| bnj1454 34947 | First-order logic and set ... |
| bnj1459 34948 | First-order logic and set ... |
| bnj1464 34949 | Conversion of implicit sub... |
| bnj1465 34950 | First-order logic and set ... |
| bnj1468 34951 | Conversion of implicit sub... |
| bnj1476 34952 | First-order logic and set ... |
| bnj1502 34953 | First-order logic and set ... |
| bnj1503 34954 | First-order logic and set ... |
| bnj1517 34955 | First-order logic and set ... |
| bnj1521 34956 | First-order logic and set ... |
| bnj1533 34957 | First-order logic and set ... |
| bnj1534 34958 | First-order logic and set ... |
| bnj1536 34959 | First-order logic and set ... |
| bnj1538 34960 | First-order logic and set ... |
| bnj1541 34961 | First-order logic and set ... |
| bnj1542 34962 | First-order logic and set ... |
| bnj110 34963 | Well-founded induction res... |
| bnj157 34964 | Well-founded induction res... |
| bnj66 34965 | Technical lemma for ~ bnj6... |
| bnj91 34966 | First-order logic and set ... |
| bnj92 34967 | First-order logic and set ... |
| bnj93 34968 | Technical lemma for ~ bnj9... |
| bnj95 34969 | Technical lemma for ~ bnj1... |
| bnj96 34970 | Technical lemma for ~ bnj1... |
| bnj97 34971 | Technical lemma for ~ bnj1... |
| bnj98 34972 | Technical lemma for ~ bnj1... |
| bnj106 34973 | First-order logic and set ... |
| bnj118 34974 | First-order logic and set ... |
| bnj121 34975 | First-order logic and set ... |
| bnj124 34976 | Technical lemma for ~ bnj1... |
| bnj125 34977 | Technical lemma for ~ bnj1... |
| bnj126 34978 | Technical lemma for ~ bnj1... |
| bnj130 34979 | Technical lemma for ~ bnj1... |
| bnj149 34980 | Technical lemma for ~ bnj1... |
| bnj150 34981 | Technical lemma for ~ bnj1... |
| bnj151 34982 | Technical lemma for ~ bnj1... |
| bnj154 34983 | Technical lemma for ~ bnj1... |
| bnj155 34984 | Technical lemma for ~ bnj1... |
| bnj153 34985 | Technical lemma for ~ bnj8... |
| bnj207 34986 | Technical lemma for ~ bnj8... |
| bnj213 34987 | First-order logic and set ... |
| bnj222 34988 | Technical lemma for ~ bnj2... |
| bnj229 34989 | Technical lemma for ~ bnj5... |
| bnj517 34990 | Technical lemma for ~ bnj5... |
| bnj518 34991 | Technical lemma for ~ bnj8... |
| bnj523 34992 | Technical lemma for ~ bnj8... |
| bnj526 34993 | Technical lemma for ~ bnj8... |
| bnj528 34994 | Technical lemma for ~ bnj8... |
| bnj535 34995 | Technical lemma for ~ bnj8... |
| bnj539 34996 | Technical lemma for ~ bnj8... |
| bnj540 34997 | Technical lemma for ~ bnj8... |
| bnj543 34998 | Technical lemma for ~ bnj8... |
| bnj544 34999 | Technical lemma for ~ bnj8... |
| bnj545 35000 | Technical lemma for ~ bnj8... |
| bnj546 35001 | Technical lemma for ~ bnj8... |
| bnj548 35002 | Technical lemma for ~ bnj8... |
| bnj553 35003 | Technical lemma for ~ bnj8... |
| bnj554 35004 | Technical lemma for ~ bnj8... |
| bnj556 35005 | Technical lemma for ~ bnj8... |
| bnj557 35006 | Technical lemma for ~ bnj8... |
| bnj558 35007 | Technical lemma for ~ bnj8... |
| bnj561 35008 | Technical lemma for ~ bnj8... |
| bnj562 35009 | Technical lemma for ~ bnj8... |
| bnj570 35010 | Technical lemma for ~ bnj8... |
| bnj571 35011 | Technical lemma for ~ bnj8... |
| bnj605 35012 | Technical lemma. This lem... |
| bnj581 35013 | Technical lemma for ~ bnj5... |
| bnj589 35014 | Technical lemma for ~ bnj8... |
| bnj590 35015 | Technical lemma for ~ bnj8... |
| bnj591 35016 | Technical lemma for ~ bnj8... |
| bnj594 35017 | Technical lemma for ~ bnj8... |
| bnj580 35018 | Technical lemma for ~ bnj5... |
| bnj579 35019 | Technical lemma for ~ bnj8... |
| bnj602 35020 | Equality theorem for the `... |
| bnj607 35021 | Technical lemma for ~ bnj8... |
| bnj609 35022 | Technical lemma for ~ bnj8... |
| bnj611 35023 | Technical lemma for ~ bnj8... |
| bnj600 35024 | Technical lemma for ~ bnj8... |
| bnj601 35025 | Technical lemma for ~ bnj8... |
| bnj852 35026 | Technical lemma for ~ bnj6... |
| bnj864 35027 | Technical lemma for ~ bnj6... |
| bnj865 35028 | Technical lemma for ~ bnj6... |
| bnj873 35029 | Technical lemma for ~ bnj6... |
| bnj849 35030 | Technical lemma for ~ bnj6... |
| bnj882 35031 | Definition (using hypothes... |
| bnj18eq1 35032 | Equality theorem for trans... |
| bnj893 35033 | Property of ` _trCl ` . U... |
| bnj900 35034 | Technical lemma for ~ bnj6... |
| bnj906 35035 | Property of ` _trCl ` . (... |
| bnj908 35036 | Technical lemma for ~ bnj6... |
| bnj911 35037 | Technical lemma for ~ bnj6... |
| bnj916 35038 | Technical lemma for ~ bnj6... |
| bnj917 35039 | Technical lemma for ~ bnj6... |
| bnj934 35040 | Technical lemma for ~ bnj6... |
| bnj929 35041 | Technical lemma for ~ bnj6... |
| bnj938 35042 | Technical lemma for ~ bnj6... |
| bnj944 35043 | Technical lemma for ~ bnj6... |
| bnj953 35044 | Technical lemma for ~ bnj6... |
| bnj958 35045 | Technical lemma for ~ bnj6... |
| bnj1000 35046 | Technical lemma for ~ bnj8... |
| bnj965 35047 | Technical lemma for ~ bnj8... |
| bnj964 35048 | Technical lemma for ~ bnj6... |
| bnj966 35049 | Technical lemma for ~ bnj6... |
| bnj967 35050 | Technical lemma for ~ bnj6... |
| bnj969 35051 | Technical lemma for ~ bnj6... |
| bnj970 35052 | Technical lemma for ~ bnj6... |
| bnj910 35053 | Technical lemma for ~ bnj6... |
| bnj978 35054 | Technical lemma for ~ bnj6... |
| bnj981 35055 | Technical lemma for ~ bnj6... |
| bnj983 35056 | Technical lemma for ~ bnj6... |
| bnj984 35057 | Technical lemma for ~ bnj6... |
| bnj985v 35058 | Version of ~ bnj985 with a... |
| bnj985 35059 | Technical lemma for ~ bnj6... |
| bnj986 35060 | Technical lemma for ~ bnj6... |
| bnj996 35061 | Technical lemma for ~ bnj6... |
| bnj998 35062 | Technical lemma for ~ bnj6... |
| bnj999 35063 | Technical lemma for ~ bnj6... |
| bnj1001 35064 | Technical lemma for ~ bnj6... |
| bnj1006 35065 | Technical lemma for ~ bnj6... |
| bnj1014 35066 | Technical lemma for ~ bnj6... |
| bnj1015 35067 | Technical lemma for ~ bnj6... |
| bnj1018g 35068 | Version of ~ bnj1018 with ... |
| bnj1018 35069 | Technical lemma for ~ bnj6... |
| bnj1020 35070 | Technical lemma for ~ bnj6... |
| bnj1021 35071 | Technical lemma for ~ bnj6... |
| bnj907 35072 | Technical lemma for ~ bnj6... |
| bnj1029 35073 | Property of ` _trCl ` . (... |
| bnj1033 35074 | Technical lemma for ~ bnj6... |
| bnj1034 35075 | Technical lemma for ~ bnj6... |
| bnj1039 35076 | Technical lemma for ~ bnj6... |
| bnj1040 35077 | Technical lemma for ~ bnj6... |
| bnj1047 35078 | Technical lemma for ~ bnj6... |
| bnj1049 35079 | Technical lemma for ~ bnj6... |
| bnj1052 35080 | Technical lemma for ~ bnj6... |
| bnj1053 35081 | Technical lemma for ~ bnj6... |
| bnj1071 35082 | Technical lemma for ~ bnj6... |
| bnj1083 35083 | Technical lemma for ~ bnj6... |
| bnj1090 35084 | Technical lemma for ~ bnj6... |
| bnj1093 35085 | Technical lemma for ~ bnj6... |
| bnj1097 35086 | Technical lemma for ~ bnj6... |
| bnj1110 35087 | Technical lemma for ~ bnj6... |
| bnj1112 35088 | Technical lemma for ~ bnj6... |
| bnj1118 35089 | Technical lemma for ~ bnj6... |
| bnj1121 35090 | Technical lemma for ~ bnj6... |
| bnj1123 35091 | Technical lemma for ~ bnj6... |
| bnj1030 35092 | Technical lemma for ~ bnj6... |
| bnj1124 35093 | Property of ` _trCl ` . (... |
| bnj1133 35094 | Technical lemma for ~ bnj6... |
| bnj1128 35095 | Technical lemma for ~ bnj6... |
| bnj1127 35096 | Property of ` _trCl ` . (... |
| bnj1125 35097 | Property of ` _trCl ` . (... |
| bnj1145 35098 | Technical lemma for ~ bnj6... |
| bnj1147 35099 | Property of ` _trCl ` . (... |
| bnj1137 35100 | Property of ` _trCl ` . (... |
| bnj1148 35101 | Property of ` _pred ` . (... |
| bnj1136 35102 | Technical lemma for ~ bnj6... |
| bnj1152 35103 | Technical lemma for ~ bnj6... |
| bnj1154 35104 | Property of ` Fr ` . (Con... |
| bnj1171 35105 | Technical lemma for ~ bnj6... |
| bnj1172 35106 | Technical lemma for ~ bnj6... |
| bnj1173 35107 | Technical lemma for ~ bnj6... |
| bnj1174 35108 | Technical lemma for ~ bnj6... |
| bnj1175 35109 | Technical lemma for ~ bnj6... |
| bnj1176 35110 | Technical lemma for ~ bnj6... |
| bnj1177 35111 | Technical lemma for ~ bnj6... |
| bnj1186 35112 | Technical lemma for ~ bnj6... |
| bnj1190 35113 | Technical lemma for ~ bnj6... |
| bnj1189 35114 | Technical lemma for ~ bnj6... |
| bnj69 35115 | Existence of a minimal ele... |
| bnj1228 35116 | Existence of a minimal ele... |
| bnj1204 35117 | Well-founded induction. T... |
| bnj1234 35118 | Technical lemma for ~ bnj6... |
| bnj1245 35119 | Technical lemma for ~ bnj6... |
| bnj1256 35120 | Technical lemma for ~ bnj6... |
| bnj1259 35121 | Technical lemma for ~ bnj6... |
| bnj1253 35122 | Technical lemma for ~ bnj6... |
| bnj1279 35123 | Technical lemma for ~ bnj6... |
| bnj1286 35124 | Technical lemma for ~ bnj6... |
| bnj1280 35125 | Technical lemma for ~ bnj6... |
| bnj1296 35126 | Technical lemma for ~ bnj6... |
| bnj1309 35127 | Technical lemma for ~ bnj6... |
| bnj1307 35128 | Technical lemma for ~ bnj6... |
| bnj1311 35129 | Technical lemma for ~ bnj6... |
| bnj1318 35130 | Technical lemma for ~ bnj6... |
| bnj1326 35131 | Technical lemma for ~ bnj6... |
| bnj1321 35132 | Technical lemma for ~ bnj6... |
| bnj1364 35133 | Property of ` _FrSe ` . (... |
| bnj1371 35134 | Technical lemma for ~ bnj6... |
| bnj1373 35135 | Technical lemma for ~ bnj6... |
| bnj1374 35136 | Technical lemma for ~ bnj6... |
| bnj1384 35137 | Technical lemma for ~ bnj6... |
| bnj1388 35138 | Technical lemma for ~ bnj6... |
| bnj1398 35139 | Technical lemma for ~ bnj6... |
| bnj1413 35140 | Property of ` _trCl ` . (... |
| bnj1408 35141 | Technical lemma for ~ bnj1... |
| bnj1414 35142 | Property of ` _trCl ` . (... |
| bnj1415 35143 | Technical lemma for ~ bnj6... |
| bnj1416 35144 | Technical lemma for ~ bnj6... |
| bnj1418 35145 | Property of ` _pred ` . (... |
| bnj1417 35146 | Technical lemma for ~ bnj6... |
| bnj1421 35147 | Technical lemma for ~ bnj6... |
| bnj1444 35148 | Technical lemma for ~ bnj6... |
| bnj1445 35149 | Technical lemma for ~ bnj6... |
| bnj1446 35150 | Technical lemma for ~ bnj6... |
| bnj1447 35151 | Technical lemma for ~ bnj6... |
| bnj1448 35152 | Technical lemma for ~ bnj6... |
| bnj1449 35153 | Technical lemma for ~ bnj6... |
| bnj1442 35154 | Technical lemma for ~ bnj6... |
| bnj1450 35155 | Technical lemma for ~ bnj6... |
| bnj1423 35156 | Technical lemma for ~ bnj6... |
| bnj1452 35157 | Technical lemma for ~ bnj6... |
| bnj1466 35158 | Technical lemma for ~ bnj6... |
| bnj1467 35159 | Technical lemma for ~ bnj6... |
| bnj1463 35160 | Technical lemma for ~ bnj6... |
| bnj1489 35161 | Technical lemma for ~ bnj6... |
| bnj1491 35162 | Technical lemma for ~ bnj6... |
| bnj1312 35163 | Technical lemma for ~ bnj6... |
| bnj1493 35164 | Technical lemma for ~ bnj6... |
| bnj1497 35165 | Technical lemma for ~ bnj6... |
| bnj1498 35166 | Technical lemma for ~ bnj6... |
| bnj60 35167 | Well-founded recursion, pa... |
| bnj1514 35168 | Technical lemma for ~ bnj1... |
| bnj1518 35169 | Technical lemma for ~ bnj1... |
| bnj1519 35170 | Technical lemma for ~ bnj1... |
| bnj1520 35171 | Technical lemma for ~ bnj1... |
| bnj1501 35172 | Technical lemma for ~ bnj1... |
| bnj1500 35173 | Well-founded recursion, pa... |
| bnj1525 35174 | Technical lemma for ~ bnj1... |
| bnj1529 35175 | Technical lemma for ~ bnj1... |
| bnj1523 35176 | Technical lemma for ~ bnj1... |
| bnj1522 35177 | Well-founded recursion, pa... |
| nfan1c 35178 | Variant of ~ nfan and comm... |
| cbvex1v 35179 | Rule used to change bound ... |
| dvelimalcased 35180 | Eliminate a disjoint varia... |
| dvelimalcasei 35181 | Eliminate a disjoint varia... |
| dvelimexcased 35182 | Eliminate a disjoint varia... |
| dvelimexcasei 35183 | Eliminate a disjoint varia... |
| exdifsn 35184 | There exists an element in... |
| srcmpltd 35185 | If a statement is true for... |
| prsrcmpltd 35186 | If a statement is true for... |
| axsepg2 35187 | A generalization of ~ ax-s... |
| axsepg2ALT 35188 | Alternate proof of ~ axsep... |
| dff15 35189 | A one-to-one function in t... |
| f1resveqaeq 35190 | If a function restricted t... |
| f1resrcmplf1dlem 35191 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 35192 | If a function's restrictio... |
| funen1cnv 35193 | If a function is equinumer... |
| xoromon 35194 | ` _om ` is either an ordin... |
| fissorduni 35195 | The union (supremum) of a ... |
| fnrelpredd 35196 | A function that preserves ... |
| cardpred 35197 | The cardinality function p... |
| nummin 35198 | Every nonempty class of nu... |
| r11 35199 | Value of the cumulative hi... |
| r12 35200 | Value of the cumulative hi... |
| r1wf 35201 | Each stage in the cumulati... |
| elwf 35202 | An element of a well-found... |
| r1elcl 35203 | Each set of the cumulative... |
| rankval2b 35204 | Value of an alternate defi... |
| rankval4b 35205 | The rank of a set is the s... |
| rankfilimbi 35206 | If all elements in a finit... |
| rankfilimb 35207 | The rank of a finite well-... |
| r1filimi 35208 | If all elements in a finit... |
| r1filim 35209 | A finite set appears in th... |
| r1omfi 35210 | Hereditarily finite sets a... |
| r1omhf 35211 | A set is hereditarily fini... |
| r1ssel 35212 | A set is a subset of the v... |
| axnulg 35213 | A generalization of ~ ax-n... |
| axnulALT2 35214 | Alternate proof of ~ axnul... |
| r1omfv 35215 | Value of the cumulative hi... |
| trssfir1om 35216 | If every element in a tran... |
| r1omhfb 35217 | The class of all hereditar... |
| prcinf 35218 | Any proper class is litera... |
| fineqvrep 35219 | If all sets are finite, th... |
| fineqvpow 35220 | If all sets are finite, th... |
| fineqvac 35221 | If all sets are finite, th... |
| fineqvacALT 35222 | Shorter proof of ~ fineqva... |
| fineqvomon 35223 | If all sets are finite, th... |
| fineqvomonb 35224 | All sets are finite iff al... |
| omprcomonb 35225 | The class of all finite or... |
| fineqvnttrclselem1 35226 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem2 35227 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem3 35228 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclse 35229 | A counterexample demonstra... |
| fineqvinfep 35230 | A counterexample demonstra... |
| axreg 35232 | Derivation of ~ ax-reg fro... |
| axregscl 35233 | A version of ~ ax-regs wit... |
| axregszf 35234 | Derivation of ~ zfregs usi... |
| setindregs 35235 | Set (epsilon) induction. ... |
| setinds2regs 35236 | Principle of set induction... |
| noinfepfnregs 35237 | There are no infinite desc... |
| noinfepregs 35238 | There are no infinite desc... |
| tz9.1regs 35239 | Every set has a transitive... |
| unir1regs 35240 | The cumulative hierarchy o... |
| trssfir1omregs 35241 | If every element in a tran... |
| r1omhfbregs 35242 | The class of all hereditar... |
| fineqvr1ombregs 35243 | All sets are finite iff al... |
| axregs 35244 | Derivation of ~ ax-regs fr... |
| gblacfnacd 35245 | If ` G ` is a global choic... |
| onvf1odlem1 35246 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem2 35247 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem3 35248 | Lemma for ~ onvf1od . The... |
| onvf1odlem4 35249 | Lemma for ~ onvf1od . If ... |
| onvf1od 35250 | If ` G ` is a global choic... |
| vonf1owev 35251 | If ` F ` is a bijection fr... |
| wevgblacfn 35252 | If ` R ` is a well-orderin... |
| zltp1ne 35253 | Integer ordering relation.... |
| nnltp1ne 35254 | Positive integer ordering ... |
| nn0ltp1ne 35255 | Nonnegative integer orderi... |
| 0nn0m1nnn0 35256 | A number is zero if and on... |
| f1resfz0f1d 35257 | If a function with a seque... |
| fisshasheq 35258 | A finite set is equal to i... |
| revpfxsfxrev 35259 | The reverse of a prefix of... |
| swrdrevpfx 35260 | A subword expressed in ter... |
| lfuhgr 35261 | A hypergraph is loop-free ... |
| lfuhgr2 35262 | A hypergraph is loop-free ... |
| lfuhgr3 35263 | A hypergraph is loop-free ... |
| cplgredgex 35264 | Any two (distinct) vertice... |
| cusgredgex 35265 | Any two (distinct) vertice... |
| cusgredgex2 35266 | Any two distinct vertices ... |
| pfxwlk 35267 | A prefix of a walk is a wa... |
| revwlk 35268 | The reverse of a walk is a... |
| revwlkb 35269 | Two words represent a walk... |
| swrdwlk 35270 | Two matching subwords of a... |
| pthhashvtx 35271 | A graph containing a path ... |
| spthcycl 35272 | A walk is a trivial path i... |
| usgrgt2cycl 35273 | A non-trivial cycle in a s... |
| usgrcyclgt2v 35274 | A simple graph with a non-... |
| subgrwlk 35275 | If a walk exists in a subg... |
| subgrtrl 35276 | If a trail exists in a sub... |
| subgrpth 35277 | If a path exists in a subg... |
| subgrcycl 35278 | If a cycle exists in a sub... |
| cusgr3cyclex 35279 | Every complete simple grap... |
| loop1cycl 35280 | A hypergraph has a cycle o... |
| 2cycld 35281 | Construction of a 2-cycle ... |
| 2cycl2d 35282 | Construction of a 2-cycle ... |
| umgr2cycllem 35283 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 35284 | A multigraph with two dist... |
| dfacycgr1 35287 | An alternate definition of... |
| isacycgr 35288 | The property of being an a... |
| isacycgr1 35289 | The property of being an a... |
| acycgrcycl 35290 | Any cycle in an acyclic gr... |
| acycgr0v 35291 | A null graph (with no vert... |
| acycgr1v 35292 | A multigraph with one vert... |
| acycgr2v 35293 | A simple graph with two ve... |
| prclisacycgr 35294 | A proper class (representi... |
| acycgrislfgr 35295 | An acyclic hypergraph is a... |
| upgracycumgr 35296 | An acyclic pseudograph is ... |
| umgracycusgr 35297 | An acyclic multigraph is a... |
| upgracycusgr 35298 | An acyclic pseudograph is ... |
| cusgracyclt3v 35299 | A complete simple graph is... |
| pthacycspth 35300 | A path in an acyclic graph... |
| acycgrsubgr 35301 | The subgraph of an acyclic... |
| quartfull 35308 | The quartic equation, writ... |
| deranglem 35309 | Lemma for derangements. (... |
| derangval 35310 | Define the derangement fun... |
| derangf 35311 | The derangement number is ... |
| derang0 35312 | The derangement number of ... |
| derangsn 35313 | The derangement number of ... |
| derangenlem 35314 | One half of ~ derangen . ... |
| derangen 35315 | The derangement number is ... |
| subfacval 35316 | The subfactorial is define... |
| derangen2 35317 | Write the derangement numb... |
| subfacf 35318 | The subfactorial is a func... |
| subfaclefac 35319 | The subfactorial is less t... |
| subfac0 35320 | The subfactorial at zero. ... |
| subfac1 35321 | The subfactorial at one. ... |
| subfacp1lem1 35322 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 35323 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 35324 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 35325 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 35326 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 35327 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 35328 | Lemma for ~ subfacp1 . By... |
| subfacp1 35329 | A two-term recurrence for ... |
| subfacval2 35330 | A closed-form expression f... |
| subfaclim 35331 | The subfactorial converges... |
| subfacval3 35332 | Another closed form expres... |
| derangfmla 35333 | The derangements formula, ... |
| erdszelem1 35334 | Lemma for ~ erdsze . (Con... |
| erdszelem2 35335 | Lemma for ~ erdsze . (Con... |
| erdszelem3 35336 | Lemma for ~ erdsze . (Con... |
| erdszelem4 35337 | Lemma for ~ erdsze . (Con... |
| erdszelem5 35338 | Lemma for ~ erdsze . (Con... |
| erdszelem6 35339 | Lemma for ~ erdsze . (Con... |
| erdszelem7 35340 | Lemma for ~ erdsze . (Con... |
| erdszelem8 35341 | Lemma for ~ erdsze . (Con... |
| erdszelem9 35342 | Lemma for ~ erdsze . (Con... |
| erdszelem10 35343 | Lemma for ~ erdsze . (Con... |
| erdszelem11 35344 | Lemma for ~ erdsze . (Con... |
| erdsze 35345 | The Erdős-Szekeres th... |
| erdsze2lem1 35346 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 35347 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 35348 | Generalize the statement o... |
| kur14lem1 35349 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 35350 | Lemma for ~ kur14 . Write... |
| kur14lem3 35351 | Lemma for ~ kur14 . A clo... |
| kur14lem4 35352 | Lemma for ~ kur14 . Compl... |
| kur14lem5 35353 | Lemma for ~ kur14 . Closu... |
| kur14lem6 35354 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 35355 | Lemma for ~ kur14 : main p... |
| kur14lem8 35356 | Lemma for ~ kur14 . Show ... |
| kur14lem9 35357 | Lemma for ~ kur14 . Since... |
| kur14lem10 35358 | Lemma for ~ kur14 . Disch... |
| kur14 35359 | Kuratowski's closure-compl... |
| ispconn 35366 | The property of being a pa... |
| pconncn 35367 | The property of being a pa... |
| pconntop 35368 | A simply connected space i... |
| issconn 35369 | The property of being a si... |
| sconnpconn 35370 | A simply connected space i... |
| sconntop 35371 | A simply connected space i... |
| sconnpht 35372 | A closed path in a simply ... |
| cnpconn 35373 | An image of a path-connect... |
| pconnconn 35374 | A path-connected space is ... |
| txpconn 35375 | The topological product of... |
| ptpconn 35376 | The topological product of... |
| indispconn 35377 | The indiscrete topology (o... |
| connpconn 35378 | A connected and locally pa... |
| qtoppconn 35379 | A quotient of a path-conne... |
| pconnpi1 35380 | All fundamental groups in ... |
| sconnpht2 35381 | Any two paths in a simply ... |
| sconnpi1 35382 | A path-connected topologic... |
| txsconnlem 35383 | Lemma for ~ txsconn . (Co... |
| txsconn 35384 | The topological product of... |
| cvxpconn 35385 | A convex subset of the com... |
| cvxsconn 35386 | A convex subset of the com... |
| blsconn 35387 | An open ball in the comple... |
| cnllysconn 35388 | The topology of the comple... |
| resconn 35389 | A subset of ` RR ` is simp... |
| ioosconn 35390 | An open interval is simply... |
| iccsconn 35391 | A closed interval is simpl... |
| retopsconn 35392 | The real numbers are simpl... |
| iccllysconn 35393 | A closed interval is local... |
| rellysconn 35394 | The real numbers are local... |
| iisconn 35395 | The unit interval is simpl... |
| iillysconn 35396 | The unit interval is local... |
| iinllyconn 35397 | The unit interval is local... |
| fncvm 35400 | Lemma for covering maps. ... |
| cvmscbv 35401 | Change bound variables in ... |
| iscvm 35402 | The property of being a co... |
| cvmtop1 35403 | Reverse closure for a cove... |
| cvmtop2 35404 | Reverse closure for a cove... |
| cvmcn 35405 | A covering map is a contin... |
| cvmcov 35406 | Property of a covering map... |
| cvmsrcl 35407 | Reverse closure for an eve... |
| cvmsi 35408 | One direction of ~ cvmsval... |
| cvmsval 35409 | Elementhood in the set ` S... |
| cvmsss 35410 | An even covering is a subs... |
| cvmsn0 35411 | An even covering is nonemp... |
| cvmsuni 35412 | An even covering of ` U ` ... |
| cvmsdisj 35413 | An even covering of ` U ` ... |
| cvmshmeo 35414 | Every element of an even c... |
| cvmsf1o 35415 | ` F ` , localized to an el... |
| cvmscld 35416 | The sets of an even coveri... |
| cvmsss2 35417 | An open subset of an evenl... |
| cvmcov2 35418 | The covering map property ... |
| cvmseu 35419 | Every element in ` U. T ` ... |
| cvmsiota 35420 | Identify the unique elemen... |
| cvmopnlem 35421 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 35422 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 35423 | A covering map is an open ... |
| cvmliftmolem1 35424 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 35425 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 35426 | A lift of a continuous fun... |
| cvmliftmo 35427 | A lift of a continuous fun... |
| cvmliftlem1 35428 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 35429 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 35430 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 35431 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 35432 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 35433 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 35434 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 35435 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 35436 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 35437 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 35438 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 35439 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 35440 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 35441 | Lemma for ~ cvmlift . Dis... |
| cvmlift 35442 | One of the important prope... |
| cvmfo 35443 | A covering map is an onto ... |
| cvmliftiota 35444 | Write out a function ` H `... |
| cvmlift2lem1 35445 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 35446 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 35447 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 35448 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 35449 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 35450 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 35451 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 35452 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 35453 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 35454 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 35455 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 35456 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 35457 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 35458 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 35459 | A two-dimensional version ... |
| cvmliftphtlem 35460 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 35461 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 35462 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 35463 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 35464 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 35465 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 35466 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 35467 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 35468 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 35469 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 35470 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 35471 | A general version of ~ cvm... |
| snmlff 35472 | The function ` F ` from ~ ... |
| snmlfval 35473 | The function ` F ` from ~ ... |
| snmlval 35474 | The property " ` A ` is si... |
| snmlflim 35475 | If ` A ` is simply normal,... |
| goel 35490 | A "Godel-set of membership... |
| goelel3xp 35491 | A "Godel-set of membership... |
| goeleq12bg 35492 | Two "Godel-set of membersh... |
| gonafv 35493 | The "Godel-set for the She... |
| goaleq12d 35494 | Equality of the "Godel-set... |
| gonanegoal 35495 | The Godel-set for the Shef... |
| satf 35496 | The satisfaction predicate... |
| satfsucom 35497 | The satisfaction predicate... |
| satfn 35498 | The satisfaction predicate... |
| satom 35499 | The satisfaction predicate... |
| satfvsucom 35500 | The satisfaction predicate... |
| satfv0 35501 | The value of the satisfact... |
| satfvsuclem1 35502 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 35503 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 35504 | The value of the satisfact... |
| satfv1lem 35505 | Lemma for ~ satfv1 . (Con... |
| satfv1 35506 | The value of the satisfact... |
| satfsschain 35507 | The binary relation of a s... |
| satfvsucsuc 35508 | The satisfaction predicate... |
| satfbrsuc 35509 | The binary relation of a s... |
| satfrel 35510 | The value of the satisfact... |
| satfdmlem 35511 | Lemma for ~ satfdm . (Con... |
| satfdm 35512 | The domain of the satisfac... |
| satfrnmapom 35513 | The range of the satisfact... |
| satfv0fun 35514 | The value of the satisfact... |
| satf0 35515 | The satisfaction predicate... |
| satf0sucom 35516 | The satisfaction predicate... |
| satf00 35517 | The value of the satisfact... |
| satf0suclem 35518 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 35519 | The value of the satisfact... |
| satf0op 35520 | An element of a value of t... |
| satf0n0 35521 | The value of the satisfact... |
| sat1el2xp 35522 | The first component of an ... |
| fmlafv 35523 | The valid Godel formulas o... |
| fmla 35524 | The set of all valid Godel... |
| fmla0 35525 | The valid Godel formulas o... |
| fmla0xp 35526 | The valid Godel formulas o... |
| fmlasuc0 35527 | The valid Godel formulas o... |
| fmlafvel 35528 | A class is a valid Godel f... |
| fmlasuc 35529 | The valid Godel formulas o... |
| fmla1 35530 | The valid Godel formulas o... |
| isfmlasuc 35531 | The characterization of a ... |
| fmlasssuc 35532 | The Godel formulas of heig... |
| fmlaomn0 35533 | The empty set is not a God... |
| fmlan0 35534 | The empty set is not a God... |
| gonan0 35535 | The "Godel-set of NAND" is... |
| goaln0 35536 | The "Godel-set of universa... |
| gonarlem 35537 | Lemma for ~ gonar (inducti... |
| gonar 35538 | If the "Godel-set of NAND"... |
| goalrlem 35539 | Lemma for ~ goalr (inducti... |
| goalr 35540 | If the "Godel-set of unive... |
| fmla0disjsuc 35541 | The set of valid Godel for... |
| fmlasucdisj 35542 | The valid Godel formulas o... |
| satfdmfmla 35543 | The domain of the satisfac... |
| satffunlem 35544 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 35545 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 35546 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 35547 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 35548 | The domain of an ordered p... |
| satffunlem2lem2 35549 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 35550 | Lemma 1 for ~ satffun : in... |
| satffunlem2 35551 | Lemma 2 for ~ satffun : in... |
| satffun 35552 | The value of the satisfact... |
| satff 35553 | The satisfaction predicate... |
| satfun 35554 | The satisfaction predicate... |
| satfvel 35555 | An element of the value of... |
| satfv0fvfmla0 35556 | The value of the satisfact... |
| satefv 35557 | The simplified satisfactio... |
| sate0 35558 | The simplified satisfactio... |
| satef 35559 | The simplified satisfactio... |
| sate0fv0 35560 | A simplified satisfaction ... |
| satefvfmla0 35561 | The simplified satisfactio... |
| sategoelfvb 35562 | Characterization of a valu... |
| sategoelfv 35563 | Condition of a valuation `... |
| ex-sategoelel 35564 | Example of a valuation of ... |
| ex-sategoel 35565 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 35566 | The value of the satisfact... |
| 2goelgoanfmla1 35567 | Two Godel-sets of membersh... |
| satefvfmla1 35568 | The simplified satisfactio... |
| ex-sategoelelomsuc 35569 | Example of a valuation of ... |
| ex-sategoelel12 35570 | Example of a valuation of ... |
| prv 35571 | The "proves" relation on a... |
| elnanelprv 35572 | The wff ` ( A e. B -/\ B e... |
| prv0 35573 | Every wff encoded as ` U `... |
| prv1n 35574 | No wff encoded as a Godel-... |
| mvtval 35643 | The set of variable typeco... |
| mrexval 35644 | The set of "raw expression... |
| mexval 35645 | The set of expressions, wh... |
| mexval2 35646 | The set of expressions, wh... |
| mdvval 35647 | The set of disjoint variab... |
| mvrsval 35648 | The set of variables in an... |
| mvrsfpw 35649 | The set of variables in an... |
| mrsubffval 35650 | The substitution of some v... |
| mrsubfval 35651 | The substitution of some v... |
| mrsubval 35652 | The substitution of some v... |
| mrsubcv 35653 | The value of a substituted... |
| mrsubvr 35654 | The value of a substituted... |
| mrsubff 35655 | A substitution is a functi... |
| mrsubrn 35656 | Although it is defined for... |
| mrsubff1 35657 | When restricted to complet... |
| mrsubff1o 35658 | When restricted to complet... |
| mrsub0 35659 | The value of the substitut... |
| mrsubf 35660 | A substitution is a functi... |
| mrsubccat 35661 | Substitution distributes o... |
| mrsubcn 35662 | A substitution does not ch... |
| elmrsubrn 35663 | Characterization of the su... |
| mrsubco 35664 | The composition of two sub... |
| mrsubvrs 35665 | The set of variables in a ... |
| msubffval 35666 | A substitution applied to ... |
| msubfval 35667 | A substitution applied to ... |
| msubval 35668 | A substitution applied to ... |
| msubrsub 35669 | A substitution applied to ... |
| msubty 35670 | The type of a substituted ... |
| elmsubrn 35671 | Characterization of substi... |
| msubrn 35672 | Although it is defined for... |
| msubff 35673 | A substitution is a functi... |
| msubco 35674 | The composition of two sub... |
| msubf 35675 | A substitution is a functi... |
| mvhfval 35676 | Value of the function mapp... |
| mvhval 35677 | Value of the function mapp... |
| mpstval 35678 | A pre-statement is an orde... |
| elmpst 35679 | Property of being a pre-st... |
| msrfval 35680 | Value of the reduct of a p... |
| msrval 35681 | Value of the reduct of a p... |
| mpstssv 35682 | A pre-statement is an orde... |
| mpst123 35683 | Decompose a pre-statement ... |
| mpstrcl 35684 | The elements of a pre-stat... |
| msrf 35685 | The reduct of a pre-statem... |
| msrrcl 35686 | If ` X ` and ` Y ` have th... |
| mstaval 35687 | Value of the set of statem... |
| msrid 35688 | The reduct of a statement ... |
| msrfo 35689 | The reduct of a pre-statem... |
| mstapst 35690 | A statement is a pre-state... |
| elmsta 35691 | Property of being a statem... |
| ismfs 35692 | A formal system is a tuple... |
| mfsdisj 35693 | The constants and variable... |
| mtyf2 35694 | The type function maps var... |
| mtyf 35695 | The type function maps var... |
| mvtss 35696 | The set of variable typeco... |
| maxsta 35697 | An axiom is a statement. ... |
| mvtinf 35698 | Each variable typecode has... |
| msubff1 35699 | When restricted to complet... |
| msubff1o 35700 | When restricted to complet... |
| mvhf 35701 | The function mapping varia... |
| mvhf1 35702 | The function mapping varia... |
| msubvrs 35703 | The set of variables in a ... |
| mclsrcl 35704 | Reverse closure for the cl... |
| mclsssvlem 35705 | Lemma for ~ mclsssv . (Co... |
| mclsval 35706 | The function mapping varia... |
| mclsssv 35707 | The closure of a set of ex... |
| ssmclslem 35708 | Lemma for ~ ssmcls . (Con... |
| vhmcls 35709 | All variable hypotheses ar... |
| ssmcls 35710 | The original expressions a... |
| ss2mcls 35711 | The closure is monotonic u... |
| mclsax 35712 | The closure is closed unde... |
| mclsind 35713 | Induction theorem for clos... |
| mppspstlem 35714 | Lemma for ~ mppspst . (Co... |
| mppsval 35715 | Definition of a provable p... |
| elmpps 35716 | Definition of a provable p... |
| mppspst 35717 | A provable pre-statement i... |
| mthmval 35718 | A theorem is a pre-stateme... |
| elmthm 35719 | A theorem is a pre-stateme... |
| mthmi 35720 | A statement whose reduct i... |
| mthmsta 35721 | A theorem is a pre-stateme... |
| mppsthm 35722 | A provable pre-statement i... |
| mthmblem 35723 | Lemma for ~ mthmb . (Cont... |
| mthmb 35724 | If two statements have the... |
| mthmpps 35725 | Given a theorem, there is ... |
| mclsppslem 35726 | The closure is closed unde... |
| mclspps 35727 | The closure is closed unde... |
| rexxfr3d 35781 | Transfer existential quant... |
| rexxfr3dALT 35782 | Longer proof of ~ rexxfr3d... |
| rspssbasd 35783 | The span of a set of ring ... |
| ellcsrspsn 35784 | Membership in a left coset... |
| ply1divalg3 35785 | Uniqueness of polynomial r... |
| r1peuqusdeg1 35786 | Uniqueness of polynomial r... |
| problem1 35808 | Practice problem 1. Clues... |
| problem2 35809 | Practice problem 2. Clues... |
| problem3 35810 | Practice problem 3. Clues... |
| problem4 35811 | Practice problem 4. Clues... |
| problem5 35812 | Practice problem 5. Clues... |
| quad3 35813 | Variant of quadratic equat... |
| climuzcnv 35814 | Utility lemma to convert b... |
| sinccvglem 35815 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 35816 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 35817 | The circumference of a cir... |
| elfzm12 35818 | Membership in a curtailed ... |
| nn0seqcvg 35819 | A strictly-decreasing nonn... |
| lediv2aALT 35820 | Division of both sides of ... |
| abs2sqlei 35821 | The absolute values of two... |
| abs2sqlti 35822 | The absolute values of two... |
| abs2sqle 35823 | The absolute values of two... |
| abs2sqlt 35824 | The absolute values of two... |
| abs2difi 35825 | Difference of absolute val... |
| abs2difabsi 35826 | Absolute value of differen... |
| 2thALT 35827 | Alternate proof of ~ 2th .... |
| orbi2iALT 35828 | Alternate proof of ~ orbi2... |
| pm3.48ALT 35829 | Alternate proof of ~ pm3.4... |
| 3jcadALT 35830 | Alternate proof of ~ 3jcad... |
| currybi 35831 | Biconditional version of C... |
| antnest 35832 | Suppose ` ph ` , ` ps ` ar... |
| antnestlaw3lem 35833 | Lemma for ~ antnestlaw3 . ... |
| antnestlaw1 35834 | A law of nested antecedent... |
| antnestlaw2 35835 | A law of nested antecedent... |
| antnestlaw3 35836 | A law of nested antecedent... |
| antnestALT 35837 | Alternative proof of ~ ant... |
| axextprim 35844 | ~ ax-ext without distinct ... |
| axrepprim 35845 | ~ ax-rep without distinct ... |
| axunprim 35846 | ~ ax-un without distinct v... |
| axpowprim 35847 | ~ ax-pow without distinct ... |
| axregprim 35848 | ~ ax-reg without distinct ... |
| axinfprim 35849 | ~ ax-inf without distinct ... |
| axacprim 35850 | ~ ax-ac without distinct v... |
| untelirr 35851 | We call a class "untanged"... |
| untuni 35852 | The union of a class is un... |
| untsucf 35853 | If a class is untangled, t... |
| unt0 35854 | The null set is untangled.... |
| untint 35855 | If there is an untangled e... |
| efrunt 35856 | If ` A ` is well-founded b... |
| untangtr 35857 | A transitive class is unta... |
| 3jaodd 35858 | Double deduction form of ~... |
| 3orit 35859 | Closed form of ~ 3ori . (... |
| biimpexp 35860 | A biconditional in the ant... |
| nepss 35861 | Two classes are unequal if... |
| 3ccased 35862 | Triple disjunction form of... |
| dfso3 35863 | Expansion of the definitio... |
| brtpid1 35864 | A binary relation involvin... |
| brtpid2 35865 | A binary relation involvin... |
| brtpid3 35866 | A binary relation involvin... |
| iota5f 35867 | A method for computing iot... |
| jath 35868 | Closed form of ~ ja . Pro... |
| xpab 35869 | Cartesian product of two c... |
| nnuni 35870 | The union of a finite ordi... |
| sqdivzi 35871 | Distribution of square ove... |
| supfz 35872 | The supremum of a finite s... |
| inffz 35873 | The infimum of a finite se... |
| fz0n 35874 | The sequence ` ( 0 ... ( N... |
| shftvalg 35875 | Value of a sequence shifte... |
| divcnvlin 35876 | Limit of the ratio of two ... |
| climlec3 35877 | Comparison of a constant t... |
| iexpire 35878 | ` _i ` raised to itself is... |
| bcneg1 35879 | The binomial coefficient o... |
| bcm1nt 35880 | The proportion of one bino... |
| bcprod 35881 | A product identity for bin... |
| bccolsum 35882 | A column-sum rule for bino... |
| iprodefisumlem 35883 | Lemma for ~ iprodefisum . ... |
| iprodefisum 35884 | Applying the exponential f... |
| iprodgam 35885 | An infinite product versio... |
| faclimlem1 35886 | Lemma for ~ faclim . Clos... |
| faclimlem2 35887 | Lemma for ~ faclim . Show... |
| faclimlem3 35888 | Lemma for ~ faclim . Alge... |
| faclim 35889 | An infinite product expres... |
| iprodfac 35890 | An infinite product expres... |
| faclim2 35891 | Another factorial limit du... |
| gcd32 35892 | Swap the second and third ... |
| gcdabsorb 35893 | Absorption law for gcd. (... |
| dftr6 35894 | A potential definition of ... |
| coep 35895 | Composition with the membe... |
| coepr 35896 | Composition with the conve... |
| dffr5 35897 | A quantifier-free definiti... |
| dfso2 35898 | Quantifier-free definition... |
| br8 35899 | Substitution for an eight-... |
| br6 35900 | Substitution for a six-pla... |
| br4 35901 | Substitution for a four-pl... |
| cnvco1 35902 | Another distributive law o... |
| cnvco2 35903 | Another distributive law o... |
| eldm3 35904 | Quantifier-free definition... |
| elrn3 35905 | Quantifier-free definition... |
| pocnv 35906 | The converse of a partial ... |
| socnv 35907 | The converse of a strict o... |
| elintfv 35908 | Membership in an intersect... |
| funpsstri 35909 | A condition for subset tri... |
| fundmpss 35910 | If a class ` F ` is a prop... |
| funsseq 35911 | Given two functions with e... |
| fununiq 35912 | The uniqueness condition o... |
| funbreq 35913 | An equality condition for ... |
| br1steq 35914 | Uniqueness condition for t... |
| br2ndeq 35915 | Uniqueness condition for t... |
| dfdm5 35916 | Definition of domain in te... |
| dfrn5 35917 | Definition of range in ter... |
| opelco3 35918 | Alternate way of saying th... |
| elima4 35919 | Quantifier-free expression... |
| fv1stcnv 35920 | The value of the converse ... |
| fv2ndcnv 35921 | The value of the converse ... |
| elpotr 35922 | A class of transitive sets... |
| dford5reg 35923 | Given ~ ax-reg , an ordina... |
| dfon2lem1 35924 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 35925 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 35926 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 35927 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 35928 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 35929 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 35930 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 35931 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 35932 | Lemma for ~ dfon2 . A cla... |
| dfon2 35933 | ` On ` consists of all set... |
| rdgprc0 35934 | The value of the recursive... |
| rdgprc 35935 | The value of the recursive... |
| dfrdg2 35936 | Alternate definition of th... |
| dfrdg3 35937 | Generalization of ~ dfrdg2... |
| axextdfeq 35938 | A version of ~ ax-ext for ... |
| ax8dfeq 35939 | A version of ~ ax-8 for us... |
| axextdist 35940 | ~ ax-ext with distinctors ... |
| axextbdist 35941 | ~ axextb with distinctors ... |
| 19.12b 35942 | Version of ~ 19.12vv with ... |
| exnel 35943 | There is always a set not ... |
| distel 35944 | Distinctors in terms of me... |
| axextndbi 35945 | ~ axextnd as a bicondition... |
| hbntg 35946 | A more general form of ~ h... |
| hbimtg 35947 | A more general and closed ... |
| hbaltg 35948 | A more general and closed ... |
| hbng 35949 | A more general form of ~ h... |
| hbimg 35950 | A more general form of ~ h... |
| wsuceq123 35955 | Equality theorem for well-... |
| wsuceq1 35956 | Equality theorem for well-... |
| wsuceq2 35957 | Equality theorem for well-... |
| wsuceq3 35958 | Equality theorem for well-... |
| nfwsuc 35959 | Bound-variable hypothesis ... |
| wlimeq12 35960 | Equality theorem for the l... |
| wlimeq1 35961 | Equality theorem for the l... |
| wlimeq2 35962 | Equality theorem for the l... |
| nfwlim 35963 | Bound-variable hypothesis ... |
| elwlim 35964 | Membership in the limit cl... |
| wzel 35965 | The zero of a well-founded... |
| wsuclem 35966 | Lemma for the supremum pro... |
| wsucex 35967 | Existence theorem for well... |
| wsuccl 35968 | If ` X ` is a set with an ... |
| wsuclb 35969 | A well-founded successor i... |
| wlimss 35970 | The class of limit points ... |
| txpss3v 36019 | A tail Cartesian product i... |
| txprel 36020 | A tail Cartesian product i... |
| brtxp 36021 | Characterize a ternary rel... |
| brtxp2 36022 | The binary relation over a... |
| dfpprod2 36023 | Expanded definition of par... |
| pprodcnveq 36024 | A converse law for paralle... |
| pprodss4v 36025 | The parallel product is a ... |
| brpprod 36026 | Characterize a quaternary ... |
| brpprod3a 36027 | Condition for parallel pro... |
| brpprod3b 36028 | Condition for parallel pro... |
| relsset 36029 | The subset class is a bina... |
| brsset 36030 | For sets, the ` SSet ` bin... |
| idsset 36031 | ` _I ` is equal to the int... |
| eltrans 36032 | Membership in the class of... |
| dfon3 36033 | A quantifier-free definiti... |
| dfon4 36034 | Another quantifier-free de... |
| brtxpsd 36035 | Expansion of a common form... |
| brtxpsd2 36036 | Another common abbreviatio... |
| brtxpsd3 36037 | A third common abbreviatio... |
| relbigcup 36038 | The ` Bigcup ` relationshi... |
| brbigcup 36039 | Binary relation over ` Big... |
| dfbigcup2 36040 | ` Bigcup ` using maps-to n... |
| fobigcup 36041 | ` Bigcup ` maps the univer... |
| fnbigcup 36042 | ` Bigcup ` is a function o... |
| fvbigcup 36043 | For sets, ` Bigcup ` yield... |
| elfix 36044 | Membership in the fixpoint... |
| elfix2 36045 | Alternative membership in ... |
| dffix2 36046 | The fixpoints of a class i... |
| fixssdm 36047 | The fixpoints of a class a... |
| fixssrn 36048 | The fixpoints of a class a... |
| fixcnv 36049 | The fixpoints of a class a... |
| fixun 36050 | The fixpoint operator dist... |
| ellimits 36051 | Membership in the class of... |
| limitssson 36052 | The class of all limit ord... |
| dfom5b 36053 | A quantifier-free definiti... |
| sscoid 36054 | A condition for subset and... |
| dffun10 36055 | Another potential definiti... |
| elfuns 36056 | Membership in the class of... |
| elfunsg 36057 | Closed form of ~ elfuns . ... |
| brsingle 36058 | The binary relation form o... |
| elsingles 36059 | Membership in the class of... |
| fnsingle 36060 | The singleton relationship... |
| fvsingle 36061 | The value of the singleton... |
| dfsingles2 36062 | Alternate definition of th... |
| snelsingles 36063 | A singleton is a member of... |
| dfiota3 36064 | A definition of iota using... |
| dffv5 36065 | Another quantifier-free de... |
| unisnif 36066 | Express union of singleton... |
| brimage 36067 | Binary relation form of th... |
| brimageg 36068 | Closed form of ~ brimage .... |
| funimage 36069 | ` Image A ` is a function.... |
| fnimage 36070 | ` Image R ` is a function ... |
| imageval 36071 | The image functor in maps-... |
| fvimage 36072 | Value of the image functor... |
| brcart 36073 | Binary relation form of th... |
| brdomain 36074 | Binary relation form of th... |
| brrange 36075 | Binary relation form of th... |
| brdomaing 36076 | Closed form of ~ brdomain ... |
| brrangeg 36077 | Closed form of ~ brrange .... |
| brimg 36078 | Binary relation form of th... |
| brapply 36079 | Binary relation form of th... |
| brcup 36080 | Binary relation form of th... |
| brcap 36081 | Binary relation form of th... |
| lemsuccf 36082 | Lemma for unfolding differ... |
| brsuccf 36083 | Binary relation form of th... |
| dfsuccf2 36084 | Alternate definition of Sc... |
| funpartlem 36085 | Lemma for ~ funpartfun . ... |
| funpartfun 36086 | The functional part of ` F... |
| funpartss 36087 | The functional part of ` F... |
| funpartfv 36088 | The function value of the ... |
| fullfunfnv 36089 | The full functional part o... |
| fullfunfv 36090 | The function value of the ... |
| brfullfun 36091 | A binary relation form con... |
| brrestrict 36092 | Binary relation form of th... |
| dfrecs2 36093 | A quantifier-free definiti... |
| dfrdg4 36094 | A quantifier-free definiti... |
| dfint3 36095 | Quantifier-free definition... |
| imagesset 36096 | The Image functor applied ... |
| brub 36097 | Binary relation form of th... |
| brlb 36098 | Binary relation form of th... |
| altopex 36103 | Alternative ordered pairs ... |
| altopthsn 36104 | Two alternate ordered pair... |
| altopeq12 36105 | Equality for alternate ord... |
| altopeq1 36106 | Equality for alternate ord... |
| altopeq2 36107 | Equality for alternate ord... |
| altopth1 36108 | Equality of the first memb... |
| altopth2 36109 | Equality of the second mem... |
| altopthg 36110 | Alternate ordered pair the... |
| altopthbg 36111 | Alternate ordered pair the... |
| altopth 36112 | The alternate ordered pair... |
| altopthb 36113 | Alternate ordered pair the... |
| altopthc 36114 | Alternate ordered pair the... |
| altopthd 36115 | Alternate ordered pair the... |
| altxpeq1 36116 | Equality for alternate Car... |
| altxpeq2 36117 | Equality for alternate Car... |
| elaltxp 36118 | Membership in alternate Ca... |
| altopelaltxp 36119 | Alternate ordered pair mem... |
| altxpsspw 36120 | An inclusion rule for alte... |
| altxpexg 36121 | The alternate Cartesian pr... |
| rankaltopb 36122 | Compute the rank of an alt... |
| nfaltop 36123 | Bound-variable hypothesis ... |
| sbcaltop 36124 | Distribution of class subs... |
| cgrrflx2d 36127 | Deduction form of ~ axcgrr... |
| cgrtr4d 36128 | Deduction form of ~ axcgrt... |
| cgrtr4and 36129 | Deduction form of ~ axcgrt... |
| cgrrflx 36130 | Reflexivity law for congru... |
| cgrrflxd 36131 | Deduction form of ~ cgrrfl... |
| cgrcomim 36132 | Congruence commutes on the... |
| cgrcom 36133 | Congruence commutes betwee... |
| cgrcomand 36134 | Deduction form of ~ cgrcom... |
| cgrtr 36135 | Transitivity law for congr... |
| cgrtrand 36136 | Deduction form of ~ cgrtr ... |
| cgrtr3 36137 | Transitivity law for congr... |
| cgrtr3and 36138 | Deduction form of ~ cgrtr3... |
| cgrcoml 36139 | Congruence commutes on the... |
| cgrcomr 36140 | Congruence commutes on the... |
| cgrcomlr 36141 | Congruence commutes on bot... |
| cgrcomland 36142 | Deduction form of ~ cgrcom... |
| cgrcomrand 36143 | Deduction form of ~ cgrcom... |
| cgrcomlrand 36144 | Deduction form of ~ cgrcom... |
| cgrtriv 36145 | Degenerate segments are co... |
| cgrid2 36146 | Identity law for congruenc... |
| cgrdegen 36147 | Two congruent segments are... |
| brofs 36148 | Binary relation form of th... |
| 5segofs 36149 | Rephrase ~ ax5seg using th... |
| ofscom 36150 | The outer five segment pre... |
| cgrextend 36151 | Link congruence over a pai... |
| cgrextendand 36152 | Deduction form of ~ cgrext... |
| segconeq 36153 | Two points that satisfy th... |
| segconeu 36154 | Existential uniqueness ver... |
| btwntriv2 36155 | Betweenness always holds f... |
| btwncomim 36156 | Betweenness commutes. Imp... |
| btwncom 36157 | Betweenness commutes. (Co... |
| btwncomand 36158 | Deduction form of ~ btwnco... |
| btwntriv1 36159 | Betweenness always holds f... |
| btwnswapid 36160 | If you can swap the first ... |
| btwnswapid2 36161 | If you can swap arguments ... |
| btwnintr 36162 | Inner transitivity law for... |
| btwnexch3 36163 | Exchange the first endpoin... |
| btwnexch3and 36164 | Deduction form of ~ btwnex... |
| btwnouttr2 36165 | Outer transitivity law for... |
| btwnexch2 36166 | Exchange the outer point o... |
| btwnouttr 36167 | Outer transitivity law for... |
| btwnexch 36168 | Outer transitivity law for... |
| btwnexchand 36169 | Deduction form of ~ btwnex... |
| btwndiff 36170 | There is always a ` c ` di... |
| trisegint 36171 | A line segment between two... |
| funtransport 36174 | The ` TransportTo ` relati... |
| fvtransport 36175 | Calculate the value of the... |
| transportcl 36176 | Closure law for segment tr... |
| transportprops 36177 | Calculate the defining pro... |
| brifs 36186 | Binary relation form of th... |
| ifscgr 36187 | Inner five segment congrue... |
| cgrsub 36188 | Removing identical parts f... |
| brcgr3 36189 | Binary relation form of th... |
| cgr3permute3 36190 | Permutation law for three-... |
| cgr3permute1 36191 | Permutation law for three-... |
| cgr3permute2 36192 | Permutation law for three-... |
| cgr3permute4 36193 | Permutation law for three-... |
| cgr3permute5 36194 | Permutation law for three-... |
| cgr3tr4 36195 | Transitivity law for three... |
| cgr3com 36196 | Commutativity law for thre... |
| cgr3rflx 36197 | Identity law for three-pla... |
| cgrxfr 36198 | A line segment can be divi... |
| btwnxfr 36199 | A condition for extending ... |
| colinrel 36200 | Colinearity is a relations... |
| brcolinear2 36201 | Alternate colinearity bina... |
| brcolinear 36202 | The binary relation form o... |
| colinearex 36203 | The colinear predicate exi... |
| colineardim1 36204 | If ` A ` is colinear with ... |
| colinearperm1 36205 | Permutation law for coline... |
| colinearperm3 36206 | Permutation law for coline... |
| colinearperm2 36207 | Permutation law for coline... |
| colinearperm4 36208 | Permutation law for coline... |
| colinearperm5 36209 | Permutation law for coline... |
| colineartriv1 36210 | Trivial case of colinearit... |
| colineartriv2 36211 | Trivial case of colinearit... |
| btwncolinear1 36212 | Betweenness implies coline... |
| btwncolinear2 36213 | Betweenness implies coline... |
| btwncolinear3 36214 | Betweenness implies coline... |
| btwncolinear4 36215 | Betweenness implies coline... |
| btwncolinear5 36216 | Betweenness implies coline... |
| btwncolinear6 36217 | Betweenness implies coline... |
| colinearxfr 36218 | Transfer law for colineari... |
| lineext 36219 | Extend a line with a missi... |
| brofs2 36220 | Change some conditions for... |
| brifs2 36221 | Change some conditions for... |
| brfs 36222 | Binary relation form of th... |
| fscgr 36223 | Congruence law for the gen... |
| linecgr 36224 | Congruence rule for lines.... |
| linecgrand 36225 | Deduction form of ~ linecg... |
| lineid 36226 | Identity law for points on... |
| idinside 36227 | Law for finding a point in... |
| endofsegid 36228 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 36229 | Deduction form of ~ endofs... |
| btwnconn1lem1 36230 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 36231 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 36232 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 36233 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 36234 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 36235 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 36236 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 36237 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 36238 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 36239 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 36240 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 36241 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 36242 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 36243 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 36244 | Connectitivy law for betwe... |
| btwnconn2 36245 | Another connectivity law f... |
| btwnconn3 36246 | Inner connectivity law for... |
| midofsegid 36247 | If two points fall in the ... |
| segcon2 36248 | Generalization of ~ axsegc... |
| brsegle 36251 | Binary relation form of th... |
| brsegle2 36252 | Alternate characterization... |
| seglecgr12im 36253 | Substitution law for segme... |
| seglecgr12 36254 | Substitution law for segme... |
| seglerflx 36255 | Segment comparison is refl... |
| seglemin 36256 | Any segment is at least as... |
| segletr 36257 | Segment less than is trans... |
| segleantisym 36258 | Antisymmetry law for segme... |
| seglelin 36259 | Linearity law for segment ... |
| btwnsegle 36260 | If ` B ` falls between ` A... |
| colinbtwnle 36261 | Given three colinear point... |
| broutsideof 36264 | Binary relation form of ` ... |
| broutsideof2 36265 | Alternate form of ` Outsid... |
| outsidene1 36266 | Outsideness implies inequa... |
| outsidene2 36267 | Outsideness implies inequa... |
| btwnoutside 36268 | A principle linking outsid... |
| broutsideof3 36269 | Characterization of outsid... |
| outsideofrflx 36270 | Reflexivity of outsideness... |
| outsideofcom 36271 | Commutativity law for outs... |
| outsideoftr 36272 | Transitivity law for outsi... |
| outsideofeq 36273 | Uniqueness law for ` Outsi... |
| outsideofeu 36274 | Given a nondegenerate ray,... |
| outsidele 36275 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 36276 | Outside of implies colinea... |
| funray 36283 | Show that the ` Ray ` rela... |
| fvray 36284 | Calculate the value of the... |
| funline 36285 | Show that the ` Line ` rel... |
| linedegen 36286 | When ` Line ` is applied w... |
| fvline 36287 | Calculate the value of the... |
| liness 36288 | A line is a subset of the ... |
| fvline2 36289 | Alternate definition of a ... |
| lineunray 36290 | A line is composed of a po... |
| lineelsb2 36291 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 36292 | Reflexivity law for line m... |
| linecom 36293 | Commutativity law for line... |
| linerflx2 36294 | Reflexivity law for line m... |
| ellines 36295 | Membership in the set of a... |
| linethru 36296 | If ` A ` is a line contain... |
| hilbert1.1 36297 | There is a line through an... |
| hilbert1.2 36298 | There is at most one line ... |
| linethrueu 36299 | There is a unique line goi... |
| lineintmo 36300 | Two distinct lines interse... |
| fwddifval 36305 | Calculate the value of the... |
| fwddifnval 36306 | The value of the forward d... |
| fwddifn0 36307 | The value of the n-iterate... |
| fwddifnp1 36308 | The value of the n-iterate... |
| rankung 36309 | The rank of the union of t... |
| ranksng 36310 | The rank of a singleton. ... |
| rankelg 36311 | The membership relation is... |
| rankpwg 36312 | The rank of a power set. ... |
| rank0 36313 | The rank of the empty set ... |
| rankeq1o 36314 | The only set with rank ` 1... |
| elhf 36317 | Membership in the heredita... |
| elhf2 36318 | Alternate form of membersh... |
| elhf2g 36319 | Hereditarily finiteness vi... |
| 0hf 36320 | The empty set is a heredit... |
| hfun 36321 | The union of two HF sets i... |
| hfsn 36322 | The singleton of an HF set... |
| hfadj 36323 | Adjoining one HF element t... |
| hfelhf 36324 | Any member of an HF set is... |
| hftr 36325 | The class of all hereditar... |
| hfext 36326 | Extensionality for HF sets... |
| hfuni 36327 | The union of an HF set is ... |
| hfpw 36328 | The power class of an HF s... |
| hfninf 36329 | ` _om ` is not hereditaril... |
| rmoeqi 36330 | Equality inference for res... |
| rmoeqbii 36331 | Equality inference for res... |
| reueqi 36332 | Equality inference for res... |
| reueqbii 36333 | Equality inference for res... |
| sbceqbii 36334 | Formula-building inference... |
| disjeq1i 36335 | Equality theorem for disjo... |
| disjeq12i 36336 | Equality theorem for disjo... |
| rabeqbii 36337 | Equality theorem for restr... |
| iuneq12i 36338 | Equality theorem for index... |
| iineq1i 36339 | Equality theorem for index... |
| iineq12i 36340 | Equality theorem for index... |
| riotaeqbii 36341 | Equivalent wff's and equal... |
| riotaeqi 36342 | Equal domains yield equal ... |
| ixpeq1i 36343 | Equality inference for inf... |
| ixpeq12i 36344 | Equality inference for inf... |
| sumeq2si 36345 | Equality inference for sum... |
| sumeq12si 36346 | Equality inference for sum... |
| prodeq2si 36347 | Equality inference for pro... |
| prodeq12si 36348 | Equality inference for pro... |
| itgeq12i 36349 | Equality inference for an ... |
| itgeq1i 36350 | Equality inference for an ... |
| itgeq2i 36351 | Equality inference for an ... |
| ditgeq123i 36352 | Equality inference for the... |
| ditgeq12i 36353 | Equality inference for the... |
| ditgeq3i 36354 | Equality inference for the... |
| rmoeqdv 36355 | Formula-building rule for ... |
| rmoeqbidv 36356 | Formula-building rule for ... |
| sbequbidv 36357 | Deduction substituting bot... |
| disjeq12dv 36358 | Equality theorem for disjo... |
| ixpeq12dv 36359 | Equality theorem for infin... |
| sumeq12sdv 36360 | Equality deduction for sum... |
| prodeq12sdv 36361 | Equality deduction for pro... |
| itgeq12sdv 36362 | Equality theorem for an in... |
| itgeq2sdv 36363 | Equality theorem for an in... |
| ditgeq123dv 36364 | Equality theorem for the d... |
| ditgeq12d 36365 | Equality theorem for the d... |
| ditgeq3sdv 36366 | Equality theorem for the d... |
| in-ax8 36367 | A proof of ~ ax-8 that doe... |
| ss-ax8 36368 | A proof of ~ ax-8 that doe... |
| cbvralvw2 36369 | Change bound variable and ... |
| cbvrexvw2 36370 | Change bound variable and ... |
| cbvrmovw2 36371 | Change bound variable and ... |
| cbvreuvw2 36372 | Change bound variable and ... |
| cbvsbcvw2 36373 | Change bound variable of a... |
| cbvcsbvw2 36374 | Change bound variable of a... |
| cbviunvw2 36375 | Change bound variable and ... |
| cbviinvw2 36376 | Change bound variable and ... |
| cbvmptvw2 36377 | Change bound variable and ... |
| cbvdisjvw2 36378 | Change bound variable and ... |
| cbvriotavw2 36379 | Change bound variable and ... |
| cbvoprab1vw 36380 | Change the first bound var... |
| cbvoprab2vw 36381 | Change the second bound va... |
| cbvoprab123vw 36382 | Change all bound variables... |
| cbvoprab23vw 36383 | Change the second and thir... |
| cbvoprab13vw 36384 | Change the first and third... |
| cbvmpovw2 36385 | Change bound variables and... |
| cbvmpo1vw2 36386 | Change domains and the fir... |
| cbvmpo2vw2 36387 | Change domains and the sec... |
| cbvixpvw2 36388 | Change bound variable and ... |
| cbvsumvw2 36389 | Change bound variable and ... |
| cbvprodvw2 36390 | Change bound variable and ... |
| cbvitgvw2 36391 | Change bound variable and ... |
| cbvditgvw2 36392 | Change bound variable and ... |
| cbvmodavw 36393 | Change bound variable in t... |
| cbveudavw 36394 | Change bound variable in t... |
| cbvrmodavw 36395 | Change bound variable in t... |
| cbvreudavw 36396 | Change bound variable in t... |
| cbvsbdavw 36397 | Change bound variable in p... |
| cbvsbdavw2 36398 | Change bound variable in p... |
| cbvabdavw 36399 | Change bound variable in c... |
| cbvsbcdavw 36400 | Change bound variable of a... |
| cbvsbcdavw2 36401 | Change bound variable of a... |
| cbvcsbdavw 36402 | Change bound variable of a... |
| cbvcsbdavw2 36403 | Change bound variable of a... |
| cbvrabdavw 36404 | Change bound variable in r... |
| cbviundavw 36405 | Change bound variable in i... |
| cbviindavw 36406 | Change bound variable in i... |
| cbvopab1davw 36407 | Change the first bound var... |
| cbvopab2davw 36408 | Change the second bound va... |
| cbvopabdavw 36409 | Change bound variables in ... |
| cbvmptdavw 36410 | Change bound variable in a... |
| cbvdisjdavw 36411 | Change bound variable in a... |
| cbviotadavw 36412 | Change bound variable in a... |
| cbvriotadavw 36413 | Change bound variable in a... |
| cbvoprab1davw 36414 | Change the first bound var... |
| cbvoprab2davw 36415 | Change the second bound va... |
| cbvoprab3davw 36416 | Change the third bound var... |
| cbvoprab123davw 36417 | Change all bound variables... |
| cbvoprab12davw 36418 | Change the first and secon... |
| cbvoprab23davw 36419 | Change the second and thir... |
| cbvoprab13davw 36420 | Change the first and third... |
| cbvixpdavw 36421 | Change bound variable in a... |
| cbvsumdavw 36422 | Change bound variable in a... |
| cbvproddavw 36423 | Change bound variable in a... |
| cbvitgdavw 36424 | Change bound variable in a... |
| cbvditgdavw 36425 | Change bound variable in a... |
| cbvrmodavw2 36426 | Change bound variable and ... |
| cbvreudavw2 36427 | Change bound variable and ... |
| cbvrabdavw2 36428 | Change bound variable and ... |
| cbviundavw2 36429 | Change bound variable and ... |
| cbviindavw2 36430 | Change bound variable and ... |
| cbvmptdavw2 36431 | Change bound variable and ... |
| cbvdisjdavw2 36432 | Change bound variable and ... |
| cbvriotadavw2 36433 | Change bound variable and ... |
| cbvmpodavw2 36434 | Change bound variable and ... |
| cbvmpo1davw2 36435 | Change first bound variabl... |
| cbvmpo2davw2 36436 | Change second bound variab... |
| cbvixpdavw2 36437 | Change bound variable and ... |
| cbvsumdavw2 36438 | Change bound variable and ... |
| cbvproddavw2 36439 | Change bound variable and ... |
| cbvitgdavw2 36440 | Change bound variable and ... |
| cbvditgdavw2 36441 | Change bound variable and ... |
| mpomulnzcnf 36442 | Multiplication maps nonzer... |
| a1i14 36443 | Add two antecedents to a w... |
| a1i24 36444 | Add two antecedents to a w... |
| exp5d 36445 | An exportation inference. ... |
| exp5g 36446 | An exportation inference. ... |
| exp5k 36447 | An exportation inference. ... |
| exp56 36448 | An exportation inference. ... |
| exp58 36449 | An exportation inference. ... |
| exp510 36450 | An exportation inference. ... |
| exp511 36451 | An exportation inference. ... |
| exp512 36452 | An exportation inference. ... |
| 3com12d 36453 | Commutation in consequent.... |
| imp5p 36454 | A triple importation infer... |
| imp5q 36455 | A triple importation infer... |
| ecase13d 36456 | Deduction for elimination ... |
| subtr 36457 | Transitivity of implicit s... |
| subtr2 36458 | Transitivity of implicit s... |
| trer 36459 | A relation intersected wit... |
| elicc3 36460 | An equivalent membership c... |
| finminlem 36461 | A useful lemma about finit... |
| gtinf 36462 | Any number greater than an... |
| opnrebl 36463 | A set is open in the stand... |
| opnrebl2 36464 | A set is open in the stand... |
| nn0prpwlem 36465 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 36466 | Two nonnegative integers a... |
| topbnd 36467 | Two equivalent expressions... |
| opnbnd 36468 | A set is open iff it is di... |
| cldbnd 36469 | A set is closed iff it con... |
| ntruni 36470 | A union of interiors is a ... |
| clsun 36471 | A pairwise union of closur... |
| clsint2 36472 | The closure of an intersec... |
| opnregcld 36473 | A set is regularly closed ... |
| cldregopn 36474 | A set if regularly open if... |
| neiin 36475 | Two neighborhoods intersec... |
| hmeoclda 36476 | Homeomorphisms preserve cl... |
| hmeocldb 36477 | Homeomorphisms preserve cl... |
| ivthALT 36478 | An alternate proof of the ... |
| fnerel 36481 | Fineness is a relation. (... |
| isfne 36482 | The predicate " ` B ` is f... |
| isfne4 36483 | The predicate " ` B ` is f... |
| isfne4b 36484 | A condition for a topology... |
| isfne2 36485 | The predicate " ` B ` is f... |
| isfne3 36486 | The predicate " ` B ` is f... |
| fnebas 36487 | A finer cover covers the s... |
| fnetg 36488 | A finer cover generates a ... |
| fnessex 36489 | If ` B ` is finer than ` A... |
| fneuni 36490 | If ` B ` is finer than ` A... |
| fneint 36491 | If a cover is finer than a... |
| fness 36492 | A cover is finer than its ... |
| fneref 36493 | Reflexivity of the finenes... |
| fnetr 36494 | Transitivity of the finene... |
| fneval 36495 | Two covers are finer than ... |
| fneer 36496 | Fineness intersected with ... |
| topfne 36497 | Fineness for covers corres... |
| topfneec 36498 | A cover is equivalent to a... |
| topfneec2 36499 | A topology is precisely id... |
| fnessref 36500 | A cover is finer iff it ha... |
| refssfne 36501 | A cover is a refinement if... |
| neibastop1 36502 | A collection of neighborho... |
| neibastop2lem 36503 | Lemma for ~ neibastop2 . ... |
| neibastop2 36504 | In the topology generated ... |
| neibastop3 36505 | The topology generated by ... |
| topmtcl 36506 | The meet of a collection o... |
| topmeet 36507 | Two equivalent formulation... |
| topjoin 36508 | Two equivalent formulation... |
| fnemeet1 36509 | The meet of a collection o... |
| fnemeet2 36510 | The meet of equivalence cl... |
| fnejoin1 36511 | Join of equivalence classe... |
| fnejoin2 36512 | Join of equivalence classe... |
| fgmin 36513 | Minimality property of a g... |
| neifg 36514 | The neighborhood filter of... |
| tailfval 36515 | The tail function for a di... |
| tailval 36516 | The tail of an element in ... |
| eltail 36517 | An element of a tail. (Co... |
| tailf 36518 | The tail function of a dir... |
| tailini 36519 | A tail contains its initia... |
| tailfb 36520 | The collection of tails of... |
| filnetlem1 36521 | Lemma for ~ filnet . Chan... |
| filnetlem2 36522 | Lemma for ~ filnet . The ... |
| filnetlem3 36523 | Lemma for ~ filnet . (Con... |
| filnetlem4 36524 | Lemma for ~ filnet . (Con... |
| filnet 36525 | A filter has the same conv... |
| tb-ax1 36526 | The first of three axioms ... |
| tb-ax2 36527 | The second of three axioms... |
| tb-ax3 36528 | The third of three axioms ... |
| tbsyl 36529 | The weak syllogism from Ta... |
| re1ax2lem 36530 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 36531 | ~ ax-2 rederived from the ... |
| naim1 36532 | Constructor theorem for ` ... |
| naim2 36533 | Constructor theorem for ` ... |
| naim1i 36534 | Constructor rule for ` -/\... |
| naim2i 36535 | Constructor rule for ` -/\... |
| naim12i 36536 | Constructor rule for ` -/\... |
| nabi1i 36537 | Constructor rule for ` -/\... |
| nabi2i 36538 | Constructor rule for ` -/\... |
| nabi12i 36539 | Constructor rule for ` -/\... |
| df3nandALT1 36542 | The double nand expressed ... |
| df3nandALT2 36543 | The double nand expressed ... |
| andnand1 36544 | Double and in terms of dou... |
| imnand2 36545 | An ` -> ` nand relation. ... |
| nalfal 36546 | Not all sets hold ` F. ` a... |
| nexntru 36547 | There does not exist a set... |
| nexfal 36548 | There does not exist a set... |
| neufal 36549 | There does not exist exact... |
| neutru 36550 | There does not exist exact... |
| nmotru 36551 | There does not exist at mo... |
| mofal 36552 | There exist at most one se... |
| nrmo 36553 | "At most one" restricted e... |
| meran1 36554 | A single axiom for proposi... |
| meran2 36555 | A single axiom for proposi... |
| meran3 36556 | A single axiom for proposi... |
| waj-ax 36557 | A single axiom for proposi... |
| lukshef-ax2 36558 | A single axiom for proposi... |
| arg-ax 36559 | A single axiom for proposi... |
| negsym1 36560 | In the paper "On Variable ... |
| imsym1 36561 | A symmetry with ` -> ` . ... |
| bisym1 36562 | A symmetry with ` <-> ` . ... |
| consym1 36563 | A symmetry with ` /\ ` . ... |
| dissym1 36564 | A symmetry with ` \/ ` . ... |
| nandsym1 36565 | A symmetry with ` -/\ ` . ... |
| unisym1 36566 | A symmetry with ` A. ` . ... |
| exisym1 36567 | A symmetry with ` E. ` . ... |
| unqsym1 36568 | A symmetry with ` E! ` . ... |
| amosym1 36569 | A symmetry with ` E* ` . ... |
| subsym1 36570 | A symmetry with ` [ x / y ... |
| ontopbas 36571 | An ordinal number is a top... |
| onsstopbas 36572 | The class of ordinal numbe... |
| onpsstopbas 36573 | The class of ordinal numbe... |
| ontgval 36574 | The topology generated fro... |
| ontgsucval 36575 | The topology generated fro... |
| onsuctop 36576 | A successor ordinal number... |
| onsuctopon 36577 | One of the topologies on a... |
| ordtoplem 36578 | Membership of the class of... |
| ordtop 36579 | An ordinal is a topology i... |
| onsucconni 36580 | A successor ordinal number... |
| onsucconn 36581 | A successor ordinal number... |
| ordtopconn 36582 | An ordinal topology is con... |
| onintopssconn 36583 | An ordinal topology is con... |
| onsuct0 36584 | A successor ordinal number... |
| ordtopt0 36585 | An ordinal topology is T_0... |
| onsucsuccmpi 36586 | The successor of a success... |
| onsucsuccmp 36587 | The successor of a success... |
| limsucncmpi 36588 | The successor of a limit o... |
| limsucncmp 36589 | The successor of a limit o... |
| ordcmp 36590 | An ordinal topology is com... |
| ssoninhaus 36591 | The ordinal topologies ` 1... |
| onint1 36592 | The ordinal T_1 spaces are... |
| oninhaus 36593 | The ordinal Hausdorff spac... |
| fveleq 36594 | Please add description her... |
| findfvcl 36595 | Please add description her... |
| findreccl 36596 | Please add description her... |
| findabrcl 36597 | Please add description her... |
| nnssi2 36598 | Convert a theorem for real... |
| nnssi3 36599 | Convert a theorem for real... |
| nndivsub 36600 | Please add description her... |
| nndivlub 36601 | A factor of a positive int... |
| ee7.2aOLD 36604 | Lemma for Euclid's Element... |
| weiunlem1 36605 | Lemma for ~ weiunpo , ~ we... |
| weiunlem2 36606 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 36607 | Lemma for ~ weiunfr . (Co... |
| weiunpo 36608 | A partial ordering on an i... |
| weiunso 36609 | A strict ordering on an in... |
| weiunfr 36610 | A well-founded relation on... |
| weiunse 36611 | The relation constructed i... |
| weiunwe 36612 | A well-ordering on an inde... |
| numiunnum 36613 | An indexed union of sets i... |
| dnival 36614 | Value of the "distance to ... |
| dnicld1 36615 | Closure theorem for the "d... |
| dnicld2 36616 | Closure theorem for the "d... |
| dnif 36617 | The "distance to nearest i... |
| dnizeq0 36618 | The distance to nearest in... |
| dnizphlfeqhlf 36619 | The distance to nearest in... |
| rddif2 36620 | Variant of ~ rddif . (Con... |
| dnibndlem1 36621 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 36622 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 36623 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 36624 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 36625 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 36626 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 36627 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 36628 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 36629 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 36630 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 36631 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 36632 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 36633 | Lemma for ~ dnibnd . (Con... |
| dnibnd 36634 | The "distance to nearest i... |
| dnicn 36635 | The "distance to nearest i... |
| knoppcnlem1 36636 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 36637 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 36638 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 36639 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 36640 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 36641 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 36642 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 36643 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 36644 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 36645 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 36646 | Lemma for ~ knoppcn . (Co... |
| knoppcn 36647 | The continuous nowhere dif... |
| knoppcld 36648 | Closure theorem for Knopp'... |
| unblimceq0lem 36649 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 36650 | If ` F ` is unbounded near... |
| unbdqndv1 36651 | If the difference quotient... |
| unbdqndv2lem1 36652 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 36653 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 36654 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 36655 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 36656 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 36657 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 36658 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 36659 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 36660 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 36661 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 36662 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 36663 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 36664 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 36665 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 36666 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 36667 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 36668 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 36669 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 36670 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 36671 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 36672 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 36673 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 36674 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 36675 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 36676 | Lemma for ~ knoppndv . (C... |
| knoppndv 36677 | The continuous nowhere dif... |
| knoppf 36678 | Knopp's function is a func... |
| knoppcn2 36679 | Variant of ~ knoppcn with ... |
| cnndvlem1 36680 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 36681 | Lemma for ~ cnndv . (Cont... |
| cnndv 36682 | There exists a continuous ... |
| bj-mp2c 36683 | A double _modus ponens_ in... |
| bj-mp2d 36684 | A double _modus ponens_ in... |
| bj-0 36685 | A syntactic theorem. See ... |
| bj-1 36686 | In this proof, the use of ... |
| bj-a1k 36687 | Weakening of ~ ax-1 . As ... |
| bj-poni 36688 | Inference associated with ... |
| bj-nnclav 36689 | When ` F. ` is substituted... |
| bj-nnclavi 36690 | Inference associated with ... |
| bj-nnclavc 36691 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 36692 | Inference associated with ... |
| bj-jarrii 36693 | Inference associated with ... |
| bj-imim21 36694 | The propositional function... |
| bj-imim21i 36695 | Inference associated with ... |
| bj-peircestab 36696 | Over minimal implicational... |
| bj-stabpeirce 36697 | This minimal implicational... |
| bj-syl66ib 36698 | A mixed syllogism inferenc... |
| bj-orim2 36699 | Proof of ~ orim2 from the ... |
| bj-currypeirce 36700 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 36701 | Peirce's axiom ~ peirce im... |
| bj-animbi 36702 | Conjunction in terms of im... |
| bj-currypara 36703 | Curry's paradox. Note tha... |
| bj-con2com 36704 | A commuted form of the con... |
| bj-con2comi 36705 | Inference associated with ... |
| bj-nimn 36706 | If a formula is true, then... |
| bj-nimni 36707 | Inference associated with ... |
| bj-peircei 36708 | Inference associated with ... |
| bj-looinvi 36709 | Inference associated with ... |
| bj-looinvii 36710 | Inference associated with ... |
| bj-mt2bi 36711 | Version of ~ mt2 where the... |
| bj-ntrufal 36712 | The negation of a theorem ... |
| bj-fal 36713 | Shortening of ~ fal using ... |
| bj-jaoi1 36714 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 36715 | Shortens ~ consensus (110>... |
| bj-dfbi4 36716 | Alternate definition of th... |
| bj-dfbi5 36717 | Alternate definition of th... |
| bj-dfbi6 36718 | Alternate definition of th... |
| bj-bijust0ALT 36719 | Alternate proof of ~ bijus... |
| bj-bijust00 36720 | A self-implication does no... |
| bj-consensus 36721 | Version of ~ consensus exp... |
| bj-consensusALT 36722 | Alternate proof of ~ bj-co... |
| bj-df-ifc 36723 | Candidate definition for t... |
| bj-dfif 36724 | Alternate definition of th... |
| bj-ififc 36725 | A biconditional connecting... |
| bj-imbi12 36726 | Uncurried (imported) form ... |
| bj-falor 36727 | Dual of ~ truan (which has... |
| bj-falor2 36728 | Dual of ~ truan . (Contri... |
| bj-bibibi 36729 | A property of the bicondit... |
| bj-imn3ani 36730 | Duplication of ~ bnj1224 .... |
| bj-andnotim 36731 | Two ways of expressing a c... |
| bj-bi3ant 36732 | This used to be in the mai... |
| bj-bisym 36733 | This used to be in the mai... |
| bj-bixor 36734 | Equivalence of two ternary... |
| bj-axdd2 36735 | This implication, proved u... |
| bj-axd2d 36736 | This implication, proved u... |
| bj-axtd 36737 | This implication, proved f... |
| bj-gl4 36738 | In a normal modal logic, t... |
| bj-axc4 36739 | Over minimal calculus, the... |
| prvlem1 36744 | An elementary property of ... |
| prvlem2 36745 | An elementary property of ... |
| bj-babygodel 36746 | See the section header com... |
| bj-babylob 36747 | See the section header com... |
| bj-godellob 36748 | Proof of Gödel's theo... |
| bj-genr 36749 | Generalization rule on the... |
| bj-genl 36750 | Generalization rule on the... |
| bj-genan 36751 | Generalization rule on a c... |
| bj-mpgs 36752 | From a closed form theorem... |
| bj-2alim 36753 | Closed form of ~ 2alimi . ... |
| bj-2exim 36754 | Closed form of ~ 2eximi . ... |
| bj-alanim 36755 | Closed form of ~ alanimi .... |
| bj-2albi 36756 | Closed form of ~ 2albii . ... |
| bj-notalbii 36757 | Equivalence of universal q... |
| bj-2exbi 36758 | Closed form of ~ 2exbii . ... |
| bj-3exbi 36759 | Closed form of ~ 3exbii . ... |
| bj-sylggt 36760 | Stronger form of ~ sylgt ,... |
| bj-sylgt2 36761 | Uncurried (imported) form ... |
| bj-alrimg 36762 | The general form of the *a... |
| bj-alrimd 36763 | A slightly more general ~ ... |
| bj-sylget 36764 | Dual statement of ~ sylgt ... |
| bj-sylget2 36765 | Uncurried (imported) form ... |
| bj-exlimg 36766 | The general form of the *e... |
| bj-sylge 36767 | Dual statement of ~ sylg (... |
| bj-exlimd 36768 | A slightly more general ~ ... |
| bj-nfimexal 36769 | A weak from of nonfreeness... |
| bj-alexim 36770 | Closed form of ~ aleximi .... |
| bj-nexdh 36771 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 36772 | Uncurried (imported) form ... |
| bj-hbxfrbi 36773 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 36774 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 36775 | Distribute quantifiers ove... |
| bj-exalimi 36776 | An inference for distribut... |
| bj-exalims 36777 | Distributing quantifiers o... |
| bj-exalimsi 36778 | An inference for distribut... |
| bj-ax12ig 36779 | A lemma used to prove a we... |
| bj-ax12i 36780 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 36781 | Closed form of ~ nfim and ... |
| bj-cbvalimt 36782 | A lemma in closed form use... |
| bj-cbveximt 36783 | A lemma in closed form use... |
| bj-eximALT 36784 | Alternate proof of ~ exim ... |
| bj-aleximiALT 36785 | Alternate proof of ~ alexi... |
| bj-eximcom 36786 | A commuted form of ~ exim ... |
| bj-ax12wlem 36787 | A lemma used to prove a we... |
| bj-cbvalim 36788 | A lemma used to prove ~ bj... |
| bj-cbvexim 36789 | A lemma used to prove ~ bj... |
| bj-cbvalimi 36790 | An equality-free general i... |
| bj-cbveximi 36791 | An equality-free general i... |
| bj-cbval 36792 | Changing a bound variable ... |
| bj-cbvex 36793 | Changing a bound variable ... |
| bj-df-sb 36796 | Proposed definition to rep... |
| bj-ssbeq 36797 | Substitution in an equalit... |
| bj-ssblem1 36798 | A lemma for the definiens ... |
| bj-ssblem2 36799 | An instance of ~ ax-11 pro... |
| bj-ax12v 36800 | A weaker form of ~ ax-12 a... |
| bj-ax12 36801 | Remove a DV condition from... |
| bj-ax12ssb 36802 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 36803 | Special case of ~ 19.41 pr... |
| bj-equsexval 36804 | Special case of ~ equsexv ... |
| bj-subst 36805 | Proof of ~ sbalex from cor... |
| bj-ssbid2 36806 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 36807 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 36808 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 36809 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 36810 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 36811 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 36812 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 36813 | Closed form of ~ spimvw . ... |
| bj-spnfw 36814 | Theorem close to a closed ... |
| bj-cbvexiw 36815 | Change bound variable. Th... |
| bj-cbvexivw 36816 | Change bound variable. Th... |
| bj-modald 36817 | A short form of the axiom ... |
| bj-denot 36818 | A weakening of ~ ax-6 and ... |
| bj-eqs 36819 | A lemma for substitutions,... |
| bj-cbvexw 36820 | Change bound variable. Th... |
| bj-ax12w 36821 | The general statement that... |
| bj-ax89 36822 | A theorem which could be u... |
| bj-cleljusti 36823 | One direction of ~ cleljus... |
| bj-alcomexcom 36824 | Commutation of two existen... |
| bj-hbalt 36825 | Closed form of ~ hbal . W... |
| axc11n11 36826 | Proof of ~ axc11n from { ~... |
| axc11n11r 36827 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 36828 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 36829 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 36830 | Alternate proof of ~ bj-ax... |
| bj-sb 36831 | A weak variant of ~ sbid2 ... |
| bj-modalbe 36832 | The predicate-calculus ver... |
| bj-spst 36833 | Closed form of ~ sps . On... |
| bj-19.21bit 36834 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 36835 | Closed form of ~ 19.23bi .... |
| bj-nexrt 36836 | Closed form of ~ nexr . C... |
| bj-alrim 36837 | Closed form of ~ alrimi . ... |
| bj-alrim2 36838 | Uncurried (imported) form ... |
| bj-nfdt0 36839 | A theorem close to a close... |
| bj-nfdt 36840 | Closed form of ~ nf5d and ... |
| bj-nexdt 36841 | Closed form of ~ nexd . (... |
| bj-nexdvt 36842 | Closed form of ~ nexdv . ... |
| bj-alexbiex 36843 | Adding a second quantifier... |
| bj-exexbiex 36844 | Adding a second quantifier... |
| bj-alalbial 36845 | Adding a second quantifier... |
| bj-exalbial 36846 | Adding a second quantifier... |
| bj-19.9htbi 36847 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 36848 | Strengthening ~ hbnt by re... |
| bj-biexal1 36849 | A general FOL biconditiona... |
| bj-biexal2 36850 | When ` ph ` is substituted... |
| bj-biexal3 36851 | When ` ph ` is substituted... |
| bj-bialal 36852 | When ` ph ` is substituted... |
| bj-biexex 36853 | When ` ph ` is substituted... |
| bj-hbext 36854 | Closed form of ~ hbex . (... |
| bj-nfalt 36855 | Closed form of ~ nfal . (... |
| bj-nfext 36856 | Closed form of ~ nfex . (... |
| bj-eeanvw 36857 | Version of ~ exdistrv with... |
| bj-modal4 36858 | First-order logic form of ... |
| bj-modal4e 36859 | First-order logic form of ... |
| bj-modalb 36860 | A short form of the axiom ... |
| bj-wnf1 36861 | When ` ph ` is substituted... |
| bj-wnf2 36862 | When ` ph ` is substituted... |
| bj-wnfanf 36863 | When ` ph ` is substituted... |
| bj-wnfenf 36864 | When ` ph ` is substituted... |
| bj-substax12 36865 | Equivalent form of the axi... |
| bj-substw 36866 | Weak form of the LHS of ~ ... |
| bj-nnfbi 36869 | If two formulas are equiva... |
| bj-nnfbd 36870 | If two formulas are equiva... |
| bj-nnfbii 36871 | If two formulas are equiva... |
| bj-nnfa 36872 | Nonfreeness implies the eq... |
| bj-nnfad 36873 | Nonfreeness implies the eq... |
| bj-nnfai 36874 | Nonfreeness implies the eq... |
| bj-nnfe 36875 | Nonfreeness implies the eq... |
| bj-nnfed 36876 | Nonfreeness implies the eq... |
| bj-nnfei 36877 | Nonfreeness implies the eq... |
| bj-nnfea 36878 | Nonfreeness implies the eq... |
| bj-nnfead 36879 | Nonfreeness implies the eq... |
| bj-nnfeai 36880 | Nonfreeness implies the eq... |
| bj-dfnnf2 36881 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 36882 | New nonfreeness implies ol... |
| bj-wnfnf 36883 | When ` ph ` is substituted... |
| bj-nnfnt 36884 | A variable is nonfree in a... |
| bj-nnftht 36885 | A variable is nonfree in a... |
| bj-nnfth 36886 | A variable is nonfree in a... |
| bj-nnfnth 36887 | A variable is nonfree in t... |
| bj-nnfim1 36888 | A consequence of nonfreene... |
| bj-nnfim2 36889 | A consequence of nonfreene... |
| bj-nnfim 36890 | Nonfreeness in the anteced... |
| bj-nnfimd 36891 | Nonfreeness in the anteced... |
| bj-nnfan 36892 | Nonfreeness in both conjun... |
| bj-nnfand 36893 | Nonfreeness in both conjun... |
| bj-nnfor 36894 | Nonfreeness in both disjun... |
| bj-nnford 36895 | Nonfreeness in both disjun... |
| bj-nnfbit 36896 | Nonfreeness in both sides ... |
| bj-nnfbid 36897 | Nonfreeness in both sides ... |
| bj-nnfv 36898 | A non-occurring variable i... |
| bj-nnf-alrim 36899 | Proof of the closed form o... |
| bj-nnf-exlim 36900 | Proof of the closed form o... |
| bj-dfnnf3 36901 | Alternate definition of no... |
| bj-nfnnfTEMP 36902 | New nonfreeness is equival... |
| bj-nnfa1 36903 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 36904 | See ~ nfe1 . (Contributed... |
| bj-19.12 36905 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 36906 | One of four lemmas for non... |
| bj-nnflemee 36907 | One of four lemmas for non... |
| bj-nnflemae 36908 | One of four lemmas for non... |
| bj-nnflemea 36909 | One of four lemmas for non... |
| bj-nnfalt 36910 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 36911 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 36912 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 36913 | Statement ~ 19.21t proved ... |
| bj-19.23t 36914 | Statement ~ 19.23t proved ... |
| bj-19.36im 36915 | One direction of ~ 19.36 f... |
| bj-19.37im 36916 | One direction of ~ 19.37 f... |
| bj-19.42t 36917 | Closed form of ~ 19.42 fro... |
| bj-19.41t 36918 | Closed form of ~ 19.41 fro... |
| bj-sbft 36919 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 36920 | Version of ~ pm11.53v with... |
| bj-pm11.53v 36921 | Version of ~ pm11.53v with... |
| bj-pm11.53a 36922 | A variant of ~ pm11.53v . ... |
| bj-equsvt 36923 | A variant of ~ equsv . (C... |
| bj-equsalvwd 36924 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 36925 | Variant of ~ equsexvw . (... |
| bj-sbievwd 36926 | Variant of ~ sbievw . (Co... |
| bj-axc10 36927 | Alternate proof of ~ axc10... |
| bj-alequex 36928 | A fol lemma. See ~ aleque... |
| bj-spimt2 36929 | A step in the proof of ~ s... |
| bj-cbv3ta 36930 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 36931 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 36932 | A theorem close to a close... |
| bj-hbsb3 36933 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 36934 | A theorem close to a close... |
| bj-nfs1t2 36935 | A theorem close to a close... |
| bj-nfs1 36936 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 36937 | Version of ~ axc10 with a ... |
| bj-spimtv 36938 | Version of ~ spimt with a ... |
| bj-cbv3hv2 36939 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 36940 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 36941 | Version of ~ cbv2h with a ... |
| bj-cbv2v 36942 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 36943 | Version of ~ cbvald with a... |
| bj-cbvexdv 36944 | Version of ~ cbvexd with a... |
| bj-cbval2vv 36945 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 36946 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 36947 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 36948 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 36949 | Version of ~ cbvex4v with ... |
| bj-equsalhv 36950 | Version of ~ equsalh with ... |
| bj-axc11nv 36951 | Version of ~ axc11n with a... |
| bj-aecomsv 36952 | Version of ~ aecoms with a... |
| bj-axc11v 36953 | Version of ~ axc11 with a ... |
| bj-drnf2v 36954 | Version of ~ drnf2 with a ... |
| bj-equs45fv 36955 | Version of ~ equs45f with ... |
| bj-hbs1 36956 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 36957 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 36958 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 36959 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 36960 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 36961 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 36962 | Biconditional version of a... |
| bj-hbaeb 36963 | Biconditional version of ~... |
| bj-hbnaeb 36964 | Biconditional version of ~... |
| bj-dvv 36965 | A special instance of ~ bj... |
| bj-equsal1t 36966 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 36967 | Inference associated with ... |
| bj-equsal1 36968 | One direction of ~ equsal ... |
| bj-equsal2 36969 | One direction of ~ equsal ... |
| bj-equsal 36970 | Shorter proof of ~ equsal ... |
| stdpc5t 36971 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 36972 | More direct proof of ~ std... |
| 2stdpc5 36973 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 36974 | Proof of ~ 19.21t from ~ s... |
| exlimii 36975 | Inference associated with ... |
| ax11-pm 36976 | Proof of ~ ax-11 similar t... |
| ax6er 36977 | Commuted form of ~ ax6e . ... |
| exlimiieq1 36978 | Inferring a theorem when i... |
| exlimiieq2 36979 | Inferring a theorem when i... |
| ax11-pm2 36980 | Proof of ~ ax-11 from the ... |
| bj-sbsb 36981 | Biconditional showing two ... |
| bj-dfsb2 36982 | Alternate (dual) definitio... |
| bj-sbf3 36983 | Substitution has no effect... |
| bj-sbf4 36984 | Substitution has no effect... |
| bj-eu3f 36985 | Version of ~ eu3v where th... |
| bj-sblem1 36986 | Lemma for substitution. (... |
| bj-sblem2 36987 | Lemma for substitution. (... |
| bj-sblem 36988 | Lemma for substitution. (... |
| bj-sbievw1 36989 | Lemma for substitution. (... |
| bj-sbievw2 36990 | Lemma for substitution. (... |
| bj-sbievw 36991 | Lemma for substitution. C... |
| bj-sbievv 36992 | Version of ~ sbie with a s... |
| bj-moeub 36993 | Uniqueness is equivalent t... |
| bj-sbidmOLD 36994 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 36995 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 36996 | Curried (exported) form of... |
| bj-dvelimv 36997 | A version of ~ dvelim usin... |
| bj-nfeel2 36998 | Nonfreeness in a membershi... |
| bj-axc14nf 36999 | Proof of a version of ~ ax... |
| bj-axc14 37000 | Alternate proof of ~ axc14... |
| mobidvALT 37001 | Alternate proof of ~ mobid... |
| sbn1ALT 37002 | Alternate proof of ~ sbn1 ... |
| eliminable1 37003 | A theorem used to prove th... |
| eliminable2a 37004 | A theorem used to prove th... |
| eliminable2b 37005 | A theorem used to prove th... |
| eliminable2c 37006 | A theorem used to prove th... |
| eliminable3a 37007 | A theorem used to prove th... |
| eliminable3b 37008 | A theorem used to prove th... |
| eliminable-velab 37009 | A theorem used to prove th... |
| eliminable-veqab 37010 | A theorem used to prove th... |
| eliminable-abeqv 37011 | A theorem used to prove th... |
| eliminable-abeqab 37012 | A theorem used to prove th... |
| eliminable-abelv 37013 | A theorem used to prove th... |
| eliminable-abelab 37014 | A theorem used to prove th... |
| bj-denoteslem 37015 | Duplicate of ~ issettru an... |
| bj-denotesALTV 37016 | Moved to main as ~ iseqset... |
| bj-issettruALTV 37017 | Moved to main as ~ issettr... |
| bj-elabtru 37018 | This is as close as we can... |
| bj-issetwt 37019 | Closed form of ~ bj-issetw... |
| bj-issetw 37020 | The closest one can get to... |
| bj-issetiv 37021 | Version of ~ bj-isseti wit... |
| bj-isseti 37022 | Version of ~ isseti with a... |
| bj-ralvw 37023 | A weak version of ~ ralv n... |
| bj-rexvw 37024 | A weak version of ~ rexv n... |
| bj-rababw 37025 | A weak version of ~ rabab ... |
| bj-rexcom4bv 37026 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 37027 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 37028 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 37029 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 37030 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 37031 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 37032 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 37033 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 37034 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 37035 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 37036 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 37037 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 37038 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 37039 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 37040 | Remove from ~ spcimdv depe... |
| elelb 37041 | Equivalence between two co... |
| bj-pwvrelb 37042 | Characterization of the el... |
| bj-nfcsym 37043 | The nonfreeness quantifier... |
| bj-sbeqALT 37044 | Substitution in an equalit... |
| bj-sbeq 37045 | Distribute proper substitu... |
| bj-sbceqgALT 37046 | Distribute proper substitu... |
| bj-csbsnlem 37047 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 37048 | Substitution in a singleto... |
| bj-sbel1 37049 | Version of ~ sbcel1g when ... |
| bj-abv 37050 | The class of sets verifyin... |
| bj-abvALT 37051 | Alternate version of ~ bj-... |
| bj-ab0 37052 | The class of sets verifyin... |
| bj-abf 37053 | Shorter proof of ~ abf (wh... |
| bj-csbprc 37054 | More direct proof of ~ csb... |
| bj-exlimvmpi 37055 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 37056 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 37057 | Lemma for theorems of the ... |
| bj-exlimmpbir 37058 | Lemma for theorems of the ... |
| bj-vtoclf 37059 | Remove dependency on ~ ax-... |
| bj-vtocl 37060 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 37061 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 37062 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 37063 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 37064 | A version of ~ vtoclg with... |
| bj-rabeqbid 37065 | Version of ~ rabeqbidv wit... |
| bj-seex 37066 | Version of ~ seex with a d... |
| bj-nfcf 37067 | Version of ~ df-nfc with a... |
| bj-zfauscl 37068 | General version of ~ zfaus... |
| bj-elabd2ALT 37069 | Alternate proof of ~ elabd... |
| bj-unrab 37070 | Generalization of ~ unrab ... |
| bj-inrab 37071 | Generalization of ~ inrab ... |
| bj-inrab2 37072 | Shorter proof of ~ inrab .... |
| bj-inrab3 37073 | Generalization of ~ dfrab3... |
| bj-rabtr 37074 | Restricted class abstracti... |
| bj-rabtrALT 37075 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 37076 | Proof of ~ bj-rabtr found ... |
| bj-gabss 37079 | Inclusion of generalized c... |
| bj-gabssd 37080 | Inclusion of generalized c... |
| bj-gabeqd 37081 | Equality of generalized cl... |
| bj-gabeqis 37082 | Equality of generalized cl... |
| bj-elgab 37083 | Elements of a generalized ... |
| bj-gabima 37084 | Generalized class abstract... |
| bj-ru1 37087 | A version of Russell's par... |
| bj-ru 37088 | Remove dependency on ~ ax-... |
| currysetlem 37089 | Lemma for ~ currysetlem , ... |
| curryset 37090 | Curry's paradox in set the... |
| currysetlem1 37091 | Lemma for ~ currysetALT . ... |
| currysetlem2 37092 | Lemma for ~ currysetALT . ... |
| currysetlem3 37093 | Lemma for ~ currysetALT . ... |
| currysetALT 37094 | Alternate proof of ~ curry... |
| bj-n0i 37095 | Inference associated with ... |
| bj-disjsn01 37096 | Disjointness of the single... |
| bj-0nel1 37097 | The empty set does not bel... |
| bj-1nel0 37098 | ` 1o ` does not belong to ... |
| bj-xpimasn 37099 | The image of a singleton, ... |
| bj-xpima1sn 37100 | The image of a singleton b... |
| bj-xpima1snALT 37101 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 37102 | The image of a singleton b... |
| bj-xpnzex 37103 | If the first factor of a p... |
| bj-xpexg2 37104 | Curried (exported) form of... |
| bj-xpnzexb 37105 | If the first factor of a p... |
| bj-cleq 37106 | Substitution property for ... |
| bj-snsetex 37107 | The class of sets "whose s... |
| bj-clexab 37108 | Sethood of certain classes... |
| bj-sngleq 37111 | Substitution property for ... |
| bj-elsngl 37112 | Characterization of the el... |
| bj-snglc 37113 | Characterization of the el... |
| bj-snglss 37114 | The singletonization of a ... |
| bj-0nelsngl 37115 | The empty set is not a mem... |
| bj-snglinv 37116 | Inverse of singletonizatio... |
| bj-snglex 37117 | A class is a set if and on... |
| bj-tageq 37120 | Substitution property for ... |
| bj-eltag 37121 | Characterization of the el... |
| bj-0eltag 37122 | The empty set belongs to t... |
| bj-tagn0 37123 | The tagging of a class is ... |
| bj-tagss 37124 | The tagging of a class is ... |
| bj-snglsstag 37125 | The singletonization is in... |
| bj-sngltagi 37126 | The singletonization is in... |
| bj-sngltag 37127 | The singletonization and t... |
| bj-tagci 37128 | Characterization of the el... |
| bj-tagcg 37129 | Characterization of the el... |
| bj-taginv 37130 | Inverse of tagging. (Cont... |
| bj-tagex 37131 | A class is a set if and on... |
| bj-xtageq 37132 | The products of a given cl... |
| bj-xtagex 37133 | The product of a set and t... |
| bj-projeq 37136 | Substitution property for ... |
| bj-projeq2 37137 | Substitution property for ... |
| bj-projun 37138 | The class projection on a ... |
| bj-projex 37139 | Sethood of the class proje... |
| bj-projval 37140 | Value of the class project... |
| bj-1upleq 37143 | Substitution property for ... |
| bj-pr1eq 37146 | Substitution property for ... |
| bj-pr1un 37147 | The first projection prese... |
| bj-pr1val 37148 | Value of the first project... |
| bj-pr11val 37149 | Value of the first project... |
| bj-pr1ex 37150 | Sethood of the first proje... |
| bj-1uplth 37151 | The characteristic propert... |
| bj-1uplex 37152 | A monuple is a set if and ... |
| bj-1upln0 37153 | A monuple is nonempty. (C... |
| bj-2upleq 37156 | Substitution property for ... |
| bj-pr21val 37157 | Value of the first project... |
| bj-pr2eq 37160 | Substitution property for ... |
| bj-pr2un 37161 | The second projection pres... |
| bj-pr2val 37162 | Value of the second projec... |
| bj-pr22val 37163 | Value of the second projec... |
| bj-pr2ex 37164 | Sethood of the second proj... |
| bj-2uplth 37165 | The characteristic propert... |
| bj-2uplex 37166 | A couple is a set if and o... |
| bj-2upln0 37167 | A couple is nonempty. (Co... |
| bj-2upln1upl 37168 | A couple is never equal to... |
| bj-rcleqf 37169 | Relative version of ~ cleq... |
| bj-rcleq 37170 | Relative version of ~ dfcl... |
| bj-reabeq 37171 | Relative form of ~ eqabb .... |
| bj-disj2r 37172 | Relative version of ~ ssdi... |
| bj-sscon 37173 | Contraposition law for rel... |
| bj-abex 37174 | Two ways of stating that t... |
| bj-clex 37175 | Two ways of stating that a... |
| bj-axsn 37176 | Two ways of stating the ax... |
| bj-snexg 37178 | A singleton built on a set... |
| bj-snex 37179 | A singleton is a set. See... |
| bj-axbun 37180 | Two ways of stating the ax... |
| bj-unexg 37182 | Existence of binary unions... |
| bj-prexg 37183 | Existence of unordered pai... |
| bj-prex 37184 | Existence of unordered pai... |
| bj-axadj 37185 | Two ways of stating the ax... |
| bj-adjg1 37187 | Existence of the result of... |
| bj-snfromadj 37188 | Singleton from adjunction ... |
| bj-prfromadj 37189 | Unordered pair from adjunc... |
| bj-adjfrombun 37190 | Adjunction from singleton ... |
| eleq2w2ALT 37191 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 37192 | Alternate proof of ~ clel3... |
| bj-pw0ALT 37193 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 37194 | Quantitative version of ~ ... |
| bj-unirel 37195 | Quantitative version of ~ ... |
| bj-elpwg 37196 | If the intersection of two... |
| bj-velpwALT 37197 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 37198 | Alternate proof of ~ elpwg... |
| bj-vjust 37199 | Justification theorem for ... |
| bj-nul 37200 | Two formulations of the ax... |
| bj-nuliota 37201 | Definition of the empty se... |
| bj-nuliotaALT 37202 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 37203 | Alternate proof of ~ vtocl... |
| bj-elsn12g 37204 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 37205 | Biconditional version of ~... |
| bj-pwcfsdom 37206 | Remove hypothesis from ~ p... |
| bj-grur1 37207 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 37208 | The extension of a predica... |
| bj-dfid2ALT 37209 | Alternate version of ~ dfi... |
| bj-0nelopab 37210 | The empty set is never an ... |
| bj-brrelex12ALT 37211 | Two classes related by a b... |
| bj-epelg 37212 | The membership relation an... |
| bj-epelb 37213 | Two classes are related by... |
| bj-nsnid 37214 | A set does not contain the... |
| bj-rdg0gALT 37215 | Alternate proof of ~ rdg0g... |
| bj-evaleq 37216 | Equality theorem for the `... |
| bj-evalfun 37217 | The evaluation at a class ... |
| bj-evalfn 37218 | The evaluation at a class ... |
| bj-evalval 37219 | Value of the evaluation at... |
| bj-evalid 37220 | The evaluation at a set of... |
| bj-ndxarg 37221 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 37222 | Closed general form of ~ s... |
| bj-rest00 37225 | An elementwise intersectio... |
| bj-restsn 37226 | An elementwise intersectio... |
| bj-restsnss 37227 | Special case of ~ bj-rests... |
| bj-restsnss2 37228 | Special case of ~ bj-rests... |
| bj-restsn0 37229 | An elementwise intersectio... |
| bj-restsn10 37230 | Special case of ~ bj-rests... |
| bj-restsnid 37231 | The elementwise intersecti... |
| bj-rest10 37232 | An elementwise intersectio... |
| bj-rest10b 37233 | Alternate version of ~ bj-... |
| bj-restn0 37234 | An elementwise intersectio... |
| bj-restn0b 37235 | Alternate version of ~ bj-... |
| bj-restpw 37236 | The elementwise intersecti... |
| bj-rest0 37237 | An elementwise intersectio... |
| bj-restb 37238 | An elementwise intersectio... |
| bj-restv 37239 | An elementwise intersectio... |
| bj-resta 37240 | An elementwise intersectio... |
| bj-restuni 37241 | The union of an elementwis... |
| bj-restuni2 37242 | The union of an elementwis... |
| bj-restreg 37243 | A reformulation of the axi... |
| bj-raldifsn 37244 | All elements in a set sati... |
| bj-0int 37245 | If ` A ` is a collection o... |
| bj-mooreset 37246 | A Moore collection is a se... |
| bj-ismoore 37249 | Characterization of Moore ... |
| bj-ismoored0 37250 | Necessary condition to be ... |
| bj-ismoored 37251 | Necessary condition to be ... |
| bj-ismoored2 37252 | Necessary condition to be ... |
| bj-ismooredr 37253 | Sufficient condition to be... |
| bj-ismooredr2 37254 | Sufficient condition to be... |
| bj-discrmoore 37255 | The powerclass ` ~P A ` is... |
| bj-0nmoore 37256 | The empty set is not a Moo... |
| bj-snmoore 37257 | A singleton is a Moore col... |
| bj-snmooreb 37258 | A singleton is a Moore col... |
| bj-prmoore 37259 | A pair formed of two neste... |
| bj-0nelmpt 37260 | The empty set is not an el... |
| bj-mptval 37261 | Value of a function given ... |
| bj-dfmpoa 37262 | An equivalent definition o... |
| bj-mpomptALT 37263 | Alternate proof of ~ mpomp... |
| setsstrset 37280 | Relation between ~ df-sets... |
| bj-nfald 37281 | Variant of ~ nfald . (Con... |
| bj-nfexd 37282 | Variant of ~ nfexd . (Con... |
| copsex2d 37283 | Implicit substitution dedu... |
| copsex2b 37284 | Biconditional form of ~ co... |
| opelopabd 37285 | Membership of an ordere pa... |
| opelopabb 37286 | Membership of an ordered p... |
| opelopabbv 37287 | Membership of an ordered p... |
| bj-opelrelex 37288 | The coordinates of an orde... |
| bj-opelresdm 37289 | If an ordered pair is in a... |
| bj-brresdm 37290 | If two classes are related... |
| brabd0 37291 | Expressing that two sets a... |
| brabd 37292 | Expressing that two sets a... |
| bj-brab2a1 37293 | "Unbounded" version of ~ b... |
| bj-opabssvv 37294 | A variant of ~ relopabiv (... |
| bj-funidres 37295 | The restricted identity re... |
| bj-opelidb 37296 | Characterization of the or... |
| bj-opelidb1 37297 | Characterization of the or... |
| bj-inexeqex 37298 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 37299 | If the intersection of two... |
| bj-opelid 37300 | Characterization of the or... |
| bj-ideqg 37301 | Characterization of the cl... |
| bj-ideqgALT 37302 | Alternate proof of ~ bj-id... |
| bj-ideqb 37303 | Characterization of classe... |
| bj-idres 37304 | Alternate expression for t... |
| bj-opelidres 37305 | Characterization of the or... |
| bj-idreseq 37306 | Sufficient condition for t... |
| bj-idreseqb 37307 | Characterization for two c... |
| bj-ideqg1 37308 | For sets, the identity rel... |
| bj-ideqg1ALT 37309 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 37310 | Characterization of the co... |
| bj-elid3 37311 | Characterization of the co... |
| bj-elid4 37312 | Characterization of the el... |
| bj-elid5 37313 | Characterization of the el... |
| bj-elid6 37314 | Characterization of the el... |
| bj-elid7 37315 | Characterization of the el... |
| bj-diagval 37318 | Value of the functionalize... |
| bj-diagval2 37319 | Value of the functionalize... |
| bj-eldiag 37320 | Characterization of the el... |
| bj-eldiag2 37321 | Characterization of the el... |
| bj-imdirvallem 37324 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 37325 | Value of the functionalize... |
| bj-imdirval2lem 37326 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 37327 | Value of the functionalize... |
| bj-imdirval3 37328 | Value of the functionalize... |
| bj-imdiridlem 37329 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 37330 | Functorial property of the... |
| bj-opelopabid 37331 | Membership in an ordered-p... |
| bj-opabco 37332 | Composition of ordered-pai... |
| bj-xpcossxp 37333 | The composition of two Car... |
| bj-imdirco 37334 | Functorial property of the... |
| bj-iminvval 37337 | Value of the functionalize... |
| bj-iminvval2 37338 | Value of the functionalize... |
| bj-iminvid 37339 | Functorial property of the... |
| bj-inftyexpitaufo 37346 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 37349 | An element of the circle a... |
| bj-inftyexpiinv 37352 | Utility theorem for the in... |
| bj-inftyexpiinj 37353 | Injectivity of the paramet... |
| bj-inftyexpidisj 37354 | An element of the circle a... |
| bj-ccinftydisj 37357 | The circle at infinity is ... |
| bj-elccinfty 37358 | A lemma for infinite exten... |
| bj-ccssccbar 37361 | Complex numbers are extend... |
| bj-ccinftyssccbar 37362 | Infinite extended complex ... |
| bj-pinftyccb 37365 | The class ` pinfty ` is an... |
| bj-pinftynrr 37366 | The extended complex numbe... |
| bj-minftyccb 37369 | The class ` minfty ` is an... |
| bj-minftynrr 37370 | The extended complex numbe... |
| bj-pinftynminfty 37371 | The extended complex numbe... |
| bj-rrhatsscchat 37380 | The real projective line i... |
| bj-imafv 37395 | If the direct image of a s... |
| bj-funun 37396 | Value of a function expres... |
| bj-fununsn1 37397 | Value of a function expres... |
| bj-fununsn2 37398 | Value of a function expres... |
| bj-fvsnun1 37399 | The value of a function wi... |
| bj-fvsnun2 37400 | The value of a function wi... |
| bj-fvmptunsn1 37401 | Value of a function expres... |
| bj-fvmptunsn2 37402 | Value of a function expres... |
| bj-iomnnom 37403 | The canonical bijection fr... |
| bj-smgrpssmgm 37412 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 37413 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 37414 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 37415 | Monoids are semigroups (el... |
| bj-cmnssmnd 37416 | Commutative monoids are mo... |
| bj-cmnssmndel 37417 | Commutative monoids are mo... |
| bj-grpssmnd 37418 | Groups are monoids. (Cont... |
| bj-grpssmndel 37419 | Groups are monoids (elemen... |
| bj-ablssgrp 37420 | Abelian groups are groups.... |
| bj-ablssgrpel 37421 | Abelian groups are groups ... |
| bj-ablsscmn 37422 | Abelian groups are commuta... |
| bj-ablsscmnel 37423 | Abelian groups are commuta... |
| bj-modssabl 37424 | (The additive groups of) m... |
| bj-vecssmod 37425 | Vector spaces are modules.... |
| bj-vecssmodel 37426 | Vector spaces are modules ... |
| bj-finsumval0 37429 | Value of a finite sum. (C... |
| bj-fvimacnv0 37430 | Variant of ~ fvimacnv wher... |
| bj-isvec 37431 | The predicate "is a vector... |
| bj-fldssdrng 37432 | Fields are division rings.... |
| bj-flddrng 37433 | Fields are division rings ... |
| bj-rrdrg 37434 | The field of real numbers ... |
| bj-isclm 37435 | The predicate "is a subcom... |
| bj-isrvec 37438 | The predicate "is a real v... |
| bj-rvecmod 37439 | Real vector spaces are mod... |
| bj-rvecssmod 37440 | Real vector spaces are mod... |
| bj-rvecrr 37441 | The field of scalars of a ... |
| bj-isrvecd 37442 | The predicate "is a real v... |
| bj-rvecvec 37443 | Real vector spaces are vec... |
| bj-isrvec2 37444 | The predicate "is a real v... |
| bj-rvecssvec 37445 | Real vector spaces are vec... |
| bj-rveccmod 37446 | Real vector spaces are sub... |
| bj-rvecsscmod 37447 | Real vector spaces are sub... |
| bj-rvecsscvec 37448 | Real vector spaces are sub... |
| bj-rveccvec 37449 | Real vector spaces are sub... |
| bj-rvecssabl 37450 | (The additive groups of) r... |
| bj-rvecabl 37451 | (The additive groups of) r... |
| bj-subcom 37452 | A consequence of commutati... |
| bj-lineqi 37453 | Solution of a (scalar) lin... |
| bj-bary1lem 37454 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 37455 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 37456 | Barycentric coordinates in... |
| bj-endval 37459 | Value of the monoid of end... |
| bj-endbase 37460 | Base set of the monoid of ... |
| bj-endcomp 37461 | Composition law of the mon... |
| bj-endmnd 37462 | The monoid of endomorphism... |
| taupilem3 37463 | Lemma for tau-related theo... |
| taupilemrplb 37464 | A set of positive reals ha... |
| taupilem1 37465 | Lemma for ~ taupi . A pos... |
| taupilem2 37466 | Lemma for ~ taupi . The s... |
| taupi 37467 | Relationship between ` _ta... |
| dfgcd3 37468 | Alternate definition of th... |
| irrdifflemf 37469 | Lemma for ~ irrdiff . The... |
| irrdiff 37470 | The irrationals are exactl... |
| iccioo01 37471 | The closed unit interval i... |
| csbrecsg 37472 | Move class substitution in... |
| csbrdgg 37473 | Move class substitution in... |
| csboprabg 37474 | Move class substitution in... |
| csbmpo123 37475 | Move class substitution in... |
| con1bii2 37476 | A contraposition inference... |
| con2bii2 37477 | A contraposition inference... |
| vtoclefex 37478 | Implicit substitution of a... |
| rnmptsn 37479 | The range of a function ma... |
| f1omptsnlem 37480 | This is the core of the pr... |
| f1omptsn 37481 | A function mapping to sing... |
| mptsnunlem 37482 | This is the core of the pr... |
| mptsnun 37483 | A class ` B ` is equal to ... |
| dissneqlem 37484 | This is the core of the pr... |
| dissneq 37485 | Any topology that contains... |
| exlimim 37486 | Closed form of ~ exlimimd ... |
| exlimimd 37487 | Existential elimination ru... |
| exellim 37488 | Closed form of ~ exellimdd... |
| exellimddv 37489 | Eliminate an antecedent wh... |
| topdifinfindis 37490 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 37491 | This is the core of the pr... |
| topdifinffin 37492 | Part of Exercise 3 of [Mun... |
| topdifinf 37493 | Part of Exercise 3 of [Mun... |
| topdifinfeq 37494 | Two different ways of defi... |
| icorempo 37495 | Closed-below, open-above i... |
| icoreresf 37496 | Closed-below, open-above i... |
| icoreval 37497 | Value of the closed-below,... |
| icoreelrnab 37498 | Elementhood in the set of ... |
| isbasisrelowllem1 37499 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 37500 | Lemma for ~ isbasisrelowl ... |
| icoreclin 37501 | The set of closed-below, o... |
| isbasisrelowl 37502 | The set of all closed-belo... |
| icoreunrn 37503 | The union of all closed-be... |
| istoprelowl 37504 | The set of all closed-belo... |
| icoreelrn 37505 | A class abstraction which ... |
| iooelexlt 37506 | An element of an open inte... |
| relowlssretop 37507 | The lower limit topology o... |
| relowlpssretop 37508 | The lower limit topology o... |
| sucneqond 37509 | Inequality of an ordinal s... |
| sucneqoni 37510 | Inequality of an ordinal s... |
| onsucuni3 37511 | If an ordinal number has a... |
| 1oequni2o 37512 | The ordinal number ` 1o ` ... |
| rdgsucuni 37513 | If an ordinal number has a... |
| rdgeqoa 37514 | If a recursive function wi... |
| elxp8 37515 | Membership in a Cartesian ... |
| cbveud 37516 | Deduction used to change b... |
| cbvreud 37517 | Deduction used to change b... |
| difunieq 37518 | The difference of unions i... |
| inunissunidif 37519 | Theorem about subsets of t... |
| rdgellim 37520 | Elementhood in a recursive... |
| rdglimss 37521 | A recursive definition at ... |
| rdgssun 37522 | In a recursive definition ... |
| exrecfnlem 37523 | Lemma for ~ exrecfn . (Co... |
| exrecfn 37524 | Theorem about the existenc... |
| exrecfnpw 37525 | For any base set, a set wh... |
| finorwe 37526 | If the Axiom of Infinity i... |
| dffinxpf 37529 | This theorem is the same a... |
| finxpeq1 37530 | Equality theorem for Carte... |
| finxpeq2 37531 | Equality theorem for Carte... |
| csbfinxpg 37532 | Distribute proper substitu... |
| finxpreclem1 37533 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 37534 | Lemma for ` ^^ ` recursion... |
| finxp0 37535 | The value of Cartesian exp... |
| finxp1o 37536 | The value of Cartesian exp... |
| finxpreclem3 37537 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 37538 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 37539 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 37540 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 37541 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 37542 | The value of Cartesian exp... |
| finxp2o 37543 | The value of Cartesian exp... |
| finxp3o 37544 | The value of Cartesian exp... |
| finxpnom 37545 | Cartesian exponentiation w... |
| finxp00 37546 | Cartesian exponentiation o... |
| iunctb2 37547 | Using the axiom of countab... |
| domalom 37548 | A class which dominates ev... |
| isinf2 37549 | The converse of ~ isinf . ... |
| ctbssinf 37550 | Using the axiom of choice,... |
| ralssiun 37551 | The index set of an indexe... |
| nlpineqsn 37552 | For every point ` p ` of a... |
| nlpfvineqsn 37553 | Given a subset ` A ` of ` ... |
| fvineqsnf1 37554 | A theorem about functions ... |
| fvineqsneu 37555 | A theorem about functions ... |
| fvineqsneq 37556 | A theorem about functions ... |
| pibp16 37557 | Property P000016 of pi-bas... |
| pibp19 37558 | Property P000019 of pi-bas... |
| pibp21 37559 | Property P000021 of pi-bas... |
| pibt1 37560 | Theorem T000001 of pi-base... |
| pibt2 37561 | Theorem T000002 of pi-base... |
| wl-section-prop 37562 | Intuitionistic logic is no... |
| wl-section-boot 37566 | In this section, I provide... |
| wl-luk-imim1i 37567 | Inference adding common co... |
| wl-luk-syl 37568 | An inference version of th... |
| wl-luk-imtrid 37569 | A syllogism rule of infere... |
| wl-luk-pm2.18d 37570 | Deduction based on reducti... |
| wl-luk-con4i 37571 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 37572 | Inference rule. Copy of ~... |
| wl-luk-a1i 37573 | Inference rule. Copy of ~... |
| wl-luk-mpi 37574 | A nested _modus ponens_ in... |
| wl-luk-imim2i 37575 | Inference adding common an... |
| wl-luk-imtrdi 37576 | A syllogism rule of infere... |
| wl-luk-ax3 37577 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 37578 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 37579 | This theorem, called "Asse... |
| wl-luk-com12 37580 | Inference that swaps (comm... |
| wl-luk-pm2.21 37581 | From a wff and its negatio... |
| wl-luk-con1i 37582 | A contraposition inference... |
| wl-luk-ja 37583 | Inference joining the ante... |
| wl-luk-imim2 37584 | A closed form of syllogism... |
| wl-luk-a1d 37585 | Deduction introducing an e... |
| wl-luk-ax2 37586 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 37587 | Principle of identity. Th... |
| wl-luk-notnotr 37588 | Converse of double negatio... |
| wl-luk-pm2.04 37589 | Swap antecedents. Theorem... |
| wl-section-impchain 37590 | An implication like ` ( ps... |
| wl-impchain-mp-x 37591 | This series of theorems pr... |
| wl-impchain-mp-0 37592 | This theorem is the start ... |
| wl-impchain-mp-1 37593 | This theorem is in fact a ... |
| wl-impchain-mp-2 37594 | This theorem is in fact a ... |
| wl-impchain-com-1.x 37595 | It is often convenient to ... |
| wl-impchain-com-1.1 37596 | A degenerate form of antec... |
| wl-impchain-com-1.2 37597 | This theorem is in fact a ... |
| wl-impchain-com-1.3 37598 | This theorem is in fact a ... |
| wl-impchain-com-1.4 37599 | This theorem is in fact a ... |
| wl-impchain-com-n.m 37600 | This series of theorems al... |
| wl-impchain-com-2.3 37601 | This theorem is in fact a ... |
| wl-impchain-com-2.4 37602 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 37603 | This theorem is in fact a ... |
| wl-impchain-a1-x 37604 | If an implication chain is... |
| wl-impchain-a1-1 37605 | Inference rule, a copy of ... |
| wl-impchain-a1-2 37606 | Inference rule, a copy of ... |
| wl-impchain-a1-3 37607 | Inference rule, a copy of ... |
| wl-ifp-ncond1 37608 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 37609 | If one case of an ` if- ` ... |
| wl-ifpimpr 37610 | If one case of an ` if- ` ... |
| wl-ifp4impr 37611 | If one case of an ` if- ` ... |
| wl-df-3xor 37612 | Alternative definition of ... |
| wl-df3xor2 37613 | Alternative definition of ... |
| wl-df3xor3 37614 | Alternative form of ~ wl-d... |
| wl-3xortru 37615 | If the first input is true... |
| wl-3xorfal 37616 | If the first input is fals... |
| wl-3xorbi 37617 | Triple xor can be replaced... |
| wl-3xorbi2 37618 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 37619 | Equivalence theorem for tr... |
| wl-3xorbi123i 37620 | Equivalence theorem for tr... |
| wl-3xorrot 37621 | Rotation law for triple xo... |
| wl-3xorcoma 37622 | Commutative law for triple... |
| wl-3xorcomb 37623 | Commutative law for triple... |
| wl-3xornot1 37624 | Flipping the first input f... |
| wl-3xornot 37625 | Triple xor distributes ove... |
| wl-1xor 37626 | In the recursive scheme ... |
| wl-2xor 37627 | In the recursive scheme ... |
| wl-df-3mintru2 37628 | Alternative definition of ... |
| wl-df2-3mintru2 37629 | The adder carry in disjunc... |
| wl-df3-3mintru2 37630 | The adder carry in conjunc... |
| wl-df4-3mintru2 37631 | An alternative definition ... |
| wl-1mintru1 37632 | Using the recursion formul... |
| wl-1mintru2 37633 | Using the recursion formul... |
| wl-2mintru1 37634 | Using the recursion formul... |
| wl-2mintru2 37635 | Using the recursion formul... |
| wl-df3maxtru1 37636 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 37638 | A version of ~ ax-wl-13v w... |
| wl-cleq-0 37639 |
Disclaimer: |
| wl-cleq-1 37640 |
Disclaimer: |
| wl-cleq-2 37641 |
Disclaimer: |
| wl-cleq-3 37642 |
Disclaimer: |
| wl-cleq-4 37643 |
Disclaimer: |
| wl-cleq-5 37644 |
Disclaimer: |
| wl-cleq-6 37645 |
Disclaimer: |
| wl-df-clab 37648 | Disclaimer: The material ... |
| wl-isseteq 37649 | A class equal to a set var... |
| wl-ax12v2cl 37650 | The class version of ~ ax1... |
| wl-mps 37651 | Replacing a nested consequ... |
| wl-syls1 37652 | Replacing a nested consequ... |
| wl-syls2 37653 | Replacing a nested anteced... |
| wl-embant 37654 | A true wff can always be a... |
| wl-orel12 37655 | In a conjunctive normal fo... |
| wl-cases2-dnf 37656 | A particular instance of ~... |
| wl-cbvmotv 37657 | Change bound variable. Us... |
| wl-moteq 37658 | Change bound variable. Us... |
| wl-motae 37659 | Change bound variable. Us... |
| wl-moae 37660 | Two ways to express "at mo... |
| wl-euae 37661 | Two ways to express "exact... |
| wl-nax6im 37662 | The following series of th... |
| wl-hbae1 37663 | This specialization of ~ h... |
| wl-naevhba1v 37664 | An instance of ~ hbn1w app... |
| wl-spae 37665 | Prove an instance of ~ sp ... |
| wl-speqv 37666 | Under the assumption ` -. ... |
| wl-19.8eqv 37667 | Under the assumption ` -. ... |
| wl-19.2reqv 37668 | Under the assumption ` -. ... |
| wl-nfalv 37669 | If ` x ` is not present in... |
| wl-nfimf1 37670 | An antecedent is irrelevan... |
| wl-nfae1 37671 | Unlike ~ nfae , this speci... |
| wl-nfnae1 37672 | Unlike ~ nfnae , this spec... |
| wl-aetr 37673 | A transitive law for varia... |
| wl-axc11r 37674 | Same as ~ axc11r , but usi... |
| wl-dral1d 37675 | A version of ~ dral1 with ... |
| wl-cbvalnaed 37676 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 37677 | A more general version of ... |
| wl-exeq 37678 | The semantics of ` E. x y ... |
| wl-aleq 37679 | The semantics of ` A. x y ... |
| wl-nfeqfb 37680 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 37681 | If ` y ` is not free in ` ... |
| wl-equsalvw 37682 | Version of ~ equsalv with ... |
| wl-equsald 37683 | Deduction version of ~ equ... |
| wl-equsaldv 37684 | Deduction version of ~ equ... |
| wl-equsal 37685 | A useful equivalence relat... |
| wl-equsal1t 37686 | The expression ` x = y ` i... |
| wl-equsalcom 37687 | This simple equivalence ea... |
| wl-equsal1i 37688 | The antecedent ` x = y ` i... |
| wl-sbid2ft 37689 | A more general version of ... |
| wl-cbvalsbi 37690 | Change bounded variables i... |
| wl-sbrimt 37691 | Substitution with a variab... |
| wl-sblimt 37692 | Substitution with a variab... |
| wl-sb9v 37693 | Commutation of quantificat... |
| wl-sb8ft 37694 | Substitution of variable i... |
| wl-sb8eft 37695 | Substitution of variable i... |
| wl-sb8t 37696 | Substitution of variable i... |
| wl-sb8et 37697 | Substitution of variable i... |
| wl-sbhbt 37698 | Closed form of ~ sbhb . C... |
| wl-sbnf1 37699 | Two ways expressing that `... |
| wl-equsb3 37700 | ~ equsb3 with a distinctor... |
| wl-equsb4 37701 | Substitution applied to an... |
| wl-2sb6d 37702 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 37703 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 37704 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 37705 | Version of ~ sbcom2 with a... |
| wl-sbalnae 37706 | A theorem used in eliminat... |
| wl-sbal1 37707 | A theorem used in eliminat... |
| wl-sbal2 37708 | Move quantifier in and out... |
| wl-2spsbbi 37709 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 37710 | This theorem provides a ba... |
| wl-lem-nexmo 37711 | This theorem provides a ba... |
| wl-lem-moexsb 37712 | The antecedent ` A. x ( ph... |
| wl-alanbii 37713 | This theorem extends ~ ala... |
| wl-mo2df 37714 | Version of ~ mof with a co... |
| wl-mo2tf 37715 | Closed form of ~ mof with ... |
| wl-eudf 37716 | Version of ~ eu6 with a co... |
| wl-eutf 37717 | Closed form of ~ eu6 with ... |
| wl-euequf 37718 | ~ euequ proved with a dist... |
| wl-mo2t 37719 | Closed form of ~ mof . (C... |
| wl-mo3t 37720 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 37721 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 37722 | Substitution of variable i... |
| wl-sb8eutv 37723 | Substitution of variable i... |
| wl-sb8mot 37724 | Substitution of variable i... |
| wl-sb8motv 37725 | Substitution of variable i... |
| wl-issetft 37726 | A closed form of ~ issetf ... |
| wl-axc11rc11 37727 | Proving ~ axc11r from ~ ax... |
| wl-clabv 37728 | Variant of ~ df-clab , whe... |
| wl-dfclab 37729 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 37730 | Using class abstraction in... |
| wl-clabt 37731 | Using class abstraction in... |
| wl-eujustlem1 37732 | Version of ~ cbvexvw with ... |
| rabiun 37733 | Abstraction restricted to ... |
| iundif1 37734 | Indexed union of class dif... |
| imadifss 37735 | The difference of images i... |
| cureq 37736 | Equality theorem for curry... |
| unceq 37737 | Equality theorem for uncur... |
| curf 37738 | Functional property of cur... |
| uncf 37739 | Functional property of unc... |
| curfv 37740 | Value of currying. (Contr... |
| uncov 37741 | Value of uncurrying. (Con... |
| curunc 37742 | Currying of uncurrying. (... |
| unccur 37743 | Uncurrying of currying. (... |
| phpreu 37744 | Theorem related to pigeonh... |
| finixpnum 37745 | A finite Cartesian product... |
| fin2solem 37746 | Lemma for ~ fin2so . (Con... |
| fin2so 37747 | Any totally ordered Tarski... |
| ltflcei 37748 | Theorem to move the floor ... |
| leceifl 37749 | Theorem to move the floor ... |
| sin2h 37750 | Half-angle rule for sine. ... |
| cos2h 37751 | Half-angle rule for cosine... |
| tan2h 37752 | Half-angle rule for tangen... |
| lindsadd 37753 | In a vector space, the uni... |
| lindsdom 37754 | A linearly independent set... |
| lindsenlbs 37755 | A maximal linearly indepen... |
| matunitlindflem1 37756 | One direction of ~ matunit... |
| matunitlindflem2 37757 | One direction of ~ matunit... |
| matunitlindf 37758 | A matrix over a field is i... |
| ptrest 37759 | Expressing a restriction o... |
| ptrecube 37760 | Any point in an open set o... |
| poimirlem1 37761 | Lemma for ~ poimir - the v... |
| poimirlem2 37762 | Lemma for ~ poimir - conse... |
| poimirlem3 37763 | Lemma for ~ poimir to add ... |
| poimirlem4 37764 | Lemma for ~ poimir connect... |
| poimirlem5 37765 | Lemma for ~ poimir to esta... |
| poimirlem6 37766 | Lemma for ~ poimir establi... |
| poimirlem7 37767 | Lemma for ~ poimir , simil... |
| poimirlem8 37768 | Lemma for ~ poimir , estab... |
| poimirlem9 37769 | Lemma for ~ poimir , estab... |
| poimirlem10 37770 | Lemma for ~ poimir establi... |
| poimirlem11 37771 | Lemma for ~ poimir connect... |
| poimirlem12 37772 | Lemma for ~ poimir connect... |
| poimirlem13 37773 | Lemma for ~ poimir - for a... |
| poimirlem14 37774 | Lemma for ~ poimir - for a... |
| poimirlem15 37775 | Lemma for ~ poimir , that ... |
| poimirlem16 37776 | Lemma for ~ poimir establi... |
| poimirlem17 37777 | Lemma for ~ poimir establi... |
| poimirlem18 37778 | Lemma for ~ poimir stating... |
| poimirlem19 37779 | Lemma for ~ poimir establi... |
| poimirlem20 37780 | Lemma for ~ poimir establi... |
| poimirlem21 37781 | Lemma for ~ poimir stating... |
| poimirlem22 37782 | Lemma for ~ poimir , that ... |
| poimirlem23 37783 | Lemma for ~ poimir , two w... |
| poimirlem24 37784 | Lemma for ~ poimir , two w... |
| poimirlem25 37785 | Lemma for ~ poimir stating... |
| poimirlem26 37786 | Lemma for ~ poimir showing... |
| poimirlem27 37787 | Lemma for ~ poimir showing... |
| poimirlem28 37788 | Lemma for ~ poimir , a var... |
| poimirlem29 37789 | Lemma for ~ poimir connect... |
| poimirlem30 37790 | Lemma for ~ poimir combini... |
| poimirlem31 37791 | Lemma for ~ poimir , assig... |
| poimirlem32 37792 | Lemma for ~ poimir , combi... |
| poimir 37793 | Poincare-Miranda theorem. ... |
| broucube 37794 | Brouwer - or as Kulpa call... |
| heicant 37795 | Heine-Cantor theorem: a co... |
| opnmbllem0 37796 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 37797 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 37798 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 37799 | The difference between two... |
| mblfinlem4 37800 | Backward direction of ~ is... |
| ismblfin 37801 | Measurability in terms of ... |
| ovoliunnfl 37802 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 37803 | Demonstration of ~ ovoliun... |
| voliunnfl 37804 | ~ voliun is incompatible w... |
| volsupnfl 37805 | ~ volsup is incompatible w... |
| mbfresfi 37806 | Measurability of a piecewi... |
| mbfposadd 37807 | If the sum of two measurab... |
| cnambfre 37808 | A real-valued, a.e. contin... |
| dvtanlem 37809 | Lemma for ~ dvtan - the do... |
| dvtan 37810 | Derivative of tangent. (C... |
| itg2addnclem 37811 | An alternate expression fo... |
| itg2addnclem2 37812 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 37813 | Lemma incomprehensible in ... |
| itg2addnc 37814 | Alternate proof of ~ itg2a... |
| itg2gt0cn 37815 | ~ itg2gt0 holds on functio... |
| ibladdnclem 37816 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 37817 | Choice-free analogue of ~ ... |
| itgaddnclem1 37818 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 37819 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 37820 | Choice-free analogue of ~ ... |
| iblsubnc 37821 | Choice-free analogue of ~ ... |
| itgsubnc 37822 | Choice-free analogue of ~ ... |
| iblabsnclem 37823 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 37824 | Choice-free analogue of ~ ... |
| iblmulc2nc 37825 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 37826 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 37827 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 37828 | Choice-free analogue of ~ ... |
| itgabsnc 37829 | Choice-free analogue of ~ ... |
| itggt0cn 37830 | ~ itggt0 holds for continu... |
| ftc1cnnclem 37831 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 37832 | Choice-free proof of ~ ftc... |
| ftc1anclem1 37833 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 37834 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 37835 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 37836 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 37837 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 37838 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 37839 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 37840 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 37841 | ~ ftc1a holds for function... |
| ftc2nc 37842 | Choice-free proof of ~ ftc... |
| asindmre 37843 | Real part of domain of dif... |
| dvasin 37844 | Derivative of arcsine. (C... |
| dvacos 37845 | Derivative of arccosine. ... |
| dvreasin 37846 | Real derivative of arcsine... |
| dvreacos 37847 | Real derivative of arccosi... |
| areacirclem1 37848 | Antiderivative of cross-se... |
| areacirclem2 37849 | Endpoint-inclusive continu... |
| areacirclem3 37850 | Integrability of cross-sec... |
| areacirclem4 37851 | Endpoint-inclusive continu... |
| areacirclem5 37852 | Finding the cross-section ... |
| areacirc 37853 | The area of a circle of ra... |
| unirep 37854 | Define a quantity whose de... |
| cover2 37855 | Two ways of expressing the... |
| cover2g 37856 | Two ways of expressing the... |
| brabg2 37857 | Relation by a binary relat... |
| opelopab3 37858 | Ordered pair membership in... |
| cocanfo 37859 | Cancellation of a surjecti... |
| brresi2 37860 | Restriction of a binary re... |
| fnopabeqd 37861 | Equality deduction for fun... |
| fvopabf4g 37862 | Function value of an opera... |
| fnopabco 37863 | Composition of a function ... |
| opropabco 37864 | Composition of an operator... |
| cocnv 37865 | Composition with a functio... |
| f1ocan1fv 37866 | Cancel a composition by a ... |
| f1ocan2fv 37867 | Cancel a composition by th... |
| inixp 37868 | Intersection of Cartesian ... |
| upixp 37869 | Universal property of the ... |
| abrexdom 37870 | An indexed set is dominate... |
| abrexdom2 37871 | An indexed set is dominate... |
| ac6gf 37872 | Axiom of Choice. (Contrib... |
| indexa 37873 | If for every element of an... |
| indexdom 37874 | If for every element of an... |
| frinfm 37875 | A subset of a well-founded... |
| welb 37876 | A nonempty subset of a wel... |
| supex2g 37877 | Existence of supremum. (C... |
| supclt 37878 | Closure of supremum. (Con... |
| supubt 37879 | Upper bound property of su... |
| filbcmb 37880 | Combine a finite set of lo... |
| fzmul 37881 | Membership of a product in... |
| sdclem2 37882 | Lemma for ~ sdc . (Contri... |
| sdclem1 37883 | Lemma for ~ sdc . (Contri... |
| sdc 37884 | Strong dependent choice. ... |
| fdc 37885 | Finite version of dependen... |
| fdc1 37886 | Variant of ~ fdc with no s... |
| seqpo 37887 | Two ways to say that a seq... |
| incsequz 37888 | An increasing sequence of ... |
| incsequz2 37889 | An increasing sequence of ... |
| nnubfi 37890 | A bounded above set of pos... |
| nninfnub 37891 | An infinite set of positiv... |
| subspopn 37892 | An open set is open in the... |
| neificl 37893 | Neighborhoods are closed u... |
| lpss2 37894 | Limit points of a subset a... |
| metf1o 37895 | Use a bijection with a met... |
| blssp 37896 | A ball in the subspace met... |
| mettrifi 37897 | Generalized triangle inequ... |
| lmclim2 37898 | A sequence in a metric spa... |
| geomcau 37899 | If the distance between co... |
| caures 37900 | The restriction of a Cauch... |
| caushft 37901 | A shifted Cauchy sequence ... |
| constcncf 37902 | A constant function is a c... |
| cnres2 37903 | The restriction of a conti... |
| cnresima 37904 | A continuous function is c... |
| cncfres 37905 | A continuous function on c... |
| istotbnd 37909 | The predicate "is a totall... |
| istotbnd2 37910 | The predicate "is a totall... |
| istotbnd3 37911 | A metric space is totally ... |
| totbndmet 37912 | The predicate "totally bou... |
| 0totbnd 37913 | The metric (there is only ... |
| sstotbnd2 37914 | Condition for a subset of ... |
| sstotbnd 37915 | Condition for a subset of ... |
| sstotbnd3 37916 | Use a net that is not nece... |
| totbndss 37917 | A subset of a totally boun... |
| equivtotbnd 37918 | If the metric ` M ` is "st... |
| isbnd 37920 | The predicate "is a bounde... |
| bndmet 37921 | A bounded metric space is ... |
| isbndx 37922 | A "bounded extended metric... |
| isbnd2 37923 | The predicate "is a bounde... |
| isbnd3 37924 | A metric space is bounded ... |
| isbnd3b 37925 | A metric space is bounded ... |
| bndss 37926 | A subset of a bounded metr... |
| blbnd 37927 | A ball is bounded. (Contr... |
| ssbnd 37928 | A subset of a metric space... |
| totbndbnd 37929 | A totally bounded metric s... |
| equivbnd 37930 | If the metric ` M ` is "st... |
| bnd2lem 37931 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 37932 | If balls are totally bound... |
| prdsbnd 37933 | The product metric over fi... |
| prdstotbnd 37934 | The product metric over fi... |
| prdsbnd2 37935 | If balls are totally bound... |
| cntotbnd 37936 | A subset of the complex nu... |
| cnpwstotbnd 37937 | A subset of ` A ^ I ` , wh... |
| ismtyval 37940 | The set of isometries betw... |
| isismty 37941 | The condition "is an isome... |
| ismtycnv 37942 | The inverse of an isometry... |
| ismtyima 37943 | The image of a ball under ... |
| ismtyhmeolem 37944 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 37945 | An isometry is a homeomorp... |
| ismtybndlem 37946 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 37947 | Isometries preserve bounde... |
| ismtyres 37948 | A restriction of an isomet... |
| heibor1lem 37949 | Lemma for ~ heibor1 . A c... |
| heibor1 37950 | One half of ~ heibor , tha... |
| heiborlem1 37951 | Lemma for ~ heibor . We w... |
| heiborlem2 37952 | Lemma for ~ heibor . Subs... |
| heiborlem3 37953 | Lemma for ~ heibor . Usin... |
| heiborlem4 37954 | Lemma for ~ heibor . Usin... |
| heiborlem5 37955 | Lemma for ~ heibor . The ... |
| heiborlem6 37956 | Lemma for ~ heibor . Sinc... |
| heiborlem7 37957 | Lemma for ~ heibor . Sinc... |
| heiborlem8 37958 | Lemma for ~ heibor . The ... |
| heiborlem9 37959 | Lemma for ~ heibor . Disc... |
| heiborlem10 37960 | Lemma for ~ heibor . The ... |
| heibor 37961 | Generalized Heine-Borel Th... |
| bfplem1 37962 | Lemma for ~ bfp . The seq... |
| bfplem2 37963 | Lemma for ~ bfp . Using t... |
| bfp 37964 | Banach fixed point theorem... |
| rrnval 37967 | The n-dimensional Euclidea... |
| rrnmval 37968 | The value of the Euclidean... |
| rrnmet 37969 | Euclidean space is a metri... |
| rrndstprj1 37970 | The distance between two p... |
| rrndstprj2 37971 | Bound on the distance betw... |
| rrncmslem 37972 | Lemma for ~ rrncms . (Con... |
| rrncms 37973 | Euclidean space is complet... |
| repwsmet 37974 | The supremum metric on ` R... |
| rrnequiv 37975 | The supremum metric on ` R... |
| rrntotbnd 37976 | A set in Euclidean space i... |
| rrnheibor 37977 | Heine-Borel theorem for Eu... |
| ismrer1 37978 | An isometry between ` RR `... |
| reheibor 37979 | Heine-Borel theorem for re... |
| iccbnd 37980 | A closed interval in ` RR ... |
| icccmpALT 37981 | A closed interval in ` RR ... |
| isass 37986 | The predicate "is an assoc... |
| isexid 37987 | The predicate ` G ` has a ... |
| ismgmOLD 37990 | Obsolete version of ~ ismg... |
| clmgmOLD 37991 | Obsolete version of ~ mgmc... |
| opidonOLD 37992 | Obsolete version of ~ mndp... |
| rngopidOLD 37993 | Obsolete version of ~ mndp... |
| opidon2OLD 37994 | Obsolete version of ~ mndp... |
| isexid2 37995 | If ` G e. ( Magma i^i ExId... |
| exidu1 37996 | Uniqueness of the left and... |
| idrval 37997 | The value of the identity ... |
| iorlid 37998 | A magma right and left ide... |
| cmpidelt 37999 | A magma right and left ide... |
| smgrpismgmOLD 38002 | Obsolete version of ~ sgrp... |
| issmgrpOLD 38003 | Obsolete version of ~ issg... |
| smgrpmgm 38004 | A semigroup is a magma. (... |
| smgrpassOLD 38005 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 38008 | Obsolete version of ~ mnds... |
| mndoisexid 38009 | A monoid has an identity e... |
| mndoismgmOLD 38010 | Obsolete version of ~ mndm... |
| mndomgmid 38011 | A monoid is a magma with a... |
| ismndo 38012 | The predicate "is a monoid... |
| ismndo1 38013 | The predicate "is a monoid... |
| ismndo2 38014 | The predicate "is a monoid... |
| grpomndo 38015 | A group is a monoid. (Con... |
| exidcl 38016 | Closure of the binary oper... |
| exidreslem 38017 | Lemma for ~ exidres and ~ ... |
| exidres 38018 | The restriction of a binar... |
| exidresid 38019 | The restriction of a binar... |
| ablo4pnp 38020 | A commutative/associative ... |
| grpoeqdivid 38021 | Two group elements are equ... |
| grposnOLD 38022 | The group operation for th... |
| elghomlem1OLD 38025 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 38026 | Obsolete as of 15-Mar-2020... |
| elghomOLD 38027 | Obsolete version of ~ isgh... |
| ghomlinOLD 38028 | Obsolete version of ~ ghml... |
| ghomidOLD 38029 | Obsolete version of ~ ghmi... |
| ghomf 38030 | Mapping property of a grou... |
| ghomco 38031 | The composition of two gro... |
| ghomdiv 38032 | Group homomorphisms preser... |
| grpokerinj 38033 | A group homomorphism is in... |
| relrngo 38036 | The class of all unital ri... |
| isrngo 38037 | The predicate "is a (unita... |
| isrngod 38038 | Conditions that determine ... |
| rngoi 38039 | The properties of a unital... |
| rngosm 38040 | Functionality of the multi... |
| rngocl 38041 | Closure of the multiplicat... |
| rngoid 38042 | The multiplication operati... |
| rngoideu 38043 | The unity element of a rin... |
| rngodi 38044 | Distributive law for the m... |
| rngodir 38045 | Distributive law for the m... |
| rngoass 38046 | Associative law for the mu... |
| rngo2 38047 | A ring element plus itself... |
| rngoablo 38048 | A ring's addition operatio... |
| rngoablo2 38049 | In a unital ring the addit... |
| rngogrpo 38050 | A ring's addition operatio... |
| rngone0 38051 | The base set of a ring is ... |
| rngogcl 38052 | Closure law for the additi... |
| rngocom 38053 | The addition operation of ... |
| rngoaass 38054 | The addition operation of ... |
| rngoa32 38055 | The addition operation of ... |
| rngoa4 38056 | Rearrangement of 4 terms i... |
| rngorcan 38057 | Right cancellation law for... |
| rngolcan 38058 | Left cancellation law for ... |
| rngo0cl 38059 | A ring has an additive ide... |
| rngo0rid 38060 | The additive identity of a... |
| rngo0lid 38061 | The additive identity of a... |
| rngolz 38062 | The zero of a unital ring ... |
| rngorz 38063 | The zero of a unital ring ... |
| rngosn3 38064 | Obsolete as of 25-Jan-2020... |
| rngosn4 38065 | Obsolete as of 25-Jan-2020... |
| rngosn6 38066 | Obsolete as of 25-Jan-2020... |
| rngonegcl 38067 | A ring is closed under neg... |
| rngoaddneg1 38068 | Adding the negative in a r... |
| rngoaddneg2 38069 | Adding the negative in a r... |
| rngosub 38070 | Subtraction in a ring, in ... |
| rngmgmbs4 38071 | The range of an internal o... |
| rngodm1dm2 38072 | In a unital ring the domai... |
| rngorn1 38073 | In a unital ring the range... |
| rngorn1eq 38074 | In a unital ring the range... |
| rngomndo 38075 | In a unital ring the multi... |
| rngoidmlem 38076 | The unity element of a rin... |
| rngolidm 38077 | The unity element of a rin... |
| rngoridm 38078 | The unity element of a rin... |
| rngo1cl 38079 | The unity element of a rin... |
| rngoueqz 38080 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 38081 | Negation in a ring is the ... |
| rngonegmn1r 38082 | Negation in a ring is the ... |
| rngoneglmul 38083 | Negation of a product in a... |
| rngonegrmul 38084 | Negation of a product in a... |
| rngosubdi 38085 | Ring multiplication distri... |
| rngosubdir 38086 | Ring multiplication distri... |
| zerdivemp1x 38087 | In a unital ring a left in... |
| isdivrngo 38090 | The predicate "is a divisi... |
| drngoi 38091 | The properties of a divisi... |
| gidsn 38092 | Obsolete as of 23-Jan-2020... |
| zrdivrng 38093 | The zero ring is not a div... |
| dvrunz 38094 | In a division ring the rin... |
| isgrpda 38095 | Properties that determine ... |
| isdrngo1 38096 | The predicate "is a divisi... |
| divrngcl 38097 | The product of two nonzero... |
| isdrngo2 38098 | A division ring is a ring ... |
| isdrngo3 38099 | A division ring is a ring ... |
| rngohomval 38104 | The set of ring homomorphi... |
| isrngohom 38105 | The predicate "is a ring h... |
| rngohomf 38106 | A ring homomorphism is a f... |
| rngohomcl 38107 | Closure law for a ring hom... |
| rngohom1 38108 | A ring homomorphism preser... |
| rngohomadd 38109 | Ring homomorphisms preserv... |
| rngohommul 38110 | Ring homomorphisms preserv... |
| rngogrphom 38111 | A ring homomorphism is a g... |
| rngohom0 38112 | A ring homomorphism preser... |
| rngohomsub 38113 | Ring homomorphisms preserv... |
| rngohomco 38114 | The composition of two rin... |
| rngokerinj 38115 | A ring homomorphism is inj... |
| rngoisoval 38117 | The set of ring isomorphis... |
| isrngoiso 38118 | The predicate "is a ring i... |
| rngoiso1o 38119 | A ring isomorphism is a bi... |
| rngoisohom 38120 | A ring isomorphism is a ri... |
| rngoisocnv 38121 | The inverse of a ring isom... |
| rngoisoco 38122 | The composition of two rin... |
| isriscg 38124 | The ring isomorphism relat... |
| isrisc 38125 | The ring isomorphism relat... |
| risc 38126 | The ring isomorphism relat... |
| risci 38127 | Determine that two rings a... |
| riscer 38128 | Ring isomorphism is an equ... |
| iscom2 38135 | A device to add commutativ... |
| iscrngo 38136 | The predicate "is a commut... |
| iscrngo2 38137 | The predicate "is a commut... |
| iscringd 38138 | Conditions that determine ... |
| flddivrng 38139 | A field is a division ring... |
| crngorngo 38140 | A commutative ring is a ri... |
| crngocom 38141 | The multiplication operati... |
| crngm23 38142 | Commutative/associative la... |
| crngm4 38143 | Commutative/associative la... |
| fldcrngo 38144 | A field is a commutative r... |
| isfld2 38145 | The predicate "is a field"... |
| crngohomfo 38146 | The image of a homomorphis... |
| idlval 38153 | The class of ideals of a r... |
| isidl 38154 | The predicate "is an ideal... |
| isidlc 38155 | The predicate "is an ideal... |
| idlss 38156 | An ideal of ` R ` is a sub... |
| idlcl 38157 | An element of an ideal is ... |
| idl0cl 38158 | An ideal contains ` 0 ` . ... |
| idladdcl 38159 | An ideal is closed under a... |
| idllmulcl 38160 | An ideal is closed under m... |
| idlrmulcl 38161 | An ideal is closed under m... |
| idlnegcl 38162 | An ideal is closed under n... |
| idlsubcl 38163 | An ideal is closed under s... |
| rngoidl 38164 | A ring ` R ` is an ` R ` i... |
| 0idl 38165 | The set containing only ` ... |
| 1idl 38166 | Two ways of expressing the... |
| 0rngo 38167 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 38168 | The only ideals in a divis... |
| intidl 38169 | The intersection of a none... |
| inidl 38170 | The intersection of two id... |
| unichnidl 38171 | The union of a nonempty ch... |
| keridl 38172 | The kernel of a ring homom... |
| pridlval 38173 | The class of prime ideals ... |
| ispridl 38174 | The predicate "is a prime ... |
| pridlidl 38175 | A prime ideal is an ideal.... |
| pridlnr 38176 | A prime ideal is a proper ... |
| pridl 38177 | The main property of a pri... |
| ispridl2 38178 | A condition that shows an ... |
| maxidlval 38179 | The set of maximal ideals ... |
| ismaxidl 38180 | The predicate "is a maxima... |
| maxidlidl 38181 | A maximal ideal is an idea... |
| maxidlnr 38182 | A maximal ideal is proper.... |
| maxidlmax 38183 | A maximal ideal is a maxim... |
| maxidln1 38184 | One is not contained in an... |
| maxidln0 38185 | A ring with a maximal idea... |
| isprrngo 38190 | The predicate "is a prime ... |
| prrngorngo 38191 | A prime ring is a ring. (... |
| smprngopr 38192 | A simple ring (one whose o... |
| divrngpr 38193 | A division ring is a prime... |
| isdmn 38194 | The predicate "is a domain... |
| isdmn2 38195 | The predicate "is a domain... |
| dmncrng 38196 | A domain is a commutative ... |
| dmnrngo 38197 | A domain is a ring. (Cont... |
| flddmn 38198 | A field is a domain. (Con... |
| igenval 38201 | The ideal generated by a s... |
| igenss 38202 | A set is a subset of the i... |
| igenidl 38203 | The ideal generated by a s... |
| igenmin 38204 | The ideal generated by a s... |
| igenidl2 38205 | The ideal generated by an ... |
| igenval2 38206 | The ideal generated by a s... |
| prnc 38207 | A principal ideal (an idea... |
| isfldidl 38208 | Determine if a ring is a f... |
| isfldidl2 38209 | Determine if a ring is a f... |
| ispridlc 38210 | The predicate "is a prime ... |
| pridlc 38211 | Property of a prime ideal ... |
| pridlc2 38212 | Property of a prime ideal ... |
| pridlc3 38213 | Property of a prime ideal ... |
| isdmn3 38214 | The predicate "is a domain... |
| dmnnzd 38215 | A domain has no zero-divis... |
| dmncan1 38216 | Cancellation law for domai... |
| dmncan2 38217 | Cancellation law for domai... |
| efald2 38218 | A proof by contradiction. ... |
| notbinot1 38219 | Simplification rule of neg... |
| bicontr 38220 | Biconditional of its own n... |
| impor 38221 | An equivalent formula for ... |
| orfa 38222 | The falsum ` F. ` can be r... |
| notbinot2 38223 | Commutation rule between n... |
| biimpor 38224 | A rewriting rule for bicon... |
| orfa1 38225 | Add a contradicting disjun... |
| orfa2 38226 | Remove a contradicting dis... |
| bifald 38227 | Infer the equivalence to a... |
| orsild 38228 | A lemma for not-or-not eli... |
| orsird 38229 | A lemma for not-or-not eli... |
| cnf1dd 38230 | A lemma for Conjunctive No... |
| cnf2dd 38231 | A lemma for Conjunctive No... |
| cnfn1dd 38232 | A lemma for Conjunctive No... |
| cnfn2dd 38233 | A lemma for Conjunctive No... |
| or32dd 38234 | A rearrangement of disjunc... |
| notornotel1 38235 | A lemma for not-or-not eli... |
| notornotel2 38236 | A lemma for not-or-not eli... |
| contrd 38237 | A proof by contradiction, ... |
| an12i 38238 | An inference from commutin... |
| exmid2 38239 | An excluded middle law. (... |
| selconj 38240 | An inference for selecting... |
| truconj 38241 | Add true as a conjunct. (... |
| orel 38242 | An inference for disjuncti... |
| negel 38243 | An inference for negation ... |
| botel 38244 | An inference for bottom el... |
| tradd 38245 | Add top ad a conjunct. (C... |
| gm-sbtru 38246 | Substitution does not chan... |
| sbfal 38247 | Substitution does not chan... |
| sbcani 38248 | Distribution of class subs... |
| sbcori 38249 | Distribution of class subs... |
| sbcimi 38250 | Distribution of class subs... |
| sbcni 38251 | Move class substitution in... |
| sbali 38252 | Discard class substitution... |
| sbexi 38253 | Discard class substitution... |
| sbcalf 38254 | Move universal quantifier ... |
| sbcexf 38255 | Move existential quantifie... |
| sbcalfi 38256 | Move universal quantifier ... |
| sbcexfi 38257 | Move existential quantifie... |
| spsbcdi 38258 | A lemma for eliminating a ... |
| alrimii 38259 | A lemma for introducing a ... |
| spesbcdi 38260 | A lemma for introducing an... |
| exlimddvf 38261 | A lemma for eliminating an... |
| exlimddvfi 38262 | A lemma for eliminating an... |
| sbceq1ddi 38263 | A lemma for eliminating in... |
| sbccom2lem 38264 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 38265 | Commutative law for double... |
| sbccom2f 38266 | Commutative law for double... |
| sbccom2fi 38267 | Commutative law for double... |
| csbcom2fi 38268 | Commutative law for double... |
| fald 38269 | Refutation of falsity, in ... |
| tsim1 38270 | A Tseitin axiom for logica... |
| tsim2 38271 | A Tseitin axiom for logica... |
| tsim3 38272 | A Tseitin axiom for logica... |
| tsbi1 38273 | A Tseitin axiom for logica... |
| tsbi2 38274 | A Tseitin axiom for logica... |
| tsbi3 38275 | A Tseitin axiom for logica... |
| tsbi4 38276 | A Tseitin axiom for logica... |
| tsxo1 38277 | A Tseitin axiom for logica... |
| tsxo2 38278 | A Tseitin axiom for logica... |
| tsxo3 38279 | A Tseitin axiom for logica... |
| tsxo4 38280 | A Tseitin axiom for logica... |
| tsan1 38281 | A Tseitin axiom for logica... |
| tsan2 38282 | A Tseitin axiom for logica... |
| tsan3 38283 | A Tseitin axiom for logica... |
| tsna1 38284 | A Tseitin axiom for logica... |
| tsna2 38285 | A Tseitin axiom for logica... |
| tsna3 38286 | A Tseitin axiom for logica... |
| tsor1 38287 | A Tseitin axiom for logica... |
| tsor2 38288 | A Tseitin axiom for logica... |
| tsor3 38289 | A Tseitin axiom for logica... |
| ts3an1 38290 | A Tseitin axiom for triple... |
| ts3an2 38291 | A Tseitin axiom for triple... |
| ts3an3 38292 | A Tseitin axiom for triple... |
| ts3or1 38293 | A Tseitin axiom for triple... |
| ts3or2 38294 | A Tseitin axiom for triple... |
| ts3or3 38295 | A Tseitin axiom for triple... |
| iuneq2f 38296 | Equality deduction for ind... |
| rabeq12f 38297 | Equality deduction for res... |
| csbeq12 38298 | Equality deduction for sub... |
| sbeqi 38299 | Equality deduction for sub... |
| ralbi12f 38300 | Equality deduction for res... |
| oprabbi 38301 | Equality deduction for cla... |
| mpobi123f 38302 | Equality deduction for map... |
| iuneq12f 38303 | Equality deduction for ind... |
| iineq12f 38304 | Equality deduction for ind... |
| opabbi 38305 | Equality deduction for cla... |
| mptbi12f 38306 | Equality deduction for map... |
| orcomdd 38307 | Commutativity of logic dis... |
| scottexf 38308 | A version of ~ scottex wit... |
| scott0f 38309 | A version of ~ scott0 with... |
| scottn0f 38310 | A version of ~ scott0f wit... |
| ac6s3f 38311 | Generalization of the Axio... |
| ac6s6 38312 | Generalization of the Axio... |
| ac6s6f 38313 | Generalization of the Axio... |
| el2v1 38364 | New way ( ~ elv , and the ... |
| el3v1 38365 | New way ( ~ elv , and the ... |
| el3v2 38366 | New way ( ~ elv , and the ... |
| el3v12 38367 | New way ( ~ elv , and the ... |
| el3v13 38368 | New way ( ~ elv , and the ... |
| el3v23 38369 | New way ( ~ elv , and the ... |
| anan 38370 | Multiple commutations in c... |
| triantru3 38371 | A wff is equivalent to its... |
| biorfd 38372 | A wff is equivalent to its... |
| eqbrtr 38373 | Substitution of equal clas... |
| eqbrb 38374 | Substitution of equal clas... |
| eqeltr 38375 | Substitution of equal clas... |
| eqelb 38376 | Substitution of equal clas... |
| eqeqan2d 38377 | Implication of introducing... |
| disjresin 38378 | The restriction to a disjo... |
| disjresdisj 38379 | The intersection of restri... |
| disjresdif 38380 | The difference between res... |
| disjresundif 38381 | Lemma for ~ ressucdifsn2 .... |
| inres2 38382 | Two ways of expressing the... |
| coideq 38383 | Equality theorem for compo... |
| nexmo1 38384 | If there is no case where ... |
| eqab2 38385 | Implication of a class abs... |
| r2alan 38386 | Double restricted universa... |
| ssrabi 38387 | Inference of restricted ab... |
| rabimbieq 38388 | Restricted equivalent wff'... |
| abeqin 38389 | Intersection with class ab... |
| abeqinbi 38390 | Intersection with class ab... |
| rabeqel 38391 | Class element of a restric... |
| eqrelf 38392 | The equality connective be... |
| br1cnvinxp 38393 | Binary relation on the con... |
| releleccnv 38394 | Elementhood in a converse ... |
| releccnveq 38395 | Equality of converse ` R `... |
| opelvvdif 38396 | Negated elementhood of ord... |
| vvdifopab 38397 | Ordered-pair class abstrac... |
| brvdif 38398 | Binary relation with unive... |
| brvdif2 38399 | Binary relation with unive... |
| brvvdif 38400 | Binary relation with the c... |
| brvbrvvdif 38401 | Binary relation with the c... |
| brcnvep 38402 | The converse of the binary... |
| elecALTV 38403 | Elementhood in the ` R ` -... |
| brcnvepres 38404 | Restricted converse epsilo... |
| brres2 38405 | Binary relation on a restr... |
| br1cnvres 38406 | Binary relation on the con... |
| elec1cnvres 38407 | Elementhood in the convers... |
| ec1cnvres 38408 | Converse restricted coset ... |
| eldmres 38409 | Elementhood in the domain ... |
| elrnres 38410 | Element of the range of a ... |
| eldmressnALTV 38411 | Element of the domain of a... |
| elrnressn 38412 | Element of the range of a ... |
| eldm4 38413 | Elementhood in a domain. ... |
| eldmres2 38414 | Elementhood in the domain ... |
| eldmres3 38415 | Elementhood in the domain ... |
| eceq1i 38416 | Equality theorem for ` C `... |
| ecres 38417 | Restricted coset of ` B ` ... |
| eccnvepres 38418 | Restricted converse epsilo... |
| eleccnvep 38419 | Elementhood in the convers... |
| eccnvep 38420 | The converse epsilon coset... |
| extep 38421 | Property of epsilon relati... |
| disjeccnvep 38422 | Property of the epsilon re... |
| eccnvepres2 38423 | The restricted converse ep... |
| eccnvepres3 38424 | Condition for a restricted... |
| eldmqsres 38425 | Elementhood in a restricte... |
| eldmqsres2 38426 | Elementhood in a restricte... |
| qsss1 38427 | Subclass theorem for quoti... |
| qseq1i 38428 | Equality theorem for quoti... |
| brinxprnres 38429 | Binary relation on a restr... |
| inxprnres 38430 | Restriction of a class as ... |
| dfres4 38431 | Alternate definition of th... |
| exan3 38432 | Equivalent expressions wit... |
| exanres 38433 | Equivalent expressions wit... |
| exanres3 38434 | Equivalent expressions wit... |
| exanres2 38435 | Equivalent expressions wit... |
| cnvepres 38436 | Restricted converse epsilo... |
| eqrel2 38437 | Equality of relations. (C... |
| rncnv 38438 | Range of converse is the d... |
| dfdm6 38439 | Alternate definition of do... |
| dfrn6 38440 | Alternate definition of ra... |
| rncnvepres 38441 | The range of the restricte... |
| dmecd 38442 | Equality of the coset of `... |
| dmec2d 38443 | Equality of the coset of `... |
| brid 38444 | Property of the identity b... |
| ideq2 38445 | For sets, the identity bin... |
| idresssidinxp 38446 | Condition for the identity... |
| idreseqidinxp 38447 | Condition for the identity... |
| extid 38448 | Property of identity relat... |
| inxpss 38449 | Two ways to say that an in... |
| idinxpss 38450 | Two ways to say that an in... |
| ref5 38451 | Two ways to say that an in... |
| inxpss3 38452 | Two ways to say that an in... |
| inxpss2 38453 | Two ways to say that inter... |
| inxpssidinxp 38454 | Two ways to say that inter... |
| idinxpssinxp 38455 | Two ways to say that inter... |
| idinxpssinxp2 38456 | Identity intersection with... |
| idinxpssinxp3 38457 | Identity intersection with... |
| idinxpssinxp4 38458 | Identity intersection with... |
| relcnveq3 38459 | Two ways of saying a relat... |
| relcnveq 38460 | Two ways of saying a relat... |
| relcnveq2 38461 | Two ways of saying a relat... |
| relcnveq4 38462 | Two ways of saying a relat... |
| qsresid 38463 | Simplification of a specia... |
| n0elqs 38464 | Two ways of expressing tha... |
| n0elqs2 38465 | Two ways of expressing tha... |
| rnresequniqs 38466 | The range of a restriction... |
| n0el2 38467 | Two ways of expressing tha... |
| cnvepresex 38468 | Sethood condition for the ... |
| cnvepima 38469 | The image of converse epsi... |
| inex3 38470 | Sufficient condition for t... |
| inxpex 38471 | Sufficient condition for a... |
| eqres 38472 | Converting a class constan... |
| brrabga 38473 | The law of concretion for ... |
| brcnvrabga 38474 | The law of concretion for ... |
| opideq 38475 | Equality conditions for or... |
| iss2 38476 | A subclass of the identity... |
| eldmcnv 38477 | Elementhood in a domain of... |
| dfrel5 38478 | Alternate definition of th... |
| dfrel6 38479 | Alternate definition of th... |
| cnvresrn 38480 | Converse restricted to ran... |
| relssinxpdmrn 38481 | Subset of restriction, spe... |
| cnvref4 38482 | Two ways to say that a rel... |
| cnvref5 38483 | Two ways to say that a rel... |
| ecin0 38484 | Two ways of saying that th... |
| ecinn0 38485 | Two ways of saying that th... |
| ineleq 38486 | Equivalence of restricted ... |
| inecmo 38487 | Equivalence of a double re... |
| inecmo2 38488 | Equivalence of a double re... |
| ineccnvmo 38489 | Equivalence of a double re... |
| alrmomorn 38490 | Equivalence of an "at most... |
| alrmomodm 38491 | Equivalence of an "at most... |
| ineccnvmo2 38492 | Equivalence of a double un... |
| inecmo3 38493 | Equivalence of a double un... |
| moeu2 38494 | Uniqueness is equivalent t... |
| mopickr 38495 | "At most one" picks a vari... |
| moantr 38496 | Sufficient condition for t... |
| brabidgaw 38497 | The law of concretion for ... |
| brabidga 38498 | The law of concretion for ... |
| inxp2 38499 | Intersection with a Cartes... |
| opabf 38500 | A class abstraction of a c... |
| ec0 38501 | The empty-coset of a class... |
| brcnvin 38502 | Intersection with a conver... |
| ssdmral 38503 | Subclass of a domain. (Co... |
| xrnss3v 38505 | A range Cartesian product ... |
| xrnrel 38506 | A range Cartesian product ... |
| brxrn 38507 | Characterize a ternary rel... |
| brxrn2 38508 | A characterization of the ... |
| dfxrn2 38509 | Alternate definition of th... |
| brxrncnvep 38510 | The range product with con... |
| dmxrn 38511 | Domain of the range produc... |
| dmcnvep 38512 | Domain of converse epsilon... |
| dmxrncnvep 38513 | Domain of the range produc... |
| dmcnvepres 38514 | Domain of the restricted c... |
| dmuncnvepres 38515 | Domain of the union with t... |
| dmxrnuncnvepres 38516 | Domain of the range Cartes... |
| ecun 38517 | The union coset of ` A ` .... |
| ecunres 38518 | The restricted union coset... |
| ecuncnvepres 38519 | The restricted union with ... |
| xrneq1 38520 | Equality theorem for the r... |
| xrneq1i 38521 | Equality theorem for the r... |
| xrneq1d 38522 | Equality theorem for the r... |
| xrneq2 38523 | Equality theorem for the r... |
| xrneq2i 38524 | Equality theorem for the r... |
| xrneq2d 38525 | Equality theorem for the r... |
| xrneq12 38526 | Equality theorem for the r... |
| xrneq12i 38527 | Equality theorem for the r... |
| xrneq12d 38528 | Equality theorem for the r... |
| elecxrn 38529 | Elementhood in the ` ( R |... |
| ecxrn 38530 | The ` ( R |X. S ) ` -coset... |
| relecxrn 38531 | The ` ( R |X. S ) ` -coset... |
| ecxrn2 38532 | The ` ( R |X. S ) ` -coset... |
| ecxrncnvep 38533 | The ` ( R |X. ``' _E ) ` -... |
| ecxrncnvep2 38534 | The ` ( R |X. ``' _E ) ` -... |
| disjressuc2 38535 | Double restricted quantifi... |
| disjecxrn 38536 | Two ways of saying that ` ... |
| disjecxrncnvep 38537 | Two ways of saying that co... |
| disjsuc2 38538 | Double restricted quantifi... |
| xrninxp 38539 | Intersection of a range Ca... |
| xrninxp2 38540 | Intersection of a range Ca... |
| xrninxpex 38541 | Sufficient condition for t... |
| inxpxrn 38542 | Two ways to express the in... |
| br1cnvxrn2 38543 | The converse of a binary r... |
| elec1cnvxrn2 38544 | Elementhood in the convers... |
| rnxrn 38545 | Range of the range Cartesi... |
| rnxrnres 38546 | Range of a range Cartesian... |
| rnxrncnvepres 38547 | Range of a range Cartesian... |
| rnxrnidres 38548 | Range of a range Cartesian... |
| xrnres 38549 | Two ways to express restri... |
| xrnres2 38550 | Two ways to express restri... |
| xrnres3 38551 | Two ways to express restri... |
| xrnres4 38552 | Two ways to express restri... |
| xrnresex 38553 | Sufficient condition for a... |
| xrnidresex 38554 | Sufficient condition for a... |
| xrncnvepresex 38555 | Sufficient condition for a... |
| dmxrncnvepres 38556 | Domain of the range produc... |
| dmxrncnvepres2 38557 | Domain of the range produc... |
| eldmxrncnvepres 38558 | Element of the domain of t... |
| eldmxrncnvepres2 38559 | Element of the domain of t... |
| eceldmqsxrncnvepres 38560 | An ` ( R |X. ( ``' _E |`` ... |
| eceldmqsxrncnvepres2 38561 | An ` ( R |X. ( ``' _E |`` ... |
| brin2 38562 | Binary relation on an inte... |
| brin3 38563 | Binary relation on an inte... |
| elrels2 38565 | The element of the relatio... |
| elrelsrel 38566 | The element of the relatio... |
| elrelsrelim 38567 | The element of the relatio... |
| elrels5 38568 | Equivalent expressions for... |
| elrels6 38569 | Equivalent expressions for... |
| dfadjliftmap2 38571 | Alternate definition of th... |
| blockadjliftmap 38572 | A "two-stage" construction... |
| dfblockliftmap2 38574 | Alternate definition of th... |
| dfsucmap3 38576 | Alternate definition of th... |
| dfsucmap2 38577 | Alternate definition of th... |
| dfsucmap4 38578 | Alternate definition of th... |
| brsucmap 38579 | Binary relation form of th... |
| relsucmap 38580 | The successor map is a rel... |
| dmsucmap 38581 | The domain of the successo... |
| dfsuccl2 38583 | Alternate definition of th... |
| mopre 38584 | There is at most one prede... |
| exeupre2 38585 | Whenever a predecessor exi... |
| dfsuccl3 38586 | Alternate definition of th... |
| dfsuccl4 38587 | Alternate definition that ... |
| dfpre 38589 | Alternate definition of th... |
| dfpre2 38590 | Alternate definition of th... |
| dfpre3 38591 | Alternate definition of th... |
| dfpred4 38592 | Alternate definition of th... |
| dfpre4 38593 | Alternate definition of th... |
| suceqsneq 38596 | One-to-one relationship be... |
| sucdifsn2 38597 | Absorption of union with a... |
| sucdifsn 38598 | The difference between the... |
| ressucdifsn2 38599 | The difference between res... |
| ressucdifsn 38600 | The difference between res... |
| sucmapsuc 38601 | A set is succeeded by its ... |
| sucmapleftuniq 38602 | Left uniqueness of the suc... |
| exeupre 38603 | Whenever a predecessor exi... |
| preex 38604 | The successor-predecessor ... |
| eupre2 38605 | Unique predecessor exists ... |
| eupre 38606 | Unique predecessor exists ... |
| presucmap 38607 | ` pre ` is really a predec... |
| preuniqval 38608 | Uniqueness/canonicity of `... |
| sucpre 38609 | ` suc ` is a right-inverse... |
| presuc 38610 | ` pre ` is a left-inverse ... |
| press 38611 | Predecessor is a subset of... |
| preel 38612 | Predecessor is a subset of... |
| dfcoss2 38615 | Alternate definition of th... |
| dfcoss3 38616 | Alternate definition of th... |
| dfcoss4 38617 | Alternate definition of th... |
| cosscnv 38618 | Class of cosets by the con... |
| coss1cnvres 38619 | Class of cosets by the con... |
| coss2cnvepres 38620 | Special case of ~ coss1cnv... |
| cossex 38621 | If ` A ` is a set then the... |
| cosscnvex 38622 | If ` A ` is a set then the... |
| 1cosscnvepresex 38623 | Sufficient condition for a... |
| 1cossxrncnvepresex 38624 | Sufficient condition for a... |
| relcoss 38625 | Cosets by ` R ` is a relat... |
| relcoels 38626 | Coelements on ` A ` is a r... |
| cossss 38627 | Subclass theorem for the c... |
| cosseq 38628 | Equality theorem for the c... |
| cosseqi 38629 | Equality theorem for the c... |
| cosseqd 38630 | Equality theorem for the c... |
| 1cossres 38631 | The class of cosets by a r... |
| dfcoels 38632 | Alternate definition of th... |
| brcoss 38633 | ` A ` and ` B ` are cosets... |
| brcoss2 38634 | Alternate form of the ` A ... |
| brcoss3 38635 | Alternate form of the ` A ... |
| brcosscnvcoss 38636 | For sets, the ` A ` and ` ... |
| brcoels 38637 | ` B ` and ` C ` are coelem... |
| cocossss 38638 | Two ways of saying that co... |
| cnvcosseq 38639 | The converse of cosets by ... |
| br2coss 38640 | Cosets by ` ,~ R ` binary ... |
| br1cossres 38641 | ` B ` and ` C ` are cosets... |
| br1cossres2 38642 | ` B ` and ` C ` are cosets... |
| brressn 38643 | Binary relation on a restr... |
| ressn2 38644 | A class ' R ' restricted t... |
| refressn 38645 | Any class ' R ' restricted... |
| antisymressn 38646 | Every class ' R ' restrict... |
| trressn 38647 | Any class ' R ' restricted... |
| relbrcoss 38648 | ` A ` and ` B ` are cosets... |
| br1cossinres 38649 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 38650 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 38651 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 38652 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 38653 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 38654 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 38655 | The domain of cosets is th... |
| dmcoss2 38656 | The domain of cosets is th... |
| rncossdmcoss 38657 | The range of cosets is the... |
| dm1cosscnvepres 38658 | The domain of cosets of th... |
| dmcoels 38659 | The domain of coelements i... |
| eldmcoss 38660 | Elementhood in the domain ... |
| eldmcoss2 38661 | Elementhood in the domain ... |
| eldm1cossres 38662 | Elementhood in the domain ... |
| eldm1cossres2 38663 | Elementhood in the domain ... |
| refrelcosslem 38664 | Lemma for the left side of... |
| refrelcoss3 38665 | The class of cosets by ` R... |
| refrelcoss2 38666 | The class of cosets by ` R... |
| symrelcoss3 38667 | The class of cosets by ` R... |
| symrelcoss2 38668 | The class of cosets by ` R... |
| cossssid 38669 | Equivalent expressions for... |
| cossssid2 38670 | Equivalent expressions for... |
| cossssid3 38671 | Equivalent expressions for... |
| cossssid4 38672 | Equivalent expressions for... |
| cossssid5 38673 | Equivalent expressions for... |
| brcosscnv 38674 | ` A ` and ` B ` are cosets... |
| brcosscnv2 38675 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 38676 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 38677 | Cosets by the converse ran... |
| cosscnvssid3 38678 | Equivalent expressions for... |
| cosscnvssid4 38679 | Equivalent expressions for... |
| cosscnvssid5 38680 | Equivalent expressions for... |
| coss0 38681 | Cosets by the empty set ar... |
| cossid 38682 | Cosets by the identity rel... |
| cosscnvid 38683 | Cosets by the converse ide... |
| trcoss 38684 | Sufficient condition for t... |
| eleccossin 38685 | Two ways of saying that th... |
| trcoss2 38686 | Equivalent expressions for... |
| cosselrels 38687 | Cosets of sets are element... |
| cnvelrels 38688 | The converse of a set is a... |
| cosscnvelrels 38689 | Cosets of converse sets ar... |
| dfssr2 38691 | Alternate definition of th... |
| relssr 38692 | The subset relation is a r... |
| brssr 38693 | The subset relation and su... |
| brssrid 38694 | Any set is a subset of its... |
| issetssr 38695 | Two ways of expressing set... |
| brssrres 38696 | Restricted subset binary r... |
| br1cnvssrres 38697 | Restricted converse subset... |
| brcnvssr 38698 | The converse of a subset r... |
| brcnvssrid 38699 | Any set is a converse subs... |
| br1cossxrncnvssrres 38700 | ` <. B , C >. ` and ` <. D... |
| extssr 38701 | Property of subset relatio... |
| dfrefrels2 38705 | Alternate definition of th... |
| dfrefrels3 38706 | Alternate definition of th... |
| dfrefrel2 38707 | Alternate definition of th... |
| dfrefrel3 38708 | Alternate definition of th... |
| dfrefrel5 38709 | Alternate definition of th... |
| elrefrels2 38710 | Element of the class of re... |
| elrefrels3 38711 | Element of the class of re... |
| elrefrelsrel 38712 | For sets, being an element... |
| refreleq 38713 | Equality theorem for refle... |
| refrelid 38714 | Identity relation is refle... |
| refrelcoss 38715 | The class of cosets by ` R... |
| refrelressn 38716 | Any class ' R ' restricted... |
| dfcnvrefrels2 38720 | Alternate definition of th... |
| dfcnvrefrels3 38721 | Alternate definition of th... |
| dfcnvrefrel2 38722 | Alternate definition of th... |
| dfcnvrefrel3 38723 | Alternate definition of th... |
| dfcnvrefrel4 38724 | Alternate definition of th... |
| dfcnvrefrel5 38725 | Alternate definition of th... |
| elcnvrefrels2 38726 | Element of the class of co... |
| elcnvrefrels3 38727 | Element of the class of co... |
| elcnvrefrelsrel 38728 | For sets, being an element... |
| cnvrefrelcoss2 38729 | Necessary and sufficient c... |
| cosselcnvrefrels2 38730 | Necessary and sufficient c... |
| cosselcnvrefrels3 38731 | Necessary and sufficient c... |
| cosselcnvrefrels4 38732 | Necessary and sufficient c... |
| cosselcnvrefrels5 38733 | Necessary and sufficient c... |
| dfsymrels2 38737 | Alternate definition of th... |
| dfsymrels3 38738 | Alternate definition of th... |
| elrelscnveq3 38739 | Two ways of saying a relat... |
| elrelscnveq 38740 | Two ways of saying a relat... |
| elrelscnveq2 38741 | Two ways of saying a relat... |
| elrelscnveq4 38742 | Two ways of saying a relat... |
| dfsymrels4 38743 | Alternate definition of th... |
| dfsymrels5 38744 | Alternate definition of th... |
| dfsymrel2 38745 | Alternate definition of th... |
| dfsymrel3 38746 | Alternate definition of th... |
| dfsymrel4 38747 | Alternate definition of th... |
| dfsymrel5 38748 | Alternate definition of th... |
| elsymrels2 38749 | Element of the class of sy... |
| elsymrels3 38750 | Element of the class of sy... |
| elsymrels4 38751 | Element of the class of sy... |
| elsymrels5 38752 | Element of the class of sy... |
| elsymrelsrel 38753 | For sets, being an element... |
| symreleq 38754 | Equality theorem for symme... |
| symrelim 38755 | Symmetric relation implies... |
| symrelcoss 38756 | The class of cosets by ` R... |
| idsymrel 38757 | The identity relation is s... |
| epnsymrel 38758 | The membership (epsilon) r... |
| symrefref2 38759 | Symmetry is a sufficient c... |
| symrefref3 38760 | Symmetry is a sufficient c... |
| refsymrels2 38761 | Elements of the class of r... |
| refsymrels3 38762 | Elements of the class of r... |
| refsymrel2 38763 | A relation which is reflex... |
| refsymrel3 38764 | A relation which is reflex... |
| elrefsymrels2 38765 | Elements of the class of r... |
| elrefsymrels3 38766 | Elements of the class of r... |
| elrefsymrelsrel 38767 | For sets, being an element... |
| dftrrels2 38771 | Alternate definition of th... |
| dftrrels3 38772 | Alternate definition of th... |
| dftrrel2 38773 | Alternate definition of th... |
| dftrrel3 38774 | Alternate definition of th... |
| eltrrels2 38775 | Element of the class of tr... |
| eltrrels3 38776 | Element of the class of tr... |
| eltrrelsrel 38777 | For sets, being an element... |
| trreleq 38778 | Equality theorem for the t... |
| trrelressn 38779 | Any class ' R ' restricted... |
| dfeqvrels2 38784 | Alternate definition of th... |
| dfeqvrels3 38785 | Alternate definition of th... |
| dfeqvrel2 38786 | Alternate definition of th... |
| dfeqvrel3 38787 | Alternate definition of th... |
| eleqvrels2 38788 | Element of the class of eq... |
| eleqvrels3 38789 | Element of the class of eq... |
| eleqvrelsrel 38790 | For sets, being an element... |
| elcoeleqvrels 38791 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 38792 | For sets, being an element... |
| eqvrelrel 38793 | An equivalence relation is... |
| eqvrelrefrel 38794 | An equivalence relation is... |
| eqvrelsymrel 38795 | An equivalence relation is... |
| eqvreltrrel 38796 | An equivalence relation is... |
| eqvrelim 38797 | Equivalence relation impli... |
| eqvreleq 38798 | Equality theorem for equiv... |
| eqvreleqi 38799 | Equality theorem for equiv... |
| eqvreleqd 38800 | Equality theorem for equiv... |
| eqvrelsym 38801 | An equivalence relation is... |
| eqvrelsymb 38802 | An equivalence relation is... |
| eqvreltr 38803 | An equivalence relation is... |
| eqvreltrd 38804 | A transitivity relation fo... |
| eqvreltr4d 38805 | A transitivity relation fo... |
| eqvrelref 38806 | An equivalence relation is... |
| eqvrelth 38807 | Basic property of equivale... |
| eqvrelcl 38808 | Elementhood in the field o... |
| eqvrelthi 38809 | Basic property of equivale... |
| eqvreldisj 38810 | Equivalence classes do not... |
| qsdisjALTV 38811 | Elements of a quotient set... |
| eqvrelqsel 38812 | If an element of a quotien... |
| eqvrelcoss 38813 | Two ways to express equiva... |
| eqvrelcoss3 38814 | Two ways to express equiva... |
| eqvrelcoss2 38815 | Two ways to express equiva... |
| eqvrelcoss4 38816 | Two ways to express equiva... |
| dfcoeleqvrels 38817 | Alternate definition of th... |
| dfcoeleqvrel 38818 | Alternate definition of th... |
| brredunds 38822 | Binary relation on the cla... |
| brredundsredund 38823 | For sets, binary relation ... |
| redundss3 38824 | Implication of redundancy ... |
| redundeq1 38825 | Equivalence of redundancy ... |
| redundpim3 38826 | Implication of redundancy ... |
| redundpbi1 38827 | Equivalence of redundancy ... |
| refrelsredund4 38828 | The naive version of the c... |
| refrelsredund2 38829 | The naive version of the c... |
| refrelsredund3 38830 | The naive version of the c... |
| refrelredund4 38831 | The naive version of the d... |
| refrelredund2 38832 | The naive version of the d... |
| refrelredund3 38833 | The naive version of the d... |
| dfblockliftfix2 38836 | Alternate definition of th... |
| dmqseq 38837 | Equality theorem for domai... |
| dmqseqi 38838 | Equality theorem for domai... |
| dmqseqd 38839 | Equality theorem for domai... |
| dmqseqeq1 38840 | Equality theorem for domai... |
| dmqseqeq1i 38841 | Equality theorem for domai... |
| dmqseqeq1d 38842 | Equality theorem for domai... |
| brdmqss 38843 | The domain quotient binary... |
| brdmqssqs 38844 | If ` A ` and ` R ` are set... |
| n0eldmqs 38845 | The empty set is not an el... |
| qseq 38846 | The quotient set equal to ... |
| n0eldmqseq 38847 | The empty set is not an el... |
| n0elim 38848 | Implication of that the em... |
| n0el3 38849 | Two ways of expressing tha... |
| cnvepresdmqss 38850 | The domain quotient binary... |
| cnvepresdmqs 38851 | The domain quotient predic... |
| unidmqs 38852 | The range of a relation is... |
| unidmqseq 38853 | The union of the domain qu... |
| dmqseqim 38854 | If the domain quotient of ... |
| dmqseqim2 38855 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 38856 | Elementhood in the domain ... |
| eldmqs1cossres 38857 | Elementhood in the domain ... |
| releldmqscoss 38858 | Elementhood in the domain ... |
| dmqscoelseq 38859 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 38860 | Two ways to express the eq... |
| brers 38865 | Binary equivalence relatio... |
| dferALTV2 38866 | Equivalence relation with ... |
| erALTVeq1 38867 | Equality theorem for equiv... |
| erALTVeq1i 38868 | Equality theorem for equiv... |
| erALTVeq1d 38869 | Equality theorem for equiv... |
| dfcomember 38870 | Alternate definition of th... |
| dfcomember2 38871 | Alternate definition of th... |
| dfcomember3 38872 | Alternate definition of th... |
| eqvreldmqs 38873 | Two ways to express comemb... |
| eqvreldmqs2 38874 | Two ways to express comemb... |
| brerser 38875 | Binary equivalence relatio... |
| erimeq2 38876 | Equivalence relation on it... |
| erimeq 38877 | Equivalence relation on it... |
| dffunsALTV 38881 | Alternate definition of th... |
| dffunsALTV2 38882 | Alternate definition of th... |
| dffunsALTV3 38883 | Alternate definition of th... |
| dffunsALTV4 38884 | Alternate definition of th... |
| dffunsALTV5 38885 | Alternate definition of th... |
| dffunALTV2 38886 | Alternate definition of th... |
| dffunALTV3 38887 | Alternate definition of th... |
| dffunALTV4 38888 | Alternate definition of th... |
| dffunALTV5 38889 | Alternate definition of th... |
| elfunsALTV 38890 | Elementhood in the class o... |
| elfunsALTV2 38891 | Elementhood in the class o... |
| elfunsALTV3 38892 | Elementhood in the class o... |
| elfunsALTV4 38893 | Elementhood in the class o... |
| elfunsALTV5 38894 | Elementhood in the class o... |
| elfunsALTVfunALTV 38895 | The element of the class o... |
| funALTVfun 38896 | Our definition of the func... |
| funALTVss 38897 | Subclass theorem for funct... |
| funALTVeq 38898 | Equality theorem for funct... |
| funALTVeqi 38899 | Equality inference for the... |
| funALTVeqd 38900 | Equality deduction for the... |
| dfdisjs 38906 | Alternate definition of th... |
| dfdisjs2 38907 | Alternate definition of th... |
| dfdisjs3 38908 | Alternate definition of th... |
| dfdisjs4 38909 | Alternate definition of th... |
| dfdisjs5 38910 | Alternate definition of th... |
| dfdisjALTV 38911 | Alternate definition of th... |
| dfdisjALTV2 38912 | Alternate definition of th... |
| dfdisjALTV3 38913 | Alternate definition of th... |
| dfdisjALTV4 38914 | Alternate definition of th... |
| dfdisjALTV5 38915 | Alternate definition of th... |
| dfeldisj2 38916 | Alternate definition of th... |
| dfeldisj3 38917 | Alternate definition of th... |
| dfeldisj4 38918 | Alternate definition of th... |
| dfeldisj5 38919 | Alternate definition of th... |
| eldisjs 38920 | Elementhood in the class o... |
| eldisjs2 38921 | Elementhood in the class o... |
| eldisjs3 38922 | Elementhood in the class o... |
| eldisjs4 38923 | Elementhood in the class o... |
| eldisjs5 38924 | Elementhood in the class o... |
| eldisjsdisj 38925 | The element of the class o... |
| eleldisjs 38926 | Elementhood in the disjoin... |
| eleldisjseldisj 38927 | The element of the disjoin... |
| disjrel 38928 | Disjoint relation is a rel... |
| disjss 38929 | Subclass theorem for disjo... |
| disjssi 38930 | Subclass theorem for disjo... |
| disjssd 38931 | Subclass theorem for disjo... |
| disjeq 38932 | Equality theorem for disjo... |
| disjeqi 38933 | Equality theorem for disjo... |
| disjeqd 38934 | Equality theorem for disjo... |
| disjdmqseqeq1 38935 | Lemma for the equality the... |
| eldisjss 38936 | Subclass theorem for disjo... |
| eldisjssi 38937 | Subclass theorem for disjo... |
| eldisjssd 38938 | Subclass theorem for disjo... |
| eldisjeq 38939 | Equality theorem for disjo... |
| eldisjeqi 38940 | Equality theorem for disjo... |
| eldisjeqd 38941 | Equality theorem for disjo... |
| disjres 38942 | Disjoint restriction. (Co... |
| eldisjn0elb 38943 | Two forms of disjoint elem... |
| disjxrn 38944 | Two ways of saying that a ... |
| disjxrnres5 38945 | Disjoint range Cartesian p... |
| disjorimxrn 38946 | Disjointness condition for... |
| disjimxrn 38947 | Disjointness condition for... |
| disjimres 38948 | Disjointness condition for... |
| disjimin 38949 | Disjointness condition for... |
| disjiminres 38950 | Disjointness condition for... |
| disjimxrnres 38951 | Disjointness condition for... |
| disjALTV0 38952 | The null class is disjoint... |
| disjALTVid 38953 | The class of identity rela... |
| disjALTVidres 38954 | The class of identity rela... |
| disjALTVinidres 38955 | The intersection with rest... |
| disjALTVxrnidres 38956 | The class of range Cartesi... |
| disjsuc 38957 | Disjoint range Cartesian p... |
| dfantisymrel4 38959 | Alternate definition of th... |
| dfantisymrel5 38960 | Alternate definition of th... |
| antisymrelres 38961 | (Contributed by Peter Mazs... |
| antisymrelressn 38962 | (Contributed by Peter Mazs... |
| dfpart2 38967 | Alternate definition of th... |
| dfmembpart2 38968 | Alternate definition of th... |
| brparts 38969 | Binary partitions relation... |
| brparts2 38970 | Binary partitions relation... |
| brpartspart 38971 | Binary partition and the p... |
| parteq1 38972 | Equality theorem for parti... |
| parteq2 38973 | Equality theorem for parti... |
| parteq12 38974 | Equality theorem for parti... |
| parteq1i 38975 | Equality theorem for parti... |
| parteq1d 38976 | Equality theorem for parti... |
| partsuc2 38977 | Property of the partition.... |
| partsuc 38978 | Property of the partition.... |
| disjim 38979 | The "Divide et Aequivalere... |
| disjimi 38980 | Every disjoint relation ge... |
| detlem 38981 | If a relation is disjoint,... |
| eldisjim 38982 | If the elements of ` A ` a... |
| eldisjim2 38983 | Alternate form of ~ eldisj... |
| eqvrel0 38984 | The null class is an equiv... |
| det0 38985 | The cosets by the null cla... |
| eqvrelcoss0 38986 | The cosets by the null cla... |
| eqvrelid 38987 | The identity relation is a... |
| eqvrel1cossidres 38988 | The cosets by a restricted... |
| eqvrel1cossinidres 38989 | The cosets by an intersect... |
| eqvrel1cossxrnidres 38990 | The cosets by a range Cart... |
| detid 38991 | The cosets by the identity... |
| eqvrelcossid 38992 | The cosets by the identity... |
| detidres 38993 | The cosets by the restrict... |
| detinidres 38994 | The cosets by the intersec... |
| detxrnidres 38995 | The cosets by the range Ca... |
| disjlem14 38996 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 38997 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 38998 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 38999 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 39000 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 39001 | Lemma for ~ disjdmqseq via... |
| disjdmqs 39002 | If a relation is disjoint,... |
| disjdmqseq 39003 | If a relation is disjoint,... |
| eldisjn0el 39004 | Special case of ~ disjdmqs... |
| partim2 39005 | Disjoint relation on its n... |
| partim 39006 | Partition implies equivale... |
| partimeq 39007 | Partition implies that the... |
| eldisjlem19 39008 | Special case of ~ disjlem1... |
| membpartlem19 39009 | Together with ~ disjlem19 ... |
| petlem 39010 | If you can prove that the ... |
| petlemi 39011 | If you can prove disjointn... |
| pet02 39012 | Class ` A ` is a partition... |
| pet0 39013 | Class ` A ` is a partition... |
| petid2 39014 | Class ` A ` is a partition... |
| petid 39015 | A class is a partition by ... |
| petidres2 39016 | Class ` A ` is a partition... |
| petidres 39017 | A class is a partition by ... |
| petinidres2 39018 | Class ` A ` is a partition... |
| petinidres 39019 | A class is a partition by ... |
| petxrnidres2 39020 | Class ` A ` is a partition... |
| petxrnidres 39021 | A class is a partition by ... |
| eqvreldisj1 39022 | The elements of the quotie... |
| eqvreldisj2 39023 | The elements of the quotie... |
| eqvreldisj3 39024 | The elements of the quotie... |
| eqvreldisj4 39025 | Intersection with the conv... |
| eqvreldisj5 39026 | Range Cartesian product wi... |
| eqvrelqseqdisj2 39027 | Implication of ~ eqvreldis... |
| fences3 39028 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 39029 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 39030 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 39031 | Lemma for the Partition-Eq... |
| mainer 39032 | The Main Theorem of Equiva... |
| partimcomember 39033 | Partition with general ` R... |
| mpet3 39034 | Member Partition-Equivalen... |
| cpet2 39035 | The conventional form of t... |
| cpet 39036 | The conventional form of M... |
| mpet 39037 | Member Partition-Equivalen... |
| mpet2 39038 | Member Partition-Equivalen... |
| mpets2 39039 | Member Partition-Equivalen... |
| mpets 39040 | Member Partition-Equivalen... |
| mainpart 39041 | Partition with general ` R... |
| fences 39042 | The Theorem of Fences by E... |
| fences2 39043 | The Theorem of Fences by E... |
| mainer2 39044 | The Main Theorem of Equiva... |
| mainerim 39045 | Every equivalence relation... |
| petincnvepres2 39046 | A partition-equivalence th... |
| petincnvepres 39047 | The shortest form of a par... |
| pet2 39048 | Partition-Equivalence Theo... |
| pet 39049 | Partition-Equivalence Theo... |
| pets 39050 | Partition-Equivalence Theo... |
| dmqsblocks 39051 | If the ~ pet span ` ( R |X... |
| prtlem60 39052 | Lemma for ~ prter3 . (Con... |
| bicomdd 39053 | Commute two sides of a bic... |
| jca2r 39054 | Inference conjoining the c... |
| jca3 39055 | Inference conjoining the c... |
| prtlem70 39056 | Lemma for ~ prter3 : a rea... |
| ibdr 39057 | Reverse of ~ ibd . (Contr... |
| prtlem100 39058 | Lemma for ~ prter3 . (Con... |
| prtlem5 39059 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 39060 | Lemma for ~ prter2 . (Con... |
| brabsb2 39061 | A closed form of ~ brabsb ... |
| eqbrrdv2 39062 | Other version of ~ eqbrrdi... |
| prtlem9 39063 | Lemma for ~ prter3 . (Con... |
| prtlem10 39064 | Lemma for ~ prter3 . (Con... |
| prtlem11 39065 | Lemma for ~ prter2 . (Con... |
| prtlem12 39066 | Lemma for ~ prtex and ~ pr... |
| prtlem13 39067 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 39068 | Lemma for ~ prtex , ~ prte... |
| prtlem400 39069 | Lemma for ~ prter2 and als... |
| erprt 39072 | The quotient set of an equ... |
| prtlem14 39073 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 39074 | Lemma for ~ prter1 and ~ p... |
| prtlem17 39075 | Lemma for ~ prter2 . (Con... |
| prtlem18 39076 | Lemma for ~ prter2 . (Con... |
| prtlem19 39077 | Lemma for ~ prter2 . (Con... |
| prter1 39078 | Every partition generates ... |
| prtex 39079 | The equivalence relation g... |
| prter2 39080 | The quotient set of the eq... |
| prter3 39081 | For every partition there ... |
| axc5 39092 | This theorem repeats ~ sp ... |
| ax4fromc4 39093 | Rederivation of Axiom ~ ax... |
| ax10fromc7 39094 | Rederivation of Axiom ~ ax... |
| ax6fromc10 39095 | Rederivation of Axiom ~ ax... |
| hba1-o 39096 | The setvar ` x ` is not fr... |
| axc4i-o 39097 | Inference version of ~ ax-... |
| equid1 39098 | Proof of ~ equid from our ... |
| equcomi1 39099 | Proof of ~ equcomi from ~ ... |
| aecom-o 39100 | Commutation law for identi... |
| aecoms-o 39101 | A commutation rule for ide... |
| hbae-o 39102 | All variables are effectiv... |
| dral1-o 39103 | Formula-building lemma for... |
| ax12fromc15 39104 | Rederivation of Axiom ~ ax... |
| ax13fromc9 39105 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 39106 | Axiom to quantify a variab... |
| sps-o 39107 | Generalization of antecede... |
| hbequid 39108 | Bound-variable hypothesis ... |
| nfequid-o 39109 | Bound-variable hypothesis ... |
| axc5c7 39110 | Proof of a single axiom th... |
| axc5c7toc5 39111 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 39112 | Rederivation of ~ ax-c7 fr... |
| axc711 39113 | Proof of a single axiom th... |
| nfa1-o 39114 | ` x ` is not free in ` A. ... |
| axc711toc7 39115 | Rederivation of ~ ax-c7 fr... |
| axc711to11 39116 | Rederivation of ~ ax-11 fr... |
| axc5c711 39117 | Proof of a single axiom th... |
| axc5c711toc5 39118 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 39119 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 39120 | Rederivation of ~ ax-11 fr... |
| equidqe 39121 | ~ equid with existential q... |
| axc5sp1 39122 | A special case of ~ ax-c5 ... |
| equidq 39123 | ~ equid with universal qua... |
| equid1ALT 39124 | Alternate proof of ~ equid... |
| axc11nfromc11 39125 | Rederivation of ~ ax-c11n ... |
| naecoms-o 39126 | A commutation rule for dis... |
| hbnae-o 39127 | All variables are effectiv... |
| dvelimf-o 39128 | Proof of ~ dvelimh that us... |
| dral2-o 39129 | Formula-building lemma for... |
| aev-o 39130 | A "distinctor elimination"... |
| ax5eq 39131 | Theorem to add distinct qu... |
| dveeq2-o 39132 | Quantifier introduction wh... |
| axc16g-o 39133 | A generalization of Axiom ... |
| dveeq1-o 39134 | Quantifier introduction wh... |
| dveeq1-o16 39135 | Version of ~ dveeq1 using ... |
| ax5el 39136 | Theorem to add distinct qu... |
| axc11n-16 39137 | This theorem shows that, g... |
| dveel2ALT 39138 | Alternate proof of ~ dveel... |
| ax12f 39139 | Basis step for constructin... |
| ax12eq 39140 | Basis step for constructin... |
| ax12el 39141 | Basis step for constructin... |
| ax12indn 39142 | Induction step for constru... |
| ax12indi 39143 | Induction step for constru... |
| ax12indalem 39144 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 39145 | Alternate proof of ~ ax12i... |
| ax12inda2 39146 | Induction step for constru... |
| ax12inda 39147 | Induction step for constru... |
| ax12v2-o 39148 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 39149 | Derive ~ ax-c15 from a hyp... |
| axc11-o 39150 | Show that ~ ax-c11 can be ... |
| fsumshftd 39151 | Index shift of a finite su... |
| riotaclbgBAD 39153 | Closure of restricted iota... |
| riotaclbBAD 39154 | Closure of restricted iota... |
| riotasvd 39155 | Deduction version of ~ rio... |
| riotasv2d 39156 | Value of description binde... |
| riotasv2s 39157 | The value of description b... |
| riotasv 39158 | Value of description binde... |
| riotasv3d 39159 | A property ` ch ` holding ... |
| elimhyps 39160 | A version of ~ elimhyp usi... |
| dedths 39161 | A version of weak deductio... |
| renegclALT 39162 | Closure law for negative o... |
| elimhyps2 39163 | Generalization of ~ elimhy... |
| dedths2 39164 | Generalization of ~ dedths... |
| nfcxfrdf 39165 | A utility lemma to transfe... |
| nfded 39166 | A deduction theorem that c... |
| nfded2 39167 | A deduction theorem that c... |
| nfunidALT2 39168 | Deduction version of ~ nfu... |
| nfunidALT 39169 | Deduction version of ~ nfu... |
| nfopdALT 39170 | Deduction version of bound... |
| cnaddcom 39171 | Recover the commutative la... |
| toycom 39172 | Show the commutative law f... |
| lshpset 39177 | The set of all hyperplanes... |
| islshp 39178 | The predicate "is a hyperp... |
| islshpsm 39179 | Hyperplane properties expr... |
| lshplss 39180 | A hyperplane is a subspace... |
| lshpne 39181 | A hyperplane is not equal ... |
| lshpnel 39182 | A hyperplane's generating ... |
| lshpnelb 39183 | The subspace sum of a hype... |
| lshpnel2N 39184 | Condition that determines ... |
| lshpne0 39185 | The member of the span in ... |
| lshpdisj 39186 | A hyperplane and the span ... |
| lshpcmp 39187 | If two hyperplanes are com... |
| lshpinN 39188 | The intersection of two di... |
| lsatset 39189 | The set of all 1-dim subsp... |
| islsat 39190 | The predicate "is a 1-dim ... |
| lsatlspsn2 39191 | The span of a nonzero sing... |
| lsatlspsn 39192 | The span of a nonzero sing... |
| islsati 39193 | A 1-dim subspace (atom) (o... |
| lsateln0 39194 | A 1-dim subspace (atom) (o... |
| lsatlss 39195 | The set of 1-dim subspaces... |
| lsatlssel 39196 | An atom is a subspace. (C... |
| lsatssv 39197 | An atom is a set of vector... |
| lsatn0 39198 | A 1-dim subspace (atom) of... |
| lsatspn0 39199 | The span of a vector is an... |
| lsator0sp 39200 | The span of a vector is ei... |
| lsatssn0 39201 | A subspace (or any class) ... |
| lsatcmp 39202 | If two atoms are comparabl... |
| lsatcmp2 39203 | If an atom is included in ... |
| lsatel 39204 | A nonzero vector in an ato... |
| lsatelbN 39205 | A nonzero vector in an ato... |
| lsat2el 39206 | Two atoms sharing a nonzer... |
| lsmsat 39207 | Convert comparison of atom... |
| lsatfixedN 39208 | Show equality with the spa... |
| lsmsatcv 39209 | Subspace sum has the cover... |
| lssatomic 39210 | The lattice of subspaces i... |
| lssats 39211 | The lattice of subspaces i... |
| lpssat 39212 | Two subspaces in a proper ... |
| lrelat 39213 | Subspaces are relatively a... |
| lssatle 39214 | The ordering of two subspa... |
| lssat 39215 | Two subspaces in a proper ... |
| islshpat 39216 | Hyperplane properties expr... |
| lcvfbr 39219 | The covers relation for a ... |
| lcvbr 39220 | The covers relation for a ... |
| lcvbr2 39221 | The covers relation for a ... |
| lcvbr3 39222 | The covers relation for a ... |
| lcvpss 39223 | The covers relation implie... |
| lcvnbtwn 39224 | The covers relation implie... |
| lcvntr 39225 | The covers relation is not... |
| lcvnbtwn2 39226 | The covers relation implie... |
| lcvnbtwn3 39227 | The covers relation implie... |
| lsmcv2 39228 | Subspace sum has the cover... |
| lcvat 39229 | If a subspace covers anoth... |
| lsatcv0 39230 | An atom covers the zero su... |
| lsatcveq0 39231 | A subspace covered by an a... |
| lsat0cv 39232 | A subspace is an atom iff ... |
| lcvexchlem1 39233 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 39234 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 39235 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 39236 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 39237 | Lemma for ~ lcvexch . (Co... |
| lcvexch 39238 | Subspaces satisfy the exch... |
| lcvp 39239 | Covering property of Defin... |
| lcv1 39240 | Covering property of a sub... |
| lcv2 39241 | Covering property of a sub... |
| lsatexch 39242 | The atom exchange property... |
| lsatnle 39243 | The meet of a subspace and... |
| lsatnem0 39244 | The meet of distinct atoms... |
| lsatexch1 39245 | The atom exch1ange propert... |
| lsatcv0eq 39246 | If the sum of two atoms co... |
| lsatcv1 39247 | Two atoms covering the zer... |
| lsatcvatlem 39248 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 39249 | A nonzero subspace less th... |
| lsatcvat2 39250 | A subspace covered by the ... |
| lsatcvat3 39251 | A condition implying that ... |
| islshpcv 39252 | Hyperplane properties expr... |
| l1cvpat 39253 | A subspace covered by the ... |
| l1cvat 39254 | Create an atom under an el... |
| lshpat 39255 | Create an atom under a hyp... |
| lflset 39258 | The set of linear function... |
| islfl 39259 | The predicate "is a linear... |
| lfli 39260 | Property of a linear funct... |
| islfld 39261 | Properties that determine ... |
| lflf 39262 | A linear functional is a f... |
| lflcl 39263 | A linear functional value ... |
| lfl0 39264 | A linear functional is zer... |
| lfladd 39265 | Property of a linear funct... |
| lflsub 39266 | Property of a linear funct... |
| lflmul 39267 | Property of a linear funct... |
| lfl0f 39268 | The zero function is a fun... |
| lfl1 39269 | A nonzero functional has a... |
| lfladdcl 39270 | Closure of addition of two... |
| lfladdcom 39271 | Commutativity of functiona... |
| lfladdass 39272 | Associativity of functiona... |
| lfladd0l 39273 | Functional addition with t... |
| lflnegcl 39274 | Closure of the negative of... |
| lflnegl 39275 | A functional plus its nega... |
| lflvscl 39276 | Closure of a scalar produc... |
| lflvsdi1 39277 | Distributive law for (righ... |
| lflvsdi2 39278 | Reverse distributive law f... |
| lflvsdi2a 39279 | Reverse distributive law f... |
| lflvsass 39280 | Associative law for (right... |
| lfl0sc 39281 | The (right vector space) s... |
| lflsc0N 39282 | The scalar product with th... |
| lfl1sc 39283 | The (right vector space) s... |
| lkrfval 39286 | The kernel of a functional... |
| lkrval 39287 | Value of the kernel of a f... |
| ellkr 39288 | Membership in the kernel o... |
| lkrval2 39289 | Value of the kernel of a f... |
| ellkr2 39290 | Membership in the kernel o... |
| lkrcl 39291 | A member of the kernel of ... |
| lkrf0 39292 | The value of a functional ... |
| lkr0f 39293 | The kernel of the zero fun... |
| lkrlss 39294 | The kernel of a linear fun... |
| lkrssv 39295 | The kernel of a linear fun... |
| lkrsc 39296 | The kernel of a nonzero sc... |
| lkrscss 39297 | The kernel of a scalar pro... |
| eqlkr 39298 | Two functionals with the s... |
| eqlkr2 39299 | Two functionals with the s... |
| eqlkr3 39300 | Two functionals with the s... |
| lkrlsp 39301 | The subspace sum of a kern... |
| lkrlsp2 39302 | The subspace sum of a kern... |
| lkrlsp3 39303 | The subspace sum of a kern... |
| lkrshp 39304 | The kernel of a nonzero fu... |
| lkrshp3 39305 | The kernels of nonzero fun... |
| lkrshpor 39306 | The kernel of a functional... |
| lkrshp4 39307 | A kernel is a hyperplane i... |
| lshpsmreu 39308 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 39309 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 39310 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 39311 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 39312 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 39313 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 39314 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 39315 | The set ` G ` defined by h... |
| lshpkr 39316 | The kernel of functional `... |
| lshpkrex 39317 | There exists a functional ... |
| lshpset2N 39318 | The set of all hyperplanes... |
| islshpkrN 39319 | The predicate "is a hyperp... |
| lfl1dim 39320 | Equivalent expressions for... |
| lfl1dim2N 39321 | Equivalent expressions for... |
| ldualset 39324 | Define the (left) dual of ... |
| ldualvbase 39325 | The vectors of a dual spac... |
| ldualelvbase 39326 | Utility theorem for conver... |
| ldualfvadd 39327 | Vector addition in the dua... |
| ldualvadd 39328 | Vector addition in the dua... |
| ldualvaddcl 39329 | The value of vector additi... |
| ldualvaddval 39330 | The value of the value of ... |
| ldualsca 39331 | The ring of scalars of the... |
| ldualsbase 39332 | Base set of scalar ring fo... |
| ldualsaddN 39333 | Scalar addition for the du... |
| ldualsmul 39334 | Scalar multiplication for ... |
| ldualfvs 39335 | Scalar product operation f... |
| ldualvs 39336 | Scalar product operation v... |
| ldualvsval 39337 | Value of scalar product op... |
| ldualvscl 39338 | The scalar product operati... |
| ldualvaddcom 39339 | Commutative law for vector... |
| ldualvsass 39340 | Associative law for scalar... |
| ldualvsass2 39341 | Associative law for scalar... |
| ldualvsdi1 39342 | Distributive law for scala... |
| ldualvsdi2 39343 | Reverse distributive law f... |
| ldualgrplem 39344 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 39345 | The dual of a vector space... |
| ldual0 39346 | The zero scalar of the dua... |
| ldual1 39347 | The unit scalar of the dua... |
| ldualneg 39348 | The negative of a scalar o... |
| ldual0v 39349 | The zero vector of the dua... |
| ldual0vcl 39350 | The dual zero vector is a ... |
| lduallmodlem 39351 | Lemma for ~ lduallmod . (... |
| lduallmod 39352 | The dual of a left module ... |
| lduallvec 39353 | The dual of a left vector ... |
| ldualvsub 39354 | The value of vector subtra... |
| ldualvsubcl 39355 | Closure of vector subtract... |
| ldualvsubval 39356 | The value of the value of ... |
| ldualssvscl 39357 | Closure of scalar product ... |
| ldualssvsubcl 39358 | Closure of vector subtract... |
| ldual0vs 39359 | Scalar zero times a functi... |
| lkr0f2 39360 | The kernel of the zero fun... |
| lduallkr3 39361 | The kernels of nonzero fun... |
| lkrpssN 39362 | Proper subset relation bet... |
| lkrin 39363 | Intersection of the kernel... |
| eqlkr4 39364 | Two functionals with the s... |
| ldual1dim 39365 | Equivalent expressions for... |
| ldualkrsc 39366 | The kernel of a nonzero sc... |
| lkrss 39367 | The kernel of a scalar pro... |
| lkrss2N 39368 | Two functionals with kerne... |
| lkreqN 39369 | Proportional functionals h... |
| lkrlspeqN 39370 | Condition for colinear fun... |
| isopos 39379 | The predicate "is an ortho... |
| opposet 39380 | Every orthoposet is a pose... |
| oposlem 39381 | Lemma for orthoposet prope... |
| op01dm 39382 | Conditions necessary for z... |
| op0cl 39383 | An orthoposet has a zero e... |
| op1cl 39384 | An orthoposet has a unity ... |
| op0le 39385 | Orthoposet zero is less th... |
| ople0 39386 | An element less than or eq... |
| opnlen0 39387 | An element not less than a... |
| lub0N 39388 | The least upper bound of t... |
| opltn0 39389 | A lattice element greater ... |
| ople1 39390 | Any element is less than t... |
| op1le 39391 | If the orthoposet unity is... |
| glb0N 39392 | The greatest lower bound o... |
| opoccl 39393 | Closure of orthocomplement... |
| opococ 39394 | Double negative law for or... |
| opcon3b 39395 | Contraposition law for ort... |
| opcon2b 39396 | Orthocomplement contraposi... |
| opcon1b 39397 | Orthocomplement contraposi... |
| oplecon3 39398 | Contraposition law for ort... |
| oplecon3b 39399 | Contraposition law for ort... |
| oplecon1b 39400 | Contraposition law for str... |
| opoc1 39401 | Orthocomplement of orthopo... |
| opoc0 39402 | Orthocomplement of orthopo... |
| opltcon3b 39403 | Contraposition law for str... |
| opltcon1b 39404 | Contraposition law for str... |
| opltcon2b 39405 | Contraposition law for str... |
| opexmid 39406 | Law of excluded middle for... |
| opnoncon 39407 | Law of contradiction for o... |
| riotaocN 39408 | The orthocomplement of the... |
| cmtfvalN 39409 | Value of commutes relation... |
| cmtvalN 39410 | Equivalence for commutes r... |
| isolat 39411 | The predicate "is an ortho... |
| ollat 39412 | An ortholattice is a latti... |
| olop 39413 | An ortholattice is an orth... |
| olposN 39414 | An ortholattice is a poset... |
| isolatiN 39415 | Properties that determine ... |
| oldmm1 39416 | De Morgan's law for meet i... |
| oldmm2 39417 | De Morgan's law for meet i... |
| oldmm3N 39418 | De Morgan's law for meet i... |
| oldmm4 39419 | De Morgan's law for meet i... |
| oldmj1 39420 | De Morgan's law for join i... |
| oldmj2 39421 | De Morgan's law for join i... |
| oldmj3 39422 | De Morgan's law for join i... |
| oldmj4 39423 | De Morgan's law for join i... |
| olj01 39424 | An ortholattice element jo... |
| olj02 39425 | An ortholattice element jo... |
| olm11 39426 | The meet of an ortholattic... |
| olm12 39427 | The meet of an ortholattic... |
| latmassOLD 39428 | Ortholattice meet is assoc... |
| latm12 39429 | A rearrangement of lattice... |
| latm32 39430 | A rearrangement of lattice... |
| latmrot 39431 | Rotate lattice meet of 3 c... |
| latm4 39432 | Rearrangement of lattice m... |
| latmmdiN 39433 | Lattice meet distributes o... |
| latmmdir 39434 | Lattice meet distributes o... |
| olm01 39435 | Meet with lattice zero is ... |
| olm02 39436 | Meet with lattice zero is ... |
| isoml 39437 | The predicate "is an ortho... |
| isomliN 39438 | Properties that determine ... |
| omlol 39439 | An orthomodular lattice is... |
| omlop 39440 | An orthomodular lattice is... |
| omllat 39441 | An orthomodular lattice is... |
| omllaw 39442 | The orthomodular law. (Co... |
| omllaw2N 39443 | Variation of orthomodular ... |
| omllaw3 39444 | Orthomodular law equivalen... |
| omllaw4 39445 | Orthomodular law equivalen... |
| omllaw5N 39446 | The orthomodular law. Rem... |
| cmtcomlemN 39447 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 39448 | Commutation is symmetric. ... |
| cmt2N 39449 | Commutation with orthocomp... |
| cmt3N 39450 | Commutation with orthocomp... |
| cmt4N 39451 | Commutation with orthocomp... |
| cmtbr2N 39452 | Alternate definition of th... |
| cmtbr3N 39453 | Alternate definition for t... |
| cmtbr4N 39454 | Alternate definition for t... |
| lecmtN 39455 | Ordered elements commute. ... |
| cmtidN 39456 | Any element commutes with ... |
| omlfh1N 39457 | Foulis-Holland Theorem, pa... |
| omlfh3N 39458 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 39459 | Analogue of modular law ~ ... |
| omlspjN 39460 | Contraction of a Sasaki pr... |
| cvrfval 39467 | Value of covers relation "... |
| cvrval 39468 | Binary relation expressing... |
| cvrlt 39469 | The covers relation implie... |
| cvrnbtwn 39470 | There is no element betwee... |
| ncvr1 39471 | No element covers the latt... |
| cvrletrN 39472 | Property of an element abo... |
| cvrval2 39473 | Binary relation expressing... |
| cvrnbtwn2 39474 | The covers relation implie... |
| cvrnbtwn3 39475 | The covers relation implie... |
| cvrcon3b 39476 | Contraposition law for the... |
| cvrle 39477 | The covers relation implie... |
| cvrnbtwn4 39478 | The covers relation implie... |
| cvrnle 39479 | The covers relation implie... |
| cvrne 39480 | The covers relation implie... |
| cvrnrefN 39481 | The covers relation is not... |
| cvrcmp 39482 | If two lattice elements th... |
| cvrcmp2 39483 | If two lattice elements co... |
| pats 39484 | The set of atoms in a pose... |
| isat 39485 | The predicate "is an atom"... |
| isat2 39486 | The predicate "is an atom"... |
| atcvr0 39487 | An atom covers zero. ( ~ ... |
| atbase 39488 | An atom is a member of the... |
| atssbase 39489 | The set of atoms is a subs... |
| 0ltat 39490 | An atom is greater than ze... |
| leatb 39491 | A poset element less than ... |
| leat 39492 | A poset element less than ... |
| leat2 39493 | A nonzero poset element le... |
| leat3 39494 | A poset element less than ... |
| meetat 39495 | The meet of any element wi... |
| meetat2 39496 | The meet of any element wi... |
| isatl 39498 | The predicate "is an atomi... |
| atllat 39499 | An atomic lattice is a lat... |
| atlpos 39500 | An atomic lattice is a pos... |
| atl0dm 39501 | Condition necessary for ze... |
| atl0cl 39502 | An atomic lattice has a ze... |
| atl0le 39503 | Orthoposet zero is less th... |
| atlle0 39504 | An element less than or eq... |
| atlltn0 39505 | A lattice element greater ... |
| isat3 39506 | The predicate "is an atom"... |
| atn0 39507 | An atom is not zero. ( ~ ... |
| atnle0 39508 | An atom is not less than o... |
| atlen0 39509 | A lattice element is nonze... |
| atcmp 39510 | If two atoms are comparabl... |
| atncmp 39511 | Frequently-used variation ... |
| atnlt 39512 | Two atoms cannot satisfy t... |
| atcvreq0 39513 | An element covered by an a... |
| atncvrN 39514 | Two atoms cannot satisfy t... |
| atlex 39515 | Every nonzero element of a... |
| atnle 39516 | Two ways of expressing "an... |
| atnem0 39517 | The meet of distinct atoms... |
| atlatmstc 39518 | An atomic, complete, ortho... |
| atlatle 39519 | The ordering of two Hilber... |
| atlrelat1 39520 | An atomistic lattice with ... |
| iscvlat 39522 | The predicate "is an atomi... |
| iscvlat2N 39523 | The predicate "is an atomi... |
| cvlatl 39524 | An atomic lattice with the... |
| cvllat 39525 | An atomic lattice with the... |
| cvlposN 39526 | An atomic lattice with the... |
| cvlexch1 39527 | An atomic covering lattice... |
| cvlexch2 39528 | An atomic covering lattice... |
| cvlexchb1 39529 | An atomic covering lattice... |
| cvlexchb2 39530 | An atomic covering lattice... |
| cvlexch3 39531 | An atomic covering lattice... |
| cvlexch4N 39532 | An atomic covering lattice... |
| cvlatexchb1 39533 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 39534 | A version of ~ cvlexchb2 f... |
| cvlatexch1 39535 | Atom exchange property. (... |
| cvlatexch2 39536 | Atom exchange property. (... |
| cvlatexch3 39537 | Atom exchange property. (... |
| cvlcvr1 39538 | The covering property. Pr... |
| cvlcvrp 39539 | A Hilbert lattice satisfie... |
| cvlatcvr1 39540 | An atom is covered by its ... |
| cvlatcvr2 39541 | An atom is covered by its ... |
| cvlsupr2 39542 | Two equivalent ways of exp... |
| cvlsupr3 39543 | Two equivalent ways of exp... |
| cvlsupr4 39544 | Consequence of superpositi... |
| cvlsupr5 39545 | Consequence of superpositi... |
| cvlsupr6 39546 | Consequence of superpositi... |
| cvlsupr7 39547 | Consequence of superpositi... |
| cvlsupr8 39548 | Consequence of superpositi... |
| ishlat1 39551 | The predicate "is a Hilber... |
| ishlat2 39552 | The predicate "is a Hilber... |
| ishlat3N 39553 | The predicate "is a Hilber... |
| ishlatiN 39554 | Properties that determine ... |
| hlomcmcv 39555 | A Hilbert lattice is ortho... |
| hloml 39556 | A Hilbert lattice is ortho... |
| hlclat 39557 | A Hilbert lattice is compl... |
| hlcvl 39558 | A Hilbert lattice is an at... |
| hlatl 39559 | A Hilbert lattice is atomi... |
| hlol 39560 | A Hilbert lattice is an or... |
| hlop 39561 | A Hilbert lattice is an or... |
| hllat 39562 | A Hilbert lattice is a lat... |
| hllatd 39563 | Deduction form of ~ hllat ... |
| hlomcmat 39564 | A Hilbert lattice is ortho... |
| hlpos 39565 | A Hilbert lattice is a pos... |
| hlatjcl 39566 | Closure of join operation.... |
| hlatjcom 39567 | Commutatitivity of join op... |
| hlatjidm 39568 | Idempotence of join operat... |
| hlatjass 39569 | Lattice join is associativ... |
| hlatj12 39570 | Swap 1st and 2nd members o... |
| hlatj32 39571 | Swap 2nd and 3rd members o... |
| hlatjrot 39572 | Rotate lattice join of 3 c... |
| hlatj4 39573 | Rearrangement of lattice j... |
| hlatlej1 39574 | A join's first argument is... |
| hlatlej2 39575 | A join's second argument i... |
| glbconN 39576 | De Morgan's law for GLB an... |
| glbconxN 39577 | De Morgan's law for GLB an... |
| atnlej1 39578 | If an atom is not less tha... |
| atnlej2 39579 | If an atom is not less tha... |
| hlsuprexch 39580 | A Hilbert lattice has the ... |
| hlexch1 39581 | A Hilbert lattice has the ... |
| hlexch2 39582 | A Hilbert lattice has the ... |
| hlexchb1 39583 | A Hilbert lattice has the ... |
| hlexchb2 39584 | A Hilbert lattice has the ... |
| hlsupr 39585 | A Hilbert lattice has the ... |
| hlsupr2 39586 | A Hilbert lattice has the ... |
| hlhgt4 39587 | A Hilbert lattice has a he... |
| hlhgt2 39588 | A Hilbert lattice has a he... |
| hl0lt1N 39589 | Lattice 0 is less than lat... |
| hlexch3 39590 | A Hilbert lattice has the ... |
| hlexch4N 39591 | A Hilbert lattice has the ... |
| hlatexchb1 39592 | A version of ~ hlexchb1 fo... |
| hlatexchb2 39593 | A version of ~ hlexchb2 fo... |
| hlatexch1 39594 | Atom exchange property. (... |
| hlatexch2 39595 | Atom exchange property. (... |
| hlatmstcOLDN 39596 | An atomic, complete, ortho... |
| hlatle 39597 | The ordering of two Hilber... |
| hlateq 39598 | The equality of two Hilber... |
| hlrelat1 39599 | An atomistic lattice with ... |
| hlrelat5N 39600 | An atomistic lattice with ... |
| hlrelat 39601 | A Hilbert lattice is relat... |
| hlrelat2 39602 | A consequence of relative ... |
| exatleN 39603 | A condition for an atom to... |
| hl2at 39604 | A Hilbert lattice has at l... |
| atex 39605 | At least one atom exists. ... |
| intnatN 39606 | If the intersection with a... |
| 2llnne2N 39607 | Condition implying that tw... |
| 2llnneN 39608 | Condition implying that tw... |
| cvr1 39609 | A Hilbert lattice has the ... |
| cvr2N 39610 | Less-than and covers equiv... |
| hlrelat3 39611 | The Hilbert lattice is rel... |
| cvrval3 39612 | Binary relation expressing... |
| cvrval4N 39613 | Binary relation expressing... |
| cvrval5 39614 | Binary relation expressing... |
| cvrp 39615 | A Hilbert lattice satisfie... |
| atcvr1 39616 | An atom is covered by its ... |
| atcvr2 39617 | An atom is covered by its ... |
| cvrexchlem 39618 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 39619 | A Hilbert lattice satisfie... |
| cvratlem 39620 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 39621 | A nonzero Hilbert lattice ... |
| ltltncvr 39622 | A chained strong ordering ... |
| ltcvrntr 39623 | Non-transitive condition f... |
| cvrntr 39624 | The covers relation is not... |
| atcvr0eq 39625 | The covers relation is not... |
| lnnat 39626 | A line (the join of two di... |
| atcvrj0 39627 | Two atoms covering the zer... |
| cvrat2 39628 | A Hilbert lattice element ... |
| atcvrneN 39629 | Inequality derived from at... |
| atcvrj1 39630 | Condition for an atom to b... |
| atcvrj2b 39631 | Condition for an atom to b... |
| atcvrj2 39632 | Condition for an atom to b... |
| atleneN 39633 | Inequality derived from at... |
| atltcvr 39634 | An equivalence of less-tha... |
| atle 39635 | Any nonzero element has an... |
| atlt 39636 | Two atoms are unequal iff ... |
| atlelt 39637 | Transfer less-than relatio... |
| 2atlt 39638 | Given an atom less than an... |
| atexchcvrN 39639 | Atom exchange property. V... |
| atexchltN 39640 | Atom exchange property. V... |
| cvrat3 39641 | A condition implying that ... |
| cvrat4 39642 | A condition implying exist... |
| cvrat42 39643 | Commuted version of ~ cvra... |
| 2atjm 39644 | The meet of a line (expres... |
| atbtwn 39645 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 39646 | There exists a 3rd atom ` ... |
| atbtwnex 39647 | Given atoms ` P ` in ` X `... |
| 3noncolr2 39648 | Two ways to express 3 non-... |
| 3noncolr1N 39649 | Two ways to express 3 non-... |
| hlatcon3 39650 | Atom exchange combined wit... |
| hlatcon2 39651 | Atom exchange combined wit... |
| 4noncolr3 39652 | A way to express 4 non-col... |
| 4noncolr2 39653 | A way to express 4 non-col... |
| 4noncolr1 39654 | A way to express 4 non-col... |
| athgt 39655 | A Hilbert lattice, whose h... |
| 3dim0 39656 | There exists a 3-dimension... |
| 3dimlem1 39657 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 39658 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 39659 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 39660 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 39661 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 39662 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 39663 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 39664 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 39665 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 39666 | Construct a 3-dimensional ... |
| 3dim2 39667 | Construct 2 new layers on ... |
| 3dim3 39668 | Construct a new layer on t... |
| 2dim 39669 | Generate a height-3 elemen... |
| 1dimN 39670 | An atom is covered by a he... |
| 1cvrco 39671 | The orthocomplement of an ... |
| 1cvratex 39672 | There exists an atom less ... |
| 1cvratlt 39673 | An atom less than or equal... |
| 1cvrjat 39674 | An element covered by the ... |
| 1cvrat 39675 | Create an atom under an el... |
| ps-1 39676 | The join of two atoms ` R ... |
| ps-2 39677 | Lattice analogue for the p... |
| 2atjlej 39678 | Two atoms are different if... |
| hlatexch3N 39679 | Rearrange join of atoms in... |
| hlatexch4 39680 | Exchange 2 atoms. (Contri... |
| ps-2b 39681 | Variation of projective ge... |
| 3atlem1 39682 | Lemma for ~ 3at . (Contri... |
| 3atlem2 39683 | Lemma for ~ 3at . (Contri... |
| 3atlem3 39684 | Lemma for ~ 3at . (Contri... |
| 3atlem4 39685 | Lemma for ~ 3at . (Contri... |
| 3atlem5 39686 | Lemma for ~ 3at . (Contri... |
| 3atlem6 39687 | Lemma for ~ 3at . (Contri... |
| 3atlem7 39688 | Lemma for ~ 3at . (Contri... |
| 3at 39689 | Any three non-colinear ato... |
| llnset 39704 | The set of lattice lines i... |
| islln 39705 | The predicate "is a lattic... |
| islln4 39706 | The predicate "is a lattic... |
| llni 39707 | Condition implying a latti... |
| llnbase 39708 | A lattice line is a lattic... |
| islln3 39709 | The predicate "is a lattic... |
| islln2 39710 | The predicate "is a lattic... |
| llni2 39711 | The join of two different ... |
| llnnleat 39712 | An atom cannot majorize a ... |
| llnneat 39713 | A lattice line is not an a... |
| 2atneat 39714 | The join of two distinct a... |
| llnn0 39715 | A lattice line is nonzero.... |
| islln2a 39716 | The predicate "is a lattic... |
| llnle 39717 | Any element greater than 0... |
| atcvrlln2 39718 | An atom under a line is co... |
| atcvrlln 39719 | An element covering an ato... |
| llnexatN 39720 | Given an atom on a line, t... |
| llncmp 39721 | If two lattice lines are c... |
| llnnlt 39722 | Two lattice lines cannot s... |
| 2llnmat 39723 | Two intersecting lines int... |
| 2at0mat0 39724 | Special case of ~ 2atmat0 ... |
| 2atmat0 39725 | The meet of two unequal li... |
| 2atm 39726 | An atom majorized by two d... |
| ps-2c 39727 | Variation of projective ge... |
| lplnset 39728 | The set of lattice planes ... |
| islpln 39729 | The predicate "is a lattic... |
| islpln4 39730 | The predicate "is a lattic... |
| lplni 39731 | Condition implying a latti... |
| islpln3 39732 | The predicate "is a lattic... |
| lplnbase 39733 | A lattice plane is a latti... |
| islpln5 39734 | The predicate "is a lattic... |
| islpln2 39735 | The predicate "is a lattic... |
| lplni2 39736 | The join of 3 different at... |
| lvolex3N 39737 | There is an atom outside o... |
| llnmlplnN 39738 | The intersection of a line... |
| lplnle 39739 | Any element greater than 0... |
| lplnnle2at 39740 | A lattice line (or atom) c... |
| lplnnleat 39741 | A lattice plane cannot maj... |
| lplnnlelln 39742 | A lattice plane is not les... |
| 2atnelpln 39743 | The join of two atoms is n... |
| lplnneat 39744 | No lattice plane is an ato... |
| lplnnelln 39745 | No lattice plane is a latt... |
| lplnn0N 39746 | A lattice plane is nonzero... |
| islpln2a 39747 | The predicate "is a lattic... |
| islpln2ah 39748 | The predicate "is a lattic... |
| lplnriaN 39749 | Property of a lattice plan... |
| lplnribN 39750 | Property of a lattice plan... |
| lplnric 39751 | Property of a lattice plan... |
| lplnri1 39752 | Property of a lattice plan... |
| lplnri2N 39753 | Property of a lattice plan... |
| lplnri3N 39754 | Property of a lattice plan... |
| lplnllnneN 39755 | Two lattice lines defined ... |
| llncvrlpln2 39756 | A lattice line under a lat... |
| llncvrlpln 39757 | An element covering a latt... |
| 2lplnmN 39758 | If the join of two lattice... |
| 2llnmj 39759 | The meet of two lattice li... |
| 2atmat 39760 | The meet of two intersecti... |
| lplncmp 39761 | If two lattice planes are ... |
| lplnexatN 39762 | Given a lattice line on a ... |
| lplnexllnN 39763 | Given an atom on a lattice... |
| lplnnlt 39764 | Two lattice planes cannot ... |
| 2llnjaN 39765 | The join of two different ... |
| 2llnjN 39766 | The join of two different ... |
| 2llnm2N 39767 | The meet of two different ... |
| 2llnm3N 39768 | Two lattice lines in a lat... |
| 2llnm4 39769 | Two lattice lines that maj... |
| 2llnmeqat 39770 | An atom equals the interse... |
| lvolset 39771 | The set of 3-dim lattice v... |
| islvol 39772 | The predicate "is a 3-dim ... |
| islvol4 39773 | The predicate "is a 3-dim ... |
| lvoli 39774 | Condition implying a 3-dim... |
| islvol3 39775 | The predicate "is a 3-dim ... |
| lvoli3 39776 | Condition implying a 3-dim... |
| lvolbase 39777 | A 3-dim lattice volume is ... |
| islvol5 39778 | The predicate "is a 3-dim ... |
| islvol2 39779 | The predicate "is a 3-dim ... |
| lvoli2 39780 | The join of 4 different at... |
| lvolnle3at 39781 | A lattice plane (or lattic... |
| lvolnleat 39782 | An atom cannot majorize a ... |
| lvolnlelln 39783 | A lattice line cannot majo... |
| lvolnlelpln 39784 | A lattice plane cannot maj... |
| 3atnelvolN 39785 | The join of 3 atoms is not... |
| 2atnelvolN 39786 | The join of two atoms is n... |
| lvolneatN 39787 | No lattice volume is an at... |
| lvolnelln 39788 | No lattice volume is a lat... |
| lvolnelpln 39789 | No lattice volume is a lat... |
| lvoln0N 39790 | A lattice volume is nonzer... |
| islvol2aN 39791 | The predicate "is a lattic... |
| 4atlem0a 39792 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 39793 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 39794 | Lemma for ~ 4at . (Contri... |
| 4atlem3 39795 | Lemma for ~ 4at . Break i... |
| 4atlem3a 39796 | Lemma for ~ 4at . Break i... |
| 4atlem3b 39797 | Lemma for ~ 4at . Break i... |
| 4atlem4a 39798 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 39799 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 39800 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 39801 | Lemma for ~ 4at . Frequen... |
| 4atlem9 39802 | Lemma for ~ 4at . Substit... |
| 4atlem10a 39803 | Lemma for ~ 4at . Substit... |
| 4atlem10b 39804 | Lemma for ~ 4at . Substit... |
| 4atlem10 39805 | Lemma for ~ 4at . Combine... |
| 4atlem11a 39806 | Lemma for ~ 4at . Substit... |
| 4atlem11b 39807 | Lemma for ~ 4at . Substit... |
| 4atlem11 39808 | Lemma for ~ 4at . Combine... |
| 4atlem12a 39809 | Lemma for ~ 4at . Substit... |
| 4atlem12b 39810 | Lemma for ~ 4at . Substit... |
| 4atlem12 39811 | Lemma for ~ 4at . Combine... |
| 4at 39812 | Four atoms determine a lat... |
| 4at2 39813 | Four atoms determine a lat... |
| lplncvrlvol2 39814 | A lattice line under a lat... |
| lplncvrlvol 39815 | An element covering a latt... |
| lvolcmp 39816 | If two lattice planes are ... |
| lvolnltN 39817 | Two lattice volumes cannot... |
| 2lplnja 39818 | The join of two different ... |
| 2lplnj 39819 | The join of two different ... |
| 2lplnm2N 39820 | The meet of two different ... |
| 2lplnmj 39821 | The meet of two lattice pl... |
| dalemkehl 39822 | Lemma for ~ dath . Freque... |
| dalemkelat 39823 | Lemma for ~ dath . Freque... |
| dalemkeop 39824 | Lemma for ~ dath . Freque... |
| dalempea 39825 | Lemma for ~ dath . Freque... |
| dalemqea 39826 | Lemma for ~ dath . Freque... |
| dalemrea 39827 | Lemma for ~ dath . Freque... |
| dalemsea 39828 | Lemma for ~ dath . Freque... |
| dalemtea 39829 | Lemma for ~ dath . Freque... |
| dalemuea 39830 | Lemma for ~ dath . Freque... |
| dalemyeo 39831 | Lemma for ~ dath . Freque... |
| dalemzeo 39832 | Lemma for ~ dath . Freque... |
| dalemclpjs 39833 | Lemma for ~ dath . Freque... |
| dalemclqjt 39834 | Lemma for ~ dath . Freque... |
| dalemclrju 39835 | Lemma for ~ dath . Freque... |
| dalem-clpjq 39836 | Lemma for ~ dath . Freque... |
| dalemceb 39837 | Lemma for ~ dath . Freque... |
| dalempeb 39838 | Lemma for ~ dath . Freque... |
| dalemqeb 39839 | Lemma for ~ dath . Freque... |
| dalemreb 39840 | Lemma for ~ dath . Freque... |
| dalemseb 39841 | Lemma for ~ dath . Freque... |
| dalemteb 39842 | Lemma for ~ dath . Freque... |
| dalemueb 39843 | Lemma for ~ dath . Freque... |
| dalempjqeb 39844 | Lemma for ~ dath . Freque... |
| dalemsjteb 39845 | Lemma for ~ dath . Freque... |
| dalemtjueb 39846 | Lemma for ~ dath . Freque... |
| dalemqrprot 39847 | Lemma for ~ dath . Freque... |
| dalemyeb 39848 | Lemma for ~ dath . Freque... |
| dalemcnes 39849 | Lemma for ~ dath . Freque... |
| dalempnes 39850 | Lemma for ~ dath . Freque... |
| dalemqnet 39851 | Lemma for ~ dath . Freque... |
| dalempjsen 39852 | Lemma for ~ dath . Freque... |
| dalemply 39853 | Lemma for ~ dath . Freque... |
| dalemsly 39854 | Lemma for ~ dath . Freque... |
| dalemswapyz 39855 | Lemma for ~ dath . Swap t... |
| dalemrot 39856 | Lemma for ~ dath . Rotate... |
| dalemrotyz 39857 | Lemma for ~ dath . Rotate... |
| dalem1 39858 | Lemma for ~ dath . Show t... |
| dalemcea 39859 | Lemma for ~ dath . Freque... |
| dalem2 39860 | Lemma for ~ dath . Show t... |
| dalemdea 39861 | Lemma for ~ dath . Freque... |
| dalemeea 39862 | Lemma for ~ dath . Freque... |
| dalem3 39863 | Lemma for ~ dalemdnee . (... |
| dalem4 39864 | Lemma for ~ dalemdnee . (... |
| dalemdnee 39865 | Lemma for ~ dath . Axis o... |
| dalem5 39866 | Lemma for ~ dath . Atom `... |
| dalem6 39867 | Lemma for ~ dath . Analog... |
| dalem7 39868 | Lemma for ~ dath . Analog... |
| dalem8 39869 | Lemma for ~ dath . Plane ... |
| dalem-cly 39870 | Lemma for ~ dalem9 . Cent... |
| dalem9 39871 | Lemma for ~ dath . Since ... |
| dalem10 39872 | Lemma for ~ dath . Atom `... |
| dalem11 39873 | Lemma for ~ dath . Analog... |
| dalem12 39874 | Lemma for ~ dath . Analog... |
| dalem13 39875 | Lemma for ~ dalem14 . (Co... |
| dalem14 39876 | Lemma for ~ dath . Planes... |
| dalem15 39877 | Lemma for ~ dath . The ax... |
| dalem16 39878 | Lemma for ~ dath . The at... |
| dalem17 39879 | Lemma for ~ dath . When p... |
| dalem18 39880 | Lemma for ~ dath . Show t... |
| dalem19 39881 | Lemma for ~ dath . Show t... |
| dalemccea 39882 | Lemma for ~ dath . Freque... |
| dalemddea 39883 | Lemma for ~ dath . Freque... |
| dalem-ccly 39884 | Lemma for ~ dath . Freque... |
| dalem-ddly 39885 | Lemma for ~ dath . Freque... |
| dalemccnedd 39886 | Lemma for ~ dath . Freque... |
| dalemclccjdd 39887 | Lemma for ~ dath . Freque... |
| dalemcceb 39888 | Lemma for ~ dath . Freque... |
| dalemswapyzps 39889 | Lemma for ~ dath . Swap t... |
| dalemrotps 39890 | Lemma for ~ dath . Rotate... |
| dalemcjden 39891 | Lemma for ~ dath . Show t... |
| dalem20 39892 | Lemma for ~ dath . Show t... |
| dalem21 39893 | Lemma for ~ dath . Show t... |
| dalem22 39894 | Lemma for ~ dath . Show t... |
| dalem23 39895 | Lemma for ~ dath . Show t... |
| dalem24 39896 | Lemma for ~ dath . Show t... |
| dalem25 39897 | Lemma for ~ dath . Show t... |
| dalem27 39898 | Lemma for ~ dath . Show t... |
| dalem28 39899 | Lemma for ~ dath . Lemma ... |
| dalem29 39900 | Lemma for ~ dath . Analog... |
| dalem30 39901 | Lemma for ~ dath . Analog... |
| dalem31N 39902 | Lemma for ~ dath . Analog... |
| dalem32 39903 | Lemma for ~ dath . Analog... |
| dalem33 39904 | Lemma for ~ dath . Analog... |
| dalem34 39905 | Lemma for ~ dath . Analog... |
| dalem35 39906 | Lemma for ~ dath . Analog... |
| dalem36 39907 | Lemma for ~ dath . Analog... |
| dalem37 39908 | Lemma for ~ dath . Analog... |
| dalem38 39909 | Lemma for ~ dath . Plane ... |
| dalem39 39910 | Lemma for ~ dath . Auxili... |
| dalem40 39911 | Lemma for ~ dath . Analog... |
| dalem41 39912 | Lemma for ~ dath . (Contr... |
| dalem42 39913 | Lemma for ~ dath . Auxili... |
| dalem43 39914 | Lemma for ~ dath . Planes... |
| dalem44 39915 | Lemma for ~ dath . Dummy ... |
| dalem45 39916 | Lemma for ~ dath . Dummy ... |
| dalem46 39917 | Lemma for ~ dath . Analog... |
| dalem47 39918 | Lemma for ~ dath . Analog... |
| dalem48 39919 | Lemma for ~ dath . Analog... |
| dalem49 39920 | Lemma for ~ dath . Analog... |
| dalem50 39921 | Lemma for ~ dath . Analog... |
| dalem51 39922 | Lemma for ~ dath . Constr... |
| dalem52 39923 | Lemma for ~ dath . Lines ... |
| dalem53 39924 | Lemma for ~ dath . The au... |
| dalem54 39925 | Lemma for ~ dath . Line `... |
| dalem55 39926 | Lemma for ~ dath . Lines ... |
| dalem56 39927 | Lemma for ~ dath . Analog... |
| dalem57 39928 | Lemma for ~ dath . Axis o... |
| dalem58 39929 | Lemma for ~ dath . Analog... |
| dalem59 39930 | Lemma for ~ dath . Analog... |
| dalem60 39931 | Lemma for ~ dath . ` B ` i... |
| dalem61 39932 | Lemma for ~ dath . Show t... |
| dalem62 39933 | Lemma for ~ dath . Elimin... |
| dalem63 39934 | Lemma for ~ dath . Combin... |
| dath 39935 | Desargues's theorem of pro... |
| dath2 39936 | Version of Desargues's the... |
| lineset 39937 | The set of lines in a Hilb... |
| isline 39938 | The predicate "is a line".... |
| islinei 39939 | Condition implying "is a l... |
| pointsetN 39940 | The set of points in a Hil... |
| ispointN 39941 | The predicate "is a point"... |
| atpointN 39942 | The singleton of an atom i... |
| psubspset 39943 | The set of projective subs... |
| ispsubsp 39944 | The predicate "is a projec... |
| ispsubsp2 39945 | The predicate "is a projec... |
| psubspi 39946 | Property of a projective s... |
| psubspi2N 39947 | Property of a projective s... |
| 0psubN 39948 | The empty set is a project... |
| snatpsubN 39949 | The singleton of an atom i... |
| pointpsubN 39950 | A point (singleton of an a... |
| linepsubN 39951 | A line is a projective sub... |
| atpsubN 39952 | The set of all atoms is a ... |
| psubssat 39953 | A projective subspace cons... |
| psubatN 39954 | A member of a projective s... |
| pmapfval 39955 | The projective map of a Hi... |
| pmapval 39956 | Value of the projective ma... |
| elpmap 39957 | Member of a projective map... |
| pmapssat 39958 | The projective map of a Hi... |
| pmapssbaN 39959 | A weakening of ~ pmapssat ... |
| pmaple 39960 | The projective map of a Hi... |
| pmap11 39961 | The projective map of a Hi... |
| pmapat 39962 | The projective map of an a... |
| elpmapat 39963 | Member of the projective m... |
| pmap0 39964 | Value of the projective ma... |
| pmapeq0 39965 | A projective map value is ... |
| pmap1N 39966 | Value of the projective ma... |
| pmapsub 39967 | The projective map of a Hi... |
| pmapglbx 39968 | The projective map of the ... |
| pmapglb 39969 | The projective map of the ... |
| pmapglb2N 39970 | The projective map of the ... |
| pmapglb2xN 39971 | The projective map of the ... |
| pmapmeet 39972 | The projective map of a me... |
| isline2 39973 | Definition of line in term... |
| linepmap 39974 | A line described with a pr... |
| isline3 39975 | Definition of line in term... |
| isline4N 39976 | Definition of line in term... |
| lneq2at 39977 | A line equals the join of ... |
| lnatexN 39978 | There is an atom in a line... |
| lnjatN 39979 | Given an atom in a line, t... |
| lncvrelatN 39980 | A lattice element covered ... |
| lncvrat 39981 | A line covers the atoms it... |
| lncmp 39982 | If two lines are comparabl... |
| 2lnat 39983 | Two intersecting lines int... |
| 2atm2atN 39984 | Two joins with a common at... |
| 2llnma1b 39985 | Generalization of ~ 2llnma... |
| 2llnma1 39986 | Two different intersecting... |
| 2llnma3r 39987 | Two different intersecting... |
| 2llnma2 39988 | Two different intersecting... |
| 2llnma2rN 39989 | Two different intersecting... |
| cdlema1N 39990 | A condition for required f... |
| cdlema2N 39991 | A condition for required f... |
| cdlemblem 39992 | Lemma for ~ cdlemb . (Con... |
| cdlemb 39993 | Given two atoms not less t... |
| paddfval 39996 | Projective subspace sum op... |
| paddval 39997 | Projective subspace sum op... |
| elpadd 39998 | Member of a projective sub... |
| elpaddn0 39999 | Member of projective subsp... |
| paddvaln0N 40000 | Projective subspace sum op... |
| elpaddri 40001 | Condition implying members... |
| elpaddatriN 40002 | Condition implying members... |
| elpaddat 40003 | Membership in a projective... |
| elpaddatiN 40004 | Consequence of membership ... |
| elpadd2at 40005 | Membership in a projective... |
| elpadd2at2 40006 | Membership in a projective... |
| paddunssN 40007 | Projective subspace sum in... |
| elpadd0 40008 | Member of projective subsp... |
| paddval0 40009 | Projective subspace sum wi... |
| padd01 40010 | Projective subspace sum wi... |
| padd02 40011 | Projective subspace sum wi... |
| paddcom 40012 | Projective subspace sum co... |
| paddssat 40013 | A projective subspace sum ... |
| sspadd1 40014 | A projective subspace sum ... |
| sspadd2 40015 | A projective subspace sum ... |
| paddss1 40016 | Subset law for projective ... |
| paddss2 40017 | Subset law for projective ... |
| paddss12 40018 | Subset law for projective ... |
| paddasslem1 40019 | Lemma for ~ paddass . (Co... |
| paddasslem2 40020 | Lemma for ~ paddass . (Co... |
| paddasslem3 40021 | Lemma for ~ paddass . Res... |
| paddasslem4 40022 | Lemma for ~ paddass . Com... |
| paddasslem5 40023 | Lemma for ~ paddass . Sho... |
| paddasslem6 40024 | Lemma for ~ paddass . (Co... |
| paddasslem7 40025 | Lemma for ~ paddass . Com... |
| paddasslem8 40026 | Lemma for ~ paddass . (Co... |
| paddasslem9 40027 | Lemma for ~ paddass . Com... |
| paddasslem10 40028 | Lemma for ~ paddass . Use... |
| paddasslem11 40029 | Lemma for ~ paddass . The... |
| paddasslem12 40030 | Lemma for ~ paddass . The... |
| paddasslem13 40031 | Lemma for ~ paddass . The... |
| paddasslem14 40032 | Lemma for ~ paddass . Rem... |
| paddasslem15 40033 | Lemma for ~ paddass . Use... |
| paddasslem16 40034 | Lemma for ~ paddass . Use... |
| paddasslem17 40035 | Lemma for ~ paddass . The... |
| paddasslem18 40036 | Lemma for ~ paddass . Com... |
| paddass 40037 | Projective subspace sum is... |
| padd12N 40038 | Commutative/associative la... |
| padd4N 40039 | Rearrangement of 4 terms i... |
| paddidm 40040 | Projective subspace sum is... |
| paddclN 40041 | The projective sum of two ... |
| paddssw1 40042 | Subset law for projective ... |
| paddssw2 40043 | Subset law for projective ... |
| paddss 40044 | Subset law for projective ... |
| pmodlem1 40045 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 40046 | Lemma for ~ pmod1i . (Con... |
| pmod1i 40047 | The modular law holds in a... |
| pmod2iN 40048 | Dual of the modular law. ... |
| pmodN 40049 | The modular law for projec... |
| pmodl42N 40050 | Lemma derived from modular... |
| pmapjoin 40051 | The projective map of the ... |
| pmapjat1 40052 | The projective map of the ... |
| pmapjat2 40053 | The projective map of the ... |
| pmapjlln1 40054 | The projective map of the ... |
| hlmod1i 40055 | A version of the modular l... |
| atmod1i1 40056 | Version of modular law ~ p... |
| atmod1i1m 40057 | Version of modular law ~ p... |
| atmod1i2 40058 | Version of modular law ~ p... |
| llnmod1i2 40059 | Version of modular law ~ p... |
| atmod2i1 40060 | Version of modular law ~ p... |
| atmod2i2 40061 | Version of modular law ~ p... |
| llnmod2i2 40062 | Version of modular law ~ p... |
| atmod3i1 40063 | Version of modular law tha... |
| atmod3i2 40064 | Version of modular law tha... |
| atmod4i1 40065 | Version of modular law tha... |
| atmod4i2 40066 | Version of modular law tha... |
| llnexchb2lem 40067 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 40068 | Line exchange property (co... |
| llnexch2N 40069 | Line exchange property (co... |
| dalawlem1 40070 | Lemma for ~ dalaw . Speci... |
| dalawlem2 40071 | Lemma for ~ dalaw . Utili... |
| dalawlem3 40072 | Lemma for ~ dalaw . First... |
| dalawlem4 40073 | Lemma for ~ dalaw . Secon... |
| dalawlem5 40074 | Lemma for ~ dalaw . Speci... |
| dalawlem6 40075 | Lemma for ~ dalaw . First... |
| dalawlem7 40076 | Lemma for ~ dalaw . Secon... |
| dalawlem8 40077 | Lemma for ~ dalaw . Speci... |
| dalawlem9 40078 | Lemma for ~ dalaw . Speci... |
| dalawlem10 40079 | Lemma for ~ dalaw . Combi... |
| dalawlem11 40080 | Lemma for ~ dalaw . First... |
| dalawlem12 40081 | Lemma for ~ dalaw . Secon... |
| dalawlem13 40082 | Lemma for ~ dalaw . Speci... |
| dalawlem14 40083 | Lemma for ~ dalaw . Combi... |
| dalawlem15 40084 | Lemma for ~ dalaw . Swap ... |
| dalaw 40085 | Desargues's law, derived f... |
| pclfvalN 40088 | The projective subspace cl... |
| pclvalN 40089 | Value of the projective su... |
| pclclN 40090 | Closure of the projective ... |
| elpclN 40091 | Membership in the projecti... |
| elpcliN 40092 | Implication of membership ... |
| pclssN 40093 | Ordering is preserved by s... |
| pclssidN 40094 | A set of atoms is included... |
| pclidN 40095 | The projective subspace cl... |
| pclbtwnN 40096 | A projective subspace sand... |
| pclunN 40097 | The projective subspace cl... |
| pclun2N 40098 | The projective subspace cl... |
| pclfinN 40099 | The projective subspace cl... |
| pclcmpatN 40100 | The set of projective subs... |
| polfvalN 40103 | The projective subspace po... |
| polvalN 40104 | Value of the projective su... |
| polval2N 40105 | Alternate expression for v... |
| polsubN 40106 | The polarity of a set of a... |
| polssatN 40107 | The polarity of a set of a... |
| pol0N 40108 | The polarity of the empty ... |
| pol1N 40109 | The polarity of the whole ... |
| 2pol0N 40110 | The closed subspace closur... |
| polpmapN 40111 | The polarity of a projecti... |
| 2polpmapN 40112 | Double polarity of a proje... |
| 2polvalN 40113 | Value of double polarity. ... |
| 2polssN 40114 | A set of atoms is a subset... |
| 3polN 40115 | Triple polarity cancels to... |
| polcon3N 40116 | Contraposition law for pol... |
| 2polcon4bN 40117 | Contraposition law for pol... |
| polcon2N 40118 | Contraposition law for pol... |
| polcon2bN 40119 | Contraposition law for pol... |
| pclss2polN 40120 | The projective subspace cl... |
| pcl0N 40121 | The projective subspace cl... |
| pcl0bN 40122 | The projective subspace cl... |
| pmaplubN 40123 | The LUB of a projective ma... |
| sspmaplubN 40124 | A set of atoms is a subset... |
| 2pmaplubN 40125 | Double projective map of a... |
| paddunN 40126 | The closure of the project... |
| poldmj1N 40127 | De Morgan's law for polari... |
| pmapj2N 40128 | The projective map of the ... |
| pmapocjN 40129 | The projective map of the ... |
| polatN 40130 | The polarity of the single... |
| 2polatN 40131 | Double polarity of the sin... |
| pnonsingN 40132 | The intersection of a set ... |
| psubclsetN 40135 | The set of closed projecti... |
| ispsubclN 40136 | The predicate "is a closed... |
| psubcliN 40137 | Property of a closed proje... |
| psubcli2N 40138 | Property of a closed proje... |
| psubclsubN 40139 | A closed projective subspa... |
| psubclssatN 40140 | A closed projective subspa... |
| pmapidclN 40141 | Projective map of the LUB ... |
| 0psubclN 40142 | The empty set is a closed ... |
| 1psubclN 40143 | The set of all atoms is a ... |
| atpsubclN 40144 | A point (singleton of an a... |
| pmapsubclN 40145 | A projective map value is ... |
| ispsubcl2N 40146 | Alternate predicate for "i... |
| psubclinN 40147 | The intersection of two cl... |
| paddatclN 40148 | The projective sum of a cl... |
| pclfinclN 40149 | The projective subspace cl... |
| linepsubclN 40150 | A line is a closed project... |
| polsubclN 40151 | A polarity is a closed pro... |
| poml4N 40152 | Orthomodular law for proje... |
| poml5N 40153 | Orthomodular law for proje... |
| poml6N 40154 | Orthomodular law for proje... |
| osumcllem1N 40155 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 40156 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 40157 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 40158 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 40159 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 40160 | Lemma for ~ osumclN . Use... |
| osumcllem7N 40161 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 40162 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 40163 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 40164 | Lemma for ~ osumclN . Con... |
| osumcllem11N 40165 | Lemma for ~ osumclN . (Co... |
| osumclN 40166 | Closure of orthogonal sum.... |
| pmapojoinN 40167 | For orthogonal elements, p... |
| pexmidN 40168 | Excluded middle law for cl... |
| pexmidlem1N 40169 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 40170 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 40171 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 40172 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 40173 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 40174 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 40175 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 40176 | Lemma for ~ pexmidN . The... |
| pexmidALTN 40177 | Excluded middle law for cl... |
| pl42lem1N 40178 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 40179 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 40180 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 40181 | Lemma for ~ pl42N . (Cont... |
| pl42N 40182 | Law holding in a Hilbert l... |
| watfvalN 40191 | The W atoms function. (Co... |
| watvalN 40192 | Value of the W atoms funct... |
| iswatN 40193 | The predicate "is a W atom... |
| lhpset 40194 | The set of co-atoms (latti... |
| islhp 40195 | The predicate "is a co-ato... |
| islhp2 40196 | The predicate "is a co-ato... |
| lhpbase 40197 | A co-atom is a member of t... |
| lhp1cvr 40198 | The lattice unity covers a... |
| lhplt 40199 | An atom under a co-atom is... |
| lhp2lt 40200 | The join of two atoms unde... |
| lhpexlt 40201 | There exists an atom less ... |
| lhp0lt 40202 | A co-atom is greater than ... |
| lhpn0 40203 | A co-atom is nonzero. TOD... |
| lhpexle 40204 | There exists an atom under... |
| lhpexnle 40205 | There exists an atom not u... |
| lhpexle1lem 40206 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 40207 | There exists an atom under... |
| lhpexle2lem 40208 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 40209 | There exists atom under a ... |
| lhpexle3lem 40210 | There exists atom under a ... |
| lhpexle3 40211 | There exists atom under a ... |
| lhpex2leN 40212 | There exist at least two d... |
| lhpoc 40213 | The orthocomplement of a c... |
| lhpoc2N 40214 | The orthocomplement of an ... |
| lhpocnle 40215 | The orthocomplement of a c... |
| lhpocat 40216 | The orthocomplement of a c... |
| lhpocnel 40217 | The orthocomplement of a c... |
| lhpocnel2 40218 | The orthocomplement of a c... |
| lhpjat1 40219 | The join of a co-atom (hyp... |
| lhpjat2 40220 | The join of a co-atom (hyp... |
| lhpj1 40221 | The join of a co-atom (hyp... |
| lhpmcvr 40222 | The meet of a lattice hype... |
| lhpmcvr2 40223 | Alternate way to express t... |
| lhpmcvr3 40224 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 40225 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 40226 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 40227 | Specialization of ~ lhpmcv... |
| lhpm0atN 40228 | If the meet of a lattice h... |
| lhpmat 40229 | An element covered by the ... |
| lhpmatb 40230 | An element covered by the ... |
| lhp2at0 40231 | Join and meet with differe... |
| lhp2atnle 40232 | Inequality for 2 different... |
| lhp2atne 40233 | Inequality for joins with ... |
| lhp2at0nle 40234 | Inequality for 2 different... |
| lhp2at0ne 40235 | Inequality for joins with ... |
| lhpelim 40236 | Eliminate an atom not unde... |
| lhpmod2i2 40237 | Modular law for hyperplane... |
| lhpmod6i1 40238 | Modular law for hyperplane... |
| lhprelat3N 40239 | The Hilbert lattice is rel... |
| cdlemb2 40240 | Given two atoms not under ... |
| lhple 40241 | Property of a lattice elem... |
| lhpat 40242 | Create an atom under a co-... |
| lhpat4N 40243 | Property of an atom under ... |
| lhpat2 40244 | Create an atom under a co-... |
| lhpat3 40245 | There is only one atom und... |
| 4atexlemk 40246 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 40247 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 40248 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 40249 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 40250 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 40251 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 40252 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 40253 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 40254 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 40255 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 40256 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 40257 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 40258 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 40259 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 40260 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 40261 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 40262 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 40263 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 40264 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 40265 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 40266 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 40267 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 40268 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 40269 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 40270 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 40271 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 40272 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 40273 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 40274 | Whenever there are at leas... |
| 4atex 40275 | Whenever there are at leas... |
| 4atex2 40276 | More general version of ~ ... |
| 4atex2-0aOLDN 40277 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 40278 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 40279 | Same as ~ 4atex2 except th... |
| 4atex3 40280 | More general version of ~ ... |
| lautset 40281 | The set of lattice automor... |
| islaut 40282 | The predicate "is a lattic... |
| lautle 40283 | Less-than or equal propert... |
| laut1o 40284 | A lattice automorphism is ... |
| laut11 40285 | One-to-one property of a l... |
| lautcl 40286 | A lattice automorphism val... |
| lautcnvclN 40287 | Reverse closure of a latti... |
| lautcnvle 40288 | Less-than or equal propert... |
| lautcnv 40289 | The converse of a lattice ... |
| lautlt 40290 | Less-than property of a la... |
| lautcvr 40291 | Covering property of a lat... |
| lautj 40292 | Meet property of a lattice... |
| lautm 40293 | Meet property of a lattice... |
| lauteq 40294 | A lattice automorphism arg... |
| idlaut 40295 | The identity function is a... |
| lautco 40296 | The composition of two lat... |
| pautsetN 40297 | The set of projective auto... |
| ispautN 40298 | The predicate "is a projec... |
| ldilfset 40307 | The mapping from fiducial ... |
| ldilset 40308 | The set of lattice dilatio... |
| isldil 40309 | The predicate "is a lattic... |
| ldillaut 40310 | A lattice dilation is an a... |
| ldil1o 40311 | A lattice dilation is a on... |
| ldilval 40312 | Value of a lattice dilatio... |
| idldil 40313 | The identity function is a... |
| ldilcnv 40314 | The converse of a lattice ... |
| ldilco 40315 | The composition of two lat... |
| ltrnfset 40316 | The set of all lattice tra... |
| ltrnset 40317 | The set of lattice transla... |
| isltrn 40318 | The predicate "is a lattic... |
| isltrn2N 40319 | The predicate "is a lattic... |
| ltrnu 40320 | Uniqueness property of a l... |
| ltrnldil 40321 | A lattice translation is a... |
| ltrnlaut 40322 | A lattice translation is a... |
| ltrn1o 40323 | A lattice translation is a... |
| ltrncl 40324 | Closure of a lattice trans... |
| ltrn11 40325 | One-to-one property of a l... |
| ltrncnvnid 40326 | If a translation is differ... |
| ltrncoidN 40327 | Two translations are equal... |
| ltrnle 40328 | Less-than or equal propert... |
| ltrncnvleN 40329 | Less-than or equal propert... |
| ltrnm 40330 | Lattice translation of a m... |
| ltrnj 40331 | Lattice translation of a m... |
| ltrncvr 40332 | Covering property of a lat... |
| ltrnval1 40333 | Value of a lattice transla... |
| ltrnid 40334 | A lattice translation is t... |
| ltrnnid 40335 | If a lattice translation i... |
| ltrnatb 40336 | The lattice translation of... |
| ltrncnvatb 40337 | The converse of the lattic... |
| ltrnel 40338 | The lattice translation of... |
| ltrnat 40339 | The lattice translation of... |
| ltrncnvat 40340 | The converse of the lattic... |
| ltrncnvel 40341 | The converse of the lattic... |
| ltrncoelN 40342 | Composition of lattice tra... |
| ltrncoat 40343 | Composition of lattice tra... |
| ltrncoval 40344 | Two ways to express value ... |
| ltrncnv 40345 | The converse of a lattice ... |
| ltrn11at 40346 | Frequently used one-to-one... |
| ltrneq2 40347 | The equality of two transl... |
| ltrneq 40348 | The equality of two transl... |
| idltrn 40349 | The identity function is a... |
| ltrnmw 40350 | Property of lattice transl... |
| dilfsetN 40351 | The mapping from fiducial ... |
| dilsetN 40352 | The set of dilations for a... |
| isdilN 40353 | The predicate "is a dilati... |
| trnfsetN 40354 | The mapping from fiducial ... |
| trnsetN 40355 | The set of translations fo... |
| istrnN 40356 | The predicate "is a transl... |
| trlfset 40359 | The set of all traces of l... |
| trlset 40360 | The set of traces of latti... |
| trlval 40361 | The value of the trace of ... |
| trlval2 40362 | The value of the trace of ... |
| trlcl 40363 | Closure of the trace of a ... |
| trlcnv 40364 | The trace of the converse ... |
| trljat1 40365 | The value of a translation... |
| trljat2 40366 | The value of a translation... |
| trljat3 40367 | The value of a translation... |
| trlat 40368 | If an atom differs from it... |
| trl0 40369 | If an atom not under the f... |
| trlator0 40370 | The trace of a lattice tra... |
| trlatn0 40371 | The trace of a lattice tra... |
| trlnidat 40372 | The trace of a lattice tra... |
| ltrnnidn 40373 | If a lattice translation i... |
| ltrnideq 40374 | Property of the identity l... |
| trlid0 40375 | The trace of the identity ... |
| trlnidatb 40376 | A lattice translation is n... |
| trlid0b 40377 | A lattice translation is t... |
| trlnid 40378 | Different translations wit... |
| ltrn2ateq 40379 | Property of the equality o... |
| ltrnateq 40380 | If any atom (under ` W ` )... |
| ltrnatneq 40381 | If any atom (under ` W ` )... |
| ltrnatlw 40382 | If the value of an atom eq... |
| trlle 40383 | The trace of a lattice tra... |
| trlne 40384 | The trace of a lattice tra... |
| trlnle 40385 | The atom not under the fid... |
| trlval3 40386 | The value of the trace of ... |
| trlval4 40387 | The value of the trace of ... |
| trlval5 40388 | The value of the trace of ... |
| arglem1N 40389 | Lemma for Desargues's law.... |
| cdlemc1 40390 | Part of proof of Lemma C i... |
| cdlemc2 40391 | Part of proof of Lemma C i... |
| cdlemc3 40392 | Part of proof of Lemma C i... |
| cdlemc4 40393 | Part of proof of Lemma C i... |
| cdlemc5 40394 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 40395 | Lemma for ~ cdlemc . (Con... |
| cdlemc 40396 | Lemma C in [Crawley] p. 11... |
| cdlemd1 40397 | Part of proof of Lemma D i... |
| cdlemd2 40398 | Part of proof of Lemma D i... |
| cdlemd3 40399 | Part of proof of Lemma D i... |
| cdlemd4 40400 | Part of proof of Lemma D i... |
| cdlemd5 40401 | Part of proof of Lemma D i... |
| cdlemd6 40402 | Part of proof of Lemma D i... |
| cdlemd7 40403 | Part of proof of Lemma D i... |
| cdlemd8 40404 | Part of proof of Lemma D i... |
| cdlemd9 40405 | Part of proof of Lemma D i... |
| cdlemd 40406 | If two translations agree ... |
| ltrneq3 40407 | Two translations agree at ... |
| cdleme00a 40408 | Part of proof of Lemma E i... |
| cdleme0aa 40409 | Part of proof of Lemma E i... |
| cdleme0a 40410 | Part of proof of Lemma E i... |
| cdleme0b 40411 | Part of proof of Lemma E i... |
| cdleme0c 40412 | Part of proof of Lemma E i... |
| cdleme0cp 40413 | Part of proof of Lemma E i... |
| cdleme0cq 40414 | Part of proof of Lemma E i... |
| cdleme0dN 40415 | Part of proof of Lemma E i... |
| cdleme0e 40416 | Part of proof of Lemma E i... |
| cdleme0fN 40417 | Part of proof of Lemma E i... |
| cdleme0gN 40418 | Part of proof of Lemma E i... |
| cdlemeulpq 40419 | Part of proof of Lemma E i... |
| cdleme01N 40420 | Part of proof of Lemma E i... |
| cdleme02N 40421 | Part of proof of Lemma E i... |
| cdleme0ex1N 40422 | Part of proof of Lemma E i... |
| cdleme0ex2N 40423 | Part of proof of Lemma E i... |
| cdleme0moN 40424 | Part of proof of Lemma E i... |
| cdleme1b 40425 | Part of proof of Lemma E i... |
| cdleme1 40426 | Part of proof of Lemma E i... |
| cdleme2 40427 | Part of proof of Lemma E i... |
| cdleme3b 40428 | Part of proof of Lemma E i... |
| cdleme3c 40429 | Part of proof of Lemma E i... |
| cdleme3d 40430 | Part of proof of Lemma E i... |
| cdleme3e 40431 | Part of proof of Lemma E i... |
| cdleme3fN 40432 | Part of proof of Lemma E i... |
| cdleme3g 40433 | Part of proof of Lemma E i... |
| cdleme3h 40434 | Part of proof of Lemma E i... |
| cdleme3fa 40435 | Part of proof of Lemma E i... |
| cdleme3 40436 | Part of proof of Lemma E i... |
| cdleme4 40437 | Part of proof of Lemma E i... |
| cdleme4a 40438 | Part of proof of Lemma E i... |
| cdleme5 40439 | Part of proof of Lemma E i... |
| cdleme6 40440 | Part of proof of Lemma E i... |
| cdleme7aa 40441 | Part of proof of Lemma E i... |
| cdleme7a 40442 | Part of proof of Lemma E i... |
| cdleme7b 40443 | Part of proof of Lemma E i... |
| cdleme7c 40444 | Part of proof of Lemma E i... |
| cdleme7d 40445 | Part of proof of Lemma E i... |
| cdleme7e 40446 | Part of proof of Lemma E i... |
| cdleme7ga 40447 | Part of proof of Lemma E i... |
| cdleme7 40448 | Part of proof of Lemma E i... |
| cdleme8 40449 | Part of proof of Lemma E i... |
| cdleme9a 40450 | Part of proof of Lemma E i... |
| cdleme9b 40451 | Utility lemma for Lemma E ... |
| cdleme9 40452 | Part of proof of Lemma E i... |
| cdleme10 40453 | Part of proof of Lemma E i... |
| cdleme8tN 40454 | Part of proof of Lemma E i... |
| cdleme9taN 40455 | Part of proof of Lemma E i... |
| cdleme9tN 40456 | Part of proof of Lemma E i... |
| cdleme10tN 40457 | Part of proof of Lemma E i... |
| cdleme16aN 40458 | Part of proof of Lemma E i... |
| cdleme11a 40459 | Part of proof of Lemma E i... |
| cdleme11c 40460 | Part of proof of Lemma E i... |
| cdleme11dN 40461 | Part of proof of Lemma E i... |
| cdleme11e 40462 | Part of proof of Lemma E i... |
| cdleme11fN 40463 | Part of proof of Lemma E i... |
| cdleme11g 40464 | Part of proof of Lemma E i... |
| cdleme11h 40465 | Part of proof of Lemma E i... |
| cdleme11j 40466 | Part of proof of Lemma E i... |
| cdleme11k 40467 | Part of proof of Lemma E i... |
| cdleme11l 40468 | Part of proof of Lemma E i... |
| cdleme11 40469 | Part of proof of Lemma E i... |
| cdleme12 40470 | Part of proof of Lemma E i... |
| cdleme13 40471 | Part of proof of Lemma E i... |
| cdleme14 40472 | Part of proof of Lemma E i... |
| cdleme15a 40473 | Part of proof of Lemma E i... |
| cdleme15b 40474 | Part of proof of Lemma E i... |
| cdleme15c 40475 | Part of proof of Lemma E i... |
| cdleme15d 40476 | Part of proof of Lemma E i... |
| cdleme15 40477 | Part of proof of Lemma E i... |
| cdleme16b 40478 | Part of proof of Lemma E i... |
| cdleme16c 40479 | Part of proof of Lemma E i... |
| cdleme16d 40480 | Part of proof of Lemma E i... |
| cdleme16e 40481 | Part of proof of Lemma E i... |
| cdleme16f 40482 | Part of proof of Lemma E i... |
| cdleme16g 40483 | Part of proof of Lemma E i... |
| cdleme16 40484 | Part of proof of Lemma E i... |
| cdleme17a 40485 | Part of proof of Lemma E i... |
| cdleme17b 40486 | Lemma leading to ~ cdleme1... |
| cdleme17c 40487 | Part of proof of Lemma E i... |
| cdleme17d1 40488 | Part of proof of Lemma E i... |
| cdleme0nex 40489 | Part of proof of Lemma E i... |
| cdleme18a 40490 | Part of proof of Lemma E i... |
| cdleme18b 40491 | Part of proof of Lemma E i... |
| cdleme18c 40492 | Part of proof of Lemma E i... |
| cdleme22gb 40493 | Utility lemma for Lemma E ... |
| cdleme18d 40494 | Part of proof of Lemma E i... |
| cdlemesner 40495 | Part of proof of Lemma E i... |
| cdlemedb 40496 | Part of proof of Lemma E i... |
| cdlemeda 40497 | Part of proof of Lemma E i... |
| cdlemednpq 40498 | Part of proof of Lemma E i... |
| cdlemednuN 40499 | Part of proof of Lemma E i... |
| cdleme20zN 40500 | Part of proof of Lemma E i... |
| cdleme20y 40501 | Part of proof of Lemma E i... |
| cdleme19a 40502 | Part of proof of Lemma E i... |
| cdleme19b 40503 | Part of proof of Lemma E i... |
| cdleme19c 40504 | Part of proof of Lemma E i... |
| cdleme19d 40505 | Part of proof of Lemma E i... |
| cdleme19e 40506 | Part of proof of Lemma E i... |
| cdleme19f 40507 | Part of proof of Lemma E i... |
| cdleme20aN 40508 | Part of proof of Lemma E i... |
| cdleme20bN 40509 | Part of proof of Lemma E i... |
| cdleme20c 40510 | Part of proof of Lemma E i... |
| cdleme20d 40511 | Part of proof of Lemma E i... |
| cdleme20e 40512 | Part of proof of Lemma E i... |
| cdleme20f 40513 | Part of proof of Lemma E i... |
| cdleme20g 40514 | Part of proof of Lemma E i... |
| cdleme20h 40515 | Part of proof of Lemma E i... |
| cdleme20i 40516 | Part of proof of Lemma E i... |
| cdleme20j 40517 | Part of proof of Lemma E i... |
| cdleme20k 40518 | Part of proof of Lemma E i... |
| cdleme20l1 40519 | Part of proof of Lemma E i... |
| cdleme20l2 40520 | Part of proof of Lemma E i... |
| cdleme20l 40521 | Part of proof of Lemma E i... |
| cdleme20m 40522 | Part of proof of Lemma E i... |
| cdleme20 40523 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 40524 | Part of proof of Lemma E i... |
| cdleme21b 40525 | Part of proof of Lemma E i... |
| cdleme21c 40526 | Part of proof of Lemma E i... |
| cdleme21at 40527 | Part of proof of Lemma E i... |
| cdleme21ct 40528 | Part of proof of Lemma E i... |
| cdleme21d 40529 | Part of proof of Lemma E i... |
| cdleme21e 40530 | Part of proof of Lemma E i... |
| cdleme21f 40531 | Part of proof of Lemma E i... |
| cdleme21g 40532 | Part of proof of Lemma E i... |
| cdleme21h 40533 | Part of proof of Lemma E i... |
| cdleme21i 40534 | Part of proof of Lemma E i... |
| cdleme21j 40535 | Combine ~ cdleme20 and ~ c... |
| cdleme21 40536 | Part of proof of Lemma E i... |
| cdleme21k 40537 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 40538 | Part of proof of Lemma E i... |
| cdleme22a 40539 | Part of proof of Lemma E i... |
| cdleme22b 40540 | Part of proof of Lemma E i... |
| cdleme22cN 40541 | Part of proof of Lemma E i... |
| cdleme22d 40542 | Part of proof of Lemma E i... |
| cdleme22e 40543 | Part of proof of Lemma E i... |
| cdleme22eALTN 40544 | Part of proof of Lemma E i... |
| cdleme22f 40545 | Part of proof of Lemma E i... |
| cdleme22f2 40546 | Part of proof of Lemma E i... |
| cdleme22g 40547 | Part of proof of Lemma E i... |
| cdleme23a 40548 | Part of proof of Lemma E i... |
| cdleme23b 40549 | Part of proof of Lemma E i... |
| cdleme23c 40550 | Part of proof of Lemma E i... |
| cdleme24 40551 | Quantified version of ~ cd... |
| cdleme25a 40552 | Lemma for ~ cdleme25b . (... |
| cdleme25b 40553 | Transform ~ cdleme24 . TO... |
| cdleme25c 40554 | Transform ~ cdleme25b . (... |
| cdleme25dN 40555 | Transform ~ cdleme25c . (... |
| cdleme25cl 40556 | Show closure of the unique... |
| cdleme25cv 40557 | Change bound variables in ... |
| cdleme26e 40558 | Part of proof of Lemma E i... |
| cdleme26ee 40559 | Part of proof of Lemma E i... |
| cdleme26eALTN 40560 | Part of proof of Lemma E i... |
| cdleme26fALTN 40561 | Part of proof of Lemma E i... |
| cdleme26f 40562 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 40563 | Part of proof of Lemma E i... |
| cdleme26f2 40564 | Part of proof of Lemma E i... |
| cdleme27cl 40565 | Part of proof of Lemma E i... |
| cdleme27a 40566 | Part of proof of Lemma E i... |
| cdleme27b 40567 | Lemma for ~ cdleme27N . (... |
| cdleme27N 40568 | Part of proof of Lemma E i... |
| cdleme28a 40569 | Lemma for ~ cdleme25b . T... |
| cdleme28b 40570 | Lemma for ~ cdleme25b . T... |
| cdleme28c 40571 | Part of proof of Lemma E i... |
| cdleme28 40572 | Quantified version of ~ cd... |
| cdleme29ex 40573 | Lemma for ~ cdleme29b . (... |
| cdleme29b 40574 | Transform ~ cdleme28 . (C... |
| cdleme29c 40575 | Transform ~ cdleme28b . (... |
| cdleme29cl 40576 | Show closure of the unique... |
| cdleme30a 40577 | Part of proof of Lemma E i... |
| cdleme31so 40578 | Part of proof of Lemma E i... |
| cdleme31sn 40579 | Part of proof of Lemma E i... |
| cdleme31sn1 40580 | Part of proof of Lemma E i... |
| cdleme31se 40581 | Part of proof of Lemma D i... |
| cdleme31se2 40582 | Part of proof of Lemma D i... |
| cdleme31sc 40583 | Part of proof of Lemma E i... |
| cdleme31sde 40584 | Part of proof of Lemma D i... |
| cdleme31snd 40585 | Part of proof of Lemma D i... |
| cdleme31sdnN 40586 | Part of proof of Lemma E i... |
| cdleme31sn1c 40587 | Part of proof of Lemma E i... |
| cdleme31sn2 40588 | Part of proof of Lemma E i... |
| cdleme31fv 40589 | Part of proof of Lemma E i... |
| cdleme31fv1 40590 | Part of proof of Lemma E i... |
| cdleme31fv1s 40591 | Part of proof of Lemma E i... |
| cdleme31fv2 40592 | Part of proof of Lemma E i... |
| cdleme31id 40593 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 40594 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 40595 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 40596 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 40597 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 40598 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 40599 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 40600 | Part of proof of Lemma E i... |
| cdlemefr29exN 40601 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 40602 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 40603 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 40604 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 40605 | TODO fix comment. (Contri... |
| cdlemefr29clN 40606 | Show closure of the unique... |
| cdleme43frv1snN 40607 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 40608 | Part of proof of Lemma E i... |
| cdlemefr32fva1 40609 | Part of proof of Lemma E i... |
| cdlemefr31fv1 40610 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 40611 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 40612 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 40613 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 40614 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 40615 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 40616 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 40617 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 40618 | Show closure of the unique... |
| cdleme43fsv1snlem 40619 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 40620 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 40621 | Part of proof of Lemma E i... |
| cdlemefs32fva1 40622 | Part of proof of Lemma E i... |
| cdlemefs31fv1 40623 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 40624 | Value of f(r) when r is an... |
| cdlemefs44 40625 | Value of f_s(r) when r is ... |
| cdlemefr45 40626 | Value of f(r) when r is an... |
| cdlemefr45e 40627 | Explicit expansion of ~ cd... |
| cdlemefs45 40628 | Value of f_s(r) when r is ... |
| cdlemefs45ee 40629 | Explicit expansion of ~ cd... |
| cdlemefs45eN 40630 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 40631 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 40632 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 40633 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 40634 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 40635 | Show closure of ` [_ R / s... |
| cdleme32fva 40636 | Part of proof of Lemma D i... |
| cdleme32fva1 40637 | Part of proof of Lemma D i... |
| cdleme32fvaw 40638 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 40639 | Part of proof of Lemma D i... |
| cdleme32a 40640 | Part of proof of Lemma D i... |
| cdleme32b 40641 | Part of proof of Lemma D i... |
| cdleme32c 40642 | Part of proof of Lemma D i... |
| cdleme32d 40643 | Part of proof of Lemma D i... |
| cdleme32e 40644 | Part of proof of Lemma D i... |
| cdleme32f 40645 | Part of proof of Lemma D i... |
| cdleme32le 40646 | Part of proof of Lemma D i... |
| cdleme35a 40647 | Part of proof of Lemma E i... |
| cdleme35fnpq 40648 | Part of proof of Lemma E i... |
| cdleme35b 40649 | Part of proof of Lemma E i... |
| cdleme35c 40650 | Part of proof of Lemma E i... |
| cdleme35d 40651 | Part of proof of Lemma E i... |
| cdleme35e 40652 | Part of proof of Lemma E i... |
| cdleme35f 40653 | Part of proof of Lemma E i... |
| cdleme35g 40654 | Part of proof of Lemma E i... |
| cdleme35h 40655 | Part of proof of Lemma E i... |
| cdleme35h2 40656 | Part of proof of Lemma E i... |
| cdleme35sn2aw 40657 | Part of proof of Lemma E i... |
| cdleme35sn3a 40658 | Part of proof of Lemma E i... |
| cdleme36a 40659 | Part of proof of Lemma E i... |
| cdleme36m 40660 | Part of proof of Lemma E i... |
| cdleme37m 40661 | Part of proof of Lemma E i... |
| cdleme38m 40662 | Part of proof of Lemma E i... |
| cdleme38n 40663 | Part of proof of Lemma E i... |
| cdleme39a 40664 | Part of proof of Lemma E i... |
| cdleme39n 40665 | Part of proof of Lemma E i... |
| cdleme40m 40666 | Part of proof of Lemma E i... |
| cdleme40n 40667 | Part of proof of Lemma E i... |
| cdleme40v 40668 | Part of proof of Lemma E i... |
| cdleme40w 40669 | Part of proof of Lemma E i... |
| cdleme42a 40670 | Part of proof of Lemma E i... |
| cdleme42c 40671 | Part of proof of Lemma E i... |
| cdleme42d 40672 | Part of proof of Lemma E i... |
| cdleme41sn3aw 40673 | Part of proof of Lemma E i... |
| cdleme41sn4aw 40674 | Part of proof of Lemma E i... |
| cdleme41snaw 40675 | Part of proof of Lemma E i... |
| cdleme41fva11 40676 | Part of proof of Lemma E i... |
| cdleme42b 40677 | Part of proof of Lemma E i... |
| cdleme42e 40678 | Part of proof of Lemma E i... |
| cdleme42f 40679 | Part of proof of Lemma E i... |
| cdleme42g 40680 | Part of proof of Lemma E i... |
| cdleme42h 40681 | Part of proof of Lemma E i... |
| cdleme42i 40682 | Part of proof of Lemma E i... |
| cdleme42k 40683 | Part of proof of Lemma E i... |
| cdleme42ke 40684 | Part of proof of Lemma E i... |
| cdleme42keg 40685 | Part of proof of Lemma E i... |
| cdleme42mN 40686 | Part of proof of Lemma E i... |
| cdleme42mgN 40687 | Part of proof of Lemma E i... |
| cdleme43aN 40688 | Part of proof of Lemma E i... |
| cdleme43bN 40689 | Lemma for Lemma E in [Craw... |
| cdleme43cN 40690 | Part of proof of Lemma E i... |
| cdleme43dN 40691 | Part of proof of Lemma E i... |
| cdleme46f2g2 40692 | Conversion for ` G ` to re... |
| cdleme46f2g1 40693 | Conversion for ` G ` to re... |
| cdleme17d2 40694 | Part of proof of Lemma E i... |
| cdleme17d3 40695 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 40696 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 40697 | Part of proof of Lemma E i... |
| cdleme48fv 40698 | Part of proof of Lemma D i... |
| cdleme48fvg 40699 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 40700 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 40701 | TODO: fix comment. TODO: ... |
| cdleme48b 40702 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 40703 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 40704 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 40705 | TODO: fix comment. (Contr... |
| cdleme4gfv 40706 | Part of proof of Lemma D i... |
| cdlemeg47b 40707 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 40708 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 40709 | Value of g_s(r) when r is ... |
| cdlemeg49le 40710 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 40711 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 40712 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 40713 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 40714 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 40715 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 40716 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 40717 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 40718 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 40719 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 40720 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 40721 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 40722 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 40723 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 40724 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 40725 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 40726 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 40727 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 40728 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 40729 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 40730 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 40731 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 40732 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 40733 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 40734 | TODO: fix comment. (Contr... |
| cdleme48gfv1 40735 | TODO: fix comment. (Contr... |
| cdleme48gfv 40736 | TODO: fix comment. (Contr... |
| cdleme48fgv 40737 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 40738 | Part of proof of Lemma D i... |
| cdleme50lebi 40739 | Part of proof of Lemma D i... |
| cdleme50eq 40740 | Part of proof of Lemma D i... |
| cdleme50f 40741 | Part of proof of Lemma D i... |
| cdleme50f1 40742 | Part of proof of Lemma D i... |
| cdleme50rnlem 40743 | Part of proof of Lemma D i... |
| cdleme50rn 40744 | Part of proof of Lemma D i... |
| cdleme50f1o 40745 | Part of proof of Lemma D i... |
| cdleme50laut 40746 | Part of proof of Lemma D i... |
| cdleme50ldil 40747 | Part of proof of Lemma D i... |
| cdleme50trn1 40748 | Part of proof that ` F ` i... |
| cdleme50trn2a 40749 | Part of proof that ` F ` i... |
| cdleme50trn2 40750 | Part of proof that ` F ` i... |
| cdleme50trn12 40751 | Part of proof that ` F ` i... |
| cdleme50trn3 40752 | Part of proof that ` F ` i... |
| cdleme50trn123 40753 | Part of proof that ` F ` i... |
| cdleme51finvfvN 40754 | Part of proof of Lemma E i... |
| cdleme51finvN 40755 | Part of proof of Lemma E i... |
| cdleme50ltrn 40756 | Part of proof of Lemma E i... |
| cdleme51finvtrN 40757 | Part of proof of Lemma E i... |
| cdleme50ex 40758 | Part of Lemma E in [Crawle... |
| cdleme 40759 | Lemma E in [Crawley] p. 11... |
| cdlemf1 40760 | Part of Lemma F in [Crawle... |
| cdlemf2 40761 | Part of Lemma F in [Crawle... |
| cdlemf 40762 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 40763 | ~ cdlemf with additional c... |
| cdlemftr3 40764 | Special case of ~ cdlemf s... |
| cdlemftr2 40765 | Special case of ~ cdlemf s... |
| cdlemftr1 40766 | Part of proof of Lemma G o... |
| cdlemftr0 40767 | Special case of ~ cdlemf s... |
| trlord 40768 | The ordering of two Hilber... |
| cdlemg1a 40769 | Shorter expression for ` G... |
| cdlemg1b2 40770 | This theorem can be used t... |
| cdlemg1idlemN 40771 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 40772 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 40773 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 40774 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 40775 | This theorem can be used t... |
| cdlemg1idN 40776 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 40777 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 40778 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 40779 | Version of ~ cdleme51finvt... |
| ltrniotaval 40780 | Value of the unique transl... |
| ltrniotacnvval 40781 | Converse value of the uniq... |
| ltrniotaidvalN 40782 | Value of the unique transl... |
| ltrniotavalbN 40783 | Value of the unique transl... |
| cdlemeiota 40784 | A translation is uniquely ... |
| cdlemg1ci2 40785 | Any function of the form o... |
| cdlemg1cN 40786 | Any translation belongs to... |
| cdlemg1cex 40787 | Any translation is one of ... |
| cdlemg2cN 40788 | Any translation belongs to... |
| cdlemg2dN 40789 | This theorem can be used t... |
| cdlemg2cex 40790 | Any translation is one of ... |
| cdlemg2ce 40791 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 40792 | Part of proof of Lemma E i... |
| cdlemg2fvlem 40793 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 40794 | ~ cdleme42keg with simpler... |
| cdlemg2idN 40795 | Version of ~ cdleme31id wi... |
| cdlemg3a 40796 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 40797 | TODO: Replace this with ~... |
| cdlemg2fv 40798 | Value of a translation in ... |
| cdlemg2fv2 40799 | Value of a translation in ... |
| cdlemg2k 40800 | ~ cdleme42keg with simpler... |
| cdlemg2kq 40801 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 40802 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 40803 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 40804 | TODO: Is there a simpler ... |
| cdlemb3 40805 | Given two atoms not under ... |
| cdlemg7fvbwN 40806 | Properties of a translatio... |
| cdlemg4a 40807 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 40808 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 40809 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 40810 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 40811 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 40812 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 40813 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 40814 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 40815 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 40816 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 40817 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 40818 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 40819 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 40820 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 40821 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 40822 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 40823 | Value of a translation com... |
| cdlemg7aN 40824 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 40825 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 40826 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 40827 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 40828 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 40829 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 40830 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 40831 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 40832 | The triples ` <. P , ( F `... |
| cdlemg9 40833 | The triples ` <. P , ( F `... |
| cdlemg10b 40834 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 40835 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 40836 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 40837 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 40838 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 40839 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 40840 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 40841 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 40842 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 40843 | The triples ` <. P , ( F `... |
| cdlemg12c 40844 | The triples ` <. P , ( F `... |
| cdlemg12d 40845 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 40846 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 40847 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 40848 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 40849 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 40850 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 40851 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 40852 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 40853 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 40854 | Eliminate the ` ( F `` P )... |
| cdlemg15 40855 | Eliminate the ` ( (... |
| cdlemg16 40856 | Part of proof of Lemma G o... |
| cdlemg16ALTN 40857 | This version of ~ cdlemg16... |
| cdlemg16z 40858 | Eliminate ` ( ( F `... |
| cdlemg16zz 40859 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 40860 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 40861 | Part of proof of Lemma G i... |
| cdlemg17dN 40862 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 40863 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 40864 | TODO: fix comment. (Contr... |
| cdlemg17f 40865 | TODO: fix comment. (Contr... |
| cdlemg17g 40866 | TODO: fix comment. (Contr... |
| cdlemg17h 40867 | TODO: fix comment. (Contr... |
| cdlemg17i 40868 | TODO: fix comment. (Contr... |
| cdlemg17ir 40869 | TODO: fix comment. (Contr... |
| cdlemg17j 40870 | TODO: fix comment. (Contr... |
| cdlemg17pq 40871 | Utility theorem for swappi... |
| cdlemg17bq 40872 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 40873 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 40874 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 40875 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 40876 | Part of Lemma G of [Crawle... |
| cdlemg18a 40877 | Show two lines are differe... |
| cdlemg18b 40878 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 40879 | Show two lines intersect a... |
| cdlemg18d 40880 | Show two lines intersect a... |
| cdlemg18 40881 | Show two lines intersect a... |
| cdlemg19a 40882 | Show two lines intersect a... |
| cdlemg19 40883 | Show two lines intersect a... |
| cdlemg20 40884 | Show two lines intersect a... |
| cdlemg21 40885 | Version of cdlemg19 with `... |
| cdlemg22 40886 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 40887 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 40888 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 40889 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 40890 | ~ cdlemg16zz restated for ... |
| cdlemg27a 40891 | For use with case when ` (... |
| cdlemg28a 40892 | Part of proof of Lemma G o... |
| cdlemg31b0N 40893 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 40894 | TODO: Fix comment. (Cont... |
| cdlemg27b 40895 | TODO: Fix comment. (Cont... |
| cdlemg31a 40896 | TODO: fix comment. (Contr... |
| cdlemg31b 40897 | TODO: fix comment. (Contr... |
| cdlemg31c 40898 | Show that when ` N ` is an... |
| cdlemg31d 40899 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 40900 | TODO: Fix comment. (Cont... |
| cdlemg33c0 40901 | TODO: Fix comment. (Cont... |
| cdlemg28b 40902 | Part of proof of Lemma G o... |
| cdlemg28 40903 | Part of proof of Lemma G o... |
| cdlemg29 40904 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 40905 | TODO: Fix comment. (Cont... |
| cdlemg33b 40906 | TODO: Fix comment. (Cont... |
| cdlemg33c 40907 | TODO: Fix comment. (Cont... |
| cdlemg33d 40908 | TODO: Fix comment. (Cont... |
| cdlemg33e 40909 | TODO: Fix comment. (Cont... |
| cdlemg33 40910 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 40911 | Use cdlemg33 to eliminate ... |
| cdlemg35 40912 | TODO: Fix comment. TODO:... |
| cdlemg36 40913 | Use cdlemg35 to eliminate ... |
| cdlemg38 40914 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 40915 | Eliminate ` =/= ` conditio... |
| cdlemg40 40916 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 40917 | Convert ~ cdlemg40 to func... |
| ltrnco 40918 | The composition of two tra... |
| trlcocnv 40919 | Swap the arguments of the ... |
| trlcoabs 40920 | Absorption into a composit... |
| trlcoabs2N 40921 | Absorption of the trace of... |
| trlcoat 40922 | The trace of a composition... |
| trlcocnvat 40923 | Commonly used special case... |
| trlconid 40924 | The composition of two dif... |
| trlcolem 40925 | Lemma for ~ trlco . (Cont... |
| trlco 40926 | The trace of a composition... |
| trlcone 40927 | If two translations have d... |
| cdlemg42 40928 | Part of proof of Lemma G o... |
| cdlemg43 40929 | Part of proof of Lemma G o... |
| cdlemg44a 40930 | Part of proof of Lemma G o... |
| cdlemg44b 40931 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 40932 | Part of proof of Lemma G o... |
| cdlemg47a 40933 | TODO: fix comment. TODO: ... |
| cdlemg46 40934 | Part of proof of Lemma G o... |
| cdlemg47 40935 | Part of proof of Lemma G o... |
| cdlemg48 40936 | Eliminate ` h ` from ~ cdl... |
| ltrncom 40937 | Composition is commutative... |
| ltrnco4 40938 | Rearrange a composition of... |
| trljco 40939 | Trace joined with trace of... |
| trljco2 40940 | Trace joined with trace of... |
| tgrpfset 40943 | The translation group maps... |
| tgrpset 40944 | The translation group for ... |
| tgrpbase 40945 | The base set of the transl... |
| tgrpopr 40946 | The group operation of the... |
| tgrpov 40947 | The group operation value ... |
| tgrpgrplem 40948 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 40949 | The translation group is a... |
| tgrpabl 40950 | The translation group is a... |
| tendofset 40957 | The set of all trace-prese... |
| tendoset 40958 | The set of trace-preservin... |
| istendo 40959 | The predicate "is a trace-... |
| tendotp 40960 | Trace-preserving property ... |
| istendod 40961 | Deduce the predicate "is a... |
| tendof 40962 | Functionality of a trace-p... |
| tendoeq1 40963 | Condition determining equa... |
| tendovalco 40964 | Value of composition of tr... |
| tendocoval 40965 | Value of composition of en... |
| tendocl 40966 | Closure of a trace-preserv... |
| tendoco2 40967 | Distribution of compositio... |
| tendoidcl 40968 | The identity is a trace-pr... |
| tendo1mul 40969 | Multiplicative identity mu... |
| tendo1mulr 40970 | Multiplicative identity mu... |
| tendococl 40971 | The composition of two tra... |
| tendoid 40972 | The identity value of a tr... |
| tendoeq2 40973 | Condition determining equa... |
| tendoplcbv 40974 | Define sum operation for t... |
| tendopl 40975 | Value of endomorphism sum ... |
| tendopl2 40976 | Value of result of endomor... |
| tendoplcl2 40977 | Value of result of endomor... |
| tendoplco2 40978 | Value of result of endomor... |
| tendopltp 40979 | Trace-preserving property ... |
| tendoplcl 40980 | Endomorphism sum is a trac... |
| tendoplcom 40981 | The endomorphism sum opera... |
| tendoplass 40982 | The endomorphism sum opera... |
| tendodi1 40983 | Endomorphism composition d... |
| tendodi2 40984 | Endomorphism composition d... |
| tendo0cbv 40985 | Define additive identity f... |
| tendo02 40986 | Value of additive identity... |
| tendo0co2 40987 | The additive identity trac... |
| tendo0tp 40988 | Trace-preserving property ... |
| tendo0cl 40989 | The additive identity is a... |
| tendo0pl 40990 | Property of the additive i... |
| tendo0plr 40991 | Property of the additive i... |
| tendoicbv 40992 | Define inverse function fo... |
| tendoi 40993 | Value of inverse endomorph... |
| tendoi2 40994 | Value of additive inverse ... |
| tendoicl 40995 | Closure of the additive in... |
| tendoipl 40996 | Property of the additive i... |
| tendoipl2 40997 | Property of the additive i... |
| erngfset 40998 | The division rings on trac... |
| erngset 40999 | The division ring on trace... |
| erngbase 41000 | The base set of the divisi... |
| erngfplus 41001 | Ring addition operation. ... |
| erngplus 41002 | Ring addition operation. ... |
| erngplus2 41003 | Ring addition operation. ... |
| erngfmul 41004 | Ring multiplication operat... |
| erngmul 41005 | Ring addition operation. ... |
| erngfset-rN 41006 | The division rings on trac... |
| erngset-rN 41007 | The division ring on trace... |
| erngbase-rN 41008 | The base set of the divisi... |
| erngfplus-rN 41009 | Ring addition operation. ... |
| erngplus-rN 41010 | Ring addition operation. ... |
| erngplus2-rN 41011 | Ring addition operation. ... |
| erngfmul-rN 41012 | Ring multiplication operat... |
| erngmul-rN 41013 | Ring addition operation. ... |
| cdlemh1 41014 | Part of proof of Lemma H o... |
| cdlemh2 41015 | Part of proof of Lemma H o... |
| cdlemh 41016 | Lemma H of [Crawley] p. 11... |
| cdlemi1 41017 | Part of proof of Lemma I o... |
| cdlemi2 41018 | Part of proof of Lemma I o... |
| cdlemi 41019 | Lemma I of [Crawley] p. 11... |
| cdlemj1 41020 | Part of proof of Lemma J o... |
| cdlemj2 41021 | Part of proof of Lemma J o... |
| cdlemj3 41022 | Part of proof of Lemma J o... |
| tendocan 41023 | Cancellation law: if the v... |
| tendoid0 41024 | A trace-preserving endomor... |
| tendo0mul 41025 | Additive identity multipli... |
| tendo0mulr 41026 | Additive identity multipli... |
| tendo1ne0 41027 | The identity (unity) is no... |
| tendoconid 41028 | The composition (product) ... |
| tendotr 41029 | The trace of the value of ... |
| cdlemk1 41030 | Part of proof of Lemma K o... |
| cdlemk2 41031 | Part of proof of Lemma K o... |
| cdlemk3 41032 | Part of proof of Lemma K o... |
| cdlemk4 41033 | Part of proof of Lemma K o... |
| cdlemk5a 41034 | Part of proof of Lemma K o... |
| cdlemk5 41035 | Part of proof of Lemma K o... |
| cdlemk6 41036 | Part of proof of Lemma K o... |
| cdlemk8 41037 | Part of proof of Lemma K o... |
| cdlemk9 41038 | Part of proof of Lemma K o... |
| cdlemk9bN 41039 | Part of proof of Lemma K o... |
| cdlemki 41040 | Part of proof of Lemma K o... |
| cdlemkvcl 41041 | Part of proof of Lemma K o... |
| cdlemk10 41042 | Part of proof of Lemma K o... |
| cdlemksv 41043 | Part of proof of Lemma K o... |
| cdlemksel 41044 | Part of proof of Lemma K o... |
| cdlemksat 41045 | Part of proof of Lemma K o... |
| cdlemksv2 41046 | Part of proof of Lemma K o... |
| cdlemk7 41047 | Part of proof of Lemma K o... |
| cdlemk11 41048 | Part of proof of Lemma K o... |
| cdlemk12 41049 | Part of proof of Lemma K o... |
| cdlemkoatnle 41050 | Utility lemma. (Contribut... |
| cdlemk13 41051 | Part of proof of Lemma K o... |
| cdlemkole 41052 | Utility lemma. (Contribut... |
| cdlemk14 41053 | Part of proof of Lemma K o... |
| cdlemk15 41054 | Part of proof of Lemma K o... |
| cdlemk16a 41055 | Part of proof of Lemma K o... |
| cdlemk16 41056 | Part of proof of Lemma K o... |
| cdlemk17 41057 | Part of proof of Lemma K o... |
| cdlemk1u 41058 | Part of proof of Lemma K o... |
| cdlemk5auN 41059 | Part of proof of Lemma K o... |
| cdlemk5u 41060 | Part of proof of Lemma K o... |
| cdlemk6u 41061 | Part of proof of Lemma K o... |
| cdlemkj 41062 | Part of proof of Lemma K o... |
| cdlemkuvN 41063 | Part of proof of Lemma K o... |
| cdlemkuel 41064 | Part of proof of Lemma K o... |
| cdlemkuat 41065 | Part of proof of Lemma K o... |
| cdlemkuv2 41066 | Part of proof of Lemma K o... |
| cdlemk18 41067 | Part of proof of Lemma K o... |
| cdlemk19 41068 | Part of proof of Lemma K o... |
| cdlemk7u 41069 | Part of proof of Lemma K o... |
| cdlemk11u 41070 | Part of proof of Lemma K o... |
| cdlemk12u 41071 | Part of proof of Lemma K o... |
| cdlemk21N 41072 | Part of proof of Lemma K o... |
| cdlemk20 41073 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 41074 | Utility lemma. (Contribut... |
| cdlemk13-2N 41075 | Part of proof of Lemma K o... |
| cdlemkole-2N 41076 | Utility lemma. (Contribut... |
| cdlemk14-2N 41077 | Part of proof of Lemma K o... |
| cdlemk15-2N 41078 | Part of proof of Lemma K o... |
| cdlemk16-2N 41079 | Part of proof of Lemma K o... |
| cdlemk17-2N 41080 | Part of proof of Lemma K o... |
| cdlemkj-2N 41081 | Part of proof of Lemma K o... |
| cdlemkuv-2N 41082 | Part of proof of Lemma K o... |
| cdlemkuel-2N 41083 | Part of proof of Lemma K o... |
| cdlemkuv2-2 41084 | Part of proof of Lemma K o... |
| cdlemk18-2N 41085 | Part of proof of Lemma K o... |
| cdlemk19-2N 41086 | Part of proof of Lemma K o... |
| cdlemk7u-2N 41087 | Part of proof of Lemma K o... |
| cdlemk11u-2N 41088 | Part of proof of Lemma K o... |
| cdlemk12u-2N 41089 | Part of proof of Lemma K o... |
| cdlemk21-2N 41090 | Part of proof of Lemma K o... |
| cdlemk20-2N 41091 | Part of proof of Lemma K o... |
| cdlemk22 41092 | Part of proof of Lemma K o... |
| cdlemk30 41093 | Part of proof of Lemma K o... |
| cdlemkuu 41094 | Convert between function a... |
| cdlemk31 41095 | Part of proof of Lemma K o... |
| cdlemk32 41096 | Part of proof of Lemma K o... |
| cdlemkuel-3 41097 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 41098 | Part of proof of Lemma K o... |
| cdlemk18-3N 41099 | Part of proof of Lemma K o... |
| cdlemk22-3 41100 | Part of proof of Lemma K o... |
| cdlemk23-3 41101 | Part of proof of Lemma K o... |
| cdlemk24-3 41102 | Part of proof of Lemma K o... |
| cdlemk25-3 41103 | Part of proof of Lemma K o... |
| cdlemk26b-3 41104 | Part of proof of Lemma K o... |
| cdlemk26-3 41105 | Part of proof of Lemma K o... |
| cdlemk27-3 41106 | Part of proof of Lemma K o... |
| cdlemk28-3 41107 | Part of proof of Lemma K o... |
| cdlemk33N 41108 | Part of proof of Lemma K o... |
| cdlemk34 41109 | Part of proof of Lemma K o... |
| cdlemk29-3 41110 | Part of proof of Lemma K o... |
| cdlemk35 41111 | Part of proof of Lemma K o... |
| cdlemk36 41112 | Part of proof of Lemma K o... |
| cdlemk37 41113 | Part of proof of Lemma K o... |
| cdlemk38 41114 | Part of proof of Lemma K o... |
| cdlemk39 41115 | Part of proof of Lemma K o... |
| cdlemk40 41116 | TODO: fix comment. (Contr... |
| cdlemk40t 41117 | TODO: fix comment. (Contr... |
| cdlemk40f 41118 | TODO: fix comment. (Contr... |
| cdlemk41 41119 | Part of proof of Lemma K o... |
| cdlemkfid1N 41120 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 41121 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 41122 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 41123 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 41124 | TODO: is this useful or sh... |
| cdlemky 41125 | Part of proof of Lemma K o... |
| cdlemkyu 41126 | Convert between function a... |
| cdlemkyuu 41127 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 41128 | Part of proof of Lemma K o... |
| cdlemk19ylem 41129 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 41130 | Part of proof of Lemma K o... |
| cdlemk19y 41131 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 41132 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 41133 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 41134 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 41135 | The value of the tau funct... |
| cdlemk35s 41136 | Substitution version of ~ ... |
| cdlemk35s-id 41137 | Substitution version of ~ ... |
| cdlemk39s 41138 | Substitution version of ~ ... |
| cdlemk39s-id 41139 | Substitution version of ~ ... |
| cdlemk42 41140 | Part of proof of Lemma K o... |
| cdlemk19xlem 41141 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 41142 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 41143 | Part of proof of Lemma K o... |
| cdlemk11tc 41144 | Part of proof of Lemma K o... |
| cdlemk11t 41145 | Part of proof of Lemma K o... |
| cdlemk45 41146 | Part of proof of Lemma K o... |
| cdlemk46 41147 | Part of proof of Lemma K o... |
| cdlemk47 41148 | Part of proof of Lemma K o... |
| cdlemk48 41149 | Part of proof of Lemma K o... |
| cdlemk49 41150 | Part of proof of Lemma K o... |
| cdlemk50 41151 | Part of proof of Lemma K o... |
| cdlemk51 41152 | Part of proof of Lemma K o... |
| cdlemk52 41153 | Part of proof of Lemma K o... |
| cdlemk53a 41154 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 41155 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 41156 | Part of proof of Lemma K o... |
| cdlemk54 41157 | Part of proof of Lemma K o... |
| cdlemk55a 41158 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 41159 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 41160 | Part of proof of Lemma K o... |
| cdlemkyyN 41161 | Part of proof of Lemma K o... |
| cdlemk43N 41162 | Part of proof of Lemma K o... |
| cdlemk35u 41163 | Substitution version of ~ ... |
| cdlemk55u1 41164 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 41165 | Part of proof of Lemma K o... |
| cdlemk39u1 41166 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 41167 | Part of proof of Lemma K o... |
| cdlemk19u1 41168 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 41169 | Part of Lemma K of [Crawle... |
| cdlemk56 41170 | Part of Lemma K of [Crawle... |
| cdlemk19w 41171 | Use a fixed element to eli... |
| cdlemk56w 41172 | Use a fixed element to eli... |
| cdlemk 41173 | Lemma K of [Crawley] p. 11... |
| tendoex 41174 | Generalization of Lemma K ... |
| cdleml1N 41175 | Part of proof of Lemma L o... |
| cdleml2N 41176 | Part of proof of Lemma L o... |
| cdleml3N 41177 | Part of proof of Lemma L o... |
| cdleml4N 41178 | Part of proof of Lemma L o... |
| cdleml5N 41179 | Part of proof of Lemma L o... |
| cdleml6 41180 | Part of proof of Lemma L o... |
| cdleml7 41181 | Part of proof of Lemma L o... |
| cdleml8 41182 | Part of proof of Lemma L o... |
| cdleml9 41183 | Part of proof of Lemma L o... |
| dva1dim 41184 | Two expressions for the 1-... |
| dvhb1dimN 41185 | Two expressions for the 1-... |
| erng1lem 41186 | Value of the endomorphism ... |
| erngdvlem1 41187 | Lemma for ~ eringring . (... |
| erngdvlem2N 41188 | Lemma for ~ eringring . (... |
| erngdvlem3 41189 | Lemma for ~ eringring . (... |
| erngdvlem4 41190 | Lemma for ~ erngdv . (Con... |
| eringring 41191 | An endomorphism ring is a ... |
| erngdv 41192 | An endomorphism ring is a ... |
| erng0g 41193 | The division ring zero of ... |
| erng1r 41194 | The division ring unity of... |
| erngdvlem1-rN 41195 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 41196 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 41197 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 41198 | Lemma for ~ erngdv . (Con... |
| erngring-rN 41199 | An endomorphism ring is a ... |
| erngdv-rN 41200 | An endomorphism ring is a ... |
| dvafset 41203 | The constructed partial ve... |
| dvaset 41204 | The constructed partial ve... |
| dvasca 41205 | The ring base set of the c... |
| dvabase 41206 | The ring base set of the c... |
| dvafplusg 41207 | Ring addition operation fo... |
| dvaplusg 41208 | Ring addition operation fo... |
| dvaplusgv 41209 | Ring addition operation fo... |
| dvafmulr 41210 | Ring multiplication operat... |
| dvamulr 41211 | Ring multiplication operat... |
| dvavbase 41212 | The vectors (vector base s... |
| dvafvadd 41213 | The vector sum operation f... |
| dvavadd 41214 | Ring addition operation fo... |
| dvafvsca 41215 | Ring addition operation fo... |
| dvavsca 41216 | Ring addition operation fo... |
| tendospcl 41217 | Closure of endomorphism sc... |
| tendospass 41218 | Associative law for endomo... |
| tendospdi1 41219 | Forward distributive law f... |
| tendocnv 41220 | Converse of a trace-preser... |
| tendospdi2 41221 | Reverse distributive law f... |
| tendospcanN 41222 | Cancellation law for trace... |
| dvaabl 41223 | The constructed partial ve... |
| dvalveclem 41224 | Lemma for ~ dvalvec . (Co... |
| dvalvec 41225 | The constructed partial ve... |
| dva0g 41226 | The zero vector of partial... |
| diaffval 41229 | The partial isomorphism A ... |
| diafval 41230 | The partial isomorphism A ... |
| diaval 41231 | The partial isomorphism A ... |
| diaelval 41232 | Member of the partial isom... |
| diafn 41233 | Functionality and domain o... |
| diadm 41234 | Domain of the partial isom... |
| diaeldm 41235 | Member of domain of the pa... |
| diadmclN 41236 | A member of domain of the ... |
| diadmleN 41237 | A member of domain of the ... |
| dian0 41238 | The value of the partial i... |
| dia0eldmN 41239 | The lattice zero belongs t... |
| dia1eldmN 41240 | The fiducial hyperplane (t... |
| diass 41241 | The value of the partial i... |
| diael 41242 | A member of the value of t... |
| diatrl 41243 | Trace of a member of the p... |
| diaelrnN 41244 | Any value of the partial i... |
| dialss 41245 | The value of partial isomo... |
| diaord 41246 | The partial isomorphism A ... |
| dia11N 41247 | The partial isomorphism A ... |
| diaf11N 41248 | The partial isomorphism A ... |
| diaclN 41249 | Closure of partial isomorp... |
| diacnvclN 41250 | Closure of partial isomorp... |
| dia0 41251 | The value of the partial i... |
| dia1N 41252 | The value of the partial i... |
| dia1elN 41253 | The largest subspace in th... |
| diaglbN 41254 | Partial isomorphism A of a... |
| diameetN 41255 | Partial isomorphism A of a... |
| diainN 41256 | Inverse partial isomorphis... |
| diaintclN 41257 | The intersection of partia... |
| diasslssN 41258 | The partial isomorphism A ... |
| diassdvaN 41259 | The partial isomorphism A ... |
| dia1dim 41260 | Two expressions for the 1-... |
| dia1dim2 41261 | Two expressions for a 1-di... |
| dia1dimid 41262 | A vector (translation) bel... |
| dia2dimlem1 41263 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 41264 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 41265 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 41266 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 41267 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 41268 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 41269 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 41270 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 41271 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 41272 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 41273 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 41274 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 41275 | Lemma for ~ dia2dim . Eli... |
| dia2dim 41276 | A two-dimensional subspace... |
| dvhfset 41279 | The constructed full vecto... |
| dvhset 41280 | The constructed full vecto... |
| dvhsca 41281 | The ring of scalars of the... |
| dvhbase 41282 | The ring base set of the c... |
| dvhfplusr 41283 | Ring addition operation fo... |
| dvhfmulr 41284 | Ring multiplication operat... |
| dvhmulr 41285 | Ring multiplication operat... |
| dvhvbase 41286 | The vectors (vector base s... |
| dvhelvbasei 41287 | Vector membership in the c... |
| dvhvaddcbv 41288 | Change bound variables to ... |
| dvhvaddval 41289 | The vector sum operation f... |
| dvhfvadd 41290 | The vector sum operation f... |
| dvhvadd 41291 | The vector sum operation f... |
| dvhopvadd 41292 | The vector sum operation f... |
| dvhopvadd2 41293 | The vector sum operation f... |
| dvhvaddcl 41294 | Closure of the vector sum ... |
| dvhvaddcomN 41295 | Commutativity of vector su... |
| dvhvaddass 41296 | Associativity of vector su... |
| dvhvscacbv 41297 | Change bound variables to ... |
| dvhvscaval 41298 | The scalar product operati... |
| dvhfvsca 41299 | Scalar product operation f... |
| dvhvsca 41300 | Scalar product operation f... |
| dvhopvsca 41301 | Scalar product operation f... |
| dvhvscacl 41302 | Closure of the scalar prod... |
| tendoinvcl 41303 | Closure of multiplicative ... |
| tendolinv 41304 | Left multiplicative invers... |
| tendorinv 41305 | Right multiplicative inver... |
| dvhgrp 41306 | The full vector space ` U ... |
| dvhlveclem 41307 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 41308 | The full vector space ` U ... |
| dvhlmod 41309 | The full vector space ` U ... |
| dvh0g 41310 | The zero vector of vector ... |
| dvheveccl 41311 | Properties of a unit vecto... |
| dvhopclN 41312 | Closure of a ` DVecH ` vec... |
| dvhopaddN 41313 | Sum of ` DVecH ` vectors e... |
| dvhopspN 41314 | Scalar product of ` DVecH ... |
| dvhopN 41315 | Decompose a ` DVecH ` vect... |
| dvhopellsm 41316 | Ordered pair membership in... |
| cdlemm10N 41317 | The image of the map ` G `... |
| docaffvalN 41320 | Subspace orthocomplement f... |
| docafvalN 41321 | Subspace orthocomplement f... |
| docavalN 41322 | Subspace orthocomplement f... |
| docaclN 41323 | Closure of subspace orthoc... |
| diaocN 41324 | Value of partial isomorphi... |
| doca2N 41325 | Double orthocomplement of ... |
| doca3N 41326 | Double orthocomplement of ... |
| dvadiaN 41327 | Any closed subspace is a m... |
| diarnN 41328 | Partial isomorphism A maps... |
| diaf1oN 41329 | The partial isomorphism A ... |
| djaffvalN 41332 | Subspace join for ` DVecA ... |
| djafvalN 41333 | Subspace join for ` DVecA ... |
| djavalN 41334 | Subspace join for ` DVecA ... |
| djaclN 41335 | Closure of subspace join f... |
| djajN 41336 | Transfer lattice join to `... |
| dibffval 41339 | The partial isomorphism B ... |
| dibfval 41340 | The partial isomorphism B ... |
| dibval 41341 | The partial isomorphism B ... |
| dibopelvalN 41342 | Member of the partial isom... |
| dibval2 41343 | Value of the partial isomo... |
| dibopelval2 41344 | Member of the partial isom... |
| dibval3N 41345 | Value of the partial isomo... |
| dibelval3 41346 | Member of the partial isom... |
| dibopelval3 41347 | Member of the partial isom... |
| dibelval1st 41348 | Membership in value of the... |
| dibelval1st1 41349 | Membership in value of the... |
| dibelval1st2N 41350 | Membership in value of the... |
| dibelval2nd 41351 | Membership in value of the... |
| dibn0 41352 | The value of the partial i... |
| dibfna 41353 | Functionality and domain o... |
| dibdiadm 41354 | Domain of the partial isom... |
| dibfnN 41355 | Functionality and domain o... |
| dibdmN 41356 | Domain of the partial isom... |
| dibeldmN 41357 | Member of domain of the pa... |
| dibord 41358 | The isomorphism B for a la... |
| dib11N 41359 | The isomorphism B for a la... |
| dibf11N 41360 | The partial isomorphism A ... |
| dibclN 41361 | Closure of partial isomorp... |
| dibvalrel 41362 | The value of partial isomo... |
| dib0 41363 | The value of partial isomo... |
| dib1dim 41364 | Two expressions for the 1-... |
| dibglbN 41365 | Partial isomorphism B of a... |
| dibintclN 41366 | The intersection of partia... |
| dib1dim2 41367 | Two expressions for a 1-di... |
| dibss 41368 | The partial isomorphism B ... |
| diblss 41369 | The value of partial isomo... |
| diblsmopel 41370 | Membership in subspace sum... |
| dicffval 41373 | The partial isomorphism C ... |
| dicfval 41374 | The partial isomorphism C ... |
| dicval 41375 | The partial isomorphism C ... |
| dicopelval 41376 | Membership in value of the... |
| dicelvalN 41377 | Membership in value of the... |
| dicval2 41378 | The partial isomorphism C ... |
| dicelval3 41379 | Member of the partial isom... |
| dicopelval2 41380 | Membership in value of the... |
| dicelval2N 41381 | Membership in value of the... |
| dicfnN 41382 | Functionality and domain o... |
| dicdmN 41383 | Domain of the partial isom... |
| dicvalrelN 41384 | The value of partial isomo... |
| dicssdvh 41385 | The partial isomorphism C ... |
| dicelval1sta 41386 | Membership in value of the... |
| dicelval1stN 41387 | Membership in value of the... |
| dicelval2nd 41388 | Membership in value of the... |
| dicvaddcl 41389 | Membership in value of the... |
| dicvscacl 41390 | Membership in value of the... |
| dicn0 41391 | The value of the partial i... |
| diclss 41392 | The value of partial isomo... |
| diclspsn 41393 | The value of isomorphism C... |
| cdlemn2 41394 | Part of proof of Lemma N o... |
| cdlemn2a 41395 | Part of proof of Lemma N o... |
| cdlemn3 41396 | Part of proof of Lemma N o... |
| cdlemn4 41397 | Part of proof of Lemma N o... |
| cdlemn4a 41398 | Part of proof of Lemma N o... |
| cdlemn5pre 41399 | Part of proof of Lemma N o... |
| cdlemn5 41400 | Part of proof of Lemma N o... |
| cdlemn6 41401 | Part of proof of Lemma N o... |
| cdlemn7 41402 | Part of proof of Lemma N o... |
| cdlemn8 41403 | Part of proof of Lemma N o... |
| cdlemn9 41404 | Part of proof of Lemma N o... |
| cdlemn10 41405 | Part of proof of Lemma N o... |
| cdlemn11a 41406 | Part of proof of Lemma N o... |
| cdlemn11b 41407 | Part of proof of Lemma N o... |
| cdlemn11c 41408 | Part of proof of Lemma N o... |
| cdlemn11pre 41409 | Part of proof of Lemma N o... |
| cdlemn11 41410 | Part of proof of Lemma N o... |
| cdlemn 41411 | Lemma N of [Crawley] p. 12... |
| dihordlem6 41412 | Part of proof of Lemma N o... |
| dihordlem7 41413 | Part of proof of Lemma N o... |
| dihordlem7b 41414 | Part of proof of Lemma N o... |
| dihjustlem 41415 | Part of proof after Lemma ... |
| dihjust 41416 | Part of proof after Lemma ... |
| dihord1 41417 | Part of proof after Lemma ... |
| dihord2a 41418 | Part of proof after Lemma ... |
| dihord2b 41419 | Part of proof after Lemma ... |
| dihord2cN 41420 | Part of proof after Lemma ... |
| dihord11b 41421 | Part of proof after Lemma ... |
| dihord10 41422 | Part of proof after Lemma ... |
| dihord11c 41423 | Part of proof after Lemma ... |
| dihord2pre 41424 | Part of proof after Lemma ... |
| dihord2pre2 41425 | Part of proof after Lemma ... |
| dihord2 41426 | Part of proof after Lemma ... |
| dihffval 41429 | The isomorphism H for a la... |
| dihfval 41430 | Isomorphism H for a lattic... |
| dihval 41431 | Value of isomorphism H for... |
| dihvalc 41432 | Value of isomorphism H for... |
| dihlsscpre 41433 | Closure of isomorphism H f... |
| dihvalcqpre 41434 | Value of isomorphism H for... |
| dihvalcq 41435 | Value of isomorphism H for... |
| dihvalb 41436 | Value of isomorphism H for... |
| dihopelvalbN 41437 | Ordered pair member of the... |
| dihvalcqat 41438 | Value of isomorphism H for... |
| dih1dimb 41439 | Two expressions for a 1-di... |
| dih1dimb2 41440 | Isomorphism H at an atom u... |
| dih1dimc 41441 | Isomorphism H at an atom n... |
| dib2dim 41442 | Extend ~ dia2dim to partia... |
| dih2dimb 41443 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 41444 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 41445 | Ordered pair member of the... |
| dihvalcq2 41446 | Value of isomorphism H for... |
| dihopelvalcpre 41447 | Member of value of isomorp... |
| dihopelvalc 41448 | Member of value of isomorp... |
| dihlss 41449 | The value of isomorphism H... |
| dihss 41450 | The value of isomorphism H... |
| dihssxp 41451 | An isomorphism H value is ... |
| dihopcl 41452 | Closure of an ordered pair... |
| xihopellsmN 41453 | Ordered pair membership in... |
| dihopellsm 41454 | Ordered pair membership in... |
| dihord6apre 41455 | Part of proof that isomorp... |
| dihord3 41456 | The isomorphism H for a la... |
| dihord4 41457 | The isomorphism H for a la... |
| dihord5b 41458 | Part of proof that isomorp... |
| dihord6b 41459 | Part of proof that isomorp... |
| dihord6a 41460 | Part of proof that isomorp... |
| dihord5apre 41461 | Part of proof that isomorp... |
| dihord5a 41462 | Part of proof that isomorp... |
| dihord 41463 | The isomorphism H is order... |
| dih11 41464 | The isomorphism H is one-t... |
| dihf11lem 41465 | Functionality of the isomo... |
| dihf11 41466 | The isomorphism H for a la... |
| dihfn 41467 | Functionality and domain o... |
| dihdm 41468 | Domain of isomorphism H. (... |
| dihcl 41469 | Closure of isomorphism H. ... |
| dihcnvcl 41470 | Closure of isomorphism H c... |
| dihcnvid1 41471 | The converse isomorphism o... |
| dihcnvid2 41472 | The isomorphism of a conve... |
| dihcnvord 41473 | Ordering property for conv... |
| dihcnv11 41474 | The converse of isomorphis... |
| dihsslss 41475 | The isomorphism H maps to ... |
| dihrnlss 41476 | The isomorphism H maps to ... |
| dihrnss 41477 | The isomorphism H maps to ... |
| dihvalrel 41478 | The value of isomorphism H... |
| dih0 41479 | The value of isomorphism H... |
| dih0bN 41480 | A lattice element is zero ... |
| dih0vbN 41481 | A vector is zero iff its s... |
| dih0cnv 41482 | The isomorphism H converse... |
| dih0rn 41483 | The zero subspace belongs ... |
| dih0sb 41484 | A subspace is zero iff the... |
| dih1 41485 | The value of isomorphism H... |
| dih1rn 41486 | The full vector space belo... |
| dih1cnv 41487 | The isomorphism H converse... |
| dihwN 41488 | Value of isomorphism H at ... |
| dihmeetlem1N 41489 | Isomorphism H of a conjunc... |
| dihglblem5apreN 41490 | A conjunction property of ... |
| dihglblem5aN 41491 | A conjunction property of ... |
| dihglblem2aN 41492 | Lemma for isomorphism H of... |
| dihglblem2N 41493 | The GLB of a set of lattic... |
| dihglblem3N 41494 | Isomorphism H of a lattice... |
| dihglblem3aN 41495 | Isomorphism H of a lattice... |
| dihglblem4 41496 | Isomorphism H of a lattice... |
| dihglblem5 41497 | Isomorphism H of a lattice... |
| dihmeetlem2N 41498 | Isomorphism H of a conjunc... |
| dihglbcpreN 41499 | Isomorphism H of a lattice... |
| dihglbcN 41500 | Isomorphism H of a lattice... |
| dihmeetcN 41501 | Isomorphism H of a lattice... |
| dihmeetbN 41502 | Isomorphism H of a lattice... |
| dihmeetbclemN 41503 | Lemma for isomorphism H of... |
| dihmeetlem3N 41504 | Lemma for isomorphism H of... |
| dihmeetlem4preN 41505 | Lemma for isomorphism H of... |
| dihmeetlem4N 41506 | Lemma for isomorphism H of... |
| dihmeetlem5 41507 | Part of proof that isomorp... |
| dihmeetlem6 41508 | Lemma for isomorphism H of... |
| dihmeetlem7N 41509 | Lemma for isomorphism H of... |
| dihjatc1 41510 | Lemma for isomorphism H of... |
| dihjatc2N 41511 | Isomorphism H of join with... |
| dihjatc3 41512 | Isomorphism H of join with... |
| dihmeetlem8N 41513 | Lemma for isomorphism H of... |
| dihmeetlem9N 41514 | Lemma for isomorphism H of... |
| dihmeetlem10N 41515 | Lemma for isomorphism H of... |
| dihmeetlem11N 41516 | Lemma for isomorphism H of... |
| dihmeetlem12N 41517 | Lemma for isomorphism H of... |
| dihmeetlem13N 41518 | Lemma for isomorphism H of... |
| dihmeetlem14N 41519 | Lemma for isomorphism H of... |
| dihmeetlem15N 41520 | Lemma for isomorphism H of... |
| dihmeetlem16N 41521 | Lemma for isomorphism H of... |
| dihmeetlem17N 41522 | Lemma for isomorphism H of... |
| dihmeetlem18N 41523 | Lemma for isomorphism H of... |
| dihmeetlem19N 41524 | Lemma for isomorphism H of... |
| dihmeetlem20N 41525 | Lemma for isomorphism H of... |
| dihmeetALTN 41526 | Isomorphism H of a lattice... |
| dih1dimatlem0 41527 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 41528 | Lemma for ~ dih1dimat . (... |
| dih1dimat 41529 | Any 1-dimensional subspace... |
| dihlsprn 41530 | The span of a vector belon... |
| dihlspsnssN 41531 | A subspace included in a 1... |
| dihlspsnat 41532 | The inverse isomorphism H ... |
| dihatlat 41533 | The isomorphism H of an at... |
| dihat 41534 | There exists at least one ... |
| dihpN 41535 | The value of isomorphism H... |
| dihlatat 41536 | The reverse isomorphism H ... |
| dihatexv 41537 | There is a nonzero vector ... |
| dihatexv2 41538 | There is a nonzero vector ... |
| dihglblem6 41539 | Isomorphism H of a lattice... |
| dihglb 41540 | Isomorphism H of a lattice... |
| dihglb2 41541 | Isomorphism H of a lattice... |
| dihmeet 41542 | Isomorphism H of a lattice... |
| dihintcl 41543 | The intersection of closed... |
| dihmeetcl 41544 | Closure of closed subspace... |
| dihmeet2 41545 | Reverse isomorphism H of a... |
| dochffval 41548 | Subspace orthocomplement f... |
| dochfval 41549 | Subspace orthocomplement f... |
| dochval 41550 | Subspace orthocomplement f... |
| dochval2 41551 | Subspace orthocomplement f... |
| dochcl 41552 | Closure of subspace orthoc... |
| dochlss 41553 | A subspace orthocomplement... |
| dochssv 41554 | A subspace orthocomplement... |
| dochfN 41555 | Domain and codomain of the... |
| dochvalr 41556 | Orthocomplement of a close... |
| doch0 41557 | Orthocomplement of the zer... |
| doch1 41558 | Orthocomplement of the uni... |
| dochoc0 41559 | The zero subspace is close... |
| dochoc1 41560 | The unit subspace (all vec... |
| dochvalr2 41561 | Orthocomplement of a close... |
| dochvalr3 41562 | Orthocomplement of a close... |
| doch2val2 41563 | Double orthocomplement for... |
| dochss 41564 | Subset law for orthocomple... |
| dochocss 41565 | Double negative law for or... |
| dochoc 41566 | Double negative law for or... |
| dochsscl 41567 | If a set of vectors is inc... |
| dochoccl 41568 | A set of vectors is closed... |
| dochord 41569 | Ordering law for orthocomp... |
| dochord2N 41570 | Ordering law for orthocomp... |
| dochord3 41571 | Ordering law for orthocomp... |
| doch11 41572 | Orthocomplement is one-to-... |
| dochsordN 41573 | Strict ordering law for or... |
| dochn0nv 41574 | An orthocomplement is nonz... |
| dihoml4c 41575 | Version of ~ dihoml4 with ... |
| dihoml4 41576 | Orthomodular law for const... |
| dochspss 41577 | The span of a set of vecto... |
| dochocsp 41578 | The span of an orthocomple... |
| dochspocN 41579 | The span of an orthocomple... |
| dochocsn 41580 | The double orthocomplement... |
| dochsncom 41581 | Swap vectors in an orthoco... |
| dochsat 41582 | The double orthocomplement... |
| dochshpncl 41583 | If a hyperplane is not clo... |
| dochlkr 41584 | Equivalent conditions for ... |
| dochkrshp 41585 | The closure of a kernel is... |
| dochkrshp2 41586 | Properties of the closure ... |
| dochkrshp3 41587 | Properties of the closure ... |
| dochkrshp4 41588 | Properties of the closure ... |
| dochdmj1 41589 | De Morgan-like law for sub... |
| dochnoncon 41590 | Law of noncontradiction. ... |
| dochnel2 41591 | A nonzero member of a subs... |
| dochnel 41592 | A nonzero vector doesn't b... |
| djhffval 41595 | Subspace join for ` DVecH ... |
| djhfval 41596 | Subspace join for ` DVecH ... |
| djhval 41597 | Subspace join for ` DVecH ... |
| djhval2 41598 | Value of subspace join for... |
| djhcl 41599 | Closure of subspace join f... |
| djhlj 41600 | Transfer lattice join to `... |
| djhljjN 41601 | Lattice join in terms of `... |
| djhjlj 41602 | ` DVecH ` vector space clo... |
| djhj 41603 | ` DVecH ` vector space clo... |
| djhcom 41604 | Subspace join commutes. (... |
| djhspss 41605 | Subspace span of union is ... |
| djhsumss 41606 | Subspace sum is a subset o... |
| dihsumssj 41607 | The subspace sum of two is... |
| djhunssN 41608 | Subspace union is a subset... |
| dochdmm1 41609 | De Morgan-like law for clo... |
| djhexmid 41610 | Excluded middle property o... |
| djh01 41611 | Closed subspace join with ... |
| djh02 41612 | Closed subspace join with ... |
| djhlsmcl 41613 | A closed subspace sum equa... |
| djhcvat42 41614 | A covering property. ( ~ ... |
| dihjatb 41615 | Isomorphism H of lattice j... |
| dihjatc 41616 | Isomorphism H of lattice j... |
| dihjatcclem1 41617 | Lemma for isomorphism H of... |
| dihjatcclem2 41618 | Lemma for isomorphism H of... |
| dihjatcclem3 41619 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 41620 | Lemma for isomorphism H of... |
| dihjatcc 41621 | Isomorphism H of lattice j... |
| dihjat 41622 | Isomorphism H of lattice j... |
| dihprrnlem1N 41623 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 41624 | Lemma for ~ dihprrn . (Co... |
| dihprrn 41625 | The span of a vector pair ... |
| djhlsmat 41626 | The sum of two subspace at... |
| dihjat1lem 41627 | Subspace sum of a closed s... |
| dihjat1 41628 | Subspace sum of a closed s... |
| dihsmsprn 41629 | Subspace sum of a closed s... |
| dihjat2 41630 | The subspace sum of a clos... |
| dihjat3 41631 | Isomorphism H of lattice j... |
| dihjat4 41632 | Transfer the subspace sum ... |
| dihjat6 41633 | Transfer the subspace sum ... |
| dihsmsnrn 41634 | The subspace sum of two si... |
| dihsmatrn 41635 | The subspace sum of a clos... |
| dihjat5N 41636 | Transfer lattice join with... |
| dvh4dimat 41637 | There is an atom that is o... |
| dvh3dimatN 41638 | There is an atom that is o... |
| dvh2dimatN 41639 | Given an atom, there exist... |
| dvh1dimat 41640 | There exists an atom. (Co... |
| dvh1dim 41641 | There exists a nonzero vec... |
| dvh4dimlem 41642 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 41643 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 41644 | There is a vector that is ... |
| dvh3dim 41645 | There is a vector that is ... |
| dvh4dimN 41646 | There is a vector that is ... |
| dvh3dim2 41647 | There is a vector that is ... |
| dvh3dim3N 41648 | There is a vector that is ... |
| dochsnnz 41649 | The orthocomplement of a s... |
| dochsatshp 41650 | The orthocomplement of a s... |
| dochsatshpb 41651 | The orthocomplement of a s... |
| dochsnshp 41652 | The orthocomplement of a n... |
| dochshpsat 41653 | A hyperplane is closed iff... |
| dochkrsat 41654 | The orthocomplement of a k... |
| dochkrsat2 41655 | The orthocomplement of a k... |
| dochsat0 41656 | The orthocomplement of a k... |
| dochkrsm 41657 | The subspace sum of a clos... |
| dochexmidat 41658 | Special case of excluded m... |
| dochexmidlem1 41659 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 41660 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 41661 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 41662 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 41663 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 41664 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 41665 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 41666 | Lemma for ~ dochexmid . T... |
| dochexmid 41667 | Excluded middle law for cl... |
| dochsnkrlem1 41668 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 41669 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 41670 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 41671 | A (closed) kernel expresse... |
| dochsnkr2 41672 | Kernel of the explicit fun... |
| dochsnkr2cl 41673 | The ` X ` determining func... |
| dochflcl 41674 | Closure of the explicit fu... |
| dochfl1 41675 | The value of the explicit ... |
| dochfln0 41676 | The value of a functional ... |
| dochkr1 41677 | A nonzero functional has a... |
| dochkr1OLDN 41678 | A nonzero functional has a... |
| lpolsetN 41681 | The set of polarities of a... |
| islpolN 41682 | The predicate "is a polari... |
| islpoldN 41683 | Properties that determine ... |
| lpolfN 41684 | Functionality of a polarit... |
| lpolvN 41685 | The polarity of the whole ... |
| lpolconN 41686 | Contraposition property of... |
| lpolsatN 41687 | The polarity of an atomic ... |
| lpolpolsatN 41688 | Property of a polarity. (... |
| dochpolN 41689 | The subspace orthocompleme... |
| lcfl1lem 41690 | Property of a functional w... |
| lcfl1 41691 | Property of a functional w... |
| lcfl2 41692 | Property of a functional w... |
| lcfl3 41693 | Property of a functional w... |
| lcfl4N 41694 | Property of a functional w... |
| lcfl5 41695 | Property of a functional w... |
| lcfl5a 41696 | Property of a functional w... |
| lcfl6lem 41697 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 41698 | Lemma for ~ lcfl7N . If t... |
| lcfl6 41699 | Property of a functional w... |
| lcfl7N 41700 | Property of a functional w... |
| lcfl8 41701 | Property of a functional w... |
| lcfl8a 41702 | Property of a functional w... |
| lcfl8b 41703 | Property of a nonzero func... |
| lcfl9a 41704 | Property implying that a f... |
| lclkrlem1 41705 | The set of functionals hav... |
| lclkrlem2a 41706 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 41707 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 41708 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 41709 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 41710 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 41711 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 41712 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 41713 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 41714 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 41715 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 41716 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 41717 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 41718 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 41719 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 41720 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 41721 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 41722 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 41723 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 41724 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 41725 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 41726 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 41727 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 41728 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 41729 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 41730 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 41731 | The set of functionals hav... |
| lclkr 41732 | The set of functionals wit... |
| lcfls1lem 41733 | Property of a functional w... |
| lcfls1N 41734 | Property of a functional w... |
| lcfls1c 41735 | Property of a functional w... |
| lclkrslem1 41736 | The set of functionals hav... |
| lclkrslem2 41737 | The set of functionals hav... |
| lclkrs 41738 | The set of functionals hav... |
| lclkrs2 41739 | The set of functionals wit... |
| lcfrvalsnN 41740 | Reconstruction from the du... |
| lcfrlem1 41741 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 41742 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 41743 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 41744 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 41745 | Lemma for ~ lcfr . The se... |
| lcfrlem6 41746 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 41747 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 41748 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 41749 | Lemma for ~ lcf1o . (This... |
| lcf1o 41750 | Define a function ` J ` th... |
| lcfrlem10 41751 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 41752 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 41753 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 41754 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 41755 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 41756 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 41757 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 41758 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 41759 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 41760 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 41761 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 41762 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 41763 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 41764 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 41765 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 41766 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 41767 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 41768 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 41769 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 41770 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 41771 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 41772 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 41773 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 41774 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 41775 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 41776 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 41777 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 41778 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 41779 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 41780 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 41781 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 41782 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 41783 | Lemma for ~ lcfr . Elimin... |
| lcfr 41784 | Reconstruction of a subspa... |
| lcdfval 41787 | Dual vector space of funct... |
| lcdval 41788 | Dual vector space of funct... |
| lcdval2 41789 | Dual vector space of funct... |
| lcdlvec 41790 | The dual vector space of f... |
| lcdlmod 41791 | The dual vector space of f... |
| lcdvbase 41792 | Vector base set of a dual ... |
| lcdvbasess 41793 | The vector base set of the... |
| lcdvbaselfl 41794 | A vector in the base set o... |
| lcdvbasecl 41795 | Closure of the value of a ... |
| lcdvadd 41796 | Vector addition for the cl... |
| lcdvaddval 41797 | The value of the value of ... |
| lcdsca 41798 | The ring of scalars of the... |
| lcdsbase 41799 | Base set of scalar ring fo... |
| lcdsadd 41800 | Scalar addition for the cl... |
| lcdsmul 41801 | Scalar multiplication for ... |
| lcdvs 41802 | Scalar product for the clo... |
| lcdvsval 41803 | Value of scalar product op... |
| lcdvscl 41804 | The scalar product operati... |
| lcdlssvscl 41805 | Closure of scalar product ... |
| lcdvsass 41806 | Associative law for scalar... |
| lcd0 41807 | The zero scalar of the clo... |
| lcd1 41808 | The unit scalar of the clo... |
| lcdneg 41809 | The unit scalar of the clo... |
| lcd0v 41810 | The zero functional in the... |
| lcd0v2 41811 | The zero functional in the... |
| lcd0vvalN 41812 | Value of the zero function... |
| lcd0vcl 41813 | Closure of the zero functi... |
| lcd0vs 41814 | A scalar zero times a func... |
| lcdvs0N 41815 | A scalar times the zero fu... |
| lcdvsub 41816 | The value of vector subtra... |
| lcdvsubval 41817 | The value of the value of ... |
| lcdlss 41818 | Subspaces of a dual vector... |
| lcdlss2N 41819 | Subspaces of a dual vector... |
| lcdlsp 41820 | Span in the set of functio... |
| lcdlkreqN 41821 | Colinear functionals have ... |
| lcdlkreq2N 41822 | Colinear functionals have ... |
| mapdffval 41825 | Projectivity from vector s... |
| mapdfval 41826 | Projectivity from vector s... |
| mapdval 41827 | Value of projectivity from... |
| mapdvalc 41828 | Value of projectivity from... |
| mapdval2N 41829 | Value of projectivity from... |
| mapdval3N 41830 | Value of projectivity from... |
| mapdval4N 41831 | Value of projectivity from... |
| mapdval5N 41832 | Value of projectivity from... |
| mapdordlem1a 41833 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 41834 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 41835 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 41836 | Lemma for ~ mapdord . Ord... |
| mapdord 41837 | Ordering property of the m... |
| mapd11 41838 | The map defined by ~ df-ma... |
| mapddlssN 41839 | The mapping of a subspace ... |
| mapdsn 41840 | Value of the map defined b... |
| mapdsn2 41841 | Value of the map defined b... |
| mapdsn3 41842 | Value of the map defined b... |
| mapd1dim2lem1N 41843 | Value of the map defined b... |
| mapdrvallem2 41844 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 41845 | Lemma for ~ mapdrval . (C... |
| mapdrval 41846 | Given a dual subspace ` R ... |
| mapd1o 41847 | The map defined by ~ df-ma... |
| mapdrn 41848 | Range of the map defined b... |
| mapdunirnN 41849 | Union of the range of the ... |
| mapdrn2 41850 | Range of the map defined b... |
| mapdcnvcl 41851 | Closure of the converse of... |
| mapdcl 41852 | Closure the value of the m... |
| mapdcnvid1N 41853 | Converse of the value of t... |
| mapdsord 41854 | Strong ordering property o... |
| mapdcl2 41855 | The mapping of a subspace ... |
| mapdcnvid2 41856 | Value of the converse of t... |
| mapdcnvordN 41857 | Ordering property of the c... |
| mapdcnv11N 41858 | The converse of the map de... |
| mapdcv 41859 | Covering property of the c... |
| mapdincl 41860 | Closure of dual subspace i... |
| mapdin 41861 | Subspace intersection is p... |
| mapdlsmcl 41862 | Closure of dual subspace s... |
| mapdlsm 41863 | Subspace sum is preserved ... |
| mapd0 41864 | Projectivity map of the ze... |
| mapdcnvatN 41865 | Atoms are preserved by the... |
| mapdat 41866 | Atoms are preserved by the... |
| mapdspex 41867 | The map of a span equals t... |
| mapdn0 41868 | Transfer nonzero property ... |
| mapdncol 41869 | Transfer non-colinearity f... |
| mapdindp 41870 | Transfer (part of) vector ... |
| mapdpglem1 41871 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 41872 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 41873 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 41874 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 41875 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 41876 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 41877 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 41878 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 41879 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 41880 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 41881 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 41882 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 41883 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 41884 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 41885 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 41886 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 41887 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 41888 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 41889 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 41890 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 41891 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 41892 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 41893 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 41894 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 41895 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 41896 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 41897 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 41898 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 41899 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 41900 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 41901 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 41902 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 41903 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 41904 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 41905 | Part 1 of proof of the fir... |
| baerlem3lem1 41906 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 41907 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 41908 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 41909 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 41910 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 41911 | Lemma for ~ baerlem5b . (... |
| baerlem3 41912 | An equality that holds whe... |
| baerlem5a 41913 | An equality that holds whe... |
| baerlem5b 41914 | An equality that holds whe... |
| baerlem5amN 41915 | An equality that holds whe... |
| baerlem5bmN 41916 | An equality that holds whe... |
| baerlem5abmN 41917 | An equality that holds whe... |
| mapdindp0 41918 | Vector independence lemma.... |
| mapdindp1 41919 | Vector independence lemma.... |
| mapdindp2 41920 | Vector independence lemma.... |
| mapdindp3 41921 | Vector independence lemma.... |
| mapdindp4 41922 | Vector independence lemma.... |
| mapdhval 41923 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 41924 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 41925 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 41926 | Lemmma for ~~? mapdh . (C... |
| mapdheq 41927 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 41928 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 41929 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 41930 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 41931 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 41932 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 41933 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 41934 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 41935 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 41936 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 41937 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 41938 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 41939 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 41940 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 41941 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 41942 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 41943 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 41944 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 41945 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 41946 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 41947 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 41948 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 41949 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 41950 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 41951 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 41952 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 41953 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 41954 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 41957 | Map from nonzero vectors t... |
| hvmapfval 41958 | Map from nonzero vectors t... |
| hvmapval 41959 | Value of map from nonzero ... |
| hvmapvalvalN 41960 | Value of value of map (i.e... |
| hvmapidN 41961 | The value of the vector to... |
| hvmap1o 41962 | The vector to functional m... |
| hvmapclN 41963 | Closure of the vector to f... |
| hvmap1o2 41964 | The vector to functional m... |
| hvmapcl2 41965 | Closure of the vector to f... |
| hvmaplfl 41966 | The vector to functional m... |
| hvmaplkr 41967 | Kernel of the vector to fu... |
| mapdhvmap 41968 | Relationship between ` map... |
| lspindp5 41969 | Obtain an independent vect... |
| hdmaplem1 41970 | Lemma to convert a frequen... |
| hdmaplem2N 41971 | Lemma to convert a frequen... |
| hdmaplem3 41972 | Lemma to convert a frequen... |
| hdmaplem4 41973 | Lemma to convert a frequen... |
| mapdh8a 41974 | Part of Part (8) in [Baer]... |
| mapdh8aa 41975 | Part of Part (8) in [Baer]... |
| mapdh8ab 41976 | Part of Part (8) in [Baer]... |
| mapdh8ac 41977 | Part of Part (8) in [Baer]... |
| mapdh8ad 41978 | Part of Part (8) in [Baer]... |
| mapdh8b 41979 | Part of Part (8) in [Baer]... |
| mapdh8c 41980 | Part of Part (8) in [Baer]... |
| mapdh8d0N 41981 | Part of Part (8) in [Baer]... |
| mapdh8d 41982 | Part of Part (8) in [Baer]... |
| mapdh8e 41983 | Part of Part (8) in [Baer]... |
| mapdh8g 41984 | Part of Part (8) in [Baer]... |
| mapdh8i 41985 | Part of Part (8) in [Baer]... |
| mapdh8j 41986 | Part of Part (8) in [Baer]... |
| mapdh8 41987 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 41988 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 41989 | Lemma for part (9) in [Bae... |
| hdmap1ffval 41994 | Preliminary map from vecto... |
| hdmap1fval 41995 | Preliminary map from vecto... |
| hdmap1vallem 41996 | Value of preliminary map f... |
| hdmap1val 41997 | Value of preliminary map f... |
| hdmap1val0 41998 | Value of preliminary map f... |
| hdmap1val2 41999 | Value of preliminary map f... |
| hdmap1eq 42000 | The defining equation for ... |
| hdmap1cbv 42001 | Frequently used lemma to c... |
| hdmap1valc 42002 | Connect the value of the p... |
| hdmap1cl 42003 | Convert closure theorem ~ ... |
| hdmap1eq2 42004 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 42005 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 42006 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 42007 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 42008 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 42009 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 42010 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 42011 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 42012 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 42013 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 42014 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 42015 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 42016 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 42017 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 42018 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 42019 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 42020 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 42021 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 42022 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 42023 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 42024 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 42025 | Map from vectors to functi... |
| hdmapfval 42026 | Map from vectors to functi... |
| hdmapval 42027 | Value of map from vectors ... |
| hdmapfnN 42028 | Functionality of map from ... |
| hdmapcl 42029 | Closure of map from vector... |
| hdmapval2lem 42030 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 42031 | Value of map from vectors ... |
| hdmapval0 42032 | Value of map from vectors ... |
| hdmapeveclem 42033 | Lemma for ~ hdmapevec . T... |
| hdmapevec 42034 | Value of map from vectors ... |
| hdmapevec2 42035 | The inner product of the r... |
| hdmapval3lemN 42036 | Value of map from vectors ... |
| hdmapval3N 42037 | Value of map from vectors ... |
| hdmap10lem 42038 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 42039 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 42040 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 42041 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 42042 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 42043 | Part of proof of part 12 i... |
| hdmapnzcl 42044 | Nonzero vector closure of ... |
| hdmapneg 42045 | Part of proof of part 12 i... |
| hdmapsub 42046 | Part of proof of part 12 i... |
| hdmap11 42047 | Part of proof of part 12 i... |
| hdmaprnlem1N 42048 | Part of proof of part 12 i... |
| hdmaprnlem3N 42049 | Part of proof of part 12 i... |
| hdmaprnlem3uN 42050 | Part of proof of part 12 i... |
| hdmaprnlem4tN 42051 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 42052 | Part of proof of part 12 i... |
| hdmaprnlem6N 42053 | Part of proof of part 12 i... |
| hdmaprnlem7N 42054 | Part of proof of part 12 i... |
| hdmaprnlem8N 42055 | Part of proof of part 12 i... |
| hdmaprnlem9N 42056 | Part of proof of part 12 i... |
| hdmaprnlem3eN 42057 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 42058 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 42059 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 42060 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 42061 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 42062 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 42063 | Part of proof of part 12 i... |
| hdmapf1oN 42064 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 42065 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 42066 | Prior to part 14 in [Baer]... |
| hdmap14lem1 42067 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 42068 | Prior to part 14 in [Baer]... |
| hdmap14lem3 42069 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 42070 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 42071 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 42072 | Case where ` F ` is zero. ... |
| hdmap14lem7 42073 | Combine cases of ` F ` . ... |
| hdmap14lem8 42074 | Part of proof of part 14 i... |
| hdmap14lem9 42075 | Part of proof of part 14 i... |
| hdmap14lem10 42076 | Part of proof of part 14 i... |
| hdmap14lem11 42077 | Part of proof of part 14 i... |
| hdmap14lem12 42078 | Lemma for proof of part 14... |
| hdmap14lem13 42079 | Lemma for proof of part 14... |
| hdmap14lem14 42080 | Part of proof of part 14 i... |
| hdmap14lem15 42081 | Part of proof of part 14 i... |
| hgmapffval 42084 | Map from the scalar divisi... |
| hgmapfval 42085 | Map from the scalar divisi... |
| hgmapval 42086 | Value of map from the scal... |
| hgmapfnN 42087 | Functionality of scalar si... |
| hgmapcl 42088 | Closure of scalar sigma ma... |
| hgmapdcl 42089 | Closure of the vector spac... |
| hgmapvs 42090 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 42091 | Value of the scalar sigma ... |
| hgmapval1 42092 | Value of the scalar sigma ... |
| hgmapadd 42093 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 42094 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 42095 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 42096 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 42097 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 42098 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 42099 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 42100 | Part of proof of part 16 i... |
| hgmap11 42101 | The scalar sigma map is on... |
| hgmapf1oN 42102 | The scalar sigma map is a ... |
| hgmapeq0 42103 | The scalar sigma map is ze... |
| hdmapipcl 42104 | The inner product (Hermiti... |
| hdmapln1 42105 | Linearity property that wi... |
| hdmaplna1 42106 | Additive property of first... |
| hdmaplns1 42107 | Subtraction property of fi... |
| hdmaplnm1 42108 | Multiplicative property of... |
| hdmaplna2 42109 | Additive property of secon... |
| hdmapglnm2 42110 | g-linear property of secon... |
| hdmapgln2 42111 | g-linear property that wil... |
| hdmaplkr 42112 | Kernel of the vector to du... |
| hdmapellkr 42113 | Membership in the kernel (... |
| hdmapip0 42114 | Zero property that will be... |
| hdmapip1 42115 | Construct a proportional v... |
| hdmapip0com 42116 | Commutation property of Ba... |
| hdmapinvlem1 42117 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 42118 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 42119 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 42120 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 42121 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 42122 | Involution property of sca... |
| hgmapvvlem2 42123 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 42124 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 42125 | Value of a double involuti... |
| hdmapglem7a 42126 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 42127 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 42128 | Lemma for ~ hdmapg . Line... |
| hdmapg 42129 | Apply the scalar sigma fun... |
| hdmapoc 42130 | Express our constructed or... |
| hlhilset 42133 | The final Hilbert space co... |
| hlhilsca 42134 | The scalar of the final co... |
| hlhilbase 42135 | The base set of the final ... |
| hlhilplus 42136 | The vector addition for th... |
| hlhilslem 42137 | Lemma for ~ hlhilsbase etc... |
| hlhilsbase 42138 | The scalar base set of the... |
| hlhilsplus 42139 | Scalar addition for the fi... |
| hlhilsmul 42140 | Scalar multiplication for ... |
| hlhilsbase2 42141 | The scalar base set of the... |
| hlhilsplus2 42142 | Scalar addition for the fi... |
| hlhilsmul2 42143 | Scalar multiplication for ... |
| hlhils0 42144 | The scalar ring zero for t... |
| hlhils1N 42145 | The scalar ring unity for ... |
| hlhilvsca 42146 | The scalar product for the... |
| hlhilip 42147 | Inner product operation fo... |
| hlhilipval 42148 | Value of inner product ope... |
| hlhilnvl 42149 | The involution operation o... |
| hlhillvec 42150 | The final constructed Hilb... |
| hlhildrng 42151 | The star division ring for... |
| hlhilsrnglem 42152 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 42153 | The star division ring for... |
| hlhil0 42154 | The zero vector for the fi... |
| hlhillsm 42155 | The vector sum operation f... |
| hlhilocv 42156 | The orthocomplement for th... |
| hlhillcs 42157 | The closed subspaces of th... |
| hlhilphllem 42158 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 42159 | Lemma for ~ hlhil . (Cont... |
| hlathil 42160 | Construction of a Hilbert ... |
| iscsrg 42163 | A commutative semiring is ... |
| rhmzrhval 42164 | Evaluation of integers acr... |
| zndvdchrrhm 42165 | Construction of a ring hom... |
| relogbcld 42166 | Closure of the general log... |
| relogbexpd 42167 | Identity law for general l... |
| relogbzexpd 42168 | Power law for the general ... |
| logblebd 42169 | The general logarithm is m... |
| uzindd 42170 | Induction on the upper int... |
| fzadd2d 42171 | Membership of a sum in a f... |
| zltp1led 42172 | Integer ordering relation,... |
| fzne2d 42173 | Elementhood in a finite se... |
| eqfnfv2d2 42174 | Equality of functions is d... |
| fzsplitnd 42175 | Split a finite interval of... |
| fzsplitnr 42176 | Split a finite interval of... |
| addassnni 42177 | Associative law for additi... |
| addcomnni 42178 | Commutative law for additi... |
| mulassnni 42179 | Associative law for multip... |
| mulcomnni 42180 | Commutative law for multip... |
| gcdcomnni 42181 | Commutative law for gcd. ... |
| gcdnegnni 42182 | Negation invariance for gc... |
| neggcdnni 42183 | Negation invariance for gc... |
| bccl2d 42184 | Closure of the binomial co... |
| recbothd 42185 | Take reciprocal on both si... |
| gcdmultiplei 42186 | The GCD of a multiple of a... |
| gcdaddmzz2nni 42187 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 42188 | Adding a multiple of one o... |
| gcdnncli 42189 | Closure of the gcd operato... |
| muldvds1d 42190 | If a product divides an in... |
| muldvds2d 42191 | If a product divides an in... |
| nndivdvdsd 42192 | A positive integer divides... |
| nnproddivdvdsd 42193 | A product of natural numbe... |
| coprmdvds2d 42194 | If an integer is divisible... |
| imadomfi 42195 | An image of a function und... |
| 12gcd5e1 42196 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 42197 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 42198 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 42199 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 42200 | Calculate the least common... |
| 12lcm5e60 42201 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 42202 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 42203 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 42204 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 42205 | Useful equation to calcula... |
| lcm1un 42206 | Least common multiple of n... |
| lcm2un 42207 | Least common multiple of n... |
| lcm3un 42208 | Least common multiple of n... |
| lcm4un 42209 | Least common multiple of n... |
| lcm5un 42210 | Least common multiple of n... |
| lcm6un 42211 | Least common multiple of n... |
| lcm7un 42212 | Least common multiple of n... |
| lcm8un 42213 | Least common multiple of n... |
| 3factsumint1 42214 | Move constants out of inte... |
| 3factsumint2 42215 | Move constants out of inte... |
| 3factsumint3 42216 | Move constants out of inte... |
| 3factsumint4 42217 | Move constants out of inte... |
| 3factsumint 42218 | Helpful equation for lcm i... |
| resopunitintvd 42219 | Restrict continuous functi... |
| resclunitintvd 42220 | Restrict continuous functi... |
| resdvopclptsd 42221 | Restrict derivative on uni... |
| lcmineqlem1 42222 | Part of lcm inequality lem... |
| lcmineqlem2 42223 | Part of lcm inequality lem... |
| lcmineqlem3 42224 | Part of lcm inequality lem... |
| lcmineqlem4 42225 | Part of lcm inequality lem... |
| lcmineqlem5 42226 | Technical lemma for recipr... |
| lcmineqlem6 42227 | Part of lcm inequality lem... |
| lcmineqlem7 42228 | Derivative of 1-x for chai... |
| lcmineqlem8 42229 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 42230 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 42231 | Induction step of ~ lcmine... |
| lcmineqlem11 42232 | Induction step, continuati... |
| lcmineqlem12 42233 | Base case for induction. ... |
| lcmineqlem13 42234 | Induction proof for lcm in... |
| lcmineqlem14 42235 | Technical lemma for inequa... |
| lcmineqlem15 42236 | F times the least common m... |
| lcmineqlem16 42237 | Technical divisibility lem... |
| lcmineqlem17 42238 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 42239 | Technical lemma to shift f... |
| lcmineqlem19 42240 | Dividing implies inequalit... |
| lcmineqlem20 42241 | Inequality for lcm lemma. ... |
| lcmineqlem21 42242 | The lcm inequality lemma w... |
| lcmineqlem22 42243 | The lcm inequality lemma w... |
| lcmineqlem23 42244 | Penultimate step to the lc... |
| lcmineqlem 42245 | The least common multiple ... |
| 3exp7 42246 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 42247 | First inequality in inequa... |
| 3lexlogpow5ineq2 42248 | Second inequality in inequ... |
| 3lexlogpow5ineq4 42249 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 42250 | Combined inequality chain ... |
| 3lexlogpow2ineq1 42251 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 42252 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 42253 | Result for bound in AKS in... |
| intlewftc 42254 | Inequality inference by in... |
| aks4d1lem1 42255 | Technical lemma to reduce ... |
| aks4d1p1p1 42256 | Exponential law for finite... |
| dvrelog2 42257 | The derivative of the loga... |
| dvrelog3 42258 | The derivative of the loga... |
| dvrelog2b 42259 | Derivative of the binary l... |
| 0nonelalab 42260 | Technical lemma for open i... |
| dvrelogpow2b 42261 | Derivative of the power of... |
| aks4d1p1p3 42262 | Bound of a ceiling of the ... |
| aks4d1p1p2 42263 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 42264 | Technical step for inequal... |
| dvle2 42265 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 42266 | Inequality lift to differe... |
| aks4d1p1p7 42267 | Bound of intermediary of i... |
| aks4d1p1p5 42268 | Show inequality for existe... |
| aks4d1p1 42269 | Show inequality for existe... |
| aks4d1p2 42270 | Technical lemma for existe... |
| aks4d1p3 42271 | There exists a small enoug... |
| aks4d1p4 42272 | There exists a small enoug... |
| aks4d1p5 42273 | Show that ` N ` and ` R ` ... |
| aks4d1p6 42274 | The maximal prime power ex... |
| aks4d1p7d1 42275 | Technical step in AKS lemm... |
| aks4d1p7 42276 | Technical step in AKS lemm... |
| aks4d1p8d1 42277 | If a prime divides one num... |
| aks4d1p8d2 42278 | Any prime power dividing a... |
| aks4d1p8d3 42279 | The remainder of a divisio... |
| aks4d1p8 42280 | Show that ` N ` and ` R ` ... |
| aks4d1p9 42281 | Show that the order is bou... |
| aks4d1 42282 | Lemma 4.1 from ~ https://w... |
| fldhmf1 42283 | A field homomorphism is in... |
| isprimroot 42286 | The value of a primitive r... |
| isprimroot2 42287 | Alternative way of creatin... |
| mndmolinv 42288 | An element of a monoid tha... |
| linvh 42289 | If an element has a unique... |
| primrootsunit1 42290 | Primitive roots have left ... |
| primrootsunit 42291 | Primitive roots have left ... |
| primrootscoprmpow 42292 | Coprime powers of primitiv... |
| posbezout 42293 | Bezout's identity restrict... |
| primrootscoprf 42294 | Coprime powers of primitiv... |
| primrootscoprbij 42295 | A bijection between coprim... |
| primrootscoprbij2 42296 | A bijection between coprim... |
| remexz 42297 | Division with rest. (Cont... |
| primrootlekpowne0 42298 | There is no smaller power ... |
| primrootspoweq0 42299 | The power of a ` R ` -th p... |
| aks6d1c1p1 42300 | Definition of the introspe... |
| aks6d1c1p1rcl 42301 | Reverse closure of the int... |
| aks6d1c1p2 42302 | ` P ` and linear factors a... |
| aks6d1c1p3 42303 | In a field with a Frobeniu... |
| aks6d1c1p4 42304 | The product of polynomials... |
| aks6d1c1p5 42305 | The product of exponents i... |
| aks6d1c1p7 42306 | ` X ` is introspective to ... |
| aks6d1c1p6 42307 | If a polynomials ` F ` is ... |
| aks6d1c1p8 42308 | If a number ` E ` is intro... |
| aks6d1c1 42309 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 42310 | Polynomial evaluation buil... |
| aks6d1c2p1 42311 | In the AKS-theorem the sub... |
| aks6d1c2p2 42312 | Injective condition for co... |
| hashscontpowcl 42313 | Closure of E for ~ https:/... |
| hashscontpow1 42314 | Helper lemma for to prove ... |
| hashscontpow 42315 | If a set contains all ` N ... |
| aks6d1c3 42316 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 42317 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 42318 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 42319 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 42320 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 42321 | If the number of elements ... |
| hashnexinjle 42322 | If the number of elements ... |
| aks6d1c2 42323 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 42324 | Special case related to ~ ... |
| idomnnzpownz 42325 | A non-zero power in an int... |
| idomnnzgmulnz 42326 | A finite product of non-ze... |
| ringexp0nn 42327 | Zero to the power of a pos... |
| aks6d1c5lem0 42328 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 42329 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 42330 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 42331 | Lemma for Claim 5, contrad... |
| aks6d1c5 42332 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 42333 | Degree multiplication is a... |
| deg1pow 42334 | Exact degree of a power of... |
| 5bc2eq10 42335 | The value of 5 choose 2. ... |
| facp2 42336 | The factorial of a success... |
| 2np3bcnp1 42337 | Part of induction step for... |
| 2ap1caineq 42338 | Inequality for Theorem 6.6... |
| sticksstones1 42339 | Different strictly monoton... |
| sticksstones2 42340 | The range function on stri... |
| sticksstones3 42341 | The range function on stri... |
| sticksstones4 42342 | Equinumerosity lemma for s... |
| sticksstones5 42343 | Count the number of strict... |
| sticksstones6 42344 | Function induces an order ... |
| sticksstones7 42345 | Closure property of sticks... |
| sticksstones8 42346 | Establish mapping between ... |
| sticksstones9 42347 | Establish mapping between ... |
| sticksstones10 42348 | Establish mapping between ... |
| sticksstones11 42349 | Establish bijective mappin... |
| sticksstones12a 42350 | Establish bijective mappin... |
| sticksstones12 42351 | Establish bijective mappin... |
| sticksstones13 42352 | Establish bijective mappin... |
| sticksstones14 42353 | Sticks and stones with def... |
| sticksstones15 42354 | Sticks and stones with alm... |
| sticksstones16 42355 | Sticks and stones with col... |
| sticksstones17 42356 | Extend sticks and stones t... |
| sticksstones18 42357 | Extend sticks and stones t... |
| sticksstones19 42358 | Extend sticks and stones t... |
| sticksstones20 42359 | Lift sticks and stones to ... |
| sticksstones21 42360 | Lift sticks and stones to ... |
| sticksstones22 42361 | Non-exhaustive sticks and ... |
| sticksstones23 42362 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 42363 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 42364 | Every primitive root is ro... |
| aks6d1c6lem3 42365 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 42366 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 42367 | Lemma to construct the map... |
| aks6d1c6isolem2 42368 | Lemma to construct the gro... |
| aks6d1c6isolem3 42369 | The preimage of a map send... |
| aks6d1c6lem5 42370 | Eliminate the size hypothe... |
| bcled 42371 | Inequality for binomial co... |
| bcle2d 42372 | Inequality for binomial co... |
| aks6d1c7lem1 42373 | The last set of inequaliti... |
| aks6d1c7lem2 42374 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 42375 | Remove lots of hypotheses ... |
| aks6d1c7lem4 42376 | In the AKS algorithm there... |
| aks6d1c7 42377 | ` N ` is a prime power if ... |
| rhmqusspan 42378 | Ring homomorphism out of a... |
| aks5lem1 42379 | Section 5 of ~ https://www... |
| aks5lem2 42380 | Lemma for section 5 ~ http... |
| ply1asclzrhval 42381 | Transfer results from alge... |
| aks5lem3a 42382 | Lemma for AKS section 5. ... |
| aks5lem4a 42383 | Lemma for AKS section 5, r... |
| aks5lem5a 42384 | Lemma for AKS, section 5, ... |
| aks5lem6 42385 | Connect results of section... |
| indstrd 42386 | Strong induction, deductio... |
| grpods 42387 | Relate sums of elements of... |
| unitscyglem1 42388 | Lemma for unitscyg. (Cont... |
| unitscyglem2 42389 | Lemma for unitscyg. (Cont... |
| unitscyglem3 42390 | Lemma for unitscyg. (Cont... |
| unitscyglem4 42391 | Lemma for unitscyg (Contri... |
| unitscyglem5 42392 | Lemma for unitscyg (Contri... |
| aks5lem7 42393 | Lemma for aks5. We clean ... |
| aks5lem8 42394 | Lemma for aks5. Clean up ... |
| exfinfldd 42396 | For any prime ` P ` and an... |
| aks5 42397 | The AKS Primality test, gi... |
| jarrii 42398 | Inference associated with ... |
| intnanrt 42399 | Introduction of conjunct i... |
| ioin9i8 42400 | Miscellaneous inference cr... |
| jaodd 42401 | Double deduction form of ~... |
| syl3an12 42402 | A double syllogism inferen... |
| exbiii 42403 | Inference associated with ... |
| sbtd 42404 | A true statement is true u... |
| sbor2 42405 | One direction of ~ sbor , ... |
| sbalexi 42406 | Inference form of ~ sbalex... |
| nfexhe 42407 | Version of ~ nfex with the... |
| nfalh 42408 | Version of ~ nfal with an ... |
| nfe2 42409 | An inner existential quant... |
| nfale2 42410 | An inner existential quant... |
| nfexa2 42411 | An inner universal quantif... |
| 19.9dev 42412 | ~ 19.9d in the case of an ... |
| 3rspcedvd 42413 | Triple application of ~ rs... |
| sn-axrep5v 42414 | A condensed form of ~ axre... |
| sn-axprlem3 42415 | ~ axprlem3 using only Tars... |
| sn-exelALT 42416 | Alternate proof of ~ exel ... |
| ss2ab1 42417 | Class abstractions in a su... |
| ssabdv 42418 | Deduction of abstraction s... |
| sn-iotalem 42419 | An unused lemma showing th... |
| sn-iotalemcor 42420 | Corollary of ~ sn-iotalem ... |
| abbi1sn 42421 | Originally part of ~ uniab... |
| brif2 42422 | Move a relation inside and... |
| brif12 42423 | Move a relation inside and... |
| pssexg 42424 | The proper subset of a set... |
| pssn0 42425 | A proper superset is nonem... |
| psspwb 42426 | Classes are proper subclas... |
| xppss12 42427 | Proper subset theorem for ... |
| elpwbi 42428 | Membership in a power set,... |
| imaopab 42429 | The image of a class of or... |
| eqresfnbd 42430 | Property of being the rest... |
| f1o2d2 42431 | Sufficient condition for a... |
| fmpocos 42432 | Composition of two functio... |
| ovmpogad 42433 | Value of an operation give... |
| ofun 42434 | A function operation of un... |
| dfqs2 42435 | Alternate definition of qu... |
| dfqs3 42436 | Alternate definition of qu... |
| qseq12d 42437 | Equality theorem for quoti... |
| qsalrel 42438 | The quotient set is equal ... |
| supinf 42439 | The supremum is the infimu... |
| mapcod 42440 | Compose two mappings. (Co... |
| fisdomnn 42441 | A finite set is dominated ... |
| ltex 42442 | The less-than relation is ... |
| leex 42443 | The less-than-or-equal-to ... |
| subex 42444 | The subtraction operation ... |
| absex 42445 | The absolute value functio... |
| cjex 42446 | The conjugate function is ... |
| fzosumm1 42447 | Separate out the last term... |
| ccatcan2d 42448 | Cancellation law for conca... |
| c0exALT 42449 | Alternate proof of ~ c0ex ... |
| 0cnALT3 42450 | Alternate proof of ~ 0cn u... |
| elre0re 42451 | Specialized version of ~ 0... |
| 1t1e1ALT 42452 | Alternate proof of ~ 1t1e1... |
| lttrii 42453 | 'Less than' is transitive.... |
| remulcan2d 42454 | ~ mulcan2d for real number... |
| readdridaddlidd 42455 | Given some real number ` B... |
| 1p3e4 42456 | 1 + 3 = 4. (Contributed b... |
| 5ne0 42457 | The number 5 is nonzero. ... |
| 6ne0 42458 | The number 6 is nonzero. ... |
| 7ne0 42459 | The number 7 is nonzero. ... |
| 8ne0 42460 | The number 8 is nonzero. ... |
| 9ne0 42461 | The number 9 is nonzero. ... |
| sn-1ne2 42462 | A proof of ~ 1ne2 without ... |
| nnn1suc 42463 | A positive integer that is... |
| nnadd1com 42464 | Addition with 1 is commuta... |
| nnaddcom 42465 | Addition is commutative fo... |
| nnaddcomli 42466 | Version of ~ addcomli for ... |
| nnadddir 42467 | Right-distributivity for n... |
| nnmul1com 42468 | Multiplication with 1 is c... |
| nnmulcom 42469 | Multiplication is commutat... |
| readdrcl2d 42470 | Reverse closure for additi... |
| mvrrsubd 42471 | Move a subtraction in the ... |
| laddrotrd 42472 | Rotate the variables right... |
| raddswap12d 42473 | Swap the first two variabl... |
| lsubrotld 42474 | Rotate the variables left ... |
| rsubrotld 42475 | Rotate the variables left ... |
| lsubswap23d 42476 | Swap the second and third ... |
| addsubeq4com 42477 | Relation between sums and ... |
| sqsumi 42478 | A sum squared. (Contribut... |
| negn0nposznnd 42479 | Lemma for ~ dffltz . (Con... |
| sqmid3api 42480 | Value of the square of the... |
| decaddcom 42481 | Commute ones place in addi... |
| sqn5i 42482 | The square of a number end... |
| sqn5ii 42483 | The square of a number end... |
| decpmulnc 42484 | Partial products algorithm... |
| decpmul 42485 | Partial products algorithm... |
| sqdeccom12 42486 | The square of a number in ... |
| sq3deccom12 42487 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 42488 | 4 times 5 equals 20. (Con... |
| 3rdpwhole 42489 | A third of a number plus t... |
| sq4 42490 | The square of 4 is 16. (C... |
| sq5 42491 | The square of 5 is 25. (C... |
| sq6 42492 | The square of 6 is 36. (C... |
| sq7 42493 | The square of 7 is 49. (C... |
| sq8 42494 | The square of 8 is 64. (C... |
| sq9 42495 | The square of 9 is 81. (C... |
| rpsscn 42496 | The positive reals are a s... |
| 4rp 42497 | 4 is a positive real. (Co... |
| 6rp 42498 | 6 is a positive real. (Co... |
| 7rp 42499 | 7 is a positive real. (Co... |
| 8rp 42500 | 8 is a positive real. (Co... |
| 9rp 42501 | 9 is a positive real. (Co... |
| 235t711 42502 | Calculate a product by lon... |
| ex-decpmul 42503 | Example usage of ~ decpmul... |
| eluzp1 42504 | Membership in a successor ... |
| sn-eluzp1l 42505 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 42506 | The sum of ` N ` constant ... |
| fz1sump1 42507 | Add one more term to a sum... |
| oddnumth 42508 | The Odd Number Theorem. T... |
| nicomachus 42509 | Nicomachus's Theorem. The... |
| sumcubes 42510 | The sum of the first ` N `... |
| ine1 42511 | ` _i ` is not 1. (Contrib... |
| 0tie0 42512 | 0 times ` _i ` equals 0. ... |
| it1ei 42513 | ` _i ` times 1 equals ` _i... |
| 1tiei 42514 | 1 times ` _i ` equals ` _i... |
| itrere 42515 | ` _i ` times a real is rea... |
| retire 42516 | A real times ` _i ` is rea... |
| iocioodisjd 42517 | Adjacent intervals where t... |
| rpabsid 42518 | A positive real is its own... |
| oexpreposd 42519 | Lemma for ~ dffltz . For ... |
| explt1d 42520 | A nonnegative real number ... |
| expeq1d 42521 | A nonnegative real number ... |
| expeqidd 42522 | A nonnegative real number ... |
| exp11d 42523 | ~ exp11nnd for nonzero int... |
| 0dvds0 42524 | 0 divides 0. (Contributed... |
| absdvdsabsb 42525 | Divisibility is invariant ... |
| gcdnn0id 42526 | The ` gcd ` of a nonnegati... |
| gcdle1d 42527 | The greatest common diviso... |
| gcdle2d 42528 | The greatest common diviso... |
| dvdsexpad 42529 | Deduction associated with ... |
| dvdsexpnn 42530 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 42531 | ~ dvdsexpnn generalized to... |
| dvdsexpb 42532 | ~ dvdssq generalized to po... |
| posqsqznn 42533 | When a positive rational s... |
| zdivgd 42534 | Two ways to express " ` N ... |
| efsubd 42535 | Difference of exponents la... |
| ef11d 42536 | General condition for the ... |
| logccne0d 42537 | The logarithm isn't 0 if i... |
| cxp112d 42538 | General condition for comp... |
| cxp111d 42539 | General condition for comp... |
| cxpi11d 42540 | ` _i ` to the powers of ` ... |
| logne0d 42541 | Deduction form of ~ logne0... |
| rxp112d 42542 | Real exponentiation is one... |
| log11d 42543 | The natural logarithm is o... |
| rplog11d 42544 | The natural logarithm is o... |
| rxp11d 42545 | Real exponentiation is one... |
| tanhalfpim 42546 | The tangent of ` _pi / 2 `... |
| sinpim 42547 | Sine of a number subtracte... |
| cospim 42548 | Cosine of a number subtrac... |
| tan3rdpi 42549 | The tangent of ` _pi / 3 `... |
| sin2t3rdpi 42550 | The sine of ` 2 x. ( _pi /... |
| cos2t3rdpi 42551 | The cosine of ` 2 x. ( _pi... |
| sin4t3rdpi 42552 | The sine of ` 4 x. ( _pi /... |
| cos4t3rdpi 42553 | The cosine of ` 4 x. ( _pi... |
| asin1half 42554 | The arcsine of ` 1 / 2 ` i... |
| acos1half 42555 | The arccosine of ` 1 / 2 `... |
| dvun 42556 | Condition for the union of... |
| redvmptabs 42557 | The derivative of the abso... |
| readvrec2 42558 | The antiderivative of 1/x ... |
| readvrec 42559 | For real numbers, the anti... |
| resuppsinopn 42560 | The support of sin ( ~ df-... |
| readvcot 42561 | Real antiderivative of cot... |
| resubval 42564 | Value of real subtraction,... |
| renegeulemv 42565 | Lemma for ~ renegeu and si... |
| renegeulem 42566 | Lemma for ~ renegeu and si... |
| renegeu 42567 | Existential uniqueness of ... |
| rernegcl 42568 | Closure law for negative r... |
| renegadd 42569 | Relationship between real ... |
| renegid 42570 | Addition of a real number ... |
| reneg0addlid 42571 | Negative zero is a left ad... |
| resubeulem1 42572 | Lemma for ~ resubeu . A v... |
| resubeulem2 42573 | Lemma for ~ resubeu . A v... |
| resubeu 42574 | Existential uniqueness of ... |
| rersubcl 42575 | Closure for real subtracti... |
| resubadd 42576 | Relation between real subt... |
| resubaddd 42577 | Relationship between subtr... |
| resubf 42578 | Real subtraction is an ope... |
| repncan2 42579 | Addition and subtraction o... |
| repncan3 42580 | Addition and subtraction o... |
| readdsub 42581 | Law for addition and subtr... |
| reladdrsub 42582 | Move LHS of a sum into RHS... |
| reltsub1 42583 | Subtraction from both side... |
| reltsubadd2 42584 | 'Less than' relationship b... |
| resubcan2 42585 | Cancellation law for real ... |
| resubsub4 42586 | Law for double subtraction... |
| rennncan2 42587 | Cancellation law for real ... |
| renpncan3 42588 | Cancellation law for real ... |
| repnpcan 42589 | Cancellation law for addit... |
| reppncan 42590 | Cancellation law for mixed... |
| resubidaddlidlem 42591 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 42592 | Any real number subtracted... |
| resubdi 42593 | Distribution of multiplica... |
| re1m1e0m0 42594 | Equality of two left-addit... |
| sn-00idlem1 42595 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 42596 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 42597 | Lemma for ~ sn-00id . (Co... |
| sn-00id 42598 | ~ 00id proven without ~ ax... |
| re0m0e0 42599 | Real number version of ~ 0... |
| readdlid 42600 | Real number version of ~ a... |
| sn-addlid 42601 | ~ addlid without ~ ax-mulc... |
| remul02 42602 | Real number version of ~ m... |
| sn-0ne2 42603 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 42604 | Real number version of ~ m... |
| sn-remul0ord 42605 | A product is zero iff one ... |
| resubid 42606 | Subtraction of a real numb... |
| readdrid 42607 | Real number version of ~ a... |
| resubid1 42608 | Real number version of ~ s... |
| renegneg 42609 | A real number is equal to ... |
| readdcan2 42610 | Commuted version of ~ read... |
| renegid2 42611 | Commuted version of ~ rene... |
| remulneg2d 42612 | Product with negative is n... |
| sn-it0e0 42613 | Proof of ~ it0e0 without ~... |
| sn-negex12 42614 | A combination of ~ cnegex ... |
| sn-negex 42615 | Proof of ~ cnegex without ... |
| sn-negex2 42616 | Proof of ~ cnegex2 without... |
| sn-addcand 42617 | ~ addcand without ~ ax-mul... |
| sn-addrid 42618 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 42619 | ~ addcan2d without ~ ax-mu... |
| reixi 42620 | ~ ixi without ~ ax-mulcom ... |
| rei4 42621 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 42622 | A number that sums to itse... |
| sn-mul01 42623 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 42624 | ~ negeu without ~ ax-mulco... |
| sn-subcl 42625 | ~ subcl without ~ ax-mulco... |
| sn-subf 42626 | ~ subf without ~ ax-mulcom... |
| resubeqsub 42627 | Equivalence between real s... |
| subresre 42628 | Subtraction restricted to ... |
| addinvcom 42629 | A number commutes with its... |
| remulinvcom 42630 | A left multiplicative inve... |
| remullid 42631 | Commuted version of ~ ax-1... |
| sn-1ticom 42632 | Lemma for ~ sn-mullid and ... |
| sn-mullid 42633 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 42634 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 42635 | The multiplicative inverse... |
| remulcand 42636 | Commuted version of ~ remu... |
| redivvald 42639 | Value of real division, wh... |
| rediveud 42640 | Existential uniqueness of ... |
| sn-redivcld 42641 | Closure law for real divis... |
| redivmuld 42642 | Relationship between divis... |
| redivcan2d 42643 | A cancellation law for div... |
| redivcan3d 42644 | A cancellation law for div... |
| sn-rereccld 42645 | Closure law for reciprocal... |
| rerecid 42646 | Multiplication of a number... |
| rerecid2 42647 | Multiplication of a number... |
| sn-0tie0 42648 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 42649 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 42650 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 42651 | ~ ltaddneg without ~ ax-mu... |
| reposdif 42652 | Comparison of two numbers ... |
| relt0neg1 42653 | Comparison of a real and i... |
| relt0neg2 42654 | Comparison of a real and i... |
| sn-addlt0d 42655 | The sum of negative number... |
| sn-addgt0d 42656 | The sum of positive number... |
| sn-nnne0 42657 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 42658 | ~ elznn0nn restated using ... |
| nn0addcom 42659 | Addition is commutative fo... |
| zaddcomlem 42660 | Lemma for ~ zaddcom . (Co... |
| zaddcom 42661 | Addition is commutative fo... |
| renegmulnnass 42662 | Move multiplication by a n... |
| nn0mulcom 42663 | Multiplication is commutat... |
| zmulcomlem 42664 | Lemma for ~ zmulcom . (Co... |
| zmulcom 42665 | Multiplication is commutat... |
| mulgt0con1dlem 42666 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 42667 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 42668 | Lemma for ~ mulgt0b1d and ... |
| mulgt0b1d 42669 | Biconditional, deductive f... |
| sn-ltmul2d 42670 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 42671 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 42672 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 42673 | ~ ltp1 without ~ ax-mulcom... |
| sn-recgt0d 42674 | The reciprocal of a positi... |
| mulgt0b2d 42675 | Biconditional, deductive f... |
| sn-mulgt1d 42676 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 42677 | Negative one is a negative... |
| sn-reclt0d 42678 | The reciprocal of a negati... |
| mulltgt0d 42679 | Negative times positive is... |
| mullt0b1d 42680 | When the first term is neg... |
| mullt0b2d 42681 | When the second term is ne... |
| sn-mullt0d 42682 | The product of two negativ... |
| sn-msqgt0d 42683 | A nonzero square is positi... |
| sn-inelr 42684 | ~ inelr without ~ ax-mulco... |
| sn-itrere 42685 | ` _i ` times a real is rea... |
| sn-retire 42686 | Commuted version of ~ sn-i... |
| cnreeu 42687 | The reals in the expressio... |
| sn-sup2 42688 | ~ sup2 with exactly the sa... |
| sn-sup3d 42689 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 42690 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 42691 | ~ suprubd without ~ ax-mul... |
| sn-base0 42692 | Avoid axioms in ~ base0 by... |
| nelsubginvcld 42693 | The inverse of a non-subgr... |
| nelsubgcld 42694 | A non-subgroup-member plus... |
| nelsubgsubcld 42695 | A non-subgroup-member minu... |
| rnasclg 42696 | The set of injected scalar... |
| frlmfielbas 42697 | The vectors of a finite fr... |
| frlmfzwrd 42698 | A vector of a module with ... |
| frlmfzowrd 42699 | A vector of a module with ... |
| frlmfzolen 42700 | The dimension of a vector ... |
| frlmfzowrdb 42701 | The vectors of a module wi... |
| frlmfzoccat 42702 | The concatenation of two v... |
| frlmvscadiccat 42703 | Scalar multiplication dist... |
| grpasscan2d 42704 | An associative cancellatio... |
| grpcominv1 42705 | If two elements commute, t... |
| grpcominv2 42706 | If two elements commute, t... |
| finsubmsubg 42707 | A submonoid of a finite gr... |
| opprmndb 42708 | A class is a monoid if and... |
| opprgrpb 42709 | A class is a group if and ... |
| opprablb 42710 | A class is an Abelian grou... |
| imacrhmcl 42711 | The image of a commutative... |
| rimrcl1 42712 | Reverse closure of a ring ... |
| rimrcl2 42713 | Reverse closure of a ring ... |
| rimcnv 42714 | The converse of a ring iso... |
| rimco 42715 | The composition of ring is... |
| ricsym 42716 | Ring isomorphism is symmet... |
| rictr 42717 | Ring isomorphism is transi... |
| riccrng1 42718 | Ring isomorphism preserves... |
| riccrng 42719 | A ring is commutative if a... |
| domnexpgn0cl 42720 | In a domain, a (nonnegativ... |
| drnginvrn0d 42721 | A multiplicative inverse i... |
| drngmullcan 42722 | Cancellation of a nonzero ... |
| drngmulrcan 42723 | Cancellation of a nonzero ... |
| drnginvmuld 42724 | Inverse of a nonzero produ... |
| ricdrng1 42725 | A ring isomorphism maps a ... |
| ricdrng 42726 | A ring is a division ring ... |
| ricfld 42727 | A ring is a field if and o... |
| asclf1 42728 | Two ways of saying the sca... |
| abvexp 42729 | Move exponentiation in and... |
| fimgmcyclem 42730 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 42731 | Version of ~ odcl2 for fin... |
| fidomncyc 42732 | Version of ~ odcl2 for mul... |
| fiabv 42733 | In a finite domain (a fini... |
| lvecgrp 42734 | A vector space is a group.... |
| lvecring 42735 | The scalar component of a ... |
| frlm0vald 42736 | All coordinates of the zer... |
| frlmsnic 42737 | Given a free module with a... |
| uvccl 42738 | A unit vector is a vector.... |
| uvcn0 42739 | A unit vector is nonzero. ... |
| psrmnd 42740 | The ring of power series i... |
| psrbagres 42741 | Restrict a bag of variable... |
| mplcrngd 42742 | The polynomial ring is a c... |
| mplsubrgcl 42743 | An element of a polynomial... |
| mhmcopsr 42744 | The composition of a monoi... |
| mhmcoaddpsr 42745 | Show that the ring homomor... |
| rhmcomulpsr 42746 | Show that the ring homomor... |
| rhmpsr 42747 | Provide a ring homomorphis... |
| rhmpsr1 42748 | Provide a ring homomorphis... |
| mplmapghm 42749 | The function ` H ` mapping... |
| evl0 42750 | The zero polynomial evalua... |
| evlscl 42751 | A polynomial over the ring... |
| evlsscaval 42752 | Polynomial evaluation buil... |
| evlsvarval 42753 | Polynomial evaluation buil... |
| evlsbagval 42754 | Polynomial evaluation buil... |
| evlsexpval 42755 | Polynomial evaluation buil... |
| evlsaddval 42756 | Polynomial evaluation buil... |
| evlsmulval 42757 | Polynomial evaluation buil... |
| evlsmaprhm 42758 | The function ` F ` mapping... |
| evlsevl 42759 | Evaluation in a subring is... |
| evlvvval 42760 | Give a formula for the eva... |
| evlvvvallem 42761 | Lemma for theorems using ~... |
| selvcllem1 42762 | ` T ` is an associative al... |
| selvcllem2 42763 | ` D ` is a ring homomorphi... |
| selvcllem3 42764 | The third argument passed ... |
| selvcllemh 42765 | Apply the third argument (... |
| selvcllem4 42766 | The fourth argument passed... |
| selvcllem5 42767 | The fifth argument passed ... |
| selvcl 42768 | Closure of the "variable s... |
| selvval2 42769 | Value of the "variable sel... |
| selvvvval 42770 | Recover the original polyn... |
| evlselvlem 42771 | Lemma for ~ evlselv . Use... |
| evlselv 42772 | Evaluating a selection of ... |
| selvadd 42773 | The "variable selection" f... |
| selvmul 42774 | The "variable selection" f... |
| fsuppind 42775 | Induction on functions ` F... |
| fsuppssindlem1 42776 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 42777 | Lemma for ~ fsuppssind . ... |
| fsuppssind 42778 | Induction on functions ` F... |
| mhpind 42779 | The homogeneous polynomial... |
| evlsmhpvvval 42780 | Give a formula for the eva... |
| mhphflem 42781 | Lemma for ~ mhphf . Add s... |
| mhphf 42782 | A homogeneous polynomial d... |
| mhphf2 42783 | A homogeneous polynomial d... |
| mhphf3 42784 | A homogeneous polynomial d... |
| mhphf4 42785 | A homogeneous polynomial d... |
| prjspval 42788 | Value of the projective sp... |
| prjsprel 42789 | Utility theorem regarding ... |
| prjspertr 42790 | The relation in ` PrjSp ` ... |
| prjsperref 42791 | The relation in ` PrjSp ` ... |
| prjspersym 42792 | The relation in ` PrjSp ` ... |
| prjsper 42793 | The relation used to defin... |
| prjspreln0 42794 | Two nonzero vectors are eq... |
| prjspvs 42795 | A nonzero multiple of a ve... |
| prjsprellsp 42796 | Two vectors are equivalent... |
| prjspeclsp 42797 | The vectors equivalent to ... |
| prjspval2 42798 | Alternate definition of pr... |
| prjspnval 42801 | Value of the n-dimensional... |
| prjspnerlem 42802 | A lemma showing that the e... |
| prjspnval2 42803 | Value of the n-dimensional... |
| prjspner 42804 | The relation used to defin... |
| prjspnvs 42805 | A nonzero multiple of a ve... |
| prjspnssbas 42806 | A projective point spans a... |
| prjspnn0 42807 | A projective point is none... |
| 0prjspnlem 42808 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 42809 | Any vector is equivalent t... |
| prjspner01 42810 | Any vector is equivalent t... |
| prjspner1 42811 | Two vectors whose zeroth c... |
| 0prjspnrel 42812 | In the zero-dimensional pr... |
| 0prjspn 42813 | A zero-dimensional project... |
| prjcrvfval 42816 | Value of the projective cu... |
| prjcrvval 42817 | Value of the projective cu... |
| prjcrv0 42818 | The "curve" (zero set) cor... |
| dffltz 42819 | Fermat's Last Theorem (FLT... |
| fltmul 42820 | A counterexample to FLT st... |
| fltdiv 42821 | A counterexample to FLT st... |
| flt0 42822 | A counterexample for FLT d... |
| fltdvdsabdvdsc 42823 | Any factor of both ` A ` a... |
| fltabcoprmex 42824 | A counterexample to FLT im... |
| fltaccoprm 42825 | A counterexample to FLT wi... |
| fltbccoprm 42826 | A counterexample to FLT wi... |
| fltabcoprm 42827 | A counterexample to FLT wi... |
| infdesc 42828 | Infinite descent. The hyp... |
| fltne 42829 | If a counterexample to FLT... |
| flt4lem 42830 | Raising a number to the fo... |
| flt4lem1 42831 | Satisfy the antecedent use... |
| flt4lem2 42832 | If ` A ` is even, ` B ` is... |
| flt4lem3 42833 | Equivalent to ~ pythagtrip... |
| flt4lem4 42834 | If the product of two copr... |
| flt4lem5 42835 | In the context of the lemm... |
| flt4lem5elem 42836 | Version of ~ fltaccoprm an... |
| flt4lem5a 42837 | Part 1 of Equation 1 of ... |
| flt4lem5b 42838 | Part 2 of Equation 1 of ... |
| flt4lem5c 42839 | Part 2 of Equation 2 of ... |
| flt4lem5d 42840 | Part 3 of Equation 2 of ... |
| flt4lem5e 42841 | Satisfy the hypotheses of ... |
| flt4lem5f 42842 | Final equation of ~... |
| flt4lem6 42843 | Remove shared factors in a... |
| flt4lem7 42844 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 42845 | Strengthening of Fermat's ... |
| fltltc 42846 | ` ( C ^ N ) ` is the large... |
| fltnltalem 42847 | Lemma for ~ fltnlta . A l... |
| fltnlta 42848 | In a Fermat counterexample... |
| iddii 42849 | Version of ~ a1ii with the... |
| bicomdALT 42850 | Alternate proof of ~ bicom... |
| alan 42851 | Alias for ~ 19.26 for easi... |
| exor 42852 | Alias for ~ 19.43 for easi... |
| rexor 42853 | Alias for ~ r19.43 for eas... |
| ruvALT 42854 | Alternate proof of ~ ruv w... |
| sn-wcdeq 42855 | Alternative to ~ wcdeq and... |
| sq45 42856 | 45 squared is 2025. (Cont... |
| sum9cubes 42857 | The sum of the first nine ... |
| sn-isghm 42858 | Longer proof of ~ isghm , ... |
| aprilfools2025 42859 | An abuse of notation. (Co... |
| nfa1w 42860 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 42861 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 42862 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 42863 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 42864 | Replace ~ ax-10 , ~ ax-11 ... |
| cu3addd 42865 | Cube of sum of three numbe... |
| negexpidd 42866 | The sum of a real number t... |
| rexlimdv3d 42867 | An extended version of ~ r... |
| 3cubeslem1 42868 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 42869 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 42870 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 42871 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 42872 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 42873 | Lemma for ~ 3cubes . This... |
| 3cubes 42874 | Every rational number is a... |
| rntrclfvOAI 42875 | The range of the transitiv... |
| moxfr 42876 | Transfer at-most-one betwe... |
| imaiinfv 42877 | Indexed intersection of an... |
| elrfi 42878 | Elementhood in a set of re... |
| elrfirn 42879 | Elementhood in a set of re... |
| elrfirn2 42880 | Elementhood in a set of re... |
| cmpfiiin 42881 | In a compact topology, a s... |
| ismrcd1 42882 | Any function from the subs... |
| ismrcd2 42883 | Second half of ~ ismrcd1 .... |
| istopclsd 42884 | A closure function which s... |
| ismrc 42885 | A function is a Moore clos... |
| isnacs 42888 | Expand definition of Noeth... |
| nacsfg 42889 | In a Noetherian-type closu... |
| isnacs2 42890 | Express Noetherian-type cl... |
| mrefg2 42891 | Slight variation on finite... |
| mrefg3 42892 | Slight variation on finite... |
| nacsacs 42893 | A closure system of Noethe... |
| isnacs3 42894 | A choice-free order equiva... |
| incssnn0 42895 | Transitivity induction of ... |
| nacsfix 42896 | An increasing sequence of ... |
| constmap 42897 | A constant (represented wi... |
| mapco2g 42898 | Renaming indices in a tupl... |
| mapco2 42899 | Post-composition (renaming... |
| mapfzcons 42900 | Extending a one-based mapp... |
| mapfzcons1 42901 | Recover prefix mapping fro... |
| mapfzcons1cl 42902 | A nonempty mapping has a p... |
| mapfzcons2 42903 | Recover added element from... |
| mptfcl 42904 | Interpret range of a maps-... |
| mzpclval 42909 | Substitution lemma for ` m... |
| elmzpcl 42910 | Double substitution lemma ... |
| mzpclall 42911 | The set of all functions w... |
| mzpcln0 42912 | Corollary of ~ mzpclall : ... |
| mzpcl1 42913 | Defining property 1 of a p... |
| mzpcl2 42914 | Defining property 2 of a p... |
| mzpcl34 42915 | Defining properties 3 and ... |
| mzpval 42916 | Value of the ` mzPoly ` fu... |
| dmmzp 42917 | ` mzPoly ` is defined for ... |
| mzpincl 42918 | Polynomial closedness is a... |
| mzpconst 42919 | Constant functions are pol... |
| mzpf 42920 | A polynomial function is a... |
| mzpproj 42921 | A projection function is p... |
| mzpadd 42922 | The pointwise sum of two p... |
| mzpmul 42923 | The pointwise product of t... |
| mzpconstmpt 42924 | A constant function expres... |
| mzpaddmpt 42925 | Sum of polynomial function... |
| mzpmulmpt 42926 | Product of polynomial func... |
| mzpsubmpt 42927 | The difference of two poly... |
| mzpnegmpt 42928 | Negation of a polynomial f... |
| mzpexpmpt 42929 | Raise a polynomial functio... |
| mzpindd 42930 | "Structural" induction to ... |
| mzpmfp 42931 | Relationship between multi... |
| mzpsubst 42932 | Substituting polynomials f... |
| mzprename 42933 | Simplified version of ~ mz... |
| mzpresrename 42934 | A polynomial is a polynomi... |
| mzpcompact2lem 42935 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 42936 | Polynomials are finitary o... |
| coeq0i 42937 | ~ coeq0 but without explic... |
| fzsplit1nn0 42938 | Split a finite 1-based set... |
| eldiophb 42941 | Initial expression of Diop... |
| eldioph 42942 | Condition for a set to be ... |
| diophrw 42943 | Renaming and adding unused... |
| eldioph2lem1 42944 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 42945 | Lemma for ~ eldioph2 . Co... |
| eldioph2 42946 | Construct a Diophantine se... |
| eldioph2b 42947 | While Diophantine sets wer... |
| eldiophelnn0 42948 | Remove antecedent on ` B `... |
| eldioph3b 42949 | Define Diophantine sets in... |
| eldioph3 42950 | Inference version of ~ eld... |
| ellz1 42951 | Membership in a lower set ... |
| lzunuz 42952 | The union of a lower set o... |
| fz1eqin 42953 | Express a one-based finite... |
| lzenom 42954 | Lower integers are countab... |
| elmapresaunres2 42955 | ~ fresaunres2 transposed t... |
| diophin 42956 | If two sets are Diophantin... |
| diophun 42957 | If two sets are Diophantin... |
| eldiophss 42958 | Diophantine sets are sets ... |
| diophrex 42959 | Projecting a Diophantine s... |
| eq0rabdioph 42960 | This is the first of a num... |
| eqrabdioph 42961 | Diophantine set builder fo... |
| 0dioph 42962 | The null set is Diophantin... |
| vdioph 42963 | The "universal" set (as la... |
| anrabdioph 42964 | Diophantine set builder fo... |
| orrabdioph 42965 | Diophantine set builder fo... |
| 3anrabdioph 42966 | Diophantine set builder fo... |
| 3orrabdioph 42967 | Diophantine set builder fo... |
| 2sbcrex 42968 | Exchange an existential qu... |
| sbcrexgOLD 42969 | Interchange class substitu... |
| 2sbcrexOLD 42970 | Exchange an existential qu... |
| sbc2rex 42971 | Exchange a substitution wi... |
| sbc2rexgOLD 42972 | Exchange a substitution wi... |
| sbc4rex 42973 | Exchange a substitution wi... |
| sbc4rexgOLD 42974 | Exchange a substitution wi... |
| sbcrot3 42975 | Rotate a sequence of three... |
| sbcrot5 42976 | Rotate a sequence of five ... |
| sbccomieg 42977 | Commute two explicit subst... |
| rexrabdioph 42978 | Diophantine set builder fo... |
| rexfrabdioph 42979 | Diophantine set builder fo... |
| 2rexfrabdioph 42980 | Diophantine set builder fo... |
| 3rexfrabdioph 42981 | Diophantine set builder fo... |
| 4rexfrabdioph 42982 | Diophantine set builder fo... |
| 6rexfrabdioph 42983 | Diophantine set builder fo... |
| 7rexfrabdioph 42984 | Diophantine set builder fo... |
| rabdiophlem1 42985 | Lemma for arithmetic dioph... |
| rabdiophlem2 42986 | Lemma for arithmetic dioph... |
| elnn0rabdioph 42987 | Diophantine set builder fo... |
| rexzrexnn0 42988 | Rewrite an existential qua... |
| lerabdioph 42989 | Diophantine set builder fo... |
| eluzrabdioph 42990 | Diophantine set builder fo... |
| elnnrabdioph 42991 | Diophantine set builder fo... |
| ltrabdioph 42992 | Diophantine set builder fo... |
| nerabdioph 42993 | Diophantine set builder fo... |
| dvdsrabdioph 42994 | Divisibility is a Diophant... |
| eldioph4b 42995 | Membership in ` Dioph ` ex... |
| eldioph4i 42996 | Forward-only version of ~ ... |
| diophren 42997 | Change variables in a Diop... |
| rabrenfdioph 42998 | Change variable numbers in... |
| rabren3dioph 42999 | Change variable numbers in... |
| fphpd 43000 | Pigeonhole principle expre... |
| fphpdo 43001 | Pigeonhole principle for s... |
| ctbnfien 43002 | An infinite subset of a co... |
| fiphp3d 43003 | Infinite pigeonhole princi... |
| rencldnfilem 43004 | Lemma for ~ rencldnfi . (... |
| rencldnfi 43005 | A set of real numbers whic... |
| irrapxlem1 43006 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 43007 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 43008 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 43009 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 43010 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 43011 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 43012 | Dirichlet's approximation ... |
| pellexlem1 43013 | Lemma for ~ pellex . Arit... |
| pellexlem2 43014 | Lemma for ~ pellex . Arit... |
| pellexlem3 43015 | Lemma for ~ pellex . To e... |
| pellexlem4 43016 | Lemma for ~ pellex . Invo... |
| pellexlem5 43017 | Lemma for ~ pellex . Invo... |
| pellexlem6 43018 | Lemma for ~ pellex . Doin... |
| pellex 43019 | Every Pell equation has a ... |
| pell1qrval 43030 | Value of the set of first-... |
| elpell1qr 43031 | Membership in a first-quad... |
| pell14qrval 43032 | Value of the set of positi... |
| elpell14qr 43033 | Membership in the set of p... |
| pell1234qrval 43034 | Value of the set of genera... |
| elpell1234qr 43035 | Membership in the set of g... |
| pell1234qrre 43036 | General Pell solutions are... |
| pell1234qrne0 43037 | No solution to a Pell equa... |
| pell1234qrreccl 43038 | General solutions of the P... |
| pell1234qrmulcl 43039 | General solutions of the P... |
| pell14qrss1234 43040 | A positive Pell solution i... |
| pell14qrre 43041 | A positive Pell solution i... |
| pell14qrne0 43042 | A positive Pell solution i... |
| pell14qrgt0 43043 | A positive Pell solution i... |
| pell14qrrp 43044 | A positive Pell solution i... |
| pell1234qrdich 43045 | A general Pell solution is... |
| elpell14qr2 43046 | A number is a positive Pel... |
| pell14qrmulcl 43047 | Positive Pell solutions ar... |
| pell14qrreccl 43048 | Positive Pell solutions ar... |
| pell14qrdivcl 43049 | Positive Pell solutions ar... |
| pell14qrexpclnn0 43050 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 43051 | Positive Pell solutions ar... |
| pell1qrss14 43052 | First-quadrant Pell soluti... |
| pell14qrdich 43053 | A positive Pell solution i... |
| pell1qrge1 43054 | A Pell solution in the fir... |
| pell1qr1 43055 | 1 is a Pell solution and i... |
| elpell1qr2 43056 | The first quadrant solutio... |
| pell1qrgaplem 43057 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 43058 | First-quadrant Pell soluti... |
| pell14qrgap 43059 | Positive Pell solutions ar... |
| pell14qrgapw 43060 | Positive Pell solutions ar... |
| pellqrexplicit 43061 | Condition for a calculated... |
| infmrgelbi 43062 | Any lower bound of a nonem... |
| pellqrex 43063 | There is a nontrivial solu... |
| pellfundval 43064 | Value of the fundamental s... |
| pellfundre 43065 | The fundamental solution o... |
| pellfundge 43066 | Lower bound on the fundame... |
| pellfundgt1 43067 | Weak lower bound on the Pe... |
| pellfundlb 43068 | A nontrivial first quadran... |
| pellfundglb 43069 | If a real is larger than t... |
| pellfundex 43070 | The fundamental solution a... |
| pellfund14gap 43071 | There are no solutions bet... |
| pellfundrp 43072 | The fundamental Pell solut... |
| pellfundne1 43073 | The fundamental Pell solut... |
| reglogcl 43074 | General logarithm is a rea... |
| reglogltb 43075 | General logarithm preserve... |
| reglogleb 43076 | General logarithm preserve... |
| reglogmul 43077 | Multiplication law for gen... |
| reglogexp 43078 | Power law for general log.... |
| reglogbas 43079 | General log of the base is... |
| reglog1 43080 | General log of 1 is 0. (C... |
| reglogexpbas 43081 | General log of a power of ... |
| pellfund14 43082 | Every positive Pell soluti... |
| pellfund14b 43083 | The positive Pell solution... |
| rmxfval 43088 | Value of the X sequence. ... |
| rmyfval 43089 | Value of the Y sequence. ... |
| rmspecsqrtnq 43090 | The discriminant used to d... |
| rmspecnonsq 43091 | The discriminant used to d... |
| qirropth 43092 | This lemma implements the ... |
| rmspecfund 43093 | The base of exponent used ... |
| rmxyelqirr 43094 | The solutions used to cons... |
| rmxypairf1o 43095 | The function used to extra... |
| rmxyelxp 43096 | Lemma for ~ frmx and ~ frm... |
| frmx 43097 | The X sequence is a nonneg... |
| frmy 43098 | The Y sequence is an integ... |
| rmxyval 43099 | Main definition of the X a... |
| rmspecpos 43100 | The discriminant used to d... |
| rmxycomplete 43101 | The X and Y sequences take... |
| rmxynorm 43102 | The X and Y sequences defi... |
| rmbaserp 43103 | The base of exponentiation... |
| rmxyneg 43104 | Negation law for X and Y s... |
| rmxyadd 43105 | Addition formula for X and... |
| rmxy1 43106 | Value of the X and Y seque... |
| rmxy0 43107 | Value of the X and Y seque... |
| rmxneg 43108 | Negation law (even functio... |
| rmx0 43109 | Value of X sequence at 0. ... |
| rmx1 43110 | Value of X sequence at 1. ... |
| rmxadd 43111 | Addition formula for X seq... |
| rmyneg 43112 | Negation formula for Y seq... |
| rmy0 43113 | Value of Y sequence at 0. ... |
| rmy1 43114 | Value of Y sequence at 1. ... |
| rmyadd 43115 | Addition formula for Y seq... |
| rmxp1 43116 | Special addition-of-1 form... |
| rmyp1 43117 | Special addition of 1 form... |
| rmxm1 43118 | Subtraction of 1 formula f... |
| rmym1 43119 | Subtraction of 1 formula f... |
| rmxluc 43120 | The X sequence is a Lucas ... |
| rmyluc 43121 | The Y sequence is a Lucas ... |
| rmyluc2 43122 | Lucas sequence property of... |
| rmxdbl 43123 | "Double-angle formula" for... |
| rmydbl 43124 | "Double-angle formula" for... |
| monotuz 43125 | A function defined on an u... |
| monotoddzzfi 43126 | A function which is odd an... |
| monotoddzz 43127 | A function (given implicit... |
| oddcomabszz 43128 | An odd function which take... |
| 2nn0ind 43129 | Induction on nonnegative i... |
| zindbi 43130 | Inductively transfer a pro... |
| rmxypos 43131 | For all nonnegative indice... |
| ltrmynn0 43132 | The Y-sequence is strictly... |
| ltrmxnn0 43133 | The X-sequence is strictly... |
| lermxnn0 43134 | The X-sequence is monotoni... |
| rmxnn 43135 | The X-sequence is defined ... |
| ltrmy 43136 | The Y-sequence is strictly... |
| rmyeq0 43137 | Y is zero only at zero. (... |
| rmyeq 43138 | Y is one-to-one. (Contrib... |
| lermy 43139 | Y is monotonic (non-strict... |
| rmynn 43140 | ` rmY ` is positive for po... |
| rmynn0 43141 | ` rmY ` is nonnegative for... |
| rmyabs 43142 | ` rmY ` commutes with ` ab... |
| jm2.24nn 43143 | X(n) is strictly greater t... |
| jm2.17a 43144 | First half of lemma 2.17 o... |
| jm2.17b 43145 | Weak form of the second ha... |
| jm2.17c 43146 | Second half of lemma 2.17 ... |
| jm2.24 43147 | Lemma 2.24 of [JonesMatija... |
| rmygeid 43148 | Y(n) increases faster than... |
| congtr 43149 | A wff of the form ` A || (... |
| congadd 43150 | If two pairs of numbers ar... |
| congmul 43151 | If two pairs of numbers ar... |
| congsym 43152 | Congruence mod ` A ` is a ... |
| congneg 43153 | If two integers are congru... |
| congsub 43154 | If two pairs of numbers ar... |
| congid 43155 | Every integer is congruent... |
| mzpcong 43156 | Polynomials commute with c... |
| congrep 43157 | Every integer is congruent... |
| congabseq 43158 | If two integers are congru... |
| acongid 43159 | A wff like that in this th... |
| acongsym 43160 | Symmetry of alternating co... |
| acongneg2 43161 | Negate right side of alter... |
| acongtr 43162 | Transitivity of alternatin... |
| acongeq12d 43163 | Substitution deduction for... |
| acongrep 43164 | Every integer is alternati... |
| fzmaxdif 43165 | Bound on the difference be... |
| fzneg 43166 | Reflection of a finite ran... |
| acongeq 43167 | Two numbers in the fundame... |
| dvdsacongtr 43168 | Alternating congruence pas... |
| coprmdvdsb 43169 | Multiplication by a coprim... |
| modabsdifz 43170 | Divisibility in terms of m... |
| dvdsabsmod0 43171 | Divisibility in terms of m... |
| jm2.18 43172 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 43173 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 43174 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 43175 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 43176 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 43177 | Lemma 2.19 of [JonesMatija... |
| jm2.21 43178 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 43179 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 43180 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 43181 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 43182 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 43183 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 43184 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 43185 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 43186 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 43187 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 43188 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 43189 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 43190 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 43191 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 43192 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 43193 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 43194 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 43195 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 43196 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 43197 | Lemma for ~ rmydioph . Us... |
| rmydioph 43198 | ~ jm2.27 restated in terms... |
| rmxdiophlem 43199 | X can be expressed in term... |
| rmxdioph 43200 | X is a Diophantine functio... |
| jm3.1lem1 43201 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 43202 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 43203 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 43204 | Diophantine expression for... |
| expdiophlem1 43205 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 43206 | Lemma for ~ expdioph . Ex... |
| expdioph 43207 | The exponential function i... |
| setindtr 43208 | Set induction for sets con... |
| setindtrs 43209 | Set induction scheme witho... |
| dford3lem1 43210 | Lemma for ~ dford3 . (Con... |
| dford3lem2 43211 | Lemma for ~ dford3 . (Con... |
| dford3 43212 | Ordinals are precisely the... |
| dford4 43213 | ~ dford3 expressed in prim... |
| wopprc 43214 | Unrelated: Wiener pairs t... |
| rpnnen3lem 43215 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 43216 | Dedekind cut injection of ... |
| axac10 43217 | Characterization of choice... |
| harinf 43218 | The Hartogs number of an i... |
| wdom2d2 43219 | Deduction for weak dominan... |
| ttac 43220 | Tarski's theorem about cho... |
| pw2f1ocnv 43221 | Define a bijection between... |
| pw2f1o2 43222 | Define a bijection between... |
| pw2f1o2val 43223 | Function value of the ~ pw... |
| pw2f1o2val2 43224 | Membership in a mapped set... |
| limsuc2 43225 | Limit ordinals in the sens... |
| wepwsolem 43226 | Transfer an ordering on ch... |
| wepwso 43227 | A well-ordering induces a ... |
| dnnumch1 43228 | Define an enumeration of a... |
| dnnumch2 43229 | Define an enumeration (wea... |
| dnnumch3lem 43230 | Value of the ordinal injec... |
| dnnumch3 43231 | Define an injection from a... |
| dnwech 43232 | Define a well-ordering fro... |
| fnwe2val 43233 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 43234 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 43235 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 43236 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 43237 | A well-ordering can be con... |
| aomclem1 43238 | Lemma for ~ dfac11 . This... |
| aomclem2 43239 | Lemma for ~ dfac11 . Succ... |
| aomclem3 43240 | Lemma for ~ dfac11 . Succ... |
| aomclem4 43241 | Lemma for ~ dfac11 . Limi... |
| aomclem5 43242 | Lemma for ~ dfac11 . Comb... |
| aomclem6 43243 | Lemma for ~ dfac11 . Tran... |
| aomclem7 43244 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 43245 | Lemma for ~ dfac11 . Perf... |
| dfac11 43246 | The right-hand side of thi... |
| kelac1 43247 | Kelley's choice, basic for... |
| kelac2lem 43248 | Lemma for ~ kelac2 and ~ d... |
| kelac2 43249 | Kelley's choice, most comm... |
| dfac21 43250 | Tychonoff's theorem is a c... |
| islmodfg 43253 | Property of a finitely gen... |
| islssfg 43254 | Property of a finitely gen... |
| islssfg2 43255 | Property of a finitely gen... |
| islssfgi 43256 | Finitely spanned subspaces... |
| fglmod 43257 | Finitely generated left mo... |
| lsmfgcl 43258 | The sum of two finitely ge... |
| islnm 43261 | Property of being a Noethe... |
| islnm2 43262 | Property of being a Noethe... |
| lnmlmod 43263 | A Noetherian left module i... |
| lnmlssfg 43264 | A submodule of Noetherian ... |
| lnmlsslnm 43265 | All submodules of a Noethe... |
| lnmfg 43266 | A Noetherian left module i... |
| kercvrlsm 43267 | The domain of a linear fun... |
| lmhmfgima 43268 | A homomorphism maps finite... |
| lnmepi 43269 | Epimorphic images of Noeth... |
| lmhmfgsplit 43270 | If the kernel and range of... |
| lmhmlnmsplit 43271 | If the kernel and range of... |
| lnmlmic 43272 | Noetherian is an invariant... |
| pwssplit4 43273 | Splitting for structure po... |
| filnm 43274 | Finite left modules are No... |
| pwslnmlem0 43275 | Zeroeth powers are Noether... |
| pwslnmlem1 43276 | First powers are Noetheria... |
| pwslnmlem2 43277 | A sum of powers is Noether... |
| pwslnm 43278 | Finite powers of Noetheria... |
| unxpwdom3 43279 | Weaker version of ~ unxpwd... |
| pwfi2f1o 43280 | The ~ pw2f1o bijection rel... |
| pwfi2en 43281 | Finitely supported indicat... |
| frlmpwfi 43282 | Formal linear combinations... |
| gicabl 43283 | Being Abelian is a group i... |
| imasgim 43284 | A relabeling of the elemen... |
| isnumbasgrplem1 43285 | A set which is equipollent... |
| harn0 43286 | The Hartogs number of a se... |
| numinfctb 43287 | A numerable infinite set c... |
| isnumbasgrplem2 43288 | If the (to be thought of a... |
| isnumbasgrplem3 43289 | Every nonempty numerable s... |
| isnumbasabl 43290 | A set is numerable iff it ... |
| isnumbasgrp 43291 | A set is numerable iff it ... |
| dfacbasgrp 43292 | A choice equivalent in abs... |
| islnr 43295 | Property of a left-Noether... |
| lnrring 43296 | Left-Noetherian rings are ... |
| lnrlnm 43297 | Left-Noetherian rings have... |
| islnr2 43298 | Property of being a left-N... |
| islnr3 43299 | Relate left-Noetherian rin... |
| lnr2i 43300 | Given an ideal in a left-N... |
| lpirlnr 43301 | Left principal ideal rings... |
| lnrfrlm 43302 | Finite-dimensional free mo... |
| lnrfg 43303 | Finitely-generated modules... |
| lnrfgtr 43304 | A submodule of a finitely ... |
| hbtlem1 43307 | Value of the leading coeff... |
| hbtlem2 43308 | Leading coefficient ideals... |
| hbtlem7 43309 | Functionality of leading c... |
| hbtlem4 43310 | The leading ideal function... |
| hbtlem3 43311 | The leading ideal function... |
| hbtlem5 43312 | The leading ideal function... |
| hbtlem6 43313 | There is a finite set of p... |
| hbt 43314 | The Hilbert Basis Theorem ... |
| dgrsub2 43319 | Subtracting two polynomial... |
| elmnc 43320 | Property of a monic polyno... |
| mncply 43321 | A monic polynomial is a po... |
| mnccoe 43322 | A monic polynomial has lea... |
| mncn0 43323 | A monic polynomial is not ... |
| dgraaval 43328 | Value of the degree functi... |
| dgraalem 43329 | Properties of the degree o... |
| dgraacl 43330 | Closure of the degree func... |
| dgraaf 43331 | Degree function on algebra... |
| dgraaub 43332 | Upper bound on degree of a... |
| dgraa0p 43333 | A rational polynomial of d... |
| mpaaeu 43334 | An algebraic number has ex... |
| mpaaval 43335 | Value of the minimal polyn... |
| mpaalem 43336 | Properties of the minimal ... |
| mpaacl 43337 | Minimal polynomial is a po... |
| mpaadgr 43338 | Minimal polynomial has deg... |
| mpaaroot 43339 | The minimal polynomial of ... |
| mpaamn 43340 | Minimal polynomial is moni... |
| itgoval 43345 | Value of the integral-over... |
| aaitgo 43346 | The standard algebraic num... |
| itgoss 43347 | An integral element is int... |
| itgocn 43348 | All integral elements are ... |
| cnsrexpcl 43349 | Exponentiation is closed i... |
| fsumcnsrcl 43350 | Finite sums are closed in ... |
| cnsrplycl 43351 | Polynomials are closed in ... |
| rgspnid 43352 | The span of a subring is i... |
| rngunsnply 43353 | Adjoining one element to a... |
| flcidc 43354 | Finite linear combinations... |
| algstr 43357 | Lemma to shorten proofs of... |
| algbase 43358 | The base set of a construc... |
| algaddg 43359 | The additive operation of ... |
| algmulr 43360 | The multiplicative operati... |
| algsca 43361 | The set of scalars of a co... |
| algvsca 43362 | The scalar product operati... |
| mendval 43363 | Value of the module endomo... |
| mendbas 43364 | Base set of the module end... |
| mendplusgfval 43365 | Addition in the module end... |
| mendplusg 43366 | A specific addition in the... |
| mendmulrfval 43367 | Multiplication in the modu... |
| mendmulr 43368 | A specific multiplication ... |
| mendsca 43369 | The module endomorphism al... |
| mendvscafval 43370 | Scalar multiplication in t... |
| mendvsca 43371 | A specific scalar multipli... |
| mendring 43372 | The module endomorphism al... |
| mendlmod 43373 | The module endomorphism al... |
| mendassa 43374 | The module endomorphism al... |
| idomodle 43375 | Limit on the number of ` N... |
| fiuneneq 43376 | Two finite sets of equal s... |
| idomsubgmo 43377 | The units of an integral d... |
| proot1mul 43378 | Any primitive ` N ` -th ro... |
| proot1hash 43379 | If an integral domain has ... |
| proot1ex 43380 | The complex field has prim... |
| mon1psubm 43383 | Monic polynomials are a mu... |
| deg1mhm 43384 | Homomorphic property of th... |
| cytpfn 43385 | Functionality of the cyclo... |
| cytpval 43386 | Substitutions for the Nth ... |
| fgraphopab 43387 | Express a function as a su... |
| fgraphxp 43388 | Express a function as a su... |
| hausgraph 43389 | The graph of a continuous ... |
| r1sssucd 43394 | Deductive form of ~ r1sssu... |
| iocunico 43395 | Split an open interval int... |
| iocinico 43396 | The intersection of two se... |
| iocmbl 43397 | An open-below, closed-abov... |
| cnioobibld 43398 | A bounded, continuous func... |
| arearect 43399 | The area of a rectangle wh... |
| areaquad 43400 | The area of a quadrilatera... |
| uniel 43401 | Two ways to say a union is... |
| unielss 43402 | Two ways to say the union ... |
| unielid 43403 | Two ways to say the union ... |
| ssunib 43404 | Two ways to say a class is... |
| rp-intrabeq 43405 | Equality theorem for supre... |
| rp-unirabeq 43406 | Equality theorem for infim... |
| onmaxnelsup 43407 | Two ways to say the maximu... |
| onsupneqmaxlim0 43408 | If the supremum of a class... |
| onsupcl2 43409 | The supremum of a set of o... |
| onuniintrab 43410 | The union of a set of ordi... |
| onintunirab 43411 | The intersection of a non-... |
| onsupnmax 43412 | If the union of a class of... |
| onsupuni 43413 | The supremum of a set of o... |
| onsupuni2 43414 | The supremum of a set of o... |
| onsupintrab 43415 | The supremum of a set of o... |
| onsupintrab2 43416 | The supremum of a set of o... |
| onsupcl3 43417 | The supremum of a set of o... |
| onsupex3 43418 | The supremum of a set of o... |
| onuniintrab2 43419 | The union of a set of ordi... |
| oninfint 43420 | The infimum of a non-empty... |
| oninfunirab 43421 | The infimum of a non-empty... |
| oninfcl2 43422 | The infimum of a non-empty... |
| onsupmaxb 43423 | The union of a class of or... |
| onexgt 43424 | For any ordinal, there is ... |
| onexomgt 43425 | For any ordinal, there is ... |
| omlimcl2 43426 | The product of a limit ord... |
| onexlimgt 43427 | For any ordinal, there is ... |
| onexoegt 43428 | For any ordinal, there is ... |
| oninfex2 43429 | The infimum of a non-empty... |
| onsupeqmax 43430 | Condition when the supremu... |
| onsupeqnmax 43431 | Condition when the supremu... |
| onsuplub 43432 | The supremum of a set of o... |
| onsupnub 43433 | An upper bound of a set of... |
| onfisupcl 43434 | Sufficient condition when ... |
| onelord 43435 | Every element of a ordinal... |
| onepsuc 43436 | Every ordinal is less than... |
| epsoon 43437 | The ordinals are strictly ... |
| epirron 43438 | The strict order on the or... |
| oneptr 43439 | The strict order on the or... |
| oneltr 43440 | The elementhood relation o... |
| oneptri 43441 | The strict, complete (line... |
| ordeldif 43442 | Membership in the differen... |
| ordeldifsucon 43443 | Membership in the differen... |
| ordeldif1o 43444 | Membership in the differen... |
| ordne0gt0 43445 | Ordinal zero is less than ... |
| ondif1i 43446 | Ordinal zero is less than ... |
| onsucelab 43447 | The successor of every ord... |
| dflim6 43448 | A limit ordinal is a non-z... |
| limnsuc 43449 | A limit ordinal is not an ... |
| onsucss 43450 | If one ordinal is less tha... |
| ordnexbtwnsuc 43451 | For any distinct pair of o... |
| orddif0suc 43452 | For any distinct pair of o... |
| onsucf1lem 43453 | For ordinals, the successo... |
| onsucf1olem 43454 | The successor operation is... |
| onsucrn 43455 | The successor operation is... |
| onsucf1o 43456 | The successor operation is... |
| dflim7 43457 | A limit ordinal is a non-z... |
| onov0suclim 43458 | Compactly express rules fo... |
| oa0suclim 43459 | Closed form expression of ... |
| om0suclim 43460 | Closed form expression of ... |
| oe0suclim 43461 | Closed form expression of ... |
| oaomoecl 43462 | The operations of addition... |
| onsupsucismax 43463 | If the union of a set of o... |
| onsssupeqcond 43464 | If for every element of a ... |
| limexissup 43465 | An ordinal which is a limi... |
| limiun 43466 | A limit ordinal is the uni... |
| limexissupab 43467 | An ordinal which is a limi... |
| om1om1r 43468 | Ordinal one is both a left... |
| oe0rif 43469 | Ordinal zero raised to any... |
| oasubex 43470 | While subtraction can't be... |
| nnamecl 43471 | Natural numbers are closed... |
| onsucwordi 43472 | The successor operation pr... |
| oalim2cl 43473 | The ordinal sum of any ord... |
| oaltublim 43474 | Given ` C ` is a limit ord... |
| oaordi3 43475 | Ordinal addition of the sa... |
| oaord3 43476 | When the same ordinal is a... |
| 1oaomeqom 43477 | Ordinal one plus omega is ... |
| oaabsb 43478 | The right addend absorbs t... |
| oaordnrex 43479 | When omega is added on the... |
| oaordnr 43480 | When the same ordinal is a... |
| omge1 43481 | Any non-zero ordinal produ... |
| omge2 43482 | Any non-zero ordinal produ... |
| omlim2 43483 | The non-zero product with ... |
| omord2lim 43484 | Given a limit ordinal, the... |
| omord2i 43485 | Ordinal multiplication of ... |
| omord2com 43486 | When the same non-zero ord... |
| 2omomeqom 43487 | Ordinal two times omega is... |
| omnord1ex 43488 | When omega is multiplied o... |
| omnord1 43489 | When the same non-zero ord... |
| oege1 43490 | Any non-zero ordinal power... |
| oege2 43491 | Any power of an ordinal at... |
| rp-oelim2 43492 | The power of an ordinal at... |
| oeord2lim 43493 | Given a limit ordinal, the... |
| oeord2i 43494 | Ordinal exponentiation of ... |
| oeord2com 43495 | When the same base at leas... |
| nnoeomeqom 43496 | Any natural number at leas... |
| df3o2 43497 | Ordinal 3 is the unordered... |
| df3o3 43498 | Ordinal 3, fully expanded.... |
| oenord1ex 43499 | When ordinals two and thre... |
| oenord1 43500 | When two ordinals (both at... |
| oaomoencom 43501 | Ordinal addition, multipli... |
| oenassex 43502 | Ordinal two raised to two ... |
| oenass 43503 | Ordinal exponentiation is ... |
| cantnftermord 43504 | For terms of the form of a... |
| cantnfub 43505 | Given a finite number of t... |
| cantnfub2 43506 | Given a finite number of t... |
| bropabg 43507 | Equivalence for two classe... |
| cantnfresb 43508 | A Cantor normal form which... |
| cantnf2 43509 | For every ordinal, ` A ` ,... |
| oawordex2 43510 | If ` C ` is between ` A ` ... |
| nnawordexg 43511 | If an ordinal, ` B ` , is ... |
| succlg 43512 | Closure law for ordinal su... |
| dflim5 43513 | A limit ordinal is either ... |
| oacl2g 43514 | Closure law for ordinal ad... |
| onmcl 43515 | If an ordinal is less than... |
| omabs2 43516 | Ordinal multiplication by ... |
| omcl2 43517 | Closure law for ordinal mu... |
| omcl3g 43518 | Closure law for ordinal mu... |
| ordsssucb 43519 | An ordinal number is less ... |
| tfsconcatlem 43520 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 43521 | The concatenation of two t... |
| tfsconcatfn 43522 | The concatenation of two t... |
| tfsconcatfv1 43523 | An early value of the conc... |
| tfsconcatfv2 43524 | A latter value of the conc... |
| tfsconcatfv 43525 | The value of the concatena... |
| tfsconcatrn 43526 | The range of the concatena... |
| tfsconcatfo 43527 | The concatenation of two t... |
| tfsconcatb0 43528 | The concatentation with th... |
| tfsconcat0i 43529 | The concatentation with th... |
| tfsconcat0b 43530 | The concatentation with th... |
| tfsconcat00 43531 | The concatentation of two ... |
| tfsconcatrev 43532 | If the domain of a transfi... |
| tfsconcatrnss12 43533 | The range of the concatena... |
| tfsconcatrnss 43534 | The concatenation of trans... |
| tfsconcatrnsson 43535 | The concatenation of trans... |
| tfsnfin 43536 | A transfinite sequence is ... |
| rp-tfslim 43537 | The limit of a sequence of... |
| ofoafg 43538 | Addition operator for func... |
| ofoaf 43539 | Addition operator for func... |
| ofoafo 43540 | Addition operator for func... |
| ofoacl 43541 | Closure law for component ... |
| ofoaid1 43542 | Identity law for component... |
| ofoaid2 43543 | Identity law for component... |
| ofoaass 43544 | Component-wise addition of... |
| ofoacom 43545 | Component-wise addition of... |
| naddcnff 43546 | Addition operator for Cant... |
| naddcnffn 43547 | Addition operator for Cant... |
| naddcnffo 43548 | Addition of Cantor normal ... |
| naddcnfcl 43549 | Closure law for component-... |
| naddcnfcom 43550 | Component-wise ordinal add... |
| naddcnfid1 43551 | Identity law for component... |
| naddcnfid2 43552 | Identity law for component... |
| naddcnfass 43553 | Component-wise addition of... |
| onsucunifi 43554 | The successor to the union... |
| sucunisn 43555 | The successor to the union... |
| onsucunipr 43556 | The successor to the union... |
| onsucunitp 43557 | The successor to the union... |
| oaun3lem1 43558 | The class of all ordinal s... |
| oaun3lem2 43559 | The class of all ordinal s... |
| oaun3lem3 43560 | The class of all ordinal s... |
| oaun3lem4 43561 | The class of all ordinal s... |
| rp-abid 43562 | Two ways to express a clas... |
| oadif1lem 43563 | Express the set difference... |
| oadif1 43564 | Express the set difference... |
| oaun2 43565 | Ordinal addition as a unio... |
| oaun3 43566 | Ordinal addition as a unio... |
| naddov4 43567 | Alternate expression for n... |
| nadd2rabtr 43568 | The set of ordinals which ... |
| nadd2rabord 43569 | The set of ordinals which ... |
| nadd2rabex 43570 | The class of ordinals whic... |
| nadd2rabon 43571 | The set of ordinals which ... |
| nadd1rabtr 43572 | The set of ordinals which ... |
| nadd1rabord 43573 | The set of ordinals which ... |
| nadd1rabex 43574 | The class of ordinals whic... |
| nadd1rabon 43575 | The set of ordinals which ... |
| nadd1suc 43576 | Natural addition with 1 is... |
| naddass1 43577 | Natural addition of ordina... |
| naddgeoa 43578 | Natural addition results i... |
| naddonnn 43579 | Natural addition with a na... |
| naddwordnexlem0 43580 | When ` A ` is the sum of a... |
| naddwordnexlem1 43581 | When ` A ` is the sum of a... |
| naddwordnexlem2 43582 | When ` A ` is the sum of a... |
| naddwordnexlem3 43583 | When ` A ` is the sum of a... |
| oawordex3 43584 | When ` A ` is the sum of a... |
| naddwordnexlem4 43585 | When ` A ` is the sum of a... |
| ordsssucim 43586 | If an ordinal is less than... |
| insucid 43587 | The intersection of a clas... |
| oaltom 43588 | Multiplication eventually ... |
| oe2 43589 | Two ways to square an ordi... |
| omltoe 43590 | Exponentiation eventually ... |
| abeqabi 43591 | Generalized condition for ... |
| abpr 43592 | Condition for a class abst... |
| abtp 43593 | Condition for a class abst... |
| ralopabb 43594 | Restricted universal quant... |
| fpwfvss 43595 | Functions into a powerset ... |
| sdomne0 43596 | A class that strictly domi... |
| sdomne0d 43597 | A class that strictly domi... |
| safesnsupfiss 43598 | If ` B ` is a finite subse... |
| safesnsupfiub 43599 | If ` B ` is a finite subse... |
| safesnsupfidom1o 43600 | If ` B ` is a finite subse... |
| safesnsupfilb 43601 | If ` B ` is a finite subse... |
| isoeq145d 43602 | Equality deduction for iso... |
| resisoeq45d 43603 | Equality deduction for equ... |
| negslem1 43604 | An equivalence between ide... |
| nvocnvb 43605 | Equivalence to saying the ... |
| rp-brsslt 43606 | Binary relation form of a ... |
| nla0002 43607 | Extending a linear order t... |
| nla0003 43608 | Extending a linear order t... |
| nla0001 43609 | Extending a linear order t... |
| faosnf0.11b 43610 | ` B ` is called a non-limi... |
| dfno2 43611 | A surreal number, in the f... |
| onnog 43612 | Every ordinal maps to a su... |
| onnobdayg 43613 | Every ordinal maps to a su... |
| bdaybndex 43614 | Bounds formed from the bir... |
| bdaybndbday 43615 | Bounds formed from the bir... |
| onno 43616 | Every ordinal maps to a su... |
| onnoi 43617 | Every ordinal maps to a su... |
| 0no 43618 | Ordinal zero maps to a sur... |
| 1no 43619 | Ordinal one maps to a surr... |
| 2no 43620 | Ordinal two maps to a surr... |
| 3no 43621 | Ordinal three maps to a su... |
| 4no 43622 | Ordinal four maps to a sur... |
| fnimafnex 43623 | The functional image of a ... |
| nlimsuc 43624 | A successor is not a limit... |
| nlim1NEW 43625 | 1 is not a limit ordinal. ... |
| nlim2NEW 43626 | 2 is not a limit ordinal. ... |
| nlim3 43627 | 3 is not a limit ordinal. ... |
| nlim4 43628 | 4 is not a limit ordinal. ... |
| oa1un 43629 | Given ` A e. On ` , let ` ... |
| oa1cl 43630 | ` A +o 1o ` is in ` On ` .... |
| 0finon 43631 | 0 is a finite ordinal. Se... |
| 1finon 43632 | 1 is a finite ordinal. Se... |
| 2finon 43633 | 2 is a finite ordinal. Se... |
| 3finon 43634 | 3 is a finite ordinal. Se... |
| 4finon 43635 | 4 is a finite ordinal. Se... |
| finona1cl 43636 | The finite ordinals are cl... |
| finonex 43637 | The finite ordinals are a ... |
| fzunt 43638 | Union of two adjacent fini... |
| fzuntd 43639 | Union of two adjacent fini... |
| fzunt1d 43640 | Union of two overlapping f... |
| fzuntgd 43641 | Union of two adjacent or o... |
| ifpan123g 43642 | Conjunction of conditional... |
| ifpan23 43643 | Conjunction of conditional... |
| ifpdfor2 43644 | Define or in terms of cond... |
| ifporcor 43645 | Corollary of commutation o... |
| ifpdfan2 43646 | Define and with conditiona... |
| ifpancor 43647 | Corollary of commutation o... |
| ifpdfor 43648 | Define or in terms of cond... |
| ifpdfan 43649 | Define and with conditiona... |
| ifpbi2 43650 | Equivalence theorem for co... |
| ifpbi3 43651 | Equivalence theorem for co... |
| ifpim1 43652 | Restate implication as con... |
| ifpnot 43653 | Restate negated wff as con... |
| ifpid2 43654 | Restate wff as conditional... |
| ifpim2 43655 | Restate implication as con... |
| ifpbi23 43656 | Equivalence theorem for co... |
| ifpbiidcor 43657 | Restatement of ~ biid . (... |
| ifpbicor 43658 | Corollary of commutation o... |
| ifpxorcor 43659 | Corollary of commutation o... |
| ifpbi1 43660 | Equivalence theorem for co... |
| ifpnot23 43661 | Negation of conditional lo... |
| ifpnotnotb 43662 | Factor conditional logic o... |
| ifpnorcor 43663 | Corollary of commutation o... |
| ifpnancor 43664 | Corollary of commutation o... |
| ifpnot23b 43665 | Negation of conditional lo... |
| ifpbiidcor2 43666 | Restatement of ~ biid . (... |
| ifpnot23c 43667 | Negation of conditional lo... |
| ifpnot23d 43668 | Negation of conditional lo... |
| ifpdfnan 43669 | Define nand as conditional... |
| ifpdfxor 43670 | Define xor as conditional ... |
| ifpbi12 43671 | Equivalence theorem for co... |
| ifpbi13 43672 | Equivalence theorem for co... |
| ifpbi123 43673 | Equivalence theorem for co... |
| ifpidg 43674 | Restate wff as conditional... |
| ifpid3g 43675 | Restate wff as conditional... |
| ifpid2g 43676 | Restate wff as conditional... |
| ifpid1g 43677 | Restate wff as conditional... |
| ifpim23g 43678 | Restate implication as con... |
| ifpim3 43679 | Restate implication as con... |
| ifpnim1 43680 | Restate negated implicatio... |
| ifpim4 43681 | Restate implication as con... |
| ifpnim2 43682 | Restate negated implicatio... |
| ifpim123g 43683 | Implication of conditional... |
| ifpim1g 43684 | Implication of conditional... |
| ifp1bi 43685 | Substitute the first eleme... |
| ifpbi1b 43686 | When the first variable is... |
| ifpimimb 43687 | Factor conditional logic o... |
| ifpororb 43688 | Factor conditional logic o... |
| ifpananb 43689 | Factor conditional logic o... |
| ifpnannanb 43690 | Factor conditional logic o... |
| ifpor123g 43691 | Disjunction of conditional... |
| ifpimim 43692 | Consequnce of implication.... |
| ifpbibib 43693 | Factor conditional logic o... |
| ifpxorxorb 43694 | Factor conditional logic o... |
| rp-fakeimass 43695 | A special case where impli... |
| rp-fakeanorass 43696 | A special case where a mix... |
| rp-fakeoranass 43697 | A special case where a mix... |
| rp-fakeinunass 43698 | A special case where a mix... |
| rp-fakeuninass 43699 | A special case where a mix... |
| rp-isfinite5 43700 | A set is said to be finite... |
| rp-isfinite6 43701 | A set is said to be finite... |
| intabssd 43702 | When for each element ` y ... |
| eu0 43703 | There is only one empty se... |
| epelon2 43704 | Over the ordinal numbers, ... |
| ontric3g 43705 | For all ` x , y e. On ` , ... |
| dfsucon 43706 | ` A ` is called a successo... |
| snen1g 43707 | A singleton is equinumerou... |
| snen1el 43708 | A singleton is equinumerou... |
| sn1dom 43709 | A singleton is dominated b... |
| pr2dom 43710 | An unordered pair is domin... |
| tr3dom 43711 | An unordered triple is dom... |
| ensucne0 43712 | A class equinumerous to a ... |
| ensucne0OLD 43713 | A class equinumerous to a ... |
| dfom6 43714 | Let ` _om ` be defined to ... |
| infordmin 43715 | ` _om ` is the smallest in... |
| iscard4 43716 | Two ways to express the pr... |
| minregex 43717 | Given any cardinal number ... |
| minregex2 43718 | Given any cardinal number ... |
| iscard5 43719 | Two ways to express the pr... |
| elrncard 43720 | Let us define a cardinal n... |
| harval3 43721 | ` ( har `` A ) ` is the le... |
| harval3on 43722 | For any ordinal number ` A... |
| omssrncard 43723 | All natural numbers are ca... |
| 0iscard 43724 | 0 is a cardinal number. (... |
| 1iscard 43725 | 1 is a cardinal number. (... |
| omiscard 43726 | ` _om ` is a cardinal numb... |
| sucomisnotcard 43727 | ` _om +o 1o ` is not a car... |
| nna1iscard 43728 | For any natural number, th... |
| har2o 43729 | The least cardinal greater... |
| en2pr 43730 | A class is equinumerous to... |
| pr2cv 43731 | If an unordered pair is eq... |
| pr2el1 43732 | If an unordered pair is eq... |
| pr2cv1 43733 | If an unordered pair is eq... |
| pr2el2 43734 | If an unordered pair is eq... |
| pr2cv2 43735 | If an unordered pair is eq... |
| pren2 43736 | An unordered pair is equin... |
| pr2eldif1 43737 | If an unordered pair is eq... |
| pr2eldif2 43738 | If an unordered pair is eq... |
| pren2d 43739 | A pair of two distinct set... |
| aleph1min 43740 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 43741 | ` aleph ` is a strictly or... |
| alephiso3 43742 | ` aleph ` is a strictly or... |
| pwelg 43743 | The powerclass is an eleme... |
| pwinfig 43744 | The powerclass of an infin... |
| pwinfi2 43745 | The powerclass of an infin... |
| pwinfi3 43746 | The powerclass of an infin... |
| pwinfi 43747 | The powerclass of an infin... |
| fipjust 43748 | A definition of the finite... |
| cllem0 43749 | The class of all sets with... |
| superficl 43750 | The class of all supersets... |
| superuncl 43751 | The class of all supersets... |
| ssficl 43752 | The class of all subsets o... |
| ssuncl 43753 | The class of all subsets o... |
| ssdifcl 43754 | The class of all subsets o... |
| sssymdifcl 43755 | The class of all subsets o... |
| fiinfi 43756 | If two classes have the fi... |
| rababg 43757 | Condition when restricted ... |
| elinintab 43758 | Two ways of saying a set i... |
| elmapintrab 43759 | Two ways to say a set is a... |
| elinintrab 43760 | Two ways of saying a set i... |
| inintabss 43761 | Upper bound on intersectio... |
| inintabd 43762 | Value of the intersection ... |
| xpinintabd 43763 | Value of the intersection ... |
| relintabex 43764 | If the intersection of a c... |
| elcnvcnvintab 43765 | Two ways of saying a set i... |
| relintab 43766 | Value of the intersection ... |
| nonrel 43767 | A non-relation is equal to... |
| elnonrel 43768 | Only an ordered pair where... |
| cnvssb 43769 | Subclass theorem for conve... |
| relnonrel 43770 | The non-relation part of a... |
| cnvnonrel 43771 | The converse of the non-re... |
| brnonrel 43772 | A non-relation cannot rela... |
| dmnonrel 43773 | The domain of the non-rela... |
| rnnonrel 43774 | The range of the non-relat... |
| resnonrel 43775 | A restriction of the non-r... |
| imanonrel 43776 | An image under the non-rel... |
| cononrel1 43777 | Composition with the non-r... |
| cononrel2 43778 | Composition with the non-r... |
| elmapintab 43779 | Two ways to say a set is a... |
| fvnonrel 43780 | The function value of any ... |
| elinlem 43781 | Two ways to say a set is a... |
| elcnvcnvlem 43782 | Two ways to say a set is a... |
| cnvcnvintabd 43783 | Value of the relationship ... |
| elcnvlem 43784 | Two ways to say a set is a... |
| elcnvintab 43785 | Two ways of saying a set i... |
| cnvintabd 43786 | Value of the converse of t... |
| undmrnresiss 43787 | Two ways of saying the ide... |
| reflexg 43788 | Two ways of saying a relat... |
| cnvssco 43789 | A condition weaker than re... |
| refimssco 43790 | Reflexive relations are su... |
| cleq2lem 43791 | Equality implies bijection... |
| cbvcllem 43792 | Change of bound variable i... |
| clublem 43793 | If a superset ` Y ` of ` X... |
| clss2lem 43794 | The closure of a property ... |
| dfid7 43795 | Definition of identity rel... |
| mptrcllem 43796 | Show two versions of a clo... |
| cotrintab 43797 | The intersection of a clas... |
| rclexi 43798 | The reflexive closure of a... |
| rtrclexlem 43799 | Existence of relation impl... |
| rtrclex 43800 | The reflexive-transitive c... |
| trclubgNEW 43801 | If a relation exists then ... |
| trclubNEW 43802 | If a relation exists then ... |
| trclexi 43803 | The transitive closure of ... |
| rtrclexi 43804 | The reflexive-transitive c... |
| clrellem 43805 | When the property ` ps ` h... |
| clcnvlem 43806 | When ` A ` , an upper boun... |
| cnvtrucl0 43807 | The converse of the trivia... |
| cnvrcl0 43808 | The converse of the reflex... |
| cnvtrcl0 43809 | The converse of the transi... |
| dmtrcl 43810 | The domain of the transiti... |
| rntrcl 43811 | The range of the transitiv... |
| dfrtrcl5 43812 | Definition of reflexive-tr... |
| trcleq2lemRP 43813 | Equality implies bijection... |
| sqrtcvallem1 43814 | Two ways of saying a compl... |
| reabsifneg 43815 | Alternate expression for t... |
| reabsifnpos 43816 | Alternate expression for t... |
| reabsifpos 43817 | Alternate expression for t... |
| reabsifnneg 43818 | Alternate expression for t... |
| reabssgn 43819 | Alternate expression for t... |
| sqrtcvallem2 43820 | Equivalent to saying that ... |
| sqrtcvallem3 43821 | Equivalent to saying that ... |
| sqrtcvallem4 43822 | Equivalent to saying that ... |
| sqrtcvallem5 43823 | Equivalent to saying that ... |
| sqrtcval 43824 | Explicit formula for the c... |
| sqrtcval2 43825 | Explicit formula for the c... |
| resqrtval 43826 | Real part of the complex s... |
| imsqrtval 43827 | Imaginary part of the comp... |
| resqrtvalex 43828 | Example for ~ resqrtval . ... |
| imsqrtvalex 43829 | Example for ~ imsqrtval . ... |
| al3im 43830 | Version of ~ ax-4 for a ne... |
| intima0 43831 | Two ways of expressing the... |
| elimaint 43832 | Element of image of inters... |
| cnviun 43833 | Converse of indexed union.... |
| imaiun1 43834 | The image of an indexed un... |
| coiun1 43835 | Composition with an indexe... |
| elintima 43836 | Element of intersection of... |
| intimass 43837 | The image under the inters... |
| intimass2 43838 | The image under the inters... |
| intimag 43839 | Requirement for the image ... |
| intimasn 43840 | Two ways to express the im... |
| intimasn2 43841 | Two ways to express the im... |
| ss2iundf 43842 | Subclass theorem for index... |
| ss2iundv 43843 | Subclass theorem for index... |
| cbviuneq12df 43844 | Rule used to change the bo... |
| cbviuneq12dv 43845 | Rule used to change the bo... |
| conrel1d 43846 | Deduction about compositio... |
| conrel2d 43847 | Deduction about compositio... |
| trrelind 43848 | The intersection of transi... |
| xpintrreld 43849 | The intersection of a tran... |
| restrreld 43850 | The restriction of a trans... |
| trrelsuperreldg 43851 | Concrete construction of a... |
| trficl 43852 | The class of all transitiv... |
| cnvtrrel 43853 | The converse of a transiti... |
| trrelsuperrel2dg 43854 | Concrete construction of a... |
| dfrcl2 43857 | Reflexive closure of a rel... |
| dfrcl3 43858 | Reflexive closure of a rel... |
| dfrcl4 43859 | Reflexive closure of a rel... |
| relexp2 43860 | A set operated on by the r... |
| relexpnul 43861 | If the domain and range of... |
| eliunov2 43862 | Membership in the indexed ... |
| eltrclrec 43863 | Membership in the indexed ... |
| elrtrclrec 43864 | Membership in the indexed ... |
| briunov2 43865 | Two classes related by the... |
| brmptiunrelexpd 43866 | If two elements are connec... |
| fvmptiunrelexplb0d 43867 | If the indexed union range... |
| fvmptiunrelexplb0da 43868 | If the indexed union range... |
| fvmptiunrelexplb1d 43869 | If the indexed union range... |
| brfvid 43870 | If two elements are connec... |
| brfvidRP 43871 | If two elements are connec... |
| fvilbd 43872 | A set is a subset of its i... |
| fvilbdRP 43873 | A set is a subset of its i... |
| brfvrcld 43874 | If two elements are connec... |
| brfvrcld2 43875 | If two elements are connec... |
| fvrcllb0d 43876 | A restriction of the ident... |
| fvrcllb0da 43877 | A restriction of the ident... |
| fvrcllb1d 43878 | A set is a subset of its i... |
| brtrclrec 43879 | Two classes related by the... |
| brrtrclrec 43880 | Two classes related by the... |
| briunov2uz 43881 | Two classes related by the... |
| eliunov2uz 43882 | Membership in the indexed ... |
| ov2ssiunov2 43883 | Any particular operator va... |
| relexp0eq 43884 | The zeroth power of relati... |
| iunrelexp0 43885 | Simplification of zeroth p... |
| relexpxpnnidm 43886 | Any positive power of a Ca... |
| relexpiidm 43887 | Any power of any restricti... |
| relexpss1d 43888 | The relational power of a ... |
| comptiunov2i 43889 | The composition two indexe... |
| corclrcl 43890 | The reflexive closure is i... |
| iunrelexpmin1 43891 | The indexed union of relat... |
| relexpmulnn 43892 | With exponents limited to ... |
| relexpmulg 43893 | With ordered exponents, th... |
| trclrelexplem 43894 | The union of relational po... |
| iunrelexpmin2 43895 | The indexed union of relat... |
| relexp01min 43896 | With exponents limited to ... |
| relexp1idm 43897 | Repeated raising a relatio... |
| relexp0idm 43898 | Repeated raising a relatio... |
| relexp0a 43899 | Absorption law for zeroth ... |
| relexpxpmin 43900 | The composition of powers ... |
| relexpaddss 43901 | The composition of two pow... |
| iunrelexpuztr 43902 | The indexed union of relat... |
| dftrcl3 43903 | Transitive closure of a re... |
| brfvtrcld 43904 | If two elements are connec... |
| fvtrcllb1d 43905 | A set is a subset of its i... |
| trclfvcom 43906 | The transitive closure of ... |
| cnvtrclfv 43907 | The converse of the transi... |
| cotrcltrcl 43908 | The transitive closure is ... |
| trclimalb2 43909 | Lower bound for image unde... |
| brtrclfv2 43910 | Two ways to indicate two e... |
| trclfvdecomr 43911 | The transitive closure of ... |
| trclfvdecoml 43912 | The transitive closure of ... |
| dmtrclfvRP 43913 | The domain of the transiti... |
| rntrclfvRP 43914 | The range of the transitiv... |
| rntrclfv 43915 | The range of the transitiv... |
| dfrtrcl3 43916 | Reflexive-transitive closu... |
| brfvrtrcld 43917 | If two elements are connec... |
| fvrtrcllb0d 43918 | A restriction of the ident... |
| fvrtrcllb0da 43919 | A restriction of the ident... |
| fvrtrcllb1d 43920 | A set is a subset of its i... |
| dfrtrcl4 43921 | Reflexive-transitive closu... |
| corcltrcl 43922 | The composition of the ref... |
| cortrcltrcl 43923 | Composition with the refle... |
| corclrtrcl 43924 | Composition with the refle... |
| cotrclrcl 43925 | The composition of the ref... |
| cortrclrcl 43926 | Composition with the refle... |
| cotrclrtrcl 43927 | Composition with the refle... |
| cortrclrtrcl 43928 | The reflexive-transitive c... |
| frege77d 43929 | If the images of both ` { ... |
| frege81d 43930 | If the image of ` U ` is a... |
| frege83d 43931 | If the image of the union ... |
| frege96d 43932 | If ` C ` follows ` A ` in ... |
| frege87d 43933 | If the images of both ` { ... |
| frege91d 43934 | If ` B ` follows ` A ` in ... |
| frege97d 43935 | If ` A ` contains all elem... |
| frege98d 43936 | If ` C ` follows ` A ` and... |
| frege102d 43937 | If either ` A ` and ` C ` ... |
| frege106d 43938 | If ` B ` follows ` A ` in ... |
| frege108d 43939 | If either ` A ` and ` C ` ... |
| frege109d 43940 | If ` A ` contains all elem... |
| frege114d 43941 | If either ` R ` relates ` ... |
| frege111d 43942 | If either ` A ` and ` C ` ... |
| frege122d 43943 | If ` F ` is a function, ` ... |
| frege124d 43944 | If ` F ` is a function, ` ... |
| frege126d 43945 | If ` F ` is a function, ` ... |
| frege129d 43946 | If ` F ` is a function and... |
| frege131d 43947 | If ` F ` is a function and... |
| frege133d 43948 | If ` F ` is a function and... |
| dfxor4 43949 | Express exclusive-or in te... |
| dfxor5 43950 | Express exclusive-or in te... |
| df3or2 43951 | Express triple-or in terms... |
| df3an2 43952 | Express triple-and in term... |
| nev 43953 | Express that not every set... |
| 0pssin 43954 | Express that an intersecti... |
| dfhe2 43957 | The property of relation `... |
| dfhe3 43958 | The property of relation `... |
| heeq12 43959 | Equality law for relations... |
| heeq1 43960 | Equality law for relations... |
| heeq2 43961 | Equality law for relations... |
| sbcheg 43962 | Distribute proper substitu... |
| hess 43963 | Subclass law for relations... |
| xphe 43964 | Any Cartesian product is h... |
| 0he 43965 | The empty relation is here... |
| 0heALT 43966 | The empty relation is here... |
| he0 43967 | Any relation is hereditary... |
| unhe1 43968 | The union of two relations... |
| snhesn 43969 | Any singleton is hereditar... |
| idhe 43970 | The identity relation is h... |
| psshepw 43971 | The relation between sets ... |
| sshepw 43972 | The relation between sets ... |
| rp-simp2-frege 43975 | Simplification of triple c... |
| rp-simp2 43976 | Simplification of triple c... |
| rp-frege3g 43977 | Add antecedent to ~ ax-fre... |
| frege3 43978 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 43979 | Double-use of ~ ax-frege2 ... |
| rp-frege24 43980 | Introducing an embedded an... |
| rp-frege4g 43981 | Deduction related to distr... |
| frege4 43982 | Special case of closed for... |
| frege5 43983 | A closed form of ~ syl . ... |
| rp-7frege 43984 | Distribute antecedent and ... |
| rp-4frege 43985 | Elimination of a nested an... |
| rp-6frege 43986 | Elimination of a nested an... |
| rp-8frege 43987 | Eliminate antecedent when ... |
| rp-frege25 43988 | Closed form for ~ a1dd . ... |
| frege6 43989 | A closed form of ~ imim2d ... |
| axfrege8 43990 | Swap antecedents. Identic... |
| frege7 43991 | A closed form of ~ syl6 . ... |
| frege26 43993 | Identical to ~ idd . Prop... |
| frege27 43994 | We cannot (at the same tim... |
| frege9 43995 | Closed form of ~ syl with ... |
| frege12 43996 | A closed form of ~ com23 .... |
| frege11 43997 | Elimination of a nested an... |
| frege24 43998 | Closed form for ~ a1d . D... |
| frege16 43999 | A closed form of ~ com34 .... |
| frege25 44000 | Closed form for ~ a1dd . ... |
| frege18 44001 | Closed form of a syllogism... |
| frege22 44002 | A closed form of ~ com45 .... |
| frege10 44003 | Result commuting anteceden... |
| frege17 44004 | A closed form of ~ com3l .... |
| frege13 44005 | A closed form of ~ com3r .... |
| frege14 44006 | Closed form of a deduction... |
| frege19 44007 | A closed form of ~ syl6 . ... |
| frege23 44008 | Syllogism followed by rota... |
| frege15 44009 | A closed form of ~ com4r .... |
| frege21 44010 | Replace antecedent in ante... |
| frege20 44011 | A closed form of ~ syl8 . ... |
| axfrege28 44012 | Contraposition. Identical... |
| frege29 44014 | Closed form of ~ con3d . ... |
| frege30 44015 | Commuted, closed form of ~... |
| axfrege31 44016 | Identical to ~ notnotr . ... |
| frege32 44018 | Deduce ~ con1 from ~ con3 ... |
| frege33 44019 | If ` ph ` or ` ps ` takes ... |
| frege34 44020 | If as a consequence of the... |
| frege35 44021 | Commuted, closed form of ~... |
| frege36 44022 | The case in which ` ps ` i... |
| frege37 44023 | If ` ch ` is a necessary c... |
| frege38 44024 | Identical to ~ pm2.21 . P... |
| frege39 44025 | Syllogism between ~ pm2.18... |
| frege40 44026 | Anything implies ~ pm2.18 ... |
| axfrege41 44027 | Identical to ~ notnot . A... |
| frege42 44029 | Not not ~ id . Propositio... |
| frege43 44030 | If there is a choice only ... |
| frege44 44031 | Similar to a commuted ~ pm... |
| frege45 44032 | Deduce ~ pm2.6 from ~ con1... |
| frege46 44033 | If ` ps ` holds when ` ph ... |
| frege47 44034 | Deduce consequence follows... |
| frege48 44035 | Closed form of syllogism w... |
| frege49 44036 | Closed form of deduction w... |
| frege50 44037 | Closed form of ~ jaoi . P... |
| frege51 44038 | Compare with ~ jaod . Pro... |
| axfrege52a 44039 | Justification for ~ ax-fre... |
| frege52aid 44041 | The case when the content ... |
| frege53aid 44042 | Specialization of ~ frege5... |
| frege53a 44043 | Lemma for ~ frege55a . Pr... |
| axfrege54a 44044 | Justification for ~ ax-fre... |
| frege54cor0a 44046 | Synonym for logical equiva... |
| frege54cor1a 44047 | Reflexive equality. (Cont... |
| frege55aid 44048 | Lemma for ~ frege57aid . ... |
| frege55lem1a 44049 | Necessary deduction regard... |
| frege55lem2a 44050 | Core proof of Proposition ... |
| frege55a 44051 | Proposition 55 of [Frege18... |
| frege55cor1a 44052 | Proposition 55 of [Frege18... |
| frege56aid 44053 | Lemma for ~ frege57aid . ... |
| frege56a 44054 | Proposition 56 of [Frege18... |
| frege57aid 44055 | This is the all important ... |
| frege57a 44056 | Analogue of ~ frege57aid .... |
| axfrege58a 44057 | Identical to ~ anifp . Ju... |
| frege58acor 44059 | Lemma for ~ frege59a . (C... |
| frege59a 44060 | A kind of Aristotelian inf... |
| frege60a 44061 | Swap antecedents of ~ ax-f... |
| frege61a 44062 | Lemma for ~ frege65a . Pr... |
| frege62a 44063 | A kind of Aristotelian inf... |
| frege63a 44064 | Proposition 63 of [Frege18... |
| frege64a 44065 | Lemma for ~ frege65a . Pr... |
| frege65a 44066 | A kind of Aristotelian inf... |
| frege66a 44067 | Swap antecedents of ~ freg... |
| frege67a 44068 | Lemma for ~ frege68a . Pr... |
| frege68a 44069 | Combination of applying a ... |
| axfrege52c 44070 | Justification for ~ ax-fre... |
| frege52b 44072 | The case when the content ... |
| frege53b 44073 | Lemma for frege102 (via ~ ... |
| axfrege54c 44074 | Reflexive equality of clas... |
| frege54b 44076 | Reflexive equality of sets... |
| frege54cor1b 44077 | Reflexive equality. (Cont... |
| frege55lem1b 44078 | Necessary deduction regard... |
| frege55lem2b 44079 | Lemma for ~ frege55b . Co... |
| frege55b 44080 | Lemma for ~ frege57b . Pr... |
| frege56b 44081 | Lemma for ~ frege57b . Pr... |
| frege57b 44082 | Analogue of ~ frege57aid .... |
| axfrege58b 44083 | If ` A. x ph ` is affirmed... |
| frege58bid 44085 | If ` A. x ph ` is affirmed... |
| frege58bcor 44086 | Lemma for ~ frege59b . (C... |
| frege59b 44087 | A kind of Aristotelian inf... |
| frege60b 44088 | Swap antecedents of ~ ax-f... |
| frege61b 44089 | Lemma for ~ frege65b . Pr... |
| frege62b 44090 | A kind of Aristotelian inf... |
| frege63b 44091 | Lemma for ~ frege91 . Pro... |
| frege64b 44092 | Lemma for ~ frege65b . Pr... |
| frege65b 44093 | A kind of Aristotelian inf... |
| frege66b 44094 | Swap antecedents of ~ freg... |
| frege67b 44095 | Lemma for ~ frege68b . Pr... |
| frege68b 44096 | Combination of applying a ... |
| frege53c 44097 | Proposition 53 of [Frege18... |
| frege54cor1c 44098 | Reflexive equality. (Cont... |
| frege55lem1c 44099 | Necessary deduction regard... |
| frege55lem2c 44100 | Core proof of Proposition ... |
| frege55c 44101 | Proposition 55 of [Frege18... |
| frege56c 44102 | Lemma for ~ frege57c . Pr... |
| frege57c 44103 | Swap order of implication ... |
| frege58c 44104 | Principle related to ~ sp ... |
| frege59c 44105 | A kind of Aristotelian inf... |
| frege60c 44106 | Swap antecedents of ~ freg... |
| frege61c 44107 | Lemma for ~ frege65c . Pr... |
| frege62c 44108 | A kind of Aristotelian inf... |
| frege63c 44109 | Analogue of ~ frege63b . ... |
| frege64c 44110 | Lemma for ~ frege65c . Pr... |
| frege65c 44111 | A kind of Aristotelian inf... |
| frege66c 44112 | Swap antecedents of ~ freg... |
| frege67c 44113 | Lemma for ~ frege68c . Pr... |
| frege68c 44114 | Combination of applying a ... |
| dffrege69 44115 | If from the proposition th... |
| frege70 44116 | Lemma for ~ frege72 . Pro... |
| frege71 44117 | Lemma for ~ frege72 . Pro... |
| frege72 44118 | If property ` A ` is hered... |
| frege73 44119 | Lemma for ~ frege87 . Pro... |
| frege74 44120 | If ` X ` has a property ` ... |
| frege75 44121 | If from the proposition th... |
| dffrege76 44122 | If from the two propositio... |
| frege77 44123 | If ` Y ` follows ` X ` in ... |
| frege78 44124 | Commuted form of ~ frege77... |
| frege79 44125 | Distributed form of ~ freg... |
| frege80 44126 | Add additional condition t... |
| frege81 44127 | If ` X ` has a property ` ... |
| frege82 44128 | Closed-form deduction base... |
| frege83 44129 | Apply commuted form of ~ f... |
| frege84 44130 | Commuted form of ~ frege81... |
| frege85 44131 | Commuted form of ~ frege77... |
| frege86 44132 | Conclusion about element o... |
| frege87 44133 | If ` Z ` is a result of an... |
| frege88 44134 | Commuted form of ~ frege87... |
| frege89 44135 | One direction of ~ dffrege... |
| frege90 44136 | Add antecedent to ~ frege8... |
| frege91 44137 | Every result of an applica... |
| frege92 44138 | Inference from ~ frege91 .... |
| frege93 44139 | Necessary condition for tw... |
| frege94 44140 | Looking one past a pair re... |
| frege95 44141 | Looking one past a pair re... |
| frege96 44142 | Every result of an applica... |
| frege97 44143 | The property of following ... |
| frege98 44144 | If ` Y ` follows ` X ` and... |
| dffrege99 44145 | If ` Z ` is identical with... |
| frege100 44146 | One direction of ~ dffrege... |
| frege101 44147 | Lemma for ~ frege102 . Pr... |
| frege102 44148 | If ` Z ` belongs to the ` ... |
| frege103 44149 | Proposition 103 of [Frege1... |
| frege104 44150 | Proposition 104 of [Frege1... |
| frege105 44151 | Proposition 105 of [Frege1... |
| frege106 44152 | Whatever follows ` X ` in ... |
| frege107 44153 | Proposition 107 of [Frege1... |
| frege108 44154 | If ` Y ` belongs to the ` ... |
| frege109 44155 | The property of belonging ... |
| frege110 44156 | Proposition 110 of [Frege1... |
| frege111 44157 | If ` Y ` belongs to the ` ... |
| frege112 44158 | Identity implies belonging... |
| frege113 44159 | Proposition 113 of [Frege1... |
| frege114 44160 | If ` X ` belongs to the ` ... |
| dffrege115 44161 | If from the circumstance t... |
| frege116 44162 | One direction of ~ dffrege... |
| frege117 44163 | Lemma for ~ frege118 . Pr... |
| frege118 44164 | Simplified application of ... |
| frege119 44165 | Lemma for ~ frege120 . Pr... |
| frege120 44166 | Simplified application of ... |
| frege121 44167 | Lemma for ~ frege122 . Pr... |
| frege122 44168 | If ` X ` is a result of an... |
| frege123 44169 | Lemma for ~ frege124 . Pr... |
| frege124 44170 | If ` X ` is a result of an... |
| frege125 44171 | Lemma for ~ frege126 . Pr... |
| frege126 44172 | If ` M ` follows ` Y ` in ... |
| frege127 44173 | Communte antecedents of ~ ... |
| frege128 44174 | Lemma for ~ frege129 . Pr... |
| frege129 44175 | If the procedure ` R ` is ... |
| frege130 44176 | Lemma for ~ frege131 . Pr... |
| frege131 44177 | If the procedure ` R ` is ... |
| frege132 44178 | Lemma for ~ frege133 . Pr... |
| frege133 44179 | If the procedure ` R ` is ... |
| enrelmap 44180 | The set of all possible re... |
| enrelmapr 44181 | The set of all possible re... |
| enmappw 44182 | The set of all mappings fr... |
| enmappwid 44183 | The set of all mappings fr... |
| rfovd 44184 | Value of the operator, ` (... |
| rfovfvd 44185 | Value of the operator, ` (... |
| rfovfvfvd 44186 | Value of the operator, ` (... |
| rfovcnvf1od 44187 | Properties of the operator... |
| rfovcnvd 44188 | Value of the converse of t... |
| rfovf1od 44189 | The value of the operator,... |
| rfovcnvfvd 44190 | Value of the converse of t... |
| fsovd 44191 | Value of the operator, ` (... |
| fsovrfovd 44192 | The operator which gives a... |
| fsovfvd 44193 | Value of the operator, ` (... |
| fsovfvfvd 44194 | Value of the operator, ` (... |
| fsovfd 44195 | The operator, ` ( A O B ) ... |
| fsovcnvlem 44196 | The ` O ` operator, which ... |
| fsovcnvd 44197 | The value of the converse ... |
| fsovcnvfvd 44198 | The value of the converse ... |
| fsovf1od 44199 | The value of ` ( A O B ) `... |
| dssmapfvd 44200 | Value of the duality opera... |
| dssmapfv2d 44201 | Value of the duality opera... |
| dssmapfv3d 44202 | Value of the duality opera... |
| dssmapnvod 44203 | For any base set ` B ` the... |
| dssmapf1od 44204 | For any base set ` B ` the... |
| dssmap2d 44205 | For any base set ` B ` the... |
| or3or 44206 | Decompose disjunction into... |
| andi3or 44207 | Distribute over triple dis... |
| uneqsn 44208 | If a union of classes is e... |
| brfvimex 44209 | If a binary relation holds... |
| brovmptimex 44210 | If a binary relation holds... |
| brovmptimex1 44211 | If a binary relation holds... |
| brovmptimex2 44212 | If a binary relation holds... |
| brcoffn 44213 | Conditions allowing the de... |
| brcofffn 44214 | Conditions allowing the de... |
| brco2f1o 44215 | Conditions allowing the de... |
| brco3f1o 44216 | Conditions allowing the de... |
| ntrclsbex 44217 | If (pseudo-)interior and (... |
| ntrclsrcomplex 44218 | The relative complement of... |
| neik0imk0p 44219 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 44220 | If an interior function is... |
| ntrkbimka 44221 | If the interiors of disjoi... |
| ntrk0kbimka 44222 | If the interiors of disjoi... |
| clsk3nimkb 44223 | If the base set is not emp... |
| clsk1indlem0 44224 | The ansatz closure functio... |
| clsk1indlem2 44225 | The ansatz closure functio... |
| clsk1indlem3 44226 | The ansatz closure functio... |
| clsk1indlem4 44227 | The ansatz closure functio... |
| clsk1indlem1 44228 | The ansatz closure functio... |
| clsk1independent 44229 | For generalized closure fu... |
| neik0pk1imk0 44230 | Kuratowski's K0' and K1 ax... |
| isotone1 44231 | Two different ways to say ... |
| isotone2 44232 | Two different ways to say ... |
| ntrk1k3eqk13 44233 | An interior function is bo... |
| ntrclsf1o 44234 | If (pseudo-)interior and (... |
| ntrclsnvobr 44235 | If (pseudo-)interior and (... |
| ntrclsiex 44236 | If (pseudo-)interior and (... |
| ntrclskex 44237 | If (pseudo-)interior and (... |
| ntrclsfv1 44238 | If (pseudo-)interior and (... |
| ntrclsfv2 44239 | If (pseudo-)interior and (... |
| ntrclselnel1 44240 | If (pseudo-)interior and (... |
| ntrclselnel2 44241 | If (pseudo-)interior and (... |
| ntrclsfv 44242 | The value of the interior ... |
| ntrclsfveq1 44243 | If interior and closure fu... |
| ntrclsfveq2 44244 | If interior and closure fu... |
| ntrclsfveq 44245 | If interior and closure fu... |
| ntrclsss 44246 | If interior and closure fu... |
| ntrclsneine0lem 44247 | If (pseudo-)interior and (... |
| ntrclsneine0 44248 | If (pseudo-)interior and (... |
| ntrclscls00 44249 | If (pseudo-)interior and (... |
| ntrclsiso 44250 | If (pseudo-)interior and (... |
| ntrclsk2 44251 | An interior function is co... |
| ntrclskb 44252 | The interiors of disjoint ... |
| ntrclsk3 44253 | The intersection of interi... |
| ntrclsk13 44254 | The interior of the inters... |
| ntrclsk4 44255 | Idempotence of the interio... |
| ntrneibex 44256 | If (pseudo-)interior and (... |
| ntrneircomplex 44257 | The relative complement of... |
| ntrneif1o 44258 | If (pseudo-)interior and (... |
| ntrneiiex 44259 | If (pseudo-)interior and (... |
| ntrneinex 44260 | If (pseudo-)interior and (... |
| ntrneicnv 44261 | If (pseudo-)interior and (... |
| ntrneifv1 44262 | If (pseudo-)interior and (... |
| ntrneifv2 44263 | If (pseudo-)interior and (... |
| ntrneiel 44264 | If (pseudo-)interior and (... |
| ntrneifv3 44265 | The value of the neighbors... |
| ntrneineine0lem 44266 | If (pseudo-)interior and (... |
| ntrneineine1lem 44267 | If (pseudo-)interior and (... |
| ntrneifv4 44268 | The value of the interior ... |
| ntrneiel2 44269 | Membership in iterated int... |
| ntrneineine0 44270 | If (pseudo-)interior and (... |
| ntrneineine1 44271 | If (pseudo-)interior and (... |
| ntrneicls00 44272 | If (pseudo-)interior and (... |
| ntrneicls11 44273 | If (pseudo-)interior and (... |
| ntrneiiso 44274 | If (pseudo-)interior and (... |
| ntrneik2 44275 | An interior function is co... |
| ntrneix2 44276 | An interior (closure) func... |
| ntrneikb 44277 | The interiors of disjoint ... |
| ntrneixb 44278 | The interiors (closures) o... |
| ntrneik3 44279 | The intersection of interi... |
| ntrneix3 44280 | The closure of the union o... |
| ntrneik13 44281 | The interior of the inters... |
| ntrneix13 44282 | The closure of the union o... |
| ntrneik4w 44283 | Idempotence of the interio... |
| ntrneik4 44284 | Idempotence of the interio... |
| clsneibex 44285 | If (pseudo-)closure and (p... |
| clsneircomplex 44286 | The relative complement of... |
| clsneif1o 44287 | If a (pseudo-)closure func... |
| clsneicnv 44288 | If a (pseudo-)closure func... |
| clsneikex 44289 | If closure and neighborhoo... |
| clsneinex 44290 | If closure and neighborhoo... |
| clsneiel1 44291 | If a (pseudo-)closure func... |
| clsneiel2 44292 | If a (pseudo-)closure func... |
| clsneifv3 44293 | Value of the neighborhoods... |
| clsneifv4 44294 | Value of the closure (inte... |
| neicvgbex 44295 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 44296 | The relative complement of... |
| neicvgf1o 44297 | If neighborhood and conver... |
| neicvgnvo 44298 | If neighborhood and conver... |
| neicvgnvor 44299 | If neighborhood and conver... |
| neicvgmex 44300 | If the neighborhoods and c... |
| neicvgnex 44301 | If the neighborhoods and c... |
| neicvgel1 44302 | A subset being an element ... |
| neicvgel2 44303 | The complement of a subset... |
| neicvgfv 44304 | The value of the neighborh... |
| ntrrn 44305 | The range of the interior ... |
| ntrf 44306 | The interior function of a... |
| ntrf2 44307 | The interior function is a... |
| ntrelmap 44308 | The interior function is a... |
| clsf2 44309 | The closure function is a ... |
| clselmap 44310 | The closure function is a ... |
| dssmapntrcls 44311 | The interior and closure o... |
| dssmapclsntr 44312 | The closure and interior o... |
| gneispa 44313 | Each point ` p ` of the ne... |
| gneispb 44314 | Given a neighborhood ` N `... |
| gneispace2 44315 | The predicate that ` F ` i... |
| gneispace3 44316 | The predicate that ` F ` i... |
| gneispace 44317 | The predicate that ` F ` i... |
| gneispacef 44318 | A generic neighborhood spa... |
| gneispacef2 44319 | A generic neighborhood spa... |
| gneispacefun 44320 | A generic neighborhood spa... |
| gneispacern 44321 | A generic neighborhood spa... |
| gneispacern2 44322 | A generic neighborhood spa... |
| gneispace0nelrn 44323 | A generic neighborhood spa... |
| gneispace0nelrn2 44324 | A generic neighborhood spa... |
| gneispace0nelrn3 44325 | A generic neighborhood spa... |
| gneispaceel 44326 | Every neighborhood of a po... |
| gneispaceel2 44327 | Every neighborhood of a po... |
| gneispacess 44328 | All supersets of a neighbo... |
| gneispacess2 44329 | All supersets of a neighbo... |
| k0004lem1 44330 | Application of ~ ssin to r... |
| k0004lem2 44331 | A mapping with a particula... |
| k0004lem3 44332 | When the value of a mappin... |
| k0004val 44333 | The topological simplex of... |
| k0004ss1 44334 | The topological simplex of... |
| k0004ss2 44335 | The topological simplex of... |
| k0004ss3 44336 | The topological simplex of... |
| k0004val0 44337 | The topological simplex of... |
| inductionexd 44338 | Simple induction example. ... |
| wwlemuld 44339 | Natural deduction form of ... |
| leeq1d 44340 | Specialization of ~ breq1d... |
| leeq2d 44341 | Specialization of ~ breq2d... |
| absmulrposd 44342 | Specialization of absmuld ... |
| imadisjld 44343 | Natural dduction form of o... |
| wnefimgd 44344 | The image of a mapping fro... |
| fco2d 44345 | Natural deduction form of ... |
| wfximgfd 44346 | The value of a function on... |
| extoimad 44347 | If |f(x)| <= C for all x t... |
| imo72b2lem0 44348 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 44349 | Natural deduction form of ... |
| imo72b2lem2 44350 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 44351 | Natural deduction form of ... |
| imo72b2lem1 44352 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 44353 | 'Less than or equal to' re... |
| lemuldiv4d 44354 | 'Less than or equal to' re... |
| imo72b2 44355 | IMO 1972 B2. (14th Intern... |
| int-addcomd 44356 | AdditionCommutativity gene... |
| int-addassocd 44357 | AdditionAssociativity gene... |
| int-addsimpd 44358 | AdditionSimplification gen... |
| int-mulcomd 44359 | MultiplicationCommutativit... |
| int-mulassocd 44360 | MultiplicationAssociativit... |
| int-mulsimpd 44361 | MultiplicationSimplificati... |
| int-leftdistd 44362 | AdditionMultiplicationLeft... |
| int-rightdistd 44363 | AdditionMultiplicationRigh... |
| int-sqdefd 44364 | SquareDefinition generator... |
| int-mul11d 44365 | First MultiplicationOne ge... |
| int-mul12d 44366 | Second MultiplicationOne g... |
| int-add01d 44367 | First AdditionZero generat... |
| int-add02d 44368 | Second AdditionZero genera... |
| int-sqgeq0d 44369 | SquareGEQZero generator ru... |
| int-eqprincd 44370 | PrincipleOfEquality genera... |
| int-eqtransd 44371 | EqualityTransitivity gener... |
| int-eqmvtd 44372 | EquMoveTerm generator rule... |
| int-eqineqd 44373 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 44374 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 44375 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 44376 | SecondPrincipleOfInequalit... |
| int-ineqtransd 44377 | InequalityTransitivity gen... |
| unitadd 44378 | Theorem used in conjunctio... |
| gsumws3 44379 | Valuation of a length 3 wo... |
| gsumws4 44380 | Valuation of a length 4 wo... |
| amgm2d 44381 | Arithmetic-geometric mean ... |
| amgm3d 44382 | Arithmetic-geometric mean ... |
| amgm4d 44383 | Arithmetic-geometric mean ... |
| spALT 44384 | ~ sp can be proven from th... |
| elnelneqd 44385 | Two classes are not equal ... |
| elnelneq2d 44386 | Two classes are not equal ... |
| rr-spce 44387 | Prove an existential. (Co... |
| rexlimdvaacbv 44388 | Unpack a restricted existe... |
| rexlimddvcbvw 44389 | Unpack a restricted existe... |
| rexlimddvcbv 44390 | Unpack a restricted existe... |
| rr-elrnmpt3d 44391 | Elementhood in an image se... |
| rr-phpd 44392 | Equivalent of ~ php withou... |
| tfindsd 44393 | Deduction associated with ... |
| mnringvald 44396 | Value of the monoid ring f... |
| mnringnmulrd 44397 | Components of a monoid rin... |
| mnringbased 44398 | The base set of a monoid r... |
| mnringbaserd 44399 | The base set of a monoid r... |
| mnringelbased 44400 | Membership in the base set... |
| mnringbasefd 44401 | Elements of a monoid ring ... |
| mnringbasefsuppd 44402 | Elements of a monoid ring ... |
| mnringaddgd 44403 | The additive operation of ... |
| mnring0gd 44404 | The additive identity of a... |
| mnring0g2d 44405 | The additive identity of a... |
| mnringmulrd 44406 | The ring product of a mono... |
| mnringscad 44407 | The scalar ring of a monoi... |
| mnringvscad 44408 | The scalar product of a mo... |
| mnringlmodd 44409 | Monoid rings are left modu... |
| mnringmulrvald 44410 | Value of multiplication in... |
| mnringmulrcld 44411 | Monoid rings are closed un... |
| gru0eld 44412 | A nonempty Grothendieck un... |
| grusucd 44413 | Grothendieck universes are... |
| r1rankcld 44414 | Any rank of the cumulative... |
| grur1cld 44415 | Grothendieck universes are... |
| grurankcld 44416 | Grothendieck universes are... |
| grurankrcld 44417 | If a Grothendieck universe... |
| scotteqd 44420 | Equality theorem for the S... |
| scotteq 44421 | Closed form of ~ scotteqd ... |
| nfscott 44422 | Bound-variable hypothesis ... |
| scottabf 44423 | Value of the Scott operati... |
| scottab 44424 | Value of the Scott operati... |
| scottabes 44425 | Value of the Scott operati... |
| scottss 44426 | Scott's trick produces a s... |
| elscottab 44427 | An element of the output o... |
| scottex2 44428 | ~ scottex expressed using ... |
| scotteld 44429 | The Scott operation sends ... |
| scottelrankd 44430 | Property of a Scott's tric... |
| scottrankd 44431 | Rank of a nonempty Scott's... |
| gruscottcld 44432 | If a Grothendieck universe... |
| dfcoll2 44435 | Alternate definition of th... |
| colleq12d 44436 | Equality theorem for the c... |
| colleq1 44437 | Equality theorem for the c... |
| colleq2 44438 | Equality theorem for the c... |
| nfcoll 44439 | Bound-variable hypothesis ... |
| collexd 44440 | The output of the collecti... |
| cpcolld 44441 | Property of the collection... |
| cpcoll2d 44442 | ~ cpcolld with an extra ex... |
| grucollcld 44443 | A Grothendieck universe co... |
| ismnu 44444 | The hypothesis of this the... |
| mnuop123d 44445 | Operations of a minimal un... |
| mnussd 44446 | Minimal universes are clos... |
| mnuss2d 44447 | ~ mnussd with arguments pr... |
| mnu0eld 44448 | A nonempty minimal univers... |
| mnuop23d 44449 | Second and third operation... |
| mnupwd 44450 | Minimal universes are clos... |
| mnusnd 44451 | Minimal universes are clos... |
| mnuprssd 44452 | A minimal universe contain... |
| mnuprss2d 44453 | Special case of ~ mnuprssd... |
| mnuop3d 44454 | Third operation of a minim... |
| mnuprdlem1 44455 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 44456 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 44457 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 44458 | Lemma for ~ mnuprd . Gene... |
| mnuprd 44459 | Minimal universes are clos... |
| mnuunid 44460 | Minimal universes are clos... |
| mnuund 44461 | Minimal universes are clos... |
| mnutrcld 44462 | Minimal universes contain ... |
| mnutrd 44463 | Minimal universes are tran... |
| mnurndlem1 44464 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 44465 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 44466 | Minimal universes contain ... |
| mnugrud 44467 | Minimal universes are Grot... |
| grumnudlem 44468 | Lemma for ~ grumnud . (Co... |
| grumnud 44469 | Grothendieck universes are... |
| grumnueq 44470 | The class of Grothendieck ... |
| expandan 44471 | Expand conjunction to prim... |
| expandexn 44472 | Expand an existential quan... |
| expandral 44473 | Expand a restricted univer... |
| expandrexn 44474 | Expand a restricted existe... |
| expandrex 44475 | Expand a restricted existe... |
| expanduniss 44476 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 44477 | Express the predicate on `... |
| rr-grothprimbi 44478 | Express "every set is cont... |
| inagrud 44479 | Inaccessible levels of the... |
| inaex 44480 | Assuming the Tarski-Grothe... |
| gruex 44481 | Assuming the Tarski-Grothe... |
| rr-groth 44482 | An equivalent of ~ ax-grot... |
| rr-grothprim 44483 | An equivalent of ~ ax-grot... |
| ismnushort 44484 | Express the predicate on `... |
| dfuniv2 44485 | Alternative definition of ... |
| rr-grothshortbi 44486 | Express "every set is cont... |
| rr-grothshort 44487 | A shorter equivalent of ~ ... |
| nanorxor 44488 | 'nand' is equivalent to th... |
| undisjrab 44489 | Union of two disjoint rest... |
| iso0 44490 | The empty set is an ` R , ... |
| ssrecnpr 44491 | ` RR ` is a subset of both... |
| seff 44492 | Let set ` S ` be the real ... |
| sblpnf 44493 | The infinity ball in the a... |
| prmunb2 44494 | The primes are unbounded. ... |
| dvgrat 44495 | Ratio test for divergence ... |
| cvgdvgrat 44496 | Ratio test for convergence... |
| radcnvrat 44497 | Let ` L ` be the limit, if... |
| reldvds 44498 | The divides relation is in... |
| nznngen 44499 | All positive integers in t... |
| nzss 44500 | The set of multiples of _m... |
| nzin 44501 | The intersection of the se... |
| nzprmdif 44502 | Subtract one prime's multi... |
| hashnzfz 44503 | Special case of ~ hashdvds... |
| hashnzfz2 44504 | Special case of ~ hashnzfz... |
| hashnzfzclim 44505 | As the upper bound ` K ` o... |
| caofcan 44506 | Transfer a cancellation la... |
| ofsubid 44507 | Function analogue of ~ sub... |
| ofmul12 44508 | Function analogue of ~ mul... |
| ofdivrec 44509 | Function analogue of ~ div... |
| ofdivcan4 44510 | Function analogue of ~ div... |
| ofdivdiv2 44511 | Function analogue of ~ div... |
| lhe4.4ex1a 44512 | Example of the Fundamental... |
| dvsconst 44513 | Derivative of a constant f... |
| dvsid 44514 | Derivative of the identity... |
| dvsef 44515 | Derivative of the exponent... |
| expgrowthi 44516 | Exponential growth and dec... |
| dvconstbi 44517 | The derivative of a functi... |
| expgrowth 44518 | Exponential growth and dec... |
| bccval 44521 | Value of the generalized b... |
| bcccl 44522 | Closure of the generalized... |
| bcc0 44523 | The generalized binomial c... |
| bccp1k 44524 | Generalized binomial coeff... |
| bccm1k 44525 | Generalized binomial coeff... |
| bccn0 44526 | Generalized binomial coeff... |
| bccn1 44527 | Generalized binomial coeff... |
| bccbc 44528 | The binomial coefficient a... |
| uzmptshftfval 44529 | When ` F ` is a maps-to fu... |
| dvradcnv2 44530 | The radius of convergence ... |
| binomcxplemwb 44531 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 44532 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 44533 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 44534 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 44535 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 44536 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 44537 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 44538 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 44539 | Lemma for ~ binomcxp . Wh... |
| binomcxp 44540 | Generalize the binomial th... |
| pm10.12 44541 | Theorem *10.12 in [Whitehe... |
| pm10.14 44542 | Theorem *10.14 in [Whitehe... |
| pm10.251 44543 | Theorem *10.251 in [Whiteh... |
| pm10.252 44544 | Theorem *10.252 in [Whiteh... |
| pm10.253 44545 | Theorem *10.253 in [Whiteh... |
| albitr 44546 | Theorem *10.301 in [Whiteh... |
| pm10.42 44547 | Theorem *10.42 in [Whitehe... |
| pm10.52 44548 | Theorem *10.52 in [Whitehe... |
| pm10.53 44549 | Theorem *10.53 in [Whitehe... |
| pm10.541 44550 | Theorem *10.541 in [Whiteh... |
| pm10.542 44551 | Theorem *10.542 in [Whiteh... |
| pm10.55 44552 | Theorem *10.55 in [Whitehe... |
| pm10.56 44553 | Theorem *10.56 in [Whitehe... |
| pm10.57 44554 | Theorem *10.57 in [Whitehe... |
| 2alanimi 44555 | Removes two universal quan... |
| 2al2imi 44556 | Removes two universal quan... |
| pm11.11 44557 | Theorem *11.11 in [Whitehe... |
| pm11.12 44558 | Theorem *11.12 in [Whitehe... |
| 19.21vv 44559 | Compare Theorem *11.3 in [... |
| 2alim 44560 | Theorem *11.32 in [Whitehe... |
| 2albi 44561 | Theorem *11.33 in [Whitehe... |
| 2exim 44562 | Theorem *11.34 in [Whitehe... |
| 2exbi 44563 | Theorem *11.341 in [Whiteh... |
| spsbce-2 44564 | Theorem *11.36 in [Whitehe... |
| 19.33-2 44565 | Theorem *11.421 in [Whiteh... |
| 19.36vv 44566 | Theorem *11.43 in [Whitehe... |
| 19.31vv 44567 | Theorem *11.44 in [Whitehe... |
| 19.37vv 44568 | Theorem *11.46 in [Whitehe... |
| 19.28vv 44569 | Theorem *11.47 in [Whitehe... |
| pm11.52 44570 | Theorem *11.52 in [Whitehe... |
| aaanv 44571 | Theorem *11.56 in [Whitehe... |
| pm11.57 44572 | Theorem *11.57 in [Whitehe... |
| pm11.58 44573 | Theorem *11.58 in [Whitehe... |
| pm11.59 44574 | Theorem *11.59 in [Whitehe... |
| pm11.6 44575 | Theorem *11.6 in [Whitehea... |
| pm11.61 44576 | Theorem *11.61 in [Whitehe... |
| pm11.62 44577 | Theorem *11.62 in [Whitehe... |
| pm11.63 44578 | Theorem *11.63 in [Whitehe... |
| pm11.7 44579 | Theorem *11.7 in [Whitehea... |
| pm11.71 44580 | Theorem *11.71 in [Whitehe... |
| sbeqal1 44581 | If ` x = y ` always implie... |
| sbeqal1i 44582 | Suppose you know ` x = y `... |
| sbeqal2i 44583 | If ` x = y ` implies ` x =... |
| axc5c4c711 44584 | Proof of a theorem that ca... |
| axc5c4c711toc5 44585 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 44586 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 44587 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 44588 | Rederivation of ~ ax-11 fr... |
| axc11next 44589 | This theorem shows that, g... |
| pm13.13a 44590 | One result of theorem *13.... |
| pm13.13b 44591 | Theorem *13.13 in [Whitehe... |
| pm13.14 44592 | Theorem *13.14 in [Whitehe... |
| pm13.192 44593 | Theorem *13.192 in [Whiteh... |
| pm13.193 44594 | Theorem *13.193 in [Whiteh... |
| pm13.194 44595 | Theorem *13.194 in [Whiteh... |
| pm13.195 44596 | Theorem *13.195 in [Whiteh... |
| pm13.196a 44597 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 44598 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 44599 | Theorem *13.22 in [Whitehe... |
| iotain 44600 | Equivalence between two di... |
| iotaexeu 44601 | The iota class exists. Th... |
| iotasbc 44602 | Definition *14.01 in [Whit... |
| iotasbc2 44603 | Theorem *14.111 in [Whiteh... |
| pm14.12 44604 | Theorem *14.12 in [Whitehe... |
| pm14.122a 44605 | Theorem *14.122 in [Whiteh... |
| pm14.122b 44606 | Theorem *14.122 in [Whiteh... |
| pm14.122c 44607 | Theorem *14.122 in [Whiteh... |
| pm14.123a 44608 | Theorem *14.123 in [Whiteh... |
| pm14.123b 44609 | Theorem *14.123 in [Whiteh... |
| pm14.123c 44610 | Theorem *14.123 in [Whiteh... |
| pm14.18 44611 | Theorem *14.18 in [Whitehe... |
| iotaequ 44612 | Theorem *14.2 in [Whitehea... |
| iotavalb 44613 | Theorem *14.202 in [Whiteh... |
| iotasbc5 44614 | Theorem *14.205 in [Whiteh... |
| pm14.24 44615 | Theorem *14.24 in [Whitehe... |
| iotavalsb 44616 | Theorem *14.242 in [Whiteh... |
| sbiota1 44617 | Theorem *14.25 in [Whitehe... |
| sbaniota 44618 | Theorem *14.26 in [Whitehe... |
| iotasbcq 44619 | Theorem *14.272 in [Whiteh... |
| elnev 44620 | Any set that contains one ... |
| rusbcALT 44621 | A version of Russell's par... |
| compeq 44622 | Equality between two ways ... |
| compne 44623 | The complement of ` A ` is... |
| compab 44624 | Two ways of saying "the co... |
| conss2 44625 | Contrapositive law for sub... |
| conss1 44626 | Contrapositive law for sub... |
| ralbidar 44627 | More general form of ~ ral... |
| rexbidar 44628 | More general form of ~ rex... |
| dropab1 44629 | Theorem to aid use of the ... |
| dropab2 44630 | Theorem to aid use of the ... |
| ipo0 44631 | If the identity relation p... |
| ifr0 44632 | A class that is founded by... |
| ordpss 44633 | ~ ordelpss with an anteced... |
| fvsb 44634 | Explicit substitution of a... |
| fveqsb 44635 | Implicit substitution of a... |
| xpexb 44636 | A Cartesian product exists... |
| trelpss 44637 | An element of a transitive... |
| addcomgi 44638 | Generalization of commutat... |
| addrval 44648 | Value of the operation of ... |
| subrval 44649 | Value of the operation of ... |
| mulvval 44650 | Value of the operation of ... |
| addrfv 44651 | Vector addition at a value... |
| subrfv 44652 | Vector subtraction at a va... |
| mulvfv 44653 | Scalar multiplication at a... |
| addrfn 44654 | Vector addition produces a... |
| subrfn 44655 | Vector subtraction produce... |
| mulvfn 44656 | Scalar multiplication prod... |
| addrcom 44657 | Vector addition is commuta... |
| idiALT 44661 | Placeholder for ~ idi . T... |
| exbir 44662 | Exportation implication al... |
| 3impexpbicom 44663 | Version of ~ 3impexp where... |
| 3impexpbicomi 44664 | Inference associated with ... |
| bi1imp 44665 | Importation inference simi... |
| bi2imp 44666 | Importation inference simi... |
| bi3impb 44667 | Similar to ~ 3impb with im... |
| bi3impa 44668 | Similar to ~ 3impa with im... |
| bi23impib 44669 | ~ 3impib with the inner im... |
| bi13impib 44670 | ~ 3impib with the outer im... |
| bi123impib 44671 | ~ 3impib with the implicat... |
| bi13impia 44672 | ~ 3impia with the outer im... |
| bi123impia 44673 | ~ 3impia with the implicat... |
| bi33imp12 44674 | ~ 3imp with innermost impl... |
| bi13imp23 44675 | ~ 3imp with outermost impl... |
| bi13imp2 44676 | Similar to ~ 3imp except t... |
| bi12imp3 44677 | Similar to ~ 3imp except a... |
| bi23imp1 44678 | Similar to ~ 3imp except a... |
| bi123imp0 44679 | Similar to ~ 3imp except a... |
| 4animp1 44680 | A single hypothesis unific... |
| 4an31 44681 | A rearrangement of conjunc... |
| 4an4132 44682 | A rearrangement of conjunc... |
| expcomdg 44683 | Biconditional form of ~ ex... |
| iidn3 44684 | ~ idn3 without virtual ded... |
| ee222 44685 | ~ e222 without virtual ded... |
| ee3bir 44686 | Right-biconditional form o... |
| ee13 44687 | ~ e13 without virtual dedu... |
| ee121 44688 | ~ e121 without virtual ded... |
| ee122 44689 | ~ e122 without virtual ded... |
| ee333 44690 | ~ e333 without virtual ded... |
| ee323 44691 | ~ e323 without virtual ded... |
| 3ornot23 44692 | If the second and third di... |
| orbi1r 44693 | ~ orbi1 with order of disj... |
| 3orbi123 44694 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 44695 | Closed form of ~ syl5 . D... |
| impexpd 44696 | The following User's Proof... |
| com3rgbi 44697 | The following User's Proof... |
| impexpdcom 44698 | The following User's Proof... |
| ee1111 44699 | Non-virtual deduction form... |
| pm2.43bgbi 44700 | Logical equivalence of a 2... |
| pm2.43cbi 44701 | Logical equivalence of a 3... |
| ee233 44702 | Non-virtual deduction form... |
| imbi13 44703 | Join three logical equival... |
| ee33 44704 | Non-virtual deduction form... |
| con5 44705 | Biconditional contrapositi... |
| con5i 44706 | Inference form of ~ con5 .... |
| exlimexi 44707 | Inference similar to Theor... |
| sb5ALT 44708 | Equivalence for substituti... |
| eexinst01 44709 | ~ exinst01 without virtual... |
| eexinst11 44710 | ~ exinst11 without virtual... |
| vk15.4j 44711 | Excercise 4j of Unit 15 of... |
| notnotrALT 44712 | Converse of double negatio... |
| con3ALT2 44713 | Contraposition. Alternate... |
| ssralv2 44714 | Quantification restricted ... |
| sbc3or 44715 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 44716 | Closed form of ~ alrimi wi... |
| rspsbc2 44717 | ~ rspsbc with two quantify... |
| sbcoreleleq 44718 | Substitution of a setvar v... |
| tratrb 44719 | If a class is transitive a... |
| ordelordALT 44720 | An element of an ordinal c... |
| sbcim2g 44721 | Distribution of class subs... |
| sbcbi 44722 | Implication form of ~ sbcb... |
| trsbc 44723 | Formula-building inference... |
| truniALT 44724 | The union of a class of tr... |
| onfrALTlem5 44725 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 44726 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 44727 | Lemma for ~ onfrALT . (Co... |
| ggen31 44728 | ~ gen31 without virtual de... |
| onfrALTlem2 44729 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 44730 | A theorem pertaining to th... |
| onfrALTlem1 44731 | Lemma for ~ onfrALT . (Co... |
| onfrALT 44732 | The membership relation is... |
| 19.41rg 44733 | Closed form of right-to-le... |
| opelopab4 44734 | Ordered pair membership in... |
| 2pm13.193 44735 | ~ pm13.193 for two variabl... |
| hbntal 44736 | A closed form of ~ hbn . ~... |
| hbimpg 44737 | A closed form of ~ hbim . ... |
| hbalg 44738 | Closed form of ~ hbal . D... |
| hbexg 44739 | Closed form of ~ nfex . D... |
| ax6e2eq 44740 | Alternate form of ~ ax6e f... |
| ax6e2nd 44741 | If at least two sets exist... |
| ax6e2ndeq 44742 | "At least two sets exist" ... |
| 2sb5nd 44743 | Equivalence for double sub... |
| 2uasbanh 44744 | Distribute the unabbreviat... |
| 2uasban 44745 | Distribute the unabbreviat... |
| e2ebind 44746 | Absorption of an existenti... |
| elpwgded 44747 | ~ elpwgdedVD in convention... |
| trelded 44748 | Deduction form of ~ trel .... |
| jaoded 44749 | Deduction form of ~ jao . ... |
| sbtT 44750 | A substitution into a theo... |
| not12an2impnot1 44751 | If a double conjunction is... |
| in1 44754 | Inference form of ~ df-vd1... |
| iin1 44755 | ~ in1 without virtual dedu... |
| dfvd1ir 44756 | Inference form of ~ df-vd1... |
| idn1 44757 | Virtual deduction identity... |
| dfvd1imp 44758 | Left-to-right part of defi... |
| dfvd1impr 44759 | Right-to-left part of defi... |
| dfvd2 44762 | Definition of a 2-hypothes... |
| dfvd2an 44765 | Definition of a 2-hypothes... |
| dfvd2ani 44766 | Inference form of ~ dfvd2a... |
| dfvd2anir 44767 | Right-to-left inference fo... |
| dfvd2i 44768 | Inference form of ~ dfvd2 ... |
| dfvd2ir 44769 | Right-to-left inference fo... |
| dfvd3 44774 | Definition of a 3-hypothes... |
| dfvd3i 44775 | Inference form of ~ dfvd3 ... |
| dfvd3ir 44776 | Right-to-left inference fo... |
| dfvd3an 44777 | Definition of a 3-hypothes... |
| dfvd3ani 44778 | Inference form of ~ dfvd3a... |
| dfvd3anir 44779 | Right-to-left inference fo... |
| vd01 44780 | A virtual hypothesis virtu... |
| vd02 44781 | Two virtual hypotheses vir... |
| vd03 44782 | A theorem is virtually inf... |
| vd12 44783 | A virtual deduction with 1... |
| vd13 44784 | A virtual deduction with 1... |
| vd23 44785 | A virtual deduction with 2... |
| dfvd2imp 44786 | The virtual deduction form... |
| dfvd2impr 44787 | A 2-antecedent nested impl... |
| in2 44788 | The virtual deduction intr... |
| int2 44789 | The virtual deduction intr... |
| iin2 44790 | ~ in2 without virtual dedu... |
| in2an 44791 | The virtual deduction intr... |
| in3 44792 | The virtual deduction intr... |
| iin3 44793 | ~ in3 without virtual dedu... |
| in3an 44794 | The virtual deduction intr... |
| int3 44795 | The virtual deduction intr... |
| idn2 44796 | Virtual deduction identity... |
| iden2 44797 | Virtual deduction identity... |
| idn3 44798 | Virtual deduction identity... |
| gen11 44799 | Virtual deduction generali... |
| gen11nv 44800 | Virtual deduction generali... |
| gen12 44801 | Virtual deduction generali... |
| gen21 44802 | Virtual deduction generali... |
| gen21nv 44803 | Virtual deduction form of ... |
| gen31 44804 | Virtual deduction generali... |
| gen22 44805 | Virtual deduction generali... |
| ggen22 44806 | ~ gen22 without virtual de... |
| exinst 44807 | Existential Instantiation.... |
| exinst01 44808 | Existential Instantiation.... |
| exinst11 44809 | Existential Instantiation.... |
| e1a 44810 | A Virtual deduction elimin... |
| el1 44811 | A Virtual deduction elimin... |
| e1bi 44812 | Biconditional form of ~ e1... |
| e1bir 44813 | Right biconditional form o... |
| e2 44814 | A virtual deduction elimin... |
| e2bi 44815 | Biconditional form of ~ e2... |
| e2bir 44816 | Right biconditional form o... |
| ee223 44817 | ~ e223 without virtual ded... |
| e223 44818 | A virtual deduction elimin... |
| e222 44819 | A virtual deduction elimin... |
| e220 44820 | A virtual deduction elimin... |
| ee220 44821 | ~ e220 without virtual ded... |
| e202 44822 | A virtual deduction elimin... |
| ee202 44823 | ~ e202 without virtual ded... |
| e022 44824 | A virtual deduction elimin... |
| ee022 44825 | ~ e022 without virtual ded... |
| e002 44826 | A virtual deduction elimin... |
| ee002 44827 | ~ e002 without virtual ded... |
| e020 44828 | A virtual deduction elimin... |
| ee020 44829 | ~ e020 without virtual ded... |
| e200 44830 | A virtual deduction elimin... |
| ee200 44831 | ~ e200 without virtual ded... |
| e221 44832 | A virtual deduction elimin... |
| ee221 44833 | ~ e221 without virtual ded... |
| e212 44834 | A virtual deduction elimin... |
| ee212 44835 | ~ e212 without virtual ded... |
| e122 44836 | A virtual deduction elimin... |
| e112 44837 | A virtual deduction elimin... |
| ee112 44838 | ~ e112 without virtual ded... |
| e121 44839 | A virtual deduction elimin... |
| e211 44840 | A virtual deduction elimin... |
| ee211 44841 | ~ e211 without virtual ded... |
| e210 44842 | A virtual deduction elimin... |
| ee210 44843 | ~ e210 without virtual ded... |
| e201 44844 | A virtual deduction elimin... |
| ee201 44845 | ~ e201 without virtual ded... |
| e120 44846 | A virtual deduction elimin... |
| ee120 44847 | Virtual deduction rule ~ e... |
| e021 44848 | A virtual deduction elimin... |
| ee021 44849 | ~ e021 without virtual ded... |
| e012 44850 | A virtual deduction elimin... |
| ee012 44851 | ~ e012 without virtual ded... |
| e102 44852 | A virtual deduction elimin... |
| ee102 44853 | ~ e102 without virtual ded... |
| e22 44854 | A virtual deduction elimin... |
| e22an 44855 | Conjunction form of ~ e22 ... |
| ee22an 44856 | ~ e22an without virtual de... |
| e111 44857 | A virtual deduction elimin... |
| e1111 44858 | A virtual deduction elimin... |
| e110 44859 | A virtual deduction elimin... |
| ee110 44860 | ~ e110 without virtual ded... |
| e101 44861 | A virtual deduction elimin... |
| ee101 44862 | ~ e101 without virtual ded... |
| e011 44863 | A virtual deduction elimin... |
| ee011 44864 | ~ e011 without virtual ded... |
| e100 44865 | A virtual deduction elimin... |
| ee100 44866 | ~ e100 without virtual ded... |
| e010 44867 | A virtual deduction elimin... |
| ee010 44868 | ~ e010 without virtual ded... |
| e001 44869 | A virtual deduction elimin... |
| ee001 44870 | ~ e001 without virtual ded... |
| e11 44871 | A virtual deduction elimin... |
| e11an 44872 | Conjunction form of ~ e11 ... |
| ee11an 44873 | ~ e11an without virtual de... |
| e01 44874 | A virtual deduction elimin... |
| e01an 44875 | Conjunction form of ~ e01 ... |
| ee01an 44876 | ~ e01an without virtual de... |
| e10 44877 | A virtual deduction elimin... |
| e10an 44878 | Conjunction form of ~ e10 ... |
| ee10an 44879 | ~ e10an without virtual de... |
| e02 44880 | A virtual deduction elimin... |
| e02an 44881 | Conjunction form of ~ e02 ... |
| ee02an 44882 | ~ e02an without virtual de... |
| eel021old 44883 | ~ el021old without virtual... |
| el021old 44884 | A virtual deduction elimin... |
| eel000cT 44885 | An elimination deduction. ... |
| eel0TT 44886 | An elimination deduction. ... |
| eelT00 44887 | An elimination deduction. ... |
| eelTTT 44888 | An elimination deduction. ... |
| eelT11 44889 | An elimination deduction. ... |
| eelT1 44890 | Syllogism inference combin... |
| eelT12 44891 | An elimination deduction. ... |
| eelTT1 44892 | An elimination deduction. ... |
| eelT01 44893 | An elimination deduction. ... |
| eel0T1 44894 | An elimination deduction. ... |
| eel12131 44895 | An elimination deduction. ... |
| eel2131 44896 | ~ syl2an with antecedents ... |
| eel3132 44897 | ~ syl2an with antecedents ... |
| eel0321old 44898 | ~ el0321old without virtua... |
| el0321old 44899 | A virtual deduction elimin... |
| eel2122old 44900 | ~ el2122old without virtua... |
| el2122old 44901 | A virtual deduction elimin... |
| eel0000 44902 | Elimination rule similar t... |
| eel00001 44903 | An elimination deduction. ... |
| eel00000 44904 | Elimination rule similar ~... |
| eel11111 44905 | Five-hypothesis eliminatio... |
| e12 44906 | A virtual deduction elimin... |
| e12an 44907 | Conjunction form of ~ e12 ... |
| el12 44908 | Virtual deduction form of ... |
| e20 44909 | A virtual deduction elimin... |
| e20an 44910 | Conjunction form of ~ e20 ... |
| ee20an 44911 | ~ e20an without virtual de... |
| e21 44912 | A virtual deduction elimin... |
| e21an 44913 | Conjunction form of ~ e21 ... |
| ee21an 44914 | ~ e21an without virtual de... |
| e333 44915 | A virtual deduction elimin... |
| e33 44916 | A virtual deduction elimin... |
| e33an 44917 | Conjunction form of ~ e33 ... |
| ee33an 44918 | ~ e33an without virtual de... |
| e3 44919 | Meta-connective form of ~ ... |
| e3bi 44920 | Biconditional form of ~ e3... |
| e3bir 44921 | Right biconditional form o... |
| e03 44922 | A virtual deduction elimin... |
| ee03 44923 | ~ e03 without virtual dedu... |
| e03an 44924 | Conjunction form of ~ e03 ... |
| ee03an 44925 | Conjunction form of ~ ee03... |
| e30 44926 | A virtual deduction elimin... |
| ee30 44927 | ~ e30 without virtual dedu... |
| e30an 44928 | A virtual deduction elimin... |
| ee30an 44929 | Conjunction form of ~ ee30... |
| e13 44930 | A virtual deduction elimin... |
| e13an 44931 | A virtual deduction elimin... |
| ee13an 44932 | ~ e13an without virtual de... |
| e31 44933 | A virtual deduction elimin... |
| ee31 44934 | ~ e31 without virtual dedu... |
| e31an 44935 | A virtual deduction elimin... |
| ee31an 44936 | ~ e31an without virtual de... |
| e23 44937 | A virtual deduction elimin... |
| e23an 44938 | A virtual deduction elimin... |
| ee23an 44939 | ~ e23an without virtual de... |
| e32 44940 | A virtual deduction elimin... |
| ee32 44941 | ~ e32 without virtual dedu... |
| e32an 44942 | A virtual deduction elimin... |
| ee32an 44943 | ~ e33an without virtual de... |
| e123 44944 | A virtual deduction elimin... |
| ee123 44945 | ~ e123 without virtual ded... |
| el123 44946 | A virtual deduction elimin... |
| e233 44947 | A virtual deduction elimin... |
| e323 44948 | A virtual deduction elimin... |
| e000 44949 | A virtual deduction elimin... |
| e00 44950 | Elimination rule identical... |
| e00an 44951 | Elimination rule identical... |
| eel00cT 44952 | An elimination deduction. ... |
| eelTT 44953 | An elimination deduction. ... |
| e0a 44954 | Elimination rule identical... |
| eelT 44955 | An elimination deduction. ... |
| eel0cT 44956 | An elimination deduction. ... |
| eelT0 44957 | An elimination deduction. ... |
| e0bi 44958 | Elimination rule identical... |
| e0bir 44959 | Elimination rule identical... |
| uun0.1 44960 | Convention notation form o... |
| un0.1 44961 | ` T. ` is the constant tru... |
| uunT1 44962 | A deduction unionizing a n... |
| uunT1p1 44963 | A deduction unionizing a n... |
| uunT21 44964 | A deduction unionizing a n... |
| uun121 44965 | A deduction unionizing a n... |
| uun121p1 44966 | A deduction unionizing a n... |
| uun132 44967 | A deduction unionizing a n... |
| uun132p1 44968 | A deduction unionizing a n... |
| anabss7p1 44969 | A deduction unionizing a n... |
| un10 44970 | A unionizing deduction. (... |
| un01 44971 | A unionizing deduction. (... |
| un2122 44972 | A deduction unionizing a n... |
| uun2131 44973 | A deduction unionizing a n... |
| uun2131p1 44974 | A deduction unionizing a n... |
| uunTT1 44975 | A deduction unionizing a n... |
| uunTT1p1 44976 | A deduction unionizing a n... |
| uunTT1p2 44977 | A deduction unionizing a n... |
| uunT11 44978 | A deduction unionizing a n... |
| uunT11p1 44979 | A deduction unionizing a n... |
| uunT11p2 44980 | A deduction unionizing a n... |
| uunT12 44981 | A deduction unionizing a n... |
| uunT12p1 44982 | A deduction unionizing a n... |
| uunT12p2 44983 | A deduction unionizing a n... |
| uunT12p3 44984 | A deduction unionizing a n... |
| uunT12p4 44985 | A deduction unionizing a n... |
| uunT12p5 44986 | A deduction unionizing a n... |
| uun111 44987 | A deduction unionizing a n... |
| 3anidm12p1 44988 | A deduction unionizing a n... |
| 3anidm12p2 44989 | A deduction unionizing a n... |
| uun123 44990 | A deduction unionizing a n... |
| uun123p1 44991 | A deduction unionizing a n... |
| uun123p2 44992 | A deduction unionizing a n... |
| uun123p3 44993 | A deduction unionizing a n... |
| uun123p4 44994 | A deduction unionizing a n... |
| uun2221 44995 | A deduction unionizing a n... |
| uun2221p1 44996 | A deduction unionizing a n... |
| uun2221p2 44997 | A deduction unionizing a n... |
| 3impdirp1 44998 | A deduction unionizing a n... |
| 3impcombi 44999 | A 1-hypothesis proposition... |
| trsspwALT 45000 | Virtual deduction proof of... |
| trsspwALT2 45001 | Virtual deduction proof of... |
| trsspwALT3 45002 | Short predicate calculus p... |
| sspwtr 45003 | Virtual deduction proof of... |
| sspwtrALT 45004 | Virtual deduction proof of... |
| sspwtrALT2 45005 | Short predicate calculus p... |
| pwtrVD 45006 | Virtual deduction proof of... |
| pwtrrVD 45007 | Virtual deduction proof of... |
| suctrALT 45008 | The successor of a transit... |
| snssiALTVD 45009 | Virtual deduction proof of... |
| snssiALT 45010 | If a class is an element o... |
| snsslVD 45011 | Virtual deduction proof of... |
| snssl 45012 | If a singleton is a subcla... |
| snelpwrVD 45013 | Virtual deduction proof of... |
| unipwrVD 45014 | Virtual deduction proof of... |
| unipwr 45015 | A class is a subclass of t... |
| sstrALT2VD 45016 | Virtual deduction proof of... |
| sstrALT2 45017 | Virtual deduction proof of... |
| suctrALT2VD 45018 | Virtual deduction proof of... |
| suctrALT2 45019 | Virtual deduction proof of... |
| elex2VD 45020 | Virtual deduction proof of... |
| elex22VD 45021 | Virtual deduction proof of... |
| eqsbc2VD 45022 | Virtual deduction proof of... |
| zfregs2VD 45023 | Virtual deduction proof of... |
| tpid3gVD 45024 | Virtual deduction proof of... |
| en3lplem1VD 45025 | Virtual deduction proof of... |
| en3lplem2VD 45026 | Virtual deduction proof of... |
| en3lpVD 45027 | Virtual deduction proof of... |
| simplbi2VD 45028 | Virtual deduction proof of... |
| 3ornot23VD 45029 | Virtual deduction proof of... |
| orbi1rVD 45030 | Virtual deduction proof of... |
| bitr3VD 45031 | Virtual deduction proof of... |
| 3orbi123VD 45032 | Virtual deduction proof of... |
| sbc3orgVD 45033 | Virtual deduction proof of... |
| 19.21a3con13vVD 45034 | Virtual deduction proof of... |
| exbirVD 45035 | Virtual deduction proof of... |
| exbiriVD 45036 | Virtual deduction proof of... |
| rspsbc2VD 45037 | Virtual deduction proof of... |
| 3impexpVD 45038 | Virtual deduction proof of... |
| 3impexpbicomVD 45039 | Virtual deduction proof of... |
| 3impexpbicomiVD 45040 | Virtual deduction proof of... |
| sbcoreleleqVD 45041 | Virtual deduction proof of... |
| hbra2VD 45042 | Virtual deduction proof of... |
| tratrbVD 45043 | Virtual deduction proof of... |
| al2imVD 45044 | Virtual deduction proof of... |
| syl5impVD 45045 | Virtual deduction proof of... |
| idiVD 45046 | Virtual deduction proof of... |
| ancomstVD 45047 | Closed form of ~ ancoms . ... |
| ssralv2VD 45048 | Quantification restricted ... |
| ordelordALTVD 45049 | An element of an ordinal c... |
| equncomVD 45050 | If a class equals the unio... |
| equncomiVD 45051 | Inference form of ~ equnco... |
| sucidALTVD 45052 | A set belongs to its succe... |
| sucidALT 45053 | A set belongs to its succe... |
| sucidVD 45054 | A set belongs to its succe... |
| imbi12VD 45055 | Implication form of ~ imbi... |
| imbi13VD 45056 | Join three logical equival... |
| sbcim2gVD 45057 | Distribution of class subs... |
| sbcbiVD 45058 | Implication form of ~ sbcb... |
| trsbcVD 45059 | Formula-building inference... |
| truniALTVD 45060 | The union of a class of tr... |
| ee33VD 45061 | Non-virtual deduction form... |
| trintALTVD 45062 | The intersection of a clas... |
| trintALT 45063 | The intersection of a clas... |
| undif3VD 45064 | The first equality of Exer... |
| sbcssgVD 45065 | Virtual deduction proof of... |
| csbingVD 45066 | Virtual deduction proof of... |
| onfrALTlem5VD 45067 | Virtual deduction proof of... |
| onfrALTlem4VD 45068 | Virtual deduction proof of... |
| onfrALTlem3VD 45069 | Virtual deduction proof of... |
| simplbi2comtVD 45070 | Virtual deduction proof of... |
| onfrALTlem2VD 45071 | Virtual deduction proof of... |
| onfrALTlem1VD 45072 | Virtual deduction proof of... |
| onfrALTVD 45073 | Virtual deduction proof of... |
| csbeq2gVD 45074 | Virtual deduction proof of... |
| csbsngVD 45075 | Virtual deduction proof of... |
| csbxpgVD 45076 | Virtual deduction proof of... |
| csbresgVD 45077 | Virtual deduction proof of... |
| csbrngVD 45078 | Virtual deduction proof of... |
| csbima12gALTVD 45079 | Virtual deduction proof of... |
| csbunigVD 45080 | Virtual deduction proof of... |
| csbfv12gALTVD 45081 | Virtual deduction proof of... |
| con5VD 45082 | Virtual deduction proof of... |
| relopabVD 45083 | Virtual deduction proof of... |
| 19.41rgVD 45084 | Virtual deduction proof of... |
| 2pm13.193VD 45085 | Virtual deduction proof of... |
| hbimpgVD 45086 | Virtual deduction proof of... |
| hbalgVD 45087 | Virtual deduction proof of... |
| hbexgVD 45088 | Virtual deduction proof of... |
| ax6e2eqVD 45089 | The following User's Proof... |
| ax6e2ndVD 45090 | The following User's Proof... |
| ax6e2ndeqVD 45091 | The following User's Proof... |
| 2sb5ndVD 45092 | The following User's Proof... |
| 2uasbanhVD 45093 | The following User's Proof... |
| e2ebindVD 45094 | The following User's Proof... |
| sb5ALTVD 45095 | The following User's Proof... |
| vk15.4jVD 45096 | The following User's Proof... |
| notnotrALTVD 45097 | The following User's Proof... |
| con3ALTVD 45098 | The following User's Proof... |
| elpwgdedVD 45099 | Membership in a power clas... |
| sspwimp 45100 | If a class is a subclass o... |
| sspwimpVD 45101 | The following User's Proof... |
| sspwimpcf 45102 | If a class is a subclass o... |
| sspwimpcfVD 45103 | The following User's Proof... |
| suctrALTcf 45104 | The successor of a transit... |
| suctrALTcfVD 45105 | The following User's Proof... |
| suctrALT3 45106 | The successor of a transit... |
| sspwimpALT 45107 | If a class is a subclass o... |
| unisnALT 45108 | A set equals the union of ... |
| notnotrALT2 45109 | Converse of double negatio... |
| sspwimpALT2 45110 | If a class is a subclass o... |
| e2ebindALT 45111 | Absorption of an existenti... |
| ax6e2ndALT 45112 | If at least two sets exist... |
| ax6e2ndeqALT 45113 | "At least two sets exist" ... |
| 2sb5ndALT 45114 | Equivalence for double sub... |
| chordthmALT 45115 | The intersecting chords th... |
| isosctrlem1ALT 45116 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 45117 | The indexed union of conne... |
| iunconnALT 45118 | The indexed union of conne... |
| sineq0ALT 45119 | A complex number whose sin... |
| rspesbcd 45120 | Restricted quantifier vers... |
| rext0 45121 | Nonempty existential quant... |
| dfbi1ALTa 45122 | Version of ~ dfbi1ALT usin... |
| simprimi 45123 | Inference associated with ... |
| dfbi1ALTb 45124 | Further shorten ~ dfbi1ALT... |
| relpeq1 45127 | Equality theorem for relat... |
| relpeq2 45128 | Equality theorem for relat... |
| relpeq3 45129 | Equality theorem for relat... |
| relpeq4 45130 | Equality theorem for relat... |
| relpeq5 45131 | Equality theorem for relat... |
| nfrelp 45132 | Bound-variable hypothesis ... |
| relpf 45133 | A relation-preserving func... |
| relprel 45134 | A relation-preserving func... |
| relpmin 45135 | A preimage of a minimal el... |
| relpfrlem 45136 | Lemma for ~ relpfr . Prov... |
| relpfr 45137 | If the image of a set unde... |
| orbitex 45138 | Orbits exist. Given a set... |
| orbitinit 45139 | A set is contained in its ... |
| orbitcl 45140 | The orbit under a function... |
| orbitclmpt 45141 | Version of ~ orbitcl using... |
| trwf 45142 | The class of well-founded ... |
| rankrelp 45143 | The rank function preserve... |
| wffr 45144 | The class of well-founded ... |
| trfr 45145 | A transitive class well-fo... |
| tcfr 45146 | A set is well-founded if a... |
| xpwf 45147 | The Cartesian product of t... |
| dmwf 45148 | The domain of a well-found... |
| rnwf 45149 | The range of a well-founde... |
| relwf 45150 | A relation is a well-found... |
| ralabso 45151 | Simplification of restrict... |
| rexabso 45152 | Simplification of restrict... |
| ralabsod 45153 | Deduction form of ~ ralabs... |
| rexabsod 45154 | Deduction form of ~ rexabs... |
| ralabsobidv 45155 | Formula-building lemma for... |
| rexabsobidv 45156 | Formula-building lemma for... |
| ssabso 45157 | The notion " ` x ` is a su... |
| disjabso 45158 | Disjointness is absolute f... |
| n0abso 45159 | Nonemptiness is absolute f... |
| traxext 45160 | A transitive class models ... |
| modelaxreplem1 45161 | Lemma for ~ modelaxrep . ... |
| modelaxreplem2 45162 | Lemma for ~ modelaxrep . ... |
| modelaxreplem3 45163 | Lemma for ~ modelaxrep . ... |
| modelaxrep 45164 | Conditions which guarantee... |
| ssclaxsep 45165 | A class that is closed und... |
| 0elaxnul 45166 | A class that contains the ... |
| pwclaxpow 45167 | Suppose ` M ` is a transit... |
| prclaxpr 45168 | A class that is closed und... |
| uniclaxun 45169 | A class that is closed und... |
| sswfaxreg 45170 | A subclass of the class of... |
| omssaxinf2 45171 | A class that contains all ... |
| omelaxinf2 45172 | A transitive class that co... |
| dfac5prim 45173 | ~ dfac5 expanded into prim... |
| ac8prim 45174 | ~ ac8 expanded into primit... |
| modelac8prim 45175 | If ` M ` is a transitive c... |
| wfaxext 45176 | The class of well-founded ... |
| wfaxrep 45177 | The class of well-founded ... |
| wfaxsep 45178 | The class of well-founded ... |
| wfaxnul 45179 | The class of well-founded ... |
| wfaxpow 45180 | The class of well-founded ... |
| wfaxpr 45181 | The class of well-founded ... |
| wfaxun 45182 | The class of well-founded ... |
| wfaxreg 45183 | The class of well-founded ... |
| wfaxinf2 45184 | The class of well-founded ... |
| wfac8prim 45185 | The class of well-founded ... |
| brpermmodel 45186 | The membership relation in... |
| brpermmodelcnv 45187 | Ordinary membership expres... |
| permaxext 45188 | The Axiom of Extensionalit... |
| permaxrep 45189 | The Axiom of Replacement ~... |
| permaxsep 45190 | The Axiom of Separation ~ ... |
| permaxnul 45191 | The Null Set Axiom ~ ax-nu... |
| permaxpow 45192 | The Axiom of Power Sets ~ ... |
| permaxpr 45193 | The Axiom of Pairing ~ ax-... |
| permaxun 45194 | The Axiom of Union ~ ax-un... |
| permaxinf2lem 45195 | Lemma for ~ permaxinf2 . ... |
| permaxinf2 45196 | The Axiom of Infinity ~ ax... |
| permac8prim 45197 | The Axiom of Choice ~ ac8p... |
| nregmodelf1o 45198 | Define a permutation ` F `... |
| nregmodellem 45199 | Lemma for ~ nregmodel . (... |
| nregmodel 45200 | The Axiom of Regularity ~ ... |
| nregmodelaxext 45201 | The Axiom of Extensionalit... |
| evth2f 45202 | A version of ~ evth2 using... |
| elunif 45203 | A version of ~ eluni using... |
| rzalf 45204 | A version of ~ rzal using ... |
| fvelrnbf 45205 | A version of ~ fvelrnb usi... |
| rfcnpre1 45206 | If F is a continuous funct... |
| ubelsupr 45207 | If U belongs to A and U is... |
| fsumcnf 45208 | A finite sum of functions ... |
| mulltgt0 45209 | The product of a negative ... |
| rspcegf 45210 | A version of ~ rspcev usin... |
| rabexgf 45211 | A version of ~ rabexg usin... |
| fcnre 45212 | A function continuous with... |
| sumsnd 45213 | A sum of a singleton is th... |
| evthf 45214 | A version of ~ evth using ... |
| cnfex 45215 | The class of continuous fu... |
| fnchoice 45216 | For a finite set, a choice... |
| refsumcn 45217 | A finite sum of continuous... |
| rfcnpre2 45218 | If ` F ` is a continuous f... |
| cncmpmax 45219 | When the hypothesis for th... |
| rfcnpre3 45220 | If F is a continuous funct... |
| rfcnpre4 45221 | If F is a continuous funct... |
| sumpair 45222 | Sum of two distinct comple... |
| rfcnnnub 45223 | Given a real continuous fu... |
| refsum2cnlem1 45224 | This is the core Lemma for... |
| refsum2cn 45225 | The sum of two continuus r... |
| adantlllr 45226 | Deduction adding a conjunc... |
| 3adantlr3 45227 | Deduction adding a conjunc... |
| 3adantll2 45228 | Deduction adding a conjunc... |
| 3adantll3 45229 | Deduction adding a conjunc... |
| ssnel 45230 | If not element of a set, t... |
| sncldre 45231 | A singleton is closed w.r.... |
| n0p 45232 | A polynomial with a nonzer... |
| pm2.65ni 45233 | Inference rule for proof b... |
| iuneq2df 45234 | Equality deduction for ind... |
| nnfoctb 45235 | There exists a mapping fro... |
| elpwinss 45236 | An element of the powerset... |
| unidmex 45237 | If ` F ` is a set, then ` ... |
| ndisj2 45238 | A non-disjointness conditi... |
| zenom 45239 | The set of integer numbers... |
| uzwo4 45240 | Well-ordering principle: a... |
| unisn0 45241 | The union of the singleton... |
| ssin0 45242 | If two classes are disjoin... |
| inabs3 45243 | Absorption law for interse... |
| pwpwuni 45244 | Relationship between power... |
| disjiun2 45245 | In a disjoint collection, ... |
| 0pwfi 45246 | The empty set is in any po... |
| ssinss2d 45247 | Intersection preserves sub... |
| zct 45248 | The set of integer numbers... |
| pwfin0 45249 | A finite set always belong... |
| uzct 45250 | An upper integer set is co... |
| iunxsnf 45251 | A singleton index picks ou... |
| fiiuncl 45252 | If a set is closed under t... |
| iunp1 45253 | The addition of the next s... |
| fiunicl 45254 | If a set is closed under t... |
| ixpeq2d 45255 | Equality theorem for infin... |
| disjxp1 45256 | The sets of a cartesian pr... |
| disjsnxp 45257 | The sets in the cartesian ... |
| eliind 45258 | Membership in indexed inte... |
| rspcef 45259 | Restricted existential spe... |
| ixpssmapc 45260 | An infinite Cartesian prod... |
| elintd 45261 | Membership in class inters... |
| ssdf 45262 | A sufficient condition for... |
| brneqtrd 45263 | Substitution of equal clas... |
| ssnct 45264 | A set containing an uncoun... |
| ssuniint 45265 | Sufficient condition for b... |
| elintdv 45266 | Membership in class inters... |
| ssd 45267 | A sufficient condition for... |
| ralimralim 45268 | Introducing any antecedent... |
| snelmap 45269 | Membership of the element ... |
| xrnmnfpnf 45270 | An extended real that is n... |
| iuneq1i 45271 | Equality theorem for index... |
| nssrex 45272 | Negation of subclass relat... |
| ssinc 45273 | Inclusion relation for a m... |
| ssdec 45274 | Inclusion relation for a m... |
| elixpconstg 45275 | Membership in an infinite ... |
| iineq1d 45276 | Equality theorem for index... |
| metpsmet 45277 | A metric is a pseudometric... |
| ixpssixp 45278 | Subclass theorem for infin... |
| ballss3 45279 | A sufficient condition for... |
| iunincfi 45280 | Given a sequence of increa... |
| nsstr 45281 | If it's not a subclass, it... |
| rexanuz3 45282 | Combine two different uppe... |
| cbvmpo2 45283 | Rule to change the second ... |
| cbvmpo1 45284 | Rule to change the first b... |
| eliuniin 45285 | Indexed union of indexed i... |
| ssabf 45286 | Subclass of a class abstra... |
| pssnssi 45287 | A proper subclass does not... |
| rabidim2 45288 | Membership in a restricted... |
| eluni2f 45289 | Membership in class union.... |
| eliin2f 45290 | Membership in indexed inte... |
| nssd 45291 | Negation of subclass relat... |
| iineq12dv 45292 | Equality deduction for ind... |
| supxrcld 45293 | The supremum of an arbitra... |
| elrestd 45294 | A sufficient condition for... |
| eliuniincex 45295 | Counterexample to show tha... |
| eliincex 45296 | Counterexample to show tha... |
| eliinid 45297 | Membership in an indexed i... |
| abssf 45298 | Class abstraction in a sub... |
| supxrubd 45299 | A member of a set of exten... |
| ssrabf 45300 | Subclass of a restricted c... |
| ssrabdf 45301 | Subclass of a restricted c... |
| eliin2 45302 | Membership in indexed inte... |
| ssrab2f 45303 | Subclass relation for a re... |
| restuni3 45304 | The underlying set of a su... |
| rabssf 45305 | Restricted class abstracti... |
| eliuniin2 45306 | Indexed union of indexed i... |
| restuni4 45307 | The underlying set of a su... |
| restuni6 45308 | The underlying set of a su... |
| restuni5 45309 | The underlying set of a su... |
| unirestss 45310 | The union of an elementwis... |
| iniin1 45311 | Indexed intersection of in... |
| iniin2 45312 | Indexed intersection of in... |
| cbvrabv2 45313 | A more general version of ... |
| cbvrabv2w 45314 | A more general version of ... |
| iinssiin 45315 | Subset implication for an ... |
| eliind2 45316 | Membership in indexed inte... |
| iinssd 45317 | Subset implication for an ... |
| rabbida2 45318 | Equivalent wff's yield equ... |
| iinexd 45319 | The existence of an indexe... |
| rabexf 45320 | Separation Scheme in terms... |
| rabbida3 45321 | Equivalent wff's yield equ... |
| r19.36vf 45322 | Restricted quantifier vers... |
| raleqd 45323 | Equality deduction for res... |
| iinssf 45324 | Subset implication for an ... |
| iinssdf 45325 | Subset implication for an ... |
| resabs2i 45326 | Absorption law for restric... |
| ssdf2 45327 | A sufficient condition for... |
| rabssd 45328 | Restricted class abstracti... |
| rexnegd 45329 | Minus a real number. (Con... |
| rexlimd3 45330 | * Inference from Theorem 1... |
| nel1nelini 45331 | Membership in an intersect... |
| nel2nelini 45332 | Membership in an intersect... |
| eliunid 45333 | Membership in indexed unio... |
| reximdd 45334 | Deduction from Theorem 19.... |
| inopnd 45335 | The intersection of two op... |
| ss2rabdf 45336 | Deduction of restricted ab... |
| restopn3 45337 | If ` A ` is open, then ` A... |
| restopnssd 45338 | A topology restricted to a... |
| restsubel 45339 | A subset belongs in the sp... |
| toprestsubel 45340 | A subset is open in the to... |
| rabidd 45341 | An "identity" law of concr... |
| iunssdf 45342 | Subset theorem for an inde... |
| iinss2d 45343 | Subset implication for an ... |
| r19.3rzf 45344 | Restricted quantification ... |
| r19.28zf 45345 | Restricted quantifier vers... |
| iindif2f 45346 | Indexed intersection of cl... |
| ralfal 45347 | Two ways of expressing emp... |
| archd 45348 | Archimedean property of re... |
| nimnbi 45349 | If an implication is false... |
| nimnbi2 45350 | If an implication is false... |
| notbicom 45351 | Commutative law for the ne... |
| rexeqif 45352 | Equality inference for res... |
| rspced 45353 | Restricted existential spe... |
| fnresdmss 45354 | A function does not change... |
| fmptsnxp 45355 | Maps-to notation and Carte... |
| fvmpt2bd 45356 | Value of a function given ... |
| rnmptfi 45357 | The range of a function wi... |
| fresin2 45358 | Restriction of a function ... |
| ffi 45359 | A function with finite dom... |
| suprnmpt 45360 | An explicit bound for the ... |
| rnffi 45361 | The range of a function wi... |
| mptelpm 45362 | A function in maps-to nota... |
| rnmptpr 45363 | Range of a function define... |
| resmpti 45364 | Restriction of the mapping... |
| founiiun 45365 | Union expressed as an inde... |
| rnresun 45366 | Distribution law for range... |
| elrnmptf 45367 | The range of a function in... |
| rnmptssrn 45368 | Inclusion relation for two... |
| disjf1 45369 | A 1 to 1 mapping built fro... |
| rnsnf 45370 | The range of a function wh... |
| wessf1ornlem 45371 | Given a function ` F ` on ... |
| wessf1orn 45372 | Given a function ` F ` on ... |
| nelrnres 45373 | If ` A ` is not in the ran... |
| disjrnmpt2 45374 | Disjointness of the range ... |
| elrnmpt1sf 45375 | Elementhood in an image se... |
| founiiun0 45376 | Union expressed as an inde... |
| disjf1o 45377 | A bijection built from dis... |
| disjinfi 45378 | Only a finite number of di... |
| fvovco 45379 | Value of the composition o... |
| ssnnf1octb 45380 | There exists a bijection b... |
| nnf1oxpnn 45381 | There is a bijection betwe... |
| rnmptssd 45382 | The range of a function gi... |
| projf1o 45383 | A biijection from a set to... |
| fvmap 45384 | Function value for a membe... |
| fvixp2 45385 | Projection of a factor of ... |
| choicefi 45386 | For a finite set, a choice... |
| mpct 45387 | The exponentiation of a co... |
| cnmetcoval 45388 | Value of the distance func... |
| fcomptss 45389 | Express composition of two... |
| elmapsnd 45390 | Membership in a set expone... |
| mapss2 45391 | Subset inheritance for set... |
| fsneq 45392 | Equality condition for two... |
| difmap 45393 | Difference of two sets exp... |
| unirnmap 45394 | Given a subset of a set ex... |
| inmap 45395 | Intersection of two sets e... |
| fcoss 45396 | Composition of two mapping... |
| fsneqrn 45397 | Equality condition for two... |
| difmapsn 45398 | Difference of two sets exp... |
| mapssbi 45399 | Subset inheritance for set... |
| unirnmapsn 45400 | Equality theorem for a sub... |
| iunmapss 45401 | The indexed union of set e... |
| ssmapsn 45402 | A subset ` C ` of a set ex... |
| iunmapsn 45403 | The indexed union of set e... |
| absfico 45404 | Mapping domain and codomai... |
| icof 45405 | The set of left-closed rig... |
| elpmrn 45406 | The range of a partial fun... |
| imaexi 45407 | The image of a set is a se... |
| axccdom 45408 | Relax the constraint on ax... |
| dmmptdff 45409 | The domain of the mapping ... |
| dmmptdf 45410 | The domain of the mapping ... |
| elpmi2 45411 | The domain of a partial fu... |
| dmrelrnrel 45412 | A relation preserving func... |
| fvcod 45413 | Value of a function compos... |
| elrnmpoid 45414 | Membership in the range of... |
| axccd 45415 | An alternative version of ... |
| axccd2 45416 | An alternative version of ... |
| feqresmptf 45417 | Express a restricted funct... |
| dmmptssf 45418 | The domain of a mapping is... |
| dmmptdf2 45419 | The domain of the mapping ... |
| dmuz 45420 | Domain of the upper intege... |
| fmptd2f 45421 | Domain and codomain of the... |
| mpteq1df 45422 | An equality theorem for th... |
| mptexf 45423 | If the domain of a functio... |
| fvmpt4 45424 | Value of a function given ... |
| fmptf 45425 | Functionality of the mappi... |
| resimass 45426 | The image of a restriction... |
| mptssid 45427 | The mapping operation expr... |
| mptfnd 45428 | The maps-to notation defin... |
| rnmptlb 45429 | Boundness below of the ran... |
| rnmptbddlem 45430 | Boundness of the range of ... |
| rnmptbdd 45431 | Boundness of the range of ... |
| funimaeq 45432 | Membership relation for th... |
| rnmptssf 45433 | The range of a function gi... |
| rnmptbd2lem 45434 | Boundness below of the ran... |
| rnmptbd2 45435 | Boundness below of the ran... |
| infnsuprnmpt 45436 | The indexed infimum of rea... |
| suprclrnmpt 45437 | Closure of the indexed sup... |
| suprubrnmpt2 45438 | A member of a nonempty ind... |
| suprubrnmpt 45439 | A member of a nonempty ind... |
| rnmptssdf 45440 | The range of a function gi... |
| rnmptbdlem 45441 | Boundness above of the ran... |
| rnmptbd 45442 | Boundness above of the ran... |
| rnmptss2 45443 | The range of a function gi... |
| elmptima 45444 | The image of a function in... |
| ralrnmpt3 45445 | A restricted quantifier ov... |
| rnmptssbi 45446 | The range of a function gi... |
| imass2d 45447 | Subset theorem for image. ... |
| imassmpt 45448 | Membership relation for th... |
| fpmd 45449 | A total function is a part... |
| fconst7 45450 | An alternative way to expr... |
| fnmptif 45451 | Functionality and domain o... |
| dmmptif 45452 | Domain of the mapping oper... |
| mpteq2dfa 45453 | Slightly more general equa... |
| dmmpt1 45454 | The domain of the mapping ... |
| fmptff 45455 | Functionality of the mappi... |
| fvmptelcdmf 45456 | The value of a function at... |
| fmptdff 45457 | A version of ~ fmptd using... |
| fvmpt2df 45458 | Deduction version of ~ fvm... |
| rn1st 45459 | The range of a function wi... |
| rnmptssff 45460 | The range of a function gi... |
| rnmptssdff 45461 | The range of a function gi... |
| fvmpt4d 45462 | Value of a function given ... |
| sub2times 45463 | Subtracting from a number,... |
| nnxrd 45464 | A natural number is an ext... |
| nnxr 45465 | A natural number is an ext... |
| abssubrp 45466 | The distance of two distin... |
| elfzfzo 45467 | Relationship between membe... |
| oddfl 45468 | Odd number representation ... |
| abscosbd 45469 | Bound for the absolute val... |
| mul13d 45470 | Commutative/associative la... |
| negpilt0 45471 | Negative ` _pi ` is negati... |
| dstregt0 45472 | A complex number ` A ` tha... |
| subadd4b 45473 | Rearrangement of 4 terms i... |
| xrlttri5d 45474 | Not equal and not larger i... |
| zltlesub 45475 | If an integer ` N ` is les... |
| divlt0gt0d 45476 | The ratio of a negative nu... |
| subsub23d 45477 | Swap subtrahend and result... |
| 2timesgt 45478 | Double of a positive real ... |
| reopn 45479 | The reals are open with re... |
| sub31 45480 | Swap the first and third t... |
| nnne1ge2 45481 | A positive integer which i... |
| lefldiveq 45482 | A closed enough, smaller r... |
| negsubdi3d 45483 | Distribution of negative o... |
| ltdiv2dd 45484 | Division of a positive num... |
| abssinbd 45485 | Bound for the absolute val... |
| halffl 45486 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 45487 | Ordering relation for a st... |
| hashssle 45488 | The size of a subset of a ... |
| lttri5d 45489 | Not equal and not larger i... |
| fzisoeu 45490 | A finite ordered set has a... |
| lt3addmuld 45491 | If three real numbers are ... |
| absnpncan2d 45492 | Triangular inequality, com... |
| fperiodmullem 45493 | A function with period ` T... |
| fperiodmul 45494 | A function with period T i... |
| upbdrech 45495 | Choice of an upper bound f... |
| lt4addmuld 45496 | If four real numbers are l... |
| absnpncan3d 45497 | Triangular inequality, com... |
| upbdrech2 45498 | Choice of an upper bound f... |
| ssfiunibd 45499 | A finite union of bounded ... |
| fzdifsuc2 45500 | Remove a successor from th... |
| fzsscn 45501 | A finite sequence of integ... |
| divcan8d 45502 | A cancellation law for div... |
| dmmcand 45503 | Cancellation law for divis... |
| fzssre 45504 | A finite sequence of integ... |
| bccld 45505 | A binomial coefficient, in... |
| fzssnn0 45506 | A finite set of sequential... |
| xreqle 45507 | Equality implies 'less tha... |
| xaddlidd 45508 | ` 0 ` is a left identity f... |
| xadd0ge 45509 | A number is less than or e... |
| xrleneltd 45510 | 'Less than or equal to' an... |
| xaddcomd 45511 | The extended real addition... |
| supxrre3 45512 | The supremum of a nonempty... |
| uzfissfz 45513 | For any finite subset of t... |
| xleadd2d 45514 | Addition of extended reals... |
| suprltrp 45515 | The supremum of a nonempty... |
| xleadd1d 45516 | Addition of extended reals... |
| xreqled 45517 | Equality implies 'less tha... |
| xrgepnfd 45518 | An extended real greater t... |
| xrge0nemnfd 45519 | A nonnegative extended rea... |
| supxrgere 45520 | If a real number can be ap... |
| iuneqfzuzlem 45521 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 45522 | If two unions indexed by u... |
| xle2addd 45523 | Adding both side of two in... |
| supxrgelem 45524 | If an extended real number... |
| supxrge 45525 | If an extended real number... |
| suplesup 45526 | If any element of ` A ` ca... |
| infxrglb 45527 | The infimum of a set of ex... |
| xadd0ge2 45528 | A number is less than or e... |
| nepnfltpnf 45529 | An extended real that is n... |
| ltadd12dd 45530 | Addition to both sides of ... |
| nemnftgtmnft 45531 | An extended real that is n... |
| xrgtso 45532 | 'Greater than' is a strict... |
| rpex 45533 | The positive reals form a ... |
| xrge0ge0 45534 | A nonnegative extended rea... |
| xrssre 45535 | A subset of extended reals... |
| ssuzfz 45536 | A finite subset of the upp... |
| absfun 45537 | The absolute value is a fu... |
| infrpge 45538 | The infimum of a nonempty,... |
| xrlexaddrp 45539 | If an extended real number... |
| supsubc 45540 | The supremum function dist... |
| xralrple2 45541 | Show that ` A ` is less th... |
| nnuzdisj 45542 | The first ` N ` elements o... |
| ltdivgt1 45543 | Divsion by a number greate... |
| xrltned 45544 | 'Less than' implies not eq... |
| nnsplit 45545 | Express the set of positiv... |
| divdiv3d 45546 | Division into a fraction. ... |
| abslt2sqd 45547 | Comparison of the square o... |
| qenom 45548 | The set of rational number... |
| qct 45549 | The set of rational number... |
| lenlteq 45550 | 'less than or equal to' bu... |
| xrred 45551 | An extended real that is n... |
| rr2sscn2 45552 | The cartesian square of ` ... |
| infxr 45553 | The infimum of a set of ex... |
| infxrunb2 45554 | The infimum of an unbounde... |
| infxrbnd2 45555 | The infimum of a bounded-b... |
| infleinflem1 45556 | Lemma for ~ infleinf , cas... |
| infleinflem2 45557 | Lemma for ~ infleinf , whe... |
| infleinf 45558 | If any element of ` B ` ca... |
| xralrple4 45559 | Show that ` A ` is less th... |
| xralrple3 45560 | Show that ` A ` is less th... |
| eluzelzd 45561 | A member of an upper set o... |
| suplesup2 45562 | If any element of ` A ` is... |
| recnnltrp 45563 | ` N ` is a natural number ... |
| nnn0 45564 | The set of positive intege... |
| fzct 45565 | A finite set of sequential... |
| rpgtrecnn 45566 | Any positive real number i... |
| fzossuz 45567 | A half-open integer interv... |
| infxrrefi 45568 | The real and extended real... |
| xrralrecnnle 45569 | Show that ` A ` is less th... |
| fzoct 45570 | A finite set of sequential... |
| frexr 45571 | A function taking real val... |
| nnrecrp 45572 | The reciprocal of a positi... |
| reclt0d 45573 | The reciprocal of a negati... |
| lt0neg1dd 45574 | If a number is negative, i... |
| infxrcld 45575 | The infimum of an arbitrar... |
| xrralrecnnge 45576 | Show that ` A ` is less th... |
| reclt0 45577 | The reciprocal of a negati... |
| ltmulneg 45578 | Multiplying by a negative ... |
| allbutfi 45579 | For all but finitely many.... |
| ltdiv23neg 45580 | Swap denominator with othe... |
| xreqnltd 45581 | A consequence of trichotom... |
| mnfnre2 45582 | Minus infinity is not a re... |
| zssxr 45583 | The integers are a subset ... |
| fisupclrnmpt 45584 | A nonempty finite indexed ... |
| supxrunb3 45585 | The supremum of an unbound... |
| elfzod 45586 | Membership in a half-open ... |
| fimaxre4 45587 | A nonempty finite set of r... |
| ren0 45588 | The set of reals is nonemp... |
| eluzelz2 45589 | A member of an upper set o... |
| resabs2d 45590 | Absorption law for restric... |
| uzid2 45591 | Membership of the least me... |
| supxrleubrnmpt 45592 | The supremum of a nonempty... |
| uzssre2 45593 | An upper set of integers i... |
| uzssd 45594 | Subset relationship for tw... |
| eluzd 45595 | Membership in an upper set... |
| infxrlbrnmpt2 45596 | A member of a nonempty ind... |
| xrre4 45597 | An extended real is real i... |
| uz0 45598 | The upper integers functio... |
| eluzelz2d 45599 | A member of an upper set o... |
| infleinf2 45600 | If any element in ` B ` is... |
| unb2ltle 45601 | "Unbounded below" expresse... |
| uzidd2 45602 | Membership of the least me... |
| uzssd2 45603 | Subset relationship for tw... |
| rexabslelem 45604 | An indexed set of absolute... |
| rexabsle 45605 | An indexed set of absolute... |
| allbutfiinf 45606 | Given a "for all but finit... |
| supxrrernmpt 45607 | The real and extended real... |
| suprleubrnmpt 45608 | The supremum of a nonempty... |
| infrnmptle 45609 | An indexed infimum of exte... |
| infxrunb3 45610 | The infimum of an unbounde... |
| uzn0d 45611 | The upper integers are all... |
| uzssd3 45612 | Subset relationship for tw... |
| rexabsle2 45613 | An indexed set of absolute... |
| infxrunb3rnmpt 45614 | The infimum of an unbounde... |
| supxrre3rnmpt 45615 | The indexed supremum of a ... |
| uzublem 45616 | A set of reals, indexed by... |
| uzub 45617 | A set of reals, indexed by... |
| ssrexr 45618 | A subset of the reals is a... |
| supxrmnf2 45619 | Removing minus infinity fr... |
| supxrcli 45620 | The supremum of an arbitra... |
| uzid3 45621 | Membership of the least me... |
| infxrlesupxr 45622 | The supremum of a nonempty... |
| xnegeqd 45623 | Equality of two extended n... |
| xnegrecl 45624 | The extended real negative... |
| xnegnegi 45625 | Extended real version of ~... |
| xnegeqi 45626 | Equality of two extended n... |
| nfxnegd 45627 | Deduction version of ~ nfx... |
| xnegnegd 45628 | Extended real version of ~... |
| uzred 45629 | An upper integer is a real... |
| xnegcli 45630 | Closure of extended real n... |
| supminfrnmpt 45631 | The indexed supremum of a ... |
| infxrpnf 45632 | Adding plus infinity to a ... |
| infxrrnmptcl 45633 | The infimum of an arbitrar... |
| leneg2d 45634 | Negative of one side of 'l... |
| supxrltinfxr 45635 | The supremum of the empty ... |
| max1d 45636 | A number is less than or e... |
| supxrleubrnmptf 45637 | The supremum of a nonempty... |
| nleltd 45638 | 'Not less than or equal to... |
| zxrd 45639 | An integer is an extended ... |
| infxrgelbrnmpt 45640 | The infimum of an indexed ... |
| rphalfltd 45641 | Half of a positive real is... |
| uzssz2 45642 | An upper set of integers i... |
| leneg3d 45643 | Negative of one side of 'l... |
| max2d 45644 | A number is less than or e... |
| uzn0bi 45645 | The upper integers functio... |
| xnegrecl2 45646 | If the extended real negat... |
| nfxneg 45647 | Bound-variable hypothesis ... |
| uzxrd 45648 | An upper integer is an ext... |
| infxrpnf2 45649 | Removing plus infinity fro... |
| supminfxr 45650 | The extended real suprema ... |
| infrpgernmpt 45651 | The infimum of a nonempty,... |
| xnegre 45652 | An extended real is real i... |
| xnegrecl2d 45653 | If the extended real negat... |
| uzxr 45654 | An upper integer is an ext... |
| supminfxr2 45655 | The extended real suprema ... |
| xnegred 45656 | An extended real is real i... |
| supminfxrrnmpt 45657 | The indexed supremum of a ... |
| min1d 45658 | The minimum of two numbers... |
| min2d 45659 | The minimum of two numbers... |
| xrnpnfmnf 45660 | An extended real that is n... |
| uzsscn 45661 | An upper set of integers i... |
| absimnre 45662 | The absolute value of the ... |
| uzsscn2 45663 | An upper set of integers i... |
| xrtgcntopre 45664 | The standard topologies on... |
| absimlere 45665 | The absolute value of the ... |
| rpssxr 45666 | The positive reals are a s... |
| monoordxrv 45667 | Ordering relation for a mo... |
| monoordxr 45668 | Ordering relation for a mo... |
| monoord2xrv 45669 | Ordering relation for a mo... |
| monoord2xr 45670 | Ordering relation for a mo... |
| xrpnf 45671 | An extended real is plus i... |
| xlenegcon1 45672 | Extended real version of ~... |
| xlenegcon2 45673 | Extended real version of ~... |
| pimxrneun 45674 | The preimage of a set of e... |
| caucvgbf 45675 | A function is convergent i... |
| cvgcau 45676 | A convergent function is C... |
| cvgcaule 45677 | A convergent function is C... |
| rexanuz2nf 45678 | A simple counterexample re... |
| gtnelioc 45679 | A real number larger than ... |
| ioossioc 45680 | An open interval is a subs... |
| ioondisj2 45681 | A condition for two open i... |
| ioondisj1 45682 | A condition for two open i... |
| ioogtlb 45683 | An element of a closed int... |
| evthiccabs 45684 | Extreme Value Theorem on y... |
| ltnelicc 45685 | A real number smaller than... |
| eliood 45686 | Membership in an open real... |
| iooabslt 45687 | An upper bound for the dis... |
| gtnelicc 45688 | A real number greater than... |
| iooinlbub 45689 | An open interval has empty... |
| iocgtlb 45690 | An element of a left-open ... |
| iocleub 45691 | An element of a left-open ... |
| eliccd 45692 | Membership in a closed rea... |
| eliccre 45693 | A member of a closed inter... |
| eliooshift 45694 | Element of an open interva... |
| eliocd 45695 | Membership in a left-open ... |
| icoltub 45696 | An element of a left-close... |
| eliocre 45697 | A member of a left-open ri... |
| iooltub 45698 | An element of an open inte... |
| ioontr 45699 | The interior of an interva... |
| snunioo1 45700 | The closure of one end of ... |
| lbioc 45701 | A left-open right-closed i... |
| ioomidp 45702 | The midpoint is an element... |
| iccdifioo 45703 | If the open inverval is re... |
| iccdifprioo 45704 | An open interval is the cl... |
| ioossioobi 45705 | Biconditional form of ~ io... |
| iccshift 45706 | A closed interval shifted ... |
| iccsuble 45707 | An upper bound to the dist... |
| iocopn 45708 | A left-open right-closed i... |
| eliccelioc 45709 | Membership in a closed int... |
| iooshift 45710 | An open interval shifted b... |
| iccintsng 45711 | Intersection of two adiace... |
| icoiccdif 45712 | Left-closed right-open int... |
| icoopn 45713 | A left-closed right-open i... |
| icoub 45714 | A left-closed, right-open ... |
| eliccxrd 45715 | Membership in a closed rea... |
| pnfel0pnf 45716 | ` +oo ` is a nonnegative e... |
| eliccnelico 45717 | An element of a closed int... |
| eliccelicod 45718 | A member of a closed inter... |
| ge0xrre 45719 | A nonnegative extended rea... |
| ge0lere 45720 | A nonnegative extended Rea... |
| elicores 45721 | Membership in a left-close... |
| inficc 45722 | The infimum of a nonempty ... |
| qinioo 45723 | The rational numbers are d... |
| lenelioc 45724 | A real number smaller than... |
| ioonct 45725 | A nonempty open interval i... |
| xrgtnelicc 45726 | A real number greater than... |
| iccdificc 45727 | The difference of two clos... |
| iocnct 45728 | A nonempty left-open, righ... |
| iccnct 45729 | A closed interval, with mo... |
| iooiinicc 45730 | A closed interval expresse... |
| iccgelbd 45731 | An element of a closed int... |
| iooltubd 45732 | An element of an open inte... |
| icoltubd 45733 | An element of a left-close... |
| qelioo 45734 | The rational numbers are d... |
| tgqioo2 45735 | Every open set of reals is... |
| iccleubd 45736 | An element of a closed int... |
| elioored 45737 | A member of an open interv... |
| ioogtlbd 45738 | An element of a closed int... |
| ioofun 45739 | ` (,) ` is a function. (C... |
| icomnfinre 45740 | A left-closed, right-open,... |
| sqrlearg 45741 | The square compared with i... |
| ressiocsup 45742 | If the supremum belongs to... |
| ressioosup 45743 | If the supremum does not b... |
| iooiinioc 45744 | A left-open, right-closed ... |
| ressiooinf 45745 | If the infimum does not be... |
| iocleubd 45746 | An element of a left-open ... |
| uzinico 45747 | An upper interval of integ... |
| preimaiocmnf 45748 | Preimage of a right-closed... |
| uzinico2 45749 | An upper interval of integ... |
| uzinico3 45750 | An upper interval of integ... |
| dmico 45751 | The domain of the closed-b... |
| ndmico 45752 | The closed-below, open-abo... |
| uzubioo 45753 | The upper integers are unb... |
| uzubico 45754 | The upper integers are unb... |
| uzubioo2 45755 | The upper integers are unb... |
| uzubico2 45756 | The upper integers are unb... |
| iocgtlbd 45757 | An element of a left-open ... |
| xrtgioo2 45758 | The topology on the extend... |
| fsummulc1f 45759 | Closure of a finite sum of... |
| fsumnncl 45760 | Closure of a nonempty, fin... |
| fsumge0cl 45761 | The finite sum of nonnegat... |
| fsumf1of 45762 | Re-index a finite sum usin... |
| fsumiunss 45763 | Sum over a disjoint indexe... |
| fsumreclf 45764 | Closure of a finite sum of... |
| fsumlessf 45765 | A shorter sum of nonnegati... |
| fsumsupp0 45766 | Finite sum of function val... |
| fsumsermpt 45767 | A finite sum expressed in ... |
| fmul01 45768 | Multiplying a finite numbe... |
| fmulcl 45769 | If ' Y ' is closed under t... |
| fmuldfeqlem1 45770 | induction step for the pro... |
| fmuldfeq 45771 | X and Z are two equivalent... |
| fmul01lt1lem1 45772 | Given a finite multiplicat... |
| fmul01lt1lem2 45773 | Given a finite multiplicat... |
| fmul01lt1 45774 | Given a finite multiplicat... |
| cncfmptss 45775 | A continuous complex funct... |
| rrpsscn 45776 | The positive reals are a s... |
| mulc1cncfg 45777 | A version of ~ mulc1cncf u... |
| infrglb 45778 | The infimum of a nonempty ... |
| expcnfg 45779 | If ` F ` is a complex cont... |
| prodeq2ad 45780 | Equality deduction for pro... |
| fprodsplit1 45781 | Separate out a term in a f... |
| fprodexp 45782 | Positive integer exponenti... |
| fprodabs2 45783 | The absolute value of a fi... |
| fprod0 45784 | A finite product with a ze... |
| mccllem 45785 | * Induction step for ~ mcc... |
| mccl 45786 | A multinomial coefficient,... |
| fprodcnlem 45787 | A finite product of functi... |
| fprodcn 45788 | A finite product of functi... |
| clim1fr1 45789 | A class of sequences of fr... |
| isumneg 45790 | Negation of a converging s... |
| climrec 45791 | Limit of the reciprocal of... |
| climmulf 45792 | A version of ~ climmul usi... |
| climexp 45793 | The limit of natural power... |
| climinf 45794 | A bounded monotonic noninc... |
| climsuselem1 45795 | The subsequence index ` I ... |
| climsuse 45796 | A subsequence ` G ` of a c... |
| climrecf 45797 | A version of ~ climrec usi... |
| climneg 45798 | Complex limit of the negat... |
| climinff 45799 | A version of ~ climinf usi... |
| climdivf 45800 | Limit of the ratio of two ... |
| climreeq 45801 | If ` F ` is a real functio... |
| ellimciota 45802 | An explicit value for the ... |
| climaddf 45803 | A version of ~ climadd usi... |
| mullimc 45804 | Limit of the product of tw... |
| ellimcabssub0 45805 | An equivalent condition fo... |
| limcdm0 45806 | If a function has empty do... |
| islptre 45807 | An equivalence condition f... |
| limccog 45808 | Limit of the composition o... |
| limciccioolb 45809 | The limit of a function at... |
| climf 45810 | Express the predicate: Th... |
| mullimcf 45811 | Limit of the multiplicatio... |
| constlimc 45812 | Limit of constant function... |
| rexlim2d 45813 | Inference removing two res... |
| idlimc 45814 | Limit of the identity func... |
| divcnvg 45815 | The sequence of reciprocal... |
| limcperiod 45816 | If ` F ` is a periodic fun... |
| limcrecl 45817 | If ` F ` is a real-valued ... |
| sumnnodd 45818 | A series indexed by ` NN `... |
| lptioo2 45819 | The upper bound of an open... |
| lptioo1 45820 | The lower bound of an open... |
| limcmptdm 45821 | The domain of a maps-to fu... |
| clim2f 45822 | Express the predicate: Th... |
| limcicciooub 45823 | The limit of a function at... |
| ltmod 45824 | A sufficient condition for... |
| islpcn 45825 | A characterization for a l... |
| lptre2pt 45826 | If a set in the real line ... |
| limsupre 45827 | If a sequence is bounded, ... |
| limcresiooub 45828 | The left limit doesn't cha... |
| limcresioolb 45829 | The right limit doesn't ch... |
| limcleqr 45830 | If the left and the right ... |
| lptioo2cn 45831 | The upper bound of an open... |
| lptioo1cn 45832 | The lower bound of an open... |
| neglimc 45833 | Limit of the negative func... |
| addlimc 45834 | Sum of two limits. (Contr... |
| 0ellimcdiv 45835 | If the numerator converges... |
| clim2cf 45836 | Express the predicate ` F ... |
| limclner 45837 | For a limit point, both fr... |
| sublimc 45838 | Subtraction of two limits.... |
| reclimc 45839 | Limit of the reciprocal of... |
| clim0cf 45840 | Express the predicate ` F ... |
| limclr 45841 | For a limit point, both fr... |
| divlimc 45842 | Limit of the quotient of t... |
| expfac 45843 | Factorial grows faster tha... |
| climconstmpt 45844 | A constant sequence conver... |
| climresmpt 45845 | A function restricted to u... |
| climsubmpt 45846 | Limit of the difference of... |
| climsubc2mpt 45847 | Limit of the difference of... |
| climsubc1mpt 45848 | Limit of the difference of... |
| fnlimfv 45849 | The value of the limit fun... |
| climreclf 45850 | The limit of a convergent ... |
| climeldmeq 45851 | Two functions that are eve... |
| climf2 45852 | Express the predicate: Th... |
| fnlimcnv 45853 | The sequence of function v... |
| climeldmeqmpt 45854 | Two functions that are eve... |
| climfveq 45855 | Two functions that are eve... |
| clim2f2 45856 | Express the predicate: Th... |
| climfveqmpt 45857 | Two functions that are eve... |
| climd 45858 | Express the predicate: Th... |
| clim2d 45859 | The limit of complex numbe... |
| fnlimfvre 45860 | The limit function of real... |
| allbutfifvre 45861 | Given a sequence of real-v... |
| climleltrp 45862 | The limit of complex numbe... |
| fnlimfvre2 45863 | The limit function of real... |
| fnlimf 45864 | The limit function of real... |
| fnlimabslt 45865 | A sequence of function val... |
| climfveqf 45866 | Two functions that are eve... |
| climmptf 45867 | Exhibit a function ` G ` w... |
| climfveqmpt3 45868 | Two functions that are eve... |
| climeldmeqf 45869 | Two functions that are eve... |
| climreclmpt 45870 | The limit of B convergent ... |
| limsupref 45871 | If a sequence is bounded, ... |
| limsupbnd1f 45872 | If a sequence is eventuall... |
| climbddf 45873 | A converging sequence of c... |
| climeqf 45874 | Two functions that are eve... |
| climeldmeqmpt3 45875 | Two functions that are eve... |
| limsupcld 45876 | Closure of the superior li... |
| climfv 45877 | The limit of a convergent ... |
| limsupval3 45878 | The superior limit of an i... |
| climfveqmpt2 45879 | Two functions that are eve... |
| limsup0 45880 | The superior limit of the ... |
| climeldmeqmpt2 45881 | Two functions that are eve... |
| limsupresre 45882 | The supremum limit of a fu... |
| climeqmpt 45883 | Two functions that are eve... |
| climfvd 45884 | The limit of a convergent ... |
| limsuplesup 45885 | An upper bound for the sup... |
| limsupresico 45886 | The superior limit doesn't... |
| limsuppnfdlem 45887 | If the restriction of a fu... |
| limsuppnfd 45888 | If the restriction of a fu... |
| limsupresuz 45889 | If the real part of the do... |
| limsupub 45890 | If the limsup is not ` +oo... |
| limsupres 45891 | The superior limit of a re... |
| climinf2lem 45892 | A convergent, nonincreasin... |
| climinf2 45893 | A convergent, nonincreasin... |
| limsupvaluz 45894 | The superior limit, when t... |
| limsupresuz2 45895 | If the domain of a functio... |
| limsuppnflem 45896 | If the restriction of a fu... |
| limsuppnf 45897 | If the restriction of a fu... |
| limsupubuzlem 45898 | If the limsup is not ` +oo... |
| limsupubuz 45899 | For a real-valued function... |
| climinf2mpt 45900 | A bounded below, monotonic... |
| climinfmpt 45901 | A bounded below, monotonic... |
| climinf3 45902 | A convergent, nonincreasin... |
| limsupvaluzmpt 45903 | The superior limit, when t... |
| limsupequzmpt2 45904 | Two functions that are eve... |
| limsupubuzmpt 45905 | If the limsup is not ` +oo... |
| limsupmnflem 45906 | The superior limit of a fu... |
| limsupmnf 45907 | The superior limit of a fu... |
| limsupequzlem 45908 | Two functions that are eve... |
| limsupequz 45909 | Two functions that are eve... |
| limsupre2lem 45910 | Given a function on the ex... |
| limsupre2 45911 | Given a function on the ex... |
| limsupmnfuzlem 45912 | The superior limit of a fu... |
| limsupmnfuz 45913 | The superior limit of a fu... |
| limsupequzmptlem 45914 | Two functions that are eve... |
| limsupequzmpt 45915 | Two functions that are eve... |
| limsupre2mpt 45916 | Given a function on the ex... |
| limsupequzmptf 45917 | Two functions that are eve... |
| limsupre3lem 45918 | Given a function on the ex... |
| limsupre3 45919 | Given a function on the ex... |
| limsupre3mpt 45920 | Given a function on the ex... |
| limsupre3uzlem 45921 | Given a function on the ex... |
| limsupre3uz 45922 | Given a function on the ex... |
| limsupreuz 45923 | Given a function on the re... |
| limsupvaluz2 45924 | The superior limit, when t... |
| limsupreuzmpt 45925 | Given a function on the re... |
| supcnvlimsup 45926 | If a function on a set of ... |
| supcnvlimsupmpt 45927 | If a function on a set of ... |
| 0cnv 45928 | If ` (/) ` is a complex nu... |
| climuzlem 45929 | Express the predicate: Th... |
| climuz 45930 | Express the predicate: Th... |
| lmbr3v 45931 | Express the binary relatio... |
| climisp 45932 | If a sequence converges to... |
| lmbr3 45933 | Express the binary relatio... |
| climrescn 45934 | A sequence converging w.r.... |
| climxrrelem 45935 | If a sequence ranging over... |
| climxrre 45936 | If a sequence ranging over... |
| limsuplt2 45939 | The defining property of t... |
| liminfgord 45940 | Ordering property of the i... |
| limsupvald 45941 | The superior limit of a se... |
| limsupresicompt 45942 | The superior limit doesn't... |
| limsupcli 45943 | Closure of the superior li... |
| liminfgf 45944 | Closure of the inferior li... |
| liminfval 45945 | The inferior limit of a se... |
| climlimsup 45946 | A sequence of real numbers... |
| limsupge 45947 | The defining property of t... |
| liminfgval 45948 | Value of the inferior limi... |
| liminfcl 45949 | Closure of the inferior li... |
| liminfvald 45950 | The inferior limit of a se... |
| liminfval5 45951 | The inferior limit of an i... |
| limsupresxr 45952 | The superior limit of a fu... |
| liminfresxr 45953 | The inferior limit of a fu... |
| liminfval2 45954 | The superior limit, relati... |
| climlimsupcex 45955 | Counterexample for ~ climl... |
| liminfcld 45956 | Closure of the inferior li... |
| liminfresico 45957 | The inferior limit doesn't... |
| limsup10exlem 45958 | The range of the given fun... |
| limsup10ex 45959 | The superior limit of a fu... |
| liminf10ex 45960 | The inferior limit of a fu... |
| liminflelimsuplem 45961 | The superior limit is grea... |
| liminflelimsup 45962 | The superior limit is grea... |
| limsupgtlem 45963 | For any positive real, the... |
| limsupgt 45964 | Given a sequence of real n... |
| liminfresre 45965 | The inferior limit of a fu... |
| liminfresicompt 45966 | The inferior limit doesn't... |
| liminfltlimsupex 45967 | An example where the ` lim... |
| liminfgelimsup 45968 | The inferior limit is grea... |
| liminfvalxr 45969 | Alternate definition of ` ... |
| liminfresuz 45970 | If the real part of the do... |
| liminflelimsupuz 45971 | The superior limit is grea... |
| liminfvalxrmpt 45972 | Alternate definition of ` ... |
| liminfresuz2 45973 | If the domain of a functio... |
| liminfgelimsupuz 45974 | The inferior limit is grea... |
| liminfval4 45975 | Alternate definition of ` ... |
| liminfval3 45976 | Alternate definition of ` ... |
| liminfequzmpt2 45977 | Two functions that are eve... |
| liminfvaluz 45978 | Alternate definition of ` ... |
| liminf0 45979 | The inferior limit of the ... |
| limsupval4 45980 | Alternate definition of ` ... |
| liminfvaluz2 45981 | Alternate definition of ` ... |
| liminfvaluz3 45982 | Alternate definition of ` ... |
| liminflelimsupcex 45983 | A counterexample for ~ lim... |
| limsupvaluz3 45984 | Alternate definition of ` ... |
| liminfvaluz4 45985 | Alternate definition of ` ... |
| limsupvaluz4 45986 | Alternate definition of ` ... |
| climliminflimsupd 45987 | If a sequence of real numb... |
| liminfreuzlem 45988 | Given a function on the re... |
| liminfreuz 45989 | Given a function on the re... |
| liminfltlem 45990 | Given a sequence of real n... |
| liminflt 45991 | Given a sequence of real n... |
| climliminf 45992 | A sequence of real numbers... |
| liminflimsupclim 45993 | A sequence of real numbers... |
| climliminflimsup 45994 | A sequence of real numbers... |
| climliminflimsup2 45995 | A sequence of real numbers... |
| climliminflimsup3 45996 | A sequence of real numbers... |
| climliminflimsup4 45997 | A sequence of real numbers... |
| limsupub2 45998 | A extended real valued fun... |
| limsupubuz2 45999 | A sequence with values in ... |
| xlimpnfxnegmnf 46000 | A sequence converges to ` ... |
| liminflbuz2 46001 | A sequence with values in ... |
| liminfpnfuz 46002 | The inferior limit of a fu... |
| liminflimsupxrre 46003 | A sequence with values in ... |
| xlimrel 46006 | The limit on extended real... |
| xlimres 46007 | A function converges iff i... |
| xlimcl 46008 | The limit of a sequence of... |
| rexlimddv2 46009 | Restricted existential eli... |
| xlimclim 46010 | Given a sequence of reals,... |
| xlimconst 46011 | A constant sequence conver... |
| climxlim 46012 | A converging sequence in t... |
| xlimbr 46013 | Express the binary relatio... |
| fuzxrpmcn 46014 | A function mapping from an... |
| cnrefiisplem 46015 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 46016 | A non-real, complex number... |
| xlimxrre 46017 | If a sequence ranging over... |
| xlimmnfvlem1 46018 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 46019 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 46020 | A function converges to mi... |
| xlimconst2 46021 | A sequence that eventually... |
| xlimpnfvlem1 46022 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 46023 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 46024 | A function converges to pl... |
| xlimclim2lem 46025 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 46026 | Given a sequence of extend... |
| xlimmnf 46027 | A function converges to mi... |
| xlimpnf 46028 | A function converges to pl... |
| xlimmnfmpt 46029 | A function converges to pl... |
| xlimpnfmpt 46030 | A function converges to pl... |
| climxlim2lem 46031 | In this lemma for ~ climxl... |
| climxlim2 46032 | A sequence of extended rea... |
| dfxlim2v 46033 | An alternative definition ... |
| dfxlim2 46034 | An alternative definition ... |
| climresd 46035 | A function restricted to u... |
| climresdm 46036 | A real function converges ... |
| dmclimxlim 46037 | A real valued sequence tha... |
| xlimmnflimsup2 46038 | A sequence of extended rea... |
| xlimuni 46039 | An infinite sequence conve... |
| xlimclimdm 46040 | A sequence of extended rea... |
| xlimfun 46041 | The convergence relation o... |
| xlimmnflimsup 46042 | If a sequence of extended ... |
| xlimdm 46043 | Two ways to express that a... |
| xlimpnfxnegmnf2 46044 | A sequence converges to ` ... |
| xlimresdm 46045 | A function converges in th... |
| xlimpnfliminf 46046 | If a sequence of extended ... |
| xlimpnfliminf2 46047 | A sequence of extended rea... |
| xlimliminflimsup 46048 | A sequence of extended rea... |
| xlimlimsupleliminf 46049 | A sequence of extended rea... |
| coseq0 46050 | A complex number whose cos... |
| sinmulcos 46051 | Multiplication formula for... |
| coskpi2 46052 | The cosine of an integer m... |
| cosnegpi 46053 | The cosine of negative ` _... |
| sinaover2ne0 46054 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 46055 | The cosine of an integer m... |
| mulcncff 46056 | The multiplication of two ... |
| cncfmptssg 46057 | A continuous complex funct... |
| constcncfg 46058 | A constant function is a c... |
| idcncfg 46059 | The identity function is a... |
| cncfshift 46060 | A periodic continuous func... |
| resincncf 46061 | ` sin ` restricted to real... |
| addccncf2 46062 | Adding a constant is a con... |
| 0cnf 46063 | The empty set is a continu... |
| fsumcncf 46064 | The finite sum of continuo... |
| cncfperiod 46065 | A periodic continuous func... |
| subcncff 46066 | The subtraction of two con... |
| negcncfg 46067 | The opposite of a continuo... |
| cnfdmsn 46068 | A function with a singleto... |
| cncfcompt 46069 | Composition of continuous ... |
| addcncff 46070 | The sum of two continuous ... |
| ioccncflimc 46071 | Limit at the upper bound o... |
| cncfuni 46072 | A complex function on a su... |
| icccncfext 46073 | A continuous function on a... |
| cncficcgt0 46074 | A the absolute value of a ... |
| icocncflimc 46075 | Limit at the lower bound, ... |
| cncfdmsn 46076 | A complex function with a ... |
| divcncff 46077 | The quotient of two contin... |
| cncfshiftioo 46078 | A periodic continuous func... |
| cncfiooicclem1 46079 | A continuous function ` F ... |
| cncfiooicc 46080 | A continuous function ` F ... |
| cncfiooiccre 46081 | A continuous function ` F ... |
| cncfioobdlem 46082 | ` G ` actually extends ` F... |
| cncfioobd 46083 | A continuous function ` F ... |
| jumpncnp 46084 | Jump discontinuity or disc... |
| cxpcncf2 46085 | The complex power function... |
| fprodcncf 46086 | The finite product of cont... |
| add1cncf 46087 | Addition to a constant is ... |
| add2cncf 46088 | Addition to a constant is ... |
| sub1cncfd 46089 | Subtracting a constant is ... |
| sub2cncfd 46090 | Subtraction from a constan... |
| fprodsub2cncf 46091 | ` F ` is continuous. (Con... |
| fprodadd2cncf 46092 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 46093 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 46094 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 46095 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 46096 | The sequence ` S ` of fini... |
| dvsinexp 46097 | The derivative of sin^N . ... |
| dvcosre 46098 | The real derivative of the... |
| dvsinax 46099 | Derivative exercise: the d... |
| dvsubf 46100 | The subtraction rule for e... |
| dvmptconst 46101 | Function-builder for deriv... |
| dvcnre 46102 | From complex differentiati... |
| dvmptidg 46103 | Function-builder for deriv... |
| dvresntr 46104 | Function-builder for deriv... |
| fperdvper 46105 | The derivative of a period... |
| dvasinbx 46106 | Derivative exercise: the d... |
| dvresioo 46107 | Restriction of a derivativ... |
| dvdivf 46108 | The quotient rule for ever... |
| dvdivbd 46109 | A sufficient condition for... |
| dvsubcncf 46110 | A sufficient condition for... |
| dvmulcncf 46111 | A sufficient condition for... |
| dvcosax 46112 | Derivative exercise: the d... |
| dvdivcncf 46113 | A sufficient condition for... |
| dvbdfbdioolem1 46114 | Given a function with boun... |
| dvbdfbdioolem2 46115 | A function on an open inte... |
| dvbdfbdioo 46116 | A function on an open inte... |
| ioodvbdlimc1lem1 46117 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 46118 | Limit at the lower bound o... |
| ioodvbdlimc1 46119 | A real function with bound... |
| ioodvbdlimc2lem 46120 | Limit at the upper bound o... |
| ioodvbdlimc2 46121 | A real function with bound... |
| dvdmsscn 46122 | ` X ` is a subset of ` CC ... |
| dvmptmulf 46123 | Function-builder for deriv... |
| dvnmptdivc 46124 | Function-builder for itera... |
| dvdsn1add 46125 | If ` K ` divides ` N ` but... |
| dvxpaek 46126 | Derivative of the polynomi... |
| dvnmptconst 46127 | The ` N ` -th derivative o... |
| dvnxpaek 46128 | The ` n ` -th derivative o... |
| dvnmul 46129 | Function-builder for the `... |
| dvmptfprodlem 46130 | Induction step for ~ dvmpt... |
| dvmptfprod 46131 | Function-builder for deriv... |
| dvnprodlem1 46132 | ` D ` is bijective. (Cont... |
| dvnprodlem2 46133 | Induction step for ~ dvnpr... |
| dvnprodlem3 46134 | The multinomial formula fo... |
| dvnprod 46135 | The multinomial formula fo... |
| itgsin0pilem1 46136 | Calculation of the integra... |
| ibliccsinexp 46137 | sin^n on a closed interval... |
| itgsin0pi 46138 | Calculation of the integra... |
| iblioosinexp 46139 | sin^n on an open integral ... |
| itgsinexplem1 46140 | Integration by parts is ap... |
| itgsinexp 46141 | A recursive formula for th... |
| iblconstmpt 46142 | A constant function is int... |
| itgeq1d 46143 | Equality theorem for an in... |
| mbfres2cn 46144 | Measurability of a piecewi... |
| vol0 46145 | The measure of the empty s... |
| ditgeqiooicc 46146 | A function ` F ` on an ope... |
| volge0 46147 | The volume of a set is alw... |
| cnbdibl 46148 | A continuous bounded funct... |
| snmbl 46149 | A singleton is measurable.... |
| ditgeq3d 46150 | Equality theorem for the d... |
| iblempty 46151 | The empty function is inte... |
| iblsplit 46152 | The union of two integrabl... |
| volsn 46153 | A singleton has 0 Lebesgue... |
| itgvol0 46154 | If the domani is negligibl... |
| itgcoscmulx 46155 | Exercise: the integral of ... |
| iblsplitf 46156 | A version of ~ iblsplit us... |
| ibliooicc 46157 | If a function is integrabl... |
| volioc 46158 | The measure of a left-open... |
| iblspltprt 46159 | If a function is integrabl... |
| itgsincmulx 46160 | Exercise: the integral of ... |
| itgsubsticclem 46161 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 46162 | Integration by u-substitut... |
| itgioocnicc 46163 | The integral of a piecewis... |
| iblcncfioo 46164 | A continuous function ` F ... |
| itgspltprt 46165 | The ` S. ` integral splits... |
| itgiccshift 46166 | The integral of a function... |
| itgperiod 46167 | The integral of a periodic... |
| itgsbtaddcnst 46168 | Integral substitution, add... |
| volico 46169 | The measure of left-closed... |
| sublevolico 46170 | The Lebesgue measure of a ... |
| dmvolss 46171 | Lebesgue measurable sets a... |
| ismbl3 46172 | The predicate " ` A ` is L... |
| volioof 46173 | The function that assigns ... |
| ovolsplit 46174 | The Lebesgue outer measure... |
| fvvolioof 46175 | The function value of the ... |
| volioore 46176 | The measure of an open int... |
| fvvolicof 46177 | The function value of the ... |
| voliooico 46178 | An open interval and a lef... |
| ismbl4 46179 | The predicate " ` A ` is L... |
| volioofmpt 46180 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 46181 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 46182 | The Lebesgue measure of op... |
| volicofmpt 46183 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 46184 | The Lebesgue measure of a ... |
| voliccico 46185 | A closed interval and a le... |
| mbfdmssre 46186 | The domain of a measurable... |
| stoweidlem1 46187 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 46188 | lemma for ~ stoweid : here... |
| stoweidlem3 46189 | Lemma for ~ stoweid : if `... |
| stoweidlem4 46190 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 46191 | There exists a δ as ... |
| stoweidlem6 46192 | Lemma for ~ stoweid : two ... |
| stoweidlem7 46193 | This lemma is used to prov... |
| stoweidlem8 46194 | Lemma for ~ stoweid : two ... |
| stoweidlem9 46195 | Lemma for ~ stoweid : here... |
| stoweidlem10 46196 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 46197 | This lemma is used to prov... |
| stoweidlem12 46198 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 46199 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 46200 | There exists a ` k ` as in... |
| stoweidlem15 46201 | This lemma is used to prov... |
| stoweidlem16 46202 | Lemma for ~ stoweid . The... |
| stoweidlem17 46203 | This lemma proves that the... |
| stoweidlem18 46204 | This theorem proves Lemma ... |
| stoweidlem19 46205 | If a set of real functions... |
| stoweidlem20 46206 | If a set A of real functio... |
| stoweidlem21 46207 | Once the Stone Weierstrass... |
| stoweidlem22 46208 | If a set of real functions... |
| stoweidlem23 46209 | This lemma is used to prov... |
| stoweidlem24 46210 | This lemma proves that for... |
| stoweidlem25 46211 | This lemma proves that for... |
| stoweidlem26 46212 | This lemma is used to prov... |
| stoweidlem27 46213 | This lemma is used to prov... |
| stoweidlem28 46214 | There exists a δ as ... |
| stoweidlem29 46215 | When the hypothesis for th... |
| stoweidlem30 46216 | This lemma is used to prov... |
| stoweidlem31 46217 | This lemma is used to prov... |
| stoweidlem32 46218 | If a set A of real functio... |
| stoweidlem33 46219 | If a set of real functions... |
| stoweidlem34 46220 | This lemma proves that for... |
| stoweidlem35 46221 | This lemma is used to prov... |
| stoweidlem36 46222 | This lemma is used to prov... |
| stoweidlem37 46223 | This lemma is used to prov... |
| stoweidlem38 46224 | This lemma is used to prov... |
| stoweidlem39 46225 | This lemma is used to prov... |
| stoweidlem40 46226 | This lemma proves that q_n... |
| stoweidlem41 46227 | This lemma is used to prov... |
| stoweidlem42 46228 | This lemma is used to prov... |
| stoweidlem43 46229 | This lemma is used to prov... |
| stoweidlem44 46230 | This lemma is used to prov... |
| stoweidlem45 46231 | This lemma proves that, gi... |
| stoweidlem46 46232 | This lemma proves that set... |
| stoweidlem47 46233 | Subtracting a constant fro... |
| stoweidlem48 46234 | This lemma is used to prov... |
| stoweidlem49 46235 | There exists a function q_... |
| stoweidlem50 46236 | This lemma proves that set... |
| stoweidlem51 46237 | There exists a function x ... |
| stoweidlem52 46238 | There exists a neighborhoo... |
| stoweidlem53 46239 | This lemma is used to prov... |
| stoweidlem54 46240 | There exists a function ` ... |
| stoweidlem55 46241 | This lemma proves the exis... |
| stoweidlem56 46242 | This theorem proves Lemma ... |
| stoweidlem57 46243 | There exists a function x ... |
| stoweidlem58 46244 | This theorem proves Lemma ... |
| stoweidlem59 46245 | This lemma proves that the... |
| stoweidlem60 46246 | This lemma proves that the... |
| stoweidlem61 46247 | This lemma proves that the... |
| stoweidlem62 46248 | This theorem proves the St... |
| stoweid 46249 | This theorem proves the St... |
| stowei 46250 | This theorem proves the St... |
| wallispilem1 46251 | ` I ` is monotone: increas... |
| wallispilem2 46252 | A first set of properties ... |
| wallispilem3 46253 | I maps to real values. (C... |
| wallispilem4 46254 | ` F ` maps to explicit exp... |
| wallispilem5 46255 | The sequence ` H ` converg... |
| wallispi 46256 | Wallis' formula for π :... |
| wallispi2lem1 46257 | An intermediate step betwe... |
| wallispi2lem2 46258 | Two expressions are proven... |
| wallispi2 46259 | An alternative version of ... |
| stirlinglem1 46260 | A simple limit of fraction... |
| stirlinglem2 46261 | ` A ` maps to positive rea... |
| stirlinglem3 46262 | Long but simple algebraic ... |
| stirlinglem4 46263 | Algebraic manipulation of ... |
| stirlinglem5 46264 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 46265 | A series that converges to... |
| stirlinglem7 46266 | Algebraic manipulation of ... |
| stirlinglem8 46267 | If ` A ` converges to ` C ... |
| stirlinglem9 46268 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 46269 | A bound for any B(N)-B(N +... |
| stirlinglem11 46270 | ` B ` is decreasing. (Con... |
| stirlinglem12 46271 | The sequence ` B ` is boun... |
| stirlinglem13 46272 | ` B ` is decreasing and ha... |
| stirlinglem14 46273 | The sequence ` A ` converg... |
| stirlinglem15 46274 | The Stirling's formula is ... |
| stirling 46275 | Stirling's approximation f... |
| stirlingr 46276 | Stirling's approximation f... |
| dirkerval 46277 | The N_th Dirichlet Kernel.... |
| dirker2re 46278 | The Dirichlet Kernel value... |
| dirkerdenne0 46279 | The Dirichlet Kernel denom... |
| dirkerval2 46280 | The N_th Dirichlet Kernel ... |
| dirkerre 46281 | The Dirichlet Kernel at an... |
| dirkerper 46282 | the Dirichlet Kernel has p... |
| dirkerf 46283 | For any natural number ` N... |
| dirkertrigeqlem1 46284 | Sum of an even number of a... |
| dirkertrigeqlem2 46285 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 46286 | Trigonometric equality lem... |
| dirkertrigeq 46287 | Trigonometric equality for... |
| dirkeritg 46288 | The definite integral of t... |
| dirkercncflem1 46289 | If ` Y ` is a multiple of ... |
| dirkercncflem2 46290 | Lemma used to prove that t... |
| dirkercncflem3 46291 | The Dirichlet Kernel is co... |
| dirkercncflem4 46292 | The Dirichlet Kernel is co... |
| dirkercncf 46293 | For any natural number ` N... |
| fourierdlem1 46294 | A partition interval is a ... |
| fourierdlem2 46295 | Membership in a partition.... |
| fourierdlem3 46296 | Membership in a partition.... |
| fourierdlem4 46297 | ` E ` is a function that m... |
| fourierdlem5 46298 | ` S ` is a function. (Con... |
| fourierdlem6 46299 | ` X ` is in the periodic p... |
| fourierdlem7 46300 | The difference between the... |
| fourierdlem8 46301 | A partition interval is a ... |
| fourierdlem9 46302 | ` H ` is a complex functio... |
| fourierdlem10 46303 | Condition on the bounds of... |
| fourierdlem11 46304 | If there is a partition, t... |
| fourierdlem12 46305 | A point of a partition is ... |
| fourierdlem13 46306 | Value of ` V ` in terms of... |
| fourierdlem14 46307 | Given the partition ` V ` ... |
| fourierdlem15 46308 | The range of the partition... |
| fourierdlem16 46309 | The coefficients of the fo... |
| fourierdlem17 46310 | The defined ` L ` is actua... |
| fourierdlem18 46311 | The function ` S ` is cont... |
| fourierdlem19 46312 | If two elements of ` D ` h... |
| fourierdlem20 46313 | Every interval in the part... |
| fourierdlem21 46314 | The coefficients of the fo... |
| fourierdlem22 46315 | The coefficients of the fo... |
| fourierdlem23 46316 | If ` F ` is continuous and... |
| fourierdlem24 46317 | A sufficient condition for... |
| fourierdlem25 46318 | If ` C ` is not in the ran... |
| fourierdlem26 46319 | Periodic image of a point ... |
| fourierdlem27 46320 | A partition open interval ... |
| fourierdlem28 46321 | Derivative of ` ( F `` ( X... |
| fourierdlem29 46322 | Explicit function value fo... |
| fourierdlem30 46323 | Sum of three small pieces ... |
| fourierdlem31 46324 | If ` A ` is finite and for... |
| fourierdlem32 46325 | Limit of a continuous func... |
| fourierdlem33 46326 | Limit of a continuous func... |
| fourierdlem34 46327 | A partition is one to one.... |
| fourierdlem35 46328 | There is a single point in... |
| fourierdlem36 46329 | ` F ` is an isomorphism. ... |
| fourierdlem37 46330 | ` I ` is a function that m... |
| fourierdlem38 46331 | The function ` F ` is cont... |
| fourierdlem39 46332 | Integration by parts of ... |
| fourierdlem40 46333 | ` H ` is a continuous func... |
| fourierdlem41 46334 | Lemma used to prove that e... |
| fourierdlem42 46335 | The set of points in a mov... |
| fourierdlem43 46336 | ` K ` is a real function. ... |
| fourierdlem44 46337 | A condition for having ` (... |
| fourierdlem46 46338 | The function ` F ` has a l... |
| fourierdlem47 46339 | For ` r ` large enough, th... |
| fourierdlem48 46340 | The given periodic functio... |
| fourierdlem49 46341 | The given periodic functio... |
| fourierdlem50 46342 | Continuity of ` O ` and it... |
| fourierdlem51 46343 | ` X ` is in the periodic p... |
| fourierdlem52 46344 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 46345 | The limit of ` F ( s ) ` a... |
| fourierdlem54 46346 | Given a partition ` Q ` an... |
| fourierdlem55 46347 | ` U ` is a real function. ... |
| fourierdlem56 46348 | Derivative of the ` K ` fu... |
| fourierdlem57 46349 | The derivative of ` O ` . ... |
| fourierdlem58 46350 | The derivative of ` K ` is... |
| fourierdlem59 46351 | The derivative of ` H ` is... |
| fourierdlem60 46352 | Given a differentiable fun... |
| fourierdlem61 46353 | Given a differentiable fun... |
| fourierdlem62 46354 | The function ` K ` is cont... |
| fourierdlem63 46355 | The upper bound of interva... |
| fourierdlem64 46356 | The partition ` V ` is fin... |
| fourierdlem65 46357 | The distance of two adjace... |
| fourierdlem66 46358 | Value of the ` G ` functio... |
| fourierdlem67 46359 | ` G ` is a function. (Con... |
| fourierdlem68 46360 | The derivative of ` O ` is... |
| fourierdlem69 46361 | A piecewise continuous fun... |
| fourierdlem70 46362 | A piecewise continuous fun... |
| fourierdlem71 46363 | A periodic piecewise conti... |
| fourierdlem72 46364 | The derivative of ` O ` is... |
| fourierdlem73 46365 | A version of the Riemann L... |
| fourierdlem74 46366 | Given a piecewise smooth f... |
| fourierdlem75 46367 | Given a piecewise smooth f... |
| fourierdlem76 46368 | Continuity of ` O ` and it... |
| fourierdlem77 46369 | If ` H ` is bounded, then ... |
| fourierdlem78 46370 | ` G ` is continuous when r... |
| fourierdlem79 46371 | ` E ` projects every inter... |
| fourierdlem80 46372 | The derivative of ` O ` is... |
| fourierdlem81 46373 | The integral of a piecewis... |
| fourierdlem82 46374 | Integral by substitution, ... |
| fourierdlem83 46375 | The fourier partial sum fo... |
| fourierdlem84 46376 | If ` F ` is piecewise cont... |
| fourierdlem85 46377 | Limit of the function ` G ... |
| fourierdlem86 46378 | Continuity of ` O ` and it... |
| fourierdlem87 46379 | The integral of ` G ` goes... |
| fourierdlem88 46380 | Given a piecewise continuo... |
| fourierdlem89 46381 | Given a piecewise continuo... |
| fourierdlem90 46382 | Given a piecewise continuo... |
| fourierdlem91 46383 | Given a piecewise continuo... |
| fourierdlem92 46384 | The integral of a piecewis... |
| fourierdlem93 46385 | Integral by substitution (... |
| fourierdlem94 46386 | For a piecewise smooth fun... |
| fourierdlem95 46387 | Algebraic manipulation of ... |
| fourierdlem96 46388 | limit for ` F ` at the low... |
| fourierdlem97 46389 | ` F ` is continuous on the... |
| fourierdlem98 46390 | ` F ` is continuous on the... |
| fourierdlem99 46391 | limit for ` F ` at the upp... |
| fourierdlem100 46392 | A piecewise continuous fun... |
| fourierdlem101 46393 | Integral by substitution f... |
| fourierdlem102 46394 | For a piecewise smooth fun... |
| fourierdlem103 46395 | The half lower part of the... |
| fourierdlem104 46396 | The half upper part of the... |
| fourierdlem105 46397 | A piecewise continuous fun... |
| fourierdlem106 46398 | For a piecewise smooth fun... |
| fourierdlem107 46399 | The integral of a piecewis... |
| fourierdlem108 46400 | The integral of a piecewis... |
| fourierdlem109 46401 | The integral of a piecewis... |
| fourierdlem110 46402 | The integral of a piecewis... |
| fourierdlem111 46403 | The fourier partial sum fo... |
| fourierdlem112 46404 | Here abbreviations (local ... |
| fourierdlem113 46405 | Fourier series convergence... |
| fourierdlem114 46406 | Fourier series convergence... |
| fourierdlem115 46407 | Fourier serier convergence... |
| fourierd 46408 | Fourier series convergence... |
| fourierclimd 46409 | Fourier series convergence... |
| fourierclim 46410 | Fourier series convergence... |
| fourier 46411 | Fourier series convergence... |
| fouriercnp 46412 | If ` F ` is continuous at ... |
| fourier2 46413 | Fourier series convergence... |
| sqwvfoura 46414 | Fourier coefficients for t... |
| sqwvfourb 46415 | Fourier series ` B ` coeff... |
| fourierswlem 46416 | The Fourier series for the... |
| fouriersw 46417 | Fourier series convergence... |
| fouriercn 46418 | If the derivative of ` F `... |
| elaa2lem 46419 | Elementhood in the set of ... |
| elaa2 46420 | Elementhood in the set of ... |
| etransclem1 46421 | ` H ` is a function. (Con... |
| etransclem2 46422 | Derivative of ` G ` . (Co... |
| etransclem3 46423 | The given ` if ` term is a... |
| etransclem4 46424 | ` F ` expressed as a finit... |
| etransclem5 46425 | A change of bound variable... |
| etransclem6 46426 | A change of bound variable... |
| etransclem7 46427 | The given product is an in... |
| etransclem8 46428 | ` F ` is a function. (Con... |
| etransclem9 46429 | If ` K ` divides ` N ` but... |
| etransclem10 46430 | The given ` if ` term is a... |
| etransclem11 46431 | A change of bound variable... |
| etransclem12 46432 | ` C ` applied to ` N ` . ... |
| etransclem13 46433 | ` F ` applied to ` Y ` . ... |
| etransclem14 46434 | Value of the term ` T ` , ... |
| etransclem15 46435 | Value of the term ` T ` , ... |
| etransclem16 46436 | Every element in the range... |
| etransclem17 46437 | The ` N ` -th derivative o... |
| etransclem18 46438 | The given function is inte... |
| etransclem19 46439 | The ` N ` -th derivative o... |
| etransclem20 46440 | ` H ` is smooth. (Contrib... |
| etransclem21 46441 | The ` N ` -th derivative o... |
| etransclem22 46442 | The ` N ` -th derivative o... |
| etransclem23 46443 | This is the claim proof in... |
| etransclem24 46444 | ` P ` divides the I -th de... |
| etransclem25 46445 | ` P ` factorial divides th... |
| etransclem26 46446 | Every term in the sum of t... |
| etransclem27 46447 | The ` N ` -th derivative o... |
| etransclem28 46448 | ` ( P - 1 ) ` factorial di... |
| etransclem29 46449 | The ` N ` -th derivative o... |
| etransclem30 46450 | The ` N ` -th derivative o... |
| etransclem31 46451 | The ` N ` -th derivative o... |
| etransclem32 46452 | This is the proof for the ... |
| etransclem33 46453 | ` F ` is smooth. (Contrib... |
| etransclem34 46454 | The ` N ` -th derivative o... |
| etransclem35 46455 | ` P ` does not divide the ... |
| etransclem36 46456 | The ` N ` -th derivative o... |
| etransclem37 46457 | ` ( P - 1 ) ` factorial di... |
| etransclem38 46458 | ` P ` divides the I -th de... |
| etransclem39 46459 | ` G ` is a function. (Con... |
| etransclem40 46460 | The ` N ` -th derivative o... |
| etransclem41 46461 | ` P ` does not divide the ... |
| etransclem42 46462 | The ` N ` -th derivative o... |
| etransclem43 46463 | ` G ` is a continuous func... |
| etransclem44 46464 | The given finite sum is no... |
| etransclem45 46465 | ` K ` is an integer. (Con... |
| etransclem46 46466 | This is the proof for equa... |
| etransclem47 46467 | ` _e ` is transcendental. ... |
| etransclem48 46468 | ` _e ` is transcendental. ... |
| etransc 46469 | ` _e ` is transcendental. ... |
| rrxtopn 46470 | The topology of the genera... |
| rrxngp 46471 | Generalized Euclidean real... |
| rrxtps 46472 | Generalized Euclidean real... |
| rrxtopnfi 46473 | The topology of the n-dime... |
| rrxtopon 46474 | The topology on generalize... |
| rrxtop 46475 | The topology on generalize... |
| rrndistlt 46476 | Given two points in the sp... |
| rrxtoponfi 46477 | The topology on n-dimensio... |
| rrxunitopnfi 46478 | The base set of the standa... |
| rrxtopn0 46479 | The topology of the zero-d... |
| qndenserrnbllem 46480 | n-dimensional rational num... |
| qndenserrnbl 46481 | n-dimensional rational num... |
| rrxtopn0b 46482 | The topology of the zero-d... |
| qndenserrnopnlem 46483 | n-dimensional rational num... |
| qndenserrnopn 46484 | n-dimensional rational num... |
| qndenserrn 46485 | n-dimensional rational num... |
| rrxsnicc 46486 | A multidimensional singlet... |
| rrnprjdstle 46487 | The distance between two p... |
| rrndsmet 46488 | ` D ` is a metric for the ... |
| rrndsxmet 46489 | ` D ` is an extended metri... |
| ioorrnopnlem 46490 | The a point in an indexed ... |
| ioorrnopn 46491 | The indexed product of ope... |
| ioorrnopnxrlem 46492 | Given a point ` F ` that b... |
| ioorrnopnxr 46493 | The indexed product of ope... |
| issal 46500 | Express the predicate " ` ... |
| pwsal 46501 | The power set of a given s... |
| salunicl 46502 | SAlg sigma-algebra is clos... |
| saluncl 46503 | The union of two sets in a... |
| prsal 46504 | The pair of the empty set ... |
| saldifcl 46505 | The complement of an eleme... |
| 0sal 46506 | The empty set belongs to e... |
| salgenval 46507 | The sigma-algebra generate... |
| saliunclf 46508 | SAlg sigma-algebra is clos... |
| saliuncl 46509 | SAlg sigma-algebra is clos... |
| salincl 46510 | The intersection of two se... |
| saluni 46511 | A set is an element of any... |
| saliinclf 46512 | SAlg sigma-algebra is clos... |
| saliincl 46513 | SAlg sigma-algebra is clos... |
| saldifcl2 46514 | The difference of two elem... |
| intsaluni 46515 | The union of an arbitrary ... |
| intsal 46516 | The arbitrary intersection... |
| salgenn0 46517 | The set used in the defini... |
| salgencl 46518 | ` SalGen ` actually genera... |
| issald 46519 | Sufficient condition to pr... |
| salexct 46520 | An example of nontrivial s... |
| sssalgen 46521 | A set is a subset of the s... |
| salgenss 46522 | The sigma-algebra generate... |
| salgenuni 46523 | The base set of the sigma-... |
| issalgend 46524 | One side of ~ dfsalgen2 . ... |
| salexct2 46525 | An example of a subset tha... |
| unisalgen 46526 | The union of a set belongs... |
| dfsalgen2 46527 | Alternate characterization... |
| salexct3 46528 | An example of a sigma-alge... |
| salgencntex 46529 | This counterexample shows ... |
| salgensscntex 46530 | This counterexample shows ... |
| issalnnd 46531 | Sufficient condition to pr... |
| dmvolsal 46532 | Lebesgue measurable sets f... |
| saldifcld 46533 | The complement of an eleme... |
| saluncld 46534 | The union of two sets in a... |
| salgencld 46535 | ` SalGen ` actually genera... |
| 0sald 46536 | The empty set belongs to e... |
| iooborel 46537 | An open interval is a Bore... |
| salincld 46538 | The intersection of two se... |
| salunid 46539 | A set is an element of any... |
| unisalgen2 46540 | The union of a set belongs... |
| bor1sal 46541 | The Borel sigma-algebra on... |
| iocborel 46542 | A left-open, right-closed ... |
| subsaliuncllem 46543 | A subspace sigma-algebra i... |
| subsaliuncl 46544 | A subspace sigma-algebra i... |
| subsalsal 46545 | A subspace sigma-algebra i... |
| subsaluni 46546 | A set belongs to the subsp... |
| salrestss 46547 | A sigma-algebra restricted... |
| sge0rnre 46550 | When ` sum^ ` is applied t... |
| fge0icoicc 46551 | If ` F ` maps to nonnegati... |
| sge0val 46552 | The value of the sum of no... |
| fge0npnf 46553 | If ` F ` maps to nonnegati... |
| sge0rnn0 46554 | The range used in the defi... |
| sge0vald 46555 | The value of the sum of no... |
| fge0iccico 46556 | A range of nonnegative ext... |
| gsumge0cl 46557 | Closure of group sum, for ... |
| sge0reval 46558 | Value of the sum of nonneg... |
| sge0pnfval 46559 | If a term in the sum of no... |
| fge0iccre 46560 | A range of nonnegative ext... |
| sge0z 46561 | Any nonnegative extended s... |
| sge00 46562 | The sum of nonnegative ext... |
| fsumlesge0 46563 | Every finite subsum of non... |
| sge0revalmpt 46564 | Value of the sum of nonneg... |
| sge0sn 46565 | A sum of a nonnegative ext... |
| sge0tsms 46566 | ` sum^ ` applied to a nonn... |
| sge0cl 46567 | The arbitrary sum of nonne... |
| sge0f1o 46568 | Re-index a nonnegative ext... |
| sge0snmpt 46569 | A sum of a nonnegative ext... |
| sge0ge0 46570 | The sum of nonnegative ext... |
| sge0xrcl 46571 | The arbitrary sum of nonne... |
| sge0repnf 46572 | The of nonnegative extende... |
| sge0fsum 46573 | The arbitrary sum of a fin... |
| sge0rern 46574 | If the sum of nonnegative ... |
| sge0supre 46575 | If the arbitrary sum of no... |
| sge0fsummpt 46576 | The arbitrary sum of a fin... |
| sge0sup 46577 | The arbitrary sum of nonne... |
| sge0less 46578 | A shorter sum of nonnegati... |
| sge0rnbnd 46579 | The range used in the defi... |
| sge0pr 46580 | Sum of a pair of nonnegati... |
| sge0gerp 46581 | The arbitrary sum of nonne... |
| sge0pnffigt 46582 | If the sum of nonnegative ... |
| sge0ssre 46583 | If a sum of nonnegative ex... |
| sge0lefi 46584 | A sum of nonnegative exten... |
| sge0lessmpt 46585 | A shorter sum of nonnegati... |
| sge0ltfirp 46586 | If the sum of nonnegative ... |
| sge0prle 46587 | The sum of a pair of nonne... |
| sge0gerpmpt 46588 | The arbitrary sum of nonne... |
| sge0resrnlem 46589 | The sum of nonnegative ext... |
| sge0resrn 46590 | The sum of nonnegative ext... |
| sge0ssrempt 46591 | If a sum of nonnegative ex... |
| sge0resplit 46592 | ` sum^ ` splits into two p... |
| sge0le 46593 | If all of the terms of sum... |
| sge0ltfirpmpt 46594 | If the extended sum of non... |
| sge0split 46595 | Split a sum of nonnegative... |
| sge0lempt 46596 | If all of the terms of sum... |
| sge0splitmpt 46597 | Split a sum of nonnegative... |
| sge0ss 46598 | Change the index set to a ... |
| sge0iunmptlemfi 46599 | Sum of nonnegative extende... |
| sge0p1 46600 | The addition of the next t... |
| sge0iunmptlemre 46601 | Sum of nonnegative extende... |
| sge0fodjrnlem 46602 | Re-index a nonnegative ext... |
| sge0fodjrn 46603 | Re-index a nonnegative ext... |
| sge0iunmpt 46604 | Sum of nonnegative extende... |
| sge0iun 46605 | Sum of nonnegative extende... |
| sge0nemnf 46606 | The generalized sum of non... |
| sge0rpcpnf 46607 | The sum of an infinite num... |
| sge0rernmpt 46608 | If the sum of nonnegative ... |
| sge0lefimpt 46609 | A sum of nonnegative exten... |
| nn0ssge0 46610 | Nonnegative integers are n... |
| sge0clmpt 46611 | The generalized sum of non... |
| sge0ltfirpmpt2 46612 | If the extended sum of non... |
| sge0isum 46613 | If a series of nonnegative... |
| sge0xrclmpt 46614 | The generalized sum of non... |
| sge0xp 46615 | Combine two generalized su... |
| sge0isummpt 46616 | If a series of nonnegative... |
| sge0ad2en 46617 | The value of the infinite ... |
| sge0isummpt2 46618 | If a series of nonnegative... |
| sge0xaddlem1 46619 | The extended addition of t... |
| sge0xaddlem2 46620 | The extended addition of t... |
| sge0xadd 46621 | The extended addition of t... |
| sge0fsummptf 46622 | The generalized sum of a f... |
| sge0snmptf 46623 | A sum of a nonnegative ext... |
| sge0ge0mpt 46624 | The sum of nonnegative ext... |
| sge0repnfmpt 46625 | The of nonnegative extende... |
| sge0pnffigtmpt 46626 | If the generalized sum of ... |
| sge0splitsn 46627 | Separate out a term in a g... |
| sge0pnffsumgt 46628 | If the sum of nonnegative ... |
| sge0gtfsumgt 46629 | If the generalized sum of ... |
| sge0uzfsumgt 46630 | If a real number is smalle... |
| sge0pnfmpt 46631 | If a term in the sum of no... |
| sge0seq 46632 | A series of nonnegative re... |
| sge0reuz 46633 | Value of the generalized s... |
| sge0reuzb 46634 | Value of the generalized s... |
| ismea 46637 | Express the predicate " ` ... |
| dmmeasal 46638 | The domain of a measure is... |
| meaf 46639 | A measure is a function th... |
| mea0 46640 | The measure of the empty s... |
| nnfoctbdjlem 46641 | There exists a mapping fro... |
| nnfoctbdj 46642 | There exists a mapping fro... |
| meadjuni 46643 | The measure of the disjoin... |
| meacl 46644 | The measure of a set is a ... |
| iundjiunlem 46645 | The sets in the sequence `... |
| iundjiun 46646 | Given a sequence ` E ` of ... |
| meaxrcl 46647 | The measure of a set is an... |
| meadjun 46648 | The measure of the union o... |
| meassle 46649 | The measure of a set is gr... |
| meaunle 46650 | The measure of the union o... |
| meadjiunlem 46651 | The sum of nonnegative ext... |
| meadjiun 46652 | The measure of the disjoin... |
| ismeannd 46653 | Sufficient condition to pr... |
| meaiunlelem 46654 | The measure of the union o... |
| meaiunle 46655 | The measure of the union o... |
| psmeasurelem 46656 | ` M ` applied to a disjoin... |
| psmeasure 46657 | Point supported measure, R... |
| voliunsge0lem 46658 | The Lebesgue measure funct... |
| voliunsge0 46659 | The Lebesgue measure funct... |
| volmea 46660 | The Lebesgue measure on th... |
| meage0 46661 | If the measure of a measur... |
| meadjunre 46662 | The measure of the union o... |
| meassre 46663 | If the measure of a measur... |
| meale0eq0 46664 | A measure that is less tha... |
| meadif 46665 | The measure of the differe... |
| meaiuninclem 46666 | Measures are continuous fr... |
| meaiuninc 46667 | Measures are continuous fr... |
| meaiuninc2 46668 | Measures are continuous fr... |
| meaiunincf 46669 | Measures are continuous fr... |
| meaiuninc3v 46670 | Measures are continuous fr... |
| meaiuninc3 46671 | Measures are continuous fr... |
| meaiininclem 46672 | Measures are continuous fr... |
| meaiininc 46673 | Measures are continuous fr... |
| meaiininc2 46674 | Measures are continuous fr... |
| caragenval 46679 | The sigma-algebra generate... |
| isome 46680 | Express the predicate " ` ... |
| caragenel 46681 | Membership in the Caratheo... |
| omef 46682 | An outer measure is a func... |
| ome0 46683 | The outer measure of the e... |
| omessle 46684 | The outer measure of a set... |
| omedm 46685 | The domain of an outer mea... |
| caragensplit 46686 | If ` E ` is in the set gen... |
| caragenelss 46687 | An element of the Caratheo... |
| carageneld 46688 | Membership in the Caratheo... |
| omecl 46689 | The outer measure of a set... |
| caragenss 46690 | The sigma-algebra generate... |
| omeunile 46691 | The outer measure of the u... |
| caragen0 46692 | The empty set belongs to a... |
| omexrcl 46693 | The outer measure of a set... |
| caragenunidm 46694 | The base set of an outer m... |
| caragensspw 46695 | The sigma-algebra generate... |
| omessre 46696 | If the outer measure of a ... |
| caragenuni 46697 | The base set of the sigma-... |
| caragenuncllem 46698 | The Caratheodory's constru... |
| caragenuncl 46699 | The Caratheodory's constru... |
| caragendifcl 46700 | The Caratheodory's constru... |
| caragenfiiuncl 46701 | The Caratheodory's constru... |
| omeunle 46702 | The outer measure of the u... |
| omeiunle 46703 | The outer measure of the i... |
| omelesplit 46704 | The outer measure of a set... |
| omeiunltfirp 46705 | If the outer measure of a ... |
| omeiunlempt 46706 | The outer measure of the i... |
| carageniuncllem1 46707 | The outer measure of ` A i... |
| carageniuncllem2 46708 | The Caratheodory's constru... |
| carageniuncl 46709 | The Caratheodory's constru... |
| caragenunicl 46710 | The Caratheodory's constru... |
| caragensal 46711 | Caratheodory's method gene... |
| caratheodorylem1 46712 | Lemma used to prove that C... |
| caratheodorylem2 46713 | Caratheodory's constructio... |
| caratheodory 46714 | Caratheodory's constructio... |
| 0ome 46715 | The map that assigns 0 to ... |
| isomenndlem 46716 | ` O ` is sub-additive w.r.... |
| isomennd 46717 | Sufficient condition to pr... |
| caragenel2d 46718 | Membership in the Caratheo... |
| omege0 46719 | If the outer measure of a ... |
| omess0 46720 | If the outer measure of a ... |
| caragencmpl 46721 | A measure built with the C... |
| vonval 46726 | Value of the Lebesgue meas... |
| ovnval 46727 | Value of the Lebesgue oute... |
| elhoi 46728 | Membership in a multidimen... |
| icoresmbl 46729 | A closed-below, open-above... |
| hoissre 46730 | The projection of a half-o... |
| ovnval2 46731 | Value of the Lebesgue oute... |
| volicorecl 46732 | The Lebesgue measure of a ... |
| hoiprodcl 46733 | The pre-measure of half-op... |
| hoicvr 46734 | ` I ` is a countable set o... |
| hoissrrn 46735 | A half-open interval is a ... |
| ovn0val 46736 | The Lebesgue outer measure... |
| ovnn0val 46737 | The value of a (multidimen... |
| ovnval2b 46738 | Value of the Lebesgue oute... |
| volicorescl 46739 | The Lebesgue measure of a ... |
| ovnprodcl 46740 | The product used in the de... |
| hoiprodcl2 46741 | The pre-measure of half-op... |
| hoicvrrex 46742 | Any subset of the multidim... |
| ovnsupge0 46743 | The set used in the defini... |
| ovnlecvr 46744 | Given a subset of multidim... |
| ovnpnfelsup 46745 | ` +oo ` is an element of t... |
| ovnsslelem 46746 | The (multidimensional, non... |
| ovnssle 46747 | The (multidimensional) Leb... |
| ovnlerp 46748 | The Lebesgue outer measure... |
| ovnf 46749 | The Lebesgue outer measure... |
| ovncvrrp 46750 | The Lebesgue outer measure... |
| ovn0lem 46751 | For any finite dimension, ... |
| ovn0 46752 | For any finite dimension, ... |
| ovncl 46753 | The Lebesgue outer measure... |
| ovn02 46754 | For the zero-dimensional s... |
| ovnxrcl 46755 | The Lebesgue outer measure... |
| ovnsubaddlem1 46756 | The Lebesgue outer measure... |
| ovnsubaddlem2 46757 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 46758 | ` ( voln* `` X ) ` is suba... |
| ovnome 46759 | ` ( voln* `` X ) ` is an o... |
| vonmea 46760 | ` ( voln `` X ) ` is a mea... |
| volicon0 46761 | The measure of a nonempty ... |
| hsphoif 46762 | ` H ` is a function (that ... |
| hoidmvval 46763 | The dimensional volume of ... |
| hoissrrn2 46764 | A half-open interval is a ... |
| hsphoival 46765 | ` H ` is a function (that ... |
| hoiprodcl3 46766 | The pre-measure of half-op... |
| volicore 46767 | The Lebesgue measure of a ... |
| hoidmvcl 46768 | The dimensional volume of ... |
| hoidmv0val 46769 | The dimensional volume of ... |
| hoidmvn0val 46770 | The dimensional volume of ... |
| hsphoidmvle2 46771 | The dimensional volume of ... |
| hsphoidmvle 46772 | The dimensional volume of ... |
| hoidmvval0 46773 | The dimensional volume of ... |
| hoiprodp1 46774 | The dimensional volume of ... |
| sge0hsphoire 46775 | If the generalized sum of ... |
| hoidmvval0b 46776 | The dimensional volume of ... |
| hoidmv1lelem1 46777 | The supremum of ` U ` belo... |
| hoidmv1lelem2 46778 | This is the contradiction ... |
| hoidmv1lelem3 46779 | The dimensional volume of ... |
| hoidmv1le 46780 | The dimensional volume of ... |
| hoidmvlelem1 46781 | The supremum of ` U ` belo... |
| hoidmvlelem2 46782 | This is the contradiction ... |
| hoidmvlelem3 46783 | This is the contradiction ... |
| hoidmvlelem4 46784 | The dimensional volume of ... |
| hoidmvlelem5 46785 | The dimensional volume of ... |
| hoidmvle 46786 | The dimensional volume of ... |
| ovnhoilem1 46787 | The Lebesgue outer measure... |
| ovnhoilem2 46788 | The Lebesgue outer measure... |
| ovnhoi 46789 | The Lebesgue outer measure... |
| dmovn 46790 | The domain of the Lebesgue... |
| hoicoto2 46791 | The half-open interval exp... |
| dmvon 46792 | Lebesgue measurable n-dime... |
| hoi2toco 46793 | The half-open interval exp... |
| hoidifhspval 46794 | ` D ` is a function that r... |
| hspval 46795 | The value of the half-spac... |
| ovnlecvr2 46796 | Given a subset of multidim... |
| ovncvr2 46797 | ` B ` and ` T ` are the le... |
| dmovnsal 46798 | The domain of the Lebesgue... |
| unidmovn 46799 | Base set of the n-dimensio... |
| rrnmbl 46800 | The set of n-dimensional R... |
| hoidifhspval2 46801 | ` D ` is a function that r... |
| hspdifhsp 46802 | A n-dimensional half-open ... |
| unidmvon 46803 | Base set of the n-dimensio... |
| hoidifhspf 46804 | ` D ` is a function that r... |
| hoidifhspval3 46805 | ` D ` is a function that r... |
| hoidifhspdmvle 46806 | The dimensional volume of ... |
| voncmpl 46807 | The Lebesgue measure is co... |
| hoiqssbllem1 46808 | The center of the n-dimens... |
| hoiqssbllem2 46809 | The center of the n-dimens... |
| hoiqssbllem3 46810 | A n-dimensional ball conta... |
| hoiqssbl 46811 | A n-dimensional ball conta... |
| hspmbllem1 46812 | Any half-space of the n-di... |
| hspmbllem2 46813 | Any half-space of the n-di... |
| hspmbllem3 46814 | Any half-space of the n-di... |
| hspmbl 46815 | Any half-space of the n-di... |
| hoimbllem 46816 | Any n-dimensional half-ope... |
| hoimbl 46817 | Any n-dimensional half-ope... |
| opnvonmbllem1 46818 | The half-open interval exp... |
| opnvonmbllem2 46819 | An open subset of the n-di... |
| opnvonmbl 46820 | An open subset of the n-di... |
| opnssborel 46821 | Open sets of a generalized... |
| borelmbl 46822 | All Borel subsets of the n... |
| volicorege0 46823 | The Lebesgue measure of a ... |
| isvonmbl 46824 | The predicate " ` A ` is m... |
| mblvon 46825 | The n-dimensional Lebesgue... |
| vonmblss 46826 | n-dimensional Lebesgue mea... |
| volico2 46827 | The measure of left-closed... |
| vonmblss2 46828 | n-dimensional Lebesgue mea... |
| ovolval2lem 46829 | The value of the Lebesgue ... |
| ovolval2 46830 | The value of the Lebesgue ... |
| ovnsubadd2lem 46831 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 46832 | ` ( voln* `` X ) ` is suba... |
| ovolval3 46833 | The value of the Lebesgue ... |
| ovnsplit 46834 | The n-dimensional Lebesgue... |
| ovolval4lem1 46835 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 46836 | The value of the Lebesgue ... |
| ovolval4 46837 | The value of the Lebesgue ... |
| ovolval5lem1 46838 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 46839 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 46840 | The value of the Lebesgue ... |
| ovolval5 46841 | The value of the Lebesgue ... |
| ovnovollem1 46842 | if ` F ` is a cover of ` B... |
| ovnovollem2 46843 | if ` I ` is a cover of ` (... |
| ovnovollem3 46844 | The 1-dimensional Lebesgue... |
| ovnovol 46845 | The 1-dimensional Lebesgue... |
| vonvolmbllem 46846 | If a subset ` B ` of real ... |
| vonvolmbl 46847 | A subset of Real numbers i... |
| vonvol 46848 | The 1-dimensional Lebesgue... |
| vonvolmbl2 46849 | A subset ` X ` of the spac... |
| vonvol2 46850 | The 1-dimensional Lebesgue... |
| hoimbl2 46851 | Any n-dimensional half-ope... |
| voncl 46852 | The Lebesgue measure of a ... |
| vonhoi 46853 | The Lebesgue outer measure... |
| vonxrcl 46854 | The Lebesgue measure of a ... |
| ioosshoi 46855 | A n-dimensional open inter... |
| vonn0hoi 46856 | The Lebesgue outer measure... |
| von0val 46857 | The Lebesgue measure (for ... |
| vonhoire 46858 | The Lebesgue measure of a ... |
| iinhoiicclem 46859 | A n-dimensional closed int... |
| iinhoiicc 46860 | A n-dimensional closed int... |
| iunhoiioolem 46861 | A n-dimensional open inter... |
| iunhoiioo 46862 | A n-dimensional open inter... |
| ioovonmbl 46863 | Any n-dimensional open int... |
| iccvonmbllem 46864 | Any n-dimensional closed i... |
| iccvonmbl 46865 | Any n-dimensional closed i... |
| vonioolem1 46866 | The sequence of the measur... |
| vonioolem2 46867 | The n-dimensional Lebesgue... |
| vonioo 46868 | The n-dimensional Lebesgue... |
| vonicclem1 46869 | The sequence of the measur... |
| vonicclem2 46870 | The n-dimensional Lebesgue... |
| vonicc 46871 | The n-dimensional Lebesgue... |
| snvonmbl 46872 | A n-dimensional singleton ... |
| vonn0ioo 46873 | The n-dimensional Lebesgue... |
| vonn0icc 46874 | The n-dimensional Lebesgue... |
| ctvonmbl 46875 | Any n-dimensional countabl... |
| vonn0ioo2 46876 | The n-dimensional Lebesgue... |
| vonsn 46877 | The n-dimensional Lebesgue... |
| vonn0icc2 46878 | The n-dimensional Lebesgue... |
| vonct 46879 | The n-dimensional Lebesgue... |
| vitali2 46880 | There are non-measurable s... |
| pimltmnf2f 46883 | Given a real-valued functi... |
| pimltmnf2 46884 | Given a real-valued functi... |
| preimagelt 46885 | The preimage of a right-op... |
| preimalegt 46886 | The preimage of a left-ope... |
| pimconstlt0 46887 | Given a constant function,... |
| pimconstlt1 46888 | Given a constant function,... |
| pimltpnff 46889 | Given a real-valued functi... |
| pimltpnf 46890 | Given a real-valued functi... |
| pimgtpnf2f 46891 | Given a real-valued functi... |
| pimgtpnf2 46892 | Given a real-valued functi... |
| salpreimagelt 46893 | If all the preimages of le... |
| pimrecltpos 46894 | The preimage of an unbound... |
| salpreimalegt 46895 | If all the preimages of ri... |
| pimiooltgt 46896 | The preimage of an open in... |
| preimaicomnf 46897 | Preimage of an open interv... |
| pimltpnf2f 46898 | Given a real-valued functi... |
| pimltpnf2 46899 | Given a real-valued functi... |
| pimgtmnf2 46900 | Given a real-valued functi... |
| pimdecfgtioc 46901 | Given a nonincreasing func... |
| pimincfltioc 46902 | Given a nondecreasing func... |
| pimdecfgtioo 46903 | Given a nondecreasing func... |
| pimincfltioo 46904 | Given a nondecreasing func... |
| preimaioomnf 46905 | Preimage of an open interv... |
| preimageiingt 46906 | A preimage of a left-close... |
| preimaleiinlt 46907 | A preimage of a left-open,... |
| pimgtmnff 46908 | Given a real-valued functi... |
| pimgtmnf 46909 | Given a real-valued functi... |
| pimrecltneg 46910 | The preimage of an unbound... |
| salpreimagtge 46911 | If all the preimages of le... |
| salpreimaltle 46912 | If all the preimages of ri... |
| issmflem 46913 | The predicate " ` F ` is a... |
| issmf 46914 | The predicate " ` F ` is a... |
| salpreimalelt 46915 | If all the preimages of ri... |
| salpreimagtlt 46916 | If all the preimages of le... |
| smfpreimalt 46917 | Given a function measurabl... |
| smff 46918 | A function measurable w.r.... |
| smfdmss 46919 | The domain of a function m... |
| issmff 46920 | The predicate " ` F ` is a... |
| issmfd 46921 | A sufficient condition for... |
| smfpreimaltf 46922 | Given a function measurabl... |
| issmfdf 46923 | A sufficient condition for... |
| sssmf 46924 | The restriction of a sigma... |
| mbfresmf 46925 | A real-valued measurable f... |
| cnfsmf 46926 | A continuous function is m... |
| incsmflem 46927 | A nondecreasing function i... |
| incsmf 46928 | A real-valued, nondecreasi... |
| smfsssmf 46929 | If a function is measurabl... |
| issmflelem 46930 | The predicate " ` F ` is a... |
| issmfle 46931 | The predicate " ` F ` is a... |
| smfpimltmpt 46932 | Given a function measurabl... |
| smfpimltxr 46933 | Given a function measurabl... |
| issmfdmpt 46934 | A sufficient condition for... |
| smfconst 46935 | Given a sigma-algebra over... |
| sssmfmpt 46936 | The restriction of a sigma... |
| cnfrrnsmf 46937 | A function, continuous fro... |
| smfid 46938 | The identity function is B... |
| bormflebmf 46939 | A Borel measurable functio... |
| smfpreimale 46940 | Given a function measurabl... |
| issmfgtlem 46941 | The predicate " ` F ` is a... |
| issmfgt 46942 | The predicate " ` F ` is a... |
| issmfled 46943 | A sufficient condition for... |
| smfpimltxrmptf 46944 | Given a function measurabl... |
| smfpimltxrmpt 46945 | Given a function measurabl... |
| smfmbfcex 46946 | A constant function, with ... |
| issmfgtd 46947 | A sufficient condition for... |
| smfpreimagt 46948 | Given a function measurabl... |
| smfaddlem1 46949 | Given the sum of two funct... |
| smfaddlem2 46950 | The sum of two sigma-measu... |
| smfadd 46951 | The sum of two sigma-measu... |
| decsmflem 46952 | A nonincreasing function i... |
| decsmf 46953 | A real-valued, nonincreasi... |
| smfpreimagtf 46954 | Given a function measurabl... |
| issmfgelem 46955 | The predicate " ` F ` is a... |
| issmfge 46956 | The predicate " ` F ` is a... |
| smflimlem1 46957 | Lemma for the proof that t... |
| smflimlem2 46958 | Lemma for the proof that t... |
| smflimlem3 46959 | The limit of sigma-measura... |
| smflimlem4 46960 | Lemma for the proof that t... |
| smflimlem5 46961 | Lemma for the proof that t... |
| smflimlem6 46962 | Lemma for the proof that t... |
| smflim 46963 | The limit of sigma-measura... |
| nsssmfmbflem 46964 | The sigma-measurable funct... |
| nsssmfmbf 46965 | The sigma-measurable funct... |
| smfpimgtxr 46966 | Given a function measurabl... |
| smfpimgtmpt 46967 | Given a function measurabl... |
| smfpreimage 46968 | Given a function measurabl... |
| mbfpsssmf 46969 | Real-valued measurable fun... |
| smfpimgtxrmptf 46970 | Given a function measurabl... |
| smfpimgtxrmpt 46971 | Given a function measurabl... |
| smfpimioompt 46972 | Given a function measurabl... |
| smfpimioo 46973 | Given a function measurabl... |
| smfresal 46974 | Given a sigma-measurable f... |
| smfrec 46975 | The reciprocal of a sigma-... |
| smfres 46976 | The restriction of sigma-m... |
| smfmullem1 46977 | The multiplication of two ... |
| smfmullem2 46978 | The multiplication of two ... |
| smfmullem3 46979 | The multiplication of two ... |
| smfmullem4 46980 | The multiplication of two ... |
| smfmul 46981 | The multiplication of two ... |
| smfmulc1 46982 | A sigma-measurable functio... |
| smfdiv 46983 | The fraction of two sigma-... |
| smfpimbor1lem1 46984 | Every open set belongs to ... |
| smfpimbor1lem2 46985 | Given a sigma-measurable f... |
| smfpimbor1 46986 | Given a sigma-measurable f... |
| smf2id 46987 | Twice the identity functio... |
| smfco 46988 | The composition of a Borel... |
| smfneg 46989 | The negative of a sigma-me... |
| smffmptf 46990 | A function measurable w.r.... |
| smffmpt 46991 | A function measurable w.r.... |
| smflim2 46992 | The limit of a sequence of... |
| smfpimcclem 46993 | Lemma for ~ smfpimcc given... |
| smfpimcc 46994 | Given a countable set of s... |
| issmfle2d 46995 | A sufficient condition for... |
| smflimmpt 46996 | The limit of a sequence of... |
| smfsuplem1 46997 | The supremum of a countabl... |
| smfsuplem2 46998 | The supremum of a countabl... |
| smfsuplem3 46999 | The supremum of a countabl... |
| smfsup 47000 | The supremum of a countabl... |
| smfsupmpt 47001 | The supremum of a countabl... |
| smfsupxr 47002 | The supremum of a countabl... |
| smfinflem 47003 | The infimum of a countable... |
| smfinf 47004 | The infimum of a countable... |
| smfinfmpt 47005 | The infimum of a countable... |
| smflimsuplem1 47006 | If ` H ` converges, the ` ... |
| smflimsuplem2 47007 | The superior limit of a se... |
| smflimsuplem3 47008 | The limit of the ` ( H `` ... |
| smflimsuplem4 47009 | If ` H ` converges, the ` ... |
| smflimsuplem5 47010 | ` H ` converges to the sup... |
| smflimsuplem6 47011 | The superior limit of a se... |
| smflimsuplem7 47012 | The superior limit of a se... |
| smflimsuplem8 47013 | The superior limit of a se... |
| smflimsup 47014 | The superior limit of a se... |
| smflimsupmpt 47015 | The superior limit of a se... |
| smfliminflem 47016 | The inferior limit of a co... |
| smfliminf 47017 | The inferior limit of a co... |
| smfliminfmpt 47018 | The inferior limit of a co... |
| adddmmbl 47019 | If two functions have doma... |
| adddmmbl2 47020 | If two functions have doma... |
| muldmmbl 47021 | If two functions have doma... |
| muldmmbl2 47022 | If two functions have doma... |
| smfdmmblpimne 47023 | If a measurable function w... |
| smfdivdmmbl 47024 | If a functions and a sigma... |
| smfpimne 47025 | Given a function measurabl... |
| smfpimne2 47026 | Given a function measurabl... |
| smfdivdmmbl2 47027 | If a functions and a sigma... |
| fsupdm 47028 | The domain of the sup func... |
| fsupdm2 47029 | The domain of the sup func... |
| smfsupdmmbllem 47030 | If a countable set of sigm... |
| smfsupdmmbl 47031 | If a countable set of sigm... |
| finfdm 47032 | The domain of the inf func... |
| finfdm2 47033 | The domain of the inf func... |
| smfinfdmmbllem 47034 | If a countable set of sigm... |
| smfinfdmmbl 47035 | If a countable set of sigm... |
| sigarval 47036 | Define the signed area by ... |
| sigarim 47037 | Signed area takes value in... |
| sigarac 47038 | Signed area is anticommuta... |
| sigaraf 47039 | Signed area is additive by... |
| sigarmf 47040 | Signed area is additive (w... |
| sigaras 47041 | Signed area is additive by... |
| sigarms 47042 | Signed area is additive (w... |
| sigarls 47043 | Signed area is linear by t... |
| sigarid 47044 | Signed area of a flat para... |
| sigarexp 47045 | Expand the signed area for... |
| sigarperm 47046 | Signed area ` ( A - C ) G ... |
| sigardiv 47047 | If signed area between vec... |
| sigarimcd 47048 | Signed area takes value in... |
| sigariz 47049 | If signed area is zero, th... |
| sigarcol 47050 | Given three points ` A ` ,... |
| sharhght 47051 | Let ` A B C ` be a triangl... |
| sigaradd 47052 | Subtracting (double) area ... |
| cevathlem1 47053 | Ceva's theorem first lemma... |
| cevathlem2 47054 | Ceva's theorem second lemm... |
| cevath 47055 | Ceva's theorem. Let ` A B... |
| simpcntrab 47056 | The center of a simple gro... |
| et-ltneverrefl 47057 | Less-than class is never r... |
| et-equeucl 47058 | Alternative proof that equ... |
| et-sqrtnegnre 47059 | The square root of a negat... |
| ormklocald 47060 | If elements of a certain s... |
| ormkglobd 47061 | If all adjacent elements o... |
| natlocalincr 47062 | Global monotonicity on hal... |
| natglobalincr 47063 | Local monotonicity on half... |
| chnsubseqword 47064 | A subsequence of a chain i... |
| chnsubseqwl 47065 | A subsequence of a chain h... |
| chnsubseq 47066 | An order-preserving subseq... |
| chnsuslle 47067 | Length of a subsequence is... |
| chnerlem1 47068 | In a chain constructed on ... |
| chnerlem2 47069 | Lemma for ~ chner where th... |
| chnerlem3 47070 | Lemma for ~ chner - tricho... |
| chner 47071 | Any two elements are equiv... |
| nthrucw 47072 | Some number sets form a ch... |
| evenwodadd 47073 | If an integer is multiplie... |
| squeezedltsq 47074 | If a real value is squeeze... |
| lambert0 47075 | A value of Lambert W (prod... |
| lamberte 47076 | A value of Lambert W (prod... |
| cjnpoly 47077 | Complex conjugation operat... |
| tannpoly 47078 | The tangent function is no... |
| sinnpoly 47079 | Sine function is not a pol... |
| hirstL-ax3 47080 | The third axiom of a syste... |
| ax3h 47081 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 47082 | A closed form showing (a i... |
| aibandbiaiaiffb 47083 | A closed form showing (a i... |
| notatnand 47084 | Do not use. Use intnanr i... |
| aistia 47085 | Given a is equivalent to `... |
| aisfina 47086 | Given a is equivalent to `... |
| bothtbothsame 47087 | Given both a, b are equiva... |
| bothfbothsame 47088 | Given both a, b are equiva... |
| aiffbbtat 47089 | Given a is equivalent to b... |
| aisbbisfaisf 47090 | Given a is equivalent to b... |
| axorbtnotaiffb 47091 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 47092 | Given a is equivalent to (... |
| axorbciffatcxorb 47093 | Given a is equivalent to (... |
| aibnbna 47094 | Given a implies b, (not b)... |
| aibnbaif 47095 | Given a implies b, not b, ... |
| aiffbtbat 47096 | Given a is equivalent to b... |
| astbstanbst 47097 | Given a is equivalent to T... |
| aistbistaandb 47098 | Given a is equivalent to T... |
| aisbnaxb 47099 | Given a is equivalent to b... |
| atbiffatnnb 47100 | If a implies b, then a imp... |
| bisaiaisb 47101 | Application of bicom1 with... |
| atbiffatnnbalt 47102 | If a implies b, then a imp... |
| abnotbtaxb 47103 | Assuming a, not b, there e... |
| abnotataxb 47104 | Assuming not a, b, there e... |
| conimpf 47105 | Assuming a, not b, and a i... |
| conimpfalt 47106 | Assuming a, not b, and a i... |
| aistbisfiaxb 47107 | Given a is equivalent to T... |
| aisfbistiaxb 47108 | Given a is equivalent to F... |
| aifftbifffaibif 47109 | Given a is equivalent to T... |
| aifftbifffaibifff 47110 | Given a is equivalent to T... |
| atnaiana 47111 | Given a, it is not the cas... |
| ainaiaandna 47112 | Given a, a implies it is n... |
| abcdta 47113 | Given (((a and b) and c) a... |
| abcdtb 47114 | Given (((a and b) and c) a... |
| abcdtc 47115 | Given (((a and b) and c) a... |
| abcdtd 47116 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 47117 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 47118 | Operands in a biconditiona... |
| nabctnabc 47119 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 47120 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 47121 | From one negated implicati... |
| twonotinotbothi 47122 | From these two negated imp... |
| clifte 47123 | show d is the same as an i... |
| cliftet 47124 | show d is the same as an i... |
| clifteta 47125 | show d is the same as an i... |
| cliftetb 47126 | show d is the same as an i... |
| confun 47127 | Given the hypotheses there... |
| confun2 47128 | Confun simplified to two p... |
| confun3 47129 | Confun's more complex form... |
| confun4 47130 | An attempt at derivative. ... |
| confun5 47131 | An attempt at derivative. ... |
| plcofph 47132 | Given, a,b and a "definiti... |
| pldofph 47133 | Given, a,b c, d, "definiti... |
| plvcofph 47134 | Given, a,b,d, and "definit... |
| plvcofphax 47135 | Given, a,b,d, and "definit... |
| plvofpos 47136 | rh is derivable because ON... |
| mdandyv0 47137 | Given the equivalences set... |
| mdandyv1 47138 | Given the equivalences set... |
| mdandyv2 47139 | Given the equivalences set... |
| mdandyv3 47140 | Given the equivalences set... |
| mdandyv4 47141 | Given the equivalences set... |
| mdandyv5 47142 | Given the equivalences set... |
| mdandyv6 47143 | Given the equivalences set... |
| mdandyv7 47144 | Given the equivalences set... |
| mdandyv8 47145 | Given the equivalences set... |
| mdandyv9 47146 | Given the equivalences set... |
| mdandyv10 47147 | Given the equivalences set... |
| mdandyv11 47148 | Given the equivalences set... |
| mdandyv12 47149 | Given the equivalences set... |
| mdandyv13 47150 | Given the equivalences set... |
| mdandyv14 47151 | Given the equivalences set... |
| mdandyv15 47152 | Given the equivalences set... |
| mdandyvr0 47153 | Given the equivalences set... |
| mdandyvr1 47154 | Given the equivalences set... |
| mdandyvr2 47155 | Given the equivalences set... |
| mdandyvr3 47156 | Given the equivalences set... |
| mdandyvr4 47157 | Given the equivalences set... |
| mdandyvr5 47158 | Given the equivalences set... |
| mdandyvr6 47159 | Given the equivalences set... |
| mdandyvr7 47160 | Given the equivalences set... |
| mdandyvr8 47161 | Given the equivalences set... |
| mdandyvr9 47162 | Given the equivalences set... |
| mdandyvr10 47163 | Given the equivalences set... |
| mdandyvr11 47164 | Given the equivalences set... |
| mdandyvr12 47165 | Given the equivalences set... |
| mdandyvr13 47166 | Given the equivalences set... |
| mdandyvr14 47167 | Given the equivalences set... |
| mdandyvr15 47168 | Given the equivalences set... |
| mdandyvrx0 47169 | Given the exclusivities se... |
| mdandyvrx1 47170 | Given the exclusivities se... |
| mdandyvrx2 47171 | Given the exclusivities se... |
| mdandyvrx3 47172 | Given the exclusivities se... |
| mdandyvrx4 47173 | Given the exclusivities se... |
| mdandyvrx5 47174 | Given the exclusivities se... |
| mdandyvrx6 47175 | Given the exclusivities se... |
| mdandyvrx7 47176 | Given the exclusivities se... |
| mdandyvrx8 47177 | Given the exclusivities se... |
| mdandyvrx9 47178 | Given the exclusivities se... |
| mdandyvrx10 47179 | Given the exclusivities se... |
| mdandyvrx11 47180 | Given the exclusivities se... |
| mdandyvrx12 47181 | Given the exclusivities se... |
| mdandyvrx13 47182 | Given the exclusivities se... |
| mdandyvrx14 47183 | Given the exclusivities se... |
| mdandyvrx15 47184 | Given the exclusivities se... |
| H15NH16TH15IH16 47185 | Given 15 hypotheses and a ... |
| dandysum2p2e4 47186 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 47187 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 47188 | Replacement of a nested an... |
| adh-minim 47189 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 47190 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 47191 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 47192 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 47193 | Fourth lemma for the deriv... |
| adh-minim-ax1 47194 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 47195 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 47196 | Sixth lemma for the deriva... |
| adh-minim-ax2c 47197 | Derivation of a commuted f... |
| adh-minim-ax2 47198 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 47199 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 47200 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 47201 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 47202 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 47203 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 47204 | Third lemma for the deriva... |
| adh-minimp-sylsimp 47205 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 47206 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 47207 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 47208 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 47209 | Fourth lemma for the deriv... |
| adh-minimp-ax2 47210 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 47211 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 47212 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 47213 | If a class with one elemen... |
| eusnsn 47214 | There is a unique element ... |
| absnsb 47215 | If the class abstraction `... |
| euabsneu 47216 | Another way to express exi... |
| elprneb 47217 | An element of a proper uno... |
| oppr 47218 | Equality for ordered pairs... |
| opprb 47219 | Equality for unordered pai... |
| or2expropbilem1 47220 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 47221 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 47222 | If two classes are strictl... |
| eubrv 47223 | If there is a unique set w... |
| eubrdm 47224 | If there is a unique set w... |
| eldmressn 47225 | Element of the domain of a... |
| iota0def 47226 | Example for a defined iota... |
| iota0ndef 47227 | Example for an undefined i... |
| fveqvfvv 47228 | If a function's value at a... |
| fnresfnco 47229 | Composition of two functio... |
| funcoressn 47230 | A composition restricted t... |
| funressnfv 47231 | A restriction to a singlet... |
| funressndmfvrn 47232 | The value of a function ` ... |
| funressnvmo 47233 | A function restricted to a... |
| funressnmo 47234 | A function restricted to a... |
| funressneu 47235 | There is exactly one value... |
| fresfo 47236 | Conditions for a restricti... |
| fsetsniunop 47237 | The class of all functions... |
| fsetabsnop 47238 | The class of all functions... |
| fsetsnf 47239 | The mapping of an element ... |
| fsetsnf1 47240 | The mapping of an element ... |
| fsetsnfo 47241 | The mapping of an element ... |
| fsetsnf1o 47242 | The mapping of an element ... |
| fsetsnprcnex 47243 | The class of all functions... |
| cfsetssfset 47244 | The class of constant func... |
| cfsetsnfsetfv 47245 | The function value of the ... |
| cfsetsnfsetf 47246 | The mapping of the class o... |
| cfsetsnfsetf1 47247 | The mapping of the class o... |
| cfsetsnfsetfo 47248 | The mapping of the class o... |
| cfsetsnfsetf1o 47249 | The mapping of the class o... |
| fsetprcnexALT 47250 | First version of proof for... |
| fcoreslem1 47251 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 47252 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 47253 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 47254 | Lemma 4 for ~ fcores . (C... |
| fcores 47255 | Every composite function `... |
| fcoresf1lem 47256 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 47257 | If a composition is inject... |
| fcoresf1b 47258 | A composition is injective... |
| fcoresfo 47259 | If a composition is surjec... |
| fcoresfob 47260 | A composition is surjectiv... |
| fcoresf1ob 47261 | A composition is bijective... |
| f1cof1blem 47262 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 47263 | The composition of three b... |
| 3f1oss2 47264 | The composition of three b... |
| f1cof1b 47265 | If the range of ` F ` equa... |
| funfocofob 47266 | If the domain of a functio... |
| fnfocofob 47267 | If the domain of a functio... |
| focofob 47268 | If the domain of a functio... |
| f1ocof1ob 47269 | If the range of ` F ` equa... |
| f1ocof1ob2 47270 | If the range of ` F ` equa... |
| aiotajust 47272 | Soundness justification th... |
| dfaiota2 47274 | Alternate definition of th... |
| reuabaiotaiota 47275 | The iota and the alternate... |
| reuaiotaiota 47276 | The iota and the alternate... |
| aiotaexb 47277 | The alternate iota over a ... |
| aiotavb 47278 | The alternate iota over a ... |
| aiotaint 47279 | This is to ~ df-aiota what... |
| dfaiota3 47280 | Alternate definition of ` ... |
| iotan0aiotaex 47281 | If the iota over a wff ` p... |
| aiotaexaiotaiota 47282 | The alternate iota over a ... |
| aiotaval 47283 | Theorem 8.19 in [Quine] p.... |
| aiota0def 47284 | Example for a defined alte... |
| aiota0ndef 47285 | Example for an undefined a... |
| r19.32 47286 | Theorem 19.32 of [Margaris... |
| rexsb 47287 | An equivalent expression f... |
| rexrsb 47288 | An equivalent expression f... |
| 2rexsb 47289 | An equivalent expression f... |
| 2rexrsb 47290 | An equivalent expression f... |
| cbvral2 47291 | Change bound variables of ... |
| cbvrex2 47292 | Change bound variables of ... |
| ralndv1 47293 | Example for a theorem abou... |
| ralndv2 47294 | Second example for a theor... |
| reuf1odnf 47295 | There is exactly one eleme... |
| reuf1od 47296 | There is exactly one eleme... |
| euoreqb 47297 | There is a set which is eq... |
| 2reu3 47298 | Double restricted existent... |
| 2reu7 47299 | Two equivalent expressions... |
| 2reu8 47300 | Two equivalent expressions... |
| 2reu8i 47301 | Implication of a double re... |
| 2reuimp0 47302 | Implication of a double re... |
| 2reuimp 47303 | Implication of a double re... |
| ralbinrald 47310 | Elemination of a restricte... |
| nvelim 47311 | If a class is the universa... |
| alneu 47312 | If a statement holds for a... |
| eu2ndop1stv 47313 | If there is a unique secon... |
| dfateq12d 47314 | Equality deduction for "de... |
| nfdfat 47315 | Bound-variable hypothesis ... |
| dfdfat2 47316 | Alternate definition of th... |
| fundmdfat 47317 | A function is defined at a... |
| dfatprc 47318 | A function is not defined ... |
| dfatelrn 47319 | The value of a function ` ... |
| dfafv2 47320 | Alternative definition of ... |
| afveq12d 47321 | Equality deduction for fun... |
| afveq1 47322 | Equality theorem for funct... |
| afveq2 47323 | Equality theorem for funct... |
| nfafv 47324 | Bound-variable hypothesis ... |
| csbafv12g 47325 | Move class substitution in... |
| afvfundmfveq 47326 | If a class is a function r... |
| afvnfundmuv 47327 | If a set is not in the dom... |
| ndmafv 47328 | The value of a class outsi... |
| afvvdm 47329 | If the function value of a... |
| nfunsnafv 47330 | If the restriction of a cl... |
| afvvfunressn 47331 | If the function value of a... |
| afvprc 47332 | A function's value at a pr... |
| afvvv 47333 | If a function's value at a... |
| afvpcfv0 47334 | If the value of the altern... |
| afvnufveq 47335 | The value of the alternati... |
| afvvfveq 47336 | The value of the alternati... |
| afv0fv0 47337 | If the value of the altern... |
| afvfvn0fveq 47338 | If the function's value at... |
| afv0nbfvbi 47339 | The function's value at an... |
| afvfv0bi 47340 | The function's value at an... |
| afveu 47341 | The value of a function at... |
| fnbrafvb 47342 | Equivalence of function va... |
| fnopafvb 47343 | Equivalence of function va... |
| funbrafvb 47344 | Equivalence of function va... |
| funopafvb 47345 | Equivalence of function va... |
| funbrafv 47346 | The second argument of a b... |
| funbrafv2b 47347 | Function value in terms of... |
| dfafn5a 47348 | Representation of a functi... |
| dfafn5b 47349 | Representation of a functi... |
| fnrnafv 47350 | The range of a function ex... |
| afvelrnb 47351 | A member of a function's r... |
| afvelrnb0 47352 | A member of a function's r... |
| dfaimafn 47353 | Alternate definition of th... |
| dfaimafn2 47354 | Alternate definition of th... |
| afvelima 47355 | Function value in an image... |
| afvelrn 47356 | A function's value belongs... |
| fnafvelrn 47357 | A function's value belongs... |
| fafvelcdm 47358 | A function's value belongs... |
| ffnafv 47359 | A function maps to a class... |
| afvres 47360 | The value of a restricted ... |
| tz6.12-afv 47361 | Function value. Theorem 6... |
| tz6.12-1-afv 47362 | Function value (Theorem 6.... |
| dmfcoafv 47363 | Domains of a function comp... |
| afvco2 47364 | Value of a function compos... |
| rlimdmafv 47365 | Two ways to express that a... |
| aoveq123d 47366 | Equality deduction for ope... |
| nfaov 47367 | Bound-variable hypothesis ... |
| csbaovg 47368 | Move class substitution in... |
| aovfundmoveq 47369 | If a class is a function r... |
| aovnfundmuv 47370 | If an ordered pair is not ... |
| ndmaov 47371 | The value of an operation ... |
| ndmaovg 47372 | The value of an operation ... |
| aovvdm 47373 | If the operation value of ... |
| nfunsnaov 47374 | If the restriction of a cl... |
| aovvfunressn 47375 | If the operation value of ... |
| aovprc 47376 | The value of an operation ... |
| aovrcl 47377 | Reverse closure for an ope... |
| aovpcov0 47378 | If the alternative value o... |
| aovnuoveq 47379 | The alternative value of t... |
| aovvoveq 47380 | The alternative value of t... |
| aov0ov0 47381 | If the alternative value o... |
| aovovn0oveq 47382 | If the operation's value a... |
| aov0nbovbi 47383 | The operation's value on a... |
| aovov0bi 47384 | The operation's value on a... |
| rspceaov 47385 | A frequently used special ... |
| fnotaovb 47386 | Equivalence of operation v... |
| ffnaov 47387 | An operation maps to a cla... |
| faovcl 47388 | Closure law for an operati... |
| aovmpt4g 47389 | Value of a function given ... |
| aoprssdm 47390 | Domain of closure of an op... |
| ndmaovcl 47391 | The "closure" of an operat... |
| ndmaovrcl 47392 | Reverse closure law, in co... |
| ndmaovcom 47393 | Any operation is commutati... |
| ndmaovass 47394 | Any operation is associati... |
| ndmaovdistr 47395 | Any operation is distribut... |
| dfatafv2iota 47398 | If a function is defined a... |
| ndfatafv2 47399 | The alternate function val... |
| ndfatafv2undef 47400 | The alternate function val... |
| dfatafv2ex 47401 | The alternate function val... |
| afv2ex 47402 | The alternate function val... |
| afv2eq12d 47403 | Equality deduction for fun... |
| afv2eq1 47404 | Equality theorem for funct... |
| afv2eq2 47405 | Equality theorem for funct... |
| nfafv2 47406 | Bound-variable hypothesis ... |
| csbafv212g 47407 | Move class substitution in... |
| fexafv2ex 47408 | The alternate function val... |
| ndfatafv2nrn 47409 | The alternate function val... |
| ndmafv2nrn 47410 | The value of a class outsi... |
| funressndmafv2rn 47411 | The alternate function val... |
| afv2ndefb 47412 | Two ways to say that an al... |
| nfunsnafv2 47413 | If the restriction of a cl... |
| afv2prc 47414 | A function's value at a pr... |
| dfatafv2rnb 47415 | The alternate function val... |
| afv2orxorb 47416 | If a set is in the range o... |
| dmafv2rnb 47417 | The alternate function val... |
| fundmafv2rnb 47418 | The alternate function val... |
| afv2elrn 47419 | An alternate function valu... |
| afv20defat 47420 | If the alternate function ... |
| fnafv2elrn 47421 | An alternate function valu... |
| fafv2elcdm 47422 | An alternate function valu... |
| fafv2elrnb 47423 | An alternate function valu... |
| fcdmvafv2v 47424 | If the codomain of a funct... |
| tz6.12-2-afv2 47425 | Function value when ` F ` ... |
| afv2eu 47426 | The value of a function at... |
| afv2res 47427 | The value of a restricted ... |
| tz6.12-afv2 47428 | Function value (Theorem 6.... |
| tz6.12-1-afv2 47429 | Function value (Theorem 6.... |
| tz6.12c-afv2 47430 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 47431 | Corollary of Theorem 6.12(... |
| funressnbrafv2 47432 | The second argument of a b... |
| dfatbrafv2b 47433 | Equivalence of function va... |
| dfatopafv2b 47434 | Equivalence of function va... |
| funbrafv2 47435 | The second argument of a b... |
| fnbrafv2b 47436 | Equivalence of function va... |
| fnopafv2b 47437 | Equivalence of function va... |
| funbrafv22b 47438 | Equivalence of function va... |
| funopafv2b 47439 | Equivalence of function va... |
| dfatsnafv2 47440 | Singleton of function valu... |
| dfafv23 47441 | A definition of function v... |
| dfatdmfcoafv2 47442 | Domain of a function compo... |
| dfatcolem 47443 | Lemma for ~ dfatco . (Con... |
| dfatco 47444 | The predicate "defined at"... |
| afv2co2 47445 | Value of a function compos... |
| rlimdmafv2 47446 | Two ways to express that a... |
| dfafv22 47447 | Alternate definition of ` ... |
| afv2ndeffv0 47448 | If the alternate function ... |
| dfatafv2eqfv 47449 | If a function is defined a... |
| afv2rnfveq 47450 | If the alternate function ... |
| afv20fv0 47451 | If the alternate function ... |
| afv2fvn0fveq 47452 | If the function's value at... |
| afv2fv0 47453 | If the function's value at... |
| afv2fv0b 47454 | The function's value at an... |
| afv2fv0xorb 47455 | If a set is in the range o... |
| an4com24 47456 | Rearrangement of 4 conjunc... |
| 3an4ancom24 47457 | Commutative law for a conj... |
| 4an21 47458 | Rearrangement of 4 conjunc... |
| dfnelbr2 47461 | Alternate definition of th... |
| nelbr 47462 | The binary relation of a s... |
| nelbrim 47463 | If a set is related to ano... |
| nelbrnel 47464 | A set is related to anothe... |
| nelbrnelim 47465 | If a set is related to ano... |
| ralralimp 47466 | Selecting one of two alter... |
| otiunsndisjX 47467 | The union of singletons co... |
| fvifeq 47468 | Equality of function value... |
| rnfdmpr 47469 | The range of a one-to-one ... |
| imarnf1pr 47470 | The image of the range of ... |
| funop1 47471 | A function is an ordered p... |
| fun2dmnopgexmpl 47472 | A function with a domain c... |
| opabresex0d 47473 | A collection of ordered pa... |
| opabbrfex0d 47474 | A collection of ordered pa... |
| opabresexd 47475 | A collection of ordered pa... |
| opabbrfexd 47476 | A collection of ordered pa... |
| f1oresf1orab 47477 | Build a bijection by restr... |
| f1oresf1o 47478 | Build a bijection by restr... |
| f1oresf1o2 47479 | Build a bijection by restr... |
| fvmptrab 47480 | Value of a function mappin... |
| fvmptrabdm 47481 | Value of a function mappin... |
| cnambpcma 47482 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 47483 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 47484 | The sum of two complex num... |
| leaddsuble 47485 | Addition and subtraction o... |
| 2leaddle2 47486 | If two real numbers are le... |
| ltnltne 47487 | Variant of trichotomy law ... |
| p1lep2 47488 | A real number increasd by ... |
| ltsubsubaddltsub 47489 | If the result of subtracti... |
| zm1nn 47490 | An integer minus 1 is posi... |
| readdcnnred 47491 | The sum of a real number a... |
| resubcnnred 47492 | The difference of a real n... |
| recnmulnred 47493 | The product of a real numb... |
| cndivrenred 47494 | The quotient of an imagina... |
| sqrtnegnre 47495 | The square root of a negat... |
| nn0resubcl 47496 | Closure law for subtractio... |
| zgeltp1eq 47497 | If an integer is between a... |
| 1t10e1p1e11 47498 | 11 is 1 times 10 to the po... |
| deccarry 47499 | Add 1 to a 2 digit number ... |
| eluzge0nn0 47500 | If an integer is greater t... |
| nltle2tri 47501 | Negated extended trichotom... |
| ssfz12 47502 | Subset relationship for fi... |
| elfz2z 47503 | Membership of an integer i... |
| 2elfz3nn0 47504 | If there are two elements ... |
| fz0addcom 47505 | The addition of two member... |
| 2elfz2melfz 47506 | If the sum of two integers... |
| fz0addge0 47507 | The sum of two integers in... |
| elfzlble 47508 | Membership of an integer i... |
| elfzelfzlble 47509 | Membership of an element o... |
| fzopred 47510 | Join a predecessor to the ... |
| fzopredsuc 47511 | Join a predecessor and a s... |
| 1fzopredsuc 47512 | Join 0 and a successor to ... |
| el1fzopredsuc 47513 | An element of an open inte... |
| subsubelfzo0 47514 | Subtracting a difference f... |
| 2ffzoeq 47515 | Two functions over a half-... |
| 2ltceilhalf 47516 | The ceiling of half of an ... |
| ceilhalfgt1 47517 | The ceiling of half of an ... |
| ceilhalfelfzo1 47518 | A positive integer less th... |
| gpgedgvtx1lem 47519 | Lemma for ~ gpgedgvtx1 . ... |
| 2tceilhalfelfzo1 47520 | Two times a positive integ... |
| ceilbi 47521 | A condition equivalent to ... |
| ceilhalf1 47522 | The ceiling of one half is... |
| rehalfge1 47523 | Half of a real number grea... |
| ceilhalfnn 47524 | The ceiling of half of a p... |
| 1elfzo1ceilhalf1 47525 | 1 is in the half-open inte... |
| fldivmod 47526 | Expressing the floor of a ... |
| ceildivmod 47527 | Expressing the ceiling of ... |
| ceil5half3 47528 | The ceiling of half of 5 i... |
| submodaddmod 47529 | Subtraction and addition m... |
| difltmodne 47530 | Two nonnegative integers a... |
| zplusmodne 47531 | A nonnegative integer is n... |
| addmodne 47532 | The sum of a nonnegative i... |
| plusmod5ne 47533 | A nonnegative integer is n... |
| zp1modne 47534 | An integer is not itself p... |
| p1modne 47535 | A nonnegative integer is n... |
| m1modne 47536 | A nonnegative integer is n... |
| minusmod5ne 47537 | A nonnegative integer is n... |
| submodlt 47538 | The difference of an eleme... |
| submodneaddmod 47539 | An integer minus ` B ` is ... |
| m1modnep2mod 47540 | A nonnegative integer minu... |
| minusmodnep2tmod 47541 | A nonnegative integer minu... |
| m1mod0mod1 47542 | An integer decreased by 1 ... |
| elmod2 47543 | An integer modulo 2 is eit... |
| mod0mul 47544 | If an integer is 0 modulo ... |
| modn0mul 47545 | If an integer is not 0 mod... |
| m1modmmod 47546 | An integer decreased by 1 ... |
| difmodm1lt 47547 | The difference between an ... |
| 8mod5e3 47548 | 8 modulo 5 is 3. (Contrib... |
| modmkpkne 47549 | If an integer minus a cons... |
| modmknepk 47550 | A nonnegative integer less... |
| modlt0b 47551 | An integer with an absolut... |
| mod2addne 47552 | The sums of a nonnegative ... |
| modm1nep1 47553 | A nonnegative integer less... |
| modm2nep1 47554 | A nonnegative integer less... |
| modp2nep1 47555 | A nonnegative integer less... |
| modm1nep2 47556 | A nonnegative integer less... |
| modm1nem2 47557 | A nonnegative integer less... |
| modm1p1ne 47558 | If an integer minus one eq... |
| smonoord 47559 | Ordering relation for a st... |
| fsummsndifre 47560 | A finite sum with one of i... |
| fsumsplitsndif 47561 | Separate out a term in a f... |
| fsummmodsndifre 47562 | A finite sum of summands m... |
| fsummmodsnunz 47563 | A finite sum of summands m... |
| setsidel 47564 | The injected slot is an el... |
| setsnidel 47565 | The injected slot is an el... |
| setsv 47566 | The value of the structure... |
| preimafvsnel 47567 | The preimage of a function... |
| preimafvn0 47568 | The preimage of a function... |
| uniimafveqt 47569 | The union of the image of ... |
| uniimaprimaeqfv 47570 | The union of the image of ... |
| setpreimafvex 47571 | The class ` P ` of all pre... |
| elsetpreimafvb 47572 | The characterization of an... |
| elsetpreimafv 47573 | An element of the class ` ... |
| elsetpreimafvssdm 47574 | An element of the class ` ... |
| fvelsetpreimafv 47575 | There is an element in a p... |
| preimafvelsetpreimafv 47576 | The preimage of a function... |
| preimafvsspwdm 47577 | The class ` P ` of all pre... |
| 0nelsetpreimafv 47578 | The empty set is not an el... |
| elsetpreimafvbi 47579 | An element of the preimage... |
| elsetpreimafveqfv 47580 | The elements of the preima... |
| eqfvelsetpreimafv 47581 | If an element of the domai... |
| elsetpreimafvrab 47582 | An element of the preimage... |
| imaelsetpreimafv 47583 | The image of an element of... |
| uniimaelsetpreimafv 47584 | The union of the image of ... |
| elsetpreimafveq 47585 | If two preimages of functi... |
| fundcmpsurinjlem1 47586 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 47587 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 47588 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 47589 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 47590 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 47591 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 47592 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 47593 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 47594 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 47595 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 47596 | Every function ` F : A -->... |
| fundcmpsurinj 47597 | Every function ` F : A -->... |
| fundcmpsurbijinj 47598 | Every function ` F : A -->... |
| fundcmpsurinjimaid 47599 | Every function ` F : A -->... |
| fundcmpsurinjALT 47600 | Alternate proof of ~ fundc... |
| iccpval 47603 | Partition consisting of a ... |
| iccpart 47604 | A special partition. Corr... |
| iccpartimp 47605 | Implications for a class b... |
| iccpartres 47606 | The restriction of a parti... |
| iccpartxr 47607 | If there is a partition, t... |
| iccpartgtprec 47608 | If there is a partition, t... |
| iccpartipre 47609 | If there is a partition, t... |
| iccpartiltu 47610 | If there is a partition, t... |
| iccpartigtl 47611 | If there is a partition, t... |
| iccpartlt 47612 | If there is a partition, t... |
| iccpartltu 47613 | If there is a partition, t... |
| iccpartgtl 47614 | If there is a partition, t... |
| iccpartgt 47615 | If there is a partition, t... |
| iccpartleu 47616 | If there is a partition, t... |
| iccpartgel 47617 | If there is a partition, t... |
| iccpartrn 47618 | If there is a partition, t... |
| iccpartf 47619 | The range of the partition... |
| iccpartel 47620 | If there is a partition, t... |
| iccelpart 47621 | An element of any partitio... |
| iccpartiun 47622 | A half-open interval of ex... |
| icceuelpartlem 47623 | Lemma for ~ icceuelpart . ... |
| icceuelpart 47624 | An element of a partitione... |
| iccpartdisj 47625 | The segments of a partitio... |
| iccpartnel 47626 | A point of a partition is ... |
| fargshiftfv 47627 | If a class is a function, ... |
| fargshiftf 47628 | If a class is a function, ... |
| fargshiftf1 47629 | If a function is 1-1, then... |
| fargshiftfo 47630 | If a function is onto, the... |
| fargshiftfva 47631 | The values of a shifted fu... |
| lswn0 47632 | The last symbol of a nonem... |
| nfich1 47635 | The first interchangeable ... |
| nfich2 47636 | The second interchangeable... |
| ichv 47637 | Setvar variables are inter... |
| ichf 47638 | Setvar variables are inter... |
| ichid 47639 | A setvar variable is alway... |
| icht 47640 | A theorem is interchangeab... |
| ichbidv 47641 | Formula building rule for ... |
| ichcircshi 47642 | The setvar variables are i... |
| ichan 47643 | If two setvar variables ar... |
| ichn 47644 | Negation does not affect i... |
| ichim 47645 | Formula building rule for ... |
| dfich2 47646 | Alternate definition of th... |
| ichcom 47647 | The interchangeability of ... |
| ichbi12i 47648 | Equivalence for interchang... |
| icheqid 47649 | In an equality for the sam... |
| icheq 47650 | In an equality of setvar v... |
| ichnfimlem 47651 | Lemma for ~ ichnfim : A s... |
| ichnfim 47652 | If in an interchangeabilit... |
| ichnfb 47653 | If ` x ` and ` y ` are int... |
| ichal 47654 | Move a universal quantifie... |
| ich2al 47655 | Two setvar variables are a... |
| ich2ex 47656 | Two setvar variables are a... |
| ichexmpl1 47657 | Example for interchangeabl... |
| ichexmpl2 47658 | Example for interchangeabl... |
| ich2exprop 47659 | If the setvar variables ar... |
| ichnreuop 47660 | If the setvar variables ar... |
| ichreuopeq 47661 | If the setvar variables ar... |
| sprid 47662 | Two identical representati... |
| elsprel 47663 | An unordered pair is an el... |
| spr0nelg 47664 | The empty set is not an el... |
| sprval 47667 | The set of all unordered p... |
| sprvalpw 47668 | The set of all unordered p... |
| sprssspr 47669 | The set of all unordered p... |
| spr0el 47670 | The empty set is not an un... |
| sprvalpwn0 47671 | The set of all unordered p... |
| sprel 47672 | An element of the set of a... |
| prssspr 47673 | An element of a subset of ... |
| prelspr 47674 | An unordered pair of eleme... |
| prsprel 47675 | The elements of a pair fro... |
| prsssprel 47676 | The elements of a pair fro... |
| sprvalpwle2 47677 | The set of all unordered p... |
| sprsymrelfvlem 47678 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 47679 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 47680 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 47681 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 47682 | The value of the function ... |
| sprsymrelf 47683 | The mapping ` F ` is a fun... |
| sprsymrelf1 47684 | The mapping ` F ` is a one... |
| sprsymrelfo 47685 | The mapping ` F ` is a fun... |
| sprsymrelf1o 47686 | The mapping ` F ` is a bij... |
| sprbisymrel 47687 | There is a bijection betwe... |
| sprsymrelen 47688 | The class ` P ` of subsets... |
| prpair 47689 | Characterization of a prop... |
| prproropf1olem0 47690 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 47691 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 47692 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 47693 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 47694 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 47695 | There is a bijection betwe... |
| prproropen 47696 | The set of proper pairs an... |
| prproropreud 47697 | There is exactly one order... |
| pairreueq 47698 | Two equivalent representat... |
| paireqne 47699 | Two sets are not equal iff... |
| prprval 47702 | The set of all proper unor... |
| prprvalpw 47703 | The set of all proper unor... |
| prprelb 47704 | An element of the set of a... |
| prprelprb 47705 | A set is an element of the... |
| prprspr2 47706 | The set of all proper unor... |
| prprsprreu 47707 | There is a unique proper u... |
| prprreueq 47708 | There is a unique proper u... |
| sbcpr 47709 | The proper substitution of... |
| reupr 47710 | There is a unique unordere... |
| reuprpr 47711 | There is a unique proper u... |
| poprelb 47712 | Equality for unordered pai... |
| 2exopprim 47713 | The existence of an ordere... |
| reuopreuprim 47714 | There is a unique unordere... |
| fmtno 47717 | The ` N ` th Fermat number... |
| fmtnoge3 47718 | Each Fermat number is grea... |
| fmtnonn 47719 | Each Fermat number is a po... |
| fmtnom1nn 47720 | A Fermat number minus one ... |
| fmtnoodd 47721 | Each Fermat number is odd.... |
| fmtnorn 47722 | A Fermat number is a funct... |
| fmtnof1 47723 | The enumeration of the Fer... |
| fmtnoinf 47724 | The set of Fermat numbers ... |
| fmtnorec1 47725 | The first recurrence relat... |
| sqrtpwpw2p 47726 | The floor of the square ro... |
| fmtnosqrt 47727 | The floor of the square ro... |
| fmtno0 47728 | The ` 0 ` th Fermat number... |
| fmtno1 47729 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 47730 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 47731 | The second recurrence rela... |
| fmtnodvds 47732 | Any Fermat number divides ... |
| goldbachthlem1 47733 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 47734 | Lemma 2 for ~ goldbachth .... |
| goldbachth 47735 | Goldbach's theorem: Two d... |
| fmtnorec3 47736 | The third recurrence relat... |
| fmtnorec4 47737 | The fourth recurrence rela... |
| fmtno2 47738 | The ` 2 ` nd Fermat number... |
| fmtno3 47739 | The ` 3 ` rd Fermat number... |
| fmtno4 47740 | The ` 4 ` th Fermat number... |
| fmtno5lem1 47741 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 47742 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 47743 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 47744 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 47745 | The ` 5 ` th Fermat number... |
| fmtno0prm 47746 | The ` 0 ` th Fermat number... |
| fmtno1prm 47747 | The ` 1 ` st Fermat number... |
| fmtno2prm 47748 | The ` 2 ` nd Fermat number... |
| 257prm 47749 | 257 is a prime number (the... |
| fmtno3prm 47750 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 47751 | Any power of two is coprim... |
| fmtnoprmfac1lem 47752 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 47753 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 47754 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 47755 | Divisor of Fermat number (... |
| fmtnofac2lem 47756 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 47757 | Divisor of Fermat number (... |
| fmtnofac1 47758 | Divisor of Fermat number (... |
| fmtno4sqrt 47759 | The floor of the square ro... |
| fmtno4prmfac 47760 | If P was a (prime) factor ... |
| fmtno4prmfac193 47761 | If P was a (prime) factor ... |
| fmtno4nprmfac193 47762 | 193 is not a (prime) facto... |
| fmtno4prm 47763 | The ` 4 `-th Fermat number... |
| 65537prm 47764 | 65537 is a prime number (t... |
| fmtnofz04prm 47765 | The first five Fermat numb... |
| fmtnole4prm 47766 | The first five Fermat numb... |
| fmtno5faclem1 47767 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 47768 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 47769 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 47770 | The factorization of the `... |
| fmtno5nprm 47771 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 47772 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 47773 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 47774 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 47775 | The mapping of a Fermat nu... |
| prminf2 47776 | The set of prime numbers i... |
| 2pwp1prm 47777 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 47778 | Every prime number of the ... |
| m2prm 47779 | The second Mersenne number... |
| m3prm 47780 | The third Mersenne number ... |
| flsqrt 47781 | A condition equivalent to ... |
| flsqrt5 47782 | The floor of the square ro... |
| 3ndvds4 47783 | 3 does not divide 4. (Con... |
| 139prmALT 47784 | 139 is a prime number. In... |
| 31prm 47785 | 31 is a prime number. In ... |
| m5prm 47786 | The fifth Mersenne number ... |
| 127prm 47787 | 127 is a prime number. (C... |
| m7prm 47788 | The seventh Mersenne numbe... |
| m11nprm 47789 | The eleventh Mersenne numb... |
| mod42tp1mod8 47790 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 47791 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 47792 | If ` P ` is a Sophie Germa... |
| lighneallem1 47793 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 47794 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 47795 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 47796 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 47797 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 47798 | Lemma 3 for ~ lighneal . ... |
| lighneal 47799 | If a power of a prime ` P ... |
| modexp2m1d 47800 | The square of an integer w... |
| proththdlem 47801 | Lemma for ~ proththd . (C... |
| proththd 47802 | Proth's theorem (1878). I... |
| 5tcu2e40 47803 | 5 times the cube of 2 is 4... |
| 3exp4mod41 47804 | 3 to the fourth power is -... |
| 41prothprmlem1 47805 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 47806 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 47807 | 41 is a _Proth prime_. (C... |
| quad1 47808 | A condition for a quadrati... |
| requad01 47809 | A condition for a quadrati... |
| requad1 47810 | A condition for a quadrati... |
| requad2 47811 | A condition for a quadrati... |
| iseven 47816 | The predicate "is an even ... |
| isodd 47817 | The predicate "is an odd n... |
| evenz 47818 | An even number is an integ... |
| oddz 47819 | An odd number is an intege... |
| evendiv2z 47820 | The result of dividing an ... |
| oddp1div2z 47821 | The result of dividing an ... |
| oddm1div2z 47822 | The result of dividing an ... |
| isodd2 47823 | The predicate "is an odd n... |
| dfodd2 47824 | Alternate definition for o... |
| dfodd6 47825 | Alternate definition for o... |
| dfeven4 47826 | Alternate definition for e... |
| evenm1odd 47827 | The predecessor of an even... |
| evenp1odd 47828 | The successor of an even n... |
| oddp1eveni 47829 | The successor of an odd nu... |
| oddm1eveni 47830 | The predecessor of an odd ... |
| evennodd 47831 | An even number is not an o... |
| oddneven 47832 | An odd number is not an ev... |
| enege 47833 | The negative of an even nu... |
| onego 47834 | The negative of an odd num... |
| m1expevenALTV 47835 | Exponentiation of -1 by an... |
| m1expoddALTV 47836 | Exponentiation of -1 by an... |
| dfeven2 47837 | Alternate definition for e... |
| dfodd3 47838 | Alternate definition for o... |
| iseven2 47839 | The predicate "is an even ... |
| isodd3 47840 | The predicate "is an odd n... |
| 2dvdseven 47841 | 2 divides an even number. ... |
| m2even 47842 | A multiple of 2 is an even... |
| 2ndvdsodd 47843 | 2 does not divide an odd n... |
| 2dvdsoddp1 47844 | 2 divides an odd number in... |
| 2dvdsoddm1 47845 | 2 divides an odd number de... |
| dfeven3 47846 | Alternate definition for e... |
| dfodd4 47847 | Alternate definition for o... |
| dfodd5 47848 | Alternate definition for o... |
| zefldiv2ALTV 47849 | The floor of an even numbe... |
| zofldiv2ALTV 47850 | The floor of an odd number... |
| oddflALTV 47851 | Odd number representation ... |
| iseven5 47852 | The predicate "is an even ... |
| isodd7 47853 | The predicate "is an odd n... |
| dfeven5 47854 | Alternate definition for e... |
| dfodd7 47855 | Alternate definition for o... |
| gcd2odd1 47856 | The greatest common diviso... |
| zneoALTV 47857 | No even integer equals an ... |
| zeoALTV 47858 | An integer is even or odd.... |
| zeo2ALTV 47859 | An integer is even or odd ... |
| nneoALTV 47860 | A positive integer is even... |
| nneoiALTV 47861 | A positive integer is even... |
| odd2np1ALTV 47862 | An integer is odd iff it i... |
| oddm1evenALTV 47863 | An integer is odd iff its ... |
| oddp1evenALTV 47864 | An integer is odd iff its ... |
| oexpnegALTV 47865 | The exponential of the neg... |
| oexpnegnz 47866 | The exponential of the neg... |
| bits0ALTV 47867 | Value of the zeroth bit. ... |
| bits0eALTV 47868 | The zeroth bit of an even ... |
| bits0oALTV 47869 | The zeroth bit of an odd n... |
| divgcdoddALTV 47870 | Either ` A / ( A gcd B ) `... |
| opoeALTV 47871 | The sum of two odds is eve... |
| opeoALTV 47872 | The sum of an odd and an e... |
| omoeALTV 47873 | The difference of two odds... |
| omeoALTV 47874 | The difference of an odd a... |
| oddprmALTV 47875 | A prime not equal to ` 2 `... |
| 0evenALTV 47876 | 0 is an even number. (Con... |
| 0noddALTV 47877 | 0 is not an odd number. (... |
| 1oddALTV 47878 | 1 is an odd number. (Cont... |
| 1nevenALTV 47879 | 1 is not an even number. ... |
| 2evenALTV 47880 | 2 is an even number. (Con... |
| 2noddALTV 47881 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 47882 | An odd nonnegative integer... |
| nnoALTV 47883 | An alternate characterizat... |
| nn0oALTV 47884 | An alternate characterizat... |
| nn0e 47885 | An alternate characterizat... |
| nneven 47886 | An alternate characterizat... |
| nn0onn0exALTV 47887 | For each odd nonnegative i... |
| nn0enn0exALTV 47888 | For each even nonnegative ... |
| nnennexALTV 47889 | For each even positive int... |
| nnpw2evenALTV 47890 | 2 to the power of a positi... |
| epoo 47891 | The sum of an even and an ... |
| emoo 47892 | The difference of an even ... |
| epee 47893 | The sum of two even number... |
| emee 47894 | The difference of two even... |
| evensumeven 47895 | If a summand is even, the ... |
| 3odd 47896 | 3 is an odd number. (Cont... |
| 4even 47897 | 4 is an even number. (Con... |
| 5odd 47898 | 5 is an odd number. (Cont... |
| 6even 47899 | 6 is an even number. (Con... |
| 7odd 47900 | 7 is an odd number. (Cont... |
| 8even 47901 | 8 is an even number. (Con... |
| evenprm2 47902 | A prime number is even iff... |
| oddprmne2 47903 | Every prime number not bei... |
| oddprmuzge3 47904 | A prime number which is od... |
| evenltle 47905 | If an even number is great... |
| odd2prm2 47906 | If an odd number is the su... |
| even3prm2 47907 | If an even number is the s... |
| mogoldbblem 47908 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 47909 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 47910 | Lemma for ~ perfectALTV . ... |
| perfectALTV 47911 | The Euclid-Euler theorem, ... |
| fppr 47914 | The set of Fermat pseudopr... |
| fpprmod 47915 | The set of Fermat pseudopr... |
| fpprel 47916 | A Fermat pseudoprime to th... |
| fpprbasnn 47917 | The base of a Fermat pseud... |
| fpprnn 47918 | A Fermat pseudoprime to th... |
| fppr2odd 47919 | A Fermat pseudoprime to th... |
| 11t31e341 47920 | 341 is the product of 11 a... |
| 2exp340mod341 47921 | Eight to the eighth power ... |
| 341fppr2 47922 | 341 is the (smallest) _Pou... |
| 4fppr1 47923 | 4 is the (smallest) Fermat... |
| 8exp8mod9 47924 | Eight to the eighth power ... |
| 9fppr8 47925 | 9 is the (smallest) Fermat... |
| dfwppr 47926 | Alternate definition of a ... |
| fpprwppr 47927 | A Fermat pseudoprime to th... |
| fpprwpprb 47928 | An integer ` X ` which is ... |
| fpprel2 47929 | An alternate definition fo... |
| nfermltl8rev 47930 | Fermat's little theorem wi... |
| nfermltl2rev 47931 | Fermat's little theorem wi... |
| nfermltlrev 47932 | Fermat's little theorem re... |
| isgbe 47939 | The predicate "is an even ... |
| isgbow 47940 | The predicate "is a weak o... |
| isgbo 47941 | The predicate "is an odd G... |
| gbeeven 47942 | An even Goldbach number is... |
| gbowodd 47943 | A weak odd Goldbach number... |
| gbogbow 47944 | A (strong) odd Goldbach nu... |
| gboodd 47945 | An odd Goldbach number is ... |
| gbepos 47946 | Any even Goldbach number i... |
| gbowpos 47947 | Any weak odd Goldbach numb... |
| gbopos 47948 | Any odd Goldbach number is... |
| gbegt5 47949 | Any even Goldbach number i... |
| gbowgt5 47950 | Any weak odd Goldbach numb... |
| gbowge7 47951 | Any weak odd Goldbach numb... |
| gboge9 47952 | Any odd Goldbach number is... |
| gbege6 47953 | Any even Goldbach number i... |
| gbpart6 47954 | The Goldbach partition of ... |
| gbpart7 47955 | The (weak) Goldbach partit... |
| gbpart8 47956 | The Goldbach partition of ... |
| gbpart9 47957 | The (strong) Goldbach part... |
| gbpart11 47958 | The (strong) Goldbach part... |
| 6gbe 47959 | 6 is an even Goldbach numb... |
| 7gbow 47960 | 7 is a weak odd Goldbach n... |
| 8gbe 47961 | 8 is an even Goldbach numb... |
| 9gbo 47962 | 9 is an odd Goldbach numbe... |
| 11gbo 47963 | 11 is an odd Goldbach numb... |
| stgoldbwt 47964 | If the strong ternary Gold... |
| sbgoldbwt 47965 | If the strong binary Goldb... |
| sbgoldbst 47966 | If the strong binary Goldb... |
| sbgoldbaltlem1 47967 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 47968 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 47969 | An alternate (related to t... |
| sbgoldbb 47970 | If the strong binary Goldb... |
| sgoldbeven3prm 47971 | If the binary Goldbach con... |
| sbgoldbm 47972 | If the strong binary Goldb... |
| mogoldbb 47973 | If the modern version of t... |
| sbgoldbmb 47974 | The strong binary Goldbach... |
| sbgoldbo 47975 | If the strong binary Goldb... |
| nnsum3primes4 47976 | 4 is the sum of at most 3 ... |
| nnsum4primes4 47977 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 47978 | Every prime is "the sum of... |
| nnsum4primesprm 47979 | Every prime is "the sum of... |
| nnsum3primesgbe 47980 | Any even Goldbach number i... |
| nnsum4primesgbe 47981 | Any even Goldbach number i... |
| nnsum3primesle9 47982 | Every integer greater than... |
| nnsum4primesle9 47983 | Every integer greater than... |
| nnsum4primesodd 47984 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 47985 | If the (strong) ternary Go... |
| evengpop3 47986 | If the (weak) ternary Gold... |
| evengpoap3 47987 | If the (strong) ternary Go... |
| nnsum4primeseven 47988 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 47989 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 47990 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 47991 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 47992 | If the binary Goldbach con... |
| bgoldbtbndlem1 47993 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 47994 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 47995 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 47996 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 47997 | If the binary Goldbach con... |
| tgoldbachgtALTV 48000 | Variant of Thierry Arnoux'... |
| bgoldbachlt 48001 | The binary Goldbach conjec... |
| tgblthelfgott 48003 | The ternary Goldbach conje... |
| tgoldbachlt 48004 | The ternary Goldbach conje... |
| tgoldbach 48005 | The ternary Goldbach conje... |
| clnbgrprc0 48008 | The closed neighborhood is... |
| clnbgrcl 48009 | If a class ` X ` has at le... |
| clnbgrval 48010 | The closed neighborhood of... |
| dfclnbgr2 48011 | Alternate definition of th... |
| dfclnbgr4 48012 | Alternate definition of th... |
| elclnbgrelnbgr 48013 | An element of the closed n... |
| dfclnbgr3 48014 | Alternate definition of th... |
| clnbgrnvtx0 48015 | If a class ` X ` is not a ... |
| clnbgrel 48016 | Characterization of a memb... |
| clnbgrvtxel 48017 | Every vertex ` K ` is a me... |
| clnbgrisvtx 48018 | Every member ` N ` of the ... |
| clnbgrssvtx 48019 | The closed neighborhood of... |
| clnbgrn0 48020 | The closed neighborhood of... |
| clnbupgr 48021 | The closed neighborhood of... |
| clnbupgrel 48022 | A member of the closed nei... |
| clnbupgreli 48023 | A member of the closed nei... |
| clnbgr0vtx 48024 | In a null graph (with no v... |
| clnbgr0edg 48025 | In an empty graph (with no... |
| clnbgrsym 48026 | In a graph, the closed nei... |
| predgclnbgrel 48027 | If a (not necessarily prop... |
| clnbgredg 48028 | A vertex connected by an e... |
| clnbgrssedg 48029 | The vertices connected by ... |
| edgusgrclnbfin 48030 | The size of the closed nei... |
| clnbusgrfi 48031 | The closed neighborhood of... |
| clnbfiusgrfi 48032 | The closed neighborhood of... |
| clnbgrlevtx 48033 | The size of the closed nei... |
| dfsclnbgr2 48034 | Alternate definition of th... |
| sclnbgrel 48035 | Characterization of a memb... |
| sclnbgrelself 48036 | A vertex ` N ` is a member... |
| sclnbgrisvtx 48037 | Every member ` X ` of the ... |
| dfclnbgr5 48038 | Alternate definition of th... |
| dfnbgr5 48039 | Alternate definition of th... |
| dfnbgrss 48040 | Subset chain for different... |
| dfvopnbgr2 48041 | Alternate definition of th... |
| vopnbgrel 48042 | Characterization of a memb... |
| vopnbgrelself 48043 | A vertex ` N ` is a member... |
| dfclnbgr6 48044 | Alternate definition of th... |
| dfnbgr6 48045 | Alternate definition of th... |
| dfsclnbgr6 48046 | Alternate definition of a ... |
| dfnbgrss2 48047 | Subset chain for different... |
| isisubgr 48050 | The subgraph induced by a ... |
| isubgriedg 48051 | The edges of an induced su... |
| isubgrvtxuhgr 48052 | The subgraph induced by th... |
| isubgredgss 48053 | The edges of an induced su... |
| isubgredg 48054 | An edge of an induced subg... |
| isubgrvtx 48055 | The vertices of an induced... |
| isubgruhgr 48056 | An induced subgraph of a h... |
| isubgrsubgr 48057 | An induced subgraph of a h... |
| isubgrupgr 48058 | An induced subgraph of a p... |
| isubgrumgr 48059 | An induced subgraph of a m... |
| isubgrusgr 48060 | An induced subgraph of a s... |
| isubgr0uhgr 48061 | The subgraph induced by an... |
| grimfn 48067 | The graph isomorphism func... |
| grimdmrel 48068 | The domain of the graph is... |
| isgrim 48070 | An isomorphism of graphs i... |
| grimprop 48071 | Properties of an isomorphi... |
| grimf1o 48072 | An isomorphism of graphs i... |
| grimidvtxedg 48073 | The identity relation rest... |
| grimid 48074 | The identity relation rest... |
| grimuhgr 48075 | If there is a graph isomor... |
| grimcnv 48076 | The converse of a graph is... |
| grimco 48077 | The composition of graph i... |
| uhgrimedgi 48078 | An isomorphism between gra... |
| uhgrimedg 48079 | An isomorphism between gra... |
| uhgrimprop 48080 | An isomorphism between hyp... |
| isuspgrim0lem 48081 | An isomorphism of simple p... |
| isuspgrim0 48082 | An isomorphism of simple p... |
| isuspgrimlem 48083 | Lemma for ~ isuspgrim . (... |
| isuspgrim 48084 | A class is an isomorphism ... |
| upgrimwlklem1 48085 | Lemma 1 for ~ upgrimwlk an... |
| upgrimwlklem2 48086 | Lemma 2 for ~ upgrimwlk . ... |
| upgrimwlklem3 48087 | Lemma 3 for ~ upgrimwlk . ... |
| upgrimwlklem4 48088 | Lemma 4 for ~ upgrimwlk . ... |
| upgrimwlklem5 48089 | Lemma 5 for ~ upgrimwlk . ... |
| upgrimwlk 48090 | Graph isomorphisms between... |
| upgrimwlklen 48091 | Graph isomorphisms between... |
| upgrimtrlslem1 48092 | Lemma 1 for ~ upgrimtrls .... |
| upgrimtrlslem2 48093 | Lemma 2 for ~ upgrimtrls .... |
| upgrimtrls 48094 | Graph isomorphisms between... |
| upgrimpthslem1 48095 | Lemma 1 for ~ upgrimpths .... |
| upgrimpthslem2 48096 | Lemma 2 for ~ upgrimpths .... |
| upgrimpths 48097 | Graph isomorphisms between... |
| upgrimspths 48098 | Graph isomorphisms between... |
| upgrimcycls 48099 | Graph isomorphisms between... |
| brgric 48100 | The relation "is isomorphi... |
| brgrici 48101 | Prove that two graphs are ... |
| gricrcl 48102 | Reverse closure of the "is... |
| dfgric2 48103 | Alternate, explicit defini... |
| gricbri 48104 | Implications of two graphs... |
| gricushgr 48105 | The "is isomorphic to" rel... |
| gricuspgr 48106 | The "is isomorphic to" rel... |
| gricrel 48107 | The "is isomorphic to" rel... |
| gricref 48108 | Graph isomorphism is refle... |
| gricsym 48109 | Graph isomorphism is symme... |
| gricsymb 48110 | Graph isomorphism is symme... |
| grictr 48111 | Graph isomorphism is trans... |
| gricer 48112 | Isomorphism is an equivale... |
| gricen 48113 | Isomorphic graphs have equ... |
| opstrgric 48114 | A graph represented as an ... |
| ushggricedg 48115 | A simple hypergraph (with ... |
| cycldlenngric 48116 | Two simple pseudographs ar... |
| isubgrgrim 48117 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 48118 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 48119 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 48120 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 48121 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 48122 | Graph isomorphisms between... |
| grimedg 48123 | For two isomorphic graphs,... |
| grimedgi 48124 | Graph isomorphisms map edg... |
| grtriproplem 48127 | Lemma for ~ grtriprop . (... |
| grtri 48128 | The triangles in a graph. ... |
| grtriprop 48129 | The properties of a triang... |
| grtrif1o 48130 | Any bijection onto a trian... |
| isgrtri 48131 | A triangle in a graph. (C... |
| grtrissvtx 48132 | A triangle is a subset of ... |
| grtriclwlk3 48133 | A triangle induces a close... |
| cycl3grtrilem 48134 | Lemma for ~ cycl3grtri . ... |
| cycl3grtri 48135 | The vertices of a cycle of... |
| grtrimap 48136 | Conditions for mapping tri... |
| grimgrtri 48137 | Graph isomorphisms map tri... |
| usgrgrtrirex 48138 | Conditions for a simple gr... |
| stgrfv 48141 | The star graph S_N. (Contr... |
| stgrvtx 48142 | The vertices of the star g... |
| stgriedg 48143 | The indexed edges of the s... |
| stgredg 48144 | The edges of the star grap... |
| stgredgel 48145 | An edge of the star graph ... |
| stgredgiun 48146 | The edges of the star grap... |
| stgrusgra 48147 | The star graph S_N is a si... |
| stgr0 48148 | The star graph S_0 consist... |
| stgr1 48149 | The star graph S_1 consist... |
| stgrvtx0 48150 | The center ("internal node... |
| stgrorder 48151 | The order of a star graph ... |
| stgrnbgr0 48152 | All vertices of a star gra... |
| stgrclnbgr0 48153 | All vertices of a star gra... |
| isubgr3stgrlem1 48154 | Lemma 1 for ~ isubgr3stgr ... |
| isubgr3stgrlem2 48155 | Lemma 2 for ~ isubgr3stgr ... |
| isubgr3stgrlem3 48156 | Lemma 3 for ~ isubgr3stgr ... |
| isubgr3stgrlem4 48157 | Lemma 4 for ~ isubgr3stgr ... |
| isubgr3stgrlem5 48158 | Lemma 5 for ~ isubgr3stgr ... |
| isubgr3stgrlem6 48159 | Lemma 6 for ~ isubgr3stgr ... |
| isubgr3stgrlem7 48160 | Lemma 7 for ~ isubgr3stgr ... |
| isubgr3stgrlem8 48161 | Lemma 8 for ~ isubgr3stgr ... |
| isubgr3stgrlem9 48162 | Lemma 9 for ~ isubgr3stgr ... |
| isubgr3stgr 48163 | If a vertex of a simple gr... |
| grlimfn 48167 | The graph local isomorphis... |
| grlimdmrel 48168 | The domain of the graph lo... |
| isgrlim 48170 | A local isomorphism of gra... |
| isgrlim2 48171 | A local isomorphism of gra... |
| grlimprop 48172 | Properties of a local isom... |
| grlimf1o 48173 | A local isomorphism of gra... |
| grlimprop2 48174 | Properties of a local isom... |
| uhgrimgrlim 48175 | An isomorphism of hypergra... |
| uspgrlimlem1 48176 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 48177 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 48178 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 48179 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 48180 | A local isomorphism of sim... |
| usgrlimprop 48181 | Properties of a local isom... |
| clnbgrvtxedg 48182 | An edge ` E ` containing a... |
| grlimedgclnbgr 48183 | For two locally isomorphic... |
| grlimprclnbgr 48184 | For two locally isomorphic... |
| grlimprclnbgredg 48185 | For two locally isomorphic... |
| grlimpredg 48186 | For two locally isomorphic... |
| grlimprclnbgrvtx 48187 | For two locally isomorphic... |
| grlimgredgex 48188 | Local isomorphisms between... |
| grlimgrtrilem1 48189 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 48190 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 48191 | If one of two locally isom... |
| brgrlic 48192 | The relation "is locally i... |
| brgrilci 48193 | Prove that two graphs are ... |
| grlicrel 48194 | The "is locally isomorphic... |
| grlicrcl 48195 | Reverse closure of the "is... |
| dfgrlic2 48196 | Alternate, explicit defini... |
| grilcbri 48197 | Implications of two graphs... |
| dfgrlic3 48198 | Alternate, explicit defini... |
| grilcbri2 48199 | Implications of two graphs... |
| grlicref 48200 | Graph local isomorphism is... |
| grlicsym 48201 | Graph local isomorphism is... |
| grlicsymb 48202 | Graph local isomorphism is... |
| grlictr 48203 | Graph local isomorphism is... |
| grlicer 48204 | Local isomorphism is an eq... |
| grlicen 48205 | Locally isomorphic graphs ... |
| gricgrlic 48206 | Isomorphic hypergraphs are... |
| clnbgr3stgrgrlim 48207 | If all (closed) neighborho... |
| clnbgr3stgrgrlic 48208 | If all (closed) neighborho... |
| usgrexmpl1lem 48209 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 48210 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 48211 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 48212 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 48213 | ` G ` contains a triangle ... |
| usgrexmpl2lem 48214 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 48215 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 48216 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 48217 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 48218 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 48219 | The neighborhood of the fi... |
| usgrexmpl2nb1 48220 | The neighborhood of the se... |
| usgrexmpl2nb2 48221 | The neighborhood of the th... |
| usgrexmpl2nb3 48222 | The neighborhood of the fo... |
| usgrexmpl2nb4 48223 | The neighborhood of the fi... |
| usgrexmpl2nb5 48224 | The neighborhood of the si... |
| usgrexmpl2trifr 48225 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 48226 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 48227 | The graphs ` H ` and ` G `... |
| gpgov 48230 | The generalized Petersen g... |
| gpgvtx 48231 | The vertices of the genera... |
| gpgiedg 48232 | The indexed edges of the g... |
| gpgedg 48233 | The edges of the generaliz... |
| gpgiedgdmellem 48234 | Lemma for ~ gpgiedgdmel an... |
| gpgvtxel 48235 | A vertex in a generalized ... |
| gpgvtxel2 48236 | The second component of a ... |
| gpgiedgdmel 48237 | An index of edges of the g... |
| gpgedgel 48238 | An edge in a generalized P... |
| gpgprismgriedgdmel 48239 | An index of edges of the g... |
| gpgprismgriedgdmss 48240 | A subset of the index of e... |
| gpgvtx0 48241 | The outside vertices in a ... |
| gpgvtx1 48242 | The inside vertices in a g... |
| opgpgvtx 48243 | A vertex in a generalized ... |
| gpgusgralem 48244 | Lemma for ~ gpgusgra . (C... |
| gpgusgra 48245 | The generalized Petersen g... |
| gpgprismgrusgra 48246 | The generalized Petersen g... |
| gpgorder 48247 | The order of the generaliz... |
| gpg5order 48248 | The order of a generalized... |
| gpgedgvtx0 48249 | The edges starting at an o... |
| gpgedgvtx1 48250 | The edges starting at an i... |
| gpgvtxedg0 48251 | The edges starting at an o... |
| gpgvtxedg1 48252 | The edges starting at an i... |
| gpgedgiov 48253 | The edges of the generaliz... |
| gpgedg2ov 48254 | The edges of the generaliz... |
| gpgedg2iv 48255 | The edges of the generaliz... |
| gpg5nbgrvtx03starlem1 48256 | Lemma 1 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem2 48257 | Lemma 2 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem3 48258 | Lemma 3 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx13starlem1 48259 | Lemma 1 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem2 48260 | Lemma 2 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem3 48261 | Lemma 3 for ~ gpg5nbgr3sta... |
| gpgnbgrvtx0 48262 | The (open) neighborhood of... |
| gpgnbgrvtx1 48263 | The (open) neighborhood of... |
| gpg3nbgrvtx0 48264 | In a generalized Petersen ... |
| gpg3nbgrvtx0ALT 48265 | In a generalized Petersen ... |
| gpg3nbgrvtx1 48266 | In a generalized Petersen ... |
| gpgcubic 48267 | Every generalized Petersen... |
| gpg5nbgrvtx03star 48268 | In a generalized Petersen ... |
| gpg5nbgr3star 48269 | In a generalized Petersen ... |
| gpgvtxdg3 48270 | Every vertex in a generali... |
| gpg3kgrtriexlem1 48271 | Lemma 1 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem2 48272 | Lemma 2 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem3 48273 | Lemma 3 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem4 48274 | Lemma 4 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem5 48275 | Lemma 5 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem6 48276 | Lemma 6 for ~ gpg3kgrtriex... |
| gpg3kgrtriex 48277 | All generalized Petersen g... |
| gpg5gricstgr3 48278 | Each closed neighborhood i... |
| pglem 48279 | Lemma for theorems about P... |
| pgjsgr 48280 | A Petersen graph is a simp... |
| gpg5grlim 48281 | A local isomorphism betwee... |
| gpg5grlic 48282 | The two generalized Peters... |
| gpgprismgr4cycllem1 48283 | Lemma 1 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem2 48284 | Lemma 2 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem3 48285 | Lemma 3 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem4 48286 | Lemma 4 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem5 48287 | Lemma 5 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem6 48288 | Lemma 6 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem7 48289 | Lemma 7 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem8 48290 | Lemma 8 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem9 48291 | Lemma 9 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem10 48292 | Lemma 10 for ~ gpgprismgr4... |
| gpgprismgr4cycllem11 48293 | Lemma 11 for ~ gpgprismgr4... |
| gpgprismgr4cycl0 48294 | The generalized Petersen g... |
| gpgprismgr4cyclex 48295 | The generalized Petersen g... |
| pgnioedg1 48296 | An inside and an outside v... |
| pgnioedg2 48297 | An inside and an outside v... |
| pgnioedg3 48298 | An inside and an outside v... |
| pgnioedg4 48299 | An inside and an outside v... |
| pgnioedg5 48300 | An inside and an outside v... |
| pgnbgreunbgrlem1 48301 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem1 48302 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem2 48303 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem3 48304 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2 48305 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem3 48306 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem4 48307 | Lemma 4 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem1 48308 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem2 48309 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem3 48310 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5 48311 | Lemma 5 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem6 48312 | Lemma 6 for ~ pgnbgreunbgr... |
| pgnbgreunbgr 48313 | In a Petersen graph, two d... |
| pgn4cyclex 48314 | A cycle in a Petersen grap... |
| pg4cyclnex 48315 | In the Petersen graph G(5,... |
| gpg5ngric 48316 | The two generalized Peters... |
| lgricngricex 48317 | There are two different lo... |
| gpg5edgnedg 48318 | Two consecutive (according... |
| grlimedgnedg 48319 | In general, the image of a... |
| 1hegrlfgr 48320 | A graph ` G ` with one hyp... |
| upwlksfval 48323 | The set of simple walks (i... |
| isupwlk 48324 | Properties of a pair of fu... |
| isupwlkg 48325 | Generalization of ~ isupwl... |
| upwlkbprop 48326 | Basic properties of a simp... |
| upwlkwlk 48327 | A simple walk is a walk. ... |
| upgrwlkupwlk 48328 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 48329 | In a pseudograph, the defi... |
| upgrisupwlkALT 48330 | Alternate proof of ~ upgri... |
| upgredgssspr 48331 | The set of edges of a pseu... |
| uspgropssxp 48332 | The set ` G ` of "simple p... |
| uspgrsprfv 48333 | The value of the function ... |
| uspgrsprf 48334 | The mapping ` F ` is a fun... |
| uspgrsprf1 48335 | The mapping ` F ` is a one... |
| uspgrsprfo 48336 | The mapping ` F ` is a fun... |
| uspgrsprf1o 48337 | The mapping ` F ` is a bij... |
| uspgrex 48338 | The class ` G ` of all "si... |
| uspgrbispr 48339 | There is a bijection betwe... |
| uspgrspren 48340 | The set ` G ` of the "simp... |
| uspgrymrelen 48341 | The set ` G ` of the "simp... |
| uspgrbisymrel 48342 | There is a bijection betwe... |
| uspgrbisymrelALT 48343 | Alternate proof of ~ uspgr... |
| ovn0dmfun 48344 | If a class operation value... |
| xpsnopab 48345 | A Cartesian product with a... |
| xpiun 48346 | A Cartesian product expres... |
| ovn0ssdmfun 48347 | If a class' operation valu... |
| fnxpdmdm 48348 | The domain of the domain o... |
| cnfldsrngbas 48349 | The base set of a subring ... |
| cnfldsrngadd 48350 | The group addition operati... |
| cnfldsrngmul 48351 | The ring multiplication op... |
| plusfreseq 48352 | If the empty set is not co... |
| mgmplusfreseq 48353 | If the empty set is not co... |
| 0mgm 48354 | A set with an empty base s... |
| opmpoismgm 48355 | A structure with a group a... |
| copissgrp 48356 | A structure with a constan... |
| copisnmnd 48357 | A structure with a constan... |
| 0nodd 48358 | 0 is not an odd integer. ... |
| 1odd 48359 | 1 is an odd integer. (Con... |
| 2nodd 48360 | 2 is not an odd integer. ... |
| oddibas 48361 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 48362 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 48363 | The structure of all odd i... |
| nnsgrpmgm 48364 | The structure of positive ... |
| nnsgrp 48365 | The structure of positive ... |
| nnsgrpnmnd 48366 | The structure of positive ... |
| nn0mnd 48367 | The set of nonnegative int... |
| gsumsplit2f 48368 | Split a group sum into two... |
| gsumdifsndf 48369 | Extract a summand from a f... |
| gsumfsupp 48370 | A group sum of a family ca... |
| iscllaw 48377 | The predicate "is a closed... |
| iscomlaw 48378 | The predicate "is a commut... |
| clcllaw 48379 | Closure of a closed operat... |
| isasslaw 48380 | The predicate "is an assoc... |
| asslawass 48381 | Associativity of an associ... |
| mgmplusgiopALT 48382 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 48383 | Slot 2 (group operation) o... |
| intopval 48390 | The internal (binary) oper... |
| intop 48391 | An internal (binary) opera... |
| clintopval 48392 | The closed (internal binar... |
| assintopval 48393 | The associative (closed in... |
| assintopmap 48394 | The associative (closed in... |
| isclintop 48395 | The predicate "is a closed... |
| clintop 48396 | A closed (internal binary)... |
| assintop 48397 | An associative (closed int... |
| isassintop 48398 | The predicate "is an assoc... |
| clintopcllaw 48399 | The closure law holds for ... |
| assintopcllaw 48400 | The closure low holds for ... |
| assintopasslaw 48401 | The associative low holds ... |
| assintopass 48402 | An associative (closed int... |
| ismgmALT 48411 | The predicate "is a magma"... |
| iscmgmALT 48412 | The predicate "is a commut... |
| issgrpALT 48413 | The predicate "is a semigr... |
| iscsgrpALT 48414 | The predicate "is a commut... |
| mgm2mgm 48415 | Equivalence of the two def... |
| sgrp2sgrp 48416 | Equivalence of the two def... |
| lmod0rng 48417 | If the scalar ring of a mo... |
| nzrneg1ne0 48418 | The additive inverse of th... |
| lidldomn1 48419 | If a (left) ideal (which i... |
| lidlabl 48420 | A (left) ideal of a ring i... |
| lidlrng 48421 | A (left) ideal of a ring i... |
| zlidlring 48422 | The zero (left) ideal of a... |
| uzlidlring 48423 | Only the zero (left) ideal... |
| lidldomnnring 48424 | A (left) ideal of a domain... |
| 0even 48425 | 0 is an even integer. (Co... |
| 1neven 48426 | 1 is not an even integer. ... |
| 2even 48427 | 2 is an even integer. (Co... |
| 2zlidl 48428 | The even integers are a (l... |
| 2zrng 48429 | The ring of integers restr... |
| 2zrngbas 48430 | The base set of R is the s... |
| 2zrngadd 48431 | The group addition operati... |
| 2zrng0 48432 | The additive identity of R... |
| 2zrngamgm 48433 | R is an (additive) magma. ... |
| 2zrngasgrp 48434 | R is an (additive) semigro... |
| 2zrngamnd 48435 | R is an (additive) monoid.... |
| 2zrngacmnd 48436 | R is a commutative (additi... |
| 2zrngagrp 48437 | R is an (additive) group. ... |
| 2zrngaabl 48438 | R is an (additive) abelian... |
| 2zrngmul 48439 | The ring multiplication op... |
| 2zrngmmgm 48440 | R is a (multiplicative) ma... |
| 2zrngmsgrp 48441 | R is a (multiplicative) se... |
| 2zrngALT 48442 | The ring of integers restr... |
| 2zrngnmlid 48443 | R has no multiplicative (l... |
| 2zrngnmrid 48444 | R has no multiplicative (r... |
| 2zrngnmlid2 48445 | R has no multiplicative (l... |
| 2zrngnring 48446 | R is not a unital ring. (... |
| cznrnglem 48447 | Lemma for ~ cznrng : The ... |
| cznabel 48448 | The ring constructed from ... |
| cznrng 48449 | The ring constructed from ... |
| cznnring 48450 | The ring constructed from ... |
| rngcvalALTV 48453 | Value of the category of n... |
| rngcbasALTV 48454 | Set of objects of the cate... |
| rngchomfvalALTV 48455 | Set of arrows of the categ... |
| rngchomALTV 48456 | Set of arrows of the categ... |
| elrngchomALTV 48457 | A morphism of non-unital r... |
| rngccofvalALTV 48458 | Composition in the categor... |
| rngccoALTV 48459 | Composition in the categor... |
| rngccatidALTV 48460 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 48461 | The category of non-unital... |
| rngcidALTV 48462 | The identity arrow in the ... |
| rngcsectALTV 48463 | A section in the category ... |
| rngcinvALTV 48464 | An inverse in the category... |
| rngcisoALTV 48465 | An isomorphism in the cate... |
| rngchomffvalALTV 48466 | The value of the functiona... |
| rngchomrnghmresALTV 48467 | The value of the functiona... |
| rngcrescrhmALTV 48468 | The category of non-unital... |
| rhmsubcALTVlem1 48469 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 48470 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 48471 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 48472 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 48473 | According to ~ df-subc , t... |
| rhmsubcALTVcat 48474 | The restriction of the cat... |
| ringcvalALTV 48477 | Value of the category of r... |
| funcringcsetcALTV2lem1 48478 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 48479 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 48480 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 48481 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 48482 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 48483 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 48484 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 48485 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 48486 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 48487 | The "natural forgetful fun... |
| ringcbasALTV 48488 | Set of objects of the cate... |
| ringchomfvalALTV 48489 | Set of arrows of the categ... |
| ringchomALTV 48490 | Set of arrows of the categ... |
| elringchomALTV 48491 | A morphism of rings is a f... |
| ringccofvalALTV 48492 | Composition in the categor... |
| ringccoALTV 48493 | Composition in the categor... |
| ringccatidALTV 48494 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 48495 | The category of rings is a... |
| ringcidALTV 48496 | The identity arrow in the ... |
| ringcsectALTV 48497 | A section in the category ... |
| ringcinvALTV 48498 | An inverse in the category... |
| ringcisoALTV 48499 | An isomorphism in the cate... |
| ringcbasbasALTV 48500 | An element of the base set... |
| funcringcsetclem1ALTV 48501 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 48502 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 48503 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 48504 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 48505 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 48506 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 48507 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 48508 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 48509 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 48510 | The "natural forgetful fun... |
| srhmsubcALTVlem1 48511 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 48512 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 48513 | According to ~ df-subc , t... |
| sringcatALTV 48514 | The restriction of the cat... |
| crhmsubcALTV 48515 | According to ~ df-subc , t... |
| cringcatALTV 48516 | The restriction of the cat... |
| drhmsubcALTV 48517 | According to ~ df-subc , t... |
| drngcatALTV 48518 | The restriction of the cat... |
| fldcatALTV 48519 | The restriction of the cat... |
| fldcALTV 48520 | The restriction of the cat... |
| fldhmsubcALTV 48521 | According to ~ df-subc , t... |
| eliunxp2 48522 | Membership in a union of C... |
| mpomptx2 48523 | Express a two-argument fun... |
| cbvmpox2 48524 | Rule to change the bound v... |
| dmmpossx2 48525 | The domain of a mapping is... |
| mpoexxg2 48526 | Existence of an operation ... |
| ovmpordxf 48527 | Value of an operation give... |
| ovmpordx 48528 | Value of an operation give... |
| ovmpox2 48529 | The value of an operation ... |
| fdmdifeqresdif 48530 | The restriction of a condi... |
| ofaddmndmap 48531 | The function operation app... |
| mapsnop 48532 | A singleton of an ordered ... |
| fprmappr 48533 | A function with a domain o... |
| mapprop 48534 | An unordered pair containi... |
| ztprmneprm 48535 | A prime is not an integer ... |
| 2t6m3t4e0 48536 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 48537 | For any finite subset of `... |
| nn0sumltlt 48538 | If the sum of two nonnegat... |
| bcpascm1 48539 | Pascal's rule for the bino... |
| altgsumbc 48540 | The sum of binomial coeffi... |
| altgsumbcALT 48541 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 48542 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 48543 | An element of the (base se... |
| zlmodzxz0 48544 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 48545 | The scalar multiplication ... |
| zlmodzxzadd 48546 | The addition of the ` ZZ `... |
| zlmodzxzsubm 48547 | The subtraction of the ` Z... |
| zlmodzxzsub 48548 | The subtraction of the ` Z... |
| mgpsumunsn 48549 | Extract a summand/factor f... |
| mgpsumz 48550 | If the group sum for the m... |
| mgpsumn 48551 | If the group sum for the m... |
| exple2lt6 48552 | A nonnegative integer to t... |
| pgrple2abl 48553 | Every symmetric group on a... |
| pgrpgt2nabl 48554 | Every symmetric group on a... |
| invginvrid 48555 | Identity for a multiplicat... |
| rmsupp0 48556 | The support of a mapping o... |
| domnmsuppn0 48557 | The support of a mapping o... |
| rmsuppss 48558 | The support of a mapping o... |
| scmsuppss 48559 | The support of a mapping o... |
| rmsuppfi 48560 | The support of a mapping o... |
| rmfsupp 48561 | A mapping of a multiplicat... |
| scmsuppfi 48562 | The support of a mapping o... |
| scmfsupp 48563 | A mapping of a scalar mult... |
| suppmptcfin 48564 | The support of a mapping w... |
| mptcfsupp 48565 | A mapping with value 0 exc... |
| fsuppmptdmf 48566 | A mapping with a finite do... |
| lmodvsmdi 48567 | Multiple distributive law ... |
| gsumlsscl 48568 | Closure of a group sum in ... |
| assaascl0 48569 | The scalar 0 embedded into... |
| assaascl1 48570 | The scalar 1 embedded into... |
| ply1vr1smo 48571 | The variable in a polynomi... |
| ply1sclrmsm 48572 | The ring multiplication of... |
| coe1sclmulval 48573 | The value of the coefficie... |
| ply1mulgsumlem1 48574 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 48575 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 48576 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 48577 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 48578 | The product of two polynom... |
| evl1at0 48579 | Polynomial evaluation for ... |
| evl1at1 48580 | Polynomial evaluation for ... |
| linply1 48581 | A term of the form ` x - C... |
| lineval 48582 | A term of the form ` x - C... |
| linevalexample 48583 | The polynomial ` x - 3 ` o... |
| dmatALTval 48588 | The algebra of ` N ` x ` N... |
| dmatALTbas 48589 | The base set of the algebr... |
| dmatALTbasel 48590 | An element of the base set... |
| dmatbas 48591 | The set of all ` N ` x ` N... |
| lincop 48596 | A linear combination as op... |
| lincval 48597 | The value of a linear comb... |
| dflinc2 48598 | Alternative definition of ... |
| lcoop 48599 | A linear combination as op... |
| lcoval 48600 | The value of a linear comb... |
| lincfsuppcl 48601 | A linear combination of ve... |
| linccl 48602 | A linear combination of ve... |
| lincval0 48603 | The value of an empty line... |
| lincvalsng 48604 | The linear combination ove... |
| lincvalsn 48605 | The linear combination ove... |
| lincvalpr 48606 | The linear combination ove... |
| lincval1 48607 | The linear combination ove... |
| lcosn0 48608 | Properties of a linear com... |
| lincvalsc0 48609 | The linear combination whe... |
| lcoc0 48610 | Properties of a linear com... |
| linc0scn0 48611 | If a set contains the zero... |
| lincdifsn 48612 | A vector is a linear combi... |
| linc1 48613 | A vector is a linear combi... |
| lincellss 48614 | A linear combination of a ... |
| lco0 48615 | The set of empty linear co... |
| lcoel0 48616 | The zero vector is always ... |
| lincsum 48617 | The sum of two linear comb... |
| lincscm 48618 | A linear combinations mult... |
| lincsumcl 48619 | The sum of two linear comb... |
| lincscmcl 48620 | The multiplication of a li... |
| lincsumscmcl 48621 | The sum of a linear combin... |
| lincolss 48622 | According to the statement... |
| ellcoellss 48623 | Every linear combination o... |
| lcoss 48624 | A set of vectors of a modu... |
| lspsslco 48625 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 48626 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 48627 | Equivalence of a _span_ of... |
| rellininds 48631 | The class defining the rel... |
| linindsv 48633 | The classes of the module ... |
| islininds 48634 | The property of being a li... |
| linindsi 48635 | The implications of being ... |
| linindslinci 48636 | The implications of being ... |
| islinindfis 48637 | The property of being a li... |
| islinindfiss 48638 | The property of being a li... |
| linindscl 48639 | A linearly independent set... |
| lindepsnlininds 48640 | A linearly dependent subse... |
| islindeps 48641 | The property of being a li... |
| lincext1 48642 | Property 1 of an extension... |
| lincext2 48643 | Property 2 of an extension... |
| lincext3 48644 | Property 3 of an extension... |
| lindslinindsimp1 48645 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 48646 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 48647 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 48648 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 48649 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 48650 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 48651 | Implication 2 for ~ lindsl... |
| lindslininds 48652 | Equivalence of definitions... |
| linds0 48653 | The empty set is always a ... |
| el0ldep 48654 | A set containing the zero ... |
| el0ldepsnzr 48655 | A set containing the zero ... |
| lindsrng01 48656 | Any subset of a module is ... |
| lindszr 48657 | Any subset of a module ove... |
| snlindsntorlem 48658 | Lemma for ~ snlindsntor . ... |
| snlindsntor 48659 | A singleton is linearly in... |
| ldepsprlem 48660 | Lemma for ~ ldepspr . (Co... |
| ldepspr 48661 | If a vector is a scalar mu... |
| lincresunit3lem3 48662 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 48663 | Lemma 1 for properties of ... |
| lincresunitlem2 48664 | Lemma for properties of a ... |
| lincresunit1 48665 | Property 1 of a specially ... |
| lincresunit2 48666 | Property 2 of a specially ... |
| lincresunit3lem1 48667 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 48668 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 48669 | Property 3 of a specially ... |
| lincreslvec3 48670 | Property 3 of a specially ... |
| islindeps2 48671 | Conditions for being a lin... |
| islininds2 48672 | Implication of being a lin... |
| isldepslvec2 48673 | Alternative definition of ... |
| lindssnlvec 48674 | A singleton not containing... |
| lmod1lem1 48675 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 48676 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 48677 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 48678 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 48679 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 48680 | The (smallest) structure r... |
| lmod1zr 48681 | The (smallest) structure r... |
| lmod1zrnlvec 48682 | There is a (left) module (... |
| lmodn0 48683 | Left modules exist. (Cont... |
| zlmodzxzequa 48684 | Example of an equation wit... |
| zlmodzxznm 48685 | Example of a linearly depe... |
| zlmodzxzldeplem 48686 | A and B are not equal. (C... |
| zlmodzxzequap 48687 | Example of an equation wit... |
| zlmodzxzldeplem1 48688 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 48689 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 48690 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 48691 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 48692 | { A , B } is a linearly de... |
| ldepsnlinclem1 48693 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 48694 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 48695 | The class of all (left) ve... |
| ldepsnlinc 48696 | The reverse implication of... |
| ldepslinc 48697 | For (left) vector spaces, ... |
| suppdm 48698 | If the range of a function... |
| eluz2cnn0n1 48699 | An integer greater than 1 ... |
| divge1b 48700 | The ratio of a real number... |
| divgt1b 48701 | The ratio of a real number... |
| ltsubaddb 48702 | Equivalence for the "less ... |
| ltsubsubb 48703 | Equivalence for the "less ... |
| ltsubadd2b 48704 | Equivalence for the "less ... |
| divsub1dir 48705 | Distribution of division o... |
| expnegico01 48706 | An integer greater than 1 ... |
| elfzolborelfzop1 48707 | An element of a half-open ... |
| pw2m1lepw2m1 48708 | 2 to the power of a positi... |
| zgtp1leeq 48709 | If an integer is between a... |
| flsubz 48710 | An integer can be moved in... |
| nn0onn0ex 48711 | For each odd nonnegative i... |
| nn0enn0ex 48712 | For each even nonnegative ... |
| nnennex 48713 | For each even positive int... |
| nneop 48714 | A positive integer is even... |
| nneom 48715 | A positive integer is even... |
| nn0eo 48716 | A nonnegative integer is e... |
| nnpw2even 48717 | 2 to the power of a positi... |
| zefldiv2 48718 | The floor of an even integ... |
| zofldiv2 48719 | The floor of an odd intege... |
| nn0ofldiv2 48720 | The floor of an odd nonneg... |
| flnn0div2ge 48721 | The floor of a positive in... |
| flnn0ohalf 48722 | The floor of the half of a... |
| logcxp0 48723 | Logarithm of a complex pow... |
| regt1loggt0 48724 | The natural logarithm for ... |
| fdivval 48727 | The quotient of two functi... |
| fdivmpt 48728 | The quotient of two functi... |
| fdivmptf 48729 | The quotient of two functi... |
| refdivmptf 48730 | The quotient of two functi... |
| fdivpm 48731 | The quotient of two functi... |
| refdivpm 48732 | The quotient of two functi... |
| fdivmptfv 48733 | The function value of a qu... |
| refdivmptfv 48734 | The function value of a qu... |
| bigoval 48737 | Set of functions of order ... |
| elbigofrcl 48738 | Reverse closure of the "bi... |
| elbigo 48739 | Properties of a function o... |
| elbigo2 48740 | Properties of a function o... |
| elbigo2r 48741 | Sufficient condition for a... |
| elbigof 48742 | A function of order G(x) i... |
| elbigodm 48743 | The domain of a function o... |
| elbigoimp 48744 | The defining property of a... |
| elbigolo1 48745 | A function (into the posit... |
| rege1logbrege0 48746 | The general logarithm, wit... |
| rege1logbzge0 48747 | The general logarithm, wit... |
| fllogbd 48748 | A real number is between t... |
| relogbmulbexp 48749 | The logarithm of the produ... |
| relogbdivb 48750 | The logarithm of the quoti... |
| logbge0b 48751 | The logarithm of a number ... |
| logblt1b 48752 | The logarithm of a number ... |
| fldivexpfllog2 48753 | The floor of a positive re... |
| nnlog2ge0lt1 48754 | A positive integer is 1 if... |
| logbpw2m1 48755 | The floor of the binary lo... |
| fllog2 48756 | The floor of the binary lo... |
| blenval 48759 | The binary length of an in... |
| blen0 48760 | The binary length of 0. (... |
| blenn0 48761 | The binary length of a "nu... |
| blenre 48762 | The binary length of a pos... |
| blennn 48763 | The binary length of a pos... |
| blennnelnn 48764 | The binary length of a pos... |
| blennn0elnn 48765 | The binary length of a non... |
| blenpw2 48766 | The binary length of a pow... |
| blenpw2m1 48767 | The binary length of a pow... |
| nnpw2blen 48768 | A positive integer is betw... |
| nnpw2blenfzo 48769 | A positive integer is betw... |
| nnpw2blenfzo2 48770 | A positive integer is eith... |
| nnpw2pmod 48771 | Every positive integer can... |
| blen1 48772 | The binary length of 1. (... |
| blen2 48773 | The binary length of 2. (... |
| nnpw2p 48774 | Every positive integer can... |
| nnpw2pb 48775 | A number is a positive int... |
| blen1b 48776 | The binary length of a non... |
| blennnt2 48777 | The binary length of a pos... |
| nnolog2flm1 48778 | The floor of the binary lo... |
| blennn0em1 48779 | The binary length of the h... |
| blennngt2o2 48780 | The binary length of an od... |
| blengt1fldiv2p1 48781 | The binary length of an in... |
| blennn0e2 48782 | The binary length of an ev... |
| digfval 48785 | Operation to obtain the ` ... |
| digval 48786 | The ` K ` th digit of a no... |
| digvalnn0 48787 | The ` K ` th digit of a no... |
| nn0digval 48788 | The ` K ` th digit of a no... |
| dignn0fr 48789 | The digits of the fraction... |
| dignn0ldlem 48790 | Lemma for ~ dignnld . (Co... |
| dignnld 48791 | The leading digits of a po... |
| dig2nn0ld 48792 | The leading digits of a po... |
| dig2nn1st 48793 | The first (relevant) digit... |
| dig0 48794 | All digits of 0 are 0. (C... |
| digexp 48795 | The ` K ` th digit of a po... |
| dig1 48796 | All but one digits of 1 ar... |
| 0dig1 48797 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 48798 | The integers 0 and 1 corre... |
| dig2nn0 48799 | A digit of a nonnegative i... |
| 0dig2nn0e 48800 | The last bit of an even in... |
| 0dig2nn0o 48801 | The last bit of an odd int... |
| dig2bits 48802 | The ` K ` th digit of a no... |
| dignn0flhalflem1 48803 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 48804 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 48805 | The digits of the half of ... |
| dignn0flhalf 48806 | The digits of the rounded ... |
| nn0sumshdiglemA 48807 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 48808 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 48809 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 48810 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 48811 | A nonnegative integer can ... |
| nn0mulfsum 48812 | Trivial algorithm to calcu... |
| nn0mullong 48813 | Standard algorithm (also k... |
| naryfval 48816 | The set of the n-ary (endo... |
| naryfvalixp 48817 | The set of the n-ary (endo... |
| naryfvalel 48818 | An n-ary (endo)function on... |
| naryrcl 48819 | Reverse closure for n-ary ... |
| naryfvalelfv 48820 | The value of an n-ary (end... |
| naryfvalelwrdf 48821 | An n-ary (endo)function on... |
| 0aryfvalel 48822 | A nullary (endo)function o... |
| 0aryfvalelfv 48823 | The value of a nullary (en... |
| 1aryfvalel 48824 | A unary (endo)function on ... |
| fv1arycl 48825 | Closure of a unary (endo)f... |
| 1arympt1 48826 | A unary (endo)function in ... |
| 1arympt1fv 48827 | The value of a unary (endo... |
| 1arymaptfv 48828 | The value of the mapping o... |
| 1arymaptf 48829 | The mapping of unary (endo... |
| 1arymaptf1 48830 | The mapping of unary (endo... |
| 1arymaptfo 48831 | The mapping of unary (endo... |
| 1arymaptf1o 48832 | The mapping of unary (endo... |
| 1aryenef 48833 | The set of unary (endo)fun... |
| 1aryenefmnd 48834 | The set of unary (endo)fun... |
| 2aryfvalel 48835 | A binary (endo)function on... |
| fv2arycl 48836 | Closure of a binary (endo)... |
| 2arympt 48837 | A binary (endo)function in... |
| 2arymptfv 48838 | The value of a binary (end... |
| 2arymaptfv 48839 | The value of the mapping o... |
| 2arymaptf 48840 | The mapping of binary (end... |
| 2arymaptf1 48841 | The mapping of binary (end... |
| 2arymaptfo 48842 | The mapping of binary (end... |
| 2arymaptf1o 48843 | The mapping of binary (end... |
| 2aryenef 48844 | The set of binary (endo)fu... |
| itcoval 48849 | The value of the function ... |
| itcoval0 48850 | A function iterated zero t... |
| itcoval1 48851 | A function iterated once. ... |
| itcoval2 48852 | A function iterated twice.... |
| itcoval3 48853 | A function iterated three ... |
| itcoval0mpt 48854 | A mapping iterated zero ti... |
| itcovalsuc 48855 | The value of the function ... |
| itcovalsucov 48856 | The value of the function ... |
| itcovalendof 48857 | The n-th iterate of an end... |
| itcovalpclem1 48858 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 48859 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 48860 | The value of the function ... |
| itcovalt2lem2lem1 48861 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 48862 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 48863 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 48864 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 48865 | The value of the function ... |
| ackvalsuc1mpt 48866 | The Ackermann function at ... |
| ackvalsuc1 48867 | The Ackermann function at ... |
| ackval0 48868 | The Ackermann function at ... |
| ackval1 48869 | The Ackermann function at ... |
| ackval2 48870 | The Ackermann function at ... |
| ackval3 48871 | The Ackermann function at ... |
| ackendofnn0 48872 | The Ackermann function at ... |
| ackfnnn0 48873 | The Ackermann function at ... |
| ackval0val 48874 | The Ackermann function at ... |
| ackvalsuc0val 48875 | The Ackermann function at ... |
| ackvalsucsucval 48876 | The Ackermann function at ... |
| ackval0012 48877 | The Ackermann function at ... |
| ackval1012 48878 | The Ackermann function at ... |
| ackval2012 48879 | The Ackermann function at ... |
| ackval3012 48880 | The Ackermann function at ... |
| ackval40 48881 | The Ackermann function at ... |
| ackval41a 48882 | The Ackermann function at ... |
| ackval41 48883 | The Ackermann function at ... |
| ackval42 48884 | The Ackermann function at ... |
| ackval42a 48885 | The Ackermann function at ... |
| ackval50 48886 | The Ackermann function at ... |
| fv1prop 48887 | The function value of unor... |
| fv2prop 48888 | The function value of unor... |
| submuladdmuld 48889 | Transformation of a sum of... |
| affinecomb1 48890 | Combination of two real af... |
| affinecomb2 48891 | Combination of two real af... |
| affineid 48892 | Identity of an affine comb... |
| 1subrec1sub 48893 | Subtract the reciprocal of... |
| resum2sqcl 48894 | The sum of two squares of ... |
| resum2sqgt0 48895 | The sum of the square of a... |
| resum2sqrp 48896 | The sum of the square of a... |
| resum2sqorgt0 48897 | The sum of the square of t... |
| reorelicc 48898 | Membership in and outside ... |
| rrx2pxel 48899 | The x-coordinate of a poin... |
| rrx2pyel 48900 | The y-coordinate of a poin... |
| prelrrx2 48901 | An unordered pair of order... |
| prelrrx2b 48902 | An unordered pair of order... |
| rrx2pnecoorneor 48903 | If two different points ` ... |
| rrx2pnedifcoorneor 48904 | If two different points ` ... |
| rrx2pnedifcoorneorr 48905 | If two different points ` ... |
| rrx2xpref1o 48906 | There is a bijection betwe... |
| rrx2xpreen 48907 | The set of points in the t... |
| rrx2plord 48908 | The lexicographical orderi... |
| rrx2plord1 48909 | The lexicographical orderi... |
| rrx2plord2 48910 | The lexicographical orderi... |
| rrx2plordisom 48911 | The set of points in the t... |
| rrx2plordso 48912 | The lexicographical orderi... |
| ehl2eudisval0 48913 | The Euclidean distance of ... |
| ehl2eudis0lt 48914 | An upper bound of the Eucl... |
| lines 48919 | The lines passing through ... |
| line 48920 | The line passing through t... |
| rrxlines 48921 | Definition of lines passin... |
| rrxline 48922 | The line passing through t... |
| rrxlinesc 48923 | Definition of lines passin... |
| rrxlinec 48924 | The line passing through t... |
| eenglngeehlnmlem1 48925 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 48926 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 48927 | The line definition in the... |
| rrx2line 48928 | The line passing through t... |
| rrx2vlinest 48929 | The vertical line passing ... |
| rrx2linest 48930 | The line passing through t... |
| rrx2linesl 48931 | The line passing through t... |
| rrx2linest2 48932 | The line passing through t... |
| elrrx2linest2 48933 | The line passing through t... |
| spheres 48934 | The spheres for given cent... |
| sphere 48935 | A sphere with center ` X `... |
| rrxsphere 48936 | The sphere with center ` M... |
| 2sphere 48937 | The sphere with center ` M... |
| 2sphere0 48938 | The sphere around the orig... |
| line2ylem 48939 | Lemma for ~ line2y . This... |
| line2 48940 | Example for a line ` G ` p... |
| line2xlem 48941 | Lemma for ~ line2x . This... |
| line2x 48942 | Example for a horizontal l... |
| line2y 48943 | Example for a vertical lin... |
| itsclc0lem1 48944 | Lemma for theorems about i... |
| itsclc0lem2 48945 | Lemma for theorems about i... |
| itsclc0lem3 48946 | Lemma for theorems about i... |
| itscnhlc0yqe 48947 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 48948 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 48949 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 48950 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 48951 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 48952 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 48953 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 48954 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 48955 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 48956 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 48957 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 48958 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 48959 | The intersection points of... |
| itsclc0b 48960 | The intersection points of... |
| itsclinecirc0 48961 | The intersection points of... |
| itsclinecirc0b 48962 | The intersection points of... |
| itsclinecirc0in 48963 | The intersection points of... |
| itsclquadb 48964 | Quadratic equation for the... |
| itsclquadeu 48965 | Quadratic equation for the... |
| 2itscplem1 48966 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 48967 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 48968 | Lemma D for ~ 2itscp . (C... |
| 2itscp 48969 | A condition for a quadrati... |
| itscnhlinecirc02plem1 48970 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 48971 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 48972 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 48973 | Intersection of a nonhoriz... |
| inlinecirc02plem 48974 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 48975 | Intersection of a line wit... |
| inlinecirc02preu 48976 | Intersection of a line wit... |
| pm4.71da 48977 | Deduction converting a bic... |
| logic1 48978 | Distribution of implicatio... |
| logic1a 48979 | Variant of ~ logic1 . (Co... |
| logic2 48980 | Variant of ~ logic1 . (Co... |
| pm5.32dav 48981 | Distribution of implicatio... |
| pm5.32dra 48982 | Reverse distribution of im... |
| exp12bd 48983 | The import-export theorem ... |
| mpbiran3d 48984 | Equivalence with a conjunc... |
| mpbiran4d 48985 | Equivalence with a conjunc... |
| dtrucor3 48986 | An example of how ~ ax-5 w... |
| ralbidb 48987 | Formula-building rule for ... |
| ralbidc 48988 | Formula-building rule for ... |
| r19.41dv 48989 | A complex deduction form o... |
| rmotru 48990 | Two ways of expressing "at... |
| reutru 48991 | Two ways of expressing "ex... |
| reutruALT 48992 | Alternate proof of ~ reutr... |
| reueqbidva 48993 | Formula-building rule for ... |
| reuxfr1dd 48994 | Transfer existential uniqu... |
| ssdisjd 48995 | Subset preserves disjointn... |
| ssdisjdr 48996 | Subset preserves disjointn... |
| disjdifb 48997 | Relative complement is ant... |
| predisj 48998 | Preimages of disjoint sets... |
| vsn 48999 | The singleton of the unive... |
| mosn 49000 | "At most one" element in a... |
| mo0 49001 | "At most one" element in a... |
| mosssn 49002 | "At most one" element in a... |
| mo0sn 49003 | Two ways of expressing "at... |
| mosssn2 49004 | Two ways of expressing "at... |
| unilbss 49005 | Superclass of the greatest... |
| iuneq0 49006 | An indexed union is empty ... |
| iineq0 49007 | An indexed intersection is... |
| iunlub 49008 | The indexed union is the t... |
| iinglb 49009 | The indexed intersection i... |
| iuneqconst2 49010 | Indexed union of identical... |
| iineqconst2 49011 | Indexed intersection of id... |
| inpw 49012 | Two ways of expressing a c... |
| opth1neg 49013 | Two ordered pairs are not ... |
| opth2neg 49014 | Two ordered pairs are not ... |
| brab2dd 49015 | Expressing that two sets a... |
| brab2ddw 49016 | Expressing that two sets a... |
| brab2ddw2 49017 | Expressing that two sets a... |
| iinxp 49018 | Indexed intersection of Ca... |
| intxp 49019 | Intersection of Cartesian ... |
| coxp 49020 | Composition with a Cartesi... |
| cosn 49021 | Composition with an ordere... |
| cosni 49022 | Composition with an ordere... |
| inisegn0a 49023 | The inverse image of a sin... |
| dmrnxp 49024 | A Cartesian product is the... |
| mof0 49025 | There is at most one funct... |
| mof02 49026 | A variant of ~ mof0 . (Co... |
| mof0ALT 49027 | Alternate proof of ~ mof0 ... |
| eufsnlem 49028 | There is exactly one funct... |
| eufsn 49029 | There is exactly one funct... |
| eufsn2 49030 | There is exactly one funct... |
| mofsn 49031 | There is at most one funct... |
| mofsn2 49032 | There is at most one funct... |
| mofsssn 49033 | There is at most one funct... |
| mofmo 49034 | There is at most one funct... |
| mofeu 49035 | The uniqueness of a functi... |
| elfvne0 49036 | If a function value has a ... |
| fdomne0 49037 | A function with non-empty ... |
| f1sn2g 49038 | A function that maps a sin... |
| f102g 49039 | A function that maps the e... |
| f1mo 49040 | A function that maps a set... |
| f002 49041 | A function with an empty c... |
| map0cor 49042 | A function exists iff an e... |
| ffvbr 49043 | Relation with function val... |
| xpco2 49044 | Composition of a Cartesian... |
| ovsng 49045 | The operation value of a s... |
| ovsng2 49046 | The operation value of a s... |
| ovsn 49047 | The operation value of a s... |
| ovsn2 49048 | The operation value of a s... |
| fvconstr 49049 | Two ways of expressing ` A... |
| fvconstrn0 49050 | Two ways of expressing ` A... |
| fvconstr2 49051 | Two ways of expressing ` A... |
| ovmpt4d 49052 | Deduction version of ~ ovm... |
| eqfnovd 49053 | Deduction for equality of ... |
| fonex 49054 | The domain of a surjection... |
| eloprab1st2nd 49055 | Reconstruction of a nested... |
| fmpodg 49056 | Domain and codomain of the... |
| fmpod 49057 | Domain and codomain of the... |
| resinsnlem 49058 | Lemma for ~ resinsnALT . ... |
| resinsn 49059 | Restriction to the interse... |
| resinsnALT 49060 | Restriction to the interse... |
| dftpos5 49061 | Alternate definition of ` ... |
| dftpos6 49062 | Alternate definition of ` ... |
| dmtposss 49063 | The domain of ` tpos F ` i... |
| tposres0 49064 | The transposition of a set... |
| tposresg 49065 | The transposition restrict... |
| tposrescnv 49066 | The transposition restrict... |
| tposres2 49067 | The transposition restrict... |
| tposres3 49068 | The transposition restrict... |
| tposres 49069 | The transposition restrict... |
| tposresxp 49070 | The transposition restrict... |
| tposf1o 49071 | Condition of a bijective t... |
| tposid 49072 | Swap an ordered pair. (Co... |
| tposidres 49073 | Swap an ordered pair. (Co... |
| tposidf1o 49074 | The swap function, or the ... |
| tposideq 49075 | Two ways of expressing the... |
| tposideq2 49076 | Two ways of expressing the... |
| ixpv 49077 | Infinite Cartesian product... |
| fvconst0ci 49078 | A constant function's valu... |
| fvconstdomi 49079 | A constant function's valu... |
| f1omo 49080 | There is at most one eleme... |
| f1omoOLD 49081 | Obsolete version of ~ f1om... |
| f1omoALT 49082 | There is at most one eleme... |
| iccin 49083 | Intersection of two closed... |
| iccdisj2 49084 | If the upper bound of one ... |
| iccdisj 49085 | If the upper bound of one ... |
| slotresfo 49086 | The condition of a structu... |
| mreuniss 49087 | The union of a collection ... |
| clduni 49088 | The union of closed sets i... |
| opncldeqv 49089 | Conditions on open sets ar... |
| opndisj 49090 | Two ways of saying that tw... |
| clddisj 49091 | Two ways of saying that tw... |
| neircl 49092 | Reverse closure of the nei... |
| opnneilem 49093 | Lemma factoring out common... |
| opnneir 49094 | If something is true for a... |
| opnneirv 49095 | A variant of ~ opnneir wit... |
| opnneilv 49096 | The converse of ~ opnneir ... |
| opnneil 49097 | A variant of ~ opnneilv . ... |
| opnneieqv 49098 | The equivalence between ne... |
| opnneieqvv 49099 | The equivalence between ne... |
| restcls2lem 49100 | A closed set in a subspace... |
| restcls2 49101 | A closed set in a subspace... |
| restclsseplem 49102 | Lemma for ~ restclssep . ... |
| restclssep 49103 | Two disjoint closed sets i... |
| cnneiima 49104 | Given a continuous functio... |
| iooii 49105 | Open intervals are open se... |
| icccldii 49106 | Closed intervals are close... |
| i0oii 49107 | ` ( 0 [,) A ) ` is open in... |
| io1ii 49108 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 49109 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 49110 | Open neighborhood and neig... |
| sepnsepo 49111 | Open neighborhood and neig... |
| sepdisj 49112 | Separated sets are disjoin... |
| seposep 49113 | If two sets are separated ... |
| sepcsepo 49114 | If two sets are separated ... |
| sepfsepc 49115 | If two sets are separated ... |
| seppsepf 49116 | If two sets are precisely ... |
| seppcld 49117 | If two sets are precisely ... |
| isnrm4 49118 | A topological space is nor... |
| dfnrm2 49119 | A topological space is nor... |
| dfnrm3 49120 | A topological space is nor... |
| iscnrm3lem1 49121 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 49122 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem4 49123 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 49124 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 49125 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 49126 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 49127 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 49128 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 49129 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 49130 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 49131 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 49132 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 49133 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 49134 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 49135 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 49136 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 49137 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 49138 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 49139 | A completely normal topolo... |
| iscnrm3v 49140 | A topology is completely n... |
| iscnrm4 49141 | A completely normal topolo... |
| isprsd 49142 | Property of being a preord... |
| lubeldm2 49143 | Member of the domain of th... |
| glbeldm2 49144 | Member of the domain of th... |
| lubeldm2d 49145 | Member of the domain of th... |
| glbeldm2d 49146 | Member of the domain of th... |
| lubsscl 49147 | If a subset of ` S ` conta... |
| glbsscl 49148 | If a subset of ` S ` conta... |
| lubprlem 49149 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 49150 | The set of two comparable ... |
| lubpr 49151 | The LUB of the set of two ... |
| glbprlem 49152 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 49153 | The set of two comparable ... |
| glbpr 49154 | The GLB of the set of two ... |
| joindm2 49155 | The join of any two elemen... |
| joindm3 49156 | The join of any two elemen... |
| meetdm2 49157 | The meet of any two elemen... |
| meetdm3 49158 | The meet of any two elemen... |
| posjidm 49159 | Poset join is idempotent. ... |
| posmidm 49160 | Poset meet is idempotent. ... |
| resiposbas 49161 | Construct a poset ( ~ resi... |
| resipos 49162 | A set equipped with an ord... |
| exbaspos 49163 | There exists a poset for a... |
| exbasprs 49164 | There exists a preordered ... |
| basresposfo 49165 | The base function restrict... |
| basresprsfo 49166 | The base function restrict... |
| posnex 49167 | The class of posets is a p... |
| prsnex 49168 | The class of preordered se... |
| toslat 49169 | A toset is a lattice. (Co... |
| isclatd 49170 | The predicate "is a comple... |
| intubeu 49171 | Existential uniqueness of ... |
| unilbeu 49172 | Existential uniqueness of ... |
| ipolublem 49173 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 49174 | The domain of the LUB of t... |
| ipolub 49175 | The LUB of the inclusion p... |
| ipoglblem 49176 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 49177 | The domain of the GLB of t... |
| ipoglb 49178 | The GLB of the inclusion p... |
| ipolub0 49179 | The LUB of the empty set i... |
| ipolub00 49180 | The LUB of the empty set i... |
| ipoglb0 49181 | The GLB of the empty set i... |
| mrelatlubALT 49182 | Least upper bounds in a Mo... |
| mrelatglbALT 49183 | Greatest lower bounds in a... |
| mreclat 49184 | A Moore space is a complet... |
| topclat 49185 | A topology is a complete l... |
| toplatglb0 49186 | The empty intersection in ... |
| toplatlub 49187 | Least upper bounds in a to... |
| toplatglb 49188 | Greatest lower bounds in a... |
| toplatjoin 49189 | Joins in a topology are re... |
| toplatmeet 49190 | Meets in a topology are re... |
| topdlat 49191 | A topology is a distributi... |
| elmgpcntrd 49192 | The center of a ring. (Co... |
| asclelbasALT 49193 | Alternate proof of ~ ascle... |
| asclcntr 49194 | The algebra scalar lifting... |
| asclcom 49195 | Scalars are commutative af... |
| homf0 49196 | The base is empty iff the ... |
| catprslem 49197 | Lemma for ~ catprs . (Con... |
| catprs 49198 | A preorder can be extracte... |
| catprs2 49199 | A category equipped with t... |
| catprsc 49200 | A construction of the preo... |
| catprsc2 49201 | An alternate construction ... |
| endmndlem 49202 | A diagonal hom-set in a ca... |
| oppccatb 49203 | An opposite category is a ... |
| oppcmndclem 49204 | Lemma for ~ oppcmndc . Ev... |
| oppcendc 49205 | The opposite category of a... |
| oppcmndc 49206 | The opposite category of a... |
| idmon 49207 | An identity arrow, or an i... |
| idepi 49208 | An identity arrow, or an i... |
| sectrcl 49209 | Reverse closure for sectio... |
| sectrcl2 49210 | Reverse closure for sectio... |
| invrcl 49211 | Reverse closure for invers... |
| invrcl2 49212 | Reverse closure for invers... |
| isinv2 49213 | The property " ` F ` is an... |
| isisod 49214 | The predicate "is an isomo... |
| upeu2lem 49215 | Lemma for ~ upeu2 . There... |
| sectfn 49216 | The function value of the ... |
| invfn 49217 | The function value of the ... |
| isofnALT 49218 | The function value of the ... |
| isofval2 49219 | Function value of the func... |
| isorcl 49220 | Reverse closure for isomor... |
| isorcl2 49221 | Reverse closure for isomor... |
| isoval2 49222 | The isomorphisms are the d... |
| sectpropdlem 49223 | Lemma for ~ sectpropd . (... |
| sectpropd 49224 | Two structures with the sa... |
| invpropdlem 49225 | Lemma for ~ invpropd . (C... |
| invpropd 49226 | Two structures with the sa... |
| isopropdlem 49227 | Lemma for ~ isopropd . (C... |
| isopropd 49228 | Two structures with the sa... |
| cicfn 49229 | ` ~=c ` is a function on `... |
| cicrcl2 49230 | Isomorphism implies the st... |
| oppccic 49231 | Isomorphic objects are iso... |
| relcic 49232 | The set of isomorphic obje... |
| cicerALT 49233 | Isomorphism is an equivale... |
| cic1st2nd 49234 | Reconstruction of a pair o... |
| cic1st2ndbr 49235 | Rewrite the predicate of i... |
| cicpropdlem 49236 | Lemma for ~ cicpropd . (C... |
| cicpropd 49237 | Two structures with the sa... |
| oppccicb 49238 | Isomorphic objects are iso... |
| oppcciceq 49239 | The opposite category has ... |
| dmdm 49240 | The double domain of a fun... |
| iinfssclem1 49241 | Lemma for ~ iinfssc . (Co... |
| iinfssclem2 49242 | Lemma for ~ iinfssc . (Co... |
| iinfssclem3 49243 | Lemma for ~ iinfssc . (Co... |
| iinfssc 49244 | Indexed intersection of su... |
| iinfsubc 49245 | Indexed intersection of su... |
| iinfprg 49246 | Indexed intersection of fu... |
| infsubc 49247 | The intersection of two su... |
| infsubc2 49248 | The intersection of two su... |
| infsubc2d 49249 | The intersection of two su... |
| discsubclem 49250 | Lemma for ~ discsubc . (C... |
| discsubc 49251 | A discrete category, whose... |
| iinfconstbaslem 49252 | Lemma for ~ iinfconstbas .... |
| iinfconstbas 49253 | The discrete category is t... |
| nelsubclem 49254 | Lemma for ~ nelsubc . (Co... |
| nelsubc 49255 | An empty "hom-set" for non... |
| nelsubc2 49256 | An empty "hom-set" for non... |
| nelsubc3lem 49257 | Lemma for ~ nelsubc3 . (C... |
| nelsubc3 49258 | Remark 4.2(2) of [Adamek] ... |
| ssccatid 49259 | A category ` C ` restricte... |
| resccatlem 49260 | Lemma for ~ resccat . (Co... |
| resccat 49261 | A class ` C ` restricted b... |
| reldmfunc 49262 | The domain of ` Func ` is ... |
| func1st2nd 49263 | Rewrite the functor predic... |
| func1st 49264 | Extract the first member o... |
| func2nd 49265 | Extract the second member ... |
| funcrcl2 49266 | Reverse closure for a func... |
| funcrcl3 49267 | Reverse closure for a func... |
| funcf2lem 49268 | A utility theorem for prov... |
| funcf2lem2 49269 | A utility theorem for prov... |
| 0funcglem 49270 | Lemma for ~ 0funcg . (Con... |
| 0funcg2 49271 | The functor from the empty... |
| 0funcg 49272 | The functor from the empty... |
| 0funclem 49273 | Lemma for ~ 0funcALT . (C... |
| 0func 49274 | The functor from the empty... |
| 0funcALT 49275 | Alternate proof of ~ 0func... |
| func0g 49276 | The source category of a f... |
| func0g2 49277 | The source category of a f... |
| initc 49278 | Sets with empty base are t... |
| cofu1st2nd 49279 | Rewrite the functor compos... |
| rescofuf 49280 | The restriction of functor... |
| cofu1a 49281 | Value of the object part o... |
| cofu2a 49282 | Value of the morphism part... |
| cofucla 49283 | The composition of two fun... |
| funchomf 49284 | Source categories of a fun... |
| idfurcl 49285 | Reverse closure for an ide... |
| idfu1stf1o 49286 | The identity functor/inclu... |
| idfu1stalem 49287 | Lemma for ~ idfu1sta . (C... |
| idfu1sta 49288 | Value of the object part o... |
| idfu1a 49289 | Value of the object part o... |
| idfu2nda 49290 | Value of the morphism part... |
| imasubclem1 49291 | Lemma for ~ imasubc . (Co... |
| imasubclem2 49292 | Lemma for ~ imasubc . (Co... |
| imasubclem3 49293 | Lemma for ~ imasubc . (Co... |
| imaf1homlem 49294 | Lemma for ~ imaf1hom and o... |
| imaf1hom 49295 | The hom-set of an image of... |
| imaidfu2lem 49296 | Lemma for ~ imaidfu2 . (C... |
| imaidfu 49297 | The image of the identity ... |
| imaidfu2 49298 | The image of the identity ... |
| cofid1a 49299 | Express the object part of... |
| cofid2a 49300 | Express the morphism part ... |
| cofid1 49301 | Express the object part of... |
| cofid2 49302 | Express the morphism part ... |
| cofidvala 49303 | The property " ` F ` is a ... |
| cofidf2a 49304 | If " ` F ` is a section of... |
| cofidf1a 49305 | If " ` F ` is a section of... |
| cofidval 49306 | The property " ` <. F , G ... |
| cofidf2 49307 | If " ` F ` is a section of... |
| cofidf1 49308 | If " ` <. F , G >. ` is a ... |
| oppffn 49311 | ` oppFunc ` is a function ... |
| reldmoppf 49312 | The domain of ` oppFunc ` ... |
| oppfvalg 49313 | Value of the opposite func... |
| oppfrcllem 49314 | Lemma for ~ oppfrcl . (Co... |
| oppfrcl 49315 | If an opposite functor of ... |
| oppfrcl2 49316 | If an opposite functor of ... |
| oppfrcl3 49317 | If an opposite functor of ... |
| oppf1st2nd 49318 | Rewrite the opposite funct... |
| 2oppf 49319 | The double opposite functo... |
| eloppf 49320 | The pre-image of a non-emp... |
| eloppf2 49321 | Both components of a pre-i... |
| oppfvallem 49322 | Lemma for ~ oppfval . (Co... |
| oppfval 49323 | Value of the opposite func... |
| oppfval2 49324 | Value of the opposite func... |
| oppfval3 49325 | Value of the opposite func... |
| oppf1 49326 | Value of the object part o... |
| oppf2 49327 | Value of the morphism part... |
| oppfoppc 49328 | The opposite functor is a ... |
| oppfoppc2 49329 | The opposite functor is a ... |
| funcoppc2 49330 | A functor on opposite cate... |
| funcoppc4 49331 | A functor on opposite cate... |
| funcoppc5 49332 | A functor on opposite cate... |
| 2oppffunc 49333 | The opposite functor of an... |
| funcoppc3 49334 | A functor on opposite cate... |
| oppff1 49335 | The operation generating o... |
| oppff1o 49336 | The operation generating o... |
| cofuoppf 49337 | Composition of opposite fu... |
| imasubc 49338 | An image of a full functor... |
| imasubc2 49339 | An image of a full functor... |
| imassc 49340 | An image of a functor sati... |
| imaid 49341 | An image of a functor pres... |
| imaf1co 49342 | An image of a functor whos... |
| imasubc3 49343 | An image of a functor inje... |
| fthcomf 49344 | Source categories of a fai... |
| idfth 49345 | The inclusion functor is a... |
| idemb 49346 | The inclusion functor is a... |
| idsubc 49347 | The source category of an ... |
| idfullsubc 49348 | The source category of an ... |
| cofidfth 49349 | If " ` F ` is a section of... |
| fulloppf 49350 | The opposite functor of a ... |
| fthoppf 49351 | The opposite functor of a ... |
| ffthoppf 49352 | The opposite functor of a ... |
| upciclem1 49353 | Lemma for ~ upcic , ~ upeu... |
| upciclem2 49354 | Lemma for ~ upciclem3 and ... |
| upciclem3 49355 | Lemma for ~ upciclem4 . (... |
| upciclem4 49356 | Lemma for ~ upcic and ~ up... |
| upcic 49357 | A universal property defin... |
| upeu 49358 | A universal property defin... |
| upeu2 49359 | Generate new universal mor... |
| reldmup 49362 | The domain of ` UP ` is a ... |
| upfval 49363 | Function value of the clas... |
| upfval2 49364 | Function value of the clas... |
| upfval3 49365 | Function value of the clas... |
| isuplem 49366 | Lemma for ~ isup and other... |
| isup 49367 | The predicate "is a univer... |
| uppropd 49368 | If two categories have the... |
| reldmup2 49369 | The domain of ` ( D UP E )... |
| relup 49370 | The set of universal pairs... |
| uprcl 49371 | Reverse closure for the cl... |
| up1st2nd 49372 | Rewrite the universal prop... |
| up1st2ndr 49373 | Combine separated parts in... |
| up1st2ndb 49374 | Combine/separate parts in ... |
| up1st2nd2 49375 | Rewrite the universal prop... |
| uprcl2 49376 | Reverse closure for the cl... |
| uprcl3 49377 | Reverse closure for the cl... |
| uprcl4 49378 | Reverse closure for the cl... |
| uprcl5 49379 | Reverse closure for the cl... |
| uobrcl 49380 | Reverse closure for univer... |
| isup2 49381 | The universal property of ... |
| upeu3 49382 | The universal pair ` <. X ... |
| upeu4 49383 | Generate a new universal m... |
| uptposlem 49384 | Lemma for ~ uptpos . (Con... |
| uptpos 49385 | Rewrite the predicate of u... |
| oppcuprcl4 49386 | Reverse closure for the cl... |
| oppcuprcl3 49387 | Reverse closure for the cl... |
| oppcuprcl5 49388 | Reverse closure for the cl... |
| oppcuprcl2 49389 | Reverse closure for the cl... |
| uprcl2a 49390 | Reverse closure for the cl... |
| oppfuprcl 49391 | Reverse closure for the cl... |
| oppfuprcl2 49392 | Reverse closure for the cl... |
| oppcup3lem 49393 | Lemma for ~ oppcup3 . (Co... |
| oppcup 49394 | The universal pair ` <. X ... |
| oppcup2 49395 | The universal property for... |
| oppcup3 49396 | The universal property for... |
| uptrlem1 49397 | Lemma for ~ uptr . (Contr... |
| uptrlem2 49398 | Lemma for ~ uptr . (Contr... |
| uptrlem3 49399 | Lemma for ~ uptr . (Contr... |
| uptr 49400 | Universal property and ful... |
| uptri 49401 | Universal property and ful... |
| uptra 49402 | Universal property and ful... |
| uptrar 49403 | Universal property and ful... |
| uptrai 49404 | Universal property and ful... |
| uobffth 49405 | A fully faithful functor g... |
| uobeqw 49406 | If a full functor (in fact... |
| uobeq 49407 | If a full functor (in fact... |
| uptr2 49408 | Universal property and ful... |
| uptr2a 49409 | Universal property and ful... |
| isnatd 49410 | Property of being a natura... |
| natrcl2 49411 | Reverse closure for a natu... |
| natrcl3 49412 | Reverse closure for a natu... |
| catbas 49413 | The base of the category s... |
| cathomfval 49414 | The hom-sets of the catego... |
| catcofval 49415 | Composition of the categor... |
| natoppf 49416 | A natural transformation i... |
| natoppf2 49417 | A natural transformation i... |
| natoppfb 49418 | A natural transformation i... |
| initoo2 49419 | An initial object is an ob... |
| termoo2 49420 | A terminal object is an ob... |
| zeroo2 49421 | A zero object is an object... |
| oppcinito 49422 | Initial objects are termin... |
| oppctermo 49423 | Terminal objects are initi... |
| oppczeroo 49424 | Zero objects are zero in t... |
| termoeu2 49425 | Terminal objects are essen... |
| initopropdlemlem 49426 | Lemma for ~ initopropdlem ... |
| initopropdlem 49427 | Lemma for ~ initopropd . ... |
| termopropdlem 49428 | Lemma for ~ termopropd . ... |
| zeroopropdlem 49429 | Lemma for ~ zeroopropd . ... |
| initopropd 49430 | Two structures with the sa... |
| termopropd 49431 | Two structures with the sa... |
| zeroopropd 49432 | Two structures with the sa... |
| reldmxpc 49433 | The binary product of cate... |
| reldmxpcALT 49434 | Alternate proof of ~ reldm... |
| elxpcbasex1 49435 | A non-empty base set of th... |
| elxpcbasex1ALT 49436 | Alternate proof of ~ elxpc... |
| elxpcbasex2 49437 | A non-empty base set of th... |
| elxpcbasex2ALT 49438 | Alternate proof of ~ elxpc... |
| xpcfucbas 49439 | The base set of the produc... |
| xpcfuchomfval 49440 | Set of morphisms of the bi... |
| xpcfuchom 49441 | Set of morphisms of the bi... |
| xpcfuchom2 49442 | Value of the set of morphi... |
| xpcfucco2 49443 | Value of composition in th... |
| xpcfuccocl 49444 | The composition of two nat... |
| xpcfucco3 49445 | Value of composition in th... |
| dfswapf2 49448 | Alternate definition of ` ... |
| swapfval 49449 | Value of the swap functor.... |
| swapfelvv 49450 | A swap functor is an order... |
| swapf2fvala 49451 | The morphism part of the s... |
| swapf2fval 49452 | The morphism part of the s... |
| swapf1vala 49453 | The object part of the swa... |
| swapf1val 49454 | The object part of the swa... |
| swapf2fn 49455 | The morphism part of the s... |
| swapf1a 49456 | The object part of the swa... |
| swapf2vala 49457 | The morphism part of the s... |
| swapf2a 49458 | The morphism part of the s... |
| swapf1 49459 | The object part of the swa... |
| swapf2val 49460 | The morphism part of the s... |
| swapf2 49461 | The morphism part of the s... |
| swapf1f1o 49462 | The object part of the swa... |
| swapf2f1o 49463 | The morphism part of the s... |
| swapf2f1oa 49464 | The morphism part of the s... |
| swapf2f1oaALT 49465 | Alternate proof of ~ swapf... |
| swapfid 49466 | Each identity morphism in ... |
| swapfida 49467 | Each identity morphism in ... |
| swapfcoa 49468 | Composition in the source ... |
| swapffunc 49469 | The swap functor is a func... |
| swapfffth 49470 | The swap functor is a full... |
| swapffunca 49471 | The swap functor is a func... |
| swapfiso 49472 | The swap functor is an iso... |
| swapciso 49473 | The product category is ca... |
| oppc1stflem 49474 | A utility theorem for prov... |
| oppc1stf 49475 | The opposite functor of th... |
| oppc2ndf 49476 | The opposite functor of th... |
| 1stfpropd 49477 | If two categories have the... |
| 2ndfpropd 49478 | If two categories have the... |
| diagpropd 49479 | If two categories have the... |
| cofuswapfcl 49480 | The bifunctor pre-composed... |
| cofuswapf1 49481 | The object part of a bifun... |
| cofuswapf2 49482 | The morphism part of a bif... |
| tposcurf1cl 49483 | The partially evaluated tr... |
| tposcurf11 49484 | Value of the double evalua... |
| tposcurf12 49485 | The partially evaluated tr... |
| tposcurf1 49486 | Value of the object part o... |
| tposcurf2 49487 | Value of the transposed cu... |
| tposcurf2val 49488 | Value of a component of th... |
| tposcurf2cl 49489 | The transposed curry funct... |
| tposcurfcl 49490 | The transposed curry funct... |
| diag1 49491 | The constant functor of ` ... |
| diag1a 49492 | The constant functor of ` ... |
| diag1f1lem 49493 | The object part of the dia... |
| diag1f1 49494 | The object part of the dia... |
| diag2f1lem 49495 | Lemma for ~ diag2f1 . The... |
| diag2f1 49496 | If ` B ` is non-empty, the... |
| fucofulem1 49497 | Lemma for proving functor ... |
| fucofulem2 49498 | Lemma for proving functor ... |
| fuco2el 49499 | Equivalence of product fun... |
| fuco2eld 49500 | Equivalence of product fun... |
| fuco2eld2 49501 | Equivalence of product fun... |
| fuco2eld3 49502 | Equivalence of product fun... |
| fucofvalg 49505 | Value of the function givi... |
| fucofval 49506 | Value of the function givi... |
| fucoelvv 49507 | A functor composition bifu... |
| fuco1 49508 | The object part of the fun... |
| fucof1 49509 | The object part of the fun... |
| fuco2 49510 | The morphism part of the f... |
| fucofn2 49511 | The morphism part of the f... |
| fucofvalne 49512 | Value of the function givi... |
| fuco11 49513 | The object part of the fun... |
| fuco11cl 49514 | The object part of the fun... |
| fuco11a 49515 | The object part of the fun... |
| fuco112 49516 | The object part of the fun... |
| fuco111 49517 | The object part of the fun... |
| fuco111x 49518 | The object part of the fun... |
| fuco112x 49519 | The object part of the fun... |
| fuco112xa 49520 | The object part of the fun... |
| fuco11id 49521 | The identity morphism of t... |
| fuco11idx 49522 | The identity morphism of t... |
| fuco21 49523 | The morphism part of the f... |
| fuco11b 49524 | The object part of the fun... |
| fuco11bALT 49525 | Alternate proof of ~ fuco1... |
| fuco22 49526 | The morphism part of the f... |
| fucofn22 49527 | The morphism part of the f... |
| fuco23 49528 | The morphism part of the f... |
| fuco22natlem1 49529 | Lemma for ~ fuco22nat . T... |
| fuco22natlem2 49530 | Lemma for ~ fuco22nat . T... |
| fuco22natlem3 49531 | Combine ~ fuco22natlem2 wi... |
| fuco22natlem 49532 | The composed natural trans... |
| fuco22nat 49533 | The composed natural trans... |
| fucof21 49534 | The morphism part of the f... |
| fucoid 49535 | Each identity morphism in ... |
| fucoid2 49536 | Each identity morphism in ... |
| fuco22a 49537 | The morphism part of the f... |
| fuco23alem 49538 | The naturality property ( ... |
| fuco23a 49539 | The morphism part of the f... |
| fucocolem1 49540 | Lemma for ~ fucoco . Asso... |
| fucocolem2 49541 | Lemma for ~ fucoco . The ... |
| fucocolem3 49542 | Lemma for ~ fucoco . The ... |
| fucocolem4 49543 | Lemma for ~ fucoco . The ... |
| fucoco 49544 | Composition in the source ... |
| fucoco2 49545 | Composition in the source ... |
| fucofunc 49546 | The functor composition bi... |
| fucofunca 49547 | The functor composition bi... |
| fucolid 49548 | Post-compose a natural tra... |
| fucorid 49549 | Pre-composing a natural tr... |
| fucorid2 49550 | Pre-composing a natural tr... |
| postcofval 49551 | Value of the post-composit... |
| postcofcl 49552 | The post-composition funct... |
| precofvallem 49553 | Lemma for ~ precofval to e... |
| precofval 49554 | Value of the pre-compositi... |
| precofvalALT 49555 | Alternate proof of ~ preco... |
| precofval2 49556 | Value of the pre-compositi... |
| precofcl 49557 | The pre-composition functo... |
| precofval3 49558 | Value of the pre-compositi... |
| precoffunc 49559 | The pre-composition functo... |
| reldmprcof 49562 | The domain of ` -o.F ` is ... |
| prcofvalg 49563 | Value of the pre-compositi... |
| prcofvala 49564 | Value of the pre-compositi... |
| prcofval 49565 | Value of the pre-compositi... |
| prcofpropd 49566 | If the categories have the... |
| prcofelvv 49567 | The pre-composition functo... |
| reldmprcof1 49568 | The domain of the object p... |
| reldmprcof2 49569 | The domain of the morphism... |
| prcoftposcurfuco 49570 | The pre-composition functo... |
| prcoftposcurfucoa 49571 | The pre-composition functo... |
| prcoffunc 49572 | The pre-composition functo... |
| prcoffunca 49573 | The pre-composition functo... |
| prcoffunca2 49574 | The pre-composition functo... |
| prcof1 49575 | The object part of the pre... |
| prcof2a 49576 | The morphism part of the p... |
| prcof2 49577 | The morphism part of the p... |
| prcof21a 49578 | The morphism part of the p... |
| prcof22a 49579 | The morphism part of the p... |
| prcofdiag1 49580 | A constant functor pre-com... |
| prcofdiag 49581 | A diagonal functor post-co... |
| catcrcl 49582 | Reverse closure for the ca... |
| catcrcl2 49583 | Reverse closure for the ca... |
| elcatchom 49584 | A morphism of the category... |
| catcsect 49585 | The property " ` F ` is a ... |
| catcinv 49586 | The property " ` F ` is an... |
| catcisoi 49587 | A functor is an isomorphis... |
| uobeq2 49588 | If a full functor (in fact... |
| uobeq3 49589 | An isomorphism between cat... |
| opf11 49590 | The object part of the op ... |
| opf12 49591 | The object part of the op ... |
| opf2fval 49592 | The morphism part of the o... |
| opf2 49593 | The morphism part of the o... |
| fucoppclem 49594 | Lemma for ~ fucoppc . (Co... |
| fucoppcid 49595 | The opposite category of f... |
| fucoppcco 49596 | The opposite category of f... |
| fucoppc 49597 | The isomorphism from the o... |
| fucoppcffth 49598 | A fully faithful functor f... |
| fucoppcfunc 49599 | A functor from the opposit... |
| fucoppccic 49600 | The opposite category of f... |
| oppfdiag1 49601 | A constant functor for opp... |
| oppfdiag1a 49602 | A constant functor for opp... |
| oppfdiag 49603 | A diagonal functor for opp... |
| isthinc 49606 | The predicate "is a thin c... |
| isthinc2 49607 | A thin category is a categ... |
| isthinc3 49608 | A thin category is a categ... |
| thincc 49609 | A thin category is a categ... |
| thinccd 49610 | A thin category is a categ... |
| thincssc 49611 | A thin category is a categ... |
| isthincd2lem1 49612 | Lemma for ~ isthincd2 and ... |
| thincmo2 49613 | Morphisms in the same hom-... |
| thinchom 49614 | A non-empty hom-set of a t... |
| thincmo 49615 | There is at most one morph... |
| thincmoALT 49616 | Alternate proof of ~ thinc... |
| thincmod 49617 | At most one morphism in ea... |
| thincn0eu 49618 | In a thin category, a hom-... |
| thincid 49619 | In a thin category, a morp... |
| thincmon 49620 | In a thin category, all mo... |
| thincepi 49621 | In a thin category, all mo... |
| isthincd2lem2 49622 | Lemma for ~ isthincd2 . (... |
| isthincd 49623 | The predicate "is a thin c... |
| isthincd2 49624 | The predicate " ` C ` is a... |
| oppcthin 49625 | The opposite category of a... |
| oppcthinco 49626 | If the opposite category o... |
| oppcthinendc 49627 | The opposite category of a... |
| oppcthinendcALT 49628 | Alternate proof of ~ oppct... |
| thincpropd 49629 | Two structures with the sa... |
| subthinc 49630 | A subcategory of a thin ca... |
| functhinclem1 49631 | Lemma for ~ functhinc . G... |
| functhinclem2 49632 | Lemma for ~ functhinc . (... |
| functhinclem3 49633 | Lemma for ~ functhinc . T... |
| functhinclem4 49634 | Lemma for ~ functhinc . O... |
| functhinc 49635 | A functor to a thin catego... |
| functhincfun 49636 | A functor to a thin catego... |
| fullthinc 49637 | A functor to a thin catego... |
| fullthinc2 49638 | A full functor to a thin c... |
| thincfth 49639 | A functor from a thin cate... |
| thincciso 49640 | Two thin categories are is... |
| thinccisod 49641 | Two thin categories are is... |
| thincciso2 49642 | Categories isomorphic to a... |
| thincciso3 49643 | Categories isomorphic to a... |
| thincciso4 49644 | Two isomorphic categories ... |
| 0thincg 49645 | Any structure with an empt... |
| 0thinc 49646 | The empty category (see ~ ... |
| indcthing 49647 | An indiscrete category, i.... |
| discthing 49648 | A discrete category, i.e.,... |
| indthinc 49649 | An indiscrete category in ... |
| indthincALT 49650 | An alternate proof of ~ in... |
| prsthinc 49651 | Preordered sets as categor... |
| setcthin 49652 | A category of sets all of ... |
| setc2othin 49653 | The category ` ( SetCat ``... |
| thincsect 49654 | In a thin category, one mo... |
| thincsect2 49655 | In a thin category, ` F ` ... |
| thincinv 49656 | In a thin category, ` F ` ... |
| thinciso 49657 | In a thin category, ` F : ... |
| thinccic 49658 | In a thin category, two ob... |
| istermc 49661 | The predicate "is a termin... |
| istermc2 49662 | The predicate "is a termin... |
| istermc3 49663 | The predicate "is a termin... |
| termcthin 49664 | A terminal category is a t... |
| termcthind 49665 | A terminal category is a t... |
| termccd 49666 | A terminal category is a c... |
| termcbas 49667 | The base of a terminal cat... |
| termco 49668 | The object of a terminal c... |
| termcbas2 49669 | The base of a terminal cat... |
| termcbasmo 49670 | Two objects in a terminal ... |
| termchomn0 49671 | All hom-sets of a terminal... |
| termchommo 49672 | All morphisms of a termina... |
| termcid 49673 | The morphism of a terminal... |
| termcid2 49674 | The morphism of a terminal... |
| termchom 49675 | The hom-set of a terminal ... |
| termchom2 49676 | The hom-set of a terminal ... |
| setcsnterm 49677 | The category of one set, e... |
| setc1oterm 49678 | The category ` ( SetCat ``... |
| setc1obas 49679 | The base of the trivial ca... |
| setc1ohomfval 49680 | Set of morphisms of the tr... |
| setc1ocofval 49681 | Composition in the trivial... |
| setc1oid 49682 | The identity morphism of t... |
| funcsetc1ocl 49683 | The functor to the trivial... |
| funcsetc1o 49684 | Value of the functor to th... |
| isinito2lem 49685 | The predicate "is an initi... |
| isinito2 49686 | The predicate "is an initi... |
| isinito3 49687 | The predicate "is an initi... |
| dfinito4 49688 | An alternate definition of... |
| dftermo4 49689 | An alternate definition of... |
| termcpropd 49690 | Two structures with the sa... |
| oppctermhom 49691 | The opposite category of a... |
| oppctermco 49692 | The opposite category of a... |
| oppcterm 49693 | The opposite category of a... |
| functermclem 49694 | Lemma for ~ functermc . (... |
| functermc 49695 | Functor to a terminal cate... |
| functermc2 49696 | Functor to a terminal cate... |
| functermceu 49697 | There exists a unique func... |
| fulltermc 49698 | A functor to a terminal ca... |
| fulltermc2 49699 | Given a full functor to a ... |
| termcterm 49700 | A terminal category is a t... |
| termcterm2 49701 | A terminal object of the c... |
| termcterm3 49702 | In the category of small c... |
| termcciso 49703 | A category is isomorphic t... |
| termccisoeu 49704 | The isomorphism between te... |
| termc2 49705 | If there exists a unique f... |
| termc 49706 | Alternate definition of ` ... |
| dftermc2 49707 | Alternate definition of ` ... |
| eufunclem 49708 | If there exists a unique f... |
| eufunc 49709 | If there exists a unique f... |
| idfudiag1lem 49710 | Lemma for ~ idfudiag1bas a... |
| idfudiag1bas 49711 | If the identity functor of... |
| idfudiag1 49712 | If the identity functor of... |
| euendfunc 49713 | If there exists a unique e... |
| euendfunc2 49714 | If there exists a unique e... |
| termcarweu 49715 | There exists a unique disj... |
| arweuthinc 49716 | If a structure has a uniqu... |
| arweutermc 49717 | If a structure has a uniqu... |
| dftermc3 49718 | Alternate definition of ` ... |
| termcfuncval 49719 | The value of a functor fro... |
| diag1f1olem 49720 | To any functor from a term... |
| diag1f1o 49721 | The object part of the dia... |
| termcnatval 49722 | Value of natural transform... |
| diag2f1olem 49723 | Lemma for ~ diag2f1o . (C... |
| diag2f1o 49724 | If ` D ` is terminal, the ... |
| diagffth 49725 | The diagonal functor is a ... |
| diagciso 49726 | The diagonal functor is an... |
| diagcic 49727 | Any category ` C ` is isom... |
| funcsn 49728 | The category of one functo... |
| fucterm 49729 | The category of functors t... |
| 0fucterm 49730 | The category of functors f... |
| termfucterm 49731 | All functors between two t... |
| cofuterm 49732 | Post-compose with a functo... |
| uobeqterm 49733 | Universal objects and term... |
| isinito4 49734 | The predicate "is an initi... |
| isinito4a 49735 | The predicate "is an initi... |
| prstcval 49738 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 49739 | Lemma for ~ prstcnid and ~... |
| prstcnid 49740 | Components other than ` Ho... |
| prstcbas 49741 | The base set is unchanged.... |
| prstcleval 49742 | Value of the less-than-or-... |
| prstcle 49743 | Value of the less-than-or-... |
| prstcocval 49744 | Orthocomplementation is un... |
| prstcoc 49745 | Orthocomplementation is un... |
| prstchomval 49746 | Hom-sets of the constructe... |
| prstcprs 49747 | The category is a preorder... |
| prstcthin 49748 | The preordered set is equi... |
| prstchom 49749 | Hom-sets of the constructe... |
| prstchom2 49750 | Hom-sets of the constructe... |
| prstchom2ALT 49751 | Hom-sets of the constructe... |
| oduoppcbas 49752 | The dual of a preordered s... |
| oduoppcciso 49753 | The dual of a preordered s... |
| postcpos 49754 | The converted category is ... |
| postcposALT 49755 | Alternate proof of ~ postc... |
| postc 49756 | The converted category is ... |
| discsntermlem 49757 | A singlegon is an element ... |
| basrestermcfolem 49758 | An element of the class of... |
| discbas 49759 | A discrete category (a cat... |
| discthin 49760 | A discrete category (a cat... |
| discsnterm 49761 | A discrete category (a cat... |
| basrestermcfo 49762 | The base function restrict... |
| termcnex 49763 | The class of all terminal ... |
| mndtcval 49766 | Value of the category buil... |
| mndtcbasval 49767 | The base set of the catego... |
| mndtcbas 49768 | The category built from a ... |
| mndtcob 49769 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 49770 | Two objects in a category ... |
| mndtchom 49771 | The only hom-set of the ca... |
| mndtcco 49772 | The composition of the cat... |
| mndtcco2 49773 | The composition of the cat... |
| mndtccatid 49774 | Lemma for ~ mndtccat and ~... |
| mndtccat 49775 | The function value is a ca... |
| mndtcid 49776 | The identity morphism, or ... |
| oppgoppchom 49777 | The converted opposite mon... |
| oppgoppcco 49778 | The converted opposite mon... |
| oppgoppcid 49779 | The converted opposite mon... |
| grptcmon 49780 | All morphisms in a categor... |
| grptcepi 49781 | All morphisms in a categor... |
| 2arwcatlem1 49782 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem2 49783 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem3 49784 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem4 49785 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem5 49786 | Lemma for ~ 2arwcat . (Co... |
| 2arwcat 49787 | The condition for a struct... |
| incat 49788 | Constructing a category wi... |
| setc1onsubc 49789 | Construct a category with ... |
| cnelsubclem 49790 | Lemma for ~ cnelsubc . (C... |
| cnelsubc 49791 | Remark 4.2(2) of [Adamek] ... |
| lanfn 49796 | ` Lan ` is a function on `... |
| ranfn 49797 | ` Ran ` is a function on `... |
| reldmlan 49798 | The domain of ` Lan ` is a... |
| reldmran 49799 | The domain of ` Ran ` is a... |
| lanfval 49800 | Value of the function gene... |
| ranfval 49801 | Value of the function gene... |
| lanpropd 49802 | If the categories have the... |
| ranpropd 49803 | If the categories have the... |
| reldmlan2 49804 | The domain of ` ( P Lan E ... |
| reldmran2 49805 | The domain of ` ( P Ran E ... |
| lanval 49806 | Value of the set of left K... |
| ranval 49807 | Value of the set of right ... |
| lanrcl 49808 | Reverse closure for left K... |
| ranrcl 49809 | Reverse closure for right ... |
| rellan 49810 | The set of left Kan extens... |
| relran 49811 | The set of right Kan exten... |
| islan 49812 | A left Kan extension is a ... |
| islan2 49813 | A left Kan extension is a ... |
| lanval2 49814 | The set of left Kan extens... |
| isran 49815 | A right Kan extension is a... |
| isran2 49816 | A right Kan extension is a... |
| ranval2 49817 | The set of right Kan exten... |
| ranval3 49818 | The set of right Kan exten... |
| lanrcl2 49819 | Reverse closure for left K... |
| lanrcl3 49820 | Reverse closure for left K... |
| lanrcl4 49821 | The first component of a l... |
| lanrcl5 49822 | The second component of a ... |
| ranrcl2 49823 | Reverse closure for right ... |
| ranrcl3 49824 | Reverse closure for right ... |
| ranrcl4lem 49825 | Lemma for ~ ranrcl4 and ~ ... |
| ranrcl4 49826 | The first component of a r... |
| ranrcl5 49827 | The second component of a ... |
| lanup 49828 | The universal property of ... |
| ranup 49829 | The universal property of ... |
| reldmlmd 49834 | The domain of ` Limit ` is... |
| reldmcmd 49835 | The domain of ` Colimit ` ... |
| lmdfval 49836 | Function value of ` Limit ... |
| cmdfval 49837 | Function value of ` Colimi... |
| lmdrcl 49838 | Reverse closure for a limi... |
| cmdrcl 49839 | Reverse closure for a coli... |
| reldmlmd2 49840 | The domain of ` ( C Limit ... |
| reldmcmd2 49841 | The domain of ` ( C Colimi... |
| lmdfval2 49842 | The set of limits of a dia... |
| cmdfval2 49843 | The set of colimits of a d... |
| lmdpropd 49844 | If the categories have the... |
| cmdpropd 49845 | If the categories have the... |
| rellmd 49846 | The set of limits of a dia... |
| relcmd 49847 | The set of colimits of a d... |
| concl 49848 | A natural transformation f... |
| coccl 49849 | A natural transformation t... |
| concom 49850 | A cone to a diagram commut... |
| coccom 49851 | A co-cone to a diagram com... |
| islmd 49852 | The universal property of ... |
| iscmd 49853 | The universal property of ... |
| lmddu 49854 | The duality of limits and ... |
| cmddu 49855 | The duality of limits and ... |
| initocmd 49856 | Initial objects are the ob... |
| termolmd 49857 | Terminal objects are the o... |
| lmdran 49858 | To each limit of a diagram... |
| cmdlan 49859 | To each colimit of a diagr... |
| nfintd 49860 | Bound-variable hypothesis ... |
| nfiund 49861 | Bound-variable hypothesis ... |
| nfiundg 49862 | Bound-variable hypothesis ... |
| iunord 49863 | The indexed union of a col... |
| iunordi 49864 | The indexed union of a col... |
| spd 49865 | Specialization deduction, ... |
| spcdvw 49866 | A version of ~ spcdv where... |
| tfis2d 49867 | Transfinite Induction Sche... |
| bnd2d 49868 | Deduction form of ~ bnd2 .... |
| dffun3f 49869 | Alternate definition of fu... |
| setrecseq 49872 | Equality theorem for set r... |
| nfsetrecs 49873 | Bound-variable hypothesis ... |
| setrec1lem1 49874 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 49875 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 49876 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 49877 | Lemma for ~ setrec1 . If ... |
| setrec1 49878 | This is the first of two f... |
| setrec2fun 49879 | This is the second of two ... |
| setrec2lem1 49880 | Lemma for ~ setrec2 . The... |
| setrec2lem2 49881 | Lemma for ~ setrec2 . The... |
| setrec2 49882 | This is the second of two ... |
| setrec2v 49883 | Version of ~ setrec2 with ... |
| setrec2mpt 49884 | Version of ~ setrec2 where... |
| setis 49885 | Version of ~ setrec2 expre... |
| elsetrecslem 49886 | Lemma for ~ elsetrecs . A... |
| elsetrecs 49887 | A set ` A ` is an element ... |
| setrecsss 49888 | The ` setrecs ` operator r... |
| setrecsres 49889 | A recursively generated cl... |
| vsetrec 49890 | Construct ` _V ` using set... |
| 0setrec 49891 | If a function sends the em... |
| onsetreclem1 49892 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 49893 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 49894 | Lemma for ~ onsetrec . (C... |
| onsetrec 49895 | Construct ` On ` using set... |
| elpglem1 49898 | Lemma for ~ elpg . (Contr... |
| elpglem2 49899 | Lemma for ~ elpg . (Contr... |
| elpglem3 49900 | Lemma for ~ elpg . (Contr... |
| elpg 49901 | Membership in the class of... |
| pgindlem 49902 | Lemma for ~ pgind . (Cont... |
| pgindnf 49903 | Version of ~ pgind with ex... |
| pgind 49904 | Induction on partizan game... |
| sbidd 49905 | An identity theorem for su... |
| sbidd-misc 49906 | An identity theorem for su... |
| gte-lte 49911 | Simple relationship betwee... |
| gt-lt 49912 | Simple relationship betwee... |
| gte-lteh 49913 | Relationship between ` <_ ... |
| gt-lth 49914 | Relationship between ` < `... |
| ex-gt 49915 | Simple example of ` > ` , ... |
| ex-gte 49916 | Simple example of ` >_ ` ,... |
| sinhval-named 49923 | Value of the named sinh fu... |
| coshval-named 49924 | Value of the named cosh fu... |
| tanhval-named 49925 | Value of the named tanh fu... |
| sinh-conventional 49926 | Conventional definition of... |
| sinhpcosh 49927 | Prove that ` ( sinh `` A )... |
| secval 49934 | Value of the secant functi... |
| cscval 49935 | Value of the cosecant func... |
| cotval 49936 | Value of the cotangent fun... |
| seccl 49937 | The closure of the secant ... |
| csccl 49938 | The closure of the cosecan... |
| cotcl 49939 | The closure of the cotange... |
| reseccl 49940 | The closure of the secant ... |
| recsccl 49941 | The closure of the cosecan... |
| recotcl 49942 | The closure of the cotange... |
| recsec 49943 | The reciprocal of secant i... |
| reccsc 49944 | The reciprocal of cosecant... |
| reccot 49945 | The reciprocal of cotangen... |
| rectan 49946 | The reciprocal of tangent ... |
| sec0 49947 | The value of the secant fu... |
| onetansqsecsq 49948 | Prove the tangent squared ... |
| cotsqcscsq 49949 | Prove the tangent squared ... |
| ifnmfalse 49950 | If A is not a member of B,... |
| logb2aval 49951 | Define the value of the ` ... |
| mvlraddi 49958 | Move the right term in a s... |
| assraddsubi 49959 | Associate RHS addition-sub... |
| joinlmuladdmuli 49960 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 49961 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 49962 | Join AB-CB into (A-C) on L... |
| mvlrmuld 49963 | Move the right term in a p... |
| mvlrmuli 49964 | Move the right term in a p... |
| i2linesi 49965 | Solve for the intersection... |
| i2linesd 49966 | Solve for the intersection... |
| alimp-surprise 49967 | Demonstrate that when usin... |
| alimp-no-surprise 49968 | There is no "surprise" in ... |
| empty-surprise 49969 | Demonstrate that when usin... |
| empty-surprise2 49970 | "Prove" that false is true... |
| eximp-surprise 49971 | Show what implication insi... |
| eximp-surprise2 49972 | Show that "there exists" w... |
| alsconv 49977 | There is an equivalence be... |
| alsi1d 49978 | Deduction rule: Given "al... |
| alsi2d 49979 | Deduction rule: Given "al... |
| alsc1d 49980 | Deduction rule: Given "al... |
| alsc2d 49981 | Deduction rule: Given "al... |
| alscn0d 49982 | Deduction rule: Given "al... |
| alsi-no-surprise 49983 | Demonstrate that there is ... |
| 5m4e1 49984 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 49985 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 49986 | Resolution rule. This is ... |
| testable 49987 | In classical logic all wff... |
| aacllem 49988 | Lemma for other theorems a... |
| amgmwlem 49989 | Weighted version of ~ amgm... |
| amgmlemALT 49990 | Alternate proof of ~ amgml... |
| amgmw2d 49991 | Weighted arithmetic-geomet... |
| young2d 49992 | Young's inequality for ` n... |
| Copyright terms: Public domain | W3C validator |